mixture fraction

Assume a two-stream problem having one portion of the boundary the fuel stream with fuel mass fraction $Y\_F=Y\_\{F,F\}$ and another portion of the boundary the oxidizer stream with oxidizer mass fraction $Y\_\{O\}=Y\_\{O,O\}$. For example, if the oxidizer stream is air and the fuel stream contains only the fuel, then $Y\_\{O,O\}=0.232$ and $Y\_\{F,F\}=1$. In addition, assume there is no oxygen in the fuel stream and there is no fuel in the oxidizer stream. Let $s$ be the mass of oxygen required to burn unit mass of fuel (for hydrogen gas, $s=8$ and for $\backslash mathrm\{C\}\_m\backslash mathrm\{H\}\_n$ alkanes, $s=32(m+n/4)/(12m+n)$Fernández-Tarrazo, E., Sánchez, A. L., Linan, A., & Williams, F. A. (2006). A simple one-step chemistry model for partially premixed hydrocarbon combustion. Combustion and Flame, 147(1-2), 32-38.). Introduce the scaled mass fractions as $y\_F=Y\_F/Y\_\{F,F\}$ and $y\_O\; =\; Y\_O/Y\_\{O,O\}$. Then the mixture fraction is defined as

:$Z\; =\; \backslash frac\{Sy\_F-y\_\{O\}+1\}\{S+1\}$

where

:$S\; =\; \backslash frac\{sY\_\{F,F\}\}\{Y\_\{O,O\}\}$

is the stoichiometry parameter, also known as the overall equivalence ratio. On the fuel-stream boundary, $y\_F=1$ and $y\_O=0$ since there is no oxygen in the fuel stream, and hence $Z=1$. Similarly, on the oxidizer-stream boundary, $y\_F=0$ and $y\_O=1$ so that $Z=0$. Anywhere else in the mixing domain,

Within the mixing domain, there are level surfaces where fuel and oxygen are found to be mixed in stoichiometric proportion. This surface is special in combustion because this is where a diffusion flame resides. Constant level of this surface is identified from the equation $Z(\backslash mathbf\{x\},t)=Z\_s$, where $Z\_s$ is called as the stoichiometric mixture fraction which is obtained by setting $Y\_F=Y\_\{O\}=0$ (since if they were react to consume fuel and oxygen, only on the stoichiometric locations both fuel and oxygen will be consumed completely) in the definition of $Z$ to obtain

:$Z\_s\; =\; \backslash frac\{1\}\{S+1\}$.

When there is no chemical reaction, or considering the unburnt side of the flame, the mass fraction of fuel and oxidizer are $y\_\{F,u\}=\; Z$ and $y\_\{O,u\}=\; 1-\; Z$ (the subscript $u$ denotes unburnt mixture). This allows to define a local fuel-air equivalence ratio $\backslash phi$

:$\backslash phi=\; \backslash frac\{sY\_\{F,u\}\}\{Y\_\{O,u\}\}=\backslash frac\{Sy\_\{F,u\}\}\{y\_\{O,u\}\}.$

The local equivalence ratio is an important quantity for partially premixed combustion. The relation between local equivalence ratio and mixture fraction is given by

:$\backslash phi\; =\; \backslash frac\{SZ\}\{1-Z\}\; \backslash qquad\; \backslash Rightarrow\; \backslash qquad\; Z\; =\; \backslash frac\{\backslash phi\}\{S+\backslash phi\}.$

The stoichiometric mixture fraction $Z\_s$ defined earlier is the location where the local equivalence ratio $\backslash phi=1$.

In turbulent combustion, a quantity called the scalar dissipation rate $\backslash chi$ with dimensional units of that of an inverse time is used to define a characteristic diffusion time. Its definition is given by

:$\backslash chi\; =\; 2\; D\; |\backslash nabla\; Z|^2$

where $D$ is the diffusion coefficient of the scalar. Its stoichiometric value is $\backslash chi\_s\; =\; 2D\_s|\backslash nabla\; Z|^2\_s$.

Amable Liñán introduced a modified mixture fraction in 1991A. Liñán, The structure of diffusion flames, in Fluid Dynamical Aspects of Combustion Theory, M. Onofri and A. Tesei, eds., Harlow, UK. Longman Scientific and Technical, 1991, pp. 11–29Linán, A. (2001). Diffusion-controlled combustion. In Mechanics for a New Mellennium (pp. 487-502). Springer, Dordrecht. that is appropriate for systems where the fuel and oxidizer have different Lewis numbers. If $Le\_F$ and $Le\_\{O\_2\}$ are the Lewis number of the fuel and oxidizer, respectively, then Liñán's mixture fraction is defined as

:$\backslash tilde\; Z\; =\; \backslash frac\{\backslash tilde\; Sy\_F-y\_\{O\}+1\}\{\backslash tilde\; S+1\}$

where

:$\backslash tilde\; S\; =\; \backslash frac\{Le\_O\; S\}\{Le\_F\}.$

The stoichiometric mixture fraction $\backslash tilde\; Z\_s$ is given by

:$\backslash tilde\; Z\_s\; =\; \backslash frac\{1\}\{\backslash tilde\; S+1\}$.

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Category:Fluid dynamics

Category:Combustion