Mixture fraction

Mixture fraction (${\displaystyle Z}$) is a quantity used in combustion studies that measures the mass fraction of one stream (usually the fuel 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.

Definition

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. The stoichiometric mixture fraction ${\displaystyle Z_{s}}$ where fuel and oxygen meet in stoichiometric proportions 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) to obtain

${\displaystyle Z_{s}={\frac {1}{S+1}}}$.

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 ${\displaystyle y_{F,u}=Z}$ 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 ${\displaystyle \phi =1}$.

Scalar dissipation rate

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 charactersitic 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 ${\displaystyle \chi _{s}=2D_{s}|\nabla Z|_{s}^{2}}$.

Liñán's mixture fraction

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

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

References

1. Williams, F. A. (2018). Combustion theory. CRC Press.
2. Peters, N. (2001). Turbulent combustion.
3. Peters, N. (1992). Fifteen lectures on laminar and turbulent combustion. Ercoftac Summer School, 1428, 245.
4. Liñán, A., & Williams, F. A. (1993). Fundamental aspects of combustion.
5. 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.
6. 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
7. Linán, A. (2001). Diffusion-controlled combustion. In Mechanics for a New Mellennium (pp. 487-502). Springer, Dordrecht.