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Show > (1) are re-written as: flW d - - d - - d - - where (10) W = j[pi,p2, •••,PN,ti, v, w;, T ] T , (11) and AT r = + 01ri 0Vi er2 Hr 2 RT + 0K TV RT 01 u(0 + 0Kj u(0 + v(®+& w(0+%$) w(Q + N RT v(e+w> RT' W2 RT Oil Q 2 0 0 QY2 ^f + 0Ay' w(Q+%*r) Q/V T _£2 T _ £w T _ Pu T _££ T pw T PU pV PW p(Cpm - Ur) T > J (12) with Uref = min[a2,max(|V'|2,An^|2)] 1 _1_ a^ 0 = U2 , re J a. = H\e + RT -1 In the above equations, H is the enthalpy per unit mass, \V\ is the local velocity magnitude, ll^l is a reference velocity (an average inlet velocity is used here), a is the sound speed and A' is a constant set to unity for the present computations. Thermophysical Properties From the updated values of the dependent variables vector, W , of Eq. (11), the thermodynamic pressure and enthalpy are computed from Dalton's Law of partial pressures as N P=Y,P* 8=1 Once the mass fractions, Ys, are computed from the partial pressure ratios, the enthalpy of each specie is computed from hs = h°f + / CPadT, I J Ir-ef where Rm and Tref are the universal gas constant and reference temperature for thermodynamic properties; W*s, CPt, hs, and h°fa are the molecular weight, specific heat, thermodynamic enthalpy and heat of formation of species s, respectively. The species specific heats, CPi are evaluated using temperature polynomials. Cp, =co3+ClT + c2T2 + c3sT3 + c4T4 14 s = l,2,--,Ar The form of the specific heat polynomials of Eq. (14) is that of the standard database of McBride et al. (1993). The specific heat of the gas mixture is obtained by mass concentration weighting of each individual species. TV ^Pm ~~ Z ^ CPa 1 (15) 8=1 The enthalpy of the gas mixture is found in a similar manner. Species viscosities as a function of temperature were obtained using a Lennard-Jones potential function (Chapman and Cowling, 1951). The viscosity of the gas mixture is computed using Wilke's mixing rule (Wilke, 1950). For the present calculations species and thermal diffusion coefficients are obtained assuming constant Schmidt and Prandtl numbers of 0.72. Chemistry and Kinetics Modeling For a set of NR elementary reactions involving A: chemical species, the reaction equations are written in the general form N N £4. s=l far £*>< = 1,2, ••-.AT, 8=1 (16) where i/J.sand v"s are the stoichiometric coefficients for the sth specie appearing as a reactant in the rth forward and backward reactions, respectively, and ns = pY8/Ws, is the molar concentration of species s. The rate of change of molar concentration of species s due to the rth elementary reaction step is % ) =(c-4.)(*/.n»p"-^]K' P=i P=i (17) where the forward and backward reaction rates for the sth reaction step, fe# and kb, are given in terms of the pressure based equilibrium rate constant, kP3, as kfr=k0r(T)e-E*'R~T K = ^(RPT)A^ 18a, 6) with A n r = *£2s(v"s - v'rs) (s is summation over all species), and the pressure based gas constant, Rp = |