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UBC Theses and Dissertations

Conductive and convective heat transfer with radiant heat flux boundary conditions Sikka, Satish 1969

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CONDUCTIVE AND C0NVECTIVE HEAT TRANSFER WITH RADIANT HEAT FLUX BOUNDARY CONDITIONS by SATISH SIKKA B.Sc. Mech. Eng., U n i v e r s i t y of D e l h i , D e l h i , I n d i a , 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. i n the Department of Mechanical E n g i n e e r i n g We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1969 r In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e 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 b y t h e Head o f my De p a r t m e n t 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 n o t 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 . D e p a r t m e n t o f MECHANICAL ENGINEERING 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 Date A p r i l 2 5 , 1 9 6 9 ABSTRACT Some c o n d u c t i v e and c o n v e c t i v e h e a t t r a n s f e r p r o b l e m s w i t h r a d i a t i v e b o u n d a r y c o n d i t i o n s a r e a n a l y s e d t h e o r e t i c a l l y . T h r e e s p e c i f i c p r o b l e m s have b e e n . a n a l y s e d . The s t u d y h a s , t h e r e f o r e , b e e n d i v i d e d i n t o t h r e e p a r t s . In P a r t I t h e t e m p e r a t u r e d i s t r i b u t i o n p r o d u c e d i n - l o n g , s o l i d c i r c u l a r and r e c t a n g u l a r c y l i n d e r s and a s o l i d s p h e r e i n i n t e r p l a n e t a r y s p a c e i s s t u d i e d . The s o l i d b o d i e s r e c e i v e p a r a l l e l r a d i a t i o n f l u x on one s i d e and emanate r a d i a n t e n e r g y t o t h e i r s u r r o u n d i n g s a t z e r o d e g r e e R a n k i n e . S t e a d y s t a t e , -con -s t a n t p h y s i c a l and s u r f a c e p r o p e r t i e s , and no h e a t l o s s by c o n -v e c t i o n a r e a s s u m e d . S o l u t i o n o f t h e l i n e a r c o n d u c t i o n e q u a t i o n w i t h n o n l i n e a r b o u n d a r y c o n d i t i o n s i s o b t a i n e d by two a p p r o x i m a t e , s e m i - a n a l y t i c a l m e t h o d s , ( i ) p o i n t m a t c h i n g and ( i i ) l e a s t -s q u a r e s f i t t i n g . The r e s u l t s a r e c o m p a r e d w i t h e a r l i e r r e s u l t s o b t a i n e d by a v a r i a t i o n a l m e t h o d . The l e a s t - s q u a r e s f i t m e t h o d a p p e a r s t o be mos t s u i t a b l e r e g a r d i n g a c c u r a c y and s i m p l i c i t y i n . c o m p u t a t i o n . I t s a c c u r a c y d o e s n o t a p p e a r t o d e p e n d a p p r e c i a b l y e i t h e r on t h e r a d i a t i o n - c o n d u c t i o n p a r a m e t e r o r on t h e s u r f a c e a b s o r p t i v i t y . The e f f e c t o f s e m i - g r a y n e s s o f t h e r e c e i v i n g s u r f a c e i s a n a l y s e d . In P a r t I I t h e 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 o f a c i r c u -l a r f i n d i s s i p a t i n g h e a t f r o m i t s s u r f a c e by c o n v e c t i o n and r a d i a t i o n a r e a n a l y s e d . The t e m p e r a t u r e i s a s sumed u n i f o r m a l o n g t h e t b a s e o f t h e f i n and c o n s t a n t p h y s i c a l and s u r f a c e p r o p e r t i e s a r e a s s u m e d . T h e r e i s r a d i a n t i n t e r a c t i o n be tween t h e f i n and i i i i t s base. Two separate situations are considered. In the f i r s t s i t u a t i o n heat transfer from the end of the f i n i s neglected. Solution of the l i n e a r conduction equation with nonlinear boun-dary Conditions has been obtained by the least-squares f i t method. A solution has also been obtained by the f i n i t e difference method and the re s u l t s compared. Results are presented for a wide range of environmental conditions and physical and surface properties of the f i n . In the second s i t u a t i o n heat transfer from the end of the f i n i s also included i n the analysis. The solution i s obtained by a f i n i t e difference procedure. I t i s shown that neglecting heat transfer from the end i s a good approximation for long f i n s or for f i n s of high thermal conductivity material. In Part III the problem of laminar heat transfer i n a c i r c u l a r tube under radiant heat flux boundary conditions has been analysed. F u l l y developed v e l o c i t y p r o f i l e i s assumed and the tube i s considered stationary. A steady radiant energy f l u x i s being incident on one half of the tube circumference while the f l u i d emanates heat through the wall on a l l sides by radiat i o n to a zero degree temperature environment. A solution by f i n i t e difference procedure has been obtained. The temperature d i s t r i -bution and the Nusselt number va r i a t i o n are presented for a wide range of the governing physical parameters. TABLE OF CONTENTS Page ABSTRACT . . . . . . . i i LIST OF TABLES V v i i LIST OF FIGURES v i i i ACKNOWLEDGEMENTS X GENERAL INTRODUCTION . . . . . . . . . . . . x i PART I Temperature D i s t r i b u t i o n i n S o l i d Bodies i n I n t e r p l a n e t a r y Space . . . . 1 ABSTRACT 2 NOMENCLATURE 3 INTRODUCTION 4 FORMULATION AND SOLUTION OF THE PROBLEM 8 a) C i r c u l a r C y l i n d e r 9 b) Rectangular C y l i n d e r 13 c) Sphere 16 DISCUSSION OF RESULTS 19 a) C i r c u l a r C y l i n d e r 20 b) Rectangular C y l i n d e r 23 c) Sphere . . . . . 24 Temperature D i s t r i b u t i o n 25 E f f e c t of Semi-Grayness of Surface P r o p e r t i e s 26 I n c l u s i o n of Convection Heat T r a n s f e r i n the A n a l y s i s 27 CONCLUSIONS . . . . . . . . . . . 28 V Page PART II Temperature D i s t r i b u t i o n and Effectiveness of a Radiating and Convecting C i r c u l a r F i n . . 44 ABSTRACT 45 NOMENCLATURE 46 INTRODUCTION 48 FORMULATION AND SOLUTION OF THE PROBLEM 51 Section A - No Heat Transfer from the End o f the F in 52 DISCUSSION OF RESULTS 59 Section B - Heat Transfer from the End o f the Fin i s Not Neglected 63 COMPARISON OF SECTION A WITH SECTION B . . . . . . . . 66 CONCLUSIONS 67 PART III Laminar Heat Transfer i n a C i r c u l a r Tube under Solar Radiation i n Space 81 ABSTRACT . . . . . . . . . 82 NOMENCLATURE 83 INTRODUCTION . . 86 FORMULATION OF THE PROBLEM 88 METHOD OF SOLUTION 92 The F i n i t e Difference Procedure 92 Determination of Nusselt Number 94 Transformation of Coordinates . . . . . . 96 Overall Energy Balance 97 DISCUSSION OF RESULTS 100 Average Wall Temperature . 100 Nusselt Number 102 v i Page Angular Wall Temperature D i s t r i b u t i o n 107 Overall Energy Balance 108 CONCLUSIONS 109 BIBLIOGRAPHY 129 APPENDICES . 135 APPENDIX A — Evaluation of Expressions for Overall Energy Balance Error 136 APPENDIX B - Evaluation of Configuration Factor F * V A 2 . . . . . . . . 139 APPENDIX C - The F i n i t e Difference Procedure . . . . . 142 LIST OF TABLES Table Page 1-1 E f f e c t of A and a on the v a l u e s of the c o e f f i c -i e n t s a. of Equation (1-8) - - - c i r c u l a r C y l i n d e r 29 1-2 Comparison of values of the c o e f f i c i e n t a„ o b t a i n e d by d i f f e r e n t methods - - -C i r c u l a r C y l i n d e r 30 1-3 Comparison of the o v e r a l l percentage e r r o r (Eq. l-33a) obtained by d i f f e r e n t methods - - -C i r c u l a r C y l i n d e r 31 1-4 E f f e c t of A and a on the values of the c o e f f i c -i e n t s A i of Equation (1-21) -Rectangular C y l i n d e r . . . 32 1-5 Comparison of values bf the c o e f f i c i e n t A 0 o b t a i n e d by d i f f e r e n t methods - - -Rectangular C y l i n d e r . . . . . . 33 1-6 Comparison of the o v e r a l l percentage e r r o r (Eq. l-33b) o b t a i n e d by d i f f e r e n t methods - - -Rectangular C y l i n d e r . . . 34 1-7 E f f e c t of A and a on the v a l u e s of the c o e f f i c -i e n t s a^ of Equation (1-30) - - - Sphere 35 1-8 Comparison of values of the c o e f f i c i e n t a Q o b t a i n e d by d i f f e r e n t methods - - - Sphere . . . . 36 1- 9 Comparison of the o v e r a l l percentage e r r o r (Eq. l-33c) obtained by d i f f e r e n t methods - - -Sphere 37 2- 1 Comparison of f i n e f f e c t i v e n e s s v a l u e s w i t h those obtained by Sparrow and Niewerth [ 3 9 ] . . . . 68 3- 1 Values of the dimensionless forms of heat f l u x , average w a l l temperature and bulk temperature f o r y=5 110 3-2 O v e r a l l e r r o r i n energy balance (Eq.3-36) . . . . I l l LIST OF FIGURES F i g u r e Page 1-1 Coordinate System Rectangular and C i r c u l a r C y l i n d e r s . . . . 38 1-2 S p h e r i c a l Coordinate System • 39 1-3 Isothermals f o r Rectangular C y l i n d e r (B=0.5, A=0.1, a=0.5) . 40 1-4 Isothermals f o r Rectangular C y l i n d e r (B=2, A=0.1, a=0.5) 41 1-5 Isothermals f o r Rectangular C y l i n d e r (B=2, A=10~ 3, a=0.5) 42 1- 6 Isothermals f o r Rectangular C y l i n d e r (B=2, A=10 - 3, a=l) . 43 2- 1 C i r c u l a r F i n Geometry 69 2-2 V a r i a t i o n of C o n f i g u r a t i o n F a c t o r with A x i a l Length and F i n Base Radius 70 2-3 E f f e c t of A on A x i a l Temperature D i s t r i b u t i o n . . 71 2-4 E f f e c t of N on A x i a l Temperature D i s t r i b u t i o n . . 72 2-5 E f f e c t of Base Surface E m i s s i v i t y and A on F i n E f f e c t i v e n e s s 73 2-6 E f f e c t of Environment Temperatures on F i n E f f e c t i v e n e s s 74 2-7 E f f e c t o f N and A on F i n E f f e c t i v e n e s s 75 2-8 E f f e c t o f F i n Length on F i n E f f e c t i v e n e s s . . . . 76 2-9 E f f e c t o f F i n Base Radius on F i n E f f e c t i v e n e s s 77 2-10 Comparison of A x i a l Temperature D i s t r i b u t i o n o b tained f o r Two C o n d i t i o n s 78 2-11 F i n E f f e c t i v e n e s s A g a i n s t F i n Length f o r Both C o n d i t i o n s 7.9 F i g u r e Page 2- 12 F i n i t e D i f f e r e n c e R e p r e s e n t a t i o n . . . 80 3- 1 T u b e N o m e n c l a t u r e and G e o m e t r y . 112 3-2 V a r i a t i o n o f A v e r a g e W a l l T e m p e r a t u r e w i t h E, (y=0.5) 113 3-3 V a r i a t i o n o f A v e r a g e W a l l T e m p e r a t u r e • w i t h 5 (y=5) 114 3-4 C r i t i c a l R e l a t i o n s h i p b e t w e e n y and \\> 115 3-5 V a r i a t i o n o f A v e r a g e W a l l T e m p e r a t u r e w i t h 5 U=25) 116 3-6 V a r i a t i o n o f N u s s e l t Number w i t h 5 f o r Y=0.5 (<|><*cr) i 1 7 3-7 V a r i a t i o n o f N u s s e l t Number w i t h £ f o r Y=0.5 U>^cr) • • 118 3-8 V a r i a t i o n o f N u s s e l t Number w i t h E; f o r Y=0.5 (^-<Pcr) . 119 3-9 V a r i a t i o n o f D i m e n s i o n l e s s H e a t F l u x w i t h £ f o r Y=0-5 (^^^>cr) . . . . . 120 3-10 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 and B u l k T e m p e r a t u r e w i t h B, f o r Y=0 .5 (ty-ty ) . . . . . . . 121 3-11 V a r i a t i o n o f N u s s e l t Number w i t h £ f o r .Y=5 (^<^cr) 122 3-12 V a r i a t i o n o f N u s s e l t Number w i t h £ f o r y=5 (ib>\b ) 123 3-13 V a r i a t i o n o f N u s s e l t Number w i t h £ : f o r ij>=0 . . . . 124 3-14 V a r i a t i o n o f N u s s e l t Number w i t h £ f o r ^ =25 . . . 125 3-15 A n g u l a r W a l l T e m p e r a t u r e D i s t r i b u t i o n (Y=0.5, 5=0.1) 126 3-16 A n g u l a r W a l l T e m p e r a t u r e D i s t r i b u t i o n (^=25, Y=5) • • 127 3-17 A n g u l a r W a l l T e m p e r a t u r e D i s t r i b u t i o n {ip=25, £ = 0.1) 128 ACKNOWLEDGEMENTS The author wishes t o express h i s deep g r a t i t u d e t o Dr. M. I q b a l who devoted c o n s i d e r a b l e time on the guidance of t h i s study. S i n c e r e thanks are a l s o extended t o P r o f e s s o r Z. Rotem f o r h i s many u s e f u l s u g g e s t i o n s . In a d d i t i o n , the author a l s o wishes t o acknowledge the very u s e f u l advice g i v e n by Mr. A.G. Fowler and Mr. K. Teng of the Computing Centre, U n i v e r s i t y o f B r i t i s h Columbia; and Dr. B.D. Aggarwala of the Mathematics Department, U n i v e r s i t y o f C a l g a r y . F i n a n c i a l support of the N a t i o n a l Research C o u n c i l o f Canada and use of the Computing Centre f a c i l i t i e s a t the U n i v e r s i t y of B r i t i s h Columbia are g r a t e f u l l y acknowledged. GENERAL INTRODUCTION In s p a c e c r a f t a p p l i c a t i o n s conductive and c o n v e c t i v e heat t r a n s f e r problems w i t h r a d i a t i v e boundary c o n d i t i o n s are f r e q u e n t l y encountered. S p a c e c r a f t components exposed t o f r e e space d i s s i p a t e heat mostly by r a d i a t i o n and may a l s o r e c e i v e s o l a r r a d i a t i o n . Extended s u r f a c e s are o f t e n used to d i s s i p a t e energy from the s p a c e c r a f t f o r m a i n t a i n i n g a thermal e q u i l i b r i u m Some o f the extended s u r f a c e s may be i n the form of f i n - t u b e r a d i a t o r s , a c i r c u l a t i n g f l u i d b e i n g used t o c a r r y heat from w i t h i n the s p a c e c r a f t to the r a d i a t o r s . The temperature d i s t r i -b u t i o n i n and heat t r a n s f e r from these s u r f a c e s are of g r e a t e n g i n e e r i n g importance. In the p r e s e n t work three s p e c i f i c problems, a l l i n the same area, but of i n c r e a s i n g complexity, have been analyzed. The study has, t h e r e f o r e , been d i v i d e d i n t o three p a r t s . P a r t I de a l s w i t h the temperature d i s t r i b u t i o n i n long , s o l i d c i r c u l a r and r e c t a n g u l a r c y l i n d e r s and s o l i d spheres i n i n t e r p l a n e t a r y , space. P a r t I I analyzes the heat t r a n s f e r c h a r a c t e r i s t i c s o f a c i r c u l a r f i n d i s s i p a t i n g heat by c o n v e c t i o n and r a d i a t i o n . P a r t I I I analyzes the problem of laminar heat t r a n s f e r i n a c i r c u l a r tube under r a d i a n t heat f l u x boundary c o n d i t i o n s . Each p a r t c o n t a i n s i t s own nomenclature, a b s t r a c t and i n t r o d u c t i o n so t h a t i t can be read independently o f the oth e r p a r t s . References t o p u b l i s h e d l i t e r a t u r e and Appendixes a r e , however, combined together a t the end. P A R T I TEMPERATURE DISTRIBUTION IN SOLID BODIES IN INTERPLANETARY SPACE ABSTRACT The temperature d i s t r i b u t i o n produced i n l o n g , s o l i d c i r c u l a r and r e c t a n g u l a r c y l i n d e r s and a s o l i d sphere i n i n t e r -p l a n e t a r y space i s s t u d i e d . The s o l i d bodies r e c e i v e p a r a l l e l r a d i a t i o n f l u x on one s i d e and emanate r a d i a n t energy t o t h e i r surroundings at zero degree Rankine. Steady s t a t e , c o n s t a n t p h y s i c a l and s u r f a c e p r o p e r t i e s , and no heat l o s s by c o n v e c t i o n are assumed. S o l u t i o n of the l i n e a r conduction equation w i t h n o n l i n e a r boundary c o n d i t i o n s i s o b t a i n e d by two approximate, s e m i - a n a l y t i c a l methods, (i) p o i n t matching and ( i i ) l e a s t -squares f i t t i n g . The r e s u l t s are presented i n s e r i e s form and are compared w i t h e a r l i e r r e s u l t s o b t a i n e d by a v a r i a t i o n a l method. The l e a s t - s q u a r e s f i t method appears t o be most s u i t a b l e r e g a r d i n g accuracy and s i m p l i c i t y i n computation. I t s accuracy does not appear t o depend a p p r e c i a b l y e i t h e r on the r a d i a t i o n -c o nduction parameter or on the s u r f a c e a b s o r p t i v i t y . The e f f e c t of semi-grayness of the r e c e i v i n g s u r f a c e i s anal y s e d . NOMENCLATURE 2a d i a m e t e r o f c i r c u l a r c y l i n d e r , d i a m e t e r o f sphere o r w i d t h o f r e c t a n g u l a r c y l i n d e r , f t . a^, A^ c o e f f i c i e n t s i n assumed tem p e r a t u r e p r o f i l e s 3 4 4 A eaG a /k , r a d i a t i o n - 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 2 b h e i g h t o f r e c t a n g u l a r c y l i n d e r , f t . B b/a, one h a l f o f t h e d i m e n s i o n l e s s h e i g h t o f r e c t a n g u l a r c y l i n d e r 2 G i n c i d e n t r a d i a t i o n f l u x , B T U / h r . f t . k t h e r m a l c o n d u c t i v i t y o f the m a t e r i a l , BTU/hr. f t . °R. R r / a , d i m e n s i o n l e s s r a d i u s T t e m p e r a t u r e a t any p o i n t i n the s o l i d , °R. x, y r e c t a n g u l a r c o o r d i n a t e s X, Y x / a , y/a, d i m e n s i o n l e s s r e c t a n g u l a r c o o r d i n a t e s Greek Symbols a c o e f f i c i e n t o f a b s o r p t i v i t y , d i m e n s i o n l e s s e c o e f f i c i e n t o f e m i s s i v i t y , d i m e n s i o n l e s s — 8 o S t e f a n - B o l t z m a n n c o n s t a n t , 0.1714 x 10 BTU/hr.ft.2 °R4 X kT/Ga, d i m e n s i o n l e s s t e m p e r a t u r e r , 6 , 4> c i r c u l a r and s p h e r i c a l c o o r d i n a t e s INTRODUCTION With advances i n space e x p l o r a t i o n , i n t e r e s t i n problems d e a l i n g w i t h the e f f e c t of s o l a r r a d i a t i o n on s p a c e c r a f t i s becoming of i n c r e a s i n g importance. Not on l y i s the temperature l e v e l o f importance, but a l s o the temperature v a r i a t i o n through-out the body. The temperature v a r i a t i o n a f f e c t s the s p a c e c r a f t i n s e v e r a l ways. I t induces thermal s t r e s s e s and causes d e f o r -mation of the s t r u c t u r a l members of the s p a c e c r a f t , which a f f e c t s the motion and s t a b i l i t y of the s a t e l l i t e . Furthermore, i t i a f f e c t s the instrument and b a t t e r y performance and i t may i n t r o -duce high-temperature regions t h a t may cause l o c a l b o i l i n g i n c r y o g e n i c f u e l tanks. The thermal energy i n a s p a c e c r a f t w i l l be g e n e r a l l y due to two sources, s o l a r energy absorbed by d i r e c t i n c i d e n c e or r e f l e c t i o n s and r a d i a t i o n from p l a n e t s and the energy generated by e l e c t r o n i c instruments w i t h i n the s p a c e c r a f t i t s e l f . The l e v e l of temperature t h a t a body w i l l a c q u i r e under these energy sources w i l l depend upon the amount of t h i s energy and the s u r f a c e a b s o r p t i v i t y and e m i s s i v i t y of the body. The temperature v a r i -a t i o n w i t h i n the body w i l l , i n a d d i t i o n , " be a f f e c t e d by the shape o f the o b j e c t , r o t a t i o n of the body with r e s p e c t t o the source of r a d i a n t energy, the c o n d u c t i v i t y and t h i c k n e s s of the w a l l and the presence or absence of i n t e r n a l r a d i a t i o n . Knowledge of temperature d i s t r i b u t i o n w i t h i n a space-c r a f t i s , t h e r e f o r e , d e s i r e d f o r thermal c o n t r o l and/or thermal s t r e s s a n a l y s i s f o r design. 5 In free space, heat transfer by convection from the surface w i l l be n e g l i g i b l e . Therefore, the thermal problem i n -volves only conduction within s o l i d material and radiat i o n at i t s surfaces. For many s o l i d materials that can be used for the type of application under consideration and under the temper-ature range generally encountered, i t can be safely assumed that t h e i r physical properties are invariant with temperature. There-fore, the d i f f e r e n t i a l equation governing heat flow reduces to the l i n e a r form. However, the surface boundary condition w i l l be nonlinear due to the fourth-power law consideration. Exact a n a l y t i c a l solution of such a problem appears to be quite d i f f i -c u l t i f at a l l possible. A standard approach to solution of such problems i s by d i r e c t f i n i t e - d i f f e r e n c e procedure. The transient, one-dimensional conduction problem with radi a t i v e boundary conditions has received considerable attention. * Mann and Wolf [1] considered the problem of heating a semi-.. i n f i n i t e s o l i d , whose i n i t i a l temperature i s zero, by a non-linear heat-transfer process. Abarbanel [2] extended t h i s for the case of a f i n i t e i n i t i a l temperature. He employed Laplace transforms technique and solved the r e s u l t i n g i n t e g r a l equations numerically. Jaegar [3] used a series method of solution while Chambre[4] solved the Volterra i n t e g r a l equations by the method of success-ive approximations. Goodman [5] developed an approximate mathematical tech-nique introducing the concept of 'heat balance i n t e g r a l ' and applied i t to problems involving a change of phase and radiati o n heat f l u x . I t i s s i m i l a r to the well known Karman-Pohlhausen •Numbers i n square brackets re f e r to references i n the Bibliography. 6 method for boundary layer flow. Later, [6], he used the heat balance i n t e g r a l to solve the problem of transient conduction i n a semi-in f i n i t e s o l i d slab subjected to a heat f l u x which i s a function of temperature and time. He [7] then extended the method to include temperature-dependent thermal properties i n the analysis. Many authors used Goodman's method i n t h e i r work. Schneider [8] used Goodman's method to extend the work of reference [6] for f i n i t e bodies. Roberts [9] used i t to study the temperature d i s t r i b u t i o n i n cylinders heated by r a d i a t i o n . F a i r a l l , et al.[10] used f i n i t e - d i f f e r e n c e techniques to solve the one-dimensional transient problem for a slab of f i n i t e thickness. Biot [11, 12] introduced the concept of penetration depth and t r a n s i t time and applied the v a r i a t i o n a l technique to solve one-dimensional transient conduction problems with rad i a t i v e boundary conditions. Lardner [13] applied Biot's v a r i a t i o n a l p r i n c i p l e to a number of d i f f e r e n t one-dimensional heat conduc-ti o n problems with rad i a t i v e boundary conditions. Richardson [14] also employed Biot's method to solve the problem of unsteady one-dimensional conduction i n a se m i - i n f i n i t e slab with the heat th f l u x at the surface proportional to the n power of the surface temperature. Problems pertaining d i r e c t l y to solar heating have been treated by references [15-30], among others. Most of them have given approximate a n a l y t i c a l or numerical solutions of l i n e a r -ized governing equations. Charnes and Raynbr [15] used pertur-bation analysis to determine the approximate, l i n e a r i z e d solution to the problem o f s o l a r h e a t i n g o f a r o t a t i n g , t h i n - w a l l e d , c i r c u l a r c y l i n d e r . N i c h o l s [16] analysed s u r f a c e temperature d i s t r i b u t i o n i n t h i n - w a l l e d spheres, cones and c y l i n d e r s sub-j e c t e d t o s o l a r r a d i a t i o n i n space. The s o l u t i o n s b y . N i c h o l s f o r the one-dimensional problem were o b t a i n e d by numerical i n t e g r a -t i o n o f the d i f f e r e n t i a l e q u a t i o n s . Other approximate a n a l y t i c a l s o l u t i o n s of problems i n t h i s f i e l d have been presented by r e f e r e n c e s [17-24].Schmidt and Hanawalt [24] d i s c u s s e d the case of a n o n r o t a t i n g c y l i n d r i c a l s h e l l exposed t o s o l a r and e a r t h r a d i a t i o n . They p o i n t e d out the importance o f the r a t i o o f a b s o r p t i v i t y t o e m i s s i v i t y on the temperature o f the s h e l l . Z e r k l e and Sunderland [25] used a t h e r m a l - e l e c t r i c a l analogue computer t o study the t r a n s i e n t , one-dimensional temperature d i s t r i b u t i o n i n a s l a b s u b j e c t e d t o s o l a r r a d i a t i o n . S a n d o r f f and Prigge [26] and Hanel [27] i n v e s t i g a t e d the problem of temperature c o n t r o l of s a t e l l i t e s and space v e h i c l e s . In order to o b t a i n maximum s t r e s s and c u r v a t u r e produced we are i n t e r e s t e d i n the maximum temperature d i f f e r e n c e i n a member. T h i s r e s u l t s when a member i s not r o t a t i n g . Approximate a n a l y t i c a l s t u d i e s of the s t a t i o n a r y cases have been r e c e n t l y presented[28, 29]. In the p r e s e n t a n a l y s i s the temperature d i s t r i b u t i o n i n l o n g , s o l i d c i r c u l a r and r e c t a n g u l a r c y l i n d e r s and a sphere s u b j e c t t o s o l a r r a d i a t i o n i s s t u d i e d . For long, s o l i d c y l i n d e r s the problem becomes two-dimensional and hence more d i f f i c u l t than t h a t f o r long , t h i n hollow c y l i n d e r s . The problem has been analysed without l i n e a r i z i n g the boundary con-d i t i o n s by two approximate, s e m i - a n a l y t i c a l methods i ) p o i n t -matching and i i ) l e a s t - s q u a r e s f i t . FORMULATION AND SOLUTION OF THE PROBLEM Consider s t a t i o n a r y , l o n g , s o l i d c y l i n d e r s , as shown i n F i g u r e 1-1, and a s o l i d sphere, as shown i n F i g u r e 1-2. Each s o l i d body i s s u b j e c t to the f o l l o w i n g c o n d i t i o n s : 2 (i) A steady r a d i a n t energy f l u x of G BTU/hr. f t . i s b e i n g i n c i d e n t on a plane p e r p e n d i c u l a r to the l i n e of v i s i o n , the source of t h i s energy being the sun. I t i s f u r t h e r assumed here t h a t the s o l i d i s p l a c e d i n p e r f e c t vacuum at zero degrees Rankine and t h a t i t does not r e c e i v e any energy from p l a n e t s or o t h e r sources. ( i i ) The c u r v a t u r e produced i s s m a l l so t h a t the i n c i d e n t 2 r a d i a n t f l u x i n BTU/hr. f t . remains the same. I t i s a l s o assumed t h a t the c u r v a t u r e produced does not i n -v o l v e a p p r e c i a b l e amount of i n t e r a c t i o n of r a d i a t i o n i n the areas o p p o s i t e to the s i d e of i n c i d e n t r a d i a t i o n , ( i i i ) The s o l i d i s of i s o t r o p i c homogeneous m a t e r i a l , (iv) For the c y l i n d e r s , the c y l i n d e r l e n g t h i s assumed l a r g e compared w i t h i t s r a d i u s or width and s m a l l compared , w i t h the d i s t a n c e between the c y l i n d e r and the sun. Under t h i s c o n d i t i o n the heat flow from the c y l i n d e r ends can be n e g l e c t e d . For the sphere, due t o angular symmetry, the temperature d i s t r i b u t i o n w i l l be a f u n c t i o n of r and 6 o n l y . Thus f o r a l l the three cases the thermal problem reduces to a two-dimensional one. (v) A balance has been achieved between the r a d i a n t heat absorbed by the s o l i d and heat r e - r a d i a t e d i n t o space. (vi) P h y s i c a l and s u r f a c e p r o p e r t i e s of the s o l i d m a t e r i a l are i n v a r i a n t with temperature, ( v i i ) The s o l i d s u r f a c e i s a d i f f u s e e m i t t e r and d i f f u s e r e f l e c t o r . ( v i i i ) Surface p r o p e r t i e s a and e are t o t a l hemispheric v a l u e s However, i n g e n e r a l , a ^ e . The governing equations, boundary c o n d i t i o n s and t h e i r s o l u t i o n s f o r the three s o l i d s c o n s i d e r e d are d e a l t w i t h s e p a r a t e l y i n the f o l l o w i n g pages. a) CIRCULAR CYLINDER Under the f o r e g o i n g assumptions, the governing d i f f e r e n -t i a l e q u a t i o n of heat conduction and the boundary c o n d i t i o n s may be w r i t t e n as f o l l o w s : * r . - - r J •Ira of = aGcosQ-e-crtT 1 1] , 3 ^ Q 4 £ m (i-2) The energy equation and the boundary c o n d i t i o n s may be rephrased i n t o convenient dimensionless forms by i n t r o d u c i n g dimensionless v a r i a b l e s as f o l l o w s : (1-4) 10 The r e s u l t i n g equations a r e : y x , ' >^x \ y x _ o ^x = a cosS - f\ [X] , 3* 484 f ; (1-6) 1 4B 4 y ; d-7) 2 * where |\ — Ecr C^O."'/^ i s a r a d i a t i o n - c o n d u c t i o n parameter. An exact a n a l y t i c a l s o l u t i o n of the system of equations (1-5) t o (1-7) appears t o be i m p o s s i b l e . Approximate serai-a n a l y t i c a l methods are, t h e r e f o r e , used t o s o l v e these e q u a t i o n s . The v a r i a t i o n a l s o l u t i o n o f t h i s problem i s g i v e n i n r e f e r e n c e [30] . The two oth e r methods of s o l u t i o n are d e s c r i b e d i n the f o l l o w i n g s e c t i o n s . Both these methods are approximate by t h e i r very n a t u r e . However, the s o l u t i o n s are o b t a i n e d without l i n e a r -i z i n g the boundary c o n d i t i o n s , (i) P o i n t Matching S o l u t i o n The p o i n t matching method i n v o l v e s d e v e l o p i n g a p o l y -nomial w i t h a number of unknown c o e f f i c i e n t s which e x a c t l y s a t i s f i e s the d i f f e r e n t i a l e quation (1-5). The boundary c o n d i t i o n s (1-6) and (1-7) are then e x a c t l y s a t i s f i e d o n l y a t a number of p o i n t s on the s u r f a c e . The number of p o i n t s chosen on the s u r -face i s the same as the number of unknown c o e f f i c i e n t s i n the polyn o m i a l . I t can be shown t h a t the d i f f e r e n t i a l e q u a t i o n (1-5) i s s a t i s f i e d by a polynomial o f the f o l l o w i n g form: 11 oo X - X Cos-nQ • d-8) The c o e f f i c i e n t s are then obtained by i n s e r t i n g (1-8) i n (1-6) and (1-7) a t the chosen p o i n t s on the boundary ( i . e . d i f f e r e n t values of 0 ) . T h i s r e s u l t s i n a s e t of n o n l i n e a r simultaneous a l g e b r a i c equations i n a^ and the parameters A and a. The number of r e s u l t i n g a l g e b r a i c equations w i l l depend on the number of terms chosen i n (1-8). The r e s u l t i n g non-l i n e a r a l g e b r a i c equations are s o l v e d f o r a^ by the Newton Raphson method f o r g i v e n v a l u e s o f the parameters A and a. In t h i s work seven t o f i f t e e n terms of (1-8) have been t e s t e d and the r e s u l t s compared. Due t o symmetry onl y one h a l f of the boundary has been c o n s i d e r e d . ( i i ) Least-Squares F i t S o l u t i o n : While the p o i n t matching method uses the same number of boundary p o i n t s as the number of unknown c o e f f i c i e n t s i n the polynomial and then e x a c t l y s a t i s f i e s the boundary c o n d i t i o n s a t these pre-chosen p o i n t s , the l e a s t - s q u a r e s f i t method uses more p o i n t s than the unknown c o e f f i c i e n t s . The boundary c o n d i t i o n s a t the pre-chosen p o i n t s are then not e x a c t l y s a t i s f i e d but i t ensures a b e t t e r f i t to the boundary as a whole. The p o i n t s are e q u a l l y spaced on the boundary so t h a t i n t h i s problem i t e l i m i n a t e s the need to i n c l u d e w e i g h t i n g f a c t o r s i n t o the l e a s t -squares f i t t i n g technique. The e x p r e s s i o n to be minimized i s 12 where denotes summation over the r e g i o n ^ 4 | and E 2 denotes summation over the r e g i o n | is y . S u b s t i t u -t i o n o f A = Z. OL \\ Cosiw i n the above e x p r e s s i o n y i e l d s -n-D (1-10) The f u n c t i o n to be f i t t e d to zero a t v a r i o u s boundary p o i n t s i s [ 2 no.'to* 1x6 - * c o s § + K(i 0c^cos^V] in 2 and S ince the f u n c t i o n i s n o n - l i n e a r i n the unknown parameters a n , i t i s f i r s t l o c a l l y l i n e a r i z e d by T a y l o r ' s expans ion and then the l e a s t - s q u a r e s f i t c r i t e r i a a p p l i e d , i . e . , •- =0 , Tv - 0, V, 2,........oo . U s i ng o n l y m terms o f t i -the s e r i e s we get a system o f m l i n e a r a l g e b r a i c equa t i ons which i s s o l v e d to determine the c o e f f i c i e n t s a . However, t h i s n i s an i t e r a t i v e method and i n i t i a l approximate va lues o f a n have to be guessed and new values obtained. In t h i s study the process was repeated u n t i l ( a / * w _ | <V05 , * . « A . I The number of boundary points was varied from 30 to 100 with seven series c o e f f i c i e n t s and the results compared. b) RECTANGULAR CYLINDER The system of governing equations for t h i s geometry may be written as, '1 B.C.I: At X=-rCX_, 7 H B.C.2: At X - - a . via - T B.C.4: At M- -\>, £(T \ . (1-13) - £<r T B.C.3: At M = +b, W 1 = " (1-14) We introduce a dimensionless temperature as A = UT/Ga , and the dimensionless lengths as, X = x/a , - i 4 L 14 Y ^ ^ / O L , 4 Y i , where a \>|a The equations (1-11) to (1-15) can now be w r i t t e n i n the dimension-l e s s form as f o l l o w s , bx* *\ - o , (1-16) ~bX (1-17) "bX 7>X (1-18) 7>X bY (1-19) "bX "bY (1-20) where _ tixQ^o^/k -*-s t ^ i e r a d i a t i o n - c o n d u c t i o n parameter. The p o i n t matching method and the l e a s t - s q u a r e s f i t method have again been employed to s o l v e the above system of e q u a t i o n s . (i) P o i n t Matching S o l u t i o n I t i s known t h a t the r e a l p a r t of the p o l y n o m i a l , s a t i s f i e s the d i f f e r e n t i a l e quation (1-16). T h i s p o l y n o m i a l was employed and the c o e f f i c i e n t s A^, f o r v a r i o u s v a l u e s of a, A and B were obtained from the boundary c o n d i t i o n s (1-17) to 15 (1-20). The same number of c o e f f i c i e n t s A^ were taken as the number of points on the boundary. In t h i s study the e f f e c t of seven to ten points on the boundary has been examined. ( i i ) Least-Squares F i t Solution As before, more points than the unknown c o e f f i c i e n t s i n the polynomial are chosen and, instead of s a t i s f y i n g the boundary conditions exactly, the error i s minimised by the least-squares technique. The points are equally spaced on the boundary so that weighting factors do not have to be included. The expression to be minimised i s Y,l | W - » J ) (1-22) where /E( denotes summation over the region X =+l, 0 4 Y 4 ^ ' , Z. denotes summation over the region Az "1^ and denotes summation over the region Y ^  ^  >-1-4^ 4 + 1. Substitution of X •=. Real i n the above expression y i e l d s 2 t^fuul^kxC^^Tl +ft«*Mi uwfl] j .(1.23) 16 As i n t h e case o f t h e c i r c u l a r c y l i n d e r t h e n o n l i n e a r l e a s t - s q u a r e s f i t t i n g t e c h n i q u e i s employed t o d e t e r m i n e t h e unknown c o e f f i c i e n t s rV^ , "*> - ^ > ^ >2, , ~0 . ' I n t h i s i n v e s t i g a t i o n 40 t o 100 p o i n t s on t h e boundary have been c o n s i d e r e d f o r seven c o e f f i c i e n t s i n t h e s e r i e s . C) SPHERE The energy e q u a t i o n and the boundary c o n d i t i o n s f o r t h e sphere a re as f o l l o w s : V -hi-1 \ t 1 "be "b  v 7»e ' k 7^ (1-24) (1-25) (1-26) I n t r o d u c i n g d i m e n s i o n l e s s v a r i a b l e s X - W T /GOL and K = */a , t h e energy e q u a t i o n and boundary c o n d i t i o n s reduce t o t h e f o l l o w -i n g d i m e n s i o n l e s s form: \ K l s ^ *e b  N ^e ) (1-27) •ax = Co»« - h[jt) I S T 4 6 4 ? J ( 1 " 2 8 ) ; d-29) 17 where i s the r a d i a t i o n - c o n d u c t i o n parameter. The above system of equations has been s o l v e d by the p o i n t matching and the l e a s t - s q u a r e s f i t methods. (i) P o i n t Matching S o l u t i o n The s e r i e s expansion f o r A i s assumed i n terms of Legendre polynomials as f o l l o w s : The c o e f f i c i e n t s a were ob t a i n e d f o r v a r i o u s v a l u e s of A and a by the p o i n t matching method d e s c r i b e d e a r l i e r . In t h i s study the e f f e c t of f i v e t o f i f t e e n terms i n the s e r i e s has been examined. ( i i ) Least-Squares F i t S o l u t i o n The c o e f f i c i e n t s of the s e r i e s expansion f o r A have a l s o been o b t a i n e d by t a k i n g more p o i n t s on the boundary than the unknown c o e f f i c i e n t s and employing a l e a s t - s q u a r e s f i t t i n g t echnique. The e x p r e s s i o n to be minimised i s (1-30) n 2 5 (1-31) 18 where ^ denotes summation ove r the r e g i o n ^p- 4 § 4 ^ and ^ denotes summation over the r e g i o n ^ 4$ 4 £^ Us ing the s e r i e s expans ion f o r A f rom e q . (1-30) the above e x p r e s s i o n y i e l d s the f o l l o w i n g fo rm: (1-32) The n o n l i n e a r l e a s t - s q u a r e s f i t t i n g t echn ique i s employed t o determine the unknown c o e f f i c i e n t s DISCUSSION OF RESULTS In order to o b t a i n numerical values o f the d i m e n s i o n l e s s temperature X , p h y s i c a l l y r e a l i s t i c v a l u e s of A and a have t o be p r e s c r i b e d . In the r a d i a t i o n - c o n d u c t i o n parameter 3 4 4 A = eoG a /k , the s o l a r r a d i a t i o n f l u x G can be c o n s i d e r e d constant. I t s value a t a d i s t a n c e from the sun equal t o t h a t o f 2 the e a r t h i s approximately 442 BTU/hr. f t . The c o e f f i c i e n t of e m i s s i v i t y may range from 0.05 to 1.0. The r a t i o a/k may vary c o n s i d e r a b l y . A c c o r d i n g l y the r e s u l t s have been analyzed by — 16 u s i n g A i n the range of 10 to 10. The s u r f a c e a b s o r p t i v i t y , a, has been v a r i e d from 0.05 to 1.0. The r e s u l t s o b tained by the p o i n t matching method and the l e a s t - s q u a r e s f i t method are compared wi t h the r e s u l t s of the v a r i a t i o n a l method as d e s c r i b e d by r e f e r e n c e s [28-30], How-ever, s i n c e a l l the three methods of s o l u t i o n are approximate and an exact s o l u t i o n of these boundary value problems i s not a v a i l a b l e , i t i s d i f f i c u l t to judge t h e i r accuracy. I t i s p o s s i b l e , however, to examine the accuracy of the approximate s o l u t i o n s from the p o i n t of view of o v e r a l l thermal energy balance on the b o d i e s . The percentage thermal energy balance e r r o r can be w r i t t e n as (1-33) The e x p r e s s i o n (1-33) f o r the c i r c u l a r c y l i n d e r , r e c t a n g u l a r c y l i n d e r and the sphere can be w r i t t e n i n the d i m e n s i o n l e s s form as, V L ~ TT-r ^ ^ 6 *• V » , (1-33C) and r e s p e c t i v e l y . D e t a i l e d d e r i v a t i o n s of the e x p r e s s i o n s (l-33a) through (1-33c) are given i n Appendix A (pp. 136-138 ) . R e s u l t s of the v a r i o u s methods f o r the t h r e e geometries are now d i s c u s s e d . a) CIRCULAR CYLINDER In the s e r i e s expansion f o r X f o r the c i r c u l a r c y l i n d e r 2 3 (1-8) the f u n c t i o n s Rcose, R cos 26, R cos 36, e t c . , can never be g r e a t e r than one i n magnitude. T h e r e f o r e , the accuracy o f the r e s u l t s w i l l depend on the magnitude of the c o e f f i c i e n t s a^. R e p r e s e n t a t i v e values of a^, a^ f a 2 , a^/ a^, a^ and a^, as o b t a i n e d by the l e a s t - s q u a r e s f i t method, f o r v a r i o u s v a l u e s of the parameters A and a are g i v e n i n Table 1-1. T h i s t a b l e shows t h a t the c o e f f i c i e n t s a^ are h i g h l y s e n s i t i v e t o v a l u e s of A, w h i l e r e l a t i v e l y l e s s s e n s i t i v e t o ex. For very s m a l l v a l u e s of -7 A, i . e . , A of the o r d e r of 10 or s m a l l e r , the magnitudes of the c o e f f i c i e n t s are decaying extremely r a p i d l y . T h i s range o f A < 10 d e a l s w i t h m a t e r i a l s of h i g h thermal c o n d u c t i v i t y , f o r example, those of s o l i d metals. T h e r e f o r e i t appears t h a t the e x p r e s s i o n chosen f o r X i s e x c e l l e n t . T h i s i s f o r t u n a t e con-s i d e r i n g the f a c t t h a t one would be d e a l i n g w i t h metals i n most of the a p p l i c a t i o n s of the type where t h i s a n a l y s i s c o u l d be -1 -7 a p p l i e d . For values i n the range of 10 t o 10 , Table 1-1 shows t h a t the magnitude of these c o e f f i c i e n t s i s decaying s t i l l -7 -1 q u i t e r a p i d l y . The range of A between 10 and 10 roughly covers m a t e r i a l s o f low thermal c o n d u c t i v i t y , f o r example g l a s s , p o r c e l a i n , e b o n i t e , e t c . To examine the d i f f e r e n c e i n numerical v a l u e s o b t a i n e d from the v a r i a t i o n a l method with those from the ot h e r methods, one has t o take i n t o account the a c t u a l l o c a t i o n o f a p o i n t i n the c y l i n d e r . The p o i n t c o n s i d e r e d i s a t R=0, so t h a t X = a^ f o r a l l the three s e m i - a n a l y t i c a l cases. Table 1-2 g i v e s the values o f the dimensionless temperature a t R=0 f o r the th r e e methods. T h i s t a b l e shows t h a t there i s very s l i g h t d i f f e r e n c e between the c e n t r e l i n e temperature e v a l u a t e d by the t h r e e methods. As a matter of f a c t the a x i a l temperatures o b t a i n e d by the v a r i a t i o n a l method and t h a t o b t a i n e d by the l e a s t - s q u a r e s f i t , w i t h seven c o e f f i c i e n t s and seventy boundary p o i n t s , are i d e n t i c a l up to f o u r s i g n i f i c a n t f i g u r e s . T h i s i s an i n t e r e s t -i n g o b s e r v a t i o n s i n c e the v a r i a t i o n a l method i s b a s i c a l l y an o p t i m i s a t i o n method and t r i e s t o b e s t - f i t a g i v e n e x p r e s s i o n and so does the l e a s t - s q u a r e s f i t method. However, the l e a s t - s q u a r e s f i t method i s much simpl e r and more s t r a i g h t forward than the v a r i a t i o n a l method. Since the exact value of the a x i a l temperature i s not a v a i l a b l e the accuracy of the three methods i s compared by the 22 values of the error i n o v e r a l l heat balance as given by equation (l-33a). This i s presented i n Table 1-3, which shows that the least-squares f i t solution gives the lea s t error i n o v e r a l l energy balance and t h i s error reduces as the number of boundary points i s increased. I t i s also possible to increase the accuracy of the v a r i a t i o n a l and point-matching solutions by taking more terms i n the series (1-8). This i s e a s i l y done for the point-matching method while for the v a r i a t i o n a l method t h i s would, be-sides increasing the number of nonlinear algebraic equations to be solved, increase t h e i r size considerably and involve considerable labour. The point-matching solution has been t r i e d by taking seven to f i f t e e n terms i n the series (1-8) and the same number of points on one half of the cylinder boundary. The points were equally d i s t r i b u t e d over half of the circumference i n such a way that (1,0) and (1,TT) locations were not included. The decay of the magnitudes of the c o e f f i c i e n t s a^ under parameters A and a i s exactly s i m i l a r to that observed i n the lea s t squares f i t solution. The a x i a l temperature i s also very close to that obtained by the other methods (Table 1-2). As the number of points i s increased, the o v e r a l l energy balance error as per equation (l-33a) decreases (Table 1-3). However, t h i s o v e r a l l error does not appear to be affected much by the v a r i a t i o n of A and a. The least-squares f i t method u t i l i z e s a lesser number of terms of the series (1-8) and more number of points on the boundary. I t i s clear that the accuracy of the solution w i l l 23 be a f f e c t e d both by the number of c o e f f i c i e n t s chosen and the number of l o c a t i o n s s e l e c t e d on the boundary. The a x i a l temper-ature and the o v e r a l l energy balance are t a b u l a t e d i n Tables 1-2 and 1-3.. The a x i a l temperature i s very c l o s e to the v a l u e s o b t a i n e d by the o t h e r two methods. The o v e r a l l energy balance i s b e t t e r than t h a t o b t a i n e d by the point-matching method or even the v a r i a t i o n a l method. The accuracy of the l e a s t - s q u a r e s f i t t i n g method i n c r e a s e s as the number of c o e f f i c i e n t s and the number of p o i n t s on the boundary are i n c r e a s e d , which no doubt slows down the convergence r a t e , and may o f f s e t the decrease i n t r u n c a t i o n e r r o r by i n c r e a s i n g the round-off e r r o r . Beyond a c e r t a i n number of terms i n the s e r i e s there seems to be a g r e a t e r advantage i n i n c r e a s i n g the number of boundary p o i n t s r a t h e r than i n c r e a s i n g the number of s e r i e s terms. Comparing the three methods c o n s i d e r e d , i t appears t h a t the l e a s t - s q u a r e s f i t method i s the b e s t i n accuracy (gives a b e t t e r o v e r a l l energy balance) and s i m p l i c i t y . b) RECTANGULAR CYLINDER The remarks made about the c i r c u l a r c y l i n d e r are a l s o t r u e f o r the r e c t a n g u l a r c y l i n d e r w i t h the emphasis t h a t i n the l a t t e r case a l l the three methods i n v o l v e h a n d l i n g of l a r g e r e x p r e s s i o n s . The p o i n t matching and the l e a s t - s q u a r e s f i t t i n g methods i n a d d i t i o n r e q u i r e some ma n i p u l a t i o n i n the s e l e c t i o n of boundary p o i n t s . In g e n e r a l , i t i s p r e f e r a b l e to a v o i d corner p o i n t s . I t i s d e s i r a b l e to d i s t r i b u t e the p o i n t s i n such a way t h a t they are e q u i d i s t a n t and may not n e c e s s a r i l y be the same number on a l l the s i d e s . 2 4 Table 1 - 4 presents values of the c o e f f i c i e n t s A Q , A^, A 2, etc., for the rectangular cylinder for some representative values of A and a. The table again shows a rapid decay i n the magnitudes of the c o e f f i c i e n t s . Table 1 - 5 presents the values of *-Ag, i . e . , the dimensionless temperature at X=Y=0. This table shows that a l l the three methods give results which agree very w e l l . The values of the o v e r a l l energy balance error from (l-33b) are presented i n Table 1 - 6 . Again the least-squares f i t method gives the highest accuracy i n terms of the o v e r a l l error. The v a r i a t i o n a l method appears next i n accuracy and f i n a l l y the point matching method. In the point matching method, for a fixed value of B, i t becomes d i f f i c u l t to d i s t r i b u t e a small number of points on the boundary uniformly. Increasing the number of points too much to ensure equidistant points leads to d i f f i c u l t y i n convergence of the system of nonlinear equations. This may explain why there seems to be no decrease i n o v e r a l l error on increasing the points from seven to ten i n the point matching method for the rectangular cylinder as i s evident from Table 1 - 6 . c) SPHERE Similar r e s u l t s have been obtained for the sphere and they are tabulated i n Tables 1 - 7 , 1 - 8 and 1 - 9 . Table 1 - 7 gives the c o e f f i c i e n t s of the series expansion for the temperature d i s t r i b u t i o n . As observed for the other two s o l i d s , the c o e f f i c -ients are decaying i n magnitude very rapi d l y . Table 1 - 8 presents the values of the dimensionless temperature at the centre of the sphere. The values obtained by the three methods agree reason-25 a b l y w e l l . The v a l u e s of the o v e r a l l energy balance from (l-33c) are p r e s e n t e d i n Table 1-9. Although the e r r o r f o r the l e a s t -squares f i t method i n t h i s case i s h i g h e r than t h a t o b t a i n e d by the v a r i a t i o n a l method, i t i s c o n s i s t e n t and does not e x h i b i t the sharp r i s e i n i n a c c u r a c y of the v a r i a t i o n a l method f o r h i g h v a l u e s of A. Thus f o r accuracy and s i m p l i c i t y the l e a s t - s q u a r e s f i t method i s recommended f o r use i n such problems. Temperature D i s t r i b u t i o n Having determined the c o e f f i c i e n t s of the s e r i e s expan-' s i o n i t i s r e l a t i v e l y simple to c a l c u l a t e the temperature d i s -t r i b u t i o n . For example, f o r the r e c t a n g u l a r c y l i n d e r , the dimensional temperature T(x,y) i s o b t a i n e d from the d i m e n s i o n l e s s temperature A(X,Y) by combining the dimensionless q u a n t i t i e s : "3 4 4 1/4 A=T/(Ga/k) and A=(eoG a /k ) so t h a t T= A(AG/eo) . F i g u r e s 1-3 to 1-6 show the i s o t h e r m a l s p l o t t e d f o r the r e c t a n g u l a r c y l i n d e r f o r v a r i o u s values of B, A and a. A l l the f o u r p l o t s demonstrate symmetry about the X - a x i s . Comparison of F i g u r e 1-^ 3 w i t h F i g u r e 1-4 shows t h a t as B, the r a t i o of h e i g h t t o width, i n c r e a s e s the g e n e r a l l e v e l of temperature i n c r e a s e s . Comparing F i g u r e 1-4 with F i g u r e 1-5 the l a t t e r , which i s f o r a lower v a l u e of A ( i . e . , i s f o r a m a t e r i a l of h i g h e r thermal c o n d u c t i v i t y ) , shows a l e s s e r temperature v a r i a t i o n . Thus as the thermal c o n d u c t i v i t y i n c r e a s e s , t h e r e i s l e s s e r temperature v a r i a t i o n across the body. Comparison of F i g u r e 1-5 w i t h F i g u r e 1-6 i n d i c a t e s t h a t a t a h i g h value of a, the c o e f f i c i e n t of 26 a b s o r p t i v i t y , the temperature v a r i a t i o n a cross the body i n c r e a s e s s l i g h t l y , e s p e c i a l l y i n the lower temperature r e g i o n . E f f e c t o f Semi-Grayness of Surface P r o p e r t i e s In s p a c e c r a f t a p p l i c a t i o n s , the s p a c e c r a f t w i l l be a t a much lower temperature compared t o the sun which i s the source of the i n c i d e n t r a d i a n t f l u x i n t h i s a n a l y s i s . T h e r e f o r e , the maximum energy emitted by the s u r f a c e of the s p a c e c r a f t w i l l be a t r e l a t i v e l y h i g h e r wavelengths while the i n c i d e n t s o l a r energy w i l l be mainly a t lower wavelengths. The c o e f f i c i e n t o f e m i s s i v i t y o f a gi v e n s u r f a c e i s , i n g e n e r a l , a f u n c t i o n o f i t s ' temperature o n l y , w hile the c o e f f i c i e n t of a b s o r p t i v i t y i s a f u n c t i o n of both, the s u r f a c e temperature and the temperature of the source of i n c i d e n t r a d i a t i o n . T h e r e f o r e , i n the p r e s e n t problem i t w i l l not be c o r r e c t t o assume t h a t a=e. The formu-l a t i o n of equations (1-1) to (1-3), (1-16) t o (1-20) and (1-24) to (1-26) does not make the above mentioned assumption. The s o l u t i o n s of these s e t s of equations a r e , t h e r e f o r e , independent of any such c o n s t r a i n t . T h i s i s t r u e as long as we c o n s i d e r t h a t the s u r f a c e p r o p e r t i e s are t o t a l hemispheric and are i n v a r i -ant w i t h temperature d i s t r i b u t i o n over the s u r f a c e . The e f f e c t o f t h i s semi-grayness, i . e . , a^e, on the temperature d i s t r i b u t i o n has been i n v e s t i g a t e d . I t i s noted t h a t f o r a giv e n m a t e r i a l and geometry, a t a f i x e d v alue o f a, an i n c r e a s e i n the va l u e of e i n c r e a s e s the maximum temperature d i f f e r e n c e and v i c e v e r s a . 27 I n c l u s i o n of Convection Heat T r a n s f e r i n the A n a l y s i s Although the a n a l y s i s has been c a r r i e d out f o r the case o f no c o n v e c t i o n heat t r a n s f e r , i n c l u s i o n of the c o n v e c t i o n term c r e a t e s no a d d i t i o n a l d i f f i c u l t i e s i n the s o l u t i o n s of the problem by e i t h e r the point-matching method or the l e a s t - s q u a r e s f i t method f o r the three s o l i d bodies c o n s i d e r e d . The energy equation remains u n a l t e r e d and so the s e r i e s s o l u t i o n i s s t i l l v a l i d . The o n l y change i s i n the boundary c o n d i t i o n s which can e a s i l y be i n c o r p o r a t e d . v 28 CONCLUSIONS T h e t e m p e r a t u r e d i s t r i b u t i o n i n t h r e e d i f f e r e n t t y p e s o f s o l i d b o d i e s , l o n g c i r c u l a r a n d r e c t a n g u l a r c y l i n d e r s a n d a s p h e r e , s u b j e c t e d t o s o l a r r a d i a t i o n i n s p a c e h a s b e e n a n a l y s e d . T w o a p p r o x i m a t e , s e m i - a n a l y t i c a l m e t h o d s , ( i ) p o i n t - m a t c h i n g a n d ( i i ) l e a s t - s q u a r e s f i t t i n g , h a v e b e e n e x a m i n e d a n d t h e r e s u l t s c o m p a r e d w i t h t h o s e o b t a i n e d b y a v a r i a t i o n a l s o l u t i o n . T h e l e a s t - s q u a r e s f i t m e t h o d h a s p r o v e d t o b e v a s t l y s u p e r i o r t o t h e o t h e r t w o , a s i t i n v o l v e s m u c h s i m p l e r c o m p u t a t i o n a n d p r o v i d e s v e r y a c c u r a t e r e s u l t s . T h e m e t h o d s p r e s e n t e d h e r e c a n b e a p p l i e d t o t h i c k h o l l o w c y l i n d e r s a n d s p h e r e s a n d c a n b e e x t e n d e d f o r o t h e r s t r u c t u r a l s h a p e s . TABLE 1-1 E f f e c t o f A and a on t h e v a l u e s o f t h e c o e f f i c i e n t s a. o f E q u a t i o n (1-8) C i r c u l a r C y l i n d e r ( L e a s t - s q u a r e s f i t method f o r 7 s e r i e s terms and 100 boundary p o i n t s ) A a ao a l a 2 a 3 a 4 a 5 a 6 l O " 1 1 1 .0 "422 .4 0. 4985 0 .1059 -0 .1675x10" 6 -0 .1060x10" 1 0 .6362x10 -8 0 .3026x10" 2 I O " 7 1 .0 42 .23 0. 4853 0 .1044 -0 .1628x10" 4 -0 .1053x10" 1 0 .7637x10 -6 0 .3012x10" 2 I O " 1 1 .0 1 .296 0. 2551 0 .6196x10" 1 -0 .4118x10" 2 -0 .9173x10" 2 0 .2819x10 -3 0 .2704x10" 2 10 1 .0 0 .3895 0. 1301 0 .3048x10" 1 -0 .6661x10" 2 -o .6859x10" 2 0 .9572x10 -3 0 .2190x10" 2 I O " 1 1 0 .5 355 .2 0. 2496 0 .5300x10" 1 -0 .2880x10" 7 -0 .5300x10" 2 0 .2084x10 -9 0 .1513x10" 2 I O " 5 0 .5 11 .23 0. 2365 0 .5148x10" 1 -0 .2762x10" 4 -0 .5231x10" 2 0 .1285x10 -5 0 .1500x10" 2 1 0 " 1 0 .5 1 .105 0. 1586 0 .3800x10" 1 -0 .1231x10' 2 -0 .4821x10" 2 0 .6663x10 -4 0 .1405x10" 2 10 0 .5 0 .3361 0. 9077x10" 1 0 .2187x10" 1 -0 .2985x10" 2 -0 .4080x10" 2 0 .3040x10 -3 0 .1248x10" 2 1 0 " 1 1 0 .1 237 .5 0. 4997x10" 1 0 .1061x10" ] -0 .337Sxl0" 9 -0 .1060x10" 2 0 .9638x10 -10 0 .3027x10" 3 I O " 2 0 .1 1 .334 0. 4561x10" 1 0 .1007x10" 1 -0 .1469x10" 4 -0 .1037x10" 2 0 .6772x10 -6 0 .2981x10" 3 10 0 .1 0 .2340 o. 3235x10" 1 0 .7736x10" 2 0 .2304x10" 3 -0 .9681x10" 3 0 .1225x10 -4 0 .2819x10" 3 TABLE 1-2 Comparison o f va lues o f the c o e f f i c i e n t a Q o b t a i n e d by d i f f e r e n t methods C i r c u l a r C y l i n d e r A a Variational Point Matching Method Least Squares Fit Method Method 7 Points 10 Points 15 Points 7 - 30* 7 - 70 7 - 100 1CT12 1.0 0.7511 X 103 0.7479 X 103 0.7519 X 103 0.7504 X 103 0.7512 X 103 0.7511 X 103 0.7511 X 103 lO" 7 1.0 0.4223 X 102 0.4206 X 102 0.4228 X 102 0.4220 X 102 0.4224 X 102 0.4223 X 102 0.4223 X 102 10"1 1.0 0.1296 X 10 0.1289 X 10 0.1297 X 10 0.1294 X 10 0.1296 X 10 0.1296 X 10 0.1296 X 10 10 1.0 0.3895 0.3865 0.3901 0.3888 0.3895 0.3895 0.3895 10" 1 1 0.5 0.3552 X 103 0.3537 X 103 0.3556 X 3 lO13 0.3549 X 3 10 J 0.3552 X 103 0.3552 X 103 0.3552 X 3 10-3 10"5 0.5 0.1123 X 102 0.1118 X 102 0.1124 X 102 0.1122 X 102 0.1123 X 102 0.1123 X 102 0.1123 X 102 lO" 1 0.5 0.1105 X 10 0.1100 X 10 0.1106 X 10 0.1104 X 10 0.1105 X 10 0.1105 X 10 0.1105 X 10 10 0.5 0.3361 0.3340 0.3366 0.3356 0.3361 0.3361 0.3361 l O " 1 1 0.1 0.2375 X 103 0.2365 X 103 0.2378 X 103 0.2373 X 103 0.2375 X 103 0.2375 X 103 0.2375 X 103 lO" 2 0.1 0.1334 X 10 0.1329 X 10 0.1336 X 10 0.1333 X 10 0.1335 X 10 0.1334 X 10 0.1334 X 10 10 0.1 0.2340 0.2329 0.2342 0.2337 0.2340 0.2340 0.2340 * This denotes 7 unknown coefficients and 30 points on the boundary. LO o TABLE 1-3 Comparison o f the o v e r a l l pe rcen tage e r r o r (Eq. l-33a) o b t a i n e d by d i f f e r e n t me C i r c u l a r C y l i n d e r A a Variational Method Point Matching Method Least Squares Fit Method 7 Points 10 Points 15 Points 7 - 30 7 - 70 7 - 100 l O " 1 2 1 .0 - 0.0134693 1 .6842149 - 0.4123867 0.3658459 - 0 .04568/0 - 0.0083730 - 0.0940919 l O " 7 1 .0 - 0.0426800 1 .6842179 - 0.4123822 0.3658451 - 0 .0456810 - 0.0083655 - 0.0040814 lO " 1 1 .0 0.0065672 1 .6838901 - 0.4123792 0.3658444 - 0 .0455469 - 0.0082284 - 0.0039473 10 1 .0 - 0.7490059 1 .6793281 - 0.4123673 0.3657885 - 0 .0505567 - 0.0133947 - 0.0091359 l O " 1 1 0 .5 0.0166636 1 .6842179 - 0.4123822 0.3658473 - 0 .0456855 - 0.0083685 - 0.0040904 10" 5 0 .5 0.0705664 1 .6842224 - 0.4123852 0.3658451 - 0 .0456795 - 0.0083625 - 0.0040859 lO " 1 0 .5 - 0.0020015 1 .6841374 - 0.4123867 0.3658459 - 0 .04555^8 - 0.0082254 - 0.0039473 10 0 .5 - 0.2901209 1 .6826615 - 0.4123762 0.3658362 - 0 .0464052 - 0.0091076 - 0.0048324 l O " 1 1 0 .1 - 0.0450330 1 .6842179 - 0.4123852 0.3658511 - 0 .0456929 - 0.0083730 - 0.0041038 lO " 2 0 .1 - 0.1428091 1 .6842157 - 0.4123867 0.3658429 - 0 .0456706 - 0.0083596 - 0.0040710 10 0 .1 0.0819797 1 .6841456 0.4123822 0.3658481 - 0 .0455678 - 0.0082433 - 0.0039592 TABLE 1-4 E f f e c t o f A and a on the va lues o f the c o e f f i c i e n t s A^ o f Equa t i on (1-21) Rec tangu la r C y l i n d e r (B=0.25) (Least-squares f i t method w i th 7 s e r i e s terms and 100 boundary p o i n t s ) A a Ao A l A 2 A 3 • A 4 A 5 A 6 I O " 1 2 1. 0 562 .3 0 .4989 0 .1999 0 .2406x10" 3 0 .5604x10 -4 -0 .2291x10" 5 - 0 . 5 0 i l x l 0 " 5 I O " 9 1. 0 99 .92 0 .4941 0 .1992 0 .1337x10" 2 0 .3183x10 -3 -0 .1023x10" 4 - 0 . 2816x10" 4 I O " 6 1. 0 17 .69 0 .4683 0 .1955 0 .7042x10" 2 0 .1872x10 -2 0 .2099x10" 4 - 0 . 1557x10" 3 I O " 3 1. 0 3 .054 0 .3578 0 .1717 0 .2772x10" 1 0 .1093x10 -1 0 .1830x10" 2 - 0 . 5733x10" 3 I O " 1 2 0. 1 316 .2 0 .4998x10" 1 0 .2000x10" 1 0 .4287x10" 5 0 .9953x10 -6 -0 .4246x10" 7 - 0 . 8931x10" 7 I O " 8 0. 1 31 .61 0 .4981x10" 1 0 .1998x10" 1 0 .4269x10" 4 0 .9983x10 -5 -0 .3927x10" 6 - 0 . 8907x10" 6 I O " 4 0. 1 3 .154 0 .4818x10" 1 0 .1975x10" 1 0 .4101x10" 3 0 .1030x10 -3 -0 .1114x10" 5 - 0 . 8866x10" 5 1 0 " 1 0. 1 0 .5522 0 .4102x10" 1 0 .1845x10" 1 0 .1878x10" 2 0 .6212x10 -3 0 .6012x10" 4 - 0 . 4267x10" 4 I O ' 1 2 0. 05 265 .9 0 .2499x10" 1 0 .9999x10" 2 0 .1275x10" 5 0 .2960x10 -6 -0 .1267x10" 7 - 0 . 2657x10" 7 I O " 8 0. 05 26 .59 0 .2494x10" 1 0 .9993x10" 2 0 .1272x10" 4 0 .2963x10 -5 -0 .1207x10" 6 - 0 . 2649x10" 6 I O ' 4 0. 05 2 .655 0 .2445x10" 1 0 .9928x10" 2 0 .1241x10" 3 0 .3023x10 -4 -0 .7017x10" 7 -o. 2645x10" 5 to to TABLE 1-5 Comparison o f v a l ues o f the c o e f f i c i e n t A 0 o b t a i n e d by d i f f e r e n t methods Rec tangu la r C y l i n d e r A a Variational Method Point Matching Method 1 Least Squares Fit Method 7 Points 9 Points 10 Points 7 - 10 7 - 80 7 - TOO 10 " 1 2 1 .0 0 .5624 X 10 3 0. 5623 X 10 3 0.5623 X 10 3 0. 5623 X 10 3 0. 5623 X 10 3 0.5623 X 103 0.5623 X 10 3 i c r 9 1 .0 0 .1000 X 10 3 0. 9990 X 102 0.9990 X 102 0. 9991 X 102 0. 9992 X 102 0.9992 X 102 0.9992 X 102 10 - 6 - 1 •0 0 .1778 X 102 0. 1769 X 102 0.1769 X 102 0. 1769 X 102 0. 1769 X 102 .0.1769 X 102 0.1769 X 10 2 10" 3 1 .0 0 .3140 X 10 0. 3055 X 10 0.3048 X 10 0. 3055 X 10 0. 3054 X 10 0.3054 X 10 0.3054 X 10 l O " 1 2 0 .1 0 .3162 X 103 0. 3175 X 10 3 0.3177 X 10"3 0. 31 75 X 10 3 0. 3162 X 103 0.3162 X 10 3 0.3162 X 10 3 10~8 0 .1 0 .3162 X 102 0. 3161 X 102 0.3161 X 102 0. 3160 X 102 0. 3161 X 102 0.3161 X 102 0.3161 X 10 2 10~4 0 .1 0 .3163 X 10 •0". 3153 X 10 .0.3153 X 10 0. 3153 X 10 0. 3154 X 10 0.3154 X 10 0.3154 X 10 lO ' 1 0 .1 0 .5612 .0. 5522 0.5518 0. 5522 0. 5522 0.5522 0.5522 lO " 1 2 0 .05 0 .2659 X 10 3 0. 2659 X 10 3 0.2659 X 10 3 0. 2659 X To3 0. 2659 X 103 0.2659 X 10 3 0.2659 X 10 3 1 0 - 8 0 .05 0 .2659 X 10 2 0. 2659 X 102 0.2659 X 102 0. 2657 X 10 2 0. 2659 X 102 0.2659 X 102 0.2659 X 10 2 lO" 4 0 .05 0 .2660 X 10 0. 2655 X 10 0.2655 X 10 0. 2653 X 10 0. 2655 X 10 0.2655 X 10 0.2655 X 10 co co TABLE 1-6 Compar ison of the o v e r a l l pe rcentage e r r o r (Eq. l-33b) ob t a i ned by d i f f e r e n t methods Rec tangu la r C y l i n d e r A a Variational Method Point Matching Method 1 Least Squares F i t Method 7 Points 9 Points 10 Points 7 - 40 7 - 8 0 | 7 - 100 1<T12 1.0 - 0.0342488 0.0025503 0.0025697 0.0025503 0.0000454 0.0000350 0.0000291 IO" 9 1.0 0.0364594 0.0422530 0.0428043 0.0422530 0.0001647 0.0000671 0.0000559 IO" 6 1.0 0.1067489 0.0161126 0.0345334 0.0161126 0.0013083 0.0007093 0.0006542 IO ' 3 1.0 0.0058122 -0.0551924 0.8436181 -0.0473037 0.0267804 0.0235520 0.0231773 IO" 1 2 0.1 0.0364825 -1.6917363 -1.8311754 -1.6917363 0.0000313 0.0000253 0.0000216 1 0 - 8 0.1 0.0481263 0.0067174 0.0067689 0.1678914 0.0000603 0.0000298 0.0000276 IO" 4 0.1 - 0.0203714 0.0816956 0.0877619 0.0816956 0.0005998 0.0002799 0.0002384 IO" 1 0.1 - 0.0398651 0.0357717 0.3065333 0.0357717 0.0087924 0.0068709 0.0066407 IO" 1 2 0.05 0.0230677 -0.0470638 -0.0470564 -0.0470638 0.0000231 0.0000298 0.0000171 IO" 8 0.05 0.0312531 0.0056140 0.0056550 0.2616279 0.0000454 0.0000291 0.0000238 IO" 4 0.05 - 0.0515928 0.0227228 0.0248097 0.2607897 0.0003278 0.0001334 0.0001132 to TABLE 1-7 E f f e c t o f A and a on the v a l u e s o f t h e c o e f f i c i e n t s a i o f E q u a t i o n (1-30) Sphere ( L e a s t - s q u a r e s f i t method f o r 7 s e r i e s terms and 40 boundary p o i n t s ) A a a 0 a l a 2 a 3 a 4 a 5 a 6 1 0 " 1 6 0. 1 3976. 0 0 .4999x10" 1 0. 3897x10" 2 -0 .3101x10 -10 -0. 3737x10" 4 -0 .1367x10" 11 0 .1310x10" 5 1 0 " 1 2 0. 1 397. 6 0 .4999x10" 1 0. 3897x10" 2 0 .2564x10 -10 -o. 3737x10" 4 -0 .2 021x10" 11 0 .1310x10" 5 1 0 " 8 0. 1 39. 76 0 .4987x10" 1 0. 3892x10" 2 -0 .3212x10 -8 -0. 3735x10" 4 0 .8997x10" 11 0 .1310x10" 5 10" 4 0. 1 3. 976 0 .4877x10" 1 0. 3847x10" 2 -0 .3176x10 -6 -0. 3714x10" 4 0 .1201x10" 8 0 .1305x10" 5 1.0 0. 1 . 0. 3954 0 .3971x10" 1 0. 3350x10" 2 -0 .2217x10 -4 -0. 3555x10" 4 0 .8219x10" 7 0 .1263x10" 5 l O ' 1 6 0. 5 5945. 5 0 .2500 0. 1948x10" 1 -0 .3654x10 -9 -0. 1869x10" 3 0 .1288x10" 10 0 .6552x10" 5 i o " 1 2 0. 5 594. 6 0 .2498 0. 1948x10" 1 -0 .1610x10 -8 -0. 1868x10" 3 0 .1044x10" 10 0 .6551x10" 5 1 0 " 8 0. 5 59. 46 0 .2479 0. 1940x10" 1 -0 .1817x10 -6 -0. 1865x10" 3 0 .6871x10" 9 0 .6543x10" 5 l O " 4 0. 5 5. 941 0 .2304 0. 1861x10" 1 -0 .1625x10 -4 -0. 1833x10" 3 0 .5994x10" 8 0 .6465x10" 5 1.0 0. 5 0. 5776 0 .1334 0. 1167x10" 1 -0 .4999x10 -3 -0. 1637x10" 3 0 .2700x10" 5 0 .5930x10" 5 l O " 1 6 1. 0 7071. 0 0 .4999 0. 3897x10" 1 o .3344x10 -9 -o. 3737x10" 3 -0 .2225x10" 10 0 .1310x10" 4 l o " 1 2 1. 0 707. 1 0 .4993 0. 3894x10" 1 -0 .9900x10 -8 -0. 3733x10" 3 0 .1209x10" 10 0 .1308x10" 4 l O " 8 1. 0 70. 71 0 .4930 0. 3868x10" 1 -0 .1020x10 -5 -0. 3721x10" 3 0 .3873x10" 8 0 .1305x10" 4 l O " 4 1. 0 7. 056 0 .4369 0. 3593x10" 1 -0 .8397x10 -4 -0. 3620x10" 3 0 .3064x10" 6 0 .1280x10" 4 1.0 1. 0 0. 6736 0 .2040 0. 1764x10" 1 -0 .1402x10 -2 -.0. 2983x10" 3 0 .1047x10" 4 0 .1117x10" 4-to TABLE 1-8 Comparison of values of the c o e f f i c i e n t a 0 obtained by d i f f e r e n t methods Sphere Dimensionless Temperature (X =a 0) a t the Centre of the Sphere A a Point-Matching S o l u t i o n Least-Squares F i t S o l u t i o n (7-40) V a r i a t i o n a l S o l u t i o n 5 P o i n t s 7 P o i n t s 10 P o i n t s 15 P o i n t s 0 .3912 X 10 4 0 .3940 X 10 4 0.3985 X 10 4 0 .3969 X i o 4 0.3976 X i o 4 0 .3976 X 10 4 10" 16 0. 1 0 .3912 X 10 3 0 .3940 X 10 3 0.3985 X 10 3 0 .3969 X 10 3 0.3976 X i o 3 0 .3976 X 10 3 10~ 12 0. 1 0 .3912 X 10 2 0 .3940 X 10 2 0.3985 X 2 10 0 .'3969 X 10 2 0.3976 X 10 2 0 .3975 X 10 2 10" 8 0. 1 0 .3911 X 10 0 .3940 X 10 0.3984 X 10 0 .3969 X 10 0.3976 X 10 0 .3970 X 10 10" 4 0. 1 0 .5850 X 10 4 0 .5892 X i o 4 0.5958 X i o 4 0 .5935 X i o 4 0.5946 X 10 4 0 .5946 X 10 4 10" 16 0. 5 0 .5850 X i o 3 0 .5892 X i o 3 0.5958 X 10 3 0 .5935 X i o 3 0.5946 X 10 3 0 .5945 X 10 3 10" 12 0. 5 0 .5849 X 10 2 0 .5892 X 10 2 0.5958 X 10 2 0 .5935 X i o 2 0.5946 X 10 2 0 .5943 X 10 2 10~ 8 0. 5 0 .5845 X 10 0 .5888 X 10 0.5954 X 10 0 .5930 X 10 0.5941 X 10 0 .5913 X 10 10" 4 0. 5 0 .6956 X i o 4 0 .7007 X 10 4 0.7086 X 10 4 0 .7058 X i o 4 0.7071 X i o 4 0 .7071 X i o 4 10" 16 1. 0 0 .6956 X 10 3 0 .7007 X 10 3 0.7086 X i o 3 0 .7058 X 10 3 0.7071 X 10 3 0 .7070 X 10 3 10" 12 1. 0 0 .6956 X 10 2 0 .7007 X 10 2 0.7086 X 10 2 0 .7057 X 10 2 0.7071 X 10 2 0 .7065 X 10 2 10" 8 1. 0 0 .6941 X 10 0 .6992 X 10 0.7071 X 10 0 .7043 X 10 0.7056 X 10 0 .6997 X 10 10" 4 1. 0 O J TABLE 1-9 Compar ison o f the o v e r a l l percentage e r r o r (Eq . l-33c) ob t a i ned by d i f f e r e n t methods Sphere P o i n t Match ing Method Least-Squares F i t Method (7-40) V a r i a t i o n a l 5 P o i n t s 7 Po i n t s 10 Po i n t s 15 P o i n t s Method A a 6.3037 3.5264 -0.8366 0.7385 -0.0057206 52 x I O - 7 I O " 1 6 0.1 6.3037 3.5264 -0.8366 0.7385 -0.0057220 52 x 1 0 - 7 i o " 1 2 0.1 6.3037 3.5264 -0.8366 0.7385 -0.0057116 127 x 1 0 " 7 i o " 8 0.1 6.3037 3.5264 -0.8366 0.7385 -0.0057206 387 x 1 0 - 7 i o " 4 0.1 6.2973 3.5276 -0.8367 0.7390 -0.0056669 84.186 10° 0.1 6.3037 3.5264 -0.8366 0.7384 -0.0057235 67 x I O - 7 i o " 1 6 0.5 6.3037 3.5264 -0.8366 0.7384 -0 .0057086 52 x I O - 7 i o " 1 2 0.5 6.3037 3.5264 -0.8366 0.7384 -0.0057235 89 x I O - 7 i o " 8 0.5 6.3030 3.5266 -0.8366 0.7384 -0.0057012 151 x I O - 5 i o " 4 0.5 6.2462 3.5362 -0.8369 0.7392 -0.0061736 - 10° 0.5 6.3037 3.5264 -0.8366 0.7385 -0.0057206 60 x 1 0 - 7 i o " 1 6 1.0 6.3037 3.5264 -0.8366 0.7385 -0.0057146 52 x I O - 7 i o " 1 2 1.0 6.3037 3.5264 -0.8366 0.7385 -0.0057101 127 x I O - 7 i o " 8 1.0 6.3017 3.5268 -0.8366 0.7385 -0.0056922 0.2468 i o " 4 1.0. 6.1860 3.5472 -0.8371 0.7424 -0.0090599 - 10° 1.0 CO F i g u r e 1-1 C o o r d i n a t e S y s t e m C y l i n d e r s R e c t a n g u l a r and C i r c u l a r F i g u r e 1-2 S p h e r i c a l C o o r d i n a t e S y s t e m F i g u r e 1-3 I s o t h e r m a l s f o r R e c t a n g u l a r C y l i n d e r ( B = 0 . 5 , A = 0 . 1 , a=0.5) F i g u r e 1-4 I s o t h e r m a l s f o r R e c t a n g u l a r C y l i n d e r (B=2, A = 0 . 1 , a =0.5 ) F i g u r e 1-5 I s o t h e r m a l s f o r R e c t a n g u l a r C y l i n d e r (B=2, A = 1 0 ~ 3 , a = 0 . 5 ) F i g u r e 1-6 Isothermals f o r Rectangular C y l i n d e r (B=2, A=10~3, a=l) P A R T I I T E M P E R A T U R E D I S T R I B U T I O N A N D E F F E C T I V E N E S S O F A R A D I A T I N G A N D C O N V E C T I N G C I R C U L A R F I N ABSTRACT A t h e o r e t i c a l a n a l y s i s has been conducted of the heat t r a n s f e r c h a r a c t e r i s t i c s of a c i r c u l a r f i n d i s s i p a t i n g heat from i t s s u r f a c e by c o n v e c t i o n and r a d i a t i o n . The temperature i s assumed uniform along the base of the f i n and c o n s t a n t p h y s i c a l and s u r f a c e p r o p e r t i e s are assumed. There i s r a d i a n t i n t e r a c t i o n between the f i n and i t s base. Two separate s i t u a t i o n s are con-s i d e r e d . In the f i r s t s i t u a t i o n heat t r a n s f e r from the end of the f i n i s n e g l e c t e d . S o l u t i o n of the l i n e a r conduction equation w i t h n o n l i n e a r boundary c o n d i t i o n s has been o b t a i n e d by the l e a s t -squares f i t method. A s o l u t i o n has a l s o been 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 method and the r e s u l t s compared. R e s u l t s are p r esented f o r a wide range of environmental c o n d i t i o n s and p h y s i c a l and s u r f a c e p r o p e r t i e s of the f i n . In the second s i t u a t i o n heat t r a n s f e r from the end of the f i n i s a l s o i n c l u d e d i n the a n a l y s i s . The s o l u t i o n i s obtained by a f i n i t e d i f f e r e n c e procedure. I t i s shown t h a t n e g l e c t i n g heat t r a n s f e r from the end i s a good approx-im a t i o n f o r long f i n s or f o r f i n s of h i g h thermal c o n d u c t i v i t y m a t e r i a l . NOMENCLATURE a r a d i u s o f c i r c u l a r f i n , f t . 3 A E ^ a a T 0 /k, r a d i a t i o n - c o n d u c t i o n parameter, dimensionless F c o n f i g u r a t i o n f a c t o r , dimensionless 2 h heat t r a n s f e r c o e f f i c i e n t , BTU/hr. f t . °R. k thermal c o n d u c t i v i t y of f i n , BTU/hr. f t . °R. 1 f i n l e n g t h , f t . L 1/a/ dimensionless f i n l e n g t h N ha/k, c o n v e c t i o n parameter, dimensionless Q r a t e of heat l o s s from f i n , BTU/hr. r , z c y l i n d r i c a l c o o r d i n a t e s f o r f i n , f t . R, Z r / a , z/a, dimensionless c y l i n d r i c a l c o o r d i n a t e s *2 f i n base r a d i u s , f t . T temperature a t any p o i n t i n the f i n , °R. T 0 f i n base temperature, °R. T f l u i d bulk temperature, °R. 00 T* e f f e c t i v e r a d i a t i o n environment temperature, °R. Greek Symbols a c o e f f i c i e n t of a b s o r p t i v i t y , dimensionless e c o e f f i c i e n t of e m i s s i v i t y , dimensionless — 8 a Stefan-Boltzmann constant, 0.1714 x 10 BTU/hr. 2 4 f t . °R. n f i n e f f e c t i v e n e s s , dimensionless g r 2 / / a ' dimensionless ' A (T-T Q) / T Q , dimensionless temperature a t any p o i n t i n the f i n T oo/ T ° ' dimensionless f l u i d bulk temperature A* T*/T c, dimensionless e f f e c t i v e r a d i a t i o n environment temperature S u b s c r i p t s 1 f i n s u r f a c e . ! 2 base s u r f a c e ~ f l u i d bulk S u p e r s c r i p t s * e f f e c t i v e r a d i a t i o n environment INTRODUCTION The design of space v e h i c l e s has s t i m u l a t e d c o n s i d e r a b l e i n t e r e s t i n r a d i a t i o n heat t r a n s f e r . For accurate thermal con-t r o l / the s o l a r energy absorbed by a space v e h i c l e by d i r e c t .incidence o r r e f l e c t i o n from p l a n e t s and the energy generated by power p l a n t s , e l e c t r o n i c equipment or other sources i n the space v e h i c l e i t s e l f must be d i s s i p a t e d away to the surroundings. The s u r f a c e areas r e q u i r e d f o r t h i s purpose are o f t e n huge, s i n c e the Stefan-Boltzmann law s e t s an absolute upper l i m i t t o the heat t r a n s f e r r a t e per u n i t area. In order t o p r o v i d e the l a r g e s u r f a c e areas r e q u i r e d , i t i s n a t u r a l to c o n s i d e r the use of ex-tended s u r f a c e s — i . e . , f i n s . Although s e v e r a l s t u d i e s have been made on the steady s t a t e heat t r a n s f e r f o r r a d i a t i n g f i n s , most of the work has been f o r the one-dimensional model. L i e b l e i n [32] analysed the one-dimensional temperature d i s t r i b u t i o n f o r a r e c t a n g u l a r f i n u s i n g a f i n i t e - d i f f e r e n c e method while Bartas and S e l l e r s [33] used numerical methods to c a l c u l a t e the f i n e f f e c t i v e n e s s f o r r e c -t a n g u l a r f i n s . However, they analyzed i n d e t a i l only a l i m i t i n g c l a s s of r a d i a t i n g f i n s — w h e r e the r a d i a t o r i s f l a t , i . e . , there i s no s p a c i a l i n t e r f e r e n c e of r a d i a t i o n by p i p e s . Chambers and Somers [34] n u m e r i c a l l y determined the f i n e f f i c i e n c y f o r a f l a t , mushroom annular f i n and compared the f i n e f f i c i e n c i e s f o r con-v e c t i v e and r a d i a t i v e heat t r a n s f e r . E c k e r t et. a l . [35] s t u d i e d some gen e r a l c h a r a c t e r i s t i c s of r a d i a t i n g f i n s . Sparrow e t a l . [36] analyzed the e f f e c t i v e n e s s of long, plane r a d i a t i n g f i n s 49 w i t h mutual i r r a d i a t i o n . They formulated the problem as a system o f i n t e g r o - d i f f e r e n t i a l equations and s o l v e d them by i t e r a t i v e n umerical methods. L a t e r , Sparrow and E c k e r t [37] c o n s i d e r e d the e f f e c t s of mutual i r r a d i a t i o n o c c u r r i n g between a f i n and i t s a d j o i n i n g base s u r f a c e f o r both b l a c k and s e l e c t i v e l y gray s u r -f a c e s . They showed t h a t i n the range of p r a c t i c a l o p e r a t i n g c o n d i t i o n s , the f i n heat l o s s i s s i g n i f i c a n t l y reduced by the presence of the base s u r f a c e . Sparrow e t a l . [38] c a l c u l a t e d the f i n e f f e c t i v e n e s s f o r one-dimensional heat flow i n annular f i n s w i t h mutual i r r a d i a t i o n between the b l a c k r a d i a t o r f i n s . R e c ently, Sparrow and Niewerth [39] have gi v e n the numerical and l i n e a r i z e d s o l u t i o n s f o r the one-dimensional heat conduction i n c o n v e c t i n g - r a d i a t i n g f i n s . Among the very few exact a n a l y t i c a l s o l u t i o n s a v a i l a b l e , Shouman [40] has g i v e n an exact g e n e r a l s o l u t i o n f o r a constant c r o s s - s e c t i o n a l area f i n . However, he has c o n s i d e r e d a one-dimensional heat flow model and n e g l e c t e d f i n - t o - b a s e i n t e r a c t i o n . Since a minimum mass of a l l equipment i s d e s i r a b l e i n space v e h i c l e s , a number of s t u d i e s have been made t o determine the optimum f i n s i z e having the l a r g e s t r a t i o of heat t r a n s f e r t o mass. L i u [41] presented the optimum s i z e r a d i a t i n g f i n of a r e c t a n g u l a r p r o f i l e . W i l k i n s [42] and N i l s o n and Curry [43] obtained the minimum-mass t r i a n g u l a r p r o f i l e f i n . C a l l i n a n and Berggren [44] analyzed simple cases of nonconvective r a d i a t o r s and suggested means of maximizing heat r e j e c t i o n per u n i t weight. Winter and Schaberg [45] suggested an intermeshing s u r f a c e arrangement of f i n s f o r thermal c o n t r o l of a space c r a f t . 50 Apparently the l i t e r a t u r e c o n t a i n s very few s t u d i e s f o r the two-dimensional heat flow i n r a d i a t i n g f i n s and they have a l l used f i n i t e - d i f f e r e n c e techniques. H o l s t e a d and Holdredge [46] haye used a f i n i t e - d i f f e r e n c e procedure to s o l v e the problem of heat t r a n s f e r i n t r a p e z o i d a l p r o f i l e f i n s w i t h no f i n - b a s e i n t e r a c t i o n f o r both one-dimensional and two-dimensional cases. Sparrow e t a l . [47] have o b t a i n e d the s o l u t i o n f o r the two-dimen-s i o n a l heat t r a n s f e r i n f i n - t u b e r a d i a t o r s . In the p r e s e n t i n v e s t i g a t i o n a s e r i e s s o l u t i o n , whose c o e f f i c i e n t s are determined by the l e a s t - s q u a r e s f i t method, has been employed to analyze the two-dimensional heat flow i n a c i r c u l a r f i n . Heat i s d i s s i p a t e d by r a d i a t i o n and c o n v e c t i o n from i t s s u r f a c e . In a d d i t i o n , r a d i a t i o n interchange with the base i s i n c l u d e d . The r e s u l t s of the l e a s t - s q u a r e s f i t method have been compared wi t h a f i n i t e d i f f e r e n c e s o l u t i o n . FORMULATION AND SOLUTION OF THE PROBLEM C o n s i d e r a c i r c u l a r f i n o f r a d i u s ' a ' and l e n g t h ' / a s shown i n F i g u r e (2-1) s u b j e c t t o t h e f o l l o w i n g c o n d i t i o n s : ( i ) T h e t e m p e r a t u r e h a s a u n i f o r m v a l u e T 0 a l o n g t h e b a s e o f t h e f i n . ( i i ) T h e r e i s no i n c i d e n t r a d i a t i o n on t h e f i n s o t h a t t h e r e i s r o t a t i o n a l s y m m e t r y , ( i i i ) T h e p h y s i c a l and s u r f a c e p r o p e r t i e s o f t h e f i n and b a s e m a t e r i a l s a r e i n v a r i a n t w i t h t e m p e r a t u r e . ' ( i y ) The f i n i s o f i s o t r o p i c homogeneous m a t e r i a l . (v) T h e f i n has r a d i a t i o n i n t e r a c t i o n w i t h t h e b a s e b u t m u l t i p l e i n t e r a c t i o n s w i t h t h e b a s e a r e n o t c o n s i d e r e d . Two c a s e s a r e c o n s i d e r e d and t h e y a r e d e a l t w i t h s e p a r a t -e l y i n two s e c t i o n s . S e c t i o n A d e a l s w i t h t h e c a s e i n w h i c h h e a t t r a n s f e r f r o m t h e e n d o f t h e f i n i s n e g l e c t e d i n c o m p a r i s o n w i t h t h a t f r o m i t s s i d e s . I n S e c t i o n B' t h e h e a t t r a n s f e r f r o m t h e e n d i s n o t n e g l e c t e d and t h e r e s u l t s o b t a i n e d a r e c o m p a r e d w i t h t h o s e o f S e c t i o n A . SECTION A No Heat Transfer from the End of the Fin The steady state d i f f e r e n t i a l equation of heat conduction and the boundary conditions may be written as follows: Energy Equation: V l • j _ . J L . + J £ L =o . (2-D Boundary Conditions are: B.C. (i) At Z - 0 , T - T (2-2) B.C. ( i i ) At t - ^ - | L = 0 • (2-3) B.C. ( i i i ) At t - 0 , -JLL - Q . (2-4) "bt B.Ci (iv) The fourth boundary condition at r=a involves the energy balance equation of an elemental area dA^ on the f i n , The heat conducted from within the f i n to the surface may be dissipated by three modes of heat transfer. F i r s t l y , the f i n surface element has radiant interchange with space : • ^ , - ^ 6 . < r T t = a — «,<rT* k . ^ A V (2-5a) Then there i s convection interchange with space &!\ • Vi { T ^ a — X.) » (2-5b) and f i n a l l y , radiant interchange with the f i n base 53 K^^K^X^s + A V V ' ^ T ^ « (2-5 c) The c o n f i g u r a t i o n f a c t o r F i n t r o d u c e d above denotes m-»-n the f r a c t i o n o f the t o t a l energy em i t t ed by s u r f a c e 'm* t h a t i s i n t e r c e p t e d by s u r f a c e ' n 1 . Us ing the r e c i p r o c i t y theorem f o r c o n f i g u r a t i o n f a c t o r s and the summation theorem (2-6a) (2-6b) M , -=»oo (2-6c) and assuming gray body p r o p e r t i e s f o r the f i n ( i . e . ct^=e^) , the energy ba lance equa t i on reduces to Rearrangement o f the terms y i e l d s "bT _ fc, <r (2-7) The gove rn ing equa t i on f o r the temperature d i s t r i b u t i o n i n the f i n and the boundary c o n d i t i o n s may be r ephrased i n t o conven ien t d imens ion l e s s forms by i n t r o d u c i n g d i m e n s i o n l e s s v a r i a b l e s as f o l l o w s : X = ( T - T „ V X , X . = t / T . , X* = T/T. . <2-•3 / 8a) (2-8b) 8c) In terms of these new v a r i a b l e s , the energy equa t i on (2-1) and the boundary c o n d i t i o n s (2-2, 2-3, 2-4, 2-7) become -r- \ ~oX S I 1 X = 0 _bX_ <U0 - 0 = 0 (2-9) (2-10) (2-11) (2-12) (2-13) Equation (2-9) i s s o l v e d by the method of s e p a r a t i o n of v a r i -a b l e s . The method of s o l u t i o n i s o u t l i n e d below. L e t the s o l u t i o n be of the form X - H Q • C l U ) (2-14) Then eq. (2-9) becomes: Thus the two r e s u l t i n g equations are: 1 <U5 " (2-15) 4-The s o l u t i o n t o (2-15) i s of the form COS YnZ -\- C 2 5V.r> YnX , (2-17) where C/4 , are constants Equation (2-16) may be r e w r i t t e n as Comparing the above equation with the equation g i v e n by Wylie [48] i t can be e a s i l y seen t h a t the above equation i s of a type r e d u c i b l e to B e s s e l ' s equation and i t s s o l u t i o n i s of the f o l l o w i n g form: 56 (2-18) where / C 4 a r e u nknown c o n s t a n t s , I D i s the m o d i f i e d B e s s e l f u n c t i o n of the f i r s t k i n d o f order zero and K Q i s the mo d i f i e d B e s s e l f u n c t i o n of the second k i n d of order zero. Thus, from (2-14), (2-17) and (2-18), the complete g e n e r a l s o l u t i o n t o the energy equation (2-9) i s (2-19) Use of equation (2-10) i n (2-19) y i e l d s C,=0 . Then (2-20) where , Cg are new unknown c o n s t a n t s . A p p l y i n g boundary c o n d i t i o n equation (2-12) t o (2-20), we have [ C 5 T » - C t K , W ] 7 . 0 . Since (2-12) must h o l d f o r a l l Z, Now while T h e r e f o r e , (2-21) Using the c o n d i t i o n of no heat t r a n s f e r from the end, i . e . f o r a l l R, we get * W \ Cos Y A L . - 0 . 57 T h i s y i e l d s the eigen-values f o r m as -m = UT . + V . W / I L > ^ ^ , 1 , 2 , -o . Thus the g e n e r a l s o l u t i o n to (2-9) becomes X - l ^ L ^ - ^ i ^ t ^ ^ l j • « - , » The unknown c o e f f i c i e n t s C (n=0,-lf 2,...00) can be d e t e r -mined by p o i n t matching or l e a s t - s q u a r e s f i t t i n g a t a f i n i t e number of p o i n t s on the boundary, choosing e q u a l l y spaced p o i n t s along the l e n g t h of the f i n and matching or f i t t i n g t o eq.(2-13). The l e a s t - s q u a r e s f i t method has been employed here s i n c e i t has been, proven to be more e f f i c i e n t than the p o i n t matching one i n P a r t I of the t h e s i s . From boundary c o n d i t i o n (2-13) the e x p r e s s i o n to be minimized i s o b t a i n e d as where denotes summation over a l l p o i n t s i n the r e g i o n Using the s e r i e s s o l u t i o n f o r X as expressed i n (2-22) the above e x p r e s s i o n takes the f o l l o w i n g form -v-58 (2-23) The f u n c t i o n to be f i t t e d t o zero a t v a r i o u s boundary p o i n t s i s /s~ . Since the f u n c t i o n i s n o n - l i n e a r i n the parameters C n , i t i s f i r s t l o c a l l y l i n e a r i z e d and then the l e a s t - s q u a r e s c r i t e r i a a p p l i e d t o get a system of l i n e a r a l g e b r a i c equations which are s o l v e d to g i v e the c o e f f i c i e n t s C^. The process i s i t e r a t e d upon u n t i l 7 ^ ^ _ In t h i s work 7 c o e f f i c i e n t s and 100 p o i n t s on the boundary were chosen. Thus f o r known values of X , X*, A, N, L and e„ the oo ' 2. temperature d i s t r i b u t i o n can be determined i f the c o n f i g u r a t i o n f a c t o r F, a a i s known. The c o n f i g u r a t i o n f a c t o r was e v a l u a t e d 1 2 and i s g i v e n by the f o l l o w i n g e x p r e s s i o n , where <j> - ~L p + \ and — "^ /(X- i s dimensionless f i n base r a d i u s . The d e t a i l e d d e r i v a t i o n of the above e x p r e s s i o n i s g i v e n i n Appendix B (p. 139) The c o n f i g u r a t i o n f a c t o r s were f i r s t t a b u l a t e d f o r v a r i o u s v a l u e s of 3 and f o r v a r i o u s p o i n t s along the f i n l e n g t h and the r e s u l t s are shown i n F i g u r e (2-2) . The system of equations (2-9) to (2-13) has a l s o been s o l v e d by a f i n i t e - d i f f e r e n c e procedure which i s d e s c r i b e d i n Appen ix C. (p. 142). DISCUSSION OF RESULTS The r e s u l t s are presented i n the form of a x i a l temper-ature d i s t r i b u t i o n and f i n e f f e c t i v e n e s s . A x i a l Temperature D i s t r i b u t i o n : The temperature, d i s t r i b u t i o n along the a x i s (R=0) i s rep r e s e n t e d i n F i g u r e s (2-3) and (2-4). F i g u r e (2-3) shows t h a t a t any p o i n t on the a x i s the temperature decreases w i t h i n c r e a s -i n g v a l u e s of the r a d i a t i o n - c o n d u c t i o n parameter A. T h i s i s as expected s i n c e h i g h values of A s i g n i f y a low value o f thermal c o n d u c t i v i t y k or h i g h values o f f i n r a d i u s 'a' or base tempera-tu r e T 0 , each of which decreases the r a t i o T/T 0. The s l o p e a t the end i s zero, as expected, i n d i c a t i n g no heat t r a n s f e r from the c o n v e c t i o n parameter N on the a x i a l temperature d i s t r i b u t i o n . An i n c r e a s e i n N causes more heat l o s s by c o n v e c t i o n and, t h e r e -f o r e , lower s u r f a c e and a x i a l temperatures. F i n E f f e c t i v e n e s s : The heat t r a n s f e r performance of the f i n can be expressed i n terms of the f i n e f f e c t i v e n e s s d e f i n e d as the end. F i g u r e (2-4) i l l u s t r a t e s the e f f e c t of the v a r i a t i o n o f Gl > (2-25) where the i d e a l heat f l u x i s d e f i n e d by assuming t h a t the e n t i r e f i n i s a t the base temperature and r a d i a t e s without i n t e r f e r e n c e . N o n - d i m e n s i o n a l i s i n g a l l the terms, the e x p r e s s i o n f o r 60 f i n e f f e c t i v e n e s s t a k e s t h e f o l l o w i n g f o r m F i g u r e s (2-5) t© (2-9) i l l u s t r a t e t h e e f f e c t o f t h e v a r i a t i o n o f some p a r a m e t e r s on t h e f i n e f f e c t i v e n e s s . ©ue t o t h e p r e s e n c e o f s e v e n i n d e p e n d e n t p a r a m e t e r s o n l y a few r e p r e -s e n t a t i v e c a s e s have b e e n p r e s e n t e d . F i g u r e (2-5) shows t h e e f f e c t o f e m i s s i v i t y o f t h e b a s e s u r f a c e on t h e f i n e f f e c t i v e -n e s s . I n c r e a s i n g v a l u e s o f t h e b a s e s u r f a c e e m i s s i v i t y d e c r e a s e t h e f i n e f f e c t i v e n e s s s i n c e t h e a c t u a l h e a t t r a n s f e r d e c r e a s e s due t o i n c r e a s i n g i n t e r a c t i o n w i t h t h e h i g h t e m p e r a t u r e b a s e w h i l e t h e i d e a l h e a t t r a n s f e r r e m a i n s t h e same . The e f f e c t o f c h a n g e s i n X ^ and X * i s shown i n F i g u r e ( 2 - 6 ) . I n c r e a s e o f X ^ and X * d e c r e a s e s t h e f i n e f f e c t i v e -n e s s f o r mos t p a r t o f t h e l o w e r v a l u e s o f X ^ and X * r a n g e and i n c r e a s e s i n t h e end f o r t h e c a s e shown . T h i s i s due t o t h e f a c t t h a t an i n c r e a s e i n X and X * d e c r e a s e s b o t h t h e a c t u a l oo h e a t t r a n s f e r and t h e i d e a l h e a t t r a n s f e r . The e f f e c t on t h e i r r a t i o , f i n e f f e c t i v e n e s s , w i l l t h e n d e p e n d upon t h e v a l u e o f t h e o t h e r p a r a m e t e r s . T h i s minimum i n e f f e c t i v e n e s s was n o t o b s e r v e d f o r 3=1, w h i c h r e p r e s e n t s t h e c a s e o f no f i n - b a s e i n t e r a c t i o n . F i g u r e (2-7) i l l u s t r a t e s t h e e f f e c t o f t h e v a r i a t i o n o f t h e c o n v e c t i o n p a r a m e t e r N on f i n e f f e c t i v e n e s s . An i n c r e a s e i n t h e v a l u e o f N d e c r e a s e s t h e s u r f a c e t e m p e r a t u r e , t h u s g r e a t l y r e d u c i n g t h e h e a t t r a n s f e r by r a d i a t i o n . T h i s r e d u c t i o n i s normally g r e a t e r than the i n c r e a s e i n the c o n v e c t i o n term, the o v e r a l l e f f e c t thus being t o decrease the f i n e f f e c t i v e n e s s . I n c r e a s i n g values o f the f i n l e n g t h reduce the temper-ature of the f i n s u r f a c e . T h i s decreases the f i n e f f e c t i v e n e s s as i l l u s t r a t e d i n F i g u r e (2-8). High v a l u e s of $ cause h i g h e r values of the c o n f i g u r -a t i o n f a c t o r thus r e s u l t i n g i n more r a d i a n t i n t e r a c t i o n of the f i n w i t h the base. T h i s suppresses the heat t r a n s f e r t o the surroundings and, t h e r e f o r e , decreases the f i n e f f e c t i v e n e s s . However, t h i s e f f e c t i s s m a l l as can be seen from the o r d i n a t e s of F i g u r e (2-9). A comparison has a l s o been made i n Table 2-1 of the value o f f i n e f f e c t i v e n e s s obtained by t h i s work wit h those o b t a i n e d by Sparrow and Niewerth [39]. In Table 2-1 8 has been taken as one t o a v o i d b r i n g i n g i n the e f f e c t o f f i n - t o - b a s e i n t e r a c t i o n The comparison c o u l d , however, be made onl y f o r a very l i m i t e d . range of values s i n c e o n l y a few r e s u l t s were a v a i l a b l e i n r e f e r -ence [39]. Although the d i f f e r e n c e i n values i s s m a l l , s i n c e h i g h c o n d u c t i v i t y f i n s (low values o f A) are c o n s i d e r e d , the f i n e f f e c t i v e n e s s as obtained by t h i s a n a l y s i s i s lower throughout the range. T h i s i s because the pr e s e n t work c o n s i d e r s a two-dimensional model which causes lower s u r f a c e temperatures. The g r e a t e r v a r i a t i o n i n f i n body temperatures causes lower f i n e f f e c t i v e n e s s s i n c e i t re p r e s e n t s g r e a t e r departure from the i s o t h e r m a l f i n case. 6 2 Comparison of the Least-Squares F i t R e s u l t s w i t h 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 The r e s u l t s of the temperature d i s t r i b u t i o n o b t a i n e d by the two methods are i l l u s t r a t e d i n F i g u r e (2-3). The v a l u e s agree wi t h each other very c l o s e l y . I t has been observed t h a t f o r h i g h values of the r a d i a t i o n - c o n d u c t i o n parameter A there i s an agreement up to f o u r s i g n i f i c a n t f i g u r e s . However, even wi t h t h i s c l o s e agreement i n temperature d i s t r i b u t i o n , there i s an a p p r e c i a b l e d e v i a t i o n i n the values of f i n e f f e c t i v e n e s s as evidenced by F i g u r e (2-8). The d e v i a t i o n depends upon the number of s e r i e s terms and the number of boundary p o i n t s chosen i n the l e a s t - s q u a r e s f i t technique. When more s e r i e s terms and more boundary p o i n t s are taken i n the l e a s t - s q u a r e s f i t method, the f i n e f f e c t i v e n e s s values approach those 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 technique. In F i g u r e (2-8) f i n e f f e c t i v e n e s s v a l u e s o b t a i n e d by the l e a s t - s q u a r e s f i t method f o r 7 s e r i e s terms and 100 boundary p o i n t s and f o r 20 s e r i e s terms and 400 boundary p o i n t s are compared with those of the f i n i t e d i f f e r e n c e method. Thus the l e a s t - s q u a r e s f i t method r e s u l t s can approach the f i n i t e d i f f e r e n c e v a l u e s , although a t the expense of i n c r e a s e d computer time. SECTION B Heat T r a n s f e r from the End of the F i n i s Not Neglected For f i n s of s h o r t l e n g t h i t may not be very c o r r e c t t o n e g l e c t the heat t r a n s f e r from the end of the f i n as compared to t h a t from i t s s i d e s . T h i s s i t u a t i o n i s analyzed i n t h i s s e c t i o n . The energy equation and the other three boundary con-d i t i o n s remain u n a l t e r e d . Only boundary c o n d i t i o n ( i i ) (equation 2-3) i s a l t e r e d . The boundary c o n d i t i o n a t the end of the f i n now takes the f o l l o w i n g form: or Z r t Using the dimensionless parameters d e s c r i b e d e a r l i e r i n S e c t i o n A, the above equation i s rendered dimensionless and i t then takes the form Thus the energy equation and the boundary c o n d i t i o n s f o r t h i s case a r e : A . » *X . VA o \ = 0 (2-10) 2=& 64 x^ Z - _ L • - 0 , ( 2 - 1 2 ) -MV+xJ ^ X * \ V - ^ J ( 2 - 1 3 ) The g e n e r a l s o l u t i o n ( 2 - 1 9 ) d e r i v e d e a r l i e r s t i l l h o l d s b u t i n t h i s case we cannot g e t any e i g e n - v a l u e s f o r m s i n c e 4^— I T ^ 0 . To s a t i s f y t h i s c o n d i t i o n we t a k e the s o l u t i o n dz 1 Z=L J i n the f o l l o w i n g form: ( 2 - 2 9 ) Use o f e q u a t i o n ( 2 - 1 2 ) i n above e x p r e s s i o n y i e l d s =0 . o r I f we t r y t o s a t i s f y e q u a t i o n ( 2 - 1 0 ), then C N = 0 and x - i & ^ f ] . ; ^ ) But then X v a n i s h e s a t Z = L a l s o which v i o l a t e s e q u a t i o n ( 2 - 2 8 ) Thus the f u l l form o f A as e x p r e s s e d i n ( 2 - 2 9 ) i s r e t a i n e d and e q u a t i o n ( 2 - 1 0 ) i s n o t s a t i s f i e d e x a c t l y . The l e a s t - s q u a r e s f i t t i n g p r o c e d u r e i s then a p p l i e d t o the boundary c o n d i t i o n s a t 65 three boundaries, z = 0, 1 = L and R = 1 as expressed by the equations (2-10), (2-28) and (2-13). However the equations became too unstable and not e a s i l y amenable to solution. This was probably due to the severe re-s t r i c t i o n s imposed on the least-squares f i t t i n g procedure i n t h i s case. The f i n i t e difference procedure was then applied to solve the system of equations (2-9, 2-10, 2-28, 2-12, 2-13). The procedure used was si m i l a r to the one used i n Section A and described i n Appendix C (p.142). The results obtained were com-pared with the f i n i t e difference results of Section A (no heat transfer from the end) and plotted i n Figures (2-10) and (2-11). The f i n effectiveness for Section B i s re-evaluated as follows: V < W CM.+vro})i^{t-r") +UT0-T»)] ' On non-dimensionalisation the above expression takes the follow-ing form: -4- \ u \.(2-31) COMPARISON OF SECTION A WITH SECTION B F i g u r e (2-10) i l l u s t r a t e s the r e s u l t s of a x i a l tempera-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 two boundary c o n d i t i o n s s t a t e d i n S e c t i o n s A and B. For low values of A ( i . e . h i g h thermal c o n d u c t i v i t y ) agreement between the two c o n d i t i o n s i s reached a t comparatively s m a l l e r values of L . T h i s i s more c l e a r l y demon-s t r a t e d by the p l o t s f o r f i n e f f e c t i v e n e s s i n F i g u r e (2-11). Both h i g h values of L or low values of A reduce the d i f f e r e n c e i n the r e s u l t s o b tained by the two c o n d i t i o n s . However, even f o r very h i g h thermal c o n d u c t i v i t y of the f i n the marked d i f f e r e n c e i n f i n e f f e c t i v e n e s s , e s p e c i a l l y at low values of L , w i l l s t i l l remain. T h i s i s due to the i n h e r e n t d i f f e r e n c e i n d e f i n i t i o n s of f i n e f f e c t i v e n e s s f o r the two cases ( c f . equations 2-26 and 2-31). Even i f heat t r a n s f e r from the end i s n e g l i g i b l e and the term R "5^1 Z = L d R i - s n e g l e c t e d , the d i f f e r e n c e i n the 2 1 f a c t o r s ( 2L+1 ^ A N C ^ L c ^ u s e a marked d i f f e r e n c e i n the values o f f i n e f f e c t i v e n e s s , e s p e c i a l l y f o r low values of L . CONCLUSIONS The problem of two-dimensional heat flow i n a c i r c u l a r f i n having r a d i a n t i n t e r a c t i o n w i t h the f i n base has been a n a l -ysed. The s o l u t i o n i s obtained by a s e r i e s expansion and the l e a s t - s q u a r e s f i t method. R e s u l t s are presented i n terms of a x i a l temperature d i s t r i b u t i o n and f i n e f f e c t i v e n e s s . The problem i s a l s o s o l v e d by a f i n i t e d i f f e r e n c e s o l u t i o n and the r e s u l t s o b t a i n e d by the two methods are compared. For m a t e r i a l s w i t h h i g h thermal c o n d u c t i v i t y the r e s u l t s agree c l o s e l y w i t h the one-dimensional model. The i n v e s t i g a t i o n r e p o r t e d h e r e i n has a l s o c o n s i d e r e d the e f f e c t of heat t r a n s f e r from the end, an e f f e c t t h a t has g e n e r a l l y been n e g l e c t e d i n p r i o r s t u d i e s of r a d i a t i n g f i n s . T h i s problem has been s o l v e d by a f i n i t e d i f f e r e n c e procedure. The r e s u l t s demonstrate t h a t the e f f e c t of heat t r a n s f e r from the end of the f i n becomes smal l f o r f i n m a t e r i a l s of h i g h thermal con-d u c t i v i t y or f o r long f i n s . 68 TABLE 2-1 Comparison o f f i n e f f e c t i v e n e s s v a l u e s w i t h t h o s e o b t a i n e d by Sparrow and N i e w e r t h [39] (g = 1, e „ = 0.4) V a l u e s o f t h i s work Reduced t o v a l u e s o f [39] A L N N r=2AL 2 N =2NL 2 C V A =6 oo oo A*=e* n t h i s work n [39] i o " 2 3.873 0.005 0.3 0.15 . 0.7 0.7 0.691 0.728 IO" 2 3.873 0.005 0.3 0.15 0.9 0.9 0.681 0.718 IO" 3 12.25 0.005 0.3 1.5 0.7 0.7 0.552 0.580 i o " 3 12.25 0.005 0.3 1.5 0.9 0.9 0.541 0.571 i o " 2 5.477 0.01 0.6 0.6 0.7 0.7 0.539 0.577 IO" 2 5.477 0.01 0.6 0.6 0.9 0.9 0.522 0.558 IO" 3 17.32 0.006 0.6 3.6 0.7 0.7 0.391 0.420 i o " 3 17.32 0.006 0.6 3.6 0.9 0.9 • 0.380 0.409 1 F i g u r e 2-4 E f f e c t o f N on A x i a l Temperature D i s t r i b u t i o n " ~io* 7 10' 6 10' 5 10"4 IO*3 10"2 10 A F i g u r e 2-5 E f f e c t o f B a s e S u r f a c e E m i s s i v i t y and A on F i n E f f e c t i v e n e s s F i g u r e 2-6 E f f e c t o f E n v i r o n m e n t T e m p e r a t u r e s on F i n E f f e c t i v e n e s s (D t o I Hi Hi (D O rt O Hi > 0 *J H-3 W Hi Hi (D o rt < (D 01 tn . O O i FIN EFFECTIVENESS w ^ 6s 0 0 F i g u r e 2-8 E f f e c t o f F i n L e n g t h on F i n E f f e c t i v e n e s s cr. *This denotes 7 terms in the series and T O O boundary points. CO CO U J L U > o U J L i . U _ U J 5 6 7 P F i g u r e 2-9 E f f e c t o f F i n B a s e R a d i u s on F i n E f f e c t i v e n e s s 8 80 F i g u r e 2-12 F i n i t e D i f f e r e n c e R e p r e s e n t a t i o n P A R T I I I LAMINAR HEAT TRANSFER IN A CIRCULAR TUBE UNDER SOLAR RADIATION IN SPACE ABSTRACT The problem of laminar heat t r a n s f e r i n a c i r c u l a r tube under r a d i a n t heat f l u x boundary c o n d i t i o n s has been analyzed. F u l l y developed v e l o c i t y p r o f i l e i s assumed and the tube i s co n s i d e r e d s t a t i o n a r y . A steady r a d i a n t energy f l u x i s being i n c i d e n t on one h a l f of the tube circumference w h i l e t h e . f l u i d emanates heat through the w a l l on a l l s i d e s by r a d i a t i o n t o a zero degree temperature environment. A s o l u t i o n by f i n i t e -d i f f e r e n c e procedure has been o b t a i n e d . The temperature d i s t r i -b u t i o n and the N u s s e l t number v a r i a t i o n are presented f o r a wide range of the governing p h y s i c a l parameters. NOMENCLATURE r a d i u s o f t u b e , f t . 2 i n c i d e n t r a d i a t i o n f l u x , B T U / h r . f t . t h e r m a l c o n d u c t i v i t y o f f l u i d , B T U / h r . t u b e l e n g t h , f t . 1/a , d i m e n s i o n l e s s t u b e l e n g t h 2 f l u i d p r e s s u r e , l b / f t . 2 h e a t t r a n s f e r r a t e , B T U / h r . f t . r a d i a l c o o r d i n a t e , f t . r / a , d i m e n s i o n l e s s r a d i a l c o o r d i n a t e P r . R e , P e c l e t n u m b e r , d i m e n s i o n l e s s /uCp/k, P r a n d t l n u m b e r , d i m e n s i o n l e s s r / a , d i m e n s i o n l e s s r a d i u s R / Pe , d i m e n s i o n l e s s r a d i u s i n t r a n s c o o r d i n a t e s 2Ua/v , R e y n o l d ' s n u m b e r , d i m e n s i o n l e s t e m p e r a t u r e a t any p o i n t , ° R . a v e r a g e f l u i d v e l o c i t y , f t . / h r . a x i a l f l u i d v e l o c i t y , f t . / h r . 84 v r a d i a l f l u i d v e l o c i t y , f t . / h r . x a x i a l c o o r d i n a t e , f t . X d i m e n s i o n l e s s a x i a l c o o r d i n a t e Greek Symbols a c o e f f i c i e n t o f a b s o r p t i v i t y o f tube w a l l , d i m e n s i o n l e s s 3 Y eoT 0 a/k , a d i m e n s i o n l e s s parameter e c o e f f i c i e n t o f e m i s s i v i t y o f tube w a l l , d i m e n s i o n l e s s 1/3 X T/(k/aa) ' , d i m e n s i o n l e s s temperature 2 v k i n e m a t i c v i s c o s i t y o f f l u i d , f t . / h r . £ X/Pe , d i m e n s i o n l e s s a x i a l d i s t a n c e 3 p d e n s i t y o f f l u i d , l b ^ / f t . — 8 2 a S t e f a n - B o l t z m a n n c o n s t a n t , 0.1714 x 10 B T U / h r . f t . . °R. 4 ip a (G^a^a/k 4) , r a d i a t i o n - 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 6 a n g u l a r c o o r d i n a t e S u b s c r i p t s 0 a t e n t r a n c e (x = 0) b f l u i d b u l k c r i t i c a l r a d i a l a t w a l l a x i a l INTRODUCTION Heat t r a n s f e r problems r e l a t i n g t o laminar flow i n tubes have been the s u b j e c t of i n v e s t i g a t i o n f o r many y e a r s . V a r i o u s i n v e s t i g a t o r s have d e a l t w i t h v a r i o u s types of boundary c o n d i t i o n s . S o l u t i o n s i n v o l v i n g p r e s c r i b e d (although v a r i a b l e ) temperature boundary c o n d i t i o n s i n c l u d e the c l a s s i c a l work of S e l l a r s , T r i b u s and K l e i n [51]. Other s o l u t i o n s i n t h i s f i e l d are w e l l reviewed by Singh [52] who a l s o i n c l u d e d the e f f e c t s of a x i a l heat con-d u c t i o n , v i s c o u s d i s s i p a t i o n and constant heat g e n e r a t i o n . Kuga [53] co n s i d e r e d a s i n u s o i d a l w a l l temperature d i s t r i b u t i o n . S o l u t i o n s i n v o l v i n g p r e s c r i b e d (although v a r i a b l e ) h e a t - f l u x boundary c o n d i t i o n s i n c l u d e the work of S i e g e l , Sparrow and Hallman [54]. Hsu [55] con s i d e r e d a s i n u s o i d a l w a l l heat f l u x d i s t r i b u t i o n and Kuga [56] s o l v e d the problem f o r s i n u s o i d a l and e x p o n e n t i a l w a l l heat f l u x e s . There i s another c l a s s o f problems i n which n e i t h e r the w a l l temperature nor the w a l l heat f l u x i s p r e s c r i b e d . Instead, the w a l l heat f l u x i s s p e c i f i e d as a f u n c t i o n of the w a l l temper-at u r e . T h i s type of problem has only r e c e n t l y r e c e i v e d some a t t e n t i o n . T h i s i s a more d i f f i c u l t problem s i n c e the heat t r a n s -f e r equation now i n v o l v e s the unknown v a r i a b l e ( e i t h e r temper-ature o r heat f l u x ) i n an i m p l i c i t r a t h e r than e x p l i c i t form. References [57-6.0] have t r e a t e d such type of problems. Sideman et a l . [57] extended Graetz s o l u t i o n to i n c l u d e s u r f a c e r e s i s -tance to heat t r a n s f e r i n laminar flow i n c i r c u l a r tubes and f l a t c o n d u i t s . S t e i n [58] s o l v e d the Graetz problem p e r t a i n i n g to the concurrent flow double pipe heat exchangers and i n t r o d u c e d 87 a n e f f e c t i v e n e s s c o e f f i c i e n t f o r h e a t e x c h a n g e r s . C h e n [ 5 9 ] s o l v e d t h e p r o b l e m o f r a d i a n t c o o l i n g o f a f l u i d i n l a m i n a r f l o w t h r o u g h a t u b e . H e o b t a i n e d a n a p p r o x i m a t e s o l u t i o n i n t e r m s o f t h e L i o u v i l l e - N e u m a n n s e r i e s a n d a l s o o b t a i n e d a n i t e r a t i v e n u m e r i c a l s o l u t i o n . D u s s a n a n d I r v i n e [ 6 0 ] a l s o p r e s e n t e d a n a p p r o x i m a t e s o l x i t i o n f o r t h e s a m e p r o b l e m a n d v e r i f i e d t h e r e -s u l t s e x p e r i m e n t a l l y . E o w e v e r , n e i t h e r o f t h e i n v e s t i g a t o r s [ 5 9 , 6 0 ] c o n s i d e r e d t h e e f f e c t o f i n c i d e n t r a d i a t i o n f l u x o n t h e h e a t t r a n s f e r r a t e o f t h e f l u i d . T h i s p a r t i c u l a r p r o b l e m h a s a p p l i c a t i o n s i n n u c l e a r r e a c t o r s a n d i n s p a c e c r a f t . I n s p a c e c r a f t a p p l i c a t i o n s t h e p r o b l e m m a y a r i s e e i t h e r i n h e a t r e j e c t i o n s y s t e m s o r i n c o u p l i n g o f t w o s a t e l l i t e s i n s p a c e . . I n t h e p r e s e n t i n v e s t i g 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 h a s b e e n e m p l o y e d t o s o l v e t h e h e a t t r a n s f e r p r o b l e m f o 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 o f f l u i d i n a t u b e b e i n g h e a t e d b y a u n i f o r m i n c i d e n t f l u x a n d a l s o u n d e r g o i n g r a d i a t i o n c o o l i n g f r o m t h e s u r f a c e . FORMULATION OF THE PROBLEM Consider a constant p r o p e r t y f l u i d i n laminar flow through a c i r c u l a r tube of r a d i u s 'a' (Figure 3-1). A steady 2 r a d i a n t energy f l u x of G BTU/hr. f t . of p r o j e c t e d area i s being i n c i d e n t on one h a l f of the tube circumference w h i l e the f l u i d emanates heat through the w a l l on a l l s i d e s by r a d i a t i o n to a 0°R. environment. At x=0, the f l u i d i s c o n s i d e r e d t o have a f u l l y developed v e l o c i t y p r o f i l e and a uniform temperature T„. Heat t r a n s f e r a t the w a l l s t a r t s at x=0. I t i s assumed t h a t the tube i s not r o t a t i n g about any a x i s so t h a t secondary flow e f f e c t s a r i s i n g from c e n t r i f u g a l f o r c e may be n e g l e c t e d . For the p h y s i c a l s i t u a t i o n s t a t e d above the f o l l o w i n g c o n s e r v a t i o n equations w i l l apply. C o n t i n u i t y Equation The c o n t i n u i t y equation i s a u t o m a t i c a l l y s a t i s f i e d s i n c e \)^. — 0 — Momentum Equation The s o l u t i o n o f t h i s momentum equation i s w e l l known and i s give n below, where U i s the average f l u i d v e l o c i t y , 89 Energy E q u a t i o n The energy e q u a t i o n f o r t h e system i s , v^ _bT_ _ _W_ ( \ "bT \ V T V 3 . ^ ~ ?CF V + T ' ~ F ~ + ~~W) I n the above energy e q u a t i o n t h e a x i a l h e a t c o n d u c t i o n i n t h e f l u i d i s n e g l e c t e d , s i n c e i t i s known t h a t the e f f e c t o f t h i s term i n the energy e q u a t i o n i s n e g l i g i b l e f o r Pe > 100. U s i n g t h e s o l u t i o n o f t h e v e l o c i t y p r o f i l e from eq. (3-1) t h e energy e q u a t i o n reduces t o •)\\Y\ - W l ^T - k F ^ T + ' <>T x » tfT"|(3-3) « L \ w) J ^ - - - 9 C ; L - ^ F + T^T+T^\ The boundary c o n d i t i o n s f o r t h e system are as f o l l o w s . B.C. ( i ) A t X- & , T - T6 (3-4) B.C. ( i i ) A t t - 0 and § - TV/2 , — -0 . (3-5) B.C. ( i i i ) A t t" -z <X , we have the f o l l o w i n g two boundary con-d i t i o n s f o r the two r e g i o n s o f the c i r c u m f e r e n c e . k "oT "Ot 3 f • (3-7, C o n v e c t i o n l o s s e s from the s u r f a c e have been i g n o r e d , a l t h o u g h t h e i r i n c l u s i o n w i l l i n t r o d u c e no a d d i t i o n a l m a t h e m a t i c a l d i f f i c u l t y . . 90 The above boundary c o n d i t i o n s assume t h a t the tube w a l l i s very t h i n and o f low thermal c o n d u c t i v i t y so t h a t there i s no temperature drop through the tube w a l l and t h a t the a x i a l and c i r c u m f e r e n t i a l heat conduction i n the tube w a l l are n e g l i g i b l e compared to the heat t r a n s f e r normal t o the tube w a l l . Since there i s symmetry about the l i n e s 6=0 and 6=TT, we may c o n s i d e r o n l y the upper p a r t of the c i r c l e , i . e . , arc ABC (Figure 3-1) f o r boundary c o n d i t i o n ( i i i ) . T h i s symmetry a l s o leads t o the f o l l o w i n g two boundary c o n d i t i o n s . B.C. (iv) At Q - ^ , ^ , O ^ t " 4 0 v • (3-8) B.C. (v) At B=T , — 0^ , 0 4+4 0L . (3-9) The energy equation and the boundary c o n d i t i o n s may be rephrased i n t o convenient dimensionless forms by i n t r o d u c i n g d i mensionless v a r i a b l e s as f o l l o w s : K = » ^ - 77-7—v77 '(3-10) The r e s u l t i n g energy equation i s , The boundary c o n d i t i o n s a r e , a t X = 0 , X = X, ; (3-12) a t ^ x 0 ana S = TV/l , ^ = 0 J (3-13, a t 8 r ( i a t 51 =0 , cud 41 91 - y co&B _ fclV] > cue 4^ . (3-14) (3-15) (3-16) (3-17) I n e q u a t i o n ( 3 - 1 4 ) , = a ( G 3 a 4 a / k 4 ) i s a d i m e n s i o n l e s s r a d i a -t i o n - c o n d u c t i o n p a r a m e t e r , METHOD OF SOLUTION An exact a n a l y t i c a l s o l u t i o n of the system of equations (3-11) to (3-17) appears t o be i m p o s s i b l e . I t i s q u i t e p o s s i b l e t o develop a s e r i e s s o l u t i o n of the energy equation (3-11) . Such a s e r i e s w i l l be a double i n f i n i t e s e r i e s , h a n d l i n g of which appeared to be very cumbersome by a l e a s t squares f i t procedure. A f i n i t e - d i f f e r e n c e s o l u t i o n was t h e r e f o r e attempted. The F i n i t e - D i f f e r e n c e Procedure: The p a r t i a l d i f f e r e n t i a l e q uation (3-11) i s p a r a b o l i c i n X and R and a l s o i n X and 6. Using the forward d i f f e r e n c e approximation f o r the d e r i v a t i v e we o b t a i n an e x p l i c i t method of s o l u t i o n . The c e n t r a l d i f f e r e n c e approximation i s used f o r the remaining d e r i v a t i v e s appearing i n the equation (3-11) . The f i n i t e - d i f f e r e n c e approximation to the energy e q u a t i o n (3-11) i s thus as f o l l o w s : where i , j , k and a^, b, c are the s u b s c r i p t s and s t e p s i z e s f o r the R, 6 and X d i r e c t i o n s r e s p e c t i v e l y . On rearrangement of terms the above e q u a t i o n y i e l d s 93 (3-18) On the boundaries, however, we do not try to s a t i s f y the d i f f e r e n t i a l equation but s a t i s f y the boundary conditions only. The general approximation was described i n Appendix C of Part II of the thesis and i s b r i e f l y repeated here. ative of a function at a point, one obtains, i n general, f o r a function <j> at any boundary, We thus s t a r t with the known temperature d i s t r i b u t i o n at X=0, u t i l i z i n g boundary condition equation (3-12). The i n t e r i o r points i n the next section of X at X=c (c i s the step s i z e i n the X-direction) are calculated by the r e l a t i o n (3-18). For the three boundaries, whose boundary conditions are expressed by the equations (3-13), (3-16) and (3-17), the general equation (3-19) to be used i n the determination of the function at the boundary reduces to the following: Using the forward-difference approximation for the deriv-(3-19) (3-20) 94 For the boundary R=l the normal d e r i v a t i v e does not v a n i s h . F o r example, over the range 0 < 6 < 2 t n e f i n i t e d i f f e r e n c e r e l a t i o n used t o determine f u n c t i o n v a l u e s a t the boundary i s where NA i s the number of i n t e r v a l s chosen i n the R - d i r e c t i o n . The above r e l a t i o n i s n o n l i n e a r and i s s o l v e d f o r X.._ ,, . , NA+1, j , k by the Newton-Raphson method. When the f u n c t i o n v a l u e s a t one X - s e c t i o n are e v a l u a t e d we then proceed t o e v a l u a t e the f u n c t i o n a t the next X - s e c t i o n i n a s i m i l a r manner. A coarse g r i d was f i r s t used and then the g r i d s i z e made f i n e r t i l l the r e s u l t s remained a p p r e c i a b l y the same wit h any f u r t h e r i n c r e a s e i n g r i d f i n e n e s s . Determination of N u s s e l t Number Once the temperature s o l u t i o n i s ob t a i n e d the N u s s e l t number can be e v a l u a t e d as f o l l o w s . A s e m i - l o c a l N u s s e l t number Nu i s d e f i n e d as the N u s s e l t number a t a c e r t a i n X - s e c t i o n averaged over the c i r c u m f e r e n c e . V = " , 3 . . (3-22, where bar over a q u a n t i t y denotes i t s average v a l u e over a c r o s s -s e c t i o n a t a c e r t a i n v a l u e of x. T^ i s the f l u i d bulk temper-95 a tu re f o r any c r o s s - s e c t i o n . Now and X -x = k_ C *T 1 (3-23) M X f (3-24) (3-25) S u b s t i t u t i n g the va lues o f q w , and T f e f rom above i n t o e q u a t i o n (3-22), we o b t a i n 2* - 2<x in [ "b^ " lire Aft o r , i n d i m e n s i o n l e s s f o rm, 1\ IT! S ince the re i s symmetry about the l i n e s 9=0 and 0 =IT , the above r e l a t i o n becomes IX* ~ M ; I X U < I U * 96 Now | may be evaluated by using the boundary conditions (3-14) and (3-15) to y i e l d the following On s i m p l i f i c a t i o n t h i s y i e l d s Transformation of Coordinates If the equation (3-11) were to be solved i n i t s present form, for high values of Peclet numbers there i s a danger of losing a l l high-order derivatives and thus getting inaccurate r e s u l t s . To avoid t h i s we introduce the following transformation; K = ft.^L . (3-27) Then equation (3-11) takes the following form The f i n i t e difference approximation to the above equation then becomes (3-29) Boundary condition equations (3-14) and (3-15) change to "ax W ) . , f 46 4 T . ,3-3i) •ad The r e l a t i o n (3-26) for Nusselt number takes the following form Overall Energy Balance Although consistency of results with f i n e r g r i d s i z e i s normally used as a c r i t e r i o n of convergence and accuracy i n f i n i t e difference solutions, a heat balance check was also made i n t h i s case. Equating the difference i n enthalpies at two sections over a certa i n length of the tube to the net heat transfer from the tube wall, we obtain the following heat balance equation. x-.o e.6 t-o. +-o 6-0 - \ \ S C ^ X T % *<Jfi4^ (3-33) Inserting the known solution of the v e l o c i t y p r o f i l e (3-1) the above equation changes to 98 X r O 0 T O - K rf^ X^t On non-dimensionalisation of a l l the terms the above equation s i m p l i f i e s to \ \ Aft A* = k i X X* tL ^ 1 M 'X,L 2 8X Use of the boundary conditions ( 3 - 1 4 ) and ( 3 - 1 5 ) for I R = I i n the above equation y i e l d s \ \ (.Nicwft -txtUeu - J J ex1* Use of the symmetry condition y i e l d s the following heat balance equation aft e r some s i m p l i f i c a t i o n and rearrangement of the terms. f \ X\ ^ V - ^ i » 4 . \ = ^ k + ^ - L J [ X W ( 3 - 3 4 , In terms of the transformed coordinates the heat balance equation becomes ( 3 - 3 5 ) Therefore, percentage error i n thermal energy balance i s given as follows: error \ . 0 r L f i r t, 0 0 ' lUfe (3-36) DISCUSSION OF RESULTS The r e s u l t s are presented i n the form of v a r i a t i o n ©f average w a l l temperature and N u s s e l t number wi t h a x i a l d i s t a n c e and the angular w a l l temperature d i s t r i b u t i o n a t any s e c t i o n . Average Wall Temperature The average w a l l temperature T~w i s d e f i n e d by equation (3-24). The average w a l l temperature v a r i a t i o n with a x i a l d i s t a n c e i s p l o t t e d i n F i g u r e s (3-2),(3-3) and (3-5) i n terms G oT 3 a of the dimensionless parameters E,, y (= ^ — ) and 1 4 d "I /"} \l> (= ct(G a a/k ) ' ) . In F i g u r e (3-2) t h i s has been p l o t t e d f o r Tw Y=0.5 and d i f f e r e n t values of . As expected, =— i s h i g h e r o f o r a h i g h e r value of a t any value of £ . For ty=0 the problem reduces t o one w i t h no i n c i d e n t r a d i a t i o n and the temperature d i s t r i b u t i o n becomes independent of a. For t h i s case the r e s u l t s are compared with those o b t a i n e d by Chen [59]. The r e s u l t s agree f a i r l y w e l l and the maximum d e v i a t i o n i n r e s u l t s f o r the range of v a l u e s over which the comparison i s made i s l e s s than 2 per cent. I t i s d i f f i c u l t t o say whether Chen's r e s u l t s or the p r e s e n t ones are more accurate s i n c e both are i t e r a t i v e n umerical s o l u t i o n s . The r e s u l t s a l s o show t h a t a t Y=0.5 and f o r < 1.8 • the average w a l l temperature decreases w i t h £ w h i l e f o r i/> > 1.8 i t i n c r e a s e s w i t h £. For \p = 1.8 the average w a l l temperature has a very s m a l l v a r i a t i o n with K. T h i s i n d i c a t e s t h a t f o r a . p a r t i c u l a r value of y there i s a c r i t i c a l value of ip f o r which T does not vary a p p r e c i a b l y from the i n i t i a l v a lue T c . P h y s i c -w a l l y t h i s means t h a t the amount of energy r e c e i v e d and energy 101 e m i t t e d a r e a d j u s t e d i n s u c h a way t h a t 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 h a s a v e r y s m a l l d e v i a t i o n . F i g u r e (3-3) shows a x i a l v e m p e r a t u r e v a r i a t i o n f o r y-5. I n t h i s c a s e t h e c r i t i c a l v a l u e o f 4> i s i n t h e n e i g h b o u r h o o d o f 8 0 . F o r < 4>cr t h e v a r i a t i o n o f t he a v e r a g e w a l l t e m p e r a t u r e i s s i m i l a r t o t h a t i n F i g u r e ( 3 - 2 ) . F o r ii» > il> , h o w e v e r , T ^ r r c r ' ' w d e c r e a s e s w i t h £ a f t e r an i n i t i a l r i s e . T h i s h a p p e n s b e c a u s e a l t h o u g h t h e w a l l t e m p e r a t u r e i n c r e a s e s w i t h £ f o r t h e p o r t i o n o f t h e t u b e e x p o s e d t o t h e i n c i d e n t r a d i a t i o n y e t i t d e c r e a s e s w i t h £ f o r t h e o t h e r p o r t i o n o f t h e t u b e . T h i s i s c l e a r l y shown i n F i g u r e (3-16) w h i c h i s d i s c u s s e d l a t e r . T h i s h a p p e n s f o r t h e c a s e p l o t t e d i n F i g u r e (3-2) a l s o where t h e v a r i a t i o n i s s u c h as t o make T i n c r e a s e w i t h £ . I n t h e c a s e o f y-5, h o w e v e r , w t h e 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 w i t h E, a t d i f f e r e n t a n g u l a r p o s i t i o n s o f t h e t u b e i s s u c h as t o make 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 d e c r e a s e w i t h £ . T h u s F i g u r e (3-3) c o n f i r m s t h e c o n c l u s i o n s a r r i v e d a t i n t h e a n a l y s i s o f F i g u r e (3-2) r e g a r d i n g a c r i t i c a l r e l a t i o n s h i p b e t w e e n y a n d $• T h e a n a l y s i s was a l s o c a r r i e d out- f o r o t h e r v a l u e s c f y and t h e r e s u l t s a r e / p l o t t e d as a g r a p h i n F i g u r e (3-4). I t may be s t r e s s e d , h o w e v e r , t h a t t h i s i s o n l y an a p p r o x i m a t e r e l a t i o n s h i p f o r t h e c r i t i c a l v a l u e o f \p f o r a g i v e n y s i n c e i t i s n o t p o s s i b l e t o k e e p T ^ c o n s t a n t w i t h £ , e s p e c i a l l y f o r h i g h v a l u e s o f i>, b e c a u s e o f t h e n o n - l i n e a r i t y o f t h e p r o b l e m . In c h o o s i n g t h e c r i t i c a l v a l u e s o f \p an a t t e m p t h a s b e e n made t o k e e p t h e n e t v a r i a t i o n o f f"w / T 0 f r o m 1.0 a min imum o v e r t h e t u b e l e n g t h . I t h a s t o be e m p h a s i z e d t h a t t o some e x t e n t t h e 102 c r i t i c a l v a l u e s are dependent upon the tube l e n g t h L . From F i g u r e (3-4) f o r a c e r t a i n v a l u e of y, i f ^ i s g r e a t e r t h a n the c o r r e s p o n d i n g c r i t i c a l v a l u e , i t may be c o n c l u d e d t h a t the average w a l l temperature o f the f l u i d a t tube e x i t w i l l exceed the i n l e t t e m p e r a t u r e . For v a l u e s o f ij> below the optimum v a l u e the average v / a l l t e mperature w i l l c o n t i n u o u s l y d e c r e a s e w i t h a x i a l d i s t a n c e . F i g u r e (3-5) i l l u s t r a t e s t h e v a r i a t i o n o f T /T G w i t h £ w f o r t h r e e v a l u e s o f y a t a f i x e d v a l u e o f \p. A t y~0.5, th e v a l u e I|J=25 i s g r e a t e r than ip and, t h e r e f o r e , T /T c i n c r e a s e s w i t h £. For y=5 and y=50 the v a l u e \p=25 i s l e s s t h a n the c o r r e s p o n d i n g optimum v a l u e s o f and t h e r e f o r e T /T Q d e c r e a s e s w i t h E,. w N u s s e l t Number The v a r i a t i o n o f N u s s e l t number w i t h E i s p r e s e n t e d i n F i g u r e s (3-6) t o ( 3 - 8 ) . F o r y=0.5 F i g u r e (3-6) i l l u s t r a t e s the v a r i a t i o n f o r two v a l u e s o f \p. I n b o t h cases ^<^ c r. F o r ij>=0 ,_ a t any s e c t i o n , the w a l l t emperature i s l e s s t h a n the i n s i d e t e m p e r a t u r e , i e . 5T —) i s n e g a t i v e , s i n c e the f l u i d i s l o s i n g h e a t . F o r t h i s 5r r=a ^ ^ case a comparison i s a l s o made w i t h Chen's [59] v a l u e s . I t i s noted t h a t a t v a l u e s o f E < 0.01 the two s o l u t i o n s d i v e r g e . T h i s may be due t o the d i f f e r e n c e i n approach of the two s t u d i e s . The comparison of N u s s e l t number v a l u e s have been made w i t h t h o s e g i v e n i n T a b l e 1 of Chen's paper. T h i s t a b l e p r e s e n t s r e s u l t s o b t a i n -ed by an approximate s o l u t i o n g i v e n i n terms o f L i o u v i l l e - N e u m a n n s e r i e s a f t e r l i n e a r i z i n g the boundary c o n d i t i o n a t the tube w a l l . I n the p r e s e n t s t u d y the r e s u l t s are o b t a i n e d by a f i n i t e - d i f f e r -ence s o l u t i o n w i t h o u t l i n e a r i z i n g the.boundary c o n d i t i o n a t the w a l l . I t has been noted t h a t N u s s e l t number dec reases w i t h ? f o r o the r va lues o f ij> a l s o when < i|> . T h i s i s e x p l a i n e d as f o l l o w s . The w a l l temperature i s g r e a t e r than the i n s i d e temperature f o r the p o r t i o n o f the tube f a c i n g the i n c i d e n t r a d i a t i o n b u t , f o r the rema in ing p o r t i o n o f the t u b e , the w a l l temperature i s l e s s than the i n s i d e t empera tu re . However, i t has been noted t h a t the i n t e g r a t e d va lue o f over the c i r -cumference , i . e . £ | r _ a de , i s n e g a t i v e . A l s o i s g r e a t e r than T w and , t h e r e f o r e , Nu i s p o s i t i v e . With i n c r e a s -i n g va lues o f £, the net heat t r a n s f e r r a t e dec reases i n magn i -tude wh i l e (T^ - T ) i n c r e a s e s so t h a t Nu d e c r e a s e s . A l s o , f o r \JJ < ij/ , i n c r e a s e o f 41 reduces Nu a t any s e c t i o n because i t dec reases the magnitude o f the ne t heat t r a n s -f e r r a t e from the f l u i d to the s u r r o u n d i n g s . F i g u r e (3-7) shows a s i m i l a r p l o t f o r t h r ee o the r v a lues o f In t h i s case > ii f o r each va lue o f 4 1 . From the a n a l y s i s i t has been noted tha t a t any s e c t i o n , the w a l l temper -a tu re i s g r e a t e r than the i n s i d e temperature f o r the p o r t i o n o f the tube f a c i n g the i n c i d e n t r a d i a n t f l u x but f o r the r ema in ing p o r t i o n the w a l l temperature i s l e s s than the i n s i d e t empera tu re . However, the v a r i a t i o n i s i n such a manner as to make (2 i r ~ I de p o s i t i v e i n t h i s c a s e . A l s o T „ > T. so t h a t )0 o r 1 r=a * w o Nu i s p o s i t i v e . Fo r low va lues o f £, Nu i s lower f o r a h i g h e r v a l ue o f i p . With i n c r e a s i n g va lues o f £, Nu i s s u b s t a n t i a l l y reduced In ve ry s h o r t d i s t a n c e s . The decrease i n Nu i s l e s s f o r h i g h e r v a lues o f \p and Nu even i n c r e a s e s towards the end f o r ^=25 and ^=120. (The a n a l y s i s has been c a r r i e d out f o r v a l ues 104 o f £ up to 0.1 o n l y ) . T h i s i s because a l though the heat t r a n s -f e r r a t e dec reases w i th £ , the d i f f e r e n c e between T and T, w b a l s© dec reases w i th £ and the N u s s e l t ;number v a r i a t i o n depends upon the two r a t e s o f dec r ease . F i g u r e (3-8) i l l u s t r a t e s the v a r i a t i o n o f N u s s e l t number w i th £ f o r a ve ry p a r t i c u l a r s i t u a t i o n when \\>a\li a t a g i v en va lue o f y. T h i s f i g u r e i l l u s t r a t e s t h a t Nu dec reases w i th % over most o f the r e g i o n but f l u c t u a t e s w i l d l y i n the n e i g h b o u r -hood of 5=0.02. The N u s s e l t number even goes to nega t i v e v a l u e s . To o f f e r an e x p l a n a t i o n f o r t h i s behav iour we have to ana l yze the v a r i a t i o n o f q u a n t i t i e s such as net heat t r a n s f e r r a t e , average w a l l temperature and bu lk temperature f o r t h i s c a s e . These are p l o t t e d i n f i g u r e s (3-9) and (3-10). F i g u r e (3-9) shows the v a r i a t i o n of the numerator of equa t i on (3-32), which r ep r e sen t s the v a r i a t i o n of d imens ion l e s s heat f l u x w i th ! . F i g u r e (3-10) shows the v a r i a t i o n w i t h ' ! o f the two terms o f the denominator o f equa t i on (3-32) which r e p r e s e n t the d imens i on l e s s average w a l l temperature and the d imens i on l e s s f l u i d bu lk temper -a t u r e . In F i g u r e s (3-9) and (3-10) the o r d i n a t e s do not r e p r e -sent the abso lu t e v a lues of heat f l u x , T". and T. , s i n c e a numer-i c a l cons t an t common to a l l o f them has been d ropped . F i g u r e (3-9) shows t h a t f o r the i n d i c a t e d va lues o f y and and f o r the range o f ! v a lues c o v e r e d , the heat f l u x decreases from +0.29 to -0 .075 . I n i t i a l l y , f o r va lues of ! l e s s than approx imate l y 0 .02 , the net heat t r a n s f e r r a t e i s p o s i t i v e , i n d i c a t i n g ne t heat t r a n s f e r i n t o the f l u i d . A l s o , i n t h i s range o f ! , T > T^ (F igure 3-10) so t h a t Nu i s p o s i t i v e . With i n c r e a s i n g va lues 1 0 5 of £, the difference i n the values of T and T, diminishes W D and at 5 = 0 . 0 2 1 , T = TV . For £ > 0 . 0 2 1 , T < T, . Therefore, i n W D W D the region 5 = 0.020 to 0 . 0 2 3 , because of the small differences: i n T*w and T^, the Nusselt number fluctuates between very large posi-t i v e and negative values. For £ > 0 . 0 2 3 5 the heat transfer rate becomes negative (Figure 3 - 9 ) in d i c a t i n g that net heat transfer now occurs from the f l u i d to the surroundings. Also i n t h i s range of values of 5 , f > T. so that Nu i s ultimately p o s i t i v e , w b However, i t may be stressed here that the wild f l u c t u a t i o n i n Nu i s only over a very small region of the tube length and i f a mean Nusselt number for the whole length of the tube were to be eval-uated, i t would not be much affected. A good agreement i n heat balance for t h i s case i s shown i n Table ( 3 - 2 ) . For y=0.5 and values of ^ less than and greater than ^ c r » the v a r i a t i o n of Nu was described by Figures ( 3 - 6 ) and ( 3 - 7 ) re-spectively. In the following sections we now describe the v a r i a t i o n of Nu for y=5. Figure ( 3 - 1 1 ) i s a plo t of Nu with £ for three values of \pf each l e s s than the c r i t i c a l vlaue of at Y - 5 . The results show a trend si m i l a r to that of the curves i n Figure ( 3 - 6 ) . When values of Nu i n Figure ( 3 - 6 ) are compared with those i n Figure ( 3 - 1 1 ) i t i s noted that for the same value of , Nu has a lower value for Y=5 than for y=0.5. For t/j=0 there i s good agreement with the results of Chen [ 5 9 ] for £ > 0 . 0 0 4 . V ariation of Nu with £ for values of IJJ greater than the c r i t i c a l value for Y=5 i s shown i n Figure ( 3 - 1 2 ) . At low values of £ the Nusselt number values are higher for higher values of IJJ. 106 But for higher values of £ there appears a trend which may seem surprising. For example, at 5=0.1, Nu decreases with increasing values of up to 300 and then increases with increasing values of i|> above 300. The apparently surprising behaviour i s ^ e a s i l y explained when we observe the v a r i a t i o n i n heat transfer rate, average wall temperature and f l u i d bulk temperature (Table 3-1). The heat transfer rate decreases with 5 for a l l values of \> and i s higher for higher values of $ at any But the difference i n average wall temperature and f l u i d bulk temperature i s also higher at any 5 for higher values of This difference i s esp e c i a l l y very small for ^=150 and tji=200 because these values of 4» are quite close to the c r i t i c a l value for y=5. This then results i n a high value of Nu. Figures (3-13) and (3-14) i l l u s t r a t e the v a r i a t i o n of Nu with 5 . for three values of y and fixed values of . .Figure (3-13) gives the plots for ^=0, which corresponds to the case of no incident radiation flux. As expected, Nu decreases with 3 5. High values of y also decrease Nu. Since y=caTa a/k, a high value of y m a Y be caused by a low value of k or a high value of e. With a low thermal conductivity f l u i d the heat transfer rate decreases, thus lowering Nu. A high emissivity of the tube wall increases the heat transfer rate but also de-creases T and T. i n such a manner as to increase (T, - T ). w b b w The r e l a t i v e increase i n (T. - T~ ) i s higher than the increase D W i n heat transfer rate, the o v e r a l l e f f e c t being a reduction i n Nu. With the inclusion of incident radiation f l u x , however, 107 the s i t u a t i o n i s not as s i m p l e . As e x p l a i n e d e a r l i e r , the N u s s e l t number w i l l depend upon whether i s g r e a t e r o r lower than the c r i t i c a l v a lue and i t s- c l o s e n e s s to t h i s c r i t i c a l v a l u e . F i g u r e (3-14) g i v e s the p l o t s f o r ^=25. Fo r low va lues o f £, Nu i s h i ghe r f o r a lower va lue o f y but f o r va lues of 5 near 0 . 1 , Nu f i r s t dec reases and then i n c r e a s e s w i th y. The e x p l a n a t i o n f o r t h i s i s on the same l i n e s as the d i s c u s s i o n o f F i g u r e (3-12). Angu la r Wa l l Temperature D i s t r i b u t i o n The angu la r w a l l temperature v a r i a t i o n i s p r e sen t ed i n F i g u r e s (3-15) to (3-17). F i g u r e (3-15) shows the angu l a r , v a r i a t i o n of w a l l temperature f o r d i f f e r e n t va lues o f 'ifi and f i x e d va lues o f y and £. In a l l cases the temperature v a r i e s . 0 from a maximum at 9=0 to a minimum at 9=180° . The s l ope a t 0=0° and 9=180° i s z e r o , i n d i c a t i n g angu la r symmetry at the two p o i n t s on the c i r c u m f e r e n c e . A l s o , f o r i n c r e a s i n g va lues o f \p, the w a l l temperature i n the ne ighbourhood o f 9=0 i n c r e a s e s r a p i d l y wh i l e the i n c r e a s e i n the w a l l temperature i n the ne ighbourhood of 6=180° i s ve ry s low. There i s a sharp change i n the w a l l temperature near 9=90° . F i g u r e (3-16) g i v e s the p l o t s o f angu la r v a r i a t i o n o f w a l l temperature f o r d i f f e r e n t va lues of £ and a f i x e d va l ue o f \\i and y. The w a l l temperature near 6=0 i n c r e a s e s w i th E, but near 6=TT i t , d e c r e a s e s w i th £. Near 6=TT the f l u i d does no t r e c e i v e any d i r e c t i n c i d e n t f l u x (a l though i t r e c e i v e s heat by convec t i on from the f l u i d f a c i n g the i n c i d e n t f l u x ) but i t l o s e s heat a c c o r d i n g to the f o u r t h power law o f r a d i a t i o n . The w a l l t empera tu re , t h e r e f o r e , d e c r e a s e s w i th £. Near 9=0 the e f f e c t o f 1 9 8 i n c i d e n t f l u x causes a slow i n c r e a s e o f the w a l l temperature w i th £ f o r t h i s case where < . S i m i l a r p l o t s . f o r d i f f e r e n t va lues o f y and a f i x e d va lue o f ip and'Y are p l o t t e d i n F i g u r e ( 3 - 1 7 ) . An i n c r e a s e i n Y reduces the w a l l temperature a t every angu la r p o s i t i o n . A h i g h va lue of e m i s s i v i t y o f the tube w a l l reduces the w a l l temperature a t each angle by i n c r e a s i n g the r a d i a t i v e heat l o s s . O v e r a l l Energy Ba lance The percentage e r r o r i n o v e r a l l energy ba lance as g i v en by equa t i on (3-36) was c a l c u l a t e d f o r some va lues o f y and \p. T h i s i s t a b u l a t e d in . Tab l e 3-2 which shows tha t the e r r o r i s l e s s than 3 per cent f o r most c a s e s . CONCLUSIONS The e f f e c t o f i n c i d e n t r a d i a n t f l u x on l a m i n a r h e a t t r a n s f e r i n a c i r c u l a r tube has been s t u d i e d . Temperature d i s -t r i b u t i o n and N u s s e l t numbers have been e v a l u a t e d . F o r the case o f no i n c i d e n t f l u x t h e r e s u l t s have been compared w i t h a r e c e n t s o l u t i o n and have been found t o agree w e l l , e x c e p t a t s h o r t d i s t a n c e s from the tube e n t r a n c e . 110 TABLE 3-1 V a l u e s o f the d i m e n s i o n l e s s f o r m s o f h e a t f l u x , 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 f o r y=5 a) 5=0.001 D imens ion less Heat F l u x Tb Tw ' Tb 150 18.31 9.03 7.90 1.13 200 25.39 9.47 7.91 1.56 300 37.03 10.14 7.94 2.20 500 55.43 11.10 7.98 3.12 1000 90.53 12.61 8.04 4.57 b) 5=0.1 D imens ion less Heat F l u x V Tb 150 4.86 8.39 8.25 0.14 200 7.11 8.85 8.53 0.32 300 11.32 9.56 8.96 0.60 500 19.10 10.56 9.55 1.01 1000 37.70 12.12 10.51 1.61 TABLE 3-2 O v e r a l l E r r o r i n E n e r g y B a l a n c e ( E q u a t i o n 3-36) Y P e r c e n t a g e E r r o r 1.8 0.5 '""-0.44 25 0.5 -3.09 25 5 -1.97 80 5 -2.20 25 50 -1.10 9=TT/2 X=0 F i g u r e 3-1 Tube Nomenclature and Geometry IO"3 IO"2 ? 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Hanawa l t , " Sk in Temperatures o f a S a t e l l i t e , " J e t P r o p u l s i o n , V o l . 27 , No. 10, 1957, pp . 1079-1083. 25. Z e r k l e , R.D. and J . E . Sunde r l and , "The T r a n s i e n t Temperature D i s t r i b u t i o n i n a S lab Sub jec t to S o l a r R a d i a t i o n , " T r a n s . A . S . M . E . , V o l . 87, S e r i e s C , J . Heat T r a n s f e r , 1965, pp . 117-133. 26. S a n d o r f f , P .E . and J . S . P r i g g e , J r . , "Thermal C o n t r o l i n a Space V e h i c l e , " J . A s t r o , V o l . 3, No. 26, 1956, pp . 4-8. 132 ,21. Hanel, R.A., "Thermostatic Temperature Control of S a t e l l i t e s and Space Vehicles," A.R.S. Journal, Vol. 29, No. 5, 1959, pp. 358-361. 28. Iqbal, M. and B.D. Aggarwala, "Temperature D i s t r i b u t i o n i n a Two-Dimensional Rectangular Soli d i n Interplanetary. Space," Proc. A.S.M.E. Annual Aviation and Space Conference, held at Beverly H i l l s , C a l i f o r n i a , 1968, pp. 625-633. 29.. Iqbal, M. and B.D. Aggarwala, "Radiant Heating of a S o l i d Spherical S a t e l l i t e , " A.I.A.A. Journal, (in press). 30. Sikka, S., M. Iqbal and B.D. Aggarwala, "Temperature Distri-... bution and Curvature Produced i n Long S o l i d Cylinders i n Space," Journal of Spacecraft and Rockets, (in press) 31. Ojalvo, I.U. and F.D. Linzer, "Improved Point-Matching Techniques," Quart. J. Mech. App. Math., Vol. XVIII, Pt. 1, 1965, pp. 41-56. 32. L i e b l e i n , S., "Analysis of Temperature D i s t r i b u t i o n and Radiant Heat Transfer along a Rectangular Fin of Constant Thickness," N.A.S.A. TN D-196, 1959. 7 33. Bartas, J.G. and W.H. S e l l e r s , "Radiation Fin Effectiveness," Trans. A.S.M.E., Vol. 84, Series C:, J . Heat Transfer, 1960, pp. 73-75. 34. Chambers, R.L. and E.V. Somers, "Radiation F in E f f i c i e n c y for One Dimensional Heat Flow i n a C i r c u l a r F i n , " Trans. A.S.M.E., Vol. 81, Series C, J. Heat Transfer, 1959, pp. 327-329. 35. Eckert, E.R.G., T.F. Irvine, J r . and E.M. Sparrow, "Analytic Formulation for Radiating. Fins with Mutual, I r r a d i a t i o n , " A.R.S. Journal, Vol. 30, 1960, pp. 644-'646. 36. Sparrow, E.M., E.R.G. Eckert and T.F. Irvine, J r . , "The Effectiveness of Radiating Fins with Mutual I r r a d i a t i o n , " J . Aero/Space Sciences, Vol. 28, No. 10, 1961, pp. 763-772. 37. Sparrow, E.M. and E.R.G. Eckert, "Radiant Interaction Between Fin and Base Surfaces," Trans. A.S.M.E., Vol. 84, Series C, J. Heat Transfer, 1962, pp. 12-18. 38. Sparrow, E.M., G.B. M i l l e r and V.K. Jonsson, "Radiating Effectiveness of Annular Finned Space Radiators, Including Mutual Ir r a d i a t i o n Between Radiator Elements," J . Aero/ Space Sciences, Vol. 29, No. 11, 1962, pp. 1291-1299. 133 3 9 . S p a r r o w , E . M . and E .R . N i e w e r t h , " R a d i a t i n g , C o n v e c t i n g and C o n d u c t i n g F i n s : N u m e r i c a l and L i n e a r i z e d S o l u t i o n s , " I n t . J . H e a t Mass T r a n s f e r , V o l . 1 1 , 1 9 6 8 , p p . 3 7 7 - 3 7 9 . 4 0 . Shouman, A . R . , "An E x a c t G e n e r a l S o l u t i o n f o r t h e T e m p e r -a t u r e D i s t r i b u t i o n and t h e R a d i a t i o n a l Hea t T r a n s f e r a l o n g a C o n s t a n t C r o s s - S e c t i o n a l - A r e a F i n , " A . S . M . E . P a p e r 6 7-WA/HT-2 7, 19 6 7 . 4 1 . L i u , C , "On M in imum-We igh t R e c t a n g u l a r R a d i a t i n g F i n s , " J . A e r o / S p a c e S c i e n c e s , V o l . 2 7 , 1 9 6 0 , p p . 8 7 1 - 8 7 2 . 4 2 . W i l k i n s , J . E . , J r . , " M i n i m i z i n g t h e Mass o f T h i n R a d i a t i n g F i n s , " J . A e r o / S p a c e S c i e n c e s , V o l . 2 7 , 1 9 6 0 , p p . 1 4 5 - 1 4 6 . 4 3 . N i l s o n , E . N . and R. C u r r y , " T h e M in imum-We igh t S t r a i g h t F i n o f T r i a n g u l a r P r o f i l e R a d i a t i n g t o S p a c e , " J . A e r o / Space S c i e n c e s , V o l . 2 7 , 1 9 6 0 , p p . 1 4 6 - 1 4 7 . 4 4 . C a l l i n a n , J . P . and W.P. B e r g g r e n , "Some R a d i a t o r D e s i g n C r i t e r i a f o r Space V e h i c l e s , " T r a n s . A . S . M . E . , V o l . 8 1 , S e r i e s C , J . H e a t T r a n s f e r , 1 9 5 9 , p p . 2 3 7 - 2 4 4 . 4 5 . W i n t e r , F . d e . and G . J . S c h a b e r g , " I n t e r m e s h i n g F i n s as a Means o f I n c r e a s i n g R a d i a t i o n H e a t T r a n s f e r Be tween O p p o s i n g S u r f a c e s , " A . S . M . E . P a p e r 6 7-WA/HT-35, 1 9 6 7 . 4 6 . H o l s t e a d , R .D . and E . S . H o l d r e d g e . " R a d i a t i o n H e a t T r a n s f e r f o r S t r a i g h t F i n s o f T r a p e z o i d a l P r o f i l e , " A . S . M . E . P a p e r 6 7 - H T - 7 3 , 1 9 6 7 . 4 7 . S p a r r o w , E . M . , V . K . J o n s s o n and W . J . M i n k o w y c z , " H e a t T r a n s -f e r f r o m F i n - T u b e R a d i a t o r s I n c l u d i n g L o n g i t u d i n a l H e a t C o n -d u c t i o n and R a d i a n t I n t e r c h a n g e Be tween L o n g i t u d i n a l l y Non-i s o t h e r m a l F i n i t e S u r f a c e s , " N . A . C . A . TN' D-2077 , 1 9 6 3 . r d 4 8 . W y l i e , C . R . , J r . , " A d v a n c e d E n g i n e e r i n g M a t h e m a t i c s , " 3 — e d . , M c G r a w - H i l l Book Company , U . S . A . , 1 9 6 6 , p p . 3 6 3 - 3 6 4 . 4 9 . W i e b e l t , J . A . , " E n g i n e e r i n g R a d i a t i o n H e a t T r a n s f e r , " H o l t , R i n e h a r t and W i n s t o n , I n c . , U . S . A . , 19 6 5 , p . 7 8 . 5 0 . S p a r r o w , E . M . , "A New and S i m p l e r F o r m u l a t i o n f o r R a d i a t i v e A n g l e F a c t o r s , " T r a n s . A . S . M . E . , V o l . 8 5 , S e r i e s C , J . H e a t T r a n s f e r , 1 9 6 3 , p p . 8 1 - 8 8 . 5 1 . S e l l a r s , J . R . , M. T r i b u s and J . S . K l e i n , " H e a t T r a n s f e r t o L a m i n a r F l o w i n a Round Tube o r F l a t C o n d u i t - The G r a e t z P r o b l e m E x t e n d e d , " T r a n s . A . S . M . E . , V o l . 7 8 , S e r i e s C , J . H e a t T r a n s f e r , 1 9 5 6 / p p . 4 4 1 - 4 4 8 . 134 52. S i n g h , S . N . , "Heat T r a n s f e r by Laminar Flow i n a C y l i n d r i c a l T u b e , " A p p l . S c i . R e s . , S e c t i o n A , V o l . 7, 1958, pp . 325-340. 53. Kuga, 0 . , "Laminar Flow Heat T r a n s f e r i n C i r c u l a r Tubes w i th Non-Isothermal S u r f a c e s , " T r a n s . Japan Soc . Mech. E n g r s . , V o l . 31 , No. 222, 1965, pp . 295-298. 54. S i e g e l , R., E. M. Sparrow and T . M . Ha l lman , "S teady Laminar - Heat T r a n s f e r i n a C i r c u l a r Tube w i th P r e s c r i b e d Wa l l Heat F l u x , " A p p l . S c i . R e s . , S e c t i o n A , V o l . 7, 1958, pp . 386-392. 55 . Hsu , C . J . , "Heat T r a n s f e r i n a Round Tube w i th S i n u s o i d a l Wa l l Heat F l u x D i s t r i b u t i o n , " A . I . C h . E . J o u r n a l , V o l . 11 , No. 4, 1965, pp . 690-695. 56. Kuga, @. , "Heat T r a n s f e r i n a P ipe w i th Non-Uniform Heat F l u x , " T r a n s . Japan Soc. Mech. E n g r s . , V o l . 32, No. 233, 1966, pp . 83-87. 57. S ideman, S . , D. Luss and R . E . P e c k , "Heat T r a n s f e r i n Laminar Flow i n C i r c u l a r and F l a t Condu i t s w i t h Cons tan t Su r f ace R e s i s t a n c e , " A p p l . S c i . R e s . , S e c t i o n A , V o l . 14, 1964-5, pp . 157-171. 58. S t e i n , R .P . , "The Grae tz Problem i n Co-cu r r en t F low Double P ipe Heat E x c h a n g e r s , " Chem. Eng . P rogress Symposium S e r i e s , V o l . 61 , No. 58, 1965, pp . 76-87. 59. Chen, J . C . , "Laminar Heat T r a n s f e r i n Tube w i t h N o n l i n e a r Rad iant Heat-F lux Boundary C o n d i t i o n , " I n t . J . Heat Mass T r a n s f e r , V o l . 9, 1966, pp . 433-440. 60. Dussan , B.I. and T . F . I r v i n e J r . , "Laminar Heat T r a n s f e r i n a Round Tube w i th R a d i a t i n g Heat F l u x a t the Outer W a l l , " P r o c . 3 ~ I n t . Heat T r a n s f e r C o n f . , 1966, V o l . V , pp . 184-189. A P P E N D I C E S EVALUATION OF EXPRESSIONS FOR OVERALL ENERGY BALANCE ERROR EVALUATION OF CONFIGURATION FACTOR F d A ^ A THE F IN ITE D IFFERENCE PROCEDURE APPENDIX A EVALUATION OF EXPRESSIONS FOR OVERALL ENERGY BALANCE ERROR The accuracy of the present analysis i s investigated from the point of view of energy balance. The percentage error i n o v e r a l l thermal energy balance i s defined as follows: . Energy absorbed - Energy emitted „ 1 A r t Percentage error = ^ x 100 . (1-33) Energy absorbed This expression i s evaluated for the three bodies considered. a) CIRCULAR CYLINDER The energy balance i s carried out over a unit length of the cylinder as follows: Energy absorbed = 1 i\0L G BTU/hr. (A-l) Energy emitted = 1 \ fcff" BTU/hr. (A-2) Therefore, percentage error i n energy balance, from equation ( 1 - 3 3 ) . , 137 b) RECTANGULAR CYLINDER: Carrying out the energy balance over a unit length of the cylinder, we have, Energy absorbed = BTU/hr. (A-3) 1 W G Energy emitted = 1 \ £ £ € + 1 e < 5 " ^ T < k ^ £<r Ij 1^ < ^ BTU/hr. (A -4 ) Percentage error = 2WG = h - £ U% ^V^V\,«]] ( l-33b) c) SPHERE: Energy absorbed = ITO- G ^ BTU/hr. Energy emitted BTU/hr. (A-6) Percentage error - j W G<X - W o . U T J t . ^ ^ t t ^ ^ (l-33c) APPENDIX B EVALUATION OF CONFIGURATION FACTOR F d A l + A 2 The c o n f i g u r a t i o n f a c t o r , view f a c t o r o r shape f a c t o r , F dA^ ->- A2 / i s d e f i n e d as the f r a c t i o n o f energy d i r e c t l y i n -c i d e n t on s u r f a c e A2 1 from s u r f a c e dA^ assumed t o be e m i t t i n g energy d i f f u s e l y . An a n a l y s i s a l o n g the l i n e s g i v e n by W i e b e l t [49] y i e l d s ( B - l ) As suggested by Sparrow [50] the above a rea i n t e g r a l i s c o n v e n i e n t l y c o n -v e r t e d i n t o a l i n e i n t e g r a l to y i e l d F = 1 - 2 i -^y^  ^ V* i - * ^ - ( ? i-Z , H x i where 1^, m^ and n^ are the d i r e c t i o n c o s i n e s o f dA^, and t h e c o o r d i n a t e s x^, y^, / r e p r e s e n t the l o c a t i o n o f the a r e a dA^. These are a l l c o n s t a n t d u r i n g the i n t e g r a t i o n . The c o o r d i n a t e 140 axes a r e g e n e r a l l y so chosen so t h a t the normal t o d A 1 l i e s p r e -c i s e l y a l o n g a c o o r d i n a t e l i n e . Then two o f the t h r e e d i r e c t i o n c o s i n e s a re z e r o and a l a r g e p o r t i o n o f the c o n t o u r i n t e g r a l o f e q u a t i o n (B-2) v a n i s h e s . Thus i n t h i s case the element d A 1 has i t s normal p o i n t i n g i n the p o s i t i v e x - d i r e c t i o n so t h a t 1 ^ 1 , m^= n]_ = 0* A d d i t i o n a l l y , z^= 0, x^ = a , y^ = 0 . S u b s t i t u t i o n o f t h e s e v a l u e s i n (B-2) y i e l d s The c o n t o u r C wh i c h bounds the a r e a A 5 i s made up o f two p a r t s : an a r c runn ing from -e0 « e < e0 and a s t r a i g h t l i n e runn ing f rom -+ 4 $vr\9 0 4 4 V"a Si.T\ 8b . On the a r c X%-^ Co&§ , ^ := N\ Si-O^ , wh i l e on the s t r a i g h t l i n e , A l s o , - ^ + ^ i " ^ + ^ -7,) A l ong both contours 141 On i n t e g r a t i o n , the above e x p r e s s i o n y i e l d s r - Z \ ft. o . V z % A ' r -r<b?«Vv .VtaA)Un^) 1 S ince e0 i s g i ven by cos e0 = a / r 2 / t h e r e f o r e , N o n d i m e n s i o n a l i s i n g z and r 2 by Z = z/a and g = r 2 / a , the above e x p r e s s i o n changes to (2-24) where <£ - Z% + f + 1 . / APPENDIX C THE FINITE DIFFERENCE PROCEDURE The e l l i p t i c p a r t i a l d i f f e r e n t i a l e q u a t i o n (2-9) was s o l v e d u s i n g a square g r i d . U s i n g the s t a n d a r d 5 - p o i n t approx-i m a t i o n o f t h e L a p l a c i a n , the f i n i t e d i f f e r e n c e scheme f o r t h e i n t e r i o r p o i n t s i s : Xi,j = • / i . j + i ' ^ + • Hi) + Xi,|-. TUT) where i , j and a, b are t h e s u b s c r i p t s and s t e p s i z e s f o r t h e Z and R d i r e c t i o n s r e s p e c t i v e l y , ( F i g u r e 2-12, p . 8 0 ) . On the b o u n d a r i e s , however, we do not t r y t o s a t i s f y the d i f f e r e n t i a l e q u a t i o n b u t s a t i s f y the boundary c o n d i t i o n s o n l y . U s i n g t h e f o r w a r d - 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 t h e d e r i v a t i v e o f a f u n c t i o n a t a p o i n t , one o b t a i n s , i n g e n e r a l , f o r a f u n c t i o n $ a t any boundary, . k - ! U < L -u.t +\%) ( C - l ) F o r example, f o r the boundary C-D (R=l) f o r a (80x10) g r i d , A c o a r s e g r i d was f i r s t used t o o b t a i n a rough shape w h i c h was used as t h e i n i t i a l d i s t r i b u t i o n f o r a f i n e r g r i d . A number o f v a l u e s ©f t h e o v e r - r e l a x a t i o n parameter were t r i e d and a l t h o u g h t h e optimum v a l u e v a r i e d w i t h the v a r i a b l e parameters A , N, e t c . o f the problem, a v a l u e o f 1.9 seemed t o be g e n e r a l l y most s u i t a b l e . 

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