Reports of the Academy of Sciences of the USSR
1964. Volume 154, No. 6

PHYSICAL CHEMISTRY

Corresponding Member of the Academy of Sciences of the USSR V. V. VOEVODSKII, R. I. SOLOUKHIN

ON THE MECHANISM AND LIMITS OF CHAIN SELF-IGNITION OF HYDROGEN WITH OXYGEN IN SHOCK WAVES

The influence of the kinetics of an exothermic reaction on the dynamics of gas flow is clearly manifested in various detonation processes. For example, the principal cause of deviations from stationarity of the reaction zone in detonation waves is the strong dependence of the induction delays in a gas adiabatically compressed by a shock wave on pressure and temperature. Chemical factors also play an important role in the nonstationary processes of detonation formation in front of the flame in compression waves that arise during the expansion of combustion products.

In work (¹) it was shown that, as the temperature is lowered in certain gas systems when they are compressed by shock waves, a noticeable change is observed in the character of ignition and of the onset of detonation upon passage through a certain temperature region. Figure 1 (see insert, p. 1415) shows typical Schlieren photographs and oscillograms of pressure and luminosity characterizing different ignition mechanisms. At high temperatures, for detonation to arise it is sufficient for a single ignition center to form. Additional compression and heating of the layers of unburned gas adjacent to the expanding combustion products leads to a sharp acceleration of the reaction and to the rapid formation of a detonation wave (mechanism a). At lower temperatures the reaction is initiated at many points, the ignition centers multiply freely, the combustion surfaces merge and may form a single flame front without sharply expressed explosive self-ignition (mechanism b). On the basis of these data, it could be assumed that we were dealing with some new qualitative effect.

Upon a more detailed examination of the data obtained at various gas temperatures and pressures behind the reflected shock wave, it was found that the temperature of transition from one detonation regime to another in the H₂ + O₂ mixture decreases as the pressure decreases. In addition, it proved that the boundary separating these two regions in the ((p, T)) plane coincides rather closely with the theoretical curve obtained by extrapolating to the experimental conditions the so-called upper ignition limit of hydrogen with oxygen (p_2), reliably determined (²) in the region of lower temperatures and pressures. Since this limit is due to the competition between the chain-branching reaction
(\mathrm{H} + \mathrm{O}_2 \to \mathrm{OH} + \mathrm{O}) ((k_2))
and the homogeneous termolecular chain-breaking reaction
(\mathrm{H} + \mathrm{O}_2 + \mathrm{M} \to \mathrm{HO}_2 + \mathrm{M}) ((k_6)),
then crossing the curve (Fig. 2) (p = p_2 = 2k_2/k_6) at high pressures and temperatures means that we pass from a region of entirely branched chains into a region of long straight chains with rare branchings, and that precisely this transition determines the change in the character of the initiation and development of ignition. To verify and refine the proposed hypothesis concerning the possibility of describing the onset of detonation in an adiabatically heated combustible mixture by means of concepts developed in the analysis of combustion kinetics at low pressures and temperatures, it was necessary to find such a characteristic of the ignition process as could be calculated quantitatively from kinetic data. As such a characteristic it was natural to choose the delay time, i.e., the time (\tau) (see Fig. 1) from the moment the shock wave passes until the appearance of the first ignition center.

It follows from the kinetic scheme of hydrogen combustion that, as (T \to T(p_2)) (see Fig. 2), the reaction proceeds through the following sequence of elementary steps:

[
\begin{array}{ll}
(k_0)\ \mathrm{H_2 + O_2 \to 2OH}^{}; &
(k_6)\ \mathrm{H + O_2 + M \to HO_2 + M}; \[2mm]
(k_1)\ \mathrm{OH + H_2 \to H_2O + H}; &
(k_7)\ \mathrm{OH \to death\ on\ the\ wall}; \[2mm]
(k_2)\ \mathrm{H + O_2 \to OH + O}; &
(k_8)\ \mathrm{HO_2 + H_2 \to H_2O_2 + H}^{*}. \[2mm]
(k_3)\ \mathrm{O + H_2 \to OH + H}. &
\end{array}
]

The rate constants of these processes and their temperature dependences have been measured with sufficient reliability, and therefore, by writing the corresponding kinetic equations, we obtain an expression for the induction period in the form({}^{(2)}): (\tau = \frac{1}{\varphi}\ln\alpha), where (\varphi) is the coefficient of the exponent in the exponential time dependence of the rate of the leading reaction, and (\alpha) is the ratio of the current reaction rate attained by the moment of explosion to the rate of the initiation reaction.

Fig. 2. Limiting curves of chain ignition of the (\mathrm{H_2 + O_2}) mixture

In the temperature range (1100)–(1800^\circ\mathrm{K}), (\varphi) will differ little from the quantity (2k_2(\mathrm{O_2}) - k_6(\mathrm{O_2})\cdot(\mathrm{M})), while in the remaining temperature region of ignition (down to (T \sim 800^\circ\mathrm{K})) (\varphi) should be understood as the larger characteristic root of the following system of equations:

[
d(\mathrm{H})/dt = 2k_2 - k_6(\mathrm{M})
+ k_8(\mathrm{HO_2})(\mathrm{H_2}) + W_0;
]

[
d(\mathrm{HO_2})/dt = k_6(\mathrm{M})(\mathrm{O_2})(\mathrm{H})
- k_7 + k_8(\mathrm{H_2}).
]

The value of (\alpha) can be estimated by using the known rate constant of homogeneous initiation of the reaction (k_0 = 10^{-10}\exp(-45\,000/RT)\ \mathrm{cm^3/sec}). For example, at (p = 1) atm and (T = 1000^\circ\mathrm{K}), assuming that by the moment of explosion a nonequilibrium concentration of atomic hydrogen ((\mathrm{H}){\max}/e) has been reached, where ((\mathrm{H})), respectively, i.e. precisely at}) is determined from the rates of reaction ((k_2)) and recombination: ((\mathrm{H})_{\max} \simeq 6\cdot 10^{20}\exp(-18\,000/RT)\ \mathrm{cm^{-3}}) ({}^{(2)}), we obtain (\alpha = W'/W_0 \simeq 10^5). In the region of low temperatures the value of (\alpha) will be several orders of magnitude smaller, since although (W_0) decreases as (\exp(-45\,000/RT)), (W') decreases still faster as a consequence of the transition through the limit (p_2). Carrying out the corresponding calculations at various (p) and (T), we obtain the curve of the dependence of (\lg \tau) on (1/T) for various pressures. In Fig. 3 the results of these calculations are compared with experimental data on the determination of (\tau) at pressures of 1, 2, and 3 atm in the temperature interval (850)–(1650^\circ\mathrm{K}). First of all it should be noted that in all cases there is a more or less sharp drop in the experimental values of (-\lg \tau) at (T = 960), 1025, and (1070^\circ\mathrm{K

* The process of chain initiation may occur somewhat differently, for example
(\mathrm{H_2 + O_2 \to H + HO_2}); for us only the possibility of using the rate constant given by N. N. Semenov is essential.

** The reaction (\mathrm{HO_2 + H_2 \to H_2O + OH}) is also quite probable. What is essential is only the possibility of chain propagation through the radical (\mathrm{HO_2}). The value (k_8) was determined by V. N. Kondrat’ev({}^{(3)}).

temperatures at which a change in the character of the detonation is observed (see above). The same deviations, and at precisely these temperatures, are also observed in the calculated curves. From this agreement, and also from the fact that in the high-temperature region ((\tau){\mathrm{calc}}) agrees quite satisfactorily quantitatively with ((\tau)) is very similar. This is also indicated by the fact that, when the temperature is lowered outside the region bounded by the limiting curve (p_3) (Fig. 2), found experimentally ((^4)) and determined quantitatively on the basis of the same mechanism ((^5)) given above, ignition after passage of the shock wave is not observed at all. The limit corresponds to the so-called chain, third limit ((^2)). The thermal-explosion limit, calculated by D. A. Frank-Kamenetskii, lies in the region of lower temperatures and does not appear under these conditions.}})), it should be assumed that the chemical mechanism under these ignition conditions and under the conditions of ordinary kinetic experiments ((T = 750\text{–}880^\circ\ \mathrm{K},\ p = 10\text{–}760\ \mathrm{mm\ Hg

The effect of “breakdown” on passing through (p_2) is observed the less sharply, the lower the pressure region, which also corresponds to the chemical picture, since under these conditions the limiting curve (p_3) comes very close to the curve (p_2) (see Fig. 2).

The only quantitative discrepancy between the results presented in this paper and kinetic calculations based on known values of the reaction-rate constants is the magnitude of the ignition delays in the temperature interval between the curves (p_2) and (p_3). The measured values (\tau) are approximately 100 times smaller than the calculated ones (see Fig. 3, in which ((-\lg \tau)_{\mathrm{calc}}) is shown as a solid line). If, in order to explain these data, one assumes that the constant (k_8) has been determined inaccurately, then

Fig. 3. Calculation and experimental data on measurement of the self-ignition delay of the mixture (\mathrm{H_2} + \tfrac{1}{2}\mathrm{O_2}) at various pressures and temperatures.
(a) — (p = 1 = 1 \pm 0.35\ \mathrm{atm}); (b) — (p = 2 \pm 0.3\ \mathrm{atm}); 1 — pure mixture, 2 — admixture of 1% (\mathrm{CO_2}), 3 — treatment of the vessel walls with KCl; (v) — (p = 2.9 \pm 0.4\ \mathrm{atm}).

the introduction of the corresponding corrections would sharply change the picture of the limits, and the discrepancy between experiment and theory would become still greater.

It seems more probable that, in the temperature range under consideration, where ignition and the subsequent detonation have a milder, more diffuse character, the delay (\tau) may depend substantially on the specific conditions of the process in the gas heated by the shock wave, as a result of which the ignition mechanism may prove to be more complex. As one of the possible hypotheses, one may suppose that the difference from ordinary thermal reactions consists in the fact that, in our case, vibrationally excited molecules ((\mathrm{O}_2)^*), whose relaxation times are of the order of (10^{-4})—(10^{-3}) sec. (¹), may play a certain role in the course of the process. These molecules can enter into the reaction

[
\mathrm{H}+\mathrm{O}_2^* \to \mathrm{OH}+\mathrm{O}
\tag{1}
]

practically without activation energy. If it is assumed that at very high temperatures (above the temperatures corresponding to intersection with the curve (p_2) at the given pressure) (k_2) is sufficiently large, then, in view of the large values of ([(\mathrm{O}2)/(\mathrm{O}_2)^*]), the contribution of reaction (1) to the equation for (\tau) will be insignificant and the position of (p_2) will remain practically unchanged, while the quantities ((t)) sec. ((T}}) and ((\tau)_{\mathrm{calc}}) at (T \gg T_2) will coincide. With a sharp change in rates after passing through (p_2), the rate of reaction (1) may become substantial, which will lead to a decrease in (\tau) in comparison with the calculation made without taking it into account. With a comparatively small change in (\tau) in the interval from (10^{-4}) to (10^{-3