NATURAL CONVECTIVE HEAT TRANSFER

FOR VERTICAL CYLINDERS·WITH TRANSVERSE MASS fLUX

1 f

, (

NATURAL CONVECTIVEHEAT TRANSFER

FOR VERTICAL· CYLINDERS WITH TRANSVERSE MASS FLUX

by

JAGJI T KUMAR KHOSLA J B. Sc. Ch. E.,

Submitted tothe Facu1ty of Graduate Studies and Research in

partial fulfilment of the requirements for the degree of Master of

Engineering.

Dept. of Chemica1 Engineering

McGi11 University

Montrea1- April 10,1967

~ Jagjit Kumar Khosla 1968

. '4t- M. Eng • Chemical Engineering

JAGJIT KUMAR KHOSLA

NA'lURAL CONVECTIVE BEAT TRANSFER FOR

VERTICAL CYLI~ERS WITH TRANSVERSE MASS FLUX

ABSTRACT

A numerical solution· for heat transfer innatural convection around

a vertical cylinder with uniform surface temperature and uniform·transverse

mass flux was obtained for a fluid of Pr = 0.7, using a perturbation tech­

nique to the solution for a flat plate. Non linear, partial differential

~quations for momentum, energy and mass transfer were reducedto ordi~ary

differential equations by introducing transverse flux and curvature var,tables.

Heat transfer rates were found to decrease with blowing andi~crease with

suction. Increase of curvature increases the rate of heat transfer. Amethod

for calculating the transverse m~8S flow when it is linked with the driving

force causing natural convection has been suggested.

.. '~, .:

AGKNOWLEDGEMENTS

The author wishes to express his thanks and appreciation for their

contributions in many ways towards the success of this project,. to

Dr. W.J.M •. Douglas, for his guidance and constructive. criticism,

The Pulp and Paper Research Institute of Canada, for their financial

support and the use of Library and other facUities,

Dr. A. Held for his invaluable help in the solution of equations on

computer,

The Staff and Graduate Students of the Chemical Engineering Department,

for their suggestions during the course of this study.

, . . J .>

. ~.'

1.

2.

3.

4.

5.

6.

. 7.

8.

9.

'., . LIST .. :OF FIGURES·':,~

Co~ordinate system for the cylinder

9 as a func·tion of '1. f' as a function of Y)

. 9, 9' and 9" as functions of YI

f', f" and f'" as functions of 11.

Third order term as percentage of leading.terms

Heat fransfer ~esults forblowing

Heat transfer results for suct.ion

Comparison of local heat transfer results for a cylinder with flat plate heat transfer results

10. Comparison of local heat transfer results for a cylinder with and without transverse mass flux

Page 17

46

47

49

50

51

53

54

55

56

Il. Average heat transfer results 60

12. Transverse mass transfer rates with and without heat transfer 62

GENERAL INTRODUCTION

Much attention has been devoted in recent years to the development

of generalized analysis of the transport of mass, momentum and energy. However,

the interaction of two or more transport processes remains a difficult problem

to generalize.

The present study treats the case of simultaneous convective transport

of mass, momentum and energy associated with the interaction of two, mutually

interacting velocities in the presence or absence of heat transfer. This aspect

of the transport phenomena arises in a wide variety of practical applications.

Evaporation of acetone from cellulose acetate thread in dry extrusion,

the sweat cooling of the electrodes in an electric arc furnace, and the drying

of paper are sorne of the cases where interaction of natural convection and trans-

verse mass flow is important. In pulp and paper industry the rate of drying

of webs is a problem of great importance. Pressure, tension and speed of the

drying roll ers depend on the strength and hence the moisture contents of the

paper sheet.

In aerodynamics the application of s~ction on an aerofoil to increase

lift and decrease drag by preventing transition of the boundary layer from

laminer to turbulent has been investigated.

Blowing of air through molten metal helps to keep the temperature

of its surface lower than that which would cause the formation of undesirable

oxides.

-2-

In polymerisation reactions, a controlled rate of injection of one

or more of the reactants along the length of a flow reactor with porous walls

may help to obtain the desirable molecular weight of the polymer.

The structural elements of turbojects, rocket engines, exhaust nozzles,

gas turbin.e blades, the combustion chamber walls of fuel cells, and the surfaces

of spacecraft during reentry are subjected to high temperatures. Injection of

a cold fluid fromthe soUd walls may be used to obtain the 'heat blocking effect',

and thus protect the surface from thermal damage. This process is called trans-

piration cooling.

Air-conditioning, evaporation, condensation of vapours on a solid sur-

face, diffusion of reactants and products towards and away from the calatyst par-

tic les in a packed bed, distillation, adsorption and absorption and in fact most

chemical engineering operations involve in one way or the other interaction bet-

ween two perpendicular fluxes.

The orientation of the present study of naturai convective heat trans-

fer with uniform transverse mass flux around a vertical cyliner has been to ob-

tain a numerical solutions of the boundary layer type equations for momentum

and energy transport. A third order perturbation technique along wi.th the Kutta-

Simpson method was employed to solve the nonQlinear par.tial differential equations

of momentum and energy transport at a Prandtl number of 0.7 •

...-.-----------

LlTERATURE REVIEW ..

In.troduction:

Thè analogy between mass, momentum. and heat transfer is applicable only

to a very li~ited range of cases. The problem of combined transport of mass,

momentum . and energy has been solved analytically and numerically for some simple

geometrt~s and flow patterns. Since the interaction of two or more transport

processes introduces many complications, simplet sit\lations are studied first.-:

Most of the analytical and numerical solutions obtained for simultaneous

heat and mass transfer have been for a fIat plate immersed in a paraI leI flow

environment. Transport processes at the stagnation point of biunt objects have

been considered. The effect of variable fluid properties (8, 23) and as a

further complication, diffusion thermo and thermo diffusion effects (25,26,28,29~35)

have been· included in some studies. The interaction of thermal radiation and

convection has also been treatteèl by some workers (5.,14,,27). .Similarly, the effect

of curvature has been found to be significant (1,2,19,21,32).

A review of the literature concerning simultaneous heat and maS6 transfer

in natural convection has been given by Li (9). Here on1y the analytical and

numerical methods employed by various workers for cases re1event to the present

problem will be discussed.

Hest Iransfer without Mass Transfer in Natural Convection

This section gives a summary of the various approaches which have been

used for the case when there is no transverse mass flux at the surface~ This

is included for the sake of completeness since the effects of mass transfer

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have .foUowed the studiés'of pure heat transfer. lt. 18 suggested tbat the

reader may prefer to proceed to the page·~: 7where heat and Diass transfer in

n@~utal. conveet~on ,.are discussed

Pure free convective heat transfer .to a vertical flat; plate has been

inves,tigated by Schuh (17) and Ostrach (15,. 16). Ostrach (16) gave the simi-

laritysolution of themomentum. and energy equations for a large number of

Prandtl ·numbers corresponding to liquid Uletals, gases, liquids and very visco\ls

fluids (Pr = .0 • .01, .• 72, 1,2,1.0.,1.0.0,1.0.0.0). Local heat transfer coefficients

1/4 have been eJl!pressed as proportional to (Gr.Pr) ; for air tbe value of pro-

.portionaHty cOI;1stant is .0.548.

Sparrow and Gregg (2.0) studied the case of laminar pure free convective

l;I.eat transfer with uniform· heat flux dong the surface of a vertical flat plate.

Similarity so,lutions have been given for the range of Prandtl nwnbers from

. .0.1 to 1.0.0 and extrapolated to .0.01. The heat transfer resl,1lts are expressed

* 1/4 * interms of a modified Grashof number, (Gr) where Gr = (Gr)(Nu). The se

. results are in good agreement to those obtained by the Von K:arman-pohlhausen

method. It has alsobeen noted that local Nusselt numbers for uniform·heat flux

are· about ID to 15% higher thanthose for the case of uniform surface tempera-

ture.

Most of the numerical solutions assume that thef:lilid properties, except for

density, do not change much in the range of temperatures eIicountered. More

.correc.tly,the vatiati0':l of viscoBity, thermal conductivity, and specifie heat

with temperature should be included. Sparrow. and Gregg (23) proposed that the

-5-

product of density withviscosity,.and of density with:ther~l conduct1.vity

18 independent of temperàture. Eor this case' it has been. -shawn, that..;the variable

property,momentum and energy equations becolQe identical to those for constant

property fluide A re.ference temperature T has been proposed, where r

Tr ... Tw - .38 (Tw - T.,J. When properdes (with the exception of coefficient

1 of thermal expansion t3 • ~ ) are evaluated at Tr , the heat transfer results

are very close to the exact solutions for a variety of gases having different

physical property temperature relationships.

Similarity solution for a non-isothermal vertical fIat plate in pure

free convective heat transfer is given by Sparraw and Gregg (22). Two families

n of surface temperature variations are considered(i) T - T~ = N x , and w

(11) mx T - Too = Me , where T and Tooare the wall and environment temperatures

w w

respectively, x is the distance from·the leading edge, and N,n,M,m are constants.

A unified analysis giving necessary and sufficient conditions ~or aIl

possible temperature and heat flux distributions at the wall for a vertical

flat plate and a vertical cylinder for which similarity solutions exist are given

by Yang .(34). Generalsimilarity transformations are assumed as follows:

u =

'1 = y 0,(x, t)

'Y(x,y, t) f(~) =

O2

(x, t) = G(x,y,t)

G (x, t) w

where u and v are dimensionless velocities,. x and y are dimensionless

-6-

distances in., longitudinal and trans~erse direct.ions. "fis the stream funct.ion,

and G is the Grashof number. Substituting these transformations :.into non­w

dimensionalized~ unsteady state momen.tum,and energy eql,lations, two ordinary

differential equations in terms of f and, 9 are obtained (differentiation is

with respect .to ~ ). For a simUarity solution' to exist, the coefficients of

f 'aJ;ld· 9 and their deriva:tives must be constants. These constants when 'evaluated

at· the prescribed conditions of temperature and heat flux distribution

give the expressions for 01 and O2" Yang lists the sit~ations which yield

sim~larity solutions.

Millsaps and· ~ohlhausen (12) have giventhe solutions for laminar free-

convective fluid motion produced by a heated ver·tical cylinder for which the

thermal distribution at the outer surface varies linearly with distance from

the leading edge. This surface temperature distribution permits the foUowing

similarity solution:

~ = zf(r) z

,9 = 1; h(r) r

where ~ is the stream function, z is distance from the leading edge, r is

distaJ;lce in, radial direction, and, 9 is the Grashof temperature. Substitution

of the above traJ;lsformation along with r = exp(t) into the momentum and energy

equations" reduces them from non-linear partial differential equation to non-

linear ordinary equations. The first estimation of the initial values was made

by assuming parabolic temperature and v~locity profiles, with subsequent cor-

rections for additional accuracy.

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Reat and Mass Transfer in Natural Convection:

An analytical solution for laminar free convection on a vertical flat plate

for heat and mass transfer is given by Somers (18). Using the Von Karman-

Pohlhausen method, the integral forms of the mDmen,tum, diffusion and energy

equations were used to eva1uate the constants. The inertia terms were neglected

in order to simplify the mathematical treatment. Somers considered different

thicknesses for the momentum, thermal, and the concentration boundary layers.

As a simplification, he also considered the case .for which the momentum boundary

layer thickness is equal to either the concentration or to thermal boundary

layer thickness, depending upon whether the convection currents are caused by

a concentration or a temperature difference. With the a$sumption of si.milarity

of ve10city, concentration and temperature profiles in the boundary.layer, the

profiles were ~pproximated by third order polynomials of (y/'O), (y/'O'), (y/5"),

respectively where y iS distance in the transverse direction and 8,5' ,8"

are the respective boundary layer thicknesses. The variable u (maximum velocity), x

.5' and 5" were taken as functions of distance from the leading edge, x, as follows

1/2 u = al x x

5' 1/4 = a x

5" .2 1/4

= a3

x

where al,a2,a3 are the constants to be evaluated. The mass transfer Nusselt

number, Nu. , and heat transfer Nusselt number Nu were given as functions vx x

of the mass and heat transfer Grashof numbers Gr and Gr respec,tively. vx x

Nu = A(Gr + Gr (Pr/sc)1/2)1/4 vx x vx

Nu = B(Pr/sc)1/2 Nu x vx

where A and B are functions of Pr,Sc and concentration at the wall.

.. ~8-

Mathers et. al. (11), h~ve obtained .the solution ,for a flat plate by

solving the mass, momentum and energy transport equations on an analogue

computer. Only the case of zero transverse velocity ~t the surface (applicable 2

to the case of equal"molal counter,' diffusion or :to heat transfer accompanied

by small mass transfer rates) was considered. Since the 1nertia terms were

neglected,the results are vaUd only for a Prandtl number of the order of 100

or higher,',i.e. for liquids. The heat and mass transfer results were given in

the form

fj 1/2 -1/4 Nu = 0.67 L(Gr + (Pr/Sc) Gr')Pr j

Nu' = 0.67 [(Gr' + (Sc/Pr)1/2 Gr)sc]1/4

where Nu, Nu'; Gr, Gr' are the heat and mass transfer Nusselt numbers and

Grashof nuœbers.

Wilcox (33) tried to improve on the solutions of Somers (18) and Mathers

et. al. (11) who had both neglected the inertia terms. In the solution of the

Integral forms of the mass, momentum and energy equations for a vertical fIat

plate, Wilcox assumed velocity, concentration and temperature profiles of a

more simplified form than those taken by Somers (18) inorder that the inertia

terms could be included. The cases of zero and non-zero wall velocities were

considered, and for the latter, both the conditions of a constant wall tempera-

ture and an adiabalic wall were evaluated. The results for the zero wall velo-

city case agree to those given by Mathers et.al.( 11) at high values of Sc and

Pr, and when Lewis number (Sc/Pr) is approximately equal to one.

SiiniTar!1;y solutions for heat transfer to a vertical flat plate .for the

effect of mass transfer at Pr = 0.73 are given by Eichhorn(6). Similarity

conditions require that the blowing or suction,rate varY as the distance from n-l

. the leading edge, x, raised to the power (n-l}/4 , Le. v Ct x", where w

nin turnis the exponent in. the power law surface temperature._F.9~ n = 0,

this requirement implies that the blowingrate at the leadingedge should be

infinite. In practiée, injection of a fluid .through a porous plate with rea-

sonable pressure drop is likely to produce a uniform transverse flux distribu-

tion over the surface. Auniform flux corresponds to n = l, which would require

the temperature distribution to be T = TOC) + Bx. w This temperature distribution

do es not correspond to any case of practical interest. Thus a similarity solu-

tion for uniform velocity at the surface is of no interest, but for the case of

a'uniform wall temperature, a similarity solution will be valid except near

the leading edge.

Quite a different approach was tried by Nakamura (13), who used the method

of successive integration and approximation for free convection on a vertical

flat plate accompanied by vapour transfer. Th'è zeroeth approximat.ion was taken

from the solution for pure natural convective heat transfer given by Ostrach (16).

Results were presented for different rates of evaporation and condensation for

Pr = 0.72 and for Sc = 0.5, 0.6 and 0.72 •

The integral solutions given by Somers (18) and Wilcox(33) are valid only

for a cOr;lstant surface temperature. Mabuchi (lq'~ extended this approach by

accounting for the effect of Prandtl number and for the effect of the wall temperature

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distribution 'rw" Too + A xn for the range 0:s n:S 1. The Integral forms oF

oënservttti:~·r~titilFf,·"W'elte~::'SC1.1.~d; with the help of assumed dimensionless

velocity and temperature profiles expressed as a power ser.ies of y/8, the

ratio of transverse distance .to boundary layer thickness.

u / / 2 / 3 1 4 - = .(y. 8). - 3(y 8) + 3()" 8) - (y 8) u . x .

= 1 - -12 (Y/8) + 16w (y/8)2 + 2-8w -w -w 1 - w

3 (y/8) - 1 - 3w

1 - w

where ux istbe unknown non-dimensionlizing velocity, which, is eliminated in

the process of solution, and w is a parameter introduced to satisfy the wall

temperature profile boundary condition. These approximate similar solutions

impose restrictions on transverse velocity Reynolds number as follows:

For n = 0

\. vwx 1 < 0.6 [, C?:rx J 1/4 for Pr = 0.73

- "4 v "

Iv xI [ Gr ] 1/4 .~ :S0.5 4

x .for Pr=1.0

For n = 1

1 vwxl [ Grx 11/ 4 v :S 0. 8 4 . .1

for Pr = 0.73

\ v xI [Gr ] 1/4 ~ :S 0.65 4

x for. Pr = 1.0

-li-

where v is the transverse mass velocity, v is the kinematic viscosity, and w

Gr is the local Grashof number at a distance x from the leading edge. x

The heat ~ransfer solution for an isolthermal wall at Pr = 0.73 is tn

good agreement with the exact solution given by Eichhorn (6) for the case of

s'trong suction and moderate blowing. It has also been concluded that local

Nusselt number depends on the upstream history of the surface temperature at

constant blowing or suction. The ratio of 10caL.Nusselt numbers with and 4

without transverse mass flux is relatively independent of the details of the

upstream blowing or suction distribution for a constant surface temperature.

A perturbation technique has been employed by Sparrow and Cess (24) for

natural convective heat transfer, with uniform blowing or suction, to a fIat

plate having a constant wall temperature. These conditions do not have a simi-

larity solution. The stream function and dimensionless temperature were expan-

ded as a power series in terms of a blowing parameter in the form of a dimen~ 1/4

sionless injection or suction velocity vwx Only the first order 4vc

approximation has been taken into account. For Pr = 0.72, the ratio of 'heat

transfer with transverse velocity to non-blowing case is given by

1/4 q / qo = 1 - 2.77 ( v w x )

4vc

Heat transfer increases with suction while decreasœwith blowing.Comparing

these results with those given by Eichhorn (6), it is concluded that the boundary

layer has 'poor memory' i.e. it is not very sens'Ïtive to the distribution of

J

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transverse mass flux at the surface. Since the perturbation method used by

Sparrow and Ces8(a~)is vaUd on1y fo,r small values of the b10wing parametet,

t'his solution cannot give the asympl:otic .solution which. exists for large

suctionrates. Transformations were suggested for the case when the power

1aw distribution of temperat~e and of transverse ~ass flux at the surface

are independent of each other.

Brd1ik and Mochalov (3) solved the integra1 forms of momentum and energy

equations for uni.form temperature and uniform transverse mass flux at the

surface of a vertical flat plate in a still atmfJspher·e. The ve10city and

temperature profiles were assumed as fourth order po1ynomials of the ratio

of transverse distance ~o the boundary layer thickness. These results, when

compared with t.hose of Sparrow and Cess (24), are found .to be 10% higher for

strong suction and 4.5% lower for strong b.1owing. Brdlik and Mocha1ov noted

that these qiscrepancies Fight be reduced if higher order terms were added .to

the perturbation method.These also determined that the effect of· transverse

mass flux increased with.an increase in Prandt1 number.

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Cl.!rvature Effect

Keeping aU the system condition same,whellClflat plate, which may be

considered as a cylinder of infinite radius, is bent to a cylinder of finite

radi.us, heat and mass transfer rates are changed. The ratio of boundary layer

thickness to the cylinder radius, named as curvature parameter, has been used

by Seban and Bond (19) as a criterion for the effect of curvature. Solutions

for Prandtl number 0.715 have been obtained for laminar forced convection

paraUel to cylind.er axis. Some numerical correction have been given by Kelly

(7) to the solutions of Seban and Bond (19).

For 1aminar forced conve'ction solutions for heat transfer to a cylinder

for blowing or suction have been considered by Wanous and Sparrow (32) using

the approach of Seban and Bond (19). It i8 observed that the effect of

blowing or suction distribution at the surface does not influence very much

the rates of heat transfer for blowing as weIl as for suction i.e. the boundary

. layer has poor memory.

Bourne and Davies (2) have obtained the solution for the case of laminar

forced convection axial flow for very long and thin vertical cylinders. Effect

of curvature and Prandtl number on local heat transfer rates has been shawn

by comparing heat transfer rates for the fIat plate.

Cylinde~ with arbitrary cross-section have been considered by Bourne and

Wardle (l) for forced convective heat transfer. Equivalent curvature parameter

has been defined such that cross-section of elliptic cylinder 1s mapped confor­

mally into that of right circular cylinders.

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For naturalconvection Sparrow and Gregg (21) gave a 1aminar boundary

layer solution for a vertical cy1inder with uniform surface temperature~ for

Prandt1 number of 0.72 and 1.0,. using a perturbation techniq!le. Not large

derivations have been observed when comparing these resu1ts with those ob-

tained by stagnation layer ana1ysis. Limits on the curvatureparameter

have been defined within which,flat plate solutionmay be applied to acy1inder wit-

hôut more than 5 percent error.

23/2 ""';;;"---"'1/4 (x/r ) ~ .11

(Gr/4) 0

for Pr = 0.72

where x is the distance from the 1eading edge and r is the radius of o

cylinder.

In the present study the effect of uniform transverse mass flux on a

vertical cy1inder in natural convection is obtained by solving the momentum,

mass -and energy equation numerically. This solution is extended .to inc1ude the

casewhen transverse mass flux ràte is determined by the driving force causing

natural convection.

ANALYSIS OF THE SYSTEM

Introduction

When the natural convective flow around a solid ls altered by the

- ,

addition of flowin transverse direction it is necessary to distinguish whether

or not the transverse flow is directly linked 'co the heat transfer rate. The

rate of transverse flow may be controlled independently in case of pressure

injection or suction of the fluid ~hrough a solid with porous walls. In other

cases, 'however, the transverse flux depends on the driving force causing natural

convection, which would be the teniperature diffe'l'ence in case of evaporation

or condensation,~hd concentration difference in case of absorption or desorp-

tion. In case of injection or suction, the transverse flow at the surface is,

for pr-actical reasons, likely to be ur.iform, while for the case of evaporation

and absorption the flux may be a function of position because of the coupling

of the transport processes. In the present work the solution of the boundary

layer equations is obtained for the case of uniform transverse flow inter acting

with natural convective flow at the s~rface of a vertical cylinder. This solu-

tion is further extended to the case where transverse flow is controlled by the

driving force causing natural convection.

Since a Hat plate may be regarded as a cylinder with infinUe radius,

the h~at transfer solution for a cylinder may be expressed as an expansion

series around the fIat plate solution to give a third order correction for the

curvature effect. A similarity solution, in which two vdocity profiles

ü(x,i=) at two different positions of x differ only by a scale factor in ü amt r,

does not exist for the case of uniform transverse flow. The solution in the

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present case therefore contains correction terms for the curvature effect, for

the transverse flow effect, and for the interaction between the two.

Mass, Momentum and Energy Equations for Uniform Transyerse Flow.

The co-ordinates used for a semi-infinite right circular cylinder

of radius, a, suspended vertically in a constant temperature still atmosphere

are indicated on Figure 1. The fluid velocities ü and v are in the directions

X andr, respectively. The transverse velocity at the solid surface is v w

The environment and wall temperatures T and T respectively, and the fluid 00 w

properties are assumed to be constant. For the variable fluid property case,

a suitable reference temperature for their evaluation has been given by Sparrow

and Gregg(23 ).The boundary layer thickness at the leading edge is taken as

zero~ The conservation of mass, momentum and energy can be represented by

equations 1,2,3, respectively.

è (--) + 0 ( __ ) 'âi ur dr vr = 0 ( 1)

_ oii + _ 'oü g(3(T - T ) + v 0 C oü) udi v~ = 'dr r'dr

00 w r (2)

_ OT + _ OT ~ 0 C OT, U ai v'~ = di- r d'r' r

(3)

where g is the acceleration du.e to gravit y , (3 is the volumetrie coefficient

of thermal expansion, v i8 the kinematic viscosity, and a

thermal diffusivity.

( =.....!L) is the pc p

Equation 2 is a simplified form of the steady state x-component,

r----'J

FIG. 1

lo-ordlnate System for tr.e cyl1nder

-1'8-

Navier-Stokes equation for a Newtonian fluid around an axi-symmetrical .body

with zero rotationa1 ve1ocity. A1so, the Prandt1 boundary layer assumption

has been app1ied. i.e.

Equation 3 1s a simp1ified form of the steady state thermal energy

conservation equation, neg1ecting

(a) viscous dissipation

(b) compression or expansion effects

(c) thermo diffusion and diffusion thermo effects.

It has a1so been assumed that 02T

k--- is negligible which wou1d not Ox2

be true in the case·of f10w of mo1ten meta1s where thermal conductivity of the

f1uid is very high.

For the purpose of obtaining geometric and therma1similarity,

equation 1,2,3 arenon-dimensionalized with the following definitions:

r x ua Y! r =- x =- J U =- V = a a V V (4)

l'co- T 9 _.

" . T = l'co -.9(T T ) T - T 00 w

00 w (5)

where Il and x are dimension1ess 1engths, u and v, dimensionJ.ess ve10citie s

and 9, a dimension1ess temperature. Substitution of equations 4- and 5' in

equations 1,2,3 gives:

-' -19-

0 0 dx (ur) +ar- (vr) = 0 (6)

Ou + Ou g f3 3 T) o Ou Uax var- = a (Too -+ 1

ar-(r or) (7) v2 r

09 09 1 1 0 ( 09 ) (8) ua;+var- =-;ar- rar-Pr ~

Boundary conditions associated with these equations are:

At r = a: r = 1, u = u = 0 (the no slip condition ),

v = v v = v T = T , and 9 = 1 (9) w' w w

At -r =00 r =00 T = T 00 9 = 0 , and

u = u = 0 (still enviroment) (10)

Equations 7,8 are non-dimensl.onalized, non-linear partial differen-

tia1 equations representing momentum and energy transport.

Definition o~ the stream function according to equation 11 satisfies the

mass conservation equation 6

u =1.Qi rdr'

v =.1 ~ r ox (11)

Substitution of equation 11 in to the momentum and energy equations 7,B

resu1ts in equations 12,13,

where r = g f3(TOI) • T)a3

2 v

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Pr = v 0:

If G is defined as -G = [g f3(T_ . T

W>a3J 1/4

4V 2

Then 9 =

Transformation of Space Co-ordinates

(12)

(13)

A transformation of space co.-ordinates is made in order to introduce

the curvature effect of the cy1inder. The size of this curvature parameter, ~,

will faci1itate the comparison of heat transfer resu1ts for a cy1inder with those

for a f1at plate.

Transformations 14,lS p 16 are used to change the space coordinates

in the momentum and energy equations 12,13, from x and r, to sand YL

respective1y.

Y'l = G x· 1/\r2_1>

1/4 ~ = 1L-

G

• v x w

(14)

(16)

-21-

~ = 2Gr x- l / 4 ( 17)

(18)

Differentiation of * and 9 with respect to x and/or r is expressed

in terms of differentiation of f withrespect to vt and ~ in equations 17

through 26, Primes denote differentiation with respect ta 1. .

.2! = Gf l. x- l /4 + GX3/ 4[df ëw\ + df dg] _ v dX 4 ~ ox d~ ~ w

From equation 5

Too - T

Tao-T w = 9(yt » g )

- v w

(19)

(20)

(21)

(22)

(23)

-22-

. . (24)

o (r 09) = 4'G2r x~ 1/2[( 1 +1:'fI )9" + 1: 9'J -or. dr 5>·l 5> (25)

~ = 9'(- 1~) + ~ (1 i ) ox 4 X 05 4 X

(26)

Substitution of equations17 through 26 in equations12 and 13 resu1ts in equations

27 and 28, which are the transformed forms of the momentum and energy transport

equations.

Of 2(1 +;y). )f"'+ (2; +.25; di" -; Vw + 0.75f)f"

2 - 0.5(f,)2 - .25; f' 0 f + 9 = 0 d1drt (27)

:r (1 + ;yt)9" + (:r; + 1.5f - 2 ;vw + 0.5; ~:)9'

- 0.51: f' 09 0 5> af = • (28)

Substitution of equations14,15,16 in equation 11 gives longitudinal: '

and transver se ve loci tj.es in'"sp'ate co- ordilnatel!! "l and ;

( 29)

v = - ~~ = -[0.75 Gfx- 1/4 - .25 Gytx-1

/4f'

df ] 1 + .25 dI - Vw r (30)

-23-

The boundary conditions 9,10 can be written as

At 'lî. = 0 r = 1 ,

T = T w'

u = 0, hence f'(O) = 0

hence e = 1

If we choose f(O) = 0 and ~: ~= 0 = 0

then the condition v = v at r = 1 is satisfied in equation 30 w

At u = 0 . . f' (00) = 0

= 0

Perturbation Technique

(31)

A perturbation technique is used to eva1uate the heat transfer resu1ts

for a cy1inder in terms of correction factors app1ied to the solution for a

Hat plate.

From eqt.:ation 16, S 1/4

=L­G

1~1/4 [ X] = -:;;: ('4) , where = 5

is the boundary layer thickness at a distance x from the leading edge and so.·~ is

the ratio of boundary layer thickness to the radius of the cylinder. It can be

readily shown that the ratio of conduction heat transfer in a f1uid layer around

a vertical cy1inder to that on a vertical f1ate plate (of the same width as the

F1uid layer thickness periphery of the cy1inder) is a function of Radiue of Cylinder

There-

fore S = Q a

can be treated as a measure of the curvature effect caused by

-24-

the cylinder. For small values of s, the functions f and 9 can be expanded

in series by a third order perturbation technique, around the f1at plate 80-

1ution to accomodate the ctirvature effect.

(32)

(33)

91,92,93, ••• are functions created to account for the curvature effect. It

has been assumed that the fourth and higher powers of S will not make any

significant contribution.to the va1uesof f and 9 • Substitution of equat10ns

32 and 33 in equations 27 and 28, neg1ecting powers of S higher thanthree,

and arranging in order of ascending powers of S, resu1ts in equations 34

and 35.

o 2 "' s [2f'" + .75f fn - 0.5(f' ) + 9 ] + S[2f'''+ 2n f + 2f" - v fil 00 0000 00 00 1l 00 00 W 00

+ 0.75 f fil +f fil -1~25f' fli-Q J+th2fill+'2Vtfll'+ 2f" - v fil 00 1 1 '.00 00.1 1 s ·2 " 1 1 w 1

+ 75 f fil + f fil + 1 25 f fil - .75(f1,)2 l 5f' f' + 9 ] • 00 2 11 • 2 00 - .• 00 2 2

(34)

--2'5-

sOI i.. e" + 1.5f ,e' ] + sti..fe" +Y1-9"+e'L-2v e'+ l5f 9' + 1 Si e' Pr 00 . 00 00 Pr II . L- 0 OO} W 00 •. 1 00 • (JO 1

+ 5f e' Sf' 9 ] + t2[··i...he" +Yle" +. e'} - 2v· e' + 2f rH + 2.5f e' • . 1 00 - •. 00 1 ~ Pr L 2 . L 1 1 W 1 1" 1 . 2 00

For the solution to be true for a11 values of S ,. coefficients of various

powers of S shou1d individua11y be equal ~o zero~ This resu1ts in eight equa-

tions 36 through 43. Since Yt. is the on1y variable, the primes in equations

36 through 43 and for a11 subsequent equations may be taken as denoting total

differentiation.

2f'" + O.75f f" - .5(f' )2 + e = 0 00 00 00 00 00

(36)

,,," "2 2fll/+ 2Y1 f + 2f" - f + O.75f fil + f f" + 1 25f f .75(f

1') 2 . l 1 1 v w 1 00 2 1 1 .• 2 00 -

(38)

(39)

-26-

(40)

4(911 +V\ 9" +.·9' ) + Pr[- 2v,9' + 1.5f19 1 + 1.5f 9

1' + .5f

19' 1 "L 0000 . W 00 00 00 00

- .5f' 911 = 0 (41) 00

It is obvious that first eqp.1ations 36 and 40 must be solved simul-

taneous1y, after which equations 37,41; 38,42; and 39,43 may be solved succes-

sive1y .. in pairs.

Equations 36 and 40 are non-linear simultaneous ordinary differential

equations wl th tw6 point boundary conditions. They r·èpresent the solution for

heat transfer for a fIat plate without transverse mass flux. Equations 37,38,

39,41,42,43 are inhomogenousordinary differentia1 equat:ions. For example,

equation 37 i8 a third order equation in fI. Since the inhomogeneous terms

(not containiqg fI or ( 1) fo and its derivatives are now known functions of ~ ,

-27-

their values· aTesubstituted from ·the solution of equations 36 and 40. l t

can be easi1y shown from the properties of inhomogeneous equations thatif fIl

18 a solution of. a part of equation ,37 ( aU terms except v fil) when . . w 00

equated to zero i.e. equation ;44

2f"'+ 2Y\ftll +. 2f" + 0.75f f"+'i f" - 1.25fl fI + Q = 0 (44) 1 Loo 00 00 1 1 00 00 1 1

andthat ,if vwf12 is a solution of another part of equation 37 (the homogenous

part and the term vwf~) whenthat part is equated to zero, i.e. equation 45

2f"l + 0.75 f frl + f fil - L 25 f' fI + Q . - v f" = 0 ( 45 ) 1 00 1 1 00 00 1 1 w 00

then fU + Vw

f 12 is the solution of equation 37, Le. f l = fU + Vw f 12 (46)

Using this approach one may obtain for equations 38, 39, 41,42 and 43, the

(47)

(48)

(49)

(50)

-28-

... (51)

where

Substitution'of equations 46 through 51 in equations 32 and 33

resu1ts in

(53)

The substitution of equations 46 through 51 in equations 36 through 45,

combined with the requirement that, for the solution to be true for a11 values

of v, the coefficients of al1 powers of v shou1d individual1y be equa1 w w

to zero,results in a set of twenty ordinary differentia1 equations, (54)

through (73).

f lf, =-00

91f =-00

-29-

.5{O.75 f f lf - .5(f' )2 + 9 ]

00 00 00 00

l' .25[1.5(Pr) f 9 ] 00 00

+ 2 Y1 f lf' + 2 f lf

] 00 00

(54)

(55)

(56)

e" = - ,5 l P (2f e' + 1 5f e' .5f' e ) + 4 'l9" + 4e' ) 11 r 11 00 • 6.6. 11- 00 11 00 00

e" = - • 25 (Pr )[ 12

e" /.1

(57)

(1.5f .. e'12 - .5f' 912 + 2f129' ) - 29' ] (59) 00 00 00 00

(61)

fit' = -22

+ f fil - fil ] 12 12 12

(63)

(64)

(65)

(66)

(67)

":31-

(68)

(69)

f3~ = -.5 (. 75foof"33-1. 75f~of33+1.5f~of33+933) + 2 Y/.f23+2f23-f22

(71)

(72)

-32-

(73)

The boundary conditions associated with these equations are:

At 'r( = 0:

r = l, u = 0 T == T , w

f'(O) = f(O) = 0 9(0) == 1

(74)

f(O) = foo(O) = fll(O) = f12

(0) =f21

(0) •••• = f33

(O) = f34

(0) = 0

(75)

900

(0) =1, ~\l(O) =912(0) ••• =93/0) =9

34(0) =0 (76)

At ~ = co:

u = O·, T = Tco

-:33-

Numerical .Solution

Equations 54 through 73 are ordinary differential equations representing

the fundamental momentum and energy transport in the axi-symmetric free con-

vection boundary layer around a vertical cylinder with uniform transverse mass

flow. These equations, being coupled, are solved in pairs at Pr = 0.7.

Equations 54 and 55, which constitute the first pair, are non-linear equations,

while the remainder are a11 inhomogeneous. The first group of terms on right .

hand side of equations 56 through 73: is the homogeneous part, the remainder

being the inhomogeneous part. Solution of any pair of equations requires the

values of the inhomogeneous terms from some or all of the previous pairs.

The solution of equations·54 through 73 has been carried out by numerical

integl'ation using the Kutta-Simpson method. The equatioŒl in each pair are res-

pectively third and second order simultaneous differential equations. For

numerical integration they are reduced to a set of five differential equations,

each of first order. Five boundâry cOnditions, 74 through 78, are available

for the solution. However, the Kutta~Simpson method is applicable only ta initial

v~lue problems. This forced us to choose two more initial conditions (i.e.

fil (0) and e' (0» arbitrarily. Withthese arbitrarily chosen values of 00 00

fil (0) and Q' (0), equations 54 and 55 are solved. This is repeated for 00 00

different combinations of fil (0) and Q' (0) until the conditions at infinity 00 00

are matched. The numerical computation was do ne on an IBM 7044 computer.

For computation, some numerical value,~~,of the independent variable, yt ,

-34-

is chosen to represent infinity. The implication of f' (~) = 0 is that the 00

value of f' 00

at ~ ~ is zero and remains zero no matter how far beyond Y1. ~

we may go. This required that at and beyond yt~, fil' fil must al so be 00' OP

zero. Similarly e" e' 00' 00

along wi th e 00 must also be zero at and beyond ~~.

Numerical integration.with. correctly chosen values of fil ( 0) and 9 ' ( 0 ) 00 00

. f f 1 fil fil' and g1ves 00' 00' . 00' 00 e ,e' ,.9" 00 00 00

as functions of l1 values are used in the solution of the succeeding pairs of equations.

These

Solution of each pair of the inhomogeneous equations 56 through 73 1s ob-

taine~ by a linear combination of the solutions of the homogeneous part

equated to zero and the full equations. A set of values of f~b(O) and e~b(O)

(a = 1,2,3; b = 1,2,3,4) is chosen to satisfy one of the two conditions at

infinity i. e. A Hnear combination of values of fil (0) ab and.

9~b(O), obtained for thehomogeneous part and the equations containing aIl the

terms, is made to satisfy f' (~) = O. ab

The interested reader is referred to

the details of the methods of solving of non-linear and inhomogeneous equatigns

given in Appendix '. Results'have been calculated to eight significant figures

after the decimal. The worst matching of the boundary conditions· at infinity

is four places after the decimal. This result is qui te good, taking into con-

sideration that the error gets accumulated as results from one set of equations

are fed to succeeding sets.

The values of f~b(O) and 9~b(O) (where a = 0,1,2,3 9 b = 0,1,2,3,4)"

which match the solution of equations 54 through 73 at "'l~ and beyond that, are

given in Table 1.

-35-

TABLE 1

fil (0) = 0.6789105 00

Q' (0) = -.2497558 00

" fil (0) = 0.11118057 Qi.1(0) = -.22487495

11

fil (0) = -.08811011 Qi.2(0) = 0.17144587 12

fil (0) = -.01239087 Q21(O) = 0.05675445 21

fil (0) = -.02569557 Q22 (0) = 0.00243219 22

fil (0) = -.0369464 Q'I tO) = -.03582649 23 23,

fil (0) = 0.01741969 931

(0) = -,.02799335 31

fil (0) = 0.0482692 932

(0) = -.00630005 32

fil (0) = 0.05879534 933

( 0) = 0.02501713 33

fil (0) = 0.01013797 934

(0) = 0.00039481 34

-36-

Local Heat Transfer Results

Heat transfer results are presented in terms of the Nusselt ·number

Nu = i/(k/h), the ratio of a length dimension to the fluid film thickness

if the whole of the heattransfer were by, conduction.

hi Nu ='k" 1

(T - T ) co w

( k o~1 ) 25 ai_ k r=a '

Nu = x ( _ ~ (Tco - Tw) ~:I ) k(T -T ) r=l

co w

X d9 oYl Nu = - - (-) (-) = -a o~~=o orr=l

2G x3/4 e' (0)

= (NU)(Gr)-1/4 = _ 29'(0) 4

Substituting equation 53 in equation 80

= -

(79)

(80)

(81)

The product of S and v is F w w It is proportional to the

ratio of transverse flow Reynoldsnumber, v x _w_ to the Grashof number.

v

" .,.:J 7-

Therefore equation 81 can be rewritten as

= -

(82)

The first term on the right hand"side of equation 82 represents the

surface temperature gradient along a flat plate for pure heat transfer. The

second and third grou~of terms represent respectively, the third order correction

for the effect of curvature, and the correction for the transverse flow. The

fourth group of terms gives the correction for the interaction between curvature

and transverse flow effects.

For the case of pure heat transfer on a flat plate,

and thereforel

(Nu)F =0 w

;=0

= -

or (Nu) = 0.5 (4Gr )1/4 F =0 w

e=O

2(9' (0)) = 0.4995'= 0.5 00

For pure heat transfer on a circular cylinder, F = 0 w

; = 0,

(83)

-38-

Therefore the fract10na1 1ncrease 1nheat transfer due to the curvature 1s

g1ven by the second group of terms 1nthe r1ght hand s1de of equat10n (85)

(Nu)S =0

F =0 w

= 1 + [s 8il (0) + ~ 2 9' 21 (0) + s3 931(0) J 9' (0) 9' (0) 9' (0) 00 00 00·

(85)

The change 1n the rate of heat transfer due to transverse mass f10w for a

fIat plate 1s g1ven by equat10n 86

(Nu)! =0

(NU)s"O

F =0 w

~ 91' 2(0) 2 92'3(0) + F3 93'4(0) ] .. 1 + [F + F

w 9' (0) w 9' (0) w 9' (0) 00 00 00

(86)

S1m11arly, for a c1rcular cy11nder, the relatlonshlp ls:

(Nu)

(Nu)F -0 w

- 1 + [Fw912(0)+F;923(0)+F;934(0)]+ftFw922(0)+ ~2Fw932(0)+~F!933(0)

.9~0(0) +ft9il(0)+~292l(0)+t393l(0) ]

(87)

-39-

Average Heat Transfer Rates

The heat transfer resu1ts in terms of Nusse1t numbers given in the

previous .section are point values at a distance x from the 1eading edge.

Near the 1eading edge, the boundary layer being thin, the rate of heat trans-

fer is higher than farther downstream. The heat transfer coefficient is

therefore inversely proportiona1 to some power of x. By taking average over

a distance L from the leading edge, an average heat transfer coefficient,h , av

is obtained as:

h av.

h av.

h av.

(Nu) av.

... = qav

ÂT

1 = LAT

2kG = L

h L =~

k

L 1 1 J '11 = AT L q dx 0

L -1/4 J 2k ÂTG e' (0) x dx 0

L x- lI\e' (0) + se' (0) + s2e' (0) + s3e, (0) + Jo 00 11 21 31

= - 2 (Gr) 1/4 [~ e' ( 0) + (s e' ( 0) + F Q' ( 0) ) 4 3 00 11 w 12

+ sF2 e' (0) + F

3 e' (0»)1 w 33 w 34 :.lx = L

(88)

(89)

-40-

Mass Transfer with or without Heat Transfer

From an examination of the equations for heat and for mass transfer,

it is apparant that under certain conditions the mass and energy transport

equations are analogous. In the absence of a chemlcal reaction and thermo-

diffusion and diffusion-thermo effects, the mass transport equation can be

written as

_ Oc + _ Oc u ox v~

D =:

r o (_ Oc ) air~ (90)

where c is the concentration of one of the components, D ls the diffusi-

vit y, and ~ is the Schmidt number, Sc. If a dimensionless concentration

difference is defined as equation 90 can be non-dimen-

sionalized to equation 9.1.

(91)

Equation 91 is similar to equation 7 except that Sc has replaced Pro

For the case of pure mass transfer from a vertical cylinder in a

still atmosphere, the driving force is the concentration difference between

the fluid near the surface and that far from it. This concentration difference

is also responsible for the creation of free convection because of the density

difference. Therefore, in the momentum equation, the body force takes the

form g( P -Poo l'co

). If a Grashof number for mass transfer is defined as

gx3(p -p ) =[ 3 T/4 3

* * ga (p -p ) * ga (p -

Gr = w 00 G w 00 r = 2· 2 2

v Poo 4 V Poo v Poo

* 1/4

x S = * then the momentum transfer equation 92 is simi1ar to

G

equation 6.

Ou + Ou u~ var = 1 0 ( ou)

+ r or r ar

Alo)

(92)

The solution for equations 91 and 92 will be the same as for equations

6 and 7 if Pr = sc, if the surface concentration is uniform, and if the

rate of transverse mass transfer at the surface is uniform a10ng the surface.

The latter condition is not usua11y met in the mass transfer case. In the

heat transfer prob1em, the rate of f10w of f1uid at the surface in the transverse

direction may be an independent variable. This is not the case in mass trans-

fer, because the transverse ve10city at the surface is determined by the co~-

centration difference between the bu1k and at the surface. Another case in

which the transverse flux is not independent is that of evaporation of a liquid

from t.he surface to the still atmosphere of its own vapours, since the rate

of evaporation is coup1ed to the heat transfer rate by the latent heat of

evaporation.

The rate of evaporation per unit time per unit area is given by

v = él" / pl.. , w

where -II q is the rate of heat transfered from the superheated

,

1

-42-

vapours in the suroundings to the surface covered with a layer of 1iquid

at its boi1ing point, ~ is the latent heat of evaporation, and p is the

vapour density at a suitab1e temperature between that of bu1k and at the

surface.

q" = k ~=a =-2GkÂT

a . x- 1/4 e' (0)

v = _ 2(~ ) kAT w ax p~

e' (0) 2 = - sr- v k A.T e' (0)

vp~

v a --!...- = v

v w = - 2 k ) (~) I( c vp ~ e' (0)

p

Vw = - l ( !r> e' (0)

A measure of the driving force for heat transfer is given by the

dimension1ess quantity, B = c ÂT 2 which is the ratio of change ~

(93)

(94)

in

entha1py of the evaporating vapours às they are heated from the saturation

temperature to the bu1k temperature, to the latent heat of evaporation.

The equation, ana1ogous to 94, applicable to the case of mass transfer

where it is the concentration difference , t:. c, which causes free convection

is as follows:

-* B* * v a 2 v = _w_ = - ... ~' (0) (95) w v ~* Sc

* Âc where B = P 'w

-43-

lt has been shown by Sparrow and Cess (24) that the heat transfer

results for a flat plate with constant transverse flow do .not differ very

( 6 ) . - _-1/4 much·from those obtained by Eichhorn. for the case where v œ x • w

This conclusion will also be established by the present study. Sparrow and

Cess termed thus phenomena as,. 'the boundary·layer has poor memory', i. e.

the rate of heat transfer is not affected by the amount of upstream trans-

verse mass flow. lt 18 hoped that. this independenée of heat. t~a:nsfer· of the

transverse .velo.city distribution atthe surfac~ re~ain,s true for.a cylinder also.

In view of these facts, it is proposed that no great error will be intro-

duced if in equation 94,95 , the values of temperature and concentration

gradients, 9'(0) and (11'(0), are subtituted .trom the heat transfer solution

for the case of uniform v w

Therefore equations 94 and 95 can, be expanded

into equations 96 and 97 respectively.

(96)

0' ( ) *2) 3 0' (0' * n' *2 0' *3 " + 23 0 Vw +~ ( 31 0)+ 32(0)vw + 'lJ33(0)vw + 34(0)vw )J (97)

For some particular values of g and B, a value of v can be found w

-44-

which will satisfy equation 96. Thua plots of v , vs. s with B as a w

parameter will give the value of Vw

along the. length of the cyUnder at

. C.,/).T ). different values of B(L e.!:: The sameplot 18 also vaUd for

equation 97 if v , S w and

À.

* * B are replaced by Vw

' S , * B respectively.

DISCUSSION OF RESULTS

Temperature and Velocity Profiles:

Temperature and velocity profiles in a boundary layer flow reveal much

about the mechanisms governing the transport processes. This is especially

helpful in complex transport processes for which it is difficult to visualize

the effect of interaction of the processes occuring simultaneously. Equations

98,99 give the temperature and velocity profiles for heat transfer with trans-

verse mass flux around d. vertical cylinder at P:C' = 0.7.

(98)

(99)

The effect of YI. ' t'.he fllnction of distance in the transverse direction~ on e

and fi, the dimensionless temperat~re and longitud~nal flow parameters respec-

tively~ is given in figures 2 and 3. Four combinations of ~ an.d F , w

the

curvature and the transverse flow parameters respectively, are considered for

illustration.

Since suction causes a flow towards the surface, conditions in the near

vicini ty of the surface are in th:'i.s case more strongly inHuenced by the bulk

e

8

ro lE: 1

curve ~ F.

a 0'2 0'0

0"8 b 0'2 0'5

c 0'2. 0'5 d 0'3 075

'-------- --

0'6

1 Note t he change in scale

0'4

Q.; '< ~

0'2

o·o-l---. '. 0'0 2 4 6 8

'tt FIG.2

10

9 as a Function of yt

15 20

e

25 30

• :g 82 g of

1 • • ." !'I "tif ~ 0 00

f a t

• • .. ~ a .. . .. e u •

s:::--... 0

If) c 0

c:; .. .. u

(ii: §

"" .. Il ... ;..

0 b

;t 0 0 0 g ~ 0 N ...

0 0 0 0

-:..

-r48-

flllid conditions. Withblowing~ on· the other hand, the temperature and

fluid velocity at the surface influence the flow field for a considerably

greatet' distance fromthe surface. Thus the temperature gradient at the

surface is higher in case of suction, and lower.for blowing, as compared to

the gradient without transverse mass flux. Alternately~ the effect of trans-

verse mass flux May be described in,terms of a reductionor an.increase in

the boundary layer thickness caused by suction or blowing.

Corresponding effects of transverse mass flux·for the longitudinal flow

variable, fI, are shown in figure 3. The maximum value of fI and hence the

maximum axial velocity in the boundary layer decreases with suction and in-

crease with blowing. The distrubance ahownin the curve,d of.Jigure 3 becomes

more pronounced at higher values of F as the boundary layer becomes w thicker.

With excessive blowing the transverse flow will dominate the longitudinal flow

caused by the density difference.

Accuracy of the Results.

Fo~ this analysis to be valid, an important requirement is that the

boundary conditions at '1 = 00 be met. For YL ,,30 to be a satisfactory

numerical approximation for Y'\. =.oor the physical conditions require that 9 and

f' be zero and remain so for alljvalues of~ > 30, and that the higher deri­

vatives of 9 and f' follow the samerequirement. Figures 4 and 5 illustrate for

the particular value of ~ = 0.3 and F = 0.75, w the way in which the boundary

conditions for Y) = co are met at '1. = 30 or earlier. For aU other values of ~

and f, it is also true that 9 = 9 ' = 9" = 0 = f' = f" = fil' • W

----~----------------------~----~;

~ g • Ct • N .. • .= 00 1

':;'> , p !!!

• ~ 1 1/ ...

0

• 0

12 o·

2 004

" u

! v

= ~C; iD o -il 1&.

i ~

ri e

______ 1 ft

~

1

1 • • .. f 1

G ~E J -~i j s:::'"

~

~ • a 0 ... 60 U a :1 ...

11).

e ci" iL-1W

"'-a " ...

e ..

~

----~--~~----r__,r_--------_T!

• N

ID 0\

\ \ \ \ \ \ \ \ \

1 1 1 "

, 1

1 1 1 0 1 1 1 1 1 1 1 , 1 1 1 1 1 1 1 1 1 " Il

,/1 Il

III /1/ 1 ~ ~

• l

,'''IIU 8N1Gyn :10 % n l'IIIU 113CIItO GIIIH!

• o

-52-

Another factor affectingaccuracyis the repres~ntation.of the functions

f and-ein_series form.in ascending powers of ~. In this expansion it was

assumed that the sum of the.terms containing fourth and higher powers of è was less than the term .. containing the third power of ~. Because it is diffi­

cult to prove this point directly, which would require the calculation of a

large number of termsin the series, a limit was set on,the system variables

so that the third order term was small relative to the sum of the leading terms.

The dependence of the third order term, as a percentage of t~e sum of the leading

. terms of the series, .. is plotted in. figure 6 as a function of ~ and F w

Withinthe limits -1 < F < 1 and 0 < ~ < 0.85, it 1s observed that the w

third order term is never more than± 4% of the sum of the leading terms.

Because the results for cylirider have been obtai.ned by a perturbation technique,

the solution is not applicable ta cylinders of bigh curvature, i.e. for

~ > 0:85, for which the neglected terms of the series would become significant. \,. ..

6 . For a Grashof number of 4 x 10 , the limitation of ~ < 0.85 corresponds ta

cylinders of a length ta diameter ratio up· ta 27. With respect ta limit on

transverse flow, it is of interest to note that if the limits for F were extended w

ta -5 < F < 1.5 , the third arder term would become at the worst -10% of the w

sum of the leading terms. The error caused by suction is smaller than that for

the same rate of blowing beeause the boundary layer is thinner and more stable.

Local Heat Transfer Results

The local heat transfer results for blowing and for Buetion, given earlier

as equation 82 are represented graphieally on figures 7 and 8~ The transverse

mass flow parameter F is preferr:ed ta v = (val J) ), when è is used as the w w w

--r -,.....j • ...... ~I.;t '-' ......

::J Z '-'

.., e

rO'I~-----------------------------------------------------------------------------------------------,

0"8

0"6

0"4

0·2

0"0 , 1" O··, 0"2 0"3 0"4 0"5

FIG.7 F

w 0"6

Heat transfer results for blowing

• • •

"PRESENT WORK

CRe'.24:"J CRef. 10 ) att = 0

(Re'. 3 )

( Re'. 6 ) Vw .-114, t.= 0

O·SO

0·40

0·20 0·'0 00"5

~~-----------

07 0"8 0·9

0 •

0 .,

• ! w 1 • . J - _1_ 1_---_ ·r-

0 .... 10 ... ~ • ~ • • ~ - -n

1 1 • • •

~

.~ b 1

~~~~~--~--~--~~o' ~ 0 S '0

a 0 ... ... U :1 • t ... •

CD~ .C!i~ "_II &&ok

11 ... • q 11 ... .. Il =

Ct

n~i~-------------------------------------------------------------------'

?,n .P.!)

11-'-' ~ ''Ç' ",Iv -.J ........ .:J ::2 ....."

L-..I ----.. :;tr-. I~

~It '-' ........ .:J ;z -./

/-.J

r

I"2-CL_---~ "aO·'O

.·0 1 -0". -a. -(M -02.... 0·0 0·2 0"4 O·. O·.

FIG. 9 Comparison of local heat transfer results for a cylinder vith fIat plate heat transfer re.u1ta

e

o .. .r

r"' 1 ..... ~,. -..... ~ z ~ ~r-' ~ 1 .....

~I" .... .... ~ . ....::::...,.,

et

2'0

ra

,'6

,'4

"2

rO

"8

0'6

0·4

0'2

0"0 0'0

--------

01 0'3 0:4 0'5

t. FIG.IO

·tt

ro

• (Re' 21 h F.·O

------_--------r . 0'8

. Fw =-0'8· . .

-0'5

-0'3

-01

-01

0'3 0'5

0'8

0'6 0'7 0'8

ro" n

0'6 • la.

~ ï_ c';j.t ~

0'4 -; ~

~

0'2

0'0 0'85

Compairson of local heat transfer results for a cylinder with and without transverse mass flux.

-57-

second parameter, because the radius appears both in ~w and ~. but not in

The flow variable F = (v x/V)(Gr/4)-1/4 represents the ratio of the w w

F • w

transverse flow Rey~olds number to a functionof Grashof number. The choice

of F. and ~ thus facilitates determination. of the effect. of curvature. How­w ç

ever, it should be noted that ~, also represents the effect of distance from

·the leading.edge as in' figure 12·whet.e x does not appear in the ordinate.

The rate of heat transfer may ~e seen to increase for suction and decrease

for blowing. A flow of fluid outward fromthe s~rface and originating at the

surface temperature, acts .to reduce the temperature gradient at the surface, and

~hus~r.educing.the heat transfer rates. A flow of fluid towards the surface has,

naturally,the opposite effect.

The curve for ~ = 0 . in figures 7 and .8, which ·represents the heat trans-

fer rate to a flat plate, differs at .higher rates of blowing and suction from

the rates givenby Sparrow and Cess (24). Since Sparrow and Cess considered

only a first order:correction for the transverse flow, their results are lower

than those obtained in the present work, for which a third order correction

has been applied. The results obtained for a fIat plate by Mabuchi (10) and

by Brdlik. and Mochalov (3) are in good a~eement with tbe present work for

blowing.For suction, however,our results are between tbose given by Sparrow

(24) and tboœ of Mabucbi (10), Brdlik and Mocbalov (3). Mabucbis and Brdl1k and

Mocbalov assumed 'similar profiles' for velocity and temperature in the boundary

layer, and. for the transverse distance variable in both profiles,used a fourth

order polynomial in terms of the ratio of transverse distance to boundary layer

-58-

thickness. Since velocity .and temperature profiles are in fact nbt'similat'

in the boundary layer, the perturbation technique was used by Sparrow and Cess

(24) and this study. It is believed that the result:~ of the present analysis

is more accurate than those of the earlier studies.

When the fIat plate results of the present study are compared with

the exact solution given by Eichhorn (6) for the case of a transverse flux

which is inversely proportional to the on~fourth power of. the distance from

the leading edge, good agreement is found for moderate rates of blowing and

suction~ . Deviations are higher for suction than for blowing. However, the ,

important conclusion is that the local heat transfer rate is influenced very

little by the upstream transverse velocity distribution. This fact was first

noticed by Sparrow and Cess (24)

~ 1 - -1/4 The effect of curvature is indicated by the parameter, ~= (-) x(Gr/4) , . a

the ratio of boundary layer thickness to the radius of the cylinder. It may be

seen from figures 7 and 8 that for any value of F , the rate of h~at transfer incrE w

ses as the cUlVFlture variable increases, the mimimum value being for a flat plate.

A physical picture of the effect of curvature onheat transfer may be given by

considering the value of heat flux for a flat plate at ... particular surface and

environment temperatures T and Too. As the surface acquires curvature in w

becoming a cylinder,whilè the same T w

and Too are maintained, the point

values of heat flux at aIl points in the boundary layer wou1d be decreased if

the heat transfer at the wall were to remain constaQt. This reduction wou1d

resu1t from the fact that for a cylinder the area for transverse flow of heat

1

.. 59-

or mass in the boundary layer increases in proportionto the Tadial distance.

If the heat flux at aIl points in the boundary layer were to decrease, the

temperature difference (Toa - T ) . would al80 decrease. w Thus for the tempera-

tures Tao and T w

to be maintained as curvature increases, it necessarily

follaws that the heat flux at the wall must increase. The magnitude of the

increase is given on figures. 7 add 8.

The combined effects of curvature, as discussed above, and of transverse.

mass flux, as discussed earlier, are Ulustrated on figure 9 as the ratio of

heat transfer for a cylinder to that for a fiat plate. An alternate represen-

tation,shown in figure 10, gives the ratio of heat transfer to a cylinder with

transverse mass flux to that without mass transfer. It is apparent that while

blowing (positive transverse mass flux) decreases heat transfer, positive cur-

vature (convex) increases the rate. Alternately one may note that while nega-

tive transverse mass flux is suction acts to increase heat transfer, negative

curvature decreases it.

Ot In these terms therefore it can be stated that the effect transverse mass

flux is opposite to that of curvature.

Figure 10 also shows heat transfer results according to equation 84, for

a cylinder without transverse mass flux. When compared with the results of

Sparrow and Gregg (21) which are indicated, the agreement is seen. to be good.

Average Heat Transfer Results

Heat transfer results, averaged over a length x = L .according to

~ ________________ ~p-~~~ __ ---------r~

-61-

equation 89,' are given in figure 11. The average value of heat tra~sfer co-

efficient is higher than the local value at x = L, beca~se of the decrease of

the local heat transfer coefficient with distance from the leading edge as the

boundary· layer thickness increases. However, the effects of ~ and F on the w

average Nusselt number are similar to tl)at for the point values given on

Figures 7 and 8.

Mass Transfer with or without.Heat Transfer.

In the analysis presented to this point, a degree of freedom·has existed

between the variables, transversevelocity, and the heat transfer driving

force, AT. HOlrever, if the source or the sink of transverse mass flow at the

solid surface occurs by means of a phase change, then the transverse velocity

and driving force become linked. In the case of coupled heat and mass transver,

the mass flux will usually not be constant over the surface. However, it has

been pointed out earlièr that the boundary layer has 'POOl' memory' i.e. the

rate of heat transfer is not much.affected by the transverse mass flux distribution

at the surface. Thus it is proposed that where transverse mass flux distribu-

tion depends on the local rate of heat transfer, the solution for the uniform

transverse mass flux distribution can he applied without large error.

Figure 12 gives the rate of transversemass flux v as a function of ç w

* ~* * and B for heat and mass transfer and v as a function of ç and B for pure w

mass transfer. Since~, with ordinate v =(v a/V),represents longitudinal C;: w w

distance from the leading edge, figure 12 gives the rate of transverse mass flux

as a function of length of the cylinder. In.this analysis, the theoretical trans­

verse mass flux increases without limit as the leading edge i.e. ~ = 0 is

approached, since the boundary layer thickness is taken as zero at this point.

• b

• 0

G

• 0

!fi 0

~------~------~------~------r-------r------+g g

... • ."

b ....

1 ., j ~ .8 ., ... :a

N1 "":'fI se ... ... :a

• li • .. ~ a li • i Il •

1

-63-

Downstream fromthe leading edge theboundary layer thickness increases in

proportion.to the one-fourth power of the distance from the leading edge. At

higher values of ~ , the mass flux may be seen from figure 12 to approach an

asymptotic value.

Li (9) has measured the rates of evaporation of water on flat plates

in natural convection at hightemperatures. For the evaporation rate he obtained

o with a 2 inch long plate in an environment at 374 C, which gives a Grashof

5 number of 3.24 x 10 , the value of the flow variable F = 0.47 may be calculated.

w

This experimental value of F is well within the range of validity.of our w

theoretical results.

For the case when the heat transfer by radiation 18 significant, the rate

of evaporation by radiation may be added to the right hand side of equation 96,

and the value of v may be found to satisfy this equation. w

.. CONCLUSIONS

For heat transfer, by laminar natural convection with uniform trans­

verse massflux; for a vertical cylinder, a numerica1 solution has been

obtained, which depicts the effect of curvature and transverse mass flux.

Qwing to the application of perturbation technique this solution is not

valid for very long and thin cylinders.

It has beenobserved, that

(i) For aIl values of transverse mass flux, the effect of curvature,

convex to the surroundings, is to increase the rate of heat transfer,

Conversely, the rate of heat transfer is decreased if the curvature to

the environment becomes concave.

(ii) For aIl values of curvature, from'a fIat plate to the highest value

for which the solution obtained remains val id, the heat transfer rate

is decreased by blowing (positive mass flux) and increased by suction

(negative massflux).

(iii) The effect of transverse mass flux is opposite to that of curvature.

(iv) The phenomena of 'poor memory' of the boundary layer has been confirmed

for a flat plate and it i.s hoped to be true for a cylinder a1so. This

provides a method for the estimation of transverse mass flow rates for

a coupled heat and mass transfer process e.g. evaporation of a liquide

Evaporation by radiation can be taken into account as long as the mass

transfer rate conforms to the limits imposed.

BIBLIOGRAPHY

1. Bourne, D.E.,. and Ward1e, S., J. Aero, Science, 29; pp 460-67, (1962).

2. Bourne, D.E., and Davis,D.R., Quart. J. Mechanics and Applied Mathematics,

11; pp 52-66, (1958).

3. Br~k, P.M., and Mocha1ov,. V.A., Int. Chem. Engg. 5; 4, pp 603-7, (1965).

4. Brindley, J., Int. H. Heat Mass Transfer, 6; pp 1035-48, (1963).

5. Cess, R.D., Advances in Heat Transfer, vol. 1, Academie Press, New York, 1964).

6. Eichhorn, R., J. Heat Transfer, C82;3, pp 260-63, (1960).

7. Kelly, H.R., J. Aero, Science, 21; (1954).

8. Kemp·',N.H., A.I.A.A., 2; (196t..).

9. Li, F.S., 'Natural convective Heat and Mass Transfer from vertical plates in

High temperature surroundings' Ph.D. Thesis (McGi11 University, 1965).

10. Mabuchi, Ikuo, Bulletin of Japan So.c. Mech. Engrs., 6; 22, pp 223-30, (1963).

Il. Mathers, W.G., Madden, A.J., Piret, E.L., Ind. Eng. Chem., 49; pp 961-68,(1957).

12. Millsaps, K., and Poh1hausen, K., J. Aero, Science, 25; pp 357-60, (1958) •

13. Nakamura, H. , Bulletin of Japan Soc. Mech. Engr.8., 5; 18, pp 311-14, (1962) •

14. Oliver, C.C., and McFadden, P.W. , J. Heat Transfer, C 88, 2, (1966).

15. Ostrach, S. , N.A.C.A., T.N. 2635, (1952).

16. Ostrach, S. , N.A.C.A. , T.R. 1111, (1953) •

17. Schuh, H., Reports and Trans. 1007, AVA Monograph, BritishM.A.P. (A~ri1 1948).

18. Somers, E.V., J. of Applied Mechnics, 23; 2, pp 295-301, (1956).

19. Seban, R.A., and Bond, R., 18; pp 671-75, (1951).

20. Sparrow, E.M., and Gregg, J.L., Trans. Am. Soc. Mech. Engrs., 78; pp435-40

(1956).

-66-

21. Ibid., 78; :pp 1823-29, (1956).

22. Ibid., 80; pp 3.79-86, (1958).

23. Ibid., 80; pp 879-86, (1958).

24. Sparrow, E.M., and Cess, R.D., J. Heat Transfer, C 83; 3, pp 387-89,

(1961) •

25. Sparrow, E.M., Minkowycs, W.J., Eckert, E.R.G., J. Heat Transfer,

C 86; 4, (1964).

26. Sparrow, E.M., Scott, C.V. Forstrom, R.J., Ebert, W.A., J. Heat Transfer, C·87; 3, (1965).

27. Sparrow and Lin, Int. J. Heat Mass Transfer, 8; pp 437 (1965).

28. Sparrow, E.M., Yang, J.W., Scott, C.J., Int. J. Heat Mass Transfer, 9; pp 53-61 (1966).

29. Sparrow, E.M., Minkowycz, W.J., Eckert, E.R.G. J. Heat Transfer,

C 88; 3, (1966).

30. Sparrow, E.M., Starr, J.B., Int.J. Heat Mass Transfer, 9; 5,

pp 508-10, (1966).

31. Tewfik, O.E., Yang, J.W., Int. J. Heat Mass Transfer, 6; pp 915-23 1

(1963).

32. Wanous, D.J., Sparrow, E.M., J. Heat Transfer, C 87; 2, pp 317-19,

(1965).

33. Wi1cox, W.R., Chem, Engg. Science, 13; pp 113-19, (1961).

34. Yang, K.T., J. App1ied Mechanics, 27; 2, pp 230-36, (1960).

35. Zeh, D.W., Gill, W.N., Chem. Engg. Progress, Symposium Series, 61;

pp 19-35, (1965).

NOMENCLATURE

Roman Symbo1s

a radius of the cy1inder, feet

B

* B

C !:::.T P

Ac

lw

., dimension1ess

dimension1ess

C ifi h BTU/1b oF p spec c eat at constant pressure,

c

Ac

concentration of one of the components Ib/ft3

c - c , 00 w Ib/ft3

D diffusion coefficient ft2/hr

F w

b (v- -xl)) ).(Gr)1/4 <:l l' lowing parameter w 4 ' dimension ess

f dimension1ess dependent variable, defined in equation 32

f ,f b a a

G

* G

Gr

Gr*

dimension1ess dependent variables, defined in equation 32,47,48.

[

gj3{T. - Tw

)a3]1/4

dimensionless constant, 4 ))2

g a3

( lw- L) dimension1ess constant -~J)2 1'0..

gj3(T. - T )x3

Grashof number, w, dimension1ess V2

(

_3{ D p') Grashof numberfor mass transfer gx J w -.;lQ)

J),,2 '

1/4

) 2 g acceleration of gravit y ft/hr •

-68-

h heat trapsfer coefficient, Bru/hr ft2 oF

k thermal conductivity, BTU/hr ft oF

L length of the cylinder

Nu Nusselt number, hx/k, dimensionless

Pr Prandtl number, C plk, dimensionless p

q" heat transfer rate, BTU/hr ft2

r distance inradia1 direction, ft

r r dimensionless distance in radial direction, a

Sc Schmidt number, ~ , dimensionless D

T temperature, oF

AT Toc - T, oF w

u velocity in x direction, ft/hr

u dimensionless velocityin x direction, üa/v

v velocity in r direction, ft/hr

v dimensionless velocity in r direction va/v

x distance in longitudinal direction, ft

x dimensionless distance in longitudinal direction

Greek Symbols

0( thermal diffusivity, J~ ,ft2/hr p

r

coefficient of thermal expansion,

3 gt3(Too - T )a

.JI 2 , dimensionless

0-1 F

-69-

r* dimension1ess

~ dimension1ess independentvariab1e, defined in equation. 15

9 dimension1ess dependent variable, (T~ - T)/(Tf/O - T ) w

9 ,9 b dimension1ess dependent variables, defined in equations 33,49,50,51. a a

latent heat of evaporation, BTU/1b

)J. absolute viscosity, 1b/ft.hr

V kindematic viscosity, ft2/hr

~ curvature effect for heat transfer, defined in equation16

curvature effect for mass transfer,

J density 1b/ft3

dimensionless dependent variable, (coo - c )/(cao - c ) w

~ stream function, defined in equation 14.

Subscripts

w wall conditions

~ environment conditions

av. average

APPENDIX

Simultaneous differential equations of arder higher than one, with j

two point boundary conditions, may be solved by numerical integration on digital

computers. The mass, momentum, and energy transport equations of this study .

were converted to a set of first order equations by defining the dependent varia-

b1e and its derivatives (except the highest one) by new functions. Since initial

values of the functions are required for a numerical integration method, some

arbitrary but reasonab1e guess of the initial values of the functions is made,

and the integration is carried out. The process is repeated with different

initial conditions unti1 the final conditions are satisfied. In the present case

three step sizes, 0.05, 0.10 and 0.20,were used in the independent variable, Yl '

in the regions 0:S ~ < 10; 10:S "rl < 20; 20 < ~ < 30 respectively, because

at higher values: of ~ , the dependent variables ~each asymptotic values.

Since equations 74 through 78 give three initial conditions and two final

conditions, two more initial conditions, f"(O) and 9 ' (0), have to be assumf?d such

that the given final conditions, f"(ao) and 9(oa) are satisfied. Method of

steepest descent was used to minimize the difference between final values f~

and 9~ , obtained by solving equations with assumed initial conditions, and the

desired final conditions, f'(~) and ·9(~). Owing to the presence of local minima

in the case of non-linear equations, the abso1ute minima was found by locating

minima on the locus of (f~- f' (ex») 2 , such that the final condition, (9jfO- 9(00»2

is always minimum.

-71-

For the case of inhûiiio~neous equations with a homogeneous part, one of the

initial conditions, fao was fixed arbitrarily, then the other initial condi-

tion, 9ko 2 , was found so that (en~- e(~» was a minimum for the homo- .

geneous part of the equations. This procedure was repeated for the full in-

homogeneous e,uations to find el

iO ' fixing the value of fil ; so that 10

2 (e. - e(oO» was a minimum. Making a linear combination of the solutions 100

of homogenous and :inhomogenous equations, the desired initial conditions can

be obtained as follows,

f"(O) = 0{ fil + fil • ho io'

where fI

0< = ~ fI

hl»

el (a) = 0(9 1 + 9'1 ho io

and ehoo ' f hoo and eiOo, f~OCI are the final values. for homogeneous and

inhomogeneous equations respectively, obtained from the corresponding assumed

ini tial values el fil ho' ho and el fil

io' io