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]
where
It satisfies the identity
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 ( direction) steady flows, the conservation equation for the mass-flux fraction reduces to
where is the mass production rate of species i.