Biomedical Engineering Reference
In-Depth Information
From equations (8.17) we can write
J
D
+
J
v
=(
L
p
+
L
Dp
)∆
P
+(
L
pD
+
L
D
)∆Π
(8.20)
On the other hand
J
D
+
J
v
can be expressed using equations (8.10), (8.14), as
J
D
+
J
v
=
j
s
V
w
j
w
+
V
w
j
w
+
V
s
j
s
=
j
s
c
s
+
V
s
j
s
=
j
s
j
s
c
s
c
s
−
c
s
(1 +
y
s
)
≈
(8.21)
Comparing these two expressions we obtain
j
s
=
ω
∆Π +
c
s
(1
−
σ
)
J
v
(8.22)
where
ω
=
c
s
(
L
p
L
D
−
L
pD
2
)
(8.23)
L
p
is the permeation coecient. Equations (8.19) and (8.23) are known as the
practical KK equations.
8.2.3 Transport Parameters L
p
,
σ
, and
ω
Membrane properties with regard to passive transport are described by Kedem
and Katchalsky using three phenomenological transport parameters. The
reflection coecient
σ
was first defined by Staverman (1951) and is depen-
dent on both the membrane and the permeating solute. Similarly, the filtra-
tion coecient
L
p
, and the permeation coecient
ω
are defined in terms of
the Onsager coecients in equations (8.17). The practical KK equations give
physical interpretation of these coecients. If we assume ∆Π = 0 in equation
(8.19), we obtain
L
p
=
J
v
∆
P
(8.24)
∆Π=0
that is, the filtration coecient is the volume flux per unit pressure in the
absence of osmotic pressure. Similarly, putting
J
v
= 0 we get
σ
=
∆
P
∆Π
(8.25)
J
v
=0
If experimentally we obtain ∆
P
= ∆Π, then
σ
= 1, and the membrane is
selective. If ∆
P
= 0, than the membrane is osmotically inactive. Assuming
J
v
= 0 in equation (8.22) we get
ω
=
j
s
∆Π
(8.26)
J
v
=0
that is, the permeation coecient expresses the solute flux generated by the
unit osmotic pressure in the absence of the net volume flux.
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