1. Introduction

Power supply affects a lot a country’s economic activity. Currently, fossil fuel is the most important part of power supply in the world; however, for the consideration of the issue of environment pollution, renewable energy -including solar energy, wind energy, and biomass energy-is being paid more and more attention. Solar energy is the best choice for most of the areas all over the world [1] as it can be utilized through solar collector or photovoltaic (PV) cell. Many researchers have been dealing with this subject, but they almost were able to determine the optimum angle for limited locations only [1-14]. In addition, a comprehensive research like this current study is not dealt with in the related literature; In fact; this was the motive to carry out this research which extends comprehensively to present formula for the optimum tilt angle of flat solar harvesters for any location and any time.

To maximize the collected energy, proper installing of the collector is really needed. One way is to install the solar collectors in the correct tilt and orientation angle, in which they would obtain the maximum insolation over a specific period of time. The tilt angle depends mainly on the position of the sun and, therefore, is different from one location to another all over the world; in regard to the best orientation angle, it is advised to be directed towards the equator. There are already plenty of investigations dealing with this subject to optimize solar power systems according to the correct tilt angle or orientation. Many authors have provided empirical models to calculate the optimum tilt angle by searching for the maximum total solar radiation on the collector surface. In reference to a specific period of time and purpose, daily, monthly, seasonally or yearly values have been calculated [15-17]. Elsayed in [18] also presented an analytical model based on long-term averaging of solar data. He outlined values of optimum tilt angles given in different literature and conducted that value of tilt angle that can be recommended.

In fact there is a wide range of tilt (±20) which is dependent on the applied model and the location. Some authors noted the existence of a correlation between the optimum tilt angle and the latitude. Frequently, it is recommended to apply the rule of thumb, in which the yearly optimum tilt angle is about (L: latitude) and a difference of tilt with about 10° would hardly affect the performance. Authors of [19] determined monthly optimum tilt angles for Izmir, Turkey; they found the optimum tilt angle to be equal to U throughout the year, while for summer and for winter was suggested, these values were suggested also by [20]. They advised to mount the solar collector at the monthly average tilt angle. During the last decade there have also been investigations by using simulation software. This software which was developed in order to simulate an entire solar power plant, takes the most influential parameters into account. The soft ware usually possesses a database of monthly mean global radiation data and different empirical models: for example authors of [21] applied the simulation software TRNSYS to calculate the optimum tilt angle for Cairo, Egypt. What they did was calculating the monthly mean of solar radiation data and compared it with the output power of solar cells. They stated the yearly optimum tilt angle to be

2. Modelling of Solar Radiation on Tilt Surface

The moment the solar radiation, which is coming from the direction of the sun, reaches the earth’s surface without being significantly scattered 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 which the observer can see. This amount can be significant in areas in which the ground is covered with snow or reflectors. The total solar radiation on a horizontal surface is called global irradiance and it 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.

As recommended by ASHRAE (1985) and presented in all text books of solar energy [22, 23], hourly global radiation hourly beam radiation hourly diffuse radiation and hourly reflect radiation on the inclined surface on a clear day are calculated using the following expression:

(1)

(2)

(3)

(4)

Where: is the beam normal radiation A is the apparent solar-radiation constant in B is the atmospheric extinction coefficient, and C is the diffuse sky factor and there values are tabulated in table 1 for a widely range of latitudes

Table 1. ASHRAE values of clear sky model parameters A, B and C [23]

and are coefficients and is ground Albedo and was assumed to be 0.5. is the ratio between global solar energy on a horizontal surface and global solar energy on a tilt surface. Meanwhile, is the view factor between the sky and the tilt surface and is the view factor between the ground and the surface that tilted at angle from the horizontal. These coefficients evaluated from:

(5)

for isotropic diffuse:

(6)

(7)

where: is the solar incident angle and it is given by:

(8)

where: is latitude, the angular location north or south of the equator, north positive;

slope of the surface; is the surface azimuth angle, with zero due south, east negative, west positive, is the hour angle, morning negative, zero at noon and afternoon positive; and is solar declination angle, it can be found from:

(9)

where: is the Julian day. Meanwhile, is the solar zenith angle, which equal to:

(10)

3. Results and Discussions

A comprehensive computer program in FORTRAN has been created in order to calculate the hourly solar radiation incident on a surface when the tilt angle is changed by interval of The surface azimuth is either for south facing or for north facing. In regard to the daily optimization, the basis is the daily total hourly radiation, for the monthly optimization the basis is the monthly total daily and for annual optimization the basis is the annual total monthly.

It is well known that a unique surface tilt angle which exists for each time, and which corresponds to each latitude angle for a particular day through which the solar radiation is at peak value. Accordingly, this study enhances these two parameters as independent variables for the daily, monthly, seasonal and annual optimum angle prediction.

The seasonal average tilt angle was calculated by finding the average value of the tilt angle for each season, but the implementation of this condition requires the collector tilt to be changed four times a year. The process of adjusting the tilt angle to its monthly optimum values throughout the year does not seem to be practical, and at the same time rises the consideration of changing the tilt angle once seasonal.

The obtained results from the FORTRAN program treated by MATLAB in order to get out polynomials those fitted the daily, monthly, seasonal and annual optimum angles.

The general form of the polynomial that prescribed the optimum tilt angle is expressed in fifth order three dimension as:

(11)

The coefficients and the variables and are defined below in table 2.

Table 2. The definition of the coefficients P00 ...P05 and the independent variables x, y for the offered polynomial

The obtained results were plotted and tabulated for comparison with other measurements and data recommended by NASA.

Figure 1 presents the Daily optimum angles for various latitudes for a complete calendar year. The abscissa presents the Julian day The markets present the calculated results and the dash lines present the daily polynomial. Obviously, depending on figure 1, it can be stated that the polynomial is well fitted to the calculated data. The positive value depicted means that the surface was inclined towards the equator, where as the negative value means that the surface was inclined towards the North Pole. The slopes were within the range of for any location on the northern hemisphere.

Figure 1. Daily optimum angle as a function of the Julian day n and the latitude angle L. The marks refer to the calculated results and the dash lines refer to the polynomial Sopt = f(n, L)

With the adoption of the long-term solar radiation data, the optimum tilt angle of a surface by using the monthly total daily solar irradiation on diversely latitudes was simulated for 12 months. Figures 2 presents the optimum angles for monthly tracking from January to December. From figure 2, the step-like lines present the calculated results, and the dash curves illustrate the polynomial equation that fitted the calculated data. Results were tabulated in table 4, for many cities from other references beside NASA data available on the internet to ease the comparison. Table 3 shows a considerably agreement in results obtained by the offered polynomial with the local measurements specially for mid and high latitudes even in some times better than NASA's data. Unfortunately, for low latitudes the model needs to more arrangements.

Figure 2. Monthly optimum angle as a function of the Julian day n and the latitude angle L. The solid lines refer to the calculated results and the dash lines refer to the polynomial Sopt = f(n, L)

Table 3. Values of monthly optimum angles from many sources

Table 4. Description of Julian day for northern and southern hemispheres that used in polynomial (11)

For the southern hemisphere another arrangement must be taken to estimate the monthly optimum angles; one can use the same polynomial with the same latitude but the months will be reversed. That means December in the northern hemisphere will be June in the southern hemisphere. The months must be shifted every 6 months after the ordinary calendar. Table 4 described the Julian day for both hemispheres for monthly tilt angle optimization, and the optimum tilt angle for southern hemisphere is negative sing of the optimum tilt angle for northern hemisphere for the same latitude.

(12)

Sometimes the cause of the difference in some values for some sites is a result of the large step of tilt angle 10°, or because the researchers had not included angles greater than 60° or less than 10°. Anyhow, it can be obviously stated from table 3 the agreement in values obtained from the mathematical model proposed in this study compared with the those results presented in other previous studies.

The study of seasonal optimum tilt angle for maximising energy collection by the absorber has been carried out. In literatures, there are two classifications for seasons: Heating and cooling group interesting with two seasons heating season and cooling season. Another classification may be adopted for solar energy group dependent on the position of the sun in the sky, they classified seasons to four seasons in 12 months. This study adopted the second classification. Each season consists of 91 days, therefore, winter (6^th November to 4^th February), spring (5^th February to 5^th May), summer (6^th May to 5^th August), and autumn (6^th August to 5^th November).

These seasonal is valid for northern hemisphere. In the southern hemisphere the situation is reverse the summer season (6^th November to 4^th February), autumn (5^th February to 5^th May), winter (6^th May to 5^th August), and spring (6^th August to 5^th November). Figure 3 shows the optimum seasonal tilt angle based on the total daily solar radiation reaching on a tilt surface. The step-like lines present the calculated optimum tilt angle for seasonal optimization. The simulated results have agreed to the local available measurements with similar pattern to reflect the trend at the site. From the figure, one polynomial equation were proposed to depict the trend of seasonal optimum as functions of the latitude and the day. The equation shows a good fit to the output as it obvious from the figure 4, in where the dash curves present the polynomial. A comparative results are tabulated in table 5. Unfortunately, there are no enough results to include in the table; moreover, some researches classified the season into only two seasons cold and hot seasons for heating and cooling of buildings purposed.

Figure 3. Seasonal optimum angle as a function of the Julian day n and the latitude angle L. The solid lines refer to the calculated results and the dash lines refer to the polynomial Sopt = f(n, L)

Table 5. Values of seasonal optimum angles from many sources

The annual optimum tilt angle has been computed based on the estimated annual total monthly solar radiation. The results illustrated graphically in figure 4 and tabulated in table 6 for comparative purpose. Results showed that the optimum values were almost positive for the northern hemisphere and ranged facing to the south (equator). The value was similar and agreed with the optimum slope presented by other researchers and for those of NASA, specially for high latitudes even in some times better than NASA's data. Additionally, the computed results were fairly consistent with the general rule that the yearly optimal tilt angle was about the latitude of the location facing to the equator, specially for middle latitudes. For the southern hemisphere the optimum tilt angle will be negative sign of the northern hemisphere at the same latitude.

Figure 4. Annual optimum angle as a function of the latitude angle L. The marks refer to the calculated results and the dash lines refer to the polynomial Sopt = f( L)

Table 6. Values of annual optimum fixed angles from many sources and NASA

4. Conclusions

We have successfully applied the ASHRAE clear sky model to create comprehensive three dimension polynomials for daily, monthly, seasonal and annual optimum tilt angle for diversely latitudes. According to the comparison between the obtained results with local measured authoritative data and NASA published data, it can safely recommended to use of these polynomials especially in mid and high latitudes in the two hemispheres.