Stoichiometry preservation and generalization of Bilger mixture fraction for non-premixed combustion with differential molecular diffusion

The Bilger mixture fraction is a widely used parameter in non-premixed combustion when considering differential molecular diffusion, a prevalent phenomenon in hydrogen or hydrogen-blended fuel combustion. The property of stoichiometry preservation of mixture fractions is investigated. Two different Bilger mixture fraction formulations are clarified. It is found that they belong to a class of one-parameter generalized mixture fraction definitions discovered in this work. Specific definitions from the class of mixture fractions are compared for hydrocarbon fuels. The comparison shows that the difference can be significant. An optimal mixture fraction definition is sought from the general definitions by minimizing its deviation from the desired properties. The obtained optimal mixture fractions show overall better preservation of stoichiometry than Bilger's definitions. The extension of the generalized mixture fraction to other fuels that contain nitrogen (like ammonia $\mathrm{NH_3}$) or sulfur (like hydrogen sulfide $\mathrm{H_2S}$) is also demonstrated.