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Stitch weld effect on solar collector efficiency factor Lo, Andy Ka-Ming 1985

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STITCH WELD EFFECT ON SOLAR COLLECTOR EFFICIENCY FACTOR by Andy K. LO B . S c , Queen's U n i v e r s i t y a t K i n g s t o n , 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d UNIVERSITY OF BRITISH COLUMBIA September, 1985 © Andy K. LO, 1985 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British 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 or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of M.E.CMr\M\CP(L tzKJ6jfkJEER.tkgj The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date QcfoUr 9 , W{ DE-6(3/81) ABSTRACT The thermal e f f e c t s of s t i t c h welding the coolant c o n d u i t s of a water-cooled f l a t p l a t e s o l a r c o l l e c t o r to i t s absorber p l a t e have been s t u d i e d . A p h y s i c a l model of the heat t r a n s f e r process from the p l a t e to the f l u i d flowing i n s i d e the tube has been presented. The heat t r a n s f e r c o e f f i c i e n t based on the d i f f e r e n c e between bond temperature and f l u i d bulk mean temperature i s an important f a c t o r i n determining the 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'. The upper and lower l i m i t s of the a c t u a l value of F' have been p r e d i c t e d by c o n s i d e r i n g two extreme boundary c o n d i t i o n s to which the f l u i d i s s u b j e c t e d . For a t h i c k and conductive tube w a l l , F' does not depend on spot s i z e and spot spacing, and tends to an upper l i m i t of 0.883. For a t h i n and non-conductive tube w a l l , the boundary c o n d i t i o n comprises of a s e r i e s of step changes i n both the a x i a l and c i r c u m f e r e n t i a l d i r e c t i o n s of the heat f l u x . In t h i s case, the heat t r a n s f e r c o e f f i c i e n t and hence F' approach t h e i r lower l i m i t s which are determined by the welding spot c o n f i g u r a t i o n . I t was a l s o found that F' i n c r e a s e s with the f o l l o w i n g parameters: the spot angle; the percentage of tube l e n g t h being welded; and the number of spots among which the welding i s being d i s t r i b u t e d . Furthermore, the temperature d i s t r i b u t i o n i n s i d e the f l u i d has a l s o been computed f o r t h i s case. Table of Contents ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v i LIST OF SYMBOLS x ACKNOWLEDGEMENTS x i i i 1 . INTRODUCTION 1 1 .1 General 1 1.2 O b j e c t i v e of the Present Work 4 1.3 L i t e r a t u r e Review 6 1.3.1 The Entry Length Problem • 7 1.3.2 Graetz Problem 8 1.3.3 Problems with C i r c u m f e r e n t i a l V a r i a t i o n ...10 1.3.4 Conjugated Problem 12 2 . MATHEMATICAL MODEL 1 6 2.1 Governing Equations ..16 2.2 I d e a l i z a t i o n s 17 2.3 S i m p l i f i e d Equations 19 2.4 Thermal Boundary C o n d i t i o n s 20 3. ANALYTICAL SOLUTIONS 24 3.1 Overview of A n a l y t i c a l S o l u t i o n s .... 24 3.2 S e p a r a t i o n of V a r i a b l e s Method 25 3.3 S o l u t i o n f o r "Th i n " Tube using S u p e r p o s i t i o n ....28 3.4 G r a p h i c a l I l l u s t r a t i o n of A n a l y t i c a l R e s u l t s ....30 4. NUMERICAL SOLUTIONS 34 4.1 Overview of Numerical Methods 34 4.2 F i n i t e D i f f e r e n c e Formulation 35 4.3 C a l c u l a t i o n of N u s s e l t Numbers 37 4.4 P r e l i m i n a r y R e s u l t s of Numerical Procedure 38 5. APPLICATION TO COLLECTOR EFFICIENCY FACTOR 42 5.1 Heat T r a n s f e r C o e f f i c i e n t based on Bond Temperature 43 5.2 Behaviour of Nu(x) f o r a C o n t i n u o u s l y Welded Tube 45 5.3 Behaviour of Nu(x) of Spot Welded Tube 46 5.4 E f f i c i e n c y F a c t o r f o r v a r i o u s Spot C o n f i g u r a t i o n s 48 6. DISCUSSION AND CONCLUSIONS 50 6.1 D i s c u s s i o n of A n a l y t i c a l and Numerical Methods ..50 6.2 Recommendations 53 6.3 C o n c l u s i o n s 55 REFERENCES 56 APPENDIX A 93 APPENDIX B .95 APPENDIX C 97 LIST OF TABLES Page Table 2.1. Thermal boundary c o n d i t i o n s f o r developed and d e v e l o p i n g flows through s i n g l y connected d u c t s . 65 v LIST OF FIGURES Page FIG. 1.1. Cross s e c t i o n of a b a s i c f l a t p l a t e s o l a r c o l l e c t o r . 58 FIG. 1.2. Energy flow i n an o p e r a t i n g water-cooled s o l a r c o l l e c t o r . 59 FIG. 1.3. Cross s e c t i o n of t y p i c a l p l a t e and tube arrangement. 60 FIG. 1.4. Common ways of p l a t e - t u b e bonding. 61,62 FIG. 1.5. Hydrodynamic and thermal entry l e n g t h s . 63 FIG. 1.6. P h y s i c a l s i t u a t i o n f o r the Graetz problem. 64 FIG. 1.7. L i n e a r v e l o c i t y p r o f i l e assumed i n Leveque method. 64 FIG. 2.1. Two extreme cases of thermal boundary c o n d i t i o n s : ( i ) A t h i c k and conductive w a l l . ( i i ) A t h i n and non-conductive w a l l . 66 FIG. 2.2. P e r i p h e r a l d i s t r i b u t i o n of w a l l heat f l u x . 67 FIG. 2.3. A x i a l d i s t r i b u t i o n of w a l l heat f l u x . 67 FIG. 3.1. Continuously welded tube with spot angle of 45°. 68 FIG. 3.2. Spot-welded tube with 2 spots occupying 60% of i t s l e n g t h . 68 FIG. 3.3 Dimensionless w a l l temperature of a tube welded c o n t i n u o u s l y with spot angle 45° versus dimensionless a x i a l d i s t a n c e 1000x = l000X/a-Pe. 69 FIG. 3.4. L o c a l N u s s e l t no. Nu and p e r i p h e r a l average N u s s e l t no. Nu p of c o n t i n u o u s l y welded tube versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l000X/a»Pe. 70 FIG. 3.5. Dimensionless w a l l temperature versus dimensionless a x i a l d i s t a n c e 1000X = l000X/a°Pe of a spot-welded tube with 2 spots occupying 60% of i t s l e n g t h , spot angle 45°. 71 FIG. 3 . 6 ( i ) . P e r i p h e r a l average N u s s e l t number Nun versus d i m e n s i o n l e s s a x i a l d i s t a n c e lOOux = l000X/a«Pe f o r a spot-welded tube with a v i s i n g l e spot occupying 60% of i t s l e n g t h , spot angle 45°. 72 FIG. 3 . 6 ( i i ) . P e r i p h e r a l average N u s s e l t no. Nu p versus dimensionless a x i a l d i s t a n c e lOOOx = l000X/a-Pe f o r a spot-welded tube with 4 spots occupying 60% of i t s l e n g t h , spot angle 45°. ' 73 FIG. 4.1. G r i d d i v i s i o n of tube volume. 74 FIG. 4.2. Nodes appearing i n f i n i t e d i f f e r e n c e energy equation. 75 FIG. 4.3. Dimensionless temperature t of a u n i f o r m l y heated tube versus dimensionless a x i a l d i s t a n c e lOOOx = l000X/a-Pe at v a r i o u s r a d i a l d i s t a n c e s : f i n i t e d i f f e r e n c e r e s u l t s . 76 FIG. 4.4. L o c a l Nusselt no. Nu of a u n i f o r m l y heated tube versus dimensionless a x i a l d i s t a n c e 1000X = l000X/a-Pe : f i n i t e d i f f e r e n c e r e s u l t s . 77 FIG. 4.5. Angular p o s i t i o n s represented by v a r i o u s curves i n F i g u r e s 4.6 - 4.9. 78 FIG. 4.6. Dimensionless w a l l temperature versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l000X/a«Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a c o n t i n u o u s l y welded tube with spot angle 45°. 79 FIG. 4.7. Dimensionless w a l l temperature versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l000X/a«Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a spot-welded tube with 2 spots occupying 60% of i t s l e n g t h , spot angle 45°. 80 FIG. 4.8. Dimensionless temperature i n s i d e the tube at r a d i a l c o - o r d i n a t e r=4/5 versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x = 1000X/a«Pe f o r a spot-welded tube with 2 spots occupying 60% of i t s l e n g t h , spot angle 45°. 81 FIG. 4.9. Dimensionless temperature i n s i d e the tube at r a d i a l c o - o r d i n a t e r=1/5 versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l000X/a«Pe, f o r a spot-welded tube with 2 s p o t s occupying 60% of i t s l e n g t h , spot angle 45°. 82 v i i FIG. 4 . 1 0 ( D . P e r i p h e r a l average N u s s e l t no. Nu p versus d i m e n s i o n l e s s a x i a l d i s t a n c e lOODx = l 0 0 0 X / a « P e : f i n i t e d i f f e r e n c e r e s u l t s f o r a spot-welded tube with a s i n g l e spot occupying 6 0 % of i t s l e n g t h , spot angle 45°. 83 FIG. 4 . l 0 ( i i ) . P e r i p h e r a l average N u s s e l t No. Nup versus d i m e n s i o n l e s s a x i a l d i s t a n c e lOODx = l 0 0 0 X / a « P e : f i n i t e d i f f e r e n c e r e s u l t s f o r a spot-welded tube with 4 spots occupying 60% of i t s l e n g t h , spot angle 45°. 84 FIG. 5.1. Bond temperature T ] 3 ( X ) approximated by temperature i n s i d e tube T ( a , 0 , X ) . 85 FIG. 5.2. N u s s e l t no. based on bond temperature Nu^ versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l 0 0 0 X / a - P e f o r a c o n t i n u o u s l y welded tube of v a r i o u s h a l f - s p o t angles <t>0 86 FIG. 5.3. N u s s e l t no. based on bond temperature Nuj-, and mean N u s s e l t no. based on bond temperature Nu^m versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l 0 0 0 X / a « P e f o r a spot-welded tube with 2 spots occupying 60% of i t s l e n g t h , spot angle 36°. 87 FIG. 5.4. Mean Nu s s e l t no. based on bond temperature N ubm versus dimensionless a x i a l d i s t a n c e 1000X = l 0 0 0 X / a - P e f o r a spot-welded tube with 8 spots occupying 60% of i t s l e n g t h , spot angle 36°. 88 FIG. 5.5. Mean Nusselt no. based on bond temperature N ubm over a dimensionless tube l e n g t h of .088 as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a spot-welded tube with spot angle of 36°. 89 FIG. 5.6. E f f i c i e n c y F a c t o r F' as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a c o l l e c t o r with d i s t a n c e W between i t s spot-welded tubes of ,15m and spot angle of 36°. 90 FIG. 6.1. E f f i c i e n c y F a c t o r F' as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a c o l l e c t o r w i t h d i s t a n c e W between i t s spot-welded tubes of ,1m and spot angle of 36°. 91 v i i i FIG. 6.2. E f f i c i e n c y Factor. F' as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a c o l l e c t o r with d i s t a n c e W between i t s spot-welded tubes of ,2m and spot angle of 36°. 92 ix LIST OF SYMBOLS A c r o s s s e c t i o n a l area, [m 2] a tube inner r a d i u s , [m] B w a l l conductance parameter ( = k w7/ka ) b bond width, [m] bond conductance, [W/m °C] C p s p e c i f i c heat c a p a c i t y of f l u i d , [ J / k g ° C ] D tube inner diameter (=2a), [m] F f i n e f f i c i e n c y parameter F' 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 g" v e c t o r i a l body f o r c e s , [N] h c o n v e c t i v e heat t r a n s f e r c o e f f i c i e n t , [W/m 2°C] I i n t e r n a l energy per u n i t mass, [ J / k g ] k thermal c o n d u c t i v i t y of f l u i d , [W/m°C] k g thermal c o n d u c t i v i t y of f i n m a t e r i a l , [W/m°C] k w thermal c o n d u c t i v i t y of tube w a l l , [W/m°C] L tube l e n g t h , [m] m / ( U L / k s 5 ) , a f a c t o r appearing i n the f i n e f f i c i e n c y F. m f l u i d mass flow r a t e , [kg/s] M maximum r a d i a l node no. r M maximum a x i a l node no. M, maximum c i r c u m f e r e n t i a l node no. <t> N t o t a l no. of welding spots along a tube N r a d i a l node no. x N x a x i a l node no. N, c i r c u m f e r e n t i a l node no. <t> Nu l o c a l N u s s e l t no. Nu^ N u s s e l t no. based on bond temperature N ubm mean N u s s e l t no. based on bond temperature NUp p e r i p h e r a l average N u s s e l t no. Nu mean N u s s e l t no. based on Nu m p p p r e s s u r e , [N/m2] Pe P e c l e t No. (=Re.Pr) q w a l l heat f l u x i n t o f l u i d , [W/m2] ^ q p e r i p h e r a l average w a l l heat f l u x , [W/m2] q u u s e f u l energy gain per u n i t l e n g t h per tube i n the flow d i r e c t i o n , [W/m] Q s o l i n c i d e n t s o l a r heat f l u x , [W/m2] R r a d i a l c o - o r d i n a t e , [m] r di m e n s i o n l e s s r a d i a l c o - o r d i n a t e (= R/a) R w w a l l r e s i s t a n c e c o e f f i c i e n t (= k7/k wD) Re Reynolds no. S s t r e s s tensor T temperature, [°C] T% n f l u i d i n l e t temperature, [°C] T m f l u i d bulk mean temperature, [°C] T w inner w a l l temperature of tube, [°C] Twm p e r i p h e r a l average w a l l temperature, [°C] x i d i m e n s i o n l e s s temperature (= k(T-T\ n ) / q a ) w' wm d i m e n s i o n l e s s c o u n t e r p a r t s of T. , T , T , T * i n ' m' w' wm peak flow speed (=2u f o r p a r a b o l i c p r o f i l e ) a x i a l component of flow v e l o c i t y , [m/s] mean flow speed, [m/s] c o l l e c t o r o v e r a l l heat t r a n s f e r c o e f f i c i e n t , [W/m 2 0C] f l u i d v e l o c i t y [m/s] d i s t a n c e between tubes, [m] welded percentage of tube l e n g t h a x i a l c o - o r d i n a t e , [m] d i m e n s i o n l e s s a x i a l c o - o r d i n a t e (= X/a.Re.Pr) spot l e n g t h and spot s p a c i n g , [m] X,/a.Re.Pr and X /a.Re.Pr 1 s thermal d i f f u s i v i t y (= k/pc ),[m 2/s] tube w a l l t h i c k n e s s , [m] f i n t h i c k n e s s , [m] p e r i p h e r a l c o - o r d i n a t e , [ r a d i a n s ] h a l f - s p o t angle [ r a d i a n s ] dynamic v i s c o s i t y , [kg/m.s] kinematic v i s c o s i t y , [m 2/s] time, [s] d i m e n s i o n l e s s temp. ( = (T - T w ) / ( T i n ~ T w ' ' x i i ACKNOWLEDGEMENTS The author c o n s i d e r s h i m s e l f f o r t u n a t e to have had Dr. E.G. Hauptmann and Dr. M. Iqb a l as h i s s u p e r v i s o r s throughout t h i s study. T h e i r guidance and a s s i s t a n c e have been c r u c i a l towards the completion of t h i s work. The U n i v e r s i t y of B r i t i s h Columbia provided t h i s wonderful s i t e of stu d y i n g and the computing f a c i l i t i e s . F i n a l l y , the author wishes to thank h i s famil y members, some from half-way a c r o s s the world, who have given s p i r i t u a l support and encouragement throughout t h i s work. x i i i 1. INTRODUCTION 1.1 GENERAL Heat t r a n s f e r phenomena i n a water-copied s o l a r c o l l e c t o r i n c l u d e s o l a r r a d i a t i o n , c o n d u c t i v e and c o n v e c t i v e c o o l i n g to the ambient, and i n t e r n a l flow f o r c e d c o n v e c t i v e heat t r a n s f e r . The most common type of s o l a r c o l l e c t o r used in b u i l d i n g h e a t i n g and water h e a t i n g i s the f l a t p l a t e water-cooled type as shown i n F i g u r e 1.1. F i g u r e 1.2 shows the manner in which the v a r i o u s modes of heat t r a n s f e r are i n v o l v e d . The b l a c k , s o l a r energy-absorbing p l a t e has means f o r t r a n s f e r r i n g the absorbed energy to a f l u i d , u s u a l l y through welded tubes i n which f l o w i n g water c a r r i e s away the heat being absorbed. C a l c u l a t i n g the amount of heat being c a r r i e d away i s the aim of f o r c e d c o n v e c t i v e heat t r a n s f e r a n a l y s i s in i n t e r n a l flows. One or two envelopes cover the s o l a r absorber s u r f a c e . The covers are t r a n s p a r e n t to incoming s o l a r r a d i a t i o n but opaque to the thermal r a d i a t i o n from the absorber p l a t e , thus reducing c o n v e c t i v e and r a d i a t i v e l o s s e s to the atmosphere. A back i n s u l a t i o n i s i n c l u d e d to reduce co n d u c t i v e l o s s e s [ 1 ] , F l a t p l a t e c o l l e c t o r s are almost always s t a t i o n a r y and p o s i t i o n e d with an o r i e n t a t i o n o p t i m i z e d f o r the p a r t i c u l a r l o c a t i o n i n q u e s t i o n , and f o r the time of year i n which the s o l a r d e v i c e i s intended to operate. Without o p t i c a l c o n c e n t r a t i o n , the f l u x of i n c i d e n t r a d i a t i o n i s , at best, 1 2 1100 W/m2. Both beam and d i f f u s e r a d i a t i o n are being absorbed by the c o l l e c t o r , which can be designed f o r a p p l i c a t i o n s r e q u i r i n g energy d e l i v e r y at moderate temperatures. The flow of c o o l a n t i n the welded tube i s commonly i n the laminar regime. Laminar flow heat t r a n s f e r i s of great t e c h n i c a l importance s i n c e i t occurs i n many he a t i n g and c o o l i n g d e v i c e s . The heat t r a n s f e r c o e f f i c i e n t between the absorber p l a t e and the c o o l i n g water i s an important f a c t o r i n s o l a r c o l l e c t o r design s i n c e i t determines the c o l l e c t o r e f f i c i e n c y , and hence the economic v a l u e of the i n s t a l l a t i o n . The performance of a s o l a r water heater i s d i r e c t l y p r o p o r t i o n a l to i t s e f f i c i e n c y f a c t o r F', which r e p r e s e n t s the r a t i o of the a c t u a l u s e f u l energy gain to the u s e f u l energy gain that would r e s u l t i f the c o l l e c t o r absorbing s u r f a c e had been at the l o c a l f l u i d temperature. Another i n t e r p r e t a t i o n of F' i s that i t i s a measure of the e f f i c i e n c y of the design as a heat exchanger [ 2 ] . A c r o s s - s e c t i o n of the p l a t e and tube arrangement i s shown i n F i g u r e 1.3. The 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 i s W{ 1/[U T (b+(W-b)F) ] + 1/C, + 7/UDk ) + 1/UDh)} , L b w (1.1) where the dimensions are as i n d i c a t e d i n F i g u r e 1.3, and the other terms, which appeared i n the "LIST OF SYMBOLS", are repeated here : 3 U r = o v e r a l l c o e f f i c i e n t of heat t r a n s f e r from the f l a t Li p l a t e to the o u t s i d e a i r , i n c l u d i n g allowance f o r rear l o s s e s , h = c o e f f i c i e n t of heat t r a n s f e r from the tube w a l l to the water i n the tube, k ,k = thermal c o n d u c t i v i t i e s of the p l a t e and tube m a t e r i a l s r e s p e c t i v e l y , = thermal conductance of the t u b e - p l a t e bond, tanh[m(W-b)/2] F = = f i n e f f i c i e n c y ; m(W-b)/2 m = v/(U L/k s6) . In equation ( 1 . 1 ) the four terms i n the r i g h t hand denominator can be thought of as the r e l a t i v e r e s i s t a n c e t o the passage of heat from the p l a t e i n t o the water due to : ( 1 ) conduction of heat a l o n g the f l a t p l a t e towards the tubes, ( 2 ) conduction of heat from the p l a t e to the tube through the t u b e - p l a t e bond, ( 3 ) conduction throughout the tube w a l l to the inner s u r f a c e of the tube, (4) t r a n s f e r of heat from the tube inner s u r f a c e i n t o the water. The governing equations of f l u i d flow i n s i d e a c i r c u l a r tube, namely, 1) the c o n t i n u i t y e q u a t i o n , 2 ) the momentum e q u a t i o n , and 3 ) the energy equation, have been w e l l formulated and e x t e n s i v e l y s t u d i e d . Although an energy 4 equation can be w r i t t e n to d e s c r i b e the temperature of the water fl o w i n g i n s i d e the c o l l e c t o r tube, the boundary c o n d i t i o n s on the water depend on the c o n f i g u r a t i o n of the c o l l e c t o r . An important aspect i s the f a s h i o n i n which the tube i s welded onto the f i n (the absorber p l a t e ) . A number of common ways of bonding are shown and b r i e f l y d e s c r i b e d i n F i g u r e s 1.4(i) - ( i v ) . An u l t r a s o n i c spot welding process i s o f t e n used f o r economic reasons. In t h i s case the boundary c o n d i t i o n i n the a x i a l d i r e c t i o n can be c o n s i d e r e d as a s e r i e s of step changes (Figure 1 . 4 ( i v ) ) . In c o n v e n t i o n a l s t u d i e s of c o n v e c t i v e heat t r a n s f e r , the mathematical model of the problem i n c l u d e s w e l l d e f i n e d boundary c o n d i t i o n s concerning the temperature or the heat f l u x . The temperature or the heat f l u x i s c l e a r l y s p e c i f i e d mathematically at the boundary of the re g i o n of i n t e r e s t , which i s the i n s i d e tube w a l l i n the present case. In r e a l i t y , because of a x i a l and p e r i p h e r a l heat conduction i n the tube m a t e r i a l , the f l o w i n g water i s s u b j e c t to a boundary c o n d i t i o n at the w a l l which can n e i t h e r be d e s c r i b e d i n terms of w a l l temperature nor w a l l heat f l u x . The s o - c a l l e d "conjugated problem" d e a l s with t h i s s i t u a t i o n by t a k i n g i n t o account the conduction i n the s o l i d m a t e r i a l . 1.2 OBJECTIVE OF THE PRESENT WORK Although laminar c o n v e c t i v e heat t r a n s f e r has been e x t e n s i v e l y s t u d i e d f o r v a r i o u s kinds of boundary c o n d i t i o n s , no p r a c t i c a l s o l u t i o n i s a v a i l a b l e f o r flows i n 5 the spot-welded tube as shown i n F i g u r e 1 . 4 ( i v ) . The o b j e c t i v e of t h i s work i s to i n v e s t i g a t e the e f f e c t s of the spot and spacing dimensions on the heat t r a n s f e r c o e f f i c i e n t between the tube w a l l and the f l u i d . Knowing t h i s c o e f f i c i e n t , the c o n s e q u e n t i a l e f f e c t on c o l l e c t o r performance, and i n p a r t i c u l a r the 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 can be determined. The q u e s t i o n of spot spacing i s of . great economic importance s i n c e the welding process i s slow and consumes energy. By knowing how f a r apart the spots can be l o c a t e d b efore the c o l l e c t o r e f f i c i e n c y i s s i g n i f i c a n t l y reduced, an optimum set of spot l e n g t h and spot spacing can be determined ( t a k i n g welding c o s t s i n t o a c c o u n t ) . As a s t a r t , however, the heat t r a n s f e r p rocess from the spot-welded tube to the f l u i d f l o wing i n s i d e must be understood. T h i s understanding should be i n terms of the d i m e n s i o n l e s s temperature d i s t r i b u t i o n i n the f l u i d , and i n terms of the mean Nusselt number, which i s the d i m e n s i o n l e s s form of the mean heat t r a n s f e r c o e f f i c i e n t from the c i r c u l a r tube to the f l u i d . O b v i o u s l y , the heat f l u x i n t o the water i s higher through the welding spots than through the r e s t of the tube. An extreme case e x i s t s when the tube w a l l i s v a n i s h i n g l y t h i n so that p e r i p h e r a l and a x i a l c onduction w i t h i n the w a l l m a t e r i a l i s n e g l i g i b l e , and the heat f l u x from the absorber p l a t e e n t e r s the water only through the welding s p o t s . At the other extreme, i f the tube m a t e r i a l i s h i g h l y conductive 6 and the w a l l i s t h i c k , conduction w i t h i n the w a l l m a t e r i a l occurs i n a l l d i r e c t i o n s and temperature v a r i a t i o n i n the w a l l w i l l be reduced. In t h i s case, the water i s s u b j e c t to a boundary of uniform temperature. In r e a l i t y , however, n e i t h e r of these c o n d i t i o n s p r e v a i l . A n a l y t i c a l s o l u t i o n s to the c o n v e c t i v e heat t r a n s f e r process i n v o l v e d i n the above extreme cases were sought f o r v a r i o u s c o n f i g u r a t i o n s of spots and spacings and checked a g a i n s t numerical s o l u t i o n s . The r e s u l t s of the 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 of these extreme models can be used as l i m i t s of the r e a l i s t i c s i t u a t i o n . 1.3 LITERATURE REVIEW I n t e r n a l flow c o n v e c t i v e heat t r a n s f e r i n v a r i o u s geometries under v a r i o u s boundary c o n d i t i o n s has been s t u d i e d i n great d e t a i l i n the l i t e r a t u r e . A major review of work i n laminar flow f o r c e d c o n v e c t i o n has been done by Shah and London [ 3 ] , i n c l u d i n g v a r i o u s duct geometries and boundary c o n d i t i o n s . The scope of the present review i s r e s t r i c t e d to f o r c e d c o n v e c t i v e laminar flow f o r a Newtonian f l u i d with constant p r o p e r t i e s , passing through s t a t i o n a r y , s t r a i g h t , non-porous ducts of constant c i r c u l a r c r o s s s e c t i o n . The l i t e r a t u r e was reviewed under four problem c a t e g o r i e s : 1. E n t r y l e n g t h problems in duct flows; 2. The Graetz problem and i t s two c l a s s i c a l methods of s o l u t i o n ; 7 3. Problems with c i r c u m f e r e n t i a l v a r i a t i o n , with an a x i a l l y developed or d e v e l o p i n g thermal p r o f i l e ; and 4. Conjugated problems, where heat conduction i n the tube m a t e r i a l i s taken i n t o c o n s i d e r a t i o n . 1.3.1 THE ENTRY LENGTH PROBLEM In f o r c e d c o n v e c t i v e heat t r a n s f e r i n ducts, f l u i d flow i s o f t e n c a t e g o r i z e d a c c o r d i n g to i t s v e l o c i t y and temperature p r o f i l e . The temperature and v e l o c i t y p r o f i l e s of the f l u i d begin to develop at the entrance to a tube. The flow i s termed h y d r o d y n a m i c a l l y d e v e l o p e d where the v e l o c i t y p r o f i l e i s a l r e a d y f u l l y e s t a b l i s h e d and does not change as the f l u i d t r a v e l s downstream. S i m i l a r l y , where the heat t r a n s f e r c o e f f i c i e n t i s a x i a l l y i n v a r i a n t the flow i s c o n s i d e r e d to be t h e r m a l l y d e v e l o p e d . The downstream d i s t a n c e s from the entrance r e q u i r e d before the flow becomes hydrodynamically and t h e r m a l l y developed are c a l l e d the hydrodynamic and thermal e n t r y l e n g t h s , r e s p e c t i v e l y , as i l l u s t r a t e d i n F i g u r e 1.5. A t h e r m a l l y and hydrodynamically d e v e l o p i n g flow i s more co m p l i c a t e d than a t h e r m a l l y developing but hydrodynamically developed flow. I t has been shown [ 4 ] , however, t h a t i f the P r a n d t l number Pr of the f l u i d i s g r e a t e r than about 5, the v e l o c i t y p r o f i l e development lea d s the temperature p r o f i l e development and i t i s s u f f i c i e n t l y a c c u r a t e to c o n s i d e r the flow at the entrance as hydrodynamically developed, even though there i s no 8 hydrodynamic s t a r t i n g l e n g t h . 1.3.2 GRAETZ PROBLEM Graetz [5] i n 1883 c o n s i d e r e d an i n c o m p r e s s i b l e f l u i d with constant p h y s i c a l p r o p e r t i e s flowing through a c i r c u l a r tube, hydrodynamically f u l l y developed and with a developing thermal p r o f i l e . The tube i s maintained at a constant and uniform temperature. The energy equation i s 9T 3 2T 1 9T u = a( + ) , 9X 9R2 R 9R (1.2) The boundary c o n d i t i o n s are : For X < 0, T = T i n = constant, f o r X > 0 a t R = a , T = T = c o n s t a n t , w at R = 0, 9T/9R = 0. The p h y s i c a l s i t u a t i o n i s as shown i n F i g u r e 1.6, and i s now known as the Graetz problem. A review of e a r l i e r work on the Graetz problem has been done by Drew [ 6 ] . Brown [7] a l s o p r o v i d e d a comprehensive l i t e r a t u r e survey f o r the Graetz problem. The c l o s e d form s o l u t i o n to t h i s problem has been obtained p r i m a r i l y by two methods : the Graetz method and the Leveque method. The Graetz method uses the s e p a r a t i o n of v a r i a b l e s technique and as a r e s u l t the d i f f e r e n t i a l equation (1.2) i s reduced t o the S t u r m - L i o u v i l l e type. The s o l u t i o n i s then obtained i n the form of an i n f i n i t e s e r i e s expansion of eigenvalues and e i g e n f u n c t i o n s . The number of terms r e q u i r e d 9 f o r a d e s i r e d accuracy i n c r e a s e s s h a r p l y as x = X/(a.Re.Pr) approaches z e r o . The Graetz s o l u t i o n to the problem, i n terms of the temperature d i s t r i b u t i o n , i s presented i n an i n f i n i t e s e r i e s 9 = (T - T w ) / ( T i n - T w ) =l} c n R n e x p ( - X 2 x ) , (1.3) where R n are the e i g e n f u n c t i o n s i n r, X n are the e i g e n v a l u e s , and c n are c o n s t a n t s . Graetz and Nusselt obtained only the f i r s t two and three terms, r e s p e c t i v e l y , of the s e r i e s [ 3 ] . S e l l a r s et a l . [8] independently determined the f i r s t ten e i g e n v a l u e s and c o n s t a n t s , and presented asymptotic formulas f o r the higher ones. The Leveque [9] method employs the s i m i l a r i t y t r a n s f o r m a t i o n technique and i t s s o l u t i o n to the r e s u l t i n g equation i s v a l i d only near x = 0. I t employs the " f l a t p l a t e " s o l u t i o n as an asymptotic approximation near the p o i n t where the step change in temperature o c c u r s . . The v e l o c i t y d i s t r i b u t i o n i n the thermal boundary l a y e r was assumed l i n e a r and having the same slope as that at the w a l l with u = 0 at w a l l . The s i t u a t i o n i s i l l u s t r a t e d i n F i g u r e 1.7. Leveque obtained the f o l l o w i n g s o l u t i o n f o r the c i r c u l a r tube : 8 = (T - T w ) / ( T . n - T w ) = [ l / r ( 4 / 3 ) ] SQ e x p ( - z 3 ) d z , (1.4) 1 " R/a where TJ = , ( 9 x / 2 ) 1 ^ 10 and T i s the gamma f u n c t i o n : r(n) = fg(e Z z n " " 1 ) d z . Mercer [10], Worsoe-Schmidt [11] and Newman [12] have extended the Leveque s o l u t i o n by a p e r t u r b a t i o n method s o l v i n g the energy equation (equ 1.2) d i r e c t l y . The corresponding s o l u t i o n i s where £ = ( 9 x / 2 ) ^ f o r the c i r c l a r tube. The f i r s t term i n t h i s s e r i e s corresponds to Leveque's s o l u t i o n . T h i s s o l u t i o n i s v a l i d f o r i n t e r m e d i a t e v a l u e s of x where both the Graetz and Leveque s o l u t i o n s are not a c c u r a t e . G r i g u l l and T r a t z [13] s o l v e d equation (1.2) using the f i n i t e d i f f e r e n c e method with two d i f f e r e n t boundary c o n d i t i o n s : ( i ) a w a l l with uniform temperature, i . e . T w= constant; and ( i i ) a u n i f o r m l y heated w a l l , i . e . q=constant. R e s u l t s are presented on graphs which show s p a t i a l d i s t r i b u t i o n s of d i m e n s i o n l e s s temperature 0(R,X), l o c a l N u s s e l t number Nu(X), and mean N u s s e l t number Nu, (X), ' m 1.3.3 PROBLEMS WITH CIRCUMFERENTIAL VARIATION A l l the above mentioned s t u d i e s r e l a t e d t o Graetz and Leveque methods are based on the simple boundary c o n d i t i o n s where th e r e i s no c i r c u m f e r e n t i a l v a r i a t i o n at the w a l l - f l u i d i n t e r f a c e . Moreover, the f l u i d temperature at the i n l e t (the step change i n w a l l c o n d i t i o n ) i s assumed to be uniform. For example, water at uniform temperature flows i n t o a s t e p change in w a l l temperature or w a l l heat f l u x . (1.5) 11 There i s a x i a l symmetry in the tube because the boundary c o n d i t i o n s do not vary i n the c i r c u m f e r e n t i a l d i r e c t i o n . Reynolds [ 1 4 ] obtained the s o l u t i o n s (temperature d i s t r i b u t i o n s ) f o r any p r e s c r i b e d heat f l u x q{<p) around the circumference of a c i r c u l a r tube, without a x i a l v a r i a t i o n . He f i r s t o b t a ined the s o l u t i o n c orresponding to heat t r a n s f e r a c r o s s a small p o r t i o n of the circumference as a F o u r i e r s e r i e s , and then o b t a i n e d the s o l u t i o n s f o r a r b i t r a r y p e r i p h e r a l heat f l u x u s i n g s u p e r p o s i t i o n . T h i s s o l u t i o n , however, does not d e s c r i b e the temperature p r o f i l e development near the tube entrance, but only the p e r i p h e r a l d i s t r i b u t i o n of the t h e r m a l ly developed flow f u r t h e r down the tube. Bhattacharyya and Roy [ 1 5 ] took one f u r t h e r step i n the a r b i t r a r y w a l l heat f l u x problem. F i r s t , they expressed the v a r i a b l e c i r c u m f e r e n t i a l heat f l u x as a F o u r i e r s e r i e s i n <f>, the p e r i p h e r a l angle, and o b t a i n e d the temperature d i s t r i b u t i o n i n the thermal entrance r e g i o n . The s o l u t i o n i s of the form t(R,0,X) = k(T - T i n ) / ( q a ) = t,(X) + t 2 ( R ) + t 3 ( R , 0 ) + t,(R,*,X) , ( 1 . 6 ) where q i s the p e r i p h e r a l average w a l l heat f l u x , and q [ 1 + | ( a m c o s m0 + b ^ s i n m<j>)] = k 9T/9R = q (1.7) r e p r e s e n t s the w a l l heat f l u x which i s a f u n c t i o n of the angle o n l y . A p p l y i n g Duhamel's s u p e r p o s i t i o n theorem, the 12 s o l u t i o n f o r an a r b i t r a r y w a l l heat f l u x d i s t r i b u t i o n q U,X) = q ( X ) [ l + | (a m(X)cosm0 + b m ( X ) sinmtf>) ] (1.8) can be o b t a i n e d . The s o l u t i o n i s expressed i n terms of an i n f i n i t e s e r i e s of ei g e n v a l u e s and e i g e n f u n c t i o n s . T h i s work w i l l be f u r t h e r d i s c u s s e d i n S e c t i o n 3.2. 1.3.4 CONJUGATED PROBLEM Rather than using a priori f l u i d boundary c o n d i t i o n s and o b t a i n i n g the s o l u t i o n f o r the f l u i d o n l y , the conjugated problem i s formulated f o r the e n t i r e s o l i d - f l u i d medium system, and a s o l u t i o n f o r both the f l u i d and s o l i d temperature i s o b t a i n e d . Luikov et a l . [16] s o l v e d the conjugated problem f o r the c i r c u l a r tube. However, no numerical r e s u l t s were presented f o r the complicated c l o s e d - f o r m s o l u t i o n . Mori et a l . [17] c o n s i d e r e d two thermal boundary c o n d i t i o n s at the o u t s i d e w a l l of the c i r c u l a r tube : (1) constant heat f l u x , @, and (2) constant temperature, (TX 1 They assumed the w a l l - f l u i d i n t e r f a c e temperature d i s t r i b u t i o n i n the a x i a l d i r e c t i o n as a power s e r i e s with unknown c o e f f i c i e n t s . The s o l u t i o n t o the energy equation fo r the f l u i d was then o b t a i n e d by superposing the Graetz s o l u t i o n . Equating the temperature and heat f l u x e s across the i n t e r f a c e of the s o l i d and f l u i d media they o b t a i n e d the unknown c o e f f i c i e n t s f o r the power s e r i e s . A c o n c l u s i o n ^ e e Table 2.1 1 3 r e l e v a n t to the present work i s that f o r a " t h i n " w a l l with constant heat f l u x s p e c i f i e d at the o u t s i d e w a l l , the l o c a l Nu(x) approaches that f o r the c o n v e n t i o n a l @ c o n v e c t i o n problem. For a " t h i c k " w a l l , however, i t approaches t h e ® s o l u t i o n because a x i a l heat conduction tends to e q u a l i z e the temperature i n s i d e the w a l l . A w a l l may be c o n s i d e r e d t h i n when 7/L < 0.0001 f o r R w > 2X10" 7 and when 7/L < 0.001 f o r Rw' * 10- 5 [ 3 ] . F a g h r i and Sparrow [18] c o n s i d e r e d the e f f e c t of simultaneous w a l l and f l u i d a x i a l conduction i n a hydrodynamically developed laminar flow. Since only the t h i n - w a l l e d tube was c o n s i d e r e d , the r a d i a l temperature g r a d i e n t i n the w a l l was n e g l e c t e d . The problem was a l s o c o n s i d e r e d to have no c i r c u m f e r e n t i a l dependence because of symmetry around the a x i s . The c i r c u l a r tube c o n s i d e r e d had an i n s u l a t e d region (x < 0) and a region of d i r e c t h e a t i n g (x > 0) where the heat f l u x at the o u t s i d e w a l l of the tube was c o n s t a n t . S o l u t i o n s were obtained by an e l l i p t i c - f i n i t e d i f f e r e n c e method employing an i t e r a t i v e scheme which d e a l t c o n s e c u t i v e l y with the f l u i d and the tube w a l l . P l o t s of- the a x i a l d i s t r i b u t i o n of the c o n v e c t i v e heat f l u x q, w a l l and bulk mean temperatures, T w and T m r e s p e c t i v e l y , and N u s s e l t number were presented. Those graphs show that a x i a l c onduction depends on the P e c l e t number, Pe, and the dimensionless* w a l l conductance parameter B = k w7/ka. S u b s t a n t i a l amounts of c o n v e c t i v e heat t r a n s f e r can occur along the n o n - d i r e c t l y heated p o r t i o n (x < 0) of the tube 1 4 because of w a l l c o n d u c t i o n . These e f f e c t s of p r e h e a t i n g are propagated downstream by the f l o w i n g f l u i d , so there i s a s u b s t a n t i a l i n c r e a s e i n both the w a l l and bulk temperatures a l l a l o n g the tube. I t was found that the e f f e c t of w a l l conduction can r e a d i l y overwhelm the e f f e c t of f l u i d a x i a l c o n d u c t i o n . In the region of d i r e c t h e a t i n g (x > 0), the Nusselt number a t t a i n s a f u l l y developed value of 48/11, independent of a x i a l c o n d u c t i o n . B a r o z z i and P a g l i a r i m i [19] analysed the i n t e r a c t i o n between c o n v e c t i o n and a x i a l heat conduction along the tube w a l l assuming a c o n v e c t i v e boundary c o n d i t i o n at the outer face of the tube. They c o n s i d e r e d a w a l l whose t h i c k n e s s possessed p e r i o d i c s t e p v a r i a t i o n s i n the a x i a l d i r e c t i o n . An i t e r a t i e procedure was set up s t a r t i n g with guessed d i s t r i b u t i o n of l o c a l N u s selt no. and f l u i d bulk mean temperature, and temperature d i s t r i b u t i o n i n the s o l i d r e g i on was determined by the f i n i t e element method. T h e i r r e s u l t s show that a x i a l w a l l heat conduction has a d e f i n i t e i n f l u e n c e on heat f l u x q(x) and the N u s s e l t no. Nu(x) d i s t r i b u t i o n e s p e c i a l l y near the thermal i n l e t s e c t i o n . Step v a r i a t i o n s i n the w a l l t h i c k n e s s produce p e r i o d i c o s c i l l a t i o n i n the d i s t r i b u t i o n of Nu(x) and q ( x ) , whose amplitude n e v e r t h e l e s s reduces i n a r e l a t i v e l y s h o r t downstream d i s t a n c e , where t h e i r p r o f i l e s approach that of a uniform t h i c k n e s s w a l l . However, o v e r a l l heat f l u x can a c c u r a t e l y be p r e d i c t e d by o r d i n a r y methods d i s r e g a r d i n g a x i a l c o n d u c t i o n . 15 The above review, although by no means complete, p r o v i d e s an o v e r a l l idea of the a v a i l a b l e r e s u l t s r e l e v a n t to the present work. No work has been found i n the l i t e r a t u r e that d i r e c t l y i n v e s t i g a t e s the s t i t c h welding e f f e c t on the N u s s e l t number, and hence on s o l a r c o l l e c t o r performance. The next chapter d e s c r i b e s the modelling of the problem, c o n c e n t r a t i n g on the c o n v e c t i v e heat t r a n s f e r a spects f o r the two extreme cases mentioned in S e c t i o n 1.2. 2. MATHEMATICAL MODEL To analyze the heat t r a n s f e r from the s o l a r c o l l e c t o r absorber p l a t e to the water f l o w i n g through i t s spot-welded tube, a mathematical model d e s c r i b i n g the p h y s i c a l and g e o m e t r i c a l s i t u a t i o n must be developed. The a p p l i c a b l e d i f f e r e n t i a l equations and boundary c o n d i t i o n s f o r both the s o l i d and f l u i d media are d i s c u s s e d i n t h i s chapter. Enormous s i m p l i f i c a t i o n i s ob t a i n e d by r e s t r i c t i n g a t t e n t i o n to a c e r t a i n c l a s s of flows and by making c e r t a i n assumptions. Those assumptions can be j u s t i f i e d f o r the t y p i c a l f l a t p l a t e s o l a r c o l l e c t o r used i n b u i l d i n g and domestic water h e a t i n g . The s i m p l i f i e d mathematical model of the heat t r a n s f e r process i s presented, and other important terms are d e f i n e d . 2.1 GOVERNING EQUATIONS As the i n v e s t i g a t i o n i n v o l v e s heat t r a n s f e r i n a moving f l u i d medium, the three c o n s e r v a t i o n equations i n f l u i d mechanics have to be s a t i s f i e d . They are presented here i n v e c t o r form : 1. C o n s e r v a t i o n of Mass :-Dp/Dr = -p V • V (2.1) 2. Co n s e r v a t i o n of Momentum :-p -D V /Dr = p g + V «S ; (2.2) 3. Con s e r v a t i o n of I n t e r n a l Energy :-p DI/Dr = - V • q + heat sources + S( V V ) ; (2.3) 16 17 where standard n o t a t i o n s i n v e c t o r c a l c u l u s are used . D( )/Dr i s the s u b s t a n t i a l d e r i v a t i v e , where T denotes time. S i s the s t r e s s tensor and the term S( V V ) r e p r e s e n t s the complete c o n t r a c t i o n 2 . The r e s u l t a n t of a l l body f o r c e s a c t i n g on the medium i s represented by g . It should be noted that these equations are a p p l i c a b l e to any continuum, s o l i d as w e l l as l i q u i d . For a s t a t i o n a r y s o l i d , the energy equation becomes p9l/9r = - V • q + heat sources . (2.4) S o l v i n g the general equations (2.1) to (2.4) as they stand i s very d i f f i c u l t . One can only hope to o b t a i n reasonably a p p l i c a b l e s o l u t i o n s f o r s p e c i a l i d e a l i z e d c a ses. A number of assumptions have been made i n modeling the s o l a r c o l l e c t o r of i n t e r e s t , which can be j u s t i f i e d by examining the p h y s i c a l parameters i n v o l v e d . 2.2 IDEALIZATIONS A t a b l e of dimensions and r e l e v a n t p h y s i c a l p r o p e r t i e s f o r a t y p i c a l f l a t - p l a t e s o l a r c o l l e c t o r i s given i n Appendix A. The f o l l o w i n g assumptions are made based on the f i g u r e s given i n the t a b l e . 1. Since the f l u i d medium i s water and temperature v a r i a t i o n s are small (AT 10 °C), i t i s c o n s i d e r e d t o be incompressible as w e l l as Newtonian. Furthermore, other p r o p e r t i e s are assumed t o stay c o n s t a n t . 2. The water flow i s laminar because Re = uD/v 1,400 . 2See Appendix C 18 3. D i s s i p a t i o n can be n e g l e c t e d because flow v e l o c i t y i s only a few cm/s and Mach number i s low. 4. No heat source i s present i n the f l u i d as only s o l a r energy i s being absorbed by the p l a t e . 5. Although there i s no hydrodynamic s t a r t i n g l e n g t h at the tube entrance, the v e l o c i t y p r o f i l e development i s known to l e a d the thermal p r o f i l e development s i g n i f i c a n t l y . T h i s can be seen by r e a l i z i n g t h a t , f o r water at moderate temperature, the P r a n d t l number, v momentum d i f f u s i v i t y Pr = — = : * 4 . a thermal d i f f u s i v i t y T h e r e f o r e , the v e l o c i t y p r o f i l e can be assumed to be a l r e a d y f u l l y developed at the entrance [ 4 ] . T h i s ' r e s u l t s i n enormous s i m p l i f i c a t i o n because the w e l l known p a r a b o l i c v e l o c i t y p r o f i l e as shown i n F i g u r e 1.5, can be s u b s t i t u t e d d i r e c t l y i n t o the energy equation without having to so l v e equations (2.1), (2.2) and (2.3) s i m u l t a n e o u s l y . T h i s assumption r e s u l t s i n a c o n s e r v a t i v e estimate of the heat t r a n s f e r c o e f f i c i e n t . 3 6. A x i a l conduction i s n e g l i g i b l e . Dimensional a n a l y s i s [18] shows that t h i s i s the case when the P e c l e t number i s l a r g e (Pe = Re.Pr > 100). F u r t h e r d i s c u s s i o n i s given i n Appendix B. 3 The heat t r a n s f e r c o e f f i c i e n t i s g r e a t e r with a uniform v e l o c i t y p r o f i l e , which i s found i n s l u g flows and at the duct i n l e t d u r i n g simultaneous development of v e l o c i t y and temperature p r o f i l e s [20]. 19 7. Only steady s t a t e performance i s c o n s i d e r e d . Thus a l l the p a r t i a l d e r i v a t i v e s with respect to time, 9( )/9T, can be d i s c a r d e d from the governing e q u a t i o n s . 2.3 SIMPLIFIED EQUATIONS Assuming the v e l o c i t y p r o f i l e of the f l u i d to be f u l l y d eveloped at the tube entrance, the r a d i a l v e l o c i t y d i s t r i b u t i o n i s assumed to be known and i n v a r i a n t with a x i a l d i s t a n c e . Due to c i r c u l a r symmetry, the v e l o c i t y i s a l s o independent of p e r i p h e r a l a n g l e . That i s , V = u ( R ) i = U[ 1 - ( R / a ) 2 ] i , (2.5) where u i s the a x i a l component of the v e l o c i t y , and U, which can be c a l c u l a t e d from the mass flow r a t e " i s the peak v e l o c i t y of the flow. T h i s p a r a b o l i c p r o f i l e i s known as the P o i s e u i l l e flow v e l o c i t y d i s t r i b u t i o n f o r i n c o m p r e s s i b l e Newtonian f l u i d with constant p r o p e r t i e s . I t s form i s d e r i v e d i n d e t a i l in Burmeister [20]. As a consequence of the v e l o c i t y d i s t r i b u t i o n being known, the c o n t i n u i t y and momentum equations need not be c o n s i d e r e d . Only the energy equations f o r both the f l u i d and s o l i d media are of concern. For the s o l i d media, under steady s t a t e c o n d i t i o n s and with constant thermal c o n d u c t i v i t y k s V 2 T = sources . (2.6) For the absorber p l a t e , the heat source i s equal to the * See Appendix A. 20 d i f f e r e n c e between the incoming s o l a r r a d i a t i o n and the heat l o s s to the ambient through the r e s i s t a n c e 1/UT. For a Newtonian incompressible f l u i d , equation (2.3) reduces to p D H / D T = Dp/Dr + heat sources - V • q + ju* , where H = I + p/p i s the enthalpy per u n i t mass, and $ i s the d i s s i p a t i o n f u n c t i o n 5 . A p p l y i n g the assumptions that the water p r o p e r t i e s stay c o n s t a n t , that no heat source i s p r e s e n t , and that d i s s i p a t i o n i s n e g l i g i b l e , the energy equation f u r t h e r reduces to [20] : p c D T / D T = kV 2T . (2.7) S u b s t i t u t i n g equation (2.5) d i r e c t l y i n t o (2.7), the steady  s t a t e energy equation f o r the f l u i d i s found to be U[1 - ( R / a ) 2 ] 3T/3X = aV 2T (2.8) where a = k/pc i s the thermal d i f f u s i v i t y . Equation (2.8) i s to be s o l v e d under c e r t a i n boundary c o n d i t i o n s as d i s c u s s e d i n the next s e c t i o n . 2.4 THERMAL BOUNDARY CONDITIONS A set of s p e c i f i c a t i o n s d e s c r i b i n g temperature and/or heat f l u x c o n d i t i o n s at the i n s i d e w a l l of the duct must be o b t a i n e d to s o l v e the f l u i d energy equation (equ.2.8). A l a r g e v a r i e t y of these thermal boundary c o n d i t i o n s can be 5See Appendix C f o r d e t a i l s . 21 s p e c i f i e d f o r the c l a s s i c a l problem concerning the thermal p r o f i l e development. Shah and London [ 3 ] attempted to systemize the boundary c o n d i t i o n s s t u d i e d i n the l i t e r a t u r e . A few w e l l - s t u d i e d cases are shown i n Table 2 . 1 . In g e n e r a l , however, because of conduction i n the tube w a l l , w e l l - d e f i n e d temperature or heat f l u x c o n d i t i o n s are seldom encountered at the i n s i d e w a l l of the duct. Only i n extreme cases can those c o n d i t i o n s be s p e c i f i e d . Two extreme cases e x i s t f o r the water f l o w i n g through the spot-welded tube under study, determined by the " t h i c k n e s s " of the tube w a l l : 1 . For a t h i c k and h i g h l y c o n d u c t i v e w a l l , the a x i a l and p e r i p h e r a l conduction of heat from the welding spot t o the r e s t of the w a l l i s c o n s i d e r a b l e compared to the r a d i a l heat t r a n s f e r i n t o the fl o w i n g water ( F i g u r e 2 . 1 ( i ) ) . Conduction occurs u n t i l the whole tube w a l l comes to a uniform temperature. 2 . For a t h i n and low cond u c t i v e w a l l as shown i n F i g u r e 2 . 1 ( i i ) , the w a l l p e r i p h e r a l and a x i a l conductance i s n e g l i g i b l e compared with the r a i d a l conductance. T h e r e f o r e , s o l a r r a d i a t i o n being absorbed by the c o l l e c t o r p l a t e passes i n t o the water only r a d i a l l y through the welding spots. Thus the f l o w i n g water i s s u b j e c t to a w a l l boundary which c o n s i s t s of spots of heat f l u x . The d e f i n i t i o n of a " t h i n " w a l l has been given i n S e c t i o n 1 . 3 . 4 . 22 For the i d e a l i z e d case (2), the boundary c o n d i t i o n s can be d e p i c t e d by mathematical e x p r e s s i o n s q u i t e s t r a i g h t f orwardly. P e r i p h e r a l l y , the he a t i n g spots occupy the angular range -0 C < 4> < <j>0. Where # 0 < <f> < (2ir-<j>0), the heat f l u x i s zero . T h i s d i s t r i b u t i o n can be expressed i n terms of a F o u r i e r s e r i e s , f(0) = { S o 1 0, otherwise = ( Q s o l 0 o A ) [1 + m| 1 (a mcosm^) ] , (2.9) where a m = 2 ( sinm</>0 )/(m<j>0 ) . £(<t>) i s as shown i n F i g u r e 2.2. I t should be noted that Q s o lUoA) = q i s the p e r i p h e r a l mean heat f l u x through the w a l l . A x i a l l y , the spots are of f i x e d l e n g t h and are l o c a t e d at f i x e d i n t e r v a l s . The heat f l u x , as a f u n c t i o n of both x and <t>, can be expressed as f o l l o w s : q(0,x) = fU)H(x) - f ( ^ ) H ( x - x 1 ) + f UMx-U^Xg) ] - f ( ^ ) H [ x - ( 2 x 1 + x s ) ] + , (2.10) 0, x < 0 where H(x) = { 1 , x £ 0 i s the H e a v i s i d e u n i t step f u n c t i o n . The i l l u s t r a t i o n of q(x,0) where -cb0 < <t> < <f>0 i s i n F i g u r e 2.3. 23 Closed form s o l u t i o n s of reasonable a p p l i c a b i l i t y can be obtained only f o r the above two extreme cases, where the boundary c o n d i t i o n s are c l e a r l y p r e d e f i n e d . Extreme case (1) corresponds to the Graetz problem and i t s s o l u t i o n i s w e l l known. The a n a l y t i c a l method and s o l u t i o n s f o r case (2) are the s u b j e c t of the next c h a p t e r . The a n a l y t i c a l r e s u l t s were checked a g a i n s t and supplemented with a numerical procedure which w i l l be d e s c r i b e d i n Chapter 4. 3. A N A L Y T I C A L SOLUTIONS The governing energy equation and a s s o c i a t e d boundary c o n d i t i o n s f o r the heat t r a n s f e r problem under study have been presented i n Chapter 2. T h i s chapter d e s c r i b e s b r i e f l y the a n a l y t i c a l methods of s o l u t i o n s a v a i l a b l e i n the l i t e r a t u r e . The method employed i n a r r i v i n g at the s o l u t i o n f o r the extreme case (2) of S e c t i o n 2.4 i s d e t a i l e d i n t h i s c h a p t e r . The p r i n c i p l e of s u p e r p o s i t i o n i s shown to be of great u s e f u l n e s s i n t h i s kind of l i n e a r problem. 3.1 OVERVIEW OF ANALYTICAL SOLUTIONS The a n a l y t i c a l s o l u t i o n s f o r hydrodynamically developed thermal entrance flows are o b t a i n e d p r i m a r i l y by the f o l l o w i n g four kinds of methods : 1. S e p a r a t i o n of V a r i a b l e s and S i m i l a r i t y Transformation methods; 2. V a r i a t i o n a l Methods; 3. Conformal Mapping Method; and 4. S i m p l i f i e d Energy Equation Method. These methods and t h e i r sources i n the l i t e r a t u r e have been b r i e f l y d e s c r i b e d by Shah and London [ 3 ] . A t t e n t i o n w i l l be given here to the s e p a r a t i o n of v a r i a b l e s method, which was employed in s o l v i n g the present problem. 24 25 3.2 SEPARATION OF VARIABLES METHOD The s e p a r a t i o n of v a r i a b l e s method was used i n the f i r s t study by Graetz [5] of the thermal entrance r e g i o n of a c i r c u l a r tube. Many of the s o l u t i o n s of l a t e r work were obtained by s i m i l a r methods. The method i n v o l v e s s e p a r a t i n g the v a r i a b l e s i n the energy equation (2.8) and determining the e i g e n v a l u e s and c o n s t a n t s of the r e s u l t i n g o r d i n a r y d i f f e r e n t i a l equations by v a r i o u s approaches. The s o l u t i o n i s presented i n terms of an i n f i n i t e s e r i e s i n v o l v i n g e i g e n v a l u e s , e i g e n f u n c t i o n s , and c o n s t a n t s . Examples of work where t h i s method i s employed to the c i r c u l a r duct are that by Reynolds [14], S e l l a r s et a l . [ 8 ] , and by Bhattacharyya and Roy [15], which have been c i t e d i n S e c t i o n 1.3. The work by Bhattacharyya and Roy can be d e s c r i b e d i n two p a r t s . F i r s t , the temperature s o l u t i o n was obtained f o r the thermal entrance region f o r developed laminar flow i n a c i r c u l a r tube with v a r i a b l e c i r c u m f e r e n t i a l w a l l heat f l u x . The energy equation i s r e w r i t t e n here, in c y l i n d r i c a l c o - o r d i n a t e s : 3T 9 2T 1 9T 1 9 2T U( 1-R 2/a 2) = o( + + ) , 9X 9R 2 R 3R R 2 90 2 (3.1) where the a x i a l conduction term 9 2T/9X 2 has been om i t t e d . The boundary c o n d i t i o n s ares at R = a : k9T/9R = q = q [ l + | 9 m U ) ] , (3.2a) 26 where g (0) = a cosing + b sinmtf* , (3.2b) 3m m m and q i s the mean w a l l heat f l u x over the cir c u m f e r e n c e ; at X- = 0 : T = T. (3.2c) I n The d i m e n s i o n l e s s forms of the above equations were expressed as 9t 9 2 t 1 9t 1 9 2 t ( 1 - r 2 ) = + + , 9x 9 r 2 r 9r r2d<t>2 (3.3) r = 1 : 9t/9r = 1 + 1 g U ) , (3.4) m= 1 m x = 0 : t = 0 , (3.5) where t = k(T - T i n ) / q a > x = aX/Ua 2 = X/a.Pe , r = R/a . (3.6) The equations (3.3) to (3.5) were s o l v e d by s e p a r a t i o n of v a r i a b l e s and the r e s u l t was : t = 4x + r 2 - r f t/4 - 7/24 + 2, (r m/m)g ($) m=1 m - L, Cr, Rn exp(-/3 2x) - !! Ln c' R g (0)exp(-/3 2x) , s=1 Os Os ^ K 0 s m=1s=0 ms ms^m Y * Fms ,^ 7 j where R ( r . f l ) are e i g e n f u n c t i o n s , 3 are e i g e n v a l u e s , ms ms ms and c m s are the corresponding c o n s t a n t s . For d e t a i l s of the mathematics the reader i s r e f e r r e d to R e f . [ l 5 ] where u s e f u l v a l u e s of the e i g e n f u n c t i o n s and eig e n v a l u e s are g i v e n . The advantage of d i m e n s i o n a l i z i n g the v a r i a b l e s as i n equations (3.6) i s perhaps best seen from the e x p r e s s i o n of the N u s s e l t number, i . e . the dime n s i o n l e s s heat t r a n s f e r 27 c o e f f i c i e n t . By d e f i n i t i o n , the l o c a l N u s s e l t number i s give n by : NuU,X) = { q(0,X)/[T U,X) - T (X) ] }(D/k) . (3.8) W 111 The term i n braces i s the l o c a l heat t r a n s f e r c o e f f i c i e n t , commonly denoted by h. E x p r e s s i n g i n terms of the dim e n s i o n l e s s temperatures, with rhc AT = q(27ra)AX ==> t =4x , p m m equat i o n (3.8) takes the form 2 [ 1 + J i 9m<* ) ] N u U , x ) = 1 1 / 2 4 + m ? i ^m^)M] - s Si c 0 s R 0 s ( 1 ^ 0 s ) e x P ( ^ 0 I x ) " s | 0 [ c m s R m s ( 1 f P m s ) g i n ( 0 ) exp(-/3 m|x) ] The p e r i p h e r a l average N u s s e l t number, according to the d e f i n i t i o n of Shah and London [ 3 ] , i s given by: Nu p(X) = { q(X)/[T w m(X) - T m ( X ) ] }(D/k) , (3.10) where q(X) = (1/2TT) f 2 7 r q U , X ) d 0 , Twm ( X ) = (l/27r); 2 7 rT w(0,X)d0 . According to the expression of t in equation (3.7), the peripheral average Nusselt number can also be written as Nu p(x) = 11/24 - s Z 1 c O s R O s O , 0 O s > exp(-/3 0|x) . (3.11) Whereas the f i r s t p a r t of Bhattacharyya and Roy's work d e a l s with a tube with no a x i a l v a r i a t i o n i n w a l l heat f l u x 28 b e s i d e s the entrance s t e p change, the second p a r t i n v o l v e s o b t a i n i n g the s o l u t i o n f o r an a r b i t r a r i l y heated tube w a l l . The s o l u t i o n was obtained by a p p l y i n g Duhamel's s u p e r p o s i t i o n formula on equation (3.7). The r e s u l t i n g s e r i e s s o l u t i o n i n v o l v e s complicated i n t e g r a l s and i s i m p r a c t i c a l f o r e n g i n e e r i n g c a l c u l a t i o n . A simpler superposing technique i s a p p l i c a b l e to the present problem. 3.3 SOLUTION FOR "THIN" TUBE USING SUPERPOSITION Consider a boundary value problem whose governing d i f f e r e n t i a l equation i s A (t) = 0 , where A i s a l i n e a r o p e r a t o r , and has d e r i v a t i v e / boundary c o n d i t i o n s Y ( * ) t' b d r y = H 1 + H 2 + + H2N ' Suppose there e x i s t 2N f u n c t i o n s t 1 , t 2 . . . . fc2N' s a t i s f y i n g the f o l l o w i n g : A ( t . ) = 0 , t i | b d r y = H i ' 1 " 1 ' 2 2 N ' 2N then t h e i r sum, t ^ , w i l l be the s o l u t i o n t o (*) because i t s a t i s f i e s the d i f f e r e n t i a l equation as w e l l as the boundary c o n d i t i o n s : A( .2* t.) = .1" [ A ( t . ) ] = 0 , i=1 i i=1 i 29 ( i l l V lbdry" i l l ( t i i b d r y > = H 1 + H 2 + + H2N • S u b s t i t u t i n g the f o l l o w i n g operator f o r A i n (*) : 9 2 1 a 1 a 2 a [ + + ( 1 - r 2 ) — ] , 3 r 2 r 3r r23tf>2 3x and l e t t i n g H 1 = f ( 0 ) H ( x ) , H 2 = - f ( 0 ) H ( x - X ; L ) , H 3 = f ( 0 ) H [ x - ( x 1 + x s ) ] , e t c , the problem posed by equations (3.3) and (2.10) i s the same as the system (*). The s o l u t i o n i s then t ( r , 0 , x ) = t ^ r ^ x ) , (3.12) where t 1 i s i d e n t i c a l to the f u n c t i o n t i n equation (3.7), t 2 = - t 1 ( r , 0 , x - x 1 ) H ( x - x 1 ) , t 3 = t 1 ( r , 0 , x - ( x 1 + x )) H [ x - ( x 1 + x s ) ] , e t c . (3.13) t ^ r ^ j x ) of equation (3.12), with a p p r o p r i a t e c o n s t a n t s , d e s c r i b e s the temperature d i s t r i b u t i o n of a steady laminar flow e n t e r i n g a c i r c u l a r tube which i s heated over a f r a c t i o n of i t s ci r c u m f e r e n c e , a s i t u a t i o n i l l u s t r a t e d by F i g u r e 3.1. A x i a l l y , the h e a t i n g s t a r t s at x=0 and i s i n v a r i a n t with d i s t a n c e . The mathematical e x p r e s s i o n f o r t h i s boundary c o n d i t i o n i s q U , x ) = f ( 0 ) H(x) , where f(</») i s given i n equation (2.9). 30 The N u s s e l t number corres p o n d i n g to s o l u t i o n (3.12) can a l s o be found by s u p e r p o s i t i o n : For -0o ^ <t> * <t>o, 1/Nu(0,x) =.|* ( t W f i - t m f . ) / 2 = 1" [ l / N U i ( 0 , x ) ] , (3.14) where Nu 1 (0,x) i s given by equation (3.9), and 1/Nu 2 (0,x) = ( t w 2 - t m 2 ) / 2 = -H(x-x 1) / N u 1 ( 0 , x - x l ) , 1/Nu 3 (0,x) = <t 3 - t f f l 3 ) / 2 = H[x-( X ; L+x s) ] / N U 1 ( 0 , X - ( X ; L + X S ) ) , e t c . (3.15) For other v a l u e s of 0 , the N u s s e l t number i s zero s i n c e t h e r e i s no heat f l u x through the w a l l . The p e r i p h e r a l average N u s s e l t number Nu^(x) can be found by s i m i l a r s u p e r p o s i t i o n on equation (3.11). The e f f e c t on the N u s s e l t numbers of n e g l e c t i n g f l u i d a x i a l c o n d u c t i o n i s d i s c u s s e d i n Appendix B. 3.4 GRAPHICAL ILLUSTRATION OF ANALYTICAL RESULTS The boundary c o n d i t i o n (2) of S e c t i o n 2.4 d e p i c t s the extreme s i t u a t i o n where the tube w a l l i s so t h i n t h a t s o l a r energy passes i n t o the water only r a d i a l l y through the welding s p o t s . The a n a l y t i c a l s o l u t i o n f o r the temperature d i s t r i b u t i o n i n t h i s case i s given by equations (3.12) and (3.13). With 1/8 of the tube c i r c u m f e r e n c e being welded c o n t i n u o u s l y along i t s l e n g t h 6 , a m= s i n ( m 0 o ) ( 2 / m 0 o ) and k>m=0 6 The spot angle of 45°,'i.e. 0 O = TT/8 was chosen a r b i t r a r i l y to enable comparison with other c a s e s . 3 1 i n the f u n c t i o n 9 m(0) which appears i n the e x p r e s s i o n of t ^ The s o l u t i o n i s w r i t t e n here f o r r = 1 : t O , 0 , x ) = 4x + 11/24 + ? [a cosm0)/m] m= 1 m " sll c O s R O s ( 1 ' ' o s ) e x P ( ^ 0 s x ) { 3 ' 1 6 ) " nil sSo [ cms Rn,s ( 1^ms ) <V°™*> e x P ( ^ m s x ) ] ' The value of t at r = 1 was p l o t t e d a g a i n s t x f o r v a r i o u s v a l u e s of 0 i n F i g u r e 3.3. T h i s diagram shows the a x i a l development of the w a l l temperature at v a r i o u s angular p o s i t i o n s . As expected, the temperatures at those angular p o s i t i o n s which are d i r e c t l y heated i n c r e a s e s h a r p l y near the entrance. Further downstream, the temperatures at a l l angles assume a l i n e a r r e l a t i o n s h i p with d i s t a n c e , a s i t u a t i o n termed as t h e r m a l l y developed. There i s a s i g n i f i c a n t d i f f e r e n c e i n temperature between the angular p o s i t i o n which i s j u s t i n s i d e the d i r e c t l y heated region (0=7T/1O) and the one which i s j u s t o u t s i d e ( 0 = 2 T T / 1 O ) . The l o c a l and p e r i p h e r a l average N u s s e l t numbers f o r the above s i t u a t i o n were p l o t t e d a g a i n s t x i n F i g u r e 3.4. The l o c a l N u s s e l t number Nu(#,x) was only shown f o r two angular p o s i t i o n s i n s i d e the welded r e g i o n , s i n c e i t s value i s zero f o r those p o s i t i o n s o u t s i d e that r e g i o n . I t might be n o t i c e d that Nu p(x) i s i d e n t i c a l to that of a uniformly heated tube. T h i s should come as no s u r p r i s e i f the d e f i n i t i o n of Nu p(x) (equ a t i o n ( 3 . 1 0 ) ) i s understood. In any case, the thermal entrance has an e f f e c t of i n c r e a s i n g 32 s h a r p l y the heat t r a n s f e r c o e f f i c i e n t , as can be seen from the N u s s e l t numbers near x=0. To i l l u s t r a t e the e f f e c t of s u p e r p o s i t i o n on equation (3.16), a tube as i n F i g u r e 3.2 with two welding spots occupying 60% of i t s l e n g t h was c o n s i d e r e d . The spot angle was kept at 45°, i . e . <I>Q = TT/8 . The s o l u t i o n becomes, f o r the w a l l temperature, 2N t(1,0,x) = | t i ( 1 , 0 , x ) , where t 1 i s given by equation (3.16), t 2 = _ t 1 ^ 1 ' ^ ' x ~ x i ^ H(x-x 1) , t 3 = t 1 (1 ^ ^ - ( x j + X g ) ) H ( x - ( x 1 + x s ) ) , t 4 = - t 1 ( 1 , 0 , x - ( 2 x 1 + x s ) ) H ( x - ( 2 x 1 + x g ) ) , e t c , with 2x-^ = 0. 6 (tube-length) , 2x s= 0.4(tube-length) . t(1,0,x) was p l o t t e d a g a i n s t x i n F i g u r e 3.5. I t can be noted that the a x i a l temperature f l u c t u a t i o n i s much more pronounced i n the range of <j> that i s d i r e c t l y heated. Again, there i s a s i g n i f i c a n t d i f f e r e n c e i n temperature from 0=tr/lO to 0=27T/1O. The g e n e r a l trend of temperature i n c r e a s i n g downstream can a l s o be observed. I t was intended to compare the mean N u s s e l t number Nu m between d i f f e r e n t welding spot c o n f i g u r a t i o n s , given a f i x e d t o t a l weld l e n g t h and a f i x e d spot angle, and hence a f i x e d t o t a l heat i n p u t . The tube c o u l d be welded to the absorber p l a t e at any number of e q u a l l y spaced and i d e n t i c a l 33 s t i t c h e s , while a s t i t c h angle of 45° and a t o t a l weld l e n g t h of 60% of the tube l e n g t h were a r b i t r a r i l y chosen as i n v a r i a n t s . R e s u l t s of two d i f f e r e n t cases were shown i n F i g u r e s 3.6 ( i ) and ( i i ) , where the tube was welded at : ( i ) a s i n g l e s t i t c h ; and ( i i ) 4 s t i t c h e s , r e s p e c t i v e l y . I t can be observed that a higher number of s t i t c h e s r e s u l t s i n a higher mean Nu s s e l t number s i n c e they introduce more "thermal entrance e f f e c t " to the temperature p r o f i l e . The c o r r e c t n e s s of the a n a l y t i c a l r e s u l t s can be checked by s o l u t i o n o b tained through numerical methods. The next chapter d e s c r i b e s the f i n i t e d i f f e r e n c e f o r m u l a t i o n of the same problem. The f i n i t e d i f f e r e n c e r e s u l t s were compared with those d e s c r i b e d i n t h i s chapter. 4 . NUMERICAL SOLUTIONS The a n a l y t i c a l s o l u t i o n s and r e s u l t s presented i n the l a s t c h a pter are those f o r the i d e a l i z e d boundary c o n d i t i o n where the f l u i d i s bounded by spots of heat f l u x at the w a l l . T h i s chapter d e s c r i b e s a numerical approach to the same boundary v a l u e problem. The r e s u l t s serve both as a check and a supplement to the a n a l y t i c a l s o l u t i o n s . The b a s i c assumptions o u t l i n e d i n S e c t i o n 2 . 2 , and hence the s i m p l i f i c a t i o n s , are r e t a i n e d , and the same d i m e n s i o n l e s s v a r i a b l e s d e f i n e d i n equation ( 3 . 6 ) are used. 4 . 1 OVERVIEW OF NUMERICAL METHODS The thermal entrance s o l u t i o n s f o r hydrodynamically developed flows can be obtained by s e v e r a l methods. The a n a l y t i c a l methods have been o u t l i n e d i n S e c t i o n 3 . 1 , while the numerical methods can be c l a s s i f i e d as f o l l o w s : 1 . F i n i t e D i f f e r e n c e Methods; 2 . Monte C a r l o Method; and 3 . F i n i t e Element Method. A b r i e f d e s c r i p t i o n of these methods and t h e i r sources i n the l i t e r a t u r e can be found i n Shah and London [ 3 ] . The method used i n the present work s o l v e s the f i n i t e d i f f e r e n c e f o r m u l a t i o n of the d i m e n s i o n l e s s energy equation u s i n g an i t e r a t i v e technique. The next s e c t i o n i s devoted to the d e t a i l s of t h i s f o r m u l a t i o n . 3 4 35 4.2 FINITE DIFFERENCE FORMULATION A program was developed to s o l v e the 3-dimensional equation (3.3) n u m e r i c a l l y , and hence the temperature d i s t r i b u t i o n of the f l u i d i n s i d e the tube was obtained. The N u s s e l t number d i s t r i b u t i o n was then computed d i r e c t l y from the temperature val u e s using the d e f i n i t i o n s i n equations (3.8) to (3.10). The tube volume was d i v i d e d by r a d i a l , c i r c u m f e r e n t i a l , and a x i a l g r i d s , with corresponding g r i d spacing of Ar, A0, and Ax, r e s p e c t i v e l y , as shown i n F i g u r e 4.1. The l o c a t i o n of each node was d e f i n e d by a c y l i n d r i c a l c o o r d i n a t e i n terms of the g r i d numbers. Thus the temperature at a l l nodes can be s t o r e d i n a 3-dimensional a r r a y , t(N r,N^,N x), where Nr =0,1, ... M r i s the r a d i a l g r i d number, N = 0,1, ... M, i s the c i r c u m f e r e n t i a l g r i d number, 9 9 and N = 0,1, ... M i s the a x i a l g r i d number of the node. X X W r i t t e n i n f i n i t e d i f f e r e n c e form, the energy equation takes the f o l l o w i n g form: t 3 ~ 2 t 5 + t , 1 t 3 - t , 1 t 2 - 2 t 5 + t„ t 0 - t 5 + + = ( 1 - r 2 ) , ( A r ) 2 r 2Ar r 2 (A<j>)2 Ax (4.1) where t 0 , . . . , t 5 represent the temperatures at the l o c a t i o n s i n d i c a t e d i n F i g u r e 4.2. T h i s equation i s w r i t t e n f o r the p o i n t (N ,N ,N ) whose temperature i s t 0 , and r = N «Ar i s L (p X IT the dimensionless r a d i a l c o o r d i n a t e of that p o i n t . Two d i f f e r e n t cases need to be c o n s i d e r e d s e p a r a t e l y : 36 ( i ) r * 0: . t 0 = [ a , t , + a 2 ( t 2 + t « ) + a 3 t 3 + a 5 t 5 ] / a 0 , (4.2) where a 0 = ( A r ) 2 ( l - r 2 ) / Ax , a, = 1 - Ar/2r , a 2 = ( A r / r A 0 ) 2 , a 3 = 1 + Ar/2r , 35 = — 2 — 2a 2 . ( i i ) r = 0 : V 2 t = 4(t - t 5 ) / ( A r ) 2 , (4.3) where t = t(1,N ,N - 1)/(M +1) avg N =0 9 9 <p i s the average temperature at the immediate neighborhood of the tube c e n t e r - l i n e . So t 0 = ( 4 A x / A r 2 ) t + (1 - 4 A x / A r 2 ) t 5 . (4.4) cl V y An imaginary s u r f a c e N r= M r + 1 was added o u t s i d e the tube w a l l to implement the temperature g r a d i e n t at the w a l l as determined by the p r e - d e s c r i b e d heat f l u x . The temperature at a node on that s u r f a c e was determined by whether the node's p e r i p h e r a l and a x i a l l o c a t i o n was w i t h i n that of a welding spot: ( i ) w i t h i n spot : At/Ar = n/4>0 ,7 t(M r+1,N N x) = t(M r-1,N ,Nx) + 2Ar (7 r/0 o) ,* (4.5) 7 (1 + 2 amcosm</») = ir/<t>0 when -<f>0<<f><<j>0, 37 ( i i ) n o t w i t h i n spot : At/Ar = 0 t(M +1, N., N ) = t(M -1, N , N ) . r ' 6' x r ' <b' x (4.6) T h e r e f o r e , the temperature at each node can be expressed i n terms of the temperatures at i t s neighborhood nodes. A f t e r i n i t i a l i z i n g the 3-dimensional a r r a y of temperatures with a non-zero f u n c t i o n , a new value f o r each node was obtained u s i n g equations (4.2) to (4.6). E v a l u a t i o n of temperatures was done one c i r c u l a r c r o s s - s e c t i o n a f t e r another, s t a r t i n g from the tube entrance N x = ^ " T ^ e temperature at the entrance was always kept at 0 to conform with the boundary c o n d i t i o n t ( r , 0 , O ) = 0. A f t e r the temperature at a l l nodes had been updated, the process was repeated u n t i l d e v i a t i o n between two s u c c e s s i v e e v a l u a t i o n s was l e s s than 0.5% at any node. Convergence was then c o n s i d e r e d a t t a i n e d and the a r r a y t(N r,N^,N x) would c o n t a i n the s o l u t i o n temperature d i s t r i b u t i o n . 4.3 CALCULATION OF NUSSELT NUMBERS Having obtained the temperature d i s t r i b u t i o n throughout the f l u i d body, the heat t r a n s f e r c o e f f i c i e n t h, and i t s dimensionless form, the N u s s e l t number, were computed a c c o r d i n g to d e f i n i t i o n s . The d e f i n i t i o n of the p e r i p h e r a l average and mean N u s s e l t numbers are reproduced here: twm^x^~tm^x^^' * f x f a l l s w i t h i n a spot; 0 , othe r w i s e , (4.7) 38 Nu m(x) = O/xJ/^Nu (x)dx , (4.8) where t^(<t>,x)= d i m e n s i o n l e s s l o c a l w a l l temperature, t w m ( x ) = p e r i p h e r a l average w a l l temperature, = (1/2ir) SQ t w ( 0 , x ) d 0 , t m ( x ) = bulk mean temperature, = (1/UA) J A ( u t ) d A = ( 2 A a 2 ) ; A d - r 2 ) t dA , and x - o u t s i d e - s p o t i f x ^ nx,+(n-1)x and x < n(x,+x ) r i s I s where n = 1,2, , x-within-spot otherwise. (4.9) In f i n i t e d i f f e r e n c e f o r m u l a t i o n , the above i n t e g r a l s were computed as f o l l o w : t(M r,0,N x) + 2 V t(M r,N 0,N x) -t(M r,M 0,N x) , twml x' 0  2 (4.10) 0 t m ( N x ) = ( 2 / t ) { N l [ 1 [UTDA(N r,0,N x) + 2 ^ ^ UTDA(N r,N 0,N x) r • 0 + UTDA(N ,M.,N )] + t ( 0 , N . f N ) 7r(Ar) 2} , r 0 x 0 x (4.11) where UTDA(N ,N ,N ) = ( 1 - r 2 ) - t(N ,N ,N ) • ( A 0 / 2 ) ( r 2 2 - r , 2 ) , IT (p X IT (p X and r, = N r-Ar , r 2 = (N r+1)-Ar . (4.12) 4.4 PRELIMINARY RESULTS OF NUMERICAL PROCEDURE With the ba s i c a l g o r i t h m as o u t l i n e d i n the pre v i o u s two s e c t i o n s , the computational program was developed i n s e v e r a l s t a ges. The r e s u l t s of each stage were checked a g a i n s t those obtained by other methods, i f a v a i l a b l e , 39 b e f o r e another stage was developed. T h i s s e c t i o n p r e s e n t s those r e s u l t s stage by stage. F i r s t , the s i m p l e s t case of a u n i f o r m l y heated tube w a l l was taken as the boundary c o n d i t i o n to t e s t the program. T h i s problem has been w e l l s t u d i e d and r e s u l t s have been t a b u l a t e d i n Shah and London[3]. The computed temperature, as expected, does not depend on the p e r i p h e r a l angle <t>, and i t s value at v a r i o u s r a d i a l c o o r d i n a t e s r was p l o t t e d a g a i n s t x i n F i g u r e 4.3. Compared with the t a b u l a t e d v a l u e s of the w a l l temperature, the f i n i t e d i f f e r e n c e r e s u l t was found to be higher near the entrance, although the two r e s u l t s converge at l a r g e x (x>80). T h i s d i s c r e p a n c y at the entrance was found i n a l l the l a t e r r e s u l t s and i t was found t o be due to the coarseness of the g r i d system. The N u s s e l t number ( Nu(#,x) = Nu (x) i n t h i s case) i s shown i n F i g u r e 4.4, with Shah and London's val u e s p l o t t e d a g a i n s t the same axes. Next, a w a l l was chosen to have 1/8 of i t s c i r c u m f e r e n c e d i r e c t l y heated (the welding spot) and i n s u l a t e d everywhere e l s e . T h i s s i t u a t i o n i s as shown i n F i g u r e 3.1. F i g u r e 4.6 shows the w a l l temperature p l o t t e d a g a i n s t x, f o r v a r i o u s angular p o s i t i o n s at the w a l l as shown i n F i g u r e 4.5. Compared with the a n a l y t i c a l r e s u l t s f o r the same problem 8, the entrance d i s c r e p a n c y can be observed as i n the case of the u n i f o r m l y heated tube. In t h i s case as w e l l as i n that of the u n i f o r m l y heated tube, a 8Shown i n F i g u r e 3.3. 40 l i n e a r r e l a t i o n s h i p between t and x can be observed f a r downstream, a region where the flow i s d e s c r i b e d as t h e r m a l l y developed. The reader i s c a u t i o n e d that when comparing the dimensionless temperatures between d i f f e r e n t cases, f o r example between those in F i g u r e s 4.3 and 4.6, i t should be remembered that the d e f i n i t i o n of t depends on the value of q which in turn depends 9 on <j>0. F u r t h e r c o m p l i c a t i n g the boundary c o n d i t i o n s by i n t r o d u c i n g a x i a l step changes cf heat f l u x , the problem shown i n F i g u r e 3.2 was s o l v e d by the program. The w a l l temperature obtained i s shown i n Figure 4.7. The temperature p r o f i l e s agree w e l l with the a n a l y t i c a l r e s u l t s 1 0 except at a s h o r t d i s t a n c e downstream from a l l the step changes. The temperature d i s t r i b u t i o n s f u r t h e r i n s i d e the tube at v a r i o u s r a d i a l c o o r d i n a t e s are a l s o shown : F i g u r e 4.8 f o r r=4/5 and F i g u r e 4.9 f o r r = l / 5 . The inner f l u i d c l o s e to the w a l l (r=4/5) has a s u b s t a n t i a l l y lower temperature than that at the w a l l , e s p e c i a l l y at those angular p o s i t i o n s w i t h i n a welding spot (0=0 and 0 = 7 T / 1 O ) . T h i s r a d i a l temperature  g r a d i e n t i s what permits the heat f l u x from the weld to pass i n t o the inner f l u i d . Close to the tube c e n t r e - l i n e , at r=l/5, the temperature h a r d l y r i s e s or f l u c t u a t e s with flow l e n g t h . Furthermore, c i r c u m f e r e n t i a l heat t r a n s f e r i s much l e s s pronounced i n the inner f l u i d as can be seen from the s m a l l e r c i r c u m f e r e n t i a l temperature g r a d i e n t . I t should be commented that there i s no t h e r m a l l y developed region i n 9 Q=Qsol (#oA) 1 0Shown i n F i g u r e 3.5. 41 t h i s case because there i s a x i a l v a r i a t i o n of boundary c o n d i t i o n s a l l along the flow l e n g t h . F i n a l l y , the flow l e n g t h to be c o n s i d e r e d was halvened to allow f i n e r d i v i s i o n s of the g r i d s without much i n c r e a s e of memory space, and the same two cases which r e s u l t e d i n F i g u r e s 3.6(i) to ( i i ) were s o l v e d by the program. Whereas the N u s s e l t numbers were c a l c u l a t e d u sing simple a n a l y t i c a l e x p r e s s i o n s o u t l i n e d i n S e c t i o n 3.3, they were now computed d i r e c t l y from d e f i n i t i o n s , u s i n g equations (4.7) to (4.1.1). As can be seen from F i g u r e s 4.10(i) and ( i i ) , the numerical method produced the same r e s u l t s as the a n a l y t i c a l method. That the numerical s o l u t i o n s converge to the a n a l y t i c a l ones as g r i d spacing gets f i n e r r e a s s u r e s that both methods work. F u r t h e r r e s u l t s a p p r o p r i a t e to c o l l e c t o r e f f i c i e n c y e v a l u a t i o n are shown i n the next chapter. I m p l i c a t i o n of those r e s u l t s to c o l l e c t o r performance w i l l a l s o be e l a b o r a t e d . 5. APPLICATION TO COLLECTOR EFFICIENCY FACTOR As mentioned i n Chapter 1, the 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' i s given by F- = •. k W{ 1/[U T (b+(W-b)F) ] + 1/C. + 7/drDk ) + l/(irDh)} . L b w The heat t r a n s f e r c o e f f i c i e n t h a p p r o p r i a t e to t h i s , e x p r e s s i o n was determined from a d e f i n i t i o n f o r Nu(x) s l i g h t l y d i f f e r e n t than that i n equation (3.10). The r e l a t i o n s h i p between t h i s Nu and x was c a l c u l a t e d f o r v a r i o u s spot angles (2<t>0) , welded percentage w of tube l e n g t h , and t o t a l number N of spots along a tube. Choosing a spot angle as i n v a r i a n t , the r e l a t i o n s h i p between F' and spot c o n f i g u r a t i o n s ( i . e . w and N) can be i l l u s t r a t e d . The r e s u l t s f o r the " t h i c k " tube a l l tend to the uniform w a l l temperature case s i n c e the heat t r a n s f e r c o e f f i c i e n t does not depend on spot c o n f i g u r a t i o n . T h i s r e s u l t can be regarded as the upper l i m i t of a c t u a l performance. On the other hand the r e s u l t s f o r the " t h i n " tube are more i n t e r e s t i n g because spot c o n f i g u r a t i o n i s important i n determining F'. T h i s r e s u l t serves as the lower l i m i t of a c t u a l performance, and w i l l be the major concern of t h i s c h a p t e r . 42 43 5.1 HEAT TRANSFER COEFFICIENT BASED ON BOND TEMPERATURE The common d e f i n i t i o n of Nu p(x) (e.g. equation(3.10)) i s based on the d i f f e r e n c e between the p e r i p h e r a l average  w a l l temperature and the bulk mean temperature of the f l u i d . A c a r e f u l a n a l y s i s of the c l a s s i c a l f i n problem r e v e a l s that a N u s s e l t number based on the d i f f e r e n c e between the bond  temperature and the bulk mean temperature should be used i n c o n j u n c t i o n with equation (1.1). The r e s u l t of the c l a s s i c a l f i n problem a s s o c i a t e d with the f l a t p l a t e c o l l e c t o r with " t h i n " tubes i s b r i e f l y o u t l i n e d below. The u s e f u l energy gain per u n i t of l e n g t h i n the flow d i r e c t i o n f o r a c o l l e c t o r tube can be shown [1] to be q u= [(W-b)F + b ] [ Q s o l - U L ( T b - T a ) ] , (5.1) where Q s o i * s t n e s o l a r energy absorbed by u n i t area of the p l a t e , T^ i s the temperature at the p l a t e - t u b e bond, T & i s the ambient temperature, and the remaining terms are d e f i n e d i n S e c t i o n 1.1. U l t i m a t e l y , the u s e f u l gain q^ must be t r a n s f e r r e d to the f l u i d . The r e s i s t a n c e to heat flow t o the f l u i d comprises that of the bond, the tube w a l l , and the f l u i d to tube r e s i s t a n c e . The u s e f u l gain can be expressed i n terms of these t h r e e r e s i s t a n c e s as 1/Cb+ 7 / 7 r D k w + 1/TrDh . I t has been shown [2] that the bond r e s i s t a n c e 1/C, i s D n e g l i g i b l e compared with the other r e s i s t a n c e s . Furthermore, s i n c e the tube w a l l i s assumed extremely t h i n , the r a d i a l 44 r e s i s t a n c e o f f e r e d by the w a l l 7/7rDk w i s a l s o n e g l i g i b l e . The u s e f u l energy gain per u n i t of flow l e n g t h per tube i s T K _ T M b m 1/»Dh . (5.2) But Q U(X) = 27raq(X), so the f o l l o w i n g f a m i l i a r form i s obtained : h b(X) = q(X) / [ T b ( X ) - T m ( X ) ] . (5.3) At the a x i a l p o s i t i o n s where no welding i s done, a t h i n l a y e r of a i r between the tube and the f i n o f f e r s a l a r g e r a d i a l r e s i s t a n c e to heat flow, and both q and h^ i n the equation can be taken as z e r o . Thus, equation (5.3) a p p l i e s whether X f a l l s w i t h i n a welding spot or not. Furthermore, s i n c e the bond and the w a l l r e s i s t a n c e are both n e g l i g i b l e , the f l u i d temperature at R = a and </> = 0 can be taken as the bond temperature ( F i g u r e 5.1). T h e r e f o r e the mean heat t r a n s f e r c o e f f i c i e n t over the e n t i r e flow l e n g t h L i s 1 L q(X)dX hbm " L ; 0 T ( a f 0 f X ) _ T m ( X ) m (5.4) In d i m e n s i o n l e s s form, equation (5.4) becomes hbm " ( k / ° ) N u b m = (k/Dl) ; j N u b ( x ) d x , (5.5) 45 where Nu^Cx) = 2 / [ t ( 1 , 0 , x ) - 4 x ] , i f x - w i t h i n - s p o t , = 0 , i f x - o u t s i d e - s p o t , (5.6) and 1 = L/a«Pe S o l v i n g equation (5.2) f o r and s u b s t i t u t i n g i n t o equation (5.1), one o b t a i n s q = WF' [Q ,- U T (T -^u s o l L m T ) ] , where F' = W{l/[U L(b+(W-b)F)] + W(irDhbm)} (5.7) The same e x p r e s s i o n f o r F' can be o b t a i n e d from equation (1.1) i f the r e s i s t a n c e s due to the bond and the tube w a l l are n e g l e c t e d . The N u s s e l t no. based on bond temperature Nu^ as i n equation (5.6) was c a l c u l a t e d u s i n g the s e r i e s s o l u t i o n and s u p e r p o s i t i o n method d i s c u s s e d i n Chapter 3. The 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 can then e a s i l y be c a l c u l a t e d . Before comparing the value of F' f o r d i f f e r e n t welding-spot c o n f i g u r a t i o n s , one c o n f i g u r a t i o n was a r b i t r a r i l y chosen to i l l u s t r a t e the bahaviour of Nu^(x) and N u ^ m ( x ) u s i n g t h i s d e f i n i t i o n . T h i s i s i l l u s t r a t e d i n the next s e c t i o n . 5.2 BEHAVIOUR OF NU(X) FOR A CONTINUOUSLY WELDED TUBE To c a l c u l a t e Nu^(x), the d i m e n s i o n l e s s temperature d i f f e r e n c e t(1,0,x)-4x i s r e q u i r e d . For a c o n t i n u o u s l y  welded tube with spot angle 20 o, t h i s d i f f e r e n c e can be found using equation (3.16) : 46 Nu b(x) = 2/[t(1,0,x)-4x] 2 " C 1 1 / 2 4 ' -mil 3m / m " s l i c 0 s R 0 s e x P ( ^ 0 s x ) ' m l l s l o C m s R r n s a m e x P ( ^ m s x ) ] ( 5 ' 8 ) Nu^ p l o t t e d a g a i n s t 1000X appears i n F i g u r e 5.2 f o r v a r i o u s v a l u e s of <j>0. The case 4>0 = ir corresponds to the w e l l known case of a uniformly heated w a l l , where t ( l , 0 , x ) equals the p e r i p h e r a l average w a l l temperature. Thus the Nuss e l t curve f o r <t>o = n c o i n c i d e s with the accepted Nu (x) P curve f o r a uniformly heated tube 1 1 and approaches the expected asymptotic value of 48/11. As the spot angle 2(j>0 decreases, so does the N u s s e l t number based on bond temperature. T h i s i s not the case f o r the N u s s e l t number based on p e r i p h e r a l average w a l l temperature N u p ( x ) , which does not depend on <j>0 but conforms with the un i f o r m l y heated case. I t i s obvious t h a t the l a r g e r the spot angle, the higher the mean heat t r a n s f e r c o e f f i c i e n t based on bond temperature h b m and hence the e f f i c i e n c y f a c t o r F'. T h i s i n c r e a s e i n F' i s ob t a i n e d at the c o s t of more welding. 5.3 BEHAVIOUR OF NU(X) OF SPOT WELDED TUBE A l a r g e r spot angle r e s u l t s i n a higher F' f o r the spot-welded as w e l l as the c o n t i n u o u s l y welded tube. For the spot-welded tube, the higher the percentage w of tube l e n g t h 1 1Shown i n F i g u r e 3.4 47 being welded to the absorber p l a t e (from which heat flows i n through the s p o t s ) , the higher i s h ^ . Furthermore, f o r a given w and <j>0, d i s t r i b u t i n g the welding i n t o a l a r g e r number N of spots r e s u l t s i n a higher h ^ . T h i s i s because more spots i n t r o d u c e more thermal entrance e f f e c t . The e x p r e s s i o n (5.8) f o r the c o n t i n u o u s l y welded tube can be superposed to o b t a i n the N u ^ l x ) and hence h ^ f o r the spot-welded tube. F o l l o w i n g a s i m i l a r argument o u t l i n e d i n S e c t i o n 3.3, one o b t a i n s 1/Nu b(x) = .| N [1/Nu. b(x) ] , (5.9) where N u 1 b ( x ) i s given by equation (5.8), and 1/Nu 2 b(x) = - H U - x ^ / N u ^ l x - X j ) , 1/Nu 3 b(x) = H [ x - ( x 1 + x s ) ] / N u 1 b ( x - ( x l + x s ) ) , e t c . (5.10) I t should be noted that N u 1 b ( x ) i n equation (5.9) corresponds to the c o n t i n u o u s l y welded tube. To c a l c u l a t e the mean Nusselt number Nu, , the bm i n t e g r a t i o n i n equation (5.5) was computed using Simpson's O n e - t h i r d Rule. F i g u r e 5.3 shows Nu b(x) and N u ^ m ( x ) f o r the a r b i t r a r i l y chosen case of 4>0 = TI/'\§, W=60% and N=2, while F i g u r e 5.4 shows N u b m ( x ) f o r </>0 = 7 r/l 0, w=60% and N=8. Because of the d i s c o n t i n u o u s nature of N u b ( x ) , N u D r r / x ^ i n both cases possess abrupt f l u c t u a t i o n s . Nonetheless, the f l u c t u a t i o n s s t a b i l i z e downstream and N u b m ( x ) approaches an asymptotic 48 v a l u e . In r e a l i t y , the f l u c t u a t i o n s of N U j ^ x ) as w e l l as that of the temperature p r o f i l e s s t a b i l i z e at a s h o r t e r downstream d i s t a n c e than i s shown due to conduction i n the tube w a l l [19]. 5.4 EFFICIENCY FACTOR FOR VARIOUS SPOT CONFIGURATIONS The u l t i m a t e g o a l of dev e l o p i n g c l o s e d form s o l u t i o n s i l l u s t r a t e d i n the p r e v i o u s two s e c t i o n s was to p r e d i c t the lower l i m i t of the 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' of a spot-welded tube. T h i s s e c t i o n p r e s e n t s g r a p h i c a l i l l u s t r a t i o n s of F' f o r v a r i o u s spot c o n f i g u r a t i o n s . F i x i n g the tube inner r a d i u s a to be 4.5 mm, and the P e c l e t number to be 5060, 1 2 the tube l e n g t h of 2m corresponds to the di m e n s i o n l e s s d i s t a n c e x of 0.088. The N u s s e l t number averaged over the tube l e n g t h was found by e v a l u a t i n g N U f c r n / * ) at x = 0.088. The spot c o n f i g u r a t i o n i s completely determined by the h a l f - s p o t angle <t>0, welded percentage w of tube l e n g t h , and the t o t a l number N of spots along a tube. T h i s i s based on the assumption t h a t spot l e n g t h and spot s p a c i n g are i n v a r i a n t with p o s i t i o n f o r a p a r t i c u l a r set of <f>0, w and N. For a tube spacing W of 0.15 m, the value of Nu b m(.088) was p l o t t e d a g a i n s t N i n F i g u r e 5.5 f o r <t>0 = TT/10 ( i . e . spot angle = 36°) f o r v a r i o u s v a l u e s of w. The c o r r e s p o n d i n g F' was computed and was p l o t t e d i n F i g u r e 5.6. 1 2 S e e Appendix A 49 As expected, the e f f i c i e n c y f a c t o r F' decreases as w decreases, and the decrease i s f a s t e r when w i s s m a l l . The more i n t e r e s t i n g aspect i s the i n c r e a s e of F' with N. I t can a l s o be observed that i n c r e a s i n g N beyond about 40 does not i n c r e a s e F' s i g n i f i c a n t l y . 6 . DISCUSSION AND CONCLUSIONS The problem of a s e r i e s of spots of heat f l u x has been s o l v e d by two independent methods. The a n a l y t i c a l method and i t s r e s u l t s of t h i s problem have been d e s c r i b e d i n Chapter 3. The same problem has been so l v e d u s i n g a f i n i t e d i f f e r e n c e f o r m u l a t i o n with an i t e r a t i v e scheme (Chapter 4). A comparison of the two methods and t h e i r r e s u l t s i s a p p r o p r i a t e . 6.1 DISCUSSION OF ANALYTICAL AND NUMERICAL METHODS The a n a l y t i c a l s o l u t i o n to the " t h i n " spot-welded tube was obtained by superposing the a v a i l a b l e s o l u t i o n due to Bhattacharyya and Roy [15]. The energy equation (equ.(3.3)) that was to be s o l v e d i s l i n e a r . Furthermore, the a x i a l boundary c o n d i t i o n of t h i s problem assumes a f o r m 1 3 which i s a simple summation of H e a v i s i d e u n i t step f u n c t i o n s , s h i f t e d and i n v e r t e d along the x - a x i s . The s o l u t i o n can t h e r e f o r e be obtained by simply s h i f t i n g , i n v e r t i n g , and summing up a number of the same s o l u t i o n (the s o l u t i o n to the c o n t i n u o u s l y welded tube whose a x i a l boundary c o n d i t i o n c o n s i s t s of a s i n g l e u n i t step f u n c t i o n ) . To c a l c u l a t e the N u s s e l t numbers f o r the c o n t i n u o u s l y welded tube u s i n g Bhattacharyya and Roy's s o l u t i o n , only the w a l l temperature t ( l , # , x ) needs to be computed. T h i s i s because the bulk mean temperature i s simply given by 1 3 I l l u s t r a t e d i n F i g u r e 2.3 and d e p i c t e d by e q u a t i o n ( 2 . 1 0 ) . 50 51 t (x) = 4x . m The computation of t ( l,0,x) i n v o l v e d three i n f i n i t e s e r i e s which appeared i n equation (3.16) : oo ( i ) L [a mcosm0)/m] , where a m= 2sin (m0o )/(m<j>0 ) ; s I l c 0 s R 0 s { 1 ' / 3 0 s ) e x P ( ^ 0 s x ) ' mil s=0 C c m s R m s ( 1 ^ m s ) ^ m c o s m < t > ) e x P < ^ m s x ) ] • Convergence of the f i r s t s e r i e s was slow and sometimes u n c e r t a i n because of the f l u c t u a t i o n of the term cos(m</>) with m. Convergence was t e s t e d only a f t e r 200 terms had been i n c l u d e d i n the s e r i e s . Although the same problem a l s o o c c u r r e d i n the t h i r d s e r i e s , the e x p r e s s i o n i n square br a c k e t s decayed e x p o n e n t i a l l y with i n c r e a s i n g m and s and was i n s i g n i f i c a n t when m^s>l0. The number of terms r e q u i r e d by the second and t h i r d s e r i e s i n c r e a s e d when x was small ( i . e . x < 5x10 ~ 3 ) . The superposed s o l u t i o n f o r the spot-welded tube d i d not i n c r e a s e the computational time s i g n i f i c a n t l y once the s o l u t i o n f o r the c o n t i n u o u s l y welded tube had been s t o r e d as a f u n c t i o n of x. T h i s was the main advantage of the present method over d i r e c t l y a p p l y i n g Duhamel's s u p e r p o s i t i o n formula. I t should be p o i n t e d out th a t the a n a l y t i c a l s o l u t i o n i s r e l a t i v e l y easy c o m p u t a t i o n a l l y f o r the w a l l temperature because the eigenvalues and e i g e n f u n c t i o n s i n v o l v e d have 52 been t a b u l a t e d i n Bhattacharyya and Roy's paper, and can be r e a d i l y used. The temperature d i s t r i b u t i o n i n s i d e the tube ( i . e . r<l) has not been computed because those eigenvalues and e i g e n f u n c t i o n s f o r r<1 , which r e q u i r e much computation, are not t a b u l a t e d . On the other hand, the f i n i t e d i f f e r e n c e method aimed at o b t a i n i n g the temperature d i s t r i b u t i o n throughout the e n t i r e tube volume. The program developed was expensive to run i n terms of CPU time and memory space r e q u i r e d . T h i s i s because the boundary c o n d i t i o n i n v o l v e s v a r i a t i o n i n a l l three dimensions. The g r i d spacing had to be small and as a r e s u l t the number of nodes i n v o l v e d was very l a r g e (M =400, M =10, and M,=20 f o r a 2m tube), r <j> Computing the N u s s e l t numbers by d e f i n i t i o n (as i n equations (3.8) and (3.10)) and comparing with t h e i r a n a l y t i c a l c o u n t e r p a r t s o f f e r e d a check to the c o r r e c t n e s s of the numerical r e s u l t s , s i n c e the temperature i n the e n t i r e volume was i n v o l v e d i n the e x p r e s s i o n of the bulk mean temperature t . A p i c t u r e of the i n t e r n a l d i s t r i b u t i o n of temperature i n c r e a s e d the understanding of the 3-dimensional heat t r a n s f e r o c c u r r i n g i n s i d e the f l u i d . T h i s understanding was not obtained through the a n a l y t i c a l s o l u t i o n . N e v e r t h e l e s s , i t i s u n f a i r t o compare the computing c o s t r e q u i r e d by the two methods. T h i s i s because the e i g e n v a l u e s and e i g e n f u n c t i o n s r e q u i r e d by the a n a l y t i c a l s o l u t i o n were a l r e a d y given and i t s computational e f f o r t 53 cannot be accounted f o r . 6.2 RECOMMENDATIONS If m a t e r i a l cost i s not an important c o n s i d e r a t i o n i n d e s i g n i n g the f l a t p l a t e c o l l e c t o r , a t h i c k copper tube i s recommended f o r spot welding. P e r i p h e r a l and a x i a l c onduction i n s i d e the tube w a l l allows heat to be t r a n s f e r r e d i n t o the f l u i d throughout the e n t i r e tube w a l l ( F i g u r e 2 . l ( i ) ) , i n s t e a d of through the welding spots a l o n e . I t should be noted, however, that a t h i c k tube i s more prone to thermal s t r e s s e s . The mean N u s s e l t number f o r t h i s case approaches an upper l i m i t : Nu m f o r the uniform w a l l temperature case. T h i s mean N u s s e l t no. Nu m(.088) i s t a b u l a t e d i n Shah and London [3] to be 4.776, and corresponds to an e f f i c i e n c y f a c t o r F' of 0.883. The a c t u a l value of F' i s , of course, lower and depends on the t o t a l amount of welding being done, as w e l l as the t h i c k n e s s and c o n d u c t i v i t y of the tube w a l l . If a low m a t e r i a l c o s t i s d e s i r e d and the tube w a l l has to be t h i n and non-conductive, a t t e n t i o n should be p a i d t o the welding spot c o n f i g u r a t i o n . When 7/L ^ 0.0001 f o r R w^ 2x10" 7, or when 7/L. < 0.001 f o r R w> 10' 5, the heat t r a n s f e r from the f i n to the f l u i d can be thought of as being r e s t r i c t e d to the passage p r o v i d e d by the welding spots ( F i g u r e 2 . 1 ( i i ) ) . The mean N u s s e l t no. based on bond temperature N u ^ and the e f f i c i e n c y f a c t o r F' f o r t h i s case approach the lower l i m i t s given i n F i g u r e s 5.5 and 5.6, 54 r e s p e c t i v e l y . The a c t u a l value i s higher and i n c r e a s e s with the i n c r e a s e of ( 7 / L R w ) . The e f f i c i e n c y f a c t o r F' i s a l s o a st r o n g f u n c t i o n of the tube s p a c i n g , W, as i l l u s t r a t e d i n F i g u r e s 6.1 and 6.2, where the spot angle have been kept at 0O = TT/1O. In any case, spot welding should be done over a l a r g e percentage of the tube l e n g t h and of the c i r c u m f e r e n c e . T h i s becomes more important when the tube i s t h i n and non-conductive. T h i s , of course, has to be weighed a g a i n s t the c o s t of welding. Moreover, spot welding should be done over a l a r g e number of c l o s e l y separated, short spots i n s t e a d of a small number of widely separated, long s p o t s . The t o t a l number of spots should be over 40 per tube ( f o r a t y p i c a l tube l e n g t h of 2m). T h i s c o n s i d e r a t i o n , important f o r the t h i n and non-conductive tube, i s to be taken a g a i n s t the c o n t r o l l a b i l i t y of the welding mechanism, u s u a l l y an i n d u s t r i a l robot. F i n a l l y , i t should be p o i n t e d out that the spot c o n f i g u r a t i o n i s j u s t one f a c t o r determining the e f f i c i e n c y of the spot-welded s o l a r c o l l e c t o r . Other ' parameters, i n c l u d i n g o v e r a l l heat t r a n s f e r c o e f f i c i e n t U"L, p l a t e t h i c k n e s s , tube spacing and bond width, e t c , a f f e c t the spot-welded c o l l e c t o r as w e l l as the c o n v e n t i o n a l l y welded c o l l e c t o r . 55 6.3 CONCLUSIONS The o b j e c t i v e s of the present work have been achieved. Two i d e a l models f o r the heat t r a n s f e r process i n s i d e the spot-welded c o l l e c t o r tube were set up i n which the boundary c o n d i t i o n s c o u l d be formulated mathematically. A t t e n t i o n was given to the t h i n and non-conductive tube, r e s u l t s f o r which corresponded to the lower l i m i t of performance. An understanding of the d e t a i l e d heat t r a n s f e r phenomenon was obtained through the f i n i t e d i f f e r e n c e s o l u t i o n of the energy equation, which d e s c r i b e d the temperature d i s t r i b u t i o n i n s i d e the e n t i r e tube volume. The e f f e c t on 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 due to spots of heat input was i n v e s t i g a t e d u s i n g the a n a l y t i c a l s o l u t i o n . T h i s a n a l y t i c a l s o l u t i o n was obtained by superposing e x i s t i n g s o l u t i o n a v a i l a b l e i n the l i t e r a t u r e . The technique of s u p e r p o s i t i o n was demonstrated to be of great u s e f u l n e s s i n t h i s kind of l i n e a r problems. For the t h i n and non-conductive tube, an i n t e r e s t i n g f i n d i n g was that d i s t r i b u t i n g a given amount of welding over a l a r g e number of short spots r e s u l t e d i n a higher e f f i c i e n c y f a c t o r . R E F E R E N C E S 1. J.A. D u f f i e and W.A. Beckman : " S o l a r E n g i n e e r i n g of Thermal Processes", John Wiley & Sons, New York. (1980) 2. A. W h i l l i e r : "Thermal R e s i s t a n c e of the Tube-Plate Bond i n S o l a r Heat C o l l e c t o r s " , S o l a r E n e r g y Vol.8, No.3, pp95-98. (1964) 3. R.K. Shah and A.L. London : "Laminar Flow Forced Convection In Ducts", A d v a n c e s i n H e a t T r a n s f e r : S u p p . l , Academic P r e s s . (1978) 4. W.M. Kays and M.E. Crawford : "Convective Heat and Mass T r a n s f e r " , McGraw H i l l . (1966) 5. L. Graetz : "On the Thermal C o n d u c t i v i t y of L i q u i d s " , Part 1, A n n . P h y s . C h e m . Vol.18, pp79-94. (1883) 6. T.B. Drew : "Mathematical A t t a c k s On Forced Convection Problems : A Review", T r a n s . A m . I n s t . C h e m . E n g . Vol.26, pp26-80. (1931) 7. G.M. Brown : "Heat or Mass T r a n s f e r i n a F l u i d i n Laminar Flow i n a C i r c u l a r or F l a t Conduit", A m . I n s t . C h e m . E n g . J o u r n a l V o l . 6 , pp179-l83. (1960) 8. J.R. S e l l a r s , M. T r i b u s and J.S. K l e i n : "Heat T r a n s f e r to Laminar Flow i n a Round Tube or F l a t Conduit - the Graetz Problem Extended", T r a n s . A S M E Vol.78, pp441-448. (1956) 9. M.A. Leveque : "Les l o i s de l a t r a n s m i s s i o n de c h a l e u r par c o n v e c t i o n " , A n n . M i n e s . M e m . , S e r . 1 2 Vol.13, pp201-299,305-312,381-415. (1928) 10. A.McD. Mercer : "The Growth of the Thermal Boundary Layer a t the I n l e t to a C i r c u l a r Tube", A p p . S c i . R e s . , S e c t . A V o l . 9 , pp450-456. (1960) 11. P.M. Worsoe-Schmidt : "Heat T r a n s f e r i n the Thermal Entrance Region of C i r c u l a r Tubes and Annular Passages with F u l l y Developed Laminar Flow", I n t . J . H e a t M a s s T r a n s f e r Vol.10, pp541-551. (1967) 12. J . Newman : "Extension of the Leveque S o l u t i o n " , J . H e a t T r a n s f e r , T r a n s . A S M E Vol.90, pp361-363. (1969) 13. U. G r i g u l l and H. T r a t z : "Thermischer e i n l a u f i n a u s g e b i l d e t e r laminarer rohrstromung", I n t . J . H e a t M a s s T r a n s f e r V o l . 8 , pp669-678. (1965) 56 57 14. W.C. Reynolds: "Heat T r a n s f e r t o F u l l y Developed Laminar Flow i n a C i r c u l a r Tube w i t h A r b i t r a r y C i r c u m f e r e n t i a l Heat F l u x " , J . Heat T r a n s f e r , Trans. ASME Vol.82, ppl08-111. (1960) 15. T.K. Bhattacharyya and D.N. Roy : "Laminar Heat T r a n s f e r in a Round Tube with V a r i a b l e C i r c u m f e r e n t i a l or A r b i t r a r y Wall Heat F l u x " , I n t . J . Heat Mass T r a n s f e r , Vol.13, ppl057-l060. (1963) 16. A.V. Luikov, V.A. Alekasashenko and A.A. Alekasashenko : " A n a l y t i c a l Methods of S o l u t i o n of Conjugated Problems in Convective Heat T r a n s f e r " , Int. J . Heat Mass T r a n s f e r , Vol.14, ppl047-l056. (1971) 17. S. Mori, M. Sakakibara and A. Tanimoto : "Steady Heat T r a n s f e r to Laminar Flow i n a C i r c u l a r Tube with Conduction i n the Tube W a l l " , Heat T r a n s f e r - Japanese Research V o l . 3 ( 2 ) , pp37-46. (1974) 18. M. F a g h r i , E.M. Sparrow : "Simultaneous Wall and F l u i d A x i a l Conduction i n Laminar Pipe-flow Heat T r a n s f e r " , J . Heat T r a n s f e r , Trans. ASME Vol.102, pp58-63. (1980) 19. G.S. B a r o z z i and G. P a g l i a r i n i : "Conjugated Heat T r a n s f e r i n a C i r c u l a r Duct with Uniform and Non-uniform Wall T h i c k n e s s " , Heat and Technology Vol.2, pp72-89. (1984) 20. L.C. Burmeister : "Convective Heat T r a n s f e r " , Wiley I n t e r s c i e n c e . (1983) 21. C.J. Hsu : "An Exact A n a l y s i s of Low P e c l e t number Thermal Entry Region Heat T r a n s f e r i n T r a n s v e r s e l y Non-uniform V e l o c i t y F i e l d s " , Am.Inst.Chem.Eng. J o u r n a l , Vol.17, No.3, pp732-740. (1971) 58 Black absorber plate Outer cover Inner cover -U 0 0 < •> 0 0 1L r Insulation Fluid conduit Collector box FIG 1.1 Cross s e c t i o n of a b a s i c f l a t p l a t e s o l a r c o l l e c t o r 5 9 FIG. 1 . 2 . Energy flow i n an o p e r a t i n g s o l a r c o l l e c t o r . FIG 1.3 Cross s e c t i o n of p l a t e and tube arrangement 61 ( i i ) T u b e embedded i n trough formed by p l a t e FIG. 1.4. Common ways of p l a t e - t u b e bonding. ( i i i ) P l a t e i n c l u d e s conduit w i t h i n i t s e l f (iv)Tube i s welded onto p l a t e at separated spots FIG 1.4 Common ways of p l a t e - t u b e bonding 63 Laminar flow Hydrodynamic entrance region Le FIG 1 . 5 Hydrodynamic and thermal entry l e n g t h s 64 FIG 1.6 P h y s i c a l s i t u a t i o n of Graetz problem FIG 1.7 L i n e a r v e l o c i t y p r o f i l e assumed in Leveque method Designation Description Applications © © © Constant wall temperature peripherally as well as axially Constant axial wall temperature with finite normal wall thermal resistance Nonlinear radiant-flux boundary condition Constant axial wall heat flux with constant peripheral wall temperature Constant axial wall heat flux with uniform peripheral wall heat flux Condensers, evaporators, automotive radiators (at high flows), with negligible wall thermal resistance Same as those for (f) with finite wall thermal resistance Radiators in space power systems, high-temperature liquid-metal facilities, high-temperature gas flow systems Same as those for (H4)for highly conductive materials Same as those for(H4)for very low-conductive materials with the duct having uniform wall thickness © © © Constant axial wall heat flux with finite normal wall thermal resistance Constant axial wall heat flux with finite peripheral wall heat conduction Exponential axial wall heat flux Constant axial wall to fluid bulk temperature difference Same as those for(hT) with finite normal wall thermal resistance and negligible peripheral wall heat conduction Electric resistance heating, nuclear heating, gas turbine regenerator, counterflow heat exchanger with C m i „ . C m a < - 1. all with negligible normal wall thermal resistance Parallel and counterflow heat exchaneers Gas turbine reeenerator Table 2.1 Thermal boundary c o n d i t i o n s f o r developed a dev e l o p i n g flows through s i n g l y connected ducts ( i ) A t h i c k and conductive w a l l ( i i ) A t h i n and non-conductive w a l l FIG 2.1 Two extreme cases of thermal boundary c o n d i t i o n s 67 f ( 4 > ) -to *o 2TT FIG 2.2 P e r i p h e r a l d i s t r i b u t i o n of wall heat f l u x q(x) Qsol x i + x* 3 X j + ^x7 FIG 2.3 A x i a l d i s t r i b u t i o n of w a l l heat f l u x 68 FIG 3.2 Spot welded tube with 2 spots occupying 60% of i t s l e n g t h 69 Lf) 1000X/ (a .Pe ) FIG. 3 . 3 Dimensionless welded tube dimensionless wall temperature of continuously with spot angle 45° versus a x i a l distance 1000x=1OOOX/a•Pe. 70 FIG. 3.4. L o c a l N u s s e l t no. Nu and p e r i p h e r a l average N u s s e l t no. Nup of c o n t i n u o u s l y welded tube v e r s u s d i m e n s i o n l e s s a x i a l d i s t a n c e l000x=l000X/a«Pe. 71 lOOOXAa.Pe) FIG. 3.5. Dimensionless w a l l temperature versus dimensionless a x i a l d i s t a n c e 1000X lOOOX/a.Pe of a spot welded tube with 2 spots occupying 60% of i t s l e n g t h , spot angle 45°. 72 CD _ , C\J CD C O • ZD zz. O J cn ro t _ CM' — O J > cn — i ro !_ CD O J OD J= tZL — 1 S_ aj Q _ CD 30.0 r 36.0 I ©1 1 42.0 48.0 0.0 6.0 12.0 18.0 24.0 1000X / a .Pe FIG. 3 . 6 ( i ) . P e r i p h e r a l average Nusselt no. Nup versus dimensionless a x i a l d i s t a n c e 1000x=1OOOX/a«Pe f o r a spot-welded tube with a s i n g l e spot occupying 60% of i t s l e n g t h , spot angle 45°. 73 o * CD __ CM CD IQOuX/ia.Pe) FIG. 3 . 6 ( i i ) . P e r i p h e r a l average Nusselt no. Nu p versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x=1OOOX/a•Pe f o r a spot-welded tube with 4 spots occupying 60% of i t s l e n g t h , spot angle 45°. 74 FIG 4.1 G r i d d i v i s i o n of tube volume. 75 FIG 4.2 Node p o s i t i o n s appeared i n f i n i t e d i f f e r e n c e energy equation 76 0.0 4.0 8.0 12.0 16.0 20.0 24.0 28.0 32.0 36.0 40.0 44.0 48.0 1000X/a.Pe FIG. 4.3. Dimensionless temperature t of a u n i f o r m l y heated tube versus dimensionless a x i a l d i s t a n c e lOOOx = lOOOX/a-Pe at v a r i o u s r a d i a l d i s t a n c e s : f i n i t e d i f f e r e n c e r e s u l t s . 77 FIG. 4.4. L o c a l N u s s e l t no. Nu of a uniformly heated tube versus dimensionless a x i a l d i s t a n c e lOOOx = lOOOX/a-Pe : f i n i t e d i f f e r e n c e r e s u l t s . FIG. 4.5. Angular p o s i t i o n s represented by v a r i o u s curves i n F i g u r e s 4.6 - 4.9. 79 CD LT) i o o cx / (a .Pe ) FIG. 4 . 6 . Dimensionless w a l l temperature versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1 0 0 0 X l 0 0 0 X / a « P e : f i n i t e d i f f e r e n c e r e s u l t s f o r a c o n t i n u o u s l y welded tube with spot angle 4 5 ° . 80 CD rsi a. frj-loooX/(a .Pe) FIG. 4.7. Dimensionless w a l l temperature versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l000X/a-Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a spot-welded tube with 2 spots occupying 60% of i t s l e n g t h , spot angle 45°. 81 CD FIG. 4.8. D i m e n s i o n l e s s t e m p e r a t u r e i n s i d e the tube a t r a d i a l c o - o r d i n a t e r=4/5 v e r s u s d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x=1000X/a»Pe f o r a spot-we l d e d tube w i t h 2 s p o t s o c c u p y i n g 60% of i t s l e n g t h , spot a n g l e 45°. 82 CD CM ro CD' QJ 1000X/(a.Pe) FIG. 4.9. Dimensionless temperature i n s i d e the tube at r a d i a l c o - o r d i n a t e r=l/5 versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000X = l000X/a«Pe f o r a spot-welded tube w i t h 2 spots occupying 60% of i t s l e n g t h , spot angle 45°. "1 1 1 1 1 1 1 1 1 1 1—•—1 1 1 1 1 1 — 0.0 5.33 10.66 16.0 21.33 26.66 32.0 37.33 42.66 lOOOX/ia.Pe) FIG. 4 . 1 0 ( i ) . P e r i p h e r a l average N u s s e l t no. Nup versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x = 1000X/a«Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a spot-welded tube with a s i n g l e spot occupying 60% of i t s l e n g t h , spot angle 45°. 84 FIG. 4 . l O ( i i ) . P e r i p h e r a l average N u s s e l t Nup v e r s u s d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x=1DOOX/a•Pe : f i n i t e d i f f e r e n c e r e s u l t s f o r a s p o t - w e l d e d tube w i t h 4 s p o t s o c c u p y i n g 60% of i t s l e n g t h , spot a n g l e 45°. 85 FIG 5.1 Bond temperature T.(X) approximated by T(a,0,X) 86 CD l£> — i FIG. 5.2. N u s s e l t no. based on bond temperature Nu D versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x=1OOOX/a•Pe f o r a c o n t i n u o u s l y welded tube of v a r i o u s h a l f - s p o t a ngles 4>0 87 Cxi FIG. 5.3. N u s s e l t no. based on bond temperature Nu^ and mean Nu s s e l t no. based on bond temperature N u ^ versus d i m e n s i o n l e s s a x i a l d i s t a n c e 1000x=l000X/a*Pe f o r a spot-welded tube with 2 spots occupying 60% of i t s l e n g t h , spot angle 36°. 88 lOOOX/a.Pe FIG. 5.4. Mean N u s s e l t no. based on bond temperature N u D m versus d i m e n s i o n l e s s a x i a l d i s t a n c e l000x=l000X/a«Pe f o r a spot-welded tube with 8 spots occupying 60% of i t s l e n g t h , spot angle 36°. 89 FIG. 5.5. Mean Nu s s e l t no. based on bond temperature Nu b r n over a d i m e n s i o n l e s s tube l e n g t h of .088 as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a spot-welded tube with spot angle of 36°. 90 CD ro No. of S t i t ches , N . 5.6. E f f i c i e n c y F a c t o r F' as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a c o l l e c t o r with d i s t a n c e W between i t s spot-welded tubes of ,15m and spot angle of 36°. 0.0 4.0 8.0 12.0 16.0 20.0 24.0 No. of S t i tches . N 28.0 32.0 FIG. 6.1. E f f i c i e n c y F a c t o r F* as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a c o l l e c t o r with d i s t a n c e W between i t s spot-welded tubes of ,1m and spot angle of 36°. 92 . 6.2. E f f i c i e n c y F a c t o r F' as a f u n c t i o n of spot c o n f i g u r a t i o n s f o r a c o l l e c t o r w i t h d i s t a n c e W between i t s s p o t - w e l d e d tubes of ,2m and spot a n g l e of 36°. APPENDIX A S p e c i f i c a t i o n s of a T y p i c a l Two-cover S o l a r C o l l e c t o r C o l l e c t o r Dimensions Tube inner r a d i u s , a = 4.5x1U"3m Cross s e c t i o n a l area of tube, A = 6.36xl0 _ 5m 2 Tube l e n g t h , L = 2m C o l l e c t o r area = 2m x 1m Absorber p l a t e t h i c k n e s s , 6 = 2.54xl0-"m Tube w a l l t h i c k n e s s , 7 = 5xl0 _ < tm D i s t a n c e between tubes, W = 0.15m F l u i d P r o p e r t i e s (50°C) S p e c i f i c heat c a p a c i t y , c^= 4.174 kJ/kg°C Thermal c o n d u c t i v i t y , k = 0.644 W/m°C Thermal d i f f u s i v i t y , a = 1.561x10 - 7 m2/s D e n s i t y , p = 988.8 kg/m3 Dynamic v i s c o s i t y , n = 5.62x10""kg/ms Kinematic v i s c o s i t y , v = 5.68x10" 7m 2/s P r a n d t l no., Pr = v/a = 3.64 Other Parameters Coolant flow r a t e per tube, m = 5.55x10~ 3kg/s Mean flow speed, u = m/pA = 0.088 m/s Reynolds no. of c o o l a n t flow, Re = 1390 93 P e c l e t ' s no. of c o o l a n t flow, Pe = RePr = 5060 Thermal c o n d u c t i v i t y of p l a t e and tube m a t e r i a l ( c o p p e r ) , k = k = 385 W/m°C s w C o l l e c t o r o v e r a l l heat t r a n s f e r c o e f f i c i e n t , U T = 4 W / m 2 ° C APPENDIX B E f f e c t of F l u i d A x i a l Conduction F l u i d a x i a l conduction has not been accounted f o r i n the a n a l y s i s as i t s r e p r e s e n t a t i v e term 9 2 t / 3 x 2 was omitted from the energy equation at the beginning. T h i s term i s important only when P e c l e t No. i s small(Pe<100). In any case, i t s e f f e c t i s obvious only at a s h o r t d i s t a n c e from the step change in thermal boundary c o n d i t i o n , as can be seen from the r e s u l t s of H S U [ 2 1 ] f o r the u n i f o r m l y heated tube. Taking a x i a l conduction i n t o account, the N u s s e l t no. Nu(x) obtained f o r X<10" 2 i s lower than that obtained with a x i a l conduction n e g l e c t e d (which corresponds to the case where Pe=<=°) , and t h i s d i screpancy i n c r e a s e s as x decreases. For Pe=5000, an a x i a l d i s t a n c e X of one tube r a d i u s corresponds to x = X/a-Pe ~ 1/5000 = 2x10-". Since the superposed s o l u t i o n , i . e . Nu(x), f o r the " t h i n " spot-welded tube makes use, along the e n t i r e flow l e n g t h , of v a l u e s c o r r e s p o n d i n g to very small x, the e f f e c t of f l u i d a x i a l conduction has caused concerns, e s p e c i a l l y when the spot l e n g t h i s of the same order of magnitude as the r a d i u s . However, the e r r o r can be shown to be i n s i g n i f i c a n t . Let Nu(x) and Nu c o(x) represent the N u s s e l t no. d i s t r i b u t i o n of a c o n t i n u o u s l y welded tube with Pe<°° and Pe=w r e s p e c t i v e l y , and Nu (x) and Nu (x) represent the 95 96 N u s s e l t no. d i s t r i b u t i o n of a spot-welded tube with Pe<°° and Pe=<» r e s p e c t i v e l y . A l s o l e t 1/Nu(x) - 1/Nu (x) = e(x) , then 1/Nu s(x) = 1/Nu(x) - H(x-x^)/Nu(x-x^) + H(x-x^-x s)/Nu(x-x 1-x s) - .... = 1/Nu (x) + e(x) CO - H(x-x 1) [ 1 / N U O O ( X - X 1 ) + e ( x - x 1 ) ] + H(x-x,-x)[1/Nu(x~x,-x ) + e(x-x.-x )] - ... i s i s i s = 1 / N U S ( X ) + e(x) - e(x-x,)H(x-x,) + co X I e ( x - x 1 - x s ) H ( x - x 1 - x s ) - .... Thus, the e r r o r 1/Nu s(x) - l/Nu^U) = e(x) - e(x-x,)H(x-x,) + e(x-x,-x )H(x~x,-x ) - .... 1 1 I s I s When x i s s m a l l , e(x) i s small s i n c e both Nu (x) and Nu (x) CO are l a r g e . When x i s l a r g e (X>10~ 2), Nu approaches Nu^ and e(x) approaches zero . Furthermore, the i n d i v i d u a l terms i n the e r r o r s e r i e s have a c a n c e l l a t i o n e f f e c t on each o t h e r , so that there i s no accumulating e r r o r i n v o l v e d i n the superposed s o l u t i o n . In c o n c l u s i o n , the d i s c r e p a n c y between Nu and Nu o(x) e x i s t s not only at small x but along the whole tube l e n g t h . However, the l a r g e P e c l e t no. of 5060 i n the present case renders that d i s c r e p a n c y n e g l i g i b l e . APPENDIX C E x p r e s s i o n of S( V V ) and $ i n C a r t e s i a n Coordinates The s t r e s s tensor : / S = s s s \ xx xy xz \ s s s yx yy yz \ s s s /, \ zx zy zz /' where s ^ x denotes the s t r e s s a c t i n g along the x d i r e c t i o n on the s u r f a c e normal to the y - a x i s . The V e l o c i t y Deformation Tensor : V V = 9u/9x 9u/9y 9u/9z^ 9v/9x 9v/9y 9v/9z 9w/9x 9w/9y 9w/9z where u, v, w are the x-, y-, z-component, r e s p e c t i v e l y , of the v e l o c i t y V . 97 The Complete C o n t r a c t i o n : S( V V ) = + s x x 3u/3x + s y x 3u/3y + s z x 3u/3z + s x y 3v/3x + s y y 3v/3y + s z y 3v/3z + s 3w/3x + s„_ 3w/3y + s „ 3w/3z The D i s s i p a t i o n F u n c t i o n : * = 2 [ ( 3 u / 3 x ) 2 + (9v / 3 y ) 2 + (3w/3z) 2 ] - (2/3)( V • V ) 2 + ( 3u/3y + 3v/3x ) 2 + ( 3v/3z + 3w/3y ) 2 + ( 3w/3x + 3u/3z ) 2 

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