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. It includes both the convectional mass flux and 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 species i is defined as^[4]

\epsilon _{i}={\frac {\rho _{i}(v+V_{i})}{\rho v}}=Y_{i}\left(1+{\frac {V_{i}}{v}}\right)

where

$Y_i=\rho_i/\rho$ is the mass fraction
$v$ is the mass average velocity of the gaseous mixture
$V_{i}$ is the average velocity with which the species i diffuse relative to $v$
$\rho _{i}$ is the density of species i
$\rho$ is the gas density.

It satisfies the identity

\sum _{i}\epsilon _{i}=1

similar to mass fraction, but, the mass-flux fraction can take positive and negative values. This variable is used in steady, one-dimensional combustion problems in place of mass fraction.^[5] For one-dimensional ( $x$ direction) steady flows, the conservation equation for the mass-flux fraction reduces to

{\frac {d\epsilon _{i}}{dx}}={\frac {w_{i}}{\rho v}}

where $w_{i}$ 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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[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.