  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.
