Environmental Engineering Reference
In-Depth Information
where
Y
=
mass of solute
/
mass of fluid
ρ
f
=
fluid density
v
=
fluid velocity
=
/
q
mass sorbed
mass of sorbent
ρ
s
=
/
mass of sorbent
volume
G
b
=
bulk fluid mass flux = mass of fluid
/
area
·
time
L
=
bed length (see Equation (7.12))
A
=
column cross-sectional area
.
This result indicates that the velocity of the solute front is inversely proportional to the
slope of the isotherm. We can illustrate this result using a Type I isotherm (Figure 7.6).
During the adsorption step, the direction is from the lower left to upper right portion of
the curve. So, d
q
d
Y
is largest (
V
is slowest) during the initial portion of sorption. This
is the rate-limiting step so the entire front moves as a discontinuous wave (stoichiometric
front). A balance across this wave shows that d
q
/
/
d
Y
reduces to
q
/
Y
, the chord from
the initial state to the saturated state in the column.
In desorption, the result is very different. Now, the direction is from the upper right to
the lower left portion of the curve. The slope d
q
d
Y
has the smallest value (
V
is fastest)
during the initial portion and
V
becomes slower as desorption continues. This causes the
desorption wave to spread and an elongated breakthrough curve results.
Some additional system parameters can also be estimated (Equations (7.12) to (7.16)).
The minimum weight of sorbent needed to treat a given total weight of fluid is
/
Y
/
q
,
A
ρ
b
L
weight of carrier fluid processed
=
bed weight
Y
Y
f
−
Y
i
t
=
q
=
q
i
,
(7.12)
A
ρ
f
v
q
f
−
where
is the difference between the feed (f) and initial (i) column conditions.
For desorption, first recognize that d
q
H
(Henry's Law constant) as the ori-
gin of the isotherm is approached. So, the minimum amount of purge fluid needed per
weight of column sorbent equals
H
. The minimum time for breakthrough can then also be
determined:
/
d
Y
=
d
q
d
Y
=
→
H
(as you approach origin
for total desorption).
(7.13)
1
H
=
A
ρ
b
L
∴
(7.14)
A
ρ
f
v
t
weight of purge required
bed weight
=
ρ
f
v
t
ρ
b
L
=
G
b
t
ρ
b
L
.
H
=
(7.15)
Rearranging,
=
ρ
b
H
G
b
t
L
.
(7.16)
Results for various isotherms are shown in Figures 7.7 to 7.10.
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