Biomedical Engineering Reference
In-Depth Information
The velocity of the liquid water in the membrane is modeled by the Schlögl
equation (Verbrugge and Hill, 1993; Singh et al., 1999), which states that the
convection is caused by the electric potential and the pressure gradient:
m
i
U
=
ε
[(
k
/
µ
)
Z C F
∇
φ
−
(
k
/
µ
)
∇
p
]
,
(6.150)
ww
φ
wf
f
c
ww
where
k
φ
is the electric permeability
k
c
i
is the hydraulic permeability of the
i
th cation
Z
f
is the charge number of the fixed charges
C
f
is the fixed-charge concentration
The flow of charged species is related to the current density by
∑
i
i
i
=
F ZN
c
,
(6.151)
c
i
The membrane conductivity is defined as
2
F
RT
∑
i
2
i
i
κ=
(
)
(
ZDC
c
)
,
(6.152)
c
c
i
Also, the electroneutrality holds
∑
i
i
ZC
+
ZC
=
0
,
(6.153)
f
f
i
Here, we note that the only mobile ions in the membrane are the monovalent cations,
and for them we have
Z
c
i
= 1
.
Equations (6.148) through (6.153) and equation (6.140) can be recombined,
yielding
−
φκ
1
[(
i
i
−∆
= −
∇
nC
). ]
i
+
(
RT
/
F
)
∇ ∇
.[
(
nC
)]
(6.154)
c
c
Since the membrane swells due to internal hydration,
p
c
is a function of hydration
rather than a constant.
Similarly to equation (6.146), the proton concentration can be described by:
e
i
C
=
(6.155)
c
f
λ
+
1
Combining equations (6.146) and (6.155) yields
(6.156)
i
CefC
c
=−
w
