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^[1] and later by Theodore von Kármán and Sol Penner in 1954.^[2][3] The mass-flux fraction of a species i is defined as^[4]

${\displaystyle \epsilon _{i}={\frac {\rho _{i}(v+V_{i})}{\rho v}}=Y_{i}\left(1+{\frac {V_{i}}{v}}\right)}$

where

• ${\displaystyle Y_{i}=\rho _{i}/\rho }$ is the mass fraction
• ${\displaystyle v}$ is the mass average velocity of the gaseous mixture
• ${\displaystyle V_{i}}$ is the average velocity with which the species i diffuse relative to ${\displaystyle v}$
• ${\displaystyle \rho _{i}}$ is the density of species i
• ${\displaystyle \rho }$ is the gas density.

It satisfies the identity

${\displaystyle \sum _{i}\epsilon _{i}=1}$,

similar to the mass fraction, but the mass-flux fraction can take both positive and negative values. This variable is used in steady, one-dimensional combustion problems in place of the mass fraction.^[5] For one-dimensional (${\displaystyle x}$ direction) steady flows, the conservation equation for the mass-flux fraction reduces to

${\displaystyle {\frac {d\epsilon _{i}}{dx}}={\frac {w_{i}}{\rho v}}}$,

where ${\displaystyle w_{i}}$ is the mass production rate of species i.