Environmental Engineering Reference
In-Depth Information
where
Q
o
is the lake outflow (L3T−1),
3
T
−1
),
k
′ is a coefficient
accounting for decay and sedimentation (T
−1
), and
V
L
is
the volume of the lake (L
3
). It is generally prudent to
recognize that
Q
o
,
k
′, and
V
L
are variables that cannot
be specified with certainty, and hence any calculated
value of
M
will also be uncertain. If the vollenweider
model is assumed to be correct, then it is possible to use
Equation (7.48) as a basis for relating the probability
distribution of
M
to the probability distributions of
Q
o
,
k
′, and
V
L
. This approach has been shown to be feasible
in a number of cases (e.g., Langseth and Brown, 2011;
Reckhow, 1979) and enables
0.03 (dimensionless), ε is the emissivity of water that
is approximately 0.97,
c
1
is Bowen's coefficient
(= 0.47 mmHg°C
−1
),
V
w
is the wind speed measured at
a fixed distance above the reservoir surface (m/s), and
f
(
V
w
) is a wind speed function which can be taken as
f V
(
)
=
19 0 0 95
.
+
.
V
2
(7.51)
w
w
where
V
w
is measured 7 m (23 ft) above the reservoir,
T
s
is the water surface temperature (°C), and
e
s
is the
saturation (water) vapor pressure at the surface of the
reservoir (mmHg). The saturation vapor pressure of
water,
e
s
(kPa) at any temperature,
T
(°C), can be esti-
mated using the empirical relation
M
to be characterized by
its uncertainty bounds.
7.5.1.2 Conservation of Energy Model.
The conser-
vation of energy model is commonly used to estimate
the temperature variations in lakes, and the energy
(heat) balance for a well-mixed reservoir can be
expressed in the form
17 27
237 3
.
T
e
s
=
0 6108
.
exp
(7.52)
T
+
.
The actual vapor pressure,
e
Tair
(kPa) can be estimated
from either the relative humidity, RH (%), or the
dew point temperature,
T
d
(°C) using the following
relations
dT
dt
(7.49)
V c
ρ
=
Q c T Q c T A J
ρ
−
ρ
+
p
p in
p
s
RH
e
,
given RH
where
V
is the volume of the reservoir (L
3
),
ρ
is the
density of water (ML
−3
),
c
p
is the specific heat of water
(EL
−3
°C
−1
),
T
is the water temperature in the reservoir
(°C),
t
is time (T),
Q
is the inflow and outflow rate
(L
3
T
−1
),
T
in
is the temperature of the inflow (°C),
A
s
is
the surface area of the reservoir (L
2
), and
J
is the net
flux of energy into the reservoir across the surface area
(EL
−2
). The specific heat,
c
p
, of pure water is commonly
taken as a constant value of 4186 J/kg·°C, although this
value actually corresponds to a temperature of 15°C;
variations of
c
p
with temperature are slight. Equation
(7.49) is an expression of the law of conservation of
energy, where the term on the left-hand side is the rate
of change of heat within the reservoir, and the terms on
the right-hand side collectively equal the net flow of
heat into the reservoir. If
J
represents the surface heat
influx from the atmosphere, then
J
can be expressed
expressed in terms of its components as
s
100
e
=
(7.53)
Tair
17 27
237 3
.
T
d
0 6108
.
exp
,
given
T
d
T
+
.
d
In Equation (7.50), the first two terms on the right-hand
side account for the net absorbed radiation that depends
only on atmospheric conditions, while the last three
terms account for heat gains that depend on the tem-
perature of the water in the reservoir.
EXAMPLE 7.15
A lake is approximately 100 m long, 100 m wide, 10 m
deep, and has an average inflow and outflow rate of
3500 m
3
/d. Under typical conditions, the wind speed is
12 km/h, the relative humidity is 70%, the air tempera-
ture is 24°C, and the net solar (short-wave) radiation
incident on the lake is 640 W/m
2
. If the average tem-
perature of the inflow is 18°C, estimate the steady-state
temperature of the lake.
J
=
J
+
σ(
T
+
273
) (
4
A
+
0 031
.
e
)(
1
−
R
)
sn
net solar
Tair
Tair
L
atmospheric
longwave
T
273
4
ongwave
−
εσ(
+
)
−
c f V T T
1
(
)(
−
)
−
f V e
(
)(
−
e
)
s
water l
w
s
Tair
w
s
Tair
Solution
conduction
evaporation
(7.50)
From the given data:
L
= 100 m,
W
= 100 m,
d
= 10 m,
Q
= 3500 m
3
/d,
V
w
= 12 km/h = 3.33 m/s, RH = 70%,
T
Tair
= 24°C,
J
sn
= 640 W/m
2
, and
T
in
= 18°C. Under
steady-state conditions, Equation (7.49) requires that
where
J
sn
is the net solar shortwave radiation (W/m
2
),
σ
is the Stefan-Boltzmann constant (= 4.903 ×
10
−9
MJ·m
−2
k
−4
d
−1
),
T
Tair
is the air temperature (°C),
A
is
a coefficient in the range of 0.5-0.7 (dimensionless),
e
Tair
is the (water) vapor pressure in air (mmHg),
R
L
is the
reflection coefficient that is typically on the order of
Q c T Q c T A J
ρ
−
ρ
+
=
0
p in
Term
p
Term
s
Term
(7.54)
3
1
2
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