Environmental Engineering Reference
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(65), describes the local motion of interstitial fluid in the membrane. In an actual
experiment, one can only measure the total molar flux as a function of the fugacity
drop across the membrane. Hence the averaged transport equation, obtained from
the average of this equation along the thickness of the membrane:
1
x
=
1
D
f
Dc
∂
��
f
∫
j
=−
P
e
=−
dx
o
e
€
ƒ„
e
x
x
∂
l
l
f
x
(66)
x
=
0
in which is referred to as the permeability of the membrane. It should be stressed
that this definition of the permeability differs from the classical definition deriv-
ing from Darcy's law, which considers the viscous flow of a Newtonian interstitial
fluid. In the present case, for the sake of generality we define the permeability as
the transport coefficient relating the molar flux to the driving force of fugacity
gradient, as found in the literature, Assuming a constant , we deduce the overall
permeability of the membrane from Eq.(67) as:
f
D
c
d
∫
P
=
D
o
df
(67)
e
f
f
f
u
in which and are the downstream and upstream fugacities respectively. Finally,
combining these equations, the permeability is given as a function of and : [166].
Dc
�
bf
D
�
(68)
P
=
os
ln ln 1
+
€
‚
e
D
f
1
+
bf
ƒ
„
d
1.2.2.12 DERJAGUIN-BROEKHOFF-DEBOER MODEL FOR PSD
CALCULATING IN MESOPOR CARBONS
An improvement of the classical DBD theory for capillary condensation/evapora-
tion in open-ended cylindrical capillaries was presented in Ref. [42]. Here, we
reintroduce the main ideas of the DBD theory and present its extension for the
capillary condensation/evaporation in spherical mesopores. It was previously
shown that the experimental adsorption data for a reference flat silica surface can
be properly described by using the disjoining pressure isotherm in the equation:
(
)
(
)
(
)
(
)
Π −l+Π −l=−
exp
h
/
exp
h
/
RT
/
v
ln ln
p
/
p
(69)
1
1
2
2
m
0
in whichandcharacterize the strength of the surface forces field, whereas the pa-
rameters and are responsible for the range of the structural forces action. Clearly,
the first term dominates in thick ad layers, whereas the second term dominates
in thin ad layers. All of the parameters appearing in Eq. (69) were tabulated pre-
viously for the adsorption of argon and nitrogen at their boiling points on the
selected reference silica surface. The critical radius, at which a spontaneous capil-
lary condensation occurs (spinodal condensation point), is closely related to the
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