Calculation of monthly average daily insolation on tilted, variously oriented surfaces using analytically weighted factors

A new, analytic method for calculating monthly average daily insolation on surfaces with various tilts and orientations is proposed. Beside the usual geometrical expressions concerning diurnal sun path, this method also takes into account changes in atmospheric transmissivity. Results are compared with those obtained by the widely accepted Klein's method also based on daily radiation data, as well as with more detailed and more accurate hourly calculations. Our results for two climatically very different locations in Yugoslavia stand in close agreement with hourly calculations for surfaces of arbitrary orientation and tilt, whereas Klein's method leads to significant error for large azimuth angles.     

Calculation of monthly average daily insolation on tilted,variously oriented surfaces using analytically weighted Rb factors

A new, analytic method for calculating monthly average daily insolation on surfaces with various tilts and orientations is proposed. Beside the usual geometrical expressions concerning diurnal sun path, this method also takes into account changes in atmospheric transmissivity. Results are compared with those obtained by the widely accepted Klein's method also based on daily radiation data, as well as with more detailed and more accurate hourly calculations. Our results for two climatically very different locations in Yugoslavia stand in close agreement with hourly calculations for surfaces of arbitrary orientation and tilt, whereas Klein's method leads to significant error for large azimuth angles.     

Comments on “Calculations of monthly average insolation on tilted surfaces” by S. A. Klein

The amount of solar energy that is intercepted by surfaces of any orientation is estimated from a new model of the clear sky, spatial distribution of solar radiation. The model was developed from measurements made during clear sky conditions and uses direct, isotropic reflected, and anisotropic diffuse radiation. The effects of azimuth, tilt, season, latitude, atmospheric turbidity, and reflectivity of the surroundings were computed using hourly measurements of normal beam and horizontal total radiation at four stations in the United States. A transformation of the co-ordinates of orientation produced a general relationship between orientation and intercepted energy. The general relationship was tested against measurements from six locations in the Northern Hemisphere and was found to be valid. The model is also a better estimator of energy intercepted by a tilted surface than are the more commonly used models.     

Monthly average daily insolation on tilted collectors in Libya

This work presents monthly average total insolation values calculated for various locations in Libya. The insolation values are for actual sky conditions based on sunshine records. Latitudes ranging from 24 to 32°N and tilt angles from 0 to 90° are considered. The presented insolation values show that the optimum value of tilt angle for space heating systems is about latitude plus 15° while that for cooling systems is 0° A combination of horizontal roof and vertical south wall gives more or less uniform insolation throughout the whole year. The maximum annual insolation occurs for tilt angle nearly equal to the latitude.     

Calculation of monthly average insolation on a shaded surface at any tilt and azimuth

Estimation of the monthly average solar radiation on surfaces of arbitrary orientation is necessary for many solar performance prediction programs and is useful for other applications. For passive solar applications, especially, overhangs are often used to seasonally modulate the amount of radiation striking the surface.

Liu and Jordan[1] have developed a method for estimation of monthly average radiation on unshaded tilted surfaces based on horizontal surface data. This method has been extended to unshaded surfaces of arbitrary tilt and azimuth by Klein[2]. Utzinger and Klein[3] have presented a graphical method for estimating monthly average radiation on shaded vertical surfaces, while Jones [4]has offered an analytical method for the same configuration. This paper presents an analytical solution to the calculation of monthly average insolation on shaded surfaces at any tilt and azimuth. Results are comparable among the three methods when shaded vertical surfaces are analyzed.

This analytical method offers an alternative to slower and less accurate numerical integration and to less general regression of numerical integration results for use in solar performance prediction programs.     

Evaluation of models to predict insolation on tilted surfaces

An empirical study was performed to evaluate the validity of various insolation models which employ either an isotropic or an anisotropic distribution approximation for sky light when predicting insolation on tilted surfaces. Data sets of measured hourly insolation values were obtained over a 6-month period using pyranometers which received diffuse and total solar radiation on a horizontal plane and total radiation on surfaces tilted toward the equator at 37° and 60° angles above the horizon. Data on the horizontal surfaces were used in the insolation models to predict insolation on the tilted surface; comparisons of measured vs calculated insolation on the tilted surface were examined to test the validity of the sky light approximations. It was found that the Liu-Jordan isotropic distribution model provides a good fit to empirical data under overcast skies but underestimates the amount of solar radiation incident on tilted surfaces under clear and partly cloudy conditions. The anisotropic-clear-sky distribution model by Temps and Coulson provides a good prediction for clear skies but overstimates the solar radiation when used for cloudy days. An anisotropic-all-sky model was formulated in this effort which provided excellent agreement between measured and predicted insolation throughout the 6-month period.     

Calculation of monthly average global solar radiation on horizontal surfaces using daily hours of bright sunshine

Several statistical models calculating the monthly average global solar radiation on horizontal surfaces using the daily hours of bright sunshine have been extensively evaluated and compared for Canadian regions. Results show that Gariépy's model provides the best overall performance and Hay's model is rated next best, whereas Rietveld's model has been found to be the least accurate. Local performances of models have also been established throughout Canada that show that both Gariépy's and Hay's models perform quite well. Recommended models have been suggested for the studied regions.     

Calculation of monthly mean solar radiation for horizontal and inclined surfaces

The use of daily rather than hourly time intervals in the calculation of the monthly mean values of solar radiation for both horizontal and inclined surfaces has for long been a desirable objective due to a reduction in the required computational effort and more general availability of daily data. This paper presents a method for such calculations, with bright sunshine hours and surface albedo being the only required input data. The incorporation of the effects of multiple reflection between the ground and atmosphere results in relationships used to calculate the diffuse and total solar radiation for a horizontal surface which are largely independent of season and location.The present paper shows that, despite the use of a daily time interval, the solar radiation incident on both horizontal and south-facing surfaces may be calculated with an accuracy generlly associated with instrumental measurements. This result is achieved without the necessity of arbitrarily varying the empirical coefficients to account for variations in climatological conditions at the stations being studied.     

Harmonic analysis of monthly insolation levels in the United States

The purpose of this paper is to analyze independent temporal variance structures in long-term mean monthly insolation data for the conterminous United States. Harmonic analysis is applied to the annual march of insolation levels at 221 stations, and nearly all variance in the data is explained by one harmonic fit. The statistics from the harmonic analysis display coherent spatial patterns that reveal the extent to which radiation levels are affected by synoptic climatological phenomena. The results suggest a method for accurately interpolating insolation estimates using as few as three parameters.     

The two-solarimeter method for insolation on inclined surfaces

In calculating insolation values on inclined surfaces one must usually start from global insolation values on a horizontal surface. If in addition to these values the global insolation on an inclined surface is known, the inaccuracies of such calculations can be substantially reduced. In general this is the case if the global insolation on two differently inclined solarimeters is given. When these global values are given, a mathematical relation between them and the direct and diffuse components on a tilted surface can be established. This method, called the “two-solarimeter method”, is described in this paper. It is shown how the two-solarimeter method for a horizontal surface can be optimized in combination with a correlation procedure. For three vertical surfaces the calculated results are compared with measured values. A detailed error analysis suggests that the two-solarimeter method may become an attractive alternative to the more complicated traditional ways of determining the direct and diffuse components (i.e. measurement of the global and diffuse insolation on a horizontal surface).     

Analysis of monthly average daily global radiation and monthly average sunshine duration at two tropical locations

The diffuse radiation falling on a horizontal surface at Madras, India has been analysed and the results presented. Five years of data have been used for the analysis and a correlation has been established between the daily average diffuse radiation with the monthly average clearness index. It has also been shown that there is an appreciable difference in the ratio of monthly average hourly diffuse to the monthly average daily diffuse radiation with respect to day length.     

Two estimation methods for monthly mean hourly total irradiation on tilted surfaces from monthly mean daily horizontal irradiation from solar radiation data of Ajaccio, Corsica

The sizing of a photovoltaic or a thermal solar system is generally based on monthly mean values of daily solar radiation on tilted surfaces. Many authors have demonstrated that it will be better to use monthly mean values of hourly radiation, particularly taking into account the Sun's position and to predict long-term performances of solar systems. (Liu and Jordan, 1963; Clark et al., 1984). Moreover, for most of the sites around the world, only monthly mean values of daily horizontal total irradiation are available for use in such calculations. We propose, by using well-known correlations in the literature, to estimate these monthly mean values of hourly total irradiation on tilted planes from monthly mean values of daily total horizontal irradiation, using three steps:

• — determination of monthly mean value of hourly total horizontal irradiation;

• — determination of monthly mean value of hourly diffuse horizontal irradiation;

• — determination of monthly mean value of hourly total irradiation on tilted planes.

In the first step, using the Collares Pereira and Rabl correlation, the root mean square error (RMSE) between correlated and experimental calculated data is 8%. In the second step, we used two methods: the first one utilizes the Erbs correlation and the second one is based on a local correlation which has been developed in our centre. Both of them gave identical results with an RMSE lower than 9%. We calculated monthly mean values of hourly total irradiation on three tilted planes (30°, 45° and 60°) and we compared these results with the experimental ones, obtaining a RMSE respectively of less than 10%. The method is then validated by these results.     

Hourly vs daily method of computing insolation on inclined surfaces

Insolation on south-facing inclined planes has been computed using hourly values of total and diffuse radiation, obtained from experimental data. Such a computation procedure is then compared with the widely used method by Liu and Jordan for obtaining daily insolation on surfaces tilted toward the equator. Very small differences are noted between the results obtained by the two methods. These differences are mainly due to three factors; (a) Liu and Jordan's formulation uses a theoretical day-length while the hourly method uses day-length as incicated by the radiation data; (b) hourly method takes into account the asymmetries of total and diffuse radiation around solar noon while the daily method implicitly assumes symmetry of the same; (c) the daily method assumes uniform atmospheric transmissivity to beam radiation throughout the day. On the other hand, the hourly method assumes constant atmospheric transmissivity for one hour only.     

Solar climates of the United States based on long-term monthly averaged daily insolation values

The purpose of this paper is to develop and analyze an objective classification and regionalization scheme of solar climates for the conterminous United States. The use of principal components analysis and Euclidean distance cluster analysis generates 18 solar climatic types based upon long-term mean monthly data. Two of the resultant regions contain 76 per cent of the 221 stations used in this study. The results are shown to vary significantly from the previously published maps of solar climates in the United States.     

Instantaneous hourly and daily radiation on tilted surfaces

A simple model to evaluate the irradiance incident on tilted surfaces of various orientations is proposed in this paper. It is based on the knowledge of the monthly daily average values of global and diffuse radiation on horizontal surfaces and their distributions over a period of a day. These distributions are approximated by a modified Gaussian expression with two parameters, which take into account the morning-to-afternoon asymmetries. Integrating the irradiance over a period of an hour or a day, values of monthly hourly average and daily on variously oriented surfaces have been calculated for the locations of Barcelona, Genoa, and Montreal. The calculated values are generally in good agreement with the data derived by the observations.