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Show The reverse rate is obtained from equilibrium considerations, based on the requirement that CO concentration approaches equilibrium in the limit of long reaction time (weo/weo = K(T», where K(T) is the equilibrium constant for CO + 02 ~ CO2 ; (Kuo, 1986). The local instantaneous species concentrations ai are calculated from mf and r. Since we assume r = i, then it follows that: (29) where ai is the maximum (or reference) value of ai; ato and a~2 correspond to r = 0, and crt02 corresponds to r = 1. By Reynolds-averaging equations 28a and 28b over the PDF, the factors (ai/a!) can be placed outside the integral. The source term for CO can then be written as: -S M· (R R R) - rfo -CIco 1/2 rr - co = y In -1.' 2' -') ""= CIo + ",,=0 CIco 2 2 (30) where: r-x ~o -= ~ 1 P f / 2 pS (J~OaH 0 1/2(J~ 1/2 exp(-E1/RT) P(mf ) dmt p -::if aco CIo 2 o 2 2 (31a) and ~o z: ~ 1 r-x ' 1/2 (J~o a. 1/2 T-1/2 exp(-E /RT) p(m f p ~ P H 0 . 2 ) dmt ac02 2 2 (31b) 0 are essentially independent of reaction progress, r, with the exception of the terms containing instantaneous temperature. The instantaneous temperature in equations 31a and 31b depends on the instantaneous fuel mixture fraction mf' enthalpy ~, and CO reaction progress variable r. We approximate the enthalpy as a function of mf and r as follows: (32) where ~ is determined from the conservation equation for mixture enthalpy. Once instantaneous enthalpy is known, the instantaneous temperature can be determined implicitly from the equation: N S hat = X. he (T) + (1 - X. ) L a i hi ( T) i=l (33 ) using relationships for enthalpy as a function of temperature and NewtonRaphson iteration. The nonlinear relationship between T and r is treated explicitly in the solution to the finite-difference equation for aCO . 14 |