THERMAL THEORY OF COMBUSTION AND EXPLOSIONS

N. N. Semenov, Leningrad

III. THEORY OF THE NORMAL PROPAGATION OF FLAME1

§ 1. Experimental Data on the Propagation of Flame

If a combustible mixture is ignited at some point by a spark or a heated body, the combustion process spreads through the entire mass of the gas. This is what happens in methane explosions in coal mines and in the gasoline internal-combustion engine. If this phenomenon is observed in a glass tube filled with a combustible mixture and ignited by a spark at one end, it is easy to see a narrow flame front spreading along the tube, separating the gas that has already burned out (behind the front) from the fresh, as yet unburned gas (ahead).

In these cases the flame moves through the initially stationary gas. In many cases, on the contrary, the combustion process is carried out in a stream of combustible gas moving toward the ignition source. In this case a stationary flame is obtained, as occurs in forced-draft gas furnaces or air-jet engines and in ordinary Bunsen burners. In both cases we are dealing with one and the same phenomenon: the motion of the flame front relative to the unburned fresh gas. In the combustion of a homogeneous, premixed mixture the process proceeds very intensively, since the reaction rate at the flame temperature is very high. The process is considerably less intensive when the combustible gas is not completely mixed with air, as, for example, in the combustion of an oil jet ejected from a nozzle, or in the combustion of a gas burner without blast. In these cases the combustion rate is limited not by the reaction rate, but by the rate of mixing, which occurs by diffusion of molecules of the combustible substance and oxygen toward one another. In diesel engines and forced-draft furnaces, efforts are made to accelerate this kind of mixing process of the reacting gases as much as possible by such design arrangements as would ensure vigorous turbulization of the gas. Despite

various mathematical difficulties, the theory of purely diffusion flames is fundamentally clear, since the combustion rate in this case is determined by simple diffusion processes. Much less clear is the basic question of the combustion of premixed gaseous combustible mixtures, which we discussed at the beginning. In this article we shall try to clarify precisely this fundamental question, restricting ourselves to its consideration. Incidentally, let us note that in a number of cases the combustible gas is always “premixed.” This occurs when the flame is associated with the decomposition of some given endothermic substance and proceeds without oxygen or air (the decomposition of $\mathrm{Cl}_2\mathrm{O}$, ozone, vapors of explosive substances, etc.). By the term combustion we shall mean not only flame propagation in processes of combination with oxygen, but also any other chemical processes in which the necessary amount of energy is released and a flame can exist.

Fig. 1

The temperature of the gases immediately behind the combustion front can easily be calculated from thermodynamic data, if heat losses in the flame zone are small. This temperature, for different mixture compositions, usually ranges from 1500 to 3000° K. At the same time, often (as a consequence of dissociation) complete combustion does not occur in the flame front, and part of the substance may burn out only beyond the flame front. A Bunsen burner usually operates on a mixture with an excess of fuel. The main combustion process takes place in the inner cone, which is a thin cap of flame. Here the fuel burns to CO, $\mathrm{CO}_2$, $\mathrm{H}_2$, and $\mathrm{H}_2\mathrm{O}$, the relative concentrations of which are determined by the equilibrium of water gas at the combustion temperature. CO and $\mathrm{H}_2$ burn out through the diffusion of oxygen from the air in the broad outer cone of the burner. The process here is determined not by the reaction rate, but by diffusion.

The two processes in a Bunsen burner can be separated; to do this, a glass tube is placed over the burner on a test tube, as shown in Fig. 1. In this case the cones are separated. The inner cone ($a$) still represents, as it were, a thin flame cap placed over the end of the burner. Since the glass tube is filled with combustion products mixed with products of incomplete combustion and dissociation of the fuel, there is no access of atmospheric oxygen to the inner cone, and the outer flame cone is not formed here. However, upon emerging into the air, these products can burn because of mixing with oxygen, and thus the outer cone ($b$) is located at the mouth of the glass tube. With such an arrangement the cones turn out to be separated.

Thus, we shall be interested in the process of combustion in a homogeneous mixture, occurring in a very thin layer of the flame front (later we shall see that the width of this layer is of the order of $10^{-2}\ \mathrm{cm}$). In this thin layer, however, reaction processes have time to occur; the heat released in this layer will be spent on heating to a high temperature the fresh cold gas situated ahead of the front; conseq-

because, on one side of this layer, the gas consists of reaction products, while on the other it consists of the initial substances, intense diffusion processes will take place in this same layer. Such is the sum of the phenomena occurring in the flame front. Our task is to take all these phenomena into account and, on the basis of this accounting, to obtain a quantitative expression for the rate of combustion of the substance in the flame front.

Since the layer in which combustion occurs is very thin, external, artificially produced swirling, as well as gas motion arising as a result of the combustion process, will only curve the surface of the front and increase its area, but will not disturb the very structure of the layer (for example, its thickness) or the processes taking place in it (they will not, for example, complicate the processes of diffusion and heat conduction in the layer).

Thus, all kinds of currents and vortices in the gas merely increase the area of the flame front, but do not change the properties of any element of the front area. In particular, on a unit surface of the flame front of a given mixture, the same quantity \(v_m\) of combustible will always burn each second. The total quantity of combustible burning each second over the entire front will be proportional to the total area of the flame front \(S\) and equal to \(v_m S\). Thus, all kinds of curvatures of the front, due to gas flows or turbulence, will increase the total quantity of burned substance in proportion to the increase of \(S\)—the area of the front.

Thus, a certain quantity constant for each given mixture naturally enters the theory of combustion—the mass rate of combustion \(v_m\), equal to the number of grams of mixture burning each second per unit area of the flame front.

Burning a definite quantity of substance each second, the flame front moves relative to the initial gas, encompassing ever new portions of it. It is not difficult to establish the relation between the mass rate of combustion and the linear velocity of displacement of the front.

In order to ensure the combustion of \(v_m dS\) grams of mixture, it is necessary that this quantity of mixture be supplied each second to the element \(dS\) of the combustion front, either by displacement of the flame front or by blowing fresh gas toward an immobile flame front. For this purpose the linear velocity of displacement of the given element of the flame front relative to the unburned gas, in the direction of the normal to the surface of the front at the given point, must be equal to

\[ v_0=\frac{v_m}{\rho_0}, \]

where \(\rho_0\) is the density of the initial cold gas. This linear velocity \(v_0\) is called the normal velocity of flame propagation. This constant, characteristic of the combustion processes of each given mixture, does not depend on the hydrodynamic conditions accompanying the combustion process.

If in a tube the flame front were plane (a circle perpendicular to the axis of the tube), then the velocity of displacement of the flame front (upon ignition at the open end of the tube) would be exactly equal to the normal velocity of propagation. In reality, the burned gases, expanding, flow out of the tube through the open end

into the atmosphere according to the laws of laminar flow (i.e., with different velocity, at different distances from the wall). At the wall itself the velocity of outflow of the burnt gases is equal to zero. It is not hard to see that, in this case, the gases burning at the edges of the plane flame front must flow at this place at an angle to the flame front. These flows also cause flows in the fresh, as yet unburnt gas in different parts of the front. A detailed (though qualitative) analysis of the hydrodynamics of the gas flow in this case leads [see Jost¹] inevitably to the conclusion that a plane flame front proves to be unstable, that it must become curved. The stable form will be a cap of the flame front, facing with its convexity in the direction of motion of the flame. In Fig. 2 a series, separated by equal intervals of time, of instantaneous photographs of the flame front in a vertical tube is shown (\(d = 5\ \text{cm}\), mixture of \(58\%\ \mathrm{CO}\) and \(42\%\) air; photograph by Barskii). During the propagation of the flame

Fig. 2

Fig. 3. Instantaneous photographs of a flame front moving in a horizontal tube (\(d = 5\ \text{cm}\)). A mixture of \(55\%\ \mathrm{CO}\) and \(45\%\) air is burning. At point \(a\) oscillations of the flame arise (photograph by Barskii)

in a horizontal tube (if it is not very narrow), convection in the gravitational field also acts on the flame front, making the front asymmetric with respect to the axis of the tube (Fig. 3). As long as there are no changes in the hydrodynamic conditions, the form of the flame front is preserved; consequently, the quantity \(v_m S\) of gas burning per second is also preserved, and hence so is the velocity of displacement of the flame front as a whole. This velocity \(v_{\text{набл.}}\), obviously, will be equal to \(\dfrac{v_m S}{\pi r^2 \rho_0}\)¹, where \(r\) is the radius of the tube, and \(S\) is, as before, the total area of the flame front. Hence

\[ v_{\text{набл.}} = v_0 \frac{S}{\pi r^2}, \]

i.e. the observed velocity of propagation in a given tube will be greater than the normal velocity by as many times as the area of the front is greater than the cross-section of the tube.

¹ With such a displacement of the flame front with velocity \(v_{\text{набл.}}\), in 1 sec. \(v_{\text{набл.}}\rho_0\pi r^2\) grams of substance enter the flame front. On the other hand, in a front of area \(S\), \(v_m S\) grams burn per second. Hence

\[ v_{\text{набл.}}\rho_0\pi r^2 = v_m S = v_0\rho_0 S. \]

Indeed, experiment shows that over a length of 0.5–1 m (depending on the total length of the tube) the velocity of propagation of the flame along the tube remains constant. This is evident both from Figs. 2 and 3 and from Fig. 4, where a photograph is presented of the propagation of the flame, made on a rotating film. (If \(w\) is the linear velocity of motion of the film, then \(\operatorname{tg}\varphi = \frac{v_{\text{obs}}}{w}\). The constancy of the angle \(\varphi\), i.e., the rectilinear trace of the flame, proves the constancy of the propagation velocity.) Both of these photographic methods make it possible to obtain values of the observed propagation velocity.

Fig. 4. Example of uniform flame propagation recorded on a rotating film (Sokolik photograph)

Table 1

Velocity of flame propagation in 10% mixtures of methane with air in tubes of various diameters and calculation from these data of the normal propagation velocity

Tube diameter, cm	Flame propagation relative to the gravitational field	Observed propagation velocity, cm/sec	Front area, cm²	Normal velocity, cm/sec
10	Horizontally	111	300	29
10	»	71	189	29
5	»	92	66	27
5	»	61.5	48.5	25
2.5	»	71.5	12.6	28
2.5	»	63	11.0	28
2.5	»	59	10.4	28
2.5	Vertically upward	58	48	28
2.5	Same	92.5	66.5	27
2.5	Vertically downward	61	46	26
2.5	Same	58	63.5	26

The first of the methods has the advantage that it makes it possible, in addition, to find from the photograph the area \(S\) of the flame front and, by the formulas given above, to calculate the normal propagation velocity \(v_0\). The results of such a calculation are given in Table 1.

We see that for tubes of different diameters, with different shapes of the front and different observed flame-propagation velocities, the calculation gives an actually constant (within the limits of observational errors) normal propagation velocity1.

We have already indicated that uniform propagation of the flame in a tube is observed only at the beginning of the process, over the first meter of the flame path.

Fig. 5

After this, the propagation of the flame, for reasons not yet clarified, assumes a nonuniform character. The flame either begins to move forward with very great speed, or its speed decreases and even takes the opposite direction—the flame begins, as it were, to oscillate. This is clearly visible, for example, in Fig. 23 of Sokolik’s article (Uspekhi fizich. nauk, 23, 236, 1940). This is also visible in Fig. 3, where the regular flame front, after passing 50 cm along the tube, suddenly disappears and a blurred picture appears, which is the result of energetic oscillations of the flame. Sometimes these flame oscillations reach very large amplitudes, as can be seen, for example, in Fig. 5 [Sokolik’s data2]. In many cases there are other kinds of disturbances of the uniform propagation of the flame in the tube, as shown, for example, in Fig. 6, which gives an example of acceleration of flame propagation and the occurrence of a detonation wave (recorded on a moving film; a stoichiometric methane mixture is burning; photograph by Sokolik).

As can be seen, the flame at first propagates slowly with a constant velocity, then the velocity begins to increase smoothly and, finally, at a distance \(S\) from the ignition point a new regime of flame propagation arises, called detonation.

Fig. 6

Detonation combustion (like normal combustion) proceeds at a constant velocity. However, the velocity of detonation exceeds the velocity of normal combustion by hundreds and thousands of times. Detonation does not arise in every mixture, but only in mixtures with a sufficiently large thermal effect. If normal propagation is connected with the propagation of the flame by heat conduction through heating by the flame front of neighboring cold

layers of fresh gas to a high temperature, ensuring a high reaction rate, then in the detonation regime the propagation of the flame takes place by means of a hydrodynamic shock wave. At the front of this wave there occurs a rapid compression of the initial gas to a very high pressure. As a result of adiabatic compression, the temperature of the gas rises sharply and the mixture reacts rapidly. In this article we shall not deal with these phenomena, which are interesting and important from the theoretical and practical point of view. We need only dwell briefly on questions of the oscillatory regime of propagation, which often arise, disrupting the regime of uniform normal propagation. There are not infrequently cases when the amplitude of these oscillations is small. In such cases, despite the presence of oscillations, the time-averaged velocity of flame propagation retains the same value as the velocity of uniform propagation taking place at the beginning of the process. In Jost’s opinion, this fact may be regarded as proof that, for small oscillations, the flame front remains steady and preserves its form. Relative to the unburned gas, the flame propagates with the same constant velocity. The cause of the visible nonuniformity of the motion of the flame relative to the observer is the oscillation of the gas column in the tube as a whole. These oscillations arise as a result of certain unexplained hydrodynamic causes associated with the motion of gases during flame propagation. Small oscillations of the flame are noticeable in Fig. 2 (successive images of the front now become slightly denser, now rarer).

In those cases when the amplitude of the gas oscillation becomes large, this oscillatory motion of the gas, by changing the distribution of flow velocities over the cross section of the tube, leads in turn to a distortion of the flame front, increasing its area and thereby increasing the mean burning velocity.

The acceleration of the flame preceding detonation, as Shchelkin showed, is also the result of a progressive increase in the area of the flame front and is not connected with an increase in the normal velocity of propagation. He showed that this acceleration is observed only when the gas flows arising as a result of combustion have a velocity exceeding the critical Reynolds velocity, i.e., when gas turbulization may arise, leading to an increase in the area of the combustion front and, consequently, to an increase in the observed propagation velocity. If the normal propagation velocity is so small that the critical velocity of the gas flow is not reached, detonation also does not arise. If the normal velocity is sufficiently large and vortices can appear, they nevertheless do not arise immediately in a smooth tube and grow gradually. Therefore we observe at first a constant velocity of normal propagation, and only after the flame has traversed a more or less long section of the tube do flame acceleration and detonation arise. Shchelkin directly proved the correctness of his considerations by observing flame propagation in a tube with rough walls (for which purpose he inserted a wire spiral inside a glass tube). In this case the vortices had to arise much more readily. And indeed, in such tubes detonation arose in the immediate vicinity of the ignition source, at a distance of a few centimeters of the flame path, i.e., the distance from the point of ignition to the onset of detonation was reduced by more than a factor of 10 in comparison with smooth tubes.

Thus, the review we have seen of the whole range of disturbances in the observed velocity of flame propagation is connected either with oscillations of the gas column or with an increase in the area of the combustion front due to hydrodynamic causes (in laminar flows—by a change in the form of the flame surface; in turbulent flows—by covering the front surface, as it were, with fine ripples). The normal propagation velocity remains constant in all these regimes, and only detonation is a fundamentally new type of flame propagation.

However, there also exist more direct and immediate methods for determining the normal velocity of flame propagation, where

the hydrodynamic situation of the gas motion is considerably simpler than in the propagation of a flame through a tube. When a flame propagates in a spherical vessel and is ignited at the center, the whole situation is so symmetrical that only radial flows are possible and there are no grounds for distortion of the spherical flame front.

A radial flow arises because the gas burned in the flame front (thus enclosed inside the flame sphere) has a combustion temperature many times higher than the temperature of the initial gas and, consequently, occupies a larger volume. This creates a radial flow of unburned gas in front of the flame front, whose velocity, when added to the normal propagation velocity (relative to the unburned gas), leads to the observed velocity \(v_{\text{набл.}}\), considerably exceeding \(v_0\). In the case where a soap bubble, expanding with the expansion of the gas, is taken as the spherical vessel, we can conduct the combustion process at constant pressure and thereby greatly simplify the calculation relating the observed velocity \(v_{\text{набл.}}\) to the normal velocity \(v_0\).

Indeed, taking, by convention, any element of the front of unit area as fixed, we know that \(v_m = v_0 \rho_0\) grams of initial gas are supplied to it every second and, consequently, the same amount of burned gas leaves it. If the density of the burned gas at the combustion temperature is \(\rho\), then the velocity of the burned gas \(v\), multiplied by \(\rho\), must, according to the law of conservation of matter, be equal to \(v_m = v_0 \rho_0\). Hence \(v_0 \rho_0 = v \rho\), or \(v = \dfrac{v_0 \rho_0}{\rho}\), or \(v_0 = \dfrac{v \rho}{\rho_0}\). But the velocity \(v\) with which the burned gas moves relative to the flame front is equal to the velocity of the flame front relative to the burned gas, and since the burned gas under the conditions of the experiment is motionless relative to the observer, \(v = v_{\text{набл.}}\) is the observed velocity of flame propagation. It can be measured by the two photographic methods indicated above: 1) by a series of successive instantaneous photographs or 2) by photographing the motion of the flame (the expansion of the flame sphere) on a rotating film (see Fig. 16 in the previously mentioned article by Sokolik, where a soap bubble at the beginning and at the end of the process is shown schematically by a dotted line). Thus, knowing \(v_{\text{набл.}}\), we find the normal velocity

\[ v_0 = \frac{v_{\text{набл.}} \rho}{\rho_0} = v_{\text{набл.}} \frac{T_0}{T_1} = v_{\text{набл.}} \frac{r_0^3}{r_1^3}, \]

where \(T_0\) is the initial temperature of the original gases and \(T_1\) is the combustion temperature, i.e. the temperature of the burned gases. Since, according to Gay-Lussac’s law, the volume \(v = \frac{4}{3}\pi r^3\) (at constant pressure) is directly proportional to the absolute temperature \(T\), the ratio \(\dfrac{T_0}{T_1}\) will be equal to the ratio \(\dfrac{r_0^3}{r_1^3}\), where \(r_0\) is the initial radius of the bubble, and \(r_1\) is the radius of the bubble when all the combustible gas in it has burned (it can easily be found from the photographs).

Thus, we obtain an elegant and, from the theoretical point of view, the clearest method for measuring the normal velocity of propagation. If one works with closed tubes or solid closed spherical vessels, then, owing to the expansion of the gases during combustion, the pressure will increase as the flame propagates. Since the propagation velocity may depend on the pressure, and since accounting for the hydrodynamics is complicated here, the method of measuring the normal velocity under these conditions is more difficult. However, using long tubes or large spherical vessels and confining ourselves to measuring the initial phase of flame propagation (when the rise in pressure is small), we can successfully apply to the analysis of the results all the same considerations and formulas that we obtained above for tubes open at one end and for the soap bubble.

The simplest method of measuring the normal velocity is nevertheless the Bunsen-burner method, i.e., the method of a stationary flame. Its only drawback is the large consumption of combustible gas. If in a tube the flame front were plane and propagated, consequently, with the normal velocity \(v_0\), then by blowing the gas with velocity \(v_0\) toward the flame we could stop the flame and thus obtain a motionless flame front in the gas jet. Such a state, however, would be rather unstable, since with small fluctuations in the velocity of the jet the flame would either jump forward or be carried backward, depending on whether the jet velocity decreased or increased in comparison with \(v_0\). If, in the same jet, we place on the axis of the tube a point source of constant ignition in the form of a glowing point or spark, then the jet velocity can be increased considerably above \(v_0\) while preserving the stability of the flame. In this case the flame front will be a cone elongated in the direction of the flow, with its vertex at the ignition point and its base on the walls of the tube. The greater the velocity \(v\) of the jet, the more elongated the cone. The angle of the cone—the angle between the axis (or, what is the same thing, the velocity of the jet) and the generator of the cone—is such that \(v_0 = v \sin \varphi\). It is not difficult to see that in this case the amount of gas supplied by the jet per unit surface of the combustion cone will be equal to \(\rho_0 v \sin \varphi = \rho_0 v_0\), and this quantity must be equal, as we know, to the mass burning velocity \(v_m = v_0 \rho_0\). Hence it is clear that the jet velocity can considerably exceed \(v_0\) without flame blow-off. If the ignition source is not a point on the axis but a heated circular wire located near the wall of the tube, then it is not difficult to understand that the shape of the front must again be conical, with its base on the igniting circle; both cases are shown in Fig. 7.

Fig. 7

Similar experiments are much better and simpler to carry out in a Bunsen burner. At the cut-off edge of the burner tube there are formed

stagnation of the gas, in which the flow velocity is small, and therefore, when the burner is lit, these edges play the role of a constant source of ignition. The gas jet issuing from the burner (the burner usually operates in the laminar-flow regime) travels still rather far in the form of a cylindrical column with the same velocity \(v\) as inside the burner tube. Thus conditions are created for the formation of a stationary conical flame front. Knowing the flow rate of the combustible gas and the cross-sectional area of the burner, we determine the blowing velocity \(v\), and from the angle of the cone \(\varphi\) we find the normal propagation velocity

\[ v_0 = v \sin \varphi . \]

Fig. 8     Fig. 9

It is simpler to measure not the angle \(\varphi\), but the height of the inner cone \(h\).

Then the angle \(\varphi\) is determined from the equation

\[ \tg \varphi = \frac{r}{h} \]

(Fig. 8).

A simple conical form is obtained when the gas-flow velocity is the same over the entire cross section of the tube. In fact, in laminar gas flow through a tube the velocity is distributed according to a paraboloid, being maximal in the middle and decreasing toward the surface of the tube. It is not difficult to understand that in this case the stationary flame front will differ somewhat from a simple cone and will take the form of a cap, shown in Fig. 9. It is precisely this shape that the inner cone of a Bunsen burner has.

For this reason, measurement of the normal velocity from the height of the cone by the preceding formulas is associated with a certain error. A more accurate method, therefore, is to photograph the cone and determine from the photograph the surface of the cone \(S\). Knowing the burning surface and the volume \(V\) of gas supplied to the burner each second, we easily find the normal velocity \(v_0\) from the formula

\[ v_0 = \frac{V}{S}. \]

In work involving determination of the propagation velocity by the burner-cone method, it is often more convenient to use an apparatus with separated cones.

Fig. 10. Normal propagation velocity in air mixtures of carbon monoxide

Solid curve—experiments of Passauer, ×—experiments of Khitrin, ○—experiments of Barskov, □—experiments of Yan.
The dashed curve is calculated from the formulas of the new theory of flame propagation for the kinetic law (37)

The largest number of measurements of normal velocities refers to air and oxygen mixtures of carbon monoxide, hydrogen, methane, or technical combustible gases of the illuminating-gas type. Flames of higher hydrocarbons, hydrogen sulfide, etc., have been studied in less detail.

Thus, the experimental material almost exclusively concerns reactions of oxidation by free oxygen, which, incidentally, are of greatest interest for practice. Processes in other two-component systems (for example, combustion in chlorine of hydrogen and organic substances), as well as in one-component systems, i.e., cases of flame propagation during exothermic decomposition of a gaseous substance, have hardly been studied at all (except for the case of flame propagation in ozone).

For two-component systems the greatest interest is presented by the dependence of the normal propagation velocity on the percentage of combustible gas (CO,

Fig. 11. Normal velocity of flame propagation in mixtures of CO with an atmosphere of \(N_2 + O_2\). The composition of \(N_2 + O_2\) along each curve is constant (Jahn)

\(1\)—87% \(N_2\) + 13% \(O_2\);
\(2\)—83% \(N_2\) + 17% \(O_2\);
\(3\)—79% \(N_2\) + 21% \(O_2\);
\(4\)—70% \(N_2\) + 30% \(O_2\);
\(5\)—60% \(N_2\) + 40% \(O_2\);
\(6\)—40% \(N_2\) + 60% \(O_2\);
\(7\)—20% \(N_2\) + 80% \(O_2\);
\(8\)—1.5% \(N_2\) + 98.5% \(O_2\)

Fig. 12. Normal velocity of flame propagation in mixtures of hydrogen with an atmosphere of \(N_2 + O_2\). The composition of \(N_2 + O_2\) along each curve is constant (Jahn)

\(1\)—87.5% \(N_2\) + 12.5% \(O_2\);
\(2\)—85% \(N_2\) + 15% \(O_2\);
\(3\)—82.5% \(N_2\) + 17.5% \(O_2\);
\(4\)—79% \(N_2\) + 21% \(O_2\);
\(5\)—75% \(N_2\) + 25% \(O_2\);
\(6\)—70% \(N_2\) + 30% \(O_2\);
\(7\)—65% \(N_2\) + 35% \(O_2\);
\(8\)—60% \(N_2\) + 40% \(O_2\);
\(9\)—50% \(N_2\) + 50% \(O_2\);
\(10\)—40% \(N_2\) + 60% \(O_2\);
\(11\)—30% \(N_2\) + 70% \(O_2\);
\(12\)—20% \(N_2\) + 80% \(O_2\);
\(13\)—10% \(N_2\) + 90% \(O_2\);
\(14\)—1.5% \(N_2\) + 98.5% \(O_2\)

\(H_2\), \(CH_4\)) in the mixture at different ratios of oxygen and inert gas. In Figs. 10, 11, 12, 13 the corresponding curves are given for mixtures of CO, \(H_2\), and \(CH_4\). The propagation velocity reaches a maximum for CO and \(H_2\) not at the stoichiometric ratio of the components, but with a noticeable excess of fuel. Dilution of the mixture by an inert gas lowers the flame propagation velocity (see Figs. 11 and 12), which is understandable, since in this case the maximum combustion temperature is lowered. The greater the heat capacity of the inert gas, the more

the more it lowers the combustion temperature and the more strongly it reduces the rate of flame propagation. Thus, if a mixture of methane with air is diluted first with carbon dioxide and then with argon, in the first case the rate of flame propagation proves to be approximately three times smaller than in the second.

Preheating the mixture increases the rate of flame propagation. In Fig. 14 the corresponding data of Passauer are given for air mixtures of carbon monoxide containing 2.3% moisture.

The propagation of flame in mixtures of carbon monoxide depends in a surprising way on the moisture content. The rate of propagation increases with an increase in the percentage of moisture; this is shown in Fig. 15 according to the data of Fiock and King. The admixture of hydrogen and other hydrogen-containing substances acts in the same way as moisture. A completely dry air and even oxygen mixture of carbon monoxide, devoid of any hydrogen-containing substances, is not at all capable of flame propagation, i.e., is not combustible. If certain substances prone to absorbing moisture, such as, for example, CCl₄, are added to a moist CO mixture, they greatly reduce the rate of flame propagation. However, these substances exert a similar effect on the rate of propagation of hydrogen flames as well, where the admixture of moisture naturally has no effect.

Fig. 13. Normal rate of flame propagation in air mixtures of methane (Coward and Hartwell)

Fig. 14

The question of the variation of the rate of propagation with pressure is not experimentally clear, since here there are sharp contradictions in the data of different authors. In addition, in experiments with a CO mixture, where the rate of propagation strongly depends on the amount of moisture, naturally—

THERMAL THEORY OF COMBUSTION AND EXPLOSIONS

different results are obtained, depending on whether the experiments are carried out at a constant partial pressure of water or at a constant percentage of moisture. The latter case is of theoretical interest, but in practice the former is usually used. Treating Ubbelohde’s data, Zel’dovich came to the conclusion that, at a constant percentage of moisture, the normal propagation velocity in air mixtures of CO does not depend on the pressure of the mixture. In oxygen mixtures the propagation velocity increases somewhat with pressure, approximately according to the law \(v_0 \simeq \sqrt[4]{p}\). In air mixtures of hydrogen, benzene, and gasoline the propagation velocity apparently does not depend on pressure; in methane mixtures the velocity decreases with pressure approximately as \(\dfrac{1}{\sqrt{p}}\). All that can be asserted in such a state of the experiment is that the velocity of normal flame propagation in oxidation reactions varies with pressure according to the law \(v_0 \simeq p^n\), where \(n\) lies between \(0\) and \(-\dfrac{1}{2}\). Correspondingly, the mass rate of combustion \(v_m = v_0 \rho\) increases with pressure according to the law \(p^m\), where \(m\) lies between \(1\) and \(+\dfrac{1}{2}\). Only in oxygen mixtures, where the combustion temperature is very high, is a very slight increase in the propagation velocity \(v_0\) with pressure observed, and correspondingly an increase in the mass rate of combustion \(v_m\) with pressure to a power somewhat higher than unity.

Fig. 15

\(1\)—\(p_{\mathrm{CO}+\mathrm{O}_2}=100\) mm; \(2\)—\(p_{\mathrm{CO}+\mathrm{O}_2}=150\) mm;
\(3\)—\(p_{\mathrm{CO}+\mathrm{O}_2}=200\) mm; \(4\)—\(p_{\mathrm{CO}+\mathrm{O}_2}=300\) mm;
\(5\)—\(p_{\mathrm{CO}+\mathrm{O}_2}=760\) mm

§ 2. Analysis of the old theoretical views on flame propagation

We have seen that all phenomena of slow flame propagation reduce to the property of the flame front to burn, each second, per unit surface, a definite quantity \(v_m\) grams of mixture or, what is the same thing, to propagate relative to the unburned gas in the direction normal to its surface with the velocity of normal propagation \(v_0\) cm/sec. The visible motion of the flame is the result of the superposition on this fundamental property of the flame front of all kinds of hydrodynamic flows. Thus, the theory of slow flame propagation is the theory of the processes occurring in the thin combustion layer in the flame front. Since the curvature of the surface of the flame front is always small in comparison with the thickness of the front (i.e. of the preheating and reaction layer), with a sufficient degree of accuracy we may regard all quantities (for example,

(for example, temperature) within this layer are functions of one single coordinate, whose direction is perpendicular to the given area element of the front (a one-dimensional problem). In addition, by associating the origin of coordinates with the flame front, i.e., by studying the process in a coordinate system moving in space with the given element of the flame front, we may assume that the distribution within the combustion and preheating layer of all quantities of interest to us (for example, temperature) does not change with time. Thus, the question reduces to the solution of a one-dimensional stationary problem.

For the subsequent calculations we introduce the following notation:

\(T_0, T, T_1\), respectively: the initial temperature of the original gas, the current temperature in the course of combustion at various points of the flame zone, and the maximum combustion temperature without allowance for heat losses. All temperatures are expressed in degrees on the absolute scale.

\(\vartheta\) is the difference \(T_1 - T\).

\(a_0, a\)—the numbers of molecules per unit volume of combustible substance in the original cold mixture and at different places in the combustion zone in the case where a decomposition reaction takes place (\(\mathrm{Cl_2O}\), \(\mathrm{O_3}\), explosive substance).

\(n, n_0\)—the total number of molecules in the original mixture at temperatures \(T\) and \(T_0\).

\(\lambda\)—coefficient of thermal conductivity of the gas. \(\lambda\) increases with temperature.

\(\lambda^*\)—coefficient of thermal conductivity of the products of combustion at temperature \(T_1\).

\(c_p\)—heat capacity of unit mass of gas at constant pressure. \(c_p\) increases with temperature. At first we shall take \(c_p\) to be independent of temperature and equal to the mean value \(\bar c_p\) between \(T_0\) and \(T_1\).

\(c_p^*\)—heat capacity of unit mass of the reaction products at temperature \(T_1\).

\(H, H_0, H_1\)—heat content of unit mass of gas respectively at temperatures \(T, T_0\), and \(T_1\).

\(\rho_0, \rho\)—density of the gas at temperatures \(T_0\) and \(T\).

\(\rho^*\)—density of the products of combustion at temperature \(T_1\).

\(v_0\)—normal velocity of propagation of the flame in \(\mathrm{cm/sec}\); i.e., the velocity of motion of the flame front relative to the cold unburned gas along the normal to the surface of the front at the given point (or, what is the same, the component normal to the front of the velocity of motion of the original cold gas relative to the flame front);

\(v\)—velocity of the gas relative to the flame front inside the combustion zone at temperature \(T\).

\(v_m\)—mass velocity of combustion, i.e., the number of grams of gas mixture burning on \(1\ \mathrm{cm}\) of the flame front.

\(D, D_0\)—diffusion coefficients of gases at temperatures \(T\) and \(T_0\).

\(M_0\)—mass fraction of the combustible substance in the original mixture or, what is the same, the number of grams of combustible in \(1\ \mathrm{g}\) of mixture. In the case of a two-component system, for example in oxidation reactions, we shall understand by \(M_0\) the mass fraction of the fuel oxidized in the course of combustion. \(M_0\) is always \(\leq 1\).

$M$ — mass fraction of fuel, or the number of grams of fuel in $1\,g$ of mixture at various places in the combustion zone. $M$ is always $\leqslant 1$. In the initial mixture $M=M_0$.

$N$ — Avogadro’s number $=6\cdot 10^{23}$ molecules.

$Q'$ — heat of reaction of one molecule of fuel, and in the case of a two-component system, for example, oxidation of CO to CO$_2$, the heat of oxidation of CO, referred to one CO molecule.

$Q$ — heat of combustion of 1 gram-molecule of fuel (heat of decomposition of 1 gram-molecule of Cl$_2$O, heat of combustion of 1 gram-mole of CO, etc.).

$L$ — heat of combustion of $1\,g$ of the initial mixture (if a one- or two-component system diluted with an inert gas is burning, then $L$ refers to $1\,g$ of the weight of the entire mixture, including the inert gas).

$w$ — the reaction rate is expressed as the number of fuel molecules that have reacted per unit time in unit volume.

For reactions of zeroth, first, and second order,

\[ w=Se^{-\frac{E}{RT}},\qquad w=kae^{-\frac{E}{RT}},\qquad w=ka^2e^{-\frac{E}{RT}}. \]

$E$ — activation energy of the reaction.
$R$ — gas constant.
$\mu$ — molecular weight of the fuel.

$\left{\begin{array}{l}
Z\text{ — collision factor, i.e., the number of collisions per unit}\
\text{time in unit volume of two molecules, if in unit volume}\
\text{there are in all these two molecules.}\
l\text{ — mean free path.}\
u\text{ — velocity of thermal motion of a molecule.}\
\sigma\text{ — effective diameter of molecules in collisions.}\
m\text{ — mass of a fuel molecule.}\
\bar m\text{ — average mass of a molecule of the mixture.}
\end{array}\right.$

The following simple relations between the indicated quantities will be needed by us in what follows:

\[ \left. \begin{gathered} M_0\rho_0=\frac{a_0\mu}{N};\qquad \frac{\rho}{\rho_0}=\frac{T_0}{T};\qquad L=\frac{M_0Q}{\mu}=\frac{M_0Q'N}{\mu}=\frac{M_0Q'}{m}= \\ =\frac{a_0Q'}{\rho_0}=H_1-H_0;\qquad Q=NQ'.\\[4pt] \text{For constant heat capacity }\quad L=c_p(T_1-T_0).\\[4pt] \text{For variable heat capacity }\quad H=\int_0^T c_p\,dT;\qquad H_0=\int_0^{T_0} c_p\,dT;\\[4pt] H_1=\int_0^{T_1} c_p\,dT;\qquad c_p=\frac{\partial H}{\partial T};\qquad D=\frac{1}{3}lu;\qquad \lambda=\frac{1}{3}lu\rho c_p;\\[4pt] Z=\sqrt{2\pi\sigma^2u};\qquad l=\frac{1}{\sqrt{2\pi\sigma^2un}}=\frac{1}{Zn}.\\[4pt] \frac{am}{\rho}=M;\qquad \frac{a_0m}{\rho_0}=M_0;\qquad Nm=\mu;\qquad n\bar m=\rho. \end{gathered} \right\} \tag{A} \]

In conclusion, let us give the various possible definitions of the concentration of the combustible and of the temperature in the combustion zone.

1. \(a\)—the number of molecules of combustible per unit volume at various points of the combustion zone. It is precisely this concentration that we shall mainly use.

2. \(\dfrac{a}{n}=C\)—the relative molecular concentration, i.e., the fraction of combustible molecules among all molecules in the mixture at various points of the combustion zone.

\[ \frac{a}{n}\,100=C\% \]

is the percentage relative molecular concentration of the combustible in the mixture.

1. \(M\) is the mass concentration of the combustible in the course of combustion (if the molecular weights of all the gases composing the mixture and of their combustion products are close to one another, \(M=\dfrac{a}{n}=C\)),

\[ B=\frac{M}{M_0} \]

is the relative mass concentration of the combustible. If the molecular weights of all gases in the mixture are the same, then

\[ \frac{M}{M_0}=\frac{a}{n}:\frac{a_0}{n_0}=\frac{C}{C_0}. \]

1. \(T\)—the absolute temperature at various points of the combustion zone.

2. \(T-T_0\)—the increase of temperature over the initial one.

3. \(T=\dfrac{T-T_0}{T_1-T_0}\)—the relative increase of temperature over the initial one.

In the flame front, within the layer of preheating and reaction, the temperature rises from the initial temperature \(T_0\) of the original cold gas to the temperature \(T_1\)—the maximum combustion temperature, measured in thousands of degrees.

Considering the distribution of temperature inside the combustion and preheating layer, it is not difficult to arrive at the following differential equation:

\[ \frac{d}{dx}\frac{\lambda}{c_p}\frac{dH}{dx}-v_m\frac{dH}{dx}+wQ'=0. \tag{1} \]

In the case where \(\dfrac{\lambda}{c_p}\) may be regarded as a constant quantity in the temperature interval from \(T_0\) to \(T_1\) (i.e., in an interval of the order of \(1500\text{--}2500^\circ\)), the equation may be written in the form

\[ \frac{\lambda}{c_p}\frac{d^2H}{dx^2}-v_m\frac{dH}{dx}+wQ'=0. \tag{1'} \]

Taking the heat capacity \(c_p\) as constant in the interval from \(T_0\) to \(T_1\), we arrive at the equation

\[ \frac{d^2T}{dx^2}-a\frac{dT}{dx}+\frac{wQ'}{\lambda}=0, \tag{2} \]

where

\[ a=\frac{v_m c_p}{\lambda}. \]

Let us present the derivation of the equations. When heat is released in the gas phase and the process of heat transfer from some points of space to others takes place, then (even in the absence of forced motion and convection under the action of gravitational force) a process of displacement of individual elements of the gas will inevitably occur as a result of thermal expansion. Thus, even in this case, and still more in the general case, heat transfer due to gas flows will be added to thermal conduction. Let us denote the linear velocity of the latter by \(V\) (a vector varying in magnitude and direction at different points of the space occupied by the gas). In this case the heat flux per unit time through an area of \(1\ \text{cm}^2\), situated perpendicular to the direction \(V\), will be \(\rho HV\), where \(\rho H\) is the heat content of a unit volume of gas. In addition, heat will be transported by the heat-conduction flux; the magnitude of this flux is, as is known, \(-\lambda \operatorname{grad} T\). The total heat flux will be equal to

\[ \vec{\Omega}=-\lambda \operatorname{grad} T+\rho HV. \]

If the reaction proceeds with rate \(w\), then it releases, each second, \(wQ'\) cal of thermal energy per unit volume. Hence the change of the heat content of a unit volume per unit time,

\[ \frac{\partial(\rho H)}{\partial t}, \]

will be equal to

\[ \frac{\partial(\rho H)}{\partial t} = -\operatorname{div}\vec{\Omega}+wQ' = \operatorname{div}\lambda \operatorname{grad} T-\operatorname{div}(\rho HV)+wQ', \]

or, using the equation of conservation of matter

\[ -\frac{\partial \rho}{\partial t}=\operatorname{div}(\rho V) \]

and the mathematical relations

\[ \frac{\partial(H\rho)}{\partial t} = \rho \frac{\partial H}{\partial t} + H\frac{\partial \rho}{\partial t} \quad \text{and} \quad \operatorname{div}(\rho HV) = H\operatorname{div}(\rho V)+\rho V\operatorname{grad} H, \]

we can rewrite the heat-distribution equation in the form

\[ \rho \frac{\partial H}{\partial t} = \operatorname{div}\lambda \operatorname{grad} T - \rho V\operatorname{grad} H + wQ'. \]

Using the relation \(c_p=\dfrac{\partial H}{\partial T}\) and \(c_p\operatorname{grad}T=\operatorname{grad}H\ ^1)\),

\(^1\) Strictly speaking, this is not so, since in the case of a chemical reaction, where the composition of the mixture changes in time and space, the heat content \(H\) changes not only with temperature, but also with the change of \(M\). If \(c_1\) and \(c_2\) are the heat capacities per unit mass at constant pressure of the initial substances and final products, then

\[ H-H_0=\int_{T_0}^{T}\left[c_1M+(1-M)c_2\right]\,dT \quad \text{or} \quad \frac{\partial H}{\partial x} = \left(\frac{\partial H}{\partial T}\right)_M \frac{\partial T}{\partial x} + \left(\frac{\partial H}{\partial M}\right)_T \frac{\partial M}{\partial x}; \]

\[ \left(\frac{\partial H}{\partial T}\right)_M = c_p = Mc_1+(1-M)c_2; \quad \left(\frac{\partial H}{\partial M}\right)_T = \int_{T_0}^{T}(c_1-c_2)\,dT. \]

we can write

\[ \operatorname{div}\lambda \operatorname{grad} T = \operatorname{div}\frac{\lambda}{c_p}\operatorname{grad} H. \]

and rewrite the basic equation in the form

\[ \rho \frac{\partial H}{\partial t} = \operatorname{div}\frac{\lambda}{c_p}\operatorname{grad} H - \rho \mathbf{V}\operatorname{grad} H + wQ', \]

or, in the one-dimensional case, when all quantities depend only on the coordinate \(x\), and the velocity vector \(\mathbf{V}\) makes an angle \(\varphi\) with the direction \(x\),

\[ \left(\text{i.e. } \mathbf{V}\operatorname{grad} H=V\cos\varphi\,\frac{\partial H}{\partial x}\right), \]

\[ \rho \frac{\partial H}{\partial t} = \frac{\partial}{\partial x}\frac{\lambda}{c_p}\frac{\partial H}{\partial x} - \rho V\cos\varphi\,\frac{\partial H}{\partial x} + wQ'. \tag{2'} \]

In application to the flame zone, where, as we have seen, the problem reduces to a one-dimensional and stationary one \(\left(\dfrac{\partial H}{\partial t}=0\right)\), and where \(\rho V\cos\varphi\) is nothing other than the weight quantity of combustible gas mixture supplied every second per unit area of the flame front, i.e. \(\rho V\cos\varphi=v_m\), we obtain the equation given above.

From what follows we shall see that under flame conditions

\[ -\frac{\partial T}{\partial x} = \frac{T_1-T_0}{M_0}\frac{\partial M}{\partial x}, \]

where \(T_1\) is the maximum temperature developed during combustion, whence

\[ \left(\frac{\partial H}{\partial M}\right)_T \frac{\partial M}{\partial x} : \left(\frac{\partial H}{\partial T}\right)_M \frac{\partial T}{\partial x} = - \frac{ M_0 \displaystyle\int_{T_0}^{T}(c_1-c_2)\,dT }{ (T_1-T_0)c_p } = \frac{M_0}{T_1-T_0} \frac{(\bar c_2-\bar c_1)(T-T_0)}{c_p}, \]

where \(\bar c_1\) and \(\bar c_2\) are the mean values of the heat capacities in the temperature interval from \(T_0\) to \(T\). The chemical transformation, as we shall see below, occurs only at high temperatures, where the difference \(c_2-c_1\) is always small. It is not difficult to verify from tabular data that, even at temperatures of several hundred degrees, the difference \(c_2-c_1\) amounts, for most combustion reactions of interest to us, to less than 10% of the value \(c_p\). The quantity \(\dfrac{T-T_0}{T_1-T_0}M_0\) is always less than unity. Therefore we may neglect the term \(\dfrac{\partial H}{\partial M}\dfrac{\partial M}{\partial x}\) in the expression for \(\dfrac{\partial H}{\partial x}\) and assume that

\[ \frac{\partial H}{\partial x} = \left(\frac{\partial H}{\partial T}\right)_M \frac{\partial T}{\partial x} = c_p\frac{\partial T}{\partial x} \]

or, in general form, \(c_p\operatorname{grad} T=\operatorname{grad} H\), which is what was required to be shown. For the same reason one may suppose that the heat \(Q'\) or \(Q\) released in the chemical transformation is constant, independent of the temperature at which the transformation occurs. Indeed,

\[ Q_T = Q_{T_0} + \mu\int_{T_0}^{T}(c_1-c_2)\,dt \simeq Q_{T_0} \]

in the case where \(Q_{T_0}\) is sufficiently large, which always holds in combustion reactions.

Before the works of Lewis and Elbe² and of Zel’dovich and Frank-Kamenetskii, all authors—partly because of an insufficient understanding of the process, partly for the sake of mathematical simplification of the problem—introduced into the theory a certain constant, namely the autoignition temperature of the given mixture. They assumed that, as the fresh gas approaches the combustion zone, it is heated by the heat released by this hot zone up to the autoignition temperature \(T_{\mathrm{в}}\), after which it ignites, i.e., in the mixture a vigorous reaction begins to proceed with a constant reaction rate \(w\), until all the substance has burned out and the temperature has reached its maximum value.

As we have already seen above (see Part I), the autoignition temperature is not a constant, but itself depends on the conditions of the experiment, first of all on the conditions of heat release and on the induction period of the igniting layer of gas. Moreover, the idea of a constant reaction rate when the temperature rises from \(T_{\mathrm{в}}\) to \(T_1\) likewise does not withstand criticism. Therefore the introduction of the concept of the autoignition temperature into the problem of the rate of flame propagation very much resembles the pre-van ’t Hoff views on the “reaction point” in the theory of autoignition (see Part I).

We shall try, while retaining the premises of the previous authors, to set forth here these old theories in their most logical and rigorous form, without adhering punctiliously to those often confused and not always correct derivations which they used. We dwell on these old theories with dubious premises not only because scientists and engineers have used them up to the present time, but also because their analysis will help us, in the next paragraph, in presenting the new, more rigorous, thermal theory of flame propagation, the foundations of which were laid in the paper of Zel’dovich and Frank-Kamenetskii³.

In solving the problem we shall use equation (2).

Proceeding from the naive ideas set out above, we must formulate the boundary conditions under which equation (2) must be integrated.

We shall plot along the ordinate axis the temperature of the gas in the combustion and preheating layer, and along the abscissa axis the distance \(x\) from that place in the layer where the ignition temperature \(T_{\mathrm{в}}\) is reached (Fig. 16). We direct the \(x\)-axis from the cold initial gas toward the reaction products. In the coordinate system adopted by us, moving with the flame front, the flame front is stationary, while the fresh initial gas moves in the direction \(x\) with velocity \(v_0\) toward the flame front.

Fig. 16

At \(x=-\infty\) the cold gas has the initial temperature \(T_0\). Then, in the preheating zone, moving in the direction of positive

the gas is heated through thermal conduction to the temperature \(T_v\) (region \(I\)). Then the gas begins to react, continuing to heat up (reaction region, region \(II\)).

After traversing a distance \(d\) from the origin of coordinates, all the initial gas has reacted, and its temperature reaches the maximum combustion temperature \(T_1\), which, in the absence of heat losses, remains constant thereafter (region \(III\)).

As we have already indicated, the rate \(w\) of the reaction at the self-ignition temperature and above was taken by previous authors (in implicit or explicit form) to be constant.

It is rather immaterial whether one takes as constant the number \(w\) of molecules transformed per unit volume per unit time independently of the gas density, which changes as heating proceeds, or whether one regards as constant the transformation time of each given molecule, i.e. takes as constant

\[ \frac{w}{\rho}=\text{const}, \]

in other words, assumes \(w\) to be proportional to \(\rho\) (which changes as heating proceeds).

The second assumption leads, in the final expression for the propagation velocity, to a value smaller than the first in the ratio

\[ 1:\sqrt{\frac{T_0}{T_1}}, \]

i.e. approximately

\[ 1:\sqrt{\frac{1}{6}}=\sqrt{6}\simeq 2.5. \]

The first assumption is equivalent to a reaction of zero order, the second—to one of first order with respect to density or pressure. As we shall see below, for a first-order reaction it is necessary to take diffusion in the layer into account, which was not done in the old theories. Only therefore is the assumption of a zero-order reaction in the old theories free from internal contradictions, and for this reason we have adopted it.

Let us consider the motion of a small volume element \(d\omega_0\) of the initial cold gas of temperature \(T_0\), containing \(a_0\,d\omega_0\) molecules of the combustible substance capable of chemical transformation. This volume element, moving from \(x=-\infty\) in the direction of positive \(x\)’s, gradually heats up. At the same time the volume of the gas element will increase as a result of thermal expansion, so that at some \(x\) and at the temperature \(T\) corresponding to this value the volume of the gas element will be

\[ d\omega=\frac{T}{T_0}\,d\omega_0. \]

For \(T>T_v\), a chemical reaction proceeds in the volume element \(d\omega\) with rate \(w\,d\omega\), where \(w\), by assumption, is a constant quantity.

In the interval \(dx\) between \(x\) and \(x+dx\) the volume element spends a certain time \(dt\), obviously equal to

\[ \frac{dx}{v}, \]

where \(v\) is the velocity of motion of the volume element at the given place. If the cold gas moves with velocity \(v_0\), then as the temperature rises the gas, owing to expansion, will move faster, namely with velocity

\[ v=v_0\frac{T}{T_0}. \]

Thus,

\[ dt=\frac{dx}{v_0}\frac{T_0}{T}. \]

The number of fuel molecules that have reacted during the residence time of the gas volume element in the layer \(dx\) will, obviously, be equal to

\[ w\,d\omega\,dt=w\,d\omega\,\frac{dx}{v_0}\frac{T_0}{T_1} =\frac{w}{v_0}\,dx\,d\omega, \]

i.e., in each element \(dx\) the same number of molecules will react. The total number of reacted molecules of the volume element over the distance between \(x=0\) and \(x=d\) will be \(\frac{wd}{v_0}\,d\omega_0\). But by definition all \(a_0\,d\omega_0\) fuel molecules in the volume element, over the distance \(d\), prove to have reacted. Hence

\[ \frac{dw}{v_0}=a_0 \quad \text{or} \quad d=\frac{a_0v_0}{w}. \tag{3} \]

Thus, we obtain the required value of the distance \(d\).

In integrating equation (2) we divide the domain of integration into two parts. In the first region, where the temperature rises from \(T=T_0\) to \(T=T_B\) and \(x\) varies from \(-\infty\) to \(0\), the reaction, by assumption, does not proceed and, consequently, \(w=0\). In the second region, where \(T\) rises from \(T=T_B\) to \(T=T_1\) and \(x\) from \(0\) to \(x=d\), the reaction rate \(w\), by assumption, is constant. In the third region, from \(x=d\) to \(+\infty\), \(w=0\), the temperature is constant and equal to \(T_1\), and thus the solution here is obvious. Thus, for region I

\[ \frac{d^2T}{dx^2}-a\frac{dT}{dx}=0, \tag{4} \]

with

\[ \text{at } x=-\infty \quad T=T_0 \quad \text{and at } x=0 \quad T=T_B . \tag{5} \]

In region II

\[ \frac{d^2T}{dx^2}-a\frac{dT}{dx}+\frac{wQ'}{\lambda}=0, \tag{4'} \]

with

\[ \text{at } x=0 \quad T=T_B \quad \text{and at } x=d \quad T=T_1 . \tag{5'} \]

As is seen, these conditions are quite sufficient for solving the problem. However, in addition to them there is one more condition—the condition of continuity of the heat flux \(q=-\lambda \frac{dT}{dx}\) at the junction of the two regions, i.e., the condition

\[ \left(\frac{dT}{dx}\right)_{x=0;\,I} = \left(\frac{dT}{dx}\right)_{x=0;\,II}. \]

This extra condition determines the value of the parameter \(v_0\) or \(v_m\), i.e., gives the answer to the basic question of the normal velocity of flame propagation. Solving this problem, we find the equation for determining \(v_m\) or \(v_0=\frac{v_m}{\rho}\) in the following form:

\[ \frac{T_B-T_0}{T_1-T_0} = \frac{1-e^{-\xi}}{\xi}, \tag{6} \]

where

\[ \xi=\alpha d=\frac{v_m c_p d}{\lambda}=\frac{a_0}{w}\frac{v_n^2\rho_0 c_p}{\lambda}=\frac{v_0^2\rho_0 c_p \tau}{\lambda} \tag{6'} \]

and the quantity \(\dfrac{a_0}{w}\) is denoted by \(\tau\).

Let us show how the solution is obtained. According to formulas (A) and (3),

\[ \frac{wQ'}{\lambda}=\frac{wL\mu}{M_0N\lambda} =\frac{v_m c_p(T_1-T_0)}{d\lambda} =\alpha\frac{T_1-T_0}{d}. \]

Substituting this quantity into equation (4) and integrating it under conditions (5), we find:

\[ (T-T_0)_I=(T_{\mathrm{в}}-T_0)e^{\alpha x} \tag{7} \]

in region \(I\) (\(x\) from \(-\infty\) to \(0\)),

\[ (T-T_0)_{II}=\frac{T_1-T_0}{d}x+(T_{\mathrm{в}}-T_0)\frac{e^{\alpha d}-e^{\alpha x}}{e^{\alpha d}-1} \tag{7'} \]

in region \(II\) (\(x\) from \(0\) to \(d\)).

Forming \(\left(\dfrac{dT}{dx}\right)_I\) and \(\left(\dfrac{dT}{dx}\right)_{II}\) at \(x=0\) and equating them, we obtain:

\[ \alpha(T_{\mathrm{в}}-T_0)=\frac{T_1-T_0}{d} -\frac{\alpha(T_{\mathrm{в}}-T_0)}{e^{\alpha d}-1}, \]

whence we obtain formula (6).

When \(T_{\mathrm{в}}\) is close to \(T_0\), i.e. the self-ignition temperature is low, and

\[ \frac{T_{\mathrm{в}}-T_0}{T_1-T_0}\ll 1. \]

This will take place when \(e^{-\xi}\) is small, and \(\xi\) is large. Then

\[ \frac{T_{\mathrm{в}}-T_0}{T_1-T_0} =\frac{1}{\xi} =\frac{1}{\alpha d} =\frac{\lambda}{v_0^2\rho_0 c_p\tau}, \]

i.e.

\[ v_0=\sqrt{\frac{\lambda}{c_p\rho_0\tau}\frac{T_1-T_0}{T_{\mathrm{в}}-T_0}}. \tag{8} \]

When \(T_{\mathrm{в}}\) is close to \(T_1\), i.e. the self-ignition temperature is high, then

\[ \frac{T_{\mathrm{в}}-T_0}{T_1-T_0} \]

is close to unity. This will take place when \(\xi\) is small and the quantity \(e^{-\xi}\) in formula (6) can be expanded in a series. Restricting ourselves to the first three terms of the expansion, we obtain

\[ \frac{T_{\mathrm{в}}-T_0}{T_1-T_0} =\frac{\xi-\dfrac{\xi^2}{2}}{\xi} =1-\frac{\xi}{2}\simeq 1-\frac{\alpha d}{2} \]

(in doing this we neglect the term \(\dfrac{\xi^2}{6}\simeq \dfrac{\alpha^2d^2}{6}\)). Thus,

\[ \frac{\alpha d}{2}=\frac{T_1-T_{\mathrm{в}}}{T_1-T_0}, \]

\[ v_0=\sqrt{\frac{2\lambda}{c_p\rho_0\tau}\frac{T_1-T_{\mathrm{в}}}{T_1-T_0}} =\sqrt{\frac{2\lambda}{\rho_0L}\frac{w}{a_0}(T_1-T_{\mathrm{в}})}. \tag{9} \]

The formula

\[ v_0=\sqrt{\frac{\lambda}{c_p\rho_0\tau}\,\frac{T_1-T_v}{T_v-T_0}}, \tag{10} \]

at which most previous authors arrived, as is not difficult to show, is an approximate expression of the general formula (6) and, generally speaking, is inaccurate.

In what follows we shall be especially interested in the case when \(T_v\) is close to \(T_1\). Here formula (9) is applicable. Let us estimate the range of applicability of this formula. Put \(\xi=\alpha d=0.5\). Then, discarding the term \(\frac{\xi^2}{6}=\frac{0.25}{6}\simeq 0.04\), we make, in comparison with \(\frac{\xi}{2}=0.25\), an error of \(16\%\) in determining the quantity \(\frac{T_1-T_v}{T_v-T_0}\), or \(8\%\) in determining the velocity \(v_0\).

Thus, the formula is applicable with the accuracy we need up to

\[ \frac{T_1-T_v}{T_v-T_0}\leq 0.25. \]

If \(T_1-T_0=2000^\circ\), then the formula is applicable up to \(T_v-T_0=1500^\circ\), or \(T_1-T_v=500^\circ\).

In what follows, another approximate method of solution for the case when \(T_v\) is close to \(T_1\) is of essential importance for us. This method leads to the same result as formula (9) and, consequently, gives the same degree of accuracy. However, it differs in the feature that it can be applied just as accurately also to cases when the reaction rate \(w\) is not constant but varies, for example, with temperature. This method was proposed by Zel’dovich and Frank-Kamenetskii and consists in the fact that, for \(T_v\) close to \(T_1\), in equation (4′) for region \(II\) one may neglect the second term and write the system in the following form:

\[ \left. \begin{array}{ll} \dfrac{d^2T}{dx^2}+\alpha\dfrac{dT}{dx}=0 & (\text{region } I),\\[6pt] \dfrac{d^2T}{dx^2}+\dfrac{wQ'}{\lambda}=0 & (\text{region } II). \end{array} \right\} \tag{11} \]

\[ \tag{11′} \]

The validity of this neglect will be clear from consideration of the complete equation (4′), if in it the quantity \(\frac{wQ'}{\lambda}\) is replaced by an equal quantity \(a\,\frac{T_1-T_0}{d}\). Integrating the equation with respect to \(x\) from \(0\) to \(d\), we obtain:

\[ -\left(\frac{dT}{dx}\right)_d+\left(\frac{dT}{dx}\right)_0 =\left(\frac{dT}{dx}\right)_0 =-\int_0^d a\,\frac{dT}{dx}\,dx+\frac{a(T_1-T_0)}{d}\int_0^d dx \]

\[ =-a(T_1-T_v)+a(T_1-T_0). \]

Since \(T_1-T_v\ll T_1-T_0\), the first term on the right-hand side of the equality may be discarded, and this means that we may neglect

term \(a\dfrac{dT}{dx}\) in equation \((4')\), i.e., to consider the term \(a\dfrac{dT}{dx}\) small in comparison with \(\dfrac{wQ'}{\lambda}\) and \(\dfrac{d^2T}{dx^2}\) ¹).

The physical meaning of the neglect is as follows. In region \(I\) the convective heat flux, expressed by the term \(a(T_v-T_0)\), is directed toward positive \(x\)’s, while the flux due to thermal conduction, expressed by the term \(\dfrac{dT}{dx}\), is directed in the opposite direction; moreover, the two fluxes compensate each other and thereby establish a stationary state. At the boundary of the regions the heat flux by thermal conduction, expressed by the term \(\left(\dfrac{dT}{dx}\right)_0\), is made equal to \(a(T_v-T_0)\simeq a(T_1-T_0)\), i.e., it carries away, for the heating of the incoming gas, practically all the heat from the reaction zone. Since the heating in the reaction zone (from temperature \(T_v\) to \(T_1\)) is very small, in region \(II\) (although it is precisely in it that the entire reaction temperature is released) practically the heat leaves by thermal conduction and does not go into additional heating of the gas. Therefore in this region equation \((11')\) is approximately satisfied.

Integrating equation (11), we obtain

\[ \left(\frac{dT}{dx}\right)_0-\left(\frac{dT}{dx}\right)_{-\infty} =\left(\frac{dT}{dx}\right)_0 =a(T_v-T_0)\simeq a(T_1-T_0)=\frac{v_0\rho_0 L}{\lambda}, \]

since, by condition, \(T_v\) is close to \(T_1\).

Integrating equation \((11')\), we obtain

\[ -\left(\frac{dT}{dx}\right)_d+\left(\frac{dT}{dx}\right)_0 =\left(\frac{dT}{dx}\right)_0 =\sqrt{\frac{2Q'}{\lambda}\int_{T_v}^{T_1} w\,dT}. \]

Comparing the two expressions obtained for the quantities \(\left(\dfrac{dT}{dx}\right)_0\), we have an equation for determining \(a\) or \(v_0\):

\[ \frac{v_0\rho_0 L}{\lambda} = \sqrt{\frac{2Q'}{\lambda}\int_{T_v}^{T_1} w\,dT} \quad\text{or}\quad v_0=\frac{1}{\rho_0 L} \sqrt{2\lambda Q'\int_{T_v}^{T_1} w\,dT}. \tag{12} \]

Representing the same in a somewhat different form,

\[ v_0=\sqrt{ \frac{2\lambda Q'a_0}{c_p^2\rho_0^2}\, \frac{I}{(T_1-T_0)^2} }, \]

¹) Here it may seem strange why we can regard the quantity \(a\dfrac{dT}{dx}\) as small in comparison with \(\dfrac{d^2T}{dx^2}\) in region \(II\), while in region \(I\) these quantities are clearly of the same order. This is explained by the fact that at the boundary of the regions, at the point \(x=0\), the second derivative undergoes a jump, and when \(T_1-T_v\ll T_1-T_0\), in region \(II\) it is much larger than the second derivative in region \(I\), while the first derivatives in both regions are of the same order.

where

\[ I=\frac{1}{a_0}\int_{T_{\mathrm{в}}}^{T_1} w\,dT. \tag{13} \]

Since

\[ a_0Q'=\frac{a_0Q}{N}=\rho_0\frac{Qa_0}{N\rho_0} =\rho_0\frac{\mu L a_0}{M_0N\rho_0} =\rho_0L=\rho_0c_p(T_1-T_0), \]

then

\[ v_0=\sqrt{\frac{2\lambda I}{c_p\rho_0(T_1-T_0)}}= \sqrt{\frac{2\lambda I}{\rho_0L}}. \tag{14} \]

In this form the formula is also applicable for \(w\) varying with \(x\).

If \(w=\mathrm{const}\), then \(I=\dfrac{w}{a_0}(T_1-T_{\mathrm{в}})=\dfrac{T_1-T_{\mathrm{в}}}{\tau}\), and formula (14) reduces to formula (9) obtained earlier by another method, thereby showing that the adopted approximate integration method has the same accuracy as the expansion of the exact solution (6) into a series up to and including the third term.

§ 3. A New Theory of Flame Propagation

In the old theories, as we have seen, two assumptions were made: 1) the mixture begins to react upon reaching the temperature \(T_{\mathrm{в}}\), and 2) it reacts with the constant rate \(w\).

We know that the rate of all reactions, as a function of temperature, is expressed by the formula

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

where \(E\) is the activation energy, i.e., a quantity characteristic of the given fuel. For substances capable of combustion, this quantity is usually large and amounts to from 25 to 80,000 cal. Therefore we should expect a very rapid increase in the reaction rate \(w\) with temperature, and hence also with the coordinate \(x\) in the flame. Consequently, the assumption that the rate is independent of temperature in the temperature interval from \(T_{\mathrm{в}}\) to \(T_1\) is entirely unjustified. Thus, if \(E=60\,000\) cal, \(T_1=2000^\circ\), and \(T_{\mathrm{в}}=1500^\circ\), the reaction rate increases by a factor of 120 in passing from \(T_{\mathrm{в}}\) to \(T_1\). If \(E=30\,000\) cal, then under the same conditions the rate increases almost by a factor of 10. In the case where \(T_1=2000\), and \(T_{\mathrm{в}}=1700^\circ\), the rate increases almost by a factor of 15 for \(E=60\,000\), and by a factor of 3 for \(E=25\,000\) cal.

With such a sharp decrease of the rate as the temperature decreases, there is no need whatever to introduce a self-ignition temperature. Indeed, this quantity was needed only in order to establish a boundary below which the reaction does not proceed and above which it does. But in real cases, when the reaction rate falls very sharply with decreasing temperature according to the law

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

this law itself determines the region where the reaction practically does not proceed. Therefore we may introduce the notation \(T_{\mathrm{в}}=T'\).

but no longer as a physical quantity, rather as a mathematical device for approximate calculation, as a temperature below which the reaction practically has no time to proceed at all. In doing so we shall have to prove that the final result, i.e., the computed rate of propagation, will practically not depend on the choice of one or another value of this fictitious quantity. Zel’dovich’s merit consists above all in this new formulation of the problem1.

Under the new assumptions the basic equations of heat propagation (1) and (2) are retained in their former form; only the rate \(w\) will depend on the temperature according to the law \(w=Se^{-\frac{E}{RT}}\), and in the boundary conditions, instead of the quantity \(T_b\), the above-mentioned temperature \(T'\) will enter. In the case where the activation energy \(E\) is sufficiently large, namely when \(\frac{E}{RT_1}\gg 1\), the temperature \(T'\), below which the reaction rate may be neglected, will lie close to \(T_1\) (which corresponds to the case when \(\frac{T_1-T_b}{T_1-T_0}\ll 1\)).

Denoting by \(\vartheta\) the difference between \(T_1\) and some temperature \(T\) inside the reaction zone (between \(T_1\) and \(T'\)), we may (since \(\vartheta\ll T_1\)) set the quantity \(e^{-\frac{E}{RT}}\) equal to \(e^{-\frac{E}{RT_1}}e^{-\frac{E\vartheta}{RT_1^2}}\).

Thus,

\[ w_1=Se^{-\frac{E}{RT_1}}e^{-\frac{E\vartheta}{RT_1^2}}, \tag{15} \]

where \(w_1\) is a constant quantity, equal to the reaction rate at the maximum combustion temperature \(T_1\). We see that the reaction rate decreases by a factor of \(e\) when \(\vartheta=\frac{RT_1^2}{E}\), by a factor of \(7.5\) when \(\vartheta=\frac{RT_1^2}{E}\cdot 2\), and so on.

For \(E=60\,000\) cal and \(T_1=2000^\circ\), \(\frac{RT_1^2}{E}\cong 125^\circ\). Consequently, at \(\vartheta=250^\circ\), or \(T'=1750^\circ\), the reaction rate \(w\) falls almost 8-fold relative to its maximum value \(w_1\) at \(T=T_1=2000^\circ\). Thus,

if we confine ourselves to such a degree of accuracy and neglect the reaction rate at \(T<1750^\circ\), then we make an error not exceeding \(12\%\) in determining \(v_0^2\), and an even smaller one in determining \(v_0\). For such a value of \(\vartheta\) \((=250^\circ)\) we may: 1) regard \(\dfrac{\vartheta_1}{T_1}\ll 1\) and use the expansion (15), and 2) owing to the smallness of \(T_1-T'\) in comparison with \(T_1-T_0\), in solving the problem use the approximate system of equations (11), putting in them

\[ w=w_1 e^{-\frac{E\vartheta}{RT_1^2}}. \]

In this case we obtain the solution in the form of formula (14), where in the present case

\[ I=\int_{T'}^{T}\frac{w_1}{a_0} e^{-\frac{E\vartheta}{RT_1^2}}\,dT =-\frac{w_1}{a_0}\int_{\vartheta_1}^{0} e^{-\frac{E\vartheta}{RT_1^2}}\,d\vartheta =\frac{w_1}{a_0}\frac{RT_1^2}{E}\int_{0}^{\beta_1} e^{-\beta}\,d\beta, \]

where

\[ \beta_1=\frac{E\vartheta_1}{RT_1^2}. \]

If we choose \(\vartheta_1\) equal to \(\dfrac{2RT_1^2}{E}\), i.e. we neglect the existence of the reaction rate when it is 8 times smaller than the maximum \((w_1)\), then \(\beta=2\); if we put \(\vartheta=\dfrac{3RT_1^2}{E}\), i.e. neglect the rate \(w\) when it is 2 times smaller than \(w_1\), then \(\beta=3\); if \(\vartheta_1=\dfrac{4RT_1^2}{E}\), i.e. \(\dfrac{w'}{w_1}=50\), then \(\beta_1=4\).

\[ j=\int_{0}^{\beta_1} e^{-\beta}\,d\beta=1-e^{-\beta_1}. \]

If \(\beta_1=2\), \(j=0.87\); if \(\beta_1=3\), \(j=0.95\); if \(\beta_1=4\), \(j=0.98\).

Thus we see that the choice of \(\vartheta_1\), provided only that \(\vartheta_1\le \dfrac{2RT_1^2}{E}\), has practically no effect on the value of \(j\), and hence also on the magnitude of the velocity. We may therefore, with a degree of accuracy sufficient for us, put \(j=1\) in all cases. In other words, everywhere in formulas (14) and (12) we may put

\[ \int_{T'}^{T_1} w\,dT \]

equal to

\[ \int_{0}^{T_1} w\,dT. \]

This very essential conclusion allows us, in the derivation, to use the auxiliary quantity \(T'\), without assigning this quantity any fundamental significance. It drops out of the final result. Thus,

\[ I=\frac{w_1}{a_0}\frac{RT_1^2}{E}\,j =\frac{S e^{-\frac{E}{RT_1}}}{a_0}\frac{RT_1^2}{E}, \]

whence, according to formula (14),

\[ v_0=\sqrt{\frac{2\lambda}{c_p\rho_0}\frac{S e^{-\frac{E}{RT_1}}}{a_0(T_1-T_0)}-\frac{RT_1^2}{E}} =\sqrt{\frac{2\lambda}{\rho_0}\frac{S e^{-\frac{E}{RT_1}}}{a_0 L}-\frac{RT_1^2}{E}}. \tag{16} \]

This formula coincides with formula (9), if in it by the velocity \(w\) one understands the velocity \(w_1=S e^{-\frac{E}{RT_1}}\) at the maximum combustion temperature, and by the difference \(T_1-T_v\) one understands the quantity \(\frac{RT_1^2}{E}\). Thus, Zel’dovich succeeded in eliminating from the formula for the propagation velocity the doubtful constant \(T_v\) and the obscure quantity \(\tau\)—the reaction time.

However, even in this form the theory cannot be regarded as satisfactory, since in it we take the quantity \(S\) to be constant (a reaction of zero order). In reality, for all homogeneous reactions \(S\) depends in one way or another on the concentration of the reacting substances. In the case where the combustion reaction is a reaction of one substance (decomposition of azomethane, decomposition of \(Cl_2O\), etc.), the quantity \(S\) is some function of the number of fuel molecules in a unit volume of the mixture at the given temperature. In the case of a monomolecular reaction \(S=ka\), and in the case of a bimolecular reaction \(S=ka^2\).

In the case where the initial mixture consists of two reacting kinds of molecules, \(S\) may be a function of the concentration of both kinds of molecules. For a bimolecular reaction in this case \(S=kab\). The numbers of molecules \(a\) and \(b\) are connected with each other by the laws of stoichiometry and diffusion1.

The number of fuel molecules \(a\) in the reaction zone decreases in the course of the reaction because of the burning out of the product. In addition, the reaction products diffuse from the reaction zone into the unburned gas and thereby also lower the value of \(a\).

The dependence of \(a\) on the coordinate \(x\) in the flame zone can be found from the joint consideration of the equations of diffusion and heat conduction (as was done by Zel’dovich and Frank-Kamenetskii), leading to a similarity of the temperature and concentration fields. The same similarity had been postulated much earlier by Lewis. In its simplest and clearest form this was done by Zel’dovich and Frank-Kamenetskii, which authors I shall follow in the exposition of this question. Here, however, there arises the very difficult question of writing the diffusion equation in a space where a temperature gradient is present. Zel’dovich writes the expression for the diffusion flux in the usual form—\(D\frac{da}{dx}\), where \(D\) is the coefficient of diffusion of the molecules \(a\) into the reaction products. Meanwhile, if special thermodiffusion phenomena are not taken into account, then it is self-evident that when the partial pressure of molecules \(a\) is equal no diffusion—

of the flux cannot exist, whereas in the presence of a temperature gradient the number of molecules per unit volume will be different at different places1; in this case one should write the equation for the diffusion flux, expressing it in terms of the partial pressure.

When, in the reaction, the number of molecules does not change (we shall at first confine ourselves to this case), instead of partial pressures we may use the ratio of the number of molecules to the gas density, and write the expression for the diffusion flux in the form \(-D\rho\, \dfrac{d\frac{a}{\rho}}{dx}\). In this case the change in the number of fuel molecules as the gas moves through the flame front (along the coordinate \(x\)) will be described by the equation

\[ \frac{d}{dx}(D\rho)\,\frac{d\frac{a}{\rho}}{dx^2} - v\rho\,\frac{d\frac{a}{\rho}}{dx} - w = 0 \tag{17} \]

[the derivation is completely analogous to the derivation of equation (1)].

Since \(D\rho\) does not depend on the gas density, and therefore, in the first approximation, not on the temperature either \(\left(\text{with the same accuracy as } \dfrac{\lambda}{c_p}\right)\), and since \(v\rho = v_m\), we may write the equation for the diffusion of fuel molecules in the flame front in the following form:

\[ D\rho\,\frac{d^2\frac{a}{\rho}}{dx^2} - v_m\,\frac{d\frac{a}{\rho}}{dx} - w = 0. \tag{18} \]

The first term expresses the mutual diffusion of molecules of the fuel and of the reaction products under the action of the difference in concentration before and after the combustion front. The second term expresses the influx of fuel molecules under the action of the gas flow, and the third—their destruction as a result of the reaction.

For \(x=-\infty\)

\[ \left(\frac{a}{\rho}\right)_{-\infty}=\frac{a_0}{\rho_0}. \]

Let us introduce a new variable

\[ a=\frac{a_0}{\rho_0}-\frac{a}{\rho}, \]

and in equation (2) a new variable

\[ \theta=\frac{c_p}{Q'}(T-T_0). \]

Then both equations are rewritten in the following form:

\[ \frac{\lambda}{c_p}\frac{d^2\theta}{dx^2}-v_m\frac{d\theta}{dx}+w=0, \tag{19} \]

\[ D\rho\frac{d^2a}{dx^2}-v_m\frac{da}{dx}+w=0. \tag{19'} \]

From the kinetic theory of gases we know that, to a first approximation,

\[ D\rho=\frac{\lambda}{c_p}. \]

In this case both equations become identical.

As we already know, the boundary conditions for equation (19) will be

\[ (\theta)_{-\infty}=0;\qquad (\theta)_{+\infty}=\frac{c_p}{Q'}(T_1-T_0)=\frac{L}{Q'}=\frac{M_0N}{\mu}. \]

For equation (18) we have the boundary conditions:

\[ \left(\frac{a}{\rho}\right)_{-\infty}=\frac{a_0}{\rho_0}, \quad \text{i.e.}\quad (a)_{-\infty}=0 \quad \text{and}\quad (a)_{+\infty}=\frac{a_0}{\rho_0}. \]

But according to formulas (A)

\[ \frac{a_0\mu}{NM_0\rho_0}=1 \quad \text{or}\quad \frac{a_0}{\rho_0}=\frac{M_0N}{\mu}, \]

i.e.

\[ (\theta)_{-\infty}=(a)_{-\infty}=0;\qquad (\theta)_{+\infty}=(a)_{+\infty}=\frac{a_0}{\rho_0}=\frac{M_0N}{\mu}. \]

If the equation and the boundary conditions for \(\theta\) and \(a\) coincide, then \(a=\theta\) throughout the entire interval. Thus,

\[ \frac{a_0}{\rho_0}-\frac{a}{\rho} = \frac{c_pT}{Q'}-\frac{c_pT_0}{Q'}, \]

or

\[ c_pT+\frac{aQ'}{\rho} = c_pT_0+\frac{a_0Q'}{\rho_0} = c_pT_1. \tag{20} \]

\(H=c_pT\) is the store of thermal energy of unit mass of the mixture.

Since \(a\) is the number of fuel molecules per unit volume, and \(Q'\) is the energy released in the chemical reaction of one fuel molecule, \(aQ'\) is the store of potential chemical energy of unit volume of the mixture, while \(\dfrac{aQ'}{\rho}\) is the store of chemical energy of unit

mass of the mixture. Thus, equality (20) can be formulated as follows: the sum of the thermal and chemical energy of a unit mass of the mixture remains constant during the combustion process. The fact that this sum is the same before and after the combustion process seems obvious, since it is a simple consequence of the law of conservation of energy. However, since we are dealing here with phenomena of thermal conductivity and diffusion, equality (20), which refers to different stages of the combustion process, is by no means a consequence of the law of conservation of energy. Thus, if the relation \(D\rho=\dfrac{c_p}{\lambda}\) between the coefficients of thermal conductivity and diffusion were not satisfied (which in fact is the case), it would be easy to show that equality (20) is also false. In that case, in some parts of the layer the sum of the energies would be greater than \(c_p T_0+\dfrac{a_0 Q'}{\rho_0}\), and in others smaller. Therefore the proof of equality (20), given by Zel’dovich and Frank-Kamenetskii, is an essential supplement to the work of Lewis, who first postulated it. We shall see below how this law can be generalized to the case when \(D\rho\) is not equal to \(\dfrac{\lambda}{c_p}\).

Lewis did not draw clear and simple conclusions from relation (20) for the general case of flame propagation. With the aid of this equality he calculated flame propagation for a particular and very special case of combustion in the decomposition of ozone. Unfortunately, in deriving it he apparently made substantial errors here. Zel’dovich was able, from this equality, together with all the preceding results, to draw conclusions general for the theory of combustion, which we shall present here, developing and refining them further.

Thus, according to equation (20):

\[ a=\rho\left\{\frac{a_0}{\rho_0}-\frac{c_p(T-T_0)}{Q'}\right\}, \]

and since

\[ Q'=\frac{Q}{N}=\frac{L\mu}{M_0N}=\frac{c_p(T_1-T_0)\mu}{\vartheta N} \quad\text{and}\quad \frac{a_0}{\rho_0}=\frac{NM_0}{\mu}, \]

then

\[ a=\frac{NM_0\rho}{\mu}\frac{T_1-T}{T_1-T_0} =a_0\frac{\rho}{\rho_0}\frac{T_1-T}{T_1-T_0} =a_0\frac{T_0}{T}\frac{\vartheta}{T_1-T_0}. \tag{21} \]

Let us return to formula (20), which gives the connection between the concentration of combustible \(a\) and the temperature in the preheating zone and the reaction zone, rewriting it in the following form:

\[ c_p(T_1-T)=\frac{aQ'}{\rho}, \]

or

\[ \frac{am}{\rho}=M=\frac{c_p(T_1-T)}{\dfrac{Q'}{m}} =\frac{c_p(T_1-T)}{\dfrac{Q}{\mu}} =\frac{c_p(T_1-T)M_0}{L} =\frac{T_1-T}{T_1-T_0}M_0 =\left(1-\frac{T-T_0}{T_1-T_0}\right)M_0=(1-T)M_0, \]

or

\[ B=\frac{M}{M_0}=(1-T), \]

i.e., the relative weight concentration of the fuel \(B\) at any point \(x\) of the combustion zone is equal to the difference between unity and the relative temperature rise \(T\) at the same point. Since at \(x=-\infty\), \(B=1\) and \(T=0\), while at \(x=+\infty\), \(B=0\) and \(T=1\), the curves \(B-x\) and \(T-x\) are symmetrical, as is schematically shown in Fig. 17.

Fig. 17

Having thus obtained the relation between the number of fuel molecules and the temperature, we can find the flame-propagation velocity, no longer making the incorrect assumption of the independence of the reaction rate \(w\) from \(a\), which was made in deriving formula (9). This was first done by Zel’dovich and Frank-Kamenetskii1.

In the case of a monomolecular reaction

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

Substituting this expression into the integral \(I\) of formula (13), using (21), we obtain

\[ I=\int_{T'}^{T}\frac{kae^{-\frac{E}{RT}}}{a_0}\,dT = ke^{-\frac{E}{RT_1}}\int_{T'}^{T_1}\frac{T_0}{T}\, \frac{\vartheta}{T_1-T_0}\, e^{-\frac{E\vartheta}{RT_1^2}}\,dT. \]

The quantity \(\dfrac{T_0}{T}\) may be taken outside the integral sign and set equal to

\[ \frac{T_0}{T_1}, \]

since \(T'\) is close to \(T_1\). In that case

\[ I=\frac{ke^{-\frac{E}{RT_1}}}{T_1-T_0}\, \frac{T_0}{T_1} \left(\frac{RT_1^2}{E}\right)^2 j, \]

where

\[ j=\int_0^{\beta_1} e^{-\beta}\beta\,d\beta = 1-e^{-\beta_1}(\beta_1+1). \]

Choosing

\[ \vartheta_1=\frac{2RT_1^2}{E},\quad \frac{3RT_1^2}{E},\quad \frac{4RT^2}{E}, \]

we obtain for \(\beta_1\) the values \(\beta_1=2, 3, 4\), and for \(j\) the values \(0.6;\ 0.8;\ 0.9\). Thus, we shall not make a large error in the expression for the velocity by putting \(j=1\). In this case, substituting the value of \(I\)

into formula (14), we obtain\(^1\)

\[ v_0 = \sqrt{ \frac{2\lambda k}{\rho_0 c_p}\, \frac{T_0}{F_1 (T_1-T_0)^2}\, e^{-\frac{E}{RT_1}} \left(\frac{RT_1^2}{E}\right)^2 } = \sqrt{ \frac{2\lambda c_p k}{\rho_0 L^2}\, \frac{T_0}{T_1}\, \left(\frac{RT_1^2}{E}\right)^2 e^{-\frac{E}{RT_1}} } \ \text{cm/sec}. \tag{22} \]

The corresponding calculation for a bimolecular reaction \(\left(w=ka^2 e^{-\frac{E}{RT}}\right)\) gives:

\[ v_0 = \sqrt{ \frac{4\lambda (ka_0)}{\rho_0 c_p} \left(\frac{T_0}{T_1}\right)^2 \frac{e^{-\frac{E}{RT_1}}}{(T_1-T_0)^3} \left(\frac{RT_1^2}{E}\right)^3 } = \sqrt{ \frac{4\lambda (ka_0)c_p^2}{\rho_0 L^3} \left(\frac{T_0}{T_1}\right)^2 \left(\frac{RT_1^2}{E}\right)^3 e^{-\frac{E}{RT_1}} }. \tag{23} \]

Since the quantities dependent on the pressure at which the experiment is conducted are only \(a_0\) and \(\rho_0\) (both directly proportional to \(p\)), for monomolecular reactions the linear velocity \(v_0\) is inversely proportional to \(\sqrt{p}\), while for bimolecular reactions it does not depend on pressure.

In deriving these formulas we assumed that the principal part of the reaction takes place in the zone located near the maximum temperature \(T_1\)\(^2\), restricting ourselves to the temperature interval \(\vartheta = T_1 - T'\). As we have seen, the method of solution is approximately valid only so long as \(\vartheta_1 \le 0.25T_1\). Since the number of molecules \(a\) decreases as the temperature rises according to formula (21), in the zone \(\vartheta_1\) (from \(T_1-T'\) to \(T_1\)) the mixture will consist chiefly of reaction products and contain the initial regulating substances only in a small percentage. In this connection it is necessary to check to what extent, under these conditions, the assumption will be satisfied that the fastest reaction occurs within the zone, and how far one may neglect the reaction at temperatures lower than \(T'=T_1-\vartheta_1\), while retaining the condition \(\vartheta_1 \le 0.25T_1\).

\(^1\) These formulas differ from Zel’dovich’s formulas only in that under the radical there appears the quantity \(\dfrac{T_0}{T_1}\) for a mono- and \(\left(\dfrac{T_0}{T_1}\right)^2\) for a bimolecular reaction. The absence of these factors in Zel’dovich is connected with his writing the diffusion equation in a different form.

\(^2\) It may seem strange how the combustion of the entire initial gas supplied with mass velocity \(v_m\) is ensured when combustion occurs in a mixture diluted 90% with combustion products and containing only 10–15% of the initial number of fuel molecules. After all, such a depleted gas still moves with mass velocity \(v_m\). One asks, where then does the remaining 90% of the mixture burn? There is no contradiction here, and the point is explained by the fact that the remaining 90% of the fuel is delivered to the boundary zone not by the motion of the gas, but by the flow of fuel molecules diffusing to the reaction site. Thus, in the combustion zone all the fuel burns, but under conditions of strong dilution by reaction products.

The situation will be worst of all for a bimolecular reaction. Let us check precisely this case. Here

\[ w = ka^2 e^{-\frac{E}{RT}} \simeq A \frac{\vartheta^2}{T^2} e^{-\frac{E}{RT}} = A \left( \frac{\vartheta}{T_1} \right)^2 e^{-\frac{E}{RT_1}\frac{\vartheta}{T_1}/\left(1-\frac{\vartheta}{T_1}\right)} = \]

\[ = F \left( \frac{\vartheta}{T_1} \right)^2 e^{-\frac{E}{RT_1}\frac{\vartheta}{T_1}/\left(1-\frac{\vartheta}{T_1}\right)}, \]

where \(F\) is some constant depending on \(T_1\), and \(\vartheta = T_1 - T\). If the reaction rate is plotted graphically as a function of \(\dfrac{\vartheta}{T_1}\), then the form of the curve will depend on the magnitude \(\gamma = \dfrac{RT_1}{E}\). Carrying out the corresponding calculation for \(\gamma = 0.05;\ 0.1\) and \(0.2\), we shall see that, if for \(\gamma = 0.05\) the rate \(w\) may be neglected when \(\dfrac{\vartheta}{T_1} \ge 0.25\), and hence it may be assumed that practically the entire reaction occurs at \(\dfrac{\vartheta}{T_1} < 0.25\), then for \(\gamma = 0.2\) this is already incorrect. Consequently, for a bimolecular reaction the applicability of formula (23) is limited to values \(\dfrac{RT_1}{E} = 0.1\), or to activation-energy values \(E\) greater than 40,000 at a maximum temperature \(T_1 \simeq 2000^\circ\). For monomolecular reactions the situation will be better, and formula (22) may be used in all practically interesting cases.

More precisely, though less transparently, this fundamental approximation for the theory may be analyzed as follows. As we have seen, the flame-propagation velocity is determined by the integral

\[ \int_{T'}^{T_1} w\,dT = I . \]

We have already indicated that this integral must depend little on the choice of \(T'\); in other words, it must differ little from its limiting value

\[ \int_{0}^{T_1} w\,dT . \]

For this to be so, it is necessary that \(w(T')\) be sufficiently small. In this case we shall obtain the correct result only if the value \(\vartheta_1 = T_1 - T'\) determined in this way does not exceed \(0.25T_1\).

\[ I = \int_{T'}^{T} w\,dT = Ae^{-\frac{E}{RT_1}} \left(\frac{RT_1^2}{E}\right)^2 \int_{0}^{\beta_1} e^{-\frac{\beta}{1-\gamma\beta}}\beta^2\,d\beta = Fj, \tag{24} \]

where

\[ \beta = \frac{E\vartheta}{RT_1^2} \quad \text{and} \quad \beta_1 = \frac{E\vartheta_1}{RT_1^2} = \frac{1}{\gamma}\frac{\vartheta_1}{T_1}. \]

In Fig. 18, \(j\) is plotted graphically as a function of \(\beta_1\) for \(\gamma = 0.05;\ 0.1\) and \(0.2\). The arrows mark those values of \(\beta_1\) at which \(j\) differs by no more than \(10\text{–}15\%\) from its limiting value. We see that these values of \(\beta_1\) will be \(4;\ 2.5;\ 2\), and the corresponding

the values of \(\dfrac{\vartheta_1}{T_1}=\gamma\beta_1\) will be \(0.2;\ 0.25\) and \(0.4\). Thus here, too, we arrive at the same conclusion as before, i.e., that the solution is valid only up to the value \(\gamma \le 0.1\), since for larger values of \(\gamma\) the reaction zone extends into the region of temperatures \(\vartheta_1 > 0.25T_1\), and the assumption that the main part of the bimolecular reaction proceeds in the zone of temperatures close to \(T_1\) becomes incorrect. Let us note in passing that the limiting value of the integral \(j=2\), obtained above [see formula (23)], is incorrect and is connected with neglecting the quantity \(\gamma\beta\) in comparison with unity in formula (24). In reality the value of this integral depends on the parameter \(\gamma\); moreover, for the values most often encountered in experiment the limiting value of the integral is equal to unity; in other words, in formula (18) under the radical there should stand \(2\), not \(4\).

Fig. 18

In the reaction zone \(x>0\), as we have seen, \(T_1-T\) is small and, in order of magnitude, is equal to

\[ \frac{RT_1^2}{E}. \]

We shall denote the concentration of the \(a\) molecules of the combustible at \(x>0\) by \(a_{\mathrm{eff}}\). Then, according to formula (21), \(a_{\mathrm{eff}}\) will, in order of magnitude, be equal to

\[ \begin{aligned} a_{\mathrm{eff}} &=a_0\frac{T_0}{T_1}\frac{\dfrac{RT_1^2}{E}}{T_1-T_0} =\frac{a_0 c_p}{L}\left(\frac{RT_1^2}{E}\right)\frac{T_0}{T_1} =\frac{a_0 c_p\mu}{M_0Q'N}\frac{RT_1^2}{E}\left(\frac{T_0}{T_1}\right) \\ &=\frac{\rho_0 c_p}{Q'}\frac{RT_1^2}{E}\left(\frac{T_0}{T_1}\right) =\frac{\rho_0 c_p T_0}{Q'}\frac{RT_1}{E} =\frac{\rho c_p}{Q'}\frac{RT_1^2}{E}. \end{aligned} \tag{25} \]

If we assume that the concentration of the combustible in the combustion zone is constant and equal to \(a_{\mathrm{eff}}\), then we have the right to use, for solving the problem of the flame-propagation velocity, formula (16), derived for a zero-order reaction, substituting into it \(S=ka_{\mathrm{eff}}\) for a monomolecular reaction and \(ka_{\mathrm{eff}}^2\) for a bimolecular reaction. In doing so, as is not difficult to verify, we obtain the same result as in the case of the rigorous treatment, expressed by formulas (22) and (23). Thus, it is sufficient to remember formula (16) and the expressions for the effective concentration \(a_{\mathrm{eff}}\) in order to obtain an expression for the flame-propagation velocities in various cases of kinetics. In particular, this method remains valid also for the case when the reaction takes place between two components \(a\) and \(b\), for example according to the law

\[ w=kabe^{-\frac{E}{RT}}. \]

If component \(a\) is deficient and component \(b\) is in excess, then the concentration \(b'\) of the component in the reaction zone may be taken equal to its concentration in the combustion products (computed from the stoichiometri-

to the kinetic equation from the initial composition of the mixture), while the concentration \(a_{\mathrm{eff}}\) can be calculated by formula (25). After this, substituting the velocity thus calculated,

\[ w = k a_{\mathrm{eff}}^{\,\nu} b' e^{-\frac{E}{RT_1}}, \]

into formula (16), we obtain the correct result. It is also interesting to note that the effective concentration of fuel acting in the reaction zone does not depend directly on its initial concentration \(a_0\), but is determined only by the maximum combustion temperature \(T_1\). Thus, if at various \(a_0\) we keep \(T_1\) constant (for example, by replacing one inert diluent with another having a different density or heat capacity, or by changing the initial temperature of the mixture), then the effective concentration of fuel in the combustion zone remains constant.

In view of this circumstance, we practically could not determine the order of the reaction from experiments on flame propagation at constant pressure (varying \(a_0\) by dilution with an inert gas). However, since \(a_{\mathrm{eff}}\) is proportional to the density of the gas or to its pressure, from experiments at different pressures we can determine the order of the reaction. In the case of combustion of a two-component system consisting of molecules \(A\) and \(B\), we can establish the order of the reaction only with respect to that component which is present in excess, since only its concentration is completely determined by the initial concentration and is equal to the difference between the initial concentration and the number of molecules that have entered into combustion. Thus, in the case of oxidation of CO considered below, as a result of comparing experimental data with formulas for the propagation velocity, we can only assert that the reaction rate is proportional to the concentration of water vapor, does not depend on \(O_2\) in oxygen-rich mixtures, and is proportional to CO in mixtures with an excess of CO. Moreover, from the dependence on pressure one may consider the reaction to be of second order. If one assumes that the kinetic law is the same for mixtures rich in oxygen and rich in CO, then it inevitably follows from these data that the reaction rate is proportional to the product \([\mathrm{CO}]\cdot[\mathrm{H_2O}]\). However, if this assumption is not made, then the experimental data would also satisfy the following law: the reaction rate is proportional to the product of the total pressure by \([\mathrm{H_2O}]\) in excess \(O_2\), and to \([\mathrm{CO}]\cdot[\mathrm{H_2O}]\) in excess CO.

Of course, this is unlikely, but it should nevertheless be stated that, although the new theory makes it possible, with the aid of experimental data on propagation velocities, to draw very many conclusions about the kinetics of homogeneous reactions at such temperatures when the reaction time is measured by \(10^{-4}\)—\(10^{-5}\) sec. (which is unattainable by any other methods), still, because of this peculiar independence of the effective concentrations from the initial ones, a complete guarantee of the uniqueness of the solution of the question cannot always be given.

In deriving formulas (22) and (23) we made a number of limiting assumptions, which do not always correspond to real cases of flame propagation. Our further task is to free ourselves from these assumptions and to extend the formulas obtained to any real case.

These limiting assumptions were: 1) the assumption of constancy of the heat capacity; 2) the assumption of constancy of \(\frac{\lambda}{c_p}\), i.e., of the ratio of thermal conductivity to heat capacity; 3) the assumption that the number of molecules does not change during the reaction; 4) the assumption of equality of the coefficient of thermal conductivity and the coefficient of diffusion.

In the case where \(\frac{\lambda}{c_p}\) is constant while \(\lambda\) itself is variable, it is necessary to use, as the variable, not the temperature but the heat content—

by \(H\) and, consequently, by the heat-distribution equation in the form (1′). The concentration field \(\dfrac{a}{\rho}\) will be similar to the heat-content field, and equation (21) will be replaced by the equation

\[ a=a_0\frac{\rho}{\rho_0}\frac{H_1-H}{H_1-H_0}. \tag{26} \]

The method of solving the equations will be completely analogous to that used earlier, except that \(H\) will everywhere replace \(T\).

However, since the integral

\[ \int_{H_1}^{H'} w\,dH \]

contains explicitly the temperature \(\left(w=Se^{-\frac{E}{RT}}\right)\), and since \(H'\) is the heat content corresponding to some temperature \(T'\), close to \(T_1\), this integral has to be transformed to the variable \(T\) in the form

\[ \int_{T_1}^{T'} wc_p\,dT, \]

where \(c_p\) is variable. Moreover, if the reaction is monomolecular, then \(w\) is proportional to \(a\), which in turn is linearly related to \(H\). Thus, the integrand already contains \(c_p^2\), and for a bimolecular reaction \(c_p^3\).

However, since the temperature interval in which the reaction proceeds is small, within it one may take \(c_p\) as constant, corresponding to a temperature close to the combustion temperature, and to a gas composition close to the products of the reaction. We shall denote the corresponding value of \(c_p\) at \(T_1\) by \(c_p^*\), and the value of the thermal conductivity \(\lambda\) by \(\lambda^*\). Calculation shows that formulas (22) and (23), in their second form, are also preserved for variable heat capacity; only, instead of \(c_p\) and \(\lambda\), one must substitute \(c_p^*\) and \(\lambda^*\), i.e., the heat capacity and thermal conductivity of the reaction products at the maximum combustion temperature \(T_1\).

In the case of a nonconstant ratio \(\dfrac{\lambda}{c_p}\) and \(D\rho\), when integrating equation (11′), we may put

\[ \frac{\lambda}{c_p}=\frac{\lambda^*}{c_p^*}, \]

i.e., regard it as a constant owing to the smallness of the temperature interval \(T_1-T'\). As for the integration of equation (11), here, instead of \(\dfrac{c_p}{\lambda}(T_1-T_0)\), there will enter

\[ \overline{\left(\frac{c_p}{\lambda}\right)}(T_1-T_0), \]

where

\[ \overline{\left(\frac{c_p}{\lambda}\right)} \]

is defined as

\[ \frac{\displaystyle\int_{T_0}^{T_1}\frac{c_p}{\lambda}\,dT}{T_1-T_0}. \]

Thus, in the formula for flame propagation, instead of the quantities \(\lambda c_p\) for a monomolecular reaction and \(\lambda c_p^2\) for a bimolecular reaction,

enter the reaction respectively as

\[ \left(\frac{\lambda}{c_p}\right)^2 \frac{c_p^*}{\lambda^*} c_p^{*2} \quad \text{and} \quad \left(\frac{\lambda}{c_p}\right)^2 \frac{c_p^*}{\lambda^*} c_p^{*3}. \]

However, owing to the smallness of these corrections, we shall not take them into account further.

If the reaction proceeds with a change in the number of molecules in the ratio \(\dfrac{n_1}{n_2}\), then in the reaction zone, where the bulk of the gas consists of reaction products, the number of fuel molecules will not be

\[ \frac{a}{\rho}=\frac{a_0}{\rho_0}\frac{H_1-H}{H_1-H_0}, \]

but

\[ \frac{a}{\rho}=\frac{a_0}{\rho_0}\frac{n_1}{n_2}\frac{H_1-H}{H_1-H_0}, \tag{27} \]

since the gas density changes in the ratio \(\dfrac{n_1}{n_2}\). Accordingly, an additional factor \(\dfrac{n_1}{n_2}\) for a monomolecular reaction and \(\left(\dfrac{n_1}{n_2}\right)^2\) for a bimolecular reaction will enter the formula under the radical.

Finally, it would be very essential to consider the case when

\[ \frac{\lambda}{c_p} \ne D\rho, \]

and their ratio is:

\[ \frac{\lambda}{c_p}:D\rho = A:B. \]

This is especially important in cases of hydrogen combustion in air or chlorine, where diffusion is determined by the coefficient of diffusion of hydrogen, while thermal conductivity is determined by the thermal conductivity of the mixture.

A simple calculation shows that in this case an additional factor \(\dfrac{A}{B}\) for a monomolecular reaction and \(\left(\dfrac{A}{B}\right)^2\) for a bimolecular reaction must be introduced under the radical in the formulas.

In the case \(\dfrac{\lambda}{c_p}:D\rho=A:B\), the basic equations (19) are written in the form:

\[ A\frac{d^2\theta}{dx^2}-v_m\frac{d\theta}{dx}+w=0. \]

\[ B\frac{d^2a}{dx^2}-v_m\frac{da}{dx}+w=0. \]

According to the law of conservation of energy in the initial and final states,

\[ (a)_\infty=a_1=\frac{a_0}{\rho_0}=(\theta)_\infty=\theta_1=\frac{c_p(T_1-T_0)}{Q'}, \]

since

\[ L=c_p(T_1-T_0)=\frac{a_0 Q'}{\rho_0}. \]

In the reaction region \(a\) and \(\theta\) still approximately satisfy the equations

\[ B\frac{d^2 a}{dx^2}+w=0, \qquad A\frac{d^2\theta}{dx^2}+w=0. \]

Integrating over this interval, we obtain

\[ \left(\frac{d\alpha}{dx}\right)_x-\left(\frac{d\alpha}{dx}\right)_{+\infty} =\frac{d\alpha}{dx}=\frac{1}{B}\int_x^\infty w\,dx, \]

\[ \left(\frac{d\theta}{dx}\right)_x-\left(\frac{d\theta}{dx}\right)_{+\infty} =\frac{d\theta}{dx}=\frac{1}{A}\int_x^\infty w\,dx, \]

i.e.

\[ \frac{d\alpha}{dx}=\frac{A}{B}\frac{d\theta}{dx} \quad \text{and} \quad \alpha=\frac{A}{B}\theta+C, \]

since, according to the boundary conditions, at \(\theta=\theta_1\), \(\alpha=\alpha_1=\theta_1\), we find

\[ C=\left(1-\frac{A}{B}\right)\theta_1, \]

whence

\[ \alpha=\frac{A}{B}\theta-\left(\frac{A}{B}-1\right)\theta_1 . \tag{28} \]

In region \(I\) we may put \(w=0\) in equations (19), whence, integrating the first time, we obtain:

\[ A\frac{d\theta}{dx}-v_m\theta=0, \]

\[ B\frac{d\alpha}{dx}-v_m\alpha=0. \]

Integrating a second time, we obtain:

\[ \alpha=\alpha' e^{-\frac{v_m x}{B}} \quad \text{and} \quad \theta=\theta' e^{-\frac{v_m x}{A}} . \]

Here \(\theta'\) and \(\alpha'\) are the values of \(\theta\) and \(\alpha\) at \(x=0\), where regions \(I\) and \(II\) are conventionally separated.

\[ \frac{a_0}{\rho_0}-\frac{a}{\rho}=\alpha; \qquad \frac{a_0}{\rho_0}=\alpha_1; \]

whence

\[ \frac{a}{\rho}=\alpha_1-\alpha=\theta_1-\alpha \]

or, according to formula (28),

\[ \frac{a}{\rho}=\frac{A}{B}(\theta_1-\theta) \]

or

\[ a=\rho\frac{A}{B}\frac{c_p}{Q'}(T_1-T) =\frac{A}{B}\frac{\rho}{\rho_0}a_0\frac{(T_1-T)}{(T_1-T_0)} =\frac{A}{B}a_0\frac{T_0}{T}\frac{T_1-T}{T_1-T_0}. \tag{29} \]

In other words, in the region where the reaction proceeds, the values of \(a\) differ from those that were obtained for \(A=B\) [see formula (21)] only by the factor \(\frac{A}{B}\).

Finally, the formulas for the normal propagation velocity in the case of simple mono- or bimolecular reactions involving one substance (decomposition of \(\mathrm{Cl_2O}\), explosive substances) will be written as follows:

\[ v_0= \sqrt{ \frac{2\lambda^{*}c_p k}{\rho_0 L^2} \left(\frac{T_0}{T_1}\right) \left(\frac{A}{B}\right) \left(\frac{n_1}{n_2}\right) \left(\frac{RT_1^2}{E}\right)^2 e^{-\frac{E}{RT_1}} } \tag{30} \]

for a monomolecular reaction, and

\[ v_0=\sqrt{\frac{2\varkappa^{*}(ka_0)}{\rho_0 L^3} \left(\frac{T_0}{T_1}\right)^2 \left(\frac{A}{B}\right)^2 \left(\frac{n_1}{n_2}\right)^2 \left(\frac{RT_1^2}{E}\right)^3 e^{-\frac{E}{RT_1}}} \tag{31} \]

for a bimolecular reaction.

A few words must be said about the width of the preheating zone and of the reaction zone. In the preheating zone \((x\) from \(-\infty\) to 0), according to formula (7), for \(T_B-T_0 \cong T_1-T_0\),

\[ T-T_0=(T_1-T_0)e^{\alpha x}, \]

where \(\alpha=\dfrac{v_m c_p}{\lambda}\). This means that the drop of the temperature \(T-T_0\) from its maximum value \(T_1-T_0\) by a factor of \(e\) occurs over a distance

\[ \zeta=\frac{\lambda}{c_p v_m}=\frac{\lambda}{c_p\rho_0 v_0} \cong \frac{D}{v_0}\frac{c_p^{*}\rho^{*}}{c_p\rho_0} \cong \frac{D}{v_0}\frac{T_0}{T_1} \cong \frac{D_0}{v_0}\frac{T_1}{T_0}, \tag{32} \]

where \(v_0\) is the linear propagation velocity and \(D_0\) is the diffusion coefficient at the initial temperature \(T_0\).

In most cases at atmospheric pressure

\[ D_0\cong 0.2;\quad T_1\cong 2\,000^\circ;\quad v_0\cong 50\ \text{cm/sec}. \]

Hence the preheating zone is

\[ \zeta=\frac{0.2}{50}\cdot\frac{2\,000}{300}=2.6\cdot 10^{-2}\ \text{cm}\cong \text{from }0.2\text{ to }0.3\ \text{mm}. \]

The reaction zone \(\xi\) can be roughly estimated, in this way, from equation (11)

\[ \left(\frac{dT}{dx}\right)_0=\frac{v_m c_p}{\lambda_0}(T_1-T_0), \]

where \(\left(\dfrac{dT}{dx}\right)_0\) is the derivative at the boundary of the reaction zone. Note further that for \(x=+\infty\), \(\left(\dfrac{dT}{dx}\right)_\infty=0\). We shall assume that the average \(\dfrac{dT}{dx}\) in the reaction zone will be equal to \(\dfrac12\left(\dfrac{dT}{dx}\right)_0\). Hence

\[ \frac{dT}{dx}=\frac12\,\frac{v_m c_p}{\lambda}(T_1-T_0) \]

or, integrating, we obtain

\[ \xi'\cong 2\,\frac{T_1-T'}{T_1-T_0}\,\frac{\lambda}{v_m c_p}. \tag{33} \]

As we have seen, the temperature drop in the reaction zone is of the order

\[ (T_1-T')=(T_1-T_0)\cdot 0.2, \]

whence the reaction zone, in order of magnitude, is

\[ \xi'\cong 0.4\,\zeta=0.4\cdot 2.6\cdot 10^{-2}\cong 0.1\ \text{mm}=10^{-2}\ \text{cm}. \]

The free path length is measured by a value of \(10^{-5}\ \text{cm}\) under normal-

temperatures and pressure. At \(2000^\circ\) it will be approximately 10 times greater, i.e., almost \(10^{-4}\) cm. This means that the entire width of the reaction zone is measured in hundreds of mean free paths. The number of collisions during diffusion in this zone is

\[ n=\left(\frac{\xi}{l}\right)^2, \]

i.e., of the order of tens of thousands.

The residence time in the reaction zone is

\[ \tau=\frac{\xi'}{v}=\frac{\xi'}{v_0}\frac{T_0}{T_1} =\frac{10^{-2}}{50}\cdot\frac{3}{20} =3\cdot10^{-5}\ \text{sec.} \]

The time between collisions is

\[ \frac{l}{u}\cong\frac{10^{-4}}{10^5}=10^{-9}, \]

i.e.,

\[ n\cong10^4. \]

§ 4. Verification of the Theory for the Simplest Reactions.

(Theory of Combustion of Explosives and Especially of Nitroglycol)

Belyaev\(^4\) advanced a hypothesis according to which all readily vaporizing explosives burn in the gas phase. By thermal conductivity, the heat of combustion continuously heats the surface of the explosive, causing its evaporation from the surface, heated to the boiling temperature \(T_{\text{boil}}\), and thus creating a continuous vapor stream directed toward the combustion zone. For all explosives that can be evaporated without causing their ignition (and that are capable not only of detonating but also of burning), this new point of view on their combustion appears far more probable than the old ideas about combustion directly in the condensed phase. Belyaev proved his hypothesis by direct experiment. Namely, having photographed the nitroglycol flame, he showed that between the liquid meniscus and the flame zone there is a narrow dark zone, the width of which is measured in tenths of a millimeter at atmospheric pressure and reaches 1 mm during combustion under reduced pressure.

Thus, Belyaev’s theory may be considered proven by direct experiment. The mass rate \(v_m\) of combustion can be measured directly from the rate \([u]\) of lowering of the meniscus during combustion of nitroglycol.

This directly measured quantity may conventionally be called the linear rate \([u]\) of combustion of liquid nitroglycol. It would be more correct to call it the linear rate of evaporation during combustion of nitroglycol. The mass rate of evaporation \(v_m\), obviously, is equal to the mass rate of combustion of nitroglycol vapors, since all evaporating nitroglycol burns in the form of vapor. Hence the linear burning rate of nitroglycol vapors is

\[ v_0=\frac{v_m}{\rho_0}=\frac{[u][\rho]}{\rho_0}, \]

where \([\rho]\) is the density of liquid nitroglycol, and \(\rho_0\) is the density of nitroglycol vapors at the experimental pressure and initial temperature.

Belyaev applied the Zel’dovich–Frank-Kamenetskii theory to the calculation of the burning rate of nitroglycol vapors.

We shall carry out the corresponding calculation, using the heat-distribution equation in the form of equation (2).

As we did earlier, let us divide the whole process into zone \(II\), where the main part of the reaction takes place and where the term \(\vartheta_m c_p \dfrac{dT}{dx}\) may be neglected, and region \(I\), otherwise the heating zone, where the reaction rate \(w\) may be neglected. The difference from the case considered above is that here the region \(I\) itself must be divided into region \(I'\) (the dark heating zone of the vapor) and region \(I''\), the heating of the liquid explosive substance (ES) from the initial temperature \(T_0\) to the boiling temperature at the boundary between the liquid phase and the vapor (Fig. 19). In both regions one and the same equation holds,

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

but with different constants \(\lambda\) and \(c_p\). For the liquid we shall denote these quantities by \([\lambda]\) and \([c_p]\). Since at the boundary between the liquid and vapor phases of the ES evaporation takes place, involving expenditure of heat energy, the heat flux \(\lambda \left(\dfrac{dT}{dx}\right)'_{\mathrm{b}}\) at the boundary itself on the vapor side will be greater than the flux \([\lambda]\left(\dfrac{dT}{dx}\right)''_{\mathrm{b}}\) on the other side of the interface, in the liquid phase, by the amount of energy spent on evaporating \(\vartheta_m\) grams of substance, evaporated in 1 sec. from a unit surface. Hence

\[ \vartheta_m F+[\lambda]\left(\frac{dT}{dx}\right)''_{\mathrm{b}} = \lambda \left(\frac{dT}{dx}\right)'_{\mathrm{b}}, \]

where \(F\) is the latent heat of evaporation of 1 g of ES.

Fig. 19

The picture of the temperature distribution is qualitatively shown in Fig. 19.

Integrating the equation for zone \(I'\), we obtain

\[ \lambda \left(\frac{dT}{dx}\right)' = \vartheta_m c_p\left(T-T_{\mathrm{boil}}\right) + \lambda \left(\frac{dT}{dx}\right)'_{\mathrm{b}} . \]

Integrating the equation for zone \(I''\), we obtain

\[ [\lambda]\left(\frac{dT}{dx}\right)''_{\mathrm{b}} = \vartheta_m [c_p]\left(T_{\mathrm{boil}}-T_0\right), \]

whence for zone \(I'\) we obtain

\[ \lambda \left(\frac{dT}{dx}\right)' = \vartheta_m \left\{ c_p\left(T-T_{\mathrm{boil}}\right) + [c_p]\left(T_{\mathrm{boil}}-T_0\right) + F \right\}, \tag{34} \]

or, integrating, we obtain that the width of the dark preheating zone is

\[ \xi=\frac{\lambda}{v_m c_p}\ln \frac{c_p(T_1-T_{\text{boil}})+[c_p](T_{\text{boil}}-T_0)+F} {[c_p](T_{\text{boil}}-T_0)+F}. \tag{35} \]

The quantity standing in the numerator under the logarithm sign is nothing other than the total heat of combustion of 1 g of liquid explosive. Thus,

\[ \xi=\frac{\lambda}{c_p v_m}\ln \frac{L}{[c_p](T_{\text{boil}}-T_0)+F}. \tag{35'} \]

As for the flame-propagation velocity, it will be expressed by the same formulas (30) and (31) as for a gas, since in zone II the equations are the same as in the purely gaseous case, and the expression for the derivative according to formula (34) is

\[ \lambda\left(\frac{dT}{dx}\right)'_0=v_m L. \]

A study of the burning rate of nitroglycol shows that \([u]\) increases directly proportionally to the external pressure \(p\), i.e. \([u]=\frac{\rho_0}{[\rho]}v_0\), where \(\rho_0\) is directly proportional to \(p\), while the density of the liquid \([\rho]\) does not depend on pressure. \(v_0\), as we have seen from the formula, varies inversely proportionally to \(\sqrt{p}\) for mono- and does not depend on \(p\) for bimolecular reactions. Thus, in order to satisfy the experimental data on the dependence of the burning rate of liquid nitroglycol on pressure, we must assume that the reaction proceeds bimolecularly. On the other hand, the direct study of the slow decomposition of nitroglycol vapors at low temperatures and pressures, carried out by Appin, showed that the reaction rate obeys the monomolecular law

\[ w=10^{14}ae^{-\frac{35000}{RT}}. \]

Belyaev assumes that at the high temperatures corresponding to the combustion zone, the rate of activation of molecules as a result of collisions becomes insufficient to establish a stationary (calculated from statistical data) concentration of active molecules. In this case the reaction rate begins to be determined by the number of activating collisions, i.e. it becomes bimolecular. Such a phenomenon has long been known for all monomolecular reactions, which, when the pressure is lowered, exhibit a fall of the monomolecular constant and therefore at sufficiently low pressures become bimolecular. It is also known that with increasing temperature the fall of the constant with decreasing pressure occurs more sharply, and thus the pressures at which the reaction passes into a bimolecular one increase.

Perhaps this is what should explain Belyaev’s hypothesis, according to which at very high temperatures the decomposition reaction of nitroglycol is bimolecular even at atmospheric pressure. It should be noted that, according to experimental data on the study

of the kinetics of monomolecular decomposition, the reaction products, as a rule, are capable of activating the molecules of the initial substance upon collision just as well as upon collision of two molecules of the initial substance. Thus, the bimolecularity of these reactions is peculiar. If, for example, for an ordinary bimolecular reaction of the type \(2\mathrm{HJ}=\mathrm{H}_2+\mathrm{J}_2\), decomposition is possible only upon collision of two particles of the initial substance and \(w=ka^2 e^{-\frac{E}{RT}}\), then here the activated nitroglycol molecule decomposes by itself and the reaction is possible upon its collision with any particle, in particular with particles of the products of its decomposition, whence

\[ w=Za(M)e^{-\frac{E}{RT}}, \]

where \((M)\) is the number of all molecules (both nitroglycol and the products of its decomposition) per unit volume.

The number \((M)\) is determined by the initial number of nitroglycol molecules \(a_0\) and is equal to \((M)=a_0\frac{T_0}{T_1}\), whence

\[ w=(Za_0)ae^{-\frac{E}{RT_1}}\frac{T_0}{T_1}. \]

Recalling the character of the derivation of the expression for the velocity \(v_0\), we shall see that it will be determined not by the expression for bimolecular reactions, but by the expression for monomolecular reactions, with the only difference that in place of the first-order constant \(k\) \((\sim 10^{13}-10^{14})\) there will stand

\[ Za \simeq \sqrt{2\pi\sigma^2ua_0}\frac{T_0}{T_1}\simeq 10^{10}. \]

In the case where \(A=B\), we obtain

\[ v_0=\sqrt{\frac{2\lambda^{*}Za_0}{\rho_0L^2}\left(\frac{T_0}{T_1}\right)^2\left(\frac{n_1}{n_2}\right)\left(\frac{RT_1^2}{E}\right)^2 e^{-\frac{E}{RT_1}}}. \tag{36} \]

As in the case of purely bimolecular reactions, \(v_0\) does not depend on the pressure \(p\), since \(a_0\) and \(\rho_0\) are directly proportional to the pressure. Consequently, \(u\) and \(v_m\) will, in agreement with experiment, be directly proportional to the pressure. Since \(\rho_0=\frac{\mu}{N}a_0\), where \(\mu\) is the molecular weight of nitroglycol,

\[ v_m=\sqrt{\frac{2\lambda^{*}Za_0^2}{L^2}\frac{\mu}{N}c_p^{*}\left(\frac{T_0}{T_1}\right)^2\left(\frac{n_1}{n_2}\right)\left(\frac{RT_1^2}{E}\right)^2 e^{-\frac{E}{RT_1}}}. \tag{36'} \]

The principal quantity for calculating the combustion velocity is the heat effect \(L\) of the decomposition of nitroglycol in the flame when it burns in an inert atmosphere. In the case of combustion in air, as we have seen, a secondary flame arises from the afterburning in air of the products of the primary decomposition. This secondary process is not related to the combustion velocity, and therefore we should be interested only in the heat of combustion of the primary process, which is readily realizable

is carried out in an inert atmosphere. Direct measurements of \(L\) have not been made. However, Appin determined the products of the primary flame by direct analysis. This analysis gave the following overall chemical equation of decomposition:

\[ \mathrm{C}_2\mathrm{H}_4(\mathrm{ONO}_2)_2 = 2\mathrm{NO} + 1.7\mathrm{CO} + 1.7\mathrm{H}_2\mathrm{O} + 0.3\mathrm{CO}_2 + 0.3\mathrm{H}_2 . \]

This means that the greater part of the molecules decomposes according to the equation

\[ \mathrm{C}_2\mathrm{H}_4(\mathrm{ONO}_2)_2 = 2\mathrm{NO} + 2\mathrm{CO} + 2\mathrm{H}_2\mathrm{O}, \]

and the smaller part according to the equation

\[ \mathrm{C}_2\mathrm{H}_4(\mathrm{ONO}_2)_2 = 2\mathrm{NO} + 2\mathrm{CO} + 2\mathrm{H}_2 . \]

Calculating the heat of such decomposition of nitroglycol on the basis of the heats of formation recommended by A. Schmidt, Belyaev arrives at the value \(L = 450\ \mathrm{cal}/\mathrm{g}\). Knowing \(L\) and using the mean heat capacities of the reaction products given in the tables of Lewis and Elbe, Belyaev obtains the following value of the maximum combustion temperature:

\[ T_1 - T_0 = 1350^\circ \quad \text{or} \quad T_1 = 1350 + 300 = 1650^\circ \mathrm{K}. \]

The activation energy \(E\) was taken by Belyaev from Appin’s experiments on the low-temperature decomposition of nitroglycol vapors. It is equal to \(E = 35000\ \mathrm{cal}\).

The heat capacity of the reaction products at \(T_1 = 1650^\circ \mathrm{K}\) was calculated by Belyaev from the data of Lewis and Elbe:

\[ c_p^{*} = 0.35 . \]

The thermal conductivity of the products at \(T = T_1\) was calculated from the mean thermal conductivity of the products at \(0^\circ\mathrm{C}\) and then recalculated to a temperature of \(1650^\circ\mathrm{K}\) with Sutherland’s correction. The ratio of the number of initial molecules to final ones is equal to

\[ \frac{n_1}{n_2} = \frac{1}{6} \]

(according to the stoichiometric equation).

Let us carry out these calculations in a somewhat different way.

According to the kinetic theory of gases,

\[ \frac{\lambda Z}{\rho_0} = \frac{1}{3}\frac{u^2 c_p}{n_0} \]

(for the derivation see below, § 5). In the case of decomposition of vapors of pure nitroglycol \(n_0 = a_0\),

\[ u^2 \simeq \frac{3 \cdot 83 \cdot 10^6 \cdot T_1}{\mu} \simeq \frac{3 \cdot 83 \cdot 10^6 \cdot 1650}{\mu} \simeq \frac{4 \cdot 10^{11}}{\mu}. \]

Both in the number of collisions and in the thermal conductivity, the principal role in the combustion zone will be played by the molecules of the products of decomposition of nitroglycol, as the lighter ones. Therefore it is precisely their \(\mu\) that must enter the formula. Let us take \(\mu = 30\). Then

\[ u^2 = \frac{4}{3}\cdot 10^{10};\quad c_p = 0.35, \]

whence

\[ \frac{\lambda Z}{\rho_0}=\frac{1.5\cdot 10^9}{a_0};\quad e^{-\frac{E}{RT_1}}=e^{-\frac{35000}{3300}}=e^{-10.6}=2.5\cdot 10^{-5}; \]

\[ \left(\frac{T_0}{T_1}\right)^2=\left(\frac{300}{1650}\right)^2=\frac{1}{30};\quad \frac{n_1}{n_2}=\frac{1}{6}\left(\frac{RT_2}{F}\right)^2= \left(\frac{2\cdot 1650^2}{35000}\right)^2=2.34\cdot 10^4. \]

Substituting all these quantities into formula (36), we obtain

\[ v_0=\sqrt{\frac{2\cdot 1.5\cdot 10^9}{2\cdot 10^5}\, \frac{2.34\cdot 10^4\cdot 2.5\cdot 10^{-5}}{30\cdot 6}} =\sqrt{50}\simeq 7\ \text{cm/sec} \]

and, correspondingly, the mass velocity \(v_m=v_0\rho_0\), which for \(\rho\simeq 7\cdot 10^{-3}\) gives \(v_m=4.9\cdot 10^{-2}\ \text{g}/\text{sec}\cdot\text{cm}^2\). The mass burning velocity actually found by Belyaev at room temperature is \(4.5\cdot 10^{-2}\ \text{g}/\text{sec}\cdot\text{cm}^2\). Thus the agreement between theory and experiment is exceptionally good.

With an increase in the initial temperature \(T_0\), for a given thermal effect \(L\), the maximum temperature \(T_1\) increases, and hence so do the reaction rate and the flame-propagation velocity. Belyaev calculated the temperatures \(T_1\) corresponding to various values of \(T_0\) from 20 to 200°. In addition, he experimentally determined the propagation velocity of combustion \(u\) (proportional to \(v_m\)) at different \(T_0\). We give some of his data.

Table 2

\(T_0\) in °C	\(T_1\) in °K	\(T_0\) in °C	\(u=\dfrac{v_m}{\rho_0}\)
20	1650	20	0.29
50	1685	60	0.33
110	1753	100	0.44
140	1790	140	0.52
170	1828	180	0.57
200	1867

The relation between \(T_1\) and \(u=\dfrac{v_m}{\rho_0}\), as is evident from the formula, will be

\[ \ln \frac{u}{T_1}=\frac{E}{2RT_1}+C_1 \quad\text{and}\quad \lg \frac{u}{T_1}=\frac{E}{2.34\cdot 2\cdot 2T_1}+C^1). \]

Substituting for each given \(T_0\) the calculated \(T_1\) and the experimentally determined \(u\), we should obtain a linear relation between \(\lg \dfrac{u}{T_1}\) and \(\dfrac{1}{T}\) with slope

\[ \frac{E}{9.2}=\frac{35000}{9.2}=3800. \]

Figure 20 shows the dependence of the experimentally found values of \(\lg \dfrac{u}{T_1}\) on \(\dfrac{1}{T_1}\) for liquid nitroglycol (Belyaev’s data). We see that the points, with some scatter, lie on a straight line with slope equal to 4100.

\(^{1)}\) \(T_0\) does not enter, since according to formula (36), when \(T_0\) is increased, \(a_0\) simultaneously decreases, so that \((a_0T_0)^2\) remains constant.

Thus, these results also confirm the theory of flame propagation and give for the activation energy a value practically coinciding with the value 35,000 adopted by us, according to Appin’s experiments, for calculating absolute velocities.

In conclusion, let us compare the observed and calculated width of the preheating zone.

Substituting into formula (35) the values \(\lambda^* = 2 \cdot 10^{-4}\), \(c_p^* = 0.4\) and \(v_m = 4.5 \cdot 10^{-2}\), the heat of evaporation of nitroglycol \(F = 43\,000\) cal. and the boiling point \(T_{\text{boil}} = 200^\circ\text{C}\), we obtain

\[ \zeta \simeq 0.2\ \text{mm}. \]

Directly measured by Belyaev, the width of the dark zone at atmospheric pressure (microphotogram and photograph of the flame) gave values of the same order.

Since the velocity \(v_m\) varies directly proportionally to the pressure, when the pressure is decreased the width of the dark zone must increase inversely proportionally to the pressure, and, indeed, Belyaev’s experiments at pressures of \(100\) mm give a dark-zone width of the order of \(1\) mm. Thus, Belyaev’s experiments are an excellent confirmation of the new theory of flame propagation.

Fig. 20

§ 5. Verification of the theory on the example of a chain oxidation reaction. (Theory of flame propagation in air and oxygen mixtures of carbon monoxide)

One of the most important combustion reactions is the combustion of carbon monoxide. Together with the oxidation of hydrogen, this reaction also determines the combustion of hydrocarbons (since it may be thought that in the preheating zone hydrocarbons are first converted into CO and H\(_2\), which then burn).

The hydrogen oxidation reaction is a chain reaction with strongly branched chains (the question of the applicability of Zel’dovich’s theory to this type of reaction remains open for the time being). In contrast to this, the CO oxidation reaction is a chain reaction whose rate very quickly reaches a stationary value, and we may consider that, despite the small width of the flame front, the rate of this reaction is completely determined by the temperature and concentration of the substances at any given point inside the combustion zone. Such behavior of the CO oxidation reaction is connected with the fact that it proceeds only in the presence of water vapor, the decomposition products of which (OH, H) are active centers. Since the amount of OH and H is limited by the available moisture content, the reaction rate cannot exceed a certain limiting ...

of this value. In § 1 we listed in detail various properties of the velocity of flame propagation in CO mixtures. Zel’dovich recalculated data from various foreign authors, as well as data obtained by our collaborators (Barskii), and showed that all the properties of the CO flame and its absolute velocity can be obtained from our general theoretical formulas if it is assumed that the rate of the CO oxidation reaction is determined by the following kinetic laws:

\[ \frac{d[\mathrm{CO}_2]}{dt} = -2\frac{d[\mathrm{O}_2]}{dt} = -\frac{d[\mathrm{CO}]}{dt} = Ze^{-\frac{25\,000}{RT}}[\mathrm{H}_2\mathrm{O}][\mathrm{CO}]. \tag{37} \]

Let us note that \(\mathrm{H}_2\mathrm{O}\) is only a catalyst and is not consumed in the course of the reaction. To make clearer the rate of the reaction proceeding according to law (37), we give a table of the value \(\omega=e^{-\frac{25\,000}{RT}}\) (the probability of reaction, calculated per one collision of \(\mathrm{H}_2\mathrm{O}\) and \(\mathrm{CO}\) molecules) and the reaction time \(\tau\) (the time for the initial products \(\mathrm{O}_2\) and \(\mathrm{CO}\) to decrease by a factor of \(e\)).

Table 3

\(T^\circ\mathrm{K}\)	\(\omega\cdot 10^3\)	\(\tau\) sec.
1 300	0,03	\(3\cdot 10^{-3}\)
1 600	0,33	\(3\cdot 10^{-4}\)
2 000	2,8	\(3{,}5\cdot 10^{-5}\)
2 400	7	\(1{,}4\cdot 10^{-5}\)
3 000	25	\(4\cdot 10^{-6}\)

In order to calculate the flame-propagation velocity for kinetics expressed by law (37), we shall consider separately case 1, when in the initial gas CO is contained in considerable excess relative to stoichiometry, and case 2, when it is contained in deficiency.

Case 1. As we have already seen above, the reaction in the combustion zone proceeds in a gas whose composition is close to that of the final products. In the cooled final products the amount of carbon monoxide will be \([\mathrm{CO}]_1=[\mathrm{CO}]_0-[\mathrm{CO}_2]_1\), where \([\mathrm{CO}]_0\) is the number of CO molecules per unit volume of the initial mixture, and \([\mathrm{CO}_2]_1\) is the number of \(\mathrm{CO}_2\) molecules per unit volume of the final products (after they have been cooled to the temperature \(T_0\)), or, what is the same thing, the number of burned CO molecules. In the combustion zone, where the temperature is close to the maximum combustion temperature \(T_1\), the number of CO molecules per unit volume (the effective concentration of CO) will be

\[ [\mathrm{CO}]=\frac{T_0}{T_1}[\mathrm{CO}]_1. \]

The number of \(\mathrm{H}_2\mathrm{O}\) molecules in the combustion zone changes with respect to \([\mathrm{H}_2\mathrm{O}]_0\) only by the factor \(\frac{T_0}{T_1}\), since \(\mathrm{H}_2\mathrm{O}\) is not consumed in the reaction. Hence the effective concentration of \(\mathrm{H}_2\mathrm{O}\) will be

\[ [\mathrm{H}_2\mathrm{O}]=\frac{T_0}{T_1}[\mathrm{H}_2\mathrm{O}]_0. \]

Thus,

\[ w=Ze^{-\frac{E}{RT}}\left(\frac{T_0}{T_1}\right)^2[\mathrm{H_2O}]_0[\mathrm{CO}]. \tag{38} \]

We see that this expression does not include the concentration of oxygen, i.e., the only concentration that changes significantly in the combustion zone. Thus, with respect to \(\mathrm{O}_2\) the reaction is of zero order, and, consequently, the mathematical expression for the flame-propagation velocity will be analogous to formula (16).

Let us derive, however, for clarity, this expression from the general formula (12):

\[ v_0=\frac{1}{\rho_0L}\sqrt{\,2\lambda Q'\int_{T'}^{T_1} w\,dT\,} \simeq \frac{1}{\rho_0L}\sqrt{\,2\lambda Q'\int_{0}^{T_1} w\,dT\,}. \tag{39} \]

Substituting \(w\) from formula (38) and carrying out the corresponding calculations, we find

\[ v_0=\sqrt{\,1400\,\frac{T_1T_0^2}{E}\,e^{-\frac{E}{RT}} \frac{[\mathrm{H_2O}]_0[\mathrm{CO}]_1}{[\mathrm{CO}_2]_1^2}\,}, \tag{40} \]

where the value \(E=25\,000\).

Let us continue the derivation of formula (40). Substituting \(w\) from formula (38) under the integral sign in formula (39), we obtain:

\[ \int_{0}^{T_1} w\,dT = Z\,\frac{RT_1^2}{E}\left(\frac{T_0}{T_1}\right)^2 e^{-\frac{E}{RT_1}}[\mathrm{H_2O}]_0[\mathrm{CO}]_1, \tag{41} \]

\[ v_0=\sqrt{\, 2\,\frac{\lambda Z}{\rho_0}\,\frac{Q'}{L\rho_0L}\, \frac{RT_1^2}{E}\left(\frac{T_0}{T_1}\right)^2 e^{-\frac{E}{RT_1}}[\mathrm{H_2O}]_0[\mathrm{CO}]_1\,}. \tag{42} \]

According to the kinetic theory of gases, the thermal conductivity is

\[ \lambda \simeq \frac{1}{3}\,lu\rho\,\frac{C}{\mu}, \]

where \(C\) is the molecular heat capacity of the mixture (at the combustion temperature \(C\simeq 9\)).

\[ Z=V\,2\pi\sigma^2u=\frac{u}{nl}, \]

where \(n\) is the number of molecules per unit volume of the mixture at the combustion temperature and is equal to \(n=n_0\frac{T_0}{T_1}\) (where \(n_0\) is the number of molecules in the cold mixture). At \(T=T_1\) the thermal velocity of the molecules is

\[ u=u_{300}\sqrt{\frac{T_1}{300}}, \]

where \(u_{300}\) is the velocity of the molecules at room temperature, i.e., for our mixture \(u_{300}\simeq 5\cdot10^4\ \mathrm{cm/sec}\).

\[ \frac{\lambda Z}{\rho_0} = \frac{1}{3}\frac{u^2C}{\mu n_0}\frac{T_1}{T_0}\frac{\rho}{\rho_0} = \frac{1}{3}\frac{u^2C}{\mu n_0} = \frac{75\cdot10^8}{300}\frac{T_1}{\mu n_0}, \]

\[ \frac{Q'}{L\rho_0}=\frac{1}{[\mathrm{CO}_2]_1}, \]

since \(Q'[\mathrm{CO}_2]_1 = L\rho_0\) is the heat released upon combustion of a unit volume of the mixture.

\[ L=\frac{M_0 Q}{\mu}, \]

where \(M_0\) is the ratio of the weight of the burning CO molecules to the weight of all molecules in the initial mixture, i.e., in view of the closeness of the molecular weights of CO, \(\mathrm{O}_2\), and \(\mathrm{N}_2\), and since the CO that has burned is equal to \([\mathrm{CO}_2]_1\),

\[ M_0=\frac{[\mathrm{CO}_2]_1}{n_0}, \]

and, consequently,

\[ \frac{1}{L}=\frac{\mu n_0}{Q[\mathrm{CO}_2]_1}, \]

where the heat of formation of one mole of \(\mathrm{CO}_2\) is \(Q=7\cdot 10^4\) cal.

Substituting all the values obtained into formula (42), we obtain formula (40), taking the mean value \(\mu \simeq 30\).

Case 2. When there is a deficiency of CO (or at stoichiometry), its concentration in the combustion zone is small and changes rapidly along the zone. Since the reaction rate is proportional to \([\mathrm{CO}]\), here we must use the same derivation that was given above for monomolecular reactions. According to formula (21), the CO concentration in the combustion zone is

\[ [\mathrm{CO}]=[\mathrm{CO}]_0 \frac{T_0}{T_1}\frac{\vartheta}{T_1-T_0}. \]

Substituting this expression into formula (37) and integrating, we readily obtain

\[ \int_{T'}^{T_1}\omega\,dT \simeq \frac{c_p}{L} \left(\frac{T_0}{T_1}\right)^2 \left(\frac{RT_1^2}{E}\right)^2 Z e^{-\frac{E}{RT_1}}[\mathrm{CO}]_0[\mathrm{H}_2\mathrm{O}]_0. \tag{43} \]

Comparing this expression with (41), we see that it differs only in that, instead of the concentration \([\mathrm{CO}]_1\) acting there, there stands the quantity

\[ [\mathrm{CO}]_{\mathrm{eff}}=\frac{c_p}{L}\frac{RT_1^2}{E}[\mathrm{CO}]_0, \]

since, when CO is deficient,

\[ \frac{[\mathrm{CO}]_0}{L}=\frac{\rho_0}{Q'}; \]

\[ [\mathrm{CO}]_{\mathrm{eff}}=\frac{c_p\rho_0}{Q'}\frac{RT_1^2}{E}. \tag{44} \]

But, as we have seen, for monomolecular reactions this quantity, when multiplied by \(T_0/T_1\), corresponds precisely to the effective concentration of \([\mathrm{CO}]\) in the reaction zone. Thus, the formula for the flame propagation rate in mixtures with a deficiency of CO is the same as for the case of a considerable excess of CO, except that \([\mathrm{CO}]_{\mathrm{eff}}\) stands in place of \([\mathrm{CO}]\). In practice, however, the difference is considerable. If in CO-rich mixtures \([\mathrm{CO}]_1\) increases with increasing \([\mathrm{CO}]_0\), then in \(\mathrm{O}_2\)-rich mixtures \([\mathrm{CO}]_{\mathrm{eff}}\), as is evident from formula (44), does not depend on \([\mathrm{CO}]_0\), provided that the combustion temperature (for example, by means of one or another preheating of the mixture) is kept constant.

Thus, we arrive at the conclusion that the velocity of flame propagation in mixtures with the same combustion temperature for all mixtures with excess \(O_2\) remains constant, independently of the excess \(O_2\). This conclusion was checked by a colleague of our Institute, Barskii, who measured the velocities of flame propagation in mixtures with different excess oxygen and one and the same amount of \([CO]_0\) (varying the ratio \(\dfrac{O_2}{N_2}\) in the mixture and thus keeping the combustion temperature constant). The velocity of flame propagation in fact did not depend in this case on the excess oxygen. Varying in the same way the CO content in mixtures with excess CO and keeping the amount of oxygen the same (i.e., again keeping \(T_1\) constant), Barskii, in accordance with formula (40), showed that the velocity of flame propagation is in this case proportional to the square root of the excess CO, i.e. \(\sqrt{[CO]_1}\).

With simultaneous variation of both components, the combustion temperature also inevitably changes, and it is impossible in such a simple way to determine the dependence of the reaction velocity on CO and \(O_2\). Attempts of this kind made by other authors are fundamentally erroneous. However, by using one or another initial preheating of the mixture, it is possible to keep \(T_1\) constant while varying the ratio of the components. Using the data of Passauer and others\(^5\), Zel’dovich made such an attempt to verify the theory and came to the same result—the propagation velocity at constant \(T_1\) does not depend on \(O_2\) when \(O_2\) is in excess and is proportional to \(\sqrt{[CO]_1}\) when CO is in excess. According to formula (40), the propagation velocity is proportional to \(\sqrt{[H_2O]_0}\) (i.e., to the square root of the concentration or partial pressure of moisture).

Fig. 21

\(1 — p_{CO+O_2}=100\ \text{mm};\)
\(2 — p_{CO+O_2}=150\ \text{mm};\)
\(3 — p_{CO+O_2}=200\ \text{mm};\)
\(4 — p_{CO+O_2}=300\ \text{mm};\)
\(5 — p_{CO+O_2}=760\ \text{mm}\)

Replotting the data of Fiock and Marvin, shown in Fig. 15, in the coordinates \(v—\sqrt{[H_2O]_0}\), we obtained good straight lines in agreement with the theory (Fig. 21). At high moisture contents there are deviations, which is natural if only because in this case the thermal characteristic of the mixture begins to change.

As for the dependence of the propagation velocity on the pressure of the mixture, it follows from formulas (40) that at constant percentage moisture content it does not depend on the pressure (since \([CO]_1\), \([CO_2]\), \([H_2O]_0\) are proportional to \(p^1\)), and at constant partial pressure of \(H_2O\), \(v_0\) is inversely proportional to \(\sqrt{p}\). We know that precisely such dependences on pressure are observed experimentally.

It was most interesting to compare the calculated absolute values \(v_0\) for mixtures of CO with air in various ratios. The calculation led to the data shown in Fig. 10 by the dotted line. Of course, here there are certain discrepancies with experiment in the region of the maximum, but they are of almost the same order as the discrepancies in the experimental data. A similar calculation for mixtures of CO with nitrogen and oxygen in different ratios also led to good agreement with the experimental data of Yana, shown in Fig. 12. In general, it must be considered that the approximate theory gives a fair description of the experimental data in this case as well.

Next, a comparison was made of the calculated and observed propagation velocity at different initial temperatures \(T_0\) for various air mixtures of CO. It turned out that the discrepancies between theory and experiment in this case are small. A qualitative explanation was also obtained for the fact of the slight increase of the propagation velocity with pressure in oxygen mixtures. The point is that, if in air mixtures one may neglect the dissociation associated with incomplete burning, then in oxygen mixtures, where temperatures of up to \(3000^\circ\ \mathrm{K}\) and higher develop, dissociation must be taken into account. For details of the calculation see our paper\(^6\); here only the result is given.

The calculation leads to an increase of the velocity with pressure as \(p^n\), where \(n \simeq 0.12\), whereas experiment gives an increase of the velocity with pressure as \(p^m\), where \(m \simeq 0.2\). Thus, in this case (if the experiments are considered flawless) the theory so far gives only a qualitatively correct result—a weak increase of \(v_0\) with pressure in oxygen-rich mixtures.

Summing up all that has been said, we may state that the new theory of combustion makes it possible, by choosing the form of the kinetic law for the rate of oxidation of CO, containing only one arbitrary constant \((E = 25\,000)\), in the majority of cases quantitatively, and only in individual cases qualitatively, to obtain rationally all the numerous experimental results on flame propagation in mixtures of CO with air and oxygen. It is very interesting to verify directly the correctness of the kinetic law postulated by us for the rate of oxidation of CO.

Unfortunately, up to the present time there are no methods for experimentally studying the kinetics of the reaction at such high temperatures, when the reaction proceeds half-way in \(10^{-3}\)—\(10^{-4}\) sec. We shall therefore try to proceed logically and, delving into the question of the mechanism of CO oxidation, attempt from this to find the kinetic law we need.

It has long been supposed that the catalysis by moisture of the oxidation of CO is based on the reaction of conversion of water vapor

\[ \mathrm{H_2O}+\mathrm{CO}=\mathrm{CO_2}+\mathrm{H_2}. \]

The hydrogen thus obtained is rapidly oxidized, and therefore the total oxidation reaction proceeds at the rate of the conversion reaction. Assuming that conversion proceeds by the simple bimolecular law, we quite simply arrive at the regularity needed for us for the oxidation of CO

\[ w = Z e^{-\frac{E}{RT}}[\mathrm{CO}][\mathrm{H_2O}]. \]

Assuming the activation energy of the conversion reaction to be equal to 25,000 cal, we obtain the law (37) adopted by us. Unfortunately, the conversion rate calculated by the indicated formula is in complete disagreement with the scant experimental facts available on the study of the homogeneous conversion reaction. As is known, in technology this reaction proceeds on catalysts, and therefore homogeneous conversion has been studied only incidentally and very little. There are data on the rate of conversion by 1) Kondrat’ev^7 and 2) Thompson^8 in quartz vessels at \(T \simeq 1\,000^\circ \mathrm{K}\). Thompson obtains for the conversion rate under these conditions the value

\[ e^{-\frac{E}{RT}} = 2.5 \cdot 10^{-14}, \]

Kondrat’ev a value 100 times larger: \(10^{-12}\). By formula (37), however, the value obtained is \(\sim 10^{-6}\), i.e., \(10^8\) times larger than Thompson’s data and \(10^6\) times larger than Kondrat’ev’s data. Thompson’s data should be considered more correct, since he experimented in a vessel of larger radius and catalysis by the wall is less probable for him than for Kondrat’ev. In any case, we see that there can be no question of so simple an explanation of law (37).

Studies of the rate of the oxidation reaction of CO at temperatures of the order of \(1\,000^\circ \mathrm{K}\) in the presence of water vapor show that this reaction is a complex chain process, its rate being proportional to \([\mathrm{CO}]\,[\mathrm{H_2O}]\) and inversely proportional to \([\mathrm{O_2}]\). It is, however, very probable that under these conditions the primary active centers necessary for the start of the chain originate in the walls of the vessel as the result of a heterogeneous process, which of course is absent at high temperatures and especially in a flame. Therefore extrapolation of the data obtained under the above-mentioned conditions to a flame will hardly be valid. Moreover, the reaction in closed vessels has also been studied very poorly. Nevertheless, we have every reason to assert that a reaction which is a chain reaction at \(T \simeq 1\,000^\circ\) remains a chain reaction at an arbitrarily high temperature. In addition, it is well known that flames of moist CO emit the spectrum of OH radicals. Moreover, Kondrat’ev quantitatively measured (by the method of light absorption) the concentration of OH radicals in rarefied flames of moist CO having a temperature of \(1\,000\)—\(1\,200^\circ \mathrm{K}\), and showed that the concentration \([\mathrm{OH}]\) exceeds its equilibrium value by thousands of times. These facts definitely indicate that the OH radical is one of the active centers of the chain reaction of CO oxidation.

Examining all possible chain schemes of this kind, we came to the conclusion that only one of them is logically possible at high temperatures. This scheme consists of the following sequence of elementary reactions:

\[ \begin{aligned} 1.\quad & \mathrm{H} + \mathrm{O_2} = \mathrm{OH} + \mathrm{O} \\ 2.\quad & \mathrm{CO} + \mathrm{OH} = \mathrm{CO_2} + \mathrm{H} \end{aligned} \qquad\} \quad \text{chain} \]

\[ \begin{aligned} 3.\quad & \mathrm{O} + \mathrm{CO} = \mathrm{CO_2} \\ 4.\quad & \mathrm{O} + \mathrm{H_2} = \mathrm{OH} + \mathrm{H} \end{aligned} \qquad \text{chain branching} \]

\[ 5.\quad \mathrm{OH} + \mathrm{H_2} = \mathrm{H_2O} + \mathrm{H} \]

As a result of chain branching, the overall oxidation reaction will accelerate with time until the concentrations of OH and H reach such values that the reverse chain-termination reaction

\[ \mathrm{OH}+\mathrm{H}\to \mathrm{H_2}+\mathrm{O} \]

\((4')\) balances the forward reaction

\[ \mathrm{O}+\mathrm{H_2}\to \mathrm{OH}+\mathrm{H}. \]

After this, a stationary oxidation reaction is established. It is then easy to show that the forward reactions 4 and 5 will be balanced by reverse reactions, i.e., in the reaction zone the equilibria

\[ 4.\quad \mathrm{O}+\mathrm{H_2}\rightleftarrows \mathrm{OH}+\mathrm{H} \quad\text{and}\quad 5.\quad \mathrm{OH}+\mathrm{H_2}\rightleftarrows \mathrm{H_2O}+\mathrm{H}. \]

will take place.

The equilibrium constants \(K_4\) and \(K_5\) of these reactions are well known.

As for the constants \(K_1\), \(K_2\), and \(K_3\) of the rates of reactions 1, 2, and 3, matters are worse here. For the constants \(K_1\) and \(K_2\), the activation energies are approximately known:

\[ E_1 = 26\,000 \pm 4\,000,\qquad E_2 = 10\,000 \pm 3\,000. \]

The pre-exponential factor for the first reaction is unknown; for the second it is of the order \(0.01Z\). As for the constant \(K_3\), it has not been studied at all. If one assumes that reaction 3 occurs at every triple collision, then at atmospheric pressure \(K_3 \simeq 10^{-3}\).

If one takes the values of the constants most probable from the experimental and theoretical points of view, the scheme leads to a kinetic law that coincides with law (37) in the sense of the dependence of the rate on temperature and on the concentrations \([\mathrm{CO}]\) and \([\mathrm{H_2O}]\) (it gives practical independence from \([\mathrm{O_2}]\)). However, for mixtures with an excess of CO it gives an absolute reaction rate approximately 10 times smaller than the true one. Since the kinetic constants of the elementary reactions have been poorly measured, the matter is evidently connected with a not entirely correct choice of these constants.

In general, we may say that testing the new theory on the example of CO oxidation leads to satisfactory results.

REFERENCES

1. Jost, Explosions- und Verbrennungsvorgänge in Gasen, Berlin, Verlag Springer, 1939.
2. Lewis and Elbe, J. Chem. Phys., 2, 537, 1934.
3. Zel’dovich and Frank-Kamenetskii, Journal of Physical Chemistry, 12, 100, 1938.
4. Belyaev, in Collected Articles on the Theory of Explosive Substances, Oborongiz, 1940.
5. Passauer, Gas und Wasserfach, 73, 313, 1930; Z. phys. Chem., A 161, 299, 1932; Fiock and Marvin, Chem. Rev., 21, 367, 1937; Jahn, Der Zündvorgang in Gasgemischen, Oldenbourg, Berlin, 1934.
6. Semenov and Zel’dovich, JETP, 10, 1940.
7. Hadman Thompson, Hinshelwood, Proc. Roy. Soc., 137, 87, 1932; 138, 297, 1932.
8. Kondrat’ev, Journal of Physical Chemistry, 11, 336, 1937.

1. If one does not count the apparently incorrect calculation of Lewis and Elbe for the special case of flame propagation in ozone. ↩↩↩↩↩↩

2. Taken on a rotating film. ↩