Environmental Engineering Reference
In-Depth Information
Solution
c
( , 0)
=
0
(4.42)
c
for
for
t
t
≤
>
δ
δ
From the given data:
c
0
= 100 mg/L,
δ
= 1 hour = 3600
seconds,
k
= 5 d
−1
= 5.787 × 10
−5
s
−1
,
V
= 20 cm/s =
0.20 m/s,
K
L
= 30 m
2
/s, and
x
= 2 km = 2000 m. The
parameter
η
is given by Equation (4.46) as
0
c
(0, )
t
=
(4.43)
0
c
( , )
∞ = .
t
0
(4.44)
The solution to the one-dimensional ADE (Eq. 4.29)
with these boundary conditions is (Thomann and
Mueller, 1987)
−
5
kK
V
(5.787 10 )(30)
(0.20)
×
L
2
η=
=
=
0.0434
2
The contaminant concentration at the water-supply
intake is given by Equation (4.45) for
t
> 1 hour (= 3600
seconds) as
c
kx
V
x V t
K t
−
(
−
δ
)(1
+
η
)
0
c x t
( , )
=
exp
−
erf
2
4
(
−
δ
)
L
x Vt
K t
−
(1
+
η
)
c
kx
V
x V t
K t
−
(
−
δ
)(1
+
η
)
−
e
rf
0
c x t
( , )
=
exp
−
erf
4
2
4
(
−
δ
)
L
L
(4.45)
x Vt
K t
−
(1
+
η
)
−
e
rf
4
where
η
is defined by
L
100
2
(5.787 10 )(2000)
(0.20)
×
−
5
η=
kK
V
L
c
(2000, )
t
=
exp
−
(4.46)
2
2000 (0
−
.20)(
t
−
3600)(1 0.0434)
+
erf
If the duration of the spill is small relative to the travel
time, then the time of travel of the peak concentration,
t
p
, to a distance
x
from the spill is given by (Samuels
et al., 2006)
4(30)(
t
−
3600)
2000 (0.2
−
0) (1 0.0434)
4(30)
t
+
−
erf
,
t
x V
V
+
δ
(1
+
η
)
which simplifies to
t
p
=
(4.47)
(1
+
η
)
{
182.6 0.01905(
−
t
−
3600)
c
(2000, )
t
=
28.03
erf
and the peak concentration at a distance
x
is given by
substituting
t
p
for
t
in Equation (4.45). This formulation
has been used for predicting the maximum concentra-
tion in a river resulting from a finite duration spill
(Samuels et al., 2006). Such applications include assess-
ing the impact of spills on drinking water intakes.
t
−
0.0434
(4.48)
}
182.6 0.01905
−
t
−
erf
t
A plot of Equation (4.48) is shown in Figure 4.4, where
the peak concentration is 22.7 mg/L and occurs at
t
= 2.85 hours.
If the spill duration is short relative to the travel time
to the water-supply intake located 2000 m downstream,
then according to Equation (4.47), the peak concentra-
tion would occur at time
t
p
, where
EXAMPLE 4.8
A contaminant is accidentally released from an indus-
trial facility into a river such that the contaminant con-
centration is maintained at 100 mg/L in the mixing zone
for a duration of 1 hour, after which the concentration
returns to zero. The contaminant undergoes first-order
decay with a decay coefficient of 5 d
−1
, the mean flow
velocity in the river is 20 cm/s, and the longitudinal dis-
persion coefficient is 30 m
2
is Estimate the maximum
contaminant concentration at a water-supply intake
located 2 km downstream of the mixing zone. Evaluate
whether it would be appropriate to assume that the
duration of the spill is small relative to the travel time
to the water supply intake.
x V
V
+
δ
(1
+
η
)
2000 (0.20)(3600)(1 0.0434)
(0.20)(1 0.0434)
+
+
t
p
=
=
(1
+
η
)
+
=
=
13 184
3.66
,
seconds
hours
Since the actual time to peak of 2.85 hours is much dif-
ferent from the short-duration-spill time to peak of 3.66
hours, it is not appropriate to assume that the spill dura-
tion is short relative to the travel time to the downstream
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