[Law_C.K.] Combustion physics

5.2. Governing Equations 163
and
{ρ
[e
∫S + + + (v+ ) 2 ]
(v + − v I ) − ρ
[e − − + (v− ) 2 ]
(v − − v I )
I
2
2
}
+ (q + − q − ) + (v + · P + − v − · P − ) · n + dS
[ ∫
= lim ρ
V→0 V
N∑
Y i f i · (v + V i )dV − ∂ ∫
∂t
i=1
V
) ]
ρ
(e + v2
dV , (5.1.30)
2
where Eq. (5.1.27) has been used in deriving Eq. (5.1.29). The volume integrals in
the above relations vanish in the absence of source or sink at the interface, leading
to the corresponding vanishing of the integrands in the surface integrals and
consequently the differential conservation relations for the fluxes in crossing the
interface.
5.2. GOVERNING EQUATIONS
5.2.1. Conservation Equations
For ease of referencing we summarize in the following the conservation equations
derived above.
Overall Continuity:
∂ρ
∂t
+∇·(ρv) = 0 (5.2.1)
Continuity of Species:
ρ DY i
Dt
= w i −∇·(ρY i V i ), i = 1,...,N (5.2.2)
Momentum:
Energy:
ρ Dv
N∑
Dt =−∇·P + ρ Y i f i (5.2.3)
i=1
ρ De
N∑
Dt =−∇·q − P :(∇v) + ρ Y i f i · V i . (5.2.4)
i=1
5.2.2. Constitutive Relations
Derivation of the constitutive relations specifying the diffusion velocity V i , the pressure
tensor P, and the heat flux vector q can be found in, for example, Hirschfelder,
Curtiss, and Bird (1954), and Williams (1985), while the reaction rate is stated in
Chapter 2 through the law of mass action.