Environmental Engineering Reference
In-Depth Information
c
1
M
c
o
t
t
0
∆
t
0
∆
t
(a)
c
1
M
1
M
2
c
o
t
t
0
∆
t
0
∆
t
(b)
Figure 7.10.
Response of a well-mixed lake to a variable contaminant inflow: (a) mass inflow over a finite interval; (b) variable
mass inflow.
Equation (7.41) up to
t
= Δ
t
, beyond which the response
is described by
d
dt
V c M Q c kV c v A c
(7.45)
(
)
=
−
−
−
L
o
L
s
L
Q
V
where
A
L
is the surface area of the lake over which set-
tling occurs. Equation (7.45) can be slightly rearranged
into the form
o
c t
( )
=
c
exp
−
+
k t
(
−
∆
t
)
(7.43)
1
L
which is derived from Equation (7.41) for a contami-
nant mass inflow of zero and an initial concentration of
c
1
. In the case of a variable mass inflow illustrated in
Figure 7.10b, where the contaminant mass inflow rate is
equal to
M
1
up to
t
= Δ
t
, and equal to
M
2
thereafter,
the response of the lake is described by Equation (7.41)
up to
t
= Δ
t
, beyond which the concentration in the lake
is described by
d
dt
V c M Q c k V c
(7.46)
(
)
=
−
− ′
L
o
L
where
v A
V
s
L
k
′
= +
k
(7.47)
L
{
}
2
M
Q kV
Q
V
o
c t
( )
=
1
−
exp
−
+
k t
(
−
∆
t
)
Since Equation (7.46) is identical with the conservation
equation that neglects sedimentation as a removal
process (Eq. 7.37), with the decay coefficient,
k
, replaced
by an effective decay coefficient,
k
′, all the previous
results are applicable provided that
k
is replaced by
k
′.
Typical values of the settling velocity,
v
s
, are in the range
of 10-16 m/year (30-50 ft/year).
Several variations of the completely mixed model
have been used in practice. In cases where there is sig-
nificant vertical stratification, the lake can be considered
as a well-mixed epilimnion overlying a well-mixed
hypolimnion, with limited interaction between the two
zones. Also, in large lakes where
A
L
> 50-100 km
2
(20-
40 mi
2
), it may be necessary to subdivide the lake into
a number of well-mixed smaller lakes.
+
o
L
L
(7.44)
Q
V
o
+
c
exp
−
+
k t
(
−
∆
t
)
1
L
which is derived from Equation (7.41) with a mass
inflow rate,
M
2
beginning at
t
= Δ
t
with an initial con-
centration of
c
1
.
The analyses described here are also applicable to
cases where the contaminants are removed by the set-
tling of suspended solids in the lake. In this case, where
the contaminants are adsorbed onto suspended solids,
the removal rate due to sedimentation can be described
by a settling velocity,
v
s
, and the conservation of mass
equation can be written as
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