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E D I T O R I A L S T A F F

E d i t o r , J . J . J A K L I T S C H , J R .

P r o d u c t i o n E d i t o r ,

A L L E N M O R R I S O N

E d i t o r i a l P r o d . A s s t . .

K I R S T E N D A H L

H E A T T R A N S F E R D I V I S I O N

C h a i r m a n , E . F R I E D

S e c r e t a r y , A . S . R A T H B U N

; n i o r T e c h n i c a l E d i t o r . E . M . S P A R R O W

T e c h n i c a l E d i t o r . W . A U N G

T e c h n i c a l E d i t o r , B . T . C H A O

T e c h n i c a l E d i t o r . D . K . E D W A R D S

T e c h n i c a l E d i t o r . R . E I C H H O R N

T e c h n i c a l E d i t o r , P . G R I F F I T H

T e c h n i c a l E d i t o r , J . S . L E E

T e c h n i c a l E d i t o r , R . S I E G E L

P O L I C Y B O A R D ,

C O M M U N I C A T I O N S

C h a i r m a n a n d V i c e - P r e s i d e n t

I. B E R M A N

M e m b e r s - a t - L a r g e

R . C . D E A N , J R .

G . P . E S C H E N B R E N N E R

M . J . R A B I N S

W . J . W A R R E N

P o l i c y B o a r d R e p r e s e n t a t i v e s

B a s i c E n g i n e e r i n g J . E . F O W L E R

G e n e r a l E n g i n e e r i n g , S . P . R O G A C K I

I n d u s t r y , J . E . O R T L O F F

P o w e r , A . F . D U Z Y

R e s e a r c h , G . P . C O O P E R

C o d e s a n d S t d s . , P . M . B R I S T E R

C o m p u t e r T e c h n o l o g y C o m . ,

A . A . S E I R E G

N o r n . C o m . R e p . ,

A . R . C A T H E R O N

B u s i n e s s S t a f f

3 4 5 E . 4 7 t h S t .

N e w Y o r k , N . Y . 1 0 0 1 7

( 2 1 2 ) 6 4 4 - 7 7 8 9

M n g . D i r . , P u b l . , C . 0 . S A N D E R S O N

O F F I C E R S O F T H E A S M E

P r e s i d e n t , S . P . K E Z I O S

E x e c . D i r . & S e c y , R O G E R S B . F I N C H

T r e a s u r e r , R O B E R T A . B E N N E T T

EDITED and PUBLISHED quarterly at theoff ices of The American Society of

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STATEMENT from By-Laws. The Society shal l notbe responsible for statements or opinionsadvanced in papers or . . . printed in itsublication s (B 13. Par 4)OPYRIGHT © 1978 by the American Society olechanical Engineers. Reprints from thisubl ication may be made on condit ions that ful lredit be given the TRANSACTIONS OF THESME. SERIES C—JOURNAL OF HEATRANSFER, and the author and date ofubl ication statedNDEXED by the Engineering Index. Inc

t ransact ions of the flSITlE

Publ i shed Quar te r l y by

T h e A m e r i c a n S o c i e t y o f

M e c h a n i c a l E n g i n e e r s

Vo l u me 100 • Numb er 1

F E B R U A R Y 1 9 7 8

journa l ofhea t

transfer

t.

V1 Jour na l o f Heat T rans fer Refe rees , 1977

3 S h a p e o f T w o - D i m e n s i o n a l S o l i d i f ic a t i o n I n t e r l a c e D u r in g D i r e c t i o n a l S o l i d i fi c a t i o n b y C o n t i n u o u s

C a s t i n g

R . S i ege l

11 Exp er i m ents on t he Ro l e o f Natu ra l Co nve c t i on i n t he Mel t i ng o f So l i ds

E . M . S p a r r o w , R . R . S c h m i d t , a n d J . W . R a m s e y

1 7 A u g m e n t a t i o n o f H o r iz o n t a l I n -T u b e C o n d e n s a t i o n b y M e a n s o f T w i s t e d - T a p e I n s e r t s a n d I n t e r n a l l y

F i n n e d T u b e s

J . H. Roya l and A . E . Berg l es

25 Some E f f ec t s o f Mech an i ca l l y -P ro duced Uns teady Boundary Layer F l ows on Conv ec t i ve Heat T rans ferA u g m e n t a t i o n

G . H . J u n k h a n

30 F l ow and Heat T rans fer in Conve c t i ve l y Coo l ed Underground E l ec t r i c Cab l e Sys tem s: Par t 1—Velocity

D i s t r i b u t io n s a n d P r e s s u r e D r o p C o r r e l a t i o n s

J. C. Chato and R . S . Abdu l had i

3 6 F l o w a n d H e a t T r a n s f e r i n C o n v e c t i v e l y C o o l e d U n d e r g r o u n d E l e c t r i c C a b l e S y s t e m s : P a rt 2 — T e m p e r a t u r e D i s t r i b u t i o n s a n d H e a t T r a n s f e r C o r r e l a t i o n s

R . S . Abdu l had i and J . C . Chato

4 1 E x p e r i m e n t s a n d U n i v e r s a l G r o w t h R e l a t i o n s f o r V a p o r B u b b l e s W i t h M i c r o l a y e r s

T . G . T heofanous , T . H . Bohrer , M. C . Chang, and P . D . Pate l

4 9 I n f l u e n c e o f S y s t e m P r e s s u r e o n M i c r o l a y e r E v a p o r a t i o n H e a t T r a n s f e r

H. S. Fath and R. L. Judd

56 Spa t i a l D i s t r i bu t i on o f Ac t i ve S i t es and Bub b l e F l ux Dens i t y

M. Su l t an and R . L . Judd

6 3 E f f e c t o f C i r c u m f e r e n t i a l l y N o n u n i f o r m H e a t i n g o n L a m i n a r C o m b i n e d C o n v e c t i o n in a H o r i z o n t a lT u b e

S . V . P a t a n k a r , S . R a m a d h y a n i , a n d E . M . S p a r r o w

7 1 E x p e r i m e n t s o n t h e O n se t of L o n g i t u d i n a l V o r t i c e s i n H o r i z o n t a l B l a s iu s F l o w H e a t e d F r o m B e l o wR. R . G i l p i n , H . Imura , and K . C . Cheng

78 Natu ra l Con vec t i on Heat T rans te r i n Beds o f I nduc t i ve l y Heated Par t i c l es

S . J . Rhee, V . K . Dh i r , and I . Cat i on

8 6 A n E x p e r i m e n t a l a n d T h e o r e t i c a l S tu d y o f H e a t T r a n s fe r i n V e r t i c a l T u b e F l o w s

R. Grei f

92 A Sur f a ce Re j uve nat i on Mo de l f o r T urbu l e n t Heat T rans fer i n Ann u l ar F l ow Wi th H i gh P randt l Num b e r s

B . T . F . Chung, L . C . T homas, and Y . Pang

9 8 C o m b i n e d C o n d u c t i v e a n d R a d i a t i v e H e a t T ra n s f e r in a n A b s o r b i n g a n d S c a t l e r i n g I n f i n it e S l a b

J. A. Roux and A . M. Smi t h

105 A Mo de l f o r F l am e P rop agat i o n i n Low Vo l a t i l e Coa l Dus t -A i r M i x tu re s

J. L. K raz i nsk i , R . O . Buck i us , and H . K r i e r

112 An I nves t i ga t i on o l t he Lam i nar Over f i r e Reg i o n A l ong Upr i gh t Sur f ace s

T . A h m a d a n d G . M . F a e t h

1 2 0 T h e C r i t i c a l T i m e - S t e p f o r F i n i t e - E l e m e n t S o l u t i o n s t o T w o - D i m e n s i o n a l H e a t - C o n d u c t i o n T r a n

s i en t sG. E . Myers

1 2 8 A M i x t u r e T h e o r y fo r Q u a s i - O n e - D i m e n s i o n a l D i ff u s i o n i n F i b e r - R e i n f o r c e d C o m p o s i t e s

A . M a e w a l , G . A . G u r l m a n , a n d G . A . H e g e m i e r

1 3 4 E x p e r i m e n t a l H e a t Tr a n s f e r B e h a v i o r o f a T u r b u l e n t B o u n d a r y L a y e r o n a R o u g h S u r f a c e W i t h B l o w

in g

R . J . Mof f a t , J . M. Hea l zer , and W. M. Kays

143 Wa ve E f f ec t s on t he T ransp or t t o F a l l i ng Lam i nar L i qu i d F i l ms

R. A . Seban and A . F aghr i

1 4 8 T r a n s i e n t B e h a v i o r o f a S o l i d S e n s i b l e H e a t T h e r m a l S t o r a g e E x c h a n g e r

J . S z e g o a n d F . W . S c h m i d t

155 T he E f f ec t of t he Lon don -Van Der Wa al s D i spers i on F orce on I n te r l i ne Heat T rans fe r

P . C . Wayner , J r .

T E C H N I C A L N O T E S

1 6 0 P r e d i c t i v e C a p a b i l i t i e s o t S e r i e s S o l u t i o n s l o r L a m i n a r F r e e C o n v e c t i o n B o u n d a r y L a y e r H e a t T r a n s fe r

F . N . L i n and B . T . Chao

( C o n t e n t s c o n t i n u e d o n p a g e 2 4 )Downloaded 04 Feb 2011 to 194.85.80.107. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm



24 /

CONTENTS (CONTINUED)

163 Nonslmilar Laminar Free Convecllon Flow Along a Nonlsothermal Verllcal Plate

B. K. Meena and G. Nath

165 A Local Nonslmilarlty Analysis of Free Convection From a Horizontal Cylindrical Surface

M. A. Muntasser and J. C. Mulligan

167 Transient Temperature Profile of a Hot Wall Due to an Impinging Liquid Droplet

M. Sekl, H. Kawamura, and K. Sanokawa

169 Combined Conduction-Radiation Heat Transfer Through an Irradiated Semitransparent Plate

R. Vlskanta and E. D. Hlrleman

172 Transient Conduction In a Slab With Temperature Dependent Thermal Conductivity

J. Sucec and S. Hedge

VOL 100, FEBRUARY 1978 Transactions of the ASME

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R. Siegel

National Aeronautics and Space Administration,

Lewis Research Center,

Cleveland, Ohio

Shape of Two-Dimensional

Solidification Interface During

Directional Solidification by

Continuous CastingAn analysis w as made of the two-dimensional solidification of an ingot being cooled and

withdrawn vertically downward from a mold consisting of parallel w alls of finite length.

In some metallurgical solidification processes the liquid metal is maintained superheated

so that it transfers heat into the solidification interface as a means of controlling the

metal structure during solidification. This energy plus the latent heat of fusion is removed

by the coolant along the sides of the ingot. Increasing the ingot withdrawal rate or the heat

addition at the solidification interface causes the interface to move downward within the

mold and have a nonplanar shape as a result o f the curved paths for heat flow into the

coolant. This distortion of the interface in the case of directional solidification leads to

an undesirable metallic structure. The present heat transfer ana lysis shows how the flat

ness of the interface is related to the ingot thickness, the withdrawal rate, the heat addi

tion from the superheated liquid metal, and the temperature difference available for cool

ing. This provides an understanding of the conditions that will yield a maximum rate of

casting while achieving the desired flatness of the interface. The results are interpreted

with respect to the conditions for obtaining an aligned eutectic structure by directionalsolidification. In this process an additional constraint must be included that relates the

ingot withdrawal rate and the heat transfer rate from the liquid metal to the solidification

interface.

I n t r o d u c t i o n

This paper is concerned with how the heat transfer condit ions

effect the two-dimensional shape of the solidification interface during

continuous cast ing of a s lab shaped ingot. By proper control of the

conditions at the solidification interface, it is possible in some casting

processes to obtain an in ternal m etall ic s tructur e tha t has a desirable

unidirec tional charac ter . The f latness of the in terface is imp orta ntin regulating this directional structure in the solidifying material, and

the re la t ion between th is f la tness and the heat f low condit ions wil l

be determined by the analysis . In addit ion to examining the direc

t ional cast ing process , the present method of solution is of general

in terest in the area of two-dimensional solid if icat ion analysis , as i t

further i l lustra tes the use th at can be made of conformal m apping in

this type of problem.

Th ere ha ve been nume rous analys es of solid if icat ion, for the mo st

pa r t by app rox im a te ana ly t ica l t e chn iques and numer ica l me thods .

The review art ic le in reference [ l l] 1 provides some apprecia t ion of

the previous work. Most of the investigations have been for one-

dimen sional geom etries , that is , for in terfaces that are p lanar , cylin

drical , or spherical . The increased in terest in two and three-dimen

sional solid if icat ion, and the ma them atica l d iff icult ies involved, are

indicated in the conference proceedings in reference [2]. A difficulty

with the mult i-d imensional problem is that the in terface shape and

its location are unknown , thereby making i t d iff icult to se t up a so

lu tion procedure for the heat conduction equation. Three analyses,

somewhat re la ted to the present s tudy, are found in references [3] ,

[4], and [5j . In reference [3] , the in terface shape was determ ined nu

merically for a cylindrical crystal being pulled from a melt. In refer

ence [4] continuous cast ing in an infin ite ly long mold was analyzed

by the heat balance in tegral m ethod . The effect of convection in the

Con tributed by the Hea t Transfer Division for publication in the JOURNALOF HEAT TRANSFER.Manuscript received by the Heat Transfer Division July15, 1977.

Journal of Heat Transfer

1 Numbers in brackets designate References at end of paper.

FEBRUARY 1978, VOL 100 / 3Copyright © 1978 by ASME

Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



l iquid region was investigated in reference [5] using numerical

methods combined with conformal mapping. The discussions a t the

end of reference [3] and the references listed in the three papers also

contain several re la ted publications of in terest . The present analysis

will obtain an analytical solution for a two-dimensional solidification

problem arising during continuous casting of a slab-shaped ingot. The

physical conditions are described at the beginning of the analysis. For

these condit ions th e shape of the solid ificat ion in terface is found to

depen d on a s ingle d imensionless param eter .

The particular problem being studied here arose in connection with

the direct formation of composite s tructures by directional-solid if i

cation from a eutectic a l loy melt . Composites consis t ing of a re in

forcing phase within a metall ic matr ix can be formed from molten

eutectics by proper control during solid ificat ion (references [6] and

[7]) . One of the req uirem ents of the process is th at t he solid if icat ion

interface be ke pt p lanar within a g iven to lerance . A second req uire

ment is that there be a suff ic iently h igh temperature gradient in the

mol ten me ta l a t the in te r face ; the requ i red tempe ra tu re g rad ien t i s

re la ted to the ra te of cast ing. This gradient can be mainta ined by

having the molten phase superheated and by having sufficient cooling

of the solid phase . The in teraction of these heat transfer quanti t ies

will be studied here for a continuou s casting configuration. F or a given

heat addit ion and cooling ra te , the maximum casting ra te that cor

responds to maintaining the solidification interface within a specified

tolerance of f la tness can be determined. The results are in terpreted

in re la t ion to the d irectiona l solid if icat ion of eutectics .

A n a l y s i s

E n e r g y E q u a t i o n a n d B o u n d a r y C o n d i t i o n s . The geome t ry

being analyz ed is shown in Fig . 1(a). An ingot is being formed by

uniform w ithdrawa l at velocity a from an insulated mold. In the region

below the mold the sides of the ingot are being strongly cooled, such

as by forced convection of a liquid metal, and are assumed at uniform

t e m p e r a t u r e tc. The molten eutectic a l loy above the in terface is

main ta ined in a supe rhea ted condit ion and conseque ntly transfers

heat to the solid-liquid interface. The superheat is necessary to control

the crysta l s truc ture in the a l loy in the process under consideration.

I t is assumed that there is a uniform heat transfer to the in terface

either by conduction, ki(dti/dn), or by convection h(ti — t/); in the

latter case the appr oxim ation is ma de tha t the hea t transfer coefficient

is uniform over the surface of the in terface . S ince the ingot is being,

withdrawn at a uniform rate , the la tent heat of fusion removed perunit are a locally at the interface is up A dx/ds where ds is a differential

length a long the in terface . The boundary condit ion a t the in terface ,

is then

dt I _ dxk— = q(s) = qi + up\ —

dn I s ds(1)

where qi can be by con duction or convection. As will be discussed later

in more deta i l , for purposes of controll ing the crysta l s tructure , the

condition s of intere st are slow withdra wal rates (in the range of several

cm/hr) and a large temperature gradient (can be 100°C/cm or more)

in the mo lten a l loy adjacent to the in terface . Hence, the last term on

the right side of equation (1) is usually much smaller than the qi t e rm.

It is a lso necessary that the in terface be kept c lose to p lanar . This

m e a n s t h a t dx/ds in the last term of equation (1) is approximately

unity and does not cause s ignif icant varia t ion in q(s). As a result, to

a good approxim ation, for a l l the condit ions of in terest here the q (s)

can be regarded as constant over the in terface and equal to

, dt\: k — = qi + up\dn\s

(2)

In the s i tuation of low interface curvature , dx/ds ~ 1 and eq uation

(2) will be a good app rox im atio n, even for large up A. Wh en t he in

terface cu rvatu re is large, equation (2) wil l be a good ap proxim ation

only when up X « qi. A second boundary condition along the interface

is that the temperature is constant and equal to the fusion tempera

tu re ,

Us) = tf (3)

Th e other b oundary cond it ions are tha t a long the insula ted mold (see

Fig . 1(a))

^ = 0dx (x = ±a, v > —6)

and along the cooled boundary,

t • (x = ±a , y < —b)

(4)

(5)

Th e dim ension 6 in Fig . 1(a) is unkno wn an d wil l be found in the so

lu tion as a function of the heat transfer quanti t ies .

The ingot is withdrawn slowly a t a s teady ra te , and i t is assumed

tha t the ene rgy pu Cpdt/dy being transported by the movement is

small compared to the heat being conducted. From results such as

thos e found in reference [8], this is a satisfa ctory as sum ptio n if the

P ec le t number u2a/(k/pCp) is less than unity. As will be discussed

later, the u values of interest here ar e quite low, in the range of several

cm per hr. This yields Peclet numbers that are well below unity. Hence

the energy equa tion to be solved within the solid ified m ateria l is La

place 's equation,

V 2 t = 0 (6)

I t might seem that the boundary condit ions are overspecif ied s ince

equations (2) and (3) g ive both the heat f lux and temperature a t the

same boundary. However, i t is the shape of the boundary that is un-

- N o m e n c l a t u r e .

A = phys ica l pa rame te r and d imens ion le ss

length = a/y = (q i + up\)a/k{tf - tc)

a = half-width of ingot

b = dis tance from center of in terface to bot

tom edge of mold

Cp = specific heat of solid

Ci, C2, e tc = in tegration c onst ants

c, d = values in mapping a long real axis of

u -p lane

F = rat io , (dti/dn)s/u

h = convective hea t trans fer coefficient

K = complete e l l ip t ic in tegral of the f irs t

k ind

k = thermal conductiv ity of solid if ied

m e t a l

ki = thermal conductiv ity of l iquid metal

/ = length of ingot

n = normal to in terface , N = n/y

Q = heat f low per unit t ime

q = heat f lux

qi = heat flux from liquid metal to solidifi

cation in terface

s = dis tance a long in terface

t = t e m p e r a t u r e ; T = (t - tc)/(tf - tc)

ti = tem pera tu re in l iqu id me ta l

u = coord inate in u-pla ne, u = £ + it)

u = withdrawal velocity of ingot

V = in te rmed ia te p lane in mapp ing

W = po ten t ia l p lane , W = -T + i\p

x,y = co ordinates in physical p lane, X = x/y,

Y = y/y

z = complex physical p lane, z = x + iy; Z =

z/y

a,0 = inflection points on solid if icat ion in

terface

7 = length scale parameter , k(tf - tc)/(qi +

upX)

5 = deviation of interface from being flat; A

= 8/7f = complex derivative , -dT/dX + idT/dY

0 = argument of f-p lane

X = l a ten t he at of fusion per uni t mass of

solid

£, i) = coordin ates of u -plan e

p = density of metal

i< = co ordi nate alo ng hea t flow in W p lane

co = m app ing p lan e, log | f + id

S u b s c r i p t s

c = at cooled surface

/ = a t solid if icat ion te mp erat ure

1 = l iquid metal

s = a t solid if icat ion in terface

4 / VOL 100, FEBRUARY 1978 Transact ions of the ASME




known and must be found to s imultaneously sa t isfy these two con

dit ions. The solution wil l be found by using a conformal mapping

p rocedu re .

D i m e n s i o n l e s s V a r i a b l e s . Before beginning the conformal

mapping it is convenient to adopt a system of dimensionless variables.

A dimen sionless tem per atu re is defined as T = (t — tc)/(tf — tc). T o

simplify th e bou ndary condit ion in equatio n (2 ) , a length scale pa

rameter is defined as

7 :k(t f-t c) k(t/-tc)

q s qi + up\

and the dimensionless lengths then become N = n/y, X = x/y, etc .

The bounda ry cond i t ions become

dN ~

at solid if icat ion in terface

T = 0 at cooled surface

(7a)

(7b)

These condit ion s are shown in Fig . 1(6) which gives the geom etry in

d imens ion le ss coo rd ina te s . Th e numbe rs a round the bounda ry wi ll

designate corresponding points in the conformal mapping.

C o n f o r m a l M a p p i n g P r o c e d u r e . The basic ideas for applying

conformal mapping to some types of solid if icat ion problems were

developed in reference [9]. Since positive he at flow is in the d irecti on

of negative temperature gradient , the —T can be taken as a potentia l

function for heat flow, and we can let —T be the real part of an a nalytic

function of a complex variable

W= -T + ii (8)

that sa t isfies Laplace 's e quation. T he l ines of constan t T a nd \p form

an orthogonal net where the constant \p l ines are a long the di rection

of heat f low. Hence, l ines 23 and 78 in Fig . 1(6) are constan t \p l ines

as there is no heat f low normal to them because the boundary is in

sula ted . As a result , the solid if ied region maps in to a rectangle

bounded by constant T an d \!< lines as shown in F ig. 2(a). If this region

can be conformally mapped in to the physical p lane, then the known

temperature distribution in Fig. 2(a) (uniformly spaced vertical lines)

wil l g ive the temperatures a t the corresponding points in Fig . 1(6) .

In the mapping process the normal derivative condit ion a t the in

terface , dT/dN = 1, will be satisfied.

The potentia l and physical p lanes can be re la ted by use of the de

rivative of equation (8)

dZ ~ dXl

dX '

-dT . dT+i —

dX dY

(9 )

where the last re la t ion on the r ight was obtain ed by using one of the

Cauchy-Riemann equations. Defining f as f = —3T/3X + i dT/dY,

equation (9) can be in tegrated to y ie ld

-fr dW+Ci (10)

By carrying out th is in tegration, the physical (Z ) plane is re la ted to

the po ten t ia l (W ) p lane , and the known tempera tu re d i s t r ibu t ion inFig . 2(a) can be mapp ed in to Fig . 1(6). Before the in tegr ation can be

carr ied out , the f has to be re la ted to W. This re la t ion wil l in troduce

the tempera tu r e de r iva t ive bounda ry cond i t ions tha t mus t be sa t i s

fied.

The temperature derivatives a t the boundaries of the solid if ied

region are used to map the region into the f-plane shown in Fig. 2(6).

Along the solidification interface dT/dN = 1, so that (dT/dX)2 +

(dT/dY)2 = 1 and th is boun dary l ies a long a unit c irc le . Along th e

constant temperature cooled surfaces dT/dY = 0 and a long the in

sulated surfaces dT/dX = 0. Th e mappin g between Fig . 2(6) and F ig .

2(a) which is needed to in teg rate equation (10) wil l be done by mean s

of a few intermediate transformations.

Firs t , a logari thmic transformation is used:


HEAT

FLOW

WITHDRAWAL

VELOCITY, u

Fig. 1(a ) Physical geometry

6 | 5

Fig. 1(b ) Dimensionless physical plane, Z = X + iY

Fig. 1 Ingot being withdrawn continuously from mold

FEBRUARY 1978, VO L 1 0 0 / 5




co = log f = log I f j +i0 (11)

where 6 is the argu men t of fas shown in Fig. 2(6). Along the un it circle

in Fig . 2(6) , « = id, so th is becomes a vert ical l ine in Fig . 2(c) .

The boundary of Fig. 2(c) can be unfolded along the real axis so tha t

the solidified region occupies the upper half plane. The result of this

is shown in Pig . 2(d) , and the m appin g is accom plished by mean s of

the Schwarz-Chris toffe l t ransform ation

__

du

C2(u + d){u -d)

(u - l)(u + l)Vu- cVu + c

Equation (12) can be in tegrated to y ie ld

- u , (1 - cf 2)

(12)

i = Ci1 , Vu

2- c

2— "

• - lo g — 7 = =2 VTT^T

2 + u 2 V T

X l o g;2 - uV \

\ V - c 2 4 . M v T

~ e 2 " |+ C 3 (13)

The co r re spond ing po in t s are ma tched be tw een the 01 and u -p lanes

to evaluate C 2 and C3, and to f ind a re la t ion be tween c and d. P o i n t

1 a t u = 0 corresponds to a - litII, point 2 a t u = c must yield _ = iw/2,

and point 3 a t u = 1 must correspond to a = °° + J7r/2 o r » + (V de

pending on whether u = 1 is approa ched from below or above a long

the real axis . The se condit ion s lead to C% = — 1 , C3 = iir/2 and 1 - d2

= Vl — c2. Thi s last relation will be used later to locate the inflection

points (a and (3 in Fig. 1(6)) along the frozen interface as this corre

sponds to po in t s u = ± d. W ith these values <~ becomes,

o = log . =1 Vu2-7 l o g — = =

•c/-u

c'1 + u

1 \/Ti2~-~c

2- uV\ — c

2

. _ J0g — - _ _ _ _ _ _2 V ^ 2 _ T ^ 2 + u V l ~C

2

+ i- (14)

Continuing the mapping transformations to re la te W and f, which

will be done by mean s of the in term ediat e variable u , the real axis of

th e u -plane is folded in to a rectangle in the V-plane shown in Fig. 2(e).

The solid if ied region is inside th is rectangle . The mapping between

u an d V is found from the Schwarz-Chris toffe l t ransform ation as

dV__

du

C4(15)

1 - c v u + c v u ^ V u T l

Then by rota t ion and transla t ion the V-plane is transformed in to

the W -p lane to comple te the mapp ing

W-- -iV - 1 (16)

Now retu rn to the in teg ration of equation (10). This is writ ten in the

form,

r 1 dW

~ J f(«) dV

1 dWdV

dudu + Ci (17)

P rom equa t ion (14 )

1 1 /V u2-c

2+ u\ 1/2 /V u

2-

i

uV T/ V l l ' - C ' t U y K / V u ^ - c ' - UV 1 — C ' \

i \Vu2 ~c 2~u ) KVH^T^ + uVT^c^)

(18)

Su bsti tu t in g equatio ns (18) and (15), and the derivative of equatio n

(16) yields

r 1 /Vu2

~ ci + uy/2 /Vu2

- c2

- uVT^^2

y/22

- c2 + u v T - c

Vu2 ~ c2Vu 2

1zdu + Ci (19)

3/ -C OOLED

/ SURFACE

T = 0

- - T

^ ^ 7

Fig. 2(a) Potent ial plane, W = - T + i\p

| - ° ° ,_

5 J—

(-°°0)

4

(O.rri

- (0,7i /2H1

a Y

(0,0)

, . -SOLIDIF ICATION (« 71)

INTERFACE '|

* 7

2 ' - INSULATED 3

SURFACEK 0 )

— log K l

Fig. 2 (c ) Inte rmed ia te co plane , _ = log | f | + 10

-INSULATED

SURFACE

5 4

- SOLIDIFICATION

INTERFACE

6 5

_ _ _ _ ^ _ _ _ _ . _ , . _ _ ^ _ _ _ _ .

( -°°,0) 1-1,01 (-C.0I (-d,0) (0,01 «J .0J (CO) (1,0) K 0 )

Fig. 2 (d ) Inte rmed ia te u plane , u = £ + / i j

dY

(0,<*>>

, - INSULATED

SURFACE

. - SOLIDIFICATION

INTERFACE

-^ dT

? (-1,0) 6 5 4 (1,0) 3 dX

Fig. 2 (b ) Tem pera t ure der iva t ive plane , f = -dT/dX + idT/dY

6 / VOL 100, FEBRUARY 1978

44 5 6

\x

'-COOLED (0,D

SURFACE SOLID IFACTION

INTERFACE-1

2 a 1 9 p 8

F ig . 2 ( e ) I n t e r m e d ia t e V p la ne

Fig. 2 P lan es used in conformal mapping pro cedu re

Transact ions of the ASME


http://v-c2/

http://v-c2/



Equa t ion (19 ) is in teg ra ted to give

- i C 4 ( l - V l ^ 2 ) '

z = -VTJ^ c2

- uVl-

2c2 Vu

2- c 2

+ u V l - c2

+ log v^n ;- log 0u

2-l)]

The constants are evaluated by having the orig in Z = 0 at u

+ d (20)

: 0 a n d

the width of the ingot be A, which means th at the real part of Z m u s t

equal - A when u = 1. Th is g ives Ci = - A / 2 a n d C 4 = ( -A/7 r ) [c 2 / ( l

- Vl — c 2 ) ] . Th en equa tion (20) becomes

Z _ z _ _ i T Vu2 - c2 - u Vl - c2

A a 2TT L °g V ^ ^ + u V l - c

2

+ logVu

2- i

Vu2

— c 2+ u

logCu2- l)]- i (21)

Equat ions for Sol id i f i c at ion Inte r fac e . The in terface between

po in t s 8 and 9 in Fig . 1(6) corresponds to points a long the real axis

u =£ in Fig . 2(d) where 0 < £ < c. Since £ is less than c, some of the

quanti t ies in the sq uare root s in equation (21) will be negative, so tha t

a long the in terface z/a becomes ,

zsi [, iV c

2— £

2

log — , — ,

a 2x I iV c2-£

2+£Vl -c

2

2 iV c2- £

2- £

- + l o g — ; = = b i

J V C2

- J2

+ £

- l o g f - a - f 2 ) ] -

The log of a complex numbe r can be wr i t ten in t e rms ofthe inverse

tangent, so th is reduces to

z, 1 T^ = - tO IT L

Vc2

- £2

+ tanV c

2- £2

- ] 2TT; U - £

2) -

f v T - c 2£

Separating real and imaginary parts and taking in to account the fact

t h a t the in terface is symmetr ic abou t x = 0 gives the pa rame t r ic

equations a long the in terface ,

a H a n _,Vc3£g£ V T ^ c 2

1 v ^- t a n l H2

0 < £ <c

(22a)

a• i o g u - 4

2) (22b)

The value of the height 5 in Fig . 1(a) is of in terest as it shows how

far the in terface deviates from being planar . Po int 2 in Fig . 1(6) cor

responds to £ = -c in Fig . 2(d) , so from equa tion (21(b)) the value of

<5/a is

< 5 1 , ,- = - - l o g ( la 2ir

•c 2) (23)

Another quanti ty of in terest is the height 6 ofthe in terface above

the bottom edge ofthe mold which is at point 7 in Fig . 1(6) . Point 7

corresponds to u = 1 in Fig . 2 (d ) so that from equation (21) b/a =

\lim u-,i Im(z/a)\. This y ie lds

6• = — l o g

a lit

" ( 1 - v T - ,(24)

4 (1 - c 2) J

In Fig . 1(6) , it is no ted tha t the re are inf lection points a long the

in terface at po in t s a a nd /?; the se co r re spond to J = ±d in Fig . 2 (d ) .In connection with obtaining the constants in equation (14) , it was

found tha t d2

= 1 — V l — c2. Ifwe le t £ = d in equation (22(a)) and

then subs t i tu te d = (1 - V l

x s \ r1

— = ± l - - t a n _ 1a \ i n f l ec t i on [_ 7T

p o i n t

From equa t ion (22b )

CE 1 i n f l ec t i onp o i n t

— ,

u-

1

2w

c 2 ) 1 ' 2 , we obtain

1

- - t a n - H l - c 2 ) 1 ' 4

7T

l o g ( l - c 2 ) 1 / 2= i -

2 a

Hi (25a)

(25b)

wher e <5/a is given by eq uat ion (23).

E x p r e s s i o n for c in T e r m s ofP h y s i c a l C o n d i t i o n s . T h e p r e

vious results are in t e r m s ofthe quan t i ty c tha t ha s va lues be tween

0 and 1 , but ar ises during the conformal ma ppin g witho ut any par

ticular physical significance. The c will now be related to the imposed

physical condit ions. By fixing points in the physical p lane, it w as

found earlier in connection with equation (20) that C 4 = — (A/TT) [C2/( 1

— V l - c2 )] . T h e C 4 can a lso be determined from the fact that the

wid th of the rectangle in Fig . 2 (a ) is un i ty as fixed by the specified

temperature d ifference. This g ives

W(7) - W{$) = 1 ••

Insert ing equations (15) and (16)

X idWdV ,du

dV du

f ( - 0Cidu

Vu2

- c 2 Vu2

- 1 Jc Vu2

du

c V T ^ u 2

The in teg ra l is another form of the complete e l l ip t ic in tegral of th e

first kind

K(VY^72)= C

Jo

dip

V l - (1 - c2) s in 2

ip

so that C 4 = —l/K(Vl — c 2 ) . Equa t in g th i s to the previously deter

mined value for C 4 gives

A =qsa i r ( l - Vl^c2)qi + up\)a

k( tf-t c) ~k(tf-tc)~ c ^ f v T 1 ? )(26)

T h u s t h e c can be determined from the physical inputs that specify

the value for qsa/k(tf — tc).

T o t a l H e a t F l o w to C o o l a n t . A q u a n t i t y ofsome in terest is the

tota l amount of heat being transferred through the solid if icat ion in

terface and then removed by the coolant . Thi s includes both the he at

f low from the superheated l iquid metal and the la tent heat removed

at the in terface . When the in terface is planar th is is Q = 2alqs, but

as the in terface becomes dis torted , the Q increases in propo r t ion to

the increased surface area . S ince the Q at the in terface must exit

through the cooled surfaces, the Qcan be writ ten as

J-y(v) at

k — dyy(6) dx

P u t t ing th i s in d imens ion le ss fo rm and u s ing the Cauchy -R iemann

equa t ion 3T/dX = -d\p/dYyields after inte grat ion Q/2lk(tf — tc) =

i/-(6) - \H7) = </<(9) - <p(8) = Im [W(9) - W(8)}. By using equations

(15) and (16) th is can be writ ten as ,

Q ru=cdWdV ,= Im \ du

2lk(tf-t c) Ju=o dV du

= Ims :- V , « 2 - c V « 2 - lzdu

Insert ing C 4 , and noting that the integration is along the £ axis of the

u -p lane wi th 0 < £ < c < 1, yields

d£r"

f^e

2

) Jolk(t

f-t

c) K(VT^

2)Jo V c 2 - £ 2 V F - £

K(c)

where K(c) is the complete e l l ip t ic in tegral

d£ /^r/2(c)- £c 2 - £ 2 V T ^ £ 2 - I

K(VT-^c2)

dip

(27)

V l — c 2 s i n 2ip

R e s u l t s a n d D i s c u s s i o n

The solution has been obtained in parametric form in terms of the

quanti ty c which is aquanti ty in the conformal mapping without d i

rect physical s ignif icance. The solution shows that the controll ing

phys ica l pa ram e te r i s qsa/k{tf - tc) where qs is the to ta l a mou nt of

Journal of Heat T ransfer FEBRUARY 1978, VOL 1 0 0 / 7Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



heat carried away from the interface by conduction through the solid.

T h e qs is equal to the sum of the hea t supplied by the molten eutectic

al loy to the in terface by convection or conduction, p lus that due to

la tent heat . The physical parameter is re la ted to c by equation (26) ..

This equation is used to obtain the c corresponding to a q sa/k(tf —

tc) of in terest ; then th is c is used in the o ther pa rts of the solution to

obtain the in terface location and shape, and the to ta l ra te of heat

removal corresponding to the g iven qsa/k(tf — tc).

In t e r f ace S h ape . The in te rface shape i s ob ta ined f rom equa t ion

(22), and the height of the center of the in terface above the bottom

y s - b

edge of the mold is obta ined from eq uation (24) . Prom thes e results ,

Figs . 3 and 4 are obtained. T he in terface sha pe has been norm alized

in Fig. 4(c) by dividing by the length <5 which is the total variation in

interface height from centerl ine to wall .

Fig. 3 shows how the interface moves further up into the mold and

becomes more planar as qs or the width a is decreased, or k(t/ - tc)

is increased. One of the condit ions for forming the proper in ternal

s tructure during directional solid if icat ion is that the solid if icat ion

interface not deviate very far from being planar. As seen in Fig. 3, the

tolerance on interface flatness determines how large A = (qi + upX)a-

/k(tf — tc) can be. The devia t ion from being planar is shown in more

deta i l in Fig . 4(a) . I f the maximum value of A for being within theplanar l imits is ^4 ma x, then the max imum wi thd rawa l r a te is

2.2

2.0

. .. _ . _ . .L 8

1.6

— - 1A

10

a (q^ + upMaA k(t f - tc)

0.40

.4 5

.5

.55

.6

.7

.8

1 \k(tf-tc)i rk(tf-

pXt aIi (28)

For example , from Fig . 4(a) , to have an in terface f la tness such that

la) DEVIATION OF INTERFACE FROM BEING PLANAR.

2.0

1.6

.8 1.0 a

0 .4 .8 1.2_ 1.6 2.0 2.4% + up?i)a

Fig. 4(b) Heigh ! of interface above bol lom edge of mold

Fig. 3A

Location and shape of solidification interface for various values of

1.0_ 1.2 1.4

(qj + upXaA"~WW

Fig. 4(a ) Deviat ion of interface from being planar


Fig. 4(c ) Norma lized shape of solidification interface

Fig. 4 Interface characterist ics

Transact ions of the ASf




6/a < 0.05 requires that the A m a x be less th an 0.8. Fig . 4(fe) gives de

tailed information on the interface deviation from flatness and vertical

location within the mold.

I t was shown in the analysis that th e inflection point a long the in

terface is located half way both vert ical ly and horizontal ly between

the ingot centerl in e and the m old surface (see Fig . 4(c)) . This m eans

t h a t if the in terface is norma l ized to unit height , the curves for all

values ofA will pass through a common centra l point . As shown by

Fig . 4(c) , the normalized profi les are practically the same over the

entire range ofA values.

F ig . 5 shows how the to ta l heat f low through the ingot varies as a

function of the pa rame te r A. For sma l l A, the in terface is essentia l ly

flat, and Q = 2alqs, hence the curve becomes a l ine of unit s lope. F or

larger A the in terface area becom es curved so there is a greater area

for heat transfer and the curve bends upward.

A p p l i c a t i o n to D i r e c t i o n a l S o l i d i f i c a t i o n of E u t e c t i c s . By

proper control of the solidification process, it is possible to obtain an

aligned eutectic structure in an ingot solidified from a molten eutectic

al loy. This a l igned s truct ure , extending a long the length of the ingot,

provides desirable s trength characteris t ics . For proper control of th e

eu tec t ic s t ruc tu re it is necessary to fulfill two conditions (references

[10, 11]). The interface must be f la t with in a certa in to lerance. The

second condit ion is that the temperature gradient in the l iquid metal

a t the solid if icat ion in terface divided by the ingot growth ra te must

be above a ce r ta in va lue tha t is de te rmined by the microscopic be

havior at the in terface during solid if icat ion of the eutectic a l loy. If

th is ra t io iscalled F, the pa rame te r A ca n be rearranged in to

[k t(dti/dn)s/u + p\]au (k,F + p\)au

k(t/-tc) k(tf-t c)

The m ax imum cas t ing ra te of the ingot will then correspond to when

A has its maximu m va lue co r re spond ing to the required f la tness of

the in terface , and F has its min imum va lue requ i red by the me ta l

lurgical behavior at the solid if icat ion in terface . Then

AmB Xk(t f - tc)

(kiFmin + p\)a(29)

To i l lustra te th is re la t ion, it is helpful to have a few numerical ex

amples. The calculations will be made with a nickel base eutectic alloy

containing s trengthening phases of Ni3Al and Ni3Cb, and having the

following pro pert ies : k of solid = 0.277 W/cm °C ; k of liquid = 0.152

W/cm °C; p of solid = 0.00839 Kg/cm3

; A = 56.2 W h r / K g ; t, =1250° C.

From p rac t ica l expe rience in ingot cast ing, for a prope r in te rna l

s t ruc tu re it is desired to have the interface sufficiently flat so that hia

< 0.0035. Th en from F ig . 4(a) the m axim um v alue of A tha t can be

imposed is A m a x = 0.48. For proper control of the a l igned eutectic

structure the Fmj„ for th is eutectic system is 150°C hr/ cm 2 (reference

[9]) . Insert in g these values in to equation (29) y ie lds ,

u m a x ( c m / h r ) = •(0.48)(0.277)[1250 - i c( ° C ) ]

[(0.152)(150) + (0 .00839)(56.2)]a(cm)

Some sample results are ,

a,

cm

0.1

0.5

1.0

tc = 750° C

28.6

5.71

2.86

u m a x , cm/h r

tc = 250° C

57.1

11.4

5.71

These values i l lustra te how the maximum cas t ing ra te can be in

creased by decreasing tc or a. The casting configuration analyzed here

is concerned with an advanced system inwhich the ingot is d i rec t ly

cooled in the region below the mold . Cur ren t indus t r ia l p rac t ice

usually has the solidifying material entirely contained in a mold tha t

is externally cooled. Th is provides a lower heat transfer than for th e

present configuration, and cast ing ra tes of 0.6 — 1.0 cm/h r a re com

monly obtained for turbine blade cast ings about one to two cm wide.

2.4

2.0

1.6

~r 1.2

S3

//

//

,' ///

//;//

LQ = 2 a % + upX)

0 .4 _ 1 . 2

(q^ +UpX)a

1.6 2.0

w t f - y

Fig. 5 Total heal flow at th e interface as a function of the parameter A

These ra tes are consis tent with the present results .

ConclusionsAn analytical solution for the solidification of an ingot during

continuous cast ing has been obtained by using a conformal mapping

m e t h o d . The two-d imens iona l shape of the solid if icat ion in terface

was obtained, and the in terface posit ion within the mold was found.

The in terface shape and posit ion are governed by a single parameter

(qi +up\)/(k/a)(tf — tc), where the numera to r is the to ta l heat f lux

being conducted away from the interface as a result of heat flow from

the superheated liquid metal and the latent heat of fusion. The results

show how the f la tness of the in terface depends on the heat addit ion

to the in terface from the superheated l iquid metal in contact with it,

an d on the solid if icat ion ra te . An increase in either of these twoquan t i t i e s causes the in terface to become more curved. Regulating

the c ry s ta l s t ruc tu re in a eutectic a l loy to obtain a l igned growth de

pend s on the f la tness of the in terface and on the ra t io of the tem per

ature gradient in the l iquid metal at the interface to the solidification

rate. For these two quantities specified, according to the requirements

fo r a part icular eutectic a l loy, the results predic t how fast the ingot

ca n be withdrawn from the mold. The withdrawal speed can be in

creased by decreasing the width of the ingot or by lowering the tem

p e r a t u r e of the cooled boundary.

A c k n o w l e d g m e n t

The au tho r app rec ia te s the helpful suggestions of Coulson M.

S c h e u e r m a n n of the Lewis Research Center .

R e f e r e n c e s1 Muehlbauer, J.C and Sunderland, J. E . "Heat Conduction with Freezing

or Melting," Appl. Mech. Rev., Vol. 18, Dec. 1965, pp. 951-959.2 Okendon, J. R. and Hodgkins, W. R., eds., Moving Boundary Problems

in Heat Flow and Diffusion, Clarendon Press, Oxford, 1975.3 Wilcox, W. R. and Duty, R. L., "Macroscopic Interface Shape D uring

Solidification ," JOURNAL OF HEAT TRANSFER, TRANS. ASME, Series C, Vol.88, F eb. 1966, pp. 45-51.

4 Sfeir, A. A. and Clumpner, J. A., "Continuous Casting of CylindricalIngots," JOURNAL OF H E A T TRANSFER, T RANS. ASME, Series C, Vol. 99, Feb.

1977, pp. 29-34.5 Kroeger, P. G. and Ostrach, S., "The S olution of a Two-Dim ensional

Freezing Problem Including Convection Effects in the Liquid Region," Intl.Jour, of Heat and Mass Transfer, Vol. 17, no. 10, Oct. 1974, pp. 1191-1207.

6 Mollard, F. R. and Flemings, M. C, "Growth of Composites from theMelt-Part I," Trans. Metall. Soc. AIME, Vol. 239, Oct. 1967, pp. 1526-1533.

Journal of Heat Transfer FEBRUARY 1978, VOL 100 / 9




Mollard, F. R. and Flemings, M. C , "Growth of Composites from the Melt-P art 9 Siegel, R., "Conformal Mapping for Steady Two-Dimensional Solidifi-II," Trans. Metall. Soc. AIME, Vol. 239, Oct. 1967, pp. 1534-1546. cation on a Cold Surface in Flowing Liquid ," NASA TN D-4771,1968.

7 Jackson, K. A. and Hunt, J. D., "Lamellar and Rod Eutectic Growth," 10 Flemings, M. C , Solidification Processing, McGraw-Hill Book Co.,Trans. Metall. Soc. AIME, Vol. 236, Aug. 1966, pp. 1129-1142. 1974.

8 Schneider, P. J., "Effect of Axial Fluid Conduction on Hea t Transfer 11 Scheffler, K. D., et al., "Alloy and Struc tural Optimization of a Direc-in the Entra nce Regions of Parallel Plates and Tu bes, " TRANS. ASME , Vol. tionally Solidified L amellar Eutectic Alloy," P WA-5300 (NASA CR-135000),79, No. 4, May 1957, pp. 765-773. Pr att & Whitney Aircraft Co., 1976.

A N N O U N C E M E N T

The following short courses wil l be run a t the 23rd Annual In ternational Gas Turbine Conference, April 9-13,1978,

in London , England . Th ey ma y be of in terest to engineers in the f ie ld of Hea t Trans fer . All courses will be offered a t th e

Grosvenor House in London. For further information and regis tra t ion contact:

J i l l Jacobson

Professional Development Dept.

A S M E

345 E. 47th St .

N.Y. , N.Y. 10027

(212) 644-7743

Name of Course: Compac t Hea t Exchange rs

Instructors: R. K. Shah and A. C. London

Date: April 9 ,1978

Fee: $150.00

Description: A comprehensive review of the methods and problems associa ted with the design of compact heat ex

change rs .

Name of Course: Blade Des ign Deve lopmen t and F ie ld E xpe r ience

Instructors: C. L. Smith , D. J . Leone, L. E. Snyder, J . W. Tumauicus, H. S targardter , E. W. Jansen

Date: April 9 ,1978

Fee: $150.00

Description: A series of s ix lectures on G as Turb ine b lade bea ms.

Name of Course: In t roduc t ion to Gas Tu rb ine

Instructor: E. Wr igh t

Date: April 9, 1978

Fee: $50.00

Description: An overview of the Gas Turbin e cycles part icularly suited to newcome rs in th is f ie ld .

Name of Course: Fou nda t ion o f Tu r bo M ach ine ry

Instructors: Drs. Chauvin , Mikolajczak, Serovy, and Lakshminarayana

Fee: $150.00

Description: A series of four lecturers on Turb o M achinery .





E. M. Sparrow

R. R. Schmidt

J . W. Ramsey

Department of Mechanical Engineering,

University of Minnesota,

Minneapolis, Minn.

cxpenmei

Confection in the ielting of SolidsExperiments are performed whose results convey strong evidence of the dominant role

played by natural convection in the melting of a solid due to an embedded heat source.

The research encompassed both melting experiments and supplementary natural convec

tion experiments, with a horizontal cylinder as the heat source. For the melting studies,

the cylinder was embedded in a solid at its fusion temperature, whereas in the natural

convection tests it was situated in the liquid phase of the same solid. A special feature of

the experiments was the use of a grid of approximately 100 thermocouples to sense thermal events within the phase change med ium. The time history of the heat transfer coeffi

cients for melting was characterized by an initial sharp decrease followed by the attain

ment of a minimum and then a rise which ultimately led to a steady value. This is in sha rp

contrast to the monotonic decrease that is predicted by a pure conduction model. The

steady state values were found to differ only slightly from those m easured for pure natu

ral convection. This finding enables melting coefficients to be taken from results for natu

ral convection. The positions of the solid-liquid interface at successive times during the

melting process also demonstrated the strong influence of natural co nvection. These in

terfaces showed that melting primarily occurred above the cylinder. In contrast, the inter

faces given by the conduction model are concentric circles centered about the cylinder.

In t roduct ion

In this paper, experiments are described which are aimed at i l lu

minating the heat transfer processes which occur when a solid is

melted by a heat source embedded within the solid . The currently

standard analytical approach to such problems is based on heat

conduction as the sole transport mechanism. On the other hand, i t

can be reasoned that the temperature differences which necessarily

exist in the melt region will induce buoyancy forces and these, in turn,

will generate natural convection motions. I t appears , however, that

experiments have not yet been reported which specifically address

this issue, aside from those on convective instabilities involved in the

melting of horizontal ice layers (e.g., [I])1

and those presented in [2]

subsequent to the submission of this paper ([2] will be elaborated

shortly) . The need for such experiments has motivated the present

work, which is part of a larger study of energy storage via solid/liquid

phase change.

Separate experiments on melting and on natural convection were

performed to obtain a definit ive relation between the heat transfer

coefficients for the two cases. In the me lting tes ts, a solid was initially

brought to i ts fusion temperature by external heating. Then, the ex-

1 Numbers in brackets designate References at end of paper.Contribu ted by the He at Transfer Division for publication in the JOURNAL

OF HEAT TRANSFER. Manuscript received by the Heat Transfer DivisionAugust 2,1977.

periment was init ia ted when a horizontal heating cylinder embeddedin the solid was energized to give a s teady rate of heat transfer . Th e

cylinder was instrumented with thermocouples and, in addition, an

array of 92 thermocouples was deployed throughout the phase change

medium with the aid of a specially designed fixture . The cylinder-

mounted thermocouples facil i ta ted the evaluation of instantaneous

heat transfer coeffic ients , while those in the medium enabled the

posit ion of the solid-liquid interface to be identif ied. The data runs

encompassed a parametric range of heat f luxes from 1430 to 7870

W /m2.

These experiments were performed with a eutectic mixture of so

dium n itrate and sodium hydroxide, which has a melting tem perature

of about 244°C (471°F). This substance has been considered for ap

plication as an intermediate range thermal s torage medium. I t has

the virtu es of being inexpensive, non-corrosive (no corrosion of a steel

tank over a period of one year) and non-toxic .The aim of the second group of experiments was to determine

natural convection heat transfer coeffic ients with the horizontal

heating cylinder situated in a purely liquid environment, specifically,

in the l iquid phase of the aforeme ntioned eutectic . For these experi

men ts , the tempera tu re o f the l iqu id env i ronmen t was ma in ta ined

as close to the fusion temperature as possible . The same instrumen

tation and heat fluxes were employed in the natural convection studies

as in the melting experiments .

The heat transfer coeffic ients for the melting experiments are

plotted here as a function of t ime and their trend, re lative to the

monotonic t imewise decrease of those for pure conduction, provides

immediate insight into the nature of the transport processes. The


Copyright © 1978 by ASME




natural convection coefficients are also plotted as a function of lime.

The steady state coeffic ients for the two groups of experiments-

melting and natural convection—are then compared to provide fur

ther evidence about the transport processes. The final evidence is

furnished by the shapes of the solid-liquid interface w hich were de

duced from the thermocouple s'rid in (he phase change medium.

The experiments of |2j and those reported here appear to have been

concurrent and independ ent efforts a imed at a comm on goal, here

are bir th s imilari t ies and differences in the two sets of experim ents

and , mos t s ig n i f ica n t , the i r qua l i ta t ive f ind ings a re mutua l ly sup

portive. Quantita tive comparisons are diff icult to make because of

I he different media employed and because the results of [2] may have

been influenced by subcooling and end losses and w ere not suppl e

men ted by separat e natura l convection experim ents .

T h e E x p e r i m e n t s

E xp er im en ta l App ara tu s . T he appara tus tha t was des igned and

fabricated for the experim ents is shown in a vertical sectional view

in Fig. 1 . As seen there, i t is made up of a test cham ber surro unde d

by a multilayer wall consisting of heating panels and insulation layers.

Th e test c hamb er is a cubical, mild-steel ta nk, 33 cm (13 in.) on a side.

For the experiments , the test chamber was f i l led with the phase

change mate rial to a heigh t of about 25 cm (10 in . ) .

The test chamber is equipped with a horizontal heating cylinder

which consists of a thin-walled incoloy sheath housing a uniformly

wound electr ical resis tance element. The outside diameter of the

sheath is 1.9 cm (% in.) and its length is 25.4 cm (10 in.). As seen in Fig.1, the cylinder is parallel to one pair of side walls and midw ay b etween

them. Its centerline is about 5.4 cm (2 in.) above the floor of the

tank .

Two auxiliary heaters were installed in the test chamber to neu

tralize potentia l problems associated with the density change that

accompanies solid-liquid phase change. Since the l iquid phase oc

cupies a-greater volum e than the solid phase (for a given mass), i t is

necessary to provide expansion space. This was accomplished by the

use of the so-called vent heater , which passes vertically downward

through th e phase change material adjacent to one of the walls of the

chamber. When actuated, the vent heater opens a vertical channel

through the solid phase-change material , and this enables any ex

pansion fluid associated with melting around the horizontal cylinder

to escape to the top of the solid.

The second auxiliary heating device is s i tuated in the upper part

of the test cha mber an d is used to help prevent th e formation of voids

during the freezing of the phase change material which necessarily

precedes each melting run. One essential measure in the prevention

of voids is to avoid the formation of a solid skin across the top of the

freezing mass and, in general , to keep the upper par t of the mat erial

in the l iquid sta te unti l the very end of freezing. Th e accomp lishment

of these objectives was furthered by the just-mentioned heater . An

additional assis t in keeping the upper pa rt of the mate rial unfrozen

until the end was provided by the heating panel s i tuated above the

top of the test chamber.

A copper water-c arrying cooling coil affixed t o the lower edge of t he

outside walls of the test chamber was activated during the freezing

of the phase change material . I ts placement in the lower part of the

chamber en sured th at freezing would occur from bo ttom to top. This

direction of freezing is anot her im por tant m easure in the avoidance

of voids.

T he the rmocoup le in s t rumen ta t ion u t i l i zed in the te s t chamber

will now be discussed. All thermo couples were made from separa te ,

but m atched rolls of 30-gage teflon-coated chromel and alumel wire .

r iBHDUS K lS t lLAT ION_HEATING PANELS -

Fig. 1 Vertica l sectional view of the experim ental apparatus

The choice of teflon as a coating was based on i ts abil i ty to tolera te

temperatures up to about 290°C (550°F). Six thermocouples were spot

welded to the horizontal heating tube around a circumference situatedmid-way along the length of the tube. The thermocouple junctions

were separate d by an an gular spacing of 60 deg. Th e chromel and al

umel lead wires were stretched t ightly along the tube and were

brought out a t one end.

As was noted earlier , 92 thermocouples were deployed throug hout

the phase change material . These thermocouples were stretched ho

rizontally between a pair of supporting racks. The racks were re

spectively positioned at th e walls adjacent to the ends of the horizontal

cylinder so that the therm ocouples were paralle l to the cylinder. The

thermocouple junctions were si tuated in a vertical plane that en

compassed the mid-length posit ion of the heated cylinder. The

' chromel wire extended from the junction tow ard one of the racks and

the alumel wire extended from the junction to the other rack. By

suitable guides, the leads were brought vertically upward along the

side walls of the chambe r and were taken out through an access port .

Once outside the ap paratu s, the leads were drawn tightly , v ia special

springs, toward a connector panel. The tension was adjusted so that

the horizontally s tretched thermocouples in the test chamber would

no t sag du r ing the exper imen ts .

Th e test ch ambe r was surrou nded by a 5 cm (2 in. ) layer of high-

temperature spun glass insulation at the s ides and top, and by a 3%

cm (IV2 in.) thick layer of calcium silicate block ins ulati on be low.

External to the insulation, each side, the top, and the bottom were

equip ped w ith a heating panel. Each pa nel consisted of a 0 .95 cm (%

in.) th ick aluminum plate , on the outer face of which was mounted

a ring-shaped electric heater. A thermoc uple was affixed to the center

of each panel. O utside of the heating panels was anoth er layer of in

sulation, 12y2 cm (5 in.) of spun glass at the sides and to p, and l1^ cm

(3 in .) of calcium sil icate a t the bottom . At the very outsid e, the in

sulation was enclosed in plywood.

The thermocouple emf's were read and recorded by a Doric Digi-

trend 210 digital mill ivoltmeter. This instrum ent can be program med

to read and record thermocouples at preselected intervals , and this

feature was employed during the la ter s tages of the data runs.

. N o m e n c l a t u r e -

h = heat transfer coeffic ient, equations (1)

and (2)

q = heat transfer per unit t ime and area at

cylinder surface

T w = surface temp eratu re of horizonta l cyl

inde r

T „ = a m b i e n t t e m p e r a t u r e

T* = phase change tempera tu re





E x pe r im en ta l P ro ce du re . At the ve ry beg inn ing o f the exper i

ments , the consti tuents of the eutectic , 83.2 percent sodium nitrate

and 16.8 perc ent sod ium h ydroxide, in granular form, were carefully

weighed and placed in the test chamber. These granular solids were

melted, and th e l iquid was well mixed to obtain a homogeneous su b

stance. Freezing was then init ia ted , makin g use of the cooling water

to freeze from below and the immersion and top panel heaters to keep

the upper portion of the material in the l iquid phase unti l the end.

At the conclusion of the freezing process, the lower part of the solid

was in the 160°C (325°F) range, whereas the upper part was in the

230°C (450°F) range.

Th e solid was then h eated to obtain a temp eratu re close to the fu

sion value. To this end, the heating panels on all faces (sides, top and

bottom) were energized and operated under automatic regulation so

as to maintain the temperature at a control point (center of top

heati ng pan el) a t a fixed value of 260° C (500° F) . This heating regime

was continued until a th in melt layer (observed through a viewing

aperture) appe ared on th e upper surface of the solid . Normally , ab out

seven days were required to atta in this s ta te . From then onward, the

con t ro l po in t tem pera tu r e was g radua l ly reduced , and the tempera

tures on the horizontal cylinder and in the phase change material were

carefully monitored. These manipulations usually encompassed a

th ree -day pe r iod .

The solid was regarded as being ready for a data run when it had

achieved a mean temperature of 243°C (469.5°F) with spatia l varia

tions of ±0.4°C (0.75°F). This temperature is about 1°C (1.5°F) below

the fusion value. Attempts to approach closer to fusion would haveresulted in an undue amount of melting of the solid .

About an hour before the init ia t ion of a data run, the vent heater

was activated. The run i tself was begun by actuating the c urrent f low

to the horizontal cylinder. At the same time, the power to the vent

heater was diminished to about a quarter of i ts in it ia l value ( i .e . , to

about two watts), and after an hour it was turned off. The heat flow

to the ho r izon ta l cy l inder was ma in ta ined cons tan t th rou ghou t the

data run, at power levels which ranged from 20.6 to 113.6 W (i.e.,

power densit ies from 1430 to 7870 W/ m2) . Run times ranged from 4

to 26 hours, increasing with lower heating rate s . Ther moc ouple data

were collected at one-minute intervals a t early t imes when changes

were rather rapid and at lesser frequencies at la ter t imes when the

tempera tu re va r ia t ions became g radua l .

For the natural convection runs, the phase change material was

liquified by appropriate adjustment of the heating panels . Then, thecontrol point temperatu re was set a t about 245°C (474°F), and ample

time (two to three days) was allowed for the l iquid to atta in tem per

ature uniformity at th is level. The init ia t ion of a data run involved

the application of power to the horizontal cylinder and the activation

of the thermocouple reading and recording instrumentation. T he d ata

runs were completed in one to two hours.

Results and DiscussionHe at T ran s fe r Coef f ic ien ts . Fo r the me l t ing exper imen ts , hea t

transfer coeffic ients were evaluated from the defining equation

IfTw-T*

(1 )

In this equation, q is the heat f lux at the surface of the horizontal

cylinder—determined from the electr ic power input and the area of

the heated surface. Tm is the circumferential-average cylinder wall

temp erature ( the mean value of s ix surface-mounted thermocouples) ,

and T* is the melting temperature of the eutectic phase change ma

teria l . During any given data r un, both q an d T* were constant, while

Tw varied with t ime, a t least during early t imes. Therefore, equ ation

(1) was evaluated a t a succession of times spanning th e entire dura tion

of each data run.

The in stantane ous he at transfer coeffic ients from equation (1) are

plot ted in Fig. 2 as a function of time. An inset at the to p of the figure

continues the presentation for larger t imes. The results are param

eterized by the surface he at f lux q, which ranges from abo ut 1400 to

7900 W/m 2 .

Inspection of the figure reveals a characteristic pattern in the results

which is comm on to all of the cases tha t were investigated. Star t ing

with high values at early times, the heat transfer coefficient decreases

sharply and atta ins a minimum , whereafter i t increases and ult imatelyreaches a s teady state value. This behavior, in itself, provides im

portant insights into the transfer processes, especially when compared

to that which would occur if only conduction were involved. For pure

conduction p hase change, the heat transfer coefficient sta rts from high

values at early t imes and then decreases monotonically with t ime,

never atta ining a s teady state value.2

Therefore , the res ults of Fig. 2

provide clear evidence that transfer mechanisms other than mere

conduction are involved.

I t appea rs tha t conduc t ion i s the dominan t t rans por t mechan ism

only at very early t imes, that is , during th e period when the transfer

coefficient decreases. Thereafter, natural convection comes into play

2There do not appear to be any quantitative heat transfer results in the lit

erature for conduction phase change due to a cylindrical source. The qualitativebehavior of the conduction coefficients can be readily deduced by physicalreasoning supported by the information given in (3) on the movement of theliquid-solid interface.

I00r

9 0 -

80:

70 <

6 0 -

50 \

4 0 -

3 0 -

20 .

O

_ 0

A

3o D

V? A * A

- u v v

n

A

O

V

D

A

OV

50

4 0

30 4

a .

A

0

V

i

-0

V

0 0

A A

o o

V V

A

0

V

o

V

A

0

V

o o

V V

A A

0 «

V V

i

~ n ^ ^ ^E 1 1 1

500 800

3 5 7 0A A A A A A A A A

. 2220o o o o o o ^

0 0

„ „ v V V V V

v v v v q W m 2 ) = l430

i i I i

I

-

300

25 0

20 0

400 ppj

350 E

300 —JZ

25 0

200

150

50 100 150 200 250 30 0 350 40 0 450

TIME (MINUTES)

Fig. 2 Heat transfer coefficients for the melting experiments





and finally takes over. It is especially noteworthy th at a steady value

of h is ultimately atta ined even though the l iquid-solid interface

con t inues to move as mel t ing p roceeds . T h is sugges ts tha t in the

steady state regime, the processes which occur in the neighborhood

of the interface do not contribute significantly to the overall thermal

resis tance. The steady state value of h for each heating rate is delin

eated by a horizontal l ine.

I t may also be observed that the onset of natural convection and

attain me nt of the s teady state occur at earl ier t imes at higher heating

rates. Inaddition, the higher the heating rate , the larger are the heat

transfer coeffic ients . These tre nds are reasonable inasmu ch as higher

heating rates correspond to higher values of the Grashof number(based on either q or ( T w — T *)) .

Fig. 1, with th e aid of equat ion (1), also conveys inform ation on th e

time variation of the cylinder surface temperature. Since both q and

T* were evaluated as cons tan ts for a given data run, equation (1)

shows that the variation of TK with t im e is reciprocal to tha t o f h.

The finding that natural convection is a major contributor to the

melting problem motivated the no-phase-change natural convection

experiments with the cylinder situated in the liquid eutectic. The heat

transfer coeffic ients for these experimen ts were evalua ted from

(2 )

T he quan t i t ie s q and Tw have the same meaning as in equation (1).

T„ represents the average of s ix thermoc ouples s i tuate d in the l iquid

and displaced by about 1.3 cm(V2

in.) from th e surface of the cylinder.T hese the rmocoup les were dep loyed in a vertical plane at the mid -

length of the cylinder. They were posit ioned below and at the s ides

of the cylinder, so that they were washed by fluid moving toward the

boundary layer that f lows around the cylinder surface.

The natural convection heat transfer coeffic ients are plotted as a

function of t ime in Fig. 3. The various runs are param eterized by the

surface heat flux, the values of which are identical to those of th e

melting experiments . The figure shows that after a brief transient

period, a steady state he at transfer regime is established. The steady

state hea t transfer coeffic ient for each heating rat e is indicated by a

solid line. The heat transfer coeffic ients increase with increasing

heating rate , as is consistent with the corresponding increase of the

Grashof number .

The main relevance of these natural convection heat transfer

coefficients is to serve as a basis for comparison with those for th e

melting problem. The stead y state values for the melting and n atural

convection problems are brought together in Fig. 4 , where they are

plotted on logarithmic coordinates as a function of the surface heat

flux. The natural convection coefficients fall consistently on a straight

l ine and th e meltin g coeffic ients , aside from a slight scatter , can be

regarded as lying on a paralle l s tra ight l ine.

\L.

r -ft

_c

10 0

9 0

8 0

7 0

6 0

5 0

4 0

3 0

2 0,

0

a

kcrc&oo-

D

ocmKHr-

A

*W v

1

)

A

b u

- 0 — 0 -

12 0

q ! W / m 2 ) =

-O—0—o—O—O-

VV

\> v "7

1 1 1

4 0

• 0 - 0 - 7870

_ i

-rr-o 2 2 2 0

1 1

6 0 8

TIME (MINUTES)

5 5 0

5 0 0

4 5 0

4 0 0

3 5 0

3 0 0

2 50

2 0 0

150

0

„ "1£

JZ

The s teady state m elting coeffic ients are about 12 percent sm aller

than those for pure natural convection. The fact that the two sets of

values are so close is a further indicatio n th at na tura l convection plays

a dominant role in the heat transfer through the l iquid melt . Indeed,

the co mparison in Fig. 4 gives license for the use of natur al convection

coefficients for the design of melting systems involving a hor izon ta l

embedded hea te r .

The small differences between the two sets of results can readily

be rationalized. As will be illustrated shortly, relatively little melting

takes place adjacent to the lower portion of the cylinder, with the

result that the melt layer is relatively thin in that region. The thinness

of the layer may impose constraints on the natural convection f lowpattern which do not exist when the cylinder is s i tuated in a pure ly

liquid environm ent. Oth er geometrical differences (s ize, shape) and

thermal differences (stratification) may also play a role . The natural

convection coefficients themselve s may be affected by the finite height

of the test chamber. Notwithstanding all of these possibil i t ies , there

are only modest differences in the heat transfer coefficients (Fig.

4) .

Liquid-Sol id Inte r fac e s . Attention will now be turned to another

type of ev idence tha t demons t ra te s the dominance of na tu ra l con

vection as a transport vehicle in the melting problem. In Figs. 5 and

6, the posit ions of the solid-liquid interface at asuccession of t imes

are plott ed, respectively for he at fluxes of 1430 and 5720 W/m 2 . These

interfaces were deduced from the tem peratures m easured in the phase

500

4 0 0

3 00

20 0

NATURAL_ CONVECTION

MELTING

J I I 11 I

q xlO*3(W/m 2)

Fig. 4 Steady state heat transfer coefficients formelt ing and for naturalconvect ion

Fig. 5 Positions of the solid-liquid interface ata succession oft imes, q 'Fig. 3 Heat transfer coefficients for the natural conve ction experime nts 1430 W /m

2





TI M E(H R S)

Fig. 6 Positions of the solid-liquid interface at a succession of times , q =5720 W/m

2

change medium via the thermocouple grid. Specifically, by examining

the temp erature history at any point in the grid , the t ime at which the

melting front passed that point could be identif ied. Once this infor

mation was obtained at a l l points where melting had occurred,

working plots were prepared to facil i ta te the interpolations needed

to draw the posit ion of the interface at any selected instant of

t ime .

Figure 5 , which corres ponds to th e lowest heat f lux used in the ex

periments, shows successive interface positions at times ranging from

1 to 24 hour s. The h eat flux for Fig. 6 is four tim es tha t for Fig. 5 and,

correspondingly, the interfaces have been plotted at t imes that are

one-qu arter of those of Fig. 5. Th e horizontal h eating cylinder is shown

in both of the f igures, and th e cylinder diam eter (1 .9 cm, % in .) pro

vides a scale from which all o ther length s may be inferred. T he floor

of the tes t chamb er is a lso indicated in the f igures.

Figures 5 and 6 show that me lting primarily take s place above th e

heate d cylinder, with very l i t t le melting below. This is in shar p con

trast to the melting pattern that would be predicted by a solution

based on pure conduction, for which the interfaces would be a suc

cession of concentric c ircles surrounding the cylinder. The strong

upw ard thr ust of the m elting zone is caused by natu ral convection.

The primary factor in this thrust is the plume which rises from the

top of the he ated cylinder. The plum e conveys hot l iquid to the up per

part of the melt region and, in this way, perpetuates the upward

movement of the interface.

There is ample evidence that the f loor of the test chamber is suf

ficiently far from the heating cylinder so as not to affect the melting

process. Such evidence is provided by photographs presented in [2]

and by the auth ors ' own visual observations of the s ize and sh ape ofthe melt region.

A comparison of Figs. 5 and 6 may be made for interfaces that

correspond to times in the ratio of a factor of one-fourth (i.e., recip

rocal to the heat flux ratio). This comparison shows that the interfaces

for the lower heat flux tend to be fuller than those for the higher heat

flux. Aside from this , the interface posit ions at corresponding t imes

tend to be more or less coincident.

It is also of inte rest t o take n ote of the size of the melt region a t th e

time when the melting heat transfer coeffic ient approaches a s teady

value. For example , from Fig. 2, i t can be observed th at for the 5720

W /m2

heat f lux, the s tead y state is appro ached in the IV2 to 2 hour

range. At this t ime , Fig. 6 shows that t he heigh t of the m elt region is

only three diameters above the top of the heating cylinder.

O th e r I s s ues . Wi th rega rd to the s teady s ta te hea t t rans fe r

coefficients prese nted in Fig. 4 , efforts m ade to compare the m w ith

li tera ture information for natural convection about a horizontal cyl

inder did not prove fruitful . The main stumbling block was the ab

sence of the thermophysical property values needed to evaluate the

li terat ure co rrelations (e .g . , [4] , pp. 444-446). In this connection,

difficulties were encountered on various fronts. First, property values

are not available for the sodium nitrate—sod ium hydroxide eutectic .

Although the properties of the pure components are tabulated [5],

these components are solids at the temperature at which the eutectic

mixture is a l iquid.3 Finally , the absence of proven procedures forevaluating properties of l iquid mixtures from component properties

[6,7] tended t o discourage even ad hoc use of the available component

data.

While i t is regre ttable tha t the aforem entioned comparison could

not be made, i t has not deterred the major f indings of the work. In

particular , the separate experiments on melting and on natural con

vection have enabled a quantita tive relation between the two sets of

heat transfer coeffic ients to be determined.

Whe n a heating eleme nt with a finite heat capacity is employed in

a transient experiment, there is some reason for concern about the

influence of tha t heat capacity on the results being sought. The nat

ural convection results prese nted in Fig. 3 provide some p erspective

about t he t ime scale of such possible influences. Th e tran sient period

in evidence there is the result of two factors: (a) the actual f low and

thermal transients in the f luid , (6) the transient due to the heat capacity of the heater . Thus, the duration of the transient that is due

to the latter is no greater than that in evidence in Fig. 3. As can be seen

in Fig. 2 , those duration t imes are shorter than the t imes at which

natural convection begins to influence the melting process.

Another aspect of the natural convection results which merits brief

men tion is the possible presence of tem pera ture s tratif ication. Th e

measured temperature distr ibutions in the l iquid prior to the init ia

t ion of a data run indicated no stratif ication at that t ime. After an

hour 's run, the vertical variation of the temperature in the l iquid

environment, over the height of the cylinder, was estimated to be 4

to 5 percent of (T w — TJ). In view of this re latively small varia tion,

and also noting that T„ was obtained as an average, i t can be con

cluded that s tratif ication was not a factor in the results .

In order to determine the phase change temperatu re in terms of the

specifics of the instru men tation ( i .e . , thermocouples, mill ivoltmeter) ,temperature versus t ime curves for freezing were generated. The

cooling curves give the fusion tem pera ture to be 244.2°C (471.5°F),

and this value was used for the evaluation of T* in equation (1).

C o n c l u d i n g R e m a r k s

The experimental results presented here have conveyed strong

evidence of the dominant role played by natural convection in the

melting of a solid due to an embedded heat source. There are three

distinct aspects of the results which contribute testimony about the

impo rtance of natural convection. First , the t imewise behavior of the

heat transfer coeffic ient deviates markedly from the monotonic de

crease that is characteris t ic of conduction phase change. The results

show that conduction domin ates only during the earliest s tages of the

melting process, whereafter natural convection comes into play and

ultim ately gives r ise to a s teady value of the transfer coeffic ient.

Fu r the r more , v ia sepa ra te exper imen ts , the s teady va lues o f the

melting coefficient were shown to differ only slightly from those for

pure natural convection. This finding enables melting coefficients to

be taken from results for natural convection.

The final piece of evidence is provided by the positions of the li

quid-solid interface, which were dete rmin ed a t a succession of t imes

during the melting process. These results showed that melting pri

marily occurred above the heated cylinder, with very l i t t le melting

3 The eutetic mixture solidifies at a temperature lower than the solidificationtempera ture of any other mixture of the components .

Journal of Heat TransferFEBRUARY 1978, VOL 100 / 15




below. Thi s is in con t ra s t to the predictions of the pure conduction

model, according to which the moving interface is represented by a

succession of concentric c ircles centered a bout t he horizonta l cylin

der.

In view of the demonstrated importance of natural convection in

mel t ing p rob lems , it appea rs un reasonab le to con t inue the p resen t

practice of neglecting it in the analysis of such problems.


This research was performed under the auspices ofB R D A g r a n t

N o . E ( l l - l ) - 2 5 9 5 .

R e f e r e n c e s1 Seki, N., Fukusako, S., and Sugawara, M., "A Criterion of Onset of Free

Convection in a Horizontal Melted Water Layer with Free Surface," ASME,J O U R N A L O F H E A T T R A N S F E R , Vol. 99,1977, pp. 92-98.

2 White, R. D., Bathe lt, A. G., Leidenfrost, W., and Viskanta, R, "Studyof Heat Transfer and Melting Front From aCylinder Imbedded in a PhaseChange Material," ASME paper 77-HT-42, 17th National Heat TransferConference, Salt Lake City, Utah , Aug. 1977.

3 Stepha n, K., and Holzknecht, B., "Heat Conduction in Solidification ofGeometrically Simple Bodies," Warme- und Stoffubertragung, Vol. 7,1974,pp . 200-207.

4 Karlekar, B. V., and Desmond, R. M.,Engineering Heat Transfer, WestPublishing Co., St. Paul, 1977.

5 Janz, G. J., Molten Salts Handbook, Academic, New York, 1967.6 Reid, R. C , and Sherwood, T. K., The properties of Gases and Liauids,

McGraw-Hill, New York, 1958.7 Blander, M., ed., Molten Salt Chemistry, Interscience, New York,

1964.

A N N O U N C E M E N TThe following short courses will be offered at the second AIAA/A SMB T hermo physics a nd He at Transfer Conference,

May 26-2 7,197 8, a t Rickey 's Hyat t House in Palo Alto , California . For further informa tion and registra tion, contact:

J i l l Jacobson

Pro fess iona l Deve lopmen t D ep t .

A S M E

345 East 47th Street

New York, New York 10017

(212) 644-7743

Nam e o f Course : T herm al Con tac t Res is tance

Instru ctor: M. M. Yovanovich

Date : May 26 ,1978

Descriptio n: A review of the theory of therm al contact resis tance associated with heat transfer between contacting

solids.

Nam e of Course : Funda men ta ls of T herm al In su la t ion

Instru ctors : G. R. Cun ningto n and C. L. Tien

Date : May 26 ,1978

Desc r ip t ion : A review of the ca lcu la t ion and te s t me thods for different types of thermal insulation, including

powders, f ibers , foams, and Multi layers .

Nam e of Course : Hea t E xchanger E qu ipm en t fo r the Power and P rocess Indus t ry

Instru ctors: F . L . Rub in and C. F . And reone

Date : May 26-27 ,1978

Descrip tion: An examination of the various types of equip men t used primarily for the transfer of heat in the power

and process industries .





J . H. Roya lSenior Engineer,

Union Carbide Corporation,

Linde Division,

Tonawanda, N. Y.

Assoc. Mem. ASME

A. E. Bergles

Professor and Chairman,

Department ol Mechanical

Engineering and Engineering

Research Insti tute,

Iowa State University,

Ames, Iowa

Mem. ASME

Augmentation of Horizontal In-Tube

Condensation by l e a n s of Twisted-

Tape Inserts and Internally Finned

TubesLow pressure steam was condensed inside horizontal tubes of different internal geome

tries to investigate passive heat transfer augm entation techniques. A smooth tube, the

smooth tube having two twisted-tape inserts, and four internally finned tubes were tested.

The twisted-tape inserts were found to increase average heat transfer coefficients by as

much as 30 percent above smooth tube values on a nominal area basis. The best perform

ing internally finned tube increased average heat transfer coefficients by 150 percent

above the nominal smooth tube values. Techniques were developed to correlate the im

proved heat transfer performance of the two twisted-tape inserts and the four internally

finned tubes. The equations developed p rovide a reasonably accurate description of both

the sectional and the average heat transfer coefficients. The finned tube correlation was

also reasonably successful in predicting the data from the one other investigation of this

augmentation technique for which detailed data were available.


During fi lmwise in-tube condensation, vapor condenses at the

cooler tube wall , and the condensate coats the tube wall with a con

tinuou s f i lm. W hen the co ndensa te layer is symm etric , with well de

fined phase seg regation, the f low regime is said to be annular. Often,

gravitational, shear, and pressure forces redistr ibute the l iquid f i lm

and disrupt this well defined phase segregation. Interfacial waves,

entrained droplets , s tra tif ication, bubbles, and vapor s lugs can all be

prese nt and can greatly modify the f low field .

If the thermophysical properties of the condensing media are fa

vorable , in-tube f i lm condensation heat transfer coeffic ients can be

quite high. However, many industria lly i mp orta nt f luids, such as or

ganic l iquids and most f luorocarbon refrigerants , do not have such

favorable properties. The condensation heat transfer coefficients for

these fluids are often low enough to present a significant heat transfer

resis tance in a condenser. For such cases, techniq ues to aug me nt in-

tube condensation can be used to reduce the condensing-side thermal

resis tance to the extent that s ignificant reductions in condenser s ize

may be possible .

A li terature survey of the f ie ld [ l]1

disclosed a surpris ing num ber

1 Numbers in brackets designate References at end of paper.Contrib uted by the H eat Transfer Division for publication in th e JOURNAL

OF HEAT TRANSFER. Manuscript received by the Heat Transfer DivisionJanuary 24,1977.

of attempts to augment condensation. However, relatively few of these

studies were involved with in-tube condensation.

Cox, e t a l . [2] exper imen ted w ith internally grooved and knurled

tubes in single and multi-tube tests. The grooved tubes were reported

to show an improvem ent in overall performance of up to ten percent;

however, the knurled tubes demonstrated no obvious advantage.

These investigators a lso indicated that spirally corrugated tubes

matched the performance of the grooved tubes.

Princ e [3] investigated heat tran sfer surfaces for use in horizontal

tube desalination evaporators where condensation occurs inside tubes.

He tested corrugated and internally r ibbed tubes along with other

surfaces. He reported significant increases in overall heat transfer

coeffic ients for the r ibbed tu bes, but decreases in heat transfer coef

f ic ients for the corrugated tubes. The individual heat transfer coef

f ic ients from these experiments were not determined.

Reisbig [4] reporte d on the condensation of Refrigerant-12 inside

tubes with inte rnal longitudinal f ins. Detailed data are not provided,

and no correlations are presented.

Go ttzm ann an d his co-workers [5 , 6] condensed hydro carbon s

within vertical tubes containing internal longitudinal fins. They noted

increased condensation rates, but provided no condensation-side film

coefficients.

Vrable , e t a l . [7-9] reported a s tudy of the condensation of Refrig

erant-12 inside tubes with internal longitudinal f ins. Inlet reduced

pressures for these tests varied from 0.18 to 0.46. Mass fluxes varied

from 86.7 to 853 kg/m2s . 26 experiments with two internally f inned

tubes were reported. These data were correlated by modifying the

Journal of Heat Transfer FEBRUARY 1978, VOL 100 / 17Copyright © 1978 by ASME




semi-emp irical, smooth t ube correlation of Cavallini and Zecchini [10].

This correlation was modified by changing the co nstant, using as the

characteristic length twice the equivalent diameter, and incorporating

an additional term to adjust for pressure effects. The equation which

correlated the data to within approximately ±30 percent is

^ 0 . 0 2 -2d ee \ Hf I \ p c r /

(1 )

where

-[©"'•«->]G

Recently , Kroger [11] reported on studies of the laminar conden

sation of Refrigerant-12 inside smooth and internally f inned tubes.

Significant improvements in heat transfer were observed with in

ternally f inned tubes, but a t Reynolds numbers well below those of

in te res t he re .

The study reported here was undertaken to investigate two po

tentia lly useful in-tube condensation augmentation techniques:

twisted-tape inserts and internally finned tubes. The techniques were

selected because both are basic passive augmentation techniques used

extensively for single-phase flow. Twisted tapes are easily fabricated

and installed, and internally f inned tubes have recently become

readily available . The study was designed to obtain a wide range of

experimental data and to develop useful correlations of the data .

Exper imental Faci l i ty and ProcedureFa c i l i ty . An experimental facil i ty was constructed to test the

augmentation techniques and to provide the smooth tube reference

data . C ondensing steam was the medium selected for the exp eriment.

The facil i ty consisted of appropriate piping and instrumentation

designed to test s ingle , in-tube condenser tubes of various diame ters ,

inter nal geo metries , and lengths. Fig. 1 is a schem atic of the facil i

ty .

For the experiments reported here, the condensing side of the fa

cil i ty was operated in an open-loop configuration. Clean steam was

introduced into the dryer through a s trainer from a building service

steam m ain. A thrott l ing valve controlled the dry, saturated steam

flow rate to the test condenser. U p to 320 kg/hr of s team a t e ight bar

with 0-16°C of superheat was available at the test condenser inlet .

The condensate or condensate-vapor mixture exited the test con

denser through an adiabatic sight glass section. The condensate-vapor

mixture then flowed to the after-condenser to complete the conden

sation process. Th e subcooled conden sate then passed throug h th e

system pressure regulation valves to the condensate f lowmeter. The

conde nsate was f inally discharged to a f loor dra in .

Water w as used as the test condenser coolant. The coolant entered

the test condenser through a f lowmeter. After passing through the

test condenser, the coolant was discharged through the coolant drain

valve. The coolant drain valve was used for coolant flow control to

pressurize the co olant s ide of the test condenser in order to supp ress

coolant side subcooled boiling.

T es t Con den se r . T wo concen t r ic tubes fo rmed the coun te rf low

test condenser. The test condenser was divided into four, 0.91 m long

units to form the test condenser sections. These sections were ar

ranged in series, with the condensing medium uninterruptedly flowingthrough t he test tub e, while the coolant was diverted through mixing

sections before proceeding to the next section. Thermocouples were

TEST FLUIDDRAINVALVE

COOLANTINLETVALVE

AFTERCONDENSERCOOLANTF L O W M

NSATE DRAINSWING CHECKVALVE

COOLANTEXIT

VALVE

-WATER L INE

- STEAM LINE- DRAIN LINE

Fig. 1 Schem atic of experim ental facility

^Nomenclature-

A = area

b = fin heightcp = specific heat a t constant pressure

d = d iamete r

/ = friction factor

F = factor, equa tion (6), etc.

G = mass flux

G e = adjusted mass flux, equation (1)

h = heat transfer coeffic ient

k = the rmal conduc t iv i ty

L = heat transfer length of tube

m = fin parameter, equation (7)

n = number of f ins

p = pressure

p cr = crit ical pressure

Pr = P rand t l numberq = heat transfer ra te

r = radius

Re = Reyno lds number

T = tempera tu re

u = velocity

w = interfin spacing

x = quali ty

y = half-pitch-to-diameter ratio

z = axial coordinate

5 = tape thickness

f = fin efficiency

H = dynamic viscosity

p = density

S u b s c r i p t s

a = axial

c = condensation

e = based on equivalent diameter

/ = liquid or friction

i = inside

o = outside

s = sa tu ra t ion

t = t angen t ia l

tt = twisted tape

v = vapor

w = wall





T a b l e 1 Se le c te d Ge ome tr icT u b e sT u b e N u m b e rT y p e

Mate r ia lOu ts ide Diamete rIns ide Diamete rE qu iva len t

D i a m e t e rWal l T h icknessT o ta l Wet ted

P e r i m e t e r

Nomina l Wet tedP e r i m e t e rT o ta l Pe r ime te r /

Nomina lPe r ime te r

Cross-SectionalArea

T ape T h icknessF in He igh tFin Base WidthF in Ti p W i d t hN u m b e r of FinsPitch, cm/180°

A B

P a r a m e t e r s of the E x p e r i m e n t a l

C D E FSmooth T wis ted T wis ted Fin # 1 Fin # 2 Fin # 3 Fi

Cu1.5875

1.38431.3843

0.10164.3487

4.3487

1.0

1.5052

n.a.n.a.n.a.n.a.n.a.n.a.

T a p e # 1 T a p e # 2

C u . S S1.5875

1.3843

0.8252

0.10167.0419

4.3487

1.6193

1.4529

0.0378n.a.n.a.n.a.n.a.

4.57

C u , S S1.5875

1.3843

0.8252

0.10167.0419

4.3487

1.6193

1.4529

0.0378n.a.n.a.n.a.n.a.

9.65

Cu Cu1.5875 1.2776

1.4707 1.18110.8260 0.7602

0.0584 0.04907.8750 5.3310

4.6203 3.7104

1.7045 1.4368

1.6264 1.0129

n.a. n.a.0.0599 0.17350.0475 0.16690.0277 0.0478

32 630.48 17.15

C u1.2776

1.1532

0.7564

0.06225.1891

3.6228

1.4324

0.9648

n.a.0.16310.13940.04346

Gn # 4

Cu1.5900

1.3970

0.6767

0.09658.1979

4.7437

1.7279

1.3864

n.a.0.14480.12450.0462

16straight 27.94

All dimensions in cm orcm2, as appropriate,n.a. = not applicable

used to measure coo lan t tempera tu res in the mixing sections at the

en t rance and exit of each condenser section. Pressure taps were pro

vided at the condenser inlet and at the exit from each section to

moni to r the local pressure of the condens ing med ium.

Test condenser tube wall temperatures were obtained from 36

copper-constantan thermocouples, spaced axially in groups of t h r e e

a t 0.3 m intervals along the test condenser. At each station, the th ree

thermocouples were distr ibuted circumferentially , with one at the

to p of the t u b e and the o the rs at 90 deg and 180 deg.

After completion of the smoo th tube exper imen ts , the t u b e was

modified by insertion of a twis ted tape . Th e twisted tapes were

sheared from stainless steel sheet stock to a width of 1.339 ± 0.003 cm.

Several s tr ips were welded together to form each tape. The t ape was

then twisted until the desired pitch was ob ta ined .

The twisted tapes were installed in the smoo th tube by removing

the te s t condense r end f i t t ings and pulling the tape into the tube . A

clearance of 0.22 mm was requ i red for installa tion; hence, the t ape

wa s in on ly in te rmi t ten t con tac t wi th the tube wall . Dimensions of

the tapes are l is ted in T a b l e 1.

The Forge-Fin Division of Noranda Metal Industries , Inc. supplied

the four f inned tubes tested during this s tudy. These tubes were

seamless tubes with integral longitudinal f ins, three of which had

spiral f ins. Table 1 presen ts the impor tan t d imens ions of the

tubes .

Pro cedure . For a particular tube, data were obtained over a range

of inlet pressures and condensing fluid mass fluxes. Typically an inlet

pressure was established and f low rates w ere varied within the l imi

tations imposed by the equipment. Quality changes were restr ic ted

to approx ima te ly 100 percen t , and in let conditions were held near

saturation. Ranges of some of the more important dependen t variablesare l is ted in T a b l e 2. After the system had stabil ized at the selected

conditions, the raw data were recorded.

A digita l compu ter was used to process the data. Coolant flow rates

and enthalpy changes were used to de te rmine the energy transfers .

T hree se t s of three wall temperatures were averaged to obtain sec

t iona l ave rage wal l t empe ra tu res . Sa tu ra t ion tempe ra tu res for each

test section were calculated by averaging the local saturation tem

pera tu res co r respond ing to the local pressures. The sectional tem

perature difference, average saturation temperature minus average

wall t em pera tu re , was corrected for tube wall thermal resis tance and

used with the sectional coolant energy gain to calculate the sectional

heat transfer coeffic ient based upon nominal tube dimensions:

Tab l e 2 Exper imental Faci l i typendent Var iab le Ranges

De-

Mass F luxHea t F lux

Inlet Vapor VelocityAverage Condensing

H e a tTransfer Coefficient

Hea t T rans fe r R a teOut le t Qua l i ty

150-583 kg/m 2s

220,000-1,400,000W /m 2

80-220 m/s19,000-110,000 W/m

2

K

41,000-200,000 W

-0 .019 ( subcoo led ) -0.041

h c = [ 2trnAz(Ts - Tw - ^ - I n&

(2)qIA k,

Tube average heat transfer coeffic ients were obtained in an ana l

ogous manner. Additionally , the total energy loss of the condensing

m e d i u m was calculated from flow rate and en tha lpy to ob ta in the

overall heat balance required for data quali ty control.

Ex p er i m en t a l Da t aFig. 2 is rep resen ta t ive of the exper imen ta l re su l t s for all tubes at

a common inlet pressure. Curves have been fi t ted to the da ta po in ts

by l inear regression techniques to clarify trends. As can be observed

from the figure, the tubes with the twisted-tape inserts , Tubes B and

C, ou tpe r fo rm T ube A, the smoo th tube , by as much as 30 percen t .

T u b e B, with the more t ightly twisted tape, consistently d emonstrates

higher heat transfer coeffic ients than Tube C.

The internally f inned tubes showed dramatic increases in hea ttransfer coefficients. For example, Tube G out-performed the smooth

t u b e by as m u c h as 150 percen t on a nominal area basis . Among the

finned tubes, it should be noted that performance improvement is not

solely a function of increased heat transfer area. For example , T ube

E, which has 40 percent more area than a smoo th tube and a fin-

height-to-radius ratio of 0.3, is a better augmentation device than tube

D , which has 70 percen t more a rea than a smoo th tube and a fin-

he igh t - to - rad ius ra t io of 0.08.

Fig. 3 is rep resen ta t ive of the exper imen ta l p ressu re d rop for all

tubes at the same common inlet pressure as Fig. 2. The f inned tube

data generally follow the same t rends as observed with the h e a t

transfer results . Tubes E and G, with the longest spiral fins have the

Jou rna l of Heat Transfer FEBRUARY 1978, VOL 100 / 19




40 0 500 6O0 700

Fig. 2 Experimental average heat transfer coefficients versus mass flux,all tubes, inlet pressure = 4.2 bar

O TUBE A

O TUBE B

Q TUBE C

V TUBE D

O TUBE E

A TUBE F

<^> TUBE G

60 0 600 700 800 900 1000

Fig. 3 Experimental total pressure drop versus mass flux, all tubes, inletpressure = 4.2 bar

highest heat transfer and the h ighes t p ressu re d rop . A compar ison

of the data for T u b e E with the da ta of T u b e F ind ica te s the effects

of fin sp i ra l .

T he twis ted - tape p ressu re d rop da ta are in con t ra s t to the f inned

tube da ta in t h a t the pressu re d rop re su l t s are not analogous to the

heat transfer results . Much greater percentage increases in pressu re

drop than increases in hea t t rans fe r are observed.It is app aren t from b oth of these f igures that the twisted tapes are

l e ss op t imum in - tube condensa t ion augmen ta t ion dev ices than are

the internally f inned tubes. The most a t t rac t ive augmen ta ted tube

seems to be T u b e G.

Additional details of the facil i ty , operating procedures, data re

duction, and exper imen ta l re su l t s are presented elsewhere [1, 12,

13].

He at T r ansfe r C or r e la t ion for T w i s t e d - T a p e I n s e r t s

An a t tem pt was made to mode l the effects ofa twisted tape on the

in - tube condensa t ion p rocess . T he me thodo logy of th is analysis was

to identify those twisted-tape effects which contributed to the aug

m e n t a t i o n of condensation heat transfer , and then to model these

effects in such a way as to make them com pa t ib le wi th those in - tube

condensation correlations which had been found suitable for the

smoo th tube [1 , 12, 13].

Thr ee effects were considered in the twis ted - tape ana lys is : the in

crease in fr ic tional pressure drop due to the addition al wetted surface

of th e t ape , the additional heat transfer area provided due to con

duction effects through the t a p e , and the increase in the vector ve

locity due to the con t r ibu t ion of the induced tangential velocity

componen t .

T a n g e n t i a l V e l o c i t y E f f e c t s. The twisted tape induces a t a n

gen t ia l componen t of velocity which is a function of radius. This velocity comp onent, when added vectorially to the axial velocity, yields

the increased velocity due to the twis ted tape at a given radius. This

is the velocity necessary to main ta in the mass f low rate through the

longer spiral f low path created by the t a p e .

The tangential velocity component is

2yn

an d the m a g n i t u d e of the resultant velocity is given by

Subs t i tu t ing equa t ion (3) in to equation (4) then yields

r2 i r \2 10.5

"- [TQ'-r

(3)

(4)

(5)

Since th e in terfacial shear s tress is a function of the velocity near

the interface and the interfacial shear is ofpr imary impor tance to the

heat transfer process, it is reasonable to conclude that the vapor ve

locity should be evaluated at the rad ius of the in terface. Th e local

interfacial radius can be determined from the local quality and the

assumpt ions of a local flow regime and local void fraction. However,

for simplicity it is suggested that equation (5) be evaluated at the tube

inside radius. By evaluating the total velocity at the wall ra ther than

a t the in terface, the effect is over-predicte d. Th is is , however, offset

by the neglect of the twisted-tape effects on the liquid film, which

would also tend to augmen t the hea t t rans fe r .

The following velocity correction factor is thus ob ta ined :

Ft = — = — Or2

+ 4y2)

0-

5(6)

Since Ft charac te r izes the tangential effect of the twis ted tape on

velocity , th is correction factor is applied to the Reyno lds number o f

in te res t in a particular correlation.

Fin Ef fec t s . A twisted tape in contact with the tube wall increases

the active heat transfer surface because the tape behaves as a fin.

Therefore, a correction factor should be included to predict the extent

of heat transfer augmentation due to the fin effect of the tape .

The twisted tape is assumed to be in good therm al contact with the

tube w all, and the wall is assumed to be a good conductor. These tub e

wall assumptions lead to the conclusion that the t e m p e r a t u r e at the

base of the twis ted tape approaches tha t of the wall far from the fin.

T hus , wi th the twis ted - tape base tempera tu re approx ima te ly equa l

to tha t of the wall, the twisted tape can be modelled as a longitudinal

fin of rectangular profile. Assuming the fin to have a length ofhalf the

tube in s ide d iamete r and to have an insulated tip, the equa t ion for

fin effectiveness is

(7)

where

, /md:\2 tank I -)

md t

\k ttb tt'

The total heat transferred per unit length of the tube is the su m of

the hea t t rans fe r red at the unfinned wall , and the hea t t rans fe r red

th rough the two fins:

20 / VOL 100, FEBRUARY 1978 Transactions of the ASME




g = qw + 1q tt (8 )

where

qw = hcL(irdt - 2Stt)(Ts - Tw)

an d

Defining the fin effect correction factor, F tt, as the ratio of total heat

transfer to the heat transfer to be expected without the f in effect re

sults in

*diLhc(T. - Tw)

Substi tuting equation (8) into equation (9) yields

7rrf,-2<>« + 2r,fFtt= ; (10)

irdi

or

2/I S tt\

* , - l + - ( - - _ ) ( 1 1 )

Therefore, by multiplying the empty tube value of the condensation

heat transfer coeffic ient by F tt, the result will be corrected for tape

conduction effects .

Wa ll Shea r Ef fec t s . The insertion of a twisted tape into the in-

tube condensation process radically alters certain basic geometricparameters. The reduction of the cross-sectional area of flow is small

for thin tapes such as those under consideration here, but the addition

to the wetted surface area is quite large.

The addition of s ignificant wetted surface to that of the smooth

tube increases the total wall shear and, hence, the pressure loss. To

predict twisted-tape heat transfer coeffic ients , those smooth tube

correlations requ iring the explicit inclusion of a fric tion factor mu st

be supplied with the proper twisted-tape fr ic tion factor.

Lop ina and Berg les [14] pre sen ted a single phas e friction factor

correction. This correction factor was the ratio between the fr ic tion

factor of a tube containing a twisted tape and the smooth tube friction

factor:

F f = — = 2.75y-°-40 6

(12)

where y is the ratio of tape half-pitch to tube inner diameter.

The alteration in the ratio of cross-sectional area to wetted area

must a lso be incorporated in the correlating procedure for those

correlations n ot explicit ly including a fr ic tion factor. This can be ac

complished by the use of the equivalent diameter in place of the

nomina l d iamete r .

S u m m a r y . The separate effects considered above are suggested

as modifications to smooth in-tube condensation correlations to en

able the prediction of in-tube condensation he at transfer coeffic ients

for tubes with twisted-tape inserts. The modifications are summarized

as follows:

1 Th e subs ti tutio n of the equivale nt diam eter for the nomin al

diam eter for those correlations no t requiring an explicit fr ic tion fac

tor.

2 The mu ltiplication of velocity or Reynolds numb er term s by

F t, the tangential velocity correction factor, equation (6).

3 The multiplication of the resultant heat transfer coeffic ient by

the fin-effect correction factor, F tt, equation (11).

4 The multipl ication of the s ingle-phase fric tion factor, where

required, by Ff , the friction factor correction factor, equation (12).

Co m pa r iso n With Present Ex per im enta l Resul t s . The in-tube

correlation of Akers, etal. [16] and Soliman, etal. [17] were modified,

as suggested above, and used to produce a point-by-point comparison

with the ex perim ental d ata . Th e correlation of Akers, e t a l . is an av

erage correlation, providing a value for the tube-averag ed he at transfer

coefficient. The correlation of Soliman, et al. is a local correlation

providing local heat transfer coeffic ients which must be integrated

to deter min e the tube-averag ed heat transfer coeffic ients.

As modified, the higher mass flux correlation of Akers, et al.[16]

becomes

Kc = o .o 26 5^ ( £ ^ )a 8

P r / o . 4 F ( ( ( l 3)de \ Hf I

where the equ iva len t d iam ete r

dvdi2

irdi + 2(di - 5 tt)

is used since the original equati on does not explicitly contain a friction

term, and no explicit correlation for increased wall shear can be

m a d e .

The correlation of Soliman, e t a l . [17] is modified by the m ultipli

cation of a ll Reynolds number terms by F t, multiplication of the

friction shear s tress ter m in equation (36) of reference [16] by Ff , and

by multiplying the resultant heat transfer coefficients by F lt. For both

correlations, experimental values were used for the various inde

penden t pa ramete rs requ i red .

The following ranges of the twisted-tape effects were encoun

te red :

F t: 1.09-1.38

Ff . 1.66 - 2.24

F tt: 1-02 - 1.10

The results of the comparisons are shown in Figs. 4-7. As can be

observed in Figs. 4 and 5, the tube-averaged data w ere predicted by

the modified correlation of Akers, e t a l. with errors of less than ±30

percent for nearly the entire data set .

The use of the Soliman correlations to predict the sectional data

was somewhat less successful. As can be observed from Figs. 6 and 7,

approximately one third of the data fall outside of the 30 percent

bands. I t should be noted th at the da ta presented in these f igures are

the sectional heat transfer coeffic ients , that is heat transfer coeffi

cients averaged over a small quali ty range. The sectional data ex

hibited more scatter than th e average data [1]. The dat a from Section

3 of the experimental facil i ty consistently evidenced lower heat

transfer coefficients than did the following, lower quality section.

Since the Soliman correlation predicts increasing heat transfer

coefficients w ith increasing quali ty , the d ata for Section 3 were con

sis tently overpredicted. Further discussion on this point is given in

reference [1].Some error results from the assumption of good thermal contact ,

between tape and tube, which was used in the analysis . I t should be

noted th at the installa tion procedures did not guarantee good therm al

contact. This thermal resis tance would reduce the f in effect of the

twisted tape. This error was minimized by using thin s ta inless s teel

str ips for the tapes.

Heat Transfer Corre la t ion for F inned TubesCo rre la t io n Dev e lo pm ent . The physics of in-tube condensation

is dramatically altered by the presence of internal f ins. The fins

provide a large increase in active heat transfer area, which contributes

to higher heat transfer rates. However, this effect is somewhat offset

by the fin-efficiency effects and the decrease in local heat transfer

coeffic ients as the condensate f i lm thickness increases, submerging

the f ins. Additionally , the f ins contribute to higher heat transfer bypromot ing more tu rb u len t mix ing wi th in and be tween phases .

To account for these effects , a correlation found suitable for the

prediction of smooth tube data was modified. The correlation of

Akers, et al. [16] was selected for modification because it provided a

reasonable correlation of the average smooth tube data obtained

during this investigation.

A multiplicative term, based upon a dimensionless combination

of geometric parameters was determined. I t served to modify the

Akers correlation and enabled the modified correlation to predict the

experimental data . The modified equation is

kt iG d„\ 0.8 r / b2

\ 191 1h c = 0 . 0 2 6 5 ^ ( — ) P r /

0 3 3 16 0 ( f ~ ) + 1 (14)de \ nf I L \w de/ J

Journal of Heat Transfer FEBRUARY 1978, VOL 100 / 2 1




EXPERIMENTAL

Fig. 6 Calculated sectional heat transfer coefficients versus experim ental„ , , „ , , , . . . . . , „ i . , ! . „ i s ec ti on al h ea t t ra ns fe r c oe ff ic ie n ts , t ub e B , m o di fi ed c o rr el at io n o f S o li ma n,Fig. 4 Calculated average heat transfer coefficients versus experim ental . r . ,average hea l transfer coefficients, tube B, mod ified corre lation of Akers , et '

l'

al . [15]

CYPCCTIMFMTfll h W x 1 ( )3

m 2 K

Fig. 5 Calculate d average heat transfer coefficients versus experim ental

average heat transfer coefficients, tube C, modified correlation of Akers, et

al- [15]

Fig. 7 Calculated sectional heat transfer coefficients versus experim entalsectional heat transfer coefficients, tube C, modified correlation of Soliman,e t a l . [16]

where w is the average of the interfin dista nce at the base a nd t ip of

the f ins. This equation is s tr ic tly empirical , in contrast to the

twisted -tape co rrelation. Fin efficiency, wall shear effects, turb ulation ,

and en t ra inm en t a re a l l inc luded in the add i t iona l te rm.

The geometrical factor used in equation (14) differs from other

dimensionless geometric parameters which have been used to corre

late s ingle-phase data for internally f inned tubes. Watkinson, e t a l .[18] used w/d. VasiPchenko and Barbaritskaya [19, 20] used bid, an d

Orn atskii, e t al. [21, 22] used w/b. The reason for the necessity of the

b2

term in the parameter used here seems to be that fin height is more

important to heat transfer performance in condensing systems than

in single-phase systems. This is perhaps due to interfin-channel

flooding considerations.

C o m p a r i s o n W i t h E x p e r i m e n t a l R e s u l t s . As shown in Fig. 8,

equation (14) correlates the average heat transfer coefficients to within

30 percen t for 95 percent of the da ta .

An attempt was made to use equation (1) to predict the experi

men tal data r eporte d here. This correlation proved to be a very poor

predictor of the experimental data , some of the data being over-

predicte d by a factor of ten. Th e reason for the gross ov erprediction

seems to be the reduced pressure term. Vrable 's data were taken at

reduced pressures an order of magnitude greater than the reduced

pressures at which the data reported here were taken. With his em

pirically determined exponent of —0.65, th is term gave too great a

weighting at lower reduced pressures.

An attempt was made to predict Vrable 's Refrigerant-12 data with

equation (14). For the tub e with 150 percen t area increase, equation

(14) predicted the data rather well. Most of the calculated values werewithin 30 percent of the experim ental res ults . A plot of calculated

versus experim ental d ata ap pears as Fig. 9 . Th e correlation failed to

predict th e data for Vrable 's tu be which had a 275 percen t area in

crease. I t is noted that th is tube falls outside the geometrical l imits

imposed upon equation (14) by i ts data base, and attempting to pre

dict dat a obtained from this tube is an unsupp ortab le extrapol ation

of the equation.

Conclusions and RecommendationsThis s tudy has clarif ied the heat transfer performance of two dif

ferent augmentation techniques for condensation inside horizontal

smooth tubes. Twisted tapes were found to increase average heat

22 / VOL 100, FEBRUARY 1978 Trans act ions of the ASME




Fig. 8 Calculated average heat transfer coefficients versus experimental

average heat transfer coefficients, tubes D-G, equation (14) Fig. 9 Calculated average heat transfer coefficients versus experimental

average heat transfer coefficients, data of Vrable [7 | , equation (14)

transfer coefficients of condens ing s team by as much as 30 pe rcen t

above the empty tube values on a nominal area basis. Internally finned

tubes were found to increase in-tube condensation heat transfer

coeffic ients by as much as 150 percent above th e smooth tube values

on a nominal area basis .

A correlation was developed to predict in-tube condensation heat

transfer coefficients for tubes equipped with twisted tapes. The

technique involved several modifications to existing smooth in-tube

condensation correlations. No additional empirical factors are utilized

in the correlation s. Two modified c orrelations, one for sectional av

erage coefficients and the other for overall average coefficients, proved

to be fairly useful predicto rs of augm ented h eat transfer performa nce.

Because no other data appear to be available in the open l i te ra tu re ,

an independent verif ication of the correlations could not be per

formed.

A correlation was also developed for the in ternally f inned tubes.Thi s correlation, equat ion (14), proved to be a satisfactory predictor

of heat transfer coefficients for in-tube condensation inside of th e

tubes with internal longitudinal f ins. The correlation also served to

predict satisfactorily th e refrigerant data of another investigation

where conditions an d tube geometries were s imilar to those te s ted

dur ing th is p rog ram.

To further understand the augmentation of in-tube condensation,

continued testing is necessary. Experiments with other working fluids

would be of special importance to the deve lopmen t of a more com

prehensive correlation. Experiments designed to investigate the de

tails of hea t t rans fe r in the region of flooded fins will also be of in

te re s t .

Acknowledgments

T his s tudy was sponso red by the E ng inee r ing Resea rch Ins t i tu te ,Iowa State University . Honeywell Corporation provided a g r a n t for

construction of the experimental facil i ty . The work was performed

in th e Heat T rans fe r L abora to ry of t h e D e p a r t m e n t of Mechan ica l

Engineering at Iowa State University .

References1 Royal, J. H., "A ugmentation ofHorizontal In-Tube Condensation of

Steam," Ph.D. Thesis, Department of Mechanical Engineering, Iowa StateUniversity, 1975.

2 Cox, R. B., Matta, G. A., Pascale, A. S., and Stromber, K. G., "SecondReport on Horizontal-Tubes Multiple-Effect Process Pilot Plant Tests andDesign," Research and Development Report No. 592, OSW Con tract No. 14-01-0001-2247: United States Department of the Interior, May 7,1970.

3 Prince, W. J., "Enhanced Tubes for Horizontal Evaporator Desalination

Process," M.S. Thesis in Engineering, U niversity of California, Los Angeles,1971.

4 Reisbig, R. L., "Condensing Heat Transfer Augmentation Inside SplinedTubes," ASME Paper 74-HT-7, AIAA/ASME Thermophysics and HeatTransfer Conference, Boston, July, 1974.

5 Milton, R. M., and Gottzm ann, C. F., "Liquefied Na tural Gas: HighEfficiency Reboilers and Condensers," Chemical Engineering Progress, Vol.68, No. 9,1972, pp. 56-69.

6 Gottzmann, C. F„ O'Neill, P. S„ and Milton, P. E„ "High Efficiency HeatExchangers," Chemical Engineering Progress, Vol. 69, No. 7, 1973, pp. 69-75.

7 Vrable, D. L., "Condensation Heat Transfer Inside Horizontal TubesWith Internal Axial Fins," Ph.D. dissertation, University of Michigan, 1974.

8 Vrable, D. L., Yang, W. J., and Clark, J. A., "Condensation of Refriger-ant-12 Inside Horizontal Tubes with Inte rnal Axial Fins," Heat Transfer 1974,Fifth International Heat Transfer Conference, Vol. 3, Japan Society of Me

chanical En gineers, Tokyo, 1974, pp. 250-254.9 Yang, W. J., "Equivalent / and ;' Factors for Condensation Inside Horizontal Smooth and Finned T ubes," Letters in Heat and Mass Transfer, Vol.1, 1974, pp. 127-130.

10 Cavallini, A., and Zecchini, R., "High V elocity Conden sation of OrganicRefrigerants Inside Tube s," Proceedings of the Xlllth International Congressof Refrigeration, International Institute of Refrigeration, Brussels, Belgium,1971.

11 Kroger, D. G., "Lamin ar Condensation Heat Transfer Inside InclinedTubes," Paper No. AIChE-29 presented at the 16th National Heat TransferConference, S t. Lou is, Mo., Aug., 1976.

12 Royal, J. H., and Bergles, A. E., "Augmentation of Horizontal In-TubeCondensation of Steam," Heat Transfer Laboratory Technical Report HTL-9,Engineering Research Institute, Iowa State University, Ames, Iowa, 1975.

13 Royal, J. H., and Bergles, A. E., "Experimental Study of the Augmentation of Horizontal In-Tube Condensation," ASHRAE Transactions, Vol. 82,Part 1,1976, pp . 919-931.

14 Lopina, R. F., and Bergles, A. E., "Heat Transfer and Pressure Drop inTape-Generated Swirl Flow of Single-Phase Water," JOURNAL OF HEAT

TRANSFER, TRANS ASME Vol. 91,1969, pp. 434-442.15 Akers, W. W., Deans, H. A., and Crosser, O. K., "Condensing Heat

Transfer Within Horizontal Tubes," Chemical Engineering Progress Symposium Series, Vol. 55, No. 29,1959, pp. 171-176.

16 Soliman, J., Schuster, J. R., and Berenson, P. J., "A General HeatTransfer Correlation for Annular Flow Conde nsation," JOURNAL OF HEATTRANSFER, T R A N S . ASME, Vol. 90,1968, pp. 267-276.

17 Watk inson, A. P., Miletti, D. L., and Tarasoff, P., "Turbulent HeatTransfer and Pressure Drop in Internally-Finned Tubes," AIChE Preprint No.10, presented at the Thirteenth National Hea t Transfer Conference, Denver,Aug. 6-9,1972.

18 Vasil'chenko, Yu. A., and Barbaritskaya, M. S., "Resistance WithNon-Isothermal Fluid Flow in Tubes With Longitudinal Fins," Thermal Engineering, Vol. 16, No . 1,1969, pp. 28-35.

19 Vasil'chenko, Yu. A., and B arbaritskaya, M . S., "Heat Transfer in TubesWith Longitudinal Fins," Thermal Engineering, Vol. 16, No. 5, 1969, pp.105-109.

Jou rna l of Heat Transfer FEBRUARY 1978, VOL 100 / 23




20 Ornats kii, A. P., Shcherb akov, V. K., and Semena, M. G., "Investi gation 21 Ornats kii, A. P., Shche rbakov , V. K., and Semena, M. G., "Investig ation

of Veloc i ty Dis t r ibut ion in Tub es With Inte rn a l Longi tud ina l F ins ," Thermal of Hea t Transfe r in Channe ls Be tween F ins of Tub es With Inte rn a l Lon gi tu-Engineering, Vol. 17, No. 2, 1970, pp. 108-111 . dina l F ins ," Thermal Engineering, Vol. 17, No. 1 1,1970, pp . 96-100.

CONTENTS (CONTINUED)

163 Nonsimilar Laminar Free Convection Flow Along a Nonisotherm al Ver tical Plate

8. K. Meena and G. Nath

165 A Loca l Nonsimilarity Analysis of Free Conve ction From a Horizo ntal Cylindrical Surface

U. A. Rfluntasser and J. C. Mulligan

167 Transient Temperature Profile of a Hot Wa ll Due to an Impinging Liquid Droplet

U. Seki, H. Kawamura, and K. Sanokawa

169 Combined Conduction- Radiation Heat Transfer Through an Irradiated Semitransparent Plate

R. Viskanta and E. D. Hirleman

172 Transient Conduction in a Slab With Tem perature Dependent Therm al ConductivityJ. Sucec and S. Hedge

2 4 / VOL 100, FEBRUARY 1978 Transact ions of the ASME




G. H.Junkhan


fowa State University,

Ames, Iowa

Some Elects of Mechanically-

Produced Unsteady Boundary Layer

Flows on Convective Heat Transfer

Augmentation

Tests of experimental results from augmentation of conuective heat transfer coefficients

at a gas/solid interface by mechanical disruption of the boundary layer are discussed on

both an average and a local basis. The averaged data show that the augmentation ob

tained depends upon the wake effects of the mechanical device used to produce the distur

bance and upon the time required to re-establish the boundary layer after the disturbance

has occurred. The local data analysis indicates that the amount of augmentation obtained

depends on the relative velocity vector of the fluid w ith respect to the mechanical device

and upon the time behavior of this vector.

In t roduct ion

A gas/solid interface is the controll ing heat transfer resis tance in

many heat transfer devices or systems. Recently an effort to au gmen t

such convective heat transfer coeffic ients by means of mechanical

scraping of surface boundary layers was reported [ l] . 1 A heate d f la t

plate, shown in Fig. 1, with air flowing parallel to its surface and a

blade rotating in a plane paralle l to the plate surface was used as the

experimental system. Details of the plate construction are given in

reference [2]. Significantly increased convective heat tran sfer coef-

ficients were obtained for scraper clearances of 0.76 mm (0.030 in.)

to 4.06 mm (0.16 in.) and blade r otat ion al spee ds of 100 to 2000 rpm

over a plate Reynolds number range from 40,000 to 650,000. The

mea surem ents rep orted in [1] were taken a t s ix s ta tions distr ibute d

spanwise across the plate a t the same distance downstre am from the

leading edge. The data from these sta tions were averaged to give asingle , local value of Nusselt number based on distance from the

leading edge. Improvements in the local Nusselt number were at

tributed to a turbulent type of flow between the scraper blade and the

plate surface; the greatest increases were obtained for flow conditions

where a laminar or transit ional boundary layer would have existed

if the scraper were not present.

This paper pr esents further analysis of the span-averaged data and

discusses local, unaverage d da ta in an effort to gain further insight

1Numbers in brackets designate References at end of paper.

Contributed by the Heat Transfer Division for publication in the JOURNALOP"HEAT TRANSFER. Manuscript received by the Hea t Transfer D ivision May16, 1977.

into the phenomena responsible for the augmentation obtained.

Span-Averaged DataFig. 2 shows typical augm entati on obtained for one clearance, to

gether with solid lines representing well-known laminar and turbulent

boundary layer heat transfer correlations taken from Kays [3].

The effect of clearance for the range investigated is relatively small

as shown in Fig. 3 where the augm entatio n, expressed as the rati o of

scraped to unscraped Nusselt number is plotted as a function of a

dimensionless c learance 0 . The dimensionless c learance is defined

by

< t> =5x

where <5 is the scraper clearance and x is the distance from the leading

edge. For laminar boundary layer flow, <j> is the equivalent of the ratioof scraper c learance to unscraped boundary layer thickness. The

UNHEATED

NOSE PIECE

THERMOCOUPLE LOCATIONS

Fig. 1 Plan view of plate





000

too

in

TURRII I FNT Rn i lN fpaPY

L AYER C O R R EL AT I O N 1

~ A— — A — * /fr

2000 r p m * — / / A

1000 rpm # — — " * ^ s / /

—-° ^ /5 00 r pm • • — " r f > /

/ / / ^ — ^ SC RAP ER_ ^

d/ REMOVED

100 rpm • — / /

^ ^ ^~~~-~ LAMINAR BOUNDARY LAYER CORRELATION

CLEARANCE 0.00127 m (0. 050 in .)

1 ! 1 I | \ f 1 1 f f 1 1 l ! 1 1

POSITION OF SCRAPER

FLOk DIRECTION

Fig. 4(a) Blade nomenclature for wake shedding

FOR AUGMENTATION

BLADE RE-. HAKE . I.ESTA B.

BOUNDARY LAYER

START O N E BLADE PERIOD

Fig. 4(b ) Time diagram for boundary layer disruption events

Fig. 4 Time and blade posit ion sketches for span-averaged data

REYNOLDS NUMBER, Re - —

Fig. 2 Span-a veraged heat transfer results for 0.00127m (0.050 in.) clearance

SYM B O L

oA

•&De

RPM

100

2 0 0 0

1 00

2 0 0 0

1002 0 0 0

ReX

8 0 , 0 0 0

8 0 , 0 0 02 5 5 , 0 0 0

2 5 5 , 0 0 05 4 0 , 0 0 0

5 4 0 , 0 0 0

,o~ - ^ A = L 3 30.5 1 .0

DIMENSIONLESS CLEARANCE »

Fig. 3 Clearance effects for span-averaged data

maximum augmen ta t ion obse rved was approx ima te ly inve rse ly

proportional to the fourth root of the clearance, with much of the data

showing even smaller effects.

The span-averaged data in Fig. 2 for 100 rpm rotational speed and

Reynolds numbers from 40,000 up to about 180,000 fall close to an

extension of the turbulent correlation to lower Reynolds numbers.

Data for higher rotational speeds are well above the extended tur

bulent correlation l ine in this Reynolds number range, but data for

each rotational speed appear to eventually fall in to or c lose to the

turbulent correlation as the Reynolds number increases. This behavior

suggests that scraping frequency causes the majority of augmentation

above the turbulent correlation, with greater improvement obtained

for higher scraping frequency.

An objective of boundary layer scraping is to disrupt an established

layer, result ing in greater f luid mixing near th e surface and a larger

heat transfer coefficient. The frequency of scraping is obviously

intimately connected with the time that an established boundary layer

exists . I t is c lear that the blade will leave a downstrea m wake which

will affect the boundary layer a t the measuring stations located on

the sp an of the plate . Th e wake will exist as long as the blade is up

stream of the measuring station; the effective blade period for aug

mentation is thus not th e same as the blade period based on rotational

speed alone.The length of the effective blade period can be calculated with the

aid of the plan view in Fig. 4(a) showing the location of the six span-

wise thermocouples, the scraper blade and pertine nt dimensions. Th e

scrap er bla de, rotat ing in a clockwise direction in Fig. 4(a), will affect

the measuring stations from the t ime it is d irectly over them until i t

is no longer in a position where its wake will pass over them. During

this t ime interval, the scraper blade will rotate through the angle in

dicated as 6. The present apparatus geometry gives an average 0 of

abou t 51 deg for the s ix thermocouple s used, or about 28 percent of

the blade period based on rotational speed for a two-bladed scrap

er .

In addition to the t ime the f low is disturbed by the wake, t ime is

required for the bo undary layer to re-establish itself. When the blade

• N o m e n c l a t u r e -

h = convective heat transfer coefficient

k = thermal conductivity of f luid

U = free stream fluid velocity

VB = scraper blade velocity

VF/B = relative velocity of fluid with respect

to the blade

x = distance from leading edge of plate

b = scraper c learance above plate surface

6 = angular posit ion of scraper blade

n = fluid viscosity

p = fluid density

r = blade period

D i m e n s i o n l e s s g r o u p s

hxN U l = Nusse l t number based on

distance from leading edge

RUX

He x = p—Reynolds number based on

d imens ion less b lade

distance from leading edge

5x

clearance

= used over symbol in equations to indicate

a vecto r qua n t i ty

26 / VOL 100, FEBRUARY 1978 Tra nsact ions of the ASM E




Table 1 Calculated ef fect ive blade period for span-averaged data. C learance = 0.00127m (05 0 In.)

N o m i n a l

Reyno lds

N u m b e r

Re x40,000

80,000

160,000

260,000

530,000

40,00080,000

160,000

260,000

40,000

80,000

160,000

40,000

80,000

160,000

260,000

530,000

Blade

Speed

rp m

100

100

100

100

100

50 050 0

50 0

50 0

1000

1000

1000

2000

2000

2000

2000

2000

W a k e

Effect

T i m e

se c

0.0840

0.0840

0.0840

0.0840

0.0840

0.01680.0168

0.0168

0.0168

0.0084

0.0084

0.0084

0.0042

0.0042

0.0042

0.0042

0.0042

R e - e s t a b l i s h m e n t

T i m e

se c

0.0423

0.0211

0.0108

0.0066

0.0032

0.04230.0211

0.0108

0.0066

0.0423

0.0211

0.0108

0.0423

0.0211

0.0108

0.0066

0.0032

tip is in a position where it will no longer influence the measuringstation with i ts wake, there is a f ixed distance from the blade t ip to

the measuring station. Assuming a developed boundary layer exists

upstream of the blade, and that the boundary layer re-establishes

itself at the same speed as the free stream velocity, the time required

for re-establishment can be estimated.

The effective blad e period for augme ntatio n is then the total t ime

consumed for the wake effect and boundary layer re-establishment;

these t ime inc reme nts are shown in the diagram in Fig. 4(b). For the

remainder of the blade period an established boundary layer exists

over the measuring station.

Re-establishment t ime varies with the free s tream velocity , inde

pen dent of scraper rotation al speed. If the sum of the wake effect tim e

and the re-establishm ent t ime is large enough, l i t t le or no established

boundary layer will ever exist at the measurement station and a larger

increase in span-averaged heat transfer coeffic ient should be observed.

Calculations of wake effect t ime, re-establishment t ime, effective

blade period, and pe rcent t ime with established boundary layer were

made for all span-averaged data. Results for the 1.27 mm clearance

are given in Table 1. Th e last column in Table 1 contains values of the

augm entatio n obtain ed, defined as for Fig. 3 . The augm entati on

shown in Tabl e 1 is plotte d in Fig. 5 as a function of the percent of tim e

an established boundary layer existed. The augmentation range for

points for which no established boundary layer existed is shown on

the ordin ate and is well above the augmen tation obtained for poi nts

with even partia lly-e stablishe d bound ary layers. Th e impro vem ent

indicated in Fig. 5 is typical of the other c learance s used.

If the data of Tab le 1 are applied to Fig. 2, those poin ts for which

an established boundary layer could not exist are all s ignificantly

above the extended turbulent boundary layer correlation; even thosedata with short established boundary layer t imes (say 50 percent or

less) show considerable improvemen t. The im provem ent is indicated

in Fig. 2 by the filled point s, all of which ha d an establis hed boun dary

layer for 50 percent of the time or less.

Loc al Ef fe c t s of B lade Mot ion. The above analysis of span-

averaged data for re-establishment of the boundary layer does not

include effects of the direction of blade motion with respec t to the flow

because the data were averaged. Local data for the same measurement

stations have subsequently been examined and regular differences

from one side of the plate to th e other w ere found. Fig. 6 shows typical

local behavior; blade motion opposite to the free stream flow direction

results in Nusselt numbers as much as 65 percent higher than when

the blade motion is generally in the same direction as the free


Effective

Blade

P e r i o d

se c

0.1263

0.1051

0.0948

0.0906

0.0872

0.05910.0379

0.0276

0.0234

0.0507

0.0295

0.0192

0.0465

0.0253

0.0150

0.0108

0.0074

T i m e w /

E s tab l i shed

B. L ayer

%58

65

68

70

71

1.537

54

61

—

1.7

36

—

——

28

51

Augmen ta t ion

Nu

Nu*x Scraped

Unsc raped

1.75

1.53

2.10

2.95

1.10

4.363.14

2.49

3.00

6.48

4.90

3.79

9.73

7.31

6.13

4.55

1.10

-

~

0

p

~-

-

-

AUGMENTATION RANGE FOR DATAWITH NO ESTABLISHED BOUNDARYLAYER

CLEARANCE = 0.00127 m (.050 in .)

o 100 rpmD 500 rpmc 1000 rpmO 2000 rpm

O

A

° O o

D

O

O

oO o

I i . 1 , 1 1 1

PERCENT TIME BOUNDARY LAYER ESTABLISHED

Fig. 5 Effect of boundary layer establishment time on augmentation

s t ream.

Such behavior suggests that differences in boundary layer distur

bance occur because non-uniform blade wakes are shed due to variable

relative motion between the f luid and the blade. Even though both

fluid velocity and blade speed ar e constan t, the relative velocity vector

of the fluid with resp ect to the blade changes continuo usly as the blade

rotates; the continuously varying angle between the absolute blade

velocity vector and the absolute fluid velocity vector results in several

different re lative motion cases. Thes e relative motion cases can be

obtained using the vector equation

VFIB =U-VB

where Vp/s is the relative velocity , U is the free s tream velocity and

VB is the blade velocity at a given radius.

Th e first of these cases is shown in Fig. 7(a) where the blade motion

w is in a direc tion ge nerally o ppo site to the m ain fluid flow velocity

U. For 0 < S < 90, a wake will be shed dow nstream of the blade be

cause the relative velocity vector VV/B will always have a downstream

component for any blade posit ion.

Th e second case is shown in Fig. 7(b) where the blade m oves gen-

FEB RUARY 1978, VOL 100 / 27




uou

100

+10

--

-

-

-

;

-

SCRAPER ROTATION

D AGAINST FREE STR.0 WITH FREE STR.

CLEARANCE

0.00127 m (0.050 in.) /

500 rpm / ^

/ / / TURBULENT

r ^ COR R E L AT I O N

, I I * I , , , :

REYNOLDS NUMBER, Re = ^X li

F ig . 6 L oc a l h e at tra n s fe r re s u lts f o r 0 . 0 0 1 2 7 m (0 .0 5 0 In . ) c le a ra n c e

10,000

^

: 1000

100

-

: SA-

1 1 1

SCRAPER ROTATION

D AGAINST FREE STR.0 WITH FREE STR.A ACROSS

_--—""tr/

&^f— — t i - Cx"^/

/ \ T U R B U L E N T

/ BOUNDARY LAYER

/ CORRELATION

2000 rpm

CLEARANCE

0.00127 m (0.050 in.)

1 I 1 l 1 l 1 1 1

10 pU x

REYNOLDS NUMBER, v

F i g . 8 L oc a l re s u l ts f o r 2 0 0 0 rp m a n d 0 .0 0 1 2 7 m (0 .0 5 0 in . ) c le a ra n c e

BLADE

« s FLOW

BLADE ROTATION

F ig . 7 Velo c i ty d iagr am for local results wi th d i f ferent blade speed and po

s i t ion cases

erally in the same direction as the flow; for this case no wake will be

shed since V F /B lies along the blade. Boundary layer re-establishment

will occur quickly because l i t t le or no wake is sh ed.

The third case, shown in Fig. 7(c), results in a blade wake since the

VF/B vector is ahead of the moving blade; the boun dary layer is dis

turbed much as in Fig. 7(a), but with less time for the wake to cover

the measurement s ta tion and probably a shorter t ime is required for

boundary layer re-establishment s ince relatively undisturbed fluid

will immediately follow the blade.

A fourth case, depicted in Fig. 7(d), exists when the blade movesgenerally in the sam e direction as the f low bu t sufficiently faster so

tha t the w ake is shed b ehind the moving blade; no wake will affect t he

measu ring statio n prior to the arrival of the blade for this case.

Th e last three cases can occur at different po sitions along the length

of the blade at the same time, s ince blade velocity increases with

distance from the center of rotation. Calculations of Vp/s for various

posit ions on the blade at constant free s tream velocity U and blade

angle 0 show that with the proper magnitudes of U and V B , the V F / B

vector will leave a wake behind the blade near the t ip and ahead of

the blade closer to the center of rotation. Between these two positions,

VF/B l ies a long the blade , so tha t no wake is shed at t hat location. As

the blade rotates to a new 6 posit ion, the angle between V B an d U

changes; the posit ion for which no wake is shed then moves along the

blade as i t rotates . The reduced Nusselt number observed in Fig. 6

for Rex

= 81,000 and blade motion generally in the free s tream di

rection would be caused by such a phenomenon; s imilar results are

found for other blade rotational speeds and clearances where the free

stream and blade velocities tested have relative velocity vectors that

behave in the same manner.

Further evidence is found from additional data taken upstream of

the center of blade rotation; the blade motion is across the free stream

direction as it passes over these measuring positions. Typical data for

this location should be lower than either of the other two locales be

cause the relative velocity vector is nearly parallel to the blade except

for very high blade velocity; in addition, the re-establishment t ime

is very short . Typical data from these locations is shown in Fig. 8

where the upstream measuring station augmentation is considerably

below the other measuring posit ions.

C o n c l u s i o n s

An analysis of heat transfer data taken on a f la t p late when the

boundary layer was disturbed by a moving blade has shown that

considerable heat transfer augmentation may be obtained for nor

mally laminar or transit ional boundary layers . The span-averaged

data analysis shows that the effective augmentation period depends

on the wake behind the blade and the t ime required to re-establish

the boundary layer. In addition, these data have indicated that the

augmentation obtained is inversely related to the amo unt of t ime th at

an established boundary layer exists over the surface.

The local data analysis has shown that the am ount of augm entation

obtained is dependent on the direction of the blade with respect to

the main flow. Mo reover, the sim ple fluid/blade vector model has

shown th at the effect of the blade depends greatly on both the m ag-

28 / VOL 100, FEB RUARY 1978Transact ions of the ASME




nitude and direction of the relative velocity vector and the t ime tha t

this vector remains in a particular orientation with respect to the flow;

the relative velocity vector thus affects not only the blad e wake effect

t ime, but the re-establishment t ime as well .

The results obtained suggest that, at least for laminar or transitional

boundary layers , regularly unsteady flow in the region normally as

cribed to bound ary layer flow will result in considerable augm entatio n

of convective heat trans fer coefficients.


The Iowa State University Engineering Research Insti tute has

partia lly supported this work.

R e f e r e n c e s

1 Hagge, J. K., and Junkh an, G. H., "M echanical Augmentation of Convective Heat Transf er," JOURNAL OF HEAT TRANSFER, V ol. 97, Nov. 1975,pp . 516-520.

2 Hagge, J. K., and Junkhan , G. H., "Experimen tal Study of a Method ofMechanical Augmentation of Convective Heat Transfer Coefficients in Air,"Technical Report HTL-3/ISU-ERI-Ames-74158, Nov. 1974, Engineering Research Institute, Iowa State University, Ames, Iowa.

3 Kays, W. M., Corwective Heat and Mass Transfer, McGraw-Hill BookCo., New York, 1966.

Journal of Heat Transfer FEB RUARY 1978, VOL 100 / 29




J . C. Chato

Professor,

Department of Mechanical and Industrial

Engineering,

University of I l l inois at Urbana-C hampalgn

Urbana, III .

R. S. Abdulhadi

Assistant Professor,


University of Jordan,

Amman, Jordan

Flow and Heat Transfer in

Convectively Cooled Underground

Electric Cable Systems: Part 1 —

Velocity Distributions and Pressure

Drop CorrelationsVelocity distributions and pressure drop correlations have been obtained experimentally

for a wide range of physical, flow, and thermal parameters in three models of oil-cooled

underground electric cable systems. The flow can be considered as consisting of several,

interconnected, parallel channels. If one of these is significantly larger in cross section,

it will dominate the behavior of the entire system with velocities up to an order of magni

tude greater than in the other channels. Laminar, fully developed flow exists at a dimen-

sionless en trance length as low as X / D H = 100. The turbulent velocity profiles are essen

tially uniform in the larger, main channel, but not in the small channels with lower veloci

ties. Laminar friction factors can be correlated w ith an equivalent Reynolds number,

based on the main flow channel alone, as fe-Re e = 30. Turbulent friction factors increase

with increasing skid wire roughness ratios and approach the asymptotic values of 0.013

and 0.021 for roughness ratios 0.0216 and 0.0293, respectively. Cable heating had no noticeable effect on the friction factor correlations.


At the pres ent t im e, about 3 ,200 km (2,000 miles) or less than one

percent of the total e lectr ical transmission l ines in the Un ited States

are underground. The largest of these cable systems has a capacity

of approx imately 500 MW at 345 kV. Although i t is generally agreed

tha t the present technology of undergrou nd transm ission is not ca

pable of both effic iently and economically meeting antic ipated re

quirements , basic design improvements of oil-cooled underground

cable systems are needed until new and different techniques are de

veloped.

Th e curren t carrying capacity of under groun d cables is l imited by

the maximum allowable temperature of the e lectr ical insulation that

is required for long life and high reliability. This limiting temperature

is abou t 85°C (185°F) and occurs a t the inner dia mete r of the insu

lation in contact with the e lectr ical conductor. This temperature

depends primarily on the rate a t which heat generated within the

system (due to conductor losses, magnetic effects, and other factors)

can be transferred to the surrounding earth or some other cooling

Contributed by the Heat Transfer Division and presented at the WinterAnnual Meeting, New York, N. Y., December 5-10,1976. Revised manuscriptreceived by the H eat Transfer Division August 20, 1976. Paper No. 76-WA/HT-45.

medium. In forced-cooled systems for oil filled, pipe-type cable cir

cuits , chil led oil is c irculated through the pipe and most of the heat

generated in the cable and insulation is absorbed by the oil. The heat

is then transferred from the oil to the envi ronm ent a t refr igeration

stations located at predetermined intervals a long the transmission

line.

By use of a forced-cooled system, the power capacity of under

ground cables can be economically increased. In such systems, the heat

generated w ithin the conductor is transferred by conduction th rough

the cable insulation to the surface of the cable . This heat is then

transferred to the oil by a combination of forced and natural con

vection. Forced convection is established by the flow of the oil in the

pipe; natural convection is set up by the tem per atu re difference be

tween the cable surface and the oil .

Th e f low in a cable-pipe is characterized by the irregular surfaces

and cross sections resulting from the particular configurations of the

cables inside the pipe and the repeated roughness of cable surfaces

due to the skid wires wound helically around the cables.

The h eat transfer an d fric tion character is t ics of surfaces having

different types of roughness and flow geometries became the subject

of extensive analytical and experimental investigation as a result of

their increasing application in the fuel e lement design for both gas

and water-cooled reactors as well as liquid-metal, fast breeder nuclear

reactor systems. More recently , these surfaces became of increasing

30 / V OL 100, FEBRUARY 1978 T ransac t ions of the ASM ECopyright © 1978 by ASME




concern to investigators involved in the design and c o n s t r u c t io n of

oil-f i l led, p ipe-type underground electr ical cable systems.

Several investigations of rod bundles with helically wrapped wire

spacers were reported [1-4]l. It was concluded from most of th e s e

investigations that both the friction factors and N u s s e l t n u m b e r s of

rod bundles with wire spacers increase with increasing roughness

ratios and decreasing distance between wire wraps.

Detailed experimental data and correlations of the fluid mechanics

and heat transfer characteris t ics ofoil-filled, pipe-type cable systems

are scarce. Published results do not cover the entire range offlow and

thermal conditions applicable in the design and o p e r a t io n of s u c h

systems. An e x p e r im e n ta l in v e s t ig a t io n of the forced cooling for a

model pipe-type cable system w as reported in 1970 [5]. T e m p e r a t u r e

profiles from the cable surface to the pipe wall were measured and

used to calculate the oil f i lm thermal resis tance. Our calculations,

based on reported system dimensions and flow velocities, indicate that

most ofthese tests were conducted under laminar flow conditions with

Reynolds numbers below 500.

Be c k e n b a c h , et al. [6] p r e s e n te d a theoretical approach for the

prediction of the friction factor as a function of the Reynolds number

for a pipe-type cable system. Their predicted values were compared

with experimental results obtained under isothermal conditions with

oil and water used as working f luids in order to cover a wide range of

flow Reynolds numbers. Correlations for friction factors are presented

in references [7-10] . Other related works were further discussed in

references [11, 12].

T h e p u r p o s e of our work is to understand more fully the flow andthermal characteris t ics of underground cable systems. In th is art ic le ,

we are r e p o r t in g on the velocity distr ibutions and p r e s s u r e d r o p s

o b ta in e d for a wide range of p a r a m e t e r s . The t e m p e r a tu r e d is t r ib u

tions and heat transfer results are reported in a companion article [13].

Dimensionless correlations were developed for a p p l ic a t io n in the

design of full-scale underground cable systems.

T h e E x p e r i m e n t a l S e t u p

G e o m e tr ic a l p a r a m e te r s of the three different models ((1) large

cables in large pipe, (2) small cables in small pipe, and (3) small cables

in large pipe) used in th is investigation are p r e s e n te d in T a b le 1 and

the different cable configurations referred to in the t a b le are shown

in Fig. 1 . Th e schem atic diagra m of th e e x p e r im e n ta l s e tu p is shown

in Fig. 2. D e ta i l s of the c o n s t r u c t io n and i n s t r u m e n t a t i o n are given

in references [11, 12].

One test section was made of 140 mm I.D. plexiglass pipe in o r d e rto facil i ta te visual observation of the flow. The second test section

consisted ofa 45 mmI.D. steel pipe with 3.2 mm thick walls. A heatin g

str ip was wrapped around the steel portion of this test section in order

to s imulate the heat generated within the outer pipe as a result of th e

e le c t r o m a g n e t ic e d d y c u r r e n ts th a t e x is t in an actual cable system.

O ne set of simulated cables was made up of three, 38 mm O.D. thin

wall steel tubes. Sixteen-gauge copper wires (1.3 mmdia) were wound

around each of the tubes to simulate the actual skid wire arrangem ent

in a real system. Electr ic resis tance heaters were inserted along the

axis in each of the cables to p r o v id e the required heat f lux rates .

Thermocouples were soldered to the cable surface at 1.22 m in tervals ,

a n d the thermocouple leads were pulled through the inside of the

tu b e s to avoid disturbing the flow.

A n o th e r set of simulated cables was made up of three, 13 mm O . D .

steel tubes. The skid wires were s imulated by 30-gauge Teflon-insulated wires (0.53 mmO.D.) , and the heater installa tion was the s a m e

as that in the larger cables. Two thermocouple junctions were soldered

to each of the three tubes at 2 m and 8.4 m from the test section inlet.

Fig. 3 shows typical sections from an actual cable and the two simu-

T a b l e 1 P h y s i c a l p a r a m e t e r s of thr e e te s t mode l s

Configuration

D p, mmD c , mmD e , mmD s , m m

S i, mmS2 , mm

D s / D e

1

1403859.7* = DH

1.29

23.011.0

0.7760.0216

M o d e l 12

1403893.5

1.29

2.61.30.7450.0138

3

14038

102.71.29

2.61.30.7580.0126

1

451318.1* = DH

0.53

6.55.00.7540.0293

M o d e l 22

451328.5

0.53

1.10.50.7250.0187

3

451331.5

0.53

1.10.50.7430.0168

M o d e l 31

14 013

107.0* = DH

0.53

6.5119.0

0.9750.00495

P itc h/D s =* 10 and 17; wire helix angles = S 3 de ga nd 52deg; test section length = 9.14 m*AC = overall flow area inconfiguration 1 for each model.

1 N u m b e r s in br a c ke ts de signa te R e f e r e nc e s at end of pa pe r .


A e = effective or main f low cross-sectional

area as shown in Fig. 1

A p = pipe cross-sectional area

d = distance between cable surface and inside

pipe wall in any radial traverse

D c = cable outside diameter

D e = h y d r a u l ic d ia m e te r b a s e d onAe

DH = hydraulic diameter based on overall

flow area

D p = pipe inside diameter

D s = skid wire height

/ = D / / A P / 2 L p [ /2

, friction factor based on

overall flow area

fe = D eAP/2LpU e2, friction factor based on

A c

L = axial d is tance between pressure taps

A P = pressure drop between two axial loca

tions

qc = heat flux rate from the cable surface to

the oil

qw = heat flux rate from pipe wall to the oil

Re = UDnh, Re y n o ld s n u m b e r b a s e d on

overall flow area

R e e = UeDelv, Re y n o ld s n u m b e r b a s e d on

A e

S i = distanc e between two adjacent cables

S 2 = distance between cable and pipe wall

u = axial velocity in radial traverse

"max=

maximum velocity in a radial tra

verse

U = mean velocity based on overall flow

area

Ue =mean velocity based onAe

x = axial d is tance from inlet of test section

y = radial d is tance from a cable surface

p = d e n s i ty

c = kinematic viscosity





Fig. 3 Typical sections of actual (right) and simulated cables (actual cableIs shown here without skid wires)

Discussion of the Hesults

Axial Veloc ity Distributions. Typical i sothermal velocity

distributions in the vertical plane of symmetry measured at three

different axial locations for laminar flow conditions in Modell are

shown in Fig. 4. Our velocity and pressure drop results indicate that

the flow was virtually fully developed at x/ f )1l = 117.

Fig.: , shows a set of typical isothermal axial velocity distributions

at different flow rates for cable configuration :J in Model 2. These

profiles indicate a shift in the position of the point at which maximum

local velocity occurs toward the smooth wall of the outer pipe with

increasing Reynolds numbers. This is in agreement with the resultsof previous experimental investigations of flow in annular passages

with the inner surface being rough and the outer surface smooth. It

is also obvious from these results that the turbulent velocity profiles

(He = 240G) are flatter than those in the laminar range.

The maximum local velocity, 11 "" '" and the tot al radial distance,

d, between a cable and the p ipe wall were used to obtain good di

mensionless correlation of all velocity data. Fig. 6 shows this corre·

lationapplied to the results of Fig. 5 as wellas for the velocity distri·

butions for cases involving heated cables at X/DH = 473.

Velocities in the small restricted triangular channel bounded by

th e two bottom cables and the outer pipe wall in configuration 2 in

Modell were measured at different flow rates in order to determine

their magnitudes relative to the corresponding velocities in the main

Configuration 3

Configuration I

Configuration 2

lated cables used in this investigation.

Oil flow rates through the test section, heat exchanger, and cali

b ra ti on devices were measu red with a combination of calibrated

venturi and rotameter systems.

Pressure drops between different points along the test section were

measured with U-tube manometers connected between the pressure

taps.

Oil temperatures at the inlet, outlet, and intermediate stations

along the test section, as well as surface temperatures of the cables

and outer pipe, were measured hy 24-gauge copper-constantan

thermocouples.Local oil velocities and corresponding temperatures were measured

with a hot-film sensor which was connected to a constant-temperature

anemometer through a temperature switching circuit. A screw-type

traversing mechanism was designed and used to move the sensor

between the cables and outer pipe in any radial direction.

Fig. 1 Cross sections of three cable configurations. Effective flow areas arewithin dashed lines. In configuration 1, effective and total flow areas are thesame

Fig. 2 Schematic diagram of exper imental setup for Models 1 and 3. ForModel 2 rotat ing measuring stations were at 2 m and 8.4 m from the inlet ofthe test section, pressure taps were at 0.9 m intervals, and temperatures couldbe measured at 2, 6, and 8.5 m from the inlet

0.12

0.10 @ a a aa

o 0 6

II IIa

• 0...08 a 8 ..

a

";- a

O.06

0.049 symbol X/DH

a 21.46 6

a117

0.02

a •"'ll

0.00 L-_--L__- '-__.L-_--L__..L._---J

0. 0 40.0 80.0

Y (mm)

Fig. 4 Local axial velocitydistributionsfor configuration 3 inModel 1 atRe=244, qc =0

Power andInstrument

Lead s \

\

T Tempera tu reMeasurementStations

o Valve

[>< ] Venturi Flowmeter

Rotating

Measuring

,Reservoir Power and\ Instrument

( L e adS

/~ T T

y T

Heat _/"§ .s

Exchanger VThermocouple "-

(Cables Coaling(i1 Water

Typ ical Cross

Section A-A





2 . 0 0

1.60

.—

- S.203

0 . 8 0

0 . 4 0

0 . 0 0

symbol Re

» 138

• 4 0 3

" 9 2 5

a 2 4 0 6

o °°

"D

A

A

A

E3

j - l j l * * * A

^

D

s

A

y

}D

A

a

&

D D

A A

s m

A A

D

A

19

• D D

O D D

D

-

A

AA

A

A D

• • - . : .

0.0 10.0 20. 0y (mm)

30.0

Fig, 5 Axial velocity distributions for configuration 3 in Model 2 at x/DH =

473, qc = 0

A

QB

a

D

- a

a

yt

_ \ _ d a D

y

a c: A-pr o f i le

m B —profile

xi

QO

'o

E

oX)

a

1

•

D

0.00 0 .40 , 0 .80y/d

Fig. 7 Dlmenslonless velocity distributions for configuration 2 in Model 1 atx/DH = 94 , Re = 1647, qc = 0

1.00

0 . 8 0

0 . 6 0

0 . 4 0

0 . 2 0

n n n

^J

- ( M d o 1

W - f . o 18 ° 8

- . 8 S 8 »° 1

S 8&

a

98*

S t

8 .O H

- o 4

S

k

' '

M i 5U D O D

1 ® a «

5o

!* "

symbol Re

138

4 0 3

9 2 5

a 2 4 0 6

o 169

• 4 5 7

»S ° |0 36

| 0 D • 2 4 8 2

* B Q

* « al O

« •

. 81

q c ,W/m2

0

0 -

0

0

4 3 8 -

4 3 8

4 3 8

4 3 8 -

_

0.00 0.40 , 0.6

y/d

Fig. 6 Dlmensionless axial velocity distributions for configuration 3 in Model2 at x/DH = 473

f low channel. The velocit ies in the restr ic ted channel were found to

be approximately one order of magnitude lower than the corre

sponding main f low velocit ies for the entire laminar f low range. The

relative magnitud es increased, however, as transit ion into turbulence

occurred. One typical comparison of these velocit ies is presented in

Fig. 7.

A similar phenomenon was found to exist for configuration 1 in both

Mod els 1 and 2 as shown in Fig. 8 which clearly indica tes th at for low

Reynolds numbers (Re = 180) in the laminar range, the velocity in

the secondary flow region between the cable and pipe wall is no more

than ten percent of the maximum velocity in the main f low channel.

Other results, not shown here but reported in references [11,12], show

that th is ra tio is a s trong function of the f low Reynolds number and

increased to about 80 percent for Re = 6600.

P r e s s u r e Dro p R esul t s a nd Co rre la t io ns . Pressure drops were

measured for different cable configurations under laminar, transi

t ional, and turbulent f low conditions for both isothermal and non-

isother mal cases. The correlatio n of the fr ic tion factor, / , against th e

usual Reynolds number, Re, based on the entire f low cross section

produced quite different curves for the three configurations. In the

lam inar rang e for Mode l 2 th e values of / .R e were 30, 12, and 8 for

configurations 1 , 2 and 3 respectively. The tran sit ion to t urbul ence

t 1 1

B t y

/oX__lVpo) d

" T

A * y-o A—prof i le °

s B — profilea

D

• D pQ

aa

a

D

^ m ^

t

D a

° 1

pipe

D

-

a

a

D

a

0.40 , , 0 .80y/d

Fig. 8 Dlmenslonless velocity distributions tor configuration 1 in Model 1 atx/DH = 142, Re = 180, qc = 400 W/m

2

star ted a roun d Re ^ 600 for configuration 1 and Re =* 1200 for con

figurations 2 and 3 . In the tu rbu len t range at Re =; 5000, the fr ic tion

factor w as 0.023 for configuratio n 1 and 0.012 for the oth er two con

figurations with a skid w ire roughness ra tio of 0.0293.

A better correlation of these results was obtained by using an ef

fective hydraulic diameter, D e, based on the actual main flow area and

wetted perimeter for each case (neglecting small channels between

the cables and the pipe wall) as shown in Fig. 9 . However, while the

latter method gives a better correlation of the results , the former is

more practical because it eliminates the need to guess the actual cable

configuration in any particular section of a cable pipe.

I n th e la m in a r r a n g e , Re e < 1,000, the best correlation is

fe • R e e = 30

However, a more conservative design value may be

fe • R e e = 35

(1)

(2)

In the transit ion and turbulent range, 1 ,000 < Re e < 20,000,

Rehme's correlation [7] based on equation (1) can be writ ten as

V - + 2.13 en^

2.13 £n Re„ - 1.664 (3)

Th e left hand side of equa tion (3) is v ir tually a s traight l ine in the

range of in terest 8 < V2/fe < 14. Introducing a linear relation, which

fits the actual function within two percent, a l lows the s implif ication

Journal of Heat Transfer FEBRUARY 1978, V OL 100 / 33




10

3

_ l

CL

10-2

</-fe-Re

"..

-

= = 3 5- = 3 0

i i 1 1 1

r — i i - r i mi — i - i — ' • ' • • iSymbol Config. Oil

; 1

• i

o 2 Poly V is OSHD 3

* 1A 2 Nalkelyne

^ ' 3

^ ^ > * ^ r E q . (4)

-6 0 0 :

:

1 0 ' 10" I0 H 10 °

R e e = U e D e / u

Fig. 9 Friction factor results based on effective hydraulic diameter in Model1 at xlD H = 120, L = 2.87 m, qc = 400 W/m 2

of equation (3) to

le(1.783 £n R e e - 3.858)2 (4)

The curve in Fig. 9 shows tha t th is correlation f i ts the up per l imits

of the data quite well for DS/DH = 0.0216. However, as shown in Pig.

10, increasing DS/DH to 0 .0293 decreased the transit ion Reynolds

number and increased the turbulent friction factor from 0.013 to 0.021,

without affecting the laminar fr ic tion factor. Thus, Rehme's corre

lation seems to work only for small roughness ratios with configuration

1.

Friction factor results for a wide range of f low and thermal pa

ram eters are shown in Fig. 11. Thes e results indicate th at cable

heating had no noticeable effect on the friction factor correlations and

tha t fully developed flow condition s existed as low as x/D H ~ 100. Our

results for configuration 1 are in good agreement w ith those given in

reference [9]. H owever, with the othe r two configurations, we mea

s u r e d c o n s id e r a b ly h ig h e r t r a n s i t io n Re y n o ld s n u m b e r s , w h e r e a sreference [9] found no subs tanti a l d ifferences a mong the th ree con

figurations. Changing the pitch-to-height ra tio of the skid wire from

10 to 17 by changing th e helix angle of the wire from approx imately

83 deg to 52 deg did not affect the results noticeably.

Reference [10] indicates that the influence of the round skid wire

extends to a total distance of 8 times the wire diameter. Therefore the

10:1 pitch to diam eter ra tio used can be considered small enough to

insure that the entire cable surface is influenced by the skid wires with

virtually no unaffected surface.

C onc lus ions

Examination of the above results suggests the following conclu

sions:

1 Th e f low in a cable pipe becomes hydrodynam ically fully de

veloped at X/DH = 100. Shorte r entran ce length would be sufficientfor the f low to become fully developed in the lower range of lam inar

Re y n o ld s n u m b e r s .

2 Fully develop ed lamin ar, axial velocity profiles are simil ar for

both isothermal and non-iso therm al cases. Und er turbul ent condi

tions, the profiles become flatter and indicate that the point at which

the maximum velocity occurs tends to shift slightly toward the smooth

wall of the outer pipe. The local maxi mum velocity and th e dista nce

between the cable surface and the pipe wall are sufficient to correlate

the velocity distr ibution results .

3 Th e axial velocity in the restricted flow channels between thr ee

cables or two cables and the pipe wall is appro xima tely one order of

magnitude lower than the maximum velocity in the main flow channel

for the entire laminar f low range. Very l i t t le mixing seems to take

-J

Q-<

'-

•

A

AD

C*

SA

•%

-

symbol Model* l

z

%

X

D S / D H

0.0216

0.0293

-

* A A A A M M &

\0* 10° \0H 10°

Re = U 0H/v

Fig. 10 Friction factor results for cable configuration 1 in tw o different Models

(After reference [10]) . Re = Re„, ( = fe

-'

- 2

1i

symbol

0

V

a c

• s

X/D H

4 0 0

4 0 0

4 0 0

° A " 1 0 09°A

X(

-q c,w/m z

0

8 7 6

4 3 8

4 3 8

° %

1 1 1 1 _

qw ,W/rrf

0 -

3 6 2

0

0

10 ' I0 J I0 H

_Re= U DH/v

Fig. 11 Friction factor results for configuration 1 in Model 2. Re = Re a , f -

U

place between the sep arate f low channels . H owever, both mixing and

flow velocit ies in the restr ic ted channels increase with increasing

Re y n o ld s n u m b e r s in th e t r a n s i t io n a l a n d tu r b u le n t r e g im e s .

4 Tran sitio n to turbule nce for configurations 2 and 3 start s around

Re c = 2000. In the trans ition region, the friction factors increase w ith

34 / VO L 100, FEBRUARY 1978 Transact ions of the ASME




increasing Reynolds numbers and approach an asymptotic value

under fully turbulent flow conditions. The transition for configuration

1 is gradual and seems to s tart a t around Re e = 600. Th e skid wire

roughness ratio for any cable configuration has a strong influence on

the transit ion Reynolds number. In addition, the turbulent fr ic tion

factors increase with increasing roughness ratios.

5 Friction factors can be best correlated by the use of an equiva

lent Reynolds number, Re e . Laminar friction factors are independent

of the skid wire roughness ratio and can be calculated from equations

(1) or (2). Tu rb ule nt friction factors can be correlated by equ atio n (4)

for re latively low roughness ratios (<.0216) but they increase considerably with higher roughness ratios. Asymptotic values for tur

bulent friction factors were 0.013 and 0.021 corresponding to skid wire

roughness ratios of 0.0216 and 0.0293 respectively.

These results are applicable for a cable to pipe diameter ratio in the

range of 0.2 < D JD P < 0.36. For D JD P < 0.2 the friction factors will

converge on the smooth pipe data . For 0 .36 < D c/D p < m a x i m u m =

0.464 all f low cross sections have about the sam e dimensions an d the

equivalent area must be taken as the total f low area for a ll configu

rations. Th e transit ion to turbulence, as well as the m agnitude of the

turbulent friction factor, may be influenced not only by the roughness

ratio, as shown in this study, but also by the relative size of the rough

surface. These parameters should be further investigated in the ranges

not covered in this report .

A c k n o w l e d g m e n t sThis work was supported in part by Contract No. E(49-18)-1568,

(RP 7821-1) from the Electr ic Power Research Insti tute and the U.

S. Energy Research and Development Administration. Oils were

furnished by the Chevron, Conoco, and Sun Oil Companies.

R e f e r e n c e s

1 Hoffman, H . J., Miller, C. W„ Sozzi, G.L„ and Sutherland, W. A., "H eatTransfer in Seven-Rod Clusters—Influence of Linear and Spacer Geometry

on Superheat Fuel Performance," GEAP-5289 General Electric Co., San Jose,Cat, Oct. 1966.

2 Shimazaki, T. T„ and Freede, W. F., "H eat Transfer and H ydraulicCharacteristics of the SRE Fuel Element," Book 1, p. 273, Reactor HeatTransfer Conference, 1956.

3 Waters, E. D., "Fluid Mixing Experiments with a Wire-WrappedSeven-Rod Fuel Assembly," HW-70178 Rev., Hanford Atomic Products Operation, Richland, Wash., Nov. 1963.

4 Kattchee, N., and Machewicz, M. V., "He at Transfer and Fluid FrictionCharacteristics of Tube Clusters with Boundary-Layer Turbulence P romoters,"ASME Paper 63- H T-l, ASME-AIChE H eat Transfer Conference, Boston,Mass., Aug. 1963.

5 Zanona, A., and Williams, J. L., "Forced-Cooling Model Tes ts," IEEE

Trans., Vol. PAS-89, No. 3, 1970, pp. 491-503.6 Beckenbach, J. W.t Glicksman, L. R., and Rohsenow, W. M., "The

Prediction of Friction Factors in Turbulent Flow for an UndergroundForced-Cooled Pipe-Type Electrical Transmission Cable System," HeatTransfer Laboratory Report No . 80619-89, MIT, 1974.

7 Rehme, K., "Simple Method of Predicting Friction Factors of TurbulentFlow in Non-Circular Channels," International Journal H eat Mass Transfer,Vol. 16,1973, pp. 933-950.

8 Lewis, M. J., "Optimising the Thermohydraulic Performance of RoughSurfaces," International Journal H eat Mass Transfer, Vol. 18, 1975, pp.1243-1248.

9 Slutz, R. A., Glicksman, L. R., Rohsenow, W. M., and Buckweitz, M. D.,"Measurement of Fluid Flow Resistance for Forced Cooled UndergroundTransmission Lines," IEEE Trans., Vol. PAS-94, No. 5, 1975, pp. 1831-1834.

10 Abdulhadi, R. S., and Chato, J. C, "Combined Natural and ForcedConvective Cooling of Underground Electric Cables," IEEE Trans., Vol.PAS-96, No. 1,1977, pp. 1-8.

11 Abdulh adi, R. S., "Na tural and Forced Convective Cooling of Under

ground Electric C ables," Ph.D. Thesis, D epartment of Mechanical and Industrial Engineering, University of Illinois at Urbana-Champaign, Dec.1975.

12 Abdulhad i, R. S., and Chato, J. C , "Natu ral and Forced ConvectiveCooling of Underground Electric Cables," Technical Report No . ME-TR-609,UILU-ENG 75-4003, Department of Mechanical and Industrial Engineering,University of Illinois at Urbana-Champaign, Dec. 1975.

13 Abdulhadi, R. S., and Chato, J. C , "Flow and H eat Transfer in Con-vectively Cooled Underground Electric Cable Systems: Part 2—TemperatureDistributions and Heat Transfer Correlations," ASME JOURNAL OF HEATT RA N SFE R, Feb. 1978, pp. 36-40.

Journal of Heat Transfer FEBRUARY 1978, VO L 100 / 35




R. S. Abdulhadi



University of Jordan,

Amman, Jordan

J . C. Chato

Professor,

Department of Mechanical

and Industrial Engineering,

University of I l l inois at Urbana -Champalgn,

Urbana, Hi.

Flow and Heat Transfer in

Conwectifely Cooled Underground

Electric Cable Systems^ Part 2 —

Temperature Distributions and Heat

Transfer CorrelationsTemperature distributions and heat transfer correlations have been obtained experimen

tally for a wide range of physical, flow and thermal parameters in three m odels of oil-

cooled underground electric cable systems. The results show that in the laminar range,

with the oils used, the thermal boundary layer thickness around the heated cables is only

of the order of 2-3 mm over the entire length of the test section. Consequently, the best cor

relation of the heat transfer results is obtained if the Nusselt number, based on the cable

diameter, is plotted against Re-Pr0A

, where the Reynolds number is based on the overall

hydraulic diameter of the cross section of the flow. For laminar flow s, the oil temperatures

in the restricted flow channels between three cables or two cables and the pipe wall are

about 11 "C higher than corresponding bulk temperatures. As the flow becomes turbulent,

the thermal boundary layer tends to vanish and the oil temperature becomes uniform over

the entire flow cross section. Laminar Nusselt numbers are independent of the skid wireroughness ratio and the flow Reynolds number, b ut increase with increasing Rayleigh

number and axial distance from the inlet, indicating significant natural convection effect.

The range of laminar Nusselt numbers was 5-16. Turbulent Nusselt numbers increase

with increasing roughness ratios. The Nusselt numbers at Re = 3000 are 30 and 60 for

roughness ratios of 0.0216 and 0.0293, respectively.

In t roduc t i o n

The current carrying capacity of underground cables is l imited by

the maximum allowable electr ical insulation temperature requiredfor long l ife and high reliabil i ty . This l imiting temperature is about

85 °C (185 °F) and occurs at the inner dia mete r of the insulation in

contact with the electr ical conductor. This temperature depends

primarily on the rate a t which heat generated within the system due

to conductor losses, magnetic effects , and other factors can be

transfered to the surrounding earth or some other cooling medium.

In forced cooled systems for oil filled, pipe -type c able circuits, chilled

oil is c irculated through the pipe and most of the heat generated in

Contributed by the Heat Transfer Division and presented at the WinterAnnual Meeting, New York, N. Y., December 5-10,1976. Revised m anuscriptreceived by the Heat Transfer Division August 20, 1976. Paper No. 76-WA/HT-42.

the cable and insulation is absorbed by the oil. The heat is then

transferr ed from th e oil to the environ men t at refrigeration stations

located at predetermined intervals a long the transmission l ine.

Th e purpo se of our work is to unde rstan d m ore fully the f low andthermal characteris t ics of underground cable systems. The velocity

distr ibutions and pressure drop correlations have been discussed in

a companion art ic le [ l] . 1 A detailed description of the experim ental

setup and range of parameters covered in the investigation are also

reported in the above article and in references [2] and [3]. References

[4-12] are selected works on heat transfer results re lated to this in

vestigation.

Three cable configurations were used: (1) Open tr iangular, (2)

close-packed tr iangular, an d (3) cradled. The three mod els used were

(1) large cables (38 mm OD) in large pipe (140 mm ID ), (2) small cab les


36 / VOL 100, FEBRUARY 1978 T ransact ions of the ASMECopyright © 1978 by ASME




(13 mm OD) in small pipe (45 mm ID) and (3) small cables in large

pipe. The cables had "skid wires" wrapped around them (1.29 mm

and 0.53 mm OD respectively for the large and sm all cables) .

In this paper, we are reporting on the temperature distr ibutions

and heat transfer results we obtained for a wide range of param eters .

Dimensionless correlations are developed for application in the design

of full-scale underground cable systems for the prediction of their

thermal behavior.

D i s c u s s i o n of t h e R e s u l t s

T e m pe ra tu re Dis t r i bu t i ons . Oil and su rface tempera tu res weremeasured for a wide range of physical and flow parameters in order

to determine temperature distr ibutions and to calculate average heat

transfer coefficients at various cross sections along the cable pipe. Oil

tempe ratures in the restr ic ted f low channels between thr ee cables or

between two cables and the pipe wall were also measured for all flow

conditions to determine the maximum oil temperatures in the sys

tem.

Cable surface temperatures were found to be essentially uniform

around the perimeter of each cable at any cross section for both

laminar and turbulent f low conditions.

Fig. 1 shows typical temper ature d istr ibutions in the vertical plane

of symmetry for configuration 2 in model 2 at different flow rates . The

choice of the measured minimum oil temperature to correlate the

results is arbitrary and was based on the fact that th is temperature

was usually very close to the oil bulk tempe ratu re as calculated from

the f low rate , in let temperature, and heat input to the system.

Fig. 2 shows typical tem pera ture d istr ibu tions m easured in a hor

izontal plane on bo th sides of the top cable of configuration 2 in model

2. These profiles indicate relatively uniform temperatures throughout

the flow region between the cable and pipe wall, as well as similarity

of the temperature distr ibutions on both sides of the cable .

Examination of both vertical and horizontal temperature profilesseems to indicate that upward natural convection currents exist along

the vertical cen ter l ine of the pipe, while secondary cu rrents exist in

a narrow region near the pipe wall . Thi s observation is in agree ment

with analytical predictions of natural convection currents in cable

pipes [13].

The effect of natural convection currents prevailed for the entire

lam inar flow regim e as shown clearly by the re sults of Fig. 1. Th e in

tensity of natural convection currents and their augmenting influence

on the overall therm al perform ance of the cable system are increasing

functions of the difference between cable and oil tem pera ture s. Th e

results of a special experiment, which was conducted to determine

1.0

0.8

0. 6

0. 4

0.2

0.0

o A — Profile

° B - Profile

0.0 0.2 0.4 0.6y/d

1.0

Fig. 1 Dimensionless tempera ture distributions for configuration 2 In model Fig. 2 Dimensionless horizontal tempera ture distributions for configuration2 at x/DH = 473, qc = 438 W /m 2 , qw = 174 W/m 2 2 In model 2 at x/D H = 473, Re = 160, <7C = 438 W/m 2 , q„ = 174 W/m 2

N o m e n c l a t u r e .

c p = specific heat of oil

D c = cable outside diameter

D H = hydraulic diameter based on overall

flow area

D p = pipe inside diameter

D s = skid wire height

d = distance between cable surface and inside

pipe wall in any radial trave rse

Gr = gp(Tc - Tb) D cs/v

2, Grashof number

S = gravitational acceleration

h = heat transfer coeffic ient between a

heated surface and surrounding oil

k = thermal conductivity of oil

N u = hD c/k, Nusse l t numberP = axial distance between skid wires

P r = ixCp/k, P r a n d t l n u m b e r

qc = heat flux rate from the cable surface to

the oil

qw = heat flux rate from inside pipe wall to

the oil

R = Ra Pr0-4

Ra = Gr-Pr, Rayleigh number

R e = UDH/V, Reyno lds number

T = local oil temperature

Tb = o il bu lk te mper a tu re

Tc = cable surface temperature

Tmax=

maximu m o i l t emp era tu re in re

s t r ic ted subchanne ls

Tm\„ = minimum oil temperature a t any cross

section

U = oil mean velocity based on overall flow

area

x = axial distance from inlet of test section

y = radial distance from a cable surface

13 = coeffic ient of thermal expansion

p = oil density

H - dynamic viscosity of oil

v = kinematic viscosity of oil

S=(T- Tmin)/(TC - T m i n ) , d imensionless

t e m p e r a t u r e

Journal of Heat Transfer FEBRUARY 197 8, VOL 100 / 37




this relations hip, will be discussed later.

Fig. 3shows typical temperature distributions in the entrance re

gion and at the end of the test section. The uniform temperature

distribution at X/DH = 124 indicates that the effects of natural con

vection are relatively insignificant in the entrance region of the cable

pipe. However, as the oil flows downstream extracting thermal energy

from the heated cables and pipe, the effects of natural convection are

enhanced resulting in elevated oil temperatures near the to p of the

pipe at X/DH = 473.

Oil temperatures in the flow sub-channels, bounded by three cables

or two cables and the pipe wall, were measured u nder different flow

conditions in order to determine th e maximum oil temperatures in

the system. Fig. 4 shows that th e sub-channel temperatures were

higher than th e bulk temperatures for the entire laminar region, but

became nearly equal as the flow became turbulent.

The thickness of the thermal boundary layer around the cables was

of the order of 2-3 mm throughout most of the laminar flow range.

Under turbulent conditions, th e thermal boundary layer diminished

as th e cable surface temperature approached that of the oil bulk.

Heat Transfer Results and Correlations. A total of 36 forced

convective heat transfer r uns were performed unde r different flow

and thermal conditions and the results were correlated in terms of

dimensionless parameters. Th e effects of the following parameters

were investigated: Re, Ra, Pr, P/Ds, X/DH, y/d, skid wire helix angle,

and three typical configurations. Other parameters, in particular

Ds/Dc an d Dc/Dp, were fixed at values which are typical in currentpractice with underground cable systems. Among these variables,

changing P/Ds from ~10 to ~1 7 by changing the helix angle from 83

deg to 52 deg had no detectable effect on the results.

In order to correlate the results of all the heat transfer experimen ts

in dimensionless form, the Nusselt number, Nu, for each flow rate was

calculated and plotted as a function of the flow Reynolds number, Re,

and the oil Prandtl number, Pr.

Two different characteristic lengths were used in attempting to

define a Nusselt number. In one case, th e Nusselt number, Nu, was

defined in terms of the hydraulic diameter, DH , of the entire flow area.

No satisfactory correlation was obtained by using DH as the charac

teristic length.

Another correlation of the same results was obtained by substi

tuting th e cable diameter, Dc, for the hydraulic diameter, DH , in the

definition of the Nusselt number. Th e correlations based on thisdefinition are presented in Fig. 5. These curves suggest that the choice

of the cable diameter as the characteristic length in the definition of

the Nusselt number gives a better correlation of all th e data obtained

in test models having significantly diverse geometrical parameters.

The physical explanation of the above observation is assumed to be

the following:

1 The thermal boundary layer thickness around th e cable was

found to be very small, of the order of only 2- 3 mm. Since th e heat

transfer coefficient for forced convection is a function of the difference

between th e cable surface temperature and th e oil bulk temperature

outside the thermal boundary layer, it can be reasonably assumed that

the spacing between th e cable an d pipe wall, represented by the hy

draulic diameter, has little influence as long as th e thermal boundary

layer is much smaller than DH-

2 On the other hand, th e buoyancy effect, which is the driving

force for natural convection, is proportional to the third power of the

characteristic length of the heated surface. Based on the above ob

servation, it is reasonable to assume that each cable behaves like a

heated surface in an infinite environment and its diameter represents

the characteristic length.

The index for the buoyancy force in natural convection is the

Rayleigh number, which was defined as:

Ra :/g(HTc - Tb)Dci

"(?) (1)

Fig. 5 indicates that while the Nusselt number in the laminar region

was independent of both the flow Reynolds number and the skid wire

roughness ratio, it increased with increasing Rayleigh numbers. This

1.0 -

0.8 -

0.6

0.4

0.2 -

0.0

symbol

-

ai

ri •

u

_ J

A

A

D a

ADA

x /D H

124

47 3

D

A

(rMD

D a

a A A

i

M •^ 4

wa

CL

CL

a

-

a o a

p

A AA A A

, , , |,0.8 1.0.0 0.2 0.4 0.6

y/d

Fig. 3 Dimensionless tempera ture distributions for configuration 2 inmodel2 at Re = 696, qc = 438 W/m

2, qw = 0

10

10

_ r 1 1 I 1 1

~ A

o

o o o & S f tD«

0 4 i f " t I T ?

D „A " V

~Z symbol configuration

T 2

A 2

° 3o 3

L. .1 _ L L .L_L

1 l 1

A

*oV

*

fa

a •

•

%

q c , w / m2

87 6

43 8

87 6

43 8

i 1 1

r Tmin i

con-2 ~| con-3 -

Tmax

JD

o0° D

o y 0 ca> a

« o ' v ? °q w , w / m

2;

362

174

36 2

174i i i i i i i i

10

io dIO

5

Re = U D H / u

Fig. 4 Dimensionless m aximum oil temperatures inmodel 2

10

10 =

1

0

= •! • "1 ! " i r r m

: Symbol Model x/DH

I 1 142o 2 124A 3 78

=

" Ra = 2 . 4 X I 06

. D: 9 . 5 X I 0

4

^ — 9 . 3 6 X I 04

i i i i 111

1 1 i I I i i ii r i i I i i i t

, o ^ •

I°><CJ»5 -l

<Z>4 / oy>- -

- O A ^ " A A A A « V \ A A A =

:

i i i i 11 n i i

10 ' m " io \o

Re Pr°4= (U D H / D M ^ Cp/k)°

4

Fig. 5 Nusselt number results, based on cable diameter, for configuration1 in three different m odels





relationship is a lso i l lustrated in more detail by the results of Fig. 6.

If the natur al convection dominates th e heat transfer , we can assum e

tha t

N u = a • Ra°-2S(2 )

where the constant, a, range s from 0.36 to 0.46 for configurati on 1 (Fig.

5). Thes e values are s l ightly less than those in the l i t e ra tu re for free

cylinders or the values reported in reference [9]. The cons tan t for

configuration 3 (Fig. 6) is 0.364. Since the cables interfere thermally

with each other, a reduc t ion in the p ropor t iona l i ty cons tan t , a, can

be expected.

Fig. 5 also shows the effect of the skid wire rough ness ra tio, DJDH,

on bo th the t rans i t ion Reyno lds number and the tu rbu len t Nusse l t

numbers . It is obvious tha t the transit ion Reynolds num ber of model

3, which had the lowest roughness ratio (DJDH = 0.0049), was almo st

one order of magni tude h ighe r than tha t for mode l 2 (DJDH =

0.0293). It is also seen from these results that in the t rans i t ion an d

turbu lent f low regimes the Nusse lt num ber as well as the s lope of it s

l ine increase with increasing skid w ire roughness ratio . We have in

suffic ient information to give generalized correlations . The bes t fit

curves through th e da ta po in ts in the t r ans i t ion - tu rbu len t reg imes

can be expressed as follows.

For DJDH = 0.0293,1.5 X IO3

< R < 2 X 104

N u = 0.011 fl0-

85

For DJDH = 0.0216, 7 X 103

< R,< 4 X 104

N u = 0.042 flo-es

For DJDH = 0.0049, 2 X 104

< R < 6 X 104

N u = 0.127 R0A9

(3 )

(4 )

(5)

where R = Re Pr0-

4

The magnitudes of the heat fluxes on the cables or on the pipe wall

had no significant effect on th e results .

Fig. 7 shows the results of six sets of heat transfer exper imen ts for

configuration 3 in mode l 2 with three different kinds of cable oils.

Since th e transit ion from laminar to tu rbu len t cond i t ions depends

largely on the con ditions in the main flow chan nel, one correlation was

tried by plott ing the Nusselt number, based on the cable diameter,

against th e f low Reynolds numb er, based on the pipe hydraulic di

ameter. However, th e p a t t e r n of the sca t te r in the t r ans i t ion an d

turbulent regimes indicated a dependence on the physical properties

of the different oils and suggested the necessity to include the P ran dtl

number as a correlating parameter. Fig . 7 shows tha t satisfactory

correlation of the heat transfer results was obtained when the Nusselt

number, based on the cable diameter, was plotted against the quantity

Re-Pr0-

4, where th e Reyno lds number was based on the hydrau l ic

d iamete r of the entire cross section, DH -

Corre la t ions of the heat transfer results for cable configuration 2

in model 2 are shown in Fig. 8. In reference to Figs. 5-8, the following

observations are made:

1 T he Nusse l t number , based on cable diameter, in the lamina r

flow regime depends prim arily on the Rayleigh numbe r as expressed

in equation (2). Th e effect of the Reyno lds number is very slight.

Thus, natural convection effects are significant in this region an d

mixed convection is the dominan t mechan ism of heat transfer .

2 The transit ion from laminar to tu rbu len t f low s ta r t s at R e =;

400 for configurati on 1, and at R e ca 1400 for configura tions 2 and 3.

T he onse t of transit ion for the la tter two configurations was actually

observed as a sudden d rop in t h e t e m p e r a t u r e of the cable surface,

as well as in the oil temperature in the restr ic ted subchannels . Other

indications of turbulence were an instantaneous increase in pressure

d rop and an increase in mea n velocity fluctuations.

3 Configuration 1 had the largest wetted , rough perim eter of al l

three configurations and, therefore, represented the largest rough

surface area under all flow conditions. At relatively low velocities, Re

< 400, the Nusselt number was almost constant and natural convec

tion was the dominant mode of heat transfer . Beyond this point, the

Nusselt number gradually increased with increasing Reynolds num

bers due to the deve lopmen t of tu rbu lence .

4 Correlations of Nusselt number results were in good agreement

for Reynolds n umb ers below 400 and above 1400. The higher Nusselt

2

1

o

I- symbol

7

O

I A

a

o

A

:

-

!P ruP r3

P r

f

o il

N a l ky l e n e - 6 0 0 1

N a l ky l e n e - 6 0 0 1

Sun No- 42

Sun No 42

Chevron D-ioo3

Chevron D-lOO3

= 7 2

= 12 0

= 2 8 5! 1 1 1 1

qc ,w/m

4 3 8

8 7 6

4 38

8 7 6

4 3 8

8 7 6

i i i • 1 1 1 1 ] i i

2% ,W /m 2

174

362 -

174 " " o

36 2 A " ^

174 $F°y3 6 2 ^

&

^ W i S f

i L i , ! i i i i i i i i ii i i

1-

-

--

-

E

-

i..

10

Re Pr (U D H / u ) ( / L i C p / k ) '

Fig. 7= 473.

Nusselt number results for cable configuration 3 in model 2 at x/DH

(After reference [12])

IO

IO

i i i I i i 'i n ' —r —r

; Mode ls I and 2

Conf igura t ion 3

3—-*""

-

H i i

I ! rTTTT ' " I 1 1 M i l l

"

0 .2

5

-

R e = 3 5 0

i i 1 1 .1 1 1 1

10

icr icr 10°

R a = Gr x pr = ( g /3 AT Dc3/u

2) ( ^ Cp/k)

10

! 1 i 1 1 : SI !

- symbol qc ,W/m2

D 4 3 8

0 4 3 8

v 8 76

P r - 7 2-

0v vv

• 15 a0

-

! 1

q w

1 ] i 1 1 i 1 i 1 i i 1 1 ! '-

,W/m2

:

0174

3 6 2 o o o v ?

8 99

:v *

®

•V***""": 1 ' i 1 1 ! 1 1 j

10 10

Re Pr

I'ig. 6 Variation of the Nusselt number with Rayleigh number in the laminar Fig. 8 Nusselt number results for cable configuration 2 in model 2 at x/DH

region for configuration 3 = 473





num bers for configuration 1 in compariso n to those for 2 and 3 in the

range 400 < Re < 1400 were primarily due to an increased level of

turbule nce pro duced as a result of the larger exposure of the skid wires

to the main f low. The skid wires act as turbulence promoters .

5 In the higher turbul ence regime, the difference between cable

surface and oil temperatures was often less than 0.5 °C. Any tem

perature f luctuations in this range resulted in the relatively large

measurement errors , which explain the scatter of the results in this

region.

A special experiment was conducted to s tudy th e effects of natura l

convection curre nts in the lam inar range. He at transfer param eterswere measured for a wide range of Rayleigh numbers by increasing

the heat flux rate from 0 — 110 W/m2, while maintaining steady

laminar f low conditions. The results show that in the laminar range,

the Nusse l t numbers nea r the en t rance (x/Dn = 124) were slightly

lower than those near the exit {X/DH = 473) for all cases where R a >

50,000. At lower Rayleigh numbers, the entrance Nusselt numbers

were equal to or sl ightly higher than those near th e exit . Thus, a t the

higher Rayleigh numbers the development of the natural convective

circulation compensates for the thickening of the boundary layer in

the axial direction.

Hea t transfer to the pipe wall has been correlated on the basis of

the Nusselt number formed by the hydraulic diameter of the entire

pipe, as shown in Fig. 9 . The orders of magnitude of these Nusselt

numbers are the same as the ones for the cables based on the cable

diameter in Figs. 7 and 8. The early transit ion to turbulence withconfiguration 1 is clearly indicated by the increase of Nusselt num bers

s ta r t ing a t R = 2000.

Conc lu s i on s

1 Tem perat ure profiles indicate tha t the therm al bounda ry layer

thickness around a heated cable is approximately 2-3 mm at any axial

location. Vertical profiles a lso indicate that oil temperatures are es

sentially uniform in the entrance region, and that higher temperatures

develop near the top of the pipe with increasing axial downstream

distance indicating the development of natural convection effects .

2 The dom inant mode of heat transfer in the laminar range is

mixed (natural and forced) convection, and the Nusselt number

throughout this range is an increasing function of the Rayleigh

number, but is independen t of the f low Reynolds number and the skidwire roughness ratio. In addition, the results indicate that the Nusselt

number increases along the axial distance downstream from the inlet,

for Ra > 50,000, in contrast to the previously held assumption that

the Nusselt number near the end of a cable pipe would be much

smaller than that in the entrance region.

3 The skid wire roughness ratio for any cable configuration ha s

a strong influence on the transit ion Reynolds number. In addition,

the turbulent Nusselt numbers increase with increasing roughness

ratios.

4 The best correlations of the heat transfer results are obtained

if the Nusselt numbers are plotted against the quantity Re-Pr 0-4,

where the Reynolds number is based on the overall hydraulic diameter

of the entire cross section. For heat transfer to the cables, the Nusselt

number should be based on the cable diameter; whereas, for heat

transfer to the pipe wall, the overall hydraulic diameter should be used

in the Nusselt number. Based on these correlations, the Nusselt

numbers in the laminar region (Re < ~1000) were between 5-16,

depending on the R ayleigh number. In the low turbulence region (Re

=i 3000), the Nusselt numbers for the cables were between 30-80,

depending mainly on the skid wire roughness ratio; whereas, the

Nusselt numbers for the pipe wall were between 60 and 200.

z symbol cable configura tion

10

O O D

o oao

AA A A A A

Re Pr " "

Fig. 9 Nusselt number results for pipe wall at x/D H = 473, qc = 876 W/m 2 ,qw = 362 W/m 2

(Alter reference [12])

A c k n o w l e d g m e n t s

This work was supported in part by Contract No. E(49-18)-1568,

(RP 7821-1) from the Electr ic Power Research Insti tute and the U.

S. Energy Research and Development Administration. Oils were

furnished by Chevron, Conoco and Sun Oil Companies.

References1 Chato, J. C. and Abdulhadi, R. S., "Flow and He at Transfer in Convec-

tively Cooled Underground Electric Cable Systems: Part 1—Velocity Distributions and Pressure Drop Correlations," ASME JOURNAL OF HEATTRANSFER, Feb. 1978, pp. 30-35.

2 Abdulhad i, R. S., "Na tural and Forced Convective Cooling of underground Electric Cables," Ph.D. Thesis, Department of Mechnical and IndustrialEngineering, University of Illinois at Urbana-Champaign, Dec. 1975.

3 Abdulhad i, R. S. and Chato, J. C , "Natu ral and Forced ConvectiveCooling of Underground Electric Cables," Technical Report Number ME-TR-609, UILU-ENG 75-4003, Dec. 1975, Department of Mechnical and Industrial Engineering, University of Illinois at Urbana-Champaign, Urbana,111.

4 Neher, J. H., and McGrath, M. H., "The Calculation of the TemperatureRise and Load Capability of Cable Systems," AIEE Trans., Vol. 76,1954, pp.752-772.

5 Buller, F. H., "Artificial Cooling of Power Cables," AIEE Trans., Vol.71, 1952, pp. 634-641.6 Sheriff, N., and Gumley, P., "Heat-Transfer and Friction Properties

of Surfaces with Discrete Roughnesses," International Journal Heat MassTransfer, Vol. 9,1966, pp. 1297-1320.

7 Notaro, J., and Webster, D. J., "Therm al Analysis of Forced-CooledCables," Paper 70TP 518-PWR., IEEE Summer M eeting at Los Angeles, Calif.,July 1970.

8 Zanona, A., and Williams, J. L., "Forced-Cooling Model Tes ts," IEEETrans., Vol. PAS-89, No. 3, 1970, pp. 491-503.

9 Slutz, R. A., Orchard, W. P., Glicksman, L. R., and Rohsenow, M. W.,"Cooling of Underground Transmission Lines: Heat Transfer Measurements,"Heat Transfer Laboratory Report No. 80619-87,1974, Massachusetts Instit! lie

of Technology, Cambridge, Mass.10 Webb, R. L., Eckert, E. R., and Goldstein, R. J., "Heat Tran sfer and

Friction in Tubes w ith Repeated-Rib Roughness," International Journal HeatMass Transfer, Vol. 14, 1971, pp. 601-617.

11 Bahder, G„ Eager, G. S., Jr., Silver, D. A. and Turner, S. E., "550 kV and765 kV High Pressure Oil Filled Pipe Cable System," IEEE Trans., Vol.

PAS-95, No. 2, 1976, pp. 478-488.12 Abdulhadi, R. S. and Chato, J. C , "Combined N atural and ForcedConvective Cooling of Underground Electric Cables," IEEE Trans., Vol.PAS-96, No. 1, 1977, pp. 1-8.

13 Chern, S.Y. and Chato, J. C, "Numerical Analysis of Convective Coolingof Pipe-Type Electric Cables," Technical R eport No. ME-TR-712, UILU-ENG76-4006, Department of Mechnical and Industrial Engineering, University of

Illinois at Urbana-Champaign, 1976.





T. G. Theofanous

T. H. BohrerM. C. Chang

P. D. Patel

Purdue University,

West Lafaye tte, Ind.

Experiments and Uniwersal Growth

Relations for Vapor Bubbles With

iicrolawersSolutions for bubble growth rates including the effects of inertia, heat transfer, and micro-

layer evaporation were obtained in generalized coordinates. E xperimental growth rate

data, covering a wide range of superheats, were obtained under precisely controlled conditions. The theoretical development appears to provide a satisfactory a priori prediction

of these data. In particular, the T-effect predicted earlier is fully corroborated by these

data which are the first to cover an extended range of this parameter.

Intr oduc t ion

The init ia l impetus for this work was to provide experimental

verif ication of the universal representation of bubble growth rates

reported recently [ l] .1

Of particular interest is the range of conditions

for which inertia effects cannot be neglected, and the characteris t ic

vapor density ratio , T [1], is large (especially relevan t to liquid me tal

boiling and/or liquids subjected to rapid depressurization). The latter

condition implies high init ia l superheats . This , in turn, promotes

inertia limitations which, however, can also be achieved, more simply,

by imposing low am bien t pressure s ( low vapor densit ies) . Thi s inde

pendent requirement for high superheats has led to a choice of ex

perimental conditions and techniques which stimulated consideration

of the add itional pro blem of microlayer effects .

Microlayer development is typical when a bubble grows in the

imm ediate vicinity of a solid wall . Thi s s i tuation arises, for example,

in nucleate boil ing, and i t has received considerable attent ion in t he

past decade. Depending on the conditions, microlayer evaporation

due to i ts own stored sup erhea t, or due to heat transfer from th e ad

jacent wall, can significantly influence the bubble growth rates. Many

theoretical s tudies have attem pted to sort out these contributions for

the mos t common heat transfer-l imited growth regime. Some have

also considered inertia limitations. Good surveys have been presentedby Cooper and V ijuk [2] and D wyer [3]. However, the lack of definitive

experimental data does not facil i ta te a choice among these theories

or provide guidance for addit ional a nalytical efforts . Although a few

experime ntalis ts have looked at such details as space/t ime variations

in microlayer thickne ss [4], including the formation of a dry -patch

[5], most data were obtained under typical nucleate boiling conditions

1 Numbers in brackets designate References at the end of paper.Contribu ted by the Hea t Transfer Division for publication in the JOURNAL

OF HEAT TRANSFER. Manuscript received by the Heat Transfer DivisionNovember 29,1976.

[6]. The ambiguities in comparisons referred to above are due to the

nonun i fo rmi ty a nd nons ta t iona r i ty o f the tempera tu re f ie ld under

these experimental conditions. The situation is even less definit ive

when inertia contributions become significant [7] .

Wit hin t he conte xt of the pr esen t work, microlayer effects first arose

in the somew hat different topology of thin wire probes. These probes

were necessary to produce a predictable (both in space and t ime)

single nucleation event in the energetically neutral way of a s imple

touchin g. Since the successful operation of this system did n ot a llow

an unlimited reduction of the probe diameter, i t became necessary

to theoretically establish th e l imit of negligible microlayer interfer

ence. This was accomplished by a s imple extension of the analysis

presente d earlier [1], which also produced a universal representatio n

of bubble growth in terms of an approp riately normalized wire-probe

radius. At this point, only straightforward extensions in the experi

men tal and analytical tools were required to consider the problem for

microlayers typical of nucleate boil ing (plane wall geometry). This

latter extension is not meant to address the complete problem of

microlayer contributions to nucleate boil ing. I t is ra ther intended as

a f irs t s tep, providing analytical insight and clear experimental ver

if ication for an idealized but re levant system.

E x p e r i m e n t a l

A pp ar a t us and P ro ce du re . T he bas ic exper imen ta l goal was to

predictably nucleate and record the growth of a s ingle vapor bubble

in a highly metastable (superheated) l iquid. To avoid ambiguities in

local conditions associated with the transient heating and/or

depress urization tec hniqu es the "wett ing" Freon (11 and 113 )/glass

pair was chosen. With special c leansing, loading, and deactivation

procedures [8 ,9] superh eats up to 60° C were atta inable for indefinite

durations. A 2000-ml Pyrex flask, fitted with an optical window, was

util ized as the ex perim ental vessel. A layer of glycerol was employed

to "seal" the l iquid Freon in this f lask. Slow depressurization and

isolation at room temperature yielded the corresponding metastable

state . The pressure was measured with a mercury manometer. Room






and glycerol temperatures were measured by means of a mercury

thermometer and copper-constantan thermocouple respectively. Any

thermal gradients whatsoever were absent prior to nucleation.

I t was de te rmined tha t p red ic tab le nuc lea t ion in th i s sys tem may

be achieved by simply touch ing two solid surfaces within th e supe r

heated l iquid. This procedure is advantageous because i t results in

a minimum disturbance of local conditions and init ia tes a bubble in

a most "natural" manner. For the spherically symmetric bubble ex

perim ents , the two stainless s teel wire probes, shown in Pig. 1, were

utilized for this purpose. By placing their horizontal segments at right

angles (forming a cross) and a bout 0 .5 mm apart , only a s l ight, e lec-tromagnetically driven, vertical travel was required for bubble nu

cleation at the point of contact (coincident also with the flask center).

Three different pr obe sizes were uti l ized with diameter s of 0 .25 m m,

0.51 mm, and 0.81 mm. For the hemispherical bubble experiments ,

a small glass disk immersed in the center of the Freon volume, and

a thin (0 .61 mm dia) s ta inless s teel wire , its one end sharpen ed to a

point, were uti l ized. By placing the wire at a posit ion normal to the

disk, a t i ts center , and about 0 .5 mm apart , only a s l ight, e lectro-

magnetically released, vertical travel was required for the sharpene d

wire end to make contact w ith the disk and thu s produce a nucleation

event. Fig . 2 i l lustrates this m echanism , within the co ntext of the

overall experimental setup.

Photographic records were obtained at 9000 fps using either a

Dynafax model 326 high-speed framing camera or a Hyc am model

K2004 16 mm high-speed motion picture cam era. A schematic of the

overall arrangement is shown in Fig. 3. Complete details are given by

B o h r e r [ 8 ] a n d C h a n g [ 9 ] .

Da ta Re duc t ion . Smooth -wa l led sphe r ica l (and hemispher ica l )

bubbles were observed in all experiments . Two typical growth se

quences are presented in Figs. 4 and 5 . Considerable agita tion and

entrainment was noted only when the wall of the spherical bubbles

encountered the vertical segments of the two nucleation probes. For

the spherical bubbles, the diameters a long two orthogonal axis never

differed by more than 15 percent, except in the very early s tages of

growth when the bubble and the probe diameters were of the same

magnitude. The hemispherical bubbles exhibited similarly small

distortions. Unde r such conditions a s imple ari t hm etic averaging is

suffic ient to express vapor volume content. Further, a probe volume

correction, important only for the very early s tages at growth, was

applied to determine the true vapor volume.

The t ime origin was determined by extrapolation to zero radius.

The only factor that may interfere with this extrapolation technique

is the delay t im e due to surface tension effects . For the cond itions of

interest here this time was determined by numerical methods [10] and

was found t o be less than 0.003 ms. A conservative (upper bound)

estimate of the uncertainty in t ime origin was determined for some

E L E C T R O M A G N E T

D u

T J

M O V E A B L EP R O B E

F I X E DP R O B E

Fig. 1 Schem at ic diagram of the nucleat ion probes

of the experiments , as follows. Physically realis t ic upper and lower

bounds were calculated for the t ime needed to reach the size of the

firs t bubble detec ted on the f i lm. Th e Rayleigh solution, for inertia -

controlled growth, provides the lower bound. An upper bound is ob

tained from the comp lete numerical simu lation [10] with an artificially

degraded growth rate by utilizing a vaporization coefficient of 0.1 (all

comparisons with these Freon data indicate a coeffic ient of unity).

Each data point was then represented by a horizontal bar, i ts length

reflecting the total uncertainty in time origin. Clearly, as the framingrate increases, the probabili ty of recording an earlier instance of the

growth sequ ence increases. This leads to a decrease in total time-origin

uncertainty . Other relevant uncertainty l imits were established as

follows: length, ±0.0025 cm; frame separation, 0.5 percent; superheat,

2 pe rcen t .

-Nomenclature^

A* = a paramete r , (

B = a pa ramete r ,

2 P * ( T „ ) - P „ \ 1/2

Pe

(lla \m(Wt(T--T*(P„))\

c = specific heat

Ja = Jakob numberP , c , C r _ - T * ( P . ) )

X p o * ( P J


n = num ber of probe segments (from bubble

center to i ts w all)

P = pressure

P r = P r a n d t l n u m b e r

P*(T) = sa tu ra t ion p ressu re a t t empera tu re

T

Qp = heat conducted to microlayer surface in

t i m e t

r = probe radius

A -

r* = dimensionless probe radius, — rB 2

R = bubb le rad ius

A *f t* = dimensio nless bubble radius, — ft

B2

T = t e m p e r a t u r e

T*(P) = sa tu ra t ion tempera tu re a t p ressu re

P

t = t ime from growth init ia t ion

A t2

t* = d imens ion less t ime , 1B

2

a = thermal diffusivity

T = a pa ramete r ,Pu*(Ta)

Po*(P~)

A = latent heat of evaporation

P = density

p*(P) = saturation density at pressure Pp*(T ) = sa tu ra t ion dens i ty a t t empera tu re

T

Tu - T*(P„)<t> = a pa ramete r , — ——-—-

T„-T*(P„)

0 = a parameter, defined by equation (7).

S u b s c r i p t s

t = liquid

s = solid

v = vapor

<*> = conditions far away from the bubble





R a n g e o f E x p e r i m e n t a t i o n . T h e ra n g e of p a r a m e t e r s c ov e re d

in the pres ent work is compa red in Table 1 with those of the experi

ment s in Derg arabed ian [11], Florschue tz , e t a l . [12], Boar d a nd

Duffey [13], and Lien [14]. Th e param eter s compar ed are the J ako b

number Ja , the supe rhea t A T , and the cha rac te r i s t ic vapor dens i ty

ratio F.

Though experiments involving liquids besides water have also been

performed by some of the cited authors, the parameter values of these

are s imilar to those shown in Table 1 .

A n a l y t i c a l

Bub b les With P ro be M ic ro la ye rs . Fo r genera li ty le t u s cons ide rnil cylindrical probes of radius r arranged with an "effective" com

mon intersection. A bubble nucleating at th is point will grow in a

spherically symm etric fashion w ith i ts wall remaining pe rpendicular

to each probe 's axis . A smoo th l iquid microlayer is left beh ind the

advancing wall . Counting from the common origin up to the bubble

wall, ther e are n probe segments each of length R(t). The formulation

for this case is the sam e as that pre sented earlier [1] with th e exception

that the heat balance equation (over the bubble) will now be aug

mented by the probe contributions. 2 For this calculation the micro-

layer is considered "thic k" compared to the the rmal wave pen etration

distance (see Appendix). Furthermore each microlayer e lement is

assumed to be "deposited" with a uniform t empera tu re o f T*, . As

suming thermodynamic equilibrium at a ll l iquid-vapor interfaces,

the temperature driving force for heat conduction in the microlayer

is (T„—

T„ (t)). The total heat conducted to the microlayer surfaceup to t ime t is :

iV^rnkt r ( ( T « , - T „ )

Jo U / 2 oae >JQ (t - T )

With this result the heat balance equation now yields:

dR / 1 2 \ 1/2 p , c , ( T „ - r „ ( £ ) )— = 1 — at(It

(1)

(T« ) ' >^Pv

+ -

iV t

2V^rnk(

Jot (T „ - T0) /dR\

)Jo ( t - r )1 Ur) *

(2 )V ae\p u(iirR

2

Together with the s implif ied form of the momentum equation and

the definition s and n otat ions of [1] the final system in dimensio nless

form is:

2In this formula t ion, the wel l known solut ions for the asymptot ic iner t ia

(Rale igh) and hea t t ransfe r (Birkhoff, et al. , Scriven) regimes are coupled byassumin g that both rem ain valid even as actual control is shifting (with growth)

from the former to the latter, provided the driving forces in each one are adjusted to reflect actual conditions. In addition, here, we assume negligible wallshear and surface tension effects.

(r - 1)0 3 + •1

2 V t *

dR *= < f > with f l* (0 ) = 0

dt*

i

< l>2 + 4, •

2V7*

I 1 \ l / 2 Cf A , - 0J f rt* 1-2

- (—•) nr* I -!--=±=dT\ I <j>dr\ = 0

\48/ Jo v f ^ 7 Jo

(3 )

(4 )

The function <t> - <p(t*; T, nr*) is obtained for various combinations

of the parameters by means of a successive approximation scheme

applied to equation (4). An init ia l variation of <j> with t* is assumed.

The integrals of the last term in equation (4) are evaluated for severalt ime s and t he result in g cubic is solved for the same tim es. This new

set of 0 values is similarly utilized to genera te yet anothe r set of results

from th e solution of the cubic . Th e process is repeated until conver

gence is achieved. Even with a gross initial guess of <j> = 1 (true only

a t t* = 0), only a few itera tions are requ ired for convergence. Equation

(3) is finally integrated to obtain R* = R*(t*; T, nr*) . The results are

prese nted as universal plots in Figs. 6 and 7 for the range of param e

ters relevant to the conditions of the experimental program.

For clarity of presentation, the results for this two-parameter family

of curves are pre sented only over the physically interesting range of

the pa ramete r T. Similarly a correspond ing range of the nr* paramete r

was chosen. In so far as the prob e contribu tion to the bubble growth

rate is concerned, a geometric and a transport effect are discerned.

Since the microlayer area increases as dR , while the bubble surface

area increases as RdR, the geometry favors a continuous decline in

microlayer imp ortanc e. As seen in Figs. 6 and 7, by the t ime R* be

comes of order nr* the solution becomes paralle l to that of a micro-

layer-free bubble (nr* = 0). Wh at is left, however, is a relatively strong

memory effect of the probe-augmented early growth. The transport

effect is related to the degree of inertia limitations present, and hence

differences in thermal gradients between the bubble wall and the

microlayer. As inertia l l imitations increase, a l l thermal gradients di

minis h and so does the microlayer role . This effect may be observed

in terms of the differences in probe contrib utions betw een solutions

in the range of non-negligible inertia l imitation t* < 1 and those in

the range of heat-tran sfer-con trolled growth, t* > 10. Over the du

ration of a particular growth event, heat transfer l imitations increase

in s ignificance with t im e. The above discussion indicates (and Figs.

6 and 7 show) th at t he corresponding ly increased role of microlayer

contribution can only be felt if the bubble size is not large enough

compared to the probe size for the geometric effect to overshadow thetransport effect .

Bub b les Wi th Wal l (P lan a r ) Mic ro laye rs . E xcep t for geometry

differences, the deriva tion her e paralle ls the one just given. The m i

crolayer now has the sha pe of a disk with a diamete r equal to that of

the b ubbl e. As above, for nonm etall ic f luids, for the t imes of interes t

the microlayer may be considered, as far as conduction is concerned,

J

( J

i

o

Fig. 2 isometric v iew of ihe exp erimental apparatus


SURGE TANKS

i nPOTENTIOMETER

- 0

Fig. 3 Schem at ic diagram of the exp erime ntal apparatus (Dynafax syst e m )

FEBRUARY 1978, VOL 100 / 43




6

2

4

5

Flg.4(a) Flg.4(b)

7

.I1

II!

24

22

~. '". " - " " ' - _ . ~'J ~ ( . - '."" ' ~ : U

("*, ",. ••~ N f f i · · ' \ . : ~ , : z ;", 1~

Flg.4(c)

23

17 18 1 2

I) .II

19 20,-

3

21

Fig, 4 Typical frames from spherical bubble growth photographic results.(Freon·113, superheat = 50.8°C, T... = 22,2°C, probediameter = 0.51 mm.frame separation = 0.224 ms)

Fig. 5 Typlcallrames Irom hemIspherical bubble growth photographic results. (Freon-113, superheat = 18.6°C, T... =22°C, probe diameter =0.61mm, frame separation = 1.332ms)

44 /VOL 100, FEBRUARY 1978 Transactions of the ASME




T a b l e 1 A com paris on of the rang e ofe xpe r im e nta t io n by var ious author s

where:

Experiments of Liquid

MaximumSuperheat

°C

Maximu m

r

MaximumJakobNo .Ja

Fiorschuetz, et al. [12]Dergarabedian [11]Board and Duffey [13]Lien [14]Present work

WaterWaterWaterWaterFreon-

113and 11

3.95.3

20.315.759.4

1.141.192.142.54

18.45

11.715.9

173.026903195

as semi-infinite (see Appendix). The case of an extremely well-conducting microlayer can also be included by simply using the thermal-transp ort properties of the wall instead of those of the fluid. Thefinal system of equation is:

dR *

dt*

(V - 1) 03

+ 4> +

01

1

2 V F 2 V F

</>2

fl*(0)=0

- — l 4>di12 (Jo

X f P <l>(u)duJo Jo

Mr) - 03(T ) ]

Vt* -r

(5)

dr = 0 (6)

Fig. 6 Universal plot of the spherical bubb le radius versus tim e for differentprobe param eter or* (V - 2)

P(C{ \at' for liquid metal/solid system

for ordinary liquid/solid system(7 )

The function <j> = <j>(t*; T, 6) was obtained for various combinationsof the param eters by solving equation (6) in a fashion similar to t ha temployed for the solution of equation (4). Th e function R* = R*{t*;F, 8) was obtained by numerically integrating equation (5). The results

are presented as universal plots in Figs. 8 and 9 for an extensive rangeof r and 0 values.

The net effect of the microlayer contribution may be found bycomparing the res ults of Fig. 8 with those of [1]. For illustration, thesolution for microlayer-free bubbles (no plate) is included for F = 20.Contrary to the results for probe microlayers, a continuously increasing microlayer contribution with time is observed here. This isfavored by both the geometric and transport effects mentioned before.Now the bubble surface area increases at the same ra te as the micro-layer surface area. This behavior does not produce a cancellation of

R"

--~

-

-

r-2~

10 -

No p la te 2 0 -

i

/ z / /Ayy/

/0y y

/fly/ sss

/$// A A Ay

j F yyAA''

W' AAA A s ,FS i /AAA s, 11

y

y

1

>V

/

/A/

<n —

- T

i i

y

y ^

y

l

1

y

Ayyy y

AAA yyy y

'y y/ yy

\y ^y

\y

\ ^ r - 2

0s

— 10

y ^ — 20

^— Na p la te

1 1

/ y

y

s?0^--yAX / ~Ay ys y

yy

:r=2o :

i i i n i

Fig. 8 Universal p lot of the hemisp herical bubble radius versus t ime c o m pared w ith the no-plate results for r = 20. Here, 8 = 1

Fig. 9 Universal plot of the hem isphe rical bubb le radius versus tim e forFig. 7 Universal plot of the spheric al bubb le radius versus tim e for different different param eter 8.8 = 1.27 c orresponds to a liquid sodium -stainless steelprobe parameter nr" (T = 5, 20) system





the transport effect as it did for the probe microlayers. Hence for smallvalues of t* , where the growth is predom inantly inertia controlled, i.e.,R* = t* , the contribution of the microlayer, as expected, is minimal.At t* ~ 10~

2a difference of only 10 perce nt is observed. For inter

mediate values of t* (i.e., t* ~ 1) the growth is by a combination ofinertia and thermal limitations, and the microlayer contribution increases. For t* = 1 this contribution is about 25 percent. From thispoint on thermal effects become increasingly important and the mi-

t, msec

Fig. 10 A com parison of the theoret ical results with exp erim ental data fora spherical bub ble: (1) num erical solution [10] ; (2) solut ion w ith the probesaccounted for; (3) solut ion of [16] ; (4) p resent solut ion without consideringthe probe ef fects. (Freon-113, superheat = 59.4°C, T„ = 22.8°C, probe diameter = 0.25 mm)

t , msec

Fig. 11 A comp arison of the theoret ical results with exp erimen tal data fora spherical bub ble: (1) num erical solution [10] ; (2) solut ion with the probesaccounted for; (3) " inert ia"; (4) "heat-t ransfer" solut ion. (Freon-113, superheat = 34.1°C, T m = 22.8°C, probe diameter = 0.2S mm )

crolayer contribution correspondingly increases. For t* = 100 themicrolayer contributes an increase of 50 percen t in bubble size. Similartrends are found for other values of the parameter Y. These resultsare in excellent agreement w ith the previously available solutions forthis (hea t transfer control) asymptotic regime of Sernas and Hooper[6] and van Ouw erkerk [5].

The effect of the parame ter 6 is shown in Fig. 9. The value 0 = 1.27corresponds to th e liquid sodium -stainless steel system which is ofcurrent interest in nuclear applications.

R e su l t s and D isc uss ion

A set of typical spherical bubble growth results (with probe microlayers) are prese nted in Figs. 10-13. Fig. 10 represents the highestsuperheat run and clearly confirms the F-effect predicted previously[1]. It also demonstrates the excellent agreement between the experimental data and the present solutions for this case of negligiblemicrolayer effects. Fig. 11 shows a similarly good comparison between

t , msec

Fig. 12 A com parison of the theoret ical results w ith exp erimental data for

a spherica l bubb le: (1) nume rical solution [10] ; (2) solut ion with the probesacc ounted for. (Freon-113, superheat = 31.9°C, Teo = 24.7°C, probe diameter= 0 . 81 mm)

t, msec

Fig. 13 A com parison of the theoret ical results w ith exp erimental data fora spheric al bub ble: (1) num erical solution [10] ; (2) solut ion with the probesacc ounted for. (Freon-11 superheat = 17.8°C, T „ = 21.7°C, prob e diam eter= 0 . 51 mm)

46 / VOL 100, FEBRUAR Y 1978 Transact ions of the ASME




t*

F!g. 14 A com parison of the theoret ical results with experimental data fora hemispherical bubble: (1) solution without plate, F = 5.09; (2) solution withplate, F = 5 .09; (3) solution w ith plate, F = 3.58 (Freon-113, superheat =36.7°C, T™ = 22 °C, probe diameter = 0.61 m m )

experiment an d predictions, for a set of conditions for which both the

" ine r t ia " and the "hea t t rans fe r" so lu t ions g rea t ly ove rp red ic t the

data . Figs. 12 and 13 dem onst rate th e predictive capabili ty u nde r

conditions for which microlayer effects become of increasing impor

tance, i .e . , larger probe diameters and lower superheats . I t has been

shown [1] th at only slight differences exist between the app rox im ate

analytical technique uti l ized here and the numerical solution (with

a vaporization coefficient of unity) [10] which may be taken as the

"exact" one. In the comparison presented before, we have avoided

even these s l ight differences by adding th e probe contributio ns to the

bubble s ize as predict ed by the pre sent work to the result s of the nu

merical s imulation. Th e results of reference [15] were uti l ized to es

t imate the microlayer thickness. The probe surface area uti l ized inthese calculations includes an increase due to the microlayer. For t he

case of Fig. 13 the re sult is sensitive to this thick ness. How ever, the

hydrodynamic displacement thickness, commonly uti l ized for mi

crolayer thickness estim ation in nucleate boil ing, is of the same mag

nitude. For the other cases some differences between the two methods

are obtained but the results are not sensit ive to such changes.

A set of typical hemisph erical bub ble growth results (w ith plane-

wall microlayers) are presen ted in Figs . 14-16. These res ults are pre

sented in terms of the dimensionless quantit ies R* an d t* and all

analytical predictions are based on the solution of equations (5) and

(6) (neglecting the small contribution of the microlayer left on the

vertical nucleation probe, in comparison to that of the microlayer a t

the base of the bubb le, on the wall) . Due to their small m agnitu de, as

seen from the comparisons already presented, uncertainties in t ime

origin have not be en inclu ded. Fig. 14 i l lustrates a case of combined(inertia and heat transfer) growth control, and a significant microlayer

influence. The sensit ivity to the parameter F is a lso indicated. Ex

cellent agreem ent is a lso indicated by the comparison of Fig. 15 for

a lower T-value (and superheat) . However, for the lowest superheat

results of Fig. 16, the agreem ent is good only up to t* = 5. Beyond this

t ime, the da ta very quickly develop a trend indicating negligible mi

crolayer contribution. To confirm this departure an additional ex

periment was run under s imilar conditions. Reproducibil i ty was ex

cellent. Since the t* -range covered in all thr ee figures corre spon ds

to similar real t ime (5-8 ms), th is difference cannot be attr ibuted to

variations in exposure t imes for the microlayer. On the other hand

it is in teres ting to no te tha t the good agreem ent observed in Figs. 14

and 15 is also in the rang e of t* < 5. As discussed in the Appendix the

"thick microlayer" assumption cannot be responsible for this dis

crepancy since i t would cause a deviation in the wrong direction. Al

though the mechanism is not c lear a t th is t ime, this discrepancy ap

pears to indicate an unexpected microlayer behavior at low superheats

(and bubble growth rates) , or perhaps high t* values, which could be

particularly relevant in considering microlayer effects in nucleate

boiling.


t*

Fig. 15 A com parison of the theoret ical results with exp erimental data for

a hemispherical bubble: (1) solution without plate, F = 3.58; (2) solution withplate, T = 3.58. (Freon-113, superheat = 29.4°C, T „ = 21 °C, probe diameter= 0 . 61 mm)

Fig. 16 A comp arison of the theoret ical results with exp erimental data fora hem ispherical bubb le: (1) solut ion without plate, V = 2.14; (2) solution withplate, T = 2.14. (Freon-113, superheat = 18.6°C, Ta = 22°C, probe diameter= 0.61 mm)


Solutions for bubb le growth r ates including the effects of inertia ,

heat transfer , and microlayer evaporation were obtained in general

ized coordinates. Experimental growth rate data were obtained underprecisely controlled conditions, covering a wide range of super heats ,

and encompassing the complete range of t* characterizing the relative

importance of inertia versus heat transfer as control mechanisms. The

theoretical development appears to provide a satisfactory a priori

predic tion of these data . In particular , the T-effect predicted earlier

is fully c orrobora ted by these d ata which are the f irs t to cover an ex

tended range o f th i s pa ra mete r .

These results a lso indicate that kinetic effects (nonequilibrium at

the vapor-liquid interface) for Freon-11 and -113 and the range of

experim ental con ditions covered are negligible. Th e microlayer effects

are precisely predictable except for an anomalous behavior observed

for t* > 5 ( i .e . , low superheat, low F-values). This behavior may be

of importance in considerations of nucleate boil ing.

R e f e r e n c e s

1 Theofanou s, T. G., and Patel , P. D., "Universal Relations for BubbleGrowth," Int. J. Heat Mass Transfer, Vol. 19,1976, p. 425.

2 Cooper, M. G., and Vijuk, R. M., "Bubble Growth in Nucleate PoolBoiling," Pap er B2.1, 4th Int. Heat Transfer Conference, Paris 1970.

3 Dwyer, O. E., Boiling Liquid Metal Heat Transfer, American NuclearSociety, 1976.





4 Jawurek, H. H., "Simultaneous Determination ofMicrolayer Geometryand Bubble Growth in Nucleate B oiling," Int. J. Heat Mass Transfer, Vol. 12,1969, p. 843.

5 van Ouwerkerk, H. J., "The Rapid Growth ofaVapor Bubble at a Liquid-Solid Interface," Int. J. Heat Mass Transfer, Vol. 14,1971, p. 1415.

6 Sernas, V., and Hooper, F. C, "The Initial Vapor Bubble Growth on aHeated Wall During Nucleate Boiling," Int. J. Heat Mass Transfer, Vol. 12,1969, p. 1627.

7 Stewart, J. K., and Cole, R., "Bubble Growth R ates During NucleateBoiling at High Jacob Numbers," Int. J. Heat Mass Transfer, Vol. 15,1972,p. 665.

8 Bohrer, T. H., "Bubble Growth in Highly Superheated Liquids," MS

thesis, Purdue University, May, 1973.

9 Chang, M. C, "A Study of Bubble Growth in Trichlorotrifluoro E thane,"MS thesis, Purdue University, May, 1974.

10 Theofanous, T. G., Biasi, L., Isbin, H. S., and Fauske, H. K., "VaporBubble Growth in Constant and Time Dependent Pressure Fields," Chem.Engng. ScL, Vol. 24,1969, p. 885.

11 Dergarabedian, P., "Observations on Bubble Growths in Various Su-perheated Liquids," J. Fluid Mech., Vol. 9,1960, p. 39.

12 Florschuetz , L. W., Henry, C.L., andRashid Khan, A., "Growth Ratesof Free Vapor Bubbles in Liquids at Uniform Superheats Under Normal and

Zero Gravity Conditions," Int. J. Heat Mass Transfer, Vol. 12, 1969, p.1465.

13 Board, S. J., and Duffey, R. B., "Spherical Vapour Bubble Growth in

Superheated Liquids," Chem. Engng. ScL, Vol. 26,1971, p. 263.14 Lien, Y. C, "Bubble Growth Rates at Reduced Pressure," DSc thesis,

MIT Feb. 1969.15 Carroll, B. J., and Lucassen, J., "Capillarity-Controlled Entrainment

of Liquid by a Thin Cylindrical Filament Moving Through an Interface," Chem.Engng. ScL, Vol. 28,1973, p. 23.

16 Mikic, B. B., Rosenhow, W. M., and Griffith, P., "On Bubble Growth

Rates," Int. J. Heat Mass Transfer, Vol. 13,1970, p. 657.17 Cooper, M. G., and Lloyd, A. J., "The Microlayer in Nucleate Pool

Boiling," Int. J. Heat Mass Transfer, Vol. 12,1969, p. 895.

A P P E N D I X

T h e T h i c k M i c r o l a y e r A s s u m p t i o n

T h e r e are several papers that discuss heat transfer mechanisms

in the microlayer at the base of a growing bubble. Th e most compre

hensive discussion has been p resen ted by van Ouwerkerk [5j. He

conc ludes tha t a l though the micro laye r evapora te s du r ing g rowth ,

in fact to such an ex ten t tha t a subs tan t ia l dry pa tch deve lops and

grows at the expense of the microlayer, the " th ick mic ro laye r" as-

sumpt ion is a d e q u a t e for bubble growth calculations in nonmeta l l ic

fluids of suffic iently high Prandtl number. There can be some ques

tion concerning th e accuracy of van Ouwerkerk ' s ca lcu la tion to nu-

cleate boiling; however, it is very well suited for our exper imen ta l

conditions. Specifically , under nucleate boil ing conditions, and

especially in subcooled boiling, bubbles may s ta r t out with R cc t and

continue with R <* v 7 (at low superheats the first regime is negligible)

bu t u l t ima te ly the growth law is strongly affected by the non-un i

formity of the temperature f ie ld . The in it ia l growth may slow down

faster than the square-root law dictates and van Ouwerkerk 's theory

will break down. However, for an in it ia l ly uniform te mpe rature f ield ,

as has been the case for all our exper imen ts , we expect (and observe)

an a sympto t ic R °= Vt behavior developing from an in it ia l R & t

growth. It is easy to see t h a t as the t ime exponen t increases above the

asympto t ic 1/2 value the th ick microlayer approximation improves.

Hence conservative l imits of the applicabil i ty of our mode l to our

exper imen ts may be obtained uti l iz ing van Ouwerkerk ' s theo ry .

Van Ouwerkerk has considered analytically the complete problem

of microlayer heat transfer and evaporation taking into account

conduction within the wall. Two param eters are impor tan t in affecting

the accuracy of the th ick microlayer approximation: MH~1/2

=

(k spsc.Jk(peC£)1/2

and 2PrZ ( ,2. HereZ j , is the growth constant of the

microlayer thickness, 5(t), i.e., 6(t) = Z(,V2iv (ve is the k inemat ic

viscosity of the l iquid). Th e growth cons tan t has been de te rmined

theoretically as Zb = 0.9. Base d on his exper imen ta l re su l t s van Ou-

werkerk recommends a value of Zb = 0.6. T h i s is a lso suppor te d by

the values measured by Cooper and Lloyd (0.4-0.7) [17]. Fig. 3 of van

Ouwerkerk can be utilized to precisely evaluate the l imitations of the

" th ick mic ro laye r" a ssumpt ion for any par t icu la r set of cond i t ions .

Since this analysis was based on the asymptotic regime of heattransfer controlled growth the error would be overestimated by Fig.

3 for cases with non-negligible inertial effects (i.e., t* < 10). As ex-

pec ted the results of Fig. 3 also indicate that when MH~1/2

> 1 t h a t

the "thick microlayer" assumption will a lways cause an underesti

mation of the bubb le g rowth ra te .

W h e n the solid-liquid pair is such tha t MH~112

~ 1 the " th ick mi-

crolayer" assumption causes negligible error. In many p rac t ica l ap-

plications, in fact, MH~l/2

is not much g rea te r than 1, (i.e., 1.3 for

n-hep tane on Perspex, 3.3 for n-hep tane or benzene on Pyrex , 4.2 for

ca rbon te t rach lo r ide on P Y R E X ( / For the presen t expe r imen t

MH"1'2

= 4.4 and Pr = 7. Even with a value of Zb = 0.5, as Fig. 3 in-

dicates, the "thick microlayer" assumption is excellent, (i.e., less than

7 percent error in bubble growth rate and less t h a n 20 percen t in

microlayer evaporation rate) .

48/ VOL 100,

FEBRUARY1978

Transact ionsof the

ASME




H. S. FathGraduate Student

R. L. JuddAssociate Professor


McMaster University,

Hamilton, Ontario, Canada

Influence of System Pressure on

iicrolayer Ewaporation Heal

Transfer

Evaporation of the microlayer underlying a bubble during nucleate b oiling heat transfer

is experimentally investigated by boiling dichloromethane (methylene chloride) on anoxide coated glass surface using laser interferometry and high speed photography. The

influence of system pressure (51.5 kN/m2—101.3 kN/m

2) and heat flux (17 kW/m

2—65

kW/m2) upon the active site density, frequency of bubble emission, b ubble departure ra

dius and the volume of the microlayer evaporated have been studied. The results of the

present investigation indicate that the microlayer evaporation phenomenon is a signifi

cant heat transfer mechanism, especially at low pressure, since up to 40 percent of the

total heat transport is accounted for by microlayer evaporation. This contribution to the

overall heat transfer decreases w ith increasing system pressure and decreasing heat flux.

The results obtained were used to support the model propounded by Hwang and Judd for

predicting boiling heat flux incorporating microlayer evaporation, natural convection and

transient thermal conduction mechanisms.

Intr oduc t ion

In recent years , boil ing heat transfer has achieved worldwide in

terest , primarily because of i ts application in nuclear reactors , je t

a ircraft engines and rockets . The heat transfer achieved by phase

change is very high and as a consequence, boiling is used to cool de

vices requiring high heat transfer ra tes .

Much research concerning nucleate boiling during the past several

years has been conducted in order to determine the mechanisms as

sociated with bubble formation that are responsible for producing the

high heat f luxes. In recent years , the mec hanism known as m icrolayer

evaporation, in which energy is transferred away from the heating

surface into the bu bbles by evaporation of a thin layer of su perheat ed

liquid underneath them, has assumed prominence as one possible

explanation of nucleate boil ing heat transfer . Snyder and Edwards

[ l] 1 were the f irs t to suggest that evaporation might occur at the base

of a bubble from a thin liquid film or "microlayer" left on the surface

during bubble formation. Rapid evaporation from the base would have

the effect of drawing large amounts of heat directly out of the boiling

surface during bubble growth. The temperature of the surface un

dern eath such a f i lm would be expected to drop significantly after

evaporation had begun.


Contribu ted by the Hea t Transfer Division for publication in the JOURNALOFHEATTRANSFER. Manuscript received by the H eat Transfer D ivision May20,1977

Moore and Mesler [2] investigated boiling on a thermocouple which

could measure the average temp eratu re within a small local area and

had a microsecond response t im e. During nucleate boil ing, these au

thors observed that the surface temperature fluctuated with time. The

fluctuation was such th at periodically the surface tem per atu re would

drop rapidly and t hen return to i ts previous level. Moore and Mesler

did not know whether these drops occurred with the same frequency

as bubble formation, nor did they have any indication of when the

temperature drops began with respect to bubble growth. However,

the rapidity of cooling immediately suggested an evaporation phe

nomenon an d gave suppor t to Snyder and E dwards ' s hypo thes is .

The firs t indirect evidence of the microlayer was obtained by

Hen dricks and Sh arp [3], who used high speed photogra phic d ata

correlated with the surface temperature-time profiles underneath

large discrete bubbles in order to determine when the local fluctuationof surface temperature occurred in the bubble growth and departure

cycle . Th e results of their exp erime nt indicated th at the rapid tem

perature decrease was associated with bubble growth, which suggested

that the rapid temperature decrease was due to microlayer evapora

tion rather than surface quenching.

Sha rp [4] repor ted two experime nts involving direct observa tion

of the microlayer by two different optical techniques involving in

terference of monochromatic l ight and reflection of polarized l ight

examined through a f la t p late over the growing bubbles. As a result

of these experiments , the microlayer was proven to exist .

Cooper and Lloyd [5] continued the work using high speed pho

tography synchronized w ith the outpu t of several rapidly responding

surface thermometers located on the surface of a glass plate during





the growth and d e p a r t u r e of individual vapor bubbles of to luene in

pool boiling at low pressure. Th e results obtained lent support to the

microlayer theory ofboiling heat transfer . Furthe rmor e, these authors

discussed the hydrodynamics of formation of the microlayer and the

effect of evapora t ion of the microlayer upon the growth of the b u b

bles .

Fur the r d i rec t measu remen ts of the microlayer thickness were

ob ta ined by J a w u r e k [6] who used an optical technique which per

mi t ted the s imul taneous de te rmina t ion of microlayer geometry and

the bubble profile in nucleate boil ing.

T he pu rpose of the presen t inves t igat ion is to presen t an experi

mental s tudy on the influence of system pressure and heat flux uponthe microlayer evaporation phenomenon and its relative significance

and contribution to the to tal heat transfer ra te , using interferometiy

and high speed photography.

E x p e r i m e n t a l I n v e s t i g a t i o n

Similar to the investigation reported by Voutsinos and Judd [7] and

Hwang and J u d d [8], d ich lo romethane (me thy lene ch lo r ide ) was

boiled on a borosil icate glass heating surface c oated w ith a half wave

length thickness of stannic oxide which conducted electr ic current

and generated heat, causing the l iquid to boil . Detailed description

of the exper imen ta l appa ra tus a ssembled to ca r ry out th is investi

gation can be found in reference [9].

A sectional view of the t e s t a s sembly is shown in F ig . 1. The t e s t

assembly was comprised of a piece of stainless s teel pip e closed by a

base plate and cover plate resting on gaskets. A glass specimen heatingsurface was arranged within the test assembly in the base plate of the

vessel. The oxide coating on the heating surface was in contact with

the dichloromethane where the boiling took place. The test vessel was

partially filled with one l i tre of d ich lo romethane so t h a t the level of

liquid was located 25 mm above the heating surface. Th e same charge

of d ich lo romethane was used th roughou t the whole test period.

T he re sea rch inves t iga t ion requ i remen ts d ic ta ted tha t sys tem

pressure should be varied. The required levels of system pressure were

ob ta ined by evacua t ing the test vessel with a vacuum pump whose

pressure could be controlled by a manually adjusted needle valve

bleeding atmospheric air. Transfer of vapor from the test vessel to the

vacuum pump was min imized by the use of a suitably located reflux

condenser.

T h e two t empera tu re m easu remen ts requ i red were the bulk l iquid

t e m p e r a t u r e and the hea t ing su r face tempera tu re . The sensingjunc t ion of the the rmocoup le used to m e a s u r e th e heating surface

. temperature was epoxied on the underside of the heating surface, and,

based upon appropriate calculations, it was assumed tha t the t e m

perature measured represented the heating surface temperature. The

ou tpu t s ignal of th i s the rmocoup le was d i rec ted to a reco rde r so t h a t

the tempera tu re cou ld be reco rded du r ing the test period and a t i m e

average could be ob ta ined . The sensing junction of each of the th ree

the rmocoup les u sed to m e a s u r e the bu lk l iquid tempera tu re was lo

cated at a different level in the bulk liquid. The outp ut s ignals of these

thermocouples were averaged to ob ta in a rep resen ta t ive measu re

m e n t . Th e approx ima te loca t ions of the the rmocoup les measu r ing

the heating surface temperature and the bu lk l iquid tempera tu re are

indicated in Fig. 1 by the symbols Tw and T„, respectively.

TIE DOWN BOLTS

- INSULATOR GASKET

V E S S E L

SIGHT T

0 RING

HEATER II

SURFACE CLAMP

WINDOW

Fig. 1 Test assembly

A t the c o m m e n c e m e n t of each test , the cooling water valve con

troll ing the flow to the condenser was fully opened. Th e barometric

p ressu re was measured and the vacuum pressure desired was calculated using the vapor pressure curve for d ich lo romethane . T hen the

vacuum pump was started and the pressure inside the test vessel was

reduced to the desired value by manua l ope ra t ion of the pressure

control valve. The direct curren t power supply was adjusted u ntil the

potential level desired was achieved. At the same t ime , the powei

i n p u t to the heating coil was adjusted so t h a t the bu lk tempera tu re

a t ta ined the sa tu ra t ion tempera tu re acco rd ing to the pressu re es

t ab l i shed .

Dur ing the period of t ime in which steady state boil ing conditions

were being established, the cam era was loaded with f i lm, the camere

power supply w as adjusted to give the required framing rate and the

t iming mark generator was switched on. F ina l ly , the surface heating

power dissipation was measured . Th e a t t a i n m e n t of thermal equi-

„ N o m e n c l a t u r e »

A T = heating surface area

C = fluid specific heat

Co = cons tan t in equa t ion (1)

C i = cons tan t in equa t ion (4)

Cz = cons tan t in equa t ion (8)

/ = frequency of bubble emission

hfg = l a ten t hea t

ke = f luid thermal conductivity

K = cons tan t for area of influence relation

ship

N / A T = active si te density

^sat = system pressure

Qm/Ar - measured heat f lux

QME/AT=

microlayer evaporation heat

flux

QNB/AT = nucleate boil ing heat f lux

QNC/AT = natural convection heat f lux

1 P / A T = predicted heat f lux

Rb = bubb le depa r tu re rad ius

t = t i m e

T — t e m p e r a t u r e

Tw = wal l tempera tu re

Tsat = sa tu ra t ion tempera tu re

T » = bu lk tempera tu re

V M E - vo lume of microlayer evaporated

ae = fluid thermal diffusivity

pe = fluid density





l ibrium was indicated when the heating surface thermocouple outp ut

remained inva r ian t wi th in ±0 .5°C and when the manomete r mea

suring the system pressure appe ared visually unchanged. When these

conditions were fulfilled, the bulk temperature was measured and the

high speed camera was activated.

I t was found that the best framing rate for the high speed camera

was 3900 frames per second w hich gave the clearest picture s. Lower

speed led to unclear interference fringes, especially at high pressures,

due to the rapid microlayer evaporation. Higher speed gave clear

pictures, but the period of t ime available to record the phenomenon

was not sufficient for the analysis of bubble formation and departure.

For each value of system pressure, the heat f lux was changed in s ix

increments s tart ing from the highest value down to the lowest. Each

film obtained for the n ucleate boil ing tests (4 system pressur e levels

by 3 levels of heat f lux) was la ter analyzed to determine the active

nucleation site density, the frequency of vapour bubble emission, the

bubble departure radius and the volume of microlayer evaporated.

T a b l e 1 D a t a o r g a n i z a t i o n f o r p h o t o g r a p h i c t e s t s

10

Sys tem Pressu re P s a t ( k N m2)

('

Heat F lux

AT

(kW/m 2)

51 5-0.50* A t m )

64.11

44.6528.70

—

21.1

64.7(~0.65 Atm)

—

44.9827.81

19.71

26.6

80.2(~0.80 Atm)

56.56

38.4624.26

—

32.2

101.3(1.00 Atm)

—

46.6329.83

17.51

39.4

si e10 -

?kH

A P SA T = 6 4 . 7 »

• PS4T = 8 0 . 2 "

O PSAT-IOI.3«

IO O

S a t u r a t io n T e m p e r a t u r e TaBt (°C)

SUPERHEAT A T u p =< TW -T^) (C>

Fig. 2 Boiling chara cteristic curve

B e s u l t s a n d D i s c u s s i o n

The results obtained for dichloromethane boiling on a glass heating

surface at a pressure of one atmosp here an d lower show tha t the de

crease in system pressure gives a higher superheat AT s u p for a given

heat flux qm/Ar, as seen in Fig. 2.

The natu ral convection heat transfer re lationship was found to

follow:

qNC /AT = C 0 (T w - T aa t)"/3 (1 )

where the constant CG varied with the system pressure. The value of

C 0 a t P s a t = 51.5 kN /m 2 (approximately one half atmosphere) is equal

to 184, which is exactly the same value given by Hwang and Judd [8].

The value of CD increases with increasing system pressure.

The relationship between the nucleate boiling heat flux qm/A r an d

the supe rhea t AT s u p can be expressed as:

/A Ta ATS, (2 )

The value of n a t P s a t = 101.3 kN/m 2 (atmospheric pressure) is equal

to 7 .5 , and i t increases with decreasing sy stem pressure.

The re sults prese nted in Fig . 3 relating to the active s i te density

N / A T demonstrate the influence of heat f lux and system pressure

upon this parameter. It can be seen that increasing the heat flux causes

an increase in the s i te density to facil i ta te the transfe r of addi tional

heat. Moreover, the curves shown in this f igure seem to be paralle l

with a slope of about m = 0.58 which more or less agrees with the

correlations which advocate that the heat f lux is proportional to the

square root of the active s i te density .

Thre e active s i te density correlations are presented for comparison

with the results of the presen t investigation, including the correlation

of Hwang and Judd [8] for dichloromethane boiling at 51.5 kN/m 2 ,

Kirby and Westwater [10] for carbon tetrachloride boil ing at 101.3

kN/m2, and Judd [11] for Freon 113 boiling at 155.9 kN/m

2. All three

of these correlations were based upon results obtained by photo

graphing the boil ing phenome non thro ugh a glass heating surface. T he


10 —

^ J U O O [ s i ]

P« T =

l 5 5-

9% m = 0 . 7 3

© PSAT = 51 .5 u |

A PsftT = 6 4 . 7 "

• PSAT = S 0 2 "

G Ps4lT=IOI.3"

KN

' L KIRILKIRBf AND

WESTWATER CIO]

m = 0.51

A O / l-HWANG AN D JUDD C8 ]

7 / / *„•"•» 3/

2* « (JL)A T » A T '

HEAT FLUX f <S2 )

Fig. 3 Active site density results





difference between the correlation for dichloromethane boiling at 51.5

k N / m2

obtained in the present investigation and that obtained by

Judd and Hwang is most l ikely the result of differences in the glass

heating surfaces used.

The influence of heat flux and system pressure upon the frequency

of bubble emission / is presente d in Fig. 4. Th e curves presented show

that the frequency of bubble emission varies directly with heat f lux

and that bubble frequency increases with pressure unti l a maximum

value has been obtained after which bubble frequency decreases.

In terms of average values, the bubble flux density * was calculated

from the active s i te density and bubble frequency by the relation

s h ip :

I" Bu bbl e F lux",

L Density j(*); [Active Site

Dens i ty ]®_ rB u b b i e

i ,-f)L F r e q u e n c y J

(3)

From Fig. 5 i t can be seen that increasing the heat f lux and system

pressure increases the bubble f lux density . The bubble f lux density

agrees very well with th e results of Hwang and Ju dd [8] as shown in

the same figure.

The effect of heat f lux and system pressure upon the maximum

bubble ra dius f l j, is presen ted in Fig . 6 , which shows th at increasing

the heat f lux decreases the maximum bubble radius. This f inding

agrees with the results presented by Hw ang and Jud d [8]. Although

at each level of heat flux various bubble sizes were observed, the av

erage bubble s ize changed significantly with h eat f lux.The average volume of microlayer evaporated VM E was obtained

by averaging the results for a number of bubble s selected from each

of the active sites on the boiling surface within the field of view ofthe

high speed camera. The volume of the microlayer evaporated VM E

depends upon the init ia l microlayer thickness and the bubble de

parture radius. Since both of these parameters decrease with in

creasing heat f lux and system press ure, i t is not unre asonab le to ob

serve that an increase in heat f lux and system pres sure leads to a de

crease in volume ofthe microlayer evap orated as indicate d in Fig . 7 .

The results of the present investigation at P s a t = 51.5 kN/m 2 were

compared with those presented by Hwang and Ju dd [8] and Voutsinos

and Ju dd [7] in Fig. 7 as well. Th e compa rison show s little difference

between the results, presumably because the way in which the analysis

HEAT FLUX - f l ^ l

F i g . 4 B u b b l e f r e q u e n c y r e s u l t s

was performed yielded comparable values.

In the determ inatio n of bubb le emission frequency, bubble f lux

density , maximum bubble radius and volume of microlayer evapo

rated, i t is necessary to consider many individual bubbles originating

from numerous active nucleation sites located within the field of view

of the high speed camera. Consequently , averages must be used to

•e- I O -

HWANG AND JUDD [8 ]

10*

HEAT FLUX - f ( m 2 'AT

F i g . 5 Bu b b l e f l u x d e n s i t y re su l t s

HWANG AND JUDD 18)

SAT m<=

= 64.7

D PSST = 8 Q 2 "

O PMT -K>I .3«

HEAT FLUX 3 0 ( =2 )A T

F i g . 6 Bu b b l e d e p a r t u re ra d i u s re su l t s





-1010 — \\

W V0UTSIN0S AND

\ \ J U D D [ 7 ] -

\

\ O x

A t- H W A N G A N D

,A

J U D D t 8 ]

=101.3"

9 ,w~

HEAT FLUX -m (^ 2 )

T

Fig. 7 Volum e of micro layer evap orated per bubble

* T f e

1 0 -

A P SA T =64.7

Q P SA T = 8 0 2

O PSST = I0I .3

S M A L L N U M E R A L S B Y T H E D AT A P O I N T S

I N D I C A T E T H E P E R C E N T A G E OF T H E T O T A L

H E A T F L U X A C C O U N T E D F O R BY M I C R O L A Y E R

E V A P O R A T I O N H E A T T R A N S F E R ^

HEAT FLUX Jm ( - z )AT

m

Fig. 8 M i c r o l a y e r e v a p o r a t i o n h e a t t r a n s f e r r e s u l t s

represent these param eters , particularly s ince considerable variation

is frequently observed within th e data sam ple. Although it is not a p

propriate to discuss the variabil i ty of the results used in the deter

mination of the values plotted in the preceding graphs because of

space l imitations, th is matter has been investigated and reported in

reference [9].

The rate of heat transfer by microlayer evaporation QME/AT, ob

tained by multiplying the average energy transferred per bubble dueto microlayer evaporation pehfs VM E by the bubble f lux density 4> ca n

be computed by the relationship:

- - N -QME/AT = Pih fgVM E * = pehfgVM E~-f •

AT

CIVME* (4)

where C\ = pehfg is a f luid property constant. The results predicted

by equation (4) are plotted in Fig. 8 as a function of the measure d hea t

flux and system pressure.

From equation (4) i t is obvious that the microlayer heat transfer

depen ds upon the f luid prop erties , the average volume of the micro-

layer evaporated and the bubble f lux density . All these parameters

are functions of both the system pres sure and the hea t f lux. At con

stant pressure (Ci = cons tant) , the average volume of the m icrolayer

evaporated decreases and the bubble f lux density increases with in

creasing heat f lux. From the results obtained, the increase in the

bubble f lux density is muc h greater tha n the decrease in the average

volume of microlayer evaporated so that VME$ appears to increase

with increasing heat f lux.

The effect of system pressure upon microlayer heat transfer de

pends upon how much each of the parameters on the r ight hand side

of equation (4) varies with pressure. The bubble f lux density <I> in

creases with increasing system pressure. On the other hand, the fluid

density pe , the la ten t hea t h/ e and the average volume of microlayer

evaporated VM E decrease with increasing system pressure. The re

duction of the average energy transferred per bubble due to microlayer

evaporation pehfgVM E is greater than the increase in the bubble flux

dens i ty 4>, so that the energy transferred by the microlayer evapora

tion QMEIAT decreases with the increase of system pressure.

The microlayer evaporation phe nome non is definite ly a s ignificant

heat transfer mechanism at a tmospheric pressure and may be sig

nificant at pressures above atmospheric pressure depending upon the

heat flux. The percentage values obtained by the present results range

up to 40 perce nt of the measured hea t flux. Such high percentages give

support to the microlayer evaporation phenomenon as an importantheat transfer mechanism. This importance, however, decreases with

increasing system pressure and decreasing heat f lux.

Heat Tran sfer Cons iderat ionsAlthough there are many models proposed to explain the nucleate

boiling phenomenon, none of these models can adequately predict

nucleate boil ing heat transfer under all the parameters of heat f lux,

subcooling and pressure for different combinations of surface and

fluid con ditions. Also, in the models which exist in the literat ure, little

attention has been given to the existence of microlayer evaporation

heat transfer me chanism as a s ignificant contributor to the total hea t

transfer . The boiling heat transfer model which is discussed below

is that proposed by Hwang and Judd [8]. This proposed model gives

an indication of the significance of the microlayer heat transfer to the

total boil ing heat transfer process at various levels of heat f lux and

system pressure.

Hwang and Judd considered the predicted heat f lux qp/A T to be

comprised of three components , microlayer evaporation QM E/AT,

transient conduction to the superheated layer (nucleate boil ing)

QNB/AT and natural convection quel AT- The natural convection

component occurs in the regions uninvolved in bubble nucleation,

whereas the microlayer evaporation component as well as the tran

sient conduction to the superheated layer component occur in the

same region of influence. However, the heat transfer associated w ith

the displacement of the superheated layer occurs in the departure

period, whereas the heat transfer associated with microlayer evapo

ration occu rs in the growth period, and so the two mech anism s com

p lemen t each o the r .





The natural convection component qNc/A-r can be predicted by

multiplying the hea t transfer by the natural convection q^c/ANC

expressed by equation (1) by a ratio representing the portion of the

surface not involved in nucleate boiling and microiayer evaporation

[l-K7ri? 62(iV/AT)]sothat:

g N c M T = ( ^ £ ) ( ^ ) = C 0 ( T , „\ANC' \ AT '

T sa t )4/ 3

I1-**'®]

(5)

The transient conduction to the superheated layer (nucleate boil

ing) component is obtained by multiplying the average heat flux re

sulting from the extraction of energy from the heating surface to form

the superheated layer (q/A)av by a ratio representing the portion of

the surface involved in the nucleate boiling and microiayer evapora

tion KirRb2(N/A'r)- Assuming only pure conduction in the liquid in

the area of influence, this part of the problem is modeled as conduc

tion to semi-infinite body (the liquid here) with a step change in

temperature (AT = Tw — T8at ) at the surface. From any conduction

text the solution for this case is given:

q/A=-ke&T

(6)v Tiagt

The superheated layer is replaced with a frequency / where / is the

frequency of bubble emission. Hence, the average heat flux over the

area of influence would be:

q/Adt = 2kf,ATy~0 TT

f__ae

(7)

The nucleate boiling component qua/AT will be predicted by:

KwRb2

qNB/AT = (q/A)au

• 2 V^JcWeKRb2

VT(Y){TW

~

= C2KRbWf(j-) (Tw-Tsat)

)]T Bi

(8 )

where C2 = 2 V-irpfC'tke is a fluid property constant which is pri

marily dependent upon temperature and only slightly dependent

upon pressure.

In each of the equations representing t he components qNc/AT and

QNB/AT, K is a parameter greater than unity relating the area of in

fluence around a nucleation site from which energy is transpor ted by

nucleate boiling to the projected bubble area at departure. If the entire

energy content within an area of influence twice the departure di

ameter were transported by the departing bubble as claimed by Han

and Griffith [12], the n the value of K would be four. This could be

considered to be the upper limit of K. However, it should not be ex

pected that the whole energy content within the area of influence

would be transported by the departure of bubble. The value of K

which best fits the present experimental results was K = 1.8.

Summing equations (4), (5) and (8) yields the relationship:

qp/A T = C0(TU, , t )4 /3

[ l - W ( I ; ) J

+ C1VME<S> + CiKRl' • < : \ * W *- Si (9)

All the parameters have been obtained by experimental measure

ments. Fig. 9 presents the verification of equat ion (9) above using the

measurements reported in the previous section for the value of K =

1.8. The various dashed lines seen in the figure serve to indicate the

contributions of microiayer evaporation, natural convection and

nucleate boiling at each level of heat flux and system pressure, whereas

the heavy solid line represents the contribution of all three mecha

nisms. At each level of system pressure, the sum of the three compo

nents representing the predicted heat flux qp/AT closely approaches

the measured heat flux qm/AT with a discrepancy of the order of ±15

percent. This agreement does not necessarily establish the general

validity of the model, since K is an empirical constant. However, it

-INCIPIENT HEAT

FLUX

^ N U C L E A T E / < /

_ BOILING ri'./y

^9 ,7

Fig. 9 Verification o! heat transfer model. Symbolscorrespond to the levels

of pressure identified in the preceding graphs. The open symbols represent

the contribution of microiayer evaporation, the half darkened symbols represent the contribution of microiayer evaporation and natural convection, and

the fully darkened symbols represent the contribution of microiayer evapo

ration, natural convection and nucleate boi l ing.

has been demonstrated that in the range of parameters investigated,

microiayer evaporation heat transfer is an essential component of the

overall heat transfer phe nomenon.

Conc lus ions

The results obtained through this research work can be summarized

as follows:

1 The investigation presents a set of measurements for saturate) 1

nucleate boiling of dichloromethane (methylene chloride) on an oxid-1

coated glass heating surface for various combinations of heat flux

varying from 17 kW/mz

to 65 kW/m2

and system pressure varyini:

from 51.5 kN/m2

to 101.3 kN/m2.

2 The calculated results showed that the microiayer evaporationphenomenon is definitely a significant heat transfer mechanism,

especially at lower pressure, since it contributed up to 40 percent oi

the measured heat flux. The importance of microiayer evaporation

decreases with increasing system pressure, and it is expected that ai

somewhat higher pressures than those investigated in the presen1

study the contribution of microiayer evaporation will become entirely

insignificant at all levels of heat flux. Consequently, the microiayer

evaporation phenomenon is unlikely to be of any consequence in de

vices such as pressurized boiling water reactors.

3 The model for predicting the nucleate boiling heat flux proposed

by Hwang and Judd was evaluated by the present measurements. The

results of this analysis indicated that the experimental data were in

agreement with the proposed model.

54 / VOL 100,FEBRUARY

1978Transactions

of theASME





The authors wish to acknowledge the support of the National Research Council of Canada through grant A4362.

R e f e r e n c e s

1 Snyder , N. R., and Edwards, D. K., " S um m a r y of Conference on Bubble

Dynamics and Boil ing Hea t Transfe r ," Memo 20-137, Je t P r op . La b . , June1 4 - 1 5 , 1 956 .

2 Moore , F. D ., and Mesler , R. B ., "The M e a su r e m e n t of Rapid Sur face

Tempera tu re Fluc tua t ion s Dur ing Nuclea te Boi ling ofWa te r , " AICHE J.,Vol.

7, 1961, PP- 620-62 4.

3 Hendr icks , R. C, and Sharp , R. R., "Ini t ia t ion ofCooling Due to B ubb le

Growth on a Heating Sur face ," NASA TN D2290 , 1 964 .4 Sharp, R. R. , "The Nature ofLiquid Fi lm Evapora t ion Dur ing Nuclea te

Boil ing," NA SA TN D1 997 , 1 964 .

5 Cooper , M. G., and Lloyd, A. J., "T he Microlayer in Nuclea te PoolBoiling," Int. JHMT, Vol. 12,1969.

6 Jawurek, H. H., "S imultaneous Dete rmina t ion of Microlayer Geometryand Bubble Growth in Nuclea te Boi l ing," Int. JHMT, Vol. 12, 1969.

7 Vouts inos , C . M. and J u d d , R. L. , "Laser Inte r fe rometr ic Invest iga t ionof the M ic r o l a ye r Eva po r a t i on P he nom e non , " JOUR NAL OF H E A T

TR A NS F ER , T R AN S . AS M E, Vo l . 97 , S e r ie s C, No. 2 , 1 975 .

8 Judd , R. L., and Hwa ng , K. S., "A Comprehensive Model for Nuc le a t ePool Boi l ing Hea t Transfer I nc lud ing M ic r o l aye r Eva po r a t i on , " JOU R NA L

OF HE AT TR AN S F ER T R AN S . AS M E, Vol . 98 , S e r ie s C, N o. 4, Nov. 1976.

9 Fa th, H., "Laser Interferometric Investigation ofMicrolayer Evaporation

for Various Levels of P r e s su r e and He a t F lux , " M. Eng. thesis , McMaste rUnivers i ty , 1977.

10 Kirby, D. B . , and Westwate r , J. W., "Bub ble and Vapour Behaviour on

a Hea ted Hor izonta l P la te Dur ing Pool Boi l ing Near Burnout ," Chem. Eng.

Progress Symposium Series, Vol. 6 1 , 1 0 6 1 , pp . 241 .1 1 J u d d , R. L., "Influence ofAcceleration on Subcooled Nucleate Boiling,"

Doctora l Disse r ta t ion, Univers i ty of Michigan, 1968.

1 2 Ha n , C. Y„ and Grif f i th , P . , "The Mechanism of Heat Transfe r in Nu

cleate Pool Boiling," Int. JHMT, Vol. 8 ,1965.





M. SultanGraduate Student

R. L. Judd

Associate Professor

Mechanical Engineer ing Department,

McMaster University,

Hamil ton , Ontario, Canada

Spatial Distribution of Active Sites

and Hubble Flux DensityAn experimental investigation is presented concerning the boiling of water at atmospher

ic pressure on a single copper surface in which the spatial distribution of active nucleation

sites was investigated at different levels of heat flux and subcooling. The results obtained

indicated that the active sites were located randomly on the heating surface, since the dis

tribution of active sites followed the Poisson relationship. Changes in heat flux and sub

cooling did not affect the distribution of active nucleation sites although additional activesites formed among the sites which had already been activated when heat flux was in

creased. The frequency of vapor bubble emission and the bubble flux density have been

studied as well. The results obtained showed that the heat flux had a great effect on the

vapor bubble emission frequency although the influence of subcooling was of lesser signifi

cance. However, the bubble flux density was found to be non-uniformly distributed over

the heating surface contrary to what had been expected.

In t roduct ion

In the last few years , in terest in boiling heat transfer has in it ia ted

considerable research in order to study the mechanism of boiling heat

transfer and to mett the design requirements of devices such as nu

clear power plant reactors . Many different aspects of boil ing heat

transfer have been investigated, but one feature which has not re

ceived sufficient a ttenti on to th e presen t time is the role of the surface

in nucleate boil ing. I t is known that when the surface temperature

exceeds the saturation temperature of the l iquid suffic iently , vapor

bubbles will form at specific locations on the hea ting surface term ed

"nucleation sites" which are pits, scratches, and grooves that have the

abili ty to trap vapor. The heating surface contains many such po

tentia lly active nucleation sites that have trapped small amounts of

vapor which are not necessarily active nucleation sites . According to

the heat f lux applied, a num ber of potentia lly active nucleation sites

will be active and the nu mbe r of active nucleation sites will increase

as the heat f lux increases unti l the entire surface becomes "vapor

blanketted" at high rates of heat f lux. However, the mechanism de

termining the manner in which potentially active nucleation sites

become active under specified operating conditions is not understoodvery well. It is to this end that the research reported herein is directed

in an attempt to relate the spatia l d is tr ibution of active nucleation

sites to heat flux, superheat, and subcooling for boiling on a single

heating surface.

Jakob f 1 ] ' was the first to demonstrate that the rate of heat transfer


Contribu ted by the Heat T ransfer Division for publication in the JOURNALOP HEAT TRANSFER. Manuscript received by the Heat Transfer Division May31, 1977.

was depend ent upon th e surface roughne ss. Corty and F oust [2] ob

tained results which showed that both the s lope and posit ion of the

boiling curve changed with changes in surface roughness. Kurihara

and Myers [3] studied boiling with different surface finishes and found

that the number of active nucleation sites increased with increased

roughness. Hsu and S chm idt [4] a lso showed tha t heat transfer in

creased w ith increasing surface roughness for a polished stainless steel

surface. Berenson [5] investigated the effect of surface roughne ss using

different surface m aterials prepared by lapping and by emery paper.

The results which he obtained indicated that heat transfer was greater

in the case of the lapped surface than in the case of the emery paper

finished surface. Thus, heat transfer decreased with an apparent in

crease in surface roughn ess, contrary to the f indings of the previous

investigators .

Vachon, Tan ger, Davis and Nix [6] studied pool boiling on polished

and chemically etched stainless s teel surfaces with a variety of dif

ferent surface roughnesses at a tmospheric pressure using disti l led

water, and ca me to the sam e conclusion as Berenso n [5] , tha t is , that

an increase in roughness does not necessarily mean an increase in heat

transfer . S houkri and Jud d [7] s tudied pool boil ing of dist i l led, de

gassed water at atmospheric pressure on a single copper surface withdifferent surface f inishes, and showed that decreasing surface

roughness shifted t he boil ing curve to higher supe rhea t for the same

heat f lux. Nishikawa [8] observed the formation of bubbles from a

horizontal surface and observed a greater num ber of bubble columns

on a rough surface than on a smooth one.

F rom this survey of the effect of surface condition on nucleate

boiling, i t may be concluded that surface roughness cannot be used

in the characteriz ation of boiling surfaces because such a measurement-

represents the macroroughness of the surface which is only related

to the distr ibutio n of active cavit ies in a very indirect ma nner . How

ever, surface roughness measurem ents may be useful for comparison

purposes when using the same technique for the preparation of boiling

58 / VOL 100, FEBRUARY 1978 Transact ions of the ASMECopyright © 1978 by ASME




surfaces from the same mate r ia l .

In 1960, Gae rtner and Westwa te r [9] repor ted a m e t h o d to de te r

mine the distribution of active nucleation sites. The method was based

on an e lec t rop la t ing phenomena by which an impressed cu r ren t re

sulted in a nickel film being plated on the su r face . T h is techn ique

identified the location of the active si tes s ince, during the electro

plating, vapor bubbles originating at the active nucleation sites would

push the plating solution away from the surface so that very l i t t le

metal could be p la ted at tha t loca t ion . High con t ra s t pho tog raphs

were taken of the plated surface to permi t coun t ing of the p inho les

representing the active si tes .In 1967, Heled and Orell [10] proposed ano the r techn ique wh ich

greatly facilitated the identification and location of active si tes . This

technique was based on the observation ofscale deposits which formed

on a surface first polished to ahigh degree and then electroplated with

either chrome or nickel. The tap wate r or disti l led water which was

boiled contained soluble salts which formed ring shaped deposits

during nucleate boil ing.

In 1970, Yoshihiro Iida and Kiyosi Kobayasi [11] exper imen ta l ly

de te rmined the active si te density , the d is t r ibu t ion of t ime average

void fraction in three boil ing regimes, and the th ickness of super

heated liquid layer on a single horizontal heating surface in sa tu ra ted

pool boiling of wate r by the use of an electr ic probe. Yoshihiro , et al.

[11] obtained good agreement between the active site density results

an d the re su l t s ob ta ined by Gaer tne r , et al. [9], and presen ted a dis

tr ibution map of void fraction in a vertical plane which showed that

the domain of high void fraction lies above the superhea ted l iqu id

layer.

In 1974, Nail, Vachon and Moreh ouse [12] s tudied pool boil ing on

a cold rolled stainless steel test specimen. Pow er was controlled so tha t

the first 12 to 20 bubbles formed on the surface remained on the

surface sufficiently long to allow a bubble locator to be positioned over

the center of origin of the selected bubbles. The posit ions of all the

centers were recorded, thus enabling the d is t r ibu t ion of active nu

cleation sites to be de te rmined .

In the present investigation, which was carried out at low hea t flux

levels in the isolated bubble regime, an electr ical resis tance probe

technique wasused to study the distr ibution of active nucleation sites

an d to measure the frequency of bubble emission at each site as will

be explained below. This te chnique proved quite satisfactory, excep t

that it was not possible to obtain active nucleation site distr ibutions

when the active site density was extremely large, since the probe couldnot easily dist inguish active nucleation sites when the d is tance be

tween sites decreased to the orde r of one bubble diameter. However,

for those conditions which per mitted the resis tance probe technique

to be used, the n a t u r e of the techn ique was such tha t the n u m b e r of

bubbles emitted per unit area per unit t ime, termed the "bubble f lux

density ," could be de te rmined as well.

I t is desirable to know how the bubble f lux density varies over a

heating surface in orde r to u n d e r s t a n d the mechan ism of nuc lea te

boiling heat transfe r. The fact that very little bubble flux density data

exists may be due to the fact that knowledge of the frequency of vapor

bubble emission is not well established. Voutsinos [13] s tud ied the

influence of heat f lux and subcooling upon the bubble f lux density .

For dichloromethane boiling on a glass surface, Voutsinos showed tha t

the bubble f lux density increases with increasing heat f lux and de

creases with increasing subcooling. More recen tly , Ju dd and H w a n g[14] studied the influence of heat flux and subcooling on the bubb le

flux den sity as well. For the same surface/fluid combina tion, Hwang

and Judd ob ta ined re su l t s in agreemen t wi th the effect of heat flux

but contrary to the effect of subcooling when the results obtained were

compared w i th those ob ta ined by Voutsinos [13] , The different sub

cooling effect may be due to different methods of comput ing the av

erage frequency of vapor bubble emission, or due to different surface

characteristics. In addition to the study ofspatia l d is tr ibution ofactive

nucleation sites described above, the frequency of vapor bubble

emission and the spa t ia l d i s t r ibu t ion of bubble f lux density over the

heating surface are repor ted in the research described herein .

E x p er i m en t a l Ap p ara t u s And P r o c e d u r eA t the commencemen t of the research, an exper imen ta l appa ra tus

cons t ruc ted by Wiebe [15] in 1970 to measure temperature profiles

nea r the heating surface already existed. However, the opera t ing

criteria for the present investigation were considerably changed from

the previous requirements and consequently the apparatus described

below is essentially a modified design.

A sectional view of the complete boiler assembly ispresented in Fig.

1. The vessel was made from schedule 40 stainless s teel pipe. Two

stainless steel flanges were welded on the ou ts ide d iamete r at bo th

ends of the pipe and a stainless s teel cover plate was attac hed to the

flange at the top of the vessel with eight cap screws which com pressed

a rubber gasket between the flange and the cover plate. Two circular

sight glass windows were located diametrically opposite at a level

which permitted observation of the bubbles at the probe location. The

heater block was made from a copper cylinder. Thirteen cartr idge

f T T f e ^ a - ^ j - -1- ^

VESSEL

BULK L IQUID THERMOCOUPLE'

BUBBLE PROBE

GUARD HEATER

-CONDENSER COIL

PLATE SPACER

SU9COOUN0 COILS

VER M I C U L I T E

FIBER GLASS

INSULATION

Fig. 1 Sect ion of boi ler vessel

-No m en c l a t u re -

A = heating surface area

a = local surface area

Q/A= heat flux

T- , = bu lk l iqu id tempera tu re

Tw = hea t ing su r face te mpera tu re

Ts = l iqu id sa tu ra t ion tempera tu re

N/A = active si te density

Na = actual population ofactive sites in local

area "a"

Na = expected population of active si tes in

loca l area "a"

P(Na) = probabili ty that a random local area

" a " has a popu la t ion "Na"

Z/Va = n u m b e r of local areas having a popu

lation "Na"

f = surface average vapor bubble frequency





heaters were installed in symmetrically located holes in the base of

the heater block. A stainless steel skirt was brazed flush with the top

of the copper block to provide a continuous extension of the boiling

surface. The boiling surface was machine-finished after the skirt was

brazed before the heater block assembly was installed in the vessel.

The heating surface was initially prepared with a fine slow speed lath e

cut. In final finishing, the heater block assembly was ro ta ted at a

higher speed in the la the chuck and the heating surface was polished

using a group of "Diam ond G rit" papers with numbe rs 200, 400, 600,

respectively. The root mean square surface roughness was measured

in ten different directions by means of a Talysurf profilometer, giving

a 10 ix in. average value of mean square roughness for the hea t ing

surface.

The present investigation required that the subcooling in the bulk

liquid be varied. This cond ition was satisfied by the us e of a single pass

heat exchanger comprised of eight stainless steel tubes, semi-circular

in shape, brazed between two stainless s teel pipes and located ap

prox ima te ly 25 mm above the sk i r t to provide the bulk subcooling.

To condense the vapor which formed at the free surface of wate r , a

single pass vapor condenser comprised of two stainless s teel tubes

arranged in a semi-circular form between two stainless steel pipes was

posit ioned approximately 25 mm below the cover plate . The cooling

water f low rate thro ugh the heat exchanger and vapor condenser was

controlled by the use of needle valves.

An X - Y stage provided with two micrometers capable of reading

±2.54 X 10- 3

mm was fixed to the cover plate on four supports . The

stage provided a controlled and accurate travel of the bubble probe

in two normal directions. The bubble probe depicted in Fig. 1 was

mounted on an ad jus tab le suppor t to control the vertical distance

between the probe tip and the heating surface. This distance was

approx ima te ly 0.2 mm in the lower and in termediate heat f lux cases

and 0.1 mm in the higher heat flux case and was chosen to optimize

the s t reng th of the signal observed on the oscilloscope screen.

Reference [16] reported that e lectr ical probes are widely used for

obtaining information on the f low structure in two-phase gas-liquid

flows. It was claimed that conductance probes, which depend for their

operation on the fact that the electr ical conductivity of a two phase

mixture is strongly dependent on the phase distr ibution, can be used

for f low regime detection, void fraction measurement, bubble and

drop size determination and f i lm thickness measurement. This con

cept was used in the present investigation to de te rmine the locations

of the active sites on the heating surface and also the frequency ofvapor bubble emission.

The conductance probe output voltage gave an indication of the

conduc tance of the wate r or the vapor between the bubb le de tec t ion

probe and the heating surface. Fig. 2(a) which represents the probe

response at some considerable distance from the nucleation site, shows

tha t the re was a bridge of water between the bubble detection probe

an d the heating surface since the signal was inva r ian t du r ing the

sampling period. The conduc tance of the medium wi th in the probe

tip-heating surface gap decreases whenever vapor is genera ted and

thus the output voltage signal decreases intermittently . Fig. 2(b)

shows the probe response at a horizontal distance of approx ima te ly

0.25 mm from the nucleation site where the signal shown in F ig . 2(c)

was obtained. The observation of "strong" f luctuations indicated the

location of the active si te . The se f luctuations also made possible the

computa t ion of frequency of vapor bubble emission.The heat f lux was ob ta ined in the present investigation by two

methods . The f irst metho d considered the rate of heat transfer in the

neck of the heater block to be de te rmined by one-d imens iona l hea t

conduction. The second method obtained the ra te of heat flux by

accounting for the total heat loss from the system to the surroundings,

which was subtracted from the to tal e lectr ic heat inpu t to the system.

The g reatest difference was 2.37 percent, which is most l ikely due to

the a ssumpt ion of one-dimensional heat conduction in the first

me thod . The second metho d values were used to rep resen t the value

of the ra te of hea t trans fe r th roug hou t .

The vessel was filled to a d e p t h of approx ima te ly 150 mm with

deionized disti l led water containing 0.1 gm of sa l t / l i t r e . T hen the

X— y stage was manipu la ted to move the bubb le p robe in different

0 . 0 5 ,

0.00

I I I I

I I 1

- 0 . 050. 0

••••1

••II•iI

PROBE RESPONSE AT

SOME CONSIDERABLE

DISTANCE FROM THE

NUCLEATION SITE

t o )

0 5

ELAPSED TIME (S)

E

0.05

0.00

- 0 . 0 5

iiifliiiaiiiiiii• • • • • • • [ M B" • • • • • • • I B

PROBE RESPONSE AT

0.25 (mm) FROM

THE NUCLEATION SITE

(b )

0.0 0.5

ELAPSED TIME IS)

1.0

<h-

0.05

0.00

-0.05

••PfPPlMMHHiii i iHtni i i i i i t i i• • • • i i i i i i i i i i i i• • • • • • • • • i n" — • • • • I 1 M I

PROBE RESPONSE ATTHE CENTER OF THE

NUCLEATION SITE

( c )

0.0 0. 5 10

ELAPSED TIME (S)

Fig. 2 Influence of probe displacement f rom nucleat ion si te

directions to check that there was no con tac t be tween the probe and

the heatin g surface, after which the copper block heaters and guardhea te rs were tu rned on. A copper block heater sett ing of approxi

ma te ly 475 k W / m2

was established to h e a t up the boiler assembly

quickly. A h e a t up t i m e of approx ima te ly two h o u r s was allowed to

elapse in orde r to achieve steady state conditions and to ensu re tha t

the nucleation sites were properly activated at the beginning of each

test, after which the heat input was reduced to the desired level; two

hours were required for the system to regain steady state whenever

any change in heat flux or subcooling took place. The readings of all

the thermocouples were displayed by m e a n s of a recording potenti

omete r , and, in add i t ion , the thermocouple s ignals were measured

individually by means of a manually balanced potentiometer. All the

thermocouple locations were identif ied in reference [17].

T h e X—Y stage was adjusted to place the bubble detection probe

in the zero reference position. The Y micrometer was fixed at its zero

reference position and the X micrometer was moved very slowly untila significant deflection in the voltage appeared on the oscilloscope

screen. Then X and Y micrometers were moved very small increments

in both directions unti l a strong signal appeared. Th e read ing of X

and y micromete r was recorded on a da ta shee t and the location so

identif ied was deemed to rep resen t the posit ion of the particular ac

tive s i te .

E ach deflection of the oscil loscope beam represented abubble

emi t ted at tha t t ime and consequen t ly the n u m b e r of deflections

observed on the oscilloscope screen over a fixed time interval could

be interpreted as the bubble emission frequency. Ten readings of the

signal from each active site were taken and averaged to ob ta in the

bubble emission frequency at tha t s i te . Afterward, the Y micrometer

was returned to its original setting and theXmicromete r was moved





once again. The same procedure described above was continued

through a 25.4 mm scan. Afterward the Y micromete r was moved

0.635 mm and the X micrometer was moved through another 25.4 mm

scan. Thes e steps were repeate d thr ough a 25.4 mm scan in the Y d i

rection. At the conclusion of the scanning procedure, an area 25.4 mm

X 25.4 mm s quare ha d been examined by the bu bble detection p robe

and the locations of a ll the active s i tes detected had been recorded.

At the end of the test , a l l the thermocouple s ignals were measured

once again using the manually balanced p otentiom eter. The amb ient

temperature was measured using a mercury in glass thermometer and

a reading of the barometric pressure was obtained in order to computethe sa tu ra t ion tem pera tu re o f the wa te r co r respond ing to the a tm o

spheric pressure.

An error analysis associated with the experimental work is pre

sented in reference [17].

Results and D iscussionThe characteristic boiling curve represents the relationship between

the wall superheat (T,„ - T„ ) and the he at f lux Q/A. Prev ious inves

tigations of the effect of surface condition on t he surface sup erhe at

have shown that i t is not possible to predict whether roughening a

surface will produce sm aller or larger values of superhe at for the same

rate of heat flux as discussed previously. Not only might the position

of the boiling curve change with the change of surface roughness, but

the slope of the boil ing curve might change as well . Conse quently , i t

is imp ortan t to know the heat f lux/superhea t re latio nship for each

surface/fluid comb ination. F ig. 3 shows the characteristic boiling curve

for the present s tudy . The experim ental dat a obtained by W iebe [15]

and Gaertner [18] were plotted on the same graph to compare with

the results of the presen t work. Subcooling h as a s l ight effect on the

superheat a t constant heat f lux as i l lustrated in Fig . 4 . Increasing

subcooling firs t causes the superheat to increase s l ightly and then

causes the superheat to decrease. The value of subcooling corre

sponding to the maximum value of superheat increases with increasing

heat flux. This type of behavior is in quali ta tive agre emen t with the

observations of Wiebe [15] and Ju dd [19] .

Fig . 5 prese nts the di str ibuti on of the active nucleation site popu

lation over the heating surface for test S I . The d istr ibution of active

sites within a 25 square array and a 100 square array can be seen at

a glance. F ig . 6 presents th e distr ibution of bubble f lux density within

the same 100 squar e array. In accordance w ith the practice followed

in the present s tudy, the bubbles forming at an active s i te on a grid

line were considered to originate within the r ight hand sector. For

instance, one active site lies on the line X = 0.8 between Y = 0.8 and

Y = 0.9; the bubbles originating from this active site were considered

to origin ate in the secto r 0.8 X 0.8.

The active site density observed in the present investigation for the

three levels of heat flux were plotted on the same graph with the data

which had been observed by Ku rihar a and Myers [3 ] and Westw aterand G aertn er [9] as shown in Fig . 7 for the sake of compa rison. T he

present results indicate that the active site density observed increases

with the increase of heat flux in the same manner as that observed by

Kur iha ra and Myers and Wes twa te r and Gaer tne r . T he p resen t re

sults indicate a greater value of active site density than the compa rable

A--A

DARKENED SYMBOLS REPRESENT WIEBE DATA

~A~A-

Ll ^- = 64.0 ( kw/m")

O '5 .= 166.0 IkW/m2)

A 5. ; 333.1 (kW/n?)

5 10 15 20 25 30 35 40 45 50 55

S U BCO O L I N G ( T s -T = ) (°C)

Fig. 4 Variat ion of surface superheat with bulk subcool ing

TOO

6 0 0

- 100

a90

O PRESENT WORK

A GAERTNER DATA

• WfEBE DATA

40 50 60

SUPERHEAT (T _T ) (c )

Fig. 3 Character ist ic boi l ing curve for present invest igat ion

O 0

m

@©

%

• •9 •

*

@

©

I

@

•

m

9

©

0

@

©

•

5©

0

®©

S

•

9

«

®

<@

®

©

8

9

©

9

@9

@©

@

@

©

i

@

a

1 • | * I. \ * i '• :

3 #

e*

; @

• @ t1 «

i#

I -@

•9

0 • @

@

®

• -

©

: » ,

i *

®

a

®

®

@

9

9

»

®

0.2 0.4 0.6 0.8

DIMENSIONLESS X DISPLACEMENT

(T - T ) • 1.7 I t )3 ra

Fig. 5 Distribution of active nuclea tion sites





BUBBLES PER SECOND PER SQUARE

0-8

0 .6

0.0

27.7

9. 6

9. 9

9. 9

4 6 . 2

3 9 . 9

47 .4

75.9

3 7 . 8

6!

13.8

18.5

23 .7

ii. e

„.,

2 8 . 3

21.1

2 0

47.5

19.9

20.7

17.5

3 3 . 7

16

21.8

8.6

12.1

0

14.8

19 .2

56 .4

20

3 2 . 6

4 0 3

3 3 . 7

6 .8

11.5

31.7

3 6 . 7

9.2

0

29 .3

12.3

24 .6

4 5 . 8

23 . 3

0

31.5

18.2

21.2

47 .8

29.1

2 8 . 2

0

5. 5

6. 7

10.2

10

9. 2

2 8 . 6

17.2

9. 9

9. 9

2 6 . 6

0

25.2

20 .2

47 5

3 0 . !

21.4

2 0 3

0

9,4

51.1

12.1

3 3 . 5

4. 4

0

!1.6

3 i . 4

\31.6

27 8

0

16.8

8,1

15 .8

23.2

20 .7

14.1

10.9

14.9

12.8

2 3.1

18 .3

28 .5

11.2

ll.i

2 8 . 9

10.4

2 6 . 2

0 .2 0 .4 0 .6

DiMENSIONLESS X DISPLACEMENT

• i = 1 6 6 . 0 ( k W / n f )A

I T, -

T» I = 1-7 (C )

Fig. 6 Distr ibut ion of bub ble flux density

A PRESENT STUDY

O GAERTNER

D KURIHARA

I

h

H E A T F L UX Q / A ( k W / m )

Fig. 7 Active site density versus heat flux

results for the thre e different levels of hea t f lux which may be due to

a difference in the heating surface microroughness. The results of the

present investigation in Fig. 8 indicate that subcooling has much less

influence on active s i te density than heat f lux. The re are no compa

rable results known for the water/copper combination which would

enable the observed variatio n in the active s i te density to be corrob-

- O — "

D J - 6 4 . 0 [ | W „ ' |

O ^ "166,0 (kW/m)

A x •3'13

-1

(KW/m I

0 5 10 15 20 25 30 35 40 '

SUBCOOLING (T 8_T„) [°C )

Fig. 8 Variation of active site density with bulk subcooling

orated, but some similar work has been done by Judd [19] with the

F reon 113 glass/surface combination which has the same tren d as the

present investigation.

Any study of the boiling heat transfe r proce ss occurring on a heating

surface should involve the spa tia l d is tr ibution of the local active s ite

population in order to properly account for the mechanisms of boiling

heat transfer . M any natura l processes involving spatia l d is tr ibutions

are described by the Poisson equation

P(X)>-UJJX

X\(1)

where

P(X) = probabili ty of event X occurring

U = expected value of event X

Only one study exists which treats th e distr ibution of active s i tes

on the heating surface. Gaertner [20] determined that for three dif

ferent levels of heat flux, the spatial distribution of the local active

site population followed the Poisson distr ibution. The implication

of the agreement found between the data and the Poisson distribution

is that the active s i tes were distr ibuted randomly on the heating

surface.

The spatia l d is tr ibution of the active s i tes on the heating surface

has been studied in the present investigation at three different heat

flux levels and various levels of subcooling as well. The heating surface

was divided into a number of small squares of area "a" so that the

probabili ty of the local population within a square having the value

"Na" could be as calculated according to

-m

(Na)Na

P(Na) = - Z N a ••

A (Na)l

(2)

where

P(Na) = probabili ty of f inding "Na" active sites in area "a"

Na ~ actual number of active s i tes in area "a "

Na = expected number of active s i tes in area "a"

A = heating surface areaa = local surface a rea

ZNa = number of small areas having a local population Na

The probabili ty distr ibutions obtained showed that agreem ent with

the Poisson distr ibution was better when the heating surface was

divided into a 25 square array than when it was divided into 100





square array although satisfactory agreement was found in both cases.

Consequently, the probability of finding the local population "Na"

in area "a" from both theoretical and experimental considerations

was calculated by equation [2] considering a/A = 1/25. The proba

bility distr ibut ion in Fig. 9 shows a good fit with the Poisson dist ri

bution and this conclusion is confirmed by the results ofax2 test. The

findings of the present investigation are consistent with those of

Gaertner's investigation so far as the randomness of the active nu

cleation site distribu tion is concerned.

Fig. 10 shows the effect of sub cooling on the surface average fre

quency of vapor bubble emission for three different levels of heat flux.The results indicate that the frequency first increases with increase

of subcooling and then decreases again with further increase of sub

cooling. The degree of subcooling corresponding to the maximumbubble frequency increases with increasing heat flux levels. The

present investigation considers the distribution of bubble flux density

within a 100 square array representing the entire heating surface. Fig.

11 shows the percentage cumulative bubble flux density versus the

percentage cumulative area of the heating surface. I t is seen that the

effect of sub cooling is negligible but that the effect of heat flux is very

significant. As the heat flux is increased, the number of active sites

increases and the bubble frequency increases as welL Consequently,

there is a tendency for the bubble flux density distribution to become

uniform at high heat flux, in which case ten percent of the area would

contribute ten percent of the bubble flux density, for instance.

0.25

-- POISSON DISTRIBUTION

0.20 No = 6.84 9.. =166.0 Ikll/m2

)A

(Ts-T.. ):: 1.7 Ie )

0.15

..z0: r--

0.10

, - - -

0.05

0.00 I I lilli, I

a , 4 9 10 12 13 14

No

TEST 8 I

Fig. 9 Local active site distribution

24 r--------,---------,---------,---------,---------,

0 *,.64.0 (kWm2 )

0 %=166,0 I kW/..i' I

6. ..2.:: 333,1A

IkW/..i' I

10 20 30 40 50

Fig. 10 Variation of average bubble frequency with bulk subcooling


Finally, the influence of heat flux and subcooling on the pattern

of the active nucleation sites was studied in the present investigation.

An area 5.0 mm X 5.0 mm square was selected from the 25.4 mm X

25.4 mm square area to examine the change in the pattern of the active

nucleation sites with changing heat flux and sub cooling during a single

experimental test. As heat flux was increased at constant subcooling,

bubbles continued to originate from the nucleation sites already ac

tivated and some nucleation sites appeared due to the increase of heat

flux. As the subcooling was increased at constant heat flux, the active

nucleation site pat tern hardly changed at all, except for one additional

active nucleation site which appeared, perhaps as the result of theslight increase of superheat with increasing sub cooling indicated in

Fig. 4.

Conclusions

1 The results of the present investigation for water boiling on a

single copper surface at atmospheric pressure indicated that the active

nucleation sites were randomly located and that their spatial distri

bution followed the Poisson distribut ion in agreement with Gaertner'sfindings.

2 Bubble flux density was observed to be non-uniformly dis

tributed even though it was thought that the heat flux was uniform

over the heating surface so that one might have expected the bubble

nux to be uniformly distribu ted as welL

3 Changes in heat flux and sub cooling did not affect the pat tern

of active nucleation sites. Although increasing heat nux caused moreactive sites to be activated, the addit ional active sites formed among

the sites which have been already activated. Subcooling appeared to

have hardly any effect at alL

References

1 Jakob, M., Heat Transfer, VoL 1, Wiley, New York, (1949).2 Carty, C. and Foust, A. S., "Surface Variables in Nucleate Boiling,"

Chemical Engineering Progress Symposium Series, VoL 51, No. 17, (1955).3 Kurihara, H. M., and Myers, J. E.,"The Effects of Superheat and Surface

Roughness on Boiling Coefficients," AIChE Journal, VoL 6, No.1, (1960).4 Hsu, S. T., and Schmidt , F. W.,"Measured Variations in Local Surface

Temperatures in Pool Boiling of Water, " ASME JOURNAL OF HEAT TRANSFER, (1961).

100

80

60

40

x

20

!'l

!li;lw 10

"! 8

15Iiw>;:q

:3

a 2

*:: 64,0 (kW/m2

)

..9.. = 166.0 (kW/m2 1

A

9..= 333.1 lkWlrl )A

8 10

8.3 IS.7

20

CUMULATIV E PERCENTAGE AREA

28.3

40 60 80 100

Fig. 11 Cumulative percentage bubble flux density versus cumulative per-centage area





5 Berenson, P . J . , "E xper i men ts on Pool Boi ling Hea t Transfe r ," Inter

national Journal of Heat and Mass Transfer, Vol. 5, (1962).

6 Vachon, R . I . , Tanger , G. E„ Davis , D. L„ and Nix, G. H„ "Po ol Boi l ingon Pol ished and Chemica lly E tched S ta inless-S tee l Sur face ," ASME JOURNALO F H E A T T RA N S F E R , Vol. 90, No. 2, (1968).

7 Sh oukr i , M. , and Judd, R . L. , "Nuclea t ion S i te Act ivat ion in S a tura tedBoil ing," ASM E JOURNAL OF HEAT TRANSF ER, Vol . 97, No. 1 , (1975) .

8 Nishikaw a, K., Transfer Society Mechanical Engineers, Japan , No. 20,(1954).

9 Westw ater and Gaertner, "Pop ulation of Active S ites in Nucleate Boiling

Heat Transfe r ," Chemical Engineering Progress Symposium, Ser ies 30, No.56 , (1960).

10 Heled , Y., and Orell, A., "Ch arac teris tic of Active Nucleatio n Sites in

Pool Boi l ing," Journal Heat and Mass Transfer, Vol. 10, pp . 553 - 554 ,(1967).

11 Yoshihiro Iida , and Kiyosi Kobayasi , "An Exp er imen ta l Invest iga t ionof the Mechanism of Pool Boi l ing Phenomena by a P r obe M e thod , " F ou r th

Inte rna t iona l Hea t Transfe r Conference , Pa r is—Versa i l les , (1970) .12 Nail, J. P. , Vachon, R . A. , and M orehouse , J ., "An SE M S tudy of Nu

c lea te S i tes in Pool Boi l ing f rom 304 S ta inless-S tee l , " ASME JOURNAL OFH E A T T R AN S F E R , Vol. 96, No. 2, (1974).

13 Vouts inos , C . M. , "Laser Inte r fe rometr ic Invest iga t ion of Microlayer

E vapora t ion for Var ious Leve ls of Subcool ing and Hea t F lux," M. Eng. thesis ,M e c ha n ic a l E ng ine e r ing De pa r tm e n t , M c M a s te r Un ive r s i t y , C a na da ,

(1976) .

14 Jud d, R . L. , and Hwang, K. S . , "A Comprehensive M odel for Nuc lea tePool Boi l ing Hea t T ransfe r ," A S ME JOURNAL OF HEAT TRANSFE R, Vol . 98,

No. 1, (1976).

15 Wiebe , J. R. , "Tempera ture Prof i les in Subcooled Nuclea te Boi l ing,"

M. E ng. thesis , Mechanica l Engineer ing Dep ar tme nt , McMas te r Univers i ty ,Canada , (1970) .

16 AS M E He a t T ra ns f e r D iv i sion , " Two P ha se F low I n s t r um e n ta t i on , "

Aug. (1969).

17 S u l t a n , M., " S pa t i a l D i s t r i bu t ion of Active S i tes and Bubble F lux

Densi ty ," M. Eng. thesis , McMaste r Univers i ty , Hamil ton, Canada , 1977.18 Gaer tner , R. F . , " P ho tog r a ph ic S tudy of Nuclea te Pool Boi l ing on a

Horizonta l S ur face ," AS ME JOURNAL OF HEAT TRANSFE R, No. 1 , (1965) .

19 Ju dd , R. L., "Influence of Acceleration on Su bcooled Nuc leate Boiling,"Ph .D. thesis , Univers i ty of Michigan, Mich. , 1968.

20 Gaer tner , R. F . , " D i s t r i bu tion of Active S i tes in Nucleate Boiling ofLiquids ," Chemica l E ngineer ing P rogress Symp osium S ervice, Vol. 59, No. 41,

(1963) .





S. V. Patankar

S. Ramadhyan i

E. M. Sparrow


University of Minnesota,

Minneapolis, Minn.

Effect of Circumferentially

Nonuniform Heating on Laminar

Combined Convection in a

Horizontal TubeAn analytical study has been made of how the circumferential distribution of the wall

heat flux affects the forced/natural convection flow and heat transfer in a horizontal tube.

Two heating cond itions were investigated, one in which the tube was uniformly heated

over the top half and insulated over the bottom, and the other in which the heated and in

sulated portions were reversed. The results were obtained numerically for a wide range

of the governing buoyancy parameter and for Prandtl numbers of 0.7 and 5. It was found

that bottom h eating gives rise to a vigorous buoyancy-induced secondary flow, with the

result that the average Nusselt numbers are much higher than those for pure forced con

vection, while the local Nusselt numbers are nearly circumferentially uniform. A less vig

orous secondary flow is induced in the case of top heating because of temperature stratifi

cation, with average Nusselt numbers that are substantially lower than those for bottom

heating and with large circumferential variations of the local Nusselt number. The fric

tion factor is also increased by the secondary flow, but much less than the average heat

transfer coefficient. It was also demonstrated that the buoyancy effects are governed sole

ly by a modified Grashof num ber, without regard for the Reynolds number of the forced

convection flow.

I ntr oduc t ion

I t is well es tablishe d th at the he at transfer characteris t ics of hori

zontal and inclined laminar forced convection pipe flows can be sig

nificantly affected by secondary fluid motions induced by buoyancy.

In the extensive literature dealing with this subject (to be summarized

short ly) , pr imary a t tention has been focused on problems charac

ter ized by uniform addit ion of heat per unit axia l length . Another

characteristic common to the entire available body of literature is that

only c ircumferentia l ly uniform ther mal bounda ry condit ions have

been considered. These have included c ircumferentia l ly uniform

(albeit axia l ly variable) wall tem pera ture and c ircum ferentia l ly uni

form wall heat flux.

In practice , there are applications where thermal c ircumferentia l

uniformity may not be a t ta ined. For instance, the heat f lux incident

on the absorber tube of a line-focus solar collector is concentrated on

1 Numbers in brackets designate References at end of paper.Contributed by the Heat Transfer Division for publication in the JOURNA L

OF HEAT TR ANSFER . M anuscript received by the Heat Transfer DivisionApril 22,197 7.

only a portion of the tube circumference. Similarly, the fluid-carryingtubes of a flat plate solar collector receive a circumferentially non

uniform heat f lux. The bu oyancy forces are sensit ive to the c ircum

ferentia l posit ioning of the port ions of the tube wall that are more

strongly and less s trongly heated. Therefo re , i t can be expected tha t

the induced secondary flows will be significantly affected by the im

posed c ircumferentia l nonuniformities , as wil l the heat transfer pa

rame te r s ( e .g ., Nusse l t numb ers ) . Recen t expe r imen ts [ l ] 1 have

demons t ra ted tha t even a low-Reyno lds -number tu rbu len t f low in

a horizontal p ipe is sensit ive to nonuniformities in c ircumferentia l

heating, thereby heightening the expecta t ion of their importance in

a laminar f low.

The foregoing considerations have served to motivate the present

investig ation. A stu dy is ma de of fully developed, buoyancy-affected

laminar f low in a horizontal tube subjected to two l imit ing heating

condit ion s. In one case , the tu be is heated uniformly over a 180 degarc encompassing the upper half of its circumference, while the lower

port ion of the tube is adiabatic . For the second case , uniform heating

is applied over the lower half of the tube and th e upper part is adia

batic . In both cases, the heat addit ion per unit length is uniform.

The highly complex f low fie lds induced by buoyancy preclude an

analytical solution of the problem . Since solutions will be sought for

Journal of Hea i Transfer FEBRUARY 1978, VOL 100 / 63





condit ions w here the natura l convection is a f irs t-order effect ra th er

than a perturbation of the forced convection, numerical techniques

are required. The difference equations were derived and solved by

adap ting the formulatio n of Pa tan kar an d Spalding [2] , which in

corporates Spalding's scheme [3] for dealing with the issues of central

versus upwind differences. Solutions were obtained over a wide range

of values of the governing buoyancy p aram eter , such th at very large

natural convection effects were encountered a t the h igh end of the

range . The P rand t l number , wh ich appea rs a s a pa rame te r in the

equations, was assigned values of 0.7 and 5 (e.g., air and water). The

results obta ined from these solutions include average Nusselt numbers , fr ic t ion factors , c ircumferentia l Nusselt number d is tr ibutions,

and isotherm and s treamline maps. The presentat ion of the results

wil l be mad e with a v iew to i l luminatin g the differences betw een th e

top heated and bottom heated cases.

Fro m a careful search of the l i tera ture , the author s were unable to

find any prior work on laminar mixed convection ( i .e . , combined

forced and natural convection) in a tube with c ircumferentia l ly

nonuniform therm al bound ary condit ions. However, as noted earl ier ,

the c ircumferentia l ly uniform case has been treated extensively .

Analyses have been performed via f in i te d ifferences [4-6] , as per

turbations of the forced convection solution [7-11], and via a boundary

layer model a t h igh values of the buoyancy param eter [12,13]. These

references encom pass stud ies of inclined tubes as well as of horizontal

tubes. Numerous experiments designed to model c ircumferentia l ly

uniform boundary condit ions have a lso been reported [14-24].

I t is re levant to comment on a recurr ing viewpoint expressed in

many of the aforementioned analytical s tudies of fu l ly developed

laminar mixed convection in a horizontal p ipe . In those papers , the

results were param eterized by the group ReRa, where Re is the con

ventional p ipe Reynolds number and Ra is Rayleigh number based

on the axia l t empera tu re g rad ien t dT/dz, that is

then i t is easi ly shown t ha t

R e = wD/v, R a = gP(dT/dz)R4/cti> (1 )

Since the departures from pure forced convection were found to in

crease with increasing ReRa, i t appeared th at the re la t ive imp ortance

of buoyancy was magnif ied a t larger values of Re— a finding th at is

contrary to in tuit ion.

To clarify th is matter , i t may f irs t be noted t ha t dT/dz is not a tru e

measure of the cross sectional tem pera ture d ifferences th at g ive r ise

to the buoyancy effects . This is because dT/dz is not only re la ted tothe heating r ate, but also to the mean velocity of the forced convection

flow. Rather , the ra te of heat transfer per unit length Q', which is a

realis t ical ly prescribable thermal input parameter , is a true measure

of the cross sectional temp erat ure d ifferences. If a modif ied Grashof

n u m b e r G r + based on Q' is defined as

ReRa = (2Ar)Gr+(3)

Since buoyancy-induced departures from pure forced convection have

been found to depend on ReRa, i t is now seen that a buoyancy pa

r a m e t e r G r + is a measure of these effects. It is especially interesting

to note tha t i t is the ma gnitude of Gr + alone, witho ut reference to any

forced convection parameters, which governs the heat transfer results.

From this , i t fo l lows that the Nusselt numbers for fu l ly developed

lamina r mixed convection in a horizontal p ipe f low are in depen dent

of the Reynolds n umb er, as is also the case for fully developed laminarforced convection in tubes.

A n a l y s i s an d S o l u t i o n s

Th e star t in g point of the analysis is the e quat ions expressing con

servation of mass, momentum, and energy, all expressed in cylindrical

coordinates. The first task in the specialization of these equations to

fully developed, laminar mixed convection in a horizontal tube is the

formulation of the buoyancy terms.

If attention is first focused on the radial momentum equation, then

the corresponding buoyancy term can be deduced from pressure and

body forces that appear therein ,

-dp/dr - pg si n I (4)

where the r , 0 coordinates are i l lustra ted in the inset s of F ig . 1 . The

density appearing in (4) can be re la ted to the temperature v ia theBoussinesq model, according to which density varia t ions are con

sidered only insofar as they contribute to buoyancy, but are otherwise

neglected. If Tb and pb are the bulk temperature and the corre

sponding bulk density a t any axia l s ta t ion in the fu lly developed re

gime, then, according to the Boussinesq approach

:Pb ~ PbP(T - Tb) (5)

The bulk state is a logical reference state in problems where the heat

input is prescribed, as in the present problem. Furthermore, i f

• p + pbgr sin (

then (4) becomes

-dp*/dr + PbgP{T - Tb) si n 8

(6)

(7)

Th e last term in (7) is the buoyancy force . A similar manip ulat ion

of the pressure and body forces in the tangential momentum equation

yields

(-dp*iae)lr + Pbg(3(T - Tb) cos ( (8)

G r + = g / 3 Q ' f l 3 A i ' 2 (2)To further a dapt the conservation equations, the constant property

assumption is in troduced and th e subscrip t b is deleted from pb in (7)

- N o m e n c l a t u r e -

cp = specif ic heat a t constant pressure

D = tube d iame te r/ = friction factor, equation (21)

G r + = modified Grashof number,

gf}Q'R3/k,'2

g = accelerat ion of gravity

h = local hea t tran sfer coefficient,

_ ql(Tw - Tb)

h = average heat transfer coeffic ient ,

q/(Tu, - Tb)

k = the rma l conduc t iv i ty

Nu = local Nusselt number, hD/k

Nu = average Nusselt number, hD/k

Nuo = forced convection value of Nu

P = dimensionless pressure , equation (10b)

p = pressure

p * = reduced pressure , equation (6)Q' = ra te of heat transfer per unit length

q = ra te of heat transfer per unit area

R = tube rad iu s

Re = Reyno lds number , WD/v

r = radia l coordinate


Tw = loca l wa l l t empe ra tu re

Tw = average wall temperature , equation

(18)

U, V. W = dimensionless velocit ies, equation

(10a)

u,v,w = velocity components in r , 6, z

W = mean value of W

w = mean axia l velocity

a = thermal d iffusivityj3 = the rma l expansion coeff ic ient

•q = dimensionless radia l coordinate , r/R

8 = angular coordinate

p = viscosity

v = kinematic v iscosity

p = dens i ty* = d imens ion le ss t empera tu re , (T - T&)/

(Q'/k)

4/ = stream function

S u b s c r i p t

b = bu lk p rope r ty





and (8) . Th e fully developed condit ion is imp lem ente d by set t ing

du/dz = du/dz = dw/dz = 0, dp/dz = dp/dz (9a)

dT/dz = dTb/dz = Q'/PCPTTR2W (9b)

in which Q' is the rate of heat transfer to the fluid per un it axial length.

In the p re sen t p rob lem, Q' is a constant . Viscous dissipation and

compression work are dele ted from th e energy equation, as is app ro

pria te for typical laminar f lows.

As a final s tep , d imensionless variables are in t roduc ed as fo llows

U =v/R'

V< vv/R'

W-

r• • - , <t>

R

T-Tb

(-dp/dzWVn)

P*j(v/R)2

(10a)

(10b)(Q'/k)

With these inputs , the equations expressing conservation of mass,

of r, 8, and z momentum, and of energy become

dU VdUy — + - — -

d(riU)/dri + dV/dd = 0

+ V2U

(11)

U 2 dV „— + Gr+$ sm<? (12)

r,2 7,2 30

dV vev uvu— + — + — =dr] r\ 3d r,

1 3 P

r, 30

+ V2V +

2 du V- — + G r + $ cos 0 (13)

aw vawu— +

di\ r, dO

i)2 dtf if

= 1 + V2W

na* va* <ywv) ldr i i) dd tcPr P r

(14)

(15)

where V 2 is the Laplace ope rator in polar coordinates . The param eter

G r + has a lready been defined in equation (2) and W is the cross sec

tional average value of W . P r i s the P rand t l number . The equa t ions

thus con ta in two independen t pa rame te r s , Gr + and P r .

As was noted earl ier , consideration wil l be g iven to two heating

condit ions. These are i l lustra ted in the insets of P ig . 1 . In the top

heating case , there is a uniform h eat f lux q on the upper half of the

tub e wall, while the lower half is insulated. In the bottom heating case,

the uniform hea t f lux is applied a long the lower half of the tub e and

the upper half is insulated. These boundary conditions are symmetric

abou t the vert ical d iame ter of the tube . Owing to th is symmetry , the

solution domain can be confined to half of the tube cross section and,

for concreteness, the region —7r/2 < 6 < TT/2 is selected.

Wh en the heat f lux boundary condit ions are expressed in terms of

the variables of the analysis , there results

3 * 1 7T 3 * „ X ,

— = - for 0 < 0 < - , — = O f o r - - < 0 < O (16 a)dr, rr 2 dr , 2

for top heating, and

3 < J > 7T— = 0 for 0 < 0 < - ,dr, 2'

3 * 1 x— = - f o r - - < 0 < Odr, it 2

(16b)

Fig. 1 Circumferential average Nusselt numbers

for bottom h eating. Th e factor 1 /ir appearing in equations (16a) and

(16b) s tem s from th e fact tha t Q' = irRq. The other boundary condi

tions on the tube wall and those on the symmetry line are conventional

and need not be s ta ted explic i t ly here .

In addit ion to sa t isfying the governing equations and boundary

condit ions, the solution mus t be compatib le with the defin ition of the

bulk temperature , the d imensionless form of which is

J JiWvdvdd = 0 (17)

where the in tegration is extended over the solution domain.

The differentia l equations (11) through (15) and their boundarycondit io ns comp rise a h ighly coupled system involving the f ive vari

ables U, V, W, P, and <fr. The numerical scheme empolyed to solve

these equat ions is an ad apta t ion of that of [2] , which is a f in i te dif

ference procedur e for three-d imen sional parabolic f lows. I t involves

marching in the streamwise direction while solving a two-dimensional

elliptic problem at each m archin g station. Th e removal of all reference

to the m arching d irection gives rise to a procedure for tw o-dimensional

elliptic situations, and this was one of the adaptive actions taken here.

A second ada ptat i on involved the reformulation of the d ifference

equations, which had been derived in [2] for rectangular coordinates,

to accommoda te r, d polar coordinates .

In recognit ion of the fact tha t the conventional centra l d ifference

approximation can yie ld unrealis t ic results a t h igh f low rates , the

discret izat ion performed in [2] was based on the so-called Spalding

difference scheme [3] (SDS). The SDS makes use of centra l d ifferences for grid Pecle t numbers (based on the local velocity and grid

dimension) in the range - 2 < P e < 2 . For Pecle t numbers outside th is

range, the d iffusion term s are dropped and th e convection terms are

evalua ted by the upwind difference schem e. In actuali ty , the SDS is

a computationally inexpensive approximation of a smooth Pecle t-

number-w eighted superposit ion of the centra l d ifference and upwind

difference representat ions [3]. Th e true weighting function involves

computationally expensive exponentia ls . In order to a t ta in greater

accuracy in the pre sent solutions , the SDS was replaced by a power-

law representat ion which c losely approximates the true weighting

function. The increment in computational t ime associa ted with the

use of the power-law representat ion is neglig ible .

On the basis of computational experiments , i t was found that so

lu tions of h igh accuracy would be obtain ed w ith a 28 X 28 grid . For

most of the cases, the grid points were dis tr ibu ted uniformlythroughout the solution domain. At h igh values of Gr + , whe re more

complex f low patterns are encountered, and where for top heating

there are sharp varia t ions of wall temperatu re , supplemental solutions

were obtained using a nonuniform grid . The se solutions were essen

t ia l ly the same as those for the uniform grid .

Th e solution method was i tera tive in nature , and the com putational

t ime required to obtain convergence was found to increase marke dly

wi th Gr + . In th is connection, i t was advantageous to proceed from

lower to higher G r + and to extrapolate the converged solutions for two

values of Gr + to obta in s tar t ing values for the next Gr + .

For each of the two thermal boundary condit ions investigated,

solutions were obtained for Gr + ranging from 10 to approxim ately 10 7.

Th e Pr an dtl num ber wa s assigned values of 0 .7 and 5 , which, re-

Journal of H eat Transfer FEBRUARY 1978, VOL 100 / 65Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



spectively, typify fluids such as air and water.


In the presentation that follows, initial consideration will be given

to those quan ti t ies th at are of mos t d irect re levance to applic ations —

the c ircumferentia l average Nusselt number and the fr ic t ion factor .

The c ircumferentia l d is tr ibutions of the local Nusselt number wil l

then be presented. F inally , information about the f low and temper

ature f ie lds wil l be provided via s treamline and isotherm maps.

C i r c u m f e r e n t i a l A v e r a g e N u s s e l t N u m b e r s . S i n c e o n ly a

portio n of the tub e wall is hea ted, an average heat tran sfer coefficient

will be evaluated such tha t i t perta ins specif ically to th at por t ion of

the wall . An average wall temperature Tm may be defined by in te

grating over the heated arc

(1M f *Jo

TwdO, (lAr) Ctj TV

TwdB (18)

respectively for top heating and bottom heating. The average heat

flux on the heated arc is q (= constant) . With these , the average

coeff ic ient and the corresponding Nusselt number may be defined

h = q/(Tw - Tb), N u = hD/k (19)

It is relevant to com pare the values of Nu for mixed convection w ith

that for a pure forced convection f low having the same thermal

boundary condit ion ( i .e . , part ia l ly heated wall) . For the la t ter case ,

an analytical solution in the form of a series can be obtained, from

which a heat transfer coefficient and Nusselt number can be evaluated

in accordance with equation (19) (e .g . , [25]) . I f th at Nus selt num ber

is denoted by Nuo, then

Nun = 3.058 (20)

independen t o f bo th the Reyno lds and P rand t l num bers , a s is cu s

toma ry for fu lly developed, forced convection lam inar p ipe f lows.

The results for Nu/Nuo are p lotted as a function of the modif ied

Grasho f number Gr + in F ig . 1. Th e curves in the up per par t of the

figure are for top heating and are referred to the right-hand ordinate.

The long-dash curves in the lower part of the f igure correspond to

bottom heating and are keyed to the lef t-hand ordinate .

The figure shows that the Nusselt number increases with increasing

values of Gr + in accordance with the expected s trengthening of the

buoyancy-induced secondary f low. The ex tent of the Nu sselt nu mb er

increase is markedly different for the two heating arrangements as

witnessed by the distinctly different ranges of the rig ht- and left-hand

ordinate scales. In the top heated case, the values of Nu/Nuo are well

below 1.5 over the entire range inves tigated. On th e other h and, for

bo t tom hea t ing , the Nu /N u 0 curves soar to values as high as 6 and 12,

respectively, for Pr = 0.7 and 5.

This remarkable d ifference is explained by the fact that the top

heated case is a relatively stable configuration in which the secondary

flows are weak. Conduction is s t i l l the dominant mechanism of heat

transfer and, hence, the Nusselt number values do not d iffer appre

ciably from the buoyancy-free prediction. In the bottom heated case,

the configuration is much less s table . The secondary f low velocit ies

are an order of magnitude higher, and natural convection is the

dominan t hea t t r ansfe r mechan ism.It is a lso seen from the f igure that the Nusselt number is quite

sens it ive to the P ra nd t l num ber , inc reas ing a s the P r and t l numb er

increases. Th is trend is in accord with tha t for natu ral convection flows

in general . I t is a lso in terest ing to note that t he imp act of the P rand tl

number is somewhat d ifferent for the two heating arrangements . In

part icular , i f the bottom heated results were to be replotted with

G r + Pr as abscissa variable, the curves for the two Pr values would lie

quite c lose to each other except in the region where they tend to be

wavy. On the other hand, a similar replotting of the results for the top

heated case s t i l l leaves a substantia l gap between the curves.

The aforementioned waviness of the curves reflects a change in the

struc ture of the secondary f low. At low values of Gr + , the secondary

flow consists of a single eddy in each vertical half of the tub e. At higher

G r + , th is s imple s tructure breaks down into a more complex pattern

consisting of multiple eddies. The levelling off of the curves at the

onset of the waviness corresponds to the end of the single eddy mode,

and th e subse quent form ation of the mu ltiple eddies leads to a renewal

of the increasing trend of the Nusselt number.

I t appears that there are threshold values of Gr + that have to be

exceeded before the buoyancy effects cause a detectable change in

the Nusselt number re la t ive to that for pure forced convection. For

Pr = 5 , the threshold v alue of (2 /7r)Gr+ is about 100 for both bottom

heating and top h eating. On the other ha nd, for Pr = 0 .7 , the respec

t ive threshold values for bottom heating and top heating are about700 and 10,000. Th us, for bottom heating, the two thresh old values

can be represented as (2Ar)Gr+Pr a 500. The threshold values for top

heating do not appear to be corre la ted by a Grashof-Prandtl prod

uct .

In addit io n to the resu lts of the present analysis , F ig . 1 contains a

pair of short- dashed l ines taken from [5] . Thes e results correspond

to mixed convection in a tube with c ircumferentia l ly uniform wall

temperature and uniform heat addition per unit axial length (the Nuo

value for these conditions is 4.36). It is seen that these Nu/Nun curves

fal l below the present curves for the bottom heated case , and i t can

be verified th at th ey fall above the curves for the top heated case. This

quali ta t ive re la t ionship appears to be physically reasonable .

F r ic t io n F a c to r . The buoyancy - induced seconda ry f low g ives

rise to an increase of the axial pressure gradient and its dimensionless

coun terpa rt , the fr ic tion facto r/ . Wh en the defin ition of the fr ict ionfactor

/ =(-dpldz)D

(21)'kpw

2

is rephrased in terms of the variables of the analysis , there fo llows

/Re = 8/IV (22)

where Re is the conventional pipe Reynolds number given by equation

(1). A direct assessm ent of the effects of buoyancy can be obtained

f rom the va lues o f the ra t io / Re / ( /R e ) 0 , in which ( /Re) 0 is the forced

convection value (equal to 64) . This ra t io is p lotted in Fig . 2 as a

function of Gr + , both for top heating and for bottom heating.

I t m ay be seen from the f igure that the increase of /R e due to

buoyancy is much greater for bottom heating than for top heating,

as was a lso true for the Nu sselt num ber. In general , the increases in/R e are substantia l ly smaller tha n the corresponding increases in Nu.

It is also interesting to note that whereas greater increases in Nu were

susta ined by f lu ids of h igher Prandtl number, the opposite trend is

in evidence for / R e . The diminished influence of buoyancy on /Re for

2.5

2.25

2.0

1.75

5°L 5

1.25

1.0

Pr=0 .7 /

//

//

Top Heoting /Bottom Heating /

//

5 ,

/ // /

/ /. 7 2 , . ' /

/ //' /

.;' / 0.7Ref. 5 ^ — — • - ' / _,-- ^ ^ J > "

^^5t?^- """5

I 03 104

Fig. 2 Friction factors





fluids of higher Pra ndtl number may be attributed to the dampeningof the secon dary flow velocities owing to the higher viscosities of suchfluids.

Fig. 2 also contains curves which depict the analytical results of [5].These curves, which correspond to circumferentially uniform walltempera ture, fall in close proximity to those for the presen t bottomheating case.

Circum ferential Nuss elt Num ber Distributio ns. The localNusselt num ber a t any circumferential location on the heated arc wasevaluated from

Nu = riD/ft, h = q/(TUJ -T b) (23)

where Tw is the local wall tempe rature. To facilitate comparisons ofthe circumferential distributions of the Nusselt number for variouscases, i t is convenient to employ the ratio Nu/Nu, as in Figs. 3 through

5.Each of these figures contains a comparison of the circumferential

distributions of Nu/Nu for top heating and for bottom heating. Thesuccessive figures, which are arranged in the order of increasing valuesof Gr+, illustrate how the Nusselt number distributions for the twoheating conditions evolve as th e buoyancy effects grow progressivelymore imp ortant. The respective values of (2Ar)Gr+ for Figs. 3-5 are102, ~105, and ~0.5 X 107 to 107. Th e slight lack of coincidence of theGr + values in Fig. 5 has no effect on the q ualitative com parison of thedistributions for top and bottom heating, which is the main concern

of this figure as w ell as of Figs. 3 and 4.The cause of this slight m ismatch is historical. When the analysis

of the present problem was first undertaken, the choice of the di-mensionless variables was influenced by those of [5]. With thosevariables, the resulting dimensionless buoyancy parameter is theproduct of a dimensionless pressure gradient and the Rayleigh

1.5

Hi1.0

0.5

- Top Hea ting, 6

- Bottom Hea ling,

0 10 20 30 40 50 60 70 80 90

number of equation (1). It was this param eter to which values wereassigned as input to the numerical solutions. Subsequently, when itwas discovered that Gr+ was a more appropriate index of thebuoyancy effects, its values were deduced from those of the aforementioned pressure gradient—Rayleigh number product in conjunction with the /Re/(/R e) 0 results. Whereas the same set of inputparameters had been assigned for all heating conditions and P rand tlnumbers, the values of/Re/(/R e)0 differ am ong the various cases (Fig.2). This gives rise to the moderate differences among the G r+ valueswhich are in evidence at the higher Gr + .

The angular coordinates 8 and 6'which appear on the abscissa ofFigs. 3-5 are defined on the ins et of Fig. 3. The 8 coordinate is usedfor the top heating case and the 8' coordinate for the bottom heatingcase. The definitions of the solid and dashed curves, as given in Fig.3, are common to all three figures.

From an examination of Fig. 3, it is seen that for Gr + ~ 102, theNusselt number distributions for top heating and bottom heating areonly slightly different, thereby confirming the small influence ofbuoyancy. The Nus selt numb ers vary by a factor of about 2V2 alongthe heated arc, with the highest values in the neighborhood of thejunction of the heated and adiabatic portions of the tube, i.e., near8 and 8' = 0. This is a plausible result since fluid passing through theunheated portion of the tube assists in transporting heat away fromthe adjacent portions of the heated arc.

As the buoyancy effects become stronger, the Nusselt number

distributions for the two heating arrangements evolve in differentways. As can be seen in Fig. 4 (Gr+ ~ 105), the Nusselt num bers alongthe bottom heated arc are very nearly uniform, whereas those alongthe top heated arc exhibit a markedly greater variation th an t hat forpure forced convection. The trend toward uniformity for bottomheating is the resu lt of the vigorous secondary flow caused by th e instability inheren t in heating from below. On the other hand , for topheating, increases in Gr+ lead to an increasingly stratified tem peratu refield which gives rise to large circumferential variations.

The effects of increasing buoyancy on the Nuss elt numb er distributions are further confirmed by Fig. 5 (Gr+ ~ 107). In addition, thereis evidence of a change in the flow pattern for the bottom heated case,as witnessed by the peaks and troughs in the dis tribution curves. Aswill be demonstrated shortly, these undulations result from thepresence of a localized recirculation zone situated near the bottomof the tube.

The Nu/Nu distributions for bottom heating are relatively insensitive to the Prandtl number. For top heating, the effects of Prandtlnumber are accentuated w ith increasing G r+ and are most significantin the neighborhood of the junction between the heated and adiabaticportions of the tube. Overall, the variations are larger at higherPrandtl numbers. This behavior is reasonable since a lower Prandtlnumber fluid has a relatively high thermal conductivity which pro-

Fig. 3 Circum ferent ial Nusselt number distr ibut ions, (2A?r)Gr+

= 102

JJy.Nu

0 10 20 30 40 50 60 70 80 909 , 8 '

Fig. 4 Circu mferent ial Nusselt number distr ibu t ions, (2/ ir)G r+

~ 10s


Nu

Nu

iO 20 30 4 0 50 60 70 8 0 90

Fig. 5 Circu mferential Nusselt numb er distributions. Top heating results arefor (2/7r)Gr

+~ 10

7; bot tom heat ing results are for (2/7r)G r

+~ 0.4 - 0.5 X

107





become distorted as G r+ increases and then breaks up into a multipleeddy structure . To the knowledge of the authors, such a mulitple eddy Jarray has not been previously reported for mixed convection in a \horizontal tube. j

The results for bottom heating and Pr = 5 are presented in Fig. 7 ;. jwith the successive diagrams arranged in order of increasing Gr +. At |the lowest value of Gr+, heating from below produces a patter n of jisotherms and streamlines that is similar to that for heating from •above (note tha t the isotherms in Fig. 7(a) are reversed relative to ithose of Fig. 6(a)). A major difference, however, is th at the secondary '

flow velocities are an order of mag nitude higher in the bottom heated icase. I

As Gr+ increases, the isotherms in the near-wall region tend to tfollow th e contour of the tube, so that t he wall temperatur e becomesmore uniform. The secondary flow continues to be characterized bya regular, single-eddy pattern at intermediate Gr+ . At the highestvalue of Gr+ , a relatively small counterrotating eddy is in evidence jat the bottom of the tube. Its presence is the cause of the undulations •in the Nu/N u curve for Pr = 5 in Fig. 5. The upflow leg of this eddy jis believed to be similar to a thermal.2 In general, as witnessed by the Ivalues of \p/i/, the secondary flow for bottom heating is much more !vigorous than that for top heating.

It is approp riate to remark h ere that t he flow and temp erature fieldswere assumed to be steady via the governing equations and the so- j

1

2A thermal is a buoyant mass which rises from a discrete location on a hori

zontal heated surface.

Fig. 6 Isotherm and streamline maps lor top heating and Pr = 8. Parametricvalues of (2/?r)Gr

+: (a) 10, (6 ) 10* ( o) 10

8

Fig. 7 Isotherm and streamline maps (or bo ttom heating and Pr = 5. Parametric values of (2/?r)Gr

+: (a) 10, (b) 10", (c) 0.S X 10

7

motes circumferential conduction and thereby reduces the variations.

Isotherm and Stream line Maps. Information about the temperature field and the secondary flow patt ern will now be presentedby means of isotherm and streamline maps. The isotherms are expressed in terms of the dimensionless temperature

(T - Tb)/(TW - Tb) (23)

The stream function was deduced from the velocity field solutionsby evaluating the integral

4,/v = - C'vdr, (24)Jo

along lines 6 = constant, with 4> = 0 at r = 0.The isotherms and stream lines are presented in Figs. 6-8. Each of

these figures contains several representations of the tube cross-sectionshowing the isotherms and streamlines for a group of related cases.In each of the cross-sectional representations, the isotherms areplotted in the left half and the streamlines in the right half. Both theisotherms and streamlines are symmetric about the vertical diameterof the tub e.

Fig. 6 is for top heating and for a Prand tl num ber of five. The successive cross-sectional represen tations correspond to increasing valuesof Gr+ as indicated in the figure caption. The isotherms reflect anincreasing degree of stratification as Gr + increases. At the lowest value

of Gr

+

, the isotherms show some tendency to conform to the circularcontour of the wall, but at higher Gr+ they become nearly horizontal.The stream lines confirm th e expected presence of a secondary flow.The rather regular single eddy in evidence at the lowest Gr + tends to

68 / VOL 100, FEBRUARY 1978 Transactions of the ASIVIE




Fig. 8 Isotherm andstreamline m

(a ) top heating, (b) bottom heating

lu t ion method. When the resulting flow field is as complex as in Fig.

7(c), it is poss ible tha t in reali ty the flow exhibits a s teady -pe r iod ic

behavior ra ther than a to ta l ly t ime- inva r ian t one. An examina t ionof this mat ter, how ever, is beyond the scope of the present work; some

form of stabil i ty an alysis wil l be requ i red to resolve the question.

Rep resen ta t ive re su l t s for Pr =0.7 are presen ted in Fig. 8 in order

to perm it a comp arison w ith corresponding cases ( i .e ., same Gr+) for

Pr = 5 in F igs . 6and 7. In general , the shapes of the i so the rms and

streamlines are similar for the two P ran d t l numbers , a l though the re

are certain differences of de ta i l .

Concluding Re mark sThe results of th is investigation have demonstra ted that the effects

of buoya ncy on laminar forced convection in a horizontal tube d epend

markedly on the circumferentia l d is tr ibution of the wall heat flux.

When heat is added a long the bottom half of the tube while the top

half is insulated , a vigorous second ary flow is induced which gives rise

to a significant increase in the average Nusselt number relative to t h a t

• for pure forced convection. The increases in Nusse l t number are

[ greater at h ighe r P rand t l numbers . The vigorous secondary flow is

I a lso instrum ental in bringing about a re la t ively uniform distr ibution

of the local Nus selt nu mb er a long the hea ted arc.

W h e n the top half of the tube is hea ted wi th the bottom half insu

la ted , temperature s tra t if icat ion ensues and the induced secondary

flow is much weaker than that corresponding to the bo t tom hea ted

case. Asa consequence, the buoyancy-rela ted increase of the Nusselt

number is also smaller . On the o the r hand , the stra t if icat ion creates

large c ircumferentia l varia t ions of the local Nusselt number.

The secondary flow increases the friction factor relative to its forced

convection value, but the increases are substantia l ly smaller than

j those susta ined by the average Nusselt number. In common with the

I Nusselt numbe r, the friction factor is more affected by bottom heating

than by top hea t ing .

It was a l so demons t ra ted tha t the buoyancy effects are governedsolely by the modified Grashof number Gr

+, withou t r ega rd to the

Reynolds number of the forced convection flow. It is the opinion of

the au tho rs tha t Gr+

is a more apt measu re of the buoyancy effects

; than the ReRa p roduc t ( equa t ion (1)) t h a t has been employed fre-

' quently in the pas t .

Acknowledgment« This research was pe r fo rmed unde r the auspices of NSF G r a n t

I ENG-7518141.

R e f e r e n c e s

[ 1 Schmidt, R., "Experiments on Buoyancy-Affected and Buoyancy-Unaffected Turbulent Heat Transfer for Water in a Tube with Circumferen-

for Pr = 0.7 and (2/?r)Gr+

= 10".

tially N on-uniform Heating," Ph.D . thesis, Department of Mechanical Engineering, University of Minnesota, 1977.

2 Patank ar, S. V., and Spalding, D. B., "A Calculation Procedure for Heat,

Mass, andMomentum Transfer inThree-Dimensional Parabolic Flows," In-ternational Journal of Heat and Mass Transfer, Vol. 15, 1972, pp. 1787-1806.

3 Spalding, D. B., "A Novel Finite-Difference F ormulation for DifferentialExpressions Involving Both First and Second Derivatives," InternationalJournal for Numerical Methods inEngineering, Vol. 4,1972, pp. 551-559.

4 Cheng, K. C, and Hong, S. W., "Combined Free and Forced LaminarConvection in Inclined Tube s," Applied Scientific Research, Vol. 27,1972, pp.19-38.

5 Hwang, G. J., and Cheng, K. C, "Boundary Vorticity Method for Con-vective Heat Transfer with Secondary Flow—Application to the Combined Freeand Forced Laminar Convection in Horizontal Tub es," Heat Transfer, Vol.4, 1970, Paper N o. NC3.5, Elsevier Publishing Company, Amsterdam, 1970.

6 Newell, P. H., and Bergles, A. E., "Analysis of Combined Free and ForcedConvection for Fully Developed Laminar Flow inHorizontal Tu bes," ASMEJ O U R N A L OF H E A T T R A N S F E R , Vol. 90,1970, pp. 83-93,

7 Faris, G. N., and Viskanta, R., "An Analysis of Laminar CombinedForced andFree Convection Heat Transfer ina Horizontal Tube," International Journal of Heat and M ass Transfer, Vol. 12,1969, pp. 1295-1309.

8 Iqbal, M., and Stachiewicz, J. W., "Variable Density Effects in CombinedFree and Forced Convection inInclined Tu bes," International Journal of Heatand Mass Transfer, Vol. 10,1967, pp. 1625-1629.

9 Iqbal, M., and Stachiewicz, J. W., "Influence of Tube Orientation onCombined Free and Forced Laminar Convection Heat Transfer," ASMEJ O U R N A L OF H E A T T R A N S F E R , Vol. 88,1966, pp. 410-420.

10 Del Casal, E., and Gill, W. N., "A Note onNatural Convection Effectsin Fully Developed Horizontal Tube Flow," AIC hE Journal, Vol. 8,1962, pp.570-574.

11 Morton, B. R., "Laminar Convection inUniformly Heated HorizontalPipes at Low Rayleigh Nu mbers," Quarterly Journal of Mechanics an d AppliedMathematics, Vol. 12,1959, pp. 410-420.

12 Siegwarth, D. P., Mikesell, R. D., Readal, T. C, and Hanratty, T. J„"Effect of Secondary Flow on the Temperature Field andPrimary Flow in aHeated Horizontal Tube," International Journal of Heat an d Mass Transfer,Vol. 12,1969, pp. 1535-1552.

13 Mori, Y., and Futagam i, K., "Forced Convective Heat Transfer in Uniformly Heated Horizontal Tubes," International Journal of Heat and MassTransfer, Vol. 10,1967, pp. 1801-1813.

14 Sabbagh, J. A., Aziz, A., El-Ariny, A. S., and H amad, G., "Combined Freeand Forced Convection in Circular Tubes," ASME JOURNAL OF HEATTRANSFER, Vol. 98,1976, p p. 322-324.

15 Bergles, A. E„ and Simonds, R. R., "Combined Forced andFree Convection for Laminar Flow inHorizontal Tubes with Uniform Heat F lux," In-ternational Journal of Heat and Mass Transfer, Vol. 14, 1971, pp. 1989-2000.

16 Hussain, N. A., and McComas, S.T, "Experimental Investigation ofCombined Convection in a Horizontal Circular Tube w ith Uniform Heat Flux,"Heat Transfer, Vol. 4,1970, Paper No. NC3.4, Elsevier Publishing Company,Amsterdam , 1970.

17 Petukhov , B. S., Polyakov, A. F., andStrigin, B. K., "Heat Transfer inTubes with Viscous-Gravity Flow," Heat Transfer— Soviet Research, Vol. 1,1969, pp. 24-31.

18 Shannon, R. L., and Depew, C. A., "Combined Free and Forced LaminarConvection in a Horizontal T ube w ith Uniform Heat Flux,"jASME JOURNALOF H E A T T R A N S F E R , Vol. 90,1968, pp. 353-357.





19 Pe tukho v, B . S . , and Polyakov, A. F . , "Ex per im enta l Inves t iga t ion ofViscogravitational Fluid Flow in a Horizontal Tube," High Temperature, Vol.5 ,1967, pp. 75-81 .

20 Petukh ov, B. S., and Polyakov , A. F., "Effect of Free Convection on H eatTransfe r D ur ing Forced F low in a Hor izonta l P ipe ," High Temperature, Vol.5 ,1967, pp. 348-351 .

21 Mor i , Y . , Futagam i, K. , Tok uda , S . , and Naka mur a , M, "Fo rced Con-vec t ive Hea t Transfe r in Uniformly Hea ted Hor izonta l Tubes, 1s t Repor t—Experimental Study on the Effect of Buoyancy," International Journal of Heatand Mass Transfer, Vol. 9 ,1966 , pp. 453-463.

22 McCom as, S . T. , and Ecker t , E. R . G., "Combin ed Free and ForcedC onve c ti on i n a H or i z on t al C i r cu l a r Tube , " JO U R N A L O F H E A T TR A N S FER, TRANS. ASME, Ser ies C , Vol . 88,1966 , pp. 147-153.

23 Iqba l , M. , "Free Convec t ion Ef fec ts Inside Tubes of F la t - P la te Sola rCol lec tors ," Solar Energy, Vol. 10,1966 , pp. 207-211 .

24 Ede , A. J. , "T he He at Transfe r Coefficient for Flow in a Pip e," International Journal of Heat and Mass Transfer, Vol. 4 , 1 9 6 1 , pp. 105-110.

25 Reynolds , W. C , "Hea t Transfe r to Ful ly Deve loped Lamina r F low ina Circula r Tub e with Arbi t ra ry Circumferent ia l Hea t F lux," JOUR NA L OFH EA T TR A N S F E R , TR A N S . A S M E, S e r i es C , V ol . 82 , 196 0 , pp . 108 - 112 .





R. R. Gilpin

H. Imura 1

K. C. Cheng


University of Alberta,

Edmonton, Alberta, Canada

Experiments on the Onset of

Longitudinal Vortices In Horizontal

Blasius Flow Heated from BelowExperiments were performed to confirm the occurrence and growth of longitudinal vorti

ces in a laminar boundary layer developing in water over a heated horizontal flat plate

with uniform surface temperature. Photographs of the vortices, measurements of the conditions of their onset, and measurements of their wavelength are presented. Comparisons

are made with theoretical instability results for the critical Grashof number and wave

length. Temperature profiles across the boundary layer were measured for flows with and

without vortices to show qualitatively the effect that the longitudinal vortices have on the

heat transfer rate at the plate. U nder conditions of thermal instability the longitudinal

vortices were found to be the first stage of the laminar-turbulent transition process in a

boundary layer heated from below .

In t roduct ion

Th e tem pera ture profile in a laminar bound ary layer flow over a flatplate with a constant wall temp erature has been extensively analyzed

since the publication of Pohlhausen 's f irs t solution [ l] .2

I t does not

appear, however, that the conditions for the onset of thermal insta

bility in this flow have been critically investigated. Physically when

a boundary layer is heated from below or cooled from above a poten

tially unstable top-heavy situation exists due to the variation of fluid

density with temperature. The problem is apparently analogous to

the case of a lamina r bo undar y layer flowing along concave wall where

an instability due to the centrifugal force exists [2-4] . In the case of

the concave wall the instabil i ty (Gortler instabil i ty) m anifests i tself

in the form of vortex rolls aligned longitudinally with the flow. Due

to these vortex rolls the f low field develops a three-dimensional

character which alters the wall heat transfer .

Calculations have been made, for example, by Mori [5] , Sparrow

and Minkowycz [6], and Chen, Sparrow, and Mucoglu [7], of the effect

of the buoyancy term on the laminar boundary layer profile on an

isothermally heated, horizo ntal, flat plate. These calculations of mixed

convection suggest that the relative importance of free and forced

convection can be represented by the physical parame ter Gr u / R e . t5 /2 .

A perplexing practic al question arises, however, as to the l imit of

validity of the two-d imensional free an d forced convection formula-

1 On leave from the Department of Mechanical Engineering, KumamotoUniversity, Japan.


Contributed by the H eat Transfer Division for publication in the JOURNALOP HEATTRANSFER. Manuscript received by the Heat Transfer Division May23, 1977.

t ion in view of the fact that the f low may be unstable . Recen tly , Wu

and Cheng [8] have obtained theoretical results which suggest that

for Grashof numbers beyond a critical value, the flow is unstable withrespect to small dis turbances in the longitudinal vortex mode. This

analysis employed a non paralle l f low mod el in which a l inearization

was made about the basic Blasius velocity and Pohlhausen temper

ature profiles .

The present experimental investigation was init ia ted to confirm

the onset of longitudinal vortex rolls in a horizontal Blasius flow of

water over a f la t p late with constant wall temperature and to assess

its practical significance. However, from flow visualization studies

it became evident that one can clearly identify , in addition to the

initial onset of the instability in the form of vortex rolls, a subsequent

amplif ication of the disturbances an d a transit ion to regimes of more

disordered secondary f low. The effects of therma l instabil i ty on th e

temperature f ie ld in this postcri t ical regime were also investigated.

I t is of interes t to note that S parrow and Hus ar 's experim ents [9] on

longitudinal vortices in natural convection f low along an inclined

plate , apparently demonstrate a s imilar sequence of events even

thoug h the na ture of the basic flow is quite different. This suggest th at

some analogy exists between the present problem and that of Sparrow

and H usar [9] as well as Lloyd and Sparro w [10]. These experim ents

have apparently motivated subsequent theoretical s tudies of this

problem [11-14],

For complete ness, i t should be mentioned tha t for the case of hor

izontal plane Poiseuil le f low heated from below, longitudina l vortex

rolls also exist. This the rma l instability problem has been studied both

theoretically and experimentally in the thermal entrance region

[15-18] and the fully developed region [19-20].

The hydrodynamic stability of a Blasius flow has been studied very

extensively in the past; however, as has been pointed out, l i t t le in-

Journal of Heat Transfer FEB RUARY 1978, V O L 100 / 71Copyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



formation is available for the t herm al insta bil i ty of this f low. Since

the laminar boundary layer flow over a flat plate with a constant wail

temperature is very basic to the s tudy of forced convection heat

transfer the presen t therm al instabil i ty problem is of considerable

theoretical and practical in terest .

2 Exper im ental App aratus and Proce dure2.1 W at er Tun ne l, An existing closed loop water tunnel, shown

in Fig. 1, was used as the basic flow facility for the experiments. This

facili ty consists essentially of a c irculation pum p, a heat exchanger,

and the associated piping for trans porti ng tb» water around a closedloop. The heat exchanger is connected to a refrigeration system so that

the water in the tunnel can be controlled at any tempera ture between

room temperature and 0°C. The water tunnel has two test sections.

The dimensio ns of the small tes t section are 25.4 X 45.7 X 213,4 cm,

width by height, by length, and those of the large one are 64.0 X 91.4

X 182.9 cm. The contraction ratio between the large and small test

sections is 5 to 1 . These test sections have windows m ade of acrylic

resin plates in order to per mit a visual investigation of the f low. The

circulation pump of the water tunnel is driven by a variable speed d-c

drive which is capable of producing a steady water velocity in the small

test section between 0.02 and 5 m/s.

2 .2 T h e I s o t h e r m a l H e a t i n g P l a t e A s s e m b l y . T h e p r e s e n t

experimen t was performed over a horizontal heating plate which was

installed in the small working section. The leading edge of the p late

was located near the end of the converging section. The heating plateand the other atta ched equi pme nt are shown in Fig. 2. The te st surface

util ized in the experiments was fabricated from copper plate with

dimensions 6 .35 mm X 152.4 cm X 24.1 cm, thickness by length by

width. The plate was designed to provide an isothermal surface to th e

bound ary layer f low which could be either h eated or cooled from the

bottom surface by circulating water from a controllable , constant

t e m p e r a t u r e b a t h .

In order to mak e the ma in flow parallel to the plate it was necessary

to accelerate the f low under the plate . This was done using a pump

to suck a small amount of water from under the plate near its leading

edge and reinjecting it further downstream. It was, however, difficult

to obtain a flow which was perfectly parallel to the plate at its leading

edge. To avoid a separation b ubble from forming a t the leading edge

it was found tha t the f low had to be directed sl ightly downward onto

the plate in this region generating a somewhat wedge-type f low.

Varying the amount of water bypassed under the plate and thus the

angle of the flow at the leading edge appeared, however, to have a

negligible effect on the results if the flow was confined to small de

viations from parallel flow. A sharp unhealed tail plate with a length

of 15.2 cm was attached to the tra il ing edge of the plate to m inimize

the disturbances caused by the wake flow behind the plate .

2 .3 T e m p e r a t u r e M e a s u r e m e n t s . T h e t e m p e r a t u r e s of t h e

heating water were measured at the inlet of the suction pipe in the

tank and at the inlet and the outlet of the heating plate by copper-

consta ntan the rmco uples, T .C. 1 to 3 (Fig. 2) . Tem per atur es in the

water tunne l were meas ured 23 cm upstream from th e leading edge

and also 40 cm downstream from the tra il ing edge by copper-con

stantan thermocouples T.C. 4 and 5. In order to measure the tem

perature profiles across the boundary layer, copper-constantan

thermoco uples formed in a loop (Fig. 2 insert) were used. The plane

of the loop was aligned perpendicular to the f low and the thermocouple was posit ioned using a traversing mechanism that could be

Fig . 1 Schematic diagram of the water tunnel facility

Thermocoup leDesign

-Thermisror ical lyCon t ro l l ed Heare r

Fig . 2 Diagram of the constant temperature plate an d auxiliary equipment

read with an accuracy of 1/500 mm.

To check whether the thermocouple wire was actually perturbing

the boundary layer flow, three wire diameters, 127, 76, and 25 n were

tried. The hot junctions of the 127 and 76 ix thermocouples were made

by butt welding the copper and constantan wires together thus pro

ducing a very small junction volume. To ob tain a successful weld of

the 25 n wires i t was found t hat an overlapping of the wires was re

quired. This produced an effective junction diameter of 50 fi. T h e

velocit ies outside the boundary layer were normally measured using

a 3 mm OD P itot tub e. The very low velocit ies were also determin ed

by measuring the t ime required for a f loating dust partic le to pass

between two lines 50 cm apart .

2 .4 T h e F lo w Visu a l iza t ion T e chn iqu e . An e lect rochemica l

technique similar to that described by Sparrow and Husar [9] was

employed t o facil i ta te direct visual observation of longitudinal vor

tices. In the present experim ents , Phenolphth alein was used as a pH

indicator instead of Thymol blue [9]. The plate surface itself served

as the negative electrode an d the posit ive electrode was provided by

a pair of copper sheets s i tuate d on th e side walls of the test section.

When a small d-c voltage (10-20 volts) is impressed between the twoelectrodes, ionization at the pla te surface generates the tracer f luid

. N o m e n c l a tare.

Gri , GTXC = Grashof numbers, g(i(AT)x

3/v '

2

andgP(AT)x c3/v

2

g = gravitational acceleration

Pr = P rand t l number

R e x , Re Xc = Reyno lds numbers , (U ax/v) an d

(U„xJv)

T, Tw, T„„ = fluid, wall, and free stream

t e m p e r a t u r e s

[/„ = free stream velocity

x, y = distance from leading edge and normal

distance from plate surface

xc = distance from leading edge to onset

point of instabil i ty

I) = coefficient of thermal expansion

ri = similari ty variable , y(Uv,/vx)l/2

S

X

dimensionless temperature difference, (T

• T„)/AT

wavelength of vortex rolls

v = kinematic viscosity

AT = temperature difference, (T w — T„)

ATrnax = maximum peak to peak amp l i tude

of tempe ra tu re f luc tuat ion





showing the pink indicator color. Photography of the motion of this

indicator dye then gave an indication of flow patterns in the boundary

layer adjacent to the plate . Th e application of the pre sent f low visu

alization me thod is restr ic ted to relatively low velocity range (0-15

cm/s) . Further details of the method can be found in [9],

3 Expe r imental Eesul t s and Discuss ion3 .1 T h e B a s i c F l o w a n d T e m p e r a t u r e F i e l d s f o r S t a b l e C a s e .

The free stream velocity above the plate was first checked for uni

formity. When the horizon tal lam inar boun dary layer is cooled from

below, the layer is stable since the density of water decreases upwardfrom the plate surface. For a stable flow case, the temperature above

the plate was found to be uniform within ±0.2°C throughout the water

with the exception of the boundary layers of the order of 1-10 mm

thick on the side walls and th e plate . Meas urem ents of the velocit ies

showed that t he free s tream velocit ies were also very uniform being

within ±0.5 percent of the mean value. Due to the growth of side wall

and plate bou ndary layers , the free s tream flow accelerated sl ightly ,

about 15 percen t, in passing through the tes t section. No direc t

measurements of the turbulence level were made in the test section;

however, by observing the deviation of the paths of suspended par

ticles from straight lines the estimated turbulence level would appear

to be less tha n 5 pe rcent.

In order to obtain an isotherma l surface over the entire plate, a high

rate of flow of the fluid circulated from the constant temperature bath

was maintained. As a result , the temperatures at the three locations,

T.C. 1, 2, and 3 shown in Fig. 2, were almost identical. Since the in

stalla tion of a permanent thermocouple at the plate surface would

disturb the uniform temperature condition required, the surface

tem pera tures w ere obtained by lowering one of the free s tream ther

mocouples from above the p late unti l i t touched the surface. The re

sults of these measurements were usually within 0 .2°C of the tem

peratu re of the circulating f luid below the plate . A max imum differ

ence of 0.6°C was obtained with a high flow velocity when the tem

peratur e difference betw een the plate and the free s tream w as 16°C.

An average of the tem per atur e obtained from the therm ocouple

touching the plate and the water bath temperature was f inally used

for the surface temperature, T„„ since it appeared to represent a good

estimate of the plate surface temperature.

In order to confirm t he Pohlhausen temp erature profiles for s table

basic f low, a number of tem peratu re profiles were measured with the

plate cooled below the free s tream temp eratur e. Measurem ents m adeof the profile are sho wn in Fig. 3 along with th e theore tical profile for

Pr = 7. Th e curve for Pr = 10 was given for reference only. It can be

seen that the results obtained with the 76 and 25 M wires agreed while

those obtained with the 127 ir wire were somew hat lower. The mea

surements with the smaller thermocouples were at the most 10 percent

below the theoretic al profile for Pr = 7, where the pro perty values were

evaluated at the free s tream temperature T„. For the present inves

tigations, th is level of agreeme nt between theo ry and experim ent was

considered as adequate .

Table 1 Range of param eters for exper ime nts

0.5

X, mo 0.287

A 0.279

D 0.533O 1.248

fc. v 0.584

V% C ?

0.5

AT,°C

7.9

8.1

8.08.1

7.9

^ T h

1.0

UQJ , cm/s

7.5

5.8

7.65.7

9.2

R8„

2 . 1 4 x 1 0 "

1 .7 3 x1 0 *

4.28 x l O4

7.61 x l O 4

5 . 5 7 x 1 0 *

sory, Pr = 7

^ ^ T h e o r y , P r-10

1.5

Diameter ofThermocouple, f£

25

76

7676

127

Pr

7.0

6.4

6.46.4

6.6

2.0 2.5

M ' , ° C

1.8-2.0

3.5-4.17.7-8.2J 1.8-12.0

15.6-15.9

T „ , ° C

25.5-21.77.8-24.07.7-8.17.8-8.38.2-8.5

( / . , c m / s

2.7-15.22.9-7.33.0-20.22.9-16.43.3-17.2

Fig. 3 Stable boundary layer temperatu re profiles produced in the case ofcooling from below

3 .2 F l o w V i s u a l i z a t i o n S t u d i e s f o r t h e U n s t a b l e C a s e of

He a t in g f rom Be low , T he ranges of pa ramete rs of the p resen t ex

periments for the case of heating from below are listed in Table 1,

Temperature differences between the plate and free s tream were

varied in the range AT = 2-16°C and velocities of the main flow were

in the range t / „ = 2 .7-20 cm/s. The corresponding Reynolds numbe rs,

Re/.', on the plate are in the range 10 3 -1 0 6 , which is within the hy-

drodynamically s table region for isothermal f lows.

Insight into the quali ta tive nature of the formation and develop

men t of longitudinal vortices is gained by direct visual obs ervation

of flow patterns made visible by the electrochemical technique. Three

representative series of photographs of the f low field are presented

in Figs. 4(a), (6), and (c) where the top and side views of the vortices

are shown. The top views of the onset and growth of the longitudinal

vortices show a remarka ble res embla nce to those shown in Fig. 1 of

Sparrow and Husar [9] for the case of longitudinal vortices in natural

convection f low on inclined plates . This observation would suggestthat the s tructure and the physical process for the onset and growth

of the longitudin al vortices in the present exp erime nts are s imilar to

those observed by Sparrow and Husa r [9] even though the natu re of

the m ain flow is different. If the flow field were two-d imensional (basic

flow withou t longitudina l vortices) , the plat e would be blanketed by

a thin and uniform film of tracer fluid moving parallel to the surface

[9]. Observations of the stable flow over a cooled plate confirmed this

flow pat tern of the dye. Also the photog raphs in Fig. 4 showed the

existence of a s table tw o-dimen sional lam inar f low (Blasius flow) in

the region near the leading edge.

An inspection of the top view in Fig. 4(a) reveals tha t an array of

more or less regularly spaced lines, oriented parallel to the main flow,

appe ars at a certain distance (roughly 1.7 ft or 0.5 m) from the leading

edge. This indicates the onset of longitudinal vortices. As noted by

Sparro w and H usa r [9], the prese nce of the disc rete lines is indicativeof spanwise motions of a cellular nature. The lines of tracer fluid

correspond to regions of upward motion between counter-rotating

vortex rolls. The foregoing observation is confirmed by an inspection

of the s ide vien's . Exam ining Fig. 4(a) , i t can be seen that a t roug hly

the sam e point in the s ide view of the plate correspo nding to the oc

currence of paralle l l ines in the top view, the tracer dye can be seen

rising off the plate. This is dye that is being lifted off the plate by the

upflow between vortex rolls. These dye patterns were never observed

for a stable flow.

The density difference t erm , /3AT, in the Grashof num ber is 3 X

l f r4

for Fig. 4(a). For Fig. 4(b), (JAT is increased to 5.1 X 10~4

while

the main stream velocity is the same. As a result , the f low became

more susceptible to instability and the onset point has moved forward

to appro xim ately t he 1 ft (0.3 m) point. In this case it can also be seen

that further down the plate the regular longitudinal vortices begin

to break up into a highly disordered flow pattern. In Fig. 4(c) , with

the main s tream velocity decreased, th is succession of the f low pat

terns shifts toward the leading edge of the plate . This indicates tha t

the low Reynolds number f low is more susceptible to thermal insta

bil i ty . Physically , one could antic ipate this behavior in that a t h igh

Reynolds number, the buoyancy effects would tend to be washed out.

The succession from stable two-dimensional laminar f low through

the develop ment of secondary f low to disordered flow represen ts the

firs t s tages of the laminar to turbulent transit ion process. I t is ap

parently characteris t ic of s imilar unstable f lows, for example, the

na tur al conve ction flow on an inclined pla te [9, 10].

3 .3 I n s t a b i l i t y O n s e t P o i n t a n d W a v e l e n g t h of L o n g i t u d i n a l

Vo rte x Roll s , For a given experimental condition, the distance along





Itx

m

'"'>'""";;

Fig. 4(a) Photographs of dyepattern onthe heating pl/lte for U", =7.3 em/s,T", = SOC,.6.T = 7.SoC. and {J.6. T = 3 .0 X 10 -4

'"'>c-oI -

ftx

m

'"'>'""";; _. .. _--- -

0.5 0.4 0.3 0,2

Fig.4(b) Photographs of dyepattern on the heatingplate for U", = 7.6 em/s.T", = 24°C• .6.T = 4.1°C. and {J.6. T = 5.1 X 10 -4

::,'>.,

"";;

ftx + 1 - m - - - - ' - - - - - r - ' - - - - . . . . L . r - - - ' - - - o , . . ~ s - . . . . I . . . - - 0 , . . : 4 - - - - - - ' r - - - - - 0 , . . ~ 2 - ........ - - O ~ , . . l . . . . L . - - - - - l

Fig. 4{ c) Photographs of dye pattern on the heating plate for U", = 2.9 em/s.T", = SoC, .6.T = 11.SoC. and {J.6.T= 4.4 X 10 - 4

the plate at which vortices could be first observed is not a precisely

definable quantity. As can be seen from the photographs in Fig. 4, the

position varies somewhat across the plate. I t also fluctuated in time.

In a given situation, however, there are positions where there are never

any vortices and similarly positions where there are always vortices.

The zone of uncertainty between these positions is typically of the

order of 20 percent of the distance from the leading edge.

Linearstability theory [8J suggests that the onset points for insta-

bility can be correlated using the Reynolds and Grashof numbers

where the critical length Xc is the distance from the leading edge to

the onset point. Fig. 5 shows the observed onset points correlated in

this fashion where the open points indicate positions at which vortices

were only occasionally observed and the solid points indicate positions

where vortices always existed. The property values of water used in

Fig. 5 were evaluated at the free stream temperature Too. A study by

Seban [21] suggests that this may not be unreasonable for a variable





property fluid with a high Prandt l num ber s ince the therm al bou ndary

layer in that case is much thinner than the velocity boundary

layer.

The measu red onset poin ts using free s tream prop erty values were

found to be correlated by

G r J c = c x R e I c (1)

where c\ was between 46 and 110. Whe n proper ties calculated at the

film temperature were used a wider scatter in the experimental results

occurred. The correlation of the results by equation (1) using fi lm

tempera tu re p rope r t ie s re su l ted in c\ being between 3 0 and 200.

From a count of the n um ber of l ines aligned in the s tream wise di

rection distr ibuted across the width of the plate , the average wave

length of the longitudinal vortices can be determined. One wavelength

corresponds to two counter-ro tating vortices. Results for xj \ versus

R e Ic are shown in Pig. 6. Th e da ta can be correlated by

c/X = e 2 R e X c1/ 2

( o r x c A = (0.035-0.070)Gr Ic1< '

3) (2 )

where c 2 = 0.16 and 0.31 bounded the data .

3.4 C o m p a r i s o n B e t w e e n T h e o r y a n d E x p e r i m e n t f o r I n -

s tabi l i ty Re sul t s . Instabil i ty results from linear s tabil i ty analysis

[8] are also plotted in Figs. 5 and 6 for comp arison. Th e theoretica l

values for ci in equa tion (1) are 100 for Pr = 7 and 75 for Pr = 10. As

men t ioned p rev ious ly , the unce r ta in ty in the measu remen t of the

onset point was approxim ately 20 percent. Takin g into consideration

possible uncertainties in the tem peratu re and velocity meas urem ents

would result in an estimated uncertainty in ci of approximately 50

percen t . Wi th in the accu racy o f the measu remen ts the measu red

values of ci of 46 to 110 are therefore seen to agree with the theory.

The value of c 2 in equation (2) calculated from the theory is 0.28

and is not very sensit ive to variations in Pr. The scatter in the ex

perimental results was quite large for this measurement; however, the

experimen tal values of c 2 between 0.16 and 0.31 are of the right order

o f magn i tude .

The level of agreement between theory and experiment is ra ther

surpris ing and unexpected in view of the known order of magnitude

of difference between t heory an d exp erim ent for the onset of longi

tudina l vortices in natural convection boundary layers a long inclined

plates [10-14] and in the therm al entra nce region of horizontal plane

Poiseuil le f low [15,16, 18] both with heating from below. Normally

one would expect the theoretically predicted onset point to fall below

that observed in the experiments s ince the instabil i ty must grow to

a f inite amplitude before i t can be observed.

Another factor that may influence the onset of instabil i ty is the

strong variation of the properties of water (in particular its viscosity)

with temp eratu re. The th eory in reference [8] pretains s tr ic tly to the

case of constant f luid properties and in the experiments variations

through the boundary layer of up to 40 percent in viscosity existed

for some tests . The o nset poin ts were observed to be correlated ade

quately using fluid properties based on the free s tream temperature;

however, some of the remaining scatter in the results may be due to

the effects of variable properties .

The circumstances under which the theory and experiment agree

are already und er investiga tion [22]; however, due to the complex

nature of the problem it may be some time before the s i tuation can

be clarif ied. Normally , in the engineering l i terature the agreement

between theor y and experim ent is take n to confirm the the oretical

prediction. However, in this particular instance one is cautionedagainst such immediate conclusion. Notwithstanding the foregoing

rem ark, the re sults shown in Figs. 5 and 6 would suggest that the

theoretical predictions are at least quali ta tively correct .

3.5 T e m p e r a t u r e M e a s u r e m e n t s i n t h e P o s tc r i t i c a l R e g i m e .

Furt her insight into the character of the unstable f low is obtained by

examin ing the tempera tu res tha t ex is t in the boundary laye r . T he

temperature profiles in Figs. 7(a) , (6) , and (c) correspond to the

conditions in the respective ph otogra phs Figs. 4(a), (6), and (c). These

temperature profiles were measured at fixed positions along the center

line of the plate . As mentio ned previously, the vortices on the p late

constantly shifted from side to s ide across the f low direction; thus,

by averaging the tem per atur e over a period of time at a fixed position,

an approximate spanwise average of the temperature was obtained.

Several profiles measured off the center l ine showed that the mean

profiles thus obtained were essentially constant across the plate . Th equalita tive behavior observed in these temperature profiles will be

il lustrated using Fig. 7(6). In Fig. 7(b) the m easurem ent of the m ean

tem perat ure profile upstream of the onset of vortices, profile n umb er

(1), gives a result tha t is the same as tha t obtain ed from the case of

cooling from below (s table case) and is only slightly below th e th eory.

At posit ions (2) and (3) which are in the postcri t ical regime with

longitudina l vortices, i t can be seen tha t the profile has changed sig

nificantly . Both th e tempe rature gradient a t the wall and total therma l

boundary layer thickness have increased above that for the Pohl-

hausen profile .

Fig. 8(6) shows the peak to peak amplitude ATmax of the temper

ature f luctuation in the boundary layer normalized by the tempera-

X

10 '

- Solid Points

O pen Points

AT , 'C

o 2

A 4

D 8O 12

v 16

/ O ^ C^ v ^

f

a

Fully Developed Vortices

Vortices at Onset

Theory, Wu & Cheng [8] -v.

. -v- =0.31 Re' /2•?-&£-

\ s-jZ*°\ <J<r *

"VOA Ve ^

> ^ A A o > ' \^ O A - " ' V

A A - ^

^^

• ^ o

o

- ^ = 0 . 1 6 R e ^

. I

10J

Fig. 5 Correlations for the point of onset of longitudinal vortices in a planeBlasius flow heated from below

4 6 « , 0 4

R e ,

4 6 8 , 0 5

Fig . 6 Correlations for the typica l w aveleng th of the longitudinal vortices





(1) A 0.279 1.44 K 1 04

At Onset Point

(2) * 0.845 4.41 x lO 4 Vortices

(3) & 1.249 6.32 «1 0 4 Vortices

Um°7.3<:m/s, T ^ S ' C , AT = 7.8°C, £AT = 3 .0 x l O " 4

F i g . 7 ( a ) M e a n t e m p e r a t u r e p r o f i le s f o r c o n d i t i o n s in F i g . 4 ( a )

x, m Kex

0.229 1.92 « 10 4 Stable Flow

(2) o 0.584 4.81 x 10 4 Vortices

(3) o 1.249 1.0 5x ]0 5 Turbulent Flow

U a) = 7.6cm/s, TtD = 24°C, AT=4.1°C, /3AT = 5.1 *W *

F i g . 7 ( b ) M e a n t e m p e r a t u r e p r o f i l e s f o r c o n d i t i o n s in F i g . 4 ( b )

x, m Re.

(1) • 0 .076 1 .59 *10 J At Onset Point

(2) n 0.305 6.32 x lO 3 Vortices

(3) D 0.845 1.81 x 10 4 Turbulent Flow

U „ = 2.9cm/s, T<0 =8°C, A T=11.5°C, /3AT =4.4x 10"'

F i g . 7 ( c ) M e a n t e m p e r a t u r e p r o f i le s f o r c o n d i t i o n s i n F i g . 4 ( c )

ture difference AT between the plate surface and the free s tream. I t

can be seen that upstream of the observed onset point there is very

li t t le ( less than 10 percent) tem peratu re f luctuation in the bo undary

layer. However, after th e onset of vortices the amp litud e of the f luc

tuations is up to 60 percent of AT. Examining th e inserts which show

samples of the temperature records at various points in the profile ,

i t can be seen that a t posit ion (2) which is imm ediately dow nstream

from the onset point the temperature f luctuations have a relatively

long period ranging 10 to 60 s . The fluctuations at th is point were

observed to be coincident with the s low but s teady shift ing and

swaying of the vortices in a direction perpendicular to the main flow.

Furth er down stream at posit ion (3) the amplitu de of the f luctuations

remains the same; however, they are of a higher frequency indicating

a more highly disordered flow. This occurs due to a breakdown of the

regular vortices sett ing the s tage for a turbulent f low. Further tem

perature m easure ment re sults shown in Figs. 7(a ) , (c) and Figs. 8(a) ,

(c) confirm these quali ta tive observations at other f low conditions.

3 .6 P r a c t i c a l I m p l i c a t i o n s f o r F o r c e d C o n v e c t i o n . Heat

Transfer on a Horizontal Plate. I t would be antic ipated that the

existence of longitudinal vortices in the laminar boundary layer could

affect the heat transfer ra te through the layer. This effect was con-

3( a )

t 2 ^ 3(b) \

3(c)

- 1 .& * A -

A""A"- A, .

2.0 3.0 5.0

F i g . 8( a) Pea k- to-p eak tempe ratur e f luctua t ion pro f i les o f posi t ions indica ted

i n F i g . 7 ( a )

S 0.5E

2(a) 2(b) 2(c) U 3(a) 3(b) 3(c)(a) 2(b)3(b) 2(b)

a-a-d-A* *...(2)••3(a).

/ 'C )

(3) \

J? 2 (a )3 ( c )

*.( ' )

2.0 3.0

V

4.0 5.0

F i g . 8 ( 6 ) Pe a k- to - p e a k t e mp e ra tu re f lu c tu a t i o n p ro f i le s a t p o s i t i o ns I n d ica te d

i n F i g . 7 ( 6 )

F i g . 8 ( c) Pe a k- to - p e a k t e mp e ra tu re f l u c tu a t i o n p ro f i l e s a t p o s i t i o n s i n d i ca te d

i n F i g . 7 ( c )

f irmed from the measurements of temperature profiles through the

bound ary layer. As noted previously, the dev elopmen t of vortices was

accompanied by an increase in the mean temperature gradient at wall.

Thus, the conventional heat transfer calculations neglecting the ex

is tence of longitudinal vortices may be in serious error in the

postcritical regime. The unstable conditions occur when heating from

below or cooling from above exists and the value of the parameter

G r I c R e I c_ 3 /

'2

exceeds some crit ical value, c\. For practical purpose;

a value of ci of 100 would be suggested on the basis of the presen

experiments . In this connection, i t is a lso of practical in terest tc

compare the present instabil i ty results with Sparrow an d Minkowycz '

76 / VO L 100, FEBRUARY 1978 Tra nsa ctions of the ASME

-

-




results [6] for 1 and 5 perce nt increa se in local he at trans fer rat e du e

to the buoyancy effect on a forced convection flow. The results from

[6] are also plotted in Fig. 5 for reference. Co mparin g these results with

the condition for the onset of instabil i ty i t would appear that the

developmen t of longitudinal vortex rolls will be the dom inan t effect

of the buoyancy force.

4 Concluding Rem arks

The present experiments confirm that thermal instabil i ty in a

Blasius f low on a horizontal plate heated from below results in the

formation of cellular secondary flows in the form of longitudinalvortices. Further, f low visualization studies have shown that a suc

cession of flow regimes exist in a thermally unstable Blasius flow on

a flat p late . These regimes include a s table two-dimensional laminar

flow regime near the leading edge of the plate, followed by the onset

of instabil i ty and growth of longitudinal vortices, and finally , the

break-up of these vortices into a turbul ent flow. Thu s, as was observed

by Sparrow and Hus ar [9] for a natural convection flow on an inclined

plate, the longitudinal vortices are seen as the first stage of the lamin ar

to turbulent transit ion process for a thermally unstable f low.

In the present s tudy the posit ion of the onset of instabil i ty was

observed to depend on both the free s tream velocity and the tem

perature difference across the boundary layer. The condition for

thermal instabil i ty , GrXc Re. , :c"3 /2 > 100, was studie d in the Re ynold s

number range 2 X 103

to 105

and the co r respond ing Grashof num ber

range 4 X 10

6

to 3 X 10

9

. Th e therm al instability is therefore occurringat much lower Reynolds numbers than that for which hydrodynamic

instability occurs in a Blasius flow. In tha t case, where therm al effects

are absent, Tollmien-Schlichting waves consti tute the f irs t s tage of

transit ion at a Reynolds num ber around 5 X 106. Heating and cooling

are known to have an effect on this s tabil i ty l im it [24, 25]. The rm al

effects would also appear to influence the onset of Gortler vortices

on a concave wall [26]. Ano ther limiting case is tha t of no free s trea m

velocity (Re x = 0) which corresponds to natural convection above an

upward facing horizontal heating surface. The on set of instabil i ty in

this case has been studied by Pera and G ebh art [23]. The pres ent

experimental results can be seen as further defining the boundary and

mechanism of instability in the flow regime between the two limiting

cases of Re = 0, i.e., pure natural convection, and Gr = 0, i.e., pure

forced convection.

It will be noted, however, that as yet the entire range of instability

conditions has not been explored. In particular , for the f low regime

with high Reynolds and high Grashof numbers one may anticipate

that two instability modes, thermal and hydrodynamic, could coexist.

This f low regime remains, however, as a subject of further re

search.


The a uthors w ish to acknowledge the contribution s of Dr. G. S . H.

Lock in the conception and planning of the Water Tunnel Facility and

of Mr. Boochoon in i ts construction and operation. The construction

of the Water Tunnel Facil i ty and work on this project were funded

by the Natio nal Research Council of Canad a. The second author, M r.

Imura, wishes to thank Professor T . Fujii , Kyushu University , Ja pan,

for his encouragement to carry out this research.

References1 Pohlhause n, E., "Der Warmeaustausch zwischen festen Korpern und

Flussigkeiten m it kleiner Reibung und Kleiner Warmeleitung," Zeitschrift furangewandte Mathematik und Mechanik, Vol. 1,1921, pp. 115-120.

2 Gortler, H., "Uber eine Analogie zwischen den Instabilitaten laminarer

Grenzschichtstromungen an konkaven Wanden und an erwarmten Wanden,"Ingenieur-Archiu, Vol. 28, 1959, pp. 71-7 8.

3 Tani , I., "Produc tion of Longitudinal Vortices in the Boundary Layeralong a Concave Wall," J. G eophys. Res., Vol. 67, 1962, pp. 3075-3080.

4 Bippes, H., and Gortler, H., "Dreidimensionale S torungen in derGrenzschicht an einer konkaven Wand," Acta Mechanica, Vol. 14, 1972, pp.251-267.

5 Mori, Y., "Buoyancy Effects in Forced Laminar Convection Flow overa Horizontal Flat Plate," JOURNAL OF HEAT TRANSFER, TRANS. ASME,Series C, Vol. 83,1961, pp. 479-482.

6 Sparrow, E. M., and Minkowycz, W. J., "Buoyancy Effects on HorizontalBoundary Layer Flow and Heat Transfer," International Journal of Heat andMass Transfer, Vol. 5, 1962, pp . 505-511.

7 Chen, T. S., Sparrow, E. M., and Mucoglu, A., "Mixed C onvection inBoundary Layer Flow on a Horizontal Plate," JOURNAL OF HEATTRANSFER, TRANS. ASME, Series C, Vol. 99, 1977, pp. 66-71.

8 Wu, R. S., and Cheng, K. C, "Therma l Instability of Blasius Flow alongHorizontal Plate s," International Journal of Heat and Mass Transfer, Vol.19,1976, pp. 907-913.

9 Sparrow, E. M., and Husar, R. B., "Longitud inal Vortices in NaturalConvection Flow on Inclined Plate s," Journal of Fluid Mechanics, Vol. 37,1969,pp . 251-255.

10 Lloyd, J. R., and Sparrow, E. M., "On the Instability of Natural Convection Flow on Inclined Plates," Journal of Fluid Mechanics, Vol. 42,1970,pp . 465-470.

11 Haaland , S. E., and Sparrow, E. M., "Vortex Instability of NaturalConvection Flow and Inclined Surfaces," International Journal of Heat an dMass Transfer, Vol. 16, 1973, pp. 2355-2367.

12 Hwang, G. J., and Cheng, K. C, "Thermal Instability of Laminar NaturalConvection Flow on Inclined Isothermal Plates," Canadian Journal ofChemical Engineering, Vol. 51, 1973, pp. 659-666.

13 Lee, J. B., and Lock, G. S. H., "Instab ility in Boundary-Layer Free

Convection along an Inclined Pla te," Trans. Canadian Society for MechanicalEngineering, Vol. 1, 1972, pp. 197 -203.

14 Kahaw ita, R. A., and Meroney, R. N., "The Vortex Mode of Instabilityin Natural Convection Flow along Inclined Plates," International Journal ofHeat and Mass Transfer, Vol. 17, 1974, pp. 541-548.

15 Hwang, G. J., and Cheng, K. C, "Convective Instability in the ThermalEntrance Region of a Horizontal Parallel-Plate Channel Heated from Below,"JOURNAL OF HEAT T RANSFER , TRAN S. ASME, Series C, Vol. 95,1973,pp . 72-77.

16 Kam otani, Y., and Ostrach, S., "Effect of Therm al Instability onThermally Developing Laminar Channel Flow," JOURNAL OF HEATTRANSFER, TRANS. ASME, Series C, Vol. 98,1976, pp. 62-66.

17 Cheng, K. C, and Wu, R. S., "Axial Hea t Conduction Effects onThermalInstability of Horizontal Plane Poiseuille Flows Heated From Below,"JOURNAL OF HEAT TR ANSFER, TR ANS. ASME, Series C,Vol. 98,1976,pp . 564-569.

18 Hwang, G. J., and Liu, C. L., "An Experime ntal Study of ConvectiveInstability in the Thermal Entrance Region of a Horizontal Parallel-PlateChannel Heated from Below," Canadian Journal of Chemical Engineering,

Vol. 54,1976, pp. 521-525.19 Nakayam a, W., Hwang, G. J., and Cheng, K. C , "Therm al Instabil ity

in Plane Poiseuille Flow," JOURNAL OF HEAT TRANSFER, TRANS.ASME, Series C, Vol. 92,1970, pp. 61-68.

20 Akiyama, M., Hwang, G. J., and C heng,K. C, "Experiments on the Onsetof Longitudinal Vortices in Laminar Forced Convection Between HorizontalPlates," JOURNAL OF HEAT TRA NSFER, TR ANS. ASME, Series C, Vol.93, 1971, pp . 335-341.

21 Seban, R. A., The Laminar B oundary Layer of a Liquid with VariableViscosity, Heat Transfer Thermodynamics and Education, Boelter Anniversary Volume, Johnson, H. A., Ed., McGraw-Hill, 1964, pp. 3 19-329.

22 Chen, T. S., private commu nication, 1976.23 Pera, L., and Gebhart, B., "On the Stability of Natural Convection

Boundary Layer Flow over Horizontal and Slightly Inclined Surfaces," International Journal of Heat and Mass Transfer, Vol. 16, 1973, pp. 1147-1163.

24 Wazzan, A. R„ Okamura, T. T., and Smith, A. M. 0., "The Stab ility ofWater Flow over Heated and Cooled Flat Plates," JOURNAL OF HEATTRAN SFER, TRANS. A SME, Vol. 90,1968, pp. 109-114.

25 Strazisar, A. J., Prahl, J. M., and Reshotko, E., "Experim ental Study

of the Stability of Heated Laminar Boundary Layers in Water," ReportFTAS/TR-75-113, Case Western Reserve University, Dept. of MechanicalEngineering and Aerospace Engineering, 1975.

26 Kahawita, R. A., and Meroney, R. N., "The Influence of Heating on theStability of Laminar Boundary Layers along Concave, Curved Walls," ASMEJournal of Applied Mechanics, Vol. 99,1977, pp. 11-17.





S. J. RheeResearch Assistant

V. K. DhirAssistant Protessor,

M em . ASME

I.Catton

Professor,

M em . ASME

Chem ical, Nuctear , and Thermal Engineer ing

Depar tment ,

School o( Engineering and Appl ied Sc ience,

University of Cali fornia, Los Angeles,

Los Angeles, Cali f .

Natural Confection Heat Transfer in

Beds of Inductively Heated

Particles1

Experimental observations of the onset of convectiue m otion in beds of inductively heated

particles have been made. Data for the heat transfer coefficient were obtained from water-

cooled beds of steel particles. The particulate beds were formed in a 10.4 cm dia glass jar,

insulated at the bottom and on the sides. The free surface of a layer of water overlying the

bed was maintained at a constant temperature by a copper plate cooled with tap water.

In the experiments, the depths of the particulate bed and the overlying layer were var

ied.

The observations showed that increasing the depth of the liquid layer over the bed tend

ed to lower the critical internal Rayleigh number at which the onset of convection oc

curred. For overlying liquid layer-to-bed depth ratios of one or more, natural convection

was observed to begin at Ri =* 11.5. The heat transfer data for Rj > 11.5 are correlated

with N u = 0.190 Riom

However, with no liquid layer on top of the bed, natural convection

was observed to occur at Rj ^ 46, which is slightly higher than observed in earlier studies

made with Joule heating.


In recent years , considerable research has been done on the on set

of convection and heat transfer behavior in volumetrically heated

layers of l iquid or particulate beds, because of i ts applications to

certain hypothetical accident s i tuations in nuclear reactors . In this

paper, the convective transfer of heat from volumetrically heated

partic ulate beds covered with a f inite layer of l iquid is s tudied.

The earliest theoretical s tudy of the occurrence of natural con

vection in porous media f i l led with l iquid was made by Horton and

Rogers [ l ] ,2

who were interested in the distribution of sodium chloride

in subterranean sand layers . A similar problem was solved la ter by

Lapwood [2], and he obtained th e same result as Ho rton an d Rogers

did for the onset of convection in a porous m edium heate d from below

and bounded at top and bottom with r igid conducting boundaries .

Lapwood's criterion for the onset of convection is:

Bs ,mgMTLPm^

CtfVf

(1)

1This work was supported by the Reactor Safety Research Division of the

U.S. Nuclear Regulatory Commission under Agreement No. AT(04-3)-34P.A.223 M od. 1.

2 Numbers in brackets designate References at end of paper.Contributed by the Hea t Transfer Division for publication in the JOURNAL

OF HEAT TRANSFER and presented at the Winter Annual Meeting, Atlanta,Nov. 27-Dec. 2,1977. Manuscript received by the He at Transfer Division August 19,1977.

In equation (1), P is the permeability of the bed and is assumed to be

based on Ergun's equation [3]. However, the subsequent experimental

results of Morrison, Rogers, and Hort on [4] did not agree with their

own or Lapwood's prediction. The basic reason given at that time for

the discrepancy between experimental and theoretical results was that

the experiments were performed under unsteady conditions. Later ,

Ka tto and Masuo ka [5] s tudied, both theoretically and experimen

tally, the onset of convection in a porous medium subjected to a linear

temperature gradient. From their theoretical model they derived the

same criterion as that of Horton et al. and Lapwood for the occurrence

of convection; but they suggested th at the th erm al diffusivity of the

mixture, used in the Rayleigh number in equation 11], should be de

fined as the thermal conductivity of the porous medium divided by

the specific hea t and density of the f luid . Thus , their modified cri te

rion for the onset of convection is:

RE

_ k/g/3/ATLP4 r r 2 (2)

kmOtff

The data for the cri t ical Rayleigh number obtained for partic le di

ame ter-to- bed d epth ratios varying from 0.04 to about 1 , and using

glass, steel, and aluminum spheres, were found to correlate well with

equation (2) .

Analysis of heat transfer across a porous layer bounded between

two rigid walls was ma de by Gup ta and Jos eph [6]. Th e theoretical

curve maximizing the N usselt num ber as a function of the R ayleigh

number was computed numerically . The results of this numerical

analysis were shown to agree very well with the exper imenta l results

o f Bur e t ta and Berm an [71. Natural convective heat transfer data of

78 / VOL 100, FEBRUARY 1978 Transact ions of the ASMECopyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



Buretta and Berman for a porous layer heated from below indicated

a critical Rayleigh number of 33 , with a transit ion in the natu re of the

convective heat transfe r a t a Rayleigh numb er between 2 00 and 300.

The onset of instability in a porous bed with an overlying layer of

liquid has been studied bo th theoretically and experimentally by Sun

[8]. The porous bed and the overlying l iquid layer were bounded at

top and bottom with rigid conducting walls. The bed was heated from

below while the l iquid layer was cooled from above. Using p ertur ba

tion theory, Sun predi cted t he Rayleigh num ber for onset of convec

tion in terms of varying ratios, ?/, of the l iquid layer depth to the par

t iculate bed depth. Sun 's experimental observations of the cri t icalRayleigh number, for the range of IJ = 0 to 0 .21, compa red very fa

vorably with the data . His experimental data show that the cri t ical

external Rayleigh number for the bed decreased with r/, whereas the

crit ical external Rayleigh number for the overlying layer increased

with j). Sun also observed th at for a fixed value of the e xtern al R ay

leigh number for the bed, the Nusselt number increased as r) was in

creased.

The thermal instability of a volumetrically heated liquid layer with

an initially stabilizing or destabilizing temperature profile in the layer

was analyzed by Sparrow, Go ldstein , and Jonsson [9]. Fro m their

analysis , Sparrow et a l . concluded th at as the volumetric heating rate

is increased, a volumetrically heate d l iquid layer becomes more sus

ceptible to instability, and the convective motion initiates at smaller

Rayleigh numbers. Experimental observations of heat transfer, as well

as the init ia t ion of instabil i ty , in a Joule he ated layer of e lectrolyte

bounded horizontally between two rigid plates held at constant and

equal temperatures were made by Kulacki and Goldstein [10]. Kulacki

and Goldstein's experimental data indicated that the critical external

Rayleigh number, based on half the layer thickness, was 620 ± 30,

whereas the value predicted by l inear s tabil i ty theory was 560. The

heat transfer data for the uppe r and lower boundarie s with R / varying

from 580 to 3.78 X 105

and Pr varying from 5.76 to 6.35 were correlated

as

an d

N u 7 = 0.756 R / °2 5

N u 0 = 2.006 R/ '0 1

(3)

(4)

The observations of Kulacki and Goldstein have subsequently been

confirmed by the experimental work of Jahn and Reineke [11]. Theproblem of upward and downward heat transfer from a volumetrically

heated horizontal layer with a destabil iz ing tempe ratu re difference

at the upper and lower boundaries has been analyzed by Suo-Antti la

and C atton [12]. Their results are in general agreement with those of

Kulacki and G oldstein , but their magnitude s for the heat transfer a t

the boundaries are different from Kulacki and Goldstein 's observa

tions, where the bounding surfaces were held at the same tempera

tu re .

Experimental observations of the onset of natural convection and

heat transfer in volumetrically heated porous layers with r igid

bounding walls have been made by Buretta and Berman [7] and Sun

[8]. In both of these studies, the porous layer contained glass particles

and volumetric heating was accomplished by the Joule heating of

water containing 0.01 mole percent of C11SO4. The bottom of the bedwas an adiabatic surface, whereas the top r igid wall was held at a

cons tan t tem pera tu re . Fo r these boundary cond i t ions , na tu ra l con

vection was observed to occur at

R/,. a* 33 . (5)

In Bure t ta and B erman ' s da ta , a d i scon t inu i ty in Nusse l t number a t

abo ut R/ = 70 for 3 mm dia particles an d at about R/ = 200 for 6 mm

dia particles was observed. No experimental reason for this anomalous

behavior was given, and i t was postulated that bifurcation was

probably caused by the presence of two different unstable modes. The

data on the lower branch, covering the range 30 < R/ < 70 for 3 mm

dia particles and 30 < R/ < 200 for 6 mm dia particles, were found to

correlate with

log (Nu ± 0.034) = 0.237 log R/ - 0.356; (6a)

while the data in the upper branch, covering the ranges 70 < R/ < 900

for 3 mm dia particles an d 200 < R/ < 900 for 6 mm dia particles, were

found to correlate with

log (N u ± 0.070) = 0.553 log R/ - 0.871 (6b)

Sun did not observe any such discontinuity in his data and also did

not make an att em pt to correlate his data . But a curve faired throu gh

Sun's data would indicate that the heat transfer could be correlated

by

Nu = 0.116 R/0-

573(6c)

The correlation equation (6c) suggests the cri t ical Rayleigh number

is abou t 45 . This value is higher than Sun 's reported value of 33 and

the difference can be attr ibuted to variabil i ty in data and error re

sult ing from personal judgment in drawing a best l ine through thedata . The heat transfer predicted by equations (6b) and (6c) differ

from each other by only a few perce nt.

Analysis for the onset of natural convection in a porous medium

•No m en c l a t u re -

Cps, cpf = the specific heats of the particulate

material and the l iquid, respectively

D = the inner diameter of the glass jar

d = the diameter of the spherical partic les

/ = frequency

g = the acceleration of gravity

ks, kf, km = thermal conductivity of the part ic les , the l iquid, and volume averaged

thermal conductivity of the porous medi

um filled with liquid, (kf + (1 — e)ks, re

spectively.

L, Lf = the depth s of the particulate bed and

the overlying layer of liquid, respectively

m s, mj = the masses of the partic les and the

liquid in the particulate bed, respectively

Nu = the Nusselt number for the porous

layer, as defined in equation [7]

Nu/ = the Nusselt number for the overlying

layer of liquid, as defined in equation [9]

P = the permeabili ty of the porous medium,

£3d

2/150 (1 - e)

2

Q„ = the heat generation rate per unit volume

of the particulate bed, [(m scps + nifCpf)-

dTldt]IV

RE = the external Rayleigh number for the

particulate bed, as defined in equation

[2 ]R/ = the Rayleigh number for the overlying

layer of liquid, g&,(Tm - TT)Lj/af» f

R/ = the internal Rayleigh number for the

particulate bed, as defined in equation

[8]

R/c = crit ical in ternal Rayleigh number

R; = the internal Rayleigh number for a vo

lumetrically heated l iquid layer of depth

L, gt3fQ,L5/S2kfam


TB , TM , TT = the tempera tu res o f the bo t

tom of the bed, the particulate bed-liquid

layer interface, and the copper plate , re

spectively

Tmax - the max imum tempera tu re in the

bed

t = t ime

V = the total volume of the particulate bed,

(TT/4)D2L

z = distance from the bottom of the testcell

ctf = the thermal diffusivity of the liquid

fif = the coeffic ient of therm al expansion of

the l iquid

AT = the temperature difference between

any two bounding surfaces

« = the poro sity of the partic ulate bed

y] = the ratio of the l iquid layer depth to the

pa r t icu la te bed dep th , L//L

VJ = the kinem atic viscosity of the l iquid

S u b s c r i p t

c = at the center line of the test cell





with internal heat generation has recently been made by Gasser and

Kazimi [13]. In their model, the top of the par ticu late bed is assum ed

to be a free surface. For bottom surface temperatures greater or less

than the free surface temperatures, Gasser and Kazimi have obtained

a relationship between the cri t ical in ternal Rayleigh numb er and the

external Rayleigh number. More recently , Hardee and Nilson [14]

have made experimental observations s imilar to those of Sun. Their

experim ental results , obtained w ith electrolytically heate d, sodium

chloride-saturated beds of sand, agree quite well with Sun 's data .

The primary purpose of the present work is to experimentally de

termine the criteria for the onset of convection and the magnitude ofconvective heat transfer in a volumetrically heated particulate bed

when a l iquid layer exists over the bed. This configuration is more

general than those stud ied previously, and is c loser to s i tuations tha t

may occur in practice, such as during the cooling of a fuel bed after

a hypothetical core disruptive accident in a l iquid me tal fast breed er

reactor. In the present experiments , induction heating is employed

so tha t the heat is generated in the metall ic partic les rathe r than in

the coolant. The bottom surface of the bed is insulated, while the top

surface of the l iquid layer is mai ntain ed at a cons tant tem pera ture .

Exper imental MeasurementsIn this work, dist i l led water was chosen as the tes t l iquid. Prelim

inary experiments with 800-1,000 m/^ partic les showed that a t one

atmo sphere sy stem pressure , i t was very diff icult to a tta in the onset

of convection prior to the incipience of boil ing in the bed. Thu s, to

reduce the resistance of the bed, larger particles were required. In all

of the experime nts repor ted in this pa per, 304 stainless steel balls 6.35

mm in diameter and having a sphericity of 5 .08 m/i were used. Da ta

for the onset of convection and convective heat transfer were obtained

for particulate bed depths of 26 mm (d/L =* 0.244) an d 52 m m (d/L

~ 0.122), while the depth of the l iquid layer overlying the bed was

varied parametrically from 0 to 209 mm.

A p p a r a t u s a nd Pr oc ed ur e. A schematic diagram of the test cell

used in this s tud y is shown in Fig . 1. Th e partic ulate b ed was form ed

in a 104 mm ID pyrex glass jar . The pyrex jar was insulated on the

outside with a thick layer of ginned cotton. Five 2.5 mm dia holes,

equally spaced along the diameter, were drilled in the base of the jar.

Thirty-gauge chromel-alumel thermocouples were passed through

these holes and pasted to the bottom surface of the jar with epoxy

resin . The glass jar was supp orted on a 20 mm thick plexiglas block.

Inductio n he ating of the partic les was achieved by placing a 10 turn,

210 mm ID work coil around the jar, and connecting the coil to a 10

kW, 453 kHz radio frequency generator. T he work coil was made of

a 4.75 mm OD copper tube. In order to obtain uniform heating of the

particulate bed, constant clearance (=^2 mm) was maintained between

different turns of the coil by holding it in a plexiglas fixture.

The surface of the l iquid layer overlying the bed was maintained

at a constant te mpe rature by a 6 mm thick copper plate , forming the

base of a cylindrical chamb er 103 mm in diameter and 40 mm in

height. Tap water was circulated through this chambe r. The surface

of the copper plate touching the layer of liquid was finely polished,

while the chamber s ide surface had fins to increase the heat transfer

area betw een the plate and the cooling water. Six 30 gauge chromel-

alumel thermocouples were embedded on the surface of the cooling

plate a t d ifferent radii and angular posit ions. The se therm ocouples,

together with three other s imilar thermocouples used to determine

the temperature at the interface between the particulate bed and the

liquid layer, were carried through three sleeves passing through the

cooling chamber. The thermocouple outputs , after passing through

a selector switch and a f i l ter to elim inate radio-frequency noise [15],

| were read on a Houston X-Y recorder.

Prior to each experim ent, the partic les and t he glass jar were

thoroughly washed with acetone. The jar and partic les were then

dried, the glass jar weighed, and the par tic les were added to the do-

sired de pth in the jar. Th e glass jar was weighed again, and t he weight

and depth information were used to determine the porosity of the bed.The porosity of a ll particulate beds studied in this work was about

0.39, and the ma ximum error in determining the porosity is expected

to be less than ±2.5 percent. Calibration of the rate of heat generation

in the particulat e bed as a function of the fractional power o utpu t of

the radio frequency generator was made for each bed depth. For the

calibration runs, the particulate bed was saturated with dist i l led

water; s tart ing with the bed at an equilibrium reference te mpera ture

and a certain power inpu t to the work coil , the tem pera ture at one of

the locations in the bed was recorded until the water s t arted to boil .

Knowing the rate of tem pera ture increase, the thermal capacity , and

the volume of the bed (partic les and w ater) , the heat generation rate

per unit volume of the bed could be determined. The rate of tem

pera ture increase at various locations in the bed did not show a dif-

cold water

Copper

Cooling Plate

104 mm ID

Pyrex Jar

Thermocouples (1,2,3,4,5)-

Fig. 1 A schem atic diagram of the experim ental set-up





ference of more than ±5 percent. In calculating the volumetric heating

rate, a mean value of three horizontal and two vertical locations was

taken. Complete experimental details are given by Rhee [16].

After the calibration, more dist i l led water was added to m a k e an

overlying liquid layer of a b o u t the des i red dep th . The work coil was

then energized and the water was deaerated by boiling it for more than

20 min. Deaerat ion of the wate r was done to preclude the possibil i ty

of any air pockets in the bed. thereafter, the power was turned off, and

the particulate bed and overlying layer were allowed to cool. Th e

cooling chambe r was the n lowered into the glass jar and gently pressed

againstthe top of the

par t icu la tebed to

smoo thout the

in terface,which might have been disturbed during deaeration. Thereafter , the

cooling cham ber was moved to the desired height and firmly fixed with

clamps. Finally , the pos i t ions of the the rmocoup les at the in terface

between the par t icu la te bed and the l iquid layer were adjusted, and

the particulate bed and liquid layer were allowed to a t ta in the rma l

equilibrium.

Starting with the bed in a static equilibrium state , the radio fre

quency generator was switched on and a desired power applied to the

work coil. The t e m p e r a t u r e of one of the the rmocoup les at the base

of the particula te bed was recorded on the X-Y recorder. Initially, the

t empera tu re was found to increase l inearly , but la ter the ra te of

temperature increase s tarted to fall off asymptotically as steady state

was approac hed. If the heat generation rate in the bed was such th at

the internal Rayleigh number was grea te r than the crit ical Rayleigh

number, the t empera tu re in the asymptotic range started to fluctuate

with t ime. These f luctuations were an indication that f luid motion

within the bed had begun. Usually , tempe rature at one of the locations

at the base of the par t icu la te bed was monitored for about 15 minutes

to insure that pseudo-steady state had been established. Thereafter ,

tempera tu re at other locations at the base of the par t icu la te bed, at

the particulate bed-liquid layer-interface and at the cooling plate were

monitored one at a t ime for a span of about f ive minutes. The fluc

tuations in the pseudo-steady state temperatures possessed a certain

characteris t ic frequency and amplitude. Overall it took one to t h r e e

hours to reach steady state in the porous layer after the work coil

was energized. This t ime is of the same o rde r as reported earlier by

Kulacki and Goldstein [10]. When the heat generation rate in the bed

was such that heat was transferred by pure conduc t ion , no oscil

la tions in t empera tu re were obse rved when a s teady s ta te was

achieved.

After noting the s teady s ta te tempera tu re shapes at various locations, th e power was t u r n e d off, the system was allowed to come to a

static equilibrium state , and the above procedure was repea ted for

different power inputs and/or for different depths of the overlying

liquid layer.

D a t a R e d u c t i o n . G e n e ra l ly , a f ive-minute t ime-averaged tem

pera tu re was obtained from the t empera tu re - t ime t races . T hese

temperatures were then used to calculate a mean tempera tu re for the

surface of in te res t . For the average tempera tu res of b o t t o m and in

terface, an a rea -we igh ted tempera tu re of different thermocouples

placed symmetrically about the axis was taken. However, for the

cooling plate , an a r i thmet ica l ly ave raged tem pera tu re was chosen as

the mean temperature, because the maximum temperature difference

between extreme observations was relatively small. Also, as the

thermocouples on the cooling plate were placed at different radii and

angular posit ions, the extra effort needed to obtain area-averagedt e m p e r a t u r e s was not seen to yield any significantly better informa

tion. For the most severe conditions s tudied in the present experi

ments , when the m a x i m u m t e m p e r a t u r e in the par t icu la te bed was

only a few degrees below the sa tu ra t ion tempera tu re , the m a x i m u m

deviation in the coo l ing p la te temp era tu res was +3.6, —4.1 K, while

similar values for the in terface and bottom surface were +9.8, —7.5

K and +8.3, -4. 8 K, respectively.

F or the thermophysical properties required to evaluate the Nusselt

and Ray le igh numb ers of the overlying liquid layer, the m e a n of the

top cooling plate and the particulate bed-liquid layer interface tem

peratures was chosen as a reference temperature. For the particulate

bed, the average of the bo t tom and the in ter face tempera tu re s was

chosen as the re fe rence tempera tu re .

R e s u l t s

A to ta l of 27 observations for convective heat transfer in vo lume-

trically heated particulate beds with nearly insulated boundaries, and

with an overlying layer of liquid cooled at the top, were made. Out of

these, f ive observ ations fell within the region where conduction was

the sole mech anism of hea t transfer in the porous layer. Data for the

overlying l iquid layer depth-to-particulate bed dep th ra t io s , i\, of 0,

0.21,1.0 , and 3.38 were taken, where the bed d e p t h was 54 mm (LID

= 0.5), while data for rj = 8.04 were taken with a bed d e p t h of 26 mm

(LID = 0.25). These data are t abu la ted in reference [16].

The dimensionless rate of heat transfer from the bed was based onthe temperature difference between the bo t tom of the bed and the

particulate bed-liquid layer interface, and is defined as:

N n -Qu L2

%km{TB — T A(7)

In equation (7) the conductivity , km, of the mix tu re is based on the

volume-averaged conductivity of the steel and water in the particulate

bed . The in te rna l Ray le igh numbe r in the bed is defined as:

R/kglfQjSP

2kn

(8)am

while the external Rayleigh number, HE, is defined in the same

manner as suggested by K a t t o and Masuoka [5] and given by equation

(2) . The Nusse l t number for the l iquid layer is based on the t emper

ature difference betweenthe

particulate bed-liquid layer interfacean d the copper plate , and is defined as:

N u , ;QuLLf

(9 )k,(TM - TT)

T he Ray le igh number for the liquid layer is defined in the con

ventional manner. The uncer ta in t ie s in the evaluation of the Nusse l t

and the in ternal Rayleigh numbers for the bed are within ±7.1 percent

an d ±10.5 percent, respectively. Similar values for the Nusse l t and

Rayle igh numbers for the l iquid layer are ±6.2 percen t and ±6.1

percent, respectively.

O n s e t of Na tura l Co nv ec t io n and Hea t Tra ns fer . Fig. 2 shows

the dependence of the dimensionless heat transfer on the in te rna l

Ray le igh numb er of the bed for different values of ?/. The d a t a for r\

— 0 are a b o u t 5-10 percen t lower than Sun ' s [8] and B u r e t t a and

B e r m a n ' s [7]heat transfer correlation, equations (6b) and (6c). This

is within the uncer ta in ty of the presen t da ta as well as those of Sun,

and Bure t ta and Berman . H owever, th e dependence of the Nusse l t

n u m b e r on the in te rna l Ray le igh number is abou t the same as re

ported by Hardee and Nilson [14], Sun, and Bure t ta and Berman . The

crit ical Rayleigh number, obtained by no t ing the in te rcep t on the

abscissa (Nu = 1) of a straight l ine drawn through the da ta po in ts ,

shows the crit ical Rayleigh number to be abou t 46 . The values of R/c

in references [7 ,8,14], where Joule heating of the l iquid in the pres

ence of glass or sand particles was employed, were reported to be abou t

\

-

-

"

-

r, - 0.00

n = 0.21

r, = 1-00

n = 3.38

n = 8.04

—1_®J—i_J

u ^ m

$©

A

Q

V

~ K £ & y ^

- R e f_Ref

Pre

ra ' ' ' ' =[ 14 ] ^

sent Study s*s

»„ - 0.190 * ° -W

\ l / ^

«" • 0."8 ?.?^W

/SyC- 0.116 «,°-573 [7]S[3],

Fig. 2 Dependence of particulate bed heat transfer on Internal Rayleighnumber anddimensionless liquid layer depth

Journa l of Heat Transfer FEBRUARY 1978, VOL 100 / 81




33 . The higher value of R/c obtained from extrapolation of the present

data is not though t to be due to the bottom surface of the test cell not

acting as a perfectly insulated surface. Th e maxim um h eat loss from

the bottom of the test cell is expected to be less than 0.7 watts or 0.3

percent of the heat generated in the particulate bed. The conductance

of the bo ttom of the cell is calculated to be 1 W/m 2— K (Bio t number

=* 10), which is too small to be of any significance. However, the

probable cause of the higher R/ c could be the additional resis tance

to transfer of heat from the partic les to the coolant. The thermal

conductivity of s teel partic les is abou t 20 t ime s higher than t ha t of

water; the s teel partic les thus can be considered to be isothermalrelative to a water layer of dept h equal to the d iameter of the sp here.

For a given temperature drop across the l iquid layer, local heat

transfer from the partic les to the thermal layer would depend on the

natu re of f low around th e partic les . Use of volume averaged therm al

conductivity is justif ied when there is no fluid m otion in the partic

ulate bed, however at the onset of convection, the thermal resistance

at the surface of the partic les may re sult in a s l ightly higher tem per

ature drop across the bed. In all of the Joule heating studies [ 7 ,8,14 ],

the ther mal conductivity of the glass or sand partic les used was ab out

the same as that of the coolant. The partic les thus acted only as hy

draulic resistances and were thermally inactive. The volume-averaged

thermal conductivity of these porous layers under conditions of simple

conduction would not differ much from natural convection conditions.

The data of Katto and Masuoka [5] for the onset of convection in

bottom heated particulate beds also indicate that the cri t ical Rayieigh

number with steel and aluminum balls was invariably higher than with

glass balls . Thus, i t would seem that dependent on the relative con

ductivities of the coolant and the particles, some variations in R Ic ar e

possible when the cri t ical Rayieigh number is obtained by extrapo

lating the natural convection data .

For a volumetrically heated particulate bed having upper boundary

as a free surface and the lower bou ndary as a rigid surface, Gasser and

Kazimi's [13] analysis showed that for external Rayieigh number less

than 40, the cri t ical in ternal R ayieigh num ber could be a double val

ued function of R#. The higher value of cri t ical in ternal Rayieigh

num ber occurred when the free surface tem peratu re was greater tha n

the bottom temp erature. In the present work, the boundary conditions

are different tha n those trea ted in reference [13] and external Rayieigh

number cannot be varied independently of internal Rayieigh number

and vice versa. By definit ion of Nusselt number, the external and

internal Rayieigh number are equal a t the onset of convection. Al

though obtained under different bounda ry conditions, i t is surpris ing

to note that for no l iquid layer a t the top and particulate bed bo ttom

temperature greater than the free surface temperature, the cri t ical

internal Rayieigh number given in reference [13] is about 40 when the

external Rayieigh number is a lso about 40.

The heat transfer is seen to increase by about threefold when a

liquid layer of about one fifth of the bed depth lies over the particu late

bed. The rate a t which the heat transfer increases with R/ a lso in

creases as the d epth of the liquid layer increases. As i; increases to one,

further im prove men t in heat transfer is observed. But the data for

tj = 3.38 and 8.04 mer ely over lap the d ata for r/ = 1. Th is me ans th at

no further im provem ent in heat transfer is realized for r/ greater than

one . The heat transfer data for t j = 0 .21 is correlated within ±10

percent as:

Nu = 0.138 R ,° w for r, = 0.21 (10)

while the data for all other values of IJ are correlated within ±26 per

cent as:

Nu = 0.190 R/0-

690f o r i ) > l . ( 1 1 )

The heat transfer given by equation (11) is abou t four t imes h igher

than the heat transfer observed by Sun and Buretta and Berman when

no liquid layer is present over the bed. In other words, the convective

motion in the overlying liquid helps to remove more heat from the bed.

Later on we will d iscuss the l iquid layer heat transfer further.

The cri t ical in ternal Rayieigh number is plotted as a function of

i] in Fig . 3 . I t is noted tha t the R ayieigh num ber a t the onset of con-

Fig. 3 Influence of T\ on internal Rayieigh number at the onset of convectionin the bed

vective motion in the bed decreases rapidly as r\ is increased to one.

Thereafter, no further reduction is observed and the critical Rayieigh

num ber reaches an asym ptotic value of about 12; i .e . ,

RIc c± 12 for n > 1. {V.\

These results indicate that the convective motion in the overlying

layer of liquid tends to destabilize the fluid in the bed at lower volu

metric heat generation rates than are required for destabil izatiorwithout the layer.

The Nusse lt nu mbe r of the overlying l iquid layer is plotte d in Fiu .

4 as a function of the liquid layer Rayieigh number, R/. In this Fig.,

the g eneral correla tion of Hol lands, et al. [17], for natur al convection

between two paralle l p lates , is a lso plotted. I t is observed that t in-

present data for the liquid layer, in which the boundary layer at tin-

partic ulate bed-liquid layer interface is disru pted because of the in

flow and outflow of liquid to and from the porous bed, is about 2.5' u

3 times higher t han for a liquid layer boun ded between tw o rigid wall-.

However, the functional dependence of the Nusselt number on the

Rayieigh number is about the same as for Benard convection between

two paralle l p lates . The he at transfer dat a for the l iquid layer can be

divided into two groups. For the laminar range (10 4 < R/ < 4.0 X 10('i.

the heat transfer data are correlated within ±28 percent by:

Nu f = 0.715 fl,-0-24 8 for 104 < Rf < 4.0 X 10 6 (i:'.l

In the turb ulent region the data are , within ±33 percent, represented

by :

Nu/ = 0.234 Rf0-

301for 5 X 10

7< Rf < 3.5 X 10

10(M i

Th e region of tran siti on from lam inar to tub ule nt flow is seen to lie

between the Rayieigh num bers of 4 X 106 and 5 X 107. The transition

region is, again, about the same as has been observed for natur-i!

convection between two paralle l p lates [18].

The coupling of the heat transfer across the layer of liquid with tin'

in ternal heat generation rate in the particulate bed is shown in Fiu ' .

5. Here, the Nusselt number of the layer is plotted as a function of K/

for different values of >/. F o r rj = 0.21 and 1.0, the convective motion

in the liquid layer is lamina r, and in transition region and heat transfer

is seen to increase as R/0-

2. Th e Rayieigh num ber of the layer for i; =

3.38 is in the turbulent range, and the heat transfer is seen to increase

as R/0'25

. Th e da ta for 7/ = 8.04, also in the t urb ule nt r ang e, show :i

s l ightly higher dependence (R/0-

2 6) on R/. The higher slope for r) -'-

8.04 arises because of the low value of Nu/ o bserved a t R/ =* 10. Th:-

Ray leigh n um ber is close to the onset of convectio n in the b ed; als->.

the particulate bed is much shallower for ?; = 8.04 (d/L = 0.244). I'

is possible that initially the flow is affected by the proximity of th' 1

rigid wall to the liquid layer-particulate bed interface. For all of tin'

dat a plotted in Fig. 5, it is observed tha t the cooling plate temp erature

remains nearly constant (289 ± 4 K). This implies that th e p artic ular

bed interface temperature increases as Q v4 / 6

or Q„3/ 4

, depend ing on

whether the f low in the upper layer is laminar or turbulent, respei '

t ively .





• T - rTTTTt - r— r— T— r - rm T | r— T I I I I TT ] I I T M I l l |T—|—| | | ITTTO n - 0.21A n • 1.00a n = 3.387 1 " 8.04

T r TTTTT

\*— Tran si t ion Region -W

Nu , - 0.715 R

. m l i I I I i,„i ) i ,1 i i i i i n 1i i i i 1 1 | l i i i L - M - L L L i j L I i n ,

F i g . 4 C o r r e l a t i o n o f o v e r l y i n g l i q u i d l a y e r h e a t t r a n s f e r d a t a

i M I N I r~

Fig. 5 D e p e n d e n c e o f l i q u i d l a y e r h e a t t r a n s f e r o n In t e r n a l R a y l e i g h n u m b e r

o f t he bed

Knowing the Nusselt number of the l iquid layer a t R/ c from Pig.

5, the Rayleigh nu mb er of the l iquid layer a t the onset of convection

in the particulat e bed is obtained from Fig. 4. These da ta are plotted

in Fig. 6 as a function of -q. The liquid layer Rayleigh number at the

incipience of convection in the bed increases very shar ply as i) is in

creased from zero to two, but thereafter R/ reaches an asymptoticvalue of about 5.5 X 10 8 . Fig. 6 shows that for all of the values of r\

studied in the present work, convective motion already exists in the

liquid layer while conduction is s t i l l the me chanism of hea t transfer

in the pa rticula te bed. As the overlying l iquid layer has access to th e

particulate bed, the convective motion in the overlying layer tends

to induce motion in the f luid saturating the pores of the particulate

bed. The motion of the f luid in the overlying layer becomes more

vigorous as rj is increased and will tend to destabil ize the l iquid at

lower heat generation rates; i t will a lso tend to enhan ce the h eat re

moval from the bed. This is exactly the trend observed in Fig . 2 and

3. As 7} is increased to about two, R/ corresponds to a fully developed

turbulent convective motion in the l iquid layer, and no further ad

vantage either in terms of a reduction in R/ c or an increase in the

Fig. 6 Variation of overlying liquid layer Rayleigh number with ij at the timeof onset of convection in the bed

Nuss elt nu mb er of the bed is realized by increasing ?/ further. The

asym ptotic na ture of R/ a t the onse t of convection in the bed is con

sis tent with the s imilar behavior of R/ c as n is increased beyond





Fluid Flow Pattern and the Temperature Distribution in the

Particulate Bed. Th e thermocouple readings for the temperatures

across the particulate bed-liquid layer interface and at the bottom

of the particulate bed indicate that liquid generally ent ers the bed on

the sides and exits from th e central portion of the bed. However, in

two observations for 7] = 0.21 and 3,38, when the heat generation rate

in the particulate bed was such that RI was close to RIc, the thermo

couple readings indicated that the liquid entered on one side of the

bed and left from the other side. As the volumetric heating of the bed

was increased, th e one-cell mode of convection shifted to a two-cell

mode. These observations still need to be confirmed with visual observations.

As pointed ou t earlier, the steady state temperature at a particular

location in the bed or the overlying liquid layer fluctuated with time.

These fluctuations were quite regular and slow at th e base plate of

the particulate bed, but were random and faster at the particulate

bed-liquid layer interface and the top cooling plate. Th e frequency

of the oscillations in temperature at th e center of the particulate bed

base plate is listed in Table 1 for various values of 1] and R/. It is ob

served that th e frequency tends to increase with RI and 1]. This is

indicative of the close relationship between the vigor of the fluid

motion in the bed and the oscillations in the local temperatur es. The

temperature distribution at the centerline of the particulate bed and

the liquid layer when 1] = 1, the depth of the particulate bed is 52 mm,

RI = 403, and Rf = 9.3 X 107, is shown in Fig. 7. The upper and lower

bounds observed for the local tempera ture during a time period of fivemin are also marked in Fig. 7. The bottom of the particulate bed is

observed to behave as an insulated surface, whereas the maximum

temperature occurs near the middle of the bed. The thermal boundary

layers at the particulate bed-liquid layer interface and at the cooling

plate are quite thin, though precise temperature profiles in th e

boundary layers were not determined because better instrumentation

would be needed to measure at the small distances involved. Th e

temperature in the bulk of the liquid layer is observed to remain fairly

constant.

Th e dimensionless maximum temperatures in the particulate bed,

noted from· temperature profiles at the centerline of the test cell

(similar to the one shown in Fig. 7), are plotted in Fig. 8 for various

values of 1] and R/. Th e data of Fig. 8 show that for 1] = 0.21 and 1.0,

when the motion in the overlying layer of liquid is laminar in nature,

an increase in RI causes the maximum temperatur e in the bed to in

crease also. However, for 1] = 3.38, an increase in RI is seen to result

in a slightly lower maximum temperature. It is postulated that th e

highly turbulent nature of the overlying layer of liquid for 1] 2: 3.38

and RI > RIc tends to induce motion deep enough into the particulate

bed to lower the maximum temperatlll'e.

Conclusions

• The onset of natural convection and heat transfer in volumetri

cally heated particulate beds having finite overlying liquid layers has

been observed.

Table 1 Frequency of oscillations in the temperature at the center of thebase of the particulate bed

f1] RJ (Hz X 103)

69 1.667

0.21347 2.381

1,030 3.8492,510 7.680

1.0089 1.922

403 3.33389 3.030

3.38410 4.762

1,200 6.1733,250 8.333

32 3.333

8.04134 5.051468 9.804

1,180 11.905


• The presence of an overlying liquid layer tends to destabilize the

fluid in the particulate bed at lower heat generation rates than would

be required to destabilize a bed without liquid above it. For liquid

layer depth-to-bed depth ratios, 1], greater than one, natural convec_

tion is observed to occur at

for 1] 2: 1

• The presence of an overlying layer of liquid tends to enhance the

transfer of heat from th e particulate bed. The heat transfer data for

1] 2: 1 are correlated by

Nu = 0.190 R[O.690 for 1] 2: 1

• The heat transfer capacity of the overlying layer of liquid is

considerably enhanced because of the inf10w and outflow of liquid to

and from the particulate bed. The heat transfer data for the liquid

layer when an inductively heated porous bed, of permeability 4.2 X

10-2 mm 2, lies underneath, are correlated by

NUf = 0.715 RfO.248 for laminar f10w

and

NUf = 0.234 Rfo. 307 for turbulent flow.

The transition from laminar to turbulent flow was observed to occur

for Rf of between 4 X 106 and 5 X 10 7.

1.2 , ______ -______ -_....,

], 0

0. 8

u

0. 6

u...'";

t: 0.4

0.2

Bottom

I ,. '?--,!,---"'! '-..........I .

IPartiCUlate ~ Overlying

Bed Layer I Fluid Layer

~ ! n t e r F a c e

0.2 0.4 0.6 0. 8

Z/(l + Lfl

], 0

Top'. 2

Fig. 7 Temperature profile at the center line of test cell for 1] = 1 and Or "0.4447 W/cm3

, R, = 9.3 X 108, R, " 403

1.2...---------------------,n' " 0.21

n '" 1 ,0

J'\ ,,1.,

.J I-,g

1.0 L,o---.:::::...---,..lo'--------,L03:-------:lO'

Fig. 8 Dependence 01 maximum temperature at the center of the lest cellon R, and '1/

Transactions of the ASME




References

1 Hor ton, C. W., and Rogers, Jr . , F. T., "Convective Curr ents in a Porous

M e d ium , " Journal Applied Physics, Vol. 16,1945, pp. 367-370.2 Lapwood , E. R., "Convection of Fluid in a Porou s Med ium ," Proc. Camb.

Phil Soc, Vol. 44,1948 , pp. 5 0 8 - 5 2 1 .

3 Leva, M., Fluidizadon, McGraw-Hil l , New York, 1959.4 Morrison , H. L., Rogers, F. T., and Ho rton, C. W., "Conv ective Cu rrents

in Porous Media , I I : Observa t ions of Condi t ions a t Onse t of Convec t ions ,"Journal Applied Physics, Vol. 20,1949, pp. 1027-1029.

5 Ka tto, Y., and Masu oka, T., "Criterion for the Onset of Convective Flow

jn a Fluid in a Porou s Medi um, " International Journal Heat Mass Transfer,Vol. 10,1967, pp. 2 97-309.

6 Gup ta, V. P., and Joseph , D. D., "Bou nds for Heat Transp ort in a PorousLayer," Journal Fluid Mech., Vol. 57, Par t 3 ,197 3, pp. 491-514.

7 Buret ta, R. J. , and Berman , A. S., "Convective Heat Transfer in a Liquid

S a tu r a te d P o r ous L a ye r , " AS M E Journal of Applied Mechan ics, Vol. 43, No.2 ,1976.

8 Sun, W. J . , "Convec t ive Instabi l i ty in Superposed Porou s and F ree

Layers ," Ph.D. Disse r ta t ion, U nivers i ty of Minnes ota , 1973.

9 Sparrow, E. M., Goldstein, R. J. , and Jonsson , V. K., "Th ermal Instability

in a Hor izonta l F luid Layer : Ef fect of Boundary C ondi t ions and Non-L inearTempera ture Prof i le ," Journal Fluid Mechanics, Vol. 18 ,1969, pp. 513-528 .

10 Kulacki, F. A., and Goldstein, R. J. , "Therma l Convection in a Horizontal

Fluid Layer with Uniform Volumetr ic Energy Sources ," Journal Fluid Me

chanics, Vol. 55, Par t 2 , 1972 , pp. 271-28 7.

11 Jah n, M., and Reineke, H., "Fre e Convection Heat Transfer with InternalHe a t S ou r c e s— C a lc u l at i ons a nd M e a su r e m e n t s , " Proceedings 5th Interna

tional Heat Transfer Conference, Vol. I l l , Nos. 2 -8 , Tokyo, 1974.

12 Suo-An ttila, A. J. , and Catto n, I. , "Th e Effect of Stabilizing Tem perat ure

Gradient on Heat Transfer from a Molten Fuel Layer with Volumetric Heating,"ASME Paper No . 74 WA/H T-45 ,1974 .

13 Gasser, R. D., and K azimi, M. S., "Onset of Conve ction in a Porou s

Medium with Inte rna l H ea t Gene ra t ion," ASM E JOURNAL OF HEATTRANSFER, Vol. 98, No. 1,1976, pp . 49 - 54 .

14 Hardee , H. C , and Nilson, R . H. , "He a t Transfe r Regimes and Dry out

in a Porous Bed of Fue l Debr is ," Nuclear Science and Engineering, Vol. 63,

No. 2 , 1977 .

15 Dhir, V. K., and Catto n, I. , "Dry out Hea t Flux es for Inductiv ely Heate dPar t icula te Beds," ASME JOURNAL OF HEAT TRANSFER, Vol . 99, No. 2 ,

1977.

16 Rhee , S . J . , "Convec t ive Hea t Transfe r in Beds of Induc t ive ly H ea tedPar t ic les ," M.S . in Engineer ing, Univers i ty of Ca l i fornia , Los Ange les , June

1977.

17 Hollands, K. G. T. , Ra i thby, G. D. , and Konicek, L. , "Corre la t ionEqua t ions for Free Convec t ion H ea t Transfe r in Hor izonta l Layers of Air and

Wa te r , " Internal Journal He at Mass Transfer, Vol. 18, Nos. 7-8, 1975, pp.879-884.

18 Cat ton , I . , "Na tura l Convec t ion in Hor izonta l Liquid Layers ," Physics

of Fluids, Vol. 19, 1966, pp. 2522 -2522 .

Journal of H eat Transfer FEBRUARY 1978, VOL 100 / 85Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



R. Greif


University of Cali fornia,

Berkeley, Colli.

An Experimental and Theoretical

Study of Heat Transfer in Vertical

Tube FlowsAn experimental and theoretical study was carried out for the heat transfer in laminar

and turbulent tube flows with air and argon. Radial temperature profiles w ere measured

at a location 108 tube diameters from the inlet of the vertical, electrically heated test sec

tion. The temperature of the tube wall was also measured. The experimental data were

in good agreement with the results obtained from numerical solutions of the conservation

equations and from simplified, fully developed solutions. F or turbulent flows the Reyn

olds numbers varied from 10,000 to 19,500; for laminar flows the Reynolds numbers varied

from 1850 to 2100 while the Rayleigh numbers varied from 70 to 80.


The present work is a s tudy of the heat transfer to a gas f lowing

in a vertical tube (cf. Donovan [15]1) . Meas urem ents have been made

with air and argon in both turb ulen t f low and in lamin ar f low. The o

retical results have been obtained from numerical solutions of theconservation equa tions as well as from solutions of s implif ied for

mulations for both turbulent and laminar f lows.

The determination of the heat transfer in turbulent pipe f lows has

been the subject of many investigations and comprehensive reviews

and bibliographies are available which provide a summary of results .

The present work on turbulent f lows therefore serves as a check on

the exp erimenta l system and also gives a basis for appraising m odels

for tu rbu len t t r anspo r t by d i rec t compar ison wi th the measu red gas

temp eratur e profiles . Experimental results were obtained over a range

of Reyn olds num bers va rying from abo ut 10,000 to 19,500.

Measurements of gas temperature profiles were also made in lam

inar f lows for Reyno lds num ber s varying from 1850 to 2100 and for

Rayleigh numbers varying from 70 to 80. Comparisons with theoretical

predictions were made and over the range of conditions tested the

importance of natural convection was clearly demonstrated.

B a s i c E q u a t i o n s

The flows investigated in this s tudy are two dimensional, s teady

and take place in a tube that is e lectr ically heated and suspended in

the vertical direction. The governing equations for the conservation

of momentum in the axial (vertical) direction and the conservation

of energy are given by

1 Numbers in bracke ts designa te References a t end of paper .

Cont r ibuted by the Hea t T ransfe r Divis ion for publ ica t ion in the JOURNALOP HEAT TRANSFER. Manuscr i pt rece ived by the Hea t Transfe r Divis ion

January 3 , 1977.

du i>upu V pv —

dX dr

i a r du"i dP

dH m i dpu —— + pv — =

bX dr r dr

rMi r

Ple/f L

dH

• ] |( Id i+ ( P r e / / - l ) u * l

Teff L d r

These equations are valid for laminar or turbulent flow; in the formercase the subscript , eff, is dropped and the result ing variables then

correspond to the usual molecular trans port qua ntit ies (cf. equations

(2) and (4)). The conserva tion equations, including the conservation

of mass, were solved numeric ally using the m etho d of Spalding ami

Pa tan ka r [1], Th e particul ar pro gram used in this s tudy is specified

in detail by Mason [2] (cf. Donovan [15]) and will not be repea ted here.

For completeness i t is noted that H is the total enthalpy.

For turbulent f low we take

Mf p + H T = p + p(2 du

dr&

in conjunction w ith the following relations for the mixing length:

e' = K(ra-r) for K( r0 - r ) < \r 0 (3a)

£' = \r0 for K( r0 - r) > Xr0, CM

along with the van Drie st dam ping factor [3] given by

t = r [ l - e x p | - ( r 0 - r ) ( r J r i " 2 W * | ] (3 c)

wi th K = 0.44, X = 0.09, and A* = 26. For the effective thermal con

ductivity we similarly take keff = k + kr so tha t

(p e///M)Pr

keff

1 + (?-)P r

P r y

(4)

In ou r ca lcu lat ions the tu rbu len t P ran d t l num ber , P ry , has been set

equal to 0.9.

86 / V OL 100, FEBRUA RY 1978Transactions of the ASRflE

Copyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



For the turbulent flow calculations the initial condition for the axial

velocity was specified as being uniform from the center of the pipe

to the position r = 0.9 ro. Over the int erval 0.9 ro<r<r0a parabolic

velocity profile was used. The numerical values for the initial velocity

profile were chosen to agree with the measured mass flow rate ac

cording to the relation:

JT O

urdro

(5)

The inlet density , po, was obtained from the inlet tem pera ture, To,

and the inlet pressure , Po- Th e boun dary condition at th e tube wallwas that of zero velocity and specified heat flux.

For the laminar f low problem the turbulent transport terms were

omitted. The only other change is that the initial velocity profile was

taken to be a parabola consistent with the measured values of the mass

flow rate.

In addition to the numerical solutions of the complete basic equa

tions, th at is, equ atio ns (l a) and (16), simplified analyses and results

have also been used for both the turbulent flow and laminar flow

problems. In turbulent f low, the well-known results of Deissler and

Bian [4] for fully developed conditions provide a simple, direct basis

for comparison with both the experimental data and the more com

plete calculations, and will be discussed in a later section. A simplified

theoretical determination of the heat transfer may also be made for

internal laminar forced flows with buoyancy effects (cf. Ostrach [5],

Ha l lman [6], Tao [7], and M orton [8]) . In these problems the buoyan t

force, which results from variations in the density, causes a funda

mental coupling between the velocity and temperature f ie lds. This

coupling requires that the solution for the velocity and temperature

fields be carried out s imultaneously. References [5-8] s implify the

internal laminar convection problem in the vertical direction by as

suming a fully developed flow with constant properties except for the

density variation in the body force term. The result ing momentum

and energy equations are then given by:

/ 1 dp \ LI d /rdu\

\PuldXg) r o V V o W

(ig(Tw - T) = 0

d T _ a _d_ /rdT\U dX r dr\ dr I

(6a)

(66)

where the subscript , w, refers to wall conditions. Solutions have been

obtained for the constant heat f lux condition and these results arediscussed in a la ter section .

For completeness we also note the important developing flow

studies carried out by Zeldin and Schmidt in air [9] and by Lawrence

and Chato in water [10]. These included both experimental and the

oretical results . The measurements in air [9] were carried out a t a

Reynolds num ber of 500 in a vertical , isothermal tube. A com parison

of the experimental and theoretical velocity distr ibutions showed

subs tan t ia l ag reemen t . However , the exper imen ta l tempera tu re

d is t r ibu t ions showed la rge r rad ial t emper a tu re g rad ien ts than were

predicted. The measurements in water [10] were carried out a t

Reynolds numbers varying from 616 to 1232 in a vertical tube with

uniform wall heat f lux. Good agreement was obtained between the

experimental measurements and the theoretical solutions. However,

the authors found that the velocity and temperature profiles never

became "fully developed" and attr ibuted this to the nonlinear vari

ation of density and viscosity on temperature.

E x p e r i m e n t a l A p p a r a t u s

Th e ex perim ental test section (cf. Figs. 1 and 2) consists of a verticalinconel steel tube which is 22 ft (6.7 m) long, 0.049 in. (0.124 cm) thick,

and 2 in . (5 .08 cm) outsid e diam eter. T he tu be is electr ically he ated

with a 3 phase 440V power supply w hich is a ttach ed to electrodes on

both ends. The tube is surrounded by two radiation shields, 10 and

12 in. (25.4 and 30.5 cm) in diameter, which are enclosed in a casing

approximately 16 in . (40.6 cm) in diameter. The casing exterior is

covered with fiber glass insulation which is 2 in. (5.08 cm) thick. To

reduc e convec tion losses from th e exterio r of the 2 in. (5.08 cm)

stainless s teel tube, a vacuum pump is used to maintain the casing

interior pressure at a value less than 0.1 in. (0.25 cm) of Hg.

Thermocouples are tack welded to the exterior of the test tube at

various axial locations (cf . Fig . 2) . Several thermocouples are also

attached to the exterior of the radiation shields at the test location,

X = 108D. A therm ocoup le pr obe, placed in a 0 .125 in . (0 .32 cm) OD

stainless steel tube and silver soldered to the outside of a 0.75 in. (1.91

cm) OD stainless s teel tube is inserted from the up per end of the ap

paratus to a location 108 diameters from the start of the heated sec

tion.2 This probe may be rotated so that the thermocouple may pass

from the pipe cente r l ine to the region close to the wall . A perforate d

radiation shield surroun ds the thermocou ple. A detailed desc ription

of the p robe is available [11,12].

M a s s F l o w R a t e

L a mi na r F low . T o measu re the mass flow ra te in lamina r f low,

the gas l ine is disconne cted from the entra nce to the exp erim ental

apparatus and connected to a wet test meter. Since the change in head

loss caused by disconnecting the gas l ine from the experimental ap

para tus and co nnecting i t to the wet test meter is small in co mparison

to the pressure drop across the needle valves and pressu re regu lators

"upstream," the f low rate may be determined from the relation:

m = pQ (7)

T he dens i ty i s eva lua ted a t the ambien t tem pera tu re and p ressu re

2 Inser t ing a nd secur ing th e probe a dis tance of approximate ly 2 .5 f t (0 .8 m)

from the upp er end of the app arat us resulted in a test location X/D = 108. Thi sloca t ion cor responds to fully deve loped f low condi t ions .

. N o m e n c l a t u r e .

cp = specific heat

D = d iamete r

/ = friction factor

h = heat transfer coeffic ient

hD/k = dimensionless heat transfer coeffi

c ien t , Nusse l t number

H = to ta l en tha lpy


( = mixing length

m = mass f lux

P = pressure


1 = heat flux

9 loss = he at loss from tu be

Q = volume flow rate

;• = rad ial c oordinate

ro = tube radiusrs = shield radius

R( TW) = resistance per unit of length

Ra = Ray le igh number = i3gr0HdTJdX)/

av

Re = Reyno lds number


T+

— dimensionless temperature = (T,„ —

T) cp VTwpw/qm

u = velocity in axial direction

v = velocity in radial direction

X = axial coordinate

y+ = dimensionless radial coordinate = (r 0 —

w

a = ther ma l diffusivity

/S = coefficient of therm al e xpan sionf = emissivity

P = dens i ty

T = shear s tress

p : - viscosity

S u b s c r i p t s

6 = bulk condition

eff = molecu la r p lus tu rbu len t

0 = inlet conditions

s = shield

T = tu rbu len t o r eddy

w = wall

108D = evaluated at X/ D = 108

Journal of H eat Transfer FEBRUARY 1978, VOL 100 / 87




-Electrodes

Ga s

exit

Cooling

coils

J~&-

-Q> —Gos supply

(cy l inders)

Air supply

( w 5 . 8 o l r n )

©Pressure

regulator

T Volve

.5ASME Nozzle

(1.27 cm)

Cold Irap

Fig. 1 Experimental apparatus

Probe —Stainless steel lube

Eccentric cam andball bearing

2—^ Flow out

Upper electrodes

Wall

thermocouplesRadiation

s

Fixed thermocouplein gas stream

•Wire screens

Test tube

Probe path

Flow in

Probe

To vacuum pumpExpansionbellows

Lower electrodes

Fig. 2 Test section

-Perforated

radiation shield

Flow

For experime nts with air, com pressed air is available from a storagetank maintained at 85 psig by a compressor. Other gases, stored ir,pressurized cylinders, discharge into a manifold connected to a rei>.ulator which maintains a constant exit pressure to the gas line at a

value between 40 and 60 psig. This line connects with the systemupstream of the nozzle in the same manner as the air supply.

He at F lux . All calculations were made using the constant, heatflux boundary condition [15]. The value of the h eat flux was obtainedexperimentally in the following m anner. Th e radiative heat loss fromthe test section to the adjacent radiation shield was calculated ac

cording to the following equation:

Q loss ~~ "

<x(?V - 2V)

( ; - )

ro(8)

where the wall and shield tem peratures, Tw an d Ts, were m easured.An energy balance on the tube w all at th e te st section yields the following equation for the heat flux to th e gas:

irDqw = IPR(TW) - TrDqh 0)

The electrical current, /, is measured with a stand ard a-c ammeterand the electrical resistance of the test section per unit of length,R(TW), is evaluated a t the m easured wall tempe rature.

Bulk Te mp era ture . The bulk temperature, T&, at the test location, X/D = 108, was obtained in the following ma nner. T he measured

tem peratu re profile and the theoretical velocity profile were used tocalculate the test section bulk tem pera ture, (Tj,)io8D, according to:

(Tb)iJo

puTrdr

r urdr

(10)

The nondimensional heat transfer coefficient, (hD/K)WS o, is evaluated based upon this value of the bulk temperature according to:

\ K I 108D

Qw D

K( Tw-T b)\i08D(ID

An inlet bulk temperature, (T b)0, is used in the nume rical calculation

and w as determined from t he following energy balance:

Jloss = rhc p{(Tl,)io8D ~ (Tfc)o]

where

an d

-r IW(Tw)dX

J» 1 0 8 D

qiossdXo

(12)

(12a)

(12ft)

Use of this value for the inlet bulk temperature in the calculationsinsures the equality of the experimental and the theoretical bulk

tempera tures at the test location.

R e s ul t s and D is c us s ion

since these values are close to the measured values at the inlet to theapparatus.

Turbulent Flow. A standard ASME 0.5 in. (1.27 cm) radiusnozzle was used to mea sure the mass flow rate for the tu rbule nt flowexperiments. The static pressure upstream of the orifice was measuredwith a mercury filled U tube manometer. The temp erature upstreamof the orifice is measured w ith a thermocouple in the gas stream. Th eorifice static pressure drop is measured with a water filled U tubemanometer attached to ports in the gas line wall located upstream anddownstream of the flow nozzle. The te mpe rature of the gas upstreamof the flow nozzle is measured with a thermoco uple. Thes e me asurements along with the knowledge of the nozzle geometry allow the flowrate to be determined w ith the use of standard ASME tables [13].

T u r b u l e n t F l o w

Wall Tem peratures. The theoretical wall tempera ture variation(T,„ v ersus X/D) obtained from the complete numerical calculationsusing the method of Spalding and P atan kar [1] (cf. Mason [2]) isshown in Fig. 3 for several cases which are summarized in Tab le 1. Theexcellent agreement between the theoretical and the experimentalresults for the wall temperature confirms the validity of the constantheat flux assumption that was used in the numerical calculations (cf.Donovan [15]).

Radial Temperature Profiles. Typical radial temperatureprofiles for both air and argon are presented in Fig. 4. The overallagreement between the theoretical and the experimental results, including the values at th e wall, is seen to be very good. Near the center

88 / V OL 100, FEBRUA RY 1978 Transact ions of the ASME




Theory

4 0 0

(°K)3 8 0 -

3 6 0

3 4 0 -

3 2 0

3 0 0 ^

2 8 0

°° #165

8 0 100

X / D

120

Fig. 3 Experimental and theoretical wall temperatures, turbulent flow (of.Table 1)

O EXPERiMENT

-THEORY

Table 1 Turbulent flow, numerical calculations

(hD/k), (w/m!)

QO 8

° #167

o o 8 8

° #169

oo08#173

165

167

169

171

172

173

185

188

A ir

Ai r

A ir

A ir

A ir

Argon

A ir

A ir

17,110

15,340

19,520

16,510

17,000

18,490

14,690

9,750

47.9

44.5

53.0

47.3

48.5

49.3

43.6

32.8

0.69

0.69

0.70

0.70

0.70

0.68

0.70

0.70

1190

780

630

410

270

260

340

230

agreem ent with th e calculations for this same value of the Reynoldsnumber as shown in Fig. 4, run 188. The discrepancy a t this Reynoldsnumber between the data of Deissler and Eian and the p resent data(and with the complete numerical calculations) is probably due to thevalue they used for the wall shear stress, rw. The wall shear stress isrequired in the determination of the nondimensional variables y

+,

T+

, and u+

. The experimental values presented for TW, or the frictionfactor, were obtained from pressure drop measurements which may

have been inaccurate at low flow rates [4, p. 37]. For completeness,it is noted that the calculated values for the friction factor, bo th fromthe p resent study and from Deissler and Eian, are in good agreement(cf. Fig. 8).

Heat Transfer and Friction Factor. Nondimensional heattransfer coefficients for air calculated for the present experiments arepresented in Fig. 7 for various Reynolds numbers. The theoreticalresults of Deissler and Eian are also presented. The agreement between their results and the present calculations is excellent and thisis somew hat surprising in view of the disagreem ent between th e two

Fig. 4 Temperature profiles in turbulent flow (Buns 165, 173, and 188, cf.Table 1)

20 -

+

16-

12-

8 -

4 -

0 -

Theory

= = = o Experiment (Deissler

A ir

8 Eian)

/ ^ f ^

/ ^ ^Nume r i ca l

Deissler ft Eian(constant propf - j ^

^/^°°s^j?°

calculations

line, however, the theoretic al tem peratu re profile is not as "flat" asthe experimental profile. This is most noticeable in the higher heatflux run s (cf. Fig. 4, run 165) and is believed to be caused b y the limitations of the mixing length formulation in this region; i.e., u.r -*• 0

(cf. equ ation (2)).The nondimensional temperature profiles resulting from the nu

merical calculations are also compared in Figs. 5 and 6 with the fullydeveloped, constant proper ty calculations of Deissler and E ian [4] andalso with their experimental data. Since the te mpera ture differencesare small the constant property assumption should be valid. It canbe seen that their data a t a Reynolds number of 19,000 are in excellentagreement with th e re sults from the complete num erical calculationsexcept in the region nea r the pipe center line. This is also consistentwith the results of the present experiment as discussed herein.

Of further interest are the data of Deissler and Eian at lowerReynolds numbers (cf. Fig. 6 for a Reynolds number of 10,000). Thedata no longer agree with the complete num erical calculations. Thisls

W contrast to the present experimental data which are in good

Fig. 5 Nondimensional temperature profile in turbulent flow (R e19,000)

TheoryDeissler a Eian

constant propH

Fig. 6 Nondimensional temperature profile In turbulent flow (Re10,000)





theoretical nondimensional temperature profiles (cf. Pigs. 5 and 6).The calculated friction factors are given in Fig. 8 along with the the oretical constant property calculations from Deissler and Eian. Theagreem ent is seen to be good as was previously note d. A summ ary ofconditions is presented in Table 1. The maximum value of the ratioof the hea t loss to the energy ge nerated is 0.14.

L a m i n a r F l o w

Wall Te m pe ra tur es . Theoretical wall temperature variationsobtained from the nume rical calculations are shown in Fig. 9 along

with the experimental data. T he values of the wall temperatures resulting from the calculations are slightly higher than the experimentalvalues. This is because the calculated wall to bulk temperature dif-

ference, Tw - Tb , is slightly higher than the experimental value.(Recall that the theoretical bulk temperature at the test location, X= 10SD, is required to be equal to the expe rimenta l value.) N ote th atthe experimental and the theoretical wall temperatu re gradien ts overthe range 60 < X/D < 108, are in good agreement, thereby givingconfidence to the constant he at flux calculation a nd to th e a ttainm entof the fully developed conditions which are part of the simplifiedanalyses (cf. Donovan [15]).

Simplified Analysis and Complete Numerical Calculations.Typical radial temperature profiles resulting from the simplifie.,1analysis and the complete numerical calculations are given in Pig. lualong with the experimentally measured values. The agreement between the two theoretical calculations is seen to be excellent. It ispointed out tha t the latte r approach includes such effects as variableproperties, flow development, and flow acceleration. In the simplifie. 1

analysis, all transport properties are evaluated at the w all temperaturewhile the numerical calculations do not make this simplification.Variable property effects are small due to the small temperature

differences. Tables 2 and 3 contain a summary of results obtaine.1

from bo th methods . The maximum va lue of the ratio of the he at los~to the energy generated is 0.41.

Typical radial temperature profiles, Tw — T, resulting from thesimplified a nalysis

3are presen ted in Figs. 11 and 12 along with the

experimental data for air and argon. The agreement between theexperimental data and the theoretical results is excellent. For com-

3 I t i s emphasized aga in tha t th e theore t ica l resul ts obta ined f rom both I lvsimplified analys is and the more complet e numerica l calculations are in ex

ce l lent agreement .

k

u u -

5 0 -

A ir

1 ' ' ' t

' \Deissler 8 Eian,

constant property

o Present results

i i i 1

0.5x10 10" 5x10Re

Fig. 7 Nusselt numb er (XID =108) versus Reynolds number

10

6 -

4 -

A i r

Deissler 8 Eian,constant property

O Present results

3x10° 1x10"

Re

3x10

Fig. 8 Frict ion factor (X/D = 108) versus Reynolds nu mber

90 / VO L 100, FEBRUARY 1978

40 I20

Fig. 9 Experimental and theoret ical wal l temperatures, laminar flow (cf .Table 2)

T (°K)

3I 0

3 00

0.2 0.E I.O.4 0.6r/R

Fig. 10 Temperature prof i le in laminar flow (Run 180, cf . Tab les 2 and 3)

Transact ions o f the ASME


http://deissler/

http://deissler/



Tab le 2 Laminar f low, numerical calculat ions 2 0

( h D / k ) ] 0 8 DJ / 1 (w/m2)

174

175

177

179

180

A ir

A i r

A ir

A i r

Argon

1910

1930

2080

1880

1870

5.19

5.19

5.14

5.21

5.18

2.93

2.94

2.90

2.96

3.01

30.4

29.1

27.6

26.4

16.2

T w - 1

no

~~̂ ^- . a Experiment

\ Combined convectiony D ^- , . ^ \ Forced conveclion

^ ^ ^ . (N o n -b u o y a n t)

V \A t r 6 S \

w

\

\— ^ h

Tab le 3 Laminar f low, s impli f ied analysis0 0.2 0.E 1.0

<hD/k

>,o

0.4 0.6

r/ R

Fig . 11 Tem perature prof i le In laminar f low (Run 174, cf . Tab les 2 and 3)

174

175

177

179

180

A ir

A ir

A ir

A ir

Argon

1880

1910

2050

1850

1850

5.22

5.25

5.25

5.23

5.31

2.89

2.92

2.91

2.92

3.01

30.4

29.1

27.6

26.4

16.2 T „ - T

( ° K >

pleteness the forced convection, nonbuoyant results are also pre

sented. The increased transp ort du e to buoyancy results in the f la tter

temp erature profiles show n. Recall the value of 4 .364 for hD/k [14]

for the nonbuoyant constant property calculation which is about 16

percent4 less than the values with buoyancy present.


T u r b u l e n t F l o w . Numerical calculations using the method of

Spalding and Patankar [1] (cf. Mason [2]) employing a Prandtl mixing

length model with van Driest damping yields temperature profiles

which are in good agreement with the experimental data for a ir and

argon, although significant differences near the center line are noted.

The Reynolds numbers varied from 10,000 to 19,500 and the corre

sponding values of the Nusselt numbers varied from 33 to 53. The

measurements were carried out in an electr ically heated tube, 2 in .

(5.08 cm) in diamet er, which was support ed in the vertical direc

tion.

La m ina r Flo w. A simplified theoretical determina tion of the heat

transfer in inte rnal forced flows with buoyancy p resen t (cf. [5-8]) gave

good agreem ent with measured t emp eratu re profiles in air and argon

and with more complete numerical calculations. Values of the

Reynolds numbers varied from 1850 to 2100 while the Rayleigh

numbers varied from 70 to 80. Over this range of conditions, om itt ing

the effects of buoyancy results in a 16 perce nt error in the Nus selt

number .


The auth or is grateful to the Nationa l Science Fou ndat ion for

support of this research.

R e f e r e n c e s

1 Patankar, S. V., and Spalding, D B., Heat and Mass Transfer inBoundary Layers, Intertext Books, London, 2nd edition, 1970.

2 Mason, H. B., "Numerical Com putation of Steady Natu ral ConvectiveFlows with Rotation and Stratification," PhD dissertation, Dept. of MechanicalEngineering, University of California, Berkeley, Calif., 1971.

3 van Driest, E. R., "On Turb ulen t Flow Near a Wall," Journal Aeronautical Sci., Nov., 1956, pp. 1007-1011.

10

15 k ~~~~~ ~~~~~. O Experiment

( r Q _ \ Combined convection

to

5

- ^ . Q X Forced convection

" " ^ - o ^ (Non-buoyant)

N x \°\ \r go n ^ \

SV

X. , . . , >»

0 1.0.2 0.4 0.6 0.8

r / R

Fig. 12 Temperature prof i le in laminar f low (Run 180, cf . Tables 2 and 3)

4 Deissler, R. G., and Eian, C. S., "Analytic and E xperimental Investigationof Fully Developed Turbulent Flow of Air in a Smooth Tube with Heat Transferwith Variable Fluid Properties," NACA Tech. Note 2629, Feb. 1952.

5 Ostrach, S., "Combined Natural and Forced Convection Laminar Flow

and Heat Transfer of Fluids with and without Heat Sources in Channels withLinearly Varying Wall Temp eratures," NACA TN 3141, Apr. 1954.

6 Hallman, T. M., "Combined Forced and Free Convection in a VerticalTube," Ph D thesis in Mechanical Engineering, Purdue University, Lafayette,Indiana , 1958.

7 Tao, L. N., "On Combined Free and Forced Convection in Channe ls,"JOURNAL OF HEAT TRA NSFER, TRANS. ASME, Series C, Vol. 82, No.3, Aug., 1960, pp . 233-238.

8 Morton, B. R., "Lam inar Convection in Uniformly Heated V erticalPipes," Journal of Fluid Mechanics, Vol. 8,1960, p. 227.

9 Zeldin, B., and Schm idt, F. W., "Developing Flow with CombinedForced-Free Convection in an Isothermal Vertical Tube," JOURNAL OFHEAT TRANSFER, TRANS. ASM E, Series C, Vol. 92,1972, pp. 211-223.

10 Lawrence, W. T., and Chato, J. C , "Heat-T ransfer Effects on the Developing Laminar Flow Inside Vertical Tubes," JOURNAL OF HEATTRANSFER, TRANS. ASME, Series C, Vol. 88, No. 2, May 1966, pp. 214-222.

11 Habib, I. S., "Heat Transfer to a Radiating Gas Flowing Turbulentlyin a Tube: An Experimental and Theoretical Study," PhD dissertation, Uni

versity of California, Berkeley, Calif., 1968.12 Habib, I. S., and Greif, R., "Heat Transfer to a Flowing Nongray Ra

diating Gas: An Experimental and Theoretical Study," International Journalof Heat and Mass Transfer, Vol. 13,1970, p p. 1571-1582.

13 Fluid Meters, ASME, 6th edition, 1971.14 Kays, W. M., Convective Heat and Mass Transfer, McGraw-Hill, New

York, 1966.15 Donovan, T. E., "He at Transfer in Internal Flows Including Buoyancy

and The rmal Radiation ," PhD dissertation, University of California, Berkeley,Calif., Dec. 1975.

4 It is noted tha t [1 - (4.36/5.19)]100 = 16 percent.

Journal of Heat Transfer FEBRUA RY 1978, V OL 100 / 91




B. T. F. ChungL. C. Thom as

Y. Pang 1


The University ol Akron,

Akron, Ohio

A Surface Rejyfenation l o d e ! for

Turbulent Heat Transfer in Annular

Flow With High Prandtl NumbersHeat transfer for high Prandtl number fluids flowing turbulently in a concentric circular

tube annulus with prescribed wall heat flux is investigated analytically. This surface re

juvenation based analysis is restricted to thermally and hydrodynamically fully developed flow with constant properties and negligible viscous dissipation.

This formulation leads to predictions for the Nusselt Number that are in basic agree

ment with predictions obtained on the basis of earlier eddy diffusivity models for 30 < Pr

< 1000 and 10* < Re < 106.


Theoretical analyses of turbulent heat transfer in concentric annuli

has been the subject of many recent s tudies . Among them, Kays and

L eung [ l ] 2 have obtained solutions for the heat transfer coeffic ient

for turbulent annular flow with constant heat fluxes at walls for a widerange of radius ratios, Pran dtl num bers and Reynolds numb ers . Using

the same classical approach, except with different expressions for

velocity distr ibution and eddy diffusivity , Wilson and Medwell [2]

and Michiyoshi and Nakajim a [3] resolved the same problem an d

obtained results in good agreement with those reported by Kays and

Leung. Recently , Chung and Thomas [4] analyzed the same problem

based on a different modeling concept known as the principle of

surface renewal. The strength of this approach l ies in the fact that

modeling param eters , such as the mean residence t ime, 7 , are uti l ized

which have physical significance and are measurable. Predictions for

Nusselt number at both the inner and outer wall , based on both ap

proaches, have been found to be in good agreem ent with the experi

mental data for air [1],

Whereas the classical approach also has been applied to the high

Pra nd tl num ber region, the surface renewal analysis of reference [4]is confined to mode rate Pr and tl num ber f luids because of the use of

the simplifying assumption that turbulent eddies move into direct

contact with the surface . B ecause of the lack of experimenta l dat a for

high values of Pr, i t is fe lt that the su pple men tatio n of previous pre

dictions based on the classical approach by surface renewal-based

1Present address: Department of Mechanical Engineering, Stanford Uni

versity, Palo A lto, California.2

Numbers in brackets designate References at end of Paper.Contribu ted by the Heat Transfer Division for publication in the JOURNAL

OF HEATTRANSFER. Manuscript received by the Heat T ransfer Division July19,1977.

predictions would be useful. Therefore, attention now is turned to the

formulation of a surface renewal-based analysis for turbul ent annular

flow of high Prandtl number f luids . In this analysis , a more general

form of the surface renewal and penetration model known as the

surface rejuvenation model will be employed. This model accounts

for the effect of the unreplenish ed layer of fluid tha t has been reportedto reside at the surface .

The surface rejuvenation model was first proposed by Harriott [5]

for interfacial mass transfer . Thi s model is based on the hypotheses

tha t 1) eddies inter mit tently com e to within various small dis tances

H of the surface, and 2) unsteady molecular transport controls during

their brief residency at th e wall . The major difference betw een the

elementary surface renewal model and the surface rejuvenation model

is tha t the former is based on the assum ption tha t the approa ch dis

tance , H, for each eddy is zero, while the latt er ac counts for the effect

of the therm al resis tance of the unreple nished layer of the f luid near

the wall region. This effect has been shown to be of negligible signif

icance for moderate to low Prandtl number f luids but of primary

concern for large Prandtl number f luids [6],

Early adaptations of the surface rejuvenation model [6-7] to tur

bulen t mo men tum and heat transfer for tube flow were found to beextremely time consuming. A more efficient computational procedure

recently ha s been proposed by Bu llin and Du kler [8], Thi s stochastic

approach has been adapted to turbulent tube f low by Rajagopal and

Thomas [9J . In the present s tudy, a s tochastic formulation of the

surface rejuvenation model will be developed for turb ulen t hea '

transfer in a concentric annulus, with emphasis on high Prand ' l

number f luids .

A n a l y s i s

Consideration is now given to a fully developed turbule nt an nu ls

flow with constant heat f luxes at the inner and/or outer surfaces "'•

q\„, and qiw' respectively . Eddies from the bulk stream are assumed

to arrive continuously at random distances H from the wall, remainn'f!

92 / VOL 100, FEBR UARY 1978 Transactions of the AOMECopyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



f0r various lengths of t ime r.

Hea t T r an s fe r . Based on the su r face re juvena t ion concep t , the

following system of equations may be writ ten for the instantaneous

energy transport associated with an individual re juvenation cycle:

dT

dB "

d 2 T

'' dy 2

dy

8>0,0<y<

q w a t y = 0

T = T 6 a s y - °= f(y) for 0 < y < H

••Tb lory>Ha t 0 = 0

(1)

(2)

(3)

(4)

Q u i= liw f ° r th e i

nner surface and q w = q % w for the outer surface, y

is the distance from ei ther inner or outer wall extending into the fluid,

D is the instan tane ous co ntact t ime, and f(y) rep resen ts the temper

ature profiles within th e wall region at the first instan t of rejuvenation.

In the present analysis the viscous dissipation has not been taken into

account, which becomes significant only when Brinkman number is

large.

As can be seen from t he above governing equatio ns, the tem per atur e

of the eddy at the f irs t insta nt of re juvenation is assumed to be uni

form and equal to the bulk stream temperature and the curvature

effect is assumed to be negligibly small. The first assumption is re

alis t ic for high Prandtl number f luids s ince the temperature drop isconcentrated in the wall region. The second assumption is reasonable

because the residence t im e of the f luid elem ent is short an d the pen

etration dep th is shallow for the fully tu rbu lent f low. A recent s tudy

by Kakarala [10] has indicated t hat the curvature effect is impo rtant

only when flow is in laminar and tran sit ion regions. Since e quatio ns

(1-4) are l inear and nonhomogeneous, they can be solved using

super-position technique and the standard Fourier integral transform

method. This gives r ise to an instantaneous temperature profile of

the form

Tb - T(y, 6) = -q wV^/k [2 Vjfr exp (-y 2 /4afl) - y/Va' erfc

(y + m(y/2 6) + 1/VI^8 C" [Tb - /(£)] jexp [

[ex pAa t

4a 6 J

- ] ) d | (5)

With 8 set equal to r , th is equation provides a relationship for the

temperature profile a t the end of the residence t ime, i .e. , the init ia l

condition for 0 < y < H encountered by the next incoming eddy.

Following Bu llin and D ukler [8], expression s can be written for the

mean in i t ia l t empera tu re p ro fi le T(y, 7) and the mean tempera tu re

profile T(y, 7) as

Tb-T(y,r)= CPT(r) f " PH(H) C Pf(f)Jo Jo Jo

X[Tb~T(y,r)]dfdHdT (6)

and

Tb-T(y,T)= C Pe(8 ) CPH(H) C" P f(f)Jo Jo Jo

X [T b - T(y, 0)] df dH dB (7 )

7 is the mean residence t ime, P$(B), P T ( T ) , PH(H) an d Pf(f) are the

probabilis t ic functions for contact t ime, residence t ime, approach

distance and init ia l profile , respectively . In this work, Danckwerts '

rand om distr i butio n function is employed for T and H, [11] i.e.,

1 / H\P „ = = e x P ( - = )

(8)

(9)

where H is the mean approach distance. An expression will be de

veloped momentarily for 7 . From the relation (12)

there is obtained

dP e(B )PM = - 7 — ^

d8

P o(0) = r e x p ( - z )

(10)

( I D

Substi tuting equations (5) and (8)—(11) into equations (6) and (7)

yields an identical expression for T(y, 7) and T{y, 7) which is given

by

Tb - T(y, 7) = Tb - T(y, 7)

= -(q wVa^/k) exp (-y/Va7) + 1/2 - = p iaT

J 77

(-§)/>-M,H-^)+ e x p ( -=-)

\ V CVT / J

ex p

d£dH (12)


A = VPc/rfH

Sr = Fann ing fr ic tion factor

liy) = t runca ted cu rve fo r temp era tu re

giy) = trun cate d curve for velocity

H_ = mean approa ch distance

#+ = d im ens ion less mean approachdistance, Hu*lv

k - conductivity of f luid

Nu = Nusse l t number , 2h(R2-R 1)/k

P = pressure

P$ = probabilis t ic function for contact

t ime

PT = probab ilistic function for residence

time

°H = proba bilistic function for appr oach

distance

"/ = probabilis t ic function for initia l

profile

Pr = P rand t l number , via

q a = wall hea t flux

R = rad ius

Rm = radius of maximum velocity of the

flow

r* = RilRi

R e = R e y n o l d s n u m b e r , 2U b(R i -

Ri)h

Sm = RmlR-i

Tb = bulk tem pera ture of the flow

T = in s tan taneous tempera tu re p ro f i le

T = mean tempe ra tu re p rof i le

u = instan taneo us velocity profile

u = mea n velocity profile

U* = friction velocity, Vg c aw/p

Ub = bulk stream velocity

v = my = distance from either inner or outer

wall extendin g into f luid

z = HI E

a

P

M

V

T

7

0a

C/l

=====li

II

=

S u b s c

w

b

1

2

====

thermal diffusivity

dens i ty

dynamic viscosity

kinematic viscosity

residence t ime

mean residence t ime

con tac t t imeshear s tress

eddy diffusivity for momentum

eddy diffusivity for heat

ylH

r i p t s

condition at wall

bulk stream condition

at inner wall

at outer wall

Journal of Heat Transfer FEBRUAR Y 1978, VOL 100 / 93




/(£) in the above integral is defined as

7(f) = fQ" Pfif) f(& df

T ha t the mean in i t ia l t empera tu re p ro f i le T(y, 7) and the mean

tempera tu re p ro f i le , T(y, 7) are identical is independent of the

boundary conditions involved. The key point is the use of random

contact t ime distr ibution function which yields the same type of

distr ibution function for residence t ime, consequently the same ex

pression for mean init ia l and mean temperature profiles is ob

ta ined .Non-dimensionalizing the above expression and after some alge

braic manipulations, we obtain the following integral equations for

the temperature profiles within the vicinity of the inner wall

Tdv)-Tb 7?!+ 1 - /-* ^ / X , A , , A± r*

2 Jo2q \wRi AiRe

e x p ( -

— exp (-AiJi) + —fri 2

£Ti(u) - Tb

2q iwRi| ex p M i ( i j + o )

and the outer wall region

T2(v) - Tb Hf

k

A2Re( 1 - r * )

+ e x p [ l - A J ) ) •

— e x p ( - A 2 r ; )hi

I du dz (13a)

J exp ( -2 ) I0 Jo

* T2(u) - Tb

2q 2wR 2

|exp (-A 2 (i? + v)]

k

exp [ -A 2 | i ? - I du dz (13b)

where A t = (y/Pi/V7fl) H h A2 = (Vpi/Vrij H 2, r* = RJR 2, v =

y/H, z = H/H, v = £/H with H = Hi, H = H\ for the inner wall region

an d H = H 2, H = H2 for the outer wall region.

The Fanning friction factor at inner and outer wall can be expressed

in term of total friction factor, fr, [3] as

fn , . * ( ! _ r 4 -fr

fr2 = -1 - r *

fr

(14a)

(14b)

where sm is the ratio of maximum velocity radius to outer radius . T he

expression of sm proposed by Kays and Leung [1] is employed

here:

: + r *

1 + r"(15a)

This formula has been supported by various experimental data and

some theoretica l predictions [2]. Th e total fr ic tion factor, fr, in a

smooth concentric annuli, can be obtained from the following familiar

empirical formula

1 1 (R e Vf r) - 0.4

Substituting equation (14) into (13) and setting 77 equal t o zero gives

expressions for the wall tempera tures a t the inner and outer surfaces

in terms of the m ean residence t im e 7 of the forms

H i

2(?i«,ftiy 2 ( 1 - r * ) 3

A i R e r* fr(s„- + A -

Joex p

(-zJo

* Ti(u) - Tb

2q iwR x

ex p (—Aiv) dv dz (16a)

Tim ~ Tb %-V-(1 - r * ) 3

2q 2wR 2 A 2R e fr(l2)

+ A,Jo

ex p

X ( - I* T2(v) ~ T b

%Q2wR2

exp (—A2u) dv dz (16b)

The Nusselt numbers at the inner and outer surfaces are expressed

in terms of equation (16) as

(17a)

Nu t =

NU 2

1 1

T\ w — Tb

2q iwR\

k

1( 1

T2w — Tb

/2q 2w R2\

-r*

r*

- / • * )

(17b)

A relationship for 7 now will be developed in term s of mom entum

transfer given below. As will be seen, the mom entu m and energy so

lutions are coupled through the mean residence t ime.

M o m e n t u m T r a n s f e r . The governing equations for the instan

taneous momentum transfer associated with an individual surface

rejuvenation cycle are

du d 2 u 1 dp

dO dy2

p dx

u = 0 a t y = 0

(18)

(19)

(20)

a t 0 = O (21)

u = finite as y —•

u = g(y) for 0 < y < tfl

= Ub {ovy>H I

where g(y) is the tru ncate d curve. The pressure gradien t in equation

(18) may be expressed in terms of inner and outer wall shear stresses

as

dp 2aiwRi dp 2cr2wR2<J IWRI _ d p

dx ~Rm

2-R i

2°

rdx~R

2• R„

By definit ion

"im = -pU b2frV «"2w

=-pUb

2fr,

(22)

(23)

where fri an d fn have been given by equation (14) . Th e coupling of

the solution of equations (18-21) with equations (22) and (23) and

with the s tochastic process m ention ed earlier leads to the following

integral equations for the velocity profiles within the vicinities of the

inner and outer walls :

Ub ~ UIJTI)

Ub-= ex p

\ V C T l '+ -

4H ^ r * ( l - r*)

R e/ Hj \ 2 ( sm

2-

2)

' ( - z )

Ub - u2(n )

Ub

r / Hi \ 1 1 H l r°>exp ( - — = = 1 , 1 - 1 + - - — i ex pl

r Ub - UAv) f f Hi ]X Tr exp - - ^ = 0 - 7 ,

Jo Ub 1 L vj.7-1 J

- exp I - - ^= = (u + v) 11 du dz (24a)L Vi/Tl J J

("£=')•xp

R eH2 \ 2 lm

V 1 /T9 '

exp ( - — = ) , ) - 1 +-—^z\ ex p (-z) fL \ vino / J 2VVT2JO Jo

' z Ub -u2(v j

or"

94 / VOL 100, FEBRUAR Y 1978 Transact ions of the ASME




X exp -—z=\v - 171 - exp -L V[iT2 J L

H 2: (u + !j) du dz (24b)

IIT 2

These results give rise to the expressions for the dimensionless mean

wall shea r stresses at inn er and ou ter surfaces of the forms

Hi AHHi V * ( l

r*2

fr

• r*) 2

r * ( l

R e

(A) /

(A )e x p ( - 2 ) 1

X exp

* t / 6 - m ( u )

H i

mW -

(~ Ay)

\ V l 'T| /

do dz (25a)

1 - s , I—LL

* 2

7 ? .

H 24 i 7 ,

Reff.

(A )- S m

2

(A ) 7 ' xp ( - z ) jJo

X exp (

* Ub - u-2(v )

_U b

H2

VT2 " )dv dz (25b)

In the present com putation, the numerical value for H+1 and H

+2 a re

assumed to be 5. This a ssump tion is based on the expe rime ntal ob

servation with a pipe by Popovich and Hu mm el [13] . The ir results

have shown that adjacent to the wall there exists a layer of smallthickness y +

= 1.6 ± 0.4 in which a linear velocity gradient occurs at

virtually all times, but within w hich the slope of gradient changes w ith

time. This f inding is basically in agreemen t with th e surface rejuve

nation concept . The mean approach distance associated with s trong

eddies reported by these workers was H+ = 5.05. Since the experi

mental data for the velocity profiles in the sublayer region at both

inner and outer surfaces are almost the same, the assumption of H+i

= H+

2 appears to be reasonable . Therefore, the left hand side of

equation (25) is known if the Reynolds number and the dimensions

of the annu lus are f ixed. Th e param eter H/\^pT can be solved from

equations (24) and (25) s imultan eously by i teratio n. Once ? is estab

lished, the N usselt n um ber can be solved from eq uation (17) .

The i terative procedure involved in computing the temperature

profile includ es 1) the ass um ptio n of a value of H/\fvT , 2) the i tera

tive computation for (Ub - u)/Ub in the vicinity of the inner an d ou terwalls from equ ation (24) ( the f irs t two terms of the r ight ha nd side

of equa tion (24) are used as an initial guess ), 3) a com paris on of th e

wall shear s tress obtained by the substi tution of (U b — u)/Ub in to

equation (25) with the exact value of wall shear stress which is ob

tained directly from eq uations (14) for the specified Reyno lds n um ber,

4) the selection of a new estimate for H/y/H? by solving the th ird order

(in Hl\f7r) algebraic equation, equation (25) (note that the left han d

side of equation (25) is given), 5) the computation of a second ap

proximation for (Ub — u)/Ub by the substi tution of the improved

value of H/\A>T in to equation (24) . (These computational s teps are

repeated until the wall shear s tress converges to within 1 perc ent of

the correct value), and 6) the sub stitu tion of the final value of H/y/H^

into equatio n (13) and the i terative solution of this equation for th e

mean temperature profile ( the f irs t two terms of the r ight hand side

of equation (13) are used as the init ia l guess) . In evaluating double

integrations in equations (13), (24) and (25), the methods of Gaussianand Gauss ian-Lagu erre qua rdra ture are employed. I t is found tha t

the numerical technique employed in the present analysis converges

rapidly . The computer program involving the details of calculation

Mentioned above is available in reference [14].

Results and DiscussionThe num erical solution of equations (24) and (25) for H/y/Tf a re

tabulated in [14] for different values of radius ratio and Reynolds

number. It is found t ha t for given Re and r*, the mean residence t ime

°f eddies in the inner wall region is a lways shorter th an t hat in th e

outer wall region. As a consequence, the Nusselt number at the inner

surface is expected to be larger.

Based on the va lues o f H /V 7? , the Nusse l t numbers computed

from equation (17) are presented graphically in Figs . 1-6 . Nusselt

numbers at both inner and outer surfaces are plotted against Reynolds

numb er ranging from 104

to 106, wi th P rand t l number a s a pa ramete r

varying from 30 to 1000. Th e radiu s ratios are chosen as 0 .2 ,0 .5 and

0.8. As expected, the Nusselt number at the inner surface is greater

than that a t the outer surface . Unfortunately , no experimental data

x 10

10<

PRESENT ANALYSIS

HLCHIYOSHI A N D NAKAJIHA (? )

10' 10"RE

10"

Fig. 1 Nu sselt numb er at the inner surface of an annulus with uniform w allheat flux and r* = 0.2

5 x 1 0 "

104

Nu2

m3

10 '

—

r

-

A

1

P R E S E N T A N A L Y S I S

H l C H I Y O S H I A N D N A

VPR=L

^

#

X

. , . . . . (

K A J I M A

0 0 0 / /

//

(3 )

A / i

^ PR = 100

- P R = 30 _

'

1 1 1 1 1 1

I f f If/RE

10"

Fig. 2 Nu sselt number at the outer surface of an annulus with uniform w all

heat flux and r' = 0.2

Journal of H eat Trans fer FEBR UARY 1978, VOL 100 / 95




PRESENT ANALYSIS

MICHIYOSHI AND NAKAJI MA (3 )

WILSON AND MEDWELL

101

~ 1 - 0 4 ~ ~ - - ~ ~ ~ ~ ~ 1 ~ 0 5 ~ - - ~ ~ ~ ~ ~ ~ 1 ~ 0 6HE

Fig. 3 Nusselt number at the Inner surface of an annulus with uniform wall

heat flux and " = 0.5

PRESENT PNALYSI S

M I C H I Y O S ~ I AND NAKAJIMA (3 )

Fig. 4 Nusselt number at the outer surface of an annulus with uniform wall


for Nusselt number in turbulent annular flow for high Prandtl number

exist in the open literature; nor is any simple heat transfer correlation

in terms of Pr, Re and r* available. However, theoretical solutions

based on the classical eddy diffusivity model [1-3] have been available.

The recent results of Michiyoshi and Nakajima [3] which do not differ

much from those of Kays and Leung [1] are included in Figs. 1-6.Nusselt numbers at the inner surface predicted by Wilson and Med

well [2] are available for certain values of Re, Pr and r* only. They are

included here for the purpose of comparison. I t is found that the

present analysis is in reasonable agreement with that of reference [3].


PRESENT ANALYS I S

MICHIYOSHI AND NAKAJIIIA (3 )

WILSON AND MEDWcLL (2 )

101 ~ - - ~ ~ ~ ~ ~ ~ - - ~ ~ ~ ~ ~ ~

104

105

REFig. 5 Nusselt number at the Inner surface of an annulus with unifo rm wall


PRESENT !\NALYSIS

MICHIYOSHI

1 0 1 ~ ~ ~ ~ ~ ~ ~ ~ ~ __ __ ~ ~ ~10 4

l()5

REFig. 6 Nussell number at the outer surface of an annulus with uniform Viall


The results of Wilson and Medwell fall between the present resultsand those of [3].

The numerical results of two limiting cases for r* = 0 and r* '" 1.0,

which correspond to the cases of circular pipe and two parallel plates

flows, respectively, are also computed. These limiting solutions

compared well with the previous theoretical works [1,3]. Comparisons

show that the present solutions with r* = 0 agree with the experi

mental data compiled in [15] within at least 10 percent for a wide

range of Prandtl and Reynolds numbers.

As pointed out by Schlichting, at this stage of development the

Transactions of the ASME




turbulent f low problem must be solved by a semi-empirical theory .

Just as other turbulence models do, the present analysis involves

certain empirical constants . They are H+

i, H+

2 ( in the analysis we

assume / /+

1 = H+

2 - 5), the radius of maximum velocity and friction

factor correlation . Note that in this work, the mean residence t imes

TI and 7 2 are obtained from the application of surface re juvenation

model to momentum transfer plus the information of radius of max

imum velocity, sm and the friction factor fr. If the experimental data

for ?i and ?2 are available, it would be unnecessary to use sm an d

fr-

To da te , the techniques for measuring T and H are available for pipeflow. Popovich and Hummel [13] applied a f lash photolysis method

for non-disturbing turbulent f low to measure the approach distance

of the eddies . Meek a nd B aer [16] and T hom as and Gre ene [17] used

i f lush moun ted anem omet er probes for measuring the mean bursting

per iod . I t i s an t ic ipa ted t ha t the same measu remen t techn iques can

be applied to turbulent annular f low. I t should be pointed out that

the predicted 7 and velocity profile based on the surface rejuvenation

model agree reasonably well with the m easur em ent of [16] and [17].

', The comparison was presented in an earlier paper [7] which was based

j on Ha rriott 's model instead of the more effic ient s tochastic approach

j of B ullin and Duc kler [8],

i A comparison betwee n the surface rejuvenation and the eleme ntary

surface renewal models associated with f low in an annulus is i l lus

trated in Fig . 7 . Th e simple surface renewal model is a l imiting case

of surface rejuvenation model with H+

= 0, which implies that a ll

eddies are directly in contact with the wall . Obviously, the model is

not appropriate for f low with high Prandtl numbers for which the

major therm al resis tan ce is concen trated in the wall layer. I t is seen

f rom F ig . 7 tha t fo r modera te P ra nd t l num ber , the Nusse l t num ber

predicted by the two surface renewal type models is in good agree

ment; however, as the Prandtl number increases, the elementary

surface renewal model appears to over-predict the heat transfer r a te .

This is due to the thermal resistance effect of the unreplenished layer,

which becomes significant for the high Prandtl number f luids . The

numerical re sults show tha t the surface renewal model yields a relation

of Nu <= P r 1 / 2 while the surface rejuvenation model predicts Nu °°

P r1/ 3

.

Conclusion

An analysis has been presented to estimate heat transfer charac

teris t ics for turbulent fully developed annular f low, with emphasis', on high Pra nd tl num ber f luids . Th e predicted Nusselt num ber s are

graphically presente d for a wide range of Reynolds num bers, Pr and tl

numbers , and rad ius ra t io s . T he same p rob lem has been a t tacked

earlier, using the eddy diffusivity approach which involves analogical

relationship for thhmm

terms of Pr and an empirical assumption

regarding the variation of em/i> near the wall region. In the present

approac h, predictions of Nusselt numb er are made in term s of 7 and

H which are directly measurable . The alternative to measuring 7 di-

j rectly is to pred ict 7 from easily mea sured wall shear stress or friction

i factor . (Note that the eddy diffusivity models a lso rely upon the in

formations of sm an d fr). Although this provides an analogical re la-

I t ionship between Nus selt num ber and fr ic tion factor, the impor tant

point is that a physically meaningful p arame ter, 7 , provides the basis

for this analogy. It appears that the present model gives a clear picture

' in describing the exchange mechanism between a turbul ent f luid andI i ts bound ary . This mech anism is consistent with the previous ex-

; perim ental observatio ns of Fage and Town send [18], Lin et a l . [19],

1 and Popovich and Hu mm el [13].

j

i AcknowledgmentI Thi s s tud y was supp orte d in par t by the Nation al Science Foun-

' dation unde r grant GK-35 883.

I References\ 1 Kays, W. M., and Leung, E. Y., "He at Transfer in Annular Passages—I Hydrodynamically Developed Turbu lent Flow With Arbitrarily Prescribed! Heat Flux," International Journal of Heat and Mass Transfer, Vol. 6, No. 7,' (1963), pp. 537-557.


w I . . i . . i i • . . . . . . .

0.7 1.0 5.0 10 50 190 500 1,000PR

Fig. 7 Comparison of present analysis with elementary surface renewal andpenetration model and eddy diffusivity model for the inner-surface Nusselt

number of an annulus with uniform wall heat flux and i" = 0.2

2 Wilson, N. W., and Medw ell, J. O., "An Analysis of Heat Transfer forFully Developed Turbu lent Flow in Concentric A nnuli," ASME, JOURNAL OFHEAT TRANSFER, Vol. 90, No. 1,1968, pp . 43-50.

3 Michiyoshi, I., and Nakajima, T., "Heat Transfer in Turb ulen t FlowWith Internal Heat Generation in Concentric Annulus (I) Fully DevelopedThermal Situation," Journal of Nuclear Science and Technology, Vol. 5, No.9, 1968, p p. 476-484.

4 Chung, B. T. F., and Thomas, L. C, "An Analysis of Heat Transfer InTurbulent Annular Flow," Proceedings 10th Southeastern Seminar onThermal Sciences, New Orlean s, 1974, pp. 78-94.

5 Harriott, P., "A Random Eddy Modification of the Penetration Theory,"Chemical Engineering Science, Vol. 17, No. 3,1962, pp. 149-154.

6 Thomas, L. C, Chung, B. T. F., and Mahaldar, S. K., "TemperatureProfiles for Turbulent Flow of High Pra ndtl Number Fluids," InternationalJournal Heat and Mass Transfer, Vol. 14, No. 9,1971 , pp. 1465-1471.

7 Thomas, L. C , Chung, B . T. F„ and M ahaldar, S.K., "Adaptation of theSurface Rejuvenation Model and Turbulent Momentum Transfer," Proceedingsof 8th Southeastern Seminar on Thermal Sciences, Nashville, TN ., 1972, pp .139-162.

8 Bu llin, J. A. and Dukler, A. E., "Random Eddy Models for Surface Renewal, Formulation as a Stochastic Process," Chemical Engineering Science,Vol. 27,1972, pp. 439-442.

9 Rajagopal, R., and Thomas, L. C , "Adaptation of the Stochastic Formulation of the Surface Rejuvenation Model to Turbulent Convective HeatTransfer," Chemical Engineering Science, Vol. 29, No. 7, 1974, pp. 1639-1644.

10 Kakarala, C. R., Ph.D . Dissertation, Depa rtmen t of Mechanical Engineering, University of Akron, Akron, Ohio (1976).

11 Danckw erts, P. V., "Significance of Liquid Film Coefficients in GasAbsorption," I&EC, Vol. 43, No. 6,1951, pp. 1460-1467.

12 Zwieterling, T. N., "The Degree of Mixing in Continuous Flow Systems,"Chemical Engineering Science, Vol. 2, No. 1,1959, pp. 1-15.

13 Popovich, A. T., and Hummel,R. L., "Experimental Study of theViscousSublayer in Turb ulent Pipe Flow," AIChE Journal, Vol. 13, No. 5,1967, pp.854-860.

14 Pang, Y., M.S. Thesis, Depa rtmen t of Mechanical Engineering, University of A kron, Akron, Ohio (1977).

15 Hubb ard, D. W., and Lightfoot, E. N., "Correlation of Heat and MassTransfer Data for High Schmidt and Reynlds Num bers," I&EC Fundamentals,Vol. 5, No. 3,1966, pp. 370-379.

16 Meek, R. 1., and Baer, A. D., "The Periodic Viscous Sublayer in Turbulent Flow," AIChE Journal, Vol. 16, No. 5,1970, pp. 841-848.

17 Thomas, L. C , and Greene, H. L., "An Experimental and TheoreticalStudy of the Viscous Sublayer for Turbulent Tube Flow," Symposium onTurbulence in Liquids, The University of Missouri, R olla, Mo., (1973).

18 Fage, A., and Townsend, H. C. H., "Examination of Turbulent Flow withUltramiscroseope," Proceedings of Royal Society Series A, Vol. 135,1932, pp.656-662.

19 Lin, C. S., Moulton, R. N., and Putnam , C. T. L., "Mass Transfer Between Solid Wall and Fluid Streams," I&EC,Vol. 45, No. 3, 1953, pp. 636-646.

FEBR UARY 1978, VOL 100 / 97




J . A. Roux

Project Engineer

A. M. Smith

Research Supervisor

AEDC Division,

ARO, Inc.,

Arnold Air Force S tation, Term.

Combined Conductive and RadiatiweHeat Transfer in an Absorbing and

Scattering Infinite Slab

Simultaneous radiation and conduction heat transfer results are presented for an absorb

ing and isotropically scattering medium having negligible emission. The radiatively par

ticipating medium is assumed to be one-dimensional and bounded by an opaque substrate

and by a semi-transparent top interface. The boundary interfaces are assumed to reflect

and transmit radiation in accordance with Fresnel's equations and Snell's law, respec

tively. A diffuse radiative flux, along with a conductive flux, is assumed incident upon the

particpating medium at the top interface. T he Chandrasekhar solution to the transport

equation is employed, then the solution of the governing energy equ ation is formulated

for the radiatively participating medium. Heat transfer to the substrate is presented as

a function of the governing parameters: albedo, optical thickness, and substrate and me

dium refractive indices. Finally, dimensionless temperature profiles are shown. Solutions

for a convective boundary condition are also derived.

Intr oduc t ion

The transfer of heat s imultaneously by conduction and radiation

occurs in many materials such as foams, f ibers , powders, and other

semi-transparent materials of practical engineering importance. The

problem considered here is for negligible emission. Negligible emissioncan even occur in the analysis of reen try hea t shields where "sev ere

environments" [ l]1

are encountere d, i.e., where the imposed (incident)

radiation is much more intense than th at which is in ternally emitted .

Here i t is important to determine not only the heat f lux but a lso the

temp erature profile , in order to compute the the rmal s tresses induced

across the medium. In addition to these applications, the analysis

presented here is a lso important for i ts application to cryodeposits

formed in thermal vacuum chambers, formed upon cryogenically-

cooled optics , and formed upon cryogenically-cooled storage tanks

used in outer space. Here th e cryodeposit emission is negligible , but

part of the incident radiant energy is radiatively scattered and

transm itted to the substrate , while the absorbed portion is conducted

to the substrate . I t is important to be able to determine the temper

ature profile because at certain temperature levels cryodeposits are

expected to undergo structural phase changes [2] which are known

to affect the reflectance (due to increased interna l scattering) and t hus


Contributed by the Heat Transfer Division and presented at the AIChE-ASME H eat Transfer Conference, Salt Lake City, Utah, August 15-17, 1977.

Revised man uscript received at ASME H eadquarters October 6,1977. PaperNo. 77-HT-50.

affect th e tran spor t of heat to the sub strat e . Having a means of pre

dicting the temperature profile would permit a better estimation ci!'

when this pha se change and hence reflectance change will occur.

Most of the work done for combined conduction and radiation h;is

been for a medium bounded by two constant temperature plates wil ii

diffuse b oun dari es, [3, 4, 5]. An extens ive bibl iogra phy of relatedproblems dealing with s imultaneous conduction and radiation heat

transfer in a radiatively partic ipating medium is given by Viskanla

and G rosh [6]. In spite of the work tha t has b een don e, most of the

solutions have been given in terms of integral equations, or solution?

have been performed for mathematically s implif ied approaches t»

the tran spo rt equation , such as tha t of Ku belka -Mu nk [7] as used in

[8]. Little has been done toward using the Chandrasekhar [9] solution

to the transport equation in connection with the energy equation t<>

solve the coupled con duction and radi ation problem . In this work, tin1

Cha ndras ekha r solution to the trans port e quation is used in con

junc tion with th e energy equ ation to provide a solution w hich is ea.-y

to employ.

Sta te me nt o f Pr ob le m

The geometry and coordinate system associated with the s imultaneou s conduction an d radiation of heat to the substra te are shown

in Fig. 1. Th e nom encl ature employe d in Fig. 1 is essentially t he sam"

as tha t used in [10] and [11]; regions 1,2, and 3, respec tively, represent

air , radiatively-partic ipating m edium, and opaq ue substrate , io ' s tl"'

incident rad iant i ntensity which is take n to be diffusely distr ibute ' ' .

The dielectr ic coating is partia lly transparent and the opaque sub

strate is considered to be either a conductor or dielectr ic haviiin

negligible emission. The air-coating interface is assumed to reflc '1





T 0 " / (o+kwy0 T - fl ' COS 8

. : JK T , -Ml a ' T S / ( o + k ) d y C o a ' l n 9 ;

o. "•n2

Substrate: m3

Fig . 1 Coordinate system and geometry

and transmit radiant intensity according to Fresnel 's equations and

Snell's law, respectively; the subst rate is a Fresnel reflector. The ex

ternal and in ternal reflectances of th e top in terface are des igna ted

as pn an d P21, respectively. Also, the reflectance of the coa t ing -sub

strate interface isdesignated P23.

T he in te r io r of the radiatively-partic ipating coating is considered

to be abso rbing and isotropically scattering. Emission is considered

to be negligible either du e to the substrate being cryogenically cooled

or due to the rad ia t ion inc iden t upon the top in terface being that

characterized by a "severe enviro nme nt" [1] (e .g . , radiatio n emi tted

from an entry probe gas cap at 20,000 K diffusely incident on a m a

terial at 1000 K). The medium is considered to be non-gray with re

spect to rad ia t ive t ranspor t and the radiative intensity f ie ld is as

sumed to be axisym metric. A specified conductive hea t load, (jc, is also

assumed to be incident upon the top interface; th is heat load may be

considered as either posit ive (heat added) or negative (heat with

drawn) from the coating (see Appendix for the case where this heat

load is due to convection). The to tal heat load to the s u b s t r a t e isdesignated by <?o and also may be posit ive or negative.

A n a l y s i s

T he monochromat ic rad ia t ive t ranspor t equa t ion sub jec t to the

above assumptions is

di(r, p) —i(r, p) t W

Yp-I £(T, p')dp' (1 )

where i(r, p) is the local radiant intensity, I(T, p), nondimensionalized

by the diffusely incident intensity Io (Pig. 1); p is the cosine of the

in ternal polar angle 0 defining the direction of/(r , p); W = al(a + k)

is the a lbedo pa ramete r , and r is the local optical depth which is re

la ted to the posit ion coordinate y by

• = Jy

(a + k)dy' (2)

with a and k being the sca t te r ing and absorption coeffic ients , re

spectively. Regular Fresn el reflection and refraction are assumed at

the coating boundaries . Mathematically these boundary conditions

are expressed as

( ( T 0 , -M) = P2I(M)I '(TO, M) + [1 - P I 2 ( M ' ) ) " 22

(3)

for the coating-air in terface and

i(0, P) = P2 3 (M)* '(0, -H) (4)

for the coating-substrate interface where ri2 is the refractive index

of the coa ting and p' is the cos #i with d\ being the external polar angle

(Fig. 1). The directions \x' and p are related by Snell's law

M ' = [l - (1 - M W (5)

an d TO = T(V = L) where L is the coating geometrical th ickness.

The solution to the t r anspor t equa t ion , equa t ion (1), is accom

plished by the m e t h o d of C h a n d r a s e k h a r . The objective of the

C h a n d r a s e k h a r m e t h o d is computa t ion of the eigenvalues and ei

genvectors of the coeffic ient matrix associated with the system of s i

multaneous differentia l equations result ing from the use of discrete

coordinates. Once the eigenvalues and eigenvectors are determined,

the homogeneous solution is known; the integration constants are then

•evaluated from the boundary cond i t ions .

Before the eigenvalues and eigenvectors can be computed , the

tran spor t equation, equation (1), together with the boun dary condi

t ions, equa t ions (3) and (4), must first be rewr i t ten in t e r m s of the

discrete ordinates. This consists of replacing the in tegral term in

equa t ion (1) by a Gauss ian quadra tu re of the form

x ;(x)dx

p

•• E ajf(xj) (6)

where xy are the quadra tu re po in ts (d isc re te o rd ina te s ) , aj are the

quadra tu re we igh ts , and p (which is an even integer) is the order of

the quad ra tu re approx ima t ion . Rep lac ing the in teg ra l te rm in equa

tion (1) by equation (6) yields a system of p simult aneou s differen tial

equa t ions


a;, aj, ak = quadra tu re we igh ts

Cj, cp+i-j = in tegration constants in equation

(11)h = convective heat-tr ansfer coeffic ient

I - local monochromatic ra diant intensity in

coating

h = inc iden t rad ian t in tens i ty on coating;

monochromat ic fornon-gray coating, to tal

for gray coating

i = I/Io, d imens ion less monochromat ic in

tensity

K = the rma l conduc t iv i ty of coating

k = absorptio n coeffic ient of coating

L = geometrical th ickness of coating

N = conduc t ion - rad ia t ion pa ramete r

n\, n% na= refractive indices of air, coating,

and substrate , respectively

p = order of q u a d r a t u r eqc ~ conductive heat flux incident on coating

top interface

go = to tal heat transfer to s u b s t r a t e

qn = monochromatic radiative f lux

qc s = conductive heat transfer to s u b s t r a t e

qiis — to tal radiative heat transfer to sub

s t ra te

T, Ts — coa t ing tempera tu re and s u b s t r a t e

tempera tu re , re spec t ive ly

Taw = ad iaba t ic wa l l t empera tu re

W = scatter a lbedo of coating

y - posit ion coordinate in coating

0 = in terna l polar angle (see Fig. 1)

01 = external polar angle (see Fig. 1)

0C = internal critical angle (see Fig. 1)

\ , \j = eigenvaluesp ., p' = co s S and cos 0i, respectively

P i, Pj,P k, P e> Pe'=

quadra tu re po in ts (d is

crete ordinates)

v = radiation frequency

P12, P21 = external and internal reflectances,

respectively, of coating top interface

P23 = reflectance of coa t ing -subs t ra te in te r

face

a = scattering coeffic ient of coating

T, TO = local optical depth and coating optical

thickness, respectively

<j> = azim uthal angle (see Fig. 1)

Journal of H eat Transfer FEBRUARY 1978, VOL 100 / 99




di(r, i+ — T. aflir, H J ) , £~\,

dr /ig

with boundary conditions

i(ro, ~Ht) = P2l(Mf)i(T0, « ) + [1 ~ P12(M/)]«22,

(7)

' = 1, p/2 (8)

and

j ( 0 , M) = P23(M^)i(0, -m), t = l P/2 (9 )

It has been shown [12] tha t the eigenvalues of equation (7) are the

values of X which satisfy

jh (1 - X V ) ' W(10)

The eigenvalues of equation (10) occur in plus and minus pairs and

are bounded such that 0 < Ai2

< 1 V <X22< I V < . . . <A p

2/2 <

l/n p2/2 . Th us all the roots (eigenvalues) X

2 have individual bounds.

Using these bound s, the numerical solution for the roots of equatio n

(10) converges very rapidly. The solution of equation (7) for W = * 1.0

is given by

p/2 1I(T, He) = L

; = i 1~X,V2

[c;( l - \jne)exiT

+ c p + w ( l + \jne)e-^], £=1 p (11)

(The solution for W = 1.0 is pres ente d in [9], bu t is not co nsideredhere s ince the radiative and conductive heat transport become un

coupled.)

The values C J and cp+i-j[j = 1 ,. . ., p/2] in equation (11) are the

p in teg ra t ion cons tan ts de te rmined f rom the boundary cond i t ions .

Subs titution of equation (11) into equation s (8) and (9) yields a system

of p nonhom ogeneous l inear a lgebraic equations to be solved for the

p values of c ; use of the Gauss-Jordan method, Cholesky [13] method,

or other computer l ibrary routines readily allows determination of

the integration constants . Thus the solution of equation (7) is given

by equation (11) which only requires the determination of the ei

genvalues X from equa tion (10) s ince the eigenvectors are explicit ly

expressed in equation (11); then the inte gration c onsta nts are easily

found via standard techniques for solving systems of nonhomogeneous

linear a lgebraic equations. The results presented in this paper were

obtained using a 48 (p = 48) po in t s ing le Gauss ian quadra tu re .

However, i t should be noted that higher or lower quadrature orders

may be used and also different types of qua dra ture form ulas may be

used such as the double Gaussian quadrature.

Now it is necessary to derive an expression for the spectr al ra diative

flux, qn . Cons is ten t wi th the ax isymmetr ic a ssumpt ion , the d imen-

sionless radiative f lux at optical dep th T in the coating m ay be ex

pressed in terms of the Gaussian quadrature as

<?fl(r) P— — = ^ L v , mmai

Trio i=l

(12)

Subs t i tu t ion of equation (11) into equation (12) and rearranging

yields

— — = 2 1 ( 1 - \m) (c,-eV E — - —irlo ;=i \ ;=i 1 + Aj M

+ cp+ l.je-^±-^-) (13)

i=l 1 - XjHi'

The summations over index i in equatio n (13) can be simplified so as

to yield

qR(r) A(l-W) Pn (1 - x - w )= _4 _ — •£ ; ( e y e V - Cp+i-je

x'r) (14)

Trio W j - i

Having an expression for the spectral radiative f lux, the solution

to the problem of combined conduction and radiation can be formu

lated. The energy equation may be writ ten [14] as

£(-*fK(J><"*)- o (15)

where K is the the rma l conductivity of the coating, T is the temper,

ature at depth y in the coating (Pig. 1), and v is the radiation frequency

which ranges from 0 to <= . The boundary conditions for equation (15j

ar e

T(0) = T S a t y = 0 (16)

where Ts is the tempera tu re of the substrate , and performing an

energy balance at the top interface yields

K—\ =qceXy = L

dy\L .

(17)

Integration of equation (15) yields

T = T s +K X " ( S J

tlR(y')dy

' ~4 R { y = L ) y

)d v +

^ K(18)

where K has been assumed co nstant an d the bou ndary conditions in

equations (16) and (17) employed. Furt her assum ing that a an d k are

independent of temperature and posit ion, equation (18) a long v/i th

equatio n (14) allows the developm ent of an explicit expression for the

tempe ra tu re p ro f i le in the m ed ium,

T-m 4ir /*

J oh

K

(i - w) pp (i - x yo (< r + K ) W jh X;

2; ' = i

eK'T) + cp+i-j(l - e~ xir)] dv

{1-W)

X [c;( l

+M

4«

+ 4' X "

/ o

• X j / i j )

wp/2 (1

x E —; = i

(cjex'To - Cp+i-je-^o) dv

(19)

where r = (a + k)y and T O = (<r + k)L.

The expression for the heat transferred to the substrate is the sum

of the heat transferred by conduction and radiation. The heat con

ducted to the substrate is given by

.dT\ . r" r (1h

* CO (

„ . o

W) P j? (1 - XjHj)

W h Ay

X [cj(ex'T<> - 1) + C p + 1 - / (1 ™ e- x> r°) ] dv (20)

and the rad iative f lux absorbed by the sub strate is given by

<?*s=2 f" f11(0,-^)11-

Jo Jo

P2l(n)]lidndv •• f " q n ( 0 ) d vJo

which, by means of equation (14) can be expressed i

<7flS: irr I I0

Jo

(1 - W) Pg (1 - XjHj) ,

J = l[ C j - Cp+1-;J

W p i X;

The tota l heat transferred to the subs trate is given by

<?o = Qcs + <JRS

which, from equatio ns (20) and (22), can be writ ten as

\w )

(21)

( 2 2 )

( 2 3 )

<? o = <Jc + 4-IT I

Jo

' (l-W) P/2 (10

w ^ \W y=i Xj

(Cj.eA,To _ c p+ i-ye~ xJT ») dv (24)

Prom equatio ns (19), (20), (22), and (24), the tem per atur e profile

and the conductive, radiative, and total heat transfer to the substrate

can be respectively calculated provided the spectral optical properties

o f the non-g ray med ium and subs t ra te , a, k, n% and 1 1 % , are known.

For such a non-gray situation, the integration with respect to fre

quency could be numerically performed using the standard technique

of subdividing the total frequency interval in to a finite number of

sub-inte rvals . However, for the purp oses of obtaining analytical re

sults and i l lustrating the effects of the pertinent dimensionless pa-





rameters , the medium and substrate optical properties will be as

sumed gray. This yields much simpler ana lytical expressions for the

temp erature profile and conductive, radiative, and total heat transfer

to the sub s t ra te ,

T , , (1 - W) P/? (1 - Xjnj) . „ .. , , ,, , „.— = x+ Y, -

1- c,-(l - e V ) + cB+ i-;(l - e~

Xr)]

, / l ( jc ( 1 - i V ) P/ ? (1 - Xjnj)

\4TVir/o WN ;= ! Ay

[ c . e X ; r o _ C p + w e - A / T 0 A (25)

qcs Qc , 4(1 " W) P/2 (1 - Xjnj)= 1 2^

irlo Trio W j=\ Xj

X [cjieô - 1) + c p + w ( l - e-ô)] (26)

M-W^f il̂ aa>[Cj-Cp +w] (27)

<?o __ gc , 4(1 - W) P £ ( 1 - XJHJ)

X [c ;exô - Cp+i-ye-V'o] (28)

whe re TV = K(a + k)Ts/iirIo with I0 for the gray case (equations

(25-28)) related to To for the non-gray case (equations ( 19,20,22,24))

by

io(gray)Jo

Io(non-gray)di ' (29)

From equation (19) or (25) i t is seen that when the radiation is

uncoupled from the conduction (W = 1.0) the tempe ratu re profile is

linear. If in addition to being uncoupled, q0 = 0, then the temperature

of the coating is a constant and is equal to the substrate tem peratu re.

It is the radiation terms in equation (19) or (25) which give rise to

non-linear-temperature behavior. From equation (19) or (25) it is also

seen tha t for a very high therm al condu ctivity (TV —• •») the temp er

ature across ' the coating also approaches Ts - However, from equation

(20) or (26) it is seen tha t the th erm al condu ctivity does not effect t he

conductive heat transfer . This is because of the assumption that the

coating was nonemitting. The beauty of equations (25), (26), (27) and

(28) is that they are s imple to evaluate .

M e s u l t s

The primary analytical results are given by equations (25), (26),

(27), and (28). The results based on these equations are now shown

in the form of temperature profiles (equation (25)) and total heat

transfer to the substr ate (equation (28)) . Th e tempe ratu re profiles

as a function of TV and W are shown in Fig. 2. Since TV rep resen ts the

importance of conduction to radiation heat transfer , i t is seen for a

given substrate temperatu re tha t as TV increases the temperature level

is decreased. This is important when a given threshold temperature

indicates the on set of melting or a crystall ine s truc ture change which

can affect the s tructural characteris t ics of the coating. As stated

earlier, N has no effect upon heat transfer to the substrate; thus the

thermal conductivity of the coating becomes impo rtant in determ ining

structu ral effects witho ut influencing hea t transfer . Also in Fig. 2 i t

is seen tha t as W increases, the temperature level within the coating

is likewise red uced for a fixed TV. Th is is beca use larger W values

correspond to more scattering; more scattering means that more ra

diation is scattere d o ut of the coating and less energy is absorbe d, t hus

causing the te mp erat ure increase to be less . Also when th e coating is

conductively cooled on top, the tempe ratu re level may be significantly

reduced. The radiation terms in equation (25) cause the profiles to

deviate from the linear profiles which would be expected for pure

conduction.

In Fig. 3 the to tal hea t transfer to th e subst rate is shown as a

function of optical thickness with coating refractive index and albedo

taken as paramete rs . Since the conductive heat load qc/irIo will cause

the heat transfer results to be shifted by a constant amount (see

equation (28)) , results will only be shown for qc = 0.0. Results for a

nonzero conductive heat load can easily be obtained by addition or

subtraction of the value of qcUh y Fig. 3 shows th at for low albedo (W

= 0.10) the heat transfer to the substrate is essentially independent

of optical th ickness because the coating is highly absorbing a nd con

ducts the absorbed heat to the substrate , which in this case is a low

reflector (black paint). The absorption is so strong for the low albedo

coating that s ignificantly increasing optical th ickness does not ap

preciably increase the absorption of the incident radiation. For high

albedo ( W = 0.99), the effect of increasing the optical thickness is to

cause a large reduction in the amount of heat transfer to the substrate.

?. u

4.0

3,0

2.0

1.0

1 qc/TrI0= 0

_ _ _ o. 9 n3 = 1.48 - i 0.00

0.6 n2 = 1- 2— 0.3 T °" ,U

p = 48

/"""*

N = 0 . 3 - / ^

> * /

// 3—

" # /

&s *>- _^^^^^

0 0.2 0.4 0.6 0.8 1.0

Dimensionless Distance, T /T 0

Fig. 2 Nondlmensiona! temperature profile as a function of dimensionless

distance lo r various values of N and W

1 2 3 4 5

Optical Thickness, T 0

Fig. 3 Heat transfer as a function of optical thickness of various values of

n 2 an d W; ns = 1.48 - iO.OO an d qc/-w!0 = 0.0

Journal of Heat Transfer FEBRUARY 1978, VOL 100 / 101Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



As the optical th ickness increases, the coating scatterin g allows less

energy to p e n e t r a t e to the abso rb ing subs t ra te . Sca t te r ing re su l t s in

reflecting much of the inc iden t rad ia t ion out of the coating. Also

shown in Fig. 3 is the effect ofrefractive index. A coating with a small

refractive index (re2 = 1.2) will absorb more hea t at small albedo values

because of the sma l l top in terface reflectance as compared to that of

a coating with a higher refractive index (rc2 = 1.4). For high albedo

values (W = 0.99) the higher refractive index coating (n2 = 1.4) is seen

to cause more heat transfer to the substrate . This is because the larger

refractive index coating has a smaller cri t ical angle which results in

the t rapp ing of in te rna l ly sca t te red rad ia t ion . Rad ia t ion inc iden t atth e top in terface at an angle greater than the crit ical angle is to tally

internally reflected, result ing in the subs t ra te hav ing ano the r op

por tun i ty to absorb this radiation. The increase in the heat transfer

to the subs t ra te obse rved for very small optical th icknesses as com

pared to t h a t for the bare subs t ra te is deno ted by the broadened l ine

width on the vertical axis ofF i g . 3, and is du e to the relative refractive

index change [11] at the substrate-coating interface. For W = 0.99 the

two curves are seen to cross at approx ima te ly TO = 0.25; below this

optical thickness value the heat transfer appears to be domina ted by

the top-interface reflectance, and above this value the heat transfer

is dominated by in ternal scattering. The albedo and refractive index

values of W = 0.99, re2= 1.2and W = 0.99, n2 = 1.4 are characteris t ic ,

respectively, of solid H20 and solid C02 in the visible wavelength

range.

I l lu s t ra ted in Fig. 4 is the to tal heat transfer to the subs t ra te as a

function ofoptical th ickness with ra2and W t aken asparamete rs , but

as opposed to Fig. 3, the results are for a bare substrate hemispherical

reflectance of about 0 .58 (which corresponds to the reflectance cal

culated from Fresnel 's equations for metals using a refractive index

characteris t ic of stainless s teel in the visible wavelength range ris =

2.48 - (3.43). Her e the hea t transfer to the substrate for the low albedo

coating increases until it becomes independen t ofTQ. Also at very small

optical th icknesses s l ightly more heat is t r ans fe r red to the subs t ra te

for the larger refractive index coating due to the larger relative re

fractive index change. As in Fig. 3, the increase in heat transfer to the

subs t ra te for very small optical th icknesses isdeno ted by the b road

ened l ine width on the vertical axis . At larger optical th icknesses for

the lower refractive index and albedo coating (re2=

1-2, W = 0.1), the

heat transfer increases because now the top-interface reflectance is

1.0

0.9

- 0.8 -

/ /

_ / Vn2=1.2

^ V .

1 1

W - 0 . 1 0

_ ._

n2= 1.4-4

/

X . ^ 099^ - " - — .

1 1 '—

0.7

0.6

0. 5

0.4

0. 3

0 1 2 3 4 5

Optical Thickness, T 0

F i g . 4 Heat transfer as a function of optical thickness forvarious values of

n 2 and W;n3 = 2.48 - 13.43 and (fc/ir'o = 0.0

the determining factor and the lower refractive index corresponds to

a lower interface reflectance [ P I 2 ( M ' ) ] . The high albedo coating shows

the heat transfer to have a maximum. T h is phenomenon is analogous

to a crit ical th ickness of high albedo insulation. This heat transfer

m a x i m u m is due to the t r app ing of in ternally scattered radiation as

explained in detail in [11]. Sim ilar to Fig. 3 (W = 0.99), higher heat

transfer to the subs t ra te is observed for the higher refractive index

coating. Fig. 5 presen ts the heat transfer to a highly absorbing sub

s t ra te as a function ofalbedo with n2 and TO t aken as paramete rs . As

the albedo approaches zero (W -* 0.0) the results are independen t

of optical th ickness .For a

given refractive index,the

heat transfer issignificantly reduced as the albedo increases because of the greater

scattering of the incident radiation. For a given optical thickness, the

curves for the two refractive indices cross at Wx

0.6. Below W » 0.6

the substra te absorbs less heat for the higher refractive index coating

because of the greater to p interface reflectance. At albedo values above

W ~ 0.6, greater heat transfer occurs for the higher refractive index

coating because of the inc reased t rapp ing of in ternally scattered ra

d ia t ion .

Shown in Fig. 6 is the heat transfer to the subs t ra te as a function

of n2 with W and n^ t aken as paramete rs . For the low and in terme

diate a lbedo values (W = 0.10, 0.70) the re su l t s appea r to be inde

p e n d e n t of the substrate reflectance due to the large value of optical

th ickness (T O = 5.0). At large W (W =0.99) a small substr ate reflec

tance effect was noticeable. It is in teresting to no te tha t for low W (W

= 0.10) the heat transfer decreases as ra2 increases. Due to the in

creased top in terface reflectance, less radiation enters the highly

absorbing coating. For the high albedo (W = 0.99) coating the heat

transfer is seen to increase as n2 increases; this isdue to the increased

t rapp ing of in ternally scattered radiation. As n2 increases, the critical

angle decreases resulting in more t rapp ing of radiation scattered into

directions greater than the crit ical angle . At the in termediate value

of W (W = 0.7), the heat transfer is seen to ob ta in a m a x i m u m at

abou t «2 = 1-7. For ra2 < 1.7 the heat transfer increases with increasing

n 2 indicating internal trapping to be dominan t . For n2 > 1.7 the heat

transfer decreases with increasing n2. indicating the top interface

re f lec tance phenomen a to be d o m i n a n t .

Fig. 7 depicts the same parameters asFig. 6 except for a small value

of TO (TO = 0.5). He re the substrate reflectance is seen to have a sig

nificant effect on the heat transfer for all values of W with reduced

heat transfer corresponding to the higher reflecting substrate. For the

moderate reflecting substrate («3 = 2.48— J3.43) the heat transferis seen to increase as n2 increases for W = 0.99, to decrease as n2 in-

1.0

0.9

3? 0.8 -

0.7

0.6

0.5

0.4

0.3

z :^ 2 § ^

-

-

i

^§5r* -^

n 2 = 1.2

I I

n 2 • 1.47

^ / ^ 0 . 5

\ \ 5 . 0

\ \\ \\\

, o \ \

u11

1

0.2 0. 4 0. 6

Albedo, W

1.0

F i g . 5 Heat transfer as a function ofalbedo forvarious values of n2 and T O !

n 3 = 1.48 - lO.OO andqc/-rrl0 = 0.0



http://5.0.0.0/

http://5.0.0.0/



W- 0 . 10 f n 3 = 1.48-10.00

1.0 1.2 1.4 1.6 1.8 2 .0 2.2

Refractive Index, n?

Fig. 6 Heat transfer as a function of coating refractive index for various valuesof n3 and W; T<J = 5-0 and qc/irlo = 0.0

creases for W = 0.10, and to reach a m aximu m for W = 0.70. For the

low reflecting s ub str ate («3 = 1.48 - iO.00) the hea t tran sfer is seen

to decrease as ni increases for W = 0.10 due to the increased top in

terface reflectance. The heat transfer is seen to atta in a maxim um for

W = 0.99 and W = 0.70; at low ni, in ternal scattering causes the re

duction in the heat transfer and at high n-i the top interface reflectance

causes the reduction in the heat transfer .

S u m m a r y a n d C o n c l u s i o n s

A solution to the problem of combined conduction and radiation

has been presented for both nongray and gray media having either

a conductive or convective boundary condition at the top interface.

The temperature profile and heat transfer have been expressed inte rms o f the Chandrasekhar so lu t ion to the rad ia t ive t ranspor t

equatio n and a re prese nted in equations (19), (20), (22), and (24) for

a nongray medium, and in equations (25-28) for a gray medium. These

equations were shown to be easy to evaluate for the gray medium,

requiring only the s imple solution of equation (10) and requiring the

use of a l ibrary compu ter routine for solving systems of s imultaneou s

linear nonhomogeneous algebraic equations for the determination

of the integration constants c ; [i = 1 , . . . , p ] . Resu l t s were ob ta ined

employing a 48-point (p = 48) s ingle Gaussian quadrature. High

values of W and iV were shown to reduce the temp eratur e level of the

temperature profile, which can be of importance in determining if the

temp eratur e in a given mediu m is approaching a crit ical tempe rature

level. Heat transfer to the substrate for media where emission is

negligible was shown to be independent of N. The heat transfer to the

substrate was shown to be a s trong function of ni, W, TO, and ba resubstrate reflectance. Effects of top interface reflectance and internal

scattering dominated the heat transfer behavior. Also, for a moder

ately reflecting su bstrate (hemispherical reflectance = 0.58), the hea t

transfer was shown to have a maximum with respect to optical

thickness. This indicates there is a cri t ical th ickness of high albedo

insu la t ion .


The research reported herein was conducted for the Arnold Engi

neering Development Center (AEDC), Air Force Systems Command

(AFSC), by ARO, Inc. , a Sverdrup Corporation Company, operating

contractor for the AEDC.


1.0

1.0 1.2 1.4 1.6 1.8 2.0 2.2

Refractive Index, n?

Fig. 7 Heat transfer as a function of coating refractive index for various valuesof n 3 and W ; T 0 = 0.5 and qc/Tl0 = 0.0

R e f e r e n c e s

1 Howe, J. T., and Green, M. J. , "Analysis of Sublim ation-Co oled C oated

Mir rors in Convec t ive and Radia t ive Environments ," AIAA Journal, Vol. 11,No. 1, Jan. 1973, pp. 8 0-87.

2 Woo d, B. E., Sm ith, A. M., Roux, J. A., and Seib er, B. A., "Sp ectra l In

frared Reflectance of H 2 0 Condensed on LN 2-Cooled Sur faces in Vacuum,"AIAA Journal, Vol. 9, No . 9, Sept . 1971, pp. 183 6-1842.

3 Viskanta, R., "He at Transfer by Conduction and Radiation in Absorbing

and Scattering Materials ," ASM E JOURNAL OF HEAT TRANSFER, Vol. 87, No.1, Feb. 1965, pp. 143-150.

4 Weston, K. C , and Hau th, J . L. , "Unst eady, Combined Radia t ion andConduc t ion in an Absorbing, Sca t te r ing, and Emit t ing Medium," ASME

J O U R N A L O F H E A T T R A N S F E R , V ol . 9 5, N o. 3 , A u g. 1 9 73 , p p . 357-364.

5 Doornink , D. G. , and Her ing, R . G. , "Tra nsien t Combined Cond uc t ive

and Radia t ive Hea t Transfe r ," ASME J O U R N A L O F H E A T T R A N SFE R , Vol.

94, No. 4 , Nov. 1972, pp. 473-478 .6 Viskanta , R . , and Grosh, R . J . , "Rece nt Advances in Radi ant H ea t

T r a ns f e r , " Appl. Mech. Reus., Vol. 17,1964, pp. 91-100.

7 S tee le , F . A., "T he Optica l Charac te r is t ics of Pape r ," Paper TradeJournal, Vol. 100, No. 37, Mar. 1935.

8 Weston , K. C , Howe, J. T., and Green, M. J. , "App roximate tem peratu re

Distribution for a Diffuse, Highly Reflecting Material," AIAA Journal, Vol.10, No. 9, Sept. 1972, pp. 1252-1254.

9 Chand rasekhar , S . , Radiative Transfer, Dover, New York, 1960,

10 Merr iam, R. L., and V iskanta , R ., "Radia t ive Charac te r is t ics of Absorbing, Emit t ing, and Sca t te r ing Media on Opaque Substra tes ," Journal of

Spacecraft and Rockets, Vol. 5 , No. 10,19 68 , pp. 1210-1215.

11 Roux, J. A., Smith , A. M., Shakrokh i, F., "Radiativ e Transfer Propertiesof High Albedo C0 2 and H 2 0 Cryodeposi ts , " AIAA Progress in Astronauticsand Aeronautics: Thermal Control and Radiation, Vol. 31, C. L. Tien, ed., M IT

P r e s s , Cambr id ge , Mass . , 1973, pp. 369-388 .

12 Roux, J . A. , Todd , D. C , and Smith , A. M. , "Eigenva lues and Eigenvec tors for Solut ions to the Radia t ive Transpor t Equa t ion," AIAA Journal,

Vol. 10, No. 7, July 1972, pp. 973-976.

13 Fran klin , J. R., Matrix Theory. Prent ice -H al l , Englewood Cliff s , NewJersey, 1968 , p . 203.

14 Sparro w, E. M., and Cess, R. D., Radiation Heat Transfer, B e lm on t ,

Ca l i fornia , Brooks/Cole Publ ishin g Comp any, 1966.

A P P E N D I X

S o l u t i o n f o r C o n v e c t i v e B o u n d a r y C o n d i t i o n

If the heat load qc at the top interface is a result of convective

heating, the boundary condition given by equation (17) must be re

placed by

FEBRUARY 1978, VO L 100 / 103




K~ \ =h[Taldy I t

•T(y = L)\aAy=L (A- l )

where h and T a„, are the specified convective heat-tran sfer coefficient

and adiabatic wall temperature, respectively, and T( y = L) is the

med ium surface temp erat ure, which is not specified. Then solving

equatio ns (15), (16) and ( A-l) by employing the sam e general proce

dure and as sump tions used to obtain equation (19) yields the following

explicit expression for the temperature profile in the medium:

(Too, - TS) hylKT(y) = TS +

(1 + hL/K)

hy/K 4TT r°> 7Q (1 - V&sn (1 - Xjnj)

1 + hL/KK Jo (o + k) W

4l !

^

hX [CJ(1 - e V o) + c p + W ( l - e-^»)] dv + ~ f " — 9

K Jo (a +

,q~W) PZ? (1 - Xjixj)

k)

W V[cy (l - c V ) + Cp+ i- jd - e~

x'r)] dc

47T

J o "/ o

(1 - wo p/? (i - \m)

W h \j+ hL/K K

X [cjex'T» - Cp+ i-ye-V o] d„ (A-2)

The expression for the heat conducted to the substrate becomes

h(Taw - T s) (h/K)4ir

Qc s = - 1 + hL/K 1 + hL/K Jo (0 +

(1 - W) a g

X ~ ^ [cy(l - e Vo) + cp + i -y ( l - e~ Vo)] dv

V(1 - W) PJ? (1 - XJIXJ)

| 4TT p - (1 - H O p/y (i - A;

1 + hL/K J o ° W ;=i A;

X \cjexir« - cp+l -je-xJT°] dv

r - (1 - W) pn (1 - XJHJ) ,

•4TT 1 I0— — — £ -^-(cj-Cp+i-rfdv (A-3)*^ 0 W ; = i A ;

while the expression for the radiation flux absorbed by the substrate<JRS is unchanged and thus equal to equation (22). Finally , the ex

pression for the total heat transfer to the substrate takes the form

. h(Taw-T s) (h/KHw r I0 (1 - W) Pff (1 - Xjnj)9o = , . , r ,„ , , , ,Tr \ __ ;— — — 7 7 ,— ^Jo (a •+ hL/K 1 +hL/K Jo (o + k) W ;=i V

Ai rX [c;( l - e

xi

T«) + cp+ i_;(l - e~

x'r<>)] dp +

1 + hL/K Jo

(1 - W) P/2 (1 - XjHj) , , . , , ,X w r £ " — [cye^ o - Cp-n-ye-V-o] d„ (A . 4)

W ;=1 X;

Equations (A-2), (A-3), and (A-4) can be applied to a medium and

subs trate w ith gray optical propert ies by employing the same proce

dure used to obtain equatio ns (25), (26), and (28) from equations (19),

(20), and (24).

104 / VO L 100, FEBRUARY 1978 Tra nsa ction s of the ASWIE




J . L Kraz insk iGraduate Student,

Department of Aeronaut ical

and Astronautical Engineering

R. 0 . Buck ius


Department of Mechanical

and Industrial Engineering, Assoc. Mem. ASME

H. KrierAssociate Professor,

Department of Aeronaut ical

and Astronaut ical Engineering

University of Il l inois at Urbana-Champalgn,

Urbana, III.

A model for Flame Propagation

in Low Volatile Coal Dust-Air

i i i t u resA model describing steady, laminar flame propagation in low volatile coal dust-air mix

tures is presented. The model includes radiative transport, a two-phase energy equation,and heterogeneous carbon gasification of the coal particles. The differential approxima

tion (or method of spherical harmonics) was used to represent the one-dimensional radi

ant energy transport. The equations were numerically integrated and predictions of flame

structure and the adiabatic flame speed are presented. The effects of radiation properties

and coal particle size are presented and discussed. The influence on the temperature pro

files of heat released in the solid phase is also described.

In t roduct ion

Coal mine explosions have been a problem for hundreds of years .

The projected need for coal as an energy source wiil dem and increased

subsurface mining which wil l , in turn , require improved techniquesfor the prevention of dust explosions. Preventive procedures, however,

must rely on fundamental knowledge concerning the behavior of coal

dust combustion. To th is end, a mathematical model of f lame prop

agation through coal dust-air mixtures has been developed. The model

has been used to predic t burning velocit ies and to s tudy the s truc ture

of coal dust- a ir f lames.

A n a l y s i s

As sum ptio ns. Coal part ic les burn in a very complicated ma nner.

The combustion process includes heterogeneous surface reactions,

devolatil iza t ion and subsequ ent reaction of the volat i le compo nents ,

swelling, cracking, and other physiochemical changes to the particles.

In addit ion, i t is well known that coal part ic les do not a lways burn

simply as shrinki ng spheres bu t can burn in te rnally and form cenos-

pheres (hollow spheres) as well [1,2] .1 The composition of the volatiles

re leased and the combustion i tse lf depend upon the ra te of heating

in a f lame, thereby making analytical descrip tion of the process ex

tremely diff icult .

Therefore , in order to obta in a tractable model of the combustion

behavior of coal part ic les , assumptions must be made. Here i t is as

sumed that the part ic les burn only a t the surface according to the

irreversib le , heterogeneous reaction C(s) + 02—"- CO2. Swelling and

cenosphere formation are not considered. Volati les combustion is not

1 Numbers in brackets designate References at end of paper.Contributed by the Heat T ransfer Division of THE AMERICAN SOCIETY OP

MECHANICAL ENGINEERS and presented at the AIChE— ASM E HeatTransfer Conference, Salt L ake City, Utah, August 15-17, 1977. M anuscriptreceived by the Heat T ransfer Division M ay 2,1977. Paper No. 77-HT-18.

included a nd he nce the m odel is strictly applicable only to low volatile

coals . A l is t of addit io nal as sump tions used in developing the th eory

is given in Table 1.

Tw o- P h as e Reac t ive F low. The gove rn ing equa t ions for the

model are those for steady, laminar flow in a two-phase system. Theseequatio ns are s implif ied forms of the general , mult i phase f low equa

t ions g iven in the tex t by Soo [3].

T a b l e 1 Mode l assumpt ions

The f low is low speed, one-dimensional , and s teady. The f low

is area-averaged to represent a laminar two-phase system.

Body forces are neglected.

Viscous forces within t he gas phase are neglected.

The velocit ies of the two phases are equal .

Pres sure in the gas phase is constan t while the solid phase pres

sure is assum ed to be zero .

Radiative heat transfer to the gas phase is neglected.

The Dufour effect (heat f lux produced by concentra t ion gradi

ents) is neglected.The Soret effect (mass transfer due to temperature gradients)

is neglected.

The molecular weight of the gas and the specif ic heats of the

phases are taken as constant .

Although actual coal part ic les are irregular in shape, they are

treate d here as equivale nt spheres by using a spheric i ty factor .

Their density remains constant . They are of uniform size and

un i fo rm tempera tu re .

Volati les combustion is not considered.

Th e frequency of in ter-part ic le coll is ions is assumed to be neg

ligible.

M olecular d iffusion in the gas phase is not inf luenced by the

presence of coal part ic les .

10

11

12

13

Journal of Heat Transfer FEBRUARY 1978, VOL 100 / 105Copyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



In a two-phase system , there are three se ts of equations— a set for

each of the two species and the overall mixt ure equ ations. S ince the

balance equ ations for the species mu st add up to the respective mix

ture equations, only two of these se ts are independent. Here the

equations for the solid phase and for the mixture are used. The re

sult ing equations are :

Mass Balances.

p m U = M = cons tan t

dZ _ F c _ — 3p mZR c

dx M rcM

~[(l-Z)Pm UYco.A=T\D

dx dx L c o 2 ( l -Z)p,

dYCQ,

x i 3

(1 )

(2 )

(3)

Energy Balances

M - f [ZCPc(Tc - T0) + (1 - Z)CPe(Tg - T0)}dx

dT t

dx \ dx+ rcHc n

dqr

dx(4)

M - f [ZCPc(Tc - To)] = Fgc(Tg - T c ) - ^ - TMCOAL (5)dx dx

Radiation Equations.

dqr

dx+ aG = 4ao-T c

4

dx

Auxiliary Equations.

PgTg

Pg(MW)g

R•• cons tan t

y 0 2 + YN2 + YC02 = i

PcPg

Pm~ Zp g+(l-Z)pc

(6)

(7 )

(8 )

(9 )

(10)

Here the diffusion coefficient, Dco 2 , and the therm al conductiv ity

X^, are assumed to vary with temperature according to

D c o 2 • (D co2. »© '\ , = (A ,)o ( ~ )

^ T0 /

(11)

(12)

For laminar f low with U c = U g the Nusselt number for spherical

part ic les is 2 . In th is case Fgc , the heat transfer lag coefficient, be

comes

F ec =ZZ\gPn

(13)

Re a c t ion Kin e t ic s . In order to evaluate r c , a functiona l form

must be found for the regression ra te , R c. Here an expression for R c

was obtained by applying E ssenhig h 's burnin g t ime expression [1].

T h e n R c becomes

R _ (14v8FDKDrc + 2F CH KC H

The factors FD and FCH are functions of excess air, which accoun t for

the progressive vitiation of oxygen in flames. These factors have been

set equal to one for present calculations, the disappearance of oxygen

being accounted for in the parameters KD an d KCH. These parameters

are as g iven by Ess enhigh [1] except th at he re the mole fraction of

oxygen (Xr>2) varies throughout the f lame.

K n = -

Kc

3pga(Do2)o XoJ-f-)

,4pM2irRTcl

\ 3 /L(MW.)J

r„ \ 0 .75(15)

(16)X02Pg ex p (-EJRT C)

The tempera tu re in the the rma l bounda ry laye r su r round ing the

part ic les is taken to be Tg.

Notice that the reaction term r c is not as simp le as the exp ressions

used for gas-phase, homogeneous reactions. For part ic le combustion,

- N o m e n c l a t u r e -

a = abso rption coeffic ient (ml)

CPc = specific heat of coal (1.794 X 10 3 J /

kg-K)

CPg = specific heat of gas (1.14 X 10 3 J /kg -

K )

dc = coal part ic le d ia (m)

Z>02 = diffusion coefficient for O2 in air (1.81

X 1 0 ~ 5 m 2 /s at 298 K)

DCO2 = diffusion coefficient for C 0 2 in N2

(1.65 X 1 0 - 6 m 2 /s a t 298 K)

Ei = activation energy of adsorption used in

reaction ra te (54.34 M J/mo le)/1 = an iso t rop ic sca t te r ing pa ram e te r

Fgc = hea t transfer lag coeff ic ient (W /m 3 -

K )

FD = oxygen vitiation factor for reaction

under d iffusional control

FCH - oxygen vitiation factor for reaction

under chemical control

G = average in tensity (W/m 2 )

ffcHE M = hea t of reaction of coal (-2 .32 X

1 0 7 J / k g o f c o a l )

HCOAL = energy re leased in solid phase (J /

kg )KD = burning parameter for reaction under

diffusional control (s/m 2)

KCH = burning parameter for reaction under

chemical control (s /m 1 )

K = extinction coefficient (m _ 1 )

M = mass f lux (kg/m 2— s)

(M W )j = gas molecular weight (28.86 kg/

mole)

n = number dens i ty (m~ 3)

Ps = gas pressure (1 .013 X 10 6 P a )

q r = radia t ive heat f lux (W/m 2 )

Qabs - absorption efficiency (0.84)

= scattering efficiency (0.16)rc = coal part ic le radius (m)

R = universal gas constant (8314 J/mole-

K )

Rc = surface regression rate of coal particles

(m/s)

T = temperature (To is 298 K)

U = velocity (m/s)

x = dis tance (m)

XQ 2 = mole fraction of oxygen

^02, VQ O 2, YN 2 = mass fractions of oxygen,

carbon dioxide, and nitrogen, respectively

(Yo2 is initially 0.235 in the stoichiometric

case) .

Z = mass fraction of coal (Z o for the stoi

chiometric case is 8 .112 X 10 ~ 2 which cor

re sponds to 0 . 1 04 kg /m 3)

a = 2irrc/\

Fc = coal mass depletion rate per un it volume

of mixture (kg/m 3-s )

\ = wavelength (m)

\g = thermal conductivity of gas (2.52 X 10" 2

W / m - K a t 2 9 8 K )

p = density (p c was taken to be 1.25 X 10 3

kg/m 3)

<r = Stefan-Boltzmann constant (5.669 X 10_ s

W /m 2-K ' i)

<rs = scattering coefficient (m _ 1 )

$ = scatter ing phase function

to = solid angle

H = cosine of polar angle

S u b s c r i p t s

a = adiabatic

c = coal

g = gas

m = mix tu re

o = condit ions a t cold end

/ = condit ions a t hot end





j the reaction ra te includes two terms. The parameter Krj accounts for

reactions which are controlled by the rate of oxygen diffusion to the

part ic le surface . The factor KC H accounts for reactions which are

controlled by the rate of chemical reaction at the particle surface. For

completeness , both types of ra te control are included in r c .

R a d i a t i o n P r o p e r t i e s . An approximate formulation of the

' radiativ e transfe r is employ ed which is referred to as the differential

approximation. T he governing expressions given in equations (6) and

(7) are obta ined by app ropria te averages of the one-dimensional gray

equation of transfer . The scatter ing phase function has been repre-

j sented in the l inear anisotropic form1 1 1; 4>(M ',M ) = 1 + 3 / W , - - < / I < - (17)

] and the gas has been assumed to be transparent to radia t ion. The

averaging required to obtain equations (6) and (7) can be obtained

either by the spherical harmonics method [4 ,5] or the mom ent method

[5j. For a one-dimensional geometry th is method has been shown to

be quite acc urate [6, 7].

I t remains to quantify the radia t ive propert ies for coal dust . This

requires knowledge of the optical constants of the part ic le materia l ,

the scattering ph ase function of the coal particle, the size distributio n,

'' and the coal concentra t ion. The real and imaginary parts of the index

of refraction for coals , carbons, and soot hav e been rep orted [8 -10 ] .

j The spectra l characteris t ic s throu ghou t the v is ib le and infrared re-

I g ions show mark ed differences between carbon and coal both inmagnitude a nd spectra l varia t ion. These values a lso depend upon the

type of coal considered. The spectra l varia t ions of coal in the near

infrared are not great, so that constant values were used in this work.

j Th e real pa rt of the index of refraction used is 1.75 and the imagin ary

part is 0.35.

The absorption and scatter ing coeff icients for a monodisperse coal

dust c loud are defined as

a = nirrc2Q abs (a) (18)

! <rs = mrr c2Q sca (a ) (19)

ij where n is the num ber of part ic les per unit mixt ure volume, Qaos t h e

-: abso rptio n efficiency, Qsca the scattering efficiency, and a = 2-irr/\,

with X representing wavelength . For s teady f low with a m onodisperse

suspension, n can be calculated from

nU = n0U0. (2 0)

The coal dust part ic les treated here are typically larger than 10 Mm

in radius. With temperatures of the order of 1500 K and larger , the

efficiencies in the large particle limit (a large) can be used. Th us , for

constant optical propert ies , the absorption and scatter ing eff ic iencies

are constant w ith respect to wavelength ( i .e ., gray behavior) . T he

result ing absorption coeff ic ient , using the optical constants g iven

i above, is

a = 0.84mrrc2. (21)

The radia t ive transfer includes the effects of scatter ing through

the scatter ing coeff ic ient and the scatter ing phase function. In the

I large part ic le l imit , the d iffracted compo nents and the ref lected

• compone nts can be separated. The diffracted port ion is concent ra ted

into a small solid angle about th e d irection of propagatio n of the incident beam [11, 12] and, in th is work, is not sep arated from the

transm itted energy incident on the part ic le . The ref lected po rt ion of

the scattered radia t ion for an opaque part ic le is a surface phenome

non. Th e coal partic le surface is very rough so th at th e phase function

\ is assumed to be that of a d iffuse , opaque sphere . The dominant

scatter ing direction is back in to the incident d irection. This is rep

resented as

* (M ) = 1 - nn ' (22)

i .e . , / i = — (Vs). S ince the scatter ing is represen ted as a surface p he-

' nome non, the scatter in g coeffic ient is

as = 0.1G nw rc2 (23)

j J ourna l of Heat Transfer

for the coal prope rt ies g iven above.

Bo un da ry Co nd i t i ons . The n ine unknowns in th i s p rob lem a re

the coal mass fraction (Z), the t empe ra tu re s o f the phases (T c,Tg),

the gas mass fractions (Y o2,Yco2), the variables describ ing the ra

diation field (q r,G), the gas density (p g) and the velocity (£/). The nine

equations which mus t be solved are the mixture continuity equation,

equation (1) , the coal mass balance equation, equation (2) , the CO2

mass balance equa tion, equation (3) , the mixtu re and coal energy

equatio ns, equations (4) and (5) , the radia t io n equations , equatio ns

(6) and (7) , a s ta te equation, equation (8) , and the gas-phase mass

conservation equation, equation (9) .Th e boun dary con ditions are given at ±<». At — °° there is a mixture

of coal dust a nd a ir in equil ibrium at

T c = T s = 298 K. (24)

The in i t ia l values of Z and Yrj2 are known. The velocity a t -co , {7 0)

is an unknown parameter . For zero heat loss , i t is the adiabatic

burning velocity and is dependent only on the initial conditions of the

mixture . S ince part ic les are present a t —<=, the adiabatic condit ions

on the radiation field are

< ?r(-») = 0 (25)

G ( - a , ) = 4 ff T 04(-< ») (26)

At +< », for s to ichiom etric bu rning, Z and Yrj2 are zero. In addition,

the t emp era tu re s a re g iven by

TC = TS = T, a (27)

where Tf a is the (known) adiabatic f lame temperature for zero heat

loss . Th e adiabatic radia t i on bou ndary condit ion is

<?,(+<») = 0 ( 2 8 )

since the part ic les absorb a l l the energy a t x = +«• ( the part ic les d is

appear only as or - * + « > ) .

S o l u t i o n T e c h n i q u e s . This is a two-point boundary value

problem with condit ions being given a t ±°>. Of course , one cannot

in tegrate the equations over an infin ite d is tance; thus, some ap

proximations mus t be made to solve the problem. Here the two-point

bound ary value problem w as converted in to an in i t ia l value problem

in which the in tegration is s tar ted from some point x = 0 a s sumed to

be " far" from th e region of maxim um combustion. I t is assumed tha tchemical reaction in the region x = — °° to x = 0 can be neglected. In

order to insure that the reaction ra te , Tc, is zero at x = 0, the term

(Ei/RT C) is replaced by

(II) (Tf - T°)\RTfl \T C- To'

in r c , as was suggested by Fri edm an an d Burke [13]. This ma kes T c

zero at x = 0 whe re T c = Tg = To, but approximates (EJRTC) at the

hot end where T c =* Tf .

Because of diffusion, the value of Yco 2 a t x = 0 wil l not be exactly

zero but is given by

Uo(Yco2)o= DC0,^^\ . (29)ax U= o

Also, the values of \g(d T s/dx) and q r a t x = 0 will not, in gener al, beknown. Physically , these quanti t ies represent conductive and radia

t ive heat losses from the system. This is s imilar to the concept of a

f lameholder used by Hirschfelder e t a l . [14]. Thi s heat loss can be

calcula ted from the heat balance between x = 0 a n d x = +<»:

X ^ l - 9 ' I = M [C PcZ„ + C p , ( l - Z . ) ] • (T fa - T f) (30)ax \x=o 11=0

where Z„ is the m ass fraction of coal a t * = +<=. The pa ram eter Tf is

fixed for each problem and as Tf —> • T/ a, t/o becomes equivalent to the

adiabatic burning velocity .

The radia t ion boundary condit ion must a lso be modif ied for use

a t x = 0. All radia t ion arr iv ing a t x = 0 is assumed to be absorbed ( the

back scattered port ion was found to be neglig ible); thus, the point x

FEBRUARY 1978, VOL 100 / 107


http://rtfl/

http://rtfl/



= 0 is t r e a t e d as if it is a black wall. In th is case , the bounda ry cond i

t ion becomes

2a qr(0 )dq A

dx lx=o(31)

In order to star t the in tegration at x = 0, one must , in general, know

the in i t ia l values of ^ 0 0 2 . dT g/dx, and qr. In add i t ion , the burn ing

velocity, Uo , isunknown. Since dTg/dx | x=o , 9r (0 ) , and (Jo are relat ed

by equation (30) , three values would have to be assumed in orde r to

star t the in tegration. One would then have to i t e ra te on these values

un t i l the bounda ry cond i t ions for the combus t ion p roduc ts weresatisfied at a given location (the hot end ) . In prac t ice , due to the ex-

pected long prehea t zones (d iscussed further in the next section), the

heat loss by conduction is very small and all gradien ts are initially very

low. Hence , Yc o2ar>d dTg/dx are assumed to be zero at x = 0. T h e n

the solution procedure involves assuming a value of Uo and calculating

<3r(0) from equation (30) with dT g/dx\x=o = 0. The equa t ions are

solved by i tera t ing on Uo unti l the boundary condit ions at the hot en d

are satisfied. The i tera t ion was terminated when convergence on Uo

to within 2 p e r c e n t had been achieved.

Because of the re la t ively short reaction zone and long the rma l re-

laxation region, the set ofdifferentia l equations forms a stiff system

which is difficult to in teg ra te . The equations were solved using the

s u b r o u t i n e D I F S U B and developed by C. W. G ear [15]. This su b

routine uses a pred ic to r -co r rec to r me thod wi th an option for in te

grating stiff systems. It features an automatic step size selection whichallows the in tegrator to take the longest step possible while satisfying

the prescribed error cr i ter ion.

Calculated Resul t sF l a m e S t r u c t u r e and Veloc i ty . P r io r to the in troduction of the

differentia l approximation for radia t ive transport , ca lcula t ions pro

vided the classical, premix ed, flame structu res of relatively thin flames

and low propaga t ion speeds . For example , using the expression for

r c discussed previously , the f lame speed was pred ic ted to be 2.8 X

10 ~ 3m/s with a conductive heat loss of only 12.55 W /m 2

at the cold

end . E x t rapo la t ing the speed from other calcula t ions at higher heat

loss, a sto ichiometric m ixture of coal dust (0.104 kg/m 3) with particles

of 50 nm in diameter results in an adiabatic flame speed of2.82 X 10"3

m/s .

S uch low speeds were orig inally thought to be the re su l t of the

l imit ing reaction kinetics as given by the functional form of Tc, an

expression derived from burning experiments of single coal particles

[1]. However, even at gasif icat ion ra tes of 100 X rc , the ad iaba t ic

f lame speed was predic ted to be only 2.1 X 10 ~ 2m/s.

A typical profile of the t e m p e r a t u r e s of th e gas and coal part ic les

an d of the coal mass fraction, for the case without radia t ive transport ,

is shown in Fig. 1.Here the f lame is fa ir ly th in , approximately 0.1 m

in length for the predicted adiabatic speed of2.82 X 1 0 - 3 m/s. Notice

- 1.0

2 0 0 0

1000

50 0

\\

T(TC-T g) j

\

\\^^-—

""~"~""*~7950 K "

\ U 0 = 0 . 2 8 c m / s

\ d C o = 5 0 . 0 / x m

\ Stoichiometric

\ Mixture

\

*******1 ! ~~L~ : ! & J W ™ _

.-. 0.8

0. 6

INI

0 0.02 0.04 0,06 0.08 010

X ( m )0.12

0. 4

0.2

0. 0

a l so tha t the coal part ic le temperature equil ibra tes very rapidly to

t h a t of the gas . The a s sumed the rmodynam ic and chemical parame

ters specified for the coal dust f lame calcula t ions are given in the

nomenc la tu re .

W i t h the differentia l approximation uti l ized to rep re sen t the ex

pected s trong contr ibution of rad ia t ive t r anspo r t , the flame speeds

and f lame th icknesses increase dramatically For a sto ichiometric

m i x t u r e of small coal part ic les (dc(s - 30urn), the adiabatic f lame

speed is pred ic ted to be 0.716 m/s. The f lame th ickness here grows

to p ropo r t ions of several meters . Fig. 2 shows the predicted coal

particle a nd gas tem pera ture profiles as well as the fairly thick reactionzone. The heterogeneous reaction ra te term has a long tail cause d by

the diffusion-controlled ra te term in r c . In add i t ion , the radia t ive

t ranspo r t p rov ides a t empera tu re ove r - shoo t near the end of the re

action zone as well as a long (10 m) re laxation zone in which the coal

part ic le and gas t empera tu re s equ i l ib ra te . A required heat loss , by

radia t ion only , was 2.8 X 10 3 W /m 2at the cold end.

Burning velocities of this order have been observed experimentally

in large enclosed furnaces [16-18] and in small furnaces with heated

walls [19, 20 ]. S tru cture of this large scale has also been observed ex

perimentally [16-18]. These results should be compared with those

obtained from small scale burners [21, 22]. In these latter experiments,

the burning velocities are lower and the flames are much th inner than

those observed in the larger more adiabatic furnaces [16].

F ig . 3 i l lustra tes addit ional f lame structure for the case shown in

Fig. 2. He re th e coal mass fraction and pa rticle size variation are given.Notice that the predic ted varia t ion ofdc/dcl> isnot a linear function

of the d is tance , x. A varia t ion of dc/dco = 1 + bx was assumed by

B h a d u r i and Bandyopadhyay [23 ] as i n p u t to their model. This , of

course, over specifies the problem if an energy equation and species

conservation equations are being solved addit ionally .

For all of the results discussed above, it has been assumed that the

energy re leased during reaction was l iberated in the gas phase only.

In general , however, a fraction of the energy may be re leased on the

particle surface. This effect would be included in the term containing

.0

-0.8

2500

£000

21500

1000

50 0

• rc/rc - ^ Au c

m < M \ / \

'A'/\

^^ 1

A \

\ ^ T c

\ U„= 71.4 cm/s\ dc =30.0/i.m

\ Stoichiometric\ Mixture

\

0.8 §

0 . 4 .

2.0 v , . 3.0X(m)

4.0

0.2

0. 0

Fig. 2 Temperature and reaction zone prof i les predicted with radiationtransport. (Uo = 0.714 m/s)

I.O

0.8

00

? 0.60

3; o . 4

"• 0.2

^ \ \Y\

.

U0»7I.4cm/s

d = 30.0 tim°o

s - ~- °V/d r

\ ~~""~-

0.0 2.0 X ( m ! 3.0 4.0 8.0

Fig. 1 Temperature and coal mass fraction prof i les predicted withoul ra

diation transport. (U0 - 0.0028 m/s)

Fig. 3 Coal mass fraction and partic le diameter va riation, complementaryto prof i les shown in Fig. 2

10 8 / VOL 100, FEBRUARY 1978 Transactions of the ASME




HCOAL in the c o a l e n e r g y balance equation , equatio n (5) . Based on

their calcula t ions, Baum and Street have assumed t hat approximately

30 per cen t of the en ergy is released in the solid phase [24],

In order to assess the effect on flame structure of this solid phase

heat re lease , a calcula t ion was made assum ing 50 percent of the heat

re lease was in the coal . The resultant changes in the temperature

differences between th e phase s are illustrated in Fig. 4. Initially, the re

is an increase in the coal tempera ture over the gas temper ature s ince

the part ic les are being heate d by radia t io n. Th en, for the case of 50

percent of the energy re leased. in the coal (dashed l ine) , there is a

further increase in the part ic le temperature due to the heat re lease .After the maxima there is a decrease because chemical reaction is

re leasing heat in the gas and a lso because the p art ic les are losing ap

preciable amounts of energy by radia t ion. As the part ic les continue

to lose energy, their temperature slowly relaxes to the adiabatic flame

tempera tu re .

I t should be emphasized here that even though the difference (T c

- Tg) is greater in th e second case, the absolute value of T c is smaller.

In fact , the maximum value of T c for the case of 50 perce nt hea t re

lease in the solid was some 30 K less tha n i ts maxi mum value for the

case of 100 pe rcen t hea t re lease in the gas. Thi s was probably du e to

the fact that even though the chemical reaction was releasing energy

in the coal, the particles were also losing more energy and the net re

sult was a decrease in the m axim um value of Tc. It was also found tha t

with 50 percent of the energy released in the solid phase, the burning

velocity increased only 4 .8 percen t from 0 .707 m/s to 0 .741 m /s.

He a t Loss Vers us F la m e S pe ed . The ad iaba t ic p rob lem i s de

fined for a region of infinite dimensions in which as x —* — <= there is

no energy lost by e i ther conduction or radia t ion so that as x -»• +<=

the tempera tu re s app roach the ad iaba t ic f l ame tempera tu re . The

solution procedure, of course, is an initial value problem in which a

small but finite energy is lost at some given point assumed far from

the region of maximum combustion.

With radia t ion included, the preheat zones are very large when

compared to c lassical analyses of gaseous, premixed lam inar f lames.

For such large dis tances, the heat lost by conduction is very small

because the temperature gradients are very low. Therefore , to de

termine the effect of heat loss (in the direction of propagation), it is

assumed that there is only a radia t ion loss of — qr a t x = 0, the cold

end.

Fig . 5 prese nts th e predic t ion of f lame speed as a function of ra

dia t ion losses . The adiabatic f lame speed is determined by extrapo

lating to zero qr, on the curve of qr versus [/<> Calculations have been

made for flames in which the temperature difference is only 3 K from

the adiabat ic value of 1953 K. This is shown in the f igure . The hea t

losses at the cold end were small, averaging less than 2.1 X 10 3 W /m 2 .

But for cases where the final temperature was about 100 K lower than

the adiabatic temperature, the required radiation energy losses exceed

8.4 X 10 4 W /m 2 .

As one moves to larger assumed heat losses (and hence lower final

temperatures a t the hot end), the preheat zone begins to shrink. At

a temperature d ifference exceeding 200 K, the preheat zone is x/2o of

the length predic ted for the 3 K loss . Solutions are then impossible

unless one begins to assume that a heat conduction loss also becomes

important . Note , too, that for the case of the 200 K loss (Tf ^ 1750

K ), the average f lame speed is less than 0.50 m /s , while the ad iabatic

speed is about 0.70 m/s .

The effect of radia t ive transfer approximations is shown by the

three curves in Fig . 5. Curve A includes absorption and anisotropic

scatter ing. This represents an absorption eff ic iency of 0.84 and apredominant back scatter ing phase function which most accurate ly

describes coal particles. Curve B assume s tha t the coal particle is black

and thus there is no scatter ing. The remaining curve C includes the

realis t ic absorptio n eff ic iency of 0.84 and neglects scatter ing. Alter

nate ly , curve C can be in terpreted as including scatter ing as if the

scattered energy were concentra ted in to a small solid angle in the

forward direction and, therefore , impossible to d is t inguish from the

incide nt energy on a particle. It is seen tha t the maxim um difference

between these approximations for the adiabatic f lame velocity is 11

percent , with the most realis t ic (curve A) being the lowest .

The differentia l approximation being used here reduces to the

correct l imit ing expressions in the optically th in emission d omina ted

case and the optically thick case. Yet, if either of these limiting solu

tions are employed, the resulting predi ctions would be quite different.

For the optically th in case , the h igh temperature region wil l radia te

energy in both d irections. S ince the media emits and does not absorb,

there is a heat loss as x —• +°° and the tem pera ture profi les would

decay with increasing x. Thus, the adiabatic solution could not be

obtaine d. For t he optically th ick case , the profi les wil l be s imilar to

the conduction solution (Fig. 1) due to the diffusion type dependence

of the optically thick solution. The profiles would be broader and the

flame speeds higher than those obtained for the conduction solution.

Similar results re la t ing to shock s tructure have been presented by

Scala and Sam pson [25]. Therefore, for the optical depths encountered

here , neither l imit ing solution would accurate ly describe the prob

lem.

Fig, 5 Laminar flame spee d as a function of heat loss by radiation at the colden d

Fig. 4 Difference in coa! particle and gas temp eratures as a function of thedistance in a flame for two eases of solid-phase energy release

Curve A;Curve B.Curve C:

absorption and back scatteringblack particles and no scatteringabsorption and strong forward scattering

J ournal of Heat Transfer FEBRUARY 1978, VOL 100 / 109Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



F la me S peed V ersus P a r t i c l e S ize . F ig . 6 p re sents the p red ic ted

adiabatic f lame speeds as a function of coal part ic le s ize . From a

standpoint of specific surface (S/V ratio), large particles provide less

specific surface for a given fixed mixture ratio. Therefore, they show

a lower reaction ra te a t any given posit ion an d, hence, a reduction in

flame speed. Previous calculations without radiation showed a marked

varia t ion in Uo as dCo was decreased, with f /o increasing 43 percent

from 2.8 X 10 " 3 m/s to 4 .0 X 10 - 3 m/s as dCo decreased from 50 jim to

25 ura.

However with radia t ion included and for our coal deplet ion ra te

term, the reduction in flame speed with size is only slight, as shown

in Fig. 6. Decreasing th e ma gnitud e of F c (by multiplying the reaction

rate by 0.4) lowered the burning velocities, but still showed the same

relative variation of Uo with particle size. Burn ing velocities measu red

by Hatto ri using 60 jim to 140 um particles also decreased w ith particle

size although his velocities were in the 0.1 to 0.2 m/s range [26].

G rum er and B ruszak [27] have found a slight increase in flame spee d

with a decrease in par ticle size for particl es in the 1 to 44 um range .

The shape of these curves may be due to the combined effects of

combustion and radiative heat transfer. From a kinetics point of view,

increasing the part ic le s ize wil l decrease F c . Thus a decrease in

burning velocity with increased particle size (as was seen in the cases

without radia t ive transport) would be expected. The absorption

coefficient for radiation given by equation (18) also varies inversely

with particle size for a fixed loading (fixed Za). Decreasing dc would

increase the absorption coefficient, effectively making the flame op

tically thicker. Such a flame would supply less energy to the particlesin the preh eat zone and a lower burning velocity would be expected.

Therefore, the variation of f/o with particle size would depend on both

these opposing factors .

The reduction in f lame speed with part ic le radius (even though

slight) is in direct conflict with the predictions of the model presented

by Ozerova and Stepan ov [28]. Their results are shown as the d otte d

line in Fig. 6. Becau se there is too little inform ation given in referen ce

[28] on the kinetic ra te parameters , heats of reaction, and other

thermodynamic variables , our values cannot be directly compared.

However, i t appears that in order to predic t such a large increase in

f/o with dCa , either the ra te of heterogeneous gasif icat ion must be

larger tha n the one being used or addit ional gas-phase (devolati l iza-

t ion) reactions must be added. S ince Ozerova and Stepanov consid

ered carbon combustion only , they did not include devolati l iza t ion

reactions.At any ra te , i t does not appear t ha t f lame speeds of the order of 1

m/s would show such a s trong increase in speed for the range of par

ticle size shown in Fig. 6. Increasing the ra te to 10 or 50 T c will increase

the slope, but it will also give much higher values of Uo than are ob

ta ined experimentally .


Util iz ing several s implify ing assumptions for the s teady f lame

propagation of coal dust-a ir mixtures , a model has been developed

and solutions have been carried out for the flame structure and flame

speed. The f lame speeds and s tructure predic ted agree with experi

mental d ata [16]. Th e model has used the differentia l approxim ation

for radiation which represents the transport behavior of such mixtures

quite well . The model by Ozerova and Stepanov [28] used a s imilar

but s impler form of the d ifferentia l a pprox imat ion.The predic ted dimensions for the f lame zones include very long

preheat zones and large re laxation zones in which the solid and gas

phase tempera tu re s equ i l ib ra te . I t appea rs tha t add i t iona l expe r i

men tal confirm ation of th is wil l require careful tes t ing in large scale

sys tems where to ta l energy losse s a re kep t to a min imum . E xamples

of such systems are those used by How ard and E ssenhigh [17] and by

Palm er and To nkin [29], where heat losses to the surrounding s were

minimized because of the scale of the experiments .

This model can, of course , be made more descrip tive of the

physiochemical processes of coal dust burning by including a de

scrip tion of the complexit ies not taken in to account. Some of these

effects include polydisperse mixtures, two-dimensional heat transport,

and devolati l iza t ion kinetics . In summ ary, however, a model has been

1.4

1.2

1.0

0. 8

Uo

(m/s)

0. 6

0. 4

0. 2

//

//

/ ^______ "~

//

//

/1 /

/

, Predictions of Ozerova

/ and Stepanov

I 1 1 \ I

^ _ _ _ ^ _ - ~ - 5. 01 r c

rc

~ — - — 0 . 4 x r c

_ i 1 1 1 _ ^

0 20 4 0 60 80 10 0 120 140 160 ISOdCo(Mm)

Fig. 6 Predictions of the adiabatic (lame speed as a function of coal particle

size fo r three different coal mass depletion rates

presented that describes many of the fundamental features of coal

dust-air flames and provides a foundation for incorporation of futinv

modeling modifications.

R e f e r e n c e s

1 E ssenhigh, R. H., "Predicted Burning Times of Solid Particles in anIdealized Dust Flame," Journal of the Institute of Fuel, Vol. 34, 1961, pp.239-244.

2 Brookes, F. R , "The Combustion of Single Captive Particles of Silkslom-Coal," Fuel, Vol. 48,1969, pp. 139-149.

3 Soo, S. L., Fluid Dynamics of Multiphase Systems, Blaisdell P ublishinj!Co., Waltham, M ass., 1967.

4 Ozisik, M . N., Radiative Transfer, John Wiley and Sons, New York,1973.

5 Cheng, P., "Tw o-dimensional Radiating Gas Flow by a M oment M ethod," AIAA Journal, Vol. 2, No. 9, 1964, pp. 1662-1664.

6 Cess, R., "On the Differential Approximation in Radiative T ransfer,"Zeitschrift fur Angewette Mathematik und Physik, Vol. 17, 1966, pp. 776-781,

7 Cheng, P., "S tudy of the Flow of a Radiating G as by a Differential Approximation," Ph.D. Thesis in Engineering, Stanford University, 1965.

8 Foster, P. J., and Howarth, C. R., "O ptical Constan ts of Carbons and

Coals in the Infrared," Carbon, Vol. 16,1968, pp. 719-729.9 Stall, V. R. and Plass, G. N., "E missivity of Dispersed Carbon Particles,''

Journal of the Optical Society of America, Vol. 50 ,1960, pp. 121-129.10 Dalzell, W. H. and Sarofim, A. F., "Optical Constants of Soot and thei:'

Application to Heat-flux Calculation s," ASM E , JOURNAL OF HE AT TRANSFER, Vol. 91,1969, pp. 100 -104.

11 Hottel, H. C. and Sarofim, A. F., Radiative Transfer, M cG raw-Hill, Ncv.York, 1967.

12 Blokh, A. G ., M . L . M odzalevskaya, and M . N. G alitskaya, "Scatteringof Thermal Radiation by Carbon Particles in Flam es," Teploenergetika, Vol.19, 1972, pp. 29-31.

13 Friedman, R. and E. Burke, "A Theoretical M odel of a Gaseous Combustion Wave Governed by a First O rder Reaction," Journal of ChemicalPhysics, Vol. 21,1953, pp. 710-714.

14 Hirschfelder, J., C. Curtiss, and D. Camp bell, "T he Theory of Flamesand Detonations," Fourth Symposium (International) on Combustion, TheWilliams and Wilkins Co., Baltimore, M d., 1952, pp. 190-211.

15 G ear, C. W., "T he Automatic Integration of Ordinary DifferentialE quations," Comm. ACM, Vol. 14,1971, pp. 176-179 and 185-190.

16 E ssenhigh, R. H., "Combustion and Flame Propagation in Coal Systems:A Review," Sixteenth Symposium (International) on Combustion, AcademicPress, New Y ork, 1976, pp. 353-374.

17 Howard, J. B. and R. H. Essenhigh, "M echanism of Solid ParticleCombustion with Simultaneous G as-Phase Volatiles Combustion," EleventhSymposium (International) on Combustion, Academic Pre ss, New Y ork, 1967.pp. 399-408.

18 Csaba, J., "Flam e Propagation in a Fully Dispersed Coal Dust Suspension," Ph.D. Thesis, Department of Fuel Tech. and Chem. E ng., University olSheffield (E ngland), 1962.

19 Ghosh, B., D. Basu, and N . K. Roy, "Studies of Pulverized Coal Flames,''Sixth Symposium (International) on Combustion, Reinhold Publishing Corp..New York, 1956, pp. 595-60 2.

20 G hosh, B. and A. A. Orning, "Influence of Physical Factors in Igni tinpPulverized Coal," Industrial and Engineering Chemistry, Vol. 47,1955. pp.117-121.

21 Horton, M. D., G oodson, F. P., and Sm oot, L. D., "Characteristics of Flat.

110 / VOL 100, FEBRUARY 1978 Transactions of the ASW.E




Laminar Coa l Dust F lames, " Combustion and Flame, Vol. 28,1977, pp. 1 87-

195.22 ' Smoot , L . D. , Hor to n, M . D. , and W il l iams, G . A. , "Pro paga t io n of

Laminar Pulverized Coal-Air Flames," Sixteenth Symposium (International)

on C ombustion, Academic Pres s , New York, 1976, pp. 375-387.23 Bhadur i , D. and Bandyopadhyay, S . , "Combust ion in Coal Dust F lames, "

Combustion and Flame, Vol. 17 ,1971 , pp. 15-24.24 Baum , M . M . and S tree t , P . J . , "Pred ic t ing the Comb ust ion Behavior

0 f Coa l Par t ic les , " Combustion Science and Technology, Vol. 3, 1971, pp.231-243.

25 Sca la , S . M . and Samps on, P . H. , "H ea t Transfe r in Hypersonic F lowvvith Radiation and Chemical Reaction," Supersonic F low Chemica l Processesand Radiative Transfer, edited by Olfe, D. B., and Zahk ay, V., Th e M acM illan

Co. , New York, 1964, pp. 319-354.26 Hatto ri, H., "F lame Propa gation in Pulverized Coal-Air M ixtures," Sixth

Symposium ( Inte rna t iona l) on Combust ion, Re inhold Publ ishing Corp. , NewYork, 1956, pp. 590 -595.

27 G rumer , J . and Bruszak, A. E . , " Inh ibi t ion of Coa l Dust-Air F lam es, "Bureau of M ines Repor t of Invest iga t ions 7552 ,U.S .Dep ar tme ntof th e Inte r ior ,Aug. 1971 .

28 Ozerova , G . E . and S tepano v, A. M . , " E f fec t of Radia t ion on F lamePropaga t ion throug h a Gas Suspension of Sol id Fue l Par t ic les , " Combustion,

Explosion, and Shock W aves, Vol. 9 ,1973, pp. 543-549.29 Palm er, K. N. and Ton kin, P. S., "Coal Dust Ex plosions in a L arge-Scale

V e r t i c a l Tube A ppa r a tu s , " Combustion and Flame, Vol. 17, 1971, pp. 159-170.

Journal of Heat Transfer FEBRUARY 1978, VOL 100




T. AhmadG. M. F aeth

Mechanical Engineering Department,

The Pennsylvania State University,

University Park, Pa.

An Investigation of the Laminar

Overfire Region Along Upright

SurfacesNumerical calculations are presented for the laminar overfire region along an upright

burning surface under natural convection conditions. The process was also examined ex

perimentally, using wicks soaked with methanol, ethanol, and propanol to simulate the

burning portion of the surface. Predicted flame shapes and wall heat fluxes w ere within

15-20 percent of the measurements, for various wall inclination angles, except near the

tip of the flame where the reaction is quenched by the cold wall. The wall heat flux is near

ly constant in the region where the flame is present, but declines beyond the tip of the

flame. Shadow graphs were employed to determine the position of onset of turbulence. For

transition beyond the tip of the flame, the flow was laminar for Gr/ < .5 — 2 X 108, al

though transition is not completely described by this parameter and effects of fuel type,

presence or absence of combustion, distance, and wall angle were also observed.

In t roduct ion

A fire burn ing on an uprig ht surface can be divided into two regions:

(1) a pyrolysis region, where the wall material is gasif ied and partly

burn ed in the gas phase ad jacent to th e surface; (2) the overfire region,

where the combus t ion p rocess i s comple ted and the the rma l p lume

generated by the f ire decays by mixing with the ambien t gas and heat

loss to the surface. The boundary between these two regions is not

precisely defined in na tura l fires, since the rate of pyrolysis at the wall

varies in a continuous manner. Pyrolysis rates are strongly dependent

upon temperature, however, and a narrow range of temperature (and

a correspondingly short dis tance) separates locations where surface

pyrolysis is negligible from points where the surface is generating

subs tan t ia l co mbus t ib le gases.

He at transfer between th e hot gases and th e surface in the overfire

region governs heating of the wall material . Eventually , temp eratures

are reached where pyrolysis is significant. In this manner, the overfire

region plays a fundamental role in the mechanism of upward flame

spread on surfaces.

The laminar pyrolysis region of a wall fire has been investigated by

Kosdon, e t a l . [I] , 1 and K im, et a l . [2]. A variable proper ty solution

of the boundary layer equations was developed in similarity variables.

The studies assumed a one-step reaction with an infinite ly thin dif

fusion flame, no gaseous radiati on, values of unity for the Cha pm an


Contr ibuted by the Hea t Tran sfer Division of THE AMERICAN SOCIETY OFMECHANICAL ENG INEERS and presented at the AIChE-ASME Heat TransferConference, Salt Lake City, Utah, August 15-17, 1977. Manuscript receivedat ASME Headquarters November 29, 1977. Paper No. 77-HT-68.

and L ewis numb ers, equal molecular weights , specific heats , and bi

nary diffusivit ies , e tc . Burning rat e measure ments agreed reasonably

well with prediction s, particularly for fuels with low molecular weights,

where fuel properties most nearly corresponded to the assumptions

of the variable property model [2].

Using an integra l model proposed by Yang [3], Pagni and S hih [4 |

investigated the overfire region of wall fires, basing initial conditions

on the pyrolysis zone results [2], Pred ictio ns of fuel consum ption and

flame shape in the overfire region are reported for a wide range of

fuels. This model is also capable of providing other quantities as well,

e .g . , tem pera ture s, velocit ies , concen trations, e tc . [4].

The present investigation extends the work of [4], undertaking more

exact theoretical treatment of the laminar portion of the overfire re

gion and giving greater emphasis to heat transfer betw een the p lume

and the surface. Unlike the pyrolysis zone, the flow in the overfire

region is not similar, and a numerical procedure developed by Hayday,

et a l . [5] was employed to solve the governing equations. The as

sumptions of the present analysis were s imilar to those used during

the ea rlier investigation s of the pyrolysis zone [1 , 2] .

Experiments were conducted to assess the predictions. The py

rolysis zone was simulated by burning fuel-soaked wicks, following

the method used by Kim, et al. [6] and Blackshear and Murty [7], T he

heat f lux to the wall was determined by a transient heating technique.

Flame shape was determined from dark f ie ld photographs. Shadow

graphs were employed to determine the point of transit ion to turbu

lence, so that the measurements could be l imited to laminar f low.

Ex p er i m en t a l Ap p ara t u s an d P ro ced u reOnly a brief description of the app ara tus is given here. Ahm ad [8]

provides a more complete discussion. A sketch of the test arrangement

is i l lustra ted in Fig . 1 . The d imensio ns of the test walls are summ a

rized in Table 1 .

112 / VO L 100, FEBRUARY 1978 Transa ct ion s of the ASM ECopyright © 1978 by ASME




Fig. 1 Test wall assembly

Table 1 Test waUsa

variation of properties while minimizing observation difficulties and

radiation due to soot formation.

Test Wall. Transient heating of the wall was used to measure the

heaL flux in the overfire region. In order to obtain the variation of heat

flux with position, the wall was segmented into blocks. Three columns

of blocks extended along the wall; only the center column was used

for the measurements. The dimensions of the blocks are given in Table

1. Measuring the rate of change of temperature of the blocks yields

the heat flux to the wall, given the thermal capacity of the blocks. For

the heat flux values encountered during the tests, calculations indi

cated that block temperatures were essentially uniform at each instantof time (the Biot modulus at all conditions was less than 0.001);

therefore, copper-constantan thermocouples were mounted on the

rear surface of the wall. The thermocouple wires were passed along

the wall and covered with insulation in order to reduce errors resulting

from the fin-effect of the wires. The temperature of each block was

continuously recorded to indicate its heating rate. The thermal ca

pacity of the wall materials was verified by calorimetry.

Condensed material was observed on the wall in the region nea r the

end of the pyrolysis zone. The condensate disappeared, however, early

in the wall heating process so that heat flux measurements, which were

not influenced by this effect, could be made.

Glass side walls were used to help maintain two-dimensional flow

near the center of the wall, and still provide a means of observing the

flame. One layer of screening (2 mm spacing, 0.5 mm diameter wire)

was mounted across the outer edge of the side walls to reduce disturbances from the room.

Wall material

Wall height

Wall widthWall thicknessBlock heightSide wall width

Copper381127

Steel

76372.43.2

19

Flame Observations. The flames were photographed in a dark

ened room with a 4 X 5 Super Graflex Camera at a magnification of

approxim ately unity, using Polaroid film (ASA 3000).

Screen to wall distance

" All distances in mm.

9.5113870 40

The onset of turbulence was det ermined by a shadowgraph, which

employed a mercu ry arc source collimated with a 180 mm diameter,

1230 mm focal length, front surface parabolic reflector. Th e shad

owgraph was projected on a screen 4 m from the test wall in order to

provide high sensitivity.

Pyrolysis Zone. The pyrolysis zone was simulated by igniting the

front surface of a fuel-soaked wick [6,7]. Burning was limited to the

front surface by shielding the remaining sides with aluminum foil.

Wicks were constructed from various porous furnace bricks and in

sulating materials; the type of material had no apparent effect on the

measurements. The fuel burning rate was measured by weighing the

wick before and after combustion over a timed interval. Methanol,

ethanol, and propanol were used as test fuels, since they provide a wide

Experimental Accuracy. The burning rate measurements are

averages of three te sts at a given condition. The repeatability of these

measurements was within ten percent of the mean.

The heat flux measurements were obtained by averaging two tests

at a given condition. These measurements were reproducible within

15 percent, with the greatest differences encountered near the point

of transition to turbulent flow.

A typical flame photograph is illustrated in Fig. 2. To the eye, the

flame zone appeared as a thin bright blue region, surrounded by a

dulle r glowing zone. I t was difficult to resolve this detail on the pho-

B = mass transfer driving potential, equation

(18)

Cp = specific heat at constant pressure

F = buoyancy parameter, equation (20)

f = dimensionless stream functionGrx* = modified Grashof number, Lg cos

</>x3/4 CpTwV w

2

Gr./ = modified Grashof number, g cos

</>X 3/vw 2

g = acceleration of gravity

h = enthalpy

hi D = enthalpy of formation of i

L = effective heat of vaporization

Mi = molecular weight of i

IiI" = wall mass flux

Pr = Prandtl number

Q = heat of combustion per VF' moles of

fuel


q" = wall heat flux

r = stoichiometry parameter, equation (18)

T == temperature

u == velocity parallel to wall

v = velocity normal to wallW == normalized reaction rate, equation (6)

Wi == reaction rate of species ix = distance along wall

Xo := end of pyrolysis zone

y = distance normal to wall

Yi = mass fraction of species i

Ypr = fuel mass fraction in transferred gas

1] = dimensionless distance normal to wall,

equation (15)

= dimensionless distance along wall, x/xo

Oi = Shvab-Zeldovich variables, equation

(13)

A := thermal conductivity

!1 = dynamic viscosity

v := kinematic viscosity

"i', Vi" == stoichiometric coefficients

p = density

TO = dimensionless wall enthal py, hwo/LT1 = dimensionless wall enthalpy, hw/L

</> = angle of wall from vertical

if; = stream function

Subscripts

avg := average value

f = flame

P = fuel

0= oxygen

P = products of combustion

w = wall above pyrolysis zone

Wo = wall in pyrolysis zone

00 := ambient

FEBRUARY 1978, VOL 100 / 113




where Q is the heat of reaction per "F' moles of fuel.

The boundary conditions far from the wall are the same in all regions

Fig. 2 Photograph 01 a methanol wall lire, Xo = 11.1 mm

u = h = YF= 0, Yo = YOoo; y ~ 00

At the wall in all regions

u =O;y =0

(6)

(7)

(8)

tographs, and the center of the luminous region was taken as the po

sition of the flame. Measurementswere limited to conditions, where

the glowing zone extended only ±20 percent of the distance between

the center position and the wall.

Analysis

The analysis considers a smooth flat wall at an angle </> from the

vertical, with a region of length Xo undergoing pyrolysis at the base

of the wall. The flame is overventilated, and when fuel consumption

is complete the flame returns to the wall at xi > Xo under the as

sumptionof a thin diffusion flame. The combusting plume region lies

between Xo and xi; a thermal plume region is at distances greater than

xI-The similarity solutions for the pyrolysis zone [1, 2] provide initial

conditions for the overfire region. In order to make maximum use of

these results, the assumptions and much of the notation of [2] have

been employed in the present study.

The assumptions of the analysis are as follows: the flow is laminar,

two-dimensional, and steady; the boundary layer approximations

apply; the temperature and composition of the ambient gas are con

stant; viscous dissipation and radiation are negligible; the combustion

process is represented by an infinitely thin diffusion flame with no

oxygen present between the wall and the flame and no fuel present

outside the flame; the flow is an ideal gasmixture with constant and

equal specific heats, equal binary diffusion coefficients, unity Lewis

number, and constant values of P!L and PA (which implies constantPrj;

the surface temperature and energy of gasification within the pyrolysis

zone is constant. Calculations described in the Appendix indicated

that radiation provided no more than 12 percent of the heat flux to

the wall for the present tests. The validity of the remaining assump

tions is discussed in [1] and [2].

The pyrolysis zone is assumed to be well-defined, with no pyrolysis

occurring in the overfire region. These conditions are satisifed by the

experiment; for a natural fire, this behavior would require a large

activation energy of pyrolysis so that the zone where the pyrolysis rate

develops to i ts full value would be narrow.

Employing these assumptions, the conservation equations are

[9]:

Introducing equations (13)-(15) into equations (1)-(5) yields the

following

fY= x/xo, '7 = (Grx*!/4/x) Jo p/poody, 1ft/v oo = 4 G r x * ! / 4 { ( ' 1 , ~ )

(15)

(10)

(14)

8Yih = hw , u = -a; = 0; Xo x, y = 0

Th e diffusion flame approximation also provides

a1ft a1ftpu = poo -, pu = - poo -

ay ax

The transformation of variables is completed by combining a How

arth-Dorodnitzyn transformation for variable properties with a local

similarity transformation which reduces tothe form used in [2] in the

pyrolysis zone.

The wall temperature was assumed to be constant in the overfire

region, which ignores the fact that the surface temperature increases

from Too far from the pyrolysis zone, to Two near the end of the py_

rolysis zone. The effect of this simplification will be examined by

considering the two limits: Tw= Two and Tw = Too. Condensation of

fuel and product species on the wall was neglected, which implies thatthe concentration gradients of all species are zero at the wall. With

these assumptions, the remaining boundary conditions may be for

mulated as follows

however, the remaining wall boundary conditions vary from regionto region_

The wall boundary conditions for the pyrolysis zone are discussed

by Kosden, et al. [IJ and Kim, et al. [2] for both solid and liquid fuels.These conditions provide

A ah -AaYFTil" = pu =-- - pu (YFT - YF) = - - - h = h .

LCp ay ' Cp ay , wo,

o< x < xc, Y = 0 (9)

Yo =0, Xo x Xi; YF= 0, Xi x; Y = ° (ll)

where th e flame position is identified as the point where both Yo =YF = 0, and Xi is the position where the flame returns to the wall.

In all regions, the heat flux to the wall is given by

A ahq" =--; y =° (12)

Cp ay

Variable Transformation. The terms involving enthalpy and

concentrations are simplified by introducing the following Shvab

Zeldovich variables [9]:

()hO = (h/L + (Yo - YoJQ/Movo'L)/(hwo/L - YoooQ/M()I'o'!,)

()hF = (h/L + YFQ/MFvF'L)/(hwo/L + YFwoQ/MFvF'I,)

()FO = (YF/MFVF' - (Y o - YoJ/Mo"o')/(YFwo/MFVF'

+ YojMovo') (13)

While only two of the ()i are independent, it is convenient to use all

three quantities in the overfire region, since wall boundary conditions

are most easily expressed in terms of ()FO and ()hO for Xo x Xi, and

()FO and ()hF for x > xI-A stream function is defined which satisfies equation (1)

(1)

(2)

(3)

(4)

(5)

apu apu-+ -=0

ax ay

(au au ) a ( au )p u-+u- =gcos</>(poo-p)+- !L-ax ay ay ay

p (u ah + u ah ) = ~ ( ~ ah) - I : lv;hioax ay ay Cp ay i

(aY a y ) a ( A a y )

p u ax' + u ay' = ay Cpay' + Wi

where

T

h = r CpdTJToo

The reaction is represented on a molal basis, as follows

vF'(Fuel) + vo'(Oxygen) ~ ,'p" (Products)

Stoichiometry then provide\'>

114 / VOL 100, FEBRUARY 1978 Transactions of the AS ME




/'" + W " - 2(/')2

+ F = 4£ (/ ' J - /" 2 )di ay

Bt" + 3 Pr/fl; ' = 4 Pr£ / / ' — - «•' —^

(16)

(17)

T a b l e 2 P h y s i c a l p r o p e r t i e s "

at aywhere partia l derivatives with respect to ?/ are denoted by primes.

Since YF = YQ = 0 at the flame, equation (13) yields the following

criteria for the flame position

6F0/ = r(B+l)/B(r+l) (18)

where

B = Y0„Q/M 0i>o'L - TO; r = y o « M F I , F 7 Y F T M 0 i<o ' (19)

T he buoyancy te rm, JF", has different forms inside and outside the

flame, as follows:

F = B(i-e h0 ) + T Q ;e Fo > e FO f

F=(B+ T O K W A F O , ) - Be h0; 6F0 « BF0 , (20)

The boundary conditions far from the wall are

f' = ehQ = ehF=Q ; , , - » (21)

while the wall boundary conditions,! / = 0 , become

/ ' = 0, / = BB h0 '/3 Pr, flho = BhF = 8F0=1;(^1

/ = / ( 0 , l ) r 3 / 4 , / ' = 8 F O ' = 0 ; l < ^

«AO = 1 + (TO - n)/B; l<Z<Hf

BhF = T I 0 F O / / ( T O + eF0f)B); ki < £ (22)

In the pyrolysis zone, the boun dary conditions are indep ende nt of

£; and / and the 0; are only func tions of TJ. In the overfire region, the

boundary condition for / a t the w all , equation (22), depe nds upo n £

and the solution is no longer s imilar . The posit ion fy is found when

0Fo = dFo f at the wall .

The m ass burning rate per un it area in the pyrolysis zone is giver,

by

m"x/n~ = - 3 Gr^fiO); £ « 1 (23)

The heat flux to the wall is given by the following expressions

q»x-PrGrx*-l'VBL^ = -8 ho '(0, £); { « £/

g " xP r G r I * - 1 / 4 B L j i „

= (T 0 + (1 - eFOl )B)6hF '(0, t)l(B6FOf ); £/ < £ (24)

If r\ is given, y can be determined from the following equation

iylX)Grx*U* = C\\ + (LFICPT„)]dr,Jo

(25)

N u m e r i c a l S o l u t i o n . The details of the calculations are discussed

by Ahm ad [8]. Briefly, the solution of equations (16)-(22) w as ob

tained by the numerical procedure described by Hayday, et al. [5]. T h e

solution in the pyrolysis region provided the initial condition, followed

by step-wise integration in the overfire region. Present calculations

in the pyrolysis zone agreed with results rep orted by Kim, et a l . [2],

within the accuracy of their plots .

fiesults a n d D i s c u s s i o n

Methanol, e thanol, and propanol were considered in both the cal

culations and the experiments . The properties employed for these

materials, as well as the procedures used to determine property values,

are summarized in Table 2 .

F low De ve lo pm en t . T he genera l cha rac te r is t ic s o f the ove rf i re

region are illustrated for a methanol fire in Figs. 3,4, and 5. Since the

flow is s imilar dur ing pyrolysis , a s ingle set of curves represen ts this

entire region. Parametric values of £ are considered for the nonsimilar

overfire region.

Velocity profiles ar e illustra ted in Fig. 3. In the lower portion of the

overfire region, the peak in the velocity profile shifts toward the wall

P r o p e r t y

Molecular weightb

Boi l ing tempera tu re (K)UkJ/kgYCp(kJlkgKYQ ( ; n J / k g - m o l K )

d

tiw X 10HN-s/m2)

e

co' fo r vF = 1Br

TO

P r

M e t h a n o l

32.04337.7

12261.37

67 518.0

1.52.60

.154

.044

.73

E t h a n o l

46.07351.588 0

1.431278

20.83.03.41

.111

.087

.73

Propano l

60.09370.478 8

1.461889

21.64.53.71

.096

.134

.73

» Ambient conditions: air at 298 K, Ko. = .231,b Molecular weight of oxygen = 32 kg/kg-rnolc

Value for the fuel at the boiling tem peratured Lower heating value of the fuel at 298 Ke

Dynamic viscosity of air at the boiling tem perature of the fuel

15.3 X 10" 6m2/s

.6 -

<4— 4

/ / / X s . 6

I I 12

1 // v^"

I f sit-^^THERMAL

1

3 \

PLUME

METHANOL BURNING IN AIR

\ -- O - l , PYROLYSIS

COMBUSTING PLUME

\

^JN. \

i ^ s ^ ^ m l

0 1 2 3 '

V

Fig. 3 Velocity profiles within a methanol-fueled wall fire burning In air

for a time as the effect of blowing due to mass transfer in the pyrolysis

zone decays. At higher posit ions on the wall , the point of maximum

velocity moves outward with increasing distance from the pyrolysis

zone.

Con centr ation profiles are shown in Fig. 4. The position of the flame

is indicated by YF= YQ = 0. Th e flame moves tow ard t he wall as £

incre ases , reachi ng th e wall at £/• = 6.5, which en ds the co mb ust: i>

region. Above the combustion region, oxygen diffuses back to the

wall .

Te mp era tur e profiles are shown in Fig . 5 . During pyrolysis , t in :

maximum temperature occurs in the f lame, which is typical of cor/ ,

bustin g similar f lows where the tem pera ture peak generally coincides

with th e position of chemical energy release. In the nonsimilar overfireregion, however, the tempe ratur e ma ximum occurs outside the f lame

posit ion . In this region, the rate of reaction in the f la .ne decreases as

the concentration of fuel at the wall decreases; direct energy loss from

the f lame zone to the wall a lso increases as the f lame posit ion ap

proa ches th e wall. As a result of both these effects, the wake-like decay

of the high temperature region produced by reaction lower on the wall

begins to dominate the characteris t ics of the temperature profile

causing the peak temperature to be outside the f lame posit ion. Near

the tip of the flame, the temperature at the flame position is relatively

low, and q uenching of the reaction could be im portant in this region.

A chemical kinetic effect of this type is not treated in the present

model. Beyond th e t ip of the f lame the temper ature profile continues

to spread, s imilar to the velocity profile .

Journa l o f Hea l Transfer FEBRUARY 1978, VO L 100 / 115




.6 -

•y, Q

— 1 —

\ \ \ \\ s

\\\\ \

\

\ \ \\\

_ T H E R M A L P L U M E \

A N

«"" *"" *"* •** —». ——"""^ 1 \r >

1

M E T H A N O L B U R N I N G I N A I R

v _ _ . . _

Y o .

-

C O M B U S T I N G P L U M E

/ ^ S J ^ -

/ x / x 0 = 0 - 1 , P Y R O L Y S IS Z O N E

f t „

0 1 2 3 4

7)

Fig. 4 Conc entration profi les within a methano l-fueled wal l f i re burning Inair

CO_ l

|_ B .4 -

(_>

. 2 -

™

-

-

- /

t

/ /V

/x~As.6

N/

' /THERMAL PLC

/ I2__̂

. / " ^ x/x0=30

^-^—^-~~~~~

J M E \

1 ' 1

A M E T H A N O L B U R N I N G I N A I R

\ x/x = o - l , P Y R O L Y S I S Z O N E

\y C O M B U S T I N G P L U M E

/ \

\ \ \

0 1 2 3 4

V

Fig. 5 Temperature profi les within amethanol-fueled wal l f i re burning Inair

C o m p a r i s i o n o f T h e o r y a n d E x p e r i m e n t . F i g . 6 p r e s e n ts p r e

dicted and measured wall heat f lux values in the laminar region for

methanol. The experimental results consider various pyrolysis zone

lengths and wall angles up to 36.6 deg (facing downward).

The wall heat f lux was obtained directly from the experiments in

the overfire region. However, the heat f lux was determined from the

burning rate in the pyrolysis zone. During pyrolysis the wall heat f lux

is related to the burning rate as follows:

q" = Lm " (26)

Integration of equations (23) and (24) provides the following rela

t ionships between average values and values at xo

<j"Uo)/<?avg" = m " U r j ) / m a;% (27)

which is typical for natu ral convection processes. Equatio ns (26) and

(27) relate the measured average burning rate to the heat f lux at X Q,

allowing determination of the heat f lux parameter; these values have

been arb itrari l y placed in the region 0 < { < 1 s ince they are the oret

ic i..

T i t 1 —1

THEORY^ = 0 . 0 4 -1

VO.O-,/

J f END OF FLAME

,fo8A°o0 \

' I °V\

- f o \

END OF PYROLYSIS A • X̂D •

A a#

AA

1 1 l i t

1 1 1 "1 ——1

METHANOL,Y 000 = 0 . 2 3 l

x0(MM) $(DEG)

l l . l 0.0 o

16.0 0.0 O

16.0 36.6 A

-

a A A OCCASIONALLY TURBULENT

• a • T U R B U L E N T

* ° r ^ ^ ^

i i i i

10

x/x0

18 20

Fig. 6 Wal l heat flux for a methano l-fueled wal l f i re burning in air

i ca lly indepen den t o f £ in this region.

A rough correlation of the laminar burnin g rate meas urements for

methanol reported by Kim, et a l . [2] is

g"xo/nu G r* (28)

for the range 5 X 104<G r x ' <10 8. Equation (28) can be rewritten in

terms of the present heat f lux parameter as follows

<j" .xPrGr x*1/ 4

/ B L M « =0.23 (29)

where th e missing prop erty values have been supplied from Table 2 .

The value from equation (29) isabout 18 percent lower than the

pres ent m easu rem ents shown in Fig . 6 for the pyrolysis zone. This

discrepancy is well within errors expected due to property uncer

ta inties ( in particular , the value of nwo used t o reduce t he d ata in [2]

is not repor ted) an d the scatter of the exp erime nts .

Th e h eat flux is low in the pyrolysis zone, due to th e blowing effect

of fuel eva por atin g at th e wall. In the overfire reg ion, the heat flux

increases, within one pyrolysis zone length, to arelatively constantvalue which is maintained until the t ip of the f lame is approached.

Beyond the end of the f lame, the heat f lux decreases rapidly once

again. The e xperim ental h eat f lux values s tart to decrease somewhat

sooner than the predicted value of%.

Two wall condi tions are i l lustrate d for the pred ictions in Fig . 6; n

= 0.044, which implies that the entire wall is a t the temperature of

the fuel surface in th e pyrolysis zone; and TI = 0, which implies that

the wall temperature in the overfire region is equal to the ambient

temperature. There is l i t t le difference between the two cases due to

the high flame temperature and the low pyrolysis zone surface tem

perature for methanol. This finding helps justify the present transient

method for measuring wall heat flux and the assumption of a constant

wall temperature used in the analysis, since it indicates that low levels

of wall heating do not substantia lly influence the results . For n = 0 ,

the wall heat flux is theoretically infinite at £=

1 due to the s tep-change in temperature, however the region of high heat f lux values

due to th is effect is too small to be shown on t he scale of Fig. 6. Fo r n

= 0.044, the f low asymptotically approaches anatural convection

process on aheated, isothermal wall , and the heat f lux eventually

becomes negative at large |.

F igs . 7 and 8 i l lustrate s imilar results for e thanol and propanol.

Transit ion to turbulence in the combusting portions of the flow oc-

curred in a shorter distance than for methanol, and a smaller pyrolysis

zone length was required to mea sure the lam inar he at f lux at large £

for these fuels.

The comparison between measured and predicted wall heat f lu"

is s imilar for the thre e fuels . The agre eme nt between th eory and ex

periment is perhaps fortuitously good in the pyrolysis zone. In the

11 6 / VO L 100, FEBRUARY 1978 Transact ions of the ASME




0

03

0.X

0

7

6

.5

4

3

2

1

0

-

- EN[

i i i

1/

aB 9

8 AA

) O F PYROLYSIS

— i 1 f , , r^

I j - E N D O F FLAME

\\\\

A \ \

A VV<

ETHANOL, Y0( n=0.23l

x0(MM) £(OEG)

3.0 0.0 Al l . l 0 .0 O15.0 36.6 D

i i i

THEORY

r, = O. I3^T , = 0 . 0

t i

10 12 14

x /x „18 20

Fig. 7 Wal l heat f lux for an ethanol-fueled wal l fi re burning in air

O .5

a ^

1 ' I 1 1 1 1 1 t

/ ^ - • ' " " " - — ^ ^ T - - - , THEORY/ / ^ x ^ * — - T . = 0.13• END OF FLAME-^ Y V , r = 0 .0

/ A A A A A A V

b \r A v

~

-

\ s\ s

A \ ^

END OF PYROLYSIS

-

„

PR0PAN0L,YOm=O.23lA

a,

x0(MM) <£(DEG)

3.0 0.0 A

l l . l 0 .0 015.0 36.6 D

A OCCASIONALLY TURBULENT

10 12

x/x„18 20

Fig. 8 Wal l heat f lux for a propanol-fueled wal l f i re burning in air

overfire region, the measurements are generally only 15-20 percent

lower than the predictions, except near the end of the flame where the

discrepancy is somewhat greater .

Fig . 9 i l lustrates the f lame posit ion results for the three fuels . T he

data has been placed in the local similarity coordinate system in order

to facilitate comparison with the theory. The values for the pyrolysis

zone were mea sure d nea r xo; leadin g edge effects resul ted in poo rer

comparison a t lower values of x. The measurements are in reasonably

good agreement with the predictions as far as they go; however, as

il lustrated in F ig . 2 , the f lame is not observed to re turn to the wall in

the manner predicted by the diffusion flame theory. This is probably

due to quench ing of the reaction, as noted earlier. The hea t flux resu lts

also supp ort th is view, since the measure d he at f lux consistently b e

gins to decline sooner th an predicted , suggesting a shorter length of

active combustion.

There are a number of factors potentia lly responsible for the dis

crepancies between theory and experim ent. In addition to quenching

near the tip of the flame, chemical kinetic effects due to fuel decom

position, as the fuel diffuses to the reaction zone, and dissociation in

high tem pera ture regions are not treated by the m odel; all these effects

would tend to reduce the h eat flux to the wall from th e value predic ted

by the m odel.

16

14

12

10

-

-

-

-

1 1 r 1

/PRO PANOL , r, =0-0.13/ x0 = 3.0MM A

\ ^ \ /ETHANOL, r, =0-0.09\ \ / x0=4.0MM D

a \ \

D \ X

\\- METH ANO L, r, =0-0.04^ V \ x 0 = l l . l MM O

• - \ /^a \ A =16.0 MM v

qx> D ^ v \

1 1 I V 1 -

10/ y f V L g x 3 c o s < ^

l x A 4 C T vz

I

Fig. 9 Flame shapes for alcoho l-fueled wal l f i res burning In air

The present model is only a crude approximation of the variable

prop erty c haracteris t ics of the f low field of the test fuels. The trea t

ment of multicomponent diffusion, and the effect of concentration

variations on properties w ould be required for a more exact, and po

tentia lly more accurate , model.

Radiation losses from the flow could also be responsible for lower

wall heat fluxes than the predictions. It is shown in the Appendix that

the test wall absorbed only about one-fourth of the radiant energy lostby the f lame which com prised ab out 12 perce nt of the wall heat f lux

at the worst condition. Th erefore, i t is conceivable that lower f lame

temp eratures result ing from this radian t energy loss could cause wall

heat f luxes lower than those p redicted by a theory which neglects

rad ia t ion .

W a l l H e a t F l u x C a l c u l a t i o n s . W hen the behav io r of the th re e

test fuels is comp ared, it is evident tha t as B increases and r decreases,

the length of the flame and the wall heat flux in the overfire region

both increase. The effect of B an d r is i l lustrated m ore systematically

in Fig. 10. Th e major effect of reduc ing the v alue of r, for a constant

value of B, is to increase the length of the flame; although the h eat flux

values increase somewhat as well . Increasing the value of B when r

is constant causes a substantial increase in the wall heat flux and the

length of the flame.

In natural f ires , the ambient oxygen concentration for a given

surface can vary due to oxygen consumption by neighboring burning

elements . As the ambient oxygen concentration is reduced, both B

an d r decrease in value, which are counteracting effects from Fig. 10.

The net result of varying the ambient oxygen concentration for

metha nol combustion is i l lustrated in Fig . 11 . In this case, decreasing

Yo» causes the heat flux to the wall to decrease and the flame length

to increase. Even though the flame is shorter at higher values of Yo»,

the heat flux increases sufficiently so that the total heat transfer rate

to the wall in the overfire region increases as Yo« increases, e.g., the

tota l heat tra nsfer rate is approxi mate ly twice as high in air as it is for

Yo» = 0.1 . The effect of ambien t oxygen concentration w as similar

for ethanol and propanol; this is a limited survey, however, and these

trends might not be observed for all materials. The more rapid surface




£ , , ( , , , , , ,

« E N D O f F L * M £ B=2

Fig. 10 Wall heat flux lor various Ban d r values, Pr = 0.73, To = t\ = 0

heating rate a t h igher ambien t oxygen concentrations implies a faster

ra te of upward flame propagation. This is generally observed in

practice.

T r a n s i t i o n t o T u r b u l e n c e . The shadowgraphs indicated the f irs t

appearance of turbulence at the outer edge of the boundary layer, with

the turb ulent region propagating tow ard the wall . The p osit ion of the

first occurrence of turbulence oscillated up and down so that the flow

was on ly tu rbu len t a portion of the t ime at the lowest posit ions. The

criteria for t r ans i t ion is different for the pyrolysis zone, the com

busting p lume region, and th e therm al plum e region. A full survey of

transit ion conditions was not completed, however laminar f low was

generally observed in the thermal plume region for Gr / < .5 — 2 X

10 8.

W h il e Gr/ is well defined, it does not completely describe the

cond i t ions for t r ans i t ion . Forexample , t rans i t ion is deferred somew h a t for stable wall angles and occurs sooner for unstable angles,

s imilar to the obse rva t ions of Vliet and Ross [10]. Tran sit ion also

occurred at lower values of G r x ' as the point of transit ion ap proached

the t ip of the f lame. Finally B an d r control m any of the processes of

a wall fire, and the re was a tendency for high B values and low r values

to reduce the value of G r x ' for transit ion. Therefore, the above cri

terion should be t r ea ted on ly as a rough guide.

T rans i t ion in the thermal plume region was not accompan ied by

a significant change in the va r ia t ion of wall heat f lux with distance,

cf. Figs. 6and 8. However, when transition occurred in the combusting

portions of the overfire region, the wall heat flux was sharply reduced

in the turbulent region. This effect is being examined more fully in

cur ren t expe r imen ts .

ConclusionsDisc repanc ies be tween measu red and predic ted wall hea t f lux

values and flame shapes were in the range 15-20 percent. The greatest

errors were observed near the t ip of the f lame where wall quenching

effects become impor tant. F actors contributing to these discrepancies

are : uncer ta in t ie s in proper ty va lues and the approx im a t ions of th e

variable property model, decomposit ion, dissociation and finite rate

chemical effects in the flow field, and radiation. Further consideration

of these phenom ena wou ld be des i rab le .

The heat f lux to the wall increases just beyond the pyrolysis zone

as the effect ofblowing decays. However, the he at f lux rem ains rela

t ive ly cons tan t th roughou t much of the combusting plume region.

Therefore, the presence of the flame provides an extended zone of high

wall heat flux, serving to hea t the combus t ib le .

11 8 / VO L 100, FEBRUARY 1978

x/ x0

Fig. 11 Effect of ambient oxygen concentration o n wall heat flux for methanol

Reduction of the ambient oxygen concentration increases the flame

length, but for the fuels considered in this investigation, the wall heat

flux decreases suffic iently so tha t the overall ra te of preheating the

combus t ib le dec reases as YQ C is reduced . Due to the high flame

temperatures within the combusting region, there is l i t t le change in

wall heat f lux due to increased wall temperatures, at least for surface

t e m p e r a t u r e s in the pyrolysis zone in the range of the present test

fuels.

Con ditions for transition in this flow are not fully resolved, however,

for transit ion in the thermal plume region, the flow was generally

l a m i n a r for Gvx' < 0.5 - 2 X10 8. T rans i t ion is not completely de

scribed by G r / ; however, effects of fuel type, presence or absence of

combustion, distance, and wall angle were also observed.

W hile the theoretical m odel yields satisfactory results , the method

of solution is ponderous for direct use in f lame spread analysis , or

pred ic t ions of hea t t rans fe r in fires. For such applications integral

models are more convenient, and the present results should be of valuein checking their accuracy.


T his re sea rch was suppor ted by the Un i ted S ta te s Depar tm en t of

Commerce , Na t iona l Bureau ofStan dar ds, G rant No. 5-9020, under

the techn ica l managemen t o f Dr . Joh n Rocke t t of the Center for Fire

Resea rch .

R e f e r e n c e s

1 Kosdon, F. J., W illiams, F . A., and Buman, C, "Com bustion of VerticalCellulosic Cylinders inAir," Twelfth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1969, pp. 253-264.

2 Kim, J. S., deRis, J., and Kroesser, F. W ., "Laminar Free-ConvectiveBurning of F uel Surfaces," Thirteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1971, pp . 949-961.

3 Yang, K-T., "Lam inar Free-Convection W ake Above a Heated VerticalPlate," ASME Journal of Applied Mechanics, Vol.31, 1964, p p. 131-138.4 Pagni, P. J., and Shih, T-M., "Excess Pyrolyzate," Sixteenth Symposium

(International) on Combustion, The Combustion Institute, Pittsburgh, 1977,pp . 1329-1343.

5 Hayday , A. A., Bowlus, D. A., and McG raw, R. A., "F ree ConvectionFrom a Vertical Flat Plate with Step Discontinuities in Surface Temperature,''ASME J O U R N A L O F H E A T T R A N S F E R , Vol. 89,1967, pp. 244-250.

6 Kim, J. S., de Ris, J., and Kroesser, F. W ., "L aminar Burning BetweenParallel Fuel Surfaces," International Jo urnal of Heat and Mass Transfer,Vol. 17,1974, pp. 439-451.

7 Blackshear, P . L., J r., and Mu rty, K. A., "Some Effects of Size, Orientation, and Fuel Molecular W eight on the Burning of Fuel-Soaked W icks,'Eleventh Symposium (International) on Combustion, The Combustion Institute , Pittsburg h, 1967, pp. 545-552.

8 Ahm ad, T., "Investigation of the Combustion Region of Fire-InducedPlumes Along Upright Surfaces," Ph.D. Thesis, The Pennsylvania State Uni-

Transact lons of the ASME




versity, 1978.9 W illiams, F. A., Combustion Theory, Addison-W esley, Reading, Ma,

1965, Chapter 1.10 Vliet, G . C, and Ross, D. C, "Turbulent Natural Convection on Upward

and Downward Facing Inclined Constant Heat Flux Surfaces," ASME JOURNAL OF HEAT TRANSFER, Vol. 97,1975, pp . 549-555.

11 Edwards, D. K., and Balakrishnan, A., "Thermal Radiation byCombustion G ases," International Journal of Heat and Mass Transfer, Vol. 16,1973, pp. 25-40.

12 Raznjevic, K., Handbook of Thermodynamic Tables and Charts, 1st

ed., M cG raw-Hill, New York, 1976.

A P P E N D I XRadia tion effects aremost impor tan t in the region adjacent to the

top of the pyrolysis zone, since the gas temperatures arehigh and the

flow has the maximum thickness for thepyrolysis zone. Rad iation was

evaluated for the most conservative condition of the p resen t expe r i

ments which involves the methano l f lame with 16 mm pyrolysis zone

length.

The theory provides the concentrations of fuel andoxygen, and the

t empera tu re , as i l lustrated in Figs . 4 and 5. In all regions of the flow,

the concentration of nitrogen is

YN=YNM~ BB FQ /(l + B)) (30)

Two other relations are ob ta ined by no t ing tha t H2O and CO 2 are

present in stoichiometric proportions unde r the present assumptions,

and the sum of all mass fractions must be un i ty .

The flow field wasdivided into f ive increments , and the average

t e m p e r a t u r e and concen t ra t ion for each inc remen t was de te rm ined .

The emissivity of an inc remen t was computed by adding individual

band emissivities, ignoring band overlap [11]. For these calculations,

me thano l was assumed to have radiation properties equivalent to

met hane . G as absorption was ignored, s ince less than two percent of

the emitted radiation was found to be absorbed by in tervening in

c remen ts . The wall wassomewha t ta rn ished and it was t aken to be

a gray body with an emissivity of 0.55 [12].

It was found that radiation absorbed by the wall was a b o u t 12

percen t of the total wall heat flux at the top of the 16 mm pyrolysis

zone. On the other hand, the flow absorbs little radiation and its to tal

radiative loss is about four t imes larger than the increase in the wall

heat flux due to rad ia t ion .

Th e effect of gas radiation is smaller at positions higher on the wall,

du e to increased rates of convection, lower fuel and product concen

trations, and lower gas t empera tu res , cf. Figs. 4-6. Therefore, for the

conditions of this experiment, direct radiative heat transfer to the wall

is relatively small, although the high temperature portions of the flow

are losing a significant amount of energy by rad ia t ion .

J ou rn a l of Heat Transfer FEBRUARY 1978, VOL 100 / 119




G. E. Myers

Professor ol Mechanical Engineering,

The University of Wisconsin,

Madison, Wise.

Mem. ASME

The Critical Time Step for Finite-

Element Solutions to Two-

Dimensional Heat-Conduction

TransientsComputationally-useful methods of estimating the critical time step for linear triangular

elements and for linear quadrilateral elements are given. Irregular nodal-point arrange

ments, position-dependent properties, and a variety of boundary conditions can be ac

commodated. The effects of boundary conditions and element shape on the critical time

step are discussed. Numerical examples are presented to illustrate the effect of various

boundary conditions and for comparison to the finite-difference method.

Intr oduc t ion

The use of the f inite-element method in solving heat-conduction

problems is becoming widespread due to the ease with which irregular

geometries can be approximated as compared with the older, moreestablished finite-difference method. The reason for this is that in

the f inite-element method, irregular nodal-point locations can be

handled with no more diff iculty tha n uniformly-spaced n odal points .

The finite-difference method, on the other hand, is fairly convenient

to use when the nodal points are located in a regular pattern but i t

becomes rather awkward when the boundaries of the region do not

conform to the noda l-point grid or when i t is desirable to have some

portions of the region more heavily populated with nod es than other

por t ions .

The finite-element literature is fairly recent (1960s), and practically

all of i t is concerned with solving steady-state s tructural problems.

Comparatively l i t t le has been writ ten regarding the use of the f i

nite-elem ent method in solving trans ient heat-cond uction p rob

lems.

One of the well known features of the f inite-difference met hod for

heat-conduction transients is that the selection of too large a time step

can lead to a meaningless, oscil la tory solution. This problem is

touched upon in practically every heat-transfer textbook. The largest

time step for which a Euler solution will be stable (and for which a

Crank-N icolson solution will be nonoscil la tory) is called the cri t ical

t ime step (A 0C). The familiar criterion for one-dimensional transients

is that aA 0c/(Ax)2 = lk- For two-dimensional problems using a square

grid of nodal points , «A 9 c / (Ax) 2 = i/j- Th e ability to choose a stab le

time step (A0 £ Adc) in a computationally-convenient manner has

previously been obtained and will be reviewed in this paper. Irregular

nodal-point locations, posit ion-dependent properties , and a variety

of boun dary condition s can be accomm odated. A similar problem

arises in the finite-element method, but it is more severe. The criticaltime step is smaller, and the knowledge of how to choose a stable time

step in a computationally convenient manner is lacking.

Myers [ l ] 1 mentions numerically-induced oscil la tions in one-

dimensional f inite-element problems but does not give any useful

ways to cope with the problem. Lemmon and Heaton [2] derive a

stabil i ty cri terion for one-dimensional transients with uniform

nodal -point spacing and uniform properties . For the f inite-elem ent

method they found tha t aA 6c/(Ax)2 = % as compared to the f inite-

difference value of V2- One-dimensional problem s are not too excit ing

as far as the finite-element method is concerned because they can be

hand led jus t as effectively by finite differences. Th e real utility of the

finite-element method comes in treating two and three-dimensional

problems with irregular regions and their concommitant irregular

noda l -po in t a r rangemen ts .

Yalamanchili and Chu [3] use the von Neumann method to deter

mine the cri t ical t ime step for two-dimensional transients . For the

finite-element method, using square elements , they report that

aA 0c/(Ax)2

=l/12 as compared to th e f inite-difference value of

lk. As

in the one-dimensional case, a more restrictive time step is found for

the f inite-element me thod. The von Neu ma nn theory is s tr ic tly valid

only for an infinite number of uniformly-spaced nodal points and

uniform thermal properties , and i t does not consider the boundary

conditions. Yalamanchili reconsiders this problem in a more-recent

Contribu ted by the He at Transfer Division for publication in the JOURNALOF HEAT TRANSFER. Manuscript received by the Heat Transfer Division July20,1977.


120 / VOL 100, FEBRUARY 1978 Transactions of the ASWIECopyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



paper [4] but does not present any new information that would allow

trea tme n t o f more genera l p rob lems .

Up to this t ime, no computationally-useful method of estimating

the critical time step has been reported for the finite-element method

that is capable of treating general problems. The present paper pro

vides such a method for linear triangular elements (equation (20)) and

for l inear quadrila te ral e lements (equation (24)) . Fur ther mo re the

paper summarizes the theory necessary to provide a thorough un

derstanding of the cri t ical- t ime-step problem. The significance of the

critical t ime step to th e Crank-N icolson scheme is discussed. Th e von

Neum ann theory is shown to overestimate A0 C in many situations and

hence is not a safe techniqu e to use in general . Num erical examp lesare given to i l lustrate the effect of various boundary c onditions and

to make comparisons with the f inite-difference method. The theory

is used to deter min e the optim um shap e of tr iangular e lemen t.

Trans ien t Equat ions

This paper is concerned with the general , two-dimensional, heat-

conduction transient described by the partia l d ifferentia l equation

p(x,y)c(x,y)_d _

dx b{ x

'y )^ ]

+ — \k(x,y). , . . . , +g'"(x,y) (1)

Observe tha t the the rma l properties can be functions of posit ion b ut

not t ime or temperature in this paper. In addition, there will be no

restr ic tion to s imple geometrical regions.

The conditions along the boundaries of the region may be any

combination of adiabatic, convective, specified heat flux, or specified

temperature as shown by equations (2) through (5).

dtAdiaba tic : — = 0

dn

Convective: — k_di_

dn••h(t„-t)

dtSpecified heat flux: — k — = q" s

dn

Spec if ied tempera tu re : t = ts

(2)

(3)

(4)

(5 )

T h e p a r a m e t e r s h, t» , q s", an d ts can vary with posit ion around the

boundary bu t may no t depend upon t ime o r tempera tu re .

Finally , the init ia l con dition is given by

t(x,y, 0) = t'-oHx,y) (6)

Equations (1-6) represent the exact mathematical description of the

class of problems considered in this paper.

S p a t i a l l y - D i s c r e t e E q u a t i o n s . The finite-element and the f i

nite-difference met hods for solving this problem both involve using

a set of n nodal points to subdivide the region into a finite number of

discrete regions. This discretiz ing proc ess is discussed in [ l] for b oththe f inite-difference and finite-elem ent me thods. Both meth ods re

place equations (1-5) by a system of first-order ordinary differential

equations that can be compactly writ ten in matrix notation as

Ct = -St + r

The init ia l condition, equation (6), is replaced by

t(0) = t<0)

(7)

(8)

T he tem pera tu re vec tor t con ta ins the tempera tu res a t the n noda l

points . The capacitance matrix C describes the thermal-storage ability

of the discretized proble m. I t involves the density , specific heat, an d

the area of each discrete region. The conductance matrix S contains

the conductances between the various discrete regions and between

the boundary and the ambien t . Bo th C and S will be symme tric an d,

if the nodal points are num bered properly , they will be banded rather

than full matr ices. The in put vector r contains th e effects of energy

generation, convective ambient tempe rature, specified boundary heat

flux, and specified boundary temperature. The matrices C, S , and r

will be constan ts ( indep endent of t ime and temp erature) for the class

of problems considered in this paper.

The matrices C, S, and r obtained by the finite-element method will

not in general be the same as one would obtain using the finite-dif

ference method. Computationally , the most s ignificant difference is

that the finite-difference C will be diagonal, whe reas it is not diagon al

in the usual f inite-element formulation. This difference is the

underlying reason that the cri t ical- t ime-step problem must be

restudied for the f inite-element method.

. No m en c l a t u re -a = shortest s ide of tr iangular e lement; L

a = constant vector in equation (11); T

A = area of f inite e lement; L2

b = longest s ide of tr iangular e lement; L

b = constant vector in equation (11); T/Q

c = specific heat; E/MT

c = third s ide of tr iangular e lement; L

c,-; = diago nal e ntr y in row i of C; E/LT

G = capac i tance ma t r ix ; E/LT

D = diagonal of C;E/LT

E = dimension of energy; E

g'" = energy generation rate per unit volume;E/OL

3

h = heat-transfer coeffic ient; E/0L2T

H = Bio t number , hL/k; none

k = the rm al conduc t iv ity ; E/QLT

L = size of square region; L

L = dimension of length; L

M = dimension of mass; M

n - inward coord ina te no rmal to boundary ;

L

n = total number of nodes; none

q = lower bound on /3X; none

ql = specified boundary heat f lux; E/QL2

r = input vector; E/QL

sa = diagonal entry in row i of S; E/QLT

Sjj = off-diagonal entry in row i of S;

E/QLT

S = conduc tance ma t r ix ; E/QLT

t = t e m p e r a t u r e ; T

ts = speci f ied bou ndary temp era tu re ; T

t„ = a m b i e n t t e m p e r a t u r e ; T

t(0 )

= in i t ia l t empera tu re ; T

t = tempera tu re vec to r; T

j w = tempera tu re vec to r a t t ime <?'"'; Tt = t ime derivative of temperature vector;

TieT = d imens ion o f temper a tu re ; T

u = eigenvector; unspecified

v = eigenvector; unspecified

x = pos i t ion coord ina te ; L

x = eigenvector; VLT/E

X = eigenvector matrix; VT/FfE

XT

= t r anspose o f e igenvec to r ma t r ix ;

VLT/Ey = posit ion coordinate; L

a = thermal diffusivity; L2/ 6

(3 = angle b etwe en side c and x-a xis; no ne

/3 = eigenvalue; none

(3i = smallest eigenvalue of equation (35);

none

A0 = time-step size; G

Adc = crit ical t ime step; 9

A0„ = Toeplitz estimate of critical time step;

9

A0* = safe estimate of cri t ical t ime step; 9

M l = estimate of A0„; 9

0 = t ime; 9

0M = t ime a t s tep number v; 99 = dimension of t ime; 9

K = eigenvalue; 1/9

K n = largest e igenvalue of equation (33);

1/9

A; = it h

eigenvalue of equation (14); 1/9

X„ = largest eigenvalue of equation (14);

1/ 9

A = diagonal e igenvalue matrix; 1/9

Mi = 1th

entry in M; none

M = diagonal matrix in equation (11); none

v = t ime-s tep number ; none

p = dens i ty ; M /L3





T r an s i en t So lu t ions . T he so lu tion to equa t ion (7) wi ll app rox i

mate th e solution to equation (1) with bou ndary co nditions given by

equations (2-5). In practical problem s it is too much work to f ind th e

exact analytical solution to equation (7). Instead, s tep-by-step,

t ime-m arching schemes such as Euler or Crank-Nicolson are used to

approximate the solution to equation (7).

In the Euler scheme, equation (7) is replaced by

Ci<"+1> = (C - SAfl)t<"> + rA 6 (9)

The Crank-Nicolson scheme replaces equation (7) by

( c + S — ) t<"+1

> = ( c - S — ) t<"> + rA0 (10)

In the finite-difference me thod C is diagonal and e quat ion (9) can be

solved explicit ly for the new set of tem pera ture s i<"+1

' . In the finite-

eleme nt meth od C is not diagonal and equation (9) does not give an

explicit solution for t(,,+1)

. Rather, a system of algebraic equations m ust

be solved at each time step (an implicit solution). The Crank-N icolson

scheme is implicit for both methods because S (which is not diagonal)

has been introduced on the left-hand side of the equation. Conse

quently in the f inite-element method one would normally use the

more accurate Crank-Nicolson scheme since it can be carried out with

no more computations than the Euler scheme, whereas in the f inite-

difference meth od one could choose Euler and avoid an implicit so

lution.

Reference [5] shows that the exact analytical solution to equ ation(7), the Euler solution, and the Crank-Nicolson solution can all be

writ ten as

i<"+1

> = a + bfi("+1

> + XM"+1

XTC(i<°> - a) (11)

where the vector b must satisfy

Sb = 0 (12)

and the vector a must satisfy the equation

Sa = r - Cb (13)

T he ma t r ix X contains the eigenvectors (xi , x 2 , . . . , x„) of the gener

alized eigenproblem given by

SX = CXA (14)

where A is a diagonal matrix containing the eigenvalues (Xi ^ X 2 £

...< X J.The only place the three solutions represented by equation (11)

differ is in the diagonal m atrix M. The d iagonal entrie s, m, in M for the

exact analytical solution to equ ation (7) are given by

Hi = exp (-X ..-A0) (15)

If these M are substi tu ted in to equation (11) the exact analytical so

lution to eq uation (7) is seen to be the su m of n decaying exponentials

(in addition to the l inear leading terms). For the approximate Euler

scheme, equation (9),

Mi = 1 - AiAfl (16 )

For the approximate Crank-Nicolson scheme, equation (10),

Hi = 1')2 + A;A0

The Euler and C rank-Nicolson relations, equations (16) and (17), are

appro xima tions to the exact re lation, equati on (15). Thes e thre e

relations between Hi and X;A0 are compared in Fig. 1 . Observe that

the Crank-Nicolson relation is a much bette r approximation than the

Euler relation. Th us for the sam e A0 Crank-N icolson will give a bet ter

approx imatio n to the solution of equat ion (7) than Euler will . As A0

goes to zero all three curves converge, bu t this is generally too small

a t ime step to be practical .

The character of each of the three solutions is re lated to the nu

merical values of Mi- Since M is diagonal, the en tries in M"+1

in equation

(11) will be Hil'+1 - Hence for values of M; between 0 and 1, Hi "+1 will

X | A 0

Fig. 1 Com parison of Euler and Crank-Nicolson values of HI i ° exactvalues

steadily decay to zero (e.g. V2, % V8, Vi 6, . . . ) . For values of M between

0 and - 1 , Mi "+1 will also decay to zero but will alter nate sign (e.g. - ' /2llk< ~

1h> Vi6. • • -)-These alternations in sign will propagate throughout

the solution and give rise to "numerically-induced oscillations" in the

temperature until they eventually die out. Finally, for values of M; less

than - 1 , Hi,,+

* will not only altern ate in sign bu t will grow in amp litude

(e .g . —2,4, -8 ,1 6 , . . .). Thi s behavior will produ ce unstab le , numer

ically-induced oscillations in the temperature solution. Observe from

Fig. 1 that the Euler scheme can exhibit a l l three kinds of behavior

(steady decay, oscillatory decay, and oscillatory growth) if AS is large

enough. The C rank-Nicolson scheme can only have steady decay and

oscillatory decay. The exact analytical solution will only have a steady

decay. I t should be men tioned tha t in the f inite-element m ethod the

exponentials in the exact analytical solution can add together in such

a way as to produce some oscil la tory behavior a t small t im es [1] hut

this behavior should not be confused with numerically-induced os

cil la t ions.The larger the value of Afl that is used, the more serious the nu

merically-indu ced oscil la tions become. Euler solutions can go out of

control in a few time steps. Although the Crank-Nicolson scheme will

never be unstable, the oscillations can make the solution useless. Thus

the selection of a suitable t ime step is important for both the Euler

and the Crank-Nico lson schemes .

The Cri t ical Time StepThe value of Ad beyond which the Euler solution would exhibit

unstable oscillations is called the critical time step, A9C. A value of A(l

less than or equal to A0C will ensure that none of the Hi is less tha n - 1 .

Th e most negative value of Hi will be Hn - It can be seen from Fig. 1, or

equation (16), that Hn = - 1 when X„ A0 = 2. Th us th e critical time step

is given by

2A0C = — (18)

Arc

The crit ical t ime step thus depends only upon the maximum ei

genvalue of the generalized eigenproblem stated by equation (14).

Therefore only the factors that de term ine the entries in C and S are

impo rtant in determining th e cri t ical t ime step. Nodal-point locations,

the rma l p rope r t ies (h,p,c, and h) , and which nodal temper atures are

specified are impo rtant. Facto rs such g'", q"s, t„ , and t he actual value

of a specified temperature, ts, enter the problem only in the vector

r and h ence do not affect th e cri t ical t ime step. The init ia l condition,

equation (6), is a lso unim port ant.

Reference [5] discusses an im port ant exception to eq uation (18)-

Wh en ther e are specified-tem perature nodes there will be an equal





number of e igenvalues that are directly associated with these nodes

which will cancel out of the solution and hence will not influence the

stabil i ty and oscil la tion characteris t ics . If any of these special e i

genvalues is An it can be disregarded. In this case A„_i should be used

in equatio n (18) to dete rmi ne the critical time step. Tha t is, the largest

eigenvalue not associated with a specified-temperature node should

be used in equatio n (18).

I t should also be observed from Fig. 1 tha t the cri t ical t ime st ep

defined by equation (18) is a lso the t ime ste p at which num erically-

induced oscil la tions will begin to show up in the C rank-Nic olson so

lution. Thus a choice of Ad s A0 C will a lso ensure that no numerically-induced oscil la tions will occur in a Crank-Nicolson solution.

Since numerically-induced oscillations are unwelcome if they are too

large, the cri t ical t ime step is an important parameter in a Crank-

Nicolson solution even though the solution will a lways be stable .

Al though i t i s impo r tan t to unders tan d th a t the max imum e igen

value determines the cri t ical t ime step, precise evaluation of A„ re

quires a major comp utational effort . I t is not a convenient tec hnique

for findin g A0C.

Safe E s t i ma tes . A "sa fe" e s t ima te (A0*) o f the c r i tica l t ime s tep

is one which is less tha n or equal to the critical tim e step. An " uns afe"

estimate of the cri t ical t ime step is one that is larger tha n A0C. T h is

would allow the possibility of choosing AO less than the unsafe esti

mate but greater than Adc and hence obtain an unstable Euler solution

or an oscil la tory Crank-Nicolson solution. A safe estimate can be

found by mak ing an estim ate of \„ th at is guara nteed to be too largeand substi tuting i t in to equation (18) instead of A„.

Ther e is a theorem by Gerschgorin [5 ,6] that can be applied almost

imme diately in the f inite-difference m etho d since C is d iagonal. Th e

finite-difference result is that

A0' c = 2 Min£ = 1

su + E | s , ; |7 = 1

(19)

where c„- a nd su are the diagonal en tries in row i of C and S and sy is

an off-diagonal entry in S when ;' ^ i. The notation in front of the

brackets means that the computation should be done for each of the

n rows, and the minimum result should be taken as A0*. Reference

[5] shows that rows corresponding to specified-tem perature nodes m ay

be excluded from this com putatio n.

The development of a similar relation for the finite-element method

is more involved since C is not diagonal. The proof depends u pon the

symmetry of C and S and upon the particular s tructure of C in addition

to the Grsechgorin theorem as shown in the appendix at the end of

the paper. For l inear tr iangular e lements the result is that

A0* = Mini = i

su + L7 = 1

7 V i

(20)

The only difference between the finite-element relation, equation (20),

and the finite-difference result, equation (19), is the factor 2. However,s ince the entries in the f inite-element C and S will in general be dif

ferent than for finite differences, the estimates will not simply differ

by a factor of two.

Equations (19) and (20) are quite general results. They may be used

for problems with irregular nodal-point locations, position-dependent

properties , and any combination of the boundary conditions given

by equations (2-5 ). Equ ation (20) is a new result that is as convenient

for the engineer to use in finite-element com putatio ns as equation (19)

has been in the past for f inite-difference computations. In f inite-

element practice a lumped-capacitance formulation is sometimes

used. In this case C is diago nal and e qua tion (19) shoul d be used to

estimate the cri t ical t ime step.

Toe plitz Est im ate s. The Fourier method (separation of variables)

used by Lemmon and Heaton [2] to estimate the critical time step and

the von Neumann method used by Yalamanchili and Chu [3] are

closely related. Both assume there are an infinite number of uni

formly-spaced nodal points and that the thermal properties are uni

form. Both methods ignore the boundary conditions of the problem.

With all of these restr ic tions C and S will be infinite ly-large Toeplitz

matrice s [7]. Analytical ex pressions for the eigenvalues of such m a

trices can be derived. If the largest eigenvalue, A„, is substituted into

equa t ion (18) , a "T oep l i tz e s t im a te" (A0„) is obtaine d. This res ult is

the same as one ob ta ins v ia the Four ie r o r von Neuma nn me thod .

Toeplitz estima tes as applied to f inite differences an d finite e lem entsare elaborated on in [5].

Since the maximum Toeplitz e igenvalue is not necessarily greater

than An, the Toeplitz estimate (A0„) is not a safe estimate. Its relation

to A0C and A0* c will become appare nt in the num erical examples of the

next section. Toeplitz estimates can, however, be used to make in

teresting comparisons between various methods. For a square

nodal-po int grid , the Toeplitz estim ate for tr iangular f inite e lem ents

has been found [5] to be

2 - \ / 3•• 0.0774 (21)

«Afl„

(A *) 22\/j}

For square f inite e lements , Yalamanchili and Ch u [3] found t hat

«Aflm

(Ax)2•• — = 0.0833

12

For f inite differences,

*A0„ 1• = 0.2500

(22)

(23)(Ax)2 4

Observe that both f inite-element methods are much more restr ic tive

than t he f inite-difference metho d. Also note tha t tr iangular e lem ents

are s l ightly more restr ic tive than square elements .

N u m e r i c a l E x a m p l e s

The numerical examples presented in this section i l lustrate the

behavior of the critical time step as a function of the nodal-point grid

size and the boundary conditions. The relationship between A9C, Ad*c

a n d A6 „ is d iscussed. Finite-difference results are incorporated for

compar ison .

Fig. 2 describes the region that will be examine d. A uniform, squ arearrangement of nodal points will be used. Two finite-element ar

range men ts will be considered—on e will have all posit ive diagonals

and w ill be indicate d by a ffi; he other will have all negative diagonals

y

/

/////

///

//////////

J /

.

r-

1

11

L _

i A1 ' .1 /

/ . ^ -Po s i t i v e -D ia g o n a l1 / E l e m e n t , ®

V- J

Negat ive-DiagonalE l e m e n t , Q 7

f_ _, yl \ r

— -, j \ j1 ' \ '

a I t - — — — _411

^ - F i n i t e - D i f f e r e n c eS y s t e m

/ / / / / / " / / ////•//////•/////// ^

Fig. 2 Examp le problem





and be indicated by a 9. The boundaries along x = 0 and along y =0 will be adiabatic. The boundaries along x = L and along y = L willboth be adiabatic, or convective, or have a specified temperature.Thermal properties will be uniform.

Ex ac t V alues. The exact values of the critical time step are shownin Fig. 3 for the various boundary conditions and element arrangements. These values are obtained by actually computing the maximum eigenvalues for equation (14) and substituting them intoequation (18). The T oeplitz values, equations (21) and (23), are alsoshown.

Observe tha t for the finite-difference method the nondimensionalcritical time step for the completely adiabatic boundary (H = 0) isindependent of the number of nodes and is the same as the Toeplitzestimate. For H > 0 it appears tha t the Toeplitz estimate is the limiting value as the num ber of nodes becomes infinite. This is reasonablesince the assumptions behind the Toeplitz estimate are more nearlysatisfied as the numbe r of nodes increases. These observations do notappear to be true for the finite-element method however. Here itappears that the limiting value approached by all but the top curveis slightly below the T oeplitz estimate. This must be due to a boundaryeffect th at is not accounted for by the T oeplitz theory. Consequentlythe Toeplitz estimate is hardly ever a safe one to use for triangularelements.

The effect of convection, for both finite differences and finite elements, is to reduce the critical time step. Thus the Toeplitz e stimateis not a safe one to use for either method if there is a convective

boundary. As H becomes large, A0C goes to zero and an excessivelylarge number of time steps would be required to com plete a transien tsolution. However as H becomes large, the convective bound ary maybe satisfactorily modeled as a specified-temperature boundary. Observe that the critical time step for the specified-temperatureboundary is less restrictive than the completely adiabatic boundary.Thus as H becomes large, one should think about changing the modelfrom a convective bounda ry to a specified-tem perature boundary toavoid being forced to tak e very small time steps .

Although the nondimensional critical time step increases (for tf> 0) as the num ber of nodes increases, A0C itself is decrea sing sinceAx is decreasing. In the limit a constant value (approximately th e

Toeplitz value) is approached, and then A9C is proportional to(Ax)

2.

Es tim ate s. Estimates of the critical time step (and of the Toeplitzvalue) based on equation s (19) and (20) are shown in Fig. 4. The samequalitative behavior as F ig. 3 is observed. The finite-difference curvesall approach the Toeplitz limit as the number of nodes increaseswhereas all but one of the finite-element curves approaches a limitof 1/24 which is below the T oeplitz limit.

The curves in Fig. 4 all fall below the corresponding curves in Fig3 as expected, bu t it is interesting to note t hat the difference betweenA0* and A0C is much less for the finite-difference metho d tha n for thefinite-element method. Th e nondiagonal na ture of the finite-elementC has not permitted as good an estimate of A„ as when C is diagonal.

D i s c u s s i o n

Previous inve stigations [2, 3] have used the Fourier or von Neumann method to study the critical-time-step problem. These techniques are seen in this paper as being related to Toeplitz matrix theoryand simply provide a limiting-case solution as the numbe r of nodalpoints becomes infinite. A lthough these tec hniques provide an interesting comparison between various discretization methods, theydo not provide a computationally useful way to make a safe estimate

of the critical time step in a general situation.The major new result presented in this paper is equation (20) which

gives a computationally convenient safe e stimate of the critical timestep for the finite-elem ent method w hen using linear triangular elements. This result is applicable to prob lems with irregular nodal-pointspacing, position-dependent properties, and a variety of boundaryconditions. The engineer can now conveniently and safely estimatethe critical time step.

The finite-element result, equation (20), cannot be expected to hold

1.0

O.IO

x

CD

O.OI

F INITED IFFERENCES

-Spe c i f i e dTe mpe ra t u re

H=0 ~

T o e p l i t z , E q . ( 2 3 p

H = IO _ - - - "

H = IOO

0.001 L

F IN ITEE L E M E N T S

Spe c i f i e dQ X Te mpe ra t u re

1.0

4 9 16 2 5 4 9 16

n nFig. 3 Crltlcal-tlme-step comparisons

25

O.IO

O.OI

0.001

F IN ITED IFFERENCES

V^ -Spe c i f i e d\ Tempera ture

H=0 "/Toepl i tz , Eq. (23) V

H = IOO.

4 9 16 25 4 9 16n n

Fig. 4 Estimates of the critical time step





for non-triangular elements but it should be possible to derive similar

results for other elements. As an example, it can be shown that for the

linear quadrila teral e lement

A0':1 n- M i n2 ;=i

c»(24)

This equation replaces equation (20) for the quadrila teral e lements .

I t is an extension of the comments of Yalamanchili and Chu [3] re

garding square f inite e lements .

Comparisons of the critical time steps for the finite-element method

with the f inite-difference method show that the f inite-element

method is more restr ic tive. This is an interesting observation, but i t

will probab ly not govern the choice between using finite e lem ents or

finite differences. The superiority of the f inite-element method at

handling irregular geometries will probably override the disadvantage

of being forced to t ake sm aller t ime steps.

Convective boundary conditions have been shown to be an im

portant factor in determining the cri t ical t ime step. An alternative

way to study this effect is shown in Pig. 5. Observe that as Ax de

creases, aA0* c/(Ax)2

for finite differences approaches 1/4 (the Toeplitz

value) for all values of h. The corresponding l imiting value for the

finite-element curves is 1/24 (54 percen t of the finite-eleme nt Toepli tz

value, equation (21)). For large hAx/k, each curve in Fig. 5 becomeslinear. In this range A0* is inversely proportional to the heat-transfer

coefficient.

The effect of the shape of the tr iangular e lement can be studie d by

using Toeplitz estimates. The various tr iangular shapes may be par

ameterized as shown in Pig. 6. Although an analytical expression for

the Toep litz e igenvalues can be derived, i t is not an easy task in gen

eral to find the max imu m eigenvalue. Consequen tly an est ima te (A(C)>

following equation (20), will be applied to the T oeplitz m atrices. For

tr iangular e lem ents i t can be shown [5] tha t

aAOl

Ax Ay

0[

2( cT

+ 2 +K 9

c o s 3 +© l

cH ]

(25)

For compa rison, the f inite-difference result for /} = 90 deg is given

by 2

aAIL_

Ax Ay

©

•\»m(26)

It should be men tioned th at for the finite-difference result it happ ens

that A#L = A9». In both cases A(C has been normalized using the

product Ax Ay which is twice the area of the tr iangle . Pig. 7 is a plot

of equations (25) and (26). The angle /? is an imp orta nt factor for tr i

angular elements . For a fixed value of ale, A0l increases as /3 increases

until the maximum is a tta ined when the tr iangle becomes isosceles.

The optimum shape of isosceles triangle is the equilateral triangle (ale

= 1).For the tr iangular e lements used in arriving at equation (21), /3 =

90 deg, ale = 1, and Ax = Ay. From equa tion (25) then,

«A0.

(Ax)2

1 6 '0.0625 (27)

This is a more restr ic tive result th an given by equation (21) bec ause

equation (25) was derived using an overestimate of the maximum

• When /3 j ^ 90 deg, the finite-difference expression is cumbersome to obtainfor some of the same reasons that fini te differences are cumbersome to programfor irregular nodal loc ations.

O.IO

O.OI

O.OOI

i/*K

l / 2 4 %

/ F i n i t e

F inite / x \Element^1^^ .

0 \ ^

0.1 1.0 10

h Ax / k

100

Fig. 5 Effect of convection on the cri t ical t ime step

Fig. 6 Geom etrical-shape param eters for tr iangular elements

Toeplitz e igenvalue, whereas equation (21) used the precise value.

Similarly , the equ ilateral tr iangle resu lt shown in Fig. 7 is 1/8V3 =

0.0722, whereas th e precise value is 1/6V3 = 0.0962. Observe tha t this

precise value for the equilateral tr iangle is less restr ic tive than the

precise value, equation (22), which Yalaman chili and C hu [3] found

for the square element, whereas the precise value for the r ight-tr i

angula r e lement, equa tion (21), is more restr ic tiv e.

Since the f inite-element C is not diagonal, the C rank-Nicolson

scheme should be used instead of Euler because it gives more accuracy

with no additional computational effort . Since Crank-Nicolson is

always stable , the cri t ical t ime step can be exceeded without losing

control of the solution. Nume rically-indu ced oscil la tions will be in

troduced into the solution however. The larger the time step the more

serious the oscil la tions. Wood and Lewis [8] show how this "noise"

can be artif ic ially dam ped by an averaging process.


The conclusions of this paper can be summariz ed as follows:

(1) Th e critical tim e step for linear triang ular elem ents can be safely





0. 3

0. 2

<3X

<

* 8

<a

Fin i te /D i f f e r e n c e s ^ . /

/ Fini te/ E le m e n t s ^ ^ -

Isosceles.

___3—— —^3 = 0°

0.5

a /c

Fig. 7 Effect of element shape on A8'„

1.0

estimated using equation (20). For l inea r quadr i la te ra l e lemen ts

equation (24) should be used.

(2) Relative to an adiabatic boundary, convection reduces the critical

t ime step whereas a specified-temperature boundary will increase the

crit ical t ime step.

(3) The equilateral tr iangle is the bes t shape of t r iangu la r e lemen t

as far as the crit ical t ime step is conce rned .

(4) Critical time steps for the f inite-element metho d are smaller than

for the f inite-difference method.


This work was carried out under the sponso rsh ip of the Na t iona l

Science Foundation under Grant G K-43429 and the Graduate School

of the University of Wisconsin—Madison. A. Pincus of the U niversity

of Wisconsin Mathematics Research Center provided several key

insights regarding the eigenvalue pro blem in th is analysis . P. Nguyen

and D. Bachelder carried out m o s t of the computa t ions .

References1 Myers , G. E., Analytical Methods in Conduction Heat Transfer,

McGraw-Hil l , New Y ork, 1971.

2 Le m m on , E. C, and He a ton , H. S. , "Accuracy, S tabi l i ty , and Osci l la t ion

Charac te r is t ics of F in i t e E l e m e n t M e thod for Solving Hea t Conduc t ionEqua t ion , " AS M E P a pe r No . 69 - WA/HT - 35 .

3 Yalamanchi l i , R. V. S., and C hu , S. C , " S ta b i l i t y and Osci l la t ion Ch ar

ac te r is t ics of Fini te -Element , F ini te -Dif fe rence , and Weighted-Residua lsM e thods forTransient Two-Dimensiona l Hea t Conduc t ion in Solids ," ASME,

J O U R N A L OF H E A T T R A N S F E R , Vol. 95, May 1973, pp. 235-239.

4 Yalamanchi l i , R., "Accuracy, S tabi l i ty , and Osci l la t ion Charac te r is t icso f T r a ns i e n t Two- Dim e ns iona l He a t C onduc t ion , " AS M E P a pe r No. 75-

W A / H T - 8 5 .

5 Myers , G. E., "Numer ica l ly- Induced Osc i l la t ion and Stabi l i ty Charac

te r is t ics of F in i t e - E le m e n t S o lu t i ons to Two- Dim e ns iona l H e a t - C onduc t ionTran sients , " Repo r t 43, Engineer ing Expe r imen t S ta t ion, Univers i ty of Wisconsin-Madison, 1977.

6 Noble, B., Applied Linear Algebra, Prentice-H all, Inc., Englewood Cliffs,NJ, 1969.

7 Gr e na nde r , U., and Szego, G., Toeplitz Forms and Their Applications,Univers i ty of California Press, Berkeley, CA., 1958.

8 Wood, W. L., and Lewis , R. W., "A C om pa r i son of Time MarchingSchemes for the Transient Hea t Conduc t ion Equa t ions ," InternationalJounial

for Numerical Methods in Engineering, Vol. 9 ,1975, p p. 679-689.

A P P E N D I XAn a l te rna te s ta temen t of the eigenproblem given by equation (14)

Sx = XCx (28)

W h e n C is diagonal as in finite differences or in the lumped-capaci

tance f inite-element formulation, equ ation (28) can be premultiplied

by C- 1

and Gerschgorin 's theorem [6] can be app l ied to obtain a

meaningful bound on X„ . T h is bound can t h e n be substi tuted into

equation (18) to obtain equation (19). When C is not diagonal, com

p u t a t i o n of C_ 1

is to be avoided due to the extensive computations

a nd the add i t iona l compute r s to rage requ i remen ts .

The firs t s tep in ob ta in ing equa t ion (20) is to define a diagonal

m a t r i x D as the diagonal of the ma t r ix C. M a t r i x D ca n be introduced

into equation (28) to give

(D -l'2SD -l'2)D l'2x = A t D - ^ C D - ^ D ^ x (29)

T h is is a new eigenproblem with the sam e eigenvalues as equation (28)

and eigenvectors D1/2

x instead of x. The parenthesized matrices are

symmetr ic .T h e r e is a relatively well-known theorem [6] t h a t can be applied

to equation (29) to give

A„ £ ~- (30)

Pi

where K„ is the maxim um e igenva lue of

( D - ^ S D - ^ V = «u' (31)

and /3i is the min im um e igenva lue of

( D - ^ C D - ^ V = pV (32)

Premul t ip ly ing equa t ion (31) by D ~1/ 2

and in t roduc ing u = d~1/2

u'

gives

D-

x

Su = «u (33)

The Gerschgorin theorem [6] may be applied to d_1

s to give an upper

bound on K„ as

Kn s M ax j^ i - ( Sil. + £ |S ij | ) J (34)

Premul t ip ly ing equa t ion (32) by D1'2

and in t roduc ing v = d_ 1

'V

gives

Cv = |3Dv (35)

N e x t a p a r a m e t e r q will be in t roduced by sub t rac t ing 6>Dv from both

sides of equation (35) to give

( C - q D ) v = (p ' -G ' )Dv

It then follows [6] t h a t

01-q

ft £ <? +

min im um e igenva lue of (C — qD)

maximum e igenva lue of D

min imum e igenva lue of (C — qD)

maximum e igenva lue of D(36)

T he va lue ofq will be chosen as the largest value for which the mini

mum eigenvalue of (C - qO ) can be guaran teed to be non-negative.

The matrix (C - <jD) can be constructed as a su m of contributions,

(C (e )- <7D

<s)), from each finite elem ent. For trian gular finite elements,

126 / VOL 100, FEBRUARY 1978Transactions of the ASME




each (C(e )

- <?D(e)

) will have re - 3 eigenv alues th at ar e 0. Th e re

maining 3 eigenvalues will be equal to the eigenvalues of the ma

trix

pc A

12

2 - 2 ( 7 1 1

1 2 - 2 ( 7 1

1 1 2 - 2 q

(37)

These eigenvalues are pcA (l - 2q)/12, pcA(l — 2q)l\2, a n d p c A ( 4 —

2</)/12. The largest value of q that will ensure non-negative eigen

values for (C<e

> - (7D(c)

) is q = 1/2. Since, for this va lue of q, (C - qD )is the sum of matrices w ith non-neg ative eigenvalues, (C — qD ) will

also have only non- negati ve eigenvalu es [6]. Hen ce equ ation (36) will

give

1(38)

, Subs tituti on of equ atio ns (34) and (38) into equ atio n (30) will give

a bound on \n which can then be substi tuted into equation (18) to

yield equation (20).

The de velopme nt of equation (24) for quadrila teral f inite e lemen ts

is the same as for triangular elements. The only difference is that the

matr ix (37) is replaced by

pcA

36

4 - 4 q 2 1 2

2 4 - 4 ( ? 2 1

1 2 4 - 4 q 2

2 1 2 4 - 4 q

From which q = 1/4 and equation (24) will result.

Furt her details of the analysis presen ted in this paper can be foundin a 180-page repo rt by My ers [5]. Also discussed in this rep ort is a

modified form of equatio n (20) which can sometim es give better es

t imates of the cri t ical t ime step. Discussions are also presented to

explain the effects of heat-transfer boundary conditions upon the

eigenvalues and hence upon the cri t ical t ime step. This report may

be obtained by writ ing to: Inform ational Resources Office, Engi

neering Experiment Station, University of Wisconsin—Madison, 1500

Johnson Drive, Madison, Wisconsin 53706.

\ Journal of H eat T ransfer FEBRUARY 1978, VOL 100 / 127




A. Wlaewal

Postdoctoral Fel low,

Department of Applied Mechanics

and Engineering Sc ience,

University of Cali fornia, San Diego,

La Jol la, Cali l .

G. A. Gurtman

Staff S cientist,

Systems, Science and Software,

La Jol la, Cali f.

G. A. Hegemier

Professor,

Department of Applied Mechanics

and Engineering Science ,

University of C ali fornia, San Diego,

La Jol la, Cali f.

A Mixture Theory for Quasi-One-

Dimensional Diffusion in Fiber-

Reinforced Composites1

A binary mixture theory is developed for heat transfer in unidirectional fibrous compos

ites with periodic, hexagonal microstructure. The case treated concerns a class of prob

lems for which heat conduction occurs primarily in the fiber direction. Model constructionis based upon an asymptotic technique wherein the ratio of transverse-to-longitudinal

thermal diffusion times is assumed to be small. The resulting theory contains information

on the distribution of temperature and heat flux in individual components. Mixture accu

racy is estimated by comparing transient solutions of the mixture equations with finite

difference solutions of the Diffusion Equation for an initial boundary value problem. Ex

cellent correlation between "exact" and mixture solutions is observed. The construction

procedures utilized herein are immediately applicable to other diffusion problems— in

particular, moisture diffusion.


In this paper a continuum mixture theory for heat conduction

in a fiber-reinforced composite is presented . As in the earlier work

on diffusion in laminated composites [1 , 2] ,2

the technique used for

construction of the continuum model follows an asymptotic method

introdu ced by Hegem ier [3] and successfully applied to the solution

of wave-propagation problems [3-5] . Th e essential feature of the as

ymptotic technique is that i t re tains the heterogeneous character of

the material , and yields, to a desired degree of approximation, in

formation on temperature distr ibution in individual components of

the composite .

It is, of course, possible to state the balance laws for composites

directly in the form of a mixture theory by introducing partia l heat

flux quantit ies and an interaction term within the framework of

general contin uum theories of mixtures , as , for example, in [6]. Use

of such an approach, however, leads to consti tutive equations that

involve mixture thermal properties which cannot be analytically determined even if the f iber and matrix are arranged in a periodic

mann er and th eir properties are known. On the other hand, use of the

asymptotic technique described in [3-5] leads directly to a continuum

theory in a mixture form involving composite material properties

which can be determined from a knowledge of the geometrical ar-

1 Research was sponsored by Air Force Office of Scientific R esearch.2 Numbers in brackets designate References at end of paper.Contrib uted by the Heat Transfer Division for publication in the JOURNAL

OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division April4,1977.

rangement and material properties of the individual consti tuents .

In the following, a mi xtu re the ory is constr uct ed for the case in

which heat co nduction occurs prim arily in a direction paralle l to the

fiber axis. An initial boundary value problem is solved, and results

are compared using the mixture theory and a f inite difference code

solution of the Diffusion Equa tion.

A study of the problem class treat ed in this pap er is important in

many thermal protection applications, such as re-entry vehicle heat

shields and nosetips in which the primary direction of heat flow is the

fiber axis . Moreover, a l though the problem s studi ed here can be di

rectly solved by using numerical techniques, the reduction to one-

dimensional problems leads to greater computational efficiency. This

advan tage becom es even more obvious when it is desired to analyze

materials with temperature dependent properties . A study of such

nonlinear effects will be the subject of a forthcoming paper .

Although th e ensuing deve lopme nt is carried thro ugh for heat

conduction problem s, i t is equally applicable to other diffusion phe

nomena which have recently attracted much attention, e.g. diffusionof moistu re in composites [7 , 8] .

F o r m u l a t i o n

Consider a periodic hexagonal array of circular cylindrical fibers

(Con stituen t 1) emb edded in a matrix (Co nstitue nt 2) as shown in Fig-

1. With respect to a polar cylindrical coordinate system r, 1) , x, let the

composite occupy the domain 0<x<(,0<r<o> to<8<2ir. We

assume tha t the tem peratu re or heat f lux bounda ry conditions on the

boundar ie s a t x = 0, ( are such that the temperature distr ibution is

similar in each hexagonal cell. As a consequence of this, the external

boun dary of the unit cell becomes a l ine of symm etry, and the com

pone nt of the heat f lux norm al to the hexagonal b ound ary vanishes-

128 / VO L 100, FEBRUAR Y 1978 T rans action s of the ASWIE




Fig. 1 Geometry and coordinate system

We further a ssume t ha t there is no thermal resistance across the bon d

between the f iber and the matrix .

To describe heat conduction in a typical unit cell , the hexagonal

boundary is appro xim ated in the following deve lopm ent by a circle ,

as in the previous investigations of wave propagation [4,5]. As a result

of this appro xim ation th e tempe ratu re distr ibutio n within the cell

is axisymm etric , with zero heat f lux norm al to the outer c ircular

boundary of the matrix . Using the notation d t( ) = 3( )/dt et c., the

init ia l boundary value problem t hat describes heat conduction in the

cell is given by the following set of equations:

(<j) Con serva tion of Ener gy

- - d- r{fqf{a)

) - <3xfe (a )) = jZ<«>oVT<«>r

(b ) Cons t i tu t ive E qua t ions

(c) Interface Condition

TW(x, TL i) = T<2>(x, n, t)

qTM(x,ri,t) = q-r

(2

Kx,h,t)(d ) Symmetry Cond i t ion

q-rm

(x, f2, t) = 0

(1)

(2 )

(3a)

(3b)

(4 )

(e) Init ia l and boun dary data .

T he quan t i t ie s q-p(<,)

, qjM

, M'"', T'™', k<o)in foregoing are, respec

tively, the heat f lux vector components , heat capacity , temperature,

and thermal conductivity of the a (=1 or 2) consti tuent.

N o n d i m e n s i o n a l i z a t i o n

To facil i ta te the analysis , the basic equations (1-4) are placed in

a nond imens iona l fo rm suggested by a subsequen t a ssum pt ion tha t

a typ ica l "macrod imens ion" ( is much larger than a "microdimen-

sion", say ?2 . In t roduce the nond imens iona l quan t i t ie s

t = f-z/C, x = x/(, r = F/F2

TM^TM/T0,q x

M

^qjM/q 0

«9;(a ) = TfJ ,q-r(a) lqo, <7o = kT a/e (5 )

- - - &<«> Mt = t/to, to = lit2Ik, ka = ~— ,n„ = —

k M

(«)

where k, ix denote mixture conductivity and heat capacity to be de

fined la ter . I t is noted h ere th at (5) can be used to define the macr o-

dimension (in term s of characteris tic time to. Th e characteristic time

to is a lso related to observation t imes in the sense that , following the

application of a tempe rature or heat pulse to a composite, the m ixture

theory shall be able to yield meaningful results only after a t ime in

terval not too small com pared to to- Similarly , the theory shall y ield

reliable results only at observation stations whose distances from the

boundar ie s x = 0 or £ are not too small compared to t. T hus , in the

absence of any explicit choice of time scale to or length scale I in a

given application, one of these quantit ies can always be selected soas to reflect the fineness of the time or length scales to which the

resolution of the temperature history is sought. In summary, therefore,

a f inite , nonzero macrodimension I for the purpose of scaling can al

ways be chosen to be one of the following: (i) actual length of the

composite in the axial direction, ( i i) a length associated with a t ime

charac teris t ic of the in put pulse [from t 0 = n(2/k,cf. (5)], (hi) an ob

servation length and (iv) a length associated with an observation time

scale .

Using (5), the basic equatio ns (1-4) ma y be writ ten as

(a) Conserv ation of Energy

- dr(rqrM) ~ dx(<lx

{a)) = HadtTM

(b ) Cons t i tu t ive E qua t ion

(c) Interface Condition s

•k a& x ,d r)TM

T « ( i , r i , t) = T<2>(.T, n , t)

q rW(x,r1,t) = q rM(x,r l,t)

(6 )

(7 )

(8a)

(8b)


A(1 )

, J 4(2 )

= functions of x and t only

B(l

', B <2' = functions of x and t only

C'1', C

(2 )= functions of x and t only

h = an arbitrary function denoting the de

pendent variables

ft(2n) = n t h order term in the expansion of h

ka = thermal conductivity of a cons t i tuen t

ka = nond imens iona l the rma l conduc t iv i ty

of a cons t i tuen t

k = thermal conductivity of the mixture in

th e x direction

I = typical macrodimension

na = volume fraction of a cons t i tuen t

P = in te rac t ion te rm

qx («) = components of heat flux vector

in the a cons t i tuen t

fr'"'. <7x(a) = components of nondimensional

heat f lux vector in the a cons t i tuen t

Qx average heat flux in a cons t i tuen t

,("' = zeroth order terms in the? r ( 0 )w

, <7x<0)'

:pansion

<7o=

reference heat f lux

Q* = a mix ture consta nt, equation (34)

f = radial coordinate

r = nond imens iona l rad ia l coo rd ina te

f i = radius of the fiber cross-section

? 2 = radius of the equivalent cell

ri = nondimensional radius of the f iber

T*" ' = temperature of the a consti tuent

f <"» = S ee equ ati on (29)

T < a) = nondimensional temperature of the a

cons t i tuen t

f(aa) — a v e r a g e t empera tu re in a cons t i tu

en t

T o = re fe rence tem pera tu re

T(2n)(<<) = n

thorder term in expansion for

Ji(a)

T*"'* = see equation (30)

t = t ime

t = nond imens iona l t ime

to = characteris t ic t ime

x = axial coordinate

x = nondimensional axial coordinate

= 1 for fiber

= 2 for matrix

fil, fc = mixture constants , equation (37)

t = nond imens iona l sma l l pa ramete r

0 = angular coordinate

T = in te r face tempera tu re

jj(«) = specific heat per unit volume of a

cons t i tuen t

/»'"' = nondimensional specific heat of a

cons t i tuen t

Ii = mixture specific heat

Journal of Heat T ransfer FEBRUARY 1978, VO L 100 / 129




(d ) Symmetry Cond i t ion

q rm(x, rh t) = 0 (9)

A s y m p t o t i c E x p a n s i o n s

T he quan t i t ie s ix?i2/k an d n(

2/k represent, respectively, charac

teris t ic thermal diffusion t imes in the transverse and longitudinal

(fiber axis) directions. The ratio of these diffusion times is ? 2 2 /^ 2 =

e2. In wha t follows, it is assum ed t ha t th is ra tio is small compared to

un i ty .

T he p remise tha t t2

« 1 suggests the following regular a sym ptoti c

expansion for a ll dependent variables:

h(x,r,t,t)= Y. t2nh i2n)(x,r,t)n= 0

(10)

If (10) is substituted into (6) through (9), an initial boundary value

problem is obtained for each order of t2. The lowest order system is

obtained by sett ing e = 0 and placing the subsc ript (0) on all depen

dent variables in (6) through (9).

I t might be pointed out that implicit in the use of the asymptotic

expansion procedure outl ine d here is the assum ption th at the non-

d imens iona l cons t i tuen t p rope r t ie s k„ and na that appear in (6) and

(7) are of order unity. This condition can easily be satisfied for com

posites with achievable volume fractions t hroug h a proper choice of

mix tu re p rope r t ie s k an d n used for scaling in (5). This is done in a

subsequent section by using the definitions (40) for mixture thermal

conductivity and heat capacity .

M i x t u r e T h e o r y

We wish to construct a mixture theory that will , as a minimum,

incorporate the effect of microstructure to a predetermined degree

of approximation. We begin by defining averaged quantit ies such

t h a t

)(1

ni Jo

( )(2a) = i - f 12r( )<2>dr

ra 2 Jr i

(11a)

( l i b )

where ri\, n<i are the volume fractions of the two cons ti tuen ts , i .e. ,

ni = n2

, ri2 = 1 - rii (12)

Equa tion (6) is now averaged acco rding to (11), so that on using the

interface condition, equation (8b), we obtain

« i [ - < Wl a )

- M A T '1" '] = P (13a)

M 2hd Ig<2a )

-M2C) (T<2a

»] = ~ P (13b)

where P is the so-called interaction term defined by

P = 2riqr^(x, r lt t) (14)

The primary object of the subsequent development in this section is

to calculate the interaction term P in terms of T ( l a ' and T ( 2 a ) to "close"

(13a) and (13b). As in [3, 4] the calculation is based on t he two low est

order systems.

Th e firs t order system is , in part ,

- M * ( 0 ) W - - dr(rq mM) = fiadtTm^\ (IS)r

<9 rT (0)(«) = 0, (16)

qx„»<«> = ~k adx$ (17)

From equations (16) and (17), i t can be concluded that both 7 I(o) <a)

an d qx(o) (a ) are independent of the radial coordinate; thus, equation

(15) becomes

" dr(rq rW )ia) ) = -2A<«>(*, t) (18)

where A*"' is a function yet to be determined. Integration of (18)

yields

130 / VO L 100, FEBRUARY 1978

9r (0 )(a ,

= - [ i lf a

-Kx,t)r +BM(x,t)~

(19)

Equa tion (19) is now subs ti tuted into the consti tutiv e equation for

<7r(0) i.e. , in

9r(0)<") = ~kadrTmM

(20)

From (20) the following expression for the temp erat ure distr ibution

T ,2('<) is obtained in terms of arbitra ry functions of x an d t only:

r2

kaT(2)M = A<«>(*, t) — + B^Hx, t)em + C<«>(x, t) (21)

T he cond i t ion tha t tem pera tu re be f in i te a t r = 0 leads to

B<i> = o (22)

Use of the symmetry condition, equation (9), furnishes

5<2 ) = . 4 ( 2 ) ( 2 3 )

Henc e, the radial hea t f lux distr ibu tion in the cell is given by

q r (0 )(i) = -AW(x, t)r (24a)

qr(0)(2» = - A <

2> [ r - i ] (24b)

The second order temperature field in each constituent is, therefore,

given by

kxT(2)w = A<»(x, t) — + C™(x, t)

2r(2 ,< 2> = A<2>(.x, t ) \ - - £nr] + C<2>(*, 0

(25a)

(25b)

The functions AMare related to the interaction term P as follows:

from (14) and the interface condition (8b) one has

P = 2n qr^(x, n, t) = 2n qr<2Hx, n, t)

= 2n< ?r (o)<1 )

( x, n , t ) + 0 ( e2

)

= 2 r 1 ( 7r (o)(2 )

0c , r1 , t ) + 0(62)

If (24a) and (24b) are now substi tuted into (26), one obtains

J L2nx'

P_

2n2

Finally , use of (27) and (25) leads to the te mp erat ure distr ibution

T<» = f (D + e2TU>V 2 + 0 (e 4 ) (28a)

T<2> = f (2) + (2Tm* (r2 _ (n r2) + o (e4) (28b)

A «) :

(26)

(27a)

(27b)

where

f <«> = T(0)<«> + e2C<«>U, t)

P PT(2)* = -

(29)

(30)4n 2& 2' Anofci

To obtain a minimal mixture theory that includes the effect of mi

crostructure, i t is necessary to satisfy the continuity of temperature

at the interface up to and including terms of 0(e 2 ) , thus

f (1) + 6 2 T ( l ) V l 2 = f (2) + 6 2 T ( 2 ) . ( r i 2 _ (n rj 2 )

= r(x, t, t), (31)

where r is the interface temperature. Using equation (31), the tem

perature distr ibution given in (28) can be writ ten in the form

T< » = T + e2T W*(r2

- m ) (32a)

T(2) = T + £2T(2). L.2_ nx_ enLL \ (32b)

If (32) is now averaged accord ing to (11), and if (30) is used to elimi-

Transact ions of the ASME




nate T(a)

*, one obtains

T(U ) = T + e2

f (2a ) . _ f 2

_P _

Ski'

p „ .

8&2

where

2 T3 rti21

n 22L 2 2 J

(33a)

(33b)

(34)

prom (33) both the interac tion term P and the interface temperaturer can be written in terms of average tempe ratures, thus

f ( l a ) _ f (2a )

T = - l (y( la) + y(2a)) + B2

(y(la) _ y(2a))

2 I ft

where the constants ft and 02 are given by

8 U i f e2J

At'8 L/e2 fej

(35)

(36)

(37a)

(37b)

The foregoing analysis completes the construction of a binary

mixture theory for the composite. The basic equa tions of the theoryare the following:

' i i | - < W l a > - Mi<3(T<i<>>j = P,

T 2 | - W2

* > - M2rJT<2<>>) = - P ,

y(la) _ y(2a)

<2A

<J x(aa)

= - t A T " ,

(13a)

(13b)

(35)

(38)

where the last equation follows directly from (7) and (11). To be appended to the above set are appropriate boundary and initial data.

It still remains to obtain expressions for mixture conductivity andspecific heat to be used in (5). To do so, all dependent variables exceptyda) a r e eliminated from the mixture equations to obtain

\imkr + n2k2)dx2

- (mm + n2li2)dt + 0(€ 2)|T< la> = 0 (39)

Based on (39), the following definitions are introduce d for the m ixtureproperties:

k = mfe*1' + n2£<

2>

fi = rti/i(1 ' + W2M (2)

(40a)

(40b)

With the definition (40), and in the limit as e —• 0, the mixture theoryreduces to an elementary theory for heat conduction in a homogeneousmaterial.

N u m e r i c a l R e s u l t s

To test th e ability of the proposed theory to model diffusion in a

composite, calculations have been conducted to determine the evo

lution of the tem pera ture field in quiescent half-space x > 0 subjectto the boundary condition

T(0,̂ )= (\°f?^ (41)

I 0, t0 < tThis boundary value problem has been solved by using (i) a fi

nite-difference scheme for equations (13,35 and 38) and (ii) a finite-difference schem e for equa tions (1 -4). The finite-difference solutionof equations (1-4)—the 'exact' solution—has been used as the correlating norm for estimating the accuracy of mixture theory solutions.

The m aterial prop erties used for calculations are given in Table 1.Computations were carried o ut for (a)e2

= .111, (6)t2

= .25 and (c)«2

=1. Since there is no intrinsic axial length scale t in the problem, the

pulse duration t"0 can be used for scaling in (5). Thus, with the samese t of material and geometrical properties, an increase in i

2corre

sponds to a decrease in the pulse duration time and vice versa.

Figs. 2-4 illustrate the average temperature profiles in the twocons tituents after a short time following the term ination of the temperature pulse. In Figs. 5 an d 6, the evolution of average tem peratureis depicted at a given distance from the bounda ry. From these figuresit can be seen that the agreement between the mixture theory resultsand the exact solutions is excellent for cases (a) and (fa). In case (c),e

2has been chosen t o be unity, c learly in violation of the basic premise

that this parameter is much smaller than one. Thus, significant discrepancy is to be expected between mixture theory predictions andthe exact solution, especially near a boundary or at times imm ediatelyfollowing the application of a temperature pulse at the boundary.

Table I Material Properties Used for Computations

Thermal Conductivity Ratiok

(11e„_ = 5 0

Specific Heat Ratio

Volume Fractions

£<1>

• 0. 5

n, = .2 (Fiber), n2 = .8 (Matrix)

T 0. 5

r( la)

,(2o)

o o o T

A A A T

Mixture Theory

Exact

Fig. 2 Profile of average temperatures at f = 1,2 (c2

= 0.111)

T 0-5

T (lo)

,(2a)

O O O T

A A A T'

Mixture Theory

Exact

Fig. 3 Profile of average temperatures at / = 1.2 (e 2= 0.25)



http://imkr/

http://imkr/



T Cdo i 1( 2 „ ) \ M i x t u r e T h e o r y _

T IU)1

T (M JEMCt

' ^

J&*^ 1

-JZ^

1

\ XL

1

- l ^ -

t

T£>zr

2.0

F i g . 4 Profile of average temperatures at t = 1.2 (e2 = 1.0)

i

F i g . 6 Evolution of average tempwatuf* at M = f.5 («2 = 1 0 )

0. 4

0

-

-

-

/1/^i /

/

I

^ x r -

1 !

, -° " '

1

.A

T H o ) ,

T (2ol

A A A Tt 2 0 )

i ! 1

- - c r ~ "

M

E

xtur

j

- - o ^ _ 2 j

3Theo

I i

\ \\\9\

y

!

\"*

XX-,

I

F i g . 5 Evolution of average temperature at x = 0.5 (f 2 = 0.111)

T 0 - 3

Mixture Theory

O O O E x act

0.25 0-50

r

F i g . 7 Tem perature mlcrostracture at x = O.S, t = 1.2 (e2

= 0.111)

Thus this case is a very str ingent test of our theory. However, even

in this case, the mixture theory predictions become more accurate

either a t d is tances far from the bou ndary, or a t long t imes com pared

to the in put pu lse dura tion, as can be seen in Figs. 4 and 6. The l imi

tation of the theory for large values of e2 arises mainly from th e ou ter

nature of the solutions that i t y ields. However, the l imitation is not

a very serious one, s ince in most app lications a typical macr odim en-

sion is indeed much larger than the cell d imension.

T he ab i l i ty of the mix tu re theo ry to p red ic t the temp era tu re mi-

crostru cture is i l lustrated in Figs. 7 and 8. Th e radial distr ibutio n of

temperature obtained from the mixture theory is a lmost identical to

the exact solution. This feature of the theory prop osed here is absen t

from most other continuum models of heat conduction in composites,

e.g. [9].

T 0 .3

o -o~< >

Mixtu re Theory

O O O E x ac t

0.50

r

F i g . 8 Temperature microsirueture at x = 0.5, t = 1.2 (e2 = 0.25)


A mixture the ory has been constru cted for diffusion in a f iber-re

inforced composite . The significant feature of the theory is that i tconverts what is essentially a three-dimensional problem to a problem

involving a single spatial variable, without losing the essential details

of local temperature distr ibution. Even if the mixture theory equa

tions are solved numerically, their solution is much more economical

than the use of a direct numerical s trategy for the original problem.

For example, for the problems treated in the last section, the solution

of mixture equations was about f if ty t imes faster than the solution

of the original system. This computational efficiency is, of course, the

result of the reduction in the number of spatia l d imensions in the

mix tu re theo ry equa t ions . T he exce l len t ag reemen t be tween the

mixture theory predictions and the exact solution of a boundary value

problem indicates that the model proposed here can be effectively

used for other problems of interest .


The authors wish to thank Dr. Kay Lie of Systems, Science and

Software for performing the "exa ct" two-dim ensional finite-differencecalculations.

R e f e r e n c e s

1 Maewal, A., Bache, T. C , and Hegemier, G . A., "A Co ntinuum Model forDiffusion in Lam inate d Com posite M edia ," ASME JOURNAL OF HEATTRANSFER, Vol. 98, p. 133,1976.

2 Nayfeh, A. H., "A Continuum M ixture Theory of Heat Conduction inLaminated Composites," Journal Appl. Mech., Vol. 42, p. 399,1975.

3 Hegemier, G . A . , "Finite Amplitude Elastic Wave Propagation in Laminated Composites," Journal Appl. Phys., Vol. 4 5 , p. 4248,1974.

4 Hegemier, G . A . and Gurtman, G . A . , "Finite-Amplitude Elastic-PlasticWave Propagation in Fiber-Reinforced Composited," Journal Appl. W>>'s-Vol. 45, p. 4254,1974.

5 Hegemier, G. A., Gurtman, G. A„ and N ayfeh, A. H„ "A Continuum





Mixture Theory of Wave Propagation in Laminated and Fiber-ReinforcedComposites," Int. J. Solids and Struct., Vol. 9, p. 395 ,1973.

6 Green, A. E. and Naghdi, P. M., "A Theory of Mixtures ," Arch. Rati,yiech. Anal., Vol. 24, p. 243,1967.

7 Springer, G. S. and Shen, C. H., "Moistu re Absorption and Desorption0f Composite Materials," Air Force Materials Laboratory, Wright-Patterson

Air Force B ase, Ohio, 1976.8 Augl, J. M. and B erger, A.M ., "The Effect of Moisture on Carbon Fiber

Reinforced Epoxy Composites I Diffusion," Naval Surface Weapons Center,White Oak, Silver Spring, Md., 1976.

9 Ben-Amoz, M. "He at Conduction Theory for Composite Mate rials," J.Appl. Math. Phys. ZAMP, Vol. 27, p. 335,1976.

Journal of Heat T ransfer FEBRUARY 1978, VOL 100 / 133Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



Exper imental Heat Transfer

Behawior o f a Turbulent Boundary

Layer on a Rough Sur face Wi th

BlowingHeat transfer measurements were made with a turbulent boundary layer on a rough, per

meable plate with and without blowing. The plate wa s an idealization of sand-grain

roughness, comprised of 1.25 mm spherical elements arranged in a most-dense array with

their crests coplanar. Five velocities were tested, between 9.6 and 73 mis, and five values

of the blowing fraction, v0/u„, up to 0.008. These conditions were expected to produce

values of the roughness Reynolds number (ReT = uTks/v) in the "transitional" and "fully

rough" regimes (5 < ReT < 70 , Rer > 70).

With no blowing, the measured Stanton numbers were substantially independent uf

velocity everywhere downstream of transition. The data lay within ± 7 percent of the

mean for all velocities even though the roughness Reynolds number became as low as 14.

It is not possible to determine from the heat transfer data alone whether the boundary

layer was in the fully rough state down to Re = 14, or whether the Stanton number in the

transitionally rough state is simply less than 7 percent different from the fully rough

value for this roughness geometry.

The following em pirical equations describe the data from the present experiments forno blowing:

C, /(K-o.25-1 = 0.0036 (-)

St = 0.0034 l-j

In these equations, r is the radius of the spherical elem ents comprising the surface, (I is

the momentum thickness, and A is the enthalpy thickness of the boundary layer.

Blowing through the rough surface reduced the Stanton number and also the roughness

Reynolds number. The Stanton number appears to have remained independent of free

stream velocity even at high blowing; but experimental uncertainty (estimated to be

±0.0 00 1 Stanton number units) ma kes it difficult to be certain. Roughness Reynolds num

bers as low as nine were achieved.

A correlating equation previously found useful for smooth walls with blowing was found

to be applicable, with interpretation, to the rough wall case as well:

SM r/so + B n -St0 IA L B J

Here, St is the value of Stanton number with blowing at a particular value of A (the en

thalpy thickness). Sto is the value of Stanton number without blowing at the same enthal

py thickness. The symbol B denotes the blowing parameter, VQ/U„ St. The comparison

must be made at constant A for rough walls, while for smooth walls it must be made at con

stant Re&.

Contrib uted by the He at Transfer Division for publication in the JOURNALOP HEAT TRANSFER. Manuscript received by the Heat Transfer DivisionMarch 31,1977.

R. J . Mof fa t

P r o f esso r o f M echan i ca l Eng i nee r i ng ,

S t an f o r d U n i v e r s i t y ,

S t a n f o r d , Cal i f .

J. M . H e a l z e r

M anager , C on t a i nm en t M e t hods ,

B o i l in g W a t e r R e a c t o r S y s te m s D e p a r t m e n t ,

G e n e r a l E l e c t r ic C o m p a n y ,

Sa n Jo s e , Cal i f .

W . M . K a y sP r o f esso r o f M e c h a n i c a l Eng i nee r i ng ,

S t an f o r d U n i v e r s i t y ,

S t a n f o r d , Cal i f .

134 / VOL 100, FEB RUAR Y 1978 T r a n s a c t i o n s o f t h e A S M ECopyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



IThe rough plate w ith blowing undergoes transition from a laminar to a turbulent

boundary layer at approximately the same momentum thickness Reynolds numb er as

does a smooth plate without blowing.


There is considerable interest in protecting surfaces from high

tempera tu re env i ronmen ts by us ing transp i ra t ion o r ab la t ion . T heablating surface, in particu lar, frequently becom es mechanically rough

during operation. I t is important, from the designer 's s tandpoint, to

be able to predict th e effects of roughness an d blowing, in comb ina

tion, on the heat transfer and friction behavior of the boundary layer.

In addition, the co mbin ation of blowing and roughness offers a good

opportunity to investig ate some old concepts in a new situation , an d

thereby challenge our understanding.

Much of the present knowledge about rough surface hydrodynamics

is based on ideas and results of the pipe-flow exp erim ents by Nik u-

radse [l]1, who investigated flow through sand-grain-roughened tubes.

The roughness elements for his experiments consisted of selected

sand, carefully s ieved and attached in maximum density to the tube

walls. The roughness was described by a single parameter, hs, the size

of the sand-grain ele ments . Th e sand-grain m easure of roughnes s

became standard in skin fr ic tion studies; and i t is s t i l l common

practice to express the effects of an arb itrary roughness in term s of

an "equiv alent sand-gr ain roughness, ks." E quiva len t sand-g ra in

roughness is defined as that san d-grain surface which would have had

the same effect upon t he f low as did the specim en surface.

Nikuradse 's results showed three domains of behavior in terms of

the "roughness Reynolds num ber," Re T = uTks/p . Fo r values less tha n

five, the flow behaved as thou gh the surface were smooth. F or value s

greater tha n 70, the fr ic tion factor became ind epen den t of the p ipe

Reynolds number: a s ta te described as "fully rough ." The region be

tween five and 70 defined a region of "tran sitiona l roughnes s." Pr an dtl

and Schlichting [2] and von Karman [3] used the Nikuradse results

(in 1934) to predict t he friction factor behavior of boun dary lay er flows

over rough surfaces. These predictions in dicated tha t a fully rough

state of the bounda ry layer would be atta ined such that th e fr ic tion

factor would be a function only of x/ ks. These characterizations of

boundary layer flows are still used, with the same values of roughnessReynolds number.

Two types of roughness are now identif ied in the l i te ratur e: k a nd

d-type. Nikuradse 's c lassical sand-grain roughness is f t- type. Two-

dimensional roughness elements (ribs, rods, or grooves perpendicular

to the f low) are referred to as "d-t ype. " The two types have different

characteris t ics though different authors have different opinions as

to what consti t utes th e defining property . Th e presen t work is con

cerned with a k - type roughness (an idealization of sand-grain be

havior).


There have been several recent s tudies of the fr ic tion behavior of

cf-type roughness mad e by the addi tion of a regular array of transv erse

rods to a f la t surface. Moore [4] s tudi ed a turbu lent b ound ary layer

in air flowing over such an array, using a fixed ratio betwe en pitch andheight. His data showed some evidence of fully rough behavior.

Shortly thereafter, Liu [5] put a similar surface into a water table with

a variable aspect ra tio . Liu reported t hat the data could not be s imply

organized by an x -Reynolds number function. Perry and Joubert , e t

al . [6,7] studied a similar rough surface in an adverse pressure g radien t

and reported no effect on the fr ic tion behavior due to the pressure

g rad ien t . T h is same s tudy a l so a t tempted to s imu la te a change in

roughness by changing the spacing between elements without

changing their s ize . The results did not correspond as had been ex

pected. I t is the present consensus that c losely spaced, regular ,

p r i sma t ic roughness e lemen ts behave d i f fe ren t ly f rom sand

grains.

Grass [8] used a bubble technique to measure the instantaneous

velocity distr ibutions in a water tunn el above a sand-grain roughness

and described the turbulent s tructure near the wall . Wu [9] used afloating element balance to measure skin friction in an air tunnel and

showed fully rough behavior with good agreement between the mea

sured values and the Prandtl-Schlichting predictions. Most recently ,

Tsuji a nd Iida [10] showed th at th e velocity profiles observed above

a rough wall could be reasonably well predicted by a modified m ix

ing-length approach, maintaining a non-zero value at the wall . The

effect of an abrupt change from a smooth to a rough surface has been

investigated by Anton ia and Lu xton [11], who detailed the velocity

distr ibutions. Town es, e t a l . [12,13] have studied the s tructu re of the

turbulence in f low through a rough pipe.

Much less has been done in the field of heat transfer than has been

done for fr ic tion. One of the f irs t systematic rough surface heat

transfer experiments was carried out by Nun ner [14]. His experiments ,

using air flowing through rough pipes, were used to establish a simple

empirical re lationship between the increase in Nusselt number due

to roughness and the increase in the skin fr ic tion. Several important

heat transfer s tudies followed, notably by Dipprey and S abersky [15],

Owen and Tho ms on [16], and Gowen and S mith [17], Dipp rey and

Sabersky studied pipe f lows with four different roughnesses, each

tested with four f luids of different Prandtl numbers. They showed

that th e rough wall heat transfer varied with the Prandt l num ber even

in the fully rough regime where th e molecular viscosity effects seem ed

unimportant. Owen and Thomson developed the idea of a "sublayer

S tant on num ber " to correlate rough pipe heat transfer data and to

accoun t fo r the P rand t l num ber dependence . Gowen and Smi th ex

tende d the Dipprey and S abersky pipe f low heat transfer s tud y to

higher Prandtl numbers and confirmed the effect on heat transfer .

The present s ta te of the art of rough wall heat transfer is well sum-


S = b lowing pa ram ete r , u„ / (u„S t )

C//2 = friction factor, g^Tolp^uJ1

c= specific heat a t constant pressure

F = blowing fraction, v0 /u „

Sc = gravitation al c onsta nt

G = free s trea m mass velocity , uap„

"=

a measure of roughness s ize

K = the size of sand grains of equivalent

roughness

"fl" = mass flux at the wall, p0v0

Re» = mom entum th ickness Reyno lds num

ber, u«,8h

R e^ = enthalpy thickness Reynolds number,

u „ A / c

R e r = roughness Reyno lds number , uTkslv

r = radius of spherical surface elements

S t = S t a n t o n n u m b e r , h/Gc

Tw = wal l tempera tu re

T ( = tem pera ture of trans pired f luid before ,

reaching the test p late from below

uT = shear velocity , u„VCf/2

u„ = free stream velocity

vo = blowing velocity at the surface

x = distance in the s treamwise direction

A = en tha lpy th ickness

0 = momentum th ickness

TO = surface shear s tress

p„ = free s tream density

v = kinematic viscosity

J o u rn a l o f H e a t T r a n s f e rFEB RUARY 1978, VOL 100 / 135




Fig. 1 Photograph of rough surface

Fig. 3 Cross section of typical compartment

DUMP

--S- ....

\ -

'HOT WIRE' .J 'FLOW METER

I. POROUS PLATE

2. HEATER WIRES

3. THERMOCOUPLES4. SUPPORT WEB5. TRANS, AIRTHERMOCOUPLE

6. HONEY COMB7. CASTING WATER PASSAGE

8, PRE-PLATE

9. TRANS, AIR DEFLECTOR

10. CASTINGII . CASTING THERMOCOUPLE12. TRANS. AIR DELIVERY

TUBE

WATER HOLDINGTANK

Fig. 2 Schematic of rough surface wind tunnel

, ; . . . . 4:\t ~ , ·,r. 9 ''''JJ ~ . .... ·.i,....;.... - •. .,. ." I • • .,.?'

. ., . '. <\, ·.t ~ . '... ' ' ' . j ~ \••::

. • ~ . . · f '\I,t' " 9 . t ~ ~ . ~ . · ' i ~ .•:.';',"' ( ' -, ." .' , ':": . .. ___ ~ ." · · f ~ ,,'. ~ , ~ , * .•, .• ., , j , ¥i:

. . . . . ~ • • " ' " . <•• .• ~ . . t.. ,"Ii,"-. ~ ~ ~ '$. "'sf "'. <t-J.. ''':to " J . " . t " ' ~

1" .. .-" • • . ,":.t . "If" tc. . . . . ~ . r , ~ . -:4 · ~ . · · . i I "J '¢, ,...... i,- . - .' .

. . . . I ~ , . ; * ' A " " ' * · . ~ \ , ...... ··<!·f \....

~ . ' f I " ... -...... f' of .. o!- f "."'!o; 4 ~ ~ . * , . , .• • . "_ '" . . ' ..' •

.... • • \ i > . ~ ~ . . . . " . ~ J "• •:J!,.. ," . A .• ~ . ." I , l:

' . ~ . . . ~ •. . • 4-.,y ~ " \."f lii:- .. " . s , · · ~ ·~ ' t # • ., •• •" '....., •• , ~ . : ~ ••• , ~ . ~ " f ' ~ ) ,. . .... , . .".. ' . ~ " ' ~ , : : : : : ' ,

Each working plate is supported by thin strips ofphenolic to thermally

isolate it from its neighbors and from the support casting. A sintered

bronze pre-plate distributes the transpiration air beneath the porous

surface plate. Plate heater wires are cemented into grooves on the

bottom ofthe working p l a t ~ s , using a high-viscosity epoxy to redUce

bleeding ofthe cement into the porous plate.The grooves were formed

by leaving out one row of balls at each line where a heater was de-

sired.

marized by Sood and Jonsson [18], Norris [19], and Lakshman and

Jayatilleke [20], who have collected the previous works in comparative

studies and also present experimental results of their own.

Prediction methods generally follow close behind data; and rough

wall heat transfer is no exception. Integral prediction schemes which

include roughness effects have been described by Dvorak [21,22] and

Chen [23]. Nestler [24] has proposed a scheme which related the in

crease in Stanton number due to roughness to the increase in skin

friction due to roughness and other boundary layer parameters. A

recent study by Lawn [25] describes predictions of both skin friction

and heat transfer in an annulus with its center cylinder roughened.

Finite difference turbulence boundary layer prediction schemeswhich

include roughness effects have been described by Lumsdaine, et al.

[26] and by McDonald and Fish [27]. Each of these efforts modified

the mixing-length distribution to introduce roughness effects.

In all of this large base of work, little has been done on boundary

layer heat transfer with roughness and nothing on the problem in

cluding blowing.The experimental objective of the present program

was to measure the heat transfer characteristics of a k-type rough

surface with and without blowing, over a wide range of conditions.

Velocity profiles and friction factor data were sought to complement

the heat transfer data and to facilitate development ofa prediction

program.

The Rough, Porous Surface

The surface selected for this study is shown in Fig. 1. It consists of

a regular array ofspherical elements, 1.25 mm in diameter, arrangedso that the crests of the spheres are all on the same plane. This ide

alization of h-type roughness is attractive because it is reproducible,

porous, deterministic, and requires only one geometric length scale

to describe it. It is more regular than sand grains, beinggeometrically

describable, and yet is more like a sand-grain surface than are the

"machinedelement" rough surfaces such as are used in other deter

ministic studies. An impermeable counterpart of the present surface

is included in the library ofsurfaces reported by Schlichting [33], thus

a baseline of"expected behavior" is available.

I t should be borne in mind that the regularity of size and arrange

ment of the elementsmay affect the interaction between the flow and

the surface and, indeed, there seems to be evidence of this in the

mixing length variations needed to predict the data from this sur

face.

The Experimental Apparatus. The apparatus constructed forthese tests is a closed-loop wind tunnel using air at essentially ambient

conditions as both the transpired and the free-stream fluid. The basic

design ofthe roughness rigwastaken from the existing heat and mass

transfer facility at Stanford described by Moffat and Kays [28].

Description of the Apparatus. The important features of the

roughness rig are shown schematically in Fig. 2. The mainair supply

is an 8.3 m:J/s fan connected through return ducting to a heat ex

changer. This is followed by a screen box to reduce mainstream ve

locity fluctuations, and then by a nozzle which delivers air to a test

section 0.1 m high and 0.5 m wide. The main stream is held at ambient

temperature plus or minus Vz°C to minimize heat transfer within the

nozzle. This keeps the enthalpy thickness nearzero at the beginning

of the rough surface. From the test section, the mainstream air exits

through a 14 to 1 multistage diffuser back into the fan. The main

stream velocity in the test section can be varied from 9mls

to over 70mls by changing the drive pulley and belting on the supply fan. When

the tunnel is operating without transpiration, a small charging blower

at the diffuser exit is used to maintain ambient pressure in the test

section. The transpiration air system is separate. I t delivers air

through a separate heat exchanger to a header box and then to the

plates which make up the test section through individual lines, each

with its own control and metering systems.

The test section itselfconsists of fixed side walls, an adjustable top,

and the rough-surfaced test plate assembly. The test plate assembly

contains 24 separate porous plates, which together form a 2.4 m long

test section. Each plate has its own transpiration air supply, instru

mentation, and embedded electrical heater for temperature control.

Across section ofa typical plate in the testsection is shown in Fig. 3.



http://trans/



Plate temperatures are measured with f ive thermocouples em

bedded in each plate . A cross-shaped pattern is used to check the

temperature distr ibution in the center of the plate . A sixth thermo

couple is provided for each plate to measure the temperature of the

transpiration air just before i t enters the plate .

The plates are m ount ed in groups of s ix, one group in each of four

aluminum castings. Alignment of the plates with each other and with

the casting was assured by clamping th e entire assem bly to a surface

plate during the assembly operation. The base castings have water

passages for temperature control and the casting temperatures are

monitored with thermocouples embedded in every second support

web. These temperatures are used in correcting for conductive heatloss from th e working p lates .

The porou s plates which form th e test surface were fabricated by

brazing together 1 .25 mm dia O.F .H.C. copper balls , s tacked in their

most dense array. Th e final dimen sions of each plate were 45.7 X 10.16

X 1.27 cm. The c oppe r balls , 3.8 kg per plat e, were suppli ed by a

bearing manufa cturer (dead soft copper ball bearings?) . To provide

a braze materia l , the balls were plated w ith 0 .0127 mm of electroless

nickel. They were then stacked, by hand, into rows inside copper

molds in their most dense array and fired in an inert a tmosphere

furnace to jus t above the meltin g temp era ture of the plating. Thi s

brazed the pack into a uniformly porous plate . This method of fab

rication, a lthough ted ious init ia l ly , provided a well-defined an d un i

formly porous plate . No subsequent machining was required.

Qua l i f ica t ion T es t s . T he appara tus was te s ted in de ta i l for re

liability before being approved for use. The tests can be summarizedas tests of the porou s plates , tests of the mainst ream condition , and

energy-balance tests of the data-reduction program and instrumen

tation system.

Hot-wire anem omete r surveys were made above the surface of the

porous plate with only the transpiration f low present to check the

uniformity of the plates . An unexpected phenomenon was found to

exist in the flow just above the surface of the plate. Above a critical

transpiration velocity, the flow formed a striated pattern of high and

low-velocity regions. I t was feared tha t these s tr ia t ions m ight affect

the heat transfer by increasing the mixing in the boundary layer. This

possibility was examine d by a special series of tests in which S tan ton

number variatio ns were recorded as a function of blowing (all o ther

tunnel para met ers be ing fixed) in a range of blowing values arou nd

the onset of the observed str ia tions. N o discernible disco ntinuity was

found—the data passed smoothly through the "crit ical blowing ve

locity" without any anomaly. It was concluded that the striations were

mainly a nuisance, in terfering with the documentation of plate per

meabili ty . No evidence was found of any variations in plate p erm e

ability.

The mean velocity in the mainstream of the tunnel was found to

be uniform within ±0.16 percen t a t 9 m/s and ±0.1 percent a t 70 m/s.

Free s tream turbulence level was 0 .4 percent a t a l l velocit ies .

Boundary layer momentum thicknesses were measured at f ive posi

t ions across the width of the tunnel, near the exit p lane, and were

uniform within 3 percen t maximum to m inim um.

Therm al qualif ication te sts were conducted to evaluate heat losses

from each test p late by conduction and radiation, and appropriate

correction algorithms developed for each of the 24 plates . These

correction algorithms allowed S tant on nu mb er values to be calculated

by energy balance, from th e power inp ut and tem pera ture dat a .

Any failure to close the energy balance und er qualif ication condi

t ions indicates a residual uncertainty in the values of S tanton num ber

which will be recorded from that plate . Such a residual cannot be

"subtracted out " unless one knows how the residual would vary as test

conditions vary. When t ha t is known, one has a correction algorithm

and the residual disappears by definition. Residual uncertainty in the

calculated value of S tant on num ber can be expressed in absolute uni ts

by normalizing the energy balance error using "average" condi

tions.

P la te Power — m"c(T 2 — T ( ) - (calculated losses)AS t = — (1)

GtypC(7'u, — T„,)typ

Losses were calculated using analytical m odels for the processes in

volved, but using coeffic ients evaluated from separately conducted,

special tests . After the correction algorithms had been developed,

energy balance tests were conducted full-scale and repeated several

t imes—in only four instances did even one plate out of the 24 show

an energy balance closure error greater than 0.0001 S tant on num ber

un i t s , normalized with average conditions.

E xp er im en t a l Re su l t s : No Blowing . All t e s t s repor ted here were

conducted with the free-stream temperature equal to the room tem

perature , and with the test p lates uniform in temperature at a level

10 -15°C above the f ree -s t ream temp era tu re .

The firs t tests were hydrodynamic to establish the characteris t ics

of the f low. Boundary layer mom entum thickness measurements weremade by integrating mean velocity profiles at several positions along

the tunnel length. The momentum thicknesses were least-squares f i t

in log coordinates; and t he de rivative of this f i t was used to f ind the

friction factor. As a check, this procedure was first applied to the

momentum thickness measurements by Simpson [29] and Andersen

[30] on smooth plate boundary layers . Skin fr ic tion coeffic ients de

termined this way compared well with the results reported by the

exper imen te rs—in bo th cases wi th in ±10 pe rcen t .

Mean velocity profiles for sand-grain roughened surfaces are de

scribed by S chlichting [33] in term s of the depression, Au+, of the log

region for the rough profile beneath his recommended correlation for

the smooth wall . The magnitude of Au+

can be used as a measur e of

the "equivalent sand-grain roughness" of a geometrically arbitrary

surface. Measurements of Au + from mean velocity profiles on the

present apparatus yielded an equivalent sand-grain roughness s izeof between 0.60 and 0.63 times th e diam eter of the spheres com prising

the surface. Schlichting qu otes 0 .625 for spherical e lements . Thu s the

behavior of the m ean profiles is as expected .

S chlichting defined th e fully rough sta te as one in which the friction

factor would be a function of "x " alone— not of free s tream velocity .

The data from the present apparatus showed this characteris t ic for

all velocit ies above 9.6 m/s , to within th e precision of the meas ure

ments of Cf/2.

Values of the fr ic tion factor from the present test l ie about 20

percent below the correlation recommended by Schlichting for the

fully rough state ( that correlation presumedly being consistent with

his tests of spherical e lements) .

No measu rements w ere made of the characteris tics of the turbulen t

structure. The regular , deterministic nature of the present surface

may have had an effect on the structure but, if so , the effect seems notto have been pronounced, s ince most of the features of "sand-grain"

surfaces have been reproduced.

Dat a presen ted in the p resen t paper will be given in terms of mo

mentu m thickness (and enthalpy thickness) ra ther than x in deference

to the current interest in local descriptors of boundary layer behavior.

I t can be shown that if Cf/2 is a function of x alone then, for a two-

dimensional, zero-pressure gradient f low, i t can be equally well de

scribed as a function of mom entu m th icknes s alone by means of the

momentum integral equation. This choice seems more l ikely to be

useful when blowing and acceleration are introduced into the prob

lem.

Pig. 4 shows the measured momentum thicknesses versus distance

along the test section for five different velocities. Also shown, for

reference, is the expe cted b ehavior of a smoo th plate for these sam e

velocit ies . The mea sured mo me ntum thickness distr ibutio n is inde

pen den t of free s trea m velocity for a ll velocit ies above 9.6 m /s— the

signature of a fully rough flow.

Intro ducti on of the concept of a "virtu al origin ," as used in F ig . 4,

has no effect on the validity of the conclusion that dO/dx is a fu nction

of 0 alone, s ince both are local descriptors—not dependent upon the

value of .x. At first glance, the fact that the data for 9.6 m/s lie beneath

the other data seems to suggest transit ionally rough behavior, s ince

i t appea rs tha t Cf/2 increased (at a fixed value of x — x0) when [/„

increased—behavior seen only in the regime of transitional roughness.

No such conclusion can be firmly drawn without placing unjustifiable

credence in the virtual origin assignme nt for the 9.6 m/s d ata. In ter ms

of locally valid descriptors , the s lope, dd/dx, of the 9.6 m/s data is

lower for every value of 8, tha n for the da ta at h igher velocit ies . While

J o u r n a l o f H e a t T r a n s f e r FEB RUARY 1978, VOL 100 / 137




FREE STREAM VELOCITY

0 u~ =

0 " „ "

o^-Au=o -

& % , "

9

27

41

5/

72

6 ta

0 £9

7 in

0 in

6 ra

s

s

/a

/ s

Is

DISTANCE ALONG TEST SECTION FROM VIRTUAL ORIGIN. (x

Fi g . 4 M o m e n t u m th i c k n e s s ve r s u s d i s t a n c e fr o m b o u n d a r y l a y e r vi r t u a lo r i g i n

this is consistent w ith smooth-wall behavior, it is not proof. Withoutknowledge of the details of behavior of the pre sent surface in the regime of transitional roughness, one cannot assign a definite meaningto the low values shown by the 9.6 m/s da ta.

Fig. 5 shows values of C//2 plotted against 6/r, the momentumthickness of the boundary layer made dimensionless using the radiusof the balls. The data at the lowest velocity, 9.6 m/s, are somewhat low

and the re rem ains a suggestion of velocity dependence in the highervelocity da ta. But, considering the relatively unsophisticated methodof determining C/-/2, th e data strongly suggest tha t Cf/2 is a functionof 9/r only and is independent of velocity.

The present apparatus was designed and developed primarily forheat transfer studies and the heat transfer data are believed to bemore accurate than the friction factor data. No experimental baselineexists, however, against which to compare the rough surface heat

transfer da ta even with no blowing. Comparisons of the present datawith correlations recommended by Nunner [14] and Norris [19]showed good agreem ent even though b oth sources de alt with pipe flowdata, not boundary layer data.

S uccess in representing th e rough wall friction factor behavior interms of the local value of momentum thickness suggested immediately that Stanton number be investigated in terms of enthalpythickness, since for conditions of uniform free stream velocity andisothermal walls, the energy and momentum equations are similar.

F ig. 6 shows the S tanton numbe r dat a for five different velocitiesplotted against entha lpy thickness. The d ata for 9.6,27, and 41.7 m/sshow the lam inar/turbule nt trans ition in differing degrees. This facetof the data is discussed in a later section of this paper. The presentdiscussion centers around the turbulent region of the data. Of primaryimportance is the observation that the Stanton number data are wellcorrelated on the basis of enthalpy thickness alone. The data for four

of the five velocities are within ± three perce nt of the mean over theentire turbulent range, and even the 9.6 m/s data are less than tenpercen t off the m ean, at worst. The fact that the data are compact inenthalpy thickness coordinates means tha t they w ould not be compactin any Reynolds number coordinate. From the heat transfer standpoint, the present data seem characteristic of fully rough behaviorat all velocities tested including 9.6 m/s.

There is no sign of velocity dependence in the heat transfer datafor velocities of 27 m/s and greater. Thus, one may infer tha t the slightdependence seen in the friction factor data may not have been real.The d ata for 9.6 m/s lie about seven percent above the m ean near th etransition; but as the bou ndary layer gets thicker, the da ta for 9.6 m/sconverge to the mean, within three perc ent. The roughness Reynolds

-._-

-

^1

oO

°° %*

i i

>

i

¥$

j . i i i

FREE STREAM VEL0CI7Y

O u„ - 9 .6 m/ s

D u „ - 2 7 . 0 m / s

O u „ - 4 1 . 7 m / s

A u „ - 5 7 . 0 m / s

k . i _ - 7 2 . 6 m / s

1 , ,

MOMENTUM THICKNESS-BALL RADIUS RATIO, 8/r

Fig. 5 Sk in f r i c t i on versus d/r

number for the run at 9.6 m/s decreases from 29 to 23 as the boundarylayer becomes thicker. If the overshoot of the 9.6 m/s da ta were dueto an effect of transitional roughness, then the divergence should haveincreased, not decreased, as the roughness Reynolds number wentdown. The S tanton number d ata in the turbulent region are correlatedwell by the form

S t = 0.0034( ; ) •

(2)

The exponent in equation (2) was chosen, with some deference totradition, to match the usual smooth-plate correlation exponent. Infact, the data show a tendency to level out as A/ r increases, and nosingle exponent can accurately describe the entire curve.

The agreement between the 9.6 m/s data and those for higher velocities is surprising in view of the friction factor results. The roughness Reynolds number for the d ata a t 9.6 m/s lies between 22 and 30,far below the usual boundary of fully rough behavior. Th e discoverythat the Stanton number data correlated with the fully rough behavior, even at such low roughness Reynolds numbers, led to a specialtest aimed at creating a very low roughness Reynolds number. Thefree stream velocity was reduced to 5.6 m/s and the boundary layerthickness augm ented by strong blowing (F = 0.004) through the first0.6 m of the test section. Stanton number d ata tak en in the las t 1.8

138 / VOL 100, FEB RUAR Y 1978 T r a n s a c t i o n s o f t h e A S M E




m along the plate a re comp ared w ith a reference set for 27 m/ s in Fig.

7, Within 0 .1 m after the blowing term inate d, the S tant on n um ber

had risen to the unblown values corresponding to the existing

boundary layer enthalp y thickness. I t is notewo rthy that th e S tanto n

number data for the entire 1.8 m of unblown length on this run at 5.6

m/s l ie exactly on the unblown data for 27 m/s. The roughness

Reynolds numb er at the dow nstream end w as only 14, as determ ined

from hot-wire anemometer measurements of the shear s tress near the

surface, nearly "s mo oth" b y the usual criteria. It seems clear tha t fully

rough heat transfer behavior persis ts to much lower roughness

Reynolds numbers, for the present surface geometry, than had been

expected.

E x p e r i m e n t a l R e s u l t s : E f f e c t s of B l o w i n g . T h e i d e n ti f yi n g

characteris t ic of the S tant on num ber for a fully rough tu rbu len t

boundary layer is its lack of dependence on free stream velocity. The

present data clearly show this independence to be preserved, with

blowing, down to values of the roughness Reynolds number at least

as low as 17, and p ossib ly preserv ed even low er, for the surfa ce used

in these tests. The evidence for this is found in Fig. 8.

F ig . 8 shows the S tanton number data w ith blowing up to F = 0.008

and with velocities from 9.6 to 74 m/ s. To simplify the figure, only t he

turbulent portions are shown. The coherence is excellent, with data

for all five velocities lying within ± five percent of the mean for each

blowing level up to an d inc luding F - 0.002. The rough ness R eynolds

number was 17 for the last data point at F = 0.002 and 9.6 m/s, based

on fr ic tion factors meas ured by mea ns of the mo me ntum integral

equation with blowing (Healzer [31]).

No measurements were made of the fr ic tion factor for blowing

greater than 0.002. Roughness Reynolds numbers for the conditions

s

a/

of heat transfer data of F = 0.004 and F = 0.008 can be estimated,

however, by noting the relationship betw een C//2 and St in the p resen t

data . By either method, the lowest roughness Reynolds numbers for

the dat a at 9 .6 m/s were approxim ately 14 at F = 0.004, and 9 at F =

0.008. Exam inatio n of the S tant on num ber dat a for F = 0.004 shows

a suggestion of velocity depende nce. The " best f i t" l ine throug h t he

27 m/s d ata l ies 20 perce nt below the l ine through th e 9 .6 m/s da ta ,

as would be predicted by a "smooth wall" correlation based on en

thalpy thickn ess Reynolds nu mbe r. To offset th is , however, the data

in Fig. 7 showed no sign of velocity dependence, again at a roughness

Reynolds num ber of 14. Th e differences in measu red S tant on n um

ber s , in Fig. 8, for F = 0.004, are of the order of the unc ertainties in

energy balance closure (0 .0001 S tanto n num ber units ) . This suggests

caution in assigning firm significance to the apparent velocity de

pendence . At F = 0.008, the d ata again show a 20 percen t drop going

from 9.6 to 27 m/s, as predicted by a smooth wall correlation. No other

data were taken at roughness Reynolds number of nine.

There is s trong evidence that the s ta te of fully rough behavior

persis ts to very low values of roughness Reynolds number for the

surface used in these tes ts . The value of ur changes only slowly with

distanc e along the tes t section; hence a l ine of constant velocity and

cons tan t F can be characterized reasonably well by a single "average"

uT. There are larger differences in roughness Reynolds numbers be

tween runs at the different velocit ies shown in F ig . 8 than there are

along the individual lines; yet the data for all five velocities tested lie

nicely along the same line. The roughness Reynolds numbers corre

spond ing to the la s t S tan ton numb er da ta po in t a re in p ropor t ion to

the free stre am velocities—a 14-fold increase from 10 to 136 covering

the rang e normally referred to as smooth up to fully rough, yet with


0

•

oA

L\

u - 9 .6 m/ s

u = 27 .0 a/s

u = 41 . 7 m/ s

u - 57 .0 m/ s

u - 72 .6 m/ s

NOTE: LINES SHOWN AS VISUAL AID ONLY

1 I 1 1 1 — 1 — l _ l _

ENTHALPY THICKNESS-BALL RADIUS RATIO, A/r

Fig. 6 S t a n t o n n u m b e r v e r s u s A/r

Gc)

00 1

_

:

1 1 1


0 » „ - 5 - 6 » / =

1 1 1

1 1 1 1 1 | I I

/ s

^ ^ *

1 I I 1 1 1 ! 1 1

1 1 1 1 | 1 ' 1 .

_

11 1

* % % »

11

l

1 I

ENTHALPY THICKNESS-BALL RADIUS RATIO. A/r

Fig. 7 S t a n t o n n u m b e r v e r s u s A / r a t um = 5.6 m / s

Journa l o f Hea t T rans fe r FEBRUARY 1978, VOL 100 / 139Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



the he at transfer behavior seemingly independe nt of velocity, at eachvalue of F.

The effects of blowing on the S tanton number for a rough wall canbe predicted by using an equation developed for smooth wall heattransfer (Kays and Moffat [32]) with only slight modifications.Blowing reduces the Stanton number of a turbulent boundary layeron a rough plate just as on a smooth pla te; and the following e quationpredicts the effect within a few percent.

_St_

S t,

. f r lU+B) "11-25

A L B J(1 + B)° (3)

In this equation, S t is the value of Stanton number with blowing, andS t0 is the value without blowing, at the same enthalpy thickness, A.Note that the fully rough prediction is to be made at constant A,whereas the smooth plate prediction is done at constant ReA- Th eagreement between predictions and da ta is good for all da ta lying m orethan 30 boundary layer thicknesses beyond the transition peak.

Expe rime ntal Re sults: The Effects of Roughn ess and Blowingon Tran sit ion . The unblown laminar boundary layers at 9.6 and 27

m/s began their transitions to turbulence at momentum thicknessReynolds numbers of about 400, as did most of the other data. Thetransition was marked by a hump in S tanton num ber which could be

regarded either as an "overshoot" of the turbulent correlation or asa low Reynolds number effect. The peak value of S tanton numberoccurred at a momentum thickness Reynolds number of about 700for both velocities.

Blowing through this rough surface does not drastically alter the

onset of transition in momentum thickness Reynolds number coordinates, but it does advance the location markedly in x -Reynoldsnumber.

Pig. 9 shows Sta nton number data in the trans ition region for 9.6m/s w ith five different values of blowing. Num bers near the onsetsof transition are momentum thickness Reynolds number values.

Of primary importance is the fact that blowing through a rough

(h/Gc)

1—r-T-T-r

5b o

o f t . °"0 0 8

1.0 10

Enthalpy Thickness/Ball Radius Ratio, A/r

Fig . 8 Stanton number versus A/ r a t F = 0, .001 , .002, .004, and .008

~i— i — T T -T ~

H

R e g = 392 M ^

Re e - 612 m BLOWING FRACTION

© F = 0

Q F = .0009

<3> F = .00 18

A F = .0041

k F - . 0078

bfc.1\ f c

{ Note : L i l i es s hown as v i s ua l a id on ly )

___! I I I I I I . I L

x-REYNOLDS NUMBER (u ^x /v )

Fig . 9 Stanton numbers In the transition region with b l o w i n g , u „ = 9 .6

m /s

140 / VOL 100, FEB RUAR Y 1978 T r a n s a c t i o n s o f th e A S M E




surface does not make i t more nearly smooth; in fact , b lowing ad

vances the transit io n in;«-Reynolds num ber coordinates and exposes

surface area to turbulent heat transfer which otherwise would have

remained in the laminar regime. Turbul ent he at transfer coeffic ients,

even with strong blowing, are larger than the laminar, unblown values.

As a result, the heat load to a structure could be increased drastically,

instead of decreased, by premature blowing. The area enclosed be

tween the curves for F = 0.000 and F = 0.001 represents the added

heat load on a s tructure caused by blowing into an otherwise lam inar

region.

R o u g h n e s s R e y n o l d s N u m b e r . H e a t t r a ns f e r d a t a f ro m t h e

present surface show fully rough behavior ( i .e . , S tant on nu mb er afunction of enthalpy thickness only) a t roughness Reynolds num bers

as low as 14, far below the usually accepted threshold value of 70.

This may reflect the particular geometry used in these tests (reg

ularly arranged spherical e lements) , a l though this geometry was in-

eluded in the l ibrary of surfaces tested by Schlichting [33] and de

scribed by "equivalent sand grain" behavior. The region of transi

t ionally rough behavior may be compressed when the roughness ele

men ts are of a single size and, in additio n, the behav ior of St an d Cf/2

within the transit ional region may be a function of the roughness

geometry. For example, in C//2 versus Re x coord ina te s , S ch l ich ting

[33] shows Cf/2 decreasing along the smooth l ine as Re x increases,

with a line of constant Cf/2 representing the fully rough state . For

sand grain roughnesses, the values of Cf/2 in the transit ional range

are shown to dip below the fully rough asymptote an d then approach

it from below. For "commercially rough pipes," with less angularroughness elements , Cf/2 is shown to approach the fully rough as

ympt ote from above, not from below, in the transit ionall y rough re

gion. This la tter characteris t ic has also been noted for d-type

roughnesses.

If , in the present case, the transit ion range is compressed by the

regularity , even if i t were of the "approach from above" type of

transit ion, i t might well be impossible to detect tra nsit iona lly rou gh

behavio r wi th measu rem en ts l imi ted to ±0 .0001 S tan ton number

units . The observed behavior might seem to be fully rough or fully

smooth, but nothing in between. With the range of x -Reyno lds

num bers in the pr esent te st, the choices migh t seem to be, in fact, fully

rough or laminar.

The local value of S ta nton num ber is affected by the processes

throughout the entire thickness of the boundary layer. As such,

S tanto n num ber is not the most cri t ical test of the s ta t e of theboundary layer; i t tends to average. I t may well be that the low

Reynolds number layers described here (Re T from 14 to 30) do, in fact,

show some elements of transit ionally rough behavior in their s truc

tures, but not in their S tanto n num ber values.

The re is another possible explanation for the appare nt persis tence

of the fully rough behavior, not necessarily related to the specific

geometry tested. If the threshold value of roughness Reynolds number

is not constant but has even a s l ight dependence on boundary layer

thickness, then a possibil i ty exists that a bounda ry layer f low which

is fully rough n ear th e leading edge of the surface (or a t trans it ion)

might remain fully rough as it progressed down the plate. Schlichting's

[33] predictions of Cf/2 versus x -Reynolds number for sand-grain

roughened surfaces show only a very sl ight convergence of the l ines

of constant roughness Reynolds number with the l ines of constant

u „ ks/v . If these two lines were parallel, instead of weakly convergent,

then a fully rough state would be persistent regardless of plate length.Alternatively, if the threshold value of roughness Reynolds number

decreased as VcJ/2 , then too, a fully rough stat e would be pe rsisten t.

In the fully rough state , \Cf/2 is a function of 0 ~-26, as shown by th e

present data . If the threshold value for the fully rough state varied

with 0 -- 12 0 , then the fully rough state would be persis tent. Such a

weak dependence could be detected only by tests covering a wide

range of boundary layer thicknesses.

F r ic t ion F ac t o r Cor r e la t i on . T he p resen t da ta se t o ffers much

evidence of a fully rough sta te for heat transfer and shows an excellent

correlation in term s of S tanto n numb er versus entha lpy thickn ess.

This suggests that a similar correlation can be offered for the friction

factor behavior even though those data alone might not justify the

form. I t is proposed that:

Cf / 0 \ -O. 2B

-1

= 0.0036 I-J

Once again, the ex ponent w as chosen to agree with a tradit ional value

for this form. Th e presen t data set does not perm it refinement of this

equation because this general form was presumed to be valid in the

da ta in te rp re ta t ion scheme used .


Th e conclusions which follow are based on experiments conducted

on a permeable, rough surface of deterministic geometry in a variable-speed wind tunnel. The test conditions are summarized here to

avoid tedious repeti t ion of s ta tements l imiting the range of validity

of the conclusions.

Turb ulent boundary layer experiments were conducted on a rough,

permeable plate made from spherical e lements , each 1.25 mm in di

ameter, arranged in a most-dense array w ith the crests of the spheres

cop lana r . T h is p la te was ma in ta ined a t a un i fo rm tempera tu re ,

10-15°C above the free s tream. The thermal and velocity boundary

layers originated very nearly at the upstream end of the rough surface.

The test p late was 2 .5 m long and 0.5 m wide. Two-dimensional be

havior was verified. St anto n num ber values were calculated by energy

balance means with corrections for heat loss. S kin friction values were

calculated from measured values of the momentum thickness at se

quential stations down the plate. Free stream velocity varied between

9.6 and 74 m/s . Transp iratio n values from v0 /u „ = 0.0 to 0.008 weretested, with v0 /u ^ held uniform along the plate .

The conclusions which follow refer to this data set for the limits of

their esta blished va lidity bu t will likely allow some ge neralization.

• Fo r the p resen t su r face , the S tan ton num ber has been shown to

be substantia lly indep ende nt of free s tream velocity for values of the

roughness R eynolds num ber as low as 14, withou t blowing. S tanto n

number values without blowing can be represented as a function of

enthalpy thickness alone by the following equation, within ± seven

percen t , wi th in the tu rbu len t reg ion :

/Ax - 0 . 25 / A \

S t = 0.0034 (-) , ( l < - < 1 0 j

F riction factor data w ith no blowing, though displaying more s catter ,

can be represented by:

C, /0X-O.25 / 8 \~

L= 0.0036 {-) , / l < - < 1 0 j

In these equa t ions , r is the rad ius of the sph eres comprising the sur

face, 6 is the momentum thickness, and A the enthalpy thickness. For

convenience, the forms were chosen to match traditional smooth-plate

correlations.

» Blowing reduces the S tan ton num ber and the roughness Reyn

olds number; but, again, for values of Re T grea te r than 14 , S tan ton

number remains substantia lly independent of free s tream velocity

for the present surface. The data are reasonably well correlated by

the following equation:

S t . r ^ i ± 5 ) ] - ( 1 + B )o .2a

S t o l A L B J

Here , S t i s the S tan ton n umbe r wi th b lowing and S t 0 i s the S tan ton

num ber wi thou t b lowing , eva lua ted a t the same en tha lpy th ickness .

The symbol B represents the blowing parameter, v0 /u „ St. This same

form is used for smooth plate correlation bu t requires comparison a t

cons tan t en tha lpy th ickness Reyno lds number .

• T he measu red S tan ton num bers are subs tan t ia l ly independen t

of free-stream velocity for roughness Reynolds numbers as low as 14,

both with and without blowing. This may be evidence that the fully

rough state, for the present surface, persists to very low values of Re T .

On the other hand, i t may be that the boundary layer was in a

t rans i t iona l ly rough s ta te , bu t tha t the S tan ton number was no t

mea surab ly different from the fully rough value. Th e choice betwee n

these two canno t be made w i thou t s tud ies of the tu rbu len t s t ruc tu re

within the boundary layers .

Journal o f Heat Transfer FEBRUARY 1978, VOL 100 / 141Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



• B l o w i n g t h r o u g h a r o u g h , p e r m e a b l e s u r f a c e a d v a n c e d t he x-

l o c a ti o n of t r a n s i t i o n ; b u t t h e m o m e n t u m t h i c k n e s s R e y n o l d s n u m b e r

a t w h i c h t r a n s i t i o n o c c u r r e d w a s s u b s t a n t i a l l y t h e s a m e as the e x

p e c t e d v a l u e for a s m o o t h p l a t e w i t h no b l o w i n g .

• B l o w i n g t h r o u g h a r o u g h s u r f a c e w h i c h w o u l d o t h e r w i s e h a v e

b e e n c o v e r e d by a l a m i n a r b o u n d a r y l a y e r c a n g r e a t ly i n c re a s e t he

h e a t l o a d o n a s t r u c t u r e b y p r o m o t i n g an e a r ly t r a n s i t i o n . T u r b u l e n t

b o u n d a r y l a y e r h e a t t r a n s f e r on a r o u g h s u r f a c e , e v e n w i t h m o d e r a t e

b l o w i n g , is m u c h h i g h e r t h a n l a m i n a r h e a t t r a n s f e r w i t h no b l o w

i n g .


T h e w o r k r e p o r t e d h e r e w a s s u p p o r t e d b y t h e D e p a r t m e n t of t h e

N a v y ( R e s e a r c h C o n t r a c t N 0 0 1 2 3 - 7 1 - 0 - 0 3 7 2 ) an d by t he Of f i c e of

N a v a l R e s e a r c h ( C o n t r a c t N 0 0 0 1 3 - 6 7 - A - 0 1 1 2 - 0 0 7 2 ) . T h e a u t h o r s w i s h

t o t h a n k D r . W . H . T h i e l b a h r , of C h i n a L a k e N a v a l W e a p o n s C e n t e r ,

f o r h i s k n o w l e d g e a b l e a s s i s t a n c e in p l a n n i n g a n d e v a l u a t i n g t h e w o r k ,

a n d M r . J a m e s P a t t o n , of t he Of f i c e of N a v a l R e s e a r c h , f o r h i s c o n

t i n u e d s u p p o r t .

A c c e s s t o t h eD a t a . T h e w o r k s u m m a r i z e d h e r e is r e p o r t e d in

d e t a i l in t he d i s s e r t a t i o n of J. H e a l z e r , S t a n f o r d U n i v e r s i t y , J u n e ,

1 9 7 4 . C o p i e s a r e a v a i l a b l e as R e p o r t H M T - 1 8 , t h r o u g h t h e o f fi ce o f

t h e T h e r m o s c i e n c e s D i v i s i o n , S t a n f o r d U n i v e r s i t y .

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sheft, No. 361, Engl ish t ransla t io n, NACA TM , No. 1292,1933.2 Pran dt l , L., and Schl icht ing, H. , "Das Widers tandsge se tz rauher P la t-

t e n , " Werft, Reederei, and Hafen, 1934, p. 1.

3 von Karman, Th. , "Turbulence and Skin F r ic t ion," Jour, Aero. Sciences,Vol. 1, No. 1, Jan., 1934.

4 Moore , W. L. , "An Exper im enta l Invest iga t ion of Bou ndary Layer Development along aRough S urface," State Unive rsity of Iowa, Ph.D . dissertation,

Aug., 1951.

5 Liu, C . K. , Kl ine , S . J . , and Johns on, J . P . , "An Expe r imenta l S tu dy ofTur bu le n t B ound a r y La ye r s on R ough Wa l l s , " R e po r t M D- 15 , De p t . of M e

chanica l Engineer ing, S tanford Univers i ty , July , 1966.

6 Per ry, A. E„Schof ie ld , W. H. , and Jouber t , P . H. , "Rough Wall Tur bu l e n t B ounda r y La ye r s , " Journal of Fluid Mechanics, Vol. 37, 1969, pp.

383-413.

7 Per ry, A. E. , and Jou ber t , P . H. , "Roug h Wall Boundary Layers in Adverse Pressure Gradients ," Journal ofFluid Mechanics, Vol. 17, 1963, pp.193-211 .

8 Grass , A. J . , "S truc t ura l Fe a tures of Tur bul ent F low over Sm ooth andR ough B ounda r i e s , " Journal of Fluid Mechanics, Vol. 50, 1971, pp. 23 3-

255.

9 Wu, J., " F low in Turbulent Wall Layer over Uniform Roughness ,"

AS M E P a pe r 72 - AP M - U, to be publ ished in journal ofApplied Mechanics,TR ANS . AS M E.

10 Tsuj i , Y., and I ida , S . , "Veloc i ty Dis t r ibut ions of Rough Wall Turb ulen tBoundary Layers Without Pressure Gradient ," Transactions of Japan Society

of Aerospace Science, Vol. 16, No. 31,1973, pp . 60-70.

11 Antonio, R. A., and Lux ton , R. T., " The R e sponse of a T u r b u l e n t

B ounda r y La ye r to a S t e p C ha nge inSur face Roughn ess . Par t 1 . Sm ooth to

R ough , " Journal of Fluid Mechanics, Vol. 48, Par t 4 , 1 9 7 1 , p p . 721-761 .

12 Tow nes, H. W„ Gow, J. L., Powe, R. E., and Web er, H., "Tu rbu lent F low

in S m oo th a nd R ough P ipe s , " Journal of Basic Engineering, T R A N S . A S M E ,Ser ies D, Vol . 94, No. 2 , June 1972, pp. 35 3-362.

13 Town es, H. W. , and Powe, R . E. , "Tur bulen ce S truc ture for Ful ly De-

veloped Flow in Rough Pipes," Journal of Fluids Engineering, TRANS. ASMES eries I , Vol. 95, No. 2, pp. 255-261 .

14 Nun ner , W. , "He a t Transfe r and Pressure Drop in R ough Tube s , "

V DI-Forschungsheft 455, Series B, Vol. 22,195 6, pp. 5-39. English translation ,A.E.R.E. Library, Transactions 786, 1958.

15 Dipprey, D. F . , and Sabersky, R . H. , "He a t and Momen tum Transfer

in S m oo th a nd R ough Tube s at Va r ious P r a nd t l Num be r s , " InternationalJournal of Heat an d Mass Tra nsfer, Vol. 6 ,196 3, pp. 329-353.

16 Owen, P . R . , and Thom son, W. R. , "H ea t Transfe r ac ross Rough Sur faces," Journal of Fluid Mechanics, Vol. 15,1963, pp. 321-334.

17 Gowen, R. A., and S mith, J. W., "Tu rbu lent He at Transfer from Smoothand Rough Sur faces ," International Journal of Heat and Mass Transfer, Vol.

11,1968, pp. 1657-1673.18 Sood, H. S . , and Jonsson, V. K. , "So me Corre la t ions for Resis tances toHe a t a nd M om e n tum Tr a ns f e r in theViscous Sublayer at Rough Walls ,"

J O U R N A L OF H E A T T R A N S F E R , T R A N S . ASME, Series C, Nov. 1969, pp .

488-499.

19 Norr is , R . H. , "S ome S imple Approximate Hea t Transfe r Cor re la t ions

for Turbulent F low in Ducts with Sur face Roughness ," Augmentation ofConvective Heat and Mass Transfer, publ ished by the ASME, New York,1971.

20 Laksh man, C , and Jaya t i l leke , V. , "T he Inf luence of Pran dt l Num berand Sur face Roughness on the Resis tance of t he La m ina r S ub la ye r to Mo

m e n tum a nd He a t T r a ns f e r , " Progress in Heat and Mass Transfer, Vol. ],

Pergam on P ress , 1969, pp. 193-330.

21 Dvorak, F . A. , "Ca lcula t ion of Tur bule nt Boun dary Layers on Rough

Surfaces in Pressure Grad ient," AIAA Journal, Vol. 7, No. 9, pp. 1752-1758, Sept.1969.

22 Dvorak, F. A., "Calcula t ions of Compressible Turbulent Boundary

Layers with Ro ughn ess and H eat Tran sfer," AIAA Jou rnal, Vol. 10, No. 11, pp .

1447-1451, Nov. 1973.

23 Chen, Karl K., "Compressible Turbu lent Bound ary Layer Hea t Transfer

to Rough Sur faces in Pressure Gradient ," AIAA Journal, Vol. 10, No. 5, May,

1972, pp. 623-629.

24 Nestler, D. E., "Compressible Turbu lent Bound ary Layer Hea t Transfer

to Rough Sur faces ," AIAA Journal, Vol. 9, No . 9, S ept. 1971.25 Law, C. J . , "The Use of an Eddy Viscosi ty Model to Predi c t the Hea t

Transfe r and Pressure Drop Per formance of Roughened Sur faces ," International Journal of Heat and Mass Transfer, Vol. 17,1974, pp. 421-428.

26 Lum sda ine , E. , Wen, H. W., and King, F. K., " Inf luence of Surface

Roughn ess and Mass Transfe r on Boundary L ayer and F r ic t ion Coef fic ient ,"Developments in Mechanics, Vol. 6,Proceedings of the 12th Midwestern

Mechanics Conference, pp. 305-318. *

27 McDonald, H. , and F ish, R . W., "Prac t ica l Ca lcula t ions of Transi t iona lB ounda r y La ye r s , " International Journal ofHeat and Mass Transfer, Vol.

16,1973, pp. 1729-1744.

28 Moffa t , R . J . , and Kays, W. M. , "T he Tur bule nt Boun dary Layer on aPorous Plate: Experimental Heat Transfer with Uniform Blowing and Suction,"

International Journal ofHeat and Mass Transfer, Vol. 11,1968, pp. 1547-1566.

29 S im pson , R. L., Moffat, R. J., and Ka ys , W. M., " The Tur bu le n t

Boundary Layer on aPorous P la te : Exper imenta l Skin F r ic t ion with Var iableInjection and Suction," International Journal of Heat and Mass Transfer, Vol.12 , 1969, pp. 77 1-789.

30 Andersen, P. S., Ka ys , W. M„ and Moffat, R. J., " The Tur bu le n tBoundary Layer on a Porous P la te : An Expe r imenta l S tu dy of the F luid Me

chanics for Adve r se F r e e - S t r e a m P r e s su r e Gr a d i e n t s , " De pa r tm e n t of M echanica l Engineer ing, S tanford Univers i ty , May, 1972.

31 Healzer, J. M., Moffat, R. J„ and Kays, W. M., "T he Turb ulen t Boundary

Layer on a Rough Porous P la te : Exper imenta l Hea t Transfe r with UniformBlowing," Repor t HMT-18, Dept . of Mechanica l Engineer ing, S tanford University, 1974.

32 Moffa t, R . J. , and Kays, W. M. , "The Behavior of Transpired TurbulentB ounda r y La ye r s , " inStudies in Convection, V ol. 1: Theory, Measurement

and Applications, Academic Press, London, 1975. (Also published as HMT-20,Depar tment of Mechanica l Engineer ing, S tanford Univers i ty , Apr . 1975. )

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ed„ 1968, pp.5 78-5 89, 610-620.

14 2 / VOL 100, FEBR UARY 1978 T r a n s a c t io n s o f t h e A S M E




R. A. SebanUniversity of Cali fornia at Berkeley,

Berkeley, Cali f .

A, FaghriArya Mehr University of Technology,

Tehran, Iran

Wawe Elects on the Transport to

Falling Laminar Liquid FilmsExisting data for the transport to falling liquid films are reviewed to show the effect of

augmented transport by surface wa ves in the cases of heating of the film, absorption by

the film from the surrounding gas, and evaporation from the film. It is shown that, while

the effect of the waves can be partially rationalized, the nature of the waves themselves

is not known exactly enough to provide a consistent specification for all of the available

data.


The effect of the waves that exist on the surface of a falling film in

lamina r mot ion has been demons t ra ted by numerous exper imen ts

for the cases of absorption of a gas, evaporation from the surface of

the film, and of heating of the film by the wall. Here some of these

results are reviewed to show th e different effects of the waves on these

three cases of transp ort . T his is done by a compariso n of the experi

mentally determined Nusselt number, (hS/k), to the theoretical value

for a Nusselt f i lm of constant thickness, d = 0.91 (AYlii)ll3

(v2lg)

xli,

with velocity distr ibution (u/um) = 3((y/5) — %(y/5)2) , these being

the quantit ies associated with the Nusselt solution for the f i lm. Thetheoretical dependence of Nusselt number on generalized length, f

= (x/6)(a/um5) is available for absorption and for heating by the wall,

but to obtain sufficient detail for values of £, for which the available

eigenfunctions are insuffic ient, the relation was derived again by a

numerical solution of the energy equation. This was done also for

evaporation; this result has not so far been indicated.

Comparisons of the experimental and theoretical values of the

trans port should be suppo rted by a specification of the natu re of the

waves, for i t is through the i nteractio n of the norm al velocit ies pro

duced by the waves and the distr ibution of the transported quantity

that the additional transport arises. But this is complicated by the

existence of an init ia l wave-free length, depending upon the way in

which the f i lm was formed by the l iquid supply, and by a region of

wave development which ostensibly culminates in a s teady regime

at a distance far enough dow n the heigh t of the f i lm. As an instanc e,Braue r [ l ] 1 made measurements on water and mixtures of water and

diethylene glycol at a distanc e of 1.3 m from th e poin t of film initiation

and found sinusoidal waves to begin at a Reynolds number, (AYIn),

of 1.2/(Ka)1/10

, where Ka = (i»4p

3g/<7

3) i s the Kap i tza number . T he


Contrib uted by the He at Transfer Division for publication in the JOURNALOP HEATTRANSFER. Manuscript received by the Heat Transfer Division May13,1977.

ratio of the crest height of the waves to the mean fi lm thickness in

creased with Reynolds number to about twice thisjReyno lds 'numbe r,

then for higher Reynolds number this ra tio remained the same up to

a Reynolds number of 140/(Ka)1 /1 0

. Between these Reynolds numb ers

the waves were no longer s inusoidal but were distorted; above the

higher Reynolds number were secondary, capil lary waves on the

surface. Throu ghou t, up to 417/1 = 1600, the average f i lm thickn ess

remains essentially that given by the Nusselt analysis. But in the range

where the ratio of the crest height to the mean film thickness is almost

constant, the wave frequency increases with Reynolds number to

indicate a basis for a proportio nal increase in transpo rt . Furth er, the

realization of a s teady wave regime at a given flow rate is s t i l l not

defined. With w ater, Brauer [1] indicated, for one Reynolds num ber,

a l inear increase in the crest height w ith distance up to 1 .8 m, so tha t

a s teady regime was not a tta ined, and this development for other

Reynolds and Kapitza numbers has apparently not yet been defined.

Wilke [3] ma de similar m easur eme nts a t a height of 2 .2 m, with dif

ferent K apitza nu mbe rs, and his results for crest height do not check,

relatively, with Brauer 's measurements made at a height of 1 .3 m.

Hea t i n gResu lts for tran spo rt from the wall to the f ilm have been obtain ed

by Bays and M cAd ams ' [2] and by W ilke [3] for heating, and by Ir i-

barne, Gosman, and Spalding [4] for e lectrolytic mass transfer . Fig .

1 contains results for the heating case and shows the theoretical so

lution for the average Nuss elt num ber for the cases of an is otherm al

wall and of constan t hea t flux at t he wall; and for small values of £, thestraight l ines indicate the Leveque solution for these two cases.

Th e results of Bays and M cAdam s [2], obtain ed with f ilm he ights

of 0 .61 and 0.124 m for a Pr an dtl n um ber of 1400 and w ith heights of

1.84 and 0.124 m for a Prandtl number of 51 , are partia lly shown by

repres entati ve po ints on Fig. 1 . The re they are s l ightly below the

isothermal wall prediction, which condition should have been realized

approximately by the steam heating used in these experiments . For

£ > 0.10, however, B ays ' resul ts, extend ing to J = 0.5, are in the region

of the Leveque l ine extended and are unaccountably low. These

anomalous results involve, partially, the greater film heights for both

experiments, while the results for J < 0.10 are, in the main, associated

with the smaller heights . For these runs, the Reynolds numbers ranged





4

3

2

^ = ^^ C ^ ~ ~ ~ ~ \

7 1 \D

D ^

--—

__

"̂

-

52C

^

540 H20

3 (7"~~~

28 0A

140 20 0

a

-- M M - rEF 2 D

--

- 4 4 0 _

y

L O A <7 -t~

T! "i

i Ji2,oo

_

1 32 0

• 4 - 3 —-5 6

80 0+

60

— —

72 0+

10" 8 10"

Fig. 1 Average Nusselt numbers for healing: The upper curve Is the theoretical value for constant heat rate, the lower is for constant temperature. Thetangent lines are the Leveque approximation. The Reynolds number for thedata of reference [3] are indicated adjacent to the points

from 3.6 to 32 for the high Prandtl number and from 80 to 1000 for

the low Prandtl number. The surface tension of the oils is unknown

so tha t the Kapi tza num ber ca nnot be evaluated, bu t in view of thehigh viscosity of the oils it is expected t o be large, so that by the Brau er

specification waves should have existed. In view of the subsequent

results that are indicated for heating, i t is possible that in this case

the waves did not have suffic ient t ime to develop.

Iriba rne, et al. [4] obta ined re sults for average Nuss elt (S herwood )

num bers for P ran dtl ( Schm idt) num bers from 1400 to 18400, using

a vertical tube w ith an "adia bat ic" section 1 m long before the tes t

section, which i tself was relatively short . This , and the high Prandtl

num bers, produced small values of £. At Pr = 1400, (AYlii) = 400, the

value of £ was 10~4, so tha t , wi th a P ra nd t l numb er l ike tha t o f Bays

and M cAdam s' , the value of {was much shorter . At a Prand tl nu mber

of 6350, lower Reynolds numb ers were atta ine d, yielding a m aximu m

£ o f .0015. But in all cases the average Nusselt numbers agreed with

the Leveque result, even though waves were observed. The implication

is that there must have been a purely conduction layer near the wallthrough which the developing concentration layer could not pen etrate

to a s ignificant degree, despite the fact that the Leveque penetra tion

distance is (y/d) = 0.14 at £ = .0015.

Wilke [3] obt ain ed he ati ng resu lts for 9.4 < P r < 210 for a film on

the outsid e of a 2 .4 m high vertical tube, heate d by water in c ounter

flow on the inside of the tube. Local coeffic ients were evaluated by

a measurement of the local mixed mean temperature along the f i lm.

These were averaged in the region in which the coefficient was nearly

asympto tic , the results in an init ia l f low developm ent length and part

of the thermal entry length apparently not being considered. For

suffic iently low Reynolds numbers, Wilke checked the predicted as

ymptotic value of 1 .8 for the Nusselt number, but above a Reynolds

number which he specified as (4r//z)« = 2460 (a/i>)°-B5, the asymptotic

Nusselt number was higher, being given as

This increase in the asymptotic value was implied as due to the waves,

and the details of the initial local behavior were not given except in

one case. It can be noted that for the fluids used by Wilke, the valuesof (4F//i)u , which range from 20 to 143 for the Prandtl numbers from

210 to 9.4, can also be expressed as (4l7 M )u = 9.6 (Ka) - 0 - 1 8 . This in

dicates far greater spread of Reynolds number than implied by

Brauer 's formulation for the change in wave form and implies that

i t is the m agnitude of an appar ent diffusivity th at is important. Chun

[5] has noted that the Weber numbers, (V^bpJ!,) = ( l /3 .8 ) (Ka)1/ 6

(4I7M)6 /6

are more nearly consta nt for this transit ion po int, changing

from 1.4 to 1 in the above range.

Fig. 1 contains the Nu sselt num bers associated w ith (ATln) >

(4T/n)u plotted with £ evalua ted for the 2.4 m test leng th of the Wilke

apparatus. The most conservative estimate of the undefined local

behavior w ould be of a depa rture from the the oretical value of the local

Nusselt number at the asymptotic value, and values of £ for such a

departure can be estimated by equating the local Nusselt number

given by the Leveque solution to tha t specified b y equation (1), togive:

18.5 X 103

(2)

hab /4T\O-M I-^ = .0292 I

k (7) w (1)

For a given Reynolds number, this relation indicates smaller values

of £ as the P ran dtl num ber increases. But th is view is l imited, for

equation (2) implies a wave effect at very low £ for very large Prandtl

numbers when no such effect was found by Ir ibarne.

Absorpt ion

Fig. 2 contains the theoretical result for the average Nusselt number

for gas absorption and also contains a line for the penetration theory

result (heat conduction into a semi-infinite medium) based on the

interfacial velocity of the film.

• N o m e n c l a t u r e .

D = diffusion coefficient

g = gravity acceleration

h = local transfer coefficient; ha, average

coefficient


um = mean velocity in the film

x = distance along the f i lm height

a = thermal diffusivity

T = fi lm mass f low rate , per unit width

& = film thickness

M = dynamic viscosity

v = kinetic viscosity

P = dens i ty

a = surface tension

144 / VOL 100, FEBRUARY 1978 Transa ct ions of the ASME




100

8

*£10

|X" ' . 1

I ' 1 ' ~ I ' - —

- - 1 [ | [ ' i i •

. _ _ . 1 • 1

j

l)

A

a i $

1D

REF. 6 367 46 2D D

_ REF. 8 23 6 813 w o t e r

8 2 2 1 5 4 0 2 7 0 0- I- rj x g l y c o l

REF: 9

K x i

a

r ^ ^ r

i i '

! e

o

a

r<v^

3

_ _r3__ j ._

- —

D

N

c

© "~~~-

>3 O

- -

K

—

\ ] 9

D

•A

8 -

8 10"

(-2

X D _

8 u m8

10"'

Fig. 2 Average Sherwood num bers for absorption: The curve is the theoretical solution w ith tangent A as the pe netration solution and tangent B asthe asym ptotic value. Line K represents the Kamal results for Schm idt num bersfrom 280 to 813

Em mert and Pigford [6] used a water film 1.14 m high for the ab sorption of O2 an d CO2 and obtained the average Sherwood num bersthat are shown by points on the figure. They were the first to showdefinitely th e aug mentation by the waves, by adding a wetting agentto the water, whereby the results were reduced to values near thetheoretical line, actually below it and near the penetration solutionextended.

Kamei and Oishi [7] absorbed CO2 into water film 2.50 m high, andbyvarying the water temperature obtained average Sherwood numbers for Schmidt numbers from 280 to 813. For Reynolds numbersfrom 200 to 2000, these re sults gave values of the Sherwood numbe rthat are represented fairly by the line shown on the figure. For £ >0.20, this line turn s downward toward an intersection w ith the theoretical curve. There is no Schmidt number effect within the accuracyof the data, and in this respect the Em mert d ata also show little effect.But the results of Kamei are substantially higher than those of Emmert.

Chung and Mills [8] absorbed CO2 into water and water-ethyl-ene-glycol films 2.0 m high and obtained a small amo unt of data inthe laminar range. Some of these are shown as points on Fig. 2, andthey are reasonably close to the Kamei results.

All of the foregoing results a re in the region of distorted waves asindicated by the Brauer specification, and even the break point forthe Kamei data, which occurs at a Reynolds number of about 200, isnot associated with the Brauer specification for the end of the si

nusoidal wave regime, which is at 2.8/(K a)1/10, or a Reynolds numberof about 35. Rather, this magnitude is associated with the few dataof Kamei th at are near th e theoretical line defined by the intersectionof the extension of the line on Fig. 2.

The W eber numbers at the break in the Kamei curve are on theorder of 0.35, and all data at lower values of £ involve higher valuesof the Weber num ber. Chung's data is at Weber numbers greater than1.8.

The Kamei data that are correlated fairly well by the line on Fig.2 can also be correlated in terms of the Reynolds number, and thePrandtl number, because the film thickness <5 that occurs in the abscissa of Fig. 2 depends upon the Reynolds number and upon theviscosity, and the Pr and tl number depends primarily upon the vis

cosity. Thus these data can also be correlated by the relation

D

/ 4 r \ 0 . 5 3 / „ \ 0.475

0.0368 ( - ) ( - ) (3 )

This serves also to specify fairly well the re sults of Chung and Mills.If it is estimated t ha t the local transfer coefficient was fairly constantover the entire height of the film, then equation (3) might be expectedto have some relation to equation (1), despite the difference inboundary conditions. The exponent of the Reynolds number is thesame, but that of the P randtl number differs. The exponent of thePrandtl number in equation (1) is of the order of expectation for alinear velocity distribu tion n ear th e wall, and tha t of equation (3) isnear the value of one-half th at is associated with the relatively uniformvelocity distribution in the outer part of the film.

The res ults of Emm ert are not correlated by equation (3), but theydemonstrate a similar Reynolds number dependence but are 20 percent lower for a Schmidt number of 462 and show a slightly greaterPr andtl number dependence. The difference is possibly due to theshorter film height. In this respect, the assumption of truly laminarbehavior initially, followed by an imme diate transition to a consta ntcoefficient, as given by equation (3), enables the specification of average Nusselt numbers like those found by Emmert by assuming atransition at about one-third of the film height. With such a length,for a film height of 2.5 m, like tha t of Kamei, there would be predictedresults only 6 percent lower than given by equation (3). Thus the

difference in the Kamei and the Emmert results might be ascribedto an initial wave-free region. But this interp retation is tenuous, andthere is still no assurance that equation 3 defines truly asymptoticperformance.

Fig. 2 also contains data obtained by Javdani [9] for the absorptionof CO2 into oil, the R eynolds numbe r being on the order of 4 and theSchm idt num ber of the order of 90000, with nearly sinusoidal wavesproduced by the vibration of a wire in the film. The Weber numberwas about 0.25 and the local limit for sinusoidal waves is indicated byBrauer to be at a Reynolds number of 2.8.

Javdani indicated th at a hydrodynamic solution for the sinusoidalwaves could be used to produc e an eddy diffusivity which varied approximately linearly with distance inward from the free surface, and




he used this in a penetration model of the transport equation to pre

dict approximately the results for values of { on the order of those for

which his resul ts are shown in Fig. 2. Th e theo ry fails for larger values

of £ and t end s to give too large a depen dence of the transfer coefficient

on the Reynolds numbe r, but i t is representativ e of what can be done

theoretically when the waves are s inusoidal.

E v a p o r a t i o n

Chu n [10] obtain ed d ata for evaporation from a f i lm of water,

heated at a constan t ra te , f lowing down the outside of a tub e 0 .61 m

high, with the initial 0.30 m being adiabatic, with local coefficients

measured at 0.076,0.13,0.18, and 0.23 m along the heated length. The

trend of the local values implied tha t those at the f inal s ta tions were

asym ptotic and these were reporte d by Chun and Seba n [5] and show n

to agree with a "wavy lamina r" prediction derived from condensation

results. In Pig. 3 all of the local values are shown in comparison to the

theoretical prediction for the local Nusselt numbers.

In some of these runs, the subcooling of the feed was as much as

4.8°C, but by a calculation of the heating of the liquid by con densation

of the vapor according to the theoretical line of Pig. 2, the subcooling

of the mean temperature at the posit ion where heating began should

have been about 0 .20°C, and less if any augmentation due to waves

is considered. But the theory indicates that the subcooling of the fluid

at the wall would have been greater than that of the mean tempera

tu re , and could have been as much as 0 .26°C. Thus the high init ia lNusselt nu mbers may have been due to init ia l subcooling and, taking

the ratio of the feed subcooling to the wall to saturation tem pera ture

difference at the asy mpto tic location, th is ra tio was on the order of

unity for runs 1 through 4 and on the order of one-half for runs 5 and

6. The higher local coefficients in the thermal development region for

the former runs indeed m ay have been due to l iquid heatin g prior to

evaporation, and the local coefficients for runs 5 and 6, which tend

to depart from the theoretical line, with a subsequent almost constant

values, at f ~ 0.02, may be the only ones that truly reveal the effect

of the waves.

The inset of Pig. 3 is a plot of the asymptotic values of the Nusselt

number, this being a repetition of the representation of reference [6]

with more detail , and containing points a t h igher Reynolds number

than does the basic f igure. I t shows the wavy laminar prediction and

the turbulent correlation of [5] for Prandtl numbers of 2 .9 , 5 .1 , and

5.9, and by solid lines it shows, for P ran dtl num bers of 2.9 and 5.1, the

Wilke relation, equation (1), with its coefficient changed to 0.0255

to f i t the few available data for a Prandtl number of 5.1 that appar

ently indicate a transit ion from the wavy laminar to the turbulent

regime. The data that are shown for the other Prandtl numbers is too

sparse to be definit ive, but the point for a Prandtl number of 2.9, at

a Reyn olds num ber of 1450, does indicate a wavy laminar s i tuation

at just abo ut the location at which the tran sit ion to the solid line is

indicated. This intersection occurs at Weber numbers of 1.0 and 1.13

for the respective Prandtl numbers of 5.1 and 2.9, and the R eynolds

num bers at th e intersection are less than those indicated by Re = 140

K a- 1

'1 0

, so that the flow is implied to be within t he regime of distorted

waves. Fur ther , the interse ction of the solid lines with the turbulent

prediction s is in the region of a Reynolds nu mb er of 1600.

The values of the Nusselt number for Reynolds numbers truly in

the wavy laminar regime correspond to s i tuations for which Wilke

measured asymptotic Nusselt numbers for heating that indicated no

augmentation. Th us the augm ented N usselt numbers for evaporation

imply an additional transport effect localized on the outer part of the

film where the effect of the additional transport for heating would be

small because of the reduced heat flux in the outer part of the film in

the case of heating.

in

6

4

2

i!

• * * ^

-

!

! ^ .

RUN' 1 2 3 4 5 6a £. o v + x

< <

—-

^

_O

^

—

&

4

h8k

2

W L

o 5.S

+ 5.

A 2 . 5 1

o

I

£~

[

,s

I 4 6 8 I0 3 2

S7D

o

o<

% "V0

a

h + .A+•

D

8 10"

8 umS

8 I

Fig. 3 Local Nusselt num ber for Evaporation: The curve is the theoreticalsolution for constant heat rate. A is the Levequ e solution for heating with thisw al l cond ition. Values for the runs areRun 1 2Be 320 574Pr 5.9 S.9Feed Subcooling, °C 1.9 4.8

The inset shows the final Nu sselt num ber, taken to be asym ptotic, for theindicated Prandtl num bers. The dashed l ines are the wavy laminar p redictionand the turbulent c orrelation, reference [5], and the sol id l ines are equ ation(2), with a coefficient of 0.0255.

3

696

5 .94 .5

4

860

5. 94. 7

5626

5 .12 .5

6800

5 .1

2 .1

146 / VOL 100, FEBRUARY 1978 Transa ct ions of the ASME




S u m m a r y

T he augmen ta t ion of t r ans fe r to laminar falling films has been

demons t ra ted by a review of existing results for the cases of hea t ing ,

absorption, and evapora t ion , to indicate the relative effects which

must be explained by the as yet unavailable models for such transport

when the waves are distorted and are no longer of sinusoidal form. The

comparisons of the data with expectation for a film of cons tan t

thickness are impeded, moreover, both by the lack of d a t a for local

Nusse lt num bers and by the inabil i ty to characterize definitively the

na tu re of the waves, particularly in regard to the height required for

wave development. N evertheless , the examination of the da ta for thethree cases does provide some estimates about the n a t u r e of the

t r anspor t . «

With heating, when the Reyno lds numbers are sufficiently large,

there is apparen t ly no effect of the waves until the temperature f ie ld

penetrates a region near the wall in which there is l i t t le augmentation

of the t r a n s p o r t , but d a t a for loca l Nusse l t num bers are lacking for

the precise definit ion of the end of th is region. When it ends the re is

apparently a rapid change of the Nusselt number to a nearly constant,

value which depend s on the Reynolds number, because the t r a n s p o r t

augmen ta t ion , where it exists in the region away from the wall, is so

dependen t .

With abso rp t ion the re is always an augmen ta t ion of the t r a n s p o r t

because the n a t u r e of the t r a n s p o r t in the region near the wall is rel

a t ive ly un impor tan t in th is case. For the regime of distorted waves,

those of the existing results that have been obtained with large heights

imply an ea r ly a t ta inmen t of asympto t ic She rwood numbers wh ich

have a Reyno lds number dependence l ike the asympto t ic Nusse l t

numbers found for hea t ing .

With evapora t ion , the one set of loca l Nusse l t numbers tha t are

available appear to be affected by in it ia l subcooling, but they do in

dicate an ultimate asymptotic behavior. These asymptotic values are

a b o u t 50 percen t g rea te r than for a film of cons tan t th ickness in the

domain where the heating results reveal no wave effect, and in the

domain where the heating results do record such an effect, the limited

results for evaporation indicate an increased augmentation which then

follows the t r a n d of the re su l t s for hea t ing .

R e f e r e n c e s

1 Brauer, H., "Stromung and Warmenbergang bei Rieselfilmen," VDI

Forschunsheft, 457, Dusseldorf 1956.2 Bays, G., and McAdams, W., "He at Transfer Coefficients in Falling Film

Heaters: Streamline Flow," Ind. Eng. Chemistry, Vol. 29,1973 pp. 1240-46.3 Wilke, W., "Warm enbergang an Rieselfilme," VDI Forschunsheft 490,

Dusseldorf 1962.4 Iribarn e, A., Gosman, A., andSpalding, D., "A Theoretical and Exper

imental Investigation of Diffusion Controlled Electrolytic Mass Transfer between a Falling Liquid Film and a Wall," International Journal of Heat andMass Transfer, Vol. 10,1967 pp. 1661 to 1676.

5 Chun, K., andSeban, R., "Heat Transfer to Evaporating Liquid Films"ASME, JOURNAL OF HEAT TRANSFER, Vol. 91,1971, pp. 391-396.

6 Emmert, R., and Pigford, R., "A Study of Gas Absorption in FallingLiquid Films," Chem. Eng. Progress, Vol. 50,1954, p. 87.

7 Kamei, S., andOishi, J., "Mass andHeat Transfer in a Falling LiquidFilm of Wetted W all Tower," Mem . Fac. of Engineering, Kyoto Univ., Vol. 171956, pp. 277-289.

8 Chung, D., and Mills, A., "Experimental Study of Gas Absorption intoTurbulent Falling Films of Water and Ethylene Glycol Mixtures, InternationalJournal of Heat and Mass Transfer, Vol. 19,1976, pp. 51-60.

9 Javdani, K.,"Mass Transfer in Wavy Liquid Films," Chemical Engineering Science, Vol. 29,1974, pp. 61-69.

10 Chun, K., "Evaporation From Thin Liquid Films," Ph.D. dissertation,University of California at Berkeley, 1969.

Journa l of Heat Trans fer FEBRUARY 1978, VOL 100 / 147




J , Szego

W. Schmid t

Mechanical Engineering Department,

The Pennsylvania State University,

University Park, Pa.

Transient Behaiior .of a Solid

Sensible Heat Thermal Storage

ExchangerThe transient response characteristics of a solid sensible heat storage exchanger which in

teracts with two energy transporting fluids are presented. The storage unit is composed

of a series of large aspect ratio rectangular channels for the fluids, separated by slabs ofthe heat storage material. The hot and cold fluids flow in counter current fashion, in alter

nate channels so that each slab of storage material is in contact with both fluids. The en

tire system is considered to be initially in equilibrium at a uniform temperature, a step

change in the inlet temperature of one of the fluids is imposed, and the thermal response

of the unit is predicted until steady state conditions are reached. The response of the stor

age exchanger to an arbitrary time variation of one of the fluids' inlet temperature may

be obtained using superposition.

In t ro d u c t i o n

The design of an economically attractive industria l or commercial

energy system require s tha t a ll possible sources of energy, includingwaste heat, be effectively uti l ized to satisfy the energy demands of

the system. The matching of energy supplies and demands may im

pose diff icult ies because, frequently , they don ' t coincide t imewise.

Consequently , a more effic ient uti l ization of the available energy is

achieved if therm al energy storage capabili t ies are incorporated into

the sys tem.

The importance of energy storage was emphasized when the Energy

Resea rch and Deve lopmen t Admin is t ra t ion (E RDA) reques ted tha t

the National Research Council conduct a s tudy on the potentia l of

energy storage systems. The report [ l]1

recognized the wide range of

potential applications and indicated that the s torage system selected

would be depende nt on i ts abil i ty to respond to the deman ds required

by the specific application.

The thermal energy is usually transported from one location to

anot her by stream s of l iquids or gases. A heat exchanger can be used

to enable the thermal energy to be transferred between the f lowing

streams. The amount of heat transfer , for s teady-state operation, is

easily calculated using conventional techniques described in basic heat

transfer textbooks. Standard heat exchangers employ thin walls to

separate the f luid s treams, and thus have negligible heat s torage

capabili t ies . If i t is desired, a t certain periods, to remove a larger

1 Numbers in brackets designate References at end of paper.Contributed by the Heat Transfer Division for publication in the JOURNAL

OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division April7, 1977.

amount of the energy from the hot s tream than required by the

thermal load, or conversely, the thermal energy demand is higher than

the energy being added to the system at that particular period of time,

i t becomes necessary to incorporate s torage capabili t ies into thesystem . The easiest way to accomplish this is by replacing the usual

thin metallic wall in the hea t exchanger with a thick layer of a material

having good heat storage characteristics. A sketch of a typical sensible

heat s torage exchanger is given in Fig . 1(a) . The outlet temperatu res

of both fluid streams will be time dependent, and in many applications

the inlet temperatures of one or both f luids may vary with t ime.

The transient response of heat exchangers has been the topic of a

num ber of pape rs . A sum mar y of these s tudies has been presented

by Schm idt [2]. Mo st investigators were mainly concerned with the

response of the energy transporting f luids, and the heat capacity of

the separating wall was either neglected or serious restr ic tions were

imposed when accounting for i ts contribution to the response of the

hea t exchanger .

Studies that have taken the heat capacity of the wall in to consid

eration are described by C ima and London [3] and London, e t a l . [4] .

Th e hea t exchange r 's walls were subdivided in the direction of flow.

Each subdivision was considered to be at a uniform t emp eratu re and

to have a heat capacity equal to its mass times the specific heat of the

materi al . The conductive resis tance to the transfer of heat offered by

the material was divided in half and each half was added to the re

sis tance offered by the convective f i lm at the exchanger 's surfaces.

Axial conduction in the metal was neglected. Solutions obtained using

an electr ical analog system were presented for the tran sient response

of a direct transfer counterflow regenerator for gas turbine applica

t ions .

A rec ent stu dy by Schm idt an d Szego [5] used a finite difference

techniq ue to pred ict the tra nsien t response of a two fluid he at s torage

exchanger for a solar energy system; but no general analysis of the

148 / VOL 100, FEBRUARY 1978 Transact ions of the ASMECopyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



COL DFLUIDINL ET

fly

5

HOT FLUID CHANNEL

COLD FLUID CHANNEL

L

(b )

Fig. 1 Heat storage exchanger; (a ) schematic of typical heat storage exchanger unit (6 ) cross section considered in the analysis

response characteris t ics of these units was reported. The presen t s tudy

will u ti l ize these techniques to determine the response of a counter-

f low heat s torage exchan ger uni t under a variety of operating condi

tions.

A n a l y s i s o f t h e R e s p o n s e o f t h e H e a t S t o r a g eE x c h a n g e r

The heat s torage exchanger system to be considered is composed

of a series of large aspect ra tio rec tangular cross-sectional cha nnels

for the energy tr ansp orti ng f luids. Th e hot an d cold f luids f low in a

counter current fashion, in alternate channels, so that each slab of the

storage material is in contact w ith both f luids (Pig. 1(a)) . Plane s of

symmetry were assumed to exist a t the mid-locations of the f lowchannels , restr ic ting the analysis to the section shown in F ig . 1(b).

The following idealizations were used in establishing the mathe

matical model describing the physical system under consideration:

(a) const ant f luid and mat erial properties

(b) uniform convec tive film coefficients

(c) two-dim ensional conduc tion within the s torage mater ial

(d) const ant f luid mea n velocit ies

(e) uniform init ia l tem pera ture distr ibuti on in the s torage ma

teria l

(f) process init ia ted by a s tep change in the inlet tem pera ture of

the hot f luid .

Based upon these assum ptions, the differentia l equations relating

the temp eratures in the system are the two-dimensional transient heat

conduction equation for the s torage material , coupled to the one-

dimensional conservation of energy equations for the energy trans

porting f luids. The result ing equations express the temperatures in

the f luids and the storage material as a function of the two spacial

coordinates, x and y, and t ime , 6:

cold fluid

_ , fa t c dt ciPc C cAc — - vc — = hcPc(tu

L dd dxJ

storage material

•t c)

1 dtm = dH m | dH m

a d8 dx2

dy2

and hot fluid

Vd th dth-]PhthAh — + uh — = hhPi

I dd dx J

(twh ~ th)

The associated init ia l and boundary conditions are:

6 = 0 tm = tc = th = t0

dt„

(1)

(2 )

(3 )

I > 0 x = 0 th = thidx

0 for 0 < y < w

dt „x = L tc = tci = 0 for 0 < y < w

dx

fity = 0 k —-=hc(tm ~t c) f o r O < x < L

dy

a n d y = w k —— = - h h ( t m - th) for 0 < x < L.

For greater generali ty the equations were nondimensionalized by

introducing the following dimensionless parameters:

hhiv _ . hcwBio t numb ers : Bi ;, = - B ic


A = flow cross-sectional area

C = specific heat a t constant pressure

d = semi-width of f low channels

E = fluid heat capacity

h = conv ective film coefficient

k = the rm a l conduc t iv i ty o f s torage ma te r i

al

K = constant defined in equation (10)

L — length of unit

m = mass rate of flow

P = heated perimeter of f low channel

Q = amoun t o f hea t s to red

R = convective resis tance ratio

S = heat transfer surface area

t = t e m p e r a t u r e

tw = m ean tem pera tu re o f the wa l l ave raged

o::

over i ts th ickness

T = d imens ion less tempera tu re

Tm = mean s to rage ma te r ia l d imens ion less

tem pera tu re ave raged ove r i t s vo lume

U = overall hea t transfer coeffic ient

v = fluid velocity

V = volume of s torage material

w = th ickness of s torage material

x = axial coordinate

y = t r ansve rse coord ina te

p = dens i ty

B = t ime

dwell t ime = —v

the rm al diffusivity of s torage mate rial

S u b s c r i p t s

c = cold fluid

ci = cold f luid entering

co = cold fluid leaving

h = hot fluid

hi = hot f luid entering

ho = hot f luid leaving

m = s to rage ma te r ia l

o = init ia l condition

wc = wall temperature cold f luid s ide

wh = wall temperature hot f luid s ide

S u p e r s c r i p t

ss = s teady s ta te cond i t ion





aOFour ie r number : Fo = —-;

d imens ion less tempera tu re : T=thi tc,

dimensionless f low length: X = —;

L

d imens ion less t ransve rse coord ina te : Y =;

th ickness / leng th ra t io : V+

= —;Li

capacity rate ratio for fluids: C4

convective resis tance ratio: R+

=

PcucA cC c mcCc Ec

PhVhAhCh l rhhCh Eh

1

PcLhc = Bi hP h = Bi£_

1 " B i c P c " Bi c '

PhLh h

A r + -P c k

P + -Phk

T he equa t ions for the f luids become:

a 9TC | dTc Gc+

Bic

wv cV+ 2dF o dX V+[Tc - Tw

and

a dTh dTh G c+ Bi eR+C +. + _ _ + yih _ j ^ _ o.

wu hV+2

dF o dX

(4)

(5)

ay= -Bih(Tuh - Th) f o r O < X < l .

Equations (6), (7) , and (8) form a coupled set and must therefore

be solved simultaneously. This is accomplished by m e a n s of a nu

merical solution using a digita l computer. The thermal s torage ma

teria l is subdivided in the X and Y directions, establishing a grid

pa t te rn . E qua t ion (7) is rewr i t ten in finite-difference form, using

backward differences for the t ime derivatives and central differences

fo r the spacial derivatives. Third order orthogonal polynomials are

fitted by least squares to Twh and Twc along the flow chan nels. These

polynominals are then substi tuted into equations (6) and (8) allowingfor an exact solution of the result ing equations for the fluids. The

conduction equation is solved by finite difference techniques, but an

iterative procedure is needed to match equa t ions and boundary

conditions. A detailed outline of th is numerical solution is presented

in the append ix to [6]. The reade r mus t keep in m i n d t h a t in the

present investigation two fluid equations m ust be solved rather than

th e one equation used in [6] for the single-fluid case; the general

procedu re is , however, essentially the same .

D i s c u s s i o n of R e s u l t s a n d C o n c l u s i o n s

T r a n s i e n t R e s p o n s e P a r a m e t e r s . The t r ans ien t re sponse of a

thermal s torage exchanger, in it ia l ly in equilibrium at a uniform

t e m p e r a t u r e , to a step change in the in le t t empera tu re of on e of the

fluids will be discussed. Results will be presente d for the case ofa step

change in the in le t t empera tu re of the hot fluid, although they areequally valid if the s tep change in in le t t empera tu re occu r red in the

cold fluid. This would require the rep lacemen t of "hea t s to red" by

"hea t re t r ieved , " "ho t" for "cold", and vice-versa in the final results.

Three new time dependent dimensionless parameters are introduced

as follows:

F r o m an orde r of magni tude ana lysis to eva lua te the impor tance

of each term in the above equations (4) and (5), it was concluded tha t

a negligible error is in t roduced if the t r ans ien t te rm in the fluid

equa t ion is abandon ed . T h is a ssumpt ion , wh ich imp l ie s neg lec ting

the hea t capac i t ies of the f luids, was previously discussed in [6] and

ca n be fu r the r subs tan t ia ted by the f indings of Cima and L ondon

[3].

T he comple te set of governing equations in nondimensional form

becomes :

cold fluid

dT c

dX

Gc+Bic

V+

[T o ~ Twc] = 0,

s to rage ma te r ia l

y + 2d

2Tm . d

2Tm

dX2

9T m

dY2

dFo, and

(6)

(7)

and

[ra te of heat transfer from hot"]

fluid at time 6 J _ Eh[thi ~ ho]

[heat transfer ra te from hot"] Eh [thi ~ tho

ssl

f luid at s teady sta te J

[ r a t e of heat transfer from c ol d!

fluid at time 0 J Ec [t ci — tco]

[heat transfer ra te from cold"! ^c [tci~ tCoS3\

f luid at s tead y state J

[amoun t of hea t s to red at t i m e 8}

[amoun t of heat s tored at steady state]

_ Q =Tm

Qs s f ss •

hot fluid

3T h G c+ m cR+C +_ _ + _ [Th-Twh]-0

with the in it ia l condition

F o = 0, Tm = Tc = Th = T0

(8)

an d the boundary cond i t ions

dT mX = 0 Th = Thi=l ^ f = 0 for 0 < Y < 1

ax3 T

X = 1 Tc = tci = 0 — - = 0 for 0 < Y < 1

axY = 0

^ 7 = B i c ( T u ) i : - r c ) f o r O < X < ldY

and

T h i s set of dependent parameters was formulated since it has simple

physical significance and is conven ien t forgraphical presentation of

the t rans ien t behav io r of the heat s torage exchanger. The basic

analysis was carried out with the complete set of governing equations,

equations (6-8), and associated boundary conditions. The results

obtained indicated that for the cases ofpractical in terest , the effects

of longitudinal heat conduction are negligible and the f irs t term on

the left hand side ofequation (7) can be neglected. Thu s it is possible

to combine G+

and V+ in to one single parameter, namely G

+/V

+,

since these parameters appear only as the indicated ratio . The

s teady-s ta te quan t i t ie s tha t are used in the definit ions of th, <=c, and

Q +can be obtained exactly , as shown in the Append ix . For all the

cases under consideration in this study the intial uniform temperature

of the storage material is equa l to the incoming tempera tu re of the

cold fluid, T0 = TCi - 0,since the s t ep change is imposed on the inlet

t e m p e r a t u r e of th e hot stream alone.

Performing an overall energy balance for the entire system without

heat losses to the surroundings gives:

15 0 / VOL 100, FEBRUARY 1978 Transact ions of the ASfVSE




dQ

d8 'Eh(thi ~ tho) + E c(tci — tco) (9)

Rearranging this equation and introducing the previously defined

parameters yield a s imple expression relating the three dependent

variables:

KdVo-th (10)

where

V W c \ 1 - Th„")

fmss

an d Th 0ss axe given by equations (A9) and (Al). T he dependen t

variables e;„ tc, and Q+

are each functions of the f ive independent

variables Bi e , ( G + / V + ) c , C+, R+ a n d F o .

In order to allow a reasonably compact graphical presentation of

the results, an additional restriction became necessary. It was assumed

that both the energy transporting fluids have equal physical properties

and that the f low channels have the same dimensions. Under these

conditions and considering turb ulent f low, the convective resis tance

ratio reduces to the r atio of the mea n velocities of the two fluids raised

to some exponen t . T h e expone n t commonly employed by mos t co r

relations is 0 .8 , and t he relation obtain ed is

R* (C+

)- (11)

Therefore only four independent dimensionless parameters are re

quired.

A more con venien t way of presentin g results was obtained by

plotting th, ec and Q+

aga ins t Fo / (G+

/ V+

) m a x . T he max imum G+/V

+

was selected for th e pres entati on of the results , s ince i t involves th e

minimum fluid heat capacity . In addition, the commonly used

steady-state parameter NTU proved to be useful as one of the vari

ables. The final results show eh, ec and Q + as functions of Bic , C + ,

N T U , F o / ( G+

/ V+

) m a x . The following relationships between the

variables hold:

B i fc :Bic

C + 0 . 8 '

N T U :S U

•&min

a n d -

\V + /m ax LBic + Bh l + 1 J

F o En

( G + / V+

) „ (PCV )n

(12)

(13)

(14)

where k has cancelled out in equation (14). I t is notew orthy th at t he

latter two variables have already been employed by Cima and London

[3] to correla te their results .

R e s u l t s — S t e p Cha ng e in One Inle t F luid Tem pera ture . From

the definit ions of ec and Q + i t is evident that both quantit ies range

from a magnitude of zero at the beginning of the process, 8 = 0, to a

value of unit y as s tead y stat e operatio n is approache d. Moreover, e /,

s tart ing from some value larger than one will a lso approach unity at

s teady s ta te . T he ma themat ica l mode l desc r ibed p rev ious ly in the

analysis does not account for the events which take place from the

sudden increase of the inlet temperature of the hot f luid to the mo

ment when the f irs t heated partic les leave the system. However, th is

t ime interval is typically very small compared to characteris t ic t ime

for the overall process. Having in mind the specified initial conditions,

i t is app are nt t ha t the di sturba nce will be felt a t the outlet of the hot

stream exactly a dwell t ime, 84, la ter , and tha t the f irs t fluid partic les

will have viewed a constant te mpe rature wall as they moved throug h

the unit . Solving equation (8) with a constant wall tempe ratu re, Twh

= T0 = 0, gives:

Tho = e x p T - B i c (^j C +°2

1 for 8 = Bd, (15)

while Th o = 0 for 0 < 8 < 8j. Thus i t follows that an abrupt change in

the hot f luid ou tlet tem per atu re will occur. A lis ting of Th o a t 8 = 8d

is presen ted in Tab le 1 . Fig . 2 shows a typical plot of the results to i l

lustrate the foregoing conclusions. The area bounded by the curves

fo r eh and ec, a nd F o / ( G + / y + ) m a x varying from zero and some speci-

0,1

1.0

10,

NT U

0. 1

1,0

3. 0

0. 1

1.0

3.0

0, 1

1,0

3. 0

O 8A

o.f1

1.010.

0. 11.010,

0, 11.010.

0. 11.010.

100.

0. 11.010.

100.

0. 11.010.

100.

0. 11.010.

100.

0. 1

1,010,

100,

0. 11.010.

100.

C O

0,095

0.619

0.939

0.091

0.500

0,750

0.0095

0.062

0.091

• | " 8 8

• h o

0.990

0.938

0.906

0.909

0.500

0.250

0,905

0,381

0.061

SB 8 8

• m0.8380,7060,557

0.8710.7S50.688

0,9200,8760,827

0. 5

0.1390.1750,3330.155

0.1110.1380.2110.320

0.0720.0810.1360.177

Tho<%>

0.9210.8720.707

0.1520,2560,001

0.0920.0170,000

0,8100,7110,3010.000

0.1220.0500.0000.000

0.0020.0000.0000.000

0,8890.8760.7600.182

0.3080.2670.0610.000

0.0290.0190.0000.000

Table 1 Operating chara cteristics of heat storage exch angers

n6-eh ot0=o.o

I O -

v e h at e= ed

I 2 3 4 5 6 7

M G V V I , . ,

Fig. 2 Typical representat ion ol the results





fied value is given by K Q + / ( G + / V + ) m a s , as can be inferred by inte

grating equation (10). Since at steady state conditions Q+

approaches

the value of one, the sh aded area in F ig . 2 is equal to Kl -

( G + / \ / + ) m a x .

F igs . 3-10 represent the f inal results of the present investigation.

Fig. 3 shows €/, and ec for C + = 1.0. The transient response of the hot

fluid , repre sente d by «/, is shown to be strongly dep ende nt up on Bi c

for NTU = 0.1 , but as the NTU is increased this dependen cy te nds

to disappe ar com pletely. In fact, for NT U = 3, the curves for Bic equal

to 0.1,1.0 and 10 coincide. It will also be noted that for specified values

o f C+ and NT U, the va lue o f the absc is sa , Fo / (G+/V+) m a x , wheresteady stat e is reached is nearly inde pend ent of B ic . The response of

the cold fluid, cc seems to be a weak function of Bi c for C+

= 1.0. Figs.

4 and 5 show analogous curves for C + = 0.1 and C + = 10. Whi le for

C+

= 10 the t ren ds are similar to those for C+

= 1.0, the results for C+

= 0.1 show tha t as the NT U is changed, the curves tend to keep their

general shape indicating that the influence of Bi c is equally detectable

for the range of NTUs examined.

The results for a constant NTU, varying Bi c and C + , are shown in

F igs . 6 and 7 . The trends are s imilar for the two NTUs considered

N TU = 1.0 and NT U = 3.0 . The l ines represe nting ic for the various

cases tend to overlap, bu t there is a clear distinctiun between th e lines

for th, namely that larger values of th weie obtained for the smaller

C+

s. The influence of the Bi c. over these results is similar to that

discussed previously in conjunction w ith Figs. 3-5.

The next three Figs. , 8 , 9 , and 10, are the representation of the

fraction of the s teady-state heat s torage as a function of Fo/(G+/

y+)max- For all cases, the consta nt N TU curves shift to the r ight as

the N TU decreases. As the B iot numbe r increases, the curves shiftto the left. It is im po rta nt to realize tha t thes e results cover a wide

range of possible physical s i tuations . The am oun t of heat s tored and

the transient f luid outlet temperatures can be evaluated employing

the steady-state quantit ies used in the definit ion of e / , , ( c and Q+,

Tabl e 1 shows these steady- state pa rame ters for the range of variables

covered in this s tudy.

1

10" IOu

Fo /(G7V)mox

Fig. 3 Response of heat storage exchang er - C+

= 1.0

IO2

- NTU = 3.0

O"1

I0U

Fo/iGVV)m(,»

Fig. 5 Response of heat storage exchang er — C+

= 10

IO2

C* = 0.1

IO "1

I0l

Fig. 4 Response of heat storage exchan ger - C+

= 0.1

io2

IO'1 I0 U

f y ( G V V ) m „

F i g . 8 Re sp o n se o f h e a t s to ra g e e xch a n g e r — NTU = 1 .0

152 / VOL 100, FEBRUARY 1978 Transactions of the ASiVlE




1 0

\ \ L~ -o.i

\ U 10

m -

|Q

\ \ \\ \ \ ~ ™

1N T U = 3.0 \

"l 0

~\ °iii°i| a

V

°j[XA~™I---L- \ T

" . ^ ^

T ' .—r-r

""

— C+=0.l

• - C4

= l.0

_

-

- • - - s -

1.0 ; 1 0 .

O io

2 4 6 8

I0 ~3

I0"z

IO '1 10 °

Fo / ( GVV1mo J

F i g . 7 Response of heat storage exchanger — NTU = 3.0

F o / f G y v )m ox

F i g . 8 The fraction of the steady-state heat storage — C+

= 1.0

F o/ f e w < ) „ „ »

F i g . 9 The fract ion of the steady-state heat storage - C+

= 0.1

Arb itrar y Time Variat ions in Fluid Inlet Te mp era ture . A

very important extension of these results is possible because of the

linear nature of the governing differential equat ions and associated

boundary conditions. The procedure, usually referred to as Duhamel's

Method, can be employed to predict the transient response of a heat

storage exchanger to an arbi trary time variation of the hot fluid inlet

o 0 3

F9fG*/V*),n0M

F i g . 10 The fraction of the steady-state heat storage — C+

= 10

0.9

0.8

w 0.7

=> 0.6

h -<i t 0.5U !

Q .

2 0.4

in

• " 0.3

0.2

O.I

0.0

-

-

-

=

-

0

8

i i i

NTU=I.O

Bi c= I.O

C * =0. l

I—

•

a

V

V

J eVG

r-J V 0

• V O

J v O

_QJ2_I I _ L _

•

\7

O

t

|D '—

•

V V y

/

— ! —

V

^

GO G

GO

i

V

G

™ ™

, L_

— I —

(J

•V

V

1 I I

I Q I ,

•-

k u - iv v

V „V V

fc -0 G O Q G | |

G o 0 ° IQ

Tho

Tco

Thi

_ ™ L _

"

-

1 1 i

2. 3. 4. 5. 6. 7 8. 9. 10 II. 12.

F i g . 11 Response of the heat storage exchanger to a time variation In the

h o t fluid inlet temperature

temperature. In order to use the present step function solutions for

superposition, the arbitrary variation of the hot fluid inlet tempera

ture must be approximated by the sum of a finite number of steps.

Next, the steady-state quantities have to be calculated to enable us

to evaluate T/,0 and Tco utilizing the values of th and tc obtained for

a single step change, recalling that the inlet temperature of the cold

fluid remains constant throughout the p rocess. The actual transi ent

response, T/,0, Tco and Tm, to the specified time dependent hot fluid

inlet temperature will be, at each instant of time, the sum of the

contribution of all the step changes up to that moment. Fig. 11 is an

illustration of this procedure for the case of C+

= 0.1, Bic = 1.0 and

NTU = 1.0.

Conclusions

A wide range of possible practical applications for heat storage

exchangers exist, however there must be a compromise between the

amount of available energy to be transferred to the cold fluid and the

desired amount stored in the storage material. If it is desired to store

a considerable portion of the energy being transported by the hot fluid

it is necessary to operate at low values of C+. On the other hand, for

small C+, Tho tends to be large, indicating that only a small fraction

of the available energy has been used. As we go to larger C+s the

amount of heat stored, for a constant geometry, decreases, but better

use of the available energy is made. It becomes apparent t hat each

proposed application has to be examined on its own merits, and the

results of this present investigation should be used as a guide to allow





the designer to establish the best configuration suitable for the ap

plication in hand.


The authors wish to acknowledge that th is work was performed

under NS F- RAN N Gran t A E R 75 -16216 . T he techn ica l ove rs igh t of

this grant is provided by ERDA, Division of Conservation, Oak Ridge,

T ennessee .

R e f e r e n c e s

1 Comm it tee on Advanced Energy S torage Systems, "Cr i te r ia for EnergyStorage R & D," ISBN 0-309-02530-3, Nat iona l Academy of Sc ience , Wash

ington, D. C , 1976.

2 Schm idt , F . W. , "Num er ica l S imula t ion of the Therma l Behavior of

Convec t ive Hea t Transfe r Eq uipm ent ," Heat Exchanger: Design and TheorySourcebook, N. Afgan and E. U. Schlunder ed., McGraw Hill, New York, 1974,

pp. 491-524.

3 Cima, R . M. , and London, A. L. , "Th e Transient Response of a Two FluidCounte r f low H ea t Exchange r—Th e Gas Turb ine Regen era tor ," TRANSACTIONS ASME, Vol. 80, 1958, pp. 1169-1179.

4 Londo n, A. L. , Sampse l l , D. F . , and McGow an, J . G. , "Th e T ransie ntResponse of Gas Turbine P lant Hea t Exchangers—Addit iona l Solut ions for

Regenera tors of the Per iodic and Direc t Transfe r Ty pes," Journal of Engineering for Power, TRANS. ASM E, Ser ies A, Vol. 86,1964, pp. 127-135.

5 Schm idt, F. W., and Szego, J. , "Transie nt Behavior of Solid Sensible Hea t

Thermal Storage Units for Solar Energy Systems," 1975 International Seminar,"Future Energy Produc t ion—Heat and Mass Transfe r Problems," Yugoslavia ,

1975.

6 Schmidt, F . W., and Szego, J., "Transient Response of Solid Sensible Heat

Thermal Storage Units—Single F luid," ASME JOURNAL OF HEAT TRANSFER,August 1976, pp. 471-477.

cold fluid

dt cE c-

1-R cP c(t l:-tw) = 0;ax

(A2)

s to rage ma te r ia l

RhPh(th-tw) + RcPc (tc-tw) = 0; (A3)

and for the entire system

-R cP cL( tc - tw) = RhPhL(th - tw) = US(th - tc). (A4)

E l i m i n a t i n g tw from the above equations gives:

dth US ,Eh-— + — (th- tc) = 0, and

ax L

„ dt c US ,E c-

1 + -—(th- tc) = 0.ax L

(A5)

(AG)

I n t r o d u c i n g t h e d i m e n s i o n l e s s p a r a m e t e r s d e f i n e d i n t h e t e x t a nd

s o l v i n g t h e r e s u l t i n g e q u a t i o n s y i e l d t h e s t e a d y - s t a t e t e m p e r a t u r e

distr ibutions in the f luids and the storage material . This la tter must

be integr ated over the volume of the s torage ma terial to obtain Tm".

The final results are:

C+ eM

- eM

Tho's

= —^-. — for C+ ^ 1;C

+-e

M

Th„s1

N T U + 1 for C+ = 1 (A7)

l-eM

for C+ * 1;

A P P E N D I X

Evaluat ion of S teady S ta te ParametersIn order to calculate the numerical values of the dependent vari

ables, eu , ec, and Q+

, used to present the f inal results of this s tudy,

th ree s teady-s ta te quan t i t ie s mus t be eva lua ted . Under the a s

sumptions of steady-state and negligible longitudinal heat conduction,

the tem peratu re distr ibution in the s torage material in the y directionbecomes l inear a t each location x along the flow channels (Fig. 1(b)).

Henc e, a mea n wall tem per atu re („, is used and effective heat-tran sfer

coefficient, Rh a nd R c, are specified by combining heat transfer

coeffic ient with the convective contributions of each of the energy

transporting f luids. The energy conservation equations become:

hot fluid

dthEh-r + RhPh(th-t„)--

ax0; (Al)

N T U

N T U + 1for C

+= 1 (A8)

1 L \M I Ml

where

- for C+ * 1;

2 L N T U + 1 Jfor C+ = l (A9)

M = NT U (1 - C+) for C+ < 1

M = N T U ( ^ — 7 - ) for C+ > 1

N = (C+ + 1) + (C+ - 1)N*

_1 1_

Bi c Bi/jN*

1 1

— + + 1Bi c Bih

154 / VOL 100, FEBRUARY 1978 Tra nsa ctions of the ASME




P. C. Wayner, Jr.

Department of Chemical and Envi ronmental

Engineering,

Rensselaer Polytechnic Institute,

Troy, N. Y.

The Elect of the London-Van Der

Waals Dispersion Force on Interline

Heat TransferA theoretical procedure to determine the heat transfer characteristics of the interline re

gion (junction of liquid-solid-vapor) from the macroscopic optical and thermophysical

properties of the system is outlined. The analysis is based on the premise that the interline transport processes are controlled by the London-van der Waals dispersion force be

tween condensed phases (solid and liquid). Numerical values of the dispersion constant

are presented. The procedure is used to compare the relative size of the interline heat sink

of various systems using a constant heat flux mode. This solution dem onstrates the im

portance of the interline heat flow number, Ahfgv~l, which is evaluated for various sys

tems.


It is well known that the vapor pressure over an adsorbed fi lm

of a nonpolar l iquid is a function of the tem per atur e and the Lon

don-van der Waals dispersion force of a ttraction between the solid

and liquid (e.g., references [1 -4 ]1

are of particular re levance). This

fact can be used to predict the interl ine hea t transfer coeffic ient and

the heat sink capability of a stable evaporating wetting film of liquid

[5-7]. These heat transfer characteris t ics are of major importance to

the analysis of the rewetting of a hot spot, the heat transfer rate from

an evaporating m eniscus, fa ll ing thin f ilm evapo ration, and boiling.

The analyses presented in [6 , 7] demonstrate the importance of the

group of properties , Ah fgh, which we call the interline heat flow

number, in which A is a dispersion consta nt, hfg is the he at of vapor

ization of an adsorbed film and v is the kinem atic viscosity . For ex

ample, a constant heat f lux model of the evaporating interl ine [6]

shows tha t the the oretical in terl ine he at s ink for a unit length of in

terline, Q, is given by

Q = [2(7 In vf- 5[hfgAhf* (1 )

in which q is the he at f lux and i] is a dimensionless thickness. H erein,methods for predicting the value of the interl ine heat f low number

and, therefore, the s ize of the interl ine he at s ink are presen ted and

used. In essence, a theoretical procedure to determine the interl ine

heat sink capability of an evaporating thin film from the macroscopic

optical properties and thermophysical properties of the system is

outlined.

E v a l u a t i o n o f A

Assuming that bulk values can be used to approximate the values

of the heat of vaporization and the kinematic viscosity , the problem

of predicting the value of the interl ine heat f low number, A hfgh, is

reduc ed to the critical one of evaluating a dispersion cons tant, A. Since

A is important to many fie lds, considerable l i terature concerning i tsdetermination is available . Basically , there are two methods of cal

culating the dispersion force between condensed bodies. The micro

scopic approach s tarts with the interaction between individual a toms

or molecules and postulates their additivity . The attractive force

between the liquid and solid is then calculated by integration over all

a toms and molecules [8]. The result is a purely geometrical term and

the Hamaker cons tan t , Asev, which is a function of the interacting

materials . The macroscopic approach uses the macroscopic optical

properties of the interacting materials and calculates the London-van

der Waals dispersion force from th e imaginary p art of the complex

dielectric "con stan ts" [1]. It is generally accepted in the literature tha t

the macroscopic approach is more satisfying than the microscopic

approach. Presently we are concerned with a first order approximation

of the effect of the dispersion force on heat transfer . Therefore, we

will use the results of both m etho ds in order to obtain m axim um use

of the prior l i terature .

A good physical description of the microscopic approach is given

by Shelud ko [2] . Using the microscopic proced ure, the equation for

the tempera tu re independen t d ispe rs ion cons tan t i s

j_Ats- A e Aslv

6TT(2)

with


Contributed by the He at Transfer Division for publication in the JOURNAL°P HEAT TRANSFER. Manuscript received by the Heat Transfer DivisionSeptember 22,1977 .

Ae e = - n r ~a n d

Ae s = -zrz~ve* V£ vs

where U; ~ l is the nu mb er of a toms or molecules per unit volume and

ft/ is the constant in London's equation for the attraction between

Journal of Heat Transfer FEBRUARY 1978, V OL 100 / 155Copyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



two atoms. If the force of a ttraction between the l iquid and solid

(represented by Ae s) is greater th an th e force of a ttrac tion between

two liquid molecules (represented by Aee) the l iquid wets the solid

and a decrease in the vapor pressure occurs.

Using the macroscopic procedure, an equivalent Ham aker constant,

As(u can be obtained from the Lifshitz-van der Waals constant,

3h a sev

4i r(3 )

Using this definit ion to relate both approache s through an equivalentHam aker consta nt, the two procedures can be unified and compared.

For the purpose of this report the Lifshitz-van der Waals constant

can be calculated using the following approxim ate formula [1]:

UsCo '• s :M « 'f )- «. (» { )] M « | ) - i ]

di (4 )M i ? ) + tsdO] M * £ ) + i ]

where es, e„, ti are the frequency dependent dielectr ic constants of

the solid , vapor (t v = 1), and l iquid evaluated along the imaginary

frequency axis , i£ . (The ass ump tions associated w ith deriving this

equation from a more general formulation are presented in [1] . ) In

turn these imaginary frequency dependent dielectr ic constants can

be calculated from th e imaginary p art of the complex dielectr ic con

stant which can be experimentally measured. The dielectric constant,

e(o>), is a complex function of the frequency, u> .

e(o>) = t'(w) + it"(oi) (5 )

Th e function e(w) is related to the com plex index of refraction, n, the

refractive ind ex, n, and the abso rp t ion cons tan t , k, by

V7(wj = n = n(w) + ik(w)

t'(ai) = n2(«>) - k

2(w )

e"(co) = 2n{u>)k{io)

(6)

(7)

(8)

For the purpose of analysis, it is useful to consider the frequency, a,

to be a complex variable:

: w' + i£ (9)

As presented above, the dispersion force depends on the value of

the dielectr ic constant on the imaginary frequency axis , e(i£) . The

function is a lways posit ive and is re lated to the dissipation of energy.in an electromagnetic wave propagated in the medium. e( i£) is a rea l

qua ntity w hich decreases monotonically from eo (the elec trostatic

dielectr ic cons tan t, £o = n2 > 1) for dielectrics or from +<» for ideal

metals a t a) = iO to l ata> = t<= for all ma teri als [9]. Witho u t the need

for a specific model, equation (10) can be used to calculate e(i|) from

the imaginary part of the dielectr ic constant, e"{a), [9].

l o g 1 0 ( - h f )

Fig . 1 Plot o( lo g 10 [«(/£) - 1] as a function of lo g 10 (h£) for various materials

eU O = 1 +IT JO

Xe"(X)dX (10)

( X2

+ £2)

where £ and X are angular frequencies of the elec tromagn etic f ield.

Therefo re, «(i£) can be obtained directly from e xperim ental d ata on

the refractive index and a bsorption consta nt. However, due to the lack

of suffic ient data for the frequency dependent optical constants ,

semi-empirical models are usually used.

Equation (4) demonstrates that a thin wetting f i lm of adsorbed

liquid on a solid surface will be stable if the dielectric c ons tant of thesolid along the imaginary frequency axis is greater than that of the

liquid, e s(i£) > ^>((f). Therefore, it is instructive t o com pare the values

of ({iO for vario us m ater ials. In F ig. 1, th e logio [e(i'£) - 1] is presented

as a function of the energy associated with an electro magnetic wave

in electron volts, logio (ft?). The source of the individual values are

given in the Appendix. For nonpolar dielectr ic materials , the s ta tic

J ^ o m e n c l a t u r e , .

A = dispersion constant [ J]

A = Hamaker constant [ see equation (2)]

[J ]D - mean separation between close packed

planes [m ]

h2-w = Planck ' s cons tan t [J-s]

hjg = heat of vaporization [J-kg*1]

k = the rma l conduc t iv ity [W-m~ l-K _1]

k = abso rp t ion cons tan t

n = refractive index

0 = heat transferred [W-m -1]

q = heat f lux [W-m-2

]

s = solid thickness [ml

T - t e m p e r a t u r e [K ]

U = overall heat transfer coefficient, [W -

m - 2 - i f -1 ]

/3 = see eq uati on (2)

7 = interfacial free energy, [J-m~2]

5 = fi lm thickness [m ]

i] = dielectr ic constant, t = e' + it"

e = dim ensio nless film thick ness , 6/<5o

v = kinem atic viscosity [m2

^- 1

]

£ = imaginary part of complex frequency

[ rad -s -1 ]

w = defined by equation (4) [ rad-s-1

]

w = angular frequency [ rad- s -1 ]

S u b s c r i p t s

£ = l iquid phase

£v = l iquid-vapor interface

s = solid

o = evaluated at in terl ine

v = vapor phase

S u p e r s c r i p t s

d = dispersion

- = averagedA = complex index of refraction

156 / VOL 100, FEBRUARY 1978 Transactions of the ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



dielectr ic constant, eo , can be obtained using the Maxwell re lation,

m = n2, in which the index of refraction, n, can be obtained in the

visible range of frequencies. It is a measure of the ind uced polarization.

Th e n -alkanes and diatomic molecules are representative of these

materials. Curves for pentane and helium are presented in Fig. 1. For

comparison, l imiting values of nitrogen, oxygen, methane and car-

bon-tetrafluoride are also given.

Polar molecules have, in addition to the induced polarization, an

orientation polarization. As a result , the dielectr ic constant is not

constant in the frequency range below the visible . The alcohols and

water are represe ntative of this group. Th e curve for water in Fig. 1

stops at hi, = 1. At £ = 0, the dielectr ic co nstan t for water has a value

of (o ~ 81 . At hi = 3.16, e ~ 1.75. At most frequ encies t he d ielect ric

constant for pentane is less than that for water. Experimentally ,

pentane has been observed to form a thin wetting f i lm on a water

substrate [10] . Films of octane on water are marginally s table . A

theoretical s tudy of hydrocarbon adsorption on water surfaces that

describes and confirms this phenomenon has been done using Lif-

shitz's theo ry [11 ]. Th e values of t(i£) for ideal metals (sm ooth and

clean surfaces free from oxides and contaminants) are also presented

in Fig. 1. Since the com plex pa rt of the dielectric consta nt, 2nk, is large

for metals, e(t'£) also has a relatively large value. In practical heat

transfer equipment these values cannot be reached since nonidealities

(oxides and adsorbed contaminates) are usually present. However,

they do represent an upper l imit for e(i£) . Experimentally , the im

portant value is the m easure d value of Ink = t" which characterizes

the surface being used. These results a lso emphasize the fact tha tmetals are high energy surfaces which are easily wetted and easily

contaminated. The dielectr ic properties of polytetrafluoroethylene

(Teflon) fall above but close to those of pent ane. Pe nta ne w ets Teflon

since its surface tension (yd

= 0.016 Jim2) is less than the cri t ical

surface tension for the wetting of Teflon (0.0185) Jim2).

Churaev used equation (4) to calculate the Hamaker constant for

various systems [14] which are presented in Table 1 . Since the di

electric constant s for metals are larger tha n th at for water, which, in

turn, is larger than that for pentane, the absolute value of the Ha

maker constants for the alkane ^ metal systems are larger than that

for the pent ane ==: water system . For comparison, the H am ake r con

stant for the He-CaF2 system is As(u * -3 .8 X 10"21

J. [15]. The lower

values of the dielectr ic constants for helium and CaF2 result in a

relatively low value of AS£D. Th e refractive index of CaF2 is approxi

mately n « 1.4 . The approximate value of the Ham aker constan t, Aseu,can be obtained using

A sto - Ae (11)

A good approximation for the Hamaker constant, A se, is (A ss Aee)06

[16]. Therefore, we can calculate the approxim ate value of AS£U from

the more readily available values of Ass an d Aee- Calculated values

for Asgv obtained using equation (11) are presented in Table 2 . The

values for Ae e were obtained using

Aee = 2 4 T T D27 (12)

Israelachvili [17] successfully c om pare d the surface ten sion of a

Table 1

C a l c u l a t e d v a l u e s o f AS£V f o r w e t t i n g f i l m s [ 1 4 ]

- / W 1 02 0

L iqu id , subs t ra te -As£u-W20

J JLiquid, substrate

Water on quartzOctane on quartzDecane on quartzTetradecane on

quartzBenzene on quar tzC C l 4 on qua r tz

1.12 Pen tane on water 0 .741.67 He xan e on wat er 0.631.57 Dec ane on water* 0.341.34 Oct ane on steel 19.2

0.91 Dec ane on steel 18.90.56 Dec ane on gold 23.1

Wat er on gold 11.2

number of saturated hydrocarbons calculated on the basis of equation

(12) with experim ental data . In Tab le 3, a list of values for A ss is given.

Due to l imitations in the availabil i ty , in terpretation and use of the

optical data , t here is a degree of inconsistency in the values of these

constants . For example, recent data on the adsorption of a lkanes on

Teflon [4] indicate the value of A ss obtained by Vassilieff and Ivanov

[12] is probably too high. As a result of these inconsistencies, the

theoretical results based on these init ia l values of the Hamaker con

stant are obviously approximate. However, the relative values cal

culated using the same set of assumptions are very instructive. The

refinement of the procedures and the availabil i ty of more extensive

optical data will lead to more consistent results in the future .

P r e d i c t e d V a l u e s of t h e I n t e r l i n e H e a t F l o w N u m b e r ,A high

Pred icted values of the interl ine hea t f low numbe r obtained using

bulk values for the heat of vaporization and the kinematic viscosity

are presented in Tab le 4 . Th e gold-alkane system and the Teflon-

alkane system are two ideal systems for comparison since the solids

Table 2

V a l u e s o f AS£V c a l c u l a t e d u s i n g e q u a t i o n s ( 1 1) a n d ( 1 2 )

/U- io20

references

[16 & 12]Gold : 45Gold : 45G o l d : 4 5G o l d : 4 5G o l d : 4 5G o l d : 4 5G o l d : 4 5Teflon : 10.5Teflon : 10.5Teflon : 10.5

Ae e 1 0 2 0

JM e t h a n eP e n t a n eH e x a n e :H e p t a n eO c t a n e :

:3 .955.04

5.48:5 .823.08

rc-Dodecane : 6.75Ni t rogenM e t h a n eOctane :Ni t rogen

:2 .49:3 .953.08:2 .49

-A sev -1020

J9.4

10.010.210.410.510.7

8.12.51.92.6

Table 3

C a l c u l a t e d v a l u e s o f An f r o m [ 1 2 ]

Mater ia l

DecaneD i a m o n dHydroca rbon I IPo lye thy lenePo lys ty rene

A,-, -1020

J

6.741.4

7.710.511.1

Mate r ia l

Po lyp ropy leneTeflonQuar tzW a t e r

4 , ' , -10 2 0J

9.0510.5

8.656.05

Table 4

T h e o r e t i c a l v a l u e s o f t h e i n t e r l i n e h e a t fl o w n u m b e r ,

hfgAv-1

L i q u i d — S u b s t r a t e

M e t h a n e — G o l dP e n t a n e — G o l dHexane—Gold

H e p t a n e — G o l dOctane—GoldDodecane—GoldNi t rogen—GoldDecane—Stee lDecane—Stee lDecane—GoldMethane—T ef lonOctane—T ef lonNi t rogen—T ef lonC C l 4 — Q u a r t zC CI4—Quar tzO c t a n e — Q u a r t z

(h fg

9.765.264.22

3.332.621.124.172.904.143.502.600.261.530.1050.4970.418

A e - V - 1 09

W a t t s

(111 .4K)2

(293K)(293K)

(293K)(293K)(293K)(77K)(293K)(447K) 3

(293K)(111.4K)(293K)(77K)(293K)(350K) 3

(293K)

ASource

Table (2)Table (2)Table (2)

Table (2)Table (2)Table (2)Table (2)Ref.Ref.Ref.

141414

Table (2)Table (2)Table (2)Ref.Ref.Ref.

141414

Th e results in [10] indicate that decane does not form a stable film on water.

2 hfg an d v ar e evaluated at this temperature .3

Reference [14] values for A were adjusted for temperature using equations (11)and (12) .

Journal of H eat Tran sfer FEBRUARY 1978, V OL 100 / 157




have been extensively studied and the alkanes are s imple nonpolar

fluids. More importantly, these systems represent a broad range, since

gold is a very high surface energy material and Teflon a very low

surface energy mate rial . The value of the interl ine heat f low n umb er

for methane-gold is 9.76 X 1CT9

W versus 2.6 X 1CT9

W fo r me th

ane-Teflon. Increasing the molecular weight of the fluid gives h/ g Ah

= 2.62 X 10~ 9 W or octane-gold. This decrease occurs because the

ratio h/g/n decreases faster than A increases. For all practical purposes

these numbers bracket the range for simple fluids. The inconsistencies

present in the table values for the interl ine heat f low number result

from the inconsistencies in th e values of A in the literatur e. For a finite

contact angle system, the "interline region" as defined herein vanishes

and the t hin f i lm starts w ith a f inite thickness. Polar f luids and l iquid

metals are not addressed herein .

T h e T h e o r e t i c a l I n t e r l i n e H e a t S i n k , Q, a n d I n t e r l i n eL e n g t h , x .

The above results can be used to estimate various characteris t ics

of the interl ine evaporation process. The theoretical in terl ine heat

sink for a unit length of interl ine, Q, is obtained using equation (1).

The length of the interl ine region of thickness range i\ > 1, x, is [6]

x = [ 2 £ n T,]0-

6[ q ] - ° '

5[hSgAhf

b(13)

These equations are presented in Figs. 2 and 3 for r\ - 10. In Fig. 2

some of the values of Ah /g/v l is ted in Table (4) are marked on the

theoretical curve. Obviously an increase in the interl ine heat f low

number gives an increase in the theoretical heat s ink, Q, and the

length of the interl ine region, *. Two of the systems, decane-steel an d

carbon tetrachlo ride-glass are given in more detail in Fig. 3. Curve (l a)

represents the decane-steel system at 293K, and Curve (lfe) represents

the system at 447K. Curves (2a and 26) represent the carbon tetra

chloride system st 293K and 350K. The results predict that the in

terline heat sink, Q, for the decane-steel system is approximately three

times larger than th at for the carbon tetrachloride system at the sam e

heat flux and P„ = 1.01 X 10 5N-m~

2. This results from the increased

length of the evaporating thin f i lm, x, of th ickness 1 < JJ < 10, which

is a function of the interline h eat flow num ber. Th e increase of Q with

temp era tu re re su l t s from the subs tan t ia l dec rease in the k inem at ic

viscosity in this temperature range. This increase will not continue,

since the dispersion constant will decrease substantia lly at h igher

tempe ra tu res . Fo r the ca rbon te t rach lo r ide -quar tz sys tem, bo th th e

dispersion constant and the reciprocal of the kinematic viscosity increase with temperature at th is temperature level (note the effect of

the relative size of At e on Aseu in equation (11)) .

As presented in [6] , the equa tion for the interl ine he at s ink can be

wr i t ten a s

Q = [2 In (r,)ka(Ts - Tv) S - T -5

[A h fg ^ p ( l4 )

thereby defining an effective overall heat transfer coefficient, Ua[

as

fo r

U ef[ =[2kaAh fghS(T s-T u)]^

Q = U el{ [£ n u]°-6

[T a - Tu]

(15)

(16)

If the temperature in the solid, T s, at the distance, S, from the surface

is known, equa tion (14) can be used to obta in Q using the values of

A hf gh presen ted he re in .

D i s c u s s i o n

A theoretical procedure to determine the interl ine heat s ink ca

pabili ty of an evaporating thin f i lm using the macroscopic optical

properties a nd the therm ophysical proper ties of the system h as been

outlined above. This procedure significantly increases our under

standing of the interline and shows how the level of the interline

tran spo rt processes is controlled by the dispersio n forces result ing

from th e interaction of the solid and liquid. Th e proce dure can be used

to predict the relative s ize of the interl ine heat s ink and, thereby,

enhanc e the developme nt of heat transfer e quip me nt which use the

evaporating interline region. Due to the basic nature of the equations,

their c urre nt level of develop ment and th e l imited availabili ty of

optical data, the use of ideal systems (e.g., neglecting surface rough

ness) has been emphasized. How ever, th is does not de trac t from the

significance of the results, since ideal systems give an order of mag

nitude approximation and enhanced understanding of real systems,

and reveal the basic mechanisms involved in the transport pro

cesses.

The constant heat f lux model explicit ly demonstrated the impor

tance of the in terl ine he at f low nu mb er [6]. Using theo retical values

of the interline he at flow num ber, th e relative size of the interline heat

sink and the interl ine length can be determine d. The absolute upper

limit of the interl ine heat s ink cannot be determ ined using only these

procedures because the imp ortan t questions of interl ine s tabili ty and

the transition from the thin film to the meniscus region have not been

addressed herein . However, these concepts do describe why the in

terline heat sink of a system like decane-steel should be greater thantha t of a system like carbon tetrachlo ride-quartz . A procedure is now

available whereby various systems can be theoretically compared.

By necessity, there are aspects of the abo ve theoretical results which

2 5 -

a3 2 0

ic r

X10

fO "

O

_

V

1 1

© >

/ ©

1 1

1

©

t

j

©©©©©©

1

( t ) .

METHANE

PENTANE

NITROGEN

DECANE

METHANE

c c i 4

! i

- GOLD

- GOLD

- GOLD

- STEEL

- TEFLON

- QUARTZ

\

\ i

( I I I .4K)

(293K)

( 77 K )

(293K)

(I I I .4K)

(350K)

1 1

20 4 0 6 0

K8 0 100 120 140 160 180

/ I / ] - I 01 0

, wafts

F ig . 2 Graphical representation of equations (1) and (13) for ij = 10

80 I60 240 320 400 48 0 560 640 720

qC" °-5

, w a t t s 0 - 5 / m

F ig . 3 Heat sink capability, 0, and interline length, x, for ?j= 10; (1a> ' ' o c 'ane-steel at 293 K; (1b) decane-steel at 447 Kj (2a) CCI4-quartz at ,'

J,'3 Ki

(2b) CCI4.quartz at 350 K

158 / V OL 100, FEBRUARY 1978 Tra nsa ction s of the AS'ME




n e ed e x p e r i m e n t a l e v a l u a t i o n , r e f i n e m e n t , a n d e x t e n s i o n b e c a u s e o f

t h e a s s u m p t i o n s i n t h e m o d e l s . T h e s e r e s u l t s i n d i c a t e t h a t t h i s c a n

b e a c c o m p l i s h e d b y m e a s u r i n g t h e i n t e r l i n e s t a b i l i t y a n d t h e i n t e r l i n e

l e n g t h , x, a s a f u n c t i o n o f h e a t f lu x a n d t h e i n t e r l i n e h e a t fl o w n u m

be r .

R e f e r e n c e s

1 Dzyaloshin skii, I . E., Lifshitz, E. M., and Pitaevsk ii, L. P., "The General

Theory of Van der W aals Forces ," Adv. Physics vol . 10,1959, pp. 165 -208.2 She ludk o, A. , Colloid Chemistry, Elsevie r Pub . Co. , New York, N .Y. ,

1966.

3 Deryagin, B. V., and Zorin, A. M., "Opt ical Study of the Adsorptio n and

Surface Condensa t ion of Vapors in the Vic ini ty of Sa tura t ion on a SmoothSurface," in Proc. 2nd Int. Congr. Surface Activity London, Vol . 2 ,1957 , pp.145-152.

4 Hu, P . , and Adamson, A. W. , "Adsorpt ion and Contac t Angle S tudies ,"

J. Colloid and Interface Science, Vol. 59, No. 3 ,1977 , p p. 605 -613.

5 Potash , M. L. , J r . , and Wayner , P . C . Jr . , "Evapora t ion f rom a Two-

dimensiona l Extended Meniscus ," International Journal of Heat and MassTransfer vol. 15, 1972, pp. 1851-1863.

6 Wayn er , P . C , Jr . , "A Constan t Hea t F lux Model of the Evapo ra t ing

Inte r l ine Region," to be publ ish ed in the International Journal of Heat andMass Transfer.

'; 7 Wayner , P . C , Jr . , Kao, Y. K. , and LaCroix, L. V., The Inte r l in e Hea t-

Transfer Coefficient of an Evapo ratin g Wettin g Film, International JournalofHeat and Mass Transfer vol . 19,1976, pp. 4 8 7 - 4 9 1 .

8 Ha m a ke r , H . C , Physica (Utrecht) vol . 4 ,1058 (1937).

9 Lan dau, L. D., and Lif shi tz , E. M. , Electrodynamics of ContinuousMedia, Pergamo n Pres s , New York, N.Y. 1960, pp. 247 -268.

', 10 Hauxwell , F . , and Ottewil l , R . H. ," A S tudy of the Sur face of Water by; Hydrocarbon Adso rpt ion," J. Colloid Interface Sci. vol. 34,1970, p. 473.

; 11 Richm ond, P . , Ninh am, W. , and Ottewil l , R . H„ "A Theore t ica l S tudyof Hydrocarbon Adsorpt ion on Water Sur faces Using Lif shitz Theory ," J.Colloid Interface Sci. Vol. 45 ,19 73, pp. 69-80.

12 Vassilieff, Chr . S t . , and Ivanov, I . B . , "S imple S emiempir ica l Meth od

of Calculating van der Waals Interac tions in Thin F ilms from Lifshitz Th eory ,"

; Zeit. fur Naturforsch. Vol. 31a, 1976, pp. 1584-1588.

13 Alexander , G. , and Rut te n, G. A. F . M. , "Sur face Charac te r is t ics of

! Treated Glasses for the Prep arati on of Glass Capillary Colum ns in Gas-L iquid! Chromatography," Journal of Chroma tography Vol. 99,1974 , pp . 8 1 - 1 0 1 .

j 14 Churaev, N. V. , "Molecula r Forces in Wett in g F i lms of Non-Po la r Liq-

| uids," Kolloidnyi Zhurnal Vol. 36,1974 , pp. 323-327 .

j 15 Richm ond, P . , and Ninh am, B. W. , "Calcula t ions of Van der Waals

J Forces Across Films of Liqu id Helium Using Lifshitz The ory, " Journal of Low

I Temperature, Physics Vol. 5 ,1971 , pp. 177 -189.

16 Krup p, H. , Schnabe l , W„ and Walte r , G„ "The Lif shitz -Van der Waals

i Constant , Com puta t ion of the Lifshi tz -Van der Waals Constant on the Basisj of Optical Da ta, " Journal Colloid and Interface Science Vol. 39, 1972, pp.

421-423.[ 17 I srae lachvi li , J . N. , "Van der Waals Dispers ion Forces Contr ib ut ion to

! Works of Adhesion and Con tac t Angle on the Basis of Macroscopic Theo ry,"! Journal of Chemical Soc, Faraday Trans . I I Vol . 69,1973, pp. 1729-1738.

18 Krupp , H., "Par t ic le Adhesion Theory and Exper iment ," Advan. ColloidInterface Sci. Vol. 1,1967, pp. 111 -239.

i 19 Mulle r , V. M. , and Churaev, N. V. , "Use of the Macroscopic The ory of

Molecular Forces to Calculate the Interaction of Particles in a Metal Hydrosol,"Kolloidnyi Zournal Vol. 36,1974 , pp. 492-497 .

AcknowledgmentThe support of NASA(AMES) P.O. #A-38597-B(DG) entitled

Table A-1

Constants and references for equat ion (A-1)

Subs tance

G r a p h i t eD i a m o n dPo lye thy lenePo lypropy leneTeflonPo lys ty rene

a

0.9480.7050.4200.3900.3170.433

fa-1017

, s/rad

3.91.52.12.11.22.1

Reference

181212121212

" T h e E f f e c t o f t h e L i q u i d - S o l i d S y s t e m P r o p e r t i e s o n t h e I n t e r l i n e

H e a t T r a n s f e r C o e f f i c i e n t " is g r a t e f u ll y a c k n o w l e d g e d .

A P P E N D I X

D e t e r m i n a t i o n o f V a l u e s o f e ( i£ ) U s e d i n F i g . ( 1 )

T h e r e a r e v a r i o u s p r o c e d u r e s t o o b t a i n s p e c t r a l g r a p h s o f e (j £) f r o m

t h e l i m i t e d e x p e r i m e n t a l d a t a . O n e o f t h e m o s t u s ef u l m a k e s u s e o f

t h e f o l l o w i n g e m p i r i c a l f o r m u l a [ 1 8 ] :

6-^~ = a exp (-b& ( A - 1 )

e(i£) + 1

T h i s e q u a t i o n w a s u s e d t o o b t a i n t h e c u r v e s f or g r a p h i t e , d i a m o n d

a n d T e f l o n . T h e c o n s t a n t s a n d r e f e r e n c e s a r e g iv e n in T a b l e A - 1 .

T h e c u r v e f or p e n t a n e w a s o b t a i n e d f r o m [ 1 1 ] . T h e d i e l e c t r i c c o n

s t a n t i s e s s e n t i a l l y c o n s t a n t ( e = n2) f r o m z e r o f r e q u e n c y t h r o u g h t o

a n u l t r a v i o l e t f r e q u e n c y , w u „ , w h i c h t h e y c h o s e t o b e t h e f i r s t i o n

i z a t i on po t e n t i a l . F o r f r e que nc i e s l e s s t h a n o>u„ t he y u se d t he f o l l owing

e q u a t i o n :

t W ) = 1 + 7 ^ L z i _ (A-2)l + ( £ / « w r

F o r f r e q u e n c i e s h i g h e r t h a n t h e p l a s m a f re q u e n c y , wp, t h e y u s e d

e q u a t i o n ( A - 3 ) w i t h s t r a i g h t l i n e i n t e r p o l a t i o n i n b e t w e e n

£( ;*) = l + u p 2 / * 2 < A -3 )

T h e c u r v e fo r h e l i u m w a s o b t a i n e d f r o m [ 1 5 ] , i n w h i c h a s i m i l a r p r o

c e d u r e w a s u s e d .

T h e d a t a fo r n i t r o g e n , o x y g e n , m e t h a n e a n d c a r b o n t e t r a f l u o r i d e

w e r e o b t a i n e d u s i n g

« = n2

( A - 4 )

a n d s t a n d a r d v a l u e s f o r t h e i n d e x of r e f r a c t i o n . T h e v a l u e s f o r g o l d

a n d s i l v e r w e r e o b t a i n e d f r o m [ 1 9 ] .

Journal of Heat Transfer FEBRUARY 1978, V OL 100 / 159




tediiicel nets

This section consists of contributions of 1500 words or equ ivalent. In computing equivalen ce, a typical one-co lumn figure

or table is equal to 250 wo rds. A one -line equation Is equal to 30 w ords. The use of a b uilt-up fraction or an Integral signor summation sign in a sentence will require additional space equal to 10 words. Technical notes will be reviewed andapproved by the specific division's reviewing co mmittee prior to publication. After approva l such contributions will bepublished as soon as possible, normally in the next Issue of the journal.

Pred ic t ive Capabi l i t ies o f Ser iesSolu t ions fo r Laminar FreeCo n v ect io n Bo u n d a ry L ay e rH e a t T r a n s f e r

F. N. Lin 1 and B. T. Chao 2

No m en c l a t u rea,b = semidiameters of e ll ip tical cylinder

g = gravitational acceleration

Gr = Grashof number , g$\ Tw - T „ | L 3 / * 2


L = reference length

Nu = Nusse l t number , qwL/(Tw - T„)k


qw = wall heat flux

r = radial distance from axis of symmetry to a surface element, r =

r/L

T = t e m p e r a t u r e ; Tm and T„ refer to surface and undisturbed am

b ien t tempera tu re , re spec tive ly

U = velocity component in x- direction; u = UL/(v G r1/ 2

)

v = velocity component in y- direction; v = vL/(u G r1/ 4 )

x = streamwise coordinate; x = x/Ly = coordinate normal to x; y = G r 1 /4 y /L

a = angle defined in Fig. 1

/3 = therm al expansion coeffic ient

9 = d imens ion less tempera tu re , (T — T«,)/(TW — T„,)

v = kinetmatic viscosity

4> = si n a

i< = stre am fu nction in (x,y) coord ina te s ; ru = dtp/dy and rv =

-df/x

In t roduct ionA number of series solutions for the prediction of laminar, free

convection boundary layer heat transfer over isothermal and non-

isothermal bodies are available in the literature. Chiang and Kaye [l] 3

used a Blasius series expansion to determine local Nusselt number

for horizontal c ircular cylinders with prescribed surface tem peratu reor surface heat f lux expressed as polynom ials of distances from the

stagnation point. The same procedure was used by Chiang, Ossin, and

Tien [2] for spheres. Koh an d Price [3] devised transfo rmati ons for

the cylinder equations used in [1] to remove their depend ence on the

1 Formerly Principal Engineer, Planning Research Corp., Kennedy SpaceCenter, Fla. Now at NASA, White Sands Test Facility, Las Cruces, N.Mex.

2Professor, University of Illinois at U rbana-Champaign, Urbana, 111. Fellow,

ASME.3 Numbers in brackets designate References at end of technical note.Contributed by the Heat Transfer Division of THE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS. Manuscript received by the Heat Transfer DivisionJuly 20,1977.

coefficients t ha t appe ar in the polynom ial surface con ditio n. Fox [4]

prop osed a further gener alizatio n of the re sult s given in [2, 3] for

treating bodies of arbitrary shape. More recently, Harpole and Catton

[5] extended the Blasius series method to low Prandtl number

fluids.

Saville and Churchill [6] used a Gortler-type series for analyzing

two-dimensional, p lanar and axisymmetrical boundary layer f lows

over smooth isothermal bodies of fair ly arbitrary contours. When

applied to horizon tal circular cylinders or sph eres, the series converge

faster tha n the correspon ding Blasiu s series. Th e Gortle r-type serieshas been used by Wilks [7] and by Lin [8] for two-dimensional

boundary layers over bodies of uniform heat flux. Kelleher and Yang

[9] also employed a Gortler-type series for treating free convection

over a nonisothermal vertical plate. The class of problems considered

in [6] was trea ted by Lin and Chao [10] using a series expansi on of the

form proposed by Merk [11] and later c orrected by Chao an d Fagbenle

[12].

When assessing the usefulness of the various proposed methods,

the investigators usually made comparisons with experim ents and/or

prior published calculations. However, such comparison was almost

always l imited to horizo ntal c ircular cylinders or spheres. The main

purpose of this note is to show that conclusions based on circular

cylinders or spheres cannot be assumed to have general validity. The

radius of convergence of the various series is s trongly depen dent on

the body configuration and, to a much less extent, on the Prandtl

number and the coordinate system involved.

Go rtler Series , S-I and S-IIInvestigations of two Mer k-typ e series by the pre sent a uthors [10,

13] showed that their radius of convergence depends, among other

things, on the coordinate system used for the transformed momentum

and energy bounda ry layer equations. Since the Gortler series has the

advantage of requiring significantly less tabulation of universal

functions than the other series solution methods, i t was decided to

develop the Gortler series, using the same two sets of coordinate

system employed in [10, 13]. One of them closely resembles that of

Saville and Churchill [6].

To be definite , we consider tw o-dimensio nal or rotationally sym

metrical bodies of uniform surface temperature Tw immersed in an

infinite ambie nt f luid of und isturb ed tem per atur e T„ as i llustrated

in Fig. 1. The governing conservation equations are well known andwere given in dimensionless form by equations (10), (11), and (12) in

[10]. Th e analysis of [13] mak es use of the coord inate pair :

I = 4 f" (rU)m

dx, r, = (r<t>)m

l-l'*y, (la,b)

a stream function /(£,?;) = I 3/4 \p(x,y) and a temperature function

S(hv) = B(x,y) to obtain

V" + 3 / /" - X( / ' )2

+ 8 = 4?d&v)

am

(2)

(3)

160 / VOL 100, FEBRUARY 1978Transactions of the ASME




a) Tw > T<D

Two -D imens ional Body Axisymmetrical Body

b) T W< T (0

u t g

Two - Dimensional Body Axisymmetrical Body

Fig. 1 Physical model and coordinate system

with boundary conditions:

T&O) = J'&O) = 0, 0(1,0) = 1 (4a,b,c)

J'{1°>) - 0 , § (? ,» ) - 0 . ( 5a ,b )

In the foregoing, the p rime denotes d/dy an d d( , )/3(£,)j) is the Jaco-

bian. The configuration function X is given by

d2 + - -

£• £n (</>/r

2) (6 )

3 ( r 4 0 ) 1 / 3 dx

The Gortler series solution based on (2) and (3) is herein designa ted

as S-I. Alterna tely , the coordina te-pair employed in [10] is:

CXr2Udx, v = rU(2^)-1'2yJo

where

U> [2/;*,]".

(7a,b)

(7c)

Upon introducing a stream function /(£,?;) = (2£)_1/2\/<(*,y) and a

temperature function 0(£,i;) identically defined as d(l,ri), the mo

mentum and energy equations become

/ ' " + / / " + A [ < ? - ( / ' ) 2 ] = 2 £d(Z,n)

P r ^ 1 0" + f6' = 2

wherein the configuration function is

A = 2£ — £n U =

my

r2U3-

(8)

(9)

(10)

The boundar y conditions are the sam e as those prescribed by (4) and

(5). The Gortler series based on (8) and (9) is designated as S-II. Wh en

the body contour is such that X or A is a constant, the bound ary layer

flow becomes self-s imilar . Under such condition, X = 3A as has b een

pointed out in [13]. In the more gen eral case of nonsimilar flows, X an d

A are expand ed in series as follows:

Ml)--

HO - i. M ir

1= 0

(11)

(12)

T he quan t i t ie s Xo and 7 d epen d only on the class of body shape, n ot

on the details of the contour. So do Ao and r . For round-nosed cylin

der s , Xo = 3, 7 = % and A 0 = 1, r = 1 . For round-nosed axisymmetric

bodies, X0 = %, 7 = %, and A 0 = V2, T = lk-The appropriate S-I series solutions are

Iil~n) = Z Jiiv)¥-1= 0

Hhv) = L Hn)lh

i- 0

(13)

(14)

from which the following equation sets can be readily established:

/ ' " o + 3 / o / " o - X o ( f ' o ) 2 + 8 o = 0

P r - ^ ' o + 3 /o§'o = 0

(15)

(16)

wi th

1(17a,b,c)

(18a,b)

/o(0) = /'o(0) = 0, g0(0 )

/ 'o («) -» 0 , g 0 (») -» 0

// /

' i + 3 / o / " i - 2 ( X 0 + 2 7 ) / o / ' i

+ (3 + 47) /"0/1 + h = Ai(/ 'o)2

(19)

P r "1

^ + 3/ofli' - 4 T / 'o§ i + (3 + 47)§'o/i = 0 (20)

with

7i(0) = /'( 0) = 0, 8i(0) = 0 (21a,b,c)

/ ' i (» ) — 0 , ? !<») — 0, (22a ,b)

e tc .

The equation pair for the zeroth -order functions, (15) and (16), is

identical to th at of the Me rk series reported in [13] when Xo is replaced

by X. Th e equa tions for the higher order functions can be made independent of the X,'s ((' > 1) by introducing

h(v)~MFi(v))

§1(ij) = \ 1 5 i ( n ) i

Mv) = M2F2i(v) + MF 22(v ) •

Mn) = M2H 2i(v) + X2i?2 2(i;)J

The functions, Ft , Hi, F2i, etc . , are universal in the sense tha t th ey

are independent of parameters associated with the details of the body

contours, since 7 = % for all round-nosed cylinders and 7 = % for al l

round-nosed axisymmetric bodies as has already been pointed out.

The local heat transfer coefficient can be calculated from

Nu _ (r 0)1/

3

Gri/4_ | i /

where

3d - -— (£,0) = fl'o(0) + XiH'i(O) ?

etc.

(23)

(24)

d' 6 ,- "1(25a)

+ [\i2

H' 2i(0) + \2H'22(0)]~e~> + • • (25b)

Similar expressions have been developed for the S-II series but we

shall refrain from presenting the details .

The firs t three sets of coupled ordina ry differentia l equatio ns

governing the universal functions /o , §0, Fi, H\, F11, H21, F2% an d ff 22

for S-I and a similar set for S-II have been numerically integrated for

Pr rangin g from 0.004 to 100 and for both round-no sed axisym metr ic

and two-dimensional bodies. Tables of their wall derivatives are

available .





Comparison of ResultsRecently , Merkin [14] presented data for local heat transfer coef-

ficient over horizontal cylinders of circular and elliptical cross section

for Pr = 0.70 and 1.0. Thes e data w ere calculated by a finite difference

numerical technique. A comparison of the dimensionless heat transfer

parameter, expressed in terms of Nu G r - 1 ' 4 and obtained from various

met hods for a c ircular cylinder, is shown in Tab le 1 . Since only th e

first two terms of the Blasius series for Pr = 0.70 are available, all

series expansion results are based on two terms. In the case of both

Merk a nd G ortler series , the second term is very small , being two to

four orders of magnitude smaller than the first. The Blasius series has

a constant f irs t term which corresponds to the value at the s tagn ation

point. Th e da ta of Table 1 indicate tha t the M erk series of [10] gives

the best agreement with the f inite difference calculation and the

Blasius series is better than either Gortler series. It may also be stated

tha t a ll series expansion results are acceptable , perh aps with the ex

ception of the Gortler series at d = 150 degrees. Ha d three ter ms be en

used, the Gortler series would yield 0.280 for S-I and 0.311 for S-II;

t h u s , some improveme n t i s ob ta ined .

From a purely theoretical s tandpoint, i t may be argued that the

validity of a series solution should be examined within i ts radius of

convergence and no t be judged from th e trunca ted series . In practice,

however, when only a l imited number of terms can be calculated as

are all cases considered in the paper, the metho d of comparison used

is quite acceptable on the basis of the semidivergent nature of the

series and the relative magnitude of the available terms.

The results for an ell iptical cylinder in s lender configuration are

shown in Fig. 2. The Merk series based on the coordinate system used

in [10]4

gives excellent agreement with Merkin 's data although some

deterioration is observable when the ell ipse eccentric angle meas ured

from the stagnation point S exceeds 140 deg (x/2a = 0.9230). On the

other hand, both Gortler series , S-I and S-II , and the Blasius series

are completely useless except in the region very close to S. For ex

ample, a t x/2a = 0.09624 which corresponds to the eccentric angle of

30 deg, the f irs t thre e term s of the S-I series for Nu Gr~1/ 4

are 0.6298

+ 0.1043 — 0.7189 and the f irs t two terms of the Blasius series are

0.8898 - 1.3983.

Fig. 3 summarizes the results for an ell iptical cylinder in blunt

configuration. T he Merk series in [10] again agrees well with the finite

difference data . Some minor discrepancies are noted in the region

where the curve exhibits the maximum. The two Gortler series give

good predictions only in the region when the eccentric angle is lessthan 70 deg (x/2a = 1.027), beyond which their usefulness rapidly

deteriorates . The two-term Blasius series yields a monotonously in

creasing value of Nu G r~1/ 4

as x/2a increases and is judged to be un

accep tab le .

The pre dictive cap abili ty of the Gortler series , when applied to

ell iptical cylinders of both slender (a/6 = 4) and blunt (a /6 = 0.5)

configuration, has also been extensively studied for Pr = 0 .001,0.01,

4Results obtained by Merk series based on the coordinate system of [13] are

not as satisfactory.

Table 1 Local heat transfer parameter, Nu/G r1/4

, for isotherm al, horizontal,circular cylinder evaluated by various predictive methods (Pr = 0.70)

0,deg.

0

30

60

90

120

150

F in i te

Difference

[14]

0.4402

0.4348

0.4190

0.3922

0.3532

0.2986

M e r k

Series

[10]

0.4402

0.4346

0.4186

0.3913

0.3508

0.2923

Blasius

Series

[1.3]0.4402

0.4350

0.4192

0.3930

0.3563

0.3091

Gortler

S- I

0.4402

0.4351

0.4187

0.3898

0.3449

0.2740

Series

S-II

0.4402

0.4396

0.4241

0.3992

0.3664

0.3273

0 is the angle measured from the stagnation point; the reference length

in Nu and Gr is cylinder diameter. Data under S-I are identical to

those evaluated from [6].

° '9< r

0. 8

0. 7

0.6

%°*

0. 4

0. 3

0.2

0.0

° Finite D ifferen ce, [14]

Merk Series, [10]

GOrtler Series, S - IGortler Series, S- H

Blasius Series, [l,3]

0.0 0.2 0.4 0.6

x/2a

0. 8

Fig. 2 Heat transfer over isothermal, horizontal, elliptical cylinder in slenderconfiguration (refere nce length in Nu and Gr is 2a )

0.5

0.0

Finite Difference, [l4]

Merk Series, [10]

Gortler Series, S - I

Gortler Series, S - I

Blasius Series, [l ,3]

o.o 0.5 i.o

x /2a

2.0

Fig. 3 Heat transfer over isotherm al, horizonta l, elliptical cylinder in bluntconfiguration (reference length In Nu and Gr Is 2 a)

162 / VOL 100, FEBRUARY 1978 Tra nsa ction s of the ASME




10, a n d 1 0 0 . T h e s e r i o u s h a n d i c a p j u s t n o t e d f o r P r = 0 . 7 2 a n d i l l u s

t r a t e d i n F ig s . 2 a nd 3 ge ne r a l l y pe r s i s t s . De t a i l s w i l l no t be g ive n he r e

s i nc e f i n i t e d i f f e r e n c e d a t a a r e n o t a v a i l a b l e f o r c o m p a r i s o n . B a s e d

0 n t h e e x a m i n a t i o n o f t h e f i r s t t h r e e t e r m s o f t h e s e r i e s , i t c a n b e i n

f e rr e d t h a t t h e i n a b i l i t y o f b o t h G o r t l e r s e r i e s t o b e a p r e d i c t i v e t o o l

w o r s e n s p r o g r e s s i v e l y a s P r a n d t l n u m b e r d e c r e a s e s f r o m 0 . 7 2 t o 0 . 0 1

a n d 0 . 0 0 4 . S o m e i m p r o v e m e n t i s n o t e d f o r P r = 1 0 0 b u t n o t e n o u g h

t o j u s t i f y t h e i r u s e .

W e h a v e a l s o e x a m i n e d t h e p r e d i c t i v e c a p a b i l i t y o f t h e B l a s i u s

s e r ie s f o r P r = 0 . 0 0 4 a n d 0 . 0 1 u s i n g t h e w a l l d e r i v a t i v e s o f t h e a p

p r o p r i a t e u n i v e r s a l f u n c t i o n s r e c e n t l y g i v e n b y H a r p o l e a n d C a t t o n

[5). G o o d r e s u l t s a r e o b t a i n e d f o r ci r c u l a r c y l i n d e r s . W h e n a p p l i e d

t o e l l i p t i c a l c y l i n d e r s , t h e r e s u l t s a r e w o e f u l l y d i s a p p o i n t i n g . T o i l

l u s t r a t e , w e c i t e h e r e s o m e c o m p u t e d d a t a f o r a n e l l i p t i c a l c y l i n d e r

of alb = 4 . A t t h e e c c e n t r i c a n g l e of 3 0 d e g ( x / 2 a = 0 . 0 9 6 2 4 ) , w h i c h

i s n o t f a r fr o m t h e s t a g n a t i o n p o i n t , t h e f i r s t t h r e e t e r m s o f t h e s e r i e s

f or Nu G r ~ 1 / 4 ( w i t h 2a a s t h e r e f e r e nc e l e ng th ) r e a d : 0 . 08965 - 0 . 12238

+ 0 . 80699 f o r P r = 0 . 004 , a nd 0 . 1393 8 - 0 . 19207 + 1.2859 for P r =

0.01.

Th e un ive r sa l f unc t io ns a s soc i a t e d w i th t he M e r k s e r i e s i n [10 ] ha v e

n o t b e e n c o m p u t e d f o r P r = 0 . 0 1 a n d 0 . 0 0 4 . C o n s e q u e n t l y , d e f i n i t i v e

c o n c l u s i o n s w i l l n o t b e s t a t e d a t t h i s t i m e . H o w e v e r , j u d g i n g f r o m t h e

i n f o r m a t i o n o b t a i n e d i n t h i s a n d e a r l i e r s t u d i e s , t h e a u t h o r s s e e n o

o b v i o u s r e a s o n w h y i t w i l l n o t p e r f o r m w e l l e v e n t h o u g h t h e p r e d i c

t i o n s m a y n o t b e a s g o o d a s t h o s e f or m o d e r a t e o r l a r g e P r a n d t l

n u m b e r s .

References1 Chang, T., and Kaye, J. , "On Lamin ar Free Convection from a Horizontal

Cylinder," Proceedings, Fourth National Congress of Applied Mec hanics,University of California, B erkeley, Ju ne 1962, pp. 1213-121 9.

No n s imi la r Lamin a r F reeConvect ion FlowAlong a Noniso thermal Ver t ica lP l a t e

B. K. Meena

1

G. Na th 2

N o m e n c l a t u r e

/ = d i m e n s i o n l e s s s t r e a m f u n c t i o n

g = d i m e n s io n l e s s t e m p e r a t u r e

j g = g r a v i t a t i o n a l a c c e l e r a t i o n

I e'it 0 ) ' = s u r ^ a c e n e a t t r a n s f e r p a r a m e t e r s

< Gm = G r a s h o f n u m b e r

| G„,o = va lu e of Gw a t t h e l e a d i n g e d g e

dux - l o c al G r a s h o f n u m b e r

h, hs = l o c a l c o n v e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t s

ho = va lue o f h b a s e d o n u n i f o r m s u r f ac e t e m p e r a t u r e

k = t h e r m a l c o n d u c t i v i t y

L = c h a r a c t e r i s t i c l e n g t h o f t h e p l a t e

P = s u r f a ce t e m p e r a t u r e v a r i a t i o n e x p o n e n t

N u y = l o ca l N u s s e l t n u m b e r


j ' Research S tudent , Depar tment of Appl ied Mathematics , Indian Inst i tu te of! Science, Bang alore, Indi a.

2 Assoc ia te Professor, D epar tm ent of Appl ied Mathem atics , Indian In st i tu te

\ °f Science, Bangalore, India.

Contrib uted b y the H eat Tr ansfer Division of THE AMERICAN SOCIETY OF

i MECHANICAL ENGINEERS. Ma nus crip t received by the Heat Transfer D ivisionf June 30,1977 .

I


2 Chiang, T. , Ossin , A. , and Tien, C . L. , "Lam inar Free Convec t ion f rom

a Sphere ," ASME, JOURNAL OF HEAT TRANSFER, Vol . 86, 1964, pp. 537-542.

3 Koh, J . C. Y., and Pr ice , J . F . , "Lam inar Free Convec t ion from Nonisothermal Cyl inder ," ASM E, JOURNAL OF HEAT TRANSFER, Vol. 87,1965, pp.237-242.

4 Fox, J. , Discussio n of References [2] and [3].5 Harpole , G. M. and Cat ton , I . , "Lam inar Natura l Convec t ion about

Downw ard Fac ing Hea ted B lunt Bodies to Liquid Meta ls ," ASME , JOURNALO F H E A T T R A N S F E R , Vol. 98,1976, pp . 208-212.

6 Savi l le , D. A., and Churchi l l , S . W. , "Lam inar Free Convec t ion in

Boundary Layers near Hor izonta l Cyl inders and Ver t ica l Axisymmetr ic Bodies," Journal Fluid Mech., Vol. 29,1967, pp. 391-399.

7 Wilks, G., "Ext ernal Natura l Convection abou t Two-dim ensional Bodieswith Constant Hea t F lux," Internal Journal Heat Mass Transfer, Vol. 15,1972,pp . 351- 354 .

8 Lin, F. N., "Lam inar Free Convection over Two-dim ensional Bodies withUniform Sur face Hea t F lux," Letters in Heat and Mass Transfer, Vol. 3,1976,pp . 59- 68 .

9 Kelleher, M., and Yang, K. T., "A Gortler-type Series for Lam inar FreeConvec t ion a long a Non- isothermal Ver t ica l P la te ," Quart. J. Mech. and Appl.Math., Vol. 25,197 2, pp. 447-457. ,

10 Lin, F . N. and Chao, B . T, , "Lamin ar Free Convec t ion over Two-Di-mensiona l and Axisymmetr ic Bodies of Arbi t ra ry Co ntour ," ASME , JOURNALO F H E A T T R A N S F E R , Vol. 96,1974 , pp. 435-442.

11 Merk, H. J . , "Rap id Ca lcula t ions for Bounda ry-Layer Transfe r us ingWe dge S o lu t i ons a nd Asym pto t i c Expa ns ions , " Journal of Fluid Me chanics,Vol. 5 ,1959, pp. 460-480.

12 Chao, B . T. , and Fagbenle , R . O. , "On M erk ' s Method of Ca lcula t ing

B ounda r y La ye r T r a ns f e r , " Int. J. Heat Mass Transfer, Vol. 17, 1974, pp.223-240.

13 Chao, B . T. , and Lin, F . N. , "Loca l S imila r i ty Solut ions for Free Con

vection Boundary Layer Flows," ASME, JOURNAL OF HEAT TRANSFER, Vol.97 , 1975 , pp . 294 - 296 .

14 Merkin, J . H. , "Free Convec t ion Boun dary Layers on Cyl inders of El

liptic Cross Section," submitted for publication in ASME, JOURNAL OF HEATTRANSFER, Vol . 99,1977, pp. 453 -457.

R = f unc t ion o f s

Ri = r a t i o o f h e a t t r a n s f e r c o e f f i c i e n t s , h/h 0

s = r a t i o o f G r a s h o f n u m b e r s


Tw = s u r fa c e t e m p e r a t u r e d e p e n d i n g o n x

T„ = a m b i e n t t e m p e r a t u r e

u, v = d i m e n s i o n l e s s v e l o c i t y c o m p o n e n t s a l o n g x a n d y d i r e c t i o n s ,

r e s p e c t i v e l y

u, v = d i m e n s i o n a l v e lo c i t y c o m p o n e n t s a l o n g x a n d y d i r e c t i o n s ,


x, y = d i s t a n c e s a l o n g a n d p e r p e n d i c u l a r t o t h e p l a t e , r e s p e c t i v e l y

x, y = d i m e n s i o n l e s s d i s t a n c e s a l o n g a n d p e r p e n d i c u l a r t o t h e p l a t e ,


fS = p a r a m e t e r d e p e n d i n g o n s

fix = c o e f f i c i e n t o f v o l u m e t r i c e x p a n s i o n

e = a p o s i t i v e c o n s t n a t

i?, £ = t r a n s f o r m e d c o o r d i n a t e s

v — k i n e m a t i c v i s c o s i t y

4> = d i m e n s i o n a l s t r e a m f u n c t i o n

S u b s c r i p t s

s, x, y = p a r t i a l d e r i v a t i v e s

w = s u r f a c e c o n d i t i o n s


' ( p r i m e ) = d i f f e r e n t i a t i o n w i t h r e s p e c t t o r\

In t roduct ionT h e f re e c o n v e c t i o n p r o b l e m o n a n o n i s o t h e r m a l v e r t i c a l p l a t e

u n d e r b o u n d a r y l a y e r a p p r o x i m a t i o n s h a s b e e n s t u d i e d b y s e v e r a l

a u t h o r s [ 1 - 3 ]3 u s i n g a p p r o x i m a t e m e t h o d s s u c h a s i n t e g r a l a n d s e r i e s

3 Numbers in bracke ts designa te References a t end of technica l note .

FEBRUARY 1978, VOL 100 / 163




10 , a n d 1 0 0 . T h e s e r i o u s h a n d i c a p j u s t n o t e d f o r P r = 0 . 7 2 a n d i l l u s

t r a t e d i n F ig s . 2 a nd 3 ge ne r a l l y pe r s i s t s . De t a i l s w i l l no t be g ive n he r e

s i nc e f i n i t e d i f f e r e n c e d a t a a r e n o t a v a i l a b l e f o r c o m p a r i s o n . B a s e d

0 n t h e e x a m i n a t i o n o f t h e f i r s t t h r e e t e r m s o f t h e s e r i e s , i t c a n b e i n

f e rr e d t h a t t h e i n a b i l i t y o f b o t h G o r t l e r s e r i e s t o b e a p r e d i c t i v e t o o l

w o r s e n s p r o g r e s s i v e l y a s P r a n d t l n u m b e r d e c r e a s e s fr o m 0 . 7 2 t o 0 . 0 1

a n d 0 . 0 0 4 . S o m e i m p r o v e m e n t i s n o t e d f or P r = 1 0 0 b u t n o t e n o u g h

t o j u s t i f y t h e i r u s e .

W e h a v e a l s o e x a m i n e d t h e p r e d i c t i v e c a p a b i l i t y o f t h e B l a s i u s

s e r ie s f o r P r = 0 . 0 0 4 a n d 0 . 0 1 u s i n g t h e w a l l d e r i v a t i v e s o f t h e a p

p r o p r i a t e u n i v e r s a l f u n c t i o n s r e c e n t l y g i v e n b y H a r p o l e a n d C a t t o n

[5) . G o o d r e s u l t s a r e o b t a i n e d f o r ci r c u l a r c y l i n d e r s . W h e n a p p l i e d

t o e l l i p t i c a l c y l i n d e r s , t h e r e s u l t s a r e w o e f u l l y d i s a p p o i n t i n g . T o i l

l u s t r a t e , w e c i t e h e r e s o m e c o m p u t e d d a t a f o r a n e l l i p t i c a l c y l i n d e r

of alb = 4 . A t t h e e c c e n t r i c a n g l e o f 3 0 d e g ( x / 2 a = 0 . 0 9 6 2 4 ) , w h i c h

i s n o t f a r fr o m t h e s t a g n a t i o n p o i n t , t h e f i rs t t h r e e t e r m s o f t h e s e r i e s

f or Nu G r ~ 1 / 4 ( w i t h 2a a s t h e r e f e r e n c e l e n g t h ) r e a d : 0 . 0 8 9 6 5 - 0 . 1 2 2 3 8

+ 0 . 8 0 6 9 9 f o r P r = 0 . 0 0 4 , a n d 0 . 1 3 9 3 8 - 0 . 1 9 2 0 7 + 1.2859 for P r =

0 . 0 1 .

T h e u n i v e r s a l f u n c t i o n s a s s o c i a t e d w i t h t h e M e r k s e r i e s i n [ 1 0 ] h a v e

n o t b e e n c o m p u t e d f o r P r = 0 . 0 1 a n d 0 . 0 0 4 . C o n s e q u e n t l y , d e f i n i t i v e

c o n c l u s i o n s w i l l n o t b e s t a t e d a t t h i s t i m e . H o w e v e r , j u d g i n g f r o m t h e

i n f o r m a t i o n o b t a i n e d i n t h i s a n d e a r l i e r s t u d i e s , t h e a u t h o r s s e e n o

o b v i o u s r e a s o n w h y i t w i l l n o t p e r f o r m w e l l e v e n t h o u g h t h e p r e d i c

t i o n s m a y n o t b e a s g o o d a s t h o s e f or m o d e r a t e o r l a r g e P r a n d t l

n u m b e r s .

References1 Chang, T. , and Kaye, J. , "On Lam inar Free Convection from a Horizontal

Cylinder ," Proceedings, Fourth National Con gress of Applied Mechan ics,

Univers i ty of Ca l i fornia , Berke ley , June 1962, pp. 1 213-1 219.

No n s imi la r Lamin a r F reeConvect ion FlowAlong a Noniso thermal Ver t ica lP l a t e

B. K. Meena

1

G. Na th 2


/ = d i m e n s i o n l e s s s t r e a m f u n c t i o n

g = d i m e n s io n l e s s t e m p e r a t u r e

j g = g r a v i t a t i o n a l a c c e l e r a t i o n

I e'it 0 ) ' = s u r ^ a c e n e a t t r a n s f e r p a r a m e t e r s

< Gm = G r a s h o f n u m b e r

| G„,o = va lu e of Gw a t t h e l e a d i n g e d g e

dux - l o c al G r a s h o f n u m b e r

h, hs = l o c a l c o n v e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t s

ho = va lue o f h b a s e d o n u n i f o r m s u r f ac e t e m p e r a t u r e

k = t h e r m a l c o n d u c t i v i t y

L = c h a r a c t e r i s t i c l e n g t h o f t h e p l a t e

P = s u r f a ce t e m p e r a t u r e v a r i a t i o n e x p o n e n t

N u y = l o ca l N u s s e l t n u m b e r


j ' Research S tudent , Depar tment of Appl ied Mathematics , Indian Inst i tu te of! Science, Bang alore, Indi a.

2 Assoc ia te Professor, D epar tm ent of Appl ied Mathem atics , Indian Inst i tu te

\ °f Science, Bangalore, India.

Contrib uted b y the H eat Tr ansfer Division of THE AMERICAN SOCIETY OF

i MECHANICAL ENGINEERS. Ma nus crip t received by the Heat Transfer D ivisionf June 30 , 1 977.

I


2 Chiang, T. , Ossin , A. , and Tien , C . L. , "Lam inar Free Convec t ion f rom

a Sp here , " ASM E, JOURNAL OF HEAT TRANSFER, Vol. 86 , 1964, pp. 5 37-542.

3 Koh, J . C. Y. , and Pr ice , J . F . , "Lam inar Free Convec t ion f rom N onisothermal Cyl inder , " ASM E, JOURNAL OF HEAT TRANSFER, Vol . 87 , 1 965 , pp.237-242.

4 Fox, J. , Discussio n of References [2] and [3].5 Harpole , G. M. and Cat ton , I ., "Lam inar Natura l Convec t ion about

Downw ard Fac ing Hea ted Blu nt Bodies to Liquid Meta ls , " ASME, JOURNALO F H E A T T R A N S F E R , Vol. 98 , 1 976 , pp . 2 0 8 - 21 2 .

6 Savi l le , D. A., and Churchi l l , S . W. , "Laminar F ree Convec t ion in

Boundary Layers near Hor izonta l Cyl inders and Ver t ica l Axisymmetr ic Bodies," Journal Fluid Mech., Vol . 29 , 1 967 , pp . 391 - 399 .

7 Wilks, G., "Ext ernal Natura l Convection abou t Two-dim ensional Bodieswith Constant Hea t F lux , " Internal Journal Heat Mass Transfer, Vol. 1 5 , 1 972,pp . 3 5 1 - 3 5 4 .

8 Lin, F. N., "Lam inar Free Convection over Two-dim ensional Bodies withUniform Sur face Hea t F lux , " Letters in Heat and Mass Transfer, Vol. 3 , 1976,pp . 5 9 - 6 8 .

9 Kelleher , M., and Yang, K. T. , "A Gortle r-type Series for Lam inar FreeConvec t ion a long a Non- isothermal Ver t ica l P la te , " Quart. J. Mech. and Appl.Math., Vol . 2 5 , 1 972 , pp . 447 - 45 7 . ,

1 0 Lin , F . N. and Chao, B . T , , "Lam inar Free Convec t ion over Two-Di-mensiona l and Axisymmetr ic Bodies of Arbi t ra ry Contour , " ASM E, JOURNALO F H E A T T R A N S F E R , Vol. 96 , 1 974 , pp . 435 - 442 .

11 Merk , H. J . , "Rap id Ca lcula t ions for Bounda ry-Layer Transfe r us ingWe dge S o lu t i ons a nd Asym pto t i c Expa ns ions , " Journal of Fluid Mechan ics,

Vol . 5 , 1 9 5 9 , pp . 460 - 480 .

12 Chao, B . T. , and Fagbenle , R . O. , "On M erk ' s Method of Ca lcula ting

B ounda r y La ye r T r a ns f e r , " Int. J. Heat Mass Transfer, Vol. 17 , 1974, pp.223 - 240 .

13 Chao, B . T. , and Lin , F . N. , "Loca l S imila r i ty Solut ions for Free Con

vection Boundary Layer Flows," ASME, JOURNAL OF HEAT TRANSFER, Vol.97 , 1 975 , pp . 294 - 296 .

14 Merkin , J . H. , "Free Convec t ion Bound ary Layers on Cyl inders of El

liptic Cross Section," submitted for publication in ASME, JOURNAL OF HEATTRANSFER, Vol . 99 , 1977, pp. 453-457.

R = f unc t ion o f s

Ri = r a t i o o f h e a t t r a n s f e r c o e f f i c i e n t s , h/h 0

s = r a t i o o f G r a s h o f n u m b e r s


Tw = s u r fa c e t e m p e r a t u r e d e p e n d i n g o n x

T„ = a m b i e n t t e m p e r a t u r e

u, v = d i m e n s i o n l e s s v e l o c i t y c o m p o n e n t s a l o n g x a n d y d i r e c t i o n s ,


u, v = d i m e n s i o n a l v e lo c i t y c o m p o n e n t s a l o n g x a n d y d i r e c t i o n s ,


x, y = d i s t a n c e s a l o n g a n d p e r p e n d i c u l a r t o t h e p l a t e , r e s p e c t i v e l y

x, y = d i m e n s i o n l e s s d i s t a n c e s a l o n g a n d p e r p e n d i c u l a r t o t h e p l a t e ,


fS = p a r a m e t e r d e p e n d i n g o n s

fix = c o e f f i c i e n t o f v o l u m e t r i c e x p a n s i o n

e = a p o s i t i v e c o n s t n a t

i?, £ = t r a n s f o r m e d c o o r d i n a t e s

v — k i n e m a t i c v i s c o s i t y

4> = d i m e n s i o n a l s t r e a m f u n c t i o n

S u b s c r i p t s

s, x, y = p a r t i a l d e r i v a t i v e s

w = s u r f a c e c o n d i t i o n s


' ( p r i m e ) = d i f f e r e n t i a t i o n w i t h r e s p e c t t o r\

In t roduct ionT h e f re e c o n v e c t i o n p r o b l e m o n a n o n i s o t h e r m a l v e r t i c a l p l a t e

u n d e r b o u n d a r y l a y e r a p p r o x i m a t i o n s h a s b e e n s t u d i e d b y s e v e r a l

a u t h o r s [ 1 - 3 ]3 u s i n g a p p r o x i m a t e m e t h o d s s u c h a s i n t e g r a l a n d s e r i e s

3 Numbers in bracke ts designa te References a t end of technica l note .

FEBRUARY 1978, VOL 100 / 163Copyright © 1978 by ASMEDownloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



expansion methods. Recently , Sparrow and co-workers [4-6] pre

sented an approximate method known as the local nonsimilari ty

meth od, for solving nonsimilar bou ndary-layer problems, which gives

more accurate results than the series and integral methods. More

recently , Kao and Blrod [7] have developed anoth er a pprox ima te

method for the solution of nonsimilar boundary layer problems. The

results are found to be sl ightly more accura te tha n those of the local

nonsim ilari ty me thod. Fur ther , Kao, e t al . [8] have applied this

technique to study the free convection problem on a vertical flat plate

with s inusoidal and exponential wall temperature variations and

linearly varying and exponentially increasing heat f lux. They havecomp ared their results with those of the difference-differentia l

method and found them in excellent agreement.

In the present note , an implicit f inite-difference techn ique [9] has

been applied to s tudy laminar free convection boundary-layer f low

along a vertical f la t p late with power-law distr ibution of wall tem

perature . I t may be noted tha t Kao, e t a l . [8] have not s tudied this ty pe

of wall temperature variation. The results have been compared with

those of the difference-differentia l , series , in tegral , and local non-

similari ty methods. Furthermore , the comparison has also been made

with the results obtained by use of the asym ptotic metho d developed

by Kao and Elrod [7] and K ao, e t a l . [8] .

G o v e r n i n g E q u a t i o n s . T he equa t ions govern ing the s teady

lamin ar boun dary-la yer free convection f low along a vertical non u-

niformly h eated f la t p late can be expressed in the dim ensionless form

as [ 3, 8]

/" ' + [3 - 2/3(s)]//" - 2/ '2 + g = 4R(f'fs' - f"fs) (1)

P r - ' g " + [3fg' - 2/3(s)(fe' + 2gf')\ = 4R(f'gs - g% ) (2)

wi th boundary cond i t ions

f(s, o) = f'(s, o) = f'(s, co) = g(s, =») = 0, g(s, o) = 1 (3)

Here , the surface temperature has been assumed to be of the form

[3]

G,„/G,„0 = s = 1 + txP (4)

where

G,„ = g0iL3(Tw - T „ ) /„2 ( 5 a )

i = C Gwdx = (s- 1 ) ! /P (p + s)(l/eY'p(p + D - K ^ o ( 5 b)Jo

v = y[ G w2/m }

i/4, + •= Gwm/Gj)^m, „> (5c)

g(S, r,) = (T - T„)/(TW - T„), R = s0(s) (5d)

d = (dGJdx)G u>-2 JXGlodx = p(p + s)(p - l ) - i ( s - l ) s - 2

(5e)

u = ULh = tpy.v = vL/t> = —if/x (5f)

The local Nusselt number is given by [3]

N u ^ G ^ )1

'4 = -[xGJ(m\ u*g'(k, 0 )

= - ( 2 ) " 1 / 2 [ s ( p + l){p + s)-l'*V.g'(s, o) (6)

where

N u ? = hx/k, h = -k dT/dy)J(Tw - T„ ) (7a)

G uS = g(3iX3(T,„ - Ta)/v\ x = x/L, y = y/L (7b)

It may be remarked t hat equations (1 ) and (2) have been writ ten after

correcting some minor printing mistakes in the equations given in

reference [3] and they are sam e as those in reference [8] ,


Equ ation s [1 ] and [2] und er the bounda ry condition s of equation

(3) have been solved nume rically using an implicit f inite-difference

scheme. Since this method is treated in great detail in reference [9] ,

for the sake of brevity , i t is not rep eate d here. The velocity and tem -

164 / VOL 100, FEBRUARY 1978

perature profiles are shown in Fig. 1 . In accordance with the usual

practice in free convection problems, the results in the form of the

Nusse lt num ber have been depicted in Fig. 2 which also contains the

correspon ding results of the integral solution [1 ] , series solution [8]

and local nonsim ilari ty (3-equa tion model) solution [ 4-6 , 1 0 ] . In ad

dit ion , Fig. 2 also contains the results obta ined by use of the asymp

totic meth od developed by Kao and Elrod [7] and K ao, e t a l . [8]. It is

evident from Fig. 2 tha t the Nus selt num ber N uj increases with s and

p . The results are found to be in good agreement with those of the

series method, local nonsimilari ty method, and asymptotic method

of references [7-8] . We have also com pared our results w ith those of

the difference-differentia l meth od [ 11 -12] and found that they agree

at lea st up to 3-decimal place. He nce , i t is not shown in Fig. 2 . The

finite-differe nce or differen ce-differ ential me tho d gives accurate

solution for larger values of s as compared to other approximate

methods. It may be noted that the local nonsimilarity solution method

gives more accurate results than the series method whereas the as

ymp totic m ethod of references [7-8] is found to give sl ightly better

results than the local nonsimilari ty method for large distances from

the leading edge of the f la t p lat e .

In order to compare our results w ith those of the integral m ethod

of reference [1 ] , we introduce th e ratio hlh„ given by [3]

h/h s = [sV*Nus/G WIV4]/[0.508 |P r2/ (0 . 9 5 2 + P r ) !

1^ ] (8)

where hs is g iven by the integral m etho d of reference [1 ] , Ri is also

known from reference [1 ] , and N uf /G m j

1 / 4

is given by equation (6).T he ra t io h/h s is shown in Fig. 2. As in referen ce [3], it is found tha t

the integral solution overestimates the rates of heat transfer .

1w\www\ / - s - 1 . 0

" \

Avs=1.4 \

\ ^ - - A^ v ^ s ^ . "*.// \ ^^>^/7 \ ^ 5 ^ *

1 I i

p-1.0,Pr= 0.72

f'g

l ^ I

0 1.0 2.0 3.0 4.0 5.0 6.0

Fig. 1 Velocity and temperature profiles

Pr-0.72

Fig. 2 Heat transfer distribution

Transact ions of the A^ME




ConclusionsThe h eat transfer increases along the f la t p late . This conclusion is

based upon limited study of the wall temperature distr ibution. The

heat transfer may not increase along the plate if o ther wall tem pera

ture distr ibutions are considered. The finite-difference results are

almost the sam e as those of the difference-differentia l me thod and

they are also in good agreemen t with those of the local no nsimi lari ty

method and the asymptotic method of Kao and Elrod. However , the

local nonsimilarity method gives more accurate results than the series

method and the asymptotic method is found to be more accurate tha n

the local nonsimilari ty method for large distances from the leadingedge. The integral solution overestimates the rates of heat trans

fer.

AcknowledgmentsOne of the authors (G. Nath) would l ike to thank the Council for

International Exchange of Scholars , Washington , for the award of a

fellowship and the Ohio State University for providing the facili

ties.

R e f e r e n c e s

1 Sparrow, E. M., "Lam inar Free Convection on a Vertical Plate withPrescribed Nonuniform Wall Heat Flux or Prescribed Nonuniform WallTemperature," NACA Technical Note, No. 3508,1955.

2 Kuiken, H. K., "General Series Solution for Free Convection Past aNonisothermal Vertical Flat Plate," Applied Scientific Research, Vol. 20A,

Nos. 2-3, Feb. 1969, pp. 205 -215.3 Kelleher, M., and Yang, K. T., "A Gortler-Type Series for Laminar Free

Convection Along a Nonisothermal Vertical Plate," Quarterly Journal ofMechanical and Applied Math., Vol. 25 , Part 4, Nov. 1972, pp . 447-457.

4 Sparrow, E. M., Quack, H. and Boerner, C. J., "Local NonsimilarityBoundary-Layer Solutions," AIAA Journal, Vol. 8, No. 1 1, Nov., 1970, pp.1936-1942.

5 Sparrow, E. M., and Yu, H . S., "Local Nonsimilarity Thermal BoundaryLayer Solutions," ASME, JOURNAL OF HEAT TRANSFER, Vol. 93, No. 4, Nov.1971, pp. 328-334.

6 Minkovyez, W. J. and Sparrow, E. M., "Local Nonsimilarity Solutionfor Natural Convection on a Vertical Cylinder," ASME, JOURNAL OF HEATTRANSFER, Vol. 96, No. 2, May 1974, pp. 1 78-183.

7 Kao, T., and Elrod, H. G., "Laminar Shear Stress Patt ern in NonsimilarBoundary Layers," AIAA Journal, Vol. 12, No. 10, Oct. 1974, pp. 1 401 -1408.8 Kao, T., Domoto, G. A., and Elrod, H . G., "Free Convection Along a

Nonisothermal Vertical Flat Plate," ASME, JOURNAL OF HEAT TRANSFER,Vol. 99, No. 1, Feb. 1977, pp. 72-78.

9 Marvin, J. G., and Sheaffer, Y. S., "A Method for Solving NonsimilarBoundary Layer Equations Including Foreign Gas Injection," NASA TechnicalNote, No. D-551 6, 1969.

10 Nath , G., "Nonsimilarity Solutions of a Class of Free ConvectionProblems," Proceedings of Indian Academ y of Science, Vol. 84A, No. 3, Sept.1976, pp. 11 4-123.

11 Smith, A. M. O. and Clutter, D. W., "Solution of Incompressible LaminarBoundary Layer Equations," AIAA Journal, Vol. 1, No. 9, Sept. 1 963, pp.2062-2071.

12 Smi th, A. M. O., and Clutter, D. W., "Machine Calculation of Compressible Laminar Boundary Layers," AIAA Journal, Vol. 3, No. 4, Apr. 1965,pp . 639-647.

A L o c a l N o n s i m i l a r i t y A n a l y s i so f F r e e C o n v e c t i o n F r o m aH o r i z o n t a l C y l i n d r i c a l S u r f a c e

M. A. Wluniasser1 and J. C. Mull igan 1

NomenclatureGr, , , = Grashof number , r 3g /3(T „, - T „ ) / « 2

Nu = Nusse l t number , h r /k

Tw = temperature of cylinder surfaceT„ = ambient temperature of the f luid

x* = actual coordinate along surface of cylinder

x = dimensionless angle from lower s tagnation point , x*/r

y* = actual coordinate normal to cylinder surface

y = dimensionless normal coordinate , y*Gvwl^/r

<t>\(x),4>2(x) = azimuthal scaling functions , equation (1 )

i** = actual s trea m function

4> = dimensionless s tream function , \j/*/vGrw1 / 4

In t roduct ion

The laminar , s teady-state , free-convective heat transfer from an

isothermal horizontal cylinder has been treated analytically in the

li terature many times with varying degrees of success. Approximate

similari ty transformations [ l] ,2

approximate series and integral

procedur es [2, 3, 4], and nu meri cal finite difference m etho ds [5] haveall been employed in obtaining solutions of the thermal boundary-

layer equations in this particular nonsimilar application. Unfortu

nately , the num erical results of these s tudies are not entirely consis

tent. In this paper , an analysis of this problem using the method of

local nonsimilari ty [6 , 7, 8] is prese nted in an effort to reconcile t he

1Mechanical and Aerospace Engineering Departm ent, N orth Carolina State

University, Raleigh, N. C.2 Numbers in brackets designate References at end of technical note.Contributed by the Heat Transfer Division of THE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS. M anuscript received by th e H eat Tra nsfer DivisionJune 13,1977.

differences in these previous solutions , as well as to dem onstr ate the

util i ty of the method in nonsimilar applications characterized by

streamwise curvature. The technique of local nonsimilarity, in general,

offers to nonsimilar convective heat transfer problems many of the

advantages of s imilari ty analysis while , a t the same time, providing

a direct and accurate alternative to the more laborious series methods.

Also, in the present application, the technique facilitates a systematic

elimination of the basic approx imatio ns of local s imilari ty and coef-

f ic ients invariance involved implicit ly in Herm ann 's [1 ] classical ap

proximate s imilari ty procedure , while avoiding altogether the alge

braic approximations necessarily uti l ized by Hermann in uniquely

evaluating coordinate scaling functions.


The basic conservation equations for the free convection from a

horizontal cylinder are f irs t s implif ied by introducing the similari ty

type t rans fo rmat ions

V = y<t>i(x), * = <t>2(x)F{r,,x), e( v,x) = (T- T„)/(TW - T „ ) ( 1 )

where <f>i and 0 2 are unknow n scaling functions. In term s of these

va r iab les , the t rans fo rmed momentum and ene rgy equa t ions be

come

d (— r 1 + c F ( — ; ) - {b + c) I — 1/d

3F\

\dr,2/

+ 9

W: d(H/<t>i)

a2e de

— + a ( P r ) F — = P r (W 0 i )Or]'* Oi]

jcxF d2F dFo2F)

I dii oxon ox Or)2)

d ^ d e _ d O dF l

. d? ; ox dt] OX J

(2)

(3)

with F = oF/dii = 0 and 9 = 1 a t ij = 0 , and dF/dt, = 0 and 0 = 0 as r,

—• •». Th e coefficient fun ctions are defined by

4>2'l<t>i = a, <l>22<tn<ln' = 6 • si n x

4» i2<t>2<t'2' - c • si n x, 0 2 0 1

3 =d • si n x

(4)

(5 )

where 4>\(x = 0 ) = cons tan t and <f>2(x = 0 ) = 0 . The la tter conditions

follow from the flow pattern which must exist a t x = 0 .

I t can be shown tha t of the functions a, b, c, and d, only one com

binatio n of two will y ield a solution of equations (4) and (5 ) which is

Journ al of Heat Transfer FEBRUARY 1978, VOL 100 / 16S




ConclusionsThe h eat transfer increases along the f la t p late . This conclusion is

based upon limited study of the wall temperature distr ibution. The

heat transfer may not increase along the plate if o ther wall tem pera

ture distr ibutions are considered. The finite-difference results are

almost the sam e as those of the difference-differentia l me thod and

they are also in good agreemen t with those of the local no nsimi lari ty

method and the asymptotic method of Kao and Elrod. However, the

local nonsimilarity method gives more accurate results than the series

method and the asymptotic method is found to be more accurate tha n

the local nonsimilari ty method for large distances from the leadingedge. The integral solution overestimates the rates of heat trans

fer.

AcknowledgmentsOne of the authors (G. Nath) would l ike to thank the Council for

International Exchange of Scholars , Washington, for the award of a

fellowship and the Ohio State University for providing the facili

ties.

R e f e r e n c e s

1 Sparrow, E. M., "Lam inar Free Convection on a Vertical Plate withPrescribed Nonuniform Wall Heat Flux or Prescribed Nonuniform WallTemperature," NACA Technical Note, No. 3508,1955.

2 Kuiken, H. K., "General Series Solution for Free Convection Past aNonisothermal Vertical Flat Plate," Applied Scientific Research, Vol. 20A,

Nos. 2-3, Feb. 1969, pp. 205-215.3 Kelleher, M., and Yang, K. T., "A Gortler-Type Series for Laminar Free

Convection Along a Nonisothermal Vertical Plate," Quarterly Journal ofMechanical and Applied Math., Vol. 25, Part 4, Nov. 1972, pp . 447-457.

4 Sparrow, E. M., Quack, H. and Boerner, C. J., "Local NonsimilarityBoundary-Layer Solutions," AIAA Journal, Vol. 8, No. 11, Nov., 1970, pp.1936-1942.

5 Sparrow, E. M., and Yu, H. S., "Local Nonsimilarity Therm al B oundaryLayer Solutions," ASME, JOURNAL OF HEAT TRANSFER, Vol. 93, No. 4, Nov.1971, pp. 328-334.

6 Minkovyez, W. J. and Sparrow, E. M., "Local Nonsimilarity Solutionfor N atural Convection on a V ertical Cylinder," ASME, JOURNAL OF HEATTRANSFER, Vol. 96, No. 2, May 1974, pp. 178-183 .

7 Kao, T., and Elrod, H. G., "Laminar Shear Stress Patte rn in NonsimilarBoundary Layers," AIAA Journal, Vol. 12, No. 10, Oct. 1974, pp. 1401-1408.8 Kao, T., Domoto, G. A., and Elrod, H . G., "Free Convection Along a

Nonisotherm al V ertical Flat Pl ate, " ASME, JOURNAL OF HEAT TRANSFER,Vol. 99, No. 1, F eb. 1977, pp. 72-78.

9 Marvin, J. G., and Sheaffer, Y. S., "A Method for Solving NonsimilarBoundary Layer Equations Including Foreign Gas Injection," NASA TechnicalNote, No. D-5516, 1969.

10 Nath , G., "Nonsim ilarity Solutions of a Class of Free ConvectionProblems," Proceedings of Indian Academ y of Science, Vol. 84A, No . 3, Sept.1976, pp. 114-123.

11 Smith, A. M. O. and Clutter, D. W., "Solution of Incompressible LaminarBoundary Layer Equations," AIAA Journal, Vol. 1, No. 9, Sept. 1963, pp.2062-2071.

12 Smi th, A. M. O., and Clut ter, D. W., "Machine Calculation of Compressible Laminar Boundary Layers," AIAA Journal, Vol. 3, No. 4, Apr. 1965,pp . 639-647.

A L o c a l N o n s i m i l a r i t y A n a l y s i so f F r e e C o n v e c t i o n F r o m aH o r i z o n t a l C y l i n d r i c a l S u r f a c e

M. A. Wluniasser1 and J. C. Mull igan 1

NomenclatureGr,,, = Grashof number, r 3g/3(T„, - T„ )/ « 2

Nu = Nusse l t number , h r /k

Tw = temperature of cylinder surfaceT„ = ambient temperature of the f luid

x* = actual coordinate along surface of cylinder

x = dimensionless angle from lower s tagnation point, x*/r

y* = actual coordinate normal to cylinder surface

y = dimensionless normal coordinate , y*Gvwl^/r

<t>\(x),4>2(x) = azimuthal scaling functions, equation (1)

i** = actual s trea m function

4> = dimensionless s tream function, \j/*/vGr w1 / 4

In t roduct ion

The laminar, s teady-state , free-convective heat transfer from an

isothermal horizontal cylinder has been treated analytically in the

li terature many times with varying degrees of success. Approximate

similari ty transformations [ l] ,2

approximate series and integral

procedur es [2, 3, 4], and num erical finite difference m etho ds [5] haveall been employed in obtaining solutions of the thermal boundary-

layer equations in this particular nonsimilar application. Unfortu

nately , the num erical results of these s tudies are not entirely consis

tent. In this paper, an analysis of this problem using the method of

local nonsimilarity [6, 7, 8] is presented in an effort to reconcile the

1Mechanical and Aerospace Engineering Departm ent, North Carolina State

University, Raleigh, N. C.2 Numbers in brackets designate References at end of technical note.Contributed by the Heat Transfer Division of THE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS. M anuscript received by th e H eat Tra nsfer DivisionJune 13,1977.

differences in these previous solutions, as well as to dem onstr ate the

util i ty of the method in nonsimilar applications characterized by

streamwise curvature. The technique of local nonsimilarity, in general,

offers to nonsimilar convective heat transfer problems many of the

advantages of s imilari ty analysis while , a t the same time, providing

a direct and accurate alternative to the more laborious series methods.

Also, in the present application, the technique facilitates a systematic

elimination of the basic approx imatio ns of local s imilari ty and coef

f ic ients invariance involved implicit ly in Herm ann 's [1] c lassical ap

proximate s imilari ty procedure, while avoiding altogether the alge

braic approximations necessarily uti l ized by Hermann in uniquely

evaluating coordinate scaling functions.


The basic conservation equations for the free convection from a

horizontal cylinder are f irs t s implif ied by introducing the similari ty

type t rans fo rmat ions

V = y<t>i(x), * = <t>2(x)F{r,,x), e( v,x) = (T- T„)/(TW - T„) (1)

where <f>i and 02 are unknow n scaling functions. In term s of these

va r iab les , the t rans fo rmed momentum and ene rgy equa t ions be

come

d (— r 1 + c F ( — ; ) - {b + c) I — 1/d

3F\

\dr,2/

+ 9

W: d(H/<t>i)

a2e de

— + a ( P r ) F — = P r (W 0 i )Or]'* Oi]

jcxF d2F dFo2F)

I dii oxon ox Or)2)

d ^ d e _ d O dF l

. d?; ox dt] OX J

(2)

(3)

with F = oF/dii = 0 and 9 = 1 at ij = 0, and dF/dt, = 0 and 0 = 0 a s r,

—• •». Th e coefficient fun ctions are defined by

4>2'l<t>i = a, <l>22<tn<ln' = 6 • si n x

4» i2<t>2<t'2' - c • si n x, 0201

3 =d • si n x

(4)

(5)

where 4>\(x = 0) = constant and <f>2(x = 0) = 0 . The la tter conditions

follow from the flow pattern which must exist a t x = 0.

I t can be shown tha t of the functions a, b, c, and d, only one com

bination of two will yield a solution of equations (4) and (5) which is

Journ al of Heat Transfer FEBRUARY 1978, VOL 100 / 16SCopyright © 1978 by ASME




consistent with the above boundary conditions on <f>\ and <fe- In terms

of a and c/a, the result is

l~4 r*x

~l™i/4

4>i= (c/a)1'* (s in x)

1 / a- a {c/a)

1'3

J ( s i n ^ d H (6)

[4 rx

13 /4

- a ( c/ a )1 / 3 J ( s i n f ) i /3 dH , d = c /a (7)

f a( x) = - ( c / a ) a ( s i n x ) " 4 / 3

X c o s i I ( si n ^ ) 1 / 3 d ^ - - a ( c / a ) ( 8 )J o 3

To be consistent with the analysis of Hermann, the arbitrary functions

a an d (c/a) are chosen as 3 and 1, respectively. Thu s, equation s (6-8)

determine a consistent set of values for <t>i(x), <j>i(x), and b(x). T h e

overall solution then follows from equations (2) and (3).

In Hermann ' s c la ss ica l app rox ima te s imi la r i ty p rocedure , the

right-hand sides of equations (2) and (3) are neglected at the outset

an d b(x) assumed to be —1 for all x. The functions <pi and 02 are then

determined approximately . From equation (8) i t can be shown that

b(x) is actually - 1 only at x = ir /2 , and tha t a rapid and substa ntia l

decrease occurs beyond x = ir/2. Interesting ly, even though b(x) varies

considerably, the values of the scaling functions 4>l a nd <j>2, computed

from equatio ns (6) and (7), actually deviate only sl ightly from Her

mann's a lgebraically approximate evaluation.

Local Similari ty and Local Nonsimilari ty AnalysisLocal similarity is not new in boun dary layer theory and , in essence,

involves suppressing streamwise derivatives such as bF/bx,

d2F/dxi>T], and dQ/dx, which occur in the r ight hand term s of equa

tions (2) and (3). Such an analysis of the pr esen t problem thu s involves

the equa t ions

F'" + 3 FF " - [b(x) + 3] (F')2 + 9 = 0 (9)

0" + 3 (Pr) FQ' = 0 (10)

where the primes denote differentia tion with respect to TJ and b(x)

is deter mine d from eq uation (8). If b(x) is chosen as —1 for all x, th e

equa t ions become the same as Herman n ' s , thus i l lu s t ra t ing the two

d is t inc t ly sepa ra te and pa r t ia l ly compensa t ing approx ima t ions in

vo lved in Herm ann ' s p rocedure .

When the co r rec t b(x) function is employed along with local s imi

lari ty , a high degree of error can be expected at large streamwise

locations because of the dele tions in the governing equa tions. To avoid

this , the deletions are systematically transferred to coupled higher

order equations by changes of variable . The proced ure is , inherently ,

one of successive approximation and is known as the method of local

nonsimilari ty . In the present problem the technique avoids the b(x)

approximation as well as the local s imilari ty approximation. First ,

the new functions E(= dF/bx) an d H(= dQ/dx) are defined and sub

stitu ted in to the right hand sides of equations (2) and (3), yielding two

equations involving the four unknowns F, 9, E, a nd H. A second set

of two equati ons, of higher level, are then obta ined by differentia ting

with respect to x and subsequen t ly de le t ing the te rms (4><il<l>i)b(F'E'

- F"E)/dx and (P r ) (0 2 / < h ) d ( F ' H - Q'E)/dx. T hus , the comple te

system of equations for the local nonsim ilari ty , second level approx

imation become

F' " + 3FF" - (b(x) + 3 ) ( F ' )2

+ 9 = (fa/faWE' - F"E) (11)

9" + 3(Pr) F& = PrtfofoiKF'H - Q'E) (12)

E'" + 3FE" - (b(x) + 9)F'E'

- (b(x) - 6) F"E - (F')2b'{x) +H=0 (13)

H" + 3 Pr FH ' + Pr(6(x) - 3) F'H - Pr(h(x) - 6) Q'E = 0 (14)

wi th the boundary cond i t ions F~F' = E = E' = H=0 and 9 = 1

when j ; = 0, and F' = E' = G = H - 0 as ij -* °°.

A third level analysis involving a total of s ix equations could be

developed in the same manner, thus displacing the approximation

even further from the fundamental energy and momentum equations.

This added computation, however, was not found to be necessary j n

the present application.

Th e num erical evaluati on of <h(x), <ta(x)> and b(x) from equations

(6 , 7), and (8) is straig htfor war d. A n implicit finite difference nu

merical method [9] known as "accelerated successive replacement"

which has been found to be particularly suite d for the solution of some

types of ordinary differentia l equations, was used in solving the

equations comprising both the first level (local similarity) as well as

the second level ( local nonsimilari ty) systems. Hermann's results

provided a convenient s tart ing point for the implicit computation.

Resu lts and Discussion

Numerical results were obtained for Prandtl numbers of 0.72,0.733

1.0, 5 .0 , and 10.0. Th e ratio of the local Nu sset t num ber Nu x and

G r^1'4, N u ^ G r , , ,

1'4

= - 0 j(x) Q'(x,v = 0), are plo tted in Fig. 1 for a

Pr an dt l num ber of 1.0 and various loca tions x, along with local values

obtained by other investigators using other methods. The firs t level

results agree very well with Merkin 's f inite difference data on the

lower side of the cylinder and up to x = 120 deg, where the variation

is not more than 1.87 percent. But, as expected, the computations

begin to deviate appreciably at x > 120 deg, with the error reaching

30 perc ent a t 150 deg.

It can be seen that t he difference between the f irs t and th e second

level computations are negligible on the forward side of the cylinder.

The second level solution corrected the rear-side deviation from

Me rkin 's d ata to less than one percent a t 150 deg. Th e deviation be

tween th e second level and H erm ann 's result , however, is as high as

nine percent a t the larger angles (x - 150 deg), and as low as 0.04

percen t a t x = 90 deg. Hermann's result is c learly most accurate in

the neighborhood of x =90 deg.

The difference between th e second level and th e Gortler series so

lution is as high as nine percent at x = 150 deg althou gh negligible on

the forward side. It can also be seen that the second level solution is

closer to the f inite-difference solution tha n any of the other approx

imat e results pres ented in Fig. 1, whereas H erm ann 's results are low

by as much as eight percent. Surprisingly, the Blasius series solution

appears to be the most accurate of the previously available solu

tions.

Table 1 shows the results of computations of Nu x /Gr„ ,1/ 4

for various

0. 5

0 . 4 -

0.3

0.2

J> o

t- +

_ °+

~ X

- A

1

— FIRST LEVEL NONSIMILARITYQF(T\Mr\ i cu c i k in wc i u io tA^Ur lU L tV L L r iUr io lWl

O

++ ^ v

+ c Ng

+ Ss+

+HERMANN [l]

INTEGRAL EQUATION [2 ]

BLASIUS SERIES [3 ]

GORTLER SERIES [4 ]

-FIN ITE DIFFERENCE [5 ]

1 1 I I 1 1

uARl

cK

+

I

rY

A \

o+

t

4

1

30 60 90x, DEGREES

120 150

Fig. 1 Periphe ral variation of the local conv ective heat transfer from a horizon tal cylinder for Pr = 1.0

166 / VOL 100, FEBRUARY 1978 Tra nsac tions of the ASME




Table 1 The ratio of the local Nusselt number and Grashof number, Nu„/Grw

1/4, for various azimuthal angles x and Prandtl numbers.

^ P r ^

0

30

60

90

1Z0

150

Ar i t h m e t i c

Jodlbauer [ 4 ] '

C h u r c h i l land Chu [ l l ]

(torgan [50]

Heraann [ I ] *

0.72

F i r s t

0,3743

0.3697

0.3565

0.3347

0.3062

0.3424

0.3473

SecondLevel

0.3745

0.3700

0.3567

0.3344

0.3020

0.Z59B

0.3329

0.3330

0.3040

0.371B

0.3106

0.733

F i r s t

0.376B

0.3722

0.3588

0.3368

0.3081

0.3442

0.3500

SecondLevel

0.3770

0.3724

0.3590

0.3365

0.3039

0.2611

0.3350

0.3354

0.3060

0.3734

0.3128

1.0

F i r s t

0.4216

0.4164

0.4013

0.3762

0.3427

0.3704

0.3881

SecondLevel

0.4214

0.4162

0.4010

0.3753

0.3379

0.2853

0.3729

0.3739

0.3397

0.4036

0.3534

5.0

F i r s tLevel

0.7169

0.7075

0.6798

0.6330

0.5665

0.5276

0.6385

SecondLevel

0.7142

0.7051

0.6790

0.6325

0.5617

0.4485

0.6235

0.6512

0.5813

0.5035

0.6073

10.0

F i r s t

0.8321

0.8703

0.8356

0.7765

0,6912

0.6163

0.7737

SecondLevel

0.8822

0.8599

0,8350

0.7753

0.6377

0.5435

0.7656

0.7435

0.7150

0.7173

0 .740 !

'Da ta co r rected fo r Prand t l nunbe r us ing in te rpo la t ion fo rmu la of Merk and Pr ins [2 ] ,

values of t h e P r a n d t l n u m b e r and at various locations x. Also, the

a r i thmat ic mean , (Num/Gru ,1 '4), is compared wi th the mean value

computed from the empirical correlations of Morgan [10], Churchill

and Chu [11], and the corrected d ata of Jodl baue r [4]. Interestin gly,

the present results for the arithmetic mean computed from the second

level are as muc h as nine percent higher th an those of Churchill a nd

Chu, deviate by as muc h as ten percent from M organ 's , yet are only

four percent from Jodlbauer 's corrected data . Thus, in the in te rmediate Prandtl number range of interest here , it appears the empirical

data of Jodlbauer, corrected using the Prandtl number interpolation

formula of Merk an d Prins [2], is the most accurate . Her ma nn's an

alytical result for the mean value of the ratio Nu/Gru,1 '' 4, corrected

for Prandtl number, deviates from the second level by as much as 6.6

percent.

Conclusions

The meth od of local nonsimilari ty was found to be a simple, direct ,

and accurate procedure in reducing the partia l d ifferentia! eq uations

of s treamwise nonsimilar free convection to a format of ord ina ry

differentia l equations. When coupled with a compute r rou t ine for

solving ordina ry differentia l eq uation s, such as the me thod of accel

erated successive replacement, accurate and inexpensive boundary-

layer computations are readily carried out. Moreover, the technique

prese rves man y of the advantages inherent in s imilari ty analysis . In

the present formulation, forexample, the functions <j>\, 4>i, an d b(x)

are only geometry dependent and, after the geometry is specified,

cases of various bou ndar y condition s with various conditions of dis

sipation or radiation can be considered with little modification. Local

comp utations are readily carried out without the need for s treamwise

numerical i teration (discretization).

For the case of the isothermal horizontal cylinder, it was found that

only two levels of computation produced results s ignificantly more

accurate than either series or in tegral procedures and with in one

percen t of those ob ta ined by f inite difference computation, thus

corroborating for the f irs t t ime the correctness of these da ta . In ad

dition, it was shown tha t the commonly he ld no t ion tha t Herm ann ' s

classical solution is accurate everywhere except at the up per an d lower

s tagna t ion po in ts isnot entirely correct . It is, essentially, a solution

which is accurate only in the immed ia te ne ighborhood of x = 9 0

deg.


This research was supported by the National Science Foundation,

N S F G r a n t N o . ENG75-14616, and the s u p p o r t is gratefully ac

knowledged.

References1 Hermann, R., "Heat Transfer by Free Convection from Horizontal

Cylinders in Diatomic Gases," NACA Technical M ember, 1366, 1954.2 Merk, H. J., and Prins, J. A., "Thermal Convection in Lam inar Boundary

Layers, III," Journal of Applied Scientific Research, Sec. A, Vol. 4,1953, pp.207-220.

3 Chiang, T., and Kaye, J., "On Laminar Free Convection from a Horizontal Cylinder," Proceedings of the Fourth National Congress of AppliedMechanics, 1962, pp. 1213-1219.

4 Saville, D. A., and Churchill, S. W., "Laminar Free Convection inBoundary Layers Near Horizontal Cylinders and Vertical AxisymmetricBodies," Journal Fluid Mechanics, Vol. 29,1967, pp. 391-399.

5 Merkin, J. H., "Free Convection Boundary Layer on an IsothermalHorizontal Cylinder," ASME Paper No. 76-HT-16, Presented at ASME-AICHEHeat T ransfer Conference, St. Louis, Mo., Aug. 9-11,1976.

6 Sparrow, E. M., Quack, H., and Boerner, C. J., "Local NonsimilarityBoundary-Layer Solutions," A1AA Journal, Vol. 8,1970, pp. 1936-1942.

7 Sparrow, E. M., and Yu, H. S., "Local Nonsim ilarity Thermal Boundary-Layer Solutions," JOURNAL OP HEAT TRANSFER, T RANS. ASME, Series

C, Vol. 93,1971, pp. 328-334.8 Minkowycz, W. J., and Sparrow, E. M., "Local Nonsimilar Solutions forNatu ral Convection on a Vertical C ylinder," JOURNAL OF HEAT TRANSFER,TRANS ASME, Series, C, Vol. 96,1974, pp. 178-183.

9 Lew, H. G., "Method of Accelerated Successive Replacemen t Appliedto Boundary-Layer E quations," AIAA Journal, Vol. 6,1968, pp. 929-931.

10 Morgan, V. T., "The Overall Convective Heat Transfer from SmoothCircular Cylinders," Advances in Heat Transfer, Vol. 11,1975, pp . 199-264.

11 Churchill, S. W., and C hu, H. H. S., "Correlating Equations for Laminarand Turbulent Free Convection from a Horizontal Cylinder," InternationalJournal Heat Mass Transfer, Vol. 18,1975, pp . 1049-1053.

T r a n s i e n t T e m p e r a t u r e P r o fi le

of a Hot Wall Due to anImping ing Liqu id Drople t

M. Sek i1

H. Kawamura 2

K. Sanokawa 3

An experiment was made to investigate theheat transfer toa liquid

drop impinging on a hot surface. The transient temperature of the

heater surface was measured by a thin-film thermometer. The sur

face temperature fell to a contact temperature imm ediately after

contact with the drop. The contact temperature increased with in-

creasing initial surface temperature TQ. In the case of the water drop,however, it was approximately constant for200°C 5 T oS 300"C;

an d it increased again for To 2; 300°C. The surface temperature at

the turning point, i.e., To ~ 300°C, roughly coincided with the

Leidenfrost point.

In t roduct ionKnowledge of heat transfer from a hot wall to an impinging l iquid

droplet is required foranalysis of spray cooling, quenching, cool down

of a cryogenic pump, etc .

W h e n a drop falls on a solid surface at a t empera tu re h ighe r than

the Leidenfrost point, a vapor layer forms beneath the drop imme

diately . T he drop is then said to be in the spherod ia l s ta te . Many

s tud ies ( [ l ] ,4

for example) have been made of the heat transfer to a

d r o p in the spherodial s ta te , but only a few studies [2, 3] have bee n

mad e of the formation process of the vapor layer. The purpose of the

p resen t s tudy is to measure the rapid change of the tempera tu re of

the hot surface during the formation of the vapor layer below a liquid

d rop .

1 Research scientist, Japan Atomic Energy Research Institute, Tokai-mura,Ibaraki-ken, Japan.

2 Research scientist, Japan Atomic Energy Research Institute, Tokai-mura,Ibaraki-kem, Japan .

3Principal scie ntist, Chief of Heat Transfer Lab, Japan Atomic Energy Re

search Institute, Tokai-mura, Ibaraki-ken, Japan.4

Numbers in brackets designate References at end of technical note.Contributed by the Heat Transfer Division of THE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS. Manuscript received by the Heat Transfer Division.





Table 1 Th e ratio of the local Nusselt number an d Grashof number, Nu„/Grw

1/4, for various azimuthal angles x an d Prandtl numbers.

^ P r ^

0

30

60

90

1Z0

150

Ar i t h m e t i c

Jodlbauer [ 4 ] '

C h u r c h i l land Chu [ l l ]

(torgan [50]

Heraann [ I ] *

0.72

F i r s t

0,3743

0.3697

0.3565

0.3347

0.3062

0.3424

0.3473

SecondLevel

0.3745

0.3700

0.3567

0.3344

0.3020

0.Z59B

0.3329

0.3330

0.3040

0.371B

0.3106

0.733

F i r s t

0.376B

0.3722

0.3588

0.3368

0.3081

0.3442

0.3500

SecondLevel

0.3770

0.3724

0.3590

0.3365

0.3039

0.2611

0.3350

0.3354

0.3060

0.3734

0.3128

1.0

F i r s t

0.4216

0.4164

0.4013

0.3762

0.3427

0.3704

0.3881

SecondLevel

0.4214

0.4162

0.4010

0.3753

0.3379

0.2853

0.3729

0.3739

0.3397

0.4036

0.3534

5.0

F i r s tLevel

0.7169

0.7075

0.6798

0.6330

0.5665

0.5276

0.6385

SecondLevel

0.7142

0.7051

0.6790

0.6325

0.5617

0.4485

0.6235

0.6512

0.5813

0.5035

0.6073

10.0

F i r s t

0.8321

0.8703

0.8356

0.7765

0,6912

0.6163

0.7737

SecondLevel

0.8822

0.8599

0,8350

0.7753

0.6377

0.5435

0.7656

0.7435

0.7150

0.7173

0 .740 !

'Da ta co r rected fo r Prand t l nunbe r us ing in te rpo la t ion fo rmu la of Merk and Pr ins [2 ] ,

values of t h e P r a n d t l n u m b e r and at various locations x. Also, the

a r i thma t ic mean , (Num/Gru ,1 '4), is compared wi th the mean value

computed from the empirical corre la t ions ofMorgan [10], Churchil l

and Chu [11], and the corrected d ata of Jodl baue r [4] . In terest in gly ,

the present results for the arithmetic mean computed from the second

level are as muc h as n ine percent h igher th an those of Churchil l a nd

Chu, deviate by as muc h as ten percent from M organ 's , yet are only

four percent from Jodlbauer 's corrected data . Thus, in the in te rmediate Prandtl number range of in terest here , it appears the empirical

data ofJodlbauer, corrected using the Prandtl number in terpolat ion

formula ofMerk an d Prins [2] , is the most accurate . Her ma nn 's an

alytical result for the mean value of the ra t io Nu/Gru,1 '' 4, corrected

for Prandtl number, deviates from the second level by as much as 6.6

percent .

ConclusionsThe meth od of local nonsimilari ty was found to be a simple , d irect ,

and accurate procedure in reducing the part ia l d ifferentia! eq uations

of s treamwise nonsimilar free convection to a format of ord ina ry

differentia l equations. When coupled with a compu te r rou t ine for

solving ordina ry differentia l eq uation s, such as the me thod of accel

era ted successive replacement, accurate and inexpensive boundary-

layer computations are readily carr ied out . Moreover, the technique

p rese rves man y ofthe advantages inherent in s imilar i ty analysis . In

the present formulation, for example , the functions <j>\, 4>i, an d b(x)

are only geometry dependent and, af ter the geometry is specified,

cases of various bou ndar y condit ion s with various condit ions ofdis

sipation or radiation can be considered with little modification. Local

comp utations are readily carr ied out without the need for s treamwise

numerical i tera t ion (discret izat ion).

For the case of the isothermal horizontal cylinder, it was found that

only two levels of computation produced results s ignif icantly more

accurate than e i ther series or in tegral procedures and with in one

pe rcen t of those ob ta ined by f in i te d ifference computation, thus

corroborating forthe f irs t t ime the correctness ofthe se da ta . In ad

dit ion, it was shown tha t the commonly he ld no t ion tha t He rm ann ' s

classical solution is accurate everywhere except at the up per an d lower

s tagna t ion po in t s isnot entire ly correct . It is, essentially, a solution

which is accurate only in the immed ia te ne ighbo rhood of x =90

deg.


This research was supported by the National Science Foundation,

N S F G r a n t N o . ENG75-14616, and the s u p p o r t is gratefully ac

knowledged.

References1 Hermann, R., "Heat Transfer by Free Convection from Horizontal

Cylinders in Diatomic Gases," NACA Technical Member, 1366, 1954.2 Merk, H. J., and Prins, J. A., "Thermal Convection in Laminar Boundary

Layers, II I," Journal of Applied Scientific Research, Sec. A, Vol. 4,1953, pp.207-220.

3 Chiang, T., and Kaye, J., "On Laminar Free Convection from a Horizontal Cylinder," Proceedings of the Fourth National Congress of AppliedMechanics, 1962, pp. 1213-1219.

4 Saville, D. A., and Churchill, S. W., "Laminar Free Convection inBoundary Layers Near Horizontal Cylinders and Vertical AxisymmetricBodies," Journal Fluid Mechanics, Vol. 29,1967, pp. 391-399.

5 Merkin, J. H., "Free Convection Boundary Layer on an IsothermalHorizontal Cylinder," ASME Pa per No. 76-HT-16, Presented at ASME-AICHEHeat T ransfer Conference, St. Louis, Mo., Aug. 9-11,1976.

6 Sparrow, E. M., Quack, H., and Boerner, C. J.,"Local NonsimilarityBoundary-Layer Solutions," A1AA Journal, Vol. 8,1970, pp. 1936-1942.

7 Sparrow, E. M., and Yu, H. S., "Local Nonsimilarity Therma l Boundary-Layer Solutions," JOURNAL OP H E A T TRANSFER, T RANS. ASME, Series

C, Vol. 93,1971, pp. 328-334.8 Minkowycz, W J., and Sparrow, E. M., "Local Nonsimilar Solutions forNatu ral Convection on a Vertical C ylinder," JOURNAL OF HEAT TRANSFER,TRANS ASME, Series, C, Vol. 96,1974, pp . 178-183.

9 Lew, H. G., "Method of Accelerated Successive Replacemen t Appliedto Boundary-Layer E quations," AIAA Journal, Vol. 6,1968, pp. 929-931.

10 Morgan, V. T., "The Overall Convective Heat Transfer from Sm oothCircular Cylinders," Advances in Heat Transfer, Vol. 11,1975, pp . 199-264.

11 Churchill, S.W., and C hu, H. H. S., "Correlating Equa tions for Laminarand Turbulent Free Convection from a Horizontal Cylinder," InternationalJournal Heat Mass Transfer, Vol. 18,1975, pp . 1049-1053.

T r a n s i e n t T e m p e r a t u r e P r o fi le

of a Hot Wall Due to anImpinging Liquid Drople t

M. S ek i1

H. Kawamura 2

K. Sanokawa 3

An experiment was made to investigate theheat transfer toa liquid

drop impinging on a hot surface. The transient temperature of the

heater surface was measured bya thin-film thermometer. The sur

face temperature fell to a contact temperature imm ediately after

contact with the drop. The contact temperature increased with in

creasing initial surface temperature TQ. In the case of the water drop,however, it was approximately constant for200°C 5 ToS 300"C;

an d it increased again for To 2; 300°C. The surface temperature at

the turning point, i.e., To ~ 300°C, roughly coincided with the

Leidenfrost point.

In t roduct ionKnowledge of heat transfer from a hot wall to an impinging l iquid

droplet is required for analysis ofspray cooling, quenching, cool down

of a cryogenic pump, e tc .

W h e n a drop fa l ls on a solid surface at at empera tu re h ighe r than

the Leidenfrost point , a vapor layer forms beneath the drop imme

diate ly . The drop is then said to be in the sphe rod ia l s ta te . Many

s tud ie s ( [ l ] , 4for example) have been made ofthe heat transfer to a

d r o p in the spherodial s ta te , but only a few studies [2, 3] have been

mad e of the formation process ofthe vapor layer . Th e purpose of the

p re sen t s tudy is to measure the rapid change of the t empera tu re of

the hot surface during the formation of the vapor layer below a liquid

d rop .

1 Research scientist, Japan Atomic Energy Research Institute, Tokai-mura,Ibaraki-ken, Japan.

2 Research scientist, Japan Atomic Energy Research Institute, Tokai-mura,Ibaraki-kem, Japan .

3 Principal scientist, Chief of Heat Transfer Lab, Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki-ken, Japan.

4 Numbers in brackets designate References at end of technical note.Contributed by the Heat Transfer Division of THE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS. Manuscript received by the Heat Transfer Division.

Journa l of Heat Transfer FEBRUARY 1978, VOL 100 / 167Copyright © 1978 by ASME




E x p e r i m e n t

The hot surface was an upper surface of the stainless steel cylinder

32 mm in diam eter (see Fig. 1). It was finished w ith num ber 1000

emery papers . Heat was supplied by a sheathed heater wound around

the cylinder . A liquid drop of d is t i l led wa ter or e thanol was dro pped

from a needle held 20 mm over the surface. The weight of a drop was

10.8 mg for water and 3 .75 mg for e thanol. The in i t ia l temperature

of a drop was nearly 20° C. A th in-f i lm therm om eter was employed

to measure the rapid change of the surface temp erat ure . A shape of

the evaporated film is shown in Fig. 1. Silicon monoxide (SiO) was

evaporated on the hot surface as an electric insulator; then nickel was

deposited on i t . S iO was evaporated again over the n ickel f i lm to

protect i t f rom oxidation a t h igh temperatures . The th ickness of the

SiO layer was on the order of 0.1 Mm arid tha t of the n ickel was abo ut

0.5 iim. Th e drop m ade contact with in a very small area; so the n ickel

f ilm was designed to meas ure the local tem per atur e just beneat h th e

d rop . P o ten t ia l t e rmina ls C and D were f i t ted in addit ion to current

te rmina ls A and B. A constant current was supplied from A to B.

Resistance change was detected by the change of the potentia l dif

ference between C and D. The n , the sur face temp era tu re in the a rea

bounded by.C and D (0.9 mm X 1.2 mm) could be measured.

R e s ul t s and D is c us s ion

Typical tem peratu re traces are shown in Fig . 2. Imm ediate ly af ter

the contact with a l iquid drop, the hot surface tem pera ture suddenly

d ropped to the "con tac t" t empera tu re (T c ) . The sudden d rop was

followed by an oscil la tory varia t ion. The se oscil la tory changes pro b

ably indicate nucleate boil ing. Liquid-solid contact was s tudied by

Bradfie ld [4] and other investigators . I t was found that the contact

was nearly periodic for a bouncing droplet. However, the period of the

p re sen t t emper a tu re o sc i l l at ion was random and much sho r te r than

that of the bouncing. Therefore , the oscil la t ion is a t tr ibutable to the

boiling.

The boil ing became more vigorous as the surface temperature in

creased. When the in i t ia l surface supe rhea t ATo reached ab out 170

K, the water drop obtained the spheroidal s ta te . For 170 K < T 0 <

200 K, however, some oscil la t ion is st i l l in evidence suggesting th at

the drop then wets the surface in term ittentl y . For To £ 200 K, the

tem peratu re drop sudden ly and recovers smoothly; no oscil la tion can

be observed in the traces (see Fig . 2-a-4) . In th is case , the min imu m

surface temp eratu re was reached within ten ms. For an e thanol dro p,

the ampli tude of temperature oscil la t ion was smaller and the frequency higher than for the water drop.

The behavior of the l iquid drop changed with the in i t ia l surface

tem pera ture . When To was low, the drop was quiescent and the for

mation of a large s ingle bubble was repea ted (Fig . 2-a-l) . The boil ing

became vigorous as the increase of the surface temperature and ex

plosive vapor generation sometimes scattered the drop (Figs. 2-a-2

n e e d l e -

( N i )r es i s t anc e —

t he rm om et e r

r

insu lator '

( S i O )

-T protec tor

/SiO)

1.0 J

- 3 2 *

CA sheathed

thermocouple

and 2-b-2) . The bouncing occurred a t a re la t ively high temperature

near the Leidenfrost point (F ig . 2-a-3) . When the temperature ex

ceeded the Leidenfrost point , the drop obtained the spheroidal s ta te

and moved about.

Total vaporization t ime for To S 200°C was measured with a

stopw atch using a s l ightly concave heating surface. The effect of the

SiO fi lm on the vaporization t ime was tested , and the results are

shown in Fig. 3. Th e vaporiza tion tim e obtaine d here agrees well with

Bau mei ster 's corre la t ion [ l j . Th e data indicate that the L eidenfrost

W A T E R E T H A N O L

2 - b - i

0 0.5 1.0 1.5 sec

Tc \

2 - b - 2

V v ^ ^ / V ^ "

0 0.1 0.2 0.3 0.4 sei

0 0.1 0.2 sec

2 - a - 4

°c

19 0

1 8 0

n o

! 6 0

• ^

•

Tc

I

2 - b - 3

°c20 0

19 0

18 0

17 0

ZOO

h Tc

0 0.0 5 0.1

2 - b - 4

0 0.1 0.2 sec 0 0.05 0.1 sec

Fig. 2 Typical temperature traces, surface temperature versus time

s

60

E

o

oN

oQ. 20

\ 1

0 o * 3 8 -o o

0 ®

®

O

O ®

^ 1I , ®

Ba u m e is te r (1 ) " ~ ^ % C

© w i t h S i O

o w i t h o u t S i 0

' # _

1 0 0 2 0 0 3 0 0

A T 0

4 0 0

Fig. 1 Experimental apparatus

Fig. 3 The total vaporization time of a water drop on the surfaces with andwithout the SiO film

168 / VOL 100, FEBRUARY 1978Transact ions of the ASME




t e m p e r a t u r e in th i s s tudy is about 300°C and that the SiO f i lm has

' only a small effect on the vaporization t ime.

The con tac t t empera tu re is plotted against the in i t ia l surface su

pe rhea t inF ig . 4 . The con tac t t em pera tu re can beexpressed in "su

pe rhea t" , ATc = Tc ~ Tsap This wil l be called "contact superheat"

hereafter . Let us assume that two solids , in i t ia l ly at different tem-

> pe ra tu re s Ti and T2 , are sudden ly b rough t in to con tac t . Then the

' t empera tu re at the contact surface is given by Tc* = (Tj/3i +

f T2@2)l(0\ + P2 ) [5]. Here , fiisVpck , an d p,c, an d k represent density ,

; specific hea t, and thermal conductiv ity , respectively . The con tac t

superheat ATc*=

(Tc* ~ T'sat) is shown by the broken lines for waterand e thano l inFig. 4.

W a t e r D r o p . For AT0 5 70 K, the con-tact superheat ATc in-

> creases with the increasing in i t ia l surface superheat . It is in fair

1 agreem ent with the theoretical AT c*. So, the boil ing does not occur

' at the mo men t of the in i t ia l contact . For 100 K < A T 0 < 200 K, AT C

is approximately constant (about 50 K) and is far below ATc*. The

nucleate boil ing s tar ts during the in i t ia l decrease of the surface

t e m p e r a t u r e . W h e n A T 0 £ 20 0 K, ATc increases again with the in-

j creasing in i t ia l supe rheat . In th is case , the vapor layer forms at Tc

i (Fig. 2-a-4); film boiling occurs; and the drop ob ta in s the s tab le

' spheroidal state . Th e initial surface tem pera ture when this just occurs,

roughly coincides with the Leidenfrost point .

E t h a n o l D r o p . The experimental contact superheat ATc agrees

roughly with the theoretical ATc* forall the surface superheat tested.

i Th e cont act sup erh eat increases linearly with increasing 7'n; no range1 of To is observed where ATc isheld constant . Nucleate boil ing ap

parently d id not occur immediate ly on contact in the present exper-

i imental range; there was a lways some delay. Th ere was, however, no

' observable delay in the case of water. The difference is similar to th e

1 onefound in the s t u d y of t ransient boil ing [6] . The reason for the

difference isno t known for the transient boil ing e i ther .


The th in - f i lm the rmomete r was manufac tu red to measu re the

t r ans ien t t empera tu re va r ia t ion of the hot surface when the l iquid

d rop impac ted it.

F or the wate r d rop , the con tac t t empera tu re was app rox ima te ly

cons tan t for 100 K < AT0 S 20 0 K; for A T 0 & 200 K, it increased

again. The turn ing point , tha t is AT 0 ~ 200 K, roughly coincided with

the Le iden f ro st t em pera tu re .

F o r the e thano l d rop , the contact temperature increased l inearly

with the increasing To.

References1 Baum eister, K. J., and Simon, F. F., "Leidenfrost Te mpe rature Its Cor

relation for Liquid M etals, Cryogens, Hydro carbons, and W ater," JOURNALOP HE A T TRANSFER, T R A N S . ASME, Series, C. Vol. 95,1973, pp. 166-173.

2 Wachters, L. H. J., and Westering, N. A. J., "The H eat Transfer from aHot Wall to Impinging Water Drops in the Spheroidal State," Chemical En -

ETHANOL

0 WATER

• ETHANOL

' ®

m®

— WATER i 00

^ ^ oS,°0o° °°ao8> ^w

CTOO

O

O

O

O

%

\' > " I . 1 . I

0 100 200 300A T 0

Fig. 4 The contact temperature versus Initial surface temperature for water

and ethanol drops

gineering Science, Vol. 21,1966, pp . 1047-1056.3 Nishi-o, S., and Hirata, K., " A Study on the Leidenfrost Tem perature,"

Proceedings ofJSME Meeting, No. 760-19,1976, pp. 115-122.4 Bradfield, W. S., "Liquid-Solid C ontact in Stable Film Boiling," I&EC

Fundamentals, Vol. 5, 1966, pp. 200-204.5 Carslaw, H. S., and Jaeger, J. C, Conduction of Heat in Solids, Oxford

University Press, Oxford, second ed., 1959, p. 88.6 Sympson, H. C, and Walls, A. S., "A Study of Nucleation Phenomena

in Transient Pool Boiling," Proceedings of Institution o f MechancialEngineers,Vol. 180, Pt 3G, 1965-66.

Combined Conduct ion-Rad ia t ion Hea t T rans fe rT h r o u g h an I r r ad ia t ed Semi -t r a n s p a r e n t P l a t e

R. V iska nta1

and E. D.Hi r leman1


Bi = Bio t number , ht/k

F = radia t ive f lux

F = incident f lux

h = effective (convective + radia t ive) heat transfer coeff ic ients


n = index of refraction

q = to ta l (conductive plus radia t ive) heat f lux

q* = dimensionless heat f lux, q/[k(Ta2 — T „ i ) / t ]


t = p la te th ickness

T = d imens ion le ss pa rame te r , {n°Fb0

+ Fd°)/[k(Ta2 - Tal)/t]

8 = d imens ion le ss t empera tu re , (T — T ai)/(T<,2 - Ta\)

K = absorption coeff ic ient

\i = cosfl, wher e d is the refracted angle in the ma te r ia l

£ = dimensionless d is tance, x/t

p = ref lectiv i ty ofan in terfaceT = t r ansmiss iv i ty ofan in terface

T0 = optical th ickness ofthe med ium, at

$ = dimensionless radia t ive f lux, F/(n°F),0+ Fd°)

S u b s c r i p t s

a = refers to ambien t cond i t ions

b = refers to the beam (coll imated) component

d = refers to the diffuse component

X = refers to the wavelength

1 = refers to side 1


S u p e r s c r i p t0

= refers to the med ium of incidence (a ir)


T h e r e are a l a rge number of physical and technological systems

which use sem i-tra nspa rent solids such as g lass , p last ics , and others

a s the rma l / s t ruc tu ra l e lemen ts . Examples of such systems include

architectural windows [ l] , 2 cover plates for flat-plate solar collectors

[2], windows for metallurgical and other furnaces [3], cover tube s for

1 School of Mechanical Engineering, Purdue University, West Lafayette,Ind.

2 Numbers in brackets designate References atend of technical note.Contributed by the Heat Transfer Division of THE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS. M anuscript received by the H eat Transfe r Division,March 28, 1977.





t e m p e r a t u r e in th i s s tudy is about 300°C and that the SiO f i lm has

' only a small effect on the vaporization t ime.

The con tac t t empera tu re is plotted against the in i t ia l surface su

pe rhea t inF ig . 4 . The con tac t t em pera tu re can beexpressed in "su

pe rhea t" , ATc = Tc ~ Tsap This wil l be called "contact superheat"

hereafter . Let us assume that two solids , in i t ia l ly at different tem-

> pe ra tu re s Ti and T2 , are sudden ly b rough t in to con tac t . Then the

' t empera tu re at the contact surface is given by Tc* = (Tj/3i +

f T2@2)l(0\ + P2 ) [5]. Here , fiisVpck , an d p,c, an d k represent density ,

; specific hea t, and thermal conductiv ity , respectively . The con tac t

superheat ATc*=

(Tc* ~ T'sat) is shown by the broken lines for waterand e thano l inFig. 4.

W a t e r D r o p . For AT0 5 70 K, the con-tact superheat ATc in-

> creases with the increasing in i t ia l surface superheat . It is in fair

1 agreem ent with the theoretical AT c*. So, the boil ing does not occur

' at the mo men t of the in i t ia l contact . For 100 K < A T 0 < 200 K, AT C

is approximately constant (about 50 K) and is far below ATc*. The

nucleate boil ing s tar ts during the in i t ia l decrease of the surface

t e m p e r a t u r e . W h e n A T 0 £ 20 0 K, ATc increases again with the in-

j creasing in i t ia l supe rheat . In th is case , the vapor layer forms at Tc

i (Fig. 2-a-4); film boiling occurs; and the drop ob ta in s the s tab le

' spheroidal state . Th e initial surface tem pera ture when this just occurs,

roughly coincides with the Leidenfrost point .

E t h a n o l D r o p . The experimental contact superheat ATc agrees

roughly with the theoretical ATc* forall the surface superheat tested.

i Th e cont act sup erh eat increases linearly with increasing 7'n; no range1 of To is observed where ATc isheld constant . Nucleate boil ing ap

parently d id not occur immediate ly on contact in the present exper-

i imental range; there was a lways some delay. Th ere was, however, no

' observable delay in the case of water. The difference is similar to th e

1 onefound in the s t u d y of t ransient boil ing [6] . The reason for the

difference isno t known for the transient boil ing e i ther .


The th in - f i lm the rmomete r was manufac tu red to measu re the

t r ans ien t t empera tu re va r ia t ion of the hot surface when the l iquid

d rop impac ted it.

F or the wate r d rop , the con tac t t empera tu re was app rox ima te ly

cons tan t for 100 K < AT0 S 20 0 K; for A T 0 & 200 K, it increased

again. The turn ing point , tha t is AT 0 ~ 200 K, roughly coincided with

the Le iden f ro st t em pera tu re .

F o r the e thano l d rop , the contact temperature increased l inearly

with the increasing To.

References1 Baum eister, K. J., and Simon, F. F., "Leidenfrost Te mpe rature Its Cor

relation for Liquid M etals, Cryogens, Hydro carbons, and W ater," JOURNALO P H E A T TRANSFER, T R A N S . ASME, Series, C. Vol. 95,1973, pp. 166-173.

2 Wachters, L. H. J., and Westering, N. A. J., "The H eat Transfer from aHot Wall to Impinging Water Drops in the Spheroidal State," Chemical En -

ETHANOL

0 WATER

• ET HANOL

' ®

m®

— WATER i 00

^ ^ oS,°0o° °°ao8> ^ w CTOO

O

O

O

O

%

\' > " I . 1 . I

0 100 200 300A T 0

Fig. 4 The contact temperature versus Initial surface temperature for water

and ethanol drops

gineering Science, Vol. 21,1966, pp. 1047-1056.3 Nishi-o, S., and Hirata, K., " A Study on the Leidenfrost Tem perature,"

Proceedings ofJSME Meeting, No. 760-19,1976, pp. 115-122.4 Bradfield, W. S., "Liquid-Solid C ontact in Stable Film Boiling," I&EC

Fundamentals, Vol. 5, 1966, pp. 200-204.5 Carslaw, H. S., and Jaeger, J. C, Conduction of Heat in Solids, Oxford

University P ress, Oxford, second ed., 1959, p. 88.6 Sympson, H. C, and Walls, A. S., "A Study of Nucleation Phenomena

in Transient Pool Boiling," Proceedings of Institution o f Mechancial Engineers,Vol. 180, Pt 3G, 1965-66.

Combined Conduct ion-Rad ia t ion Hea t T rans fe rT h r o u g h an I r r ad ia t ed Semi -t r a n s p a r e n t P l a t e

R. V iskanta 1and E. D.Hi r l e ma n1


Bi = Bio t number , ht/k

F = radia t ive f lux

F = incident f lux

h = effective (convective + radia t ive) heat transfer coeff ic ients


n = index of refraction

q = to ta l (conductive plus radia t ive) heat f lux

q* = dimensionless heat f lux, q/[k(Ta2 — T „ i ) / t ]


t = p la te th ickness

T = d imens ion le ss pa rame te r , {n°Fb0

+ Fd°)/[k(Ta2 - Tal)/t]

8 = d imens ion le ss t empera tu re , (T — T ai)/(T<,2 - Ta\)

K = absorption coeff ic ient

\i = cosfl, wher e d is the refracted angle in the ma te r ia l

£ = dimensionless d is tance, x/t

p = ref lectiv i ty ofan in terfaceT = t r ansmiss iv i ty ofan in terface

T0 = optical th ickness ofthe med ium, at

$ = dimensionless radia t ive f lux, F/(n°F),0

+ Fd°)

S u b s c r i p t s

a = refers to ambien t cond i t ions

b = refers to the beam (coll imated) component

d = refers to the diffuse component

X = refers to the wavelength



S u p e r s c r i p t0

= refers to the med ium of incidence (a ir)


T h e r e are a l a rge number of physical and technological systems

which use sem i-tra nspa rent solids such as g lass , p last ics , and others

a s the rma l / s t ruc tu ra l e lemen ts . Examples of such systems include

architectural windows [ l] , 2 cover plates for flat-plate solar collectors

[2], windows for metallurgical and other furnaces [3], cover tube s for

1 School of Mechanical Engineering, Purdue University, West Lafayette,Ind.

2 Numbers in brackets designate References atend of technical note.Contributed by the Heat Transfer Division of THE AMERICAN SOCIETY OF

MECHANICAL ENGINEERS. M anuscript received by the H eat Transfe r Division,March 28, 1977.

Journa l of Heat Transfer FEBRUARY 1978, VOL 100 / 169Copyright © 1978 by ASME




parabolic solar concentra tors [4], and m any others . In these systems,

the semi- t ranspa ren t ma te r ia l s are i r radia ted from some thermal

radia t ion source at h igh tempera tu re and hea t t r ans fe r though the

materia l occurs by combined conduction and radia t ion. The problem

is of cu r ren t p rac t ica l in te re s t as hea t t r ans fe r th rough semi t rans -

pa ren t ma te r ia l s is re levan t to solar energy uti l iza t ion systems [2] .

Heat transfer through semitransparent materia ls at low tem pera tu re

on which there is incident therm al radia t ion is predic ted by ignoring

the in te rac t ion of conduction with radia t ion [2-4] . T h i s is accom

plished by assuming tha t conduc t ion p redomina te s ove r r ad ia t ion ,

and in a p lane layer of materia l with a constant th ermal conductiv ity ,

for examp le , the tem per atur e profi le is l inear . This is an oversimp lification of a physical s i tuation and may result in errors when pre

dic t ing the heat transfer ra te . An extensive review ofheat transfer

th rough rad ia t ing and nonrad ia t ing semi- t ranspa ren t so l id s is

available [5] .

The purpose of this note is to present an analytical solution for the

s teady -s ta te t empera tu re d i s t r ibu tion in a plate of semi-transparent

solid which is irradi ated on one side by beam (collimated) a nd diffuse

fluxes. The valid i ty of neglecting the in teraction of conduction with

radiation in the solid is examined by comparing the heat transfer rates

p red ic ted wi th those based on an app rox ima te app roach .

A n a l y s i s

P h y s i c a l M o d e l a n d A s s u m p t i o n s . C o n s id e r s t e a d y - s ta t e h e a t

transfer by combined conduction and rad ia t ion throu gh a p lane layer

o f semi- t ranspa ren t ma te r ia l such as glass which is i r r ad ia ted un i formly on one side from some external source. The physical thickness

of the layer t ismuch greater than the wavelength of radia t ion A so

that wave in terference effects can be neglected. The in terfaces are

assumed to be optically smoo th and paralle l to each other . Thus , the

radia t ion ref lection and transmission characteris t ics are predic ted

from Fresn el 's equations. The m ateria l is assumed to be isotropic and

homogeneous with neglig ible scatter ing compared to absorption. Th e

temperature of the medium is suff ic iently low so that in ternal emis

sion can be neglected in compar i son to the abso rp t ion of ex te rna l

radia t ion. For the sake ofconvenience and ease of analysis , the ra

diation field incident on the layer is resolved into a beam (collimated)

(Ff,°) and a diffuse ( iv°) component.

The s teady -s ta te conse rva t ion equa t ion for the p rob lem is

• ( - » £ • ' ) - o

dTq(x) = —k 1- F(x) = c o n s t a n t

dx

where the local radiative flux F(x)3 is given by [5]

F (x ) = f ° \T lx(na)n°Fbx

0e-^^{\ - P 2 X ( M ) e - 2 < ™ - « ^ ) / " ] /Jo

(1 )

(2)

where

0(roX ,p.) + 2F dx°[T 3U xx) - R 3 (2 T O X - K Xx)])dX (3)

T3(x)= Cl nWe-'HuT-tdn'/lXTo^) (4)

Jo

n3(x) = J ' T1(M ')P2(M)e-l/"(M /)" -2dMV/3(T 0X ,M ) (5)

an d

0( TOX , M) = [1 - pi(ii)p2(n)exp{-2T0\/\n\)] (6)

The factor fi accounts for multip le in terref lections between the two

interfaces. Sett ing th is factor to unity is not a lways just if iable . Th is

is expected to be part icularly true for oblique angles of incidence (n

-* 0) when both p%(p) an d P2(M) approach unity and w hen the optical

p a t h , T0X/|M| , is small such th at e x p ( — 2 T 0 \ ) / | / I | ) « 1.

S o l u t i o n . If dimensionless variables are in t roduced , the energy

equation (2) becomes

d 6+ r $ ( £ ) = q* (6)

Assuming that the total (convective plus radiative) heat transfer from

the p la te to a fluid at a m b i e n t t e m p e r a t u r e Ta can be expressed by

Newton's cooling law, energy balances at the two interfaces yield the

bounda ry cond i t ions

dO— + B i i = 0 a t f ' 0

an d

d 6

' d f+ Bi 2(G - 1) = 0 at $ = 1 (7)

The solution of equation (6) with the imposed boundary condit ions

is

9 = (1 - B i i £ ) ( l / B i i ) / D + r j[( l - £) - l / B i a K l / B h W O )

+ (1 - B i i J ) ( l / B i i B i2 ) * ( l ) + D f $(v)dvJo

-(I- BhSXl/Bh) J ^ *(7,)di?}/£) (8)

where

D = [ (1 /Bii) + (1 /Bi2) - I ] " 1(9)

Equation (8) shows that , as expec ted , in the absence of in teraction

between conduction and radia t ion (r =0) the tem per atu re profi le is

de te rmined by pu re conduc t ion . The in teg ra t ion of<J>(£) over £ (i.e.,

dum my integ ration variable i?) indicated th at equatio n (8)ca n be

carr ied out analytically , and there results

f **<")di = ( * n o ) f" \nx(n°h°Fbx° (-?-)

J o V Ff,° + Fd°I Jo { \To X/

[1 - e- <

r

»^» f + p2X (n)e-

2T

°>-'»[l - e<™/*>f)]//3(T oX,/*)

+ 2{Fdx°lT 0X)\T 4(0 ) - f4(T„xf) + R4(2 roX )

- R 4 ( 2 T O X - T OX £ )] dX (10)

where

a n d

T 4 ( x ) • J T1(M')M'Me-l/,'dM7)3(T 0,M)

RtW = f n(n')P2(n)n'ne~x,)

'd fi'/(3(T0,n)

(I D

(12)

3 The error in equations (55) and (58) of reference [5] has been corrected.

The sum ofthe conductive and radia t ive f luxes at the surface 1, i,

= 0, is obta ined from equa tions (6) and (8) ,

g i* = [ - ^ + * ( ? ) l = 1/23 + r [*(0)/Bii

L d£ Jf=o L

+ * ( 1 ) / B i 2 - C1 Hv)dv)/D (13)

Jo

Bec ause of th e rath er com plex form of equ atio n (13), it is difficult to

draw g eneral conclusions regarding the effect of in ternal absorption

of radia t ion on the to ta l hea t transfer ra te . The various specia l cases

follow imm ediate ly from the general solution prese nted and wil l he

examined nex t .

S p ec i a l Cas es . F o r the spec ia l ca se ofnegligible convective resis

tance at the two faces of the plate (1/Bit = 1/Bi2 = 0) the tem perature

d is t r ibu t ion , eq ua t ion (8 ) , r educes to





e = Z + rJo Jo J

(14)

The tempera tu re g rad ien t at the surface (£ = 0) is obtained from

equations (14) and (10) resulting in

dJ-\ - 1 +

d£ lo rf°) Jop°Fb°+Fd°)

n A ° ) p ° f | , x 0 1 - p 2 x ( M ) e - 2 W "

- (^j (1 - e - W » . ) ( i - p2 x

0 i )e- W e ) U(

ToX,n)

+ 2F dh°[T a(0) - R 3 ( 2 r o J ) ] - 2F dk°[T4(0 ) - f 4 ( r 0 J

+ R4(2roX) - R 4 ( T „ X ) ] / T „ X } dX (15)

Examina t ion of th is equation reveals that radia t ion a lways increases

the temp erature gradient and hence the conductive f lux at the surface

£ = 0. The to ta l heat f lux at the surface follows from equation (13)

yie lding

<?r - i + r J $(y)drio

(16)

Since the in tegra l in th is equation is a lways posit ive , the resu lt indi

cates that in ternal absorption of radia t ion a lways decreases the he at

transfer rate at the irradiated surface when T2- T\ > 0 and increases

the ra te when T2 - Ti < 0.F or a gray med ium, KX = « = constant , the wavelength independent

radiation surface characteristics, and the integration over wavelength

indicated in equations (12) and (16) are e l iminated.

For example , when only a beam componen t ofthe radia t ive f lux

(Fd° = 0) is incide nt on the plate , the med ium is gray with wavelength

independen t t r ansmiss ion and ref lection characteris t ics of th e in

terfaces and ther e is neglig ible convective resis tance at the surfaces

(1/Bii = 1/Bi2 = 0) , the t empera tu re d i s t r ibu t ion becomes

6 = £ + TTI(P°)(P/T0)\(1 - e-('»/">*) + p 2(p . )e- 2r ° /«(l - e(V»){ )

/A°"I .O

- £[(l - e-"'») + p a M e - ^ U - e^")}]/0(To ,n) (17)

and the dimensionless surface heat f lux at £ = 0 simplifies to

9 l * = - l - T\[T1(^)/0{To,n)][y(T„,n) - <MT„,M)]I (18)

where

7 ( T „ , M ) = 1 - p2(M)e-2^"

a nd

- ( - ) (1 - e - » / " ) [ l - P2(n)e-^"} (19)

<Mw«) = [1 - P2Me- ir° l") (20)

E x a m p l e

S ample re su l t s are presen ted in Fig. 1 and T a b l e 1 for a semi-

t ranspa ren t ma te r ia l hav ing a wave leng th independen t abso rp t ion

coeffic ient and neglig ible convective resis tances a t the surfaces such

tha t Ttt\ = T\ and T a 2 = T 2 . In the analysis the surfaces were assumed

to be optically smooth and the radia t ion characteris t ics were calcu

lated from the Fresnel [6] equations. The angle of incident 9° outside

the medium was re la ted to the angle of refraction inside the m edium

by Snell's law, n° sin0° = n sinf). Th e results of Fig. 1 show tha t TI>//?

increases with the optical th ickness T„ = Ht and app roaches an as

ymptotic value as T 0 becomes large. The results of this figure together

with equation (18) indicate that absorption of radiation in the material

increases the conductive heat flux at the surface . S ince the param eterT can become greater than unity when ei ther (p°Fb° + Fd°) is large

and/or k(T 2 — Ti)/t is small, one can envision may physical situat ions

where presence of radia t ion would increase s ignif icantly the con

ductive flux. As a practical example, consider a plate of glass (k = 0.75

W/mK) having th ickness t = 0.3 cm and a temperature d ifference T2

— T\ - 1 K across it which is irradia ted with a total flux \i°F]y° + Fd°

= 500 W/m 2 . The in te rac t ion pa rame te r F has a value of 2.

T h e p a r a m e t e r TI0//3 represents the d imensionless radia t ive f lux

inside the medium at £ = 0. S ince the ref lectiv i ty , p2(p), is relatively

sma l l , the dependence of n ^ / / } on the optical th ickness T„ is m u c h

weake r than tha t on thedirection (p°) of the inc iden t beam of ra

diation. At p" - 0 al l of the incide nt radia t ion is ref lected an d hence

T10//3 = 0.

If it is assumed tha t the rad ia t ion abso rbed by a p la te of semi-

transparent material is distributed uniformly, then for the special caseof gray materia l and a beam of rad ia t ion (Fd° = 0) incident on the

surface an approximate expression for the heat f lux can be obtained.

Th e analysis y ie lds th at the d im ensionless he at f lux a t the surface (£

= 0) is given by

Q l . approx = - 1 - r [ n ( M ° ) / , ? ( T 0 , M ) ]

1(1 •r » / " ) [ l + P 2 ( M ) e - T » / " ] | (21)

Fig. 1 Variat ion ol function r^y/ft with the optical thickness T „

Exa minatio n of Tab le 2 reveals tha t a significant loss in accuracy will

result in the p red ic t ion of the heat transfer ra te if equation (18) is

used. There is even a change in sign of the ratio <?i*A/*i,approxfor large

values of the optical th ickness r0 of the ma te r ia l .

References1 "Solar Radiatio n Consid erations in Bui ld ing P la nn ing and De sign ,"Na t iona l Ac a de my ofSciences, Washington, D. C, 1976.

Table 1 Variation of function Ti$//3 with the optical thickness T 0 an d the

cosine of the angle of Incidence p. deg

roV°

0.01

0.1

1.0

10

100

1.0

0.9238

0.9298

0.9550

0.9600

0.9600

n/n°0.5

0.8381

0.8525

0.9044

0.9108

0.9108

= 1.5

0.25

0.5863

0.6195

0.7253

0.7358

0.7358

0.10

0.2786

0.3213

0.4210

0.4284

0.4284

1.0

0.8018

0.8163

0.8770

0.8889

0.8889

n/n"0.5

0.7245

0.7458

0.8262

0.8386

0.8386

= 2.0

0.25

0.5211

0.5535

0.6664

0.6816

0.6816

0.10

0.2612

0.2992

0.3974

0.4076

0.4076

Journal ofHeat Transfer FEBRUARY 1978, VOL 100 / 171Downloaded 21 Dec 2010 to 193.140.21.150. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cf



Table 2 Comparison of heat transfer rates Qi'/Qi', approx tor Fd° =0, n/n" = 1.5, 1/BI, = 1/B1 2 = 0, and T = 10

T „ V °

0.010.11.02.04.0

10.0100.0

1.0

1.0000.9990.9110.7100.3560.000

- 0 . 2 2 5

0.5

0.9850.9850.8680.6150.254

- 0 . 0 4 6- 0 . 2 3 0

0.25

0.9590.9620.8440.5790.223

- 0 . 0 5 9- 0 . 2 3 1

0.1

0.9610.9640.8350.5660.214

- 0 . 0 6 3- 0 . 2 3 1

2 Duffie, J. A., and Beckm an, W. A., Solar Thermal Processes, John W ileyand Sons, New York, 1974.

3 Siegel, R., and Hussain , N. A., "Combined R adiation, Convection andConduction for a System w ith a Partially T ransmitting Wall," NASA TechnicalNote TN D-7985, Washington, D. C, 1975.

4 Ahmazadeh, J., and Gascoigne, M., "Efficiency of Solar Collectors,"Energy Conversion, Vol. 16,1976, pp. 13-21.

5 Viskanta, R., and Anderson, E. E., "Hea t Transfer in Sem i-transparentSolids," in Advances in Heat Transfer, Vol. 11, Academic Press, New York,1975, pp . 317-441.

6 Siegel, R., and Howell, J. R., Thermal Radiation Heat Transfer,McGraw-H ill, New York, 1972.

Transien t Conduct ion in a SlabW i t h T e m p e r a t u r e D e p e n d e n tTherma l Conduc t iv i ty

J. S u c e c 1

S. Hedge 2

Nom enc l a t u reB; = hL/ke Bio t number

F = ct et/L2 F our ie r number

h = surface coefficient of heat transferj = subsc r ip t ind ica t ing x location of nodes

k, k it ke = the rm a l conduc t iv i ty a t any tempera tu re , a t in i ti a l t em

p e r a t u r e Ti , and a t f lu id t empera tu re Te, respectively

K j-i, K j+1 = 1 + 7 ( 0 , - 1 " + 0 j") /2 , 1 + 7(<t>j+in + <t>jn)/2, respec

tively

L = half th ickness of s lab

n = number o f t ime inc remen ts

N = denotes node a t surface

t = t ime

T, Ti, Te = slab temperature , in i t ia l s lab temperature , and f lu id

temp era tu re , r e spec tive ly

x = space coordinate measured from slab center

X = x/L nond imens iona l space coo rd ina te

ae = thermal d iffusivity of solid a t temperature Te

y = defined by equation (6)<t> = (T — Te)/(Ti — T e) t empe ra tu re excess ra t io

0c , 4>x - value of 0 a t center and a t any nondimensional posit ion,

respectively

1 Professor of Mechanical Engineering, University of Me., Orono, Maine.Mem. ASME.

2 Engineer, West Coast Paper Mills, Dandeli, India. Formerly, GraduateStudent, University of Maine at Orono.

Contribu ted by the He at Transfer Division for publication in the JOURNALOF HEAT TRANSFER Manuscript received by the Heat T ransfer Division May27, 1977.

ABi = hbx/ke

AF = « e A t / A x 2

Ax = node spacing in x direction

At = t ime increment


Considerable a t tention has been devoted to the problem of tran

sient heat conduction in solids with variable thermal conductiv ity .

Most of the efforts employ a thermal conductiv ity which l inearly

depends upon temperature because th is is a realis t ic approximation

for many materia ls of in terest . Friedmann [ l] 3 summarizes the work

don e prior to 1958, gives linearized s olutio n bou nds for specified

boundary temperature , and presents some analog solutions for con

stant temperature and constant heat flux boundaries. Yang [2] effects

a solution to the semi-infin ite s lab by use of a s imilar i ty transforma

tion, while Goodman [3] and Koh [4] u t i l ize approximate in tegral

methods. Dowty and Haworth [5] solve the infinite slab case by finite

difference methods, but the results are limited to certain temperature

ranges and the case of heating only . A two-dimensional transient is

dealt with by Hays and Curd [6] by use of a restr ic ted varia t ional

principle . A f in i te e lement formulation of the general problem and

a few represen tat ive resu lts are given by Aguirre-Ram irez and Oden

[7], while Vujanovic [8] and Ma stana iah and M uthu naya gam [9] apply

the method of optimal l inearization. In tegral method techniques are

applied by Krajewski [10] and by Chung and Yeh [11] while, most

recently, Mehta [12] used an iterative technique, based on a modified

constan t prope rty solution, to y ie ld solutions to the variable thermalconduc t iv i ty p rob lem.

In a l l of the previous work on the variable thermal conductiv ity

problem , numeric al results are pres ented only for the case of specified

temperature or specified flux on the surface of the solid, and not for

the more frequently encountered s i tuation in practice , namely, the

convective type boundary condition. In addition, none of the available

numerical results are in a comprehensive, easy to use , form for the

designer. I t is the in te nt of th is technical note to solve the problem

by f in i te d ifference methods with a convective type boundary con

dition when the thermal conductivity varies linearly with temperature,

and to present these results in such a form that they serve as ap

proxim ate correction factors (with a certa in error band ) for the con

stant property Heisler charts , as g iven, for instance, in [13] .

AnalysisTh e problem to be considered is a s lab , of half th ickness L, whose

thermal conductiv ity is a l inear function of the temperature , which

is in i t ia l ly a t constant temperature T; throughout, when, suddenly ,

i t is subjected to a f lu id a t constant temperature Te with a constant,

surface coefficient of heat transfer h between th e f lu id and the s lab .

In the absence of generation and radia t ion, the s lab 's temperature

distr ibution in space x and t ime t i s r equ i red . The ma thema t ica l

s ta tement of the problem, in nondimensional form, is g iven as fo l

lows:

— \[l + y<t>\—\=— (1dXl dXl dF

0 = l a t F = O f o r O < X < l

—- = 0 a t X = 0 for F > 0

ax- [1 + 70] — = Bs0 a t X = 1 for F > 0

dXThe nonlineari ty in equation (1) , when coupled with the type of

boun dary co ndit ions for the s lab , precludes, a t present , an exact an

alytical solution; so f in i te d ifference methods were chosen to solve

equatio n (1). Explicit difference equa tions were derived for the various

interior nodes by making an energy balance on the material associated

with the nodes, and th e result is , for the J t h in ter ior node,

3 Numbers in brackets designate References at end of technical note.





Table 2 Compa rison of heat transfer rates Qi'/Qi', approx tor Fd° =0, n/n" = 1.5, 1/BI, = 1/B12 = 0, and T = 10

T „ V °

0.010.11.02.04.0

10.0100.0

1.0

1.0000.9990.9110.7100.3560.000

- 0 . 2 2 5

0.5

0.9850.9850.8680.6150.254

- 0 . 0 4 6- 0 . 2 3 0

0.25

0.9590.9620.8440.5790.223

- 0 . 0 5 9- 0 . 2 3 1

0.1

0.9610.9640.8350.5660.214

- 0 . 0 6 3- 0 . 2 3 1

2 Duffie, J. A., and Beckm an, W. A., Solar Thermal Processes, John Wileyand Sons, New York, 1974.

3 Siegel, R., and Hussain , N. A., "Combined R adiation, Convection andConduction for a System with a Partially Transmitting Wall," NASA TechnicalNote TN D-7985, Washington, D. C, 1975.

4 Ahmazadeh, J., and Gascoigne, M., "Efficiency of Solar Collectors,"Energy Conversion, Vol. 16,1976, pp. 13-21.

5 Viskanta, R., and Anderson, E. E., "Hea t Transfer in Sem i-transparentSolids," in Advances in Heat Transfer, Vol. 11, Academic Press, New York,1975, pp . 317-441.

6 Siegel, R., and Howell, J. R., Thermal Radiation Heat Transfer,McGraw-H ill, New York, 1972.

Transien t Conduct ion in a SlabW i t h T e m p e r a t u r e D e p e n d e n tTh e rma l Co n d u c t iv i ty

J . S u c e c 1

S. Hedge 2

No m en c l a t u reB; = hL/ke Bio t number

F = ct et/L2

Four ie r number

h = surface coefficient of he at tran sfer

j = subsc r ip t ind ica t ing x location of nodes

k, k it ke = the rm al conduc t iv i ty a t any tempera tu re , a t in i tia l t em

p e r a t u r e Ti , and a t f lu id tempera tu re Te, respectively

Kj-i, K j+1 = 1 + 7 ( 0 , - 1 " + 0 j")/2 , 1 + 7(<t>j+in+ <t>j

n)/2, respec

tively

L = half th ickness of s lab

n = number o f t ime inc remen ts

N = denotes node at surface

t = t ime

T, Ti, Te = slab temperature, in it ia l s lab temperature, and fluid

temp era tu re , re spec tive ly

x = space coordinate measured from slab center

X = x/L nond imens iona l space coord ina te

ae = thermal diffusivity of solid at temperature Te

y = defined by equation (6)<t> = (T — Te)/(Ti — T e) t empe ra tu re excess ra t io

0c , 4>x - value of 0 at center and at any nondimensional posit ion,

respectively

1Professor of Mechanical Engineering, University of Me., Orono, Maine.

Mem. ASME.2

Engineer, West Coast Paper Mills, Dandeli, India. Formerly, GraduateStudent, University of Maine at Orono.

Contribu ted by the He at Transfer Division for publication in the JOURNALOF HEAT TRANSFER. Manuscript received by the H eat Transfer Division May27, 1977.

ABi = hbx/ke

AF = « e A t / A x2

Ax = node spacing in x direction

At = t ime increment


Considerable attention has been devoted to the problem of tran

sient heat conduction in solids with variable thermal conductivity .

Most of the efforts employ a thermal conductivity which l inearly

depends upon temperature because this is a realis t ic approximation

for many materials of interest . Friedmann [l]3

summarizes the work

don e prior to 1958, gives linearized s olutio n bou nds for specified

boundary temperature, and presents some analog solutions for con

stant temperature and constant heat flux boundaries. Yang [2] effects

a solution to the semi-infinite s lab by use of a s imilari ty transforma

tion, while Goodman [3] and K oh [4] uti l ize approxim ate integral

me thod s. Dowty and H aw orth [5] solve the infinite slab case by finite

difference methods, but the results are limited to certain temperature

ranges and the case of heating only. A two-dimensional transient is

dealt with by Hays and Curd [6] by use of a restr ic ted variational

principle . A finite e lement formulation of the general problem and

a few represen tative resu lts are given by Aguirre-Ram irez and Oden

[7], while Vujanovic [8] and Ma stana iah and M uthu naya gam [9] apply

the method of optimal l inearization. Integral method techniques are

applied by Krajewski [10] and by Chung and Yeh [11] while, most

recently, Mehta [12] used an iterative technique, based on a modified

constan t prope rty solution, to yield solutions to the variable thermalconduc t iv i ty p rob lem.

In all of the previous work on the variable thermal conductivity

problem , numeric al results are pres ented only for the case of specified

temperature or specified flux on the surface of the solid, and not for

the more frequently encountered situation in practice, namely, the

convective type boundary condition. In addition, none of the available

numerical results are in a comprehensive, easy to use, form for the

designer. I t is the inte nt of this technical note to solve the problem

by finite difference methods with a convective type boundary con

dition when the thermal conductivity varies linearly with temperature,

and to present these results in such a form that they serve as ap

proxim ate correction factors (w ith a certain error ba nd) for the con

stant property Heisler charts , as given, for instance, in [13].

AnalysisTh e problem to be considered is a s lab, of half th ickness L, whose

thermal conductivity is a l inear function of the temperature, which

is init ia l ly at constant temperature T; throughout, when, suddenly,

i t is subjected to a f luid at constant temperature Te with a constant,

surface coefficient of heat transfer h between th e f luid and the slab.

In the absence of generation and radiation, the s lab 's temperature

distr ibution in space x and t ime t i s requ i red . T he ma themat ica l

statement of the problem, in nondimensional form, is given as fol

lows:

— \[l + y<t>\—\=— (1dXl dXl dF

0 = l a t F = O f o r O < X < l

—- = 0 at X = 0 for F > 0

ax- [1 + 70 ] — = Bs0 at X = 1 for F > 0

dX

The n onlinearity in equation (1), when coupled w ith the type of

boun dary co nditions for the s lab, precludes, a t present , an exact an

alytical solution; so f inite difference methods were chosen to solve

equatio n (1). Explicit difference equatio ns were derived for the various

interior nodes by making an energy balance on the material associated

with the nodes, and th e result is , for the Jt h

in terior node,

3Numbers in brackets designate References at end of technical note.





^ .»+i = AFKj-itj-i" + AFK j+ l<t > j+ in

+ [1 - AFKj-x - A F K J + 1 ]0 , " (2 )

However, an explicit equation de rived in this fashion for the surface

node N led to , in some cases, a severe s tabil i ty cri terion w hich w ould

cause excessive mac hine ru n tim e. This was overcome by implicitizing

at the boundary node, so that the s tabil i ty l imit shif ted back to the

interior nodes. This gives the following boundary node equation

0jv"+1

= [4>Nn

+ 2 A F K w _ i0 w _ i ' > + 1 i / | l + 2K N- tAF + 2AFA Bi| (3)

Because of the one-dimensionali ty of the problem, equation (3) isactually implicit in 4>N

n+l, since one can solve for 0w_t" + 1 from

equation (2) and use i t in equation (3).

I t was found that equations (2) and (3) were consistent with the

governing partia l d ifferentia l equation and with the convective type

boundary condition as long as At/A x —- 0, as At and A* approa ch zero.

The fact that t he f inite difference la tt ice mus t be refined in this re

stricted fashion does not present any difficulty, since the conditional

stabil i ty requirement for the set of equations (2) and (3) is that

AF^l/iKj-t + Kj+i) (4 )

Since AF is proportional to At/Ax2, the usual ty pe of la tt ice refine

ment which takes place at constant AF insures that At/Ax —» 0. Nu

merical experimentation indicated that the use of 31 nodes was suf-

ficient for convergence of the finite difference solution.

The finite difference solution was checked by comparison toavailable l imiting cases. For the constant conductivity s i tuation, the

results were compared to those of Heisler [14] and satisfactory

agreement noted. Such was also the case when a comparison with

Krajewski's [10] approximate solutions for Bj —• °° was made. Grap hs

of the response functions for the slab center are given in [15], from

which much of the present work was drawn.

When this work was begun, i t was reasoned that the most logical

form of the l inear thermal conductivity relation should be the fol

lowing:

k = ki\l + t(l ~ 4,)\ (5 )

the reason for this init ia l choice being that the s lab at some time,

namely t = 0, would be at a nondimensional temperature which would

allow it to have the reference conductivity ft; . Solutions were found

and graphed for 0 versus F when equation (5) was used, and i t wasnoted tha t the curves for different values of c exhibited quite a bit of

spread among themselves. This spread seemed also to be thwarting

efforts to correlate the curves with the constant property results of

Heisler's charts. Because of this , the following alternate representation

of the conductivity versus temperature relation was tr ied:

ft = fte[l + 70] (6)

It was found th at the use of equation (6) greatly reduced t he sp read

of the curves of different 7 and hence was used in all of the work being

presented here. To i l lustrate , the use of equation (6) rathe r tha n (5)

reduced the spread of the curves over much of the F range by a factor

of at le ast tw o, when B j = °> and X = 1.

D e v e l o p m e n t of A p p r o x i m a t e R e l a t i o n s f o r D e s i g nU s e

The finite difference solution results, even when put into the form

of charts , are cumbersome, to say the least , because of the large

numb er of charts nee ded. For every curve on Heisler 's [14] chart for

X = 0.0 (which is a curve of 7 = 0), one would need a minim um of four

other curves from t he prese nt work, namely, curves for 7 = + 0.5, +0.3,

- 0 . 3 , and —0.5. In addition, prelim inary at tem pts to f ind a s imple

equivalent of Heisler 's auxiliary, or posit ion correction ratio , chart

for the variable conductivity case were unsuccessful. Hence, effort

was directed toward the establishm ent of an easy to use appro xima te

correlation of these finite difference results with the results of

Heisler 's charts . Sought f irs t was a relationship between the tem-.

perature excess ratio at any X,0x, and the temperature excess ratio

at the center , 0 C , a t the same time, by attempting to collapse the

curves of (j>x/4>c, at various 7, on to the 7 = 0 curves of Heisler's aux

il iary chart . Thi s collapsing factor mu st equal zero when 7 = 0, must

approach unity as F gets large for all 7, and would, in general, depend

on Bj and on X. Guided by these constra ints , a tr ia l and error process

resulted in the following two factors. For 7 g 0

1 - 7 exp | -1 .5 5 ( l - 7 ) + 12X/(4 .05 + 1.0/Bj)]

F~ 1 6 e x p | 3 . 1 F ( 0 . 5 - 0 . 1 / B i ) |

For 7 £ 0

1 - 7 exp | - 1 .55 7 + 12X7(4 .05 + 1,0/Bj)|

"~ 4 3 e x p { 3 . 1 F ( 0 . 5 - 0 . 1 / B i ) j

The collapsing factors, Cp, are the ratio of the ordinate of Heisler's

auxiliary (posit ion correction) chart , a t the value of X and B ; of in

terest , to the actual temperature excess ratio <px/<l>c forhe 7 of in

terest in the variable condu ctivity case. Or, viewed another way, the

ordinate of Heisler 's posit ion correction chart is in terpreted as being

C F [0 X / 0 C ] where the 0x /0c in this product is the one for the variable

thermal conductivity s i tuation. These correction factor correlations

were made using the following parameter values; 7 = 0 .5 to -0 .5 in

0.1 increm ents , X = 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0, B; = » , 20 .0,10.0,

5.0, 2.4, 1.0, 0.625, 0.4, and 0.2, F = 0.30 to w hatev er valu e is nee ded

for all <j> to be less than 0.01. Th e error in 4>xl<t>c due to use of this

correction factor is , in most cases, less than three percent with a

maximum error of five percent in a few cases. For Bj < 0.2 and —0.5

< 7 < 0.5, Heisler's charts can be used directly with little error; for

then the temp erat ure distr ibutio n is a lmost lumped in the space

coordina te, and the precise value of the the rma l conductivity is of little

consequence because of the convection controlled heating or cool

ing.

The next s tep was to attempt a correlation between the center

tempe rature excess ratio , </ > c, for 7 ^ 0 and the corresponding qua ntity

for 7 = 0, as portrayed in Heisler's main chart. Guided by some of the

same consid erations as were used to develop the C F expressions and

also by the form of the solution function for certain s teady state

problems with variable conductivity, the following generalized center

temperature excess ratios, 0 C* , were dev ised . Fo r 7 ^ 0

0C* = <t>c\l + 0.5757(1 - 0 C)[1 - exp( -0.5 705 Bj)]] (9)

F or 7 S O

0C* = 0C |1 + 0.5957(1 - <j>c + 0.45 0C 2 )[1 - exp (-0 .79 Bj)]) (10)

Th e correlatio ns (9) and (10) have the following m eaning: <j>c* is the

ordin ate of Heisler 's main c hart a t th e values of F and Bj of interest ,

while <t>c is the actual temperature excess ratio at the s lab center a t

the value of 7 for the problem. Alternately , the r ight hand sides of

equations (9) and (10) can be viewed as the ordinate value of Heisler's

main c hart for the s lab. For F a 0 .30, the equ ations (9) and (10) are

accurate to within 3 .8 percent, with the usual accuracy being closer

to two percent, except for some cases at low values of 0 C where small

errors , in the temperature excess ratio , such as 0 .001 to 0 .002, get

magnified on a percent basis because of division by the small 0C.

Hence, as long as small error can be tolerated, equations (7) through

(10), in conjunction with Heisler's [14] main chart and auxiliary chart

for the s lab, a llow a rapid, easy solution to the transie nt conductio n

problem with thermal conductivity a l inear function of tempera

tu re .

Sample ProblemConsider a two percent tungsten steel s lab, in it ia l ly at 730° C

throughout, which is to be cooled in a 26.7°C fluid environment. If

the cooling is done such that Bj = 5, it is required to find the nondi

mensional t im e F for the center tem peratu re excess ratio , <j>c, to reach

0.237 and the surface temperature excess at th is t ime. A plot of the

therm al conduc tivity values from [16] yields a s traight l ine tha t can

be represented as follows for the specific conditions of this prob

lem:

k = M l + 7 0 ) = 61.25(1 - 0.50) W/m°C (11)

(It is imp ortan t to bear in mind tha t 7 is no t the slope of the k versus





T curve but, ra ther, is the s lope multiplied by (T; — Te)/k e).

This sam ple problem will be solved using the average conductivity

k with Heisler 's charts ( the solution mode normally used for variable

conductivity problems by the engineer in industry), and then by using

the appropriate correlation equations from the set (7)-(10) with

Heisler 's charts . Both solutions will be compared w ith the f inite dif-

ference solution of [15l_0 = (1 +_0.237)/2 = 0.6185, henc e, using (11),

k = 42.3 W/m"C and B; = hL/k = 7.239. At 1/Bj = 0.138 and 4>c =

0.237, one read s from H eisler 's cha rts, as given in [17], th at F = 0.88

and </>x=i/</>c = 0.18. Changing the basis of F from k to ke, for later

comparison, gives the following solution to the problem using the

method o f "ave rage" conduc t iv ity :

F = 1.27 and 4>x=il<t>c = 0.18

Now, solving by the correlation eq uations given in this paper, one

gets from equation (10), using B, = 5, y = —0.5, a nd 4> c = 0.237, that

0C* = 0.182. Using this as the ordina te on Heisler 's main ch art a long

with 1/Bi = 0.2 gives, from [17], F = 1.12. From equation (8), CF = 1.08

so that the ordinate of Heisler 's posit ion correction chart is taken to

be 1.0&[<t>x=i/<l>c]- This ordinate value from the auxiliary chart in [17]

is 0.26 at 1/Bi = 0.2, hence 1.08[<j>x=i/<t>c] = 0.26 or <t>x=i/4>c = 0.241.

T h u s , the solution to this problem using the correlation equations of

this paper gives, F = 1.12 and <px=il<t>c — 0.241. Th e finite difference

solution, for this case, in [15] yields F = 1.10 and <l>x=i/<Pc=

0.237. So

the error in F and in the surface tem perat ure excess ratio by using the

average conductivity te chniq ue is 15.8 and 24 percent, respectively.

The corresponding errors using the correlation equations of this work

are 1.82 and 1.6 percent.


nite series solution numerically for these, and would employ finite

differences for the variable conductivity s i tuations. Also presented

are the f inite difference equatio ns whose solution forms th e basis of

the result ing correlation equations and which can be programed for

situations in which additional accuracy is deemed important.


The authors thank the Computing and Processing Services De

partment of the University of Maine at Orono for their donation of

compute r t ime .

R e f e r e n c e s

1 Friedmann, N. E., "Quasilinear Heat Flow," TRANS. ASME, Vol. 80,1958, pp. 635-645.

2 Yang, K. T., "Transien t Conduction in a Semi-Infinite Solid withVariable Thermal Conductivity," Journal of Applied Mechanics, TRANS.ASME, Vol. 80,1958, pp . 146-147.

3 Goodman, T. R., "The Heat Balance Integral—F urther Considerationsand Refinem ents," JOURNAL OF HEAT TRANSFER, TRANS. ASME, Series C,Vol. 83, Feb., 1961, pp. 83-86.

4 Koh, J. C. Y., "One-Dimensional H eat Conduction with ArbitraryHeating Rate and Variable Properties," Journal of the Aerospace Sciences,Vol. 28,1961, pp. 989-990.

5 Dowty, E. L., and Haworth, D. R., "Solution Charts for Transien t HeatConduction in Materials with Variable Thermal Conductivity," ASME PaperNo. 65-WA/Ht-29,1965.

6 Hays, D. F., and Curd, H. N., "Heat Conduction in Solids: TemperatureDependent Thermal Conductivity," International Journal of Heat and Mass

Transfer, Vol. 11, 1968, pp. 285-295.7 Aguirre-Ramirez, G., and Oden, J. T., "Finite Elem ent Technique Applied to Heat Conduction in Solids with Tem perature Dependent ThermalConductivity," International Journal for Numerical Methods in Engineering,Vol. 7,1973, pp. 345-355.

8 Vujanovic, B., "Application of the Optimal Linearization Method to the

j.heat.transfer.1978.vol.100.n1

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