Page 106 - gas transport in porous media

Chapter 6: Conservation Equations
that can be arranged in the form
⎛ ⎞ 99
⎜ ⎟
K ⎜ ⎟
⎜ γ 2
γ ⎟
v γ =− · ⎜∇ p γ − ρ γ g − µ γ ∇ v γ ⎟ − F · v γ (6.152)
µ γ ⎜ ⎟
⎝ ⎠
Brinkman Fochheimer
correction correction
It is important to recognize that the Brinkman correction (Brinkman, 1947) appears
naturally in terms of the intrinsic average velocity and not the superficial average
velocity that appears in Darcy’s law and the Forchheimer correction. For application
purposes, one needs to work with the superficial velocity so that Eq. (6.152) takes
the form

⎡ ⎤
⎢ ⎥
⎢ ⎥
γ 2 ⎥
K ⎢
v γ =− ⎢∇ p γ − ρ γ g − µ γ / ε γ ∇ v γ ⎥ − F · v γ (6.153)
µ γ ⎢
⎥
⎣ ⎦
Brinkman Forchheimer
correction correction
in which µ γ /ε γ is sometimes referred to as the Brinkman viscosity. When values of
the Brinkman viscosity different than µ γ /ε γ are encountered, it means that this term
is being used as an empirical correlating factor.

6.4 CLOSURE FOR MASS TRANSFER
Given a means of determining the superficial mass average velocity, we are ready to
return to Eq. (6.134) and list some of the results that are available. For dilute solutions,
the development given by Eqs. (6.88) through (6.98) provides

γ
∂ c Aγ γ + ,
ε γ +∇ · v γ c Aγ = ∇· D Am ∇c Aγ +∇ · ˜v γ ˜c Aγ (6.154)
∂t
dispersive
transport
∂ c As γκ
− a v + a v R As γκ
∂t

Here we have discarded the homogeneous reaction rate term since homogeneous
reactions are generally unimportant in porous media processes. Use of the averaging
theorem and the decomposition given by the first of Eqs. (6.131) allows us to express