Untitled - Aerobib - Universidad Politécnica de Madrid

40 CHAPTER 2. TRANSPORT PHENOMENA IN GAS MIXTURES
Formula (2.7) shows that there are three causes for diffusion corresponding to the three
terms in the right hand side of the equation. Namely, the differences in composition,
pressure and temperature. 6
first of these three causes. 7
Elementary Theory of Diffusion only acknowledge the
In Aerothermochemical problems it is advantageous to eliminate mole fraction
X 1 from Eq. (2.7). Furthermore, by doing so a simplified expression is obtained.
Elimination is immediately attained by keeping in mind the following relations 8
Thus obtaining
X 1 = M m
Y 1 ,
M 1
(2.9)
1
= Y 1
+ Y 2
= 1 ( 1
+ − 1 )
Y 1 ,
M m M 1 M 2 M 2 M 1 M 2
(2.10)
( )
M1 − M 2
¯f 1 = −ρD 12 ∇Y 1 + ρ D 12 Y 1 (1 − Y 1 ) ∇p
M m p − D ∇T
T 1
T . (2.11)
The first term in the right hand side of this expression gives the diffusion flux corresponding
to differences in composition. This term express Fick’s law. The second
term gives the diffusion flux arising from differences in pressure. Since D 12 is always
positive, formula (2.11) shows that pressure diffusion tends to carry the heavier
molecules towards the higher pressures. The third term gives the diffusion flux arising
from differences in temperature. Thermal diffusion coefficient D T 1 can be positive,
negative or zero. Therefore, no general rules can be giving regarding the sens of this
flux.
Formula (2.11) shows that diffusion’s behaviour in a binary mixture of gases
is determined by the values of two coefficients: D 12 and D T 1 . Frequently D T 1 is
substituted by
k T =
M 2 m
M 1 M 2
D T 1
ρD 12
. (2.12)
Systematic measurements on diffusion coefficients of gas mixtures are not
available. Therefore theoretical computations become necessary. They can be performed
when knowing the interaction potential between molecules. If forces between
molecules are independent from their relative orientation, the interaction potential
takes the form
ϕ = ε 12 f
( r
σ 12
)
, (2.13)
6 Where selective fields of forces exists, that is, fields acting with different strength over the two species
of the mixture, they originate a new cause of diffusion. See Ref. [1], p. 143. Such occurs for example,
when there exist ionized molecules and the mixture is submitted to the action of an electromagnetic field.
7 See Ref. [5], p. 198.
8 See chapter 1, pp. 12-13.