Biomedical Engineering Reference
In-Depth Information
is the hydration index (defined as the number of moles of water per
equivalent sulfonic or carboxylic acid groups SO
3
-
or COOH-
in the membrane)
and the numerical values of 2.5 and 22 correspond to the number of water molecules
dragged per migration of cations (Li+, Na+, H+, etc.), ion, and the possible maximum
hydration index, respectively. Springer et al.
also present an empirical formula
relating
C
w
to
where
λ
λ
as follows:
e
λ
C
=
,
(6.146)
w
f
λ
+
1
Here,
f
is an experimentally determined swelling coefficient for the membrane,
and
e
is expressed as
dry
ρ
m
e
=
(6.147)
E
m
with
ρ
m
dry
corresponding to the density of the dry membrane, and
E
m
the equivalent
weight of the membrane. Rearranging equations (6.144) through (6.147) yields:
2
m
2
m
−∇ −
DCC k
ε
(/)
µ
∇ −
p
ε
(/ ( .
k
µ
∇
p
∇+
C
)
w
w
w
w
h
l
l
w
h
l
l
w
(6.148)
e
efC
5
(
∇
CiF
. /
)
=
0
w
2
44
(
−
)
w
Given distributions of pressure
p
l
and current density , we can solve equation
(6.148) to obtain
C
w
. To investigate a given case with a specific partial hydration
level,
C
w
can be prescribed at the anode-side membrane border, while prescribing
the water content at the cathode-membrane interface such that water balance con-
ditions described by equations (6.140) and (6.141) are satisfied on a point-by-point
basis along the length of the membrane.
The flux of protons through the membrane is described by the Nernst-Planck
equation:
i
i
i
i
i
i
i
i
NZF
=−
(/
TDC DCCU
)
∇ −
φ
∇
+
,
(6.149)
c
c
c
c
c
c
c
w
where
the terms on the right-hand side represent migration due to the electric field,
diffusion, and convection of the dissolved protons
Z
c
i
is the charge number of the
i
th cation
D
c
i
is the diffusion coefficient pertaining to the
i
th cation
C
c
i
is the molar concentration of the
i
th cation
φ
is the electrical potential
is the convective velocity of the liquid water
U
w
