Solar energy

The sun is the main source of renewable energy available on the earth. We define as solar energy every form of energy directly derived from the sun, i.e. the photovoltaic one. However, the other renewable sources are also indirectly connected with the sun. Hydraulic energy, for example, is linked to the water cycle in nature through evaporation and the subsequent precipitation, which is caused by solar heating. Similarly, the winds, from which wind energy derives, are generated by changes in atmospheric pressure linked to solar heating. Finally, also the chemical potential energy of biomass is a form of solar energy storage captured by plants through photosynthesis. From this point of view, it is legitimate to say that fossil fuels also owe their existence to the sun, as they were formed over the geological eras by the anaerobic transformation of organic substances [1]. It is therefore clear how all the forms of energy that we have at out disposal are directly or indirectly connected to the presence and energy of the sun. This chapter is devoted to understanding the nature of the energy irradiated by the sun and its movements. The first section is devoted to the solar geometry, in which we will go through the definitions of the angles that characterize the position of the sun with respect to a point on the earth and the direction and angle of incidence of a solar solar ray. In the second section we will examine the characteristics of the extraterrestrial radiation: it is important in understanding and using solar radiation data, since it should represent an upper limit of the amount of solar radiation available on the earth. More details on these arguments can be found in [2].

1 The Sun

The sun is a sphere with a diameter of 1.39 x \(10^{9}\) m and on average its distance from the earth is \(1.5 \times 10^{11} m\). Seen from the earth it rotates on its axis every 4 weeks, however the rotation is not perceived equal from all the point of the earth but is about 27 days at the equator and 30 days on polar regions. The sun has an effective black body temperature1 of 5777 K, but on its interior the temperature is much higher, then the energy produced inside is transferred to the surface and irradiated into space till the earth.

Sun-Earth geometry.

However, in reality the sun does not work, as a blackbody radiator at a fixed temperature, rather the emitted solar radiation is the composite result of the several layers that emit and absorb radiation of various wavelengths. The final extraterrestrial solar radiation and its spectral distribution have now been measured by various methods in several experiments. The solar spectrum changes throughout the day and with location. Standard reference spectra are defined to allow the performance comparison of photovoltaic devices from different manufacturers and research laboratories. We can distinguish between three main standards:

  • AM0: is the standard spectrum for space applications, it has an integrated power of \(1366.1 \; W/m^2\).
  • AM1.5 Global spectrum: is designed for flat plate modules and has an integrated power of \(1000 \; W/m^2\).
  • AM1.5 Direct spectrum: is defined for solar concentrator work, it includes the direct beam from the sun plus the circumsolar component in a disk 2.5 degrees around the sun. The direct plus circumsolar spectrum has an integrated power density of \(900 \; W/m^2\).

The data for these three spectra are available for the download at the website2. Assuming that the sun behaves as a blackbody, we could compare these spectra with the theoretical distribution obtained by the Planck’s law. Denoting as \(T = 5777 \; K\) the temperature of the sun, with \(R_s = 6.955 \cdot 10^{8} \; m\) its radius and with \(R_{se} = 1.495 \cdot 10^{11} \; m\) the mean distance sun-earth, then the Planck’s Law allow to write the emittivity power of the sun as:

\[E_{\lambda} = \biggl( \frac{R_{s}}{R_{se}} \biggl)^{2} \frac{C_1}{\lambda^{5} [exp(C_{2}/\lambda T)-1]} \qquad(1)\]

In Equation 1, \(\lambda\) is the wavelength expressed in micrometers, \(C_1\) and \(C_2\) are two constants given by

\[\begin{array}{ll} C_1 = 3.742 \cdot 10^8 (W \mu^4)/m^2 && C_2 = 1.439 \cdot 10^4 \mu K \end{array}\]

As consequence \(E_{\lambda}\) will be measured in \([W/(m^2 \mu m)]\). In Figure Figure 1, the four spectra are compared: it is interesting to note the differences among the red and the blue and green curves, that represent the real and the adjusted spectral distribution of the sun, respectively. The air mass zero (AM0), or extraterrestrial spectrum used to generate the terrestrial reference spectra was developed by Gueymard [3] and is a synthesis of several air mass zero data sets. The comparison between AM0 and AM1 highlights by difference the effects of atmospheric attenuation on the direct component arriving on a surface orthogonal to the sun’s rays. These effects are particularly accentuated in the absorption bands of ozone (300-400 \(nm\)), water vapor (800-2000 \(nm\)) and carbon dioxide (above 2500 \(nm\)).

Figure 1: Spectal distribution of the sun under different standards (Source: NREL).

1.1 Radiation Definitions

As the solar radiation goes through the atmosphere it suffers different processes of absorption, dispersion or scattering that result in lower levels of solar radiation being received at the Earth’s surface. These are due to the atmosphere components, such as ozone or \(CO_2\), and solid and liquid particles in suspension like aerosols or water vapour. However, the main source of attenuation is the cloud cover. In general, the processes of absorption and attenuation differently affect the wavelengths of solar radiation, therefore the spectral distribution of the solar radiation at ground level differs from the extraterrestrial one. We can distinguish between four quantities that in general are used to express solar irradiance:

  • Extraterrestrial Radiation: it represents the amount of radiation that could be theoretically available if there were no atmosphere. Even if it could present some randomness, in general it is considered deterministic. Once the radiation enters in the atmosphere we could make two further distinctions in beam and diffuse radiation. The notations involved are several: \(G_{sc}\) is the solar constant, \(G_n\) is the value of \(G_{sc}\) corrected for the \(n\)-th day of the year. Moreover, \(G_{o,n}\) represents \(G_{n}\) computed for an horizontal surface. Finally \(H_{o,n}\) is the daily integral of \(G_{o,n}\). More details will be given below.

  • Beam Radiation: it represents the solar radiation received from the sun without being scattered by the atmosphere. Consequently to its definition, the Beam radiation is maximum when the sky is clear3. In theory the Beam radiation can be zero: even if this could seem strange, in days when the sky is fully covered by clouds, the instruments devoted to the measurement4 could register a value of zero. It is also denoted as Direct Normal Irradiance or DNI.

  • Diffuse Radiation: it represents the solar radiation received from the sun after its direction has been changed and scattered by the atmosphere. In application, usually, this measure is considered with respect to an horizontal surface (Diffuse Horizontal Irradiance or DHI).

  • Global Solar Radiation: it represents the sum of the beam and the diffuse radiations. A common practice is to consider it with respect to an horizontal surface (Global Horizontal Irradiance or GHI).

Figure 2: Different definitions types of solar irradiance.

1.2 Solar constant

The solar constant \(G_{sc}\) is defined as the energy received from the sun per unit of time (typically a second), on a unit area of surface perpendicular to the direction of propagation of the radiation at the mean earth-sun distance outside the atmosphere. The most recent estimate from the Wold Radiation Center in Switzerland gives a mean value of \(\overline{G}_{sc} = 1367 \, W/m^2\). In order to prove this result, we have to consider the total energy irradiated from the sun per \(m^2\), given by: \(q_0 = 6.316 \cdot 10^7 W/m^2\). Denoting as \(R_s\) the radius of the sun and with \(R_{se}\) the mean earth-sun distance, the following simple equation with \(G_{sc}\) unknown can be solved:

\[q_{0} 4 \pi R_{s}^2 = G_{sc} 4 \pi R_{se}^2 \, \, \Rightarrow \, \, \, G_{sc} = q_{0} \frac{R_{s}^2}{R_{se}^2} = 1367 \, W/m^2\]

However, \(G_{sc}\) is actually not constant. For practical purposes it is possible to neglect the variations due to sunspots5 and to assume that the flow of energy radiated by the sun, \(q_0\), is constant. Consequently, \(G_{sc}\) will change according to the variation of the earth-sun distance. Counter intuitively the solar constant assumes its maximum value (around 1420 \(W/m^2\)) during the lasts days of December and its minimum (around 1320 \(W/m^2\)) during the summer in the late June. Spencer (1971), as cited by Iqbal [4], recovered an accurate equation (\(\underline{+} 0.01 \%\)) that can be used for the correction of \(G_{sc}\) during the year. Denoting as \(G_{n}\) the extraterrestrial radiation incident on the plane normal to the radiation on the \(n\)-th day of the year:

\[\begin{equation} G_{n} = G_{sc} \bigl( 1.000110 + 0.034221 cosB + 0.001280 sinB +0.000719 cos2B + 0.00077 sin2B \bigl)\end{equation} \qquad(2)\]

where \(B\) represents an angle in radians and is computed as:

\[\begin{equation} B = (n-1) \frac{2 \pi}{365} \end{equation} \qquad(3)\]

Figure 3: Values of the solar costant in Bologna.

1.3 Equatorial coordinates

The position of the sun with respect to a point \(P\) on the Earth’s surface can be identified through an equatorial coordinate system. Each point on the Earth’s surface is identified by a pair of angular coordinates:

Figure 4: Relations between declination and latitude, and hour angle and longitude.
  • Latitude \(\phi\): it is the angular location at the north or at the south of the equator, where by convention it is considered a positive sign in the north with respect to the equator and negative in the south. It varies in the range [-90°,90°]. In practice, it is the angular distance of \(P\) from the equator measured along the meridian passing through \(P\).
  • Longitude \(\psi\): it is the angular distance, measured along the equator, of the meridian passing through \(P\) from the meridian of Greenwich \(G\). It is considered positive at the East of \(G\) and negative at West. It varies in the range [-180°,180°].
    Based on the definitions, latitude and longitude are equatorial coordinates: they refer exclusively to the equatorial plane and to the Earth’s axis6. By tracing the junction between the center of the Earth and that of the sun, it is possible to find the point of intersection S. In this context we define:
  • Solar Declination \(\delta\): homolog of latitude, it is the angular position of the sun when it is situated on the local meridian with respect to the plane of the equator. As the latitude is positive in the North and negative in the South. It varies in the range [-23.50°,23.50°]. By definition it is equal for every point on the Earth.
  • Hour Angle \(\omega\): homolog of longitude, it is the angular displacement of the sun at the East or West of the local meridian due to the rotation of the Earth on its axis. It is negative in the morning, zero at the solar noon and positive in the afternoon. It is defined as the difference between the longitude of \(P\) (fixed) and the longitude of \(S\) (variable). It is equal for all the points on the same meridian.

1.4 Solar declination \(\delta\)

Given \(n\), that represents the number of the day during the year, the solar declination can be approximated with a well known equation proposed by Cooper [5]:

\[\begin{equation}\delta = 23.45 \text{sin} \biggl ( 360 \frac{284+n}{365} \biggl)\end{equation} \qquad(4)\]

In 1971, Spencer, as cited by [4], found a more accurate specification, that gives an error less \(\underline{+} 0.035°\%\).

\[\begin{equation}\begin{split} \delta = \frac{180}{\pi} \bigl( & {\small{0.006918}} - {\small{0.399912}} \text{cos}(B) + {\small{0.070257}} \text{sin}(B) - {\small{0.006758}} \text{cos}(2B) + \\ & + {\small{0.000907}} \text{sin}(2B) - {\small{0.002697}} \text{cos}(3B) + {\small{0.00148}} \text{sin}(3B) \bigl) \end{split}\end{equation} \qquad(5)\]

where \(B\) is the same as Equation 3. During the revolving movement, the Earth’s axis is constantly inclined towards the North Star at an angle of 23.45 with respect to the plane of the orbit. This inclination is responsible for the variation of the solar constant during the year and consequently for the number of hours per day and the changing of the seasons.

1.4.1 Example

\[\texttt{solar_time_declination}(\text{start_date}, \text{ end_date})\]

1.5 Hour angle \(\omega\)

During a movement of rotation of the Earth around its axes the hour angle \(\omega\) changes of 360°. Therefore in a day of around 24 hours the hour angle vary of 15° degrees each hour. However the duration of the mean day is not exactly 24 hours, given the clock hour \(\overline{\theta}\) (eventually without the Legal Hour) the relation with the solar hour \(\theta_s\) is the following: \[\tag{2.7} \theta_s - \overline{\theta} = 4 \, \bigl( \, \psi_P - \psi_{ref} \, \bigl) + E_{n}\]

The difference between the two hours is expressed in minutes. \(\psi_P\) represents the longitudes of the meridian of the observer, while \(\psi_{ref}\) denote the longitude of the meridian where the local time zone is setted (for Italy and Central EU the reference is \(\psi_{ref} = 15\)°). The last element \(E(n)\) represents a time correction in minutes:

\[\tag{2.8} E_{n} = 229.2 \bigl( 0.00075 + 0.001868 cosB - 0.032077 sinB - 0.014615cos2B -0.04089 sin2B \bigl)\]

where \(n\) is the number of the day and \(B\) is the same as Equation 3. Once that the solar hour is known, it is possible to recover the hour angle \(\omega\) as:

\[\tag{2.9} \omega = \psi_{ref} \, \bigl( \, \theta_s - 12 \bigl)\] where \(\theta_s\) is expressed in hours.

2 Solar time adjustment

\[\texttt{solar_time_adjustment}(\text{start_date}, \text{ end_date})\]

\[E(n) = 229.2 \bigl( 0.00075 + 0.001868 cosB -0.032077 sinB - 0.014615cos2B -0.04089 sin2B \bigl) \qquad(6)\]

Figure 5: Seasonal time adjustment in Bologna.

3 Solar time constant

\[\texttt{solar_time_constant}(\text{start_date}, \text{ end_date})\]

4 Solar angle minmax

\[\texttt{solar_angle_minmax}(\text{latitude}, \text{start_date}, \text{ end_date})\]

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References

[1]
Comini, G. and Savino, S. (2013). La captazione dell’energia solare. CISM.
[2]
Duffie, J. A. and Beckman, W. A. (2013). Solar engineering of thermal processes. New York: John Wiley; Sons.
[3]
Gueymard, C. (2004). The sun’s total and spectral irradiance for solar energy applications and solar radiation models. Solar Energy 76 423–53.
[4]
Iqbal, M. (1983). An introduction to solar radiation. Elsevier.
[5]
Cooper, P. I. (1969). The absorption of radiation in solar stills. Solar Energy 12 333–46.

Footnotes

  1. The effective black body temperature of 5777 K is the temperature of a black body radiating the same amount of energy as does the sun.↩︎

  2. https://www.nrel.gov/grid/solar-resource/spectra-am1.5.html↩︎

  3. Clear sky is the condition where the visibility is greater or equal to 23km.↩︎

  4. Pyrheliometers are used for the measure of Beam normal radiation, while pyrometers are used for the measure of the global radiation on horizontal surface↩︎

  5. Areas of the sun where the temperature is about 5000 K.↩︎

  6. As is known, the Earth’s axis intersects orthogonally the equatorial plane at the point O, center of the Earth.↩︎