Environmental Engineering Reference
In-Depth Information
usually in estimating
c
(
x
,0). Furthermore, in ocean envi-
ronments, it is well known that the (turbulent) diffusion
coefficient
D
y
increases with increasing size of the
plume, and so
D
y
is commonly taken as a function of the
width of the plume,
L
, where
0 5
2 3 2 0 10
.
( )(
0 15
0 1
)
6
2
.
( . )(
0 1
0 1 1
erfc
c
(
15 6
, )
=
)
exp
(
×
)( .
.
5
)
=
0 114
.
kg/m
3
=
114
mg/L
Therefore, the expected concentration at a location
15 m downstream of the confluence and 9 m from the
left bank is 114 mg/L. Immediately downstream of the
confluence, the concentration 9 m from the left bank is
approximately equal to zero, since this is 6 m into the
clean stream.
= 2 3σ ( )
(3.128)
L
x
y
where
σ
y
(
x
) is the standard deviation of the concentra-
tion distribution across the plume at a distance
x
from the source. The centerline concentrations corre-
sponding to various functional forms of the diffusion
coefficient are given in Table 3.1. of particular note is
the case in which the diffusion coefficient depends
on the length scale of the plume to the 4/3 power,
which is commonly assumed to represent the turbulence
characteristics in the ocean. Although the “4/3-law”
for the diffusion coefficient can be derived from
turbulence theory (Tennekes and Lumley, 1972), it has
also been shown that the 4/3 relationship exists in ocean
environments where shear dispersion dominates the
mixing process (Chin and roberts, 1985a). Based on all
these considerations, the 4/3 relationship for
D
y
is com-
monly used as the default assumption
Rectangular Source.
In the rectangular planar source
shown in Figure 3.19b, the source of width
b
is between
y
= ±
b
/2 and
z
= ±∞. In the case of a continuous release
with a source concentration equal to
c
0
(ML
−3
), the
steady-state concentration distribution downstream of
the source is given by (Brooks, 1960)
c
kx
V
y b
0
2
c x y
( ,
)
=
exp
+
/
2
V
D x
−
y b
−
/
2
V
D x
in ocean
erf
erf
2
2
environments.
y
y
(3.127)
EXAMPLE 3.12
This equation was originally developed and is widely
used to estimate contaminant concentrations down-
stream of ocean outfalls of length
b
. In these applica-
tions, the maximum concentration at any given distance
x
downstream of the outfall occurs on the plume cen-
terline (where
y
= 0), and so most of the interest is
Wastewater is discharged into the ocean from a 35-m-
long diffuser such that the contaminant concentration
at the diffuser after initial mixing is 100 mg/L. The ocean
current at the diffuser is 10 cm/s in a direction perpen-
dicular to the diffuser, and the diffusion coefficient
TABLE 3.1. Centerline Concentrations for Various Diffusion Coefficient Formulations
Diffusion Coefficient,
D
y
Centerline Concentration,
c
(
x
,0)
Plume Width,
L
a
/
b
D
y
0
kx
V
b V
D x
y
24
D x
Vb
exp
y
c
erf
1
+
0
4
2
L
b
12
D x
Vb
D
y
0
2
1
+
y
0
kx
V
3 2
/
c
exp
erf
0
2
12
D x
Vb
−
y
0
2
1
+
1
4
3
+
D x
Vb
8
L
b
y
0
2
1
D
kx
V
3 2
/
y
0
c
exp
erf
0
2
8
D x
Vb
−
y
0
2
1
+
1
a
L
is defined as
2 3σ
y
.
Search WWH ::
Custom Search