Finite Difference Analysis of Forced-convection Heat Transfer in Entrance Region of a Flat...

8/11/2019 Finite Difference Analysis of Forced-convection Heat Transfer in Entrance Region of a Flat Rectangular Duct

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A p p l . s c i. R e s . S e c t i o n A , Vo l . 1 3

FINITE DIFFERENCE ANALYSIS OF FORCEDCONVECTION HEAT TRANSFER IN ENTRANCE

REGION OF A FLAT RECTANGULAR DUCT

by CHING LAI HWANG and LIANG TSENG FAN

Kansas Stare Univers i ty, Ma nhat tan , Kansas, U.S .A.

S u m m a r y

T h e l a m i n a r f o r c ed c o n v e c t i o n h e a t t r a n sf e r i n t h e e n t r a n c e r e g i o n of af l a t r e c t a n g u l a r d u c t i s s t ud i e d . I n t h i s re g i o n t e m p e r a t u r e a n d v e l o c i t yp r o f il e s a r e s i m u l t a n e o u s l y d e v e l o p e d . T h e b a s i c g o v e r n i n g e q u a t i o n s o fm o m e n t u m , c o n t i n u i t y , a n d e n e r g y a re e x p r e ss e d i n f i n it e d i f f r e n ce f o r ma n d s o l v e d n u m e r i c a l l y b y u s e o f a h i g h s p e ed c o m p u t e r f or a m e s h n e t w o r ks u p e r i m p o s e d o n t h e f l o w fi el d . A l l f l u id p r o p e r t i e s a r e a s s u m e d t o b ec o n s t a n t . T h e c a s e s o f u n i f o r m c o n s t a n t w a l l t e m p e r a t u r e a n d o f u n i f o r mc o n s t a n t h e a t f l u x fr o m w a l l to f lu i d a re c o n s id e r ed . N u s s e l t n u m b e r s a r er e p o r t e d f o r P r a n d t l n u m b e r s i n t h e r a n g e o f 0 .0 1 t o 5 0. T h e e x a c t s o l u t i o no f t h e e n e r g y e q u a t i o n o b t a i n e d b y m e a n s o f t h e n u m e r i c a l m e t h o d isc o m p a r e d w i t h t h e r e s u l ts o f a p p r o x i m a t e s o lu t io n s .

Nomenc l a t u r e

A s u r f a c e a r e a o f c h a n n e l w a l l s t h r o u g h w h i c h h e a t i s b e i n g t r a n s f e r r e da d u c t h a l f - h e i g h tC ~ s p e c if i c h e a t a t c o n s t a n t p r e s s u r eD e e q u i v a l e n t d i a m e t e r f o r a d u c t , 4 6G z G r a e t z n u m b e r ,R e aP r / (x /De )h h e a t - t r a n s f e r c o e f f ic i e n t ,Q/{A(At)}k t h e r m M c o n d u c t i v i t y o~ t h e f lu i dN u m a v e r a g e N u s s e l t n u m b e r ,hmDe/kN u z l o c a l N u s s e l t n u m b e r ,hxDe/kP r P r a n d t l n u m b e r ,C~iz/kp f l u i d p r e s s u r eP o p r e s su r e a t c h a n n e l m o u t hP d i m e n s i o n l e s s p r e s s u r e ,(p -- po)/p~~o(2 h e a t - t r a n s f e r r a t e

R e R e y n o l d s n u m b e r ,Ouoct/l~R e a d i a m e t e r R e y n o l d s n u m b e r ,puoD/ ~ puo4a/Ft

4 0 1


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40 2 CHING-LAI HW ANG AND LIANG-TSENG FAN

t t e m p e r a t u r eto t e rnpera tu re o f f lu id a t en t rance sec t ion o f channe lt l con s tan t wa l l e rape ratu re

tw wa l l t em pera tu reu f lu id velo ci ty in x-di rec t ionu0 f lu id velo ci ty a t in le tU dimensionless u veloci ty,u / u ov f lu id velo ci ty in y-di rec t ionV dimensionless ve loci ty,avp/ t~x coord ina te a long channe lX dimensionless x-coordin ate ,# x / ( O a 2 u o ) = ( x / a ) / R e X dilnellsionless x-co ord inate defin d asI~x / (oD~euo) ~ (x /De) /Ree ~X / 1 6y coord inate across chann el

Y dimel ls ionless y-coord inate ,y / a

the rm al d i f Ius iv i ty of f lu id ,k / p C pv k inema t i c v i scos i ty o f f lu idp f luid densiyB dyn amic v i scos i tyof f lu id0 d imension less t emp era tu re , de f ined by 8 ) ,( t - t o ) / ( h - t o ) for

cons tan t wa l l t empera tu re ,k ( t - - t o ) / ( a g )fo r cons tan t hea t f lux caseO,x d imensionless bu lk tem pe ra tu re a t an y locat ion x , def ined by 15)Ow dimensionless wal l tem pe ra tu re def ined by 8)

1 . I n t r o d u c t i o n . M o s t o f t h e p u b l i s h e d p a p e r s o n t h e a n a l y t i c a ls t u d i e s o f l a l n i n a r f o r c e d - c o n v e c t i o n h e a t t r a n s f e r i n t u b e s o r d u c t sh a v e b e e n c o n c e r n e d p r i m a r i l y w i t h t h e e a se o f a fu l l y e s t a b l i sh e dv e l o c i t y p r o fi le , o r h a v e b e e n b a s e d o n a n a s s u m p t i o n o f a u n i f o r mv e l o c i t y p ro f il e . F o r f l u id s w i t h h i g h P r a n d t l n u m b e r , s u c h a s o ils ,a n a s s u m p t i o n o f a f u l ly e s ta b l i s h e d p ~ r a b o l i c v e l o c i t y p r o f il et h r o u g h a n e n t i r e c o n d u i t d o e s n o t l e a d t o s i g n i fi c a n t e r r o r b e c a u s et h e v e l o c i t y p r o f il e is e s ta b l i s h e d I n u c h I n o re r a p i d l y t h a n t h et e l n p e r a t u r e p r o fi le . I n c o n t r a s t , f o r f lu i d s w i t h v e r y lo w P r a n d t ln u l n b e r , s u c h a s l i q u i d I n e ta l s , th e t e m p e r a t u r e p r o f i l e i s e s t a b l i s h e dr a u c h I n o r e r a p i d l y t h a n t h e v e l o c i t y p r o fi le , a n d t h e a s s u m p t i o no f u n i f o r m v e l o c i t y m a y n o t i n v o l v e l a rg e e r r o r f o r t h e t y p i c a la p p l i c a t io n . H o w e v e r , fo r f lu i d s w i t h P r a n d t l n u l n b e r s n e a r u n i t y ,a s f o r g a se s, b o t h t h e t e m p e r a t u r e a n d v e l o c i t y p r o fi le s d e v e l o p a ta si m i la r r a t e a l o n g a t u b e o r a d u c t i f b o t h t h e v e l o c i t y a n dt e m p e r a t u r e a r e a s s u l n e d t o b e u n i f o r m a t t h e e n t r a n c e , a n dn e i t h e r t h e a s s u l n p t i o n o f f u l l y d e v e l o p e d p a r a b o l i c v e l o c i t yp r o f il e n o r u n i f o r m v e l o c i t y i s s a ti s f a c to r y.

D u r i n g t h e l as t d e c a d e m o r e a n d m o r e a t t e n t i o n h a s b e e n g i v en t o


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I IEAT TRAN SFER IN ENTRANCE REGION OF A DUCT 40 3

t h e c a se o f s i m u l t a n e o u s d e v e l o p m e n t o f v e l o c i t y an d t e m p e r a t u r ep ro f i l e s i n t he en t r anc e r eg ion o f a t ub e o r a duc t . T he l i t e r a tu r e

be fo re 1958 i s w e l l sum m ar i zed i n Re f . 1). Fo r a c i r cu l a r t u be t hee n e r g y e q u a t i o n w a s s o l v e d b y n u m e r i ca l i n t e g r a t io n b y K a y s 9 ) f ort he ca se o f r = 0 .7 a n d b y G o l d b e r g a ) u si ng K a y s m e t h o d o na c o m p u t e r f o r o t h e r m a g n i t u d e s o fr in the rang e of 0 .5 to 5 .0 .T h e y u s e d t h e v e l o c i t y p r o fi le s i n t h e e n t r a n e e r e g i o n , w h i c h w e r ede t e rm ine d ana ly t i c a l l y b y L a n g h a a r 4 ).

F o r f l a t r e c t a n g u l a r d u c t s S p a r r o w 5) w a s t h e fi rs t i n v e s t ig a t o rt o u se t h e K a r m a n - P o h l h a u s e n m e t h o d t o o b t ai n t h e s o lu ti onf o r t h e c a s e o f u n i f o r m c o n s t a n t w a l l t e m p e r a t u r e . T h e a p p r o a c h o f

S p a r r o w s v e l o c i t y p r of il e t o t h e f u l ly d e v e lo p e d p a ra b o l ic o n e isn o t a s y m p t o t i c 5) b e c a u s e S c h i l l e r s v e l o c i t y p r o f il e a) in t h eb o u n d a r y l a y e r is as s u m e d . W i t h o u t t h is a s s u m p t i o n o f t h e v e l o c i t yp ro fi le , th e s i m p l i ci t y of S p a r r o w s a n a ly s i s f or t h e t e m p e r a t u r ep r of il e w o u l d n o t b e r ea li ze d . S p a r r o w i n d i c a te d t h a t t h e f in a lc h e c k f o r t h e i n f l u e n c e o f t h e a p p r o x i m a t e v e l o c i t y p r o fi le o n t h er e s u l t s o f h e a t - t r a n s f e r a n a l y s i s c a n b e m a d e o n l y w h e n a m o r ee x a c t s o l u t i o n t o t h e e n e rg y e q u a t i o n i s a v a i l a b l e .

T h e s a m e m e t h o d w a s e m p l o y e d f o r t h e c a s e o f c o n s t a n t h e a t f l u x

b e t w e e n t h e w a ll s a n d t h e f lu id b y S i e g e l a n d S p a r r o w 6 ) .T h e r e a r e t w o t h e o r e t i c a l w o r k s p u b l i s h e d r e c e n t l y f o r t h i s

s i m u l t a n e o u s d e v e l o p m e n t o f v e l o c i ty a n d t e m p e r a t u r e d is tr i-b u t i o n s i n t h e e n t r a n c e r e g io n o f t u b e s a n d d u c t s 7) 8). H o w e v e r,t h e y d o n o t p r e s e n t e x a c t so l u ti o n s o f m o m e n t u m a n d e n e r g yequa t ions spec i f i c a l l y.

S t e p h a n 9) em p lo yed app rox im a te s e ri e s so lu t i on fo r t he con s t an twa~ll t em pe ra t u r e ca se . Th e c o r r e l a t i on eq ua t i o n g iven i n h i s pa pe ra p p e a r s t o b e s i m p l e a n d r e a s o n a b l y c o r r e c t .

R e c e n t l y H a n l ) p r o p o s e d a n a p p r o x i m a t e m e t h o d w h i c h is t h ee x t en s i on o f t h e L a n g h a a r m e t h o d 4) t o t h e t h e rm a l p r ob l em .H a n 10 ) a p p l i e d t h e m e t h o d t o t h e c a se w i t h c o n s t a n t h e a t f l u x ina f l a t d u c t. T h e r a t e o f a p p r o a c h o f t h e l o c a l N u s s e k n u m b e r t o i tsa s y m p t o t i c v a l u e i s o b v i o n s l y q u i t e d i f f e r e n t i n t h e t w o d i f f e r e n ta p p r o x i m a t e m e t h o d s 6 ) 1 0 ) .

T h e p r o b l e m o f s i m u l t a n e o u s d e v e l o p m e n t o f v e l o c i t y a n d t h e r m a lp r o Ii le s i n t h e e n t r a n c e r e g io n i n v o l v e s t h e s o l u t i o n o f h i g h l y n o n -l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s . T h e r e I o r e , m a n y d i f f e r e n t a p -p r o x i m a t e m e t h o d s h a v e b e e n d e v e lo p e d f o r t h e s o lu t io n . T h e s e


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4 4 C H I N G L A I H WA N G A N D L I A N G T S E N G FA N

a p p r o x i m a t e m e t h o d s a r e n o t a p p li c ab l e t o a l l t h e s im i l a r n o n -l i n e a r p a r t i M d i f t e r e n t i a l e q u a t i o n s . F o r i n s t a n c e , t h e v i s co u s di s-s i p a t i o n t e r m s i n t h e e n e rg y e q u a t i o n s a r e n e g l e c t e d i n a ll t h ep a p e r s c i t e d p r e v i o u s l y. I f o n e c o n s i d e r s t h e c a s e s w i t h a p p r e c i a b l ev i sc o u s d is s ip a ti o n , a d i f f e re n t m a t h e m a t i c a l m e t h o d t a u s t b e e m -p l o y e d .

T h i s p a p e r p r e s e n t s r e s u l t s o f t h e n u m e r i c a l a n a l y s i s o f t h ep r o b l e m . T h i s w o r k m a y b e c o n s i d e re d a s a n e x a m p l e o f t h es u c c e s s I u l a p p l i c a t i o n o f t h e f i n i t e d i f f e r e n c e a n a l y s i s i n c o m b i -n a t i o n w i t h t h e u s e o f a h i g h s p e e d d i g i t a l c o m p u t e r i n t h e s o l u t i o no f h i g h l y n o n - l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s .

T h e s a m e m e t h o d w a s a p p l ie d t o s o lv e o t h e r h i g h l y n o n - l i n e a rp a r t i a l d i f f e r e n t i a l e q u a ti o n s11 ) . H o w e v e r, m e t h o d s f o r e r r o r e s ti -m a r i o n o f t h e n u m er _ic al m e t h o d a s a p p l i e d t o t h e s o l u t i o n o f n o n -l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s h a v e y e t t o b e e s t a b l i s h e df i r m l y. T h e r e f o r e , t h e a d e q u a c y o f t h e n u m e r i c a l m e t h o d u s e d i nt h i s w o r k w a s d e m o n s t r a t e d b y c o m p a r i n g t h e r e s u l t s w i t h t h er e s u l t s o b t a i n e d b y o t h e r m e t h o d s , w h e n e v e r t h e y a r e a v a i l a b l e .I n e m p l o y i n g t h e n u m e r i c a l a n a l y si s , t h e s e l e ct io n o f t h e p r o p e rm e s h s i ze s i n o r d e r t o a c h i e v e r a p i d c o n v e rg e n c e o f t h e s o l u t i o n

t o t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n s w i t h i n t h e c a p a c i t y o f t h ec o m p u t e r is a di f i c u lt h u t v e r y im p o r t a n t t a sk . T h e p r o p e r m e s hs iz es w e r e d e t e r m i n e d s e m i - t h e o r e t i c a l l y i n t h is w o r k .

T h r e e s e p a r a t e c a s e s r e p r e s e n t i n g t h e f o l l o w i n g b o u n d a r y c o n -d i t i ons a r e cons ide red .

1. C o n s t a n t w a l l t e m p e r a t u r e , p a r a b o l i c v e l o c i t y p r o f il e t h r o u g h -o u t , t e m p e r a t u r e u n i f o r m a t t h e e n t r a n c e . T h e n u m e r i c a l s o lu t io n f o rt h is c a s e is p r e s e n t e d a s a c h e c k o n t h e a d e q u a c y o f t h e n u m e r i c a lm e t h o d , sin ce i t c a n b e c om p a r e d w i th N o r r i s a n d S t r e i d ' sresu l t s l2) .

2. C o n s t a n t w a l l t e m p e r a t u r e , v e l o c i t y a n d t e m p e r a t u r e u n i fo r ma t d u c t e n t r a n c e . T h e v e l o c i t y p ro f il e is d e v e l o p i n g i n t o a p a r a b o l i co n e . T h e e x a c t s o l u t io n b y t h e f i n it e d i ff e r e n c e a n a l y s i s is c o m p a r e dw i th th a t b y S p a r r o w 5) a n d w i th t h a t b y S t e p h a n g ) .

3 . C o n s t a n t h e a t f l u x f r o m w a l l t o fl u id , v e l o c i t y a n d t e m p e r a -t u r e u n i f o r m a t t h e e n t r a n c e . T h e y d e v e l op s i m u l ta n e o u s l y. T h er e su lt s a r e c o m p a r e d w i t h th o s e o f S i e g e l a n d S p a r r o w ~ ) .

T h e s o l u t i o n f o r t h e r a n g e s o f G r a e t z n u m b e r f r o m I 0 t o 1 0,0 00a n d o f P r a n d t l n u m b e r f r o m 0.01 t o 5 0 f o r c o n s t a n t w a l l t e m p e r a -


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H E AT T R A N S F E R I N E N T R A N C E R E G I O N O F A D U C T 4 0 5

t u t e i s p r e s e n t e d . F o r t h e c a s e o f c o n s t a n t h e a t f l u x f r o m w a l l t of iu i d , t h e s o l u t io n f o r t h e r a n g e o fx /De) /Reaf rom 10-4 to 1 .0 ando f P r a n d t l n u m b e r f r o m 0 .1 t o 5 0 is p r e s e n t e d .

T h e f i n i t e d i f f e r e n c e a n a l y s i s o f t h e m o m e n t u m a n d c o n t i n u i t ye q u a t i o n s i s e m p l o y e d f i rs t t o o b t a i n t h e t w o - d i m e n s i o n a l v e l o c i t yp ro fi le s l3 ) 14). Th e va lues o f t he ve lo c i ty p ro f i le s a r e su bs t i t u t edi n to t h e e n e rg y e q u a t i o n t o s o lv e th e t e m p e r a t u r e p ro f ile s . T h ef i n it e d if f er e n c e t e e h n i q u e e m p l o y e d in o b t a i n i n g t h e t e m p e r a t u r ep r o fi le s a n d t h e h e a t t r a n s f e r p a r a m e t e r s , is d e s c r i b e d in s o m ed e t a il . A n I B M 1 62 0 d i g i ta l c o m p u t e r w a s u s e d .

2. Th e basic equations. Cons ide r a s em i - in f in i t e pa ra l l e l p l a t e ,a s s h o w n i n f ig . 1, i n w h i c h a s t e a d y , t w o - d i m e n s i o n a l , l a m i n a r f lo wo f a f l u i d w i t h c o n s t a n t p h y s i c a l p r o p e r t i e s i s t a k i n g p l a c e u n d e rthe cond i t i ons o f neg l ig ib l e v i scos i ty d i s s ipa t ion e ff ec t and neg l i -g i b l e l o n g i t u d i n a l h e a t f l o w. Wi t h t h e u s u a l P r a n d t l b o u n d a r yl a y e r a s s u m p t i o n s t h e l a w s o f c o n s e r v a t i o n o f m o m e n t u m , m a s s ,a n d e n e rg y a r e a s f o ll o w s :

u Ou 1 dp 02u

u - - ~ v ~ y p d x v - - 1 )

~ ~v

x + ~ y o , 2 )

~t Ot 02tu - - + v - - = ~ - - 3 )

x Oy Oy2

t y ~ ~

_ ~ ~ / ~~_~~ ...... _~/z / / / , [ .~/ z/ / / / ~ z / .z /~

t~

F ig . 1 . G e o m e t r y o f p a r a l le l p l a t e c h a n n e l a t e n t r a n c e r e g io n .

T h e c o n t i n u i t y p r i n c i p l e c a n a l s o b e e x p r e s s e d i n i n t e g r a l f o r m b y

gg

2uoa = 2 f u dy. 4)


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406 C H I N G L A I t t WA N G A N D L I A N G T S E N G F A N

In t roduc ing the fo l lowing d imens ion less parameters fo r the mo-m e n t u m a n d c o n t i n u i t y e q u a t i o n s

x = ~ x / p a 2 u o ) = x / a ) / R ~ ,

Y = y /a ,

U = u i u o ,

V = a v p / f f ,

P = p - po)/p ~~.

(1), (2), and (4) become in dimensionless form,

8 U 8 U d P 8 2UU -~ X - + V 8 Y - dX + --Y~- (5)

8 U 8 V- - + - - 0 , 6 )8 X 8 Y

I - ~ f U d l / . 7 )0

Fo r the ener gy equ at ion, (3), the fol lowing different forms ofd imens ion less t em per a tu re a re in t roduced . For the cases o f cons tan twa l l t empe ra tu r e

o = t - t o ) / h - t0 ) , Sa)

where t i s f lu id temper a ture , h i s the cons tan t wal l t empera~ure ,and t0 i s the un i fo rm f lu id temp era ture a t the en t ra nce of the duc t .For the case of cons tan t hea t f lux f rom wal l to f lu id

0 = k( t - - to) / (ag ) , (8b)

where t i s a var iab le represen t ing the tempera ture o f wal l as wel l

as f lu id at any pos i tion , to the temp era t ure o f the wal l an d tha tof the f lu id a t the en t rance of the duc t andg t he cons t an t hea tflux at wall .

The ener gy equa t ion, (5), in dimensionless form then becom es

80 80 1 820u + v 9)

8 X 8 Y P r 8 Y 2

The con di t ion of cons tan t wal l t em pera ture occurs no t on ly incondensors and evapora tors , hu t i s a l so offen approximated inparal le l - f low hea t exchangers , especial ly wher e the two f luid ca-


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H E T T R N S F E R I N E N T R N C E R E G I O N O F D U C T 407

paci ty ra tes and hea t t ransfer res i s tances a re near ly the same. Thebou nd ary condi t ions uncler cons idera tion a re as fo llows:

a t

~U 80for X > O , Y = O : ~ Y - - O , V = O , a Y O,

for

X = O , O G Y < I U ~ - I , V = O , O~ - ( t o - - t o ) / ( t l - - t o ) = O ,

10)

X ~ 0, y = 1 U ~- 0, V = 0 , 0 = (tl -- to) / ( t l - - to) ~ 1 .

The condi t ion of con s tan t he a t f lux f rom wal l to f lu id occursin e lec t r ica l res i s tance hea t ing , or when hea t ing by o ther methodsprovides a cons tan t hea t f lux . At the en t rance the wal l and f lu idt empera tu re s a r e equa l . Downs t r eam the wa l l t empera tu re a t f i r s tincreases rap id ly. T hen the wal l t emp era tu re grad ien t(t/x)wanbe-comes cons t an t and equa l t o t he cons t an t f lu id t emp era tu re g rad ien t(3t/~x)~l~ia as th e local Nussel t nu m be r appro aches a c ons tantmag ni tude . T he bo un dar y condi t ions a re cons idered to be asfollows:

at

for

f; r

X = 0 , 0 < y < I U = I, V- - O ,

Y = - I U = 0 , V = 0 ,

0 G Y G I 0 - - 0 ,

~U 0 ~ 0 ~ ) _ 0~ 0 , Y = O Y

X > O , Y = 1 U : 0 , V = 0 , ( B ) - : 1.a t Y ~ l

11)

3. Solution o/ the equation. Since al l the f luid propert ies areas sumed to be cons t an t , t he two-d imens iona l ve loc i ty componen tean be obtained from (5) , (6) , and (7) . The resul ts are then su5st i -tu ted in to th e energy equat ion , (9), to so lve the tem per a tu re profi le .Th e finit e differenc e anal yses of (5), (6), and (7) are discu ssedpreviou sly 13)14). The values of U a nd V obta ined are dire ct lyemployed in solving (9).

To obta in the te m per a tu re prof iles by so lv ing the ene rgy equat ion ,(9), i t is ap pro xim ate d by the fol lowing f ini te difference equat ion s(see the me sh n etw or k of f ig . 2) :


8/22

4 0 8 C H I N G - L A I H WA N G A N D L I A N G - T S E N G FA N

~20

~ y 2

U = g a , U j l , ,2

V - - V j ,~ V j l , k2

O 0j+l , 0i ,

X z t X

0 O j , ~ + , - O j , k - 1

O y 2 A Y

12)

0 ] + l , k + 1_ _ 2 0 1 + 1 , 2 ff0]+1 , /~ .1 ) @_ 0 ] , /g+ l 2 0 ] , 1 c q - O J , l c -1

2 d Y ) 2Y

C ? rT r

~ '" -- '2 o ~ J i , I ~ + , ~ - - xx ~ o

Fig. 2. Mesh netw ork fo r d i fference representa t ions .

S u b s t i t u t i n g (1 2) in t o ( 9) , c o n s i d e r i n g t h e v a l u e s o f 0 w i t h s u b -s c r i p t ] + 1 a s u n k n o w n s , a n d t a k i n g k f r o m 1 t o n , o n e o b t a i n s ns i m u l t a n e o u s e q u a t i o n s w i t h n u n k n o w n s ( 0 j+ l , 1 ; 0 j + l, 2 ; . . . ; 0 j + l, ,).T h e s e l i n e a r s i m u l t a n e o u s e q u a t i o n s a r e s o l v e d b y u s i n g T h o m a s 'sm e t h d 15) ( c.f . A p p e n d i x ) . T h e b o u n d a r y c o n d i t i o n s f o r t h e s e d i f f e r -e n c e e q u a t i o n s a r e a s f o ll o w s : f or th e c a se o f c o n s t a n t w a l l te m p e r a -t u t e , f r o m ( 1 0 ) ,

X = o , o < y < 1 , o = o ,fo r X > 0 , Y = 0 :0 y + 1 ,2 = 03 +1,0, (13)

fo r X ~ O, Y = 1 Oj+l,n l ~--- 1.0.

F o r t h e c a s e o f c o n s t a n t h e a t f l u x f r o m t h e w a l l t o t h e f l ui d , f r o m(11),


9/22

H E AT T R A N S F E R I N E N T R A N C E R E G I O N O F A D U C T 4 0 9

a s

X = 0 ' 00, k = 0 , k = 1 , 2 , . . , n + 1,

for X > 0, Y = 0: 0j+l, 2 Oj+l, 0 fo r any ] ,

f o r X > 0 , Y ~ - - 1 O j + l , n + l = O 3 + l , n + d Y.

Oj l,n l - - On i, n

A Y= 1.0.

(14)

4 . H eat - t rans[er param eters .A f t e r t h e te m p e ; a t u r e 0 =O X , Y )i s e v a l u a t e d , t h e b u l k t e m p e r a t u r e ( of m i x e d m e a n t e m p e r a t u r e ) isd e t e r m i n e d b y

1

f U j + l , k d Y = 10

a n d1

Obulk = f Oj+l, ~U j+I, ~ d Y . (15)0

F o r t h e c a s e of c o n s t a n t w a l l t e m p e r a t u r e t h e l o c a l N u s s e l tn u m b e r , N u x , is o f s e c o n d a r y i m p o r t a n c e , a n d t h e a v e r a g e N u s s e l tn u m b e r , N u m , i s d e s ir e d , so t h a t t h e t e m p e r a t u r e o f t h e f l u i d

l e a v i n g t h e h e a t e x c h a n g e r m a y b e e v a l u a t e d . T h e r e f o r e , t h e a v e r -a g e N u s s e l t n u m b e r ,N u m , w h i c h i s d e f i n e d i n t h e u s u a l w a y a s

N u m = h m D e / k , (16)

i s p r e s e n t e d . F o r a d u c t w i t h s p a e i n g 2 a , t h e e q u i v a l e n t d i a m e t e r ,De, i s equa l t o 4a . The ave rage hea t - t r ans I e r coe I f i c i en t f o r al eng th o f du c t X ( and un i t w id th ) i s , t he r e fo re , g iven by

h m - A { A t) - -2x (1)(d t) ' (17)

w h e r e t h e t e m p e r a t u r e d i f f e r e n c e ,At, i s t a k e n a s t h e l o g - m e a nt e m p e r a t u r e

t ~ - t o ) - t ~ - t , x )d t ) 1 8 )

l n E t w - t o ) / t , ~ - t h ,~ ~

T h e t o t a l h e a t f ] u x Q f r o m t h e e n t r y t o x i s

~~)2 = - t - 2 k d x . 1 9 )

0


10/22

410 C H I N G L A I H WA N G A N D L I A N G T S E N G FA N

S u b s t i t u t i n g 19) a n d 17) i n t o 1 6), t h e m e a n N u s s e l t n u m b e r,i n d i m e n s i o n l e s s f o r m , i s o b t a i n e d

x4 80

N u m - - X AO ) -i-~ ).v= ldX .2 O )

T h e t r a p e z o i d a l r u l e i s u s e d f o r t h i s c a l c u l a t i o n a s f o l l o w s :

4 0j, n+ l -- 0j, n) + 0j+l, n+l -- 0j+l,n ) A XN u m - Z - - . 2 1 )

X A O ) ~ 2 ~ Y 2

F o r t h e c a s e o f c o n s t a n t h e a t f l u x fr o m t h e w a l l to t h e f iu i d ,eh e m e a n N u s s e l t n u m b e r ,N u m , h a s n o p a r t i c u l a r u s e , a n d t h el o c a l N u s s e l t n u m b e r,N u z , is d e si r ed s o t h a t t h e w a l l t e m p e r a t u r ec a n b e e v a l u a t e d a t a n y p o i n t i n t h e h e a t e x c h a n g e r. T h e r e f o r e ,t h e l o c a l N u s s e l t n u m b e r ,N u z , which i s de f ined a s

N u . : h ,,De /k , 22)

i s p r e sen t ed .S ince t he l oea l hea t coe ff i c i en t , hx , i s g iven by

h x - - A A t ) - - 1 ) A x ) A t 23)

a n d Q x i s g i v e n b y

? ~= + k ~ x ) 1 ) , 2 4 )y-a

t h e l o c a l N u s s e l t n u m b e r,N u x , i n d i m e n s i o n l e s s f o r m , i s g i v e n b y

_ 8

Nu~ AO)

T h e l a s t b o u n d a r y c o n d i t i o n o f 11) e x p r e s s e s t h e e o n d i t i o n t h a tt h e s l o p e o f t h e t e m p e r a t u r e p r o fi le a t t h e w a l l is m a i n t a i n e d a tu n i t y f o r a ll X a n d t h u s t h e h e a t f i u x a t th e w a l l i s c o n s t a n tt h r o u g h o u t th e d u c t. T h e l o c al w a l l t e m p e r a t u r e g r a d i e n t a n dl o c a l t e m p e r a t u r e d i I f e r en c e a r e u s e d i n c o m p u t i n g t h e l o c a lN u s s e l t n u m b e r. T h e r e f o r e , t h e l o c a l N u s s e l t n u m b e r i s

Nux = 4 /AO,


11/22


12/22

412 C H I N G - L A I H W A N G A N D L I A N G - T S E N G F A N

The values of AX are fixed by the velocity data weich are ob-ta ined by solving (5), (6), an d (7). U is in che fange of 0 < U < 1.5.

TEe Prandtl number, Pf, is in the range of 0.01 to 50. Therefore,different values of A Y are chosen depending on the values of Pfand X. The mesh sizes, weich depend on velocit y dat a ob taine dby solving (5), (6), and (7), are shown in Tab le I, where N is the

T A B L E I

M e s h si ze s f o r t h e m o m e n t u m a n d c o n t i n u i t y e q u a t i o n s ~4X A X A Y N

]0 . 0 0 0 1 ]

0 .000 5

0 .00I ]

0 .0 I ]

0.1 ]

f u l l y d e v e l o p e d ]

0 . 0 0 0 1

0 . 0 0 0 2

0 .000 25

0.001

0 .005

0.01

0 .0125

0 .025

0 .025

0 .05

0.1

0 . I

8

4

4 0

2 0

10

10

mesh number of y14). Comparing some results obtained from this

numerical method with the known results, it is felt that takingU -- 1.0 and at a short distance near the entry, when the value ofPrU AY)2/ 12AX) is less than 0.05, the solution would convergeto the exact one. Table II shows, as an example, the values of X,mesh sizes, mesh numbers, and PrU AY)2/ 12AX) used in thecomputer calculation when Pr = 1.0.

To obtain the velocity correctly at the duct entrance, when Xvaries Irom 0 to 0.001, smal l AX is used as shown in Table I. How-ever, in this range of X, when using the energy equation, AX istak en as shown in Table II in order to keep t he val ue of PrU A y)2/(12AX) as small as possible within th e co mput er capacit y. Fo r Xgreater than 0.001, AX is the same in Table I and Table II, how-ever, A Y is much smaller in Table II th an it is in Table I. F orlinear interpolation of tee values of veloeity on the computer, themesh size of Y can be ob tai ned as shown in Table II. SinceT h o m a s s met hod 15) is used, the numb er of mesh points for Y,(equal to the number of simultaneous linear equations) can be aslarge as 200, for an IBM 1620 compute r wi th a 60,000 storagecapacity, when solving the energy equation, (9).


13/22

F IE AT T R A N S F E R I N E N T R A N C E R E G I O N O F A D U C T 413

TA B L E I I

Mesh sizes for the energy equationPrU A y s

X AX A Y N

o0.001 ]

0.01

0.1 ]

0.5 ]

2.0 1

0.000,5

0.001

0.005

0.01

0.05

0 00625

0.0125

0.025

0.05

0. I

12A X

160 ] 0.0055

80 0.013

40 0.01

20 0.02

10 0.017

As mention ed there is difficulty in obtai ning aecurate values atthe very vicinity of the entrance where the Graetz number RePr/x/De) is large; therefore, Kays2), in his numerical solution for

tube flow, used the P o hl h a u se n s flat plate solution until RePr/x/De) = 1,000, and the n sta rte d bis numeri cal solution for RePr/x/De)


14/22

414 C H I N G - L A I H WA N G A N D L I A N G - T S E N G FA N

1 .6 1 I I i , i i i y i

- -P re sen t work j - - - 9ch[ i ch t ing . -- - - - -Spar row ~ =0

0,7

0 . 6

0.4

0 o.o~ 0.04 o.~ o. o. Io o.h ' ' . 1 4 OJ6X R e a

0.5

9 1. . . . .

0 , 1 8 0 . 2 0

Fig. 3. Comp arison of veloc ity profiles used in prese nt wo rk and inS p a r r o w s 5) an d S c h l i c h t i n g s l S ) .

I00

70

40

~ 2 0

I075

Io

i i i i l l l l l , i i i i III H I 1 1 i f i t

The curv e: The result of Ref. 12}he dots: Present w o r k

p r nnnnpntn~50 , , , , i r r, n T r r uJ_20 4.10 ~G - Reclp5xIOZ 103

4

Fig. 4. Comparison of results of present work for parabolic velocity profilethroughout duct lengh with results of Norris and Streid12 .

TA B L E I I l

N u m e r i c a l e o m p a r i s o n b e t w e e n t h e r es ul ts o f t h e p r e s e nt w o r k

a n d t h o s e b y N o r r i s a n d S t r e i d f or t h e c a s e o f f ul ly d e v e l o p e d

pa rab o l i e ve loc i ty p rof i ]e s

G z P r es e n t Wo r k

104090

200

4001,0002,000

8,00010.000

Num

N o r r i s a n d

St re id l z )7.868.629.67

11.614.4

19.62 3 . 9

3 6 . 7

3 9 . 9

7 77

8.479.64

11.6114.13

19.1023.3436.1840.32


15/22

H E AT T R A N S F E R I N E N T R A N C E R E G I O N O F A D U C T 415

duct length are shown by dots on fig. 4. The solution for this caseis obtained in order to check the adequacy of the present numerical

meth0d. This case has a lre ady been tr ea ted by N o r r is a nd S t r e i d 12)using Leveque s me tho d for Graetz numbers of about 400, and adirect numerical integration of the energy equation for Graetznumbers below this value. Fig. 4, and Table I II show th at agreementbetween the results of the present work and those of N or r is andS t r e i d is excellent with the maxi mum deviation of 3 percent inthe range of Graetz number from 10 to 10,000. This small dis-crepancy provides some measures of confidence in the numericaltechnique employed.

Fig. 5 shows the surface and bul k tem pera ture varia tion forvarious Prandtl numbers for the case with constant wall temper-ature.

i o/ I I Surface -~ 1 l I

0 . 6

~;oo70

0 . 2

0 0 0. 2 0.4 0,6 0.1~ I.O 1.2 1.4 1.6x / o

X = e

F i g 5 F l u i d a n d s u r f ac e t e m p e r a t u r e s Va ri a t i o n f o r v a r i o u s P r a n d t l n u m b e r sat entrance region; COlastant wall ternperature.

Fig. 6 shows the variation of the average Nusselt number Numwith Graetz number, RePr/ x/De), for the case with constantwalI temperature. Num is the average Nusselt number based on thelog-mean temperature difference.

For the case of constant wall temperature the deviation of thepresent results Irom those b y S p a r t o w 5), when t he y are expressedin terms of Num, is less th an 5 percent in t he fange of Graetznumber from 10 to 10,000, except for the cases in which Pr = O. 1and 0.01 for GZ less than 200. The present work shows that, for auniform velocity profile and any Pr number, the average Nusselt


16/22

416 C H I N G L A I H WA N G A N ] 9 L I A N G T S E N G FA N

70 .

O

50

4O

z O

IC

0lO

f I I I I II II 1 1 I I I I III .. .. I

I I I I IL LI I I I i I l. ll lI i40 I0 ~ 4~I0 ~ 10~

~R @d rG z = ~ / Oe

1 I I I I4 10 ~ 10 4

Fig. 6. Variation. of average Nusselt numberNumwith Graetz number forf la t rectangular duct for constant wal l temperature ,Num is based on log-

mean temperature difference, defined by (18).

n u m b e r b a s e d o n t h e l o g - m e a n te m p e r a t u r e d i ff e re n c e is a s y m p t o t i c

t o 10 w h e n G z i s l e ss t h an 10. I t i s a l so sho w n tha t , f o rP r = 0.1a n d 0.0 1 , t h e c u r v e s f o r t h e a v e r a g e N u s s e l t n u m b e r ,Num, a r ea s y m p t o t i c t o 7 . 6 a n d b e c o m e a s y m p t o t i c a t v a l u e s o fG z w h i c ha re l es s t h a n t h o se p r e d i c t e d b y t h e f a ir e d lin e in S p a r r o w sw o r kS ) . H o w e v e r , S p a r r o w s r e su lt s a gr e e r e a s o n a b l y w e ll w i t ht h e r e s u l ts o f t h e p r e s e n t w o r k f o r o t h e r P r a n d t t n u m b e r s . W h e nt h e P f i s l es s t h a n 0 .1 , n e g l e c t i n g t h e h e a t c o n d u c t i o n i n t h e f l o wd i re c ti o n in th e p r e s e n t w o r k a n d i n S p a r r o w s w o u l d c a u s es o m e d e v i a t i o n i n t h e h e a t t r a n s f e r p a r a m e t e r f r o m t h e t r u e r e s u lt s.T h e r e f o r e , th e a d e q u a c y o f t h e p r e s e n t w o r k sh o u l d b e c o m p a r e dw i t h t h e r e s u l ts o f th e s o l u ti o n o f t h e e n e r g y e q u a t i o n w h i c h i n -c l u d e d t h e l o n g i t u d i n a l c o n d u c t i o n t e r m ,k(O20/~X2).I t m a y b ec o n c l u d ed t h a t t h o u g h S p a r r o w s a p p r o x i m a t e v e l o c i ty p ro fi le sd o n o t a p p r o a c h a s y m p t o t i c a l l y t o a p a r a b o li c s h ap e , a s s h o w n i nf ig . 3 , t h e y d o n o t g i v e a n a p p r e c i a b l e e r r o r in t h e h e a t - t r a n s f e rp a r a m e t e r s o b t a i n e d b y s o lv i n g t h e e n e r g y e q u a t i o n . T h e p r e s e n tw o r k a ls o c o n f ir m s t h e a d e q u a c y o f t h e f a i r e d c u r v e s d r a w n b yS p a r r o w f o rG z l e s s t han 200 .

T h e a g r e e m e n t o f t h e p r e s e n t r e su l ts w i t h th o s e o f S t e p h a n ~) is


17/22

H E AT T R A N S F E R I N E N T R A N C E R E G I O N O F A D U C T 4 7

e x c e l le n t , T h e a s y m p t o t i c a l v a l u e o f a v e r a g e N u s s e l t n u m b e r b yS t e p h a n 9) f o r a n y v e l o c i t y p r o f il e e x c e p t u n i f o r m o n e s i s 7 . 55 .

T h i s i s s li g h t l y l o w e r t h a n t h e e x p e c t e d v a l u e o f 7 . 6 w h i c h i s a ls oo b t a i n e d i n t h e p r e se n t w o r k . S t e p h a n ' s a s y m p t o t i c a l av e ra g eN u s s e l t n u m b e r f o r t h e u n i f o r m v e l o c i t y p r o f i l e i s 9 . 8 7 w h i c h i s1 .3 l o w e r t h a n t h e co r r e s p o n d i n g v a l u e o f 1 0 .0 o b t a i n e d in t h ep r e s e n t w o r k . S t e p h a n g a v e t h e I o l l o w i n g c o r r e l a t io n I o r m u l a f o rh i s r esu l t s .

O . 0 2 4 ( P r / X ) l . 14N ~ m = 7.55 ~- (31)

1 ~- O.0358(Pr )O.81 /(X )O .64

T h e m a x i m u m d e v i a t i o n o f th e p r e s e n t r e su l t s f r o m t h i s c o rr e la t io ni s 3 p e r c e n t . B e c a u s e o f i ts s i m p l i c i ty a n d a g r e e m e n t w i t h t h ep r e s e n t w o r k , (3 1) i s r e c o m m e n d e d f o r e n g i n e e r i n g u s e . Ta b l e I Vs h o w s t h e n u m e r i c a l c o m p a r i s o n b e t w e e n t h e r e s u l t s o b t a i n e d i n t h ep r es en t w o r k w i t h t ho 8 e b y S t e p h a n 9 ) .

TA B L E I V

N u m e r i e a l c o m p a r i s o n b e t w e e n t h e r e s u lt s o b t a in e d b y S t e p h a n 9 a n d t h o s e i nt h e p r e s e n t w o r k f o r t h e c a s e w i t h d e v e l o p i n g v e l o c i t y a n d t e m p e r a t u r e p r o fi le s

a n d w i t h c o n s t a n t w a l l t e m p e r a t u r e

P r 0.72P r e s en t W o r k

Gz

1.15 2 x 104

2.304 10 a1.0 47 10 a

9.6 10

5.236 10

2.304 10

1.152 10

6.583

Gz

X '

6.25 X 10 5

3.125 x 10 .4

6.8 75 10 -4

7.5 10-a

1.37 x 10 -2

3.1 25 10 -2

6.25 10 -2

1.094 X 10 -I

P r = 10

X

N~t m

S t e p h a n 9eq. 31)

78.37

35.71

24.61

10.23

9.083

8.235

7.885

7.735

N ~ m

S t e p h a n 9eq, 31)

72.87

35.09

24.91

10.42

9.124

8.225

7.865

7.707

P r e s e n t Wo r k

4.0 X 104

2.0 104

1.0 104

4.0 10 a

2.0 10 a1.0 103

5.0 IOs1.6 102

8.0 10

1.524 10

2.5 10 -4

5.0 10 -4

1.0 10 -a

2.5 10 -a

5.0 10 -3

1.0 10 -2

2.0 X 10 -2

6.25 10 -2

1.25 10 -1

6.56 10 -1

9 6 3 0

6 9 5 8

5 0 6 3

3 3 7 6

2 5 2 4

1 9 2 3

1 5 0 4

1 0 8 6

9.44

7.961

98.65

70.69

51.53

34.37

25.73

19.25

15.44

11.01

9.40

7.878


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418 C H I N G - L A I H WA N G A N D L I A N G - T S E N G FA N

Fig. 7 shows the variations of fluid and surface temperature fordifferent Prandtl numbers for the case of constant heat flux fromwall to fluid. As expected the surface temp erat ure a nd fluidternperature are increased rapidly for the smaller Prandtl numbersay Pr = 0.1 and ve ry slowly for larger Pr and tl numb er sayPr ~ 10. After a short dimensionless distance X, from the entrythe temperature difference between the surface of the wall and thefluid become s pract ical ly consta nt. Therefore this solution for thecase of constant heat flux can be applied for the case of constanttemperature difference between the wall and the fluid after a shortdistance from the entry. This distance can generally be estimated

from fig. 7.2.0 / / ~

Fig 7. Fluid ar~d surface temperature variation for various Prandtl numbersin the entrance region; for consant heat flux.

Fig. 8 shows the local Nusselt numbers Nux, with x/De)/Re asabscissa and the Prandtl number as a parameter. They are plottedon semi-log scale.

In the case of constant wall temperature Nusselt numbers areplotted against Graetz numbers. In this plot there is one curve forthe case of the parabolic velocity profile and an othe r curve for theuniform veloc ity profile and each of these two curves is applicablefor all values of Pr. However in the case of const ant heat fluxthere does not exist a single curve for all values of the Prandtlnumb er for the case of a parabolic velocity profile thro ugho ut noris there anoth er single curve for the case of uniform velocit y profile.Therefore the results of the con stan t heat flux case are plot ted as


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H E AT T R A N S F E R I N E N T R A N C E R E G I O N O F A D U C T 4 9

in fig. 8 , th at is, the local Nus sel t nu m ber is plo t ted against thed imens ion less d i s tance f rom the duc t en t ry.

eo

5o

. \ \ \ g o (o

20 t

l a - ~.a z

f l l l l l t l q l l l l t t t l l010-4 4 I0 3 4 LO z 4 I0 1 4 I~i

x/DeRed

F i g . 8 . Va r i a t i o n o f l o c a l N u s s e l t n u m b e r N u z w i t h x / D e ) / R e a f o r f l a tr e c t a n g u l a r d u c t w i t h c o n s t a n t h e a t f l u x . N u z i s b a s e d o n t h e t e m p e r a t u r e

d i f f e r e n c e tw -- th), w h e r e t w, t 0 a r e w a l l a n d b u l k t e m p e r a t u r e s r e s p e c t i v l y .

TA B L E V

T h e l o c a l N u s s e l t n u m b e r f o r t h e e a s e w i t h t h e c o n s t a n t h e a t I l u x

a n d f o r d e v e l o p i n g v e l o e i t y p r o f i l e

Nu

x/De

Re

5 x 10 -4

1.25 10 3

4 . 3 7 5 1 0 - a

7 .5 10 -a2 10 -2

6 ,25 x 10 -2

0,1

P r = 0 . 7 P r = 1

2 1 . 9 815 .11

1 0 . 0 3

8 . 9 0 l8 . 2 3 98 .221

8 . 2 2 1

2 4 . 3 41 6 . 6 2

10 .79

9 . 3 0 78 . 3 0 78 . 2 3 48 . 2 2 5

Pc = 10

5 0 . 7 43 4 . 0 7

2 0 . 6 6

1 7 . 0 31 2 . 5 5

9 .5018 . 5 3 9

As expec ted the loca l Nusse l t num ber a pproaches 8 .23 asym pto t ic -a l ly l~ . The smal ler the va lue of the Pr and t l num ber, the smal le ri s the va lue of X a t which the Nusse l t num ber becom es equa l tothe asympto t ic va lue of 8 .23 .

Table V presen ts some se lec ted numer ica l resu l t s ob ta ined in th i swerk .


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4 2 0 C H I N G L A I H WA N G A N D L I A N G T S E N G FA N

H a n ' s r es u lt s 1) d if fe r a p p r ec i a b ly f r o m S i e g e l a n d S p a r r o w ' sr e su lt s 6) w i th r e spec t t o t he ways the y approach the a sym pto t i cva lue in te rms of the loca l Nusse l t number. The resu l t s of thepresent w ork agree c lose ly wi th the la t te r.

S in ce S i e g e l a n d S p a r r o w 6 ) , i n t h e ir w o r k o f t h e c o n s t a nthea t f lux case , used the appr oxim ate ve loc i ty prof iles , the curvesfor wal l t empera ture var ia t ion have changed in s lope a t the endof the thermal en t rance reg ion in cont ras t to the resu l t s of thepresent w ork . How ever, the i r resu l t s agree reason ably wel l wi th thepresent resu l t s in te rms of local Nusse l t numb ers .

A c k n o w l e d g e m e n t . D r. W i l s o n T r i p p , P ro fe ss or of M e-

chan ical Eng inee r ing a t Kansa s S t a t e Un ive r s i t y, con t r ibu ted toth is work . This s tudy was par t ly suppor ted by the Air Force Off iceo f Sc ien ti fi c Resea rch Gran t A F-A FO SR -463 -64 .

A P P E N D I X

Fro m subs t i tu t io n of the d i ffe rence equat ions (12) in to the e nergyeq ua tio n (9), orte can ob tai n (32) in wh ich th e 0's wi th th e ~ + 1subsc r ip t a r e t he unknowns , and the O s with the 1' subscr ip t a r e

known ,IC/el 0]+l,/c-1 -~ [Ak] 0j+l,/c -~- [Bk] 0j+l,/c+l = [D/c], (32)

[ ]C-~ Pr 2 ( Ay ) 2 '

I U j , / C + U j + ~ , ~ 2Ak = - 2A X + Pr 2(AY) 2

[ ]B ~ = P r 2(AY)2 'BUj, k + U,+I ,~ 0 Vj ,~ + V~+I,k (O,,~+, - - O~, -1 )

D = L 2 dX ~ ~ - 2 2 AYl Oj./ c+ l 20j,/c -}- 0t,/c-1

+. Pr 2(AY) 2

Sub st i tu t ing k = 1, 2 , 3 . . . . , n into (32) w ith the bo un da ry con-di t ions given by (13) or (14) , n unknowns and n s imultaneousequa t ions a r e ob ta ined . In m a t r ix fo rm these equa t ions a r e

where


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22/22

422 H E A T T R A N S F E R I N E N T R A N C E R E G I O N O F A D U C T

T h e s e t r a n s f o r m 3 3) i n t o

0]+ 1 , n = G~z

03 '+1, r : G r Q r 0 j + l , r + l . r : 1 , 2 , . . , n - - 1 3 5 )

I f W , Q and G a r e ca l cu l a t ed i n o rde r o f i nc r ea s ing r, i t fo l l ows t h a t35 ) can be u sed t o ca l cu l a t e t heO ~ t , r i n o rde r o f dec rea s ing r,

t ha t i s , 0 3 1 , n , 0 ~ 1 , ~ - 1 , . . . , 0 3 1 , 2 , 0 ~ ~ ,

R e c e i v e d 1 7 t h J u n e , 1 96 3.

R E F E R E N C E S

1) R o h s e n o w , W. M . a n d H . C h o i , H e a t , M a ss , a n d M o m e n t u m T r a n s f e r , p p . 1 6 3 -1 67 , P r e n t i c e H a l l , I n c . , E n g l e w o o d C l i f f, N . J . , 1 9 61 .

2 ) K a y s , W . M . , Tr a n s . A S M E . 7 7 (1 95 5) 1 2 65 .3 ) G o d l b e r g , P. , M . S . T h e s i s , M e c h . E n g r. , M a s s a e h u s e t t s I n s t . o f Te e h . (1 9 58 ).4 ) L a n g h a a r , H . L . , Tr a n s . A S M E , 6 4 , A - 5 5 ( 19 4 2) .5 ) S p a r r o w , E . M . , N A S A Te e h . N o t e 3 3 31 ( 19 5 5) .6 ) S i e g e l , R . a n d E . M . S p a r r o w , A . I . C h . E . J o u r n a l 5 ( 19 59 ) 7 3 - 7 5 .7) M u r a k a w a , K . , B u l l. J a p . S e c . M e e h . E n g r s . 3 ( 19 60 ) 3 4 0.8 ) S l e z k i n , N . A . , J . o f A p p l . M a t h . a n d M e e h . 2 3 (1 95 9) 4 7 3 .9 ) S t e p h a n , K . , C h e m . I n g . Te c h . 3 1 (1 95 9) 7 7 3 - 7 7 8 .

10) H a n , L . S . , I n t e r n a t i o n a l D e v e l o p m e n t s i n H e a t T r a n s f e r , 1 961 I n t e r n a t i o n a lH e a t Tr a n s f e r C o n f e r e n c e , S e p t e m b e r ( 19 6 1) , C o l o ra d o , U S A , p p . 5 9 1- -5 9 7.

11) H w a n g , C . L . , P h . D . T h e s i s , K a n s a s S t a t e U n i v e r s i t y, J u l y ( 19 6 2) .12) N o r r i s , R . H . a n d P. P. S t r e i d , T r a n s . A S M E ,6 (1940) 525-533 .13) B o d o i a , J . R . a n d J . F. O s t e r l e , A p p l . S ei. R e s . , A 1 0 (1 96 1) 2 65 .1 4) H w a n g , C . L . a n d L . T. F a n , A p p I . S e i. R e s . , B 1 0 ( 19 6 3) 3 2 9 .15} L a p i d u s , L e o n , D i g i t a l C o m p u t a t i o n f o r C h e m i c aI E n g i n e e r s , p p . 2 5 4 -2 5 5 ,

M c G r a w - H i l l B o o k C o m p a n y, I n c ., N e w Yo r k , ( 19 62 ).1 6) R i c h t m y e r , R . D ., D i f f e r e n c e M e t h o d s f o r I n i t i a l Va l u e P r o b l e m s , p p . 9 1 - 1 0 1 ,

I n t e r s c i e n e e P u b l i s h e r s , N e w Yo r k , ( 19 5 7) .17) G r a n d y, R . A . , A F S W C T N - 6 1- 2 9 , P a r t I , 8 3 -9 5 , A F S W C S e e o n d H y d r o -

d y n a m i e C o n f e r en c e N u m e r i e a l M e t h o d s o f F l u i d F l o w P r o b l e m s , M a y ( 19 61 ).1 8) S c h l i e h t i n g , H . , Z . a n g e w. M a t h . M e c h . 1 4 (1 93 4) 3 68 .

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