Physical and Chemical Mechanisms of Reactions that Proceed at Extremely High Rates in a Propagating Flame and the Detonation of Gas

Abstract

The characteristic periods of the reactions in a propagating flame and the detonation of most gases are found to be less than 10⁻⁴ and 10⁻⁶ s, respectively. The extreme rates and accelerations of reactions responsible for such short times correspond to the law of the dependence of the rate of the combustion reaction on temperature, which is an exponent containing a Boltzmann factor in a positive power index. The kinetics of the processes is determined mainly by a sharp increase in the concentrations of free atoms and radicals as they approach those of the initial reagents. The conclusions of the theory agree with experimental results.

INTRODUCTION

The great interest in the processes of combustion, explosion, and detonation of gases is due to their important role in technology, power engineering, and everyday life, and by features of their patterns associated with topical problems of the theory of chemical kinetics and combustion. Years of research have led to the development of the dynamic gas theory of explosion and detonation. At the same time, fundamental problems of the chemical and physicochemical aspects of the theory of combustion, explosion, and detonation started to be solved only in the last two to three decades. The aim of this work was to study the kinetic mechanism of extreme rates and accelerations of reactions in the modes of flame propagation, explosion, and detonation of gases. Their relevance is due to the importance of the theory and the problem of controlling these processes.

Depending on the chemical mechanism, the combustion of gases can be considered a result of either progressively growing self-heating of the reaction mixture or the avalanche multiplication of active intermediate products, accompanied by self-heating at a high rate of the process. Combustion caused self-heating only is referred to as thermal combustion. Semenov [1, 2] showed that such ignition occurs if the rate of the release of heat (q₊) is greater than that of its removal (q_–), and if the release of heat is accelerated more than its removal upon an increase in temperature (T):

$${{q}_{ + }} \geqslant {{q}_{-}},$$

(1)

$$d{{q}_{ + }}{\text{/}}dT \geqslant d{{q}_{-}}{\text{/}}dT.$$

(2)

The rate of the release of heat is equal to the product of the rate of the reaction (W) and the heat effect ($\bar {Q}$):

$${{q}_{ + }} = W\bar {Q}.$$

(3)

The rate of heat removal is:

$${{q}_{ - }} = \alpha S(T-{{T}_{0}}){\text{/}}V,$$

(4)

where α is the coefficient of heat transfer; S and V are the surface and volume of the reactor, respectively; and T₀ is the wall temperature.

Relation (1) refers only to self-heating. Relation (2) refers to the mode of the process, in which q₊ grows more than q_– as the temperature rises. Simultaneous fulfillment of conditions (1) and (2) results in the progressive accumulation of heat in the system and more intense acceleration of the reaction in the ignition mode. Intense self-heating can result in combustion transitioning to explosion and detonation. The theory of combustion, which considers self-heating as the sole factor in the self-acceleration of the reaction, is usually referred to as thermal.

In contrast to the thermal combustion model, which contains reactions of only valence-saturated compounds, chain combustion proceeds with the key participation of free atoms and radicals, which are regenerated in alternating reactions with the initial reagents. A series of alternating reactions forms a reaction chain. In one type of reaction, active particles multiply and the reactions between the initial reagents and these particles accelerate. This type of process is described by the scheme

$${\text{y}} + {\text{B}} \to 3{\text{x}},$$

$$3({\text{x}} + {\text{A}})3{\text{y}} + {\text{P}},$$

where x and y (free atoms and radicals) are reaction chain carriers (CCs), A and B are the initial molecules, and P is the final product.

Because of the free valences in free atoms and radicals, the activation energies of their reactions with valence-saturated compounds are tens of kcal/mol less than those of intermolecular reactions. The rate constants for CC reactions are thus thousands of times higher. Chain combustion therefore proceeds at a rapid rate even at pressures tens and hundreds of times lower than atmospheric. At such pressures, combustion is essentially not accompanied by self-heating. In order to clarify the chemical aspects of chain combustion, the theory of chain reactions considers their isothermal course.

At pressures above hundredths of atmospheric pressure, combustion is accompanied by notable self-heating. This is the main reason such combustion was considered the result of self-heating only, while the chemical process of combustion and detonation was identified with a one-stage reaction between the initial valence-saturated compounds (e.g., [1–8]). The dependence of the temperature and the rate of the reaction were attributed to the Arrhenius law [4–9], even though it applies only to the rate constant. For a mathematical description, the process was considered a one-step reaction [3–8]. The effective rate constant was determined from the measured rate of the process, using an arbitrarily accepted reaction equation of the first (and less often second) kinetic order. This allowed us to describe (but not explain) only the observed kinetics of the process under narrow conditions corresponding to the calculations. Works that ignore the chain-like nature of combustion can be encountered even today [10–12].

However, a series of theoretical and experimental studies showed that due to their very high activation energies, the reactions between valence-saturated compounds are thousands of times slower than combustion reactions and are essentially not accompanied by self-heating. Regardless of pressure and temperature regime (in both explosion and detonation), the combustion reactions of gases are chain reactions and proceed according to the specific laws of non-isothermal ones (e.g., [13–16]).

In contrast to scientific research, which considers the reactions of atoms and radicals, numerical modeling is (with rare exceptions) used only to describe specific known laws. The kinetics of important heterogeneous stages is not considered in these calculations because they have no mechanism or kinetic parameters. In a number of modeling studies, ignition was considered the result of a thermal avalanche instead of a chain [17, 18]. Equations are often solved numerically with no allowance for the chain nature of the process, even though intermolecular reactions are extremely slow and, even at very high temperatures, do not support combustion initiated by an external source. Combustion was also modeled in [19] with no allowance for the reactions between free atoms and radicals. The combustion of 2H₂ with O₂ and 2CO with O₂ are in this case considered one-stage trimolecular reactions, to which rate constants of second kinetic-order reactions are attributed with obvious violation of dimensionality, and the calculated rates of the reactions are overestimated by many thousands of times. Such studies clearly contradict both the obvious laws of chemical kinetics and the very fact of combustion.

In this work, we clarify the reasons for the extreme rates and accelerations of reactions using the example of H₂ oxidation. A link in the reaction chain (i.e., a periodically repeating set of several stages) consists of the reactions [2, 3]

$${{{\text{H}}}_{2}} + {{{\text{O}}}_{2}} = {\text{H}} + {{{\text{O}}}_{2}},$$

(0)

$${\text{H}} + {{{\text{O}}}_{2}} = {\text{OH}} + {{{\text{O}}}_{2}},$$

(I)

$${\text{OH}} + {{{\text{H}}}_{2}} = {{{\text{H}}}_{{\text{2}}}}{\text{O}} + {\text{H}},$$

(II)

$${\text{O}} + {{{\text{H}}}_{2}} = {\text{OH}} + {\text{H}}.$$

(III)

The limit stage is reaction (I) due to the multiplication of free valences and the higher activation energy relative to stages (II) and (III), and thus the lower rate constant [16].

Along with act (I), H atoms participate in the formation of low-activity products not involved in chain combustion. The chain is terminated in such reactions, which include the heterogeneous recombination of active particles and the trimolecular reaction

$${\text{H}} + {{{\text{O}}}_{2}} + {\text{M}} = {\text{H}}{{{\text{O}}}_{2}} + {\text{M}}.$$

(IV)

At moderately high temperatures, the rate of the reaction for the regeneration of atomic hydrogen,

$${\text{H}}{{{\text{O}}}_{2}} + {{{\text{H}}}_{2}} = {{{\text{H}}}_{{\text{2}}}}{{{\text{O}}}_{2}} + {\text{H}}$$

(V)

is extremely low due to the very high activation energy (26 kcal/mol) [20].

HO₂ radicals are adsorbed on the reactor walls and also react with an inhibitor (In):

$${\text{H}}{{{\text{O}}}_{2}} + {\text{In}} \to {\text{termination}}{\text{.}}$$

(VI)

To consider the patterns of chain combustion, partial quasi-stationary concentrations [2] are used to reduce the system of kinetic equations of active particles to one equation associated with a CC, the reaction of which branches out and thereby limits the rate of combustion. As noted above, the particles in this process are H atoms. For the sake of generality, O₂ is denoted as B, and the concentration of H atoms becomes n. Based on above reactions (0)–(VI), the change in n is then expressed by the equation [2]

$$\frac{{dn}}{{dt}} = {{\omega }_{0}} + (f - g)n = {{\omega }_{0}} + \varphi n,$$

(5)

where f and g are the rates of reactions (I) and (IV) of the CC at their unit concentrations. These rates are equal:

$$\begin{gathered} f = 2{{k}_{1}}{\kern 1pt} \text{[}{\text{B}}] = 2k_{1}^{0}{{e}^{{-~{{E}_{1}}/RT}}}[{\text{B}}], \\ g = {{k}_{{{\text{eff}}}}}. \\ \end{gathered} $$

(6)

The rate of O₂ consumption (i.e., the rate of the process) corresponds to the equation

$$\begin{gathered} W = d[{\text{B}}]{\text{/}}dt = {{\omega }_{0}} + \{ {{k}_{1}} + {{k}_{5}}{\kern 1pt} [{\text{M}}]\} [{\text{B}}]{\kern 1pt} n \\ = {{\omega }_{0}} + {{k}_{c}}{\kern 1pt} [{\text{B}}]{\kern 1pt} n, \\ \end{gathered} $$

(7)

where [M] is the concentration of the mixture.

The role of chain termination reactions with the participation of active OH and O particles is negligible, due to the very high values of the rate constants of stages (II) and (III).

It follows from Eq. (5) that a very low stationary concentration of active particles, determined by the ratio of the rates of their formation and loss in a very slow reaction (0), is established if f < g under conditions. If f > g (i.e., CCs multiply faster than they are lost), there is an avalanche multiplication of active particles. The consumption of the initial reagents reacting with these CCs in reactions (I)–(III) is correspondingly accelerated as an avalanche. A condition for chain ignition is thus the inequality

$$f > g,{\text{ i}}{\text{.e}}.,{\text{ }}\varphi > 0.$$

(8)

A feature of the acceleration of the combustion reaction caused by a rise in temperature is that it is much stronger than it should be according to Arrhenius function ${{e}^{{-E/RT}}}$. For example, when heating a stoichiometric mixture of H₂ with air from 823 to 853 K at 1 atm, the rate constant of the intermolecular reaction characterized by the activation energy of 225 kJ/mol, grows by only ≈5%. The rate of this reaction (0) will rise by the same small amount. The increase in the rate of О₂ consumption in reaction (0) is only 0.5%. In an experiment, the mixture ignites with such heating and burns out in small fractions of a millisecond.

The extremely strong temperature dependence of the rate of the combustion reaction is due to a characteristic of the change in the CC concentrations. At each time and for each value of f and g, the rate of change in the CC concentration correlates with their concentration, as can also be seen from Eq. (5). If f > g, the correlation is positive, and the n value grows exponentially over time even at a constant temperature:

$$n = \frac{{{{\omega }_{0}}}}{{f - g}}{{e}^{{(f-g)t}}}.$$

(9)

The f value includes rate constant k₁ with its own Boltzmann factor, in accordance with Eq. (6). Concentration n and rate W thus depend on temperature when f > g, according to an exponential law with a positive exponent. Since this functional dependence applies at each given temperature, it is obeyed during combustion and is obviously much stronger than the exponent in the Arrhenius law.

At times of t > t₀ ≅ 2.5/φ, where we can already ignore the ω_o value in Eqs. (5) and (7), the integration of Eq. (5) with allowance for expression f according to formula (7) and the temperature dependence of k₁ yields the dependence of n on temperature and time:

$$n = {{n}_{0}}\exp \mathop \smallint \limits_{{{t}_{0}}}^t \left[ {{{f}_{0}}\exp \left( {-\frac{{{{E}_{{\text{b}}}}}}{{RT}}} \right) - g} \right]dt.$$

(10)

Substituting n from expression (10) into the rate of Eq. (5), we obtain [21, 22]

$$\frac{W}{{[{\text{B}}]}} = {{k}_{1}}{{n}_{0}}\exp \mathop \smallint \limits_{{{t}_{0}}}^t \,[{{f}_{0}}\exp (-E{\text{/}}RT)-g]{\kern 1pt} dt.$$

(11)

The sharp acceleration of the process as the temperature rises is due primarily to a strong increase in the rates of reactions between active particles of H and OH with O₂ and H₂, respectively. Because of the high values of rate constants k₁ and k₂ of these reactions, the rates of growth of their absolute values with temperature are also high. There is accelerating multiplication of active particles according to law (10), so the consumption of the initial reagents is accelerated just as strongly; i.e., the reaction’s rate of combustion is in accordance with Eq. (11). According to expressions (10) and (11), the exponent also grows over time. This strong temperature dependence of the rate ensures rapid propagation of the flame, an easy transition of combustion into a chain-thermal explosion, and self-acceleration of the reaction in the modes of a chain-thermal explosion and detonation. The exponential law with a positive Boltzmann factor in the positive power index has no known analogs.

With an inhibitor, the g value includes the reaction’s rate of inhibition at a unit CC concentration. The very strong effect inhibitors have on combustion is due to the exponential dependence of rate W on g.

The correlation found in [21, 22] between n_i and $\partial {{n}_{i}}{\text{/}}\partial t$ generally follows from a system of equations that considers different types of reactions between active particles [23]. In these equations, the products of concentrations and rate of the reaction constants are included in rates W_j of elementary reactions related to all components. Particle diffusion is also considered. The diffusion term does not eliminate this correlation. Systems of equations that consider the reactions of all components and one active particle are generally not linear. At each point in space and time, however, the behavior of its solutions is determined by the properties of a linearized system that includes the contribution from diffusion terms. The concentrations of all active particles change according to law $\exp {\text{(}}\Phi t{\text{)}}$, where value ${{\Phi }}$ plays the role of φ in formula (5) and is equal to the difference between the f₁ and g₁ values, which are analogs of f and g. The products of the rate constants and the concentrations related to branching and termination of chains are included in quantity ${{\Phi }}$, respectively, with positive and negative signs. Of course, the rate constants in ${{\Phi }}$ (and thus ${{\Phi }}$ itself) change along with temperature. However, the exponential character of the dependence of the concentrations of active particles on $\Phi t$ remains. The rate of branching (and thus value f₁ with its own Boltzmann factors) depend on temperature much more than the rate of termination. The rate of the chain reaction therefore actually grows along with temperature when f₁ > g₁, according to a double exponent similar to the one displayed by Eq. (11).

EXPERIMENTAL VERIFICATION OF Eqs. (10) AND (11)

The kinetics of the combustion reaction of a stoichiometric mixture of H₂ with O₂ in a thermostatted quartz reactor was studied at constant temperatures of 773 and 768 K and an initial pressure of 2.25 Torr. A diagram of the setup is shown in Fig. 1. The microthermocouple was placed in a quartz capillary washed with hydrofluoric acid, which reduced the efficiency of heterogeneous atomic recombination and its heating. As can be seen in Fig. 2, the difference between heating at the specified initial temperatures did not exceed 0.15°C (i.e., it was less than 3% of the difference between the initial temperatures). The recorded maximum rise in temperature (4°C) was mainly due to the recombination heating of the thermocouple, which can also be seen from the highest heating being reached later than the strongest chemiluminescence and the greatest drop in pressure (i.e., the concentration of O₂). Due to the heterogeneous chain termination that removes the energy of recombination, low pressures close to P₁, and a high rate of conductive heat removal, the process was virtually isothermal at each specified temperature. The time needed to reach the maximum rate of the reaction after the induction period was 0.36–0.5 s, which was thousands of times longer than that of heat removal.

Fig. 1.

Block diagram of the setup for kinetic studies at low pressures: (1) temperature controller, (2) potentiometric temperature recorder, (3) vacuum setup, (4) membrane pressure gauge, (5) pressure transducer, (6) vacuum gauge, (6 '–8) DC amplifiers, (9) photomultiplier, and (10) multichannel oscilloscope.

Fig. 2.

Oscillograms of (1, 1 ') pressure, (2, 2 ') chemiluminescence, and (3, 3 ') temperature at (1–3) 768 K and (1 '–3 ') 773 K.

We considered shallow depths of conversion at which О₂ consumption can be ignored (i.e., the integrand in expression (10) at each specified temperature is constant). Since self-heating was negligible and virtually the same at the specified initial temperatures, its contribution can be ignored with no loss of accuracy when considering the ratio of rates. This allows us to write Eq. (11) in the form

$$\begin{gathered} W{\text{/}}{{[{{{\text{O}}}_{2}}]}_{0}} = \frac{{{{k}_{1}}{{{[{{{\text{H}}}_{2}}]}}_{0}}}}{{2\left( {1-\frac{{{{P}_{1}}}}{{{{P}_{0}}}}} \right)}} \\ \times \;{\text{exp}}\left\{ {[2k_{1}^{0}{{{[{{{\text{O}}}_{2}}]}}_{0}}\exp \left( {-\frac{{{{E}_{1}}}}{{RT}}} \right)-{{k}_{{{\text{het}}}}}](t-{{t}_{0}})} \right\}. \\ \end{gathered} $$

(12)

In this expression, E₁ is the activation energy of the limit stage (i.e., branching), which is 70 kJ/mol, and k_het is the rate constant of heterogeneous loss of H atoms, which is 2k₁ [O₂]₁, where [O₂]₁ is the concentration of O₂ at the first flammability limit measured in the same experiments. This formula was used to determine the ratio of the rates of the chain reaction at these two different temperatures. Results from our calculations and experiments are given in Table 1. The last column shows the calculated increase in rates that would occur if the temperature dependence were expressed by the Arrhenius law.

Table 1.   Ratio of the rates of reactions at 773 and 768 K

As we can see from the table, the recorded increase in the rate caused by raising the initial temperature agrees with the one calculated according to the equation within a measuring error of t − t₀.

The measuring results allow us to determine the contribution from an increase in the concentration of CCs to the observed anomalous dependence of the rate on temperature. We therefore compare the increase in the rate of the reaction and that of the rate constant of the branching stage, which corresponds to the generally accepted temperature dependence of the rate according to the Arrhenius law. According to the stoichiometry of the process,

$$2{{{\text{H}}}_{2}} + {{{\text{O}}}_{2}} = 2{{{\text{H}}}_{{\text{2}}}}{\text{O}},$$

the loss in the number of moles of the mixture is equal to the reduction in the number of moles of O₂, and thus the drop in pressure in a closed reactor. The ratio of the maximum values of the slopes of kinetic curves 1 and 1 ' is therefore equal to that of the maximum rates of O₂ consumption at 773 and 768 K. If the temperature dependence of the rate of the reaction is conventionally expressed by the Arrhenius law, this ratio and the set temperatures are related by the familiar expression

$${\text{ln}}\frac{{W{\kern 1pt} '}}{W} = \frac{{E({{T}_{2}}-{{T}_{1}})}}{{R{{T}_{1}}{{T}_{2}}}}.$$

(13)

Substituting a 1.29 ratio of maximum rates and the given values of Т₁ and Т₂ into this expression, we obtain the effective value of the activation energy of the chain process: 60.0 kcal/mol. At the same time, the activation energy in the rate constant of rate-limiting reaction (I) is only 16.7 kcal/mol [20]. Based on these data and the expression for the rate of O₂ consumption,

$$W = {{k}_{1}}[{\text{H}}][{{{\text{O}}}_{2}}],$$

(14)

it is obvious that the difference between the values obtained above for the activation energy of the chain process and that of rate constant k₁ equal to 43.3 kcal/mol corresponds to the temperature dependence of the growth rate of the concentration of H. Even at a specified temperature difference of only 5°C, the contribution from an increase in the concentration of atomic hydrogen to the acceleration of the chain process is thus much greater than the one from the rate constant of the limit stage, which corresponds to the traditional hypothesis that the temperature dependence of the rate corresponds to the Arrhenius law.

Our results allow us to estimate the concentration of H that ensures the observed excess of the increase in the rate of the reaction over that of the rate constant of the limit stage. Dividing the maximum value of W by the corresponding concentration of О₂ (Fig. 2), we obtain the concentration of Н at the moment of the maximum rate: 1.3 × 10¹⁴ atom/cm³. This value and the above activation energy are in good agreement with the concentration of H measured via EPR for a flame of the same mixture at 930 K and 3 Torr [24], which is close to 3 × 10¹⁵ atom/cm³. Based on the given values of the concentration and activation energy of 43 kcal/mol, a concentration of 2.3 × 10¹⁵ atoms/cm³ is obtained at 930 K.

DIFFERENCE BETWEEN THE TEMPERATURE DEPENDENCES OF A REACTION’S RATE OF COMBUSTION AND THE RATE CONSTANT OF THE LIMIT STAGE IN A PROPAGING FLAME

The specificity of the temperature dependence of the rate of the reaction and the role of active particles is even more pronounced at greater differences in temperature. Below, we compare the rate of reactions between H₂ and O₂ at 843 K (the autoignition temperature of the stoichiometric mixture is in the immediate vicinity of the third ignition limit at 1 atm in a closed quartz reactor with a diameter of 7.4 cm) and the temperature of flame propagation under conditions that preclude the transition from combustion to explosion (i.e., with no adiabatic compression or its effect on temperature). The rate of the reaction near the third ignition limit given in [3] is 7.2 × 10¹⁵ molecule/(cm³ s). The characteristic period of heat removal is thus three orders of magnitude shorter than that of the reaction, so there is virtually no self-heating.

Flames propagated in a mixture of 15% H₂ with air in a molybdenum-sealing glass tube 1.2 cm in diameter and 500 cm long. A diagram of the setup is shown in Fig. 3. Photomultipliers, the signals of which were transmitted to a multichannel oscilloscope connected to a computer, were placed along the tube. The tube was wrapped in black paper to eliminate stray light interference. The tube was washed with boric acid to reduce the efficiency of the heterogeneous recombination of atoms and radicals. Based on data from a chromatographic analysis, hydrogen (a small component of the mixture) burned out almost completely by the time the mixture left the reaction zone. In accordance with the reaction’s stoichiometry, half as much O₂ was consumed.

Fig. 3.

Diagram of the setup: (1) high-voltage power supply, (2) oscilloscope, (3) photosensors, (4) reactor, (5) valve, (6) vacuum gauge, (7) vacuum pump, and (8) sampler valve.

The rate of the reaction was determined as the ratio of the concentration of O₂ consumed in the flame to the period the combustible mixture was in the flame zone. The period of the reaction was determined as a fraction of the flame’s speed, which was determined from the slope of the x–t diagrams of the flame’s travel after establishing an almost constant speed on the diagrams with virtually the same slope (Fig. 4) corresponding to one of 20 m/s. A flame zone width close to 0.15 cm was determined from a profile of temperature along a flame that was known for a mixture of specific composition [25] and corresponded to the value given in [8]. This width at a given flame speed corresponded to a period no longer than 10⁻⁴ s. The adiabatic heating of this mixture was 1070 K. Assuming a 15% loss, the heating was 910 K. Allowing for an initial temperature of 293 K, the flame temperature was thus 1203 K. The average rate of the reaction was determined as the ratio of the amount of oxygen to the time the mixture was in the reaction zone. Since the initial concentration of H₂ at this temperature was 2.2 × 10¹⁷ molecule/(cm³ s) and the period of the reaction was 10⁻⁴ s, the rate of hydrogen combustion in the reaction was 2.2 × 10²¹ molecule/(cm³ s), and the rate of the reaction with respect to oxygen was 1.1 × 10²¹ molecule/(cm³ s).

Fig. 4.

x–t diagrams of a mixture of 15% H₂ and air.

The extreme nature of the rate of the reaction is apparent from virtually all 2.2 × 10¹⁷ good-sized hydrogen molecules and an equivalent amount of oxygen with even larger molecules being consumed in each cm³, in less than ten thousandths of a second.

Another sign of the extreme rate of the reaction is its acceleration. At 843 K, the rate of the reaction in a stoichiometric mixture of H₂ and O₂ was 7.2 × 10¹⁵ molecule/(cm³ s) [3]. To compare the rates at the two considered temperatures, we recognized that the molar fractions of H₂ and O₂ in the experiment with flame propagation were smaller than those in the stoichiometric mixture in experiments at 843 K. With corrections, the rate of the reaction at 1203 K was 4.87 × 10²² molecule/(cm³ s), so the chain process accelerated by a factor of 6.7 × 10⁶ as the temperature rose from 843 to 1203 K. At the same time, the rate constant of the limit act (reaction (I)) grew by only 20 times with the same increase in temperature (i.e., it was 3.3 × 10⁵ times lower). Since O₂ was consumed only in the reaction with atomic hydrogen, it is obvious that this difference between the increases in W and k₁ was due to a rise in the concentration of H. The increase in the rate during flame propagation was determined essentially by another in the concentration of atomic hydrogen. This increase corresponds to Eq. (10) and obviously not the Arrhenius law.

Dividing the rate by k₁ and the concentration of O₂, we obtain the concentration of atomic hydrogen in the flame: 4.8 × 10¹⁵ atom/cm³, which is 0.1% of the entire mixture and 0.5% of the (Н₂)₀. This concentration is close to the one measured for a flame of mixtures of a similar composition via mass spectrometry in [25] and exceeds the equilibrium values by several orders of magnitude.

A flame velocity of 20 m/s in a hydrogen-poor mixture inside a tube of small diameter was thus achieved, due to the tremendous rate and extreme acceleration of the reaction, along with the high concentration of atomic hydrogen multiplying according to a positive exponential law in accordance with Eq. (10).

RATE OF THE SELF-ACCELERATION OF THE REACTION AND THE CONCENTRATION OF ATOMIC HYDROGEN IN A DETONATION WAVE

The detonation velocity of a mixture of 30% H₂ and air is close to 2 × 10⁵ cm/s [26]. According to [8], the width of the flame zone is close to 0.15 cm, so the time the mixture is in the flame zone is close to 10⁻⁶ s. The extreme nature of the rate and acceleration of the reaction in detonation is even more pronounced. In a period on the order of a microsecond, the mixture has time to heat up to the temperature of ignition, transition to the detonation mode, and react almost completely. The adiabatic temperature is 2390 K. Assuming a 10% heat loss, the temperature of the flame is 2178 K. At this temperature, the concentration of O₂ in the detonation wave is 5 × 10¹⁸ molecule/cm³. The rate obtained by dividing this concentration by a period of 10⁻⁶ s is 5 × 10²⁴ molecule/(cm³ s), which is 7 × 10⁸ times higher than the one at 843 K. The reaction is thus accelerated hundreds of millions of times per million fractions of a second. At the same increase in temperature, rate constant k₁ of the limit stage grows by only 450 times. It is also obvious from our results that the acceleration of the combustion reaction is actually due only to a sharp increase in the concentration of atomic hydrogen, in accordance with Eq. (10). These results show especially clearly the inapplicability of the Arrhenius law to the rate of the reaction and testify to the need to revise earlier conclusions reached with this law.

Dividing the rate of the reaction by a 4.9 × 10¹⁸ molecule/cm³ product of the concentration of O₂ and a 6.5 × 10^–12 molecule/(cm³ s) value of k₁, we obtain a 1.35 × 10¹⁷ atom/cm³ concentration of atomic hydrogen. This means that around 7% of the initial hydrogen was converted into H atoms by the chain mechanism during detonation, which was around 25% of the existing concentration of H₂.

CONCLUSIONS

An answer was obtained to the fundamental question of the reasons for the extreme rates and accelerations of combustion, explosion, and detonation of gases. Due to the internal energy of the initial reagents, the reaction system produces very high concentrations of free H atoms and radicals. These are regenerated and multiplied in their rapid reactions with the initial reagents, resulting in progressive self-acceleration of the combustion reaction. The extreme rates and accelerations of reactions achieved upon the rise in temperature during combustion, explosion, and detonation are determined not by the Arrhenius law, as was assumed earlier, but by an increase in the atom and radical concentrations according to an exponential law containing the Boltzmann factor in a positive power index. This law is also responsible for a very rapid increase in the rate of heat release. Condition (2) is satisfied at high rates, and an avalanche-like accumulation of heat in the system occurs alongside an already developing chain avalanche. Combustion transitions to the explosion mode and, with a strong explosion, to detonation. Such rates and accelerations of the chain process are the reasons for the destructive effect of explosions and the detonation of combustible gases. At the same time, the chain nature of the reactions allows us to control the processes of combustion, explosion, and detonation using inhibitors and promoters [14, 15, 26].