ISSN 19907931, Russian Journal of Physical Chemistry B, 2013, Vol. 7, No. 2, pp. 154–160. © Pleiades Publishing, Ltd., 2013. Original Russian Text © S.A. Gubin, V.A. Shargatov, 2013, published in Khimicheskaya Fizika, 2013, Vol. 32, No. 4, pp. 80–86.
COMBUSTION, EXPLOSION, AND SHOCK WAVES
An Efficient Approximate Method for Solving the Problem of the Establishment of Chemical Equilibrium in the Products of Explosion of Gas Mixtures S. A. Gubin and V. A. Shargatov National Research Nuclear University “MEPhI”, Moscow, Russia email:
[email protected] Received March 26, 2012
Abstract—A method for calculating the changing composition of the explosion products in the case where the chemical equilibrium is absent but the bimolecular reactions are in quasiequilibrium is developed. At each time step of numerical integration, the change in the total number of molecules in the system is calcu lated using a single differential equation written based on the selected kinetic mechanism. Then, the mixture composition is calculated under the assumption that the entropy of the mixture reaches its maximum value at a given internal energy, mass density, and molar mass. It is shown that the proposed method can be used to calculate the characteristics of explosive transformation processes after the induction period. Keywords: chemical kinetics, chemical equilibrium, explosion products, the numerical simulation of reacting flows, stiff system of equations DOI: 10.1134/S1990793113020139
The chemical reactions that occur in the explosion products change their composition and, consequently, thermodynamic parameters. During cooling, the internal energy and temperature change, whereas dur ing expansion, the temperature, internal energy, and specific volume change. The changed thermodynamic parameters correspond to a different chemical equilib rium composition of the mixture. The time evolution of the composition is calculated using a stiff system of chemical kinetics equations. This is the most general approach. At a large number of individual species in the explosion products and a large number of chemical reactions involved, calcula tion of the mixture composition requires a long com putation time. Typically, the time it takes for calculat ing the mixture composition is much longer than that spent on solving the gas dynamics equations. However, one can do without solving the system of stiff differential equations by using one of the two extreme models of composition change. The first model assumes that the composition of the explosion products remains constant while their thermodynamic parameters change, whereas the second model postu lates that the mixture exists in the state of chemical equilibrium at any point in time. The second model is applicable if the chemical reactions occur so rapidly that the deviation from the equilibrium composition at any moment of time is negligibly small. If the calculation results within the framework of these two models differ insignificantly, then the
change in the composition can be ignored and con sider it constant. However, this is often not the case. Here is an example. Consider the detonation prod ucts of a stoichiometric hydrogen–oxygen mixture. Let the initial mixture be kept at a pressure of 1 bar and a temperature of 298 K. Let the composition of the detonation products at the given time be as specified in Table 1. This composition is chemically equilibrium at a density of 0.9028 kg/m3 and a temperature of 3682 K. The specific heat at constant volume calculated on the assumption that the mixture composition remains constant is 2719 J kg–1 K–1. The same specific heat cal culated for the chemical equilibrium mixture is 13298 J kg–1 K–1. That is, the values of the specific heat for the two limiting models differ almost by a fac tor of 5. Let now the cooling of the mixture be character ized by a heat loss of 5000 kJ kg–1. For simplicity, we assume that density of the mixture remains unchanged in the process of cooling. The characteristics of cooling at constant composi tion are given in the “Frozen” row in Table 1, whereas those belonging to cooling under chemical equilib rium conditions, in the “Equilibrium 2” row. Table 1 shows that the constantcomposition model predicts a temperature of 1691 K, whereas the chemicalequilib rium model, 3229 K. In these cases, both the density and specific internal energy are identical. The pressure in the mixture calculated under the assumptions of constant composition and chemical equilibrium are
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Table 1 Mixture composition, kmol/kg mixture H2
H2O
H2O2
HO2
O2
OH
H
O
ρ, μ, T, K P, MPa kg/m3 kg/kmol
U, kJ/kg 723.2
Equilibrium 1 0.0111
0.03698 1.46E–06 1.40E–05 0.0033
0.0092
0.00556 0.0026
0.9028 3682 1.904
14.51
Frozen
0.03698 1.46E–06 1.40E–05 0.0033
0.0092
0.00556 0.0026
0.9028 1691 0.87434
14.51
–4276
0.04643 7.82E–07 5.58E–06 0.002
0.0045
0.0015
0.9028 3229 1.488
16.35
–4276
0.0111
Equilibrium 2 0.0061
0.874 and 1.488 MPa, respectively. Thus, the two lim iting models give the results so different that it is worth spending effort on correctly describing changes in the composition of the explosion products caused by changes in their thermodynamic parameters. We performed thermodynamic calculations using our own thermodynamic code, which is part of a soft ware package for calculating gasdynamic flows. The equations used to calculate the composition and ther modynamic parameters are given in the following sec tion. The applicability of the chemical equilibrium mix ture model to the products of explosion of fuel–oxy gen and fuel–air mixtures was discussed in detail in [1–6]. The applicability of the chemical equilibrium model to hydrogen–oxygen mixtures in the case of heat supply at constant density was assessed in [1]. In [2, 5], a sufficient condition for the applicability of the chemical equilibrium mixture model to gasdynamic calculations was obtained. The authors of [6] com pared the results of gasdynamic calculations per formed using the system of equations of chemical kinetics for calculating the composition of the explo sion products with the results of gasdynamic calcula tions within the framework of the chemical equilib rium mixture model and the constant composition mixture model. The chemical equilibrium mixture model has been previously used in gasdynamic calculations, see, e.g., [7–9]. The authors of [1] proposed an approximate model for calculating the equilibrium composition of the explosion products and estimated the time it takes to restore equilibrium by using an approximate kinetic equation [10]. It was demonstrated [2–6] that the process of establishment of chemical equilibrium in the products of explosion of gas mixtures proceeds in several stages. First, partial chemical equilibrium is established, without changes in the total number of molecules in the system. This is due to the fact that the rapid bimo lecular reactions that ensure the establishment of such partial equilibrium occur much faster than the dissoci ation and recombination reactions, which are respon sible for establishing full chemical equilibrium. That is, even in cases where there is no chemical equilib rium, it suffices to know the molar mass of the mixture RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
0.0007
and two required thermodynamic parameters (for example, temperature and density) to determine the composition of the mixture. The existence of quasiequilibrium in exchange reactions was discussed in [11]. The authors of [11] used this assumption to perform approximate calcula tions of the composition of a hydrogen–oxygen mix ture. This same assumption underlies an approximate model for calculating the molar mass of gas mixtures at high temperatures [12]. The author of [10, 12] pro posed an approximate kinetic equation for calculating the molar mass of the mixture and then the thermody namic parameters. No method for calculating the mixture composition has been proposed. This method is described in the present paper. The aim of the present work is to develop a method for calculating the changing composition of the explo sion products in the case where there is no chemical equilibrium, but the bimolecular reactions are in equi librium. According to [1–7, 10–12], such a method should include only one differential equation for cal culating the change in the number of molecules per unit mass of the mixture due to the reactions of recom bination and dissociation. This equation is used instead of dozens of equations of chemical kinetics. In the present work, we propose to supplement this equa tion with an algebraic equation for calculating the equilibrium composition at given values of the internal energy, density, and molar mass. The equation is derived based on the characteristic function extremum method. To verify the proposed method, its predictions were compared with the results of integration of the com plete system of chemical kinetics equations for model problems. At present, such kinetic calculations for simple mixtures of hydrocarbons and oxygen are in good agreement with experimental data. Formally, the method with one kinetic equations for the number of molecules per kg of mixture is simi lar to that proposed in [10, 11], which uses a single equation for calculating the change in the molar mass. However, the approach to calculating the righthand side of the equation, i.e., the rate of change of the given quantity is different. In [10, 12], the rate of change of the molar mass was calculated using an Vol. 7
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approximate equation containing some constants, specific for each initial composition of the mixture. In the present work, the rate of change of the num ber of molecules in the mixture is calculated using a selected set of elementary chemical reactions. In cal culating the rate of change of the number of molecules in the mixture, a detailed composition of the latter is used, calculated on the assumption that, for given molar mass, density, and internal energy, the system has maximum entropy. That the number of molecules in the system changes is the result of the dissociation and recombination reactions, which have much lower rates than the bimolecular reactions that control the process of establishing a local chemical equilibrium without changing the number of particles in the sys tem. In other words, as the number of molecules in the system changes, the chemical composition of the mix ture very quickly becomes chemically equilibrium for the available number of molecules in the system at the given time. Thus, a detailed kinetic mechanism is used, but without the need to solve the stiff system of equations. The core of our approach is to replace the complete system of chemical kinetics equations of by an equiva lent system composed of a differential and a few alge braic equations. This physically sound approach reduces the computation time. What we offer is not an approximate kinetic equation, but an approximate method for solving the complete system of equations of chemical kinetics. This method is always consistent with the detailed mechanism and can be used together with solution of the stiff rigid system, replacing it at the stage when bimolecular reactions are in quasiequilib rium state. The proposed method is based on a physi cally reasonable assumption that equilibrium in the bimolecular reactions is established much faster than the full chemical equilibrium, being applicable when this assumption holds. This paper is based on the assumption that, under certain conditions, the systems of kinetic equations can be solved as follows. At each time step of numeri cal integration, the change in total number of mole cules in a system is calculated using the single differen tial equation based on the chosen kinetic mechanism. Then, the mixture concentration is calculated assum ing that the entropy of the mixture reaches its maxi mum value at a given internal energy, density, and number of molecules per unit volume.
The entropy of a mixture of ideal gases with known dependence of the heat capacity of the temperature is reads as k
S=
∑ (S
− R ln ( RTM i v P0 )) *M i ,
i
(1)
i =1
where k is the number of individual compounds in the mixture, Mi is the number of moles of the ith substance per 1 kg of mixture, R is the universal gas constant, T is the temperature of the mixture, v is the specific internal volume of the mixture, Si is the standard entropy, and P0 = 1 atm. The internal energy of the mixture is a preset value. This condition leads to the equation k
U0 −
∑U M i
i
= 0,
(2)
i =1
where U0 is the preset specific internal energy of the mixture, and Ui is the internal energy of one mole of mixture component i at a given temperature T. In addition, the amount of each chemical element in the mixture is also known, since it is determined by the initial composition of the mixture, remaining unchanged during chemical reactions. For chemical element j, k
∑n M ij
i
− n j = 0; j = 1,..., m.
(3)
i =1
Here, m is the number of different chemical elements in the mixture, nj is the number of kgatoms of the jth element in 1 kg of mixture, nij is the number of atoms of the jth element in a molecule of the ith substance. If the number of molecules in the system is con stant, then the concentrations obey the equality k
∑M
i
− M 0 = 0,
(4)
i =1
where M0 is the total number of moles in 1 kg of mix ture. Following the Lagrange procedure, we write the function k
L=
∑ (S
i
− R ln ( RTM i v P0 )) M i
i =1
⎛ k ⎞ (5) ⎜ nij M i − n j ⎟ λ j ⎜ ⎟ ⎠ j =1 ⎝ i =1 k k ⎛ ⎞ + ⎜U 0 − U i M i ⎟ λ u + ( M i − M 0 )λ µ, ⎜ ⎟ ⎝ ⎠ i =1 i =1 where λj, λu, and λµ are the Lagrange multipliers, which, along with Mi and T, are the independent vari ables of the Lagrangian. At equilibrium, the derivatives m
EQUATION FOR CALCULATING THE CHEMICAL EQUILIBRIUM COMPOSITION OF THE MIXTURE AT CONSTANT MOLAR MASS Let us apply the wellknown principle that the chemical equilibrium system has the maximum entropy for given specific internal energy and density to a mixture consisting only of gases.
+
∑∑
∑
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∑
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AN EFFICIENT APPROXIMATE METHOD FOR SOLVING THE PROBLEM
of the Lagrangian with respect to the independent variables must be equal to zero. This condition allows us to write k + m + 2 algebraic equations for k + m +2 unknowns: m
S i − R ln ( RTM i v P0 ) +
∑n λ ij
j
− H i T + λ µ = 0;
j =1
(6)
i = 1,..., k, k
∑n M ij
i
− n j = 0; j = 1,..., m,
(7)
i =1
k
U0 −
∑U M
= 0,
(8)
− M 0 = 0.
(9)
i
i
i =1
k
∑M
i
where ωi is the rate of consumption and the formation of species i per unit volume; cj are the molar concen trations of the mixture components; μi is the molar mass of species i; kj is the reaction rate calculated according to the law of mass action, χi is the chemical formula of species i participating in the jth reaction; ν 'ij and ν ''ij are the stoichiometric coefficients of the ith product or reactant, respectively in the jth chemical reaction; K cj is the equilibrium constant; T is the tem perature; k jf is the rate constant of the jth reaction in the forward direction; Aj is the preexponential factor, nj is the temperature exponent; Eaj is the activation energy; and J is the number of reactions in the kinetic scheme. Let с denote the total number of moles of all mix ture components per unit volume. Then,
( )
∂c = ∂t
i =1
This system of algebraic equations has no analytical solution, so it is solved approximately by the Newton method. Note that excluding Eq. (9) from system (6)– (9) and setting λµ = 0 leads to a system of equations corresponding to the state of chemical equilibrium without assuming the constancy of the molar mass, i.e., to the traditional UV problem. This enables to readily use the above equations and the respective computer code for both determining the chemical equilibrium composition and solving the problem of partial equilibrium (i.e., at constant molar mass). To calculate Ui, Si, and enthalpy Hi at a given tempera ture, we used polynomials from [13]. EQUATION FOR CALCULATING CHANGES IN THE NUMBER OF MOLECULES IN THE SYSTEM When the chemical kinetics equations are applied to calculating the composition of the mixture, the fol lowing a system of equations is used
∂ci = ωi , ∂t N
∑
N
ν 'ij χ i
i =1
∑ ν''χ ; ij i
j = 1,..., J ,
i =1
J
ωi =
∑ k (ν'' − ν' ); i = 1,..., N, j
ij
ij
j =1
kj =
k jf
⎛ ⎜ ⎜ ⎝
N
∏ i =1
N
ν' ci ij
− 1c Kj
∏c i =1
ν "ij i
⎞ ⎟ ; j = 1,..., J , ⎟ ⎠
n ⎛ E ⎞ f k j = A jT j exp ⎜ − a j ⎟ , ⎝ RT ⎠
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J
∑ω . i
(10)
i =1
Formally, in (10), the summation is over all reactions. However, bimolecular reactions do not change the number of molecules, so the summation can be restricted to the recombination and dissociation reac tions. This makes redundant the question of the valid ity of the rate constants of a lot of reactions. These bimolecular reactions have no effect on the molar mass of the mixture, and therefore, there is no point in ensuring a high accuracy of their rate constants. Given the above, Eq. (10) can be recast as
( )
∂c = ∂t
J
∑ω , i
(11)
i =1
where it is assumed that the reactions that change the number of molecules in the system have numbers from 1 to J, the number of recombination reactions in the mechanism. The number of moles per kg of mixture is given by (12) M 0 = c ρ. Now, to calculate how the composition of the explo sion products changes, we can use a single differential equation (11) together with algebraic equations (6)– (9), (12) instead of a stiff system of differential equa tions of chemical kinetics. Differential equation (11) and Eq. (12) make it possible to determine the molar mass of the mixture, and then by solving the algebraic system of equations (6)–(9), to calculate the mixture composition. Calculations are performed at each time step of solution of differential equation (11). CALCULATION RESULTS The proposed approximate method for solving stiff systems of kinetic equations was applied to a model problem. Consider the chemical equilibrium explo Vol. 7
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Table 2 Mixture composition, vol % H2O
H2O2
HO2
Equilibrium 1 15.58
56.07
0.0013
0.01404
Frozen
15.58
56.07
0.0013
0.01404
8.82
79.26
0.00071 0.00552
H2
Equilibrium 2 Partial equi librium
11.70
O2
16.5 16.0
U, kJ/kg
7.57
3.51
0.4847
3505
0.9602
14.71
–206
4.81 12.43
7.57
3.51
0.4847
1476
0.4098
14.71
–5206
2.96
2.00
0.84
0.4847
3050
0.7405
16.60
–5206
3.10 13.64
2.95
0.4847
1636
0.4482
14.71
–5206
6.12
2 3
μ, kg/kmol
4.81 12.43
kg/kmol
1
P, MPa
O
sion products of a hydrogen–oxygen stoichiometric mixture. The parameters of the mixture given in the “Equilibrium 1” row of Table 2 correspond to the so called instantaneous explosion of the initial mixture. The composition is calculated for an initial mixture at 0.1 MPa and 298 K. Reducing the specific internal energy of the mix ture by 5000 kJ/kg at constant volume and constant composition decreases its pressure and temperature. This thermodynamic state is presented in the “Fro zen” row of Table 2. Later, at constant volume and constant internal energy, the mixture relaxes to the state of chemical equilibrium (the “Equilibrium 2” row of Table 2). The characteristics of the process of establishment of chemical equilibrium were calculated using the mechanisms presented in [14, 15]. The mechanism from [14] consists of 15 exchange reactions and 6 recombination reactions, whereas the mechanism presented in [15] includes 12 exchange reactions and 6 recombination reactions. It was assumed that the density and specific internal energy of the mixture
17.0
T, K
H
61.59 1.41E–09 9.04E–07 7.01
OH
ρ, kg/m3
4
15.5 15.0 14.5 0.E+00 1.E–05 2.E–05 3.E–05 4.E–05 5.E–05 t, s Fig. 1. Time history of the molar mass of the mixture dur ing the establishment of chemical equilibrium: (1) Gear method and mechanism from [14], (2) proposed method and mechanism from [14], (3) Gear method and mecha nism from [15], and (4) proposed method and mechanism from [15].
remains constant, i.e., expansion and heat loss are absent. The calculations were performed by solving the system of equations of chemical kinetics correspond ing to the selected mechanism with the Gere method [16]. It was necessary to solve a stiff system of nine dif ferential equations (8 for the composition and one for the temperature of the mixture). In addition, we conducted calculations within the framework of the proposed method. In this case, a sin gle differential equation for the number density of the mixture and a system of 11 algebraic equations are solved for six recombination–dissociation reactions to calculate the molar concentrations. The bimolecular reactions are disregarded. If a partial equilibrium is established for the bimolecular reactions, the quasi equilibrium composition of the products is exactly the same as the composition of the mixture resulting from the occurrence of the bimolecular reactions of this kinetic mechanism. At each step of numerical integra tion of the differential equation, the system of alge braic equations is solved. With the problem solved by the Newton method, a fast convergence is observed if the initial approximation values are set equal to the values of the composition and temperature of the mix ture at the previous step of integration. Figure 1 shows the time evolution of the molar mass of the mixture. The calculation was performed until the state of the mixture was close to the equilib rium. For both kinetic mechanisms, a good agreement between the results obtained by the proposed method and the Gear method. For the kinetic mechanism from [15], curves 3 and 4 in Figs. 1 and 2 merge. Figure 2 shows the dependence of the temperature of the mixture on the time. It is seen that, for both kinetic mechanisms, the proposed approximate method (curves 2 and 4) also provides good accuracy in calculating the temperature, with the results being consistent with those obtained from the full kinetic calculation (curves 1 and 3). Figure 3 provides a closer look at the temperature change in the initial stage. The proposed method is based on the assumption that the mixture instanta neously reaches chemical equilibrium without chang
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T, K
1700 2
T, K 2
1650
1
3
4
159
1
1600 1550
2000
1500
1500
1450
1000
1400 0.E+00 1.E–08 2.E–08 3.E–08 4.E–08 5.E–08 6.E–08 t, s
1.00E–05 3.00E–05 5.00E–05 2.00E–05 4.00E–05 0.00E+00 t, s
Fig. 3. Time evolution of the temperature of the mixture in the course of establishment of chemical equilibrium: (1) Gear method and (2) proposed method; the mechanism from [15].
Fig. 2. Time history of the temperature of the mixture dur ing the establishment of chemical equilibrium. The nota tions are the same as in Fig. 1.
ing its molar mass. Therefore, when the approximate method is used, a jump in temperature is observed at the initial time (curve 2 in Fig. 3). The parameters of this state are given in the “Partial equilibrium” row of Table 2. In solving the system of equations of chemical kinetics, the process of establishing a partial equilib rium takes a finite, though short, time. After that, curves 1 and 2 in Fig. 3 converge and eventually merge, as assumed in developing of the proposed method.
peratures above 2200 K, curves 1 and 2 merge. Note that, only over a time constituting a few percent of the process duration, the calculation is performed by the Gear method. The rest of the calculation is performed without loss of accuracy by the proposed more effi cient calculation method.
Note that the approximate method error shown in all these graphs is much smaller than the differences in the results obtained with the different kinetic mecha nisms. In fact, in Figs. 1 and 2, the difference is only noticeable between the curves corresponding to the different kinetic mechanism (curves 1 and 3).
Based on the assumption of the existence of a par tial chemical equilibrium in the explosion products, we developed a model to calculate their composition and thermodynamic parameters. Another assumption underlying the model is that, at a given density, inter
The proposed model was successfully applied to calculating the thermodynamic parameters and com position of the mixture after the induction period for an explosion. Consider as an example, the problem of the chemical transformation of a hydrogen–oxygen stoichiometric mixture at constant density. It was assumed that initially the mixture had a density of 0.4847 kg m–3 and a temperature of 2000 K. The cal culation was performed by the Gear method using the kinetic mechanism from [15]. Figure 4 (curve1, thin line) show the temperature time history. The second calculation was performed as follows. Until the moment when the temperature reached 2200 K, the calculation was carried out by the Gear method. This first time interval includes the induction period. Dur ing the induction period, the bimolecular reactions are not in equilibrium, so the complete system of kinetic equations should be solved. After this point, the calcu lation is carried out using the method proposed in the present work, a method with a single differential equa tion. The results of this calculation are represented in Fig. 4 by curve 2 (thick line). As can be seen, at tem RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY B
CONCLUSIONS
T, K
4000 3500 3000
2 2500 1 2000
1.E–08
1.E–07
1.E–06
t, s
1500 1.E–05
Fig. 4. Time evolution of the temperature of the mixture during the explosion of a hydrogen–oxygen stoichiometric mixture: (1) Gear method and (2) proposed method; the mechanism from [15]. Vol. 7
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nal energy, and molar mass, the entropy of the explo sion products reaches its maximum value. It is believed that if this assumption holds at any given time, the composition of the mixture can be calculated using a single differential equation for the number of mole cules per unit mass of the mixture. Without significant loss in accuracy, the respective stiff system of differen tial equations can be replaced by a single differential equation and a system of algebraic equations. This method is always consistent with the detailed mecha nism and can be used separately or in conjunction with the stiff system, replacing it when the bimolecular reactions reach the quasiequilibrium state. The constituent equations of the model were derived and the respective computer code written. The applicability of the model was demonstrated by solving a model problem. It was shown that the proposed model can be used to calculate the characteristics of the explosive transformation process after the induc tion period. REFERENCES 1. Yu. A. Nikolaev and P. A. Fomin, Fiz. Goreniya Vzryva 18 (1), 66 (1982). 2. A. A. Borisov, S. A. Gubin, and V. A. Shargatov, Progr. Astron. Aeron. 134, 138 (1991). 3. U. F. Bryakina, S. A. Gubin, A. M. Tereza, and V. A. Shargatov, Russ. J. Phys. Chem. B 4, 969 (2010).
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