Untitled - Aerobib - Universidad Politécnica de Madrid

3.9. THE CASE OF ONLY TWO CHEMICAL SPECIES 77
which by integration gives
where i is an integration constant.
p + mv − 4 3 µ dv = i, (3.72)
dx
Energy Equation
In this case it is convenient to make use of the total energy equation (3.54) which by
integration gives
mh + ρ ∑ i
Y i h i v di + 1 2 mv2 − λ dT
dx − 4 dv
µv = e, (3.73)
3 dx
where e is an integration constant.
The two first terms of this equation give the enthalpy flux m ∑ ε i h i of the
i
mixture per unit surface normal to x, as can easily be proved by keeping in mind
(3.61) and (3.68). Therefore Eq. (3.73) may also be written in the form
(
1
m
2 v2 + ∑ )
ε i h i − λ dT
dx − 4 dv
µv = e, (3.74)
3 dx
i
whose physical interpretation is obvious.
Diffusion Equations
Equations (3.56) remain valid
∑
j
Y j
M j
[
vdi − v dj
D ij
+
( 1 dY i
Y i dx − 1 dY j
Y j dx
( 1 dp
p dx
+ M j − M i
M m
)
)]
= 0, (i = 1, 2, . . . , l),
∑
Y i v di = 0.
i
(3.75)
3.9 The case of only two chemical species
Let us now assume that in the preceding case the number of chemical species of the
mixture reduces to two, A 1 and A 2 . Then, a single chemical variable Y 1 , denoted Y ,
is enough to define the composition of the mixture, since Y 2 = 1 − Y . Similarly the
flux fraction ε i of A 1 will be denoted ε and one has ε 2 = 1 − ε.