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INT. O3MM. HE~T ~ S S TRANSFE R 0735-1933/86 $3.00 + .00Vol. 12, pp. 3-10, 1986 @Pergamon Press Ltd. Printed in the Unite d States

A SIMPLE HEAT TRANSFER CORRELATION FORCROSS FLOW IN TUBE BUNDLES

T.H. Hwang and S.C. YaoDepartment of Mechanical EngineeringCarnegie - Mellon University

Pittsburgh, Pennsylvania 15213U.S.A.

( C ~ n i c a t e d b y J .P . H a r tn e t t a n d W . J . M in ko wy cz )

ABSTRACTThe empirical method of Biery [1] for the estimation of crossflow heat transferin various geometries tube bundles has been simplified. At high Reynolds numberconditions, intense turbulent mixing exists such that the heat transfer is notsensitive to the variation of the detailed geometries of the bundles. A simple heattransfer correlation based on tube diameter can be deduced as N-'u = 0.366 Re 0.6d dPr I/3. This simple correlation fi ts the experimental data well within + 20% for4000 < Re < 2 x 10~. Good agreements between this simple correlation anddexisting correaltions are also observed.

I n t r o du c t i o n

Extensive studies [1-7] have been conducted for the heat transfer in tube bundles atcrossflow due to the wide applications of shel l-and- tube exchangers. A large quantity of datawas compiled in reference [7] for heat transfer over smooth tube bundles. Severalcorrelations [3,4,5] were proposed for d ifferent geometries of bundles at some Reynoldsnumber range. However, from the designer's view point, there is still a need for simple andgeneral correlations on the prediction of crossflow heat transfer coefficients for tube bundleswith various geometries.

Some attempts in this direction have been made by previous investigators. Recently, Biery

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4 T . H . } { w a n g a n d S . C . Y a o V o l . 1 2 , N o . I

[ 1 ] p r o p o s e d a m e t h o d o f c o r r e l a ti n g h e a t t r a n s f e r d a ta b y t r a n s f o r m i n g th e a c t u al t u b eb u n d l e l a y o u t to a n e q u i l a te r a l t r ia n g u l a r p i tc h i n f i n i t e l y s m o o t h t u b e b a n k ( E T P - I - S T B ) w h i c hha s t he s a m e h y d r a u l i c - d i a m e t e r - t o - t u b e - d i a m e t e r r a t i o ( d / D ) a s t he o r i g i na l bund l e . F i g . 1s how s the t yp i c a l s ta gge r e d a r r a y a nd i n - l i n e a r r a y t ube bu nd l e w i t h t he f l ow no r m a l t o t het ube s. I n t yp i c a l i ndus t r i a l s he l l - a n d - t ub e he a t e xc ha nge r bund l e s , t he p i t c h - t o - - d i a me t e r r a t io sa r e i n t he r a nge o f 1 . 25 t o 2 .5 . T he he a t t r a ns f e r c o r r e l a t i on i s e xp r e s se d [ 1 ] i n t he f o r m o f

N-'-uD = C a Re t)6 pr m ( I )

A s i n g le c u r v e o f C h vs. ( X T / d ) A ( t r a n s v e r s e - p i t c h - t o - d i a m e t e r r a t i o o f E T P - I - S T B ) w a so b t a i n e d f o r b u n d l e s w i th p i t c h - t o - d i a m e t e r r a ti o s o f 1 .2 5 to 2 .5 0 a n d R e y n o l d s n u m b e r s , R eda bove 4000 . E qu a t i on ( 1 ) g i ve s a s a t i s f a c to r y p r e d i c t i on f o r s t a gge re d t u be bu nd l e s bu t g i ve s 20p e r c e n t o v e r - p r e d i c t i o n f o r i n - l i n e b u n d l e s . I t is i m p o r t a n t to p o i n t o u t t h a t th e h y d r a u li cd i a m e t e r i n B i e r y ' s s t udy [ 1 ] w a s de f i n e d u s i ng t he mi n i m um f l ow c r os s s e c t i ona l a r e a i n ab u n d l e a s th a t o f K a y s & L o n d o n [ 7 ]

4 ( m i n i m um f l ow c r os s s e c t i ona l a r e a ) ( l ong i t ud i na l p i t c h )D = (2)( he a t t r a ns f e r a r e a )

A t a f i r s t l ook , t he na t u r e o f f l u i d f l ow a nd he a t t r a ns f e r f o r bu nd l e s w i t h s t agge r e d o ri n - l i n e a r r ay s m a y b e h a v e v e ry d i f f e r e n t l y . H o w e v e r , a t h i g h R e y n o l d s . n u m b e r s i t i s li k e lyt h a t t h e t u r b u l e n t m i x i n g i s s o i n t e n s iv e t h a t a s i m p l e r f o r m u l a t i o n t h a n t h a t o f B i e r y [ 1 ] c a nb e e s t ab l is h e d f o r g e n e r a l c o n f ig u r a t io n s . T h e o b j e c t i v e o f t h is c o m m u n i c a t i o n i s t o p r e s e n t as i m p l e c o r r e l a t io n d e d u c e d f r o m B i e r y 's t ra n s f o r m a t i o n m e t h o d f o r b u n d l e s o f v a r i o u sc o n f i g u r a t i o n s .

A n a l y s i s a n d D i s c u s s i o n

T he c oe f f i c i e n t C i n e qua t i on ( 1) w as c o r r e l a t e d by B i e r y [ I ] a s :h

0 . 0 8 2 63 [ 1 . 2 - ( X T / d ) ~ ]C h = 0 . 2 8 1 8 ( X r / d ) A - 0 . 1 2 82 - [ ( X T / d ) a - 0 . 8 9 24 ] ( 3)

w h e r e ( X T / d ) A = 0 . 5 [ 3 . 6 28 ( D / d ) + 1 ] I/2 i s t h e t r a n s ve r s e -p i t c h- w - - d ia m e t e r r a t i o o f t h e

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V o l . 1 3, N o . 1 C O R R E L A T I O N F O R C R O S S F L O W I N ~ i ~ E B L % D L E S 5

e q u i v a l e n t E T P - I - S T B , t h e R e y n o l d s n u m b e r ( R e D ) i s b as e d o n t h e h y d r a u l i c d i a m e t e r ( D )de f i n e d i n e qua t i on ( 2 ), a nd t h e f l ow ve l oc i t y a t t he m i n i m um f l ow c r os s s e c ti on i n bund l e s .

I n f a c t , e qua t i ons ( 1 ) a nd ( 3 ) c a n be r e a r r a nge d i n t o a s i mp l e e xp r e s s i on a s :

d- - Re 0.6 p r m (~_~_)o4Nuj = C h d DD--- [[ 0.0953 3 + 0.1409 ( 3.628 + i //2]F(

0.0343 08 R e O.6 i/3 d o4[0.362 8 D / d + I] '''~ - 0.9848 } d Pr [--~-] " (4)

w h e r e N u i s t h e a v e ra g e N u s s e lt n u m b e r a r o u n d a t u b e i n t h e b u n d l e .d

F u r t he r e xa mi n a t i on o f e q ua t i on ( 4 ) s how s a n i n t e r e s t i n g re s u l t t ha t C ( d / D ) '4 i s ve r yh

c l o s e t o a s ing l e va l ue o f 0 .366 f o r s t a gge re d a nd i n - l i ne bund l e s w i t h p i t c h - t o - d i a m e t e r r a t i o so f 1 . 25 t o 2 .50 . T he c o r r e l a t i on i n r e f e r e n c e [ 1 ] ha s be e n r e p l o t t e d i n F i g . 2 , w he r e / i st he i nve r s e o f bu nd l e s o l i d i t y a nd c a n be e xp r e s s ed a s

p = ( I ) 1:2 (5)

Th e voidage of a tube bundle, ~, can be also related to the tube pitches as

X X - v d 2 / 4~ = "f l,X XT L

B y s u b s t it u t i n g th e s i n g l e v a l u e o f C ( d / D ) 4 , e q u a t i o n ( 4 ) b e c o m e sh

(6 )

N u = 0.366 Re o.e prl /3 (7)d d

E qu a t i on ( 7) g i ve s a s i mp l e e xp r e s s i on f o r t he he a t t r a ns f e r c oe f f i c i e n t o f t ub e bund l e s i n s p i t eo f t h e d e t a i l s o f t u b e s p a c i n g a n d t u b e l a y o u t i n t h e p r e v i o u s l y d e s c r i b e d r a n g e o f c o n d i t i o n s .T h e o n l y i m p o r t a n t e f f e c t o f l a y o u t g e o m e t r y c o m e s f r o m t h e v e l o c it y a t t h e m i n i m u m c r o sss e c t io n i n t h e b u n d l e u s e d i n t h e R e y n o l d s n u m b e r b a se d o n t u b e d i a m e t e r . E q u a t i o n ( 7) isc om pa r e d w i t h va r i ous e xpe r i me n t a l da t a i n F i g . 3 . T he da t a f a l l w i t h i n + 20% o f t h i s

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6 T . H . H w a n g a n d S . C . Y a o V o l . 1 3 , N o . 1

c o r r e l a ti o n f o r R e g r e a te r t h a n 4 0 0 0 . T h e r e f o r e , at hi g h R e y n o l d s n u m b e r c o n d i t io n s ( R e >d d

4000), e qua t i on (7 ) c a n be u se d ge ne r a ll y . A t h i gh R e yn o l ds nu m be r c r o s sf l ow , v i o l e n tt u r bu l e n t m i x i ng e x is t s i n t he bun d l e a nd dom i na t e s he a t t r a ns f e r s o t ha t t he de t a i l e d ge om e t r yo f bu nd l e be c ome s l es s i mp or t a n t . A s e xpe c t e d , t h is phe nom e na b r e a ks dow n a t i n t e r me d i a t eR e yn o l ds num be r c on d i t i ons (R e < 4000) a s show n i n F i g . 3.

d

A l t houg h e qua t i on ( 7) is i n good a g r e e me n t w i th t he e xpe r i me n t a l da t a f o r R e > 4000 a sd

s how n i n F ig . 3 , s ome l i m i t a t i on o f t he R e y no l ds nu m be r r a nge s hou l d be c ons i de r e d he re . A si nd i c a t e d i n r e f e r e nc e [ 3 , 4 , 5 ] , w he n t he f l ow R e yno l ds num be r i s ve r y h i gh , s a y R e > 2 x

dl 0 s t h e d e p e n d e n c e o f R e y n o l d s n u m b e r b e c o m e s m o r e s t r o n g e r b e c a u se t h e f lo w t h r o u g hbun d l e be c ome s inc r e a s i ng l y t u r bu l e n t . T he N us s e l t nu m be r i s, a s s ugge s te d by r e f e r e nc e [ 4 , 5 ] ,p r op o r t i on a l to t he R e yn o l ds nu m be r t o the e xpone n t o f 0 .84 i n s t e a d o f 0 .6 . T he r e f o r e ,e qua t i on ( 7 ) , t he s i mp l e c o r r e l a t i on de duc e d f r om B i e r y ' s t r a ns f o r ma t i on me t hod , i sr e c om me n de d f o r 4000 < R e < 2 x 10: .

d

E q u a t i o n ( 7) m a y b e c o m p a r e d t o e q u a t io n ( 8) w h i c h is r e c o m m e n d e d b y C o l b u r n [ 2 ] f o rs moo t h t ube bu nd l e i n t he R e yno l ds nu m be r r a nge o f 103 t o 4 x 10~

BN u = 0.33 Re o.6 pri /3 (8)d d

I t i s i n t e r e s t i ng t o s e e t ha t t he on l y s l igh t l y d i f f e r e nc e be t w e e n e q ua t i on ( 7 ) a nd e qua t i on ( 8 ) i st he c ons t a n t . T o t e s t the ge ne r a l i t y a nd s i mp l i c i t y o f e qua t i on (7 ), s ome da t a t a ke n f r om o t he rc o r r e l a t i ons [ 3 , 4 , 5 ] w h i c h w e r e p r opos e d f o r d i f f e r e n t bund l e ge ome t r i e s , a r e a l s o c om pa r e dw i t h e qua t i on ( 7) i n Fi g . 4. T he e xc e l l e n t a g r e e m e n t s be t w e e n t he c o r r e l a t i ons a nd e qu a t i on(7) a re observed.

C o n c l u s i o n s

T h e e m p i r i c a l f o r m u l a t i o n o f B i e ry [ 1 ] , w h i c h e m p l o y e d g e o m e t r ic t r a n s f o r m a t i o n s , fo rt he c r o s s f l ow he a t t r a ns f e r i n t ube bund l e s ha s be e n e x t e nde d . A si mp l e he a t t r a ns f e rc o r r e l a t i on ha s be e n de d uc e d a s s how n i n e qua t i on (7 ). T h i s s i mp l i c i t y r e s u l t s be c a us e o f t hei n t e n s i v e tu r b u l e n t m i x i n g w h i c h e x is ts i n t h e b u n d l e a t h i g h R e y n o l d s n u m b e r i n c r o ss f io w .T h e h e a t t r a n s f e r b e c o m e s a l o c a l p h e n o m e n o n i n w h i c h b u n d l e g e o m e t r ie s a r e n o l o n g e ri m p o r t a n t . T h e e f f e c t o f b u n d l e c o n f i g u r a t i o n a p p e a r s o n l y i n t h e v e lo c i ty t e r m i n th e

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V o l . 1 3, N o . I O O R R E I A T I O N F O R C R O S S F I iY ~ I N ~ B U N I X a ~ 7

R e yno lds num be r ba se d on tube d i a m e te r . Equa t ion ( 7 ) p r e d ic t s e xpe r im e n ta l r e su l t s w i th in +20% for 400 0 < Re < 2 x 105 Go od agreeme nts be tween equa t ion (7) and exis ting

dc o r r e l a t ions a r e a l so obse r ve d .

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

The a u tho r s a r e g r a t e f u l f o r t he suppor t o f t he Of f i c e o f Na va l R e se a r c h ( Gr a n t No .N00014-79--C-0623) in per fo rm ing th is s tudy.

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

C nond im e ns iona l c oe f f i c i e n t , R e 0.4 pr2t3 t i th D

d tube d i a m e te rD hydr a u l i c d i a m e te r , de f ine d in e qua t ion ( 2 )

he a t t r a ns f e r c oe f f i c i e n tkN u DN u dP r

the r m a l c on duc t iv i ty o f f l u ida ve r a ge Nusse l t num be r ba se d upon hydr a u l i c d i a m e te r Da ve r a ge Nusse l t num be r ba se d upon tube d i a m e te rP r a n d t l n u m b e r

R e d

R e DSS t

R e y n o l d s n u m b e r . Ud/vR e y n o l d s n u m b e r , UD/vs o l id i t y , v d : / [ 4 X T X L ] o rS ta n ton num be r

1 / /

f r e e s t r e a m ve loc i tyU f low ve loc i ty a t min imu m cross-- sec tion in bund lesX LX T

l ong i tud ina l p i t c ht r a nsve r se p i t c h

G r e e k s y m b o l s

DE

squa r e r oo t o f t he inve r se o f bund le so l id i tyvoidage , 1 - S

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8 T . H . H w 'o .n g a n d S . C . Y a o

R e f e r e n c e s

1. J.C. Biery, J o! Heat Transfer, A S M E , 10__33, 705 (198 1).2. A .P . Co lburn , Trans . o f the A me r i can Ins t . o f C hem . Eng ., 29, 174 (1933) .3 . V . Gn 'e l ins k i , A . Zukauskas and A . Skrinska, Heat Exchanger Des ign Handbook,

sec t ion 2 .5 .3 , He misp here Publ ica t io n Co. , New Y ork (1983).4. A. Zukauskas, Heat Exchanger, Hemisphere Publ ica t ion Co. New York, p . 49 (1981) .5. A. Zukauskas, Advanced in Heat Transfer, 8, 93 (1972)6. T.H. Hwang, Ph.D. Thesis, M e c h a n i c a l E n g i n e e r i n g D e p a r t m e n t C a r n e g i e - M e l l o n

Univers i ty , P i t t sburgh (1985)7. W . Ka ys an d A.L. L ondo n, Compact Heat Exchanger, M c G r a w - H i l l , N e w Y o r k

(1964).

V o l . 1 3 , N o . I

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V o l. 1 3, N o . 1 ( X ) R R E I A T I O NO R C R O S S F L O W I N ~ B U N I X J ~ 9

~ - x , -4T O 0 0X~,- L o 0 0

0 0 0o - ! - o o

t

0 0 ~ ) ~ O ~0 0 ----~0 0 0 00 0 0o o - 4 - - o o ~

( a ) I n - L i n e ( b ) S t a g g e r e d

F I G . 1Typica l Layout o f Tube Bundle s .

0 . 5 0

0 . 4 5

0 . 4 00 . 3 6 C

0 . 3 5

0 , 3 0

0 . 2 5

S y m b o l s 1 L/dJ0 1.250 1 , 5 i" 1 2 " 0 121 2 .5

xT,~1 . 2 5 1 . 5 1 .7 5 2 . 0 2 . 2 5 2 . 5 0

\L ~ L t l ~ l l | ~ { l 1 1 | | l l i i * | i J l i | i l l l J -

2 3 4 5 6 7 8

F I G . 2R e p l o t o f t h e H e a t T r an s fe r C o r r e la t i o n o f T u b e B u n dl es [ i ]v s , T u b e S p a c i n g a t H i g h R e y n o l d s N u m b e r s .

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I 0 T . H . H w a n g a n d S . C . Y a o V o l . 1 3 , N o . 1

1 0 3 ,

1 0 2

.... u dp r 1 / 3

1 0

1 . C

N'--ud = 0 . 3 6 6 R e d 0 - 6 P r ' ~ / 3

~ ~ , ~~-~ ~ * [ 7 ] , 5 0 - 1 . ~ '

+ ~ [ ] [ 7 ] . 5 0 - 1 . 2 [ 7 ] 2 . 0 - 1 . 2 ![ ][ ] ~ , [ 7 ] ~ . 0 - 2 . 0 [ 7 ] 1 . 5 - 2 . 0[ 7 ] 2 . 0 - 1 . 5-P [ 7 ] : 2 . 0 - 1 . 5

t * [ 7 - ] 2 . 0 - 2 . 0L ~ C 6 1 . ~ - I . 5

, i | i i i i i i 1 | i t I i i J i l l1 0 3 4 0 0 0 1 0 4 1 0 sR e d

B u n d l eA r r a ~ ,

SII

SSSItI

F I G . 3H e a t T r a n s f e r i n T u b e B u n d l e s a t H i g h R e y n o l d s N u m b e r s .

1 . 2

1 . 1

, . ~ 1 . 0

, ; ~ o .~~ gO. ca 0 .8

I Z Z 0 . 7

0 . 8

o l _10 3

. . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .. . . ~ . . . . . ~ 1 5 %

X T X L, s l l ~ l - l ~ - I R e , . | 1.25-1.25 C3~O @ | 1.5-1.5 ~s~

[ ] n n ~ 2 , 0 - 2 . 0 [ 3 ]v ~ ' / 2 . 5 - 2 . 5 [ 3 ]

/ d p t a b a n d o f [ 4 , 5 Ji L i ~ I i t J / j j L h i , , l l i ~ i J , i , , L1 0 4 R e d 1 0 5 1 0 5

P I G . 4C o m p a r i s o n o f E q u a t i o n ( 7 ) w i t h O t h e r s C o r r e l a t i o n s [ 3, 4, 5 ].