Biomedical Engineering Reference
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where
A
is the active membrane surface, ∆
n
i
is the number of moles of the
i
th component, ∆
V
is the volume of sol
u
tion
p
ermeating the membrane in
time ∆
t
,∆
x
is the membrane thickness,
c
i
=
∆
n
i
∆
V
is the mean concentration,
and
v
i
=
∆
x
∆
t
is the permeation speed of
i
th component molecules across the
membrane. Equation (8.12) can be rewritten as
j
s
c
s
−
V
w
j
w
=
v
s
−
v
w
=
J
D
(8.14)
and it expresses the diffusion flux
J
D
of solute (
s
), that is, the diffusion speed
of this solute relative to the solvent. Equation (8.9) can now be interpreted as
ψ
=
J
v
∆
P
+
J
D
∆Π
(8.15)
In a system in near-equilibrium, every flux is a linear function of all driving
forces:
n
J
i
=
L
ik
X
k
(8.16)
k
=1
where
L
ik
is the proportionality coecients. Having identified the fluxes and
driving forces present in the system in equation (8.15), we can write this
condition (8.16) as
J
v
=
L
p
∆
P
+
L
pD
∆Π
J
D
=
L
Dp
∆
P
+
L
D
∆Π
(8.17)
These equations are known as the phenomenological KK equations. The
four terms in the equations represent processes of filtration (
L
p
∆
P
), osmo-
sis (
L
pD
∆Π), ultrafiltration (
L
Dp
∆
P
), and diffusion (
L
D
∆Π). Accordingly,
the coecients are called the coecients of filtration, osmotic permeation,
ultrafiltration, and diffusion. Additionally, the Onsager relation requires
L
Dp
=
L
pD
(8.18)
8.2.2 Practical KK Equations
Equation (8.17) for a volume flux in a membrane system shown in Figure 8.1
can be rewritten as
J
v
=
L
p
∆
P
−
L
p
σ
∆Π
(8.19)
where
σ
denotes the so-called reflection coecient, defined as
σ
=
−
L
pD
/L
p
(Staverman 1951). It may assume values 0
1. If
σ
= 1, the membrane
is semipermeable, that is, it fluxes only the solvent and generates maximal
osmotic pressure. For 0
≤
σ
≤
1 the membrane is selective, that is, it fluxes
solvent better than the solute, and for
σ
= 0 the membrane is called perme-
able. The solute and the solvent permeate equally, and such a membrane is
osmotically inactive.
≤
σ
≤
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