# Eddy Dissipation Model

## Nonpremixed flames

$$ \begin{align} &F + \nu Ox \rightarrow (1+\nu)P, \ &R_F = -A\bar{\rho}\frac{\epsilon}{k}\min\left(y_F, ,\frac{y_{Ox}}{\nu}\right), \ &R_{Ox} = \nu R_F, \ &R_{P} = -(1+\nu) R_F. \end{align} $$

- $A$ is an empirical constant with a value of 4 in Magnussen (1977).
- The expression for $R_F$ is based on the concentration of the limiting reagent.
- $y_F= y_{Ox}/\nu$ at stoichiometric conditions.

## Premixed flames

$$ \begin{align} &R_F = -A\bar{\rho}\frac{\epsilon}{k}\min\left(y_F, ,\frac{y_{Ox}}{\nu},,B\frac{y_P}{1+\nu}\right), \ \end{align} $$

- $B$ is an empirical constant with a value of 0.5 in Magnussen (1977).

## Note

- Magnussen says that the above formulation of $R_F$ for premixed flames applies to diffusion or premixed flames. But in that case, in a diffusion flame, if we feed pure fuel and oxidizer without any products, then the reaction rate will be zero. Many literature sources ignore this when presenting the model. The StarCCM+ Theory Guide and the Fluent User Guides do make these distinctions.
- Variations on this model allow more general reaction schemes than given above, but the differences only account for the stoichiometry; reaction kinetics are not included.
- In Fluent, in LES, $\epsilon/k$ is replaced with $\tau^{-1} = \sqrt{2S_{i,j}S_{i,j}}$, where $S_{i,j}$ is the rate of strain tensor.

# Eddy Dissipation Concept (EDC) Model

This model allows for detailed chemistry.
Reactions are assumed to take place in fine structures where dissipation occurs.
The mean reaction rate of species $i$ is given by
$$\tilde{R}_i = \frac{\bar{\rho}\gamma^2}{\tau^*(1-\gamma^3)}(\tilde{Y}_i-Y^*).$$

- Here, $\gamma$ is the fraction of the flow occupied by the fine scale structures.
- This expression combines equations 5, 6, and 19 in Gran (1996).
- $\tau^
*$ is modeled as (Gran 1996 equation 4) $$\tau^*= \left(\frac{C_{D2}}{3}\right)^{1/2}\tau_\eta = \left(\frac{C_{D2}}{3}\right)^{1/2}\left(\frac{\nu}{\epsilon}\right)^{1/2},$$ - $\gamma$ is modeled as (Gran 1996 equation 1) $$\gamma = \left(\frac{3C_{D2}}{4C_{D1}^2}\right)^{1/4}\left(\frac{\nu^*\tilde{\epsilon}}{\tilde{k}^2}\right)^{1/4}.$$
- $C_{D1}$ and $C_{D2}$ are model constants taken as 0.134 and 0.5, respectively.
- The concentrations $Y_i^*$ are computed as the composition in adiabatic, constant pressure homogeneous reactors.
- Gran (1996) uses steady flow reactors with residence time $\tau^
*$, with $Y_i^*$ computed by solving to steady state the equation $dY_i^*/dt = (Y_i^m-Y_i^*)/\tau^* +\dot{m}_i^{\prime\prime\prime}/\rho$. The reactor inlet state $Y_i^m$ is not specified explicitly, but could be taken as the cell average composition. - Fluent assumes reaction in an adiabatic constant pressure batch reactor for time $\tau^*$ with the initial cell composition as the initial batch reactor composition.

- Gran (1996) uses steady flow reactors with residence time $\tau^