Prediction of Symmetrical and Asymmetrical of Diurnal Global Solar Irradiance Distribution — New Approach

A simple formula to predict the received global solar irradiance q(t), W/m for clear days is suggested on pure theoretical basis. It is expressed in terms of the length of the local day time td which is well defined in literatures on meteorological basis. The introduced distribution is also a function of the maximum value of the daily received irradiance qmax. which in turn is expressed in term of the solar constant. This renders the trial to be a closed system. Thus the obtained distribution is not a semi empirical one. Both cases of symmetrical and asymmetrical distributions for q(t) are considered. For its simplicity it can be easily integrated along the length of the day to get the daily totals of solar energy received by unit horizontal area. This is important for practical applications. Comparison between computed according to the present model and published experimental meteorological data in Barcelona (Spain), Hong Kong (China), Jeddah and Makkah (Saudi Arabia) is given as illustrative examples. Better fitting relative to the published trials for the same locations are obtained. The introduced model itself gives good fitting for the intermediate intervals points of the local day time which is the more effective region. The estimated relative error is 12% for Hong Kong, and it is 7% for Barcelona, Jeddah and Makah.


Introduction
The prediction of the diurnal global solar radiation q(t) W/m 2 is needed as one important input parameter to study theoretically the design and performance es-Optics and Photonics Journal timation of solar systems for solar energy exploitation, for example, the performance and efficiency of a solar cell, flat plate collector, water heating and treatment, pool heating, space heating, solar cookers, (Heating, Ventilation and air conditioning) (HVAC) technological systems.It is also required to study the production of electricity using molten salt technologies in which the liquid salt is pumped through panels in a solar collector for further heating.
Analysis of solar radiation measurements has aroused the interest of many investigators.As an example, different distributions for solar irradiance with different fitting degrees are given [1]- [6].A lot of experimental meteorological data for many locations are published [7]- [14].Trials to introduce governing formulae are given [5] [10] [11] [15].
The received solar energy is a function of several variables [2] such as the nature and extent of cloud cover, the aerosol and other atmospheric constituents such as O 2 , N 2 , CO 2 , O 3 , dust etc.Such a function depends also on other parameters such as the sunshine hours, the solar declination angle, the latitude, the altitude and the relative humidity [2].
As a result of these challenges, it is not always possible to predict theoretically the actual shape of such a function to get accurate values of the received irradiance for a given location.Different trials are given by different authors with different degrees of fitting accuracy [1] [18].Most of such trials are either semi empirical or incomplete to form a closed system or they are difficult to be integrated.
The need for more accurate trials with better fitting degrees is still required.
El-Adawi et al. [2] introduce a power expression for such a function, the parameters of which were determined through the least fitting technique.The given expression is not easy to be integrated.
Good fitting with published experimental meteorological data is obtained with maximum relative error 11%.Other trials expressed the required distribution in the form of polynomial [in(t − t max )] [16]  [4] with maximum relative error 15%.
The present trial represents a new approach to introduce a suggested formula based on well-established solar data such as the length of the solar day "t d " in hours, which is well defined in [15], and is also expressed through the maximum value of the daily solar irradiance q max W/m 2 .The expression for (t d ) is well defined in literatures on meteorological basis [15].
To get a closed system, the value of q max is suggested in terms of the extraterrestrial solar constant adjusted for the variation of the distance between the sun and the earth along the time of year [2] [3].Thus the introduced distribution is not a semi-empirical one.This is an advantage of the present trial.Moreover, it can be easily integrated and thus it is feasible for practical applications.
A comparative study between the experimental meteorological published Optics and Photonics Journal data of the received global solar irradiance in different locations [7] [9] [12] [13] and that computed using the present suggested model is given.The relative errors are indicated.

Theory
The experimental measured meteorological values of the global solar irradiance q(t), W/m 2 received on a horizontal surface as measured by different authors [1] [2] [7] revealed a symmetrical distribution about a maximum average value q max acquired at midday time (t max between sunrise t r and sunset (t s ) i.e. max 2 2 .
This symmetrical behavior is shown to be true for the whole solar year [7].
Moreover, the behavior of this function for different locations reveals its uni- However, some authors [4] discussed the case of asymmetrical distribution for which q max occurs at "t max " shifted from the midday times i.e. max 2 This case will be considered in the present trial.
In the present trial the suggested model to predict the function q(t) W/m 2 is given in the form: ( ) Shifted time scale is considered for which t r = 0.This distribution satisfies the following conditions: ii) At iii) At This gives: This gives: For symmetrical distribution, max 2 This gives: Finally, one gets for symmetrical distribution the following expression: Optics and Photonics Journal For asymmetrical distribution: For symmetrical distribution the total daily solar energy received per unit area of a horizontal surface is given as: ( ) ( ) Authors of different trials obtained for the same integral the following values: max 0.565 d q t [1]   max 0.517 d q t [4]   max 0.533 d q t [16]   max 0.557 d q t [17]   While for asymmetrical distribution the obtained value is: max 0.4715 d q t [4].
This shows that the daily totals of the global solar irradiance on a horizontal surface depends on the degree of symmetry about the point t = t max , at which the received solar irradiance attains its maximum value [4] It is worth to note that the length of the day "t d " can be expressed in terms of the latitude L and the solar declination "δ" [15] as follows: ( ) 24 cos tan tan 180 where, 284 23.45sin 360 365 and "n" is the day number of the year starting from 1 January i.e., (1 ≤ n ≤ 365).
To get a closed system of equations, the physical quantity q max is suggested on theoretical basis to be in the form [2] [3] [5] [10] [11]: where, s is the extraterrestrial solar constant adjusted for the variation of the distance between the sun and the earth and along the time of the year [11]: And, s = 1353 W/m 2 [10] is the solar constant.It is worth to note that q max is computed [3] according to Equation (14) for Jeddah and Makkah.
The estimated value q max (Jeddah) = 856.8W/m 2 while the experimental value is 915 W/m 2 and q max (Makkah) = 878 W/m 2 , while the experimental value is 938 W/m 2 .The obtained relative error is 6% [3].
And "α" is a correction factor.It is well defined in the literatures [2] [10] Its Optics and Photonics Journal value is given by the relation: Its value is estimated [3] to be 0.65 and 0.63 for Jeddah and for Makkah respectively.
This value depends on the optical thickness, the reflectivity of the underlying terrain and depends also on the solar zenith angle [2] [3].

Computations
The function q(t) is computed according to Equation ( 9) for different locations.
The obtained results are compared with the corresponding meteorological published data as illustrative examples.
The measure of fitting is taken as .
exp cal exp q q q ε − = This step is summarized as follows: 1) The considered data for Barcelona (Spain) (41˚23'N, 2˚7'E) January 1973 [9] are given in Table 1 and are illustrated graphically (as shown in Figure 1).
4) The data for Makkah (Saudi Arabia) (38.5˚E, 21.5˚N) March 1983 [13] are given in Table 4 and are illustrated graphically (as shown in Figure 4).
The relative errors obtained according to our model are compared with the corresponding published meteorological values obtained for the same locations and at the same local day time as shown in the corresponding tables.
It is revealed that our model is promising and gives better fitting relative to some other trials as [4] [17] irrespective of the extreme points.The model itself gives good fitting for the intermediate points.This is the more effective one.The      2) The introduced formula can easily be integrated along the local day time to get the total energy received per day per unit area.

4 . Conclusions 1 )
The introduced trial to predict the daily global solar irradiance for clear days is promising.It gives good fitting ( ) 12% ≅ when compared with the corresponding measured data.