The mass-flux fraction (or Hirschfelder-Curtiss variable or Kármán-Penner variable) is the ratio of mass-flux of a particular chemical species to the total mass flux of a gaseous mixture. It includes both the convectional mass flux and the diffusional mass flux. It was introduced by Joseph O. Hirschfelder and Charles F. Curtiss in 1948 and later by

Theodore von Kármán
Theodore von Kármán ( hu, ( szőllőskislaki) Kármán Tódor ; born Tivadar Mihály Kármán; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fields of aeronau ...

and Sol Penner in 1954. The mass-flux fraction of a species i is defined as
:$\backslash epsilon\_i\; =\; \backslash frac\; =\; Y\_i\backslash left(1+\backslash frac\backslash right)$
where
*$Y\_i=\backslash rho\_i/\backslash rho$ is the mass fraction
*$v$ is the mass average velocity of the gaseous mixture
*$V\_i$ is the average velocity with which the species i diffuse relative to $v$
*$\backslash rho\_i$ is the density of species i
*$\backslash rho$ is the gas density.
It satisfies the identity
:$\backslash sum\_i\; \backslash epsilon\_i\; =1$
similar to mass fraction, but, the mass-flux fraction can take both positive and negative values. This variable is used in steady, one-dimensional combustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustio ...

problems in place of mass fraction.Penner, S. S. (1957). Chemistry problems in jet propulsion (Vol. 1). Pergamon Press. For one-dimensional ($x$ direction) steady flows, the conservation equation for the mass-flux fraction reduces to
:$\backslash frac\; =\; \backslash frac$
where $w\_i$ is the mass production rate of species i.
References

{{reflist, 30em Chemical properties Dimensionless numbers of chemistry Combustion