Page 17 - gas transport in porous media

Webb
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2.2.1.1 Fick’s law
Fick’s law is actually two laws. Fick’s first law is the relationship of the diffusive
flux of a gas component as a function of the concentration gradient under steady-state
conditions. Fick’s second law relates the unsteady diffusive flux to the concentration
gradient. Both laws were originally derived for clear fluids (no porous media).
First Law

Clear Fluids
Fick’s first law for a binary system basically states that the mole or mass flux is
proportional to a diffusion coefficient times the gradient of the mole or mass concen-
tration. For the mole flux formulation, Fick’s first law of diffusion for the mole flux
M
of component A, J , in one dimension in a clear fluid (no porous medium) is
A
M
J
A =−cD AB,CF ∇x A
where c is the concentration of the gas, D AB,CF is the diffusion coefficient in a clear
fluid, and x A is the mole fraction of component A. The above form of Fick’s first
law is commonly misused. The M superscript on the mole flux denotes that the mole
flux is relative to the molar-average velocity, NOT to stationary coordinates (BSL,
pg. 502). The mole flux equation relative to stationary coordinates for a binary system
is given by

D D D
N − x A N + N
A A B =−cD AB,CF ∇x A
where N is relative to stationary coordinates. The second term on the LHS is the
molar-average velocity. The mass flux form relative to stationary coordinates is

F A − ω A (F A + F B ) =−ρ g D AB,CF ∇ω A
where F is the mass flux and ω A is the mass fraction of component A.
Fick’s first law and a number of equivalent forms (mole and mass forms, relative
to mole or mass velocities or stationary coordinates) are discussed in great detail by
BSL (Chapter 16). The relationships between the various fluxes are also discussed
in detail in BSL (Chapter 16). However, many applications that use Fick’s first law
overlook the coordinate system issues. In particular, many applications use Fick’s
law for the molar-average velocity and incorrectly apply it to stationary coordinates
as discussed later in this chapter.

Porous Media
The above forms of Fick’s law are appropriate for clear fluids. For application to
porous media, Fick’s first law is often modified by the introduction of a porous media
factor, β,or
D ∗ = β D AB,CF
AB