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Related Symbols
MassTransportPDEComponent
MassTransferValue
MassConcentrationCondition
NDSolveValue
Related Categories
Chemistry
Partial Differential Equations
Finite Element Method
Numeric Cyclic Voltammetry
Solve a coupled reaction model
Example Notebook
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Define a coupled reaction model:
m
a
s
s
t
r
a
n
s
p
o
r
t
m
o
d
e
l
︷
∂
c
A
(
t
,
x
)
∂
t
+
∇
·
(
-
d
A
∇
c
A
(
t
,
x
)
)
=
m
a
s
s
t
r
a
n
s
f
e
r
b
o
u
n
d
a
r
y
︷
Γ
x
=
0
k
c
c
A
(
t
,
x
)
-
k
a
c
B
(
t
,
x
)
m
a
s
s
t
r
a
n
s
p
o
r
t
m
o
d
e
l
︷
∂
c
B
(
t
,
x
)
∂
t
+
∇
·
(
-
d
B
∇
c
B
(
t
,
x
)
)
=
m
a
s
s
t
r
a
n
s
f
e
r
b
o
u
n
d
a
r
y
︷
Γ
x
=
0
k
a
c
B
(
t
,
x
)
-
k
c
c
A
(
t
,
x
)
The underlying reaction model is given by
c
A
+
e
⇌
c
B
and Butler–Volmer kinetics.
Set up the mass transport model variables
v
a
r
s
:
I
n
[
1
]
:
=
v
a
r
s
=
{
{
c
A
[
t
,
x
]
,
c
B
[
t
,
x
]
}
,
t
,
{
x
}
}
;
Set up a region
Ω
:
I
n
[
2
]
:
=
Ω
=
L
i
n
e
[
{
{
0
}
,
{
1
/
4
0
}
}
]
;
Specify material parameters:
I
n
[
3
]
:
=
c
A
b
u
l
k
=
1
;
d
A
=
1
0
^
-
5
;
d
B
=
1
0
^
-
5
;
Specify the electrochemical rate constants:
I
n
[
4
]
:
=
k
0
=
1
0
^
-
5
;
α
=
1
/
2
;
β
=
1
/
2
;
e
f
0
=
0
;
r
t
b
y
f
=
2
5
.
7
*
1
0
^
-
3
;
k
c
[
t
_
]
:
=
k
0
E
x
p
[
-
α
/
r
t
b
y
f
(
e
[
t
]
-
e
f
0
)
]
k
a
[
t
_
]
:
=
k
0
E
x
p
[
β
/
r
t
b
y
f
(
e
[
t
]
-
e
f
0
)
]
Specify the potential:
I
n
[
5
]
:
=
t
s
=
1
;
ν
=
-
1
;
e
1
=
1
/
2
;
e
[
t
_
]
:
=
P
i
e
c
e
w
i
s
e
[
{
{
e
1
+
ν
t
,
0
≤
t
≤
t
s
}
,
{
e
1
+
2
ν
t
s
-
ν
t
,
t
s
≤
t
≤
2
t
s
}
}
]
S
p
e
c
i
f
y
m
a
s
s
t
r
a
n
s
p
o
r
t
m
o
d
e
l
p
a
r
a
m
e
t
e
r
s
,
m
a
s
s
d
i
f
f
u
s
i
v
i
t
y
d
A
a
n
d
d
B
.
T
h
e
c
o
n
c
e
n
t
r
a
t
i
o
n
o
f
c
A
a
t
t
h
e
f
a
r
e
n
d
i
s
s
e
t
t
o
t
h
e
b
u
l
k
c
o
n
c
e
n
t
r
a
t
i
o
n
a
n
d
t
h
e
c
o
n
c
e
n
t
r
a
t
i
o
n
o
f
c
B
i
s
s
e
t
t
o
0
:
I
n
[
6
]
:
=
p
a
r
s
=
"
D
i
f
f
u
s
i
o
n
C
o
e
f
f
i
c
i
e
n
t
"
{
{
d
A
,
0
}
,
{
0
,
d
B
}
}
,
"
B
o
u
n
d
a
r
y
C
o
n
d
i
t
i
o
n
1
"
"
A
m
b
i
e
n
t
C
o
n
c
e
n
t
r
a
t
i
o
n
"
{
k
a
[
t
]
/
k
c
[
t
]
c
B
[
t
,
x
]
,
k
c
[
t
]
/
k
a
[
t
]
c
A
[
t
,
x
]
}
,
"
M
a
s
s
T
r
a
n
s
f
e
r
C
o
e
f
f
i
c
i
e
n
t
"
{
k
c
[
t
]
,
k
a
[
t
]
}
,
"
B
o
u
n
d
a
r
y
C
o
n
d
i
t
i
o
n
2
"
"
M
a
s
s
C
o
n
c
e
n
t
r
a
t
i
o
n
"
{
c
A
b
u
l
k
,
0
}
;
Set up the model:
I
n
[
7
]
:
=
e
q
n
=
{
M
a
s
s
T
r
a
n
s
p
o
r
t
P
D
E
C
o
m
p
o
n
e
n
t
[
v
a
r
s
,
p
a
r
s
]
M
a
s
s
T
r
a
n
s
f
e
r
V
a
l
u
e
[
x
0
,
v
a
r
s
,
p
a
r
s
,
"
B
o
u
n
d
a
r
y
C
o
n
d
i
t
i
o
n
1
"
]
,
M
a
s
s
C
o
n
c
e
n
t
r
a
t
i
o
n
C
o
n
d
i
t
i
o
n
[
x
1
/
4
0
,
v
a
r
s
,
p
a
r
s
,
"
B
o
u
n
d
a
r
y
C
o
n
d
i
t
i
o
n
2
"
]
}
O
u
t
[
7
]
=
∇
{
x
}
·
-
1
1
0
0
0
0
0
.
∇
{
x
}
c
A
[
t
,
x
]
+
(
1
,
0
)
c
A
[
t
,
x
]
,
∇
{
x
}
·
-
1
1
0
0
0
0
0
.
∇
{
x
}
c
B
[
t
,
x
]
+
(
1
,
0
)
c
B
[
t
,
x
]
N
e
u
m
a
n
n
V
a
l
u
e
0
.
0
0
0
0
1
-
1
9
.
4
5
5
3
0
.
5
-
1
.
t
0
.
≤
t
≤
1
.
-
1
.
5
+
t
1
.
≤
t
≤
2
.
0
.
T
r
u
e
2
.
7
1
8
2
8
-
1
.
c
A
[
t
,
x
]
+
3
8
.
9
1
0
5
0
.
5
-
1
.
t
0
.
≤
t
≤
1
.
-
1
.
5
+
t
1
.
≤
t
≤
2
.
0
.
T
r
u
e
2
.
7
1
8
2
8
c
B
[
t
,
x
]
,
x
0
,
N
e
u
m
a
n
n
V
a
l
u
e
0
.
0
0
0
0
1
1
9
.
4
5
5
3
0
.
5
-
1
.
t
0
.
≤
t
≤
1
.
-
1
.
5
+
t
1
.
≤
t
≤
2
.
0
.
T
r
u
e
2
.
7
1
8
2
8
-
3
8
.
9
1
0
5
0
.
5
-
1
.
t
0
.
≤
t
≤
1
.
-
1
.
5
+
t
1
.
≤
t
≤
2
.
0
.
T
r
u
e
2
.
7
1
8
2
8
c
A
[
t
,
x
]
-
1
.
c
B
[
t
,
x
]
,
x
0
,
D
i
r
i
c
h
l
e
t
C
o
n
d
i
t
i
o
n
{
c
A
[
t
,
x
]
1
,
c
B
[
t
,
x
]
0
}
,
x
1
4
0
Set up initial conditions such that only
c
A
is in bulk solution:
I
n
[
8
]
:
=
i
c
s
=
{
c
A
[
0
,
x
]
c
A
b
u
l
k
,
c
B
[
0
,
x
]
0
}
;
Solve the model:
I
n
[
9
]
:
=
t
m
a
x
=
2
t
s
;
{
c
A
f
u
n
,
c
B
f
u
n
}
=
N
D
S
o
l
v
e
V
a
l
u
e
[
{
e
q
n
,
i
c
s
}
,
{
c
A
,
c
B
}
,
{
t
,
0
,
t
m
a
x
}
,
{
x
}
∈
Ω
]
O
u
t
[
9
]
=
I
n
t
e
r
p
o
l
a
t
i
n
g
F
u
n
c
t
i
o
n
D
o
m
a
i
n
:
{
{
0
.
,
2
.
}
,
{
0
.
,
0
.
0
2
5
}
}
O
u
t
p
u
t
:
s
c
a
l
a
r
,
I
n
t
e
r
p
o
l
a
t
i
n
g
F
u
n
c
t
i
o
n
D
o
m
a
i
n
:
{
{
0
.
,
2
.
}
,
{
0
.
,
0
.
0
2
5
}
}
O
u
t
p
u
t
:
s
c
a
l
a
r
Visualize the concentrations at various points in time:
I
n
[
1
0
]
:
=
M
a
n
i
p
u
l
a
t
e
P
l
o
t
{
c
A
f
u
n
[
t
,
x
]
,
c
B
f
u
n
[
t
,
x
]
}
,
{
x
,
0
,
1
/
2
0
}
,
,
,
R
u
l
e
[
]
O
u
t
[
1
0
]
=
t
1
.
9
c
A
c
B
Visualize the cyclic voltammogram at various points in time:
I
n
[
1
1
]
:
=
M
a
n
i
p
u
l
a
t
e
P
a
r
a
m
e
t
r
i
c
P
l
o
t
{
-
(
e
[
t
]
-
e
f
0
)
,
k
c
[
t
]
(
c
A
f
u
n
[
t
,
0
]
-
(
k
a
[
t
]
c
B
f
u
n
[
t
,
0
]
)
/
k
c
[
t
]
)
}
,
{
t
,
0
,
t
m
a
x
}
,
,
,
R
u
l
e
[
]
O
u
t
[
1
1
]
=
d
t
1
See Also
Transient Gray–Scott Model
Transient Landau–Ginzburg Model
Solute Diffusion Under Centrifugation
Fokker–Planck Mass Transport Model
Smoluchowski Diffusion Equation
Related Symbols
MassTransportPDEComponent
MassTransferValue
MassConcentrationCondition
NDSolveValue
Publisher Information
Contributed by:
Wolfram Staff