Environmental Engineering Reference
In-Depth Information
EXAMPLE 7.14
M
2
= × . .
kg/day, and the concentration
as a function of time is given by Equation (7.44)
as
2 0 16
0 32
The average concentration of TP in a lake is 30
µ
g/L,
and an attempt is to be made to reduce the phosphorus
level in the lake by reducing phosphorus inflows into
the lake. The target phosphorus concentration is 15
µ
g/L.
(a) If the discharge from the lake averages 0.09 m
3
/s, the
first-order decay rate for phosphorus is 0.01 d
−1
, and the
volume of the lake is 300,000 m
3
, estimate the maximum
allowable phosphorus inflow in kg/yr. (b) If this loading
is maintained for 3 years but suddenly doubles in the
fourth year, estimate the phosphorus concentration in
the lake 1 month into the fourth year.
2
{
}
M
Q kV
Q
V
o
c t
( )
=
1
−
exp
−
+
k t
(
−
∆
t
)
+
o
L
L
Q
V
o
+
c
exp
−
+
k t
(
−
∆
t
)
1
L
where
c
1
= 15
µ
g/L = 15 × 10
−6
kg/m
3
, Δ
t
= 1095
days, and after 1 month (= 30 days)
t
= 1095 + 30 =
1125 days. The concentration in the lake is then
given by
0 32
.
Solution
c
(
1125
)
=
7776
+
( .
0 01 300 000
)(
,
)
{
}
+
(a) From the data given,
Q
o
= 0.09 m
3
/s = 7776 m
3
/d,
k
= 0.01 d
−1
, and
V
L
= 300,000 m
3
. For an ultimate
concentration,
c
∞
, of 15
µ
g/L (= 15 × 10
−6
kg/m
3
),
Equation (7.42) gives
7776
300 000
1
−
exp
−
+
0 01
.
(
1125 1095
−
)
15
,
7776
300 000
−
6
×
10
exp
−
+
0 01 1
.
(
125 1095
−
)
,
M
Q kV
=
25
µg/L
c
∞
=
+
o
L
Hence, the lake concentration rebounds to almost the
original concentration within 1 month. This is a reflection
of the relatively short detention time in the lake.
which rearranges to
M Q kV c
−
6
=
[
+
]
=
[
7776
+
( .
0 01 300 000
)(
.
)](
15 10
×
)
o
L
kg/d
∞
Although the completely mixed model cannot predict
the specific change of water quality at individual loca-
tions within lakes and reservoirs, this type of model is
particularly useful in estimating the behavior and water-
quality trends if the temporal scale of the simulations is
sufficiently long, such as months or years (kuo and
yang, 2002). In cases where a water body has been
loaded with phosphorus for a number of years, large
quantities of phosphorus might have accumulated in the
lake bottom sediments, adding years to the estimated
time to reach an equilibrium concentration (Jørgensen
et al., 2005).
The steady-state vollenweider model given by Equa-
tion (7.37) is commonly applied to predict nutrient
levels in lakes that correspond to various loading rates,
and is particularly useful in determining the nutrient
loading that corresponds to a desired nutrient level in a
lake (e.g., Langseth and Brown, 2011). Desired nutrient
levels in lakes can be derived from a desirable trophic
state, a regulatory water-quality standard, or a target
concentration that is less than the water-quality stan-
dard to account for uncertainties in the vollenweider
model. In all of these applications, the loading rate
M
(MT
−1
) required to achieve a target concentration
c
∞
(ML
−3
) is given by
=
0 16
.
=
59
k
g/yr
(b) Maintaining a mass loading of 0.16 kg/d for 3 years
(= 1095 days), with
c
o
= 30
µ
g/L = 30 × 10
−6
kg/m
3
yields a concentration at time
t
given by Equation
(7.41) as
{
}
M
Q kV
Q
V
o
c t
( )
=
1
−
exp
−
+
k t
+
o
L
L
Q
V
o
+
c
exp
−
+
k
t
o
L
which for
t
= 1095 days gives
0 16
.
c
(
1095
)
=
7776
+
( .
0 01 300 000
)(
,
)
{
}
7776
300 000
1
−
exp
−
+
0 01
.
(
1095
)
,
7776
300 000
−
6
+
30 10
×
exp
−
+
0 01 1095
.
(
)
,
= 15 µg/L
Hence, after 3 years, the phosphorus level has
already decreased to the target level of 15
µ
g/L.
In the fourth year, the mass flux doubles to
M Q k V c
(7.48)
=
(
+ ′
)
o
L
∞
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