HEAT OF

TRANSFER

COEFFICIENT

HEAT-RELEASING D.

A.

IN T U R B U L E N T

FLOW

LIQUID

Kazenin

In K u t a t e l a d z e ' s w e l l - k n o w n m o n o g r a p h [1] the e x p r e s s i o n a i ao -- t + A Z '

dqv Z - - 4qw

(1)

was obtained a n a l y t i c a l l y f o r the c o r r e c t i o n f a c t o r allowing for i n t e r n a l heat r e l e a s e o r heat a b s o r p t i o n in c a l c u l a t i o n of the heat t r a n s f e r coefficient i n a r o u n d tube. H e r e Z is the r e l a t i v e d e n s i t y of the i n t e r n a l heat s o u r c e . The c o n s t a n t coefficient A i s e x p r e s s e d in the following way in t e r m s of q u a d r a t u r e s of the d i s t r i b u t i o n s of the d i m e n s i o n l e s s v e l o c i t y w (})=w/< oJ > and the t u r b u l e n t t h e r m a l c o n d u c t i v i t y ht(}) o v e r the r a d i u s } = 2 r / d 1

I

1

1

d~

(2)

o

N u m e r i c a l v a l u e s of the coefficient A w e r e obtained in [1] for s p e c i a l c a s e s of l a m i n a r flow with a p a r a b o l i c v e l o c i t y p r o f i l e (A---0. 272) and t u r b u l e n t flow with a v e l o c i t y d i s t r i b u t i o n c o n f o r m i n g to a 1/7 law and P r a n d t l n u m b e r P =0 (A=0. 0834). We give the r e s u l t s of c a l c u l a t i o n of this coefficient for the case of a t u r b u l e n t flow with P ~ 0. I n t e g r a t i n g by p a r t s in the n u m e r a t o r and d e n o m i n a t o r of E q . (2) we obtain 1

1

o

0

A = t -- T

(1 + Xt / X) ~ d~

(3)

The m a i n difficulty in r e d u c i n g Eq. (3) to a f o r m c o n v e n i e n t for n u m e r i c a l c a l c u l a t i o n s l i e s in the choice of an a p p r o x i m a t i n g r e l a t i o n s h i p giving the d i s t r i b u t i o n of ?~t/?~ over the tube r a d i u s . It should be noted that

~f#O .0f30 fffZO

~t

~ffO ~flTt7

P ~t -- Pt ~

(4)

w h e r e P t i s the t u r b u l e n t P r a n d t l n u m b e r .

10 #

Fig. 1 MOSCOW. T r a n s l a t e d f r o m Z h u r n a l P r i k l a d n o i Mekhaniki i T e k h n i c h e s k o i F i z i k i , Vol. 9, No. 4, pp. 161-162, J u l y - A u g u s t , 1968. O r i g i n a l a r t i c l e s u b m i t t e d J a n u a r y 3, 1968.

9 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

520

We u s e the P r a n d t l e x p r e s s i o n for the t u r b u l e n t t a n g e n t i a l s t r e s s in t e r m s of the m i x i n g length l and also its r e p r e s e n t a t i o n in t e r m s of the t u r b u l e n t v i s c o s i t y coefficient Pt dw~ ~ ~ -----p l -~-r) '

dw

~t = -- ~t -~-

(5)

By c o m p a r i s o n we have ~_L

2_ in dw = - - ~ d--F

(6)

A c c o r d i n g to I. N i k u r a d z e ' s m e a s u r e m e n t s , at R =p< w > d / # > 10 ~ the d i s t r i b u t i o n of the m i x i n g length o v e r the tube r a d i u s is i n d e p e n d e n t of R. The w e l l - k n o w n i n t e r p o l a t i o n f o r m u l a [1] gives in this case 2l / d = 0A4--0.08 ~2 _ 0.06 ~4

(7)

It is a s s u m e d in what follows that the v e l o c i t y d i s t r i b u t i o n o v e r the c r o s s s e c t i o n of the tube c o n f o r m s to a 1/7 law co = w l ( w )

= 1.22 (1-- ~)'/,

(8)

U s i n g (4), (6), (7), and (8) and p e r f o r m i n g the n e c e s s a r y t r a n s f o r m a t i o n s we put Eq. (3) in a f o r m s u i t a b l e f o r n u m e r i c a l c a l c u l a t i o n s on an e l e c t r o n i c digital c o m p u t e r . The r e s u l t s of t h e s e c a l c u l a t i o n s a r e given in the f o r m of a r e l a t i o n s h i p b e t w e e n the coefficient A and the c r i t e r i o n R * - R P / P t i n Fig. 1. It should be r e m e m b e r e d that t h e s e r e s u l t s a r e valid for R > 105. LITERATURE 1.

CITED

S~ S. Kutateladze, F u n d a m e n t a l s of Heat T r a n s f e r T h e o r y [in R u s s i a n ] , Mashgiz, M o s c o w - L e n i n g r a d , 1962.

521