Page 301 - Introduction to Computational Fluid Dynamics

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APPENDIX A. DERIVATION OF TRANSPORT EQUATIONS
Under certain restricted circumstances of interest in this book, the diffusion flux
is given by Fick’s law of mass diffusion
∂ρ k
m i,k =− D , (A.20)
∂x i
2
5
where D (m /s) is the mass diffusivity. Substituting Equations A.19 and A.20 in
Equation A.18, we can show that
∂(ρ k ) ∂(ρ k u 1 ) ∂(ρ k u 2 ) ∂(ρ k u 3 ) ∂ ∂ρ k ∂ ∂ρ k
+ + + = D + D
∂t ∂x 1 ∂x 2 ∂x 3 ∂x 1 ∂x 1 ∂x 2 ∂x 2
∂ ∂ρ k
+ D + R k . (A.21)
∂x 3 ∂x 3
It is a common practise to refer to species k via its mass fraction ω k defined as
ρ k
ω k = ω k = 1. (A.22)
ρ m
all species
Using this definition, Equation A.21 can be compactly written as
∂(ρ m ω k ) ∂(ρ m u j ω k ) ∂ ∂ω k
+ = ρ m D + R k . (A.23)
∂t ∂x j ∂x j ∂x j
Note that when the mass transfer equation is summed over all species of the
mixture, the mass conservation equation for the bulk fluid (Equation A.5) is re-

trieved. This is because R k = 0. That is, when some species are generated by
a chemical reaction, others are destroyed so that there is no net mass generation in
the bulk fluid.

A.5 Energy Equation
3
The first law of thermodynamics, when considered in rate form (W/m ), can be
written as
˙
˙
˙
˙
˙
˙
E = Q conv + Q cond + Q gen − W s − W b , (A.24)
where
˙
E = Rate of change of energy of the CV,
˙
Q conv = Net rate of energy transferred by convection,
˙
Q cond = Net rate of energy transferred by conduction,
5 The mass diffusivity is defined only for a binary mixture of two fluids 1 and 2 as D 12 . In mul-
ticomponent gaseous mixtures, however, diffusivities for pairs of species are nearly equal and a
single symbol D suffices for all species. Incidentally, in turbulent flows, this assumption of equal
(effective) diffusivities has even greater validity.