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Appendix A: Thermochemical data

Thermochemical properties, such as enthalpies of formation, entropies, and heat capacities are very important to ensure the consistency between the rate parameters of the forward and reverse elementary reactions. They are also useful for estimating the heat release rate. Thermochemical data for all molecules or radicals have been estimated and stored as 14 NASA polynomial coefficients, according to the McBride et al. (1993) formalism. The NASA polynomials take the following form: where a_i, i ∈  [1,7] , are the numerical NASA coefficients for the fourth-order polynomial. Each species is characterized by fourteen numbers. The first seven numbers are for the high-temperature range, generally from 1000 to 5000 K, and the following seven numbers are the coefficients for the low-temperature range, generally from 300 to 1000 K. When these parameters are not available in the literature (McBride et al. 1993) or in databases⁸, which is the most frequent case for species present in automotive fuels, they have to be estimated. In this case, these data were automatically calculated using the software THERGAS (Muller et al. 1995), which was developed in the LRGP laboratory and is based on the group and bond additivity methods proposed by Benson (1976) and updated based on the data of Domalski & Hearing (1996). The enthalpies of formation of alkyl radicals have been also updated according to the values of bond dissociation energies published by Tsang & Hampson (1986) and by Luo (2003) and following the recommendations of Benson & Cohen (1997).

An elementary reversible reaction i involving L chemical species can be represented in the general form (A.4)where are the forward stoichiometric coefficients, and are the reverse ones. χ_l is the chemical symbol of the lth species.

The kinetic data associated to each reaction are expressed with a modified Arrhenius law where T is the temperature, E_a the activation energy of the reaction, A the pre-exponential factor, and n a coefficient that allows the temperature dependence of the pre-exponential factor. If the rate constant associated to the forward reaction is k_fi(T), then the one associated to the reverse reaction is k_ri(T), verifying (A.5)where K_pi is the equilibrium constant, when the activity of the reactants is expressed in pressure units (Benson 1976): (A.6)Here, and are the variation in entropy and enthalpy occurring when passing from reactants to products in the reaction i, P⁰ is the standard pressure (P⁰ = 1.01325 bar), k_B is the Boltzmann’s constant, and ν_l are the stoichiometric coefficients of the L species involved in reaction i: . Combined with Eqs. (A.2) and (A.3), and can be calculated with the NASA coefficients: (A.7)Finally, we can calculate the reverse reaction rate for the reaction i: (A.8)

Appendix B: Chemical equilibrium calculation

To compute the equilibrium abundance of the species in a definite system considered as an ideal gas, we have developed a thermodynamical equilibrium calculator TECA. TECA is software that allows equilibrium calculation for a complex mixture. More specifically, for a given initial state of an ideal-gas mixture, the chemical-equilibrium program is able to determine the gas composition at a defined temperature and pressure. This calculation is based on the principle of the minimization of Gibbs energy (e.g. Gibbs 1873; White et al. 1958; Eriksson & Rosen 1971; Smith & Missen 1982; Reynolds 1986): (B.1)where L is the total number of species, the partial free energy of the species l, and N_l the number of moles of the species l.

The partial free energy of a compound l, behaving as an ideal gas, is given by (B.2)where g_l(T,P) is the free energy of the species l at the temperature T and the pressure P of the system and R is the ideal gas constant.

For an ideal gas, g_l(T,P) is given by (B.3)where and are respectively, the standard-state enthalpy and entropy of the species l at the temperature T of the system.

The enthalpy and the entropy are expressed as NASA polynomials as described above.

Appendix C: Pressure-dependent reactions

Under some conditions, several reactions do not have the same rate constant depending on whether they occur under low or high pressure (respectively k₀(T) and k_∞(T)). In this case, between these two limits what is called a fall-off zone appears. This is typically the case in reactions requiring a collisional body to proceed, such as thermal dissociation or recombination (three-body) reactions. In the present kinetic model, we have different types of reactions with pressure-dependent rate constants (Table C.1). In some cases, some species act more efficiently as collisional bodies than do others. Then, when available from the literature, collisional efficiencies are used to specify the increased efficiency of the lth species in the ith reaction (see for example reaction (2) in Table C.1).

For the pressure-dependent reactions, the rate constant at any pressure is taken to be (C.1)where the reduced pressure P_r is given by (C.2)and  [M]  is the concentration of the mixture, weighted by the efficiency of each compound, α_l, in the reaction studied: (C.3)where  [X_l]  is the concentration of the species k.

As shown in Table C.1, three methods of representation of the rate expression in the fall-off region are used (enhanced collisional body efficiencies of certain species are presented below the reaction):

• the Lindemann et al. (1922)formulation, illustrated by reaction (1) inTable C.1;

• the Troe (1983) formulation, see for example reaction (2) in Table C.1;

• the SRI formulation proposed by Stewart et al. (1989), illustrated by reaction (3) in Table C.1.

In the Lindenman form, F is unity (F = 1).

In the Troe form F is given by (C.4)with and (C.5)the four parameters a, T***, T* and T** must be specified but it is often the case that the parameter T** is not used because of the lack of data.

The approach taken at the Stanford Research Institute (SRI) by Stewart et al. (1989) is in many ways similar to that taken by Troe, but the blending function F is approximated differently. Here, F is given by (C.6)where (C.7)

Appendix D: Photodissociations

© ESO, 2012