Solar Energy 74 (2003) 417–427

T he effective size of the solar cone for solar concentratingsystems

*D. Buie , C.J. Dey, S. BosiSolar Energy Group, School of Physics A28, University of Sydney, Sydney, NSW 2006, Australia

Received 15 December 2002; received in revised form 10 April 2003; accepted 21 April 2003

Abstract

In this paper we define a virtual solar cone, whose principle axis is aligned with the solar vector, having a radial angulardisplacement containing a pre-defined proportion of the terrestrial solar radiation. By simulating various sunshape profiles,the angular extent of the energy distribution is established to give the effective size of the solar cone for a range ofatmospheric conditions. Then, by simulating the reflection of these solar distributions off a set of non-ideal mirroredsurfaces, accounting for non-specular reflection and mirror shape errors, the combined effect of sunshape and mirrorproperties on the solar image is obtained. Clear trends are presented that show the dependence of the effective size of thesolar image on the accuracy of a mirrored surface for different sunshapes. We then identify the effective size of the solarimage at the absorber plane that must be accommodated in the design and optimisation of solar concentrating systems. 2003 Elsevier Ltd. All rights reserved.

Keywords: Insolation; Concentrating systems; Sunshape

1 . Introduction reflected off a mirrored surface in a generic solar concen-trating system. To estimate this, an accurate model of the

The broadening of the solar image when sunlight travels radial energy distribution of the sun must be established.from the near vacuum of space to the Earth’s surface is Originally, work conducted by the Lawrence Berkeleydue to interactions of the sunlight with atmospheric Laboratories (LBL) (Grether et al., 1975; Noring et al.,particulates. Where the radius of the particles is large in 1991) in the late 1970s and early 1980s measured thecomparison to the wavelength of the sunlight, small angle degree to which sunshapes varied from location to loca-forward scattering occurs, forming the solar aureole (Mie, tion. Eleven sites were chosen across the United States1908). The amount of energy in this circumsolar region is exhibiting different atmospheric characteristics such asimportant for two reasons (Noring et al., 1991; Buie et al., altitude, proximity to sources of large particulates, and2003). Firstly, the radial solar energy profile (sunshape) humidity. Grether, Evans, Hunt and Wahlig collated overplays a non-negligible role in determining the overall flux 200 000 individual solar profiles into what was called thedistribution at the focal region of imaging concentrators. reduced database (RDB). Between them, they publishedSecondly, depending on the acceptance angle of a solar over 15 articles relating to their methodology of collectingconcentrating system, overestimation of the power output and the spectral nature and distribution of circumsolarmay occur if all the power is assumed to fall within the radiation (a full reference list of their contribution isextent of the solar disc only. provided byNoring et al., 1991). This work was an

This paper is concerned with the second of these issues: important step toward the creation of a terrestrial sunshapeexamining the amount of energy that is available in a given model.radial displacement about the solar vector, that has been Work carried out byRabl and Bendt (1982)in defining a

sunshape using the RDB was followed bySchubnell et al.(1991), Schubnell (1992a,b), Steinfeld and Schubnell*Corresponding author. Tel.:161-2-9351-5979; fax:161-2-(1993), Neumann and Schubnell (1992), Neumann and9451-7725.

E-mail address: buie@physics.usyd.edu.au(D. Buie). Groer (1996), Neumann et al. (1998)and Neumann and

0038-092X/03/$ – see front matter 2003 Elsevier Ltd. All rights reserved.doi:10.1016/S0038-092X(03)00156-7

418 D. Buie et al. / Solar Energy 74 (2003) 417–427

Nomenclature

x Circumsolar ratioF Energy contained within the circumsolar region of the terrestrial solar beamcs

F Total terrestrial solar beami

u Angular displacement from the solar vectoru Radial displacement of the solar diskd

u Radial displacement of the circumsolar regionD

u Radial displacement defining the size of the solar coneL

f Radial solar energy profile (sunshape) per steradiank Sunshape variableg Sunshape variableL Percentage of the terrestrial solar beamP Probability distribution of the combined surface errors Standard deviation of the probability distribution

Witzke (1999).All of these authors made contributions to displacement containing a pre-defined proportion of thethe description of the radial energy distribution of the sun terrestrial solar radiation. The effective size of the solarand the effect this distribution has on the ultimate flux cone is then described for a range of solar conditions. Thedistribution in the focal region of concentrating systems. broadened distribution, or solar image caused by the

Neumann et al. (2002)described six sunshape profiles reflection of the solar cone off a mirrored plane arethat are indicative of a typical range of atmospheric calculated. Clear trends are presented that show theconditions. These profiles, combined with their statistical dependence of the effective size of the solar image on theweights, represent a numerical database for calculating the accuracy of a mirrored surface for different sunshapes. Theinfluence of variable conditions of the sunlight scattering effective size of the solar image at the absorber plane thaton solar concentrating systems.Buie et al. (2003)extended must be accommodated in the design and optimisation ofthe analysis of the LBL sunshape data, describing an solar concentrating systems is then discussed.empirical sunshape model that illustrates little variationover a range of geographic locations and presenting analgorithm that can be used to recreate sunshapes for 2 . The sunshapesimulating solar concentrators. In summary, work de-scribed in the literature to date involves simulation of the Buie et al. (2003)showed that terrestrial sunshapesradial energy distribution of the sun for clear skies show little variation from location to location when theyshowing good agreement with a vast amount of observed are described in terms of their circumsolar ratio (CSR).data. The CSR (x) is defined as the radiant solar flux contained

Concentrating systems have the added complexity of within the circumsolar region of the skyF , divided bycsreflecting sunlight off at least one mirrored surface. The the incident radiant flux from the direct beam and cir-reflection causes a broadening of the solar beam. The cumsolar regions combinedF ,ibroadening is due to both the dispersion effects of non-specular surfaces, and surface normal deviations from Fcs

]x 5 . (1)mirror shape errors. Although tracking errors are equally Fiimportant for estimating the size of the flux distribution inthe imaging plane of a solar concentrator, they do not If u is the radial displacement about the solar vector, thebroaden the solar image and therefore are excluded from angular extent of the circumsolar region (u ) is set to beD

these calculations.Johnston (1995)showed that these two 43.6 milliradians (mrad), complementary to an activesources of errors, both surface normal deviation and cavity radiometer having an acceptance half angle of 2.58.dispersion, could be combined for a mirrored surface, and The inner limit of the circumsolar region is the edge of thethe broadening of the solar image could be simulated by solar discu and has a generally accepted value of 4.65d

considering a probability distribution of errors. Therefore, mrad (0.2668) as calculated byPuliaev et al. (2000)at theif sufficient statistical information is gathered characteris- Observatorio Nacional in Brazil. Since we are consideringing mirror surfaces, the typical reflected solar distribution concentrating collectors only and therefore exclude insola-off that surface is predictable. tion falling outside the circumsolar region, for brevity we

This paper defines asolar cone whose principle axis is will refer to F as the incident radiation. The incidenti

aligned with the solar vector and that has a radial angular radiation and the radiation contained in the circumsolar

D. Buie et al. / Solar Energy 74 (2003) 417–427 419

region can be well-approximated by considering the inte- and that uses only one parameter, the CSR, based on thegral of the sunshapef(u ) according to, LBL reduced database andNeumann et al. (2002)sun-

shape profiles. Typical CSR values vary from zero to 0.3,uD

but have been recorded close to unity (Noring et al., 1991).F 5 2p E f(u ) sinu du , (2)i,cs Of the solar profiles collected byNeumann et al. (2002),

0,ud 71.5% had a CSR less than 0.05 and 89.5% of thesunshapes had a CSR less than 0.15; a complete statisticaluD

breakdown is presented in their work.¯2p E f(u )u du , (3) Using a charged coupled device camera, 63 normalised

0,ud sunshapes with a CSR of or about 5% were correlated byNeumann et al. (2002)from three sites across Europe. Thefor small u.comparison between this data and sunshapes created usingIf the sunshape is known accurately, then the angularEq. (4) with CSRs of 4%, 5% and 6% can be seen inFig.distribution of the incident radiation is well defined. The1. Clearly, Eq. (4) is a good representation of independent-sunshape described byBuie et al. (2003),ly observed data.

Using Eqs. (1–6), the proportionL of the incidentcos(0.326u )]]] for hu [Ru0#u # 4.65 mradj, radiation bounded by a specific radial displacementucos(0.308u ) Lf(u )55 k g about the central solar vector can be estimated from,e u for hu [Ruu .4.65 mradj,

uL

(4) Ef(u,x)u du0]]]]L(x)5 , (7)uD

wherek andg are given by, Ef(u,x)u du20.3

k 5 0.9 ln(13.5x)x , (5) 0

20.43 for any given CSR.g 52.2 ln(0.52x)x 2 0.1 , (6)Using Eq. (7) the effective size of the solar cone was

represents a sunshape model that is independent of location estimated for solar radiation with CSRs ranging from 0.01

Fig. 1. Comparison between Eq. (4) and observed data collated byNeumann et al. (2002).

420 D. Buie et al. / Solar Energy 74 (2003) 417–427

Fig. 2. The effective size of a solar cone containing various percentages of the incident radiation.

to 0.8 (Fig. 2). The six curves represent various per- Johnston (1995)showed that by considering as acentages of the incident radiation of 70%, 80%, 85%, 90%, systemic mass both the slope error (defined as the angular95% and 98%. Obviously the upper asymptotic limit to the deviation of the actual surface normal vectors from theireffective size of the solar cone is 43.6 mrad, encompassing ideal directions, measured in milliradians (Johnston, 1998))100% of the incident radiation, as defined here. and surface dispersion effects, the combined error could be

By definition, a solar profile with a CSR of 0.3 has 70% treated by the probability distribution (P),of its energy contained within the solar disc. For each of

dP u 2 2the CSR sunshapes created using Eq. (4) the effective edge 2u / 2s] ]5 e , (8)2duof the solar disc was determined (Fig. 3). The value of the sangular displacement for all of the simulations should be4.65 mrad, the accepted limit of the solar disc. This implies wheres is the standard deviation of the combined surfacethat the error associated with the prediction of the size of slope and dispersion error, andu is the radial displacementthe solar disc is61% for sunshapes with a CSR of less of a reflected beam from the specular direction. A realthan 0.2 and64% for the remainder of the sunshapes. surface can then be characterised by the standard deviation

of the probability distribution. As an example of different2 .1. Reflection of a mirrored surface surfaces, a high quality optical mirror has a probability

distribution with a standard deviation of about 0.2 mrad,If the terrestrial solar image is reflected off a perfect whereas poorer quality solar reflectors could have a

(planar) surface, the reflected image would be identical to standard deviation as high as 8 mrad.the original. In real systems though, the reflection of an To investigate the reflection of the solar image off aimage off a surface causes a distortion in that image. Two non-ideal mirrored surface, the convolution of the solarprinciple effects cause this distortion in solar concen- image created from Eqs. (3)–(6) and the probabilitytrators. Firstly, real surfaces interact with the reflected distribution from Eq. (8) was conducted (Rabl, 1985,pp.radiation, causing the expected specular reflection to form 202–205). The convolution in principal involves redistri-a dispersive cloud. Secondly, mirror shapes are not perfect, buting the vectors representing the solar image accordingthat is to say, the variations in the surface normals from the to a normalised two-dimensional representation of a radialideal mirror shape are an additional influence on the Gaussian distribution. Initially a solar image consisting ofreflected image. one hundred thousand appropriately weighted vectors

D. Buie et al. / Solar Energy 74 (2003) 417–427 421

Fig. 3. Demonstration of the accuracy of Eq. (4) at predicting the edge of the solar disc for a range of atmospheric conditions. The angulardisplacement of the solar disc should be 4.65 mrad.

Fig. 4. The effective size of the solar cone after reflection off a surface with a standard deviation of surface errors of 3.5 mrad.

422 D. Buie et al. / Solar Energy 74 (2003) 417–427

Fig. 5. The angular displacement of the reflected solar cone for different mirror errors for a CSR of 0.05. Curves are shown for thecollection limits of different percentages of the incident radiation.

Fig. 6. The angular displacement of the reflected solar cone for different mirror errors for a CSR of 0.1. Curves are shown for the collectionlimits of different percentages of the incident radiation.

D. Buie et al. / Solar Energy 74 (2003) 417–427 423

Fig. 7. The angular displacement of the reflected solar cone for different mirror errors for a CSR of 0.2. Curves are shown for the collectionlimits of different percentages of the incident radiation.

Fig. 8. The angular displacement of the reflected solar cone for different mirror errors for a CSR of 0.3. Curves are shown for the collectionlimits of different percentages of the incident radiation.

424 D. Buie et al. / Solar Energy 74 (2003) 417–427

Fig. 9. The angular displacement of the reflected solar cone as a function of the circumsolar ratio for different mirror tolerances for anincident radiation collection efficiency of 80%.

Fig. 10. The angular displacement of the reflected solar cone as a function of the circumsolar ratio for different mirror tolerances for anincident radiation collection efficiency of 90%.

D. Buie et al. / Solar Energy 74 (2003) 417–427 425

Fig. 11. The angular displacement of the reflected solar cone as a function of the circumsolar ratio for different mirror tolerances for anincident radiation collection efficiency of 95%.

extending over the angular radius of the circumsolar region conditions where the concentrator has certain predefinedwas created. A normalised two dimensional representation parameters (Figs. 9–11).of a radial Gaussian distribution was also created, extend-ing to an angular displacement of three standard deviationsfrom each solar vector. The weight of each of the solar 3 . Discussionvectors was then redistributed according to the normalisedGaussian about the position of each vector, simulating a Figs. 5–8illustrate the dependence of the effective sizereal surface with a standard slope error ofs. of the reflected solar cone on the quality of the reflecting

It must be pointed out that the convolution of a radial surface, for a generic solar concentrating system. As inerror distribution and a solar image is only truly repre- Figs. 2 and 4,a range of incident radiation proportions (L)sentative of a reflected beam at normal incidence. For was chosen so as to get a clear indication of the opticalnon-normal reflection, being the majority of reflections, the performance of concentrating systems. Of the four solarerror distribution would be asymmetric (or elliptical). conditions ranging from a CSR of 0.05 to 0.3, the effectiveTherefore this method is only an approximation of a real size of the reflected solar cone increases with the degra-system. dation of the optical quality of the mirrored surface

The energy distribution of various solar images was (defined by the standard deviation of the combined surfacecalculated including the effective size of the solar cone, errors). Also as the incident beam becomes broaderafter the reflection off assumed planar mirrors of differing (defined by a larger CSR), the effective size off the solaroptical qualities.Fig. 4 illustrates a typical result similar to cone increases substantially, particularly for small surfaceFig. 2, except that the original solar image was reflected errors.off a surface with a standard deviation of 3.5 mrad. The An approximately linear relationship exists between theresults of the simulations have been presented in two ways: angular size of the reflected solar cone and the standardfirstly, the effective size of the solar cone was compared deviation of errors, as the standard deviation of errorsprimarily to the optical quality of mirrors for particular increases. This is qualitatively different to the uppersolar conditions (Figs. 5–8) and secondly, the size of the asymptotic limit in the effective size of the solar cone thatsolar image was primarily compared to various solar exists in reflecting the solar image off an ideal surface. As

426 D. Buie et al. / Solar Energy 74 (2003) 417–427

the standard deviation of the surface error increases, the Bendt (1982),the general relationship between the cir-influence of the sunshape distribution diminishes, effec- cumsolar ratio, mirror accuracy, absorber acceptance angletively reducing the sunshape distribution to a point source. and optical efficiency has been established in design curvesTherefore, the resulting reflected energy distribution is covering a wide range of the important parameters. Furthereffectively described by a radial Gaussian distribution work will examine commercial realities such as thewhereu ~ s. efficiency of conversion devices, their costs, the non-linearL

Figs. 9–11illustrate the effective size of the solar cone dependence of mirror accuracy and mirror cost, in additionfor various solar conditions, described by an increasing to the parameters investigated here.CSR from 0.01 to 0.8. The higher the quality of thereflecting surface, the smaller the size of the angular spreadof the solar image; analogous to a smaller absorber

4 . Conclusionaperture that would be required in a generic solar concen-trator. For locations in the world where there is little

The influence of sunshape, described by the circumsolaratmospheric scattering, represented by a low CSR, theratio, and mirror tolerance, described by the standardoptical quality of the mirror surface becomes more critical.deviation of the mirror errors incorporating non-specularityFor environments where there is generally large atmos-and non-idea mirror shapes, on the angular distribution inpheric scattering (signified by average CSRs of betweenthe focal region of a generic solar concentrating system has0.2 and 0.3), very little improvement in the collectionbeen investigated. The results, obtained by detailed ray-efficiency can be achieved by using mirrors with an opticaltracing, and presented as a series of graphs covering thequality better than between 4.5 or even 3 mrad in theparameters over their typical ranges, allow broad designstandard deviation of the surface errors.considerations to be made in the design of solar concen-Ideally, the size of the absorber aperture in solartrating systems.concentrating systems should be approximately the size of

the reflected solar image in the focal plane, plus anallowance to account for the typical tracking errors of thereflector modules. If all the power of the reflected beam

R eferenceswas thought to be within the confines of the direct beamradiation (4.65 mrad about the solar vector) then an

B uie, D., Monger, A.G., Dey, C.J., 2003. Sunshape distributionsoverestimation of the collector efficiency would occur.for terrestrial solar simulations. Solar Energy 74 (2), 113–122.

For the design of solar concentrating systems at aG rether, D.F., Evans, D., Hunt, A.J., Wahlig, M., 1975. Measure-

particular site, the size of the reflected solar image can be ment of circumsolar radiation. In: Proceedings of Symposiumdefined by two parameters: the CSR and the standard of Optics in Solar Energy Utilization, 19th Annual Technicaldeviation of the surface error of the mirrors. If enough Symposium of The Society of Photo-Optical Instrumentationsunshape information about a site is known then average, Engineers. San Diego, CA, pp. 18–22.

J ohnston, G., 1995. On the analysis of surface error distributionstypical, or effective maximum CSRs may be predicted andon concentrated solar collectors. J. Solar Energy Eng. 117,used as a design parameter (seeRabl and Bendt, 1982).294–296.The solar distribution chosen for the design process may

2J ohnston, G., 1998. Focal region measurements of the 20 m tiltedbe related to improvements in the seasonal or daily outputdish at the Australian National University. Solar Energy 63,characteristics.117–124.The size of the absorber can then be determined by

M ie, G., 1908. Beitrage zur optik truber medien, speziell kol-accurate knowledge of the standard deviation of the liodaler metallo sungen. Ann. Physik 25, 377–445.surface errors usingFigs. 4–11.Alternatively, given an N eumann, A., Groer, U., 1996. Experimenting with concentratedoptimum absorber aperture and therefore acceptance angle, sunlight using the DLR solar furnace. Solar Energy 58, 181–the quality of the reflective surfaces can be chosen to 190.ensure that a certain percentage of the incident radiation N eumann, A., Schubnell, M., 1992. Irradiance and sunshape

measurements for the Cologne site. In: Proceedings of the 8thstrikes the absorber. The absorber aperture may be de-International Solar Forum, pp. 1173–1183.termined by thermal considerations leading to a trade-off

N eumann, A., von der Au, B., Heller, P., 1998. Measurements ofbetween optical efficiency and the total conversion ef-circumsolar radiation at the Plataforma Solar (Spain) and atficiency of the concentrating system (Steinfeld and Schub-DLR (Germany). In: Proceeding of the International Solarnell, 1993). For concentrating photovoltaic systems, theEnergy Conference Solar Engineering, pp. 429–438, ISBN

sunshape spectral dependence could also be incorporated0-7918-1856-X.

with a similar method, as previously described bySchub- N eumann, A., Witzke, A., 1999. The influence of sunshape on thenell (1992b). DLR solar furnace beam. Solar Energy 66 (6), 447–457.

Whilst the current paper does not outline an analytical N eumann, A., Witzke, A., Jones, S.A., Schmitt, G., 2002. Repre-design procedure for incorporating CSR considerations sentative terrestrial solar brightness profiles. J. Solar Energyinto solar concentrators, as for example given byRabl and Eng. 124, 198–204.

D. Buie et al. / Solar Energy 74 (2003) 417–427 427

N oring, J.E., Grether, D.F., Hunt, A.J., 1991. Circumsolar radia- S chubnell, M., 1992a. Influence of circumsolar radiation ontion data: the Lawrence Berkeley Laboratory reduced data base. aperture, operating temperature and efficiency of a solar cavityIn: National Renewable Energy Laboratory. NREL/TP-262- receiver. Solar Energy Mater. Solar Cells 27, 233–242.4429.http: / / rredc.nrel.gov/solar /pubs/circumsolar / title.html S chubnell, M., 1992b. Sunshape and its influence on the flux

P uliaev, S., Penna, J.L., Jilinski, E.G., Andrei, A.H., 2000. Solar distribution in imaging solar concentrators. J. Solar Energy Eng.diameter observations at Observatorio Nacional in 1998–1999. 114, 260–267.Astronomy & Astrophysics Supplement Series 143, 265–267. S chubnell, M., Keller, J., Imhof, A., 1991. Flux density dis-

R abl, A., 1985. Active Solar Collectors and Their Applications. tribution in the focal region of a solar concentrator system. J.Oxford University Press. Solar Energy Eng. 113, 112–116.

R abl, A., Bendt, P., 1982. Effect of circumsolar radiation on S teinfeld, A., Schubnell, M., 1993. Optimum aperture size andperformance of focusing collectors. J. Solar Energy Eng. 104, operating temperature of a solar cavity-receiver. Solar Energy237–250. 50 (1), 19–25.