Chemistry: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]
- [math]\displaystyle{ \epsilon_i = \frac{\rho_i (v+ V_i)}{\rho v} = Y_i\left(1+\frac{V_i}{v}\right) }[/math]
where
- [math]\displaystyle{ Y_i=\rho_i/\rho }[/math] is the mass fraction
- [math]\displaystyle{ v }[/math] is the mass average velocity of the gaseous mixture
- [math]\displaystyle{ V_i }[/math] is the average velocity with which the species i diffuse relative to [math]\displaystyle{ v }[/math]
- [math]\displaystyle{ \rho_i }[/math] is the density of species i
- [math]\displaystyle{ \rho }[/math] is the gas density.
It satisfies the identity
- [math]\displaystyle{ \sum_i \epsilon_i =1 }[/math],
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 ([math]\displaystyle{ x }[/math] direction) steady flows, the conservation equation for the mass-flux fraction reduces to
- [math]\displaystyle{ \frac{d\epsilon_i}{dx} = \frac{w_i}{\rho v} }[/math],
where [math]\displaystyle{ w_i }[/math] 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.
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