Mass-flux fraction

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]


  • is the mass fraction
  • is the mass average velocity of the gaseous mixture
  • is the average velocity with which the species i diffuse relative to
  • is the density of species i
  • is the gas density.

It satisfies the identity

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 problems in place of 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.


  1. Hirschfelder, J. O., & Curtiss, C. F. (1948, January). Theory of propagation of flames. Part I: General equations. In Symposium on Combustion and Flame, and Explosion Phenomena (Vol. 3, No. 1, pp. 121-127). Elsevier.
  2. von Karman, T., & Penner, S. S. (1954). Fundamental approach to laminar flame propagation.
  3. von Karman, T., & Penner, S. S. (1954). The thermal theory of constant-pressure deflagration for first-order global reactions.
  4. Williams, F. A. (2018). Combustion theory. CRC Press.
  5. Penner, S. S. (1957). Chemistry problems in jet propulsion (Vol. 1). Pergamon Press.
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