Part 10

1. Calculation of clear-sky solar radiation, Rso
Class Material: Masahiro Tasumi
Most of the following equations are from “FAO56 Crop Evapotranspiration”, occasionally with minor
modifications.
1. Calculation of extraterrestrial solar radiation (Ra) in W/m2.
Ra =
×
Where Gsc is solar constant 1367 W/m2, d is relative distance between earth and sun (no unit, or
astronomical unit AU), θ is solar incident angle (rad). In case Ra is calculated as negative, Ra value
should be set as zero because it is at night having no extraterrestrial solar radiation.
cosθ is calculated as;
cosθ = sinδ sinφ + cosδ cosφ cosω
Where δ is solar declination angle (rad), which is the angle between
solar beam and the earth’s equatorial plane (If expressed in degree for
convenience, 0o for spring and fall equinoxes, plus or minus 23.43o (=
the inclination of the earth’s axis) in summer or winter solstice,
respectively).
φ is latitude of target area (rad; North=Plus); in this homework, use
latitude of the weather station (31o56.3’ N = “0.5574 rad.”). ω is hour
angle (rad; south = 0).
δ = 0.409 × sin
2
365
− 1.39
where DOY is Day of Year. DOY for June 13, 2014 is 164. (31+28+31+30+31+13=164)
ω =
12
+
−
15
+ − 12
where t is time with unit in hour; e.g. for t for 11:00 is 11.0, t for 14:36 is 14.60. Lz is longitude of the

2. center of local time zone (East = Plus), not in rad but in degrees. In case of Japan, “135”. Lm is
longitude of the target area in degrees. In case of this homework, use longitude of the weather station
(131o24.8’ E = “131.4133”). Sc is seasonal correction for solar time (hr) and is calculated as:
Sc = 0.1645 ×
4 ( − 81)
364
− 0.1255 ×
2 ( − 81)
364
− 0.025 ×
2 ( − 81)
364
d2 is calculated as:
d =
1
1 + 0.033
2
365
2. Calculation of clear-sky at-surface solar radiation (Rso) in W/m2.
Rso = τ × Ra
Where τ is air transmittance (unitless). The following two methods available for calculating τ:
Method A (Do not use this method in this homework): τ is calculated by altitude as:
τ = 0.75 + 2 × 10 ×
where z is altitude in meter. In this homework, z is altitude of the weather station, 9.2m.
Method B (Use this method in this homework): τ is calculated in the following equation:
τ = K + K
where KB and KD are portion of atmospheric transmittance for beam and diffuse radiation.
K = 0.98
−0.00146
− 0.075
.
Where, P is air pressure (kPa), KT is turbidity coefficient that represent air turbidity from zero to one.
In this homework, use KT = 1 (clean air). W is precipitable water (mm) calculated as:
W = 0.14 + 2.1

3. Where ea is near-surface (e.g. 1.25m – 2.0m) vapor pressure (kPa).
KD is calculated as:
If KB ≥ 0.15, Then KD = 0.35 – 0.36KB
If KB < 0.15, Then KD = 0.18 + 0.82KB
Note that KB (and then KD) cannot be calculated at night. In EXCEL calculation, you can leave this
problem and set Rso as zero at night (when Ra is zero).
Comments for Methods A and B: Generally, Method A is regarded as more simple and easy, and
Method B is regarded as more complicated but accurate. However, both are quite empirical equations
based on many assumptions in atmospheric conditions. The best method is to test both methods for
your area of interest, and evaluate which method works better for you.
3. Supplemental Equations
In this homework, air pressure P (kPa) has been given (you need to convert the unit from hPa to kPa).
In case P is not available, P can be estimated by elevation as:
P = 101.3
293 − 0.0065
293
.
In case ea (kPa) is not given, ea is calculated by dew point temperature Tdew (oC) as:
e = 0.6108
17.27
× 237.3
Or, ea is calculated by air temperature Tair (oC) and relative humidity RH (%) as:
e = 0.6108
17.27
× 237.3
×
100