Environmental Engineering Reference
In-Depth Information
where
Θ
' =
d
Θ
'/
dy
. Substitution of these two equations in the differential
equation we need to solve gives:
ε
u
B
=−
(
)
1
+
RT
1
− ε ρ
H
CO
2
A
We indeed have a solution for our equation. The boundary condition and
initial condition are given as:
p
(0, )
t
=
p
and
p
( ,0)
z
=
0
CO
0
CO
2
2
We can now calculate the breakthrough curve, without having any knowl-
edge about the form of
Θ
. Surprised? Let's see how this works. For the
breakthrough curve we only need to know the pressure as a function of
time at position
z
=
L
. Why does this help us to fi nd the solution? Let us
assume we would like to know the solution at a time
t
' and position
L
.
From the form of the solution and boundary condition, we have:
(
)
(
)
(
)
(
)
p
L t
,
′
=Θ
L
+
Bt
′
=Θ
0
+
Bt
=
p
0,
t
=
p
CO
CO
0
2
2
Or, the solution at
z
=
L
, is the same as the solution at
z
= 0, at a time
t
',
if the arguments of
Θ
are the same:
LBt
+=
′
+
Bt,
which tells us that we have the same solution at
z
=
L
as we have at
z
=
0, not at time
t
but at
t
':
(
)
1
+
RT
1
−
ε ρ
H
1
=−
CO
t
′
t
Lt
=+
A
2
L
ε
Therefore, if we start injecting CO
2
in the adsorber at
t
= 0, the CO
2
will
simply arrive at length
L
a little later. The coeffi cient
B
can be interpreted
as the velocity at which this front is traveling:
B
u
ε
u
(
)
(
)
u
′
=
≈ ε
u
1
−
RT
1
− ε ρ
H
,
(
)
CO
1
+
RT
1
−
ε ρ
H
A
2
CO
A
2
where the last step is only valid if the adsorption is small. Hence, the
profi le that we inject will move with an effective velocity that depends on
the Henry coeffi cient. In addition, it will come out of the column in exactly
the same shape, as is illustrated in
Figure 6.3.5
.
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