Ratio of mass-flux of chemical species to total mass flux
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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de1156aeca1b24ef4d81fe3999cb728684cb80dc)
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 ![{\displaystyle v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597)
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.[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.
- ^ 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.