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17.2.1 Calculation of Scalar Quantities

Density weighted mean scalars (such as species fraction and temperature), denoted by $\overline{\phi}$, are calculated from the probability density function (PDF) of $f$ and $c$ as

 \overline{\phi} = \int^1_0 \int^1_0 \phi(f, c) p(f,c) \, df \, dc (17.2-1)

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

 \overline{\phi} = \overline{c} \int^1_0 \phi_b(f) p(f) \, df \,\, + (1-\overline{c}) \int^1_0 \phi_u(f) p(f) \, df (17.2-2)

where the subscripts $b$ and $u$ denote burnt and unburnt, respectively.

The burnt scalars, $\phi_b$, are functions of the mixture fraction and are calculated by mixing a mass $f$ of fuel with a mass $(1-f)$ of oxidizer and allowing the mixture to equilibrate. When non-adiabatic mixtures and/or laminar flamelets are considered, $\phi_b$ is also a function of enthalpy and/or strain, but this does not alter the basic formulation. The unburnt scalars, $\phi_u$, are calculated similarly by mixing a mass $f$ of fuel with a mass $(1-f)$ 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, $\overline{\phi}_u$, are functions of $\overline{f}$ only. The unburnt density, temperature, specific heat, and thermal diffusivity are fitted in FLUENT to third-order polynomials of $\overline{f}$ using linear least squares:

 \overline{\phi}_u = \sum^3_{n=0} c_n \overline{f}^n (17.2-3)

Since the unburnt scalars are smooth and slowly-varying functions of $\overline{f}$, these polynomial fits are generally accurate. Access to polynomials is provided in case you want to modify them.

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