8.4.3 Flamelet Generation

The laminar counterflow diffusion flame equations can be transformed from physical space (with $x$ as the independent variable) to mixture fraction space (with $f$ as the independent variable) [ 273]. In ANSYS FLUENT, a simplified set of the mixture fraction space equations are solved [ 272]. Here, $N$ equations are solved for the species mass fractions, $Y_i$,

and one equation for temperature:

The notation in Equations  8.4-6 and 8.4-7 is as follows: $Y_i$, $T$, $\rho$, and $f$ are the $i$th species mass fraction, temperature, density, and mixture fraction, respectively. $c_{p,i}$ and $c_p$ are the $i$th species specific heat and mixture-averaged specific heat, respectively. $S_i$ is the $i$th species reaction rate, and $H_i$ is the specific enthalpy of the $i$th species.

The scalar dissipation, $\chi$, must be modeled across the flamelet. An extension of Equation  8.4-2 to variable density is used [ 163]:

where $\rho_{\infty}$ is the density of the oxidizer stream.