Basic Equations for Sunshine Duration and Extraterrestrial Solar Irradiation

Extraterrestrial solar irradiation (H0) and length of the day (So) can be estimated deterministically by taking into consideration basic geographic and astronomic quantities including latitude, (ф), declination, (5), surface azimuth angle, (7), hour angle, (ю), zenith angle, (9z), solar altitude angle, (as), solar azimuth angle, (7) and solar constant (Isc). Interrelationships among these variables are presented either in the form of equations or tables in many solar energy books (Iqbal 1983; Duffie and Beckman 2006). Figure 6.1 presents the astronomical configuration of these quantities.

In the following, the physical determinations and valid equations for such a configuration are given.

Latitude (ф): The angular location north (positive) or south (negative) of the equator, (-90° < ф < 90°).

where 8o is the sun’s position angle, which depends on day of the year, dn that is zero in January first. In that case declination angle could be estimated according to

5 = 0.006918 – 0.399912cos 8° + sin8° – 0.006759cos28°

+ 0.000907sin28° — 0.002697cos 8° + 0.00148sin3 8° (6.2)

Surface azimuth angle (7): The deviation of the projection on a horizontal plane of the normal to the surface from the local meridian, with zero due south, east negative and west positive, (-180° < у < 180°).

Hour angle (ю): The angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 15° per hour as morning negative and afternoon positive.

Zenith angle (8z): The angle between the vertical and the line to the sun i. e., the angle of incidence of beam radiation on a horizontal surface. At solar noon zenith angle is zero, in the sunrise and sunset this angle is 90°.

Cos(8z) = cos($) cos(5) cos(®) + sin^) sin(5) (6.3)

Solar altitude angle (as): The angle between the horizontal and the line to the sun.

Solar azimuth angle (7): The angular displacement from south of the projection of beam radiation on the horizontal plane. Displacements east of south are negative and west of south are positive. Solar azimuth angle can be estimated as,

Length of the day (S0): This term is described as time duration between sunrise and sunset, and it is one of the extraterrestrial variables of classical Angstrom (1924) equation and could be estimated as,

cosS0 = – tanфtan5

Length of the day (S0) could be estimated in hours by dividing degree value of S0 to 15 that represents timely one as degree.

c. Arc{cosSo) ^ ^

So = —15— (6.6)

Solar constant (ISC): It is equivalent to the energy from the sun, per unit time, re­ceived on a unit area of surface perpendicular to the direction of propagation of the radiation, at mean earth-sun distance, outside of the atmosphere. Solar constant has been adopted as 1.94cal/cm2/sec or 1367W/m2 (Duffie and Beckman 2006). Recently, this value is considered with its physical meaning and estimated as 1366.1W/m2 that covers solar spectrum between 0 and 1000 p. m, (Gueymard 2004).

Extraterrestrial solar irradiation (H0): Extraterrestrial solar irradiation is important not only for solar engineering calculation but also for energy balance of the earth. It is a function of solar constant, sun-earth distance ratio and declination angle. Sun – earth distance ratio changes with time of the year and it is expressed as,

where R is the mean value of sun-earth distance which is equal to 1.49 x 1011 m; R is the actual sun-earth distance; n is the day number in the year and Y is the total day of year. All these angles and aforementioned equations take part in the determination of H0 and S0. Consequently, H0 received from the sun on a unit area of surface perpendicular to the radiation direction of propagation, at any earth-sun distance, outside of the atmosphere, can be estimated as,

as the total extraterrestrial solar irradiation from sunrise to sunset becomes

It is obvious that for solar irradiation estimation extraterrestrial solar irradiation and sunshine duration have the same importance, i. e., they are directly and function­ally related to each other. Equation (6.9) indicates total extraterrestrial solar irradi­ation intensity change depending on the length of day and sun-earth astronomical position.

Updated: August 2, 2015 — 9:58 am