Environmental Engineering Reference
In-Depth Information
The initial condition (I.C.) states that no solute is initially in the system. The boundary
condition (B.C.) at
z
0 is the input pulse. The second boundary condition means that
the solution must remain finite for any axial length of the bed.
Take the Laplace transform:
=
d
2
c
d
z
2
D
0
α
V
0
α
d
c
dz
−
→
+
−
s
c
(
z
,
t
=
0)
=
0
where
c
(
z
=
0
,
s
)
=
c
0
(
s
)
c
(
z
→∞
,
s
)
=
finite
k
e
λ
z
Assume
c
=
.
The characteristic equation is:
D
0
α
V
0
α
λ
−
2
λ
−
s
=
0
V
0
α
2
α
2
D
0
V
0
α
4
s
D
0
α
use only negative root
λ
=
±
+
k
=
c
0
(
s
)
.
The solution for
c
is:
c
0
(
s
)e
λ
z
c
(
z
,
s
)
=
,
where
λ
is defined above. We can now obtain an equation for various moments:
lim
s
→
0
c
0
(
s
)e
λ
z
m
0
(
z
)
=
=
c
0
d
c
d
s
.
m
1
(
z
)
=−
lim
s
→
0
To calculate
m
1
(
z
)
α
D
0
/
V
0
V
0
z
(
−
1
/
2)4
d
c
d
s
=
d
c
0
d
s
e
λ
z
c
0
(
s
)e
λ
z
+
2
D
0
1
α
D
0
/
V
0
+
4
s
e
λ
z
1
−
α
V
0
lim
s
→
0
d
c
0
d
s
e
λ
z
1
d
c
d
s
lim
s
→
0
=
+
m
0
.
−
m
1
(
z
)
−
m
1
(
z
=
0)
Rearranging, and using the definitions for the normalized moment:
+
α
z
V
0
µ
1
(
z
)
=
µ
1
(
z
=
0)
=
α
L
V
0
=
L
v
µ
1
=
µ
1
(
L
)
−
µ
1
(
z
=
0)
=
time for pulse to travel from
z
=
0to
z
=
L
(time of travel)
Search WWH ::
Custom Search