Environmental Engineering Reference
In-Depth Information
Q
L
C
0
Q
g
Q
g
Q
L
C
w
H
w
FIGURE 6.19
Schematic of submerged bubble aeration in a wastewater lagoon.
where
C
w
is the concentration in the aqueous phase that would be in equilibrium
with the concentration associated with the bubble,
C
g
. Note that the effect of external
pressure on the bubble size has been neglected in deriving the above equation. The
equilibrium concentration in the vapor phase of the bubble is given by the air-water
partitioning constant,
C
g
=
K
,aw
C
w
.
In an aeration apparatus, it is more convenient to obtain the specific air-water
interfacial area,
a
v
(m
2
per m
3
of total liquid volume). Hence
A
b
/V
b
=
6
/D
b
=
a
v
(V
w
/V
a
)
, where
D
b
is the average bubble diameter,
V
w
is the total liquid volume,
and
V
a
is the total air volume in the reactor at any time. The ratio
V
a
/V
w
is called the
gas
hold-up,
ε
g
. Using the above definitions,
we can rewrite the equation for the rate of change of concentration associated with
the bubble as
ε
g
. Therefore, we can write
a
v
=
(
6
/D
b
)
(K
w
a)
V
w
V
a
1
K
aw
(K
w
a)
V
w
V
a
C
w
.
d
C
g
d
t
+
C
g
=
(6.96)
a few seconds), a rea-
sonable assumption is that during this time,
C
w
is a constant. This means we can
integrate the above equation to obtain the concentration of the pollutant associated
with a single bubble.
Using the initial condition,
C
A
(
Since the rise time of a single bubble is really small (
τ ≈
τ =
0
)
=
0, we have
C
g
(
τ
)
=
K
aw
C
w
1
e
−
((K
w
a)(V
w
/V
a
)(
1
/K
aw
)
τ
)
.
−
(6.97)
The rate of change of pollutant concentration within the aqueous phase is given by
Q
g
V
w
C
g
(
d
C
w
d
t
=−
τ
)
,
(6.98)
where
Q
g
is the volumetric flow rate of air. The initial condition for the aqueous phase
is
C
w
(
0
)
C
w
for a batch reactor.We also note that the residence time of a gas bubble
=
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