Journal of Engineering Physics and Thermophysics, Vol. 87, No. 5, September, 2014

HEAT AND MASS TRANSFER IN COMBUSTION PROCESSES

NUMERICAL MODELING OF THE IGNITION OF HYDROGEN–OXYGEN MIXTURESUNDER NONEQUILIBRIUM CONDITIONS

G. Ya. Gerasimov and O. P. Shatalov UDC 541.126:127; 536.46

A numerical study has been made of the intensifi cation of combustion in hydrogen-oxygen mixtures with a low-tem-perature nonequilibrium plasma, when the concentration of the active particles in the mixture (atoms, radicals, ions, and excited particles) is much higher than their equilibrium concentrations. Primary emphasis has been placed on the infl uence of higher-than-average concentrations of electronically excited O2(a 1Δg) molecules and O( 1Δ) atoms on the acceleration of the process of ignition. Electron-beam irradiation of the fuel mixture and the action of a high-voltage nanosecond discharge on it were selected for comparison of the effi ciency of different types of plasma initiation of combustion.

Keywords: combustion, excited oxygen, kinetic mechanism, induction time, electron-beam irradiation, gas discharge.

Introduction. The design of hypersonic passenger airplanes requires that new effi cient engines operating on hy-drogen fuel be developed [1]. Since such engines are characterized by a short time of residence of the fuel mixture in the combustion chamber (of the order of 1 ms), practical implementation of the process of combustion under hypersonic-fl ight conditions calls for the improvement of the mixing of reactants, reduction in the ignition length, and increase in the fl ame stability [2]. Therefore, development of novel methods for intensifi cation of combustion in the H2–O2 mixture is attracting increased attention of researchers at present.

Both thermal and nonequilibrium methods are employed to accelerate the process of ignition of a fuel mixture. In the case of nonequilibrium excitation and formation of a low-temperature plasma the concentrations of active particles in the mixture (atoms, radicals, ions, and excited particles) substantially exceed their equilibrium concentrations. This is observed in la-ser-induced excitation [3], in electric discharges of various types [4], and also in electron-beam irradiation of the mixture [5].

A theoretical study of the ignition under nonequilibrium conditions calls for a detailed consideration of the physics, hydrodynamics, and chemical kinetics of the initiating process. Many of these phenomena are quite complicated and not completely understood. The involvement of excited and charged particles in the process of combustion offers a possibility for the new and effi cient reactions accelerating the ignition to appear. The particles that have attracted increased recent attention of the researchers are O2(a 1Δg) molecules [6]. These molecules are of low excitation energy (0.98 eV) and are formed in any oxygen-containing plasma, signifi cantly intensifying reactions of chain branching in combustion.

Despite the considerable progress made in the understanding of the ignition of the H2–O2 system under nonequilibri-um conditions, there are a great number of uncertainties in the kinetics of this process. In particular, rate constants of chemical reactions involving singlet oxygen O2(a 1Δg) in different kinetic models may differ by more than several orders of magnitude [7–9]. In the present work, we have made an effort to give a description most adequate as for now for the process of ignition of the mixture in question under nonequilibrium conditions based on an analysis of the available kinetic data.

Kinetic Mechanism of the Process. A numerical study of the process of ignition of the H2–O2 system under non-equilibrium conditions was carried out with the kinetic model [10] of combustion of the mixture in question, in which mol-ecules and atoms in the ground electron state are involved. This model was supplemented with the corresponding reactions involving electronically excited O2(a 1Δg) molecules and O(1Δ) atoms, which are of paramount importance in various appli-cations [11]. In fi nal form, the sets of reactions involving electronically excited O2(a 1Δg) molecules and O(1Δ) atoms is sup-plied in Table 1 describing both the chemical reactions of conversion of electronically excited components and the reactions

0062-0125/14/8705-1063 ©2014 Springer Science+Business Media New York 1063

Institute of Mechanics of M. V. Lomonosov Moscow State University, 1 Michurin Ave., Moscow, 119192, Russia; email: gerasimov@imec.msu.ru. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 87, No. 5, pp. 1022–1028, September–October, 2014. Original article submitted December 5, 2013.

DOI 10.1007/s10891-014-1108-z

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of quenching of these components on collision with other particles. All the reactions in the indicated model are reversible. The rate constants of back reactions are computed from the rate constants of forward reactions and from constants using the thermodynamic data of [22]. Reactions of dissociation and recombination involving different inert collision partners M are considered as individual reactions with corresponding rate constants.

TABLE 1. Parameters of the Reactions in the H2–O2 System Involving Electronically Excited Components of Oxygen

Reaction number Reaction M log A n E, K T, 103 K ∆ log k

Literature source

1 O2(a 1Δg)+H2=HO2+H 13.04 0 17475 0.3–2 [12]

2 O2(a 1Δg)+M=O2+M H2 12.01 0 3850 0.5–1 [7]

O2 6.26 0 200 0.1–0.5 0.2 [13]

N2 4.92 0 0 0.3 [13]

Ar 5.11 0 0 0.3 [14]

He 3.78 0 0 0.3 [15]

H2O 6.48 0 0 0.3 0.3 [13]

HO2 13.30 0 0 0.3 0.2 [16]

O, H 8.08 0 0 0.3 [17]

3 O2(a 1Δg)+H=OH+O 13.60 0 2530 0.3–1 0.1 [7]

4 O+O+M= O2(a 1Δg)+M O2 17.42 –1 0 0.3–5 0.4 [18]

N2 16.82 –1 0 0.3–5 0.4 [18]

H2O 17.22 –1 0 0.3–5 0.4 [18]

Ar 16.52 –1 0 0.3–5 0.4 [18]

O 17.98 –1 0 0.3–5 0.4 [18]

5 O2(a 1Δg)+ O2(a 1Δg)=2O2 7.73 0 560 0.3–0.5 [17]

6 2O2(a 1Δg)=O2+O2(b1Sg+) –3.38 3.8 –700 0.3–2 0.1 [19]

7 O2(a 1Δg)+O3=2O2+O 13.49 0 2840 0.3–0.4 0.1 [13]

8 O(1D)+H2=H+OH 13.86 0 0 0.2–0.4 0.1 [13]

9 O(1D)+O2=O+O2(a 1Δg) 11.98 0 –67 0.2–0.4 0.2 [13]

10 O(1D)+O2=O+ O2(b1Sg+) 13.19 0 –67 0.2–0.4 0.1 [13]

11 O(1D)+M=O+M O2 12.46 0 –67 0.2–0.4 0.1 [13]

N2 13.08 0 –130 0.2–0.5 0.05 [13]

Ar 11.48 0 0 0.3 [20]

H2 12.56 0 0 0.3 0.1 [13]

H2O 11.60 0 0 0.3 [13]

12 O(1D)+H2O=OH+OH 13.99 0 –60 0.3-0.4 [21]

13 O(1D)+H2O=H2+O2 12.12 0 0 0.3 [13]

14 O(1D)+O3=O+O+O2 13.86 0 0 0.1–0.4 0.2 [13]

15 O(1D)+O3=O2+O2 13.86 0 0 0.1–0.4 0.2 [13]

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One basic stage accelerating the ignition of the mixture in question is the reaction of initiation of the chain O2(a 1Δg) + H2 → H + HO2. The rate constant of the reaction reverse to it was measured in [12] and was used subsequently for description of measuring the velocity of propagation of a laminar fl ame [23]. In the recent work [24], this rate constant has been evaluated theoretically on the basis of G3 theory. At low temperatures, calculation results agreed with experimental data [12] and have been recommended in [25] for kinetic calculations. A smaller value of the indicated rate constant was obtained in [26] with the bond-energy method. Figure 1 compares the considered kinetic data and the rate constant of the quenching reaction O2(a 1Δg) + H2 → O2 + H2. The dependence of the quenching-rate constant on temperature has been obtained from the experimental data of [27]. We notice that the rate constant of the quenching reaction is much larger than the corresponding quantity for the initiation reaction. Nonetheless, the reaction of initiation involving O2(a 1Δg) is the leading reaction in the acceleration of the ignition of the mixture.

The second basic reaction accelerating the ignition of the hydrogen–oxygen mixture is the reaction of branching of the chain O2(a 1Δg) + H → O + OH. The rate constant of the overall reaction, which also includes the mechanism of quench-ing, was measured in [28, 25] at temperatures of 300–431 K and 520–930 K respectively. As follows from the results obtained in [28], the reaction between the electronically excited molecule O2(a 1Δg) and the hydrogen atom proceeds mainly by the mechanism of splitting of the O atom from the molecule. Approximation of the results of measuring the rate constant of this reaction, performed in [7], is entered in Table 1 and is shown in Fig. 1.

Rate constants of the recombination reaction O + O + M → O2(a 1Δg) + M for different inert collision partners M have been obtained using the method of [30], which is based on the assumption that the probability of formation of an excited molecule is proportional to the multiplicity of degeneracy of the corresponding electron state. In this case the rate constant of the reaction of formation of O2(a 1Δg) is equal to the product of the general rate constant, which has been taken from [18] in the case in question and for a multiplicity of degeneracy of the state a 1Δg of 0.33.

Calculation Results and Their Discussion. A computational procedure is based on the well-known application package CHEMKIN [31]. The calculations have been carried out in the approximation of an adiabatic process at constant pressure, which enjoys wide application in modeling the process of ignition of fuel mixtures [18].

The infl uence of O2(a 1Δg) additions on the increase in the velocity of propagation of the fl ame in the H2–O2 mixture has been studied in the recent experiment [32], where use was made of a method based on the organization of steady-state combustion in a low-pressure fl ow reactor to whose inlet one fed gaseous hydrogen and oxygen. Oxygen molecules were excited to a singlet state by the glow electric discharge in a cell placed in the O2-feed line. The experiments have shown that the presence of even a small number of excited oxygen molecules in the discharge-activated fl ux (of the order of several percent) enables one to considerably reduce the length of the induction zone and to simultaneously effect ignition at a lower temperature.

Fig. 1. Rate constants of basic reactions involving O2(a 1Δg): 1 and 2) theoretical esti-mates of [24] and [26]; 3) calculation from the data of [12]; 4 and 5) approximation of experimental data in [7]; symbols, experimental data.

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A numerical 2D analysis of all experimental stages has been made in [33], where a new kinetic scheme of reactions involving O2(a 1Δg) molecules and based on the matching of the calculated data to experimental ones was proposed. Figure 2 compares the dependence of the induction time on the content of O2(a 1Δg) in the mixture, calculated using the proposed kinetic scheme, and the corresponding calculation from the kinetic scheme of [33] for conditions similar to those of the ex-periment in [32]. We notice that in the region O2(a 1Δg) ≥ 1% where the decisive role in initiating ignition is played by the O2(a 1Δg) molecules, the results of calculating from the two kinetic schemes differ little. The major difference is observed at low concentrations of O2(a 1Δg), where the process is initiated by nonexcited components. As the performed analysis shows, this difference is attributed to the use, in [33], of a rate constant of the initiation reaction H2 + O2 → OH + OH several orders of magnitude larger than the corresponding value adopted in the present work. At equal values of the rate constant of this reaction, both kinetic schemes yield virtually identical results.

The agreement of the results of calculating by the two kinetic schemes, one of which has been obtained in [33] from the matching of the calculated data to experimental ones, allows the assumption that the second kinetic scheme pro-posed in the present work must also correctly describe experimental data [32]. Nonetheless, the kinetic scheme of [33] has the abnormally low value of the rate constant of the reaction of quenching of O2(a 1Δg) by H2 molecules compared to the corresponding value adopted in the present work. Agreement with the experimental data of [32] was attained in [33] by decreasing the rate constant of the reaction O2(a 1Δg) + H → O + OH (used as the adjustable parameter) by one order of magnitude.

The reaction of quenching of O2(a 1Δg) by H2 molecules is of paramount importance in the kinetic scheme proposed in the present work. Figure 3 shows the dependence of the mixture′s temperature on time, which has been calculated for conditions close to those of the experiment in [32]. We notice that the induction time at O2(a 1Δg)/O2 = 0 is equal to 42 ms. Account taken of the quenching reaction for the case of O2(a 1Δg)/O2 = 6% leads to rapid growth of 50 K in the temperature, which in turn reduces the induction time to 23 ms. Computation from the complete kinetic scheme, which includes all the reactions involving O2(a 1Δg), repeats the change in the temperature at the initial stage of the process, but the induction time becomes much shorter. This fact is attributed to the different behavior of the active components in these cases. The concen-trations of the O2(a 1Δg) molecules at the initial stage of the process are equal in both cases, but the presence of the reaction of branching of the chain O2(a 1Δg) + H → O + OH in the complete kinetic scheme leads to a much high concentration of the H atoms and other active components, which accelerates ignition.

The excited atoms O(1D) can also make a signifi cant contribution to the initiation of ignition. They are formed in the oxygen-containing mixture, in particular, on irradiating it with electrons [34]. Kinetic data on chemical reactions involving

Fig. 2. Delay time of ignition in the O2–H2 mixture vs. content of O2(a 1Δg) at p = 0.0132 atm, T0 = 775 K, H2 = 71.4%, and O2 = 28.6%: solid curve, calculation by the proposed kinetic scheme: dotted curve, calculation from the kinetic scheme of [33].

Fig. 3. Infl uence of different reactions of quenching on the behavior of the temperature of the O2–H2 mixture at p = 0.0132 atm, T0 = 775 K, H2 = 71.4%, and O2 = 28.6%: 1) O2(a 1Δg)/O2 = 0, 2) O2(a 1Δg)/O2 = 6% with account of just the quenching of O2(a 1Δg) molecules, and 3) O2(a 1Δg)/O2 = 6% calculations using the complete kinetic scheme.

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O(1D) atoms are quite limited and belong mainly to the atmospheric chemistry. Since O(1D) is a metastable particle with a radiation lifetime of the order of 100 s [35], its loss in the gas is controlled by collisions with other particles. Figure 4 shows the effect of reduction in the induction time from the addition of O(1D) and compares it to the corresponding effect from the action of O2(a 1Δg). The calculation has been performed for a hydrogen–air mixture under the assumption that N2 is an inert component. We notice that the addition of 6% O2(a 1Δg) is equivalent to adding nearly 0.01% O(1D). Nonetheless, analogous computations with adding O(3P) instead of O(1D) yield the same results. This points to the rapid quenching of O(1D) fol-lowed by the initiation of combustion in chemical reactions involving O atoms in the ground state.

Under actual conditions, physical action on a fuel gas mixture gives rise to all kinds of active particles (atoms, radi-cals, ions, and excited particles) in it. Low-temperature nonequilibrium plasmas formed by different methods have different compositions. Therefore, it seems of interest to compare the effi ciencies of different types of plasma initiation of combustion. For this purpose we selected the electron-beam irradiation of a hydrogen–oxygen mixture and the action on it by a high-volt-age nanosecond discharge. In the fi rst case we used the kinetic model [38] that considers a continuous action of radiation on the mixture. In the second case the composition of the mixture and its temperature after the discharge were used as the initial data for further computations [39].

The formation of active components in the model of [38] is described with primary radiation yields Gij which de-termine the number of particles of the intermediate component j generated by the macrocomponent i in absorbing 100 eV energy. The quantities Gij are virtually independent of the kind of ionizing radiation, the energy of its particles, and the density of their fl ux. The overall radiant yield of the active component j is determined by the expression Gj = SjGijxi. Numerical val-ues of the quantities Gij for the hydrogen–air mixture that have been obtained by analysis of the available experimental data[34, 40, 41] are equal to

2 2

12 2

22 2

5.6H 2.4H 0.4H 2.8e 6.0H ,

5.3O 2.1O 1.2O 3.3e 2.8O 2.4O( D) ,

5.4N 2.3N 0.7N 3.0e 3.1N 2.4N( D) .

+ +

+ +

+ +

→ + + +

→ + + + +

→ + + + +

Fig. 4. Infl uence of O2(a 1Δg) and O(1D) additions on the reduction in the induction time in the stoichiometric hydrogen–air mixture at p = 1 atm: 1) calculation without the ignition initiation, 2) calculation with the 6% O2(a 1Δg) added, and 3) calculation with the 0.01% O(1D) added; symbols, experimental data for the mixture without initiating additions [36, 37].

Fig. 5. Comparison of different kinds of initiation of ignition in the stoichiometric hy-drogen–oxygen mixture at p = 1 atm and W = 4·10–3 J/cm3: 1) self-ignition (W = 0), 2) thermal heating, 3) action of a gas discharge [39], and 4) electron irradiation; symbols, experimental data for self-ignition from [36, 37].

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Such representation of the radiation action of the mixture contains a number of simplifi cations. In particular, the dis-sociative state O2(B3 Su

− ) having a high radiation yield is considered as the state forming O atoms directly. The source term in the kinetic equations which describes the formation of active components on exposure to ionizing irradiation is equal to Si = 1.036·10–10ρIGi. We assume the observance of the principle of kinetic independence of the elementary stages of secondary chemical reactions and the homogeneity of radiation-energy absorption.

Numerical calculations of the ignition of the hydrogen–air mixture on exposure to ionizing radiation have been performed as applied to the conditions of high-voltage nanosecond discharge: p = 1 atm and W = 4·10–3 J/cm [39]. The absorbed radiation dose D is defi ned as D = 3.92·103 WT/p in this case. The results of these calculations and their comparison to the corresponding calculations for the mixture′s self-ignition in thermal heating at a prescribed value of W, and ignition under the action of the gas discharge are given in Fig. 5. An analysis of this fi gure shows that both methods of initiation of ignition yield nearly identical results at equal absorbed energies. Appreciable differences only appear at low temperatures (T ≤ 900 K) where continuous irradiation of the mixture is more effi cient. The delay time of ignition of the fuel mixture in the case of thermal action, when all degrees of freedom are heated equally, differs little from the time of self-ignition delay. This demonstrates that the nonequilibrium excitation of the gas particles is more effi cient at a prescribed value of the absorbed energy.

Conclusions. We have proposed a kinetic mechanism of initiation of ignition in the H2–O2 system, which includes chemical reactions involving electronically excited O2(a 1Δg) molecules and O(1D) atoms. It has been shown that the pres-ence of even a small number of the excited oxygen molecules in the fuel mixture (of the order of several percent of the O2 concentration) enables us to considerably reduce the ignition-delay time. The excited O(1D) atoms also make a signifi cant contribution to the initiation of ignition. As the calculations have shown, the addition of 0.01% O(1D) is equivalent to adding 6% O2(a 1Δg). Nonetheless, analogous calculations adding O(3P) instead of O(1D) yield the same results, which points to the rapid quenching of O(1D) followed by the initiation of combustion in chemical reactions involving O atoms in the ground state.

Under actual conditions, physical action on the fuel mixture gives rise to all kinds of active particles (atoms, radicals, ions, and excited particles). To compare the effi ciency of various types of plasma initiation of combustion, we have considered electron irradiation of the hydrogen-air mixture and the action of a high-voltage nanosecond discharge on it. It has been shown that both methods of initiation of ignition yield nearly identical results at equal absorbed energies.

NOTATION

A, preexponential factor in the reaction-rate constant, (cm3/mole)m–1·s·K–n; D, absorbed radiation dose, J/kg; E, activation energy of the reaction, K; Gij, radiation-chemical yield of the jth component under the radiation action on the mac-rocomponent i of the gas, particles/100 eV; Gj, overall yield of the jth component of the gas, particles/100 eV; I, irradiation intensity, J·kg–1·s–1; k = ATn exp (–E/T), rate constant; ki, rate constant of the ith reaction, (cm3/mole)m–1·s–1; m, order of reaction; n, exponent on the temperature in the reaction-rate constant; p, pressure, atm; Si, source term in the kinetic equation for the ith gas component, mole/(cm3·s); t, time, ms; T, temperature, K; W, absorbed energy, J/cm3; xi, mole fraction of the ith gas component; Δ log k, error; ρ gas density, g/cm3; τ, ignition-delay time (induction time), ms. Subscripts: i, component or reaction number; j, component number; 0, initial value of the parameter.

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