Solar EnergyVol 32, No. 5. pp. 677 679. 1984
0038 092X/84 $3.00+ .00 ~ 1984PergamonPress Ltd.
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TECHNICAL NOTE Comparison of two diffuse sky radiation models W. D. TURNERt a n d M . SALIM~ Mechanical Engineering Department, Texas A & M University, College Station, TX 77843, U.S.A.
(Received 17 February 1983; accepted 15 July 1983) INTRODUCTION Many models have been developed to predict the diffuse sky solar radiation since the original work of Liu and Jordan[I]. A review of several of the older models and some of the newer models is contained in Duffle and Beckman. The diffuse solar radiation models may be hourly, daily, or monthly models, and are typically based on the measurement of global solar radiation (total solar radiation on a horizontal surface). The other parameter typically used in the model is usually based on extraterrestrial solar radiation on a horizontal surface. Recent models of this type include the Erbs Model , the Orgill-Hollands Model, the Collares-Pereira and Rabl Model, and the Blytheville, A R Model developed by Mujahid and Turner . The Erbs, Orgill-Hollands, and Blytheville Models are hourly models using kr, the ratio of hourly measured global to hourly extraterrestrial solar radiation on a horizontal surface, as a parameter in the model. The Collares-Pereira and Rabl Model is a daily prediction model. Comparisons of several of these models with recorded solar radiation data at Blytheville, AR, are made in [6, 7]. Instead of using extraterrestrial solar radiation as a parameter, some models will use some form of clear day global radiation. The Bugler Model, and the Stauter and Klein Model are two recent hourly models which use this approach. Comparisons of the Bugler Model with recorded data from Blytheville, AR, were made in . The Stauter-Klein Model was not compared, however, and the purpose of this technical note is to compare the accuracy of the Stauter-Klein Model with recorded data from Blytheville, AR and to present a slightly different model which better matches the recorded data.
could predict the long-term performance of the MCCC project in simulation programs. These models have been reported previously [6, 7, 9, 10]. COMPARISON OF STAUTER-KLEIN MODEL WITH BLYTHEVILLE DATA The Stauter-Klein Model for diffuse solar radiation presented in  is based on Hottel's clear sky model . The plotting parameters are Id/l VS I/lc, defined as follows (the notation used is from Duffle and Beckman): where I d is diffuse sky radiation (predicted), I is measured hourly global radiation, and I~ is calculated "clear day" hourly global radiation from Hottel's model. The correlation equations for the Stauter-Klein Model are as follows 
1.0-0.11/1 c 1.11 + 0.0396 (I/Ic) -0.789 (1/1,.)2 0.20
for 0<_1/lc<0.48 for 0.48 < I/1~ < 1.10 for I/Ic>_ 1.10.
The use of kr (based on extraterrestrial solar radiation) for modelling purposes is generally preferred because it eliminates having to define a "clear day". There is obviously going to be a difference between clear days from one location to another, and some knowledge of the local climate as well as geographical variations, such as latitude and altitude, will affect the standard clear day. Hottel's is one such approach. ASH RAE has another. In any event, it is important to verify the models to determine their accuracy. In this model validation, the approch used was to calculate the diffuse component, la, from measurements of the beam and global solar radiation. The equation used was
SOLAR RADIATION MEASUREMENTS AT BLYTHEVILLE, ARIZONA From April 1978 to April 1980, a weather station was maintained at Blytheville, Arkansas site of the 240 KW~ Mississippi County Community College (MCCC) Solar/Thermal Photovoltaic Project. The station was maintained by the University of Arkansas-Fayetteville. Details of the weather station are included in [9, 10], and will not be repeated here. It is sufficient to state that measurements were made each minute of direct normal, global, and diffuse sky radiation, using, respectively, an Eppley Normal Incidence Pyrheliometer (NIP), an Eppley Precision Spectral Pyranometer (PSP), and an Eppley PSP with a shadow band. The data were recorded on cassette tapes and were analyzed and transferred to nine-track tapes at the University of Arkansas. The sensors were all class 1 devices, and the instrumentation and data package was a sophisticated system built by E.G. & G., Las Vegas, NV, and provided by the United States Department of Energy for the project. Although the recording period was only two years, it is felt that the data form a very reliable base. In addition to monitoring and analyzing the solar data, models were developed which tAssociate Professor and Member ISES. ;~Graduate Student. 677
ld = / - lb, cos 0.
where I a is diffuse solar radiation (calculated), 1 is measured global, lb, is measured direct normal, and, 0. is solar zenith angle. The data were screened for obvious errors such as a misaligned NIP, and the calculated diffuse data were then compared to the diffuse radiation predicted by the Stauter-Klein Model, based on the measured global radiation. The only data used were from 9 a.m. to 3 p.m. (solar time) to eliminate the very low sun angles in the winter. The measured diffuse data were not used in the model comparisons because of the errors introduced in the shadow band correction factors. Figure 1 shows the average difference between 1Jl calculated from the recorded data and 1Jl predicted from the Stauter-Klein Model plotted vs 1/I C. There were large plus and minus differences at some values of I/I~, but the best curve through the data produced very small average inaccuracies. The average error difference up to I/I,. of 0.55 was less than 0.025. The Stauter-Klein Model generally underpredicted the diffuse component at Blytheville up to I/I,. values of about 1.05. The maximum average error difference occurred at an 1/1,. value of about 0.75, but it was only about 0.035. The Staute>Klein Model fit the Blytheville data very well.
Technical Note 05
Fig. 1. Comparison of the recorded diffuse and predicted diffuse from Stauter-Klein Model.
02 01 0
110 ,i, ~12
DETERMINATION OF A DIFFUSE MODEL BASED ON HOTTEL'S STANDARD CLEAR DAY
In order to determine where the differences were and why the Stauter-Klein Model slightly underpredicted the diffuse, a similar model based on the Blytheville data was developed. The initial approach was to attempt to curve tit the entire region from I/I C= 0 to I/Ic = 1.I, but a single polynomial equation could not model the nearly fiat regions of Id/l for both low and high values ofl/l~. Linear regions were selected (after viewing all computer-plotted data) at the beginning and end values of I/lo and various polynomials were plotted to determine the "best fit" in the non-linear region. The result is the following set of equations for Id/L
t l lull =
for I/Ic < 0.48
Fig. 2. Comparison of Stauter-Klein Model with Blytheville model.
well and is a reasonable model to use based on a standard clear day. An alternative approach would be to use the model developed herein from the Blytheville solar radiation data. Additional model validations are necessary before a "best" model can be selected, based either on a clear day standard or on exterrestrial solar radiation on a horizontal surface. It is important, however, to validate the models against the various data bases in an attempt to verify their accuracy or place limitations on their usage.
+ 2.638(1/1,,)2 - 1.574(1/I~)3 for 0.48 < I/I,. < 1.10
Acknowledgements--The data for this project were orginally
obtained by the senior author while at the University of Arkansas-Fayetteville under Subcontract to the Mississippi County Community College, Dr. Harry V. Smith, President. The MCCC solar project was funded by DOE Grant No. EG-77-05-5565. Computer time for the modeling in this technical note was provided by the Mechanical Engineering Department, Texas A & M University.
The Stauter-Klein Model, eqn (1), looks very different from the above set of equations, but the two models are, in fact, very similar. Figure 2 is a comparison of the two models. The reasons for the underprediction of the diffuse component can be readily seen. The Stauter-Klein Model is lower than the Blytheville Model except for very clear days, corresponding to I/l~ > 1.05. The discrepancies in the linear portions of the curve are partially explainable. From diffuse models developed in , based on kr as the parameter, the diffuse component was determined to be a function of solar altitude angle, a logical result when one considers the difference in air mass between a solar altitude angle of 20° as opposed to 60 °. In , the horizontal value of la/l for very clear days was much higher for low sun angles than for high sun angles. For this particular study, the lower sun angles have been omitted, since only the three hours on either side of solar noon were considered. It is very likely that the value of 0.18 for fall could, in fact, increase to around 0.20. The higher straight line at the lower values of I/I c cannot be explained. There is an overwhelming amount of data which indicates la/l recorded is approximately 1.0 for very low values of I/I c (which indicates no beam component). Based on the Blytheville data, it would appear that the Stauter-Klein equation should be slightly higher for small values of I/1,.. Comparisons were also made between the Stauter-Klein Model and the Blytheville data for each month. The Stauter-Klein Model was extremely good at modeling the Blytheville data from May through September, and especially good for the partly clear to clear days, i.e. I/I,. from 0.6 to 0.9. The model performed the worst in the winter months, December through February. CONCLUSIONS The diffuse sky radiation predicted from the Stauter-Klein Model matched the Blytheville, Arkansas diffuse data quite
1. B. Y. H. Liu and R. C. Jordan, The interelationship and characteristic distribution of direct, diffuse and total solar radiation. Solar Energy 3, 1 (1960). 2. J. A. Duffle and W. A. Beckman, Solar Engineering of Thermal Processes. Wiley, New York (1980). 3. D. G. Erbs, R. C. Stauter, and J. A. Duffle, The basis and effects of inaccuracies in diffuse radiation correlations. Proc. AS/ISES 1980 Ann. Meet. pp. 1429-1433. Phoenix, Arizona (1980). 4. J. F. Orgill and K. G. T. Hollands, Correlation equation for hourly diffuse radiation on a horizontal surface. Solar Energy 19, 357 (1977). 5. M. Collares-Pereira and A. Rabl, The average distribution of solar radiation-correlations between diffuse and hemispherical and between daily and hourly values. Solar Energy 22, 155 (1979). 6. A. M. Mujahid and W. D. Turner, Diffuse sky measurements and determination of corrected shadow band multiplication factors. ASME Paper 80-WA/SOL-6 (1980). 7. A. M. Muahid, Analysis and modeling of solar energy data from the Mississippi County Community College Site, Blytheville, Arkansas. PhD Thesis, University of Arkansas-Fayetteville (1981 ). 8. J. W. Bugler, The determination of hourly insolation on an inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation. Solar Energy 19, 477 (1977).
Technical Note 9. W. D. Turner and A. Muijahid, Solar Energy measurements and solar radiation model for Blytheville, Arkansas. Presentedat ISES Meeting, Atlanta, Georgia (1979). 10. A. Mujahid and W. D. Turner, Diffuse sky measurements and model. ASME Paper 79-WA/SOL-5 (1979).
11. H. C. Hottel, A simple model for estimating the transmittance of direct solar radiation through clear atmospheres. Solar Energy 18, 129 (1976). 12. Handbook of Fundamentals. American Society of Heating. Ventilating and Air Conditioning Engineers (1977).