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).     

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.     

A brief review of models for computing solar radiation on inclined surfaces

Modelling of solar energy systems requires estimation of the hourly radiation incident on surfaces of different tilts and orientations. Most meteorological stations report radiation values on a daily rather than hourly basis. Again, the value of the diffuse component may or may not be reported. Hence, different strategies may be encountered, for each location, with the common goal of computing hourly radiation on inclined surfaces.

In this study, five different schemes are presented to achieve this goal. In each scheme, certain correlations are required which are discussed in detail. The study is divided into five sections, each dealing with a specific type of correlation. These sections are: (i) correlations between daily diffuse and global radiation; (ii) correlations between hourly/daily global radiation; (iii) correlations between hourly/daily diffuse radiation; (iv) correlations between hourly diffuse and global radiation; and (v) models for computing diffuse sky radiation on inclined surfaces. The last section deals extensively with the anisotropic nature of sky diffuse radiation.

The important aspects of all correlation studies are highlighted, and the relative merits and demerits of their results are brought to light.

Mathematical expressions, where available, for models/correlations are provided so that the reader will have access to a comprehensive study. This information should be useful for modelling purposes in which computation of radiation on surfaces of different orientations and tilts is required.     

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.     

Calculation of monthly average insolation on tilted surfaces

Several simplified design procedures for solar energy systems require monthly average meteorological data. Monthly average daily totals of the solar radiation incident on a horizontal surface are available. However, radiation data on tilted surfaces, required by the design procedures, are generally not available. A simple method of estimating the average daily radiation for each calendar month on surfaces facing directly towards the equator has been presented by Liu and Jordan [1]. This method is verified with experimental measurements and extended to allow calculation of monthly average radiation on surfaces of a wide range of orientations.     

The frequency distribution of daily insolation values

Frequency distributions of insolation values are needed in order to derive simple correlations for predicting the performance of solar energy systems. In this paper the frequency distribution of daily total hemispherical solar irradiation values on the horizontal surface is derived from measured data for 90 locations in the U.S. The results can be approximated by generalized distribution curves which depend only on the clearness index, defined as ratio of terrestrial over extraterrestrial insolation. The results agree well with the curves originally derived by Liu and Jordan, except for a correction at high insolation values. The deviation of individual locations from the generalized curves is examined. A breakdown according to time of year reveals some seasonal variation. The theoretical distribution corresponding to random insolation sequences is also derived; it agrees very well with the generalized frequency distribution curves.     

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.     

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.     

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.     

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.     

Vortex instability of natural convection flow on inclined surfaces

The linear stability of laminar natural convection flow adjacent to a heated, inclined, upwardfacing plate is investigated for disturbances having the form of longitudinal vortices. The stability problem is formulated with account being taken of the fact that the basic flow and temperature fields depend on the streamwise coordinate. One of the demonstrated consequences of retaining the transverse velocity of the basic flow is the so-called bottling effect, wherein the disturbance vorticity and temperature are contained within the respective boundary layers of the basic flow. The calculated neutral stability curves exhibit an altogether different character depending upon whether the streamwise dependence of the basic flow and temperature fields is taken into account or suppressed; the magnitude of the critical Grashof numbers from the two models differs by several orders of magnitude. The results also show that the greater the inclination of the plate from the vertical, the more susceptible is the flow to the vortex-type instability.The relationship of the analytical results to available experimental information is discussed.     

Estimation of daily total and diffuse insolation in India from weather data

Insolation and weather data for a large number of cities in India is analysed and correlated. Correlations based on a citywise regression analysis indicate that daily total insolation correlates best with sunshine duration, all clouds and precipitation. However these relations are not useful for predicting insolation at locations where this data is not measured. Monthwise correlations which are valid over a region are more useful. Hence such correlations have been developed for Indian conditions. In order to increase the accuracy of prediction of these correlations, India is divided into two regions on the basis of the climatic characteristics of the winter monsoon.Finally the Liu and Jordan model for predicting daily diffuse radiation from daily total radiation has been tested and found to be applicable for Indian conditions. However the numerical values obtained are very different from those obtained for conditions in the United States.     

Droplet behaviors on inclined surfaces with dynamic contact angle

Dynamic contact angle (DCA) is of crucial importance in the numerical investigation of water management problems in proton exchange membrane fuel cells (PEMFCs). In order to well predict the gas-liquid two-phase flow in the complex flow field in PEMFCs, first, it is very critical to build a robust DCA model that is capable of simulating droplet behaviors on a single surface under various conditions. In our previous research work, an advancing-receding DCA (AR-DCA) model is developed and has been successfully validated against a series of experiments from the available literature for droplet impact on surfaces. In this study, the AR-DCA model is further applied to simulate droplet behaviors on inclined surfaces with different droplet impact velocities, impact angles and viscosities. It is found that the droplet spreading and deformation from the simulations have excellent agreement with those captured in the corresponding experiments. The results also indicate that higher impact velocity and impact angle can facilitate the spreading trend at the droplet trailing edge and have no notable effects on the leading edge. In addition, the increase of droplet viscosity leads to a transition from droplet deposition phenomenon to partial rebound on the surface.     

Predicting daylight illuminance on inclined surfaces using sky luminance data

Daylight illuminance, particularly on vertical surfaces, plays a major role in determining and evaluating the daylighting performance of a building. In many parts of the world, however, the basic daylight illuminance data for various vertical planes are not always readily available. The usual method to obtain diffuse illuminance on tilted planes would be based on inclined surfaces models using data from the horizontal measurements. Alternatively, the diffuse illuminance on a sloping plane can be computed by integrating the luminance distribution of the sky ‘seen’ by the plane. This paper presents an approach to estimate the vertical outdoor illuminance from sky luminance data and solar geometry. Sky luminance data recorded from January 1999 to December 2001 in Hong Kong and generated by two well-known sky luminance models (Kittler and Perez) were used to compute the outdoor illuminance for the four principal vertical planes (N, E, S and W). The performance of this approach was evaluated against data measured in the same period. Statistical analysis indicated that using sky luminance distributions to predict outdoor illuminance can give reasonably good agreement with measured data for all vertical surfaces. The findings provide an accurate alternative to determine the amount of daylight on vertical as well as other inclined surfaces when sky luminance data are available.     

Predicting solar irradiance on inclined surfaces using sky radiance data

The availability of more comprehensive solar irradiance data is invaluable for the reduction of cooling load in buildings and for the evaluation of the performance of photovoltaic plants. In many parts of the world, however, the basic solar irradiance data are not always readily available. This paper presents an approach to calculate the solar irradiance on sloped planes by integrating the sky radiance distribution. Sky radiance data recorded from January 1999 to December 2001 in Hong Kong were used to estimate the solar irradiance for the horizontal and four principal vertical surfaces (N, E, S and W). The performance of this approach was assessed against data measured in the same period. Statistical results showed that using sky radiance distributions to predict solar irradiance can give reasonably good agreement with measured data for both horizontal and vertical planes. The prediction approach was also employed to compute the solar irradiance of a 22.3° (latitude angle of Hong Kong) inclined south oriented surface. The findings indicated that the method can provide an accurate alternative to determine the amount of solar irradiance on inclined surfaces facing various orientations when sky radiance data are available.