Optimum solar flat-plate collector slope: Case study for Helwan, Egypt

Transcription

1 Energy Conversion and Management 47 (2006) Optimum solar flat-plate collector slope: Case study for Helwan, Egypt Hamdy K. Elminir a, *, Ahmed E. Ghitas a, F. El-Hussainy b, R. Hamid a, M.M. Beheary b, Khaled M. Abdel-Moneim a a National Research Institute of Astronomy and Geophysics, Solar and Space Department, Marsed Street, Helwan, Cairo, Egypt b Al-Azhar University, Faculty of Science, Nasr City, Cairo, Egypt Received 10 July 2004; received in revised form 20 November 2004; accepted 27 May 2005 Available online 14 July 2005 Abstract This article examines the theoretical aspects of choosing a tilt angle for the solar flat-plate collectors used in Egypt and make recommendations on how the collected energy can be increased by varying the tilt angle. The first objective in this investigation is to perform a statistical comparison of three specific anisotropic models (Tamps Coulson, Perez and Bugler) to recommend one that is general and is most accurate for estimating the solar radiation arriving on an inclined surface. Then, the anisotropic model that provides the most accurate estimation of the total solar radiation has been used to determine the optimum collector slope based on the maximum solar energy availability. This result has been compared with the results provided by other models that use declination, daily clearness index and ground reflectivity. The study revealed that PerezÕs model shows the best overall calculated performance, followed by the Tamps Coulson then Bugler models. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Anisotropic model; Optimum collector slope; Perez model; Bugler model; Tamps Coulson model; Clearness index; Declination * Corresponding author. Tel.: ; fax: address: hamdy_elminir@hotmail.com (H.K. Elminir) /$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi: /j.enconman

2 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) Nomenclature G H hourly global solar radiation incident on horizontal surface, W/m 2 G T hourly global solar radiation incident on inclined surface, W/m 2 I BH hourly beam radiation incident on horizontal surface, W/m 2 I BT hourly beam radiation incident on inclined surface, W/m 2 I DH hourly diffuse radiation incident on horizontal surface, W/m 2 I DT hourly sky diffuse radiation incident on inclined surface, W/m 2 I GH hourly ground reflected radiation incident on horizontal surface, W/m 2 I GT hourly ground reflected radiation incident on inclined surface, W/m 2 R T ratio of beam radiation on tilted surface to that on horizontal surface h incidence angle, degrees b surface slope, degrees c surface azimuth angle, degrees / latitude angle, degrees q ground albedo x hour angle, degrees d declination, degrees 1. Introduction The amount of solar energy incident on a solar collector in various time scales is a complex function of many factors including the local radiation climatology, the orientation and tilt of the exposed collector surface and the ground reflection properties. In the case of limited site area available for exposing the solar collector arrays, as well as in large scale industrial applications, the economical shaping and exposure of collectors with respect to the available insolation and site area may become a matter of primary importance. The amount of insolation may also be diminished due to shading of the collector surface by nearby objects or adjacent collectors arranged in an array. The best way to collect maximum daily energy is to use solar tracking systems. The choice to use a tracker hinges on a tradeoff between the cost of the tracking system and the savings provided by using fewer solar collectors to obtain a given amount of power. The trackers are expensive, need energy for their operation and are not always applicable. Therefore, it is often practicable to orient the solar collector at an optimum tilt angle and to correct the tilt from time to time. Over the last few years, many authors have presented models with which to predict solar radiation on inclined surfaces. Some of these models apply to specific cases; some require special measurements and some are limited in their scope. These models use the same method of calculating beam and ground reflected radiation on a tilted surface. The only difference exists in the treatment of the diffuse radiation. The approximation commonly used for converting the diffuse component value for a horizontal surface to that for a tilted one is that sky radiation is isotropically distributed at all times [1 3]. However, theoretical as well as experimental results have shown that this simplifying assumption is generally far from reality [4]. Thus, it appears that sky radiance should

3 626 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) be treated as anisotropic, particularly because of the strong forward scattering effect of aerosols [5 8]. Reviews on transforming data recorded by horizontal pyranometers to data that would have been received by tilted surfaces are given by Refs. [9 15]. From the previous reviews, one concludes that there is a wide range of models recommended by different investigators for year round application. This paper deals with the optimum slope and orientation of a surface receiving solar radiation. A comparative study on several inclined models is presented, and the implication for determination of the optimum tilt angle is discussed. We begin with measured hourly global and diffuse radiation received on a horizontal surface. These quantities are then transposed onto an inclined plane by a mathematical procedure. The accuracies of the models are then compared on the basis of statistical error tests, and the most accurate model is recommended. The optimum tilt angle was computed by searching for the values for which the total radiation on the collector surface is a maximum for a particular day or a specific period. The analyses are also extended to predict the optimum tilt angle based on declination, daily clearness index and ground reflectivity. 2. Measuring station and data corrections The measuring station is located on the roof of the National Research Institute of Astronomy and Geophysics (latitude N and longitude E) where it is located on a hilltop site about 30 km south of Cairo in desert surroundings. All sensors are installed on the roof top in a position relatively free from any external obstruction and readily accessible for inspection and general cleaning. In this study, the broadband filter method was used to measure the quantities of normal radiation at different bands. The filters used in this study are Schott filters (2 mm thick) whose cutoff wavelengths were determined using a spectrophotometer. These filters were arranged on a rotatable disk and mounted on an Eppley normal incidence pyrheliometer. Their main characteristics are given in Table 1. Solar radiation on a horizontal plane was monitored with a high precision pyranometer that is sensitive in the wavelength range from 300 to 3000 nm. The measurement of ground reflected radiation involves a pyranometer mounted horizontally and facing downwards at two meters from the ground. The diffuse radiation was measured by a pyranometer equipped with a special shading device to exclude direct radiation from the sun. The shadow ring is painted black to minimize the effect of multiple reflections. Additional data recorded on a vertical surface facing south every 10 min during daylight from January to December 2003 in Colorado, USA, is used to reinforce the results from the primary data set. All data sets were subjected to various quality control tests. Three types of data checks were performed to identify missing data, data that clearly violates Table 1 Filter characteristics Old name Filter reference Interval bands (lm) Filter factor OG1 OG RG2 RG RG8 RG Clear GG

4 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) physical limits and extreme data. Hours when the data was known to be bad or missing were omitted. Secondly, any hour with an observation that violated a physical limit or conservation principle was eliminated from the data set, including: reported hours with a diffuse fraction greater than 1 or beam radiation exceeding the extraterrestrial beam radiation. To eliminate the uncertainty associated with radiation measurements at large incidence angles, hours with a zenith angle larger than 80 were eliminated. The final data set was constructed from the measured data that passed all of the quality control checks. 3. Estimation methodology 3.1. Estimation of beam radiation on an inclined surface The models discussed here all share the same formulation for the beam and ground reflected components. For a surface oriented in any direction with respect to the meridian, the trigonometric relation for the incidence angle h has been given by Kondratyev [16] and in detail by Coffari [17]. This relation can be written in the following form: cos h ¼ ðsin / cos b cos / sin b cos cþsin d þ ðcos / cos b þ sin / sin b cos cþcos d cos x þ cos d sin b sin c sin x; ð1þ where the meanings of the different symbols are as given in the nomenclature. There are several commonly occurring cases for which Eq. (1) is simplified. For a fixed surface sloped toward the south or north, that is, with a surface azimuth angle c of 0 or 180 (a very common situation for fixed flat-plate collectors), the last term drops out. cos h ¼ ðsin / cos b cos / sin bþsin d þ ðcos / cos b þ sin / sin bþcos d cos x ¼ sin d sin ð/ bþþcos d cos x cosð/ bþ. ð2þ For horizontal surfaces, the angle of incidence, h, is the zenith angle of the sun, h z. Its value must be between 0 and 90 when the sun is above the horizon. For this situation, b = 0 and Eq. (2) becomes cos h z ¼ cos / cos d cos x þ sin / sin d. ð3þ The geometric factor, R T which is the ratio of the beam radiation on the tilted surface, I BT to that on a horizontal surface at any time, can be calculated exactly by appropriate use of Eq. (1). R T ¼ cos h cos h z ¼ sin d sinð/ bþþcos d cos x cosð/ bþ. ð4þ cos / cos d cos x þ sin / sin d Therefore, the hourly beam radiation received on an inclined surface can be expressed as I BT ¼ I BH R T ¼ðG H I DH ÞR T. It should be noted that at the grazing angles (i.e., just at sunrise or at sunset), R T can change rapidly and may approach infinity or zero because both the numerator and the denominator are small numbers. This depends on the slope, latitude and date. ð5þ

5 628 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) Estimation of ground reflected radiation on an inclined surface In this section, we are going to focus on ground reflected radiation, which is also significant and which can sometimes reach values of the order of 100 W/m 2 for a vertical plane [18 20]. Depending on the earthõs cover, the albedos for beam and diffuse radiation, q B and q D, respectively, may not be identical. The total radiation reflected by the ground cover into an entire hemisphere can be written as (I BH Æ q B + I DH Æ q D ). We address here two particular cases: (1) isotropic reflection and anisotropic reflection. Isotropic reflection: Isotropic reflection usually occurs when the ground cover is a perfectly diffuse reflector, such as a concrete floor, and the horizon is unobstructed. Under the isotropic condition, the fraction of solar radiation incident on the inclined surface is given by the above quantity multiplied by the configuration factor from the ground to the inclined surface. Thus I GT ¼ 1 2 ði BH q B þ I DH q D Þð1 cos bþ. ð6þ When the reflectances for beam and diffuse radiation are identical, we can use a common albedo, q. Under such a condition, Eq. (6) reduces to the following: I GT ¼ 1 2 qð1 cos bþg H. ð7þ Anisotropic reflection: Other authors have proposed anisotropic ground reflectance models [21,22], but lack of experimental data has hampered their validation. Under an anisotropic condition, Eq. (7) should be multiplied by the following factor 1 þ sin 2 ðh z =2Þ ðjcos DjÞ, where D is the azimuth of the tilted surface with respect to that of the sun. Consequently, with anisotropic reflection produced under clear skies, the ground reflected radiation incident on an inclined surface can be written as follows: 1 I GT 2 qð1 cos bþg H ½1þsin 2 ðh z =2ÞŠ ðjcos DjÞ. ð8þ However, the matter is complicated if the reflections are specular. To handle such a situation, recourse has to be made to specialized literature such as Siegel and Howell [23]. Instead of the complex rules and mathematical routines mentioned above, the ground reflected radiation in the present study is assumed to be isotropic Estimation of sky diffuse radiation on an inclined surface Description of the Tamps and Coulson s model Diffuse irradiance is difficult to determine accurately with the simple parameterization methods that were used to calculate direct normal irradiance in the previous section, since its spatial distribution is generally unknown and time dependent. Three diffuse subcomponents are used to approximate the anisotropic behavior of diffuse radiation. The first is an isotropic part received uniformly from the entire sky dome. The second is circumsolar diffuse radiation resulting from forward scattering of solar radiation and concentrated in the part of the sky around the sun. The third, referred to as horizon brightening, is concentrated near the horizon and is most

6 pronounced in clear skies. Several models have been proposed to estimate the diffuse radiation on a tilted surface (not all of which account for these three diffuse subcomponents). Tamps and Coulson [6] developed a model from readings taken in clear skies. They introduced geometrical terms into the isotropic model to take into account the brightening of the sky in the region of the sun and at the horizon. I DT ¼ I DH 1 þ cos b 2 1 sin 3 b 2 ð1 þ cos 2 h sin 3 ð90 cþþ. From the previous equation, one concludes that Tamps and Coulson observed that the clear sky condition can be depicted by modifying the basic isotropic formulation by two factors: the factor ð1 þ cos 2 h sin 3 ð90 cþþ to take into account the increased diffuse radiation in the circumsolar region and the factor ð1 sin 3 ðb=2þþ to take into account the brightening of the sky near the horizon. It should be pointed out that a weakness of this model is that for a horizontal collector, the expression does not reduce to I DH Description of the Perez s model PerezÕs model is composed of three distinct elements: (1) A geometrical representation of the sky dome, (2) A parametric representation of the insolation conditions and (3) A statistical component linking the two. The governing equation is I DT ¼ I DH H.K. Elminir et al. / Energy Conversion and Management 47 (2006) þ cos b a 1 ð1 F 1 ÞþF 1 2 a 2 a 1 ¼ maxð0; cos hþ; a 2 ¼ maxðcos 85 ; cos h z Þ; þ F 2 sin b ð9þ ; ð10þ where a 1, a 2 are the solid angles occupied by the circumsolar region, weighted by its average incidence on the slope and the horizontal, respectively and F 1, F 2 are the coefficients of circumsolar and horizon brightness, respectively (dimensionless).these multiplicative factors set the radiance magnitude in the two anisotropic regions relative to that in the main portion of the dome. The degree of anisotropy of the model is a function of only these two terms. The model can go from an isotropic configuration (F 1 = F 2 = 1) to a configuration incorporating circumsolar and/or horizon brightening. n h p io F 1 ¼ max 0; F 11 þ F 12 D þ F 13 h z ; ð12þ h F 2 ¼ F 21 þ F 22 D þ F 23 h z p 180 ð11þ i 180 ; ð13þ D ¼ m I DH ; ð14þ I on where m is the air mass (dimensionless) and I on is the extraterrestrial irradiance at normal incidence (W/m 2 ). The required coefficients, F ij are obtained from Perez et al. [15]. For a vertical surface (b = 90 ), Eq. (10) becomes 1 I T¼90 ¼ I DH 2 ð1 F a 1 1ÞþF 1 a 2 þ F 2. ð15þ

7 630 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) Description of the Bugler model The governing equation given by Bugler is I DT ¼ 0.5 I DH 0.05 I BNðbÞ ð1þcos bþþ0.05 I BN ðbþcos h; cos h z ð16þ where I BN (b) is the beam irradiation normal to a plane of slope b and is easily calculated by Iqbal [19] in W/m Models evaluation 4.1. Methods of statistical comparison To evaluate the accuracy of the estimated data from the three models described above, we used three statistical indicators: normalized mean bias error, NMBE, normalized root mean square error, NRMSE and correlation coefficient, CC defined as ( NRMSE ¼ 1 1 X N )1 2 ðy i X i Þ 2 X N ; ð17þ i¼1 where N is the number of data points, Y i is the predicted data point and X i is the observed data point. An error of zero would indicate that all the output patterns computed by the model perfectly match the expected values. NMBE ¼ 1 X 1 N X N i¼1! ðy i X i Þ. ð18þ The NRMSE provides information on the short term performance of a model by allowing a term by term comparison of the actual difference between the estimated value and the measured value. The NMBE provides information with respect to over or under estimation of the estimated data. A positive value indicates an over estimation in the values, while a negative one indicates under estimation, and a low NMBE value is desired. Finally, we used the CC factor to test the linear relation between calculated and measured values. P N i¼1 CC ¼ Y i Y X i X nhp ih N i¼1 ðy i Y Þ 2 PN i ; ð19þ i¼1 ðx i X Þ 2 o1 2 where Y is the predicted mean value and X is the measured mean value Results of models validation The results for south facing surfaces are given in Tables 2 4. It is noted that the NRMSE for all three models increase as the slope of the collector increases but remain in a domain of errors for

8 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) Table 2 Normalized root mean square errors for global radiation received on inclined surfaces Perez modelõs [NRMSE, in (%)] Bugler modelõs [NRMSE, in (%)] Tamps and CoulsonÕs [NRMSE, in (%)] S S S January February March April May June July August September October November December Table 3 Normalized mean bias errors for global radiation received on inclined surfaces Perez modelõs [NMBE, in (%)] Bugler modelõs [NMBE, in (%)] Tamps and CoulsonÕs [NMBE, in (%)] S S S January February March April May June July August September October November December which these relations can be applied with good accuracy. Inspecting the results, it is apparent that the models agree quite well with each other during the summer months. They deviate from each other in the winter months when the effects of the difference in the diffuse radiation parameterization are at their maximum. The NRMSE results indicate that the Perez and Tamps anisotropic models show similar performance on an overall basis, but BuglerÕs model exhibits much larger error. The NMBE results show that PerezÕs and BuglerÕs models substantially under predict the irradiance incident on an inclined surface, and the Tamps model considerably over predicts the irradiance incident on an inclined surface on an overall basis. Our findings confirm the observation that PerezÕs model describes the irradiance on inclined planes more accurately than BuglerÕs or TampsÕs models.

9 632 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) Table 4 Correlation coefficients values as a function tilt angles Perez modelõs correlation coefficient Bugler modelõs correlation coefficient Tamps and CoulsonÕs correlation coefficient S S S January February March April May June July August September October November December Optimizing the tilt angle of solar collectors 5.1. Optimum tilt angle based on geometric factor Many investigations have been conducted to determine, or at least estimate, the best tilt angle for solar collectors. Some of these are, for example, / + 20 [24], / + (10! 30 ) [25] and / + 10 [26], whereas some researchers suggest two values for the tilt angle, one for summer and the other for winter, such as / ±20 [27] where / is the latitude, + for winter and for summer. In the past few years, computer programs have been used, and the results have shown that the optimum tilt angle is almost equal to the latitude [28]. It is clear from the previous review that there is no definite value of the tilt angle that can be recommended. In Section 4, some results concerning the accuracy of models to estimate irradiance on inclined planes is tested by comparing the predictions to measurements of various tilt and azimuth angles. PerezÕs model is found to perform significantly better than the other two. The equations describing the instantaneous total insolation on a tilted surface under the assumption of anisotropic distribution of the sky diffuse radiation was presented in the following form: 1 þ cos b G T ¼ðG H I DH ÞR T þ I DH 2 a 1 ð1 F 1 ÞþF 1 a 2 þðf 2 sin bþ þ 1 2 q ð1 cos bþg H. ð20þ After computing the total insolation on an inclined plane, it is useful to determine the effect of slope, albedo and orientation by calculating the geometric factor, R b, where R b ¼ Total radaiation on the tilted surface Total radiation on a horizontal surface ¼ G T G H. ð21þ

10 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) We have calculated the geometric factor, R b, for south facing surfaces inclined at (5! 90 ) from the horizontal position. All calculations are based on the anisotropic sky diffuse radiation model, and ground reflection is assumed perfectly diffuse. From Table 5, we observe that at high tilt angles and during summer months, R b may be less than one. On the other hand, the monthly average daily values of b Opt are predicted and listed in Table 6. The optimum tilt angle was found by searching for values for which R b is a maximum for a specific period. Since changing the tilt angle to its daily and monthly optimum values throughout the year does not seem to be practical, Table 5 Monthly values of the geometric factor R b as a function tilt angle Solar collectorõs tilt angles Panel A January February March April May June July August September October November December Panel B January February March April May June July August September October November December Table 6 Predicted values of the optimum tilt angle for the Helwan site January February March April May June July August September October November December

11 634 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) another possibility, such as changing the tilt angle once per season was considered. Using the predicted values given by Table 6 the optimum tilt of a flat-plate collector for use during the winter season is approximately for the Helwan site. The optimum tilt of a flat-plate collector used during the summer months is 15 for the same site. Finally, the optimum tilt angle of a flat-plate collector used continuously throughout the year is and oriented towards the south for the Helwan site Optimum tilt angle based on clearness index The clearness index, K I, defined as the ratio of earthõs surface global irradiance over the extraterrestrial global irradiance, was introduced as a norm to characterize the optimum tilt of south facing collectors at a given point in time when only the global irradiance is known. According to Elsayed [29], one can show that the optimum tilt angle depends on several parameters as indicated below b Opt ¼ f ð/; N; c; K I ; qþ; ð22þ where N is the day that represents the month under consideration. The ground reflectivity q is about 0.2 for ground without snow and 0.7 for ground covered by snow as recommended by Duffie and Beckman [2]. The monthly average clearness index K I hardly ever falls outside the range in most locations around the world. The following correlation is developed and found to predict the values of b Opt with an accuracy of about 6 for / =20 40 and an accuracy of about 10 for / = [29]. b Opt ¼ð6 4.8K I þ 0.86K 0.27 I / þ / 2 Þ þð31k 0.37 I þ 0.094K 0.46 I / þ K 1.7 I / 2 Þcos N þ 11.5 ; ð23þ where N is the Julian day of the mean day of each month. Using the previous formula, the predicted optimum tilt angle for each month at the Helwan site was tabulated in Table 7. It is noted from Table 7 that the optimum tilt angle for the month of March is approximately equal to the latitude angle, /. For this month, a solar collector tilted at an angle equal to the latitude will receive solar radiation nearly normally. Similarly, the optimum angle recorded for September is approximately equal to the latitude of the location. The yearly optimum tilt angle is 26.7 for the Helwan site and oriented towards the south. These results seem to agree quite well with the predicted values obtained from the previous section on a monthly and yearly basis. Table 7 Predicted values of the optimum tilt angle for the Helwan site January February March April May June July August September October November December N K I b Opt

12 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) Table 8 Predicted values of the optimum tilt angle for the Helwan site January February March April May June July August September October November December Eq. (24) Eq. (25) Eq. (26) Eq. (27) Optimum tilt angle based on declination factor Various equations listed in Tiris and Tiris [30] were tested here to recommend the one that is most accurate for estimating the optimal collector slope for the Helwan site based on the declination factor. b ¼ ðdþ; b ¼ ðdþ ðdþ 2 ; b ¼ ðdþ ðdþ ðdþ 3 ; b ¼ þ 25.79ðK 1 Þ 1.49 ðdþ. ð24þ ð25þ ð26þ ð27þ Table 8 shows the results for the above mentioned equations, and we can observe that the agreement is good between the results obtained by Eqs. (24), (25) and (27) with respect to the previous results obtained in Sections 5.1 and 5.2. Also, it is noticeable that the optimum tilt angle for June is negative; the negative sign determines the orientation of the solar collector, which means that the solar collector is faced towards the north. A positive sign indicates that the solar collector is directed toward the south. 6. Experimental verification In this section, experimental verification was presented based on weekly measurements obtained at a multi-position test facility [31]. The tests were started at noon time when the PV module is vertical and facing north (i.e., the tilt angle is 90 ). The tests continue to the horizontal mode and are completed to south facing with the tilt angle of +90. At every five degrees variation Table 9 Comparison between predicted and measured values of the optimum tilt angle January February March April May June July August September October November December R b K I Eq. (24) Eq. (25) Eq. (27) Experiment

13 636 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) in the tilt angle, the PV module parameters (i.e., current and voltage) were recorded. Using a computer program for fitting the data smoothly for each day of the measurements, the optimum tilt angles are determined. Here, the optimum tilt angle is the angle at which the maximum average output is obtained. Regarding the results obtained from Table 9, the experimental optimum tilt angles seem to agree quite well with the predicted values obtained from the previous sections except in the winter where the variation of diffuse fraction is a maximum. 7. Conclusions Irradiation data recorded on vertical surfaces facing south and at 40 every 10 min during the daylight hours from January to December 2003 have been compared with the estimated solar radiation from inclined surface models. The results show that PerezÕs model most accurately reproduces the variation in irradiation on all vertical surfaces. Therefore, we used Perez model to determine the best inclination for a south facing solar collector. The results of computations show that the optimum tilt angles with respect to the maximum daily insolation amounts incident on the collector surface exhibit a strong seasonal trend. During the winter months, the maximum daily insolation is received on a south facing collector with tilt angles around 43.33, whereas during the summer, the maximum daily insolation is incident on a nearly horizontal surface. The maximum yearly solar radiation can be achieved using a tilt angle approximately equal to a siteõs latitude. On a daily basis, the optimum tilt angles of south facing collectors may vary within relatively wide limits. In conclusion, summarizing the previous considerations, the optimum tilt angle of solar radiation collection systems located in Helwan follow the general rule applied by many researchers that yearly optimum tilt is about (/ ±15 ) where / is the latitude of the location and where plus and minus signs are used in the winter and summer, respectively. References [1] Liu B, Jordan R. Daily insolation on surfaces tilted towards the equator. Trans ASHRAE 1962;67. [2] Duffie J, Beckman W. Solar engineering of thermal processes. New York: Wiley; [3] Gopinathan K. Solar radiation on inclined surfaces. Solar Energy 1990;45. [4] Dave J. Validity of the isotropic distribution approximation in solar energy estimations. Solar Energy 1977;19. [5] Bugler J. The determination of hourly insolation on a tilted plane using a diffuse irradiance model based on hourly measured global horizontal insolation. Solar Energy 1977;19. [6] Tamps C, Coulson L. Solar radiation incident upon slopes of different orientation. Solar Energy 1977;19. [7] Hay J. Calculation of monthly mean solar radiation for horizontal and tilted surfaces. Solar Energy 1979;23. [8] Klucher M. Evaluation of models to predict insolation on tilted surfaces. Solar Energy 1979;23: [9] Page J. Prediction of solar radiation on tilted surfaces. In: Solar energy R&D in the European Community, Series F. 1986;(3). [10] Hay J, McKay D. Calculation of solar irradiances for inclined surfaces: verification of models use hourly and daily data, IEA, Solar Heating and Cooling IEC Standards 891 and 1215, Bureau Central de la Commission Electrotechnique Internationale, Genéve, [11] Reindl D, Beckman A, Duffie A. Evaluation of hourly tilted surface radiation models. Solar Energy 1990;45. [12] Perez R, Stewart R, Arbogast C, Seals J, Scott J. An anisotropic hourly diffuse radiation model for sloping surfaces: description, performance validation, site dependency evaluation. Solar Energy 1986;36.

14 H.K. Elminir et al. / Energy Conversion and Management 47 (2006) [13] Perez R, Seals R, Ineichen P, Stewart R, Menicucci D. A new simplified version of the Perez diffuse irradiance on tilted surfaces. Solar Energy 1987;39. [14] Perez R, Seals R, Stewart R. Modeling irradiance on tilted planes: a simpler version of the Perez model US wide climate/environmental evaluation, Advances in Solar Energy Technology. In: Proc ISES Solar Word Congress, Hamburg, FRG, vol. 4, [15] Perez R, Ineichen P, Seals R, Michalsky J, Stewart R. Modeling daylight availability and irradiance components from direct and global irradiance. Solar Energy 1990;44. [16] Kondratyev Y. Radiation in the atmosphere. New York: Academic Press; [17] Coffari E. The sun and celestial vault, solar energy engineering. New York: Academic Press; 1977 [Chapter 2]. [18] Robinson N. Solar radiation. New York: Elsevier Publishing Company; [19] Iqbal M. An introduction to solar radiation. Canada: Academic Press; [20] Ineichen P, Perez R, Seals R. The importance of correct albedo determination for adequately modeling energy received by tilted surfaces. Solar Energy 1987;39:221. [21] Gardner C, Nadeau C. Estimating south slope irradiance in the Arctic a comparison of experimental and modeled values. Solar Energy 1988;41. [22] Gueymard C. An anisotropic solar irradiance model for tilted surfaces and its comparison with selected engineering algorithms. Solar Energy 1987;38. [23] Siegel R, Howell J. Thermal radiation heat transfer. New York: McGraw-Hill; [24] Hottel C. Performance of flat-plate energy collectors. Space heating with solar energy. Proc course symp. Cambridge: MIT Press; [25] Löf G, Tybout A. Cost of house heating with solar energy. Solar Energy 1973;14. [26] Kern J, Harris I. On the optimum tilt of a solar collector. Solar Energy 1975;17. [27] Yellott H. Utilization of sun and sky radiation for heating and cooling of buildings. ASHRAE J 1973;15. [28] El-kassaby M. The optimum seasonal and yearly tilt angle for south-facing solar collectors. ISES Solar World Congress, Hamburg, Germany, [29] Elsayed M. Optimum orientation of absorber plates. Solar Energy 1989;42. [30] Tiris M, Tiris C. Optimum collector slope and model evaluation: case study for Gebze, Turkey. Energy Conver Manage 1997;39. [31] Mosalam Shaltout M, Hassan A, Ghitas A. Optimum tilt of PV systems for electric power generation at Cairo. In: 11th E.C. PV Solar Energy Conference, October, Switzerland, 1992.