Regression

In order to simulate a multi-cycle analysis, the products of combustion are moved into the exhaust gas recirculation (EGR) and the fuel in the burnt gases is moved into fuel in the unburnt gases. This process ensures that the next cycle begins with unburnt gases.

ECFM-3Z Regression

In multiple cycle simulations, the progress variable should be reset to zero, otherwise this would mean that the main combustion occurs unconditionally in diffusion (post-flame) mode.

A method to regress the progress variable back to zero is implemented whereby the limiting line between unburnt and burnt gases is shifted towards the burnt gases whenever the gas temperature decreases below a certain value

T_{c u}

(default 1200K). The effect of this is to have only unburnt gases in the local mixture. Species that exist only in the burnt gases (O, H, OH, N, F_b) are recombined into species that exist in the unburnt region (O2, H2, N2, F_u, respectively). The result is that the progress variable returns to zero and that what were before products of combustion are now EGR constituents. The mass fractions of species

i

and their tracers

T_{i}

are changed at each time step according to:

图 1. EQUATION_DISPLAY

Y_{u, i} \leftarrow Y_{u, i} + D D (Y_{i}^{S} - Y_{u, i})

(3932)

where:

图 2. EQUATION_DISPLAY

D D = 0.5 (1 - 1 \times 10^{- 5 / \max [40, 0.6666 / (R P M d t) 1}) (1 + \tanh [\frac{(T_{c u} - 200 - T_{b})}{300}])

(3933)

$Y_{i}^{S}$ is the mass fraction of species $i$ after recombination of the species that exist only in the burnt gas region. For example, O and OH are recombined into O₂, H and OH into H₂, and N into N₂. For the fuel species:

图 3. EQUATION_DISPLAY

Y_{u, f} \leftarrow Y_{u, f} + D D (Y_{f u} + Y_{f b} + Y_{u, f})

(3934)

图 4. EQUATION_DISPLAY

Y_{f b} \leftarrow Y_{f b} - Y_{f b} D D

(3935)

图 5. EQUATION_DISPLAY

Y_{f u} \leftarrow Y_{f u} + Y_{f b} D D

(3936)

This model does not essentially change the mean mixture composition (apart from the fractions of O, H, N, OH which are usually small); it just moves the products of combustion into EGR and the fuel in the burnt gases into fuel in the unburnt gases (thus the return of the progress variable $c$ to zero). This is particularly useful in a multi-cycle analysis where the mixture must be at the unburnt state before the next cycle commences.

Model Extension to Multiple and Multi-Component Fuels

The ECFM-3Z model extension for multiple or multi-component fuels is based on the assumption that the single flame front concept with a single definition of flame surface density and progress variable remains valid. The fuel taking part in the reaction process is now a mean fuel C_nH_mO_p whose mass fraction is equal to:

图 6. EQUATION_DISPLAY

Y_{F} = \underset{i c}{Σ} Y_{f, i c}

(3937)

This fuel has a molecular weight of:

图 7. EQUATION_DISPLAY

θ = \frac{\underset{i c}{Σ} Y_{f, i c}}{Σ Y_{f, i c} / M_{f, i c}}

(3938)

a mean C, H, O content of:

图 8. EQUATION_DISPLAY

{n, m, p} = \frac{\underset{i c}{Σ} Y_{f, i c} {n, m, p}_{i c} / M_{f, i c}}{\underset{i c}{Σ} Y_{f, i c} / M_{f, i c}}

(3939)

and a mean enthalpy at 0K of:

图 9. EQUATION_DISPLAY

h_{f}^{0} = \frac{\underset{i c}{Σ} Y_{f, i c} h_{f, i c}^{0} / M_{f, i c}}{\underset{i c}{Σ} Y_{f, i c} / M_{f, i c}}

(3940)

Reconstruction of individual fuel burning rates based on the assumption of a unique progress variable leads to [815]:

图 10. EQUATION_DISPLAY

\frac{{\dot{ω}}_{i c}}{Y_{f, i c}} = \frac{\dot{ω}}{Y_{F}} i c = 1, 2, ...., N C

(3941)

where ${\dot{ω}}_{i c}$ is the fuel consumption rate in the different zones of fuel component $i c$ , and $\dot{ω}$ is the overall fuel consumption rate in the same zones (see Eqn. (3931) and Eqn. (3846)). For the auto-ignition / knock stage, complex chemistry libraries are used for the auto-ignition combustion process. Such libraries are of the form:

图 11. EQUATION_DISPLAY

{\dot{ω}}_{A I} = f (T_{u}, p, X_{e g r}, Φ, T_{u}^{'}, F F)

(3942)

where

T_{u}

is the conditional temperature in the unburnt gases,

p

is the thermodynamic pressure, and

X_{e g r}

is the EGR mole fraction:

图 13. EQUATION_DISPLAY

X_{e g r} = 1 - \frac{4.76 Y_{O 2}^{u} / W_{O 2}}{Σ_{i \neq f u e l}^{N_{s p e c i e s}} \frac{Y_{i}^{u}}{W_{i}}}

(3944)

Φ

is the equivalence ratio, and

T_{u}^{'}

is the conditional temperature fluctuations in the unburnt gases FF is the fraction:

图 14. EQUATION_DISPLAY

F F = \frac{Y_{F, 1}}{Y_{F, 1} + Y_{F, 2}}

(3945)

Libraries are also available for determining the laminar flame speed $U_{l}$ :

图 15. EQUATION_DISPLAY

U_{l} = f (T_{u}, p, X_{e g r}, Φ, T_{u}^{'}, F F)

(3946)

Additionally, the conditional-averaged fractions of species in the unburnt gases are introduced by solving equations of the form:

图 16. EQUATION_DISPLAY

\frac{\partial ρ T_{i}}{\partial t} + \nabla\cdot (ρ v T_{i}) - \nabla\cdot (Γ \nabla T_{i}) - 2 Γ \nabla T_{i} \cdot \frac{\nabla b}{b} = 0

(3947)

where $b$ is the regress variable $b = 1 - c$ and $Γ$ is the effective diffusivity given by:

图 17. EQUATION_DISPLAY

Γ = ρ D + μ_{t} / {S c}_{t}

(3948)

ECFM-CLEH Regression

In order to simulate a multi-cycle analysis, the ECFM-CLEH regression model resets values of progress variable to zero every cycle—at a specified time or crank angle.

The following steps occur:

Combustion flags are re-initialized—these allow a better efficiency of the model.
Products of combustion are converted into Exhaust Gas Recirculation (EGR).
The EGR composition is updated in order to obtain a global conservation of species and mixture molecular weight.
In every cell for which the global combustion progress variable is greater than zero, O2 and fuel mass fractions are set to zero.
The flame surface density (FSD) variable $Σ$ is set to 0 in the entire domain.
Pressure, Enthalpy, and/or Temperature are updated—depending upon the conservation options that are selected.
Combustion progress variables are set to zero by forcing unburnt fuel mass fraction to equal fuel mass fraction—with this reset approach, this is equal to zero.

Specified Burn Rate Regression

In the Specified Burn Rate combustion model, a conservative scalar approach is used to track the mixture composition by defining three streams: fuel, EGR and oxidizer. The unburnt state of the mixture composition is defined with these three streams. In order to simulate a multi-cycle analysis, the products of combustion are moved into the exhaust gas recirculation (EGR). By resetting the progress variable from 1 to 0, the fuel in the burnt gases is moved to the fuel in the unburnt gases. To ensure that the next cycle begins with unburnt gases, this process converts the products of combustion to the EGR stream, and resets the fuel and oxidizer stream mass fractions based on remaining fuel/oxidizer mass fractions computed from fuel-lean/rich/stoichiometric conditions.