8.2.1 Mixture Fraction Theory

Definition of the Mixture Fraction

The basis of the non-premixed modeling approach is that under a certain set of simplifying assumptions, the instantaneous thermochemical state of the fluid is related to a conserved scalar quantity known as the mixture fraction, $f$. The mixture fraction can be written in terms of the atomic mass fraction as [ 319]

where $Z_i$ is the elemental mass fraction for element, $i$. The subscript ox denotes the value at the oxidizer stream inlet and the subscript fuel denotes the value at the fuel stream inlet. If the diffusion coefficients for all species are equal, then Equation  8.2-1 is identical for all elements, and the mixture fraction definition is unique. The mixture fraction is thus the elemental mass fraction that originated from the fuel stream.

If a secondary stream (another fuel or oxidant, or a non-reacting stream) is included, the fuel and secondary mixture fractions are simply the elemental mass fractions of the fuel and secondary streams, respectively. The sum of all three mixture fractions in the system (fuel, secondary stream, and oxidizer) is always equal to 1:

This indicates that only points on the plane ABC (shown in Figure  8.2.1) in the mixture fraction space are valid. Consequently, the two mixture fractions, $f_{\rm fuel}$ and $f_{\rm sec}$, cannot vary independently; their values are valid only if they are both within the triangle OBC shown in Figure  8.2.1.

Figure 8.2.1: Relationship of $f_{\rm fuel}$, $f_{\rm sec}$, and $f_{\rm ox}$

Figure 8.2.2: Relationship of $f_{\rm fuel}$, $f_{\rm sec}$, and $p_{\rm sec}$

ANSYS FLUENT discretizes the triangle OBC as shown in Figure  8.2.2. Essentially, the primary mixture fraction, $f_{\rm fuel}$, is allowed to vary between zero and one, as for the single mixture fraction case, while the secondary mixture fraction lies on lines with the following equation:

where $p_{\rm sec}$ is the normalized secondary mixture fraction and is the value at the intersection of a line with the secondary mixture fraction axis. Note that unlike $f_{\rm sec}$, $p_{\rm sec}$ is bounded between zero and one, regardless of the $f_{\rm fuel}$ value.

An important characteristic of the normalized secondary mixture fraction, $p_{\rm sec}$, is its assumed statistical independence from the fuel mixture fraction, $f_{\rm fuel}$. Note that unlike $f_{\rm sec}$, $p_{\rm sec}$ is not a conserved scalar. This normalized mixture fraction definition, $p_{\rm sec}$, is used everywhere in ANSYS FLUENT when prompted for Secondary Mixture Fraction except when defining the rich limit for a secondary fuel stream, which is defined in terms of $f_{\rm sec}$.

Transport Equations for the Mixture Fraction

Under the assumption of equal diffusivities, the species equations can be reduced to a single equation for the mixture fraction, $f$. The reaction source terms in the species equations cancel (since elements are conserved in chemical reactions), and thus $f$ is a conserved quantity. While the assumption of equal diffusivities is problematic for laminar flows, it is generally acceptable for turbulent flows where turbulent convection overwhelms molecular diffusion. The Favre mean (density-averaged) mixture fraction equation is

The source term $S_m$ is due solely to transfer of mass into the gas phase from liquid fuel droplets or reacting particles (e.g., coal). $S_{\rm user}$ is any user-defined source term.

In addition to solving for the Favre mean mixture fraction, ANSYS FLUENT solves a conservation equation for the mixture fraction variance, $\overline{f^{'2}}$ [ 152]:

where $f' = f- \overline{f}$. The default values for the constants $\sigma_t$, $C_g$, and $C_d$ are 0.85, 2.86, and 2.0, respectively, and $S_{\rm user}$ is any user-defined source term.

The mixture fraction variance is used in the closure model describing turbulence-chemistry interactions (see Section  8.2.2).

For a two-mixture-fraction problem, $\overline{f_{\rm fuel}}$ and $\overline{f_{\rm fuel}^{'2}}$ are obtained from Equations  8.2-4 and 8.2-5 by substituting $\overline{f_{\rm fuel}}$ for $\overline{f}$ and $\overline{f_{\rm fuel}^{'2}}$ for $\overline{f^{'2}}$. $\overline{f_{\rm sec}}$ is obtained from Equation  8.2-4 by substituting $\overline{f_{\rm sec}}$ for $\overline{f}$. $\overline{p_{\rm sec}}$ is then calculated using Equation  8.2-3, and $\overline{p_{\rm sec}^{'2}}$ is obtained by solving Equation  8.2-5 with $\overline{p_{\rm sec}}$ substituted for $\overline{f}$. To a first-order approximation, the variances in $\overline{p_{\rm sec}}$ and $\overline{f_{\rm sec}}$ are relatively insensitive to $\overline{f_{\rm fuel}}$, and therefore the equation for $\overline{p_{\rm sec}^{'2}}$ is essentially the same as $\overline{f_{\rm sec}^{'2}}$.

The equation for $\overline{p_{\rm sec}^{'2}}$ instead of $\overline{f_{\rm sec}^{'2}}$ is valid when the mass flow rate of the secondary stream is relatively small compared with the total mass flow rate.

The Non-Premixed Model for LES

For Large Eddy Simulations, transport equation is not solved for the mixture fraction variance. Instead, it is modeled as

where
	$C_{\rm var}$	=	constant
	$L_{\rm s}$	=	subgrid length scale (see Equation  4.11-16)

The constant $C_{\rm var}$ is computed dynamically when the Dynamic Stress option is enabled in the Viscous dialog box, else a constant value (with a default of 0.5) is used.

If the Dynamic Scalar Flux option is enabled, the turbulent Sc ( $\sigma_t$ in Equation  8.2-4) is computed dynamically.

Mixture Fraction vs. Equivalence Ratio

The mixture fraction definition can be understood in relation to common measures of reacting systems. Consider a simple combustion system involving a fuel stream (F), an oxidant stream (O), and a product stream (P) symbolically represented at stoichiometric conditions as

where $r$ is the air-to-fuel ratio on a mass basis. Denoting the equivalence ratio as $\phi$, where

the reaction in Equation  8.2-7, under more general mixture conditions, can then be written as

Looking at the left side of this equation, the mixture fraction for the system as a whole can then be deduced to be

Equation  8.2-10 allows the computation of the mixture fraction at stoichiometric conditions ( $\phi$ = 1) or at fuel-rich conditions (e.g., $\phi$ $>$ 1), or fuel-lean conditions (e.g., $\phi$ $<$ 1).

Relationship of $f$ to Species Mass Fraction, Density, and Temperature

The power of the mixture fraction modeling approach is that the chemistry is reduced to one or two conserved mixture fractions. Under the assumption of chemical equilibrium, all thermochemical scalars (species fractions, density, and temperature) are uniquely related to the mixture fraction(s).

For a single mixture fraction in an adiabatic system, the instantaneous values of mass fractions, density, and temperature depend solely on the instantaneous mixture fraction, $f$:

If a secondary stream is included, the instantaneous values will depend on the instantaneous fuel mixture fraction, $f_{\rm fuel}$, and the secondary partial fraction, $p_{\rm sec}$:

In Equations  8.2-11 and 8.2-12, $\phi_i$ represents the instantaneous species mass fraction, density, or temperature. In the case of non-adiabatic systems, the effect of heat loss/gain is parameterized as

for a single mixture fraction system, where $H$ is the instantaneous enthalpy (see Equation  5.2-7).

If a secondary stream is included,

Examples of non-adiabatic flows include systems with radiation, heat transfer through walls, heat transfer to/from discrete phase particles or droplets, and multiple inlets at different temperatures. Additional detail about the mixture fraction approach in such non-adiabatic systems is provided in Section  8.2.3.

In many reacting systems, the combustion is not in chemical equilibrium. ANSYS FLUENT offers several approaches to model chemical non-equilibrium, including the finite-rate (see Section  7.1.2), EDC (see Section  7.1.2), and PDF transport (see Chapter  11) models, where detailed kinetic mechanisms can be incorporated.

There are three approaches in the non-premixed combustion model to simulate chemical non-equilibrium. The first is to use the Rich Flammability Limit (RFL) option in the Equilibrium model, where rich regions are modeled as a mixed-but-unburnt mixture of pure fuel and a leaner equilibrium burnt mixture (see this section in the separate User's Guide). The second approach is the Steady Laminar Flamelet model, where chemical non-equilibrium due to diffusion flame stretching by turbulence can be modeled. The third approach is the Unsteady Laminar Flamelet model where slow-forming product species that are far from chemical equilibrium can be modeled. See Sections  8.4 and 8.6 for details about the Steady and Unsteady Laminar Flamelet models in ANSYS FLUENT.