Critical behaviour in chemically reacting systems—III. An analytical criterion for insensitivity

COMBUSTION A N D F L A M E 24, 43-52 (1975)

43

Critical Behaviour in Chemically Reacting Systems-Ill. An Analytical Criterion for Insensitivity B. F. GRAY School o f Chemistry, University o f Leeds, Leeds LS2 9JT, England

A form of practical stability, based on experimental reproducibility, is defined, and called insensitivity, to emphasise the most important physical characteristic shown by chemicallyreacting systems possessing this property. The necessaryand sufficient conditions for the insensitivity of an exothermic first order decomposition reaction are obtained rigorously and it is shown without approximation for the well-stirred ease that the maximumsubcritical dimensionless temperature rise is unity, regardless of the value of parameter e (the inverse of the dimensionless heat of reaction).

Introduction

The theory of thermal explosions for constant concentrations of reactants as developed by Semenov for a homogeneous medium, offers no conceptual difficulties. The solutions of the energy conservation equation are clearly divided into two classes: (a) those where the temperature is bounded for all time; and, (b) those where the temperature becomes unbounded after a finite time. The latter class of solutions are often referred to as supercritical and the former class as subcritical. The critical solution is defined as the subcritical solution attaining the highest temperature, yet remaining bounded. All this is well known and best discussed on a thermal diagram [1 ]. In terms of the dimensionless temperature rise, the critical solution is characterised by 0 max= 1. When the realistic case of a time dependent reactant concentration is considered, the definition of criticality becomes much more difficult since the temperature always reaches a maximum and finally decays to 0 = 0 as t ~ oo. Some attempts have been made to bypass this problem by simply obtaining numerical solutions of the equations in question [2-4] and examining the latter, which are "exact", with a view to choosing a critical solution. This is extremely difficult to do since the nature of the time dependence of the dimensionless temperature is extremely sensitive to parameter variation in the critical region and the ensuing results are necessarily arbitrary and

subjective. At best, the results of such computations and arbitrary definitions of criticality indicate [1] that the maximum subcritical temperature rise is somewhere in the region 0~lor2. Thomas and Bowes [5] adopted a different approach. They made the empirical postulate that criticality was signaled by the appearance of an inflection in the temperature time curve. Again critical maximum dimensionless temperature rises between 1 and 2 are indicated, and as the heat of reaction rises (reactant consumption becomes less important), the maximum critical dimensionless temperature rise approaches unity continuously as one would expect of a meaningful definition, i.e., it should approach the Semenov limit continuously if the latter is to have any meaning at all. Another approximate method has been devised by Thomas [6], who assumed that the reactant was all consumed at a fixed "average" temperature, thus integrating the equation for fuel concentration and then using the approximate reactant concentration to discuss the energy conservation equation. With a quadratic approximation also made to the temperature dependence of the rate constant, it is not easy to see the nature of the errors in the results but the numerical values are in fair agreement with other approximate methods. Frank-Kamenetskii [7] performed a similar analysis at an earlier date, and his results (and

Copyright ©1975 by the Combustion Institute Published by American Elsevier Publishing Company, Inc.

44 numerical errors) are adequately discussed by Gray and Lee [1]. Adler and Enig [8] have contributed yet another arbitrary definition of criticality in this type of system. They take the critical curve as that one which first exhibits an inflection in the 0~. curve before the maximum in 0, i.e., they eliminate the time variable and work in the phase (0;~) plane. They showed, analytically, that this first possible inflection always occurs at 0 = 2 for a first order reaction regardless of other parameter values in the system. The Adler-Enig criterion is open to serious criticism [9], in particular it does not pass over uniformly to the Semenov limit as the small parameter e ~ 0 +, which is necessary if the classical thermal theory of explosions with e = 0 is to be accepted as meaningful. Gray and Sherrington [9, 10] have applied the techniques frequently used in Liapounov stability theory to this problem and obtained criticality criteria similar to those of Thomas which pass over continuously in the limit e ~ 0 + to the results of the Semenov theory. More recently, Gray [ 11, 12] has shown that the Semenov theory, where e = 0, often referred to as "explosion theory without fuel consumption" has not been properly formulated owing to unwitting use of an infinitely stretched time coordinate. The previously used solution (with constant fuel concentration) is in fact only valid over a very short time period and is completely analogous to the inner solution well known in the boundary layer theory. The complete solution (inner and outer) can be obtained exactly for the case e = 0 and this does in fact include the consumption of the fuel, which decays to zero. The cricital condition is unchanged from the one originally derived by Semenov but the theory is put on a firm basis and corrections to it (for small e) are no longer to be regarded as anything more than the rounding off of corners on the uniformly valid combination of inner and outer solutions. In this region of small e, it is possible to use the techniques of singular perturbation theory to calculate corrections to the continued Semenov solution, but at the moment this would not appear to be of pressing importance as it has been shown that the best currently available thermocouples [12] are incapable of distinguishing between the

B.F. GRAY zeroth order and first order corrected solutions for e ~ 10 -2 . Of more interest in the region where e is appreciably greater than the above value, as for sawdust and fertilizer piles, bales etc., and here a perturbation expansion may not converge very rapidly if at all. Even if the expansion does converge, one still has the problem of defining the critical condition in a nonarbitrary manner. In this paper, this problem is solved exactly using stability theory without obtaining an explicit form for the solution of the equations. The definition of stability is an operational one, distinct from that of Liapounov, and also more stringent than the latter. Stability and Sensitivity It seems desirable to relate the notion of the physical stability of a system to the experimental errors inherent in attempts to set up the system in a given initial condition repeatedly. Clearly this cannot be done exactly, but one tends to regard a combustion system as stable if it behaves reproducibly in successive runs when the initial conditions are the same within experimental error. One speaks of a critical limit when initial conditions, which are the same within experimental error, give rise to subsequent behaviour, which is not the same to within experimental error, i.e., the system becomes sensitive. This notion of stability would apply to a system well beyorid its explosion limit, where it would reproducibly explode every time giving the same temperature maximum. It would thus perhaps be better to refer to insensitivity rather than stability, as reference to a consistently explosive situation as 'stable' would cause confusion. We shall thus refer to a system as insensitive if fluctuations of the initial condition (within the experimental error) do not give rise to measurable variations in the subsequent trajectory. We shall refer to the critical limit(s) as those bound. aries in parameter space where insensitivity ceases to hold and the 'same' experiment can give different results. Mathematically, we can define insensitivity in two ways, closely related, depending on whether we choose to use time as the independent variable or whether we choose to work in the phase plane for two variable autonomous systems. In the former case, let the system be

CRITICAL BEHAVIOUR IN REACTING SYSTEMS described by n dependent variables x 1(t) . . . Xn(t ), with initial values xl ( 0 ) . . . Xn(O ). We assume that we can set up the initial condition within a hyper-

sphere of radius R "~i~=l ~ (O) ) ~, where ~i(O) is the error expected in setting up variable x i at t = 0. We shall call the trajectory, x I ( t ) . . . Xn(t), insensitive if all trajectories which originate at t = 0, within the hypersphere of radius R, remain within a cylinder (with a time-like axis) of the same radius for all subsequent times. With just two dependent variables the hypersphere becomes a circle and the trajectory becomes the axis of a cylinder of radius R. The trajectory is insensitive if all neighbouring trajectories originating within the circle at t = 0 remain within the cylinder for all subsequent times. We shall illustrate this definition with application to the Semenov equation below. If the neighbouring trajectories to the one in question are denoted by x 1 (t) + ~1 (t) . . . . Xn(t ) + ~n(t), then the necessary and sufficient condition that the trajectory, x 1( t ) . . . Xn(t), be insensitive is that [13] :

A(R)=Maximum

1 d (

~)

~<0, (1)

t,~ i

the maximization being subject of course to n (z ~ - = R = constant. i=1 In the case of an autonomous system with only two variables, it may be more convenient to define a different type of insensitivity especially if one is working in the phase plane. We replace the time by one of the two dependent variables, and choose or redefine the latter so that it increases with time. If the two variables are y(t) and x(t), we now define a trajectory as the functiony(x), and call this insensitive in the phase plane if all traj[ctories originating within R = (~(0) 2)½ at x = 0 remain within R for 0 ~
45

Maxixmum Id~x x/~- 1 ~"=

R

x =Maximum

(2)

[X/~'~d1~]=R
where neighbouring trajectories are denoted by y (x) + ~"(x). Insensitivity in the phase plane means that curves in the x - y plane which originate close together will remain close together and will not 'spread' apart in the x-y space (as they do near an unstable node for instance). Insensitivity in the phase plane does not imply insensitivity since the velocity at which the phase plane curves are traversed does not enter into the definition of the former. Further work on the relationship between the two forms of insensitivity would be desirable from a mathematical point of view, but neither seems to have been extensively discussed in the mathematical literature [ 13]. However, it appears that the converse statement is true, i.e., that insensitivity implies insensitivity in the phase plane. In spite of its weaker character, insensitivity in the phase plane is much easier to handle in general and will be used below to discuss the explosion of a fuel decomposing by a first order reaction, giving a simple analytical result for the insensitivity limits. Sensitivity will be discussed in a later paper, where more than two variables will be considered. The Semenov Theory: e = 0 Although this model, characterised by the single energy conservation equation, has recently been solved exactly [12] to exhaustion of fuel, it is a useful illustrative example of the meaning of insensitivity and its coincidence with stability in special cases. Consider the single equation:

dO = eO dr a 0,

(3)

for a trajectory 0(0. Neighbouring trajectories, 0 ( 0 + ~'(r) will satisfy the same equation, thus:

dO + d_~ eO + ~ d~ dr = - aO - a~,

(4)

46

B.F. GRAY

so substituting from (3), d ~ = e 0 ( e ~ - l ) -a~, dr

(5)

and the necessary condition (1) for insensitivity becomes: M a x i m u m [ ~ d~1

=

Maximum [e0 (e ~"- 1)~"- a~"2]

[R <~0

dzd--~O=exPIl+~]-aO

dr

aOmax = 0 .

(12)

(6)

subject to ] ~1 = R. The maximisation with respect to r in this case simply gives 0 = 0 max' where 0 max satisfies d0_0= e0max _

In a real situation, R would be chosen to be a measure of the accuracy with which the initial conditions could be set up experimentally (see discussion). It is interesting, as a second example to treat the Semenov problem with the Arrhenius term included fully, i.e., without the exponential approximation, replacing Eq. (3) by

(7)

where/a <<1. This equation allows the possibility of steady states beyond the critical region for ignition, i.e., fully ignited states. This upper region is well known to exhibit extinction and should therefore show a limit where insensitivity breaks down a given value of the maximum dimensionless temperature. Following the same procedure as above, we obtain

below

Substituting in the inequality (6) and noting that [ ~ = R, we obtain 0max ( e R - 1 ) - R < ~

0

(8)

as the necessary and sufficient condition for insensitivity within a tube of radius R. Rearranging (8), we find

0max<

R eR------~ '

(10)

identical with the Semenov condition for stability. If we do not take the limit R ~ 0, but nevertheless, assume R < < 1, then condition (9) becomes: 0max < 1 - R

xp

l+/a0)[l+/a(0+~)l

(9)

and the maximum value of the right hand side of (9) is unity as R -~ 0. So if we assume arbitrarily high precision in setting up our experiments, the necessary and sufficient condition for insensitivity is that:

0 max ~< 1

×

(11)

and, finally, the condition for insensitivity is seen to be:

Omax~xPl(l+gOmax)[1R+ld(Omax+R)])I1
(14)

0 max and R cannot be separated explicitly in the above inequality, but the only physical interest is in the case when R < < 1. Expanding the exponential and retaining terms to first order in R, we obtain

CRITICAL BEHAVIOUR IN REACTING SYSTEMS

tion about insensitivity even in the absence of any singularities, a point which has caused difficulty in many previous discussions.

0max

[ 1 + btOmax ] 2

l+g0max )

47

2(l+/a0max)2

..

(15)

The Case e 4= 0: Finite Heat of Reaction In this case, we have two first order differential equations:

~<1, so in the limit of infinite precision (the parallel of all previous discussions of the subject where this has been assumed implicitly)

0 max [ 1 + I.tOmax ] 2

dX_ ~<1.

either Omax ~<1 + 2# +0(#2),

(17)

eXexp(-

dr

(16)

Rearranging this inequality and isolating 0 max' we can quickly show that the inequality (16) implies

(19)

dO-~-=hexP(l-~go~-aO'dr

0

1 + laO

)

'

(20)

if we restrict ourselves to first order reactions. Since these equations are autonomous, it is both possible and convenient to work in the phase (0 - ;k) plane. However, in discussing questions of sensitivity and stability, it is convenient to use the variable z = 1 - X as the independent variable as it increases with time. Our basic equation in the (0 - z) plane is then

or

o

max

/> __i bt2

_2 +o(1), /.t

(18)

as the necessary and sufficient conditions for insensitivity. Condition (17) is identifiable as the subcritical condition, the equality sign denoting the ignition limit. Condition (18) represents the condition for stable burning, and the equality sign gives the extinction condition (which of course is lost if the exponential approximation # = 0 is made). From the inequality (16), it is easily shown that the system is insensitive globally, i.e., under all conditions, for/~ t> ¼. Thus, in this region, critical (sensitive) behaviour is not shown. This result is not new, being well known since the original formulation of Semenov. However, the above method throws a new light on the well known theory and it is interesting to note that it is not a localised treatment, around a singular point, for instance. Information is given about both the ignition and extinction limits from the single inequality (16). We shall see in the next section that Eq. (16) is capable of giving similar informa-

aO exp ( -

dO = 1 _ dz

e

0 ) 1 + O0 e (1 -Z)

(21)

with the initial condition 0 = 0, z = 0. The interval is 0 ~
a(O + ~ ' ) e x p ( =1

dz ~

-(0 +~')~ 1 + U(0 + ~ _ d__0_0, e(1 -z) dz

(22)

or finally, ~'= a e x P ( l ~ O ~

dz

e (1 - z)

~ - (0 + {')exp([ 1 + p0][1 ~+/~(0+~-)]

1" (23)

48

B.F. GRAY

Applying the condition (2) for insensitivity in the phase plane to Eq. (23), we obtain the condition:

being points A and B. These points are given approximately as 0"4"- la12

Oc

xp [ l + # 0 c ] [ l + / a ( 0 e + R )

-

2 "''' la

(30)

0B - 1 + 2/a + . . . .

(31)

~
where 0 C is the value of 0 which maximises the function d

(Vt~--). Straight-forward analysis

shows that 0 c is not the maximum value of 0 attained (as it was in the previous example), but the value of 0 where the trajectory O(z) intersects the curve

i.e., the roots of Eq. (29). Returning to the inequality (24) and taking the limit R ~ 0, we obtain the corresponding inequality 0c/[ 1 +/a0c 12 <~ 1

~=a0exp ( _ 1 + ~ ) _

e(0-D 1 - 0 +f(1 -/.Ig') '

(25)

where

l

_R

Rexp, [1 +U0] [1 +/a(0 +R)] f = 1-exp

f

as the necessary and sufficient condition for insensitivity in the phase plane for this system. Inequality (32) holds for

(26)

[l+gO] [I+~(O+R)I

and g=

(0 + R) (2 + 2/a0 +/.tR)

(27)

(1 + uO) 2 [1 + u(O + R)] 2 •

In the limit as R ~ 0, i.e., our experimental accuracy becomes (or is assumed to become) arbitrarily high, Eq. (25), (26) and (27) reduce to

e [o - (1 + uo)21 1 - 0 + (1 + IsO) 2 [1 - 2/a0/(1 + / a 0 ) 3 ]

(28)

This curve crosses the maximum locus at two points, the roots of the equation 0 = (1 + ~u0) 2

.................. S

(29)

In Fig. 1, the curve (28) is shown in relation to the maximum locus, the two solutions of Eq. (29)

(32)

i_

c

~ z=(~-~)

Fig. 1. The intersection of curve 28 (which has two branches) with the maximum locus curve.

49

CRITICAL BEHAVIOUR IN REACTING SYSTEMS 0 c >t 0 A ,

(33)

and 0c < 0B ,

(34)

and in the limiting case/2 ~ 0 (33) becomes meaningless and (34) becomes Oc
The Case with Finite R,/2 --, 0 Since the exponential approximation, (/2 -, 0), has been the most extensively discussed case in the literature, it is worthwhile to present the exact results for this case for insensitivity in the phase plane with nonzero R as well as the limiting case R --, O.

In this case we wish to find the conditions for insensitivity in the phase plane of the equation

(35)

It remains to show that 0 c can be replaced by the experimentally measurable 0 max in all these inequalities. This can be shown without difficulty (see appendix) and we can then conclude that the necessary and sufficient conditions for insensitivity in the phase plane of Eq. (19) is that either

If we denote neighbouring trajectories by O(z) + ~ (z) straightforward substitution in Eq. (38) leads us to: d-[ - ae-O

az

Omax>~ ~

-2/2 . . . . .

(38)

dO = 1 _ aOe -0 dz e e(l - z)

e(1-z)

[0(1 - e-~') - ~"e-~'l , (39)

(36) and the necessary and sufficient condition (2) for insensitivity in the phase plane leads to the condition

or

0 max < 1 + 2/2 . . . . .

(37) Oc(1 - e - R ) - R e - R < 0

the former condition representing fully fledged and reproducible ignition, and the latter representing fully fledged and reproducible slow combustion. Experiments or runs yielding maxima in the range 1 + 2/2 < 0 max < ~ - 2 will not be /2 /2 reproducible in a real experimental system except when: (a) The roots of Eq. (29) merge and the region of sensitivity vanishes. This is easily seen to occur when/2 I> 0.25 exactly as in the Semenov case. (b) The value of e is so large that the intercept of the curve (28) on the X axis occurs at a value I> 1, i.e., beyond the initial condition. Then again all integral curves are in the insensitive range and sensitive behaviour is again no longer possible. This occurs when e ) 2, regardless of the value of /2. This would represent a material with very low heat of reaction, whereas the condition/2 ) ¼ would represent a material with very low activation energy. Since it is necessary that 0 max/22.

(40)

where R is the radius of the 'tube' with O(z) as its axis within which we require neighbouring solutions to remain for 0 < z < 1.0 c is the value of 0 where the integral curve crosses the curve (41)

X = aOe "0 + e (0 - f )

0-1-f

'

where f = R e -R ](1 - e - R ) .

(42)

Again (see appendix), we can prove that 0 c can be replaced by 0 max and so the rigorous condition for insensitivity in the exponential approximation is Oma x < . R l ( e R - 1)

(43)

The function on the right hand side of inequality (43) is always < 1 and, in particular, for small but finite R we have

B. F. GRAY

50 0 max ~< 1 - R/2

(44)

This is in agreement with recent previous work on the zeroth order problem (e = 0), where it was shown that the solution of (38) depends discontinuously on a , at the value a = e, when 0ma x = 1 (11, 12). Of course, discontinuity implies sensitivity, and the present work passes over smoothly to the case 6 = 0 in the limit e ~ 0, since interestingly, the conditions (43) and (44) do not involve e, provided the latter is <0.5, when the system is always insensitive and the phenomenon of criticality disappears.

Discussion The present work gives an exact treatment of the classical problem of "critical conditions including fuel consumption in a well-stirred system", misnamed as this problem has turned out to be (11, 12). It is given on the basis of a mathematical formulation of what is meant by the experimentalist when referring to criticality, i.e., criticality is the condition when the system ceases to be insensitive to the inherent errors involved in its repetition, and the 'same' experiment no longer gives reproducible subsequent behaviour. We require, for insensitivity, that trajectories lie within a 'tube' of constant radius around a given trajectory as axis for all times of interest provided they start within it initially. A previous discussion (9, 10) on the basis of stability in the sense of Liapounov indicated a similar region of parameter ( a - 6) space for this type of stability also, using however fairly restricted types of IAapounov function. It now appears, that in relation to the concept of insensitivity as used by the experimentalist (often unfortunately referred to as stability) stability in the sense of Liapounov is too general in so far as it effectively considers a tube centred on a given trajectory, but this tube does not necessarily have to have a width given by the spread of initial values, it can be much greater than this. A choice of relatively simple Liapounov functions, such as quadratic, has the effect of strictly limiting the ratio of the tube width to the experimental error, but clearly the ratio which corresponds with the experimental notion of stability is unity, i.e., the definition used here. The correction to a from its zeroth order value

e has already (i2) been shown to be within the experimental error of the best thermocouples for e ~- 10 -2 . In the spirit of this discussion, i.e., relation of the conditions for insensitivity to experimental reproducibility, it is natural to examine the accuracy with which a given experiment can be set up, reproducibly, and its effect on the parameters a and e. Under most conditions, the main lack of precision occurs in the ambient temperature setting, this rarely being controllable to better than 1 K. The ambient temperature dependence of a and 6 is given by

a o~T OeR/RT o

(45)

e ~x T~0,

(46)

and it is easily shown that the relative uncertainties caused by an uncertainty 6T o are given by:

6--ga 6 =T (--T-oo ° ' -T'-Ra-~-oo) 6e _ 6

26T o

(47)

(48)

ro

For a typical explosive material, one may take E 40 kcal, T 0 ~ 500 K, 6 T o ~ 1 K giving 6a

~, 0.08 ,

(49)

a

16e/el being completely negligible. For e <

10 -2 , the corrections to the critical value of a due to "fuel consumption" are in fact less than the uncertainty given by (49), and the theory developed here for criticality is only necessary for values of e > 10-2 with present day experimental techniques. Appendix: Proof that 0 c ~< 1 - R implies 0 max < I - R We shall prove that if 0 e satisfies the inequality (40), where 0 e is defined as the intersection of the integral curve with the curve C

CRITICAL BEHAVIOUR IN REACTING SYSTEMS

X=aOe-O + e(O-1")

(41)

O- 1 -f

then the same is true of 0 max' and we can then replace inequality (40) by the more convenient one (43). The proof is given explicitly for the case /a = 0, but can be repeated for any/a without significant change. The proof amounts to showing that at any intersection x between the integral curve I and the curve C we must have

51 ". 0max < 1 - R, if 0c < 1 - R , which is what we would like to prove. It remains therefore to prove (50). Differentiating the curve (41) to obtain the left hand side of (50), and using the differential equation (38) directly to obtain the right hand side, we get

a(l -0~ )e -°c +

e

o~ -1 - f

e(0c / ) . > - (0 c - 1 _f)2,

eX~ Xc _aOce-Oc

'

which can be rearranged to give i.e., the intersection x must be as depicted in Fig. 2. If (50) is satisfied once the integral curve

a( l - Oc )e -Oc

e

(Oc - 1 _f)2 >aOc (0c - 1 - f ) e-Oc

©

-6,

( f - o~ ) and since 0e < f e 1 - R , this inequality becomes, after multiplying through by ( f - 0c) and rearranging,

M I UM LOCUS

afe-Oe > e(f - Oc)

,~~

CURVE~C'

... I-R1!-_2-_-_--_-_-_-_-_-_22 ................ 7 2 2 .-.2. .____x~_ ___ -

,urm~*~

gfO+f) A =(l-z) Fig. 2. The intersection of the curve C (Eq. 41) with the integral curve of Eq. 38. has crossed C at x, it cannot re-cross C as that would involve an intersection which violated (50). The integral curve must therefore cross the maximum locus below the point z, where 0 = 1 - R

t

-(~c - - f T--f)

,

since 0c < f , the numerator o f the square bracket will be negative and the inequality will be satisfied. Hence inequality (5 0 ) is proved, and we can use 0 max rather than 0c in the insensitivity condition.

The author wishes to acknowledge helpful discussions with Professor P. Gray. References 1. Gray, P., and Lee, P. R., Comb. and Oxid. Rev. 2, 1 (1967). 2. Rice, O. IL, et al., J. Am. CherrL $oc. 57, 2212 (1935). 3. Todes, O. M.,Acta Phys. U. R. S. S. 5, 785 (1936). 4. Squire, W., Combust. Flame 7, 1 (1963). 5. Thomas, P. H., and Bowel, P. G., Brit. J. App. Phys. 12, 222 (1961). 6. Thomas, P. H.,Proc. Roy. Soc. (London} A 262, 192 (1961). 7. Frank-Kamenetskii, D. A., Zh. Phys. Khim. 20, 139 (1946). 8. Adler, J. W., and Enig, J. W., Combust. F/ame 8, 97 (1964).

52

B . F . GRAY

9. Gray, B. F., and Sherrington, M. E., Combust. Flame 19, 435 (1972). 10. Gray, B. F., and Shenington, M. E., Combust. Flame 19, 445 (1972). 11. Gray, B. F., Combust. Flame, 20, 313 (1973). 12. Gray, B. F., Combust. Flame, 20, 317 (1973). 13. Hahn, W., Stability of Motion, Springer-Verlag, New York (1967), pps. 269, 278.

Received May 14, 1974 Nomenclature 0

dimensionless temperature rise =

E ( T - T o ) / R ~o.

TO T

ambient temperature. reactant temperature. dimensionless concentration.

u

Rro/E

E a

activation energy. dimensionless effective heat transfer coefficient. dimensionless time.

r

e

cpRTao/QEW0

c p Q W0

heat capacity. density. heat o f reaction. initial concentration o f fuel (~ = W/WO).