Untitled - Aerobib - Universidad Politécnica de Madrid

3.7. ONE-DIMENSIONAL MOTIONS 73
reduces to
ρ ∂v ∂v
+ ρv
∂t ∂x = − ∂p
∂x + 4 3
∂
∂x
(
µ ∂v )
. (3.52)
∂x
Energy equation
Equation (3.34) reduces to
ρ ∂u ∂u
+ ρv
∂t
∂x = − p ∂v
∂x + 4 ( ∂v
3 µ ∂x
− ∂ (
λ ∂T
∂x ∂x
) 2
)
− ∂
∂x
(
ρ ∑ i
Y i h i v di
)
Or if it is decided to use the total energy equation (3.38) it takes the form
ρ ∂ (h + 1 )
∂t 2 v2 + ρv ∂ (h + 1 )
∂x 2 v2 = ∂p
∂t + 4 (
∂
µv ∂v )
3 ∂x ∂x
+ ∂ (
λ ∂T ) (
− ∂ ρ ∑ )
Y i h i v di .
∂x ∂x ∂x
i
.
(3.53)
(3.54)
The system of equations (3.49), (3.50), (3.52) and (3.53), or (3.54), must be
completed with the following equations.
Total mass fraction equation
∑
Y i = 1. (3.55)
i
Diffusion equations
The system of equations (3.8) which gives the diffusion velocities reduces to
∑
[ (
Y j vdi − v dj 1 ∂Y i
+
M j D ij Y i ∂x − 1 )
∂Y j
Y j ∂x
j
+ M j − M i
M m
( )] 1 ∂p
= 0, (i = 1, 2, . . . , l),
p ∂x
∑
Y i v di = 0.
i
(3.56)
State equation
The state equation is, by assumption, that of ideal gases
p
ρ = R mT. (3.39)