# 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]

where

- 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.

## References

- 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.
- von Karman, T., & Penner, S. S. (1954). Fundamental approach to laminar flame propagation.
- von Karman, T., & Penner, S. S. (1954). The thermal theory of constant-pressure deflagration for first-order global reactions.
- Williams, F. A. (2018). Combustion theory. CRC Press.
- Penner, S. S. (1957). Chemistry problems in jet propulsion (Vol. 1). Pergamon Press.