Environmental Engineering Reference
In-Depth Information
Combining Equations (3.145) and (3.146) gives the gov-
erning equation as
Simplifying this equation by replacing
τ
by
x
/
V
yields
the steady-state solution at a distance
x
downstream of
the continuous sources as
2
2
∂
∂
c
∂
∂
c
∂
∂
c
z
=
D
+
D
(3.147)
y
z
τ
y
2
2
2
2
m
x D D
Vy
D x
Vz
D x
c x y z
( ,
, )
=
exp
−
−
(3.152)
4
4
4
π
y
z
y
z
which is in the form of the two-dimensional diffusion
equation (even though this is a three-dimensional
problem), and
τ
is the average travel time from the
source to any distance
x
downstream of the source. For
a contaminant source with a steady mass release rate of
m
, the solution to Equation (3.147) can be visualized in
Figure 3.21, where a “slab of fluid” moving with velocity
V
intersects the source and is injected with a mass,
M
,
of fluid given by
This equation describes the steady-state concentration
in cases where lateral diffusion is unrestricted, however,
in cases where lateral boundaries exist the principle of
superposition can be used to determine the locations of
image sources such that the lateral concentration gradi-
ents are equal to zero at the boundaries.
EXAMPLE 3.15
M m
w
V
=
(3.148)
A tracer is released into a deep ocean at a rate of
10 kg/s. The mean ocean current is 25 cm/s, and the dif-
fusion coefficient is isotropic and equal to 0.1 m
2
is Esti-
mate the concentration 100 m downstream from the
source under the following conditions: (a) the source is
far below the ocean surface and the tracer is conserva-
tive; (b) the source is far below the ocean surface and
the tracer undergoes first-order decay with a rate con-
stant of 0.1 min
−1
; and (c) the source is 5 m below the
ocean surface and the tracer undergoes first-order decay
with a rate constant of 0.1 min
−1
.
where
w
is the width of the slab of fluid. once the slab
of fluid has passed the source, the injected mass then
diffuses in the transverse directions within the slab. In
the limit that the width of the slab,
w
, is very small, then
the solution to Equation (3.147) must be sought for the
following boundary and initial conditions,
M
w
(3.149)
c y z
( ,
, )
0 =
δ
( , )
y z
Solution
c
(
±∞ ± ∞
,
, )
τ
=
0
(3.150)
From the given data:
m
= 10 kg/s
,
V
= 25 cm/s = 0.25 m/s,
D
x
=
D
y
=
D
z
= 0.1 m
2
/s, and
x
= 100 m.
(a) For a conservative tracer dispersing in an unbounded
environment, the concentration distribution can be
approximated by Equation (3.152). At a location
100 m downstream of the source,
y
= 0 m,
z
= 0 m,
and Equation (3.152) gives
Finding the solution to Equation (3.147) subject to the
conditions given in Equations (3.149) and (3.150) is
exactly the same as the fundamental diffusion problem
in two dimensions, and the solution is therefore given
by
m
w
V
w D D
y
D
2
z
D
2
c y z
( ,
, )
τ
=
exp
−
−
(3.151)
2
2
m
x D D
Vy
D x
Vz
D x
4
τ
4
τ
4
π τ
c x y z
( ,
, )
=
exp
−
−
y
z
y
z
4
4
4
π
y
z
y
z
slab downstream of source
slab upstream of source
continuous source
z
V
x
y
tracer in slab
tracer plume
w
w
Figure 3.21.
Steady continuous source.
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