Biomedical Engineering Reference
In-Depth Information
The potential distribution:
8
<
nþ
1
cosh
s
!
9
=
2
p
2
D
h
þ
ð
2
n
1
Þ
D
h
l
D
y
ð
1
Þ
1
cos
ð
z
2
N
4
ð
W
=l
D
Þ
2
n
ÞpD
h
2
W
1
p
cosh
s
!
2
p
2
D
h
n¼
1
þ
ð
2
n
1
Þ
H
l
D
ð
2
n
1
Þ
1
2
4
ð
W
=
l
D
Þ
J
¼
4
z
1
þ
ð
s
!
:
;
2
p
2
D
h
2
n
Þ
1
D
h
n
þ
1
cosh
l
D
x
ð
1
Þ
cos
ð
y
2
4
ð
H
=
l
D
Þ
2
n
1
Þ
pD
h
þ
p
cosh
s
!
2
H
2
p
2
D
h
þ
ð
2
n
1
Þ
W
l
D
ð
2
n
1
Þ
1
2
4
ð
H
=
l
D
Þ
is obtained by solving the Poisson-Boltzmann Eqn
(2.162)
, with the boundary conditions
x
¼
0
¼
y
¼
0
J
vz
J
vy
v
v
J
ð
D
h
Þ¼J
ð
z
:
z
;
y
Þ¼
H
=
W
=
D
h
;
2.6.1.6 Ohmic model for electrolyte solutions
In this section, a model for electrolyte solutions is derived. This model is useful for formulating mixing
problems in an electrokinetic system. The model was formulated by Chen et al.
[37]
, who followed the
approach of Levich
[38]
. We consider here a monovalent binary electrolyte (
j
z
þ
j¼j
z
j¼
1), where the
subscripts
þ
and
denote the cation and anion, respectively. The local charge density and conduc-
tivity
s
el
are determined as:
r
el
¼ Fðc
þ
c
Þ:
F
2
c
m
Þ:
where
F
is the Faraday constant,
m
is the ionic mobility, and
c
is the concentration. Electro-neutrality
can be evaluated based on the ratio between the concentration difference of cations and anions and the
total concentration of ions.
s
el
¼
ð
c
þ
m
þ
þ
r
el
c
þ
c
Q ¼
Fm
þ
s
el
¼
c
:
(2.165)
c
þ
þð
m
=
m
þ
Þ
While the concentration difference contributes to the charge density, the total ion concentration
contributes to the electrical conductivity. Thus, electroneutrality can be assumed if the above ratio is
very small,
Q
1. Under electroneutrality,
the concentration of both ion types is the same,
c
þ
¼
c
¼
c
, which is called the reduced concentration. The conductivity is then:
s
el
¼
F
2
ð
m
þ
þ
m
Þ
c
:
(2.166)
The conservation of species can be formulated for the ions as:
D
c
D
t
2
c
D
V
:
(2.167)
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