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A study of some problems arising from combustion and microwave heating

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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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,
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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
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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)
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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].
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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
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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.
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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 .
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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
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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 )
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(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.
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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.
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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)
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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)
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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.
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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
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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 ,
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(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
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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.
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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
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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.
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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.
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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,
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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.
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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,
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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,
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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.
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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,
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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
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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
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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,
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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)
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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
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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
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(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
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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
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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 >
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(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)
■
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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 .
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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
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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)
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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,
=
+
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( 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
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( 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.
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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® )
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(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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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(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)
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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]
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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)
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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 .
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(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.
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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 £ .
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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).
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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 .
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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.
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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
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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
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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.
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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
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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 .
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(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
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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.
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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
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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
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(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)
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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
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#
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 ) =
.
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(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.
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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.
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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.
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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.
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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
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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
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[9] K. K. Tam and Andonowati. Numerical study of a problem in the combustion of
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[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.
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