INT. COMM. HEAT MASS TRANSFER 0735-1933/84 $3.00 + .00 Vol. ii, pp. 467-476, 1984 ©Pergamon Press Ltd. Printed in the United States

NATURAL CONVECTION EFFECTS ON HEAT TRANSFER TO POWER-LAW FLUIDS FLOWING UNDER TURBULENT

CONDITIONS IN VERTICAL PIPES

A.V. Shenoy Chemical Engineerin~ Division

National Chemical Laboratory, Pune ~ii008, India

(Cc, t~t~nicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT A t h e o r e t i c a l a n a l y s i s o f t h e e f f e c t o f buoyancy on t h e h e a t t r a n s f e r t o non-Newtonian power- law f l u i d s f o r upward f low i n v e r t i c a l p ipes unde r t u r b u l e n t c o n d i t i o n s has been p r e s e n t e d . A c r i t e r i a f o r l i m i t i n g 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 t o l e s s t h a n 5% has been developed for varying pseudoplasticity index. The equation for quantitative evaluation of the natural convection effect on the forced convection has been suggested to be applicable for upward as well as downward flow of the power-law fluids by a change in the sign of the controlling term.

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

Mixed c o n v e c t i o n h e a t t r a n s f e r t o non-Newtonian f l u i d s

has been r e c e i v i n g c o n s i d e r a b l e a t t e n t i o n i n t h e r e c e n t y e a r s

as n o t e d i n t h e r e c e n t s u r v e y a r t i c l e by Shenoy and Mashe lkar [ 1 ] .

T h i s i s an outcome o f t h e obvious r e a l i s a t i o n t h a t i n most h e a t

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

c o n d i t i o n s as b e i n g w h o l l y i n t h e f o r c e d c o n v e c t i o n regime or

w h o l l y i n t h e f r e e c o n v e c t i o n reg ime as t r u l y speak ing b o t h

mechanisms a lways o p e r a t e s i m u l t a n e o u s l y . Even though i n some

p r a c t i c a l s i t u a t i o n s , i f one mode o f c o n v e c t i o n domina te s t h e

other, it is still the combined effect of the two that actually

determines the heat transfer characteristics.

NCL C o m m ~ t i o n No. 3~12

467

468 A.V. Shenoy Vol. ii, No. 5

Laminar mixed convection heat transfer to non-Newtonian

fluids in external flow situations such as the vertical flat

plate and horizontal cylinder, has been theoretically analyzed

by Shenoy [2,3] and a predictive equation for estimating the

combined effect of forced and free convection in power-law fluids

as well as viscoelastic fluids has been provided. In internal

flow situations, such as flows through heated or cooled tubes,

mixed convection in non-Newtonian fluids has been studied for the

case of horizontal as well as vertical tubes. The problem of

laminar heat transfer to non-Newtonian fluids in horizontal tubes

has been analyzed by Metzner and Gluck [ 4] and Oliver and Jonson [5]

who have provided useful correlations. In the case of vertical

tubes, laminar mixed convection heat transfer to non-Newtonian

fluids has been studied theoretically and experimentally for both

upward and downward flows under constant heat flux as well as

constant temperature conditions by a number of investigators [6-9]

The situation is not the same in the case of turbulent flow heat

transfer to non-Newtonlan fluids in vertical pipes and no reported

work exists on the theoretical aspects or the experimental data

to date. The purpose of the present work is to provide a

theoretical analysis for the turbulent flOw situation when both

forced and free convection effects becume important in the case

of non-Newtonian power-law fluids.

Theoretical Analysis

We consider a power-law non-Newtonian fluid, with a fully

developed velocity profile and at uniform temperature Ti, to be

flowing upward in a vertical, circular tube of radius R and

maintained at a constant temperature T w. The externally

controlled pressure difference is such that forced turbulent

convection condition exists. Due to the temperature difference

(Tw-Ti) , the fluid in the boundary layer near the heated surface

experiences a buoyancy force due to reduced density. The

buoyancy force acts in the direction of motion leading to a

drop in the shear stress across the buoyant region resulting in

reduced turbulence. As a consequence of the increased

laminarisation, the turbulence structure can be considered to

be akin to a buoyancy free flow at some reduced value of Reynolds

n~ber.

Vol. ii, NO. 5 HEAT TRANSFER TO POWER-LAW FLUIDS IN PIPES 469

Assuming t h a t the f l u i d s unde r c o n s i d e r a t i o n have a g r a d u a l

v a r i a t i o n o f d e n s i t y w i t h t e m p e r a t u r e , we have the buoyan t l a y e r

i d e n t i c a l l y e q u a l t o t h e t h e r m a l l a y e r . Now, assuming t h a t the

t e m p e r a t u r e g r a d i e n t i n the t he rma l l a y e r can be a p p r o x i m a t e l y

w r i t t e n a s ( T - T b ) / 6 T , _ we can w r i t e the r e d u c t i o n i n s h e a r s t r e s s

across the buoyant layer as

% AT = (~Tw_Tb) Tb

Le t 1 l l w ~ dT = ~ , t h e n e q u a t i o n (1) can be

b

w r i t t e n a s

D e n o t i n g the combined l a m i n a r s u b - l a y e r and the b u f f e r

l a y e r t h i c k n e s s as 6M, we a t t e m p t t o f i n d a r e l a t i o n s h i p be tween

6M and 6 T by an o r d e r o f magni tude a n a l y s i s o f the g o v e r n i n g

momentum and e n e r g y e q u a t i o n s f o r the f l o w under c o n s i d e r a t i o n .

For power - l aw non-Newton ian f l u i d s , the momentum and e n e r g y

e q u a t i o n s i n the s i m p l i f i e d forms a r e g i v e n as f o l l o w s :

Momentum e q u a t i o n

(V2z) ÷ .i ~ (rVrVz) = _~b ~ z r ~ r

Jp + ~ 1 ~ Ir~V.~nl _ ~= ~ r ~r ~'~'-i/j ~g

(3)

Ener e~ e o u a t i o n

(VzT).. ,i ~ (rVrT) ~ z r J r

k 1 ,:% [~ ~_._..~ ~b Cp r ~ r L'~rJ

Assuming

V z " O(Vm), V r ~ O( ~Vm or 6 , Vm ) 1 c l c

0+)

T ~ 0 (Tw)~ r ~ O(a)~ z ~ 0 (I c) (5)

470 A.V. Shenoy Vol. ii, No. 5

~r 6 T ; 6z

We get by ordering of the terms in equations (3) and (4)

~M I ~T 1 ~ and ~

--~c I I c 1 Re n+i ~'- n+l

c Pr c Re c

where 2-n n 2 l-n 3(n-l)

Rec : m c and Pr = P [-~] I c V c k m

(6>

(7)

for the present case, assuming i c : D (diameter of the vertical

tube) we have

~T ~ B 1

Pr ~-

Thus e c u a t i o n (2) can be r e w r i t t e n as

A'r_. = I

er T

L e t us now i n t r o d u c e a d i m e n s i o n a l w a i l - l a y e r t h i c k n e s s ~M +

defined for power-law fluids as

2-n n n

"Cw ~ ~M ~w T gM*

K w

To f i n d t he boundary be tween the b u f f e r l a y e r and t he t u r b u l e n t

c o r e we t a k e t he i n t e r s e c t i o n o f the e q u a t i o n s p r o v i d e d by

Clapp [10] f o r the b u f f e r l a y e r and t u r b u l e n t c o r e d u r i n g t h e

t u r b u l e n t f l o w o f power- law f l u i d s .

(9)

(lo~

Vol. Ii, No. 5 HEAT TRANSFER TO POWER-LAW FI;JIDS IN PIPES 471

Thus we have 3.05n * 3.8

2 . 2 2 ~.~ - e

Combining equations (i0) and (II) we have

1 ..2,_

gM = ~e 3"°Sn*3"s'~n2"22 ) ~ n i, 2 -n

2n 2

(xz)

(~)

Thus the fractional reduction in shear stress due to the buoyancy term can be expressed as follows

1 .1 K " ~ (~b_~) g

= 2"~ Pr ~- ~'~ ~ ~- ~ 2n

(]-3)

The above equation (13) can be modified using

Or =

Re =

~b ( ~ b o ~ ) g D

2 ;~ K b

~b V 2-nDn m

%

2n+l Vm2-2n

W

1 ~- ~b V~

(1%)

(~5)

(Z6)

Thus we have

! ~x: ~ (3-O'n+3.8)In

rv 2 . 2 2 Or

I 2n.i ~r 2 me m

.i (. 1 2*n

2n

(~

472 A.V. Shenoy Vol. ii, No. 5

Using the Blasius type of expression provided by Dodge and

Metzner [ii], namely

We have i..

2~ n

~wnt = [(a2__) 2 e (3.05n*3.8)~2.22 A/

(is)

n

p r ~ J . . . . . . . (] .9)

The t u r b u l e n c e s t r u c t u r e o f t he c o r e f low i s d e t e r m i n e d by t h e

s h e a r s t r e s s i n t he w a l l l a y e r and because o f t he r e d u c t i o n o f

s t r e s s a c r o s s t h i s l a y e r dUe t o buoyancy t h e s i t u a t i o n i s s i m i l a r

t o t h a t o f a b u o y a n c y - f r e e f low a t some r e d u c e d v a l u e o f Reynolds !

number R e .

The r e l a t i o n s h i p be t ween t h e n o m i n a l and t he r e d u c e d

s h e a r s t r e s s w i t h t h e r e s p e c t i v e Reyno lds number can be w r i t t e n

as 2

-6 w -

UW

(2o)

Again using f - ~ w • write

w

~Cw

2-~ (2- n)

_ I~ 1 --2-n (21)

We now use the empirical form of Clapp [i0] for relating

Nuseelt nuaber to Reynolds number for power-law fluids as

Nu a Re ~ • Thus we have

(22)

Vol. Ii, No. 5

where B :

HEAT TRANSFER TO POWER-LAW FLUIDS IN PIPES

0.8(2-n) n

n [2"~ ( 2 - n ) ]

473

(23)

AS AT = ~ - ~' W W

eq. (19) as

f l I~ ~2--'-/2÷~ 2 Nu'

Nu

we can r e w r i t e equa t ion (22) us ing

i _

(3.09n÷3.8~ n Qr 2.22 ] 1 ~n÷2-~ (2+n)

Pr Z- Re 2n '

B

(2%)

R e s u l t s and Discuss ion

The v a r i a t i o n of B as Riven by e q u a t i o n (23) wi th r e s p e c t to changes i n the p s e u d o p l a s t i c l t y index n i s shown In Table 1.

TABLE 1

n c ~ B C

1. o o. o791 o. 25o o. ~ 7 3 . 6 o ~ o -5 0.9 0.0770 0.255 0 .~63 124tx10 -5 0.8 0.0760 0.263 0.681 9.7~xi0 "6

-6 0.7 0.0752 0.270 0.810 3.92xi0 0.6 0.07%0 0.281 0.9~7 1. ]57/0 "6

0.~ 0.0723 0.290 1.08~ 2.0~x10 -7

0.~ 0.0710 0.307 1.22~ l.~Tx10 -8 0.3 0.0683 0.32~ 1.3%8 2.01xl0 "I0

0.2 O. 06%6 O. 3~9 1. ~+8 3.07xlO "i ~

Thus, f~om equa t ion (22) u s ing these va lues of B i t can be concluded t h a t a 10~ r e d u c t i o n in shear s t r e s s induced by buoyancy would lead to a 5~ r e d u c t i o n i n hea t t r a n s f e r c o e f f i c i e n t .

Using equa t ion (19) , a c r l t e r £ a f o r a r e d u c t i o n of 5~ or l e s s i n the hea t t r a n s f e r c o e f f i c £ e n t can be s e t up as fo l lowss

474 A.V. Shenoy Vol. Ii, No. 5

Pr

i_ i_._

< c (25)

2*n . { ~ ) wbe re 2 n 2.22n (26)

c = 0.1 (~-~-) •

The va lues of C f o r v a r y i n g p s e u d o p l a s t t c i t y index a re shown

In Table 1, i n d i c a t i n g c l e a r l y t h a t wi th i n c r e a s i n g non-Newtonian

behaviour , I t i s i n c r e a s i n g l y d i f f i c u l t t o e a i n t a i n the r e d u c t i o n

in h e a t t r a n s f e r due to the e f f e c t of buoyancy to l e s s than 5~.

The e x p r e s s i o n given by equa t ion (2%) gives the Nusse l t

number f o r mixed c o n v e c t i o n when fo rced convec t ion i s s l i g h t l y

impaired by f r e e c o n v e c t i o n in case of upward power-law f l u i d flow in hea ted p ipes . However, i t must be noted t h a t the simple

e q u a t i o n (2~) d e s c r i b i n g the buoyancy - i n f l u e n c e d h e a t t r a n s f e r

In v e r t i c a l tubes c~old a l s o be used f o r downward Flow by

i n t r o d u c i n g a p o s i t i v e s i g n i n p lace of the n e g a t i v e s ign because

buoyancy f o r c e s in such c i r cums tances would cause an l n c r e a s e in

s t r e s s a c r o s s the buoyant l a y e r .

Cp

D

f

f t

g

Or

h

k

% 1 c n

s p e c l f i c h e a t a t c o n s t a n t p r e s su re ( E J / ~ )

tube d i a m e t e r (m)

f r i c t i o n f a c t o r as d e f i n e d by e q u a t i o n (16)

reduced f r i c t i o n f a c t o r as d e f i n e d by e q u a t i o n (16) us ing the reduced wal l shea r s t r e s s .

a c c e l e r a t i o n due to g r a v i t y (m/sac 2)

C~ashof number f o r power-law f l u i d s as de f i ned by equa t ion

h e a t t r a n s f e r c o e f f i c i e n t (EW/~K) (1~)

the rmal c o n d u c t i v i t y (KN/mK) c o n s i s t e n c y index f o r bulk of the Fluid (K~/m sac 2"n)

c o n s i s t e n c y index f o r f l u i d a t the wal l (Eg/m see 2"n)

c h a r a c t e r i s t i c l e n g t h (m) pseud o p l a s t i c i t y index

Vol. Ii, No. 5 HFATTRANSFERTOPOWER-LAWFI/YlDS IN PIPES 475

Nu'

P Pr

Prc r

R

Re !

Re

Rec T

T b T t T

W V

111 g

r V

Z

&B

6 M q

%

~w !

r-,w Az:

N u s s e l t number ( = h D/k)

r educed N u s s e l t number ob ta ined due to the e f f e c t s o f buoyancy and d e f i n e d as above

p r e s s u r e (~II~) Prandtl number as defined by equation (7) for i c = D

c h a r a c t e r i s t i c P r a n d t l number as d e f i n e d by e q u a t i o n (7) r a d i a l c o - o r d i n a t e (m) tube r a d i u s (m)

Reynolds number as d e f i n e d by e q u a t i o n (15) r educed Reynolds number o b t a i n e d due to the e f f e c t s of buoyancy and d e f i n e d by e q u a t i o n (15) w i t h the

a p p r o p r i a t e va lue of V m- c h a r a c t e r i s t i c Reynolds number as d e f i n e d by e q u a t i o n (7) t e m p e r a t u r e (°C or OK) t e m p e r a t u r e o f the bu lk of the f l u i d (°C or OK) i n i t i a l t e m p e r a t u r e (°C or OK) w a l l t e m p e r a t u r e (°C or OK)

average velocity (m/sec)

velocity in the radial direction (m/see)

v e l o c i t y i n t he a x i a l d i r e c t i o n (m/sec) axial c o - o r d i n a t e (m)

c o e f f i c i e n t s whose v a l u e s f o r v a r y i n g n a r e a v a i l a b l e in Ref. (II).

boundary layer thickness (m)

buoyant boundary layer (m)

thickness of boundary sub-layer plus buffer layer (m)

d i m e n s i o n l e s s t h i c k n e s s of boundary s u b - l a y e r p l u s buffer layer (m)

t h e rma l boundary layer (m) density (kg/m 3)

i n t e g r a t e d d e n s i t y (Kg/m 3) g iven by e q u a t i o n (2) d e n s i t y o f bu lk o f t he f l u i d (Kg/m 3)

d e n s i t y o f f l u i d a t the wa l l (Kg/m 3)

w a l l shea r s t r e s s (N/m 2)

r educed wall s h e a r s t r e s s (N/m 2)

change o f shea r s t r e s s a c r o s s bouyant l a y e r (N/m 2)

476 A.V. Shenoy Vol. ii, No. 5

i.

e

3.

4.

5.

6.

7.

8.

9.

i0.

11.

References

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D.R. Oliver and V.G. Jenson, Chem.Eng.~ci., 19, 115 (1964).

S.H. De Young and G.F. Scheele, A.I.Ch.E.J. 9 16, 712 (1970).

W.J. Marner and R.A. Rehfuss, Chem.Eng.J., 3, 294 (1979-).

W.J. Marner and H.K. McMillan, Chem.Eng.Sci., 27~ 473 (1979-)

G.F. Scheele and H.L. GTeene, Ind.Eng.Chem.Fundam., i0, 102 (1971).

R.M. ClaPp, In~gamatAonal Developments in Heat Transfer, Part III, 652-~I; D.159; D-211-~; A,~.M.E. New York (1961).

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