Eddy Dissipation Concept

Eddy Dissipation Model

Magnussen 1977

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

Gran 1996

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.