Environmental Engineering Reference
In-Depth Information
EXAMPLE 9.2
z
=
3 5
.
L
=
3 5 9 25
. ( .
)
=
32 4
.
m
m
N
Wastewater is discharged at a rate of 1 m
3
is from a port
that is 60 m below the ocean surface. Field measure-
ments indicate that the ocean is typically stratified, with
a density of 1025.0 kg/m
3
at the discharge depth and
1023.5 kg/m
3
at the ocean surface. The density of the
freshwater discharge is 998 kg/m
3
. Estimate the dilution,
rise height, and wastefield thickness of the plume. How
would these plume characteristics be affected by a
current of 15 cm/s?
1.6
L
L
=
1.6(9.25) 14.8
=
m Daviero and Roberts (2006)
,
N
h
=
n
1.7
=
1.7(9.25) 15.7
=
m Wong (
,
1985
)
N
Hence the maximum rise height is approximately 32 m
and the thickness of the (trapped) plume is in the range
of 14.8-15.7 m, which can be approximated as 15 m.
If an ambient current of 15 cm/s exists, then
u
a
= 0.15 m/s and the plume/crossflow length scale,
L
b
,
can be estimated by Equation (9.11) as
Solution
B
u
0.0271
0.15
0
3
L
=
=
=
8.02
m
From the given data:
Q
0
= 1 m
3
/s, Δ
z
= 60 m,
ρ
0
= 1025.0 kg/m
3
,
ρ
1
= 1023.5 kg/m
3
, and
ρ
e
= 998
kg/m
3
. The buoyancy frequency,
N
, and the discharge
buoyancy flux,
B
0
, are estimated as
b
3
a
Therefore, the ocean current has a significant effect
on the plume dynamics after a rise height of around
8 m. The plume dilution can be estimated by Equation
(9.23), and, based on the experimental dilution
coefficients in Table 9.3, a conservative estimate of
plume dilution,
S
, can be obtained using
C
SSF
= 1.3,
which gives
g d
dz
ρ
9.81
1025.0
1023.5 1025.0
60
−
a
N
= −
≈ −
=
0.0155
Hz
ρ
0
ρ
−
ρ
0
e
B Q g Q
=
′ =
g
0
0
0
ρ
e
(1)
1025.0 998
998
−
SQ N
u B
4 3
/
=
(9.81)
0
=
C
SSF
1 3
/
2 3
4 3
/
a
=
0.0271
m /s
4
3
/
(1)(0.0155)
(0.15)
S
= 1.3
1 3
/
2 3
/
(0.0271)
The stratification length scale,
L
N
, is given by Equation
(9.21) as
which yields
S
= 16. The corresponding rise height,
z
m
,
is given in Table 9.4 as
B
N
1 4
/
(0.0271)
(0.0155)
1 4
/
0
L
N
=
=
=
9.25
m
3 4
/
3 4
/
1 9
/
1 9
/
L
L
8.02
9.25
b
z
=
2.9
L
=
2.9
(9.25)
=
26.4
m
m
N
N
Therefore stratification will significantly affect the
plume dynamics after a rise height of around 9.25 m.
The plume dilution can be estimated using Equation
(9.21) with values of
C
SSS
given in Table 9.3. The more
conservative estimate of plume dilution,
S
, is obtained
by taking
C
SSS
= 0.90, which gives
which indicates a maximum rise height of around 26 m,
which is less than the 32 m rise height in a stagnant
environment. There is no experimental relationship
given for the plume thickness; however, the plume
thickness can be estimated using the conservation of
mass relationship,
SQ N
B
5 4
/
0
=
C
SSS
3 4
/
0
Q c Q c
0 0
=
p p
5 4
/
S
(1)(0.0155)
0.0271
=
0.90
3 4
/
where
c
0
and
c
p
are the tracer concentrations at the
discharge and terminal height of the plume, and
Q
p
is
the volumetric flow rate of the plume at the terminal
height. Taking
c
0
/
c
p
=
S
and
Q
p
=
u
a
πD
2
/4, where
D
is the
plume diameter at the rise height, gives
which yields
S
= 11. The rise height,
z
m
and plume thick-
ness,
h
n
, can be estimated using the empirical relations
in Table 9.4, which yield
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