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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