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 1948Hirschfelder, 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. and later by Theodore von Kármán and Sol Penner in 1954.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. The mass-flux fraction of a species i is defined asWilliams, F. A. (2018). Combustion theory. CRC Press.
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 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.Penner, S. S. (1957). Chemistry problems in jet propulsion (Vol. 1). Pergamon Press. 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