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Solar radiation basics

The purpose of this section is to acquaint the reader with the basic terminology, concepts, and formulas needed to understand and utilize solar radiation data. This is accomplished in the context of explaining the parameters needed to calculate the position of the sun in the sky. Later, the spectral nature of solar radiation will be briefly described.

Extraterrestrial Radiation

Solar radiation incident outside the earth's atmosphere is called extraterrestrial radiation. On average the extraterrestrial irradiance is 1367 Watts/meter2 (W/m2). This value varies by ±3% as the earth orbits the sun. The earth's closest approach to the sun occurs around January 4th and it is furthest from the sun around July 5th. The extraterrestrial radiation is

Eqn SR.1
  Io= 1367 * (Rav / R)2 W/m2
   
where Rav is the mean sun-earth distance and R is the actual sun-earth distance depending on the day of the year. An approximate equation for the effect of the earth-sun distance is

Eqn SR.2
  (Rav / R)2 = 1.00011
+ 0.034221 * cos(b)
+ 0.001280 * sin(b)
+ 0.000719 * cos(2b)
+ 0.000077 * sin(2b)
   
where b = 2pn / 365 radians [NREL. SR1] and n is the day of the year. For example, January 15 is year day 15 and February 15 is year day 46. There are 365 or 366 days in a year depending if the year is a leap year.
Fig SR.1

Fig SR.1
   
The earth's axis is tilted approximately 23.45 degrees with respect to the earth's orbit around the sun. As the earth moves around the sun, the axis is fixed if viewed from space (Figure SR.1). In June the orientation of the axis is such that the northern hemisphere is pointed towards the sun. In December the earth is on the other side of the sun and the earth's axis in the northern hemisphere is pointing away from the sun. During the spring and fall equinoxes the earth's axis is perpendicular to an imaginary line drawn between the earth and the sun.

As viewed from earth, the sun is higher in the sky during summer and lower in the sky as winter approaches. (Note that summer in the northern hemisphere is winter in the southern hemisphere and vise versa.) The declination of the sun is the angle between a plane perpendicular to a line between the earth and the sun and the earth's axis. An approximate formula for the declination of the sun is

Eqn SR.3   d = 23.45p / 180 * sin(2p * (284 + n) / 365)
   
[Duffie and Beckman. SR2]. A more accurate algorithm can be found in [Joe Michalsky. SR3]. A C program based on the Michalsky algorithm can also be found on the NREL RREDC web site.

Zenith, Azimuthal, and Hour Angles

To describe the sun's path across the sky one needs to know the angle of the sun relative to a line perpendicular to the earth's surface—this is called the zenith angle (q)—and the sun's position relative to the north-south axis, the azimuthal angle (a). The hour angle (w) is easier to use than the azimuthal angle because the hour angle is measured in the plane of the "apparent" orbit of the sun as it moves across the sky. Since the earth rotates approximately once every 24 hours, the hour angle changes by 15 degrees per hour and moves through 360 degrees over the day. Typically, the hour angle is defined to be zero at solar noon, when the sun is highest in the sky.

Solar and Local Standard Time

To describe the position of the sun in local standard time1, one needs to know the relationship between solar time and local standard time. Local time is the same in the entire time zone whereas solar time relates to the position of the sun with respect to the observer, and that is different depending on the exact longitude where solar time is calculated. To adjust solar time for longitude one must subtract (Longlocal Longsm)/15 (units are hours) from the local time. Longlocal is the longitude of the observer in degrees and Longsm is the longitude for the standard meridian for the observer's time zone.

Equation of Time

As the earth moves around the sun, solar time changes slightly with respect to local standard time. (This is mainly related to the conservation of angular momentum as the earth moves around the sun.) This time difference is called the equation of time and can be an important factor when one is at sea, navigating by the sun or stars. It is also important when determining the position of the sun for solar energy calculations. An approximate formula for the equation of time (Eqt) in minutes is
Eqn SR.4a   Eqt = -14.2 sin(p(n + 7) / 111)
for year day n between 1 and 106
Eqn SR.4b   Eqt = 4.0 sin(p(n - 106) / 59)
for year day n between 107 and 166
Eqn
        SR.4c   Eqt = -6.5 sin(p(n - 166) / 80)
for year day n between 167 and 246
Eqn SR.4d   Eqt = 16.4 sin(p(n - 247) / 113)
for year day n between 247 and 365
   
[Watts. SR4]. A more precise equation of time can be found in the solar position program 'solpos', available for downloading on the NREL web site.

Using the longitude correction and the equation of time, the relationship between the solar time and local standard time is

Eqn SR.5   Tsolar = Tlocal
+ Eqt / 60
+ (Longsm - Longlocal) / 15
   
Values are in hours. Since equations use sine and cosine functions it is conceptually easier to calculate using the hour angle (w) instead of time. The relationship between hour angle and time is
Eqn SR.6   w = p * (12 - Tsolar) / 12
   
The hour angle is in units of radians.

With the above information, one can now calculate the cosine of the zenith angle:
Eqn SR.7   cos(Z) = sin(l)sin(d)
+ cos(l)cos(d)cos(w)
   
where l is the latitude of the location of interest.

Sunrise and Sunset Times

The calculation of sunrise and sunset times provides an easy exercise to test one's understanding of the information presented so far. Sunrise and sunset occur when the sun is at the horizon and hence the cosine of the zenith angle is zero.

Setting the cosine of the zenith angle to zero in Equation SR.7 results in the following equation:
Eqn SR.8   wsr,ss = arccos(-tan(l)tan(d))
   
where wsr is the sunrise hour angle and wss is the sunset hour angle.

The sunrise and sunset hour angles are not exactly the same value as the sunrise and sunset times that appear in the local paper. The sunrise reported in the paper will be earlier and the sunset times will be later. The reason for this difference is that the sunlight is refracted as it moves through the earth's atmosphere and the sun appears slightly higher in the sky than simple geometrical calculations indicate. This is the same effect that makes a stick appear to bend when is it placed in water. During the middle of the day the effect is small, but during the sunrise or sunset periods, the effect can change the apparent solar time by about 5 minutes.

Global, Beam, and Diffuse Irradiance

Near noon on a day without clouds, about 25% of the solar radiation is scattered and absorbed as it passes through the atmosphere. Therefore about 1000 w/m2 of the incident solar radiation reaches the earth's surface without being significantly scattered. This radiation, coming from the direction of the sun, is called direct normal irradiance (or beam irradiance).

Some of the scattered sunlight is scattered back into space and some of it also reaches the surface of the earth. The scattered radiation reaching the earth's surface is called diffuse radiation. Some radiation is also scattered off the earth's surface and then re-scattered by the atmosphere to the observer. This is also part of the diffuse radiation the observer sees. This amount can be significant in areas in which the ground is covered with snow.
Fig SR.2

Fig SR.2
   
The total solar radiation on a horizontal surface is called global irradiance and is the sum of incident diffuse radiation plus the direct normal irradiance projected onto the horizontal surface. If the surface under study is tilted with respect to the horizontal, the total irradiance is the incident diffuse radiation plus the direct normal irradiance projected onto the tilted surface plus ground reflected irradiance that is incident on the tilted surface.

Solar Radiation on Tilted Surfaces

The amount of direct radiation on a horizontal surface can be calculated by multiplying the direct normal irradiance times the cosine of the zenith angle. On a surface tilted T degrees from the horizontal and rotated g degrees from the north-south axis, the direct component on the tilted surface is determined by multiplying the direct normal irradiance by
Eqn SR.9   cos(q) = sin(d)sin(l)cos(T)
- sin(d)cos(l)sin(T)cos(g)
+ cos(d)cos(l)cos(T)cos(w)
+ cos(d)sin(l)sin(T)cos(g)cos(w)
+ cos(d)sin(T)sin(g)sin(w)
   
Spectral Nature of Solar Radiation

Fig SR.3

Fig SR.3

Fig SR.4

Fig SR.4
   
Solar radiation is the result of fusion of atoms inside the sun. Part of the energy from this fusion process heats the chromosphere, the outer layer of the sun that is much cooler than the interior of the sun, and the radiation from the chromosphere becomes the solar radiation incident on earth. The solar radiation is not much different from the radiation of any object that is heated to about 5800 Kelvin except that the 'surface' of the sun is heated by the fusion process. The radiation spans a large range of wavelengths from 200 nm to more that 50000 nm with the peak around 500 nm (Figure SR.3 and Figure SR.4). Approximately 47% of the incident extraterrestrial solar radiation is in the visible wavelengths from 380 nm to 780 nm. The infrared portion of the spectrum with wavelengths greater than 780 nm account for another 46% of the incident energy and the ultraviolet portion of the spectrum is with wavelengths below 380 nm accounts for 7% of the extraterrestrial solar radiation.

As the sunlight passes through the atmosphere, a large portion of the UV radiation is absorbed and scattered. Air molecules scatter the shorter wavelengths more strongly than the longer wavelengths. This scatters more blue light and is the reason the sky appears blue. Water Vapor and atmospheric dust further reduce the amount of direct sunlight passing through the atmosphere. On a clear day approximately 75% of the extraterrestrial direct normal irradiance passes through the atmosphere without being scattered or absorbed.
Tbl SR.1  
Absorption and scattering
under typical clear sky conditions
Factor Percent absorbed Percent scattered Percent of total passing through the atmosphere
Ozone 2% 0%  
Water vapor 8% 4%
Dry air 2% 7%
Upper dust 2% 3%
Lower dust 0% 0%
Total absorbed or scattered 87% 87% 76%
   
Many applications are concerned with specific regions of the solar spectrum. For example, building designers are interested in lighting for the human eye, which is sensitive only to the visible part of the spectrum. The eye's response to various wavelengths is called a photopic curve. Daylighting illuminance is the sum, over all wavelengths, of the product of solar irradiance at a given wavelength times the photopic response at that wavelength. The SI unit for illuminance is the lux (lumen/m2).

References

SR1    Partridge, G. W. and Platt, C. M. R. 1976. Radiative Processes in Meteorology and Climatology.
SR2    Duffie, J. A. and Beckman, W. A. 1991. Solar Engineering of Thermal Processes, 2nd edn. J. Wiley and Sons, New York.
SR3    Joe Michalsky, J. J. 1988. The astonomical almanac's algorithm for approximate solar position (1950-2050), Solar Energy, 40, 227-235.
SR4    Watt Engineering Ltd. On the Nature and Distribution of Solar Radiation US Government Printing Office Stock No. 016-000-00044-5, March 1978
SR5    Wehrli, C. Extraterrestrial Solar Spectrum, Publication no. 615, Physikalisch-Meteorologisches Observatorium + World Radiation Center (PMO/WRC) Davos Dorf, Switzerland, July 1985.
SR6    American Society for Testing and Materials (ASTM) Standards (E-891) and (E-892).

Notes

[1] Local Standard Time (LST) is different from clock time because clock time is typically shifted by one hour in the summer to provide Daylight Savings Time. LST is equal to clock time in the winter and is not changed to Daylight Savings Time in the summer.

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© 2002, UO Solar Radiation Monitoring Laboratory.
Last revised: March 16, 2002.


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