Environmental Engineering Reference
In-Depth Information
where
c
0
(
x
,
t
) is the fundamental solution to the one-
dimensional diffusion problem and given by
2
M
x
D t
c x t
( , )
=
exp
−
0
4
A
4
π
D t
x
x
2
M
x
D t
100
10
4 50 30
(
−
)
2
c x t
( , )
=
exp
−
(3.95)
c
0
(
−
10 30
,
)
=
exp
−
0
4
A
4
π
D t
(
)(
)
(
250
π
02865
)
4
(
50 30
)(
)
x
x
3
=
0 0
.
kg/m
= .
2 865
mg/L
Although Equation (3.94) contains an infinite summa-
tion of image contributions, the image contributions to
the tracer concentration within the bounded domain
generally decrease to zero as the distance of the image
sources from the bounded (real) domain increases. The
superposition approach shown here can be used simi-
larly to derive solutions to diffusion in other bounded
domains.
x
n
, can be calculated using
Equations (3.92) and (3.93), and the corresponding con-
centrations at the center of the lake, at
x
= −10 m and
t
= 30 days, are calculated by
The image locations,
x
n
and
′
2
M
(
x
−
x
D t
)
n
c x
(
−
x t
, )
=
exp
−
0
n
4
A
4
π
D t
x
x
EXAMPLE 3.6
M
(
x
− ′
x
D t
)
2
n
c x
(
−
′
x t
, )
=
exp
−
0
n
4
A
4
π
D t
A lake that is 50 m wide, 50 m long, and 5 m deep is
treated with an algaecide using a small boat that tra-
verses the lake from one side to the other. The boat
dumps 100 kg of algaecide across a path the is 15 m
from one side of the lake, as shown in Figure 3.13. The
diffusion coefficient of the algaecide in the lake is esti-
mated as 50 m
2
/d. If the algaecide is delivered approxi-
mately instantaneously and is well mixed over the depth
of the lake, estimate the concentration in the middle of
the lake after 30 days.
x
x
and the results of these calculations are summarized in
the following table:
−
′
x
n
c
0
(
x
−
x
n
,
t
)
x
n
′
c x x t
n
0
(
, )
n
(m)
(mg/L)
(m)
(mg/L)
1
30
2.232
−70
1.599
2
80
0.755
−120
0.388
3
130
0.111
−170
0.041
4
180
0.007
−220
0.002
Solution
5
230
0.000
−270
0.000
Total:
3.105
2.030
From the given data:
L
= 50 m,
W
= 50 m,
d
= 5 m,
M
= 100 kg,
D
x
= 50 m
2
/d,
t
= 30 d,
d
1
= 35 m, and
d
2
= 15 m. The area of the lake cross section is
A
=
(50)(5) = 250 m
2
, and if the initial tracer location is at
x
= 0, then at the center of the lake
x
= −10 m. The con-
centration at the center of the lake after 30 days con-
sidering only the real source and neglecting the lake
boundaries is given by
It is apparent from these results that beyond the
4th image set, the contribution to the concentration
at the center of the lake is negligible. using Equation
(3.94) with the tabulated results gives the concentration
at the center of the lake after 30 days as 2.865 mg/L +
3.105 mg/L + 2.030 mg/L = 8.000 mg/L.
3.3.1.3 Continuous Plane Source.
Consider the case
in which a tracer is released continuously from a source
that covers the entire cross-sectional area, this is com-
monly referred to as a continuous plane source. Two
distinct boundary conditions are encountered in prac-
tice: fixed concentration at the source, and fixed mass
flux at the source. An example of the former case is
where a wastewater discharge mixes with a stream flow
to yield a constant concentration of contaminant in the
mixture, while an example of the latter case is when
wastewater is discharged uniformly over the cross
section of a stream. The theoretical concentrations
resulting from these two scenarios are different and are
presented separately below.
x
= 0
reservoir
tracer at
t
= 0
x
15 m
50 m
Figure 3.13.
Lake.
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