The utilizability method with hourly vs daily insolation data

The significant difference between using hourly vs daily insolation data in the utilizability method is demonstrated for both the specific case of Bet Dagan, Israel and the more general case of utilizability curves that are generated from hourly and daily insolation correlations. For the annual utilizability curves for flat-plate collectors, the curves based on daily insolation data are shown to be significantly different from the corresponding curves based on hourly data, with the differences between the curves increasing with increasing threshold. A simple explanation for this observation is presented and the importance of the need for accurate utilizability curves at high thresholds is discussed.     

On the calculation of solar utilizability for south oriented flat plate collectors tilted to an angle equal to the local latitude

The utilizability or -curve method is a design method for some types of solar energy systems that use flat-plate solar collectors. It can be a very useful design tool, for the preliminary design or the design of small systems, cases where methods easy to use and apply are required.

In this paper, a simple method to calculate monthly average hourly and daily flat plate collector utilizability is presented. It is based on empirical correlation of with independent variables that separate meteorological and collector parameters, making possible quick evaluations of changes in collector design and collector inlet temperature. In general, the method reduces the calculations required to determine , and gives good results compared to long-term hourly simulations as well as to existing hourly and daily utilizability calculations methods.     

The utilizability function—II Validation of theory against data-based correlations

A new theoretical approach to the calculation of the utilizability function is developed and applied to the case of the annual performance of solar collectors of high concentration ratio. The utilizability is calculated in mathematical closed form as an integral over the contributions of the daylight hours of one representative day, equinox. The daily variations in the instantaneous solar radiation (due, for example, to varying degrees of cloudiness) are represented by random fluctuations superimposed on the clear day radiation pattern. The results are in good agreement with the corresponding utilizability curves that are based on detailed meteorological data. A simple and practical analytic expression for the utilizability is derived for the limiting case of clear climates, and turns out to provide rather accurate predictions for the utilizability for all climates as well. The presentation is divided into two separate papers: the current paper, in which the details of the theory and derivation are presented, and the following paper, for the reader who may not not be interested in all the details of the derivation, in which the solved numerical examples and the comparisons with results based on detailed meteorological data are presented.     

Derivation of method for predicting long term average energy delivery of solar collectors

Based on the utilizability concept of Hottel, Whillier, Liu and Jordan, an analytical model has been developed to predict the long term average energy delivery of almost any solar collector. The presentation has been split into two separate papers: a users guide (without explanation of the origin of the formulas) and the present paper (which derives these formulas and documents the validation). The model is applicable whenever the average operating temperature of the collector (receiver surface, fluid inlet, fluid outlet or mean fluid) is known. If the operating temperature is not known explicitly the model will give adequate results when combined with the , f-chart of Klein and Beckman. By contrast to the alternative of hour-by-hour simulation, prediction methods such as the present model and the f-chart offer the advantages of automatically averaging over year-to-year weather fluctuations and of being sufficiently simple to permit hand calculation of long term performance. In a comparison with hourly summations of insolation data, the present model has been found to have an error of less than 3 per cent for the radiation available to a solar collector and an error of about 5 per cent for the heat delivery of solar thermal collectors.     

Analysis of short-term solar radiation data

Solar radiation data are available for many locations on an hourly basis. Simulation studies of solar energy systems have generally used these hourly values to estimate long-term annual performance, although solar radiation can exhibit wide variations during an hour. Variations in solar radiation during an hour, such as on a minute basis, could result in inaccurate performance estimates for systems that respond quickly and non-linearly to solar radiation. In addition, diffuse fraction regressions and cumulative frequency distribution curves have been developed using hourly data and the accuracy of these regressions when applied to short-term radiation has not been established. The purpose of this research is to investigate the inaccuracies caused by using hourly rather than short-term (i.e., minute and 3 min) radiation data on the estimated performance of solar energy systems. The inaccuracies are determined by examination of the frequency distribution and diffuse fraction relationships for short-term solar radiation data as compared to existing regressions and by comparing calculated radiation on tilted surfaces and utilizability based on hourly and short-term radiation data.     

The determination of hourly insolation on an inclined plane using a diffuse irradiance model based on hourly measured global horizontal insolation

Using only measured hourly values of global insolation on a horizontal surface, a method has been developed for computing the corresponding hourly values of insolation on a surface inclined at any angle and oriented in any direction. The method uses a solar radiation model in which the diffuse component is calculated from global horizontal radiation using three different relationships; the appropriate equation is selected according to the value of the ratio of measured hourly global insolation to hourly global insolation computed for clear sky conditions. The method has been checked using measured hourly values in Melbourne over a 5-yr period of insolation on both a horizontal surface and a plane inclined at 38° to the horizontal facing north. The differences between the computed hourly values and the measured hourly values are found to be approximately normally distributed about zero with a standard deviation of 0.16 MJ m⁻². This method is particularly useful for predicting the heat output of inclined solar flat plate collectors when only measured global horizontal insolation is available, which is often the case. Good agreement was found between the predicted output of a typical collector using measured 38° insolation and the computed hourly values using this method. Since the method has been checked only against Melbourne data it should be applied elsewhere with caution, but it is believed to have general application.     

A new look at long term collector performance and utilizability

A simple technique has been developed to calculate monthly collection efficiency or monthly utilizability for solar thermal flat-plate collectors. It is applicable to south facing tilted collectors operating with a fixed fluid inlet temperature although extensions to other more generalized uses of utilizability are discussed. The heart of the technique is an empirically determined performance map that makes possible quick evaluations of changes in collector design, geographic location and collector inlet temperature. The collector input variables are those that are commonly measured in most thermal test procedures; geographic input variables are the mean monthly temperature and K_T (the Liu and Jordan clearness factor). The procedure was developed for monthly optimum fixed tilts but a simple correction can be made to incorporate arbitrary monthly fixed tilts. The method, in general, gives good results compared to long term hourly simulation. The technique also allows one to determine under what operating conditions collector performance begins to depend on site-to-site solar radiation/weather variability and what uncertainties can be expected from its use.     

Comparison of long-term flat-plate solar collector performance calculations based on averaged meteorological data

The use of averaged meterological data for collector performance calculations is studied. To this end, a steady state, two-dimensional, nodal, heat transfer analysis is developed for a flat-plate solar collector. The analysis accounts for the temperature gradients in the fluid flow and vertical directions in the collector, the physical and thermodynamic properties of the materials in the collector, the collector location, the orientation and dimensions of the collector, the number of cover plates and any thin film selective coatings on the cover plates or absorber. Also accounted for are the time dependent variations in the meteorological conditions, insolation, and collimated and diffuse solar irradiation. The spectral nature of radiation heat transfer in the collector is modeled by two spectral bands, solar and thermal, with 3.0 μm as the cutoff frequency between the solar and thermal bands of radiation. The results indicate that long term collector performance calculations based on averaged meteorological data will not correlate with calculations based on hourly data if the weather is highly variable. When the weather variations are mild, averaged data can give results very close to those based on hourly data.     

The frequency distribution of instantaneous insolation values

The frequency distribution of instantaneous terrestrial insolation values based on measurements carried out over a one-year period has been evaluated. The resulting fractional time distribution curves for instantaneous radiation are found to differ from that obtained previously for daily values. The instantaneous fractional time curves exhibit a distinct step associated with the predominance of two radiation states, clear and cloudy. A model derived from the concept of Boltzmann statistics is shown to be in good agreement with the experimental measurements over a wide range of air masses and average clearness indices.     

Incident solar radiation on Argentina from the geostationary satellite GOES: Comparison with ground measurements

Daily values of the solar radiant energy incident on earth's surface constitute a quantity of increasing importance, not only in nonconventional energy development but also for agricultural, weather, and climate monitoring and predictions. In Argentina, the solar global radiation has been measured since 1978 through a network of pyranometers distributed all over the country. Simultaneously, in the area limited by 21° and 41°S latitude and 53° and 67°W longitude, insolation determinations have been made using the geostationary satellite GOES from May 1982 to June 1984. Therefore, it was possible to compare the simultaneous hourly and daily values of solar global radiation corresponding to the northern part of the country during 1982 and 1983. In this article, the results of the comparison are presented. It is shown that the standard error of the satellite-derived hourly and daily insolation values when compared against pyranometers is about 25%, and in the range of 15 to 20% of the mean values, respectively. Therefore, operational estimation of surface insolation in the region using GOES data appears normally feasible.     

Estimating the performance of a photovoltaic pumping system

A simple algorithm for estimating the long-term performance of a photovoltaic pumping system without battery storage is presented. This methodology uses the standard solar utilizability correlation equation to calculate the flow rate of a system with an insolation threshold and a pumping rate that has a nonlinear dependence on insolation. The algorithm's estimates of total monthly pumped volume compare well with those of an hourly simulation using Typical Meteorological Year data for four U.S. locations.     

A non-linear flat-plate collector model

The analysis of a non-linear flat-plate collector is presented in which the overall loss coefficient is assumed to be a linear function of the temperature difference between the fluid in the collector and the environment. The instantaneous performance of collectors calculated using linear and non-linear models is presented in terms of three dimensionless numbers. For an assumed parabolic solar radiation profile, the average daily performance for a constant fluid inlet temperature can be presented in terms of three further dimensionless numbers. The resulting expressions for both instantaneous and daily performance are shown graphically. A method of calculating linear collector performance characteristics from non-linear curve fitting to experimental results is shown so that the predicted daily performance from non-linear and linear curve fits can be compared. In most circumstances, a linear fit is adequate.     

Estimation of half-hour solar radiation values from hourly values

A statistical Markovian insolation model for predicting the time-sequence of half-hour solar radiation values on a horizontal surface which uses the hourly insolation values is developed. The hourly transition density function, governing the diurnal evolution of the hourly solar radiation values, is used for obtaining the half-hour transition density function. A transition density function is a measure of the probability of the event at the next immediate hour of interest when the event at the present hour is given. The estimation of half-hour transition density function is done through the fundamental decomposition theorem for the density function. This assumes a set of well defined intermediate states. As a first approximation, the half-hour transition density function is assumed to be temporally stationary. Furthermore, it is assumed that the cumulative probability distribution functions of the normalised initial hour solar radiation value and the normalised initial half-hour solar radiation values are not significantly different; the hourly, or half-hourly solar flux values are normalised by the corresponding extraterrestrial solar flux values. The validity of these assumptions is established through the successful time-sequence predictions of the half-hour insolation values. The time-sequence aspect of solar radiation values is proved by comparing the predicted joint cumulative distribution functions for several successive normalised half-hour values, with the corresponding distribution function for the recorded values. In order to prove that the predicted and actual distribution functions are from the same set, the non-parametric statistical test proposed by Kolmogorov and Smirnov has been used.     

Model evaluation and optimum collector slope for a tropical country

Various models proposed in the literature, viz. the Beam method, proposed by Morse & Czarnecki; two methods proposed by Liu & Jordan, using hourly values and average daily values of solar radiation: and Klucher's method are compared in this paper using measured hourly values of global and diffuse radiation on a horizontal surface and the total radiation on a tilted surface. It was observed that the model proposed by Klucher estimates the radiation more consistently with the experimental observations for clear as well as cloudy sky conditions.

This method was used for predicting the year round radiation availability on (i) a fixed collector inclined at an optimum tilt, (ii) a partially tracking collector in which the slope is fixed but following the sun's azimuth angle, and (iii) a fully tracking collector. In this analysis, measured values of hourly global and diffuse solar radiation on a horizontal plane for four widely separated Indian stations, viz. Delhi (28°38′N), Poona (18°29′N), Calcutta (22°36′N), and Madras (13°8′N) as per India Meteorological Department classification for climatic analysis, have been used.

With the model proposed, the radiation availability on different slopes for all the twelve months of the year for a fixed and partially tracking collector have been evaluated and optimum tilts for various seasons of a year for different solar energy applications for all the four Indian stations have been determined. For a typical winter month at New Delhi in the semi-arid zone of India, a fixed collector, a partially tracking collector and a fully tracking collector receive 60, 80 and 95% more radiation compared to that received on a horizontal plane respectively. Another observation is that the variation of the insolation received on inclined surfaces is not at all sensitive to the angle of tilt in the neighbourhood of the optimum angle of operation.     

Study of diffuse and global radiation characteristics in India

Recently, the Indian Meteorological Department has made available, for 13 locations in India, pyranometric data for total and diffuse radiation on an hourly and daily basis. The period of observation is from 1957 to 1975. This data is analysed to reexamine the correlations between monthly-average daily values of diffuse and total insolation and between hourly and daily insolation. The relationship between monthly diffuse/total and total/extraterrestrial ratios is found to be linear. The present correlation implies that the diffuse component is significantly larger than that predicted by other correlations. No noticeable effect of location or seasonal variation was found. Comparison of the present data with earlier studies for India indicates a trend of increasing diffuse radiation with the lapse of time. The ratio of hourly to daily insolation agrees with the Liu and Jordan correlation for total insolation but differs significantly for diffuse radiation. The need for refinement of the Liu and Jordan correlation between hourly and daily diffuse radiation is pointed out.     

Field performance and operation of a flat-glass solar heat collector

A “Base-Line,” flat-glass solar heat collector has been designed and constructed that can be manufactured economically for commercial use. Four of the collectors, 34 by 76 in. (approximately 18 ft²), were installed to provide hot water to a private home in Melbourne, Florida.The details of the collector are described, including coverplates, solar absorber, absorber coating, spacers, seals and glazing.A simple relationship has been established between the collector efficiency, the collector temperature and the rate of insolation for constant rates of flow of circulating fluids.The theoretical and field performance curves have been correlated for collector efficiency, collector temperatures, incident solar radiation and ambient air temperatures. The effect of fluid flow on collector temperatures for various collector parameters has also been presented.     

On the optimum exposure of flat-plate fixed solar collectors

With the increasing range and scale of applications of solar energy conversion systems a detailed knowledge of the available solar fluxes on surfaces of various tilt and slope aspect is required. The widely used engineering routines for siting and exposing of solar collector arrays may not always be consistent with the maximum available solar energy income. In the present study a computational algorithm is proposed suitable for the calculation of optimum tilt angles of a solar collector, receiving the maximum insolation for given values of direct beam, global and diffuse radiation, and given surface reflectivity. Formulas are derived also for the calculation of optimum tilt angles yielding maximum daily insolation for each month of the year. The dependence of the optimum tilt angle on the diffuse-to-global radiation ratio, as well as on the surface reflectivity is evaluated.     

Calculation of flat-plate collector utilizability

The “utilizability” or φ-curve method developed by Whillier [1] and later generalized by Liu and Jordan[2, 3], can be a very useful design tool for some types of solar energy systems which utilize conventional flat-plate solar collectors. The generalized φ-curve method in its original form, however, has several drawbacks. The calculation effort required to assess long-term collector performance is significant. The calculations can not be completely automated on a computer or hand calculator because many graphs are needed for which analytical representations are not available.In this paper, the φ-curve method is reviewed and situations for which the method is useful are described. Then, an easier method, using daily -charts, rather than the original hourly φ-curves, is presented. The -charts greatly reduce the calculations required to determine flat-plate collector utilizability. -charts can be easily implemented on a programmable hand calculator. A comparison of the original hourly and the new daily calculation methods is presented. An example demonstrating the use and utility of the -charts is also included.     

One-minute global irradiance probability density distributions conditioned to the optical air mass

Knowledge of the temporal variability of the solar irradiance is important to study solar energy systems involving thermal and photovoltaic processes. The differences between hourly and instantaneous values of the clearness index considerably affect the utilizability of photovoltaic systems. In this work, we analyzed the probability density distributions of one-minute values of global irradiance, conditioned to the optical air mass, considering those as an approximation to the instantaneous distributions. The study reveals that the bimodality that characterizes these distributions increases with optical air mass. We propose the use of a functional form based on Boltzmann's statistics in order to describe these distributions. This function can be used for the generation of synthetic radiation data. Expressing the distribution as a sum of two functions provides an appropriate modeling of the bimodality feature that can be associated with the existence of two levels of irradiation corresponding to two extreme atmospheric situations, cloudless and cloudy conditions.