Mixture fraction

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 (for hydrogen 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

${\displaystyle Z={\frac {Sy_{F}-y_{O}+1}{S+1}}}$

where

${\displaystyle S={\frac {sY_{F,F}}{Y_{O,O}}}}$

is the stoichiometry parameter, also known as the overall equivalence 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, ${\displaystyle 0<Z<1}$. 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 local fuel-air equivalence ratio ${\displaystyle \phi }$

${\displaystyle \phi ={\frac {sY_{F,u}}{Y_{O,u}}}={\frac {Sy_{F,u}}{y_{O,u}}}.}$

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

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

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

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

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

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

Amable Liñán introduced a modified mixture fraction in 1991[6][7] that is appropriate for systems where the fuel and oxidizer have different Lewis numbers. If ${\displaystyle Le_{F}}$ and ${\displaystyle Le_{O_{2}}}$ 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

${\displaystyle {\tilde {S}}={\frac {Le_{O}S}{Le_{F}}}.}$

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

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