Biomedical Engineering Reference
In-Depth Information
By dimensional analysis, it is easy to show flux
J
=
vC
, and Equation 5.27 is rewritten as
K D
C
x
∂
∂
−1
J
= −
(5.28)
So far, the analysis is on a single-solute molecule (Brownian motion). To understand the
macroscopic diffusion behavior, we consider the average flux in the pore. The average flux
averaged over the pore cross section is
1
−
∫
λ
J
=
2
J
d
(5.29)
β β
0
By combining Equations 5.28 and 5.29, the relationship between the fl ux and the concentra-
Equations 5.28 and 5.29, the relationship between the fl ux and the concentra-
s 5.28 and 5.29, the relationship between the fl ux and the concentra-
5.28 and 5.29, the relationship between the fl ux and the concentra-
5.29, the relationship between the fl ux and the concentra-
the fl ux and the concentra-
lux and the concentra-
the concentra-
concentra-
tion could be established:
1
−
∫
λ
K D
C
x
∂
∂
−
1
J
=
2
−
d
(5.30)
β β
0
Now we need an expression for
C
so as to substitute
C
in Equation 5.30. The two-dimensional
concentration distribution along the
z
direction was found to be a function of
x
,
g
(
x
), and the
long range interaction between solute and the pore wall,
E
(
β
), which is
E
kT
(
β
C g x
=
( )exp
−
(5.31)
From Equations 5.30 and 5.31, a local flux equation is obtained:
K D
C
z
∂
∂
J
= −
d
1
−
∫
λ
E
kT
( )
β
−
1
K
exp
−
d
(5.32)
β β
where
K
=
0
d
1
−
∫
λ
E
kT
( )
β
ex
p
−
d
β β
0
To solve the above partial differential equation, we need two boundary conditions, which
are concentrations at both ends of the pore
C
0
and
C
L
. Because of the exponential distribu-
t
ion
of concentration in the
z
direction, the average number of solutes enter/leave the pore
(
C
b
) is less than the number of solutes available in the bulk (
C
b
) by a factor of
K
p
1
−
∫
λ
C
C
E
kT
(5.33)
K
=
b
b
=
2
exp
−
d
β β
p
0
