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 \mathrm{C}_m\mathrm{H}_n alkanes, s=32(m+n/4)/(12m+n)). 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 = \frac{Sy_F-y_{O}+1}{S+1} where :S = \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, 0. The mixture fraction is a function of both the spatial coordinates \mathbf{x} and the time t, i.e., Z=Z(\mathbf{x},t). 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(\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 = \frac{1}{S+1}. ==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 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 \phi :\phi= \frac{sY_{F,u}}{Y_{O,u}}=\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 :\phi = \frac{SZ}{1-Z} \qquad \Rightarrow \qquad Z = \frac{\phi}{S+\phi}. The stoichiometric mixture fraction Z_s defined earlier is the location where the local equivalence ratio \phi=1. ==Scalar dissipation rate==

In turbulent combustion, a quantity called the scalar dissipation rate \chi with dimensional units of that of an inverse time is used to define a characteristic diffusion time. Its definition is given by :\chi = 2 D |\nabla Z|^2 where D is the diffusion coefficient of the scalar. Its stoichiometric value is \chi_s = 2D_s|\nabla Z|^2_s. ==Liñán's mixture fraction==

Amable Liñán introduced a modified mixture fraction in 1991 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 :\tilde Z = \frac{\tilde Sy_F-y_{O}+1}{\tilde S+1} where :\tilde S = \frac{Le_O S}{Le_F}. The stoichiometric mixture fraction \tilde Z_s is given by :\tilde Z_s = \frac{1}{\tilde S+1}. ==References==