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Solar Engineering of Thermal Processes, Manuais, Projetos, Pesquisas de Engenharia Elétrica

energia renovável, energia solar, livro básico da área

Tipologia: Manuais, Projetos, Pesquisas

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Baixe Solar Engineering of Thermal Processes e outras Manuais, Projetos, Pesquisas em PDF para Engenharia Elétrica, somente na Docsity! Tp SOLAR ENGINEERING OF THERMAL PROCESSES SOLAR ENGINEERING OF THERMAL PROCESSES Second Edition JOHN A, DUFFIE Emeritus Professor of Chemical Engineering WILLIAM A. BECKMAN Professor of Mechanical Engineering Solar Energy Laboratory University of Wisconsin-Madison Se2z GNR Es “ plbtoteco Contrate BCulta ingsgneria pur do 63) A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Brisbane Toronto Singapore xdy Prefaçe to the First Edition several drafis of the book which have been typed by our student helpers at the laboratory: it has often been difficult work, and their persistence, skill and good humor have been tremendous. Not the least, we thank our patient families for their forbearance during the lengthy process of putting this book together. JOHN A. DUFFIE WILLIAM A. BECKMAN Madison, Wisconsin June 1980 CONTENTS | PART 1 - FUNDAMENTALS 1 Introduction, 1 1. SOLAR RADIATION , 3 1] TheSun,3 [ 1.2 The Solar Constant, 5 ' 1.3 Spectral Distribution of Extraterrestrial Radistion, 7 1.4 Variation of Extraterrestrial Radiation, 9 | 1.5 Definitions, JO 1.6 Direction of Beam Radiation, 13 1.7 Angles for Tracking Surfaces, 21 1.8 Ratio of Beam Radiation on Tilted Surface tothat on Horizontal Surface, 25 ! 1.9 Shading, 32 1.10 Extraterrestrial Radiation on a Horizontal Surfice, 39 111 Summary, 44 References, 44 | 2. AVAILABLE SOLAR RADIATION : 46 2.1 Definitions, 46 ; 22 Pytheliometers and Pyrheliomenrio Scales, 47| 2.3 Pyranometers, 51 2.4 Measurements of Duration of Sunshine, 57 2.5 Solar Radiation Data, 58 2.6 Atmospheric Attenuation of Solar Radiation, 2.7 Estimation of Average Solar Radiation, 68 28 Estimation of Clear Sky Radiation, 73 2.9 Distribution of Clear and Cloudy Days and flours, 77 1 2.10 Beam and Diffuse Components of Hourly Radiation, 80 2.11 Beam and Diffuse Components of Daily Rafiation, 83 | 2.12 Beam and Diffuse Components of MonthlyRadiation, 8 2.13 Estimation of Hourly Radiation from Daily Data, 87 em no Contente 2.14 Radiation on Sloped Surfaces, 91 2.15 Radiation on Sloped Surfaces - Isotropic Sky, 94 2.16 Radiation on Sloped Surfaces - Anisotropic Sky, 96 2.17 Radiation Augmentation, 102 2.18 Beam Radiation on Moving Surfaces, 107 2.19 Average Radiation on Sloped Surfaces - Isotropic Sky, 109 2.20 Average Radiation on Sloped Surfaces - The K-T Method, 113 2.21 Effects of Receiving Surface Orientation, 119 2.22 Utilizability, 122 2.23 Generalized Utiizability, 125 2.24 Daily Utilizability, 134 225 Summary, 141 References, 141 SELECTED HEAT TRANSFER TOPICS 147 3.1 The Electromagnetic Spectrum, 147 3.2 Photon Radiation, 148 3.3 The Blackbody - a Perfect Absorber and Emitter, 148 3.4 Planck's Law and Wien's Displacement Law, 149 3.5 Stefan-Bolizmann Equation, 151 3.6 Radiation Tables, 15) 3.7 Radiation Intensity and Flux, 154 3.8 Infrared Radiation Exchange Between Gray Surfaces, 156 3.9 Sky Radiation, 157 3.10 Radiation Heat Transfer Coefficient, 158 3.11 Natural Convection Between Flat Parallel Plates, 160 3.12 Convection Suppression, 164 3.13 Vee-Corugated Enclosures, 167 3.14 Heat Transfer Relations for Internal Flow, 169 3.15 Wind Convection Coefficients, 173 3.16 Heat Transfer and Pressure Drop in Packed Beds, 176 3.17 Effectiveness - NTU Calculations for Heat Exchangers, 178 References, 181 RADIATION CHARACTERISTICS OF OPAQUE MATERIALS 184 4.1 Absorptance and Emittance, 185 4.2 Kirchotfs Law, 187 43 Reflectance of Surfaces, 189 4.4 Relationships Among Absorptance, Emittance, and Reflectance, 193 4.5 Broadband Emíttance and Absorptance, 194 4.6 Calculation of Emittance and Absorptance, 196 47 Measurement of Surface Radiation Properties, 198 48 Selective Surfaces, 199 Contents xvit 4.9 Mechanisms of Selectivity, 204 4.10 Optimum Properties, 208 4.tt Angular Dependence of Sotar Absorptance, 209 4.12. Absorptance of Cavity Receivers, 210 4.13 Specularly Reflecting Surfaces, 211 References, 213 5. RADIATION TRANSMISSION THROUGH GLAZIN ABSORBED RADIATION 216 5.1 Reflection of Radiation, 216 5.2 Absorption by Glazing, 220 5.3 Optical Properties of Cover Systems, 221 5.4 Transmittance for Diffuse Radiation, 226 5.5 Transmittance-Absorptance Product, 228 5.6 Angular Dependence of (to), 230 5.7 — Spectral Dependence of Transmittance, 231 5.8 Effects of Surface Layers on Transmittance, 234 5.9 Absorbed Solar Radiation, 235 S.J0 Monthly Average Absorbed Radiation, 239 5.11 Absorptance of Rooms, 246 5.12 Summary, 248 References, 248 6. FLAT-PLATE COLLECTORS 250 6.1 Description of Fiat-Plate Collectors, 250 6.2 Basic Flat-Plate Encrgy Balance Equation, 251 6.3 Temperature Distributions im Flat-Plate Collectors, 252 64 Collector Overall Heat Loss Coefficient, 254 6.5 Temperature Distribution Between Tubes and the Collector Efficiency Factor, 268 6.6 Temperature Distribution in Flow Direction, 276 6.7 | Coltector Heat Removal Factor and Flow Factor, 277 6.8 Critical Radiation Level, 281 6.9 Mean Fluid and Plate Temperature, 282 6.10 Effective Transmittance-Absorptance Product, 283 6.11 Effects of Dust and Shading, 286 6.12 Heat Capacity Effects in Flat-Plate Collectors, 287 6:13 Liquid Heater Plate Gcometries, 291 6.14 Air Heaters, 296 6.15 Measurements of Collector Performance, 301 6.16 Collector Characterizations, 302 6.17 Collector Tests: Efficiency, Incidence Angle Madifier, and Time Constant, 303 6.18 Test Data, 312 Contents 6.19 Thermal Test Data Conversion, 316 6.20 Flow Rate Corrections to Fa(To), and FpU,, 317 6.21 Flow Distribution in Collectors, 320 6.22 In Situ Coliector Performance, 322 6.23 Practical Considerations for Flat Plate Collectors, 323 6.24 Summary, 326 References, 327 CONCENTRATING COLLECTORS 330 7.1 Collector Configurations, 331 7.2 Concentration Ratio, 333 7.3 Thermal Performance of Concentrating Collectors, 336 7.4 Optical Performance of Concentrating Collectors, 341 7.5 Cylindrical Absorber Arrays, 343 7.6 Optical Characteristics of Nonimaging Collectors, 345 7.7 Orientation and Absorbed Energy for CPC Collectors, 351 7.8 Performance of CPC Collectors, 356 7.9 Linear Imaging Concentrators + Geometry, 358 7.10 Images Formed by Perfect Linear Concentrators, 362 7.11 Images from Imperfect Linear Concentrators, 367 7.12 Ray-Trace Methods for Evaluating Concentrators, 369 7.13 Incidence Angle Modifiers and Energy Balances, 370 7.14 Paraboloidai Concentrators, 376 7.15 Central Receiver Collectors, 377 7.16 Practical Considerations, 378 References, 379 ENERGY STORAGE 382 8.1 Process Loads and Solar Collector Outputs, 383 8.2 Energy Storage in Solar Process Systems, 384 8.3 Water Storage, 385 8.4 —Stratification in Storage Tanks, 388 8.5 Packed Bed Storage, 393 8.6 Storage Walls, 400 8.7 Seasonal Storage, 403 8.8 Phase Change Energy Storage, 405 8.9 Chemical Energy Storage, 410 References, 412 SOLAR PROCESS LOADS 414 9.1 Examples of Hourly Loads, 415 9.2 Hot Water Loads, 416 9.3 Space Heating Loads; Degree Days; Balance Temperature, 417 Contents ata LO. 1. 9.4 Building Loss Coefiicients, 420 9.5 Building Energy Storage Capacity, 422 9.6 Cooling Loads, 423 9,7 Swimming Pool Heating Loads, 423 References, 426 SYSTEM THERMAL CALCULATIONS 427 10.1 Component Modeis, 428 10.2 Collector Heat Exchanger Factor, 429 10.3 Duct and Pipe Loss Factors, 43! 10.4 Controls, 434 10.5 Collector Arrays; Series Connections, 436 10.6 Performance of Partially Shaded Collectors, 439 10.7 Series Atrays with Sections with Different Orientation, 441 10.8 Use of Modified Collector Equations, 444 10.9. System Models, 447 10.10 Solar Fraction, 450 10.11 Summary, 45! References, 452 SOLAR PROCESS ECONOMICS 453 1.1 Costs of Solar Process Systems, 454 11.2 Design Variables, 457 11.3 Economic Figures of Merit, 458 11.4 Discounting and Inflation, 461 1.5 Present Worth Factor, 463 11.6 Life Cycle Savings Method, 466 11.7 Evaluation of Other Economic Indicators, 471 118 The ?,, P; Method, 474 11.9 Uncertainties in Economic Analyses, 480 11.10 Summary, 483 References, 484 PART II - APPLICATIONS 12. Introduction, 485 SOLAR WATER HEATING - ACTIVE AND PASSIVE 487 12.1 Water Heating Systems, 487 12.2 Freezing and Boiling, 492 12.3 Auxiliary Energy, 494 Part I FUNDAMENTALS In Part I, we Ireat the basic ideas and calculation procedures that must be understood in order to appreciate how solar processes work and how their performance can be predicted. The first five chapters are basic to the material in Chapter 6. In Chapter 6 we develop equations for a collector which give the usefu! output in terms of the available solar radiation and the losses. An enesgy balance is developed which says, in essence, that the useful gain is the (positive) difference between the absorbed solar cnergy and the thermal losses. The first chapter is concemed with the nature of the radiation emitted by the sun and incident on the earth's atmosphere. This includes geometric considerations, i.e., the direction from which beam solar radiation is received and its angle of incidence on various surfaces and the quantity of radiation received over various time spans. The next chapter covers lhe effects of the atmosphere on the solar radiation, the radiation data that are available, and how those data can be processed to get the information thai we ultimately want — the radiation incident on surfaces of various orientations. Chapter 3 notes a set of heat transfer problems that arise in solar energy processes and is part of the basis for analysis of coltectors, storage units, and other components. The next two chapters treat interaction of radiation and opaque and transparent materials, i.e., emission, absorptíon, reflection, and transmission of solar and long wave radiation. These first five chapters lead to Chapter 6, a detailed discussion and analysis of the performance of flat-plate collectors. Chapter 7 is concemed with concentrating collectors and 8 with energy storage in various media. Chapter 9 is a brief discussion of the loads imposed on solar processes and the kinds of information that must be known in order to analyze the process. Chapter 10 is the point a1 which the discussions of individual components are brought together to show how solar process systems function and how their long- term performance can be determined by simulations. The object is to be able to quantitatively predict system performance; this is the point at which we proceed from components to systems and see how transient system behavior can be calculated. The last chapter in Part 1 is on solar process economics. ft concludes with a method for combining the large number of economic parameters into two which can be used to optimize thermal design and assess the eifects of uncertainties in an economic analysis. Chapter 1 SOLAR RADIATION The sun's structure and characteristics determine the nature of the energy it radiates into space, The first major topic in this chapter concemns the characteristics of this energy outside of the earth's atmosphere, its intensity and spectrat distribution. We will be concemed primariiy with radiation in à wavelength range of 0.25 to 3.0 um, the portion of the electromagnetic radiation that includes most of the energy radiated by the sun. “The second major topic in this chapter is solar geometry, i.e., the position of the sun in the sky, the direction in which beam radiation is incident on surfaces of various orientations, and shading. The third topic is extraterrestrial radiation on a horizontal surface, which represents the theoretical upper limit of solar radiation available at the earth's surface. An understanding of the nature of extraterrestriat radiation, the effects of orientation of a receiving surface, and the theoretically possible radiation at the earth's surface is important in understanding and using solar radiation data. 11 THE SUN The sun is a sphere of intensely hot gaseous matter with a diameter of 1.39 x 10º m and is, on the average, 1.5 x 10!! m from the earth. As seen from the carth, the sun rotates on its axis about once every four weeks. However, it does not rotate as a solid body; the equator takes about 27 days and the polar regions take about 30 days for cach rotation. “The sun has an effective blackbody temperature of 5777 K.! The temperature in the central interior regions is variously estimated at 8 x 106 to 40 x 108 K and the density is estimated to be about 100 times that of water. The sun is, in effect, a continuous fusion reactor with its constituent gases as the “containing vessel" retained by gravitational forces. Several fusion reactions have been suggested to supply the Urhe effective blackbody temperature of 5777 K is the temperature of a blackbody radiating the same amount of energy as does the sun. Other effective temperatures can be defined, for example, that comesponding to the blackbody temperature giving the same wavelengih of maximum radiation as solar radiation (about 6300 K). ; : 4 Solar Radiation energy radiated by the sun. The one considered the most important is a process in which hydrogen (i.e., four protons) combines to form helium (i.e., one helium nucleus): the mass of the helium nuclcus is less than that of the four protons, mass having been lost in the reaction and converted to energy. “The energy produced in the interior of the solar sphere at temperatures of many millions of degrees must be transferred out to the surface and then be radiated into space, A succession of radiative and convective processes occur with successive emission, absorption, and reradiation; the radiation in the sun's core is in the x-ray and gamma-ray paris of the spectrum, with the wavelengths of the radiation increasing as the temperature drops at larger radial distances. A schematic structure of the sun is shown ir Figure 1.1.1. It is estimated that 90% of the energy is generated in the region of O to 0,238 (where R is the radius of the sun), which contains 40% of the mass of the sun. At a distance 0.7R from the center, the temperature has dropped to about 130,000 K and the density has dropped to 70 kg/m?; here convection processes begin to become important, and the zone from 0.7 to 1.0R is known as the convective zone. Within this zone the temperature drops to about 5000 K and the density to about 105 kg/n?, “The sun's surface appears to be composed of granules (irregular convection cells), with dimensions from 1000 to 3000 km and with cell lifetime of a few Corana T = — 108k 2 = very low Chromosphere 7 = 5000 K + Reversing layer Hundred ot = SA Phatosphere iupper laver of the convective zone, saures of most solar / TT âo% ot mass radiation) 18% 6! vamo So at rear | gerettud 0.23R 7 O a OR T=-B40XIOK p= 10kgim? mu Figure 1.1.1 The structure of the sun. 1.2 The Solar Constant 5 minutes. Other features of the solar surface are small dark arcas called pores, which are cf the same order of magnitude as the convective cells, and larger dark areas called sunspots, which vary in size. The outer layer of the convective zone is called the photosphere. The edge of the photosphere is sharply defined, even though it is of low density (about 10-4 that of air at sea level). It is essentially opaque, as the gases of which it is composcd are strongly ionized and able to absorb and emit a continuous spectrum of radiation. The photosphere is the source of most solar radiation. Outside of the photosphere is a more or less transparent solar atmosphere, observable during total solar eclipse or by instruments that occult the solar disk. Above the photosphere is a layer of cooler gases several hundred kilometers deep called the reversing layer. Outside of that is a layer referred to as the chromo- sphere, with a depth of about 10,000 km. This is a gaseous layer with temperatures somewhat higher than that of the photosphere but with lower density. Still further out is the corona, à region of very low density and of very high (108 K) temper- ature. For further information on the sun's structure see Thomas (1958) or Robinson (1966). This símplified picture of the sun, its physical structure, and its temperature and density gradients will serve as a basis for appreciating that the sun does not, in fact, function as a blackbody radiator at à fixed temperature. Rather, the emitted solar radiation is the composite result of the several layers that emit and absorb radiation of various wavelengths. The resulting extraterrestrial solar radiation and its spectral distribution have now been measured by various methods in several experiments; the results are noted in the following two sections. 12 THE SOLAR CONSTANT Figure 1.2.1 shows schematically he geometry of the sun-carth relationships. The eccentricity of the carth's orbit is such that the distance between the sun and the earth Sun 127x10'm 7900 mi Sojar constant | =1367 Wim? Gocl = 483 But hr 92 Mm? hr = 1495 x 10!!m =9.3x10/mi Distance al 17% Figure 1.2.1 Sun-earth relationships. 10 Solar Radiation Variation of the earth-sun distance, however, does lead to variation of extraterrestrial radiation flux in the range of + 3%. The dependence of extraterrestrial radiation on time of year is indicated by Equation 1.4.1 and is shown in Figure 14.1. Gon = Gec[1+ 0.033 cos 3604) (1.4.1) 365 where G,, is the extraterrestrial radiation, measured on the plane normal to the radiation on the ath day of the ycar. 15 DEFINITIONS Several definitions will be useful in understanding the balance of this chapter. Air Mass m The ratio of the mass of atmosphere through which beam radiation passes to the mass it would pass through if the sun were at the zenith (i directly overhead). Thus at sea level, m = 1 when the sun is at the zenith, and m = 2 for à zenith angle 8, of 60º. For zenith angles from 0º to 70º at sea level, to a close approximation, m= ljcos 8, (1.5.1) For higher zenith angles, the effect of the earth's curvalure becomes sigaificant and must be taken into account. For a more complete discussion of air mass, see Robinson (1966), Kondratyev (1969), or Garg (1982). Beam Radiation The solar radiation received [rom the sun without having been scattered by the atmosphere. (Beam radiation is often referred to as dircct solar radiation; to avoid confusion between subscripts for direct and dilfuse, we use the term beam radiation) Diffuse Radiation The solar radiation received from the sun after its direction has been changed by scattering by the atmosphere. (Diffuse radiation is referred to in some meteorological literature as sky radiation or solar sky radiation: the definition used here wil) distinguish the diffuse solar radiation from infrared radiation emitted by the atmosphere.) Total Solar Radiation The sum of the beam and the diffuse solar radiation on a surface.3 (The most common measurements of solar radiation are total radiation on a horizontal surface, often referred to as global radiation on the surface.) Irradiance, W/m? The rate at which radiant energy is incident on à surface, per unit area of surface, The symbol G is used for solar irradiance, with appropriate subscripts for beam, diffuse, or spectral radiation. 3 Total solas radiation is sometimes used to indicate quantities integrated over all wavelengths of the solar spectrum. 1.5 Definitions u Irradiation or Radiant Exposure, Jjm? The incident energy per unit area on a surface, found by integration of irradiance over à specified time, usually an hour or a day. Insolation is a term applying specifically to solar energy irradiation. The symbol /f is used for insolation for a day. The symbol / is used for insolation for an hour (or other period if specified). The symbols H and ! can represent beam, diffuse, or total and can be on surfaces of any orientation. Subscripts on G, H, and 4 are as follows: o refers to radiation above the carth's atmosphere, referred to as extraterrestrial radiation; b and d refer to beam and diffuse radiation; T and n refer to radiation on a tilted plane and on a plane normal to the direction of propagation. If neither 7 nor » appear, the radiation is on a horizontal plane. Radiosity or Radiant Exitance, W/m? The rate at which radiant energy leaves a surface, per unit area, by combined emission, reflection, and transmission. Emissive Power or Radiant Seif-Exitance, W/m? The rate at which radiant energy leaves a surface per unit area, by emission onty. Any of these radiation terms, except insolation, can apply to any specified wavelength range (such as the solar energy spectrum) or to monochromatic radiation. Insolation refers only to irradiation in the solar energy spectrum. Solar Time Time based on the apparent angular motion of the sun across the sky, wilh solar noon the time the sun crosses the meridian of the observer. Solar time is the time used in al] of the sun-angle relationships; it does not coincide with local clock time, Iris necessary to convert standard time to solar time by appiying two corrections. First, there is a constant correction for the difference in longitude between the observer's meridian (longitude) and the meridian on which the local standard time is based.é The sun takes 4 minutes to transverse 1º of longitude, The second correction is from the equation of time, which takes into account the perturbations in the earth's rate of rotation which affect the time the sun crosses the observer's meridian. The difference in minutes between solar time and standard time is Solar time — standard time = 4(Lg — Lic) 4 E (1.5.2) where Ly i» lhe standard meridian for he local time zone. L,, is the longitude of the location in question. and longitudes are in degrees west. i.e.. 0º <L < 360º, The equation of time E (in minutes) is determined from Figure 1.5.1 or Equation 1.5.35 [from Spencer (1971). as cited by Igbal (19833). E = 229,20.000075 + 0.001868 cos B — 0.032077 sin B — 0014615 cos 28 — 0.04089 sin 28 ) (1.5.3) 2 Standard meridians for continental U.S. time zones are: 105ºW; and Pacific, 1209W. 5 AI equations use degrees, not radians. ste. 75ºW; Central, 90ºW; Mountain. Solar Radiation Equation of time, min o | —0 NA DJ FE MA MOIS ASOÕÃGNÕD Month Figure 15.1. The equation of time £ in minutes, às à function of time of year where B=(n- 38 (1.536) 365 and n = day of the year. Thus 1 Sn < 365. Note that the corrections for equations of time and displacement from the standard meridian are in minutes and that there is a 60 minute difference between daylight saving time and standard time. Time is usually specified in hours and minutes. Care must be exercised in applying the corrections, which can total more than 60 minutes. Example 1.5.1 At Madison, WI, what is the solar time corresponding to 10:30 AM central time on February 3? Solution In Madison, where the longitude is 89 4º, Equation 1.5.2 gives tandard time + 4(90 — 89.4) + E tandard time + 2.4 + E Solar time 1.6 Direction of Beam Radiation 13 On February 3. n = 34, and from Equation 1.5.3 or Figure 1.5.1, E is —13.5 minutes, so the correction to standard time is —11 minutes. Thus 10:30 AM central standard time is 10:19 AM solar time, In this book all times are assumed to be solar times unless indication is given otherwise. 1.6 DIRECTION OF BEAM RADIATION The geometric relationships between a plane of any particular orientation relative to the earth at any time (whether that plane is fixed or moving relative to the earth) and the incoming bcam solar radiation, that is, the position of the sun relative to thai Plane, can be described in terms of several angles [Benford and Bock (1939)]. Some of the angles arc indicated in Figure 1.6.1. The angles are as follows: é Latitude, ihe angular location north or south of the equaior, north positive; 90 < gs 90º. ô — Declination, the angular position of the sun at solar noon (i.e,, when the sun is on the local meridian) with respect to the plane of the equator, north positive; -23,45º 5 6< 2345º, B Slope, the angle between the plane of the surface in question and the horizontal; OS Bs 180º. (> 90º means that the surface has a downward facing component.) Y Surface azimuth angle, the deviation of the projection on a horizontal plane of the normal to the surface from the local meridia, with zero due south, east negative, and west positive; —180º < y< 180º «Hour angle, the angular displacement of the sun east or west of the local meridian due to rotation of the earth on its axis at 15º per hour, morming negative, aftemoon positive, 8 — Angle of incidence, the angle between the beam radiation on a surface and the normal to that surface. Additional angles are definc that describe the position of the sun in the sky: 8. Zenith angle, the angle between the vertical and the line to the sun, angle of incidence of beam radiation on a horizontal surface, &, Solar altitude angle, the angle between the horizontal and the line to the sun, ie. the complement of the zenith angle. Y Solar azimuth angle, the angular displacement from south of the projection of beam radiation on the horizontal plane, shown in Figure 1.6.1. Displacements east of south are negative and west of south are positive. .e., the The deciination 8 can de found from the equation of Cooper (1969): - 8=23.45 sin(a60268.2-0) (1.6.1) a Solar Radistion f Zenith Normaito horizonta! surtace Sun Figure 1.6.1 (a) Zenith angle, slope, surface azimuth angle, and solar azimuth angle for a tilted surface, (o) Plan view showing solar azimuth angle. Table 1.6.1 Recommended Average Days for Months and values of n by Monthst For the Average Day of the Month a forith Month DayofMonth Date mn Dayof Year — à Declination January i 17 7 -209 February a+i 16 47 30 March s9+i 16 7 -24 April 90+1 15 105 94 Mey 10+i 15 135 18.8 June 151415 n 162 Bl July 18141 17 198 212 August n2+i 16 228 13,5 September 24341 is 258 October mm+i 15 288 November 30 +1 14 318 December 3341 10 344 à From Klein (1977) The day of the year n can be conveniently obtained with the help of Table 1,6.1. Note that declination is a continuous fanction of time. The maximum rate o change of declination is at the equinoxes, when it is about 0,5º/day. For most engineering calculations, the assumption of an integer n to represent a day results in a satisfactory calculation of declination. 8.6 Direction of Beam Radiation 15 There is à set of usefu) relationships among these angles. Equations relasing the angle of incidence of beum radiation on a surface, 8, to the other angles are cos 8=sin ôsin gcos -sin 8cos gsin fcos y + cos Sos gos fcos w +cos Gsin sin ficos ycos o +cos 6sin Bsin ysin o (1.6.2) and cos 8 = cos &, cos + sin 8, sin B cos( -y (1.6.3) Example 1.6.1 Calculate the angie of incidence vf beam radiation on a surface located at Madison, Wi at 10:30 (selar time) on February 13, if the surface is tilted 45º from the horizontal and pointed 15º west of south. Solution Under these conditions, n is 44, lhe declination 8 from Equation 1.6.3 is —14º, the hour angle co is -22.5º (15º per hour times 1.5 hours before noon), and the surface azimuth angle is 15º. Using a slope 8 of 45º and the latitude & of Madison of 43ºN, Equation 1.6.2 is cos 8= sint-t4) sin 43 cos 45 —sin(-14) cos 43 sin 45 cos 15 + cos(-14) cos 43 cos 45 cos(-22.,5) + COS(-14) sin 43 sin 45 cos 15 cos(-22.5) + cos(-14) sin 45 sin 15 sin(-22,5) cos 8=- 0.117 +0.121 + 0,464 + 0418 — 0.068 =0.817 B=35º E There are several commonty occurring cases for which Equation 1.6,2 is simplificd. For fixed surfaces sloped toward the south or north, that is, with à surface azimuth angle of 0º or 180º (a very common situation for fixed flat-plate coltectors), the last term drops out. For vertical surfaces, = 90º and the equation becomes cos 6 sin Écos cos y+ cos ôsin pros ycos « +cos ôsin ysin «o (1.6.4) ; Solar Radiation ” declination leads directly to times of sunrise and sunset and day length, for either isphere. . , pemisçãe additional angle of interest is the profile angle of beam radiation on à receiver plane R that has a surface azimuth angle of y. It is the projection s the solar altitude angle on a vertical plane perpendicular to the plane in Susie Expressed another way, it is the angle through which a plane that is initially À is in the plane of the surface in question in horizontal must be rotated about an axis 1% 1| e order to include ihe sun. The solar altitude angle o, tie, £BAC), and the ei angle a, (ie., ZDEF), for the plane R are shown in Figure 1.6.4. The plane A 8, includes the sun. Note that the solar altitude and profile angle are the same when u E sun is in a plane perpendicular to the surface R (e.g., at solar noon for a surfáce wi à surface azimuth angle of 0º or 180º). The profile angle is useful in calculating shading by overhangs. It can be calculated from tino, (1.6.12) cos(k— Example 1.6.3 ; O west of Calculate the solar altitude, zenith, and profile angles for a surface facing 25º west of i o. south at 4:00 PM solar time on March 16 at a latitude of 43º. Solution o The solar altitude angle «3, is a function only of time of day and dectination. E March 16, from Equation 1.6.1 (or Table 1.6.1), dis 2.49, At 4:00 PM, qo = 60º. From Equation 1.6.5, recognizing that cos 8, = sin(90 — 8,) = sin o, sin &, = cos 43 cos(-2.4) cos 60 + sin 43 sin(-2.4) = 0,337 a, = 19.7º a EF) fe Figure 2.6.4 The solar altitude angio x, (LBAC), and the profile angie a, (DEF) for a surface R, 1.7 Angles for Tracking Surfaces a The solar azimuth angle for this time can be caiculated with Equation 1.6.6: tangj= im =2330, Do “= 66,80 —cos 43 tan (15.4) % = 66.8 COS Mew=tan(-2.46)ftan 43 = 0045, ag = 9260 Thus C,, Cy and Cy are all Land y, = 3 = 66.89, The profite angie for the surface with y= 25º is calculated with Equation 1.6.12: tan co = — lan 19, =25.7º E TOs(66.8 = 23) 0.480, Op = 25.7 in various types nf charts. Examples of these are the Sun Angie Calculator (1954) and the solar position charts (plots of «x, or 8, vs. % for various &, 8, and 09) in Section 1.9 and Appendix H. Care is necescary in interpreting information from other sources, since nomenclature, defini itions, and sign conventions may vary from those used here, 17 ANGLES FOR TRACKING SURFACES Some soar collectors "track" the sun by moving in prescribed ways to minimize the angle of incidence of beam radiation on their surfaces and thus maximize the incident beam radiation. The angtes of incidence and the surface azirmuth angles are needed for these coliectors. The relationships in this section will be useful in radiation calculations for these moving surfaces. For further information see Eibling et al. (1953) and Braun and Mitchel] (1983). Tracking systems are classified by their morions, Rotation can be about a single axis (which could have any ocientation but which in practice is usually horizontal cast-west, horizontal north-south, vertical, or parallel 10 the earth's axis) or it can be about two axes, The following sets of equations (except for Eguations 1.7.4) are for surfaces that rotate on axes that are parallel to the surfaces. For a plane rotated about a hor izontal east-west axis with a single daily adjustment so that the beam radiation is normal to the surface at noon each day, cos6= sin? 8 + cos28 cos q (17.19) nz Solar Radiation The slope of this surface wil] be fixed for each day and will be B=ho-d (1.7.1b) The surface azimuth angle for a day will be 0º or 180º depending on the latitude and declination: H(4-9>0,7=0 GO H(g-9)<0, y= 180º For a plane rotated about a horizontal east-west axis with continuous adjustment to minimize the angle of incidence, cosg=(1--cos26 sina)? (1.7.28) The stope of this surface is given by tan = tan 8. fos 4] (1.7.2b) The surface azimuth angle for this mode of orientation will change between 0º and 180º if the solar azimuth angle passes through £ 90º, For either hemisphere, Fbd<90, y=0º (1.7.20) Hlj>90, y= 180º For a plane rotated about a horizontal north-south axis with continuous adjustment to minimize the angle of incidence, cos B=(cos28, + cos? Bsin? q)? (1.732) The slope is given by tan B=1an ô.jcos (7— 4) €1.7.3b) The surface azimuth angle y will be 90º or -90º depending on the sign of the solar azimuth angle: Hx>0, 7=90º (1.730) IE 4<0, y=-90º For a plane with a fixed slope rotated about a vertical axis, the angle of incidence is minimized when the surface azimuth and solar azimuth angles are equal. 17 Angles for Tracking Suríaces From Equation 1,6.3, the angle of incidence is cos = cos 8. cos B + sin 8. sin 8 The slope is fixed, so B= constant The surface azimurh angfe is Y=% (1.7.43) (1.7.4b) €1.7.40) For a plane rotated about a north-south axis parallel to the earth's axis with continuous adjustment (o minimize 8, cos B=cos é The slope varies continuously and is tan p= DO cos y The surface azimuth angle is gs p= tam! CSM A, gore, cos 6'sin q where cos = cos 8. cos À + sin 8. sin q b if [ue Dest), %=0 c- cos &' sin q, 1 otherwise 1 if%20 Ge Nha = ifk<o (17.59) (17.5) (17.5) (17.59) (17.5) (1.758) Fora plane that is continuously tracking about two axes to minimize the angle of incidence, cos 8 (1.769) 7 “1 "| | Mu Solar Radintion za B=a (1.7.6b) v=% (1.7.60) Example 1.7.1 Caleulate the angle of incidence of beam radiation, the slope of the surface, and the surface azimuth angle for a surfaceat a = 40º, 8= 21º, and q = 30º (2:00 PM) and b 4=40º, 8=21º, and 0= 100º if it is continuously rotated about an east- west axis to minimize 6. Solution a Use Equations 1.7.2 for a surface moved in this way, First caloulate the angle of incidence: 8=cost(l- cos? 21 sin? 30)? = 27.8? Next calculate 6, from Equation 1.6.5: 8. = cos-t(cos 40 cos 21 cos 30 + sin 40 sin 21)=31.8º we now need 1, Using Equations 1.6.6g, cos (2, = (tan 21)/tan 40) and 6d, = 62.80. Thus C; = 1,C)= 1.C3= 1, and y,=y. So = cm! Si 30.205 21 = 62,49 HE ingl é Then fom Equation 1.7.2b B = tan-I(tan 31.8 cos 62.4) = 16.0º From Equation 1.7.2, with X, < 90,7=0. b The procedure is the same as in a: 6 = cosl - cos? 21 sin? 100)? = 66.89 8, = cos-Ifcos 40 cos 21 cos 100 + sin 40 sin 21) = 83.9º The value of 09, is still 62.8º, from part a. In this case, the constanis in Equation dem L66aareC;=-1,C,= 1, and C;=1. net si 100 cos 21 =1124º = sin! sin 100 GO 2 + 180 18 Ratio of Beam Kadiatlon on Tiied Surface to That on Horizontal Surface 2 The slope is lhen B=tan-! tan 83.9kos 12.4/= 74.30 And since Lg] > 90, 7 will be 180º. (Note that these results can be checked using Equation 1.6.5.) 1.8 RATIO OF BEAM RADIATION ON TILTED SURFACE TO THAT ON HORIZONTAL SURFACE For purposes of solar process design and performance calculations, it is often necessary to calculate the hourly radiation on a tilted surface of a collector from meusurements or estimates of solar radiation on a horizontal surface. The most commonly available data are total radiation for hours or days on the horizontal surface, whereas the need is for beam and diffuse radiation on the plane of à collector. The geometric factor R,, the ratio of beam radiation on the tiited surface to that on a horizontal surface at any time, can be calculated exactly by appropriate use of Equation 1.6.2. Figure 1.8.1 indicates the angle of incidence of beam radiation on the horizontal and tilted surfaces. The ratio G,4/G, is given byó - Cor p= LE Gh Ghacos Goncos 6 cos6 (1.8.1) cos 8, and cos 8 and cos 8, are both determined from Equation 1.6.2 (or from equations derived from Equation 1.6.2). Sb Sim » Figure 1.8.1 Beam radiation on horizontal and tilted surfaces. É The symbol G is used in this book to denote rates, while 7 is used for energy quantities integrated over an tour. The original development of Ry by Hoitel and Woeriz (1942) was for hourly periods; for an hour (using angles at the midpoint of the hout), Rp = 174/p- je Solar Radiation Two situations arise, for positive values or for negative values of (q — f). For positive values, the charts are used directly. If (6 P) is negative (which frequently occurs when collectors are sloped for optimum performance in winter or with vertical collectors), the procedure is modified. Determine cos 8, as before. Determine cos 6 from the appropriate absolute value of ( — £) using the curve for the other hemisphere, that is, with the sign on the dectination reversed. Example 1.8.2 Calculate R, for a surface at tatitude 40ºN at a tilt 30º toward the south for the hour 9to 10 solar time on February 16. Solution Use Figure 1.8.2c) for the hour £2.5 hours from noon as representative of the hour from 9 to 10. To find cos 6, enter at a latitude of 40º for the north latitude date of February 16. Cos 6, = 0.45. To find cos 8, enter at a latitude of 6 — B= LOS tor the same date. Cos 8=0.73. Then Ry = 08 bÊ cos 8, The ratio can also be calculated using Equation 1.8.2. The declination on February 16is-13º: Rey = 28 10 cos 13) cos(37.5) sin 0 sini3) , gy E “cos 40 cos(-13) cos(-37.5) + sin JO sin(-IB) — Example 1.8.3 Calculate R, for a latitude 40ºN at a tilt of 50º toward the south for the hour 9 to LO solar time on February 16. Solution Cos 8, is found as in the previous example and is 0.45, To find cos 8, enter at an abscissa of 10º, using the curve for February 16 for south latitudes. The value of cos 8 from the curve is 0.80. Thus R, = 0.80/0,45 = 1.78. Equation 1.8.2 can also be used: = cos(-10) cos(-3) cos(-37.5) + sint-10) sin(-13) | cos 40 cos(-13) cos(-37.5) + sin 40 sin(-13) ro Má 1 is possible, using Equation 1.8.2 05 Figures 1.8.2, to construct plots showing the effects of coilector tilt on &, for various times of the year and day. Figure 1.8.3 shows such a plot for a latitude of 40º and a slope of 50º. Ht illustrates 1.8 Ratio of Beam Radiation on Tiited Surface to That on Horizontal Surface cosbicos 8. Ro = Declination, degrees Figure 1.8.3 Ratio &, for a surface jo from solar noon. > dit slopo 50 to south at latitude 46º for various hours that very large gains in incident bea: Im radiati ti a surface toward the equator. ion are to be had by tilting a Teceiving for a plane cotated continuousty beam radiation on th: from E: 7.2a, the ratio of beam radiation on the e plane, from Equation 1.7.2a, the rati j e «7. 2a, the ratie ati Plane to that on a horizontal surface at any time is Ry= A co8ô sinta)'? £Os fcos deos + sin Gsin é (1.8.4) a | | | | 2 Solar Radiation 19 SHADING Three types of shading problems occur so frequently that meshods are needed to cope with them. The first is shading of a collector, window, or ather receiver by near-by trees, buildings or other obstructions. The geomenies may be irregular, and systematic calculations of shading of the receiver in question may be difficult. Recourse is made to diagrams of the position of the sun ia the sky, e.g., plots of solar altitude a, vs. solar azimuth , on which shapes of obstructions (shading profiles) can be superimposed to determine when the path from the sun to the point in question is blocked. The second type includes shading of collectors in other than the first row of muitirow arrays by the collectors on the ajoining row. The third includes shading of windows by overhangs and wingwalls. Where the geometries are regular, shading is amenabie to calculation, and the results can be presented in general form. This will be treated in Chapter 14. At any point in time and at a particular latitude, q, 6, and 6 are fixed. From the equations in Section L.6, the zenith angle 8, or solar altitude angle 0x, and the solar azimuth angle 7, can be calculated. A solar position plot of 8, and q, vs. 7, for latitudes of £45º is shown in Figure 1.9.1. Lines of constant declination are labeled by dates of mean days of the months from Table 1,6.1. Lines of constant hour angies labeled by hours are also shown. Plots for tatitudes from O to *70º are included in Appendix H. The angular position of buildings, vingwalls, overhangs or other obstructions can be entered on the same plot. For example, as observed by Mazria (1979) and Solar Altitude Angla ais so 100 st O o Solar Azimuth Angle, 7s Figure 1.9.1 Sotar position plot for £ 45º latitude. Solar attitude angle and solar azimuth angle are functions of declination and hour angle, indicated on the plots by dates and times. The dates shown are for norihem hemisphere; for southern hemisphere use the corresponding dates as indicated in Figure 1.8.2, See Appendix H for other latitudes. 19 Shuding 33 A - . nderson (1982), if à building or other obstruction of known dimensions and orient, q cleo rio ea FM Me it time ie te ecc amd O poe é e angular coordinates corresponding to altitude and azimuth amgle a) can be caleulated fo oo CORCE seimuth angle 7, and object aftinde Example 5 1.9.1 and 192 a trigonometric considerations. This is illustrated in azimuih angles may be made : po cn Vely, measurements of object altitude and 'y be made at the site of à Proposed receiver and the angles plorted On the solar position j : ã Plot. A variet s ; angles. 'y Of instruments are available to measure the Example 1.9.1 Om to the north of à long wall that shades itwhen is af uniform height of 2.5 m above the center of this wall on a solar position chart with a the wal] all oriented on a southeast to northwest axis displaced À proposed collector site a S is 10: the sun is low in the sky. The wall the proposed colector area. Show Oriented cast-west and b the w 20º from east-wesr. Solution In each case, we pi i + We pick several points on the to) of a i i it piso P Of the wall to establish the coordinates & — Take three points indicated bs YA, B, and C on the di i and Band C westor A, Points B' and C' are taken lo the eus DER vi eo object altitude angles as B a; ih bee ai angina Same pri nd C and with object azimuth angles changed only in Solar Radiation For point A, the object azimuth %, is 0º. The object altitude angle is tan 0,4 = 25/10, Gy = 140º For point B, SB =(102+ 102)? = 14.Lm, tan Gg = 2.5/14.1, Gog = 10.0º tan %g = 1010, og = 450º For point C, SC =(102+302)2 =31.6m, =4.,52º 1.6º im = 25816, tan %ç=3000, Ke = There are points coresponding to Band C butto the cast of A; these will tao bject azimuth angles except with negative signs. The shading prof e teinod b je, Ttis shown by the solid i inates dent of latitud determined by these coordinates is independes line on the plot for à = 45º. Note that at object azimuth angles of 90º, the object distance becomes infinity and the object altitude anglo becomes 0? o E 10 8) 20 7o| E 30 s E ao <s a É s És ç o E 3 ê 70 20) 6a so 10 so -50 o E 100 west Solar Azimuth Angle 75 The sun is obscured by the wall only during times shown on the diagram. The wall does not cast a shadow on point S at any time of day from tate March to mid- September. For December 10, it casts a shadow on point S before 9:00 AM and after 3:00 PM. 19 Shading 5 b. The obstruction of the sky does not show cast-west symmetry in this case, so five points have been chosen as shown to cover the desirable range. Point A is the same as before, ie., 0, = 1409, % = 0º. Arbitrasily select points on the wall for the caleulation. In this case the caleulations are easier if we select values of the object azimuth angle and calculate from them the corresponding distances from the point to the site and the corresponding €,. In this case we can select values of 7, for points B, €, D, and E of 45º, 90º, 30º, and —60º. For point B, with £,g = 45º, the distance SB can be calculated from the law of sines: sin 79 — Sin (18045 - 70) SB 10 + SB=104m tan Og = 2.5/10.4, Gon = 13.5º For the point D, with %,p = -30º, the calculation is sin LIO HSC180- NO 30) SD , To D=146m tan Gp = 2.5/14.6, mp =9.7º The caleulations for points C and E give O,c =5.2ºat fc = 9º ando, = 2.6º ut Yg = 600º. The shading profile determined by these coordinates is plotted on the solar position chart for É = 45º and is shown as the dashed line. In this case, the object altitude angle goes to zero at azimuth angles of -70º and 110º. In cither case, the area under the curves represents the wall, and the times when the wall would obstruct the beam radiation are those times (declination and hour angles) in the areas under the curves. E There may be some freedom in sclecting points to be used in plotting object coordinates, and the calculation may be made easier (as in the preceding example) by selecting the most appropriate points. Applications of trigonometey wil! always provide the necessary information. For obstructions such as buildings, the points selected must.jnclude comers or limits that define the extent of obstruction. It may or may not be necessary to select intermediate points to fully define shading. This is illustrated in the following example. Example 1.9.2 It is proposed to install a solar collector at a level 4.0 m above the ground, A rectangular building 30 m bigh is focated 45 m to the south, has its long dimension om an east-west axis, and has dimensions shown in the sketch, The latitude is 45º. a Sotar Radiation At any point in time, the solar radiation incident on a horizontal plane outside of the atmosphere is the normal incident solar radiation as given by Equation 1.4.1 divided by R,; Go= Get + 0.033 cos “ln cos 8. (40.1 where G. is the solar constant and 2 is the day of the year. Combining Equation 1.6.5 for cos 8, with Equation 1.10.1 gives G, for a horizontal surface at any time between sunrise and sunset. G= Gl + 0.033 cos 38ônicos ócos ôcos o + sin ésin 5) (1.10.2) It is often necessary for calculation of daily solar radiation to have the integrated daily extraterrestriaf radiation on a horizontal surface, H,. This is obtained by integrating Equation 1.10.2 over the period from suniise to sunset. IF G, is in watts per square meter, H, in joules per square meter is 24X 3600Gx e 7 (1 40.033 cos 60 x (cos cos sin co, + Ho sin Qin) (1103) where 1», is the sunser hour angle, in degrees, from Equation 1.6.10. The monthly mean! daily extraterrestrial radiatiom, Ma, is à uscful quantity. For latitudes in the range +60 to -60 it can be calculated with Equation 1.10.3 using n and 6 for the mean day of the month!! from Table 1.6.1. 1, is plotted as a function of latitude for the northem and southem hemispheres in Figure 1.10.1. The. curves are for dates that give the mean radiation for the month and thus show fo. Values of H, for any day can be estimated by interpotation. Exact values of H, for all latitudes are given in Table L.10.1, Example 1.10. What is H,, the day's solar radiation on a horizontal surface in the absence of the atmosphere, at latitude 43ºN on April 15? Solution For these circumstances, n = 105 (from Table 1.6.1), 6 = 9,4º (from Equation 1.6.1), and = 43º, From Equation 1.6.10 cos «3, = tan 43 tan 9.4 and (,= 98.9º 10 An overbar is used throughout the book to indicate a monthly average quantity. 1lThe men day is the day having H, closest to FZ,. LO Extraterrestrial Radiation on u Horizontal Surface so “0 430 DU É Pd q ia nes 3 1 O Sep. 15 q Ma 16d no Oct. 15 O o - Alia AS SINE 20 30 0 50 so 70 so so North latitude, degrees so Ho. daily Mim? É 8 8 June 19 Juy 17 May 35 Aug. 16 30 40 50 o 7 ao so South latitude, degrees Figure 1.10.1 Extraterrestrial daily radiation on a horizontal surface. mean days of the month from Table 1.6.1. The curves are for the 41 í | 1.10 Extraterrestrial Radiation on a Horizontal Surface “ Solar Radiation a Table 1.10.1 Monthly Average Daily Extraterrestrial Radiation, MJ/m? Then from Equation 110,3, with Go, = 1367 Wim? é Jan Feb Ma Apr May Jun Jul Aug Sep Ot Nov De TOTO | 24x 3600 x 1367 H, = AX 3600 x 136 360 x 105 % 00 00 12 193 372 448 54 00 00 00 | e x -[1+0.033 cos 365 — [ss 00 00 22 192 370 47 64 00 00 00 i fm 00 00 47 196 366 442 90 06 00 00 i ) xx 989 : b y “ ” ! 1 X(cos 43 cos Es s 00 07 78 210 359 433 nº 22 00 006 ! cos 9.4 sin 98.9 + ão SD 43 sin 9, 7 O 27 109 231 353 421 148 49 03 00 i 6 12 54 139 254 357 410 7 78 20 04 3.8 MJm? O 35 83 169 2716 366 410 38A 309 205 108 45 23 From E 5 62 13 198 296 376 394 26 241 134 73 48 T le tgure 1.10.a, for the curve for April, we read H = 34.0 MJ/m?, and from 50 91 144 225 315 385 415 400 341 255 167 103 77 i able L.10.1 we obtain 44, = 33.8 MJjm? by interpotation. ! ” as 122 174 251 332 292 417 404 353 278 196 133 102 i . 4 153 203 274 346 397 417 406 364 208 224 164 137 | His also of interest to caleulate the extraterrestrial radiation on horizontal ce a : . a a horizontal 35 183 231 296 358 400 415 406 373 7 250 193 168 | sur face for an hour Peciod. Integrating Equation 1.10.2 for a period between hour 3 213 257 315 368 400 411 404 378 332 274 222 199 angles co, and 4 which define an hour (where “% is the larger), 25 22 282 332 315 398 404 400 382 246 296 250 229 . 1% 20 305 347 379 393 305 393 382 356 316 217 25% 12x 3600 i 15 296 326 359 380 385 384 383 380 364 334 301 285 lo= z — Gs (1 + 0.033 cos 380 a) 10 320 M4 368 379 375 370 31 375 370 350 324 31 | 5 3M2 360 375 IA 363 353 356 367 372 363 345 335 d . . . n(a - q O 362 374 3718 367 348 335 340 357 372 373 363 357 Picos ôsin 0» —sin mn) + a sin êsin 3) €1.10.4) -5 380 385 379 356 350 34 321 344 369 380 379 376 0 395 393 377 345 3LL 292 299 329 363 385 303 394 (The limits «o, and co, may define a time other than an hour.) -I5 408 398 372 330 289 268 276 31 354 387 404 409 . -20 418 400 364 313 266 22 252 201 33 386 412 d21 Example 1.10,2 -25 425 400 354 293 21 25 26 2710 329 382 dit 431 Wiat is che solar radiati . DO 430 397 340 W2 HA 187 199 246 H2 6 420 48 . a e o ar radiation on a horizontal surface in the absence of the atmosphere at 35 432 391 325 248 186 158 170 221 293 366 420 442 atitude 43ºN on April [5 between the hours of 10 and nº “0 431 382 306 223 158 120 142 194 272 355 417 445 Solution -A5 428 371 286 196 129 100 113 166 249 340 412 445 -S0 423 357 263 168 100 72 84 138 224 324 405 443 The deelination is 9.49 (from th ious . -55 417 341 239 139 72 45 57 109 198 305 306 440 Equation [.10.4 with o nd o pg PD. Bor April 15,15 105, Using IO a -S0 410 324 212 109 45 22 31 80 470 284 387 437 30º and q, = 15º, 65 405 306 185 79 21 03 10 52 141 262 378 437 O 408 288 156 50 04 00 00 26 11 240 314 449 ho, 75 49 W6 126 24 00 00 04 OR 80 29 361 462 -80 427 274 97 06 DO 00 O 00 50 206 388 471 | x eos 43 cos 9.445; a z 85 432 77 72 00 00 00 00 00 24 203 393 476 cos 43 cos 9.4 sin (-15) — sin (-30)] +7 -0 433 28 62 00 00 00 00 00 14 W4 394 478 - 12 x 3600 x 567 360 x 105 x pedia ( + 0.033 cos 5 15 —(-30) ao » sin 43 sin 9. =3,79 MJm? n The hourty extraterrestrial radiation can also be approximated by writing A tion 1.10.2 in terms of 4, evaluating « at the midpoint of the hour. For te Sircumstançes 9f Example 1.10.2, the hour's Tadiation so estimated is 3 80 Mk ê Differences between the hourly radiation calcutated by these two methods ill bo 44 Solar Radiation slightly larger at times near sunrise and sunset but are stitl small. For larger time spans, the differences become larger. For example, for the same circumstances as in Example 1.10.2 but for the two-hour span from 7:00 to 9:00, the use of Equation 1.104 gives 4.58 MJ/m?, and Equation 1.16.2 for 8:00 gives 4.61 MJ/m?, 111 SUMMARY In this chapter we have ouilined the basic characteristics of the sun and the radiation it emits, noting that the solar constant, the mean radiation flux density outside of the earth's atmosphere, is 1367 W/m? (within +1%), with most of the radiation in a wavelength range of 0.3 to 3 um. This radiation has directional characteristics that are defined by a set of angles that determine the angle of incidence of the radiation on a surface. We have included in this chapter those topics that are based on extraterrestrial radiation and the geometry of the earth and sun. This is background information for Chapter 2, which is concerned with effects of the atmosphere, radiation measurements, and data manipulation. REFERENCES Anderson, E. E. Fundamentais of Solar Energy Conversion, Addison-Wesley Publishing Co.. Reading, MA (1982). Benford, F. and J. E. Bock, Trans. cf the American Hlumination Engineering Soc. 34, 200 (1939), “A Time Analysis of Sunshine.” Braun, J. E. and . C. Mitchell, Solar Energy. 31, 439 (1983). "Solar Geometry for Fixed and Tracking Surfaces." Cooper, PL. Solar Energy, 12, 3 (1969). "The Absorprion of Solar Radiation in Solar Stills” Coulson, K. L., Solar and Terrestrial Radiation, Academic Press, Nes York (1975). Duncan, €. H., R. €. Wilson, 3. M. Kendall, R. G. Harsison, and J. R. Hickey, Solar Energy, 28, 385 (1982). “Latest Rocket Measurements of the Solar Constant.” Eibling, 1. A R. E. Thomas, and B. A. Landry. Report to the Office of Saline Water, U.S. Department of the Interior (1953). “An Investigation of Multiple-Effect Evaporation of Saline Waters by Steam from Solar Radiatior Froblich. C.. in The Solar Output and tts Variation (O. R. White, ed.), Colorado Associated University Press, Boulder (1977). “Contemporary Measures of the Solar Constant” Garg. H. P, Treatise on Solar Energy, Vol. 1, Wiley-Interscience, Chichester (1982). Hickey, 5. Ro B. M. Ailton, F. 4, Griffin, H. Jacobowitz, P. Pelligrino, R. E. Maschhoff, E. A. Smith, and T. H. Vonder Haar, Solar Energy, 28, 443 (1982). “Extraterrestrial Solar Irradiance Variability: Two and One-Half Years of Measurements from Nimbus 7." Hotel, H. C. and B. B. Woertz, Trans. ASME, 64, 91 (1942). "Performance of Flat-Plate Solar Heat Collectors." t Igbal, M., An Introduction to Solar Radiation, Academic Press, Toronto (1983). References as Jobnson, E. + 4. 0f Meteorology, Li, 431 (1954), "The Solar Constant.” Jones, R. E., Solar Energy, 24, 305 (1980), "Eltecis of Overhang Shading of Windows having Arbitrary Azimuth,” Klein, 8. A., Solur Energy 19, 325 (1977) Tilted Surfacos “Calculation of Monthly Average Insolation on Kendraiyev, K. Y., Radiation in the Atmosphere, Academic Press, New York and London (1969), Mucria, E., The Passive Solar Energy Book, Rondale Press, Emmaus, PA (1979), NASA SP-8055, National Aeronautics and Space Administration, May (1971). Electromagnetic Radiation.” "Solar Robinson, N. (ed.), Solur Radiation, Isevier, Amsterdam (1966). Sun Angie Calculator, Libby-Owens-Ford Glass Company (1951). Spencer, 3. W. Search, 2 (5), 172 (1971), "Fourier Series Representation of the Position of the Sun” Thekaekara, M. P. and A. 3. Drummond, National Physical Science, 229, 6 ([971). “Standard Values for the Solar Constant and lts Spectral Components.” Thekackara, M. P., Solar Energy, 18, 309 (1976). "Solar Radiation Measurement: Techniques and Instrumentation.” Thomas. R. N., Trans. ef the Conference on Use of Solar Energy, 1, 1, University of Arizona Press (1958). "Features of the Solar Spectrum as Iimposed by the Physics of the Sun." U.S. Hiydrographic Office Publications No. 214 (1940). "Tables of Computed Altitude and Avimuth,, Whillier, A., Notes on Solar Energy prepared at MeGill University (19654). Whillier, A... Solar Energy, 9, 164 (1965b). “Solar Radiation Graph.” Wihillier, A., Personal communication (1975 and 1979), Wilson, R. €.. S. Gulkis, M. Janssen, H, S, Hudson, and G. A. Chapman, Science, 211, 700 (1981). "Observations of Solar Irradiance Variability. so Available Solar Radiution Output connector Window je Filter detemt Model EG wire-wouna thermopile Figure 22.2 Cross section of the Eppley Normal Incidence pyrheliometer. Courtesy The Eppley Laboratory. cross-section of a recent model of the Eppley is shown in Figure 2.2.2. The instrument mounted on a tracking mechanism is shown in Figure 2.2.3. The detector is at the end of the collimating tube, which contains several diaphragmis and which is blackened on the inside. The detector is a mukijunction thermopile couted with Parsons optical black. Temperature compensation to minimize sensitivity to vanations in ambient temperature is provided. The aperture angie of the instrumem ts 5.7º, so the detector receives radiation from the sun and from an area of the circumsotar sky two orders of magnitude larger than that of the sim. The Kipp & Zonen actinometer is based on the Linke-Feussacr design and uses a 40-junction constantan-manganin thermopile with hot junctions heated by radiation and cold junctions in good thermal contact with the case. In this instrumem the assembly of copper diaphragms and case has very large thermal capacity, orders of magnitude more than the hot junctions. On exposure to solar radiation the hot junctions rise quickly to temperatures above the cold junction; the difference in lhe temperatures provides a measure of the radiation. Other pyrheliometers were designed by Moll-Gorczynski, Yanishevskiy, and Michelson. The dimensions of the cellimating systems are such that the detectors are exposed to radiation from the sun and from a portion of the sky around the sun. Since the detectors do not distinguish between forward-scattered radiation, which comes from the circumsolar sky, and beam radiation, the instruments are, in effect, defining beam radiation. An experimenta! study by Jeys and Vant-Hul! (1976), which utilized several lengths of coltimating tubes so that the aperturc angles were reduced in step from 5.72º ta 2,02º, indicated that for cioudiess conditions this reduction in aperture angle resulted in insignificant changes in the measurements of beam radiation. On a day of thin uniform cloud cover, however, with solar altitude angle of less than 32º, as much as 11% of the measured intensity was received from the circumsolar sky between aperture angles of 5,72º and 2.02º. 1 ' 23 Pyranometers es — * Pyrheliometer (NIP) ou an altazimuth tracking Figure 2.2.3 An Eppley Normal incidenc mount. Courtesy The Eppley Laboratory, generalize from the few data available, bot it appcars th: affect the angular distribution of radiation within the Pyrheliometers. The World Meteorological Organization Salibration of pyrheliometers only be undertaken on days in mects or exceeds a minimum value. at thin clouds or haze can field of view of standard (WMO) recommends that whtich atmospheric clarity 23 PYRANOMETERS Instruments for measuring total (beam plus diffuse) radi Pyranometers, and it is from these instruments that most of ti tadiation arc obtained. The detectors for these instrume: independent of Wavelengih of radiation over the solar ene: they should have a response independent of the angle of incidence of the sola: radintion, The detectors of most Pyranometers are covered with one or tuo hemispherical glass covers to Protect them from wind and other extraneous effects, te covers must be very uniform in thickness so as not to Cause uneven distribution of dons om the detectors. These factors are discussed in more detail by Coulson Commonty used pyranometers in the United States are the E ley and Spectrolab insttuments, in Europe the Moll-Gorezynski, in the USSR m Yanishevskiy, and in Australia the Trickeit- Norris (Groiss) pyra ne ation are referred to as lhe available data on solar DIS must have 2 response T8y spectrum. In addition, jometer. si | | | | s Availahte Solar Radiation The Eppley 180º pyranometer was the most common instrument in ihe United States. Tt used a detector consisting of two concentric silver rings; the outer ring was coated with magnesium oxide, which has a high reflectance for radiation in the solar energy spectrum, and the inner ring was coated with Parson's black. which has a very high absorptance for solar radiation. The temperature difference between these tings was detected by a thermopile and was à measure of absorbed solar radiation. The circular symmetry of the detector minimized the effects of the surface azimuth angle on instrument response. The detector assembly was placed in a nearly spherica! glass bulb, which has a transmiftance greater than (1.90 over most of the solar radiation spectrum, and the instrument response was nearly independent of wavelength except at the extremes of the spectrum. The response of this Eppley was dependent on ambient temperature, with sensitivity decreasing by 0.05 to 0,15%/C [Coulson (1975)); much of the published data taken with these instruments was not corrected for temperature variations. It is possible to add temperature compensation to the external circuit and remove this source of error. It is estimated that carefully used Eppleys of this type could produce data with less lhan 5% errors but that esrors of twice this could be expected from poorly maintained instruments. The theory df this instrument has been carefully studied by MacDonald (1951). The Eppley 180º pyranometes is no longer manufactured and has been replaced by other instruments. The Eppley Black and White pyranometer utilizes Parson's-- biack- and barium-sulfate-coated hot and cold thermopile junctions and has better angular (cosine) response. It uses an optically ground glass envelope and temperature compensation to maintain cahibration within £1.5% over a temperature range of -20 to +40 €. Itis shown in Figure 2.3.1. The Eppley Precision Spectral Pyranometer (PSP) utilizes a thermopile detector. two concentric hemispherical optically ground covers, and temperature compensation Figure 2.3.1 The Eppley Black and White pyranometer. Courtesy The Eppley Laboratory. terms pe crimes meme 23 Pyranometers 5 that resuls in temperature dependence of 0.54% from -20 to +40 C. (Measurements of irradiance in spectral bands can be made by usc of bandpass filters; the PSP can be fitted with hemispherical domes of fiter glass for (hi purpose, See Stewart et al (1985) for information and references.) It is shown in Figure 2.3.2. The Moll-Gorczynski pyranometer uscs a Moll thermopile to measure the temperature difference of the black detector surface and the housing of the instrument. The thermopile assembly is covered with two concentric glass hemisphericat domes to protect it from weather and is rectangular in configuration with the thermocouples aligned in a row (which results in some sensitivity to the azimuth angle of the radiation). Pyranometers are usually calibrated against standard pyrheliometers, A standard method has been set forth in the Annals of the Intemational Geophysical Year HGY, (1958)), which tequires that readings be taken at times of clear skies, with the pyranometer shaded and unshaded at the same time as readings are taken wilh the pyrhetiometer. Shading is recommended to be accomplished by means ofa dlise held 1 m from the pyranometer with the disc just large enough to shade the glass envelope. The calibration constant is then the ratio of the difference in the output of She shaded and unshaded pyranometer 10 the cuxtpat of the pyrheliometer multiplied by lhe calibration constant of the pycheliometer and cos 8., the angle of incidence of beam radiation on the horizontal pyranometer. Care and precision are required in thesc calibrations. It is also possible, as described by Norris (1973), tu calibrate pyranometers against a secondary standard pyranometer suçh as he Eppley precision pyranometer. This secondary standard pyranometer is thought to be good to +1% when calibrated Against a standard pyrheliometer. Dircet comparison of the precision Eppley and field instruments car be made to determine the calibration constant of the fiefd instruments. Figure 2.3.2 The Eppley Precision Spectral pyranometer. Couresy The Eppley Laboratory. E Avuilable Solar Radiation A pyranometer (or pyrheliometer) produces a voltage from the thermopile detectors that is a function of the incident radiation. is necessary to use a potentiometer to detect and record this output. Radiation data usuaily must be integrated over some period of time, such as an hour or a day. Integration can be done by means of planimetry or by electronic integralors. Tt has been estimated that with careful use and reasonably frequent pyranometer calibration, radiation measurements should be good within +5%; integration errors would increase this number. Much of the available radiation data prior to 1975 is probabiy not this good, largely because of infrequent cafibrution and in some instances because of inadequate integration procedures. Another class of pyranometers, originally designed by Robitzsch, utilizes detectors that are bimetallic elements heated by solar radiation; mechanical motion of the element is transferred by a linkage to an indicator or recorder pen. These instruments have the advantage of being entirely spring driven and thus require no electrical energy. Variations of the basic design are manufactured by several European fisms (Fuess, Caselia, and SIAP). They are widely used in isolated stations and are a major source of the solar radiation data that are availabte for locations outside ví Europe, Australia, Japan, and North America. Data from thesc instruments are generalty not as accurate as that from thermopile-type pyranometers. Another type of pyranometer is based on photovoltaic (solar cell) detectors. Examples are the LICOR LI-200SA pyranometer and the Yellor Solarimeter. They are less precise instraments than the thermopile instruments and have some limitations on their use, They are also less expensive than lhermopile instruments and are easy to use. The main disadvantage of photovoliaic detectors is their spectrally selectivg response. Figure 2.3.3 shows à typical terrestrial solar spectrum and the spectral response of a silicon solar cell. If the spectral distribution of incidem radiation was T T T T T Salar radiation € 2000 + ã Solar cel E response Ê ã E 1000 + & ê E 8 ot I 4 03 05 07 10 20 20 Wavelength, gm Figure 2.43 Spectral distribution of extraterrestrial solar radiation and spectral response of a silicon solar cell. From Coulson, Solar and Terrestrial Radiation, Academic Press, New York 1975). 23 Pyranometers ss fixed, a calibration could be established that would remain constant; however, there are some variations in spectral stribution? with clouds and atmospheric water vapor. LI-COR estimates that the ertor introduced because of spectral response is +59 maximum under most conditions of natural dayliglt and is 3% under typical conditions, . Photovoltaic detectors have additional characteristics of interest. Their response to changing radiation levels is essentially instantancous and is linear with radiation. The temperature dependence is +0.15%/C maximum. The LI-COR instrument is fited with an acrylic dilfuscr which substantially removes the dependence of response on the angle of incidence of the radiation. The response of the detectors is independent of its orientation. but reflected radiation from the ground or other sutroundings will in general have a different spectral distribution than global horizontal radiation, and measurements on surtaces receiving significant amonnts of reflected radiation will be subject to additional errors. The preceding discusston dealt entirely with measurements of total radiation on a horizontal surface. Two additional kinds of measurements arc made with pyranometers: measurements of diffuse radiation on horizontal surfaces and measurement of solar radiation on inclined surfaces. Measurements of diffuse radiation can be made with pyranometers by shading the instrument from beam radiation. This is usually done by means of a shading ring, as shown in Figure 2.3.4, The ring is used to allow continuous recording of difluse Figure 2.3.4 Pyranometer with shading ring 10 eliminate beam radiation. Courtesy The Eppley rt 2 This will be discussed in Section 2.6. so Available Solar Radiation all data are based on the WRR pyrheliometric scale. The Atlas includes” tables that show averages, maxima, minima, cxlraterrestria! radiation, and sunshine hours. Appendix G includes some data from the Atlas. Average daily solar radiation data are also available from maps that indicate general trends. For example, a world map is shown in Figure 2.5.2 |Lóf et al, (1966a,b)].9 In some geographical areas where climate does not change abrupuy with distance (i.e., away from major influences such as mountains or large industria! cities), maps can be used as a source of average radiation if data are not available. However, large-scale maps must be used with care becausc they do not show local physical or climatological conditions that may greatly affect local solar energy availability. For calculating the dynamic behavior of solar energy equipment and processes and for simulations of long-term process operation, more detailed solar radiation data (and related meteorological informatiun) are needed. An example of this type of dala (hourly integrated radiation, ambient temperature, and wind speed) is showr im Table 2.5.) for a January week in Boulder, Colorado. Additional information may also be included in these records, such as wet bulb temperature and wind direction. In the United States there has been a network of stations recording solar radíation on a horizontal surface and reporting it as daily values. Some of these stations also reported hourly radiation. In the 1970s, the U.S. National Occanic and Atmospheric Administration (NOAA) undertook a program to upgrade the number and quality of the radiation measuring stations, to rehabilitate past data (to account for sensor deterioration, calibration errors, and changes in pyrhcliometric scales), and to make these data available (with related meteorological data) on magnetic tapes. In 1978, corrected data tapes of hourly meteorolegical information (including soar radiation on a horizontal surface based on the Solar Constant Reference Scale) for 26 stations over a period of 23 years became available. These tapes are reterved to as the SOLMET tapes and are described in detail in the SOLMET Manual (1978). Fhe monthly average data for the United States stations shown in Appendix G are derived from the SOLMET data. In the jate 1970s, the U.S. federal govemment funded thc development and operation of a national sotar radiation network (SOLRAD), Measurements of hourly total horizontal and direct normal radiation were made at the 38 stations that were part ofthe network. Eleven of the stations also measured diffuse radiation. Data for 1977 ta 1980 were checked for quality and are available from the National Climatic Data Center. Funding for much of the program was reduced in 1981, and by 1985 the network was shut down. Since then, some additional funding has become available to upgrade the insuumentation at many of the stations to automate data acquisition and recalibrate pyranometers. 1 É Monthly average daily radiation on surfaces other than horizontal are in Volume IL of the Atas. é Figure 2.5.2 is reproduced fram deJong (1973), who redrew maps originally published hy Lóf et al. (19663). deJang has compiled maps and radiation Gata from many sources. Solor ) ontar Daity Means of Tojal --— Radiation (Begm + D Incident en q Horia Surtaca, catsemt E Z i able 2.51 Hourly Radiation for the Hour Ending at the Indicated Time, Air Temperature, and Wind Speed Data for a January Week, Bouider, Colorado. MEN Day Hour 1 Ta Y Day Tour 1 To Y — ——Nnn—N—— km? e ms Mr c ms Day Hour ' Ta v lt — Hm Cm 8 L o 8 õ nos E 2 o 8 va 1252 H $ : n 19 o +47 4 4 ot s& “ q 5 pas ss E $ 5 o 2 u 46 foto no 56 54 un Do Sa 2 5 5 a 6 non 38 56 94 no 9 89 36 o R » u 12 163 se Bo 2 9 4 40 8 ? u ss 19 0 . no o «NI 3 8 8 17 8 2» q 2 9 4 su o Ro So 740 po o 3 s8 8 10 33 s 2 o b i o e a Dou am 2a - .z 12 y RR: Lo Pliml Lia as x 12 5 0 167 63 12 7 3 22 9 1 o 9 Bo ng ú éo o a sá 12 18 o E , =I72 6 12 19 ; is tm PLS Ea 29 no 3 Ra > +0 9 4 o 9 16 389 RW 3 2.2 0 59 180 RR! 8.9 ' ros is E O : 5 9 POR BL 56 76 5 ê ss 9 7 o 2019 a 2.2 o sa 9 3 4 920 0 3 92 u a o ! 5 1 “ 40 Voo 1926 56 54 2 10 155 PR”: o i 3 : 9 54 13 MIO 72 45 9 43 su o i 0 4s 3 o 15 130 83 9 12 402 9º u q : á Ê 9 3 3º 16 %3 is õ 5 o 3.6 2» 5 é a . 13 6 o 31 7 54 10 1 o 10 13 1872 13 7 0 6 5 v 44 36 10 2 o Do 7 13 $ 17 40 a po º no 36 10 3 0 o 15 138 13 9 314 5 » 9 oo 31 10 4 o 1 16 Ts 3 to a 63 5 b º 22 67 10 5 o 1 1” 205 3 u 1809 sa 5 B 9 28 72 10 6 o 10 18 4 1 1 2299 63 o 17 80 10 7 o 10 19 o o DB 2 o 17º 58 10 8 33 10 20 o : 14 ' o 42 14 a 10 9 419 10 2 o : 14 2 o 16 14 a 1968 s7 10 10 1047 o» 2 0 i 4 à o és 1733 67 10 u 1570 10 n uv i 14 4 o 7 14 15 1331 72 10 1 1805 0 2 o i 14 5 14 16 837 67 í és 5 42 so % 2 u 1 a " B 138 50 69 já ; 0 é 8 14 18 4 3327 i n 2 º ti 14 96 39 61 À 14 k x o ú 19 o 0 36 | “u 3 o " 15 o 44 76 14 9 452 49 1 20 o 1954 | u 4 o Dow 2 38 63 Modo no as ú 0 3» 36 | nos o ; non 4 50 63 oO ar 168 8 M La Do 39 58 | n 6 o n 18 0 56 45 Mo 12º 18 36 0 6154 | Mot 383 “ou 0 +47 63 o : es Available Solar Radiation 26 ATMOSPHERIC ATTENUATION OF SOLAR RADIATION Solar radiation at normal incidence received at the surface of the earth is subject to variations due to change in the extraterrestriai radiation as noted in Chapter 1 and to two additional and more significant phenomena. (t) atmospheric scatiering by air molecules, water, and dust and (2) atmospheric absorption by Os. H,O, and CO. Igbal (1983) reviews these matters in considerable detail. Scattering of radiation as it passes ihrough the atmosphere is caused by interaction of the radiation with air molecules, water (vapor and droplets), and dust. “The degree to which scattering occurs is a function of the number of partictes through which the radiation must pass and of the size of thc particles relative to À, the wavelength of the radiation. The path length of the radiation through air molecules is described by the air mass. The particles of water and dust encountered by the radiation depends on air mass and on the time- and location-dependent quantities of dust and moisture present in the atmosphere. Air molecules are very small relative to the wavelength of the solar radiation, and scattering occurs in accordance with the theory of Rayleigh (i.c., ihe scanering coefficient varies with 4-4). Rayleigh scattering is significant only at short wavelengths; abavc À = 0.6 jum it has little effect or atmospheni transmittance. Dust and water in lhe atmosphere (end to be in larger particle sizes due to aggregation of water molecules and condensation of water on dust particles of various sizes, These effects are more difficult to treat than the effects of Rayleigh scattering by air molecules, as the nature and extent of dust and moisture particles in the atmosphere are highly variabie with location and time. “Two approaches have been used to treat this problem. Moon (1940) developed a transmission cocfficient for precipitable water [the amount of water (vapor plus liquid) in the air column above ihe observer] that is a function of 72 and a coefficient for dust that is a function of A-2:75, Thus these transmittances are less sensitive to wavelength than is the Rayleigh scaitering. The overall transmittance due 10 seattering 1s the product of lhree transmitiances, which are three different functions of À. The second approach lo estimation of cffcets of scattering by dust und water is by use of Ângstrôm's turbidity equation. An equation for atmospheric transmittance due to aerosols, based on this equation, can be written as Wa expl-pa"Sm) (26) where is the Ângstrôm turbidity coefficient, « is a single lumped wavelength exponent, and À is the wavelengih in micrometers. Thus there are two parameters, Band o, that describe the atmospheric turbidity and its wavelength dependence; 8 varies from O to 0.4 for very clean to very turbid atmospheres, «a depends on the size distribution of the aerosols (a value of 1.3 is commonly used), and 8 and & will vary wilh time as atmospheric conditions change. More detailed discussions of scatiering are provided by Fritz (1958) who included effects of clouds, by Thekaekara (1974) in a review, and by Igbal (1983). 2.6 Almospheric Attenuation of Solar Radistion 65 À increases above 0.29 à sa -29 um, until at 0,35 i i weak ozone ahsorption band near À = 06um ste no abrovptim, eres also a Water vapor absorhs stron; nds à . R igly in bands in the infrared pa Egeu, with Sirong absorption bands centered at 10,14, and um von 25 dm me rtemision Of the atmosphere is very low due to absorption by Ho 2 He energy in the extraterrestrial spect à i 5 Of the total solar spectrum, am ve a te groaoa a o tan 5 ate Pe 4 AM energy received at the ground at À > 2.5 um is very n =» . R and E ffects of Rayleigh Seattering by air molecules and absorption by O,, H,0, ne > one spectral distribution of beam irradiance are shown in Figure 26 1 R : am atmosphere with = O and 2 em of preci entra here g precipitable water, w. TI Do ni sistibution is shown as a reference. The Rayleigh Seeing ira Y the difterence berween the extraterrestrial cu ) e s d the curve a shaded arcas; its effect become: seleng o D lise s small at wavelengths greater th The cara absorption bands arc shown by the shaded areas “e about Dó um, "he eflcct of air mass is ilustrated in Fi, vi . e s Ir mas in Figure 2.6.2, which shows th distribution of beam imadiance for air masses of O (the extraterrestrial NDT and 5 for an atmosphere of low urbidity. Teto 3 The spectral distribution of 1 ati The 1 otal radiation depends aiso on th S distribution of the diffuse radiation. Some measurements are avaiablo im ta sale and visible portions of the spectrum [see Robinson (1966) and Kondra ' (1969), which have ted to the conclusion that in the wavelength range (0,35 to 080 / | | | | Clean Atmosphera, & = 0 Air Mass = 1 r Extraterrestrial | y Rayieigh Attenuation » 8 ê 250] Direct Normal Spectra! Irradiance, Wim? um B s 10 15 20 25 30 as 40 Wavelengih, um Figure 24 a sure 2.6.1 An example of lhe cffecis of Rayleigh Scaltering and atmos) the spectral distribution oFbeam irrudianee, Adapted From Igbal (1983), Pen abeerprca on Available Sotar Radiation Values of F, can be calculated from Equation L. 10,3 using day umbers om Table 1.6.1 for the mean days of the month, or it can be obtaincd em eiti pia 110.1 or Figure 1.10.1. The average day length À cun be calculated tom ut n 1.6.41 or it can be obtained from Figure 1.6.3, for the meu day of the no Vas indicated in Table 1.6.1. Lof et al. (19663) developed sets ot o a a a various climate types and locations based on radiation data then avail table. 5 given in Table 2.7.2. Table 2.72 Climatic Constants for Use in Equation 2.7.2 Sunshine Hours in Percentage of Possible Location Climated Veg Range A bo Location Cmã Do 7 os 03 jue, NM BsBW E 6sas 4 Auta GA cr M 4811 E 0.38 9 % Blue Hill, MA Dr D 42-60 2 º 2 os Brownsville, TX Bs GDsp 4780 6 3 om Buenos Aires, Arg cr q atos so 0.26 É Charleston, SC cr E 6075 e7 dam 009 Darien. Manchuria Dw D 58-81 s7 03% 0,23 El Paso, TX sw Dã 78.88 sa 0.54 020 Ely NV Bw Bi 61-89 n 0.54 o 8 am 22 Hamburg, Germany cr D va x : “ o Eu I Ar G 71 $ gt Dt M 40.72 ss “Io o a q ou Malange, Angola AwBS GD aits 58 o. % om Miami, FL Av EGD 5671 65 q 2 Nice, France €s SE 49-76 61 o q : x 0,30 Poona, India (Monsoon) Am s 25.49 a” sam (Dm a5-49 81 a Kisanganá, Zaire Af B 34-56 a : Tamanrasset, Algeria BW Dp Te . à Climatic classification bused on Trewartha's map (1954, 1961), where climate types are: AÉ Tropical forest climate, constantly mois; reinfal all through der Am Tropical forest climate, monsoon rain; shom dry seasoo, but total rainfatl suflhcient to support rain forest. Aw— “ropical forest climate, dy season ia winter BS — Steppe or semiarid climate. BW Desert or arid climate. , | CS Mesothermal forest climate: constartly mois raintal alt through the year. K winter. Cs Mesotherma! forest climate: dry season in . Dt Microthermal snow forest climate: constanely moist; rainfdl al through the year. Dow Microtherma! snow forest climate; dry season jm winter 2.7 Estimation of Average Solar Radintion n The following example is based on Madison data (although the procedure is not recommended for a station where there are data) and includes comparisons of the estimated radiation wilh SOLMET data and estimates for Madison based on the Blue Hill constants (those which might have been used im the absence of constants for Madison). Example 2.7.1 Estimate the monthly averages of lotal solar radiation on a horizontal surface for Madison, Wisconsin, latitude 43º, based on the average duration of sunshine hour duta of Table 2.7.1. Solution The estimates re based on Equation 2.7.2 using constants a = 0.30 and 5 = 0.34 from Table 2.7.2. Values of H, are obtained from Figure 1.10.1 and day lengths from Equation 1.6.1], each for the mean days of the month. The desired estimates are obtained in the following table, which shows daily 4 in MI/m2. (For Footnotes for Table 2.7.2 (continued) D Vegetation classification based on Kiichler's map, where vegetation Iypes are: B Droadicafevergreentees. Broudleaf evergreen, shrubform. minimom height 3 feet, growth singly or in groups or patches. D Broadlleaf deciduous trees. Ds Broadfeaf deciduous. shrubform. minimum height 3 feet, plants sufficiently far apar that they frequently do not touch. Dsg — Broadleat deciduous, sbrubform, minimum height 3 feet, growth singhy or in proups or patches. E Noedleicaf exergreen trees. G Grass and other herbaccous plants, SD Grass and ather herbacevus plants; broadizaf deciduous trees. CiDsp Grass and other herbaceous plants; broadieaf deciduous, shrubforms, minimum height 3 feet, growth singly ox in groups or patches M Mixed broadleaf deciduous and necdleleaf evergreen trees. S — Semideciduous: broadleaf evergreen and broadleaf deciduous trees. . SE — Semideciduous: broadleat evergreen and broadleaf deciduous trees: needielear evergreen trees. Note: These constants are based ou radiation data available before 1966 and do not reflect mprovements in data processing and interpretation made since then. The results of estimations for United States stations will be at variance with SOLMET data. Iris recommended that these correlations be uscd only when there are no radiation data available. - Available Sokar Radiation comparison, data for Madison from Appendix G are shown, and in (he last colums estimates of Madison radiation determined by using constants « und & for Blue Hill) mesh Ha N nJN Month Mm? dr Mt? Mutmé MJjm? 1136 43 0.49 nã so 2 18.80 10.2 055 92 91 9a 26.05 17 sy 138 129 1,4 335 12 057 1646 154 4 May 3939 144 63 19,8 no June 4174 154 0.07 221 2 tuiy 4052 14,8 0.66 224 na Aug 358% = 073 19.4 2 Sept. 2877 3 oa ue 16.4 Ou 4 10% 047 10.3 nó Nov 146! “s vas 57 às Dee UEM 29 n44 aa 53 & From SOLMET data * Using constants for Blue Bill n The agreement between measured and caleulated radiation is reasonably good, even though the constants a and 4 for Madison were derived from a different data base ftom the measured data. If we did not have constants for Madison and had to choose a climate close to that of Madison, Blue Hill would be a reasonabte choice. The estimated averages using the Blue Hilf constants are shown in the last column. The trends are shown, but the agreement is not as good. This js the more typical situation in the use of Equation 2.7.2. Data are also available on mean monthly cloud cover É, expressed as tenths of the sky obscured by clouds. Empiricat relationships have been derived to relate monthly average daily radiation /7 to monthly average cloud cover C. These are usually of the form Ecasro 213) o Norris (1968) reviews several attempts to develop such a correlation. Bennett (1965) compared correlations of // / dt, with C, with 4/N, and with a combination of the two variables and found the best correlation to be with Hi /N, that is, Equation 2.7.2. Cloud cover data arc estimated visualty, and there is not necessarily a direct relationship between the presence of partial cloud cover and solar radiation at any particular time. Thus (here may not be as gol a stalistical relationship between H1 Ho and C as there is between 7 (ff, and n/N. Many surveys of solar radiation data [e.g., Bennett (1965) and Lôf et al. (1966a,b)) bave been based on corretations 28 Esti tation of Clear Sky Radiation 3 Of radiation with sunshine h "our data, However, Paltri used cloud cover data to modify clear sky data si tolo (1976 have Monthly averages of 7£, which are in 90d agreement with meas vê ta. ges of Ff, whicl goad agy nt with measured average da 28 ESTIMATION OF CLEAR SKY RADIATION Phere and for four climate types. The atmospherie transmittançe fe ati i 6, Ti) e ce for beam radiation t, is ConlGon (OF GyriG,y) and is given in th o form W=a0+ arexpl-ticos 8) (2.8.1) The constants “or 44, and X for the standard at are toa Ra 'mospherc with 23 km visibility and kº, which are given for altitudes less than 2.5 km by ap=0,4237- D.00821(6 AP (28.10) ai = 0.5055 +0.00595(6.5 - Ap Qêto) Ate E =02711+00185825 AP (28.19) where À is the altitude ofihe observer in kilometers. (Hotic] al: 20: 1, and º for a standard atmosphere with 5 km visibilit Correction factors are applied to aa, oo types. The correction factors r= ne 2.8.1. Thus, the transmittance of SO gives equations for and &º to allow for changes | 0 ai, and 4 ges in climate “potão ri=ai/al, and sy = kk are given in Table |s standard atmosphere for beam radiation can be + ' able 28.1 Correction Factors for Climate Types? Climate Ty Yre + “ rk Tropical Midiatitude summer 6» 055 ia Subarctic summer 0.99 095 vor Mixitatitude winter 1 vol 1 103 LOL 109 * From Hotel (1976). Availabie Solar Radintion 74 determined for any zemith angle and any altitude up to 2.5 fam. The clear sky beam normal radiation is then (2.8.2) Grab = Gon i M ar sky horizontal beam radiation where G,, is obtained from Equation 1.4.1. The clear sky is 28.3) Goo = Gonto cos 8; ( For periods of an hour, the clear sky horizontal beam radiation is (2.84) leo = Ion th COS 8: Example 2.8.1 beam radiation of the standard clear atmosphere at ansmittance for e ] a tlítuo st 22 at 11:30 AM solar time. Estimate the intensity Madison (altitude 270 m) on Augu: r ' of beam radiation at that time and its component ema horizontal surface. — , wo: HS drãco a : Vora * Solution Psi hs TOS N On August 22, n is 234, the deelination is 1].4º, and from Equation 1.6.4 the cosin of the zenith angle is 0,846. fficients for Equation 2.8.1. First, the values for * ae oba ns 2.8.1b to 2.8.1d for an altitude the standard atmosphere are obtained from Equatioi of 0.27 km: ap = 0.4237 - 0.00821(6 — 027P=0.154 af = 0.5055 + 0.00595(6.5 — 0.27P = 0.736 k* = 0.2711 + 0.01858(2.5 -0.27) = 0.363 The climate-type correction factors are obtained from Table 2.8.! for midiatitude summer. Equation 2.8.1 becomes 1 = 0.51440.97) + 0:735(0.99) exp(-0.363 x 1.02/0.846) = 0.62 “The extraterrestrial radiation is 1339 Wim? from Equation 3.4.1. The beam radiation is then Gony = 1339 x 0.62 = 830 Wim? cem 2.8 Estlmation of Clear Sky Radintion 7] “The component on a horizontal plane is 830 x 0,846 = 702 Wim? n His also necessary to estimate the clear sky diffuse radiation on à horizontal surface to get the total radiation, Liu and Jordan (1960) developed an empirical relationship between the transmission coefficient for beam and diffuse radiation for clear days: u=(4=0 271 0.294% (2.8.5) where Ty is GG, (ot 1/f,), the ratio of diffuse radiation to the extraterrestrial (beam) radiation on the horizontal plane, The equation is based on data for three stations. The data used by Liu and Jordan predated that used by Hottel and may not be entirely consistent with it; until better information becomes available, it is suggested that Equation 2.8.5 be used to estimate diffuse clear sky radiation, which can be added to the beam radiation predicted by Hottel's method to obtain a clear hours total, These calculations can be repeated for each hour of the day, based on the midpoints of the hours, to obtain a standard clear day's radiation, H,. Example 2.8.2 Estimate the standard clear day radiation en a horizontal surface for Madison on August 22. Solution For cach hour, based on the midpoints of the hour, the transmittances of the atmosphere Tor beam und diffusc radiation are estimated. The calculation of £, is illustrated for the hour [1 to 12 (ie, at 11:30) in Example 2.8.1, and the bcam radiation for a horizontal surface for the hour is 2.53 MJfm? (702 Wim? for the hour). The calculation of 7, is based on Equation 2.8.5: = 0.271 - 0.29440.62) = 0.089 Next Gu Calculated by Equation 1.4.5, às 1339 Wim? so that Then Ga is Gucos 0, Gog= 1339 x 0.846 x 0.089 = 101 Wim? Then the diffuse radiation for the hour is 0.36 MJ/m?, The total radiation on a horizontal plane for the hour is 2.53 + 0.36 = 2.89 MJ/m?. These calculations are s Available Solar Radiation Solving for y in this cquation is not convenient, and Herzog (1985) gives an explicit equation for 7 from a curve fit. -2 - 1.184E-- 27.182 expl- 2.56) 25.64) 1.498 + Kr, mas — Kr.min and (2.9.6b) A value 0f Ky cin Of 0.05 was recommended by Bendt ex al. Hollands and Huget (1983) recommend that & ax be estimated from Krmax = 0.6313 + 0.267k7 — 11.9(Ky-- 0.758 (2.9.6) The universality of the Liu and Jordan distributions has becn questioncd, particularly as applied to tropical climates. Saunier et al. (1987) propose an alternative expression for the distributions for tropical climates, A brief review of papers on the distributions is included in Knight et al. (1991). Símilar distribution functions have been developed for houriy radiation. Whillier (1953) observed that when the houriy and daily curves for a location are ploited, the curves are very similar, Thus the distribution curves of daily occurrences of Ky can also be applied ta hourly cleamess indexes. The ordinate in Figure 2.9.2 can be replaced by kp and the curves will approximate the cumulative distribution of hourly cleamess. Thus for a climate with K7 = 0.4, 0.493 of the hours will have kz equai to or less than 0.40. 2.10 BEAM AND DIFFUSE COMPONENTS OF HOURLY RADIATION In this and the following two sections we review methods for estimation of the fractions of total horizontal radiation that are diffuse and beam. The questions of the best methads for doing these calculations are not fully settled. A broader data base and improved understanding of the data wil! probably cad to improved methods. In each section we review methods that have been published and then suggest one for use. The suggested correlations are in substantial agreement with other correlations, and the set is mutually consistent. The split cf total solar radiation on a horizontal surface into its diffuse and beam components is of interest in twa contexts. First, methods for calculating total radiation on surfaces of other orientation from data on a horizontal surface require separate treatments of beam and diftuse radiation (see Section 2.15), Second, estimates of the Jong-time performance of most concentrating collectors must be based on estimates of availability of beam radiation, The present methods for 2-H Beam and Diffuse Components of Houriy Radiation 81 10 08 06 tatt os 02 [o o a 00 o2 04 [a 08 Figure 2,10.€ À sample of difíuso fra etion vs, cl ads go sleamess index data irom Cape Canaveral, FL, b é » Wilh kr, the houriy ch i 10.1 shows a plot of diffuse fraction /,/1 vs, kr Far Cape Cana nt =» Lo In order 19 obtain 1,/ vs. & o 1º! Vs. ky correlations, data fr i NA ! 1 ms, om many locations simila E qua conure 2.10.1 are divided into “bins," or ranges of values ER a a dean ese "bin are averaged ta obtain a point on the plot. A set of these oito o ne correlation. Within each of the bins therc is a distibutiom of poi is ints; ak of (15 may be i k produced by skies with hi i Siffuse fraction, or by skies thai ave clear for O Bs e gh Part of the hour, teading to a low diffuse Fepresent a particular hour very closely adequately represents the diftuse fraction, ' Eis Oreilt and Hollands (1977) have used data of this “bs etal. (1982) have used data from four U. S Reindl et al. (19904) have used do Europe, The three correlation a identical, although they were deri correlation, Figure 2.10,3, is10 type from Canadian Stations, one Australian station, and am independent data set from United States and Cpo in Figure 2.10,2, They are essentialiy vet from three separate data bases. The Erts et al + produces results thas are for ractica - ANA is represemted by the following equation Ee 10- u J O 0.249k7 forkr<0 7 oa Lbdiy for 035 <ky < 0,75 o.t77 forky>0.75 Available Solar Radiation tab! DAL Orgil& Haliands —- - Etbs, etal, oz Reid, ota. kr i it 2.18.2 The ratio 4,17 as a function of hourly cleamess index, kr. Showing the Drgi lado (97, Erbs et al. (1982), and Reindt et at. (1990) correlarions. os 06 04 o2 99 Figure 2.10.3 The satio / (1 as à f n of hourly clearness index, ky, from Erbs et al. sato 4/1 nei y clearnes: 1, from Erbs gt (1982). 1.0- 0.09% forkr<0.22 09511 — 0.1604ky + 4.388k7 — 16.638 kj + 12.336 k$ (210.1) = for 022<ky 50.80 1 0.165 for kz > 0.80 For values of kr greater than 0.8, there arc very few data. Some S ne qa that are available show increasing diffuse fraction as ky increases nove no ps ise 1 iffuse fraction is probably due to reflection of radiati apparent rise in the diffuse r din from ing ti cured but when there are clou: s during times when the sun is unobscure: e ) o de sun to the observer. The use of a diffuse fraction of 0.165 às recommended in this region. erp 2:11 Beam and Diffuse Components of Daily Radintlon as In a related approach described by Boes (1975), values of 4,14 from corre- Iations are modified by a restricted random number that adds a statístical variation tg the corretation. 211 BEAM AND DIFFUSE COMPONENTS OF DAILY RADIATION Studies of available daily radiation data have shown thai the average fraction which is ditfuse, Hy/H, is à function of Ky, the day's cleamess index. The ariginal Somelation of Liu and Jordan (1960) is shown in Figure 2.11.1; the data vero for Blue Hill, Massachusetts. Also shown un the graph are plots of data for Israet from Stanhil (1966), for New Delhi from Choudhury (1963), for Canadian stations from Ruth and Chant (1976) and Tuller (1976), for Highett (Melbournc), Australia, ftom Bamnister (1969), and from Collares-Pereira and Rabi (19793) for feur US, stations. There is some disagrecmen, with differences probably due in part to instrumental Figure 2.10.2) is shown in Figure 2.11,2. A seasonal dependence is shown; the spring, summer, and fal] data are essentially the same, while the winter data shows somewhat lower diffusc fractions for hígh values of Ky. The season is indicated by as ê giÊ 08 ii ES =" Liu & Jordan ÉS gafo - Ruth & Chant E = — — Highot > -— Stanhi) * ==" Tuer I 02 ee Choudhuy Coliares-Perisra & Fabi º . º 02 04 08 oa 1.0 Key - Daiiytotatradiaton my "É Daily extratomastrial? E, Figure 2.111 Correlations of daity ditíuse fraction with daily cleamess index. Adapted from Kiein and Duffic (1978). Available Solar Radintion E 2) Figure 2.1.2 Suggested correlation of daily diffuse fraction with Kg. From Erbs et ab. (1982) the sunset hour angle O,. Equations representing this set of correlations are as follows'!: For q, <814º | 10-0.2727K7 +2M95KF Kr<0715 19H14 KG + 9.3879 Ky (ta) (ou for Kr 20.715 and for ay, > 814º 2 10+02832K7-25557K? rg, 0702 + 0 8448Kj Ha. (2.11.1b) H los for Ky 20.722 1 The Collares-Pereira and Rabi correlation is 0.99 for kr 0.17 VABB=227KrA DADA (ar 017 <K7< 0.75 Ha 2) -N865K)+ 14 64BKT —0.54K7 + 0.632 tor 075 <Kr< 0.80 02 for kr 20.80 2.42 Beam and Diftuse Components of Monthly Radiation ss Example 2.11.1 The day's total radiation on a horizontal surface for St, Louis. Missouri (latitude 38.6º) on September 3 is 23,0 MJ/m?. Estimate the fraction and amount that is diffuse, Solution For September 3, the declination is 7º. From Equation 1.6.10, the sunset hour angle is 95.6º. From Equation 1.10.3, the day's extraterrestria] radiation is 33.3 MJ/m', Then From Figure 2.11.2 or Equation 2.11.1b, Ha/H is 0.26, so an estimated 26% of the day's radidtion is diffuse, The diffuse energy is 0.26 x 23.0 = 6.0 MJfm?, " 2.12 BEAM AND DIFFUSE COMPONENTS OF MONTHLY RADIATION Charts similar to Figures 2.11.] and 2.11.2 have becn derived to show the distribution of monthly average daily radiation into its beam and diffuse components. En this case, the monthly fraction that is diffuse, Hy/H, is plotted as a function of monthly average clearness index, K7 (= j7/H.). The data for these plots can be obtained from daily data in cither of two ways. First, monthly data can be plotted by summing (he daily data diffusc and total radiation. Second, as shown by Liu and Jordan, a gencralized daily 4,/H versus Ky curve can be used with a knowledge of the distribution of good and bad days (the cumulative distribution curves of Figure 2.9.2) to develop the manthly average relationships, Figure 2,121 shows several correlations of 4q/!! versus K7. The curves of Page (1964) and Collares-Pereira and Rabl (19793) are based on summations of daily total and diflusc radiation. The original curve of Liu and Jordan (modified to correct for a small error in H4/H at low X7) and those labeled Highett, Stanhil!, Choudhury. Ruth and Chant, and Tuller are based on daily correlations by the various authors (as in Figure 2.11.1) and on the distribution of days with various Ky as shown in Figure 2.9.2. The Collares-Pereira and Rabl curve in Figure 2.12.1 is for their all-year correlation; they found a seasonal dependence of the relationship which they expressed in terms of the sunset hour angle of the mean day of the month. There is significant disagreement among the various correlations of Figure 2.12.1. Instrumental problems and atmospheric variables (air mass, season, or other) may contribute to the differences. Erbs et al. developed monthly average diffuse fraction correlations from the daily diffuse correlation of Figure 2.11.2. As with the daily correlations. there is a %” Available Solar Radialion 0.20 Í 218 6 Hour from solar neon o 3 0.14 3 zo 2 E De A ia 5 é ao DS ma) == i o 5 oo É jp Ê So " > 006] Too 1 2 004 “3 LT 002 ELA o 8 8 10 " 12 13 ta 15 16 Hours from sunrise to sunset Lpjid!2isdLrrLLos + 3 7 so 105 120 Sunset hour angt, «u,, Degreos Figure 2.132 Relationship berwecn hourly diffuse and daily diffuse radiation on a horizontal surface as a function of day length. Adapted form Liu and Jordan (1960). Figure 2.13.2 shows a retated set of curves for r,, the ratio of hourly diffuse 10 daily diffuse radiation, as a function of time and day length. In conjunction with Figure 2.11.2, it can be used to estimate hourly averages of diffusc radiation if the average daily total radiation is known: nus fé (2.133) These curves are based on the assumption of Liu and Jordan (1960) bat 4,4Hy is the 2.14 Radiation on Sloped Surfnces 9 same as (,//f,, and are represented by the following equation NE cos o o A (213,4) “E sin am AEE cos ax Example 2.13.3 From Appendix G, the average daily June total radiation on a horizontal plane in Madison is 22.1 MJ/m?. Estimate the average diffuse, the average beam, and the average total radiation for the hours 10 to LI and Lto 2. Solution The mean daily extraterrestrial radiation JT, for June for Madison is 41.7 MJ/m2 (from Table 1.10.1 or Equation 1.10.3 with 2 = 162), 0) = 113º, and the day length is 15.1 hours (from Equation 1.6.11). Then (as in Example 2.12.1), Ry = 0.53. From Equation 2.12.1, Ay/H' = 0.40, and the average daily diffuse radiation is (0.40 x 22.1 = 8.84 MJ/m?, Entering Figure 2.13.2 for an average day length of 15.1 hours and for 1.5 hours from solar noon, we find rq = 0.102. (Or Equation 2.13.3 can be used with «o = 22.5º and 6), = 113º to obtain r; = 0.102.) Thus the average diffuse for those hours is 0.102 x 8.84 = 0.90 MJ/m?. From Figure 2.13.1 (or from Equations 2.13,1 and 2.13.2) from the curve for 1.5 hours from solar tioon, for an average day length of 15,1 hours, r, = 00.108, and average hourly total radiation is 0.108 x 22,1 = 2.38 MJfm?. The average beam radiation is the difference between the total and diffuse, or 2.38 — 0.90 = [.48 MIfm?. = 214 RADIATION ON SLOPED SURFACES We tum next to the gencrat problem of calculation of radiation on tilted surfaces when only the total radiation on a horizontal surface is known, For this we need the directions from which the beam and diffuse components reach the surface in question. Section [.8 deait with the geometric probtem of the direction of beam radiation. “The direction from which diffuse radiation is received, ie, its distribution over the sky dome, is à function of conditions of cloudiness and atmospheric clarity, whích are highly variable, Some data are available, for example, from Kondratyev (1969) and Coulson (1975). Figure 2.14.1, from Coulson, shows profiles of diffuse radiation across the sky as à function of angular elevation from the horizon in a plane that includes the sun, The data are for clear sky and for smog conditions. Clear day data such as that in Figure 2.14.1 have led to à description of the diffuse radiation as being composed of three parts. The first is an isotropic par, teceived uniformiy from all of the sky dome. The second is circumsolar diffuse, resulting from forward scartering of solar radiation and concentrated in the part of the ” Available Solar Radiation = 0.965 um ve intensity am l l l L L l l E o 40 20 o 2 40 60 BO Zenith angle, degrees. Figure 214.1 Kelative intensity of solar sadiation (at À = 0.365 um) as a foncrion of elevation angle in the principal plane that includes the sun, for Los Angeles, for clear sky and for smog, Adapted from Conison. Solar and Terrestrial Radiation, Academic Pross, Nes York (1975). sky around the sun. The third, referred to as horizon brightening, is concentrated near the horizon, and is most pronounced in clear skies. Figure 2.14.2 shows schematically these three parts of ihe diffuse radiation. . The angular distribution of diffuse is to some degree a function of the reflectance py (the albedo) of the ground. A high reflectance (such as that of fresh snow, with pg approximately 0.7) results in reflection of solar radiation back to the sky, which in tum may be scattered to account for horizon brightening. Gircumsolar O “- Horizon “Area!, Apz Figure 2.142. Schematio of the distribution of diffuse radiation over the sky dome, showing the circumsolar and horizon brightening components added to the isotropic component. Adapted from Pescz et al. (1988). 2.14 Radiation on Sloped Suríaces 3 Sky models, in the context used here, are mathematical representations of the diffuse radiation. When beam and reflected radiation are added, they provide the means of calculating radiation on a tilied surface from measurements on the horizontal. Many sky models have been devised. A review of some of them is provided by Hay and McKay (1985), Since 1985, others have been developed. For purposes of this book, three of the most useful of these models are presented. The isotropic model is in Section 2.15, and two anisotropic models are in Section 2.16. The differences among them are in the way they treat the three parts of the diffuse radiation. K is necessary to know or to be able to estimate the solar radiation incident on tilted surfaces such as flat-plate collectors, windows, or other passive system receivers. The incident solar radiation is the sum of a set of radiation sueams including beam radiation, the three components of diffuse radiation from the sky, and radiation reilected from the various surfaces “secn” by the tilted surface. The total incident radiation vo this surface can be written as? Ir= try + Iráiso + Erdes + Irah: + Irrep (2.14.1) where the subscripts iso, cs. hz, and ref? refer to the isotropic, circumsolar, horizon, and reflected radiation streams. For a surface (a collecior) of area A, the total incident radiation can be expressed in terms of the bcam and diffusc radiation on thc horizontal surface and the total radiation on the surfaçes that reflect to the Lilted surface, The terms in Equation 2.14.] become Adr = 1Ryà ct Laço As e + les ÃO + Lapa po hoc + DhpiAiFre (Quis) The first tem is the beam contribution. The second is the isotropic diffuse term which includes the product of sky area A, tan undefined area) and the radiation view factor from lhe sky to the collector F,... The third is the circumsolar difluse, which is treated as coming from the same direction as the beam. The fourth term is the contribution of the diffuse from the horizon from a band with another undefined area Ay; The fifth term is the set of reflected radiation streams from the buildings, fields, etc., to which the tilted surface is exposed. The symbol i refers to each of the reflected streams: f, is lhe solar radiation incident on the ith surface, p; is the diffuse reflectance of that surface, and Fis lhe vicw factor from the ith surface to the bilted surface, Tt is assumed that the reflecting surfaces are diffuse reflectors; specular reflectors require a different treatment. 13 This and following equations are written in terms of / for an hour. They could also be serilten in terms of G, the irradiance. Ea Available Solar Radiation In general, it is not possible (o calculate the reflected encrgy term in detail, to account for buildings, trees, etc. the changing solar radiation incident on thern, and their changing reflectances. Standard practice is to assume thal there is one surface, a horizontal, diffusely reflecting ground, large in extent, contributing to this term. In this case, is simply / and p; becomes p,, à composite "ground" reflectance. Equation 2.14,2 can be rewritten in a useful form by interchanging areas and view factors (since the view factor reciprocity relation requires that, for example, Asc = AF os). This eliminales the undefined arcas A, and Ag, The arcaA, appears in each term in the equation and cancels. The result is an equation that gives Ein terms cf parameters that can be determined either thecretically or empirically. tr= Rb + laisoP os + lgsRo + lapeFeha + IPgD og (2,14.3) This equation, with variations, is the basis for methods of caleulating 4, that are presented in the following sections. When 4, has been determined, the ratio of total radiation on the tilted surface to that on the horizontal surface can be determined. By definition, - - Total radiation on the tilted surface 17 dia ! (2.14.4) Total radiation on a horizontal surface 1 Many models have been developed, vf varying complication, as the basis for calculating /r. The differences are largely in the way that the difíuse terms are treated. The simplest model is based on the assumptions tha! the beam radiation predominates (when it matters) and that the diffusc (and ground-reflected radiation) are effectively concentrated in the area of the sun. Then R = R, and all radiation is treated as beam. This leads to substantial overestimation of £,, and the procedure is not recommended. Preferred methods are given in the following two sections. 2.15 RADIATION ON SLOPED SURFACES - ISOTROPIC SKY It can be assumed fas suggested by Hotel and Woeriz (1942)] that ihe combination of diffuse and ground-reflected radiation is isotropic. With this assumption, the sum of the diffuse from the sky and the ground-reflected radiation on the tilted surface is the same regardless of orientation, and the total radiation on the tilted surface is the sum of the beam contribution calculated as /,R, and the diffuse on a horizontal surface, Fy. This represents an improvement over the assumption that all radiation can be treated as beam, but better methods are available, An improvement on these models, thc isotropic diffuse model, was derived by Liu and Jordan (1963). The radiation on the tilted surface was considered to include three components: beam, isotropic diffuse, and solar radiation diffuscly reflected from the ground. The third and fourth terms in Equation 2.14.3 arc taken as zero as all diffuse is assumed to be isotropic. A surface tiltcd at slope B from the 2.15 Radiation on Sloped Surfaces--Isotropic Sky 95 horizontal has a view factor to the sky ,., which is given by (1 + cos f)/2. (If the diffuse radiation is isotropic, this is also R,, lhe ratio of diffuse on the tilted surface ta that on the horizontal surface.) The surface has a view factor to the ground Fegof (1 cos By/2, and if the surroundings have a diffuse reflectance of p, for the total solar radiation, the reflected radiation from the surroundings on the surface will be tpg(1 - cos f)/2. Thus Equation 2.14.3 is modificd to give the total solar radiation on the tilted surface for an hour as the sum of three terms: L+e I- tes tuto + SEE tpg tro f (2.15.1) and by the definition of &, f, ujl+cosB i-cos atado eo fr NR E (2.152) Example 2.15.1 Using the isotropic diffuse model, estimate the beam, diffuse, and ground-reflected components of solar radiation and the total radiation on a surface sloped 60º toward the south at a latitude of 40ºN for the hour 9 to 10 AM on February 20. Here ! = 1.04 MI/m? and p, = 0.60. Solution For this hour, 4, = 2.3] MJjm2, so kz = 1.04/2,31 = 0.45. From the Erbs correlation, Equation 2.10.1, 14] = 0.757. Thus 14=0.757x 1.04 = 0.787 MJ/m? 1,=0.243x LOM = 0.253 MJjm? The hour angle «o for the midpoint of the hour is -37.5º. The declination ô is —11.6º. Then — cos(40 - 60) cos(- 1.6) cos(-37.5) + sin(40 — 60) sinf-11.6) Rh » cos 40 cos(-11.6) cos(-37.5) + sin 40 sin(- 11.6) = 979 0.466 tm Equation 2.15.1 gives the three radiation streams and the total: Ir=0.253x 1.71 + 0787(LE59860) + 1,04 x 0.60/=sos 60) = 0.433 + 0.590 + 0.156 = 1.18 Mm? | 100 Available Solar Radi Table 2.16.1 Brighiness Coefficients for Perez et al, Anisotropic Sky> Range of e tu to ta fm ta fa 0-1.665 0,196 1084-0006 Olá O180 0019 1.065 - 1.230 0.236 Os19 -0.180 00] 002 0038 1.330 - 1.500 qas4 032] 0255 UM 008 0046 1.500 - 1.950 0866 DIB] 035 0203 0403 0.049 1.950 - 2.800 1026 OH 0426 0273 0602 0061 2.800 - 4.500 0978 0986 -0350 0280 0915 DOM 4.500 - 6.200 0748 0913 0,236 0173 1045 0065 6200 -1 O318 -0757 0103 0062 1698 023% & From Perez et al, (1988). This set of equations altows calculation of all of the three diffuse components on the tilted surface. It remains to add the bcam and ground-reflected contributions. The total radiation on the tilted surface includes five terms: the beam, the isotropic diffuse, the cireumsolar diffuse, the diffusc from the horizon, and the ground- reflected term (which parallel the terms in Equation 2.14.3): 1+ redes tato (216.14) +laFasinB+ ted Equations 2.16.8 through 2.16.14, with Table 2.16.1, constitute 4 working version of the Perez et al. model, Tts use is ilustrated in the following example. Example 2.16.2 Do Example 2.15.1 using the Perez et al. method, Solution From Example 2.15.1: 4, = 2.31 MJ/m?, £ = 1.04 MJjm?, 1, = 0.253 Mym?, lg = 0.787 MJ/m2, cos O = 0.799, B = 37.0º, cos 8, = 0.466, 6, = 62.2º, and R$ = 171. To use Equation 2.16.14, we need 4,b,E, und 4 in addition to the quantities already calculated: a = maxf0, cos 37.0] = 0.799 b=maxicos 85, cos 62.2] = 11.466 and atb = 0.799/0.466= 1.71 (the same as R, in Example 2.15.1) 2.16 Radintion on Sloped Surfaces-- Anisotropie Sky ro Next calcuiate 4. The air mass , from Equation 1.5.1, is ma= Ifcos 62.2 = 1/0466 = 2.144 We also need Usc Equation 14.t withn= 51: ln = asa + 0033006 Ei SU s005 Mim? From the defining equation for 4, Equation 2.16.11, 4=0.787 x 2.144/5.025 = 0,336 We next calculate € from Eguation 2.16.10. Thus Z, = Iyjcos 8, é cabe ns s O; = 0.253% = 0.544 MJm?, and E 3jcos 62.2 8. Sis0 ide , 5.535 x 1046227 —Uudd. > > =130 145.535 x 104622) With this we can go to the table of coefficients needed in the calculation of F, and Fo. These are, for the third & range, fi =0454 to 0.321 fa =-0.255 fu=0072 fo=-0008 fo =-0046 So Fr maxlO, (0454 + 0.321 x 0,336 + 62.2/7-0.255)/180)] = 0,285 Fo = 0.072 + (0.098) x 0,336 + 62.274-0,046)/180 = 0.011 We now have everything needed to use uation 2.16.14 4 i the sloped surface: Eq O ger the total radiation on 1r=0.253x 1,7140787 - 0.285/I-teos 60) +0.787x 0.285 x 1.71 =0.,011 x 0.787 sin 60 + 1.04x 0.60 (t-ses co) 2 =0.433+0.422 +0.384- 0.007+0.156= 1.39MJfm? This is about 8% higher than the result of the H ibout 8º IDKR model and ab i dlian the isotropie model for this example. né about 16% higher uz Available Soiar Radiation The next question is which of these models for total radiation on the tilted surface should be used. The isotropic model is the simplest, gives the most conservative estimates of radíation on the tilted surface, and has been widely used. The HDKR model is almost as simple to use as the isotropic and produces results that are closer to measured values. For surfaces sloped toward the equator, these models are suggested. The Perez model is more complex to use and generaliy predicts slightly higher total radiation on the tiltcd surface; it is lhus the least conservative of the three methods. K agrees the best by a small margin with measurements. !é For surfaces with y far from 0º in the northern hemisphere or [80º in the southerm hemisphere, the Perez model is suggested. (In examples to be shown in later chapters, the isotropic and HDKR methods will be used, as they are more amenable to hand calculation.) 2.17 RADIATION AUGMENTATION K is possible to increase the radiation incident on an absorber by use of planar reflectors. In the models discussed in Sections 2.15 and 2.16, ground-reflected radiation was taken into account in the last term, with the ground assumed to be a horizonial diffuse reflector infinite in extent, and there was only one term in the summation in Equation 2.15.2. With ground reflectances narmally of the order of 0.2 and low collector siopes, the contributions of ground-refiected radiation are smali. However, with ground reflectances of 0.6 to 0.7 typical of snow and with high siopes,!? the contribution of reflected radiation of surfaces may be substantial. In this section we show a more general casc of the effects vt diffuse reflectors of finite size. Consider the geometry sketched in Figure 2.17.1. Consider two intersecting planes, the receiving surface c (i.e., a solar collector or passive absorber) and à diffuse reflector r. The angle between the planes is y. The angle y is 180º — £ if the reflector is horizontal, but the analysis is not restricted to à horizontal reflector. The length of the assembly is m. The other dimensions of the coltector and reflector are n and p, as shown. If the reflector is horizontal, Equation 2. 14.3 becomes dr = 16Rh + hdP ist Be cr + IpgE cg (7) where F.., is again (1 + cos f)/2. The view factor F,. is obtained from Figure 16 The HDKR method yields slightiy bexer results than either the isotropic mode! or the Perez model in predicting utilizable radiation when the critical radiation levels are significant. See Sections 2.20 10 2.22 for notes on utilizable energy. FT Ata slope of 459, a flat surface sees 85% sky and 15% ground. Al a stope of 90º, it sees 50% sky and 50% ground. 2.17 Rodintion Augmentatiun 103 Figure 2.471 Gicometric relationship of an energy receiving surface c and reflecting surface 1. 217.2, Fis obtained from he reciprocity relationship A, san be obtained from the summation rule, Ft E, + EE Fc is showa in Fig a Of the ratios 908, 120º, and 150º, =AF and Fo, . 1. The view factor ure 2.17.2 as à function of the ratios t/m and plm for W of Example 2.17.1 À vertical window receiver in a passive heating system is 3.0 m high and 6.0 m k There is deployed in front of it a horizontal, diffuse reflector of the same Tema extending out2.4m. What is the view factor from the reflector to the window? What is the view factor from the window to the reflector? What is the view f: tor from the window to the ground beyond the reflector? certos Solution For the given dimensions, nm is 3 0/6.0 à : o 30/60 = 0,5, py 4/60 = Figure 2.17.2a, the view factor E, , is 0.27. Pim bs 24/60 = 0 and tom The area of the window is 18,0 m? and the F e s 18.0 m?, e area of the reflcctor i 2 From the reciprocity relationship, P., = (14,4 x 0.27)/18.0 = 0.22, Ts tt The view factor from window 19 sky, E ) . “ecsviS (1 + cos 90)/2, View factor from collector o ground is then 1 = (0.50 4 0.22) = om or tdo ES am Hthe surfaces e and r are very long in extent (ie., m is large relative to n p as might be the case with long arrays of collectors for large-scale solar applications), Hottel's "crossed-string” method gives the view factor as Fa=ttP 1 2» (2.172) Available Solar Radiation ç é B s E z ê s “01 02 03 05 10 23.5 10 20 - nfm 01 02 03 05 0.06, e & 04 06 View factor, Fr-c e 2 0.08 im = 1.0 o 0702 05 10 23 5 1 20 ter Figure 2.172 View factor £,., as a function of the relative dimensions of the collecting and reflecting surfaces. Adapted from Hamilton and Morgan (1952). ion Angmentulion 105 re 2.17,3 Section of reflector and colector surfaces, where 5 is the distance from the upper edge af the collector to the vuter edge of the refletor, measured in a plane perpendicular to planes c and 7, as shown in Figure 2.17.3, This can be determined from (n2+p2-2npcos 2 (2173) [For a collector array as in Example 2.17.1 but very long in extent, » = (302 + 24205 =3.84 mand F,, = (342.4 3.84/4.8 = 0.33] It is necessary to know the incident radiation on the plane of the reflector. The team component is calculated by use of Rpy for the orientation of the reflector Surface, The ditlusc component must be estimated from the view factor F,,. For any orientation of the surface », Est Fic +Eng= (2174) where the vicw factors are from surface r to sky, to surface «:, and to ground. The vicw factor F, , is determined as noted abyvc and E v.g Will be zero for a horizontal reflector and will be small for collectors that ure long in extent. Thus as a first approximation, F,.,= | F,. for many practical cases (where there are no other obstructions). There remains the questions of the angle of incidence of radiation reflecicd from surface 1 on surface c. As an approximation. an average angle of incidence can be taken as that of the radiation from the midpoint of surface to the midpoint of surface Sa shown in Figure 2.17.3,/8 The average angle of incidence 8, is piven by sin 8, Y (2.475) The total radiation reflected from surfaçe r with area A,to surface c with area 4, if 18 As the refletor area becomes vers; large, the angle of incidence becomes that given by the ground- Teflectunce curve of Pigure 5.4.1, where the angle y beiween the reflector and the coltector is /, the abscissa om the figure, e-B=18 35 45 ss 25 35 as ss Latitude Latitude & 1 1) 50 pre reeeeotieoemerreererem E vertical / ã asp s0F 3 E Dec. / / 4 s Jon > ba E 30Ê 7 F 25 29f 15E 106 LAILA A | = CASADA EA uu e o aee sut 2s 38 25 38 E tomado é Latitude é e to Figure 2.19. Estimated Rh for surtaces facing the equator as à function of latitude tor various tg - f). by months. tb B=1SPiqbi to = 0% (e) (4 B)= 158: (4) B=909, For the soulhem hemisphere, interchange months as shown on Figure 1.8.2, and use lhe absolute value of latitude, From Beckman etal. (1977). nr Po 112 Available Solar Radiation For surfaces in the southem hemisphere sloped toward the equator, with 7= 180º, the equations are R= costg-+ f) cos ôsin ai + (2/180) oi sinhg-+ B) sin & (2.19.49) cos ficos Bsin ay + (17/180)0 sin Qsin Ô cosil- tan êtan à) and o; = min (2.19.4b) [aos Blum à) | “The numerator of Equation 2.19.3a or 2.19.4a is the extraterrestrial radiation on the tilted surface, and the denominator is that on the horizontal surface. Fach of these is obtained by integration of Equation 1.6.2 over the appropriate time period, from true sanrise to sunset for the horizonta! surface and from apparent sunrise Lo apparent sunset an the tilted surface. For convenience, piots of R, as à function of latitude for various slopes for = 0º (or 180º in the southem hemisphere) are showa in Figure 2.19.1, and corresponding tables of Ry are in Appendix D. These values of É, can be used for surface azimuth angtes of 0º (or RO?) + 15º with lintte error. The following example illustrates the kind of calculations that will be used in estimating monthly radiation on collectors as part Of heating system design procedures. Example 2.19.1 A collector is to be installed in Madison, latitude 43º, at a slope of 60º to the south. Average daily radiation data are shown in Appendix G. The ground rellectance is 0.2 for ali months exccpt December and March (g, = 0.4) and January and February (pg = 8.7. Using the isorropic diffuse assumption, estimate the monthly average radiation incident on the collector. Solution “The caleulation is detailed below for January, and the results for the year are indicuted in a table. The basic equation to be used is Equation 2.19.1, The first steps are to obtain HH and Rb. The ratio Hyff is à function of Ky and can be obtained from Equation 2.12.1 or Figure 2.12.2. For the mean January day, the 17th, from Table 1.6.1, 1 = 17, O = 20.99 The sunset hour angle is caleulated trom Equation 1.6.10 and is 69.19. With n = 17 and «3, = 69.1º, from Equation 1.10.3 (or Figure 1.10.1 or Table 1.10.1), Ho is 13.36 MJ/m?. Then Ky = 5.85/13.36 = 0.44. The Etbs correlation, Equation 2.12.1a. is used to calculate HH com Ky and aa, and gives HH = 0.45. The calculation of R, requires the sunset hour angle on the sloped collector. From Equations 2.19.3 cosH-tan(43 — 60) tan(-20.8)] = 96.7º 2.20 Average Radiation on Sloped Surfaces-—The K-T Method n3 The angle q), was calculated as 69.1º and is less than 96.7º. so 3, = 69.1º. Then p= vost 17) cost-20.9) sin 69.1 + (7 x 69.1/180) sin(-17) sin(-20.9) p = =2,79 cos 43 cost-20.9) sin 69.1 + (rx 69.1/180) sin 43 sinç--20.9) The equation for Jfy, Equation 2.19.1, can now be solved: Hir= 58501 - 0.45)2.79 + 585 x 045(L+ c05 00) + 5.85 x 07! cos 80) =8.87 + 2.004 1.02= 11.89 Mm? The results for the 12 months are shown in the table. Energy quantities are in megajoules per square meter. The effects of sloping the receiving plane 60º to the south on the average radiation (and thus on the total radiation through the winter season) are large indeed. The monthly results are shown in the table. The 7, values are shown to à tenth of a megajoule per square meter. The last place is uncertain due to the combined uncertainties in lhe data and the correlations for H3/H and R. TU is difficult to put limits of accuracy on them; they are probably no better than 10%. | Month " Ho Kr Haftt Ro p» ty Jan. ses 1337 Uda 0.46 2.79 07 o Feb, 9.13 1881 0.49 0.41 2.04 87 155 Mar. 12,89 2603 o.so u43 142 os 15% Apr. 1588 2378 047 146 1.96 82 us May 19.79 142 est UA om “2 153 Jume 221 4178 0:53 0.40 0.62 “2 159 July 2196 40.55 (54 11.39 0.66 02 164 Aug. 19,39 as? 0.54 0.39 0.84 [ 16.6 Sep. 14.75 2880 0.51 va 1] o2 158 Out 16,34 20.90 0.0 11.40 E] 02 149 Nov sn 14.63 039 1.54 2.56 “2 96 Dec. as ER 0.37 0.54 3.06 ns £s n 2.20 AVERAGE RADIATION ON SLOPED SURFACES - THE K-T METHOD An alternative approach to calculation of average radiation on sloped surfaces has been developed by Klein and Theilacker (1981). K is a bit more cumbersome to use but shows improved results over the isotropic method when compared with hourly caleulations based on many years of radiation data. The method is first outlined 114 Available Solar Radiation below in a form restricted to surfaces facing the equator and then in general form for surfaces of any oricniation, As with Equations 2.19.1 and 2.19.2, it is based on the assumption that both diffuse and ground-reflected radiation streams are isotropic. The long-term value of R can be calculated by integrating Gy and G from sunrise to sunset for all days over many years of data for a single month and summing. (For example, data for all days in January for 10 years should represent the long-term average for January.) $ [orar Tate =dyrtt o (2.20.1) $ fica day=1 tor The denominator is NH. To evaluate the numerator, it is convenient to replace Gy by 1; and exchange the order of lhe integration and summation. Using Equation 2,15.1, the radiation at any time of the day (i.e., for any hour) for N days is nir= nf- abs + ESB) tp; [I-c8 | (2.20.2) where the 7 and 1; are long-term averages of the total and the diffusc radiation, obtained by summing the values of f and 1, over N days for cach particular hour and dividing by A. Equation 2.20.1 then becomes?! [leiam dese deseo o (2.20.3) or Equations 2.13,1 and 2.13,3 define thc ratios of hourly to daily total and hourty to daity difíuse radiation, and Equations 2.13.2 and 2.13,4 relate r, and r; to time «» and sunset hour angle 0, Combining these with Equation 2.0.3 leads, for the case of south-facing surfaces in the northem hemisphere, to (2.20.4a) 21 The developracnt of this equation assumes that the day length does not change during the month, 220 Average Radiation on Sloped Surfuces—The K-T Method uns where qo, is again given by comi | Cosan gtan 6) aj= no) (2.20.4b) cos-[an(g - Ban 6] and 3) = cos-[an(g - 8) tan 6] (2.20.40) Also, « and b are given by Equarions 2.13.2b and 2.13.2€, and dis given by d=sin 0, - Es sin 03 FR cos (3, (2.20,49) Equations 2.20.4 can be used in thc southem he phere for north-faci by mis y ' s) for north-facing surfaces Example 2.20.1 Redo Example 2.19.1 for the month of January. using the Klein-Theilacker method. Solution For January. from Example 2.19.1, Ff, = 13.36 MUm?, FLgH = 0.45. and for the mean day of the month (x = 17), q = 6); = 69.1º, Forthe mean day, a = 0.409 + 0.5016 sin(69.1 — 60) = 0.488 & = 0.6609 — 0.4767 sin(69.1 — 60) = 0.586 d=sin 69,1 EX62 Tão “Os 69.1 = 0.504 cos L-tan(43 — 60) tan(-20.9)] = 96.7º Using Equation 2,20.4a, R = Cos43 60) nx691 0504 cos 43 [o-se - vasifin 69.1 -— ao Sos x) Tx 1 + 586 erga + sin 69.Icos 69.1 --2 cos 96 ) +045[ 1 +cos 60 1 -cos 60 500) + 0.7(L 280660 4.449 40.338 +0.075= 195 So monthly average radiation on the colector would be He=IÍR=585x195= 114 Mm? 120 Available Solar Radiation 20 3 'r. MU/m? per day HandH LiILSILAILIILAS dan. Feb. Mar. Apr. May Jun. Jul Aug Sept Oct. Nov, Dec Figure 2.244 Variation in estimated average daily radission on surfaces of various stopes às a function of time of ycar for a latitude of 45º, Ky of 0.50 surface azimuth angle of 0º, and a ground reflectançe 090.20. temperate climates. Figure 2.2].1 shows the varialions of Hr (and H) ihrough the year and shows the marked differences in energy received by surfaces of various in summer and winter. . do 2.214,24) shows the total annual energy received as a function of slope and indicates a maximum at approximately 8 =$. The maximum is a broad one, and the changes in total annual energy are less than 5% for slopes of 20º more or less than the optimum. Figure 2.21.2(a) also shows total “winter” energy, taken as the total energy for the months of December, January, February, and March, which would represent the time of the year when most space heating loads would occur. The siope corresponding to the maximum estimated total winter energy is approximately 60º, or 6 + 15º. A 15º change im the slope ot the collector frara the optimum means a reduction of approximately 5% in the incident radiation. The dashed portion of the winter total curve is estimated assuming that there is substantial snow cover in January and February that results in a mean ground reftectance of 0.6 for those 2 months. Under this assumption, the total winter energy is less sensitive to slope than with p, = 0.2. The vertical surface receives 8% less energy than does the 60º surínce if p, = 0.6, and 11% lessifp,=0.2. — Calculation of total anmual energy for $= 45º, Er = 0.50, and p, = 0.20 for surfaces of slopes 30º and 60º are shown as a functios of surface azimuth angle in 2.21 Effects of Receiving Surface Orientation ui 8 E E g 3 é ê 5» 6% ê É ê 5 s 2 3 5 q ê FÊ 5 E 5a g z g é 2 B& so o 15 30 as Slope Surface azimuth angie, y ta (o) Figure 2.21.2 (a) Variation of tutal annual energy and total winter (December to March) energy 35 a funelion of surface slope for a Iutitude of 45º, Kr of 0.50, and surface aziminh angle of 0º Ground reflectance is 0.20 except for the dashed curve where it is taken às 0.60 for January and February, (b) Variation of total annual energy with surface azimuth angie for slopes of 30º and 60º, Tatilude of 45º, Kr0f 0.50, and ground reflectance of 0.20, Figure 2.21.2(b). Note the expanded scale. The reduction in annual energy is small for thesc examples, and the generalization can be made ihat facing collectors 10º to 20º cast or west of south should make little difference in the annual energy received. (Not shown by annual radiation figures is the effect of azimuth angle on the diumal distribution of radiation on the surface, Each shift of y of 15º will shift the daily maximum of available energy by roughly an hour toward moming if y is negative and toward afternoon if y is positive, This could affect the Performance of a system for Which there are regular diurnal variations in energy demands on the process.) Note that there is implicit in these calculations the assumption that the days are symmetrical about sotar noon. Similar conclusions have been reached by others, for example, Morse and Csamecki (3958), who estimated the relative total annual beam radiation on surfaces of variable slope and azimuth angle. From studies of this kind, gencral “rules of thumb” can be stated, For maximum annual energy avaitability, a surface slope equal to atitude is best. For maximum summer availability, slope should be approximately 20º to 15º Jess than the latitude. For maximum winter energy availability, slope should be approximately 10º to 15º more than the latitude. The slopes are not critical, deviations of 15º result in reduction of (he order of 5%, The expected presence of a reflective ground cover such as snow leads to higher slopes for maximizing wintertime energy availability. The best surface azimuth angles for maximum incident radiation are 0º in the northem hemisphere or 180º in the southem hemisphere, that is, the surfaces should face the equator. Deviations in azimuth angles of 10º or 20º have small effect on total annual energy availabilicy. m Available Solar Radiation 222 UTILIZABILITY qu No he concepts of utilizabffity arc developed. The atlove a critical or threshold intensity is. useful, then we can define a radiation statistic, called utilizability, as he fraction of the total sadiation that is received at an intensity higher than the critical evel. We can then oz the period by this fraction to find the total ulizable multiply the average radiation £ energy. In these sections we define utlizability and show for any critical level how it can be calculated from radiation data or estimated from K7. Tu this section we present the concept of monthly average hourly utilizabitity (the $ concept) as developed by Whillier (1953) and Hotel and Whillier (1958). Then in Section 2.23 we show how Liu and Jordan (1963) gencralized Whillier's É curves. Tn Section 2.24 we show an extension of the hourly utilizability to montbly average daily utilizability (the é concept) by Klein (1978). Collares-Pereira and Rabl (1979a,b) independently extended hourly utilizability to daily utilizability. Evans et al. (1982) have developed a modified and somewhat simplified general method for calculating monthly average daily utilizability. ta Chapter 6 we develop in detail an cnergy balance equarion to represent the performance of a solar collector. The energy balance says, in essence, that the useful gain at any time is the difference between ihe solar energy absorbed and the thermat losses from the collector. The losses depend on the difference in temperature between the collector plate and the ambient temperature and on a heat loss coefficient. Given a coefficient, a coilector temperature, and an ambient temperature (Le., a loss per unit area), there is a value of incident radiation that is just enough so that the avsorbed radiation equals the losses. This value of incident radiation is the critical radiation level, fr., for that colector operating under those conditions. H€ the incident radiation on the tilted surface of the collector /r is equal ta fr. all of the absorbed energy will be lost and there will be no useful gain. Tí the incident radiation excecds Ir, there will be useful gain and the collector should be operated. JE Ly às Jess than Jr. no useful gain is possible and the collector should not be operated. The utilizable energy for any hou” is thus (17 = Iro)*, where the superscript + indicates that the utilizable energy can be zero or positive but not in this and the following two sections basis is a simple one; if only radiation negative. The fraction of an hour's total en: utilizability for that particular hour: = fr=ind. 222.1) Ei ergy that is above the critical level is the where $, can have values from zera to unity. The hour's utilizability is the ratio of the shaded area (Ly = 1rç) to the total avea (Fr) under the radiation curve for the hour as shown in Figure 2.22.1. (Utilizability contd be defined on the basis of rates, ie. using Gy and Gr.. but as à practical mater, radiation data are available on an hourly basis and that is the basis in use.) 222 Utilizability 13 6r Time Figure 2.224 Gy vs. time for a day. For wn, E is the area under the G7 curve, ' ti y the hour shown, fp T otris tro is the area under the constant critical radiation tevel curve. , ie uiizabity for à single hour is not useful, However, utilizability for a articular hour for a month of N days (e.g. 10 to 1i in Jam in wii , average radiation is /7 is useful. It can be found from ver) a th “ 0= 15 Vr-irt N z Ih (2.22.2) The monthly average utilizabic enei i f . " 1 rgy for the hour is the product / cateuaion can be done for individual hours (10 to Li, LI to 12, di )or Eno and the result summed to get the months utilizable enet e app ] d to g le energy. If the application i that the conditions of critical radiatioi | rat n level and incident radiation are s; i about solar noon, the calculations can be done for hour-pairs (e.g., 10 to and Tn ; Zo0r9 to EO and 2 to 3) and the amount of calculations halved. º " Ge dou average radiation data by months and a critical radiation level, the ext step is to determine q. This is done by processin) ation da ! 3 the h a ; las outlined by Whillier (1953) as follows: Me holy slim a ir fai Dor a given location, hour, month. and collector orientation, plot a cumulative a tuo curve of /r/fr. An example for a vertical south-facing surface at Blue : Ea A. or January is shown in Figure 2.22.2 for the hour-pair 11 ta 12 and 12 to doi is provides E picture of the frequency of occurrence of clear, party cloudy, or ly skies in that hour for the month. For example, fi o i a À - s he hour- f Fi 2.22.2, for f of 0.20, 20% of the da) jaion char a 2a 10% of ho 1 .20, ys have radiation that is less than 10% average. and for f of 0.80, 20% of the days h: jon ii E a Acuede OE ie aver y5 have radiation in that hour-pair that A dimensionless critical radiation is defined as (2.223) Ansemple is shown as the horizontal line in Figure 2.22.2 where X, = 0.75 and f. 04. The shaded area represents the monthly utilizability, that is, the fraction of nthly energy That is above (he enitical level. Integrating hourly utilizability over a 14 Available Solar Radiation 25 T T T T Biue Hi Observatory South-facing vertical surface Januaty, 1953-1056 zo HI-12 am. ars 12-1 pum, x - Critical ad. * Average raú. fe I Q .2 0.4 85 [E] 19 Fractlonal time, f, during which radiation < 3, Figure 2,222 Cumutative distribution curve for hourly radiation on a south-facing vertical surface in Blue Hill, MA. Adupted from Liu and Jordan (1963). all values off, gives é for that critical radiation level: 1 *[ du df (2.224) Hc e As the critical radiation level is varicd, f, varies, and graphical integrations of the curve give utilizability À as a function of critical radiation ratio X,.. An example derived from Figure 2.22.2 is shown ir Figure 2.223. Wihillier (1953) and later Liu and Jordan (1963) have shown that in a particular location for a I-month period, à is essentially the same for all hours. Thus, although the curve of Figure 2.22.3 was derived for the hour-pair 11 to 12 and 12 t0 1, it is useful for all hour-paírs for the vertical surface at Blue Hill, The line labeled "imiting curve of identical days” in Figure 2.22.3 would result from a cumulative distribution curve that is à horizontal line at a value of the ordinate of 1.0 in Figure 2.22.2. In other words, every day of the month looks like the average day. The difference between the actual q curve and this limiting case represents the error in ulilizable energy that would be made by using a single average day to represent a whole month, Example 2.22.1 Caleulate the utilizabie energy on a south-facing vertical sofar collcctor in Blue Hil!, MA, for the month of January when the critical radiation level on the collector is 1.97 Mimi, The averages of January solar radiation on a vertical surface are 1.52, 1,15, and 0.68 MJ/m? for the hour-pairs 0.5, 1.5, and 2.5 hours from solar noon. 2.23 Generalized Utilizabitity ns 10 oa o a Usilizabinty, é o3| Limeting curve of identical days. 04 08 12 16 O. Critical radiation ratio, X, = yu/fy Figure 2.22.3 Utilizability curve derived by icalhy à i y numericaliy inté F a iz y Integrating Figure 2.2.2. Adapted Solution For the hour-pair 11 to 12 and 1210 1, the dimensionless critical radiation ratio X is e 107. 1.52 9:70 x and the utilizability, from Figure 2.22.3, is 0.54. The uti , 22.3, is 0,54, The utiliz: collector during this hour is tabs energy om the Irô= 1.52 x 0.54 = 0.82 MJjmê For the hour-pair 10 to 11 and 1 to 2 the valve of X, is 0.93, gs 0.43, and heis DAS. For the hour-pair 9 to 10 and 2 to 3, X, is 1.57, is 0.15, and rg is 0.10 The average utilizable energy for the month of January is then , ND ir$=31x2(0.82+04940.10)=875 MI/m? E hes 223 GENERALIZED UTILIZABILITY We now have a way of calculating q for specific locations and specific orientations, For most locations the necessary data are not available, but it is possible to make use of lhe observed statistical nature of solar radiation to develop generalized q curves Ihat depend enly on É, latitude, and collector slope. As noted above, $ curves are Do Available Solar Radíntion sunset hour angle and the day length of February 16, the mean day of the month, are 78.9º and 10.5 hours, respectively. The monthly average ratio /Ly/H is 0.39 from Figure 2.12.2, and ) = 7.5º. The fatios +, and rg from Figures 2.13. and 2.13.2 are 0.158 and 0.146. For the mean day in February and from Equation 1.8.2, R, is 1.62. Then fr ftom Equation 2.23.4 is E = 0.5 x 20.5[(0.158 — 0.39 x 0.146)1.62 + 0.39 x 0.146 + 0.88 + 0.740.158 + 0.12] = 2.33 MJm? The critical radiation rate for this hour-pair is eim 1280 “kh 233 0.55 From the figure of Example 2.23.1, é is 0.50. The utilizable energy for the month for this hour-pair is UE=233x 0.50x 2x 28 = 65.2 Mm? n Liu and Jordan (1963) have generalized the calculations of Example 2.231. They found that the shape of the À curves was not strongly dependent on the ground reflectance or the view factors from the collector to the sky and ground. Consequently, they were able to construct a set of $ curves for a fixed value of Er. The effect of tilt was taken into account by using the monthly average ratio of beam radiation on a tilted surface to month!y average beam radiation on a horizontal surface Ro as a parameter. The generalized é curves arc shown in Figures 2.23.1 for values of Er of 0.3, 0.4, 0.5, 0.6, and 0.7. The method of constructing these curves is exactly like Example 2.23.1, except thal the tilt used in their calculations was 47º and tie ground reflectance was 0.2. À comparison of the é curve from Example 2.23.1, in which the tilt was 40º and the ground reflectance was 0.7 with the gencralized é , curve for Er = 0.5 and R$ = 1.79, shows that the two are nearly identical. With the generalized à curves, it is possible to predict the utilizable energy ata constant critical level by knowing only the long-term average radiation. This procedure was illustrated (for one hour-pair) in Example 2.23.2. Rather than use the é curve calculated in Example 2.23.1, the generalized 6 curve could have been uscd. The onty additionat calculation is determining R, so that the proper curve can be selected. In Example 2.23.2, X, = 0.55. From Table D-7 in Appendix D, Ry = 179. Figure 2.23.1e is used to obtain 6; it is approximately 0.50. Xt is convenient for computations to have an analytical representation of the utilizability function. Clark et al. (1983) have developed a simple algorithm to represent the generalized É functions. Curves of q vs. X, derived from long-term 2.23 Generalized Udliizability m 1 Usilizabitity, o o 0 Usilizabilicy, é Figure (1363). “ANTT To | : 1 Beco Tifted surface | po Bs H Ny 1 TT — 1 E itenbca dava" Horizon ERR q [SE 2 E Limitingcune gado = oc liia N bias EE 0 04 08 12 16 20º 24 28 32 Critica radiation ratio, X, (a) NUTTTO T TTT DB T | + 05 — N | /-Limiting curve, Rg + os | q ! Titgd surto, à, O | a! E [7-2 30,6 + oa | N + ting curve “ a ntical days] 4 oa 0 04 08 12 1.6 zo 24 28 32 Critical radiation ratio, Xp ca 2.231 Generalized q curves for south-facing surfaces. Adapted from Liu and Jordan | 1» Available Solar Radintion 223 Generalized Utilizabilicy 133 | VETO TT weather data are representeu by | Rç=05 4 Va i > os 4 | o EX 2X | - 3 05 (1-XolXmP EX =2 (2.23.54) ! HH | . tas Ling cure em 4 | lgt-Le2+(1 +24) 2.74)? otherwise ZE — Tilted suríace, É, 2 E 29 4 wire 2X -DQ-X,) (2.23.5b) 5 mu F Horizontal surface A F | NK 3 , Xm = 1.85 + 0.169RyR? - 0.0696 cos B/k? - 0.981Ír/c0528 (2.23.5€) o2- + q o E Tente da] 3 The monthty average hourly cicamess index E is defined as Guliuid Lupilisiiiss o % 0 06 12 18 20 24 28 a? Rel (2.23.6) Cica radiation ati, Xe i teh é Kt can be estimated using Equations 2.13.2 and 2. 13.4: . nà KoF, . GL = Eir= Kofa + heos 0) (2237) where a and b are given by Equations 2.13.2b and 2.13.2e, The remaining term in Equation 2.23.5 is Ry, the ratio of monthly averae hously radiation on the tilted surface 10 that on a horizontal surtace. Ra=bil=tri(ttr) (2.23.8) “OTTO TOTO NOT TITTITO | Rr=08 R=:070 + Example 2.23.3 - t + 08) 4 | 4 Repeat Example 2.23.2 using the Clark et al. equations. E 4 I . É N | 4 Solution = 08 Limiting curve Limiting curve . E p>e | Ryu q The calculations to be made are Ry, É. XX, 8, and finally 9. Intermediate Ê , 5 results from Example 2.23.2 that are useful here are / = 2.33 MJ/m2; 1, = 0.158; E surteco a 00, = 789%: w= 7.59; and X, = 0.549: 02 Montana boricomal Ryetro 2.33 =144 [O identical dove 4 a NI 4 nH o 0.158x20.3x 0.50 Limitir oETEPSTL NO LUSA 110 o | , 0 04 0.8 12 0 04 08 12 16 To calculate kr we need the constants q and b in Equation 2.23.7; ta Critical radiation catio, X, te) ás à = 0.409 + 0.5016sin(78.9 - 60) = 0.571 Figure 223.1 (continued) and b=0,6609 - 0,4767sin(78.9 — 60) = 0.506 14 Available Solar Radiatlon Thus Er = 0.50(0.571 + 0.506 cos 7.5)= 0.536 Next calculate X,, with Equation 2.23.8e; 0 (0.981 x 0.536 = 0.169 Lé4. 0.0696 €05 = 1,942 Xm= 1.85 + 0.536? 0.536? co The last steps are to calculate g and é with Equations 2.23.5b and 2.23.5a: g=(1.942- 12 1.942) = 16.24 5492 Then o=[1624 - [16.24 +(1+2x 16.24)1- assa ] =0.52 a This is neasly the same $ as that from Example 2.23.2. “The é charts graphicaliy illustrate why a single average day should not be used to predict system performance under most conditions, The difference in utilizability as indicated by the limiting curve of identical days and the appropriate $ curve is the extor that is incurred by basing performance on an average day. Only if Kr is high or if the critical level is very low do all 6 curves approach the limiting curve. For many siluations the error in using one average day to predict performance is substantial. The é curves must be used hously, even though a single 6 curve applies for a given collector orientation, critical level, and month. This means that three to six hourly calculations must be made per month if hour-pairs are used. For surjaces facing the equator, where hour-pairs can be used, the concept of monthly average daily utilizability, 9, provides a more convenient way of calculating useful energy. However, for processes that have critical radiation levels that vary in repeatabie ways through the days of a month and for surfaces that do not face the equator, the generalized é curves must be used for each hour. 2.24 DAILY UTILIZABILITY The amount of calculations in the use of é curves led Klein (1978) ta develop the concept of monthly average daily utilizabiliy, É. This daily utifizabiliy is defined as the sum for a month (over all hours and days) of the radiation on a tilted surface that is above a critical level divided by the monthly radiation. In equation form, DL lir-ir Era (2.24.1) HiN 224 Daily Utilizability 135 where the critical level is similar to that used in the $ concept?! The monthly utilizable energy is then the product HjN' The concept of daily utilizability is illustrated in Figure 2.24.1, Considering either of the two sequences of days, Gis the ratio of the sum of the shaded arcas to Lhe total areas under the curves. The value of 6 for a month depends on the distribution of hourly values of radiation in the month. 1fit is assumed that all days are symmetrical about solar noon and that the hourly distributions are as shown in Figures 2.13.1 and 2.13.2, then & depends on the distribution of daily total radiation, that is, on the relative frequency of occurrence of below average, average, and above average daily radiation values.” Figure 2.24.1 illustrates this point. The days in the top sequence are all average days; for the low critical radiation level represented by the solid horizontal line, the shaded areas show utilizable energy, whereas for the high critical level represented by the dotted line, there is no utilizable energy. The bottom sequence shows three days of varying radiation with the same average as before; utilizable energy for the low critical radiation level is nearly the same as for the first set, but there is utilizable energy above the high critical level for the nonuniform set of days and none for the Sequence À Radiation Day 1 Dy2 Day3 Figure 2.24. Two sequences of days with the same average radiation levels on the plane of the colector. From Klein (1978). 24 The critical level for & is based on monthly average "optical efficiency” and temperatures rather than on values for particular hours, This wil! be discussed in Chapter 21. 25 Klein assumed symmetrical days in his development of $. It can be shown that depanure from Symmetry within days (e.g., if aftenoons are brighter than mormings) wil! lead tg increases in 8; thus a & calculateá from the corretations of this section is somewhat conservative. References 42 Available Solar Radiation ; : 140 From Equation 1.8.2, Ry, is 1.38, Then R, can be calculaicd using Equation irom Equation 1.8.2, Ryn 2.24.2: | 9134 062) 1384 0.146 rosie fiz 0.134 x 062 (1 sos 60) Toi 1 Equation 2.19.2 is used to calculate R. From Figure 2.19.1 or Appendis » Ri . 1.42. From Figure 2.12.2 HalH is 0.43 at Ky = 0,50, (See Example 2.19. more details.) Then 1 - cos 60 60 J=1.23 R (041.42 + 043(-+60860) + 04f o and Ra!R = 1.12/1.23 = 0.91 From Equation 2.24.3 the dimensionless average critical radiation level is 245 x 3600 - 0.146 x 1.12 x 12.89 x 109 $ i ions 2.24.4, We can now get the utilizability 9 from Figure 2.24.2c or from Equatons 24 With Er = 0.50, a = 0.685, b = 0.819, ande = 0.411, $= 0.68. The months utilizable energy is thus EN $= HRNô= 1289x 1.23x 31 x 0.68 = 334 MJjm? E The $ depend on R and R,, which in tum depend on the aivisioa of toi radiation into beam and diffuse components. As noted in Section z A ace sustenta! uncertainties in determining this division, The correlation 5 a Mes ii IC N Er of Li 1960) was used by Klein (1978) to genorate the 1 a nos of Pá aa (1976), which indicates significantly Dieter actions of diffuse radiation, was also used to generate & charts, and de real Is vers a significantly different from those of Figure 2.24.2. A ground rei e ane o a ed in generating the charts, but a value of 0.7 was also use: : made no E ificant difference. Conscquentiy, even if the diffuse-to-total correi ion 5 cam ea a result of new experimental evidence, the É curves will remain vali a i pia using different correlations will change the predictions of radiation on a tiltes course, j ' à MI change the performance estimates. . o tio em te thought Of as a radiation statistic that has built into i cia ? be applied to a variety ef design iation levels. The $ and q concepts can e ema for heating systems, combined sotar energy-heat pump Systems, and a dibers The concept of utilizability has been extended to en to pessive sã ei p ii that cannot ildi excess energy (unutilizable energy) a ii sue can be estimated. The unutilizability ídea can also apply 10 uildin) photovoltaic systems with limited storage capacity. 215 SUMMARY In this chapter we have described the instruments (pyrheliometers and Pyranometers) used to measure solar radiation, Radiation data are available in several forms, with the most widoty available being pyranometer measurements of total (beam plus diffuse) radiation on horizontal surfaces, These data are available on an hourly basis from a limited number ot Stations and on a daily basis for Many stations. Solar radiation information is needed in severa! different forms, depending on the kinds of calculations that are to be done, These calculations fall into two major categories. First (and most detailed), we may wish to calculate on an hous-by-hour basis Lhe long-time performance of a solar Process system; for this we want honrly information of solar radiation and other meteorological measurements, Second, monthly average solar radiation is useful in estimating long-term performance of some kinds of solar processes. 1 not possible to predict what solar radiation will be in the future, and recourse is made to use of past data to predict what solar processes will do. time distribution of radiation ir a day, and radiation on surfaces other than horizontal. We introduced the concept of utilizabitity, a solar radiation statistic based on levels of sadiation available above critical levels, Determination of eriti collectors will be treated in Chapters 6 and 7, and the utilizability concepts will be the dasis for most of Part LI], on desiga of solar energy processes. REFERENCES Bameror, R. B. and 1. R. Howell. Sofar Energy. 22, 229 (1979). “Predicted Daily and Yearly Average Radiation Performance of Optimal Trapezoidal Groove Solar Energy Collectors.” Banmister, 1, W Solar Radiation Records, Division SÊ Mechanical Engineering, Commonwealth Scientifir and Industrial Research Organization, Australia (1966-1969) Beckman, W. A., S.A, Klein, and J. À, Duffie, Solar Hfeating Design by the AChart Method, Wiley-Imerscience, New York (1977), Bendt, P., M. Coltares.Pereira, and A. Rabl, Solar Energy, 27,1 (1981). "The Frequency Distriburion of Daily Radiation Values.” Bene, 1, Solar Energy, 9, 145 (1965), "Monthly Maps of Mean Daily Insolation for the United States,” Benseman, R. F. and F. W. Cook, New Zealand Í Sciences, 12, 696 (1969), "Solar Radiation inNew Zealand The Standard Year anq Radiation on Inclined Slupes.” * Boes, E. C., Sandia Repor SAND 75.0565 (1975). “Estimating the Direct Component of Solar Radiation." Chiam, FL. F., Solar Energy, 26, 503 (1981), “Planur Concentrators for Flat-Plate Collectors. Available Solar Radiation 142 “Stationary Reflector- Augmenied Flat-Plate + Chiam, H. F. Solar Energy, 29, 65 (1982). Colleciurs. 963) .N.K.D., Solar Energy, 7,44 (196 Comqena V..J, R. Owenby, and R. G. Baldwin, NOAA report to Department of Energy (Nov, Cinquemani. V.. JR. , and R. 1978). "Input Data for Solar Systems. no D.R.S. A, Klein, and WA. Beckman. Trans. ASME, / Solar Energy as “Adgorishm for Evaluating the Houriy Radiation Utilizabiity Function: colicas te Average Distributiun of "Soler Radiation at Now Delhi.” Engrg.. NUS, 281 Collares-Pereira, M. ar w Energy, 22, 155 (19794), . and A. Rabi, Solar Eneryy. 22, Solar Radiation - Correlations between Diffuse and Hemispherical and between Daily and Honrly olar Radiati Insolation Values. (1979), ' - Rabl. Solar Energy. 23, 235 (19 Podia Long E Performance of Nonconcentrating and of Concentrating Solar Predicting 'Simple Procedure for Collectors.” no it European Solur Radiation Atlas, Vol. 1, iss European Communities (CEC), Lu Globo Radio o Horizontal Surfaces, Vol. 2, Inclined Surfaces (W. Palz, ed), Vertag TÚ jobal Ra o Rheinland, Koln, (1984). c f 751. Coutson, K.L.. Solar and Terrestrial Radiation, Academic Press. New York (1975) . B.. Nei Radiation Received by a Horizontal Surface at the Earth, monograph published by dejong, B.. Net Def University Press, Netherlands (1973). sus do À. 3. Archiv far Meteorologis Geophysik Bioktimatologic. Series B, 7. 5 Drummond, À. 3.. K adiation “On the Measurement of Sky Ra | dd, A. 3. 1 of Applied Meteorology, 3, 810 (1964). "Comments on Sky Radiation Drummond, A. 3, Jd) Measurement and Corrections.' e, Solar Energy. 28, 293 (1982), . GS. À. Klein, and J. A. 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Sheridan, Solar Energy, 19, 663 (1977), “The Use of Planar Reflectors Srassie, 8, L. .L j $ for Increasing the Energy Yield of Flat-Plate Collectors. au immer, D. P., K. G. Zínn, K. C, Herr, and B. E, Wood, Report Laden St Ls atames Seia Lab (1978), "Augmented Solar Energy Collection Using Various Pin Scientil . 1 á Surfaces: Theoretical Caleulations and Experimental Results. . D.C. and W. R. Morgan, National Advisory Committee for Aesonautics, Technical Not and W R. o) a asse (is Radiant Interchange Configuration Pacto. References 143 + in Proc, Firsr Canadian Solar Radiation Data Workshop (J. E, Hay * Of Supply and Services Canada, 59 (1980), “Caleulation of the Solar Radiation Incident on an Inclined Surtaçe” Hay, 1. F. and D.C. McKay, 48 3, Solar Energy, 3, 203 (1985), “Estimating Solar Irradiance en inclincd Surfaces: A Review and Assessment DF Methodologies.” Herzog, M. E, MLS. 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Solar Energy, 22. 87 (1979), “A 54 tudy of Canadian Diffuse and "Total Solar Radiation Data - Tt Monthly Average Hourly Radiation” Jabal, M., An Introduetion ta Solar Radiation, Academic Press, Toronto (1983) Jeys TH. and E. E, Vant-Hull, Solar Energy, 18, 343 (1976). “The Contribution of the Solar Aureole to the Measurements af Pyrhetiometers ” Klein, S. A. Solar Energy, 19, 325 (1977). Tilted Surfaces” Elein, 8, “Calculation of Monthly Average Insolation on Solar Energy, 21, 393 (1978). "Calcutations ot Flut-Plate Colicctor Utitizabitity.” Klein, S. A. and 3. A, Duffie, in Proc. SÍ 197% Annual Meeting American Section, International Solar Energy Society, 2,2, 672 (1978). “Estimation of Monthly Average Diffuse Radiation.” Klein, S. À. and 3. C Theilackes, Tras. ASME, 4. Solar Energy Engra, 103, 29 tI981). “An Algorithm for Calculating Monthly-Average Radiation on Inclined Surfaces * Knight, K. MS. A. Klein, and 1. A. Duffie, Solar Energy, 46, 109 (1991). “A Methodology for the Synthesis of Houriy Weather Data,” Kondrayev, K. Y., Acrinometey (iranslated from Russian), NASA TT P.9712 (1965); also Radiation in the Armosphere, Academio Press, New York (1969). Xiucher, T. M.. Solar Energy. 23,111 (1979). “Evaluating Models to Predica Insolation on Tilted Surfaçes.” Lutimer, 3. R.. in Proc. First Canadian Solar Radiation Data Workshop (3, E. Hay and T. K Won, eós.) Ministry ur Supply and Services Canada, 81 (1980). "Canadian Procedures for Monitoring Solar Radiation.” Liu, B. Y. H. and R, C. Jordan, Solar Energy, 4(3), E (1960) Chagpeteristie Distribution of Direct, Diffuse and Total Solar Radiat - “The Interrelationship and 14 Available Solar Radiation Liu, B. Y. H. and R. €. Jordan, ASHRAE tournal, 3 (0), 53 (19625. “Daily Insolatioo on Surfaces Tilted Toward the Equator.* Liu, B. Y. H. and R. C. 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Franklin Institute, 230, 583 (1940), “Proposed Standard Solar Radiation Curves for Engineering Use" Commonwealth Scientific Flal-Plate Solar Absorbers: Morse, R. N. and 3. T. Cramexki, Repor F.E.6 of Engineering Secti and Industrial Research Organization, Melbourne, Australia (1958). The Effect on Incident Radiation of Inclinatior and Ortentation,” Norris, D, J.. Solar Energy, 12, 107 (1968). "Correlation of Solar Radiation with Clouds.” Noris, D. ).. Sofar Energy, 14, 99 (1973). “Calibration of Pyranometess.” Norris, D. 5. Sotar Energy, 16, 53 (1974). "Calibration of Pyranometers in Inclined amil Tivverted Positions, Olseth. 1. A. and A. Skartveit, Solar Energy, 339, 343 (1987). "A Probability Density Modei for Hourly Total and Beam Irradiance on Arbiltarily Oriented Planes.” Orgill, 1. F. and K. G. T. Hollanús, Sotar Energy, 19, 357 (1977). "Correlation Equation for Howrly Diffuse Radiation on a Horizontal Surface.” Page, 3. K., Proc. of the UN Conference on New Sources of Energy. 4, 378 (1964). "The Estimation of Monthly Mean Values of Daily Total Short-Wave Radiation of Vertical and Inelined Surfaces from Sunshine Records for Latitudes 400N-4008 " Page, .K.. Prediction of Solar Radiation on Inclined Surfaces, Reidel (for lhe Commission af the European Communities), Dordrecht, Holland (1986). Paltrídge, G. W, and D. Proctor, Solar Energy, IR, 235 (1976). "Monthiy Mean Solar Radíation Statistics for Australia.” Perez, R., R. Seais, P. Ineichen, R. Stewart, and D. Menicucci, Sofar Energy. 39, 221 (1987). "A New Simplificd Version of the Perez Diffuse Irradiance Model for Tiltcd Surfaces.” Perez, R., R. Stewart, R. Seals, and T. Guertin, Sandia National Laboratories Contractor Report SAND$8-7030 (Oer. 1988), "The Development and Verification of the Perez Diffuse Radiation Model.” References 145 Reindi, DT. MS. Thesis, Mech. Ê E R 1 M$. Thesis, Mechanical Engineering, U. of Wisconsin-Mad Eutimating Diffusc Radiation on Horizontal Surfaces and Total Radiation om Tilted Surfe, com Reindl, D. T., W. À. Beckman, and 4. A. Duffic, Solar Ene Correlation: Reindi. D.'T. W, A. Beckman, and 3. A, WA and 3. A, Duffie, Sotar 4 Hourly Tílted Surface Radiation Models. Pe Solto Energy, 45, 9 (19906). "Evaluation of vo 45, 1 (1990), * iffuse Fraction Robinson, N fed), Solar Radiation, Elsevier. Amsterdam (1966) Ruth, D, W. and R. E, Chen, Solar Energy. 18, 153 (1976). Radiation 10 Total Radiation in Cara The Relatos ot Diuse Suunier, G. Y., T. À. Reddy. and S. Kumar, Solar Eno |: . N 38. 169 (1987). Probabiity Distribution Function vf D ) TA Monthly aily Global lradiation Values : ama Tempra Loca 'y Global Irratiation Values Appropriate for Both Tropical Skarveit, A. and 1. A. Olseth, Solur Energy, 36, 333 (1986), Semvei Au "Modelling Sloped Irrudiance at Skartveit, A! and 3. A. Olseth, Solar En e , Sul erry, 38, Y "a Su AGA 1 A Ol 8) 38, 277 (1987). "A Model for the Diffuse Fraction SOLMET Manual, U.S, National Climatic Center, Ashvilo, NC, Vols, 1 and 2 (1978) Stanhill, G., Solar Energy, LO (23,69 (1966), "Diffuse Sky and Cloud Radiation in Israel” Stewart, R., D. W. Spencer, and R, Perez, 1 . . . rez, in Advances in Solar Energy, 2 Duífie, eds), ! (1985). “The Measurement of Solar Rudiation." CURA He Boerand A Temps, R. €. and K. 1. Coulson, Salar Energy, 19, 179 (1977 Radiation dr upon . son, Solar Energy, 19, 1 S ç : 8 €1977). “Solar Radiation Incident upor Thekackara, MP Supplement 1a the Proc. of 204 Anntal Mein, Environmental! Science, 21 (1974), “Data on Incident Solar Energy. ef he dna or Thekuekara, M. P., Solar Energy, 28, 309 (1976) “Solar Rad Treiutera Mm adiation Measurement: Techniques Tremunha, G. T., 4n introduction to Climate, 3rd ed., McGraw-Hill, New York (1954). Trem . . Tewantha, G. T.. The Earth's Problem Climatrs, University of Wisconsin Press, Madison (1961). Tuller, 8. E, Solar Energy, 18, 259 (1976). “The Relation: ip between Tn d . 8 (1976). fionship between Difíuse, Total am Whillior, A... Ph.D. Thesis, MIT. Cambridge, MA (1953), ia tu Solar Energy Collection and lts Witllior, A., Arch Met. Geoph, Biokt. Series B, 7, 197 (1956). are Dei Values of Total Radiation from Daily Summations,” Pe Dremintion of Nous Waúllier, A. Solar Energy, 9, 164 (1965). “Solar Radiation Graphs.” Wiebelt, 1. A. and 3, B. Henderson. Trans. ASME, J, Ileai Transfer, LOM, 101 (1979), Ordinátes fo Total Solar Radiation Property Evaluation ftom Spectral Data.” tested World Meteorolagical On i 'ganization (WMO), Guide to Meteorotogi Practices, 3rd ed, WMO and T.P.3, WMO, Geneva (1969 ), CR men na Ob 150 Selected Neat Transfer Toplos equating to zero, the wavelength corresponding to the maximum of the distribution can be derived. This leads to Wien's displacement law, which can be written as AqmasT = 2897.8 umk 43 Planck's law and Wien's displacement law are ilustrated in Figure 34.1, which shows spectral radiation distribution for blackbody radiation from sources at 6000, 1000, and 400 K. The shape of the distribution and the displacement of the wavelengih of maximum intensity is clearly showa. Nose that 6000 K represents an approximation of the surface temperature of the sun so the distribution shown for that temperature is an approximation of the distribution of solar radiation outside the carih's atmosphere. The other two temperatures are representative of those encountered in low- and high-temperature solar-heated surfaces. The same information shown in Figure 3.4.1 has been replotted on à normalized linear scale in Figure 3.4.2. The ordinate on this figure, which ranges from sero to one, is the ratio of the spectral emissive power to the maximum value at he same temperature. This clearly shoves the wavelength division between à 6000 K suurce and lower temperature sources at 1000 and 400 K. o! Wimê um Yefocus ot maximo Spectral emissive pomer, Exa. 5 Wovelength, À, um Figure 3.41. Spectral distribution of blackbody radiation. 36 Radiation Tables Is VT ART 1 : [= oeliia IS 3 % HI8 E «06|[ | É A oa 1 oz ol ) am ra RR RR RR RR Wavelength, am Figure 3.4.2 Normalized spectral distribution of blackbody radiation. 3.5 STEFAN-BOLTZMANN EQUATION Pane 5 law gives the spectral distribution of radiation from a blackbody, but i ngineering calculations the total energy is often of more interest. By integrar Planck's law over all wavelengths, the total energy es | fe s, tha otal energy emitted by a blackbody is fornd E =| En dA=or! [ à (3.5.1) mer Gis the Stefan-Boltzmann constant and is equal ta 5.6697 x 108 Wim2K4. ls constant appears in cssentially all radiation equations. TE 36 RADIATION TABLES Starti ! 5 ing with Planck's law (Equation 3.4.1) of the spectral distribution of blackbody radiation, Dunkle (1954) has mn, presented i i aenfadons Pancho Ro va Doom a method for simplifying blackbody Ep=— A A'Texplcyar)-1] Gsm Equation 3.6.1 can be intej i i . grated to give the radiation betw imii The total cmitted from zero to any wavelength À is given dy amy vaveng im A Ea dà [ (3.6.2) Eoab | o Selected Heat Transter Toplos 36 Radintion Tables a i Fº, the Substituting Equation 3.6.1 into 3.6.2 and noting that by dividing by O) Table 3.6.1b Fraction af Blackbudy Radiation Energy between Zero und | | integral can be made to be only a function of 27. AT for Even Fraclional Increments Ara ! | ar F 1 Tm 0-Ar 7, mk midpoint b ' fo. AT.umK — midpoint ' foro É = adam 0.63 — o focdr o Mtmk miópin art o otarflexplca anj-1l 0.05 1880 1660 oss 4410 “425 0J0 200 2050 0.60 4740 “sm 5 » am 2320 065 5130 4930 able 36.1a Fraction of Blackbody Radiont Energy betweca Lero a so 2560 am o ss and àT for Even Increments of AT 030 3120 ata ao Si50 5850 DO RMR fom AGHMK four . a 0.80 ea6o caso AT,umk fo.ar AT.pmk fo-ar ATumkK fo.ar ass 3350 3230 085 7850 310 070 [00 [0700 40 3580 3460 0.90 9380 as10 todo 0.0003 4000.5643 8000 — 0.856% | 5 a 3830 sro 095 12500 1 1100 0.0009 s600 0.5793 g100 O.8601 ? ano 3970 100 oo iesoo 1200 09021 4700 05937 amo 08639 1300 00043 4800 0.605 s300 08676 | 1400 0.0077 4900 0.6209 8400 0871 The value of this integral is the fraction of the blackbody energy between zero and 1500 00128 so 0.637 Bs gas AT. Sargent (1972) has caleulated values for convenient intervals and the results are 1600 0.0197 s100 6461 8600 — 0.8778 l given in able 3.6.1. (Note that when the upper limit of integration of Equation voo 005 5200 046579 so om 3.63 is infinity, dhe value of the integral is unity) 1508 qs21 Sm Ee eso 0887 36 For use in a computer, the following polynomial approximations to Equation 2000 0.0667 5500 04909 s000 08899 e inte been given by Pivovonsky and Nagel (1961). For y greater than or 2100 0.0830 5600 0010 9100 08927 equal to 2200 0.1009 sm 07107 9200 08954 2300 0.1200 sso0 0.720 9300 0.8980 fre Foaro dá em 2400 0.1402 syga 0.729] 9490 0.9005 Car as a La at [rm 74 3)m pr6]my+ 6) (3.6.4) 2500 01613 6000 07378 9500 09030 2600 0.1831 sumo 07461 9600 09054 For less 2700 0.2053 e200 07541 go 090% Yless than 2 2800 0.2279 6300 07618 900 09099 2900 0.2506 6ao 07692 9900 09120 5 5 sl f y 3000 om esmo Om 10000 09141 DS SOaO * SIETOO 13305 600) 3100 02958 00 0981 nojo 0938 sv 3200 03181 6700 07897 12000 09450, . 6. 3300 0.340 6800 0.796 13000 09550 here vo 3400 03617 eo 08022 14000 09628 where y=Cy/AT. 3500 0.3829 00 08080 15000 09689 3600 0.4036 7100 08137 16000 0.937 Example 3,6.1 a 8191 17000 09776 o . Í E gua 0 08244 iodo 09807 Assume that he sun is a blackbody at 8777 K. a What is the wavelengrh at which the 3900 0.4624 7400 08295 19000 0.9833 4 maximum monochromatic emissive power occurs? b What is the energy from this | 4900 0.4809 7500 0.8343 20000 09855 source that is in the visible part of the electromagnetic spectrum (0.38 to 0.78 um)? 4 É ; a100 0.4987 7600 083% 30000 08952 ! 2. Po 4200 0.5160 mo 0836 40000 09978 Solution | 4300 0.5327 7800 08479 500 09988 ! 4400 o.sus8 m00 08521 o a Thé value of AT at which thc maximum monochromatic emissive power occurs is 2897.8 um K, so the desired wavelength is 2897.8/5777, or 0.502 um. b From MTL ——"T 184 Selected Hent Transfer Topics Table 3.6.1 the fraction of encrgy between zero and AT = 0.78 x 5777 = 4506 pm is 56%, and the fraction of the energy between zero and AT = 0.38 x 5777 = 2195 pmk is 10%. The fraction of the energy in the visible is then 565% mínus 10%, or 46%. These numbers are close to the values obtained from the actual distribulion of energy from the sun as calculated in Example 1.3.1. n 3.7 RADIATION INTENSITY AND FLUX Thus far we have considered the radiation leaving a black surface in all directions; however, it is often necessary to describe the dircctional characteristics of a general radiation fieid in space. The radiation intensity is used for this purpose and is defined as the energy passing through an imaginary plane per unit area per unit time and per unit solid angle whose central direction is perpendicular to the imaginary plane. Thus, in Figure 3.7.1, if AE represents the energy per unit time passing through AA and remaining within Aq, then intensity is? I= lim E. (3.71) MA 50 AAAW Aw0 “The intensity 1 has both a magnitude and a direction and can be considered as a vector quantity. For a given imaginary plane in space, we can consider two intensity vectors that are in opposite directions. These two vectors are often distinguished by the symbo! ** and 1. The radiation flux is closely related to the intensity and is defined as the energy passing through an imaginary plane per unit area per unit time and in all directions on one side of the imaginary plane. Note that the difference between intensity and flux is that the differential area for intensity is perpendicular ta the direction of propagation, whereus the differential area for flux lies in a plane that forms the base of a hemisphere through which the radiation is passing. The intensity can be used to determine the flux through any plane. Consider an elemental area AA on an imaginary plane covered by a hemisphere of radius 7 as aa NORMAL TO PLANE | du IMAGINARY PLANE Figure 3.7.1 Schematic of radiation intensity, 3 The symbol 1 is used for intensity when presenting basic radiation hes€ transfer ideas and for solar tadiatios integrated over an hour period when presenting sotar radiation ídeas. The Iwo will seldom be used together. Plane. Second, the radiation Flux witl hay two possible directions of the normal to th emphasize which of the two the superscript + or — directions, 3.7 Radiation Intensity and Flux +72 Schematic of radiation Flux. shown in Figure 3,7.2, The energy per unit ti E t the surface of the hemisphere from th equal o UE Sm tea A4 on e arga AM is equal to AQ=IAA( cos dA” 4 (3.7.2) Where AAr? is the solid angle between AM am Perpendicujar to the intensity vector. f direction can then be defined as id AA! and SA cos Bis the arca The energy fiux per unit sotid angle in the 6, Sg= im Lico 6 MA r SAS0 AA (3.73) The radiation flux is then found b; , integrati isphe) ee a Y inlegrating over the hemisphere, The sphere n terms of the angles and óso that í nt2 g= Icos O si j [ cos 8 sin 6 dg dg (3.7.4) Kis convenient to define d=cos 8su that ar q= taud Í [ duda Q.7.5) Two important Points concerring the radiation flux j ddiation flux is, in general, a functi nO red Pi ion Of the orientation of the chosen imaginary e two valucs corresponding to cach of the e Imaginary plane. When it is necessary to té vo possible values of the radiation flux is being considere an be used along with a definition of the posiive and negarive Selected Hezt Transfer Topics 160 3.1t NATURAL CONVECTION BETWEEN FLAT PARALLEL PLATES The rate of heat transfer between twa plates inclined at some angle to he nero pa obvious importance in the performance of flat-plate coilectors. Free comes Ton het transfer data arc usually correlated in terms of two or three dimension! es pura e as the Nusselt number Ny; the Rayleigh number Ra, and the Pranda pum W dr Some authors correlate data in terms of the Grashof number, which is the ratio Rayleigh number to the Prandil number. ” Ea Nusselt, Rayleigh, and Prandel numbers are given byé 3.1.1) Nu AL t esfrart? 41.7) Re= a 311.3) Pr=g é where ha = heat transfer coefficient L late spacing ermal conductivity = E tant pe nano coefficiem of expansion (for an ideal gas, B' = 1/7) AT = temperature difference between plates v= kinematic viscosity mr = thermal diffusivity For parallel plates the Nusselt number is the ratio of a pure comtuction resistance to a convection resistance [ie., Nu = (L/k)/(1/h)] so that a Nussel it resents pure conduction. . do (1958) examinça the published results of à number of investi tioos má concluded that the most reliablc data for usc in solar colector calculations “ 35) were contained in Repon 32 published by the U.S. Home Finance is É : e In a more recent experimental study using air. Hollands et al. y pel e relationship between the Nusselt number and Rayleigh number tor tilt angtes froi to 75º as 13 á no Ra cos 8 7-8 ) fe = RR Eh | SO | Ga ' il It at the mean 5 Fluid properties in the convection relationships of this chapter should be evaluated at temperature. 3-H Natural Convectio belween Flaz Paralle! Plates 161 where the meaning of the + exponent is that only positive values of the terms in the square brackets are to be used (ie. use zero if the term is negative). For horizontal surfaces, the results presented by Tabor compare favorably with the correlation of Equation 3.11.4. For vertical surfaces the data from Tabor approximate the 75º tilt data of Hollands et a]. Actual callector performance will always ditter from analysis, but a consistent set of data is necessary to predict the trends to be expected from design changes. Since a common purpose of this type of data is to evaluate collector desiga changes, the correlation of Hollangs e; al is considered to be the most reliable. Equation 3.114 is plotted in Figure 3.11.1. Tn addition to the Nusselt number, there às a second scale on the ordinate giving the value of the hear transfer coefficient times the plate spacing for a mean temperature of 10 C, The scale of this ordinate is not dimensionless but is mmWim2C, For temperatures other than 10 €, a factor E, the ratio of the thermal) conductivity of air at 10 C to that at any other temperature, has been plotted as a function of temperature in Figure 3.112. Thus to find hi at any temperature other thar 10 €, it is only necessary to divide Eshl as read from the chart by E) at the appropriate temperature,6 The abscissa also has an extra scale, FATE, To find ATP at temperatures other than 10 C, it is only necessary 10 divide FLAT by Es, where Ey is the ratio of L/Tya at the desired temperature to 1/Tverat 10 C. The ratio Fi is also plotted in Figure 3.11.2. ESTE mm! C taie ontyj not 10 108 ETTA Trama Correlation nf Holiands et al. (19764 AAA no É Ê pr “| 4 a f o É í % ê o É é z ? T 5 = E | Ê concuetio td 30 1 ot not od 108 fayleigh number Figure 3.211 Nusselt Number às a fonction of Rayieigh number for free convection heat transfer between paralel flat plates at various slopes. É The towercase letter ! is used as a reminder that the units are millimeters instead of meters. E Selected Heat Transfer Topics 19 os og ED] 02] a 2 Zo a & os 04 L es | Ra n6 85 50 TEMPERATURE, C Figure 3.112 Air propesty corrctions F, and > for use with Figure 3 1L.L. From Tabor (1958). Example 3.11.1 Find the convection heat transfer coefficient between two parallel plates separated by 25 mm with a 45º till. The lower plate is at 70 € and the upper plate is at 50 €. Solution 029 WimkK, T=333K mis. (Property data At the mean air temperature of 60 C air properties are k= so |' = 1/333, v= 1.88 x 10-5 mêfs, and q = 2.69 x 107 are from Appendix E.) The Rayleigh number is 9.81x 20x 00 Ra=— = -= 1.82x 10º 333 x 1.88 x 105 x 2.69 x 105 From Equation 3.11.4 or Figure 3.11.1 he Nusselt number is 2.4. The heat transfer coefficient is found from 2.4 x 0.029 enuk-tIÊ US 22,78 Wim2kK h=Nut=""0025 As an alternative, the dimensional scales of Figure 3.11.! cam be used with the property corrections from Figure 3.11.2. At 60 C, F| = 6. 49 and Fa = 0.86. Therefore, F;ATÉ = 0.49 x 20x 25º = 1.53 x 10º mmiC. From the 45º curve in Figure 3.1.1, Fab! = 59. Finally, h = 5940.86 x 25) = 2.74 Wink. Even with the substantially reduced radiation heat transfer resulting from the low emittance in Example 3.10.1, the radiation heat transfer is about one-half of the convection heat transfer. 3.11 Natural Convection between Flat Parallel Plates 163 Fra TP, mar? E (air only) o 10% E TT pr TT qi TT” Reporr3z St —-— Raithby etal;A = 5, 60, « Randall tal, | 4 z af —..— De Grafi and Van Der Hetd 100 5 E Vertical A £ ês 41. LÊ So E Ê E é IE 2 3)» ã / o é . 4 .< A » PA, | Ji Í 1º ot 1 E Ravicigh number Lit Figure 3.113 Nusselt number as a fanction of Rayleigh number for free canvection heat transfer between vertical flat plates, Mis recommended that the 75º correlation of Figure 3.11.1 be used for vertical surfaces. The correlation given by Equation 3.11.1 does not cover the range from 75 to 90º, but comparisons with other correlations suggest that the 75º represents the vertical case adeguately. Raithby ex al. (1977) have examined vertical surface convection data from a wide range of experimental investigations. They propose a correlation that includes the influence of aspect ratio À, that is, the ratio of plate height to spacing. Their correlation is plotted in Figure 3.11.3 for aspect ratios of 5, 60, and infinity. For comparison, other correlations that do not show an aspect ratio effect are also plotted on this figure and correspond approximately to the Raithby et al. correlation with an aspect ratio of between 10 and 20. Most of the experiments utilize a guarded hat-plate technique that measures the heat transfer only at the center of the test region. Consequently the end cffects arc largely excluded. However, Randall ct al. (1977) uscd un interferometric technique that allowed determination of local heat transfer coefficients from which averages were determined; they could not find an aspect ratio effect although a range of aspect ratios from 9 to 36 was covered. The Raithby et al. (1977) correlation also includes an angular correction for angles from 70 to 110º which shows a slight increase in Nusselt number over this range of tilt angles consistent with the trends of Figure 3.11.1 [see Randall et al. 1977]. tis unusual to find a collcctor sloped at angles between 75 and 90º; if they are to be that stcep, they will probably be vertical. Windows and collector-storage walis are essentially always vertical. For vertical surfaces the four correlation shown in Selected Heat Transfer Topics jure 3,11,3 (with A & 15 for the Raithby et al. result) agree within approximately with the 75º correlation of Hollands et al. in Figure 3.11.1. Vertical solar lectors will have an aspect ratio on the order of 60, but at this aspect ratio the fithby et af. result fails well below other correlations. Conseguently, the 75º krelation of Figure 3,11.1 will give reasonable or conservative predictions for rtical surfaces. 12 CONVECTION SUPPRESSION me of the objectives in designing solar collectors is to reduce the heas loss through 1e covers. This has led to studies of convection suppression by Hollands (1965), idwards (1969), Buchberg et al. (1976), Amold ct as. (1977, 1978), Meyer ct al. 1978), and others. In these studies the space between two plates, with one plate ieated, is filed with a transparent or specularty reflecting honeycomb to suppress the »nset of fluid motion. Without fluid motion the heat transfer between the plates is by zonduction and radiation. Care must be exexcised since improper desig can lead to increased rather than decreased convection losses, as was first shown experimentally by Charters and Peterson (1972) and later verified by others. For slats, as shown in Figure 3.12.1, the results of Meyer et al. (1978) can be expressed as the maximum of two numbers as Nu= max[1.1Ci Co Ra$38, 3] (3.321) Figure 342.1 Siats for suppression of convection. From Meyer et aí. (1978) 312 Convection Suppression 165 where Cj and C, ure given in Figure 3 2.2 and the subscript £ indicutes that the a ure given in Pigure 3.12, à cul a Plaie spacing Z is the characteristic engih. Note that the coefficient € has a nhas maximum near 2n aspect ratio of 2. To assess the ma i e con ses e magnido SE he convection suppression with stars, tis possible compare E ton 5 “L vit the correlation of Randall ez a). (1977) obtained form dita to Same equipment. Although the Randall correfation use; 22-29 on the Raylcigh number, the correlation cam be slightly modified o have a e ave an exponent of 0,28. The ratio of the two correiations is then Sto o mexÊttcyCotapos 1] Mtmo stais max foi3rap “lcos(g- as)P 1] oug * Mean value T Rongs as 3a [ 2.3 Aspect rata, /f, E, Ga , E o 76 TILT ANGLE £, degrees ” (312.23 Figure 3,122 Coéfficiems € Bare 2.12.2 Coêfficiens C' and Cy for use in Equation 2.12.1. From Meyer et al, (1978). Selected Heat Transfer Topics vo Their data is well represented by an equation of the form alRe Pr Dill! (3.14,5) De = Mim ty (Re Pr Di iven in Fable 3.19.1. ale 1958) e copored by Rocnon and Chi (196). present ae Nussele numbers for the case of constant wall temperature. e a E nu numbers of 0.7, 5, and infinity arc shown in Figure 314.1, “ s ri nu also be represented by an equation of the form of Equation 3.14. dra b,m,n, and Nic, given in Table 3.14.2. Example 3.14.1 e! h he What is the heat transfer coefficient inside the tubes of a solar collector in unica e tubes are 10 mm in diameter and separated by a distance 100 mm. The cm E 15 m wide and 3 m long, and has total flow rate of water of 0,075 kg/s. The wat isat80C. ion 3.14.5 for Table 3.14.1 Constants Tor Equation 3. : Calcuiation of Local Nu for Circular Tubes with Constant Hest Rate. Prandd Number a bom 07 000398 ootId L66 112 10 0.00236 000857 166 113 o 000172 001 166 129 Nu = 44 TI 3 TT Timm 1 Laminar low Circular tubes Constant wall temperature + A "| Rm 1000 pi tos Lito , Re PrDhIL Figure 3,14.1 Average Nusselt numbers in short tubes for various Prandil numbers. 314 Heat Transfer Relations for Internal Flow 1m Table 3.242 Constants for Equation 3.145 for Culeulation of Average Nu for Circular Tubes with Constant Wall Temperuture. Prandil Number a dom A =— [——— 8? 0.079] 00331 Lis 0.82 s 00534 00335 115 082 o 0.0461 ema 115 08 Solution The coliector has 15 tubes so that the flow rate per tube is 0.005 kgfs. The Reynolds number is which indicates laminas Now, The Prandtl number is 2.2 so that RePrDIL=1800x22x001/3= 13 From Figure 3,14.1 the average Nusselt number is 4.6 s0 the average heat transfer coefficient is h=NukiD=46 x 0.600] =3 10 Wing ” In the study of solar air heaters and colieetor-storage walls it is necessary to know Lhe forced convection heat transfer coefficient berween two flat plates. For air the following correlation cam be derived from the datu of Kays and Crawford (1980) for fully developed turbulent flow syith One side heated and the other side insulated: Na =0.0158 Re9:8 (.14.6) where the characteristic Tength is the hydraulic diameter (twice the plate spacing). For flow situations in which LD is 10, Kays indicates that the average Nusselt number is approximately 16% higher than that given by Equation 3.14.6. At Z/D, equal to 30, Equation 3, 14.6 still underpredicis by 5%. At 1/D, equal to 100, the effect of be entrance region has largcly disappeared. Tan and Charters (1970) have experimentally studied flow of air between parallel plates with smalt aspect ratios for use in solar air heaters, Their results give higher heat Iransfer coefficients by about 10% than those given by Kays with an infinite aspect ratio. Selected Hest Transfer Fopies m Table 4.143 Constants for Equation 3.14.5 for Calculation of Local Nu for Infinite Flat Plates, One Side Insulated and Constant Heat Flux on Other Side, Prande Number a o " 07 000190 0.00563 17 117 1 0.00041 0.00156 212 1.59 o 0.00021 0.00060 224 1. Nu, 254 3 [TRT Tr[Trem LS Ri 20 Laminar fiow Parallel fiat plates. One wall constant temperature Other adiabatio. T Average Nusselt Number os TT 8 j pare As + a sli pts pila + 3 10 109 900 RePrDpfL Figure 3.142 Average Nusselt numbers in short ducis with onc side insulated and one side at constant wall temperature for various Prandtt numbers. The loca! Nusselt number for laminar flow between two flat Plates with one side insulated and the other subjected to a constant hcai flux has been obtained by e etal. (1964). The results have becn correlated in the form of Equation 3.14.5 witl the constants given in Table 3.14,3. . For the case of parallel plates with constant temperature on one side and insulated on the other side, Mercer et al. (1967) obtained the average Nus numbers shown in Figure 3.14.2. They also correlated these data into the form o! Equation 3,14.7 for (LI < Pr< 10; 49 0.0606(Re Pr DyWL)? (3.147) Mu=dd+, + 0.0909(Re Pr Dyt 7 Pr? The results of Sparrow (1955) indicate that for Re Pr Dy/L < 1000, the Pr = 10 Nusseft numbers are essentially the same as for the case when the hydrodynamic profile is fully developed. 218 Wind Convection Coefficients 13 Example 3.14,2 à Determine the convective heat transfer coefficient for air flow in a channel | m side by 2 m long. The channel thickness is 15 mm and the air flow rate is 0.03 kg/s. The average air temperature is 35 C. b If the channel thickness is halved, what is the heat transfer coefficient? € Jfthe Now rate is halveg, what is the heat transfer coefi ficient? Solution 2 Ata temperature of 35 C: the viscosity is 1.88 x 10-5 m2/s and the thermal conductivity is 0.0268 WimC. The hydruulio diameter D, is twice the plate spacing 4 and the Reynolds number can be expressed in terms of the flow tate per umit width mfW. The Reynolds number is then 2x 0,03 —ÍX00S 3200 1x 1.88 x 105 so that the flow is tubulent. From Equation 3.14.6 the Nusselt number is Nu = 0.0158(3200)08 = 10,1 and the heat transfer coefficient is h = Nu k/D, = 9 Win?C. Since DL is less than 100, 9 Wim2C is probably a few percent too low. b If the channel thickness is halved, the Reynolds number remains the same but the heat transfer coefficient will double 10 18 Win? € Ff the flow rate is halved, the Reynolds number will be 1600, indicating laminar ow. Equation 3.14.7 or Figure 13,14.2 should be used. The value of Re Pr DyfL is 1600 x 0.7 x 0.03/2 = 16,8 so the Nusselt number is 6.0 and the heat transfer coefficient is 6.2 Wim2C. “ 315 WIND CONVECTION COEFFICIENTS The heat loss fram flat plates exposed to outside winds is important in the study of solar collectors. Sparrow et al. (1979) did wind tunnel studies on rectangular plates “! various orieutations and found the following correlation over the Reynolds number range of 2x 101109 x 108: Nu = 0,86 RelV2ppi/3 (3.15,1) where the characterístic length is four times the Plate area divided by the plate Perimeter. For laminar flow (ie., Reynolds numbers less than 106, the critical Re for flow over a flat plate) aver a very wide flat plate at zero angie of attack, the 14 Selected Heat Transfer Topics analysis of Pohlhausen [see Kays and Crawford (1980)] yields a coefficient for Equation 3.15.1 of 0,94.º This agreement at low Reynolds numbers suggests that Equation 3.15. may be valid at Reynolds numbers up to 106 where direct experimental evidence is lacking. This extrapolation is necessary since u solar collector array 2 m by 5 m has a characteristic length of 2.9 m and Reynolds number of 9.4 x 105 in a 5 m/s wind. From Equation 3.15.1, the heat transfer coefficient under these conditions is approximately 7 WimiC. McAdams (1954) reports the data of Jurges for a 0,5 m? plate in which the convection coefficient is given by the dimensional equation h=5.7+3.8V (3.15.2) where V is the wind speed in m/s and À is in W/m2C, It is probable that the effects of free convection and radiation are included in this equation. For this reason Watmuff et al. (1977) report that this equation should be h=28+3.0V (3.15.3) For a 0.5 mê plate, Equation 3.15.1 yields a heat transfer coefficient of 16 Wim?C at a 5 m/s wind speed and a temperature of 25 C. Equation 3.15.3 yields a value of 18 Wim2C at these conditions. Thus there is agreement berwcen the two at a characteristic length of 0.5 m. It is not reasonable to assume that Equation 3.15,3 is valid at other plate lengths. The flow over a collector mounted on a house is not always well] represented by wind tunnel tests of isolated plates. The collectors will sometimes be exposcd directly to the wind and other times will be in thc wake region. The roof itself will certainly influence the flow pattems. Also, nearby trees and buildings will greatly effect local flow conditions. Mitchell (1976) investigated the heat transfer from various shapes (actually animal shapes) and showed that many shapes were well represented by a sphere when the equivalent sphere diameter is the cube root of the volume. The hcat transfer obtained in this manner is an average that includes stagnation regions and wake regions. A similar situation might be anticipated to occur in solar systems. Mitchell suggesis that lhe wind tunnel results of thesc animal tests should be increased by approximately 15% for outdoor conditions. Thus, assuming a house to be a sphere, the Nusseit number can be expressed as Nu =0.42 Ret é (3.15.4) where the characteristic length is the cube root of the house volume. 9 To be consistent with Equation 3.15.1 the characteristic length in the Poblhausen solution must be changed to twice the plate Jength. “This changes lhe familiar coefficient of 0.564 to 0.94 3.15 Wind Convection CoetTcients 175 When the wind speed is very low, free convecti ai i h 5 , ection conditions may dominate. Free convection data for hot inclined flat plates facing upward are not available However results are available for horizontal and vertical flat plates. For hoi orizuntal flat plates with aspect ratios uy : 4) gi p to 7:!, Lloyd and following eguations: ” Moran (1978) pve the Nie = 0.76 Ra for Ii Ra IO? (3.15.5) Nu =0.15Ra!8 for 10] Ra <3x 10º (3.15.6) where the characteristic Jength às four times the arca divi i her a ided by the perimeter. (The original reference used A/P.) For vertical plates McAdams gives Ni Ni = 0.59 Ra 8 for 103< Ra <t09 (3.157) Nu =0.13 Ra! for 109 < Re <JQ!2 (.15.8) where the characteristic length is the plate height. For large Rayleigh numbers, as would be found in most solar coliector systems, Equations 3.15.6 and 3.15,8 apply and the characteristic Tength drops out of the calculation of the heat transfer coefficient. The heat transfer coefficients from these two cquations are nearly the same since the coefficient on the Rayleigh numbers difter only slightly. This megas that horizontal and vertical collectors have a minimum heat transfer coefficient (i.c., under free convection conditions) of about S Wim2C for a 25 € temperature difference and a valuc of about 4 Wim2C ata temperature difference of 10C. From the preceding discussion it is apparent that thc calculation of wind- induced heat transfer coefficients is not well established. Until additional experimental evidence becomes available, the following guidelines arc recommended. When frec and forced convection occur simultancousiy, Me Adams recommends that bath values be calculated and the larger value used in calculations. Counseguenty, it appears that a minimum value of approximately 5 Wim? occurs in solar collectors under still air conditions. For forced conve ici ildii I ' ection conditions over buildings the 1: of Mitchel] can be expressed as ve results hu = 8.600.8 £94 (3.15.9) The feat transfer coefficient Cir Wim2C) f *C) for flush-mounted collectors Pi besta rs can then be - 600.6 ds max, á6408 (3.15.10)
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