Page 90 - gas transport in porous media

Chapter 6: Conservation Equations
At this point we can see that the last term in Eq. (6.69) can be discarded on the basis of
A=N A=N 83

1 2
r A u r A h A (6.73)
2 A
A=1 A=1
It is convenient to express the last term in Eq. (6.72) in terms of the partial molar
enthalpy and the molar rate of reaction. These are given by
r A
H A = h A M A , R A = (6.74)
M A
and these definitions allow us to express the last term in Eq. (6.72) as
A=N A=N

r A h A = R A H A (6.75)
A=1 A=1
For the special case of a single independent reaction, the molar rates of reaction can
be expressed in terms of the stoichiometric coefficients and the molar rate of reaction
of the pivot species, N, to obtain

A=N A=N

R A H A = ν A H A R N = H reaction R N (6.76)
A=1 A=1
Use of this result in Eq. (6.72) provides
DT Dp
ρc p =−∇ · q + Tβ +∇v : τ (6.77)
Dt Dt
A=N

− ρ A u A ·∇h A − H reaction R N
A=1
When there are multiple independent reactions (Cerro et al., 2004), one must
identify those reactions and the heats of reaction associated with them. The diffusive
flux of enthalpy, ρ A u A ·∇h A , can be ignored for most problems; however, combustion
processes are a special case that must be considered with care. In addition, care must
be taken to remember that q represents both the conductive and radiative transport,
and for an isotropic medium we would express this quantity as
R
q =−λ∇T + q (6.78)
In general, we wish to use the axioms for energy to determine a single temperature,
T, and this means that the species energy equations represented by Eqs. (6.57) are not
needed. Instead, it is the total energy equation given by Eq. (6.77) that provides us
with information about the temperature. However, what is true for energy is not true
for either mass or momentum, that is, we must make use of the species continuity
equation given by Eq. (6.11) or by Eq. (6.19) and the species momentum equation
given by Eq. (6.29).