Untitled - Aerobib - Universidad Politécnica de Madrid

48 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
Making use of relations (1.33) and (1.36) in chapter 1, it is obtained
∑
Y j
D ij
j≠i
M j
[
¯vdi − ¯v dj
+ ∇Y i
− 1 ρ
− ∇Y j
Y j
Y i
(
DT j
Y j
+ M j − M i ∇p
M m p
]
= ¯0, (i = 1, 2, . . . , l).
− D T i
Y i
) ∇T
T
(2.29)
Diffusion of a species whose mass fraction is very small is particularly important
in certain problems. For example, diffusion of active particles which propagate
chemical reactions through a combustion wave. Then it becomes possible to give an
explicit approximate expression for the diffusion flux of such species. For diffusion
arising from differences in composition this expression is
(
∑ Y j ∇Yj
− ∇Y )
i
j≠iM j Y j Y i
¯f i = ρY i¯v di = ρY i
∑ Y j
. (2.30)
M j D ij
j≠i
System (2.29) is, generally, too complicated to be used in the solution of many
of the Aerothermochemistry problems. Therefore, in some of its applications, diffusion
fluxes of the species are substituted by approximate expressions under the assumption
that diffusion flux due to differences in composition ¯f iY of each species is
proportional to its gradient of molar fraction. That is, by adopting for such flux an
expression of the form
¯f iY = −ρ M i
M m
D im ∇X i . (2.31)
Here D im is a binary diffusion coefficient of the species through the mixture formed
by the rest. This expression is exact only when binary diffusion coefficients and molar
masses of the species are all equal. If mean molar mass of the mixture is also constant,
one has
¯f iY = −ρD im ∇Y i , (2.32)
by virtue of Eqs. (1.33) and (1.36) in chapter 1. This expressions will be used in
chapter 13. In aerothermochemical problems diffusion fluxes arising from differences
in pressure and temperature are, in general, negligible. C. R. Wilke [14] has given an
explicit approximate expression, empirically deduced, similar to (2.31) for fluxes due
to differences in composition.