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Conjugate natural convection between two concentric spheres. Lau, Meng Hooi 1971

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CONJUGATE NATURAL CONVECTION BETWEEN TWO CONCENTRIC SPHERES . . by LAU MENG HOOI A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF . MASTER OF APPLIED SCIENCE i n the Department of Mec h a n i c a l 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 to t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA November, 19 71 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 of the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the 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 Study. I f u r t h e r agree 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 copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . LAU MENG HOOI Department of,-Mechanical E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date /<?7/ ABSTRACT T h i s work c o n s i d e r s the conjugate c o n v e c t i v e heat t r a n s f e r between a sphere c o n t a i n i n g heat sources and a c o n c e n t r i c envelope maintained at a s p e c i f i e d constant temperature. The space between the two i s f i l l e d w ith an e s s e n t i a l l y i n c o m p r e s s i b l e f l u i d . Steady, laminar and r o t a t i o n a l l y symmetrical f r e e c o n v e c t i o n i s assumed to take p l a c e over the gap width and conduction i s the s o l e t r a n s p o r t mechanism c o n s i d e r e d i n s i d e the core. Two l i m i t i n g c ases, of an i n n e r sphere o f i n f i n i t e l y l a r g e r e l a t i v e heat c o n d u c t i v i t y , l e a d i n g t o an i s o t h e r m a l core to f l u i d i n t e r f a c e ; and of the converse case of s m a l l c o n d u c t i v i t y l e a d i n g to a c o n s t a n t f l u x i n t e r f a c e are c o n s i d e r e d s e p a r a t e l y . The a n a l y s i s o f heat t r a n s p o r t leads to the s o l u t i o n of the governing equations through r e g u l a r p e r t u r b a t i o n expansions with the Grashof number as main parameter. The r a t i o of c o n d u c t i v i t i e s , r a d i u s r a t i o and P r a n d t l number appear as secondary parameters. S t r e a m l i n e s , i s o v o r t i c i t y curves and isotherms are o b t a i n e d f o r v a r i o u s combinations of the parameters. The v e l o c i t y d i s t r i b u t i o n i s determined and both l o c a l and o v e r a l l v a l u e s o f the N u s s e l t number are ob t a i n e d . A flow v i s u a l i z a t i o n t e s t was undertaken and the core s u r f a c e temperature d i s t r i b u t i o n was determined e x p e r i -m e n t a l l y . Reasonable q u a l i t a t i v e agreement with the a n a l y s i s i s found. i i i ACKNOWLEDGEMENTS The author wishes to express h i s g r a t i t u d e and s i n c e r e thanks t o P r o f e s s o r Zeev Rotem f o r h i s guidance and advice throughout a l l the stages o f the program. S p e c i a l thanks are due to Drs. E.G. Hauptmann and M. I q b a l f o r t h e i r s uggestions and d i s c u s s i o n s . A l s o , the author wishes to thank the e n t i r e s t a f f of the Mechanical E n g i n e e r i n g Depart-ment, U n i v e r s i t y o f B r i t i s h Columbia, f o r t h e i r a s s i s t a n c e . S p e c i a l thanks are due to Messrs. J . Hoar and P. Hurren ( c h i e f t e c h n i c i a n s ) and L. Dery f o r t h e i r a s s i s t a n c e d u r i n g the c o n s t r u c t i o n and o p e r a t i o n of the experiment. Support of t h i s program was pro v i d e d by the N a t i o n a l Research C o u n c i l of Canada through Grant No. 67-2772 f o r which thanks are due. The author i s g r a t e f u l f o r the award of a U n i v e r s i t y of B r i t i s h Columbia Graduate F e l l o w s h i p from 1969 to 1971. i v TABLE OF CONTENTS Page ABSTRACT i i ACKNOWLEDGEMENTS i i i LIST OF TABLES v i LIST OF FIGURES v i i NOMENCLATURE x Chapter 1 INTRODUCTION . . . . . 1 2 ANALYSIS 8 2.1 Formulation o f Conjugate Problem . . . . 8 2.2 Governing Equations . • !0 2.3 Method of S o l u t i o n . 1 5 2.4 S o l u t i o n 2 1 2.5 S o l u t i o n o f Constant F l u x Problem . . . . 30 3 EVALUATION OF ANALYTICAL RESULTS 33 3.1 Range of V a l i d i t y of S o l u t i o n s 34 3.2 Streamlines 38 3.3 V e l o c i t y D i s t r i b u t i o n 47 3.4 V o r t i c i t y Contours 52 3.5 Temperature D i s t r i b u t i o n and Contours 59 3.6 Heat-Transfer Rates 7-L 4 EXPERIMENT 8 3 4.1 Experimental Apparatus 83 V Chapter Page 4.2 Experimental Procedure 94 4.3 Experimental R e s u l t s 96 5 CONCLUSIONS 103 REFERENCES . . APPENDIX I - C a l c u l a t i o n of Vi s c o u s D i s s i p a t i o n E f f e c t APPENDIX I I - Conjugate Problem w i t h Inner Sphere C o n t a i n i n g D i s t r i b u t e d Sources . . APPENDIX I I I - A B r i e f Review on Small Grashof Numbers N a t u r a l Convection About a Heated Sphere . v i LIST OF TABLES Table Page I O v e r a l l N u s s e l t Numbers f o r Conjugate Case, 3 = 1.15, 2.0, and 3.0 82 II Experimental Surface Temperature D i s t r i b u t i o n of Two C o n c e n t r i c Spheres, G = 1.32xl0 4 101 I I I Experimental Surface Temperature D i s t r i b u t i o n of Two C o n c e n t r i c Spheres, G = 3.72xl0 5 101 IV Experimental Surface Temperature D i s t r i b u t i o n of Two C o n c e n t r i c Spheres, G = 5.78xl0 5 102 V R a t i o of Viscous D i s s i p a t i o n Term t o Con-d u c t i o n Term; R a t i o of Viscous D i s s i p a t i o n Term to Convection Term (Air) 109 VI R a t i o of V i s c o u s D i s s i p a t i o n Term to Con-d u c t i o n Term; R a t i o o f Viscous D i s s i p a t i o n Term to Convection Term (Water) . 109 v i i LIST OF FIGURES F i g u r e Page 2.1.1 P h y s i c a l C o n f i g u r a t i o n 10 .3.1.1 Approximate Upper Bound o f G f o r v a r i o u s a/3 37 3.2.1 Streamlines f o r Conjugate Case, to=10, 6=2.0, a=.72, G=103 41 3.2.2 Streamlines f o r Conjugate Case, co=10l^, 3=2.0, a=.72, G=103 42 3.2.3 Streamlines f o r Constant F l u x Case, 3=2.0, a=.72, G=103 43 3.2.4 Streamlines f o r Conjugate Case, co=10, 6=1.15, a=.72, G=103 44 3.2.5 Streamlines f o r Conjugate Case, to=10, 3=2.0, o= 10 G=72 45 3.2.6 Streamlines f o r Conjugate Case, w=10, 3=2.0, a=.72, G=2100 . . . 46 3.3.1 R a d i a l V e l o c i t y , V R vs. Radius (Various Angular P o s i t i o n s ) 50 3.3.2 T a n g e n t i a l V e l o c i t y , V Q vs. Radius (Various Angular P o s i t i o n s ) . 51 3.4.1 V o r t i c i t y Contours, Conjugate Case, w=10, 3=2.0, o=.72, G=103 55 3.4.2 V o r t i c i t y Contours, Conjugate Case, w=10 1 5, 3=2.0, a=.72, G=103 56 3.4.3 V o r t i c i t y Contours, Constant F l u x Case, 3=2.0,, o=.72, G=103. 57 3.4.4 V o r t i c i t y Contours, Conjugate Case, w=10, 3=2.0, o=10, G=72 58 3.5.1 Temperature D i s t r i b u t i o n , Conjugate Case,,'; co=10, 3=2.0, a=.72, G=103 62 v i i i F i g u r e Page 3.5.2 Temperature D i s t r i b u t i o n o f Inner Sphere, Conjugate Case, UJ=10, 6=2.0, a=.72, G=103 6 3 3.5.3 Temperature D i s t r i b u t i o n , Conjugate Case, w=10, 3=2.0, a=10, G=72 6 4 3.5.4 Temperature D i s t r i b u t i o n , Conjugate Case, w=10l5, 3=2.0, a=.72, G=103 65 3.5.5 Temperature D i s t r i b u t i o n , Constant F l u x Case, 3 = 2.0, a=.72, G=103 66 3.5.6 Isotherms, Conjugate Case, u)=10, 3=2.0, a=.72, G=103 67 3.5.7 Isotherms, Conjugate Case, w=10 , 3=2.0, a=10, G=72 . 68 3.5.8 Isotherms, Conjugate Case, w=10l5, 3=2.0, a=.72, G=10 3 69 3.5.9 Isotherms, Constant F l u x Case, 3=2.0, a=.72, G=103 70 3.6.1 N u s s e l t Number A g a i n s t Angular P o s i t i o n , Conjugate Case, OJ=10, 3=2.0, a=.72, G=103 . . 77 3.6.2 N u s s e l t Number A g a i n s t Angular P o s i t i o n , Conjugate Case, oo=10, 3=2.0, a=10, G=72 . . . 78 3.6.3 N u s s e l t Number A g a i n s t Angular P o s i t i o n , Conjugate Case, w=10l 5, 3=2.0, a=.72, G=103 79 3.6.4 N u s s e l t Number A g a i n s t Angular P o s i t i o n , Conjugate Case, OJ=10 , 3=2.0, c=.72, G=1400 . . 8 0 3.6.5 O v e r a l l N u s s e l t Number as Fu n c t i o n of Ray l e i g h Number, Conjugate Case 81 4.1.1 Experimental Apparatus 8 4 4.1.2 P l e x i g l a s Inner Sphere 8.5 4.1.3 L o c a t i o n s of Thermocouples on Surface o f Inner Sphere 87 i x F i g u r e Page 4.1.4 Support Stem w i t h G l a s s Hemisphere 8 8 4.1.5 Top View of Support Stem and Glass Hemisphere 89 4.1.6 T y p i c a l Thermocouple C a l i b r a t i o n Curve . . . . 91 4.1.7 Platinum-Wire Temperature-Sensing Probe, C a l i b r a t i o n Curve 92 4.1.8 Layout o f Experimental Apparatus 9 3 4.4.1 Smoke P a t t e r n at G=1.32xl0 4, AT=.024°C . . . . 97 4.4.2 Smoke P a t t e r n a t G=3.72xl0 5, AT=7.21°C . . . . 9 8 4.4.3 Smoke P a t t e r n a t G=5.78xl0 5, AT=10.3°C . . . . 99 X NOMENCLATURE A(R,6,<f>) dimensionless heat source f u n c t i o n (eq. 2.2.3) B(R') heat source f u n c t i o n per u n i t volume (eq. B-l) c s p e c i f i c heat o f the f l u i d at constant P c . pr e s s u r e V v i s c o u s d i s s i p a t i o n number, -- v— — R , 2K.AT _ f r e f 2 4 . . E , E o p e r a t o r s 3 2 6 Grashof number, g y A T r e f R ' / v g a c c e l e r a t i o n of g r a v i t y k thermal c o n d u c t i v i t y Nu^ l o c a l N u s s e l t number f o r the i n n e r sphere, / 3-1 ,• 30 i • " ( ~3~ > X T m l R = 1 Nu l o c a l N u s s e l t number f o r the outer sphere, o i 3-1 \ 8 0 1 Q2 K-p Q r e f e r e n c e r a t e of heat f l u x 3 Ra R a y l e i g h number, ^ Y ^ T r e f R i / ^ v a ^ R r a d i a l c o o r d i n a t e T temperature a. Q x ( 3 - D T r- r e f e r e n c e temperature, — r e f tr r V dime n s i o n l e s s v e l o c i t y 4TT3K ^ R ' . f I X I a thermal d i f f u s i v i t y 6 r a d i u s r a t i o , R' / R! ' o ' 1 y e x p a n s i v i t y a t constant p r e s s u r e e r a t i o o f v i s c o u s d i s s i p a t i o n term t o conduction term 5 dimensionless v o r t i c i t y v e c t o r , [V x V ] ^ n cos 0 2 V Laplace o p e r a t o r , s p h e r i c a l c o o r d i n a t e s 0 angular c o o r d i n a t e 0 dimensionless temperature X r a t i o of v i s c o u s d i s s i p a t i o n term to c o n v e c t i o n term u dynamic v i s c o s i t y v k i n e m a t i c v i s c o s i t y p d e n s i t y of f l u i d a P r a n d t l number, v/a cj) c i r c u m f e r e n t i a l c o o r d i n a t e (longitude) $ d i s s i p a t i o n f u n c t i o n v c ¥ d i m e n s i o n l e s s stream f u n c t i o n , V'/vR'. ' ' I OJ thermal c o n d u c t i v i t y r a t i o , k g / k f S u p e r s c r i p t s , s u b s c r i p t s ' r e f e r s t o dimensional q u a n t i t i e s r e f e r s t o average f r e f e r s t o f l u i d 1 , ~ r e f e r s t o i n n e r sphere o r e f e r s t o outer sphere r e f /"refers to r e f e r e n c e v a l u e s s r e f e r s to s o l i d m a t e r i a l 1 1. INTRODUCTION The study of t h e r m a l l y induced f l u i d motion i n en-c l o s e d spaces has been r e c e i v i n g i n c r e a s i n g a t t e n t i o n over r e c e n t years. The reason f o r t h i s seems t o be t h a t such flows p l a y a r o l e i n a wide area of t e c h n o l o g i c a l a p p l i c a t i o n , ranging from c o n v e c t i o n i n the annular space between a n u c l e a r r e a c t o r core and i t s p r e s s u r e v e s s e l t o a d v e c t i o n i n l a k e s and t o thermal i n s u l a t i o n of d o u b l y - g l a z e d windows. F u r t h e r -more, the mathematical d e s c r i p t i o n of such a problem leads t o a s e t of n o n - l i n e a r coupled p a r t i a l d i f f e r e n t i a l equations which i s of i n t e r e s t i n i t s e l f . B a t c h e l o r [1], o r i g i n a l l y motivated by the problem of thermal i n s u l a t i o n of b u i l d i n g s ( p a r t i c u l a r l y d o u b l y - g l a z e d windows) i n v e s t i g a t e d a n a l y t i c a l l y the flow regimes i n a r e c t a n g u l a r c a v i t y . The model he c o n s i d e r e d had two v e r t i c a l w a l l s of a c a v i t y a t d i f f e r e n t temperatures, with a narrow a i r - s p a c e between them. The c a v i t y i s c l o s e d at the top and bottom and i n f i n i t e i n the t h i r d d i r e c t i o n (the b r e a d t h ) . F l u i d motion i s generated by buoyancy and i t i s assumed to be e s s e n t i a l l y two-dimensional. B a t c h e l o r estimated the heat f l u x f o r v a r i o u s flow regimes i n the c a v i t y . He a l s o p o s t u l a t e d t h a t a t a s u f f i c i e n t l y h i g h v a l u e of the R a y l e i g h number the flow would c o n s i s t of a core of constant v o r t i c i t y and temperature surrounded by a continuous boundary l a y e r . Poots [ 2 ] , u s i n g a doubly o r t h o g o n a l expansion f o r the stream f u n c t i o n and the temperature, computed B a t c h e l o r ' s model f o r s e v e r a l v a l u e s of R a y l e i g h number and to some ext e n t v e r i f i e d the c o n c l u s i o n s p r e v i o u s l y a r r i v e d a t . Subsequent i n v e s t i g a t o r s are Wilkes and C h u r c h i l l [3], E l d e r [4 ] and de V a h l Davis [5 ]. They a l l employed v a r i o u s f i n i t e d i f f e r e n c e schemes to o b t a i n numerical r e s u l t s . In both [3] and [4 ] i n s t a b i l i t y of the n u m e r i c a l scheme was a problem. Wilkes' d i f f i c u l t i e s arose a p p a r e n t l y from the r e t e n t i o n of the v o r t i c i t y as an e x p l i c i t unknown i n the boundary c o n d i t i o n s . E l d e r , i n order t o achieve numerical s t a b i l i t y of h i s scheme, had t o approximate by s e t t i n g the normal g r a d i e n t of the v o r t i c i t y at the h o r i z o n t a l boundaries to zero. The j u s t i f i c a t i o n f o r t h i s assumption i s not immed-i a t e l y obvious. On the other hand de Vahl Davis, i n an approach s i m i l a r to t h a t of Wilkes, overcame the i n s t a b i l i t y problem by c a l c u l a t i n g the v o r t i c i t y from the stream f u n c t i o n at the boundary. At s u f f i c i e n t l y l a r g e values of the R a y l e i g h number, he found [5 ] t h a t there was indeed a core of approx-i m a t e l y constant v o r t i c i t y , but not one o f constant temperature too as p r e v i o u s l y p o s t u l a t e d by B a t c h e l o r . The papers reviewed above do not c o n s i d e r the presence of a s o l i d core i n s i d e the e n c l o s e d space. Recently Mack et a l . [6,7,8,9,10] i n v e s t i g a t e d the cases of steady laminar n a t u r a l c o n v e c t i o n of f l u i d i n a gap between c o n c e n t r i c c y l i n d e r s and c o n c e n t r i c spheres, both e x p e r i m e n t a l l y and 3 a n a l y t i c a l l y . The c y l i n d e r s or spheres are kept at c o n s t a n t temperature with the i n n e r c y l i n d e r or sphere b e i n g the h o t t e r . Flow v i s u a l i z a t i o n f o r both c o n f i g u r a t i o n s and heat t r a n s f e r d ata f o r the two c o n c e n t r i c spheres were obtained. The authors r e p o r t e d t h a t the flow regime i n the gap between the two c o n c e n t r i c spheres depended upon the diameter r a t i o of the spheres. S i m i l a r r e s u l t s were found f o r the case of c o n c e n t r i c c y l i n d e r s . Three types of flow-regimes were observed; (a) a 'kidney-shaped eddy 1 type (b) a 'crescent-eddy' type (c) ' f a l l i n g v o r t i c e s ' ; these correspond r e s p e c t i v e l y to diameter r a t i o s of 3.14, 1.72, and 1.19. Mack and Bishop [ 9 ] , Mack and Hardee [10] obtained a n a l y t i c a l s o l u t i o n s i n the form of s e r i e s expansions i n powers of the R a y l e i g h number f o r c o n c e n t r i c c y l i n d e r s and spheres. In t h e i r expansions the c o e f f i c i e n t s of h i g h e r - o r d e r terms a l s o depend upon the P r a n d t l number. A proof of convergence of t h e i r p e r t u r b a t i o n expansions was not g i v e n . T h e i r a n a l y t i c a l s o l u t i o n s f o r c o n c e n t r i c spheres seem t o y i e l d convergent r e s u l t s up to a maximum value of the R a y l e i g h number of 1600 (based on gap w i d t h ) , while i n t h e i r e x perimental s t u d i e s the range of R a y l e i g h numbers used i s r a t h e r higher, than 1600. Thus d i r e c t comparison between exp e r i m e n t a l and a n a l y t i c a l r e s u l t s i s perhaps not meaningful. I t i s customary i n heat t r a n s f e r work to s p e c i f y i d e a l i z e d boundary c o n d i t i o n s f o r both the temperature and flow v e l o c i t y i n order to a r r i v e at a w e l l - s e t problem. That i s the boundary c o n d i t i o n s are u s u a l l y constant; temperature, c o n s t a n t f l u x or a combination of these. How-ever, i n a c t u a l p r a c t i c e the boundary c o n d i t i o n s a t the conducting e n c l o s i n g s u r f a c e s are r a r e l y known beforehand. They depend upon the coupled mechanisms o f conduction i n the s o l i d boundary and conv e c t i o n of the f l u i d over the boundary. In t e c h n o l o g i c a l l y i n t e r e s t i n g geometries the heat t r a n s f e r ('film') c o e f f i c i e n t at the s u r f a c e i s v i r t u a l l y never uniform. T h i s v a r i a t i o n leads to a r e d i s t r i b u t i o n of heat f l u x i n the s o l i d , i n some balance w i t h the c o n v e c t i v e motion which takes p l a c e i n the f l u i d . The e f f e c t of the coupled c o n d u c t i o n / c o n v e c t i o n mechanism i s most pronounced i n f r e e c o n v e c t i v e heat t r a n s f e r : not only are the momentum and energy f i e l d s of the f l u i d coupled but i n a d d i t i o n the energy f i e l d s of the f l u i d and the s o l i d are coupled as w e l l . Due t o t h a t c o m p l i c a t i o n problems of t h i s type have on l y r e c e i v e d scant a t t e n t i o n . The f o l l o w i n g i s a review o f p u b l i s h e d work. Perelman HH c o n s i d e r e d two examples o f two dimensional flow around a body with l i n e and plane heat sources and used the term 'conjugate' f o r t h i s type o f problem. In h i s a n a l y s i s the v e l o c i t y p r o f i l e s were assumed t o be e i t h e r l i n e a r or o f the s l i p - f l o w type, u n i n f l u e n c e d by c o n v e c t i o n . In view o f t h i s assumption, the momentum f i e l d i n the f l u i d i s uncoupled from the energy f i e l d . Hence the s o l u t i o n to the momentum equation i s known s e p a r a t e l y of the c o n v e c t i o n of heat. Some time l a t e r Rotem [12] developed an approximate method f o r the e v a l u a t i o n of i n t e r f a c e temperature p r o f i l e s and the t r a n s f e r c o e f f i c i e n t f o r heat t r a n s f e r t o a f o r c e d laminar boundary l a y e r with w a l l conduction. Again the method a p p l i e s t o the case of uncoupled equations only. K e l l e h e r and Yang [13] employed a G o e r t l e r s e r i e s f o r the conjugate problem f o r the f r e e c o n v e c t i o n of f l u i d over a two-dimensional conducting body with i n t e r n a l heat sources. Here the v e l o c i t y can indeed not be s p e c i f i e d i n advance: i t i s coupled to the temperature d i s t r i b u t i o n on the s u r f a c e of the s o l i d , which i n t u r n depends upon both the heat source d i s t r i b u t i o n and c o n v e c t i o n . The r e g i o n of i n t e r e s t .e of t h e i r a n a l y s i s was p l a c e d near the l e a d i n g edge. Lock and Gunn [14] c o n s i d e r e d the laminar f r e e con-v e c t i o n from a downward p r o j e c t i n g f i n , but s o l v e d the convec-t i o n and conduction p a r t s o f the problem s e p a r a t e l y . The s o l u t i o n was completed by matching the i n t e r f a c i a l temperature and heat f l u x . The study of a conjugate heat t r a n s f e r problem i n a * confined space i s apparently not a v a i l a b l e i n the l i t e r a t u r e . A l s o , p r e v i o u s i n v e s t i g a t o r s o f conjugate heat t r a n s f e r problems, both i n e x t e r n a l and i n i n t e r n a l flows [15], have c o n s i d e r e d the conjugate e f f e c t s i n one or two dimensions o n l y . The p r e s e n t study c o n s i d e r s steady conjugate laminar t h r e e -dimensional n a t u r a l c o n v e c t i o n (with r o t a t i o n a l symmetry) i n * An e x c e p t i o n t o t h i s i s the r e c e n t a n a l y s i s of Rotem [16] of the conjugate c o n v e c t i o n f o r the case of h o r i z o n t a l c o n c e n t r i c c y l i n d e r s . 6 a f l u i d between two c o n c e n t r i c spheres. The s o l u t i o n s t o the q u a s i - l i n e a r coupled governing equations of energy and v o r t i c i t y t r a n s p o r t are obtained i n the form of an asymptotic expansion of the v a r i a b l e s stream f u n c t i o n and temperature i n terms of ascending powers of the Grashof number. The other parameters which are of importance i n t h i s problem are the P r a n d t l number, the r a d i u s r a t i o , and the thermal c o n d u c t i v i t y r a t i o . The range of each parameter over which the expansion scheme i s v a l i d i s e v a l u a t e d i n d e t a i l . The cases of a constant f l u x or an i s o t h e r m a l i n n e r sphere with an i s o t h e r m a l outer envelope are a l s o c o n s i d e r e d i n the p r e s e n t work: these are l i m i t i n g cases of the conjugate problem. An experiment was designed f o r the v i s u a l i z a t i o n of the flow p a t t e r n i n the gap between two conjugate c o n c e n t r i c spheres. The experimental r e s u l t s p r o v i d e f u r t h e r i n s i g h t i n t o the nature of the c o n v e c t i v e flow and supplement the a n a l y t i c a l r e s u l t s . An a p p l i c a t i o n which t h i s a n a l y s i s would model i s : g e o p h y s i c a l c i r c u l a t i o n s . For g l o b a l g e o p h y s i c a l c i r c u l a t i o n s the v e c t o r a c c e l e r a t i o n of g r a v i t y i s everywhere normal to the boundary and c o n v e c t i o n occurs o n l y when a c e r t a i n R a y l e i g h number exceeds a c r i t i c a l v a l u e . However, i n s i g h t to the l o c a l c o n v e c t i o n regime may be gained by c o n s i d e r i n g the g r a v i t a t i o n a l f i e l d to be o r i e n t e d everywhere' p a r a l l e l t o the v e r t i c a l a x i s as shown i n F i g u r e 2.1.1, i n t h i s t h e s i s . For i n some g e o p h y s i c a l models, s m a l l e r regions of c l o s e d c i r c u -l a t i o n are shown to e x i s t [ 7 ] , i n which the c o n d i t i o n s would not be d i s s i m i l a r t o those c o n s i d e r e d here. Another a p p l i c a t i o n i s to the design of a s p h e r i c a l i n s u l a t i n g f l a s k to minimize the heat l o s s : f o r l a r g e s p h e r i c a l c o n t a i n e r s i t i s o f t e n not f e a s i b l e to have a vacuum between the c o n c e n t r i c s h e l l s , as t h i s would i n t r o d u c e b u c k l i n g problems and bending s t r e s s , i n s t e a d of membrane s t r e s s e s , p a r t i c u l a r l y near the supports. Thus a s p h e r i c a l gap f i l l e d with gas i s r e t a i n e d between the s p h e r i c a l envelope and a c o n f i g u r a t i o n i s obtained f o r which the p r e s e n t a n a l y s i s i s a p p l i c a b l e . 8 2. ANALYSIS 2.1 Fo r m u l a t i o n of Conjugate Problem The a n a l y s i s c o n s i d e r s steady conjugate laminar t h r e e - d i m e n s i o n a l n a t u r a l c o n v e c t i o n i n a f l u i d c o n tained between two c o n c e n t r i c spheres. The p h y s i c a l c o n f i g u r a t i o n i s t h a t shown i n F i g u r e 2.1.1. A s o l i d i n n e r sphere i s heated e i t h e r by a constant source at i t s centr e o r by d i s t r i b u t e d s o urces. I t i s cooled by laminar n a t u r a l c o n v e c t i o n of the f l u i d i n s i d e the gap between the c o n c e n t r i c spheres. The out e r sphere i s kept at a constant temperature. Thus heat t r a n s f e r a t the i n t e r f a c e between the s o l i d i n n e r sphere and the f l u i d depends upon the coupled mechanisms of conduction i n s i d e the i n n e r sphere and c o n v e c t i o n i n the f l u i d . By d e f i n i t i o n the t r a n s f e r o f heat i s t h e r e f o r e of the conjugate type. Some p r e v i o u s i n v e s t i g a t o r s [10] have c o n s i d e r e d the n a t u r a l c o n v e c t i v e problem f o r the s p h e r i c a l c o n f i g u r a t i o n without the a d d i t i o n a l c o m p l i c a t i o n of conductive e f f e c t s along the boundaries and t h i s work w i l l be d i s c u s s e d l a t e r . The temperature and heat f l u x d i s t r i b u t i o n s on the s u r f a c e of the i n n e r sphere are of g r e a t importance i n the numerous a p p l i c a t i o n s f o r which the problem t r e a t e d here i s a model. The o b j e c t i v e s of the p r e s e n t i n v e s t i g a t i o n are as f o l l o w s : (i) d e t e r m i n a t i o n of the temperature and heat f l u x v a r i a t i o n s on the i n n e r sphere; ( i i ) of the heat f l u x v a r i -a t i o n on the s u r f a c e of the ou t e r sphere; ( i i i ) and l a s t l y of the temperature, v e l o c i t y and v o r t i c i t y d i s t r i b u t i o n s o f the f l u i d i n the gap between the c o n c e n t r i c spheres. I t w i l l be shown t h a t the parameters o f importance i n the a n a l y s i s are (a) G , the Grashof number; (b) a, the P r a n d t l number of the f l u i d ; (c) GO, the r a t i o o f the thermal c o n d u c t i v i t y o f the s o l i d i n n e r sphere t o t h a t o f the f l u i d ; (d) 6, the r a t i o of the r a d i u s of the ou t e r sphere to t h a t o f the i n n e r sphere. S p h e r i c a l c o o r d i n a t e s are used, the angular c o o r d i n a t e 8 b e i n g measured cl o c k w i s e from the apex as shown i n F i g u r e 2.1.1. A l l p h y s i c a l p r o p e r t i e s are assumed co n s t a n t , except the d e n s i t y o f the f l u i d i n as f a r as i t s dependence on the temperature i s concerned. T h i s assumption and i t s i m p l i c a t i o n s were f i r s t i n t r o d u c e d by Boussinesq. I t i s assumed t h a t the compression work and v i s c o u s d i s s i p a t i o n are a l s o n e g l i g i b l e . Hence the corresponding terms w i l l be d e l e t e d from the energy e q u a t i o n . The l a t t e r assumption i s checked i n Appendix I of t h i s t h e s i s . An axisymmetrical c o n v e c t i v e flow p a t t e r n i s assumed i . e . there i s no azimuthal s w i r l . Hence a l l q u a n t i t i e s are taken to be independent of the l o n g i t u d e , c£> . 10 CONSTANT TEMPERATURE OUTER ENVELOPE SOLID INNER SPHERE CENTRAL HEAT SOURCE MATCHING OF TEMPERATURE AND FLUX AT SOLID/FLUID INTERFACE F i g u r e 2.1.1 P h y s i c a l C o n f i g u r a t i o n 2.2 Governing Equations A l l dimensional q u a n t i t i e s used (except f o r pr o p e r t y values) are primed i n what f o l l o w s and a l l dimensionless q u a n t i t i e s are unprimed. L e t R| be the r a d i u s of the i n n e r sphere. R = R'/R- and 3 = R'/R' a ^ e the dimensionless c 1 o 1 11 r a d i a l c o o r d i n a t e and dimensionless r a d i u s o f the ou t e r sphere r e s p e c t i v e l y . T and T are the temperatures of the i n n e r and outer spheres r e s p e c t i v e l y ; g, the a c c e l e r a t i o n o f g r a v i t y ; Y, the v o l u m e t r i c c o e f f i c i e n t o f thermal expansion a t constant p r e s s u r e ; v, the k i n e m a t i c v i s c o s i t y ; and a, the thermal d i f f u s i v i t y . A r e f e r e n c e temperature and temperature d i f f e r e n c e are i n t r o d u c e d : = AT r e f and AT r e f = Q x ( 3 - D / ( 4 7 T 3 R ; f K f ) where Q i s r a t e of heat f l u x , assumed constant. Let Ra = gyAT r e f i denote the R a y l e i g h number; a = v / a the P r a n d t l number; G = gyAT r e f i 2 the Grashof number; 9 = (T-T J / A T c r e f ' r e f the dimensionless f l u i d temperature; 12 0 = (T-T -)/AT -r e f ' r e f the d i m e n s i o n l e s s i n n e r sphere temperature. The equations of c o n t i n u i t y , motion and energy f o r a Newtonian, i n c o m p r e s s i b l e constant p r o p e r t y f l u i d w i l l not be s t a t e d here; they may be found i n many textbooks. Stokes stream f u n c t i o n ¥ w i l l be i n t r o d u c e d , so t h a t the e q u a t i o n of c o n t i n u i t y i s f u l f i l l e d i d e n t i c a l l y . Then i n the dimensionless q u a n t i t i e s i n t r o d u c e d above the steady s t a t e governing equa-t i o n s reduce to the f o l l o w i n g , 2 P 4U/ = L. 9 ( ¥ ' E ^ + 2E 2 f , n 31 . i I I . R2 >(R,n) + R2 {1.T]2) R 3n ; 0 9(Rxn,0) + ( l - n ) G 3(R,n) (2.2.1) 2 a 8 ( W ' 0 ) V z0 = ~ (2.2.2) R -3(R,n) V 20 = A(R,9,(j)) . (2.2.3) The boundary .conditions s u b j e c t to which (2.2.1) to (2.2.3) have t o be s o l v e d w i l l be 3 ¥ ^ = 3R = 0 a t R = 1, p (2.2.4) 3 m y = JQ = 0 a t 6 = 0, 7T (2.2.5) o r , a l t e r n a t i v e l y s t a t e d <F = - ( l - n 2 ) I | I = o a t n = ± 1 (2.2.5a) 0 = © a t R = 1 (2.2.6) 30 39 w 3 R = 9R a t R = 1 (2.2.7) 0 = -1 a t R = 3 . (2.2.8) Here n = cos 6 and t h e o p e r a t o r s E 2 , E 4 and V 2 a r e d e f i n e d as f o l l o w s : -E 2 = + ( i - n 2 ) 3 2 3R 3n 4 2 2 E = E [E^] a 4 4 ( i - n 2 ) , n 3 3 . 3R R" ) + 4 ( i - n 2 ) 3 2 R 3 n 3 3 R 3 n 2 R 4 3 n 2 , 2 ( i - n 2 ) 3 4 - _ R 3R 3n d - n 2 ) 3 4 R 3R 14 v 2 = a 2 I 2 d . ( l - n 2 ) 3 2 2 n a_. 9 R 2 R 8 R ~ ^ r - 3 n 2 - ^ 9 n = E 4 + 2 ^ ( n/R) | ^ + §=; ( n / R ) | ^ ) . The l e f t - h a n d s i d e of the e q u a t i o n ( 2 . 2 . 1 ) r e p r e s e n t s the d i f f u s i o n of v o r t i c i t y . The f i r s t two terms on the r i g h t -hand s i d e o f the equation ( 2 . 2 . 1 ) give the c o n v e c t i o n of v o r t i c i t y w h i le the t h i r d term r e p r e s e n t s the buoyancy e f f e c t . The l e f t - h a n d s i d e of the e q u a t i o n ( 2 . 2 . 2 ) r e p r e s e n t s the conduction e f f e c t s , the r i g h t - h a n d s i d e the c o n v e c t i v e e f f e c t s . The e f f e c t s o f v i s c o u s d i s s i p a t i o n and compression work were n e g l e c t e d i n the energy e q u a t i o n ( 2 . 2 . 2 ) . In Appendix I i t w i l l be shown t h a t f o r c l o s e d spaces t h i s i s always a p e r m i s s i b l e assumption. L a s t l y , equation ( 2 . 2 . 3 ) i s the conduction e q u a t i o n f o r the s o l i d i n n e r sphere. A(R,6,c)>) i s the source f u n c t i o n : For the case of a s i n g l e heat source a t the c e n t r e of the i n n e r sphere, the source f u n c t i o n may be s e t to zero everywhere i * except a t the c e n t e r p o l e . The boundary c o n d i t i o n s ( 2 . 2 . 4 ) s t a t e t h a t t h e r e w i l l be no flow along and through the s u r f a c e s of the two spheres. C o n d i t i o n s ( 2 . 2 . 5 ) ensure t h a t the flow i s axisymmetrical * The case of the i n n e r sphere c o n t a i n i n g u n i f o r m l y d i s t r i b u t e d heat sources i s c o n s i d e r e d i n Appendix I I . 15 about the v e r t i c a l a x i s Q = 0 , IT ( i . e . a t n = ± 1). Boundary c o n d i t i o n s (2.2.6) and (2.2.7) s t a t e the energy c o n s e r v a t i o n p r i n c i p l e , i . e . the e q u a l i t y o f temperature and f l u x on both s i d e s o f the i n t e r f a c e s o l i d / f l u i d . The f a c t t h a t both temperature and f l u x are s p e c i f i e d w i l l be seen to le a d t o a problem of the conjugate type. (2.2.8) i s a n o r m a l i z a t i o n c o n d i t i o n f o r the temperature. 2.3 Method of S o l u t i o n In the present a n a l y s i s i t i s assumed t h a t the i n n e r sphere has a c e n t r a l heat source o n l y i . e . A(R,0,(f>) i s zero i n e q uation (2.2.3). In s p h e r i c a l geometry the f l u i d cannot be i n e q u i l i b r i u m when there i s a non-zero temperature d i f f e r e n c e between the i n n e r sphere and the bulk of the f l u i d , no matter how s m a l l t h a t d i f f e r e n c e may be as the v e c t o r a c c e l e r a t i o n of g r a v i t y i s everywhere i n c l i n e d to the s o l i d to f l u i d i n t e r f a c e . T h i s i s due to the t a n g e n t i a l component of the buoyancy f o r c e s i n the f l u i d . I n i t i a l l y the f l u i d may be at r e s t , but buoyancy f o r c e s w i l l cause c o n v e c t i v e motion which a f t e r some time w i l l become qu a s i - s t e a d y . The governing time-independent equations w i l l then apply to t h i s steady motion. I t should be noted t h a t the c o n v e c t i v e flow i s the only source o f v o r t i c i t y a v a i l a b l e . As mentioned above, the energy equations (2.2.2) and (2.2.3) f o r the f l u i d and the s o l i d i n n e r sphere r e s p e c t i v e l y are coupled through the boundary c o n d i t i o n s (2.2.6) and (2.2.7). On the o t h e r hand 16 the v o r t i c i t y t r a n s p o r t e q u a t i o n and the energy equation f o r the f l u i d are a l s o coupled through the buoyancy term i n (2.2.1). As the motion o f the f l u i d i s set-up by buoyancy f o r c e s o n l y , i t w i l l be n a t u r a l to seek s o l u t i o n s to equations (2.2.1), (2.2.2) .-and (2.2.3) f o r the dependent v a r i a b l e s V, 0,0 i n terms of p e r t u r b a t i o n expansions i n the Grashof number. Th i s i s an ad hoc assumption i n the sense t h a t any s u i t a b l e parameter c h a r a c t e r i z i n g the r a t i o of buoyancy to v i s c o u s f o r c e s might have been used w i t h s u i t a b l e s c a l i n g of the v a r i a b l e s , e.g. Ra. That i s , an expansion of the present type i s not n e c e s s a r i l y unique [18] . Moreover, the problem d e a l t w i t h here i s a multi-parameter one i n v o l v i n g some other parameters, a, 3 and OJ . The p e r t u r b a t i o n s o l u t i o n s are o b t a i n e d f o r a range of Grashof numbers. For each value of G, combinations of v a r i o u s values of the other parameters a, O J and 3 are c o n s i d e r e d . There i s a l i m i t a t i o n on the range of r a d i u s r a t i o a d m i s s i b l e : o b v i o u s l y 3>1; as 3 tends to i n f i n i t y the i n n e r sphere becomes a s i n g l e sphere i n an unbounded expanse of f l u i d . To t h i s l a t t e r case the r e g u l a r p e r t u r b a t i o n theory to be out-l i n e d w i l l not apply [19,20]. On the other hand, i f o  tends to zero or i n f i n i t y the i n n e r sphere has e i t h e r a constant f l u x or an i s o t h e r m a l envelope r e s p e c t i v e l y . The former of these cases leads to s i n g u l a r i t y and must be s o l v e d s e p a r a t e l y . The case of heat t r a n s f e r between two c o n c e n t r i c isothermal spheres has 17 been s o l v e d a n a l y t i c a l l y by Mack and Hardee [10]. In s e c t i o n 2.5 the case of the constant f l u x i n n e r sphere s u r f a c e w i t h an i s o t h e r m a l outer envelope (the s o l u t i o n f o r which had not been a v a i l a b l e ) w i l l be c o n s i d e r e d . p r e s e n t no p a r t i c u l a r d i f f i c u l t i e s . However, as a °° the energy equation tends to become s i n g u l a r and not a l l boundary c o n d i t i o n s f o r the temperature p r o f i l e may be f u l f i l l e d . T h e r e f o r e a d i f f e r e n t parameter p e r t u r b a t i o n s t a r t i n g with V/a r a t h e r than 1 / v as the dimensionless stream f u n c t i o n , has to be used. (2.2.2) and (2.2.3) f o r conjugate n a t u r a l - c o n v e c t i v e heat t r a n s f e r between c o n c e n t r i c spheres w i t h n e i t h e r of the parameters G, a, 3 and GO i n f i n i t e l y s m a l l or very l a r g e . The The i n f l u e n c e of a i s as f o l l o w s : s m a l l values I t i s proposed here to s o l v e equations (2.2.1), r e g u l a r p e r t u r b a t i o n expansions f o r the v a r i a b l e s ¥,0 and 0 are assumed to be oo 0 0 (2.3.1) 0 0 0 0 (R, n) (2.3.2) 1=0 m=0 0 0 0 0 0 (2.3.3) 1=0 m=0 18 I t may be shown t h a t f o r the case c o n s i d e r e d here, terms t r a n s c e n d e n t a l i n the expansion parameter G do not occur; they i n v a r i a b l y g ive r i s e t o homogeneous equations with homo-geneous boundary c o n d i t i o n s which w i l l y i e l d o n l y a t r i v i a l s o l u t i o n . Thus such terms would at most be a s s o c i a t e d with complex eigenmotions which are not c o n s i d e r e d here any f u r t h e r . S u b s t i t u t i n g equations (2.3.1), (2.3.2) and (2.3.3) i n t o the governing equations the f o l l o w i n g expansions f o r *F, 0 and 0 are obtained. ¥ = G\ (R,n) + G2¥° (R,n) + G2a^2 ( R , n) + , (2.3.4) 0 = 0° (R) + Ga0^ (R,n) + G 2a02 (R,n) + G 2 a 20 2 (R,n) , (2.3.5) 0 0° (R) + Ga0^ (R,n) + G 2a02 (R,n) + G 2 a 20 2 (R,n) (2.3.6) E q u a t i n g c o e f f i c i e n t s of equal powers of G Ja , the governing equations reduce to an i n f i n i t e s e t of uncoupled l i n e a r d i f f e r e n t i a l e q u a t i o n s , 0 (2.3.7) R2 1 3R 3n 9n 3R i a 4 ° (2.3.8) 9 3 - ( R x n , 0 o ) 3 ( R , n ) (2.3.9) (2.3.10) ! 3 ( ^ 1 / 6 0 ) R2 3 ( R , n ) (2.3.11) 2 E ¥ + i _ <JI R ' l - n 2 . 3 n R 3n 1 (2.3.12) (2.3.13) (2.3.14) a 3 ( ^ 2 / 0 0 ) R 2 3 ( R , n ) (2.3.15) (2.3.16) 1 3<Y° , oj) a 3 < ^ , 6°) R 3 ( R , n ) R 3 ( R , n ) (2.3.17) 20 The above s e t of equations i s s o l v e d i n sequence s u b j e c t to the f o l l o w i n g boundary c o n d i t i o n s : 0, a t R = 1 (2.3.18) -1 at R = 3 (2.3.19) 0^ must not have a s i n g u l a r i t y at the centre of the i n n e r sphere g r e a t e r than the zeroeth term, namely 1/R, f o r 1, m > 1 (2.3.20) .m a t R = 1; 1, m > 1 (2.3.21) 0' m 90 m 30 m 3R 3R 3 ^ j 3R 1 k 2 i 3 V i a t R = 3; 1, m _> 1 (2. 3.22) at R = 1; 1, m > 1 (2.3.23) at R = 1, 3; j , k > 1 (2.3.24) at n = ± 1 ; j , k > 1 (2.3.25) 21 2.4 S o l u t i o n The equations (2.3.7) and (2.3.8) are s o l v e d f i r s t . They are the conduction equations f o r the f l u i d assumed immobile, and f o r the s o l i d i n n e r sphere. The s o l u t i o n s are the temper-ature d i s t r i b u t i o n f o r pure conduction s u b j e c t t o boundary c o n d i t i o n s (2.3.18) and (2.3.19). 0°o = ( i - 1) x (^y) (2.4.2) Upon s u b s t i t u t i o n o f (2.4.2) i n t o (2.3.9), E 4 ¥ ° 1 = ( l - n 2 ) / R x (^j) , (2.4.3) T h i s i s a 1 c r e e p i n g flow' equation. A s o l u t i o n i s stream f u n c t i o n t ¥° = ( 1 - n 2 ) ( ~ + B XR + C^R 2 -• | i + D ] L R 4 ) X ( ^ T ) (2.4.4) which has to s a t i s f y the boundary c o n d i t i o n s (2.3.25). The constants of i n t e g r a t i o n A^, B-^  , and are determined by a p p l y i n g c o n d i t i o n s (2.3.24) and are given here i n c l o s e d form, 22 h = ( 6 9 - 4 6 8 + 6 6 ? - 4 B 6 + 3 5 ) / 8 A B 1 = ( - 3 B 9 + 8 6 8 - 5 3 7 - 5 B 5 + 8 3 4 - 3 3 3 ) / 8 A C± = ( 2 3 9 - 1 2 3 7 + 1 0 3 6 + 1 0 3 5 - 1 2 3 4 + 2 3 2 ) / 8 A D1 = ( 2 3 7 - 6 3 6 + 4 3 5 •+ 4 3 4 - 6 3 3 + 2 3 2 ) / 8 A where A = 4 3 8 - 9 3 7 + 1 0 3 5 - 9 3 3 + 4 3 2 , The next two equations (2.3.10), (2.3.11) are s o l v e d s i m u l t a n e o u s l y a f t e r s u b s t i t u t i n g (2.4.2) and (2.4.4) i n t o (2.3.11). They are V 2 0 ^ = 0 ( 2 . 4 . 5 ) V ^ G J = - 2 n ( - 4 + -4 + -4 - — + D 1)x(-^-) , ( 2 . 4 . 6 ) R R R 8R x 3-1 the s o l u t i o n s o f which are F 2 0 , = n (E,R + ) , ( 2 . 4 . 7 ) x . R 3 - 1 and 1 A F B D R 2 07 = n (—K + -4 + — + C + E R i — + — In R ) x ( - 2 - ) . 1 2R R R 1 x 2 12 3-1 ( 2 . 4 . 8 ) The boundary c o n d i t i o n s ( 2 . 3 . 2 0 ) to ( 2 . 3 . 2 3 ) are used to determine the constants o f i n t e g r a t i o n ; E^, E^, and F., . They are no l o n g e r given e x p l i c i t l y here but were c a l c u l a t e d d i r e c t l y from the boundary c o n d i t i o n s by computer. The next e q u a t i o n to be s o l v e d i s o b t a i n e d by s u b s t i t u t i n g ( 2 . 4 . 4 ) i n t o ( 2 . 3 . 1 2 ) , , o o 1 2 A , B . ( 1 2 B 2 + A,) 1 2 B , C , E % = n d - n 2 ) ( — ^ + 3 + • — H R R R B 2 - — + 1 2 B 1 D 1 + C1 - | + D I R 2 ) X ( ^ | T ) . ( 2 . 4 . 9 ) 2R The s o l u t i o n f o r i - s o = n ( i - n ) (-4- - - = - ^ + i r + .(12B: + A, ) R 2 R 2 2 R 2 24 1 1 ^1^1 2 a 3 1 4 + R + H a R - ^4 (12B 1D 1 + R a 5 D 1 r 6 B l 3 R 5 + G2 R + "nT + 6T R l n R " IW l n R) 8 2 X (F rT ) ' ( 2 . 4 . 1 0 ) which s a t i s f i e s boundary c o n d i t i o n s (2.3.25). The constants of i n t e g r a t i o n s G a, H a, I a and J a are determined by a p p l y i n g c o n d i t i o n s (2.3.24). The next h i g h e r - o r d e r term i n the expansion of ¥ can now be determined upon s u b s t i t u t i n g e q u a t i o n (2.4.8) i n t o 24 (2.3.13): — + —~ + C + R R 4 1 2 2 A 1 3 F 1 2 B 1 D 1 R ' R 6 2 (2.4.11) The s o l u t i o n i s , ^ = n ( i - n 2 ) (% + l5 - — R + + t£ R 3 - ^L-^ R 12 8 z 24 , , D,R6 B ^ 3 R 5 + -G, R + -= = — l n R l n R) ^ 288 15 840 B 2 ' (2.4.12) which s a t i s f i e s boundary c o n d i t i o n s (2.3.25). The constants of i n t e g r a t i o n G^/ H2/ I ^ an<3 a r e determined by a p p l y i n g c o n d i t i o n s (2.3.24). I t i s now p o s s i b l e to determine the h i g h e r - o r d e r temperature terms. The next two equations (2.3.14) and (2.3.15) are s o l v e d s i m u l t a n e o u s l y upon s u b s t i t u t i n g (2.4.2) and (2.4.10) i n t o (2.3.15). The equations are V 2 e ^ = 0 (2.4.13) 9 1 ••• 7 D 1 r 2 a (12B D + C ) v 2 0 ^ = ( 3 n 2 - D (-4^ - G ; R + ^ - i -• H2 B 1 C 1 ( 1 2 B i + A i > z2 A 1 B 1 J 2 ' R 2R 2 24R 3 R 4 2R 5 R 6 25 + _ m R _ _ m R j x ^ ) . (2.4.14) The s o l u t i o n s are and L a 9* = (3n 2 - 1) (K* R 2 + - | ) X ( ^ T ) , (2.4.15) R 02 = <3^ " { K5 R 2 + ^ - 2 W - Hr + 2W~)R3 B B.C. (12B 2 + A,) l a + (T=- + R + -4^=- + — — + 4 320' 12 IT4R 4 R 2 B i (12B 1D 1 + C 1) 2 6R T + 2 W R l n R + m R In R R 3 A l B l ) Q 3 — In R i - 4 l n R x ( i ) . (2.4.16) 3360 ~" 1 Q R 3 ( ~ l 8 - l The constants of i n t e g r a t i o n are , ^ , L 2 a n < ^ L 2 * I n v ^ e w of the h i g h e r - o r d e r s i n g u l a r i t y which would g i v e r i s e t o a t the c e n t r e of the i n n e r sphere, L a w i l l be s e t t o zero f o l l o w i n g c o n d i t i o n (2.3.20). The remaining unknown constants are determined u s i n g boundary c o n d i t i o n s (2.3.21) to (2.3.23). The l a s t terms of the temperature expansions "2 2 c a l c u l a t e d are and • T n e equations are (2.3.16) and (2.3.17). 26 0 , ( 2 . 4 . 1 7 ) • 2 ( A l " A i F i ) 3 A ! B 1 ( 3 n^ - i ) { - i 2 1 1 + 1 1 ( 2R 7 R 6 2 R 5 ( I 2 + B 1 F 1 - 2 A 1 C 1 > , 5 A 1 2 1 , g _ _ - ( A E + B ^ + 2 C 1 F r ) i T R 4 8 1 1 1 1 1 R 3 3 A D F . 3 B _ . ( 3 B C - — L I - - 1 ) 1 - ( H * - - J : + B - E . + 2 C ? . 2 4 • 2 8 1 1 1 f 4 D L F L ) J 7 B D 5 C , 3 E ( — ± - ± i ) - ( G ° + 3 C n D , - ) R 2 1 2 Z - 1 X 8 D± + 4 D l E l ) R 2 - 6 i l n R A l n R 1 2 R 1 0 9 D l 2 ) ft 3 M R l n R - r R l n R ( x <FT> 2 ( 4 A 2 6 A i F ] _ 2 A 1 B 1 ( 2 B]_ F]_ + A i C i ) n j ~1 6 3 4" ' R R R° R A i + ( 4 B C - 2 A D 1) L. + ( 4 B . E . - -A 6R J 1 1 4 R 1 1 3 + 4 C 1 + 4 D 1 F 1 ^ ( 3 B 1 D 1 - ^ i , + ( 4 C 1 D 1 - E 1 - ^ ) R + ( « D , + 1 0 D 1 E 1 ) R 2 27 + - In R - ^  In R + (^1) R 2 In R I x ( J L ) 3 3R 6 l P —-L + j 2 C l E ; L - 2D 2 R 3 + ^1 i n R | X ( J ^ - ) 3 . (2 . 4.18) The s o l u t i o n s are and e 2 = (3n 2 - i ) I nh2 R 2 + 4 + — i g . - (-? 3LL-) i I R 84R 5 6 R ( 3 I 2 + 5 B 1 F 1 " 5 A l C l ) A l i n 12R^ 16 1 1 1 1 1 6R 36 1 2 2 240 1 1 R 3 E + 2C 2 + 8 D l F l ) " ~ OG$ + 5 C l D l + - g ^ , 28 4 73D A B, A + §r ( - D E ) ~ - In R ~ l n R 448 6R 72R 17B B D 2C ( L) R in R - (-^-i - _ i ) R in R + ( \ ' ) R J I N R 720 2 45 6 0 4 8 0 , D1 . R4 . )• ' . 3 ,3 + j A l A 1 F 1 A 1 B 1 ( ) R l n R x ( ^ ) + j 5 T 3 252 \ p x < 15R 6R 9R J (2B nF + A,C ) 5B 1 X X 1 + (4B. E, - + 4C; + 4D,F.) £ 6 R 2 — i " i 6 - i - i r e 2 C 3 + — (B D + 2C..E,) + — (4C,D 1 - E + — ) 6 1 X 3 1 1 36 1 X X 36 4 D D 2R F l , + £i_ ( _ i + D..E..) (4B..C - 2A D -) 6 24- 1 1 15 1 1 X 4 3 B l C l 2 r 3 °1 4 + ( — ) R In R + ( — ) R l n R l n R + ( — ) R l n R 18 36 432 72 A i b Q 2 / B 3 - In R + (P° + •-=•) } x . ( 2 . 4 . 2 0 ) R 2 R B _ X 18  N 2 i s s e t to zero s i n c e a s i n g u l a r i t y o f t h i r d order i s not p e r m i s s i b l e at the centr e of the i n n e r sphere (2.3.20). With-'"b out l o s s of g e n e r a l i t y P 2 i s a l s o s e t to zero s i n c e from the temperature matching c o n d i t i o n s at the i n t e r f a c e o n l y one of the c o n s t a n t s , or P^ need be r e t a i n e d . The other *"b b b b b '"b remaining c o n s t a n t s , M 2, M 2, N 2, P 2 / Q 2 and Q 2 are determined by the a p p l i c a t i o n of boundary c o n d i t i o n s (2.3.21) t o (2.3.23). I t i s seen t h a t the f u n c t i o n s o f n appearing i n the e x p r e s s i o n s f o r 0™ or 0™ are the s e t of s p h e r i c a l harmonics, while i n Gegenbauer polynomials appear. 30 2 , 5 S o l u t i o n of Constant F l u x Problem In t h i s s e c t i o n , the case of f r e e c o n v e c t i v e heat t r a n s f e r between two c o n c e n t r i c spheres with a p r e s c r i b e d constant f l u x on the s u r f a c e of the s o l i d i n n e r sphere and an i s o t h e r m a l outer sphere i s c o n s i d e r e d . The heat f l u x , x 4TTR! 2 o - - k 3 T R! 1 on the s u r f a c e o f the i n n e r sphere i s s p e c i f i e d i n advance to be constant. Thus the conjugate e f f e c t a t the i n t e r f a c e between the s o l i d i n n e r sphere and the f l u i d i s l o s t . The governing e q u a t i o n s , (2.2.1) and (2.2.2) as given i n s e c t i o n 2.2 are a p p l i c a b l e f o r t h i s constant f l u x case. The energy f i e l d s of the f l u i d and the s o l i d i n n e r sphere are no longer coupled. Thus the temperature equation (2.2.3) f o r the i n n e r sphere i s not c o n s i d e r e d here. T h i s case i s s t i l l a multi-parameter one i n v o l v i n g parameters of G, o and 3. I t has one parameter, namely OJ , l e s s than the conjugate case. However, the l i m i t a t i o n s on the ranges o f a and 3 o u t l i n e d i n s e c t i o n 2.3 s t i l l h o l d . The n o n - l i n e a r coupled governing equations (2.2.1) and (2.2.2) f o r t h i s constant f l u x case are s o l v e d by a p e r t u r b a t i o n scheme i n the s i m i l a r manner as f o r the con-jugate case. The expansions f o r ¥ and 0 are those of (2.3.4) and (2.3.5) r e s p e c t i v e l y . The s i m p l i f i e d boundary c o n d i t i o n s s u b j e c t to which the governing equations have t o be s o l v e d w i l l be 31 90, 9R = 1 a t R = (2.5.1) 90 ra 9R = 0 at R = 1; 1, m > 1 (2.5.2) 1 a t R = B (2.5.3) 0 a t R = 3; 1, m > 1 (2.5.4) = 0 at R = 1, 6; j , • k >_ 1 (2.5.5) K 2 i 3 ¥ i ¥ . = - (1-n ) 2 — = 0 a t n = ± l ; j , k > l . (2.5.6) 3 3n The boundary c o n d i t i o n (2.5.1) s t i p u l a t e s t h a t the f l u x on the s u r f a c e of t h e ' i n n e r sphere remains constant i . e . u n i t y a f t e r n o r m a l i z a t i o n . The constant f l u x c o n d i t i o n i s f u l f i l l e d by the zeroeth order term of the temperature, 0, expansion. T h e r e f o r e no fhigher order terms i n 0 should g i v e a c o n t r i b u t i o n to the f l u x on the s u r f a c e of the i n n e r sphere i . e . boundary c o n d i t i o n (2.5.2). The boundary c o n d i t i o n (2.5.3) i s a normal-i z a t i o n c o n d i t i o n of the temperature at the outer sphere. (2.5.4) s t a t e s t h a t the h i g h e r order terms i n the 0 expansion are zero at the outer envelope. The boundary c o n d i t i o n s (2.5.5) and (2.5.6) have the same meaning as p r e v i o u s l y . 0 m j 9R The s e t of uncoupled l i n e a r d i f f e r e n t i a l equations g i v e n i n s e c t i o n 2.3 (with the e x c e p t i o n t h a t the equations f o r 0 are not c o n s i d e r e d here) are s o l v e d s u b j e c t to the o boundary c o n d i t i o n s given above. The s o l u t i o n f o r 0 O i s o° - 1 (B+D ~~ R ~ 6 which f u l f i l l e d the boundary c o n d i t i o n s (2.5.1) and (2.5.3). The s o l u t i o n s f o r a l l the h i g h e r order terms i n the 0 and ¥ expansions are those g i v e n i n s e c t i o n 2.4 except the R B 2 R 3 constant f a c t o r s (g^y) / (3^1") a n c ^ ^8^T^ w i l l not appear i n the s o l u t i o n s . The constants of i n t e g r a t i o n s i n the s o l u t i o n s are determined d i r e c t l y u s i n g the boundary c o n d i t i o n s (2.5.1) to (2.5.6) i n the computer. 33 3. EVALUATION OF ANALYTICAL RESULTS In the p r e v i o u s chapter, the a n a l y t i c a l s o l u t i o n s f o r the conjugate heat t r a n s f e r between c o n c e n t r i c spheres has been c o n s i d e r e d . Su b ject t o the l i m i t a t i o n s o u t l i n e d b e f o r e , these s o l u t i o n s may be assumed to be v a l i d f o r v a r i o u s combinations of the f o u r c h a r a c t e r i s t i c parameters G, a , w, and 6 . I t i s a n t i c i p a t e d t h at there w i l l be l i m i t i n g v a l u e s f o r each of the parameters beyond which the p e r t u r b a t i o n expansions w i l l no longer converge. Other f a c t o r s which may l i m i t the range of a p p l i c a b i l i t y of the s o l u t i o n s are as f o l l o w s : the steady laminar a x i s y m m e t r i c a l flow assumed i n s e c t i o n 2.1 may become u n s t a b l e l e a d i n g to a flow of more complicated nature and to t u r b u l e n c e . The c r i t e r i o n used here i n determining approximately the r a d i u s of convergence of the p e r t u r b a t i o n expansions i s o u t l i n e d i n s e c t i o n 3.1. The l i m i t a t i o n s on the a p p l i c a b i l i t y o f the s o l u t i o n s as mentioned above w i l l a l s o be d i s c u s s e d . Contours of s t r e a m l i n e s , v o r t i c i t y l i n e s and isotherms p l o t t e d with the a i d of the computer were obt a i n e d . The r a d i a l and t a n g e n t i a l v e l o c i t y components o f the f l u i d i n the gap between the two c o n c e n t r i c spheres were c a l c u l a t e d by d i f f e r e n t i a t i n g the stream f u n c t i o n a n a l y t i c a l l y and computing the r e s u l t s . F i n a l l y , the temperature d i s t r i b u t i o n s and the heat t r a n s f e r r a t e s (Nusselt Numbers) are g i v e n . 3•1 Range of V a l i d i t y o f S o l u t i o n s The p e r t u r b a t i o n expansions f o r the v a r i a b l e s o f V, 0 and 0 are obtained up to the second order terms. The s e r i e s expansions f o r f , 0 and 0 are equations (2.3.4) through (2.3.6). The e r r o r made by t r u n c a t i n g such asymptotic expansions i s of the order of the f i r s t term n e g l e c t e d . T h i s term w i l l tend t o zero r a p i d l y as the parameter 6 i s reduced t o zero. In the nume r i c a l e v a l u a t i o n s f o r f i n i t e , non zero G , . a l l the terms f o r V, 0 and 0 d e r i v e d were r e t a i n e d f o r a c t u a l computation. Due to the complexity of h i g h e r order.terms no e x t r a term was d e r i v e d t o check convergence f o r each of the expansions. Thus the numerical convergence o f the expansions cannot be guaranteed. An a l t e r n a t i v e c r i t e r i o n i s proposed here to determine the p r a c t i c a l upper bound of the convergence o f the s e r i e s expansions as f o l l o w s : For given values of G, a, to and 3 and a t a ' t y p i c a l ' l o c a t i o n the value o f any h i g h e r -order terms f o r 0 and 0 con s i d e r e d must not be g r e a t e r than the p r e v i o u s term. In a d d i t i o n the sum of any two or more h i g h e r - o r d e r terms must not have a value g r e a t e r than o o the p r e v i o u s term or the fundamental terms, i . e . ^ , 0 O and ~ o 0 o . The c r i t e r i o n o f convergence d e f i n e d here i s d i f f e r -ent from t h a t used by Mack e t al_. [9, 10], These workers c o n s i d e r e d t h a t as long as the maximum magnitude o f any 35 higher order term i n e i t h e r s e r i e s f o r ¥ and 0 d i d not exceed the maximum value of the appropriate lowest order term at a ' t y p i c a l ' l o c a t i o n , convergence could be assumed. On t h i s b a s i s , they found from t h e i r a n a l y t i c a l s o l u t i o n s at low a that i t was p o s s i b l e t o have double c e l l s i n the flow f i e l d . Their s e r i e s f o r ¥ and 0 were obtained by expanding i n ascending powers of Rayleigh number r a t h e r than i n the Grashof number. Thus the c o e f f i c i e n t expressions f o r the various terms i n the s e r i e s are dependent on a : they in c l u d e constants w i t h c x . Double c e l l s cannot occur on the b a s i s on which the rad i u s of convergence i s define d i n t h i s t h e s i s . Now, the exis t e n c e of a secondary c e l l i n a d d i t i o n t o the primary c e l l had been observed experimentally by Bishop et a l . [7] f o r a Rayleigh number (as define d i n t h i s t h e s i s ) of approxi-3 mately 45 x 10 at a value of B of 1.19. However, the range of Rayleigh numbers (for which the s e r i e s w i l l 'converge') considered both by Mack et al_. and i n the present t h e s i s i s below that quoted by Bishop f o r the occurrence of a secondary c e l l . I f the, ra d i u s of convergence proposed by Mack had been adopted here a secondary c e l l would indeed e x i s t f o r a value of Ra large enough (but not n e c e s s a r i l y at low a or B). An example of t h i s i s i l l u s t r a t e d i n s e c t i o n 3.2. There-fore i t can be concluded t h a t a numerical 'demonstration of double c e l l s i s r e l a t e d to the f a c t t h a t the s e r i e s i s no longer convergent i n the work c i t e d . 36 For both the conjugate and constant f l u x cases con-s i d e r e d , numerical i n v e s t i g a t i o n s show t h a t the upper bound on G or Ra decreases m o n o t o n i c a l l y w i t h a i n c r e a s i n g from 0.01 to 100 and/or 3 i n c r e a s i n g from 1.15 to 2.5. T h i s i s shown i n F i g u r e 3.1.1. The upper bound on G i s lower i n the c o n s t a n t f l u x case than i n the conjugate case f o r f i x e d v alues of a and 3. The lowest value of O J ( a p p l i c a b l e o n l y f o r the conjugate case) f o r which the upper bound of G i s s u f f i c -i e n t l y h i g h t o be of i n t e r e s t i s 3. I f the r a t i o of thermal c o n d u c t i v i t y us should v a n i s h then the boundary c o n d i t i o n s i n the temperature f i e l d ( i . e . 2.3.21 and 2.3.23) can no longer be f u l f i l l e d s i m u l t a n e o u s l y . T h i s i n e f f e c t c o r r e s -ponds to the constant f l u x case and the 'conjugateness' of the problem i s l o s t . The other p o s s i b l e l i m i t a t i o n s t o the range of a p p l i c a b i l i t y of the s o l u t i o n s are the n o n - c o n s i d e r a t i o n of compressive work and v i s c o u s d i s s i p a t i v e e f f e c t s ; more over, the flow may not be steady laminar and a x i s y m m e t r i c a l . In Appendix I, Tables V and VI show the r a t i o s of the magnitude of the v i s c o u s d i s s i p a t i o n term to e i t h e r the conduction term (e) or the c o n v e c t i o n term (A) i n the energy equation (2.2.2) of the f l u i d -in the gap between the two c o n c e n t r i c spheres at both r a d i a l and angular p o s i t i o n s . I t i s seen from the t a b l e s t h a t the r a t i o o f the v i s c o u s d i s s i p a t i o n term t o e i t h e r term i n the energy equation i s at l e a s t of the order — 8 of 10 . Hence n e g l e c t i n g v i s c o u s h e a t i n g as assumed i n 1.0 1.5 3 2.0 2.5 F i g u r e 3.1.1 Approximate Upper Bound of G f o r Var i o u s a/3 38 s e c t i o n 2.2 i s j u s t i f i e d . I t w i l l be shown l a t e r through the e xperimental i n v e s t i g a t i o n t h a t the flow i n the gap between the two c o n c e n t r i c spheres i s steady laminar and axisymmetrical about the v e r t i c a l a x i s . 3.2 S t reamlines In view of the q u a l i t a t i v e s i m i l a r i t y f o r a l l combin-a t i o n s of G, a, to and 3 f o r which the expansions are convergent, contour p l o t s of s t r e a m l i n e s , i s o v o r t i c i t y l i n e s and isotherms are given f o r a f i x e d value of Ra=720 f o r both conjugate and the constant f l u x cases. The s t r e a m l i n e s as g i v e n by ¥ equation (2.3.4) f o r a=.72, G=1000, 3=2.0 and OJ=10 are p r e -sented i n F i g u r e 3.2.1. A s i n g l e c e l l o f the ' c r e s c e n t -eddy' type (observed p r e v i o u s l y [10] f o r the case of an i s o -thermal i n n e r sphere) i s o b t a i n e d . The f l u i d flows upward i n the immediate v i c i n i t y of the i n n e r sphere and downward along the outer sphere. The flow i n the f l u i d space i s ' t o r o i d a l ' with r o t a t i o n a l symmetry. There i s a j e t l i k e flow p o l a r i z -a t i o n above the top of the i n n e r sphere. The lower r e g i o n of the f l u i d space acts l i k e a r e s e r v o i r w i t h f l u i d withdrawn from i t i n the v i c i n i t y o f the i n n e r sphere and f l u i d f l o w i n g i n t o i t from the boundary o f the o u t e r sphere. The c e n t e r o f the c i r c u l a t o r y motion, where *F has a maximum v a l u e , i s i n the upper h a l f of the flow r e g i o n a t 6 - 82° and s l i g h t l y p a s t the mid p o i n t of the annular space. 39 S i m i l a r s t r e a m l i n e c o n f i g u r a t i o n s are o b t a i n e d w i t h values o f Ra=720 ,. 6=2.0 f o r u>=10^ (approximately i s o t h e r m a l i n n e r sphere) and the constant f l u x case as shown i n F i g u r e s 3.2.2, 3.2.3 r e s p e c t i v e l y . F i g u r e 3.2.4 shows the st r e a m l i n e s f o r a=.72, G=1000, OJ=10 and 6=1.15 with the annular space between the two c o n c e n t r i c spheres p l o t t e d here s t r e t c h e d i n t o a r e c t a n g u l a r form f o r c l a r i t y . I t i s seen t h a t the center of the s t r e a m l i n e s i s a t 8=90°. As 6 tends to u n i t y , the flow i s e s s e n t i a l l y 'creeping f l o w 1 i n a narrow o gap. The c r e e p i n g flow s o l u t i o n ( i . e . f^) p l a c e s the cente r of the s t r e a m l i n e s at 8=90°. I n c r e a s i n g the value of a t o 10 w i t h the values of Ra=720, 6=2.0 f i x e d , f o r the conjugate case (with oo=10 , 10"^) and f o r the constant f l u x case, the s t r e a m l i n e c o n f i g u r a t i o n s remain e s s e n t i a l l y unchanged. The values o f the s t r e a m l i n e contours are lower than the p r e v i o u s values as the buoyancy e f f e c t s are r e l a t i v e l y reduced (G decreases when Ra i s maintained constant w h i l e a i s i n c r e a s e d ) , F i g u r e 3.2.5, The center o f the eddy c r o s s s e c t i o n remains approximately i n the same p o s i t i o n as bef o r e f o r a l l cases. The upward and outward displacements o f the cente r depends on the value of Ra when 6 i s l a r g e r than 1.15 ( i . e . h i g h e r order terms i n W then become im p o r t a n t ) . A l s o the angular p o s i t i o n of the center of the s t r e a m l i n e s depends on the value of OJ i . e . OJ=10 , 8 - 82°; and f o r the constant f l u x case 8 = 75°. 40 As s t a t e d p r e v i o u s l y i n s e c t i o n 3.1, f o r convergence of the asymptotic expansion i n ¥ (with a l l the terms a c t u a l l y o d e r i v e d used i n the computations) the fundamental term must always be the predominant term. I f m u l t i p l e c e l l s should occur i n the annular space, the adjacent c e l l s would have values of the stream f u n c t i o n of op p o s i t e a l g e b r a i c s i g n However, the fundamental term i s always p o s i t i v e throughout the flow f i e l d . Thus i n order t o have m u l t i p l e c e l l s the h i g h e r order terms would have t o become larger than the fundamental term. F i g u r e 3.2.6 shows the presence of a secondary c e l l a t the lower r e g i o n of the gap f o r the conjugate case w i t h OJ=10, a=.72, G=2100 and 6=2.0. The secondary c e l l does, not extend across the gap and i s much weaker than the primary c e l l . I t may be concluded, as e x p l a i n e d above, t h a t here the expansion f o r ¥ i s no longer convergent. Moreover, Bishop e t a l . [7] observed e x p e r i m e n t a l l y t h a t the f i r s t secondary c e l l was formed near the top of the i n n e r sphere. Hence, i n the present a n a l y s i s the occurrence of a secondary c e l l cannot be a genuine f e a t u r e . 42 45 F i g u r e 3.2.5 Streamlines f o r Conjugate Case, co=10, 3=2.0, a=10, G=72 3.3 V e l o c i t y D i s t r i b u t i o n s I t was assumed p r e v i o u s l y t h a t the c o n v e c t i v e flow p a t t e r n i s ax i s y m m e t r i c a l . Thus there i s no l a t i t u d i n a l v e l o c i t y , V^. The r a d i a l and t a n g e n t i a l v e l o c i t y components of the f l u i d i n the R- and 8- d i r e c t i o n s r e s p e c t i v e l y are obt a i n e d a n a l y t i c a l l y by d i f f e r e n t i a t i n g the stream f u n c t i o n , e q u a t i o n (2.3.4): V. 1 3¥ R R s i n e 38 1 3¥ R Z 3n (3.3.1) ( A l B l R G x 2n _ ± . + _ ± + C , - - + D I R R 1 8 i*2) x <^r + G 2 x (3n2 - 1) G a R 3 + H a R + ~% +' % * R R A-i B. -j 9 B C. 2 + ( 1 2 B £ + A 1 ) + - i - i - — (12B 1D 1 + C±) 2R 24R 2 24 D R - i B i ) R 2 + _ i £_ R l n R + —— R l n R > x (^-) 144 560 60 ) P + G 2 a x (3n2-D I b J b R 3 + H* R + ^ + 2 K K 48 A F C R 2 D R B + - ± i + —— R In R 12R 8 24 288 15 3 \ 2 - — l n R x (J^r) (3.3.1a) 840 P V f i = _ _ L _ l i = - ik^Lll 3 1 , (3.3.2) RsinO 3R R 3R = ( l - n ) ? ,3 B, ± - 2C, + R" R H - 4 D l R 2 8 X + G x n - 5G: R 3 - 3H a R + 2 J 2 R^~ A 1 B 1 2R 3 (12BJ + A x) 2 4R - B 1 C 1 + R (12B 1D 1 + C±) -D^R 24 R 3 B 1 R ) R + 2 — (1 + 5 l n R) ±— (1 + 3 l n R) x (^fy) 560 60 + G 2a x n \ - 5G^ R 3 - 3H^ R + — £ + — I R 12R P C R D R 4 B R — + — + - ± - (1 + 3 In R) 4 . 6 48 15 49 + R-840 (1 + 5 l n R) x (giiy) (3.3.2a) The r a d i a l and t a n g e n t i a l v e l o c i t i e s were e v a l u a t e d n u m e r i c a l l y u s i n g the e x p r e s s i o n s (3.3.1a) and (3.3.2a). For GJ=10, 6=2.0, G=1000, a=.72; the r a d i a l v e l o c i t y V R i s p l o t t e d a g a i n s t r a d i u s R f o r v a r i o u s angular p o s i t i o n s 6 as shown i n F i g u r e 3.3.1. At a given r a d i a l p o s i t i o n the i magnitude of the r a d i a l v e l o c i t y i n c r e a s e s with d e c r e a s i n g 0 . Each p r o f i l e has e i t h e r a maximum or a minimum value o c c u r i n g at an approximate r a d i a l p o s i t i o n between 1.4 j< R <^  1.55. For 0°<6<.60°, the f l u i d has a r a d i a l outflow and a r a d i a l i n f l o w when 90° < 6 < 180°. I t i s seen t h a t the r a d i a l outflow a t 0=0° i s about twice t h a t o f the r a d i a l i n f l o w at 0=180°. F i n a l l y , the r a d i a l v e l o c i t y g r a d i e n t near the i n n e r sphere's s u r f a c e ( i . e . R=1.0) i s h i g h e r than t h a t near the outer sphere's s u r f a c e ( i . e . R=2.0). For the same values of O J , 6/ G and a as b e f o r e , the p r o f i l e s of the t a n g e n t i a l v e l o c i t y , V Q, a g a i n s t the r a d i a l p o s i t i o n , R, f o r v a r i o u s 0 are shown i n F i g u r e 3.3.2. Each p r o f i l e shows a maximum value f o r upward flow and a minimum value of downward flow. The upward flow speed near the i n n e r sphere i s h i g h e r than the downward flow speed near the outer sphere. For 0=90°, the t a n g e n t i a l v e l o c i t y , V Q, i s zero at R=1.5 i n d i c a t i n g t h a t the upward and the downward flow are approximately e q u a l . However, the upward v e l o c i t y i s s t i l l g r e a t e r then the downward v e l o c i t y . The c r o s s o v e r p o i n t from upward flow t o downward flow occurs i n the r e g i o n 1.4 <_ R <_ 1.55. T h i s c r o s s o v e r p o i n t moves towards the i n n e r sphere as 0 i n c r e a s e s . v e l o c i t y p r o f i l e s are s i m i l a r f o r other combinations of to, 3, G and a . 3.4 V o r t i c i t y Contours As mentioned i n s e c t i o n 2.3, the c o n v e c t i v e motion of the f l u i d due t o buoyancy f o r c e s i s the only source of v o r t i c i t y a v a i l a b l e i n the gap between the c o n c e n t r i c spheres. With r o t a t i o n a l symmetry, the v o r t i c i t y v e c t o r i n the p o s i t i v e (J) - d i r e c t i o n i s given by: The g e n e r a l f e a t u r e s of the r a d i a l and t a n g e n t i a l ? = [ V x V ] 9R 9V 6 R 2 I = ( l - n )2 G 2B •T- ~ 1 0 D l R + I x (3^T ) L R 53 1 2 2B,C, R + - ~ (12B^ + A ) + — - + - (12B,D + C.) 4RZ 1 1 R 4 1 1 1 D 1 R R 2 "1 - ± — + £ — (9 + 14 In R) + c x 560 B. 12 K O 6 I 0 A T F 1 C 1 R D 1 r 3 (_ 1 4 G ^ R 2 + - i - i + JL + - i L _ R 2R 2R 4 12 B R -+ _ ± + (9 + 14 i n R) ) 3 840 (3.4.1) The o v e r a l l f e a t u r e s o f the v o r t i c i t y contours shown i n F i g u r e s 3.4.1 through 3.4.4 are s i m i l a r . The v o r t i c i t y f i e l d may be c o n v e n i e n t l y d i v i d e d i n t o t h r e e r e g i o n s which may vary i n r a d i a l e x t e n t . The m e r i d i o n a l length of the i s o v o r t i c i t y l i n e s and t h e i r r a d i a l p o s i t i o n depend upon the fou r parameters G, o, O J and 6. These three r e g i o n s are d e f i n e d as: the immediate v i c i n i t y o f the i n n e r sphere where the v o r t i c i t y i s generated; the c e n t r a l v o r t e x r e g i o n where the v o r t i c i t y from the i n n e r sphere i s t r a n s f e r r e d by d i f f u s i o n and c o n v e c t i o n ; the boundary at the outer sphere where v o r t i c i t y o f the f l u i d i s d i s s i p a t e d . Note t h a t t h i s arrange-ment i n t o three r e g i o n s resembles q u a l i t a t i v e l y t h a t d e s c r i b e d 54 by B a t c h e l o r f o r f r e e c o n v e c t i o n between p a r a l l e l plane boundaries. In the immediate v i c i n i t y of the i n n e r sphere the v o r t i c i t y of the f l u i d i s generated by the c o n v e c t i v e motion se t up by the heated i n n e r sphere. Thus the i s o v o r t i c i t y l i n e s have the h i g h e s t magnitude and i n angular e x t e n t here i n comparison to those i n the other two r e g i o n s . The f l u i d p a r t i c l e s are r o t a t i n g i n the clockwise d i r e c t i o n . The v o r t i c i t y v e c t o r d e f i n e d by equation (3.4.1) i s at r i g h t angles and i n t o the plane of the paper i n each of the f i g u r e s f o r t h i s r e g i o n . The c e n t r a l v o r t e x r e g i o n i s separated from the o t h e r two r e g i o n s by the two i s o v o r t i c i t y l i n e s of value zero. The most notable f e a t u r e i n t h i s r e g i o n i s the ' t o r o i d a l -shaped' s u r f a c e s of the v o r t i c i t y sheets. With r o t a t i o n a l symmetry, each of these t o r i extends around the whole annular space between the c o n c e n t r i c spheres. They can be c o n s i d e r e d as 'vortex tubes' and the whole of the c e n t r a l r e g i o n as a 'vortex r i n g ' . The centre l i n e of the vortex i s i n the upper r e g i o n of the flow f i e l d and q u i t e near to the mid-point of the gap. T h i s p o s i t i o n i s d i f f e r e n t from t h a t of the c e n t r e o f the s t r e a m l i n e s ( s e c t i o n 3.2), f o r each of the corresponding cases. Near the centre of the v o r t e x i s a r e g i o n of nearly constant v o r t i c i t y e.g. F i g u r e 3.4.3. The magnitude of the v o r t i c i t y i s the h i g h e s t i n t h i s r e g i o n . The sense of r o t a t i o n of the f l u i d i s o p p o s i t e ( i . e . counter-56 F i g u r e 3.4.3 V o r t i c i t y Contours, Constant F l u x Case, 6=2.0, o=.72, G=10 3 ' F i g u r e 3.4.4 V o r t i c i t y Contours, Conjugate Case, OJ=10 , 3=2.0, a=10, G=72 59 clockwise) to t h a t i n the r e g i o n at the i n n e r sphere s u r f a c e as the v o r t i c i t y i s t r a n s f e r r e d from t h e r e . The v o r t i c i t y i s t r a n s f e r r e d t o and d i s s i p a t e d at the boundary r e g i o n o f the outer sphere. Hence the v o r t i c i t y v e c t o r i s i n the same d i r e c t i o n as t h a t i n the immediate v i c i n i t y o f the inn e r sphere. The f e a t u r e s o f the v o r t i c i t y contours are th e r e q u a l i t a t i v e l y s i m i l a r t o those i n the r e g i o n near the i n n e r sphere. But f o r each corresponding magnitude of the v o r t i c i t y the i n d i v i d u a l contour does not have the same angular e x t e n t i n t h i s r e g i o n , a t the out e r sphere. As i n the s t r e a m l i n e c o n f i g u r a t i o n , the f e a t u r e s o f the v o r t i c i t y contours change l i t t l e w i t h a change of a, p r o v i d e d Ra., 8 and O J be kept constant. While the p o s i t i o n of the ce n t r e o f the vo r t e x does not change with a, i t i s a f f e c t e d by Ra., 8 and O J . 3.5 Temperature D i s t r i b u t i o n and Contours The temperature d i s t r i b u t i o n of the f l u i d i n the annular space between the c o n c e n t r i c spheres i s given by eq u a t i o n (2.3.5). For the conjugate case? equation (2.3.6) y i e l d s the temperature d i s t r i b u t i o n i n the s o l i d i n n e r sphere with a s i n g l e heat source at the c e n t r e . C a l c u l a t i o n of the temperature i n s i d e the i n n e r sphere i s i r r e l e v a n t f o r both the constant f l u x case and when O J i s very l a r g e 60 15 e.g. OJ=10 ( i . e . an i s o t h e r m a l i n n e r sphere) even i n the conjugate case. I t i s seen from the F i g u r e s 3.5.1 and 3.5.3 t h a t f o r .72 <_ o <_ 10 and with Ra, 3 and O J f i x e d , a change i n a does not a l t e r the temperature p r o f i l e s or the contours, F i g u r e s 3.5.6 and 3.5.7. T h i s i n d i c a t e s t h a t the temperature d i s t r i b u t i o n and the heat t r a n s f e r r a t e s are q u i t e independent of a, a t l e a s t i n t h a t range. A l t e r n a t i v e l y expressed, the influence of a i s a higher order effect on both the temperature d i s t r i b u t i o n and heat t r a n s f e r r a t e s . A proof o f t h i s i s g i v e n l a t e r i n s e c t i o n 3.6. The g e n e r a l f e a t u r e s of the temperature p r o f i l e s are s i m i l a r f o r the conjugate and.constant f l u x cases, F i g u r e s 3.5.1 to 3.5.5. The temperature d i s t r i b u t i o n due t o pure o conduction ( i . e . 0 O) i s given by the dashed l i n e f o r comparison. For the conjugate case, a t any p a r t i c u l a r r a d i a l p o s i t i o n 1 _< R <_ 3 r the temperature (as seen from the f i g u r e s ) i n c r e a s e s with d e c r e a s i n g 9. The p r o f i l e s f o r 0=0° and 60° l i e below the curve f o r pure conduction w h i l e the p r o f i l e s f o r 90° and 150° l i e above t h i s curve. The p r o f i l e s f o r 120° and 150° are r e l a t i v e l y c l o s e together compared to any two p r o f i l e s . T h i s i n d i c a t e s t h a t the l o c a l heat t r a n s f e r r a t e s f o r both the i n n e r sphere and the outer sphere i s q u i t e independent of 0 f o r 120° 1 0 f . 180°. In t h e i r experimental work, Scanlan e t a l . [21] p o s t u l a t e d t h a t f o r the occurrence of m u l t i c e l l u l a r flow, a r e v e r s a l of the o r d e r i n g of the temperature p r o f i l e s would occur (see t h e i r F i g u r e s 8 to 10). D i r e c t comparison of t h e i r r e s u l t s with those given here i s perhaps not meaningful s i n c e t h e i r range of Ra used i n t h e i r experiments was c o n s i d e r -ably h i g h e r and w i t h a s m a l l e r 3 than t h a t c o n s i d e r e d here. However, i t i s seen t h a t the temperature p r o f i l e s i l l u s t r a t e d here, there i s no r e v e r s a l of t h e i r o r d e r . Hence the e x i s t e n c e of m u l t i c e l l u l a r flow does not occur here i n accord with the d i s c u s s i o n i n s e c t i o n 3.2 above. The temperature on the s u r f a c e of the i n n e r sphere v a r i e s with angular p o s i t i o n except i n the conjugate case where O J=10 X^ ( i . e . an i s o t h e r m a l i n n e r sphere). The temperature v a r i a t i o n on the i n n e r sphere's s u r f a c e i s most pronounced f o r the constant f l u x case. For i t i s i n t h i s case t h a t i t i s assumed the heat f l u x on the s u r f a c e of the i n n e r sphere i s uniform even though c o n v e c t i o n i s t a k i n g p l a c e i n the f l u i d . On the other hand f o r the conjugate case, there i s a r e d i s t r i -b u t i o n of heat f l u x i n s i d e the i n n e r sphere when there i s c o n v e c t i o n i n the f l u i d . A l s o the temperature of the f l u i d at the i n t e r f a c e must be matched with i n n e r sphere temperature i . e . at R=l, F i g u r e s 3.5.1, 3.5.2 and 3.5.6. The temperature contours w i l l be e s s e n t i a l l y those . of c o n c e n t r i c c i r c l e s f o r pure conduction of heat from the s i n g l e source, at the c e n t r e of the c o n c e n t r i c spheres. The r e s u l t i n g c o n v e c t i v e motion of the f l u i d d i s p l a c e s the isotherms upwards, F i g u r e s 3.5.6 to 3.5.9. T h i s i s due to the r e l a t i v e l y c o l d f l u i d from the bottom being heated up by both conduction and c o n v e c t i o n as i t r i s e s t o the top of the F i g u r e 3.5.6 Isotherms, Conjugate Case, w=10, 3=2.0, a=.72, G=10 3 6 8 gap. The shape o f the isotherms i s s t i l l s i m i l a r t o c i r c l e s s i n c e pure conduction i s predominant. In s e c t i o n 2.2, i t i s shown t h a t a t the beg i n n i n g of the s o l u t i o n of the energy equation the i n n e r sphere's s u r f a c e i s i s o t h e r m a l and/or of const a n t f l u x thus f o r pure conduction alone, the zero isotherm c o i n c i d e s with the s u r f a c e of the i n n e r sphere. When con v e c t i v e motion takes p l a c e , the p o r t i o n of the zero i s o -therm approximately i n the upper h a l f r e g i o n i s d i s p l a c e d upwards i n t o the annular space. The lower p o r t i o n i s d i s p l a c e d i n t o the s o l i d sphere where there i s a r e d i s t r i b u t i o n o f f l u x , F i g u r e 3.5.6. When w i s very l a r g e the s u r f a c e o f the i n n e r sphere can be con s i d e r e d t o be i s o t h e r m a l even though there i s a r e d i s t r i b u t i o n of heat f l u x . The temperature contours are ' d i s t o r t e d ' from t h a t of c o n c e n t r i c c i r c l e s around the r e g i o n f o r 6=120° approximately e.g. F i g u r e 3.5.9. T h i s shows t h a t i n t h i s r e g i o n v i g o r o u s c o n v e c t i o n i s t a k i n g p l a c e i n the immediate v i c i n i t y of the i n n e r sphere. 3.6 Heat T r a n s f e r Rates The l o c a l N u s s e l t numbers KT /o\ (3-D 30 Nu. (8) = - — x — i 8 3R R=l 72 and Nu o(0) (6-1) Q2 80 — — - x 6 x — 6 8R R=6 are d e f i n e d f o r the i n n e r sphere and out e r sphere r e s p e c t i v e l y . The N u s s e l t numbers d e f i n e d here have the r a d i u s o f the i n n e r sphere as the common len g t h s c a l e . The o v e r a l l heat t r a n s f e r r a t e i s then o b t a i n e d by i n t e g r a t i n g the l o c a l N u s s e l t numbers over the e n t i r e s u r f a c e o f the i n n e r sphere or the outer sphere. The o v e r a l l N u s s e l t numbers f o r the i n n e r sphere and outer sphere are gi v e n r e s p e c t i v e l y by: 1 26 J 8R o s i n 6 d9 (3.6.1) R=l Nu~ = - - f c i i f 3 2 — ° 26 J 8R s i n 8 d6 (3.6.2) R=6 I t i s seen t h a t Nu^ must be equal to N U q . E v a l u a t i n g s e p a r a t e l y both Nu^ q , w i l l serve as a check on the value of the o v e r a l l N u s s e l t number. The N u s s e l t numbers, Nu^ Q , aire dependent upon 0, Ra, O J and 6 and as shown i n F i g u r e s 3.6.1 t o 3.6.4. The curves o f Nu^ Q show t h a t i n comparison t o the pure conduction value 73 (shown as a dashed l i n e i n the f i g u r e s ) the heat t r a n s f e r r a t e has been i n c r e a s e d f o r about the lower t w o - t h i r d s of the i n n e r sphere and n e a r l y h a l f of the upper outer sphere and reduced elsewhere. The g r e a t e s t d e v i a t i o n of the heat t r a n s f e r r a t e from the pure conduction value i s a t 0=0° i n each case. There i s an angular p o s i t i o n i n the upper h a l f of the spheres where the heat t r a n s f e r r a t e from both the i n n e r and outer spheres i s the same. T h i s angular p o s i t i o n moves towards the top o f the spheres i . e . 0=0° as OJ decreases and as Ra i n c r e a s e s , F i g u r e s 3.6.1, 3.6.3 and 3.6.4. For 120° 0 <_ 180° and Ra=720, Nu. do not vary much with 0 as i n f e r r e d i n s e c t i o n i ,o -* 3.5 from the temperature p r o f i l e s f o r 120° and 150°. However, i n t h i s r e g i o n there e x i s t s a maximum value f o r Nu^ and a minimum value f o r N U q . The l o c a t i o n s of these t u r n i n g p o i n t s depend on the values o f Ra, a and the conjugateness of the i n n e r sphere i . e . O J . F i g u r e 3.6.5 shows the t u r n i n g p o i n t s at 0-130° and 125° f o r and N U q r e s p e c t i v e l y w i t h Ra-1000, OJ=10 . As i n f e r r e d i n s e c t i o n 3.5 from the temperature con-tours i n t h i s r e g i o n near the i n n e r sphere, there i s vigorous c o n v e c t i v e motion t a k i n g p l a c e i n the f l u i d . Thus t h i s i s the l o c a t i o n where the heat t r a n s f e r r a t e (or f l u x ) from the i n n e r sphere i s a maximum. As mentioned b e f o r e i n s e c t i o n 3.3, the bottom of the annular space between the c o n c e n t r i c spheres a c t s as i f i t were a r e s e r v o i r . The hot f l u i d i s cooled as i t flows downwards along the outer sphere. Hence, the heat t r a n s f e r r a t e decreases as the f l u i d flows i n t o the r e s e r v o i r . I n t h e r e g i o n a r o u n d .0-130° t h e r e i s v i g o r o u s c o n v e c t i o n o f t h e f l u i d n e a r t h e i n n e r s p h e r e . So some o f t h e ' c o l d ' f l u i d n e a r t h e o u t e r s p h e r e w i l l be e n t r a i n e d by t h e r e l a t i v e l y h o t t e r r i s i n g f l u i d n e a r t h e i n n e r s p h e r e . The r e m a i n i n g c o l d f l u i d w i l l f l o w i n t o t h e ' r e s e r v o i r 1 . T h i s w i l l r e d u c e t h e h e a t t r a n s f e r r a t e t o a v a l u e lower t h a n t h a t o f p u r e c o n d u c t i o n . Nu w i l l be a minimum n e a r 0=125° s i n c e most o f o t h e h e a t f l u x i s c o n v e c t e d away by t h e f l u i d n e a r t h e i n n e r s p h e r e . T h i s r e s u l t s i n t h e d i s t o r t i o n o f t h e i s o t h e r m s , see s e c t i o n 3.5. The i n f l u e n c e o f a on Nu. i s n o t v e r y s i g n i f i c a n t i ,o 1 ^ f o r t h e r a n g e o f Ra c o n s i d e r e d i n t h i s t h e s i s . F o r a f i x e d O J , low Ra c o r r e s p o n d s t o low c o n v e c t i o n t h u s c o n d u c t i o n p l a y s t h e p r e d o m i n a n t r o l e . Hence f o r a f i x e d v a l u e o f O J and Ra=720, a change o f a does n o t a l t e r Nu. s i g n i f i c a n t l y , F i g u r e s 3.6.1 and 3.6.2. F o r a v a l u e o f a=.72, N U q i s g r e a t e r t h a n t h a t f o r a=10. I t i s u s u a l t o c o n s i d e r a f l u i d t o be more c o n d u c t i v e as a d e c r e a s e s . Thus f o r a f i x e d low Ra, as a d e c r e a s e s , G (buoyancy e f f e c t s ) i n c r e a s e s . T h i s w i l l d i s p l a c e t h e i s o t h e r m s f u r t h e r upwards i n t h i s c a s e . T h e r e -f o r e N U q w i l l i n c r e a s e n e a r t h e t o p o f t h e o u t e r s p h e r e as a d e c r e a s e s . I t w i l l now be shown t h a t t h e i n f l u e n c e o f a i s a h i g h e r o r d e r e f f e c t f o r t h e o v e r a l l N u s s e l t numbers. 75 Le t 1®. = - § + TNI X cos 0 + (TN2 + TN3 )x(3 c o s 2 0 - l ) 3R ( 0 - 1 ) "+ TN4 30° where TNI x cos 0 = (6a) x ~=-a R 2 0 8 9 0 TN2 x (3 c o s ' 0 - 1 ) = (G^a) x 2 TN3 x (3 c o s 2 0 - 1) + TN4 = (G 2a 2) x « i o R 30 30° 30°- o 30^ 80 2 i . e . — = + Ga — - + G a — - + G o — -3R 3R 3R 3R 8R Hence Nu. l ( 6 - 1 ) 26 30 3R s i n 0 d0 R=l = 1 - TN4 R=l (3.6.3) 76 S i m i l a r l y N U Q = 1 - 6 2 x TN4 R=3 Equations (3.6.3) and (3.6.4) show t h a t the values o f the o v e r a l l N u s s e l t numbers are a l t e r e d from t h a t o f pure 2 conduction alone by TN4. The term TN4 comes from and i s independent of 8 i . e . the zeroeth s p h e r i c a l harmonic. There-f o r e Nu. are f u n c t i o n s of Ra only s i n c e the c o e f f i c i e n t of i , o •* 2 2 i s Ra . Nu. ^ i n c r e a s e s w i t h i n c r e a s e o f 3 and Ra, and as i , o ' OJ decreases, F i g u r e 3.6.5 and Table I. For f i x e d values o f 3 and Ra, and as OJ tends to u n i t y , t h i s i m p l i e s t h a t the f l u i d i n the gap of the c o n c e n t r i c spheres becomes as conductive as the s o l i d i n n e r sphere. T h e r e f o r e the heat t r a n s f e r r a t e i n c r e a s e s . I f 3 i s s m a l l i . e . 1 _< 3 ^ 1.25, the o v e r a l l heat t r a n s f e r r a t e i s e s s e n t i a l l y t h a t o f pure conduction as there i s r e l a t i v e l y no c o n v e c t i v e motion i n the f l u i d . The same value of 1.12 f o r the o v e r a l l N u s s e l t number as t h a t g i v e n by Mack and Hardee f o r 3=2.0, Ra=10 0 0 i s 15 obt a i n e d here f o r computing the conjugate case when OJ=10 0° 30° 60° 90° 120° 150° 180° e F i g u r e 3.6.1 N u s s e l t Number A g a i n s t Angular P o s i t i o n , Conjugate Case OJ=10 , 3=2.0, a=.72, G=10 00 F i g u r e 3.6.4 Nu s s e l t Number Ag a i n s t Angular P o s i t i o n , Conjugate Case, u>=10, 6=2.0. a=.72, G=1400 ' ' ' 00 o 500 R" 1000 1500 F i g u r e 3.6.5 O v e r a l l N u s s e l t Number as F u n c t i o n of R a y l e i g h Number, Conjugate Case • 00 TABLE I OVERALL NUSSELT NUMBERS FOR CONJUGATE CASE, 3 = 1.15, 2.0, and 3.0 * Ra = Ra x ( 3 ) s - r O V E R A L L N U S S E L T N U M B E R 8=1. 15 8=2.0 8= 3.0 * Ra G 0 co=10 OJ=10 w=10 OJ=5 co=10 w=5 10 10 0.01 1.000 1.000 1.00 1.00 1.00 1.00 1.00 100 10 0.10 • 1.000 1.000 1.00 1.00 1.00 1.06 1.08 500 10 0.50 1.000 1.000 1.01 1.01 1.01 720 10 0.72 1.000 1.000 1.02 1.02 1.03 1000 10 0.10 1.000 1.000 1.02 1.04 1.06 1000 10 10 1.000 1.000 1.03 1.04 1.06 2000 200 10 1.001 1.002 1.12 1.17 1.26 2000 2800 0.72 1.001 1.002 1. 12 1.17 1.26 3000 300 10 1.001 1.004 1.26 1.36 1.57 3000 4000 0.72 1.001 1.004 1.26 1.36 1.57 5000 1000 5 1.004 1.008 1.74 oo t o 83 4. EXPERIMENT The purpose of the experimental i n v e s t i g a t i o n was a flow v i s u a l i z a t i o n study of the conjugate n a t u r a l c o n v e c t i o n p a t t e r n between two c o n c e n t r i c spheres. The experimental equipment was designed so t h a t i t resembled as c l o s e l y as p o s s i b l e the a n a l y t i c a l model o u t l i n e d i n the p r e v i o u s c h a p t e r s . The experimental r e s u l t s were expected t o g i v e a b e t t e r understanding of the flow p a t t e r n s and temperature d i s t r i -b u t i o n s on the s u r f a c e o f the i n n e r sphere. They were t o c o n f i r m as w e l l t h a t the flow i n the gap between the c o n c e n t r i c spheres was steady, laminar and a x i s y m m e t r i c a l . Experiments of t h i s type had been performed by Bishop e t a l . [6,7] on the c o n v e c t i o n p a t t e r n s between c o n c e n t r i c i s o t h e r m a l spheres. 4.1 Experimental Apparatus The experimental apparatus, shown i n F i g u r e 4.1.1-, c o n s i s t e d of.,;two c o n c e n t r i c spheres. The i n n e r sphere was made out of two 6" diameter p l e x i g l a s hemispheres. Each o f the hemispheres had a 3" diameter h e m i s p h e r i c a l c a v i t y . F i g u r e 4.1.2 shows the arrangement of the c y l i n d r i c a l h e a t i n g element w i t h the f o u r thermocouples i n s i d e the s p h e r i c a l c a v i t y and the e i g h t thermocouples on the s u r f a c e of the i n n e r sphere. The p o s i t i o n s o f the thermocouples are shown F i g u r e 4.1.1 Experimental Apparatus F i g u r e 4.1.2 P l e x i g l a s Inner Sphere 86 i n F i g u r e 4.1.3. The i n n e r sphere was supported by a p l e x i -g l a s stem, a c e n t r a l h o l e was d r i l l e d through the support stem f o r the p a s s i n g of two .066" diameter p o l y e t h y l e n e tubes (used f o r the i n t r o d u c t i o n of s i l i c o n e o i l i n t o the s p h e r i c a l c a v i t y ) . The s i l i c o n e o i l was r e q u i r e d t o t r a n s f e r heat from the h e a t i n g element evenly t o the envelope of the c a v i t y . The h e a t i n g element had an output of up t o 50 watts. The p l e x i g l a s sphere was sprayed w i t h a f l a t b l a c k p a i n t t o reduce the r e f l e c t i o n o f l i g h t from i t and thus f a c i l i t a t e the photographic work. The outer sphere c o n s i s t e d o f two 10" diameter Corning pyrex g l a s s hemispheres with a .2 81" w a l l t h i c k n e s s . The lower h a l f o f the outer sphere had a 2" diameter h o l e . Through t h i s h o l e a 2" diameter c o l l a r of 1/8" h i g h , machined on the 3" outer support stem, was f i t t e d , glued and s e a l e d with s i l i c o n e compound. As shown i n F i g u r e 4.1.4, p a r t of the p o l y s t y r e n e support stem extended i n t o the gap between the c o n c e n t r i c spheres. T h i s p a r t was 1 3/4" l o n g , 1" i n diameter and threaded i n t e r n a l l y with a 1/2" diameter h o l e d r i l l e d a l ong the whole l e n g t h of the support stem. The thermocouple wires and the connecting leads to the h e a t i n g element were passed through the c e n t r a l hole of both support stems. The i n n e r stem w i t h a 1" diameter, 1/4" h i g h c o l l a r was then screwed onto the ou t e r stem. T h i s arrangement ensured the c o n c e n t r i c i t y o f the two spheres. 87 gure 4.1.3 L o c a t i o n s of Thermocouples on Surface of Inner Sphere F i g u r e 4.1.4 Support Stem with Glass Hemisph F i g u r e 4.1.5 Top View of Support Stem and Glass Hemisphere 90 Smoke c o u l d be i n t r o d u c e d to the gap between the c o n c e n t r i c spheres through one o f the two 5/8" diameter h o l e s running through the whole l e n g t h of the o u t e r sphere support stem, F i g u r e 4.1.5. The Pyrex hemispheres were s e a l e d t o -gether with s i l i c o n e g l u e . The s u r f a c e temperature of the outer sphere was measured by f o u r thermocouples a t p o s i t i o n s of 0°, 30°, 90° and 120°. The outer sphere support stem was mounted onto a p l e x i g l a s s base. The base was h e l d i n p o s i t i o n by two b r a s s b o l t s t o the frame of a water tank (30" x 28" x 30") as shown i n F i g u r e 4.1.1. The water tank was used to p r o v i d e a constant temperature bath. A temperature c o n t r o l l e r designed by the ;: * Geophysics Department, U n i v e r s i t y of B r i t i s h Columbia, was used t o c o n t r o l the water temperature i n the tank. The platinum wire temperature s e n s i n g probe of the temperature c o n t r o l l e r was taped to the s i d e of the outer sphere support stem. A ' B r i s k e t ' h e a t i n g tape (output c a p a c i t y 60 watts) connected to the temperature c o n t r o l l e r was p l a c e d around the p l e x i g l a s base, F i g u r e 4.1.1. The water i n the tank was c i r c u l a t e d from the top to the bottom of i t by a 'Randolph' pump which was connected to the o u t l e t and i n l e t p ipes a t the aluminium b a c k - p l a t e of the tank. The thermocouple wires used were of 36 gauge Chromel-Alumel type. F i g u r e 4.1.6 shows a t y p i c a l c a l i b r a t i o n curve f o r the thermocouples. As the c a l i b r a t i o n data f o l l o w e d the The h e l p of Dr. R u s s e l and Mr. B. Goldberg i n d e s i g n -i n g t h i s item i s g r a t e f u l l y acknowledged. 9 Manufacturer's data v Experimental data I I 1 40 60 80 TEMPERATURE / °C T y p i c a l Thermocouple C a l i b r a t i o n Curve gure 4.1.8 Layout of Experimental Apparatus 94 manufacturers' data very c l o s e l y , o n l y three of the thermo-couples were c a l i b r a t e d . The platinum wire s e n s i n g probe was a l s o c a l i b r a t e d , F i g u r e 4.1.8. The s i d e s o f the tank were e i t h e r p a i n t e d b l a c k or l i n e d w i t h b l a c k c o n s t r u c t i o n paper so t h a t the whole e n c l o s u r e was dark t o f a c i l i t a t e photographic work. A c e n t r a l narrow s l i t of 1/8" width was cut i n t o the b l a c k paper l i n i n g at one s i d e of the tank f o r c o l l i m a t i n g the i l l u m i n a t i n g l i g h t . Two 500-watts 35 mm s l i d e p r o j e c t o r s were p o s i t i o n e d i n f r o n t of the s l i t so t h a t a v e r t i c a l plane of l i g h t t r a v e r s e d the c e n t r e plane of the c o n c e n t r i c spheres. On the adjacent s i d e of the tank a r e c t a n g u l a r opening was cu t i n t o the b l a c k paper l i n i n g to allow v i s u a l and photo-g r a p h i c o b s e r v a t i o n of the flow p a t t e r n i n the i l l u m i n a t e d v e r t i c a l plane between the c o n c e n t r i c spheres. F i g u r e 4.1.8 shows the whole experimental apparatus assembled. 4.2 Experimental Procedure Owing t o the low thermal c o n d u c t i v i t y of p l e x i g l a s the i n n e r sphere was heated up s l o w l y i n s m a l l increments of h e a t i n g - r a t e s u n t i l the d e s i r e d value was achieved. S u f f i c i e n t time was allowed to e l a p s e b e f o r e the h e a t i n g r a t e was i n c r e a s e d . A L-C f i l t e r u n i t was used to smooth out the h e a t i n g c u r r e n t ( s u p p l i e d by a H e a t h k i t r e g u l a t e d low v o l t a g e D.C. power s o u r c e ) . At the same time, the water temperature 95 i n the tank was kept c o n s t a n t . Before each e x p e r i m e n t a l run, a p e r i o d o f approximately two days was allowed t o achieve s t e a d y - s t a t e , such t h a t the thermocouples' readings were cons t a n t over t h i s p e r i o d . C i g a r smoke was g e n t l y i n t r o d u c e d i n t o the gap between the c o n c e n t r i c spheres through e i t h e r one of the two tubings (Figure 4.1.1) which connected the two 1/8" h o l e s i n the outer sphere support stem t o the two openings at the back p l a t e of the tank. S u f f i c i e n t time was allowed f o r the s t a b i l i z a t i o n o f the flow p a t t e r n b e f o r e photographs were taken. A 35 mm camera and Kodak TRI-X f i l m w i t h a speed o f 400 ASA were used f o r photography. The l i g h t source from the two pro-j e c t o r s was switched on only when photographs were taken. T h i s avoided supplementary h e a t i n g of the i n n e r sphere s u r f a c e through a b s o r p t i o n of r a d i a t i o n from the l i g h t source. Focusing the camera d i r e c t l y on the i l l u m i n a t i n g plane was d i f f i c u l t and u n s u c c e s s f u l . So an i n d i r e c t method was used whereby the camera was focused onto a f i n e l y p r i n t e d s u r f a c e p l a c e d adjacent t o the outer sphere, i n the plane of the path of the i l l u m i n a t i n g l i g h t . A p o l a r i z i n g f i l t e r was attached to the camera l e n s to e l i m i n a t e most of the r e f l e c -t i o n s from the inn e r sphere. Exposure times were e i t h e r 1/2 second or 1 second w i t h lens a p e r t u r e s s e t t i n g a t F5.6 and F8 r e s p e c t i v e l y . The exposed f i l m was developed u s i n g Kodak D-19 developer f o r 10 minutes so as to r a i s e the f i l m speed t o 800 ASA. The photographs were p r i n t e d on hi g h c o n t r a s t paper f o r b e t t e r c o n t r a s t o f the smoke p a t t e r n s . 96 4.4 Experimental R e s u l t s F i g u r e s 4.4.1 to 4.4.3 are photographs of the flow p a t t e r n i n the gap between the c o n c e n t r i c spheres o b t a i n e d under three o p e r a t i n g c o n d i t i o n s . Due to the t h i c k n e s s of the outer g l a s s sphere, i t s i n n e r s u r f a c e acted as a r e f l e c -t o r . T h i s caused the b r i g h t and dark r e g i o n s over the s u r f a c e of the outer sphere. There are two dark l i n e s a c r o s s the gap i n the photographs. These are the shadows of the h o r i z o n t a l s e a l between the two h a l v e s o f the outer sphere. There are some dust p a r t i c l e s on the s u r f a c e o f the outer sphere which appear as b r i g h t s c a t t e r spots i n the photographs. S i m i l a r l y , the white l i n e s t a r t i n g from the top of gap i s the r e f l e c t i o n of the outer sphere s u r f a c e thermocouples. The r e f l e c t i o n of l i g h t by the outer sphere support stem appears as a white r e g i o n i n the bottom of the gap. From both v i s u a l and photographic o b s e r v a t i o n s i t i s seen t h a t the flow p a t t e r n i n the gap between the c o n c e n t r i c spheres i s steady and axisymmetrical about the v e r t i c a l a x i s . At the diameter r a t i o used (3=1.67) and f o r a l l three temperature d i f f e r e n c e s , the flow p a t t e r n i s of the ' c r e s c e n t -eddy 1 type as p r e d i c t e d i n the a n a l y s i s . The flow p a t t e r n s can e s s e n t i a l l y be d i v i d e d i n t o two r e g i o n s as f o l l o w : (a) the r e g i o n i n the immediate v i c i n i t y of each sphere where there i s a t h i n - l a y e r of h i g h v e l o c i t y flow; (b) the c e n t r a l eddy r e g i o n where the f l u i d i s moving r e l a t i v e l y s l o w l y . The f l u i d v e l o c i t y near the i n n e r sphere 98 100 i s higher than that near the outer sphere. At the top of the inner sphere there i s a j e t - l i k e flow p o l a r i z a t i o n . The f l u i d separates from the inner sphere near the top and flows downward along the outer sphere into the bottom of the gap. This region i s r e l a t i v e l y stagnant and acts as i f i t were a reservoir. The centre of the eddy i s i n the upper half of the gap between the concentric spheres. Its position remains r e l a t i v e l y stationary for a l l three temperature differences tested. Bishop et a l . [7] had reported that i n t h e i r experi-ment on natural convention between concentric isothermal spheres, as the high speed flow separated from the top of the inner sphere, a corner eddy was observed i n the 'corner' formed by the in t e r s e c t i o n of the surface of the inner sphere with the v e r t i c a l axis of symmetry. However, t h i s phenomenon i s not observed here for natural conjugate convection. Tables I I , III and IV show the temperature d i s t r i b u t i o n s on the surface of the inner sphere, the temperature of the o i l i n various positions (Figure 4.1.2) i n the cavity inside the inner sphere and the surface temperature of the outer sphere. In each case, the temperature on the surface of the inner sphere decreases as 6 increases, as predicted t h e o r e t i c a l l y . The following Tables (II to IV) show the temperature d i s t r i b u t i o n on the surfaces of the concentric spheres at the positions; indicated i n Figure 4.1.2. 101 TABLE I I 6 = 1. 4 watts AT = „024°C G = 1.32 x 10 4 P o s i t i o n Temperature P o s i t i o n Temperature (1) 0° 28.3 (9) A 33°C (2) 30° 28°C (10) B 34.2°C (3) 45° 28°C (11) C -(4) 60° 27.5°C (12) D 36.6°C (5) 90° 26.5°C (13) * 0° 24.5°C (6) 120° - (1.4) * 30° 24.5°C ( ) 150° 25.5°C (15) * 90° 24.5°C (8) 165° 25.3°C (16) 120°* 24.5°C * Outer Sphere. TABLE I I I • Q = 4. 29 watts AT = 7.21°C / G = 3.72 x 10 5 P o s i t i o n Temperature P o s i t i o n Temperature (1) 0° 35.5°C (9) A 48.5°C (2) 30° 35.4°C (10) B 49.5°C (3) 45° 35.1°C (11) C 36.2°C (4) 60° 32.5°C (12) D 54 °C (5) 90° 31°C (13) * 0° 25.3°C (6) 120° - (14) * 30° 25.3°C (7) 150° 28.8°C (15) * 90° 25.3°C (8) 165° 28.3°C (16) 120°* 25.3°C * Outer Sphere. 1 0 2 TABLE IV • Q = 6.14 watts AT = 10. 3°C , G = 5.78 x 10'5 P o s i t i o n Temperature P o s i t i o n Temperature (1) 0° 37.5°C (9) A 61°C (2) 30° 37.0°C (10) B 68.5°C (3) 45° 37.0°C (11) C -(4) 60° 36.8°C (12) D 72.5°C (5) 90° 36.5°C (13) * 0° 24.5°C (6) 120° - (14) * 30° 24.5°C (7) 150° 31.4°C (15) * 90° 24.5°C (8) 165° 31.2°C (16) * 120° 24.5°C * Outer sphere. Although the experimental r e s u l t s o b tained here are f o r the values o f w=15, 6=1.67, a=0.72 and a p p r e c i a b l y h i g h e r G numbers than those f o r which the p e r t u r b a t i o n expansions w i l l be v a l i d , the flow p a t t e r n and the temperature d i s t r i b u t i o n s on the s u r f a c e of the i n n e r sphere are q u a l i t a t i v e l y s i m i l a r to those p r e d i c t e d by the a n a l y s i s . Moreover, the experimental r e s u l t s c o n f i r m t h a t the flow i n the gap between the c o n c e n t r i c spheres i s steady, laminar and ax i s y m m e t r i c a l , assumptions used i n the a n a l y s i s . 103 5. CONCLUSIONS The p r e s e n t i n v e s t i g a t i o n has l e d to the f o l l o w i n g r e s u l t s : 1. T h e o r e t i c a l s o l u t i o n s f o r the governing equations f o r steady laminar a x i s y m m e t r i c a l conjugate n a t u r a l c o n v e c t i o n between two c o n c e n t r i c spheres were o b t a i n e d . The case of a constant f l u x i n n e r sphere with an i s o t h e r m a l outer sphere was s o l v e d s e p a r a t e l y . 2. The l i m i t s of the a p p l i c a b i l i t y of the s o l u t i o n s were d e f i n e d . 3. The s t r e a m l i n e c o n f i g u r a t i o n was found to be of the crescent-eddy type. The e x i s t e n c e o f secondary c e l l s was found not to be a genuine f e a t u r e of e i t h e r the conjugate or the non-conjugate cases c o n s i d e r e d here. 4. Contours o f i s o v o r t i c i t y l i n e s , isotherms and d i s t r i b u t i o n s o f v e l o c i t y and temperature of the f l u i d i n the gap between the c o n c e n t r i c spheres were ob t a i n e d and d i s c u s s e d . 5. L o c a l heat t r a n s f e r r a t e s from both i n n e r and outer spheres were determined. 6. The i n f l u e n c e of P r a n d t l number on the o v e r a l l heat t r a n s f e r r a t e was found to be a h i g h e r - o r d e r e f f e c t . 10 4 7. The flow p a t t e r n o b t a i n e d e x p e r i m e n t a l l y i s steady, laminar, axisymmetrical and of the crescent-eddy type. 8. The e x p e r i m e n t a l r e s u l t s support the a n a l y s i s a t the very l e a s t q u a l i t a t i v e l y . REFERENCES G.K. B a t c h e l o r , Heat t r a n s f e r by f r e e c o n v e c t i o n across a c l o s e d c a v i t y between v e r t i c a l boundaries at d i f f e r e n t temperatures, Quart. J . A p p l i e d Math. 3, 1954, 209-233. G. Poots, Heat t r a n s f e r by laminar f r e e c o n v e c t i o n i n en c l o s e d plane gas l a y e r s , Quart. J . Mech. A p p l i e d Math. 11, 1958, 257-273. J.O. Wilkes and S.W. C h u r c h i l l , The f i n i t e - d i f f e r e n c e computation o f n a t u r a l c o n v e c t i o n i n a r e c t a n g u l a r e n c l o s u r e , A.I. Ch. E. 12_, 161-166. J.W. E l d e r , Numerical experiments with f r e e c o n v e c t i o n i n a v e r t i c a l s l o t , J . F l u i d Mech. 24, 1966, 823-843. G. de Vahl D a v i s , Laminar N a t u r a l c o n v e c t i o n i n an e n c l o s e d r e c t a n g u l a r c a v i t y , I n t e r n . J . Heat/Mass T r a n s f e r 11, 1968, 1675-1693. E.H. Bishop, R.S. K o l f a t , L.R. Mack and J.A. Scanlan, Photographic s t u d i e s o f c o n v e c t i o n p a t t e r n s between c o n c e n t r i c spheres, S.P.I.E. J . 3^ 1964, 47-49. E.H. Bishop, R.S. K o l f a t , L.R. Mack and J.A. Scanlan, Convective heat t r a n s f e r between c o n c e n t r i c spheres, Proc. 1964 Heat T r a n s f e r / F l u i d Mech. I n s t . , ( S t a n f o r d U.P. 1964), 69-80. E.H. Bishop, L.R. Mack and J.A. S c a l a n , Heat t r a n s f e r by n a t u r a l c o n v e c t i o n between c o n c e n t r i c spheres, I n t e r n . J . Heat/Mass T r a n s f e r 9_, 1966, 6949-662. L.R. Mack-and E.H. Bishop, N a t u r a l c o n v e c t i o n between h o r i z o n t a l c o n c e n t r i c c y l i n d e r s f o r low R a y l e i g h numbers, Quart. J . Mech. A p p l i e d Math. 2_1, 19 68, 223-24.1. L.R. Mack and H.C. Hardee, N a t u r a l c o n v e c t i o n between c o n c e n t r i c spheres a t low R a y l e i g h numbers, I n t e r n . J . Heat/Mass T r a n s f e r 11_, 1968, 387-396. T.L. Perelman, On Conjugated problems of heat t r a n s f e r , I n t e r n . J . o f Heat/Mass T r a n s f e r 3, 1961, 293-303. 106 12. Z. Rotem, The e f f e c t o f thermal conduction o f the w a l l upon c o n v e c t i o n from a s u r f a c e i n a laminar boundary l a y e r . I n t e r n . J . Heat/Mass T r a n s f e r 10, 1967, 461-466. 13. M.D. K e l l e h e r and K-T. Yang, A steady conjugate heat t r a n s f e r problem w i t h conduction and f r e e c o n v e c t i o n , A p p l i e d S c i . Res. 17, 1967, 249-269. 14. G.S.H. Lock and J.C. Gunn, Laminar f r e e c o n v e c t i o n from a downward-projecting f i n , J . Heat T r a n s f e r ASME S e r i e s C, 1968, 63-70. 15. E . J . Davis and W.N. G i l l , The e f f e c t s of a x i a l conduc-t i o n i n the w a l l on heat t r a n s f e r w i t h laminar flow, I n t e r n . J . Heat/Mass T r a n s f e r 13, 1970, 459-470. 16. Z. Rotem. Conjugate f r e e c o n v e c t i o n from h o r i z o n t a l conducting c i r c u l a r c y l i n d e r s , I n t . J . of Heat/Mass T r a n s f e r 19, 1971, i n p r e s s . 17. L. Knopoff, The c o n v e c t i o n c u r r e n t h y p o t h e s i s , Reviews of Geophysics 2_, 1964, 89-122. 18. M. Van Dyke, P e r t u r b a t i o n methods i n f l u i d mechanics, Academic. Press 196 4. 19. F.E. F e n d e l l , Laminar n a t u r a l c o n v e c t i o n about an i s o t h e r m a l heated sphere at smal l Grashof number, J . F l u i d Mech. 34, 1968> 163-176. 20. C.A. Hieber and B. Gebhart, Mixed c o n v e c t i o n from a s m a l l sphere a t s m a l l Reynolds and Grashof numbers, J . F l u i d Mech. 3_8, 1969 , 137-159. 21. J.A. Scanlan, E.H. Bishop and R.E. Powe, N a t u r a l con-v e c t i o n heat t r a n s f e r between c o n c e n t r i c spheres, I n t e r n . J . Heat/Mass T r a n s f e r 1_3, 1970 , 1857-1872. 22. J . Proudman and J.R.A. Pearson, Expansions at s m a l l Reynolds: numbers f o r the flow p a s t a sphere and a c i r -c u l a r c y l i n d e r . J . F l u i d Mech. 2_, 1957, 237-262. 23. J . J . Mahoney, Heat t r a n s f e r a t s m a l l Grashof number. Proc. Roy. Soc. A238, 1957, 412-423. 24. Md. A. Hossain and B. Gebhart, N a t u r a l c o n v e c t i o n about a sphere a t low Grashof number, F o u r t h I n t e r n . Heat T r a n s f e r Conference, P a r i s , 1970, V o l . 4, 1-12. 107 APPENDIX I CALCULATION OF VISCOUS DISSIPATION EFFECT The energy e q u a t i o n o f the f l u i d i n c l u d i n g the v i s c o u s d i s s i p a t i o n terms (with r o t a t i o n a l symmetry) expressed i n s p h e r i c a l c o o r d i n a t e s i s V K V 2T - pC (V« ^ ~ + -1 21 ) + y $ = 0 (A-l) f P R 3R' R« 36 V where y $ v i s the d i s s i p a t i o n term of mechanical energy by v i s c o u s e f f e c t s per u n i t volume. The d i s s i p a t i o n f u n c t i o n , I 8 V R 2 V R i 3 V e 2 V R v e $ = 2 < ( — + ( J * + ± S.) + (_£. + J± c o t 6) I 3R R' • R' 38 R R' R' 3R 1 V R1 R' ^ R 38 The d i m e n s i o n l e s s r a d i a l and t a n g e n t i a l components of v e l o c i t y are i n t r o d u c e d as f o l l o w s , VR = - T VR ; v 1 V, R! V 1Rendering the energy equation (A-l) d i m e n s i o n l e s s , 10 8 y 2 0 - a (V_ l i + l i 1®) + = 0 (A-2) 3R R 86 V where the v i s c o u s d i s s i p a t i o n number i s V = 2 y v R 2 Kj, AT _ i f r e f Le t e = v V 20 denote the dimensionless r a t i o of the v i s c o u s d i s s i p a t i o n term to the conduction term. L e t X = v a (V 30_ ^0 30) 3R R 39 denote the dimensionless r a t i o of the v i s c o u s d i s s i p a t i o n term to the c o n v e c t i o n term. For a i r with G=1000, a=0.72, R|=l f t , 3=2.0, OJ=10 and P=1.146 x 10 ^ ° , the values o f |e| and |x| at both angular and r a d i a l p o s i t i o n s i n the gap between the two c o n c e n t r i c spheres are giv e n i n Table V. 109 TABLE V RATIO OF VISCOUS DISSIPATION TERM TO CONDUCTION TERM: RATIO OF VISCOUS DISSIPATION TERM TO CONVECTION TERM (AIR) R 1.1 1.3 1.5 1.8 e 1 e 1 6. 5 8 x l 0 - 1 1 3. 56X10" 1 1 3. 16X10" 1 1 3. 2 7 x l 0 _ 1 1 15° |X| . 2. 7 2 x l 0 ~ 9 9. 2 5 x l 0 - 1 0 1. 8 4 x l 0 ~ 8 5. OOxlO" 1 0 15° 1 e 1 1. 8 9 x l 0 ~ 9 9. l O x l O " 1 0 7. 9 5 x l 0 ~ 1 0 8. I S x l O - 1 0 60° 1 A 1 5. 42xl0"- 1 0 2. 2 8 x l 0 ~ 1 0 3. 5 4 x l 0 - 9 1. 8 0 x l 0 ~ 1 0 60° 1 e 1 3. 2 3 x l 0 ~ 1 0 1. 6 4 x l 0 ~ 1 0 1. 4 5 x l 0 ~ 1 0 1. 4 9 x l 0 " 1 0 135° 1 x 1 8. 0 0 x l 0 ~ 1 0 4. 9 4 x l 0 ~ 1 0 1. 4 3 x l 0 ~ 9 4. l O x l O " 1 0 135° For water w i t h G=62, a=11.6, R|=l f t , 6=2.0, to=10 and -12 t?=4.42x10 , the values of |e| and | x | a t both angular and r a d i a l p o s i t i o n s i n the gap between the two c o n c e n t r i c spheres are given i n Table VI. TABLE VI RATIO OF VISCOUS DISSIPATION TERM TO CONDUCTION TERM: RATIO OF VISCOUS DISSIPATION TERM TO CONVECTION TERM (WATER) R 1.1 . 1.3 1.5 1.8 e 1 e 1 6. -13 33x10 X J 6. 7 8 x l 0 ~ 1 4 4. 21x10 1 4 3. 0 7 x l 0 ~ 1 4 15° 1 X 1 8. -12 98x10 4. 1 4 x l 0 ~ 1 2 8. -12 99x10 x 9. -12 0 5x10 15° 4. -13 65x10 J 1. -12 22x10 9. -13 45x10 x 1. -12 17x10 X ' 60° 1 x 1 2. 1 9 x l 0 - 1 1 6. -12 63x10 x z 1. 3 3 x l 0 ~ 9 2. 8 0 x l 0 " 1 2 60° 1 e 1 5. -13 12x10 • 2. 5 7 x l 0 - 1 3 1. 8 1 x l 0 ~ 1 3 1. 3 7 x l 0 - 1 3 135° | x\ 2. -11 58x10 X- L 1. 0 5 x l 0 _ 1 1 7. 6 0 x l 0 - 1 1 5. -12 62x10 135° 110 Tables V and VI show t h a t the r e l a t i v e o r d e r of magnitude of the v i s c o u s d i s s i p a t i o n term t o e i t h e r the con-d u c t i o n term (| e | ) or the c o n v e c t i o n t e r m ( I ^ I ) i s at l e a s t ~8 of the order of 10 . Thus the assumption t h a t the v i s c o u s d i s s i p a t i o n o f energy i n the f l u i d assumed i n s e c t i o n 2.1 to be n e g l i g i b l e i s found t o be v a l i d . I l l APPENDIX I I CONJUGATE PROBLEM WITH INNER SPHERE CONTAINING DISTRIBUTED SOURCES Consider now the i n n e r sphere o f the conjugate problem i n s e c t i o n 2 wi t h uniform d i s t r i b u t e d heat sources i n s t e a d of a s i n g l e heat source a t i t s c e n t r e . The energy equation of the i n n e r sphere then becomes K x V 2 T. = B (R) (B-l) s i where 8(R) i s the heat source f u n c t i o n per u n i t volume assumed cons t a n t . The r e f e r e n c e temperature, ^ T r e f ' d e f i n e d i n s e c t i o n 2.2, wit h Q r e p l a c e d here by 8 x — TT R1 3 3 i AT ^ r e f B x % TT R!3 x ( 3 - D 3 0; 4TT 3 R[ K f B R|2 ( 3 - D 38 K f Rendering the energy equation (B-l) d i m e n s i o n l e s s , i t becomes ~ 36 y 2 ~Q = u ( 3 - D ( B ~ 2 ) = c o n s t a n t . f o r given v a l u e s of w and 6 . 112 Then the energy equation (2.2.3) given i n s e c t i o n 2.2 f o r the i n n e r sphere i s r e p l a c e d here by equation (B-2). The boundary c o n d i t i o n s and the method of s o l u t i o n f o r t h i s case are e x a c t l y the same as p r e v i o u s l y d e s c r i b e d i n s e c t i o n 2. A l s o the s e t of uncoupled l i n e a r d i f f e r e n t i a l equations given i n s e c t i o n 2.3 remains e s s e n t i a l l y the same with the s o l e e x c e p t i o n t h a t the equation (2.3.7) i s r e p l a c e d here by Equation (B-3) i s s o l v e d simultaneously with equation (2.3.8) s u b j e c t t o the boundary c o n d i t i o n s (2.3.18) and (2.3.19). The s o l u t i o n s are 38 (B-3) O J (6-D ~ o O J (6-1) o (6-1) As the equations f o r ^  a n d the h i g h e r order terms of the 0, and 0 expansions and the boundary c o n d i t i o n s remain unchanged; t h e i r s o l u t i o n s are those given i n s e c t i o n 2.4 . 113 Hence f o r the conjugate problem the stream f u n c t i o n , Y and the temperature d i s t r i b u t i o n 0 of the f l u i d i n the gap of the two c o n c e n t r i c spheres do not depend on what form of heat source d i s t r i b u t i o n i n s i d e the i n n e r sphere. However, the i n n e r sphere temperature d i s t r i b u t i o n does depend on the heat source d i s t r i b u t i o n i n s i d e the in n e r sphere. 1 1 4 APPENDIX I I I A BRIEF REVIEW ON SMALL GRASHOF NUMBERS NATURAL CONVECTION ABOUT A HEATED SPHERE In the a n a l y s i s o f conjugate n a t u r a l c o n v e c t i o n between c o n c e n t r i c spheres, i t i s seen t h a t as the r a d i u s r a t i o , 3"*00 the problem reduces to a s i n g l e heated sphere i n an unbounded expanse of f l u i d . The r e s u l t i n g flow f i e l d i s e s s e n t i a l l y c o n f i n e d to a v e r t i c a l plume above the heat source. The temperature vanishes o u t s i d e the plume and as R->-°° the v e l o c i t y should v a n i s h everywhere except i n the narrow wake r e g i o n above the sphere, wherein i t should be bounded. The r e g u l a r p e r t u r b a t i o n expansion scheme employed i n t h i s t h e s i s i s inadequate except i n the r e g i o n near the sphere. ° 3 For i t i s i n the s o l u t i o n f o r ¥^ (eq. 2.4.4), as R-+°° the 0 (R ) o term of ¥^ corresponds to an 0(R) behaviour i n V R (eq. 3.3.1a). Thus the v e l o c i t y boundary c o n d i t i o n a t i n f i n i t y i s not s a t i s f i e d . T h i s i s analogous to the Whitehead paradox [22] f o r s m a l l Reynolds number flow past a f i n i t e s i z e t h r e e -dimensional body ( i . e . the c o n v e c t i v e e f f e c t must be c o n s i d e r e d a t the d i s t a n t r e g i o n away from the sphere, although d i f f u s i v e e f f e c t i s predominant near the sphere). T h e r e f o r e , there e x i s t s an o u t e r r e g i o n i n which the con-115 v e c t i v e , d i f f u s i v e and bouyancy e f f e c t s are of the same order of magnitude. An inner-and-outer matched asymptotic expansions w i l l be r e q u i r e d f o r the s o l u t i o n of the flow f i e l d . Note t h a t a t a l a r g e d i s t a n c e away from the sphere, although there i s conjugate e f f e c t a t the s u r f a c e o f the sphere, i t w i l l appear as a heat p o i n t source. In o r d e r to o b t a i n the governing equations i n the o u t e r r e g i o n where the c o n v e c t i v e and d i f f u s i v e e f f e c t s are o f . t h e same o r d e r , Mahoney [23] i n t r o d u c e d an a p p r o p r i a t e l e n g t h -1 s c a l e ( i . e . R=R.G 2) t o g e t h e r with an asymptotic expansion - l 1 i n terms o f G 2. He noted the complexity o f o b t a i n i n g exact s o l u t i o n s f o r the outer r e g i o n . Instead he obtained s i m i l a r -i t y s o l u t i o n s t o the equations by assuming the e x i s t e n c e o f a v e r t i c a l plume i n t h i s r e g i o n . However, i t i s i m p o s s i b l e to match the outer s o l u t i o n t o the r e g u l a r p e r t u r b a t i o n . expansion i n the i n n e r r e g i o n . R e c e n t l y , F e n d e l l [19] o b t a i n e d an approximate s o l u t i o n , f i r s t by seeking a s i m i l a r i t y s o l u t i o n , and then by l i n e a r i z -i n g the equations i n the outer r e g i o n i n the manner of Oseen's equation. The magnitude o f the assumed uniform stream i s based upon the c o o r d i n a t e p e r t u r b a t i o n s o l u t i o n i n the plume. Hence the v e l o c i t y above the sphere i s reduced from unbounded growth wi t h d i s t a n c e from the sphere to a. constant magnitude. T h i s procedure can be expected t o y i e l d q u a l i t a t i v e r e s u l t s o n l y . In t h e i r c o n j e c t u r e s , Hieber and Gebhard [20] showed t h a t i t seems p l a u s i b l e to assume t h a t 116 v e l o c i t y i n the wake behave as a uniform stream o n l y i n the matching r e g i o n . T h i s i s based on t h e i r r e s u l t s on f r e e and f o r c e d c o n v e c t i o n from a sphere a t low Reynolds and Grashof numbers. Hossain and Gebhart [24] employed a s i n g l e parameter p e r t u r b a t i o n scheme i n t h e i r a n a l y s i s to t h i s problem. The Grashof number i s used as the expansion parameter i n combin-a t i o n with e x p o n e n t i a l l y decaying f u n c t i o n s . These decaying f u n c t i o n s w i l l ensure the v e l o c i t y and temperature v a n i s h as R->°°. However, the disadvantages o f t h i s scheme are (a) i t i s o n l y v a l i d f o r very s m a l l G, (b) the temperature boundary c o n d i t i o n at the s u r f a c e of the i n n e r sphere i s not s a t i s f i e d completely. The d i f f i c u l t i e s i n the a n a l y s i s o f s m a l l Grashof numbers from a heated sphere seem t o be t h a t the flow f i e l d i s d i v i d e d i n t o r e g i o n s where the predominance of p a r t i c u l a r p h y s i c a l e f f e c t s v a r i e s f o r each r e g i o n i . e . (a) i n the v i c i n i t y of the sphere the d i f f u s i v e e f f e c t i s predominant, (b) i n the matching r e g i o n of the i n n e r and ou t e r flow f i e l d the c o n v e c t i v e , d i f f u s i v e and bouyancy e f f e c t s are of equal importance, (c) a t i n f i n i t y the v e l o c i t y should be zero. The l i m i t s o f these r e g i o n s i n the flow f i e l d are a p p a r e n t l y very complicated. Hence not a l l s c a l i n g lengths or v e l o c i t i e s are known. The techniques [20] which are employed s u c c e s s -f u l l y f o r the f o r c e d c o n v e c t i o n flows are not s u c c e s s f u l here except q u a l i t a t i v e l y . T h i s i s due to the e n t i r e l y d i f f e r e n t flow f i e l d b e i n g encountered. T h e r e f o r e a study of conjugate n a t u r a l convection from a s i n g l e heated sphere would be i n t e r e s t i n g and c h a l l e n g i n g . 

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