## 17.2.1 Calculation of Scalar Quantities

Density weighted mean scalars (such as species fraction and temperature), denoted by , are calculated from the probability density function (PDF) of and as

 (17.2-1)

Under the assumption of thin flames, so that only unburnt reactants and burnt products exist, the mean scalars are determined from

 (17.2-2)

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:

 (17.2-3)

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.

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