Mixture fraction

Mixture fraction ([math]\displaystyle{ Z }[/math]) 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.

## Contents

* 1 Definition
* 2 Relation between local equivalence ratio and mixture fraction
* 3 Scalar dissipation rate
* 4 Liñán's mixture fraction
* 5 References

## Definition

Assume a two-stream problem having one portion of the boundary the fuel stream with fuel mass fraction [math]\displaystyle{ Y_F=Y_{F,F} }[/math] and another portion of the boundary the oxidizer stream with oxidizer mass fraction [math]\displaystyle{ Y_{O}=Y_{O,O} }[/math]. For example, if the oxidizer stream is air and the fuel stream contains only the fuel, then [math]\displaystyle{ Y_{O,O}=0.232 }[/math] and [math]\displaystyle{ 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]\displaystyle{ s }[/math] be the mass of oxygen required to burn unit mass of fuel (for hydrogen gas, [math]\displaystyle{ s=8 }[/math] and for [math]\displaystyle{ \mathrm{C}_m\mathrm{H}_n }[/math] alkanes, [math]\displaystyle{ s=32(m+n/4)/(12m+n) }[/math][5]). Introduce the scaled mass fractions as [math]\displaystyle{ y_F=Y_F/Y_{F,F} }[/math] and [math]\displaystyle{ y_O = Y_O/Y_{O,O} }[/math]. Then the mixture fraction is defined as

[math]\displaystyle{ Z = \frac{Sy_F-y_{O}+1}{S+1} }[/math]

where

[math]\displaystyle{ 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]\displaystyle{ y_F=1 }[/math] and [math]\displaystyle{ y_O=0 }[/math] since there is no oxygen in the fuel stream, and hence [math]\displaystyle{ Z=1 }[/math]. Similarly, on the oxidizer-stream boundary, [math]\displaystyle{ y_F=0 }[/math] and [math]\displaystyle{ y_O=1 }[/math] so that [math]\displaystyle{ Z=0 }[/math]. Anywhere else in the mixing domain, [math]\displaystyle{ 0\lt Z\lt 1 }[/math]. The mixture fraction is a function of both the spatial coordinates [math]\displaystyle{ \mathbf{x} }[/math] and the time [math]\displaystyle{ t }[/math], i.e., [math]\displaystyle{ 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]\displaystyle{ Z(\mathbf{x},t)=Z_s }[/math], where [math]\displaystyle{ Z_s }[/math] is called as the stoichiometric mixture fraction which is obtained by setting [math]\displaystyle{ 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]\displaystyle{ Z }[/math] to obtain

[math]\displaystyle{ 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]\displaystyle{ y_{F,u}= Z }[/math] and [math]\displaystyle{ y_{O,u}= 1- Z }[/math] (the subscript [math]\displaystyle{ u }[/math] denotes unburnt mixture). This allows to define a local fuel-air equivalence ratio [math]\displaystyle{ \phi }[/math]

[math]\displaystyle{ \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]\displaystyle{ \phi = \frac{SZ}{1-Z} \qquad \Rightarrow \qquad Z = \frac{\phi}{S+\phi}. }[/math]

The stoichiometric mixture fraction [math]\displaystyle{ Z_s }[/math] defined earlier is the location where the local equivalence ratio [math]\displaystyle{ \phi=1 }[/math].

## Scalar dissipation rate

In turbulent combustion, a quantity called the scalar dissipation rate [math]\displaystyle{ \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]\displaystyle{ \chi = 2 D |\nabla Z|^2 }[/math]

where [math]\displaystyle{ D }[/math] is the diffusion coefficient of the scalar. Its stoichiometric value is [math]\displaystyle{ \chi_s = 2D_s|\nabla Z|^2_s }[/math].

## 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 [math]\displaystyle{ Le_F }[/math] and [math]\displaystyle{ Le_{O_2} }[/math] are the Lewis number of the fuel and oxidizer, respectively, then Liñán's mixture fraction is defined as

[math]\displaystyle{ \tilde Z = \frac{\tilde Sy_F-y_{O}+1}{\tilde S+1} }[/math]

where

[math]\displaystyle{ \tilde S = \frac{Le_O S}{Le_F}. }[/math]

The stoichiometric mixture fraction [math]\displaystyle{ \tilde Z_s }[/math] is given by

[math]\displaystyle{ \tilde Z_s = \frac{1}{\tilde S+1} }[/math].

## 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.

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