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(b) What is the concentration at the release point after
24 hours?
(
)
g
ξ η ζ
,
,
d d d
ξ η ζ
x
y
z
2
2
2
∫
∫
∫
(
)
=
c x y z t
,
,
,
3
2
x
y
z
(
)
1
1
1
4
t
D D D
x
y
z
2
2
2
(
)
(
)
(
)
x
−
ξ
y
−
η
z
−
ζ
Solution
exp
−
−
−
4
D t
4
D t
4
D t
x
y
z
(a) The concentration as a function of time is given
by
(3.139)
where the contaminant source is located in the region
x
∈ [
x
1
,
x
2
],
y
∈ [
y
1
,
y
2
],
z
∈ [
z
1
,
z
2
]. Superposition in time
can also be applied to yield the concentration distribu-
tion,
c
(
x
,
y
,
z
,
t
), resulting from a continuous mass input
m
( )
as
M
x
D t
2
y
D t
2
z
D t
2
c x y z t
( ,
,
, )
=
exp
−
−
−
3
2
4
4
4
x
y
z
(
4
π
t
)
D D D
x
y
z
In this case,
M
= 1 kg,
D
x
= 10 m
2
/s,
D
y
= 15 m
2
/s,
D
z
= 0.1 m
2
/s, and therefore, the concentration as a
function of time at
x
= 100 m,
y
= 100 m,
z
= 10 m,
is given by
( )
m d
τ
τ
t
∫
(
)
=
c x y z t
,
,
,
3
2
0
[
]
(
)
4
π
t
−
τ
D D D
x
y
z
x
D t
2
y
2
z
D t
z
2
exp
−
)
−
)
−
1
(
(
(
)
4
−
τ
4
D
t
−
τ
4
−
τ
c
(
100 100 10
,
,
, )
t
=
x
y
3
2
(
4
π
t
)
10 15 0 1
100
4 10
(
)(
)( . )
(3.140)
2
2
2
100
4
10
4 0 1
exp
−
−
−
where the transient source is located at
x
= 0,
y
= 0,
z
= 0. In the case of a distributed transient source,
m x y z t
(
)
t
(
15
)
t
( . )
t
0 00580
.
667
kg/m
( , , , )
, the resulting concentration distribution,
c
(
x
,
y
,
z
,
t
), is given by
=
exp
−
3
3
2
t
t
ξ η ζ τ ξ η ζ τ
π
(
)
m
,
,
,
d d d d
t
x
y
z
2
2
2
∫
∫
∫
∫
(
)
=
c x y z t
,
,
,
(b) The concentration,
c
0
, at the release point,
x
= 0,
y
= 0,
z
= 0, is given by
3
2
[
]
4
(
t
−
τ
)
D D D
x
0
x
y
z
1
1
1
y
z
(
)
−
2
(
)
−
2
(
)
−
2
x
D t
−
ξ
y
D t
−
η
z
D t
−
ζ
exp
−
)
−
)
−
(
(
(
)
4
τ
4
τ
4
τ
1
0 00580
.
x
y
z
c
=
=
kg/m
3
0
3
2
3
2
(3.141)
(
4
π
t
)
(
10 15 0 1
)(
)( . )
t
3.3.3.2 Instantaneous Point Source in Shear Flow.
Diffusion of a tracer released instantaneously into a
water environment where the mean velocity varies spa-
tially in one or more of the coordinate directions is
commonly encountered in coastal waters. Consider a
three-dimensional flowing environment in which the
velocity,
V
, is oriented along the
x
axis and described by
the relation
and at
t
= 24 hours = 86,400 seconds, the concentra-
tion at the release point,
c
0
, is given by
0 00580
.
−
10
3
−
4
c
0
=
=
2 29 10
.
×
kg/m
=
2 29 10
.
×
µ
g/L
3
2
86 400
,
Concentrations at this level would not be detect-
able, and the contaminant concentration can be
taken as zero, for all practical purposes.
V V t
=
0
( ) λ
+
y
+
λ
z
(3.142)
y
z
where
λ
y
and
λ
z
are the rates of change of velocity with
distance in the
y
and
z
coordinate directions, respec-
tively, and
V
0
(
t
) is the time-varying magnitude of the
velocity at the origin of the coordinate system. The con-
centration distribution resulting from the instantaneous
release of a mass,
M
, at the origin is given by (okubo
and Karweit, 1969)
3.3.3.1 Spatially and Temporally Distributed
Sources.
The principle of superposition can be applied
to the fundamental solution of the three-dimensional
diffusion equation to yield the concentration distribu-
tion,
c
(
x
,
y
,
z
,
t
), resulting from an initial mass distribu-
tion (per unit volume),
g
(
x
,
y
,
z
), as
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