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Analysis of a water-trickle solar collector Wong, Ken Y.F. 1978

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ANALYSIS OF A WATER-TRICKLE SOLAR COLLECTOR by KEN Y.F.JWONG B.A. Sc., U n i v e r s i t y o f B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES i n i n the Department o f M e c h a n i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g to t he r e q u i r e d s t a n d a r d The U n i v e r s i t y o f B r i t i s h Columbia June, 1978 © Ken Y.F. Wong In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Brit ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Ken Y.F. Wong Department of mechanical e n g i n e e r i n g The University of Brit ish Columbia 2 0 7 5 Wesbf-ook P l a c e V a n c o u v e r , C a n a d a V 6 T 1W5 Date August 2 1 s t , 1978. i i ABSTRACT A t h e o r e t i c a l s t u d y o f the performance o f w a t e r - t r i c k l e c o l l e c t o r s was c a r r i e d o u t i n t h i s t h e s i s , where both the f l a t -p l a t e and c o r r u g a t e d - p l a t e w a t e r - t r i c k l e c o l l e c t o r s have been c o n s i d e r e d . In o r d e r to p r e d i c t -the performance o f w a t e r - t r i c k l e c o l l e c t o r s , t h e d i f f e r e n t modes o f heat l o s s from the c o l l e c t o r s must be taken i n t o a c c o u n t . These modes o f heat l o s s a r e r a d i a t i o n , i c o n v e c t i o n , and e v a p o r a t i o n - c o n d e n s a t i o n . A t h e o r i t i c a l model f o r w a t e r - t r i c k l e c o l l e c t o r s has been d e v e l o p e d p r e v i o u s l y . In t h i s model, the heat l o s s due to e v a p o r a t i o n -c o n d e n s a t i o n i s p r e d i c t e d by assuming e v a p o r a t i o n i s by means o f m o l e c u l a r d i f f u s i o n o f water v a p o r t h r o u g h a s t a g n a n t a i r l a y e r . In the new model d e v e l o p e d , t he e v a p o r a t i o n - c o n d e n s a t i o n l o s s i s p r e d i c t e d by means o f the heat-mass t r a n s f e r a n a l o g y h m / h = < P C p L e 2 ' 3 ) -\ By e v a l u a t i n g t h i s mode o f heat l o s s i n a d i f f e r e n t way, the s e n s i t -i v i t y o f the r e s u l t s based on the assumption o f the p r e v i o u s model i s t e s t e d . A t an average p l a t e temperature o f 50°C o r h i g h e r , the heat l o s s due to e v a p o r a t i o n - c o n d e n s a t i o n was found to a c c o u n t f o r a p p r o x i m a t e l y 50 p e r c e n t o r more o f the t o t a l l o s s . T h i s r e s u l t i s i i i a l s o p r e d i c t e d by the p r e v i o u s model. Thus both methods can be used f o r e v a l u a t i n g e v a p o r a t i v e heat l o s s from w a t e r - t r i c k l e c o l l e c t o r s . The heat-mass t r a n s f e r a n a l o g y was then extended t o c o r r u -g a t e d - p l a t e w a t e r - t r i c k l e c o l l e c t o r s . The heat l o s s due t o e v a p o r a t i o n -c o n d e n s a t i o n was e v a l u a t e d and was found t o be l e s s than l o s s e s from f l a t - p l a t e w a t e r - t r i c k l e c o l l e c t o r s . A new pr o c e d u r e was d e v e l o p e d i n the p r e s e n t work t o e v a l u a t e (xa), the f r a c t i o n o f i n c i d e n t energy t h a t i s u l t i m a t e l y absorbed by a c o r r u g a t e d w a t e r - t r i c k l e c o l l e c t o r . T h i s p r o c e d u r e was d e r i v e d based on p e r f o r m i n g r a d i a t i v e heat b a l a n c e s o v e r s u r f a c e s i n s i d e a water-t r i c k l e c o l l e c t o r . Based on t h i s p r o c e d u r e , v a l u e s o f (TCX) f o r c o r r u g a t e d s u r f a c e s were o b t a i n e d and were found t o be h i g h e r than v a l u e s f o r f l a t s u r f a c e s . T h i s r e s u l t i s r e a s o n a b l e s i n c e t h e r e i s a l a r g e r a r e a exposed to r a d i a t i v e heat i n p u t on a c o r r u g a t e d s u r f a c e than on a f l a t s u r f a c e . E x p r e s s i o n s f o r the temperature d i s t r i b u t i o n a l o n g the c o r r u g a t e d c r o s s - s e c t i o n o f the a b s o r b i n g s u r f a c e o f a w a t e r - t r i c k l e c o l l e c t o r were d e v e l o p e d . These e x p r e s s i o n s were d e r i v e d based on s t a n d a r d heat t r a n s f e r c o r r e l a t i o n s f o r f i n s . With t h e s e e x p r e s s i o n s , temperature p r o f i l e s o f the c r o s s - s e c t i o n o f a c o r r u g a t e d water-t r i c k l e c o l l e c t o r was o b t a i n e d by u s i n g r a d i a t i o n d a t a a v a i l a b l e f o r Vancouver. With t h e s e temperature p r o f i l e s , an e x p r e s s i o n f o r the r a t e o f u s e f u l heat g a i n o f a c o r r u g a t e d w a t e r - t r i c k l e c o l l e c t o r was o b t a i n e d . i v An attempt has been made to d e r i v e s t a n d a r d c o l l e c t o r f a c t o r s such as F' and F R , w h ich, i f s u c c e s s f u l , would s i m p l i f y f u t u r e e f f o r t i n o b t a i n i n g v a l u e s o f the u s e f u l heat g a i n o f a c o r r u g a t e d w a t e r - t r i c k l e c o l l e c t o r e x t e n s i v e l y . The e x p r e s s i o n s d e r i v e d i n t h i s t h e s i s , however, a r e c o m p l i c a t e d and a r e d e t e r m i n e d t o be o f l i t t l e p r a c t i c a l s i g n i f i c a n c e . The r e a s o n b e i n g t h a t e x t e n s i v e computations a r e u s u a l l y i n v o l v e d i n o b t a i n i n g v a l u e s f o r t h e s e f a c t o r s and such e f f o r t s a r e seldom w o r t h w h i l e . V ACKNOWLEDGEMENT The work d e s c r i b e d i n the t h e s i s was con d u c t e d under the s u p e r v i s i o n o f Dr. M. I q b a l . The a u t h o r wishes to e x p r e s s h i s g r a t i t u d e f o r Dr. I q b a l ' s guidance and a d v i c e d u r i n g t h i s r e s e a r c h . Thanks a l s o go t o Dr. N. E p s t e i n f o r h i s a d v i c e i n the mass t r a n s f e r a s p e c t o f the o v e r a l l a n a l y s i s . Above a l l , I thank God f o r g i v i n g me the o p p o r t u n i t y t o b e t t e r m y s e l f a c a d e m i c a l l y and s p i r i t u a l l y d u r i n g the c o u r s e o f the r e s e a r c h . T h i s r e s e a r c h i s s u p p o r t e d by the N a t i o n a l Research C o u n c i l o f Canada under Grant No. 67-2283. v i TABLE OF CONTENTS S e c t i o n Page ABSTRACT i i ACKNOWLEDGEMENT i v TABLE OF CONTENTS v LIST OF TABLES x i LIST.OF FIGURES x NOMENCLATURE x i v 1. GENERAL INTRODUCTION 1 2. PREVIOUS WORK 3 2.1 Work R e l a t e d t o F l a t - P l a t e Water-T r i c k l e C o l l e c t o r s 3 2.2 Work R e l a t e d t o C o r r u g a t e d - P l a t e W a t e r - T r i c k l e C o l l e c t o r s 5 2.3. R e l a t e d Work on C o n v e c t i v e Heat T r a n s f e r 8 2.3.1 Free C o n v e c t i v e Heat T r a n s f e r . . . 8 2.3.2 Fo r c e d C o n v e c t i v e Heat T r a n s f e r . . 11 2.3.3 Combined Free and F o r c e d C o n v e c t i o n 13 v i i S e c t i o n Page PART I. FLAT-PLATE WATER-TRICKLE SOLAR COLLECTOR MODEL . . 15 3. DEVELOPMENT OF PRESENT FLAT-PLATE MODEL 16 3.1 Combined Free and F o r c e d C o n v e c t i o n . . . . 16 3.2 Flow o f T h i n Water F i l m Down An I n c l i n e d Plane 17 3.3 Heat T r a n s f e r A c r o s s a Laminar T h i n F i l m Flow 19 3.4 Heat and Mass T r a n s f e r A c r o s s the A i r L a y e r 21 3.4.1 Method o f S o l u t i o n 21 3.4.2 An Open-Ended W a t e r - T r i c k l e C o l l e c t o r . 22 4. DEVELOPMENT OF COLLECTOR FACTORS 27 4.1 Heat T r a n s f e r A c r o s s a T h i n L i q u i d F i l m 28 4.1.1 Fo r c e d C o n v e c t i v e Heat T r a n s f e r . . . 28 4.1.2 R a d i a t i v e Heat T r a n s f e r C h a r a c t e r i s t i c s 30 4.2 Heat and Mass T r a n s f e r C o e f f i c i e n t s f o r Ai r 31 4.3 D e r i v a t i o n o f F 1 and F R 32 5. COMPARISON OF RESULTS BASED ON BEARD'S MODEL AND THE PRESENT FLAT-PLATE WATER-TRICKLE COLLECTOR MODEL 37 v i i i S e c t i o n Page PART I I . CORRUGATED-PLATE WATER-TRICKLE SOLAR COLLECTOR MODEL 42 6. INTRODUCTION 43 7. PRESENT WORK 45 7.1 C o r r u g a t i o n Geometry 45 7.2 E f f e c t o f P l a t e Geometry on Heat A b s o r p t i o n C h a r a c t e r i s t i c s 45 7.3 Heat T r a n s f e r A n a l y s i s 48 7.3.1 D e r i v a t i o n o f Flow F a c t o r s f o r a C o r r u g a t e d - P l a t e C o l l e c t o r 48 7.3.2 D e r i v a t i o n o f Heat T r a n s f e r C o e f f i c i e n t s 54 8. DISCUSSION OF RESULTS 8.1 Comparison o f R e s u l t s 60 8.2 D i s c u s s i o n o f the Models 62 8.3 E f f e c t o f Gap S p a c i n g on Heat T r a n s f e r Rates 63 9. CONCLUSIONS 66 10. RECOMMENDATIONS 69 REFERENCES 71 APPENDICES A. ITERATION PROCEDURE FOR DETERMINING HEAT TRANSFER CHARACTERISTICS OF A WATER-TRICKLE COLLECTOR 76 B. DETAILED HEAT TRANSFER ANALYSIS FOR FLUIDS INSIDE A WATER-TRICKLE COLLECTOR 80 B . l Temperature D i s t r i b u t i o n o f the T h i n Water F i l m . . . 81 i x S e c t i o n Page B.2 C o n v e c t i v e Heat T r a n s f e r A c r o s s the A i r L a y e r s 88 C. DERIVATION OF THE MASS TRANSFER CORRELATION FOR SURFACES WITH A UNIFORM VELOCITY PROFILE 95 D. DERIVATION OF (xa) FOR A CORRUGATED-PLATE SURFACE 99 D.I Dry C o r r u g a t e d - P l a t e 99 D.1.1 D e r i v a t i o n o f (xa) 99 D.1.2 D e r i v a t i o n o f Shape F a c t o r s 107 D.2 C o r r u g a t e d - P l a t e w i t h T h i n Water F i l m s i n V a l l e y s of. the C o r r u g a t i o n 109 D.2.1 D e r i v a t i o n o f (xa) 109 D.2.2 F r a c t i o n s o f I n c i d e n t F l u x Re-f l e c t e d and T r a n s m i t t e d f o r an N-Cover System . 115 D.3 E v a l u a t i o n o f the Monthly Averaged (xa)'s o f a C o r r u g a t e d - P l a t e f o r Vancouver . . . . 117 E. DERIVATION OF USEFUL HEAT GAIN EXPRESSION FOR A CORRUGATED-PLATE SOLAR COLLECTOR . 119 LIST OF TABLES T a b l e Page 1. Mass T r a n s f e r C o e f f i c i e n t vs. Reynolds Number . . 26 2. Heat T r a n s f e r C h a r a c t e r i s t i c s o f a S i n g l e -Cover C o r r u g a t e d C o l l e c t o r a t D i f f e r e n t Average P l a t e Temperatures 59 x i LIST OF FIGURES F i g u r e Page 1. S k e t c h o f a Tube- a n d - P l a t e C o l l e c t o r 129 2. S k e t c h o f a C o r r u g a t e d - P l a t e W a t e r - T r i c k l e . C o l l e c t o r . 130 3. Development o f A i r and Water V e l o c i t y P r o f i l e s Down an I n c l i n e d P l a n e 131 4. P r o b a b l e A i r and Water V e l o c i t y P r o f i l e s I n s i d e an A i r - T i g h t C o l l e c t o r 132 5. T h i n R i p p l i n g F i l m Flow Down an I n c l i n e d P l a n e 133 6. D i m e n s i o n l e s s Heat T r a n s f e r C o e f f i c i e n t s as a F u n c t i o n o f the Reynolds Number 4 x/u 134 7. P r o b a b l e V e l o c i t y P r o f i l e s f o r A i r and Water Under the I n f l u e n c e o f Combined Free and F o r c e d C o n v e c t i o n 135 8. Open-Ended W a t e r - T r i c k l e C o l l e c t o r 136 9. E n l a r g e d View o f the Flow P a t t e r n I n s i d e an Open-Ended C o l l e c t o r ^37 10. Energy B a l a n c e f o r t he T h i n Water F i l m F l o w i n g Down a n - I n c l i n e d Plane 138 11. R i s e o f Temperature o f Water Al o n g the C o l l e c t o r 139 12. F" as a F u n c t i o n o f G C p w / U t F ' 140 13. F r o n t a l Heat Loss as a F u n c t i o n o f Water S u r f a c e Temperature f o r a S i n g l e - C o v e r C o l l e c t o r 141 14. F r o n t a l Heat Loss as a F u n c t i o n o f Water S u r f a c e Temperature f o r a Two-Cover C o l l e c t o r 142 x i i F i g u r e Page 15. F r o n t a l Heat Loss as a F u n c t i o n o f Water S u r f a c e Temperature f o r a T h r e e - C o v e r C o l l e c t o r 143 16. Cover S u r f a c e Temperature as a F u n c t i o n o f W a t e r - S u r f a c e Temperature f o r a S i n g l e - C o v e r C o l l e c t o r "144 17. Cover S u r f a c e Temperature as a F u n c t i o n o f W a t e r - S u r f a c e Temperature f o r a Two-Cover C o l l e c t o r "145 18. Cover S u r f a c e Temperature as a F u n c t i o n o f -Water-Surface Temperature f o r a T h r e e - C o v e r System "146 19. F r o n t a l Heat Loss C o e f f i c i e n t as a F u n c t i o n o f Average Water S u r f a c e Temperature, • e -=0.95, wind = 0 147 20. F r o n t a l Heat Loss C o e f f i c i e n t as a F u n c t i o n •of Average Water S u r f a c e Temperature, e p = 0.95, wind = 5 m/s 148 21. F r o n t a l Heat Loss C o e f f i c i e n t as a F u n c t i o n o f Average Water S u r f a c e Temperature, e = 0.95, wind = 10 m/s . 149 r 22. F r o n t a l Heat Loss C o e f f i c i e n t as a F u n c t i o n o f Average Water S u r f a c e Temperature, e =0.1, wind = 0 150 23. F r o n t a l Heat Loss C o e f f i c i e n t as a F u n c t i o n o f Average Water S u r f a c e Temperature, e p = 0.1, wind = 5 m/s 151 24. F r o n t a l Heat Loss C o e f f i c i e n t as a F u n c t i o n o f Average Water S u r f a c e Temperature, e = 0.1, wind = 10 m/s I 5 2 25. Heat L o s s e s from a C o l l e c t o r P l a t e f o r a One-Cover, Two-Cover, and T h r e e - C o v e r System, £ p = .95, 153 26. Heat L o s s e s from a C o l l e c t o r P l a t e f o r a One-Cover, Two-Cover, and T h r e e - C o v e r System, e = 0.1 154 x i i i F i g u r e Page 27. C o l l e c t o r E f f i c i e n c y as a F u n c t i o n o f F l u i d I n l e t Temperature 155 28. F l a t - P l a t e W a t e r - T r i c k l e C o l l e c t o r Performance C o r r e l a t i o n 156 29. A S o u t h - F a c i n g C o r r u g a t e d P l a t e With S h o r t -wave Beam R a d i a t i o n I n c i d e n t a t an Ang l e 6 . . . 157 30. D e t a i l s o f the C o r r u g a t e d C r o s s - S e c t i o n 158 31. Energy Weighted Monthly Averaged ( x a ) f o r 3 = 30° I 5 9 32. Energy Weighted Monthly Averaged (TO) f o r • 8 = 50° 1 6 0 33. Energy Weighted Monthly Averaged (TO) f o r B = 70° 161 34. (TO) as a F u n c t i o n o f the A n g l e o f I n c i d e n c e 9 f o r a R e c t a n g u l a r F i n n e d C o l l e c t o r P l a t e , •3 = 70° 1 6 2 35. Q u a l i t a t i v e S k e t c h o f the Temperature P r o f i l e A l o n g t he C o r r u g a t e d C r o s s - S e c t i o n 163 36. F r o n t a l Heat Loss Rate vs. Average P l a t e Temperature f o r a S i n g l e - C o v e r C o r r u g a t e d W a t e r - T r i c k l e C o l l e c t o r 164 37. F r o n t a l Heat Loss Rate vs. Average P l a t e Temperature f o r a Two-Cover; C o r r u g a t e d W a t e r - T r i c k l e C o l l e c t o r 165 38. F r o n t a l Heat Loss Rate vs. Average P l a t e Temperature f o r a T h r e e - C o v e r C o r r u g a t e d W a t e r - T r i c k l e C o l l e c t o r 166 39. F r o n t a l Heat Loss. C o e f f i c i e n t as a F u n c t i o n . .1 o f the Average P l a t e Temperature o f a C o r r u g a t e d W a t e r - T r i c k l e C o l l e c t o r , Wind V e l o c i t y = 0 167 x i v F i g u r e Page 40. F r o n t a l Heat Loss C o e f f i c i e n t as a F u n c t i o n o f the Average P l a t e Temperature o f a C o r r u g a t e d W a t e r - T r i c k l e C o l l e c t o r , Wind V e l o c i t y =5 m/s. . . 168 41. F r o n t a l Heat Loss C o e f f i c i e n t as a F u n c t i o n o f the Average P l a t e Temperature o f a C o r r u g a t e d W a t e r - T r i c k l e C o l l e c t o r , Wind V e l o c i t y =10 m/s . . 169 42. C o r r u g a t e d W a t e r - T r i c k l e C o l l e c t o r Performance C o r r e l a t i o n 170 43. F r o n t a l Heat Loss Rate as a F u n c t i o n o f Gap S p a c i n g 171 44. E v a p o r a t i v e Heat Loss Rate as a F u n c t i o n o f Gap S p a c i n g 172 45. Net R a d i a t i o n B a l a n c e f o r a Dry C o r r u g a t e d S u r f a c e 173 46. S k e t c h o f an N-Cover System 174 47. Net R a d i a t i o n B a l a n c e f o r a Wetted C o r r u g a t e d S u r f a c e 175 ' X V NOMENCLATURE a a i r l a y e r t h i c k n e s s i n s i d e the c o l l e c t o r (m) A s u r f a c e a r e a o f c o l l e c t o r (m ) b a i r l a y e r t h i c k n e s s i n s i d e the c o l l e c t o r (m) C p w s p e c i f i c heat o f water ( J k g - 1 ° C _ 1 ) C^ c o n c e n t r a t i o n o f s o l u t e A (kgm~ ) d a i r gap t h i c k n e s s from w a t e r s u r f a c e to the a d j a c e n t c o v e r s u r f a c e (m) D-j w i d t h o f water f i l m i n the v a l l e y s o f a c o r r u g a t e d p l a t e (m) D2 wi d t h o f the top p a r t o f a c o r r u g a t e d p l a t e (m) Dg c o r r u g a t i o n h e i g h t (m) 2 1 D^ B d i f f u s i v i t y o f s o l u t e A i n s o l v e n t B (m S" ) 2 -1 Dy d i f f u s i v i t y (m s ) E Q 8T / 8 X 1 /2 f a reduced stream f u n c t i o n , \Jj(x,y)/(2vU o ox) f g d i m e n s i o n l e s s v e l o c i t y u/U-j F s t a n d a r d f i n e f f i c i e n c y F 1 c o l l e c t o r e f f i c i e n c y f a c t o r F^ c o l l e c t o r h e a t removal f a c t o r F 5 1 ' F 5 2 ' F 5 3 ' F 5 4 s h a p e f a c t o r s F c V F c 2 ' F c 3 ' F c 4 f 1 " 9 0 * 1 0 0 5 °f s o l a r energy absorbed i n a c o r r u g a t e d s u r f a c e xv i F f r a c t i o n o f t o t a l s o l a r energy absorbed by w a t e r g g r a v i t a t i o n a l c o n s t a n t (ms ) 1 -2 G water f l o w r a t e p e r u n i t a r e a ( k g s " m ) 3 2 Gr G r a s h o f number, gBgATd /v -2 -1 h heat t r a n s f e r c o e f f i c i e n t (Wm °C ) -2 -1 h c c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t (Wnf °C" ) h mass t r a n s f e r c o e f f i c i e n t (ms~^) m h o v e r a l l f o r c e d c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t from w ater s u r f a c e t o the a d j a c e n t c o v e r s u r f a c e 2 1 i n a w a t e r - t r i c k l e c o l l e c t o r (Wm °C~ ) H monthly average d a i l y t o t a l r a d i a t i o n r e c e i v e d on a 2 -1 h o r i z o n t a l s u r f a c e (Mdrrf day" ) p I i n c i d e n t s o l a r f l u x (Wm ) k thermal c o n d u c t i v i t y (Wm~^°C~^) k-j,k£ c o n s t a n t s Le Lewis number, k/pCpD^g L l e n g t h o f c o l l e c t o r (m) m ( U t / k S ) } ^ ( r r f 1 ) P p r e s s u r e (Nm~ ) P^ m log-mean p r e s s u r e d i f f e r e n c e ( P - j ) / ^ n ( P - j / P 2 ) » (Nm~ 2) Pr P r a n d t l number (yC /k) q heat t r a n s f e r r a t e (Wm ) q i , c l ' q i , c 2 ' q i , c 3 ' q i , c 4 f r a c t i o n s o f r a d i a n t f l u x i n c i d e n t on a c o r r u g a t e d s u r f a c e xv i i a,0 c i .Pg c 2 » f r a c t i o n s o f o u t g o i n g r a d i a n t f l u x from a c o r r u g a t e d %,c3>%,c* s u r f a c e R u n i v e r s a l gas c o n s t a n t (8314.4 J Kmole" 1 ° K _ 1 ) Ra R a l e i g h number (Gr • Pr) R . gas c o n s t a n t f o r a i r (287.045 J k g ' ^ K " 1 a i r ) K W o^ Re v Reynolds number, ux/v Re L Reynolds number, uL/v RF r e d u c t i o n f a c t o r f o r i n c i d e n t s o l a r f l u x p a s s i n g through condensed water d r o p l e t s (N) R ^ y f r a c t i o n o f t o t a l i n c i d e n t f l u x r e f l e c t e d a t the (2N)th s u r f a c e f o r an N-cover c o l l e c t o r ws t o t a l f r a c t i o n o f i n c i d e n t f l u x r e f l e c t e d from a wat e r l a y e r S i n c i d e n t s o l a r f l u x a f t e r p a s s i n g through the c o v e r systems (Wm~2) S 1 amount o f energy absorbed by water (Win ) Sc Schmidt number, v/D^g S f s h a d i n g f a c t o r f o r beam r a d i a t i o n on c o r r u g a t e d s u r f a c e S h Q v g average Sherwood number, h m _ a v g L / D A B S h l o c a l 1 o c a l S h e ™ 0 0 d ™ m b e i " » h m - l o c a l x / D A B t TIME T temperature (°C) i n l e t f l u i d t emperature (°C) f r a c t i o n o f t o t a l i n c i d e n t f l u x t r a n s m i t t e d a f t e r p a s s i n g through the (2N)th s u r f a c e , f o r an N-cover system v e l o c i t y a l o n g a c o l l e c t o r s u r f a c e (ms~^) c o n s t a n t v e l o c i t y (ms~^) -2 -1 t o t a l f r o n t a l heat l o s s c o e f f i c i e n t (Wm °C ) wind v e l o c i t y (ms~^) v e l o c i t y normal to a c o l l e c t o r s u r f a c e (ms~^) f l u i d f l o w r a t e (kg s"^) a dimension f o r a c o r r u g a t e d s u r f a c e (m) d i s t a n c e a l o n g the water fl o w d i r e c t i o n (m) d i s t a n c e normal to a c o l l e c t o r s u r f a c e (m) wi d t h o f a c o l l e c t o r s u r f a c e (m) thermal d i f f u s i v i t y (m s~ ) an g l e o f i n c l i n a t i o n o f the t i l t e d p a r t o f a c o r r u g a t e d p l a t e c o r r u g a t i o n a n g l e a n g l e o f i n c l i n a t i o n o f a c o l l e c t o r from the h o r i z o n t a l v o l u m e t r i c c o e f f i c i e n t o f e x p a n s i o n f o r a i r (°C~^) f l o w r a t e p e r u n i t w i d t h (kg m~^s~^) x v i x 6 d e c l i n a t i o n ( t h e a n g u l a r p o s i t i o n o f the sun a t s o l a r noon w i t h r e s p e c t t o the p l a n e - o f - t h e e q u a t o r ) 6^. t h i c k n e s s o f f i n (m) 6-j t h i c k n e s s o f the water f i l m (m) 6' p e n e t r a t i o n depth o f mass t r a n s f e r (m) X l a t e n t heat o f v a p o r a t i o n ( J kg~^) u dynamic v i s c o s i t y (kg m~^s~^) 2 -1 v k i n e m a t i c v i s c o s i t y (m s ) C G r x / R e x 2 - 5 h GV R ex 2 i i y/2 / v t " n-j y /u/2x 0 a n g l e o f i n c i d e n c e o f beam r a d i a t i o n 6-| d i m e n s i o n l e s s temperature e e m i s s i v i t y a Stephen-Boltzmann c o n s t a n t ( 5 . 6 0 9 7 x l O " 8 W m " 2 ° K " 4 ) a-| s u r f a c e t e n s i o n (Nm~^) (j) l a t i t u d e *1 y/2 / ~ u D A B X ijj s t r e a m f u n c t i o n u hour angle x t r a n s m i s s i v i t y p r e f l e c t i v i t y (xa) S u b s c r i p t s a b c d ec f 9 i % 9 i 0 P r w W-| ws t bO pc XX _3 d e n s i t y o f w a t e r , (kg m ) transmi t t a n c e - a b s o r p t a n c e p r o d u c t a i r b u l k , beam c o l l e c t o r d i f f u s e e v a p o r a t i o n - c o n d e n s a t i o n f l u i d i n n e r c o v e r s u r f a c e o u t e r c o v e r s u r f a c e c o v e r i n i t i a l e x t r a - t e r r i s t r i a l pi a t e r a d i a t i o n w ater w a l l water s u r f a c e tube, t o t a l bond p l a t e to c o v e r ANALYSIS OF A WATER-TRICKLE SOLAR COLLECTOR 1 1. GENERAL INTRODUCTION E v e r s i n c e the f u e l c r i s i s became a r e a l i t y , p e o p l e i n the i n d u s t r i a l i z e d w o r l d have e x p e r i e n c e d the i n c o n v e n i e n c e o f energy s h o r t a g e . V a r i o u s a l t e r n a t i v e energy s o u r c e s have been propos e d , and one o f t h e s e i s s o l a r energy. The i d e a o f h a r n e s s i n g s o l a r energy i s not new. In f a c t , many d i f f e r e n t d e s i g n s o f s o l a r c o l l e c t o r s were b u i l t and t e s t e d s e v e r a l decades ago, but no major e f f o r t has been made i n the f i e l d u n t i l r e c e n t l y . S o l a r energy has many advantages. I t i s c l e a n , i n e x h a u s -t i b l e and a v a i l a b l e almost anywhere on e a r t h . However, the i n t e n s i t y o f s o l a r r a d i a t i o n i s low and u s u a l l y l a r g e c o l l e c t o r areas are r e q u i r e d . T h i s f a c t o f t e n poses problems f o r p l a c e s where open space i s s c a r c e . I f one i s c o n s i d e r i n g u t i l i z i n g s o l a r energy f o r r e s i d e n -t i a l h e a t i n g p u r p o s e s , h i g h temperatures o f t e n are n o t r e q u i r e d . T h i s l e a d s t o a f i e l d o f s o l a r energy u t i l i z a t i o n c a l l e d f l a t - p l a t e c o l l e c t o r t e c h n o l o g y . The most common form o f f l a t - p l a t e c o l l e c t o r i s the c o n v e n t i o n a l t u b e - a n d - p l a t e type ( F i g u r e . ! ) . T h i s type o f c o l l e c t o r i s being, used e x t e n s i v e l y and much work has been done i n a n a l y z i n g i t s thermal performance. In o r d e r to s i m u l a t e the p e r f o r m -ance o f t h i s type o f c o l l e c t o r , v a r i o u s models have been proposed. The most common are those o f H o t t e l and Woertz [1], H o t t e l and w h i l l i e r [ 2 ] , W h i l l i e r [ 3 ] , B l i s s [ 4 ] and K l e i n [ 5 ] . The f i r s t f o u r 2 models neglect the effects of collector thermal capacitance because of the amount of extra computation involved. Klein suggests that these effects should be taken into account because they may be significant. Another type of flat-plate collector is called the water-trickle collector (Figure 2). A water-trickle collector, as the name implies, involves water trickling down the collector plate surface and absorbing part of the insolation. The plate itself can either be flat or corrugated. In this thesis, both the flat-plate and corrugated-plate trickling flow collectors will be modeled and the heat transfer characteristics analyzed. Each type will be treated separately; Part I deals with the flat-plate and Part II deals with the corrugated plate. 3 2. PREVIOUS WORK 2.1 Work R e l a t e d t o F l a t - P l a t e W a t e r - T r i c k l e C o l l e c t o r s A l i t e r a t u r e s e a r c h has shown t h a t t h e r e i s no e x p e r i m e n t a l work r e l a t i n g to f l a t - p l a t e w a t e r - t r i c k l e c o l l e c t o r s . There i s , however, one a n a l y t i c a l model f o r t h i s type o f c o l l e c t o r . T h i s model was d e v e l o p e d by Beard [ 6 ] d u r i n g h i s work w i t h the Thomason ' S o l a r i s ' w a t e r - t r i c k l e c o l l e c t o r s . A Thomason c o l l e c t o r i s a c o r r u g a t e d - p l a t e w a t e r - t r i c k l e c o l l e c t o r i n v e n t e d by H.E. Thomason. Because o f i t s p o p u l a r i t y , Beard d e c i d e d t o i n v e s t i g a t e i t s performance experimen-t a l l y . In o r d e r to c o r r e l a t e the e x p e r i m e n t a l d a t a , Beard d e v e l o p e d a s i m p l e model f o r t h i s type o f c o l l e c t o r . The major assumptions he used i n h i s a n a l y s i s were: 1. The l o c a l water temperature i s the same as the l o c a l p l a t e t e m perature. 2. The c o l l e c t o r a b s o r b i n g p l a t e i s f l a t . 3. The c o n v e c t i v e h e a t t r a n s f e r f o r the a i r l a y e r i n s i d e the c o l l e c t o r i s a p u r e l y f r e e c o n v e c t i v e t r a n s f e r . 4. The dropwise c o n d e n s a t i o n e f f e c t i s n e g l e c t e d . 5. The e v a p o r a t i v e mechanism o f water vapour from the w a t e r s u r f a c e i s due t o m o l e c u l a r d i f f u s i o n o f vapour m o l e c u l e s a c r o s s a s t a g n a n t a i r l a y e r . 4 The t h r e e modes o f heat t r a n s f e r from the w a t e r s u r f a c e , t h e r e f o r e , a re r a d i a t i o n , c o n v e c t i o n and e v a p o r a t i o n - c o n d e n s a t i o n . Assuming i n f i n i t e p a r a l l e l p l a t e s , the r a d i a t i v e l o s s i s q r l a ( T 4 - T 4 ) v ws g r (2.1) The c o n v e c t i v e l o s s i s g i v e n by H o l l a n d ' s c o r r e l a t i o n f o r f r e e c o n v e c t i o n [ 7 ] : % 1 k«A,s-V 1+1.44 1 1708 Ra cos 3 . ', 1 7 0 8 ( s i n 1.83) 1.6 Ra cos 3 Ra cos 3 5830 1/3 •(2.2) where [ ] i s s e t to zero i f the q u a n t i t y i n s i d e the b r a c k e t i s n e g a t i v e . The mass t r a n s f e r l o s s i s g i v e n by the s t a n d a r d d i f f u s i o n e q u a t i o n : ADvPa(P - P •) v ws g r 1 e c d RT P, (2.3) £ m The t o t a l h e a t l o s s i s t h e r e f o r e 5 q q r l + q c l + q e c . . . .(2.4) T h i s t o t a l h eat l o s s i s assumed equal to the heat t r a n s f e r e d through the c o v e r , The heat t r a n s f e r r e d to the ambient due to c o n v e c t i v e and r a d i a t i v e e f f e c t s i s c o v e r Thomason c o l l e c t o r , Beard c a l c u l a t e d the f r o n t a l heat l o s s e s a t d i f f e r e n t average p l a t e t e m p e r a t u r e s . A l t h o u g h Beard s t a t e s t h a t h i s c a l c u l a t e d r e s u l t s are i n good agreement w i t h the e x p e r i m e n t a l r e s u l t s , i t i s f e l t t h a t an improved model t o p r e d i c t the performance o f a w a t e r - t r i c k l e c o l l e c t o r can be d e v e l o p e d by removing some o f h i s a s sumptions. q . . . .(2.5) q - MVV + V ( TgO- Ta) . . . .(2.6) By u s i n g the i t e r a t i o n p r o c e d u r e i n Appendix A f o r the one-2.2 Work R e l a t e d t o C o r r u g a t e d - P l a t e W a t e r - T r i c k l e C o l l e c t o r s A l i t e r a t u r e s e a r c h i n t o the work r e l a t e d to w a t e r - t r i c k l e c o l l e c t o r s has shown t h a t a l l i n v e s t i g a t i o n s so f a r have been e x p e r i -6 mental in nature. Although Beard proposed a simple theoretical model, his work was s t i l l primarily experimental. Some of these investigations will be described in the following paragraphs. In order to compare performances, San Martin [ 8 ] carried out experiments on three types of collectors. They were a water-trickle collector, a thermal trap collector, and a tube-and-plate collector. The three collectors were erected side by side in outdoors. All data were recorded simultaneously for the collectors under a variety of different operating conditions. The collector of interest in this case is the water-trickle collector. In constructing this collector, San Martin used a corrugated aluminum roofing material with a sine-wave cross-sectional shape as his absorber plate. This collector had 2 a collection area of about 15 ft and was roughly square in shape. Based on his results, San Martin concluded that a water-trickle collector should be operated at temperatures below 125°F(62°C) in order to maintain a good operating efficiency. He did not clearly state what this efficiency should be,.but based on one set of his data, an efficiency of thirty-five percent was obtained when the collector was operated at about 150°F. Smith [9] presented a method of numerical modelling of water-trickle collectors in an attempt to predict San Martin's experimental results. Generally, he obtained results which showed the same trend as San Martin's data but the values were somewhat higher. Because he did not take into account dropwise-condensation 7 effects, he attributed the differences among the results to the un-certainty of the solar flux incident on.the absorber plate surface. This uncertainty was caused by the random formation of condensed water vapour droplets on the inside cover surface adjacent to the absorber plate. Bush [10] constructed a home-made corrugated water-trickle collector and evaluated its performances under a variety of operating conditions. In his work, both experimental and analytical studies were done. His analytical work was basically for a tube-and-plate collector since he did not take into account the mass transfer effects. Therefore, although his experimental results are useful, his theoretical results are inapplicable.One interesting feature of Bush's collector is that the ends are open. Thus atmospheric air can enter and exit the collector under the influences of the surface drag of the trickling water film and the wind. The analysis presented in Section 3.4.2 can therefore be used to predict the performance of such a collector. A very.popular type of water-trickle collector is the Thomason 'Solaris' collector. In order to evaluate its performances, Beard [6] carried out outdoor tests on two such collectors. Generally, he concluded that the Thomason collectors had high efficiencies at low operating temperatures. He also concluded that under the same kind of operating conditions, a Thomason collector was not as efficient as a tube-and-plate collector. This conclusion is logical since there 8 i s an e x t r a mode o f e v a p o r a t i v e h e a t t r a n s f e r i n h e r e n t i n w a t e r - t r i c k l e c o l l e c t o r s . Beard then d e v e l o p e d a s i m p l e mass d i f f u s i o n model i n an attempt t o p r e d i c t performances o f w a t e r - t r i c k l e c o l l e c t o r s . The l i m i t a t i o n s o f t h i s model have been d i s c u s s e d i n S e c t i o n 2.1 o f t h i s t h e s i s . From the f o r e g o i n g r e v i e w , i t i s c l e a r t h a t most s t u d i e s on w a t e r - t r i c k l e c o l l e c t o r s are e x p e r i m e n t a l i n n a t u r e . In the f o l l o w i n g s e c t i o n , a d e t a i l e d r eview o f the c o n v e c t i v e heat t r a n s f e r between p a r a l l e l s u r f a c e s w i l l be d i s c u s s e d . Free and f o r c e d con-v e c t i v e heat t r a n s f e r w i l l be reviewed s e p a r a t e l y w i t h the aim o f a r r i v i n g a t an a c c u r a t e heat t r a n s f e r a n a l y s i s f o r a c o r r u g a t e d w a t e r - t r i c k l e c o l l e c t o r . 2.3 R e l a t e d Work on C o n v e c t i v e Heat T r a n s f e r 2.3.1 Free C o n v e c t i v e Heat T r a n s f e r P u r e l y f r e e c o n v e c t i v e h e a t t r a n s f e r between two h o r i z o n t a l plane s u r f a c e s p a r a l l e l to each o t h e r has been, s t u d i e d e x t e n s i v e l y [11-14]. R e c e n t l y , some heat t r a n s f e r s t u d i e s between two i n c l i n e d p l a n e s u r f a c e s p a r a l l e l to each o t h e r have been c a r r i e d o u t . These s t u d i e s may be u s e f u l i n s o l a r energy a p p l i c a t i o n s . Because o f the e x i s t e n c e o f f r e e c o n v e c t i o n i n w a t e r - t r i c k l e c o l l e c t o r s , a b r i e f r e v iew o f the work r e l a t e d to t h i s mode o f h e a t t r a n s f e r w i l l be o u t l i n e d below. 9 In f r e e c o n v e c t i o n , the two l i m i t i n g cases o f v e r t i c a l and h o r i z o n t a l l a y e r s have been i n v e s t i g a t e d [11-17] both e x p e r i m e n t a l l y and a n a l y t i c a l l y . Thus the heat t r a n s f e r mechanisms and c o r r e l a t i o n s a r e w e l l documented. Globe and/Dropkin [11] s t u d i e d f r e e c o n v e c t i v e h eat t r a n s f e r i n l i q u i d s c o n f i n e d by two h o r i z o n t a l p l a t e s , w i t h the bottom p l a t e h e a t e d . Plows [16] d i d a n u m e r i c a l s t u d y o f the s t e a d y s t a t e f r e e c o n v e c t i v e heat t r a n s f e r by i n v e s t i g a t i n g the e f f e c t s o f the f o r m a t i o n o f the s o - c a l l e d B e r n a r d c e l l s . E c k e r t and C a r l s o n [17] s t u d i e d the f r e e c o n v e c t i o n i n an a i r l a y e r c o n f i n e d between two v e r t i c a l p l a t e s o f d i f f e r e n t temperatures.. The above are j u s t a few o f the many examples o f s t u d i e s r e l a t i n g to. n a t u r a l c o n v e c t i o n f o r both h o r i z o n t a l and v e r t i c a l s u r f a c e s . These s t u d i e s were extended to f r e e c o n v e c t i v e heat t r a n s f e r between p a r a l l e l s u r f a c e s i n c l i n e d a t v a r i o u s a n g l e s . Dropkin and Sommerscales [18] s t u d i e d f r e e c o n v e c t i v e h e a t t r a n s f e r i n l i q u i d s between i n c l i n e d s u r f a c e s . W i t h i n the range o f 4 8 R a l e i g h numbers they c o n s i d e r e d ( 5 x 1 0 t o 7 . 1 7 x 1 0 ) , they assumed t h a t the f l o w was p u r e l y t u r b u l e n t . T h i s assumption was s u p p o r t e d by the work o f Globe and D ropkin f o r . the h o r i z o n t a l c a s e , where t u r b u l e n t f l o w was found to e x i s t f o r Ra > 5 x l 0 4 . However, s i n c e the f l o w regimes o f the i n c l i n e d f l u i d l a y e r may be d i f f e r e n t from t h o s e o f the h o r i z o n t a l l a y e r , such an assumption can o n l y be viewed as b e i n g a p p r o x i m a t e l y t r u e . 10 An experimental study of laminar free convection in an inclined rectangular channel was done by Ozoe et al. [19]. In their experiment, they used both silicone oil and air as their transfer media. They varied the aspect ratio, of the channel and measured the 3 heat transfer rates for the range of Raleigh numbers from 3x10 to 5 1 xlO . By fixing the heat input at the bottom plate, and varying the angle of inclination, the authors found a minimum and a maximum rate of heat transfer at some critical angles. These angles were found to be a strong function of the aspect ratio and a weak function of the Raleigh number. They concluded that a transition in the mode of circulation was responsible for the minimum rate of heat transfer but did not explain the occurrence of the maximum rate. Hollands et al.. [7] studied free convective heat transfer across inclined air layers.. By careful measurements of the heat transfer rates, the authors derived an empirical relation for free convective heat transfer as a function of the angle of inclination and the Raleigh number. The range of Raleigh numbers examined was from subcritical.( < 1708/cos 3 ) to about 1 xlO^ while the range of the angle of inclination was from 0 to 70 degrees measured from the horizontal. The correlation developed was used by Beard in his mass diffusion model.. Buchberg et al. [20] summarized the major studies related to free convection in enclosed spaces. By comparing some of the correlations for predicting free convective heat transfer, and 11 discussing their limitations, the authors recommended that any of the mentioned correlations in their paper could be used in design calculations. In light of the recent inclined layer data, the authors recommended that the spacing between the absorber plate and successive covers should be in the range of 4 to 8cm to insure minimum gap conductance. The exact value of this spacing would depend on factors such as the heat loss rate, edge shading of the absorber plate and cost. The spacing used in conventional collector designs is approximately 1 to 2.5 cm. This range of values is based on Hottel's original work with flat-plate solar collectors [1]. In his paper, Hottel stated that by increasing the air space between covers beyond.half an inch, there was l i t t l e change in the heat transfer rate. Thus by recommending a new range for the air gaps, Buchberg et al. moved away from conventional design procedure. Their arguments seem reasonable and are strongly supported by the experiments of many researchers [7,19,21]. In the following section, a brief review of studies relating to forced convection over surfaces will be presented. 2.3.2 Forced Convective Heat Transfer Studies related to forced convective heat transfer are also extensive. In order to make them familiar, some of these works will be outlined below. 12 Tan and Charters [22] investigated experimentally the forced convective heat transfer for fully developed turbulent flow in a rectangular duct. The top wall was heated electrically and provided a uniform heat flux into the air stream. The bottom wall was insulated. With such an arrangement, the authors attempted to simulate a solar air heater. By measuring the air and wall tempera-tures, they were able to evaluate the local Nusselt numbers along the duct at different Reynolds numbers. By comparing their results with those of Sparrow et al. [23], they recommended that for fully-developed regions in flat-plate solar collectors, the relation to be used was Nu = 0.018 Re v"- 8Pr 0 , 4 . . . .(2.7) A where 9500 < Rev < 22,000.. Ostrach and Kamotani [24] studied heat transfer character-istics of air in a laminar, fully-developed horizontal channel flow with the bottom plate heated. Their experiments covered a range of Raleigh numbers between 100 and 13,500 and with Reynolds numbers less than 100. When the Raleigh number was below the critical point ( <1708/cos 3), the temperature profile from top to bottom of the channel was found to be linear and heat was transported by conduction only through the layer of air. Above the critical point, the flow became unstable and vortex rolls were observed in the channel. The 13 heat transfer rate was also increased due to this unstability. Mori and Uchida [25] did an analytical study for the same flow conditions. Using the energy integral method, and solving the governing set of non-linear equations, they obtained a correlation between the Nusselt number and the Raleigh number. Hwang and Chang [26] solved the same set of governing equations by numerical methods. Ostrach and Kamotani compared all these results and found that the agreements among them were very good. At Ra> 8000, the authors observed another change in the flow regime. They attributed this change to the appearance of the so-called 'second-type vortex rolls.' This new type of flow phenomenon was not studied in any detail. Ostrach and Kamotani then performed their experiments under confined conditions. The flow inside the channel was then by free convection. They found that for the range of Raleigh numbers considered, the heat transfer rates were identical to those of the fully-developed case. 2.3.3 Combined Free and Forced Convection Chen et al. [27] studied mixed convection for a boundary layer flow over a horizontal plate. By considering the buoyancy effects, they solved the governing.non-rlinear equations by means of local similarity and local non-similarity methods. By considering a 2 5 buoyancy parameter £ = Gr / Re ' , numerical results were obtained A A for the local surface heat transfer rates, velocity and temperature distributions for gases having a Prandtl number value of 0.7. 14 When £ was positive, a negative pressure gradient along the plate was induced. The initial flow, which was purely forced convective, was therefore, aided by this favourable gradient. The opposite was true when the buoyancy parameter was negative. The heat transfer was increased by.the favourable pressure gradient and decreased by the unfavourable pressure gradient. Numerical results for the local heat transfer rate were obtained for £ between -0.03 and 1.0. Lloyd and Sparrow [28] studied combined free and forced convective flows over vertical surfaces. They analyzed the heat transfer characteristics of vertical surfaces by using a similarity solution for the set of governing differential equations. So far a literature search relating to free and forced convection has been presented. In a water-trickle collector, the air circulation pattern above the corrugated plate is due to the com-bined effect of the buoyancy of air and the gravity-induced surface dragging of the trickling water firl.m. The net effect, .then, is very much like a purely forced convective flow superimposed by free con-vection. In the following chapter, a refined flat-plate model based on heat transfer coefficients derived in this thesis and from known correlations will be presented. 15 P A R T I FLAT-PLATE WATER-TRICKLE SOLAR COLLECTOR MODEL 16 3. DEVELOPMENT OF PRESENT FLAT-PLATE MODEL 3.1 Combined Free and Forced Convection Presently, most commercially produced water-trickle collec-tors are of the corrugated type. In this part of the thesis, a simple flat-plate model will be developed because of its simpler geometry and consequently simpler heat transfer analysis. The differences between the water-trickle collector and the conventional tube-and-plate collector are the absence of tubes and the fact that water is exposed on the plate surface. Other than these differences, the overall heat transfer analysis will be similar to that of a tube-and-plate collector. As discussed earlier, the model developed by Beard is a simple one. Since the flow between the inner cover surface and the collector plate of a water-trickle collector is a special case of forced convection, the mass transfer process is a convective mass transfer instead of the diffusional one assumed by Beard. Because of the inclination of the collector and the temperature difference, there is also a free convection effect. The net effect, then, is a combination of forced and free convection. The determining factors as to which mode of the transfer processes is more dominant are: 17 1. the inclination, 2. the gap distance between the inner cover surface and the plate, 3. the plate temperature, 4. the mass flow rate of water, 5. the cover temperature, and 6. collector plate geometry. These factors can be summarized by dimensionless variables. They are the Reynolds number Re and the Grashof number Gr. A In the following section, the analysis of the various aspects of a trickling flow will be presented. 3.2 Flow of Thin Water Film Down an Inclined Plane The amount of work that has been done on thin film flow is very extensive. Most work centers on film condensation and flow of a 1iquid film.down the outside or the inside of a circular tube for applications to distillation columns. The earliest work and probably the pioneering work on filmwise condensation on flat plate was formulated by Nusselt [29]. By equating the forces of gravity and viscosity, and assuming linear temperature profile, he obtained expressions for temperature distribution along the plate and heat transfer coefficients as a function of the Reynolds number 4r /u . Experiments by Rohsenow [30] and numerous other authors also gener-ally show good agreement with each other. 18 For applications to a water-trickle flat-plate collector, Figure 3 shows the gradual development of the velocity gradients of both air and water as they flow down the collector plate inclined at an angle 8 from the horizontal. Generally, the water film approaches a fully developed gradient much faster than the air layer due to its lower kinematic viscosity. With this in mind, i t is safe to assume that the thin film is fully developed over the whole length of the collector. As pointed out by Seban and Faghri [31], for a Reynolds number 4T/u of 400, the flow should be fully developed at x/6-] =5.8, with x being the distance along the collector and 5-| the film thickness. In analyzing the thin film flow, a force balance on a small increment dx along the plate in Figure 3 gives the well-known velocity expression p g 2 U = — ( 6 v y - £-) sin 3 . . . .(3.1) y 1 2 To analyze the velocity gradient development of the air, the collector is assumed to be air-tight. The induced air circulation pattern should then be as shown in Figure 4.. It must be emphasized that the pattern shown is the time-averaged profile, which is dependent on variables mentioned earlier. To determine the heat flux across this layer, two simple mathematical expressions for the velocity profiles of air will-be assumed in the next section. 19 3 . 3 Heat Transfer Across a Laminar Thin Film Flow The amount of work on heat transfer across a thin liquid film is also extensive, with most of the recent work being semi-analytical or experimental. In actual thin film flow, the formation of surface waves or ripples are possible, depending on the Reynolds number 4 r / y . Numer-ous researchers attempted to determine the ripple formation criterion with results showing general agreement with each other. For example, Kapitza [ 3 2 ] predicted that capillary waves would form on the laminar layer when the Reynolds number 4 r / y exceeds the value of where y g/pa^  is called the Kapitza number. Generally, i t is agreed that Reynolds number [ 3 2 ] . , Weber number [ 3 5 ] , and Froude number [ 3 3 ] are determining variables of wave incipience. flowing down a vertical surface at all but the smallest of Reynolds number displays random characteristics. Some of these characteristics are the wave velocity, amplitude, and wave shape. They and numerous other researchers [ 3 4 , 3 5 , 3 6 ] have shown that the film structure consists essentially of two portions: a base film portion next to the wall and the waves (Figure 5 ) . The waves on the film can carry 4 3 As pointed out.by Telles and Dukler [ 3 3 ] , a thin liquid film 20 a significant portion of the total flow (approximately three-tenths to six-tenths in the range of Reynolds number 4r./.u from 400 to 6,000) and travel faster than the base film. As a wave overtakes the fluid in the base film in front of i t , i t mixes with this fluid and the base film loses identity. An equivalent amount of fluid is continu-ously left behind and relaminarizes to make up the laminar base film in back of this wave. Figure 6 summarizes the major results in the research of thin film liquid flow.. In this figure, the classical work of Nusselt is shown as curve (A) along with numerous other curves at different Prandtl numbers. The curve of Nusselt gives consistently lower heat transfer coefficients, the reason being that Nusselt did not take into account the wavy effect-of the thin film surface. Chun and Seban [35] developed an expression for the heat transfer rate which takes into account the waviness: h c = 0.606 ( k 2 g ) 1 / 3 ( r / y ) ~ ' 2 2 . ... .(3.3) This equation is.shown as curve (B) ,in Figure.6. Due to the statistical nature of the flow, and the associated turbulent mixing within the ripples, the results generally have the same trend, but the extent of scatter is considerable. However, given experimental accuracies, the range of scattering is considered to be acceptable. 21 3.4 Heat and Mass Transfer Across the Air Layer 3.4.1 Method of Solution If the space between the first cover and the plate of a water-trickle collector is pneumatically sealed, the velocity will be as shown in Figure 4. Assuming the flow is fully developed, at steady state, the resulting one-dimensional energy equation can be written as: Referring to Figure 4, the bottom part of the profile is approximated as being linear and the top part parabolic. However, in the actual flow, the velocity profile is influenced by both free and forced con-vections. In this application, free and forced convections oppose each other. Free convection becomes more significant i f the temperature difference between the plate and cover is sufficiently high. For a fixed inclination, the actual velocity profiles should look roughly like those in Figure 7. To analyze the overall convective heat transfer, i t is assumed that there are two separate air layers with thicknesses a and b inside the collector. Based on the above assumptions, the energy equation is solved for both gaps a and b, and the net effect is calculated (Appendix B). 3T 8x a. u 3 T . . . .(3.4) 22 Under the same flow conditions and in the same flow geometry, mass and heat transfer are analogous and a direct relationship exists. The ratio of mass to heat transfer coefficient is given by the well-known equation hmi/h = [ p y L e ) 2 / 3 ] 1 • • • • -( 3- 5) Using the above equation, the mass transfer coefficient can be estimated from the heat transfer coefficient. By knowing the latent heat of vaporization for water, the heat loss due to evaporation is then found. The water evaporated subsequently condenses on the inner cover surface because of its lower temperature. 3.4.2 An Open-Ended Water-Trickle Collector If the ends of a water-trickle collector are open, ambient air will enter and leave the collector as shown in Figure 8. In ana-lyzing the velocity gradient development in a steady state-situation, the problem is mathematically equivalent to a problem in Couette flow. That is, in an open-ended collector, all vapour diffusion takes place within 6', the vapour penetration depth. It is assumed that the velocity is constant at this depth, and that the state of water vapour between the cover and the boundary represented by 6' is at ambient conditions. 23 To solve for the mass and heat transfer, the steady state diffusion equation is used: 9C U l 1 7 = °AB 2 A 2 r c A .(3.6) Assuming transverse diffusion is much greater than axial diffusion, 2 3y » 2 9 C\ 3x< Equation (3.6) can be written as 9C, 9x 2 9 C = D A AB „ 2 9y £ . .(3.7) with boundary conditions: x = 0 , y y = 6 -AO for all y, 0 , C A = C A W S for all x >0, C^A = C^ Q for all x. Figure 9 is a sketch of the above flow situation. With the above assumptions, the penetration theory of mass transfer developed by Higbie [37] can be used. In this theory, the solution of Equation (3.7) can be written in terms of an error function as = C Ai <CAi " W •Hi t 2 dt sa J-o (3.8) 24 where . . . .(3.9) The local mass flux of A at surface is <NA> local AB 3C„ 3y AB y=o dC, dy y=0 for a fixed value of x. The local Sherwood number can be shown to be ^ l o c a l = Tl R e x 1 / 2 S c ] / 2 (3.10) The average Sherwood number over the whole length of the collector is therefore (Sh) a u n = ^-Re.1/2 S c 1 / 2 . . . . .(3.11) a v g / F L The details of the derivations are in Appendix C. The mass transfer coefficient is therefore h m = 1.129 Re^/2 S c 1 / 2 . . . . .(3.12) 25 Table 1 shows the results of the mass transfer coefficient as a function of the Reynolds number. In the same table values of Ii for J m an air tight collector are also given. In the air-tight case, h^ is calculated for a steady-state, fully developed., pure forced convective flow.(Appendix B). By comparing the two mass transfer coefficients, one observed that for an open-ended collector, the heat and mass transfer coefficient is a strong function of the Reynolds number, whereas, in an air-tight collector, they are independent of the Reynolds number. No:,detailed analysis was done for an open-ended collector in this thesis. However, i f one wishes to perform such.an analysis, the pro-cedure is exactly the same as for other water-trickle collectors. 26 TABLE 1 Mass T r a n s f e r C o e f f i c i e n t vs. Reynolds Number, L = 5m, D A B = 2.6 x 10" 6m 2s, d = 0.0254 m Open-Ended C o l l e c t o r A i r - T i g h t C o l l e c t o r Sc = 'AB Re L = VL/v h m ( m / s ) ( E q . 3.12) h m ( m / s ) ( E q . 3.5)_ 0.65 1 x l O 3 1.51 X I O " 4 5 x 1 0 3 3.36 X I O " 4 1 x 1 0 4 4.75 X I O " 4 5 x 1 0 4 1.06 X 1 0 " 3 1 x 1 0 5 1.50 X I O " 3 2 x I O 5 2.12 X 1 0 ~ 3 3 x I O 5 2.60 X I O " 3 4 x 1 0 5 3.00 X I O " 3 5 x 1 0 5 3.36 X 1 0 " 3 1.18 x 10' 27 4. DEVELOPMENT OF COLLECTOR FACTORS The usual analysis of tube-and-plate collectors is based on a number of collector factors called: 1. collector efficiency factor, 2. collector heat removal factor, and 3. collector flow factor. In the following sections, mathematical expressions for corresponding factors applicable to water-trickle collectors are developed. The following assumptions are made in the process of develop-ing the collector factors: 1. The flow is fully developed both thermally and dynamically. 2. All conditions are at steady state. 3. The collector plate is heated uniformly. 4. The flow is laminar, with all fluid properties • constant. The heat transfer from the plate to the water surface and from the water surface to the air are analyzed separately. Then the heat lost to the ambient from the top cover surface is balanced with the losses, inside the collector. In doing so, all the modes of heat 28 transfer must be taken into account. The iteration procedure in Appendix A will be used in the computations. Various flow factors can then be developed based on the derived heat transfer coefficients. 4.1 Heat Transfer Across a Thin Liquid Film The heat transfer into the liquid film is by convection from .the absorber plate and by radiation. The former will be treated first. . 4.1.1 Forced Convective Heat Transfer For fully developed laminar flow, the steady state energy equation is i l . ja.!L. + i | L . ... .(4.D For a thin liquid film flowing down an incline, v = 0, and Equation (4.1) reduces to = ± f . . . . -(4.2) 3y2 % 9 X In Figure 10, a small control volume is drawn indicating the heat balance of the thin flow. The heat balance of the control volume can be written as 29 3T o r 3T bw bw wl Hw ""pw 8x S ' + V l '% = 0 , 9x 3T 9x = E WC Ow . .(4.3) • .(4.4) pw Combining E q u a t i o n s (4.2) and (4.4) y i e l d s 2 I L L = J L E 2 Ow .(4.5) 3y where u i s g i v e n by E q u a t i o n ( 3 . 1 ) . The s o l u t i o n o f E q u a t i o n (4.5) i s (Appendix B ) , where T(x, y ) T(x) 'Ow •w 12 + T(x) , -U tx [ T f i - T a - U 7 ] e p a uV (4.6) •(4.7) T h i s e x p r e s s i o n can be s i m p l i f i e d i f a l i n e a r temperature p r o f i l e i n the y - d i r e c t i o n i s assumed. For such a p r o f i l e , T ( x , y ) -w + T(x) (4.8) E q u a t i o n (4.8) i s p l o t t e d i n F i g u r e 11 f o r d i f f e r e n t mass f l o w r a t e s . F or a t h i n water f i l m , the f l o w i s f u l l y d e v e l o p e d p r a c t i c -a l l y o v e r the whole c o l l e c t o r . Thus the a n a l y s i s p r e s e n t e d s h o u l d be u s e f u l i n e s t i m a t i n g the h e a t t r a n s f e r from the c o l l e c t o r p l a t e to the ambient. 30 The above a n a l y s i s i s based on s o l v i n g the energy e q u a t i o n f o r a f o r c e d c o n v e c t i o n s i t u a t i o n . In the next s e c t i o n , a b r i e f d i s c u s s i o n o f the r a d i a t i v e heat t r a n s f e r c h a r a c t e r i s t i c s i n the t h i n w a t e r f i l m w i l l be p r e s e n t e d . 4.1.2 R a d i a t i v e Heat T r a n s f e r C h a r a c t e r i s t i c s The sun's e f f e c t i v e temperature i s i n the neighbourhood o f 6,000°K. A t t h i s t e m p e r a t u r e , the major p o r t i o n o f the b l a c k body s p e c t r a l e m i s s i v e power i s i n the range o f s h o r t w a v e l e n g t h s . Thus wavelengths o f importance i n s o l a r energy a p p l i c a t i o n s a r e i n the u l t r a v i o l e t to n e a r - i n f r a r e d range (about 99 p e r c e n t o f s o l a r energy i s w i t h i n the range 0.2 ym< A<4.0 ym). A f t e r p a s s i n g through the atmosphere, the r a d i a t i o n t h a t r eaches the ground i s i n the range o f 0.29 t o 3.0 ym. On the o t h e r hand, a b s o r b e r s u r f a c e temperatures a r e t y p i c a l l y around 400°K. At t h i s t e m p e r a t u r e , the major p o r t i o n o f the s p e c t r a l e m i s s i v e power o f a b l a c k body i s i n the l o n g wavelength r e g i o n ( > 3 y m ) . Thus i t i s p o s s i b l e t o a n a l y z e . t h e c o l l e c t o r h eat t r a n s f e r i n t e r a c t i o n s w i t h the i n c i d e n t short-wave r a d i a t i o n s e p a r a t e l y from the c o l l e c t o r h eat l o s s e s . In o t h e r words, i n a n a l y z i n g the performance o f a c o l l e c t o r , one can f i r s t c o n s i d e r the i n c i d e n t s o l a r , f l u x i n the r e g i o n of. h i g h c o v e r s u r f a c e t r a n s p a r e n c y , then compute.the l o s s e s based on r e - e m i s s i o n i n the long-wave r e g i o n i n which the c o v e r s u r f a c e s may o r may not be opaque. 31 4.2 Heat and Mass Transfer Coefficients for Air Assuming that the air circulation pattern is as shown in Figure 4, the forced convective heat transfer coefficient is constant for all plate temperatures. Based on the gap distance d.from the water surface to the inside cover surface, the overall forced convec-tive heat transfer coefficient is h x s = 1 •/(l/h 1+l/h 2) = ka/(a + b) . . . . .(4.9) The details of the above derivation are in Appendix B. The ratio of the mass to heat transfer is hm/hxs = ( P a C p a ( L e ) 2 / 3 r 1 » ' • ' ' ( 4 - 1 0 ) where h m is correspondingly the overall mass transfer coefficient. By definition, hm = N A Z / ( C w s " C g i } • * ' ' ' ( 4 J 1 ) where C = concentration = P / RT for ideal gas,- and NAZ = rate of mass transfer. The mass transfer is therefore NAZ = h (p C ( L e ^ V V — - (4.12) P RT1IC RT • ws gi 32 Thus the heat transfer due to evaporation is h (p C ( L e ) 2 / 3 ) - 1 P P . q e c = X S a P a U ^ - T ^ X ....(4.13) e c R 'ws 'gi and the evaporative transfer coefficient is h (p C ( L e ) 2 / 3 ) _ 1 P P . h = x s ^ a pa^  L _ ( W S _ J 1 ) A (4.14) e c R(T -T .) Tws Tgi v ws gv 3 With the expressions for the various heat transfer coefficients, i t is now possible to develop some flow factors which are useful in design calculations. 4.3 Derivation of F' and F R F', by definition, is the ratio of the actual useful energy gain to the useful energy gain i f the collector absorbing surface were at the average fluid temperature. This factor is essentially a constant for any collector design and fluid flow rate. F R, the heat removal factor, is the ratio of the actual useful energy gain of a collector to the useful energy gain i f the whole collector surface were at the fluid inlet temperature. As will be seen later, these two factors are extremely use-ful in calculating the useful heat gain of a collector. Following Bliss [ 4 ] , the useful energy gain per unit area is, by definition, given by 33 v = S-VW • .(4.15) also, q v = h (T - T_) Mux c p f (4.16) where is the average fluid temperature at any section. Eliminating V nux rrir ( s - u t(T f-T a)) c t (4.17) Then the average rate of heat gain per unit area is s - u t ( i rL JO Tfdx - T a) (4.18) But by definition, F1 is the ratio of the actual useful heat gain to the gain when the plate temperature is equal to the average fluid tempera-ture. Thus, F' hc + U t (4.19) Since h c » U t > F1 is usually close to unity. Combining Equations (4.17) and (4.19) yields lux = F ' ( S - u t ^ f V)- (4.20) 34 Taking the differentials of both sides of the above equation, dq u x = -F' Ut dTf . . . . .(4.21) For a width Az and length Ax of collector, the heat balance can be written as quxAzAx = W(ATf) C p w . . . .(4.22) where W = water flow rate (kg/s). Let G be the water flow rate per unit area, G = W/L(Az) , then, Axq u x = GL(AT f)C p w . . . . .(4.23) Combining (4.21) and (4.23), and letting Ax approach dx, da " F ' U t d x °qux _ * . . . . .(4.24) %x Pw Integrating Equation (4.24) with boundary condition q =q n at x = 0, "ux - q u 0 e " < F ' U t X / G C p w L ) • • • • 35 Integrating Equation (4.25) between limits 0 and L, % - V ^ O - e - ^ V V ) . . . . .(4.26) At x = 0, T.p = T^ ., thus, quO = F'(S-U t(T f i-T a)) ' • • • -<4-27) Substituting Equation (4.27) into Equation (4.26), q u = > ^ 0 -e"(F,Ut(GCpwV'{S-Ut(Tf.-Ta)) . . .(4.28) By definition, is the ratio of the actual heat gain to the gain i f the plate temperature is at the initial bulk fluid temperature. Therefore, FR * 0 - e " < F ' W ) , . . . .(4.29) which is identical to the expression for a tube-and-plate collector. Thus Equation (4.28) can be written as % - FR(s - ¥ T f i - V ) • • • • -(4-30> 36 The expressions for F' and F R are those developed by Bliss [4] in his paper for a dry flat plate with water flowing beneath. In a water-trickle collector, the water flows on top. However, the expressions are s t i l l the same but with new values of h £ and U^. developed in this work. Defining F", the collector flow factor, as the ratio F R to F', a plot of F" verses GCp/u^F1 is shown in Figure 12. This plot is used when one wants to calculate F R quickly by knowing F 1. For design calculations, F' and F R are used extensively; by knowing just the inlet fluid and the air temperature, the useful heat gain can be calculated directly. With the heat and mass transfer coefficients developed, i t is now possible to analyze the performance of a flat-plate water-trickle collector. The next chapter will present results based on Beard's work discussed earlier and those developed in this thesis. 37 5. COMPARISON OF RESULTS BASED ON BEARD'S MODEL AND PRESENT FLAT-PLATE WATER-TRICKLE COLLECTOR MODEL With the two models, i t is now possible to compute various heat transfer characteristics of a simple flat-plate water-trickle collector. By using the iteration scheme in Appendix A, and the heat and mass transfer coefficients developed, the thermal performance of a simple flat^plate water-trickle collector is analyzed and compared to the experimental results. Figures 13 to 18 show various character-istics associated with a water-trickle collector based on the two models at various environmental conditions. Generally, the two models give results which are in good agreement with each other, with the present model giving slightly higher losses. The mass transfer in all cases accounts for a large percentage of the total heat loss, and this percentage increases at higher water surface temperatures. The number of covers on the collector seems to have l i t t l e or no effect on this percentage. Thus a water-trickle collector should be operated at lower plate temperatures than conventional tube-and-plate collectors in order to reduce the mass transfer effect. San Martin [8] concluded that in order to maintain high efficiency, the operating temperature of a water-trickle collector should be less than 145°F (62°C). 38 Figures 19 to 21 are prepared based on the heat-mass transfer analogy (Equation 3.5). They show the top loss coefficient as a function of the water surface temperature for a water surface emissivity of 0.95. Figures 22 to 24 show the same kind of plots for an emissivity of 0.1. The latter figures are useful when a selective plate is used, with most of the plate surface exposed because of dry patches. Generally, ambient temperature and wind velocity have a strong influence on the top loss coefficient for a fixed plate temperature. This influence is least significant for a three-cover collector. When the ambient air temperature is higher than the plate temperature, there are reversals of temperature gradients in the collector. The mass transfer in this case will be zero because the cover temperature is higher than the water surface temperature, and the driving force for the mass transfer is negative. Thus a sharp decrease in the values of the top loss coefficient is observed. Again, this effect is most pronounced for a single-cover collector. As pointed.out by Duffie and Beckman [38], the heat transfer mechanism in this case is due to radiation and conduction only. It should be added that conduction was assumed because the Nusselt number for the operating conditions considered was sufficiently close to unity. Some interesting results have come from the two models. The fact that a large portion of the heat loss is due to mass trans-fer leads to a conclusion that a water-trickle collector is ineffic-ient at higher plate temperatures. The effects of ambient con-39 ditions are most pronounced for a single-cover collector. These effects diminish with the increasing number of covers on a collector. For example, the ambient air has practically no effect on the top loss coefficient of a three-cover collector, whereas the effect is quite significant for a single-cover system. Generally, both models give results which are very close to each other, and one can use either model for prediction purposes . Figures 25 to 26 show heat losses from a water surface under some specified, conditions. The losses, from a one-cover, two-cover, and three-cover system are shown along with the contributions from different modes of heat transfer. Figure 25 is based on a surface emissivity of 0.95 and Figure 26 is based on a value of 0 . 1 . In both cases radiation is the dominant mode of heat transfer between the covers.. When a selective surface is used, e =0 . 1 , the radiation loss is somewhat reduced but s t i l l dominates between the covers. The loss to the environment due to a 5 m/s wind .is about five times the radiation heat loss. Between the water surface and the first cover, the dominant mode of heat loss is due to evaporation-condensation. This loss accounts for more than half of the total loss for all three systems under consideration. The net effect, by using a selective surface„is lower total heat loss because of the reduced radiation loss. This effect is most apparent for a one-cover system but is relatively insignificant for a three-cover system. 40 Figure 27 shows collector efficiency as a function of the fluid inlet temperature. The solid lines are for results obtained in this work and the dashes for those of a tube-and-plate collector. The efficiency plotted is the instantaneous efficiency, since the insolation value is fixed. From this plot, a three-cover system is shown to be the most efficient at high fluid inlet temperatures for both types of collectors. However, a water-trickle collector has lower efficiency at all values of fluid inlet temperature than the tube-and plate collector. This is. expected since there is an added mode of heat transfer inherent in water-trickle collectors. Figure 28 shows a different kind of efficiency plot. The abscissa shows the ratio of (T^-T a).to the incident solar flux 1^ . The solid line represents a least square f i t of the actual experi-mental, data obtained by Beard [6]. These data are based on the actual performance of the Thomason 'Solaris 1 water-trickle collector. The dashes represent the calculated results based on the heat-mass transfer analogy. In the present calculations, a twenty percent reduction of the incident flux is assumed due to the effect of drop-let formation on the inside cover surface. This value is reasonable since Beard observed experimentally a ten to twenty percent reduction of energy transmitted due to droplet formation. The experimental curve is higher than the calculated curve, but its efficiency also drops off faster. The experimental curve represents 41 heat transfer from a corrugated surface whereas the calculated one is for a flat plate. In the present case, the mass transfer area for the corrugated plate is about one-half of that assumed for a flat-plate. Thus higher mass transfer rate is expected from the modelled collector, which leads to a lower efficiency. As the plate tempera-ture rises, the effect of dropwise-condensation is stronger and the transmission of energy is further reduced. In the modelled curve, the reduction factor is assumed constant at twenty percent. Thus the experimental efficiency drops off faster because of this increasing reduction at higher temperatures. Generally, the same trend is observed between the modelled and the experimental curves. Part II of this thesis, the section immediately following, will produce results which will simulate the experimental results even better. P A R T I I CORRUGATED-PLATE WATER-TRICKLE SOLAR COLLECTOR MODEL 43 6. INTRODUCTION In Part I of this thesis, a simple model called the flat-plate water-trickle collector model was developed. A heat transfer analysis based on this model was then performed and results were compared to results calculated from Beard's mass diffusion model, and also to data from Beard's experiments on the Thomason "Solaris" collectors. As mentioned earlier, the results obtained based on the two models were in close agreement, with the flat-plate water-trickle collector model giving slightly higher heat losses. When compared to experimental data, the calculated efficiency was slightly lower than the actual efficiency. This difference was attributed to the larger surface area of mass transfer assumed in the model. In this part of the thesis, the heat-mass transfer analogy will be extended to a corrugated absorber plate. Results will then be compared to experimental data and to theoretical results based on Beard's model. There are many possible geometrical configurations for the absorber corrugation. Most existing water-trickle collectors, however, have cross-sectional shapes as shown in Figure 2. Other shapes, such as those approximating sine waves, can be used but are less popular. The present work will therefore concentrate on a corrugation geometry as shown in Figure 2. Generalized correlations based on this geometry will be developed. Heat transfe calculations will be done by assigning chosen values to the dimen sions of a hypothetical corrugated collector plate. 4 5 7. PRESENT WORK 7.1 Corrugation Geometry For a Thomason collector, the cross-sectional shape of the absorber plate is shown in Figure 2. Another type of corrugation shape possible is that used by San Martin [8].. The absorber plate used in San Martin's collector was made of aluminum roofing, with its cross-section roughly resembling a sine curve. Other shapes are certainly possible; however, a.general analysis of all possible shapes of corrugation is beyond the scope of this thesis. Therefore, a general analysis of only the Thomason corrugation geometry will be presented and simulated results compared to Beard's data. 7.2 Effect of Plate Geometry on Heat Absorption Characteristics In this work, the analysis will be limited to collectors inclined towards the equator. Because of the corrugation, the absorber plate will reflect the incident flux differently from that reflected from a flat plate. This difference is also influenced by the surface azimuth of the plate normal. Figure 29 shows a south facing corrugated surface, 0, the angle of incidence of beam radiation is given by cos 0 = (cos(cj) - B)cos 6 cos co+ sin (<j) - B)sin 6) . . . .(7.1) 46 Figure 30 shows a detailed sketch of the corrugated cross-section. The short-wave radiation, after striking the collector plate, will either be absorbed or reflected. Most refelected radiation will be diffuse because of the surface roughness of the plate. For design purposes, one is required to evaluate (xa), the fraction of the total incident energy that is absorbed by the collector plate. (xa) is given by (xa) = q 0,2N- ( F15 c'0,cl + F35 q0,c3 + F45q0,c4) 0^,c2 F25 + ( 1 -pwspcTw }  pws + p c ( 1 - 2 pws ) Tw 2 r pws +p c ( 1 -2pwsV (7.2) Detailed development of Equation (7.2) is given in Appendix D. In the above equation the F's are shape factors between surfaces of the corrugated plate and the adjacent surface of the innermost cover. If there is negligible absorption by water, the sum of the terms inside the square bracket is zero. Thus (xa) for a dry corrugated-plate system is (xa) = q 0,2N- ( F15%,cl + F25 C'0,c2 + F35 q0,c3 + F45 q0,c4 )- • ' - ( 7' 3 ) The (xa) expressions given by Equations (7.2) and (7.3) include the effect of both beam and diffuse radiation. Klein [39] and Klein 47 et al. [40] have presented a procedure to determine the energy-weighted (xa) for the whole day or the monthly average daily value. Following this procedure, (xa) values were computed for slopes of 30°, 50°, and 70° for Vancouver. These values are plotted in Figures 31 to.33. Using the net radiation method outlined in Appendix D, (xa) values can be calculated for different corrugation geometries. For comparison, a flat-plate, a V-shape and a rectangular fin corrugation geometry are also plotted in the figures mentioned, Figure 31 shows a plot for collectors inclined at 30° slope. As can be seen, a flat-plate gives the lowest values of (TO) for every month, and the rec-tangular-fin corrugation.gives the highest values. The Thomason corrugation geometry, with a corrugation angle of roughly 15° lies some-where between the two extremes. Generally (xa) is highest in the summer months and lowest in the winter. Figure 32 shows how (xa) varies in a year for collectors sloped at 50°.; The overall trends are the same as for collectors at 30° slope except that values of (xa)'s are almost constant, with greater differ-ences in the winter months. Figure 33 shows results for collectors at 70° slope. For a flat-plate, values of (xa) are higher in winter months than those in summer months. However, for other geometries, this effect is some-what reduced and is actually reversed in some cases (the V-shape, for example). A possible explanation, is that for corrugated surfaces, the angle of incidence of beam radiation changes with the geometry. 48 In some cases, (xa) actually increases slightly at higher angles of incidence. Figure 34 shows a plot of (xa) vs. angle of incidence for a rectangular finned collector sloped at 70°. As seen from the curves, (xa) actually increases slightly, then decreases as the angle of incidence increases.. Thus the net values of (xa) for a corrugated plate are higher than those for a flat plate. Although water-trickle collectors are less efficient because of the higher heat losses, this effect is off-set somewhat by the higher values of (xa). In the following section, a heat transfer analysis will be presented based on the corrugation geometry. 7.3 Heat Transfer Analysis (Appendix E) 7.3.1 Derivation of Flow Factors for a Corrugated-Plate Collector The cross-section of a corrugated-plate collector is composed of two parts, (a) the portion exposed to air, and (b) the portion exposed to water. These two portions will be treated separately. The temperature distribution along the cross-section exposed to air can be derived independently from the flow direction gradient since the flow direction gradient is usually small compared to the former. Figure 35 shows a qualitative sketch of the temperature distribution along the cross-section. 49 Considering the portion of the cross-section exposed to air, the temperature distribution can be determined from standard calculations for fins. Following Duffie and Beckman's procedure [38], the tempera-ture distribution for the surface bounded by points 1 to 2 in Figure 35 is given by T-T a-S/U t VV S ' U t cosh(mx) cosh |m ~Y (7.4) where 1%, is the temperature at point 2. The temperature distribution from point 2 to 3 is given by IT1D3 f ( T w s - V V V - ( V V y U t ) c 0 S h MS« 2 ^ mD, sinh cos a,-sinh mx + (T 2 - T a - s i / u t ^ c o s h m x (7.5) where and S1 = I 3 b ( ^ ) b ( S f ) 1 + I p d ( T a ) d , T 2 is given by r mD„ mD9 T 2 = [(tanh - ^ ( s i n h (S 4 + U ^ ) , ( S ] -Ut(Tws-T a)J mDo i r mD~ + cosh -^4- (S1 +UtTa)JL(cosh : ^ - ) U cos a 2 mD? mD~ + (tanh -L)(sinh ^ cosa2 t (7.6) 50 Similarly, temperature distribution between points 6 and 5 is T - T a - V U t W V U t cosh mx mD0 (7.7) cosh and between points 5 and 4. T - T a " IL mD, ( T w s - T a - S 3 / U t ) - ( T 5 - T a - W c ^ h Hoit mD, sinh mx sinh cosou + (Tg - T Q - S 3/U t)cosh mx , .(7.8) where and S3 = V T a ) b ( S f ) 3 + V T a ) d mD, mD, < t a n h ^ s i n h ^ ( S 4 + U t T a ) - ( S 3 - U t ( T w s - V ) mD, + cosh — — (S~ + U.T ) cosou 3 t a' mD, <cosh ad? U t mD9 mD~ + (tanh - / ) ( s inh ^ ) U t •1 (7.9) The total amount of heat gain for the surface bounded by points 1 and 6 is qtotal = qwater + q3 + q4 » . . .(7.10) 51 where Wer = ¥ S 2 - ¥ T w s " Ta^ • (7.11) and q3 1 m f Sl -MTws " V ^ l - U t ( T 2 -Ta)]cosh ^ mD, sinh mD, cosh cosa 2 mD, + ^ S l - U t ( W l S i n h colt 1_ m mD, f f S3 ~ ¥Tws - V<h " MT5. - V^ c o s h cl^ T 1 mD, sinh cosa. cosa. (7.12) mD, cosh cosa. + [S 3 - U t(T 5 -T a)] sinh mD, cosa. (7.13) (All derivations are in Appendix E.) By examining the pertinent variables, one concludes that a flow factor F' for a corrugated plate seems difficult to be derived in a form that is useful for design calculations. The main difficulty lies with the fact that the values of the S's for different portions of the corrugated cross-section are different. 52 It is possible to check that the foregoing formulation reduces to that of a tube-and-plate collector by making the following assumptions: (i) S = S] = S2 = S 3 = S 4 , D3 (ii) D9/2 is to be replaced by 0, and — ^ — is to be replaced L. cos ot^  D9 D„ by Tzzih) i n Equations (7.5) to (7.13) (excluding (7.7)), C. cosct^ By doing so, the expressions in Equations (7.5) to (7.13) (excluding (7.7)) simplify extensively. With the above assumptions, one is in effect assuming continuous fins from points 1 to 3, and from points 6 to 4, where the incident flux is S. For such a case, Equation (7.5) becomes T - T - S/U. cosh mx a L _ T - T - S/IL D9 D, w s a ' t „„. m i 2 ,. 3 % cosh m (-s- + > — — — ) v 2 cosa2 (7.14) Equation (7.6) becomes T -T -S/U. ws a ' t T2 " T a - S / U t = -^—t D — ' • • • - ( 7 J 5 ) cosh [4- + T~—) 2 cosou and Equations (7.8) to (7.9) become identical to (7.14) and (7.15), respectively. Equations (7.12) and (7.13) both become = - i s - U. (T e - 1 ) \ tanh m {-S + ~~-)]• m \_ t v ws a'J v 2 cosa2'J 53 D 2 D 3 D„ D„ f 1 t a n h m(-T+ccTa m(-3-+-2 cosa. = (- cosa, -) S - U t(T, ws - V. (7.17) Thus , E q u a t i o n (7.10) becomes " t o t . 1 = [ D l + 2 F 4 + a l ^ ) ] [ S - U t ( T w s - T a » • o r 2D~ q t o t a l = < V D 2 + F , L S - M T w s - T a n ' ( 7 J 8 ) where 2D~ D, + F ( D 0 + — ) r - 1 2 c ° s p a 2 . . . . . ( 7 . 1 9 ) D l + < D2 + coslj) Fo r a t u b e - a n d - p l a t e c o l l e c t o r , F' i s F' 1/U4 " r 1 1 1 1 ( W p + DtH U t ( D t + W F) + C b 0 + u D t h f i J (7.20) I f one n e g l e c t s t h e bond and f l u i d - t o - t u b e r e s i s t a n c e i n E q u a t i o n ( 7 . 2 0 ) , t h e e q u a t i o n becomes 54 which is in the same form as Equation (7.19). However, since the S's for a corrugated surface are seldom equal, i t appears difficult to derive the expression for F' in a useful form. The expression for F R, the heat removal factor, is the same as for a flat-plate water-trickle collector since F R is indepen-dent of the plate geometry. For a corrugated surface, F R, the heat removal factor, is the same as for a flat plate water-trickle collector since F R is independent of the plate geometry. For a corrugated surface, F R is not useful since i t is dependent on F1. 7.3.2 Derivation of Heat Transfer Coefficients The three modes of heat transfer for a water-trickle collector are radiation, convection, and evaporation-condensation. Although these three modes exist for both the flat-plate and the corrugated-plate models, the detailed mechanisms are different. For example, the net energy absorbed by a corrugated plate is different from that absorbed by a flat plate because of the different reflective behaviours. In order to estimate the amount of radiative heat transfer, the pro-jected area of the corrugated surface onto the glazing surface is assumed to be the actual heat transfer surface. This practice is used commonly in flat-plate solar collector calculations. Because of the corrugation, the actual flow regimes are very complicated. The circulation pattern on top of the fluid film is due to the combined effects of free and forced convection, and 55 the pattern on the top part of the corrugation should approximate a free convection. There are also the edge effects between the two regions mentioned. On top of these main patterns, there are always the possibilities of occurrences of 'the first and second type vortices' as observed by Hollands [7] and many others [19-24]. Thus the actual circulation patterns are due to the combined effects of all the mentioned flow regimes, and a purely theoretical solution to the problem is formidable. A semi-analytical approach based on known experimental correlations will therefore be used in determining the convective and evaporative heat transfer rates. Assuming there is no mixing between the top and the valley parts of the corrugation, the top parts can be viewed as regions of free convection, and the valley parts are assumed to be regions of combined free and forced convections. The analysis for the regions of free convection is quite straightforward since experimental correl-ations are readily available. The combined convective regions are much more complicated. As mentioned earlier, works on combined free and forced convections were limited to laminar flows, and to either horizontal or vertical surfaces. Ordinarily, most solar collectors are operated at fixed inclinations somewhere between the horizontal and the vertical. The effects, of the inclination to the heat transfer rates are indeterminate at present. Since the flow regimes on top of the trickling fluid film are functions of the Raleigh number, the angle of inclination, the corrugation geometry, 56 and the velocity of the water film, they would be free and forced convection-dominated, and the extents of the contributions from each are again indeterminate. Thus i t is impossible to.simply pick out a correlation from known literature and apply i t to this case. A different approach will therefore have to be used. As seen from Figure 7, and except for the case when the ambient temperature is higher than the absorber plate temperature, free and forced convections oppose each other. For the case when the ambient temperature is higher, there will be almost no free convection and the circulation of air is that due to the moving water film (Figure 4). Assuming steady state, the heat transfer for the latter case is the same as for pure conduction. This statement is supported by Ostrach's work [24]. Since for most cases free and forced convections oppose each other, the actual amount of heat transferred, will depend on the resultant forces. For design purposes, a simple way to estimate the heat transfer rate is by using the value calculated from the dominant mode only. This will lead to an overestimate of the actual amount of heat transferred. However, this method offers the advantage of ease of computation. In the present work, heat losses due to both free and forced convections are calculated and the larger of the two is used in estimating the top loss coefficients. The mass transfer rate is therefore also overestimated since i t is coupled to the convective transfer rate (Equation 4.10). For the present 57 study, free convection turns out to be the larger of the two in all cases. Thus the heat transfer correlations between the absorber plate and the inner surface of .the cover are: 1. Radiation: q = g ( ^ S " l 4 c l ) . . . . .(7.22) 2. Convection (Tabor's correlation for free convection [41]): hp r l = (1 - .0018(T -10)) 1- 1 4 ( A T^ Q'!' 1 for 6 = 45°, ci ws ( a + b ) U - U / 104<Gr< 107. (7.23) 3. Evaporation-condensation: h (o C Le 2 / 31~^ P P , . " " ' " ' V - 6 ' ( ^ S - l a l ) X (7.24) e c R ws 'gi For heat transfer between covers, the two modes of heat transfer are: 1. Radiation: q = — ^ ~ • . . . .(7.25) r i l / e r t + 1/e -1 g ws 58 2. Convection (Tabor's correlation for free convection [41]: 1.14(AT)0'31 hp c 1 = (1 - .0018(Tws-10)) 0 > 0 7 for 8 = 45% l a + D ; 104 < Gr < 107. (7.26) The losses to the ambient are: 1. Radiation: % - V ^ . n " Ta) ' ( 7- 2 7) 2. Convection: q c = (5.7 + 3.8 V w)(T c n - \) . . . . .(7.28) Assuming no change in temperature inside covers, the iteration procedure outlined in Appendix A is used to find the total frontal heat losses under different operating conditions. Table 2 shows some of the results obtained for a single-cover collector under a 5 m/s wind and with T =10°C. a 59 TABLE 2 Heat Transfer Characteristics of a Single-Cover Corrugated Collector At Different Average Plate Temperatures T a = 10°C, Wind Velocity =5 m/s, e = 0.95 Tp(°C) hrl(W/m2oC) hcl(W/m2oC) hec(W/m2oC) q(W/m2) 20 4.61 2.02 1.34 61.5 30 4.94 2.43 2.91 14.88 40 5.29 2.69 3.66 244.0 50 5.68 2.85 5.53 370.4 60 6.12 2.95 7.87 521.9 70 6.62 2.99 11.17 708.5 80 7.18 3.00 15.29 925.6 60 8. DISCUSSION OF RESULTS FOR WATER-TRICKLE COLLECTORS 8.1 Comparison of Results Between Flat-Plate and Corrugated-Plate  Water-Tri ckle Col 1ectors In the last chapter, correlations for the different modes of heat loss have been listed. Using the iteration method in Appendix A, heat transfer characteristics under different operating conditions have been calculated. Figure 36 to Figure 41 have been prepared show-ing these characteristics for a water-trickle collector of arbitrary dimensions. Figure 36 to 38 show the frontal heat loss as a function of the average plate temperature, and Figure 39 to 41 give the top loss coefficient as functions of the average plate temperature under different operating conditions. Figure 36 shows the heat loss as a function of the average temperature for a single-cover collector, and Figure 37 to Figure 38 are for two and three-cover collectors respectively. From these figures, i t can be observed that the losses increase as the average plate temperature increases. This is expected since there should be more loss i f the temperature difference between the absorber plate and the ambient is large. The losses predicted by the flat-plate model are also shown in the figures. By comparing the flat-plate and corrugated-plate models, i t is immediately obvious that the flat-plate model predicts higher losses. The main reason for this 61 observation is that in the flat-plate model, the mass transfer area is larger than that assumed in the corrugated-plate model. In the flat-plate model, the trickling fluid film is assumed to be covering the whole collector absorbing surface, whereas in the corrugated-plate model, water flow exists only in the valleys of the corrugation. Also in the corrugated-plate model, the gap spacing from the valleys of the corrugation to the adjacent glazing surface is arbitrarily chosen as 1.5 inches, whereas in the flat-plate model, this value is 1 inch. Thus there may be an added effect due to this difference. The exact influence of the gap spacing on the heat losses will be discussed in Section 8.3 of this chapter. Figure 39 shows the top loss coefficient as a function of the average plate temperature for a wind velocity of 0, and Figure 40 to Figure 41 are for a 5 and a 10 m/s wind velocity respectively. Generally, the resultant coefficients based on,the corrugated-plate model are lower than those predicted by the flat-plate model. Other than these differences, the two models predict results showing the same trends under the influence of different conditions. The de-tailed discussion of the results predicted by the flat-plate model in Section 5 of this thesis therefore applies. In Figure 42 the predicted results based on. the flat-plate model and the corrugated-plate model.are compared with Beard's [6] data for Thomason-type collectors. In preparing the present results, a reduction factor of 0.8 is assumed arbitrarily. This factor 62 accounts for the reduction of the incident solar flux after i t passes through the layer of condensed vapour droplets. This factor is reasonable since Beard observed experimentally that the incident solar flux was reduced ten to twenty percent, depending on the operating conditions. From the figure, .the average Thomason curve gives the highest efficiency. The flat-plate model gives the lowest efficiency and the corrugated-plate model gives results somewhere in between. Thus the.corrugated-plate model predicts results which are closer to the actual experimental data than the flat-plate model. If a reduction factor of 0.9 were assumed,, calculations show that the corrugated-plate model predicts an efficiency curve higher than the actual Thomason data. Since the reduction factor is a function of many variables, one cannot predict accurately what this reduction factor should be under different conditions. Therefore a simple approach is to assume a constant value of reduction for the operating conditions considered. So far two models have been developed in this thesis. The next section will discuss the two models in terms of their limitations and the assumptions used. Beard's model will also be included. 8.2 Discussion of the Models Two models have been developed. The flat-plate model was developed similarly to Beard's model in that both assume a flat plate as their transfer surface. In calculating the evaporative heat 63 loss, Beard assumes a mass diffusion process between the absorbing plate and the adjacent cover surface. The flat-plate model predicts the evaporative loss by using the heat-mass transfer analogy. In both the flat-plate and the. corrugated-plate models the convective heat loss terms are first evaluated and the mass transfer terms are estimated by using this analogy. Since in both the models developed, the convective term is coupled to the evaporative term, the inaccuracy in calculating one term will lead to the inaccuracy of the other. When evaluating heat transfer characteristics of a corrugated-plate water-trickle collector, the corrugated model developed in this work should be a better model to use. As seen from Figure 42, results based on this model are closest to the actual. Thomason data. Thus the corrugated-plate model should give better results than both the flat-plate model and Beard's model for predicting heat losses of a corrugated surface water-trickle, solar collector. 8.3 Effect of Gap Spacing on Heat Transfer Rates As mentioned in Section 2.3.1, Hottel stated that gap spacings between surfaces inside flat-plate solar collectors have negligible effects on frontal heat losses i f they were increased beyond half an inch. However, Buchberg [20] recommended that values from 4 to 8 cm.should be used to avoid the occurrence of maximum or minimum conductances due to vortex rolls. Because of 64 these two contradictory statements, the effects of gap spacing on heat transfer rates in water-trickle collectors will be investigated. By varying the values of d, the gap spacing between the absorber plate and the adjacent cover surface, the evaporative heat losses are calculated. Figure 43 and Figure 44 show evaporative heat losses as functions of the gap spacings d for different temperatures. A curve based on Beard's mass diffusion model and the curve based on the heat-mass transfer analogy are both plotted for comparison. From Beard's model, the evaporative loss decreases with an increasing d. The heat-mass transfer analogy, however, predicts an almost constant evaporative loss for any value of d. In order to explain the contradictory results between Beard's model and the present work, the two equations used to predict the mass transfer losses will be examined in details. The mass diffusion equation was initially developed to- calculate the steady state mass diffusion rate through a stagnant gas layer. This equation pre-dicts that the mass diffusion rate is inversely proportional to the gap layer thickness. Thus by increasing the layer thickness, the mass diffusion rate is reduced accordingly. This effect may be offset somewhat by the change in the difference of the partial vapour pressures and the temperatures but, as can be seen from the plots, the net effect is a decreasing mass diffusion rate with an increasing gas layer thickness. Based on the heat-mass transfer analogy, the mass transfer rate is almost constant for any layer thickness. This 65 result is predicted by the free convective heat transfer rate since the heat and mass transfer rates are coupled. In his work with free convection, Buchberg [20] showed that the free convective heat transfer rate in an inclined air layer approached an asymptotic value for a gap thickness greater than 4 cm. Thus in water-trickle collectors, the mass transfer rate should approach an asymptotic value at about the same gap thickness. In actuality, the convective transfer rates inside water-trickle collectors are due to combined free and forced convections. By estimating the heat and mass transfer rates based on Tabor's correlation, the result is an overestimate of the actual heat loss, since for most cases free and forced convection oppose each other. Beard's model underestimates the actual heat loss since his equation predicts a steady state diffusion of water vapour molecules through a stagnant air layer, which is true only when the ambient temperature is higher than the average absorber plate temperature. The corres-ponding heat transfer mechanism in this case will be by pure conduction only. For most cases, both the mass and heat transfer rates are influenced by the combined free and forced convections. Therefore, Beard's model underestimates the evaporative losses. The actual heat loss rate, then, should lie somewhere in between the two modelled curves. 66 9. CONCLUSIONS In this thesis, a theoretical study of water-trickle collectors has been carried out. Based on this study, two theoretical models have been developed. The first one is called the flat-plate water-trickle collector model and the second one is called the corrugated-plate water-trickle collector model. Computer-simulated results based on these two models have been obtained and were compared to results obtained independently by Beard [6]. Beard carried out experiments on the Thomason water-trickle collectors, and based on his experi-mental data, he developed.a theoretical model. The correspondence among the present computer-simulated results, Beard's experimental data and results from his model is good. The first model developed is the flat-plate model. In this model, the collector absorbing surface is assumed to be flat. In order to predict the frontal heat loss rate, losses contributed by different modes of heat transfer must be taken into account. The radiative loss term is evaluated by calculating the radiative heat exchange between two. infinite, parallel gray surfaces. Tabor's correlation [39] for free convective heat transfer between two inclined parallel surfaces is used to estimate the. convective heat loss from the absorber plate to the adjacent cover. The evaporative loss term is estimated by the heat-mass transfer analogy correlation. 67 This model is very similar to Beard's model. The only difference is that the evaporative loss is calculated differently in the present model. .. The second model is the corrugated-plate model. Although the heat loss terms from the collector absorbing plate are evaluated similarly to. the first model, the second model is more refined since i t takes into account the effects of the corrugation geometry. A new expression of ( x a ) for a corrugated-plate surface has been developed by using the net radiation method. Monthly averaged values of ( x a ) for different corrugation geometries have been evaluated and results were plotted in Figures 31 and 33. A rectangular fin-shaped type of corrugation was found to have the highest values of ( x a ) for any month. It was concluded that Beard's model underestimates the actual experimental results, and the present models overestimate them. The corrugated-plate model was found to be the most accurate model since i t predicts results closest to Beard's experimental data. Based on the predicted results, a water-trickle collector is found to be less efficient than ordinary tube-and-plate collectors. This is expected since water-trickle collectors have an extra mode of heat loss. Although water-trickle collectors are less efficient, they have been used extensively, the major reason being that water-trickle 68 collectors cost less than conventional tube-and-plate collectors. A quantitative economic analysis was not done in the present work but price figures should be readily available from distributors. If one were to operate water-trickle collectors at relatively low temperatures ( <62°C), the collector efficiencies would be comparable to other types of collectors. Thus for a small sacrifice in efficiency, one may be able to reduce the initial capital investment substantially. 69 10. ' RECOMMENDATIONS Two models for water-trickle collectors have been developed in this thesis. These two models, however, contain limitations which lead to inaccuracies of predicted results. As discussed in Section 3, the convective flow pattern between the absorber surface and the cover of a water-trickle collector is a combined free and forced convection, where the two effects usually oppose each other. By using Tabor's correlation for free convection [39], the resultant convective loss term will be an overestimate of the actual value. The evaporative loss term will therefore for the same reason also be higher than the actual loss term. Thus as a further improvement to the present work, either a theoretical or experimental investigation of the combined free and forced convection between inclined parallel surfaces is necessary. Since evaporation represents a significant portion of the total heat loss, a possible area of further study is the suppression of such a loss. By suppressing the convective flow in the collectors, the evaporative loss should be reduced accordingly. The ideal situation is when the layer of air inside the collector becomes stagnant. In such a case, the total loss will, consist of radiation, conduction, and molecular diffusion of water vapour. The analysis for such a case is straightforward since all heat transfer correlations are wel1-documented. In order to optimize the performance of water-trickle collectors, values of (TO) should be optimized. The present work shows that a rectangular fin-shaped corrugation gives the highest monthly average (TO). However, the cost of production of such a shape is not known at present. Thus, a detailed economic analysi of a system using such a collector would be worthwhile in future studies. 71 R E F E R E N C E S 72 REFERENCES 1. Hottel, H.C. and Woertz, B.B., "Performance of Flat Plate Solar Heat Collectors." Trans. ASME, Vol. 64, 1942, pp. 91-104. 2. Hottel, H.C. and Whillier, A., "Evaluation of Flat-Plate Solar Collector Performance," Trans, of the Conference on the Use  of Solar Energy,Vol. II, Thermal Processes, University of Arizona, 1955, pp. 74-104. 3. Whillier, A., "Solar Energy Collection, and Its Utilization for House Heating," Sc. D. Thesis, M.I.T., 1955. 4. Bliss, R.W., "The Derivation of Several 'Plate-Efficiency Factors' Useful in the Design of Flat-Plate Solar Heat Collectors," Solar Energy, Vol. 3, No. 4, 1959, pp. 55-64. 5. Klein, S.A., "The Effects of Thermal Capacitance Upon the Perform-ance of-Flat-Plate Solar Collectors," M.Sc. Thesis, Univer-sity of Wisconsin, 1973.-6. Beard, J.T. et al., "Performance and Analysis of 'Solaris' Water-Trickle Solar Collector," Proceedings of 1976 Joint Solar  Energy Conference: Sharing the Sun, Solar Technologies in the 70's. Published by ISES, Winnipeg, Canada, August 15-20, 1976. 7. Holland, K.G.T. et al., "Free Convection Heat Transfer Across Inclined Air Layers," Journal of Heat Transfer, Vol. 98, No. 2, May 1976, pp. 189-193. 8. San Martin, R.L. and Fjeld, G.J.-, "Experimental Performance of Three Solar Collectors," Solar Energy, Vol. 17, No. 6, 1975, pp. 345-349. 9. Smith, P.R., "Numerical Modelling of Thermal-Trap and Water-Trickle Solar Collectors," Proceedings of the Workshop on  Solar Collectors for the Heating and Cooling of Buildings, NSF-RA-N-75-109, New York City, Nov. 21-23, 1974, Univer-sity of Maryland, May 1975, pp. 315-321. 10. Bush, G.E., "Evaluation of Home Solar Heating System," UCRL-51711 Lawrence Livermore Lab., January 9, 1975. 73 11. Globe, S. and Dropkin, D., "Natural Convection Heat Transfer in . Liquids Confined by Two Horizontal Plates and Heated from Below," Journal of Heat Transfer, Trans. ASME, Ser. C, Vol. 81,. February 1959, p. 24. 12. Schmidt, E. and Silverston, P.L., "Natural Convection in Horizontal Liquid Layers," Chem. Eng. Prog. Symp. Series, No. 29, Vol. 55, 1959, p. 163. 13. Chandra, K., "Instability of Fluids Heated from Below," Proceedings of the Royal Society (London), Ser. A, Vol. 164, 1958, pp. 231-242. 14. Ogura, Y. and Yagihasi, A., "A Numerical Study of Convection Rolls in a Flow Between Horizontal Parallel Plates," Journal of  the Meterological Society of Japan, Vol. 47, 1969, pp. 205-217. 15. Quintiere, J. and Mueller, W.K.,. "An Analysis of Laminar Free and Forced Convection between Finite Vertical Parallel Plates," ••••• Journal of Heat Transfer, Trans. ASME, Ser. C, Vol. 95, 1973, pp. 53-59. 16. Plows, W.H., "Some Numerical Results for Two-Dimensional Steady Laminar Bernard Convection," The Physics of Fluids, Vol. kk, 1968, pp. 1593-1599. 17. Ekert, E.R.G. and Carlson, W.O., "Natural Convection in an Air Layer Enclosed Between Two Vertical Plates with Different Temperatures," International Journal of Heat and Mass  Transfer, Vol. 2, 1961, p. 106. 18. Dropkin, D. and Sommerscales, E., "Heat Transfer by Natural Con-vection in Liquids Confined by Two Parallel Plates Which Are Inclined at Various Angles With Respect to the Horizon-tal," Journal of Heat Transfer, Trans. ASME, Ser. C, Vol. 87, 1965, pp. 77-84. 19. Ozoe, H. et al., "Natural Convection in an Inclined Rectangular Channel at Various Aspect Ratios and Angles - Experimental Measurements," International Journal of Heat and Mass  Transfer, Vol. 18, 1975, pp. 1425-1431. 20. Buchberg, H. et al., "Natural Convection in Enclosed Spaces -A Review of Application to Solar Energy Collection, Journal  Heat Transfer, Trans ASME, Ser. C, May 1976, pp. 182-. 188. 74 21. Arnold, J.N. et al., "Experimental Investigation of Natural." Convection in Inclined Rectangular Regions of Differeng Aspect Ratios," ASME, Paper No. 75-HT-62, 1975. 22. Tan, H.M. and Charters, W.W.S., "An Experimental Investigation of Forced-Convective Heat Transfer for fully-Developed Turbulent Flow in a Rectangular Duct with Asymmetric Heating, Solar Energy, Vol. 13, 1970, pp. 121-125. 23. Sparrow, E.M. et al., "Experiments on Turbulent Heat Transfer in an Asymetrically Heated Rectangular Duct," Trans. ASME, Ser. C, Vol. 88, 1966, p. 170. 24. Ostrach, S. and Kamotani, Y., "Heat Transfer Augmentation in Laminar Fully Developed Channel Flow by Means of Heating from Below," Journal of Heat Transfer, Trans. ASME, Ser. c, May 1975, pp. 220-225. 25. Mori, Y. and Uchida, Y., "Forced Convective Heat Transfer Between Horizontal Plates," International of Journal of Heat Mass Transfer, Vol. 9, 1966, pp. 803-817. 26. Hwang, G.J. and Chang, K.C., "A Boundary Vorticity Method for Finite Amplitude Convection in Plane Poisueille Flow," Developments in Mechanics, Proceedings of the 12th Mid- western Mechanics Conference, Vol. 6, 1971, pp. 207-220. 27. Chen, T.S. et al., "Mixed Convection in Boundary Layer Flow on a Horizontal Plate," Trans. ASME, Ser. C, February 1977, pp. 66-71. 28. Lloyd, J.R.'and Sparrow, E.M.-, "Combined Forced and Free Convec-tion Flow on Vertical Surfaces," International Journal of  Heat Mass Transfer, Vol. 13, 1970, pp. 434-438. 29. Nusselt, W., "Die Oberflachen Kondensation des wasserdamfes," Zeitschrift des Vereines Deutscher Ingenieure, Vol. 60, 1916, pp. 541-569. 30. Rohsenow, W.M., "Heat Transfer and Temperature Distribution in Laminar-Film Condensation," Trans. ASME, Vol. 78, 1956, pp. 1645-1648. 31. Seban, R.A. and Faghri, A., "Evaporation and Heating with Turbulent Falling Liquid Films," Journal of Heat Transfer, May 1976, pp. 315-317. 75 32. Kutadeladze, S.S., Fundamentals of Heat Transfer, Edward Arnold, London, 1963, p. 307. 33. Telles, A.S. and Dukler, A.E., "Statistical Characteristics of Thin, Vertical, Wavy, Liquid Films," Ind. Eng. Chem. Fund., Vol. 9, No. 3, 1970, pp. 412-421. 34. Brumfiel.d, K.L. and Theofanous, T.G., "On the Prediction of Heat Transfer Across Turbulent Liquid Films," Trans. ASME, Ser. C, August 1976, pp. 496-502. 35. Chun, K.R. and Seban, R.A., "Heat Transfer to Evaporative Liquid Films," Journal of Heat Transfer,. Trans ASME, Ser. C, Vol. 93, 1971, pp. 391-396. 36. Dukler, A.E., "Fluid Mechanics and Heat Transfer in Vertical Falling-Film Systems," Chem. Eng. Prog. Symp. Sen'., Vol.. 56, No. 30, 1960, pp. 1-10. 37. Higbie, R. Trans. AIChE, 31, 1935, pp. 368-389. 38. Duffie, J.A. and Beckman, W.A., Solar Energy Thermal - Processes, John Wiley and Sons, 1974. 39. Klein, S.A., "A Design Procedure for Solar Heating System," Ph. D. Thesis, University of Wisconsin, 1976. 40. Klein et al., "A Design Procedure for Solar Heating Systems," Solar Energy, Vol. .18, 1976, p. 113. 41. Tabor, H., "Radiation, Convection and Conduction Coefficients in Solar Collectors," Bulletin of the Research Council  of Israel, 6C(3), 1958. 42. Siegel, R. and Howell, J.R., Thermal Radiation Heat Transfer, Vols. I and II, Washington, D.C.: NASA Sp-164, 1968. 76 A P P E N D I C E S 77 APPENDIX A . ITERATION PROCEDURE FOR DETERMINING HEAT TRANSFER CHARACTERISTICS OF A WATER-TRICKLE SOLAR COLLECTOR The heat loss from a water-trickle solar c o l l e c t o r consists of radiation, convection, and evaporation-conden-sation. Assuming bottom and edge losses are ne g l i g i b l e , the problem is to find the frontal rate of heat loss. Hottel and Woertz CD have shown that for an N-cover f l a t - p l a t e c o l l e c t o r , there exists a non-linear system of N+1 equations in N+1 unknowns. The N+1 equations are developed by carrying out heat balances between successive surfaces in the c o l l e c t o r . Hottel and Woertz noted that the thermal resistance within a glass cover is negligible compared to the cover-to-cover resistance, and a mean temperature can be assumed for each for heat transfer analysis. They also.assumed that there was no absortion in the covers so that the energy loss rate from the absorber plate is the same as the energy loss rate into the atmosphere. The N+1 equations are: A<TP - T > q top (h + h )A(T - T ,) c ec p g l ' = 0 l / e g + l / e p - 1 A(T 4 ... T 4 . ) a g i - l q i ' q top - T •) = 0 l / e + 1/e - 1 9 P q top e <JA(T4M - T 4 ) g gN sky' h A(T ,. - T ) = 0 w gN a • . . . .(A.l ) 78 The convective c o e f f i c i e n t s h's can be evaluated c based on known experimental c o r r e l a t i o n s . Beard C6) used Hollands' correlation for free convection in his work whereas Klein (5) used Tabor's correlations. Between the absorber plate and the cover, the evaporative loss c o e f f i c i e n t is evaluated by the heat-mass transfer analogy (Equation (3.4.2)). For a given average plate temperature, the N+l equations can be solved i t e r a t i v e l y to determine the top heat loss rate q t 0p a n d the mean cover temperatures. The top 1oss c o e f f i c i e n t U^, by d e f i n i t i o n , i s ut • <«top>/A<TP - V-It also can be evaluated i t e r a t i v e l y . The procedure for a single-cover c o l l e c t o r is outlined below. By heat balance, ( h n + h , + h )(T - T ) = U.(T - T ) . . . . ( A . 3 ) v cl r l ec p gl t p a v ; Rearrangi ng, U t ( T D - T a } T = j t P - . . . . (A.4) g 1 p (h , + h n + h ) cl r l ec The f i r s t step is to guess T ^ so that the heat transfer c o e f f i c i e n t s can be evaluated. U. for a single-cover c o l l e c t o r 1 s 79 + -1 . . . .(A.5) ) • h cl h r l + h h +h w ec ra Knowing 1 ^, right hand side of Equation (A.5) is known. A new value of T .j can therefore be calculated from Equation (A.4). The procedure is repeated until the value of T -j converges. cover c o l l e c t o r s . The complexity of the i t e r a t i o n , however, increases rapidly with N, the number of covers. For a t r i p l e -cover c o l l e c t o r , the i t e r a t i o n involves polynomial equations of the fourth power. For N>3, a complex i t e r a t i o n scheme such as the Newton-Raphson method is probably necessary. Fortunately, for most practical applications, three covers or less are required for f l a t - p l a t e c o l l e c t o r s . The same procedure also applies to double and t r i pie-APPENDIX B 8 Q DETAILED HEAT TRANSFER ANALYSIS FOR THE FLUIDS INSIDE A WATER-TRICKLE SOLAR COLLECTOR To analyze the heat transfer c h a r a c t e r i s t i c s of a water-trickle c o l l e c t o r , the problem is pictured as follows: AIR -WATER As shown in the diagram, water t r i c k l e s down the inclined absorber plate, and induces an a i r c i r c u l a t i o n pattern. Since the a i r layer immediately above the water surface is being dragged along, i t flows down the inclined plane. Assuming that the c o l l e c t o r is pneumatically sealed, the volume of a i r that reaches the end of the c o l l e c t o r w i l l be reversed and forced up the i n c l i n e . Thus there exists a 'pseudo-interface'between the two layers of a i r . The velocity of a i r at this interface should therefore be zero. The velocity of a i r immediately above the water layer is the same as the water surface v e l o c i t y . The three regions, the water layer and the two a i r layers, w i l l be analyzed separately in terms of their heat transfer c h a r a c t e r i s t i c s . B. 1 Temperature Distribution of the Thin Water Film 81 The major assumptions used in the following analysis are: 1. Steady state, laminar, one-dimensional flow. 2. Fully developed velocity and temperature p r o f i l e s 3. Constant heat flux from the absorber plate into the water. For the water layer, a heat balance of a small i n -crement along the flow is shown in Figure 10. This heat balance is (assuming S' = 0) bw W:C T-. ••+ q .dx = W;C (T, +• dx) + q dx, pw bw Mwl pw bw » ' Hw a X 3T. q . - q bw ^wl w and . .'(B.l) . .(B.2) 3x Wc pw The one dimensional energy equation is 9 2T , ^ _ 2L • . . . . (B . 3) a y w Assuming 9T b w/9x = 9T/9x = E Q w , and substitute into Equation (B.3), 9^T U . 2 a 9y w Ow ' 82 (B.4) For the t r i c k l i n g flow, the velocity p r o f i l e is 2 u = — ( 6 n y )sinB, u 1 2 w (B.5) where y is the distance measured normal from the plate surface Rearranging Equation (B.5), one obtains u = ^ y + k 2y' (B.6) where k 1 p g si nf3 ( — ) u w p g w3 s i n 8 2y w Substitute Equation (B.6) into (B.4), 9 T 1 2 — 2 = — (k y + k 2y ) E Q w ^ aw (B.7) The boundary conditions are: (D A t y = firqw 9T 'ay y = 5-83 (ii) At y = 0, q , = -k -J Mwl w ( i i i ) At x = y = 0, T = T ST 3y f i y=0 Assuming that the variation of the water temperature along the flow direction can be treated independently of that normal to the flow ( y - d i r e c t i on), integrating Equation (B.7) gives 3T E0w 3y a w 2 3 •) + f-,(x), (B . 8 ) where f-,(x) is a function of x only. Integrating once more, , 3 k 9y' 1 .4 h0w k . y d k2^ f T(x,y) = (—! + ) + / f (x)dx + f 9 ( x ) . . . (B.9) aw 6 1 2 J0 At y = 0, 8T — = -q /k = f,(x) , . „ Mw w 1 v ' dy E0w k l y 3 k 2 y 4 q w l y and T(x,y) = - — ( + ) - + f„(x) (B.10) a 6 12 k c w w D i f f e r e n t i a t i n g Equation (B.10) with respect to x, one finds f 2 ( x ) = E n. d x + C 0 , . Ow 3 84 . . ( B . l l ) and EOw k l y 3 k 2 y 4 q w l y T(x,y) = ( + ) -aw 6 1 2 w ^ EOw d x + C3. . (B.l2) From Equation (B.2), qwl qw -E Ow WC. pw and q = U.(T - T ), w^ t ws a (B.13) substitute Equation (B.13) into (B.2), q , U.(T - T ) wl t ws a Ow WC WC pw pw (B.14) Rearranging Equation (B.2) yields 9T ws WC (q , - q ) = 0, pw V Mwl Mw' o X (B.15) where 9T /9x = 9T. /9x = 9T/9x. ws bw 85 Dividing Equation (B.15) by U^, one obtains WC 3T q . - pw ws wl + (T - T ) = 0. . . . . (B.16) Solving the above linear d i f f e r e n t i a l equation yields T - T - q ,/U\ -U.x/WC ws a Mwl t t pw = e . . . . .(B.17) T_. - T - q ,/U. f i a ^wl t T h u s -U\x/WC t pw T = (T.. - T - q 1/U.)e ws v f i a ^wl t + T a + q w l/U t. . . . ,(B.18) Substitute Equation (B.18) into (B.16) a T 1.1 ut - V / W C P „ T-'Tr ZT ( < T ^ • T » " q w l / U t ) e 3x WC WC pw pw + WV- . . . . (B.19) Differentiate Equation (B.10) w i t n r e S p e c t to x, and compare to Equation (B.19), one finds 86 qwlX Ut -Utx/WCpw Vx) =- — <<Tf1 " Ta " qwl/Ut>e wc wc pw pw -WC pw x ( ) + q w l X/U t) + C 3. • • • -(B .20) Ut From Equation (B.10), E0w V 3 V 4 T ( x , y ) = _ ( _ _ + — ) - q w l y / k w + qwlx/WC a w 6 12 u\ -U.x/wC t t pw WC pw ^ T f 1 " T a " q w l / U t > e X < - W C p w / U t ) + q w l x / U t ) + C 3 ( B - 2 1 ) From boundary condition ( i i i ) , T = T^. at x = y = 0, then, T n = (T,. - T - q ,/U.) + C Q x=y = 0 v f i a \1 t 3 " T f i , • • • •(B .22) t h u s C3 = T a + qwl/Uf ' * • -(B-23) Therefore, 87 T(x,y) = _ ( — _ + _ ) - q y/k a w 6 12 •U + x/WC t pw a + q w l/U t. . • • .(B.24) The above expression is based on a unit width of c o l l e c t o r surface. 88 B.2 Convective Heat Transfer across Air Layers Assuming the a i r layer between the water surface and the f i r s t cover surface consists of two regions of c i r c u l a t i o n as shown below, the heat transfer c h a r a c t e r i s t i c s are analyzed for each region . •water The layer thicknesses for the regions are chosen a r b i t -r a r i l y as a and b, and the heat transferred across each layer is calculated. The flows are assumed to be purely!forced convective, one-dimensional, 1 aminar, fully-developed, and at steady state. For the layer thickness b, the velocity p r o f i l e is assumed to be l i n e a r : u = Uy/b. . . . .(B.25) Let = 9T b a/3x = = constant, 8T/3x . (B.26) 89 the governing energy equation is i f l =— E- . . . . (B.27) • o ua . 9y 2 a a A simple heat balance shows that q - qi w 1 E_ = . . . . . (B.28) U a W C a pa Thus 3 2T Uy ~2 = ~ b E 0 a , • • • -(B.29) with boundary conditions: i .) At y = 0, q = -k 9T/3yl a |y=o i i . ) At x = 0, T b a = T b a 1 = constant Integrating Equation (B.29) yields 3T UE y 2 oaJ + f n (x)., . . . . (B.30) 90 again, .3 UE y' 0a T(x,y) = + f 1 ( x ) y + f 2 ( x ) . . . . .(B.31) 6 a b a Differentiate with respect to x, 3T 3f ] 3 f 2 — = E n = y + . . . . .(B.32) 3x u 3x 3x Since q = -k 3T/9yj , and from Equation (B.30), w a y=o 3T/3y I =•• f, (x) |y=o- 1 •q„/k . . . . . (B.33) w a Thus f, (x) = -q/k . . . . (B .34) i w a Combining Equation (B.32) and (B.34) yields 3x " a w f 9 ( x ) = E n x + C ? - ( -y )/k . . . . . (B.35) In order to obtain an approximate solution, assume q w = constant Then, Equation (B.35) becomes 91 f 2 ( x ) = E Q a x + C 2 . . . . . (B , 3 5 a ) S u b s t i t u t i n g E q u a t i o n ( B . 3 5 a ) i n t o ( B . 3 1 ) y i e l d s U E n y 3 q T ( x , y ) = — ; y + E Q a x + C g . . . . . ( B . 3 6 ) 6 a b k a a By d e f i n i t i o n , t h e b u l k t e m p e r a t u r e o f a i r i s b ,uT dx b a ' dx S u b s t i t u t i n g E q u a t i o n s ( B . 2 5 ) a n d ( B . 3 6 ) i n t o ( B . 3 7 ) y i e l d s U E 0 a b 2 T b a = — + E 0 a x + C 2 + 2 C l b / 3 - . . . .(B.38) a A t x = 0, T b a T b a l U E 0 a b 2 1 5a a + C 2 + 2 ^ b/3 . . . . . (B .39) Rearranging Equation ( B . 3 9 ) , 92 U E n b 2 Oa Co = T k . 2 C . b / 3 . . . . . ( B . 4 0 ) 2 bal 1 1 Substituting Equation ( B . 4 0 ) into ( B . 3 8 ) yields Tu = T, . + E n x. . . . . ( B . 4 1 ) ba bal Oa Substituting Equation ( B . 4 1 ) into ( B . 3 6 ) yields U E 0 a * 3 % U E o a b 2 6a b k, 15 a . a a t» • • • ( B . 4 2 ) The local heat transfer c o e f f i c i e n t is given by ka 9 T / 9 y | y = o h = • • • - ( B . 4 3 ) T - T . w ba U E 0 a b 2 where T = E n x + T 2 C , b / 3 . . . • .(B . 4 4 ) w Ua bal ^ g a I Substituting Equations ( B . 4 1 ) and ( B - . 4 4 ) into ( B . 4 3 ) yields 93 'xl U E 0 a b - 2± 9 T (B.45) 1 5 a . 3y Simpli fy i ng, h -I b 2 2 q, f - q, , ->LL_ = ( ( _ w l j j - l . . . . ( B . 4 6 ) k a 3 1 5 qw For an approximate s o l u t i o n , assume q = q, , then W I h -b/k = 3 / 2 . . . . . (B.47) x I a S i m i l a r l y , h x 2 can be shown to be k a q 1 - q h x 2 = — ( 1/3 - ( )/15 )"! . . . . (B.48) b q For q 1 = q, h / 2 b / k a = 3. S imi lar analysis for the a i r layer with thickness a has been performed and resul ts obtained. Assuming a parabol ic ve loc i ty p r o f i l e , and Eg a = 0, 94 = .'"2 (B .49) Thus the overall heat transfer c o e f f i c i e n t from the water surface to the adjacent cover surface is 1 hxs = : r -< 1 / h x l + 1 / h x 2 ) + ( 1 / h x 3 + 1 / h x 4 ) = k / ( a + b) . . . . . ( B . 5 0 ) a Equation ( B . 5 0 ) is va l i d only when there is no net heat gained by the a i r layers. Ostrach (24] showed experimentally that under constant heat flux conditions, the heat transfer rate for a fully-developed, laminar flow is almost identical to the rate for conduction. The above analysis is for an idealized laminar, purely forced convective flow confined in p a r a l l e l plates. In a c t u a l i t y , however, the flow patterns above the c r i t i c a l Raleigh number are complicated and consist of several flow regimes (as discussed in Section 3.1). Thus for design calculations, Tabor's exper-imental correlations for free convection are used in obtaining the design plots (Figure 39 to 41) under di f f e r e n t operating conditions. 95 APPENDIX C DERIVATION OF THE MASS TRANSFER CORRELATION FOR SURFACES WITH A UNIFORM PROFILE To derive the mass transfer c o e f f i c i e n t s for the water vapor region in an open-e.nded water-tri ckl e c o l l e c t o r , the s i t - " uation is shown in Figure 8. The flow is assumed to be at steady state, laimnar, fully-developed, and one-dimensional. Inside the small penetration depth 5 ' , the velocity u is assumed to be constant (37) . Thus the one-dimensional mass transfer equation is 8 C A 9 2 C A U = D A B — . . . . ( C l ) 3x 3y with boundary conditions: i.) At x = 0, C A = C A Q, for a l l y, i i . ) At y = 0, C A = C A i for a l l x>0, i i i . ) At y = 6 ' , Cft = C A Q for a l l x. The solution to the above equation is CA = C A i " ( C A i " C A 0 ) e r f c h ' ( C 2 ) 96 and erfck 77 J e dt . . . (C .3) 0 By d e f i n i t i o n , the local mass flux of solute A at the surface i s 3C N A ) l o c a l "DAB ay dC, AB y=o dy y=o for a fixed x. But dC A ^ dC A d<|)1 dy d(j)1 dy d ( e r f ^ ) = " ( C A i " C A 0 ^ d&. 0.5 . /u/DABx (C.4) 2 -<j>n T h - e f 0 - N A ) l o c a l = D A B ( C A . - C A 0 ) - e 1 O.S^ uT^ T y = 0 <CAi " C A 0 V ABU (C5) TTX B u t N A ) l o c a l = h m ) l o c a l ( C A i " C A 0 ) 97 therefore hm)local =J DAB U / 7 T X' • ' ' - ( C - 6 ) By d e f i n i t i o n , NA)avg = 7j N A ) l o c a l d x ' • ' ' -<C-7> 1 ° - X NA)avg =7 | h m ) l o c a l ( ' C A i " C A 0 ) d x - ( C ' 8 ) Substituting Equation (C.6) into (C . 8 ) yields N nx = 2h n ,(C. • - C n n ) . . . . . (C.9) A)avg m)1ocal Ai AO' ' B u t NA)avg = h m ) a v g ( C A i " C A 0 J » t h u s hm)avg = 2 hm)local> • ' ' and Sh „ = 2Sh, „ a l avg 1ocal 2 n m ) l o c a l x / D A B D A B J <fx 98 = l _ / I l I I , - l . R e 0 . 5 S c Npv/ v D A B Therefore 0 5 0 5 S h a v g = 1.129 Re u , 3Sc -° (C.12) 99 APPENDIX D DERIVATION OF (xa) FOR A CORRUGATED SURFACE The transmittance-absorptance product (xa), by def-i n i t i o n , is the fraction of the incoming energy that is u l t -imately absorbed by the absorber plate of a c o l l e c t o r . The expression for a f l a t plate is well-known. For a corrugated plate, however, i t does not appear to have been derived. Therefore an attempt w i l l be made in this appendix to derive the expression for (xa) for a corrugated surface. In the f i r s t part of this derivation, a dry corrugated surface is assumed, and the expression for (xa) is derived accordingly. In the second part, an expression for (xa) is again derived by assuming that water is present in the valleys of the corr-ugation. D . 1 Dry Corrugated Plate D.1.1 Derivation of (xa) . The situation for a dry corruagated plate is shown in Figure 45. As seen fron the figure, there are four d i s t i n c t parts of the corrugated surface which absorb parts of the incident radiation,- In deriving (xa), the net radiation method (N) (N ) is used. Let R ' and T ' be the overall reflectance and transmittance of an N-cover system; for such a system, there 100 are are a total of 2N cover surfaces (Figure 46). For a c o l l e c t o r with N covers, heat balances show that (Figure 45) : F c l = q i , c l " V,cl> • • • '(D- 1- 1) Fc2 = q i , c 2 " q0,c2> ' • • -(D--1.2) Fc3 = q i , c 3 " q0,c3' ' ' • '(D-1-3) Fc4 = q i , c 4 " q0,c4> • * • '(D- 1- 4) Ignoring long wave radiation emitted by the plate, the out-going fluxes are: q 0 , c l = P c l q i , c l ' ' * * - C 3 - 1 - 5 ) q0,c2 = P C2 q i,c2' ' ' * - O - 1 - 6 ) q0,c3 = Pc3 qi,c3, ( D J ' 7 ) "0,c4 = p c 4 q i , c 4 ' • ' * '(D-1-8) where p'd = exp(-.0255 - 6 .683cosei + 5 .947 (cose i) 2 - 2.484(cose . ) 3 ) . 101 For a c o l l e c t o r with N covers, F , the fraction of rad-iation reflected back to the ambient by the entire system (N) (N) is comprised of two portions: R^  and q.. 2N^2N f i r s t portion is the fraction of incident radiation on the top surface that is reflected by the entire N-cover system (excluding the (N) absorber p l a t e ) . The second portion, q^ 2NT2N ' s t' i e f r a c t l ' o n of the incident radiation on the 2Nth surface that is trans-mitted through the covers and goes into the atmosphere. Thus Also, qg 2N' t n e outgoing flux from surface 2N, is comprised of the transmitted flux from the top and the reflected flux from the 2Nth surface. Thus, V ? N = T i N > + *i.2**n • ( D : 1 - 1 0 ) Suppose there are n corrugations along, the width of the co l l e c t o r as shown below, I 2Nth surface n corrugations 102 a net radiation balance over each of the corrugation is as fol1ows: qi,2N q0,2N qi,c4. q 0 , c 4 si/ i ,c,I q0,c3 0,cl q i , c3 q i ,c2 q0,c2 From the absorber plate to the (2Nth) surface, qi,2N = F15 q0,cl + F25 q0,c2 + F35 q0,c3 + F45 q0,c4, .(D.l .11 ) and for each section of the corrugation, 'i ,cl F21 q0,c 2 + F31 q0,c3 + F41 q0,c4 + F51 q0,2N' .(D.l .12) 103 q i , c 2 = F12 q0,cl 1 + F32 q0,c3 + F c o q n 0 M , . . . .(D.l .13) F42 q0,c4 + F52 q0,2N' q i , c 3 = F13 q0,cl + F23 q0,c2 + F43 q0,c4 + F53 q0,2N' \" " • (^  • ^  -14) q i , c 4 = F14 q0,cl + F24 q0,c2 + F34 q0,c3 + F 5 4 q 0 5 2 N . • • • -(D.l .15) Substituting Equaiion (D.l.11) into (D.l.10) yields a = T (N) + R(N) + R(N) + q0,2N ' l K2N h15 q0,cl K2N h25 q0,c3 R2N ) F45 q0,c4- • • • '( D-V-16) Combining Equations (D.l.5) to (D.l.8) with Equations (D.l.12) to (D .1 .15) yields - q 0 , c l / p c l + F21 q0,c2 + F31 q0,c3 + F41 q0,c4 + F51 q0,2N - °» ' * ' '(D- 1- 1 7) F12 q0,cl ~ q0,c2 / pc2 + F32 q0,c3 + F42 q0,c4 + F52q0,2N'<= ° 104 (D.l .18) F13 q0,cl + F23 q0,c2 " q0,c3 / pc3 + F43 q0,c4 + F53 q0,2N = °> (D.l .19) F14 q0,cl + F24 q0,c2 + F34 q0,c3 " q0,c4 / pc4 + F54 q0,2N = °- (D.l .20) Rearranging Equations (D.l.16) to (D.l.20) in matrix form, 1 / p c l F21 F31 F41 F51 F , o - l ; / p r o F 0 0 F , „ F , 1 2 'c2 32 '42 52 F13 F23 " 1 / p c 3 F43 F! 14 F24 F 34 _ 1 / P r " F-c4 54 (N) R(N) - ( N ) F R ( N . ) K2N h15 K2N h25 2N r35 K2N h45 q 0 , c l 0 q0,c2 0 q0,c3 — 0 q0,c4 0 s ^q0,2N^ [ 1 ) (D.l .21) The above matrix equation contains 5 equations in 5 unknowns and is therefore solvable. 105 Summing Equations (D.l.l) to (D.l.4), one obtains. Fcl + Fc2 + Fc3 + Fc4 = (qi,cl + q i , c 2 + q i , c 3 + q i , c 4 } " (Vcl + q0,c2 + q0,c3 + q0,c4> • • ' • (D. 1 .22) Summing Equations (D.l.12) to (D.l.15), one obtains (qi,cl + + q i , c 4 } = ( 1- F15 ) q0,cl + ( 1- F25 ) q0,c2 + ( 1 - F 3 5 ) q 0 , c 3 + « 1 " F 4 5 ) < » 0 , c 4 + q 0 , 2 N ( D . l . 2 3 ) Substituting Equation (D.l.23) into (D.l.22), one obtains Fcl + Fc2 + Fc3 + Fc4 = qO,2N- ( F15 q0,cl + F25 q0,c2 + F35 qO,c3 + F 4 5 q 0 j C 4 ) . (D.l.24) Combining Equation (D.l.l 1) with (D.l.24), one obtains F c l + F c 2 + F c 3 + Fc4 = q0,2N-qi.,2N' ' " ' ' ( D - 1 - 2 5 ) But by d e f i n i t i o n , F c-| + F c2 + F c 3 + F c 4 = ^ T 0 ^ thus (xa) = q Q j 2N " qi,2N* * * ' -C3-1-26) 106 For a f ixed corrugation geometry, the shape factors can be c a l -cu lated . Therefore the problem is to solve the matrix equation (D. l .21) and obtain values for the q 0 ' s . Subst i tut ing Equation (D. l .11) into ( D . l . 2 6 ) , one obtains (,xa) = Q 0 j 2 N " ( F 1 5 q 0 , c l + F 2 5 q 0 , c 2 + F 3 5 q 0 , c 3 + F 4 5 q 0 , c 4 ) -. . . . (D. l .27) Equation (D. l .27) w i l l be used in obtaining values of (xa) for d i f f erent corrugation geometry. 107 D.l .2 Derivation of Shape Factors The following diagram i11ustrates a detailed sketch of a corrugation cross-section: F 5 E 4 The l e t t e r s A, B, C etc. designate the dimensions as shown. From standard shape factor calculations (4 2} , shape factors for the above sketch are as follows: F15 = F16 =(E/A)0.5(1-F 2 6B/E), F25 = F26 = 1-2((A+B-C)/2B) , F 3 5 = F 1 5 ' 108 F 4 5 = 1 ' F51 = F56 F61 = (E/(E + B))0.5(1-F 2 6B/E) , F =F F r52 r56 62 =(E/(E+B))B F 2 g/E, F53 = F51 ' F 5 4=B/(E+B). The dimensions of the corrugation are related as follows: C=(A2+B2-2AB c o s ( 9 0 + a 2 ) ) ^ / 2 E = B + 2D,3 tan(a 2), A = D = D o / c o s ( a 0 ) . 109 D. 2 Corrugated Plate with Thin Water Film in Valleys of the Corrugation D.2.1 Derivation of (xa ) Figure 47 shows the corrugated plate's cross-section with water in the v a l l e y s . Using the net radiation method, and balancing the heat fluxes as before, one obtains the following system of equations: F c l q i , c l " q 0 , c l ' Fc2 = ( q i , c 2 ' q0,c2 } " Fwa' Fc3 = q i ,c3 " q0,c3' Fc4 = q i ,c4 " q0,c4' q 0 , c l = p c l q i ,cl ' q0,c2 = Rws qi,c2' q0,c3 = p c 3 q i , c 3 ' q0,c4 p c 4 q i , c 4 ' (D.2.8) w + a. „..TI!P F r = R l " ' + q i , 2 N T 2 N ' '0,2N 1 = T^N> + q- „..R (N) i ,2N"2N no • (D.2 .9) . (D.2 .10) From the absorber plate to the 2Nth surface q i , 2 N = F 1 5 q 0 , c l + F 2 5 q 0 , c 2 + F 3 5 q 0 , c 3 + F 4 5 q 0 , c 4 - - ( D - 2 . H ) Following a s imi lar procedure as for a dry corrugated p la te , the resultant matrix equation i s : - 1 / p c l F21 F31 F41 F51 F12 " 1 / R w s F32 F42 F52 F13 F23 " 1 / p c 3 F43 F53 F14 F24 F34 " 1 / p c 4 F54 ( N ) F R ( N ) F R ( N ) F R(N) , K2N h15 2N 2^5 2N 35 K2N h45 1 '0 ,cl q 0 ,c2 '0,c3 '0,c4 q 0,2N •T (N) 1 (D.2.12) Since water absorbs a portion of the incoming r a d i a t i o n , the net (xa) product for the present case wi l l be s l i g h t l y smaller than for a dry corrugated p la te . Let 1 = F - F c2 h c2 wa* Ill where F is the fraction of the incident flux absorbed by water wa F' 0 can be shown to be 0 c c ( 1 - P W J ( 1 - P _ ) T p i ws c w ( D o 1 3 ) F 2 - - 2 q i , c 2 ' • ' ' ^ u - 6 ' ^ ) Pws + p c ( 1 - 2 p w s ) T w 2 and R = • . . . .(D.2.14) w s ^ P w s P ^ w If"absorption.by water is assumed to be small (T =1), and p w s = p c , then 2p R = L. . . . .(D.2.15) W S 1 + P C Thus the net ( x a ) product for a wetted corrugated plate is <™> = F c l +Fc2 + F c 3 + F c 4 -Summing Equation (D.2.1) to (D.2.4) yields F c l + F c 2 + F c 3 + Fc4 = ( q i , c l + q i , c 2 + c ' i , c 3 + q i , c 4 ) ( q 0 , c l + q 0 , c 2 + q 0 , c 3 + q o , c 4 ) - F w a -(D .2 .1 6) 112 Following the same derivation procedure as for a dry corrugated plate, ( x a ) is found to be ( x a ) = q0,2N- ( F15 q0,cl + F25 q0,c2 + F35 q0,c3 + F45 q0,c4 ) F w a , . . . .(D.2.17) ( 1" pws ) ( 1- pc ) Tw pws + pc ( 1- 2 pws ) xw a n d Fwa = q i , c 2 ( 1 ~ ~ >• ^ w s ^ w 1- pws pc Tw S i n c e q0,c2 = Rws q i , c 2 pws + pc ( 1- 2 pws ) Tw 2 2 i , c 2 ' 1- pws pc Tw substituting Equation (D.2.19) into (D.2 .18) yields F qo,c2 ( 1- pws pc Tw 2 ) wa T p +p (l-2p ) x ws Hc ws w (D.2.18) q. 9 , . . . . (D.2.19) 113 2 (1-p )(l-p )T p +p (l-2p JTW Kws c w ws c ws (1 ; ) ^PwsP^w 2 ^PwsPJw 2 ( 1-Pws pc Tw 2 ) q0,c2 ( — Pws + pc ( 1" 2 pws ) Tw ^ - P w s ^ T - P c ) ^ p +p (1 -2p ) x 2  Mws M c Mws w - 1 ) . . . .(D.2.20) Substituting Equation (D.2.20 ) into (D.2.17) yields ( r a ) = q 0,2N- ( F15 q0,cl + F35 q0,c3 + F45 q0,c4 ) ,C2 ( F25 + C-1 _ pws pc Tw' Pws +Pc ( 1- 2 pws ) Tw' ^-Pws^^Pc^w Pws + pc ( 1- 2 pws ) Tw 2 ']> (D.2.21) Note that when there is no absorption by water, the sum of the terms inside the square bracket is zero, and the ( x a ) expression is the same as for a dry corrugated plate. 114 The next section summarizes the results of the fractions reflected and transmitted by an N-cover system. 115 D.2.2 Fractions of Incident Flux Reflected and Transmitted by an N-cover System For a single-cover c o l l e c t o r , m P g - P g ( 1 - 2 P g K g 2  R U ) = . . . . .(D.2.22) L , 2 2 g g (l-p 2)x 1 1 i 2 2 1 -P - T g g For a double-cover system, p +p (l-2p ) T 2 ( 2 ) _ g g g g K22 " 2 2 g g (1-P ) 2 T p +p (1-2p ) T 2 g g 2 g g g g ( i + ( ) 2 ( i - ( ) d) ') i 2 2 i n 2 T 2 1 - P T l-p T g g g g * Results obtained from R. Siegel's unpublished notes on solar energy u t i l i z a t i o n . 116 0-PJ 2 T p+p„(1-2P„)T 2 (?) g g ? g g g g 2 - 1 T(2) = ( ) 2 ( 1 _ ( )d) 1_ . (D.2.25) 1 0 0 0 0 , 2 2 . 2 2 1-Pg 1-Pg ^ g A special case that yields simple results is a system of N covers that has negligible absorption and has the same r e f l e c t i v i t y throughout the system. For such a case, T =1 , and R (N) _ 2 N Pg 1+(2N-l)p g . (D.2.26) T ( N ) ]" P9 . . . . . (D.2.27) l+(2N-l)p g 117 D. 3 Evaluation of the Monthly Averaged ( x a ) of a Corrugated  Plate Co l l ec tor . In order to evaluate (TO ) of a coorugated plate , .one has to take into account the beam and the diffuse portions of the incoming r a d i a t i o n . The general expression for ( x a ) i s ss (~ ) = • • • • (D.3.1) % The numerator of the above equation can be approximated as: SS — SS — J . ( x a ) b ! 8b + ^ r ( T a ) d ! B d , . . . . (D.3.2) where the f i r s t term is for beam r a d i a t i o n , and the second term is for dif fuse r a d i a t i o n . Since monthly average hourly total radiat ion data on horizontal surfaces are usually a v a i l a b l e , the f i r s t term of Equation (D.3.2) can be estimated by ss , — ss — . E (xa) b I b = E ( T a ) b I b R b (D.3.3) sr p sr where a l l terms are ca lculated for each hour and summed from sunrise to sunset of the monthly representative day. 118 The second term of Equation (D.3.2) can be estimated by using values calculated at an angle of incidence of 60 degrees. Taking into account both the sky and the ground diffuse radiation, this term can be written as ( r a ) d I p d = ( x a ) s d H 6 d(l +cos6),/2 + (TO)_H 1Tb(1-COS3)P/.2. . . . .(D.3.4-) Thus Equation (D.3.1) can be written as _ 11 ( x a ) b l b K H3d (xa) = — = + (xa ) s d - ^ z (l+cos3)/2 H3 H3 + (xa) r d(l-cos3)p/2. . . . .(D.3..5) Assuming ( Tra) r c| = ( ~ ) s c | ' a n c ' u s l' n9 monthly average hourly and daily solar radiation data, monthly average values of (xa) can be calculated. 119 APPENDIX E DERIVATION OF USEFUL HEAT GAIN EXPRESSION FOR A CORRUGATED-PLATE SOLAR COLLECTOR To derive the expression for the useful heat gain for a corrugated-plate c o l l e c t o r , the situation is pictured as foilows: D2/2 D2/2. At the centrelines of the tops and bottoms of the corrugation, there is no heat flow because of symmetry. Since the absorber plate is usually a good conductor, one can assume one-dimensional heat flow along the cross-section and treat the above problem as a c l a s s i c a l f i n s i t u a t i o n . Neglecting the corner e f f e c t s , the corrugated cross-section from points 1 to 3 is stretched out into a linear f i n : 120 insulated end - S. D 2 /2 I . P 3/cos a 2 u WS The heat balance for the surface enclosed by points 1 and 2 yields d 2x U + (T - T. ) / k f i n 5 f i n (E.l) with boundary conditions dT x = 0 0, ( 1 1 ) , T x=D 2/2 Defining m =7 u t / k f i n 6 f 1 n and 4,i = T - T a - S 4/U t > 121 Equation (E.l) becomes 1 2 dx' -. m = 0, with boundary conditions ( i ) d^ . dx 0, x = 0 ( i i ) 1(1 • = T 2 - Tfl - S 4/U t x=D2/2 The general solution is 1^ = sinh mx + C 2 cosh mx • • .(E.2) Solving for the constants with the boundary conditions yields T-T a-S 4/U t cosh mx T 2 " T a " S 4 / U t c o s h m ( D 2 / 2 ) (E.3) 122 By Fourier's Law of conduction, the heat conducted to point 2 is dT 2 f i n f i n dx x=D2/2 tanh mD2/2 = ( S 4-U t(T 2-T a) ) — -. . . . . (E.4) m Solving for the temperature d i s t r i b u t i o n bounded by points 2 and 3, Equation (E.2) also applies but with boundary condi t i ons: ( i ) T| = T , x = 0 L T | / = Tws' x=D 3/cosa 2 where x is now measured from point 2. The solution is T-T -S,/U. = C, sinh mx + (T 9-T -S./U.) cosh mx, (E.5) a i t i c. a i x where C-jis given by C1 = (T w s-T a-S 1/U t)-(T 2-T a-S 1/U t)cosh(mD 3/cosa 2) si nh(mD Q/cosa 0) 3 2 . . . .(E.6) 123 By Fourier's of conduction, the heat conducted through point 2 is dT (E.7) x = 0 Solving Equation (E.7), mD. ( V V T w s - V M s r V V V > C 0 S h — q 9 = (1/m) ; C 0 S a2 sinh mD^/cos a 2 (E.8) Equating Equations (E.4) and (E.8) yields ( S 4-U t(T 2-T a) )tanh mD2/2 S r U t ( T w s - T a ) - ( S r U t ( T 2 - T a ) ) c o s h m D 3 / c 0 S a2 sinh mD^/cos . . (E.9) Solving Equation (E . 9) f or ~T * yi el ds 124 T 2 = ( (tanh mD 2/2)(sinh inD3/cos a 2 ^ S 4 + U t T a ^ - ( S 1 - U t ( T w s - T a ) ) + cosh mD 3/cosa 2 ( S ^ t V } x.(.(cosh mD 3/cos a 2)U t+(tanh mU^/2 ) ( s i nh mD 3/cos a ^ ) 1 x U t ) • • • • (E .10) Due to the diurnal movement of the sun, except at solar noon, T 2 is usually d i f f e r e n t from T^. Therefore, i t is required to repeat the above process starting from point 6. The temperature d i s t r i b u t i o n from point 6 to point 5 is ws J 4 ^ 6; i D 3/cos a 2 D2/2 . X X insulated end T - V V u t cosh mx T 5 " T a " S 4 / U t C O s h m D 2 / 2 . . ( E . l l ) The tempetature d i s t r i b u t i o n between points 5 and 4 is T-T -S-/U. = C„sinh mx +(T C-T -S 0/U.)cosh mx, • . (E.12) a o t c o a o t where x is now measured from point 5, and C 9 is given by 125 (Tws-Ta-S3 /Ut ,- (VTa-S3 /Ut )cosh m D 3 / c 0 S a2 C = : ..(E.13) sinh mD0/cos '3' a2 Tgis given by T 5 = ( (tanh mD2/2)(sinh mD3/cos a 2^ S4 + Ut Ta^ "(W^s-V^Wa) cosh m D 3 / c o s a2 > x ( U t cosh mD3/cos a 2 + u"t(tanh mD2/2). (sinh mD3/cos ) ) J (E .14) The rate of heat gained by the surface bounded by points 1 and 3 is also the rate of heat conducted through point 3, M3 f i n f i n dT dx x=D3/cos a 2 k.p. Sf. ( m C,cosh mx + m(T0-T -S,/U.)sinh mx ) ti n t i n i ^ a i T . at x = '.D3/cos a2 126 = (1/m) ( ( S 1 - U t ( T 2 - T a ) ) s i n h mD3/cos a 2 + (cosh mD3/cos a 2 ) ( s i n h mD3/cos a 2 ) _ 1 x (cosh mD3/cos a 2 ) ) ) . . . . .(E.15) Sim i l a r l y , the rate of heat gain through point 4 is q 4 = (1/m) ( ( S3~ Ut( T5" T- a) ) s i n h mD3/cos a 2 + (cosh mD3/cos a 2 ) ( s i n h mD3/cos a 2 ) 1 * < S3- Ut< T Ws- Ta>-< S3- Ut'V Ta>> x (cosh mD3/cos a , , ) ) ) . . . .(E.16) The rate of heatgained from the surface bounded by points 3 and 4 is q . = D,( S 0 - U.(T - T ) ) , . . . . (E .17) Mwater 1 v 2 t ws a where S 2 = 1 6 b ( x a ) b ( S f ) 2 + I 0 d ( T a ) d . Thus the total rate of heat gain is 127 q t o t a l = q3 + q4 + qwater' where the q's are given by Equations (E.15) to (E.17). By d e f i n i t i o n , F' is the ratio of the actual heat gain to the heat gain i f the whole c o l l e c t o r were at the local water temperature. If the whole c o l l e c t o r were at the local water temperature, the heat gain is q = (S 4D 2 + S1D3/co-^ ag+SgD^SgDg/cos a 2 - (Dg + D^Dg/cos a 2 ) U t ( T w s " T a ) ) ' ' ' " - ( E - 1 9 ) Thus r = < q3 + q4 + qwater > x ( S 4 D 2 + S 1 D 3 / c O S c ^ + S ^ + S 3D 3/cos a 2 - ( D 2 + D1+2D3/cos a 2 ) U t ( T w s " T a ^ ) ^ ....(E.20) 128 F I G U R E S Figure 1. Sketch of a tube-and-plate c o l l e c t o r . VO cover Figure 2 . Sketch of a corrugated>plate water^-trickl c o l l e c t o r . 2o* u Figure 7. Probable v e l o c i t y p r o f i l e s for a i r and water under the influence of combined free and forced convection. Right hand figure i s the pattern at higher temperature d i f f e rence oo e n between water surface and the cover. water Figure 8 . Open-ended water-trickle c o l l e c t o r . A i r enters from the top of the c o l l e c t o r and exits at the bottom. Figure 9 . Enlarged view of the flow pattern inside an open-ended water-trickle c o l l e c t o r . Water vapor i s assumed to diffuse through a small penetration depth 6 Q and encounters ambient conditions. cover plate for the thin water f i l m flowing down an i n c l i n e d plane. 139 5 0 . r 1. 2. 3, 4. DISTANCE ALONG FLOW (m) Figure 11. Rise of temperature of water along the c o l l e c t o r . S=680 W/m2, T =10 l OC f T f ±=30 0 C, wind velocity=5 m/s, ' C =-4190 J/kg°C, F*=l,, £ w s=-95, A=5.0 m2. Figure 13. Fro n t a l heat loss rate as a function of water surface temperature for a single-cover c o l l e c t o r . A i r temperature=10 0 C, wind velocity=0. — I 1 1 1 1 r Beard model — — — Present f l a t - p l a t e 20. 30. 40. 50. 60. 70. WATER SURFACE TEMPERATURE ( 0 C) Figure 14. Frontal heat loss rate as a function of water surface temperature for a two-cover c o l l e c t o r . T =10 C, wind v e l o c i t y =0. _, a ^ ro 1 1 1 1 1 r B e a r d m o d e l — — — « — P r e s e n t f l a t - p l a t e 20. . W . 4 0 . 5 0 . 6 0 . 7 0 7 WATER SURFACE TEMPERATURE ( ° C) F i g u r e 15. F r o n t a l h e a t l o s s r a t e as a f u n c t i o n o f w a t e r s u r f a c e t e m p e r a t u r e f o r a t h r e e - c o v e r c o l l e c t o r . T = 10° C, w i n d v e l o c i t y = 0. Beard model Present f l a t - p l a t e m model T J : J o . k> 20. 30! 40. 5 WATER SURFACE TEMPERATURE ( °C) Figure 16. Cover surface temperature as a function of water surface temperature for a single-cover c o l l e c t o r . T =10 C, wind v e l o c i t y = 0 3 . Beard model Present f l a t - p l a t e . model 1 1 I I I » 20. 30. 40. 50. 60. 70. WATER SURFACE TEMPERATURE ( 0 C) Figure 17. Cover surface temperature as a function of water surface temperature for a two-cover c o l l e c t o r . T & = 10 °C, wind v e l o c i t y = 0, I I I I I I I 20. 30. 40. 50. 60. 70. 80. WATER SURFACE TEMPERATURE ( DEGREE C) Figure 18. Cover surface temperature as a function of water surface temperature for a three-cover c o l l e c t o r . T a = 10 0 C, wind v e l o c i t y = 0. 147 T = -20°C 2 0 - 30. 40. 50. 60. 70. 80. WATER SURFACE TEMPERATURE (DEGREE C) F i g u r e 1 9 , F r o n t a l h e a t l o s s c o e f f i c i e n t as a f u n c t i o n o f w a t e r s u r f a c e t e m p e r a t u r e . T = - 2 0 , 1 0 , 40 0 C , e = 0 . 9 5 , w i n d v e l o c i t y = 0 . a ^ 148 12 10. u o C N 8 . EH 53 W H U H pH PM §6-u cn cn o En rt! Hi < EH 53 T = -20 °C 10 °C 40 °C one-cover two-cover three-cover 1 1 30, 40. 50. 60. 70. WATER SURFACE TEMPERATURE ( 0C) Figure 20. Frontal heat loss c o e f f i c i e n t as a function of wa?er surface temperature. T a = -20, 10, 40 o C , £ p =0.95, wind v e l o c i t y = 5 m/s. 149 I I I I I I I 30. 40. 50. 60. 70. WATER SURFACE TEMPERATURE ( °C) Figure 21. Frontal heat loss c o e f f i c i e n t as a function of water surface temperature. T = -20, 10, 40 °C, cl e =0.95, wind v e l o c i t y = 10 m/s. P 150 l j 1 r r T = -20 u C I 1 1 1 I | 30. 40. 50. 60. 70. WATER SURFACE TEMPERATURE ( 0 C) Figure 22. Frontal heat loss c o e f f i c i e n t as a function of water surface temperature. T = -20, 10, 40 0 C, e =0.1, wind v e l o c i t y =0-151 3 0 . 4 0 . 5Q. 6 0 . 7 0 . WATER SURFACE TEMPERATURE ( ° C ) F i g u r e 2 3 . F r o n t a l h e a t l o s s c o e f f i c i e n t as a f u n c t i o f w a t e r s u r f a c e t e m p e r a t u r e . T = - 2 0 , 1 0 , 40° C, e _ = 0 . 1 , w i n d v e l o c i t y = 5 m / s . a 152 | I 1 I I I 30, 40. 50., 60, 70. WATER SURFACE TEMPERATURE ( Q C) Figure 24. Frontal heat loss c o e f f i c i e n t as a function of water surface temperature. T = -20, 10, 40 0 C, a en = 0.1, wind v e l o c i t y = 10 m/s. 153 q q ^cs rs q = 334 = 351 + 33 U = 9.6 W/m 0 C T . = 24.2 °C c l . v . ' . 1 4 7 ' . q g i . " . 4 1 : . v . : 1 9 6 T - s o ° c Wo q q cs ^rs 172 = 145 + 27 U = 4.4 W/m 2°C q r 2 = 1 1 5 ' qc2 = 5 7 .T o = 1 5 . 9 ° C c2 •T . = 39.7UC c l q r l = 6 1 ' q c l = 1 6 ' q e c = 9 5 / l * 9 T = 50. C ws q q cs rs 115 = 9 7 + 1 8 q r 3 = 7 8 ' q c 3 = 3 6 q r 2 = 7 9 ' q c 2 = 3 5 U, = 2.8 w/m 2°C c3 T q r l = 3 8 ' q c l = 1 0 ' q e c = 6 7 c2 "cl 13.9°C o 28.9 C 4 4. °C / / 7 7 » 7—7 / t /—r-Tws - 50. C Figure 25. Heat losses from a c o l l e c t o r plate predicted by the present f l a t - p l a t e model for a one, two, and three-cover system. T = 50.°C, T = 10.oc, c = 0.95, wind v e l o c i t y = WS cl 5 m/s, d = 2.54 cm., t i l t = 45 degrees. q q ^cs rs q = 278 = 254 + 24 q ,=19, q -,=47, q =212 ^ r l ^ c l ' ^ec U. 7.0 W/m °C 154 T =20.3 °C c l / * r w r ft •T = 50.°C ws q n q _ = 3.9 W/m2 9: lcs rs t q = 158 = 133 + 25 T _ = 15.4° C c2 q r 2=103, q c 2=54 T = 37.0 C q =9, q ,=21, q =128 C ^ r l ^ c l ' ^ec 1 ' ' ' 1 ' ' ' ' ' T = 50.°C ws U = 2.2 W/m 2°C 'cs J r s 94 = 51 +43 q r 3 = 6 4 ' * c 3 = 3 ° q r 2 = 6 3 ' q c 2 = 3 1 c 2 T _ = 19.°C c3 T n = 31.°C q n=5, q =77 ^ r l ^ec T . = 43.1°C c l / * i r / r t » / » t f / / 7 / / 'T - 50.°C ws Figure 26. Heat losses from a c o l l e c t o r plate predicted by the present f l a t - p l a t e model fcjr a one, two, and three-cover system. T = 50.°C, T = 10. C, e = 0.1, wind v e l o c i t y J ws a P = 5 m/s, d = 2.54 cm., t i l t = 45 degrees. 0.8 0.6 w M u H £ 0.4 w 0.2 0.0 1 1 F l a t - P l a t e mode l T u b e - a n d - p l a t e c o l l e c t o r 20. 40. 60. FLUID INLET TEMPERATURE (°C) 100. Figure 27. Collector e f f i c i e n c y as a function of 2 water i n l e t temperature for I 3 = 1000 w/m . F R = 0.9, F , = 1.0, wind v e l o c i t y = 5 m/s, T & = 10.° C e = 0.95, KL = 0., (TO )=0.92(one-cover), (Pxa ) = 0.84(two-cover), ( xa ) = 0.78(three-cover) 156 T T T Average c o r r e l a t i o n for Thomason 'Solaris' c o l l e c t o r Present f l a t - p l a t e model 0.8 0.01 0.02 0,03 0.04 "T> - T „ — i (m °C/YI) IB Figure 28. Water-Trickle c o l l e c t o r performance c o r r e l a t i o n . RF (reduction factor) = 0.8, T = 10,° C, 2 wind v e l o c i t y = 5 m/s, G = 0.01 kg/m s. Figure 29. South-facing corrugated plate with short-wave beam ra d i a t i o n incident at angle 6 . cn N covers Figure 30. Details of the corrugated cross-section. rectangular finned V-shaped corrugated plate f l a t plate one-cover rectangular finned V-shaped corrugated plate f l a t plate 0.8 0.7 s 0.6 0.5 one-cover two-cover three-cover JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 32. Monthly averaged ( T a ) for c o l l e c t o r s sloped at 50 degrees, l a t i t u d e = 49.2 degrees, KL = 0.0375. o rectangular finned V-shaped corrugated plate f l a t plate one-cover 0.8 0.7 0.6 0.5 two-cover JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Figure 33. Monthly averaged ( T o t ) for c o l l e c t o r s sloped at 70 degrees, l a t i t u d e = 49.2 degrees^ K L = 0.0375. 162 one-cover 1 ' HI 0. 15. 30. 45. 60. 75. 90. ANGLE OF INCIDENCE Figure 34. (Ta ) as a function of the angle of incidence for a rectangular finned c o l l e c t o r p l a t e . Slope = 70 degrees, la t i t u d e = 49.2 degrees, KL = 0.0375. F i g u r e 3 5 . Q u a l i t a t i v e s k e t c h o f t h e t e m p e r a t u r e p r o f i l e a l o n g t h e c o r r u g a t e d c r o s s - s e c t i o n . CO 4 0 0 , CN e s w £ 3 0 0 , CO cc o EH £ J 2 0 0 . a <c EH 53 PH 1 0 0 . 7 0 AVERAGE PLATE TEMPERATURE ( UC) Frontal heat loss rate as a function of average plate temperature Figure 3 6 . for a single-cover, corrugated-plate wate r - t r i c k l e c o l l e c t o r . T 1 0 UC -T 1 1 1 r corrugated-plate model I — — — f i a t - p l a t e model 400|-CN 20. 30. 40, 50. 60. 70, AVERAGE PLATE TEMPERATURE ( C) Figure 37. Frontal heat loss rate as a function of average plate temperature for a two-cover c o l l e c t o r . T =10 C, wind velocity=0., £ = £ = . 9 5 . a P ws 400 300. EH 02 § 200, EH < W g i o o . EM T 1 r corrugated-plate model •f l a t - p l a t e model evaporative loss 70. 30. 40. 50. AVERAGE PLATE TEMPERATURE ( Figure 38. Frontal heat loss rate as a function of average plate temperature for a three-cover c o l l e c t o r . T =10°C, wind=0., e = e =0 a ws 95 167 1 1 1 i r T = - 2 0 C a I 1 1 1 I t 3 0 . 4 0 , 5 0 , 6 0 , 7 0 . AVERAGE PLATE TEMPERATURE ( °C) F i g u r e 3 9 . F r o n t a l h e a t l o s s c o e f f i c i e n t as a f u n c t i o n o f a v e r a g e p l a t e t e m p e r a t u r e f o r a c o r r u g a t e d - p l a t e w a t e r -t r i c k l e c o l l e c t o r . e =e = 0 . 9 5 , w i n d v e l o c i t y = 0 . 168 30. 40. 50. 60. 70. AVERAGE PLATE TEMPERATURE ( UC) Figure 40. Frontal heat loss c o e f f i c i e n t as a function of average plate temperature for a corrugated-plate water-t r i c k l e c o l l e c t o r . e =e =0.95, wind v e l o c i t y ^ 5 m/s. p ws J ' 169 30. 40. 50. 60. 70. AVERAGE PLATE TEMPERATURE ( C) Figure 41. Frontal heat loss c o e f f i c i e n t as a function of average plate temperature for a corrugated-plate water-t r i c k l e c o l l e c t o r . £=e =0.95, wind velocity= 10 m/s. p ws 170 U w H u H fa 0.2 T T T average c o r r e l a t i o n for Thomasor 'Solaris' c o l l e c t o r f l a t - p l a t e model corrugated-plate model 1 0.01 0.02 "T , - T f a 0.03 (m2 °c/W) 0.04 Figure 42. Water-trickle c o l l e c t o r performance c o r r e l -ation. = average water temperature, RF (reduction factor) = 0.8, T 2 a G = 0.01 kg/m s. = 10 °C, w i nd v e l o c i t y = 5 m/s, I I I Beard model (mass d i f f u s i o n ) heat-mass transfer analogy T =30°C, wind=0., T =10UC P a P I 1 1 I I | | I 1. 2. 3. 4. 5. 6. 7. GAP SPACING FROM ABSORBER PLATE TO COVER SURFACE (cm) Figure 43. Evaporative heat loss rate as a function of the gap spacing from the absorber plate to i t s adjacent cover surface. I I r heat-mass transfer analogy — — Re arc: model (mass diffusion) T =50UC, T =10 C, wind = 0. 3. 4. 5. 6. 7. GAP SPACING FROM ABSORBER PLATE TO COVER (cm) Figure 44. Evaporative heat loss rate as a function of the gap spacing from the absorber plate to the adjacent cover surface. Figure 45- Net radiation balance for a dry corrugated surface. F i g u r e 45 . S k e t c h o f a n N - c o v e r s y s t e m . F i g u r e 47. N e t r a d i a t i o n b a l a n c e f o r a w e t t e d c o r r u g a t e d s u r f a c e . 

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