Conditional Enthalpy

The Conditional Enthalpy model solves an equation for the (static) enthalpy conditioned in the unburnt gases. The Conditional Enthalpy model also solves for the unburnt enthalpy variance which is used to calculate the unburnt temperature.

The following equation for the conditional unburnt enthalpy is used with the unburnt mass fractions $Y_{u}$ to determine the unburnt temperature $T_{u}$ which is used by the ECFM-3Z and ECFM-CLEH models.

Figure 1. EQUATION_DISPLAY

\frac{\partial (ρ h_{u})}{\partial t} + \nabla\cdot (ρ v h_{u}) - \nabla\cdot (ρ D_{t} \nabla h_{u}) - ρ D_{t} \frac{\nabla b \cdot \nabla h_{u}}{b} + \nabla\cdot (\frac{ρ}{ρ_{u}} q_{u}) + \frac{ρ}{ρ_{u}} \frac{\nabla b}{b} \cdot q_{u} = \frac{ρ}{ρ_{u}} (\frac{\partial p}{\partial t} + v \cdot \nabla p) + ρ ε

(3902)

Note that currently Simcenter STAR-CCM+ neglects the last two terms on the left-hand-side of Eqn. (3902).

The temperature variance is determined from the enthalpy variance, which is solved by the following equation:

Figure 2. EQUATION_DISPLAY

\frac{\partial ρ {\bar{h_{u}}}^{' 2}}{\partial t} + \nabla \cdot (ρ {\bar{h_{u}}}^{' 2} v) - \nabla \cdot [(\frac{μ}{\Pr} + \frac{μ_{t}}{\Pr_{t}}) \nabla {\bar{h_{u}}}^{' 2}] = 2 \frac{μ_{t}}{\Pr_{t}} \nabla h_{u} \cdot \nabla h_{u} - ρ \frac{C ({Re}_{t})}{τ} {\bar{h_{u}}}^{' 2}

(3903)

using the enthalpy to temperature conversion:

Figure 3. EQUATION_DISPLAY

{\bar{T_{u}}}^{' 2} = \frac{{α \bar{h_{u}}}^{' 2}}{C_{p}^{2}}

(3904)

$h_{u}$ is the conditional enthalpy in the unburnt gases.
$T_{u}$ is the conditional temperature in the unburnt gases.
$ρ_{u}$ is the conditional density in the unburnt gases.
$p$ is the thermodynamic pressure
$D_{t}$ is the turbulent diffusivity.
$S_{h}$ represents a contribution to the equation from the enthalpy.
$v$ is the mean flow velocity.
$ρ$ is the mean density.
$ε$ is the turbulent dissipation.
$b$ is the premixed combustion regress variable.
$q_{u}$ is the boundary heat flux.
$α$ is a multiplier for the unburnt temperature variance ${T^{'}}_{u}^{2}$ .
$C_{p}$ is the heat capacity of unburnt gas at unburnt temperature.

Segregation Factor

Unburnt Temperature: The segregation factor in the unburnt temperature $s_{T}$ is calculated by:
Figure 4. EQUATION_DISPLAY

$s_{T} = \frac{{\bar{θ}'}^{2}}{θ (1 - θ)}$

(3905); where:
Figure 5. EQUATION_DISPLAY

$θ' = \frac{{T'}_{u}^{2}}{(T_{u} - T_{\min}) (T_{\max} - T_{u})}$

(3906); Figure 6. EQUATION_DISPLAY

$θ = \frac{T_{u} - T_{\min}}{(T_{\max} - T_{\min})}$

(3907); $T_{\max}$ and $T_{\min}$ are either the maximum and minimum temperatures in the library, or of the domain—as specified.
Mixture Fraction: The mixture fraction segregation factor (normalized mixture fraction variance) is calculated by:
Figure 7. EQUATION_DISPLAY

$s_{Z} = \frac{\bar{{Z'}^{2}}}{Z (1 - Z)}$

(3908); where the mixture fraction $Z$ is determined by Eqn. (3489)—or for ECFM-CLEH, $Z = Z_{F}$ determined by Eqn. (3878).