Page 103 - gas transport in porous media

Whitaker
96
In addition, there is a class of problems for which both the normal and tangential
components of the mass average velocity are zero at the fluid solid interface leading
to the boundary condition given by
BC. v γ = 0 , at the γ –κ interface (6.139)
This condition requires that the mean free path be small compared to the pore diameter
and that the effect of adsorption and reaction at the γ –κ interface be neglected. When
these two conditions are not satisfied, one is confronted with Knudsen flow, slip
owing to multicomponent effects (Graham, 1833; Kramers and Kistemaker, 1943),
and coupling of the adsorption/reaction process with the momentum transfer process.
Within the framework of the method of volume averaging, some of these conditions
have been examined by Whitaker (1987) and more recently by Altevogt et al. (2003a,
2003b). Detailed closure problems have not been developed for either case, thus a
comparison between theory and experiment in the absence of adjustable parameters
has not been made. In the absence of this type of comparison between theory and
experiment, the dusty gas remains as a popular model of this complex process (Mason
and Malinauskas, 1983).
The superficial average of Eq. (6.138) can be expressed as
) *
∂
(ρ γ v γ ) + ∇ · (ρ γ v γ v γ ) =− ∇p γ + ρ γ g + µ γ ∇· ∇v γ (6.140)
∂t
and if we neglect variations of the density in both the body force and the inertial
terms, we can express this result as
) *
∂v γ
ρ γ + ρ γ ∇ · (v γ v γ ) =− ∇p γ + ε γ ρ γ g + µ γ ∇ · ∇v γ (6.141)
∂t
Here we see the need to interchange integration and differentiation in every term
except the one representing the gravitational force. Use of the general transport
theorem, the averaging theorem, and the second of Eqs. (6.131) eventually leads to
γ
∂ v γ γ γ −1
ρ γ +ρ γ v γ ·∇ v γ +ρ γ ε γ ∇· ˜v γ ˜v γ (6.142)
∂t

1
γ 2 γ
=− ∇ p γ +ρ γ g+µ γ ∇ v γ + n γκ · −I˜p γ + µ γ ∇˜v γ dA
V γ
A γκ
To obtain this result, one must impose the length scale constraints given by Eq.
(6.117) and a step-by-step development is given by Whitaker (1997). Although we
seek a governing equation for v γ , the form given by Eq. (6.142) is a necessary step
since we can use it in conjunction with the decomposition
γ
v γ = v γ +˜v γ (6.143)
and Eq. (6.138) to determine the spatial deviation velocity, ˜v γ . The boundary value
problem for ˜v γ and ˜p γ is known as the closure problem and the development of this