^{1}

^{*}

^{2}

^{3}

One of the important parameters that affects the performance of a solar collector is its tilt angle with the horizon. This is because the variation of tilt angle changes the amount of solar radiation reaching the collector surface. Meanwhile, is the rule of thumb, which says that solar collector should be orientated towards the Equator with a tilt equal to latitude, valid for high latitudes region? Thus, it is required to determine the optimum tilt for Equator facing collectors. In addition, the question may rise: how much adjustments of Equator facing solar collector tilt angle is reasonable to do during a year? A mathematical model was used for estimating the solar radiation on a tilted surface, and to determine the optimum tilt angle and orientation (surface azimuth angle) for the solar collector at any latitude. This model was applied for determining optimum tilt angle in the high latitudes zone in the Southern Hemisphere, on a daily basis, as well as for a specific period. The optimum angle was computed by searching for the values for which the radiation on the collector surface is a maximum for a particular day or a specific period. The results reveal that changing the tilt angle 12 times in a year ( i.e. using the monthly optimum tilt angle) maintains approximately the total amount of solar radiation near the maximum value that is found by changing the tilt angle daily to its optimum value. This achieves a yearly gain in solar radiation up to 1.8 times of the case of a horizontal surface while the daily gain reaches 60 times approximately. Moreover, general formulas are proposed for predicting daily optimum tilt angle and optimum tilt angle over any number of days.

The performance of a solar collector is highly influenced by its orientation (with respect to the Equator) and its angle of tilt with the horizon (with respect to the ground). This is due to the fact that both the orientation and tilt angle change the solar radiation reaching the surface of the collector. Designing an installation to yield maximum annual energy minimizes the necessary installed capacity and reduces the cost of equipment. To achieve this, solar collector must be mounted at right angles to the Sun’s rays. The best way to collect maximum daily energy is to use tracking systems. A tracker is a mechanical device that follows the direction of the Sun on its daily sweep across the sky. The most effective tracking could be achieved by mounting the collector on a two-axis tracker that continuously tracks the Sun by the hour and through the seasons. As the trackers are expensive, and need energy for their operation, this method of tracking is quite cumbersome and inconvenient practically. Thus, the majority of installations are with fixed mountings. Therefore, it is often practicable to orient the solar collector at an optimum tilt angle, B_{opt} and to correct the tilt from time to time. For this purpose, one should be able to determine the optimum slope of the collector at any latitude, for any surface azimuth angle, and on any day or any period of the year. Various schemes have been proposed for optimizing the tilt angle and orientation of solar collectors designed for different geographical latitudes or possible utilization periods. However, as the goal of this work is to treat this question regarding the high latitudes region in Southern Hemisphere, it is reasonable to restrict ourselves to main available literature concerning this zone directly or indirectly.

In this context, Soulayman [_{opt} for south facing collector at any latitude from 0˚ to 60˚. Soulayman and Sabbagh [_{opt} at any latitude, L, and for any direction (surface azimuth angle, G). This algorithm could be used for treating B_{opt} in the high latitudes regions. Stanciu, and Stanciu [_{opt} theoretically and experimentally.

The main objective of this study is to develop a simple and easy way for finding daily, monthly, seasonally, half-yearly and fixed optimum tilt angles for any location in the high latitudes regions and to determine the yearly energy gain. As experimental data concerning the treated question in the studied region are not available to authors, the results of the present study could not be compared with the experimental results. However, the comparison with theoretical results of other researchers will be provided.

Radiation data are the best source of information for estimating average incidence radiation. Lacking these data, it is possible to use empirical relationships to estimate radiation from hours of sunshine or cloudiness, relative humidity and ambiance temperature, which are widely available from many hundreds of stations in many countries. The main part of empirical relationships is restricted to hours of sunshine or cloudiness. However, these relationships could be written as:

where H = monthly average daily radiation on a horizontal surface, H_{0} = the monthly average daily extraterrestrial solar radiation on a horizontal surface, n = monthly average daily hours of bright sunshine, N = monthly average of the maximum possible daily hours of bright sunshine (i.e., the day length of the average day of the month), C = monthly average daily cloud cover, RH = relative humidity and T = ambiance temperature. Supposing Equation (1) is applicable for daily values, it is to demonstrate that daily optimum tilt angle, B_{opt}_{,d} is the solution of the following nonlinear algebraic equation:

in relation to B where the summation should be ignored when dealing with daily period. In Equation (2) W_{ss} and W_{sr} are the sunset and sunrise hour angle on tilted surface:

C(N) is the daily correction factor for Sun-Earth average distance:

δ is the solar declination angle which could be calculated as below:

A_{1}, A_{2}, A_{3} and A_{4} are functions of solar and collector angles:

The geometric tilt factor R_{b}, the ratio of beam radiation on the optimum tilted surface to that on a horizontal surface at any time or period of time, can be calculated exactly in the case of extraterrestrial radiation by appropriate use of solar incidence angle on tilted surface and on horizon. A similar factor, R_{b}, could be introduced to express the solar energy gain on the optimum tilted surface to that tilted at angle equal to the latitude. For B = 0˚, B = B_{opt}, B = L daily extraterrestrial solar radiation is:

and summation by N should cover the length of period in consideration. For daily values no summation by N is used.

Soulayman [_{opt}_{,d} = L − δ proposed in [_{opt}_{,d} at latitudes from −40˚ to −66˚ the results presented in _{opt}_{,w}, monthly tilt angle B_{opt}_{,m} (see _{opt}_{,s} (see

work, concerning monthly optimum tilt angle at different latitudes, with those of [

Nijegorodov et al. [

On the other hand, in order to evaluate the possible solar energy gain using tilt angle adjustment the total yearly extraterrestrial solar radiation at B = 0˚, B = B_{opt}, B = L should be calculated on a daily, monthly, seasonally as well as on a half-yearly and yearly basis. The results, for one example L = 55˚, are given on _{2} stands for solar radiation at B = B_{opt}, H_{1} stands for solar radiation at B = L and H_{0} stands for solar radiation at B = 0˚. It is seen from _{2} with H_{1} it is seen from _{2} is greater than H_{1} remarkably over the period started from 22/9 to 21/3 while these values are near to each other during half a year started from 22/3 to 21/9.

By taking the ratio between the values related to surface with optimum tilt to those on a horizontal one and to

those tilted by latitude angle for the same period of time, the corresponded tilt factors, R_{b} = H_{2}/H_{0} and R_{b}_{1} = H_{2}/H_{1}, could be calculated on daily, monthly, seasonally, half-yearly and yearly basis’s. These results are given in Figures 6-8.

It is seen from

When applying the algorithms proposed in [

It was found that daily optimum tilt angle, B_{opt}_{,d}, can be calculated effectively using the following equation:

where “a”, “b” and “c” are functions of latitude L. These functions are of the following form:

El-Kassa by [_{opt}_{,d} at latitudes up to 60˚ in Northern Hemisphere with verifying the applicability of his formula during the period starting from 22/9 to 21/3 but his formula suffers from uncertainty during period starting from 22/3 to 21/9 (see [_{opt}_{,d} for any chosen day at any latitude in Northern Hemisphere but he repeated the same formula of El-Kassaby [_{opt}_{,d} for 60˚S latitudes and Equation (2), the algorithm of [

When applying Equation (12), that of [_{opt}_{,d} for L = −50˚ the

obtained results are given in _{opt}_{,d} remarkably during the period starting from 22/9 to 21/3 and overestimates slightly B_{opt}_{,d} during the period starting from 22/3 to 21/9 with an absolute deviation < 5˚; and b) Equation (12) gives the results of the algorithm with an absolute deviation < 1˚; So Equation (12) could be applied with a very good accuracy with regard to the algorithm [

When integrating Equation (12) for obtaining optimum tilt angle B_{opt}_{,p} at any period of time one obtains the following formula:

where N_{1} and N_{2} are the day numbers of the period beginning and ending respectively.

When calculating monthly optimum tilt angle 60˚S latitude using Equation (16) and comparing the obtained results with those of set of equations in [

It could be noted that, the methodologies used in [

Statistical indicators used in

a) The mean bias error (MBE) given by:

Date | B_{opt}_{,d} (˚) for latitude L = −50˚ | ||
---|---|---|---|

[ | [ | Equation (12) | |

1/1 | −27.0 | −8.3 | −8.6 |

15/1 | −28.7 | −12.7 | −12.2 |

1/2 | −32.5 | −20.3 | −19.9 |

15/2 | −36.7 | −28.3 | −28.0 |

1/3 | −41.7 | −37 | −37 |

15/3 | −43.2 | −45.7 | −45.6 |

1/4 | −54.0 | −56.1 | −56.0 |

15/4 | −59.4 | −63.2 | −63.0 |

1/5 | −64.9 | −69.4 | −69.3 |

15/5 | −68.8 | −73.3 | −73.2 |

1/6 | −72.0 | −76.3 | −76.2 |

15/6 | −73.3 | −77.4 | −77.3 |

1/7 | −73.1 | −77.2 | −77.1 |

15/7 | −72.8 | −76.7 | −76.8 |

1/8 | −67.9 | −72.3 | −72.4 |

15/8 | −63.8 | −68 | −68.1 |

1/9 | −57.7 | −60.8 | −60.9 |

15/9 | −52.2 | −53.2 | −53.5 |

1/10 | −45.8 | −43.2 | −43.8 |

15/10 | −40.4 | −34.5 | −34.7 |

1/11 | −34.6 | −24.6 | −24.1 |

15/11 | −30.9 | −16.3 | −16.6 |

1/12 | −27.9 | −10.4 | −10.5 |

15/12 | −26.7 | −7.8 | −7.8 |

Month | B_{opt}_{,m} (˚) for latitude L = −60˚ | ||
---|---|---|---|

Equation (16) | [ | [ | |

1 | −22.3 | −24.4 | −23.7 |

2 | −36.6 | −41.2 | −38.5 |

3 | −54.6 | −56 | −56.3 |

4 | −69.9 | −70 | −72.6 |

5 | −78.9 | −79.8 | −81.9 |

6 | −82.1 | −86.2 | −85.6 |

7 | −80.7 | −83.4 | −83.9 |

8 | −74.0 | −75.2 | −76.6 |

9 | −60.8 | −62 | −62.6 |

10 | −42.8 | −48 | −44.0 |

11 | −25.9 | −30.8 | −27.4 |

12 | −17.6 | −18.2 | −17.7 |

MBE | 2.41766 | 1.425014 | |

RMSE | 2.98281 | 2.234652 | |

t | 4.58982 | 2.745675 | |

R^{2} | 0.994 | 0.999 |

b) The root mean square error (RMSE) given by:

c) The t-statistic is given as:

n is the number of data pairs; ^{th} predicted and the i^{th} referenced data.

A mathematical model was applied for determining the optimum tilt angle of the solar collector at any latitude of the interval [−40˚, −66˚] in Southern Hemisphere. The optimum tilt angle was computed by searching for the values for which the radiation on the collector surface is a maximum for a particular day or a specific period. For an equator facing flat solar collector:

It is sufficient to adjust solar collectors tilt angle weekly (once/week) as this adjustment leads to the daily gain approximately.

It is sufficient to adjust solar collectors tilt angle 12 times (once/month) as this adjustment leads to the daily gain approximately.

It is practically sufficient to adjust solar collectors tilt angle twice a year: once from 22/3 to 21/9 and the other from 22/9 to 21/3 as the losses in the energy gain are not very important.

It is practically sufficient to orientate the solar collectors at tilt angle equal to the latitudes during the period started from 22/3 to 21/9 as the losses in the energy gain are less than 0.2.

The first part of the rule of thumb, which says that solar collector should be orientated towards the Equator is true for high latitudes in Southern Hemisphere while the second part, which says that solar collectors should be tilted at angle equal to latitude, is not valid during the started from 22/9 to 21/3 for high latitudes region.

General formulae were proposed for determining optimum daily tilt angle and optimum tilt angle for any number of days.

The applicability of the proposed formulae was verified.

This study will be hopefully expanded to cover northern high latitudes zone.

Authors thank Dr. R. Jabra for useful discussions and English reviewing.

Soulayman Soulayman,Alhelou Mohammad,Nouredine Salah, (2016) Solar Receivers Optimum Tilt Angle at Southern Hemisphere. Open Access Library Journal,03,1-11. doi: 10.4236/oalib.1102385

B: Tilt angle (˚).

B_{opt}_{,d}: Daily optimum tilt angle (˚).

B_{opt}_{,w}: Weekly optimum tilt angle (˚).

B_{opt}_{,m}: Monthly optimum tilt angle (˚).

B_{opt}_{,s}: Seasonally optimum tilt angle (˚).

B_{opt}_{,y}: Yearly optimum tilt angle (˚).

d: Solar declination angle (˚).

G: Collector azimuth angle (˚).

H_{0}: Extraterrestrial solar radiation (J/m^{2}).

H: Solar radiation on Earth surface (J/m^{2}).

L: Latitude angle (˚) which is positive for Northern Hemisphere and negative for Southern Hemisphere.

N: Day number in the year.

W_{sr}: Sunrise hour angle on a tilted surface (rad).

W_{ss}: Sunset hour angle on a tilted surface (rad).

R_{b}: Energy gain in relation to B = 0˚ case.

R_{b}_{1}: Energy gain in relation to B = L case.