**1. Introduction**

110 Solar Radiation

National Radiological Protection Board ,2002, Health effects from ultraviolet radiation:

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topical Chilean Andes. *Solar Energy*, 57, 133-140.

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correcting for wavelength shifts, slit function differences and defining a spectral reference, in The Nordic Intercomparison of Ultraviolet and Total Ozone Instruments at Izana, October 1996, edited by B. Kjeldstad, B. Johnson, and T.

CERES surface and atmospheric radiation budget: Methodology. *J. Geophys. Res.*,

The Sun is a sphere of intense hot gaseous matter with a diameter of 1.39 x 109m and is about 1.5 x 1011m away from the Earth. A schematic representation of the structure of the Sun is shown in Figure 1.1. The Sun's core temperature is about 16 million K and has a density of about 160 times the density of water. The core is the innermost layer with 10 percent of the Sun's mass, and the energy is generated from nuclear fusion. Because of the enormous amount of gravity compression from all the layers above it, the core is very hot and dense.

Fig. 1.1. The structure of the Sun

The layer next to it is the radiative zone, where the energy is transported from the sunspot interior to the cold outer layer by photons. Other features of the solar surface are small dark areas called pores, which are of the same order of magnitude as the convective cells and larger dark areas called sunspots, which vary in size. The outer layer of the convective cells is called the photosphere. The photosphere is the layer below which the Sun becomes opaque to visible light. Above the photosphere is the visible sunlight which is free to propagate into space, and its energy escapes the Sun entirely. The change in opacity is due to the decreasing amount of H− ions, which absorb visible light easily. The next layer referred to as the chromospheres, is a layer of several thousand kilometers in thickness, consisting of transparent glowing gas above the photosphere. Many of the phenomena occurring in the photosphere also manifest in the chromospheres. Because the density in the chromospheres continues to decrease with height and is much lower than in the photosphere, the magnetic field and waves can have a greater influence on the structure. Still further out is the corona which is of very low density and has a high temperature of about 1×106oK to 2×106oK.

The radiation from the sun is the primary natural energy source of the planet Earth. Other natural energy sources are the cosmic radiation, the natural terrestrial radioactivity and the geothermal heat flux from the interior to the surface of the Earth, but these sources are energetically negligible as compared to solar radiation. When we speak of solar radiation, we mean the electromagnetic radiation of the Sun. The energy distribution of electromagnetic radiation over different wavelength is called Spectrum. The electromagnetic spectrum is divided into different spectral ranges (Figure 1.2).

Fig. 1.2. Spectral ranges of electromagnetic radiation

The layer next to it is the radiative zone, where the energy is transported from the sunspot interior to the cold outer layer by photons. Other features of the solar surface are small dark areas called pores, which are of the same order of magnitude as the convective cells and larger dark areas called sunspots, which vary in size. The outer layer of the convective cells is called the photosphere. The photosphere is the layer below which the Sun becomes opaque to visible light. Above the photosphere is the visible sunlight which is free to propagate into space, and its energy escapes the Sun entirely. The change in opacity is due to the decreasing amount of H− ions, which absorb visible light easily. The next layer referred to as the chromospheres, is a layer of several thousand kilometers in thickness, consisting of transparent glowing gas above the photosphere. Many of the phenomena occurring in the photosphere also manifest in the chromospheres. Because the density in the chromospheres continues to decrease with height and is much lower than in the photosphere, the magnetic field and waves can have a greater influence on the structure. Still further out is the corona which is of very low density and has a high temperature of

The radiation from the sun is the primary natural energy source of the planet Earth. Other natural energy sources are the cosmic radiation, the natural terrestrial radioactivity and the geothermal heat flux from the interior to the surface of the Earth, but these sources are energetically negligible as compared to solar radiation. When we speak of solar radiation, we mean the electromagnetic radiation of the Sun. The energy distribution of electromagnetic radiation over different wavelength is called Spectrum. The electromagnetic

spectrum is divided into different spectral ranges (Figure 1.2).

Fig. 1.2. Spectral ranges of electromagnetic radiation

about 1×106oK to 2×106oK.

Solar radiation as it passes through the atmosphere undergoes absorption and scattering by various constituents of the atmosphere. The amount of solar radiation finally reaching the surface of earth depends quite significantly on the concentration of airborne particulate matter gaseous pollutants and water (vapour, liquid or solid) in the sky, which can further attenuate the solar energy and change the diffuse and direct radiation ratio (Figure 1.3).

Fig. 1.3. Radiation balance of the atmosphere

The global solar radiation can be divided into two components: (1) diffuse solar radiation, which results from scattering caused by gases in the Earth's atmosphere, dispersed water droplets and particulates; and (2) direct solar radiation, which have not been scattered. Global solar radiation is the algebraic sum of the two components. Values of global and diffuse radiations are essential for research and engineering applications.

Global solar radiation is of economic importance as renewable energy alternatives. More recently global solar radiation has being studied due to its importance in providing energy for Earth's climatic system. The successful design and effective utilization of solar energy systems and devices for application in various facets of human needs, such as power and water supply for industrial, agricultural, domestic uses and photovoltaic cell largely depend on the availability of information on solar radiation characteristic of the location in which the system and devices are to be situated. This solar radiation information is also required in the forecast of the solar heat gain in building, weather forecast, agricultural potentials studies and forecast of evaporation from lakes and reservoir. However, the best solar radiation information is obtained from experimental measurement of the global and its components at the location. The use of solar energy has increased worldwide in recent years as direct and indirect replacements for fossil fuel, motivated to some degree by environmental concerns such were expressed in the Kyoto Protocol. As a result, a complete knowledge and detailed analysis about the potentiality of the site for solar radiation activity is of considerable interest.

#### **1.2 Radiation fluxes at horizontal surface**

The energy balance on a horizontal surface at the ground or on a solid body near the ground is given by

$$Q + K + H + L + W + P = 0\tag{1.1}$$

Each term in this equation stands for an energy flux density or power density in Wm-2. The vectorial terms in equation (1.1) are counted positive when they are directed towards the surface from above or below. The parameters have the following meaning.

*Q* =net total radiation=sum of all positive and negative radiation fluxes to the surface

*K* Heat flux from the interior of the body (ground) to its surface

*H* Sensible heat flux from the atmosphere due to molecular and convective heat conduction (diffusion and turbulence)

*L* Latent heat flux due to condensation or evaporation at the surface.

W= Heat flux due to advection that is heat transported by horizontal air current. W is set zero if:


P = Heat flux brought to the surface by falling precipitation. P is often not taken into consideration because the measurements are confined to times without precipitation (Kasten, 1983).

The net total radiation *Q* is at daytime, to be compensated by the heat fluxes K, H and L the net total radiation Q in equation (1.1) given

$$
\underline{Q} = \left(G - R\right) + \left(A - E\right) \tag{1.2}
$$

*Q* is called the total radiation balance.

G= global radiation = sum of direct and diffuse solar radiation on the horizontal surface

R= reflected global radiation = fraction of G which is reflected by the body (ground)

A= atmospheric radiation = downward thermal radiation of the atmosphere (from atmosphere gases, mainly water vapour and from clouds)

E= terrestrial surface radiation = upward thermal radiation of the body (ground). G and R are solar or shortwave radiation fluxes therefore

$$Q\_s = G - R \tag{1.3}$$

Is called net solar or net global radiation, or short wave radiation balance. A and E are terrestrial or long wave radiation fluxes so that

$$Q\_t = A - E \tag{1.4}$$

Is called the long wave radiation balance and

$$-\underline{Q}\_t = E - A\tag{1.5}$$

the (upward) net terrestrial surface radiation.

The short-wave radiation fluxes exhibit a pronounced variation during day light hours; the long-wave radiation fluxes vary but slightly because the temperature of atmosphere and ground vary during the day.

The ratio

114 Solar Radiation

the forecast of the solar heat gain in building, weather forecast, agricultural potentials studies and forecast of evaporation from lakes and reservoir. However, the best solar radiation information is obtained from experimental measurement of the global and its components at the location. The use of solar energy has increased worldwide in recent years as direct and indirect replacements for fossil fuel, motivated to some degree by environmental concerns such were expressed in the Kyoto Protocol. As a result, a complete knowledge and detailed analysis about the potentiality of the site for solar radiation activity

The energy balance on a horizontal surface at the ground or on a solid body near the

Each term in this equation stands for an energy flux density or power density in Wm-2. The vectorial terms in equation (1.1) are counted positive when they are directed towards the

*H* Sensible heat flux from the atmosphere due to molecular and convective heat

W= Heat flux due to advection that is heat transported by horizontal air current. W is set

P = Heat flux brought to the surface by falling precipitation. P is often not taken into consideration because the measurements are confined to times without precipitation

The net total radiation *Q* is at daytime, to be compensated by the heat fluxes K, H and L the

G= global radiation = sum of direct and diffuse solar radiation on the horizontal surface R= reflected global radiation = fraction of G which is reflected by the body (ground)

E= terrestrial surface radiation = upward thermal radiation of the body (ground).

A= atmospheric radiation = downward thermal radiation of the atmosphere (from

a. the measuring surface is located at a horizontal and homogeneous plane of

*Q* =net total radiation=sum of all positive and negative radiation fluxes to the surface

sufficient extension so that the so called Katabatic flow is negligible b. the measuring time is small compared to time of an air mass exchange.

surface from above or below. The parameters have the following meaning.

*K* Heat flux from the interior of the body (ground) to its surface

*L* Latent heat flux due to condensation or evaporation at the surface.

*QKHLW P* 0 (1.1)

*Q GR AE* (1.2)

*Q GR <sup>s</sup>* (1.3)

is of considerable interest.

ground is given by

zero if:

(Kasten, 1983).

**1.2 Radiation fluxes at horizontal surface** 

conduction (diffusion and turbulence)

net total radiation Q in equation (1.1) given

atmosphere gases, mainly water vapour and from clouds)

G and R are solar or shortwave radiation fluxes therefore

*Q* is called the total radiation balance.

$$
\underline{Q}\_s = \frac{R}{G} \tag{1.6}
$$

is called short-wave radiation of the body

Terrestrial surface radiation E is composed of two terms:

1. The thermal radiation emitted by the body ground i.e.

$$E\_t = \alpha\_t \sigma T^4 \tag{1.7}$$

where *<sup>t</sup>* is called effective long-wave absorptance of the surface, slightly depending on temperature T. is called Stefan Boltzman constant = 5.6697 x 108 Wm-2K-4.

2. Reflected atmosphere radiation

$$E\_2 = Q\_t \cdot A \tag{1.8}$$

where 1 *Qt t* = effective long-wave reflectance of the surface. Thus E is strictly given by

$$E = E\_1 + E\_2 = a\_t \cdot \sigma T^4 + (\mathbf{l} - a\_t) \cdot A \tag{1.9}$$

#### **1.3 Solar declination angle**

The angle that the Sun's makes with equatorial plane at solar noon is called the angle of declination. It varies from 23.45o on June 21 to 0o on September 21 to -23.45o on December 21, to 0o on March 21. It also defined as the angular distance from the zenith of the observer at the equator and the Sun at solar noon.

The axis of rotation is tilted at an angle of 23.45° with respect to the plane of the orbit around the Sun. The axis is orientated so that it always points towards the pole star and this accounts for the seasons and changes in the length of day throughout the year. The angle between the equatorial plane and a line joining the centres of the Sun and the Earth is called the *declination angle* ( ) Because the axis of the Earth's rotation is always pointing to the Pole Star the declination angle changes as the Earth orbits the Sun (Figure 1.4)

Fig. 1.4. Orbit of the Earth around the Sun

On the summer solstice (21st June) the Earth's axis is orientated directly towards the Sun, therefore the declination angle is 23.45° (Figure 1.4). All points below 66.55° south have 24 hours of darkness and all point above 66.55° north have 24 hours of daylight. The sun is directly over head at solar noon at all points on the Tropic of Cancer. On the winter solstice (21st December) the Earth's axis is orientated directly away from the Sun, therefore the declination angle is -23.45° (Figure 1.4). All points below 66.55° south have 24 hours of daylight and all point above 66.55° north have 24 hours of darkness. The sun is directly over head at solar noon at all points on the Tropic of Capricorn. At both the autumnal and vernal equinoxes (23rd September and 21st March respectively) the Earth's axis is at 90° to the line that joins the centres of the Earth and Sun, therefore the declination angle is 0° (Figure 1.4).

Fig. 1.5. The celestial sphere. Declination angle ( ) is the declination angle which is maximum at the solstices and zero at the equinoxes.

The equation used to calculate the declination angle in radians on any given day is

$$\mathcal{S} = 23.45 \frac{\pi}{180} \sin \left[ 2\pi \left( \frac{284 + n}{365.25} \right) \right] \tag{1.10}$$

where:

116 Solar Radiation

On the summer solstice (21st June) the Earth's axis is orientated directly towards the Sun, therefore the declination angle is 23.45° (Figure 1.4). All points below 66.55° south have 24 hours of darkness and all point above 66.55° north have 24 hours of daylight. The sun is directly over head at solar noon at all points on the Tropic of Cancer. On the winter solstice (21st December) the Earth's axis is orientated directly away from the Sun, therefore the declination angle is -23.45° (Figure 1.4). All points below 66.55° south have 24 hours of daylight and all point above 66.55° north have 24 hours of darkness. The sun is directly over head at solar noon at all points on the Tropic of Capricorn. At both the autumnal and vernal equinoxes (23rd September and 21st March respectively) the Earth's axis is at 90° to the line that joins the centres of the Earth and Sun, therefore the

Star the declination angle changes as the Earth orbits the Sun (Figure 1.4)

) Because the axis of the Earth's rotation is always pointing to the Pole

the *declination angle* (

Fig. 1.4. Orbit of the Earth around the Sun

declination angle is 0° (Figure 1.4).

= declination angle (rads)

n = the day number, such that n = 1 on the 1st January and 365 on December 31st.

Fig. 1.6. The variation in the declination angle throughout the year.

The declination angle is the same for the whole globe on any given day. Figure 1.6 shows the change in the declination angle throughout a year. Because the period of the Earth's complete revolution around the Sun does not coincide exactly with the calendar year the declination varies slightly on the same day from year to year.

#### **1.4 Solar hour angle**

The hour angle is positive during the morning, reduces to zero at solar noon and increasingly negative when the afternoon progresses. The following equations can be used to obtain the hourly angle when various values of the angles are known.

$$\sin w = -\frac{\cos \alpha \sin A\_z}{\cos \delta} \tag{1.11}$$

$$\sin w = \frac{\sin \alpha - \sin \delta \sin \phi}{\cos \delta \cos \phi} \tag{1.12}$$

Where

 = altitude angle w = the hour angle Az = the solar azimuth angle = observer angle = declination angle

The hour angle is equals to zero at solar noon and since the hour angle changes at 15° per hour, the hour angle can be calculated at any time of day. The hour angles at sunrise (negative angle) and sunset (ws) is positive angle. They are important parameters and can be calculated from

$$\cos \mathbf{w} \mathbf{s} = -\tan \phi \tan \delta \tag{1.13}$$

$$
\cos \mathbf{s} = \cos^{-1} \left( -\tan \phi \tan \delta \right) \tag{1.14}
$$

$$L = \frac{2}{15} \cos^{-1} \left( -\tan \phi \tan \delta \right) \tag{1.15}$$

This L is known as the Length of the day also known as the maximum number of hour of insolation.

#### **1.5 Solar constant**

The solar constant is defined as the quantity of solar energy (W/m²) at normal incidence outside the atmosphere (extraterrestrial) at the mean sun-earth distance. Its mean value is 1367 W/m². The solar constant actually varies by +/- 3% because of the Earth's elliptical orbit around the Sun. The sun-earth distance is smaller when the Earth is at perihelion (first week in January) and larger when the Earth is at aphelion (first week in July). Some people, when talking about the solar constant, correct for this distance variation, and refer to the solar constant as the power per unit area received at the average Earth-solar distance of one "Astronomical Unit" or AU which is 1.49 x 108 million kilometres (IPS and Radio Services).

### **2. Empirical equations for predicting the availability of solar radiation**

#### **2.1 Angstrom-type model**

118 Solar Radiation

The declination angle is the same for the whole globe on any given day. Figure 1.6 shows the change in the declination angle throughout a year. Because the period of the Earth's complete revolution around the Sun does not coincide exactly with the calendar year the

The hour angle is positive during the morning, reduces to zero at solar noon and increasingly negative when the afternoon progresses. The following equations can be used

cos sin sin

sin sin sin sin cos cos *<sup>w</sup>* 

The hour angle is equals to zero at solar noon and since the hour angle changes at 15° per hour, the hour angle can be calculated at any time of day. The hour angles at sunrise (negative angle) and sunset (ws) is positive angle. They are important parameters and can

cos tan tan *ws*

<sup>1</sup> *ws* cos tan tan

This L is known as the Length of the day also known as the maximum number of hour of

The solar constant is defined as the quantity of solar energy (W/m²) at normal incidence outside the atmosphere (extraterrestrial) at the mean sun-earth distance. Its mean value is 1367 W/m². The solar constant actually varies by +/- 3% because of the Earth's elliptical orbit around the Sun. The sun-earth distance is smaller when the Earth is at perihelion (first week in January) and larger when the Earth is at aphelion (first week in July). Some

15 *L*

<sup>2</sup> <sup>1</sup> cos tan tan

> >

(1.14)

(1.15)

cos *Az <sup>w</sup>* 

> 

 

(1.11)

(1.12)

(1.13)

declination varies slightly on the same day from year to year.

to obtain the hourly angle when various values of the angles are known.

**1.4 Solar hour angle** 

Where 

 = altitude angle w = the hour angle

= observer angle

be calculated from

insolation.

**1.5 Solar constant** 

= declination angle

Az = the solar azimuth angle

Average daily global radiation at a specific location can be estimated by the knowledge of the average actual sunshine hours per day and the maximum possible sunshine hour per day at the location. This is done by a simple linear relation given by Angstrom (1924) and modified by (Prescott, 1924).

$$\frac{G}{G\_O} = a + b \left(\frac{S}{S\_{\text{max}}}\right) \tag{2.1}$$

In Nigeria, the hourly global solar radiation were obtained through Gun Bellani distillate, and were converted and standardized after Folayan (1988), using the conversion factor calculated from the following equations.

$$G = (1.35 \pm 0.176)H\_{GB}KJ \text{ / m}^2 \tag{2.2}$$

Where G is the monthly average of the daily global solar radiation on a horizontal surface at a location (KJ/m2-day), G0 is the average extraterrestrial radiation (KJ/m2-day). S is the monthly average of the actual sunshine hours per day at the location. Smax monthly average of the maximum possible sunshine hours per day, n is mean day of each month.

$$G\_o = \frac{24 \times 3600}{\pi} G \text{sc} \left( 1 + 0.033 \text{Cos} \frac{360n}{365} \right) \left( \text{Cos} \phi \text{Cos} \delta \text{Sin} \, W\_S + \frac{2 \pi W\_S}{360} \text{Sin} \phi \text{Sin} \, \delta \right) \tag{2.3}$$

$$S\_{\text{max}} = \frac{2}{1\,\text{S}} \cos^{-1}\left(-\tan\phi \tan\delta\right) \tag{2.4}$$

Several researchers have determined the applicability of the Angstrom type regression model for predicting global solar irradiance (Akpabio et al., 2004; Ahmad and Ulfat, 2004; Sambo, 1985; Sayigh, 1993; Fagbenle, 1990; Akinbode, 1992; Udo, 2002; Okogbue and Adedokun, 2002; Halouani et al., 1993; Awachie and Okeke, 1990; El –Sebaii and Trabea; 2005, Falayi and Rabiu, 2005; Serm and Korntip, 2004; Gueymard and Myers, 2009; Skeiker, 2006; Falayi et al., 2011 ). Of recent (Akpabio and Etuk 2002; Falayi et al., 2008; Bocco et al., 2010; Falayi et al., 2011) have developed a multiple linear regression model with different variables to estimate the monthly average daily global. Also, prognostic and prediction models based on artificial intelligence techniques such as neural networks (NN) have been developed. These models can handle a large number of data, the contribution of these in the outcome can provide exact and adequate forecast (Krishnaiah, 2007; Adnan, 2004; Lopez, 2000; Mohandes *et al*., 2000).

#### **2.2 Method of model evaluation**

### **2.2.1 Correlation coefficient (r)**

Correlation is the degrees of relationship between variables and to describe the linear or other mathematical model explain the relationship. The regression is a method of fitting the linear or nonlinear mathematical models between a dependent and a set of independent variables. The square root of the coefficient of determination is defined as the coefficient of correlation *r*. It is a measure of the relationship between variables based on a scale 1. Whether *r* is positive or negative depends on the inter-relationship between *x* and *y*, i.e. whether they are directly proportional (*y* increases and *x* increases) or vice versa (Muneer, 2004).

#### **2.2.2 Correlation of determination (r2 )**

The ratio of explained variation, (*G*pred - *G*m)2, to the total variation, (*G*obser - *G*m)2, is called the coefficient of determination. *G*m is the mean of the observed *G* values. The ratio lies between zero and one. A high value of *r*2 is desirable as this shows a lower unexplained variation.

#### **2.2.3 Root mean square error, mean bias error and mean percentage error**

The root mean square error (RMSE) gives the information on the short-term performance of the correlations by allowing a term-by-term comparison of the actual deviation between the estimated and measured values. The lower the RMSE, the more accurate is the estimate. A positive value of mean bias error (MBE) shows an over-estimate while a negative value an under-estimate by the model. MPE gives long term performance of the examined regression equations, a positive MPE values provides the averages amount of overestimation in the calculated values, while the negatives value gives underestimation. A low value of MPE is desirable (Igbai, 1983).

$$MBE = \frac{1}{n} \left[ \sum \left( G\_{pred} - G\_{obs} \right) \right] \tag{2.5}$$

$$RMSE = \left\{ \left[ \frac{1}{n} \sum \left( G\_{pred} - G\_{obs} \right)^2 \right] \right\}^{\frac{1}{2}} \tag{2.6}$$

$$MPE = \left[\sum \left(\frac{G\_{obs} - G\_{pred}}{G\_{obs}} \times 100\right)\right] / n \tag{2.7}$$

#### **3. Monthly mean of horizontal global irradiation**

Monthly mean global solar radiation data leads to more accurate modelling of solar energy processes. Several meteorological stations publish their data in terms of monthly-averaged values of daily global irradiation. Where such measurements are not available, it may be

Correlation is the degrees of relationship between variables and to describe the linear or other mathematical model explain the relationship. The regression is a method of fitting the linear or nonlinear mathematical models between a dependent and a set of independent variables. The square root of the coefficient of determination is defined as the coefficient of correlation *r*. It is a measure of the relationship between variables based on a scale 1. Whether *r* is positive or negative depends on the inter-relationship between *x* and *y*, i.e. whether they are directly proportional (*y* increases and *x* increases) or vice versa (Muneer,

The ratio of explained variation, (*G*pred - *G*m)2, to the total variation, (*G*obser - *G*m)2, is called the coefficient of determination. *G*m is the mean of the observed *G* values. The ratio lies between zero and one. A high value of *r*2 is desirable as this shows a lower unexplained

The root mean square error (RMSE) gives the information on the short-term performance of the correlations by allowing a term-by-term comparison of the actual deviation between the estimated and measured values. The lower the RMSE, the more accurate is the estimate. A positive value of mean bias error (MBE) shows an over-estimate while a negative value an under-estimate by the model. MPE gives long term performance of the examined regression equations, a positive MPE values provides the averages amount of overestimation in the calculated values, while the negatives value gives underestimation. A low value of MPE is

100 / *obs pred*

*obs*

Monthly mean global solar radiation data leads to more accurate modelling of solar energy processes. Several meteorological stations publish their data in terms of monthly-averaged values of daily global irradiation. Where such measurements are not available, it may be

*G G MPE <sup>n</sup> <sup>G</sup>*

*RMSE G G pred obs <sup>n</sup>*

**3. Monthly mean of horizontal global irradiation** 

1 <sup>2</sup> <sup>2</sup>

<sup>1</sup> *MBE G G pred obs <sup>n</sup>* (2.5)

1

(2.6)

(2.7)

**)** 

**2.2.3 Root mean square error, mean bias error and mean percentage error** 

**2.2 Method of model evaluation 2.2.1 Correlation coefficient (r)** 

**2.2.2 Correlation of determination (r2**

2004).

variation.

desirable (Igbai, 1983).

possible to obtain them from the long-term sunshine data via models presented in Chapter 2.
