Untitled - Aerobib - Universidad Politécnica de Madrid

2.2. DIFFUSION 47
coefficients. 14 The values of k T are generally small. Normally they do not exceed 0.1.
Thereby thermal diffusion is of secondary importance in Aerothermochemistry and its
influence is, usually, disregarded.
In Refs. [8], [9], [10] and [11] recent information on theoretical and experimental
values of transport coefficients can be found.
For mixtures containing more than two components, the problem has been
solved by Curtiss and Hirschfelder [2], [12]. In such case, the aforementioned three
causes of diffusion, namely, differences in composition, pressure and temperature, still
hold. Yet in this case diffusion flux of each species depends not only on the gradient
of its molar fraction but is a linear function of the gradients of all molar fractions.
Diffusion flux ¯f i of species A i (i = 1, 2, . . . , l) is
¯f i = ρY i¯v di = ρ M i
M 2 m
∑
M j D ij
′
j≠i
(
∇X j + (X j − Y j ) ∇p
p
)
− D T i
∇T
T . (2.26)
Here D ′ ij is the diffusion coefficient of species A i and A j in the mixture. For a binary
mixture, D ′ ij is diffusion coefficient D ij previously defined. For the case of more than
two components it differs. D T i is the thermal diffusion coefficient of species A i in
the mixture.
The l equations (2.26) are not independent from each other. In fact, diffusion
fluxes of l species must satisfy the condition
∑
¯f i = ¯0. (2.27)
i
Diffusion coefficients D ij ′ in system (2.26) can be expressed as a function of
binary diffusion coefficients D ij of the species taken in couples. Ref. [2], pp. 541
and 543, and Ref. [12] contain expressions for these coefficients as well as for those
of thermal diffusion. Expressions for D ′ ij and D T i are too complicated for practical
computations. Then it is advantageous to eliminate D ij ′ by substituting system (2.26)
with the following that makes explicit the influence of binary diffusion coefficients 15
∑ X i X j
(¯v dj − ¯v di ) =∇X i + (X i − Y i ) ∇p
D ij
p
j≠i
− ∑ (
X i X j DT j
− D ) (2.28)
T i ∇T
ρD ij Y j Y i T . j≠i
As was done for the case of binary diffusion, it is also convenient to eliminate
molar fractions of the species by introducing their corresponding mass fractions.
14 A critical study on the subject will be found in Ref. [6], p. 492 and following, as well as bibliography.
15 See Ref. [2]. p. 517.