'''Mixture fraction''' (<math>Z</math>) is a quantity used in combustion studies that measures the mass fraction of one stream of a mixture formed by two feed streams, one the fuel stream and the other the oxidizer stream.<ref>Williams, F. A. (2018). Combustion theory. CRC Press.</ref><ref>Peters, N. (2001). Turbulent combustion.</ref> Both the feed streams are allowed to have inert gases.<ref>Peters, N. (1992). Fifteen lectures on laminar and turbulent combustion. Ercoftac Summer School, 1428, 245.</ref> The mixture fraction definition is usually normalized such that it approaches unity in the fuel stream and zero in the oxidizer stream.<ref>Liñán, A., & Williams, F. A. (1993). Fundamental aspects of combustion.</ref> The mixture-fraction variable is commonly used as a replacement for the physical coordinate normal to the flame surface, in nonpremixed combustion.
==Definition==
Assume a two-stream problem having one portion of the boundary the fuel stream with fuel mass fraction <math>Y_F=Y_{F,F}</math> and another portion of the boundary the oxidizer stream with oxidizer mass fraction <math>Y_{O}=Y_{O,O}</math>. For example, if the oxidizer stream is air and the fuel stream contains only the fuel, then <math>Y_{O,O}=0.232</math> and <math>Y_{F,F}=1</math>. In addition, assume there is no oxygen in the fuel stream and there is no fuel in the oxidizer stream. Let <math>s</math> be the mass of oxygen required to burn unit mass of fuel (for hydrogen gas, <math>s=8</math> and for <math>\mathrm{C}_m\mathrm{H}_n</math> alkanes, <math>s=32(m+n/4)/(12m+n)</math><ref>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.</ref>). Introduce the scaled mass fractions as <math>y_F=Y_F/Y_{F,F}</math> and <math>y_O = Y_O/Y_{O,O}</math>. Then the mixture fraction is defined as
:<math>Z = \frac{Sy_F-y_{O}+1}{S+1}</math>
where
:<math>S = \frac{sY_{F,F}}{Y_{O,O}}</math>
is the stoichiometry parameter, also known as the overall equivalence ratio. On the fuel-stream boundary, <math>y_F=1</math> and <math>y_O=0</math> since there is no oxygen in the fuel stream, and hence <math>Z=1</math>. Similarly, on the oxidizer-stream boundary, <math>y_F=0</math> and <math>y_O=1</math> so that <math>Z=0</math>. Anywhere else in the mixing domain, <math>0<Z<1</math>. The mixture fraction is a function of both the spatial coordinates <math>\mathbf{x}</math> and the time <math>t</math>, i.e., <math>Z=Z(\mathbf{x},t).</math>
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 <math>Z(\mathbf{x},t)=Z_s</math>, where <math>Z_s</math> is called as the stoichiometric mixture fraction which is obtained by setting <math>Y_F=Y_{O}=0</math> (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 <math>Z</math> to obtain
:<math>Z_s = \frac{1}{S+1}</math>.
==Relation between local equivalence ratio and mixture fraction==
When there is no chemical reaction, or considering the unburnt side of the flame, the mass fraction of fuel and oxidizer are <math>y_{F,u}= Z</math> and <math>y_{O,u}= 1- Z</math> (the subscript <math>u</math> denotes unburnt mixture). This allows to define a local fuel-air equivalence ratio <math>\phi</math>
:<math>\phi= \frac{sY_{F,u}}{Y_{O,u}}=\frac{Sy_{F,u}}{y_{O,u}}.</math>
The local equivalence ratio is an important quantity for partially premixed combustion. The relation between local equivalence ratio and mixture fraction is given by
:<math>\phi = \frac{SZ}{1-Z} \qquad \Rightarrow \qquad Z = \frac{\phi}{S+\phi}.</math>
The stoichiometric mixture fraction <math>Z_s</math> defined earlier is the location where the local equivalence ratio <math>\phi=1</math>.
==Scalar dissipation rate==
In turbulent combustion, a quantity called the scalar dissipation rate <math>\chi</math> with dimensional units of that of an inverse time is used to define a characteristic diffusion time. Its definition is given by
:<math>\chi = 2 D |\nabla Z|^2</math>
where <math>D</math> is the diffusion coefficient of the scalar. Its stoichiometric value is <math>\chi_s = 2D_s|\nabla Z|^2_s</math>.
==Liñán's mixture fraction==
Amable Liñán introduced a modified mixture fraction in 1991<ref>A. 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. 11–29</ref><ref>Linán, A. (2001). Diffusion-controlled combustion. In Mechanics for a New Mellennium (pp. 487-502). Springer, Dordrecht.</ref> that is appropriate for systems where the fuel and oxidizer have different Lewis numbers. If <math>Le_F</math> and <math>Le_{O_2}</math> are the Lewis number of the fuel and oxidizer, respectively, then Liñán's mixture fraction is defined as
:<math>\tilde Z = \frac{\tilde Sy_F-y_{O}+1}{\tilde S+1}</math>
where
:<math>\tilde S = \frac{Le_O S}{Le_F}.</math>
The stoichiometric mixture fraction <math>\tilde Z_s</math> is given by
:<math>\tilde Z_s = \frac{1}{\tilde S+1}</math>.
==References== {{reflist|30em}}
Category:Fluid dynamics Category:Combustion