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F u rth e r re p ro d u c tio n p ro h ib ite d w ith o u t p e rm is sio n . o A STUDY OF SOME PROBLEMS ARISING FROM COMBUSTION AND MICROWAVE HEATING by Andonowati D epartm ent of M athem atics and Statistics McGill University Montreal, Quebec Canada May 1995 A DISSERTATION SU B M IT TE D T O T H E FA CU LTY O F G R A D U A T E S T U D IE S AND R E S E A R C H of M c G il l U n i v e r s i t y IN PARTIAL FU LFILLM EN T OF T H E R EQ U IR EM EN T S FO R the D e g r e e o f D o c t o r o f P h il o s o p h y Copyright © 1995 by Andonowati Reproduced with permission of the copyright owner. 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Further reproduction prohibited without permission. McGill U niversity FACULTY O F GRADUATE-STUDIES AND RESEARCH W ftT A U TH O R'S NAME: DEPARTMENT: 1____________________________________________________________________________ M A T H E M A T I C S AM STATISTICS TITLE O F TH ESIS: A flVPY OF M M P^ - P.__________________ D EG R EE SOU G H T: p fip g L E -M lS A R lS fN fr f W CflM glijTlPN AKP M M O w rtrg P e rm a n e n t A d d re ss of A uthor: A u th o rization is h e re b y g iven to McGill U niversity to m a k e this th e s is av a ilab le to r e a d e r s in th e McGill U niversity Library o r o th e r library, e ith e r in its p r e s e n t form o r in re p ro d u c tio n . T h e a u th o r re s e rv e s o th e r p u b lic atio n rights, a n d n e ith e r th e th e s is n o r e x te n siv e e x tra c ts from it m a y b e p rin te d o r o th e rw ise b e re p r o d u c e d w ithout th e a u th o r’s w ritten p erm issio n . T h e a u th o riz a tio n is to h a v e effect o n th e d a t e given a b o v e u n le s s a jdeferral of o n e y e a r is r e q u e s te d by th e a u th o r ory su b m jltin g th e th e sis. LAVAL AV^ t i I :------ “ T >-,,: r *. f ‘'U'V <*V'All - fj 1995 Recor;a-: __ S ig n a tu re of A uthor: D ate: 3 k 95~ 9 / ®%>/ 9 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ttE A D N fi- A cknow ledgm ents I am very grateful to Professor K. K. Tam for his invaluable inspiration during the writing of this thesis. W ithout his help and the contribution on his side, this thesis could not be presented as it is. I wish to express my sincere thanks for his patience and his kindness. I would like to thank Professor A. Evans and Professor J. J. Xu, from whom I learned numerical analysis and developed a skill of numerical simulations. I owe Professor C. Roth a debt of g ratitute for all he has done for me and sincerely thank him for his constant encouragement, which has been present since I arrived at McGill University. I would like to thank Ms. Wenqun Mao, Ms. Reem Yassawi, Dr. Christopher Anand, Dr. W illiam Anglin, Ms. Marie-Ange Paget, and Mr. Chonghui Liu for the help th at they willingly provided through my stay at McGill University. I wish to thank Professor I. Klemes, Professor G. Schmidt, Ms. V. Mcconnell, and the rest of the staff at the D epartm ent of M athematics and Statistics a t McGill University for their kindness. I wish to express my sincere thanks to my parent whom I owe my success. To Jacques, I thank him deeply for his constant and loving support. ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A bstract A model for microwave heating is considered. The behaviour of the solution is ana lysed and a procedure to calculate the solution is presented. This procedure is based on an eigenfunction expansion. Combining both analytical argum ent and numerical results, it is then shown th at the qualitative behaviour of the solution can be deduced from the fundamental-mode. An extension to a model of porous medium combus tion is presented. It is again shown th a t the qualitative behaviour of the solution is captured by the first eigenmode. T he corresponding model for traveling combustion waves is examined and a numerical solution is sought. The algorithm for com puta tion is based on a shooting m ethod used in an existence proof. The idea is to cut the infinite domain into two semi-infinite intervals and apply the shooting technique on both sides. This two-sided shooting m ethod is then applied to compute traveling combustion waves of a solid material. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R esum e Un modele de rechauffement par micro-onde est considere. Le com portem ent de la solution est analysee et une procedure pour la calculer est presentee. C ette procedure est basee sur une expansion des fonctions propres. En combinant l'argum ent analytique et les resultats numeriques, il est m ontre que le com portement qualitatif de la solution peut etre reduit par le mode fondamental. Une extension a un model de combustion pour une m atiere poreuse est presentee. Dans ce cas aussi, le premier mode peut decrire le comportement qualitatif de la solution. Le model correspondant pour les ondes de combustion est examine et une solution numerique est donnee. L'algorithme pour resoudre ces calculs est base sur une m ethode de “shooting” , deja utilisee dans une preuve d'existence d'une solution. L'idee est de decouper le domaine infini en intervals semi-finis et d' appliquer la technique de “shooting” des deux cotes. C ette methode est ensuite appliquee pour calculer les ondes de combustion d'une m atiere solide. iv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C ontents A c k n o w le d g m e n ts ii A b s tr a c t iii R esum e iv i 1 2 I n tr o d u c tio n 1 A S tu d y of a M o d e l fo r M icro w av e H e a tin g 2.1 In tro d u c tio n ...................... 7 . 7 2.2 Prelim inary R e s u lts ....................................................................................... 9 2.3 Properties of the Solution .......................................................................... 13 2.4 Fundamental-M ode A p p ro x im a tio n .......................................................... 18 2.5 Numerical R e su lts........................................................................................... 23 2.5.1 The S p h e r e ......................................................................................... 24 2.5.2 The Finite C y lin d er............................................................................ 24 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5.3 3 The Rectangular B lo c k ..................................................................... 24 2.6 Concluding R e m a r k s ...................................................................................... 26 P o ro u s M e d iu m C o m b u stio n : S o lu tio n b y an E ig e n fu n c tio n E x p a n sio n In tro d u c tio n ....................................................................................................... 36 3.2 Solution by Eigenfunction E x p a n s io n ......................................................... 39 3.3 T he Isolated Fundam ental-M ode.................................................................. 41 3.4 An Analysis on the Steady State Solution ................................................. 47 3.5 Numerical Results for the Truncated Multi-Mode S y s t e m s .................. 52 3.6 Concluding R e m a r k s ....................................................................................... 55 3.1 4 36 P o ro u s M e d iu m C o m b u stio n : C o m p u ta tio n o f T ra v e lin g W av e So lu tio n s 64 4.1 In tro d u c tio n ....................................................................................................... 64 4.2 Frelim inary R e s u lts .......................................................................................... 67 4.3 ......................................................................... 70 4.4 Numerical R esu lts............................................................................................. 71 4.5 Concluding R e m a r k s ....................................................................................... 75 Algorithm of Com putation • vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 A n A p p lic a tio n o f th e T w o -S id ed S h o o tin g M e th o d in C o m p u ta tio n of T ra v e lin g C o m b u stio n W av es of a S olid M a te ria l 8 6 5.1 In tro d u c tio n .................................................................................................... SG 5.2 Behaviour of the Solution 89 5.3 T he Algorithm of C o m p u ta tio n ........................ 91 5.4 Numerical R esu lts........................................................................................... 93 5.5 Concluding R e m a r k s ............................................................. 95 ...................................................... B ib lio g ra p h y 1 0 0 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List o f Figures 2.1 maxx C/(x,m) vs m for different 6 15 2 .2 maxx ^ ( x , m ) vs m 16 2.3 A and 1(A) vs A ......................................................................................... 2.4 /( / l) for a sphere with 2.5 .................................................................................... 7 = 0.2, p = 1.0, and a = 8.0, 10.0, 12.0. 7(A) for a sphere with a = 10.0, p = 1.0, and 2.6 /(A ) for a sphere with a = 10.0, 7 7 = . , 0 .2 , 0.3. 0 1 . . ... 7 = 0.2, p = 1.0, and a = 8.0, 10.0, 12.0. 1(A) for a finite cylinder with a = 10.0, 7 7 = 0.1, . , 0 2 0.3. 7 = 0.2, p = 1.0, and a = 8.0, 10.0, 2.11 1(A) for a rect. block with a = 10.0, p — 1.0, and 2.12 1(A) for a rect. block with a = 10.0, 7 30 31 = 0.1, 0.2, 0.3, and p = 0.05, 0.10, 1.00.................................................................................................... 2.10 1(A) for a rect.. block with 28 29 2.8 /(A ) for a fin. cylinder with a = 10.0, p. = 1.0, and 2.9 27 = 0.1, 0.2, 0.3, and p = 0.05, 0.10, 1.00 2.7 /(A ) for a fin. cylinder with 21 7 = 0.1, 0.2, 32 12.0. 33 0.3. . 34 = 0.1, 0.2, 0.3, and p = 0.05, 0.10, 1.00............................................................................................................. viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 • 3.1 G ( X , U ) ^ vs U for different t, with X(Q) > X ...................................... 3.2 U(i) as a function of t for t / ( 0 ) = 2 0 2 0 .0 3.3 U(i) as a function of t for t/( 0 ) = 8 .0 with fi = 0.1. 3.4 X ( t ) as a function of t for U(0) = 2 0 .1. . . 57 3.5 u (z ,t) with fi = 0.1 and U(0) = 2.0 for N = l, 3, 5, and 9........................ 5S 3.6 u (z ,t) with n = 0.1 and t/( 0 ) — 8.0 for N = l, 3, 5, and 9......................... 59 3.7 u ( z , t ) with pt = 0.2 and C/(0) = 4.0 for N = l, 3, 5, and 9......................... 60 3.8 n( 2 ,f) with pt = 0.4 and C/(0) = 4.0 for N = l, 3, 5, and 9 ......................... 61 3.9 u(0.5,t) with U(0) = 4.0 and ft = 0.1 for N = l, 3, 5, 8 , and 13............... 62 3.10 x(0-5,f) with i7(0) = 4.0 and [i = 0.1 for N = l, 3, 5, 9, and 13. ... 62 3.11 u { z ,t) w ith u ( z , 0 ) = ... 63 4.1 $ (x ), w (x), and u(x) for ft = 3.5, A = 1.0.................................................. 76 4.2 ip(x) for two different values of u(0): u(0) = 2.41011122 and u(0) = . , 8 .0 , and . . : ..................... . , 8 .0 , and 20.0 with . sin 2 2n z and fi = 8 0 with [i = 0.1. . . 0 .2 /j = 0 for N = l, 3 and 9. 2.41011123 with pi = 3.5, A = 1.0, 0(0) = 2.5, w{0) = 1.08960011 4.3 56 . . 77 . . 78 u(x) for two different values of u(0): u(0) = 2.41011122 and u(0) = 2.41011123 with fi = 3.5, A = 1.0, 0 (0 ) = 2.5, w(0) = 1.08960011 4.5 56 w (x ) for two different values of u ( 0 ); it(0) = 2.41011122 .and ^(0) = 2.41011123 with fi = 3.5, A = 1.0, 0(0) = 2.5, u>(0) = 1.08960011 4.4 44 . . 79 0 (x ) for different values of fi\ pL = 3.5 and pt = 4.0; A = 1.0, 0(0) = 2.5. 80 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 .6 u;(x) for different values of /x: /x = 3.5 and fi = 4.0; A = 4.7 ix(x) for different values of y.: fi = 3.5 and fi = 4.0; A = 1.0, 0(0) = 2.5. 82 4 .8 . , 0(0) = 2.5. 81 1 0 0 (x ) for different values of 0(0): 0(0) — 2.5, 2.4, 2.3, 2 .2 ; A = 1.0, /x = 3.5............................................................................................................... 83 4.9 iu(x) for different values of 0(0): ^(0) = 2.5, 2.4, 2.3, 2.2; A = 1.0, /x = 3.5............................................................................................................... 84 4.10 u(x) for different values of 0(0): 0(0) = 2.5, 2.4, 2.3, 2 .2 ; A = 1.0, /x = 3.5............................................................................................................... 85 5.1 The phase plane O' vs 0............................................................................ 91 5.2 0(£) and F(0(£)) vs f. The choice of 6 in the algorithm is such th at H/e. — 8 > 0m...................... 5.3 The solution 0 for a = 10.0, 16 < c < 30, with H =1.0. 96 Note th at c-(10) = 13.4.............................................. 5.4 The solution 0 for a = 20.0, 1200 < c < 1400, w ith H=1.0. 97 Note th at c*(20) = 910.0........................................................................................... 5.5 98 ^ vs f ^or different c: c = 15, 20, 25, and 30, where H = 1.0 and a = 10.0........................................................................................................... ... . 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 1 Introduction This study begins by considering a model for microwave heating. The main purpose of the study is to see if a fundamental-mode demonstrates the qualitative behaviour of the solution. This fundamental-mode approximation was described in [1] for a model of a combustion problem. T he model for microwave heating we consider can be found in [2]. The equation governing the model is the decoupling of a reaction-diffusion type equation for tem pe rature of th e m edium coupled with the Maxwell equation for the electric field which causes th e heating and is w ritten as J t = V ■(k(0) V 0) + 6E ( x ) m . (1-1) where 8 is th e tem perature; k(6) is the diffusivity with the properties k(8) > 0 , &'(0) > 0; 6 is a positive param eter incorporating the geometry of the medium; E (x) is the heating source due to the electric field; and f ( 0) is the auto-catalytic chemical heating source with the properties f ( 8) > 0, / ' ( 0) > 0. As observed by Smyth [3] it is qB realistic to take f{0 ) to be of Arrhenius type, th a t is f(0 ) = e a+° for some a > 0. In 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 Introduction this study, we consider equation (1.1) in a bounded domain D subject to th e initial condition and the boundary condition 0 (x, 0 ) = h(x) > 0 , ( 1 .2 ) 5 = 0 on d D . (1.3) As in [4], we show th a t the behaviour of the solution to the above IBV P may be deduced from the solution of ^ = V 2u + S E ( x ) F ( u ) , u(x, 0) = H (x) > 0 , where u — (1.4) u (x, t) = 0on dD , (1.5) k(s)ds. We then analyse the tem poral behaviour of m a x xu ( x ,t) . Let (pn and An be the normalized eigenfunctions and eigenvalues of the boundaryvalue problem V V » = -A \(pn , tpn = 0 ( 1 .6 ) on d D , (1-7) Ai < A2 < A3 < .... Let Ck be the k th coefficient of the series H ( x ) =: Ck<Pit(x). We adopt an approximation procedure based on an eigenfunction expansion. We show sj(x , t) = A(t)<pi(x), where A (t) is obtained from the integro-differential equation dA r ^ = —AjA + S f D , A(0) = C , , (1.8) (1.9) is dominant. The critical phenomenon of the solution u, moreover, can be captured from this fundamental-mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Introduction 3 T he above study is presented in Chapter 2. We extend our previous study to a model for porous medium combustion proposed by Norbury and S tuart in [5]. The governing equations for this model are da (1.10) g l — Xr, du d ,., , ,.du. ‘’ T t ^ T z ^ + % ) + “ - “ + '•. dw H— = u - w , „ (i-u ) ( 1 .1 2 ) 1 - 7 ' (U3) with r = f l ( a —aa)H(u — uc) g } ^ g f( w ) , (1.14) where the non-dimensionalised quantities c , u, and w are the heat capacity of the solid, the solid tem perature, and the gas tem perature, respectively; g is proportional to th e product of oxygen concentration and gas tem perature; t is the time variable and z is the space f(w ) is variable. H(£) = 0 if £ < 0 and //(£ ) — 1 otherwise. The function usually taken to be proportional to w 2. The param eter fiisproportional to the inlet gas velocity while the param eter A is linearly related to the specific heat of th e combustible solid. The param eter a measures the rate of depletion of variable g\ it is proportional to the ratio of the gas consumption to the solid consumption. The param eter oa satisfies 0 < aa < 1 , and uc denotes the critical switching tem perature related to the burning zone, th a t is a region in z —plane where r > 0 . This model is based on the asym ptotic consideration of a large-activation-cnergy lim it concept developed by Frank-Kamenetskii [6 ]. This concept became a fundamen tal step in understanding the chem istry underlying the combustion proccess. Further, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 Introduction in 1979, Matkowsky and Sivashinsky [7] provided a rigorous justification for diffusiontherm al models of porous medium combustion based on a general asym ptotic argu ment on the large-activation-energy-limit. As in [8 ], we consider the combustion of a porous slab occupying 0 < z < 1, using the following initial and boundary conditions a(z, 0 ) = <j3 , a(z, oc) = aa , (1-15) ,u ( z , 0 ) = u0(z) , (1*16) u>(0,f)=0, (1.17) $ ( ( M) = < 7a , (1*18) u( 0 ,f) = u ( l ,i ) = 0 where u and w have been normalized so th at the am bient tem perature is zero. Fol lowing [8 ], we replace the reaction rate r in (1.14) by r = (a - att)fi1/2gw2 , (1.19) and take d — 0. Using a modified Oseen-type linearization, to simplify the problem, in [8 ] Tam argued th a t the information regarding the ignition and the qualitative dependence of the solution on param eters can be deduced from the first term in the eigenfunction expansion. In this study, we by-pass the Oseen-type linerization. Instead, we expand the so lution in a series of eigenfuntions to obtain an infinite system of ordinary differential equations. As it was argued in [8 ], th a t the first term of the expansion dominates the solution, we firstly focus on th e isolated fundam ental mode, giving some analysis regarding the ignition and param eter dependence of the solution. We, then, analyse th e dependence of the steady-state solution on a param eter for th e infinite system Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Introduction 5 of ordinary differential equations as well as the truncated multi-mode systems. The numerical results for truncated multi-mode systems confirm the validity of the qual itative behaviour derived from using a single-mode. A large part of this study is described in [9] by Tam and Andonowati and we present it in Chapter 3. In further study of porous medium combustion, we seek numerical solution of traveling combustion waves. We noted th a t a large number of combustion phenomena are modelled as a propagation of waves. Matkoswky and Sivashinky [7], in particular, provided a model of traveling combustion waves for solidfuel. In [10],Norbury and Stuart analysed a one-dimensional, time-dependent model for traveling combustion waves for porous medium derived from [5]. An existence proof was, then, derived by reducing the problem into a two-point free boundary problem overa finite interval and applying local bifurcation theory. In this study, we consider the following model c-^— — Ar — 0 , d?u _ + OT_ du „ + I„ _ B + r = 0 , dw (i—— ax = (1.20) (1.21) ( 1 .2 2 ) (1.23) derived from the previous model in C hapter 3, where the param eter a is taken to be zero [10] and x = z — ct, where c is the wave speed. The appropriate boundary conditions for this model are u (± o o ) = u a , (1.24) u;(±oo) = wa , (1.25) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Introduction 6 0 < cr(—oo) = Co < , (1.26) a(oo) = 1 . (1.27) 1 A technique in computing traveling combustion wave solution for porous medium was introduced by Tam and Andonowati in [11]. We present this com putation in C hapter 4. The algorithm of this com putation was developed from an existence proof by using a shooting method [1 2 ]. The idea of the shooting m ethod can be illustrated as follows. The position x = 0 is chosen to be the point where ^ ( 0 ) = 0 . In [12] Tam proved th a t there is a set of values of u( 0 ) and w( 0 ) such th at equations ( 1 .2 1 ) and (1.22) have a solution satisfying the limiting behaviour at —oo . It was proved th a t a subset of such solutions can be extended to the right to satisfy boundary condition at oo . Based on this result, we developed an algorithm to com pute such u(0) and u>(0 ), and so, to establish a numerical solution. For a set of param eter values (i and A, numerical results strongly suggest th a t there is a unique solution to the problem. It is also found th a t there is a lim it of the inlet gas velocity /z, say p*, below which no numerical solution can be constructed. Finally in C hapter 5, we apply the shooting technique developed in C hapter 4 to compute traveling combustion wave solution of a solid m aterial. This study is described by Andonowati in [13]. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 2 A S tu d y o f a M odel for M icrowave H eating 2.1 In tro d u ctio n We consider a model for microwave heating. The purpose of this study is to demon strate th a t th e qualitative behaviour of the solution can be approximated by a funda mental mode. We start the study by presenting a model for microwave heating. The equations governing the model consist of a reaction-difTusion type equation for the tem perature of the medium coupled with the Maxwell equation for the electric field which causes the heating. After some simplification [2], the equations are decoupled and the relevant equation then has the form f t = V - (fc(«)V«) + 6E ( x ) m • (2 .1 ) Here, $ is th e tem perature; fc(0) is the diffusivity with the properties lc(0) > 0, k'{6) > 0. In this study we take k(0) — fie 18, The param eter 6 is a positive param eter 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 8 incorporating the geometry of the medium; j5 (x ) is the heating source due to the electric field; and f ( 0) is the auto-catalytic chemical heating source w ith the properties f(0 ) > 0, f ( 0 ) > 0. As observed by Smyth [3] it is realistic to take f(0 ) to be of qQ Arrhenius type, th at is f( 0 ) = e°+« for some a > 0. We note th a t the microwave heating problem has been recently considered by a number of authors such as [2], [3], and [14]. For k(6) — 1 and E ( x ) = 1 the equation (2 . 1 ) is a central problem in the combustion theory and has been studied by many authors such as [1], [6 ], [7]. For fc(0) ^ 1 the equation (2.1) has also been studied extensively in [4], [15] and elsewhere. Microwave heating has become increasingly im portant in industry with applica tions in a variety of drying and processing facilities. More detailed discussion of these applications can be found in [2], [3], and [14]. In this note, we consider equation (2.1) in an open bounded domain D subject to the initial condition and the boundary condition , (2 .2 ) 0(x, t) = 0 on dD . (2.3) 0 (x,O) = h(x) > We first study the behaviour of solution. 0 We then develop an approxim ation procedure based on an eigenfunction decomposition. From this approxim ation, we focus on its fundamental-mode. Let Scr be the critical value of 5, where the steady state solution of (2 . 1 ) is 0 (e“ ) for 6 > S„, calculated from using the fundam ental mode. We show th a t this critical value approximates th e critical value of the param eter of solution u. Thus the critical phenomena of th e solution can be predicted from the equation for this single mode. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 9 In the next section, we present some preliminary results. We investigate the behaviour of the solution in Section 2.3. A procedure to calculate the solution as well as some justification of the fundamental-mode approximation are described in Section 2.4. In Section 2.5 numerical results for the fundamcntal-modc approximation are carried for some simple configurations, viz, a sphere, a finite cylinder, and a rectangular block. We present the concluding remark in the last section. 2.2 P relim in a ry R e su lts We introduce the transformation u= J0 k(s)ds. (2.4) W ith this transform ation, Equation (2.1) becomes ^ where K (u ) = = I< (u){V2u + 8E ( x ) F ( u ) } , fc(0(u))and F ( u ) = (2.5) . Since u(0)is monotonic increasing, we observe th a t both K (u ) and F ( u ) have the same features ask(0) and f(0 ) respectively. The initial condition (2.2) and the boundary condition (2.3) become ti(x, 0) = i / ( x ) , (2.6) u (x ,t) = 0 on dD , (2.7) and where H { x ) = k(s)ds . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 10 The results in this section can be found in [4]. We present them for the sake of completeness but in a different manner. First we observe th at f ( 0 ) = F ' ( u ) ~ = F'(u)k{0), and so F'(u) = j^ j-. Since f ( 0 ) — e°+®, then a2 f ( ° ) = W ) 7(q Z T+70) m > 0 • (2-8) From the fact th at k(0) > 0, we obtain F'(u) > 0. L em m a 1 Let u be the solution o f the following initial boundary valued problem ^ = K {V 2u + S E ( x ) F (u )) , subject to the conditions u (x, 0) = H (x ) > 0, u(x, t) = 0 (2.9) on dD, where K is a positive constant and H (x ) satisfies V 2 /f(x ) + 6E F ( H ( x ) ) > 0. Then j f ( x , t ) > 0 fo r x € D and t > 0, before u reaches its equilibrium. P r o o f 1 Let v = . Then u (x ,0 ) > 0 and v satisfies ~ = I< {V2v + S E {x)F '(u)v} . (2 . 1 0 ) Suppose the Lemma is not true. Let D\ = {x € Z?|u(x,tx) = 0} fo r the first instant t = fj > 0, if there is one, otherwise the Lemma is obviously true. We may assume that D\ D, since if D i = D then the equilibrium has been reached. From equation (2.10), if ( x ,t) € D x [0, fi] then v satisfies - ^ - I < V 2v = 6K E ( x ) F ’{ u )v > 0 , (2.11) o(x ,0 ) > 0 , v = 0 on dD x (0 ,ti], (2.12) and the conditions Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11 Microwave Heating where D = D U d D . By the minimum principle u(x, t) > 0 in D x [0,fi]. Since D\ is not empty, there is an interior point, say Xi, o f D such that — 0 giving u (x ,ii) = 0 fo r all x , x £ f l . We thus obtain a contradiction and the Lemma follows. [See [16], p. 173.] L e m m a 2 Let L be the parabolic operator m = T l - K ( u ) { V u + «£ (x )F (« )}. (2.13) Let u, Z , and z be solutions o f dZ | sup K ( Z ( x , t )) j 9T_ \ x e D , o < i < r J dz f inf K ( z ( x , t )) d t ~ \ x e D , 0<t<T L(u) = 0 , (2.14) { V 2Z + 6E ( x ) F { Z ) } = 0 , (2.15) {V2z + 6E ( x ) F (z )} = 0 , (2.16) subject to the the initial condition u (x , 0 ) = Z ( x , 0 ) = z(x, 0 ) = 7Y(x) > 0 and the boundary condition u(x.,t) = Z ( x , t ) = z ( x , t ) = 0 on dD , where H (x ) satisfies V 2# + 6E F ( H ) > 0 and T > 0. Then we have z < u < Z in D x [0, T\. Before we proceed to the proof, we note th a t by writing /f(it), K ( Z ) , and K (z), we have been using the transformations u = f] k(s)d s, Z = f] k(s)ds, and z = f ] k(s)ds. Assuming th a t the tem perature 0 > 0 , we implicity assume th a t the solutions u, Z, and z for above differential equations are non-negative. P r o o f 2 Let K l = supx€£,.0<KT K ( Z ( x , t ) ) , then K 1 > 0. By the first Lemma, BZ > 0. For 0 < t < T , we have |jf dt L (Z ) = d7 K ( Z ) { V 2 + 8E ( x ) } F ( Z ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.17) 12 Microwave Heating = 1 Fi7 ^ ( Z ) { - ^ ( x ) F ( Z ) + <5£(x)F(Z)} F)7 H K {Z) dZ I< 1 ' & n 1 > 0. We 77iu,y also show that L ( z ) < 0 for 0 < t < T . The result follows from the maximum principle [16], p. 187. L e m m a 3 Z and z, and hence u, tend to the same asymptotic state as t —*• oo. P r o o f 3 For fixed T, let G = supx6£).0<i<;r K ( Z ( x , t)) and r be a transformation such that r(t) = Gt. With this transformation equation (2.15) becomes ^ - = V 2Z + 8E ( x ) F ( Z ) . (2.18) OT Similarly ifg = infX6 D;0 < t < r 2)) andrj is a transformation such thatr](i) = gt, the equation (2.16) becomes £ _ = v 2* 4- 8E ( x ) F ( z ) . arj (2.19) Since T is arbitrary, Z and z tend to the same asymptotic state and since z < u < Z on [0 ,T ], so does u: The influences of the param eters on the steady state solution of equation (2.1) can thus be observed through the following equation ^ = V 2u + 6E { x ) F { u ) . (2.20) For E (x ) = 1, using an eigenfunction decomposition for u, Tam in [1] and [4] dem onstrated th at the feature of u can be obtained from its fundam ental mode. Considering th a t £ ( x ) in this problem is positive and bounded, we shall show th at the similar treatm ent would be adequate. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 Microwave Heating 2.3 P r o p e r tie s o f th e S olu tion We consider the following IBVP fill % = V 2u + 6E ( x ) F ( u ) . u(x, 0) = H (x) > 0 , u(x, t) = 0 on dD . (2.21) (2.22) We wish to investigate the behaviour of solution u(x, I). In doing so, we first consider the following IBVP ^ = V 2U + S E ( x ) F ( m ) , U (x, 0) = H (x ) > 0 , for some m > 0 U (x, i) = 0 on d D , (2.23) (2.24) . L e m m a 4 L e t U ( x , t , m ) be the solution o f (2.23) and (2.2/,). lf'V 2H + S E (x )F (m ) > 0 m , then W 3t (x , t , m ) > 0 , fo r all t > 0 . P r o o f 4 Let s = ^ . Then s satisfies ! = v2s s (x ,0 ) > 0 f o r x € D , s ( x ,t) = 0 on dD . <2-25> (2.26) B y the minimum, principle s > 0, thus the Lemma follows. From now on, we use the upper bar to denote the steady state solution, u, for example, denotes the steady state solution of u. We also assume, as we have been doing, th a t the initial condition H (x ) is such th at V 2/ / + S E ( x ) F (H ) > 0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 Microwave Heating L e m m a 5 Let u be the steady state solution o f (2.21) and (2.22). Let U be the steady state solution of (2.23) and (2.24) w m P ro o f 5 = m a x xii{x). Then m a x x U ( x , m ) > m. Since, from Lemma 1, u { x ,t) increases as t increases it follows that u ( x , t ) < m . Let V ( x , l , m ) = U [ x ,t,m ) - u ( x ,t) , then ^ at - V 2V = 8E ( x ) F ( m ) - 8E { x )F (u ) > 0 , V (x , 0) = 0 , (2.27) V (x, t) = 0 on 8 D , (2.28) thus by minimum principle V > 0 giving U { x , t , m ) > u (x , i). Taking the limit as t oo and then m a x X) we obtain m a x x U (x ,m ) > m . The steady state solution for (2.23) is U (x, m ) = 6F (m ) ^ §V «‘(X) « (2-29) where Bi = f D E(x)<p;(x)dx and <p; and A,- are the normalized eigenfunctions and eigenvalues of the boundary-value problem VV,- = -A f a , (2.30) tpi = 0 on d D . (2.31) It follows th a t m a x x U ( x ,m ) = 6M F ( m ) ) (2.32) where M = m a x x X); |^ p ;(x ). In Figure (2.1) we show the graphs of m a x x U (x, m ) vs m, for different 5. Forthe case when 8 is such th a t m a x x U(x, m) — m = 0has a single root, say mo, it is not difficult to see th a t m a x xu (x ) < mo. [See Figure (2.1) for Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 8 = 81 and Microwave Heating 15 m 8 - 8- m Figure 2.1: maxx £ /(x ,m ) vs m for different 8 8 = 53>] Since suppose it is not true, then m ax xO(x) > m 0. B ut for any m > mo, we have m > U (x , m ) giving m axxu (x ) > f/(x ,m ). This contradicts Lemma 5 and thus m ax xti(x) < mo. Let 8 be such th a t m a x x U (x,m ) — m = 0 has three roots, say n ij, m i, and m 3 , where m 5 < m 2 < m 3. Then for those 8 we have th e following properties. P r o p e r t y 1 Let m i, m 2, a n d m 3, where m i < m 2 < m3} be the roots of m a x x U (x ,m ) —m = 0. Then either 0 < m ax xu (x ) < m i or m 2 < maxxu (x ) < m 3 . P ro o f o f P ro p e rty 1 Suppose it is not true, then m axxu (x ) > m 3 o r m i < m axxii(x) < m 2. B ut fo r any m such that m > m 3 o r m i < m < m 2 we obtain m a x x U (x ,m ) < m giving m a x x U (x , m ) < m ax xtt(x) which again contradicts Lemma 5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Microwave Heating m m Figure 2.2: maxx i/(x ,m ) v s m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 17 The following property is stronger than the Property P ro p e rty 2 1 . Let m i, m 2 , and m 3, wherem\ < m 2 < m 3, be the roots of m a x x0 (x ,m ) — m = 0. Then 1. m a x xu ( x , t ) lies on [0 , mi] all the time or 2. m a x x u ( x , t ) lies on [m 2 ,m 3] all the time. P ro o f of P ro p e rty 2 Suppose the properly is not true. Then there is a t0 > 0 such that m i < m a x xu ( x , t 0) < m 2 or m a x xu ( x , t 0) > m 3. Consider the first case: there is a to > 0 such that m\ < m a x xu(x,ta) < m-t. Let mo = m a x xu(x,to). Since u ( x , t ) increases with t then I I (x ) < m 0 and so V 2H + 6E ( x ) F { m Q) > 0 . (2.33) Using Lemma 4 and (2.33) there is an interval (a0, ai) C (m 1 ,m 2 ) such that mo € (ao, a i) and for any m € (doi<*i) U ( x , t , m ) is a non-decreasing function of t. This gives U (x ,t,m ) < m axxU(x,m) < m (2.34) fo r m e (a 0 ,a i). Now consider the following IB V P ^ = V 2 t/ + S E ( x ) F { m 0) , £7(x,0) = H ( x ) > 0 , U (x ,t ) = 0 on d D , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.35) (2.36) 18 Microwave Heating For t < *0, u(x, t) < m a x xu(x, i0) = m Q. Thus U ( x , t , m n) > u(x, t) fo r t < t 0 and so U ( x , t 0, m 0) > u ( x , t 0) giving m a x x U ( x , t o , m 0) > m a x xit ( x ,t0) = m 0 . (2.37) This contradicts (2.3J,). The same idea applied for the second case, there is a to > 0 such that m a x xu ( x , t 0) > m 3, leads to a contradiction and thus proving Property 2. Let 8UcT and 8ucr be the biggest and the smallest value of 6 such th a t the line m is tangent to the lower and upper portion of the graph m a x x U ( x , m ) v s m , respectively. Let 8 be such th a t 8(j„ < 8 < 8Vcr. For these 8 the Properties 1 and 2 follow. Since u ( x , t ) increases as t increases (Lemma 1) we infer from Property 2 th a t if the initial condition H ( x ) is small then u (x ,i)is small all the time. 2.4 F u n d a m en ta l-M o d e A p p ro x im a tio n We consider the following IBVP ^ = V 2u + 8 E ( x ) F { u ) . at u(x, 0) = H (x) > 0 , u (x, t) = 0 on dD . (2.38) (2.39) Let tpn and An be the normalized eigenfunctions and eigenvalues of the boundary-value problem VVn = ipn = 0 , (2.40) on d D . (2.41) Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Microwave Heating 19 Let Ai < A2 < A3 < .... Let Ck be the kth coefficient of the series II (x) = Eg,. We adopt the following approximation procedure which can be attributed to Galerkin. Let » * ( x , 0 = E S M ' ' )(0 v * (x ), (2.42) where >4^ (() is the solution of the integro-differential equation = -A ? a !n > + S / D £ ( x ) F ( E i z f ^ » V i(x )V i(x )< f» (x ), = C i, (2.43) (2.44) for 1 < i < N . Equations (2.43) and (2.44) constitute N equations with N unknowns. We show th a t if S ^ a ^ y p ^ x ) is the eigenfunction expansion of u(x, t) and th at .A ^ ( t) converges as N —►oo, then A j^ (f) —» a,-(t) as N -» oo. Thus if .4 ^ ( 1 ) converges, then S ff( x, t) converges to u ( x , t ) as N —►oo in the mean square. L em m a 6 Let u ( x , t ) be the solution of (2.38) and (2.39) and let E £,a,(i)y?i(x) be theeigenfunction expansion of u ( x , t ) . Let s ^ ( x , t ) = A ’^ M z5 where calculated from equations (2.43) and (2.44), for 1 < i < N . I f A \N\ l ) converges, then s ^ ( x , t ) converges to u ( x , i ) as N —►oo in mean square. P ro o f 6 By substituting into equation (2.38), multiplying by <Pi(x), then integrating over D, it is not difficult to see that ^ = —A?a; + S f £ (x ) F (E g r< rW (x ))W(x)</v(x), (2.45) u,(0) = C i, (2.46) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 20 Since F is a bounded function, the solution u(x, t) of (2.38) and (2.39) is unique. By the uniqueness of the eigenfunction expansion of u(x, t), the solution to the system (2.45) and (2.46) is unique. Let A.W = ■••>-'4^) 0 ,0 ,0 ,...), where - = -A *a \N) + 8 JD E ( x ) F ( ^ M N)fp{(x ))tp i( x )d v { x ) , 1 < i < N . (2.47) ASN)(0) = C i . (2.48) / / A W -* A as N oo, where A = ( A \ , A 2, . . . , A n , 4/v + i,...) then A{ is a solution o f (2.45) and (2.46) f o r Vi. B y the uniqueness of the solution of (2-45) and (2.46), A{ — a{, Vi. Inspired by the behaviour of m a x xu(x, t) described in the previous section, we seek the first approximation s i( x ,t) . In this first eigenfunction approxim ation Si(x, t) = A(f)y?i(x), A(t) is obtained from the integral equation M = - X \ A + 8 J^E(x)F(Aipi(x))tpi(x)dv(x), A(0) = Cx . (2.49) (2.50) Let 1(A) = f E(x)F(A<pi(x))(pi(x)dv(x). (2.51) JD The equilibrium values of A can be obtained graphically from the intersection of the straight line ^ vs A and the S-shape curve of the graph 1(A) vs A. Let So- be the critical value, where the steady state of (2.43) and (2.44) undergoes a rapid transition from being 0 (1 ) to 0 ( e “ ). This critical value 8„ is obtained when th e straight line A2j4 vs A is tangent to the lower portion of the S-shape curve, Figure 2.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating' Sc 1(A) Figure 2.3: A and 1(A) vs A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 Microwave Heating If for some 8 , < s i,s i > is of the same order of m agnitude as < s n ,$ n > for any N > 1, then for th a t 8, < s i , s \ > approximates the steady state solution u. Here s n is the steady state solution of s n( x , t ) and < /(x ),g r(x ) > is defined as < / ( x ) , t f ( x ) > = f /( x ) p ( x ) d x . (2.52) JD This argument is based on the convergence of spi to u in the mean square. Further if this is true for any values of 8 close to the value of <$CT, then critical param eter 8„ obtained from the single mode approximates the critical param eter of th e solution u. We carry out the computation for a unit sphere, assuming spherical sym m etry with E(r) = e~T. We show th a t < s i,s i > is of the same order of m agnitude as < s n , s n > , for N =2,3,4, ... ,15, as long as 6 is not very close to 5cr = 7.256. [See Table 2.1 as examples.] Note th at this is calculated using the fundamental-mode. sn SN > N=10 N=15 0.256368 0.254733 0.254732 0.254732 <5 = 6.0 1.068883 1.061605 0.061586 1.061584 0.56899810® 0.57207010® 0.57207810® 0.57207810® 0% II OO 0 N=5 II N=1 0 < Table 2.1 The m agnitude of the smallest steady-state solution s n for different S with a = 10, 7 — 2.0, and ft = 1.0. In Table 2.2, we consider the following example. T he critical param eter £&■ calcu lated from the fundamental-mode for th e sphere configuration with a = 10 , 7 = . , 1 0 and /j = 1.0 is 6 a- — 7.256. We want to check, for 8 near £«., the m agnitude of the smallest steady-state for different N . It is shown in Table 2.3, th a t < s i ,s i > is of the same order of m agnitude as < s/v, s n > for N =5 and 10 , except for 8 — 7.2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 23 which is very close to the critical point 8„ = 7.256. This indicates th a t the critical param eter obtained from the single mode approximates the critical param eter of the solution u within a small error. We repeat similar calculations for different a and 7 and obtain similar results. Thus, it is not only th a t the first mode is dom inant but also th at the critical value 6Cr, obtained by using a single mode, approximates the critical value of 8 of u. < SfltSN > N=1 N=5 N=10 5 = 7.0 2.847926 3.058751 3.058550 rH hIt 3.360060 3.915313 3.914835 5 = 7.2 4.243676 0.385138106 0.385141 10® 5 = 7.3 0.404318106 0.406070106 0.406072 10® 5 = 7.4 0.42560810® 0.427681 106 0.427685 10® Table 2.2 The m agnitude of the smallest steady-state solution 7 2.5 sn for 6 near 6er with a = 10, ~ 2.0, and fi = 1.0. N u m erica l R e su lts Let 8„ be th e critical value of 6 obtained from the single mode (2.43). Suppose that A i is the smallest A satisfying /(A i) = &sc then _d_ dA { / ( A ) “ f r L A = 0 - (2,53) We m ay thus calculate 8„ and the corresponding smallest steady state solution of A from the following: /(A t ) — A \ I ' { A \ ) = 0 , Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (2.54) 24 Microwave Heating and (2.55) For several values of (i, a , and 7 , in Table 2.3, 2.4, and 2.5 we sumraerize the critical values of 8 and the corresponding smallest steady-state solution A. 2.5.1 T h e S phere Let D be a sphere of unit radius. Assuming the spherical symmetry, we have A2 = 7r2 in and the corresponding normalized eigenfunction is <^i(r) = —11_ ssin^ r . Let E ( x ) = e~r. ttt We calculate 1(A) for different values of fi, a , and 1(A) for 7 = . Figure (2.4) shows the graph 0.2 with a = 8.0, 10.0, 12.0 and fi = 1.0, while Figure (2.5) shows the graph 1(A) for oc — 10.0 with 7 = 0.1, the graph of 1(A) for a = 10 w ith 2.5.2 7 7 . , 0.3 and fi — 1.0. In Figure (2.6) we show 0 2 = 0.1, 0.2, 0.3 and fi = 0.05, 0.10, 1.00. T h e F in ite C y lin d e r Let D be a cylinder with a unit circle base and a hight equal to one. Assuming an axial symmetry, the eigenvalue A2 = 7r 2 + c2, where c is the first zero of the Bessel Function of order 0, Jo(c), (c=2.405). The corresponding normalized eigenfunction ^ \ J \ s in ^ J o (c r ). For th e case of E ( x ) = e~z, we repeat the calculation as in (2.5.1). The results are shown in Figures (2.7), (2.8), and (2.9). 2.5.3 T h e R e c ta n g u la r B lock Let D = { ( s ,y ,z ) |0 < x < l , 0 < y < aandO < z < 6 ). We have Af = 7t2(1 + ^ + ^-) and tp = ^ sin irxsin ^ sin For the case of E (x ) = e~x , a = 1, and b = 1, we Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 25 repeat the calculation as in (2.5.1). The results are shown in Figures (2.10), (2.11), and (2 . 1 2 ). o = 8.0 sphere fin. cyl. rec. block 7 = 0.1 A 7 = 0.2 A 7 = 0.3 ficr A fi 0.05 0 .1 1 1 0.348 0.148 0.384 0.259 0.436 0.10 0.221 0.696 0.295 0.768 0.516 0.817 1.00 2.205 6.962 2.948 7.681 5.156 8.713 0.05 0.095 0.501 0.129 0.555 0.237 0.633 0.10 0.190 1.002 0.257 1.110 0.473 1.266 1.00 1.898 10.023 2.565 11.098 4.728 12.662 0.05 0.055 1.027 0.074 1.138 0.137 1.300 0.10 0.109 2.055 0.147 2.276 0.273 2.599 1.00 1.083 20.546 1.467 22.762 2.729 25.994 *cr Scr Table 2.3: Critical points of fi and the corresponding steady-state solutions A. a = 10.0 sphere fin. cyl. rec. block 7 = 0.1 A 7 = 0.2 A 7 = 0.3 A fi 0.05 0.098 0.333 0.123 0.363 0.168 0.402 0.10 0.196 0.665 0.246 0.726 0.336 0.803 1.00 1.957 6.654 2.451 7.256 3.352 8.033 0.05 0.084 0.478 0.106 0.523 0.147 0.581 0.10 0.168 0.956 0.212 1.046 0.293 1.162 1.00 1.676 9.563 2.115 10.459 2.925 11.621 0.05 0.048 0.980 0.061 1.072 0.084 1.192 0.10 0.096 1.960 0.121 2.144 0.168 2.384 1.00 0.956 19.599 1.208 21.444 1.673 23.838 5cr Scr ficr Table 2.4: Critical points of fi and the corresponding steady-state solutions A. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 26 a = 12.0 sphere fin. cyl. rec. block 7 = 0.2 7 = 0.1 7 = 0.3 fi 0.05 A Scr A Scr A Scr 0.092 0.324 0.112 0.351 0.144 0.385 0.10 0.183 0.647 0.223 0.702 0.516 0.288 1.00 1.828 6.473 2.228 7.017 2.877 7.696 0.05 0.079 0.465 0.096 0.505 0.125 0.556 0.10 0.157 0.929 0.192 1.010 0.250 1.111 1.00 1.563 9.293 1.916 9.466 2.492 11.113 0.05 0.045 0.952 0.055 1.035 0.072 1.140 0.10 0.090 1.904 0.110 2.071 0.143 2.279 i.o r 0.891 19.043 1.093 20.708 1.424 22.792 Table 2.5: Critical points of 6 and the corresponding steady-state solutions A. 2.6 C on clu d in g R em ark s We have considered a model for microwave heating. The behaviour of the solution is analysed and a procedure to calculate the solution is presented. This procedure is based on an eigenfunction expansion. An ordinary integro-differential equation for the am plitute of its fundam ental mode is presented. It is shown th a t besides the fundamental mode dominance, the critical phenom ena of the solution can also be deduced. Having confirmed th e fundam ental-mode dominance, num erical results are derived for this single mode for some simple geometries, viz, a sphere, a finite cylinder, and a rectangular block. It is found th a t the critical param eter S„ decreases as a increases and it increases as 7 increases. This critical param eter the steady-state solution undergoes a rapid transition from being determ ines when 0 ( 1 ) to Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 (e°). Microwave Heating CM 27 a -12.0 a - 8.0 a - 10,0 © ■ 200 400 600 80 0 1000 A Figure 2.4: 1(A) for a sphere w ith 7 = 0.2, fi = 1 .0 , and a = 8.0, 10.0, 12.0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 Microwave Heating <\l o ■ 0 200 400 600 80 0 1000 A Figure 2.5: 1(A) for a sphere w ith a = 10.0, fi = 1.0, and 7 = 0.1, 0.2, 0.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 29 I I 7 O' 0 200 = 0.1 600 400 1000 A 7 0 200 600 400 = 0.2 1000 A 0 200 400 1000 A Figure 2.6: /(.A) for a sphere with a = 10.0, 1.00. 7 = 0.1, 0.2, 0.3, and fi — 0.05, 0.10, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 Microwave Heating CM o 0 200 600 400 Figure 2.7: I ( A ) for a fin. cylinder witli 7 800 1000 = 0.2, y = 1.0, and a = 8.0, 10.0, 12.0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating CM — y - 0 .1 o• 200 400 600 800 1000 A Figure 2.8: 1 (A ) for a fin. cylinder with a = 10.0, fi = 1.0, and 7 = 0.1, 0.2, Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. 32 Microwave Heating I § 7 © 0 200 400 = 0.1 600 600 1000 600 7 = 0.2 600 1000 A § § § o 0 200 400 A 7 0 200 400 600 600 = 0.3 1000 A Figure 2.9: 1(A) for a finite cylinder w ith a = 10.0, 0.10, 1.00. 7 = 0.1, 0.2, 0.3, and (i = 0.05, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 Microwave Heating o ■ 0 200 400 600 1000 8 00 A Figure 2.10: I {A) for a rect. block with 7 = 0.2, fi = 1.0, and a = 8.0, 10.0, 12.0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating § § o 0 200 400 1000 600 A Figure 2.11: 1(A) for a rect. block with a = 10.0, = 1.0, and 7 = 0.1, 0.2, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Microwave Heating 35 00 I § 7 o 0 300 = 400 0.1 1000 A 7 0 €00 200 = BOO 0.2 1000 A 7 0 200 400 100 = 0.3 BOO 1000 A Figure 2 . 1 2 : 1 ( A ) for a rect. block with a = 10.0, 0.10, 1.00. 7 = . , 0 .2 , 0.3, and fi = 0.05, 0 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 3 Porous M edium C om bustion: Solution by an E igenfunction E xpansion 3.1 In tro d u ctio n In [5], Norbury and Stuart constructed a model for porous m edium combustion based on asym ptotic analysis on th e concept of a large-activation-energy-limit. As the energy-limit E —* oo, the porous medium combustion undergoes three different length scales. First is the scale in which the tem perature of solid reactant is raised from its equilibrium state ue to a threshold tem perature u c. At this level the chemical reaction become significant. The second scale is th at in which th e equilibrium tem perature is almost the same as the threshold tem perature, u e ~ uc. Third is the scale when the equilibrium ue > uc, which is the state when the combustion takes place. Here, the threshold tem perature uc acts as a switching tem perature below which the reaction 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous M edium Combustion 37 is not significant and above which the reaction is limited by the gas production and the depletion of the medium. T he basic model for porous medium combustion is based on a reaction of a burn ing porous solid reactant and oxygen carried by gas through its pori. This can be described as Solid Reactant + O2 —»Heat + C O 2 + Ash. (3.1) T he basic goal for modelling isto determ ine the total heat produced per unit mass of reactant. T he model in [5] is formulated as follows Ar, 4 = + + dw V‘~dz = U ~ W ’ (3.2) <3 ' 3 > (3<4) = -= , p. (3-5) with r = H{<j~crn)H{u — uc)pl^2g f { w ) . (3.6) oz The non-dimensionalised quantities <r, u, and w are the heat capacity of the solid, the solid tem perature, and the gas tem perature, respectively; g is proportional to the product of oxygen concentration and gas tem perature; i is the tim e variable and z is the space variable. = 0 if f < 0 and H(£) = 1 otherwise. The function f ( w ) is usually taken to be proportional to w2. The param eter p is proportional to the inlet gas velocity while the param eter A is linearly related to the specific heat of the combustible solid. The param eter a measures the ratio of gas consumption to that of solid. The param eter aa satisfies 0 < <Ja < 1, and uc denotes the critical switching tem perature related to the burning zone, th a t is a region in z —plane where r > Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0 . Eigenfunction Expansion for Porous Medium Combustion 38 In [17], Tam considered the combustion of a porous slab occupying 0 < z < 1, using the following initial and boundary conditions <t(z,0) = cr4 ,cr(z, oo) = cra , (3.7) u (0 ,t) = u ( l,f ) = 0 ,u (z ,0 ) = u 0 (z) , (3.8) xu(0, t) — 0 , (3.9) ( M ) = 0«, (3-10) 0 where u and w have been normalized such th a t the ambient tem perature is zero. The reaction rate r in (3.6) was also replaced by r = (a —<Ta)p}l2gw2 . (3-11) The reason for doing so is th a t from (3.4), as u > ru, w increases w ith z. Thus we expect th a t if u is large, so is w. The switching function H ( o —aa), on th e otherhand, is to obtain a more manageable structure. The rate of decay r of the solid reactant, however, is proportional the heat capacity a of the solid being considered. To simplify the problem, in [17] Tam used a modified Oseen-type linearization. An ordinary differential equation was then derived. It was argued th a t the infor mation regarding the ignition and the qualitative dependence of th e solution on the param eters can be deduced from the ordinary differential equation. Instead of using an Oseen-type linearization, in this study we expand the solution in a series of eigenfunctions. An infinite system of ordinary differential equations is derived in Section 3.2. As it was argued in [17], th a t the first term of th e expansion dominates the solution, in Section 3.3 we focus on the isolated fundam ental mode, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous A fed iu m Combustion 39 giving some analysis regarding the ignition and param eter dependence of the solu tion. An analysis on the steady state solution of the general system as well as the truncated multi-mode systems is presented in in Section 3.4. In Section 3.5, numer ical computations for truncated multi-mode systems are carried out, and the results confirm the validity of the qualitative behaviour derived from using a single-mode. The concluding rem arks are presented in the last section. We note th a t a large part of this study is described by Tam and Andonowati in [9]. To lessen the complexity of the problem we neglect the eflect of the non-linear radiation and take d = 3.2 0 . S o lu tio n b y E ig en fu n ctio n E x p a n sio n Let x = <t —&a- Using (3.11) and integrating equation (3.5)with respect to z we have g{z,t) = gae x p { - a p l/2 f x ( M ) w 2{s,t)ds} . Jo (3.12) From (3.4), we obtain w(z,t) = p. f u(s,f)e*^*ds. Jo (3.13) Thus if we solve for u and x> the above results give g and w. Let {<^n} = { \/2 sin m r z } be a set of normalized eigenfunctions corresponding to eigenvalues {7 n} = {n 7r}. We expand u and x in term s of {<£>„}: u (z ,t) = S ? U n(i)tpn( z ) , (3.14) X(*,i) = E ? % i ( % n(2) , (3-15) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 Eigenfunction Expansion for Porous Medium Combustion keeping in mind the convergence of the expansion to x is non-uniform at 1. z — 0 and We should recall th at u(z ,t) is the solid tem perature and x >sproportional to the heat capacity of the solid. Using the notations f V?n(z) = tpn(z) = V>n(s)ds, (3.16) <pn{s)e3/,1ds , (3.17) Jo f JO and substituting (3.15) and (3.14) into (3.2) and (3.3) we obtain , OaEiipiUl + Ei'ZjXupUPjU'j -S .-7 (3.18) = ftpiUi + n - ' e - ’^ZitpiUi - S itpiUi - A"aS ^ X j , (3.19) where the notation S n denotes Multiplying equations (3.18) and (3.19) by <pn and integrating from 0 to 1 with respect to z we obtain X n‘ £ vm & V ne-^^dz = —p~5t 2aE>i'Ej'£k%iX,i XjUkUi <jaU'n + Zi'ZjXiUj f Jo f Jo ifinpj(fndz < p w $ k<ptyne~2:sltldz (3.20) = - 7 n2Un + n - ' U U i I 1 Vi<pne - ^ d z - U n - A-1* ; . (3.21) Jo Equations (3.IS) and (3.19) constitute an infinite dimensional dynamical system. While it is not possible to solve such a system in general, a great deal of effort has Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous M edium Combustion 41 gone into its study. (See for examples [IS], [19], and [2 0 ]). To reduce it to a finite dimensional system so th a t a t least some numerical work can be done, simplifying assumptions, which usually depend on some knowlcgde of the underlying problem, m ust be made. In the present case, experience in dealing with combustion of solid m aterials suggests th at the first eigenmode is dominant. We therefore adopt finite truncation and consider only the interaction of the first N modes, discarding all term s involving modes of order (N + l) and higher. The case of N = 1 is studied botli analytically and numerically. Numerical work on N =3, 5, 9, 13 does lend support to the conjecture of the first mode dominance. 3 .3 T h e Iso la ted F u n d a m en ta l-M o d e Let X n and Un equal zero for n > 2. From the equations (3.20) and (3.21), dropping the subscript 1 on A'i and U\, we obtain X ' = - f i - ^ 2\ g aC i { p ) X U 2 - p - W a C i i r i X ' X U 2 , (3.22) Wa + C M X } ! / 1 = - ( l + T r 2)U + i i - ' C 4{ i i ) U - \ - l X ‘ , (3.23) where CM = f‘ C M = f Co0 0 = / (3.26) C M = [ <Pme~z/,idz . (3.27) , Jo Jo ^\<p\'p\e~2zl>ld z , Jo Jo Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.24) (3.25) 42 Eigenfunction Expansion for Porous M edium Combustion Supposing th a t the conjecture of the first mode dominance is valid, we may then consider th at X approximates the am plitude of the heat capacity of the solid minus the am plitude of the ash. U, on the other hand, approximates the am plitude of the solid tem perature. Thus, the tem poral evolution of u and x and an analysis regarding param eter dependence can be simply deduced from (3.22) and (3.23). From (3.22), XV* !? !' + a/i-l C2([i)XU2 ' ’ 1 implying th at the heat capacity of solid decreases as i increases. Substitute (3.28) into (3.23) and we have { f t * + ap~l C2{p )X U 2} K + Cz{p)X}U' = U[a(fi){\ + p T ^ a C M X U 2} + gaC M X U ] , (3.29) where a{jx) = - (1 + tt2) /4 • (3.30) Stationary points for U are obtained from the equation U[a{fi){\ + p ^ 2aC2( p ) X U 2} + gaC M X U ] = 0 , (3.31) which gives U{1) = 0, (3.32) ,.5 /2 (3-33) Calculating C4(fi) — we have a(p) < 0 1+v2*2); ft is not difficult to show th a t for p > 0 , , implying th e stationary points £/l2,3) are both positive provided Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 Eigenfunction Expansion for Porous M edium Combustion th at ^ ( i ) (3'34) or W ' (3'35) Let us examine the behaviour of U(t) for p > 0. 1 . Since (3.28) implies th a t X (t) is a strictly decreasing function of lim e, it is clear th a t if X (t) is sufficiently small, f/ 12'3) become complex conjugates and equation (3.31) has only the real solution U^K If X is given by - _ " W ( /i ) C 2( p ) /^MsaCM/*)}2’ (3‘36) Figure (3.1) show G(X, U) ^ vs V for different t, for X ( 0 ) > X , where G { X , U ) = {p2/z + a p - 1C2{ p ) X U 2}{<ra + C3{p)X } > 0 . (3.37) 2. If X (0) < X , U (t) has only one stationary value U = 0. Thus, no m atter how large (7(0) is, U(t) decreases to zero as t increases, indicating a diffusion type process. 3. For X (0) > X , let Ui(t) and UT(i), where Ui(t) < UT{i), denote £/*2 ,3 )(f). Sup pose we start w ith £/(0) such th at £//(0) < 17(0) < f/r (0 ), then ^ ( 0 ) > 0, and U(t) increases toward UT(t). Meanwhile -?f(i) is decreasing until at some t , say t = t0y X (t0) = X , and Ui(to) and Ur(to) coalesce. For t > tQ, is the only stationary point, and so U(t) decreases to zero as t increases. This procccss Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 Eigenfunction Expansion for Porous Medium Combustion I i It U 0 5 1C 15 25 U 5 10 u 15 25 Figure 3.1: G (X , U ) ^ vs U for different t, with X (0) > X . is typical of ignition, where the tem perature of the solid starts to increase to attain a. maximum value and then decreases to zero due to the depletion of the medium. 4. For X (0) > X , if t/{ 0 ) < £/j(0), then U{t) is monotonically decreasing to zero; if 1/(0) > t/r (0), U (t) decreases to !/,-(/) while Ur{t) decreases and Ui(t) increases. After some tim e t, say t > ti, UT(t) and Ui[t) disappear and U(t) decreases to zero. In the first case, the solid tem perature is too low to start an ignition for a combustion to take place. In the latter, th e solid tem perature is already high enough th at the burning of the medium does not have a boosting effect on the tem perature. Both processes are of the diffusion type. From the above observation, it is clear th a t a necessary condition for ignition is 4ao?{ii)C2{ii) m > Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.38) 45 Eigenfunction Expansion for Porous M edium Combustion It can be said th a t for an ignition to take place, the amount of combustible medium should exceed a certain am ount which depends on some param eters. We may calculate ct(fi), and Ciip) to obtain / \ 1/2 r 2 l i 4 7T2( l + “ O') = <■’' { C M = + u ( l + U2 7T2 ) ’-(l+ S M , "( 1 + /» ,» ) » (3.39) £ jtV 7 ( 1 - f 1* ) (1 + irV )3 (1 32irV 7(l + « - ' / “) /i ‘ ( f t ’1 + 3) + i r V ) 3(l + 9ir7 /i7) 2 (1 + it 7,.7 ) 7 ’ l ) and Ciip) = [ ip\<p\ip\e~2zllldz Jo 4y/2n3p 7 r (1 — e "2^ ) (1 + ir2p 2) 2 4(1 + 7r2p 2) 32v/27rV (l + e -1/M) ( 1 + 7TV ) 3 ( l + 9 ttV 2 ) 4(1 e_2^ ‘) . (4 + 7r2/r )(4 + 9tt2//2) ^ 1 6 \/2 7 r V (l - e"1^ ) ( 1 + 7T2/X2 ) 2 ( 1 + 3 1 6 tT2 ^ 2 ) ' 1 + 47T2 / ! 2 ' + . 3 S/i2 4v/2//'1 7 r ( l -j-TT2/ / 2 ) 2 8 15/i 7r2 * ^ 8 ) Even though the expression for ct(p), C i(/i), and Ci(p) is rather cumbersome, we can readily obtain the following approximation. For p « 1 , we have ct{p) = - 7r 2 /x3 / 2 •}- 0 { p 9/2), (3.42) CiOO= fz*2+ 0 (A (3-«) = H For p » 1 ^ + 0{ii<). (3.44) , we obtain « (/0 = “ (! + * V /2 + 0 ( p 1/2), (3.45) CM « jlj , (3.46) CM ~ | f (3.47) ■ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous M edium Combustion 46 Thus the condition (3.38) can be approximated by * ( ° ) > ~TjT~2 i for A* « 1, (3-48) and x m > (3.49) 9a Expressions (3.48) and (3.49) give approxim ate conditions regarding to the amount of the combustible medium for the ignition to take place. The following results are obtained by taking a = 0.001, ga = 1.0, A = 1.0, oa — 0.001, and X (0) = 20y/2fv. Table (2.1) shows £f,(0), (7r (0), and X = j y 2° g g f g j 2 , for different values of p. For p = 0.1, Figure 3.2 shows U(t) for £/(0) = 2.0,8.0, and 20.0. In Figure 3.3, we magnify U(t) with 17(0) = 8.0 for a small tim e interval after t = 0. From Table 1, we see th a t for p = 0.1, 2.0 < (7*(0), (7j(0) < 8.0 < (7r (0), and 20.0 > I7r (0). Thus for (7(0) = 2.0, u(z, i) decreases monotonically to zero as t increases. For (7(0) = 8.0, u(z, t) increases toward its maximum and then decreases to zero as t increases. Finally for U(0) = 20.0, u(z, t) decreases monotonically to zero as t increases. In Figure 3.4, we present the graphs of X (i) for those values of U(0) = 2.0,8.0, and 20.0. It shows th a t a bigger (7(0) gives rise to a smaller lim ^oo X ( t ) , suggesting more complete burning of the medium. Finally, we note th a t th e existence of U and its correct lim iting behaviour as t —►oo hold for all p > 0 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous Medium Combustion 47 X p 0.1 W ) 3.217676 MO) 16.530144 4.911745 0.2 2.808179 33.551849 2.566539 0.3 3.093760 48.851337 2.017108 0.4 3.567548 63.556698 1.812289 0.5 4.141251 77.963486 1.724810 0.6 4.784901 92.188492 1.689272 0.7 5.484778 106.288223 1.680449 0.8 6.233309 120.293671 1.686748 0.9 7.025729 134.223526 1.702170 1.0 7.858753 148.089752 1.723341 2.0 18.056311 284.350067 2.021881 3.0 31.078306 417.475647 2.322278 4.0 46.497520 548.121887 2.595876 5.0 64.074455 676.576660 2.845966 6.0 83.650375 803.015564 3.076996 7.0 105.111572 927.558899 3.292484 8.0 128.372925 1050.296265 3.495076 9.0 153.368866 1171.294922 3.686780 10.0 180.047897 1290.607910 3.869149 Table 3.1: C/j(0), f/r (0), and X for different values of p. 3 .4 A n A n a ly sis on th e S te a d y S ta te S o lu tio n Let Un = /im t_.0 0 C/n(i) and denote a\n\ p ) = /J <pntpie~z^ d z . Since X'n(t) and U'n(t) —> 0 as t —►oc, we obtain from (3.21) + (3.50) Thus the system governing the am plitude of the steady state solution u contains a single param eter p. An im m ediate question arises whether u is a function of p or Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 Eigenfunction Expansion for Porous M edium Combustion not. In this section we show th a t indeed lim {^ 0 O« ( 2 ,t ) = 0, independently of the param eter p. Thus, we shall show th at Un(t), n > 1, tends to zero as t —►oo, independently of the param eter p. Calculating a,•"*(/*) — Jo <Pn<Pie~z^Mdz we obtain 1-f-it?ir2n2 if n is odd and l+^ir2!-* P n (!+/**&) + ( i + i f e 5) if n is odd and n = * if n is even and n ^• i 2tii ir, ( l ~ e ~ 1’,‘)7i 2 22 l+**2 ir2 n 2 l+ ir iP i* (i+lA«»»i + ( 1 + I . W (3.51) if ” is even and n = > ) Let tin , ) - f M I - | * * * £ » > « (3.52) ” i s ° dd if n i s e v e n (3.53) and (3.54) , ( " 1' 0 = 1 + , . W In term of these functions / , h, and g, the equation (3.50) can be w ritten as p (ll + l)t/„ = I M E if(n ,ii)h (i,[ i)U i — g (n ,ii)U n , (3.55) . (3.56) + 1) - J ( » ,0 ) } 0 . = Dividing both sides by f ( n , p), we obtain M + 1) z £ n ^ ) l u n = K;h{i,p)Ui . / ( n 5^) (3.57) The left hand-side of (3.15) is a function of n and p while the right-hand side is a function of p. We thus obtain (3.58) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 Eigenfunction Expansion for Porous Medium Combustion where for brevity, C(p) is used to denote the sum. Since f ( n , p ) > 0 for any p > 0 and n > 1, we may define F M = . (3 .5 9 ) It is not difficult to see th a t F( n, p) > 0 for p > 0. From (3.58) we obtain C{p) = F( n ,p )U n . (3.60) Since C(p) is independent of n, we can then write C(p) = F ( l , p ) U x = F(2,p)U2 - (3.61) Considering th a t F ( n , p ) has different expressions when n is even and odd, we may thus write u« = { for p > I .r . lf n 1SeVen> p - 62> 0. Weconclude th a t if both U\ (t) and t/2(t) tend to 0 as i tends to oo, so does Un(t) for any n > 1. From (3.57) we have F ( n , p ) U n = E i h ( i ,p ) U i . (3.63) Substituting (3.62) for each {/,• in the right hand side of the above equation, we obtain for p > 0, = + Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 3 -6 4 ) Eigenfunction Expansion for Porous Medium Combustion 50 Using the notations (3.66) and (3.67) we obtain /F (l,ri-A Q i)F (l,ri -B(p)F(2,ft) \-A (r)F (l,ii) F(2,fi) — BF (2, fi) = 0 (3.68) Let S be the 2 x 2 m atrix in equation (3.68), then (3.69) D e ts ip ) = F { l , p ) F { 2 , p ) { l - A( p) - B { p ) } . If D ets( p) ^ 0 then U\ = O2 = 0 and so Iim<_o0 (Jn(t) — 0 for any n > 1. It is not difficult to compute th a t, for p > 0, Det$(p) is a decreasing function of p and lim^-Mjo Dets(fi) = 0. Thus D et s(p) > 0, for p > 0, giving limi_ 0o Un(t) = 0 for any As p —►0, it follows from (3.51) th at (3.70) and from (3.55), (3.71) and hence, by taking the lim it as p —►0, we obtain 7„f/n = 0, giving Un = 0 for n > 1. We, thus, conclude th a t limt_oo u(z^t) = 0, independently of param eter p. For th e truncated multi-mode system K =-ii-v'xg*E E EXiVkv, f' j=i k=1 1=1 -p~5/2a'£i'£Y,HX' iX3Ui‘U‘J0I i i=l Jk=l =1 , 1=1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ( 3.72) Eigenfunction Expansion for Porous M edium Combustion 51 .=i j=i ~1&Un + P- 1 XI i= l / Vi¥»«c” */ '‘rfz ~ ~ A_iX^ , (3.73) */ 0 where JV > 1, equation (3.55) becomes = (3.74) Following the analysis for the general system, for /x > 0, we obtain for the trun cated multi-mode system f (i , / 0 - a 6 o f ( i ,<0 - £ f ( 2 , / 0 -/tW F ( l,r t W F ( 2 ,r t- /? ( V ) F ( 2 ,r t I P. = 0, (3.75) l0 i where { y 'tf SfaK r m + S ) jr • jj ‘fN is even and B(- ) = / S g , , ^ ifM iso rid if N is e v e n ’ K = { N - l)/2 , L = N / 2 - 1, and M = JV/2. If is the 2 x 2 m atrix in the equation (3.75), then D e t s M = F ( l t p ) F { 2 ,p ) { l - A(p) - B ( p )} . (3.78) For p > 0, th e D e t s ^ p ) > 0, for any N. Thus Un(t) tends to 0 as t tends to infinity, for any n > 1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 Eigenfunction Expansion for Porous M edium Combustion 3.5 N u m erica l R e su lts for th e T ru n ca ted M u ltiM o d e S y stem s In this section we calculate numerical solutions of the truncated m ulti-mode systems to support the conjecture of first-mode dominance. We consider the truncated multimode systems as follows N K N N I E = - f i - 3/2\ g a £ XjVkUi vm<piv>ne~2*/,ldz j=1k=l i=l J0 i=i j=ik=ii-1 N + - 7 iUkU, [ $wjipwi<pne~2zf>idz , J0 (3.79) N EE «=i j~ i <pi<pj<pndz = l u n + /!_! £ Ui [ 1 W n e - ^ d z - U n ,= 1 , (3.80) -to where W = 3, 5, and 9. Since the computing tim e increases rapidly by increasing N , the com putation for N = 13 is carried out only in a few cases. These results are then compared with the results for N = 1 to show th at the qualitative behaviour of solution is captured by the first eigen-mode. The algorithm of computation is the following. 1. Let X ' and U' be 1 X [ (t) ' X ’(t) X it® ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.81) Eigenfunction Expansion for Porous M edium Combustion 53 and ‘ £W ) ' v m (3.82) \ um! respectively. Then X ' and U' are the solutions of X ' = A{X ,U )X' + b{X ,U ), (3.83) U' = C ( X , U)U' + d{ X, U) , (3.84) for some NxN m atrixes A and C and vectors b and d in which each of their elements is a function of X ( t ) and U(t), where * (() = ( M i) \ X 2(t) (3.85) \ X N(t) and ( W) \ U{i) = W ) (3.86) \ Unit) ) Given the values of X (t,) and U(U), - ^ ( ti) and Uf(ti) can be found by solving (3.83) and (3.84). 2. Having X '( tj ) and U‘(ti), we calculate X (f,-+ t) and U{U+j), where U+i = U+Ati, using R unge-K utta of order 4. 3. We repeat step 1 and then step 2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous Medium Combustion 54 In the following computation we have taken a = 0.001, ga = 1.0, A = 1.0, and oa = 0.001. The initial value is taken as x(*>0) = 10. By expanding x in th e Sine Fourier series, we obtain its ith coefficient A,(0) = We use these values of A',-(0) in the computation. We also take u(z, 0) = U(0)tpi(z). Figures 3.5 and 3.6 show u(z, t) where p = 0.1 and N = 1, 3, 5, and 9, for 17(0) = 2.0, and 8.0, respectively. Note th a t we use different scales on t for each (7(0) to get a clear comparison. We can see th a t for (7(0) = 2.0, u ( z ,t ) is monotonically decreasing to zero as t increases N — 1, 3, 5, and 9. This phenomenon was as predicted by the isolated fundam ental mode solution since 0.2 < (7/(0). For (7(0) = 8.0, on the other hand, u(z, t ) is monotonically increasing to its m axim um value and then monotonically decreasing to zero as t increases. This was again predicted by the isolated fundamental mode solution since (7(0) = 8.0 is in between (7/(0) and (7r (0). W ith (7(0) — 4.0, Figures 3.7 and 3.8 show u ( z , t ) for N = 1, 3, ,5, and 9, where p = 0.2 and 0.4, respectively. From Table 3.1 we see th a t for either p = 0.2 or p = 0.4, Ki(0) < 4.0 < (7r (0). For this case all those figures show th a t u (z ,t) is monotonically increasing toward its maximum and then monotonically decreasing to zero, as predicted by the isolated fundamental mode solution. For p = 0.1 and (7(0) = 4.0, we compare u (0 .5 ,t) and x (0 .5 ,i) for N = 1, 3, 5, 9, and 13. Figures 3.9 present ti(0.5,<) on the interval 0 < t < 0.65. It is shown th a t for N =9 and 13 the graphs of u(0.5,i) are really close to each other. W hile there is a quantitative difference with the graph for N — 1, they share the same feature th a t u is monotonically increasing toward its maximum and then monotonically decreasing to zero. The sam e feature is also shared by th e graphs of x(0.5, t) for N = l, 3, 5, 9, and 13 presented in Figure 3.10. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous Medium Combustion 55 We repeat some of the above computations using the initial condition u (r, 0) = 8.0 sin2 2irz, which is distinctly different from the first eigenfuction. Figure 3.11 shows th a t except in the tim e interval 0 < i < 0.15, the first eigenmode again captures the essential behaviour of the solution as given by the nine-mode approximation. 3.6 C on clu d in g R em ark s T he solution by eigenfunction expansion for the combustion of a porous slab occupying the region 0 < z < 1 is considered. By a heuristic argum ent, it was believed that the behaviour of u { z , t) can be deduced from the first term of its Fourier series. (See [4]). A bundant numerical results show th a t, indeed, this conjecture is justified. The features such as ignition and tem poral solution are adequately approximated by the single mode. An analysis of the p. param eter dependence of the steady state solution is also considered. with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous Medium Combustion © . IA ■ 1.0 0.5 0.0 1.5 Figure 3.2: U(t) as a function of i for £7(0) = 2.0, 8.0, and 20.0 with ft = 0. © . 0.0 0.02 0 .0 4 0 .0 6 0 .0 8 0.10 t Figure 3.3: t/(<) as a function of t for U(0) = 8.0 with p = 0.1. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous Medium Combustion to CSJ o 0.0 0.S t 1.0 1.5 Figure 3.4: X ( t ) as a function of t for C/(0) = 2.0, 8.0, and 20.0 with p = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous M edium Combustion Figure 3.5: u ( z , t ) with p — 0.1 and ?7(0) = 2.0 for N = l, 3, 5, and 9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 Eigenfunction Expansion for Porous M edium Combustion N =5 N = 9 Figure 3.6: u { z , t ) with fi = 0.1 and f/(0) = 8.0 for N = l, 3, 5, and 9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous M edium Combustion Figure 3.7: u ( z , t ) with p. = 0.2 and £/(0) = 4.0 for N = l, 3, 5, and 9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 60 Eigenfunction Expansion for Porous M edium Combustion Figure 3.8: u ( z , t ) with p = 0.4 and £/(0) = 4.0 for N = l, 3, 5, and 9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 u» s’« cv 0.0 0.1 0.2 0 .4 0 .3 0 .5 0.6 t Figure 3.9: u(0.5, i) with t/(0 ) = 4.0 and ft = 0.1 for N = l, 3, 5, 9, and 13. N .1 N*3 N -S N -» N .1 J (V o■ 0.0 0.1 0.2 0 .3 0 .4 0 .5 0.6 t Figure 3.10: x(0.5,<) with £/(0) = 4.0 and fi — 0.1 for N = l, 3, 5, 9, and 13. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eigenfunction Expansion for Porous M edium Combustion 0 < t < 1.0 0.1 < t < 1.0 63 0.15 < t < 1.0 N = 1 0 < t < 1.0 0.1 < t < 1.0 0.15 < t < 1.0 N = 3 0 < t < 1.0 0.1 < t < 1.0 0.15 < t < 1.0 N = 9 " Figure 3.11: u ( z , t ) with u (z,0 ) = 8.0sin227rz and p. = 0.2 for N = l, 3 and 9. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 4 Porous M edium C om bustion: C om putation o f Traveling W ave Solutions 4.1 In tro d u ctio n In this chapter, we study porous medium combustion modelled as traveling com bustion waves. We wish to introduce a new technique to com pute traveling wave solutions. We consider a one-dimensional, tim e-dependent model for porous medium combustion proposed by Norbury and Stuart in [10]. The dependent variables in this model are the heat capacity of th e solid, the solid tem perature, and the gas tem pe rature w ritten as a , u and to, respectively. Let t be the tim e variable and z be the space variable. The governing equations are | = -A r, du d 2u aTt = d ? + m - u + r ' 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.1) . . ( 4 ' 2 ) Traveling Wave Solution for Porous M edium Combustion ^ dw 65 = u-w , (4.3) with r = /it/2i/(cr - < tq)H{ u - uc) f ( w ) , (4.4) where H(£) = 1 if £ > 0 and H(£) = 0 otherwise. The function f { w ) is usually taken to be proportional to w2 . The param eter ft is proportional to inlet gas velocity while the param eter A is linearly related to the specific heat of the combustible solid. <7o and uc denote the critical switching param eters related to the burning zone, that is a region in z —plane where r > 0 . The param eter <7o is the solid heat capacity of the burnt medium and uc is the threshold tem perature for the reaction to start. 0 < Co < 1 . T he boundary conditions are ti(± o o ,t) = u0 < < uc , (4.5) w (± o o ,t) = wa , (4.6) 0 < cr(z, oo) = < tq < 1 , (4.7) cr(z, -o o ) = 1 . (4.8) Norbury and Stuart sought a traveling wave solution for the model in term of x = z — ct , where c is the wave speed. W ith this transform ation of independent variables, the equations (4.1), (4.2), and (4.3) become c ^ -A r ^ = °, (4.9) + c c r ~ + iw - tt + r = 0 , (4.10) dw n li-j- + w —u = 0 . dx /. . . , (4-11) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous Medium Combustion 66 T he boundary conditions are u (± o o ) = ua , (4-12) u>(±oo) = wa , (4-13) 0 < er(—oo) = a 0 < 1 , (4-14) <7 ( 0 0 ) = 1 . (4-15) Assuming th a t u > uc on a finite intervals (—£ , L ) for some L (to be determined later) and u < uc elsewhere, Norbury and Stuart divided the interval (—0 0 , 0 0 ) into three intervals (—0 0 , —£], (—L, L) , and [ £ , 0 0 ) . On both (—0 0 , —L] and [ £ , 0 0 ) the system of ordinary differential equations (ODEs) (4.9), (4.10), and (4.11) is reduced to a linear system of ODEs. Thus, th e problem is reduced to a two-point free boundary problem over a finite domain. The position L is to be determ ined from th e m atching of the solutions on ( —£ ,£ ) and the solutions on both (—0 0 , —£] and [ £ , 0 0 ). Norbury and Stuart in [10] proved the existence of solutions for th e above two-point free boundary problem over a finite interval using local bifurcation theory. A modification of the model for porous medium combustion above was considered by Tam [12]. Instead of using r as it is in (4.4), he defined r = f p l 2H ( u ) w 2. To have homogeneous boundary conditions on u and w, let u = u — ua and w = w — wa . Dropping the ’tilde’ we still have the same system of ODEs (4.9), (4.10), and (4.11) with boundary conditions u(± o o ) = 0 and io(±oo) = 0 . In [12], Tam gave an existence proof of the above system by using a two-sided shooting m ethod. The proof is based on an a priori bound, u m , of th e m aximum of u. Tam divided the infinite interval (—0 0 , 0 0 ) into two semi infinite intervals (—0 0 ,0] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous Medium Combustion 67 and (0, oo) where x = 0 is chosen to be th e point where ^f(O) — 0 . Tam proved th a t there is a set of values ofs u(0) > 0 and u>(0) > 0 such th a t equations (4.10) and (4.11) have a solution satisfying the lim iting behaviour a t —oo . It was proved then, th a t such solutions can be extended to the right to satisfy the boundary condition at oo . It should we noted th a t the results in this Chaper are presented in [11] by Tam and Andonowati. 4.2 P r elim in a ry R e su lts We begin by recapitulating some results from [12] and considering the asym ptotic behaviour of the solution. In what follows let ” ' ” denote differentiation with respect to x. Let x = 0 be chosen such th a t u'(0) = 0. Based on the proof th a t a solution for equations (4.9), (4.10), and (4.11) exists for some io(0) > 0 and u(0) > 0, the following results can be found in [12]. L e m m a 1 te(0) < u(0) and u'(x) > 0 fo r —oo < x < 0. L e m m a 2 The wave speed c satisfies f < c < for o0 < a < 1. L em m a 3 Let te(0) < 2/x1/2, (4.16) arid 1/A < u(0) < 2ji1^2( l + p ) . (4.17) = co. Equations (4.9), (4.10), and (4.11) become (4.18) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous M edium Combustion u" + 0 u ' + uj —u + r = 68 0, (4-19) fiw' 4- w —u = 0. (4.20) The boundary conditions become u(±oo) = 0 , (4-21) w(±oo) = 0 , (4.22) 0 < 0 ( —oo) = c<T0 , (4.23) 0(oo) = c . (4.24) Integrating (4.19) from —oo to 0 and using equations (4.18) and(4.20) we have f = H kp . J —o o + ^(0) - (4.25) M A A Since from (4.18) 0 is increasing while from Lem m a(l) v! > 0 on (—oo, 0), we obtain ■0(—oo) f u'dx < I J —o o ijju'dx < 0(0) f «/-*o o u'dx . (4.26) • / —oo Substituting to (4.25) we have A/xtn(0) < 0(0) < A/zto(0) — A0(—oo)[u(0) — 1/A ]. Au(0) + 1 (4.27) Using u(0) > 1/A we find < *(0) < Since 0 < tn(0) < \„ w (0). (4.28) the above result gives 0 < 0(0) < 2A/i 3/2 . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (4.29) Traveling Wave Solution for Porous M edium Combustion 69 We further note th at the system of differential equations (4.9), (4.10), and (4.11) above has two critical points ( a ,u ,u ', w ) = (1,0,0,0) and (<r0 , 0,0,0) . If we linearize equations (4.9), (4.10), and (4.11) at x = —oo, we have v!" -j- {co0fi + l)u " + (ccr0 - /i)u' = (4.30) 0 This gives characteristic values <*1,2 a3 = --^(c<T0/l + 1) ± ^ ( s/icOQfl+ l ) 2- 4/1(070 ~ (>■)) (4.31) =0 (4.32) In order to satisfy the the boundary condition of u at x = —oo, at least one of the characteristic values m ust be greater than zero. This gives ccr0 — ft < 0 or c < /i/er0. Since of one the characteristic values is negative, the critical point (<t0, 0,0,0) is unstable. Similarly, we may linearize equations (4.9), (4.10), and (4.11) at x = oo to have v!" + (c/i + 1)u" + (c - fi)u' = 0, (4.33) which gives characteristic values 71,2 = 73 = ~ ^ Cfi + ^ ~~ 4/^ C “ ^ 0 (4.35) Since it was proved th at c > | , where <Tq < d < the characteristic values + 7 1 ,2 1 , we thus have c > /i , implying are both negative. This shows th at the critical point ( 1 , 0 , 0 , 0 ) is stable. with permission of the copyright owner. Further reproduction prohibited without permission. 70 Traveling Wave Solution for Porous M edium Combustion 4.3 A lg o rith m o f C o m p u ta tio n We summarize the previous section as follows. 1. Let x = 0 is chosen to be a point where v! = 0, then Au(0) + 1 0 < iu(0) < «(0) , (4.36) u>(0) < 2/i1/2 , (4.37) 1/A < u(0) < 2//1/2(l + \l ) , (4.38) < t/>(0) < A/ru;(0) , and so0 < ij)(0) < 2A/z3/2 . (4.39) 2. The critical point (a0, 0,0,0) is unstable and (1 ,0 ,0 ,0 ) is stable. A lg o rith m . First we fix A and n . Since 0 < ^/>(0) < 2Afi3^2 , we fix i/>(0) in th a t range. We then allow the values u(0) and w(0) to change within the bounds 1/A < u(0) < 2/xx^2(l + n) and 0 < w(0) < 2fj}/2 such th at iu(0) < n(0) and (4.28) is satisfied. Knowing th a t the critical point at x = —oo is unstable, we integrate to the left first. We choose u(0) and zw(0) such th at the integration can go to a reasonable distance, say to —L, where u{—L) and w ( —L) are reasonably close to zero. We obtain a set S i of values of (u(0),tn(0)) as a result of this search. According to the existence proof in [12], there is a subset of say S']?, such th a t the solution can be extended to the right to satisfy the boundary condition at x = oo . Our objective is to find th at subset S £ . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous M edium Combustion 4.4 71 N u m e r ic a l R e su lts Since <t(x) —►1 as x —►oo , we have ij>(x) = cct[ x ) —► c as x —>oo. (4.40) and thus we obtain the wave speed c from the integration to the right. Furtherm ore ij;(x) = co(x) —> c<Tq as x —v —oo. (4.41) Suppose th a t we are able to integrate both to the left and to the right to reasonably long distances —L and M , respectively, such th at w ( —L), u(—L ) and u;(A/), u (M ) are reasonably close to zero, then we obtain c« ^ > (M ), (4.42) <t0 * (4.43) Numerical com putation shows th a t for ft = 3.5, A = 1.0, and ^(0 ) = 2.5, with tw(0) = 1.08960011 and u(0) = 2.41011122, we are able to integrate to the left to a distance —L = 12.2, where ^>(—12.2) = 0.19702243, u>(—12.2) = 0.02288569, and « (-1 2 .2 ) = 0.02303857. We accept the values to(—12.2) and u (—12.2) as reasonably close to zero. T he integration to the right shows th a t as x — ►oo, ip(x) —» 35.48 , w ith w(0) and tt(0) < 0.02 which, again, is reasonably close to 0. Thus, we calculate c = 0(oo) « 35.48 and so <r0 « « 0.0055531, a(0) « ^ « 0.0704622 [See Tables(4.1) and (4.2)]. The related graph is shown in Figure (4.1). For fi = 3.5, A = 1.0, and t/>(0) = 2.5, Figures (4.2), (4.3), and (4.4) show how a small difference in the initial values of tn(0) and u(0) produces a totally different Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 Traveling Wave Solution for Porous M edium Combustion result when we integrate the equations (4.18), (4.19), and (4.20) to th e left. This result is due to the stiffness of the problem. X w u 0.000 2.50000000 1.08960011 2.41011122 -0.500 1.57640547 0.89245308 2.29659511 -1.000 0.98406208 0.69769115 1.99387898 -1.500 0.63385046 0.52728728 1.60670353 -2.000 0.43793539 0.39030959 1.23004233 -2.500 0.33170634 0.28608103 0.91196477 -3.000 0.27476942 0.20944742 0.66374435 -3.500 0.24411638 0.15428830 0.47867709 -4.000 0.22730903 0.11510651 0.34439058 -4.500 0.21780320 0.08750062 0.24855391 -5.000 0.21219204 0.06814639 0.18086225 -5.500 0.20869978 0.05461408 0.13336123 -6.000 0.20639100 0.04516216 0.10016396 -6.500 0.20476388 0.03855755 0.07701863 -7.000 0.20354255 0.03393402 0.06089979 -7.500 0.20257093 0.03068623 0.04967489 -8.000 0.20175792 0.02839273 0.04184991 -8.500 0.20104870 0.02676093 0.03638213 -9.000 0.20040928 0.02558827 0.03254511 -9.500 0.19981804 0.02473511 0.02983229 -10.000 0.19926088 0.G2410611 0.02788821 -10.500 0.19872830 0.02363773 0.02645858 -11.000 0.19821368 0.02329055 0.02535220 -11.500 0.19771194 0.02304579 0.02440830 -12.000 0.19721855 0.02290626 0.02346233 -12.100 0.19712045 0.02289310 0.02325563 -12.200 0.19702243 0.02288569 0.02303857 Table 4.1: ^ ( s ) , u^*), aQd u (*) for M = 3.5, A = 1.0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous M edium Combustion $ X 73 Ul u 2.41011122 0 .0 0 0 2.50000000 1.08960011 0.500 3.80113963 1.26060574 2.30967912 1 .0 0 0 5.45015364 1.38538716 2.07979811 1.500 7.35470585 1.46013558 1,82103348 2.000 9.40114156 1.49156724 1.58209021 2.500 11.48563832 1.48914030 1.37493300 3.000 13.52673141 1.46161933 1.19898802 3.500 15.46701103 1.41625854 1.05051705 4.000 17.27025904 1.35880507 0.92536034 4.500 18.91713761 1.29370814 0.81972430 5.000 20.40085537 1.22435655 0.73036378 10.000 28.20497556 0.61868158 0.31509818 15.000 30.19381467 0.33223720 0.21179856 20.000 30.87758591 0.22234345 0.17659515 25.000 31.24447245 0.17915679 0.15985192 30.000 31.50922196 0.15927116 0.14917242 35.000 31.72846791 0.14764305 0.14094052 35.000 31.72846791 0.14764305 0.14094052 40.000 31.92051769 0.13922902 0.13397426 45.000 32.09276264 0.13231805 0.12783257 50.000 32.24904550 0.12629285 0.12231567 100.000 33.28818793 0.08833332 0.08652895 Table 4.2: i/>(x), w(x), and u(z) for fj = 3.5, A = 1.0. c <r(0) o0 5.0 10.44 0.2394636 0.1057076 4.0 20.41 0.1224890 0.0333605 3.5 35.48 0.0704622 0.0055052 Table 4.3: c, <7(0), and ffo for different values of /<, with ^(0) = 2.5, A = 1.0. Table 4.3 shows some approxim ate values of c, <r(0), and < tq for different values of p , where A = 1 and ^(0) = 2.5 . It is shown th a t there is a substantial dependence of Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 Traveling Wave Solution for Porous M edium Combustion co n p : c increases as decreases with fi fi decreases. It is also shown th a t the term inal heat capacity er0 . It should be noted th a t the param eter f i measures the completeness of the burning of the solid. T he more complete the burning, the smaller the param eter fi. Thus, it can be concluded th a t the more complete the burning the higher the wave speed and the smaller o0 . Figures(4.5), (4.6), and (4.7) show x ), tw(x), and u(x) for different values of /1, with A = 1 and ^(0 ) = 2.5 . Note th a t c is calculated from 0 (x ) as x —* oo. c (7(0) (70 2.5 35.48 0.0704622 0.0C55052 2.4 30.09 0.0797607 0.0038503 2.3 24.49 0.0939159 0.0029288 2.2 20.88 0.1053640 0.0001189 2.1 17.76 0.1153640 0.0000796 m Table 4.4: c, <7(0), and <To for different values of ^(0), with fi = 3.5, A = 1.0. In Table 4.4 we calculate th e approxim ate values of the wave speed c for different values of t/>(0) = co-(O) , where A = 1 and fi — 3.5 . Having c, we calculate the heat capacity of solid u at the point where the solid tem perature u attains its maximum, th a t is, a t x — 0 . Furthermore, from th e integration to the left and the obtained value of c we may calculate the term inal heat capacity of the m aterial <7o . It shows, again, th a t the higher the wave speed c, the smaller the term inal heat capacity cr0. Figure (4.8) depicts for different values of tp(0) as shown in Table(4.4). The related graphs for w (x) and u (x ) can be seen in Figures (4.9) and (4.10). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous Medium Combustion 4.5 75 C on clu d in g R em ark s We have considered a procedure of computation for a system of differential equations defined on an infinite interval. This computation procedure is based on an existence proof using a two-sided shooting method. Out of the unknown quantities i>{x), u(x), and tw(x) we elim inate ij>{x). Choosing i = 0 as a point such th a t it'(O) = 0, the com putation procedure is then developed to find (u(0),u>(0)) over a rectangle such th a t the numerical solution meets the conditions at x = ±oo. Since the system of equations is unstable as x —* —oo the integration to the left prevents us from going to a long distance. However, the integration to the right is stable. For a set of values fi and A, numerical results strongly suggest th at there is a unique solution to the problem. It is found th a t there is a limit of gas velocity f1, below which no numerical solution can be constructed. Further, numerical results show th a t there is a substantial dependence of the wave speed c on the param eter fi, c increases as fi decreases. Since fi measures the completeness of the burning of the solid, in which the more complete the burning the smaller the param eter fi, it can be concluded th a t the more complete the burning the higher the wave speed. Furtherm ore, it is found th at the higher the wave speed c the smaller the terminal heat capacity c tq . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous Medium Combustion Ho 3 3 15 - lO 20 M Figure 4.1: w (x), and u(:r) for fi = 3.5, A = 1.0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous Medium Combustion 77 3 a - n (0 ) s 2,41011123 i(0 ) - 2.41011123 O •IB H Figure 4.2: for two different values of u(0): u(0) = 2.41011122 and u(0) = 2.41011123 w ith fi = 3.5, A = 1.0, t£(0) = 2.5, tn(0) = 1.08960011 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 Traveling Wave Solution for Porous M edium Combustion u (0 ) = 2.41011122 « 3| O u (0 ) = 2.41011123 •5 O x Figure 4.3: u?(a:) for two different values of u(0): u(0) = 2.41011122 and u(0) = 2.41011123 with fi = 3.5, A = 1.0, ^ (0 ) = 2.5, to(0) = 1.08960011 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous M edium Combustion 79 u ( 0 ) = 2.41011123 u (0 ) = 2.410111: ■B o Figure 4.4: u (z) for two different values of u(0): u(0) = 2.41011122 and u(0) = 2.41011123 w ith p = 3.5, A = 1.0, tf(0) = 2.5, u>(0) = 1.08960011 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave So/ution for Porous M edium Combustion 80 a - fi = 4.0 —1® 3 —© O 8 Figure 4.5: 0 (x ) for different values of ft: fi = 3.5 and ft = 4.0; A = 1.0, 0 (0) = 2.5. Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous Medium Combustion 81 m fi = 3.5 in -10 •5 0 5 10 15 20 Figure 4.6: io(z) for different values of fi: fi = 3.5 and fi = 4.0; A = 1.0, 1^(0) = 2.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous Medium Combustion CvJ o c si ■ CM in U) 3 3.5 i - 10 i - i S O • ■ ■ 5 10 15 20 X Figure 4.7: u(x) for different values of fi: fi = 3.5 and fi — 4.0; A = 1.0, ^>(0) = Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous M edium Combustion .2.3 a 2 Vo -10 -5 0 5 10 15 20 X Figure 4.8: 0 (x ) ior different values of ^>(0): ^(0 ) = 2.5, 2.4, 2.3, 2.2; A = 1.0, /x Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 Traveling Wave Solution for Porous Medium Combustion tn o tn o o o Figure 4.9: tu(a;) for different values of ^(0): ip(0) = 2.5, 2.4, 2.3, 2.2; A = 1.0, fi = 3.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Traveling Wave Solution for Porous M edium Combustion IA CM CM in o in O O o 2.5 2.4 2.3 - 2.2 Figure 4.10: u(x) for different values of V’(O): V’(O) = 2.5, 2.4, 2.3, 2.2; A = 1.0 3.5. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 5 A n A pplication o f th e Tw o-Sided Shooting M ethod in C om putation o f Traveling C om bustion W aves of a Solid M aterial 5.1 In tro d u ctio n We apply the numerical technique we developed in Chapter 4 to compute the trav eling combustion wave solution of a solid medium. The model we considered can be form ulated as follows % = dt dt V 2e + H x f { 6), (5.1) s x m , (5.2) = 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 Solid M edium Combustion where 9 and x are the tem perature and the concentration of th e combustible m aterial, x and t are independent variables for space and time. H is a positive num ber related to the chemical properties of the combustible material, the external tem perature, and the geometrical dimension of the medium. The param eter a , considerably bigger than one, is proportional to the activation energy of the medium and e = e-0r . The governing equations for a traveling wave for the above problem may be ob tained byletting x = x , —oo < x < oo and £ = x + cf, where c is the wave speed, equations (5.1) and (5.2) become 9" - cff + H Xf(0 ) = 0, (5.4) x" - ex' - t x m (5.5) = 0, where the prim e denotes the derivative with respect to £. The relevant boundary conditions to the problem are 0 (-o o ) = 0, 0(oo) = 9max, x ( - o o ) = 1, x(oo) = 0, (5.6) where 9max is the maximum tem perature reached after combustion, whose value has to be determined. It should be noted th at no solution to Equations (5.4) and (5.5) subject to the boundary conditions (5.6) is possible. This can be seen by taking the lim it on Equation (5.4) as £ —* —oo; the left hand side of this equation gives 1, while th e right hand side equals 0. In [21], Tam replaced the function f{9 ) by I 0 “ *-? otherwise . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.7) Solid M edium Combustion SS By assuming th a t ip = £0 + H x is bounded and satisfies the boundary conditions imposed on 0 and x >we have H = eO + H x and 0max = H /e . Thus, instead of using the Equations (5.4), (5.5), and the boundary conditions above, Tam in [21] considered 9" - c9' 4- {H - e0)g{6) = 0 (5.S) subject to boundary conditions 0 (-o o ) = 0, 0{oo) = H ie. (5.9) Tam, then, proved the existence of a solution using phase-plane m ethod. Previously, a num ber of authors proved the existence of a solution for such problems. Aronson and Weinberger [22], for example, gave a detailed proof of the existence of a solution for a more general function of F[9) such th at 0 satisfies the equation 0" - c9' + F{0) = 0 (5.10) subject to conditions 0(f) € [0,1], 0(f) £ 0 , and l i m ^ co0(f) = 0 . Also see [23] for a discussion of such problems. The com putation algorithm is based on a shooting m ethod as follows. Since the problem is invariant under translation we may choose the location of f = 0 to be such th a t 0(0) = H /e —6, for a small positive S. For each c, we derive an a priori bound for 0'(O). Assuming th a t the solution exists for some c, for those c, a set of values of 0'(O) such th a t the solution can be extended to the right to satisfy the boundary condition at f = oo is not empty. Furthermore, there m ust be a subset of such values such th a t the solution can extended to the left to satisfy the boundary condition at f = —oo. Indeed, Tam in [21] showed th a t the solution to (5.8) and (5.9) exists for c > y /id , where a — m a This m ethod of com putation was motivated by the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Solid M edium Combustion 89 existence proof in [12] and it was employed in [11] by Tam and Andonowati. The content of this section can be seen in [13]. In the next section we present the properties of solutions 0(f). We derive an a pri ori bound for 0#(O) in Section 5.3. The algorithm of com putation is then constructed. In Section 5.4 numerical results are obtained and concluding remarks are presented in the last section. 5.2 B eh aviou r o f th e S o lu tio n We note th a t the differential equation (5.8) above has two critical points (0,0') = (0,0) and (0,0') = (H /e , 0). By linearizing the equation near the critical points, we should have c2 > AH in order to satisfy the boundary condition at f = —oo . For this c2 > AH the critical point (0,0) is stable while the critical point (H /e , 0) is saddle. The following properties of 0 can be derived easily from (5.8) and (5.6) and by examining the direction field of the phase plane 0' v s 0. These properties of the solution 0 are to be used in constructing the algorithm of com putation in th e next section. P r o p e r ty 1 //'0 (f) is a solution o f Equation (5.8) with the boundary conditions (5.9) then 0 < 0(f) < H /e and 0(f) is monotonically increasing. P r o p e r ty 2 Let F (6(f)) = (H —e9(Q )g(9(£)), then F (0 (f)) has exactly one extreme which is a relative maximum, say, at f = f m. For f < f m, F (0 (f)) is monotonically increasing, and fo r f > fm, F (0 (f)) is monotonically decreasing. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Solid M edium Combustion 90 P r o p e r ty 3 0(£)) has exactly one inflection point, Ze, such that fo r Z < Ze, 0'(Z) is monotonically increasing and fo r Z > Zc Q'iZ) is monotonically decreasing. P r o p e r ty 4 I f Ze is the inflection point o f 6(Z) and is the point such that F(0(£m)) is maximum then Ze < Zm • P ro o f X and 2 The proof o f Property 1 follows directly from (5.8) and (5.9), while the proof o f Property 2 follows directly from the behaviour o f F. P ro o f 3 and 4 The rest o f the proof is obtained by examining the direction field of the phase plane O' vs 6 as follows. Let p = then dp _ f d d O d e ~ d z 2d z ' Since 0(Z) is monoionocally increasing, then ^ > 0, and so ^ has the same sign as . From (5.8), < 0 fo r 0 < p < F ( 6)/c and > 0 fo r p > F (0 )jc, and thus >0 i f 0 < p < F ( 6)(c <0 i f p > F (0 )jc { ‘ ; Now we consider Figure 5.1. Since the solution to (5.8) subject to (5.6) exists, there must be a trajectory connecting the critical point (0,0) and (H/e, 0) which obeys the direction field (5.12). Clearly 6(Z) has exactly one inflection point, Ze, such that fo r Z < Ze, 0'(Z) ls monotonically increasing and fo r Z > Ze, 8'(Z) Is monotonically decreasing, and Ze < Zm , where Zm is the point such that F(0(Zm)) *s maximum. 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 91 Solid M edium Combustion m p 9 = Hft 9 Figure 5.1: T he phase plane 0' vs 9. 5.3 T h e A lg o rith m o f C o m p u ta tio n For fixed numbers a and H , let 8 be a small positive num ber relative to 9max — H /e such th at H /e - 8 > 9m , where F (0m) = (H - eBm)g{9m) is m axim um . Such a 9m satisfies F'(0m) = 0 or (H - e9m){— ^ — )2e £ % = Qt + 9m - 1 ), 0 < 9 m < H / e . (5.13) We choose £ = 0 to be a point such th a t 0(0) = H / e — 8 > 9m (5*14) Given 0(0) = H / e — 8, an a priori bound for 0'(Q) can be obtained as follows. From the equation (5.8) 0'(O) = c + ~ { H - £0(O)}$(0(O)). c Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.15) Solid M edium Combustion 92 Using Properties 1 and 4, we have 0(0) > 0(fm) > 0(£e), where F(0(£m)) is maximum and £e is the inflection point of 0(£). From Property 3, we conclude th at 0"(O) < 0 (See Figure (5.2)). Thus, 0 < 0'(O) < ~ { H - e0(0)}g(0(0)). 0 _ _ , (5.16) afl(O ) Substituting 0(0) = H / e — 8 into (5.16) and knowing th a t £(0(0)) = —1 < e“ = 1/e, we have 0 < 0'(O) < ~ (5.17) Note th a t we choose f = 0 to be a point such th a t 0(0) is close to H/e. The reason for doing this is th at the critical point (H/e, 0) is unstable. Thus the integration to the right of Equation (5.8), using Runge-K utta of order four, will accumulate a high truncation error for a reasonably long distance of £ from the initial point. The critical point (0,0), on the other hand, is stable and so the truncation error of the integration to the left will not be m ultiplied at each step of the integration. The com putation algorithm is as follows. Since the solution of (5.8) exists only for some c such th a t c > y / i H , we start with some c, with c > y /iH . For a fixed c we calculate Considering th a t the critical point (H/e, 0) is unstable and and (0,0) is stable, we integrate Equation (5.8) to the right w ith 0(0) = H / e — 8 and a fix 0'(O) € (0, ^). We then allow the value of 0'(O) to change within the bounds 0 < 0'(O) < | , until we find th a t the integration to the right can go to a reasonable distance, say L\, such th a t 0(£) is monotonically increasing to H / e and 0'(£) is monotonically decreasing to 0. We change c and repeat the same procedure to find the corresponding *'(0). Thus for each a , we obtain a set Aq = {(c,0'(O)) | 0(0) = H / e —8 such th a t the integration of Equation (5.8) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 93 Solid Medium Combustion to the right can go far enough where 0(f) is monotonically increasing to Hj e and 0'(f) is monotonically decreasing to 0}. (5.18) For each (c, 0'(O)) € AQ, we integrate Equation (5.8) to the left to verify th a t 0(f) tends to zero. 5.4 N u m erica l R e su lts We dem onstrate the above algorithm with a = 10 and H = 1.0. Calculating 8max = H je = H * e“ and 0m from (5.13) we obtain 9max — 22026.46579481 and 0m = 1425.31026371. By choosing S = 26.46579481, we have 0(0) = 22000.0 > 0m. As an example, let c = 16. This gives | = 1.65411218 and so 0 < 0^0) < 1.65411218. We integrate Equation (5.8) to th e right with 0'(O) = N \ *0.1, where N \ runs from 1 to 16 and found th a t 0'(O) = 1.6 is the best candidate for a refinement. We pursue th e same integration to th e right with 0'(O) = 1.6 + N? * 0.01, where N 2 runs from -10 to 10 to find 0'(O) = 1.64 is the candidate for a further refinement. The proccess continues until we found 0'(O) = 1.64018585, where the integration to the right can go as far as f = 26.0 in which 0(f) is monotonically increasing toward 6max = 22026.46579481 while 0'(f) is monotonically decreasing to 0. For different values of c we repeat the same process to find the corresponding 0'(O). From the above procedure, we obtain Aa=,io = {(c,0'(O))28] "(I) = 22000.0 } . For each value of (c,0'(O)) € Ao_ i0, we then integrate Equation (5.8) to the left to check whether th a t value yields a solution. In th e case of c = 16 w ith 0'(O) = 1.64018585 found, the integration to the left can go as far as f = —200.0, where 0(—200.0) = 0.00472871. We consider this value of 0(—200.0) is close to 0, and thus numerical solution for a = 10, H = 1.0, and c = 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. # 84 Solid M edium Combustion is established. We repeat the same procedure for different values of a. It is found th a t as c decreases, the integration to the left toward the stable node becomes increasingly difficult. It is then conceivable th a t there is a limit for c, say c = c*, below which no traveling wave solution can be constructed. This c* is a function of a , c* increases with a . We note th at the param eter a is proportional to the activation energy of the m aterial. Larger values of a correspond to more combustible m aterial, and there result larger critical values c*. Numerical results give a strong indication th at solutions of (5.8) subject to the boundary conditions (5.9) exist for c > c’ (a). Thus it is suggested, numerically, th a t c* is the minimum speed for the combustion waves. For a = 10.0 and H = 1.0, Figure(4.2) shows the solution 0(£, c) for some c, c > c*(10) . Numerical solution 0(£, c) for a = 20.0 is presented in Figure(4.3). We noted th a t in [2] Tam derived a sufficiency condition for the solution to exist, th a t is c > (5.19) where ct = max 9 = £ max , 6 ’ 0 < 0 < H /e 5.20) and F( 0( t , c) ) = ( H - ee(t,c))g(e(t, c)) (5.21) (See Figure(4.4)). Let cJ(a ) = . Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (5.22) 95 Solid M edium Combustion We present in Table(5.1) a comparison of the values of c5(a) and c“(a ) for some a . We find th a t c’ (a) is considerably larger than c*(o) and thus the numerical result gives a better lower bound of c for the solution to exist than the one found analytically in [2]a c3 <7 c’ 10.0 90.629 19.0399 13.4 12.0 455.597 42.6894 30.4 14.0 2438.343 98.7592 70.8 20.0 470302.493 1371.5721 910.0 Table 5.1: The comparison of lower bounds for the wave speed c derived analitically, c% and calculated numerically, c*, for different values of a with H = 1.0 5.5 C on clu d in g R em a rk s We seek a numerical solution for traveling combustion waves of a solid m aterial. The m athem atical model is presented in Equation (5.8) subject to th e boundary conditions (5.9). Since the boundary conditions are prescribed a t £ = ± o o , to find a numerical solution for this problem we ask where an integration should be carried out from. We answer the question by presenting a com putation algorithm based on a two-sided shooting method. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 96 Solid M edium Combustion # H/e m 6 Figure 5.2: 6(£) and F( 9(Q) vs f . T he choice of 6 in the algorithm is such th at H / e - 6 > 0 m. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 97 Solid M edium Combustion c — 16 Figure 5.3: The solution 6 for a = 10.0, 16 < e < 30, w ith H=1.0. Note th a t c*(10) = 13.4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Solid M edium Combustion 98 c = 1200 Figure 5.4: The solution 9 for a = 20.0, 1200 < c < 1400, with H=1.0. Note th at c*(20) = 910.0. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 99 Solid M edium Combustion c = 30 c = 25 c = 20 c = 15 o -250 Figure 5.5: a = 10. 0 . -200 *150 *100 -50 0 vs £ for different c: c = 15, 20, 25, and 30, where H — 1.0 and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Bibliography [1] K. K. Tam. C riticality dependence on d ata and param eters for a problem in combustion theory. J. Austrl. Math. Soc. Ser. B, 27:416-441, 1986, [2] J. M. Hill and Pincombe A. H. Some sim ilarity tem perature profiles for the microwave heating of a half-space. J. Austrl. Math. Soc. Ser. B, 33:290-320, 1992. [3] Smyth N. F. T he effect of conductivity on hotspots. J. Austrl. Math. Soc. Ser. B, 33:403-413, 1992. [4] K. K. Tam. Criticality dependence on d ata and param eters for a problem in combustion theory, with tem perature-dependent conductivity. J. Austrl. Math. Soc. Ser. B, 31:76-80, 1989. [5] Norbury J. and Stuart A. M. Models for porous medium combustion. Quart. J. Mech. and Appl. Math., 42:154-178, 1989. [6] D. A. Frank-Kamenetskii. Diffusion and Heat Transfer in Chemical Kinetics. Plenum Press, New York, 1959. 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Solid M edium Combustion 101 [7] Matkowsky B. J. and Sivashinsky G. I. Propagation of a pulsating reaction front in solid fuel combustion. Siam J. Appl. M ath., 39:465-478, 1979. [8] K. K. Tam. Porous medium combustion: Ignition, tem poral evolution, and extinction. J. Austrl. Math. Soc. Ser. B , 31:76-80, 1989. [9] K. K. Tam and Andonowati. Numerical study of a problem in the combustion of a porous medium. Submitted to J. Austrl. Math. Soc. Ser. B, December, 1994. [10] Norbury J. and Stuart A. M. Travelling combustion waves in a porous medium, part i - existence. Siam J. Appl. Math., 48(1):155-169, 1988. [11] K. K. Tam and Andonowati. Computation of travelling combustion waves in a porous medium. Studies in Applied Mathematics, 91:179-187, 1994. [12] Tam K. K. Traveling wave solutions for combustion in a porous medium . Studies in Applied Mathematics, 81(3):249-263, 1989. [13] Andonowati. Two-sided shooting m ethod in com putation of traveling combustion waves of a solid m aterial. Accepted fo r a publication in J. Austrl. Math. Soc. Ser. B. January, 1995. [14] C. J. Coleman. On the microwave hotspot problem. J. Austrl. Math. Soc. Ser. B, 33:1-8, 1987. [15] A. Lacey and Wake G. C. Therm al ignition with variable therm al conductivity. IM A J. Appl. Math., 28:23-39, 1982. [16] P rotter M. H. and Weinberger H. F. Maximum Principles in Differential Equa tions. Springer-Verlag, New York, 1984. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Solid M edium Combustion [17] K. K. Tam. 102 Porous medium combustion: Ignition, temporal evolution, and param eter dependence. J. Austrl. Math. Soc. Ser. B, 33:16-26, 1991. [18] Eckhaus W. Studies in Nonlinear Stability Theory. Springer, Berlin, 1965. [19] Newell A. C. Rand D. A. Broomhead D. S., Indik R. Local adaptive Galerkin bases for large-dimensional dynamical systems. Nonlinearity, 42:159-178, 1991. [20] R. Tem am . Infinite Dimensional System s in Mathematics and Physics. Springer Applied M athem atics Series, Springer, Berlin, 1988. [21] Tam K. K. Travelling wave solutions for a combustion problem. Studies in Applied Mathematics, 81:117-124, 1989. [22] Aronson D. G. and Weinberger H. F. Multidimensional nonlinear diffusion arising in population genetics. Advance in Mathematics, 30:33-76, 1978. [23] B arenblatt G. I. Librovich V. B. Makhiviladze G. M. Zeldovich, YA. B. The Mathematical Theory of Combustion and Explosions. Consultants Bureau, Plenum , New York, 1985. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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