Page 303 - Introduction to Computational Fluid Dynamics

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APPENDIX A. DERIVATION OF TRANSPORT EQUATIONS
The chemical energy is positive for exothermic reactions and negative for en-
˙
dothermic reactions. Evaluation of Q chem depends on the chemical reaction model
˙
employed in a particular situation. The Q rad term represents the net radiation ex-
change between the control volume and its surroundings. Evaluation of this term, in
general, requires solution of integro-differential equations [48]. However, in certain
˙
restrictive circumstances, the term may be represented analogous to Q cond with k
replaced by radiation conductivity k rad as
16σ T 3
k rad = , (A.31)
a + s
where σ is the Stefan–Boltzmann constant and a and s are absorption and scattering
coefficients, respectively.
Work Done by Surface and Body Forces
Following the convention that the work done on the CV is negative, it can be shown
that
∂ ∂
˙
−W s = [σ 1 u 1 + τ 12 u 2 + τ 13 u 3 ] + [τ 21 u 1 + σ 2 u 2 + τ 23 u 3 ]
∂x 1 ∂x 2
∂
+ [τ 31 u 1 + τ 32 u 2 + σ 3 u 3 ] , (A.32)
∂x 3

˙
−W b = ρ m (B 1 u 1 + B 2 u 2 + B 3 u 3 ). (A.33)
Adding these two equations and making use of Equations A.11–A.14 can show
that
D V 2
˙ ˙
−(W s + W b ) = ρ m + µ v − p · V, (A.34)
Dt 2
2
where V /2 is the mean kinetic energy and the viscous dissipation function is given
by

2 2 2

∂u 1 ∂u 2 ∂u 3
v = 2 + +
∂x 1 ∂x 2 ∂x 3
2 2 2

∂u 1 ∂u 2 ∂u 1 ∂u 3 ∂u 3 ∂u 2
+ + + + + + . (A.35)
∂x 2 ∂x 1 ∂x 3 ∂x 1 ∂x 2 ∂x 3
Combining Equations A.24–A.35 therefore leads to

o
∂ρ m e o ∂(ρ m u j e ) ∂ ∂T ∂ m j,k h k
+ = k m −
∂t ∂x j ∂x j ∂x j ∂x j
D V 2
˙
˙
+ − p . V + µ v + Q chem + Q rad .
Dt 2
(A.36)