Environmental Engineering Reference
In-Depth Information
Surface concentration balance:
∂
c
s
∂
3
R
(1
−
α
)
t
=
k
p
(
c
−
c
s
)
−
kc
s
−→
first-order reaction.
α
Initial conditions:
c
(
t
=
0
,
z
)
=
0
=
,
=
.
c
s
(
t
0
z
)
0
Boundary conditions:
c
(
t
≤
0
,
z
=
0)
=
c
0
(
t
)
c
s
(
t
≥
0
,
z
=
0)
=
0
c
(
t
,
z
→∞
)
=
finite
.
Laplace transform:
2
c
D
0
α
∂
−
v
∂
c
3
R
(1
−
α
)
3
R
(1
−
α
)
z
−
s c
−
k
p
(
c
−
c
s
)
−
s c
+
k
p
(
c
−
c
s
)
−
k c
s
=
0
,
∂
z
2
∂
α
α
yielding:
k
p
s
k
p
(1
−
α
)
(1
−
α
)
3
R
3
R
c
s
=
+
+
k
α
α
v
0
α
.
v
=
D c
−
v
c
−
∴
A c
=
0
,
3
R
k
p
+
(1
−
α
)
3
R
(1
−
α
)
α
from
∂
c
s
k
p
where
A
=
k
p
1
−
s
term
.
3
R
(1
−
α
)
α
∂
t
+
k
+
α
Boundary conditions:
c
(
z
=
0)
=
c
0
(
s
)
→∞
=
.
c
(
z
)
finite
Solution to part (b)
4
D
0
A
1
2
D
0
c
0
e
λ
z
2
c
(
z
)
=
where
λ
=
v
0
±
v
0
+
(lim
s
→
0
e
λ
z
)
m
0
(
z
)
=
lim
s
→
0
c
(
z
)
=
(
lim
s
→
0
c
0
)
m
0
(
z
=
0)
4
D
0
lim
s
→
0
A
1
2
D
0
0
lim
s
→
0
λ
=
v
0
−
v
+
≡
λ
0
≤
0
k
p
3
R
(1
−
α
)
3
R
(1
−
α
)
.
α
lim
s
→
0
A
=
k
p
1
−
3
R
(1
−
α
)
α
k
+
k
p
α
0)e
λ
z
∴
m
0
(
z
)
=
m
0
(
z
=
.
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