Chapter II Summary of literature review in the fields of solar energy and related
2.1 Solar energy and its availability
2.1.2 Solar Radiation Geometry with Respect to Earth and Collector Surface
2.1.1.2 Solar radiation detection and measurement
There are various types of photodetectors for detecting and measuring light, with different spectral responsivities. These include photodiodes, photoresistors, thermopiles, CCDs, solar cells, etc. For measuring the components of solar radiation (beam, diffuse or global as well as other related quantities), there are special instruments (built from elementary photodetectors) classified as radiometers or sun photometers. The key instruments are the pyrheliometer (sun photometer) for measuring beam, and the pyranometer (radiometer) for measuring either global or diffuse radiations (Duffie, et al., 2006) (Brooks, 2006).
2.1.1.3 Available clear sky solar radiation
The direct radiation (Ib) incident on a surface (say, a two axis tracking collector) with known latitude and time of day can be estimated through the equation of Hu and White as cited by Alata (Alata, et al., 2005):
VW? XYZ[ \. ]^\._]` (2.1.1.4)
where Gon is the solar constant (as in eq.2.1.1.2 and 2.1.1.2.1) and m (as in eq.2.1.1.3 and 2.1.1.3.1) is the air mass number. This magnitude is important when evaluating the efficiency of the tracking and concentrating systems for clear sky conditions.
2.1.2 Solar Radiation Geometry with Respect to Earth and Collector Surface
Figure 5, shows the main geometric relations that are used to describe the way that the sun rays, the collector surface and earth surface positions, relate to each other. Additional relations are depicted in Figure 6. All these and additional variables and constants are listed and discussed in detail in next section, while details about collectors and tracking techniques are discussed in section 2.1.3.
2.1.2.2 Sun-Earth-Collector Geometry and Other Related Variables and Constants There are many variables and constants that describe different geometric and time magnitudes. These magnitudes can be found in different cited sources in a dispersed fashion. So, to address a concise and clear approach for the present work, we collect together the ones that are immediately relevant. Other quantities are derived, calculated or otherwise well known. Some basic quantities and definitions (many of them as found in Duffie, et. al., 2006) are:
(1) Sun-earth mean distance (SE) =1.495x1011 m (= 1 astronomical unit);
(2) Sun diameter (SD) =1.39x109 m;
(3) Earth diameter (ED) =1.27x107 m;
(4) Apparent diameter of the sun: maximum angle between 2 sun rays that hit the earth’s surface is ε=32’. This can be obtained from the above, as:
! ? 2ab8;< 'dScc 4 e 0.53 e 32g e 9.3 ;h;bai;a<8 (2.1.1.4) This becomes larger in December solstice and smaller in June solstice.
This value should be taken into account when determining the acceptance angle of a solar collector and hence in determining the necessary sun tracking resolution and accuracy. This quantity represents the nominal (minimal) acceptance of any collecting surface.
(5) Earth’s polar axis declination: δ0 = 23.450;
(6) Solar apparent angular speed (due to earth’s rotation):
ωs = 3600/24h = 150/h = 0.250/min = 15”of arc/s (2.1.1.5)
&s ? 2j bai/24l ? @dm bai/l ? Pdm bai/ ;< ? 72.72nbai/8 (2.1.1.5.1) (7) Local meridian(Lloc): is the longitude of the observer (ex: at Durban/UKZN
plant: Lloc = 300 56’40.0”E = 30.940E);
(8) Standard meridian (Lst): is the longitude of the standard meridian for the local time zone. Example: Durban/UKZN plant is in GMT+2 time zone, which corresponds to 300E;
p ? 15 H "q$ qrs# (2.1.1.6)
For UKZN (and entire RSA): p ? 15 H 2 ? 30 (2.1.1.6.1) (9) Solar noon: is the time when the sun is over the local meridian. Then, the
projection of zenith line is collinear with the N-S axis; hour and azimuth angles are then null.
(10) Local or standard time (Ltime): the local clock time (politically/economically determined);
(11) Day number (d): is the day of the year, being d=1 at 1st January. In 31th December d=365 (= 366 for leap years);
(12) Equation of time (E): is an equation that accounts for perturbations on the earth’s rotation speed and hence on the apparent sun angular speed (this also makes solar time a varying axis). E in minutes is:
t ? 229.2"0.000075 ) 0.001868 78 9 $ 0.0320778;< 9 $ 0.0146157829 $ 0.040849 8;<29#(2.1.1.7)
where B is the fractional year. See equation 2.1.1.2.2
(13) Solar time (Stime): time based on the above sun’s apparent angular motion:
v ? p) 4"p $ p# ) t (in minutes) (2.1.1.8) For Durban/UKZN solar plant Stime becomes:
v ? p) 4 H "30 $ 30.94# ) t ? $3.76 ;< ) t"min# ) p (2.1.1.9) (14) Zenith line: the normal to the horizontal surface which is also the line from
the centre of the earth, that crosses (the point of) the desired geographic location (see also Figure 4 and Figure 6);
(15) Zenith angle (θz): the angle of incidence of beam radiation with respect to the zenith line on the desired geographic location (see also Figure 4 and Figure 6);
(16) Incidence Angle(θ): the angle of beam of radiation with respect to the normal to a desired surface - examples: the normal to the receiving face of a flat plate collector or (the axis of) a paraboloidal concentrator;
(17) Surface azimuth angle(γ): If a desired surface is denoted by σ and the normal to that surface by Nσ: Then, the surface azimuth angle is the angle of the horizontal plane projection of Nσ with respect to local meridian;
(18) Solar azimuth angle(γs): the beam radiation’s horizontal projection angle with respect to south, where angular displacements eastwards are negative;
It is worth mentioning that, there are other used conventions in the definition of the azimuth angle, for example the one whose zero degrees reference is north (See Figure 6);
(19) Solar altitude or elevation angle (α): It is the complement to the zenith angle (θz);
(20) Slope or surface tilt angle (β): It is the angle of the desired surface with respect to the horizontal;
(21) Declination angle(δ): It is the angle between the sun beam radiation and the equatorial plan. This magnitude varies from - δ0 to + δ0 (-23.450 to +23.450) governed by the following equations [by Cooper(1969) and Spencer (1971) respectively, as cited by Duffie, et al., 2006]:
? 8z< '360dR{O2013 4 (2.1.1.10) Where d is the current day of the year. For more accuracy:
δ = 0.006918 – 0.399912 cos B + 0.070257 sin B – 0.006758 cos 2B +
0.000907 sin 2B - 0.002679 cos 3B + 0.00148 sin 3B (2.1.1.10.1) where B is the fractional year. See equation 2.1.1.2.2.
(22) Latitude angle(φ): It is the angle between the zenith line and the equatorial
plane; it is the latitude of the observer (At UKZN solar plant: φ = 290 49’2.0”S).
(23) Tracking angle (ρ): It is the angle that is produced when a tracking axis rotates about itself. This angle (or angles) is determined by the tracking type adopted. For example, in one axis tracking (see section 2.1.3.3.1.2) the tracking angle ρ corresponds to the hour angle.
(24) Longitude angle (λc): It is the longitude of the observer(λ = Lloc).
(25) Hour angle(ω): It is the angular displacement of sun beam from the local meridian, being negative eastward and positive westward. ω (in degrees) relates to Stime (in minutes) as:
ω = Stime * 0.25 -180º (2.1.1.11)
(26) Sunrise angle: It is the hour angle at sunrise;
ω
sunrise = - cos-1(-tan(φ) * tan(δ))(27) Sunset angle: It is the hour angle at sunset. It is numerically the symmetric of the sunrise angle.
Figure 6 - Representation of solar angles (Stine, et al., 2008).
Note that the azimuth angle (γ) depicted in the Figure 6, is the one whose zero degrees reference is pointing due north, while the same angle defined formerly, references to south; however both the angle defined formerly and the one depicted assume the clockwise angular motion as positive.
Of all these magnitudes, the most important ones in the point of view of sun tracking for maximum power collection, namely α, γ, δ, ω (and other), will be further discussed in the following chapters/sections.