Environmental Engineering Reference
In-Depth Information
The behavior of a bubble plume is controlled by the
following nondimensional parameters (Helfer et al.,
2011; McDougall, 1978),
Solution
From the given data:
Q
0
= 5.0 m
3
/s,
d
= 10.0 m,
T
= 15°C,
N
= 0.2 Hz, and
p
a
= 101 kPa = 1.01 × 10
5
Pa. At 15°C,
the density of water can be calculated using the
Gill (1982) formula, Equation (7.22), which yields
ρ
w
= 999.13 kg/m
3
and hence
γ
w
=
ρ
w
g
= (999.13)
(9.81) = 9801 N/m
3
. Based on the given data:
2
Q p
λ
πα ρ
(
+
1
)
0
a
M
=
(7.34)
M
4
2
H u
2
3
w
B
N H
gQ
3
4
H
h
=
C
(7.35)
M
0
a
5
p
1 01 10
9801
.
×
a
h
=
=
=
10 3
.
m
a
γ
w
where
Q
0
is the rate at which air is injected into the
water at atmospheric pressure (m
3
/s),
p
a
is atmospheric
pressure (Pa),
H
is the absolute pressure head at the
diffuser level (m),
h
a
is atmospheric pressure expressed
as an equivalent head of water, which is typically around
10.2 m (33.5 ft),
N
is the buoyancy frequency (s
−1
),
defined by Equation (7.29),
u
B
is a velocity scale (m/s)
defined by
H h
=
+ =
d
10 3 10 0
.
+
.
=
20 3
.
m
a
Assuming typical values of
u
s
= 0.3 m/s,
λ
= 0.3, and
α
= 0.083, the following values can be calculated using
Equations (7.36), (7.34), and (7.35), respectively,
λ
2
2
u
=
u
(
+
1
)
=
( . )( .
0 3 0 3
+
1
)
=
0 327
.
m/s
B
s
λ
2
u
=
u
(
+
1
)
(7.36)
Q p
λ
πα ρ
(
2
+
1
)
B
s
0
a
M
=
M
2
2
3
4
H u
w
B
where
u
s
is the slip velocity of the bubbles relative to
the liquid in the plume, which is typically around
0.3 m/s (1 ft/s),
λ
is the dispersion term, which is typi-
cally taken as 0.3,
ρ
w
is the average density of the water
(kg/m
3
), and
α
is the entrainment coefficient, which is
typically taken as 0.083. The dimensionless parameter
M
M
given by Equation (7.34) measures the air source
strength relative to the pressure head of the surround-
ing fluid, and
C
M
measures the strength of the stratifi-
cation relative to the strength of the source. A high
value of
C
M
represents a case where stratification is
strong compared with the source strength. On the
other hand, a low value of
C
M
represents a high source
strength compared with the level of stratification and
favors the bubble plume reaching the surface without
any internal detrainments. Aeration systems with same
values of
M
M
and
C
M
are expected to promote the
same mixing patterns in the water. Identification of the
optimal values of
M
M
and
C
M
is best done by field
investigation; however, lower vales of
C
M
are usually
preferable.
−
5
2
( . )( .
5 0 1 01 10
×
)( .
0 3
+
1
)
=
4
π
( .
0 083
)
2
(
999 13 20 3
.
)(
. ) ( .
2
0 327
)
3
=
442
N H
gQ
3
4
H
h
=
( . ) (
0 2 20 3
9 81 5 0
3
. )
4
20
3
10 3
.
.
=
=
C
55
M
( .
)( . )
0
a
Hence, for the present case,
M
M
= 442 and
C
M
= 55.
These values can be used as a basis to assess the perfor-
mance of the proposed system under similar conditions
in other lake systems.
7.5 WATER-QUALITY MODELS
Water-quality models are commonly utilized to assess
various lake management options. Ideally, the modeling
of lake water quality simulates lake processes and their
interconnected and independent relationships. Several
water-quality models that are commonly used in lakes
are described below.
EXAMPLE 7.13
7.5.1 Zero-Dimensional (Completely Mixed) Model
It is proposed to use bubble plumes generated by the
injection of 5.0 m
3
/s of air to destratify a 10.0-m-deep
lake that has an average temperature of 15°C and a
buoyancy frequency of 0.2 Hz. Atmospheric pressure at
the lake site is 101 kPa. Test results for various aeration
systems are available for various values of
M
M
and
C
M
.
What are the values of
M
M
and
C
M
in the present case?
The response of lakes to the input of contaminants
can sometimes be estimated by assuming that the lake
is well mixed. This approximation is justified in the
following cases: (1) when wind-induced circulation is
strong, and (2) when the time scale of the analysis is
sufficiently long (on the order of a year) that seasonal
mixing processes yield a completely mixed
lake.
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