UNIFICATION OF THERMAL AND CHAINTHEOR~ES OF EXPLOSION LIMITS
2747
On the Unification of the Thermal and Chain Theories of Explosion Limits
by B. E’. Gray and C. H. Yang Defense Research Corporation, Santa Barbara, Cdifornia
(Received March 17, 1966)
A theory of explosions is developed which contains, as special cases, both the thermal theory and the isothermal chain theory of Semenoff. It contains qualitative features not present in either theory but under special circumstances reduces to each. Two illustrative applications are discussed in a general manner. Phase plane analysis, particularly suited to systems with only one important intermediate, is used.
1. Introduction It is well established in the literature that two alternative mechanisms of explosion exist for gaseous systems. In the thermal theories the heat generated in the system by chemical reaction or other means is compared with the heat lost by the system to its surroundings by means of conduction or radiation in a closed system. The explosion limit is given by1 @(T,P) = Z(T,P,d), where T and P are the temperature and pressure in the gaseous system and d is a parameter connected with the physical size and shape of the system. @ is usually an Arrheniuslike function of temperature. This type of theory does not need detailed knowledge of chemical kinetics. It is therefore incapable of yielding any chemical kinetic data or explainingpurely chemical effects such as sensitization, inhibition, etc. On the other hand, the alternative type of theory, the isothermal chemical kinetic or branched-chain theory, originally devised by Semenoff , 2 ignores the energy balance in the system. No account is taken of the fact that the chain branching and termination reactions are not isothermal and must, in fact, raise the temperature of the system above that of its surroundings, which in turn causes losses to the latter. In fact, if the heat produced exceeds the losses before branching exceeds termination, it is conceivable that a branched chain reaction may explode for purely thermal reasons. In the Hz-Oz reaction it has been noted that the determination of the chain limit is complicated by the tendency of the gases to self-heating,athus causing thermal explosion. Purely isothermal theories in the form discussed by Semenoff cannot account for the effect of added chain carriers on the explosion limits; i e . , they predict that the explosion limit is inde-
pendent of the initial condition. In fact, the limits are very sensitive to the initial concentration of chain carriers. The present paper presents a unified treatment of the thermal and chain mechanisms of explosion by considering, simultaneously, the kinetic equation of the chain reaction and the energy conservation equation of the system. The explosion phenomenon is analyzed by studying the behavior of the solutions of these equations in the phase plane.
II. The Unified Mathematical Model To illustrate a unified theory for both the thermal and the chain mechanisms, let us consider a simple reaction scheme with firsborder chain branching and termination. xf x
... +kbx
-I-
...
+ . . . +inert products kt
AHb
AHt
where z represents the concentration of the chain carrier and Icb and kt are the rate constants for branching and termination, respectively. The initiation of the chain carrier is neglected as only the solutions with finite initial concentration of the chain carrier are considered, The kinetic equation, which completely characterizes the isothermal chain theory, is dx -= dt
(kb
- kt)z
(1) C. H. Yang, Combust. Flame, 6 , 215 (1962). (2) N. N. Semenoff, “Chemical Kinetics and Chain Reactions,” Pergamon Press, Ltd., Oxford, 1953. (3) G . J. Minkoff and C. F. H. Tipper, “Chemistry of Combustion Reactions,” Butterworth, 1962, p. 13.
Volume 69,Number 8 August 1966
B. F. GRAYAND C. H.YANG
2748
whereas the energy conservation equation, which completely characterizes the thermal theory, is
where 1(T - To) is the thermal loss term and c is an average spe,cific heat for the system. We assume fixed pressure throughout this discussion and also spatial homogeneity of the system; ie., T and x are mean values across the reaction vessel. Now let us put A H b / c = hb, etc., so we have to deal simultaneously with the two equations
dT - = (kbhb f ktht)x dt
-1
(3)
As neither of these equations contains the independent variable explicitly on the right-hand side, we can make use of the methods of the phase (2’ - x ) plane4in this case. (The present method is particularly suited to systems with only one important intermediate. If more than one has to be considered, the same type of analysis can be used with somewhat greater complication.) Clearly simultaneous integration of eq. 3 to give x and T as explicit functions of time is extremely difIicult owing to the temperature dependence of the rate constants (k Although numerical computation of the trajectory is possible when a proper set of initial values is given, these computations usually fail to give an over-all picture of the physical behavior. To study eq. 3 in the phase plane, the first step is to eliminate the independent variable
-
+
dT - -- (kbhb ktht)x dx ( k b - kt)X
-I
then apply the well-known stability theorems of Liapounov to determine the behavior of the integral curves in the ( T - x ) plane near the singularities and possibly the qualitative nature of the integral curves in the whole phase plane. Retaining only linear terms, the transformed equation is easily shown to be
+
dT’ - - - x’R(TJ2(ktht f kbhb) T’(XekbEbhb - a) dx’ X’R(TJ2(kb- k t ) T’(X&bEb)
- Ax‘
+
BT’ Cx’ -I- DT’
(7)
where a = RTs2(bl//bT)(,= x. and T , are used here without superscript since we have not yet specified which singularity is involved. To derive this equation we have written kb = Abe-Eb’RT and kt = At where Ab and At are independent of temperature, and we have assumed the activation energy of the recombination reaction to be zero although, in actual practice, it will usually be a small negative number. The nature of the singularity is found by examining the characteristic equation associated with (7) , Le. X2 - ( B
+ C)X + BC - AD = 0
(8)
the roots of which are = ‘/z(B
+ C)
f
‘ / z d ( B - C)2
+ 4AD
(9)
If X , are both real and differing in sign, that particular singularity is a saddle point. The condition for this is A D > BC, ie.
+
(kbhb ktht)X,k& > (kb - kt)(XakbEbhb - a) (10) Equation 10 can be rewritten as
(4)
The next step is to find the singularities of this equation which are given by solving the simultaneous equations
(6)
if all the positive terms in eq. 3 are represented by (R as the total heat release rate. If we assume dkt/bT = 0, the left-hand side of (11) is always positive. Inequality (11) will be satisfied with either da/bT 2 bl/bT or kt 2 kb. The former relationship is the condition for explosion in the thermal theory while the latter one is the condition for explosion in the chain theory. One solution of (5) is obviously kb = kt so (11) is satisfied for this singularity, which is therefore a saddle point. Tel is identical with the explosion limit temperature as deduced from Semenoff’s isothermal
to enable eq. 4 to be linearized in the neighborhood of each of the singular points. Having done this, we can
(4) H. T. Davis, “Introduction to Nonlinear Differential and Integral Equations,” Dover Publications, New York, N. Y., 1962.
X(kb
X(kbhb
- kt)
=:
0
+ k&) - 1 = 0
(5)
We denote the coordinates of the singular points by ( x i , T d ) , and in this case there are only two so j = 1, 2. The third step is to transform to new dependent variables
2’ T’
- 2: T -Ti
= x =
The Journal of Physical Chemistry
UNIF~CATION OF THERMAL AND CHAINTHEORIES OF EXPLOSION LIMITS
chain theory. The second singularity is obviously x8@)= 0, T8(2)= To, which turns out to be a stable nodal point with no reactive intermediate and the temperature of the system coinciding with the temperature Toof the heat bath. As there are only these two singularities in the phase plane we can draw a qualitative diagram of the integral curves; see Figure 1.
---
I I / /
w X
Figure 1.
The saddle point (1) is cut by two dotted curves, the separatrices, which divide the positive quadrant of the phase plane into four regions (A, B, C, and D). The slope of the separatrix which cuts the T axis is negative owing to the presence of the heat loss term in the energy conservation equation. Obviously, any initial condition in quadrants A or B will result in explosion, while any initial condition in quadrants C or D will result in the system approaching the stable nodal point ( 2 ) without any increase in temperature. In the limit, as all the heats of reaction get smaller and smaller, the separatrix gets almost parallel to the x axis, and the result is that the explosive quadrants A and B are separated from C and D by the line k, = Jcb; i.e., it reduces to Semenoff’s isothermal result from the chain theory. Thus, the main qualitative effect of removing the restrictive assumption of isothermal processes is that the explosion limit (separatrix) becomes dependent upon the initial concentration of the chain carrier, as occurs inexperimental resutts. At very low x concentrations it is obvious, from eq. 4, that the slope dT/dx of any integral curve, as well as the separatrix, is largely determined by the heat loss term. The slope dT/dx of the separatrix reflects the sensitivity of the explosion limits to the initial concentration of x. The nonisothermal theory provides a framework for the explanation of the effects of vessel coatings on explosion limits insofar as the coatings are carrier sinks. In view of the negative slope of the separatrix, it can be concluded that the weaker the carrier sink, the lower
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the explosion temperature. The detailed application to surface effects in specific reactions will be discussed elsewhere. Examination of Figure 1 indicates some interesting types of behavior; e.g., in some explosions one should observe an initial spontaneous decrease in temperature just before the large and rapid increase characterizing the explosion itself. However, the chain carrier itself is increasing all through the process. On the other hand, if we have an initial condition over to the right of quadrant B, the temperature will increase very slowly at first, then suddenly increase rapidly (highly nonlinear behavior). However, the near-discontinuity in the temperature variation is well characterized in the carrier concentration behavior since, during the slow temperature rise, it is in fact decreasing and starts to increase rapidly with the temperature. Similarly, from the phase diagram one can discuss various types of stable behavior in an equally simple manner.
m. Application to a Pure Thermal Explosion:
The Reaction A --t x +.B We have shown in the previous section that the unified model of explosions reduces to the chain theory when heat release and loss are neglected. Now we will examine the case of a pure thermal explosion. Consider the reaction scheme A +-x + B, in which there is no radical chain, although an intermediate radical x is involved. The conservation equations are (in the initial stages) dx - = kl - kzx dt
and the singular points are given by kl(h1
+ hz) = I
x8 =
ki
-
kz
(13)
Obviously if ’ > 0, the reaction must be exothermic; otherwise (13) would have no solution. The transformed equation is easily shown to be
+
dT’ - - - R(T.)2kzhzx’ (kihiE1 - a)Tf dx’ -R(TJ2k2xf klE1T’
+
(14)
From the previous discussion it is obvious that for the system to exhibit explosive behavior it must have at least one saddle point. The necessary and condition for this is
R(TA2hhzklE1> (klh&
-
a)R(Ts)2k2
Volume 69,Number 8 August 1966
R. GARY,R. G. BATES,AND R. A. ROBINSON
2750
IV. Conclusion
i.e. (klhlEl -
or
klEl(h1
a!>
> hZklE1
+ h2) >
(15)
a!
+
i.e., the reaction must be exothermic (hl hz > 0) and sufficiently so that (15) is satisfied, or the singularity is a stable nodal point, no explosive behavior being possible. Obviously, in an adiabatic system ( I = 0, a! = 0) inequality 15 is satisfied, and explosive behavior will be shown. Again the separatrix which cuts the T axis has a nonzero slope (negative in most cases), and the intermediate concentration will determine if the explosion is to occur. Thus, in a nonchain explosion of this type, vessel coating with materials which affect the value of 2 will alter the explosion limits.
By consideration of the kinetic and energy conservation equations simultaneously, one can deduce a considerable amount of semiquantitative and qualitative information about potentially explosive materials. The restrictive assumptions of both the isothermal chain and purely thermal theories are removed, and the resultant generaliied theory naturally explains phenomena not amenable to explanation by either theory separately. For example, vessel coating, addition of chain carriers and intermediates, etc., all affect the explosion limit explicitly within the framework of the theory. Also, as the separatrix is not parallel to the 2 axis, it may be possible to account for the irreproducibility of results encountered with certain coatings if these coatings produce x concentrations in a region where the slope of the separatrix is large.
Dissociation Constant of Acetic Acid in Deuterium Oxide from 5 to 50". Reference Points for a pD Scale
by Robert Gary, Roger G. Bates, and R. A. Robinson National Bureau of Standards, Washington, D . C. (Received March 19, 1966)
Electromotive force measurements of a cell without liquid junction have been used to determine the dissociation constant of acetic acid in deuterium oxide from 5 to 50". The enthalpy, entropy, and heat capacity changes on dissociation of acetic acid have been calculated. Values of -log (aD+ycl-) and the conventional ~ U Dvalues for the equimolal (0.05 m) acetic acid-sodium acetate buffer solutions have been determined. These provide a second k e d point for standardizing the pD scale, supplementing data for the equimolal mixture of KDzPOl and NkDP04 established in an earlier investigation.
Introduction
P t ; D2(g) at 1 atm., CHaCOOD (m), The measurement of the second dissociation conCHaCOONa (m), NaCl (m'), AgCl; Ag stant of deuteriophosphoric acid in deuterium oxide at 5" intervals from 5 to 50". The dissociation conhas been repohed recently,l along with of p(aDycl) and paD for equimolal K D ~ ~ o ~ - Nstant ~ ~of Dacetic ~ ~acid ~ has been derived over this tembuffer solutions. (1) R. Gary, R. G. Bates, and R. A. Robinson, J. Phys. Chem., 68, We have now measured the e.m.f. of the cell 3806 (1964). The Journal of Physical Chemistry
