Environment and Solar Radiation
In contrast to the environment radiation, the solar radiation is modeled by the directional energy flux. Therefore, the solar radiation is defined via its power flow (intensity) and its directional vector. In addition to the solar radiation from the computational domain boundaries, a solar radiation source emitting directional radiation can be specified.
The Sun directional vector can be calculated using the following formula:
 | (1) |
|---|
where 
is
the zenith direction, 
is
the north direction, 
is
the east direction;
θs is the solar altitude angle. That is, the angle between the direction of the geometric center of the sun’s apparent disk and the (idealized) horizon;
ϕs is the solar azimuth angle. That is, the angle from due north in a clockwise direction.
The solar altitude angle θs can be calculated, to a good approximation, using the following formula:
 | (2) |
|---|
where Φ is the local latitude, δ is the current Sun declination,
is
the hour angle of the present time t in
s (for example, at solar noon 12:00AM, t = 43200 s
and h = 0).
The solar azimuth angle ϕs can be calculated, to a good approximation, using the following formulas:
 | (3) |
|---|
Location and Time
The current Sun declination δ is approximately given by the following expression:
 | (4) |
|---|
where N is the day of the year beginning with N = 0 at midnight Coordinated Universal Time as January 1 begins (i.e. the days part of the ordinal date -1). The number 10, in (N + 10), is the approximate number of days after the December solstice to January 1. The number 23.44 is the Earth’s axial tilt.
The solar radiation intensity can be calculated using the following formula:
 | (5) |
|---|
where Cm = 0.092 and m1 = 1.029/(sinθs + 0.029) are empirical values;
n is the cloudiness, i.e. the presence of clouds in the sky and ranges from 0 (no clouds in the sky) to 1 (all the sky is covered by clouds);
Q*s0 = 1360 W/m2 is the solar constant (i.e. the energy reaching the Earth from the Sun at the top of the atmosphere).
Location, Time and Turbidity
The irradiance normal to the beam is calculated using the Linke Turbidity factor as follows (Ref. 2):
 | (6) |
|---|
where Bn is the irradiance normal to the beam (W∙m-2), Qs0 = 1367 W/m2 is the solar constant, εj is the correction to mean solar distance on Julian day number J (see Equation 7), m is the relative optical air mass corrected for station atmospheric pressure (usually known as the air mass), TLK is the Linke turbidity factor (for air mass m = 2) and δr(m) is the Rayleigh optical depth of clear blue dry sky at air mass m.
The correction factor to allow for the varying solar distance due to eccentricity of Earth’s orbit (εj) is calculated from:
 | (7) |
|---|
where εj is the correction factor for solar distance, J' is the day angle in degrees (see comments for Equation 12).
The Rayleigh optical depth is a function of the air mass. It’s reciprocal is given by:
 | (8) |
|---|
where m is the optical air mass corrected for elevation. (The polynomial limit for reliability is an air mass less than 20. For air mass m > 20, then 1⁄δr(m) = 10.4 + 0.718m.)
The sea level air mass must be corrected for station atmospheric pressure, as follows:
 | (9) |
|---|
where m is the optical air mass corrected for station atmospheric pressure, P is the atmospheric pressure of the station and P0 is the atmospheric pressure at sea level.
The sea level pressure correction can be calculated as follows:
 | (10) |
|---|
where h is the Height above the sea level in meters.
The relative optical air mass at sea level is approximately equal to (1/sinθs) but is calculated accurately as:
 | (11) |
|---|
where mSL is the relative optical air mass at sea level and θs is the solar altitude calculated from Equation 2.
The formula used in the European Solar Radiation Atlas (Ref. 18) to compute representative values of the Sun declination (δ) based on a 365-day year:
 | (12) |
|---|
where J' is the day angle in radians, which may be calculated as: J'=J·(360/365)·(π/180) in rad, where J is a Julian day number. Days are numbered continuously through the year to produce a Julian day number: January 1 = 1, February 1 = 32, March 1 = 57 in a non-leap year. A calendar year is always considered as a common (non-leap) year. The formula Equation 12 is more precise than Equation 4 and used for the Location, Time and Turbidity model only.
The cloudiness (n) can be taken into account the same way and multiplied Equation 6 by (1 – n).
It is recommended to use the Location, Time and Turbidity model since it is more accurate than the Location and Time model.