AIM-87-1 640 Eddy Diffusivity of Heat for Drag Reducing Turbulent Pipe Flows H.K. Yoon and A.J. Ghajar, Oklahoma State Univ., Stillwater, OK

AlAA 22nd Thermophysics Conference June 8-10, i987/Honolulu, Hawaii

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics 1633 Broadway, New York, NY 10019

EDDY DIFFUSIVIn OF HEAT FOR DRAG REDUCING TURBULENT PIPE FLOWS

Hyung K. Ywn* and Afshin J. majar** School of Mechanical and Aerospace Engineer ing

Oklahoma State U n i v e r s i t y Stillwater, Oklahoma 74078

!+ECL Experiments were cnnducted t o v e r i f y t h e

assumptions and g e n e r a l a p p l i c a b l l i t y of a new

semi-empir ical equa t ion for e d d y d i f f u s i v i t y of h e a t proposed p r e v i o u s l y for v i s c o e l a s t i c t u r b u l e n t

p i p e flows. The experiments were performed for

Separan I\P-273 and Polyox WSR-301 s o l u t i o n s with

c o n c e n t r a t i o n s r ang ing from 10 t o 1000 ppm and

Separan AP-30 wi th ConcenLration of 3000 ppm i n

thermally f u l l y developed t u r b u l e n t flow i n p ipes

w i t h diameters of 1 . 1 1 and 1.88 cm I . D . under

c o n s t a n t wall h e a t f l u x . The experiments V e P i f i e d

t h e assumptions made i n r ega rd t o t h e un ive r sa l iLy

of t h e minimum asymptotes far fr iCLinn and h e a t

t r a n s f e r . The p r e d i c t i o n of h e a t transfer- c o e f f i c i e n t s wiLh t h e u s e of t h e proposed equa t ion

for a l l of t h e experimental da ta is wi th in a maximum d e v i a t i o n of 30%. -

c

D f

fP

f S

FR

h

j h k

kS

N o m e ~ ~ l ~ t ~ c o n s t a n t s used i n Eq. (7) c o n s t a n t Lhat c h a r a c t e r i z e s t h i c k n e s s of

wall l a y e r

S p e c i f i c h e a t of f l u i d

i n s i d e diameter of t e s t s e c t i o n

f r i c t i o n f a c t o r . f = r W / ( p u z / z )

f r i c t i o n facLor for polymer s o l u t i o n

f r i c t i o n f a c t o r for s o l v e n t

f r i c t i o n d rag r e d u c t i o n ra t io ,

FR = ( f , - f p ) / f s

h e a t transfer c n e f f i c i e n t

Colburn 1 - f a c t o r , j h S t Pr;l3

thermal Conduct ivi ty of f l u i d

thermal Conduc t iv i ty of p i p e

* Presen t a d d r e s s , Korea I n s t i t u t e of Energy and ReSnUrCes, Daejeon, Chungnam, Korea.

_, ** Assnc ia t e P r o f e s s o r , Member AIAA.

1

K L

Lh

LP P

+ P

P . 1

P.

P"

Rea

R

S t

Tb

Te Tf

T i

Tw i

U

U T U +

U

V ws

"ch

X

Y

Y +

von Karman C O n s t B n t , K = 0 . 4

lengLh of test s e c t i n n

hea ted l eng th O f t e s t s e c t i o n

d i s t a n c e between p res su re t a p s

p r e s s u r e

appa ren t P r a n d t l number, Pva = qac/k

h e a t flow r a t e

h e a t f l u x

r ad ia l c o o r d i n a t e d i r ecL ion from t h e

c e n t e r 1 ine

i n s i d e r a d l u s of test s e c t i o n

d imens ion le s s i n s i d e r a d i u s of Lest s e c t i o n , r; = riuT/va

Outside radius of test s e c t i n n

e l e c t r i c a l r e s i s t a n c e of p ipe

appa ren t Reynolds number, Re, pUD/na

S t an tnn number, S t = h/pcU

l o c a l f l u i d hulk temperaLure

hulk e x i t temperature

f i l m t empera tu re , T~ E ( T ~ + T ~ ) / z

bulk i n l e t temperature

i n s i d e wall s u r f a c e t empera tu re

o u t s i d e wall m r f a c e temperatuve

s t reamwise mean v e l o c i t y , u = u ( y ) s h e a r v e l o c i t y . uT = ( ~ ~ / p )

nondImensLona1 mean v e l o c i t y , ut = u/u,

mass ave rage v e l o c i t y

e l e c t r i c a l v o l t a g e ( p o t e n t i a l )

Weissenberg number, ws = lU/D

c r i t i c a l Weissenberg number for h e a t

t r a n s f e r

l o c a l a x i a l d i s t a n c e

c o o r d i n a t e d i r e c t i o n nnrmal t o w a l l ,

y - r i - r

nondimensional d i s t a n c e normal t o wall ,

112

Y + = YU,/Va

t u r b u l e n t eddy d i f f u s i v i t y of h e a t

t u r b u l e n t eddy d i f f u s i v i t y of mpmentum

appa ren t v i s c o s i t y a t t h e wall

z e r o s h e a r r a t e appa ren t v i s c o s i t y

n, + s h e a r r a t e

i f l u i d L i m e s c a l e

"a P d e n s i t y of f l u i d

TW

i n f i n i t e s h e a r r a t e appa ren t ViScoSitY

appa ren t kinematic v i s c o s i t y , va - n a I p

wall s h e a r s t r e s s

Introd!J!?!ron It h a s been e s t a b l i s h e d t h a t sma l l a d d i t i o n o f

c e r t a i n polymers t o t u r b u l e n t p ipe f lows Could

r e s u l t i n a d r a s t i c r e d u c t i o n i n f r i c t i o n d r a g and

heat, t r -ansfer ."2 T h i s f i n d i n g i n i t i a t e d a number

o f s t u d i e s i n t h e p o s s i b l e use o f polymer a d d i t i v e s

i n p r a c t i c a l eng inee r ing systems. However, r e c e n t

r e v i e v s of t h e r e l e v a n t works suggested t h a t most

of t h e p rev ious experimental and a n a l y t i c a l SLudies

have been c a r p ied o u t under inadequate expel.imenLa1

condiLLons and i n a c c u r a t e experimental r e s u l t s . I "

P a r l i c u l a r l y , Reynolds analogy. which has been used

t n cnrx-elnte momentum and h e a t t r a n s f e ? phenomena

i n most a n a l y t i c a l s t u d i e s , was v e ~ i f i e d Lo be

i n v a l i d f o r visCoelSStiC t u r b u l e n t pipe f lows. '-' To remedy t h e inadequacy O f tile e x i s l i n g h e a t

t r a n S f e r models, i n a pe rv ious paper Yoon and

Chajar ' p re sen ted a new h e a t eddy d i f f u s i v i t y

equa t ion f o r v i ~ c o e l a s t i c t w b u l e n t p ipe f lows

which was developed with t h e a i d o f Kwack's"

expe r imen ta l d a t a and was v e r i f i e d wi th l i m i t e d

experimental d a t a . The aim o f L h i s Study is t o

conduct new experiments i n o rde r Lo v e r i f y t h e

assumptions made i n t h e e v a l u a t i o n o f t h e

a d j u s t a b l e c o n s t a n t s i n t h e proposed h e a t eddy

d i f f u s i v i t y equa t inn and Lo e s t a b l i s h its g e n e r a l

a p p l i c a b i l i t y t o va r ious experimental d a t a .

t e s t s e c t i o n s Cons i s t o f copper p l a t e s Which were

s i l v e r - a r c so lde red t o t h e ends of t h e tesL s e c t i o n

t o s e c u r e a well def ined e l e c t p i c c i r c u i t through

t h e end p l a t e s . The upstream and downsLream

s e c t i o n s were e l e c t r i c a l l y i n s u l a t e d from one

a n o t h e r . The two-sect ion arrangement allows t h e

o p e r a t o r Lo chonse t h e l o c a t i o n where he d e s i r e s

t h e Lemperature p r o f i l e t o begin deve lop ing .

W

The t e s t s o l u t i o n s were prepared i n t h e 2 . 2 1

m3 mixing tank and g r a v i t y d ra ined t o t h e 2.21 m3

s t o r a g e t a n k . To minimize mechanical deg rada t ion

of polymer s o l u t i o n s f o r t hese experiments , t he

2.27 m3 upstream s t a i n l e s 8 s t e e l s t o r a g e tank was

p r e s s u r i z e d , fo rc ing the f l u i d through the test s e c t i o n . While t h e o v e r a l l flow system was

opera t ed w i t h p r e s s u r i z e d a i r (up t o 80 p s i s ) using

t h e once-through mode, t h e experiments were

conducted by ma in ta in ine t h e p re s su re i n t h e tank

from 30 to 40 p s i g . The 13.63 m 3 t ank was used Lo

s t o r e t a p water which was needed to clean t h e

b u i l t - u p of polymer on t h e i n n e r s u r f a c e O f t h e

t e s t s e c t i o n a f t e r each experiment . The flow r a t e s

which WePe obtained by t h i s system, even though

seve re ly diminished due t o cons ide rab le f r i c t i o n

d rag i n t h e t e s t s e c t i o n of sma l l diameter and long

l e n g t h , Covered Reynolds numbers based On t h e

apparenL viscosity of up to 1 . 2 x 105.

-,

The Constant wall h e a t f l u x boundary Condition

was maintained by a Lincoln DC-600 welder . I t can

operaLe in t h e cons t an t v n l t a g e OP cons tan t currenL

mode, and has a 100% duty c y c l e r a t i n g a t 600 amps

and 4 4 v o l t s . The t e s t s e c t i o n s were i n s u l a t e d

from t h e environment using f i b e r g l a s s pipe

i n s u l a t i o n and vapor-proof p ipe t a p e . Double

C M T R O L "*WE C L I U W S E C W

T€sr BECnOn ,.I, om '0, LiD . > o w

The p r e s e n t experimenLs were conducted i n t h e

f l u i d dynamics l a b o r a t o r y a t Oklahoma S t a t e

U n i v e r s i t y . R schematic diagram of t h e f low

c i x u l a t i n n system is shown i n Fig. 1 . C*TC* 011 UlXlNO TANK l,oI cDN,('oL

" I L V S S

WESSUPE RELEASE VALVE

m e t e s t s e c t i o n s used consist of two seamless

s h i n l e s s s t e e l p ipes (Type 304) w i t h i n s i d e

d i ame te r s o f 1.88 cm ( L I D = 6 1 1 ) and 1 . 1 1 cm ( L I D =

10461. These t e s t s e c t i o n s ensu re the the rma l ly

f u l l y developed cond i t ion f o r v i s c o e l a s t i c f l u i d s

which r e q u i r e 400 t o 500 d i ame te r s f o r t h e minimum

h e a t t r a n s f e r asymptote . ' m e end connec t ions o f system.

TO ORAlN

Fig . I Schematic diagram of t h e f l o u c i r c u l a t i o n c

2

wrapping o f f i b e r g l a s s i n s u l a t i o n was deemed enough

t o produce t h e w e l l - i n s o l a t e d cond i t ion .

I n t h e p r e s e n t f low system, e i t h e r t h e

hydpodynamic and thermal e n t r a n c e r e g ions can

develop s lmul t aneous ly from t h e beginning of t h e

t e s t s e c t i o n , or t h e v e l o c i t y p r o f i l e can be f u l l y

developed before h e a t t r a n s f e r starts. The

measurements of p r e s s u r e drop and h e a t t r a n s f e r

were taken a t t h e same time in t h e the rma l ly f u l l y

developed r eg ion w i t h t h e use o f one U-tube mercury

manometer and I 30 gauge copper-constantan

thePmOcoUples connected to a 40 channel Monitor

Labs Datalogger Model 9302. To enSUPe f u l l y

developed f low, numerous p r e s s u r e t a p s and

thermocouples were d r i l l e d and a t t a c h e d t o t h e t e s t

s e c t i o n s . FOP b e s t p r e s s u r e drop r e s u l t s , t h e

p re s su re t a p s were d r i l l e d such t h a t t h e r a t i o of wall t h i c k n e s s t o t a p h o l e d i ame te r f o r both test s e c t l o n s was g r e a t e r than 1 . 5 and less than 2.

P re s su re drop i n t h e e n t r a n c e r eg ion was measured

with a multi-column water manometer. To e l i m i n a t e

t h e e f f e c t of e l e c t r i c a l c u r r e n t f lowing through

the test s e c t i o n on t h e thermocouple r e a d i n g s ,

copper oxide cement vas used as t h e adhes ive . To

o b t a i n t h e bulk t empera tu re a t t h e end of t h e t e s t

s e c t i o n , a temperature well which c o n s i s t e d o f f i v e

b a f f l e s was i n s t a l l e d j u s t downstream o f each t e s t

s e c t i o n . S ince t h e inlet temperature o f t h e f l u i d

vas uniform a c r o s s t h e test s e c t i o n , i t was measured by means o f a thermncouple probe inserted

i n t h e calming s e c t i o n .

I

The flow r a t e was measured by a one-inch

t u r b i n e meter l o c a t e d upstream from t h e t e s t

s e c t i o n . T h i s t u r b i n e meter mnnitnred by a Hewlett-Packard frequency counter Can produce

i n s t a n t or t ime-averaged r e a d i n g s so t h a t i t

enab le s one not on ly t o o b t a i n t h e ave rage Plow

Pate b u t a l s o t o check t h e f low s t a b i l i t y .

Apparent v i s c o s i t i e s of s o l u t i o n s Were

measured a t wide range o f s h e a r r a t e s (0.36 t o 2 x

1O4sec-’) w i th t h e u s e o f two Coue t t e v i scomete r s

(Brookf i e ld Synchro -E lec t r i c Model LVT wi th U L

adap to r and a Fann Model VC) and a c a p i l l a r y tube

viscometer (0.94 mm I.D. and l / d = 325). S o l u t i o n

samples were taken from t h e downstream head tank

du r ing each run or immediately a f t e r each run .

me f r i c t i o n factor was determined from t h e 1

f q l l o v i n g equa t ion :

Thus t h e de t e rmina t ion o f f r i c t i o n f a c t o r r e q u i r e s

measurements of t h e mean Ve loc i ty tu) and t h e

p r e s s u r e drop ( A P ) a c r o s s t h e d i s t a n c e between two

p r e s s u r e t a p s ( L p ) when the f l u i d d e n s i t y ( p ) and

t h e i n s i d e diameter o f test s e c t i o n ( D ) are known.

The h e a t t r a n s f e r C o e f f i c i e n t was ob ta ined

from t h e fo l lowing expres s ion :

(2)

The de te rmlna t inn o f h e a t t r a n s f e r coefficient

r e q u i r e s measurements o f t h e h e a t f l o w r a t e ( i ) , t h e i n s i d e wall s u r f a c e t empera tu re (TWi) and t h e

bulk mean t empera tu re (Tb). m e h e a t f low rate can

be measured i n two ways: 1 ) from t h e en tha lpy r i s e

of the f l u i d determined from measut-ements o f t h e

flow r a t e and t h e d i f f e r e n c e between i n l e t and

o u t l e t bulk t empera tu res , and 2 ) from t h e

e l e c t r i c a l power supp l i ed a c r o s s t h e t e s t

s e c t i o n . me second method was used because o f its good r e l i a b i l i t y . The h e a t flux Pate c a l c u l a t e d

from t h e p o t e n t i a l drop c a n be w r i t t e n as

(3)

where V is t h e p o t e n t i a l , R is t h e p i p e r e s i s t a n c e ,

and Lh is the hea ted p ipe l e n g t h . me i n s l d e wall

temperature can be determined u s i n g t h e F o u r i e r ‘ s

law o f h e a t conduct ion from measurements o f t h e

o u t s i d e wall temperature (Two). For a f l u i d

f lawing i n s i d e a hollow cyl inder wi th uniform h e a t

g e n e r a t i o n i n t h e tube wall and n e g l i g i b l e h e a t

l o s s t o t h e SurPOundings. t h e fo l lowing equa t ion is v a l i d :

where ks is t h e thermal c o n d u c t i v i t y o f t h e p ipe .

The l o c a l bu lk mean t e m p e r a t w e (Tb) a t t h e d e s i r e d

l o c a t i o n x is determined from measurement o f t h e

e x i t bulk temperature (Te ) . S ince t h e heat

generatim in the tube is considered to be fairly uniform ~CPOSS the tube. the following equation holds:

T = T - (T ~ T.) . (L - x)/Lh ( 5 ) h e e l h

where Ti is the inlet bulk temperature. In sumary. the determination of heat transfer coefficient requires measurements of the heat flow rate, the inlet and nutlet bulk temperatures, and the Outside wall temperatwe when the dimensions of the test sections. the properties of the Pipe and the solutions are known. The resistance and thermal conductivity of the pipe Were determined at the outside wall temperature. The density, specific heat. and L'lermal cnnductivity of the solution were taken to be the same as those of water and were evaluated at the film temperature (Tf) with the exception of thermal conduztiviLy Which was evaluated at tbe exiL bulk temperature. The equations used for the variation of pipe pmperLies and physical prOperLie3 of the Solution with temperature are given elsewhere. 'I

me reliability of the flow circulation system and Lhe enperlmental procedures Mere checked with several caliSration runs for measurements of friction factors and heat Lransfer cnefficients for a Newtonian fluid (tap water) by comparing the experimental results with well-established Newtonian correlations. '-I2 The uncertainty analyses of the overall experimental procedures for Newtonian and viscoelsstic fluids shoved that there

is 5-81 uncerlainty for friction factors and 8.12% uncertainty fo r heat transfer coefficients. More detailed description of the experimental apparatus

and procedures are presented elsewhere."

The viscoelastic fluids used were the well- mPxed~ homogeneous^ aqueous solutions of polyacrylamide (Separan AP-273) with concentrations of 10, 50, 100, 300, 500, and 1000 ppm and polyethylene oxide (Polyox WSR-301 ) with concentrations of 100, 300, 500, and 1000 ppm. The apparent viscosity of each polymer Solution at wide range of shear rates was measured and the results for Separan AP-273 solutions are presented in Figs. 2 and 3. Similar results for Polyox WSR-301

solutions may be found elsewhere." The measured

viscosity data were used to estimate the fluid time

scale by the Powell-Eyring model, which has the fnllowing expression

h e fluid time scale was determined by a linear regression method wiLh the use of all the viscosity data for each solution.

The measurements of pressure drop and heat transfer are pre8enLed in terms of friction facto7 and Colbum j-factor in Figs. 4 and 5 for Separan AP-273 and Figs. 6 and 7 for Polyox WSR+301, respectively. it is obsewed fvom these figwe3 that the reduction in friction factors and heat transfer coefficients for the smaller pipe (1 . l l cm test section) is more pronounced than that for the larger one (1.88 cm test section). This can be

.- Y. mc-'

Fig. 2 Apparent ViScosiLy v s . Shear rate fop Separan AP-273 solutinns in the 1.88 cm test section.

Fig. 3 Apparent viscosity vs. shear rate for Separan AP-213 solutions in the 1 . l l cm test section. L

Fig . 4 Friction f a c t o r vs. appa ren t Reynolds ~ ~ g . 6 ~ r i c t i n n factor YS. appa ren t Reynolds number fop Separan AP-273 s n l u t i n n s i n t h e number for Polyox WSR-301 s o l u t i o n s i n t h e 1.88 and 1.11 Cm test s e c t i o n s . 1.88 and 1.11 cm test s e c t i o n s .

Fig. 5 Colburn j - f a c t o r VS. apparent Reynolds number for SepaPan AP-273 s 0 1 u t l n n ~ i n the 1.88 and 1.11 cm test s e c t i o n s .

exp la ined by t h e fo l lowing i n t e r p r e t a t i o n : t h e

pnlymer molecules are considered t o i n f l u e n c e t h e

boundary l a y e r close t o the p i p e wa l l . This

i n f l u e n c e should be seen i n t h e smlleep p i p e before

t h e l a r g e r one s l n c e t h e boundary l a y e r would f o r m

a l a r g e r p o r t i o n of t h e total flow i n t h e small p ipe . ‘ However. t h e minimum asymptotes for

f r i c t i o n f a c t o r s and h e a t t r a n s f e r c o e f f i c i e n t s

remain t h e same, independent of p i p e d i ame te r .

Comparison of F igs . 4-7 also i n d i c a t e s t h a t Polyox

WSR-301 s o l u t i n n is much less e f f e c t i v e i n reducing

f r i c t i n n d rag and h e a t t r a n s f e r than the Separan

AP-273 s o l u t i o n . This is due t o t h e f a c t t h a t

F i g . 7 Colburn j-factor vs. appa ren t Reynolds numbber Por Polyox WR-301 a o l u t i n n s i n t h e 1.88 and 1.11 cm test s e c t i o n s .

u n l i k e Separan AP-273 solution. Polyox WSR-301

s o l u t l o n is weakly shear dependent. I n a d d i t i o n .

t h e ave rage m l e c u l a r weight oP Separan AP-273 (H =

6 m i l l i o n ) 1s m c h greater than that of Polyox YSR-

301 (H - 4 m i l l i o n ) .

Baat EdQy DlffUsI~Ity Equation

In o r d e r t o s o l v e the time-mean ene rgy

e q u a t i o n for drag r educ ing t u r b u l e n t p l p e flow, an

expres s ion for eddy d t f f u s l v i t y of h e a t . ~ h . *iCh

t a k e s i n t o accnun t p a r t i c u l a r c h a r a c t e r i s t i c s of

viscoelastic f l u i d s is r e q u i r e d . Most a n a l y t i c a l

s tud ie s use e i t h e r a n expres s ion for eddy

5

d i f f u s i v i t y o f h e a t which is v a l i d f o r a p a r t i c u l a r

polymer c o n c e n l r i t i o n , mostly for t h e maximum h e a t

t r a n s f e r r e d u c t i o n asymptot ic c a s e , o r a d i r e c t

analogy between eddy d i f f u s i v i t i e s o f h e a t and

momentum. Ne i the r one of t h e s e schemes have

g e n e r a l p r e d i c t i v e c a p a b i l i t y f o r wide r anges o f

polymer c o n c e n t r a t i o n s throughout t h e flow

f i e l d . ' The l a t t e r c a s e is even l e s s d e s i r a b l e

s i n c e s e v e r a l s t u d i e s have shown t h a t t h e t u r b u l e n t

P r a n d t l number of concen t r a t ed v i s c o e l a s t i c f l u i d s

is no t u n i t y , e s p e c i a l l y neap t h e wal l , where it is impor t an t to have a c c u r a t e va lues o f h e a t eddy

d i f f u s i v i t y f o r h e a t t r a n s f e r c a l c u l a t i o n s . ' " ' 6

To remedy t h e inadequacy o f t h e e x i s t i n g

a n a l y t i c a l s t u d i e s f o r h e a t t r a n s f e r i n

v i s c o e l a s t i c t u r b u l e n t p ipe flows, t h e p rev ious

work o f the au thor s ' Formulated a semi-empir ical

e q u s t i o n f o r eddy d i f f u s i v i t y o f h e a t i n terms o f

f r i c t i o n d r a g r e d u c t i n n r a t i o (FR) and Weissenberg

number (Ws). Tnese two important d imens ion le s s

pa rame te r s f o r v i s c o e l a s t i c f l u i d s can be

determined from t h e experimental measurements o f

p r e s s u r e d rop and f l u i d r h e o l o g i c a l p r o p e r t i e s .

The use o f f r i c t i o n d rag r e d u c t i o n ra t io and

Weissenberg number p l a y s an important r o l e i n

c o r r e l a t i n g f v i c t i o n f a c t o r with h e a t t r a n s f e r

c o e f f i c i e n t and can account For s e v e r a l important

f a c t o r s i n f l u e n c i n g t h e f r i c t i o n and h e a t t r a n s f e r

behavior of v i s c o e l a s t i c t u r b u l e n t p i p e f lows , such

a s p ipe diametep, s o l v e n t chemis t ry , d e g r a d a t i o n ,

as well as t h e t y p e and t h e c o n c e n t r a t i n n of t h e

polymer. me proposed equa t ion h a s t h e Following

form:

where a = 0.37. b = 0.15, c = 3.0, and W s c h - 200. F u r t h e r d e t a i l s on the development o f Eq. ( I ) are given i n Ref. 7 .

The proposed equa t ion is i n q u a l i t a t i v e

agreement w i t h t h e experimental o b s e r v a t l o n s , and

is presen ted i n a gene ra l i zed form which is

a p p l i c a b l e t o non-Newtonian v i s c o e l a s t i c as well as

Newtonian f l u i d s . For Newtonian f l u i d s (FR - 0 and

us = 0) the ra t io o f eddy d i f f u s i v i t i e s is e q u a l to

u n i t y which I s i n agreement wi th Eq. (7). For viscoelastic f l u i d s t h e deg ree o f r e d u c t i o n i n h e a t

t r a n s f e r is even mre drast ic than t h a t i n

f r i c t i o n . Th i s behavior is accounted f o r through

t h e use of t h e two dimensionless parameters FR a n d

W s . P a r t i c u l a r l y , even a f t e r t h e f r i c t i o n f a c t o r

o f a S o l u t i o n r eaches t h e minimum d r a g asymptote ,

t h e r e is s t i l l a dec rease i n t h e h e a t t r a n s f e r

c o e f f i c i e n t up t o t h e c r i t i c a l Weissenberg number

f o r h e a t t r a n s f e r . Th i s behavior is aCcOunLed f o r

by t h e second term i n Eq. ( I ) . Even though t h e

r e d u c t i o n i n h e a t t r a n s f e r is l imited by Lne

minimum h e a t t r a n s f e r asymptote , f u r t h e r polymer

a d d i t i o n r e s u l t s i n an i nc reased Weissenberg number beyond t h e c r i t i c a l value. The proposed e q u a t i o n

can n o t e x p l a i n t h i s phenomenon. For t h e case o f

W s > W s c h , t h e term (Ws/Wsch) i n Eq. (7) is s e t t o

u n i t y .

In "UP p r e v i o u s work'. t h e c o n s t a n t a was

d e t e m i n e d from the Newtnnian f l u i d behav io r , and

t h e c o n s t a n t s b and c from t h e minimum asymptot ic

c o n d i t i o n s for h e a t t r a n s f e r * ( jh = 0.03

and f r i c t i o n " ( F = 0.20 Re,0.48), r e s p e c t i v e l y .

It was assumed tha t t h e s e maximum r e d u c t i n n

asymptotes a r e gene ra l and independent of t h e

experimental a p p a r a t u s , procedures employed, and

t h e types o f polymer used. In o r d e r to e s t a b l i s h

t h e g e n e r a l i t y o f t h e c o n s t a n t s i n Eq. ( I ) , t h i s

assumption should be v e r i f i e d . As compared wi th

t h e we l l - e s t ab l i shed Newtonian Fluid behavior .

cons ide rab ly d i f f e r e n t va lues have been Suggested

f o r t h e minimum asymptot ic c a s e s , e s p e c i a l l y f o r

h e a t t r a n s f e r . S ince t h i s d i f f e r e n c e is cons ide red t o be caused by t h e inadequate

experimental c o n d i t i o n s such as s h o r t test s e c t i o n ,

s e v e r e mechanical deg rada t ion and in su f fLc ien t

s t u d y o f polymer ?. i t is q u i t e va luab le t o

i n v e s t i g a t e t h e minimum asymptot ic cases for

f r i c t i o n and h e a t t r a n s f e r based on t h e

experimental d a t a o f p rev ious s tud ie s"" and t h i s

s tudy . These s t u d i e s Were designed t o minimize t h e

above mentioned experimental d e f i c i e n c i e s . F igu re

8 shows t h a t t h e minimum asymptotes o f t h i s Study

a r e i n agreement wi th t h o s e oP Refs. 8 and 1 4

w i th in t h e u n c e r t a i n t y range o f t h e experimental

a p p a r a t u s o f both works. This conf i rms t h a t t h e

minimum asymptot ic c o n d i t i o n s f o r h e a t t r a n s f e r and

f r i c t i o n are g e n e r a l , independent o f the

expe r imen ta l appa ra tus , p rocedures employed, and

t h e polymer t y p e s used.

1

Fig. 8 Comparison of the maximum reduction asymptotes n f t h i s s tudy with c o r r e l a t i o n s of Refs. 8 and 1 4 .

One of t h e key Paramete73 i n Eq. ( 7 ) is t h e

c r i t i c a l Weissenberg numbe? f o r h e a t t r a n s f e r

( W s c h ) . Based on the experimental r e s u l t s of

Kwack6 t h e v a l u e Of Wsch was Laken t o be

approximately equa l t o .200. I n o u r p r e v i o u s wopk'

i t was assumed t h a t t h i s value is u n i v e r s a l . To

v e r i f y t h e g e n e r a l i t y of Eq. ( 7 1 , t h i s assumption

should be v e r i f i e d . In a d d i t i o n , t h e c r i t i c a l

Weissenberg number can sugges t t h e Optimum

concen t r a t ion compromislng t h e Performance and the

economics of polymer a d d i t i o n . The re fo re , i t is

very important Lo make an a c c u r a t e de t e rmina t ion of

t h i s value. S e v e r a l s t u d i e s r e p o r t e d t h e c r i t i c a l

Weissenberg number f o r h e a t t r a n s f e r t o be of t h e

o rde r of 200 - 250 Over Reynolds number range of

20,000 to 30,000 f o r . aqueous pnlyacrylamide

s o l u t i o n s (Separan AP-~273) ."9 '" 'L The c u r r e n t

s t u d y is planned t o confirm t h e P rev ious r e s u l t s

and Lo v e r i f y t h e g e n e r a l a p p l i c a b i l i t y o f them t o

othe? s o l u t i o n s . I n t h i s s t u d y , a h igh ly

concen t r a t ed polymer s o l u t i o n was d i l u t e d to

i n v e s t i e a t e t h e flow ~ h a r a ~ t e r i s t i c s over wide

range o f c o n c e n t r a t i o n s . For t h i s purpose, an

aqueous Separan AP-30 so lu t ion ' of 3000 ppm was

-'

o f Weissenberg numbers. The a p p a r e n t v i s c o s i t y

d a t a and t h e f l u i d t ime scale f o r each

c o n c e n t r a t i o n are p resen ted i n F ig . 9 . T h i s f i g u r e

shows t h a t t h e v i s c n s i t y d a t a fo r Separan AP-30 s o l u t i o n f o l l o w t h e similar t r e n d t o thaL f o r

Separan AP-213 ( s e e F i g s . 2 and 3). T h i s Snlut iOn is a l s o s t r o n g l y s h e a r dependent . The

d imens ion le s s h e a t t r a n s f e r c o e f f i c i e n t s a t a n

appa ren t Reynolds number of approximately 10,000 a s a f u n c t i o n o f Weissenberg number a r e p re sen ted i n

Fig. 10. The c r i t i c a l Weissenberg number f o r h e a t

t r a n s f e r is d e f i n e d a t a va lue o f Weissenberg

number f a r which d imens ion le s s heat t r a n s f e r

c o e f f i c i e n t i n c r e a s e s from the minimum h e a t

t r a n s f e r asymptote wi th d e c r e a s i n g Weissenberg

number. From F ig . 10 t h e c r i t i c a l Weissenberg

number f o r h e a t t r a n s f e r is e s t i m a t e d t o be nf the order of 200. Th i s r e s u l t is i n agreement wi th

t h o s e of t h e p rev ious works conducted f o r Separan

AP-273 so1utiOns. '"5"6 Th i s i n d i c a t e s t h a t the

c r i t i c a l Weissenberg number fa r h e a t t r a n s f e r is g e n e r a l , independent o f t h e expe r imen ta l a p p a r a t u s ,

p rocedures enblnyed. and the polymer t y p e s used.

Now t h a t t h e assumptions made i n t h e

e v a l u a t i o n o f t h e a d j u s t a b l e c o n s t a n t s i n t h e

proposed h e a t eddy d i f f u s i v i t y equa t ion have been

v e r i f i 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 e q u a t i o n

should be f u r t h e r v a l i d a t e d w i t h expe r imen ta l d a t a

nf t h i s s tudy . T h i s w i l l be p r e s e n t e d i n t h e n e x t

s e c t i o n .

Redict ions __-_ An a c c u r a t e p r e d i c t i o n of v e l o c i t y p r o f i l e is

e s s e n t i a l n o t only to i n v e s t i g a t e t h e mechanism of

momentum t r a n s p e r bu t also to p r e d i c t t h e

phennmenon of heat t r a n s f e r . In t h i s s t u d y , t h e

momentum eddy d i f f u s i v i t y model proposed by Cess'" was used to 301ve the time-averaged Navier-Stokes

e q u a t i o n . For a f u l l y developed p i p e f l o w , t h e

e x p r e s s l o n is

prepared f n r d i l u t i o n . For each c o n c e n t r a t i o n , the 2 +2

h e a t t r a n s f e r d a t a were taken a t v a r i o u s flow rates E m l r 2 2 2

toge the? w i t h t h e appa ren t v i s c o s i t i e s a t wide Va I 1 - = 211 + % L1 - (T) 1 C l 2 ( E - ) 1

range of s h e a r r a t e s . For d i l u t i o n , t h e Proper

amount o f t a p water was added to o b t a i n t h e d e s i r e d

c o n c e n t r a t i o n . This p rocess Of d i l u t i o n con t inued

u n t i l t h e h e a t t r anSfeP data covered a w i d e range

- ( 1 - r/ri) ),2)1'2 . 1 (8) xC1 - e x p ( - T r 2

A /pi

7

experiments . '* I n a l l c a s e s t h e p r e d i c t i o n s show

e x c e l l e n t agreement wi th expe r imen ta l ly measured

p r o f i l e s In t u r b u l e n t v i s c o e l a s t i c flows. I n o r d e r

t o f u r t h e r v e r i f y t h e Computational scheme, t h i s '-

t echn ique was used t o p r e d i c t t h e v e l o c i t y p r o f i l e s

and t h e momentum eddy d i f f u s i v i t i e s i n bo th

Newtonian and v i s c o e l a s t i c p ipe flows. As for

comparison d a t a , t h e measurements of Mizushina and

U s u i ' were adopted. They used a laser-Doppler

anemometer t echn ique , which can produce accuraLe

measurements of t h e flow f i e l d wi th t h e lemt

amount o f d i s t u r b a n c e t o t h e flow. As shown i n

Fig. 11. t h e p r e d i c t i n n s show e x c e l l e n t agreement

w i t h expe r imen ta l ly measured v e l o c i t y p r o f i l e s i n

both Newtonian and drag-reducing f l o v s . F igu re 1 2

. Flg . 9 Apparent v i s c o s i t y VS. s h e a r rate f o r i l l u s t r a t e s t h a t t h e cess model can be u s e d

s u c c e s s f u l l y i n p r e d i c t i n g t h e eddy d i f f u s i v i t i e s Separan AP-30 i n t h e 1.88 cm test s e c t i o n .

TRANSFER ASYMPTOTE

100 Mt 102 w3 w4 10-41 ' " " ' " " ' " ' I ' " ' I ' " J w. 10-1

Fig. 10 Colburn J - f a c t o r v s . Weissenberg number for

When t h e Cess mndel is a p p l i e d t o t h e v i s c o e l a s t i c

flows. the parameter A+ t h a t c h a r a c t e r i z e s t h e

t h i c k n e s s of t h e near-wal l r eg ion ( v i s c o u s s u b l a y e r

and b u f f e r r e g i o n ) should be determined such that.

i t can p r o p e r l y account f o r t h e v a r l a t l o n s of t h e

laminar l a y e r t h i c k n e s s wi th t h e . a d d i t i o n of polymer. I n t h e c u r r e n t s t u d y . t h e parameter A'

was determined u s i n g t h e i t e r a t i v e computat lonal

scheme proposed by Tiederman and Reischman" f o r

c a l c u l a t i o n o f v e l o c i t y p r o f i l e i n v i s c o e l a s t i c

t u r b u l e n t flows. The procedure r e q u i r e s only

p r e s s u r e d rop and flow rate informatlon. The

p r e d i c t i o n s based on t h i s scheme have been compared

wi th s e v e r a l experimental v e l o c i t y p r o f i l e s for channe l flows" and l i m i t e d p i p e flow

Separan AP-30 i n t h e 1 .88 cm t e s t s e c t i o n .

of momentum for t h e maximum d r a g r e d u c t i o n c a s e as

well a s t h e Newtonian case.

20

0 1 10 102 3

Fig . 11 Time-mean v e l o c i t y p r o f i l e f o r Newtmian and v i s c o e l a s t i c Plows: p r e d i c t e d and experimental . '

Wlth t h e known v e l o c i t y p r o f i l e , t h e h e a t

t r a n s f e r c o e f f i c i e n t can be determined by s o l v i n g

t h e time-mean energy equa t ion wi th t h e use of t h e

proposed equa t ion for eddy d l f f u s i v i t y of h e a t and the computat ional scheme proposed by Yoon and

c h a j a r . ' F igu re 13 shows t h a t t h e p r e d i c t i o n s of heat t r a n s f e r c o e f f i c i e n t s with t h e use of the

proposed equa t ion are i n good agreement with

expe r imen ta l data o f t h i s s tudy and Knack.' It

should be noted t h a t even though t h e form of t h e

proposed equa t ion and t h e two a d j u s t a b l e c o n s t a n t s L

8

t h a t t h e proposed ef luat inn f o r eddy d i f f u s i v i t y of

h e a t is c a p a b l e of p r e d i c t i n g expe r imen ta l heat

t r a n s f e r c o e f f i c i e n t s f o r v i s c o e l a s t i c t u r b u l e n t

p ipe f lows wi th wide r anges of polymer

c o n c e n t r a t i o n s provided t h e expe r imen ta l

measurements of t h e P r e s s u r e d rop and t h e f l u i d --PREm"IW OF

THIS WORK I 103

Fig. 1 2 Momentum eddy d i f f u s i v i t i e s f o r Newtonian and v i s c o e l a s t i c f lows: p red ic t ed and experimental . '

F i g . 13 Comparison of t h e p r e d i c t e d C o l b u r n j - f a c t o r s u s i n g t h e proposed h e a t eddy d i f f u s i v i t y e q u a t i o n w i t h measurements.

b and c Were e s t a b l i s h e d based on t h e experimental

d a t a of Kwack f o r t h e minimum d r a g and h e a t

asymptot ic c o n d i t i o n s , t h e proposed equa t ion was

a b l e t o p r e d i c t comparably t h e h e a t t r a n s f e r

coe f f i c ienLs for a l l polymer c o n c e n t r a t i o n s

Obtained from our r e c e n t experiments and Kwack's

experiments without any ad jus tmen t of t h e

c o n s t a n t s . Furthermore, it should be a lso remarked

t h a t both experimental r e s u l t s were ob ta ined under

d i f f e r e n t experimental c o n d i t i o n s such as d i f f e r e n t

p ipe d i ame te r s . l e v e l of d e g r a d a t i o n , s o l v e n t ,

e t c . The p r e d i c t i o n s f o r a l l Concen t r a t ions r e su l t i n a maximum o f 30% d e v i a t i o n from t h e

..-, measuremepts. The r e s u l t s o f t h i s s tudy i n d i c a t e

time s c a l e are a v a i l a b l e .

Conc lus ions

?,e g e n e r a l a p p l i c a b i l i t y of the proposed h e a t

eddy d i f f u s i v i t y eQuation' was f u r t h e r v e r i f i e d

with OUP vecen t expe r imen t s f o r Separan AP-273 and

Poiyox WSR-301 s o l u t i o n s and Kwack'9' experiments

f a r Separan AP-273 s o l u t i o n . me g e n e r a l i t y o f t h e

assumptions made i n t h e e v a l u a t i o n of t h e c o n s t a n t s

i n t h e proposed e q u a t i o n were e s t a b l i s h e d based on

Lhe f i n d i n g s t h a t t h e minimum asymptotes f o r

f r i c t i o n and h e a t t r a n s f e r and t h e Cr i t i ca l Weissenberg number for h e a t t r a n s f e r are u n i v e r s a l

a n d independent of t h e expe r imen ta l a p p a r a t u s .

Procedures employed, and t h e t y p e s of Polymer

used. me f i n d i n g s of t h i s s t u d y confirmed t h a t

t h e s i n g l e e q u a t l o n proposed for h e a t eddy

d i f f u s i v i t y can be used to p r e d i c t w i th good

accuracy t h 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 f o r

d i p r e r e n t polymer s o l u t i o n s with wide r ange of c o n c e n t r a t i o n .

A c k n o v l e d d

T h i s work was p a r t i a l l y sponsored by t h e

U n i v e r s i t y Cen te r for Energy Research (UCER) a t Oklahoma S t a t e U n i v e r s i t y .

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