Page 300 - Introduction to Computational Fluid Dynamics

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APPENDIX A. DERIVATION OF TRANSPORT EQUATIONS
5. Inincompressibleflows,thedensityρ m isexternallyspecifiedasaconstantorasa 13:6 279
function of temperature and the sum of partial densities of mixture components.
In compressible flow, however, the density is recovered from an equation of
4
state. Thus, according to the law of corresponding states, for reduced pressure
p r < 0.5 and reduced temperature T r > 1.5, the density is calculated from the
perfect gas relation
p pM g
ρ m = = , (A.17)
R g T R u T
where M g is the molecular weight of the gas and R u is the universal gas constant.

A.4 Equation of Mass Transfer
The conservation of mass for species k of the mixture is stated as

˙
˙
Rate of accumulation of mass (M k,ac ) = Rate of mass in (M k,in )
˙
− Rate of mass out (M k,out )
+ Rate of generation within CV (R k ).

To apply this principle, let ρ k be the density of the species k in a fluid mixture
2
of density ρ m . Similarly, let N i,k be the mass transfer flux (kg/m -s) of species k in
the i direction. Then

∂(ρ k V )
˙
M k,ac = ,
∂t
˙ ,
M k,in = N 1,k x 2 x 3 | x 1 + N 2,k x 3 x 1 | x 2 + N 3,k x 1 x 2 | x 3
˙
M k,out = N 1,k x 2 x 3 | x 1 + x 1 + N 2,k x 3 x 1 | x 2 + x 2 + N 3,k x 1 x 2 | x 3 + x 3 .
Dividing each term by V and letting x 1 , x 2 , x 3 → 0, we get

∂(ρ k ) ∂(N 1,k ) ∂(N 2,k ) ∂(N 3,k )
+ + + = R k . (A.18)
∂t ∂x 1 ∂x 2 ∂x 3

Now, the total mass transfer flux N i,k is the sum of convective flux due to bulk fluid
motion (with each species having the same velocity as the bulk fluid) and diffusion

flux (m ). Thus,
i,k

N i,k = ρ k u i + m . (A.19)
i,k

4 Reduced pressure and temperature are defined as p r = p/p cr and T r = T/T cr , where the suffix cr
stands for the critical point.