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Large activation energy analysis of the ignition of self-heating prorous bodies
COMBUSTION
AND FLAME
57:247-253 (1984)
247
Large Activation Energy Analysis of the Ignition of Self-Heating Porous Bodies JOSl~ M. V E G A a n d A. L I I ~ i , N E. T.S. Ingenieros Aeronduticos, Universidad Polit6cnica de Madrid, Madrid, Spain
A large activation energy analysis of the problem of thermal ignition of self-heating porous bodies is carried out by means of a regular perturbation method. A correction to the well-known Frank-Kamenetskii estimate of the ignition limit is calculated, for symmetric bodies, by using similarity properties of the equations giving higher order terms in an expansion in powers of I/E (E = activation energy). Our estimate compares well with numerical results, and differs from others in the literature, which are not better than Frank-Kamenetskii's one from an asymptotic point of view. Dirichlet and Robin type of boundary conditions are considered. A brief analysis of the extinction problem for no reactant consumption is also presented.
I. I N T R O D U C T I O N The problem of thermal ignition of porous catalysts has received considerable attention in the literature. Frank-Kamenetskii's pioneering work [1] took advantage of the fact that the nondimensional activation energy E, defined below, is usually a large parameter. His solution for the ignition regime is the leading order term o f an expansion in powers of E ~ of the solution o f the exact equation and satisfies the so-called Frank-Kamenetskii (F-K) equation. This equation has known closed-form solutions in one and two dimensions, and possesses similarity properties which make it possible to express the solution in three dimensions in terms o f some canonical functions which can be calculated from a second order initial value problem (see Chandrasekhar and Wares [2] and Chambr6 [3]). It is seen, in particular, that no solution of the F-K equation (satisfying the appropriate boundary conditions) exists if the Damk6hler number exceeds a certain critical value 6t 0, which is a first approximation, as E ---' oo, of the ignition limit 81. Copyright © 1984.by The Combustion Institute Published by Elsevier Science Publishing Co.. Inc. 52 Vanderbilt Avenue. New York. NY 10017
More recently, a number of workers tried to determine the dependence of 6t on E. Parks [4] and Shouman et al. [5, 61 calculated 6t by numerical computations of the exact problem; Shouman and Donaldson [7] used a power series expansion on the spatial variable to describe the solution of the exact problem. Bowes and Thomas [8] assumed that the temperature inside the body is uniform (see Thomas [9]) and made a large activation energy analysis of the simple resulting problem. Hardee et al. [10] replaced the Arrhenius term by a polynomial. Takeno [ 11 ] used the m a x i m u m temperature inside the body (which is an unknown), instead of the temperature at its surface, to make nondimensionalizations, and solved the leading order problem in the limit E ~ oo. Comparison with numerical results showed that T a k e n o ' s approximation is qualitatively good, at least for zero-order reactions; nevertheless the estimates of 8~ given in [1 1] are zero-order approximations as E ---, e~ and, therefore, they are not better than the original Frank-Kamenetskii estimate from an asymptotic point o f view. T a k e n o ' s idea has been used also by Bazley and Wake [12], Takeno and Sato
0010-2180/84/$03.00
248
JOSI~ M. VEGA and A. Lllq.~N
[13], and Gill et al. [14, 15]. For a variational method for calculating 6t see Fradkin and Wake [16]. In this note, we shall calculate higher order approximations, as E ~ ~ , of the ignition limit. Such approximations will be given by problems having similarity properties which will allow them to be reduced to canonical initial value problems. We shall take into account the reactant consumption and the effect of external heat and mass transfer resistances. The dimensionless heat and mass conservation equations and the boundary conditions to be considered are
\ dx2+x
considered here will require only algebraic changes in the analysis below. 2. T H E
DIRICHLET
Let us consider first the problem
/ d2r J d~x)
r'(0)=c'(0)=0;
(1)
E 7:'(0)=e'(0)=0;
v(1 -7:(1))=7:'(1),
--
.~ - I/~--, oo,
a - ~ E = O(1).
1,
(6)
+---
+6
1-
(2)
where x, 6, and 7: are the nondimensional space coordinate, concentration, and temperature; n is the reaction order and (5, L', and 3;- ~ are the Damk6hler number, the nondimensional activation energy, and the nondimensional maximum adiabatic temperature rise; j = 0, 1, and 2 for symmetric bodies in 1,2, and 3 dimensions; and v and a are the Biot numbers for external heat and mass transfer. Bars over variables and parameters will be used when considering boundary conditions of the type (2), while they will be omitted when using boundary conditions of the Dirichlet type. We shall make the realistic hypothesis /~--, oo,
{5)
which reduces (4) and (5) to a problem in r.This problem may be written as
dx 2 x dx
,r(l - e(l)) = e ' ( 1 ) ,
r ( 1 ) = c ( 1 ) = I.
(4)
An integration of the first equation in (4) leads to the relation
dx 2 x dx
exp (~(7:- 1)/7:),
d2c j dc
6 = 3' ~ c" e x p ( E ( r - 1)/r), E
3'r+c=7+
='~ = e"
PROBLEM
.,7 - I = O ( 1 ) ,
(3)
The first two of these are necessary for the F-K equation to give an approximate solution of the problem. The remaining two are not essential in the analysis; values of'); and a smaller than those
× exP(l+2/E)
(7)
=0,
{8)
,~'(0) = , ~ ( 1 ) = 0 in terms of the variable 4~= E ( r - 1).
The simplest asymptotic analysis, as E --, ~ , of (7) and (8), requires the introduction of an expansion of the type
3'
.y2
1
(~)= (~0 "3rE (~I + E~ (~2 + E ( ~ 3 + O
(,,y ,y3) E2 , E 3
into (7) and (8) to get a sequence of recursive problems giving ~i, i = 0, 1 . . . . . (E I and 3,E- 1 are treated as independent small parameters to account for the case 1,~ 3' "~ E). Then, the problems giving ~i, i > 1, are singular at the ignition limit. To avoid this difficulty, the Damk6hler number will be expanded: 3 = 6 0 +
249
IGNITION OF POROUS BODIES (2JE)5~ + • • • • To define the u n k n o w n s 51, 6z, • . . , an a r b i t r a r y a d d i t i o n a l c o n d i t i o n must be i m p o s e d on the p r o b l e m giving 4~i, for any i > 1. Such c o n d i t i o n s can be chosen in such a way that (i) no s i n g u l a r i t y a p p e a r s in the higher o r d e r p r o b l e m s and (ii) 4~i, i > 1, can be e x p r e s s e d in terms of canonical functions. A redefinition o f 4~ will allow us to obtain 4~, 4'2, and 4'3 at the same time. Let us introduce g = (1 - n T / E ) O ,
(9)
A = (1 - n2//E)6
and the e x p a n s i o n s g=go+(E
l+nTZ/2Ea)g 1+...,
A=Ao+(E-I
+nTZ/2EZ)AI+ ....
(10)
The p r o b l e m s giving go and g~ and found to be d2g(I - -
j dgo + - --
dx 2
+ AoegO =
O;
x dx
go'(O) = g o ( l ) = O,
write
as
(11)
d w o l d s = 1 - j + Oolcoo,
(15)
dOolds = - 0o + 200/6o0,
lim COo~S=lim 0o/(j + 1)s= 2 as
Oo(g) = Ao,
at
(16)
s~O~, coo(g) = - go' ( 1)
.~= go(0).
(17)
F o r j = 0 and 1, (15) and (16) have c l o s e d form solutions, while f o r j = 2, 0o = Oo(s) and COo = COo(S)may be obtained by n u m e r i c a l c o m p u t a tions on (15) and (16). O b s e r v e [see (17)] that the functions 0o = Oo(s) and COo = COo(S)give Ao and - g o ' (1) in terms o f g0 (0); it is seen, in particular, that no solution o f ( l l ) exists i f A o > Aot = sup,>o Oo(s) (Aot = 0.879, 2.000, and 3.322 f o r j = 0, 1, and 2, r e s p e c t i v e l y ) . The differential equation and the b o u n d a r y condition a t x = 0 in (12) are invariant under the g r o u p of t r a n s f o r m a t i o n s given by (13) and
(11) g l - - ' g l + 2c~(1 + go) + o~2.
d2gl
j dgl
As an additional condition tbr defining At, we select
--+---+AoegO
dx 2
x dx
(A, ) ×
gl + ~ 0 0 - g 0 "
gl(0) = go(0)[2 + go(0)] + 2 - At/Ao,
which is invariant under the group. W e introduce the new v a r i a b l e s
=0;
gL'(O)=g~(l)=O.
(12)
The equation and the b o u n d a r y condition at x = 0 in (11) are invariant under the group o f transformations go-*go+cO,
x--*3x,
Ao~Ao/32e%
(13)
This invariance a l l o w s us to obtain an analytical solution o f (11) (see [2,3]). F o l l o w i n g C h a m b r e [3], we define the new v a r i a b l e s coo = - xgo'
(18)
0o = Aox 2e g o,
s = go(0) - go, (14)
which are invariant under the group (13), to
01 = go[2 + go] - gl + 2 - AI/Ao, COl= x[2go' + 2gogo' - g l ' 1,
(19)
which are invariant under the group too, to write (12), (18) in the form d w l / d s = [(1 -J)col +
26Oo 2 -
OoOl]/coo,
dOl/ds = col~coo,
lim col/4s 2 = lira 01/s 2 = 2/(3 + j ) as
s ~ O ",
01(s-) = 2 - Ai/Ao,
coL(s-)= 2go' (1) + gl '(1),
(20)
250
JOSt~ M. V E G A and A. LI/qAN
where ~is defined in (17). The value o f g t a t x = 0 is readily obtained, in terms o f g, from (18) and (20): gl(0) = 01(s-) + 2 g + ~2.
(21)
N o w , from (9), (I0), (I4), (17), (19), (20), and (21), we get
(1 - n y / E ) 6 = Oo(s'){ 1 + ( 1 / E + ny 2/2E 2) × [2 - 0~(s-)] + O(/~)},
son o f the estimate (25) with numerical results given by Parks [4] is made in Fig. 1. The temperature profiles are easily obtained in terms o f the functions 00 = Oo(s) and 0~ = 0t(s). From the definitions (14) and (19), go(x) and g~(x) are seen to be given by
x = ~/Oo(s)/Ao exp[(s - s3/2], go=~-s,
(22)
E(1 - n y / E ) [ r ( 0 ) - 1]
gl = 2 - AI/Ao + (.~- s)[2 + g - s] - Ol(s), in terms o f the parameter s, for 0 _< s _ g.
= g + ( I / E + n 7 2 / 2 E 2) x
[0~(g) + 2~+ ~2] + O(g),
(23)
E ( I - n T / E ) ~ b / ( j + 1)3,
When using the new variables and parameters
= wo(S-) + ( l / E + n y 2 / 2 E 2) × [wl(s-) + 2wo(s-)] + O(#),
r = ~1~, (24)
where # = m a x { 7 / E 2, "y3/E3}, and ff = ( j + 1)(dc/dx)x= ~ is the observable reaction rate per unit volume. Equations (22)-(24) give parametrically, through .q, r(0), and ff in terms o f 6. The ignition limit, fir, corresponds to the first maximum of 6; it is reached at $ = -~x, where
( 1 / E + n 3, 2/2E 2)Wo(O~)wl (o~) +
+ O(t,), [2 - w0(o0] 2 _ 211 - j + Oo(oO/wo(a)]
E = F_.I~.
"Y= "~G/e~, 8c~"
/
G-I)
6 = __--7exp~ E rs \
es=e(l),
"rs
,
(26)
cs= ~(I),
Eqs. (1) and (2) become Eqs. (4) and (5), which have been considered in the previous section. The boundary conditions (2) at x = 1 and the relation (6) lead to ~s = [1 + ¢ / o ( j + 1)] -1,
Oo(a) { 1 + ( 1 / E + n7 2/E 2) (25)
where Or(a) = 0.9519, 0.910, and 0.869 f o r j = 0, 1, and 2, respectively. Since 2 - 0~(t~)>0, 6 I increases as E - J, n, or 3' increases. A compari-
(27)
When taking into account hypothesis (3), Eq. (24), and the definitions (26), Eqs. (27) become e~ = 1 + O(/~ - 2), 1 - n -E
(1 - n y /E)61
x [2 - 0,(a)l + O(g)},
c = clcs,
fs = 1 + e # / ~ / v ( j + 1).
with c~ = 1.187, 1.386, and 1.608, 00(o0 = 0.879, 2.000, and 3.322, and Wj(a) = 3.250, 2.580, and 1.980 for j = 0, 1, and 2, respectively, and Wo(a) = 2 [a is the value o f s at the first maximum of the function 00 = 0o(S)]. Then
=
3. T H E E F F E C T O F H E A T A N D M A S S TRANSFER RESISTANCES
___--£ E
( f s - 1)
+-E
+
x
\E
2~ ~
+ O(g)
v
'
IGNITION OF POROUS BODIES
251
,6 r I 3.5
-
i 8t
3.4 0
0
F
0
0
0
o
3.322
3.3 L 10
2o
#o
io
5~
: 30
t 40
±. SO
30
40
50
6'0
,~
8L
9o
6t
O 2
i. 10
~_ 20
10
20
.
0 0
o
o
6'0
--70
80
90
60
70
80
90
E
Fig.
1. A c o m p a r i s o n
between
the present
estimate
of
the ignition limit
(
) and
n u m e r i c a l r e s u l t s o f [4] ( O ) f o r n = 0 a n d n o e x t e r n a l h e a t a n d m a s s t r a n s f e r r e s i s t a n c e s .
w h e r e # = m a x { # / E 2, , ~ 3 / j ~ 3 } . we get
T h e n , f r o m (26)
A 2 = Ao[2 - 01 -- (20:0 + o:l)/~l,
F = v(l - n # / E ) .
(1 - n'~/L~)~= Ao(s0 + (2/~')AI(S-)
+ (1/E + n# 2 / E 2)A2(s-') + O(/5.), (28)
The ignition limit corresponds to the first maximum of 6 = 6(s-), which is reached at a value of # which, in the first approximation, is given by the equation
where #Ao = 0o exp[ - ~oo/~7)], Al = Ao(~Oo/~7) exp[ - wo/~7)],
(30)
(1 -j)o:o + Oo (31) 2 - - CO0
(29)
Remember that in (29)-(31), o~i and Oi are known
252
JOSI~ M. VEGA and A. LI/q,AN
Ao{
A2]
r
All
,I .1
L .1
2
3
~
5
7
1
F i g . 2. A0,, A . , a n d A2, v e r s u s # = u ( 1 ignition limit.
2
3
5
-n'f/E),
functions of g; therefore, (28) and (31) give a parametric representation, through ~, of the function 6i = (51(z7). A01, All, and A2, are plotted versus z~ in Fig. 2 for the spherical geometry (j = 2). The function A0, = A0ffzT),which gives a first approximation of the ignition limit, was obtained by Thomas [ 171. The temperature profiles may be obtained as in the previous section. 4. A N O T E ON T H E E X T I N C T I O N The analysis given before describes the response of the porous body in a nearly frozen reacting mode, or ignition regime, which exists only for 6 < 61. For 6 larger than an extinction value 6E (6E < 60, there is an additional fast reaction response mode, which is to be considered briefly. In the limit E ~ oo, 6E = finite, the effect of reactant consumption must be taken into account. If n > - 1 and the reaction term is assumed to vanish for c = 0, three distinguished zones appear in the porous body: (a) an internal core without reactant, (b) an external region where the chemical reaction is frozen, and (c) a
7
10
2
3
~.
5
7
100
to b e u s e d in E q . ( 2 8 ) to c a l c u l a t e t h e
thin reaction region, which is placed at the common boundary of regions (a) and (b), where the reactant arriving by diffusion from region (b) is completely consumed. A singular perturbation analysis of the problem is possible in this case, and provides an asymptotic approximation to the extinction limit 6E (see [18-20]). Here we shall consider only the case of no reactant consumption, whose analysis differs from those of Ref. [ 18-20]. If E ~ Qo but 7 E 0, the concentration is c = l in first approximation, and the conservation equation for the enthalpy takes the form d2T
--
j dT
dx 2
+ ---+8' x clx
exp(- 1/7)=0,
= 0,
,
x=0
in terms of 6
7" T =-
E
and
=
exp(
.
(32)
IGNITION
OF POROUS
253
BODIES
For the sake of brevity, we consider boundary conditions of the Dirichlet type in (32); an analysis of the Robin problem could be made by means of the same ideas as those used in Section 3. For sufficiently large values of E, there is an interval, [hE’, 6,‘], of multiplicity of solutions of (32). The upper multiplicity bound is
Then 6E’ = bOE’ + 0( 1 /E). For an analysis (33) in cylinders (j = 1) see Parter [21]. REFERENCES I.
Frank-Kamenetskii,
Transfer
Chandrasekhar. 109:551
exp(E),
Chambre.
Parks. I. R., J. Chem.
7. 8.
A. R
Flame 23: 17 (1974).
Shouman.
A. R..
Shouman.
A. R..
and Donaldson.
Bowes, C. P.. and Thomas.
A. B.. Cornbust. A.
B.. Combusr.
P. H..
Combusf.
Flame
Thomas,
P. H.,
Trans. Faraday Sot. 56:833 (1960).
Hardee,
H. C..
Lee.
D. 0..
and Donaldson.
A. B..
in the form
l/T,)=O,
15.
Flame 181403 (1972). Takeno. T., Cornbust. Flame 29:209 (1977). Bazley. N. W.. and Wake. G. C.. Combusl. Flame 33: 161 (1978). Takeno. T.. and Sate. K.. Cornbust. Flume 38:7S (1980). Gill.
W..
Donaldson,
16.
T,(l)=O.
(33)
B.. and Shouman.
Flame 36~217 (1979).
Gill,
Shouman.
W..
A.
R..
and Donaldson.
A.
R
A.
B..
,
Flume 41:99 (1981).
Fradkin.
L. Ju., and Wake.
G. C..
J. Insr.
Math.
Appl. 20~47
I (1977).
17.
Thomas,
18.
Urrutta.
Trans. Faraday Sot. 54:60 (1958). An Anal.viical Model of Decomposition Morors, Part III. Insti-
P. H..
J. L. .and Liitan.A..
Hydrazine
A straightforward numerical analysis of (33) shows that it has two solutions if 60’ > 60~’ , and no solution if 60’ < dOE’, where
A.
Combusl. Cornbust.
tuto
National
de
Tecnica
Aeroeapacial,
Madrtd,
1975. 19.
Kapila.
A. K.. and Matkowsky.
B.
I SIAM J. Appl.
Math. 36:373 (197). 20.
Kapila,
21.
SIAM J. Appl. Moth. 38:382 (1980). Parter. S. V.. SIAM J. Appl. Math. 26:687 (1974).
16.837, and 29.565 1, and 2.
and Donaldson.
9.
14.
j=O,
A. B.. and Tsao. H. Y..
IO.
j dTO
for
. Donaldson.
Shouman,
Cornbust.
IO:221 (1966).
13.
80E’ = 6.966,
Phys. 19:1795 (1952). Phys. 34:46 (1961).
Flame 29:213 (1977).
12.
to write (32), in first approximation,
=o,
J.
Flame 241203 (1975).
11.
T=T,,+E-IT,+...
+&’ exp(-
Asfrophys.
P. L., J. Chem.
3.
6.
G. W..
(1949).
Combust.
A’=&,‘+E-‘6,‘+.**,
d2To --+,z dx=
1969.
S.. and Wares.
4. 5.
where 6, is the ignition limit calculated in Section 2. Observe that the temperature profiles of the ignition regime which were calculated in Section 2 correspond to the solution T = 0 of (32) in the 1imitE-t 00. To calculate the lower multiplicity bound, 6E’, we introduce the expansions
A.. Diffusion and Hear Kinerics, 2nd Ed.. Plenum
D.
in Chemical
Press. New York, 2.
6,’ -f
of
A. K..
Received January
Matkowsky.
B. J.. and Vega.
1983; revised December
1983
J. M..
AND FLAME
57:247-253 (1984)
247
Large Activation Energy Analysis of the Ignition of Self-Heating Porous Bodies JOSl~ M. V E G A a n d A. L I I ~ i , N E. T.S. Ingenieros Aeronduticos, Universidad Polit6cnica de Madrid, Madrid, Spain
A large activation energy analysis of the problem of thermal ignition of self-heating porous bodies is carried out by means of a regular perturbation method. A correction to the well-known Frank-Kamenetskii estimate of the ignition limit is calculated, for symmetric bodies, by using similarity properties of the equations giving higher order terms in an expansion in powers of I/E (E = activation energy). Our estimate compares well with numerical results, and differs from others in the literature, which are not better than Frank-Kamenetskii's one from an asymptotic point of view. Dirichlet and Robin type of boundary conditions are considered. A brief analysis of the extinction problem for no reactant consumption is also presented.
I. I N T R O D U C T I O N The problem of thermal ignition of porous catalysts has received considerable attention in the literature. Frank-Kamenetskii's pioneering work [1] took advantage of the fact that the nondimensional activation energy E, defined below, is usually a large parameter. His solution for the ignition regime is the leading order term o f an expansion in powers of E ~ of the solution o f the exact equation and satisfies the so-called Frank-Kamenetskii (F-K) equation. This equation has known closed-form solutions in one and two dimensions, and possesses similarity properties which make it possible to express the solution in three dimensions in terms o f some canonical functions which can be calculated from a second order initial value problem (see Chandrasekhar and Wares [2] and Chambr6 [3]). It is seen, in particular, that no solution of the F-K equation (satisfying the appropriate boundary conditions) exists if the Damk6hler number exceeds a certain critical value 6t 0, which is a first approximation, as E ---' oo, of the ignition limit 81. Copyright © 1984.by The Combustion Institute Published by Elsevier Science Publishing Co.. Inc. 52 Vanderbilt Avenue. New York. NY 10017
More recently, a number of workers tried to determine the dependence of 6t on E. Parks [4] and Shouman et al. [5, 61 calculated 6t by numerical computations of the exact problem; Shouman and Donaldson [7] used a power series expansion on the spatial variable to describe the solution of the exact problem. Bowes and Thomas [8] assumed that the temperature inside the body is uniform (see Thomas [9]) and made a large activation energy analysis of the simple resulting problem. Hardee et al. [10] replaced the Arrhenius term by a polynomial. Takeno [ 11 ] used the m a x i m u m temperature inside the body (which is an unknown), instead of the temperature at its surface, to make nondimensionalizations, and solved the leading order problem in the limit E ~ oo. Comparison with numerical results showed that T a k e n o ' s approximation is qualitatively good, at least for zero-order reactions; nevertheless the estimates of 8~ given in [1 1] are zero-order approximations as E ---, e~ and, therefore, they are not better than the original Frank-Kamenetskii estimate from an asymptotic point o f view. T a k e n o ' s idea has been used also by Bazley and Wake [12], Takeno and Sato
0010-2180/84/$03.00
248
JOSI~ M. VEGA and A. Lllq.~N
[13], and Gill et al. [14, 15]. For a variational method for calculating 6t see Fradkin and Wake [16]. In this note, we shall calculate higher order approximations, as E ~ ~ , of the ignition limit. Such approximations will be given by problems having similarity properties which will allow them to be reduced to canonical initial value problems. We shall take into account the reactant consumption and the effect of external heat and mass transfer resistances. The dimensionless heat and mass conservation equations and the boundary conditions to be considered are
\ dx2+x
considered here will require only algebraic changes in the analysis below. 2. T H E
DIRICHLET
Let us consider first the problem
/ d2r J d~x)
r'(0)=c'(0)=0;
(1)
E 7:'(0)=e'(0)=0;
v(1 -7:(1))=7:'(1),
--
.~ - I/~--, oo,
a - ~ E = O(1).
1,
(6)
+---
+6
1-
(2)
where x, 6, and 7: are the nondimensional space coordinate, concentration, and temperature; n is the reaction order and (5, L', and 3;- ~ are the Damk6hler number, the nondimensional activation energy, and the nondimensional maximum adiabatic temperature rise; j = 0, 1, and 2 for symmetric bodies in 1,2, and 3 dimensions; and v and a are the Biot numbers for external heat and mass transfer. Bars over variables and parameters will be used when considering boundary conditions of the type (2), while they will be omitted when using boundary conditions of the Dirichlet type. We shall make the realistic hypothesis /~--, oo,
{5)
which reduces (4) and (5) to a problem in r.This problem may be written as
dx 2 x dx
,r(l - e(l)) = e ' ( 1 ) ,
r ( 1 ) = c ( 1 ) = I.
(4)
An integration of the first equation in (4) leads to the relation
dx 2 x dx
exp (~(7:- 1)/7:),
d2c j dc
6 = 3' ~ c" e x p ( E ( r - 1)/r), E
3'r+c=7+
='~ = e"
PROBLEM
.,7 - I = O ( 1 ) ,
(3)
The first two of these are necessary for the F-K equation to give an approximate solution of the problem. The remaining two are not essential in the analysis; values of'); and a smaller than those
× exP(l+2/E)
(7)
=0,
{8)
,~'(0) = , ~ ( 1 ) = 0 in terms of the variable 4~= E ( r - 1).
The simplest asymptotic analysis, as E --, ~ , of (7) and (8), requires the introduction of an expansion of the type
3'
.y2
1
(~)= (~0 "3rE (~I + E~ (~2 + E ( ~ 3 + O
(,,y ,y3) E2 , E 3
into (7) and (8) to get a sequence of recursive problems giving ~i, i = 0, 1 . . . . . (E I and 3,E- 1 are treated as independent small parameters to account for the case 1,~ 3' "~ E). Then, the problems giving ~i, i > 1, are singular at the ignition limit. To avoid this difficulty, the Damk6hler number will be expanded: 3 = 6 0 +
249
IGNITION OF POROUS BODIES (2JE)5~ + • • • • To define the u n k n o w n s 51, 6z, • . . , an a r b i t r a r y a d d i t i o n a l c o n d i t i o n must be i m p o s e d on the p r o b l e m giving 4~i, for any i > 1. Such c o n d i t i o n s can be chosen in such a way that (i) no s i n g u l a r i t y a p p e a r s in the higher o r d e r p r o b l e m s and (ii) 4~i, i > 1, can be e x p r e s s e d in terms of canonical functions. A redefinition o f 4~ will allow us to obtain 4~, 4'2, and 4'3 at the same time. Let us introduce g = (1 - n T / E ) O ,
(9)
A = (1 - n2//E)6
and the e x p a n s i o n s g=go+(E
l+nTZ/2Ea)g 1+...,
A=Ao+(E-I
+nTZ/2EZ)AI+ ....
(10)
The p r o b l e m s giving go and g~ and found to be d2g(I - -
j dgo + - --
dx 2
+ AoegO =
O;
x dx
go'(O) = g o ( l ) = O,
write
as
(11)
d w o l d s = 1 - j + Oolcoo,
(15)
dOolds = - 0o + 200/6o0,
lim COo~S=lim 0o/(j + 1)s= 2 as
Oo(g) = Ao,
at
(16)
s~O~, coo(g) = - go' ( 1)
.~= go(0).
(17)
F o r j = 0 and 1, (15) and (16) have c l o s e d form solutions, while f o r j = 2, 0o = Oo(s) and COo = COo(S)may be obtained by n u m e r i c a l c o m p u t a tions on (15) and (16). O b s e r v e [see (17)] that the functions 0o = Oo(s) and COo = COo(S)give Ao and - g o ' (1) in terms o f g0 (0); it is seen, in particular, that no solution o f ( l l ) exists i f A o > Aot = sup,>o Oo(s) (Aot = 0.879, 2.000, and 3.322 f o r j = 0, 1, and 2, r e s p e c t i v e l y ) . The differential equation and the b o u n d a r y condition a t x = 0 in (12) are invariant under the g r o u p of t r a n s f o r m a t i o n s given by (13) and
(11) g l - - ' g l + 2c~(1 + go) + o~2.
d2gl
j dgl
As an additional condition tbr defining At, we select
--+---+AoegO
dx 2
x dx
(A, ) ×
gl + ~ 0 0 - g 0 "
gl(0) = go(0)[2 + go(0)] + 2 - At/Ao,
which is invariant under the group. W e introduce the new v a r i a b l e s
=0;
gL'(O)=g~(l)=O.
(12)
The equation and the b o u n d a r y condition at x = 0 in (11) are invariant under the group o f transformations go-*go+cO,
x--*3x,
Ao~Ao/32e%
(13)
This invariance a l l o w s us to obtain an analytical solution o f (11) (see [2,3]). F o l l o w i n g C h a m b r e [3], we define the new v a r i a b l e s coo = - xgo'
(18)
0o = Aox 2e g o,
s = go(0) - go, (14)
which are invariant under the group (13), to
01 = go[2 + go] - gl + 2 - AI/Ao, COl= x[2go' + 2gogo' - g l ' 1,
(19)
which are invariant under the group too, to write (12), (18) in the form d w l / d s = [(1 -J)col +
26Oo 2 -
OoOl]/coo,
dOl/ds = col~coo,
lim col/4s 2 = lira 01/s 2 = 2/(3 + j ) as
s ~ O ",
01(s-) = 2 - Ai/Ao,
coL(s-)= 2go' (1) + gl '(1),
(20)
250
JOSt~ M. V E G A and A. LI/qAN
where ~is defined in (17). The value o f g t a t x = 0 is readily obtained, in terms o f g, from (18) and (20): gl(0) = 01(s-) + 2 g + ~2.
(21)
N o w , from (9), (I0), (I4), (17), (19), (20), and (21), we get
(1 - n y / E ) 6 = Oo(s'){ 1 + ( 1 / E + ny 2/2E 2) × [2 - 0~(s-)] + O(/~)},
son o f the estimate (25) with numerical results given by Parks [4] is made in Fig. 1. The temperature profiles are easily obtained in terms o f the functions 00 = Oo(s) and 0~ = 0t(s). From the definitions (14) and (19), go(x) and g~(x) are seen to be given by
x = ~/Oo(s)/Ao exp[(s - s3/2], go=~-s,
(22)
E(1 - n y / E ) [ r ( 0 ) - 1]
gl = 2 - AI/Ao + (.~- s)[2 + g - s] - Ol(s), in terms o f the parameter s, for 0 _< s _ g.
= g + ( I / E + n 7 2 / 2 E 2) x
[0~(g) + 2~+ ~2] + O(g),
(23)
E ( I - n T / E ) ~ b / ( j + 1)3,
When using the new variables and parameters
= wo(S-) + ( l / E + n y 2 / 2 E 2) × [wl(s-) + 2wo(s-)] + O(#),
r = ~1~, (24)
where # = m a x { 7 / E 2, "y3/E3}, and ff = ( j + 1)(dc/dx)x= ~ is the observable reaction rate per unit volume. Equations (22)-(24) give parametrically, through .q, r(0), and ff in terms o f 6. The ignition limit, fir, corresponds to the first maximum of 6; it is reached at $ = -~x, where
( 1 / E + n 3, 2/2E 2)Wo(O~)wl (o~) +
+ O(t,), [2 - w0(o0] 2 _ 211 - j + Oo(oO/wo(a)]
E = F_.I~.
"Y= "~G/e~, 8c~"
/
G-I)
6 = __--7exp~ E rs \
es=e(l),
"rs
,
(26)
cs= ~(I),
Eqs. (1) and (2) become Eqs. (4) and (5), which have been considered in the previous section. The boundary conditions (2) at x = 1 and the relation (6) lead to ~s = [1 + ¢ / o ( j + 1)] -1,
Oo(a) { 1 + ( 1 / E + n7 2/E 2) (25)
where Or(a) = 0.9519, 0.910, and 0.869 f o r j = 0, 1, and 2, respectively. Since 2 - 0~(t~)>0, 6 I increases as E - J, n, or 3' increases. A compari-
(27)
When taking into account hypothesis (3), Eq. (24), and the definitions (26), Eqs. (27) become e~ = 1 + O(/~ - 2), 1 - n -E
(1 - n y /E)61
x [2 - 0,(a)l + O(g)},
c = clcs,
fs = 1 + e # / ~ / v ( j + 1).
with c~ = 1.187, 1.386, and 1.608, 00(o0 = 0.879, 2.000, and 3.322, and Wj(a) = 3.250, 2.580, and 1.980 for j = 0, 1, and 2, respectively, and Wo(a) = 2 [a is the value o f s at the first maximum of the function 00 = 0o(S)]. Then
=
3. T H E E F F E C T O F H E A T A N D M A S S TRANSFER RESISTANCES
___--£ E
( f s - 1)
+-E
+
x
\E
2~ ~
+ O(g)
v
'
IGNITION OF POROUS BODIES
251
,6 r I 3.5
-
i 8t
3.4 0
0
F
0
0
0
o
3.322
3.3 L 10
2o
#o
io
5~
: 30
t 40
±. SO
30
40
50
6'0
,~
8L
9o
6t
O 2
i. 10
~_ 20
10
20
.
0 0
o
o
6'0
--70
80
90
60
70
80
90
E
Fig.
1. A c o m p a r i s o n
between
the present
estimate
of
the ignition limit
(
) and
n u m e r i c a l r e s u l t s o f [4] ( O ) f o r n = 0 a n d n o e x t e r n a l h e a t a n d m a s s t r a n s f e r r e s i s t a n c e s .
w h e r e # = m a x { # / E 2, , ~ 3 / j ~ 3 } . we get
T h e n , f r o m (26)
A 2 = Ao[2 - 01 -- (20:0 + o:l)/~l,
F = v(l - n # / E ) .
(1 - n'~/L~)~= Ao(s0 + (2/~')AI(S-)
+ (1/E + n# 2 / E 2)A2(s-') + O(/5.), (28)
The ignition limit corresponds to the first maximum of 6 = 6(s-), which is reached at a value of # which, in the first approximation, is given by the equation
where #Ao = 0o exp[ - ~oo/~7)], Al = Ao(~Oo/~7) exp[ - wo/~7)],
(30)
(1 -j)o:o + Oo (31) 2 - - CO0
(29)
Remember that in (29)-(31), o~i and Oi are known
252
JOSI~ M. VEGA and A. LI/q,AN
Ao{
A2]
r
All
,I .1
L .1
2
3
~
5
7
1
F i g . 2. A0,, A . , a n d A2, v e r s u s # = u ( 1 ignition limit.
2
3
5
-n'f/E),
functions of g; therefore, (28) and (31) give a parametric representation, through ~, of the function 6i = (51(z7). A01, All, and A2, are plotted versus z~ in Fig. 2 for the spherical geometry (j = 2). The function A0, = A0ffzT),which gives a first approximation of the ignition limit, was obtained by Thomas [ 171. The temperature profiles may be obtained as in the previous section. 4. A N O T E ON T H E E X T I N C T I O N The analysis given before describes the response of the porous body in a nearly frozen reacting mode, or ignition regime, which exists only for 6 < 61. For 6 larger than an extinction value 6E (6E < 60, there is an additional fast reaction response mode, which is to be considered briefly. In the limit E ~ oo, 6E = finite, the effect of reactant consumption must be taken into account. If n > - 1 and the reaction term is assumed to vanish for c = 0, three distinguished zones appear in the porous body: (a) an internal core without reactant, (b) an external region where the chemical reaction is frozen, and (c) a
7
10
2
3
~.
5
7
100
to b e u s e d in E q . ( 2 8 ) to c a l c u l a t e t h e
thin reaction region, which is placed at the common boundary of regions (a) and (b), where the reactant arriving by diffusion from region (b) is completely consumed. A singular perturbation analysis of the problem is possible in this case, and provides an asymptotic approximation to the extinction limit 6E (see [18-20]). Here we shall consider only the case of no reactant consumption, whose analysis differs from those of Ref. [ 18-20]. If E ~ Qo but 7 E 0, the concentration is c = l in first approximation, and the conservation equation for the enthalpy takes the form d2T
--
j dT
dx 2
+ ---+8' x clx
exp(- 1/7)=0,
= 0,
,
x=0
in terms of 6
7" T =-
E
and
=
exp(
.
(32)
IGNITION
OF POROUS
253
BODIES
For the sake of brevity, we consider boundary conditions of the Dirichlet type in (32); an analysis of the Robin problem could be made by means of the same ideas as those used in Section 3. For sufficiently large values of E, there is an interval, [hE’, 6,‘], of multiplicity of solutions of (32). The upper multiplicity bound is
Then 6E’ = bOE’ + 0( 1 /E). For an analysis (33) in cylinders (j = 1) see Parter [21]. REFERENCES I.
Frank-Kamenetskii,
Transfer
Chandrasekhar. 109:551
exp(E),
Chambre.
Parks. I. R., J. Chem.
7. 8.
A. R
Flame 23: 17 (1974).
Shouman.
A. R..
Shouman.
A. R..
and Donaldson.
Bowes, C. P.. and Thomas.
A. B.. Cornbust. A.
B.. Combusr.
P. H..
Combusf.
Flame
Thomas,
P. H.,
Trans. Faraday Sot. 56:833 (1960).
Hardee,
H. C..
Lee.
D. 0..
and Donaldson.
A. B..
in the form
l/T,)=O,
15.
Flame 181403 (1972). Takeno. T., Cornbust. Flame 29:209 (1977). Bazley. N. W.. and Wake. G. C.. Combusl. Flame 33: 161 (1978). Takeno. T.. and Sate. K.. Cornbust. Flume 38:7S (1980). Gill.
W..
Donaldson,
16.
T,(l)=O.
(33)
B.. and Shouman.
Flame 36~217 (1979).
Gill,
Shouman.
W..
A.
R..
and Donaldson.
A.
R
A.
B..
,
Flume 41:99 (1981).
Fradkin.
L. Ju., and Wake.
G. C..
J. Insr.
Math.
Appl. 20~47
I (1977).
17.
Thomas,
18.
Urrutta.
Trans. Faraday Sot. 54:60 (1958). An Anal.viical Model of Decomposition Morors, Part III. Insti-
P. H..
J. L. .and Liitan.A..
Hydrazine
A straightforward numerical analysis of (33) shows that it has two solutions if 60’ > 60~’ , and no solution if 60’ < dOE’, where
A.
Combusl. Cornbust.
tuto
National
de
Tecnica
Aeroeapacial,
Madrtd,
1975. 19.
Kapila.
A. K.. and Matkowsky.
B.
I SIAM J. Appl.
Math. 36:373 (197). 20.
Kapila,
21.
SIAM J. Appl. Moth. 38:382 (1980). Parter. S. V.. SIAM J. Appl. Math. 26:687 (1974).
16.837, and 29.565 1, and 2.
and Donaldson.
9.
14.
j=O,
A. B.. and Tsao. H. Y..
IO.
j dTO
for
. Donaldson.
Shouman,
Cornbust.
IO:221 (1966).
13.
80E’ = 6.966,
Phys. 19:1795 (1952). Phys. 34:46 (1961).
Flame 29:213 (1977).
12.
to write (32), in first approximation,
=o,
J.
Flame 241203 (1975).
11.
T=T,,+E-IT,+...
+&’ exp(-
Asfrophys.
P. L., J. Chem.
3.
6.
G. W..
(1949).
Combust.
A’=&,‘+E-‘6,‘+.**,
d2To --+,z dx=
1969.
S.. and Wares.
4. 5.
where 6, is the ignition limit calculated in Section 2. Observe that the temperature profiles of the ignition regime which were calculated in Section 2 correspond to the solution T = 0 of (32) in the 1imitE-t 00. To calculate the lower multiplicity bound, 6E’, we introduce the expansions
A.. Diffusion and Hear Kinerics, 2nd Ed.. Plenum
D.
in Chemical
Press. New York, 2.
6,’ -f
of
A. K..
Received January
Matkowsky.
B. J.. and Vega.
1983; revised December
1983
J. M..