2.2 Theoretical background
2.2.4 Solar Radiation on tilted surface
As sunlight is smoothly distributed over whole areas, a mere figure for intensity is never sufficient without knowledge of the orientation of the surface in question. Typically, the orientation of a surface is described by the zenith angle, the angle between the sunbeam and the normal of the area. If the surface area is not perpendicular to the sunbeam (i.e.
zenith angle is not zero), a larger area is required to catch the same flow as the cross section of the sunbeam; therefore the intensity in tilted surfaces in higher for the same sun flow. These areas are represented in Figure 2.2.17.
Figure 2.2.17. Incidence of sun’s rays in horizontal and tilted surfaces.
If Ib denotes the irradiation on a surface with the sun in its zenith, the irradiation on an area where the sun is observed under the zenith angle θz (in the figure is expressed as plain θ), Ib,n, which means on an horizontal surface, the irradiation is reduced to
/@,, = /@( )
Equation 2.2-17
(Green Rhino Energy, 2010)
Thus, the expression of the irradiation on a tilted surface, Ib,T, expressed as a function of the incidence angle θ, nor the zenith angle θz, is
/@,A = /@( )
Equation 2.2-18
Where the incident angle of sun beam radiation over the tilted surface, θ, is obtained as follows
= − (
Equation 2.2-19
For better understanding, Figure 2.2.18 shows those angles and the named irradiations.
Figure 2.2.18. Irradiation on a tilted surface.
2.2.4.1 Optimum Surface Orientation
After all these statements can be derived that, in order to obtain the maximum quantity of solar radiation per surface unit, is needed to tilt the collector till the solar beam radiation reaches the surface perpendicularly.
In the northern hemisphere, if the latitude φ, is bigger than the declination δ, the collector must be tilted heading south. Vice versa for the southern hemisphere.
The optimum slope β of the collector, βopt, is the difference between the latitude and declination, and consequently varies along the year. The expression of this optimum tilting depending on n, which is the day of the year, is:
(+B>:8<8() = − ()
Equation 2.2-20
(Departamento de Ingeniería Térmica y Fluidos, 2004)
Figure 2.2.19. Annual insolation improvement by tilting compared to horizontal situation.
There are different ways of taking care of this seasonal variation that is needed to face for obtaining the maximum irradiance on the tilted surface.
20 2.2. Theoretical background Tracking
In order to maximize the direct beam insolation on a surface, it is possible to rotate the surface around two axes, namely the tilt and the azimuth angle, which requires two motors. Tracking collectors can be one-axis tracking or two-axis tracking.
Figure 2.2.20. Two-axis tracking collector.
(Allbiz)
Typically, the marginal energy gains from tracing the azimuth angle are low. Hence, the second best option is to keep the slope flexible, but facing due south.
Fixed Tilt
In case there is no possibility to move the surface at all, it is considered south direction as optimal orientation, and the optimal tilt angle, βopt, for receiving the maximum amount of direct beam radiation, depends on the period of use of the solar installation:
- Constant annual Consumption: tilting must be equal to the latitude, β = φ.
- Preferential winter consumption: tilting should be the latitude increased in 10°, β = φ + 10º.
- Preferential summer consumption: tilting should be the latitude decreased in 10°, β = φ - 10º.
(IDAE, 2009)
However, as tilting the surface up causes the diffuse light portion to decrease, another consideration must be taken for humid climates: decrease the tilting by setting 10 – 25% less than the latitude. In Germany, for instance, at 48°Ν, a tilt angle of 30°
would be optimal, whereas in Spain, it could be up to 40°.
Seasonal Tilt
In regions where most of the irradiance occurs in summer, it may be beneficial to adjust the tilt angle for winter and summer. For example, in Germany, 75% of solar irradiance is experienced from April to September. The optimal angle for the summer would be 27° and for winter 50°, rather than 30° if the modules couldn't be tilted at all. However, the case in Spain is dissimilar as seasonal differences are less pronounced (summer accounts for 60%), making a seasonal tilt less critical. In Finnish case, it should be done as in Germany.
(Green Rhino Energy, 2010) 3A Sun tracking system
A new sun-tracking concept was proposed last year (accepted on the 3rd of December 2010) in Yunnan Normal University, Kunming 650500, PR China; by Yi Ma, Guihua Li and Runsheng Tang.
The optical performance of solar panels with such sun-tracking system was theoretically investigated based on the developed mathematical method and monthly horizontal radiation. The mechanism of the proposed sun-tracking technique is that the azimuth angle of solar panels is daily adjusted three times at three fixed positions:
eastward, southward and westward in the morning, noon, and afternoon, respectively, by rotating solar panels about the vertical axis (3A sun-tracking, in short).
Figure 2.2.21. Geometry of three azimuth angles tracked solar panels.
The analysis indicated that the tilt-angle of solar panels, β3A, the azimuth angle of solar panels in the morning and afternoon from due south, φa, and the solar hour angle when the azimuth angle adjustment was made in the morning and afternoon, ωa, were three key parameters affecting the optical performance of such tracked solar panels.
Figure 2.2.22. Three orientations of 3A tracked solar panels (top view).
Calculation results showed that, for 3A tracked solar panels with a yearly fixed tilt-angle, the maximum annual collectible radiation was above 92% of that on a solar panel with full 2-axis sun-tracking; whereas for those with the tilt-angle being seasonally adjusted, it was above 95%. Results also showed that yearly or seasonally optimal values of β3A, φaand ωafor maximizing annual solar gain were related to site latitudes, and empirical correlations for a quick estimation of optimal values of these parameters were proposed based on climatic data of 32 sites in China.
(Ma, Li, & Tang, 2011)
2.2.4.2 Total radiation on tilted surfaces
According to Liu and Jordan’s model (1963), which is an improvement from the isotropic model, the global radiation can be decomposed in three components: beam radiation, diffuse solar radiation, and solar radiation diffusely reflected from the ground.
22 2.2. Theoretical background
Figure 2.2.23. Components of total radiation on a tilted surface.
The complexity of deducing how much radiation belongs to each component is that flat-plate solar collectors absorb both beam and diffuse components of solar radiation;
and solar radiation data is usually registered for a horizontal surface, without distinguishing between beam and diffuse components.
Consequently, for using horizontal total radiation data to estimate radiation on the tilted plane of a collector of fixed orientation, it is necessary to know the geometric factor Rb, which is the ratio of beam radiation on the tilted surface, Gb,T, to that on a horizontal surface at any time, Gb. This ratio is given by
C@ =*@,A
*@ = *@,
*@, =
Equation 2.2-21
Where cos θ and cos θz are the incidence and zenith angles respectively, and can be determined from Equations 2.2-6 and 2.2-9.
As the global radiation, IT, is the sum of the three radiation components:
/A = /@,A+ /D,A+ /E,A
Equation 2.2-22
Each component is defined as:
• Beam radiation, Ib,T.
Represents the direct part of solar radiation.
/@,A = /@C@
Equation 2.2-23
• Diffuse radiation, Id,T.
Assuming an isotropic distribution of the diffuse radiation over the hemisphere, the diffuse part is only dependent on the horizontal tilt angle β and the diffuse radiation of the horizontal surface. The correction for the diffuse component depend on the distribution of diffuse radiation over the sky, which depends on the type, extent and location of clouds, and also on the amounts and spatial distribution of other atmospheric components that scatter solar radiation. Then, if a surface tilted at slope β from the horizontal has a view factor to the sky given by F#GHIJ KL M, the diffuse radiation can be expressed as
/D,A = /D1 + (
2
Equation 2.2-24
• Reflected radiation, Ir,T.
Some solar radiation may be reflected from the ground to the surface. The energy of the reflected light is dependent on the ground’s ability to reflect, a property which is expressed by the albedo factor ρ. The albedo, or reflectance, varies from 0,15 to 0,85, as can be seen in Table 2.2.1. (Green Rhino Energy, 2010)
Table 2.2.1. Albedo range.
Asphalt 0,15
Naked Ground 0,17
Lawn 0,21
Untilted Field 0,26
Weather-beaten concrete 0,30
Old snow 0,58
Fresh snow 0,85
Albedo ρ
The surface has a view factor to the ground of F#"HIJ K
L M, and if those surroundings have reflectance of ρ for the total solar radiation, the reflected radiation from the surroundings on the surface from the solar radiation is
/E,A = (/@+ /D)N 1 − (
2
Equation 2.2-25
Subsequently, the total solar radiation on the tilted surface for an hour is, dependant by the radiation on a horizontal surface is:
/A = /@C@+ /DF#G-+ OL M + (/@+ /D)N F#"-+ OL M
Equation 2.2-26
(Duffie & Beckman, 1980)
It can be shown with the help of the above formulas that tilting up a surface can increase the irradiance incident. The actual amount depends on numerous factors such as latitude, day in the year, albedo and clearness index as well as both the tilting angle and the surface azimuth.
Figure 2.2.24. Irradiation components.
24 2.2. Theoretical background In Figure 2.2.24 are plotted the irradiation components relative to the global irradiation on a horizontal surface facing due south on 20th March at 50°Ν with an albedo of 30% on a reasonably clear day, clearness index 0,5. Under these circumstances, the optimal tilt angle would be around 40°. This can be deduced by looking at the maximum of the curve “total” (the highest curve) and its position at the horizontal axis of the graph, which refers to the horizontal tilt angle.
Intuitively, the tilting effect is more pronounced for higher latitudes, as it happens to one of the cases of this thesis: Finland.
(Green Rhino Energy, 2010)