A MODEL FOR THE ANGULAR DISTRIBUTION OF SKY RADIANCE CONFERENCE PAPER PRESENTED AT THE JOINT ASME/AIC~E 18th NATIONAL HEAT TRANSFER CONFERENCE SAN DIEGO, CALIF., AUGUST 6 - 8, 1979 F.C. HOOPER AND A.P. BRUNGER AUGUST, 1979 MASTER UNIVERSITY OF TORONTO DEPARTMENT OF MECHANICAL ENGINEERING TORONTO, CANADA M5S 1A4 PREPARED FOR THE U.S. DEPARTMENT OF ENERGY SOLAR ENERGY UNDER CONTRACT EY-76-C-02-2939 - DISCLAIMER This bwk war re pared sensaounr of wknanrorsd by an mew of me United Statn Govanmart. Neither the United Smm Government mr any agew lhareof. 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BRUNGER ABSrnCT Z - zenith angle of a point on the sky dome = tilt angle of slope with respect to the A flexible mathematical model is introduced - horizontal which describes the radiance of the dome of the sky under various conditions. This three-component e,@ - zenith angle and azimuthal angle in a spherical co-ordinate system with ?he continuous distribution (TCCD) model is compounded by observer at the origin the superposition of three separate terms, the iso- tropic, circumsolar 'and horizon brightening terms, INTRODUCTION each representing the contribution of a particular There is very little general appreciation amongst sky characteristic. In use a particular sky condition designers of solar collector systems of the magnitude is characterized by the values of the coefficients of of the diffuse fraction of the total solar radiation each of these three terms, defining the distribution incident on a collector, nor of the distribution of of the total diffuse component. The TCCD model has its origin over the hemisphere of the sky. The been demonstrated to fit both the normalized clear sky diffuse fraction is, in fact, substantial, especially data and the normalized overcast sky data with an RMS in more northern latitudes where the air mass is rela- error of about ten percent of the mean overall sky . tively high. For instance, for eleven locations in radiance. By extension the model could describe Canada, computations based on recorded radiation data variable or partly clouded sky conditions. The model have shown that typically forty percent of the annual can aid in improving the prediction of solar collector incident short wave solar radiation on a flat plate performance. collector tilted at the latitude angle is diffuse. NO%NCLATUF,,E Obviou~lythis is an important fraction of the total, yet most design procedures for solar collection weighting parameters of the TCCD model A0.A1,A2 - systems, after elaborate computation of the angular AZ - hour angle of the sun with respect to the direction of the beam radiation component then make naridian plane defined by the zenith and ' one or the other of two gross assumptions. They the normal vector to the sloping surface assume either that, all of the diffuse radiation comes c,d - positive constants from the direction of the sun, the circumsolar region, diffuse sky radiation incident on a or that it originates uniformly from over the sky, the Dh - horizontal surf ace - isotropic assumption. These tvo assumptions lead to diffuse sky radiation incident on a substantially different results, and it can be argued DS - sloping surface that both are quite vrong. There is clearly a need i - angle of incidence for a better understanding of the spatial distribution I - direct beam solar radiation intensity at of the diffuse component if design computations of the the earth's surface total radiation incident on solar collectors are to be direct beam solar radiation intensity at I0 - rational and reliable. the top of the atmosphere A study of the variation in the intensity of anisotropy index short vave radiance over the dome of the s@ has led kS - to a three-component continuous distribution (TCCD) m optical air mass = l/cos (2) - model which describes the sky radiance distribution s angular distance from the solar position - under various conditions. This nev rnode3. can be to a general point on the sky hemisphere Z zenith angle of the sun made to describe quite closely the observed distri- - butions of diffuse radiation for both clear and over- cast conditions. Extensions to the model could permit (1). have assumed that the diffuse sky radiation is . description of time variable and partially clouded uniformly distributed across the dome of the sky sky conditions, such as frequently occur. ( eg . Garnier and Ohmura ( 2) , Temps and Coulson ( 3 ) ) . This isotropic assumption leads to the equation Fundamental Equations . The distribution in the intensity of the diffuse D~ = D~ (1 + cosa)/2 (3) radiation coming rrcan the dome of the sky is difficult to model accurately since it depends on a number of describing'the relationship betveen D , the diffuse factors, the most camplicating,being the type and irradiance on a tilted flat surface msaking an angle distribution of cloud cover at any one time. The flux a vith the horizontal, and Dh,the diffuse irradiance of energy incident on a tilted plane (see Fig. 1) due on a horizontal surface. to diffuse radiation from an elemental area of sky in The isotropic model of equation (3) leads to the direction (g,h) is underestimations of the insolation.on south-facing slopes and overestiuiations of the insolation on ms = d(g,h) cos(i) cos(g) dg dh, ' (1) north-facing slopes. E~uation(3) is most accurate under conditions approximating complete overcast. where d(g.h) is the intensity of the sky radiation in Garnier and Ohmura (4) reported that under varying sky conditions at Mont St. Hilaire, Quebec, the Zenith isotropic assumption yielded estimates of diffuse radiation on sloping surfaces that were in disagree- ~mlto the sbve ment vith the measured values by as much as fifty-six t percent. However, with respect to values of total global radiation, the relative errors introduced by using equation (3) seem to be more generally of the order of about five percent. The sun-centered, or directional, model assumes that all the diffuse sky radiation can be treated as if coming from the direction of the sun. This assumption leads to. the equation which can greatly overestimate the incoming diffuse radiation depending on the position of the sun rela- tive to the receiving surface. Hay (5) compared the isotropic and sun-centered models with a combinational model and an anisotropic --------- model. The combinational model treats one ha].? of the sky-duffuse radiation as being isotropically distributed and the remaining half as originating from the sun's disk. The anisotropic model uses the anisotropy index defined by equation (k) to divide Elementthe direction of sky (g,h) in the diffuse radiation into direct and isotropic portions. kS = ] cos(i) Fig.1 - Co-orilinate system for calculating the flux of diffuse energy on a tilted plane The anisotropy index is a measure of the effective the direction (g,h) and i is the angle betveen the atmospheric transmissivity to the direct solar beam, direction (g,h) and the normal to the tilted plane. corrected for the optical air mass, m, and the In the cc-ordinate system of Fig. 1 it can be shown incidence angle, i. The anisotropic model is an that improvement over the other models in terms of both bias error and RMS error. There have been several tabulations made of The total sky-diffuse radiant energy intercepted by average sky luminance and irradiance values in order the- surface can then be found by integrating equation to obtain standard distributions of diffuse sky (1) over Che part of the sky visible to the tilted radiation. As early as 1922 Kimbol and Hand (6) surface. That is, for a tilted planar surface, tabulated distributions of average sky luminance for several sun altitude angles, and for cloudy and clear sky conditions, at Chicago and at Washington, D.C. D = d(g,h) cos2(g) cos(h) dg dh (2) f 1n ' They then used these tables to calculate the daylight -- -- 2 2 illumination on vertical and sloping surfaces of various aspect angles. Tome and Norman (71, as If the function d(g,h) is the actual distribution of reported by Robinson (81, tabulated average sky sky-diffuse radiance, the result of the integration luminance data as a function of the amount of cloud.