Environmental Engineering Reference
In-Depth Information
−1
c
c
c
c
3
3
e
e
o
=
⋅
=
6.37
erf
(9.70)
c x
( , 0)
=
c e
−
kx u
/
a
erf
(9.72)
o
3
3
c
c
2
3
β
x
L
2
3
β
x
L
max
o
max
2 1
+
−
1
2 1
+
−
1
where
The decay coefficient,
k
(T
−1
), is frequently expressed in
terms of the time,
T
90
(T), required for 90% reduction
in mass due to decay, where
=
12
o
ε
β
u L
(9.71)
ln
10
2.30
(9.73)
T
=
≈
The diffusion coefficient at the diffuser,
ε
o
, can be esti-
mated using the diffuser length,
L
= 39.2 m = 3920 cm
and the Okubo relation (Eq. 9.59) as
90
k
k
Fecal coliforms and
Escherichia coli
are nonconserva-
tive tracers that are frequently used as indicator organ-
isms in far-field dispersion models, particularly with
regard to assessing the potential contamination of
beaches by pathogenic microorganisms. Inactivation of
fecal coliforms in the ocean is caused primarily by solar
radiation, with secondary causes of inactivation being
nutrient deficiency, temperature, salinity, and predation
by natural microbiota. Field studies from around the
world have reported
T
90
values for fecal coliform in the
range of 0.6-24 hours in daylight and 60-100 hours at
night (Wood et al., 1993). A study in Mamala Bay,
Hawaii, showed that during the daytime hours, a con-
servative estimate of
T
90
for
E. coli
was 6.9 hours, and
during the night, negligible decay could be assumed
(Roberts, 1999). Decay is seldom considered in near-
field models, since the time scale of near-field mixing is
typically on the order of minutes, which is much shorter
than the time scale of decay processes. In far-field
models, it is reasonable to neglect decay if the travel
time is less than the nighttime duration. Worst-case sce-
narios for assessing the impact of ocean outfalls on
beaches correspond to assuming a persistent onshore
(wind-induced) current during the nighttime hours.
Such worst-case scenarios have been observed in the
vicinity of ocean outfalls (e.g., Stewart, 1973).
ε
o
=
0 0103
.
L
1 15
.
=
0 0103
.
(
392
0
)
1 15
.
=
140
cm /s
2
=
0 014
.
m /s
2
The parameter,
β
, is therefore given by Equation (9.71)
as
12
ε
12(0.014)
(0.11)(39.2)
o
u L
β
=
=
=
0.0390
Substituting into Equation (9.70) to find the distance,
x
,
from the diffuser, where the dilution is 100, gives
−1
3
100
=
6.37
erf
3
2
3
0.0390
39.2
x
2 1
+
−
1
which simplifies to
3
2(1 0.000663 )
0.0637
=
erf
+
x
3
−
1
Using the error function tabulated in Appendix D gives
x
=
10 200
,
m
=
10 2
.
km
EXAMPLE 9.7
Hence, the dilution reaches 100 at a distance of about
10.2 km downstream of the diffuser.
An ocean outfall discharges domestic wastewater 6 km
offshore, and under worst-case conditions, an onshore
wind causes surface currents of 22 cm/s to move the
contaminant plume directly toward recreational beaches.
A near-field model indicates an initial dilution of 20, and
application of the Brooks model for a conservative con-
taminant indicates a dilution of 110 between the beaches
and the zone of initial dilution (ZID). If the fecal coli-
form concentration in the wastewater effluent is
40,000 CFU per 100 ml, and the time for 90% decay
in seawater is 6 hours, determine the fecal coliform
The Brooks model of far-field dispersion assumes that
the contaminant in the discharged wastewater is conser-
vative. In cases where the contaminant is not conserva-
tive and the decay process can be approximated by a
first-order reaction with decay parameter
k
(T
−1
), the
concentration distribution along the plume centerline
given by Equation (9.67) can be multiplied by
e
kx u
−
/
a
to
give
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