Environmental Engineering Reference
In-Depth Information
near
eld
c
=
f c u D
( ,
,
,
ρ ρ
,
,
g u y
,
,
)
ocean surface
(9.1)
near-
eld dilution,
S
,
measured here
1
e
e
e
a
a
x
n
This functional relationship assumes that the port dis-
charge is fully turbulent and hence the viscosity of the
port discharge can be neglected. Assuming that the
density difference between the effluent and seawater is
small compared with their absolute densities, the kine-
matics of the effluent plume does not depend explicitly
on the absolute density of the effluent plume and
ambient seawater but on the difference in densities and
the resultant buoyancy effect. This approximation is
called the
Boussinesq approximation
. The buoyancy
effect is measured by the effective gravity,
g
′ (lT
−2
),
defined by the relation
h
n
ocean current
u
a
y
plume
port
bottom of ocean
diffuser pipe
Figure 9.5.
Plume in unstratified ocean.
ρ ρ
ρ
−
a
e
g
′ =
g
(9.2)
e
9.2.1 Near-Field Mixing
The dynamics of near-field mixing or
initial dilution
depends on whether the effluent plumes originating at
the diffuser ports merge prior to either being trapped
or reaching the ocean surface. In cases where adjacent
plumes do not merge, each plume behaves (approxi-
mately) independently and identically, and near-field
mixing can be inferred from the analysis of a single
plume. In cases where adjacent plumes merge well in
advance of either being trapped or reaching the ocean
surface, the (merged) plume behaves as if the effluent
were discharged from a long slot or line. Such plumes
are called
line plumes
. In this section, the near field
region is defined as the region in which the discharge
structure and/or effluent properties have a significant
effect on the dilution of the effluent.
and the functional expression for the contaminant con-
centration in the effluent plume at a distance
y
above
the discharge port, Equation (9.1), can be written as
c
=
f c u D g u y
( ,
,
,
′
,
,
)
(9.3)
2
e
e
a
This relationship can be simplified by defining a
volume
flux,
,
Q
0
(l
3
T
−1
),
specific momentum flux,
,
M
0
(l
4
T
−2
), and
specific buoyancy flux,
,
B
0
(l
4
T
−3
), by the relations
2
D
(9.4)
Q u
=
e
π
0
2
2
D
(9.5)
M Q u
=
=
u
2
e
π
0
0
e
2
9.2.1.1 Single Plumes.
Consider the case of a single
effluent plume as illustrated in Figure 9.5. The effluent
is discharged through a port of diameter
D
(l) at veloc-
ity
u
e
(lT
−1
), where the effluent has a density
ρ
e
(Ml
−3
)
and contains a contaminant at concentration
c
e
(Ml
−3
).
Consider further that the ambient ocean water has a
density
ρ
a
(Ml
−3
) (assuming unstratified conditions), a
depth-averaged velocity
u
a
(lT
−1
), and we are interested
in calculating the contaminant concentration
c
(Ml
−3
)
at the end of the near field, where the discharge port is
located at a distance
y
(l) below the ocean surface. Key
geometrical characteristics of the effluent plume shown
in Figure 9.5 are the thickness of the wastefield,
h
n
, on
leaving the near field, and the distance,
x
n
, of the near-
field boundary from the discharge port.
The relationship between the contaminant concen-
tration,
c
, and the parameters controlling the dilution of
the effluent plume can be written in the following func-
tional form:
2
D
(9.6)
B Q g
=
′ =
u
e
π
g
′
0
0
2
The variables
Q
0
,
M
0
, and
B
0
involve only
u
e
,
D
, and
g
′
and can therefore be used instead of
u
e
,
D
, and
g
′ in
Equation (9.3) to yield the following functional expres-
sion for the contaminant concentration in the effluent
plume:
c
=
f c Q M B u y
( ,
,
,
,
,
)
(9.7)
3
e
0
0
0
a
Based on the Buckingham pi theorem, Equation (9.7)
can be expressed as a relationship between four dimen-
sionless groups, and the following groupings are particu-
larly convenient:
Q c
u L c
L
L
L
L
y
L
0
e
M
M
=
f
,
,
(9.8)
4
2
a
b
Q
b
b
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