Density weighted mean scalars (such as species fraction and temperature), denoted by , are calculated from the probability density function (PDF) of and as
Under the assumption of thin flames, so that only unburnt reactants and burnt products exist, the mean scalars are determined from
where the subscripts and denote burnt and unburnt, respectively.
The burnt scalars, , are functions of the mixture fraction and are calculated by mixing a mass of fuel with a mass of oxidizer and allowing the mixture to equilibrate. When non-adiabatic mixtures and/or laminar flamelets are considered, is also a function of enthalpy and/or strain, but this does not alter the basic formulation. The unburnt scalars, , are calculated similarly by mixing a mass of fuel with a mass of oxidizer, but the mixture is not reacted.
Just as in the non-premixed model, the chemistry calculations and PDF integrations for the burnt mixture are performed in FLUENT, and look-up tables are constructed.
Turbulent fluctuations are neglected for the unburnt mixture, so the mean unburnt scalars, , are functions of only. The unburnt density, temperature, specific heat, and thermal diffusivity are fitted in FLUENT to third-order polynomials of using linear least squares:
Since the unburnt scalars are smooth and slowly-varying functions of , these polynomial fits are generally accurate. Access to polynomials is provided in case you want to modify them.