Environmental Engineering Reference
In-Depth Information
surface current of 4.5 cm/s and currents on the bottom
of the lake are negligible, assess whether significant
wind-induced mixing is likely to occur in the lake.
Typically, well-stratified lakes have Fr
D
<< 0.1, weakly
stratified lakes have 0.1 < Fr
D
< 1, and well-mixed lakes
have Fr
D
> 1. The average velocity,
V
, and the change in
density, Δ
ρ
, are sometimes expressed in the forms
Solution
QL
V
V
=
(7.32)
From the given data: Δ
z
= 10.0 m,
T
= 13.5°C,
Δ
ρ
= 0.500 kg/m
3
, and Δ
u
= 4.5 cm/s = 0.045 m/s. For the
given average lake temperature (13.5°C), the density of
water calculated using the Gill (1982) formula, Equa-
tion (7.22), is
ρ
= 999.34 kg/m
3
. Using these data, the
following estimates can be made:
L
∆ρ =
d
(7.33)
where
Q
is the volumetric discharge through the
impoundment (lake or reservoir) (L
3
T
−1
),
L
is the length
of the impoundment (L),
V
L
is the volume of the
impoundment (L
3
), and
β
is the average density gradi-
ent, defined as the change in density per unit depth
(ML
−4
). For freshwater impoundments,
β
and
ρ
0
are
commonly approximated by 10
−3
kg/m
4
and 1000 kg/m
3
respectively.
g
∂
∂
ρ
g
∆
∆
9 81
999 34
.
0 500
10
.
N
= −
≈
=
=
0 0222
.
Hz
ρ
z
ρ
z
.
∂
∂
u
z
∆
∆
u
z
0 045
10
.
≈
=
=
0 0045
.
Hz
EXAMPLE 7.12
and hence the Richardson number is given by Equation
(7.30) as
Measurements in a 10-m-deep lake show a mean veloc-
ity of 10 cm/s and a density difference between the top
and bottom of the lake of 4.1 kg/m
3
. If the mean density
of the lake water can be taken as 998 kg/m
3
, estimate
the strength of the stratification.
N
u
z
2
N
u
z
2
( .
0 0222
0 0045
)
2
Ri =
≈
=
=
24
2
2
2
∂
∂
∆
∆
( .
)
Since Ri >> 0.25, wind-induced mixing in the lake is
unlikely to occur under the given conditions.
Solution
From the data given,
d
= 10 m,
V
= 10 cm/s = 0.1 m/s,
Δ
ρ
= 4.1 kg/m
3
,
ρ
0
= 998 kg/m
3
, and hence the densimet-
ric Froude number is given by Equation (7.31) as
7.4.4.2 Densimetric Froude Number.
The stability of
a lake is sometimes measured by the
densimetric Froude
number
, Fr
D
(dimensionless), which is a measure of the
ratio of inertial to buoyancy forces defined as
V
0 1
.
Fr
D
=
=
=
0 16
.
∆ρ
ρ
0
4 1
998
.
gd
( .
9 81 10
)(
)
inertia
buoyancy
V
Fr
D
=
=
(7.31)
∆ρ
ρ
0
gd
Since 0.1 < Fr
D
< 1, the lake should be classified as
weakly stratified.
where
V
is the average velocity in the lake (LT
−1
), Δ
ρ
is
the change in density over the depth (ML
−3
),
ρ
0
is the
average density of the water (ML
−3
),
g
is the accelera-
tion due to gravity (LT
−2
), and
d
is the average depth of
the lake (L). Low values of Fr
D
indicate that buoyancy
forces are dominant in controlling the motion of the
fluid, and high values of Fr
D
indicate that buoyancy
forces have a relatively small effect on the motion of the
fluid. Measurements by Long (1962) in laboratory chan-
nels indicated that if Fr
D
<< 0.32, the impoundment is
likely to become stratified; if Fr
D
≈ 0.32, the impound-
ment is likely to be weakly stratified; and if Fr
D
>> 0.32,
the impoundment is likely to be vertically mixed.
7.4.5 Artificial Destratification
Mechanical stirrers, water pumps, and air-bubble plume
systems are all used to destratify lakes and reservoirs.
The most common approach is the air-bubble plume
system in which compressed air is pumped through a
perforated pipe at the bottom of the reservoir. The flow
induced by the rising air bubbles carries cold hypolim-
netic water from the bottom of the reservoir toward the
top of the reservoir. The efficiency of this system depends
on the rate at which air is injected into the water and
the strength of the stratification.
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