Environmental Engineering Reference
In-Depth Information
c
o
High initial concentration
M
Q
o
+
kV
L
M
Low initial concentration
c
o
Time
Time
0
0
(a)
(b)
Figure 7.9.
Response of a well-mixed lake to a constant contaminant inflow: (a) mass inflow; (b) lake response.
Completely mixed models are frequently called
zero-
dimensional models
, since they do not have any spatial
dimension.
which simplifies to the following differential equation
that describes the contaminant concentration in the
lake as a function of time:
7.5.1.1 Conservation of Mass Model.
Contaminant
mass fluxes into a lake can come from a variety of
sources, including municipal and industrial waste dis-
charges, inflows from polluted rivers, direct surface
runoff, contaminant releases from sediments, and con-
taminants contained in rainfall (atmospheric sources).
Denoting the rate of contaminant mass inflow to a lake
by
M
(MT
−1
), assuming that the lake is well mixed, and
assuming that the contaminant undergoes first-order
decay with a decay factor,
k
(T
−1
), the law of conserva-
tion of contaminant mass requires that
dc
dt
Q
V
M
V
=
o
+
+
k c
(7.39)
L
L
Taking the mass inflow rate,
M
, to be constant and
beginning at
t
= 0, and taking the initial condition as
c
=
c
o
at
t
=
0
(7.40)
yields the following solution to Equation (7.39)
{
}
M
Q kV
Q
V
o
c t
( )
=
1
−
exp
−
+
k t
d
dt
V c M Q c kV c
+
(7.37)
(
)
=
−
−
o
L
L
(7.41)
L
o
L
Q
V
o
+
c
exp
−
+
k
t
0
L
where
V
L
is the volume of the lake (L
3
),
c
is the average
contaminant concentration in the lake (ML
−3
), and
Q
o
is the average outflow rate (L3T−1).
3
T
−1
). The first-order decay
factor,
k
, is the sum of the decay factors of all first-order
decay processes by which the contaminant is removed
from the water, including chemical, and biological trans-
formations. Equation (7.37) states that the rate of
change of contaminant mass in the lake,
d
(
V
L
c
)/
dt
, is
equal to the mass influx,
M
, minus the mass outflow
rate,
Q
o
c
, minus the rate at which mass is removed by
first-order decay,
kV
L
c
. It is emphasized here that the
mass influx,
M
, includes the contaminant influx from all
sources, including direct discharges from outfalls and
releases from sediments. Equation (7.37) is sometimes
referred to as a vollenweider model after vollenweider
(1968, 1975, 1976). Assuming that the volume of the
lake,
V
L
, remains constant, Equation (7.37) can be put
in the form
where the first term on the right-hand side of Equation
(7.41) gives the buildup of concentration due to the
continuous mass input,
M
, and the second term
accounts for the dieaway of the initial concentration,
c
0
. The mass inflow rate,
M
, and contaminant concen-
tration,
c
, in Equation (7.41) as a function of time are
illustrated in Figure 7.9 for cases in which the initial
concentration,
c
0
, is less than and greater than the
asymptotic concentration given by Equation (7.41).
Taking
t
→ ∞ in Equation (7.41) gives the asymptotic
concentration,
c
∞
, as
M
Q kV
c
∞
=
(7.42)
+
o
L
Equation (7.41) can also be used to calculate the lake
response to a mass inflow over a finite interval, Δ
t
. This
case is illustrated in Figure 7.10a for a mass inflow rate,
M
, over an interval Δ
t
. The response is described by
V
dc
dt
(7.38)
+
(
Q kV c M
+
)
=
L
o
L
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