Page 302 - Introduction to Computational Fluid Dynamics

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APPENDIX A. DERIVATION OF TRANSPORT EQUATIONS
˙
Q gen = Net volumetric heat generation within the CV, 13:6 281
˙
W s = Net rate of work done by surface forces, and
˙
W b = Net rate of work done by body forces.
Each term will now be represented by a mathematical expression.
Rate of Change
The equation for the rate of change is
o
∂(ρ m e ) V 2 p V 2
˙ o
E = , e = e + = h − + , (A.25)
∂t 2 ρ m 2
2
where e represents specific energy (J/kg), h is specific enthalpy (J/kg), and V =
2
2
o
2
u + u + u . In the expression for e , contributions from other forms of energy
1 2 3
(potential, chemical, electromagnetic, etc.) are neglected.
Convection and Conduction
Following the convention that heat energy flowing into the CV is positive (and vice
versa), it can be shown that
o
∂ (N j,k e )
˙ k
Q conv =− , (A.26)
∂x j
where N j,k is given by Equation A.19. Now, since all species have the same
velocity,
∂
˙ 2
Q conv =− N j,k (h k − p k /ρ k + V /2) , (A.27)
∂x j
where p k is the partial pressure of species k. After some algebra, it can be shown
that

o
∂(ρ m u j e ) ∂ m j,k h k
˙
Q conv =− − . (A.28)
∂x j ∂x j
The conduction contribution is given by Fourier’s law of heat conduction, so
that
∂ ∂T
∂q j
˙
Q cond =− = k m . (A.29)
∂x j ∂x j ∂x j
Volumetric Generation
Two principal components of volumetric energy generation are chemical energy
˙
˙
(Q chem ) and radiative transfer (Q rad ). Thus,
˙
˙
˙
Q gen = Q chem + Q rad . (A.30)