UBC Theses and Dissertations

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UBC Theses and Dissertations

Combined free and forced convection through vertical non-circular ducts with and without peripheral wall… Khatry, Abdul Kader 1970

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COMBINED FREE AND FORCED CONVECTION THROUGH VERTICAL NON-CIRCULAR DUCTS WITH AND WITHOUT PERIPHERAL WALL CONDUCTION by ABDUL KADER KHATRY B.E.(Mech.), U n i v e r s i t y o f S i n d , Jamshoro, W. P a k i s t a n , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. i n the Department o f M e c h a n i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA September, 1970 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced d e g r e e at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa r tment The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada i i ABSTRACT A n a l y s i s o f combined f r e e and f o r c e d c o n v e c t i o n through v e r t i c a l n o n - c i r c u l a r d u c t s has been c a r r i e d o u t u s i n g v a r i a t i o n a l t e c h n i q u e . F u l l y d e v e l o p e d f l o w w i t h u n i f o r m a x i a l h e a t i n p u t i s assumed. A l l f l u i d p r o p e r t i e s a r e c o n s i d e r e d i n v a r i a n t w i t h t e m p e r a t u r e e x c e p t the v a r i a t i o n o f d e n s i t y i n the buoyancy term o f the e q u a t i o n o f mot i o n . A g e n e r a l s t u d y o f the problem has been made i n t h r e e s t a g e s : ( i ) F o r c e d c o n v e c t i o n w i t h o u t c i r c u m f e r e n t i a l w a l l c o n d u c t i o n . For t h i s c a s e , a known v e l o c i t y e x p r e s s i o n i s used and a p a r t i -c u l a r l y s i m p l e v a r i a t i o n a l e x p r e s s i o n has been p r e s e n t e d . N u s s e l t numbers a r e c a l c u l a t e d f o r r e c t a n g u l a r , rhombic, i s o s c e l e s t r i a n g u l a r , and r i g h t - a n g l e d t r i a n g u l a r d u c t s . R e s u l t s compared w i t h the a v a i l a b l e s o l u t i o n s have shown e x c e l l e n t agreement. ( i i ) Combined f r e e and f o r c e d c o n v e c t i o n w i t h o u t c i r c u m f e r e n t i a l w a l l c o n d u c t i o n . N u s s e l t numbers have been computed f o r r e c t a n g u l a r and rhombic d u c t s . A f i n i t e - d i f f e r e n c e p r o c e d u r e has a l s o been c a r r i e d o u t and the r e s u l t s a r e p r e s e n t e d f o r rhombic d u c t o n l y . In both o f the above cases ( i ) and ( i i ) , the c o n d i t i o n o f u n i f o r m i i i p e r i p h e r a l h e a t f l u x has been d i r e c t l y u t i l i z e d i n d e r i v i n g the v a r i a -t i o n a l e x p r e s s i o n , thus r e l e a s i n g the thermal boundary c o n d i t i o n f r o m s a t i s f y i n g e x a c t l y the c o n d i t i o n a t the w a l l . The c o n d i t i o n o f u n i f o r m c i r c u m f e r e n t i a l h e a t f l u x r e s u l t s i n lower v a l u e s o f N u s s e l t numbers as compared t o t h a t o f u n i f o r m c i r c u m f e r -e n t i a l w a l l t e m p e r a t u r e . T h i s d i f f e r e n c e i n N u s s e l t number v a l u e s de-c r e a s e s w i t h the i n c r e a s e i n R a y l e i g h number. A t h i g h e r v a l u e s o f R a y l e i g h number, both the c o n d i t i o n s tend t o produce about the same r e s u l t s . ( i i i ) C o n j u g a t e problem o f combined f r e e and f o r c e d c o n v e c t i o n when p e r i p h e r a l w a l l c o n d u c t i o n i s i n c l u d e d . The e q u a t i o n s c o u p l i n g h e a t c o n d u c t i o n i n the w a l l s w i t h the con-v e c t i o n i n s i d e the f l u i d a r e s o l v e d to e s t a b l i s h the i n f l u e n c e o f p e r i -p h e r a l w a l l c o n d u c t i o n . The problem has been s o l v e d i n a g e n e r a l i z e d way and the r e s u l t s have been p r e s e n t e d f o r r e c t a n g u l a r d u c t s . I t i s found t h a t l a r g e v a l u e s o f the f r e e c o n v e c t i o n e f f e c t s and/or o f the con-d u c t i o n parameter tend t o m i n i m i z e the asymmetries i n c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e . i v TABLE OF CONTENTS Page INTRODUCTION AND BRIEF REVIEW OF LITERATURE 1 FORMULATION OF THE PROBLEM AND ASSUMPTIONS 7 GENERAL VARIATIONAL FORMULATION AND SOLUTION 13 DISCUSSION OF RESULTS 22 F o r c e d C o n v e c t i o n 22 Combined Free and F o r c e d C o n v e c t i o n 27 C o n j u g a t e Heat T r a n s f e r 31 CONCLUSION 33 LITERATURE CITED 57 APPENDICES A. Development of Heat F l u x E x p r e s s i o n f o r C o n j u g a t e Heat T r a n s f e r 63 B. F i n i t e D i f f e r e n c e A p p r o x i m a t i o n 67 C. V a r i a t i o n a l S o l u t i o n f o r Laminar F o r c e d C o n v e c t i o n . . 74 D. V a r i a t i o n a l F o r m u l a t i o n f o r C o n j u g a t e Problem . . . . 79 V LIST OF TABLES T a b l e Page 1. F o r c e d C o n v e c t i o n N u s s e l t Numbers f o r R e c t a n g u l a r Ducts A g a i n s t A s p e c t R a t i o s from F o r m u l a t i o n (C5) 35 2. N u s s e l t Numbers f o r Combined F r e e and F o r c e d C o n v e c t i o n . Comparison o f the R e s u l t s from F o r m u l a t i o n (34) w i t h those o f Ref. [16] f o r the Square Duct 36 3. V a l u e s o f the P r e s s u r e Drop Parameters L A g a i n s t R a y l e i g h Numbers Ra f o r R e c t a n g u l a r Ducts o f A s p e c t R a t i o s 1 t o 10 and R e f e r e n c e P o i n t a t X = a, Y = b 37 4. N u s s e l t Numbers f o r Rhombic Duct. Comparison o f R e s u l t s from the V a r i a t i o n a l F o r m u l a t i o n (34) w i t h those from the F i n i t e - D i f f e r e n c e A p p r o x i m a t i o n , Appendix B 38 5. Comparison o f P r e s s u r e Drop Parameters f o r Rhombic Duct from the V a r i a t i o n a l F o r m u l a t i o n (34) w i t h those from the F i n i t e - D i f f e r e n c e A p p r o x i m a t i o n , Appendix B 39 6. V a l u e s o f the P r e s s u r e Drop Parameters L A g a i n s t R a y l e i g h Numbers Ra f o r Rhombic D u c t s . R e f e r e n c e P o i n t a t X = 0, Y = b 40 7. The C o n d u c t i o n Parameter K and R a y l e i g h Numbers Ra which G i v e V a l u e s o f N u s s e l t Numbers c l o s e t o those o f Han [2] 41 VI LIST OF FIGURES F i g u r e Page 1. Flow Through V e r t i c a l N o n - c i r c u l a r Duct 8 2. C o - o r d i n a t e System 20 3. F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t A s p e c t R a t i o f o r R e c t a n g u l a r Duct 42 4. F o r c e d C o n v e c t i o n . R e s u l t a n t Normal Heat F l u x on the Boundary A g a i n s t i t s P r e s c r i b e d V a l u e f o r R e c t a n g u l a r Duct 43 5. F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t Duct A n g l e a f o r Rhombic Duct 44 6. F o r c e d C o n v e c t i o n . R e s u l t a n t Normal Heat F l u x on the Boun-dar y A g a i n s t i t s P r e s c r i b e d V a l u e f o r Rhombic Duct . . 45 7. F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t Duct A n g l e a f o r I s o s c e l e s T r i a n g u l a r Duct 46 8. F o r c e d C o n v e c t i o n . R e s u l t a n t Normal Heat F l u x on the Boun-dar y A g a i n s t i t s P r e s c r i b e d V a l u e f o r I s o s c e l e s T r i a n g u -l a r Duct . 47 9. F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t Duct A n g l e a f o r R i g h t - a n g l e d T r i a n g u l a r Duct . . . . 48 10. Combined F r e e and F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t A s p e c t R a t i o f o r R e c t a n g u l a r Duct . . . 49 v i i F i g u r e Page 11. Combined F r e e and F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t Duct A n g l e a f o r Rhombic Duct . . . . 50 12. Comparison o f Temperature D i s t r i b u t i o n on the Boundary o f a Rhombic D u c t , a = 9 0 ° , f r o m the V a r i a t i o n a l F o r m u l a t i o n (34) w i t h F i n i t e D i f f e r e n c e A p p r o x i m a t i o n , Appendix B . 51 13. Comparison o f R e s u l t a n t Normal Heat F l u x f o r Rhombic D u c t , a = 9 0 ° , f r o m V a r i a t i o n a l F o r m u l a t i o n s (34) and(C5) A g a i n s t i t s P r e s c r i b e d V a l u e 52 14. Temperature D i f f e r e n c e <j> f o r D i f f e r e n t v a l u e s o f K A l o n g the Boundary o f a Square Duct a t Ra = 1000 53 15. N u s s e l t Number V a r i a t i o n s o f a Square Duct f o r D i f f e r e n t V a l u e s o f K A g a i n s t R a y l e i g h Numbers 54 16. N u s s e l t Number V a r i a t i o n s o f a R e c t a n g u l a r Duct w i t h A s p e c t R a t i o = 2 f o r D i f f e r e n t V a l u e s o f K A g a i n s t R a y l e i g h Numbers 55 17. N u s s e l t Number V a r i a t i o n s o f a R e c t a n g u l a r Duct w i t h A s p e c t R a t i o = 3 f o r D i f f e r e n t V a l u e s o f K A g a i n s t R a y l e i g h Numbers , 56 18. C o - o r d i n a t e System and A n a l y t i c a l Model 65 19. R e c t a n g u l a r G r i d System 69 ACKNOWLEDGEMENTS The a u t h o r wishes t o e x p r e s s h i s s i n c e r e a p p r e c i a t i o n f o r the a d v i c e and encouragement o f Dr. M. Iqbal whose con-s t a n t i n t e r e s t and g u i d a n c e f a c i l i t a t e d the e x e c u t i o n o f the p r e s e n t i n v e s t i g a t i o n . Acknowledgements a r e a l s o due t o Dr. B. D. Aggarwala o f the Mathematics Department, U n i v e r s i t y o f C a l g a r y , f o r h e l p f u l d i s c u s s i o n s . Use o f the Computing C e n t r e f a c i l i t i e s a t The U n i v e r -s i t y o f B r i t i s h Columbia and the f i n a n c i a l s u p p o r t o f the N a t i o n a l R e search C o u n c i l o f Canada a r e g r a t e f u l l y acknow-l e d g e d . i x NOMENCLATURE A = a r e a o f c r o s s - s e c t i o n Cp = s p e c i f i c h e a t o f the f l u i d a t c o n s t a n t p r e s s u r e C-| = 3T/3z t e m p e r a t u r e g r a d i e n t i n f l o w d i r e c t i o n C 2 - 0.25 (1-F) = h y d r a u l i c d i a m e t e r = (4«cross-sectional a r e a ) / ( h e a t t r a n s f e r p e r i m e t e r ) F = Q/pCpC-jU, h e a t g e n e r a t i o n p a r a m e t e r , d i m e n s i o n l e s s g = g r a v i t a t i o n a l a c c e l e r a t i o n K = viK^/d^Kf, c o n d u c t i o n p a r a m e t e r , d i m e n s i o n l e s s L = p r e s s u r e drop p a r a m e t e r , d i m e n s i o n l e s s = hD^/Kf, N u s s e l t number, d i m e n s i o n l e s s 2 4 (p gC C-|BD^J/K^y, R a y l e i g h number, d i m e n s i o n l e s s Nu Ra _ P 0 = heat g e n e r a t i o n r a t e = heat g e n e r a t i o n r a t e i n w a l l per u n i t a r e a S = c i r c u m f e r e n c e o f d u c t T = temperature u = a x i a l v e l o c i t y U = ave r a g e a x i a l v e l o c i t y V = u/U, d i m e n s i o n l e s s a x i a l v e l o c i t y w = w a l l t h i c k n e s s z = a x i a l c o o r d i n a t e i n f l o w d i r e c t i o n p U C D C l D h ( T - T r e f . ) / [ - ] , d i m e n s i o n l e s s t e m p e r a t u r e K f f u n c t i o n a r e the f l u i d p r o p e r t i e s i n s t a n d a r d n o t a t i o n f l u i d thermal c o n d u c t i v i t y w a l l thermal c o n d u c t i v i t y x i COMBINED FREE AND FORCED CONVECTION THROUGH VERTICAL NON-CIRCULAR DUCTS WITH AND WITHOUT PERIPHERAL WALL CONDUCTION I INTRODUCTION AND BRIEF REVIEW OF LITERATURE The s u b j e c t o f h e a t t r a n s f e r i n f l o w through n o n - c i r c u l a r d u c t s has r e c e n t l y drawn i n c r e a s e d a t t e n t i o n . Ducts o f n o n - c i r c u l a r s h a p e s , which l a c k the r o t a t i o n a l symmetry p r e v a i l i n g i n c i r c u l a r ones, f r e q u e n t -l y c o n t a i n pronounced c o r n e r s . A t c o r n e r r e g i o n s a d j a c e n t t o the apex o f a n o n - c i r c u l a r d u c t , poor h e a t c o n v e c t i o n c o n d i t i o n s e x i s t i n the f l u i d due t o n e i g h b o u r i n g low v e l o c i t y r e g i o n s . High t e m p e r a t u r e s can t h e r e f o r e be e x p e c t e d t o o c c u r when heat f l u x i n t o the f l u i d o f such r e g i o n s f r o m t he s u r r o u n d i n g w a l l s i s h i g h . In e n g i n e e r i n g p r a c t i c e , p a r t i c u l a r l y when l e v e l s o f t e m p e r a t u r e and h e a t f l u x a r e h i g h , i t becomes i m p o r t a n t to i n v e s t i g a t e the hot s p o t s and t h e i r i n f l u e n c e on the o v e r a l l h e a t t r a n s f e r b e h a v i o u r o f a n o n - c i r c u l a r d u c t . F or such an i n v e s t i g a t i o n , one needs a model amen-a b l e t o m a t h e m a t i c a l a n a l y s i s . T h i s model may be chosen i n such a way t h a t the h e a t f l u x between the d u c t w a l l s and the f l u i d i s p e r i p h e r a l l y c o n s t a n t . S i n c e i t i s known t h a t r o t a t i o n a l symmetry i n a n o n - c i r c u l a r d u c t i s l a c k i n g both f o r v e l o c i t y and te m p e r a t u r e f i e l d s , h e a t conduc-t i o n a l o n g the p e r i p h e r y i n s i d e the d u c t w a l l s can f r e q u e n t l y e x i s t . The above model i s t h e r e f o r e a l i m i t i n g c a s e which o c c u r s when the p e r i p h e r a l w a l l h e a t c o n d u c t i o n becomes z e r o , w i t h v a n i s h i n g h e a t con-d u c t a n c e o f the w a l l s and when a t the same time the h e a t g e n e r a t i o n i n s i d e the w a l l s i s d i s t r i b u t e d e v e n l y around the p e r i p h e r y . The boun-dary v a l u e problems f a l l i n g i n t h i s c a t e g o r y a r e g e n e r a l l y s a i d t o have Neumann c o n d i t i o n s . The second model i s w i t h h i g h enough h e a t c o n d u c t a n c e o f the w a l l such t h a t the p e r i p h e r a l w a l l h e a t c o n d u c t i o n i s h i g h and t h a t any non-u n i f o r m i t y i n w a l l t e mperatures i s r e a d i l y s u p p r e s s e d , g i v i n g r i s e t o c o n s t a n t w a l l t e m p e r a t u r e around the p e r i p h e r y o f the d u c t . The boundary v a l u e problems f a l l i n g i n t h i s c a t e g o r y a r e g e n e r a l l y s a i d t o have D i r i c h l e t c o n d i t i o n s . These two models d i s c u s s e d above can be c o n s i d e r e d t o r e p r e s e n t the two l i m i t i n g s i t u a t i o n s o f n o n - c i r c u l a r d u c t h e a t t r a n s f e r . In p r a c -t i c a l a p p l i c a t i o n s , where both w a l l t e m p e r a t u r e and w a l l h e a t f l u x d i s -t r i b u t i o n s around t he d u c t p e r i p h e r y a r e not c o n s t a n t , the o v e r a l l h e a t t r a n s f e r c h a r a c t e r i s t i c s w i l l l i e between t h o s e f o r the two l i m i t i n g c a s e s . The e x t e n t o f t h i s r o t a t i o n a l asymmetry o f t e m p e r a t u r e and h e a t f l u x would depend upon the d u c t c o n f i g u r a t i o n , the w a l l thermal conduc-t i v i t y and i t s t h i c k n e s s , and the thermal c o n d u c t i v i t y o f the f l u i d . I t i s i m p o r t a n t then t o u n d e r s t a n d the d i f f e r e n c e between the o v e r a l l h e a t t r a n s f e r b e h a v i o u r o f t h e s e two l i m i t i n g cases and the e f f e c t o f p e r i p h e r -a l w a l l c o n d u c t i o n upon them. In many l a m i n a r h e a t t r a n s f e r p r o c e s s e s , i t i s known t h a t the e f f e c t s o f buoyancy c r e a t e d by the g r a v i t a t i o n a l f o r c e f i e l d and d e n s i t y d i f f e r e n c e s c a n n o t be i g n o r e d . In a i d i n g f l o w s (when the body f o r c e e f f e c t and f l u i d f l o w a r e i n the same d i r e c t i o n ) t h e h e a t t r a n s f e r r a t e s 3 can i n c r e a s e by as much as t h r e e t i m e s . I t i s known t h a t f o r a i d i n g f l o w i n v e r t i c a l t u b e s , the e f f e c t o f f r e e c o n v e c t i o n i s t o i n c r e a s e t h e v e l o c i t y and te m p e r a t u r e g r a d i e n t s a t the w a l l . A t the c o r n e r r e g i o n s o f n o n - c i r c u l a r t u b e s , where i n g e n e r a l the f l u i d slows down, the e f f e c t o f buoyancy i s t o a c c e l e r a t e the f l o w and r a t e o f heat t r a n s f e r . I t , t h e r e f o r e , appears t h a t f o r l a m i n a r f l o w t hrough v e r t i c a l n o n - c i r c u l a r d u c t s , i n a d d i t i o n t o w a l l thermal c o n d u c t i v i t y , the buoyancy r a t e w i l l a l s o i n f l u e n c e the r o t a t i o n a l asymmetries o f te m p e r a t u r e and h e a t f l u x . We w i l l now p r e s e n t a b r i e f l i t e r a t u r e s u r v e y i n the a r e a . The n o n - c i r c u l a r d u c t l i t e r a t u r e on the t o p i c o f f o r c e d c o n v e c t i o n as w e l l as combined f r e e and f o r c e d c o n v e c t i o n f o r the c o n d i t i o n o f u n i f o r m c i r c u m f e r e n t i a l h e a t f l u x i s r a t h e r l i m i t e d . Major a t t e n t i o n has been d i r e c t e d to the case o f u n i f o r m c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e . A b r i e f l i t e r a t u r e c i t e d on combined f r e e and f o r c e d c o n v e c t i o n w i t h u n i f o r m p e r i p h e r a l w a l l t e m p e r a t u r e i s as f o l l o w s : Hallman [1] a n a l y t i c a l l y i n v e s t i g a t e d t h e problem o f l a m i n a r f l o w through a c i r c u l a r p i p e w i t h u n i f o r m h e a t i n g o r c o o l i n g a t the w a l l . The problem i s s o l v e d i n terms o f B e s s e l f u n c t i o n s . An e x p e r i m e n t a l i n v e s t i -g a t i o n was a l s o c a r r i e d o u t t o v e r i f y t h e t h e o r e t i c a l a n a l y s i s . Han [ 2], u s i n g d o u b l e F o u r i e r s e r i e s , s o l v e d the probl e m o f Numbers i n s q u a r e b r a c k e t s i n d i c a t e r e f e r e n c e s a t the end o f the t h e s i s . 4 l a m i n a r f l o w through v e r t i c a l r e c t a n g u l a r c h a n n e l s . A complex f u n c t i o n was u t i l i z e d by Tao [3] t o d i r e c t l y r e l a t e the v e l o c i t y and t e m p e r a t u r e p r o f i l e s . From the a n a l o g y o f membrane v i b r a t i o n e q u a t i o n s , Aggarwala and Iqbal [4] have p r e s e n t e d e x a c t s o l u t i o n s o f the problem f o r a s e t o f s t r a i g h t v e r t i c a l t r i a n g u l a r d u c t s w h i l e Lu [5] has i n d i c a t e d t h a t the problem o f combined f r e e and f o r c e d c o n v e c t i o n through v e r t i c a l r e c -t a n g u l a r tubes under u n i f o r m a x i a l t e m p e r a t u r e g r a d i e n t s can a l s o be s o l v e d by a n a l o g y t o p l a t e t h e o r y . A v a r i a t i o n a l f o r m u l a t i o n f o r a r b i -t r a r y shaped d u c t s i s c o n t a i n e d i n [ 6 ] . Some s t u d i e s o f f o r c e d c o n v e c t i o n w i t h c o n s t a n t p e r i p h e r a l h e a t f l u x have been r e p o r t e d i n the l i t e r a t u r e . H. M. Cheng [ 7 ] , u s i n g s e p a r a t i o n o f v a r i a b l e s , o b t a i n e d a s o l u -t i o n f o r l a m i n a r f l o w i n a r e c t a n g u l a r channel t h a t was u n i f o r m l y h e a t e d on a l l f o u r s i d e s and e v a l u a t e d the w a l l t e m p e r a t u r e s f o r a s p e c t r a t i o s o f 1, 2 and 4. An e x a c t s o l u t i o n f o r c i r c u l a r s e c t o r was o b t a i n e d by Yen [8] and E c k e r t e t a l . [ 9 ] . They [9] found t h a t the a v e r a g 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 may be d i f f e r e n t by an o r d e r o f magnitude f o r the two l i m i t i n g c a s e s o f boundary c o n d i t i o n s . A c a r d i o d d u c t was t r e a t e d by Tao [10] who p r e s e n t e d an e x a c t s o l -u t i o n f o r a r b i t r a r y shaped d u c t s by the method o f c o n f o r m a l mapping. Sparrow and H a j i S h e i k h [11] have p r e s e n t e d a somewhat s i m p l e r c l o s e d form s o l u t i o n w h i c h , however, r e q u i r e s c o n s t r u c t i o n o f o r t h o n o r m a l f u n c t i o n s . 5 Sparrow and S i e g e ! [ 1 2 ] , w h i l e d e s c r i b i n g a p p l i c a t i o n o f v a r i a -t i o n a l method f o r d u c t f l o w h e a t t r a n s f e r have a l s o p r e s e n t e d a f o r m u l a -t i o n f o r the c o n d i t i o n o f u n i f o r m c i r c u m f e r e n t i a l h e a t f l u x . T h i s f o r m u l a t i o n , however, r e q u i r e s d e v e l o p i n g 'a p r i o r i 1 a t e m p e r a t u r e f u n c t i o n t o s a t i s f y the boundary c o n d i t i o n e x a c t l y . While they have s u c -c e s s f u l l y d e v e l o p e d such a f u n c t i o n f o r a r e c t a n g u l a r d u c t , i t w i l l be, however, v e r y d i f f i c u l t t o use t h e i r f o r m u l a t i o n f o r many o t h e r d u c t shapes o f e n g i n e e r i n g i n t e r e s t . The a u t h o r s o f r e f e r e n c e [12] have a l s o o b t a i n e d a s o l u t i o n w i t h u n i f o r m h e a t i n g on o n l y the two broad w a l l s f o r an a s p e c t r a t i o o f 10. S o l u t i o n s f o r r e g u l a r p o l y g o n a l d u c t s a r e g i v e n by K. C. Cheng [ 1 3 ] . S a v i n o and S i e g e l [14] p r o v i d e d the a n a l y t i c a l s o l u t i o n f o r the c a s e i n which the s i d e s o f the r e c t a n g u l a r channel a r e u n i f o r m l y h e a t e d , but the h e a t f l u x .on the s h o r t s i d e s i s an a r b i t r a r y f r a c t i o n between 0 and 1 o f the f l u x on the broad s i d e s . In r e f e r e n c e [ 1 5 ] , a s i t u a t i o n was a n a l y s e d w h e r e i n the h e a t i n g , which c o u l d be u n i f o r m o r n o n - u n i f o r m , o c c u r r e d o n l y on the broad w a l l s b u t was removed v a r i o u s s h o r t d i s t a n c e s from the s i d e w a l l s . The c o n d i t i o n o f u n i f o r m c i r c u m f e r e n t i a l h e a t f l u x f o r the prob-lem o f combined f r e e and f o r c e d c o n v e c t i o n through v e r t i c a l n o n - c i r c u l a r d u c t s makes the s o l u t i o n o f the problem v e r y d i f f i c u l t . The o n l y i n f o r -mation a v a i l a b l e i n p u b l i s h e d l i t e r a t u r e i n t h i s c a s e i s by Iqbal e t a l . [16] who p r e s e n t e d a p o i n t matching s o l u t i o n o f the problem f o r r e g u l a r p o l y g o n a l ducts.; 6 In n o n - c i r c u l a r d u c t s , when the w a l l o f the channel has a non-z e r o f i n i t e thermal c o n d u c t i v i t y and t h i c k n e s s , the a n a l y s i s o f h e a t t r a n s f e r becomes v e r y complex. In t h i s s i t u a t i o n , n e i t h e r the tempera-t u r e nor the t e m p e r a t u r e g r a d i e n t s a t the w a l l s can be p r e s c r i b e d , and as such i t i s c a l l e d t he c o n j u g a t e problem, [17, 18, 1 9 ] . An e x a c t s o l u t i o n o f f u l l y d e v e l o p e d l a m i n a r f o r c e d c o n v e c t i o n through r e c t a n g u l a r c h a n n e l s w i t h h e a t g e n e r a t i o n i n the c o n d u c t i n g broad w a l l s , and unheated n o n - c o n d u c t i n g s h o r t w a l l s has been p r e s e n t e d by S i e g e l and S a v i n o [20]. They o b t a i n e d t h e s o l u t i o n s f o r the t e m p e r a t u r e d i s t r i b u t i o n s a l o n g the channel w a l l s f o r s e v e r a l w a l l h e a t i n g d i s t r i b u -t i o n s . An e x a c t t r e a t m e n t would become e x t r e m e l y c o m p l i c a t e d i f the c o n j u g a t e c o n d i t i o n s were imposed on a l l f o u r s i d e s . A d d i t i o n a l i n c l u s i o n o f buoyancy e f f e c t s when the v e l o c i t y f i e l d g e t s c o u p l e d w i t h the temper-a t u r e f i e l d would f u r t h e r c o m p l i c a t e the a n a l y s i s . The aim o f the p r e s e n t s t u d y i s ( i ) t o d e v e l o p a v a r i a t i o n a l f o r m u l a t i o n f o r the problem o f combined f r e e and f o r c e d c o n v e c t i o n which d i r e c t l y t a k e s i n t o a c c o u n t the l i m i t i n g s i t u a t i o n o f c o n s t a n t p e r i p h e r a l h e a t f l u x boundary c o n d i t i o n , and ( i i ) t o extend t h i s f o r m u l a t i o n t o i n c l u d e the e f f e c t s o f p e r i p h e r a l w a l l c o n d u c t i o n . In a d d i t i o n , ( i i i ) a p a r t i c u l a r l y s i m p l e v a r i a t i o n a l f o r m u l a t i o n f o r pure f o r c e d c o n v e c t i o n w i l l be p r e s e n t e d . A g e n e r a l f o r m u l a t i o n o f the problem i s p r e s e n t e d i n the next s e c t i o n . II FORMULATION OF THE PROBLEM AND ASSUMPTIONS C o n s i d e r f u l l y d e v e l o p e d l a m i n a r f l o w through a s t r a i g h t v e r t i c a l d u c t o f a r b i t r a r y c r o s s - s e c t i o n , F i g u r e 1, o f u n i f o r m w a l l t h i c k n e s s w and thermal c o n d u c t i v i t y K . The f l o w i s i n the v e r t i c a l upwards d i r -e c t i o n a l o n g the p o s i t i v e z - a x i s , p a r a l l e l t o the f o r c e o f g r a v i t y . The d u c t r e c e i v e s u n i f o r m h e a t i n p u t per u n i t l e n g t h i n the f l o w d i r e c t i o n . T h i s energy may be c o n s i d e r e d as through u n i f o r m l y d i s t r i b u t e d h e a t s o u r c e s i n the w a l l . A l l f l u i d p r o p e r t i e s a r e assumed c o n s t a n t e x c e p t f o r the v a r i a t i o n o f d e n s i t y i n the buoyancy term o f the momentum e q u a t i o n . V i s c o u s d i s s i p a t i o n and p r e s s u r e work terms i n the energy e q u a t i o n a r e n e g l e c t e d . The f l u i d may c o n t a i n u n i f o r m volume h e a t s o u r c e s . Under the above i d e a l i z a t i o n s , the g o v e r n i n g e q u a t i o n s o f motion and energy can be w r i t t e n a s , (1) u (2) F i g u r e 1. Flow Through V e r t i c a l N o n - c i r c u l a r Duct. 9 A l l terms o f the d i f f e r e n t i a l f o r m o f the c o n t i n u i t y e q u a t i o n a r e i d e n t i c a l l y z e r o . For s m a l l v a r i a t i o n o f d e n s i t y , the e q u a t i o n o f s t a t e i n the l i n ^ e a r form can be w r i t t e n a s , P = Po I 1 - j3 ( T - T o ) ] > (4) where s u b s c r i p t o r e f e r s t o the c o n d i t i o n o f f l u i d a t some r e f e r e n c e p o i n t a t the w a l l . For u n i f o r m h e a t f l u x i n the f l o w d i r e c t i o n , the w a l l t e m p e r a t u r e i s g i v e n by, T w a i i = T o t z -§Y » where To i s a g a i n the r e f e r e n c e t e m p e r a t u r e a t z = 0 and = C 1 , Q b e i n g a c o n s t a n t t e m p e r a t u r e g r a d i e n t i n the f l o w d i r e c t i o n . N o n - d i m e n s i o n a l i z i n g e q u a t i o n s (1) and ( 2 ) by the f o l l o w i n g non-d i m e n s i o n a l i z i n g p a r a m e t e r s : X =• -^r- > Y = ' d i m e n s i o n l e s s c o - o r d i n a t e s , (5a) V = -jj- ' d i m e n s i o n l e s s v e l o c i t y , (5b) <±> — TL—Trei — , d i m e n s i o n l e s s t e m p e r a t u r e , (5c) P CpUC, Df/kf we g e t the g o v e r n i n g momentum and energy e q u a t i o n s i n d i m e n s i o n l e s s 10 form a s , 2 V V + Ra $ - L (6) V 4> - V = - F . (7) In ( 6 ) , L i s the d i m e n s i o n l e s s p r e s s u r e drop parameter and i s g i v e n by, JUL U In the above e q u a t i o n s (6) and ( 7 ) , R a y l e i g h number Ra and the heat g e n e r a t i o n parameter F a r e p r e s c r i b e d q u a n t i t i e s , V and <j> a r e depen-dent v a r i a b l e s w h i l e L i s an unknown c o n s t a n t . C o n t i n u i t y e q u a t i o n i n n o n - d i m e n s i o n a l form can be w r i t t e n a s , jj V dA = . / / o / A . (8) In o r d e r t o s o l v e f o r V, cf>, and L, we need one more e q u a t i o n which i s p r o v i d e d by e v a l u a t i n g the buoyancy f o r c e w i t h r e f e r e n c e t o the tem-p e r a t u r e a t a p o i n t on the w a l l where the temperature d i f f e r e n c e i s con^ s i d e r e d z e r o , t h a t i s , <p — 0 on a p r e s c r i b e d p o i n t a t the w a l l . (9) The c o n d i t i o n o f no s l i p a t the w a l l g i v e s , V = O a t the w a l l . (10) In the p r e s e n t s t u d y , e q u a t i o n s (6) t o (8) w i t h c o n d i t i o n s (9) and 11 (10) have to be s o l v e d f o r the f o l l o w i n g two s e t s o f thermal boundary c o n d i t i o n s : Case I U n i f o r m c i r c u m f e r e n t i a l h e a t f l u x (when c i r c u m f e r e n t i a l conduc-t i o n i n the w a l l i s z e r o ) . The thermal c o n d i t i o n o f c o n s t a n t h e a t f l u x on the s u r f a c e r e -s u l t s i n the d i m e n s i o n l e s s f o r m a s , = C 2 = 0-25 ( 1 - F ) a c o n s t a n t a t the w a l l . (11) C o n t i n u i t y c o n s i d e r a t i o n s , e q u a t i o n ( 8 ) , have been employed i n d e r i v i n g e q u a t i o n ( 1 1 ) . E q u a t i o n (8) i s r e d u n d a n t i n t h i s sense and can-not be u t i l i z e d as an a d d i t i o n a l e q u a t i o n . However, e q u a t i o n (8) can be used i n v e r i f y i n g the a c c u r a c y o f the v e l o c i t y e x p r e s s i o n , once a s o l u -t i o n o f i t has been o b t a i n e d . The v a l u e o f C^ = 0.25(1-F) has been o b t a i n e d by making a s i m p l e energy b a l a n c e a c r o s s two s e c t i o n s o f the d u c t . Case II C o n j u g a t e problem (when c i r c u m f e r e n t i a l c o n d u c t i o n i n the w a l l has n o n - z e r o f i n i t e v a l u e ) . - O-25(I-F; + K i ^ . a t t h e w a 1 1 * ( ] 2 ) * The thermal boundary c o n d i t i o n (12) has been d e v e l o p e d from the thermal energy b a l a n c e o v e r a s m a l l e l e m e n t o f the w a l l and the d e t a i l s a re c o n t a i n e d i n Appendix A. In e q u a t i o n s ( 9 - 1 2 ) , by the term " a t the w a l l " i t i s meant i n n e r s i d e o f the wal1. 12 Once the s o l u t i o n o f e q u a t i o n s (6) and (7) i s o b t a i n e d , f u n c t i o n s V and cf> w i l l be known and N u s s e l t number can be e v a l u a t e d . N u s s e l t num-ber based on the d i f f e r e n c e between t h e a v e r a g e w a l l t e m p e r a t u r e and b u l k t e m p e r a t u r e can be w r i t t e n a s , 1- F N u s s e l t number = — j = — — - — > (13) where, . _ ff<f> VC/A I'm x J/VctA i s t h e b u l k t e m p e r a t u r e o f the f l u i d . A g e n e r a l v a r i a t i o n a l f o r m u l a t i o n and s o l u t i o n o f e q u a t i o n s (6) and (7) w i t h thermal boundary c o n d i t i o n ( T l ) i s g i v e n i n t h e next s e c t i o n . I l l GENERAL VARIATIONAL FORMULATION AND SOLUTION We w i l l here d e v e l o p a v a r i a t i o n a l e x p r e s s i o n f o r the s o l u t i o n o f e q u a t i o n (6) and (7) f o r the thermal boundary c o n d i t i o n under cas e I ; the u n i f o r m p e r i p h e r a l h e a t f l u x , w h i l e a f i n i t e d i f f e r e n c e p r o c e d u r e i s g i v e n i n Appendix B. C o n s i d e r the e x p r e s s i o n where X and Y a r e i n -dependent v a r i a b l e s , V(X,Y) and 4>(X,Y) a r e the v e l o c i t y and tem-p e r a t u r e f u n c t i o n s t o be d e t e r m i n e d and V , <j> , X X Vy and <|>y a r e the p a r -t i a l d e r i v a t i v e s w i t h r e s p e c t t o X and Y r e s p e c t i v e l y . Y b (14) F u n c t i o n f and the i n t e g r a t i o n domain R a r e known from the s t a t e -ment o f the problem. S i s the boundary o f the d u c t , as shown i n the ad-j o i n i n g diagram. 14 In the neighbourhood o f V and <j>, V and 4> can be r e p l a c e d by V(X,Y) + e-jV(X,Y) and cp(X,Y) + e 2^(X,Y) r e s p e c t i v e l y , where V and * a r e c o n t i n -u o u s l y d i f f e r e n t i a b l e f u n c t i o n s o f V and $ and e-j and a r e parameters ( s m a l l r e a l c o n s t a n t s ) . Then i n terms o f t h e s e f u n c t i o n s , the i n t e g r a l o f e q u a t i o n (14) can be wr i t t e n a s , R + V y + €,VT , 4>y+*2.<£Y) d x d y - |&(^ ,Y>43 + ^ ) ^ s . (15) s For s t a t i o n a r y v a l u e o f I =«.=<> = Qk\,-u-o = °- <16> T h i s g i v e s the n e c e s s a r y c o n d i t i o n s : _ 2 -(31} _ M.) - n (17) & - ! * ( & > - fc<l& = ° , n R - ( , 8 ) where n i s a u n i t v e c t o r i n the d i r e c -t i o n o f the normal , to the boundary, as shown i n the a d j o i n i n g d i a gram. We a r e i n t e r e s t e d i n s o l v i n g the e q u a t i o n s (6) and ( 7 ) , which can be wri t t e n a s , fed*) + l y ( ^ ) + *-4> + L = O » (20) I x ( ^ ) + IY(^) " V + F = 0 , (21) or ^ + R o-^C^) - R<xV + RcxF = O- (21a) Comparing e q u a t i o n (20) w i t h e q u a t i o n ( 1 7 ) , we g e t | i = - R c ^ c b - L , (22) a n d ^1 = _ v x • M. = ^ = Vv . (23) T h e r e f o r e , j = -Ra.4>v - L V + ^ - V x + - i - V * + H C X J Y , * , ^ ^ ) ' (24)' where H ( X S Y , < } > ) i s the " c o n s t a n t " o f i n t e g r a t i o n , x y From e q u a t i o n (24) % =• ~ R a V + ^ ' (25) Comparing e q u a t i o n (21a) w i t h e q u a t i o n ( 1 8 ) , we g e t | l = - R a . V + R c F = - R c c V -t by e q u a t i o n (25) (27) by e q u a t i o n (26) > 16 (28) and by e q u a t i o n ( 2 6 ) , T h e r e f o r e , H where J(X,Y ) i s the " c o n s t a n t " o f i n t e g r a t i o n . (29) Rcc F cj> - R e <$1 - RQ . ^ £ + J C x 7 Y ) , (30) Now we can w r i t e , ^ R j - - ^ ( V x + V y ) - ^(<fc? + <^) - R ^ V + RcLCbF - L V + JCX,Y) . (31) The p r e s e n c e o f J(X,Y) i n (30) o n l y adds a " c o n s t a n t " t o I and has no e f f e c t on i t s b e i n g s t a t i o n a r y under the c o n d i t i o n (16). S u b s t i t u t i n g the v a l u e s o f a n d ~d4>y from e q u a t i o n s (28) and (29) i n t o e q u a t i o n (19), we g e t 0 on S, (32) o r - RQ . 0. T h e r e f o r e , G = - R a . C 2 4> (32a) (33) Hence the f u n c t i o n s f and G are known, and we can w r i t e , - 2 L v | C/KOIY + 2. Ro. C z | 4> d s . (34) A d m i s s i b l e f u n c t i o n s V and tj> which make I s t a t i o n a r y w i l l g i v e 17 the s o l u t i o n to our boundary v a l u e problem. A c c o r d i n g t o R i t z method, the t r i a l f u n c t i o n s chosen s h o u l d s a t i s f y the boundary c o n d i t i o n s , o t h e r w i s e the c h o i c e o f the f u n c t i o n s i s , t o a l a r g e e x t e n t , a r b i t r a r y . The a p p r o x i m a t e s o l u t i o n becomes c l o s e r t o the e x a c t s o l u t i o n when the number o f a p p r o x i m a t i o n s i s i n c r e a s e d . A g e n e r a l i z e d v a r i a t i o n a l f o r m u l a t i o n w i t h o u t i n c l u s i o n o f the f u n c t i o n G i s p r e s e n t e d i n [ 6 ] , S i n c e i t seems to be e x t r e m e l y d i f f i c u l t t o d e v e l o p 'a p r i o r i 1 a t e m p e r a t u r e f u n c t i o n whose normal d e r i v a t i v e i s a c o n s t a n t on the boundary o f a r b i t r a r y shaped d u c t s , an a d d i t i o n a l f u n c t i o n G i s i n t r o d u c e d t o t a k e c a r e o f t h i s c o n d i t i o n . T h i s f u n c t i o n G i s e v a l u a t e d , e q u a t i o n ( 3 3 ) , u s i n g a n e c e s s a r y c o n d i t i o n on the p e r i -phery o f the d u c t , e q u a t i o n (19) and the boundary c o n d i t i o n ( 1 1 ) . T h e r e -f o r e , the t r i a l f u n c t i o n <j> does not n e c e s s a r i l y have to s a t i s f y e x a c t l y the c o n d i t i o n ^ ^ = : C 2 a t the boundary, but i t must not v a n i s h a t the boundary as w e l l . F u n c t i o n V has t o be chosen i n such a way t h a t i t v a n i s h e s a t the boundary i n o r d e r t o s a t i s f y e q u a t i o n ( 1 0 ) . A p a r t i c u l a r l y s i m p l e v a r i a t i o n a l f o r m u l a t i o n f o r pure f o r c e d c on-v e c t i o n and s o l u t i o n s f o r r e c t a n g u l a r , rhombic, i s o s c e l e s t r i a n g u l a r and r i g h t - a n g l e d t r i a n g u l a r d u c t s i s c o n t a i n e d i n Appendix C, w h i l e a g e n e r a l v a r i a t i o n a l e x p r e s s i o n f o r c o n j u g a t e problem, when the e f f e c t s o f p e r i p h e r a l w a l l c o n d u c t i o n a r e a l s o i n c l u d e d , i s p r e s e n t e d i n Appendix D. The g e n e r a l forms o f V and <j> can be w r i t t e n a s , V = fb ( A . + A,X + A z Y + A 3 x Z + A 4 X Y + A 5 Y 2 + - •• • n t e r m s ) > ( 3 5) 4> = Bo+ B, X + B 2 Y + B 3 y + B 4 X Y + B 5 Y Z + n terms , (36) where ^ = O i s the e q u a t i o n o f the boundary, and i t e n s u r e s t h a t V = 0 at the wal 1. I n s e r t i n g the above t r i a l f u n c t i o n s (35) and (36) i n t o e q u a t i o n (34) we o b t a i n the f u n c t i o n a l i n terms o f 2n p a r a m e t e r s , I = I ( A 0 , A i , A z ; , Bo, B„ B 2 , , L ) . (37) Making I s t a t i o n a r y w i t h r e s p e c t t o A's and B's, T h i s w i l l r e s u l t i n 2n l i n e a r s i m u l t a n e o u s a l g e b r i c e q u a t i o n s i n (2n + 1) unknowns, (2n + 1 ) ^ unknown b e i n g L. T h i s r e q u i r e s one more r e l a t i o n which i s p r o v i d e d by e q u a t i o n ( 9 ) . The s o l u t i o n o f the r e s u l t -i n g e q u a t i o n s p r o v i d e s the d e s i r e d unknown c o - e f f i c i e n t s A i and Bi as w e l l as L. Hence V and <$> e x p r e s s i o n s w i l l be known ,and e n g i n e e r i n g i n -f o r m a t i o n , l i k e N u s s e l t number ( 1 3 ) , can be o b t a i n e d . E q u a t i o n ( 1 1 ) , o b t a i n e d by making a s i m p l e energy b a l a n c e a c r o s s two s e c t i o n s o f the d u c t , can a l s o be deduced from v a r i a t i o n a l f o r m u l a -t i o n (34) as f o l l o w s : Making I s t a t i o n a r y w i t h r e s p e c t t o Bo w i l l r e s u l t i n 19 0 = J | C - 2 RaV + ^ R c F ^ c f x d Y + 2 R a C * [ d s , (38) J | V c / x c / y - F | | c 7 X c l y = C ^ J d s , J ^ C / A - F j ( c / A = C z J d s , (1-F) / / d A = Czjds. T h e r e f o r e r - h e \ - 1 _ F l ^ f l - 1-F T h e r e f o r e , c 2 _ &-F)-j^ - DT / J s = ^ T ~ Us i n g e q u a t i o n ( 3 4 ) , c a l c u l a t i o n s a r e made f o r r e c t a n g u l a r and rhombic d u c t s , the c o - o r d i n a t e system f o r t h e s e d u c t s b e i n g shown i n F i g u r e s 2b and 2c. Thus f o r r e c t a n g u l a r d u c t , t a k i n g even powers o f X and Y, v e l o c i t y and te m p e r a t u r e f u n c t i o n s can be chosen a s , v = fb ( A o + A | * + A 2-Y 2+ A 3 x 4 -1- A4V* + As x \ X J , (39) 4> = B o + B, X 2 + B i Y 2 + B 3 X 4 + B^YV BsxV". (40) In (39), e q u a t i o n o f the boundary can be w r i t t e n as / b = <V-x 2 ; (b a - Y x ; . (41) S i m i l a r l y f o r rhombic d u c t , V = jh ( Aa + A ( X 2 + A x Y Z + A 3 X % A< V + As x V j , (42) F i g u r e 2. C o - o r d i n a t e System. <fi - Bo + a, x + Bx y2+ Bi x + 8 4 y 4 + Bs x*y2» (43) where jy = ( Yi- m x-f) (Y~ mx-t) (Y+ + l) (Y- m x+ t) , (44) and m — tan °ty2 ? oC b e i n g the d u c t a n g l e . To s i m p l i f y the e q u a t i o n o f boundary and the r e s u l t i n g c a l c u l a -t i o n s , we have taken b = 1 i n e q u a t i o n (44). However, s i n c e e q u a t i o n s (6) and (7) have been n o n - d i m e n s i o n a l i z e d by the h y d r a u l i c d i a m e t e r , the f i n a l r e s u l t s have been a d j u s t e d by the f a c t o r b/D^. We w i l l now d i s c u s s the r e s u l t s and l i m i t a t i o n s o f t h i s a n a l y s i s. IV DISCUSSION OF RESULTS In the p r o c e e d i n g pages, r e s u l t s o f the p r e s e n t work a r e d i s -c u s s e d under t h r e e main h e a d i n g s : ( i ) F o r c e d c o n v e c t i o n when the e f f e c t s o f p e r i p h e r a l w a l l c o n d u c t i o n a r e n e g l e c t e d ; ( i i ) E f f e c t o f f r e e c o n v e c t i o n upon f o r c e d c o n v e c t i o n when the p e r i p h e r a l c o n d u c t i o n i n the w a l l i s n e g l e c t e d ; ( i i i ) C o n j u g a t e h e a t t r a n s f e r when the e f f e c t s o f p e r i p h e r a l w a l l c o n d u c t i o n are a l s o i n c l u d e d . R e s u l t s f o r each geometry t r e a t e d a r e d i s c u s s e d s e p a r a t e l y . In each c a s e , r e s u l t s a r e compared, wherever p o s s i b l e , w i t h the a v a i l a b l e s o l u t i o n s i n the p u b l i s h e d l i t e r a t u r e . An I.B.M. 360/67 computer was used and a l l c a l c u l a t i o n s were c a r r i e d out i n d o u b l e p r e c i s i o n . A l l r e s u l t s a r e p r e s e n t e d n e g l e c t i n g i n t e r n a l h e a t g e n e r a t i o n , F = 0. ( i ) F o r c e d C o n v e c t i o n U s i n g the g e n e r a l v a r i a t i o n a l e x p r e s s i o n (C5) f o r pure f o r c e d c o n v e c t i o n , N u s s e l t numbers have been c a l c u l a t e d f o r r e c t a n g u l a r , rhombic, 23 i s o s c e l e s t r i a n g u l a r and r i g h t - a n g l e d t r i a n g u l a r d u c t s . The v e l o c i t y and tem p e r a t u r e f u n c t i o n s f o r t h e s e d u c t s a r e g i v e n i n Appendix C. R e c t a n g u l a r Duct For r e c t a n g u l a r d u c t , N u s s e l t number v a l u e s f o r a s p e c t r a t i o (= b/a) v a r y i n g f r o m 1 t o 10 a r e g i v e n i n T a b l e 1. For a s p e c t r a t i o 1, when the d u c t i s s q u a r e , N u s s e l t number v a l u e s g i v e n by [12] and [13] are 3.09 and 3.096 r e s p e c t i v e l y and a r e v e r y c l o s e t o the p r e s e n t v a l u e o f 3.0968. Cheng [7] computed a N u s s e l t number o f 3.82 f o r squ a r e d u c t s , but t h i s v a l u e appears i n c o n s i s t e n t w i t h h i s r e s u l t s f o r r e c t a n g u l a r d u c t s o f h i g h e r a s p e c t r a t i o s and, a c c o r d i n g t o h i s own a d m i s s i o n , may be i n e r r o r . A c o m p u t a t i o n by I r v i n e , as r e p o r t e d i n [ 1 2 ] , y i e l d e d a . N u s s e l t number o f 2.9, a d e v i a t i o n o f 6 per c e n t f r o m the c u r r e n t r e s u l t s . A comparison o f N u s s e l t number v a l u e s w i t h [ 7 ] and [12] f o r h i g h e r a s p e c t r a t i o s i s a l s o g i v e n i n T a b l e 1. V a r i a t i o n o f N u s s e l t numbers a g a i n s t d u c t a s p e c t r a t i o s i s p l o t t e d i n F i g u r e 3, which shows t h a t the N u s s e l t numbers d e c r e a s e s l i g h t l y as the a s p e c t r a t i o i n c r e a s e s from 1 t o 5 and become a l m o s t con-s t a n t i n the a s p e c t r a t i o range o f 5 t o 10. T h i s v a r i a t i o n o f N u s s e l t numbers i s a l s o d i s c u s s e d q u a l i t a t i v e l y i n [ 1 2 ] . In o r d e r t o compare two l i m i t i n g s i t u a t i o n s , " N u s s e l t numbers a r e a l s o p l o t t e d , F i g u r e 3, f o r u n i f o r m c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e [ 2 ] . For u n i f o r m c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e , the N u s s e l t number i n c r e a s e s from 3.60 t o 6.79 as the a s p e c t r a t i o i n c r e a s e s from 1 to 10 and ap-proaches 8.23 as the a s p e c t r a t i o approaches i n f i n i t y . The N u s s e l t num-ber f o r f l o w between i n f i n i t e p a r a l l e l p l a t e s w i t h u n i f o r m a x i a l h e a t i n p u t i s a l s o 8.23. So the p a r a l l e l p l a t e channel r e p r e s e n t s the l i m i t -i n g s i t u a t i o n f o r the r e c t a n g u l a r d u c t w i t h u n i f o r m p e r i p h e r a l w a l l t e m p e r a t u r e , but 1 ,-the p a r a l l e l p l a t e model" c a n n o t be us e d ' t b p r e d i c t h o t s p o t ( c o r n e r ) t e m p e r a t u r e s when the h e a t f l u x i s u n i f o r m around the p e r i -p hery. T h i s a s p e c t i s a l s o d i s c u s s e d i n [ 1 2 ] . In o r d e r t o see how c l o s e l y the c o n d i t i o n ^ ^ = c 2 h a s been s a t i s -f i e d a t the boundary by the assumed e x p r e s s i o n f o r <j>, e q u a t i o n ( C 7 ) , the c a l c u l a t e d v a l u e s o f ^ / © N on the boundary, from ( C 7 ) , have been p l o t t e d a g a i n s t C^, e q u a t i o n ( 1 1 ) , shown as c h a i n e d d o t t e d l i n e s i n F i g u r e 4. The l i s t e d v a l u e s o f on the r i g h t hand s i d e o f t h i s d i a -gram a r e o b t a i n e d from (11) by r e p l a c i n g P h w i t h b (and b = 1 ) , g i v i n g f o r the r e c t a n g u l a r d u c t , r - -Area. 1 /Ar\ C z — — — — =s — — (45) b x. rertmeter 1 + t>/a F i g u r e 4 shows t h a t the c o n d i t i o n o f u n i f o r m h e a t f l u x has been s a t i s f i e d v e r y c l o s e l y . The y - d i r e c t i o n t e m p e r a t u r e g r a d i e n t s d e v i a t e a l i t t l e more from the d e s i r e d v a l u e s compared t o the x - d i r e c t i on g r a d i e n t s and t h i s e f f e c t i s more pronounced near the c o r n e r s . A t h i g h e r a s p e c t r a t i o s , say 10, y - d i r e c t i o n g r a d i e n t s tend t o become c o n s t a n t o v e r the e n t i r e s h o r t s i d e o f the d u c t , a l t h o u g h the v a l u e o f the f l u x i s s l i g h t l y h i g h e r than the p r e s c r i b e d v a l u e . Reason f o r t h i s i s t h a t the c o r n e r s i n f l u e n c e the s h o r t e r s i d e s more than the l o n g e r ones. 25 Rhombic Duct N u s s e l t numbers computed f o r rhombic d u c t a r e shown p l o t t e d a g a i n s t the d u c t a n g l e a i n F i g u r e 5. A t a = 90° when the d u c t a c q u i r e s the s q u a r e shape, the N u s s e l t number v a l u e i s 3.0859 as compared t o 3.0968 from r e c t a n g u l a r d u c t , a s p e c t r a t i o = 1 . Comparison o f the two l i m i t i n g boundary c o n d i t i o n s shows t h a t N u s s e l t numbers d e c r e a s e more r a p i d l y w i t h the d e c r e a s e i n a n g l e a f o r u n i f o r m h e a t f l u x than f o r u n i -form w a l l t e m p e r a t u r e . V a r i a t i o n o f the w a l l t e m p e r a t u r e g r a d i e n t s has been shown i n F i g u r e 6. T h i s f i g u r e shows t h a t boundary c o n d i t i o n has been s a t i s f i e d q u i t e c l o s e l y by the ap p r o x i m a t e e x p r e s s i o n chosen f o r <j> (C8) when the d u c t a n g l e i s l a r g e . At lower v a l u e s o f a, the c a l c u l a t e d f l u x f l u c t u a t e s r a p i d l y around the p r e s c r i b e d v a l u e Z^. The c o r n e r e f f e c t s a r e a l s o i n -c r e a s e d w i t h the d e c r e a s i n g v a l u e o f a. I s o s c e l e s T r i a n g u l a r Duct N u s s e l t numbers have been c a l c u l a t e d f o r i s o s c e l e s t r i a n g u l a r d u c t u s i n g the t e m p e r a t u r e e x p r e s s i o n (C9) and a r e p l o t t e d a g a i n s t a n g l e a, F i g u r e 7. The N u s s e l t number c u r v e r e a c h e s a maximum v a l u e when the d u c t a n g l e a = 60° and f a l l s as the a n g l e d e v i a t e s from 6 0 ° , the d e c r e a s e b e i n g more r a p i d when the a n g l e a i n c r e a s e s from 60° than when the a n g l e d e c r e a s e s . V a r i a t i o n o f N u s s e l t numbers w i t h d u c t a n g l e a i s more i n the p r e s e n t case than i n the case o f c o n s t a n t p e r i p h e r a l w a l l t e m p e r a t u r e . 26 For e q u i l a t e r a l t r i a n g u l a r d u c t , a = 6 0 ° , N u s s e l t number v a l u e i s 1.8994 as compared t o 1.892 g i v e n by [ 1 3 ] , and 1.90 g i v e n by [16] a t Ra = 1. From the v a r i a t i o n o f a l o n g the s l o p i n g s i d e AB, shown i n F i g u r e 8, we can see t h a t the boundary c o n d i t i o n o f u n i f o r m p e r i p h e r a l h e a t f l u x i s s a t i s f i e d more c l o s e l y f o r l a r g e r a n g l e s than f o r s m a l l e r a n g l e s o f the d u c t . A t a = 6 0 ° , N u s s e l t number v a l u e s have been com-pared above and show q u i t e a good agreement b u t the c a l c u l a t e d v a l u e s o f ^ " ^ / ^ f r o m (C9) a r e not v e r y c l o s e t o i n F i g u r e 8. Hence one can con-c l u d e t h a t i t i s not r e a l l y n e c e s s a r y t o have t h i s c o n d i t i o n s a t i s f i e d by the a p p r o x i m a t e e x p r e s s i o n f o r <j> i n o r d e r t o have f a i r l y a c c u r a t e v a l u e o f N u s s e l t number. I t i s p o s s i b l e t h a t n e g a t i v e and p o s i t i v e f l u c -t u a t i o n s may g e t b a l a n c e d i n a v e r a g i n g o u t $ o v e r the boundary and i n c a l c u l a t i n g b u l k t e m p e r a t u r e o v e r the e n t i r e c r o s s - s e c t i o n o f the d u c t . S i n c e no comparison o f N u s s e l t numbers i s a v a i l a b l e a t some o t h e r a n g l e s , n o t h i n g can be s a i d d e f i n i t e l y about the above s t a t e m e n t . R i g h t - A n g l e d T r i a n g u l a r Duct F i g u r e 9 shows N u s s e l t number p l o t f o r r i g h t - a n g l e d t r i a n g u l a r d u c t . The o n l y p o i n t a t which t he v a l u e can be compared w i t h i s o s c e l e s t r i a n g u l a r d u c t i s when a = 45° and the d u c t i s o f r i g h t - i s o s c e l e s t r i a n g u l a r shape. The two v a l u e s d i f f e r by 3 per c e n t , the p r e s e n t v a l u e b e i n g h i g h e r than the i s o s c e l e s t r i a n g u l a r c a s e . N u s s e l t number b e h a v i o u r i s same as the p r e v i o u s shaped d u c t e x c e p t t h a t the c u r v e i s s y m m e t r i c a l around 4 5 ° . 27 In a l l the f o u r g e o m e t r i e s t r e a t e d , i t i s found t h a t N u s s e l t number v a l u e i s lower f o r the p r e s e n t boundary c o n d i t i o n o f c o n s t a n t p e r i p h e r a l h e a t f l u x than f o r the c a s e o f u n i f o r m p e r i p h e r a l w a l l temper-a t u r e . From the comparison o f the r e s u l t s made w i t h the a v a i l a b l e s o l u -t i o n s , we can say t h a t the v a r i a t i o n a l f o r m u l a t i o n (C5) i s f a i r l y a c c u r a t e . ( i i ) Combined F r e e and F o r c e d C o n v e c t i o n W i t h o u t P e r i p h e r a l Wall C o n d u c t i o n Study o f combined f r e e and f o r c e d c o n v e c t i o n has been c a r r i e d o u t w i t h the v a r i a t i o n a l f o r m u l a t i o n ( 3 4 ) . Two d u c t shapes have been t r e a t e d under t h i s h e a d i n g , (a) r e c t a n g u l a r , and (b) rhombic. The c o - o r d i n a t e system chosen i s the same as f o r f o r c e d c o n v e c t i o n , F i g u r e 2. R e c t a n g u l a r Duct Computations were c a r r i e d o u t w i t h the v e l o c i t y and t e m p e r a t u r e e x p r e s s i o n s (39) and (40) r e s p e c t i v e l y . N u s s e l t numbers were c a l c u l a t e d f o r a s p e c t r a t i o s 1 t o 10 and R a y l e i g h numbers v a r y i n g from 0 t o 5000. N u s s e l t number v a l u e s a g a i n s t a s p e c t r a t i o s f o r v a r i o u s v a l u e s o f Ra have been shown i n F i g u r e 10. The o n l y comparison a v a i l a b l e i n the p u b l i s h e d l i t e r a t u r e i s from r e f e r e n c e [16] f o r s q u a r e d u c t . The r e s u l t s o f the p r e s e n t a n a l y s i s as 28 w e l l as o f r e f e r e n c e [16] a r e g i v e n i n T a b l e 2, which shows q u i t e c l o s e agreement between the two a n a l y s e s . R e s u l t s f o r Ra = 0 were compared w i t h t h o s e from (C5) where the v e l o c i t y e x p r e s s i o n was known e x a c t l y . The two r e s u l t s were found t o be i n e x a c t agreement w i t h i n the c o m p u t a t i o n a l l i m i t s . V a l u e s o f the p r e s s u r e drop parameter L as a f u n c t i o n o f the R a y l e i g h number and the a s p e c t r a t i o have been l i s t e d i n T a b l e 3. T h i s t a b l e shows t h a t p r e s s u r e drop parameter goes on i n c r e a s i n g w i t h the i n c r e a s e i n R a y l e i g h number as w e l l as the a s p e c t r a t i o . P r e s s u r e drop parameters l i s t e d i n the above-mentioned t a b l e have been c a l c u l a t e d by t a k i n g <j> = 0 a t the c o r n e r p o i n t (X = a, Y = b ) . Changing the r e f e r e n c e p o i n t w i l l change the p r e s s u r e drop parameter v a l u e s , as can be seen from e q u a t i o n ( 6 ) . However, knowing the tempera-t u r e f u n c t i o n and p r e s s u r e drop parameter f o r a g i v e n r e f e r e n c e p o i n t , i t s v a l u e f o r a p a r t i c u l a r R a y l e i g h number can be c a l c u l a t e d f o r any o t h e r r e f e r e n c e p o i n t . As r e s u l t s c o u l d not be compared f o r h i g h e r a s p e c t r a t i o s , i t i s d i f f i c u l t t o e s t i m a t e the a c c u r a c y o f the p r e s e n t r e s u l t s . Rhombic Duct N u s s e l t numbers f o r rhombic d u c t a r e p l o t t e d i n F i g u r e 11. These v a l u e s have been c a l c u l a t e d u s i n g e q u a t i o n s (42) and (43) i n ( 3 4 ) . 29 F i g u r e 11 shows t h a t the N u s s e l t number v a l u e i n c r e a s e s w i t h the i n c r e a s e i n the R a y l e i g h number and d e c r e a s e s w i t h the a n g l e a f o r a p a r t i c u l a r Ra. As the R a y l e i g h number i n c r e a s e s , N u s s e l t number v a l u e s a r e g i v e n f o r a l i m i t e d range o f a. The l i m i t o f a d e c r e a s e s w i t h i n c r e a s i n g Ra v a l u e s . R e s u l t s a r e g i v e n o n l y i n the r e g i o n where they a r e b e l i e v e d to be a c c u r a t e . In o r d e r t o check the a c c u r a c y o f the p r e s e n t f o r m u l a t i o n , t h e r e s u l t s have a l s o been computed u s i n g f i n i t e - d i f f e r e n c e p r o c e d u r e , de-t a i l s o f which a r e g i v e n i n Appendix B. I t was o b s e r v e d t h a t the f i n i t e -d i f f e r e n c e s o l u t i o n t a k e s about two t o t h r e e hundred times more machine time f o r e v e r y s i n g l e v a l u e o f N u s s e l t number compared to t h a t o f the v a r i a t i o n a l s o l u t i o n . F o r t h i s r e a s o n , o n l y a l i m i t e d amount o f d a t a was o b t a i n e d by t h i s method. Some n u m e r i c a l v a l u e s o f N u s s e l t number, o b t a i n e d by the f i n i t e - d i f f e r e n c e s o l u t i o n , have been compared w i t h t h o s e o f the v a r i a t i o n a l r e s u l t s and a r e g i v e n i n T a b l e 4. A comparison o f p r e s s u r e drops i s g i v e n i n T a b l e 5. These comparisons show t h a t the r e -s u l t s by the two methods a r e q u i t e c l o s e t o each o t h e r . However, the f i n i t e - d i f f e r e n c e r e s u l t s c o u l d have been improved upon a t a c o n s i d e r a b l y more c o s t o f computer t i m e . F o r squa r e d u c t , the v a r i a t i o n a l r e s u l t s o f T a b l e 2 agree w i t h t h o s e i n T a b l e 4. S i m i l a r l y , the p r e s s u r e - d r o p p a r a -meter v a l u e s i n T a b l e s 3 and 5 f o r squa r e d u c t a r e same. As the agreement o f the r e s u l t s o b t a i n e d by t h e s e two methods was good and the machine time f o r the f i n i t e - d i f f e r e n c e method was v e r y l o n g , no f u r t h e r attempt was made t o compute N u s s e l t numbers by 30 f i n i t e - d i f f e r e n c e a p p r o x i m a t i o n f o r h i g h e r R a y l e i g h numbers and lower d u c t a n g l e s o r f o r o t h e r g e o m e t r i e s . T a b l e 6 c o n t a i n s the p r e s s u r e drop parameter v a l u e s , which were used (and which can be used f o r f u r t h e r c a l c u l a t i o n s o f N u s s e l t numbers a t o t h e r a n g l e s a and Ra)as an i n i t i a l guess f o r f i n i t e - d i f f e r e n c e approx-i m a t i o n . To see how f a r the t e m p e r a t u r e d i s t r i b u t i o n , o b t a i n e d by the assumed e x p r e s s i o n f o r <j>, e q u a t i o n (43), d i f f e r s from t h a t o b t a i n e d by f i n i t e - d i f f e r e n c e a p p r o x i m a t i o n , the t e m p e r a t u r e p r o f i l e s on the boundary f o r a = 90° have been p l o t t e d i n F i g u r e 12. T h i s f i g u r e shows t h a t , a l t h o u g h the boundary c o n d i t i o n (11) i s not s a t i s f i e d e x a c t l y by the chosen f u n c t i o n the t e m p e r a t u r e p r o f i l e s a r e v e r y c l o s e t o each o t h e r . S i m i l a r comparison was o b s e r v e d f o r o t h e r d u c t a n g l e s a l s o . N u s s e l t numbers o b t a i n e d f o r s q u a r e d u c t (a = 90°) from (C5) and from (34) a t Ra = 0 a r e 3.0859 and 3.0897 r e s p e c t i v e l y . These two r e s u l t s a r e a l m o s t same. But the c u r v e s o f "ac/ 5^N on the boundary, ob-t a i n e d f r o m <j> e x p r e s s i o n s f o r t h e s e two d i f f e r e n t f o r m u l a t i o n s , d e v i a t e from each o t h e r c o n s i d e r a b l y , as shown i n F i g u r e 13. Hence one can say a g a i n t h a t i t i s n o t r e a l l y n e c e s s a r y f o r the assumed e x p r e s s i o n <j> t o s a t i s f y the boundary c o n d i t i o n (11) e x a c t l y t o have a f a i r l y good tem-p e r a t u r e d i s t r i b u t i o n and o v e r a l l N u s s e l t number v a l u e s . However, c l o s e r the s a t i s f a c t i o n o f the boundary c o n d i t i o n , more a c c u r a t e the r e s u l t s w i l l be. 31 I n c r e a s i n g the number o f terms i n v e l o c i t y and/or t e m p e r a t u r e e x p r e s s i o n i n c r e a s e s the s i z e o f the m a t r i x t o be s o l v e d . S i n c e the m a t r i x o b t a i n e d i s i l l - c o n d i t i o n e d ; s l i g h t i n c r e a s e o r d e c r e a s e i n any o f the terms produces c o n s i d e r a b l e change i n the f i n a l r e s u l t s . V e r y long e x p r e s s i o n s f o r assumed f u n c t i o n s V and $ a r e not recommended, t h e r e -f o r e . ( i i i ) C o n j u g a t e Heat T r a n s f e r In t h i s s e c t i o n , the e f f e c t o f p e r i p h e r a l w a l l c o n d u c t i o n has been s t u d i e d o n l y f o r r e c t a n g u l a r d u c t , a l t h o u g h the g e n e r a l v a r i a t i o n a l f o r m u l a t i o n (D3) can be used f o r many o t h e r d u c t - s h a p e s . Computations were c a r r i e d o u t w i t h s i x and t e n c o - e f f i c i e n t s i n V and <f> e x p r e s s i o n s r e s p e c t i v e l y . I t i s found t h a t as the c o n d u c t i o n parameter K becomes v e r y l a r g e , the w a l l t e m p e r a t u r e tends t o become u n i f o r m . On the o t h e r hand, as the v a l u e o f K approaches z e r o , the w a l l h e a t f l u x becomes c i r c u m f e r e n t i a l l y u n i f o r m . For the f o r m e r c a s e (K -> °°), the r e s u l t s o f the p r e s e n t a n a l y s i s c o u l d be compared w i t h t h o s e o f Han [ 2 ] , who p r e s e n t e d an e x a c t s o l u t i o n o f (6) t o (8) f o r r e c t a n g u l a r d u c t s under the c o n d i t i o n o f u n i f o r m c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e . Such a com-p a r i s o n o f N u s s e l t number v a l u e s f o r a s q u a r e d u c t i s c o n t a i n e d i n T a b l e 7. The v a l u e s o f K and Ra used i n t h i s t a b l e a r e t h o s e which g i v e r e s u l t s c l o s e to t h a t o f Han. I t may be noted i n t h i s t a b l e t h a t f o r f o r c e d c o n v e c t i o n , Ra = 0, a v e r y h i g h v a l u e o f the c o n d u c t i o n p a r a m e t e r , (K = 100) g i v e s r e s u l t s c l o s e t o t h a t o f Han. As R a y l e i g h number i s 32 i n c r e a s e d , s m a l l e r v a l u e s o f K a r e r e q u i r e d t o produce the e f f e c t o f almo s t u n i f o r m c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e . T h i s l a s t o b s e r v a t i o n i s i n a c c o r d a n c e w i t h the r e s u l t o f [ 1 6 ] , where i t i s shown t h a t h i g h v a l u e s o f the buoyancy parameter tend t o produce r o t a t i o n a l symmetry o f te m p e r a t u r e and h e a t f l u x i n n o n - c i r c u l a r d u c t s . The e f f e c t o f K on c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e can be demon-s t r a t e d by p l o t t i n g the v a r i a t i o n o f w a l l t e m p e r a t u r e d i f f e r e n c e <j> a g a i n s t v a r i o u s v a l u e s o f K. Such a p l o t i s shown i n F i g u r e 14 f o r a square d u c t a t Ra = 1000. I t i s c l e a r l y e v i d e n t from t h i s diagram t h a t as K i n c r e a s e s , the w a l l t e m p e r a t u r e r a p i d l y tends to become u n i f o r m . At K = 20, tj) i s a l m o s t equal t o z e r o t h r o u g h o u t the c i r c u m f e r e n c e . A t v a l u e s o f Ra g r e a t e r than 1000, <(> becomes a l m o s t z e r o on the c i r c u m f e r -ence a t v a l u e s o f K lower than t h o s e shown i n F i g u r e 14. A g e n e r a l v a r i a t i o n o f N u s s e l t number a g a i n s t R a y l e i g h number w i t h K as a parameter i s shown i n F i g u r e s 15 t o 17. These f i g u r e s a r e f o r a s p e c t r a t i o s 1, 2 and 3 r e s p e c t i v e l y . These f i g u r e s show t h a t as the a s p e c t r a t i o i n c r e a s e s , the e f f e c t o f K parameter on the N u s s e l t number becomes more i m p o r t a n t . In t h e s e t h r e e f i g u r e s , f o r K -> °°, the N u s s e l t number v a l u e s agree v e r y c l o s e l y t o t h o s e o f Han. For the s i t u a t i o n e q u i v a l e n t to K = 0 ( u n i f o r m c i r c u m f e r e n t i a l w a l l h e a t f l u x ) , p u b l i s h e d l i t e r a t u r e c o n t a i n s N u s s e l t number i n f o r m a t i o n o n l y f o r the squa r e d u c t , and t h i s has been d i s c u s s e d i n the p r e c e d i n g pages. V CONCLUSION A g e n e r a l s o l u t i o n o f the problem o f l a m i n a r f o r c e d and combined f r e e and f o r c e d c o n v e c t i o n through v e r t i c a l n o n - c i r c u l a r d u c t s has been o b t a i n e d u s i n g v a r i a t i o n a l t e c h n i q u e . Two d i f f e r e n t thermal c o n d i t i o n s have been a n a l y s e d : Case 1. C o n s t a n t c i r c u m f e r e n t i a l h e a t f l u x when the p e r i p h e r a l w a l l c o n d u c t i o n i s n e g l e c t e d ; Case 2. C o n j u g a t e h e a t t r a n s f e r when the p e r i p h e r a l w a l l c o n -d u c t i o n i s i n c l u d e d . I t can be c o n c l u d e d t h a t the u n i f o r m c i r c u m f e r e n t i a l h e a t f l u x c o n d i t i o n r e s u l t s i n lower v a l u e s o f the N u s s e l t numbers compared to those o f u n i f o r m c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e c o n d i t i o n . The c i r c u m -f e r e n t i a l w a l l c o n d u c t i o n i s more i m p o r t a n t f o r low v a l u e s o f the buoyancy parameter and low v a l u e s o f the c o n d u c t i o n parameter. H i g h e r v a l u e s o f e i t h e r o f t h e s e parameters tend t o m i n i m i z e asymmetries i n c i r c u m f e r e n t i a l w a l l t e m p e r a t u r e . I t may be added t h a t the v a r i a t i o n a l f o r m u l a t i o n (C5) g i v e s e x c e l l e n t r e s u l t s f o r pure f o r c e d c o n v e c t i o n f l o w . The v a r i a t i o n a l f o r m u l a t i o n (34) g i v e s v e r y good r e s u l t s f o r low R a y l e i g h numbers w h i l e a t h i g h e r R a y l e i g h numbers, the r e s u l t s a r e good when the d u c t geometry i s c l o s e t o a c i r c l e . The v a r i a t i o n a l s o l u t i o n (D3) p r e s e n t e d f o r c o n j u g a t e problem i s i n d e e d v e r y a c c u r a t e . 35 TABLE 1. F o r c e d C o n v e c t i o n N u s s e l t Numbers f o r R e c t a n g u l a r Ducts A g a i n s t A s p e c t R a t i o s f r o m F o r m u l a t i o n ( C 5 ) . A s p e c t N u s s e l t Ref. [7] Ref. [12] Ref. [13] R a t i o Number * H.M. Cheng Sparrow & S i e g e l K.C. Cheng 1 3.0968 3.82 3.09 3.096 2 3.0270 2.90 3 2.9702 4 2.9459 2.91 5 2.9360 6 2.9385 7 2.9403 8 2.9442 9 2.9487 10 2.9534 2.88 2.90 N u s s e l t number v a l u e s t o f o u r t h d e c i m a l p l a c e a r e g i v e n p u r e l y f o r r e c o r d . 36 TABLE 2. N u s s e l t Numbers f o r Combined Fr e e and F o r c e d C o n v e c t i o n . Comparison o f the R e s u l t s from F o r m u l a t i o n (34) w i t h t h o s e o f Ref. [16] f o r the Square Duct. R a y l e i g h Number N u s s e l t Number P r e s e n t V a r i a t i o n a l Ref [16] 0 3.09 3 23* 100 3.17 3 32 500 3.50 3 69 1000 3.88 4 14 2000 4.58 4 95 5000 6.14 6 34 V a l u e c a l c u l a t e d a t Ra = 1 TABLE 3. Values o f the P r e s s u r e Drop Parameters L A g a i n s t R a y l e i g h Numbers Ra f o r R e c t a n g u l a r Ducts o f A s p e c t R a t i o s 1 to 10, and R e f e r e n c e P o i n t a t X = a, Y •= b. R a y l e i g h Number • Ra A s p e c t R a t i o b/a 1 2 3 4 5 6 7 8 9 10 0 28.458 31.104 34.191 36.482 38.167 39.449 40.458 41.275 41 .954 42.529 100 41.384 45.861 51.449 55.927 59.461 62.335 64.759 66.878 68.788 70.551 500 90.776 98.704 107.12 113.15 117.83 121.74 125.09 128.00 130.54 132.74 1000 143.20 156.39 164.48 170.64 175.49 179.08 181.53 183.05 183.84 184.07 2000 252.56 258.27 265.07 270.34 272.49 271.59 268.59 264.43 259.77 255.02 3000 347.23 350.80 356.48 358.67 355.03 347.18 1 337.39 |327.25 i 317.58 308.70 4000 435.30 437.78 441.93 |439.05 427.97 412.48 I 396.09 |380.64 366.85 354.76 5000 518.57 520.77 522.68 1513.31 493.93 470.88 i 448.48 J428.48 i 411.28 396.65 38 TABLE 4. N u s s e l t Numbers f o r Rhombic Duct. Comparison o f R e s u l t s From the V a r i a t i o n a l F o r m u l a t i o n (34) w i t h those from the F i n i t e - D i f f e r e n c e A p p r o x i m a t i o n , Appendix B. a R a y l e i g h N u s s e l t Number Number V a r i a t i o n a l F i n i te-- D i f f e r e n c e 90° 0 3.0897 3 1228 100 3.1733 3 .1304 500 3.4992 3 4515 1000 3.8859 3 .8224 2000 4.5847 4 4811 80° 0 2.9753 2 .9632 100 3.0670 3 0268 500 3.4180 3 3718 1000 3.8239 3 7601 2000 4.5395 4 4352 70° 0 2.6530 2 6555 100 2.7679 2 7310 500 3.1896 3 1430 1000 3.6482 3 5818 2000 4.4047 4 2922 60° 0 2.1799 2 2076 100 2.3298 2 3033 500 2.8550 2 8096 1000 3.3832 3 3073 2000 4.1789 4 0514 39 TABLE 5. Comparison o f P r e s s u r e Drop Parameters f o r Rhombic Duct from the V a r i a t i o n a l F o r m u l a t i o n (34) w i t h t h o s e from the F i n i t e - D i f f e r e n c e A p p r o x i m a t i o n , Appendix B. a R a y l e i g h P r e s s u r e Drop Parameter L Number V a r i a t i o n a l F i n i t e D i f f e r e n c e 90° 0 28.458 28.458 100 41.384 41.589 500 90.775 91.544 1000 148.200 150.439 2000 252.560 257.837 80° 0 28.370 28.370 100 38.683 38.893 500 79.035 79.993 1000 127.300 129.919 2000 217.130 222.387 70° 0 28.112 28.112 100 36.284 36.679 500 69.498 71.073 1000 110.840 113.704 2000 189.980 195.143 60° 0 26.697 27.670 100 33.915 34.329 500 61.271 63.119 1000 97.528 101.033 2000 169.17 174.739 TABLE 6. V a l u e s o f the P r e s s u r e Drop Parameters L A g a i n s t R a y l e i g h Numbers Ra f o r Rhombic D u c t s . R e f e r e n c e P o i n t a t X = 0, Y = b. R a y l e i g h Number Ra Duct Ang l e a 90° 80° 70° 60° 50° 40° 30° 20° 10° 0 28.458 28.370 28.112 27.697 27.153 26.516 25.843 25.219 24.763 100 41.384 38.683 36.284 33.915 31 .319 28.146 23.840 17.772 13.583 500 90.775 79.035 69.498 61.271 53.837 47.135 41.955 40.959 52.810 1000 148.20 127.30 110.84 97.528 86.819 78.909 74.814 76.702 89.101 2000 252.56 217.13 189.98 169.17 153.78 143.58 138.61 138.91 143.99 3000 347.23 299.74 263.81 236.75 217.05 !203.74 i 195.80 191.76 189.94 i 4000 | 435.30 376.96 333.11 300.27 276.25 259.34 247.67 238.96 231.89 i 5000 | 518.58 450.11 398.82 360.40 332.01 311.28 i i 295.65 282.58 271.61 41 TABLE 7. The C o n d u c t i o n Parameter K and R a y l e i g h Numbers Ra Which Give V a l u e s o f N u s s e l t Numbers C l o s e to Those o f Han [ 2 ] . Ra C i r c u m f e r e n t i a l C o n d u c t i o n Parameter K — Han [2] 3 4 5 7 20 100 0 3 .6059 3.6078 100 3 6950 3.6986 500 4 0490 4.0491 1000 4.4589 4.4579 2000 5.1822 5.1816 3000 5.7988 5.7943 4000 6.3225 6.3158 5000 6.7632 6.7642 6000 7.1594 7.1547 U N I F O R M C I R C U M F E R E N T I A L U N I F O R M C I R C U M F E R E N T I A L H E A T F L U X W A L L T E M P E R A T U R E b o a 2 3 4 5 6 7 8 9 10 ASPECT RATIO, b/a F i g u r e 3. Fo r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t A s p e c t R a t i o f o r R e c t a n g u l a r Duct. 0'6 0-5 C 2 b 0 a -G»X C 2 b /a •5 1 0-4 CN o 0-3 0-2 ' U J I i t n i n n m i i l l l i r i l l l l T i r i ' u-' 'XLLLLLLmiiiniinnuiii,i..M\fim,,, ,Tn? i rTTTTTi iTV t i n , , , . •333 2 •25 3 • 2 4 •167 5 0-1 _ 10' • 091 10 _L 0 0-1 F i g u r e 4. 0-2 0-3 0-4 0-5 0-6 0-7 F r a c t i o n o f X o r Y 0-8 0-9 1-0 F o r c e d C o n v e c t i o n . R e s u l t a n t Normal Heat F l u x on the Boundary A g a i n s t i t s P r e s c r i b e d V a l u e f o r R e c t a n g u l a r Duct. U N I F O R M C I R C U M F E R E N T I A L H E A T F L U X . . . . . . . . . . . U N I F O R M C I R C U M F E R E N T I A L W A L L T E M P E R A T U R E 90 80 70 60 50 40 30 20 10 ANGLE F i g u r e 5. F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t Duct Ang l e a f o r Rhombic Duct. 55 -CN o >45-c 2 oc °499 5 '453 50 _ »433 60 "409 70 80 °383 80 •35 90-353 90 _ 0 F i g u r e 6. •4 •6 •8 F r a c t i o n of A B •9 1-0 Forced C o n v e c t i o n . R e s u l t a n t Normal Heat F l u x on t h e Boundary A g a i n s t i t s P r e s c r i b e d V a l u e f o r Rhombic Duct. F i g u r e 7. F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t Duct A n g l e a f o r ^ I s o s c e l e s T r i a n g u l a r Duct. o F r a c t i o n of A B ^ F i g u r e 8. F o r c e d C o n v e c t i o n . R e s u l t a n t Normal Heat F l u x on t h e Boundary A g a i n s t i t s P r e s c r i b e d V a l u e f o r I s o s c e l e s T r i a n g u l a r Duct. F i g u r e 9. F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t Duct A n g l e a f o r R i g h t - a n g l e d T r i a n g u l a r Duct. 1 2 3 4 5 6 7 8 9 ASPECT RATIO, b/a F i g u r e 10. Combined F r e e and F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t A s p e c t R a t i o f o r R e c t a n g u l a r Duct. 90 80 70 60 50 40 30 20 10 A N G L E ol g F i g u r e 11. Combined F r e e and F o r c e d C o n v e c t i o n . V a r i a t i o n o f N u s s e l t Number A g a i n s t Angle a f o r Rhombic Duct. F i g u r e 12. Comparison o f Temperature D i s t r i b u t i o n on the Boundary o f Rhombic Duct, a = 9 0 ° , from the V a r i a t i o n a l F o r m u l a t i o n (34) w i t h F i n i t e -D i f f e r e n c e A p p r o x i m a t i o n , Appendix B. 45 30 25 L-1 r3$/c3N c 2 V A R I A T I O N A L V A R I A T I O N A L F O R M U L A T I O N (34) F O R I I U L A T I O N (C5) * = 9 0 ° R a = 0 0 •2 -3 -4 -5 FRACTION OF •6 •9 1-0 F i g u r e 13. Comparison o f R e s u l t a n t Normal Heat F l u x f o r Rhombic Duct, a = 90° from V a r i a t i o n a l F o r m u l a t i o n s (34) and (C5) A g a i n s t i t s P r e s c r i b e d V a l u e . Cn 10 •3 0 K=0 i 0 •7 >8 FRACTION OF AB F i g u r e 14. Temperature D i f f e r e n c e f o r D i f f e r e n t V a l u e s o f K A l o n g the Boundary o f a Square Duct a t Ra - 1000. F i g u r e 15. N u s s e l t Number V a r i a t i o n s o f a Square Duct f o r D i f f e r e n t V a l u e s o f K A g a i n s t R a y l e i g h Numbers. 8 7 cr LU cn 5 b LU CO CO 1 1 1 1 i i i 111 l i ASPECT i i i i 111 i I I RATIO = 2 i i i 111 i i i i i i 11 — 1 — 0 -I I I I i i i i i l i i i i 1111 i I I i i i II 1 i i I i i II I 10 2 RAYLEIGH NUMBER, F i g u r e 16. N u s s e l t Number V a r i a t i o n s o f a R e c t a n g u l a r Duct w i t h A s p e c t R a t i o = 2 f o r D i f f e r e n t V a l u e s o f K A g a i n s t R a y l e i g h Numbers cn Cn F i g u r e 17. N u s s e l t Number V a r i a t i o n s o f a R e c t a n g u l a r Duct w i t h A s p e c t R a t i o = 3 f o r D i f f e r e n t V a l u e s o f K A g a i n s t R a y l e i g h Numbers. LITERATURE CITED 58 LITERATURE CITED 1. H a l l m a n , T. M . , "Combined F o r c e d and Free Laminar Heat T r a n s f e r i n V e r t i c a l Tubes w i t h U n i f o r m I n t e r n a l Heat G e n e r a t i o n . " T r a n s . ASME, V o l . 78, 1956, pp. 1831-1841. 2. Han, L. S., "Laminar Heat T r a n s f e r i n R e c t a n g u l a r C h a n n e l s . " J . Heat T r a n s f e r , T r a n s . ASME, S e r . C, V o l . 81, 1959, pp. 121-128. 3. Tao, L. N., "On Combined Free and F o r c e d C o n v e c t i o n i n C h a n n e l s . " J . Heat T r a n s f e r , T r a n s . ASME, S e r . C, V o l . 82, 1960, pp. 233-238. 4. A g g a r w a l a , B. D. and M. I q b a l , "On L i m i t i n g N u s s e l t Number From Membrane A n a l o g y For Combined F r e e and F o r c e d C o n v e c t i o n Through V e r t i c a l D u c t s . " I n t . J . Heat Mass T r a n s f e r , V o l . 12, 1969, pp. 737-748. 5. Lu, P. C , "A T h e o r e t i c a l I n v e s t i g a t i o n o f Combined F r e e and F o r c e d C o n v e c t i o n Heat G e n e r a t i n g Laminar Flow I n s i d e V e r t i c a l P i p e s With P r e s c r i b e d Wall T e m peratures." M.S. T h e s i s , Kansas S t a t e C o l l e g e , Manhattan, Kansas, 1959. 6. I q b a l , M., B. D. Aggarwala and A. G. F o w l e r , "Laminar Combined F r e e and F o r c e d C o n v e c t i o n i n V e r t i c a l N o n - c i r c u l a r Ducts Under U n i -form Heat F l u x . " I n t . J . Heat Mass T r a n s f e r , V o l . 12, 1969, pp. 1123-1139. 59 7. Cheng, H. M., " A n a l y t i c a l I n v e s t i g a t i o n o f F u l l y D e v e l o p e d Laminar Flow F o r c e d C o n v e c t i o n Heat T r a n s f e r i n R e c t a n g u l a r Ducts With U n i f o r m Heat F l u x . " M.S. T h e s i s , M a s s a c h u s e t t s I n s t , o f T e c h -n o l o g y , M a s s a c h u s e t t s , 1957. 8. Yen, J . T., " E x a c t S o l u t i o n o f Laminar Heat T r a n s f e r i n Wedge-Shaped Passages With V a r i o u s Boundary C o n d i t i o n s . " W r i g h t A i r Development C e n t r e , Tech. R e p o r t 57-224, J u l y 1957. 9. E c k e r t , E. R. G., T. F. I r v i n e , J r . , and J . T. Yen, " L o c a l Laminar Heat T r a n s f e r i n Wedge-Shaped P a s s a g e s . " T r a n s . ASME, V o l . 80, 1958, pp. 1433-1438. 10. Tao, L. N., "The Second Fundamental Problem i n Heat T r a n s f e r o f Laminar F o r c e d C o n v e c t i o n . " J . A p p l i e d M e c h a n i c s , T r a n s . ASME, Se r . E, V o l . 84, 1962, pp. 415-419. 11. Sparrow, E. M. and A. H a j i - S h e i k h , "Flow and Heat T r a n s f e r i n Ducts o f A r b i t r a r y Shape With A r b i t r a r y Thermal Boundary C o n d i t i o n s . " J . Heat T r a n s f e r , T r a n s . ASME, S e r . C, V o l . 88, 1966, pp. 351-358. 12. Sparrow, E. M. and R. S i e g e l , "A V a r i a t i o n a l Method F or F u l l y D e v e l -oped Laminar Heat T r a n s f e r i n D u c t s . " J . Heat T r a n s f e r , T r a n s . ASME, S e r . C, V o l . 81, 1959, pp. 157-167. 13. Cheng, K. C , "Laminar F o r c e d C o n v e c t i o n i n R e g u l a r P o l y g o n a l Ducts With U n i f o r m P e r i p h e r a l Heat F l u x . " J . Heat T r a n s f e r , T r a n s . ASME, S e r . C, V o l . 91, 1969, pp. 156-157. 14. S a v i n o , J . M. and R. S i e g e l , "Laminar F o r c e d C o n v e c t i o n i n Rectangu-l a r Channels With Unequal Heat A d d i t i o n on A d j a c e n t S i d e s . " I n t . J . Heat Mass T r a n s f e r , V o l . 7, 1964, pp. 733-741. 15. S a v i n o , J . M., R. S i e g e l and E. C. B i t t n e r , " A n a l y s i s o f Laminar F u l l y Developed Heat T r a n s f e r i n T h i n R e c t a n g u l a r Channels With Fuel L o a d i n g Removed From the C o r n e r s . " Amer. I n s t , o f Chem. Eng., Chem. Eng. P r o g r e s s Symposium S e r i e s , No. 60, V o l . 61, 1965, pp. 84-96. 16. I q b a l , M., S. A. A n s a r i and B. D. A g g a r w a l a , " E f f e c t o f Buoyancy on F o r c e d C o n v e c t i o n i n V e r t i c a l R e g u l a r P o l y g o n a l D u c t s . " J . Heat T r a n s f e r , T r a n s . ASME, S e r . C, V o l . 92, 1970, pp. 237-244. 17. Perelman, T. L., "On C o n j u g a t e d Problems o f Heat T r a n s f e r . " I n t . J . Heat Mass T r a n s f e r , V o l . 3, 1961, pp. 293-303. 18. Rotem, Z., "The E f f e c t o f Thermal C o n d u c t i o n o f the Wall Upon Con-v e c t i o n From a S u r f a c e i n a Laminar Boundary L a y e r . " I n t . J . Heat Mass T r a n s f e r , V o l . 10, 1967, pp. 461-466. 19. D a v i s , E. J . and W. N. G i l l , "The E f f e c t s o f A x i a l C o n d u c t i o n i n the Wall on Heat T r a n s f e r With Laminar Flow." I n t . J . Heat Mass T r a n s f e r , V o l . 13, 1970, pp. 459-470. 20. S i e g e l , R. and J . M. S a v i n o , "An A n a l y t i c a l S o l u t i o n o f the E f f e c t o f P e r i p h e r a l Wall C o n d u c t i o n on Laminar F o r c e d C o n v e c t i o n i n 61 R e c t a n g u l a r C h a n n e l s . " J . Heat T r a n s f e r , T r a n s . ASME, V o l . 87, . S e r . C, 1965, pp. 59-66. 21. Knudsen, J . G. and D. L. K a t z , " F l u i d Dynamics and Heat T r a n s f e r . " McGraw-Hill Book Co., New York, N.Y., 1958. 22. Timoshenko, S. and J . N. G o o d i e r , "Theory o f E l a s t i c i t y . " McGraw-H i l l Book Company, New York, N.Y., 2nd E d i t i o n , 1951. APPENDICES APPENDIX A 64 APPENDIX A DEVELOPMENT OF HEAT FLUX EXPRESSION FOR CONJUGATE HEAT TRANSFER In o r d e r t o d e v e l o p the second thermal boundary c o n d i t i o n , equa-t i o n ( 1 2 ) , we w r i t e thermal energy b a l a n c e o v e r a s m a l l shaded e l e m e n t , F i g u r e 18. N e g l e c t i n g w a l l c o n d u c t i o n e f f e c t s i n the f l o w d i r e c t i o n and te m p e r a t u r e g r a d i e n t s a c r o s s the w a l l t h i c k n e s s and assuming thermal con-d u c t i v i t y o f the w a l l to be i n v a r i a n t w i t h t e m p e r a t u r e , we g e t k y d ^ d s = ^ d s + w k w lids-)** 5' <A1> which g i v e s , ^ = f b t . + ^ • (A2) By making a s i m p l e energy b a l a n c e o v e r the p o r t i o n o f the f l u i d f l o w i n g between s e c t i o n s 1 and 2 o f a d u c t , as shown i n f i g u r e below, we P U Cp A ( T, - Tz) = i P A z + Q A Az. , (A 3) F i g u r e 18. C o - o r d i n a t e System and A n a l y t i c a l M o del. where T-| and a r e the b u l k t e m p e r a t u r e s a t the two s e c t i o n s , and P i s the heat t r a n s f e r p e r i m e t e r o f the d u c t . S i m p l i f y i n g e q u a t i o n ( A 3 ) , we g e t \ = T" (UF' P U 9> C , . (A4) S u b s t i t u t i n g t h i s v a l u e o f ^ and o f ^ and from equa-t i o n (5C) i n t o e q u a t i o n ( A 2 ) , we g e t 2. where 5 = ^ j N = and i s the c o n d u c t i o n parameter. K i s a p r e s c r i b e d c o n s t a n t whose v a l u e may v a r y from z e r o t o i n -f i n i t y . C o n s t a n t K c o n t r o l s the c i r c u m f e r e n t i a l v a r i a t i o n o f w a l l tem-p e r a t u r e and h e a t f l u x . T h i s c o n s t a n t i s the r a t i o o f the c i r c u m f e r e n -t i a l c o n d u c t i o n a l o n g the d u c t w a l l to t h a t o f normal c o n d u c t i o n i n t o the f l u i d and i s i n f a c t the r a t i o of the l o c a l N u s s e l t number t o the B i o t number. When the p e r i p h e r a l w a l l c o n d u c t i o n i s z e r o , e q u a t i o n (A5) r e -duces t o = 0 - 2 5 Cl-F;. (A6) APPENDIX B 68 APPENDIX B F i n i t e D i f f e r e n c e A p p r o x i m a t i o n A f i n i t e d i f f e r e n c e p r o c e d u r e t o c a l c u l a t e V and <f) from e q u a t i o n (6) and (7) i s g i v e n below. A l t h o u g h the g e n e r a l method can be a p p l i e d t o rhombic, i s o s c e l e s t r i a n g u l a r and r i g h t - a n g l e d t r i a n g u l a r d u c t s , r e c -t a n g u l a r d u c t needs o n l y a s l i g h t m o d i f i c a t i o n . D e t a i l s a r e g i v e n f o r rhombic d u c t and because o f symmetry about t h e c o - o r d i n a t e a x e s , o n l y one q u a d r a n t w i l l be c o n s i d e r e d . The problem o f i n t e r p o l a t i o n a t the s l o p i n g boundary can be a v o i d e d by u s i n g a r e c t a n g u l a r g r i d , F i g u r e 19. The h o r i z o n t a l and v e r t i -c a l s t e p s i z e s can be w r i t t e n a s , — y * h = ~ r r = _,—-7-— h o r i z o n t a l s t e p s i z e , ( B l ) Z N S i n ° % where N i s the number o f i n t e r v a l s a l o n g the X o r Y a x e s . U s i n g the s t a n d a r d t h r e e - p o i n t a p p r o x i m a t i o n f o r the second d e r i -v a t i v e o f a f u n c t i o n , the L a p l a c i a n a t the p o i n t i , j can be w r i t t e n a s , 2- N Cos */2 = h t a n0 ^ v e r t i c a l s t e p s i z e , (B2) + (B3) F i g u r e 19. R e c t a n g u l a r G r i d System. 70 where, <pifj = dp . and Xx- = i - h ; Y j = j - k . L e t u be the o v e r - r e l a x a t i o n c o n s t a n t , then e q u a t i o n s (6) and (7) become, Vi'* = 5(77^5%) + l ) +Si* v 1 + 0 - " ] V i«*' ( B 4 ) = 1 ( H " W - W + S ' " i ^ } + 0 - » ) 4 « . <B5) where, S i j V = ViM,j + + ( V i J t , + V y - i ) . The t e m p e r a t u r e p o i n t s on the boundary a r e c a l c u l a t e d u s i n g the e q u a t i o n (B6) The above e q u a t i o n i s o b t a i n e d by u s i n g t h r e e p o i n t L a g r a n g i a n D i f f e r e n t i a t i o n f o r m u l a s i n e q u a t i o n ( 1 1 ) . Any one o f the c o r n e r p o i n t can be taken as r e f e r e n c e p o i n t (<j> = 0) • The temp e r a t u r e a t the o t h e r c o r n e r p o i n t can be c a l c u l a t e d by l i n e a r l y i n t e r p o l a t i n g w i t h the n e a r e s t n e i g h b o u r i n g p o i n t s . 71 The a l g o r i t h m proceeds as f o l l o w s : S i n c e e q u a t i o n s (6) and (7) a r e two e q u a t i o n s w i t h t h r e e unknowns, an i n i t i a l v a l u e o f L i s assumed. T h i s v a l u e was taken from the v a r i a -t i o n a l r e s u l t s . A l s o an i n i t i a l d i s t r i b u t i o n o f V and <|> i s assumed. Then new v a l u e s o f V and <t> a r e c a l c u l a t e d a t each i n t e r i o r p o i n t on the g r i d u s i n g e q u a t i o n s (B4) and (B5) and on the boundary u s i n g e q u a t i o n (B6). T h i s p r o c e s s o f r e c a l c u l a t i n g v a l u e s o f V and rj> i s r e p e a t e d u n t i l on two c o n s e c u t i v e i t e r a t i o n s , the a b s o l u t e d i f f e r e n c e between the new and o l d v a l u e s o f both V and <f> a t e v e r y p o i n t i s not more than e-j ( u s u a l l y £, = i o 5 ). Once the v a l u e s o f V and $ a r e c a l c u l a t e d , the c o n t i n u i t y equa-t i o n (8) has to be checked. Thus d e f i n e In f a c t L must be chosen such t h a t f ( L ) = 0 (= z^ p r a c t i c a l l y ) . I f t h i s does not h o l d , then L has t o be m o d i f i e d . A b e t t e r v a l u e o f L was d e t e r m i n e d u s i n g Newton's method f o r f i n d i n g a z e r o o f a n o n - l i n e a r e q u a t i o n , namely where n r e p r e s e n t s i t e r a t i o n number. Usi n g t h i s new v a l u e o f L, V and <}> a r e r e c a l c u l a t e d as d e s c r i b e d above. T h i s p r o c e s s i s r e p e a t e d u n t i l the c o n t i n u i t y e q u a t i o n (B7) i s (B7) Kir,-,) 72 s a t i s f i e d ( w i t h i n r e a s o n a b l e l i m i t , t h a t i s , - f ( L ) — £ z and G ^ ^ r l o 5 t o i o ) . Comments: 20 i n t e r v a l s (N = 20) were f i r s t used t o o b t a i n a rough shape o f V and (j) which was then used as an i n i t i a l d i s t r i b u t i o n f o r N = 50. I f the i n i t i a l d i s t r i b u t i o n o f V and cfi were made z e r o , the number o f i t e r -a t i o n s r e q u i r e d t o " s e t t l e " the shape o f V and <(> were p r o h i b i t i v e . The p r a c t i c a l d i f f i c u l t y e x p e r i e n c e d i n the p r e s e n t c a s e was i n g e t t i n g c o n v e r g e n c e o f the f u n c t i o n $ as the t e m p e r a t u r e f u n c t i o n was found t o o s c i l l a t e , as a f u n c t i o n o f the number o f i t e r a t i o n s , depending upon the i n i t i a l v a l u e chosen f o r L f o r the c a s e N - 50. The r e a s o n f o r t h i s i s t h a t the t e m p e r a t u r e v a l u e s on the boundary have t o be c h a n g i n g i n o r d e r t o s a t i s f y (B6) f o r each i t e r a t i o n . T h i s i n t u r n e f f e c t e d a l l the v a l u e s i n s i d e the boundary. To speed up c o n v e r g e n c e , the v a l u e s o f V were saved when cf> was a t an extreme, then when V was a t an extreme, i t was r e p l a c e d by the p r e v i o u s saved v a l u e s . T h i s reduced the time o f c o n v e r g e n c e . However, to speed up c o n v e r g e n c e , t h e r e s t i l l remains room f o r improvement. A number o f v a l u e s o f the o v e r - r e l a x a t i o n c o n s t a n t w was t r i e d but the v a l u e which gave the b e s t r e s u l t was 1.9. A s i m i l a r t r e a t m e n t t o s o l v e e q u a t i o n (6) and (7) f o r c o n s t a n t p e r i p h e r a l w a l l t e m p e r a t u r e boundary c o n d i t i o n i s o u t l i n e d i n [ 6 ] . The computer time taken i n the p r e s e n t c a s e was about 1.5 times as i n [6] and the s l i g h t i n c r e a s e o r d e c r e a s e i n the v a l u e o f L was found t o i n -c r e a s e the time o f c o n v e r g e n c e by about a f a c t o r o f 2. APPENDIX C 75 APPENDIX C VARIATIONAL SOLUTION FOR LAMINAR FORCED CONVECTION For pure f o r c e d c o n v e c t i o n , when the buoyancy e f f e c t s a r e n e g l e c -t e d , the e q u a t i o n o f motion reduces t o V V - - L . ( C l ) Eq u a t i o n ( C l ) w i t h boundary c o n d i t i o n (10) f o r l a m i n a r f l o w i n d u c t s i s e q u i v a l e n t to t o r s i o n a l problems o f beams i n s o l i d m e c h a n i c s . S o l u t i o n s o f t h e s e e q u a t i o n s a r e a v a i l a b l e f o r many g e o m e t r i e s o f e n g i n -e e r i n g i n t e r e s t . T h e r e f o r e we can p r o c e e d w i t h the assumption t h a t the s o l u t i o n f o r the v e l o c i t y f i e l d i s known and w r i t e (7) a s , = ft C*^) = V - F , (C2) where f,, i s known and the h e a t g e n e r a t i o n parameter F i s abso r b e d i n i t . (C2) i s a P o i s s o n ' s e q u a t i o n . S o l u t i o n o f e q u a t i o n (C2) w i t h e q u a t i o n (11) i s as f o l l o w s : C o n s i d e r i n g the v e l o c i t y f u n c t i o n t o be known, we w r i t e , 1 = // f (V(,4>/4>x, dxdy - ( 4 3 ) d S , (C3) R s which g i v e s the same n e c e s s a r y c o n d i t i o n s as e q u a t i o n s (18) and (19) i n 76 R and on S r e s p e c t i v e l y . Comparing e q u a t i o n (21) w i t h e q u a t i o n (18) we g e t -f = - V ^ + F < f > - 4 ^ - ^ + J C X , Y ) . (C4) S u b s t i t u t i n g the v a l u e s o f and i n t o e q u a t i o n (19) we get Q - - cz 4>. Hence f o r l a m i n a r f o r c e d c o n v e c t i o n , v a r i a t i o n a l i n t e g r a l equa-t i o n can be w r i t t e n as I =// (-Vc£ + F4>-^ - ^ ) c J x d Y + C 2f4>ds, R 5 o r I = j j [ c ^ 2 + 4>YZ + 2 ( V - F ) 4? ] c i x d y -icj^ds. (C5) R s E q u a t i o n (C5) can a l s o be o b t a i n e d by d i v i d i n g e q u a t i o n (34) by Ra t h r o u g h o u t and l e t t i n g Ra -+ °°. E q u a t i o n (C5) w i l l be u t i l i z e d t o d e t e r m i n e N u s s e l t numbers f o r r e c t a n g u l a r , rhombic, i s o s c e l e s t r i a n g u l a r and r i g h t - a n g l e d t r i a n g u l a r d u c t s . In e q u a t i o n ( C 5 ) , the t e m p e r a t u r e f u n c t i o n <j> i s the o n l y unknown. As mentioned e a r l i e r , f u n c t i o n <j> does not have t o s a t i s f y e x a c t l y the boundary c o n d i t i o n ( 1 1 ) . I t i s t h i s l a c k o f r e s t r i c t i o n on $ which makes (C5) much e a s i e r t o han d l e compared t o t h a t g i v e n i n [ 1 2 ] . The co-o r d i n a t e system chosen i s shown i n F i g u r e 2. We w i l l now show how the f u n c t i o n s <j> were d e t e r m i n e d f o r the f o u r d u c t s . For r e c t a n g u l a r d u c t , e x a c t v e l o c i t y e x p r e s s i o n i s known [ 2 1 , 22] and can be wri t t e n a s , V = f l Z I cos cos ^ (C6) 71=1,3,5 171=1,3,5 where a^- - •—z ^hzm , 4 K a2- b2-/ L r= 5" y 1 6 b ^ (-o**^* -run A : 4 ( nSr + r£\ Because o f the symmetry o f the d u c t about the c o - o r d i n a t e a x e s , odd powers o f X and Y can be e x c l u d e d . Thus t a k i n g t e n terms i n the e x p r e s s i o n f o r <|>, we w r i t e , 2 2 4 Z 2. 4 <p = ff0 + B, X + BzY + B3 X + B4 X Y + # 5 Y * B6 -/- B7 x 4 y z * # 5 x 2y^ + 8? y6. (C7) I t may be noted t h a t the number o f terms i n <|> e x p r e s s i o n (C7) have been taken as ten compared to s i x i n e q u a t i o n ( 4 0 ) . The r e a s o n f o r t h i s i s t h a t the v a r i a t i o n a l e x p r e s s i o n (C5) i s s i m p l e r compared to e q u a t i o n ( 3 4 ) . I n s e r t i n g e q u a t i o n s (C6) and (C7) i n t o (C5) and c a r r y i n g o u t the o p t i m i z a t i o n p r o c e s s , as o u t l i n e d e a r l i e r , we g e t 10 e q u a t i o n s w i t h 11 unknowns, L b e i n g an unknown c o n s t a n t . The l l * ' 1 e q u a t i o n i s p r o v i d e d by 78 ( 9 ) . These e q u a t i o n s a r e then s o l v e d t o g e t the unknown c o - e f f i c i e n t s Bj , which a r e then used t o c a l c u l a t e N u s s e l t number g i v e n by e q u a t i o n ( 1 3 ) . S i n c e e x a c t v e l o c i t y e x p r e s s i o n s a r e not known f o r rhombic, i s o s c e l e s t r i a n g u l a r and r i g h t - a n g l e d t r i a n g u l a r d u c t s , a p p r o x i m a t e e x p r e s s i o n s from [6] f o r Ra = 0 have been used. The a p p r o x i m a t e temper-a t u r e f u n c t i o n chosen f o r rhombic d u c t has the form 4> - Bo + Bi x + B2 y2 + B3 x4 + B4Y* + Bs x y2 +• B6 X 6 + B7 y6 + BS x V + 8, x2y + B,A xV- (C8) The t e m p e r a t u r e f u n c t i o n s f o r the two t r i a n g u l a r d u c t s , each con-t a i n i n g t e n c o - e f f i c i e n t s were assumed a s , 4> = Ba + B,Y + B2X + B3Y + B4*Y + Bsy + e6 x y + B7 x y + e8 y + B9 x • (C9) f o r the i s o s c e l e s shape; w h i l e f o r the r i g h t - a n g l e d t r i a n g u l a r , 4> = Bo + Bi x + BZ y -r B3 x + B4 x y + Bsy~ + B& x3 + By xy + B& xy 2 + B? y3. (cio) An e x a m i n a t i o n o f the N u s s e l t number v a l u e s f o r the f o r c e d c o n v e c -t i o n through the f o u r d u c t shapes i s p r e s e n t e d under D i s c u s s i o n . APPENDIX D 80 APPENDIX D VARIATIONAL FORMULATION FOR CONJUGATE PROBLEM A g e n e r a l v a r i a t i o n a l e x p r e s s i o n f o r the second thermal boundary c o n d i t i o n , e q u a t i o n ( 1 2 ) , i s o b t a i n e d i n t h i s a p p e n d i x . Such an e x p r e s s i o n can be o b t a i n e d by e x t e n d i n g e q u a t i o n (34) to c o n s i d e r t h e e f f e c t s o f p e r i p h e r a l w a l l c o n d u c t i o n . Thus f o r e x p r e s s i o n ( 3 4 ) , we have Si - -2 ff ( V V + Ra.4> + L) c J v d x d Y + 2 9.o,(({vcp - v+F) cf^ d x d y F o r t he problem a t hand, i . e . , c o n j u g a t e problem, a l l terms i n 61 v a n i s h e x c e p t t he l a s t . To make t h i s term v a n i s h , we mo d i f y 61 and w r i t e i t a s , ( D l ) Si = - 2 ff ( w + &OL<P +L) cfy dxdy +2 R<x J j ( v ^ - V + F) x<\y + 2 j p v ^ d s - Z R c j ^ ^ - C . - K ^ O d s , (D2) and c o r r e s p o n d i n g l y w r i t e I a s , (D3) 

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