Environmental Engineering Reference
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where:
Equations 6.36-6.43 provide information for the solution
of the Penman-Monteith (FAO56-PM) method. The above
procedure for computing net radiation is quite demanding
and other simpler equations have been proposed for the
estimation of net radiation. Two of the simpler empirical
equations that can be used for calculating net radiation are
shown in the following sections.
10
−
9
MJK
−
4
σ
s
=
Stefan-Boltzmann constant
(
4
.
903
×
m
−
2
d
−
1
)
,
T
aK
=
average absolute daily temperature [i.e.
(T
max,
K
+
T
min,
K
)/
2]
=
◦
C
T
max,
K
=
+
daily maximum absolute temperature [K
273
.
16
)
,
=
◦
C
T
min,
K
=
daily minimum absolute temperature (K
+
6.3.16.3 Equation 1 from Irmak et al. (2003)
for Approximating Net Radiation
Irmak et al. (2003) proposed an empirical equation for the
estimation of net radiation that is based on a minimal amount
of climate data. The data required are site latitude, maximum
air temperature, minimum air temperature, and mean rela-
tive humidity. It was proposed that net radiation could be
estimated using the following equation in cases where solar
radiation is measured at the site:
273.16), and
u
air
v
=
actual vapor pressure, kPa.
Daily values for the clear sky radiation,
R
s
0
, are a func-
tion of elevation,
z
(
m
), and the extraterrestrial radiation,
R
a
(Doorenbos and Pruitt, 1977):
R
s
0
=
(
0
.
75
+
0
.
00002
) R
a
(6.39)
where:
R
n
=−
0
.
054
T
max
+
0
.
101
T
min
+
0
.
462
R
S,
measured
−
49
.
243
d
r
+
50
.
831
(6.44)
extraterrestrial radiation, MJ/m
2
/d.
R
a
=
where:
Extraterrestrial radiation can be calculated on a daily basis
as a function of the day of the year, the solar constant, the
solar inclination, and the latitude in accordance with the
following empirical equation:
net radiation, MJ/m
2
/day,
R
n
=
maximum daily air temperature,
◦
C,
T
max
=
minimum daily air temperature,
◦
C,
T
min
=
measured solar radiation, MJ/m
2
/day, and
R
S,
measured
=
G
sc
d
r
ω
s
sin
φ
sin
δ
cos
φ
cos
δ
sin
ω
s
(6.40)
1440
π
R
a
=
+
d
r
=
inverse relative distance from the earth to
the sun, ranges from 0.967 to 1.033.
where:
solar constant (i.e., 0.0820 MJ/m
2
/min),
6.3.16.4 Equation 2 from Irmak et al. (2003)
for Approximating Net Radiation
Net radiation
R
n
can also be estimated using the following
equation when measured data are not available:
G
sc
=
φ
=
latitude [rad, where 1 rad
=
π
/180
×
(decimal
degrees)],
d
r
=
inverse relative distance from the earth to sun,
ω
s
=
sunset hour angle, rad, and
δ
=
solar declination, rad.
R
n
=−
0
.
09
T
max
+
0
.
203
T
min
−
0
.
101
RH
mean
+
0
.
687
R
S,
predicted
+
3
.
97
(6.45)
The
inverse relative distance
can be computed as
0
.
033 cos
2
π
365
D
j
where:
d
r
=
1
+
(6.41)
predicted solar radiation, MJ/m
2
/day.
R
S,
predicted
=
The solar inclination
δ
can be computed as
The solar radiation can be predicted as follows:
R
S,
predicted
=
K
t
R
a
T
max
−
0
.
409 sin
2
π
1
.
39
T
min
0
.
5
(6.46)
δ
=
365
D
j
−
(6.42)
where:
where:
K
t
=
empirical coefficient equal to 0.162 for interior
regions and 0.190 for coastal regions and
D
j
=
day of the year.
extraterrestial radiation, MJ/m
2
/day.
R
a
=
The sunset hour angle can be computed as follows:
Samani (2000) used average monthly data for each year
to obtain an indication of the empirical coefficient
K
t
.Data
ω
s
=
arc cos [
−
tan
(φ)
tan
(δ)
]
(6.43)
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