# Evaluation of Empirical Formulae for Estimating Global Radiation at Jos

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2 2 1 o N cos cos85 tan tan (2) 15 o latitude of location north positive 9.95 for Jos (Gelshik, 1997) and is the solar declination given by Cooper (1969): 23.45sin 360 n 284 /365 (3) n is the day of the year: 1 n 365. Values of Angstrom Model Stm for Jos were adopted from Musa (2010) and used in equation (1) to obtain estimates of H. The Angstrom equation for H is (Angstrom, 1924): n H H o a b (4) N where H o is the monthly average of the radiation outside of the atmosphere for the same location and a and b are constants dependent on site. Their values are determined by regression. Ojosu (1990) obtained these values as a = and b = for Jos. The Angstrom equation for Jos can therefore be written as: H = (5) H o where σ is the monthly average daily value of the fraction of possible sunshine hours given by: n / N (6) n is the actual sunshine duration for the day. The extraterrestrial radiation H o is calculated from (Duffie & Beckman, 1991): H o 24 x 3600 s I sc s cos360n / 365 cos cos sin sin sin (7) I is the solar constant and equals 1367Wm 2 (Tiwari, 2006). sc Reitveld s Model Reitveld s correlation for calculating the average insolation H is (Reitveld, 1978): H / H o (8) Reitveld s equation is believed to have universal validity (Halouani, Nguyen & Vo-ngoc, 1993). IV. Results and Discussion Estimates of the average daily insolation made using the three empirical models, viz. Monteith s, Angstrom s and Rietveld s, are recorded in appendix B. Fig.1 shows the prediction of these models along with the measured values. Monteith s model has its maximum and minimum values simultaneously with the measured data (in February and July respectively) while the maximum and minimum values of Angstrom s model coincides with that of Rietveld s (in November and August respectively). It is also noteworthy that peaks of insolation always occurred in the months of February and November. Figures 2, 3 and 4 show the performance of the models in comparison with the observed values. Note the degree of sloping of these lines and the clustering of the points around the lines. A summary of this comparison is outlined below. Measured Insolation: H G in 2 1 Averages (a) Wet season: 19.2 ± 2.1, APE = 10.9% (b) Dry season: 24.7 ± 1.1, APE = 4.5% (c) 5 year: 21.9 ± 3.3, APE = 15.0% DOI: / Page

3 Monteith s Model: Angstrom s Model: Rietveld s Model: H M in Evaluation of Empirical Formulae for Estimating the Global Radiation at Jos 2 1 (i) Averages (a) Wet: 19.4 ± 2.1, 10.6% (b) Dry: ± 1.7, 6.9% (c) 5 year: 22.1 ± 3.4, 15.2% (ii) Correlation with H G : R 2 = (iii) Errors: MBE = 0.177(0.81%), RMSE = 0.420(1.9%). H A in 2 1 (i) Averages (a) Wet: 18.6 ± 2.5, 13.5% (b) Dry: 23.9 ± 1.3, 5.2% (c) 5 year: 21.2 ± 3.3, 15.7% (ii) Correlation with H G : R 2 = (iii) Errors: MBE = (-2.8%), RMSE = 1.38(6.3%) H R in 2 1 (i) Averages (a) Wet: 17.9 ± 2.5, 13.8% (b) Dry: 23.1 ± 1.2, 5.3% (c) 5 year: 20.5 ± 3.3, 16.0% (ii) Correlation with H G : R 2 = (iii) Errors: MBE = (-6.2%), RMSE = 1.38(8.4%) We note that Monteith s model has the highest value of the coefficient of determination, R 2, while Rietveld s model has the least. In terms of errors, the Rietveld s model is the least accurate whereas Monteith s model is the most accurate. In all these, Angstrom s model comes between the other two models. But while Monteith s model overestimates the average insolation, the other two models underestimate it, each in proportion to its MBE. A drawback of Monteith s model is that it is partly dependent on measurement of radiation: values of the irradiance at solar noon are needed as input before the model can be applied, and these may not be readily available. In general, values of insolation during the wet season are lower than in the dry season. The wet season values, therefore, contribute largely to errors in annual insolation. This can also be inferred from the values of the absolute percentage errors (APE) above. V. Conclusion Out of the three empirical models of insolation that were tested in this work, it has been found that Monteith s model was the most accurate while Angstrom s model was rated second best. Rietveld s model was found to be the least accurate. The values of insolation obtained in this work strongly suggest that solar energy could be utilized in Jos and its environs to meet some energy needs, at least at the domestic level. VI. Recommendations In view of the significance of radiation to our biotic environment, we should equip our observatories with the latest and most reliable instruments for measurement of solar radiation and other related quantities. Further research should be conducted on solar radiation on tilted surfaces at Jos. DOI: / Page

4 Fig.1: Variation of mean daily insolation with months at Jos: models and measured values Fig. 2: Performance of Montieth s model for Jos Fig. 3: Performance of Angstrom s model for Jos DOI: / Page

5 Fig. 4: Performance of Rietveld s model for Jos References [1]. J.L. Monteith, Principles of Environmental Physics Edward Arnold (Publishers) Limited, London [2]. N. Halouani, C.T. Nguyen and D. Vo-ngoc, Calculation of monthly average global solar radiation on horizontal surfaces using hours of bright sunshine, Solar Energy, 50 (3), 1993, [3]. M.J. Gelshik, Estimation of solar radiation using metrological parameters: sunshine hours, relative humidity and maximum air temperature, thesis, University of Jos, [4]. P.T. Cooper, Absorption of solar radiation in solar stills, Solar Energy, 12 (3), [5]. A. Musa, Estimation of average insolation and the radiation balance of the earth-atmosphere system at Jos, thesis, University of Jos, Nigeria, [6]. A. Angstrom, Solar and terrestrial radiation, Quarterly Journal of Royal Meteorological Society, 50, 1924, [7]. J.O. Ojosu, On the bounds for global solar radiation estimates from sunshine hours for Nigerian cities. Nigerian Journal of Solar Energy, 9, 1990, [8]. J.A. Duffie and W.A. Beckman, Solar Engineering of Thermal Processes 2 nd Edition. John Wiley and Sons, Chichester [9]. G.N. Tiwari, Solar Energy: Fundamentals, Design, Modeling and Applications Narosa Publishing House, New Delhi, India [10]. [M.R. Rietveld, A new method for estimating the regression coefficients in the formula relating solar radiation to sunshine hours, Agricultural Meteorology, 19, 1978, [11]. G.N. Tiwari, Solar Energy: Fundamentals, Design, Modeling and Applications Narosa Publishing House, New Delhi, India Appendix A Raw Data for Jos Used for Analysis Appendix A1: Global radiation measured in ml of Gunn-Belllani radiation integrator Months JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC Appendix A2: Records of bright sunshine hours measured in hours Months JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC DOI: / Page

6 Appendix B: Average Daily Insolation at Jos 2 1 Appendix B1: Global radiation H G in MJm day converted from ml of Gunbellani Mean Appendix B2: Monthly average daily insolation 2 1 5year mean = 21.9±3.3 MJm day, APE = 15.1%. H M in 2 1, as predicted by Monteith s model Mean year mean = 22.1±3.4 MJm day, APE = 15.5%, MBE = 0.177, 0.81%, RMSE = 0.420, 1.9%, 2 R = Appendix B3: Monthly average daily insolation H A in 2 1, as predicted by Angstrom s model Mean year mean = 21.2±3.3 MJm day, APE = 15.7%, MBE = , -2.8%, RMSE = 1.38, 6.3%, 2 R = Appendix B4: Monthly average daily insolation H R in 2 1, as predicted by Rietveld s model Mean year mean = 20.5±3.3 MJm day, APE = 16.0%, MBE = -1.38, -6.2%, RMSE = 1.83, 8.4%, 2 R = DOI: / Page

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