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 \/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

Liñán Martínez, Amable

Vega de Prada, José Manuel

English

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

COMBUSTION AND FLAME 57:247-253 (1984) 247 
Large Activation Energy Analysis of the Ignition of Self-Heating 
Porous Bodies 
JOSE M. VEGA and A. LINAN 
E.T.S. Ingenieros Aeronduticos, Universidad Politecnica 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 \/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. 
1. INTRODUCTION 
The problem of thermal ignition of porous cata-
lysts 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 be-
low, is usually a large parameter. His solution 
for the ignition regime is the leading order term 
of an expansion in powers of E ~' of the solution 
of the exact equation and satisfies the so-called 
Frank-Kamenetskii (F-K) equation. This equa-
tion has known closed-form solutions in one and 
two dimensions and possesses similarity proper-
ties which make it possible to e x p r e s s the solu-
tion in three dimensions in terms of some canoni-
cal functions which can be calculated from a 
second order initial value problem (see Chandra-
sekhar and Wares [2] and Chambre \7>]) It is 
that no solution of the 
F-K 
equation (satisfying the appropriate boundarv 
conditions) exists if the Damkohler ex-
ceeds a certain critical value <S ° which is a first 
mation F —> oo of the igni t ion l imit 
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 6[ on E. Parks [4] 
and Shouman et al. [5, 6] calculated 5, by numer-
ical 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 maximum temperature inside the body 
(which is an unknown), instead of the tempera-
ture at its surface to make nondimensionaliza-
tions and solved the leading order problem in 
the limit Comparison with numerical 
results showed that Takeno's approximation is 
qualitatively good at least for zero-order reac-
tions; nevertheless the estimates of <5 given in 
[11l are zero-order approximations as E —• oo 
and therefore they are not better than the origi-
nal Frank-Kamenetskii estimate from an asymp-
totic point of view Takeno's idea has been used 
also 
by Bazley and Wake IT 2] Takeno and Sato 
0010-2180/84/503.00 
248 JOSE M. VEGA and A. LINAN 
[13, , and Gill et al. [14, 15]. For a variational 
method for calculating Si see Fradkin and Wake 
[16.. 
In this note, we shall calculate higher order 
approximations, as E —* oo, 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 
( d2f j df\ d2c j dc 
a
772 + xJx)=7x2 + xJx 
= 7 — cn exp (-£"(7— 1)/T), (1) 
E 
T ' ( 0 ) = c ' (0) = 0; v(\ — f(\)) = T ' (X), 
a(l —(l))) = c ' ( l ) , (2) 
where x, c, and 7 are the nondimensional space 
coordinate, concentration, and temperature; n is 
the reaction order and 5, E, and 7 ~' are the 
Damkohler number, the nondimensional activa-
tion energy, and the nondimensional maximum 
adiabatic temperature rise; y = 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 parame-
ters 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 
y-]E- 7 ' = 0(1), 
o
{E=0(\). (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 7 and a smaller than those 
considered here will require only algebraic 
changes in the analysis below. 
2. THE DIRICHLET PROBLEM 
Let us consider first the problem 
( d2r j dr\ d2c j dc 2+ ~~ ~ I = 2+ ~~ ~ 
= 7 — c"exp(£"(T— 1 ) / T ) , (4) 
E 
T ' (0) = c'(0) = 0; T(l) = c ( l ) = l . (5) 
An integration of the first equation in (4) leads to 
the relation 
77 + 0 = 7 + 1, (6) 
which reduces (4) and (5) to a problem in 7.This 
problem may be written as 
d 4> j d4> / 7//> \n 
+ +511 — — 
dx x dx \ E 
x expl - — - )— (J, 
1l +<t>/E/ 
(7) 
(8) 4>'(0) = <Mi) = o 
in terms of the variable 
<t> = E(r-l). 
The simplest asymptotic analysis, as L. —> 00, 
01 (7) auu (8), requires the iiitrociuction 01 an 
expansion of trie type 
7 I 
0=0o+ 4>1+ <J>2+
 F<t>3 + O ^ ^ , ^^ 
1 / 7 7 3 
^- E 
into (7) and (8) to get a sequence of recursive 
problems giving </>,, / = 0, 1, . . . . (E~l and 
yE ~' are treated as independent small parame-
ters to account for the case 1< 7 < E). Then, the 
problems giving 0,, / > 1, are singular at the 
ignition limit. To avoid this difficulty, the 
Damkohler number will be expanded: 5 = 60 + 
6 
IGNITION OF POROUS BODIES 249 
(y/E)8\ + •• • . To define the unknowns 6], 62, 
• • • • , n arbitrary additional londition muss t b 
imposed on the problem giving 0,-, for any / > 1. 
Such conditions can be chosen in such a way that 
(i) no singularity appears in the higher order 
problems and (ii) 0,, / > 1, can be expressed in 
terms of canonical functions. 
A redefinition of </> wiil allow us to obtain <j>u 
4>2, and 03 at the same time. Let us introduce 
g = (l — ny/E)4>, A = (l— ny/E)d (9) 
and the expansions 
g = go+(E ~l + ny2/2E2)g\ + • • •, 
A = A0+ (E ~' + ny2/2E2)A] + • • •. (10) 
The problems giving g0 and g, and found to be 
a
 SO J " 5 0 
+ h A0e*o = 0; 
dx2 x dx 
go'(0) = go(l) = 0, (11) 
d g) j dg\ 
+ + A0e*o 
dx2 x dx 
x ( g\+ — -go2 ) = 0 ; 
*i'(0) = s"i(l) = 0. (12) 
The equation and the boundary condition at x 
= 0 in (11) are invariant under the group of 
transformations 
go~*go + &, x~^(3x, A0—• A0/(32ea. (13) 
This invariance allows us to obtain an analytical 
solution of (11) (see [2,3]). Following Chambre 
[3], we define the new variables 
ii3o=—xgo' 0o = AoX^e^o, s = go(0) — g0, 
(14) 
which are invariant under the group (13), to 
write (11) as 
dw0/ds = 1 —j + d0/oj0, 
d9o/ds= — 60 + 20o/uo, 
lim <J)0/S = lim $Q/(J + 1 )s = 2 
as s - > 0 + , 
8o(s) = A0, Q)0(s)= — g0'(l) 
at s = go(0). 
(15) 
(16) 
(17) 
For j = 0 and 1, (15) and (16) have closed-
form solutions, while for/' = 2, 60 = 80(s) and o>0 
= o>o(s) may be obtained by numerical computa-
tions on (15) and (16). Observe [see (17)] that the 
functions d0 = do(s) and co0 = uo(s) give A0 and 
— go' (1) in terms of go (0); it is seen, in particu-
lar, that no solution of (11) exists if A0 > A0I = 
supi>o 00(s)(A01 = 0.879, 2.000, and 3.322 fory 
= 0, 1, and 2, respectively). 
The differential equation and the boundary 
condition dXx = 0 in (12) are invariant under the 
group of transformations given by (13) and 
g\~*g\ + 2a(l + go) + a2. 
As an additional condition for defining A1; we 
select 
S,(0) = go(0)[2 + go(0)] + 2 — A1/A0, (18) 
which is invariant under the group. We introduce 
the new variables 
0. =s"o[2 + go] — g\ + 2 — A1/A0, 
coi = x[2g 0 ' +2gogo' — g\' ], (19) 
which are invariant under the group too, to write 
(12), (18) in the form 
d(jo\/ds= [(1 — /)cj | + 2a>o2 — d0d[]/o3o> 
dd\/ds = a>]/a>o> 
lim a>i/4s =l im(? , / s = 2 / ( 3 + 7 ) 
as s—>0 +, 
8$J) = 2-A1/AQ, 
U](s) = 2g0'(l) + £i'(l), (20) 
where 5 is defined in (17). The value ofgi a tx = 
0 is readily obtained, in terms of s, from (18) and 
(20): 
gl(0) = 61(s) + 2s + s2. (21) 
Now, from (9), (10), (14), (17), (19), (20), and 
(21), we get 
(1 - ny/E)8 = 80(s){ 1 + (\/E+ ny2/2E2) 
x[2-0,(5)] + O(/i)}, (22) 
E(l-ny/E)[T(0)-l] 
= §+{\/E + ny2/2E2) 
x[6](s) + 2s + s2] + 0(fi), (23) 
E(\-ny/E)^/(j+X)y 
= u>o(s) + (\/E+ny2/2E2) 
X [«,(*)+ 2a>„tf)] + OG*), (24) 
where \i = ma\{y/E2, 7 Vis 3}, and yj/ = (j + 
$(dc/dx)x=i is the observable reaction rate per 
unit volume. Equations (22)-(24) give paramet-
rically, through s, T(0) , and \p in terms of 8. The 
ignition limit, 5,, corresponds to the first maxi-
mum of 5; it is reached at s = Si, where 
S/ = a 
(l/E+ny 2/2E2)u0(a)o>, (a) 
+ + 0(/x), 
[2 - « „ ( « ) ] 2 - 2 [ l - y + O^aVudLa)] 
with a = 1.187, 1.386, and 1.608, 60(a) = 
0.879, 2.000, and 3.322, and w,(a) = 3.250, 
2.580, and 1.980 for j = 0, 1, and 2, respec-
tively, and a>o(a) = 2 [a is the value of s at the 
first maximum of the function 0Q = do(s)]. Then 
(\-ny/E)b, 
= d0(a){\+(\/E+ny2/E2) 
x[2-0 , (a) ] + O(/O}, (25) 
where 0,(a) = 0.9519, 0.910, and 0.869 fory = 
0, 1, and 2, respectively. Since 2 - 0,(a)>O, 6j 
increases as E ~', «, or 7 increases. A compari-
son of the estimate (25) with numerical results 
given by Parks [4] is made in Fig. 1. 
The temperature profiles are easily obtained in 
terms of the functions B0 = 60{s) and 6i = 0, (s). 
From the definitions (14) and (19), go(x) and 
gi(x) are seen to be given by 
x = yJd0(s)/Ao exp[(s - s)/2], 
go = s-s, 
g1=2-Ai/A0+(s-s)[2 + s-s]-el(s), 
in terms of the parameters, for 0 < 5 < s. 
3. THE EFFECT OF HEAT AND MASS 
TRANSFER RESISTANCES 
When using the new variables and parameters 
T=T/TS, c = c/cs, E = E/fs, 
y = yfs/cs, 
8cs" 
6 = expl E 
f,-\ 
(26) 
c, = c(l), fs=f{\), 
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 
fs=l+cs\l,/yvU+l)- (27) 
When taking into account hypothesis (3), Eq. 
(24), and the definitions (26), Eqs. (27) become 
cs=l+0(E-2), 
X
~
n
~E ) (fs~l) 
1 fco0(s) 2 /030(S)\ 2 
E I v E 
1 n y2\ 2ai0(s) + oii(s) 
+T;*0X—1—+ o w . 
IGNITION OF POROUS BODIES 
3.6 
3.5 
3.4 
3.322 • 
3.3 - 30 
O O 
o o o 
40 50 60 70 80 90 
2.1 
10 
20 30 40 50 
— , — 9-_ "Q—Q —O- -O—-Q 
60 70 80 90 
Fig. 1. A comparison between the present estimate of the ignition limit ( ) and 
numerical results of |4] (O) for n = 0 and no external heat and mass transfer resistances. 
whereat = max{y/E2, 7V.E3}. Then, from (26) 
we get 
(1 — ny/E)S = A0(s) + (2/E)A$s) 
+ (1 /£ + ny 2/E 2) A2(s) + 0(/I), 
(28) 
where 
An = 0o exp[ —coo/*0], 
A) = Ao(wo/*') exp[ — WQ/V)\, (29) 
A2 = Ao[2 — 6[ — (2coo + <ji$/i>], 
v = v(\ — ny/E). (30) 
The ignition limit corresponds to the first 
maximum of 5 = 5(s), which is reached at a 
value of s which, in the first approximation, is 
given by the equation 
0 ~J)<*>o + 60 
Z-co 0 
( j l ) 
Remember that in (29)—(31), w, and 0, are known 
20 
252 JOSE M. VEGA and A. LINAN 
Fig. 2. A0I, A„, and A2, versus i> = v(\ —ny/E), to be used in Eq. (28) to calculate the 
ignition limit. 
functions of s; therefore, (28) and (31) give a 
parametric representation, through s, of the 
function 5i = 8\(y). A()i, An , and A2i are plotted 
versus v in Fig. 2 for the spherical geometry (j = 
2). The function A0i = A0I(j7), which gives a first 
approximation of the ignition limit, was obtained 
by Thomas [17]. 
The temperature profiles may be obtained as in 
the previous section. 
4. A NOTE ON THE EXTINCTION 
The analysis given before describes the response 
of the porous body in a nearly frozen reacting 
mode, or ignition regime, which exists only for <5 
< h\. For h larger than an extinction value hE (§£ 
< di), there is an additional fast reaction re-
sponse mode, which is to be considered briefly. 
In the limit E -• oo, hE = finite, the effect of 
reactant consumption must be taken into ac-
count. 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 
thin reaction region, which is placed at the com-
mon 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 hE (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 -* oo but yE -• 
0, the concentration is c= 1 in first approxima-
tion, and the conservation equation for the en-
thalpy takes the form 
d2T j dT 
—— + ~ + 5' exP(~ l /7 ) = 0, 
dxz x dx 
( — ) =^' 
\ dx A=o 
T( 1) = -
1 
(32) 
in terms of 
r 
T=-
E 
and 5 '=—exp(£) . 
El 
IGNITION OF POROUS BODIES 253 
For the sake of brevity, we consider boundary 
conditions of the Dirichlet type in (32); an analy-
sis 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, [bE', b\'], of multiplicity of solutions of 
(32). The upper multiplicity bound is 
W =— expCE), 
/72 
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 
limit E —> oo. 
To calculate the lower multiplicity bound, bE', 
we introduce the expansions 
b' =5o' +E~i5[' + •••, 
luwiiic \JZ.), in HISt approximation, in me form 
U l n 1 Ulrt 
U
 " J_ ° - , . . ._, . _ „ 
dx1 x dx 
/dT0\ 
dx )x=o i ) —0, /0(1) —0. \ dx /
 x -
(33) 
A straightforward numerical analysis of (33) 
shows that it has two solutions if <50' > SOE ' . and 
no solution if So' < d0E', where 
I>OE' = 6.966, 16.837, and 29.565 
for y = 0, 1, and 2. 
Then bE' = 80E' + 0(\/E). For an analysis of 
(33) in cylinders (j = 1) see Parter [21]. 
REFERENCES 
1. Frank-Kamenetskii, D. A., Diffusion and Heat 
Transfer in Chemical Kinetics, 2nd Ed.. Plenum 
Press, New York, 1969. 
2. Chandrasekhar, S., and Wares, G. W., Astrophys. J. 
109:551 (1949). 
3. Chambre, P. L., J. Chem. Phys. 19:1795 (1952). 
4. Parks, J. R., J. Chem. Phys. 34:46 (1961). 
5. Shouman, A. R., Donaldson, A. B., and Tsao, H. Y.. 
Combust. Flame 23:17 (1974). 
6. Shouman, A. R., and Donaldson. A. B., ,ombust. 
Flame 24:203 (1975). 
7. Shouman, A. R., and Donaldson. A. B., ,ombust. 
Flame 29:213 (1977). 
8. Bowes, C. P., and Thomas, P. H., Combust. Flame 
10:221 (1966). 
9. Thomas, P. H., Trans. Faraday Soc. 56:833 (1960). 
10. Hardee, H. C Lee, D. O., and Donaldson, A. . . , 
Combust. Flame 18:403 (1972). 
11. Takeno, T., Combust. Flame 29:209 (1977). 
12. Bazley, N. W., and Wake, G. C . Combust. Flame 
33:161 (1978). 
13. Takeno, T., and Sato, K., Combust. Flame 38:75 
(1980). 
14. Gill, W.. Donaldson, A. B.. and Shouman, A. R.. 
Combust. Flame 36:217 (1979). 
15. Gill, W.. Shouman. A. R.. and Donaldson, A. B.. 
Combust. Flame41:99(1981). 
16. Fradkin, L. Ju.. and Wake, G. C J. Inst. Math. 
Appl. 20:471 (1977). 
17. Thomas, P. H., Trans. Faraday Soc. 54:60 (1958). 
18. Urrutia, J. L.,and Lifian.A., An Analytical Model of 
Hydrazine Decomposition Motors, Part III, Insti-
tuto Nacional de Tecnica Aeroespacial, Madrid, 
1975. 
19. Kapila, A. K., and Matkowsky, B. I., SIAM J. Appl. 
Math. 36:373 (197). 
20. Kapila, A. K., Matkowsky, B. J., and Vega, J. M.. 
SIAM J. Appl. Math. 38:382 (1980). 
21. Parter, S. V., SIAM J. Appl. Math. 26:687 (1974). 
Received January 1983; revised December 1983 


Large activation energy analysis of the ignition of self-heating porous bodies

Archivo Digital UPM

An Analytical Model of Hydrazine Decomposition Motors, Part III, Instituto Nacional de Tecnica Aeroespacial,

Diffusion and Heat Transfer

Faraday Soc.

revised

Large activation energy analysis of the ignition of self-heating porous bodies

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