ECFM-CLEH

ECFM-CLEH is a combustion model in which burning rates are limited by a thermodynamic equilibrium given by complex chemistry.

The ECFM-CLEH model that is similar to ECFM-3Z in that the cell sub-grid is modeled with a fixed number of zones, and the premixed flame modeling is the same for both models (flame propagation and auto-ignition), and an equation is solved for the flame surface density. The main difference from ECFM-3Z is that with ECFM-CLEH, the chemical species are represented with tracers, fuel mass fractions and then progress variables, in a similar way to flamelet models. In order to avoid complete combustion of reactants and recover the correct combustion energy release, species mass fractions are limited by thermodynamic equilibrium tables.

Turbulence mixes the fuel in the unburnt gases, which comes from spray or liquid film evaporation. Therefore, the premixed fuel

Y_{f | P M}

(and tracer

Z_{f | P M}

) increase thanks to the fuel mixing effect coming from

Y_{f | U M}

. Combustion can occur only in the premixed and diffusion zones and then in the post-oxidation zone. As the premixed combustion progresses, fuel diffusion

Y_{f | D F}

(and

Z_{f | D F}

) is generated in the burnt gases area and is then burnt. Post-oxidation combustion can then also take place when specific thermodynamic conditions are met and fuel mixing is sufficiently homogenous. Normalized mixture fraction variance (mixture fraction segregation factor)

s_{Z}

in Eqn. (3908), which is involved in the diffusion equilibrium library—and in soot or NORA NOx libraries when they are used—is provided by the ECFM-CLEH model (unless you deactivate the ZF Variance Computed property).

The transport equations governing the above quantities in the various zones are:

Figure 1. EQUATION_DISPLAY

\frac{\partial ρ Z_{f | P M}}{\partial t} + \nabla\cdot (ρ Z_{f | P M} v) - \nabla\cdot [(\frac{μ}{σ} + \frac{μ_{t}}{σ_{t}}) \nabla Z_{f | P M}] = T_{U M} - T_{P M \to D F}^{Z_{F}} - T_{P M \to P S T X}^{Z_{F}}

(3861)

Figure 2. EQUATION_DISPLAY

\frac{\partial ρ Z_{f | D F}}{\partial t} + \nabla\cdot (ρ Z_{f | D F} v) - \nabla\cdot [(\frac{μ}{σ} + \frac{μ_{t}}{σ_{t}}) \nabla Z_{f | D F}] = \tilde{c} \dot{ω} f, e v a p + T_{P M \to D F}^{Z_{F}} - T_{D F \to P S T X}^{Z_{F}}

(3862)

Figure 3. EQUATION_DISPLAY

\frac{\partial ρ Z_{f | P S T X}}{\partial t} + \nabla\cdot (ρ Z_{f | P S T X} v) - \nabla\cdot [(\frac{μ}{σ} + \frac{μ_{t}}{σ_{t}}) \nabla Z_{f | P S T X}] = T_{P M \to P S T X}^{Z_{F}} - T_{D F \to P S T X}^{Z_{F}}

(3863)

Figure 4. EQUATION_DISPLAY

\frac{\partial ρ Y_{f | U M}}{\partial t} + \nabla\cdot (ρ Y_{f | U M} v) - \nabla\cdot [(\frac{μ}{σ} + \frac{μ_{t}}{σ_{t}}) \nabla Y_{f | U M}] = (1 - \tilde{c}) \dot{ω} f, e v a p - T_{U M}

(3864)

Figure 5. EQUATION_DISPLAY

\frac{\partial ρ Y_{f | P M}}{\partial t} + \nabla\cdot (ρ Y_{f | P M} v) - \nabla\cdot [(\frac{μ}{σ} + \frac{μ_{t}}{σ_{t}}) \nabla Y_{f | P M}] = T_{U M} - T_{P M \to D F}^{Y_{F}} - T_{P M \to P S T X}^{Y_{F}} - {\tilde{\dot{ω}} f |}_{A I} - {\tilde{\dot{ω}} f |}_{P M}

(3865)

Figure 6. EQUATION_DISPLAY

\frac{\partial ρ Y_{f | D F}}{\partial t} + \nabla\cdot (ρ Y_{f | D F} v) - \nabla\cdot [(\frac{μ}{σ} + \frac{μ_{t}}{σ_{t}}) \nabla Y_{f | D F}] = \tilde{c} \dot{ω} f, e v a p + T_{P M \to D F}^{Y_{F}} - T_{D F \to P S T X}^{Y_{F}} - {\tilde{\dot{ω}} f |}_{D I F F}

(3866)

Figure 7. EQUATION_DISPLAY

\frac{\partial ρ Y_{f | P S T X}}{\partial t} + \nabla\cdot (ρ Y_{f | P S T X} v) - \nabla\cdot [(\frac{μ}{σ} + \frac{μ_{t}}{σ_{t}}) \nabla Y_{f | P S T X}] = T_{P M \to P S T X}^{Z_{F}} + T_{D F \to P S T X}^{Z_{F}} - {\tilde{\dot{ω}} f |}_{P S T X}

(3867)

The mixing term

T_{U M}

is given by:

Figure 8. EQUATION_DISPLAY

T_{U M} = ρ \frac{C (Re)}{τ} Z_{f | U M}

(3868)

The other transfer terms between the different combustion zones are functions of mixing, combustion progress, and thermodynamic conditions:

$T_{P M \to D F}^{Z_{F}}$
$T_{P M \to P S T X}^{Z_{F}}$
$T_{D F \to P S T X}^{Z_{F}}$
$T_{P M \to D F}^{Y_{F}}$
$T_{P M \to P S T X}^{Y_{F}}$
$T_{D F \to P S T X}^{Y_{F}}$

The reaction rates in the different zones are given by:

Figure 9. EQUATION_DISPLAY

{\tilde{\dot{ω}} f |}_{A I} = ρ \tilde{\dot{ω}} (Z_{f | P M}, Z_{F}, T_{m i x}, s_{T}, P, Z_{E G R})

(3869)

Figure 10. EQUATION_DISPLAY

{\tilde{\dot{ω}} f |}_{P M} = ρ U_{l} Σ (Z_{f | P M} - Y_{f | P M}^{e q})

(3870)

Figure 11. EQUATION_DISPLAY

{\tilde{\dot{ω}} f |}_{D I F F} = ρ \frac{C (Re)}{τ} \tilde{α} (Y_{f | D F} - Y_{f | D F}^{e q})

(3871)

Figure 12. EQUATION_DISPLAY

{\tilde{\dot{ω}} f |}_{P S T X} = ρ [{A r}_{1} e^{- \frac{{E a}_{1}}{T}} - {A r}_{2} e^{\frac{{E a}_{2}}{T}}] (Y_{f | P S T X} - Y_{f | P S T X}^{e q})

(3872)

where

$C (Re)$ is a tabulated turbulent diffusion constant (function of the turbulent Reynolds number $Re$ )
$τ$ is the turbulence timescale
$Σ (= ρ σ)$ is the flame surface density that is obtained from its transport equation Eqn. (3909)
$U_{l}$ is the laminar flame speed, calculated by Eqn. (3910)
$\tilde{\dot{ω}}$ is the auto-ignition rate obtained from TKI-pdf tables, see TKI Tables
${A r}_{1}$ and ${A r}_{2}$ are Arrhenius coefficients
${E a}_{1}$ and ${E a}_{2}$ are activation energies
$\tilde{c}$ is the premixed progress variable in ECFM-CLEH, defined as:
Figure 13. EQUATION_DISPLAY

$\tilde{c} = \frac{Z_{f | P M} + Z_{f | P S T X} - Y_{f | P M} - Y_{f | P S T X}}{Z_{f | P M} + Z_{f | P S T X} - Y_{f | P M}^{e q} - Y_{f | P S T X}^{e q}}$

(3873)
$\tilde{α}$ is the diffusion progress variable, defined as:
Figure 14. EQUATION_DISPLAY

$\tilde{α} = \frac{Z_{f | D F} - Y_{f | D F} + Z_{f | P M} - Y_{f | P M} + Z_{f | P S T X} - Y_{f | P S T X}}{Z_{f | D F} - Y_{f | D F}^{e q} + Z_{f | P M} - Y_{f | P M} + Z_{f | P S T X} - Y_{f | P S T X}}$

(3874)

The equilibrium fuel mass fractions

Y_{f | P M}^{e q}

Y_{f | D F}^{e q}

, and

Y_{f | P S T X}^{e q}

are tabulated functions of the form:

Figure 15. EQUATION_DISPLAY

Y_{f | P M}^{e q} = f (Z_{F}, Z_{f | P M}, T_{m i x}, s_{T}, Z_{E G R}, P)

(3875)

Figure 16. EQUATION_DISPLAY

Y_{f | P S T X}^{e q} = f (Z_{F}, Z_{f | P S T X}, T_{m i x}, s_{T}, Z_{E G R}, P)

(3876)

Figure 17. EQUATION_DISPLAY

Y_{f | D F}^{e q} = f (Z_{F}, Z_{f | D F}, T_{m i x}, s_{Z}, Z_{E G R}, P)

(3877)

where the global fuel tracer

Z_{F}

is given by:

Figure 18. EQUATION_DISPLAY

Z_{F} = Y_{f | U M} + Z_{f | P M} + Z_{f | D F} + Z_{f | P S T X}

(3878)

where:

$Z_{E G R}$ is the mass fraction
$s_{T}$ is the temperature segregation factor calculated by Eqn. (3905).
$s_{Z}$ is the mixture fraction segregation factor calculated by Eqn. (3908).
$T_{m i x}$ is the temperature of the mixing (unburnt) zone which is computed analytically from the unburnt O₂ and unburnt fuel

ECFM-CLEH also uses the one-step irreversible mechanism:

C_{x} H_{y} O_{z} N_{w} + (x + \frac{y}{4} - \frac{z}{2}) O_{2} \to x C O_{2} + \frac{y}{2} H_{2} O + \frac{w}{2} N_{2}

(3879)

which can also be written as:

Figure 19. EQUATION_DISPLAY

Y_{F} + Y_{O 2} \to Y_{C O 2} + Y_{H 2 O} + Y_{N 2}

(3880)

where

Y_{F}

is the global fuel mass fraction that is given by the sum of fuel mass fractions in the different zones:

Figure 20. EQUATION_DISPLAY

Y_{F} = Y_{f | U M} + Y_{f | P M} + Y_{f | D F} + Y_{f | P S T X}

(3881)

The amount of burnt fuel

Ω_{b f}

is computed from:

Figure 21. EQUATION_DISPLAY

Ω_{b f} = Z_{F} - Y_{F}

(3882)

The oxidizer and products are therefore:

Figure 22. EQUATION_DISPLAY

Y_{O 2} = β (1 - Z_{F} - Z_{E G R}) - υ_{O 2} Ω_{b f}

(3883)

Figure 23. EQUATION_DISPLAY

Y_{C O 2} = Y_{u, C O 2} + υ_{C O 2} Ω_{b f}

(3884)

Figure 24. EQUATION_DISPLAY

Y_{H 2 O} = Y_{u, H 2 O} + υ_{H 2 O} Ω_{b f}

(3885)

Figure 25. EQUATION_DISPLAY

Y_{N 2} = Y_{u, N 2} + υ_{N 2} Ω_{b f}

(3886)

where $υ_{O 2}$ , $υ_{C O 2}$ , $υ_{H 2 O}$ and $υ_{N 2}$ are the stoichiometric coefficients of the one-step reaction, and $β$ is a fixed value which represents the mass ratio of oxygen in pure air. $Y_{u, C O 2}$ , $Y_{u, H 2 O}$ , and $Y_{u, N 2}$ are the unburnt contributions of CO2, H2O, and N2 respectively from exhaust gas recirculation (EGR) and air.