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Show Rm/Patm is defined as the universal gas constant divided by one atmosphere of pressure. In general, the reference reaction rate, kQr(T) for the rth reaction is a function of temperature and its value is determined from an empirical correlation as follows: k0r(T) = ArTa- (19) Equation (19) is of the form found in the standard database of McBride et al. (1993). In an equilibrium state, it is known that the partial pressure ratio of the species components is equal to the equilibrium constant, kp. At equilibrium, dG\p>T = 0 and therefore only the Gibbs free energy formation for each species is needed to determine the equilibrium constant. For a simple reaction of the form a.4 -I- bB «=>• cC + dD, the equilibrium constant in terms of partial pressures (nondimensionalized by atmospheric pressure) is defined as , \PAY\PB P ~ \PC] b_ /-AG0 \pDy ~ exp\~R^F (20) where AG° is defined as the standard free energy change. A relationship can be extended to any general chemical reaction using the standard free energy change expressed in terms of molar Gibbs free energy [kcal/kg • kmol). The molar Gibbs free energy for the rth reaction is defined by the following expression. N N AG? = XXAtf, - 5>;5A<& (21) s=l 8=1 where the partial molar Gibbs free energy of the sth species in the rth reaction, Ao°s, consistent with the polynomial fits for Cp of McBride et al. (1993) is: Ag°s = Ah- TAs = C0a(l-lnT)T-Cl3y-C2ay -C3a--C4a-+C5,-C6.T (22) where, Cns, Ci,,--- C\a are identical to the constants in Eq. 14 and C^a and Cea are the 6th and 7th constants in the McBride database and represent the enthalpy, h% , and entropy, sV, of formation, respectively. The total Gibbs free energy exchange for the rth reaction is computed as follows A' AG°(T) = 5>"-i/)rsAas 0(T) (23) ?=i The equilibrium reaction rate for the rth reaction is then computed as follows, k" = exp(ntP) (24) The total rate of change of mass concentration of species j is obtained by summing up all the changes due to all the rth reaction steps and multiplying by the molecular weight of the sth species. iVfl r=l w, - o TV P=i Tl, N - ^ n (25) Implicit Numerical Method A psuedo-time marching procedure is devised for integrating Eq. (10) by using an implicit Euler difference for the psuedo-variable, r. Central differences are used to approximate the viscous terms. The implicit expression for the equations at the (p + l)th iteration in pseudotime can be represented as: Wp + 1 - W p F Ar +1* - *•£» +1* - *-Ci - [E - E, - [F - FJ IP+I lJ-h P+I fc-4 (26) + [G - Gr]p+1 - [G - Gt,]p+! = HJ+1 The fluxes are linearized in the standard fashion (see, for example, Shuen et al., 1993) and the resulting Ja-cobian matrices are given in the Appendix. A multilevel grid, implicit scheme is used to solve the system of equation [See, Edwards and Roy, (1998) and references therein]. Grid Adaption The re-distribution of grid points takes place in a line-by-line fashion. Points are allowed to move along a coordinate line like beads sliding along a fixed string, the final positions being governed by local flow properties. This technique was applied to high speed flows with shocks by Harvey et al. (1991-1993). The grid adaption technique is essentially an error equi-distribution method and involves the redistribution of grid points such that a solution error measure represented by a weighting function, w,, is equally distributed over a coordinate line. WiAsi = K where Wj is the weighting function based on gradients of flow properties and represents the spring constant with K as the resultant force. The grid interval, As,, is defined as the distance between adjacent nodal points along a line of constant computational coordinate. To prevent unacceptable grid skewness which would adversely affect the solution, contributions based on grid quality measures are added to the the contributions based on flow properties alone. Together, this adaptive grid formulation is analogous to the minimization of the energy in a system of tensional and torsional springs set between grid points [see Gnoffo (1982) and 5 |