Mixture fraction is a quantity used in combustion studies that measures the mass fraction of one stream of a mixture formed by two feed streams, one the fuel stream and the other the oxidizer stream. Both the feed streams are allowed to have inert gases. The mixture fraction definition is usually normalized such that it approaches unity in the fuel stream and zero in the oxidizer stream. 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 and another portion of the boundary the oxidizer stream with oxidizer mass fraction. For example, if the oxidizer stream is air and the fuel stream contains only the fuel, then and. In addition, assume there is no oxygen in the fuel stream and there is no fuel in the oxidizer stream. Let be the mass of oxygen required to burn unit mass of fuel. Introduce the scaled mass fractions as and. Then the mixture fraction is defined as where is the stoichiometry parameter, also known as the overall equivalence ratio. The stoichiometric mixture fraction where fuel and oxygen meet in stoichiometric proportions is obtained by setting to obtain
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 and . This allows to define a local fuel-air equivalence ratio The local equivalence ratio is an important quantity for partially premixed combustion. The relation between local equivalence ratio and mixture fraction is given by The stoichiometric mixture fraction defined earlier is the location where the local equivalence ratio.
In turbulent combustion, a quantity called the scalar dissipation rate with dimensional units of that of an inverse time is used to define a charactersitic diffusion time. Its definition is given by where is the diffusion coefficient of the scalar. Its stoichiometric value is.
Liñán's mixture fraction
introduced a modified mixture fraction in 1991 that is appropriate for systems where the fuel and oxidizer have different Lewis numbers. If and are the Lewis number of the fuel and oxidizer, respectively, then Liñán's mixture fraction is defined as where The stoichiometric mixture fraction is given by
