ECFM-3Z

The ECFM-3Z model is a combustion model based on a flame surface density transport equation and a mixing model that can describe inhomogeneous turbulent premixed and diffusion combustion. The idea is to divide the sub-grid cell taking into account the local stratification. The probability density function (PDF) of the mixture fraction is defined by three Dirac functions. The evolution of the mass included in the three mixing zones is computed and modified by using a mixing model based on local turbulent time-scale.

The mathematical structure of the ECFM-3Z model, consists of four major components:

Mixing Model

3Z stands for three zones of mixing:

the unmixed fuel zone
the mixed gases zone
the unmixed air plus EGR zone

The three zones are too small to be resolved by the mesh and are therefore modelled as sub-grid quantities. The mixed zone is the result of turbulent mixing and molecular mixing between gases in the other two zones, and is where combustion takes place.

Generally, the mass fractions of species in the mixed zone

Y_{i}^{m}

can be defined as the conditional averages of the fraction:

Figure 1. EQUATION_DISPLAY

Y_{i}^{m} = {Y_{i} |}_{Z = Z_{m}} = \int_{δ V} Y_{i} (x', t) δ [Z (x', t) - Z_{m}] d V

(3836)

$Z_{m}$ is the mean mixture fraction, $δ$ is the Dirac function, and $V$ is the cell volume.

When soot or NORA NOx libraries are used, the ECFM-3Z model also automatically accounts for the normalized mixture fraction variance (mixture fraction segregation factor) $s_{Z}$ in Eqn. (3908).

All species in the ECFM-3Z model are conditioned in this zone. The other two zones are characterized by the fuel in the unmixed fuel zone and the species in the unmixed air and EGR zone. However, as a simplification, only oxygen is solved for in the latter zone—with all other species algebraically related to it.

The equations governing the mass fractions of the unmixed fuel (

Y_{f, u m}

) and unmixed oxygen (

Y_{O 2, u m}

) are:

Figure 2. EQUATION_DISPLAY

\frac{\partial ρ Y_{f, u m}}{\partial t} + \nabla\cdot (ρ v Y_{f, u m}) - \nabla\cdot [(D + \frac{μ_{t}}{S c}) \nabla Y_{f, u m}] = - \frac{β_{mi x}}{τ_{m i x}} Y_{f, u m} (1 - Y_{f, u m} \frac{ρ}{ρ_{u}} \frac{M_{m}}{M_{f}}) + {\dot{ω}}_{e v a p}

(3837)

and

Figure 3. EQUATION_DISPLAY

\frac{\partial ρ Y_{O 2, u m}}{\partial t} + \nabla\cdot (ρ v Y_{O 2, u m}) - \nabla\cdot [(D + \frac{μ_{t}}{S c}) \nabla Y_{O 2, u m}] = - \frac{β_{mi x}}{τ_{m i x}} Y_{O 2, u m} (1 - \frac{Y_{O 2, u m}}{Y_{O 2, \inf}} \frac{ρ}{ρ_{u}} \frac{M_{m}}{M_{a i r + e g r}})

(3838)

where

M_{m}

M_{f}

, and

M_{a i r + e g r}

are the molecular masses of the mean gases, fuel, and air (with exhaust gas recirculation), respectively.

β_{mi x}

is a tuning coefficient (default 1.0), and

τ_{m i x}

is the mixing timescale—which is equal to the turbulence timescale

τ_{t u r b}

$τ_{t u r b} = \frac{k}{ε}$

(3839)

Y_{O 2, \inf}

is given by:

Figure 4. EQUATION_DISPLAY

Y_{O 2, \inf} = \frac{Y_{u, O 2}}{1 - Y_{u, f}}

(3840)

Species in the mean space are transformed to the corresponding species in the mixed space. The mass fraction of the generic species

i

is then given by:

Figure 5. EQUATION_DISPLAY

Y_{(u) i}^{m} = (Y_{(u) i} - Y_{u, i} C_{x}) \frac{ρ}{ρ - ρ^{U M}}

(3841)

Brackets “()” denote either tracer

(u)

or non-tracer

u

species.

ρ^{U M}

is the mass of the unmixed zone per unit of gas volume given by:

Figure 6. EQUATION_DISPLAY

ρ^{U M} = ρ [C_{x} (Y_{u, O 2} + Y_{u, H 2} + Y_{u, C O} + Y_{u, N O} + Y_{u, s o o t} + ....) + Y_{f, u m}]

(3842)

where

C_{x}

is a coefficient ratio of the unmixed to mean masses:

Figure 7. EQUATION_DISPLAY

C_{x} = \frac{Y_{O 2, u m}}{Y_{u, O 2}} = \frac{Y_{i, u m}}{Y_{u, i}}

(3843)

where

Y_{i, u m}

is the mass of all species in the unmixed zone per unit mass of gas.

In ECFM-3Z, all stages of combustion, flame propagation, ignition, and post-flame / emissions are calculated based on the gases in the mixed zone.

Post-Flame and Emissions Model

Any species that is present in the burnt gases can undergo further reactions. In ECFM-3Z there are five sets of reactions in the burnt gases:

Fuel Post-Oxidation Chemistry
Dissociation and Radical Formation Chemistry
$C O ⇌ C O_{2}$ Kinetics Chemistry
NO Chemistry
Soot Chemistry

The initial composition in the burnt gases

Y_{b}^{m}

is computed from:

Figure 8. EQUATION_DISPLAY

Y_{b}^{m} = \frac{Y^{m} - (1 - c) Y_{u}^{m}}{c}

(3844)

Fuel Post-Oxidation Chemistry: When fuel evaporates into burnt gases (progress variable $c$ = 1) or when there is insufficient oxygen to even burn the existing fuel partially into CO ( $Φ > Φ_{2}$ ), an additional fuel species $Y_{f b}$ is created. However, unlike $Y_{f u}$ , this will not burn in premixed mode, but behind the flame (diffusion-mode combustion). The oxidation reaction is:; Figure 9. EQUATION_DISPLAY

${υ_{F_{b}} F}_{b} + υ_{O_{2}} O_{2} \Rightarrow υ_{{C O}_{2}} C O_{2} + υ_{H_{2} O} H_{2} O$

(3845); The eddy break-up assumption is used to calculate the fuel burning rate:; Figure 10. EQUATION_DISPLAY

${\dot{ω}}_{f b} = C \frac{ε}{k} \min [Y_{f b}^{m}, Y_{O 2}^{m} / υ_{O 2}]$

(3846); where $υ_{O 2}$ is the stoichiometric coefficient for O₂ in Eqn. (3845). The progress variable $c$ is defined as:
Figure 11. EQUATION_DISPLAY

$c = 1 - \frac{Y_{f}}{Y_{u, f}}$

(3847); where $Y_{u, f}$ is the mean fuel tracer mass fraction.
Dissociation and Radical Formation Chemistry: Dissociation effects are not negligible at high temperatures and must therefore be taken into account. A set of reactions involving species that don't contain carbon is implemented, in which the species are assumed to be at equilibrium. The reactions are:
Figure 12. EQUATION_DISPLAY

$N_{2} ⇌ 2 N$
(3848)
Figure 13. EQUATION_DISPLAY

$O_{2} ⇌ 2 O$
(3849)
Figure 14. EQUATION_DISPLAY

$H_{2} ⇌ 2 H$
(3850)
Figure 15. EQUATION_DISPLAY

$2 O H ⇌ O_{2} + H_{2}$
(3851)
Figure 16. EQUATION_DISPLAY

$2 H_{2} O ⇌ O_{2} + 2 H_{2}$
(3852)
where the first three reactions model the dissociation of bi-atomic molecules into their respective atoms, which will then be used in other post-flame mechanisms—such as the NO and CO₂ oxidation kinetics.
$C O ⇌ C O_{2}$ Kinetics Chemistry: The following reaction mechanism is used:
Figure 17. EQUATION_DISPLAY

$C O + O H ⇌ {C O}_{2} + H$
(3853)
NO Chemistry: The extended 3-step Zeldovich mechanism is solved. The relevant reactions are:
Figure 18. EQUATION_DISPLAY

$O + N_{2} ⇌ N + N O$
(3854)
Figure 19. EQUATION_DISPLAY

$O_{2} + N ⇌ O + N O$
(3855)
Figure 20. EQUATION_DISPLAY

$N + O H ⇌ H + N O$
(3856)
Soot Chemistry: Soot chemistry is based on the competing principals of formation and oxidation:
Figure 21. EQUATION_DISPLAY

$\frac{d}{d t} [S O O T] = \frac{d}{d t} {[S O O T]}_{f} - {\frac{d}{d t} [S O O T]}_{o x}$

(3857)
with
Figure 22. EQUATION_DISPLAY

$\frac{d}{d t} {[S O O T]}_{f} = A_{s f e r c} [F_{b}] p^{0.5} e^{- T_{s f e r c} / T}$

(3858)
where $A_{s f e r c}$ and $T_{s f e r c}$ are tuning parameters.
Figure 23. EQUATION_DISPLAY

${\frac{d}{d t} [S O O T]}_{o x} = \frac{12}{ρ_{s o o t} D_{s o o t}} [S O O T] R_{t}$

(3859); In the above equations, [ ] indicates molar concentration, $ρ_{s o o t}$ is the soot particle density, $D_{s o o t}$ is the soot particle diameter, and $R_{t}$ is the net reaction rate of the intermediate reactive sites.; The species mass fractions are updated by:
Figure 24. EQUATION_DISPLAY

$Y^{m} = (1 - c) Y_{u}^{m} + c Y_{b}^{m}$

(3860)
Other Considerations: Apart from the reactions shown under Dissociation and Radical Formation Chemistry, the post-flame chemistry is frozen when the burnt gas temperature $T_{b}$ satisfies $T_{b} < T_{c u t e}$ , where $T_{c u t e}$ is an empirical parameter.