The Irradiation Data   

Solar Radiation.

The solar constant, which is defined as the average energy flux incident on a unit area perpendicular to the solar beam outside the Earth's atmosphere has been measured to be

S = 1.367 kW/m2

The solar radiation incident on a collector on the Earth's surface is affected by a number of mechanisms, as shown in Figure 1. Part of the incident energy is removed by scattering or absorption by air molecules, clouds and other particles in the atmosphere. The radiation that is not reflected or scattered and reaches the surface directly is called beam radiation. The scattered radiation which reaches the ground is called diffuse radiation. Some of the radiation is reflected from the ground onto the receiver; this is called albedo radiation. The total radiation consisting of these three components is called global radiation. Although it varies significantly, a solar irradiance value of 1 kW/m2 has been accepted as the standard for the Earth's surface. The spectral distribution of this standard radiation is called Global AM 1.5 solar spectrum, where AM stands for Air Mass and AM 1.5 indicates that the direct beam path of the sun's rays travels through 1.5 times the thickness of the atmosphere in a typical situation.

Figure 1 - Solar radiation in the atmosphere.

Experimental Data.

Calculation of the incident radiation for a particular site from theoretical methods is extremely difficult as it is highly dependent on variables such as local weather conditions, the composition of the atmosphere above the site, and the reflectivity of surrounding land. For this reason, the design of a photovoltaic system relies on the input of experimental data measured as close as possible to the site of the installation.
This data is generally given in the form of global irradiation on a horizontal surface for each day at a particular location, or for a representative day of every month. Since solar panels are usually tilted, a manipulation of the experimental data is necessary.

Daily Irradiation.

Solar irradiance integrated over a period of time is called solar irradiation. Of particular significance in solar panel design is the irradiation over a day. The process of determining the total irradiation incident on a tilted plane over one day consists of calculating the equivalent radiation outside of the atmosphere, comparing this with experimental data measured on a horizontal surface at the site to determine the relative components of beam and diffuse, then adjusting them for the panel angle and including the albedo radiation.

Daily Extraterrestrial Irradiation

The daily irradiation received by a unit horizontal surface above the Earth's atmosphere consists only of the beam component, B0 and is calculated as follows:

B0=(24/PI)*S*(1+0.033*cos(2*PI*dn/365))*(cos fi * cos omega_s * + omega_s * sin fi * sin delta)   (kWh/m2)   Eq. 1

where

S = solar constant = 1.367 kW/m2
dn = day number of the year from 1 to 365
fi = latitude
delta = solar declination
omega_s = sunset hour angle (see SunPath User's Guide).

To determine an average value for a particular month, a day towards the middle of the month is chosen as the basis for calculations.

Diffuse and Beam Components of the Irradiation on a Horizontal Plane.

A clearness index, KT, can be defined to describe the attenuation of solar radiation by the atmosphere at a given site during a given month, and is calculated from the proportion of the available radiation that reaches the ground:

KT = G / B0   Eq. 2

where G is the global irradiation received on a horizontal plane at the site, which is usually determined experimentally as described above.

The diffuse irradiation can be approximated as:

D = G (1 - 1.13 * KT)   (kWh/m2)   Eq. 3

and the beam irradiation by

B = G - D   (kWh/m2)   Eq. 4

Note that it is assumed that no albedo radiation reaches the horizontal surface, as would be expected from consideration of the physical situation.

Adjustment for Tilted Plane.

For a panel facing due north in the southern hemisphere, or due south in the northern hemisphere, and tilted at an angle beta to the horizontal, the calculations for beam irradiation B, diffuse irradiation D and albedo radiation R are as follows:

B(beta) = B (cos (fi - beta) * (sin omega_s_1 - omega_s_1 * cos omega_s_1)) / (sin omega_s - omega_s * cos omega_s)   (kWh/m2)   Eq. 5

D(beta) = 0.5 * (1 + cos beta) * D   (kWh/m2)   Eq. 6

R(beta) = 0.5 * (1 - cos beta) * ro * D   (kWh/m2)   Eq. 7

where

omega_s_1 = the sunset hour angle adjusted for a tilted plane to allow for the effect of the sun rising and setting behind the panel, using the formula:

omega_s_1 = - arccos (- arctan (fi - beta) * tan delta)   Eq. 8

ro = reflectivity of the surrounding area. Typical reflectivity values are 0.2 for dry bare ground, 0.3 for grassland and 0.6 for snow.

The total irradiation received by a tilted plane over one day is then found by summing the calculated values of the beam, diffuse and albedo irradiations.

Irradiation Over Short Periods.

The following calculations derive the irradiation for a short specified period, such as an hour or a minute, and are taken from [1].

Firstly, two conversion factors must be calculated, one for diffuse radiation, rd, and one for global radiation, rg.

rd = (PI/T)*(cos omega - cos omega_s)/(omega_s * cos omega_s - sin omega_s)   Eq. 9

rg = (PI/T)*(a + b cos omega) * (cos omega - cos omega_s)/ (omega_s * cos omega_s - sin omega_s)   Eq. 10

where

T = the number of time divisions in one day, for example T = 24 for a result in hours, T = 1440 for a result in minutes
omega = hour angle of the sun at the relevant time
omega_s = sunrise hour angle for a horizontal plane

and a and b are as given below:

a = 0.409 - 0.5016 * sin (omega_s + 1.1047)   Eq. 11

b = 0.6609 + 0.4767 * sin (omega_s + 1.1047)   Eq. 12

The values for irradiation over the short period can then be obtained as follows, using the daily diffuse irradiation D, determined as in Eq. 3, and the daily global irradiation G obtained from experimental data:

Gs = G * rg   Eq. 13

Ds = G * rd   Eq. 14

Bs = Gs - Ds   Eq. 15

These values are for a horizontal plane. They can be adjusted for a tilted plane as in Equations 5, 6 and 7.



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This page was updated: 15 April 1998