Page 89 - gas transport in porous media

82
A=N
A=N Whitaker

∇v A : (− I p A ) =−(∇· v)p − (∇· u A )p A (6.66)
A=1 A=1
A=N A=N

∇v A : τ A =∇v : τ + ∇u A : τ A (6.67)
A=1 A=1
A=N A=N

p A v A = pv + p A u A (6.68)
A=1 A=1
Use of Eqs. (6.64) through (6.68) in Eq. (6.63) leads to
A=N
∂ Dp
(ρh) +∇ · (ρhv) =−∇ · q + +∇v : τ + ρ A h A u A (6.69)
∂t Dt
A=1
A=N A=N A=N

1 2
+ u A ·∇p A + ∇u A : τ A − r A u
2 A
A=1 A=1 A=1
The first five terms in this result have the same form as the enthalpy transport equation
for single component systems (Whitaker, 1983), while the last four terms are all
associated with diffusive fluxes. When convective transport is significant, the rate
of work terms involving the diffusive fluxes can be neglected relative to Dp/Dt and
∇v : τ , and when convective transport is insignificant these rate of work terms can
be discarded relative to the thermal terms in Eq. (6.69). When chemical reactions
occur, we can show that the last term is negligible, thus we will discard the last three
terms in Eq. (6.69) in order to express the enthalpy transport equation as
A=N
∂ Dp
(ρh) +∇ · (ρhv) =−∇ · q + +∇v : τ + ρ A h A u A (6.70)
∂t Dt
A=1
Given that h is a function of the state of the system, we can express the total enthalpy
per unit mass in terms of a thermal equation of state given by

h = h(T, p, ω A , ω B , ... , ω N−1 ) (6.71)

In order to transform Eq. (6.70) from an enthalpy transport equation to a trans-
port equation for the temperature, one must develop the time and space derivatives
of Eq. (6.71). This becomes algebraically tedious; however, the details are given
elsewhere (Whitaker, 1989, Eqs. 50 through 67) and the result is given by
A=N A=N
DT Dp
ρc p =−∇ · q + Tβ +∇v : τ − ρ A u A ·∇h A − r A h A (6.72)
Dt Dt
A=1 A=1