Thermal Theory of Combustion and Explosions

N. N. Semenov, Leningrad

Introduction

In 1884, in his book Essays on Chemical Dynamics1, Van ’t Hoff formulated the basic laws governing the course of the simplest chemical reactions. In the case where the reaction is associated with the interaction of two particles, its rate is proportional to the number of encounters of the original molecules with one another, i.e., to the square of the number of molecules per unit volume if one kind of molecule reacts \((2\mathrm{Cl}_2\mathrm{O}=2\mathrm{Cl}_2+\mathrm{O}_2)\), and to the product of the numbers of molecules of the one and the other kind if the reaction proceeds between two different particles \((\mathrm{H}_2+\mathrm{J}_2=2\mathrm{HJ})\). Van ’t Hoff called reactions of this kind bimolecular (or reactions of the second order).

In the case where a molecule decomposes during free flight between two collisions \((\mathrm{CH}_3\mathrm{NNCH}_3=\mathrm{C}_2\mathrm{H}_6+\mathrm{N}_2)\), the reaction rate will be proportional to the first power of the number of molecules per unit volume; such reactions Van ’t Hoff called monomolecular (or reactions of the first order).

The rate constant of a reaction increases with temperature according to the law \(e^{-\frac{E}{RT}}\), where \(T\) is the absolute temperature, \(R\) is the gas constant, equal to \(2\ \mathrm{cal}\) per \(1^\circ\), and \(E\) is a quantity characteristic of each reaction (determining the magnitude of the increase in rate with temperature). Several years later Arrhenius proved the validity of the temperature law in greater detail and gave a simple kinetic interpretation of the quantity \(E\), which he called the energy of activation. He proposed that not every molecule undergoes reaction, but only a molecule in an active state, and that the energy which converts an ordinary molecule into the active state is greater than some value \(E\). In this case, according to Boltzmann’s law, the number of active molecules will constitute the fraction \(e^{-\frac{E}{RT}}\) of the inactive ones. Hence the reaction rate, expressed as the number of reacting molecules per unit time in unit volume, will be

\[ w=\frac{dx}{dt}=(a-x)^n e^{-\frac{E}{RT}}, \]

where \(n=2\) for bi- and \(n=1\) for monomolecular reactions (\(x\) is the number of molecules that have reacted by the given moment \(t\)). At the beginning

reactions, when \(x=0\),

\[ w_0 = ka^n e^{-\frac{E}{RT}} \]

(\(a\) is the number of molecules of the initial substance per unit volume at the beginning of the reaction). The constant \(k\) for bimolecular reactions is, obviously, equal to \(\sqrt{2\pi\sigma^2 u}\), where \(\sigma\) is the diameter, and \(u\) the thermal velocity of the molecule. In the case of monomolecular reactions \(k \simeq \nu\), where \(\nu\) is the number of vibrations of the molecule for the bond that is broken, i.e. a quantity of the order of \(10^{13}—10^{14}\).

Five years ago Eyring, on the basis of modern statistical methods and data on the structure and energetics of molecules, reconsidered (the theory of the activated complex) this question and arrived, for the simplest cases, practically at the same result as before. For more complex molecules entering into bimolecular reactions, he proved the possibility of a small steric factor, which considerably diminishes the value of \(k\), and gave a method for an approximate estimate of this factor. For monomolecular reactions Eyring indicated that in some (though very special) cases the constant \(k\) may exceed the values \(10^{14}\), while still remaining equal to \(10^{13}—10^{14}\) for the majority of real processes. Thus, for elementary reactions, the views of Van ’t Hoff and Arrhenius have retained their full force up to the present day. However, in the light of the modern development of chemical kinetics (in particular, the theory of chain reactions), the majority of reactions that actually proceed are connected with a sequence of mutually related elementary reactions and therefore appear to be far more complex than this seemed to the founders of chemical kinetics at the dawn of its development. The elementary stages of a complex reaction (which are reactions of various intermediate products with the initial substances) proceed according to the laws of Van ’t Hoff and Arrhenius, but the overall reaction often follows quite different dependences than simple bi- and monomolecular laws. Very common, for example, for the initial phases of conversion (20—30%) of many exothermic reactions is the regularity

\[ w = Ae^{\varphi t}, \]

where \(\varphi\) depends on the concentration of the initial substances and increases with temperature according to the law

\[ e^{-\frac{E}{RT}}. \]

In this case the reaction rate as a function of the concentration of the initial substance is

\[ \frac{dx}{dt}=w=kx, \]

i.e. the reaction rate in its first phases proves proportional to the amount of substance that has reacted by the given moment of time. In many cases this law can be generalized to the whole course of the transformation (taking burn-out into account) in the form

\[ w = k(a-x)xe^{-\frac{E}{RT}}. \]

Such laws are typical for the majority of combustion reactions in oxygen and air. In many cases the rate of a complex reaction depends on the square roots of the concentrations of one of the kinds of molecules reacting with one another. Thus, for example, for the reaction of hydrogen with bromine,

\[ w = k(\mathrm{H}_2)\sqrt{(\mathrm{Br}_2)}e^{-\frac{E}{RT}}, \]

where the number of corresponding molecules per unit volume is indicated in parentheses.

However, there are cases when a complex reaction formally leads to the same type of regularities as the simple monomolecular and bimolecular laws, although the meaning and magnitude of the constant \(k\) are different here, and the activation energy \(E\) is obtained by adding and subtracting the activation energies of the elementary processes composing the complex reaction. Thus, for example, the decomposition of many organic substances is formally described by the equation

\[ w = kae^{-\frac{E}{RT}}, \]

although there is no doubt that this process proceeds through a series of mono-, bi-, and trimolecular stages. Fortunately, for all our subsequent considerations we need only the formal expression, found experimentally, for the reaction rate as a function of the concentrations of the initial substances and of temperature, and there is no need to know the true mechanism of the reaction.

The mathematical treatment of results is simplest in those cases when we are dealing with the formal regularities of reactions of the first and second order. For these reasons we shall be concerned chiefly with them.

After these brief remarks on chemical kinetics, let us turn to the subject of our article—the present state of the thermal theory of combustion and explosions. Incidentally, these phenomena are so widespread and are used so intensively in technology (furnaces, internal-combustion engines, explosives) that the theoretical foundations of the doctrine of combustion and explosions ought to be presented in the general course of physics alongside the doctrine of heat conduction and thermodynamic cycles. I see the reason for the absence of such a chapter in the general course in the fact that, until recently, science had not managed to understand these phenomena. However, in the last ten years, as a result mainly of the work of Soviet scientists, the basic principles of the theory of combustion and explosions have been created, and the gap existing in science has begun to be filled. As we shall see below, the theory of these phenomena in its fundamental part is not complicated and can be successfully presented to students.

Unfortunately, in all the world literature there is only one attempt at a systematic exposition of the theory of the indicated phenomena on the basis of our own and foreign works—this is the recently published book by Jost\(^2\). There is not a single such attempt in the Russian language. My colleagues and I are especially to blame for this, since we contributed to a considerable degree to the creation of the theory of the indicated phenomena. This article is an attempt to rectify matters at least partially by giving a brief outline of the theory of combustion and explosions. Of course, this does not remove the question of the necessity of writing a detailed monograph on this branch of science.

The question of the nature of the spontaneous ignition of a combustible gas, arising when its temperature is raised to a certain “spontaneous-ignition temperature,” was first posed by van ’t Hoff in 1883 in the already mentioned Studies in Chemical Dynamics. There van ’t Hoff also gives, in general, a correct qualitative formulation of the cause of the phenomenon. Let us quote the corresponding excerpt from van ’t Hoff’s book.

“One of the conclusions of the preceding chapter deserves special attention, since, apparently, it is in contradiction with the phenomenon of ignition.

Indeed, the experiments just cited on the influence of temperature on the rate of transformation lead to the assumption of the continuity of this phenomenon, a continuity that is the result both of the experiments described and of the theoretical considerations that I have set forth. However, the phenomenon of ignition, suddenly occurring at a given temperature, apparently indicates that the assumed continuity permits exceptions; Meyer, for example, expresses himself on this matter as follows:

“The reaction point could in general be called the very lowest temperature at which there still takes place a definite chemical transformation, a temperature that is usually called the ignition point when one is dealing with combustible substances.”

Here there is an obvious contradiction with the conceptions established concerning the influence of temperature on the rate of transformations; indeed, these conceptions exclude any sudden accelerations and require that a transformation, if it occurs at a given temperature, should also occur at other temperatures, although its rate changes. A deeper study of this subject will show us that the phenomenon of ignition in no way obliges us to admit the existence of a temperature at which the transformation begins. In that case this phenomenon is wholly included in the system of developed conceptions.

The starting point of this argument is the three conditions that must be fulfilled in every transformation capable of producing ignition. Before setting out these conditions, I shall note that by the term “ignition” I understand not only the phenomena of explosion, but also any transformation that takes place with a local rise of temperature up to the so-called ignition temperature. In every phenomenon of this kind we see that the following conditions are fulfilled:

1. The transformation that gives ignition is accompanied by the liberation of heat.

2. This transformation already proceeds more or less slowly at temperatures lower than the ignition temperature.

3. This transformation is accelerated when the temperature is raised.

When these three conditions are fulfilled, a phenomenon similar to ignition can occur. To show this, let us suppose that at some point in a medium not capable of chemical transformation, for example in atmospheric air, the temperature has been raised from \(0\) to \(T\). When the heat source is removed, the rise of temperature will gradually spread, thus forming something like a heat wave. This wave will propagate with a definite velocity, while at the same time taking on temperatures ever closer to \(0\). The graphical representation of the relation between temperature and distance from the initial point \(O\) is represented by the line \(T_1A_1\) (Fig. 1), the temperatures being plotted along \(OT\) and the distances along \(OD\). For brevity let us denote by \(\Delta T\) the lowering of temperature in the heat wave in the first moments of its propagation.

Let us now suppose that in the medium under consideration a chemical transformation is possible and that it satisfies the three conditions given above; in a word, let us replace, for example, atmospheric air by an explosive mixture. The local rise of temperature will then form the wave described, with only this difference: as it propagates, the temperature will decrease less rapidly, especially at the beginning, because the transformation caused by the rise in temperature, in turn, causes the liberation of heat. The value of \(\Delta T\) will thus be diminished, and the graphical representation of the heat wave for this case will be represented by the line \(T_1A_1\) in Fig. 2.

Let us now determine the influence of the temperature increase in both cases. As regards a medium not subjected to chemical action (atmospheric air), the matter is simple: in the heat wave the lowering of temperature \(\Delta T\) will be more considerable (as is represented by the line

$T_2A_2$ in Fig. 1) as a result of the greater difference in temperatures between the wave and the medium in which it propagates. The assumption of the possibility of chemical transformation (an explosive mixture) introduces, alongside this effect which increases $\Delta T$, another effect which diminishes it; in fact, transformation accelerated by the higher temperature of the wave also yields, in this case, a greater quantity of heat. Thus, if the retarding effect predominates, then the value of $\Delta T$ will decrease as the temperature rises.

Having established this, we can predict that there is some temperature $T_2$ (Fig. 2) at which the magnitude $\Delta T$ is reduced to zero; in other words, a temperature at which the heat wave preserves its temperature during propagation, as represented by the horizontal line $T_2A_2$ in Fig. 2. In exactly the same way, an even higher initial temperature $T_3$ will produce a heat wave whose temperature, instead of decreasing, will rise to such a value at which complete transformation can occur. Such a process is represented by the line $T_3A_3$ in Fig. 2.

Fig. 1

Obviously, waves with decreasing temperature produce only negligible transformation, whereas waves with increasing temperature lead to complete transformation. Consequently, the temperature $T_2$, which gives a wave at constant temperature, corresponds precisely to the ignition temperature.

It would not be difficult to express everything stated above by a mathematical formula, but I prefer to express it in the following way: the ignition temperature is such a temperature at which the initial heat of the body, caused by thermal conductivity, etc., is equal to the heat produced in the same time by the transformation.

Fig. 2

It seems to me that, as a result of these considerations, the phenomenon of ignition fits completely into the accepted ideas about the influence of temperature on the rate of transformations. (Van ’t Hoff, Essays in Chemical Dynamics, pp. 111–113.)

Although Van ’t Hoff did not give here any quantitative treatment of the phenomenon of self-ignition, and although even Van ’t Hoff’s qualitative formulations are far from exact, nevertheless the physical essence of the phenomenon was established by Van ’t Hoff quite correctly. It was pointed out for the first time that self-ignition is the result of a proposed reaction which, in the event that it releases sufficiently much heat (i.e., if the mixture is heated to the corresponding temperature) so that this heat cannot be carried away in time by the walls, leads to a violent increase in temperature and reaction rate, which we observe as an explosion. Thus, there is no contradiction between the phenomenon of self-ignition and the continuity of the change in the reaction rate with increasing temperature. The phenomenon of self-ignition is a consequence of the temperature dependence of the reaction rate, if one takes into account—

that heating which arises in the course of an exothermic reaction and which leads to the temperature of the mixture changing with time.

It is very curious that over the course of forty years Van’t Hoff’s point of view on the thermal nature of self-ignition had not been mathematically formulated in a clear form. And this was despite the fact that the concept of the self-ignition temperature was used rather often, and moreover was used as a certain constant of a substance (introducing it, for example, into the theory of flame propagation), although already from Van’t Hoff’s analysis, as well as from experimental determinations of this quantity, it was obvious that the self-ignition temperature is not a constant, that it depends, for example, on the conditions of heat removal.

In 1927–1928 I succeeded³ in formulating mathematically, in the simplest possible way, Van’t Hoff’s point of view. Incidentally, at that time I knew nothing of Van’t Hoff’s work and arrived at this idea by the method of analogy, transferring into the field of combustion my and A. F. Walter’s results from the field of the breakdown of dielectrics. It is very instructive how the same physical factors and conceptions can be applied to the most diverse phenomena of nature, and therefore I shall allow myself to dwell briefly on our work on the thermal breakdown of dielectrics. The electrical conductivity of a dielectric increases with temperature according to the same law as the rate of a chemical reaction. When an electric voltage is applied to a dielectric, by Joule’s law heat is liberated in it, proportional to the electrical conductivity and to the square of the applied potential difference. As a result of the thermal conductivity of the dielectric and of the electrodes, heat is given off to the outside. However, when the voltage is increased, the amount of heat liberated may become so great that thermal equilibrium is disturbed. The dielectric will then begin to heat up progressively, as a result of which the electrical conductivity will grow and will further increase the liberation of heat, until, finally, such a large current flows that the dielectric melts and boils, i.e. is “broken down.” This constitutes the physical basis of the thermal theory of dielectric breakdown, which, as we see, is very close to the theory of thermal explosion.

An essential difference, however, is the circumstance that the dielectric usually does not melt as a whole, but the breakdown proceeds along a narrow cylindrical channel, located in the form of a thread between two electrodes. This is what led Wagner, who in 1910 was the first to advance the thermal theory of dielectric breakdown, to formulate it incorrectly (he took into account the heat removal not into the electrodes, but into the dielectric along the surface of the channel through which the breakdown proceeds). Walter, Fock, and I in the USSR, and Rogowski in Germany, independently of one another, were the first to formulate the thermal theory of breakdown correctly.

We explained the question of why a thread-like breakdown is obtained as follows: we assumed that the entire dielectric is heated by the current and that the main removal of heat goes into the electrodes. However, owing to the small temperature gradient in planes parallel to the electrodes, the disturbance of thermal equilibrium arises first of all

in the center, where the temperature, though not by much, is nevertheless higher than at the edges. As soon as a rapid rise of temperature begins in the central part of the dielectric, the electrical conductivity begins to grow so rapidly, and the current to increase so rapidly, that the voltage on the electrodes automatically falls, as a result of which the remaining parts of the dielectric remain unaffected by the breakdown. In this, and only in this, lies the difference between breakdown and explosion. This divergence, however, does not concern the question of the conditions under which thermal equilibrium is disturbed, but the question of the character of the development of the phenomenon after equilibrium has already been disturbed. A number of considerations led to the conclusion that, in the case when the initial electrical conductivity is large and breakdown occurs at low voltage, the conditions for the formation of filamentary breakdown deteriorate, and we can directly observe the disturbance of equilibrium throughout the dielectric, leading to melting and evaporation of a considerable part of it. And indeed, at sufficiently high temperatures, in the thermal breakdown of rock salt, evaporation of a considerable part of the substance occurs (bursting out in the form of a column of smoke from under the electrodes), and the result of the breakdown is a hole a finger wide. For gelatin, a phenomenon of this kind can be observed at room temperature.

As a result of the mathematical formulation of the theory, a number of consequences were obtained (the temperature coefficient of the breakdown voltage must be equal to one half of the temperature coefficient of the electrical conductivity; the breakdown voltage must decrease according to definite laws depending on the change in the thermal conductivity, the electrodes, etc.). All these consequences were rigorously confirmed by the experiments of Walther and Inge, whereby the thermal theory of breakdown was established definitively.

When in 1927 I became interested in the phenomenon of explosion, I at once grasped the analogy of this phenomenon with thermal breakdown. If in the case of breakdown the amount of heat generated by the current per unit volume is

$$ q = 0.24\sigma_0 e^{-\frac{E}{RT}} V^2, $$

where $\sigma_0 e^{-\frac{E}{RT}}$ is the electrical conductivity, $V$ is the applied voltage, then in the case of a chemical reaction the same quantity is

$$ q = Q'w = Q'kae^{-\frac{E}{RT}}, $$

where $Q'$ is the thermal effect calculated per one molecule of product, and $a$ is the number of molecules of the initial substance per unit volume (i.e. a quantity proportional, for example, to the pressure of the gas). Thus, if in the case of thermal breakdown we can find the relation between breakdown voltage and temperature, then in the case of spontaneous ignition we must obtain the relation between the temperature of spontaneous ignition and the gas pressure.

In the subsequent sections of this article we shall set forth the thermal theory of combustion and explosions, dividing it into three parts: I. The theory of thermal spontaneous ignition; II. The theory of thermal ignition; and III. The theory of thermal propagation of flame, illustrating the conclusions of the theory with admittedly not very numerous experimental material. The entire theory set forth has been developed over the course of several years at the Institute of Chemical Physics by the works of Semenov, Todes, Zel’dovich, Frank-Kamenetsky

and in part Belyaev, Appin, and Shchelkin. The experimental illustrations have been taken both from the works of our Institute (Zagulin, Neiman, Roginsky, Belyaev, Appin, and others) and from foreign works.

The whole of the present article is devoted only to thermal ignition and combustion, and in it the extensive and very interesting field of chain self-ignition and the propagation of cold flames has remained untouched; a separate article will be devoted to it.

I. THERMAL SELF-IGNITION1

If in the gas phase there proceeds a reaction with rate \(w\), measured by the number of product molecules appearing in 1 sec. per unit volume, then the amount of heat liberated every second in the entire volume of the vessel \(v\) will be equal to

\[ q_1 = v \cdot Q' \cdot w. \]

Here \(Q'\) is the heat liberated in each elementary act of the reaction, equal to \(Q'=\frac{Q}{N}\), where \(Q\) is the heat of reaction liberated in the formation of 1 gram-molecule of product, and \(N\) is Avogadro’s number \((N = 6\cdot 10^{23})\). As was indicated, the rate of reaction in the initial stage, as a function of the absolute temperature \(T\) and of the number of molecules of the initial substance \(a\) per unit volume, will be equal to \(w = k_1 a e^{-\frac{E}{RT}}\) for monomolecular reactions, and \(k_2 a^2 e^{-\frac{E}{RT}}\) for bimolecular reactions. Thus,

\[ q_1=\frac{vQka^n e^{-\frac{E}{RT}}}{N}, \tag{1} \]

where for monomolecular reactions \(n=1\), and for bimolecular reactions \(n=2\).

The amount of heat removed from the reaction space by the walls of the vessel will be

\[ q_2=\chi (T-T_0)S, \tag{2} \]

where \(\chi\) is the heat-transfer coefficient, \(T\) is the temperature of the reacting gas, \(T_0\) is the temperature of the vessel walls, specified from outside, and \(S\) is the surface of the vessel.

In Figs. 3 and 4 the dependences of \(q_1\) and \(q_2\) on temperature are shown. Fig. 3 corresponds to the case when the vessel wall is maintained at temperature \(T_0\), while the pressure of the reacting gases is varied, i.e., the magnitude \(a\) in formula (1). Curve 1 corresponds to the smallest \(a\), equal to \(a_1\); curve 2 to the mean \(a_2\), and curve 3 to the largest \(a_3\).

For \(a=a_1\), the heat input \(q_1\) is at first greater than the heat removal \(q_2\). As a result, the gas will be heated above the vessel walls. However, such a state will last only until the tempe-

the gas temperature will not reach a certain value \(T'_1\) (the intersection of the curves \(q_1\) and \(q_2\)). In this case \(q_1 = q_2\). The gas will not heat up further, since for \(T > T'_1\) the heat transfer \(q_2\) will be greater than the heat input \(q_1\), and, if for some reason the gas were even overheated above \(T'_1\), it would again cool down to the same temperature \(T'_1\). Thus, in the case under consideration, the reaction does not lead to self-ignition, and the matter is limited only to heating the gas to the temperature \(T'_1\), somewhat higher than the temperature of the vessel walls.

Fig. 3

If, at the same vessel temperature \(T_0\), we fill it with a gas capable of reaction at a sufficiently high pressure, or, what is the same thing, at a sufficiently large value \(a = a_3\), then \(q_1\) as a function of temperature will be represented by curve 3, which nowhere intersects the straight line of heat removal \(q_2\). In this case the heat input \(q_1\) is greater than the heat removal \(q_2\) at all temperatures and, consequently, the gas will heat up continuously, the reaction will proceed faster and faster, which we perceive as an explosion. Therefore, for \(a = a_3\), thermal self-ignition will take place.

Fig. 4

Curve 2 for \(a = a_2\) touches the straight line of heat removal at one point and, thus, separates the region where a stationary reaction occurs from the region where self-ignition occurs. The quantity \(a = a_2\), or the corresponding pressure \(p_2\), determines the critical pressure of self-ignition at the given vessel temperature \(T_0\).

If, keeping the gas pressure constant, we vary the temperature of the vessel walls \(T^{(1)}_0 < T_0 < T^{(3)}_0\), then the behavior of the gas is depicted—

\(^{1)}\) Only in the case where we artificially (for example, by adiabatic compression) heat the gas above the second intersection point \(T'_2\), then \(q'_2\) again becomes greater than \(q_1\), and further automatic heating of the gas becomes possible, leading to an explosion. This second intersection point does not give a stable value of the temperature, since if \(T\) is slightly less than \(T'_2\), the temperature will drop to \(T'_1\). If \(T\) is slightly higher than \(T'_2\), an explosion will occur. Thus, this second intersection point is of no significance in the theory of self-ignition and is of interest only in questions of artificial ignition associated with heating the gas.

is represented by Fig. 4. Here there is one heat-input curve \(q_1\) and three heat-removal straight lines, corresponding to three temperatures of the vessel walls. By the same reasoning as above, we can show that for \(T_0^{(1)}<T_0\) self-ignition does not occur; for \(T_0^{(3)}>T_0\) self-ignition takes place. The temperature \(T_0\), at which the curve \(q_1\) touches the straight line \(q_2\) at one point, we shall call the lowest temperature of self-ignition or simply the temperature of self-ignition at the given pressure \(p\). The point of tangency of the curves \(q_2\) and \(q_1\) corresponds to the temperature \(T_1\); the difference \(\Delta T=T_1-T_0\) we shall call the pre-explosive heating. Between the temperature of self-ignition and the pressure (or the number \(a\)) of the mixture an analytical relation can be established, using the circumstance that at the point of tangency (at \(T=T_1\)) the quantities \(q_1=q_2\) and their derivatives with respect to temperature \(\dfrac{dq_1}{dT}=\dfrac{dq_2}{dT}\) are equal, i.e.,

\[ \frac{vQkan e^{-\frac{E}{RT_1}}}{N}=\chi(T_1-T_0)S; \tag{3} \]

\[ \frac{vQkanE}{NRT_1^2}e^{-\frac{E}{RT_1}}=\chi S. \]

From these two equations we can first of all find the temperature \(T_1\) at the point of tangency as a function of the temperature of the vessel walls \(T_0\).

Eliminating \(\chi S\) from both equations, we obtain

\[ 1=\frac{E}{RT_1^2}(T_1-T_0)\quad \text{or}\quad \frac{RT_1^2}{E}-T_1+T_0=0; \]

\[ T_1=\frac{1\pm \sqrt{\,1-\frac{4RT_0}{E}\,}}{2\frac{R}{E}}. \]

Let us note that for the majority of the reactions of interest to us \(\dfrac{RT_0}{E}\) is a small quantity, usually not exceeding 0.05. The temperature of self-ignition is usually below \(1000^\circ\) K, while the activation energy \(E\) is usually greater than 20,000 cal; moreover, for a low value of \(E\) we have a low temperature of self-ignition and, conversely, for a large \(E\), a high one. The solution with the sign \(+\) must be discarded, since in this case \(T_1\) turns out to be equal to \(\dfrac{R}{E}\), i.e., has a value of the order of \(10,000^\circ\) and higher\(^1\)), and since the solution with

\(^1\) The two signs occur because the function \(e^{-\frac{E}{RT}}\) has an inflection at very high temperatures and, as \(T\to\infty\), tends to unity. Therefore the heat-removal straight lines, when extrapolated to tens of thousands of degrees, intersect the curve \(q_1\) once more (not shown in Fig. 3).

the sign —, corresponding to the point of tangency shown in Figs. 3 and 4, gives a much lower temperature \(T_1\).

Thus,

\[ T_1=\frac{1-\sqrt{1-\dfrac{4RT_0}{E}}}{2\dfrac{R}{E}} =\frac{2\left(\dfrac{RT_0}{E}\right)+2\left(\dfrac{RT_0}{E}\right)^2+4\left(\dfrac{RT_0}{E}\right)^3+\ldots}{2\dfrac{R}{E}} . \]

For \(\dfrac{RT_0}{E}<0.05\) we may discard the terms of the expansion beginning with

\[ 4\left(\frac{RT}{E}\right)^3, \]

thereby making an error of the order of

\[ 2\left(\frac{RT}{E}\right)^2 \cdot 100\% < 0.0025\cdot 2\cdot 100\% = 0.5\% \]

of the measured quantity \(T_1\). Thus,

\[ T_1=T_0+\frac{RT_0^2}{E}. \tag{4} \]

The pre-explosion heating is

\[ \Delta T_1=T_1-T_0=\frac{RT_0^2}{E}. \tag{5} \]

The heating \(\Delta T\) that arises in the case when ignition is known not to occur (the temperature \(T_0\) is below the self-ignition temperature) will always be less than \(\Delta T_1=\dfrac{RT_0^2}{E}\).

Thus, conversely, if the heating \(\Delta T\) is less than \(\dfrac{RT_0^2}{E}\), then a thermal explosion is impossible; and, on the contrary, if \(\Delta T>\dfrac{RT_0^2}{E}\), a thermal explosion must take place. For different values of \(T_0\) and activation energy \(E\), the pre-explosion heating will be different, but, as a rule, it never exceeds, in the cases of interest to us, several tens of degrees. Thus, for example, at \(T_0=700^\circ\mathrm{K}\) and \(E=30\,000\) cal, \(\Delta T_1=33^\circ\). At \(T_0=700^\circ\mathrm{K}\) and \(E=60\,000\) cal, \(\Delta T_1\sim16^\circ\). Thus we see that the ratio \(\dfrac{\Delta T_1}{T_0}\simeq\dfrac{RT_0}{E}\) is always small (of the order of a few hundredths). Therefore in what follows we shall approximately put

\[ \frac{1}{T_0+\Delta T_1}=\frac{1}{T_0}\left(1-\frac{\Delta T_1}{T_0}\right) \]

and so on.

Substituting the values found for \(T_1\) into equations (3), we obtain the self-ignition conditions:

\[ \frac{QvkanE}{NRT_0^2} \left(1-2\frac{\Delta T_1}{T_0}\right) e^{-\frac{E}{RT_0}\left(1-\frac{\Delta T_1}{T_0}\right)} =\chi S \]

or, substituting \(\dfrac{\Delta T_1}{T_0}=\dfrac{RT_0}{E}\) and neglecting in the left-hand side \(2\dfrac{\Delta T_1}{T_0}\) in comparison with unity (which corresponds to an error in the value of \(a\) smaller,

than 100%), we obtain the condition for self-ignition in the form

\[ \frac{Q\nu k a n E e}{NRT_0^2 \chi S}\, e^{-\frac{E}{RT_0}}=1 . \tag{6} \]

The number of molecules \(a\) per unit volume is related to the pressure \(p\), expressed in mm Hg, by the following relation:

\[ p=\frac{aRT}{N}\ \text{dyn}/\text{cm}^2 =\frac{aRT}{N}\cdot 10^{-6}\ \text{bar}/\text{cm}^2 =\frac{aRT}{N}\cdot 10^{-6}\cdot 750\ \text{mm Hg}/\text{cm}^2, \]

where \(R=83.15\cdot 10^6\) erg. Hence

\[ p=aT\cdot 10^{19},\qquad a=\frac{p}{T}\cdot 10^{19}. \tag{7} \]

Substituting in (6), we obtain the relation between the gas pressure \(p\) and the self-ignition temperature

\[ \frac{Q\nu k p^n E e\cdot 10^{19}}{NRT_0^{2+n}\chi S}\, e^{-\frac{E}{RT_0}}=1, \]

\[ \lg \frac{p}{T^{1+\frac{2}{n}}} = \frac{A}{T_0}+B, \qquad \text{where } A=\frac{0.217E}{n}, \tag{8} \]

\[ B=\frac{1}{n}\lg b,\qquad b=\frac{NR\chi S}{Q\nu k e E\cdot 10^{19}}. \]

For \(n=1\)

\[ \lg \frac{p}{T^3}=\frac{A}{T_0}+B, \qquad \text{where } A=0.217\,E. \tag{9} \]

For \(n=2\)

\[ \lg \frac{p}{T^2}=\frac{A}{T_0}+B, \qquad \text{where } A=0.11\,E. \tag{10} \]

In the case where a bimolecular reaction proceeds between two components of the mixture, for example hydrogen and chlorine, all the formulas are retained, and only the quantity \(a\) (the total number of molecules of \(\mathrm{H}_2\) and \(\mathrm{Cl}_2\) in \(1\ \mathrm{cm}^3\)) is multiplied by the product \(\gamma(1-\gamma)\), where \(\gamma\) is the fraction of one component and \(1-\gamma\) the fraction of the other component. In addition, the thermal conductivity will change when the composition of the mixture changes.

Let us give several examples of verification of this equation. In Fig. 5 are shown the data obtained by Zagulin\(^{6,3}\) for the decomposition of \(\mathrm{Cl}_2\mathrm{O}\). On the abscissa axis are plotted the reciprocals of the absolute temperatures of the apparatus, and on the ordinate axis \(\lg \frac{p}{T}\), where \(p_1\) is the pressure at which self-ignition occurs. [On the ordinate axis Zagulin plotted \(\lg \frac{p}{T}\), and not \(\lg \frac{p}{T^2}\). However, as is not difficult to see, in the temperature interval investigated the variation of \(\lg \frac{p}{T^2}\) as a function of \(\frac{1}{T}\) will also practically follow a straight line, and in the first approximation parallel to the straight line \(\lg \left(\frac{p}{T}\right)-\frac{1}{T}\).]

The decomposition of \(\mathrm{Cl}_2\mathrm{O}\), according to Hinshelwood,\(^7\) proceeds bimolecularly, with \(E = 21\,000\)—\(22\,000\) cal. Zatulin obtained for \(A\) [see formula (9)] the value \(2\,500\). According to the theory and Hinshelwood’s data, \(A\) should be equal to \(0.11\), \(E = 0.11 \times 22\,000 = 2\,400\), in good agreement with experiment.

Fig. 5

In Fig. 6 are given data of the same author\(^6\) for the spontaneous ignition of mixtures of hydrogen with chlorine at different ratios of the mixture components.

As was to be expected, \(\lg \dfrac{p}{T}\), and consequently, with a sufficient degree of accuracy, also \(\lg \dfrac{p}{T^2}\), is linearly related to the reciprocal absolute temperature; moreover, all straight lines are parallel to one another, i.e., they correspond to one and the same \(A\). The constant \(A\) turned out to be equal to \(2\,600\)—\(2\,900\).

Studying the kinetics of the reaction \(\mathrm{H}_2 + \mathrm{Cl}_2\) at a temperature of \(205^\circ\), Pease showed that it proceeds bimolecularly [the rate is proportional to the product \((\mathrm{H}_2)\cdot(\mathrm{Cl}_2)\)]. In that case, from Zatulin’s data we obtain \(A = 0.11E\), or

\[ E = \frac{2\,750}{0.11} = 25\,000\ \text{cal}. \]

Christiansen gives for the activation energy of this reaction the value \(25\,000\)—\(30\,000\) cal. From Pease’s data\(^8\) it follows that the activation energy is equal to \(20\,000\)—\(25\,000\) cal. In Fig. 7 is shown the curve of the dependence of \(\lg \dfrac{p}{T^3}\) on \(\dfrac{1}{T}\) for the spontaneous ignition of azomethane according to Rice and Allen.\(^9\) Curve 1 is for pure azomethane; curve 2 is for azomethane diluted with helium (76% He in the mixture).

Fig. 6

Azomethane decomposes monomolecularly; however, with increasing pressure the rate constant increases somewhat, tending toward the limiting value \(k_\infty\) at sufficiently high pressures. Therefore, in the expression for the critical pressure \(p\) one must multiply by \(\dfrac{k}{k_\infty}\) and plot not \(\lg \dfrac{p}{T_0^3}\), but

\[ \lg \frac{kp}{k_\infty T_0^3}, \]

which is what has been done in Fig. 7. The black circles in this figure correspond to experiments with pure azomethane, the crosses to mixtures of azomethane with helium (76% helium), and the circles to mixtures with nitrogen (50%). The con-

constant \(A\) turned out to be equal to 11,000. Calculating from this \(E=\dfrac{A}{0.217}\), the authors obtained \(50\,000\) cal, i.e., the same figure \((51\,000)\) that Rice and Ramsperger give from direct kinetic experiments. \(^{10}\)

As we have seen, the condition of self-ignition [formula (6)] includes both parameters determining the reaction rate (the constant \(k\), the activation energy \(E\)) and thermal quantities (the heat effect of the reaction \(Q\), the heat-transfer coefficient \(\chi\), the dimensions of the vessel). The condition of self-ignition can be rewritten in a simpler form if these parameters are appropriately grouped and reduced to macroscopic quantities that are easily determined experimentally (Todes \(^{11,4}\)). As one of such parameters determining heat transfer, let us choose the time of thermal relaxation, i.e., the time \(t_e\) during which an initially heated (but nonreacting) gas diminishes, owing to heat transfer to the walls, its excess temperature \((\Delta T=T-T_0)\) by a factor of \(e\). \(t_e\), of course, depends neither on the magnitude of \(\Delta T\) nor on the absolute temperature \(T_0\), since the coefficient \(\chi\) does not depend on these quantities. It is not difficult to verify that

Fig. 7

\[ t_e=\frac{Cav}{\chi SN}, \tag{11} \]

where \(C\) is the heat capacity of a gram-mole.

We shall characterize the reaction rate by the reciprocal quantity, namely the reaction time \(t_r\), understanding by this conditionally the time during which all the initial substance would be consumed by the reaction if the reaction proceeded at a constant rate, namely the one corresponding to the initial concentration of the combustible gas. In other words,

\[ t_r=\frac{a}{kan e^{-\frac{E}{RT}}}. \tag{12} \]

\(^{1}\) Indeed,

\[ \frac{vaC}{N}\frac{dT}{dt}=-\chi S(T-T_0). \]

Let \(T_1\) be the initial temperature of the gas. Then

\[ \ln \frac{T-T_0}{T_1-T_0}=-\frac{\chi SNt}{Cav} \]

or

\[ T-T_0=(T_1-T_0)e^{-\frac{\chi SNt}{Cav}}; \]

and hence the time

\[ t_e=\frac{Cav}{\chi SN}. \]

$t_r$ can be easily measured. For this it is sufficient to observe the initial reaction rate at a temperature not far from self-ignition, to extrapolate the value obtained to the self-ignition temperature, and to determine $t_r$ by dividing the total number of molecules of the initial substance in a unit volume by the measured initial reaction rate, expressed as the number of product molecules per second.

It is not hard to see that in this case formula (6) may be rewritten in the following simple form:

\[ \frac{t_e}{t_r}\frac{QEe}{CRT_0^2}=1, \]

or the self-ignition condition will be

\[ \frac{t_r}{t_e}=\frac{QEe}{CRT_0^2}. \tag{13} \]

If

\[ \frac{t_r}{t_e}>\frac{QEe}{CRT_0^2}, \]

then thermal self-ignition is impossible and a stationary reaction proceeds; if

\[ \frac{t_r}{t_e}<\frac{QEe}{CRT_0^2}, \]

then the gas will always self-ignite thermally.

Let us now consider in more detail the questions of heat transfer from the gas to the vessel wall.

Assuming $q_2=\chi(T-T_0)S$, we supposed that heat transfer proceeds by convection. In this case the calculation of $\chi$ is very difficult. However, there are a number of experiments on heat transfer from which it is seen that, at gas pressures below atmospheric and for the small vessel dimensions usually used in self-ignition experiments, and, finally, for small values of the pre-explosion temperature difference $\Delta T_1$, heat transfer is in practice very close to conductive. Frank-Kamenetskii $^{8}$ was the first to draw attention to this and introduced into the theory of thermal self-ignition, in place of convective heat transfer, conductive heat transfer. Referring readers interested in the exact solution of the problem to Frank-Kamenetskii’s article $^{8}$, we shall give here an approximate derivation. Our approximation will consist in the fact that, although in the case of conductive heat transfer the gas temperature will be different at different points inside the vessel, we shall nevertheless assume that the reaction rate is the same at all points of the vessel and corresponds to a temperature $T_1$, intermediate between the maximum $T_{\max}$ at the center of the vessel and the temperature $T_0$ at the vessel walls. Since the pre-explosion heating is small and is measured in one or two tens of degrees, this assumption leads to approximately correct results. For simplicity, consider an infinitely extended flat vessel whose walls are parallel to one another and are located at a distance $2r$ apart, and are at temperature $T_0$. Inside the vessel a reaction is taking place, releasing per second, per unit volume, heat $u$, the same at all points of the space between the planes (the approximation to reality indicated by us). Let us direct the $x$-axis perpendicular to the vessel walls and take the plane $x=0$ as the plane,

passing midway between the planes of the walls. In this case we can write the differential equation of heat conduction

\[ \lambda \frac{d^2 T}{dx^2}+u=0, \]

which, under the condition \(T_{x=+r}=T_{x=-r}=T_0\) and \(\dfrac{dT}{dx}=0\) at \(x=0\), makes it possible to find the temperature distribution \(T\) inside the vessel. Let \(\lambda\) be the coefficient of thermal conductivity of the reacting gas. The solution of the equation will be:

\[ T-T_0=\frac{1}{2}\frac{u}{\lambda}r^2\left(1-\frac{x^2}{r^2}\right). \]

In the middle of the vessel the temperature \(T\) will be equal to

\[ T_{\max}-T_0=\frac{1}{2}\frac{u}{\lambda}r^2. \]

Hence the heat flux through the surface \(S\) of the vessel will be:

\[ q_2=-\lambda\left(\frac{dT}{dx}\right)_r S=urS \]

or, expressing \(u\) through \(T_{\max}-T_0\),

\[ q_2=\frac{2\lambda}{r}S\left(T_{\max}-T_0\right). \]

Substituting for \(T_{\max}\) the mean value

\[ T_1=\frac{T_{\max}+T_0}{2} \]

or \(T_{\max}=2T_1-T_0\), we obtain

\[ q_2=\frac{4\lambda}{r}S(T_1-T_0)=\chi S(T_1-T_0). \]

Thus, \(\chi=\dfrac{4\lambda}{r}\), whence it is clear that the entire solution remains the same as in the case of convective heat transfer; only instead of \(\chi\) one must substitute \(\dfrac{4\lambda}{r}\). In the case of a plane vessel \(\dfrac{v}{S}=2r\). Substituting \(\chi\) and \(\dfrac{v}{S}\) into expression (6), we obtain the condition for self-ignition in a plane vessel in the form

\[ \delta_{kr}= \frac{QkanEee^{-\frac{E}{RT_0}}\,2r^2}{NRT_0^2\,4\lambda}=1 \]

or

\[ \delta_{kr}= \frac{Er^2 Qkane^{-\frac{E}{RT_0}}}{RT_0^2\lambda N} = \frac{2}{e} =0.74. \tag{14} \]

If the calculation is carried out more rigorously, taking into account the difference in reaction rates at different points inside the vessel, then, as Frank-Kamenetskii showed, the critical condition for self-ignition for a plane vessel will be:

\[ \delta_{kr}=0.88, \tag{15} \]

respectively:

for a cylindrical vessel

\[ \delta_{\mathrm{cr}} = 2.00 \tag{16} \]

and for a spherical one

\[ \delta = 3.32. \tag{17} \]

Here by \(r\) in the expression for \(\delta_{\mathrm{cr}}\) (14) is meant the radius of the cylinder or sphere. These expressions make it possible to calculate the absolute values of the self-ignition temperature from thermal and kinetic data, since the single unknown quantity \(\chi\) has been replaced by the thermal conductivity of the gas. Unfortunately, there are very few cases of reactions leading to self-ignition for which the kinetics of the corresponding reaction has been determined accurately.

Frank-Kamenetskii\(^{12}\) calculated the self-ignition temperature of azomethane at different pressures, starting from kinetic data for this reaction, the heat of reaction, and the dimensions of the vessels used; taking \(\lambda = 10^{-4}\ \mathrm{g}^{-1}\ \mathrm{sec}^{-1}\ \mathrm{cm}^{-1}\), he compared the values calculated by formula (17) with the self-ignition temperatures of azomethane observed by Rice and Allen\(^{9}\). The same was done for the decomposition of methyl nitrate (also a monomolecular reaction) according to the data of Appin and Chariton\(^{13}\). The data given below were obtained in this way.

As is seen, the agreement between the theoretical and experimental values is quite good, which apparently proves the correctness of Frank-Kamenetskii’s assumptions concerning conductive heat transfer in the pre-explosion region\(^{1}\).

On the basis of Volmer’s\(^{14}\) data on the kinetics of the decomposition of nitrous oxide, Frank-Kamenetskii predicted that this gas is capable of self-ignition, although at very high temperatures.

Zel’dovich and Yakovlev\(^{15}\) did in fact observe self-ignition of \(N_2O\), and the observed and predicted self-ignition temperatures proved to be in good agreement (see the table).

Decomposition of azomethane \((\mathrm{CH}_3)_2\mathrm{N}_2 = \mathrm{C}_2\mathrm{H}_6 + \mathrm{N}_2\) (according to Rice’s data)	Decomposition of methyl nitrate \(2\mathrm{CH}_3\mathrm{ONO}_2 = \mathrm{CH}_3\mathrm{OH} + \mathrm{CH}_2\mathrm{O} + 2\mathrm{NO}_2\) (according to Appin and Chariton’s data)	Decomposition of nitrous oxide \(N_2O\) (according to Zel’dovich and Yakovlev’s data)
\(p\) mm	\(T^\circ_{\mathrm{calc}}\ \mathrm{K}\)	\(T^\circ_{\mathrm{obs}}\ \mathrm{K}\)
191	619	614
102	629	620
67	635	626
55	638	630
38	644	636
31	647	643
23.5	653	651
18	656	659

\(^{1}\) According to the most recent values of the constant \(k\) for the decomposition of azomethane, the agreement between theoretical and experimental values for this reaction becomes somewhat worse.

We shall allow ourselves here to analyze further the question of the absolute values of self-ignition temperatures for the cases of decomposition of $\mathrm{Cl_2O}$ and the reactions $\mathrm{H_2 + Cl_2}$.

The decomposition of $\mathrm{Cl_2O}$ is not a simple bimolecular reaction; most closely, the course of this reaction is expressed by the autocatalytic law

\[ w = kx(a-x)e^{-\frac{E}{RT}}, \]

where $a$ is the number of initial molecules, and $x$ is the number that have reacted. The reaction reaches its maximum rate at $x=\frac{a}{2}$, when

\[ w_{\max}=k\frac{a^2}{4}e^{-\frac{E}{RT}}. \]

Thus, the minimum self-ignition temperature is determined not by the initial rate of the process, but by the maximum rate.

As we have seen, the maximum rate in this case varies bimolecularly. This is in agreement with Hinshelwood’s results[^7]. As we have already indicated, the activation energy $E$, according to Hinshelwood, is equal to $22\,000$ cal, and the constant $k$ is the ordinary constant of the bimolecular process, with $k = V\overline{2\pi\sigma^2 u}$.

According to Hinshelwood, $\sigma = 4.8 \cdot 10^{-8}$, and the constant $k \sim 10^{-10}$. Thus, in formula (16) it is necessary to substitute $k=10^{-10}$. Instead of $a$ one must take $\frac{a}{2}$ and $n=2$. The heat of decomposition of $\mathrm{Cl_2O}$ is $Q=22\,000$ cal; $\lambda$ may be taken approximately as $5 \cdot 10^{-5}$.

Zagulin, in his experiments on the self-ignition of $\mathrm{Cl_2O}$, used a cylindrical vessel with $r \sim 1$ cm.

According to formula (16), as applied to a cylindrical vessel, we obtain

\[ \delta_{\mathrm{cr}}=\frac{Er^2Qka^n e^{-\frac{E}{RT}}}{RT_0\lambda N}=2; \]

for $\mathrm{Cl_2O}$, accordingly, we shall have

\[ \delta_{\mathrm{cr}}= \frac{ 2.3\cdot 10^4 \cdot 2.2\cdot 10^4 \cdot 10^{-10} a^2 e^{-\frac{22\,000}{T_0}} }{ 4\cdot 2T_0^2 \, 5\cdot 10^{-5}\cdot 6\cdot 10^{23} } =2. \tag{18} \]

Since we have already established that the temperature dependence of the critical self-ignition pressure of $\mathrm{Cl_2O}$ agrees well with the kinetic data, it is sufficient for us to verify that one of the values of the critical pressure corresponding to one or another self-ignition temperature satisfies condition (18). This will prove that the absolute values of all experimental data on self-ignition can be calculated from kinetic data. Let us take one of the points obtained by Zagulin[^6]: at $T=454^\circ$ or $\frac{1}{T}=22\cdot 10^{-4}$, the critical self-ignition pressure proves to be equal to

250 mm Hg. Hence the number of molecules per unit volume is

\[ a=\frac{2.7\cdot10^{19}\cdot250\cdot273}{760\cdot454}=5.4\cdot10^{18}, \]

\[ e^{-\frac{11000}{T}}=e^{-\frac{11000}{454}}=e^{-24.2}=10^{-10.5}=3.2\cdot10^{-11}, \]

whence \(\delta_{\mathrm{cr}}\approx 1\); the last number practically coincides with the value 2, since an error in \(T\) of \(10\)—\(15^\circ\) is sufficient to compensate this difference. Moreover, a small inaccuracy in the magnitude of \(E\), or an error by a factor of two in the constant \(k\) or the coefficient of thermal conductivity \(\lambda\), can lead to the indicated discrepancy.

Let us now analyze the reaction \(\mathrm{H_2+Cl_2}\), assuming that it proceeds as a simple bimolecular reaction. According to Zagulin’s data\(^6\) on self-ignition, \(A=2750\), i.e. \(E=\dfrac{A}{0.11}=25000\) cal. Since here we are dealing with collisions of two kinds of molecules, the total number of collisions between \(\mathrm{H_2}\) and \(\mathrm{Cl_2}\) molecules in 1 sec in \(1\ \mathrm{cm^3}\) will be

\[ Z_{12}=2(\mathrm{H_2})(\mathrm{Cl_2})\left(\frac{\sigma_1+\sigma_2}{2}\right)^2 \left\{\frac{2\pi kT(m_1+m_2)}{m_1m_2}\right\}^{\frac12}. \tag{19} \]

If \(m_1\) is the mass of \(\mathrm{H_2}\), and \(m_2\) the mass of \(\mathrm{Cl_2}\), then we may take \(m_2\gg m_1\); then

\[ Z_{12}=2(\mathrm{H_2})(\mathrm{Cl_2})\left(\frac{\sigma_1+\sigma_2}{2}\right)^2 \left(\frac{2\pi kT}{m_1}\right)^{\frac12} = \]

\[ =(\mathrm{H_2})(\mathrm{Cl_2})\left(\frac{\sigma_1+\sigma_2}{2}\right)^2 \left\{\frac{2\pi RT}{\mu_1}\right\}^{\frac12}. \]

Substituting \(R=83.15\cdot10^6\) and \(\mu_1\)—the molecular weight of \(\mathrm{H_2}\), equal to 2, \(\sigma_1=2.4\cdot10^{-8}\), \(\sigma_2=5\cdot10^{-8}\), and \(T=556^\circ\mathrm{K}\), we obtain \(Z_{12}=(\mathrm{H_2})(\mathrm{Cl_2})\cdot7\cdot10^{-10}\).

Hence the reaction rate is

\[ w=7\cdot10^{-10}(\mathrm{H_2})(\mathrm{Cl_2})e^{-\frac{25000}{RT}}, \]

which at \(T=556^\circ\) gives

\[ w=7\cdot10^{-10}(\mathrm{H_2})(\mathrm{Cl_2})\cdot4\cdot10^{-10} =2.8\cdot10^{-19}(\mathrm{H_2})(\mathrm{Cl_2}). \]

According to Zagulin’s data, at \(T=556^\circ\) the explosion pressure is equal to 200 mm (100 mm \(\mathrm{H_2}\) and 100 mm \(\mathrm{Cl_2}\)):

\[ (\mathrm{H_2})=(\mathrm{Cl_2})= \frac{2.7\cdot10^{19}\cdot100\cdot273}{760\cdot556} =1.8\cdot10^{18}\ \text{molecules in }1\ \mathrm{cm^3}, \]

\[ Q=2.2\cdot10^4, \]

\[ \lambda=3\cdot10^{-4\,1)}. \]

\(^1\) For atmospheric pressure, from direct experiments and from Ishikawa’s experiments\(^ {16}\) it is seen that the heat-transfer coefficient \(\chi=5\cdot10^{-4}=\dfrac{4\lambda}{r}\), \(\lambda=1.25\,r\cdot10^{-4}\). With po-

Hence

\[ \delta_{cr}= \frac{2.5\cdot 10^{4}\cdot 2.2\cdot 10^{4}\cdot 2.8\cdot 10^{-19}\cdot 3.2\cdot 10^{36}} {2.3\cdot 10^{5}\cdot 3\cdot 10^{-4}\cdot 6\cdot 10^{23}} =4.5, \]

which is a quantity of the same order as the value 2 required by the thermal theory; in order to obtain 2 instead of 4.5, one must take the ignition \(T\) not as \(564^\circ\), but as \(540^\circ\).

The reaction \(\mathrm{H_2+Cl_2}\) is hardly a simple bimolecular reaction; both by analogy with the photochemical reaction \(\mathrm{H_2+Cl_2}\) at room temperature, and in connection with the strong effect of traces of oxygen on the thermal reaction that has been discovered, this reaction must be regarded as a chain reaction. The kinetics of this reaction have been little investigated, but, judging from the detailed work of Pease\(^8\), it formally follows a bimolecular law. If we use Pease’s data directly, putting \(E=25\,000\) cal, then in calculating \(\sigma_{cr}\) we obtain a value 5–100 times smaller than 2. In itself this would not be at all surprising, but we cannot use Pease’s data because, extrapolating Zagulin’s straight line to the temperature \(250^\circ\), at which Pease worked, we find that at this temperature explosion should occur at a pressure equal to \(500\text{–}550\) mm, whereas Pease at these temperatures recorded rate curves without any explosion at 760 mm pressure. Thus the rate of the \(\mathrm{H_2+Cl_2}\) reaction in Pease’s work was several times slower than in Zagulin’s, which is quite possible, since Pease points to large variations in the rate when, for example, traces of oxygen in the mixture are changed.

Pease supposes that the thermal reaction \(\mathrm{H_2+Cl_2}\) proceeds by a chain path according to the scheme:

1) \(\mathrm{Cl_2}+\text{wall}\longrightarrow \mathrm{Cl} + [\mathrm{Cl}]\) (adsorbed \([\mathrm{Cl}]\) subsequently reacts on the wall with \(\mathrm{H_2}\) or with the material of the wall)

\[ \text{2) }\quad \mathrm{Cl}+\mathrm{H_2}=\mathrm{HCl}+\mathrm{H}. \]

\[ \text{3) }\quad \mathrm{H}+\mathrm{Cl_2}=\mathrm{HCl}+\mathrm{Cl}. \]

\[ \text{4) }\quad \mathrm{Cl}+\text{wall}\longrightarrow [\mathrm{Cl}]. \]

The reaction rate, according to this scheme,

\[ w=\frac{2k_1k_2}{k_4}(\mathrm{H_2})(\mathrm{Cl_2}), \]

\[ k_2=k_{20}e^{-\frac{6000}{RT}}, \]

where \(k_{20}\) numerically practically coincides with that calculated by formula (19),

\[ k_1=k_{10}e^{-\frac{E}{RT}}; \]

\(k_4\) does not depend on temperature.

Putting \(E=31\,000\) cal and \(\dfrac{2k_{10}}{k_4}\) of order 1 (which is very probable, if both quantities are equal to \(\dfrac{uS}{4v}\), where \(u\) in one case corresponds to the thermal velocity with which \(\mathrm{Cl_2}\) molecules impinge on the wall, and in the other—

with an increase of temperature by \(250^\circ\)) increases, i.e. \(\lambda=3\cdot 10^{-4}\) represents the correct value. In addition, such a value will be obtained if we put \(\lambda_{\mathrm{H_2+Cl_2}}\) equal to \(\dfrac{1}{2}\lambda_{\mathrm{H_2}}\).

Cl atoms), we formally arrive at the very same simplest bimolecular law, on the application of which we based the calculation of the absolute values of the ignition temperature and compared it with experiment.

Besides the numerical calculations indicated, a whole series of qualitative facts is in agreement with the thermal theory. Thus, dilution of azomethane with nitrogen changes the critical pressure of spontaneous ignition only little, in accordance with the fact that the thermal conductivity of nitrogen is close to that of azomethane. Conversely, an admixture of helium very strongly raises the critical pressure of azomethane, owing to the increased thermal conductivity of helium. For the case of the reaction \(\mathrm{H}_2 + \mathrm{Cl}_2\), Zagulin\(^6\) has shown that the minimum of the critical pressure upon changing the composition of the mixture lies at about \(70\%\ \mathrm{Cl}_2\) in the mixture. At first sight this contradicts the bimolecular nature of the reaction. However, if one takes into account that the thermal conductivity of \(\mathrm{Cl}_2\) is much less than the thermal conductivity of \(\mathrm{H}_2\), it becomes clear that ignition is facilitated when the mixture is enriched with \(\mathrm{Cl}_2\). With increasing vessel size the critical pressure of spontaneous ignition decreases, as is to be expected from the theory, etc.

The condition for spontaneous ignition, written in the form

\[ \frac{t_r}{t_e}=\frac{QEe}{CR T_0^2} \]

[see formula (13)], is applicable with sufficient accuracy also in the case of conductive heat transfer. To prove this it is necessary to calculate the times of thermal relaxation for plane, cylindrical, and spherical vessels. In doing so one has to solve the equation for the cooling of a heated gas

\[ \frac{dT}{dx}=\frac{\lambda}{c}\Delta T, \quad \text{where} \quad \Delta T=\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2} \]

(\(\lambda\) is the thermal conductivity, and \(c\) is the heat capacity of a unit volume of gas). The solution of this equation in all three cases is represented in the form of a series, each term of which contains \(e^{\rho_i t}\), where \(\rho_i\) is a quantity that increases rapidly with increasing \(i\). Therefore the time of thermal relaxation \(t_e\) will be equal to \(\sim 1/\rho_1\), where \(\rho_1\) is the first and smallest of all the quantities \(\rho_i\). A well-known calculation\(^ {17}\) gives, for a plane vessel with distance \(2r\) between the planes,

\[ t_e=\frac{4cr^2}{\lambda \pi^2}=\frac{cr^2}{2.6\lambda}. \tag{20} \]

For a cylinder of radius

\[ t_e=\frac{4cr^2}{\lambda \pi^2}=\frac{cr^2}{5.7\lambda}. \tag{21} \]

For a spherical vessel with radius \(r\)

\[ t_e=\frac{cr^2}{10\lambda}. \tag{22} \]

\(^1\) The heat capacity of a unit volume of gas \(c\) is related to the heat capacity of a gram-mole \(C\) by the relation \(\dfrac{c}{a}N=C\).

Remembering that \(t_r=\dfrac{a}{ka n e^{-\frac{E}{RT}}}\) and substituting \(t_e\) and \(t_r\) into condition (13), we obtain

\[ \delta_{kp}=\frac{r^2 k a n e^{-\frac{E}{RT}} Q E}{N R T_0^2 \lambda}. \]

For a cylindrical vessel \(\delta_{kp}=2.1\) (instead of 2, calculated by Frank-Kamenetskii); for a spherical one \(\delta_{kp}=3.7\) (instead of 3.32, calculated by Frank-Kamenetskii) [see formulas (16), (17)].

Thus, we may consider that condition (13) does indeed retain its significance also for the case of heat transfer by conduction for any vessel shapes.

Relation (13) is of enormous importance in determining the nature of ignition. The point is that, in addition to thermal ignition, there are cases of ignition determined by an isothermal acceleration of the chemical process (chain ignition). It is not always easy to distinguish them from one another, and relation (13) is a reliable means for doing so.

Indeed, if the reaction time, measured directly at the ignition point or extrapolated to it, is

\[ t_r < \frac{QEe}{C R T_0^2}\cdot t_e, \tag{23} \]

then the ignition undoubtedly has a thermal nature. If, however,

\[ t_r > \frac{QEe}{C R T_0^2}\cdot t_e \tag{24} \]

and especially if it is much greater than this quantity, we may confidently assert that there is no thermal ignition and that the ignition has another nature \(^{1}\).

Let us show the application of formula (13) by examples.

Let us clarify the question of whether \(\mathrm{H}_2+\mathrm{Cl}_2\), illuminated by light of high but easily attainable intensity, can lead to a thermal explosion of the mixture. Let us carry out the calculation for a cylindrical vessel \(r=1\), illuminated from the end face and filled with an equimolecular mixture of \(\mathrm{H}_2+\mathrm{Cl}_2\). We know that each absorbed quantum of light (beginning with the blue and up to the deep ultraviolet part of the spectrum) creates a chain in the reaction \(\mathrm{H}_2+\mathrm{Cl}_2\) with a length from \(10^3\) to \(10^6\) molecules of HCl per quantum. Under conditions when \(\mathrm{H}_2\) and \(\mathrm{Cl}_2\) are not specially purified, no fewer than \(10^3\) molecules of HCl appear as a result of the absorption of one quantum. If \(n_0\) is the number of light quanta absorbed per unit volume, then \(w=10^3 n_0\), or

\[ t_r=\frac{a}{w}=\frac{10^{19}}{n_0\cdot 10^3}=\frac{10^{16}}{n_0}. \]

\(^{1}\) In calculating \(t_e\), we assume the heat transfer to be conductive. However, if in fact it is convective, then the true \(t_e\) will be less than the \(t_e\) appearing in the formula. Thus, if we conclude that a thermal explosion is impossible for the calculated value of \(t_e\) [inequality (24)], then it is all the more impossible if the true \(t_e\) is still smaller.

For a cylindrical vessel (taking \(\lambda_{\mathrm{H_2+Cl_2}}=10^{-4}\) and the heat capacity of unit volume \(c=\dfrac{5}{2\cdot 10^4}\))

\[ t_e=\frac{cr^2}{5{,}7\lambda}=\frac{5}{5{,}7\cdot 2\cdot 10^4\cdot 10^{-4}}\sim 0.5\ \text{sec}. \]

The activation energy \(E\) for the photochemical reaction is well known; it is equal to \(E=6000\ \text{cal}\); \(Q=22000\ \text{cal}\).

For ignition it is necessary that the condition

\[ \frac{t_r}{t_e}=\frac{QEe}{CRT_0^2} \]

be satisfied; or, substituting the numbers,

\[ \frac{10^{16}}{n_0\cdot 0.5}=\frac{2.2\cdot 10^4\cdot 6\cdot 10^3\cdot 2.7}{5\cdot 2(300)^2}, \]

whence

\[ n_0=5\cdot 10^{13} \]

quanta absorbed in \(1\) sec. in \(1\ \text{cm}^3\).

At approximately such light intensities photochemical ignition of the mixture \(\mathrm{H_2+Cl_2}\) does occur. Thus, in this case the nature of self-ignition is thermal.

As another example let us consider the case of displacement of the ignition boundaries of a detonating mixture under the action of ultraviolet light. In pressure–temperature coordinates, the region of self-ignition of a detonating mixture has a peculiar character, quite different from the cases considered above (see Fig. 8). Along with a lower limiting pressure there is an upper limiting pressure, above which ignition does not occur at the given temperature. A whole series of oxidation reactions and some decomposition reactions possess this property. The rate of reaction a little below the lower and a little above the upper limit is so small that relation (13) is not fulfilled. Thus, here we are dealing with ignition of a non-thermal character. The theory of chain reactions, as is known, has described and explained in detail and fully these cases, which fall outside the subject of our article.

Fig. 8

When a detonating mixture is illuminated with ultraviolet light of very short wavelength (1400–1800 Å), a photochemical reaction proceeds. The light dissociates oxygen into atoms, which then initiate the reaction chain. According to data of N. N. Semenov1, which we shall use below, the chain length, i.e. the number of water molecules per one oxygen atom, being equal to unity at room temperature, increases to two at \(350^\circ\) and then begins to grow rapidly. In the region from \(410\) to \(460^\circ\) this growth follows the law \(e^{-\frac{E}{RT}}\), where the activation energy \(E=30000\ \text{cal}\). This occurs in the case when the intensity

light is sufficiently small. With an increase in the intensity of the light, the chain length, beginning at a certain temperature, starts to grow faster than at low intensity, and when it reaches a certain critical value at a temperature \(T\), lower than the self-ignition temperature (at the given pressure), photochemical ignition occurs. The greater the intensity of the light, the lower the temperature at which ignition occurs. The ignition region is displaced to the left (Fig. 8, successive curves 1, 2, 3), and the more so the greater the intensity. The opinion is widespread that such a displacement of the region is connected with thermal self-ignition occurring at high light intensities. The analysis carried out by Nalbandyan and myself shows that this is not so.

As we know, for thermal self-ignition it is necessary that \(\dfrac{t_r}{t_e}\) be no greater than \(\dfrac{QEe}{CRT_0^2}\). From Nalbandyan’s data it is known that in a mixture \(2\mathrm{H}_2+\mathrm{O}_2\) at a pressure of \(18\ \mathrm{mm}\) and in the presence of illumination ignition arises at \(T=434^\circ\mathrm{C}\) (i.e., \(25^\circ\mathrm{C}\) lower than self-ignition in the absence of illumination). At a temperature \(1^\circ\) lower than \(434^\circ\mathrm{C}\), the measured reaction rate corresponds to \(t_r=66.6\) sec. The experiments were carried out in a cylindrical tube of radius \(1.12\ \mathrm{cm}\). The heat capacity \(c\) of the mixture \(2\mathrm{H}_2+\mathrm{O}_2\) under these conditions is \(c=3\cdot10^{-6}\). The coefficient of thermal conductivity of this mixture, according to tabular data, is equal to \(23.73\cdot10^{-5}\) at room temperature. Taking into account the increase of the thermal-conductivity coefficient with temperature, we may put \(\lambda_{434^\circ\mathrm{C}}=47.5\cdot10^{-5}\). Hence, according to formula (21), we find the relaxation time

\[ t_e=\frac{cr^2}{5.7\lambda} =\frac{3\cdot10^{-6}\cdot1.23}{5.7\cdot47.5\cdot10^{-5}} =1.4\cdot10^{-3}\ \text{sec}. \]

Thus,

\[ \frac{t_r}{t_e}=4.7\cdot10^4; \]

the thermal effect

\[ Q=\frac{2}{3}\cdot60\,000=40\,000, \]

whence, taking into account that the activation energy of the photochemical reaction \(E=30\,000\) and the molecular heat capacity \(C=5\),

\[ \frac{QEe}{CRT_0^2} = \frac{4.7\cdot10^4\cdot3\cdot10^4\cdot2.7}{5\cdot2\cdot5\cdot10^5} = 7.5\cdot10^2. \]

Thus,

\[ \frac{QEe}{CRT_0^2}\ll\frac{t_r}{t_e}, \]

i.e., photochemical ignition of the mixture \(2\mathrm{H}_2+\mathrm{O}_2\), just like dark self-ignition, cannot in any way arise as a result of thermal causes and must have an entirely different nature.

Up to now we have confined ourselves to the analysis of the critical condition for self-ignition and have said nothing about how, with time, the temperature of the gas and the amount of reacting substance change. Let us briefly consider this question, following Todes\(^{4}\); in doing so, we shall first make the simplifying assumption that the reaction rate does not change with time and remains the same as at the beginning. This assumption is, of course, incorrect, since the number of initial molecules decreases with the passage of time. However, we shall show below that this assumption does not introduce

into the calculations no substantial error. Under these assumptions the equation determining the change of the gas temperature with time will be:

\[ \frac{Cav}{N}\cdot\frac{dT}{dt} = \frac{kane^{-\frac{E}{RT}}Qv}{N} -\chi S(T-T_0). \tag{25} \]

At pressures below the critical pressure this equation leads to the gradual establishment of the stationary temperature \(T_1=T_0+\Delta T\).

In the case when the pressure is considerably above the critical one, the second term on the right-hand side of the equation may be neglected and the equation

\[ \frac{dT}{dt} = \frac{k\cdot a^{(n-1)}Q\cdot e^{-\frac{E}{RT}}}{C}. \]

may be integrated.

Integration of this equation gives a peculiar course of the temperature with time. In Fig. 9 the solution is shown for a monomolecular reaction with constant \(k=10^{14}\) at \(\frac{E}{RT}=40\), \(\frac{Q}{CT_0}=25\). As we see, for a rather considerable time (2–3 sec.) the temperature rises very slowly, until, finally, it reaches the value

\[ T_0+\frac{RT_0^2}{E} \left(\frac{RT_0^2}{E}\right. \]

pre-explosion heating). From this moment the further rise of temperature to the value corresponding to the explosion temperature (several thousand degrees) occurs practically instantaneously, or, more exactly, in a time constituting a negligible fraction of \(t\). After the whole initial product has burned, the gas begins to cool. The diagram in Fig. 9 is broken, since it is natural that it is impossible to fit on one drawing temperatures of the order of several thousand degrees and several degrees. The time elapsing from the beginning of the reaction to the pre-explosion heating \(t_i\) is called the induction period. It practically coincides with the time from the beginning of the reaction to the explosion (the latter is easily measurable experimentally). For \(t_i\) Todes found the approximate expression

\[ t_i= \frac{RT_0CT_0a}{EQkane^{-\frac{E}{RT}}} = t_r\frac{CRT_0^2}{EQ}; \]

since \(\frac{RT_0}{E}\) is a small quantity, not exceeding, in the cases of interest to us, \(0.05\), \(\frac{CT_0}{Q}\) is likewise a small quantity of order \(\frac{5\cdot500}{20000}\simeq 0.1\) and less, and \(\frac{CRT_0^2}{EQ}\) is of the order from \(0.01\) to \(0.001\).

Fig. 9

Thus, \(t_i = 0.01—0.001\,t_r\). Such is the order of magnitude of the induction period. If we now recall the relation \(t_r \ll \dfrac{QEe}{CRT_0^2}\,t_e\), then \(t_i \ll e t_e\), i.e., the induction period is of the same order as the time of thermal relaxation. Near the critical pressure \(t_i\) increases severalfold, but still remains of the same order of magnitude.

These considerations lead to a very important consequence: since from the beginning of the reaction to the end of the induction time only \(0.01—0.001\) of the reaction time elapses, then by the moment of the rapid rise in temperature, or by the moment of explosion, the substance has had time to be converted by no more than \(1\%\). This circumstance fully justifies the assumption that the reaction rate remains practically constant up to the explosion itself. It justifies not only our calculations of the induction period, but also all the computations of the conditions of spontaneous ignition made earlier, since there we everywhere obtained that the number of reacting molecules at the moment corresponding to the tangency of the straight line of heat loss and the curve of heat production remains the same as in the initial gas. Thereby the considerably more complicated consideration of the question of the conditions of explosion with allowance for burn-out becomes unnecessary for the cases of interest to us.

The matter changes substantially in the case of reactions associated with small \(E\) or with small \(Q\). In this case the typical picture of explosion disappears. If the activation energy is very small and the reaction proceeds at every collision of particles, then in general we cannot prepare such a gas or, if it is a reaction between two gases, we cannot mix them. They will burn at the interface during mixing. Such, for example, is the case when sodium vapor is mixed with chlorine. If the activation energy is small, but the reaction nevertheless does not proceed too rapidly (with a small steric factor in the collision of particles), then the heating \(\Delta T\) will be very large and there will be no quantitative difference between reactions occurring below or above the explosion limit.

For small \(E\), the absence of a sharp boundary between the stationary and the explosive regime is connected with large heating and, consequently, with a high reaction rate even in the stationary regime. In the case of small \(Q\), this difference is eliminated because, owing to the small thermal effect, the conditions of explosion correspond to high temperatures; thus the substance reacts sufficiently rapidly before these temperatures are reached.

Consequently, typical explosive gases must be understood as those which, along with a considerable thermal effect (20,000 and more cal/mol), also possess sufficient thermal stability, withstanding considerable heating without decomposition, i.e., at the same time have a sufficiently high activation energy (20,000 cal and more). To such cases, which in practice are satisfied by all typical explosive gases, all the results obtained are applicable. It should be noted, however, that a small value of \(E\) makes an explosion atypical only in the case when the rate of mono- or bimolecu-

... reaction is connected with abnormally small constants \(k\). When this is not the case (when, for example, for a bimolecular reaction \(k=\sqrt{2\pi\sigma^2 u}\)), a small value of the activation energy does not interfere with the sharpness of the explosive conditions and does not smear out the phenomenon of explosion. In this case the matter changes only in the sense that the explosion occurs at very low temperatures. The quantity \(\dfrac{E}{RT}\) remains large, despite the comparatively low value of \(E\).

Such cases apparently include the explosion of HBr with ozone, which occurs quite sharply, but at a temperature of the order of \(-100^\circ\mathrm{C}\), or the explosion of \(\mathrm{H}_2\) with fluorine. As for a small value of \(Q\) at normal \(E\), here the difference between the explosive and the stationary regime is always smeared out. In the work of Todes and Melent’ev\(^4\), an analysis of the character of the phenomenon for a monomolecular reaction (with constant \(k=10^{13}\)) was carefully carried out (by numerical integration), taking burnout into account.

Fig. 10

Fig. 11

Let us give two curves showing how the substance is consumed with time \(\left(\xi=\dfrac{b}{a}\right.\), where \(b\) is the number of molecules that have reacted, and \(a\) the initial number): 1) in the case of a large thermal effect (typical explosion, see Fig. 10), and 2) in the case of a small thermal effect (Fig. 11). These figures excellently illustrate the qualitative reasoning given above. Making use of the occasion, we give the curve of the course of the temperature (Fig. 12) with time, corresponding to the same conditions as Fig. 10 (typical explosion). We see that for a typical explosion a change in pressure by \(0.2^\circ\) near the self-ignition limit qualitatively changes the picture of the reaction\(^1\). Nothing of the kind is observed in the case

\(^1\) Here \(\mu=\dfrac{\chi S}{Cav}\), i.e. inversely proportional to the pressure \(p\). We see that for large \(Q\), when \(\mu\) changes from \(6.84\cdot10^4\) to \(6.83\cdot10^4\), i.e.

small thermal effects (Fig. 11; the numbers at the curves in the figure denote the values of \(\mu\)).

With this we conclude the theory of thermal self-ignition for the simplest reactions and turn to questions connected with thermal explosion in autocatalytic processes.

It was shown that, at pressures exceeding the critical pressure, the induction period is usually very small and is measured in times of the order of 1 sec. or less. Meanwhile, in many cases self-ignition occurs with a considerably greater delay. Thus, for example, self-ignition of mixtures of methane with oxygen at a temperature of \(730^\circ\mathrm{C}\) and a pressure of \(40\ \mathrm{mm}\) occurs with a delay of 4 min.; induction periods of the same order are observed in the self-ignition of other hydrocarbons, as well as in the oxidation reactions of carbon disulfide and hydrogen sulfide and in the decomposition of \(\mathrm{Cl}_2\mathrm{O}\). Especially considerable delays in self-ignition are observed in liquid and solid explosive substances, where they are often measured in tens of minutes and even hours. According to the data of Roginskii and co-workers \(^{18}\), in the self-ignition of lead azide and nitroglycerin in sealed ampoules the delays sometimes reach 5–10 hours.

Fig. 12

The study of the kinetics of all these reactions\(^{1}\) showed that they all proceed autocatalytically, i.e., the reaction rate as a function of the amount \(x\) of substance that has reacted in the first stages of the transformation is described by the equation \(\frac{dx}{dt}=\varphi x+n_0\). Here \(n_0\) is the number of molecules of the final or intermediate product generated each second per unit volume; generation may occur as the result of a bi- or monomolecular reaction in the volume or as the result of a heterogeneous process. In most cases \(n_0\) is very small and may be neglected for any appreciable values of \(\varphi\). However, if we wish to express \(x\) or \(w\) as a function of time, it is necessary to know \(n_0\), since integration of the equation gives

\[ x=\frac{n_0}{\varphi}\left(e^{\varphi t}-1\right)\quad \text{and}\quad \frac{dx}{dt}=n_0 e^{\varphi t}. \tag{26} \]

When the pressure is changed by \(0.14\%\), a sharp change occurs in the ignition pattern. For \(\psi\) equal to 6.83, a slow reaction proceeds; for \(\psi\) equal to 6.84, after a certain induction period there occurs instantaneous combustion of the substance (explosion).

\(^{1}\) The theory of all these phenomena was developed at the Institute of Chemical Physics; for the literature see Semenov, Chain Reactions.

The constant \(\varphi\) itself decreases in the course of the reaction owing to the decrease in the amount of the initial substances. In the particular case \(\varphi=\varphi_0(a-x)\); here the reaction rate as a function of \(x\) is represented by the curve in Fig. 13, and as a function of time by the curves in Fig. 14, of which curves \(1, 2, 3, 4\) correspond to pressures \(p_1>p_2>p_3>p_4\).

If, for a thermal explosion to arise, it is necessary that the reaction rate reach some critical value \(w_{\mathrm{cr}}\), then at pressure \(p_1\) the explosion will occur after a time \(\tau_1\), for pressure \(p_2\) after a time \(\tau_2\); at pressure \(p_4\) an explosion will not occur at all. The smallest pressure at which an explosion is possible will be \(p_3\), and to this pressure there corresponds the maximum possible delay \(\tau_3\). Thus, in these cases the induction period is no longer connected with the heating of the mixture, but with the time required for isothermal self-acceleration of the reaction to values at which a thermal explosion becomes possible. Therefore the induction period here may be very long.

Fig. 13

Fig. 14

Let us express mathematically the ignition conditions for the given case, limiting ourselves to the first stages of the reaction (10–20%), while we may approximately regard the constant \(\varphi\) as a constant quantity depending on the initial amounts of the initial substances and on the temperature. Usually the constant \(\varphi\) is proportional to the first or second power of the pressure of the mixture, or, what is the same, to the number of molecules of the initial substances per unit volume, and increases exponentially with temperature according to the law \(e^{-E/RT}\). We shall further restrict ourselves to cases in which the time of thermal relaxation (of the order of \(0.01\)–\(0.5\) sec.) is small in comparison with the time of autocatalytic acceleration of the reaction and may be neglected.

Under these assumptions, the entire thermal theory of self-ignition developed earlier by us remains valid; only instead of \(w=ka^n e^{-E/RT}\) we must substitute in the formulas

\[ w=fa^n x e^{-E/RT}+n_0=\varphi x+n_0 . \]

Thus, for example, under the assumption of conductive heat removal for a cylindrical vessel, the critical ignition condition [formula (16)] will be

\[ \delta_{cr}=\frac{Er^2Q(\varphi x-n_0)}{RT_0^2\lambda N}=2. \tag{27} \]

From this equation, knowing \(n_0\) and \(\varphi=fa^n e^{-\frac{E}{RT}}\), we can calculate \(x\), i.e. the depth of conversion at which the explosion will occur. Usually the more interesting quantity is the delay period, which is easily measured experimentally. As we have seen,

\[ x=\frac{n_0}{\varphi}\left(e^{\varphi\tau}-1\right). \]

Substituting this expression into formula (27), we obtain

\[ n_0\left(e^{\varphi\tau}-1\right)+n_0=n_0e^{\varphi\tau} =\frac{2RT_0^2}{Er^2Q}\lambda N, \]

where \(\tau\) is the delay period before the explosion occurs. Hence

\[ 0.434\,\varphi\tau+\lg\frac{n_0}{T_0^2} =\lg\frac{2R\lambda N}{Er^2Q}=\mathrm{const}, \tag{28} \]

since we neglect the weak dependence of \(\lambda\) on pressure and temperature.

The quantity

\[ \frac{2R\lambda N}{Er^2Q}\simeq \frac{10^{-5}\cdot 10^{23}}{10^4\cdot 10^4} \simeq 10^{10} \]

to within one order of magnitude. Hence \(\mathrm{const}\simeq 10\). In the case of small \(n_0\), for example less than \(10^8\), \(\lg\frac{n_0}{T_0^2}\simeq 2\). For smaller \(n_0\), for example \(10^4\), \(\lg\frac{n_0}{T_0^2}\simeq -2\). We see that \(0.434\,\varphi\tau\), when \(n_0\) changes by a factor of \(10^4\), changes only by 20% in either direction about the value \(0.434\,\varphi\tau=10\). Thus, in this case we may approximately assume that, when pressure and temperature vary over a certain fairly significant interval, the quantity

\[ \varphi\tau=f\tau a^n e^{-\frac{E}{RT}}=\mathrm{const}. \tag{29} \]

Hence, at constant pressure, the relation between the delay period \(\tau\) and the absolute temperature must satisfy the expression

\[ \lg\tau=\frac{A}{T}+B, \tag{30} \]

where

\[ A=\frac{0.434E}{R}=0.22\,E. \]

At constant temperature and varying pressure, the relation

\[ \lg\tau=C-n\lg p. \tag{31} \]

must hold. In the case of autocatalytic decomposition of explosives, only the temperature can vary, and only relation (30) must hold. In this case, of course, the value of the constant will be substantially different from that for gases, owing to the different heat-transfer condition.

For the first time, law (30) was established by Roginskii18 and his coworkers for trotyl and nitroglycerin in sealed ampoules (apparently, the use of sealed ampoules is necessary in order to prevent the loss of autocatalytic products by volatilization). Figure 15 gives Roginskii’s data for trotyl. The activation energy for trotyl proved to be equal to 27,000 cal, and for nitroglycerin—25,700 cal.

After we had developed the theory of the induction period, Garner and coworkers19 carried out a series of investigations on the decomposition of crystals of solid explosives suspended in an evacuated flask. In these cases he studied both the kinetics of decomposition and the relation between the delay period of explosion and temperature.

It turned out that the law \(w = n_0 e^{\varphi t} = 10^{0.434\varphi t}\) in the first stages of decomposition (up to 20–30%) is well obeyed for a number of explosives (lead styphnate, barium azide, mercury fulminate). It was found that, when \(T\) changes by \(50^\circ\mathrm{C}\), \(n_0\) is practically constant.

The quantity \(\varphi\) varies with temperature according to the law \(e^{-\frac{E}{RT}}\). The activation energy \(E\) is equal to 40,000–46,000 cal for lead styphnate and 30,000 cal for mercury fulminate.

The relation between \(\lg \tau\) and \(\frac{1}{T}\) also proved, in accordance with the theory, to be linear; moreover, the activation energy \(E\) for lead styphnate, calculated from the constant \(A\), is equal to 39,000, and for mercury fulminate—30,000 cal, i.e., practically the same as for the quantity \(\varphi\), as indeed should follow from the theory.

Fig. 15

Unfortunately, for combustible gases investigations of the induction period of this kind have been carried out only for the case of methane oxidation at high temperatures. The oxidation reaction of methane proceeds autocatalytically, satisfying the law \(w = n_0 e^{\varphi t}\) up to 30–40% conversion. Laws (30) and (31) were verified by Neiman and Egorov20; moreover, as it turned out, the dependence

\[ \tau e^{-\frac{E}{RT}} p^n = \mathrm{const} \]

was confirmed very accurately. Figure 16 shows the experimentally obtained dependence \(\lg \tau — \frac{1}{T}\) at different pressures of the methane–oxygen mixture. The activation energy \(E\) is equal to 90,000, and the exponent \(n\) is close to 2. Kinetic experiments carried out by Bone21 and Hinshelwood22 (though at much lower temperatures) give, for the constant \(\varphi\), values of \(E\) and \(n\) that are close.

Here attention should be drawn to the relation between \(\lg p\) and \(\frac{1}{T}\) at a constant delay period, and to the same relation as applied to the minimum explosive pressure at each given temperature (correspondingly, at the maximum delay time, corresponding to the maximum of the rate of the autocatalytic curve).

In the first case (constancy of \(\tau\) in the initial stages of the transformation)

\[ \lg p=-\frac{0.22E}{nT}+C=\frac{A}{T}+C . \tag{32} \]

For methane,

\[ \frac{0.22E}{n}=A\simeq 10\,000 . \]

In the second case the coefficient at \(\frac{1}{T}\) is somewhat different.

Fig. 16

In the initial stages \(\varphi\) increases with the number of molecules in the volume (\(a\) means also pressure) according to the law

\[ w=\varphi x=fa^{2}e^{-\frac{E}{RT}}. \]

When the amount of reacted substance \(x\) is large, then

\[ \varphi x=f(a-x)^{2}xe^{-\frac{E}{RT}} \]

or

\[ fa(a-x)xe^{-\frac{E}{RT}} \]

(the latter in the case where the total pressure of the mixture plays a role, as, for example, happens when chains terminate at the walls).

Accordingly, the maximum reaction rate will be

\[ w_m=f\frac{4}{27}a^{3}e^{-\frac{E}{RT}} \]

or

\[ w_m=f\frac{1}{4}a^{3}e^{-\frac{E}{RT}} . \]

In any case, the maximum rate will be proportional to the cube of \(a\), and hence also to the cube of \(p\). Hence, according to the thermal-explosion formula,

\[ \lg \frac{p_m}{T}=\frac{0.22E}{3T}+C', \]

or approximately \(\lg p_m=\frac{0.22E}{3T}+C'\), or, in the general case for any \(n\),

\[ \lg p_m=\frac{0.22E}{(n+1)T}+C'=\frac{A_1}{T}+C'. \tag{33} \]

Thus, the temperature coefficient of the minimum explosion pressure will be smaller than the temperature coefficient of the explosion pressure at constant delay (corresponding to experiments with small pre-explosion burning) in the ratio \(\frac{n}{n+1}\). In particular, for methane at \(n=2\) this ratio will be \(^{2}/_{3}\). If the quantity \(A\) in formula (32) is equal to \(10\,000\), then for the coefficient \(A_1\) we obtain \(A_1={}^{2}/_{3}A \approx 6\,600\).

Zagulin\(^6\) experimentally determined the dependence of the minimum explosion pressure on temperature and obtained straight lines analogous to Figs. 6 and 5. From these straight lines he found \(A_1=7\,000\), i.e. a value very close to the calculated value \(6\,600\).

We derived the dependence \(0.434\,\varphi\tau=\mathrm{const}\) under the assumption that \(\lg \frac{n_0}{T_0^2}\) is at least twice as small as \(P=\ln \frac{2RN}{Er^2Q}\). In the case where this quantity is close to \(P\) and \(n_0\) itself varies with temperature, the relation \(\varphi\tau=\mathrm{const}\) ceases to be justified, and one should use the relation \(\varphi\tau+\ln \frac{n_0}{T_0^2}=\mathrm{const}\), or approximately

\[ 0.434\,\varphi\tau+\lg n_0=\mathrm{const}\approx 16 \tag{34} \]

(since \(n_0\) is proportional to \(T_0\) to a power higher than the second). This formula must be used when \(\lg n_0\) is of the same order as the constant, i.e. \(\lg n_0\) is close to 16, when \(0.434\,\varphi\tau\) is noticeably smaller than \(\lg n_0\). The relation may be rewritten in the form \(n_0 e^{\varphi\tau}=\mathrm{const}\).

For small \(\varphi\tau\), corresponding to large \(n_0\), we obtain another limiting law:

\[ n_0 e^{\varphi\tau}=n_0(1+\varphi\tau)=\mathrm{const}, \]

or

\[ \varphi\tau=\frac{\mathrm{const}}{n_0}-1, \]

i.e., with increasing temperature \(\varphi\tau\) will decrease.

In the limit, when \(n_0\) is very large, \(\varphi\tau\) may be neglected altogether, and the explosion will occur without any delay, or more precisely with a delay determined by the time of thermal relaxation. It is not difficult to understand wherein the physical essence of the matter lies here.

The rate \(w\) at \(t=0\) is equal to \(n_0\). If \(n_0\) becomes so large that thermal explosion becomes possible, then there is no need for the realization of self-acceleration of the reaction due to autocatalysis. In this, apparently—

...in this, apparently, lies the main reason for the considerable reductions in self-ignition temperatures when negligible amounts of certain active admixtures of the type \(NO_2\) are added to the combustible mixture. These admixtures, reacting with the combustibles, significantly increase the rate of formation of final or active products. With an increase in \(n_0\), \(\varphi \tau\) falls, and since \(\varphi\) at the given temperature retains a constant value, this causes a considerable shortening of the induction period. Conversely, by keeping \(\tau\) fixed, we can, by adding an active admixture, considerably reduce \(\varphi\), i.e., considerably lower the explosion temperature.

II. IGNITION BY HEATED BODIES

Before proceeding to the calculation of the ignition temperatures of a combustible gas by heated wires or balls, i.e., by bodies of the sizes usually used for this purpose, let us examine, following Zel’dovich \(^{1)}\), the conditions of ignition in the case of a flat vessel containing a combustible gas, one wall of which is maintained at a high temperature \(T_1\), and the other at a low temperature \(T_0\). To eliminate convection, let us arrange the vessel so that the flat walls lie parallel to the earth’s surface, with the more heated wall on top. The problem consists in finding the minimum temperature of the hot wall \(T_1\) which (with room temperature \(T_0\) of the lower wall) will ignite the gas.

We shall still take the reaction rate to be

\[ w = ka^n e^{-\frac{E}{RT}}, \]

and the amount of heat generated each second per unit volume to be

\[ \frac{Qka^n e^{-\frac{E}{RT}}}{N}. \]

By virtue of its exponential dependence on temperature, the reaction will proceed mainly only near the hot wall, thereby disturbing in this region the linear fall of temperature with distance from the upper plate. This reduction of the temperature gradient near the hot wall will be the greater, the higher the temperature \(T_1\), since the more vigorously the reaction proceeds, the more heat is released. The corresponding temperature distributions are shown schematically for different \(T_1\) in Fig. 17. Zel’dovich rigorously proved the assertion (qualitatively expressed earlier by van’t Hoff) that ignition will occur at that temperature \(T_1\) for which the temperature gradient

\[ \left(\frac{dT}{dx}\right)_{x=0} \]

near the hot surface becomes equal to 0. In other words, when the heated wall ceases to lose heat and all the heat flowing toward the cold plate is generated by the reaction, which proceeds mainly in a comparatively narrow layer \(\xi\) near the hot plate.

Fig. 17

\(^{1)}\) In press.

Let us calculate the amount of heat generated by the reaction at \(T_1\), close to the ignition temperature, i.e., when \(\left(\dfrac{dT}{dx}\right)_0=0\). As we shall see below, the distance \(\xi\) is about \(100\, l_0\) from the distance \(d\) between the plates, and, thus, the gradient \(\dfrac{dT}{dx}\) for \(x>\xi\) will not only remain linear as before, but its magnitude will differ by no more than \(100\, l_0\) from \(\dfrac{T_1-T_0}{d}\), i.e., from the gradient that would occur if an inert nonreacting gas were located between the plates. We may therefore assume that the heat flux through \(1\ \mathrm{cm}^2\) at \(x>\xi\) will always be approximately equal to

\[ q=\lambda \frac{T_1-T_0}{d}, \]

where \(\lambda\) is the coefficient of thermal conductivity of the hot gas.

The equation determining the temperature distribution with allowance for the heat generated by the reaction will be

\[ \lambda \frac{d^2T}{dx^2}+F(T)=0, \tag{1} \]

where

\[ F(T)=\frac{Q}{N}ka^n e^{-\frac{E}{RT}}. \]

Near the hot plate, the difference \(T_1-T\) is small in comparison with \(T_1\), and we may put

\[ e^{-\frac{E}{RT}}=e^{-\frac{E}{RT_1}}e^{-\frac{E(T_1-T)}{RT_1^2}}, \]

whence

\[ \lambda \frac{d^2T}{dx^2}=-F(T_1)e^{-\frac{(T_1-T)E}{RT_1^2}}. \tag{2} \]

Integrating this equation under the condition \(\left(\dfrac{dT}{dx}\right)_0=0\) (the ignition condition) and \(T_{x=0}=T_1\), we obtain

\[ \frac{dT}{dx}= \sqrt{ \frac{2F(T_1)RT_1^2}{\lambda E} \left(1-e^{-\frac{(T_1-T)E}{RT_1^2}}\right) }. \tag{3} \]

The quantity \(\dfrac{RT_1^2}{E}\), as we have already seen, usually does not exceed several tens of degrees and, consequently, is always much less than \(T_1-T_0\).

As \(T_1-T\) changes from \(0\) to \(\dfrac{RT_1^2}{E}\), the variable quantity under the root,

\[ \left(1-e^{-\frac{(T_1-T)E}{RT_1^2}}\right), \]

changes from \(0\) to \(0.63\). Further, as \(T_1-T\) increases up to \(\dfrac{2RT_1^2}{E}\), this quantity rises almost to \(0.9\) and thereafter remains practically equal to unity.

We see that the main part of the growth of the gradient \(\dfrac{dT}{dx}\) ends already at

\[ T_1-T=\frac{RT_1^2}{E}, \]

i.e., that the main part of the reaction is concentrated—

in this zone. Subsequently the gradient retains an almost constant value

\[ \frac{dT}{dx}=\sqrt{\frac{2F(T_1)RT_1^2}{\lambda E}}. \tag{4} \]

According to what was said above, we may equate this gradient to the value \(\dfrac{(T_1-T_0)}{d}\), whence

\[ d=\sqrt{\frac{\lambda E(T_1-T_0)^2}{2F(T_1)RT_1^2}} =\sqrt{\frac{\lambda E(T_1-T_0)Ne^{\frac{E}{RT}}}{2Qkan}}, \tag{5} \]

which gives the ignition condition relating the distance between the plates and the gas pressure \(p\) to the ignition temperature \(T_1\).

The width of the zone \(\xi\), where the greater part of the reaction takes place, is determined by integrating equation (2), where, as a first approximation,

\[ \sqrt{1-e^{-\frac{(T_1-T)E}{RT_1}}}\simeq \sqrt{\frac{0.63}{2}}=\sqrt{0.30}\sim 0.5. \]

Hence we find the order of magnitude

\[ \xi=\frac{RT_1^2}{E}\, \frac{1}{0.5\sqrt{\frac{2F(T_1)RT_1^2}{\lambda E}}} = \sqrt{\frac{2RT_1^2\lambda}{EF(T_1)}} ; \tag{6} \]

the ratio

\[ \frac{\xi}{d} = \sqrt{ \frac{2RT_1^2\lambda\,2F(T_1)RT_1^2} {EF(T_1)\lambda E(T_1-T_0)^2} } = 2\frac{RT_1}{E}\frac{T_1}{T_1-T_0} \tag{7} \]

gives a value of the order of \(0.1\), i.e., the zone in which the reaction proceeds is considerably smaller than the distance between the plates.

Let us now pass to the cases of gas ignition that interest us, by small heated bodies—spheres and wires. Let a sphere of radius \(\rho\), heated to the temperature \(T_1\), be placed at the center of a sphere of very large diameter \(R\) (\(R\gg\rho\)), which is filled with reacting gas. Assuming that the zone in which the reaction proceeds is at a distance from the surface of the sphere appreciably smaller than the radius of the sphere, we reduce the problem, with an approximation sufficient for our purposes, to the plane case just considered. The ignition conditions will still be

\[ \left(\frac{dT}{dr}\right)_{r=\rho}=0. \]

The reaction zone will be at a distance \(\xi\) from the surface of the sphere, which will still be expressed by the formula

\[ \xi=\sqrt{\frac{2\lambda RT_1^2}{EF(T_1)}}. \]

The heat flux, calculated per unit surface of the boundary of the reaction zone, will be expressed approximately by formula (4), since outside the zone \(\xi\) the heat release due to the reaction may be neglected. The integral heat flux through a closed surface of radius \(r=\rho+\xi\) will then be equal to

\[ 4\pi(\rho+\xi)^2 q = 4\pi(\rho+\xi)^2 \sqrt{\frac{2F(T_1)RT_1^2\lambda}{E}} . \tag{8} \]

This integral flux is equal to the flux through any other surface of the sphere of radius \(r\).

For \(r>(\rho+\xi)\), where the reaction may be neglected, the temperature distribution will not differ from the temperature distribution in a nonreacting gas, except that instead of a ball of radius \(\rho\) we must take a ball of radius \(\rho+\xi\), since, owing to the heat release in the small zone, the temperature drop in it is negligible.

The temperature distribution around a sphere of radius \(\rho+\xi\), heated to \(T=T_1\), at a temperature at infinity equal to \(T_0\), will be

\[ T-T_0=\frac{(T_1-T_0)(\rho+\xi)}{r}, \]

and the flux

\[ q=-\lambda\frac{dT}{dr} = \lambda\frac{(T_1-T_0)(\rho+\xi)}{r^2}. \]

The integral flux through a sphere of radius \(r\) is equal to

\[ 4\pi q r^2=4\pi(T_1-T_0)(\rho+\xi)\lambda . \tag{9} \]

Equating (8) and (9), we obtain

\[ 4\pi(\rho+\xi)^2 \sqrt{\frac{2F(T_1)RT_1^2\lambda}{E}} = 4\pi(T_1-T_0)(\rho+\xi)\lambda \]

or

\[ (\rho+\xi) = \sqrt{\frac{\lambda E(T_1-T_0)^2}{2F(T_1)RT_1^2}} . \tag{10} \]

The ratio

\[ \frac{\xi}{(\rho+\xi)} = \frac{2RT_0^2}{E(T_1-T_0)} \]

is less than unity. Let us take, for example, \(T_1=600^\circ\mathrm{C}\), \(T_0=300^\circ\mathrm{C}\), \(E=30\,000\); then

\[ \frac{\xi}{(\rho+\xi)}=\frac{1}{6}. \]

Thus, we have shown that the assumptions we adopted concerning the smallness of \(\xi\) in comparison with \(\rho\) are satisfied within the limits of the accuracy required by us.

Neglecting \(\xi\) in comparison with \(\rho\), we obtain the ignition condition in the simple form:

\[ \rho= \sqrt{\frac{\lambda E(T_1-T_0)^2}{2F(T_1)RT_1^2}} . \tag{11} \]

This relation connects the temperature \(T_1\) of the ball igniting the gas with its radius \(\rho\). The smaller the radius \(\rho\) of the ball, the higher it must be heated in order for it to cause ignition of the gas.

Recalling that \(a^n\) is proportional to \(\left(\dfrac{p}{T}\right)^n\), we obtain a relation between the critical pressure and the temperature at constant \(\rho\):

\[ \lg \frac{p}{T_1}=\frac{0.217}{2T_1}E+C=\frac{A}{T_1}+C, \tag{12} \]

i.e., an expression of practically the same type as for self-ignition in a closed vessel [(9) and (10) of Section I]. Unfortunately, there is no corresponding systematized experimental material for checking these relations.

The relation between \(\rho\) and \(T_1\) can be represented in the following form:

\[ \lg \frac{\rho T_1}{T_1-T_0}=\frac{0.217}{T_1}\left(\frac{E}{2}\right)+B. \tag{13} \]

Silver\(^{1}\) studied the dependence of \(T_1\) on \(\rho\) for the ignition of pentane, illuminating gas, and hydrogen in air, using heated metal balls with diameters from 1 to 5 mm. Plotting \(\lg \dfrac{\rho}{T_1-T}\) as a function of \(\dfrac{1}{T}\), he obtained good straight lines; the value of \(E\) calculated from them was 40,000 for pentane and 45,000 for hydrogen. The activation energy obtained for hydrogen (45,000) is considerably lower than what follows from kinetic data. It is possible that Silver’s data are distorted by catalysis on the surface of the metal ball. The presence of catalysis, according to experimental and theoretical results, should strongly affect the ignition temperature. Indeed, wires of different materials ignite the same gas at different temperatures. In particular, with a very strongly catalyzing action, ignition, as is known, does not occur at all; instead, combustion proceeds by a flameless route. This happens because the reaction taking place on the catalyzing heated surface envelops the latter with a layer of reaction products, shielding the heated layers from the combustible gas. In this case the molecules of the combustible substance burn by diffusion into the hot layer, without ignition. It is a great pity that such an important question as the ignition of a gas has not been subjected to systematic experimental investigation. The latter would make it possible not only to test the theory, but also to draw very important conclusions about the character of the chemical reaction by determining \(E\) and \(n\) in the formula \(w=ka^n e^{-\frac{E}{RT}}\) over a very large temperature interval (since, by varying the gas pressure and the wire diameter from 5 mm to \(5\,\mu\), we can vary the temperature igniting the gas by several hundred degrees).

It is not difficult to analyze the question of ignition by heated wires (of radius \(\rho\)) placed coaxially in an external wide cold cylinder (of radius \(R \gg \rho\)).

\(^{1}\) Silver (cited from Jost’s book\(^{2}\)) gave an incorrect theory, as a result of which the activation energies he listed were half as large.

Arguing in exactly the same way as in the case of the sphere, we obtain the following relation between \(\rho+\xi\) and \(T_1\):

\[ (\xi+\rho)\ln\frac{R}{(\rho+\xi)} = \sqrt{\frac{E\lambda(T_1-T_0)^2}{2F(T_1)RT_1^2}}; \]

\[ \xi=\sqrt{\frac{2\lambda RT_1^2}{EF(T_1)}};\qquad \frac{\xi}{(\xi+\rho)\ln\frac{R}{(\rho+\xi)}}= \frac{2RT_1^2}{E(T_1-T_0)} = \frac{1}{6}, \]

which, for \(\dfrac{R}{(\rho+\xi)}\sim 10\), gives \(\ln\dfrac{R}{(\rho+\xi)}\sim 2\), or \(\dfrac{\xi}{(\xi+\rho)}\simeq \dfrac{1}{3}\), whence approximately

\[ \rho\ln\frac{R}{\rho} = \sqrt{\frac{E\lambda(T_1-T_0)^2}{2F(T_1)RT_1^2}}. \tag{14} \]

Until now we have considered the case of purely conductive heat transfer. Let us discuss the question of convection, since the view is quite widespread that calculations of heat transfer by incandescent wires, made on the basis of the equation of conductive heat transfer from wires, are completely incorrect. In broad outline, the role of convection reduces to the fact that the wire is “clothed” in a cylindrical layer of laminarily flowing gas, inside which conductive heat transfer takes place, while at its boundaries the temperature of the external medium \(T_0\) prevails.

According to experimental data, the convective heat transfer from cylinders (referred to unit surface area of the cylinder) is equal to \(q=a(T_1-T_0)=\dfrac{\lambda}{2\rho}A(T_1-T_0)\), where \(A\) increases with increasing wire diameter \(\rho\).

The magnitude of the coefficient \(A\), according to the theory of heat transfer, depends on the dimensionless parameter \(B=\dfrac{d^3\delta^2 g}{\mu^2}\dfrac{\Delta T}{T}\), where \(d=2\rho\), \(\delta\) is the gas density, \(g\) is the acceleration of gravity, \(\mu\) is the gas viscosity, \(\Delta T\) is the temperature difference \((T-T_0)\) (\(T\) being some mean value of the temperature between \(T_1\) and \(T_0\)). It turns out that when \(B\) changes from \(10^{-5}\) to \(10^3\), the quantity \(A\) increases from 0.45 to 2.6, i.e. only by a factor of five.

Let us calculate the value of \(B\) for a gas at atmospheric pressure, taking \(d<1\) mm, \(\delta=10^{-3}\), \(\mu=10^{-4}\), \(g=10^3\), \(\dfrac{\Delta T}{T}\sim 1\). We obtain \(B<10^2\), i.e. for different diameters from \(1\) mm down to \(d=1\) \(A\) changes from 1.7 to 0.5 (for \(r=0.1\), \(A=0.7\)).

Now we can calculate the radius \(R'\) of the laminar layer, where heat transfer proceeds by conduction. Equating the total flux from unit length of wire, \(\pi\lambda A(T_1-T_0)\), to the flux obtained by solving the conduction problem of thermal conductivity between two coaxial cylinders, we obtain

\[ \pi\lambda A(T_1-T_0) = \frac{2\pi\lambda(T_1-T_0)}{\ln\dfrac{R'}{\rho}}, \tag{15} \]

whence \(\ln \dfrac{R'}{\rho}=\dfrac{2}{A}\), \(A=1.7\), \(\ln \dfrac{R'}{\rho}=\dfrac{2}{1.7}\simeq 1.2\), or \(R'=3.2\,\rho\), which gives, for \(\rho=1\) mm, \(R'=3.2\), while for \(\rho=0.1\) mm and \(A=0.7\), \(\ln \dfrac{R'}{\rho}=\dfrac{2}{0.7}\sim 3\) and \(R'=25\,\rho=2.5\) mm.

According to formula (14), we see that, for an outer-cylinder radius \(R\), the presence of convection changes matters only insofar as, instead of \(\ln \dfrac{R}{\rho}\), there appears \(\ln \dfrac{R'}{\rho}\). For \(R=3\) cm and \(\rho=1\) mm, \(\ln \dfrac{R}{\rho}\simeq 3\), while \(\ln \dfrac{R'}{\rho}\simeq 1.2\). For \(\rho=0.1\) mm, \(\ln \dfrac{R}{\rho}\simeq 6\), while \(\ln \dfrac{R'}{\rho}=3\); i.e., with the accuracy we need (of the order of 100% in determining absolute values and tens of degrees in determining the ignition temperature), one may use the conduction solution for wires of radius \(1\) mm and less.

In conclusion, the question of the composition of the hot mixture near the igniting wire close to the ignition limit should be analyzed. Since the temperatures of very thin wires are high, the question arises whether the wire will not be surrounded by a practically reacted mixture, as occurs in the case of strongly catalyzing surfaces.

Zel’dovich\(^{1}\), in his article, gives an answer to this question. As it turns out, the percentage of reaction products in the gas near the igniting surface is always small, since these products diffuse intensively from the reaction zone \(\xi\) into the cold gas.

We give the corresponding calculation below.

The distribution of the number \(b\) of molecules of the reaction product per unit volume and the temperature distribution between the planes are determined by the equations:

\[ \lambda \frac{d^{2}T}{dx^{2}}+w\frac{Q}{N}=0. \]

\[ D\frac{d^{2}b}{dx^{2}}+w=0. \]

Replacing the variable \(T\) by the variable \(\theta=c'\dfrac{(T-T_0)N}{Q}\) and remembering that \(D=\dfrac{\lambda}{c'}\), we rewrite the first equation in the form \(D\dfrac{d^{2}\theta}{dx^{2}}+w=0\).

It follows from this that both variables \(\theta\) and \(b\) satisfy one and the same equation. Far from the hot plate \(T=T_0\) and \(b=0\); thus, for large \(x\) (for example, near the cold plate) \(\theta=0\) and \(b=0\). At the surface of the hot plate \(\dfrac{db}{dx}=0\) always, since a noncatalyzing surface neither reacts with nor absorbs the reaction products and, consequently, the flux of substance is not equal to zero. When the temperature of the hot plate is close to the ignition temperature

\(^{1}\) In press.

(and only in this case), \(\left(\dfrac{dT}{dx}\right)_0\) at the hot plate is equal to 0. Thus, when we reach the ignition condition, the temperature field becomes similar to the concentration field, since \(\theta\) and \(b\) are determined by identical equations and identical boundary conditions.

Thus, \(\theta=b\), or \(\dfrac{c'(T-T_0)N}{Q}=b\) at all points between the plates.

If the number of initial molecules of the hot mixture was \(a\), then the burnout \(b\) will correspond to a temperature \(T\) determined by the equation

\[ \frac{b}{a}=\frac{c'(T-T_0)N}{aQ}. \]

The quantity \(\dfrac{aQ}{N}\) is the total amount of heat generated by a unit volume of gas during an explosion. The quantity \(c'(T-T_0)\) is the amount of heat required to heat a unit volume of gas to the temperature \(T\). We are interested in the burnout at the wire itself, i.e., the quantity

\[ \frac{b_1}{a}=\frac{(T_1-T_0)c'N}{aQ}. \]

\(\dfrac{c'}{a}\) is the heat capacity of one molecule; \(\dfrac{c'}{a}N\) is the heat capacity of a gram-molecule, whence

\[ \frac{b_1}{a}=\frac{(T_1-T_0)C}{Q}, \quad \text{since } C\simeq 5,\ \text{then } \frac{b_1}{a}=\frac{(T_1-T_0)5}{Q}. \]

For \(T_1=500^\circ\) and \(Q=3\cdot10^4\), \(\dfrac{b_1}{a}\simeq \dfrac{1}{30}\), i.e., only \(3\%\) of the reaction products are in the mixture near the hot surface.

For \(T_1=1300^\circ\) and \(Q=3\cdot10^4\), \(\dfrac{b_1}{a}=\dfrac{5\cdot10^3}{3\cdot10^4}=0.17\), i.e., \(17\%\).

Thus, up to very high temperatures, the percentage of dilution by reaction products even at the heated surface itself is small. The same relations also hold for the case of ignition by wires or balls.

LITERATURE

1. Van’t Hoff, Essays on Chemical Dynamics, ONTI, Khimteoret, L., 1936.
2. Jost, Explosions- und Verbrennungsvorgänge in Gasen, Berlin, Verlag Springer, 1939.
3. Semenov, Chain Reactions, Goskhimtekhizdat, L., 1934.
4. Todes, Acta Physicochimica, USSR, 5, 785, 1936; Todes and Melent’ev, 11, 153, 1939.
5. Frank-Kamenetskii, Journal of Physical Chemistry, 13, 738, 1939.
6. Zagulin, Z. physik. Chem., 1, 275, 1928.
7. Hinshelwood, J. Chem. Soc., 123, 2730, 1923; 125, 1841, 1924.
8. Piece, J. Am. Chem. Soc., 56, 2388, 1934.
9. Rice and Allen, J. Am. Chem. Soc., 57, 310, 1935.
10. Kassel, Kinetics of Homogeneous Gas Reactions, Khimteoret, L., p. 186, 1934.

1. Todes, Journal of Physical Chemistry, 4, 78, 1933.
2. Loc. cit., p. 754.
3. Appin and Khariton, Journal of Physical Chemistry, 8, 866, 1936.
4. Volmer, Z. physik. Chem., 9, 141, 1930.
5. Zel’dovich and Yakovlev, DAN, 19, 699, 1938.
6. Isikawa, Z. physik. Chem., 10, 299, 1930.
7. Frank and Mises, Differential Equations of Mathematical Physics, L.—M., pp. 644, 648, 649, 1937.
8. Roginsky, Sow. Phys., 1, 640, 1932.
9. Garner and others, Trans. Farad. Soc., 29, 544, 1933; Trans. Roy. Soc., 139, 576, 1933; J. Chem. Soc., 1939, 1939.
10. Naiman and Egorov, Journal of Physical Chemistry, 3, 61, 1932.
11. Bone, Proc. Roy. Soc., 134, 578, 1932.
12. Hinshelwood, Proc. Roy. Soc., 129, 284, 1930.

1. To be published shortly. ↩↩↩