OCR Text |
Show The instantaneous values, weo and w~o' are actually a function of the fuel mixture fraction mf, the CO reaction progress variable r, and the enthalpy hm· The correct way to obtain the Reynolds-average reaction rates requires the use of a three-variable PDF. The form of this PDF, however, is not known. One usual remedy is to approximate the enthalpy as a function of mf and r. We further simplify the problem by neglecting fluctuations in r (we assume r = r). This reduces the quadrature to the form of equation 5. The mean fuel mixture fraction, mf, and mean fuel mass fraction, mfr' must match the values from the modeled conservation equations. Mathematically, the following relations are satisfied: I1- XS rof = rot P(rof ) dmf (26) o (27) o where P(mf) is a two-parameter PDF with triangular shape and a delta function to represent intermittence for mf = 0 and mf = 1-Xs . A triangular PDF was chosen and was shown to be mathematically efficient and accurate in numerical sensit~vity studies. Shape parameters for the PDF were found using equations 26 and 27 with Newton-Raphson iteration. Once the shape of the PDF is known, average values, weo and w~o' can be determined from global reaction mechanisms. The instantaneous reaction rate for CO oxidation can be written in terms of a forward rate (Howard, et al., 1972) and a reverse rate: (28a) 1/2 -1/2 (fH 0 T exp(-E2 /RT) (28b) 2 where: A1 1.3 x 1011 m3 kgmol-1 s-l E1 1.255 x 108 J kgmol-1 K-1 A2 1.233 x 1016 (m3 kgmol-1) 1/2 s-l E2 4.05 x 108 J kgmol-1 K-1 13 |