Environmental Engineering Reference
In-Depth Information
N
(60)
J
=
St
D
The permeability is then defined as the flux per unit of fugacity gradient (Df
f
=l)
across the membrane:
Jl
(61)
P
=
D
f
From the thermo dynamical point of view, a system of fluid particles sorbed in a
immobile porous medium deviates from equilibrium when a gradient in molecules
chemical potential exists. Under isothermal conditions and on the assumption that
transport mechanism is diffusive, the local molar flux ~j (number of moles of
fluid per unit surface per unit time) through the fixed porous solid satisfies the
Maxwell-Stefan equation:
cD
j
= −
o
∇m
(62)
RT
where T is the temperature, R the ideal gas constant, c the average interstitial
concentration(number of moles per material unit volume), while Do stands as the
collective diffusivity of the sorbed fluid, as previously discussed. Moreover, using
the definition of the chemical potential:
(
)
( )
(
)
m
f T
,
≡m
T
+
RT
ln ln
f
/
f
(63)
0
0
one can alternatively consider the gradient in fugacity as the driving force of fluid
motion. Hence the rearranged expression of the local molar flux:
cD
j
=− ∇
o
f
(64)
f
where f is the fluid fugacity. In order to estimate the concentration in the mi-
croporous membrane, refer to the classical Langmuir model, commonly used to
describe adsorption isotherms of fluids in microporous adsorbents:
bf
cc
=
s
1
+
bf
(65)
Here is the complete filling concentration and is an equilibrium adsorption con-
stant, whichcan be interpreted as the inverse of a characteristic filling pressure.
We stress that we use the Langmuir model for its ability to reproduce the ad-
sorption isotherms simulated in our membrane models (see supplementary infor-
mation) and its convenient analytical form. Let us now consider a microporous
membrane of thickness in the direction and separating two infinite bulk fluid res-
ervoirs exhibiting a difference in chemical potentials. Under these conditions Eq.
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