Environmental Engineering Reference
In-Depth Information
3 2
/
L
2
3
x
L
∂
∂
c
x
∂
∂
∂
∂
c
y
=
1
+
β
u
=
ε
(9.62)
(9.69)
a
y
y
with boundary conditions
The formulation for predicting pollutant concentrations
downstream of an outfall given by Equation (9.67) has
been widely used to predict the far-field mixing of
ocean outfall discharges, and its popularity is no doubt
due to its simplicity and the fact that the maximum
concentration is expressed in terms of measurable
quantities.
L
c
y
<
o
2
c
(0,
y
)
=
(9.63)
L
0
y
>
2
c
( ,
± ∞ = 0
)
(9.64)
where
x
is in the direction of flow (l),
y
is in the trans-
verse (horizontal) direction (l), and
ε
y
is the transverse
diffusion coefficient (lT−2)
2
T
−1
). Brooks (1960) assumed
that the transverse diffusion coefficient,
ε
y
, increases
with the size of the plume in accordance with the
four-
thirds law
originally proposed by Richardson (1926) and
supported by the field results of Okubo (1971). Accord-
ing to the four-thirds law, the transverse diffusion coef-
ficient,
ε
y
, can be expressed in the form
EXAMPLE 9.6
A 39.2-m-long multiport diffuser discharges treated
domestic wastewater at a rate of 5.73 m
3
/s and at a
depth of 28.2 m. If the ambient current is 11 cm/s, esti-
mate the distance downstream of the diffuser to where
the dilution is equal to 100.
Solution
4 3
/
ε
=
ε
(9.65)
Equation (9.61) estimates the initial dilution in the far-
field model as
y
o
L
where
ε
o
is the transverse diffusion coefficient (lT
−2
) at
the outfall (
x
= 0), and
is the transverse dimension of
the plume that varies with the distance,
x
, from the dif-
fuser. The transverse dimension of the plume,
, can be
defined in terms of the concentration distribution as
c
c
u Lh
Q
e
a
=
0
where
u
a
= 11
cm/s = 0.11 m/s,
L
= 39.2 m,
h
= 0.3(28.2)
= 8.46 m,
Q
= 5.73 m
3
/s, and hence
= 2 3σ
y
(9.66)
c
c
u Lh
Q
(0.11)(39.2)(8.46)
5.73
e
a
=
=
=
6.37
where
σ
y
is the standard deviation of the cross-sectional
concentration at a distance
x
from the source. An ana-
lytic expression for the resulting concentration distribu-
tion is not available, but the maximum concentration
(along the plume centerline) is given by
o
The initial plume dilution for the far-field model is
therefore estimated as 6.37. This estimate of initial dilu-
tion is certainly not as accurate as using a near-field
model to estimate the initial dilution, but this approxi-
mation is acceptable when the far-field dilution is much
greater than the near-field dilution. If a near-field model
is used to estimate the initial dilution, Equation (9.61)
is still valid, but the initial height of the waste field,
h
, is
no longer taken as 30% of the depth. Equation (9.67)
gives the far-field dilution as
3
c x
( , 0)
=
c
o
erf
(9.67)
3
2
3
β
x
L
2 1
+
−
1
where
β
is defined by
=
12
o
ε
β
u L
(9.68)
c
3
max
=
erf
3
c
2
3
β
x
L
o
2 1
+
−
1
The transverse dimension of the plume,
, at a distance
x
from the source corresponding to Equation (9.67) is
given by
and the total dilution,
c
e
/
c
max
, is given by
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