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The use of an approximate integral method to account for intraparticle conduction in gas-solid heat exchangers Riahi, Ardeshir 1985

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THE USE OF AN APPROXIMATE INTEGRAL METHOD TO ACCOUNT FOR INTRAPARTICLE..CONDUCTION ''IN GAS-SOLID HEAT EXCHANGERS. by ARDESHIR RIAHI B.Sc (Hons) , U n i v e r s i t y of Wales ,U.K. 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in FACULTY OF GRADUATE STUDIES Department of Mechanical E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard UNIVERSITY OF BRITISH COLUMBIA APRIL 1985 © ARDESHIR RIAHI, 1985 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r re f e r e n c e and study. I f u r t h e r agree that p e r m i s s i o n f o r ex t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of Mechanical E n g i n e e r i n g U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date: APRIL 1985 ABSTRACT The m a t h e m a t i c a l e q u a t i o n s d e s c r i b i n g t r a n s i e n t h e a t t r a n s f e r b e t w e e n t h e f l u i d f l o w i n g t h r o u g h a f i x e d bed and a m o v i n g b e d o f p a c k i n g were f o r m u l a t e d . The r e s i s t a n c e t o h e a t t r a n s f e r w i t h i n t h e p a c k i n g due t o i t s f i n i t e t h e r m a l c o n d u c t i v i t y was t a k e n i n t o a c c o u n t . An a p p r o x i m a t e i n t e g r a l method was a p p l i e d t o o b t a i n an a n a l y t i c a l s o l u t i o n t o t r a n s i e n t r e s p o n s e o f t h e bed p a c k i n g . R e s u l t s f o r two c a s e s o f f i x e d a n d m o v i n g b e d were o b t a i n e d . The v a l i d i t y o f t h e a p p r o x i m a t e method was c h e c k e d a g a i n s t t h e more e x a c t method e m p l o y e d by H a n d l e y a n d Heggs who o b t a i n e d t h e r e s u l t s f o r a f i x e d bed o f p a c k i n g w i t h a s t e p c h a n g e i n f l u i d i n l e t t e m p e r a t u r e . I t was c o n c l u d e d t h a t t h e a p p r o x i m a t e mehod g i v e s r e s u l t s t h a t a g r e e w e l l w i t h t h e more e x a c t m e t h o d s . The method c o n s i d e r e d h e r e p r o v i d s a q u i c k d e t e r m i n a t i o n o f t h e p a c k i n g mean t e m p e r a t u r e i n o r d e r t o o b t a i n t h e e f f e e t i v e n e s s . T h e o t h e r p e c u l i a r i t y o f t h i s m ethod i s t h a t t h e e f f e c t o f p a c k i n g t h e r m a l c o n d u c t i v i t y c a n be e x a m i n e d v e r y q u i c k l y s i n c e t h e s o l u t i o n i s i n a n a l y t i c a l f o r m . The a n a l y s i s o f t h e r e s u l t s r e v e a l e d t h a t a s t h e t h e r m a l c o n d u c t i v i t y o f t h e p a c k i n g d e c r e a s e s t h e d i f f e r e n c e b e t w e e n i t s s u r f a c e and mean t e m p e r a t u r e i n c r e a s e s . A s e r i e s o f c h a r t s s h o w i n g t h e c o m p a r i s o n b e t w e e n t h e p a c k i n g s u r f a c e a n d mean t e m p e r a t u r e s f o r d i f f e r e n t t h e r m a l c o n d u c t i v i t i e s a r e p r e s e n t e d . i i The approximate method was a moving bed of packing. I t was packing thermal c o n d u c t i v i t y i s s e r i e s of c h a r t s r e p r e s e n t i n g versus dimensionless l e n g t h c o n d u c t i v i t i e s are presented. then a p p l i e d to the case of concluded that the e f f e c t of more severe than expected. A the moving bed e f f e c t i v e n e s s for d i f f e r e n t thermal Table of Contents ABSTRACT .'. . i i L i s t of Tables . . . . v i L i s t of F i g u r e s ........ . . . . . . . . . . . . . . . v i i Acknowledgements . . . . i x Nomenclature x 1. I n t r o d u c t i o n ...1 1 . 1 General 1 1.1.1 S t a t i o n a r y matrix 7 1.1.2 Moving matrix 10 1.2 Review of previous work 15 1.2.1 Schumann model 16 1.2.2 I n t r a c o n d u c t i o n model 17 1.3 Scope of the present i n v e s t i g a t i o n 21 2. The Governing Equations 22 2.1 Dimensionless parameters 22 2.2 The mathematical models 26 2.2.1 Schumann model 27 2.2.2 I n t r a p a r t i c l e Conduction Model 30 2.3 Non-Dimensional form of governing equations 33 3. The method of s o l u t i o n 36 3.1 I n t r o d u c t i o n to t h e l n t e g r a l method 36 3.2 Planar geometry 40 3.2.1 S e m i - i n f i n i t e s l a b 40 3.2.2 Slab of f i n i t e t h i c k n e s s 43 3.3 S p h e r i c a l geometry 45 3.3.1 Sphere of i n f i n i t e r a d i u s 46 3.3.2 Sphere of f i n i t e r a d i u s 48 i v 3.4 Temperature-Dependent thermal p r o p e r t i e s 50 3 . 5 Numer i c a l procedure 51 3.5.1 Fixed bed ... 51 3.5.2 Moving bed regenerator 56 4. RESULTS AND DISCUSSION ..60 4. 1 F i x e d bed ... . .60 4.1.1 S p h e r i c a l geometry ..60 4.1.2 Planar geometry 61 4.2 Moving bed regenerator 63 5. C o n c l u s i o n 67 6. Areas of f u r t h e r r e s e a r c h 68 BIBLIOGRAPHY 84 APPENDIX A:DERIVATION AND DIMENSIONAL ANALYSIS ...87 APPENDIX B:INTEGRAL METHOD 101 APPENDIX C: EFFECTIVENESS COMPUTATION 113 APPENDIX D :SAMPLE CALCULATION 119 APPENDIX E :THE COMPUTER PROGRAM 126 v L i s t of Tables 1 . C o r r e l a t i o n s f o r the 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 . . . . . 92 v i L i s t of F i g u r e s 1. Conductance heat exchanger. 2 2. Schematic arrangement of an MHD duct 3 3. Types of a i r heaters 8 4. F i x e d bed regenerator. 9 5. Rotary regenerator flow arrangement...... 12 6. F a l l i n g c l o u d regenerator 13 7. Moving pebble bed regenerator 14 8. E f f e c t of s o l i d thermal c o n d u c t i v i t y on the temperature prof i l e 19 9. T y p i c a l dimensions of a regenerator 23 10. Schematic r e p r e s e n t a t i o n of p e n e t r a t i o n depth concept.38 11. Comparison between the numerical and. a n a l y t i c a l method employed to s o l v e the d i f f u s i o n equation (Bi=.02) 69 12. Comparison between the numerical and a n a l y t i c a l method employed to s o l v e the d i f f u s i o n equation (Bi=.25) 70 13. Comparison between the numerical and a n a l y t i c a l method employed to s o l v e the d i f f u s i o n equation (Bi = 2) 71 14. The c h a r a c t e r i s t i c S shaped curves of f l u i d o u t l e t temperature p r o f i l e f o r a f i x e d bed regenerator 72 15. The e f f e c t of thermal c o n d u c t i v i t y on the s o l i d temperature p r o f i l e (Bi=.02) 73 16. The e f f e c t of thermal c o n d u c t i v i t y on the s o l i d temperature p r o f i l e (Bi = .25) 74 17. The e f f e c t of thermal c o n d u c t i v i t y on the s o l i d temperature p r o f i l e (Bi = 2) 75 18. The e f f e c t of thermal c o n d u c t i v i t y on the s o l i d v i i t e m p e r a t u r e p r o f i l e i n a m o v i n g bed .76 19. The e f f e c t o f t h e r m a l c o n d u c t i v i t y on r e g e n e r a t o r e f f e c t i v e n e s s . 77 20. The c o m p a r i s o n b e t w e e n t h e o r i g i n a l p r o f i l e a n d t h e a l t e r n a t i v e p r o f i l e ( B i = 2) 78 21. The c o m p a r i s o n between t h e e f f e c t i v e n e s s b a s e d on t h e two p r o f i l e s ( B i = 2) 79 22. M o v i n g bed r e g e n e r a t o r e f f e c t i v e n e s s b a s e d on t h e a l t e r n a t i v e p r o f i l e ( B i = 0 . 1 , 0 . 5 , 1 , 2 ) 80 23. M o v i n g bed r e g e n e r a t o r e f f e c t i v e n e s s b a s e d on t h e a l t e r n a t i v e p r o f i l e ( B i = 4 , 8 ) 81 24. M o v i n g b e d r e g e n e r a t o r e f f e c t i v e n e s s b a s e d on t h e a l t e r n a t i v e p r o f i l e ( B i = 0 . 1 , l 0 ) 82 25. The e f f e c t o f c a p a c i t y r a t e r a t i o on e f f e c t i v e n e s s . . . . 8 3 v i i i Acknowledgements The author wishes to express h i s a p p r e c i a t i o n towards Prof.E.G.Hauptmann for a l l h i s h e l p and a d v i c e . The author would a l s o l i k e to thank Miss A f a r i n . R a d j a i e ( f o r her moral support) and Mr I r a j . R i a h i ( f o r h i s f i n a n c i a l s u p p o r t ) . T h i s work i s d e d i c a t e d to my p a r e n t s . ix Nomenclature 2 A = S o l i d s u rface area per u n i t bed volume m /m A, = Bed c r o s s - s e c t i o n a l area m b B = P o r o s i t y Bi = B i o t number=hR/K s C = F l u i d s p e c i f i c heat c a p a c i t y at constant pressure J/kg K = S o l i d s p e c i f i c heat c a p a c i t y J/kg K d = Matrix semi-thickness m D g v= Eq u i v a l e n t s p h e r i c a l diameter m F = Normalised f l u i d temperature (T -T. ) / ( T _ - T . ) f 1 f 1 1 2 f ( 0 ) = P r e s c r i b e d heat f l u x at the s o l i d s u r f a c e W/m h = Convective heat t r a n s f e r c o e f f i c i e n t W/m K i = Number of time steps K = Matrix thermal c o n d u c t i v i t y W/m K s y = Distance from the bed entrance m L = Bed length m 3 M = Bed d e n s i t y kg/m s 2 m = f l u i d mass flow r a t e / area kg/m s . 2 mg = S o l i d mass flow r a t e / area kg/m s n = Number of l e n g t h steps along the bed Nu = Nusselt number=hd/K s P = P e r i o d of f l u i d flow s Pr = P r a n d t l number Q = Heat t r a n s f e r J r = Distance from the c e n t r e of sphere m R Radius of sphere x Re = R e y n o l d s number T = S o l i d t e m p e r a t u r e K = S o l i d s u r f a c e t e m p e r a t u r e K T^ = F l u i d t e m p e r a t u r e K u = F l u i d v e l o c i t y m/s u = S o l i d v e l o c i t y m/s s - 3 V, = Bed volume m b x = D i s t a n c e from t h e s u r f a c e of t h e s l a b m z = D i m e n s i o n l e s s t h i c k n e s s or r a d i u s G r e e k symbols a = T h e r m a l d i f f u s i v i t y m / s 5 = P e n e t r a t i o n d e p t h m 6 0 = D i m e n s i o n l e s s p e n e t r a t i o n d e p t h -e = E f f e c t i v e n e s s -V = D i m e n s i o n l e s s t ime=hA(0-y/u)/M C s s -e = Time from t h e s t a r t o f t h e o p e r a t i o n s A = D i m e n s i o n l e s s bed l e n g t h = £ 3 y = L V = K i n e m a t i c v i s c o s i t y m / s I = D i m e n s i o n l e s s d i s t a n c e a l o n g t h e bed=hAy/mC n = D i m e n s i o n l e s s p e r i o d = VQ_P pt = F l u i d mass d e n s i t y kg/m p s = S o l i d mass d e n s i t y kg/m^ * = N o r m a l i s e d s o l i d t e m p e r a t u r e = ( T - T ) / ( T f ^ -T. ) -I x i S u b s c r i p t s f = F l u i d i = I n l e t or i n i t i a l m = Mean 0 = O u t l e t s = s u r f a c e < x i i 1. INTRODUCTION 1 . 1 GENERAL Heat exchange between two f l u i d streams at d i f f e r e n t temperatures represents an important t e c h n o l o g i c a l process in many branches of i n d u s t r y . There are e s s e n t i a l l y two types of heat exchangers [ 1 ] , 1 1. Recuperators or conductance heat exchangers,in which the thermal c o n d i t i o n s are assumed to be i n v a r i a n t with time.Thus the r a t e s of heat flow are steady,with the con v e c t i o n from the h o t t e r f l u i d c o n t i n u o u s l y equal to the c o n v e c t i o n to the c o l d e r f l u i d and both equal to the steady r a t e of conduction through the s e p a r a t i n g heat t r a n s f e r surface ( F i g . 1 ) . 2. Capacitance heat exchangers or regenerators which make use of the thermal c a p a c i t y of the heat t r a n s f e r s u r f a c e . Regenerators are used e x t e n s i v e l y as a i r heaters.As an example c o n s i d e r MHD (Magnetohydrodynamic) power genera t i o n which uses the i n t e r a c t i o n of an e l e c t r i c a l l y conducting f l u i d with a magnetic f i e l d to convert part of the energy of the f l u i d d i r e c t l y i n t o e l e c t r i c i t y ( F i g . 2 ) [ 2 ] . 1 numbers i n square bracket r e f e r to the B i b i 1 i o g r a p h y 1 2 1- HOT FLUID 2- COLD FLUID DISTANCE FIG.1(a) A ® CD HOT FLUID FIG.1(b) (a) C h a r a c t e r i s t i c temperature d i s t r i b u t i o n Figure . 1 (b)"Conductance heat exchanger (Counter flow) COLD FLUID 3 B-MAGNETIC FIELD I - CURRENT L - LOAD Figure 2. Schematic arrangement of an MHD duct. 4 In the f o s s i l f i r e d MHD system, the temperature of the f l u i d (combustion a i r ) at the magnetic f i e l d . i n l e t , which determines the c o n d u c t i v i t y and hence the power d e n s i t y at that p o i n t , depends on the performance of the combustion chamber. For optimum performance , i t i s e s s e n t i a l to preheat 0 the combustion a i r to a temperature greater than 1200 C. T h i s i s achieved by u t i l i z i n g r e g e n e r a t i v e a i r p r e h e a t e r s . Another area in which regenerators are f i n d i n g i n c r e a s i n g a p p l i c a t i o n i s as exhaust heat exchanger i n small gas t u r b i n e s [ 3 ] , I n these engines , i t i s e s s e n t i a l to o b t a i n a l a r g e heat t r a n s f e r area w i t h i n the s m a l l e s t p o s s i b l e volume.This r e q u i r e s passages of small diameter and the l a r g e number of these can be very c o n v e n i e n t l y i n c o r p o r a t e d in a r e generator. The p e c u l i a r i t y of a regenerator i s that the heat t r a n s f e r media ( c a l l e d the matrix) i s a l t e r n a t i v e l y heated by one gas and cooled by the o t h e r . T h i s means that e i t h e r the matrix must be moved p e r i o d i c a l l y in and out of the gases or that the matrix be a l t e r n a t i v e l y swept by the h o t t e r f l u i d , when i t absorbs heat, and then by the c o l d e r f l u i d to which the heat absorbed i s then returned.The former i s c a l l e d a c o n t i n i u o u s l y o p e r a t i n g and the l a t t e r i s c a l l e d an i n t e r m i t t e n t regenerator, r e s p e c t i v e l y . There are two main problems a s s o c i a t e d with the mechanical design of these two types of regenerators [ 1 ] : 5 1. P r o v i d i n g means f o r changing e i t h e r the gas flow or the matrix p e r i o d i c a l l y to enable the l a t t e r to be heated . and c o o l e d by c o n t a c t with both gases in turn and thus to exchange heat between them. 2. S e a l i n g the two gas flows b e f o r e , d u r i n g and a f t e r the exchange of heat to prevent e x c e s s i v e leakage. Problem 1 i s e a s i l y solved i n the case of c o n t i n u o u s l y o p e r a t i n g r e g e n e r a t o r s , but problem 2 i s more d i f f i c u l t to s o l v e . I n the case of i n t e r m i t t e n t regenerators both problems can be e a s i l y avoided. One of the major problems a s s o c i a t e d with the thermal design of r e g e n e r a t o r s i s the t r a n s i e n t response of the m a t r i x . T h i s problem a r i s e s when the regenerator i s made up of m atrices with low thermal c o n d u c t i v i t y . I f the thermal c o n d u c t i v i t y of the matrix i s s u f f i c i e n t l y low then allowance must be made for the thermal g r a d i e n t w i t h i n the matrix; t h i s i s c a l l e d the i n t r a c o n d u c t i o n e f f e c t . Consequently there i s a combined c o n v e c t i o n -conduct ion ( wi t h i n the matrix) heat t r a n s f e r i n r e l a t i o n to uniquely c o n v e c t i v e heat t r a n s f e r . T h e i n c l u s i o n of the matrix c o n d u c t i v i t y makes the thermal design of the regenerator more complicated .There are a number of s o l u t i o n s to t h i s problem.For example i t i s a common p r a c t i c e to adopt d i f f e r e n t numerical schemes (such as t r a p e z o i d a l approximation, c e n t r a l d i f f e r e n c e approximation or Crank- N i c h o l s o n method) to s o l v e the t r a n s i e n t response of the regenerator matrix. However,these numerical schemes 6 i n v o l v e s u b s t a n t i a l expenditure of computational time and computer storage [4].Heggs 'and Carpenter [5] p r e d i c t e d that the programs which n e g l e c t the i n t r a c o n d u c t i o n t y p i c a l l y r e q u i r e a s i x t h of the computing time of those i n c l u d i n g i t . The purpose of the present study i s to propose an a l t e r n a t i v e s o l u t i o n to the problem of determining t r a n s i e n t reponse of the m a t r i x . The proposed method u t i l i z e s an approximate i n t e g r a l technique to o b t a i n a s o l u t i o n i n an a n a l y t i c a l form.This avoids the use of numerical schemes and thereby enables the designer to take account of the i n t r a c o n d u c t i o n e f f e c t more e f f i c i e n t l y . The c e n t r a l f e a t u r e s of the regenerator heat exchanger are regenerator matrices.The c h o i c e of the matrix i s mainly a f f e c t e d by the way i n which the p e r i o d i c h e a t i n g and c o o l i n g of t h i s matrix i s p e r f o r m e d . l t may c o n s i s t of p l a t e s , wires ,spheres or broken s o l i d s of i r r e g u l a r shape [ 1 ] . I d e a l l y the matrix m a t e r i a l s e l e c t e d should possess high v a l u e s of s p e c i f i c h e a t , d e n s i t y and m e l t i n g point,good s t r e n g t h at e l e v a t e d temperatures;also i t should be e a s i l y a v a i l a b l e and cheap.It should not r e a c t c h e m i c a l l y with the i m p u r i t i e s present i n the h e a t i n g g a s e s ( u s u a l l y combustion g a s ) . In some cases the aerodynamic drag p r o p e r t i e s of the matrix are of great importance.Thus i t i s obvious that a l l these p o i n t s should be c o n s i d e r e d before a d e c i s i o n to s e l e c t the a p p r o p r i a t e matrix i s made. 7 The matrix can be brought i n t o contact with the heat exchanging f l u i d s (both hot and c o l d ) in d i f f e r e n t ways.The f o l l o w i n g types of regenerators may t h e r e f o r e be d i s t i n g u i s h e d ( r e f e r to F i g . 3 ) . 1.1.1 STATIONARY MATRIX In t h i s type of regenerator, the matrix i s s t a t i o n a r y while the change over from the heating to the c o o l i n g p e r i o d of the matrix i s performed by a l t e r n a t e l y changing the flow of the two f l u i d s . The rate of heat t r a n s f e r to or from a s u r f a c e i s d i r e c t l y p r o p o r t i o n a l to the area a v a i l a b l e f o r heat t r a n s f e r . T h e o v e r a l l dimension of the regenerator i s r e l a t e d to the heat t r a n s f e r area; i n c r e a s i n g the s u r f a c e area per u n i t bed volume decreases the o v e r a l l dimension r e q u i r e d f o r a p r e s c r i b e d amount of heat t r a n s f e r . I n the case of a s t a t i o n a r y matrix, the two heat exchanging f l u i d s flow a l t e r n a t e l y through one and the same matrix.Consequently the o v e r a l l dimension i s the same f o r both the h e a t i n g and the c o o l i n g s i d e ( F i g . 4 ) . Regenerators of t h i s type are s u b d i v i d e d i n t o d i f f e r e n t groups depending on the type of the matrix used.The pebble bed f o r example ( F i g . 4 a ) i s one type whose matrix elements can be of any shape and may c o n s i s t of l i g h t , inexpensive ceramic m a t e r i a l s capable I REGENERATOR AIR HEATER I 1 RECUPERATIVE Continuous Intermittent Rotary matrix Moving pebble bed Moving f l u l d l z e d bed F a l l i n g cloud Para l l e l flow Counter flow I Metal ceramic tubes Ceramic tubes with metal pins or f i ns Ceraml tubes Stat ionary pebble bed Checkers Figure 3. Types of a i r heater . 10 of w i t h s t a n d i n g high temperaturesf1].These matrices can be e a s i l y removed from t h e i r c o n t a i n e r s to be cleaned and i f neccessary r e p l a c e d . P o s s i b l e kinds of elements are spheres or other i r r e g u l a r shapes.The elements can be supported by the w a l l s of t h e i r c o n t a i n e r . F o r t h i s type of regenerator, p o r o s i t y (the r a t i o of empty volume to the t o t a l volume) i s an important f a c t o r which comes i n t o a n a l y s i s . T h e p o r o s i t y f o r a f i x e d bed i s u s u a l l y about 0.37 to 0.38. The Cowper stove i s an a l t e r n a t i v e type in which the matrix i s made of r e f r a c t o r y b r i c k [6].The b r i c k regenerator i s used e x t e n s i v e l y in g l a s s making and s t e e l making i n d u s t r i e s , f o r p r e h e a t i n g a i r to temperatures of the order of 900 to 1200 C ( F i g . 4 b ) . 1.1.2 MOVING MATRIX The c h a r a c t e r i s t i c of t h i s type of regenerator i s that the switch over from the heating to the c o o l i n g p e r i o d and v i c e - v e r s a i s due to the matrix movement.This type of regenerator i s s u b d i v i d e d i n t o d i f f e r e n t groups depending on the matrix and the movement of the matrix (Fig.3).The four common types are r o t a r y matrix,moving pebble bed, f a l l i n g l i q u i d s l a g and f a l l i n g s o l i d p a r t i c l e s [ 2 ] , The o p e r a t i o n of a r o t a r y regenerator r e l i e s on the thermal storage of a slowly r o t a t i n g matrix.With each . 1 1 r e v o l u t i o n of the matrix a c y c l e of h e a t i n g by hot a i r and c o o l i n g by c o l d a i r i s completed.Axial and r a d i a l flow are the two basic forms of matrix c o n f i g u r a t i o n ( F i g . 5 ) . One of the major design problems with regenerators of t h i s type i s to prevent the high pressure gas l e a k i n g to the low pressure a i r . T h i s i s c o n t r o l l e d with the a i d of a p p r o p r i a t e s e a l s . The second type of regenerator which i n c o r p o r a t e s a moving matrix i s a f a l l i n g c l o u d r e g e n e r a t o r ( F i g . 6 ) . T h i s type of regenerator has two chambers,nominally a s u p e r i o r chamber in which the matrix i s heated by the f l u i d and an i n f e r i o r chamber i n which the f l u i d i s heated by the matrix. The matrix c o n s i s t s of small s o l i d p a r t i c l e s of a s u i t a b l e heat t r a n s f e r medium,such as potassium sulphate,which are c o n t i n u o u s l y melted i n the upper chamber.The molten m a t e r i a l must then be p r e s s u r i z e d by a s l a g pump (not shown i n F i g . 6 ) before i n j e c t i o n to the lower chamber where i t i s atomized before f a l l i n g through the r i s i n g c o l d f l u i d . I n t h i s chamber the molten d r o p l e t s are s o l i d i f i e d and then returned e x t e r n a l l y to the top of upper chamber f o r r e c y c l i n g . T h e r e are t e c h n i c a l problems with the design of l i q u i d s l a g regenerators such as development of the s l a g pump and a t o m i z a t i o n of the molten m a t e r i a l [ 7 ] . An a l t e r n a t i v e to f a l l i n g l i q u i d s l a g i s f a l l i n g s o l i d p a r t i c l e s ( F i g . 7 ) . The matrix c o n s i s t s of broken s o l i d p a r t i c l e s e i t h e r of r e g u l a r or i r r e g u l a r shape.The AIR Figure 5(a) Schenatic diagram of rotary regenerator . DOTATION Radial flow Axial flow Figure 5(b) Rotary regenerator flow arrangement • Figure 6. F a l l i n g c loud regenerator* 14 1 5 e x t e n s i o n o f t h i s t y p e i s a m o v i n g p e b b l e bed r e g e n e r a t o r . O n e o f t h e f u n d a m e n t a l r e q u i r e m e n t s i s t o e n s u r e t h e u n i f o r m movement o f t h e p a r t i c l e s t h r o u g h t h e s y s t e m , e s p e c i a l l y a t i n l e t and o u t l e t , t o p r e v e n t t h e f o r m a t i o n of d e a d s e c t i o n s . A new c o n c e p t i n t h e d e s i g n o f t h e s e r e g e n e r a t o r s i s t h e use o f a d i v e r g e n t bed i n t h e d i r e c t i o n o f gas p a s s a g e . l t h a s been p r o v e n t h a t s u c h a d e s i g n c o n t r i b u t e s t o t h e u n i f o r m t e m p e r a t u r e p r o f i l e i n t h e g a s b e h i n d t h e bed [ 8 ] . 1.2 REVIEW OF PREVIOUS WORK R e g e n e r a t i v e h e a t e x c h a n g e i s one o f t h e most common i n d u s t r i a l p r o c e s s e s . l t i s t h e r e f o r e o f some i m p o r t a n c e t o f o r m u l a t e t h e l a w s g o v e r n i n g t h e r a t e o f h e a t t r a n s f e r i n s u c h a c a s e , a n d i f p o s s i b l e , t o o b t a i n a m a t h e m a t i c a l e x p r e s s i o n f o r t h e t e m p e r a t u r e d i s t r i b u t i o n t h r o u g h o u t s u c h a s y s t e m . Much work has been done on d e v e l o p i n g m a t h e m a t i c a l m o d e l s o f r e g e n e r a t o r s [ 4 - 1 4 ] . T h e t h e o r e t i c a l c o n s i d e r a t i o n s o f t h e f u n d a m e n t a l p h y s i c s o f h e a t t r a n s f e r a r e c o m p l i c a t e d and c e r t a i n s i m p l i f y i n g a s s u m p t i o n s must be made i n o r d e r t o o b t a i n a u s e f u l m a t h e m a t i c a l m o d e l . M a t h e m a t i c a l a n a l y s e s o f r e g e n e r a t o r s a r e d i v i d e d i n t o t h r e e g r o u p s w h i c h a r e e x p l a i n e d i n t h e f o l l o w i n g s e c t i o n s . 1 6 1.2.1 SCHUMANN MODEL The s i m p l e s t m a t h e m a t i c a l model o f a r e g e n e r a t o r was f i r s t d e v e l o p e d by Schumann i n 1929 [ 1 3 ] . He s u g g e s t e d a m o d e l i n w h i c h a f l u i d s t r e a m was a l l o w e d t o f l o w t h r o u g h a p a c k e d bed of b r o k e n s o l i d s ( F i g . 4 a ) . However h i s m o d e l c a n be e m p l o y e d f o r t h e c a s e o f f l u i d p a s s i n g t h r o u g h t h e c h a n n e l s o f b r i c k m a t r i x ( o f t e n c a l l e d c h e q u e r w o r k ) ( F i g . 4 b ) . Schumann's method o f d e v e l o p i n g an e x a c t m a t h e m a t i c a l t r e a t m e n t t o t h e h e a t t r a n s f e r p r o b l e m i n a r e g e n e r a t o r a s sumes a bed c o n s i s t i n g o f c r u s h e d m a t e r i a l a t a u n i f o r m t e m p e r a t u r e ; a f l u i d i s a l l o w e d t o p a s s l e n g t h w i s e t h r o u g h t h e p r i s m a t a u n i f o r m r a t e o f f l o w . T h e p r o b l e m i s t o f i n d t h e d i s t r i b u t i o n o f t e m p e r a t u r e i n t h e bed and i n t h e f l u i d f o r a l l t i m e , a s s u m i n g t h a t 1 . The t h e r m a l p r o p e r t i e s of the system are independent of the t e m p e r a t u r e . 2. The a x i a l c o n d u c t i o n i n e i t h e r the f l u i d phase or the s o l i d phase i s n e g l i g i b l e compared t o t h e t r a n s f e r of heat from s o l i d t o f l u i d . 3. The f l u i d f l o w r a t e does not v a r y a l o n g the bed. 4. There i s no t r a n s v e r s e t h e r m a l g r a d i e n t w i t h i n t h e p a r t i c l e s a t any i n s t a n t . 17 Based on these assumptions ,Schumann d e r i v e d a p a i r of coupled d i f f e r e n t i a l equations (given i n chapter 2) which determine the t r a n s f e r of heat. With an a p p r o p r i a t e set of boundary c o n d i t i o n s the problem i s so l v e d completely. Willmot [6] has presented a computer s o l u t i o n f o r the Schumann model.In an order of magnitude a n a l y s i s ,he has shown that the a x i a l conduction w i t h i n the matrix i s 2 n e g l i g i b l e p r o v i d e d that d/L i s small;where d i s the semi-thickness of the matrix and L i s the bed l e n g t h . T h i s r a t i o i s n e g l i g i b l e i n most p r a c t i c a l cases. 1.2.2 INTRACONDUCTION MODEL One of the major disadvantages of the Schumann model i s that i t n e g l e c t s the thermal g r a d i e n t w i t h i n the m a t r i x . T h i s s i m p l i f y i n g assumption i s j u s t i f i e d p r o v i d e d that the matrix has a very high thermal c o n d u c t i v i t y . In many cases t h i s i s not so. Glas s and ceramic (from which regenerator m a t r i c e s are o f t e n made [12]) have a s u f f i c i e n t l y low thermal c o n d u c t i v i t y that allowance must be made f o r the thermal g r a d i e n t w i t h i n the p a r t i c l e s . The matrix thermal c o n d u c t i v i t y i s taken i n t o c o n s i d e r a t i o n in terms of a dime n s i o n l e s s parameter c a l l e d B i o t number which i s d e f i n e d as 18 Bi=hR/K ( s p h e r i c a l geometry) , s =hd/Ks (planar geometry) , where h=convective heat t r a n s f e r c o e f f i c i e n t , R=radius of sphere , d=senu-thickness of matrix , K -matrix thermal c o n d u c t i v i t y . s A model which i n c l u d e s the e f f e c t of matrix thermal c o n d u c t i v i t y i s c a l l e d an i n t r a c o n d u c t i o n model.Figure.8 shows the e f f e c t of c o n d u c t i v i t y (or Bi number)on the matrix temperature p r o f i l e . P revious s t u d i e s have shown that i n t r a c o n d u c t i o n may have a s i g n i f i c a n t e f f e c t on the thermal behavior of r e g e n e r a t o r s [4-13],although the i n c l u s i o n of t h i s e f f e c t makes the model more complicated. D i f f e r e n t mathematical models have been proposed and have been s o l v e d e i t h e r n u m e r i c a l l y or a n a l y t i c a l l y . Handely and Heggs [12] a p p l i e d the Crank-Nicholson method in order to o b t a i n a s o l u t i o n to t r a n s i e n t response of the matrix i n a f i x e d bed r e g e n e r a t o r . T h e i r t h e o r e t i c a l r e s u l t s were in good agreement with t h e i r experimental observation.The authors a l s o proposed a d i m e n s i o n l e s s group which p r e d i c t s the d i v i d i n g l i n e between the Schumann and i n t r a c o n d u c t i o n models. 19 F i g u r e 8. E f f e c t o f t h e r m a l c o n d u c t i v i t y o f s o l i d o n t h e t e m p e r a t u r e p r o f i l e , (B5=hR/K) . 20 Hausen [ 1 5 ] p r o p o s e d a model i n w h i c h t h e e f f e c t o f m a t r i x t h e r m a l c o n d u c t i v i t y i s t a k e n i n t o c o n s i d e r a t i o n i n t e r m s o f an o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t . T h i s i s t h e so c a l l e d m o d i f i e d - i n f i n i t e c o n d u c t i o n m o d e l . The o v e r a l l h e a t t r a n s f e r c o e f f i c i e n t i s d e f i n e d i n t e r m s o f t h e a c t u a l c o n v e c t i v e h e a t t r a n s f e r c o e f f i c i e n t a nd a p a r a m e t e r w h i c h i s c a l l e d m o d i f i c a t i o n f a c t o r . A number of a u t h o r s [ 4 , 6 , 1 5 ] h a v e p r o p o s e d d i f f e r e n t e x p r e s s i o n s f o r t h i s m o d i f i c a t i o n f a c t o r . An a n a l y t i c a l s o l u t i o n t o t r a n s i e n t h e a t c o n d u c t i o n i n s o l i d s c a n be o b t a i n e d by t h e a p p r o x i m a t e i n t e g r a l m e t h o d . T h i s method was f i r s t e m p l o y e d by Goodman [ 1 6 ] t o s o l v e a v e r y s i m p l e p r o b l e m of u n s t e a d y h e a t c o n d u c t i o n i n a s e m i - i n f i n i t e s l a b . H i s r e s u l t s were i n good a g r e e m e n t w i t h t h e w e l l e s t a b l i s h e d r e s u l t s o f C a r s l a w and J a e g e r [ 1 7 ] . H o w e v e r , t h e e l e g a n t m e t h o d s p r o p o s e d by C a r s l a w and J a e g e r w i l l o n l y be s a t i s f a c t o r y i f t h e t h e r m a l c o n d u c t i v i t y i s i n d e p e n d e n t o f t e m p e r a t u r e ; w h e r e a s t h e i n t e g r a l method ca n be e x t e n d e d t o i n c l u d e t h e e f f e c t o f t e m p e r a t u r e d e p e n d e n t t h e r m a l c o n d u c t i v i t y . G o o d m a n a p p l i e d t h e i n t e g r a l method t o t r a n s i e n t h e a t c o n d u c t i o n i n p l a n a r g e o m e t r y . L a r d n e r and P o h l e [ 1 8 ] h a v e d e m o n s t r a t e d t h a t t h e method i s e q u a l l y a p p r o p r i a t e f o r s p h e r i c a l g e o m e t r y . A more c o m p l e t e d i s c u s s i o n o f t h i s method i s g i v e n i n t h e f o l l o w i n g c h a p t e r . 21 1.3 SCOPE OF THE PRESENT INVESTIGATION The l i t e r a t u r e s e a r c h c o n f i r m s t h e f a c t t h a t f o r a s u f f i c i e n t l y l o w t h e r m a l c o n d u c t i v i t y , t h e i n t r a c o n d u c t i o n e f f e c t s h o u l d be t a k e n i n t o c o n s i d e r a t i o n . The i n c l u s i o n o f t h i s e f f e c t makes t h e a n a l y s i s more i n v o l v e d . The s o l u t i o n t o t r a n s i e n t r e s p o n s e o f t h e m a t r i x h a s t o be d e t e r m i n e d . The p u r p o s e o f t h e p r e s e n t work i s t o p r o p o s e an a n a l y t i c a l s o l u t i o n t o t h e m a t r i x t r a n s i e n t r e s p o n s e by u t i l i z i n g an a p p r o x i m a t e i n t e g r a l t e c h n i q u e . The use o f t h i s method a v o i d s l e n g t h y c o m p u t e r p r o g r a m s ; t h e method p r o v i d e s a q u i c k d e t e r m i n a t i o n o f t h e s o l i d mean t e m p e r a t u r e i n o r d e r t o o b t a i n t h e e f f e c t i v e n e s s o f t h e r e g e n e r a t o r . The u s u a l s i m p l i f y i n g a s s u m p t i o n o f c o n s t a n t s o l i d t h e r m a l p r o p e r t i e s c a n a l s o be r e l a x e d . The f i r s t s t a g e a f t e r t h e d e v e l o p m e n t o f t h e p r o p o s e d a p p r o x i m a t i o n i s t o e x a m i n e i t s v a l i d i t y a g a i n s t more r i g o r o u s a n a l y s e s . O n c e t h e a c c u r a c y o f t h e method i s e s t a b l i s h e d , i t c a n t h e n be e x t e n d e d t o i n c l u d e d i f f e r e n t t y p e s o f r e g e n e r a t o r a n d a l s o t h e e f f e c t o f m a t r i x g e o m e t r y on t h e r e g e n e r a t o r p e r f o r m a n c e . I t s h o u l d be e m p h a s i z e d t h a t t h e r e a r e no p u b l i s h e d r e s u l t s f o r t h e m o v i n g bed r e g e n e r a t o r . T h e d e v e l o p m e n t o f t h e i n t e g r a l method p r o v i d e s t h e means t o o b t a i n d e s i g n d a t a f o r t h e m o v i n g bed r e g e n e r a t o r . 2. THE GOVERNING EQUATIONS 2.1 DIMENSIONLESS PARAMETERS It i s a common p r a c t i c e i n regenerator design to present the r e s u l t s i n terms of a number of dimensionless parameters which are introduced to s i m p l i f y the governing equations. These parameters were f i r s t i n t roduced by Hausen [15]. N u m b e r o f t r a n s f e r u n i t s ( N T U ) T h i s parameter i s a l s o termed the reduced l e n g t h (A) i n the l i t e r a t u r e . I t i s d e f i n e d i n terms of the heat t r a n s f e r c o e f f i c i e n t , h e a t t r a n s f e r area per u n i t bed volume and the f l u i d c a p a c i t y r a t e 2 ( r e f e r to Fig.9) . £=hAy/(mC), y = L, where h=convective heat t r a n s f e r c o e f f i c i e n t , A=heat t r a n s f e r a r e a / u n i t bed volume , m=fluid flow r a t e / u n i t area , C = f l u i d s p e c i f i c heat c a p a c i t y at constant p r e s s u r e . These terms are e x p l a i n e d i n more d e t a i l i n Appendix A. 2 c a p a c i t y rate=mC 22 2 3 Figure 9. Typ ica l dimensions o f a r e g e n e r a t o r . 24 D i m e n s i o n l e s s p e r i o d The p e r i o d o f e a c h h e a t i n g and c o o l i n g c y c l e i s n o n d i m e n s i o n a l i s e d as T?=hA(0-y/u)/(M .C ) , s s n=hA(P-L/u)/(M .C ) , s s where P = p e r i o d of f l u i d f l o w , L=bed l e n g t h , u = f l u i d v e l o c i t y , M =bed d e n s i t y , s C s = s o l i d s p e c i f i c h e a t c a p a c i t y . The t e r m L/u w h i c h r e p r e s e n t s t h e f l u i d r e s i d e n c e t i m e i s u s u a l l y i g n o r e d . T h i s i s p e r m i s s i b l e i n many r e g e n e r a t o r a p p l i c a t i o n s b e c a u s e i t i s n e g l i g i b l e compared t o t h e p e r i o d (p) [ 1 2 ] . In s h o r t c y c l i n g a p p l i c a t i o n s t h e r e s i d e n c e t i m e becomes o f s i m i l a r m a g n i t u d e t o t h e p e r i o d and t h e e f f e c t o f L/u c a n n o t be i g n o r e d . E f f e c t i v e n e s s T h i s i s d e f i n e d as t h e r a t i o of t h e a c t u a l r i s e i n t h e m a t r i x t e m p e r a t u r e t o i t s maximum p o s s i b l e r i s e ( r e f e r t o F i g . 9 ) , t h a t i s e=[m C (T -T .)/((mC) . ) . ( T , . - T . ) ] , s s msO s i min f i s i 25 where m = s o l i d f l o w r a t e / b e d a r e a , s (mC) . =minimum o f t h e two mC , min (m C )/(mC) . = c a p a c i t y r a t e r a t i o , s s min I t s h o u l d be e m p h a s i z e d t h a t t h e s o l i d t e m p e r a t u r e i s p r e s e n t e d a s a mean t e m p e r a t u r e , w h i c h i s d i f f e r e n t t h a n t h e s o l i d s u r f a c e t e m p e r a t u r e . T h i s i s e x p l a i n e d i n more d e t a i l i n A p p e n d i x D. D i m e n s i o n l e s s t h i c k n e s s o r r a d i u s The s l a b t h i c k n e s s and t h e s p h e r e r a d i u s a r e n o n d i m e n s i o n a l i s e d a s ( r e f e r t o F i g . 9 ) 1. F o r t h e s l a b ; z=x/d , where d = t h i c k n e s s o f t h e s l a b . 2. F o r t h e s p h e r e ; z = r/R , where R=sphere r a d i u s 26 Normalised temperatures The f l u i d and s o l i d t e m p e r a t u r e s a r e n o r m a l i z e d a s 1. F o r t h e f l u i d F=(T -T . ) / ( T , . - T .) . f SI f1 s i 2. F o r t h e s o l i d *=(T -T . ) / ( T r . - T . ) . s s i f i s i 2.2 THE MATHEMATICAL MODELS I t was e x p l a i n e d i n t h e p r e v i o u s c h a p t e r t h a t r e g e n e r a t o r s c a n be d i v i d e d i n t o d i f f e r e n t g r o u p s d e p e n d i n g on t h e t y p e o f m a t r i x e m p l o y e d . I n t h e p r e s e n t work two t y p e s o f r e g e n e r a t o r s a r e c o n s i d e r e d ( f i x e d a n d m o v i n g bed) w i t h two m a t r i x g e o m e t r i e s ( p l a n a r a n d s p h e r i c a l ) . The e q u a t i o n s a r e d e v e l o p e d f o r t h e r a t e o f h e a t t r a n s f e r b e t w e e n a g r o u p o f s o l i d p a r t i c l e s ( e i t h e r m o v i n g o r f i x e d ) a n d t h e f l u i d m o v i n g c o u n t e r c u r r e n t l y t o t h e p a r t i c l e s . T h e r e a r e two h e a t t r a n s f e r p r o c e s s e s w h i c h t a k e p l a c e i n a t h e r m a l r e g e n e r a t o r . However one may p r e d o m i n a t e d e p e n d i n g on t h e a s s u m p t i o n s made i n d e v e l o p i n g t h e m o d e l . I f i t i s assumed t h a t t h e m a t r i x t h e r m a l c o n d u c t i v i t y i s i n f i n i t e (Schumann m o d e l ) t h e d o m i n a n t h e a t t r a n s f e r p r o c e s s 27 i s h e a t t r a n s f e r a c r o s s t h e s u r f a c e of the m a t r i x ( o r h e a t t r a n s f e r t o t h e f l u i d ) . On t h e o t h e r hand i f t h e m a t r i x t h e r m a l c o n d u c t i v i t y i s assumed t o be f i n i t e ( I n t r a c o n d u c t i o n model) ,the c o n d u c t i o n a l s o becomes i m p o r t a n t . The two models a r e a n a l y s e d i n more d e t a i l i n t h e next two s e c t i o n s . 2.2.1 SCHUMANN MODEL T h i s i s t h e s i m p l e s t model of a t h e r m a l r e g e n e r a t o r and i s b a s e d on t h e f o l l o w i n g s i m p l i f y i n g a s s u m p t i o n s : a. The thermal properties of the system are independent of temperature, b. The transfer of heat by conduction in the f l u i d i t s e l f i s small compared to the heat transfer by convection from the f l u i d to the s o l i d , c. The f l u i d flow rate does not vary along the bed, d. There is no thermal gradient within the matrix. I f t h e r e i s no t r a n s v e r s e t h e r m a l g r a d i e n t w i t h i n t h e m a t r i x , t h e n t h e m a t r i x c a n be assumed t o be a t a u n i f o r m t e m p e r a t u r e and can be r e p r e s e n t e d by a s i n g l e t e m p e r a t u r e a t any p o i n t a l o n g t h e r e g e n e r a t o r ( F i g . 8 a ) . Thus t h e o n l y h e a t t r a n s f e r p r o c e s s i s t h e h e a t 28 g a i n e d / l o s t by t h e f l u i d p a s s i n g t h r o u g h t h e r e g e n e r a t o r . F l u i d phase heat t r a n s f e r equation I f m i s t h e mass r a t e o f f l u i d f l o w / u n i t bed a r e a p a s t a s e c t i o n a d i s t a n c e y f r o m t h e e n t r a n c e o f t h e r e g e n e r a t o r , t h e n b e t w e e n y and y+dy t h e h e a t t r a n s f e r r e d f r o m / t o t h e f l u i d i n t i m e dd w i l l be ( r e f e r t o A p p e n d i x A) dQ=mC[ U T , / 9 y ) + l / u . ( 9 T r / 9 0 ) ]dy.A,_ , ( 2 . 1 . a ) 1 6 f y b =mC(DT r/Dy)dy .A. , (2: K b ) i b where D/Dy= ( 9/9y) + 1 / u . (3/3(9) S o l i d phase heat t r a n s f e r equation The t o t a l h e a t f l o w t o t h e f l u i d must be e q u a l t o t h e h e a t l o s t by t h e m a t r i x , t h a t i s d Q = h A ( T - T f ) d y . A f a . ( 2 . 2 ) I f m i s t h e mass f l o w r a t e o f s o l i d / u n i t a r e a , s t h e n b e t w e e n y and y+dy t h e h e a t l o s t f r o m t h e m a t r i x w i l l be dQ=m C [ ( 3 T / 3 y ) +1/u .( 3 T / 9 0 ) ] d y . A L . ( 2 . 3 . a ) s s 6 s y b 29 The above equation can a l s o be w r i t t e n as dQ=M C [u (3T/3y) +OT/30)-'jdy.A^ (2.3.b), s s s 6 y b where M =m /u , s s s i f the bed i s s t a t i o n a r y (u s=0) then equation (2.3.b) becomes dQ=-M C (9T/30)dy.A, . (2.3.c) s s b The s i g n d i f f e r e n c e between equations (2.3.a) and (2.3.c) i s due to the change of d i r e c t i o n in which y i s measured; f o r the moving bed y i s measured from the s o l i d entrance whereas f o r the f i x e d bed y i s measured from the opposite end. A l s o i n both of the equations the matrix temperature i s represented by a s i n g l e temperature T. Th i s i s because the model assumes that at any p o i n t along the regenerator the matrix i s at a uniform temperature. Summary of the equations Combining equations (1),(2) and (3): For the moving bed 30 hA(T-T )=m C [(3T/3y) +1/U .(3T/30) ] , (2.4.a) • t s s 6 s Y =m C (DT/Dy) , s s and hA (T-T,.) =mC[(3T/3y)+1/u.(3T /3#) ] , (2.4.b) t t t) t y =mC(DTf/Dy) . For the f i x e d bed:Only the s o l i d phase equation changes, hA(T-T c)=-M C (3T/30) . (2.5) 2.2.2 INTRAPARTICLE CONDUCTION MODEL For regenerators composed of ma t r i c e s with low thermal c o n d u c t i v i t y , e g . g l a s s and ceramics,assumption (d) of the Schumann model i s i n v a l i d . Thus allowance must be made, for thermal g r a d i e n t w i t h i n the matrix. The temperature of the s o l i d at any po i n t along the regenerator can no longer be represented as one temperature. I t i s thus d e s i r a b l e to o b t a i n the temperature d i s t r i b u t i o n w i t h i n the s o l i d and represent the matrix temperature as a mean temperature. There are thus two heat t r a n s f e r processes f o r an i n t r a c o n d u c t i o n model , 1. Heat i s g a i n e d / l o s t by the f l u i d p a s s i n g through the regenerator. 2. Heat i s t r a n s f e r r e d w i t h i n the matrix. 31 1. F l u i d phase heat t r a n s f e r equation T h i s equation i s the same as that f o r the Schumann model. 2. S o l i d phase heat t r a n s f e r equation There are two stages of heat t r a n s f e r in the s o l i d phase. (a) Heat t r a n s f e r across the su r f a c e o f s o l i d : T h i s can be represented as h(T -T )=K (9T/9x) (Planar geometry) , (2.5.a) S L S X — u =-K (9T/9r) ( S p h e r i c a l geometry). (2.5.b) s r—K I t should be emphasized that i n t h i s case T s r e p r e s e n t s the s o l i d s u r f a c e temperature which might not n e c e s s a r i l y be the same as mean s o l i d temperature. The heat l o s t by the s o l i d can be represented i n terms of the rate of change of i t s i n t e r n a l energy as ( r e f e r to Appendix A) d cl dQ=m C d y . A u [ ( 3 ( J T dx)/9y)+(9(/ T dx)/90)/u 3/d , (2.6) s s b o 0 s d where (/ T dx)/d rep r e s e n t s the mean s o l i d temperature, 0 and d i s the semi-thickness of the matrix. 32 (b) Heat t r a n s f e r w i t h i n the s o l i d : The matrix thermal c o n d u c t i v i t y i s f i n i t e and there i s a temperature d i s t r i b u t i o n w i t h i n the matrix.Heat t r a n s f e r w i t h i n the matrix i s represented by the d i f f u s i o n equation.Assuming the problem i s one dimensional we have f o r p l a n a r geometry 9T/90=a(9 2T/3x 2) , (2.7.a) f o r s p h e r i c a l geometry 9T/90=a[{3 2T/9r 2)+(2/r)(9T/9r)] . (2.7.b) The above equations are coupled by the symmetry c o n d i t i o n p l a n a r (9T/9x) x = d = 0 d=thickness of s l a b (2.8.a) s p h e r i c a l (9T/9r) r = 0 = 0 R=radius o f sphere (2.8.b) 33 The d i s t r i b u t i o n of gas and s o l i d temperature along the regenerator i s obtained by s o l v i n g these equations with a p p r o p r i a t e i n i t i a l c o n d i t i o n s f o r a f i x e d bed , the i n i t i a l c o n d i t i o n i s represented by a step change i n the gas i n l e t temperature , T f=T f. f o r 9 >0 and y=0 , T=T. , f o r 0=0 and y=0 . fo r a moving bed , the i n l e t temperatures are s p e c i f i e d . 2.3 NON-DIMENSIONAL FORM OF GOVERNING EQUATIONS The governing equations can be nondimensionalised i n terms of dimensionless parameters £, rj,Biot number and normalised temperatures d e f i n e d i n s e c t i o n 1. The equations in d imensionless form are (Refer to Appendix A) F l u i d phase Equations (2.4.b) and (2.5) become ( r e f e r to Appendix A), 3F/9£=(¥ -F) , ( f i x e d bed) , (2.9.a) s =(F-¥ ) , (moving bed) . (2.9.b) Again the sign d i f f e r e n c e i s due to the change of d i r e c t i o n 34 in which £ (or y) i s measured. S o l i d phase  Planar geometry Equations (2.5.a),(2.7.a)and (2.8.a) become ( r e f e r to Appendix A) (3*/3z) = Bi(¥ -F) , (2.10.a) z = 0 s 3*/3r?=[ 3 2 * / 3 z 2 ] / B i , ( 2 . l 0 . b ) (3+/3z) =0 . (2.10.C) z= 1 S p h e r i c a l Geometry Equations (2.5.b),(2.7.b)and(2.8.b) become(refer to Appendix A) (3*/3z) =-Bi(¥ -F) , (2.11.a) z= 1 s 3¥/3r?=[ ( 3 2 * / 3 z 2 ) + (2/z) 0*/3z) ]/(3Bi) , (2.1Kb) (3*/3z) -=0 . (2.11.c) z = U The i n i t i a l c o n d i t i o n s are nondimensionalised f o r the case of a f i x e d bed as , 35 F=1 at £ = 0 and T?>0 , *=0 at 7?=0. and £>0 . 3. THE METHOD OF SOLUTION 3.1 INTRODUCTION TO THEINTEGRAL METHOD. The i n c l u s i o n of matrix thermal c o n d u c t i v i t y i n the thermal design of regenerators makes the a n a l y s i s more complicated.The d i f f u s i o n equation (equations.2.10.b,2.11.b) must be so l v e d to o b t a i n the temperature d i s t r i b u t i o n w i t h i n the m a t r i x . T h i s can be done by employing numerical techniques ( d i s c u s s e d i n the review of pr e v i o u s work) which r e q u i r e lengthy computer programs. An a l t e r n a t i v e s o l u t i o n i s the a p p l i c a t i o n of the approximate i n t e g r a l t e c h n i q u e . T h i s method was f i r s t i n t r o d u c e d by von Karman and Pohlhausen i n order to solve the boundary l a y e r problem in f l u i d mechanics.However,the method i s e q u a l l y a p p r o p r i a t e f o r unsteady heat conduction in solids.Goodman [16] employed the technique to solve the d i f f u s i o n equation,coupled with e i t h e r l i n e a r or n o n - l i n e a r boundary c o n d i t i o n s . The method makes use of two assumpt i o n s : a. The thermal p r o p e r t i e s ( i e . c o n d u c t i v i t y , d e n s i t y etc) are u s u a l l y assumed to be independent of temperature i n order to l i n e a r i z e the d i f f u s i o n e q uation. 36 3 7 b. The s o l i d i s i n i t i a l l y at a constant temperature. Goodman subsequently has developed , a technique to account for temperature dependent thermal p r o p e r t i e s . T h i s i s e x p l a i n e d i n more d e t a i l i n s e c t i o n ( 3 . 4 ) of t h i s chapter. The i n t e g r a l method i n t r o d u c e s a q u a n t i t y 5(8) c a l l e d the p e n e t r a t i o n depth.This i s d e f i n e d as a d i s t a n c e i n t o which the heat f l u x at the s u r f a c e penetrates the s o l i d , a n d beyond which there i s no heat t r a n s f e r r e d . C o n s e q u e n t l y there w i l l be a temperature g r a d i e n t i n s i d e the s o l i d up to the p e n e t r a t i o n depth ,while the s o l i d w i l l be at a uniform temperature beyond t h i s p o i n t ( F i g . 1 0 ) . T h i s i s expressed mathematically as ( 9 T / 3 x ) =0 . x = 6 The p e n e t r a t i o n depth i s analogous to the boundary l a y e r t h i c k n e s s in f l u i d mechanics. The technique adopted by the approximate i n t e g r a l method can be e x p l a i n e d as f o l l o w s ; 38 Figure 10, Schematic representat ion of penetrat ion depth concept , T s = Surface temperature, Ti= I n i t i a l temperature. 3 9 The s o l i d temperature i s represented by a polynomial in x (or r f o r s p h e r i c a l geometry) .The order of the polynomial i s l i m i t e d by the number of v a r i a b l e c o n s t r a i n t s ( i e . boundary c o n d i t i o n s , i n i t i a l c o n d i t i o n s etc) .The unknown c o e f f i c i e n t s are u s u a l l y a f u n c t i o n of time. One way to improve the accuracy of the assumed p r o f i l e i s to i n c r e a s e the order of the polynomial.Each a d d i t i o n a l parameter which i s thereby i n t r o d u c e d i s determined from an a d d i t i o n a l d e r i v e d c o n s t r a i n t . T h i s may not always be p o s s i b l e . Koh.y [ 1 9 ] has demonstrated that the temperature p r o f i l e i s b e t t e r approximated by an e x p o n e n t i a l f u n c t i o n than a polynomial, however the a n a l y s i s becomes more i n v o l v e d . There i s never a unique procedure to f o l l o w i n using the i n t e g r a l method.The u l t i m a t e c r i t e r i o n f o r determining whether or not a p a r t i c u l a r p r o f i l e i s s u c c e s s f u l must i n v o l v e an assessment of both i t s accuracy and s i m p l i c i t y . A simple polynomial p r o f i l e has been found adequate f o r most e n g i n e e r i n g purposes. 40 3.2 PLANAR GEOMETRY The s o l i d t e m p e r a t u r e i s r e p r e s e n t e d by a p o l y n o m i a l o f x,where x i s t h e d i s t a n c e f r o m t h e s u r f a c e o f t h e s o l i d . S i n c e t h e o r d e r of t h e p o l y n o m i a l i s l i m i t e d by t h e number o f c o n s t r a i n t s , i t w o u l d be a d v a n t a g e o u s t o m o d el t h e m a t r i x a s a s e m i - i n f i n i t e s l a b e x t e n d i n g i n t h e x d i r e c t i o n . I n t h i s way t h e o r d e r o f t h e p o l y n o m i a l c a n be i n c r e a s e d a s w i l l be s e e n l a t e r . 3.2.1 S E M I - I N F I N I T E SLAB The t e m p e r a t u r e d i s t r i b u t i o n T ( x , 0 ) i n s i d e t h e s l a b i s t o be c a l c u l a t e d s u b j e c t t o t h e f o l l o w i n g c o n s t r a i n t s T ( 5 , 0 ) = T . 3T(S,0)/3x=O , ( 3 . 2 . 1 ) ( 3 . 2 . 2 ) 3 T ( 0 , 0 ) / 3 x = h . ( T - T f ) / K = - f ( 0 ) . ( 3 . 2 . 3 ) T h e r e a r e t h r e e c o n s t r a i n t s so t h e p r o f i l e must be r e p r e s e n t e d by a s e c o n d - o r d e r p o l y n o m i a l . The a s s u m p t i o n t h a t t h e s l a b i s i n i t i a l l y a t a c o n s t a n t t e m p e r a t u r e c a n be u t i l i z e d i n d e r i v i n g an a d d i t i o n a l c o n s t r a i n t . E q u a t i o n ( 3 . 2 . 1 ) i s 41 d i f f e r e n t i a t e d with respect to time and then s u b s t i t u t e d i n the d i f f u s i o n equation.The r e s u l t i s 3 2T(6,0)/3x 2=O . (3.2.4) This i s u s u a l l y c a l l e d the smoothing c o n d i t i o n . The temperature d i s t r i b u t i o n can now be represented by a cu b i c p r o f i l e . T h e c o n s t r a i n t s are nondimensionalised i n terms of z (dimensonless depth) ,Biot number ,normalised temperatures and 6^, that i s ( r e f e r to Appendix B) *=0 at z=5 Q , (3.2.5) 3¥/3z=0 at z=5 Q , (3.2.6) 3¥/9z=Bi . (* -F)=-f(rj) at z = 0 , (3.2.7) s 2 2 3 */3z =0 at z=6 Q , (3.2.8) where 5^=5/6 , i s the dimensionless p e n e t r a t i o n depth. The c u b i c p r o f i l e w i l l take the form ( r e f e r to Appendix B) *=f (i?) . ( 6 0 - z ) 3 / ( 3 . 5 2 ) . (3.2.9) 42 The s u r f a c e temperature i s obtained by s e t t i n g z=0 i n equation (3.2.9) ,the r e s u l t i s •*-=f ( T J ) . 6 a / 3 . (3.2.10) s 0 Equation (3.2.1.0) expresses a r e l a t i o n s h i p between the s o l i d s u r f a c e temperature and the p e n e t r a t i o n depth.Consequently, once the p e n e t r a t i o n depth i s c a l c u l a t e d , i t w i l l only r e q u i r e a simple a l g e b r a i c manipulation to c a l c u l a t e the s u r f a c e temperature or v i c e - v e r s a . The p e n e t r a t i o n depth i s obtained by i n t e g r a t i n g the d i f f u s i o n equation (eqn.2.10.b) from z=0 to z = S Q and s u b s t i t u t i n g f o r * from equation (3.2.9),the r e s u l t i s ( r e f e r to Appendix B) S =[ 12. (SVt (17) d7j ) / ( f (77) .Bi) ] ° " 5 . (3.2.11) 0 0 The s u r f a c e temperature can then be obtained by s u b s t i t u t i n g f o r 6^  from equation (3.2.10).The r e s u l t i s * =[4.f (v) • (SV£ (T?) d i?)/(3.Bi) ] ° ' 5 . (3.2.12) s 0 43 3.2.2 SLAB OF FINITE THICKNESS I n i t i a l l y , t h e symmetry c o n d i t i o n (eqn.2.10.c) does not a f f e c t the temperature d i s t r i b u t i o n w i t h i n the matrix.The matrix can thus be modeled as though i t were s e m i - i n f i n i t e . However,at some l a t e r time the p e n e t r a t i o n depth reaches the c e n t r e of the matrix and the symmetry c o n d i t i o n comes i n t o e f f e c t . A t t h i s stage the p e n e t r a t i o n depth has no meaning and the model should be r e p l a c e d by a s l a b of f i n i t e t h i c k n e s s whose f a r su r f a c e i s i n s u l a t e d ( r e p r e s e n t i n g the symmetry c o n d i t i o n ) . The s l a b i s s u b j e c t to the f o l l o w i n g c o n s t r a i n t s 3T(d,0)/3x=O (3.2.13.a) T(0,0)=T (3.2.14.a) s S f S (3.2.15.a) The above equations i n dimensionless form are 3¥( 1 , T J ) / 3 Z = 0 (3.2.13.b) ¥(0,77)=* (3.2.14.b) 3¥(0 ,T?)/3z = Bi.(¥ -F)=-f (T?) . (3.2.15.b) 44 The second-order p r o f i l e must then take the form *=+ - f (rj) . ( z 2 - 2 . z ) / 2 . (3.2.16) s The s u r f a c e temperature ¥ w i l l be obtained by i n t e g r a t i n g the d i f f u s i o n equation (eqn.2.10.b) with respect to z.The s o l i d temperature ¥ i s r e p l a c e d by the polynomial express ion(eqn.3.3.4).As opposed to the p r e v i o u s case,the i n t e g r a t i o n extends from z=0 to z=1. The r e s u l t i s ( r e f e r to Appendix B) V * =[f(r?)/3 + J f ( v ) dr? / B i ]+constant . (3.2.17) s 0 Assume T J Q I S the time at which the p e n e t r a t i o n depth reaches the f a r surface of the s l a b .The i n i t i a l c o n d i t i o n ( i e . * ( 7 ? ^ ) ) can be obtained by s 0 e x p l i c i t l y s e t t i n g 5^=1 i n equation (3.2.10).Setting • = * ( T ? ~ ) i n equation (3.2.12) r e s u l t s i n o b t a i n i n g s s 0 7?Q. The constant of i n t e g r a t i o n i s then obtained by s e t t i n g * =+ (77.) and 77=17_ in equation ( 3 . 2 . 1 7 ) .The s s 0 0 r e s u l t i s V * =f <TJ)/3 +/ f ( r j ) d(r?)/Bi For 77>7i . (3.2.18) "0 45 In summary If T * I E N E C 3 u a t i o n (3.2.12) i s used. If 77>T7Q then equation (3.2.18) i s used. 3.3 SPHERICAL GEOMETRY Lardner and Pohle [18] have demonstrated that f o r s p h e r i c a l geometry, the polynomial r e p r e s e n t a t i o n of the temperature p r o f i l e i s i n a p p r o p r i a t e . S i n c e f o r s p h e r i c a l geometry the steady s t a t e s o l u t i o n i s p r o p o r t i o n a l to ( l / r ) , t h e y suggested a p r o f i l e of the form T=(Polynomial i n r ) / r . (3.3.1) Although Lardner and Pohle d e a l t with a s p h e r i c a l h ole, the same method i s a p p l i c a b l e to the case of a s o l i d sphere.The procedure adopted f o r s p h e r i c a l geometry i s the same as that f o r planar geometry , that i s , the o r i g i n a l p r o f i l e i n c l u d e s the p e n e t r a t i o n depth.As soon as the p e n e t r a t i o n depth reaches the centre of the sphere,a second p r o f i l e should be used. 46 3.3.1 SPHERE OF INFINITE RADIUS The p e n e t r a t i o n depth i s measured from the s u r f a c e of the sphere. The temperature p r o f i l e i s su b j e c t to the f o l l o w i n g c o n s t r a i n t s 3T(R-5,6)/dr=0 , (3.3.2.a) T(R--5,0)=T(i) , (3.3.3.a) 9T(R,0)/3r=-h.(T -T )/K , (3.3.4.a) s I S 3 2T(R-8,0)/3r 2=O . (3.3.5.a) Equ a t i o n ( 3 . 3 . 5 . a ) r e p r e s e n t s the smoothing c o n d i t i o n . The above equations i n dimensionless form are 3¥( 1 - 6 q , T ? ) / 3 Z = 0 , (3.3.2.b) *( 1 -5 , T J ) =0 , (3.3.3.b) 3¥( 1 ,77)/3z = - B i . (¥ -F)=-f (77) , (3.3.4.b) 3 2+( 1 - 8 0 , T ? ) / 3 Z 2 = 0 , (3.3.5.b) where 6^=8/R=dimensionless p e n e t r a t i o n depth.(3.3.6) 47 There are 4 c o n s t r a i n t s c o n s e q u e n t l y the polynomial w i l l be a cubi c . A d o p t i n g the suggestion by Lardner and Pohle, the r e s u l t i s 3 2 *=(A.z +B.z +C.z+D)/z . (3.3.7) Applying the c o n s t r a i n t s , t h e p r o f i l e must take the form *=-f (r?) . [z-( 1-5 ) ] 3/UQ-(3-6 ) .z) . (3.3.8) The s u r f a c e temperature i s obtained by s e t t i n g z=1,in above equation, the r e s u l t i s * =-f (77) .5-/(3-5.) . (3.3.9) s 0 0 The p e n e t r a t i o n depth i s obtained by i n t e g r a t i n g the d i f f u s i o n equation (eqn.2.11.b) a f t e r * has been r e p l a c e d by equation(3.3.8). The i n t e g r a t i o n extends from z=1-6^ to z=1,the r e s u l t i s ( r e f e r to Appendix B) [ ( 5 6 2 - 6 ^ ) / ( 3 - 6 n ) ]=20(/ 7 ?f (TJ) dr?)/( 3Bi . f (TJ) ) .(3.3.10) U u (J o The s u r f a c e temperature i s obtained by s u b s t i t u t i n g fo r 5 n from equation(3.3.9),the r e s u l t i s 48 3 * 2 [ 5 f (r ?)-2* ]=20(* - f (r?) ) 2 ( f* t (77) dr?) / ( 3Bi ) ( 3 . 3 . 1 1 ) S S S g 3.3.2 SPHERE OF FINITE RADIUS As in the case of planar geometry,the symmetry c o n d i t i o n at the c e n t r e does not come i n t o e f f e c t u n t i l the p e n e t r a t i o n depth reaches the centre.At t h i s p o i n t the p r o f i l e has to be changed to take account of symmetry c o n d i t i o n . T h e new set of c o n s t r a i n t s are 3T(0,0)/3r=O , (3.3.12.a) 3T(R,0)/3r=-h.(T -T,)/K , (3.3.13.a) s r s T(R,0)=T . (3.3.14.a) s In dimensionless form the equations are 3*(0, T?)/3Z = 0 , (3.3.12.b) 3*(1 ,T?)/3z = - B i . (* -F)=-f (T?) , (3.3.13.b) *(l , r ? ) = * . (3.3.14.b) s Again adopting the suggestion by Lardner and Pohle the p r o f i l e w i l l take the form 4 9 3 2 *=(A.z +B.z +C.z)/z It should be emphasized that s i n c e the sphere i s s o l i d , t h e term 1/z should not appear in the f i n a l e x p r e s s i o n of the temperature p r o f i l e . T h i s i s why there i s no constant term i n c l u d e d in the polynomial of z i n the above equat i o n . A p p l y i n g the c o n s t r a i n t s the p r o f i l e w i l l take the form + f ( T? ) . ( 1-z 2)/2 . (3.3.15) s I n t e g r a t i o n of the d i f f u s i o n equation a f t e r e q u a t i o n ( 3 . 3 . 1 5 ) i s s u b s t i t u t e d for + w i l l g i v e * = - f ( T j ) / 5 - ( f f ( r j ) dr? )/Bi +constant . (3.3.16) s 0 The constant of i n t e g r a t i o n i s o b t a i n e d by the procedure e x p l a i n e d i n previous s e c t i o n , t h e r e s u l t i s + =- f ( T ? ) / 5 - ( / 7 ? f (77) dr?)/Bi , (3.3.17) s 0 where 77^  d e f i n e s the end of the i n i t i a l stage and the beginning of the second stage. 50 In summary If r7< 77^  Equat ion ( 3 . 3 . 1 1 ) should be used If V-VQ Equation(3.3.17)should be used 3.4 TEMPERATURE-DEPENDENT THERMAL PROPERTIES The usual s i m p l i f y i n g assumption of constant thermal p r o p e r t i e s made in de v e l o p i n g the mathematical model of regenerators can be relaxed.When the thermal p r o p e r t i e s are temperature-dependent, the d i f f u s i o n equation i s r e p l a c e d by p.C.3T/30=3(K.3T/3x)/3x . (3.4.1) Both K and p.C are temperature-dependent.The procedure to o b t a i n the. temperature d i s t r i b u t i o n w i l l be somewhat d i f f e r e n t to that adopted f o r constant thermal properties.Goodman [20] has demonstrated that only the thermal p r o p e r t i e s at the s u r f a c e enter the problem. T h i s s i m p l i f i e s the a n a l y s i s in a way that there w i l l s t i l l be only one unknown,ie.the s u r f a c e temperature. The procedure i s e x p l a i n e d i n more d e t a i l i n Goodman's paper [20]. 51 3.5 NUMERICAL PROCEDURE The f l u i d and s o l i d phase equations d e s c r i b e the i n t r a p a r t i c l e conduction model.The f l u i d phase equation i s solv e d n u m e r i c a l l y by a f i n i t e d i f f e r e n c e approximation, while the s o l i d phase equation i s sol v e d using the approximate i n t e g r a l method.The two unknowns are the f l u i d and the s o l i d temperature throughout the bed. The c a l c u l a t i o n i s c a r r i e d out f o r two types of reg e n e r a t o r s , namely f i x e d bed and moving bed. 3.5.1 FIXED BED The regenerator bed i s represented by a 2 dimensional g r i d . The length of the bed y (or A in dimensionless form) i s d i v i d e d i n t o n equal increments of 5y(or 5£), that i s A=n.5£ . (3.5.1) The p e r i o d P (or n i n dimensionless form) d u r i n g which the f l u i d i s passed through the bed i s d i v i d e d i n t o i equal increments of 56 or ( 5TJ in dimensionless form) n=i . 5r? (3.5.2) 52 At each poin t along the bed the f l u i d and s o l i d temperatures are represented as F ( n , i ) and q»(n,i) s r e s p e c t i v e l y . At each step po i n t there are two unknown temperatures,ie F(n,i+1) and *(n,i+1) , prov i d e d the temperatures at the ( n , i ) p o i n t are known. The f l u i d phase equation (eqn.2.9.b) i s represented by a c e n t r a l - d i f f e r e n c e approximation as (1+A£/2)F(n,i + 1 ) - * ( n , i + 1)A£/2=(1-A£/2)F(n-1 , i + 1 ) + s A£.«Mn-1 , i + 1 )/2 . The s o l i d phase equations i n v o l v e / f (77) dr? .In order to represent the equations in numerical form,the i n t e g r a l term i s approximated by the area under the curve f(r?) versus T? ,that i s AT? / f (T7)di7=Ar1 (n, i + 1 ) 0 =Ar1 (n,i)+Ar?[f ( n , i + 1 ) + f ( n , i ) ]/2 , (3. 5. 3. a) and / u f (r7)d?7=Ar2(n, i + 1 ) ^0 =Ar2(n, i )+AT?[ f (n, i + 1 )+f ( n , i ) ]/2, (3.5.3. b) where i n i t i a l l y 53 Arl(n,0)=0 , and Ar 2 (n , r?Q ) =0 . ( 3 . 5 . 3 . c ) The s o l i d phase equations can now be w r i t t e n i n t h e i r numerical form. Planar geometry For r?<r?0 , 5 Q(n,i)=[12Ar1(n,i)/(Bi.f(n,i))]°' 5 , (3.5.4) * ( n , i + 1 ) = [ 4 f ( n , i + 1).Ar1(n,i + 1)/(3Bi)]°' 5 . (3.5.5) s For T?>T?0 , * (n,i+1)=Bi.f(n,i+1)/3+Ar2(n,i+1)/Bi , (3.5.6) where f(n,i)=Bi.[F(n,i)-¥ ( n , i ) ] . (3.5.7) s In order to determine which equation should be used to o b t a i n the s o l i d s u r f a c e temperature,the p e n e t r a t i o n depth should be c a l c u l a t e d f i r s t . I f the p e n e t r a t i o n depth i s l e s s than the semi-thickness of the matrix ,equation (3.5.5) should be used, otherwise,equation(3.5.6)should be used. 54 S p h e r i c a l g e o m e t r y The same procedure as for the planar geometry i s adopted.The p e n e t r a t i o n depth and s o l i d s u r f a c e temperature are represented i n t h e i r f i n i t e d i f f e r e n c e form as For T J < T J , 2 3 55 ( n , i ) - 6 ( n , i ) =20(3-5 ( n , i ) ) . A r 1 ( n , i ) / ( 3 B i . f ( n , i ) ) , (3.5.8) 3* 2 ( n , i + 1 ) (5f (n, i + 1 )-24« (n, i + 1 ) ) = s s 2 0 ( * g ( n , i + 1 ) - f ( n , i + 1 ) ) 2 . A r 1 ( n , i + 1 ) / ( 3 B i ) . (3.5.9) For 77>7?0 , * (n,i+1)=-f(n,i+1)/5-Ar1(n,i+1)/Bi , (3.5.10) where f(n,i)=Bi.[¥ ( n , i ) - F ( n , i ) ] , (3.5.11) s where rj^ r e p r e s e n t s the end of f i r s t stage and the beginning of the second stage. In order to determine which of the above equations should be used, the p e n e t r a t i o n depth at 55 each step p o i n t should be c a l c u l a t e d f i r s t . At the (n,i) step p o i n t , t h e unknowns are F(n,i+1) and * s ( n , i + 1 ) , p r o v i d e d the temperatures at the (n-1,i+1) and ( n , i ) p o i n t s are known. The s t a r t i n g values f o r the s o l u t i o n are obtained from the i n i t i a l c o n d i t i o n which in t h e i r f i n i t e d i f f e r e n c e form are F(n,i)=1 At n=0 and i>0 , (3.5.12.a) F(n,i)=0 At i=0 and n>0 , (3.5.12.b) * (n,i)=0 At i=0 and n>0 , (3.5.12.c) s 6 (n,i)=0 At i=0 and n>0 . (3.5.12.d) At the entrance (ie.n=0)the f l u i d temperature does not vary with time,that i s F(0,i+1)=F(0,i)=1 . Consequently,there i s only one unknown temperature,¥ , which can be obtained from e i t h e r equation (3.5.6) or (3.5.5). 56 3.5.2 MOVING BED REGENERATOR In the case of a moving bed reg e n e r a t o r , t h e p e r i o d of op e r a t i o n i s not an independent c h a r a c t e r i s t i c of the system.It i s d e f i n e d as the time i t takes a s o l i d p a r t i c l e to t r a v e r s e one f u l l l e n gth of the regenerator chamber. Consequently,the p e r i o d of the c y c l e i s r e l a t e d to the regenerator height.The c o n d i t i o n s are steady at the entrance and e x i t of the regenerator. Only the temperature d i s t r i b u t i o n at d i f f e r e n t p o i n t s along the regenerator bed du r i n g one c y c l e i s of i n t e r e s t . To obta i n a maximum e f f i c i e n c y , a c o n t r a - f l o w arrangement should be adopted. The bed i s d i v i d e d i n t o n equal increments of A£.The d i r e c t i o n £ i s measured from the s o l i d e n t r a n c e . T h i s i s because the p e n e t r a t i o n depth during the i n i t i a l stages of the c y c l e must be obtained i n order to determine which equation should be used f o r the s o l i d temperature c a l c u l a t i o n . The s o l i d and f l u i d temperatures at each p o i n t along the regenerator are represented by " f^ C n J a n d F(n) r e s p e c t i v e l y ; * s ( n ) r e p r e s e n t s the s o l i d i n l e t temperature,whereas F(0) repr e s e n t s the f l u i d o u t l e t temperature. The f l u i d phase equation (eqn.2.9.b)is r e p r e s e n t e d by a c e n t r a l d i f f e r e n c e approximation as 57 F(n+1 )-F(n)=A£[F(n+1 )-¥ (n+1 )+F ( n ) - ¥ ' • ' ( ri ) ]/2 . ( 3 . 5 . 1 3 ) s s The s o l i d phase equations in t h e i r numerical form are shown below, Planar geometry For 77<7} 0 0.5 (3.5.14) * (n+1)=[4.f(n+1).Ar1(n+1)/(3.Bi)] 0.5 (3.5.15) For 77>7j 0 * (n+1)=f(n+1)/3+Ar2(n+1)/Bi (3.5.16) where f(n)=Bi.[F(n)-¥ (n)] (3.5.17.a) Ar 1 (n+1) = / f ( 7 j ) dr? (3.5.17.b) 0 Ar2(n+1)=/ f (rj) drj . (3.5.17.c) 58 S p h e r i c a l geometry For r?<r?Q , 5 5 2 ( n ) - 6 Q ( n ) = 2 0 ( 3 - S 0 ( n ) ) A r 1 ( n ) / ( 3 B i . f ( n ) ) , (3.5.18) 3* 2(n+1)[5f(n+1)-2* (n+1)] = 20[* (n+1)-f(n+1) ] 2 s s s ,Ar1(n+1)/(3Bi) . (3.5.19) For 7j>7?0 , * (n+1)=-f(n+1)/5-Ar1(n+1)/Bi , (3.5.20) s where f ( n ) = B i . [ * ( n ) - F ( n ) ] , (3.5.21.a) s Ar? Ar1(n+1)=/ f(r?) dr? , (3.5.21.b) 0 Ar2(n+1)=/ 0 f(r?) dr? . (3.5.21.c) ^0 It should a l s o be emphasized that Ar?=A£mC/(M C u ) . (3.5.22) s s s At each p o i n t along the regenerator there are two unknowns,namely, * s ( n + l ) a n d F(n+1),provided that "¥ (n)and F(n) are known, s 59 The a n a l y s i s proceeds by using the average of the f l u i d and s o l i d i n l e t temperatures as a f i r s t approximation f o r the f l u i d o u t l e t temperature.The c a l c u l a t e d i n l e t temperature i s then compared with the a c t u a l given i n l e t temperature.If there i s any discrepancy the i n i t i a l approximation i s a d j u s t e d and the a n a l y s i s i s then repeated. 4. RESULTS AND DISCUSSION The f i r s t stage i n a n a l y s i n g the r e s u l t s i s to e s t a b l i s h the v a l i d i t y of the approximate i n t e g r a l method.This i s achieved by comparing the r e s u l t s obtained a g a i n s t the r e s u l t s p u b l i s h e d by d i f f e r e n t authors. U n f o r t u n a t e l y the p u b l i s h e d r e s u l t s are scarce.Most of the p r e v i o u s a n a l y s e s have d i s r e g a r d e d i n t r a c o n d u c t i o n e f f e c t s . O n e source of r e s u l t s a v a i l a b l e are those p u b l i s h e d by Handley and Heggs [12] ,who employed the Crank-Nicholson scheme to s o l v e the d i f f u s i o n equation numerically.The r e s u l t s are mainly f o r a f i x e d bed regenerator with a s p h e r i c a l m a t r i x . 4.1 FIXED BED 4.1.1 SPHERICAL GEOMETRY The d i f f u s i o n equation,coupled with the a p p r o p r i a t e boundary c o n d i t i o n s , were s o l v e d using the approximate i n t e g r a l method f o r d i f f e r e n t B i o t numbers.The r e s u l t s where then compared with those p u b l i s h e d by Handley and Heggs. Figs.11,12,13 represent the f l u i d o u t l e t temperature f o r d i f f e r e n t B i o t numbers.An e x c e l l e n t agreement between the two methods can be seen.The comparison of the r e s u l t s were made f o r a range of dimensionless parameters 1<A<40 and 0<Bi<5 which cover the design range of most i n d u s t r i a l r e g e n e r a t o r s . 60 61 Experimental s t u d i e s c a r r i e d out by the same authors r e v e a l e d an e x c e l l e n t accuracy of the r e s u l t s [12]. 4.1.2 PLANAR GEOMETRY Un f o r t u n a t e l y no r e s u l t s c o u l d be found f o r comparing the numerical and i n t e g r a l method f o r the plana r geometry.Most of the pre v i o u s s t u d i e s have c o n c e n t r a t e d on the s p h e r i c a l geometry (which i s a more p r a c t i c a l geometry). The f l u i d o u t l e t temperature p r o f i l e i s expected to f o l l o w a c h a r a c t e r i s t i c t rend (S shaped p r o f i l e ) i f the method i s c o r r e c t . I t i s apparent from Fig.14 that the f l u i d o u t l e t temperature f o l l o w s the expected trend.Thus i t can be deduced that the i n t e g r a l method i s e q u a l l y a p p l i c a b l e to the p lanar geometry.Fig.14 r e p r e s e n t s the f l u i d o u t l e t temperature p r o f i l e f o r d i f f e r e n t reduced l e n g t h s . Once the v a l i d i t y of the method i s e s t a b l i s h e d , the a n a l y s i s can be extended to examine the s e v e r i t y of i n t r a c o n d u c t i o n e f f e c t f o r d i f f e r e n t geometries. T h i s i s achieved by comparing the d i f f e r e n c e between the s o l i d s u r f a c e and mean temperature f o r the two common geometries at d i f f e r e n t B i o t numbers. 62 It i s expected that as the B i o t number i s inc r e a s e d , the d i f f e r e n c e between the s o l i d s u r f a c e and mean temperature should i n c r e a s e . T h i s i s because as the B i o t number i n c r e a s e s ( i e . K decr e a s e s ) , the s depth i n t o which the heat f l u x at the sur f a c e p enetrates the s o l i d decreases. Consequently, the surf a c e temperature w i l l be much higher than the temperature at the ce n t r e of the sphere.Thus the r e s u l t i n g mean temperature i s s m a l l e r . ( r e f e r to F i g , 8 ) . To show t h i s , t h e graph of percentage d i f f e r e n c e between the s o l i d s u r f a c e and mean temperature was p l o t t e d f o r d i f f e r e n t B i o t numbers (Figs.15,16,17). The percentage d i f f e r e n c e was obtained from D i f f e r e n c e = ( * -+ )/+ s m s E v i d e n t l y , a s the B i o t number i n c r e a s e s the maximum d i f f e r e n c e between the two temperatures i n c r e a s e s , as e x p e c t e d . l t can be seen ( F i g . 17) that f o r a moderate B i o t number (Bi=2) a maximum d i f f e r e n c e between the two temperatures i s about 9 percent f o r the planar geometry.Consequently, a model which n e g l e c t s the i n t r a c o n d u c t i o n e f f e c t overestimates the e f f e c t i v e n e s s . I t i s a l s o evident from the same ch a r t s that for a f i x e d d i m e n s i o l e s s group (A and Bi) ,the 63 d i f f e r e n c e between the sur f a c e and mean temperature i s g r e a t e r f o r the plana r geometry than that f o r the s p h e r i c a l geomtry.That i s , t h e i n t r a s p h e r e conduction mechanism has l e s s s i g n i f i c a n c e than i n t r a p l a n a r conduct i o n . T h i s i s because f o r an equal B i o t number and reduced l e n g t h (A), the sphere rad i u s has to be small e r than the s l a b t h i c k n e s s ( f o r the same m a t e r i a l ) . Carpenter and Heggs a r r i v e d at the same c o n c l u s i o n i n t h e i r a n a l y s i s [ 5 ] . Most of the i n v e s t i g a t o r s have t r i e d to p r e d i c t the d i v i d i n g l i n e between the Schumann and i n t r a c o n d u c t i o n models [ 5 ] . The d i v i d i n g l i n e i s u s u a l l y expressed i n terms of dimensionless parameters (A, n ,Bi) .The main reason f o r t h i s i s to reduce the computation time r e q u i r e d to sol v e the model n u m e r i c a l l y . T h i s of course i s not important i f the problem i s sol v e d a n a l y t i c a l l y . 4.2 MOVING BED REGENERATOR Since there are no r e s u l t s a v a i l a b l e f o r a moving bed regenerator, the v a l i d i t y of the r e s u l t s presented here are based upon the v a l i d i t y of the method. As f o r the f i x e d bed, the d i f f e r e n c e between the s o l i d s u r f a c e and mean temperature i n c r e a s e s as the Biot number in c r e a s e s ( F i g . 1 8 ) . Consequently, at a f i x e d A, the mean s o l i d temperature decreases as the B i o t number 64 i n c r e a s e s . S i n c e the regenerator e f f e c t i v e n e s s i s d e f i n e d in terms of the s o l i d mean temperature, the r e d u c t i o n i n the former i s r e f e l e c t e d i n the r e d u c t i o n of the l a t t e r . S o i t can be deduced that as the Bi o t number i s in c r e a s e d , the e f f e c t i v e n e s s should d e c r e a s e . T h i s i s shown in Fig.19. I t should be poi n t e d out that f o r a f i x e d thermal c o n d u c t i v i t y , the B i o t number reduces as the s i z e of the sphere i s reduced.So one would expect a higher e f f e c t i v e n e s s f o r a matrix of smaller s i z e . B u t the s i z e of the sphere i s l i m i t e d by i t s t e r m i n a l v e l o c i t y . T h a t i s , i f the spheres are too sm a l l , they might be blown o f f the top by the oncoming f l u i d . The a n a l y s i s r e v e a l e d that as the B i o t number i s inc r e a s e d ,the d i s t a n c e increment at which the s o l i d and f l u i d temperatures are c a l c u l a t e d must be reduced.This r e d u c t i o n i n the d i s t a n c e increment (S£) might cause a s t a b i l i t y problem at very high B i o t numbers.The s t a b i l i t y problem i s due to the form of the temperature p r o f i l e i n the s o l i d . T h e present d e r i v e d p r o f i l e i s based on Lardner and Pohle [18] suggestion that ¥O0/z. Fur t h e r a n a l y s i s r e v e a l e d that the sphere should be assumed to be s o l i d even at the i n i t i a l stages where the p e n e t r a t i o n depth concept i s a p p l i e d , then the term (constant/z) must be omitted from the f i n a l e x p r e s s i o n f o r the s o l i d temperature p r o f i l e . T h i s would r e s u l t s i n a second form of p r o f i l e which i s of the form 65 *=-f ( T J ) [ . Z - ( 1 - 6 0 ) ] 3 / ( 3 5 0 ) 2 . The i n t e g r a t i o n of the d i f f u s i o n equation w i l l r e s u l t in For 77<770 , 27* 4 + 54* 3.'f (TJ.)+45* 2 . f ( 7 ? ) 2 = 20f (r?) 3. ( / ^  f ( T J ) d r j ) / B i ' . S S S Q For 77>7?Q , * =-.9/7?0 tin) dr?/Bi - f ( r j ) / 5 -fV f(r?) drj/Bi . 0 r,0 S u r p r i s i n g l y , f o r I J>^, the s o l i d temperature i s almost i d e n t i c a l to the o r i g i n a l d e r i v e d s u r f a c e temperature based on Lardner and Pohle suggestion.The s o l u t i o n to the above equations were compared with the o r i g i n a l r e s u l t s . T h e r e s u l t s were extremely c l o s e f o r low B i o t numbers as i s evident from Figs.20,21.However,the advantage of t h i s second p r o f i l e i s that i t i s s t a b l e even f o r high B i o t numbers (Bi>8).This i s shown by p l o t t i n g the c h a r t s of e f f e c t i v e n e s s versus reduced l e n g t h f o r d i f f e r e n t B i o t numbers (Figs.22,23,24). I t i s t h e r f o r e suggested that at high B i o t numbers the second p r o f i l e must be used. I t i s evident from the c h a r t s of e f f e c t i v e n e s s versus reduced l e n g t h that the e f f e c t of B i o t number i s very s i g n i f i c a n t . F o r example, c o n s i d e r Fig.24 which shows the e f f e c t i v e n e s s versus reduced l e n g t h f o r Bi = 0.1 and Bi = 10. I t 66 can be seen that i n order to o b t a i n the same e f f e c t i v e n e s s ( f o r example 60 p e r c e n t ) , the reduced length (or the bed length) has to be almost t r i p l e d in the case of B i = l 0 . T h i s c r i t i c a l i n formation had not been a v a i l a b l e p r i o r to t h i s work. F i n a l l y ,from the d e f i n i t i o n of e f f e c t i v e n e s s , i t i s apparent that as the c a p a c i t y rate r a t i o ( R ) 3 i s reduced, the e f f e c t i v e n e s s reduces.This i s shown in Fig.25 . R=(m .C )/(m.C) s s •5. CONCLUSION The approximate i n t e g r a l method was employed to o b t a i n an a n a l y t i c a l s o l u t i o n to t r a n s i e n t response of a regenerator matrix; the f o l l o w i n g p o i n t s are concluded, 1. The approximate method g i v e s r e s u l t s that agree w e l l with the more exact methods employed f o r the case of a f i x e d bed r e g e n e r a t o r . 2. The e f f e c t of B i o t number i s much more severe than expected i n the case of a moving bed regenerator. 67 6. AREAS OF FURTHER RESEARCH The i n t r a c o n d u c t i o n model i s based on a number of s i m p l i f y i n g assumptions .The use of an i n t e g r a l method enables one to r e l a x some of these assumptions. In order to examine the e f f e c t of these assumptions the f o l l o w i n g i s suggested, 1. The s i m p l i f y i n g assumption of constant thermal p r o p e r t i e s ( f o r the s o l i d ) should be r e l a x e d . T h i s can be e a s i l y achieved with the use of the i n t e g r a l method [19]. 2. I t i s b e l i e v e d that the bed ex t e n s i o n i n the d i r e c t i o n of gas flow c o n t r i b u t e s to a uniform temperature p r o f i l e [ 8 ] . Thus the s i m p l i f y i n g assumption of uniform f l u i d v e l o c i t y should be r e l a x e d . 68 Fluid outlet temperature VS Reduced time For a fixed bed III AC Ul o.t Q. Z Ul 0.7 Ul 0.4 mi "* 0.1 o Q Ul N 0.4 O.J O.t 0.1 Spherical geometry B1=h.R/K=.02 0<> 2 4 6 9 10 12 Reduced time (h*A*t/M*C) Legend A N u m e r l c o l Method O Integral Method 14 Figure 11. Comparison between the numerical and a n a l y t i c a l method employed Fluid outlet temperature VS Reduced time fixed bed,BI=.25 Legend A Num«rlcol Method Reduced time (h*AM/M*C) F i g u r e 12. Compar i son between the n u m e r i c a l and a n a l y t i c a l method employed Fluid outlet temperature VS Reduced time fixed bed,BI=2 i Legend A N u m e r i c a l Method O Integral Method Reduced time (h*AM/M*C) F i g u r e 1 3 . Compar i son between t he nume r i c a l and a n a l y t i c a l method employed Fluid outlet temperature VS Reduced time For a fixed bed ui Reduced time (h*A*t/M*C) F i g u r e 1 4 . The c h a r a c t e r i s t i c S shaped c u r v e s o f f l u i d o u t l e t t e m p e r a t u r e p r o f i l e f o r a f i x e d bed r e g e n e r a t o r ( v a r i o u s r educed l e n g t h ) . ro % Difference between solid surface and mean Temperatures, 0.8 5 0.7 a E 0.6 0.8 S 0.4 • o.s 0 0.2 Q B i= . 02 % D 1 f f = ( V T m ) / T s F i x e d bed Reduced l e n g t h = 8 / / / m m m \ M s 6 10 Reduced time (h*A*t/M*C) 16 Legend A Plonor geometry ° SRiJsr.'sfi.iissinsJr.y F i g u r e 1 5 . The e f f e c t o f t he rma l c o n d u c t i v i t y on s o l i d t e m p e r a t u r e p r o f i l e ( B i = . 0 2 ) . — i OJ % Difference between solid surface and mean Temperatures, 3 • 2 o o Fixed bed % D1f f=(T s -T m )/T s B1=0.25/ . ^ - " ^ ^ Reduced length=5 £ * *t / f / / ""o 5 10 Reduced time (h*AM/M*C) 15 Legend A Plonor geometry O Sphericol g«om«fry Figure 16 . The e f fec t o f thermal conduct iv i ty on the s o l i d temperature p r o f i l e (Bi =0.25). % Difference between solid surface and mean Temperatures, 10 a E 2 Fixed bed Bi=2 % D1ff=(T s -T m )/Tjj Reduced 1ength=40 8 10 Reduced time (h*A*t/M*C) 18 Legend A Plonor geometry ° S p h e r l c d geometry Figure 17. The e f f e c t of thermal conduct iv i ty on s o l i d temperature p r o f i l e ( Bi=2 ), % DIFFERENCE BETWEEN SOLID SURFACE AND SOLID MEAN TEMPERATURE to - i 1 0 1 2 0 4 9 0 7 0 0 Reduced length (h*A*L/M*C) F i g u r e 1 8 . The e f f e c t o f t he rma l c o n d u c t i v i t y on s o l i d t e m p e r a t u r e p r o f i l e (Mov ing b e d ) . MOVING BED EFFECTIVENESS VS REDUCED LENGTH Reduced length (h*A*L/M*C) Figure 19. The e f fec t o f thermal conduct iv i ty on regenerator e f fec t i veness ( o r i g i n a l p r o f i l e ) SOLID SURFACE TEMPERATURE VS REDUCED LENGTH Bl=2 o CO 1000 •00 •00 700 t o o •00 400 t o o t o o 100 Moving bed regenerator Spherical geometry BI»h.1VK=2 Capacity r a t e ra t1o=l $ 6 A A A A A A A A A A A A A 6 6% o-* 0 T r 1 — i 1 — i — 1 2 » « • • Reduced length (h*A*L/M*C) Legend o, Original prof «• O Altarnatlva prolIto I Figure 20. The comparison between the o r i g i n a l p r o f i l e and the alternative p r o f i l e MOVING BED REGENERATOR EFFECTIVENESS VS REDUCED LENGTH 0-A-0 E f f = ( T s m 0 - T s 1 ) / ( T f 0 - T S m 1 ) S p h e r i c a l g e o m e t r y B i=h .R /K=2. C a p a c i t y r a t e r a t i o = l 1 2 3 4 5 6 7 Reduced length (h*A*L/M*C) 8 Legend A Original profile O Alternativ^profile F i g u r e 21 • The c o m p a r i s o n b e t w e e n t h e e f f e c t i v e n e s s b a s e d o n t h e t w o p r o f i l e s ( Bi=2 ). Effectlvness VS Reduced length (NTU) CD O > • Ul Capacity rate rat io(R)=l Spherical geometry Bi=h.R/K Legend • 1 - 0 . 1 B I - O . B • 1 - 1 . 0 B I - 2 . 0 2 8 4 8 Reduced length (NTU) Figure 22. Moving bed regenerator « effectiveness based on the alternative profile. Effectiveness VS Reduced length (NTU) 1 O f l i i i i I 0 1 2 3 4 8 0 7 Reduced length (NTU) F i g u r e 2 3 . Mov ing bed r e g e n e r a t o r e f f e c t i v e n e s s based on t h e a l t e r n a t i v e p r o f i l e . Effectiveness VS Reduced length (NTU) F i g u r e 24 . Mov ing bed r e g e n e r a t o r e f f e c t i v e n e s s based on t h e a l t e r n a t i v e p r o f i l e . Figure 25. THE EFFECT OF C A P A C I T Y RATE RATIO ON EFFECTIVENESS R=M1*C1/M2*C2 1 a I" 4 • • 7 Reduced length (h*A*L/M*C) Legend A R=1 O R=1/2 • R=1/3 . . . B R=1A X R=1^ 6_ CO OJ 84 BIBLIOGRAPHY Hryniszak.W. "Heat exchangers", 1958 Womack.H. "Open c y c l e MHD power g e n e r a t i o n " ,1969 Bayley.F.J Owen.J and Turner.A "Heat t r a n s f e r " , 1972 Carpenter,K.J. and Heggs.P.J. " P r e d i c t i o n of d i v i d i n g l i n e between conduction and convection e f f e c t s in regenerator design" , Trans.I.Chem.Eng. v o l 56,1978 Carpenter.K.J. and Heggs.P.J. "The e f f e c t s of packing m a t e r i a l , s i z e and arrengment on the performance of thermal regenerators ", Int.Heat t r n s f e r . C o n f 6-th, Toronto 1978 Willmot.A.J "The r e g e n e r a t i v e heat exchanger computer r e p r e s e n t a t i o n " J.Heat mass t r a n s f e r . V o l 12,P 997, 1969 Horn.G Sharp.A and Hryniszak.W " A i r heaters and seed recovery from MHD p l a n t " ,Phi 1.Trans.Roy.Soc. Vol 261,1967 S c h n e l l e r . J and Halvacka.V "Moving bed heat exchangers and r e g e n e r a t o r s " ,Heat exchanger design and theory source book, Chap27 Carpenter.K.J and Heggs.P.J. "A m o d i f i c a t i o n of the thermal regenerator i n f i n i t e conduction model to p r e d i c t the e f f e c t s of i n t r a c o n d u c t i o n " , Trans.I.Chem.Eng.Vol 57,1979 Cserveny.I. " C o n t r i b u t i o n s to the thermal design of moving pebble bed r e g e n e r a t i v e a i r h e a t e r s " ,Revue.roumanie des s c i e n c e s techniques, V o l 22 ,No 1, P125,1977 Furnas.C. "Heat t r a n s f e r from a gas stream to a bed of broken s o l i d s ", Indus.Eng.Chem, 1930 Handley.D and Heggs.P.J. "The e f f e c t of thermal c o n d u c t i v i t y of the packing m a t e r i a l on t r a n s i e n t heat t r a n s f e r in a f i x e d bed " ,J.heat mass t r a n s f e r . Vol 12.P549 1969 Schumann.T.E "A l i q u i d flowing through a porous prism" ,J of F r a n k l i n I n s t . V o l 208, 1929 Saez.A and Mccoy.B.J " T r a n s i e n t a n a l y s i s of packed bed thermal storage system" Int.J.Heat mass t r a n s f e r . Vol 26, NO1,1983 Hausen.H " v e r v o l l s t a n d i g a t e Berechnung des Warmeaustauches in regeneratoren". Z.Ver.Dt.Ing.No2,31-l942. Goodman.R.T "The heat balance i n t e g r a l and i t s a p p l i c a t i o n to problems i n v o l v i n g a change of phase", Heat t r a n s f e r and f l u i d mechanics I n s t i t u t e , C a l i f o r n i a i n s t i t u t e of technology, Pasadena June 1957, p383. Carslaw.H.S. and Jauger.J.C. "Conduction of heat i n s o l i d s " Oxf.Univ.Press, london 1959. Lardner.T.J and Pohle.F.V ,J.app.Mech V o l 28,1961 Goodman.R.T "The heat balance i n t e g r a l - f u r t h e r c o n s i d e r a t i o n and refinements" ,Trans.ASME. (J.heat t r a n s ) , 1961 86 20. Koh.C.Y. "One dimensional heat conduction with a r b i t r a r y heating r a t e and v a r i a b l e p r o p e r t i e s " , J . o f Aerospace S c i n c e , l 9 6 l 21. McConnachie.T and Thodos.G "Transfer process in the flow of gases through packed and dis t e n d e d bed of spheres" ,A.I.CH.E j o u r n a l 1963 22. Rowe.P.N Claxton.K.J and Lewis.T.B "Heat and mass t r a n s f e r from a s i n g l e sphere in an e x t e n s i v e f l o w i n g f l u i d " ,Trans.Inst.Chem.Eng,Vol 43,1965 23. K r e y s z i g . E "Advanced e n g i n e e r i n g mathematics" 1967 24. Nakada.T ,Nakamura.N ,Narita.Y and T a i r a . T "Studies of f a l l i n g p a r t i c l e s regenerator", 5-th Int.Conf.MHD. 1971 APPENDIX A:DERIVATION AND DIMENSIONAL ANALYSIS A . l Dimensionless parameters I t i s a common p r a c t i c e i n regenerators thermal design to represent the r e s u l t s in terms of a group of dim e n s i o n l e s s parameters. The governing equations w i l l a l s o be s i m p l i f i e d by t r a n s f o r m i n g the independent v a r i a b l e s y , 0 and x(or r ) i n t o these dimensionless parameters. Dimensionless l e n g t h T h i s i s d e f i n e d i n the f o l l o w i n g manner £=h.A.y/(m.C) . (A.1.1) The number of t r a n s f e r unit(N.T.U) i s e q u i v a l e n t to £ at y=L, that i s A=h.A.L/(m.C) . (A.1.2) The number of t r a n s f e r u n i t i s a l s o c a l l e d the reduced l e n g t h . 87 88 D i m e n s i o n l e s s time T h i s i s d e f i n e d as r j=h.A. (0-y/u)/(M .C ) . (A.1.3) The dimensionless p e r i o d of one c y c l e o p e r a t i o n , n, i s eq u i v a l e n t to rj at 6= p e r i o d , that i s For moving bed rege n e r a t o r s , t h e p e r i o d of one c y c l e i s d e f i n e d as the time r e q u i r e d by the s o l i d to t r a v e l one f u l l l e n g t h of regenerator.The dimensionless time i s expressed in terms of the dimensionless l e n g t h . The time 6 r e q u i r e d by the s o l i d to t r a v e l a f u l l l e n g t h of regenerator i s S u b s t i t u t i n g f o r 6 i n equation (A.1.3) and i g n o r i n g the f l u i d r e s i d e n c e time y/u ,the dimen s i o n l e s s time w i l l be n=h.A.(P-L/u)/(M .C ) . (A.1.4) s s 0=y/u s T j = h.A . y/(M .C .u ) , s s s (A.1.5.a) but from (A.1.1) 89 y=m.C. £/h.A (A.1.5.b) so 77= £ .m.C/(M .C .u ) . (A. 1 .5) s s s P o r o s i t y ( B ) The p o r o s i t y or v o i d f r a c t i o n i s d e f i n e d as B=(V, -n.V )/V, , (A.1.6.a) b s b where V, i s the bed volume and n i s the number of s o l i d b p a r t i c l e s in the b e d . P o r o s i t y can a l s o be d e f i n e d as B=(p -M )/p . (A.1.6.b) s s s H e a t t r a n s f e r a r e a The heat t r a n s f e r area i s d e f i n e d as A = ( p a r t i c l e s s u r f a c e area)/bed volume , or A=(n.A )/V. , (A.1.7) s b but from (A.1.6.a) V =(n.V ) / ( l - B ) , b s 90 so s u b s t i t u t e f o r V, i n equation (A.1.7),the r e s u l t i s b A=A .(1-B)/V . . (A.1.8) s s For planar geometry equation (A.1.8) reduces to A=(1-B)/d d=semi-thickness . (A.1.9.a) For s p h e r i c a l geometry equation (A.1.8) reduces to A=3.(1-B)/R . R=radius . (A.1.9.b) In p r a c t i c e the matrix i s not of r e g u l a r geometry.The regenerator i s u s u a l l y composed of broken rocks. I t i s t h e r e f o r e r e q u i r e d to express the c h a r a c t e r i s t i c s i z e of the matrix in terms of an e q u i v a l e n t s p h e r i c a l diameter.This i s d e f i n e d as 3 D =[6.net volume of rocks /7r.n] . (A.1.10) ev If the rocks are a l l of the same shape and s i z e , the above equation can be w r i t t e n as D 3 =[6.V /TT] ev s (A.1.11) 91 Heat t r a n s f e r c o e f f i c i e n t I t i s a common p r a c t i c e to represent the heat t r a n s f e r c o e f f i c i e n t in terms of flow Reynolds number.There are a number of d i f f e r e n t c o r r e l a t i o n s suggested f o r t h i s purpose,some of which are l i s t e d in Table.1 . Experimental s t u d i e s have shown that the degree of packing (or p o r o s i t y ) has a very l a r g e i n f l u e n c e on the heat t r a n s f e r c o e f f i c i e n t [9] .Consequently,the c o r r e l a t i o n s which i n c l u d e such e f f e c t are more c o n s e r v a t i v e . There are two ways to account f o r p o r o s i t y e f f e c t , t h e s e are 1.It i s suggested [8,20] that the p o r o s i t y e f f e c t should be in c l u d e d in Reynolds number c a l c u l a t i o n . R e y n o l d s number i s then r e d e f i n e d as Re = u.D /[ (1-B) . !»]• , ev Re=u.D /(B.v) 5 , ev these are so c a l l e d m o d i f i e d Reynolds number. 2.The p o r o s i t y can be taken i n t o c o n s i d e r a t i o n as an independent v a r i a b l e . Most of the c o r r e l a t i o n s l i s t e d i n Table.1 are based on such a c o n s i d e r a t i o n . " r e f 20 5 r e f 8 92 It i s r e a d i l y shown [21] that in the absence of n a t u r a l c o n v e c t i o n , the Nusselt number of a s i n g l e sphere in an e x t e n s i v e flow approaches 2 ,when the Reynolds number approaches O.The c o r r e l a t i o n s that are based on such f i n d i n g are thus more a c c u r a t e . In the present study,the c o r r e l a t i o n due to I.S.Cservery [10] was u t i l i z e d f o r heat t r a n s f e r c a l c u l a t i o n . C o r r e l a t i o n s Author Comments N u - 0 . 3 3 2 P r 1 / 3 R e 0 , 5 T h i s i s used fo r chequer work matrix N u - ( . 2 5 5 / B ) * P r 1 / 3 R e 2 / 3 Handley and Heggs Nu«(.29A/B)PrRe 0' 7 S c h n e l l e r . J Modified Reynolds number n , 0.67 D 1.3 Nu».0l6Pr Re F r a n t z . J P o r o s i t y e f f e c t ignored N u = 2 + 0 . 6 9 P r 1 / 3 R e ° * 5 Rowe.P.N for a s i n g l e sphere For Re>l00 Nu-2+0.6(3.25-2.25B)Re°' 5Pr 1 / 3 For Re<100 Nu-2+6(3.25-2.25B)Pr l / 3(Re/l00) 1 , 6 9 Cserveny.I An own i n t e r - p o l a t i o n formula Table. 1. Correlations f o r the convective heat t r a n s f e r c o e f f i c i e n t . 94 A.2 D e r i v a t i o n of Governing equations The governing equations are developed f o r a general case of moving bed regenerators.These are then m o d i f i e d f o r the case of f i x e d bed regenerator a c c o r d i n g l y . The t o t a l heat flow to the f l u i d with mass flow r a t e m, between y and y+dy (where y i s measured from f l u i d entrance) comprise two components.The f i r s t component i s the heat t r a n s f e r r e d to the mass (m.A, .d0) i n moving between y and b y+Sy, that i s dQ1=(m.d0.A).C.(3T r/3y) .dy . (A.2.1.a) b f 0 J The second component i s the heat t r a n s f e r r e d to the f l u i d e n closed between y and y+dy as i t s temperature changes with time,that i s dQ2=p.A b.dy.C.(3T f/ae) .dfl , (A.2.1.b) but m=p.u ,so dQ2=(m.A./u).C.dy.(3T,/30) .d0 . (A.2.1.c) b f y The t o t a l heat t r a n s f e r r e d to the f l u i d i s then dQ=m.C.A .[(3T /3y)+ (3T /30)/u].dy.d0 . (A.2.2) T h i s i s equal to the heat l o s t from the s o l i d , t h a t i s 95 dQ=h. (A. dy . A ) . (T -T ).d0 . (A.2.3) b s r Equating the t o t a l heat l o s s from the s o l i d to the t o t a l heat gain by the f l u i d ,the r e s u l t i s m.C.[(3T /3y)+(3T /30)/u]=h.A.(T -T f) . (A.2.4) From chain r u l e DT f/Dy=3T f/3y+(3T /30)/u , so equation (A.2.4) can be w r i t t e n as m.C.dT r/dy=h.A.(T -T,) . (A.2.5) f s f The heat l o s t from the s o l i d can a l s o be expressed i n terms of i t s r a t e of change of i n t e r n a l e nergy.Defining U as the i n t e r n a l energy / u n i t area then d U=p .C .f T dx f o r one s o l i d , (A.2.6.a) s s o d U=M .C .f T dx f o r the whole bed . (A.2.6.b) s s J 0 As f o r the f l u i d the t o t a l change i n s o l i d i n t e r n a l energy comprises two components,that i s h.A.(T -T )=(dU/d0)/d , (A.2.7) s f 96 where cl cl dU/d#=M C [u . 9 ( / T dx)/9y + 9(f T dx)/30] . (A.2.8) s s s o 0 If the bed i s s t a t i o n a r y then u^=0 and h.A(T -T,)=-(M .C / d ) . 9 ( J T dx)/30 . (A.2.9) s f s s o Note that the sign d i f f e r e n c e between equations (A.2.9) and (A.2.8) i s due to the d i r e c t i o n i n which y i s measured. It can be seen that i f the s o l i d temprature i s uniform throughout ,then equations (A.2.8 and A.2.9) reduce to the s o l i d phase equations f o r the Schumann model. In the present study, the two equations used to model the regenerator are equations (A.2.5) and the d i f f u s i o n equation,which i s 2 2 p C (3T/30)=K (3 T/9x ) Planar geometry , (A.2.10.a) =K [ 9 2 T / 9 r 2 + ( 2 / r ) . 9 T / 9 r ] S p h e r i c a l . (A.2.10.b) s The d i f f u s i o n equation i s coupled with the f o l l o w i n g boundary c o n d i t i o n s 97 l.The symmetry c o n d i t i o n (3T/3x). =0 planar , (A.2. 11.a) x = d (3T/3r) = 0 S p h e r i c a l . (A.2.11.b) r = 0 2.Heat f l u x at the s u r f a c e K .(3T/3x) = h.(T -T ) planar , (A.2.12.a) s x=0 s f K .(3T/3r) _ =h.(T -T ) s p h e r i c a l . (A.2.12.b) s r — H r s The above equations are nondimensionalised in terms of £, rj,Bi and normalised temperatures. From the d e f i n i t i o n of these parameters the f o l l o w i n g can be deduced , dT =(T -T ).dF , dT=(T..-T.).d* , f 0 l 36>=M .C .3r?/(h.A) , s s 3y=m.C.3£/(h.A) , T -T.=(* - F ) . ( T , -T.) , S f S fO l dz=dr/R For s p h e r i c a l geometry , 98 dz=dx/d F o r p l a n a r g eometry . E q u a t i o n (A.2.5) c a n now be w r i t t e n i n d i m e n s i o n l e s s form as * -F=dF/d£ . (A.2.13) s F o r moving bed r e g e n e r a t o r s , y i s measured from t h e s o l i d e n t r a n c e . C o n s e q u e n t l y , e q u a t i o n (A.2.13) must be a l t e r e d a c c o r d i n g l y , t h e r e s u l t i s F - * =dF/d£ . (A.2.14) s The d i f f u s i o n e q u a t i o n ( A . 2 . 1 0 ) can a l s o be w r i t t e n i n d i m e n s i o n l e s s form as p l a n a r g e o m e t r y p .C .h.A. (3¥/3ij)/(M .C ) = K . ( 3 2 + / 3 z 2 ) / d 2 , s s s s s from t h e d e f i n i t i o n o f p o r o s i t y (B) , M =p .(1-B) , s s from e q u a t i o n (A.9.a) A = ( l - B ) / d .So s u b s t i t u t e f o r A and M g i n above e q u a t i o n , t h e n 9*/3T7=K . ( 3 24>/3z 2)/(h.d) , s o r 3 * / 3 T ? = ( 3 2 * / 3 z 2 ) / B i . (A.2.15) 99 The c o r r e s p o n d i n g c o n s t r a i n t s a r e 1 . S y m m e t r y c o n d i t i o n U * / 3 z ) =0 (A. 2. 16) z=1 2.The h e a t f l u x a t t h e s u r f a c e K .0¥/9z)/d=h.(* -F) , s s t h a t i s ( 3 * / 3 z ) =Bi.(¥ -F) . (A.2.17) z = 0 s s p h e r i c a l g e o m e t r y The d i f f u s i o n e q u a t i o n i n d i m e n s i o n l e s s f o r m w i l l be p C h.A( 3¥/3T?)/M C =K [ ( 3 2 * / 3 z 2 ) / R 2 + ( 2 / z . R ) ( 3 * / 3 Z ) / R ] , s s s s s bu t M -p . (1-B) , s s and A=3.(1-B)/R , so 3(3*/3T7) .h/R=k /R 2[ 3 2*/9z 2+(2/z) .9¥/3z] , s o r 3*/3rj=[ 3 2*/3z 2+(2/z) . ( 9*/3z) ]/(3Bi ) . (A.2.18) 100 The c o r r e s p o n d i n g c o n s t r a i n t s are 1.Symmetry c o n d i t i o n (3¥/3z) =0•. (A.2.19) z = 0 2 .Heat f l u x at the surface (K /R).(9*/9z)=h.(F-* ) , s s that i s (3*/3z) =Bi.(F-* ) . (A.2.20) z= 1 s APPENDIX B:INTEGRAL METHOD B . l Planar geometry The matrix ( u s u a l l y chequrework) i s modelled i n two stages S e m i - i n f i n i t e s l a b The d i f f u s i o n equation i n dimensionless form i s The p e n e t r a t i o n depth,defined in chapter 3,has a c h a r a c t e r i s t i c p roperty such that the s o l i d i s at an e q u i l i b r i u m temperature ( i n i t i a l temperature) at any p o i n t beyond p e n e t r a t i o n depth. The matrix i s sub j e c t to the f o l l o w i n g d i m e n s i o n l e s s boundary c o n d i t i o n s 3*/9z=(9 2*/3z 2)/Bi (B.1.1) *(5 0,7})=*.=0 , (B.1.2) 9*(S n , T?)/9z = 0 (B.1.3) 9<M0 ,7?)/9z = Bi . (* -F)=-f (77) . (B.1.4) D i f f e r e n t i a t i n g equation (B.1.2) with respect to 77 and s u b s t i t u t i n g back i n t o the d i f f u s i o n equation r e s u l t s i n an 101 102 e x t r a boundary c o n d i t i o n c a l l e d the smoothing c o n d i t i o n , t h a t i s 3 2*( 6 0 , 7 ? ) / 3 z 2 = 0 . (B. 1 .5) There are 4 c o n s t r a i n t s . T h e matrix temprature i s thus represented by a cubic p r o f i l e , t h a t i s 2 3 *=A+Bz+Cz +Dz . (B.1.6) Appl y i n g the c o n s t r a i n t s , t h e r e w i l l be 4 equations A+B5„+C6^+D5 3=0 , 0 0 0 ' 2 B+2C.6Q+3D.6Q=0 , B=-fU) , 2C+6D.5Q=0 . The 4 unknowns are found by s o l v i n g the above simultaneous equations ,the cubi c p r o f i l e i s then *=f ( T ? ) . ( 5 0 - z ) 3 / ( 3 . 5 2 ) . (B.I.7) The s u r f a c e temprature * s i s obtained by s e t t i n g z=0 i n equation (B.1.7),the r e s u l t i s 1 03 * =f ( 7 7 ) . 5 V 3 . (B. 1 .8) s 0 The d i f f u s i o n equation i s now i n t e g r a t e d with respect to z,that i s J S o 0 ( d * / d r i ) dz=(j5Q0 ( 9 2 * / 9 z 2 ) d z ) / B i . (B.1.9) From L i e b n i t z theorem J Q 0 ( 9 * / 9 Z ) dz = d(/^0 ¥ d z ) / d r ? - * ( 6 0 , rj)(d5 /drj) , (B.1.10) but by d e f i n i t i o n ¥ ( 6^ , 77) =0 , so equation (B.1.9) becomes d(/ 0 * dz)/drj=[ (9*/9z) . - 0 * / 3 z ) „ ]/Bi . (B.1.11) 0 z = 5 n 2 = 0 Applying equations (B.1.3) and (B.1.4)to the above equation , the f o l l o w i n g i s obtained d(/ 0 * dz)/drj=f (rj)/Bi . (B.1.12) 0 S u b s t i t u t i n g f o r * from (B.1.7), the r e s u l t i s d ( / ° [ f (r?) . (5 - z ) 3 / ( 3 S 2 ) ] dz)/d 7 ?=d(f (rj) .5 2/l2)/d77 . (B.1.13) 0 (J U u So equation (B.1.12) becomes d[f (7?) .5 2/l2]/d7j=f (7?)/Bi . (B.1.14) 104 The s o l u t i o n to the above d i f f e r e n t i a l equation i s obtained by i n t e g r a t i o n with respect to 17,the r e s u l t i s 6 =[ 12. ( J ^ f (77) dr? )/ ( B i .f (77) ] ° ' 5 , (B.1.15) 0 o where 6Q = 0 , at r?=0 . The su r f a c e temperature i s computed by s u b s t i t u t i n g f o r 5^ from (B.1.8),the r e s u l t i s * = [4.f (77) . ( A (77) dTj)/(3.Bi) ] ° ' 5 . (B.1.16) s 0 Slab of f i n i t e t h i c k n e s s At time V - W ^ r the p e n e t r a t i o n depth reaches the centre of the matrix.From t h i s p o i n t on the p e n e t r a t i o n depth has no meaning and the matrix has to be remodelled.This i s achieved by mo d e l l i n g the matrix as a f i n i t e s l a b ( t h i c k n e s s d) s u b j e c t to the f o l l o w i n g dimensionless c o n s t r a i n t s *( 0 ,77) =* , (B.1.17) s 9*(0 ,7?)/3z = B i . (* -F)=-f (T ? ) , (B.1.18) 3*( 1 ,77)/9z = 0 . (B.1.19) There are 3 c o n s t r a i n t s . S o the temperature p r o f i l e must be a second-order p o l y n o m i a l , t h a t i s 105 2 , 4>=A+Bz+Cz . (B.1.20) The con s t a n t s A,B and C are found from c o n s t r a i n t s . T h e p r o f i l e w i l l then take the form +f (TJ) . ( z 2 - 2 z ) / 2 . (B. 1.21) s The d i f f u s i o n equation i s now i n t e g r a t e d with respect to z. A f t e r a p p l y i n g the L i e b n i t z theorem,the r e s u l t i s 1 d(J * dz)/dT7=[ (3*/3z) • -(3¥/3z) J / B i . (B.1.22) 0 2 = 1 z = 0 Apply i n g equations (B.1.18) and (B.1.21) to the above equation, the f o l l o w i n g i s obtained d ( * - f ( r j ) / 3 ) / d T j = f ( T j ) / B i . (B.1.23) s The s o l u t i o n to the above d i f f e r e n t i a l equation i s obtained by i n t e g r a t i o n with r e s p e c t to rj,the r e s u l t i s 7? * - f ( r j ) / 3 = ( / f(ij) drj)/Bi +Constant . (B.1.24) s 0 At time 77=r?Qf 5^=1, i e . t h e p e n e t r a t i o n depth has reached the t h i c k n e s s of the slab.So from (B.1.8) * s ( ^ 0 ) = f ( ^ 0 ) / 3 (B.1.25) 1 06 S u b s t i t u t e f o r * (71) i n e q u a t i o n (B. 1 .24) , t h e r e s u l t i s s 0 * s ( i ? 0 ) - f ( T ? 0 ) / 3 = ( / ^ 0 f (7?) d T ? ) / B i + C p n s t a n t , so C o n s t a n t = - ( f|J0f (rj) d r 7 ) / B i . ( B . 1 . 2 6 ) S u b s t i t u t e f o r C o n s t a n t i n e q u a t i o n ( B . 1 . 2 4 ) , t h e r e s u l t i s * =f ( T J ) / 3 + ( JV f ( r j ) d r ? ) / B i . ( B . I . 2 7 ) ^0 B.2 S p h e r i c a l geometry As f o r t h e p l a n a r g e o m e t r y , t h e s p h e r i c a l m a t r i x i s m o d e l l e d i n two s t a g e s Sphere of i n f i n i t e r a d i u s The d i f f u s i o n e q u a t i o n f o r s p h e r i c a l g e o m e t r y i n d i m e n s i o n l e s s f o r m i s 9*/9T?=[ 9 2 * / 3 z 2 + ( 2 / z ) . 3 * / 3 z ] / ( 3 . B i ) . ( B . 2 . 1 ) The p e n e t r a t i o n d e p t h c o n c e p t i s e m p l o y e d t o a p p r o x i m a t e t h e s o l i d t e m p e r a t u r e p r o f i l e . T h e p e n e t r a t i o n d e p t h i s m e a s u r e d f r o m t h e s u r f a c e o f t h e s p h e r e . T h e m a t r i x i s s u b j e c t t o t h e f o l l o w i n g d i m e n s i o n l e s s c o n s t r a i n t s 1 07 *( 1-6 ,r?)=* =0 , (B.2.2) > * ( 1 - 5 0 , T J ) / 9 Z = 0 , (B.2.3) 3*(1 ,r?)/3z = - B i . (* -F)=-f (TJ) . (B.2.4) s Equation (B.2.2) i s d i f f e r e n t i a t e d with respect to r? and then s u b s t i t u t e d back i n the d i f f u s i o n e q u a t i o n . T h i s r e s u l t s in an e x t r a boundary c o n d i t i o n of the form 3 2 * / 9 z 2 + ( 2 / z ) . ( 3 * /3z)=0 . (B.2.5) Applying equation (B.2.3) to the above e q u a t i o n , i t reduces to the smoothing c o n d i t i o n , t h a t i s 9 2+( 1-6 0,T?)/9Z 2 = 0 . (B.2.6) The suggested p r o f i l e i s of the form [17] "^(p o l y n o m i a l in z ) / z . (B.2.7) The reason for t h i s form of the p r o f i l e i s that i t resembles the steady s t a t e s o l u t i o n of the d i f f u s i o n equation.The steady s t a t e s o l u t i o n being * 0((1/z) 108 The polynomial of z i s a cubic.So the f i n a l p r o f i l e w i l l take the form 2 4»=Az +Bz+C+D/z . (B.2.8) Appl y i n g the c o n s t r a i n t s and s o l v i n g f o r A,B,C and D ,the f i n a l e x p r e s s i o n w i l l be *=-f ( T 7 ) . [ Z - ( 1 - 5 0 ) ] 3 / ( 5 2 . ( 3 - 5 0 ) . Z ) . (B.2.9) The s u r f a c e temprature i s obtained by s e t t i n g z=1 i n the above equation,that i s * =-f (rj) . 5 n / ( 3 - 5 - ) . (B.2. 10) s 0 0 The d i f f u s i o n equation (B.2.1) i s r e w r i t t e n as 3(*.z)/ar?=[3 2( + . z ) / 9 z 2 ] / ( 3 . B i ) . (B.2.11) The above equation i s i n t e g r a t e d with respect to z. A f t e r a p p l y i n g the L i e b n i t z theorem,the r e s u l t i s 1 1 2 2 d(J *.z dz)/dr?=(/ 9 (*.z)/3z d z ) / ( 3 . B i ) . (B.2.12) 0 0 The r i g h t hand s i d e of the above equation can be s i m p l i f i e d by a p p l y i n g equations (B.2.2),(B.2.3) and (B.2.4).Equation (B.2.12) reduces to 109 1 d( J • *.z dz)/dr? = - f ( r j ) / [ B i . ( 3 - 6 f t ) ] . (B.2.13) i - « 0 o I t can be seen that the above d i f f e r e n t i a l equation i s h i g h l y n o n - l i n e a r . The n o n - l i n e a r i t y can be reduced to some extent by r e a d j u s t i n g the d i f f u s i o n equation.Equation (B.2.1) i s now w r i t t e n as 3(*.z 2)/3rj=[ 9 ( z 2 . 9*/3z)/3z]/(3.Bi ) . (B.2.14) The above equation i s now i n t e g r a t e d with respect to z,the r e s u l t i s 1 2 2 1 d ( J *.z dz)/dr?=[z (3*/3z)] / ( 3 B i ) . (B.2.15) 1" 60 1" 50 A f t e r a p p l y i n g the c o n s t r a i n t s , t h e above equation reduces to 1 2 d ( J *.z dz)/dr?=-f (rj)/(3Bi ) . (B.2.16) ' - 6 o I t can be seen that equation (B.2.16) i s l e s s n o n - l i n e a r i n comparison with equation (B.2.13). S u b s t i t u t i n g f o r * from equation (B.2.9) and performing the i n t e g r a l , t h e r e s u l t i s d[ ( S Q - 5 6 2 ) . f (T?)/(3-6 0) ]/dT7=-20f (r?)/(3Bi) . ( B . 2 . 1 7 ) 1 10 The above d i f f e r e n t i a l equation i s so l v e d by i n t e g r a t i o n with respect to 77, with the i n i t i a l c o n d i t i o n (6Q = 0 at 17=0) the r e s u l t i s [ ( 5 5 2 - 6 3 ) / ( 3 - 6 n ) ] = 20(/ 7 ?f ( 7 7 ) dr?)/.(3Bi.-f (TJ).) . (B.2.18) u u u 0 The s u r f a c e temprature i s obtained by s u b s t i t u t i n g f o r 5Q from equation ( B . 2 . 1 0 ) , t h e r e s u l t i s 3 * 2 [ 5 f (TJ)-2* ] = 20[* - f (77) ] 2 . (/^f (77) d 7 ? ) / ( 3 B i ) . (B.2.19) s s s 0 Sphere of f i n i t e r a d i u s At some time r j ^ t h e p e n e t r a t i o n depth reaches the ce n t r e of the sphere.At t h i s time the p e n e t r a t i o n depth concept should be d i s r e g a r d e d due to the symmetry e f f e c t . T h e matrix i s now subject to the f o l l o w i n g dimensionless c o n s t r a i n t s 3*(0,7?)/3z = 0 , (B.2.20) 9*( 1 , r?)/3z = - B i . (* -F)=-f (77) , (B.2.21) * ( 1 , T J ) = * . (B.2.22) s The p r o f i l e w i l l take the form [17] •=(polynomial i n z ) / z (B.2.23) 111 It should be noted that s i n c e the sphere i s subject to a boundary c o n d i t i o n at the c e n t r e , t h e polynomial should not i n c l u d e a constant term .So • t h e • p r o f i l e . w i l l take the form 2 *=Az +Bz+C . (B.2.24) The unknowns A,B and C are found by a p p l y i n g the c o n s t r a i n t s , the r e s u l t i s *=* +f (77) . [ 1-z 2]/2 . (B.2.25) s The modified d i f f u s i o n equation (B.2.14) i s now i n t e g r a t e d with respect to z,the r e s u l t i s 1 2 1 2 d(J *.z dz)/drj=[; 3(z .9*/3z)/3z d z ] / ( 3 B i ) , 0 0 = - f ( T | ) / ( 3Bi ) . (B.2.26) S u b s t i t u t i n g f o r * from equation (B.2.25) and performing the i n t e g r a t i o n , t h e r e s u l t i s d [ * /3 + f (r?)/l5 ] / d T?=-f (rj)/(3Bi) . (B. 2.27. a) The above d i f f e r e n t i a l equation i s sol v e d by i n t e g r a t i o n with respect to 77,the r e s u l t i s * +f (7j) /5 = - ( / f(rj) d77)/Bi+Constant . (B.2.27) s 0 1 1 2 At rpn , 6=1 .So ¥ (T? ) can be obtained by s e t t i n g 0 0 s 0 6Q=1 i n equation ( B . 2 . 1 0 ) , t h e r e s u l t i s * (T?.)=-f ( T J.)/2 • ( B.2.28) S 0 0 S u b s t i t u t i n g for * S ^ Q ^ * n e Q u a t i ° n (B.2.27) the constant of i n t e g r a t i o n i s o b t a i n e d , t h a t i s Constant = -3f ( T J )/l0+( J^O f(r>) drj)/Bi . (B.2.29) The term f ^ p ) can be computed by s e t t i n g 6 =1 i n equation (B.2.18),the r e s u l t i s f (T? ) = 1 0 (/^ O f(r?) drj)/(3Bi) . (B.2.30) From equations (B.2.30) and ( B . 2 . 2 9 ) , i t can be deduced that the constant of i n t e g r a t i o n i s zero.So equation (B.2.27) becomes * =-f (i7)/5--(J T ?f ( T » ) d7?)/Bi . (B.2. 31) s 0 APPENDIX C: EFFECTIVENESS COMPUTATION In regenerators the thermodyhamically p e r f e c t s i t u a t i o n occurs when the matrix at the e x i t from the regenerator i s at the same temperature of the e n t e r i n g hot f l u i d . Obviously ,-this i s not p o s s i b l e i n p r a c t i c e because of the r e s i s t a n c e to the heat t r a n s f e r between the two media. To measure the performance of a regenerator a g a i n s t the i d e a l i s e d s i t u a t i o n , a parameter c a l l e d e f f e c t i v e n e s s i s employed.This i s d e f i n e d as the r a t i o of the a c t u a l r i s e (or drop) i n the matrix temperature to the maximum p o s s i b l e r i s e . I n mathematical! form e i s d e f i n e d as e=m .C .(T -T .)/[(m.C) . .(T..-T .)] , (C.1) s s msO msi min f i msi where (m.C) . =minimum of the two c a p a c i t y r a t e s , min and T =Mean s o l i d temperature . ms It i s apparent from equation (C.1) that the s o l i d temperature at i n l e t and o u t l e t are represented as mean temperatures.Consequently,the f i r s t step towards the e f f e c t i v e n e s s computation, i s the c a l c u l a t i o n of the s o l i d mean temperature. 1 13 114 C.1 Planar geometry The mean s o l i d temperature i n planar geometry i s d e f i n e d as d T =(/ T dx)/d , • (C.2) m o or i n dime n s i o n l e s s form 1 * =7 * dz . (C.3) m J 0 There are two expre s s i o n s f o r * depending on the p e n e t r a t i o n depth.These were obtained i n the pre v i o u s Appendix. F o r 7?<r?Q , *=f (TJ) . ( 6 0 - z ) 3 / ( 3 . 6 2 ) . (B.I.7) For T7>T?0 , *=* s + f (77) . ( z 2 - 2 . z ) / 2 . (B.1.21) The corresponding mean temperatures are then For 77<770 , * =f ( 7 7 ) . [ J 1 ( 5 n - z ) 3 d z ] / ( 3 . 5 2 ) , m Q U l) 1 1 5 b u t 1 6 n 1 / * d z = ( / u * d z ) + ( J * d z ) , 0 0 6 Q by d e f i n i t i o n t h e s e c o n d i n t e g r a l i s e q u a l t o z e r o a n d t h e mean t e m p e r a t u r e w i l l be 6 - 3 , ., „2. * =f(r?).[J P(6 - z ) J d z ] / ( 3 . 5 „ ) , t h a t i s * = f ( r ? ) . 6 2 / l 2 . ( C . 4 ) m U F o r T } > T ? 0 , 1 2 * =/ [* + f ( n ) . ( z - 2 . z ) / 2 ] dz , m o s p e r f o r m i n g t h e i n t e g r a t i o n , * =* - f ( r ? ) / 3 , ( C . 5 ) m s where f ( r j ) = B i . (F-¥ ) . ( C . 6 ) s C.2 S p h e r i c a l geometry The mean t e m p e r a t u r e i n c a r t e s i a n c o o r d i n a t e ( 3 d i m e n s i o n a l ) i s d e f i n e d a s 1 1 6 V ( ; / ; T dx ay a«)/( j j /a, a y az) . ( c . 7 ) The above equation can be transformed to s h e r i c a l c o o r d i n a t e system by a p p l y i n g the Jacobian t r a n s f o r m a t i o n [21]. D e f i n i n g x = r .cos# . sin<£ , y = r . sin0 . sin<£ , z = r.cos</> . The Jacobian transformer w i l l be J = 9x/9r dx/dd 9x/90 9y/9r 9y/90 dy/d<p 9z/9r 9z/96> 9z/9tf> 2 • * = - r sin<£ (C.9) so ///dx dy dz=///-r sin<£ dr dd d<j> (C.10) S u b s t i t u t e the above,in equation (C.7) R 2 v/2 2 n R 2 2 7 R ^Z2 T =/ (-Tr ) d r . / d0.J sin<£ dtf>// - r d r . / si n 0 d<£./ dd m 0 -TT/2 0 0 0 -TT/2 or R o R o T =/ (T.r ) d r / ( / r dr) m 0 0 (C.11) 1 17 The mean temperature i n dimensionless form i s 1 2 * = 3. J *•. z dz . (C.12) m 0 There are two expr e s s i o n s f o r 4», depending on the p e n e t r a t i o n depth,these are For T ? < T ? 0 , *=-f (77) . [z-( 1-5 ) ] 3 / [ (3-5 )5gZ] . (B.2.9) The mean temperature i s obtained by s u b s t i t u t i n g the above e x p r e s s i o n for * i n equation ( C . 1 2 ) . F i r s t i t i s noted that 1 2 1 ~ 5 n 2 * =3/ (*.z ) dz+3/ 0 (*.z ) dz , (C.13) m 1-6 0 and by d e f i n i t i o n *=0 beyond p e n e t r a t i o n depth.So the second i n t e g r a l i n equation (C.13) i s zero and (C.13) reduces to 1 2 * =3./ (*.z ) dz . (C.14) m i - 6 o S u b s t i t u t i n g f o r * from equation (B.2.9) in above equation,the r e s u l t i s * =-3.f(r?).; 1 z . [ z - ( 1 - 5 n ) ] 3 d z / [ ( 3 - 5 n ) . 5 2 ] , m -j _ § 0 0 0 0 1 18 so * =-3..f (TJ) . ( 5 6 2 - 6 3 ) / [ 2 0 . (3-5.) ] . (C.15) m 0 0 0 It i s i n t e r e s t i n g to note that from equation (B.2.18) the mean temperature can a l s o be w r i t t e n as * =-/ f (T?) drj/Bi . (C. 16) m o For rj>rj0 , 4>=¥ +f (JJ) . ( 1-z 2)/2 . (B.2.25) s S u b s t i t u t i n g f o r * in equation (C.12),the r e s u l t i s * =3/ 1[* . z 2 + f (TJ) . ( z 2 - z 4 / 2 ) ] dz , m o s that i s * =* +f ( T J ) / 5 , ( C 17) m s where f ( r j ) = B i . ( * -F) . (C. 18) s APPENDIX D :SAMPLE CALCULATION D.l Governing equations The governing equations are w r i t t e n in t h e i r numerical form as F l u i d phase (1+A$/2)F(n,i+1)-A£/2¥ (n,i+1)=A£/2¥ (n-1,i+1)+(1-A£/2)F(n-l) 5 S S o l i d phase 1. Planar geometry For TJ<TJ 0.5 (D.2) * (n,i+1)=[4f(n,i+1).Ar1(n,i+1)/(3Bi)] 0.5 (D.3) For T)>T?0 , * (n,i+1)=f(n,i+1)/3+Ar2(n,i+1)/Bi , (D.4) where 1 19 120 f (n r "i )=Bi-[F(n, i ) - * ( n , i ) ] , .(D.5) s Ar 1 (n , i + 1 )=Ar 1 (n , i ) +Arj[ f (n , i + 1 ) + f (n , i ) ]/2 , (D.6) Ar2(n, i + 1 )=Ar2(n, i )+Ar?[ f (n, i + 1 ) + f (n, i ) ]/2 , (D.7) note that Ar 1 (n , 0 ) =0 and Ar2(n,r? )-=0 . 2 . S p h e r i c a l geometry For T ? < 7 ? 0 , 5 6 2 ( n , i ) - 8 3 ( n , i ) = 2 0 [ 3 - 5 Q ( n , i ) ] A r 1 ( n , i ) / ( 3 B i . f ( n , i ) ) , (D.8) 3 * 2 ( n , i + 1 ) [ 5 f ( n , i + 1 ) - 2 * (n,i+1)]=20[* ( n , i + 1 ) - f ( n , i + 1 ) ] 2 s s s .Ar1(n,i+1)/(3Bi) . (D.9) For 7?>i70 , * (n,i+1)=-f(n,i+1)/5-Ar1(n,i+1)/Bi , (D.10) s where f ( n , i ) = B i [ * ( n , i ) - F ( n , i ) ] , (D.11) s Ar 1 (n, i + 1 )=Ar 1 (n, i )+AT?[ f (n , i + 1 ) + f (n, i ) ]/2 . (D.12) 121 D.2 Method of s o l u t i o n D.2.1 F i x e d bed The a n a l y s i s begins by f i r s t computing'5 ( n , i ) at each step p o i n t . I f 5^(n,i)<1,then equations (D.3 or'D.9) are used to compute the s o l i d temperature.If however,6^(n,i)>1 ,then equations (D.4) or(D.lO) are used to compute the s o l i d temperature. There i s a step change in the f l u i d temperature at the entrance to the regenerator ( i e . F ( 0 , i ) = F ( 0 , i + 1 ) ) .Thus e s s e n t i a l l y there i s only one unknown temperature which i s * g ( n , i + 1 ) . T h i s can be computed by s o l v i n g equations (D.3) or (D.9) .At any other p o i n t the unknown temperatures can be computed by s o l v i n g the two simultaneous equations r e p r e s e n t i n g the f l u i d phase and s o l i d phase ,these are For 77<77Q , Equations (D.1) and (D.3) or (D.9) should be s o l v e d to compute the unknowns 4» ( n , i + l ) and F (n , i +1 ) .This i s done by s u b s t i t u t i n g f o r F(n,i+1) from (D.1) in (D.3) or (D.9) and s o l v i n g f o r * ( n , i + l ) . I t should be c l e a r that * (n,i+1) l i e s s s w i t h i n the l i m i t 0<¥ (n,i+1)<1 s (D.13) 1 22 For 7?>rjQ , The l i n e a r equations (D. 1 ) and (D.4.) or (D.10) are gathered together in a matrix form U [ F ( n , i + 1),4> (n, i + 1)]=H , (D.14) where U and H are the f o l l o w i n g m a t rices l . F o r p l a n a r geometry U= 1+A£/2 -A£ -Bi/3-Ar?/2 1+Bi/3+Arj/2 (D.15.a) H= A£ (* (n - 1 , i + 1))/2+(1-A£/2)F(n-1 , i + 1 ) A r 2 ( n , i ) / B i + A 7 ? [ F ( n , i ) - * ( n , i ) ] / 2 (D.15.b) 1 23 2.For s p h e r i c a l geometry U= 1+A£/2 -A$/2 -Bi/5-A7?/2 1+Bi/5+Ar>/2 (D.16.a) H= A£( + (n-1,i+1))/2+(1-A£/2)F(n-1,i + 1 ) -Ar1 (n,i)/Bi+Arj[F(n.,i)-* ( n , i ) ]/2 (D.16.b) The s o l u t i o n i s merely the i n v e r s i o n of (D.14),that i s [F(n,i+1),+ (n,i+1)]=U .H (D.17) The s t a r t i n g values f o r the i t e r a t i o n are F(0,i)=1 For i>0 , (D.18.a) 5 Q(n,0)=0 For n>0 , (D.18.b) * (n,0)=0 For n>0 (D.18.c) The i n i t i a l f l u i d temperature at any p o s i t i o n other than entrance i s computed from 1 24 3*/9z = - f (TJ) (D.19.a) but *(n,0)=0 for a l l n, so F(n,0)=0 for n>0 (D.19.b) D.2.2 Moving bed The f l u i d phase need to be a l t e r e d f o r t h i s c a s e , s i n c e £ has to be measured from the s o l i d e n t r a n c e . A l s o f o r moving bed regenerators AT? i s expressed i n terms of A £ . Consequently, the unknowns are F(n+1) and * (n+1). The s equations w i l l be i d e n t i c a l to the previous case ( f i x e d bed), i f A£ i s r e p l a c e d by - A £ . The s o l i d and f l u i d i n l e t temperatures are known.In order to s t a r t the i t e r a t i o n ,one need to know the f l u i d o u t l e t temperature. T h i s i s i n i t i a l l y approximated as the mean s o l i d and f l u i d i n l e t temperatures,that i s The f l u i d and s o l i d temperatures at each step p o i n t are then computed using the procedure o u t l i n e d f o r the f i x e d bed.The c a l c u l a t e d f l u i d i n l e t temperature i s then compared with the a c t u a l given temperature.If there i s any dis c r e p e n c y , the i n i t i a l approximation i s r e a d j u s t e d and the procedure i s then repeated u n t i l the two values c o i n c i d e . F ( 0 ) = [ F ( A ) + * (0)]/2 . (D.22) The U and H matrices are except that moving f i x e d a l s o the problem i s one unknowns are F(n+1) and * (n+1). i d e n t i c a l to the p r e v i o u s case d i m e n s i o n a l , t h a t i s the APPENDIX E :THE COMPUTER PROGRAM Three programs j f i x e d p l a n ,fixedsph and moving were w r i t t e n f o r handl i n g the thermal design of f i x e d and moving bed regenerators r e s p e c t i v e l y . The i n t r a c o n d u c t i o n e f f e c t was i n c l u d e d i n a l l the three programs. The programs were w r i t t e n i n F o r t r a n language and were run on Amdahl 470. The f i r s t two programs were w r i t t e n f o r the purpose of examining the v a l i d i t y of the i n t e g r a l method.This was achieved by comparing the a n a l y t i c a l r e s u l t s o b t a i n e d (using the i n t e g r a l method) with the p u b l i s h e d r e s u l t s (using numerical methods).The t h i r d program was w r i t t e n i n order to o b t a i n a set of c h a r t s f o r moving bed r e g e n e r a t o r s . The input data f o r the f i r s t two programs i n c l u d e d the parameters r e q u i r e d to c a l c u l a t e the reduced l e n g t h and dimensionless p e r i o d f o r the f i x e d bed.However,for a moving bed regenerator i t was only neccessary to compute the reduced l e n g t h . A s u i t a b l e time and d i s t a n c e increments were chosen(0.2 fo r both).The f l u i d and s o l i d s u r f a c e temperatures at each ste p p o i n t were then computed. For a moving bed regenerator the f l u i d and s o l i d s u r f a c e temperatures at each l e n g t h increment along the whole le n g t h were obtained . Two e x t e r n a l s u b r o u t i n e s were used i n a l l three programs,namely 1 26 127 Z e r o l T h i s e x t e r n a l subroutine was used to solve the two n o n - l i n e a r equations for the p e n e t r a t i o n depth and s o l i d s u r f a c e temperature, when 6^<1. SLE T h i s e x t e r n a l subroutine was used to s o l v e the two l i n e a r simultaneous equations f o r the f l u i d and s o l i d s u r f a c e temperatures.The subroutine uses the matrix i n v e r s i o n technique to solve the equations The l i s t of a l l three programs are i n c l u d e d at the end t h i s s e c t i o n . 1 28 1 29 IMPLICIT REAL*8 (A-H.O-Z) REAL*8 LEN.LENU REAL*4 RELEU C C C THIS IS A PROGRAM TAKING ACCOUNT OF, C INTRAPARTICLE CONDUCTION FOR SPHERICAL COORDINATE C C THE NAME OF THIS PROGRAM IS RBGE C DIMENSION TGU(100),TSU(100),HTU(100),TSM(100) DIMENSION A1U(5,5),B1U(5),TU(5,5),X1U(5),IPERM(10) DIMENSION ARU(100),DIFF(100) DIMENSION PENEU(100),EFF(100) COMMON /UPPER/ DTU.BIU COMMON /UTERI/ DIS.RL2.RL3 COMMON /DELTA/ TG1,TG2,TS1,TS2,AR1 COMMON /SOLTE2/TG6,TS5,AR3 C c C COMMON BLOCK DELTA DEFINES THE VARIABLES USED IN SUBROUTINE FN C IN WHICH THE PENETRATION DEPTH IS CALCULATED. C C C COMMON BLOCK /SOLTE2/ DEFFINES THE VARIABLES USED IN SUBROUTINE C FCN IN WHICH TS(N,I*1),TG(N,I-M ) ARE CALCULATED. C C c c c C HT-HEAT TRANSFER AR-AREA UNDER CURVE HT VS TIME C ARE*AREA UNDER THE CURVE F*PENETRATION VS TIME C X-SOLUTION OF NON-LINEAR EQUATION C XI-SOLUTION OF LINEAR EQUATION SOLVED BY MATRIX INVERSION C A1-MATRIX OF COEFFICIENTS C Bl-MATRIX OF RIGHT HAND SIDE OF EQUATION A.X-B C DT-INCREMENT OF TIME C CONST-F(TIME) WHEN T-TO C PENE-PENETRATION DEPTH C READ(5,500)SOLK,DENSOL,CPSOL,DIASOL,PORO,SOLFLO 500 FORMAT(E7.2,F7.1,F4.1,EB.1,F4.1,F4.2) READ(5,5100)GAMTU,LENU,BEDAU,VOLFU 5100 FORMAT(F6.1,F5.2,F4.1,F5.2) CPGU-0.917+(2.58E-4)*GAMTU-(3.98E-8)*GAMTU**2 VISGU-1.4 6E-6*GAMTD* * 1.5/(110. •GAMTU) DENGU-353./GAMTU GASKU-6.16E-7*GAMTU**0.6B VKIGU-DENGU/VISGU DENU-DENSOL/DENGD-1. GASFU-VOLFU*DENGU VTERU-SOLFLO/(DENSOL*(1.-PORO)*BEDAU) UGASU-GASFU/(DENGU* BEDAU) RETNU-UGASU*DIASOL*VKIGU PRANU-VISGU*CPGU/GASKU VOID-3.25-2.25*PORO IF(REYKU.GE.100.) GO TO 8000 1 30 Y4U-200. EXTERN AX FN2U LOGICAL LZ2U C C C C X(l)ANDX(2) ARE THE INITIAL GUESSES FOR TG(N,I+1) C AND TS(N.I-M). C c TG6-TGU(NU) TS5«TSU(NU) AR3-ARU(MJ) C C THE VALUES OF TG(N,I+1),TS(N,I+1) ARE NOW CALCULATED C c CALL ZER01(X4U,Y4UtPN2U,E2,LZ2U) TGU(NU*1)-RL2*TG6/RL3-DIS*(X4U+TS5)/(2.*RL3) TSU(NU-M)»X4U ARU(NU-M )»ARU(NU)-<TGU(NU-M)*TGU(NU)-TSU(NU-H )-TSU(NU))*RLU HTU(NU)-BIU*(TGU(NU)-TSU(NU)) EXTERNAL FNU LOGICAL LZU E1-5.E-7 X2U-0. Y2U-2. TGI-TGU(NU) TG2-TGU(NU+1) TS1«TSU(NU*1) TS2-TSU(NU) ARI-ARU(NU) C c C THE VALUE OF PENETRATION DEPTH IS NOW CALCULATED C c CALL ZER01<X2U,Y2U,FNU,E1fLZU) PENEU(NU-M )-X2U RL5«(5.*PENEU(NU)**2-PENEU(NU)**3)/(3.-PENEU(NU)) TSM(NU)--3.*BIU*(TSU(NU)-TGU(NU))*RL5/20. WRITE(B,2560)PENEU(NU+1),TSU(NU+1)ITGU(NU+1),DTU,BIU,TGU(2) 2560 F0RMAT(6F12.4) IF (NU.NE.2) GO TO 2570 DIFF(NU)-0. GO TO 2560 2570 DIFF(NU)-100.*(TSU(NU)-TSM(NU))/TSU{NU) 2580 EFF(NU)-R*(TSM(NU)-TSU(2))/(TGU(NU)-TSU(2)) GO TO 901 C c c C IF T » T 0 USE MATRIX INVERSION TO SOLVE 2 LINEAR EQUATIONS C AT THIS STAGE PENETRATION DEPTH IS EQAL TO THICKNESS OF PLAT C C 2590 PENEU(NU-H)-PENEU(NU) 131 NDIKA-5 HDIMT-5 NDIMBX-5 NN1-2 KSOL-1 C C NN1-NO.OF EQUATIONS, A1-MATRIX OF COEFFICIENTS C Bl-MATRIX OF KNOWN RIGHT HAND SIDE A*X-B C c c A1U(1,1)-RL3 A1U(1,2)-DIS/2. A1D(2,1)—BIU/5.-DTU/2. A1U(2,2)-1.-*BIU/5.«-DTU/2. B1U(1)-RL2*TGU(NU)-DIS*TSU(NU)/2. B0«DTU*TSU(NU)/2. B1U(2)—ARU(NU)/BIU*DTU*TGU(NU)/2.-B0 CALL SLE(NN1,NDIMA,A1U,NSOL,NDIMBX,B1U,X1U, i IPERM.NDIMT.TU.DET.JEXB) TGU(NU*1)«X1U(1) TSU(NU*1)-X1U(2) ARU(NU-»1)-ARU(NU)-RLU«(TGU(NU)*TGU(NU-H )-TSU(NU+1)-TSU(NU)) HTU(NU)-BIU*(TGU(NU)-TSU(NU)) TSM(NU)--ARU(NU)/BIU EFF(NU)-R*(TSM(NU)-TSU(2))/(TGU(NU)-TSU(2)) DIFF(NU)«100.*(TSU(NU)-TSM(NU))/TSU(NU) 901 MU-NU-2 1000 CONTINUE DTEMP-TGU(NU)-(TGINU-TSINU) IF(DABS(DTEMP).GE.0.1) GO TO 1001 DO 1050 NU-2.NNU MU-NU-2 IF(MU.GT.0)GO TO 1601 DMU«DIS*MU GO TO 1602 1601 DKU-DMU+DIS 1602 WRITE(6,3301)DMU,EFF(NU) 3301 FORMAT(F4.2,P6.3) WRITE(7,3302)DMU,TSU(NU) 3302 FORMAT(F4.2,F8.3) 1050 CONTINUE GO TO 999 1001 1F(DTEMP.GT.0.) GO TO 1002 TGU(2)-TGU(2)*DABS(DTBMP/4.) GO TO 1 1002 TGU(2)-TGU(2)-DABS(DTEMP/4.) GO TO 1 999 STOP END FUNCTION FNU(X) C PENETRATION DEPTH IS CALCULATED IN THIS FUNCTION C IMPLICIT REAL*8 (A-H.O-2) COMMON /UPPER/ DTD,BID 1 32 COMMON /DELTA/ TGI.TG2.TS1,TS2,AR1 P2-TS1-TG2 p1.BlU*DTU*(TS1*TS2-TG1-TG2)/2.*AR1 FNU-(5.*X**2-X**3)*P2*BIU-20.*(3.-X)*P1/(3.*BIU) RETURN END FUNCTION FN2U(X) C C TS(N,I*1) AND TG(N,X+1) ARE CALCULATED IN T H I S SUBROUTIN C USING THE NON LINEAR EQUATION FOR TS C IMPLICIT REAL*8 (A-H.O-2) COMMON /UPPER/ DTU.BIU COMMON /UPER1/ DIS,RL2,RL3 COMMON /SOLTE2/ TG6,TS5,AR3 X1-RL2*TG6/RL3-DIS*(X*TS5)/(RL3*2.) P4«5.*BIU*(X-X1)-2.*X P2»X-BIU*(X-X1) P1-AR3*BIU*DTU*(X-X1*TS5-TG6)/2. IF(P2.GT.0) GO TO 10 FN2U-3.*X**2*P4-20.*(-P2)**2*P1/(3.*BIU) GO TO 20 10 FN2U-3.*X**2*P4-20.*P2**2*P1/(3.*BIU) 20 RETURN END 

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