Vibrational non-equilibrium in the hydrogen–oxygen reaction. Comparison with experiment

Abstract

A theoretical model is proposed for the chemical and vibrational kinetics of hydrogen oxidation based on consistent accounting of the vibrational non-equilibrium of the HO₂ radical that forms as a result of the bimolecular recombination H+O₂ → HO₂. In the proposed model, the chain branching H+O₂ = O+OH and inhibiting H+O₂+M = HO₂+M formal reactions are treated (in the terms of elementary processes) as a single multi-channel process of forming, intramolecular energy redistribution between modes, relaxation, and unimolecular decay of the comparatively long-lived vibrationally excited HO₂ radical, which is able to react and exchange energy with the other components of the mixture. The model takes into account the vibrational non-equilibrium of the starting (primary) H₂ and O₂ molecules, as well as the most important molecular intermediates HO₂, OH, O₂(¹Δ), and the main reaction product H₂O. It is shown that the hydrogen–oxygen reaction proceeds in the absence of vibrational equilibrium, and the vibrationally excited HO₂(v) radical acts as a key intermediate in a fundamentally important chain branching process and in the generation of electronically excited species O₂(¹Δ), O(¹D), and OH(²Σ⁺). The calculated results are compared with the shock tube experimental data for strongly diluted H₂–O₂ mixtures at 1000 < T < 2500 K, 0.5 < p < 4 atm. It is demonstrated that this approach is promising from the standpoint of reconciling the predictions of the theoretical model with experimental data obtained by different authors for various compositions and conditions using different methods. For T < 1500 K, the nature of the hydrogen–oxygen reaction is especially non-equilibrium, and the vibrational non-equilibrium of the HO₂ radical is the essence of this process. The quantitative estimation of the vibrational relaxation characteristic time of the HO₂ radical in its collisions with H₂ molecules has been obtained as a result of the comparison of different experimental data on induction time measurements with the relevant calculations.

Keywords:

• gas phase reacting mixture
• chemical kinetics
• vibrational relaxation
• vibrationally excited HO₂ radical
• electronically excited species

Notes

1. The use of the formal kinetic description was and remains relevant in terms of its use in macro-kinetic applications.

2. According to existing concepts (see, for example, [4]), if molecules or radicals AB are 2-atomics (O₂, H₂, OH, …), such a reaction proceeds in the two following elementary steps: forming vibrationally excited AB(v) and its subsequent relaxation to an equilibrium distribution of vibrational states (in this case, by deactivation in collisions). If the AB are three (or more) atomics (HO₂, H₂O, H₂O₂, …), such a reaction proceeds in the three following elementary steps: forming vibrationally excited AB(v), intramolecular energy redistribution, and deactivation in collisions. The vibrational relaxation times, τ _vib, are directly dependent on conditions; so the value of the rate constant of the vibrational-to-translational energy exchange process is proportional to pressure, p, and depends on the buffer gas type, M. If, in the course of the reaction, an equilibrium distribution of vibrational states is maintained (the ‘high pressure limit’), i.e. τ _vib ≪ τ _ch (where τ _ch is the chemical reaction characteristic time), the equilibrium approximation is correct, the reaction rate of recombination (dissociation) is determined by the rate of its first stage, independent of p or M, and the recombination rate constant is defined by the value of k_∞ (the dissociation rate constant, k′_∞ = k_∞/K). In the opposite case, i.e. when τ _vib ≫ τ _ch, (the ‘low pressure limit’) the reaction rate is determined by the rate of the vibrational relaxation stage, and the relationship k/k′ = K(T) is invalid. Really in practice, an intermediate case is realized; the reaction rate constant and efficiencies of third-bodies, M, in the framework of the formal kinetics (on the basis of an assumed mechanism), is searched for empirically as a result of fitting to the experiment, and the relationship k/k′ = K(T) is used.

3. Such as the inability of the equilibrium kinetics concept to predict the efficiencies of different third bodies, M, in recombination–dissociation reactions; the possibility of the formation of electronically excited states O₂(¹Δ), O(¹D), OH(²Σ⁺) and reactions with their participation; and the complex multi-channel character of quenching processes, including chemical quenching processes (see [2]).

4. That is, there is an overlap of the HO₂(v) lifetime and collision time statistical distributions.

5. In terms of non-equilibrium statistical mechanics, these equations are the corresponding moments of the master equations (for details, see [25]).

6. In this context, non-empirical sources of kinetic information, such as ab initio analysis of PESs and dynamic calculations of equilibrium rate constants of elementary reactions, become of particular importance.

7. In the simplest case of single-quantum vibrational-to-translational (VT) transfer, one has the relation $P_{m,m-1}\equiv P\left\{\begin{array}{c}m,m-1\\ n,n\end{array}\right\}=m{P}_{10},P_{10}\equiv P\left\{\begin{array}{c}1,0\\ 0,0\end{array}\right\}.$

8. The ‘stoichiometric part of the combustible mixture’ [33] was not more than 3%.