Environmental Engineering Reference
In-Depth Information
Fixed Source Concentration.
In this scenario, the con-
centration of the tracer at the source remains constant
for a specified period of time. The two cases considered
here are illustrated in Figure 3.14, where Figure 3.14a
shows the tracer release beginning at
t
= 0 and lasting
indefinitely, and Figure 3.14b shows the tracer release
lasting for a time
τ
and then suddenly being shut off. The
solution of the one-dimensional advection-diffusion
equation for the case of an indefinite release of a tracer
undergoing first-order decay is given by (o'Laughlin
and Bowmer, 1975)
EXAMPLE 3.7
A waste discharge mixes with the flow in a river such that
the combined flow exiting the mixing zone has a volu-
metric flow rate of 9 m
3
/s and a contaminant concentra-
tion of 100 mg/L. The river is 10 m wide and 3 m deep, and
it is anticipated that the contaminant will have a longitu-
dinal diffusion coefficient of 6 m
2
/s and a decay constant
of 130 d
−1
. A water supply intake is located 1 km
downstream of the mixing zone as shown in Figure 3.15.
(a) What is the maximum contaminant concentration that
is expected at the water supply intake? (b) How would
this maximum concentration be affected if the waste dis-
charge is limited to operating for only 1 hour per day.
c
Vx
D
x Vt
D t
−
Γ
0
2
c x t
( , )
=
exp
(
1
−
Γ
)
erfc
2
2
x
x
(3.96)
Vx
x Vt
D t
+
Γ
+
exp
(
1
+
Γ
)
erfc
Solution
2
D
2
x
x
From the given data:
Q
= 9 m
3
/s,
c
0
= 100 mg/L,
W
=
10 m,
d
= 3 m,
D
x
= 6 m
2
/s,
k
= 130 d
−1
= 0.001505 s
−1
,
and
x
= 1000 m. Based on these data,
A
= (10)(3) =
30 m
2
, and
V
=
Q
/
A
= 9/30 = 0.30 m/s. The parameter Γ
is defined by Equation (3.97) as
where Γ is defined as
kD
V
x
(3.97)
Γ =
+
1 4
2
and
k
is the first-order decay constant. In the case where
the tracer release lasts for a finite time,
τ
, as shown
in Figure 3.14b, the solution given by Equation (3.96)
is applicable for
t
≤
τ
, and for
t
>
τ
, the concentration,
c
(
x
,
t
), is given by (Chapra, 1997)
kD
V
( .
0 001505 6
0 30
)( )
=
=
x
Γ =
1 4
+
1 4
+
1 184
.
2
2
( .
)
(a) The maximum contaminant concentration occurs at
the water supply intake when the waste discharge oper-
ates continuously. In this case, the contaminant concen-
tration at the intake is given by Equation (3.96) as
c
Vx
D
x Vt
D t
−
Γ
0
2
c x t
( , )
=
exp
(
1
−
Γ
)
erfc
2
2
x
x
+
x V t
D t
−
(
−
−
τ
τ
)
Γ
Vx
D
c
Vx
D
x Vt
D t
−
Γ
−
erfc
exp
(
1
+
Γ
)
0
2
c x t
( , )
=
exp
(
1
−
Γ
)
erfc
2
2
(
)
2
2
x
x
x
x
x Vt
+
Γ
−
x V t
D t
+
(
−
−
τ
τ
)
Γ
Vx
x Vt
D t
+
Γ
erfc
erfc
+
exp
(
1
+
Γ
)
erfc
2
D t
2
(
)
2
D
2
x
x
x
x
(3.98)
100
2
( . )(
0 3
1000
2 6
)
c
(
1
km
, )
t
=
exp
(
1 1 184
−
.
)
( )
c
(
t
)
1000
−
( . ) ( .
0 3
t
t
1 184
)
c
0
erfc
2
( )
6
( . )(
0 3 1000
2 6
)
+
exp
(
1 1 184
+
.
)
0
time,
t
( )
(a)
1000
+
( . ) ( .
0 3
t
1 18
4
)
c
(
t
)
c
0
erfc
2
( )
t
6
(3.99)
{
204 1 0 07250
.
−
.
t
0
time,
t
c
(
1
km
, )
t
=
50 0 01012
( .
)
erfc
τ
t
(b)
}
204 1 0 07250
.
+
.
t
Figure 3.14.
Temporal concentration distribution for continu-
ous source. (a) Infinite duration source. (b) Finite duration
source.
+
( .
5 158 10
23
×
)
erfc
t
(3.100)
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