Untitled - Aerobib - Universidad Politécnica de Madrid

2.2. DIFFUSION 39
Hereinafter, notation in chapter 1 will be kept, adopting the vectorial and tensorial
notation defined in the Appendix to chapter 3.
2.2 Diffusion
Let ¯v i be the velocity of species A i at P . 4 ¯v i is generally different from velocity ¯v of
the mixture. The difference ¯v di is called diffusion velocity of A i
¯v di = ¯v i − ¯v. (2.2)
The following relations exist between ¯v, ¯v i and ¯v di
¯v = ∑ i
Y i¯v i , (2.3)
∑
Y i¯v di = ¯0. (2.4)
i
Flux dm i of A i through dσ during time dt, when P moves at velocity ¯v of the
mixture, is obviously
dm i = ρ i¯v di · ¯n dσ dt. (2.5)
The problem lies in calculating flux vector ¯f i ,
¯f i = ρ i¯v di . (2.6)
Let us first consider the case of a binary mixture formed by species A 1 and A 2 .
The problem has been solved by Enskog [3] and Chapman [4]. The flux of A 1 is
¯f 1 = ρY 1¯v d1 = −ρ M (
1M 2
Mm
2 D 12 ∇X 1 − (Y 1 − X 1 ) ∇p )
∇T
− D T 1
p T . (2.7)
Here D 12 and D T 1 are, respectively, the coefficients of diffusion and thermal diffusion
5 of the mixture. Flux ¯f 2 of A 2 is, from (2.4) and (2.6),
¯f 2 = − ¯f 1 . (2.8)
4¯v i is defined as follows: taking a volume element dV around P and forming the mean value of velocities
¯v i , of the molecules A i on it. Such value is ¯v i
¯v i =
1 X
¯v j i
N ,
i
where N i is the number of molecules of A i on dV . If dV is large respect to the molecular scale and
small respect to the macroscopic scale, the value ¯v i thus defined is independent from size and shape of dV .
Velocity ¯v of the mixture is, by definition
¯v = X i
j
Y i¯v i .
5 The coefficient of thermal diffusion must not be confused with the thermal diffusivity, Ed.