THE USE OF AN APPROXIMATE INTEGRAL METHOD TO ACCOUNT FOR INTRAPARTICLE..CONDUCTION ''IN GAS-SOLID HEAT EXCHANGERS. by ARDESHIR RIAHI B.Sc (Hons) , University o f 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 We a c c e p t to of M e c h a n i c a l this thesis the required as Engineering conforming standard UNIVERSITY OF BRITISH COLUMBIA APRIL 1985 © ARDESHIR RIAHI, 1985 In presenting requirements British freely that this available permission Department in partial f o r an a d v a n c e d d e g r e e Columbia, scholarly thesis I that for reference for purposes or agree by be his understood that copying financial gain shall copying her of M e c h a n i c a l U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e : APRIL 1985 Engineering Columbia of allowed permission. Department by I shall of t h i s the without agree thesis for Head this of make i t further representatives. or p u b l i c a t i o n be University the L i b r a r y granted or not the and s t u d y . extensive may at f u l f i l m e n t of t h e of my It is thesis my for written ABSTRACT The mathematical transfer moving heat equations between t h e f l u i d flowing b e d o f p a c k i n g were transfer conductivity An within analytical packing.Results obtained. against formulated. into integral solution to for through method inlet that the approximate with t h e more e x a c t methods. The considered obtain the method i s can that the effect very form. quickly increases.A packing thermal series i t s of c h a r t s surface conductivities and was other of since that surface and are presented. i i quick of this conductivity solution i s in revealed that decreases mean temperatures well i n order to peculiarity packing a concluded the temperature showing the comparison mean with a temperature the checked agree provids of the results the bed and Heggs I t was of packing thermal The a n a l y s i s between here an the bed of p a c k i n g results p a c k i n g mean the thermal conductivity difference the mehod g i v e s the of method temperature. effeetiveness.The be e x a m i n e d analytical as of thermal to obtain e m p l o y e d by H a n d l e y step fluid to and moving bed were of the approximate fora fixed determination bed and a resistance response of f i x e d who o b t a i n e d t h e r e s u l t s method The was a p p l i e d transient t h e more e x a c t method in a fixed heat account. two c a s e s The v a l i d i t y change transient t h e p a c k i n g due t o i t s f i n i t e was t a k e n approximate describing between for different The a p p r o x i m a t e method a moving bed of p a c k i n g . packing thermal series of versus was t h e n a p p l i e d t o the case I t was c o n c l u d e d that the e f f e c t c o n d u c t i v i t y i s more s e v e r e than expected. of of A c h a r t s r e p r e s e n t i n g t h e moving bed e f f e c t i v e n e s s dimensionless conductivities length are presented. for different thermal Table of Contents ABSTRACT .'. . i i List of T a b l e s List of F i g u r e s . . . .vi ........ . ..............vii Acknowledgements ....ix Nomenclature 1. x Introduction ...1 1 . 1 General 1.2 2. 3. 1 1.1.1 Stationary matrix 1.1.2 Moving m a t r i x 7 10 Review of p r e v i o u s 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 t h e p r e s e n t The Governing 2.1 Dimensionless 2.2 The investigation Equations 22 parameters mathematical 22 models 2.2.1 Schumann model 2.2.2 Intraparticle form 26 27 Conduction 2.3 Non-Dimensional The method of 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 3.2 P l a n a r geometry 3.3 Model of g o v e r n i n g solution equations 30 33 36 method 36 40 3.2.1 Semi-infinite 3.2.2 S l a b of Spherical 21 finite slab 40 thickness geometry 43 45 3.3.1 S p h e r e of infinite 3.3.2 S p h e r e of finite i v radius radius 46 48 3.4 T e m p e r a t u r e - D e p e n d e n t 3 . 5 Numer i c a l 3.5.1 50 51 F i x e d bed RESULTS AND ... bed r e g e n e r a t o r 51 56 DISCUSSION 4. 1 F i x e d bed ..60 ... . .60 4.1.1 S p h e r i c a l geometry 4.1.2 Planar 4.2 M o v i n g properties procedure 3.5.2 Moving 4. thermal ..60 geometry 61 bed r e g e n e r a t o r 63 5. Conclusion 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 APPENDIX DIMENSIONAL B:INTEGRAL METHOD ANALYSIS ...87 101 APPENDIX C: EFFECTIVENESS COMPUTATION 113 APPENDIX D :SAMPLE 119 APPENDIX E :THE COMPUTER PROGRAM CALCULATION v 126 List 1.Correlations for the of Tables convective coefficient..... heat transfer 92 vi List of Figures 1. Conductance heat exchanger. 2. Schematic 3. T y p e s of a i r h e a t e r s 8 4. F i x e d bed 9 5. Rotary 6. Falling 7. Moving pebble 8. Effect arrangement of an 2 MHD duct 3 regenerator. regenerator flow arrangement...... 12 cloud regenerator of bed 13 regenerator s o l i d thermal 14 c o n d u c t i v i t y on the temperature prof i le 19 9. Typical 10. Schematic 11. Comparison employed 12. 14. between The between The The The characteristic effect The profile of numerical effect effect S shaped for a fixed thermal profile effect temperature 18. 23 of of of and 69 analytical method equation (Bi=.25) analytical equation curves bed ( B i = 2) of fluid regenerator on the on the ( B i = .25) method 71 outlet 72 solid solid 74 conductivity on the ( B i = 2) vii 70 73 conductivity thermal method (Bi=.02) (Bi=.02) thermal profile equation and concept.38 analytical conductivity thermal profile and. numerical to s o l v e the d i f f u s i o n temperature 17. the between t h e temperature 16. numerical t o s o l v e the d i f f u s i o n temperature 15. the to s o l v e the d i f f u s i o n Comparison employed of a r e g e n e r a t o r 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 d e p t h Comparison employed 13. dimensions solid 75 conductivity on the solid temperature 19. T h e profile effect of i n a moving thermal bed .76 conductivity on regenerator effectiveness. 20. The comparison alternative 2 1 . The two profile prof il e s bed 23. M o v i n g 24. M o v i n g bed alternative effect original profile and ( B i = 2) between the e f f e c t i v e n e s s based on effectiveness based on (Bi =0 . 1,0 .5 , 1 ,2 ) effectiveness based on effectiveness rate viii the 81 (Bi=0.1,l0) of c a p a c i t y the 80 (Bi=4,8) regenerator profile the 79 regenerator profile the 78 ( B i = 2) profile bed alternative the regenerator alternative The between comparison 22. M o v i n g 25. 77 ratio based on the 82 on effectiveness....83 Acknowledgements The author wishes Prof.E.G.Hauptmann The author her moral to express his for a l l his help would a l s o appreciation and towards advice. like t o thank M i s s A f a r i n . R a d j a i e support) and Mr This i s dedicated Iraj.Riahi (for his support). work ix t o my parents. (for financial Nomenclature A = Solid surface area per unit 2 b e d volume m /m A, = Bed c r o s s - s e c t i o n a l a r e a b B = Porosity Bi = Biot C m number=hR/K = Fluid s heat specific capacity at constant pressure = Solid d specific = Matrix D g v F = J/kg K capacity Equivalent spherical fluid P r e s c r i b e d heat diameter flux = Number o f t i m e steps = Matrix conductivity y = Distance L = Bed l e n g t h M = Bed d e n s i t y s m m g n from (T -T. f 1 at the s o l i d i thermal m temperature = Convective s heat m h K J/kg K semi-thickness = Normalised f(0)= heat transfer )/(T_-T.) f1 1 surface coefficient 2 W/m W/m K W/m K t h e bed e n t r a n c e m m 3 kg/m 2 = fluid mass f l o w rate / area kg/m s . 2 = Solid mass f l o w r a t e / area kg/m s = Number o f l e n g t h s t e p s a l o n g Nu = N u s s e l t t h e bed number=hd/K s P = P e r i o d of f l u i d flow Pr = P r a n d t l number Q = Heat R r = D Ri as dt ia un s c eo f from s p h e rteh e c e n t r e o f s p h e r e transfer s J x m Re = Reynolds T = Solid temperature K = surface temperature K K T^ = Fluid temperature u = Fluid velocity m/s = Solid velocity m/s u Greek Solid number s - V, b = Bed volume x = Distance z = Dimensionless from m the s u r f a c e of the s l a b 3 m t h i c k n e s s or r a d i u s symbols a = Thermal 5 = Penetration = Dimensionless 6 0 m /s diffusivity m depth e = V = Dimensionless e = Time A = Dimensionless penetration - Effectiveness from time=hA(0-y/u)/M the start V = Kinematic I = Dimensionless n = Dimensionless t p s * s C s of the operation bed l e n g t h = £ 3 p - depth s y= L viscosity m /s distance along period = t h e bed=hAy/mC VQ_ P = Fluid mass density kg/m = Solid mass density kg/m^ = Normalised solid temperature=(T-T xi ) / ( T ^ -T. ) f I - Subscripts f = Fluid i = Inlet m = Mean 0 = Outlet s = surface or initial < xii 1. INTRODUCTION 1 . 1 GENERAL Heat exchange between temperatures in many b r a n c h e s There 1. represents are e s s e n t i a l l y an i m p o r t a n t are heat the rates convection from the hotter of heat transfer Capacitance use of process 1 be invariant with continuously fluid and b o t h the equal equal to t o the separating heat (Fig.1). heat the [1], flow a r e s t e a d y , w i t h the of c o n d u c t i o n t h r o u g h surface exchangers to fluid convection to the colder rate different e x c h a n g e r s , i n which the assumed time.Thus steady at technological two t y p e s o f h e a t R e c u p e r a t o r s or conductance the streams of i n d u s t r y . thermal c o n d i t i o n s 2. two f l u i d exchangers or r e g e n e r a t o r s which thermal capacity of a r e used extensively the heat make transfer surface. Regenerators example c o n s i d e r MHD which uses fluid w i t h a magnetic the 1 fluid numbers i n square an ( M a g n e t o h y d r o d y n a m i c ) power g e n e r a t i o n the i n t e r a c t i o n directly as a i r h e a t e r s . A s field into of an electrically to convert part electricity bracket refer 1 conducting of the energy of (Fig.2)[2]. t o the B i b i 1 i o g r a p h y 2 1- HOT FLUID 2- COLD FLUID DISTANCE FIG.1(a) A ® HOT CD FLUID FIG.1(b) (a) Figure . 1 C h a r a c t e r i s t i c temperature ( b ) " C o n d u c t a n c e heat (Counter flow) exchanger distribution COLD FLUID 3 B-MAGNETIC FIELD I - CURRENT L - LOAD Figure 2. Schematic arrangement of an MHD duct. 4 In fluid the fossil (combustion fired the that depends on chamber. F o r system, the temperature a i r ) at the magnetic determines point, MHD conductivity the and field . inlet, the which hence t h e power d e n s i t y performance optimum p e r f o r m a n c e of of the at combustion , i t is essential to preheat 0 the combustion This i s a c h i e v e d by utilizing Another in increasing gas a area turbines large large [3],In heat i s as exhaust area r e q u i r e s passages number of t h e s e can than regenerators these engines transfer greater 1200 C. regenerative a i r preheaters. which application volume.This in a i r to a temperature heat are finding exchanger , i t is essential within of the small in small to smallest obtain possible diameter and the be very conveniently incorporated a regenerator a regenerator. The peculiarity transfer media by gas one (called and t h e m a t r i x must gases or hotter fluid, fluid that is called an intermittent mechanical the m a t r i x ) c o o l e d by the be i t absorbs the heat main that the alternatively in and alternatively h e a t , and absorbed i s then o p e r a t i n g and regenerator, two is then by out either of swept the returned.The the latter heat heated t h e o t h e r . T h i s means t h a t matrix a continiuously are is be moved p e r i o d i c a l l y when t o which There of by the the colder former is called respectively. problems d e s i g n o f t h e s e two associated t y p e s of with regenerators [1]: the 5 1. P r o v i d i n g means f o r c h a n g i n g matrix . and periodically c o o l e d by Sealing the exchange of Problem operating solve.In can be design of two heat gas case gases of f l o w or latter t o be in turn and b e f o r e , d u r i n g and excessive problem conductivity solved i n the c a s e problem after 2 i s more difficult r e g e n e r a t o r s both the arises low of is when t h e thermal the transient response regenerator conductivity. matrix is If sufficiently this the intraconduction a combined called there c o n d u c t i o n ( wi t h i n practice is the convective heat conductivity more to to trapezoidal matrix) makes the adopt .There problem.For different approximation, example numerical central the regenerator matrix. the effect. in relation a it to of the of the number of design is schemes difference or C r a n k - N i c h o l s o n method) t o s o l v e the of are then within inclusion thermal up convection- transfer transfer.The complicated this heat the thermal low matrix; is of the gradient thermal i s made thermal solutions to problems the must be made f o r the regenerator the of c o n t i n u o u s l y allowance matrix thus leakage. the major problems a s s o c i a t e d w i t h of m a t r i c e s w i t h uniquely heated them. intermittent regenerators Consequently the avoided. of matrix.This both flows r e g e n e r a t o r s , but the the to prevent 1 is easily easily One between t h e gas enable contact with to exchange heat 2. to either a common (such as approximation transient However,these n u m e r i c a l response schemes 6 involve substantial computer storage the programs require of reponse solution of the approximate analytical thereby The avoids by the effect cooling of plates, way this wires of those study proposed designer to more in including i t . ,spheres an transient method u t i l i z e s solution of n u m e r i c a l take an in an schemes and account of the efficiently. the is regenerator c h o i c e of which matrix that i s to propose to obtain a use f e a t u r e s of the time the regenerator matrices.The affected [5] p r e d i c t e d and intraconduction typically present technique form.This central the time t o t h e p r o b l e m of d e t e r m i n i n g integral intraconduction the matrix. enables The neglect of of c o m p u t a t i o n a l Carpenter the computing purpose alternative are [ 4 ] . H e g g s 'and which a sixth The expenditure the heat the m a t r i x periodic performed.lt or b r o k e n solids exchanger of i s mainly heating may and consist irregular of shape [1]. Ideally high the m a t r i x v a l u e s of strength material specific selected h e a t , d e n s i t y and at e l e v a t e d temperatures;also available and cheap.It should impurities present gas). In some c a s e s matrix are of g r e a t these points select the a p p r o p r i a t e matrix not i n the h e a t i n g should be melting i t should possess point,good be easily react chemically with gases(usually the aerodynamic importance.Thus should drag i s made. combustion p r o p e r t i e s of i t i s obvious considered the before that the all a decision to 7 The matrix can exchanging fluids following types distinguished 1.1.1 brought (both hot of (refer into and contact cold) the in d i f f e r e n t regenerators to with may heat ways.The therefore be Fig.3). STATIONARY MATRIX In this stationary cooling type while period changing the The of flow related to surface area dimension exchanging the same transfer t o the heat per heat the h e a t i n g to the fluids. t h e two overall the is of unit required transfer.In matrix i s performed of h e a t the from the the m a t r i x proportional transfer.The regenerator, the change o v e r of rate directly the be for case fluids area dimension bed a of the from the area; a surface i s prescribed the amount flow a l t e r n a t e l y heat increasing a stationary of through the the overall matrix, the o v e r a l l and for regenerator i s volume d e c r e a s e s heating alternately available of transfer matrix.Consequently same f o r b o t h t o or by heat the two one and dimension cooling is side (Fig.4). Regenerators different groups used.The p e b b l e whose consist matrix of of depending bed type on f o r example elements light, this can be are the subdivided t y p e of t h e (Fig.4a) of inexpensive ceramic any is into matrix one shape and materials type may capable AIR I I HEATER 1 REGENERATOR RECUPERATIVE Intermittent Continuous I Metal tubes Rotary matrix Moving pebble bed Moving fluldlzed bed Parallel flow Falling cloud Counter flow Stationary pebble bed Figure 3. Types o f a i r heater. Checkers ceramic Ceramic tubes with metal pins or f i n s Ceraml tubes 10 of withstanding high be easily and i f neccessary are be to removed spheres or supported type of the by other the about 0.37 to is making temperatures important for a can this of empty volume factor which comes fixed bed is usually stove used i s an refractory the brick extensively industries, of alternative order in for of 900 of this type [6].The brick g l a s s making preheating to i n which 1200 C and air to (Fig.4b). MOVING MATRIX The characteristic the period type elements container.For (the r a t i o porosity of cleaned shapes.The elements their porosity i s made of regenerator kinds can 0.38. Cowper the m a t r i x that w a l l s of matrices c o n t a i n e r s t o be irregular volume) i s an analysis.The The their replaced.Possible into 1.1.2 from regenerator, total steel temperaturesf1].These s w i t c h over and of v i c e - v e r s a i s due regenerator depending on (Fig.3).The pebble The thermal the m a t r i x and falling liquid of heating to t o the m a t r i x i s subdivided f o u r common t y p e s bed, particles from the type regenerator the cooling movement.This into different the movement of are rotary slag and is the groups matrix matrix,moving falling solid [2], o p e r a t i o n of a r o t a r y r e g e n e r a t o r storage of a s l o w l y rotating relies matrix.With on the each . 1 1 revolution and of the matrix cooling by cold One regenerators pressure of of gas controlled The of superior the this type chamber heated by t h e m a t r i x . upper an inferior a before to the falling chamber the returned externally of liquid molten slag solid The m a t r i x heat high air.This i s seals. which incorporates a is heated i n which by pump the to the of s m a l l transfer medium,such a s ( n o t shown chamber rising regenerators must in i n the then Fig.6) be before i t i s atomized cold fluid.In of solidified upper this and then chamber problems with such solid where are technical the consists material top a the f l u i d i s a r e c o n t i n u o u s l y melted droplets are the for the design as development of the pump and a t o m i z a t i o n o f t h e m o l t e n m a t e r i a l [ 7 ] . An solid slag prevent with chambers,nominally chamber lower through recycling.There two molten by a s l a g and r a d i a l cloud regenerator(Fig.6).This suitable chamber.The injection to hot a i r problems t h e low p r e s s u r e has sulphate,which pressurized is i n which t h e m a t r i x and by configuration design of r e g e n e r a t o r fluid potassium to is a falling of matrix major type regenerator particles of the a i d of a p p r o p r i a t e second moving m a t r i x type forms leaking with of h e a t i n g a i r i s completed.Axial f l o w a r e t h e two b a s i c (Fig.5). a cycle alternative particles particles to f a l l i n g (Fig.7). either liquid The m a t r i x slag is falling consists of broken of r e g u l a r or i r r e g u l a r shape.The AIR Figure S c h e n a t i c diagram o f r o t a r y 5(a) regenerator. DOTATION Radial Figure 5(b) flow Axial Rotary r e g e n e r a t o r flow arrangement • flow Figure 6. F a l l i n g cloud regenerator* 14 1 extension of this regenerator.One the ensure the system , especially of these of dead inlet direction such a design contributes gas i n the gas R E V I E W OF outlet, A new to is to through the to prevent concept t h e bed the i n the design of a d i v e r g e n t the bed bed been proven uniform in that temperature [8]. P R E V I O U S WORK heat processes.lt formulate the laws case,and expression pebble requirements p a s s a g e . l t has behind industrial such a and sections. of moving the p a r t i c l e s r e g e n e r a t o r s i s the use Regenerative a at a fundamental the profile is u n i f o r m movement o f formation 1.2 of type 5 exchange i s one i s therefore governing the i f possible, f o r the temperature to of of some rate of t h e most common importance heat obtain a distribution to transfer in mathematical throughout such system. Much models of work the and certain fundamental physics simplifying a useful done groups which on assumptions a n a l y s e s of developing mathematical theoretical of heat mathematical Mathematical three been r e g e n e r a t o r s [4-14].The of obtain has transfer m u s t be considerations are complicated made i n o r d e r to model. regenerators are divided are explained i n the following into sections. 1 6 1.2.1 SCHUMANN MODEL The was simplest first developed suggested flow a model through However i n which a packed through chequer the work) Schumann's mathematical regenerator at a fluid (Fig through problem temperature assuming in i s the to 1929 stream [ 1 3 ] . He was a l l o w e d t o solids of brick of (Fig.4a). of fluid matrix (often developing transfer a fluid prism find i s at an a the exact problem of crushed allowed i na material to uniform pass rate of distribution bed and i n t h e f l u i d of for a l l time, that The t h e r m a l p r o p e r t i e s of in regenerator f o rt h e case t o t h e heat the a .4b). method treatment of bed of broken channels a uniform temperature; flow.The 2. Schumann assumes a bed c o n s i s t i n g lengthwise 1. by model h i s m o d e l c a n be e m p l o y e d passing called mathematical of the system a r e independent the temperature. The axial conduction the solid phase transfer 3. The f l u i d 4. There of heat is particles flow is i n e i t h e r the f l u i d negligible from s o l i d a t any i n s t a n t . to the to f l u i d . r a t e does not vary no t r a n s v e r s e compared phase o r along the bed. thermal gradient within the 17 B a s e d on t h e s e coupled which assumptions differential determine appropriate solved the set of ,Schumann d e r i v e d a equations (given transfer of in pair of chapter heat. With 2) an boundary c o n d i t i o n s the problem i s completely. Willmot [6] has p r e s e n t e d the Schumann m o d e l . I n an o r d e r has shown t h a t t h e a x i a l a computer solution for of magnitude a n a l y s i s conduction ,he w i t h i n the matrix i s 2 negligible provided semi-thickness length.This 1.2.2 of ratio the model is that that conductivity. ceramic of assumption matrix has In many c a s e s this regenerator [ 1 2 ] ) have a s u f f i c i e n t l y the p a r t i c l e s . L is a very i s not Biot number the the bed cases. low t h e r m a l thermal which Schumann gradient within is justified high so. thermal Glass and m a t r i c e s a r e o f t e n made conductivity be made f o r t h e t h e r m a l The m a t r i x the conductivity i s i s d e f i n e d as that gradient within i n t o c o n s i d e r a t i o n i n terms of a d i m e n s i o n l e s s called is i n most p r a c t i c a l disadvantages simplifying the must and i t n e g l e c t s the thermal (from which allowance matrix d MODEL major matrix.This provided i s small;where i s negligible INTRACONDUCTION One o f t h e the t h a t d/L taken parameter 18 Bi=hR/K s =hd/K s (spherical ( p l a n a r geometry) , where h = c o n v e c t i v e R=radius of heat K -matrix s conductivity is called matrix Previous may effect makes Handely response in an effect studies the have order effect model Heggs to of the m a t r i x matrix Bi thermal number)on the in a were observation.The which t h e Schumann and intraconduction the t h e r m a l the behavior inclusion of complicated. proposed of this Different and have been analytically. obtain experimental group that [12] a p p l i e d results between on more theoretical dimensionless (or shown n u m e r i c a l l y or and of . i n t r a c o n d u c t i o n model.Figure.8 m o d e l s have been either method the [4-13],although mathematical solved conductivity profile. have a s i g n i f i c a n t regenerators , , of c o n d u c t i v i t y temperature coefficient of m a t r i x , thermal includes shows t h e e f f e c t transfer sphere d=senu-thickness A model w h i c h geometry) , a fixed the Crank-Nicholson solution bed to transient regenerator.Their i n good agreement w i t h authors predicts also their proposed the d i v i d i n g i n t r a c o n d u c t i o n models. a line 19 Figure 8. Effect of thermal temperature conductivity profile, (B5=hR/K). of solid on the 20 Hausen matrix in is a model thermal conductivity terms the o f an so c a l l e d overall the [15] p r o p o s e d heat i s taken heat transfer modified-infinite transfer coefficient i s defined which modification of [4,6,15] authors this modification An in have proposed analytical solids can method.This method o b t a i n e d by was first a very simple problem in semi-infinite with and [17].However, the well established thermal conductivity is whereas the method can and heat Pohle equally only applied have appropriate complete discussion following chapter. a number expressions integral Goodman [16] be were in of satisfactory good Carslaw integral planar of spherical this to method include method to Lardner the method geometry.A is i f the thermal geometry. that by temperature; extended demonstrated for of dependent the to heat conduction results independent conduction in [18] be temperature conductivity.Goodman transient A the e l e g a n t methods proposed will integral by results Jaeger of and of heat conduction of u n s t e a d y and effect different employed Carslaw the factor. the approximate slab.His agreement Jaeger i n terms coefficient to transient solve a The factor. solution be This c o n d u c t i o n model. parameter of consideration coefficient. transfer is called the e f f e c t into c o n v e c t i v e heat for actual overall i n which is more given i n the 21 1.3 SCOPE OF The THE literature sufficiently effect this to low should effect thermal be taken makes the purpose analytical utilizing the relaxed. a l s o be The first approximation rigorous on of the there is regenerator regenerator are no regenerator.The the means regenerator. to work is matrix present the matrix integral the of technique. solid the solid its the accuracy performance. published of design use method thermal of of effect I t should for an by this provides in order usual properties proposed against the to of The the validity extended results development the development a l s o the The regenerator. examine and propose mean t e m p e r a t u r e constant be solution t r a n s i e n t response programs; the to a of determined. then obtain inclusion t o be the after can intraconduction has to , i t the of analyses.Once established types stage for The e f f e c t i v e n e s s of can that involved. of assumption fact c o n s i d e r a t i o n . The computer simplifying the a n a l y s i s more to determination obtain into the lengthy confirms conductivity, approximate method a v o i d s a quick of solution an INVESTIGATION search t r a n s i e n t response The to PRESENT method more is include different matrix geometry of be emphasized the the integral data for moving method the that bed provides moving bed 2. 2.1 DIMENSIONLESS It is present a the THE PARAMETERS common practice results p a r a m e t e r s which a r e equations. GOVERNING EQUATIONS in i n terms of introduced regenerator a number of to simplify These p a r a m e t e r s were f i r s t design to dimensionless the governing i n t r o d u c e d by Hausen [15]. Number of t r a n s f e r u n i t s T h i s parameter the literature. coefficient,heat fluid (NTU) i s a l s o termed the reduced It i s defined i n t e r m s of transfer per capacity rate 2 area (refer unit to Fig.9) bed the length heat volume (A) transfer and . £=hAy/(mC), y = L, where h = c o n v e c t i v e A=heat heat transfer m=fluid flow C=fluid specific transfer coefficient a r e a / u n i t bed rate/unit heat area , volume , , c a p a c i t y at constant pressure. These 2 terms a r e e x p l a i n e d capacity i n more d e t a i l rate=mC 22 i n Appendix in A. the 23 Figure 9. Typical dimensions o f a r e g e n e r a t o r . 24 Dimensionless The period period of each nondimensionalised n=hA(P-L/u)/(M s s P=period L=bed s =bed is usually (p) [12]. becomes L/u s is residence time ) , flow , heat capacity , density L/u which In short , specific because represents . the f l u i d i s permissible in many i t i s n e g l i g i b l e compared cycling applications of s i m i l a r magnitude c a n n o t be cycle ) , ignored.This applications cooling velocity , s term .C s of f l u i d C =solid The .C length u=fluid M and as T?=hA(0-y/u)/(M where heating the to the period regenerator to the period residence time and the e f f e c t of ignored. Effectiveness This i s defined matrix temperature Fig.9), that i s e=[m s C s as the r a t i o to of the a c t u a l i t s maximum possible (T -T . ) / ( ( m C ) . ) . ( T , . - T . ) ] , msO s i min f i s i rise rise in (refer the to 25 where m = s o l i d f l o w s (mC) rate/bed . =minimum min area o f t h e t w o mC (m C )/(mC) . = c a p a c i t y s s min It should presented solid in surface Appendix temperature temperature.This thickness slab rate ratio, the s o l i d ,which temperature i s different i s explained in more is than the detail radius as ( r e f e r and the sphere radius are to Fig.9) For the slab; z=x/d 2. or thickness nondimensionalised 1. that , D. Dimensionless The be e m p h a s i z e d a s a mean , For t h e sphere , where d=thickness of t h e s l a b . ; z = r/R , where R=sphere radius 26 Normalised The 1. temperatures fluid temperatures are normalized as For the f l u i d F=(T 2. and s o l i d f -T .)/(T,.-T .) . f1 s i SI For the s o l i d *=(T s -T .)/(T .-T . ) . s i f i s i r 2.2 THE M A T H E M A T I C A L MODELS It was regenerators explained the type of regenerators matrix The of matrix geometries between fixed)and the (fixed fluid that depending work and moving two types bed) with of heat spherical). developed a group of s o l i d chapter groups the present (planar and are previous different employed.In are considered equations transfer the c a n be d i v i d e d i n t o on two in for the rate particles(either moving countercurrently transfer processes moving or to the particles. There in a thermal depending If a r e two heat i t infinite regenerator. However o n t h e a s s u m p t i o n s made i s assumed that which one may i n developing the matrix thermal (Schumann model) t h e d o m i n a n t heat take place predominate the model. conductivityi s transfer process 27 is heat transfer across transfer to the thermal conductivity the surface fluid). (Intraconduction On i s model) of the matrix the other assumed ,the hand (or i f the matrix to be conduction heat finite also becomes important. The two two m o d e l s are analysed i n more detail i n t h e next of a thermal regenerator sections. 2.2.1 SCHUMANN This and i s the simplest i s based a. MODEL on t h e f o l l o w i n g s i m p l i f y i n g The thermal independent b. model properties the system are of temperature, The t r a n s f e r of heat by conduction itself i n the fluid i s small compared to the heat t r a n s f e r by convection from the f l u i d to the s o l i d , c. The f l u i d d. There i s no thermal g r a d i e n t w i t h i n the matrix. If the of assumptions: there matrix,then uniform flow r a t e does not vary a l o n g the bed, i s no t r a n s v e r s e the matrix temperature a t any p o i n t Thus only heat c a n be assumed gradient to a n d c a n be r e p r e s e n t e d temperature the thermal along process be by a the regenerator transfer within i s at a single (Fig.8a). the heat 28 gained/lost by the fluid passing through the regenerator. F l u i d phase heat t r a n s f e r e q u a t i o n If past m i s t h e mass a section a regenerator,then from/to rate distance between the f l u i d of f l u i d flow/unit y from y a n d y+dy i n t i m e dd w i l l bed area t h e entrance of the t h e heat transferred be ( r e f e r t o Appendix A) dQ=mC[ U T , / 9 y ) + l / u . ( 9 T / 9 0 ) ]dy.A,_ , 1 6 f y b (2.1.a) r = m C ( D T / D y ) d y .A. , i b (2: K b ) r w h e r e D/Dy= ( 9 / 9 y ) + 1 / u . (3/3(9) S o l i d phase heat t r a n s f e r e q u a t i o n The the total heat lost heat flow f then will m s i s t h e mass between must be e q u a l t o by t h e m a t r i x , t h a t i s dQ=hA(T-T )dy.A If to the fluid y f a . flow a n d y+dy (2.2) rate of the heat solid/unit lost from area , the matrix be dQ=m C [ ( 3 T / 3 y ) +1/u . ( 3 T / 9 0 ) s s 6 s ]dy.A . y b L (2.3.a) 29 The a b o v e e q u a t i o n can a l s o be w r i t t e n a s dQ=M C [u (3T/3y) +OT/30)-'jdy.A^ s s s 6 y b (2.3.b), where M =m /u , s s s if t h e bed i s s t a t i o n a r y ( u = 0 ) then s equation (2.3.b) becomes dQ=-M C (9T/30)dy.A, . s s b The (2.3.c) sign from f o r the entrance moving whereas equations temperature temperature T. T h i s any p o i n t a l o n g uniform the bed is i n both For equations t h e moving bed (1),(2) and ( 3 ) : from t h e is by measured a single assumes the matrix temperature. is of the equations the i s b e c a u s e t h e model Summary o f t h e e q u a t i o n s Combining i n which y bed y represented regenerator (2.3.a) and y i s measured f o r the f i x e d t h e o p p o s i t e end. A l s o matrix between i s due t o t h e change o f d i r e c t i o n measured; solid difference (2.3.c) that at i s at a 30 hA(T-T )=m C [ ( 3 T / 3 y ) +1/U .(3T/30) ] , • t s s 6 s Y (2.4.a) =m C (DT/Dy) , s s and hA (T-T,.) = m C [ ( 3 T / 3 y ) + 1 / u . ( 3 T /3#) ] , t t t) t y (2.4.b) =mC(DT /Dy) . f For the f i x e d bed:Only the s o l i d h A ( T - T ) = - M C (3T/30) regenerators composed thermal conductivity,eg.glass (d) of the must be made, f o r t h e r m a l Schumann temperature of regenerator can temperature. temperature matrix There It Heat model the no of matrices and i s invalid. gradient within Thus any longer be represented thus temperature within to the s o l i d a s a mean a r e t h u s two h e a t point desirable Heat along as obtain and the one the represent temperature. transfer processes f o r an model , i s gained/lost by t h e f l u i d p a s s i n g through the regenerator. 2. allowance t h e m a t r i x . The at is w i t h low ceramics,assumption solid distribution intraconduction 1. (2.5) INTRAPARTICLE CONDUCTION MODEL For the changes, . c 2.2.2 phase e q u a t i o n i s transferred within the matrix. 31 1. Fluid phase heat This transfer equation e q u a t i o n i s t h e same a s t h a t f o r t h e Schumann model. 2. S o l i d phase heat t r a n s f e r There equation a r e two s t a g e s o f h e a t transfer i n the solid phase. (a) H e a t across the surface of transfer solid: T h i s c a n be r e p r e s e n t e d a s h ( T -T )=K (9T/9x) L S S =-K It s (9T/9r) should represents ( P l a n a r geometry) u r—K be (Spherical emphasized the s o l i d necessarily The X— heat lost terms of the (refer t o Appendix that by t h e s o l i d solid can in this which be s might n o t energy in as A) cl 3/d , (2.6) u and case T represented r a t e of change of i t s i n t e r n a l d (2.5.b) temperature. dQ=m C d y . A [ ( 3 ( J T d x ) / 9 y ) + ( 9 ( / T d x ) / 9 0 ) / u s s b o 0 d (/ T d x ) / d r e p r e s e n t s t h e mean s o l i d 0 d i s the s e m i - t h i c k n e s s of the m a t r i x . where (2.5.a) geometry). s u r f a c e temperature be t h e same a s mean , s temperature, 32 (b) H e a t transfer The is matrix the distribution within the matrix diffusion equation.Assuming dimensional we for solid: thermal c o n d u c t i v i t y a temperature transfer within is finite within the is represented the problem by is the one have p l a n a r geometry 2 spherical (2.7.a) 2 geometry 9T/90=a[{3 T/9r )+(2/r)(9T/9r)] 2 The there matrix.Heat 9T/90=a(9 T/3x ) , for and 2 above e q u a t i o n s a r e c o u p l e d by the . (2.7.b) symmetry condition planar (9T/9x) x=d =0 d=thickness of slab R=radius sphere (2.8.a) spherical (9T/9r) =0 r=0 of (2.8.b) 33 The the distribution regenerator i s obtained with a p p r o p r i a t e for a fixed by initial 2.3 , condition i s represented temperature , f o r 0=0 and y=0 . a moving bed , t h e i n l e t temperatures are specified. NON-DIMENSIONAL FORM OF GOVERNING EQUATIONS The terms governing of temperatures dimensionless Fluid equations dimensionless normalised in equations f o r 9 >0 a n d y=0 , f T=T. these along conditions i n t h e gas i n l e t T =T . f temperature by s o l v i n g bed , t h e i n i t i a l a s t e p change for o f gas and s o l i d form be parameters defined nondimensionalised £, in section rj,Biot 1. The a r e (Refer t o Appendix in number and equations A) phase Equations (2.4.b) and (2.5) become ( r e f e r 3F/9£=(¥ -F) , s =(F-¥ Again can the sign ) , difference (fixed t o Appendix A ) , bed) , (2.9.a) (moving bed) . (2.9.b) i s due t o t h e change o f d i r e c t i o n 34 in which Solid £ ( o r y) i s measured. phase Planar geometry Equations Appendix ( 2 . 5 . a ) , ( 2 . 7 . a ) a n d (2.8.a) become (refer to A) (3*/3z) = Bi(¥ z=0 -F) , (2.10.a) s 3*/3r?=[ 3 * / 3 z ] / B i , (2.l0.b) (3+/3z) (2.10.C) 2 2 =0 z= 1 . Spherical Geometry Equations (2.5.b),(2.7.b)and(2.8.b) become(refer to Appendix A) (3*/3z) =-Bi(¥ -F) z= 1 s , 3¥/3r?=[ ( 3 * / 3 z ) + ( 2 / z ) 0 * / 3 z ) ] / ( 3 B i ) , 2 (3*/3z) The of initial a fixed 2 -=0 z=U . conditions bed as , (2.11.a) (2.1Kb) (2.11.c) a r e n o n d i m e n s i o n a l i s e d f o r the case 35 F=1 at £ = 0 and T?>0 , *=0 at 7?=0. and £>0 . 3. 3.1 INTRODUCTION TO THEINTEGRAL METHOD. The thermal inclusion design of complicated.The must matrix thermal regenerators diffusion matrix.This techniques require c o n d u c t i v i t y i n the makes equation the computer alternative approximate introduced be by distribution employing of p r e v i o u s is the solids.Goodman technique.This problem in fluid conditions. work) method in which with The order heat the technique either method of t h e was first to solve mechanics.However,the f o r unsteady [ 1 6 ] employed equation,coupled boundary within numerical application by von Karman and P o h l h a u s e n layer more programs. i s equally appropriate diffusion done solution integral boundary method can analysis (equations.2.10.b,2.11.b) ( d i s c u s s e d i n the review lengthy An in of be s o l v e d t o o b t a i n t h e t e m p e r a t u r e the the THE METHOD OF SOLUTION linear makes conduction t o solve the or non-linear use of two assumpt i o n s : a. The t h e r m a l usually order properties(ie.conductivity,density assumed to linearize to be independent the d i f f u s i o n 36 etc) of temperature equation. are in 37 b. The s o l i d is initially Goodman subsequently account f o r temperature explained The the which i n more d e t a i l integral penetration the heat beyond which will flux (9T/3x) The properties.This i s ( 3 . 4 ) of t h i s introduces a quantity is defined ,while chapter. 5(8) called as a d i s t a n c e penetrates the i n s i d e the s o l i d point solid will up is there to be a t a (Fig.10).This into solid,and transferred.Consequently the to the uniform expressed as =0 x=6 penetration thickness thermal technique in section gradient beyond t h i s mathematically dependent i s no h e a t depth temperature. developed , a at the surface be a t e m p e r a t u r e temperature has depth.This there penetration The method at a constant . depth is in fluid mechanics. technique adopted method c a n be e x p l a i n e d analogous t o the boundary by the as f o l l o w s ; approximate layer integral 38 Figure 10, Schematic r e p r e s e n t a t i o n o f depth c o n c e p t , T = Surface temperature, Ti= I n i t i a l temperature. s penetration 39 The in solid x temperature (or r f o r s p h e r i c a l polynomial to is limited geometry) by constraints (ie. boundary conditions etc) .The the time. way improve t h e accuracy i n c r e a s e the parameter which additional order of i s thereby derived the .The a polynomial order number of unknown a f u n c t i o n of to by of of the variable conditions, usually One is i s represented initial coefficients t h e assumed polynomial.Each profile additional introduced i s determined constraint.This may not demonstrated that the are from always an be possible. Koh.y [ 1 9 ] has profile is than polynomial, a involved. using There the determining successful and better is integral whether must however never an not exponential function analysis a unique or simple by the method.The i n v o l v e an simplicity.A adequate approximated of polynomial profile purposes. both more to f o l l o w i n criterion particular assessment f o r most e n g i n e e r i n g becomes procedure ultimate a temperature for profile its has is accuracy been found 40 3.2 PLANAR The GEOMETRY solid x,where x is solid.Since number o f matrix temperature the the distance order of constraints,it as a this way increased will be 3.2.1 slab the would the seen the be a polynomial surface is limited advantageous slab of the by the t o model extending the of in polynomial of the the x can be inside the later. SLAB temperature i s t o be by polynomial order SEMI-INFINITE The from semi-infinite direction.In as i s represented distribution calculated subject T(x,0) to the following constraints (3.2.1) T(5,0)=T. 3T(S,0)/3x=O (3.2.2) , 3T(0,0)/3x=h.(T -T )/K =-f(0) f There are be represented assumption temperature additional three constraints by a that the can be so second-order the profile must polynomial. The slab i s i n i t i a l l y utilized constraint .(3.2.3) in .Equation at a constant deriving an (3.2.1) is 41 differentiated substituted with respect i n the d i f f u s i o n This i s usually The represented nondimensionalised depth) that ,Biot number i s (refer *=0 profile.The in terms Q of z is the profile 5^=5/6 be are (dimensonless and 6^, (3.2.5) , Q at z=6 (3.2.6) Q , , (3.2.7) (3.2.8) , dimensionless will is B) 3¥/9z=Bi . (* - F ) = - f ( r j ) a t z = 0 s where now constraints , 3¥/3z=0 a t z = 5 2 2 3 */3z =0 can ,normalised temperatures to Appendix at z=5 result condition. distribution by a c u b i c then (3.2.4) the smoothing temperature and . 2 called time equation.The 3 T(6,0)/3x =O 2 to take the penetration form (refer *=f (i?) . ( 6 - z ) / ( 3 . 5 ) 3 0 2 . d e p t h . The t o Appendix cubic B) (3.2.9) 42 The in surface temperature equation (3.2.9) ,the r e s u l t •*-=f ( T J ) . 6 / 3 s 0 between penetration depth is The or z = and S Q S substituting result result =[ 12. (S t 0 surface once only the and the penetration require calculate depth d7j the a simple surface 6^ obtained (eqn.2.10.b) * from to Appendix ) / ( f (77) . B i ) ] ° " temperature for for i s (refer (17) is equation substituting V 0 relationship vice-versa. the d i f f u s i o n (3.2.9),the The to penetration integrating to , i twill manipulation a temperature depth.Consequently, temperature is expresses surface calculated algebraic z=0 solid z=0 (3.2.10) (3.2.1.0) the setting . a Equation i s o b t a i n e d by from can then from equation B) . (3.2.11) 5 be equation by o b t a i n e d by (3.2.10).The is =[4.f (v) * s • (S £ 0 V (T?) di?)/(3.Bi) ] ° ' 5 . (3.2.12) 43 3.2.2 SLAB OF F I N I T E THICKNESS Initially,the does not a f f e c t the it matrix.The matrix were depth symmetry some c a n t h u s be m o d e l e d penetration be later centre condition time of comes the the within as t h o u g h into replaced by is symmetry condition). following constraints penetration matrix effect.At d e p t h has no meaning whose f a r s u r f a c e and a s l a b of f i n i t e and stage the model thickness (representing The is slab the this insulated subject the to the 3T(d,0)/3x=O (3.2.13.a) T(0,0)=T (3.2.14.a) s S The (eqn.2.10.c) semi-infinite. reaches the should condition the temperature d i s t r i b u t i o n However,at the symmetry above e q u a t i o n s f (3.2.15.a) S in dimensionless form a r e 3¥( 1 , T J ) / 3 Z = 0 (3.2.13.b) ¥(0,77)=* (3.2.14.b) 3 ¥ ( 0 , T ? ) / 3 z = B i . ( ¥ - F ) = - f (T?) . (3.2.15.b) 44 The second-order *=+ The then - f (rj) . ( z - 2 . z ) / 2 the diffusion t o z.The s o l i d the polynomial the previous The take the . will equation temperature be obtained (eqn.2.10.b) ¥ result integration is (refer opposed extends to Appendix by with i s replaced express ion(eqn.3.3.4).As case,the form (3.2.16) surface temperature ¥ respect z=1. must 2 s integrating to profile from by to z=0 B) V * s = [ f ( r ? ) / 3 + J f ( v ) dr? / B i 0 Assume depth reaches initial s = * (T?~) s 0 7?Q. The setting result the time a t which far surface condition (ie. * explicitly • the TJQIS setting 5^=1 in equation constant * =+ (77.) s s 0 of and ]+constant the of ( 7 ? ^ ) ) can s 0 in equation . penetration the be 77=17_ 0 in slab obtained .The by (3.2.10).Setting (3.2.12) r e s u l t s integration (3.2.17) in i s then obtaining obtained by e q u a t i o n ( 3 . 2 . 1 7 ) .The is V * =f <TJ)/3 +/ f(rj) d(r?)/Bi "0 For 77>7i . (3.2.18) 45 In summary If T If * I E N E C 3 ation (3.2.12) i s used. u 77>T7Q t h e n e q u a t i o n (3.2.18) i s used. 3.3 SPHERICAL GEOMETRY L a r d n e r and spherical Pohle geometry, temperature geometry profile the (l/r),they the is steady same sphere.The same as profile that penetration profile solution depth is for applicable adopted the of the spherical proportional to (3.3.1) with a s p h e r i c a l to the for spherical penetration , that case hole, of a solid geometry i s the i s , the original depth.As soon r e a c h e s t h e c e n t r e of t h e s p h e r e , a s h o u l d be u s e d . for of t h e form f o r p l a n a r geometry includes representation and P o h l e d e a l t is procedure that in r ) / r . Lardner method demonstrated inappropriate.Since suggested a p r o f i l e Although have polynomial state T=(Polynomial the [18] as the second 46 3.3.1 SPHERE OF I N F I N I T E The penetration RADIUS depth surface o f t h e s p h e r e . The subject to the f o l l o w i n g 3T(R-5,6)/dr=0 T(R--5,0)=T(i) temperature from t h e profile is constraints , (3.3.2.a) , (3.3.3.a) (3.3.4.a) 3 T(R-8,0)/3r =O (3.3.5.a) . 2 Equation(3.3.5.a)represents above e q u a t i o n s q * ( 1 -5 , T J ) =0 the smoothing in dimensionless 3¥( 1 - 6 , T ? ) / 3 Z = 0 (3.3.2.b) , (3.3.3.b) 3 +( 1 - 8 , T ? ) / 3 Z = 0 2 condition. form a r e , 3¥( 1 ,77)/3z = - B i . (¥ - F ) = - f (77) 2 0 where measured 9 T ( R , 0 ) / 3 r = - h . ( T -T )/K , s I S 2 The is 6^=8/R=dimensionless , , penetration (3.3.4.b) (3.3.5.b) depth.(3.3.6) 47 There are polynomial 4 will constraints consequently the be a c u b i c . A d o p t i n g t h e s u g g e s t i o n by L a r d n e r and P o h l e , t h e r e s u l t i s 3 2 *=(A.z +B.z +C.z+D)/z Applying . (3.3.7) 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 The surface z=1,in )] /UQ-(3-6 3 temperature is The integrating penetration the * has been integration (refer obtained (3.3.8) by setting above e q u a t i o n , t h e r e s u l t i s * =-f (77) . 5 - / ( 3 - 5 . ) s 0 0 after ) .z) . . depth diffusion replaced extends t o Appendix from by (3.3.9) is equation obtained (eqn.2.11.b) equation(3.3.8). z=1-6^ t o z=1,the B) 7? n The surface temperature for 5 equation(3.3.9),the result i s n from The result i s [ ( 5 6 - 6 ^ ) / ( 3 - 6 ) ] = 2 0 ( / f (TJ) dr?)/( 3 B i . f (TJ) ) U u (J o 2 by i s o b t a i n e d by .(3.3.10) substituting 48 ]=20(* - f (r?) ) ( f* t (77) dr?) / ( 3Bi ) ( 3 . 3 . 1 1 ) 3 * [ 5 f (r )-2* 2 2 ? S S S g 3.3.2 SPHERE OF F I N I T E As i n the case condition until this account of p l a n a r geometry,the a t the c e n t r e does not the point RADIUS penetration depth t h e p r o f i l e has t o of symmetry come reaches be symmetry into effect the c e n t r e . A t changed condition.The to new take set of constraints are 3T(0,0)/3r=O , (3.3.12.a) 3 T ( R , 0 ) / 3 r = - h . ( T -T,)/K s T(R,0)=T , (3.3.13.a) r s . (3.3.14.a) s In dimensionless form 3*(0, T?)/3Z = 0 the equations a r e , 3*(1 ,T?)/3z = - B i . (* - F ) = - f (T?) *(l,r?)=* (3.3.12.b) , . (3.3.13.b) (3.3.14.b) s Again adopting t h e s u g g e s t i o n by L a r d n e r the p r o f i l e w i l l take t h e form and P o h l e 4 9 3 2 *=(A.z +B.z +C.z)/z It is should solid,the expression there of the emphasized that 1/z not term of i s no be the temperature constant z i n the should term since appear will take i n the i n c l u d e d i n the the sphere profile.This above e q u a t i o n . A p p l y i n g profile the the final is why polynomial constraints form + f ( T? ) . ( 1 - z ) / 2 2 . (3.3.15) s Integration of the equation(3.3.15)is * =-f(Tj)/5-(f s The constant procedure 0 diffusion equation s u b s t i t u t e d for + w i l l f ( r j ) dr? ) / B i + c o n s t a n t of integration explained after give . (3.3.16) i s obtained in previous by section,the the result i s + = - f ( T ? ) / 5 - ( / f (77) dr?)/Bi , s 0 7 ? where 77^ d e f i n e s t h e end the second beginning of the of the initial stage. (3.3.17) stage and 50 In summary If r7< 77^ E q u a t i o n ( 3 . 3 . 1 1 ) s h o u l d be used If V-VQ E q u a t i o n ( 3 . 3 . 1 7 ) s h o u l d be used 3.4 TEMPERATURE-DEPENDENT The usual simplifying p r o p e r t i e s made i n regenerators THERMAL assumption developing the c a n be relaxed.When temperature-dependent, PROPERTIES mathematical model e q u a t i o n i s r e p l a c e d by . (3.4.1) B o t h K and p.C a r e t e m p e r a t u r e - d e p e n d e n t . T h e to obtain the. t e m p e r a t u r e different to that properties.Goodman thermal p r o p e r t i e s simplifies only the analysis procedure paper [ 2 0 ] . has for enter i n a way t h a t surface i s explained procedure will be somewhat constant demonstrated at the surface one u n k n o w n , i e . t h e The distribution adopted [20] of the thermal p r o p e r t i e s a r e the d i f f u s i o n p.C.3T/30=3(K.3T/3x)/3x of c o n s t a n t t h e r m a l that the thermal only problem. there will the This still be temperature. i n more d e t a i l i n Goodman's 51 3.5 NUMERICAL PROCEDURE The and fluid intraparticle solved while conduction numerically the solid approximate and solid fluid a difference calculation namely 3.5.1 FIXED The carried equation is approximation, solved using the the bed. out for two types of BED grid. bed form) is n=i . 5r? by a 2 divided into n equal 5£), that i s (3.5.1) P (or which t h e f l u i d dimensionless represented . period into is The l e n g t h o f t h e bed y ( o r A i n of 5y(or A=n.5£ divided phase the bed and moving b e d . regenerator increments is throughout is dimensionless during equation fixed dimensional The finite describe method.The two unknowns a r e t h e f l u i d temperature regenerators, equations model.The phase integral the s o l i d The by phase i equal n in i s passed dimensionless through increments form) t h e bed is o f 56 o r ( 5TJ i n form) (3.5.2) 52 At each p o i n t temperatures t h e bed t h e f l u i d are represented respectively. unknown along At each step temperatures,ie provided the as F(n,i) point F(n,i+1) temperatures at and and there and solid q»(n,i) s a r e two *(n,i+1) , the ( n , i ) p o i n t a r e known. The fluid represented phase equation by a c e n t r a l - d i f f e r e n c e (eqn.2.9.b) is a p p r o x i m a t i o n as (1+A£/2)F(n,i + 1 ) - * ( n , i + 1)A£/2=(1-A£/2)F(n-1 , i + 1 ) + s A £ . « M n - 1 , i + 1 )/2 . The order to form,the under AT? / 0 solid phase represent integral equations involve the term equations / f (77) in dr? .In numerical i s a p p r o x i m a t e d by t h e a r e a t h e c u r v e f(r?) v e r s u s T? , t h a t i s f (T7)di7=Ar1 ( n , i + 1 ) =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 initially 53 Arl(n,0)=0 The solid their , phase e q u a t i o n s numerical Planar and Ar 2 (n , r? ) =0 can now be written in form. geometry For r?<r? , 0 5 (n,i)=[12Ar1(n,i)/(Bi.f(n,i))]°' , 5 Q * s (3.5.4) (n,i + 1 ) = [ 4 f ( n , i + 1).Ar1(n,i+ 1)/(3Bi)]°' For * .( 3 . 5 . 3 . c ) Q T?>T? 5 . (3.5.5) , 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 used order to obtain penetration penetration the matrix to determine the depth depth which e q u a t i o n solid should is less ,equation surface be the (3.5.5) otherwise,equation(3.5.6)should be be temperature,the calculated than should first.If the semi-thickness should used. be of used, 54 Spherical geometry The is same f o r the planar geometry a d o p t e d . T h e p e n e t r a t i o n d e p t h and solid surface temperature procedure are difference form For TJ<TJ 55 2 (n,i)-6 3 as represented in their as , (n,i) =20(3-5 (n,i)).Ar1(n,i)/(3Bi.f(n,i)) 3 * ( n , i + 1 ) (5f (n, i + 1 )-24« s s 2 2 g For 77>7? , (3.5.8) . (3.5.9) (n, i + 1 ) ) = 20(* (n,i+1)-f(n,i+1)) .Ar1(n,i+1)/(3Bi) * finite , 0 (n,i+1)=-f(n,i+1)/5-Ar1(n,i+1)/Bi , (3.5.10) where f(n,i)=Bi.[¥ s (n,i)-F(n,i)] where rj^ r e p r e s e n t s beginning In equations of the order the second to should end of first (3.5.11) stage and the stage. determine be , u s e d , the which of the above p e n e t r a t i o n depth at 55 each step point At the F(n,i+1) the should be c a l c u l a t e d (n,i) step first. point,the and * ( n , i + 1 ) , p r o v i d e d unknowns are the temperatures s at ( n - 1 , i + 1 ) and ( n , i ) p o i n t s a r e known. The starting obtained finite from values the i n i t i a l for the solution c o n d i t i o n which in are their d i f f e r e n c e form a r e At 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) 6 (n,i)=0 At i=0 and n>0 . (3.5.12.d) s the entrance does not v a r y with (ie.n=0)the Consequently,there equation . is , which c a n be (3.5.6) or temperature time,that i s F(0,i+1)=F(0,i)=1 temperature,¥ fluid (3.5.5). only obtained one from unknown either 56 3.5.2 MOVING BED REGENERATOR In t h e c a s e period of of a moving operation bed is regenerator,the not an independent characteristic of t h e s y s t e m . I t is as the time a solid t o t r a v e r s e one full i t takes length particle of the r e g e n e r a t o r period of the chamber. cycle i s related height.The c o n d i t i o n s a r e steady exit the of distribution regenerator obtain The A£.The entrance.This during the obtained be used F(n) £ is solid the n equal the stages t o determine of temperature along the contra-flow regenerator * ( ) increments from the penetration the of solid depth c y c l e must be which e q u a t i o n temperature and f l u i d respectively; points measured because initial i n order the be a d o p t e d . f o r the s o l i d The along is a t t h e e n t r a n c e and efficiency,a bed i s d i v i d e d i n t o direction to the regenerator one c y c l e i s o f i n t e r e s t . To maximum should Consequently,the Only different bed d u r i n g a arrangement regenerator. at defined should calculation. temperatures a t each are represented point by " f ^ C n J a n d represents the s o l i d inlet temperature,whereas F ( 0 ) r e p r e s e n t s the f l u i d outlet n s temperature. The fluid by a central phase e q u a t i o n (eqn.2.9.b)is d i f f e r e n c e approximation as represented 57 F(n+1 )-F(n)=A£[F(n+1 )-¥ s The solid s phase e q u a t i o n s form a r e shown Planar (n+1 )+F ( n ) - ¥ ' • ' ( ri ) ]/2 . ( 3 . 5 . 1 3 ) in their numerical below, geometry For 77<7} 0 0.5 (3.5.14) 0.5 * (n+1)=[4.f(n+1).Ar1(n+1)/(3.Bi)] For * 77>7j (3.5.15) 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) dr? (3.5.17.b) Ar 1 (n+1) = / 0 Ar2(n+1)=/ f(7j) f (rj) drj . (3.5.17.c) 58 Spherical geometry For r?<r? Q , 55 (n)-6Q(n)=20(3-S (n))Ar1(n)/(3Bi.f(n)) 2 0 , (3.5.18) 3 * ( n + 1 ) [ 5 f ( n + 1 ) - 2 * (n+1)] = 2 0 [ * ( n + 1 ) - f ( n + 1 ) ] s s s 2 ,Ar1(n+1)/(3Bi) For * s 7j>7? 0 . 2 (3.5.19) , (n+1)=-f(n+1)/5-Ar1(n+1)/Bi , (3.5.20) where f(n)=Bi.[* (n)-F(n)] , (3.5.21.a) f(r?) dr? , (3.5.21.b) s Ar? Ar1(n+1)=/ 0 Ar2(n+1)=/ f(r?) dr? . 0 (3.5.21.c) ^0 It should a l s o be e m p h a s i z e d that Ar?=A£mC/(M C u ) . s s s At two each point along (3.5.22) the r e g e n e r a t o r unknowns,namely, * ( n + l ) a n d "¥ ( n ) a n d s s F ( n ) a r e known, there are F(n+1),provided that 59 The the fluid analysis proceeds and inlet solid approximation calculated the actual discrepancy and for inlet the a n a l y s i s inlet initial i s then u s i n g the a v e r a g e temperatures fluid temperature given the the by outlet i s then as a of first temperature.The compared with temperature.If t h e r e i s any approximation is repeated. adjusted 4. RESULTS AND DISCUSSION The first establish the method.This against stage validity is the r e s u l t s previous effects.One by H a n d l e y are mainly spherical 4.1 solve available [ 1 2 ] ,who employed the diffusion for a bed obtained authors. a r e scarce.Most of intraconduction a r e those the published Crank-Nicholson equation fixed i s to integral the r e s u l t s disregarded of r e s u l t s and Heggs results by c o m p a r i n g have results approximate the published r e s u l t s source to the the p u b l i s h e d by d i f f e r e n t analyses scheme analysing of achieved Unfortunately the in numerically.The regenerator with a matrix. FIXED BED 4.1.1 SPHERICAL GEOMETRY The diffusion appropriate the boundary approximate numbers.The published represent Biot by integral can most and excellent be for different compared Heggs. the temperature agreement of regenerators. 60 those for different between the two of the r e s u l t s dimensionless which c o v e r with Biot Figs.11,12,13 seen.The c o m p a r i s o n 0<Bi<5 industrial and outlet were made f o r a r a n g e 1<A<40 method where t h e n Handley the f l u i d with c o n d i t i o n s , were s o l v e d u s i n g results numbers.An methods equation,coupled parameters the design r a n g e of 61 Experimental authors studies revealed an carried excellent o u t by t h e same accuracy of the results [12]. 4.1.2 PLANAR GEOMETRY Unfortunately comparing planar geometry.Most The fluid expected from the outlet method geometry f o r the (which that temperature i s correct. the fluid the expected trend.Thus integral planar be f o u n d f o r the p r e v i o u s s t u d i e s a characteristic i f t h e method the outlet could have is a geometry). to follow Fig.14 follows of on t h e s p h e r i c a l more p r a c t i c a l profile) results t h e n u m e r i c a l and i n t e g r a l concentrated that no method trend It it for apparent temperature can be deduced applicable to represents profile is (S shaped is outlet i s equally geometry.Fig.14 temperature profile the different fluid reduced lengths. Once t h e v a l i d i t y the analysis of intraconduction o f t h e method c a n be e x t e n d e d effect t o examine t h e s e v e r i t y fordifferent This i s a c h i e v e d by c o m p a r i n g the solid surface i s established, the d i f f e r e n c e and mean t e m p e r a t u r e common g e o m e t r i e s a t d i f f e r e n t geometries. Biot between f o r t h e two numbers. 62 It i s expected increased, that the d i f f e r e n c e and mean t e m p e r a t u r e as as the B i o t the Biot number between t h e s o l i d should increase.This number i n c r e a s e s ( i e . K is surface i s because decreases), the s depth into which penetrates surface the the solid temperature temperature resulting at flux at the surface decreases. Consequently, the w i l l be the mean heat much centre higher than the of t h e s p h e r e . T h u s t h e temperature is smaller.(refer to Fig,8). To between plotted The show t h i s , t h e the solid percentage d i f f e r e n c e Evidently,as difference the between expected.lt moderate B i o t between the the (Figs.15,16,17). was o b t a i n e d from Biot number i n c r e a s e s t h e two can be t h e maximum temperatures seen (Bi=2) increases, ( F i g . 17) a maximum i s about geometry.Consequently, intraconduction that a fora difference 9 percent f o r model which e f f e c t overestimates effectiveness. It for numbers was -+ )/+ m s number the difference and mean t e m p e r a t u r e t h e two t e m p e r a t u r e s planar neglects s of p e r c e n t a g e surface for d i f f e r e n t Biot Difference=(* as graph is a fixed also evident dimensioless from group t h e same c h a r t s (A and that B i ) ,the 63 difference between is f o r the planar greater spherical geomtry.That mechanism has less conduct i o n . T h i s and reduced smaller Most 4.2 MOVING Since line solid Biot n number has t o be ( f o r the the [ 5 ] . The in terms same i s solved time and dividing line i s dimensionless reason required of c o u r s e to predict Schumann of , B i ) .The main the computation for this i s to solve the i s not important i f analytically. BED REGENERATOR there a r e no r e s u l t s a v a i l a b l e f o r a the v a l i d i t y f o r the f i x e d (Fig.18). temperature moving of the r e s u l t s p r e s e n t e d bed here a r e o f t h e method. bed, t h e d i f f e r e n c e between t h e s o l i d and mean t e m p e r a t u r e increases intraplanar radius thickness between models (A, b a s e d upon t h e v a l i d i t y surface f o r an e q u a l conduction analysis [5]. expressed problem regenerator, As than f o r the a n d Heggs a r r i v e d a t t h e same model n u m e r i c a l l y . T h i s the that o f t h e i n v e s t i g a t o r s have t r i e d parameters reduce slab in their intraconduction to significance ( A ) , the sphere the dividing usually than temperature is,the intrasphere Carpenter conclusion and mean geometry i s because length than material). the the surface increases Consequently, decreases as as at the Biot number a f i x e d A, t h e mean the Biot number 64 increases.Since the regenerator e f f e c t i v e n e s s terms of solid former i s refelected can be the deduced effectiveness should matrix by out t h a t reduces one would of s m a l l e r of the number for a fixed as the expect size.But i t s terminal small, as the B i o t the r e d u c t i o n size the s i z e be blown thermal of it , the Fig.19. It conductivity, the sphere a higher effectiveness velocity.That t h e y might latter.So in in i nthe i s increased s h o u l d d e c r e a s e . T h i s i s shown number reduced.So i n the reduction that be p o i n t e d the B i o t mean t e m p e r a t u r e , i s defined of t h e s p h e r e is fora i s limited i s , i f the spheres a r e too off the top by the oncoming fluid. The analysis increased fluid ,the stability problem in the problem are to penetration (constant/z) solid be solid depth must profile must is t h e s o l i d and be reduced.This (S£) might numbers.The i s based even that the cause a stability profile on i nthe Lardner sphere at the i n i t i a l is applied, be o m i t t e d from which a t which number and ¥O0/z. revealed concept temperature form o f p r o f i l e Biot Biot of t h e t e m p e r a t u r e [18] s u g g e s t i o n t h a t assumed the increment at very high analysis as calculated distance present derived Further that increment i s due t o t h e form solid.The the distance temperatures reduction Pohle revealed profile.This i s o f t h e form be s t a g e s where t h e then the f i n a l would should the term expressionfor results i n a second 65 *=-f The (TJ)[.Z-(1-6 ) 0 integration For 27* 4 3 2 S identical equations tin) dr?/Bi and were reduced drj)/Bi'. 3 -f -f(rj)/5 f(r?) drj/Bi . V r, 0 the s o l i d extremely temperature surface suggestion.The compared i s almost temperature solution with the original close for low B i o t based t o t h e above results.The numbers as i s from Figs.20,21.However,the advantage of t h i s is that it i s shown reduced (Figs.22,23,24). It result in Q f o r IJ>^, Pohle were (Bi>8).This numbers 2 t o the o r i g i n a l d e r i v e d Lardner versus will , Q Surprisingly, profile . S 77>7? 7?0 evident 2 + 54* . ' f (TJ.)+45* . f ( 7 ? ) = 20f ( r ? ) . ( / ^ f ( T J ) =-.9/ 0 results 0 , 0 For on 3 of the d i f f u s i o n e q u a t i o n 77<77 S * ] /( 3 5 ) is length significant.For effectiveness for It i s therfore i s evident from that even must suggested be the c h a r t s the reduced of different effect example, c o n s i d e r versus for high by p l o t t i n g t h e c h a r t s the second p r o f i l e length stable length Biot numbers effectiveness Biot that second numbers at high Biot used. of of effectiveness Biot Fig.24 number which f o r B i = 0.1 versus i s very shows the and B i = 10. I t 66 can be seen (for example that i n order 60 p e r c e n t ) , length) has t o be a l m o s t critical information to obtain the reduced tripled t h e same length effectiveness (or i n the case of had n o t been a v a i l a b l e the bed Bi=l0.This prior to this work. Finally apparent the ,from t h e d e f i n i t i o n that as effectiveness R=(m s .C s )/(m.C) the c a p a c i t y reduces.This of rate effectiveness, ratio (R) 3 i t is i s reduced, i s shown i n F i g . 2 5 . •5. The an approximate analytical The with a 2. the method was to employed transient to obtain response following points are concluded, a p p r o x i m a t e method g i v e s r e s u l t s t h a t agree the more e x a c t fixed The integral solution regenerator matrix; 1. CONCLUSION bed effect expected methods e m p l o y e d of a well f o r the c a s e of regenerator. of B i o t number i n the case i s much more of a moving bed 67 severe than regenerator. 6. AREAS OF FURTHER RESEARCH The i n t r a c o n d u c t i o n model simplifying enables examine assumptions one t o r e l a x the .The is based use of some of t h e s e effect of these on an a integral assumptions. assumptions number of method In o r d e r to the f o l l o w i n g i s suggested, 1. The simplifying properties can assumption ( f o r the s o l i d ) be e a s i l y achieved with of constant should thermal be r e l a x e d . T h i s t h e use of the integral method [ 1 9 ] . 2. I t is believed direction of temperature assumption relaxed that gas flow profile of the bed contributes [ 8 ] . Thus uniform . 68 extension fluid to the velocity a in the uniform simplifying should be Fluid outlet temperature V S R e d u c e d time For a fixed b e d III AC Ul Q. o.t Ul 0.7 Ul mi 0.4 "* 0.1 Q geometry B1=h.R/K=.02 Z o Spherical 0.4 O.J Ul N O.t Legend 0.1 0<> 2 4 6 9 10 12 14 Reduced time (h*A*t/M*C) Figure 11. Comparison between the numerical and a n a l y t i c a l method employed A Numerlcol Method O Integral Method Fluid outlet temperature V S R e d u c e d time fixed bed,BI=.25 Legend A Reduced time (h*AM/M*C) Figure 12. C o m p a r i s o n b e t w e e n t h e n u m e r i c a l and a n a l y t i c a l m e t h o d e m p l o y e d Num«rlcol M e t h o d i Fluid outlet temperature V S Reduced time fixed bed,BI=2 Legend Reduced time (h*AM/M*C) Figure 13. C o m p a r i s o n b e t w e e n t h e n u m e r i c a l and a n a l y t i c a l m e t h o d e m p l o y e d A Numerical Method O Integral M e t h o d Fluid outlet temperature V S Reduced time For a fixed bed ui Reduced time (h*A*t/M*C) Figure 14. The c h a r a c t e r i s t i c S s h a p e d 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 e d u c e d l e n g t h ) . ro % Difference between solid surface and mean Temperatures, 0.8 5 0.7 a 0.6 % D1ff=(VT )/T Bi=.02 m s Reduced l e n g t h = 8 F i x e d bed E 0.8 S 0.4 • o.s 0 0.2 / / / \ m m m M s Legend Q A Plonor g e o m e t r y ° SRiJsr.'sfi.iissinsJr.y 6 10 16 Reduced time (h*A*t/M*C) —i Figure 15. 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 s o l i d t e m p e r a t u r e p r o f i l e (Bi=.02). OJ % Difference between solid surface and mean Temperatures, 3 • Fixed bed % D1ff=(T -T )/T s B1=0.25/ . ^ - " ^ ^ Reduced m s length=5 2 o £ * * t ""o / f / / o Legend A Plonor geometry O 5 10 Sphericol g«om«fry 15 Reduced time (h*AM/M*C) Figure 16. The e f f e c t o f 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 =0.25). % Difference between solid surface and mean Temperatures, 10 a E 2 Fixed bed Bi=2 % D1ff=(T -T )/Tjj s m Reduced 1ength=40 Legend 8 10 A Plonor geometry ° Spherlcd geometry 18 Reduced time (h*A*t/M*C) Figure 17. The e f f e c t o f thermal c o n d u c t i v i t y on s o l i d temperature p r o f i l e ( Bi=2 ), % DIFFERENCE BETWEEN SOLID S U R F A C E AND SOLID MEAN TEMPERATURE to-i 0 1 1 2 0 4 9 0 7 0 0 Reduced length (h*A*L/M*C) Figure 18. 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 s o l i d t e m p e r a t u r e p r o f i l e (Moving bed). MOVING BED E F F E C T I V E N E S S V S R E D U C E D LENGTH Reduced length (h*A*L/M*C) Figure 19. The e f f e c t o f thermal c o n d u c t i v i t y on r e g e n e r a t o r effectiveness (original profile) SOLID S U R F A C E TEMPERATURE V S R E D U C E D L E N G T H Bl=2 1000 •00 Moving bed regenerator •00 Spherical geometry 700 BI»h.1VK=2 6 Capacity r a t e too rat1o=l •00 400 too A o CO A A A A A A % 6 A too Legend o, Original prof «• 100 O o-* 0 Figure A A A A A $ 6 20. r T 1 2 1 » —i « 1 • — i Altarnatlva prolIto I — • Reduced length (h*A*L/M*C) The comparison between the o r i g i n a l profile and the a l t e r n a t i v e p r o f i l e MOVING B E D REGENERATOR E F F E C T I V E N E S S V S R E D U C E D LENGTH Eff=(T s m Spherical 0 -T s 1 )/(Tf -T 0 S m 1 ) geometry Bi=h.R/K=2. Capacity rate ratio=l Legend A Original profile O Alternativ^profile 0-A0 1 2 3 4 5 6 7 8 Reduced length (h*A*L/M*C) Figure 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 o n t h e two p r o f i l e s ( Bi=2 ). Effectlvness V S Reduced length (NTU) Capacity rate Spherical ratio(R)=l geometry Bi=h.R/K CD O > Legend • •1-0.1 Ul BI-O.B •1-1.0 BI-2.0 2 8 4 8 Reduced length (NTU) Figure 22. Moving bed regenerator « effectiveness based on the alternative p r o f i l e . Effectiveness V S Reduced length (NTU) 1 Of l 0 1 i 2 i 3 i i 4 8 0 I 7 Reduced length (NTU) Figure 23. M o v i n g bed r e g e n e r a t o r effectiveness b a s e d on t h e a l t e r n a t i v e profile. Effectiveness V S Reduced length (NTU) Figure 24. M o v i n g bed r e g e n e r a t o r effectiveness b a s e d on t h e a l t e r n a t i v e profile. Figure 25. THE EFFECT OF C A P A C I T Y RATE RATIO ON EFFECTIVENESS R=M1*C1/M2*C2 Legend A R=1 O R=1/2 • B R=1/3 ... R=1A X R=1^6_ 1 a I" 4 • • Reduced length (h*A*L/M*C) 7 CO OJ 84 BIBLIOGRAPHY Hryniszak.W. "Heat exchangers", Womack.H. "Open c y c l e MHD Bayley.F.J Owen.J and Carpenter,K.J. dividing effects vol line and power g e n e r a t i o n " Turner.A Heggs.P.J. "Heat in regenerator ,1969 transfer", 1972 " P r e d i c t i o n of between c o n d u c t i o n design" and convection , Trans.I.Chem.Eng. 56,1978 Carpenter.K.J. packing and Heggs.P.J. material, size p e r f o r m a n c e of trnsfer.Conf Willmot.A.J computer 12,P 1958 thermal 6-th, "The effects arrengment regenerators Toronto on ", of the Int.Heat 1978 r e g e n e r a t i v e heat exchanger 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 . 997, Vol 1969 Horn.G Sharp.A and recovery and "The from MHD Hryniszak.W plant" " A i r heaters and ,Phi 1 . T r a n s . R o y . S o c . seed Vol 261,1967 Schneller.J and e x c h a n g e r s and and theory Halvacka.V regenerators" source Carpenter.K.J "Moving and ,Heat bed heat exchanger book, Chap27 Heggs.P.J. "A m o d i f i c a t i o n of thermal regenerator infinite predict the of i n t r a c o n d u c t i o n " , effects Trans.I.Chem.Eng.Vol Cserveny.I. conduction bed model the to 57,1979 " C o n t r i b u t i o n s to the moving p e b b l e design thermal design regenerative a i r heaters" of ,Revue.roumanie des 1, 22 ,No to a bed P125,1977 Furnas.C. "Heat of solids broken H a n d l e y . D and conductivity heat Vol ", transfer. Hausen.H "The 1930 of thermal the p a c k i n g m a t e r i a l on transient 12.P549 bed liquid V o l 26, " ,J.heat mass 1969 flowing through Inst.Vol 208, a porous 1929 "Transient analysis s t o r a g e system" of Int.J.Heat mass Berechnung des packed NO1,1983 "vervollstandigate Warmeaustauches stream effect Mccoy.B.J thermal a gas Indus.Eng.Chem, , J of F r a n k l i n Saez.A and from in a fixed Schumann.T.E "A prism" transfer Heggs.P.J. of transfer transfer. bed sciences techniques, Vol in regeneratoren". Z.Ver.Dt.Ing.No2,31-l942. Goodman.R.T "The application 1957, balance to problems p h a s e " , Heat California heat transfer institute integral involving and fluid and i t s a change of mechanics of t e c h n o l o g y , Institute, Pasadena June p383. C a r s l a w . H . S . and solids" Jauger.J.C. Oxf.Univ.Press, Lardner.T.J and consideration trans), 1961 london Pohle.F.V Goodman.R.T "The and heat "Conduction refinements" in 1959. ,J.app.Mech V o l balance of heat 28,1961 integral-further ,Trans.ASME. (J.heat 86 20. Koh.C.Y. "One d i m e n s i o n a l arbitrary ",J.of 21. with Aerospace S c i n c e , l 9 6 l of gases through spheres" transfer "Transfer process p a c k e d and d i s t e n d e d i n the bed o f ,A.I.CH.E j o u r n a l 1963 Rowe.P.N C l a x t o n . K . J flowing conduction r a t e and v a r i a b l e p r o p e r t i e s M c C o n n a c h i e . T and Thodos.G flow 22. heating heat and L e w i s . T . B from a s i n g l e fluid" sphere "Heat and mass i n an e x t e n s i v e ,Trans.Inst.Chem.Eng,Vol 23. Kreyszig.E 24. Nakada.T ,Nakamura.N ,Narita.Y of r e g e n e r a t o r " , 5-th falling "Advanced particles Int.Conf.MHD. 1971 engineering 43,1965 mathematics" and T a i r a . T 1967 "Studies APPENDIX A.l A:DERIVATION AND Dimensionless It to the simplified and x(or r)into Dimensionless This i n regenerators thermal results dimensionless parameters. be terms these dimensionless a design group equations w i l l of also variables y,0 parameters. length number of t r a n s f e r manner . unit(N.T.U) (A.1.1) i s equivalent to £ y=L, t h a t i s A=h.A.L/(m.C) . The of by t r a n s f o r m i n g t h e i n d e p e n d e n t £=h.A.y/(m.C) at in The g o v e r n i n g i s d e f i n e d i n the following The ANALYSIS parameters i s a common p r a c t i c e represent DIMENSIONAL number o f t r a n s f e r unit i s also called length. 87 (A.1.2) the reduced 88 Dimensionless This time i s d e f i n e d as rj=h.A. (0-y/u)/(M The d i m e n s i o n l e s s equivalent period defined t o rj a t 6= p e r i o d , length required The length time 6 (A.1.4) .C ) . s period by t h e s o l i d o f one c y c l e i s to travel time one full i s expressed in length. required by the solid to travel a full s Substituting residence f o r 6 i n equation .C s from (A.1.1) (A.1.3) and i g n o r i n g t h e time y / u , t h e d i m e n s i o n l e s s Tj=h.A.y/(M but is of regenerator i s 0=y/u fluid s of regenerator.The d i m e n s i o n l e s s terms o f t h e d i m e n s i o n l e s s n, that i s moving bed r e g e n e r a t o r s , t h e a s t h e time (A.1.3) ) . o f one c y c l e o p e r a t i o n , n=h.A.(P-L/u)/(M For .C .u s ) , s time w i l l be (A.1.5.a) 89 y=m.C. £/h.A (A.1.5.b) so 77= £ .m.C/(M s P o r o s i t y .C .u ) . s s (A. 1 .5) (B) The p o r o s i t y or v o i d fraction i s defined B=(V, -n.V )/V, b s b where V, b particles as , (A.1.6.a) i s t h e b e d volume and n i s t h e number i n the bed.Porosity c a n a l s o be d e f i n e d B=(p -M )/p . s s s Heat t r a n s f e r The h e a t of solid as (A.1.6.b) a r e a t r a n s f e r area A=(particles i s defined surface as area)/bed volume , or A=(n.A but from s )/V. , b (A.1.6.a) V =(n.V ) / ( l - B ) , b s (A.1.7) 90 so substitute f o r V, i n e q u a t i o n b A=A For planar geometry spherical A=3.(1-B)/R regenerator therefore . R=radius usually matrix i n t e r m s of an defined as D If 3 ev the =[6.net rocks are above e q u a t i o n D 3 ev can =[6.V s be /TT] . (A.1.9.a) of of regular geometry.The broken rocks. It i s the c h a r a c t e r i s t i c written rocks/7r.n] the to (A.1.9.b) equivalent spherical a l l of to . composed volume of is (A.1.8) (A.1.8) r e d u c e s i s not r e q u i r e d to express result .. (A.1.8) r e d u c e s equation the m a t r i x is s d=semi-thickness geometry In p r a c t i c e .(1-B)/V equation A=(1-B)/d For s (A.1.7),the of diameter.This . same shape size the is (A.1.10) and size, the as (A.1.11) 91 Heat transfer It coefficient i s a common p r a c t i c e t o r e p r e s e n t coefficient number i n terms o f f l o w of different Reynolds Experimental packing (or p o r o s i t y ) transfer which studies 1.It number is be Re = u.D Re=u.D independent Table.1 "ref ref 5 20 8 shown large [8,20] are for a this . that the degree of influence on t h e h e a t correlations There are effect,these are that the i n R e y n o l d s number porosity effect calculation.Reynolds r e d e f i n e d as ev ev / [ (1-B) . !»]• , /(B.v) t h e s e a r e so c a l l e d 2.The i n Table.1 .Consequently,the for porosity suggested included i s then [9] suggested s u c h e f f e c t a r e more c o n s e r v a t i v e . two ways t o a c c o u n t should has a v e r y coefficient include have transfer number.There correlations purpose,some o f w h i c h a r e l i s t e d the heat porosity 5 , modified can v a r i a b l e . Most a r e b a s e d on s u c h a Reynolds number. be t a k e n into consideration of correlations the consideration. a s an listed in 92 It i s readily convection, shown the N u s s e l t [21] t h a t number o f extensive flow approaches O.The c o r r e l a t i o n s are t h u s more In the I.S.Cservery calculation. approaches 2 i n the absence a ,when that single the a r e based of sphere natural in an Reynolds number on s u c h finding accurate. present [10] was study,the utilized correlation for heat due to transfer Correlations Nu-0.332Pr Author Re 0 , 5 Nu-(.255/B)*Pr 1 / 3 1 / 3 Nu«(.29A/B)PrRe ' 0 T h i s i s used f o r chequer work m a t r i x Re Handley and Heggs 2 / 3 Schneller.J 7 , 0.67 1.3 Nu».0l6Pr Re n D Nu=2+0.69Pr For 1 / 3 Re°* 5 5 1 / 3 Re<100 Nu-2+6(3.25-2.25B)Pr Table. Modified Reynolds number Frantz.J Porosity effect ignored Rowe.P.N for a single sphere Cserveny.I An own i n t e r - polation formula Re>l00 Nu-2+0.6(3.25-2.25B)Re°' Pr For Comments 1. l / 3 (Re/l00) 1 , 6 9 C o r r e l a t i o n s f o r t h e 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 . 94 A.2 Derivation The case of G o v e r n i n g governing equations equations are developed for a general o f moving bed r e g e n e r a t o r s . T h e s e a r e t h e n m o d i f i e d f o r the case of f i x e d The total bed r e g e n e r a t o r heat between y a n d y+dy comprise two transferred flow accordingly. t o the f l u i d w i t h mass f l o w (where y i s m e a s u r e d components.The t o t h e mass first (m.A, .d0) b from fluid component i n moving rate m, entrance) i s t h e heat between y and y+Sy, t h a t i s d Q 1 = ( m . d 0 . A ) . C . ( 3 T / 3 y ) .dy . b f 0 (A.2.1.a) r J The fluid with second component is t h e heat transferred e n c l o s e d between y a n d y+dy a s i t s t e m p e r a t u r e b but f .dfl , (A.2.1.b) m=p.u ,so dQ2=(m.A./u).C.dy.(3T,/30) .d0 . b f y total heat transferred dQ=m.C.A This changes time,that i s dQ2=p.A .dy.C.(3T /ae) The to the i s equal to the f l u i d . [ ( 3 T /3y)+ t o the heat i s then (3T / 3 0 ) / u ] . d y . d 0 lost from (A.2.1.c) . the s o l i d , t h a t i s (A.2.2) 95 dQ=h. (A. dy . A ) . (T -T ).d0 . b s r Equating total heat the t o t a l gain heat by t h e f l u i d m.C.[(3T / 3 y ) + ( 3 T From c h a i n loss (A.2.3) from the solid to the ,the r e s u l t i s /30)/u]=h.A.(T - T ) f . (A.2.4) rule DT /Dy=3T /3y+(3T /30)/u , f so f equation (A.2.4) can be w r i t t e n as m.C.dT /dy=h.A.(T -T,) . f s f (A.2.5) r The heat lost terms of i t s r a t e the internal from t h e s o l i d can a l s o o f change of i n t e r n a l energy /unit area be e x p r e s s e d i n e n e r g y . D e f i n i n g U as then U=p d .C . f T dx s s o f o r one s o l i d U=M d .C . f T dx s s f o r t h e whole bed . , (A.2.6.a) (A.2.6.b) J 0 As f o r the f l u i d comprises the t o t a l change in solid internal energy two c o m p o n e n t s , t h a t i s h.A.(T -T )=(dU/d0)/d , s f (A.2.7) 96 where cl cl dU/d#=M C [u . 9 ( / T d x ) / 9 y + 9 ( f T d x ) / 3 0 ] . s s s o 0 If t h e 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 d x ) / 3 0 s f s s o Note t h a t and c a n be seen t h a t throughout solid ,then (A.2.9) between the present regenerator are equation,which p C equations (A.2.9) =K boundary temprature uniform (A.2.8 and A.2.9) r e d u c e f o r t h e Schumann study, is model. t h e two e q u a t i o n s equations t o the (A.2.5) and used t o model the diffusion , (A.2.10.a) is 2 2 (3 T/9x ) (3T/30)=K diffusion i f the s o l i d equations phase e q u a t i o n s In The difference . (A.2.8) i s due t o t h e d i r e c t i o n i n w h i c h y i s m e a s u r e d . It the the s i g n (A.2.8) [9 T/9r +(2/r).9T/9r] 2 s P l a n a r geometry 2 equation conditions is coupled Spherical with the . (A.2.10.b) following 97 l.The symmetry condition (3T/3x). x=d (3T/3r) 2.Heat flux K =0 planar =0 r=0 a t the Spherical K s The = h.(T x=0 .(3T/3r) _ =h.(T dT normalised =(T the -T . (A.2.11.b) ) planar , ) spherical -T are nondimensionalised following can be d e d u c e d , dT=(T..-T.).d* , f0 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 . (A.2.12.b) s t e m p e r a t u r e s . From t h e ).dF (A.2.12.a) f r above e q u a t i o n s these parameters -T s r —H £, rj,Bi and (A.2. 11.a) surface .(3T/3x) s , , spherical geometry , i n t e r m s of definition , of 98 dz=dx/d Equation (A.2.5) can * For solid s moving The s form as (A.2.13) is measured ,equation result from (A.2.13) the must be is (A.2.14) . equation(A.2.10) can form . in dimensionless regenerators,y =dF/d£ also be written in as geometry .C s .h.A. ( 3 ¥ / 3 i j ) / ( M the d e f i n i t i o n from equation g written Consequently from M geometry . bed diffusion dimensionless s be accordingly,the F-* planar now -F=dF/d£ entrance. altered p For planar i n above 9*/3T7=K of .C s )=K porosity (A.9.a) (B) , .So 2 M s =p 2 s , .(1-B) substitute , for A and then .( 3 4>/3z )/(h.d) 2 .( 3 + / 3 z ) / d 2 s A=(l-B)/d equation, s s 2 , or 3*/3T?=(3 */3z )/Bi 2 2 . (A.2.15) 99 The corresponding 1.Symmetry condition U*/3z) 2.The h e a t K s constraints are z=1 =0 flux (A. 2. 16) at the surface . 0 ¥ / 9 z ) / d = h . ( * -F) , s that i s (3*/3z) spherical The z=0 = B i . ( ¥ -F) s . (A.2.17) geometry diffusion equation i n dimensionless form will be p C h.A( 3¥/3T?)/M C =K [ ( 3 * / 3 z ) / R + ( 2 / z . R ) ( 3 * / 3 Z ) / R ] , s s s s s 2 but M and A=3.(1-B)/R so or s -p s 2 2 . (1-B) , , 3(3*/3T7) .h/R=k / R [ 3 * / 9 z + ( 2 / z ) . 9 ¥ / 3 z ] , s 2 2 2 3*/3rj=[ 3 * / 3 z + ( 2 / z ) . ( 9*/3z) ] / ( 3 B i ) 2 2 . (A.2.18) 100 The c o r r e s p o n d i n g 1.Symmetry constraints are condition (3¥/3z) =0•. (A.2.19) z=0 2.Heat flux a t the surface (K / R ) . ( 9 * / 9 z ) = h . ( F - * ) , s s that i s (3*/3z) =Bi.(F-* ) . z= 1 s (A.2.20) APPENDIX B:INTEGRAL METHOD B.l Planar The geometry matrix (usually chequrework) i s modelled i n two stages Semi-infinite slab The equation diffusion in dimensionless form i s (B.1.1) 3*/9z=(9 */3z )/Bi 2 The penetration characteristic equilibrium depth,defined property temperature beyond p e n e t r a t i o n The 2 matrix such in that (initial chapter the solid temperature) at 3,has is any at a an point depth. i s subject to the following dimensionless boundary c o n d i t i o n s *(5 ,7})=*.=0 , (B.1.2) 9*(Sn,T?)/9z = 0 (B.1.3) 0 9<M0,7?)/9z = B i . (* - F ) = - f (77) Differentiating substituting back into equation . (B.1.2) w i t h the d i f f u s i o n 101 (B.1.4) respect equation t o 77 and results in an 102 extra boundary condition called t h e smoothing condition,that is 3 *( 6 , 7 )/3z = 0 2 2 0 There ? (B. 1 .5) are 4 constraints.The matrix represented by a c u b i c temprature is thus profile,that is 2 3 *=A+Bz+Cz +Dz Applying . . (B.1.6) the c o n s t r a i n t s , t h e r e will be 4 e q u a t i o n s A+B5„+C6^+D5 =0 , 0 0 0 ' 3 2 B+2C.6 +3D.6 =0 , Q Q B=-fU) , 2C+6D.5 =0 . Q The 4 simultaneous unknowns are found equations ,the c u b i c *=f ( T ? ) . ( 5 - z ) / ( 3 . 5 ) 3 2 0 The equation surface temprature (B.1.7),the * s result i s by solving profile the above i s then . i s o b t a i n e d by s e t t i n g (B.I.7) z=0 i n 1 03 * =f (77) s The to diffusion . (B. 1 .8) equation i s now i n t e g r a t e d with respect z,that i s J 0(d*/dri) dz=(j 0 Liebnitz theorem S o From JQ0(9*/9Z) but .5V3 0 0 2 Q dz = d ( / ^ 0 ¥ dz)/dr -*(6 , ? * dz)/drj=[ ( 9 * / 9 z ) 0 z Applying equation d(/ . 2 equations = 5 . -0*/3z) n (B.1.3) and (B.1.9) rj)(d5 /drj) , (B.1.10) 0 ¥ ( 6^ , 77) =0 , so e q u a t i o n by d e f i n i t i o n d(/ (9 */9z ) dz)/Bi 5 (B.1.9) becomes „ ]/Bi 2 = . (B.1.11) 0 (B.1.4)to the above , the f o l l o w i n g i s obtained 0 0 * dz)/drj=f ( r j ) / B i Substituting f o r * from . (B.1.12) (B.1.7), the r e s u l t i s d ( / ° [ f (r?) . (5 - z ) / ( 3 S ) ] d z ) / d = d ( f (rj) .5 /l2)/d77 0 (J U u 3 2 2 7? So e q u a t i o n (B.1.12) becomes d [ f (7?) . 5 / l 2 ] / d 7 j = f (7?)/Bi 2 . (B.1.13) . (B.1.14) 104 The obtained solution by to integration 6 =[ 12. ( J ^ f (77) 0 o where 5^ from * s differential with respect dr? ) / ( B i . f (77) ] ° ' surface temperature (B.1.8),the equation is t o 17,the r e s u l t is , (B.1.15) 5 of f i n i t e the time achieved (77) by s u b s t i t u t i n g f o r dTj)/(3.Bi) ] ° ' 5 . (B.1.16) thickness V-W^r the p e n e t r a t i o n matrix.From no meaning i s computed result is = [ 4 . f (77) . ( A 0 At of above 6Q = 0 , a t r?=0 . The Slab the and t h e this point matrix has depth reaches the on t h e p e n e t r a t i o n to be depth has remodelled.This is by m o d e l l i n g t h e m a t r i x a s a f i n i t e d) s u b j e c t t o the f o l l o w i n g * ( 0 ,77) =* s slab , (B.1.17) , 3*( 1 ,77)/9z = 0 . are 3 c o n s t r a i n t s . S o be a s e c o n d - o r d e r (thickness dimensionless constraints 9*(0,7?)/3z = B i . (* - F ) = - f ( T ? ) There centre (B.1.18) (B.1.19) the temperature polynomial,that i s profile must 105 4>=A+Bz+Cz The c o n s t a n t s profile A,B will then 2 , (B.1.20) . and C take are found from constraints.The t h e form +f (TJ) . ( z - 2 z ) / 2 . 2 (B. 1.21) s The to diffusion z. A f t e r d(J applying the L i e b n i t z 2 equations = 1 i n t e g r a t e d with respect theorem,the r e s u l t i s -(3¥/3z) z = 0 J / B i. (B.1.18) a n d (B.1.21) t o (B.1.22) the above the f o l l o w i n g i s obtained d(* The s -f(rj)/3)/dTj=f(Tj)/Bi solution obtained i s now 1 * dz)/dT7=[ (3*/3z) • 0 Applying equation, equation . (B.1.23) to the above differential by i n t e g r a t i o n with respect t o rj,the equation i s result i s 7? * -f(rj)/3=(/ s At the t i m e 7=r?Qf 0 f(ij) drj)/Bi t h i c k n e s s of the s l a b . S o s ( ^ 0 ) = f ( ^ . 5^=1, i e . t h e p e n e t r a t i o n d e p t h h a s 7 * +Constant 0 ) / 3 from (B.1.24) reached (B.1.8) (B.1.25) 1 06 Substitute for* s * (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 0 ( i ? ) - f s 0 ( T ? ) / 3 = ( / ^ 0 f (7?) 0 dT?)/Bi+Cpnstant , so C o n s t a n t = - ( f|J0f (rj) d r 7 ) / B i Substitute f o r Constant * =f (TJ)/3+( J V . (B.1.26) i n equation f(rj) dr?)/Bi (B.1.24),the result i s . (B.I.27) ^0 B.2 Spherical As for modelled Sphere geometry the i n two of planar matrix i s radius diffusion dimensionless spherical stages infinite The geometry,the equation for spherical geometry form i s 9*/9T?=[ 9 * / 3 z + ( 2 / z ) . 3 * / 3 z ] / ( 3 . B i ) . 2 The approximate depth is (B.2.1) 2 penetration the s o l i d i s measured subject in from depth concept temperature the surface to the following i s profile.The employed to penetration of the sphere.The dimensionless constraints matrix 1 07 * ( 1-6 ,r?)=* =0 , >*(1-5 ,TJ)/9Z = 0 (B.2.2) , 0 (B.2.3) 3*(1 ,r?)/3z = - B i . (* - F ) = - f (TJ) . s Equation and then results (B.2.2) substituted is differentiated back in the 2 2 Applying equation (B.2.3) diffusion respect equation.This form . to t o r? (B.2.5) the above e q u a t i o n , i t to the smoothing c o n d i t i o n , t h a t i s 9 +( 1 - 6 , T ? ) / 9 Z = 0 0 . profile i s o f t h e form 2 The with i n an e x t r a b o u n d a r y c o n d i t i o n o f t h e 3 * / 9 z + ( 2 / z ) . ( 3 * /3z)=0 reduces (B.2.4) 2 suggested "^(polynomial The reason resembles the equation.The * 0((1/z) [17] in z)/z . for this steady steady (B.2.6) state form state of (B.2.7) the profile solution solution being of is the that it diffusion 108 will The polynomial take t h e form of z i s a cubic.So the f i n a l profile 2 4»=Az +Bz+C+D/z . Applying ,the final (B.2.8) t h e c o n s t r a i n t s and s o l v i n g expression will 3 2 0 the surface 0 temprature above e q u a t i o n , t h a t * s n 0 equation 0 The simplified . in (B.2.11) with respect theorem,the r e s u l t 2 2 9 (*.z)/3z 1 * . z dz)/dr?=(/ 0 z=1 i s r e w r i t t e n as i s integrated the L i e b n i t z 1 by s e t t i n g (B.2. 10) 2 The above e q u a t i o n d(J D (B.2.9) . (B.2.1) 2 applying . i s obtained 3(*.z)/ar?=[3 ( + . z ) / 9 z ] / ( 3 . B i ) After and is =-f (rj) . 5 / ( 3 - 5 - ) The d i f f u s i o n A,B,C be *=-f ( T 7 ) . [ Z - ( 1 - 5 ) ] / ( 5 . ( 3 - 5 ) . Z ) The for dz)/(3.Bi) to z. is . (B.2.12) 0 right by (B.2.4).Equation hand side applying of the equations (B.2.12) r e d u c e s to above equation c a n be (B.2.2),(B.2.3) and 109 1 d(J • * . z dz)/dr? = - f ( r j ) / [ B i . ( 3 - 6 ) ] i-« o . f t (B.2.13) 0 It c a n be seen t h a t t h e above d i f f e r e n t i a l highly n o n - l i n e a r . The n o n - l i n e a r i t y extent by readjusting (B.2.1) i s now the equation c a n be r e d u c e d diffusion w r i t t e n as 2 z,the t o some equation.Equation 3(*.z )/3rj=[ 9 ( z . 9*/3z)/3z]/(3.Bi ) . (B.2.14) 2 The is above equation i s now integrated with respect to result i s 1 d(J *.z 1 2 dz)/dr?=[z 2 1 (3*/3z)] " 0 6 After reduces /(3Bi) 1 applying the . (B.2.15) " 0 5 constraints,the above equation to 2 1 d(J *.z dz)/dr?=-f ( r j ) / ( 3 B i ) . (B.2.16) '- o 6 It can non-linear in Substituting integral,the d[ ( S Q - 5 6 2 ) be seen that comparison f o r * from equation with equation (B.2.16) equation is less (B.2.13). (B.2.9) and p e r f o r m i n g the result i s . f ( T ? ) / ( 3 - 6 ) ]/dT7=-20f ( r ? ) / ( 3 B i ) 0 . (B.2.17) 1 10 The above integration with differential respect t o 77, w i t h ( 6 Q = 0 a t 17=0) t h e r e s u l t 3 7 ? n The 5Q from surface equation the is initial dr?)/.(3Bi.-f (77) temprature (TJ).) i s obtained (B.2.10),the 2 result is 7 ? centre some should matrix is time rj^the penetration this be d i s r e g a r d e d now . (B.2.19) radius of t h e s p h e r e . A t concept (B.2.18) by s u b s t i t u t i n g f o r 2 At by condition . 3 * [ 5 f ( T J ) - 2 * ] = 2 0 [ * - f (77) ] . (/^f ( 7 7 ) d ) / ( 3 B i ) s s s 0 S p h e r e of f i n i t e solved is [ ( 5 5 - 6 ) / ( 3 - 6 ) ] = 20(/ f u u u 0 2 equation subject to time due the depth reaches the penetration t o t h e symmetry the following depth effect.The dimensionless constraints 3*(0,7?)/3z = 0 , (B.2.20) 9*( 1 , r?)/3z = - B i . (* - F ) = - f ( 7 7 ) *(1,TJ)=* The p r o f i l e will s (B.2.21) (B.2.22) . take , t h e form •=(polynomial in z)/z [17] (B.2.23) 111 It a should boundary include be n o t e d term 2 *=Az +Bz+C and C are take t h e form found by applying modified with . the (B.2.25) diffusion respect equation t o z,the (B.2.14) is now result i s 2 2 * . z dz)/drj=[; 3(z .9*/3z)/3z d z ] / ( 3 B i ) , 0 0 1 = - f ( T | ) / ( 3Bi ) Substituting performing d[* The for * . (B.2.26) from equation (B.2.25) and the i n t e g r a t i o n , t h e r e s u l t i s /3 + f ( r ? ) / l 5 ] / d T ? = - f ( r j ) / ( 3 B i ) above integration s should not the r e s u l t i s 1 * to (B.2.24) 2 integrated polynomial .So • t h e • p r o f i l e . w i l l *=* +f (77) . [ 1 - z ] / 2 s The i s subject . unknowns A,B constraints, d(J s i n c e the sphere c o n d i t i o n at the centre,the a constant The that with differential respect t o 77,the . (B. 2.27. a) equation is solved by result i s +f (7j) /5 = - ( / f ( r j ) d 7 7 ) / B i + C o n s t a n t 0 . (B.2.27) 11 2 At 6Q=1 rpn , 6=1 0 0 in equation * S s (T? ) c a n be 0 (B.2.10),the (T?.)=-f ( T J . ) / 2 0 0 Substituting constant .So ¥ for of i n t e g r a t i o n equation * f (T? ) = 1 0 (/^O From that the setting result is S (B.2.28) ^ Q ^ * n e Q u a t i° n (B.2.27) the i s obtained,that i s f ^ p ) c a n be (B.2.18),the by • C o n s t a n t = - 3 f ( T J ) / l 0 + ( J^O The t e r m obtained f(r>) d r j ) / B i . computed by (B.2.29) setting 6 =1 in result is f(r?) d r j ) / ( 3 B i ) equations (B.2.30) and constant of . (B.2.30) ( B . 2 . 2 9 ) , i t c a n be integration is zero.So deduced equation (B.2.27) becomes * =-f (i7)/5--(J f ( T » ) 0 T? s d7?)/Bi . (B.2. 31) APPENDIX C: EFFECTIVENESS COMPUTATION In r e g e n e r a t o r s t h e t h e r m o d y h a m i c a l l y p e r f e c t occurs at when the the matrix at the e x i t same temperature O b v i o u s l y ,-this resistance measure is not t o the heat the idealised performance rise.In s in practice a temperature hot fluid. because of the two regenerator called to media. against and T =Mean is temperature solid apparent at inlet r i s e (or the possible maximum e i s d e f i n e d as temperature from and outlet computation, are first (C.1) 1 13 rates, that the s o l i d represented step i s the c a l c u l a t i o n temperature. (C.1) . equation temperatures.Consequently,the effectiveness is of t h e a c t u a l (m.C) . =minimum of t h e two c a p a c i t y min ms To the effectiveness .C . ( T -T .)/[(m.C) . . ( T . . - T .)] , s msO msi min f i msi where mean entering as the r a t i o m a t h e m a t i c a l ! form e=m It of the regenerator i s between t h e parameter i s defined drop) i n the matrix the possible transfer situation,a employed.This of from situation as mean towards the of the solid 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 d x ) / d , m o or in dimensionless • (C.2) form 1 * =7 * dz . m There (C.3) J 0 are penetration two expressions depth.These were for * obtained depending in the on t h e previous Appendix. For , 7?<r?Q *=f (TJ) . ( 6 - z ) / ( 3 . 6 ) 3 2 0 For T7>T? 2 s corresponding For (B.I.7) , 0 * = * + f (77) . ( z - 2 . z ) / 2 The . 77<770 . mean t e m p e r a t u r e s (B.1.21) a r e then , * m =f ( 7 7 ) . [ J 1 Q (5 -z) n U 3 dz]/(3.5 ) l) 2 , 11 5 but 1 6 1 n / * dz=(/ * dz)+(J u 0 0 by d e f i n i t i o n mean t h e second temperature will 6 * dz) , Q integral i s equal to zero and the be * =f(r?).[J 6 3 J , ., - -z) P(6 „2. dz]/(3.5„) , that i s * =f(r ).6 /l2 m U . 2 ? For T}>T? 0 , 1 * =/ m performing * where (C.4) m o [* + f ( n ) . ( z 2 - 2 . z ) / 2 ] dz , s the integration, =* - f ( r ? ) / 3 s f ( r j ) = B i . (F-¥ s , (C.5) ) . (C.6) C.2 S p h e r i c a l geometry The mean dimensional) temperature i s d e f i n e d as in cartesian coordinate (3 11 6 V ( ; / ; T The above coordinate a«)/(jj/a, a az) . ay dx equation system can by (c.7) y be transformed a p p l y i n g the Jacobian to sherical transformation [21]. Defining x = r .cos# . sin<£ , y = r . s i n 0 . sin<£ , z = r.cos</> . The J a c o b i a n J= so dx/dd 9y/9r 9y/90 dy/d<p 9z/9r 9z/96> 9z/9tf> m 0 be 9x/90 ///dx dy dz=///-r R will 9x/9r Substitute T =/ transformer • 2 =-r equation 2 2 R n (C.10) (C.7) 2 d 0 . J sin<£ dtf>// - r -TT/2 0 0 v/ (C.9) sin<£ d r dd d<j> the above,in 2 * sin<£ (-Tr ) d r . / 27R ^Z 2 d r . / s i n 0 d<£./ dd 0 -TT/2 or R T =/ m o R (T.r ) d r / ( / r 0 0 o dr) (C.11) 1 17 The mean t e m p e r a t u r e in dimensionless 2 = 3. J *•. z 0 form i s 1 * m There are penetration T?<T? For dz . two (C.12) expressions for the , 0 )5gZ] . 3 above on depth,these are *=-f (77) . [ z - ( 1-5 ) ] / [ (3-5 The 4», d e p e n d i n g mean t e m p e r a t u r e expression i s obtained for * i n equation by (B.2.9) substituting (C.12).First the i t i s noted that 2 1 * =3/ m and 1-6 * 2 (*.z ) dz , for (C.13) i s z e r o and (C.13) r e d u c e s t o (C.14) o * from equation (B.2.9) in above result i s =-3.f(r?).; z.[z-(1-5 )] -j _ § 0 0 1 m (C.13) 2 (*.z ) dz . 1 equation,the 0 *=0 beyond p e n e t r a t i o n d e p t h . S o t h e s e c o n d =3./ m i-6 Substituting 5 0 i n equation * ~ n (*.z ) dz+3/ by d e f i n i t i o n integral 1 n 3 dz/[(3-5 ).5 ] , 0 0 2 n 1 18 so * It the =-3..f (TJ) . ( 5 6 - 6 ) / [ 2 0 . ( 3 - 5 . ) ] . 0 0 0 2 m is interesting 3 to note that mean t e m p e r a t u r e c a n a l s o * =-/ f (T?) drj/Bi m o For from e q u a t i o n be w r i t t e n (C.15) (B.2.18) as . (C. 16) rj>rj , 0 4>=¥ +f (JJ) . ( 1 - z ) / 2 s 2 Substituting * (B.2.25) f o r * in equation (C.12),the r e s u l t i s =3/ [* 1 m . o . z + f (TJ) . ( z - z / 2 ) ] dz , 2 s 2 4 that i s * where m =* s +f ( T J ) / 5 , f ( r j ) = B i . ( * -F) s (C . 17) (C. 18) APPENDIX D :SAMPLE CALCULATION D.l Governing equations The governing equations are written in their numerical form as Fluid phase (1+A$/2)F(n,i+1)-A£/2¥ (n,i+1)=A£/2¥ 5 Solid phase 1. P l a n a r For (n-1,i+1)+(1-A£/2)F(n-l) S geometry TJ<TJ 0.5 * For * (D.2) (n,i+1)=[4f(n,i+1).Ar1(n,i+1)/(3Bi)] T)>T? 0 0.5 (D.3) , (n,i+1)=f(n,i+1)/3+Ar2(n,i+1)/Bi where 1 19 , (D.4) 120 f (n "i )=Bi-[F(n, i ) - * ( n , i ) ] s r , .(D.5) Ar 1 (n , i + 1 )=Ar 1 (n , i ) +Arj[ f (n , i + 1 ) + f (n , i ) ]/2 , (D.6) A r 2 ( n , i + 1 ) = A r 2 ( n , i )+Ar?[ f (n, i + 1 ) + f (n, i ) ]/2 , note that Ar 1 (n , 0 ) =0 and Ar2(n,r? 2.Spherical T?<7?0 For )-=0 . geometry , 56 (n,i)-8 (n,i)=20[3-5 (n,i)]Ar1(n,i)/(3Bi.f(n,i)) 2 (D.7) 3 Q , (D.8) 3 * ( n , i + 1 ) [ 5 f ( n , i + 1 ) - 2 * (n,i+1)]=20[* ( n , i + 1 ) - f ( n , i + 1 ) ] s s s 2 .Ar1(n,i+1)/(3Bi) 7?>i7 For * s 0 . 2 (D.9) , (n,i+1)=-f(n,i+1)/5-Ar1(n,i+1)/Bi , (D.10) where f(n,i)=Bi[* s (n,i)-F(n,i )] , Ar 1 (n, i + 1 )=Ar 1 (n, i )+AT?[ f (n , i + 1 ) + f (n, i ) ]/2 . (D.11) (D.12) 121 D.2 Method o f D.2.1 Fixed The solution bed analysis step point.If to compute b e g i n s by 5^(n,i)<1,then the s o l i d equations (D.4) first computing'5 equations (D.3 ( n , i ) at each or'D.9) a r e t e m p e r a t u r e . I f however,6^(n,i)>1 or(D.lO) are used to used ,then compute t h e solid temperature. There entrance i s a s t e p change to the essentially can g (D.9) .At computed any by representing For 77<77 Q within the unknown t e m p e r a t u r e by solving two the equations simultaneous solid phase the .Thus which is (D.3) or can be equations ,these are , (D.1) and (D.3) or t h e unknowns 4» ( n , i + l ) and for * at t h e unknown t e m p e r a t u r e s t h e f l u i d phase and substituting solving be computed other point f l u i d temperature (ie.F(0,i)=F(0,i+1)) i s o n l y one solving Equations compute regenerator there * (n,i+1).This i n the for s F(n,i+1) (n,i+l).It from (D.9) s h o u l d be F (n , i +1 ) . T h i s (D.1) i n (D.3) s h o u l d be c l e a r that * solved i s done or s (D.9) (n,i+1) to by and lies limit 0<¥ s (n,i+1)<1 (D.13) 1 22 For The gathered 7?>rj , Q linear together equations in a matrix (D. 1 ) and (D.4.) o r form U [ F ( n , i + 1),4> (n, i + 1)]=H , where U and H a r e t h e f o l l o w i n g l.For (D.10) a r e (D.14) matrices p l a n a r geometry 1+A£/2 -A£ U= (D.15.a) -Bi/3-Ar?/2 A£ (* 1+Bi/3+Arj/2 ( n - 1 , i + 1))/2+(1-A£/2)F(n-1 , i + 1 ) H= (D.15.b) Ar2(n,i)/Bi+A7?[F(n,i)-* (n,i)]/2 1 23 2.For spherical geometry 1+A£/2 -A$/2 (D.16.a) U= -Bi/5-A7?/2 A£( + 1+Bi/5+Ar>/2 (n-1,i+1))/2+(1-A£/2)F(n-1,i + 1 ) (D.16.b) H= -Ar1 ( n , i ) / B i + A r j [ F ( n . , i ) - * The solution i s merely the i n v e r s i o n [F(n,i+1),+ The starting (n,i+1)]=U values of (D.14),that i s (D.17) .H f o r the i t e r a t i o n a r e F(0,i)=1 F o r i>0 , (D.18.a) 5 (n,0)=0 F o r n>0 , (D.18.b) * F o r n>0 (D.18.c) Q (n,0)=0 The than ( n , i ) ]/2 initial entrance fluid temperature i s computed from at any position other 1 24 (D.19.a) 3*/9z = - f (TJ) but *(n,0)=0 f o r a l l n, so F(n,0)=0 for n>0 (D.19.b) D.2.2 M o v i n g bed The fluid phase need t o be a l t e r e d £ h a s t o be measured from the s o l i d bed AT? is regenerators Consequently, equations bed), the will unknowns be order solid to start outlet expressed are identical and and f l u i d F(0)=[F(A)+* The fluid computed discrepency, procedure This inlet in F(n+1) to terms and of A £ . * (n+1). s the previous case using ,one need to is initially The (fixed a r e known.In know the approximated fluid as the temperatures,that i s (D.22) procedure inlet given the i n i t i a l i s then temperatures temperatures the fluid actual inlet (0)]/2 . and s o l i d bed.The c a l c u l a t e d the fluid the i t e r a t i o n temperature. mean s o l i d with e n t r a n c e . A l s o f o r moving i f A £ i s r e p l a c e d by - A £ . The then for this case,since a t each outlined temperature repeated u n t i l f o r the f i x e d i s then temperature.If approximation step point are there compared i s any i s r e a d j u s t e d and t h e t h e two v a l u e s coincide. The except U and H matrices are i d e n t i c a l to the previous case that moving also fixed the unknowns a r e problem F(n+1) and is * one (n+1). dimensional,that is the APPENDIX E :THE COMPUTER Three written bed was programs f o r handling regenerators included written examining achieved the method) input data of with step the The programs were integral the r e s u l t s obtained published only (using results (using i n order t o regenerators. two p r o g r a m s calculate of m e t h o d . T h i s was p r o g r a m was w r i t t e n f o r the f i x e d i t was f o r the purpose the included reduced l e n g t h and bed.However,for neccessary to the a moving compute t h e length. both).The point fluid increment Two effect the A s u i t a b l e time and d i s t a n c e for The i n t r a c o n d u c t i o n f o r moving bed to period regenerator reduced moving programs. f o r the f i r s t required dimensionless and the a n a l y t i c a l a s e t of c h a r t s parameters bed validity methods).The t h i r d The were l a n g u a g e and were run on Amdahl 470. by c o m p a r i n g numerical a n d moving of f i x e d two p r o g r a m s were w r i t t e n integral obtain the thermal design i n a l l the three first ,fixedsph respectively. in Fortran The the jfixedplan PROGRAM fluid and s o l i d were t h e n and s o l i d along surface surface temperatures length subroutines programs,namely 1 26 at were o b t a i n e d were chosen(0.2 temperatures computed. F o r a moving b e d t h e whole external i n c r e m e n t s were used at each regenerator each length . in a l l three 127 Zerol This two and external non-linear solid subroutine equations surface was used f o r the temperature, to solve penetration when the depth 6^<1. SLE This two The this linear external subroutine simultaneous equations solid surface matrix inversion technique list of section. was temperatures.The a l l three to used f o r the fluid subroutine solve programs a r e to s o l v e the uses the and the equations included at the end 1 28 1 29 IMPLICIT REAL*8 (A-H.O-Z) REAL*8 LEN.LENU REAL*4 RELEU C C C C C C C THIS I S A PROGRAM TAKING ACCOUNT OF, INTRAPARTICLE CONDUCTION FOR SPHERICAL COORDINATE THE NAME OF THIS PROGRAM I S RBGE 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 C C C C C C C COMMON BLOCK /SOLTE2/ DEFFINES THE VARIABLES USED IN SUBROUTINE FCN IN WHICH TS(N,I*1),TG(N,I-M ) ARE CALCULATED. c c c C C C C C C C C C C COMMON BLOCK DELTA DEFINES THE VARIABLES USED IN SUBROUTINE FN IN WHICH THE PENETRATION DEPTH I S CALCULATED. HT-HEAT TRANSFER AR-AREA UNDER CURVE HT VS TIME ARE*AREA UNDER THE CURVE F*PENETRATION VS TIME X-SOLUTION OF NON-LINEAR EQUATION XI-SOLUTION OF LINEAR EQUATION SOLVED BY MATRIX INVERSION A1-MATRIX OF COEFFICIENTS Bl-MATRIX OF RIGHT HAND SIDE OF EQUATION A.X-B DT-INCREMENT OF TIME CONST-F(TIME) WHEN T-TO PENE-PENETRATION DEPTH READ(5,500)SOLK,DENSOL,CPSOL,DIASOL,PORO,SOLFLO 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 500 1 30 Y4U-200. EXTERN AX FN2U LOGICAL LZ2U C C C C C C c X(l)ANDX(2) ARE THE I N I T I A L GUESSES FOR TG(N,I+1) AND TS(N.I-M). TG6-TGU(NU) TS5«TSU(NU) AR3-ARU(MJ) C C C c THE VALUES OF TG(N,I+1),TS(N,I+1) ARE NOW CALCULATED CALL ZER01(X4U,Y4U PN2U,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) t C c C C c THE VALUE OF PENETRATION DEPTH IS NOW CALCULATED CALL ZER01<X2U,Y2U,FNU,E1 LZU) 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) TGU(NU+1),DTU,BIU,TGU(2) F0RMAT(6F12.4) I F (NU.NE.2) GO TO 2570 DIFF(NU)-0. GO TO 2560 DIFF(NU)-100.*(TSU(NU)-TSM(NU))/TSU{NU) EFF(NU)-R*(TSM(NU)-TSU(2))/(TGU(NU)-TSU(2)) GO TO 901 f I 2560 2570 2580 C c c C C C C I F T » T 0 USE MATRIX INVERSION TO SOLVE 2 LINEAR EQUATIONS AT THIS STAGE PENETRATION DEPTH I S EQAL TO THICKNESS OF PLAT 2590 PENEU(NU-H)-PENEU(NU) 131 NDIKA-5 HDIMT-5 NDIMBX-5 NN1-2 KSOL-1 C C C C NN1-NO.OF EQUATIONS, A1-MATRIX OF COEFFICIENTS Bl-MATRIX OF KNOWN RIGHT HAND SIDE A*X-B c c i 901 1000 1601 1602 3301 3302 1050 1001 1002 999 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, 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) MU-NU-2 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 DKU-DMU+DIS WRITE(6,3301)DMU,EFF(NU) FORMAT(F4.2,P6.3) WRITE(7,3302)DMU,TSU(NU) FORMAT(F4.2,F8.3) CONTINUE GO TO 999 1F(DTEMP.GT.0.) GO TO 1002 TGU(2)-TGU(2)*DABS(DTBMP/4.) GO TO 1 TGU(2)-TGU(2)-DABS(DTEMP/4.) GO TO 1 STOP END FUNCTION FNU(X) PENETRATION DEPTH IS CALCULATED IN THIS FUNCTION 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 C C TS(N,I*1) AND TG(N,X+1) ARE CALCULATED IN T H I S USING THE NON LINEAR EQUATION FOR T S 10 20 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 FN2U-3.*X**2*P4-20.*P2**2*P1/(3.*BIU) RETURN END SUBROUTIN
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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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Title | The use of an approximate integral method to account for intraparticle conduction in gas-solid heat exchangers |
Creator |
Riahi, Ardeshir |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | The mathematical equations describing transient heat transfer between the fluid flowing through a fixed bed and a moving bed of packing were formulated. The resistance to heat transfer within the packing due to its finite thermal conductivity was taken into account. An approximate integral method was applied to obtain an analytical solution to transient response of the bed packing. Results for two cases of fixed and moving bed were obtained. The validity of the approximate method was checked against the more exact method employed by Handley and Heggs who obtained the results for a fixed bed of packing with a step change in fluid inlet temperature. It was concluded that the approximate method gives results that agree well with the more exact methods. The method considered here provides a quick determination of the packing mean temperature in order to obtain the effectiveness. The other peculiarity of this method is that the effect of packing thermal conductivity can be examined very quickly since the solution is in analytical form. The analysis of the results revealed that as the thermal conductivity of the packing decreases the difference between its surface and mean temperature increases. A series of charts showing the comparison between the packing surface and mean temperatures for different thermal conductivities are presented. The approximate method was a moving bed of packing. It was packing thermal conductivity is series of charts representing versus dimensionless length conductivities are presented. then applied to the case of concluded that the effect of more severe than expected. A the moving bed effectiveness for different thermal |
Subject |
Heat equation Heat exchangers |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080810 |
URI | http://hdl.handle.net/2429/25137 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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