Mixture fraction (${\displaystyle Z}$) is a quantity used incombustion studies that measures themass fraction of one stream of a mixture formed by two feed streams, one the fuel stream and the other the oxidizer stream.^[1][2] Both the feed streams are allowed to have inert gases.^[3] The mixture fraction definition is usually normalized such that it approaches unity in the fuel stream and zero in the oxidizer stream.^[4] The mixture-fraction variable is commonly used as a replacement for the physical coordinate normal to the flame surface, in nonpremixed combustion.

Assume a two-stream problem having one portion of the boundary the fuel stream with fuel mass fraction${\displaystyle Y_F=Y_{F,F}}$ and another portion of the boundary the oxidizer stream with oxidizer mass fraction${\displaystyle Y_O=Y_{O,O}}$. For example, if the oxidizer stream is air and the fuel stream contains only the fuel, then${\displaystyle Y_{O,O}=0.232}$ and${\displaystyle Y_{F,F}=1}$. In addition, assume there is no oxygen in the fuel stream and there is no fuel in the oxidizer stream. Let${\displaystyle s}$ be the mass of oxygen required to burn unit mass of fuel (forhydrogen gas,${\displaystyle s=8}$ and for${\displaystyle {\mathrm{C}}_m{\mathrm{H}}_n}$alkanes,${\displaystyle s=32(m+n/4)/(12m+n)}$^[5]). Introduce the scaled mass fractions as${\displaystyle y_F=Y_F/Y_{F,F}}$ and${\displaystyle y_O=Y_O/Y_{O,O}}$. Then the mixture fraction is defined as

${\displaystyle Z=\frac{S{y}_F-y_O+1}{S+1}}$

where

${\displaystyle S=\frac{s{Y}_{F,F}}{Y_{O,O}}}$

is thestoichiometry parameter, also known as the overallequivalence ratio. On the fuel-stream boundary,${\displaystyle y_F=1}$ and${\displaystyle y_O=0}$ since there is no oxygen in the fuel stream, and hence${\displaystyle Z=1}$. Similarly, on the oxidizer-stream boundary,${\displaystyle y_F=0}$ and${\displaystyle y_O=1}$ so that${\displaystyle Z=0}$. Anywhere else in the mixing domain,. The mixture fraction is a function of both the spatial coordinates${\displaystyle \mathbf{x}}$ and the time${\displaystyle t}$, i.e.,${\displaystyle Z=Z(\mathbf{x},t).}$

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${\displaystyle Z(\mathbf{x},t)=Z_s}$, where${\displaystyle Z_s}$ is called as the stoichiometric mixture fraction which is obtained by setting${\displaystyle Y_F=Y_O=0}$ (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${\displaystyle Z}$ to obtain

${\displaystyle Z_s=\frac{1}{S+1}}$.

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

${\displaystyle \varphi =\frac{s{Y}_{F,u}}{Y_{O,u}}=\frac{S{y}_{F,u}}{y_{O,u}}.}$

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

${\displaystyle \varphi =\frac{SZ}{1-Z}\Rightarrow Z=\frac{\varphi }{S+\varphi }.}$

The stoichiometric mixture fraction${\displaystyle Z_s}$ defined earlier is the location where the local equivalence ratio${\displaystyle \varphi =1}$.

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

${\displaystyle \chi =2D|\mathrm{\nabla }Z{|}^2}$

where${\displaystyle D}$ is the diffusion coefficient of the scalar. Its stoichiometric value is${\displaystyle {\chi }_s=2D_s|\mathrm{\nabla }Z{|}_s^2}$.

Amable Liñán introduced a modified mixture fraction in 1991^[6][7] that is appropriate for systems where the fuel and oxidizer have differentLewis numbers. If${\displaystyle L{e}_F}$ and${\displaystyle L{e}_{O_2}}$ are the Lewis number of the fuel and oxidizer, respectively, then Liñán's mixture fraction is defined as

${\displaystyle \tilde{Z}=\frac{\tilde{S}{y}_F-y_O+1}{\tilde{S}+1}}$

where

${\displaystyle \tilde{S}=\frac{L{e}_{O}S}{L{e}_F}.}$

The stoichiometric mixture fraction${\displaystyle {\tilde{Z}}_s}$ is given by

${\displaystyle {\tilde{Z}}_s=\frac{1}{\tilde{S}+1}}$.

• Fluid dynamics
• Combustion