Flame Speed Calculations

Laminar Flame Speed

Simcenter STAR-CCM+ incorporates three correlations for calculating the unstrained laminar flame speed. The Universal Laminar Flame Speed option is recommended for mixtures of multiple fuels—it automatically selects an appropriate laminar flame speed correlation for each individual fuel component in a mixture and then uses the Hirasawa method [771] to calculate the combined laminar flame speed of the blended fuel mixture. For pure fuels, you can use either the correlation by Metghalchi and Keck [768]—which is only valid for equivalence ratios

ϕ

that are close to 1—or, the correlation by Gülder [769] which is recommended for most general applications. When using the a Flamelet model with the Inert Stream model, the Laminar Flame Speed correlation accounts for the influence of any exhaust gas recirculation (EGR) that is present. Note that when using the Inert Stream model, the inert mass is considered as EGR.

Universal Laminar Flame Speed Option

Simcenter STAR-CCM+ automatically identifies the most appropriate laminar flame speed correlation for each individual fuel in a mixture of fuels.

The following table shows the types of fuel components in a mixture that are recognised and the respective individual laminar flame speed correlation that is used.


Fuel Component	LFS Correlation
Methane / Alcohols	Gülder See Eqn. (3579).
Hydrogen	Verhelst and UniMORE See Verhelst and UniMORE.
Ammonia	Goldman The Goldmann correlation calculates the laminar flame speed of mixtures of ammonia and air, as follows: Figure 1. EQUATION_DISPLAY $S_{l} = S_{l, 0} T_{n}^{α} P_{n}^{β} κ$ (3570) where $S_{l, 0}$ is the laminar flame speed at reference conditions, $T_{n}$ is the normalized temperature, $P_{n}$ is the normalized pressure, and $κ$ is the correction factor by experimental data.
Other Hydrocarbons	Methgalchi and Keck See Eqn. (3573).

Once Simcenter STAR-CCM+ determines the individual laminar flame speed correlations for each of the individual fuels in a mixture, it then uses the Hirasawa method [771] to calculate the combined laminar flame speed correlation for the multi-component fuel blend and air mixture as follows:

Figure 2. EQUATION_DISPLAY

S_{l, m} = \exp (\frac{- T_{a, m}}{T_{f, m}})

(3571)

where

T_{f, m}

is the adiabatic flame temperature of the mixture and

T_{a, m}

is the activation temperature of the mixture.

The equivalence ratio of a multi-component fuel blend and air mixture is calculated to represent the equivalence ratios of each of the components, regardless of their concentrations:

Figure 3. EQUATION_DISPLAY

ϕ_{m} = \frac{Σ_{i}^{n} \frac{X_{F, i}}{X_{O}}}{{(Σ_{i}^{n} \frac{X_{F, i}}{X_{O}})}_{s t}}

(3572)

where

X_{F, i}

is the mole fraction of the

i

th fuel component in the unburnt mixture, and

X_{O}

is the mole fraction of the oxidizer.

{(Σ_{i}^{n} \frac{X_{F, i}}{X_{O}})}_{s t}

represents the stoichiometric fuel blend to oxidizer molar ratio, assuming complete combustion.

Metghalchi

The correlation that is proposed by Metghalchi and Keck [768] is calculated as follows:

Figure 4. EQUATION_DISPLAY

S_{l} = S_{l 0} {(\frac{T_{u}}{T_{0}})}^{α} {(\frac{p}{p_{0}})}^{β} (1 - 2.1 Y_{E G R})

(3573)

where

p

is the pressure,

T

is the temperature, the subscripts

0

and

u

denote reference and unburnt gas properties, respectively,

S_{l}

is the laminar flame speed, and

Y_{E G R}

is the mass fraction of any exhaust gas recirculation (EGR) that is present. The default value for the reference temperature

T_{0}

298 K

and for the reference pressure

P_{0}

101325 P a

. The reference laminar flame speed

S_{l 0}

and the exponents

α

and

β

depend on the equivalence ratio

ϕ

of the fuel. The exponents are defined as:

Figure 5. EQUATION DISPLAY

α = 2.18 - 0.8 (ϕ - 1)

(3574)

and:

Figure 6. EQUATION DISPLAY

β = - 0.16 + 0.22 (ϕ - 1)

(3575)

The reference laminar speed

S_{l 0}

is a weak function of fuel type and is fit by a second-order polynomial of the form:

Figure 7. EQUATION DISPLAY

S_{l 0} = B_{m} + B_{2} {(ϕ - ϕ_{m})}^{2}

(3576)

where coefficients

ϕ_{m}

B_{m}

and

B_{2}

are specified in the following table:


Fuel	$ϕ_{m}$	$B_{m}$ [cm/s]	$B_{2}$ [cm/s]
Methanol	1.11	36.92	-140.51
Propane	1.08	34.22	-138.65
Isooctane	1.13	26.32	-84.72

If $0.4 \leq p \leq 50$ atm, $300 \leq T \leq 700$ K, and $0.8 \leq ϕ \leq 1.5$ , the authors claim that the laminar flame speed is within 10% of the measured data. $ϕ \geq 0.65$ is recommended because:

None of the combustion models can accurately predict burning in lean mixtures.
The laminar flame speed correlation function progressively diverges from the experimentally observed data in lean mixtures.

Verhelst and UniMORE

The Verhelst and UniMORE combined model calculates the laminar burning velocity of mixtures of hydrogen, air, and EGR at pressures and temperatures that are typically found in engines.

For pressures below 40 bar, Simcenter STAR-CCM+ automatically uses the Verhelst correlation [773], defined as follows:

Figure 8. EQUATION_DISPLAY

S_{l} (λ, p, T, f) = S_{l, 0} (λ, p) {(\frac{T}{T_{0}})}^{α (λ, p)} F (λ, p, T, f)

(3577)

where

λ

is the air to fuel equivalence ratio,

α

is the temperature exponent, and

f

is the residual gas content.

For pressures ranging from 40 bar to 130 bar, the UniMORE correlation from [772] is used to calculate the laminar burning velocity.

The UniMORE model provides laminar ﬂame speed values that are derived using a dedicated ﬁtting procedure.

For lean to rich mixtures (equivalence ratios

0.7 \leq ϕ \leq 1.5

) the laminar burning velocity is defined with a simple correlation method:

Figure 9. EQUATION_DISPLAY

S_{l} (ϕ) = \sum_{i} a_{i} {(\ln (ϕ))}^{i} \cdot {(\frac{T_{u}}{T_{0}})}^{\sum_{i} b_{i} {(\ln (ϕ))}^{i}} \cdot {(\frac{p}{p_{0}})}^{\sum_{i} c_{i} {(\ln (ϕ))}^{i}}

(3578)

where

a_{i}, b_{i}, c_{i}; i = 0...5

are polynomial fitting coefficients.

For ultra-lean (

0.4 \leq ϕ < 0.7

) mixtures, the laminar burning velocity is obtained by weighting pressure and temperature with scaling factors [772].

Gülder

The second laminar flame speed correlation, which Gülder proposed [769], is calculated as follows:

Figure 10. EQUATION_DISPLAY

S_{l} = Z W ϕ^{η} \exp [- ξ {(ϕ - 1.075)}^{2}] {(\frac{T_{u}}{T_{0}})}^{α} {(\frac{P}{P_{0}})}^{β} (1 - 2.1 Y_{E G R})

(3579)

where

Y_{E G R}

is the mass fraction of any exhaust gas recirculation (EGR) that is present, and

Z

W

η

ξ

α

and

β

are fuel-dependent constants that are defined in the table below:


Fuel	$Z$	$W$	$η$	$ξ$	$α$	$β$
Fuel	$Z$	$W$	$η$	$ξ$	$α$	$ϕ < 1$	$ϕ > 1$
Methane	1	0.422	0.15	5.18	2.00	–0.5	–0.5
Propane	1	0.446	0.12	4.95	1.77	–0.2	–0.2
Methanol	1	0.492	0.25	5.11	1.75	–0.2/ $\sqrt{ϕ}$	-0.2 $ϕ$
Ethanol	1	0.465	0.25	6.34	1.75	–0.17/ $\sqrt{ϕ}$	-0.17 $ϕ$
Iso-octane	1	0.4658	–0.326	4.48	1.56	–0.22	–0.22

Turbulent Flame Speed

Zimont

Zimont used the following correlation for the turbulent flame speed [805]:

Figure 11. EQUATION_DISPLAY

S_{t} = 0.5 G {(u′)}^{3 / 4} S_{l}^{1 / 2} α_{u}^{- 1 / 4} I_{l}^{1 / 4}

(3580)

Here,

u′

is the turbulent velocity,

S_{l}

is the laminar flame speed (see Laminar Flame Thickness),

α_{u}

is the unburnt thermal diffusivity of the unburnt mixture, and

I_{l}

is the integral turbulent length scale.

The stretch factor

G

takes the stretch effect into account by representing the probability of unquenched flamelets which is obtained by integrating the log-normal distribution of the turbulent dissipation rate:

Figure 12. EQUATION_DISPLAY

G = \frac{1}{2} e r f c [- \frac{1}{\sqrt{2 σ}} (ln \frac{ε_{c r}}{ε} + \frac{σ}{2})]

(3581)

where

e r f c

is a complementary error function and

σ

is the standard deviation of the distribution of

ε

computed with the following equation:

Figure 13. EQUATION_DISPLAY

σ = μ_{s t r} ln (I_{l} / η)

(3582)

where

I_{l}

is the integral turbulent length scale,

η

is the Kolmogorov micro-scale, and

μ_{s t r}

is an empirical model coefficient with a default value of 0.28.

ε_{c r}

is the turbulent dissipation rate at the critical strain rate

g_{c r}

Figure 14. EQUATION_DISPLAY

ε_{c r} = 15 ν g_{c r}^{2}

(3583)

where

ν

is the kinematic viscosity of the fluid.

A high value for

g_{c r}

suggests no occurrence of the flame stretch. One method to compute

g_{c r}

is to assume that it is proportional to the chemical time scale:

Figure 15. EQUATION_DISPLAY

g_{c r} = \frac{B S_{L}^{2}}{α_{u}}

(3584)

where value of constant $B$ is $500$ .

Peters

The Peters correlation [806] for turbulent flame speed has the following form:

Figure 16. EQUATION_DISPLAY

S_{t} = S_{l} (1 + σ_{t})

(3585)

where:

Figure 17. EQUATION_DISPLAY

\begin{matrix} σ_{t} = - A B + \sqrt{{(A B)}^{2} + C \frac{u′ I_{l}}{S_{l} δ_{l}^{0}}} \\ A = \frac{A_{4} B_{3}^{2}}{2 B_{1}} \\ \begin{matrix} B = \frac{I_{l}}{δ_{l}^{0}} c_{e w} \\ C = A_{4} B_{3}^{2} \end{matrix} \end{matrix}

(3586)

where $δ_{l}^{0}$ is the laminar flame thickness, and $c_{e w}$ is Ewald’s corrector which has a default value of 1.0.

$A_{1}$ , $A_{4}$ , $B_{1}$ , and $B_{3}$ are model constants with default settings of 0.37, 0.78, 2.0, and 1.0, respectively.

Note

A wall effect constant is used to model the quenching of the flame at walls by multiplying the constant when computing the turbulent flame speed. By modifying the constant from 0 to 1, the wall quenching effect on the flame in the vicinity of wall boundaries can be adjusted from fully extinguished to no effect.

Flame Speed Multiplier

The flame speed multiplier is a scale factor applied to $S_{l}$ obtained from any of the Laminar Flame Speed methods listed above.