Page 100 - gas transport in porous media

Chapter 6: Conservation Equations
93
Here we have used c As to represent the surface concentration of species A (moles
per unit area) and R As to represent the molar rate of production of species A owing
to heterogeneous reaction (moles per unit area per unit time). In order to obtain a
γ
solution for c Aγ , and therefore c Aγ , we require a connection between the surface
concentration, c As , and the bulk concentration, c Aγ . The two classic connections are
the linear interfacial flux relation given by

c Aγ v Aγ · n γκ = k A1 c Aγ − k −A1 c As , at the γ –κ interface, (6.122)
A = 1, 2, ... , N

and the approximation of local adsorption equilibrium. For a linear adsorption
isotherm, this latter connection is given by

c As = K eq c Aγ , at the γ –κ interface , A = 1, 2, ... , N (6.123)
While this result is strictly true at the condition of equilibrium, the general form can
often be used as a reasonable approximation for dynamic systems (Whitaker, 1986b,
1999).
We begin the averaging procedure with Eq. (6.120) and express the superficial
average of that result as

) *
∂c Aγ + ,
+ ∇· c Aγ v Aγ = R Aγ , A = 1, 2, ... , N (6.124)
∂t
For a rigid porous medium, one can use the transport theorem and the averaging
theorem to obtain

∂ c Aγ 1
+∇ · c Aγ v Aγ + n γκ · c Aγ v Aγ dA = R Aγ (6.125)
∂t V
A γκ
where it is understood that this applies to all N species. Since we seek a transport
equation for the intrinsic average concentration, we make use of Eq. (6.114) to express
Eq. (6.125) in the form

γ
∂ c Aγ 1 γ
ε γ +∇ · c Aγ v Aγ + n γκ · c Aγ v Aγ dA = ε γ R Aγ (6.126)
∂t V
A γκ
At this point, it is convenient to make use of the jump condition given by Eq. (6.121)
in order to obtain
γ
∂ c Aγ γ 1 ∂c As 1
ε γ +∇ · c Aγ v Aγ = R Aγ − dA + R As dA (6.127)
∂t V ∂t V
A γκ A γκ