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15.2.1 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 [ 336]

 f = \frac{Z_i - Z_{i,{\rm ox}}}{Z_{i,{\rm fuel}} - Z_{i, {\rm ox}}} (15.2-1)

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  15.2-1 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:

 f_{\rm fuel} + f_{\rm sec} + f_{\rm ox} = 1 (15.2-2)

This indicates that only points on the plane ABC (shown in Figure  15.2.1) 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  15.2.1.

Figure 15.2.1: Relationship of $f_{\rm fuel}$, $f_{\rm sec}$, and $f_{\rm ox}$

Figure 15.2.2: Relationship of $f_{\rm fuel}$, $f_{\rm sec}$, and $p_{\rm sec}$

FLUENT discretizes the triangle OBC as shown in Figure  15.2.2. 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:

 f_{\rm sec} = p_{\rm sec} \times (1 - f_{\rm fuel}) (15.2-3)

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. The normalized mixture fraction definition for the second scalar variable is used everywhere 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, 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

 \frac{\partial}{\partial t} (\rho \overline{f}) + \nabla \cd... ..._t}{\sigma_t} \nabla \overline{f} \right) + S_m + S_{\rm user} (15.2-4)

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, FLUENT solves a conservation equation for the mixture fraction variance, $\overline{f^{'2}}$ [ 167]:

 \frac{\partial}{\partial t} \left(\rho \overline{f^{'2}} \ri... ...- C_d \rho \frac{\epsilon}{k} \overline{f^{'2}} + S_{\rm user} (15.2-5)

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  15.2.2).

For a two-mixture-fraction problem, $\overline{f_{\rm fuel}}$ and $\overline{f_{\rm fuel}^{'2}}$ are obtained from Equations  15.2-4 and 15.2-5 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  15.2-4 by substituting $\overline{f_{\rm sec}}$ for $\overline{f}$. $\overline{p_{\rm sec}}$ is then calculated using Equation  15.2-3, and $\overline{p_{\rm sec}^{'2}}$ is obtained by solving Equation  15.2-5 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

A transport equation is not solved for the mixture fraction variance. Instead, it is modeled as

 \overline{f^{'2}} = C_{\rm var} L^2_{\rm s} {\vert \nabla \overline{f} \vert}^2 (15.2-6)

  $C_{\rm var}$ = constant
  $L_{\rm s}$ = subgrid length scale (see Equation  12.9-16)

The constant $C_{\rm var}$ is computed dynamically when the Dynamic Stress option is enabled in the Viscous panel, 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  15.2-5) 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

 {\rm F} + r\;{\rm O} \rightarrow (1+r) \; {\rm P} (15.2-7)

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

 \phi =\frac{{\rm (fuel/air)}_{\rm actual}}{{\rm (fuel/air)}_{\rm stoichiometric}} (15.2-8)

the reaction in Equation  15.2-7, under more general mixture conditions, can then be written as

 \phi \; {\rm F} + r\; {\rm O} \rightarrow (\phi +r) \; {\rm P} (15.2-9)

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

 f = \frac{\phi}{\phi + r} (15.2-10)

Equation  15.2-10 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 single mixture fraction, adiabatic systems, the instantaneous values of mass fractions, density, and temperature depend solely on the instantaneous mixture fraction, $f$:

 \phi_i = \phi_i (f) (15.2-11)

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, $p_{\rm sec}$:

 \phi_i = \phi_i (f_{\rm fuel}, p_{\rm sec}) (15.2-12)

In Equations  15.2-11 and 15.2-12, $\phi_i$ 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

 \phi_i = \phi_i (f,H) (15.2-13)

for a single mixture fraction system, where $H$ is the instantaneous enthalpy (see Equation  13.2-7).

If a secondary stream is included,

 \phi_i = \phi_i (f_{\rm fuel}, p_{\rm sec}, H) (15.2-14)

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  15.2.3.

In many reacting systems, the combustion is not in chemical equilibrium. FLUENT offers several approaches to model chemical non-equilibrium, including the finite-rate (see Section  14.1.1), EDC (see Section  14.1.1), and PDF transport (see Chapter  18) 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 Section  15.7.5). 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  15.3 and 15.5 for details about the Steady and Unsteady Laminar Flamelet models in FLUENT.

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