### 7.11.2. Equivalence Ratio Correlation

The equivalence ratio correlation follows the approach by Metghalchi and Keck [124], expressing the laminar burning velocity as a base value at reference conditions, , multiplied by correction factors for preheat and pressure dependencies:

| (7–53)

The exponents for preheat dependency and for pressure dependency are quadratic polynomials in equivalence ratio:

| (7–54)

| (7–55)

When all three coefficients are set to zero, then the preheat dependency or the pressure dependency is disabled. For reference burning velocity, the following options are available:

- Fifth Order Polynomial

- Quadratic Decay

- Beta Function

All three options specify the flammability limits for fuel-lean and for fuel-rich mixtures, and . The burning velocity is set to zero if the local equivalence ratio is out of these bounds.

#### 7.11.2.1. Fifth Order Polynomial

The laminar burning velocity may be specified as a polynomial up to fifth order:

| (7–56)

This polynomial is evaluated on a specified fit range, . Outside this range, the burning velocity is modeled to linearly decay to zero at the flammability limit.

#### 7.11.2.2. Quadratic Decay

For quadratic decay the maximum laminar burning velocity at reference conditions, , and the corresponding equivalence ratio, , are given. For smaller or larger equivalence ratio the burning velocity is modeled to decrease according to a quadratic decay coefficient, :

| (7–57)

This quadratic function is evaluated on a specified fit range, . Outside this range, the burning velocity is modeled to linearly decay to zero at the flammability limit.

#### 7.11.2.3. Beta Function

The beta function correlation sets the maximum laminar burning velocity, , and the corresponding equivalence ratio, . A beta function is used to model the velocity decay to zero at the fuel-lean or fuel-rich flammability limit:

where and .

#### 7.11.2.4. Residual Material Dependency

In order to account for the residual material, an optional correction factor is multiplied to the laminar burning velocity:

| (7–58)

where denotes the molar fraction (volume fraction) of the residual material and is a coefficient function of the equivalence ratio:

| (7–59)

In combination with the residual material model, the values of , and are calculated using the equivalence ratio conditioned to the ‘fresh’ mixture without residual material.

The residual material dependency in the laminar burning velocity correlation is optional. The default residual material dependency coefficient is constant zero, that is, no residual material dependency.

#### 7.11.2.5. Metghalchi and Keck

The correlation by Metghalchi and Keck is based on the equivalence ratio correlation with quadratic decay described above. Predefined sets of coefficients are provided for several hydrocarbon fuels. The fuel type is characterized by the number of carbon atoms in the fuel molecule, here called the fuel carbon index. Table 7.2: Fuel Dependent Coefficients for Metghalchi and Keck Laminar Burning Velocity Correlation lists the coefficients for methane, propane and iso-octane (gasoline), respectively.

Table 7.2: Fuel Dependent Coefficients for Metghalchi and Keck Laminar Burning Velocity Correlation

Carbon Index | Fuel | [m/s] | [m/s] |  |  |
--- | --- | --- | --- | --- | --- | ---
1 | Methane | 0.35 | -1.387 | 1.06 | 0.533 | 1.68
3 | Propane | 0.342 | -1.387 | 1.08 | 0.536 | 2.50
8 | Iso-octane | 0.263 | -0.847 | 1.13 | 0.601 | 3.80

For fuels with other carbon indices the coefficients are obtained by linear interpolation of the provided values. Fit range, preheat dependency and pressure dependency are modeled independent of the fuel (Table 7.3: Common Coefficients for Metghalchi and Keck Laminar Burning Velocity Correlation).

Table 7.3: Common Coefficients for Metghalchi and Keck Laminar Burning Velocity Correlation

0.7 | 1.4 | 2.98 | -0.8 | 0 | -0.38 | 0.22 | 0

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