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15.3.3 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) [ 286]. In FLUENT, a simplified set of the mixture fraction space equations are solved [ 285]. Here, $N$ equations are solved for the species mass fractions, $Y_i$,


 \rho \frac{\partial Y_i}{\partial t} = \frac{1}{2} \rho \chi \frac{\partial^2 Y_i}{\partial f^2} + S_i (15.3-6)

and one equation for temperature:


 \rho \frac{\partial T}{\partial t} = \frac{1}{2} \rho \chi \... ...partial Y_i}{\partial f} \right] \frac{\partial T}{\partial f} (15.3-7)

The notation in Equations  15.3-6 and 15.3-7 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  15.3-2 to variable density is used [ 180]:


 \chi(f) = \frac{a_s}{ 4 \pi} \frac{3 {(\sqrt{\rho_{\infty}/\... ...fty}/\rho} + 1} \exp \left( -2 [{\rm erfc}^{-1}(2f)]^2 \right) (15.3-8)

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


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