Field Guide to Solar Optics Book 2020009994, 9781510636972, 9781510636989

Citation preview

Field Guide to

Solar Optics

Julius Yellowhair

SPIE Terms of Use: This SPIE eBook is DRM-free for your convenience. You may install this eBook on any device you own, but not post it publicly or transmit it to others. SPIE eBooks are for personal use only. For details, see the SPIE Terms of Use. To order a print version, visit SPIE.

Library of Congress Cataloging-in-Publication Data Names: Yellowhair, Julius E., author. Title: Field guide to solar optics / Julius Yellowhair. Description: Bellingham, Washington : SPIE Press, [2020] | Includes bibliographical references and index. Identifiers: LCCN 2020009994 | ISBN 9781510636972 (spiral bound) | ISBN 9781510636989 (pdf) Subjects: LCSH: Solar collectors–Optical properties. | Solar energy. Classification: LCC TJ812 .Y45 2020 | DDC 621.47/2–dc23 LC record available at https://lccn.loc.gov/2020009994

Published by SPIE P.O. Box 10 Bellingham, Washington 98227-0010 USA Phone: 360.676.3290 Fax: 360.647.1445 Email: [email protected] Web: www.spie.org Copyright © 2020 Society of Photo-Optical Instrumentation Engineers (SPIE) All rights reserved. No part of this publication may be reproduced or distributed in any form or by any means without written permission of the publisher. The content of this book reflects the thought of the author. Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon. Printed in the United States of America. First printing. For updates to this book, visit http://spie.org and type “FG47” in the search field.

Introduction to the Series In 2004, SPIE launched a new book series under the editorship of Prof. John Greivenkamp, the SPIE Field Guides, focused on SPIE’s core areas of Optics and Photonics. The idea of these Field Guides is to give concise presentations of the key subtopics of a subject area or discipline, typically covering each subtopic on a single page, using the figures, equations, and brief explanations that summarize the key concepts. The aim is to give readers a handy desk or portable reference that provides basic, essential information about principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. The series has grown to an extensive collection that covers a range of topics from broad fundamental ones to more specialized areas. Community response to the SPIE Field Guides has been exceptional. The concise and easy-to-use format has made these small-format, spiral-bound books essential references for students and researchers. Many readers tell us that they take their favorite Field Guide with them wherever they go. The popularity of the Field Guides led to the expansion of the series into areas of general physics in 2019, with the launch of the sister series of Field Guides to General Physics. The core series continues as the SPIE Field Guides to Optical Sciences and Technologies. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page and provides full coverage of that topic on that page. Highlights, insights, and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships, and alternative methods. While complete in their coverage, the concise presentation may be best as a supplement to traditional texts for those new to the field. Field Guide to Solar Optics

Introduction to the Series The SPIE Field Guides are intended to be living documents. The modular page-based presentation format allows them to be updated and expanded. In the future, we will look to expand the use of interactive electronic resources to supplement the printed material. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at [email protected]. J. Scott Tyo, Series Editor The University of New South Wales Canberra, Australia

Field Guide to Solar Optics

Related Titles from SPIE Press Keep information at your fingertips with these other SPIE Field Guides: Atmospheric Optics, Second Edition, Larry C. Andrews (Vol. FG41) Nonlinear Optics, Peter E. Powers (Vol. FG29) Optomechanical Design and Analysis, Katie Schwertz and Jim Burge (Vol. FG26) Physical Optics, Daniel G. Smith (Vol. FG17) Radiometry, Barbara G. Grant (Vol. FG23) Other related titles: Energy Harvesting for Low-Power Autonomous Devices and Systems, Jahangir Rastegar and Harbans S. Dhadwal (Vol. TT108) Polymer Photovoltaics: A Practical Approach, Frederik C. Krebs (Vol. PM175) Power Harvesting via Smart Materials, Ashok K. Batra and Almuatasim Alomari (Vol. PM277) Solar Energy Harvesting: How to Generate Thermal and Electric Power Simultaneously, Todd P. Otanicar and Drew DeJarnette (Vol. SL21) The Art of Radiometry, James M. Palmer and Barbara G. Grant (Vol. PM184)

Field Guide to Solar Optics

vii

Table of Contents Glossary of Symbols and Notation Background on Energy and Solar Technologies Energy Usage and Needs Energy Resources Solar Resource Concentrating Solar Technologies Photovoltaic Solar Technologies Other Solar Technologies

xiii 1 1 2 3 4 5 6

Solar Radiation Sun Properties Earth Orbit Earth Celestial Sphere Earth–Sun Angles Sun Angular Subtense Sun Position Sun Movement Solar Radiation Energy Blackbody Radiation Solar Spectral Irradiance Terrestrial Solar Spectrum Direct and Diffuse Radiation Solar Radiation Data Solar Radiation Metrology Radiometry Quantities Radiometry Basics Geometrical Considerations Energy Transfer Example Etendue Sources and Surfaces

8 8 9 10 11 12 13 14 15 16 17 19 20 21 22 23 24 26 27 28 29

Fundamentals of Solar Optics Principles of Reflection and Refraction Vector Reflection and Refraction Reflection Coefficients Transmission Coefficients Flat Mirrors Curved Mirrors

30 30 31 32 33 34 35

Field Guide to Solar Optics

viii

Table of Contents Other Curved Surfaces Aberrations in Mirrors Astigmatism Solar Collector Basics Solar Glass Reflective Coatings Concentration Ratios Concentration Limit

36 37 38 39 41 42 43 44

Collector Optics for Solar Technologies Flat Plate Collector Linear Collectors Parabolic Trough System Linear Fresnel Collector Parabolic Collectors Heliostat Collector Heliostat Field Aimpoint Dish Concentrator Sizing a Parabolic Trough Collector Field Sizing a Power Tower Collector Field Fresnel Lens Concentrator Solar Furnace Solar Simulator Another Concentrator Type: Cassegrain Solar Multiple

45 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Optical Characterization and Analysis Mirror Surface Slope Error Mirror Shape Error Mirror Specularity Mirror Facet Canting-Alignment Error Facet Canting Adjustment On-Axis Canting Strategy Off-Axis Canting Strategy Tracking and Pointing Errors Sunshape Gravity and Wind Impacts Combined Optical Errors

61 61 62 63 64 65 66 67 68 69 70 71

Field Guide to Solar Optics

ix

Table of Contents Shading and Blocking Cosine Losses Intercept Factor Mirror Soiling Atmospheric Attenuation Collector Optical Efficiency

72 73 74 75 76 77

System Modeling Approaches Cone Optics Hermite Polynomials Ray Tracing Systems Performance Modeling

78 78 79 80 81

Metrology Tools Deflectometry Method Deflectometry Surface Determination Laser Scanning System Target Imaging Metrology Beam Characterization System Radiometer and Flux Gauge Reflectometer Emissometers

82 82 84 85 86 87 88 89 90

Other Nonimaging Optics and Solar Collectors Secondary Concentrator Other Compound Parabolic Surfaces Waveguides Free-Form Surfaces Metasurfaces Spectral Splitting Optics

91 91 92 93 94 95 96

Special Topics Solar Glint and Glare Solar Technology Interference

97 97 98

Equation Summary

99

Cited References

108

Bibliography of Further Reading

109

Field Guide to Solar Optics

x

Table of Contents Online Resources

113

Index

114

Field Guide to Solar Optics

xi

Field Guide to Solar Optics The Field Guide to Solar Optics consolidates and summarizes optical topics in solar technologies and engineering that are dispersed throughout literature. It also attempts to clarify topics and terms that could be confusing or at times misused. As with any technology area, optics related to solar technologies can be a wide-ranging field. The topics selected for this field guide are those frequently encountered in solar engineering and research for energy harvesting, particularly for electricity generation. Therefore, the selected topics are slanted toward solar thermal power, or as it is commonly called, concentrating solar power. The first section provides background on energy needs and usage, and explains where solar technologies fit into the energy mix. Section 2 covers properties of the sun and presents our basic understanding of solar energy collection. The third section introduces optical properties, concepts, and basic components. In Section 4, the various optical systems used in solar engineering are described. Optical systems used for solar energy collection are commonly referred to as collectors (e.g., a collector field)—a term that is frequently used in this field guide. Another term commonly applied in solar collectors is nonimaging optics. The fifth section introduces concepts for characterizing optical components/systems and analysis approaches. Lastly, the measurement tools commonly used in solar engineering and research are described in Section 6. The fundamentals of each topic are covered. Providing methods or approaches to designs was not the goal of this field guide. However, the fundamental understanding that can be gained from the book can be extended and used for design of components and systems. Julius Yellowhair June 2020

Field Guide to Solar Optics

xiii

Glossary of Symbols and Notation a A AOD Ap AU B BCS c c1 c2 C C CCD C max CPC CRg CRo CSP CSR d D DNI E E DNI E sun E l;sun f F Fðx; yÞ F 1!2 G2 GHI GTI h h H H n ðxÞ HPE

aperture diameter area aerosol optical depth projected area astronomical unit back focal distance beam characterization system speed of light first radiation constant in Planck’s function second radiation constant in Planck’s function center of curvature concentration charge-coupled device maximum concentration ratio compound parabolic concentrator geometrical concentration ratio optical concentration ratio concentrating solar power circumsolar ratio earth-to-sun distance distance or diameter direct normal irradiance radiation (light) energy solar direct normal irradiance solar irradiance solar spectral irradiance focal length focal point irradiance profile using Hermite polynomials view factor Star class second brightest global horizontal irradiance global tilted irradiance hour Planck’s constant heliostat tracking error operator Hermite polynomials Hermite polynomial expansion

Field Guide to Solar Optics

xiv

Glossary of Symbols and Notation HTF ˆi I Iu k L; l L LCOE M Ml n ni nˆ N p P PV Qe r; R rsun rˆ R R RMS S Sˆ SCA SCM SM ster tˆ ts T TES TMY ul V a a

heat transfer fluid incident ray radiant intensity Lambertian surface intensity Boltzmann’s constant length radiance levelized cost of energy/electricity radiant exitance spectral radiant exitance day of the year (1 to 365) index of refraction of optical material surface normal vector north direction probability density function power photovoltaic radiant energy radius of curvature sun mean radius reflected ray slant range earth-to-sun distance root mean square optical ray path sun direction unit vector solar collector assembly solar collector module solar multiple steradian transmitted ray solar time temperature thermal energy storage typical meteorological year spectral energy density dwarf star designation absorptance sun altitude angle

Field Guide to Solar Optics

xv

Glossary of Symbols and Notation b g g d ε ~ ε εd εH h hE hatm hblock hcosine hfield hoptical hreflect hshade u uB uc ui ur us ut l lpeak n p rs;p r, rt rd s s s ts;p t, tt f f fr

reflected beam angle spread sun azimuth angle intercept factor declination angle bolt thread count heliostat tracking errors directional emittance hemispherical emittance efficiency insolation weighted efficiency atmospheric attenuation efficiency heliostat blocking efficiency cosine efficiency collector field efficiency optical efficiency reflectance (soiling) efficiency heliostat shading efficiency projection angle Brewster’s angle critical angle angle of incidence angle of reflection sun half angle angle of transmission radiation (light) wavelength peak wavelength (Wien’s displacement law) radiation (light) frequency pi reflectivity in s and p light polarizations total reflectance directional reflectance mirror slope error, specularity optical error Stefan–Boltzmann constant transmissivity in s and p light polarizations total transmittance camera pixel phase angle sun latitude angle concentrator rim angle Field Guide to Solar Optics

xvi

Glossary of Symbols and Notation fs fðuÞ wðrÞ F x c v vs

sun subtended angle sunshape local material phase radiant power or flux circumsolar ratio ground cover ratio sun hour angle solid angle

Field Guide to Solar Optics

Background on Energy and Solar Technologies

1

Energy Usage and Needs The world needs energy to function and move societies forward. Access to adequate and reliable energy resources is crucial for economic growth and for maintaining the quality of our lives. As our need for energy continues to grow, one important question to ask is whether current levels of energy consumption and production are sustainable. The United States (U.S.) Energy Information Administration (EIA) publishes energy data periodically. The graph shows the world energy consumption from different energy sources, starting from 1990 and projected to 2040. This includes all of the energy consumed (fuel for transportations, electricity, etc.).

Oil continues to be the largest resource for our energy needs. However, oil supplies are finite, and it is believed that its use contributes to climate change. Coal usage has seen an increase in recent years. Coal is also finite in supply and also could be contributing to climate change. Renewable energy sources have been growing rapidly in the last decade and, according to the figure above, will continue to grow in the future and outpace other energy sources. Developed countries, including the U.S., have plans to increase renewable sources into their energy portfolios to meet some of our growing demands for energy and to produce cleaner energy. Field Guide to Solar Optics

2

Background on Energy and Solar Technologies

Energy Resources Various energy resources are currently used to meet our energy needs, particularly for electricity. Some resources are petroleum based (i.e., fossil fuels), which are considered nonrenewable, meaning that the resource is a finite supply. Nuclear energy is usually grouped with nonrenewable sources. Renewable sources are energy sources that can be regenerated or replenished. The common nonrenewable and renewable energy sources are listed below. Nonrenewable: • Fossil fuels • Natural gas • Nuclear energy

Renewable: • Wind • Hydro • Geothermal • Biomass • Solar

In 2015, the world’s energy came from fossil fuels (83%), nuclear energy (5%), and renewables and biofuel (12%). Most of the energy generated is used for heating and as fuel for transportation. Only about 18% is in the form of electricity. The chart shows the energy sources for electricity generation (IEA 2016).

Hydro contributed the most out of all of the renewable sources. Contributions from other renewables (wind, biomass, solar) were low but showed rapid increases since 2000, and EIA projects that growth will continue for clean energy production. Field Guide to Solar Optics

Background on Energy and Solar Technologies

3

Solar Resource The sun provides radiant and thermal energy that can be harvested for solar heating, lighting, and electricity generation. There is enough solar energy incident on the earth’s surface to supply the world’s energy needs several times over. In 2000, the United Nations Development Programme, as part of the World Energy Assessment, estimated that the annual solar energy incident on the earth was up to about 49,800 EJ (exajoule, which is 1018 joules), while the total energy consumed worldwide in 2016 was about 560 EJ. However, only certain regions of the earth have consistent annual sunshine to make solar renewable technologies a viable and cost-effective option for large-scale deployment.

In the U.S., for example, the southwest region consistently receives a good amount of sunlight annually, as seen in the map. This is the region where most of the large U.S. solar plants are deployed.

Field Guide to Solar Optics

4

Background on Energy and Solar Technologies

Concentrating Solar Technologies Concentrating solar power (CSP), sometimes called solar thermal power, utilizes the thermal energy of the sun, which is converted into mechanical energy to produce electricity. Solar energy is collected and concentrated on receivers using large collectors (i.e., mirrors or other reflective surfaces). Receivers composed of metal tubes flow a heat transfer fluid (HTF) that is heated by the concentrated solar power. The hot fluid (water, oil, or molten salt) is transferred to the power block, where water is heated to generate steam, which then runs a turbine to generate electricity. Some of the hot fluid can be stored for later use or cloudy days. This is called thermal energy storage. There are three main types of CSP technologies: parabolic trough, power tower, and dishengine systems. A schematic of each type is shown below.

The Solar Energy Generating Systems (SEGS) power plants near Barstow, California utilize the parabolic trough technology and are the longest-running CSP plants in the world (commissioned in the mid-1980s). There are nine plants in total, generating 361 MW. Ivanpah and Crescent Dunes plants in Nevada are large power tower plants commissioned in 2014 and 2015. Ivanpah has three power tower units producing at a capacity of 392 MW. Crescent Dunes has one central tower producing at a capacity of 110 MW with 10 h of thermal energy storage. The last parabolic dish project in the U.S. was a 1.5 MW plant in Maricopa, Arizona in 2009.

Field Guide to Solar Optics

Background on Energy and Solar Technologies

5

Photovoltaic Solar Technologies Photovoltaic technologies (often called solar panels) utilize the spectral energy of the sun to convert absorbed photons into free electrons in a semiconductor material for generating electrical current. Several different semiconductors can be utilized that have an absorptive response within the solar spectrum, with silicon (Si) being the most commonly used semiconductor. Si can be mono- or polycrystalline, or it can be a film. To utilize the solar spectrum more efficiently, double- or triple-junction semiconductors are used. To reduce the active areas (i.e., to reduce material cost and/or improve efficiencies), concentrating photovoltaic (PV) technology is used, in which collecting optics concentrate the sunlight over the smaller semiconductor material. The plot shows the solar spectrum, along with several different types of semiconductor solar cells, with responses over visible and infrared (IR) wavelengths. c-Si is crystalline silicon. The cells labeled top, middle, and bottom are examples of cells that can form triple junctions—combined they cover the solar spectrum well.

Field Guide to Solar Optics

6

Background on Energy and Solar Technologies

Other Solar Technologies There are other solar technologies that are not discussed in detail here. However, the optical designs and analyses described in the following sections can be applied to these technologies. A few of these other common solar-related technologies are listed below and on the next page. Water desalination is the process of separating the clean water from brine in salt water. The oldest desalination technique is based on the water still. Water in a large pan is heated and evaporated onto the glass (or other transparent) cover above the pan, where clean water is collected and directed to a tank by gravity assistance. The solar collector can be based on the flat plate collector described on page 45. Newer desalination techniques may use concentrated solar, where concentrating collectors heat the water faster than flat plate collectors.

In residential water heating systems, a heat transfer fluid is heated using solar thermal energy. The heat from the fluid is then transferred to water in a heat exchanger system (e.g., water tank) to provide hot water to the residence. Usually installed on the roof, the solar collector (where the transfer fluid is heated) can be based on a flat plate collector. High-solar-absorbing materials can be used to better collect solar energy.

Field Guide to Solar Optics

Background on Energy and Solar Technologies

7

Other Solar Technologies (cont.) In solar lighting applications, radiant solar energy is collected outside the building, and the collected light is piped into the building to provide natural lighting inside the building. Solar light can be collected with parabolic concentrators (see page 49) or with secondary concentrators (see page 91). The collectors can then be coupled to waveguides (see page 93) to pipe the collected solar light into and throughout the building. Alternatively, direct sunlight can be transmitted through transparent windows (commonly known as skylights) located on the roof of the buildings. The windows can be durable plastic or glass. A solar oven can be heated with solar thermal energy to cook anything that can be cooked in a regular oven. A Fresnel lens (see page 56) can be used to concentrate sunlight on heating elements such as coils. Flat reflectors (see page 34) can also be used to direct and concentrate the sunlight onto the cooking surfaces. A glass cover plate that can reflect IR light well can serve to trap the heat inside the oven. Absorptive materials can also be added to better collect the solar energy.

Field Guide to Solar Optics

8

Solar Radiation

Sun Properties The sun is a star at the heart of our solar system. It is classified as a G2 V star, with G2 standing for the second hottest stars of the yellow G class and V representing a dwarf star. The surface temperature is about 5800 K. The mean diameter of the sun is 109 times that of the earth, or 1,392,539 km. The mean distance from the earth to the sun is 1 astronomical unit [AU], which is defined as 149,597,871 km, or about 108 times the diameter of the sun. Property

Value

Mean Surface Temperature

5800 K

Mean Diameter

1,392,539 km

Mean Density

1.41 g/cm3

Mass

1.989 3 1030 kg

Solar Constant

1,367 W/m2

Luminosity

3.85 3 1026 W

Mean Distance to Earth

149,597,871 km

Field Guide to Solar Optics

Solar Radiation

9

Earth Orbit The earth orbits the sun in an elliptical path at a speed of 108,000 km/h (67,108 mph) and with eccentricity of 0.0167. It completes one orbit every 365.256 days (1 sidereal year). The extra partial day accumulates to a full day every four years, creating what is known as a leap year. The planet’s distance from the sun varies as it orbits. At perihelion, around 3 January, the earth is closest to the sun when the distance is about 147,098,074 km. At aphelion, around 3 July, it is farthest from the sun when the distance is about 152,097,701 km. The average distance between the earth and the sun is about 149.6 million km, or one astronomical unit [AU]. The changing seasons are determined by the tilt of the earth’s rotation axis, not its distance from the sun. Because of the axial tilt of the earth, the highest solar energy is incident at 23.4 deg north of the equator at the summer solstice (Tropic of Cancer) and 23.4 deg south of the equator at the winter solstice (Tropic of Capricorn).

As seen from Earth, the planet’s orbital prograde motion (i.e., earth’s rotation direction) makes the sun appear to move with respect to other stars at a rate of about 1 deg (or a sun or moon diameter) every 12 h eastward per solar day based on the earth’s orbital speed. Field Guide to Solar Optics

10

Solar Radiation

Earth Celestial Sphere The celestial sphere is an abstract sphere that is used to define and visualize some of Earth’s parameters.

North celestial pole is the northern point in the sky, where all of the stars seem to rotate around the North Star, or pole star Polaris. Autumnal equinox occurs when the sun is at the point in the southern hemisphere where the celestial equator and ecliptic intersect. Ecliptic is the sun’s path on the celestial sphere. Celestial equator is a circle on the celestial sphere that lies in the same plane as the earth’s terrestrial equator and is tilted at 23.45 deg to the ecliptic. Declination is comparable to the geographical latitude of the earth; it is projected onto the celestial sphere and is usually measured in degrees. Right ascension is the celestial equivalent of the terrestrial longitude projected onto the celestial sphere. Vernal equinox occurs when the sun is at the point in the northern hemisphere where the celestial equator and ecliptic intersect. South celestial pole is the southern point in the sky about which stars rotate. Field Guide to Solar Optics

Solar Radiation

11

Earth–Sun Angles The top figure illustrates the declination angle, which is the angle between Earth’s equatorial plane and the line to the sun. At the time of year when the northern part of the earth’s rotation axis is tilted toward the sun at þ23.45 deg, the sun will be at its highest point at solar noon. This time of year is called summer solstice, which occurs around 21 June. Conversely, the time of year when the earth’s north rotation axis is tilted away from the sun at –23.45 deg is called winter solstice, which occurs around 21 December. On this day, the sun will be at its lowest point at solar noon. Midway between the two solstices are the vernal (spring) equinox around 20 March and autumnal (fall) equinox around 22 September. At the equinoxes, the declination angle is 0 deg. The lower figure illustrates the latitude angle f and the hour angle v for an observer on Earth at point P. These parameters are used to calculate the sun position relative to the observer on Earth.

Field Guide to Solar Optics

12

Solar Radiation

Sun Angular Subtense The sun is sufficiently far away from Earth that sometimes it is approximated as a point source. However, for imaging or sun projection in solar thermal technologies, the extent of the sun must be considered when calculating the concentrated solar energy distribution. The sun angular extent determines its image size on the target plane.

The full angular extent of the sun is defined as
r  2us 5 2 tan sun d where rsun is the mean radius of the sun, and d is the distance to the sun from the earth’s surface. The sun full subtended angle is typically listed as 32 arcmin, 0.5 deg, or 9.3 mrad. Because the earth orbits around the sun in an elliptical path, the distance between the earth and the sun varies throughout the year. The variation in distance changes the sun angular extent or angle subtense as shown below.

Field Guide to Solar Optics

Solar Radiation

13

Sun Position In modeling solar technology systems, the first step is to determine the sun position in the sky. The sun position can be calculated for a particular latitude f on Earth and time (day of year and local solar time ts). From this information, the sun direction from the observer is determined. The equations defined below assume a global coordinate system with the x-axis pointing west, the y-axis pointing towards zenith, and the z-axis pointing north. Note that these equations1 are based on solar time and come from the spherical geometric relationship of the earth and sun. The angle of the earth’s rotation associated with the clock is the hour angle, where ts is specified in local solar time.  þf for north of equator Latitude: 5 f for south of equator Hour angle: vs 5 15ðts  12Þ   284 þ n Declination: d 5 23.45 sin 360 365 where n is the day of the year (1 to 365). Solar altitude angle: sin a 5 cos f cos d cos vs þ sin f sin d     cos a sin f  sin d  Solar azimuth: g 5 signðvs Þcos1  sin a cos f Solar unit vector: Sˆ 5 ½ˆx; yˆ ; zˆ  where xˆ 5 sin g cos a yˆ 5 sin a zˆ 5 cos g cos a

Field Guide to Solar Optics

14

Solar Radiation

Sun Movement The sun position varies daily as well as seasonally. The figure illustrates the sun position every hour from 6 am to 6 pm in solar time for summer and winter solstices and spring equinox days in Albuquerque, New Mexico, where the latitude = 34.96 deg from the equatorial plane.

The sun position equations can be used to calculate the sun’s angular movement across the sky. Sun vectors can be calculated for varying sun positions during the course of a day to determine its angular movement. The change in angle can be determined from the dot product of the sun vectors for the different sun positions: Dv 5 cos1 ðSˆ1 ⋅ Sˆ2 Þ where Sˆ1 and Sˆ2 are solar position vectors from the observer on Earth at two different instances in time. The sun hour-angle equation (page 13) can also be used and shows that the sun moves 15 deg every hour. From this, we can determine that the sun moves 0.25 deg (4.36 mrad) every minute, or about one-half of the sun disk angular extent. Reflective solar collectors such as heliostats have two-axis angle movements to track the sun throughout the day. To keep the sun position from changing, relative to the heliostat, by more than 0.5 mrad, the heliostat tracking must update every 13.75 s or faster.

Field Guide to Solar Optics

Solar Radiation

15

Solar Radiation Energy Radiation is emitted at the speed of light but can be emitted at different energy levels. The photon energy Qe can be defined in terms of the radiation frequency n or wavelength l: c Qe 5 hn 5 h l where h is Planck’s constant, and c is the speed of light. To find the photon energy in electronvolts [eV], the following approximation can be used with the wavelength specified in microns (mm): Qe 5

1.2398 eV l ½mm

The different energy regimes are usually depicted on the electromagnetic (EM) spectrum, as shown below, where the frequency, and therefore the energy, increases to the left. The wavelength, however, reduces to the left since it is inversely proportional to the energy.

From the solar spectrum most of the sun’s energy on Earth is between 280 nm and 2.5 mm (ultraviolet to infrared), marked by the shaded region in the EM spectrum diagram above. Over the solar spectrum, the sun emits mostly in the visible region (400 to 700 nm).

Field Guide to Solar Optics

16

Solar Radiation

Blackbody Radiation Emission, or radiant exitance, from a blackbody source with temperature T [K] is described by Planck’s function, which has units of W/m2-mm: Planck’s constant: h 5 6:626 3 1034 J · s Speed of light: c 5 2:998 3 108 m=s Boltzmann’s constant: k 5 1:381 3 1023 J=K The equation is sometimes simplified by grouping the constant terms into radiation constants: 1st radiation constant: c1 5 2phc2 5 3.742 3 1016 W · m2 2nd radiation constant: c2 5 hc=k 5 1.439 3 102 m · K The wavelength at the peak of the blackbody distribution is determined by Wien’s displacement law: lpeak T 5 2898 mm · K

The sun has a surface temperature of about 5800 K, which gives a peak emission wavelength of about 0.5 mm. Field Guide to Solar Optics

Solar Radiation

17

Solar Spectral Irradiance The solar spectrum amplitude changes throughout the day and with location. Standard reference spectra are defined to allow the performance comparison of solar technologies from different manufacturers and laboratories. The ASTM G159 solar spectrum standard was replaced in 2005 with the current ASTM G-173-03 standard. Under the current standard, two components are defined for terrestrial use: (1) the Air Mass 1.5 (AM1.5) Global spectrum is designed for flat plate PV modules and has an integrated irradiance of 1000 W/m2, and (2) the AM1.5 Direct þ Circumsolar spectrum is defined for solar concentrator work, which includes the direct beam from the sun plus the circumsolar component in a disk 2.5 deg around the sun, and has an integrated irradiance of 900 W/m2.

The SMARTS (Simple Model of the Atmospheric Radiative Transfer of Sunshine) program, available from the National Renewable Energy Laboratory (NREL), is used to generate the standard spectra and can also be used to generate other spectra as required. The extraterrestrial solar spectrum is measured at the top of the atmosphere at the mean earth-to-sun distance. Field Guide to Solar Optics

18

Solar Radiation

Solar Spectral Irradiance (cont.) Since the spectrum is measured outside the atmosphere, absorption from atmospheric constituents are not observed in the spectrum. Terrestrial solar spectra are reduced in amplitude from the extraterrestrial solar spectrum because the extraterrestrial spectrum propagates through the earth’s atmosphere where various constituents in the atmosphere absorb and/or scatter the light. Direct normal irradiance (DNI) measures only the direct solar radiation (with a 0.5 deg divergent cone). The sensor used to measure DNI points directly at the sun, tracks the sun, and reports the measurement in units of W/m2. Clouds, haze, and increased atmospheric constituents scatter the sunlight and reduce the DNI. Circumsolar spectral irradiance is typically measured within a ±2.5 deg field of view centered on the 0.5 deg diameter solar disk but excluding the radiation from the disk. The direct normal and circumsolar spectral irradiances combined gives the direct þ circumsolar irradiance. Global tilt measures spectral radiation from the solar disk plus diffuse light, with the sensor facing south and tilted 37 deg from horizontal. Air mass (AM) is the ratio of the path length through the atmosphere toward the sun position to the zenith path length.2 The ratio typically used in solar work is 1.5 (AM1.5), which is when the sun’s position is at an angle of about 48 deg relative to the normal to the earth’s surface or zenith direction.

Field Guide to Solar Optics

Solar Radiation

19

Terrestrial Solar Spectrum The terrestrial solar spectra are reduced in amplitude from the extraterrestrial spectrum because the extraterrestrial spectrum propagates through the earth’s atmosphere where various constituents in the atmosphere absorb and/or scatter the light. Ozone (O3), water (H2O), and carbon dioxide (CO2) in the atmosphere are the largest absorbers of sunlight. Absorption dips can be seen in the terrestrial solar spectrum.

When the solar spectral irradiance is integrated over the entire wavelength range, the irradiance from the sun on the earth’s surface can be determined by Z

þ∞

E sun 5 0

 E l;sun dl 5

1000 W=m2 ! global tilt 900 W=m2 ! direct þ circumsolar

The results of the integration are for the ASTM G-173-03 standard spectrum. Measurement over the solar disk is taken only as the direct normal irradiance (DNI). The solar irradiance on the earth’s surface of 1000 W/m2 is typically taken as ‘1 sun.’ Other units are sometimes used to express this quantity: 1 sun ¼ 1000 W=m2 5 1 kW=m2 5 0:1 W=cm2 Field Guide to Solar Optics

20

Solar Radiation

Direct and Diffuse Radiation Some of the solar radiation entering the earth’s atmosphere is absorbed and/or scattered. Direct beam radiation comes in a direct line from the sun. Diffuse radiation is scattered out of the direct beam by molecules, aerosols, and clouds. The sum of the direct beam, diffuse, and ground-reflected radiation arriving at the collector surface is called total or global solar radiation.

The total or global solar radiation striking a solar collector has two components: direct beam radiation and diffuse radiation. Additionally, radiation reflected by the surface in front of a collector (e.g., ground or objects) contributes to the solar radiation received. But, unless the collector is tilted at a steep angle from horizontal and the ground is highly reflective (e.g., snow), this contribution is usually small. On days with clear skies, most of the solar radiation is direct beam radiation. On overcast days, the sun is obscured by clouds, so the direct beam radiation goes to zero and the diffuse radiation is increased. Because diffuse radiation comes from all regions of the sky, it is also referred to as sky radiation. The portion of total solar radiation that is diffuse is about 10% to 20% for clear skies and up to 100% for cloudy skies. Dust and haze (e.g., from smoke and pollution) impacts are measured by the aerosol optical depth (AOD). An AOD of 0.01 corresponds to a clean atmosphere, and an AOD of 0.4 corresponds to hazy conditions. Field Guide to Solar Optics

Solar Radiation

21

Solar Radiation Data The viability of any solar technology installation depends largely on the solar radiation availability at the installation site. The solar resource data provides critical information for siting and planning. There are several resources available that can provide solar radiation data. The largest database is the National Solar Radiation Database (NSRDB), developed and maintained by the National Renewable Energy Laboratory (NREL) in collaboration with other government agencies, universities, and companies. In all, the NSRDB has more than 50 years of solar radiation and supplementary meteorological data from various sites, primarily in North and South Americas and United States territories. The latest update to the database covers 1998 to 2014 solar and meteorological data. Radiation data is also collected and made available by the U.S. National Oceanic and Atmospheric Administration (NOAA). International groups also provide weather data for worldwide locations. Some international groups include: • Commission of the European Communities (CEC) European Solar Radiation Atlas • World Radiation Data Center • Solar Radiation and Radiation Balance Data (The World Network) Typical meteorological year (TMY) data is frequently used for solar technology systems simulations in order to assess or predict the performance of the power plant. TMY is a collation of selected weather data that has been collected for a specific location during a time period much longer than a year. This data is selected so that it presents the range of weather phenomena for the location in question, while still giving annual averages that are consistent with the long-term averages for the location of interest.

Field Guide to Solar Optics

22

Solar Radiation

Solar Radiation Metrology Two instruments are primarily used to measure solar radiation for direct normal irradiance (DNI), global horizontal irradiance (GHI), and global tilted irradiance (GTI). A pyrheliometer is an instrument for measuring direct beam solar irradiance. Sunlight enters the instrument through a window and is directed onto a thermopile or thermocouple, which converts heat to an electrical signal that can be recorded. The signal voltage is converted via a formula to measure irradiance [W/m2]. A pyrheliometer is used with a solar tracking system to keep the instrument aimed at the sun.

Similarly, a pyranometer uses thermopiles or thermocouples to measure solar irradiance [W/m2] on a planar surface over a hemisphere. Direct sunlight and diffuse sunlight enter the glass hemispheric dome and arrive at an array of temperature sensors. The temperature is then converted to an electrical signal, and the signal is calibrated to provide an irradiance measurement. When the pyranometer is placed on a level surface, it measures GHI. When placed on a tilted surface, it measures GTI.

Field Guide to Solar Optics

Solar Radiation

23

Radiometric Quantities

Symbol

Radiometric Quantity

SI Units

Qe

Radiant energy

joule [J]

F, P

Radiant power or flux

watt [W 5 J/s]

I

Radiant intensity

W/ster

M

Radiant exitance

W/m2

E

Irradiance

W/m2

L

Radiance

W/m2-ster

In solar research and engineering, flux is a versatile term. It is sometimes used to refer to actual radiant flux (power), irradiance, or radiance. In this field guide, the term flux is reserved for radiant power with units in watts. Another term used in the solar industry is flux density, which is an irradiance quantity with units of W/m2. The number of “suns” is sometimes used to quantify solar concentration, where “one sun” refers to the solar irradiance on the earth’s surface, which is typically taken as 1,000 W/m2; therefore, “suns” is an irradiance quantity with units of W/m2. In this field guide, SI (the International System) units are used. For example, irradiance quantities are specified with units of W/m2. However, some solar industry professionals find it easier to read W/cm2 as the amount of solar concentration on a receiver: 1 W=m2 5 104 W=cm2

Field Guide to Solar Optics

24

Solar Radiation

Radiometry Basics Radiant exitance M [W/m2] is the radiation emitted by a source of area A, whereas irradiance E [W/m2] is radiation incident on a target surface of area A.

Irradiance E can be written as E5

dF dA

Radiant exitance is defined in the same way. The difference is in the direction of the radiation. The radiant flux or power F [W] is calculated as the integration of irradiance (or radiant exitance) over the collection area: Z F 5 E dA Radiation (emitted or incident) can be bounded by a surface of area A. Radiation can also be defined over a solid angle v. A solid angle is defined as dA 5 sin udu df r2 Z Z sin udu df v5

dv 5

f

u

The projected solid angle V is the projection of the solid angle v onto the observer plane and is defined as dV 5 cos u dv Field Guide to Solar Optics

Solar Radiation

25

Radiometry Basics (cont.) Radiant intensity I [W/ster] is the flux contained within a solid angle, where the solid angle is defined as the ratio of the differential spherical surface area to the square of the sphere radius. I5

dF dv

where dv 5

dA r2

Radiance L [W/m2/ster] is the flux per unit source projected area per unit solid angle. L5

L5

d2 F dA dV

d2 F dðA cos uÞ dv

The inverse square law relates to the reduction of irradiance [W/m2] from a point source over distance d. The cosine factor accounts for the receiving target surface normal direction. E5

I cos u d2

Field Guide to Solar Optics

26

Solar Radiation

Geometrical Considerations The projected area Ap is the intercepted area of a collecting surface and is reduced from the actual area of the collecting surface. Ap 5 A cosðuÞ where A is the area of the collecting surface, and u is the angle between the surface normal vector of the collecting surface and the direction of the incident radiation. The flux incident on the collecting surface is F 5 E · Ap 5 E · A cosðuÞ An example of this is fixed-tilt photovoltaic modules mounted on residential roofs. The collected flux is reduced by cos u; cos u will vary due to the sun’s position changing throughout the day. View factor relates to the fraction of radiation leaving one surface and intercepted by another surface.

1 F 1!2 5 A1

Z

Z A1

cos u1 cos u2 dA1 dA2 pd2 A2

The equation is similar for radiation transfer in reverse, which results in A1 F 1!2 5 A2 F 2!1 . Field Guide to Solar Optics

Solar Radiation

27

Energy Transfer Example In any optical system, radiative energy is transferred from one surface to another. An example of radiation transfer is provided below.

Radiation transfer between any two surfaces can be described by dF 5

L dA1 cos u1 dA2 cos u2 d2

where L is the radiance [W/m2-sr], and d is the distance between the surfaces. The radiance can be from either surface. The above equation is the flux received by one surface, with the other surface emitting. If the emitting surface is reflecting incident radiation, then the reflected radiation is determined by the surface reflectance r. Noting that the radiance is defined as L5

d2 F dA cos udv

and the solid angle is defined as dv 5

dA d2

the flux incident on a surface can be rewritten as dF 5 LdA cos udv

Field Guide to Solar Optics

28

Solar Radiation

Etendue Etendue is a term that is sometimes used to describe the performance of optical systems, including nonimaging optical systems. However, the term is not always well understood. Etendue is a fundamental optical property. Etendue enforces the conservation of energy as light propagates through the optical system, and as such, it is sometimes called the throughput of the optical system. Etendue is also sometimes called the geometrical extent or the AV product. The etendue, or throughput, is invariant within an optical system; therefore, the AV product is constant.

Assuming no transmission losses from element to element in an optical system, the AV relation holds throughout the optical system: A1 V 1 5 A0 V 0 5 A2 V 2 where A1 could be a source area, A2 could be the collecting surface area, and A0 is the optical element aperture area. The V terms are the projected solid angles. Using the radiation transfer example between two surfaces, the differential flux incident on one surface from the radiation emitted by the other surface is dF 5 LdA cos u dv where L is the radiance. Substituting the projected solid angle, the expression can be rewritten as dF 5 LdAdV where dAdV is the differential throughput, or the etendue.

Field Guide to Solar Optics

Solar Radiation

29

Sources and Surfaces An isotropic source is an ideal point source that emits radiation uniformly in all directions. The intensity is independent of direction. The sun is sometimes considered a point source since it appears small from Earth and emits light uniformly in all directions. Stars are also used as point sources. For a Lambertian source, the radiance is uniform in all directions, but the intensity falls off as a cosine with the viewing angle: I u 5 I 0 cos u: A surface with texture will reflect incident light diffusely, which in some cases can be approximated as Lambertian. When light strikes a Lambertian surface independent of the incidence angle ui, the surface reflects light as a Lambertian source, as illustrated below. In solar technologies, some surfaces, particularly surfaces with texture, are approximated as Lambertian surfaces.

Most surfaces are neither purely diffuse nor purely specular. These surfaces are semi-diffuse or semi-specular.

Field Guide to Solar Optics

30

Fundamentals of Solar Optics

Principles of Reflection and Refraction Light incident at the interface of two different media, such as air and glass, follows the laws of reflection and refraction. Law of reflection: ui 5 ur Law of refraction or Snell’s law: n1 sinðui Þ 5 n2 sinðut Þ where n1 and n2 are the refractive indices of the two media. Multiple reflections and refractions occur in second surface mirror reflectors that are typically used in solar thermal technologies to collect sunlight. Referring to the diagram, 1. The light ray is incident on the top glass surface. Part of the light is reflected, and the other part is transmitted. 2. The transmitted light reflects off of the bottom surface, which has a reflective coating; thus, it gives a strong reflection with little to no transmission. 3. The reflected light is incident at the glass/air interface from the bottom. Part of the light is reflected back into the glass, and the other part is transmitted through. At each interface, the light follows the law of reflection, or Snell’s law. There is a condition that occurs when light is deflected 90 deg to the optical axis (i.e., ut 5 90 deg), and the incidence angle reaches the critical angle uc, usually from the denser media side, e.g., the glass/air interface:
 n ui 5 uc 5 sin1 1 n2 Total internal reflection occurs when ui . uc. Field Guide to Solar Optics

Fundamentals of Solar Optics

31

Vector Reflection and Refraction Reflected and refracted rays can also be represented as vectors, having magnitude and direction. For reflection, three normalized vectors can be defined: • î 5 incident ray vector • rˆ 5 reflected ray vector • nˆ 5 surface normal vector at the ray incidence location The law of reflection states that the angle of incidence equals the angle of reflection, or ˆı · nˆ 5 rˆ · nˆ and all three vectors must lie in the same plane; i.e., ðˆı 3 rˆ Þ · nˆ 5 0 Then for any of these two given vectors, the third vector can be uniquely determined by the two above equations. For example, if î and rˆ are given, then nˆ is a linear combination of the first two vectors ðaˆı þ bˆrÞ due to coplanarity. The coefficients are normalized, resulting in the formula for the surface normal vector: ˆı þ rˆ nˆ 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 2ˆı · rˆ If the incident ray î and surface normal vector nˆ are known, then the reflected ray can be determined by ˆ nˆ rˆ 5 ˆı þ 2ðˆı · nÞ ˆ Similarly, the refracted ray t must be coplanar with the ˆ This can be incident ray î and surface normal vector n. written as tˆ 5 aˆı þ bnˆ The coefficients are normalized. Observing the ray path in refraction, the refracted ray vector can be calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 

2  n n1 2 ˆt 5 1 ˆı  nˆ ˆı · nˆ þ ðˆı · nÞ ˆ þ 1 n2 n2 where n1 and n2 are the refractive indices of the two media at the interface (e.g., air/glass). Field Guide to Solar Optics

32

Fundamentals of Solar Optics

Reflection Coefficients Fresnel equations specify the amplitude coefficients for reflection and transmission at the interface of two media. The reflection coefficients, referred to as reflectivity, are defined for the two orthogonal light polarizations (s and p) as rs 5

n1 cosðu1 Þ  n2 cosðu2 Þ n cosðu1 Þ  n1 cosðu2 Þ ;r 5 2 n1 cosðu1 Þ þ n2 cosðu2 Þ p n2 cosðu1 Þ þ n1 cosðu2 Þ

where u1 and u2 are the angles of incidence and transmission, respectively. The reflectance is the square of the reflectivity: r2s and r2p . The total reflectance is r2s þ r2p rt 5 2

For p polarized light, there is a particular angle of incidence for which light will not be reflected. This angle of incidence is called Brewster’s angle and is calculated by
 n2 1 uB 5 tan n1 At normal incidence, u1 5 u2 5 0, the reflectivities for both polarizations are equal. Standard glass has a refractive index n2 of about 1.5. The reflectance from a glass surface in air (n1 5 1) at normal incidence is
2 0.5 rt 5 5 0.04 5 4% 2.5

Field Guide to Solar Optics

Fundamentals of Solar Optics

33

Transmission Coefficients Transmission coefficients, or transmissivity, also exist and are defined as ts 5

2n1 cosðu1 Þ 2n1 cosðu1 Þ ;t 5 n1 cosðu1 Þ þ n2 cosðu2 Þ p n2 cosðu1 Þ þ n1 cosðu2 Þ

where u1 and u2 are the angles of incidence and transmission, respectively. The transmittance is the square of scaled by an additional factor: h i h the itransmissivity n2 cosðu2 Þ n1 cosðu1 Þ

jt2s j; and

n2 cosðu2 Þ n1 cosðu1 Þ

jt2p j. The total transmittance is 
  n cosðu2 Þ t2s þ t2p tt 5 2 2 n1 cosðu1 Þ

At Brewster’s angle, the p polarized light is completely transmitted. For the air/glass interface, Brewster’s angle is
 1 1.5 5 56.3 deg uB 5 tan 1 Some materials may also absorb light. When light is incident on the material, conservation of energy accounts for all of the light, or t þ r þ a 5 1 integrated over all wavelengths and all directions. Some materials are opaque and do not transmit the incident light at certain wavelengths. In these cases, the material absorptance aðlÞ can be estimated from the material reflectance: tðlÞ ! 0; aðlÞ 5 1  rðlÞ

Field Guide to Solar Optics

34

Fundamentals of Solar Optics

Flat Mirrors An ideal flat mirror merely changes the direction of the incident rays without modifying the ray bundle profile. A bundle of parallel rays will reflect as a bundle of parallel rays; i.e., a single flat mirror cannot concentrate or focus parallel light rays. A point source reflected by a flat mirror will appear to emanate from behind the mirror (right diagram).

A flat mirror is in the limit of a spherical or parabolic mirror when the radius of curvature becomes infinite. The focal length also becomes infinite. In solar thermal systems, large mirrors (several meters on a side) are used to collect and concentrate the sunlight onto a receiver or target surface. The surface quality of these mirrors is low, so they are referred to as nonimaging optics. It is difficult to manufacture large monolithic concentrators. Therefore, smaller flat mirrors are sometimes used to construct concentrators with large apertures in a piece-wise fashion. In this case, the series of flat mirrors on a spherical or parabolic profile can be made to have focusing abilities. The reflections from the flat mirrors overlap at the focal point or the target location. In the left diagram, the mirrors follow a parabolic profile. The right diagram illustrates a Fresnel arrangement, where the mirrors may be mounted on a flat frame structure. Surface Normal Vectors

Field Guide to Solar Optics

Fundamentals of Solar Optics

35

Curved Mirrors Spherical and parabolic surfaces are the two main types of curved surfaces used for solar collection. They can concentrate or focus an incident bundle of parallel rays. The diagram illustrates the difference between spherical and parabolic surfaces. Both surfaces have the same center of curvature C, location, radius of curvature r, and focal length f, measured from the vertex to the focal point F. The focal length is one-half the distance from the vertex to the center of curvature C, or one-half of the radius of curvature. Note that the spherical surface is indistinguishable from the parabolic surface near their vertices. Incident rays that are parallel to the optical axis will get focused at the focal point F.

The ratio of the focal length to the mirror clear aperture width D is referred to as the f-number, or f/#. A parabolic collector with 0.25 f-number will have a rim angle (discussed on page 46) of 90 deg, i.e., D 5 4f. A unique feature of spherical and parabolic surfaces is that if a ray emanates from the center of curvature C toward the surface, the ray will reflect back on itself to C. This feature, along with the vertex, can be used to boresight the collectors or align mirror facets. Field Guide to Solar Optics

36

Fundamentals of Solar Optics

Other Curved Surfaces Other curved surfaces are aspherical and include hyperbolic and elliptical surfaces. Both of these surfaces can focus light under different incident light conditions. A hyperbolic mirror surface can focus an incoming converging beam to the focal point. If the incident beam is directed at the external focal point, the reflected beam will focus at its internal focal point F. An elliptical mirror surface has two focal points. Light emanating from one point will focus at the other. These types of reflectors are typically used in solar simulator systems (discussed on page 58), where the lamp filament is lined up with the internal reflector focal point F1, and the test articles are placed at the external focal point F2. Previous examples of curved surfaces are concave surface shapes, where the surface curves toward the side of incident light. In convex surfaces, the surface curves away from the side of incident light. The illustration shows a parabolic convex surface having a virtual focal point. Reflected parallel light appears to emanate from the virtual focal point.

Field Guide to Solar Optics

Fundamentals of Solar Optics

37

Aberrations in Mirrors Optical aberrations degrade the image quality in optical systems or reduce solar concentrations in solar collectors. Some examples of typical aberrations are spherical, astigmatism, coma, distortion, and chromatic. In solar thermal systems, spherical aberrations and astigmatism are the most prominent. Spherical aberrations occur when radial zones across the aperture focus to different points along the optical axis, creating a caustic near the focal point and a beam spot that is larger than the ideal, nonaberrated spot. A perfect parabolic surface will focus incident parallel light to a point without spherical aberrations. A spherical surface with radial zones focusing at different points on the optical axis will leave a caustic at the focal point, which is referred to as a spherical aberration. Best focus is usually at the minimum spot size or at the circle of least confusion. Coma is an off-axis aberration that occurs when the light rays are incident at an angle and different radial zones across the optical aperture focus at different points radially on the image plane. Distortion occurs when the optical system mapping of the object points on the image plane is not ideal. For example, in the presence of distortion, a rectilinear grid in the object plane will be mapped to a different pattern in the image plane. The two main types of distortion are barrel and pincushion. Chromatic aberrations are present in refractive optical systems, e.g., glass lenses, where the different wavelengths of the source refract at slightly different angles due to the refractive index dependence on wavelength n(l). Field Guide to Solar Optics

38

Fundamentals of Solar Optics

Astigmatism There are two forms of astigmatism in optical systems. The first type occurs when the source is off-axis. The two orthogonal directions over the optical aperture focus the incident light beam at different points along the reflection direction, even when the mirror is rotationally symmetric. The two orthogonal directions are defined as the tangential and sagittal planes. The circle of least confusion and minimum spot size are between the sagittal and tangential focal points. The second type of astigmatism occurs when the orthogonal axes (e.g., horizontal and vertical axes) on the reflecting surface focus on-axis parallel rays at different points along the optical axis. In other words, the mirror orthogonal axes have different focal lengths, making the mirror non-rotationally symmetric. The difference in focal lengths can be due to manufacturing errors that cause slight shape differences in the two axes, or the shape design can be intentional for use with offaxis sources. Just like mirror surface slope error, these aberrations will contribute to beam spreading on the receiver or target surface, reducing the peak irradiance and increasing light spillage.

Field Guide to Solar Optics

Fundamentals of Solar Optics

39

Solar Collector Basics In solar thermal systems, solar collectors (i.e., mirrors) are the fundamental components needed to collect and concentrate solar radiation onto a receiver or heatabsorbing element(s). A typical solar collector mirror facet has the so-called sandwich construction. This refers to the way the mirror facets are constructed. Two glass sheets “sandwich” the reflective film. One glass sheet serves as the substrate, and the other serves as the front cover. The glass sheets are on the order of 1–5 mm thick. The reflective film is typically silver because silver reflects well over the solar spectrum (discussed on page 41). The mirror, which is on the order of 1–3 m2 in size, can nominally be flat or curved, depending on the solar technology application. The glass assembly is bonded to a support structure that can be a metal framework, stamped sheet metal, honeycomb panels, or another stiff structure. Mounting points are then bonded to the structure and serve as the interface points to the large collector frame. Three mounting points are ideal if two-axis mirror facet angle adjustments are needed. Additional mounting points can induce stress on the mirror facet. During facet angle adjustments, or canting, one mounting point serves as the pivot point, while the other mounting points are adjusted with actuators to tilt the mirror into alignment. A rigid collector frame, such as a heliostat frame, can hold an array of the mirror facets. The mirror array is arranged such that it will concentrate the sunlight onto a target or receiver. Field Guide to Solar Optics

40

Fundamentals of Solar Optics

Solar Collector Basics (cont.) Due to the sandwich construction, these mirrors are considered second surface reflectors, where the incident light is transmitted through the top glass and reflected off of the silver coating to transmit through the glass again. Because an incident light ray transmits through the glass twice, the glass should have low absorptance. For this reason, lowiron glass is typically used (discussed on page 41). Other mirror facet constructions use honeycomb sheetmetal enclosures with injected foam, or stamped sheet metal, providing the backing support structure. These alternative constructions can reduce the overall weight of the mirror facet but still provide sufficient stiffness. In some cases, durable reflective films have been used on certain types of mirrors. These mirrors do not require a cover glass sheet and are called first surface reflectors. The trade-off is that the glass cover protects the reflective coating from the environmental elements but may absorb some light, whereas external reflective films do not require a cover glass but are directly exposed to the environment. Another first surface reflector is polished metal such as aluminum. To keep manufacturing costs as low as possible, solar collectors are not designed to have high optical quality. Solar collectors have surface slope errors of around 1 mrad root mean square (RMS), whereas imaging optics have surface slope errors on the order of sub-microradians RMS. For this reason, solar collectors are referred to as nonimaging optics. They only serve to collect and concentrate sunlight. However, if observation distances are small (a few meters from the mirror), objects in reflection can be distinguishable.

Field Guide to Solar Optics

Fundamentals of Solar Optics

41

Solar Glass Most glass used for solar collectors is tempered float glass with a low-iron content. Glass is highly durable against outdoor environment conditions and chemicals. The float glass process, also known as the Pilkington process after Sir Alastair Pilkington, allows glass to be formed in large thin sheets. The glass materials (a mixture of silica sand, soda ash, and lime) are placed in a furnace and melted to about 1600 °C or higher. The molten glass is then moved to a float bath, where liquid glass flows evenly over molten tin to form glass with flat surfaces and of a certain width and thickness. Solar glass has thickness between 1 and 5 mm. The hot glass is annealed (slowly cooled) to minimize distortions. The controlled annealing allows glass to form with virtually parallel surfaces. Finally, the cooled glass is cut into smaller pieces on the order of 1 m2, which are used to construct solar collectors. Most glass contains iron impurities in the form of iron salts within the silicon oxide that reduce the transmittance of light. Low-iron glass is a type of high-clarity glass made from silica with very low amounts of iron. Sources for low-iron glass include low-iron sand and limestone. To produce low-iron glass, furnaces must be designed to handle higher melting temperatures compared to the standard float-glass process. The low level of iron removes the greenish-blue tint that can be seen, especially on larger and thicker glass sizes. In addition to solar applications, lowiron glass is used for aquariums, display cases, some windows, and other applications where high clarity is desired.

Field Guide to Solar Optics

42

Fundamentals of Solar Optics

Reflective Coatings Depending on the optical application, several different metal coatings can be used for reflecting light. Aluminum coatings are used as a low-cost material option for general optical applications. Aluminum reflectance is low in the visible band of light. Gold is much more expensive and is typically used in IR optical systems since it is highly reflective in the IR region. In solar technologies, silver coatings are typically used (as opposed to aluminum or gold) because silver reflects well across the solar spectrum, particularly over the peak of the solar spectrum (direct þ circumsolar AM1.5 in the figure). It is desirable to reflect as much solar energy as possible for a given mirror size so that the energy yield can be maximized.

A solar collector with a silver reflective coating and lowiron glass has a solar-weighted reflectance of 95–96%. To make a solar reflector, a thin layer of silver is deposited on the glass surface. This is followed by deposition of a thin layer of copper to protect the silver coating. In some cases, paint is applied to protect the silver and copper layers from corrosion and to ensure a mirror lifetime of .25 years.

Field Guide to Solar Optics

Fundamentals of Solar Optics

43

Concentration Ratios Two definitions are used for solar concentration: optical and geometrical concentration ratios. A concentration ratio (CR) is used to describe the amount of light energy concentration achieved by a collector. The optical concentration ratio is the averaged irradiance over the receiver area Ar divided by the insolation incident (or DNI) on the collector aperture: CRo 5

1 Ar

R

E r dAr Ea

5

E r;avg Ea

The geometric concentration ratio is the area of the collector aperture Aa divided by the surface area of the receiver Ar: A CRg 5 a Ar Note that if the insolation (irradiance) at the aperture and at the receiver are both uniform over their entire respective areas, the optical and geometric concentration ratios are equal. In practice, the geometric concentration ratio is frequently used. If higher concentration ratios are desired, multiple concentrators, such as curved mirrors or lenses, must be used. That is, the collecting aperture size must be increased for the same receiver area. With a heliostat field, to increase the concentration ratio on the receiver, more beams from heliostats are added. Field Guide to Solar Optics

44

Fundamentals of Solar Optics

Concentration Limit The maximum concentration of the sun at the earth’s surface can be determined from the second law of thermodynamics. The energy balance ensures that the radiant flux on the sun’s surface is the same as the radiant flux over a sphere centered on the sun and with radius R equal to the distance to Earth, assuming no attenuation over the distance: 4pr2s E s 5 4pR2 E e where Es and Ee are the irradiances on the sun’s surface and at the earth’s surface, respectively. The concentration of the sun on Earth then cannot exceed the sun’s surface temperature: CE e 5 sT 4e ≤ sT 4s 5 E s where C is the concentration ratio, s is the Stefan– Boltzmann constant, and the e and s subscripts are for earth and sun, respectively. The maximum concentration in three dimensions for circular or point-focus concentrators (in air) is
2 Es R 1 C max 5 5 5  46; 000 suns Ee rs sin2 us The maximum concentration in two dimensions for linear concentrators (in air) becomes C max 5

R 1 5  215 suns rs sin us

Field Guide to Solar Optics

Collector Optics for Solar Technologies

45

Flat Plate Collector The flat plate collector is the oldest type of collector. Its basic components consist of a transparent glaze cover sheet, an absorber plate, fluid-carrying tubes, insulation, and a side enclosure. The fluid can be water or a synthetic fluid.

The transparent cover should maximize the transmission of the incident solar energy to the absorber plate. Antireflective coatings or surface texturing can be used to increase the transmittance. The absorber plate is usually a highly absorptive and conductive metal sheet. It is painted to maximize absorptance. The insulation and sealed enclosure reduce the heat loss from the sides and back. Flat plate collectors are usually nontracking. The installer must decide on the azimuth and tilt angles. The obvious choice for azimuth is south-facing (for northof-equator installation). The easiest choice for the tilt angle is to set it to the location latitude angle. The projected area of a flat plate collector intercepts the incident solar radiation. Thus, this collector is susceptible to cosine losses. Photovoltaic modules can be thought of as flat plate collectors. However, instead of an absorber plate and tubes, solar cells collect the incident solar radiation through a transparent cover plate.

Field Guide to Solar Optics

46

Collector Optics for Solar Technologies

Linear Collectors Linear collectors are curved in one direction and usually have a rectangular aperture. Because they are curved only in one direction, they focus the incident sunlight to a line. The parabolic trough concentrator is an example of a linear collector. The curved cross-section follows a parabolic shape of the form x2 y5 4f where f is the focal length of the parabolic shape. The rim angle can be calculated as   8ðf =DÞ
1 fr 5 tan 16ðf =DÞ2
1  
1 D 5 sin 2r The concentration C on a flat surface at the focal point perpendicular to the optical axis can be determined by considering only the angle beam spread from the sun: C5

a 1 5 D 2 sinðus Þ

which shows half the maximum solar concentration Cmax possible for an ideal linear concentrator. A circular receiver is shown as a dashed circle. The size is just big enough to intercept rays from an ideal concentrator. Imperfections from the reflector (i.e., slope errors) will cause the reflected beam to spread farther, which will require a slightly larger receiver size to minimize light spillage. Linear Fresnel concentrators also fall into the linear concentrator category. Field Guide to Solar Optics

Collector Optics for Solar Technologies

47

Parabolic Trough System Parabolic trough systems are the most mature CSP technology. Solar Electricity Generating Systems (SEGS) near Barstow, California has been generating electricity at 394 MW (gross) capacity since the late 1980s. To collect the thermal energy, a receiver tube is positioned at the focal line of the parabolic trough and a heat thermal fluid (e.g., synthetic oil) is pumped through it. Typical receivers are made of metal tubes with a black absorptive coating enclosed in evacuated glass to reduce heat losses. An antireflective coating may be added to the outer glass to further increase efficiency. Parabolic troughs are lined up in a series and repeated for many rows. A parabolic trough unit that spans the length of the receiver tube is called a solar collector module (SCM). A linear array of modules is called a solar collector assembly (SCA). The modules are fixed together and move as a group when tracking the sun. SCA arrays are usually aligned north to south and track the sun east to west. The main limitation is the low concentrations on the receiver tubes, which results in low fluid temperatures (~400 °C) and low plant efficiencies. To increase concentration ratios and overall efficiencies, some new designs include larger collector apertures and/or a secondary concentrator around the receiver tube. Field Guide to Solar Optics

48

Collector Optics for Solar Technologies

Linear Fresnel Collector Linear Fresnel collectors follow principles similar to those of parabolic trough technology. These collectors use a receiver tube that consists of an absorber tube inside an evacuated glass tube. The differences are: (1) the steeply curved mirrors in parabolic troughs are replaced by long parallel lines of thin mirrors with a shallow curvature; (2) all of the mirrors are near the ground; and (3) the receiver tube stays fixed, while the mirrors move together to track the sun.

The stationary receiver tube is located a few meters above the field. The collector field can achieve solar concentrations of around 30. To achieve higher concentrations, a secondary reflector can be added above the receiver tube to collect the sun rays that miss or reflect off of the receiver tube and reflect them back onto the receiver tube. A main advantage of linear Fresnel reflectors is the use of low-profile reflectors to reduce wind impacts. The reduced wind impacts can minimize the amount of materials used for the structure. In addition, the mirrors are simplified, being flat or slightly curved.

Field Guide to Solar Optics

Collector Optics for Solar Technologies

49

Parabolic Collectors Ideal full paraboloidal concentrators have surfaces that are symmetrical on an axis of revolution, which is also the optical axis: x2 þ y2 z5 4f where f is the focal length of the parabolic concentrator (i.e., the distance from the concentrator vertex to its focal point F). For example, the sun rays that are parallel to the optical axis will focus or concentrate at the focal point. The highest concentration achievable for parabolic concentrators is 1 C5 sin2 ðus Þ where us is the sun half angular extent, or sun half angle. Examples of parabolic concentrators in solar thermal systems are heliostats and dish concentrators. The main differences between a heliostat and dish concentrator are the focal lengths and solar tracking schemes. Heliostats have focal lengths that range from tens to thousands of meters, depending on the plant size. In a 100 MW power tower plant, for example, the farthest heliostat from the tower can have a slant range of up to 1,600 m, almost a mile. Because heliostats have long focal lengths, they can appear flat. Dish concentrators, on the other hand, have very short focal lengths, on the order of a few meters. The focal-length/diameter ratio (f/D) or f/# is typically ,1 for dish concentrators. For solar tracking, heliostats are pointed midway between the sun and the receiver or target, whereas dish concentrators are pointed directly at the sun. Field Guide to Solar Optics

50

Collector Optics for Solar Technologies

Heliostat Collector Heliostats in power tower systems are used for reflecting and concentrating the sunlight onto a receiver located on top of a central tower. Typical heliostats have two angle drives that allow daily and yearly tracking of the sun in azimuth and elevation angles. A properly focused heliostat will reflect the sun onto a vertically oriented target plate, which could be the receiver surface.

A 5 pr1 r2 5

pf 2 tanðus þ sÞ cosðuÞ

r1 5 f tanðus þ sÞ

r2 5 f

tanðus þ sÞ cosðuÞ

The area of the reflected beam A on the target plate, defined by radii r1 and r2, includes the sun half angle us, any slope errors s in the heliostat, and the heliostat focal length f. The angle u is the reflected beam direction relative to the target plane surface. The slope errors can include heliostat surface slope errors and mirror facet canting errors. The heliostat surface shape can also be impacted by gravity and wind loads, which will contribute to the overall slope error s. The heliostat can also exhibit astigmatism, which is the focal length difference in orthogonal directions on the heliostat plane—usually horizontal and vertical directions. Field Guide to Solar Optics

Collector Optics for Solar Technologies

51

Heliostat Field CSP power tower systems use large heliostat fields to collect solar thermal energy. A heliostat field consists of many heliostats. The number of heliostats depends on the CSP power tower plant size. The heliostats reflect the solar rays to a receiver located on top of a tower. Ideally, the heliostats have parabolic shapes to focus the reflected solar rays on the receiver. The size of a heliostat ranges from 1 m2 to more than 120 m2 in reflective surface area. Developers perform trade-offs (in cost and performance) to decide on the size of the heliostat to be built and installed.

The heliostat field is configured for the receiver type, e.g., cavity, external cylindrical, or external box. Large power tower plants (.100 MW) with external receivers have heliostats surrounding the tower, so they are sometimes called a central tower. A system with a cavity-type receiver will usually have a heliostat field on the north or south side of the cavity opening, depending on whether the system is located north or south of the equator. The heliostat field is laid out to maximize optical efficiency. Parameters such as shading and blocking are minimized, and other parameters such as reflectance are maximized. The parameters that impact heliostat optical performance are discussed in the next section: Optical Characterization and Analysis (pages 61–77). Field Guide to Solar Optics

52

Collector Optics for Solar Technologies

Aimpoint Aimpoint refers to a target point on the receiver or target surface. Depending on the sun position, heliostats are oriented to “hit” the aimpoint. Aimpoints can be defined for a variety of reasons, such as maximizing the peak irradiance, reducing the peak irradiance, or beamshaping. A single aimpoint maximizes the peak irradiance. For multiple aimpoints, groups of heliostats direct the reflected beams to a different aimpoint.

The contour plots illustrate beam irradiance distribution on flat plate targets for a single aimpoint and four aimpoints (two by two) that are 1 m apart in both horizontal (x) and vertical (y) directions. The lower plots are horizontal slice plots through the beam centroid. For a single aimpoint, the peak irradiance is high. With four aimpoints, the beam is “stretched out,” peak irradiance is reduced, and beam uniformity can be achieved over an area of a certain size. Field Guide to Solar Optics

Collector Optics for Solar Technologies

53

Dish Concentrator Dish concentrators have much shorter focal lengths than heliostats. The focal ratios (f/D or f/#) of dish concentrators are typically less than 1. Dish concentrators are used with Stirling engines to generate electricity. The engine is placed at the focal point of the dish. Due to their small focal ratios, dish concentrators can concentrate the sun to high values. Concentrations of .10,000 have been demonstrated. In dish-engine systems, the parabolic dish can be a few meters in diameter. A monolithic dish surface is difficult to manufacture; therefore, the dishes are composed of smaller mirrors facets. An important aspect of dish-engine systems is accurate positioning and pointing of the mirror facets. The mirror facets must follow the dish parabolic profile. The individual facets must also point in the right direction. The surface normal of each facet must point at the 2f (or center of curvature) of the full dish. This ensures that the facets will concentrate the sun rays at the dish focal point f; i.e., incident rays follow the law of reflection, shown by u. The pointing or alignment of the mirror facets is referred to as canting. Imperfect mirror facets will dictate a modified canting strategy so as to generate a uniform irradiance distribution at the receiver surface. Additionally, the canting strategy can be tailored to accommodate the shape or arrangement of the receiver.

Field Guide to Solar Optics

54

Collector Optics for Solar Technologies

Sizing a Parabolic Trough Collector Field Some guidelines for sizing a parabolic trough field are provided using a simplified approach. The system consists of a parabolic trough collector, a receiver tube with thermal energy storage (TES) typically specified in hours, and a power block, where the electricity is generated. The power output (i.e., plant capacity) can be simplified as P 5 hplant DNI Ac where hplant is the annual plant efficiency, DNI is the direct normal irradiance, and Ac is the total aperture area of the trough collector. The collection area needed to produce the nameplate power output P is then Ac 5

P hplant DNI

If the aperture width of the collector is wa, then the collector length lc is A P lc 5 c 5 wa hplant DNI wa If thermal storage is included in the plant, then the solar multiple (SM) must be calculated. The length of the collector with thermal storage then increases to lc;total 5 SM ⋅ lc The total collector aperture area becomes Ac;total 5 wa lc;total ½m2  Since the length of the tube receiver is approximately the same as the length of the collector, lc,total can be taken as the total length of the receiver tube as well. Field Guide to Solar Optics

Collector Optics for Solar Technologies

55

Sizing a Power Tower Collector Field Some guidelines for sizing a power tower collector field are provided. Given a nameplate plant capacity and thermal storage capacity, the collector field size can be estimated. This is useful for estimating the total reflective area and/or land coverage. The diagram shows the power tower system with thermal storage. The nameplate power output P4 can be expressed in terms of the thermal power P1 incident on the receiver: P 4 5 hpb P 3 5 hpb

hpb hrec P2 5 P1 SM SM

where SM is the solar multiple, and assuming no losses. Substituting the parameters for P1, we get hpb hrec P4 5 hcoll DNI Acoll SM SM Solving for the aperture area of the collector field, we get P4 Acoll 5 hpb hrec hcoll DNI where hpb is the power block efficiency, hrec is the receiver efficiency, hcoll is the annual field optical efficiency (typically 60–65%), and DNI is the direct normal irradiance. The plant efficiency can be taken as hplant 5 hpb hrec hcoll Assuming some values for the parameters, we can estimate the total field reflective area for a 130 MW plant as Acoll 5

130 3 106 W  8.45 3 105 m2 ð0.3Þð0.9Þð0.6Þð950 W=m2 Þ

and the plant efficiency can be estimated as hplant 5 ð0.3Þð0.9Þð0.6Þ 5 0.162 5 16.2% For exact field sizing, detailed modeling of the plant is needed. Field Guide to Solar Optics

56

Collector Optics for Solar Technologies

Fresnel Lens Concentrator For concentrating photovoltaic technologies, the sizes of the solar cells are reduced. For this reason, the cover plate, which can be glass or a polymer, must be able to focus the collected sunlight onto the smaller solar cells. Fresnel lenses are a cost-effective way to focus the sunlight. A Fresnel lens has a reduced form factor compared to a regular lens of similar optical properties. Fresnel lenses consist of a series of concentric grooves etched into glass or plastic and can replace conventional optical lens. The contours act as individual refracting surfaces, bending parallel light rays to a common focal length. As a result, Fresnel lenses, although physically narrower in profile, can focus light similarly to the way conventional optical lenses focus light but have several advantages over their thicker counterparts. Their thin, lightweight construction, availability in small as well as large sizes, and excellent light-gathering ability make them useful in a variety of applications. Fresnel lenses are most often used in light-gathering applications, such as condenser systems or emitter/detector setups. Therefore, they are used with solar cells or low-temperature solar thermal systems. They can also be used as magnifiers or projection lenses in illumination systems.

Field Guide to Solar Optics

Collector Optics for Solar Technologies

57

Solar Furnace Unlike a dish concentrator with a Stirling engine, which tracks the sun, a solar furnace dish is stationary. This is achieved by combining it with a heliostat that tracks the sun and reflects the sunlight toward the dish. The furnace dish is typically housed in a building, and the heliostat is external to the building. A large attenuator or shutter is positioned just outside the dish building but in front of the heliostat. The attenuator controls the amount of light that is incident on the dish. The heliostat is nominally flat such that it reflects near-parallel rays toward the dish. The rays are also parallel to the dish optical axis. The dish then focuses the rays to its focal point. The dish focal point can be located by co-aligning a pinhole with the dish vertex that defines the dish optical axis. The pinhole is positioned at the 2f point (center of curvature) of the dish. This point can be located visually by placing your eye there, upon which the image of the eye will fill the entire dish surface. After defining the dish optical axis, a laser can be co-aligned with it. Fluxmapping can be performed through the focal zone along the dish optical axis while “on-sun” to find the minimum spot size or highest solar concentration. The solar furnace dish at Sandia is 6.2 m in diameter and has a 4.1 m focal length.

The solar furnace can be used for high-temperature, highsolar-concentration experiments, as in high-temperature chemistry or accelerated aging tests of solar materials and components. Field Guide to Solar Optics

58

Collector Optics for Solar Technologies

Solar Simulator A solar simulator can be used in place of a solar furnace. The main advantage of solar simulators is that they are weather independent, whereas a solar furnace can only operate when the sun is available. With a solar simulator, experiments can be run continuously during the day or at night. However, solar simulators are smaller in scale and usually cannot achieve the same concentration levels as a solar furnace. The main components of a solar simulator are high-intensity lamps, a current controller, reflectors, and a mounting structure to hold the lamps and reflectors. The lamps are usually xenon or metal halide types. Their output can simulate the solar spectrum very well. Xenon lamps tend to be more expensive than metal halide lamps. The reflectors are usually truncated elliptical reflectors. Elliptical reflectors have two focal points, F1 and F2. The lamp filament is aligned with focal point F1

inside the reflector. The reflector will concentrate the light at the other focal point F2, where test articles are placed. Multiple reflectors and lamps can be cascaded to increase the irradiance (or concentration) at coincident focal points of the multiple reflectors. Field Guide to Solar Optics

Collector Optics for Solar Technologies

59

Another Concentrator Type: Cassegrain The classic Cassegrain reflector has been considered as a solar collector in the past. Cassegrain is typically used in telescope designs. Cassegrain reflectors have potential as alternatives to dish concentrators in dish-engine systems. Where the Stirling engine would normally be in the dish-engine system is a secondary reflector in a Cassegrain. With this modification, in the Cassegrain design the engine can be moved behind the primary reflector at the focal point. This may provide the advantages of shifting the weight closer to the pedestal and better balancing of the weight on the structure.

The radii of curvature for the primary and secondary mirrors of the two mirrors in a classic Cassegrain system are defined, respectively, as R1 5

2DF F
B

R2 5

2DB F
B
D

where F is the system effective focal length, B is the back focal distance (i.e., the distance from the secondary to the focal point), and D is the distance between the two reflective surfaces.

Field Guide to Solar Optics

60

Collector Optics for Solar Technologies

Solar Multiple In solar thermal technologies, the solar multiple (SM) is a measure of collected solar resource beyond what is needed for a nameplate capacity. The nameplate capacity is usually based on a single design point and typical meteorological conditions for the location. SM greater than 1 could simply be an overdesign of the plant to account for variations in the solar resource from the design point (e.g., seasonal solar resource variations or clouds). The solar multiple can be defined in terms of the solar collection area: aperture of reflective area SM 5 exact aperture of reflective area where the numerator is the actual reflective area aperture (usually larger) of the plant, and the denominator is the exact reflective area aperture needed to produce the nameplate capacity at the design point. Another reason to collect more solar energy is to supply the thermal energy storage, which could be heated thermal fluid stored in a large insulated tank. Thus, the solar multiple can be approximated in terms of operation time: SM 

daily solar hours þ thermal storage hours daily solar hours

Daily solar hours is defined as the availability of the sun (in hours) in a day to run the plant at its nameplate capacity, again, usually at a design point. The numerator includes the sun availability in hours and the thermal energy storage capacity in hours. A solar multiple of 1 is the thermal input needed to operate at the nameplate capacity with no thermal energy storage. Typically, solar power plants are designed with a solar multiple .1, even without thermal storage, to account for the variation of the solar resource.

Field Guide to Solar Optics

Optical Characterization and Analysis

61

Mirror Surface Slope Error The surface quality of an optical surface can be specified by surface spatial variations (units in microns or nanometers RMS) or slope variations (units in milliradians RMS) called slope error. These errors are usually from manufacturing. Slope error is specified as the variation of the surface normal vector across the optical surface. The reflected beam spread due to slope error is then twice the surface normal variance. In solar optics, slope error is typically used because slope error in milliradians can easily be translated into angle deviations, for example, to calculate reflected beam deviations or beam spread. Two errors are usually associated with surface slope error: macroscale surface slope variations, sometimes called surface irregularity, and shape error. The surface variations refer to deviations from the ideal surface profile. To quantify the surface slope variations, low-order polynomials can be fitted to the surface variations and shape by a maximum-likelihood or least-squares fit calculation. The polynomial coefficients provide estimates of the surface variation magnitudes. A design shape can also be compared to the measured surface. The differences are the residual errors, which are the slope errors in the surface. In terms of surface spatial frequencies, the shape and slope errors can be associated with low- to mid-spatial frequencies, and the high spatial frequencies are due to specularity or microroughness. Field Guide to Solar Optics

62

Optical Characterization and Analysis

Mirror Shape Error The shape error refers to deviations from mirror design shape, such as the shape that defines the focal length.

Shape errors impact the focal length of the mirror. If the shape has a higher curvature in a concave mirror, the focal length will be shorter. The focal length sensitivity can be determined by starting with a mirror parabolic equation: r2 z5 4f dz r2 5
2 df 4f 4f 2 Dz r2 To get a sense of the sensitivity of the collector focal length to shape changes, assume a 1 m diameter mirror, a design focal length of 100 m, and a shape deviation of þ0.25 mm at the edges (from the design shape): Df 5


Df 5


4 · 1002 ð0.25 3 10
3 Þ 5
40 m 0.52

A 0.25 mm height deviation at the edge of the mirror will cause a change of 40 m (shorter) in focal length. The sensitivity is more pronounced for longer-focal-length mirrors.

Field Guide to Solar Optics

Optical Characterization and Analysis

63

Mirror Specularity Mirror specularity is defined as the diffuse reflections off of a mirror surface due to the microroughness on the surface. The specularity or surface microroughness is not specified as surface slope error because the surface slopes can be very high, which would drive the total slope error to very high values. Instead, specularity is specified as the angular spread of the reflected beam. Specularity causes diffuse reflections and is therefore specified as an angular beam spread upon reflection. Surface slopes due to specularity at the micro-scale can be very high. Due to the very high surface slopes, multiple reflections can occur within the surface. Mirror soiling (small dust particles on the surface) can contribute to specularity. The total optical error due to collector surface imperfections can be described as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ssurface−error 5 4s2slope error þ s2specularity −

The factor of 22 is due to the slope error being defined at the surface.

Field Guide to Solar Optics

64

Optical Characterization and Analysis

Mirror Facet Canting-Alignment Error Most solar collectors concentrate the sunlight, meaning that the collectors have a spherical or parabolic surface such that the reflected sunlight is concentrated at the collector focal point F. Large collectors are needed to achieve high concentration ratios. However, large monolithic mirror surfaces are difficult to manufacture at low cost. Handling large mirrors is also a challenge. Therefore, large collectors are constructed with smaller mirror facets attached to either a curved or flat frame. The mirror facets, either flat or curved, are arranged such that they follow the profile of a large parabola with a specified focal length f.

When all of the mirror facets are perfectly aligned or canted, the surface normal vectors emanating from the center of the mirror facets should all coincide at the 2F point or the center of curvature. Any deviation from this ideal alignment is referred to as canting errors; i.e., the surface normal vectors do not coincide at the 2F point. The result is a reduced-concentration beam spreading at the focal point. Each mirror facet is attached to the frame structure by multiple bolts. Ideally, three bolts are used, and the bolts are arranged such that they provide facet angle adjustments in azimuth uaz and elevation uel angles. Field Guide to Solar Optics

Optical Characterization and Analysis

65

Facet Canting Adjustment Most collectors in solar thermal applications (e.g., heliostats) have individual mirror facets that are adjustable in angle with respect to the frame. Adjusting the mirror facet angles is called facet canting adjustment. The objective is to adjust the mirror angles such that the heliostat forms a focusing shape like a parabola. The mirror facets are mounted on a rigid frame, usually using three attachment points. Two of those points provide adjustments through bolts and nuts, and the third point serves as a pivot point where the nuts on the bolt are not adjusted.

In the middle diagram, the top bolt is adjusted to move the mirror facet in elevation angle about the pivot point and the side adjustment point. Adjustment to the side bolt moves the mirror facet in azimuth angle. The amount of angle movement depends on the bolt thread count and on the adjustments to the bolt in terms of number of turns. For a known canting error, the number of bolt turns needed to fix the error is given by N 5 Duðlp 3 εÞ where Du is the measured canting error in azimuth or elevation, lp is the pivot arm length, and ε is the bolt thread count, which is usually given in threads per inch (TPI). Some facets are mounted using four bolts in a rectangular pattern. In this case, adjusting one bolt at a time can induce stress in the facet, which leads to additional slope errors on the reflective surface. These adjustments should be carefully calculated and applied.

Field Guide to Solar Optics

66

Optical Characterization and Analysis

On-Axis Canting Strategy The canting strategy refers to the mirror facet alignment strategy employed to ensure that all of the mirror facets on a heliostat concentrate the sunlight to a point on the receiver for a particular point in time. For that particular point in time when the facets were aligned, the collector will have ideal focusing. For example, if the facets were canted for Day 80 at solar noon, the collector will have ideal focusing on that day and time, and focusing will degrade slightly away from that time. The type of canting strategy determines the angles on the mirror facets and thus the overall shape of the collector. Each mirror facet must have angle adjustments in two axes. There are two types of canting strategies: on-axis and off-axis canting. On-axis canting strategy refers to the mirror facet alignment when the heliostat, the receiver, and the sun are all aligned. For most solar collectors, e.g., heliostats, this is a hypothetical case because, in reality, it is unlikely, although not impossible, that the sun will be aligned with heliostats and the receiver. In the on-axis canting scenario, the canting application merely means that the heliostat will form into an axisymmetric parabolic surface.

Field Guide to Solar Optics

Optical Characterization and Analysis

67

Off-Axis Canting Strategy Off-axis canting strategy refers to the mirror facet alignment for the sun position on a particular day and time. The day and time determine the sun position for which the heliostat will have ideal alignment (i.e., smallest spot size on the receiver). Since the sun is in an off-axis position, offaxis aberrations such as astigmatism are adjusted in the mirror facet alignment to correct for the aberrations (for this particular day/time only). Therefore, the collector (e.g., heliostat) surface will not be axisymmetric.

The choice of canting strategy depends on several factors, such as collector field size, field configuration, location, etc. The goal is to achieve high annual optical efficiency from the collector field. Several methods exist for performing mirror facet canting. The most common is the on-sun canting approach, which is an off-axis canting method. In this method, the mirror facets are adjusted while the collector is on-sun; i.e., the sun is reflected onto the target and the collector is tracking the sun. Other methods utilize inclinometers, lasers, photogrammetry, and reflected target structures.

Field Guide to Solar Optics

68

Optical Characterization and Analysis

Tracking and Pointing Errors In CSP power tower systems, the heliostats must accurately track the sun as it moves across the sky. The sun’s movement changes diurnally (daily) and seasonally. In an ideal scenario, the heliostats will always reflect the sun to a particular point (the aimpoint) on the receiver or another target. Due to optical, manufacturing, and installation errors, this is not always the case. These errors will cause the heliostat to mistrack. If the tracking error strack is large enough (e.g., $1 mrad), it is preferable for the errors to be quantified and corrected.

The errors that can cause tracking inaccuracies are pedestal tilt in the east–west (ε1) and north–south (ε2) directions, azimuth (ε3) and elevation (ε4) reference bias, azimuth (ε5) and elevation (ε6) linear error, drive axis nonorthogonality (ε7), and boresight error (ε8). These errors can be decomposed into azimuth and elevation pointing errors. The errors can be estimated from a set of on-sun tracking observations:3   Du 5 H~ ε Da 

where H 5

sin u tan a cos u

cos u tan a 1
sin u 0

0 u 1 0

0 a

tan a 0

1 cos a

0



;

~ ε 5 ð ε1 ε2 ε3 ε4 ε5 ε6 ε7 ε8 ÞT ; u and a are heliostat angles in azimuth and elevation, respectively; and T is the matrix transpose symbol. Field Guide to Solar Optics

Optical Characterization and Analysis

69

Sunshape The sunshape describes the intensity distribution across the sun’s disk. Near the center of the disk, the intensity is fairly uniform. The intensity then gradually decreases until it falls off quickly at the edge. Limb darkening is an optical effect where the center part of the sun’s disk appears brighter than the edge or limb of the image. Circumsolar radiation is the diffuse radiation surrounding the sun’s disk, and it can vary with location, climate, and time of day. The circumsolar ratio (CSR) is the fraction of the direct normal irradiance (DNI) not coming from the sun disk. By convention, the region between the outer rim of the sun’s disk (±0.26 deg) and the DNI definition angle of 2.5 deg is considered to be circumsolar. Several models are used to describe the sun’s intensity profile (see graph). The simplest model is a pillbox that extends over ±4.65 mrad. A Gaussian distribution with s 5 2.73 mrad is also sometimes used as an estimate of the sunshape. There are also more accurate models, such as the one described by Buie et al.:4  cosð0.326uÞ fðuÞ 5

cosð0.308uÞ ek ug

for for

0 ≤ u ≤ 4.65 mrad u . 4.65 mrad

where k 5 0.9 lnð13.5xÞx
0.3 ; and g 5 2.2 lnð0.52xÞx
0.43
1. The circumsolar ratio is defined as x 5 Fcs =Fi , where Fcs is the circumsolar radiant flux, and Fi is the incident radiant flux.

Field Guide to Solar Optics

70

Optical Characterization and Analysis

Gravity and Wind Impacts Both gravity and wind impact the collector structural rigidity and stability, which in turn impact the collector optical performance. The Fg gravity force vector always points down. The wind force vector mostly lies horizontally, and the pointing direcFw tion depends on the wind direction. Gravity is a constant force, whereas the wind force can vary randomly due to its changing speed and direction, as well as the movement of the collector during tracking. Gravity causes the structure to sag. The amount of sag or structural deflection depends on the structural rigidity and load. Sag can change with heliostat elevation angle. As an example, for a 100 m2 heliostat, 1 mm thick glass alone can weigh about 500 kg. The weight can more than double with support structures. During the design phase, the structure must be designed with sufficient rigidity for the load. It helps to know the structural deflections for varying elevation angles. This can be determined from finite element analysis models. If known, these deflections can be compensated for during mirror facet canting adjustments. The impacts from wind can be significant. Wind impacts can cause collector structural deflections and jitter in the reflected beam. Vortices can form near the heliostat surface, causing additional jitter in the reflected beam. Reflected beam jitter is less of an issue than the impacts on the mechanical structure. Significant wind loads can cause permanent damage to the structure. The Solar Power Tower Design Basis document5 specifies wind levels for safe heliostat operation. Operation is limited to 12 m/s (27 mph) wind gust and 8 m/s (18 mph) mean speed. Above these wind levels, heliostats must be stowed to avoid damage to the structure. Gravity and wind both impact collector optical performance. Field Guide to Solar Optics

Optical Characterization and Analysis

71

Combined Optical Errors The collector tracking error can be defined as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi stracking 5 Du2 þ Da2 where the errors in the azimuth and elevation axes are combined. All optical errors are assumed to follow normal or Gaussian distributions. That is, when the errors from all of the collectors (e.g., heliostats) are combined, the distribution of the errors will tend to follow Gaussian distributions. The probability distribution function for each error source can be defined as


p5e

ðx
x0 Þ2 2s2

The different error sources can be combined through a convolution calculation: P 5 p1 p2 : : : The error sources can be combined as a root sum square: soptical

total

5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2surface þ s2canting þ s2tracking þ s2wind þ s2gravity

The beam reflected from the collectors will spread due to these error sources and the sun angular spread. The total reflected beam angular spread can be calculated as sbeam−spread−total 5 soptical−total þ ssunshape The sunshape is sometimes included in the convolution calculation. Here, it is merely added to the result of the convolution calculation since sunshape is a systematic “error” source. Also, soptical_total is one standard deviation wide. To include more of the beam spread from the optical errors, Nsoptical_total can be used, where N is the number of standard deviations (N 5 1, 2, or 3). Field Guide to Solar Optics

72

Optical Characterization and Analysis

Shading and Blocking The top figure illustrates shading and blocking of the incident and reflected sun rays (dashed lines and shaded areas) in a power tower system. Shading occurs when the sun rays incident on a heliostat become blocked by neighboring heliostats. Blocking occurs when the rays reflected by a heliostat toward the receiver become blocked by the heliostat that is directly in front. Both shading and blocking result in energy lost. The shading and blocking efficiencies are defined as A
As hs 5 T AT hb 5

AT
Ab AT

where AT is the total heliostat area, and As and Ab are the shaded and blocked areas on the heliostat, respectively. To help maximize the solar energy incident on the receiver at any given day and time, the shading and blocking impacts must be minimized by varying the spacing between heliostats. The spacing can be made large enough to eliminate shading and/or blocking. However, this increases the land usage, which adds cost to the system. An optimization must be performed that balances the solar energy collection efficiency and cost. A heliostat field typically has an annual optical efficiency of around 60%. Field Guide to Solar Optics

Optical Characterization and Analysis

73

Cosine Losses The impacts of cosine losses are observed when the solar collector is not directly pointed at the sun. The collector will effectively have a smaller intercept area, i.e., the projected area. In power tower systems, heliostats must point halfway between the sun and the receiver to put the reflected beam on the receiver. In a surround field, the heliostats on the south side of the tower will experience higher cosine losses (for plants north of the equator).

The angle u between the incident solar ray and the heliostat surface normal vector is what limits the collection area of the heliostat. The effective collection area, or projected area, is calculated as AP 5 Ah · cosðuÞ where Ah is the actual heliostat area. For the full heliostat field, the collection area is reduced by X X Ah;i cosðui Þ 5 Ah cosðui Þ AP;total 5 i

i

where i corresponds to the ith heliostat. In CSP systems, cosine losses are difficult to optimize. To minimize the losses, usually more heliostats are placed on the north side of the surround field. For photovoltaic systems, cosine losses can be minimized by placing the modules on two-axis trackers that track the sun. The cosine efficiency is defined as dsun · nˆ ðdot productÞ hcos 5 cosðui Þ 5 ~ ~ where dsun is the vector pointed at the sun, and nˆ is the surface normal vector of the heliostat during on-sun tracking. Field Guide to Solar Optics

74

Optical Characterization and Analysis

Intercept Factor The intercept factor is the ratio of the incident concentrated solar energy intercepted by the receiver to the incident concentrated solar energy reflected by the collector field. It is preferable to have the intercept factor be a large value (e.g., .0.9). The energy not intercepted by the receiver is referred to as spillage, which should be minimized because this energy is usually lost and not recovered. The intercept factor value depends on the size of the receiver as well as the optical errors previously described, such as the collector surface errors, canting errors, and sunshape. The optical errors determine the amount of reflected beam spread. When the collector optical errors are high, the reflected beam spread will also be high, which will reduce the intercept factor. The receiver size may then need to be increased to improve the intercept factor. Increasing the receiver size may increase cost; therefore, cost–performance trade-offs need to be balanced.

The intercept factor efficiency is defined as the ratio of the irradiance [W/m2] or flux [W] intercepted by the receiver to the total irradiance or flux incident at the receiver plane: hint 5

irradiance or flux intercepted by the receiver total incident irradiance or flux

Field Guide to Solar Optics

Optical Characterization and Analysis

75

Mirror Soiling Mirror soiling refers to the accumulation of dust and other contaminants on the collector top surface over time that results in mirror reflectance reduction. These contaminants will interfere with the incoming solar radiation by scattering and absorbing the light. The impact is a decrease in the total collector reflectance r. In CSP systems, a clean collector with a silver reflective coating and low-iron glass will have reflectance of $0.95. Over time, due to soiling, this value will be reduced, which will necessitate cleaning the collectors to bring the reflectance value back up. Currently, cleaning is done with water and brushes. There is ongoing research to develop collector cleaning approaches that use less or no water. Contaminants on the top collector surface will cause the incident light to scatter or be absorbed.

For second surface reflectors, the dust and contaminants are encountered twice: once upon incidence at the top surface and again after reflection off of the bottom reflective coating. For photovoltaic modules, the dust impacts are encountered once, but they can still reduce the energy output from the modules. The forward-scattered light can still be utilized. The reflectance efficiency, which is impacted by soiling, can be defined as href 5 r, where r is the collector reflectance. Field Guide to Solar Optics

76

Optical Characterization and Analysis

Atmospheric Attenuation Atmospheric attenuation is caused by various particulates in the air (e.g., water vapor, CO2, dust, etc.) that scatter and absorb the light. Although the direct normal irradiance (DNI) measured on the earth’s surface accounts for atmospheric attenuation, the impact must also be added to the beam reflected by the heliostat collector to the receiver, which can cover distances of tens of meters to over a kilometer at the outer edge of the collector field. Various atmospheric loss models exist. Some can be applied during clear sky days and others can be applied during hazy days. Currently there is no agreedupon standard for applying specific models. It has been shown that attenuation is a function of site elevation, water vapor, scattering coefficient, tower height, and slant range. Some detailed models incorporate all of these parameters, in addition to time of year and site climate. Sun

Receiver

Heliostat

R ge, Ran t n a Sl here osp Atm

H

Tower D

Assuming that most solar technologies are deployed in areas of primarily clear skies, the following model,6 which is based on empirical data, can be used:  hatm 5

0.99321
0.0001176 R þ 1.97 · 10
8 R2 expð
0.000106 RÞ

R ≤ 1 km R . 1 km

This atmospheric attenuation model accounts for radiation losses over the slant range R and assumes a visibility distance of about 40 km. Field Guide to Solar Optics

Optical Characterization and Analysis

77

Collector Optical Efficiency The collector optical efficiency combines all of the efficiencies described and quantifies the optical performance of the collector field. It can be reported in several different ways. In most cases, it is reported as an instantaneous efficiency, or design point efficiency, for a particular day and time, or as an annualized efficiency. The instantaneous efficiency is calculated as the product of the instantaneous efficiency terms introduced previously, where hsb is shading and blocking, hcos is cosine losses, hint is the interception of sun rays at the receiver, href is the heliostat reflectance, and hatm is atmospheric attenuation: hopt 5 hsb · hcos · hint · href · hatm The annual optical efficiency (sometimes referred to as the unweighted annual optical efficiency) for the ith heliostat can be calculated as P365 R t1 n51 t0 hðtÞdt hðiÞ 5 P R t1 365 n51 t0 dt The insolation E weighted optical efficiency can be calculated as P365 R t1 n51 t0 EðtÞhðtÞdt R t1 hE ðiÞ 5 P 365 n51 t0 EðtÞdt The annual optical efficiency of an entire heliostat field can then be calculated as PN h hðiÞ hfield 5 i51 Nh where Nh is the total number of heliostats in the field. In CSP power tower systems, the optical efficiency parameter, in addition to cost parameters, is used to optimize the field layout distribution.

Field Guide to Solar Optics

78

System Modeling Approaches

Cone Optics Cone optics refers to the use of error cones and convolution of these error cones to describe light beam propagation through an optical system. Cone optics provides an efficient method to model solar collector systems to predict performance and/or perform design trade-offs. The error cone of the reflected beam is convolved with the sunshape to calculate irradiance profiles at the receiving surface. A single central ray is traced, and the reflected ray is surrounded by a cone of radiation defined by the distribution of the sunshape and optical errors. The incident central ray from the sun reflects specularly. If the collector element is an ideal surface, i.e., free of errors, the sunshape would be reflected perfectly, as indicated by the cone with dashed lines. When errors (time or space dependent) cause the alignment of nˆ to become nondeterministic, surface normal vectors are described by a distribution of directions, as ˆ The nˆ error cone is shown by the cone drawn about n. convolved with the sunshape to obtain a distribution called the effective sunshape, which is shown by the cone with solid lines in the specular reflection direction. The effective sunshape is projected onto the receiver to obtain the flux contribution from the concentrator element. The total flux on the receiver is a summation of flux contributions from all of the concentrator elements. The advantage of the cone optics approach is computational speed.

Field Guide to Solar Optics

System Modeling Approaches

79

Hermite Polynomials Hermite polynomial expansion (HPE) is an efficient analytical method for predicting irradiance profiles from the collector field. The HPE method can be more efficient in computation speed than the cone optics approach. Since the sunshape and the collector error distributions follow Gaussian-like circular distributions, in this method Hermite polynomials are used to represent the irradiance distribution calculated through a convolution of the heliostat projection on the receiver, sunshape, and optical error distributions. That is, the irradiance profiles at the image plane are approximated with truncated Hermite polynomials. Several orders of these polynomials are shown below:7
h
2
2 i Fðx; yÞ 5 2s1x sy exp  12 sxx þ syy 3 hP P

 i y x 6 6i i50 j50 Aij H i sx H j sy =i!j!

H 0 ðxÞ 5 1 H 1 ðxÞ 5 x H 2 ðxÞ 5 x2  1 H 3 ðxÞ 5 x3  3x :::

The speed of the HPE computation results from the fact that a truncated polynomial (e.g., sixth order) expansion is an accurate approximation of the flux density. This method has been shown to become more accurate in predicting the flux and spillage when (1) the errors increase, (2) the slant range increases, or (3) the size of the heliostat is reduced (either physically or effectively by focusing or canting). The ability to use a single flux calculation to predict the flux from a given heliostat design on any tower or receiver is the main strength of this method. With this capability, the method can scale the results of one detailed initial performance run during system optimization calculations and quickly calculate performances for comparison between different system designs.

Field Guide to Solar Optics

80

System Modeling Approaches

Ray Tracing The ray tracing method, as the name implies, generates rays from the source and propagates the rays through the system, following reflection and refraction laws. Rays are generated at the source based on a Monte Carlo sampling. Depending on the source type, each ray will have a starting position and propagation direction. A ray generated from a sun source will be based on the sunshape probability density function. A ray generated at source S will intersect the collector at point C. Following the law of reflection, the ray at C will be reflected to the receiver and intersect it at point T according to
!  ! T  C 5 CS  2 CS · nˆ nˆ where nˆ is the surface normal vector of the collector (e.g., heliostat) at the ray intersection point. The source strength (in watts) will be divided equally among the total number of rays generated (watts per ray), and the flux will accumulate at the target surface. Irradiance is calculated by accumulating the incident rays in rectangular bins of known dimensions on the target surface. Ray tracing can be computationally intensive since millions to billions of rays need to be generated and traced to get accurate results. With modern, fast computers, this is no longer an issue. A convergence test determines the minimum number of rays needed for the modeled system. In this test, the number of rays is increased until the results at the target surface converge to a stable value. Advanced ray tracing codes incorporate nonsequential techniques such as scattering, i.e., a parent ray generating children rays. The advantage of ray tracing methods is the accuracy, flexibility, and versatility in building complex shapes and geometries. Field Guide to Solar Optics

System Modeling Approaches

81

Systems Performance Modeling Computer codes exist [e.g., NREL’s System Advisor Model (SAM) and Sandia’s HELIOS] that can perform solar systems performance modeling to predict the utilization of solar energy by the system to produce electricity or another form of energy. Some codes have modules to perform system optimizations. Two types of optimizations can be performed: (1) optimization of the collector field and (2) overall system optimization. Collector field optimization involves maximizing the transfer of the collected solar energy to the receiver for a system size and geographic location. To maximize the energy transfer, parameters such as shading, blocking, and cosine impacts must be minimized. To achieve this, the heliostat spacing is adjusted while also tracking the ground cover ratio c, defined as the ratio of the heliostat area to the ground area. Optimum values of c range from 0.2 to 0.6. The tower height may also be adjusted to optimize the annual collector field efficiency. An optimized system design is a resulting combination of tower height, receiver size, and collector field layout that gives the lowest calculated system energy cost, or levelized cost of electricity, for a given system size, e.g., 100 MWe (megawatt electrical) and thermal storage. In order to calculate the energy cost, both system performance and system capital and operating costs must be determined. The cost parameters are input into the model to perform the optimization.

Because multiple design combinations are performed, cone optics or Hermite polynomial approaches are typically used for their computational speed. Field Guide to Solar Optics

82

Metrology Tools

Deflectometry Method Deflectometry, also referred to as fringe reflection or reverse Hartmann test, is a dynamic method of simultaneously determining the surface normal vectors at many points across a concentrator surface. Regular light interferometric techniques do not apply well here since the solar collector surface quality is far lower than in precision optics. Instead, artificial fringes or known patterns are displayed on a screen. A camera is positioned to see the reflection of the active screen in the specular surface of interest. Any known patterns such as horizontal or vertical lines can be displayed or moved across the screen, and the reflected image is collected by the camera. The mapping of the pattern after reflection is determined and is used to calculate the surface normal vectors on the surface under test. A more efficient method is to display sequentially horizontal and vertical sinusoidal fringes of increasing frequency. For curved reflectors with reasonably short focal lengths (i.e., ≤10 m), the screen can be positioned to coincide with the center of curvature, or 2f location, of the surface under test to keep the screen size sufficiently small. For flat surfaces, the focal lengths are infinitely long. In this case, placing the screen at infinite distances is impractical, so the screen size must be increased to slightly larger than twice the size of the test surface to account for the surface slope errors.

Field Guide to Solar Optics

Metrology Tools

83

Deflectometry Method (cont.) Before collecting data, careful calibrations of the setup must first be performed, including measurement of the setup geometry, the distortions in the camera lens, and the distortions in the screen, which could be a television monitor or a large wall (and a projector). A series of grayscale sinusoidal fringe patterns, or sinusoidally varying brightness patterns, is displayed on the screen, and the reflected images are recorded. Initially, a single cosine wave is displayed and then shifted three times by 90 deg per shift to get an estimate of the absolute positioning of the mapped camera pixels. The phase shifts provide four brightness levels for each pixel of the camera. Then the frequency of the fringes is increased to refine the camera pixel mapping accuracy each time the fringe frequency is increased. For each fringe image capture, the equations below are used to calculate the phase angle. Here is an example of vertical and horizonal sinusoidal fringes that are phase shifted by 90 deg in four frames:

I 1 5 A cosðfÞ þ B I 2 5 A cosðf þ 90 degÞ þ B I 3 5 A cosðf þ 180 degÞ þ B I 4 5 A cosðf þ 270 degÞ þ B The phase angle f(x,y) of each camera pixel can then be determined as I  I2 tan½fðx; yÞ 5 4 I1  I3 Field Guide to Solar Optics

84

Metrology Tools

Deflectometry Surface Determination The absolute phase angle can be determined by standard phase unwrapping algorithms that are used with interferometric techniques. However, the issue with these techniques is the modulo-2p ambiguity of the phase from the calculation; i.e., the phase value is repeated every 2p. An alternative method that has been implemented successfully with deflectometry is the so-called temporal drilldown method. In this method, a single sinusoidal fringe (dark purple line in the graph) is first displayed in horizonal and then vertical directions. Since only one fringe is displayed, there is no ambiguity in the location of the phase that is calculated. However, the positional uncertainty of the calculated phase is high, as shown by the dark purple dot and the dark purple dashed lines. To improve on the phase uncertainty, the fringe frequency is increased in succession. For example, after displaying one fringe, two, four, eight, and higher numbers of fringes can be displayed. The four-fringe case is shown by the light purple line in the figure. With each increase in fringe density, the calculated phase is refined and the uncertainty is improved, as shown by the light purple dot and light purple dashed lines. The other light purple dot is mathematically equivalent to the light purple dot that is of interest but is eliminated from one-fringe calculation. In the single-fringe case, the slope of the sinusoid is varying slowly (i.e., the grayscale range is low), which leads to increased positional uncertainty during the phase calculation. As the fringe density is increased, the slopes of the sinusoids drastically increase, thereby reducing the uncertainty on the calculated phase.

Field Guide to Solar Optics

Metrology Tools

85

Laser Scanning System A laser scanning system was developed and implemented by Sandia and NREL8 for reflector surface characterization. The system uses a laser to scan the reflector surface; the laser beam is reflected back to a target, and a camera is used to detect the location of the laser spot. Individual mirror facets or entire collector systems can be measured with this method. A test is performed by placing the reflective surface at roughly twice its focal length from the laser scanner and target board so that the reflected laser beam lands on the target board. The laser is steered by scanning mirrors. The location of the return spot is measured with a CCD camera. This process is repeated across the surface of the test article in a user-defined pattern. From the scanner aiming angles and return spot locations, the slope at each point and the shape of the mirror surface are computed using predefined coordinate definitions. Surface slope deviations from the ideal surface can be determined. To test flat reflective surfaces, the target board size must be increased to slightly larger than twice the size of the mirror under test to capture all of the reflected light. The camera calibration involves temporarily mounting a rectangular grid of spots with known spacing on the target board, and collecting images of the target grid with the camera. The centroids of each circular spot are determined to a fraction of a pixel, and a fit of the surface is performed. This surface fit is possible because the actual grid spacing in both X and Y directions is known.

Field Guide to Solar Optics

86

Metrology Tools

Target Imaging Metrology Camera target imaging metrology in some cases has been successfully implemented. Particularly, the so-called camera look-back methods with theoretical image overlay have been used to assess the canting (or alignment) quality of mirror facets on a concentrator. If the canting errors are severe, the systems can provide feedback to make corrections. Two system examples are illustrated.

The camera look-back methods rely on the collector as a mirror to perform “imaging” of a target. As such, the distance between the collector and the target structure must be carefully considered. At a few meters apart, the quality of the target image is sufficient to distinguish features on the target that can be used to make assessments of the collector canting or surface quality. At larger distances between the collector and target, the target features become indistinguishable due to the slope errors in the collector surface. In general, the measurement sensitivity is increased with distance, but it must be balanced with the quality of the reflected image. If a collector is designed with a known shape (e.g., focal length), then the theoretical target image can be determined, assuming an ideal collector surface shape. This theoretical image can be overlaid on the actual image of the target. Deviation of the actual image from the ideal image provides collector canting, shape, or surface qualities. Moving the target across the collector, by moving either the target structure or the camera, can further supply collector surface quality information over the entire collector surface. Field Guide to Solar Optics

Metrology Tools

87

Beam Characterization System A beam characterization system (BCS), sometimes referred to as a fluxmapper, is used to characterize the profile of the irradiance distribution from a concentrated solar beam on a white Lambertian target board. In addition to the target board, a camera with a suitable lens and neutral density filters is used to capture the images of the irradiance distributions. The distance between the camera and the target board can be several meters for a dish concentrator, or hundreds of meters for a heliostat field. A computer with imagecapturing software is used to collect and process the images to determine the beam size, beam shape, and centroid. Monitoring the beam centroid over time can be useful to assess the pointing and tracking performance of the collector. If tracking errors are observed, then the centroid data can be used to develop tracking error corrections. The beam size and shape inform the quality of the collector surface and canting alignment. If high solar concentrations are being put on the target board, then the target board must be water cooled to avoid damaging the surface. This makes up the basic BCS. In more elaborate systems, a flux gauge or radiometer is placed within the target board, usually at the center. This provides a direct measurement of the irradiance at a point on the irradiance distribution, which can then be used to scale the beam images or fluxmaps.

Field Guide to Solar Optics

88

Metrology Tools

Radiometer and Flux Gauge To measure the irradiance at a point on the concentrated solar beam, a radiometer or heat flux gauge can be used. The Kendall radiometer was developed primarily to measure the solar irradiance and is therefore accurate for wavelengths corresponding to the solar spectrum. The incident solar radiation is directed into a cavity that is coated with a highly absorptive coating so that it approximates a blackbody cavity. The walls of the cavity are thermally connected to a large heat sink by a thermopile, which provides a voltage proportional to the difference in temperature between the cavity wall and the heat sink. The voltage is calibrated against a voltage from a similar cavity that is connected to the heat sink in the same way but is not subject to any radiation. Kendall radiometers typically have measurement uncertainties of ,1% with a response time of a few seconds. Due to their high accuracies in measuring solar irradiance, these devices are typically used to calibrate other solar radiation measurement devices such as heat flux gauges. A heat flux gauge can consist of a thermopile or sometimes simply a pair of thermocouples in which the elements are separated by a thin layer of thermalresistant material. The size and thickness of the layer determines the range of irradiances that can be measured. Under a temperature gradient, the two thermopile junction layers will be at different temperatures and will register a voltage. The heat flux is proportional to this differential voltage. The thermal resistance layer can be made very thin to improve the sensor response time. Heat flux gauges come in different sizes to measure different levels of solar irradiances. Field Guide to Solar Optics

Metrology Tools

89

Reflectometer A reflectometer is a device that measures the reflectance of sample surfaces. There are two important parameters to consider when using reflectometers: the beam angle of incidence ui and the aperture setting at the detector (see left figure). The aperture setting (usually specified in milliradians) limits the collection of diffuse reflections to an angular extent of f. The right figure shows the basic configuration of a reflectometer. The source can be an LED or laser diode at wavelength l. For solar applications, the detector signal is calibrated to provide a solar-weighted reflectance over the aperture setting.

Advanced reflectometers have integrating spheres to collect the reflected light. Reflectometers can distinguish total, diffuse, and specular reflections. A small port in the integrating sphere can open and close to allow the specularly reflected light to exit. This allows the total and diffuse light to be measured, from which the specularly reflected light can be estimated. In addition, reflectometers can use a white light source to measure the reflectance over several wavelength bands in the visible and IR regions, and calculate the solarweighted reflectance from those measurements. Generally, the directional reflectance can be expressed as R 2p R p=2 L sin ur cos ur dur dfr rd ðu; fÞ 5 R0 R0p=2 r 2p Li sin ui cos ui dui dfi 0 0 where Li and Lr are the incident and reflected radiances, respectively. The beam is defined over its solid angle. Field Guide to Solar Optics

90

Metrology Tools

Emissometers An emissometer works according to the same principles as a reflectometer but measures the in-band reflectance on samples that are all in the IR, which it then uses to calculate the thermal emittance of the samples. The conservation of energy states that the sum of the transmitted, reflected, and absorbed light in all directions and at all wavelengths must equal unity, or tþrþa51 The directional emittance becomes εd 5 1  rd  ts for transmissive surfaces, where rd is the measured directional reflectance. For opaque surfaces, the transmittance component ts is zero. The total directional emittance of an opaque surface at a given temperature is R∞ r ðlÞuðl; T Þdl εT ðu; fÞ 5 1  0 R ∞d 0 uðl; T Þdl where u is Planck’s function in terms of spectral energy density at a given temperature: uðl; T Þ 5

8phc l5 ðehc=lkT  1Þ

When the sample reflectance is measured over several incidence angles, the total hemispherical emittance can be estimated by Z εH 5 2

p=2

0

εT ðuÞ sin u cos u du

Emissometers provide measurements at room temperature but calculate the emittance at the specified temperature. Like commercial reflectometers, portable emissometers are also commercially available.

Field Guide to Solar Optics

Other Nonimaging Optics and Solar Collectors

91

Secondary Concentrator A secondary concentrator has the function of further concentrating the incident light. The most common secondary concentrator is the compound parabolic concentrator (CPC). A CPC is formed by two parabolic surfaces with offset optical axes.

The focal length of both parabolic surfaces is determined by aex f5 2ð1 þ sin ui Þ where aex is the diameter of the exit aperture, and ui is the maximum incidence angle of the incoming light, where 2ui determines the acceptance angle. The overall length of the CPC is determined by a ð1 þ sin ui Þ cos ui L 5 ex sin2 ui The diameter of the entrance aperture can be calculated as a aen 5 ex sin ui By combining the above two equations, the length of the CPC can be simplified to L 5 ðaen þ aex Þ cot ui These parameters can be varied to design a CPC that fits the design requirements. Field Guide to Solar Optics

92

Other Nonimaging Optics and Solar Collectors

Other Compound Parabolic Surfaces Parabolic collectors (left diagram) are ideal concentrators of incident parallel rays. A small section of the parabola can be cut out to create an off-axis parabola (middle diagram), and it will still focus the incident light to a point at F. The full parabola can also be truncated (right diagram). Truncated parabolas can be combined to form compound parabolic concentrators.

The general parabolic profiles are described by z5

r2 4f

where f is the focal length, the z axis lines up with the optical axis, and r2 5 x2 þ y2. The figure below illustrates an example of a compound parabolic concentrator formed from two parabolic profiles (shown as a 2D projection). The dashed lines are the optical axes of the individual parabolic surfaces. The parabolas are tilted in opposite directions. The tilts and positions can be optimized for the application. This type of concentrator can be used within a parabolic trough system as a secondary concentrator near the receiver tube, where the focal points of the parabolas are coincident on the receiver surface. Field Guide to Solar Optics

Other Nonimaging Optics and Solar Collectors

93

Waveguides A waveguide has the function of collecting, confining, and propagating light along its extent. Waveguides can be rectangular or cylindrical. Rectangular waveguides are referred to as slab waveguides. The most common cylindrical waveguide is the optical fiber.

At the entrance aperture, the incident light follows Snell’s law as n1 sin ui 5 n2 sin ut where n1 and n2 are the refractive indices of the outer and inner materials, respectively, and ui and ut are the light incident and transmitted angles, respectively. The incident light will propagate inside the waveguide if the incident angle of the light on the waveguide sidewall is larger than the critical angle uc. The critical angle is determined by applying Snell’s law and occurs when no light is transmitted into the outer material, i.e., when ut 5 90° at the n2–n1 interface: n2 sin uc 5 n1 sin 90° 5 n1
 n1 1 uc 5 sin n2 For light propagation inside the waveguide, the angle must be greater than the critical angle. This requirement can be used to determine the acceptance angle of the waveguide. Knowing that uc . 90° – ut, the first equation above can be written as n1 sin ui , n2 sinð90°  uc Þ Solving for ui determines the acceptance angle. A properly designed parabolic dish can be used to inject light into an optical fiber. Field Guide to Solar Optics

94

Other Nonimaging Optics and Solar Collectors

Freeform Surfaces Freeform surfaces are a relatively new class of optical surfaces that do not necessarily follow axial symmetry. “Freeform” can include smooth or discontinuous surfaces, which allows greater design flexibility for maximizing optical performance. The flexible surfaces are described by polynomials such as Zernike or Q-polynomials. In nonimaging applications, accuracy of the surfaces is not as important. Nonuniform rational B-splines (NURBS), a simultaneous multiple surface (SMS) process, or the Miñano–Benitez design method can be used. The figure shows an example of the SMS process for surface generation, where rays or wavefronts, W1 and W2, from two sources, S1 and S2, are mapped to points on the receiving plane, R1 and R2. The rays can also be mapped in reverse. The freeform surfaces are generated by optimizing the locations of the intersection points, Ti and Bi, where i 5 1, 2, 3, etc. Two strategies are employed in the design of freeform optical surfaces: multiparameter optimization and direct mapping. The advantage of the freeform method is that the surfaces can be formed to minimize aberrations or provide a specific light pattern at the output. Freeform surfaces can be used for a secondary concentrator in a linear Fresnel concentrator system that recuperates rays that miss the receiver. Field Guide to Solar Optics

Other Nonimaging Optics and Solar Collectors

95

Metasurfaces A micro- or nanostructured surface is usually referred to as a metamaterial or metasurface because its interaction with light (namely, ε and m) can be modified from its natural behavior. The parameters ε and m characterize the response to the electric and magnetic fields of light, respectively. Metamaterials initially gained interest at longer wavelengths (e.g., microwaves), where negative index and cloaking have been demonstrated. Eventually, applications moved into the optical regime, where, for example, a planar metasurface can be designed to concentrate the reflected light, as shown in the diagram. Fermat’s principle states that the path taken by a ray of light between two points, a and b, is that of least time and is defined as Z b Z b S5 dwðrÞ 5 nðrÞdr a

a

where w(r) is the local light phase, and n(r) is the local material index of refraction. Snell’s law can be derived from this equation. A gradient of a phase jump at the interface of two media will change the optical path but will still follow Fermat’s principle. The result is a generalized Snell’s law for reflection and refraction:9 Z b S5 dwðrÞ þ wðrs Þ a

It is this additional light path phase shift w(rs) that is localized at the interface between two media (denoted by subscript s) and can be implemented with metasurfaces, which is the key to light manipulation at the interface. With this flexibility, solar collector designs can be improved to increase collection efficiencies.

Field Guide to Solar Optics

96

Other Nonimaging Optics and Solar Collectors

Spectral Splitting Optics A standard glass beamsplitter will split the incident beam in two, but this split is only in power. A spectral splitting device will split a broadband source into two or more spectral bands. The simplest spectral splitting device is a colored glass that reflects and/or absorbs light within a spectral band and transmits the rest of the light. More-sophisticated spectral splitters are based on holographic elements, dispersive optical materials, or metasurfaces. The figure below (left) is a schematic of a spectral splitting device separating a broadband source into several spectral bands. This technique can be used to build hybrid solar systems (e.g., a combination of different PV cells and a solar thermal element).

For example, a solar spectral splitting device can be designed to have cutoffs, such as those shown in the right figure. Spectral band 1 can be coupled to one PV cell type that has a strong response in that spectral band, spectral band 2 can be coupled to a different PV cell type, and spectral band 3 (mostly IR) can be collected for thermal energy. Such a hybrid system can have higher solar-toelectric efficiencies because the spectral bands can be tuned (or matched) to responses that are specific to the PV cell types. Otherwise, if a single PV cell type is used, only the portion of the spectral band to which the solar system has a strong response will be utilized, and the rest of the spectral energy will be wasted, thus lowering the overall system efficiency.

Field Guide to Solar Optics

Special Topics

97

Solar Glint and Glare Solar glint and glare are unwanted reflections off of specular surfaces. Glint is defined as a momentary flash of light. Glare is defined as a continuous source of brightness. Both glint and glare can result from solar fields (CSP and PV). Solar glint and glare can be a nuisance but can also become hazardous if they are intense enough to inhibit the safe operation of vehicles and aircraft. In CSP systems, the array of mirrors gives strong reflections of sunlight. In PV systems, the smooth cover glass reflects sunlight by Fresnel reflections.

Solar glint and glare are observed when the observer is in the vicinity of the sunlight’s reflection direction from the solar field. b specifies the reflected beam angular spread caused by the sun’s angular extent fs and any surface irregularities on the reflecting surface. The magnitude of the reflected light from the reflecting surface depends on the reflectance r as a function of incidence angle ui, which will vary depending on the surface smoothness (or roughness) and the sun irradiance and position. When a human observer views glint or glare, the irradiance on the retina can be defined as10
Er 5

rðui ÞE DNI b2




d2p t f2



where the first group of parameters depends on the source (i.e., target reflectance, source irradiance, and source subtended angle), and the second group comprises the ocular parameters (i.e., pupil diameter, ocular transmittance, and focal length). Depending on the glare intensity, a momentary glare exposure can cause a temporary after-image or debilitating glare, which can be hazardous. Field Guide to Solar Optics

98

Special Topics

Solar Technology Interference In solar technologies, particularly CSP, components such as collectors and receivers can become hot. In power tower systems, for example, the receiver surface temperatures can reach 600 °C and above. This could potentially interfere with any IR or thermal sensors that are used in the vicinity of a solar plant. Guidelines for estimating the thermal emissions from solar receivers are provided here. The Planck blackbody emission from an object with temperature T is given as   c1 W i h Ml 5 2 c2 l5 expðlT Þ  1 m mm where the optical constants c1 and c2 were defined on page 16. The power emitted from the receiver is P l;rec 5 M l Arec where Arec is the surface area of the receiver. If the size of the receiver is unknown, it can be estimated from the treatment developed in Section 4 (pages 45–60) by knowing the plant capacity, thermal storage capacity, and efficiencies. An irradiance limit of 600 kW/m2 is usually assumed for power tower receivers and can be taken as the average irradiance on the receiver surface. Through conservation of energy and neglecting atmospheric attenuation, the power emitted by the receiver will emit into a sphere. The spectral irradiance at the sensor location at a distance r from the receiver can be calculated as   P l;rec Arec W 5 Ml E l;sensor 5 Asphere Asphere m2 mm where Asphere is the surface area of the sphere that has radius r equal to the source to sensor distance. The same process can be used for other solar technologies. Field Guide to Solar Optics

99

Equation Summary Atmospheric attenuation (Schmitz):
0.99321
0.0001176 R þ 1.97 ⋅ 10
8 R2 R ≤ 1 km hatm 5 expð
0.000106 RÞ R . 1 km Cassegrain system parameters (radii of curvature): R1 5

R2 5

2DF F
B

2DB F
B
D

Circle surface equation: ðy
rÞ2 þ x2 5 r2 Collector field efficiency: PN h hfield 5

hðiÞ Nh

i51

where N h is the total number of heliostats in the field. Collector optical efficiency: hopt 5 hsb ⋅ hcos ⋅ hint ⋅ href ⋅ hatm P365 R t1 hðiÞ 5

n51 t hðtÞdt P365 0R t1 n51 t0 dt

P365 R t1 hE ðiÞ 5

n51 t EðtÞhðtÞdt P365 0R t1 n51 t0 EðtÞdt

Concentration limits: Linear concentrators : C max 5

R 1 5 rs sin us

 2 R 1 Point concentrators : C max 5 5 rs sin2 us

Field Guide to Solar Optics

100

Equation Summary Concentration ratio:

Optical concentration : CRo 5

1 Ar

R

E r dAr Ea

Geometrical concentration : CRg 5

5

E r; avg Ea

Aa Ar

Conservation of light: tþrþa51 Cosine efficiency: dsun · nˆ ðdot productÞ hcos 5 cosðui Þ 5 ~ where ~ dsun is the vector pointed at the sun, and nˆ is the surface normal vector of the heliostat. CPC aperture: aen 5

aex sin ui

CPC focal length: f5

aex 2ð1 þ sin ui Þ

CPC length: L5

aex ð1 þ sin ui Þ cos ui 5 ðaen þ aex Þ cot ui sin2 ui

Critical angle:  uc 5 sin
1

n1 n2



CSP power output: P 5 hplant DNI Acoll

Field Guide to Solar Optics

101

Equation Summary Directional emittance: εd 5 1
rd
ts Directional reflectance: R 2p R p=2

L sin ur cos ur dur dfr rd ðu; fÞ 5 R0 R0p=2 r 2p Li sin ui cos ui dui dfi 0 0 Emittance: R∞ r ðlÞuðl; T Þdl εT ðu; fÞ 5 1
0 R ∞d 0 uðl; T Þdl Energy balance (sun source): 4pr2s E s 5 4pR2 E e Fermat’s principle: Z

b

S5

Z

b

dwðrÞ 5

a

nðrÞdr

a

Focal length variation (parabolic concentrator): dz r2 5
2 df 4f

Df 5


4f 2 Dz r2

f/# (for infinite object distance): f =# 5

f D

Generalized Snell’s law: Z S5 a

b

dwðrÞ þ wðrs Þ

Field Guide to Solar Optics

102

Equation Summary Heliostat tracking observations:   Du 5 H~ ε Da  H5

sin u tan a cos u

cos u tan a
sin u

1 0

0 1

u 0 0 a

tan a 0

1  cos a

0

~ ε 5 ðε1 ε2 ε3 ε4 ε5 ε6 ε7 ε8 ÞT , where T represents transpose. Hemispherical emittance: Z p=2 εT ðuÞ sin u cos u du εH 5 2 0

Intensity: I5

dF ½W=ster dv

Intercept factor: g5

irradiance or flux intercepted by the receiver total incident irradiance or flux

Irradiance at sensor location: E l;sensor 5

P l;rec Arec 5 Ml Asphere Asphere

Irradiance profile Hermite (truncated) polynomials:  h 2  2 i Fðx; yÞ 5 2s1x sy exp
12 sxx þ syy 3 hP P     i y x 6 6
i i50 j50 Aij H i sx H j sy =i!j!

Law of reflection: ui 5 ur

Field Guide to Solar Optics

H 0 ðxÞ 5 1 H 1 ðxÞ 5 x H 2 ðxÞ 5 x2
1 H 3 ðxÞ 5 x3
3x :::

103

Equation Summary Mirror surface error: ssurface

error

5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4s2slope error þ s2specularity

Optical error: soptical

total

sbeam

5

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s2surface þ s2canting þ s2tracking þ s2wind þ s2gravity

spread total

5 soptical

total

þ ssunshape

Parabolic equation (general): z5

r2 4f

where r2 5 x2 þ y2 , and f is the focal length. Parabolic surface rim angle:  fr 5 tan
1

  8ðf =DÞ
1 D 5 sin 2r 16ðf =DÞ2
1

Phase shift and phase angle equations: I 1 5 A cosðfÞ þ B I 2 5 A cosðf þ 90°Þ þ B I 3 5 A cosðf þ 180°Þ þ B I 4 5 A cosðf þ 270°Þ þ B tan½fðx; yÞ 5

I4
I2 I1
I3

Photon energy: E 5 hn 5 h

c l Field Guide to Solar Optics

104

Equation Summary Planck’s function (spectral radiant exitance): 2phc2 l ðe
1Þ c1 i Ml 5 h c2 5 l expðlT Þ
1

M ðl; T Þ 5

hc=lkT

5

1st radiation constant: c1 5 2phc2 5 3.742 3 10
16 W ⋅ m2 2nd radiation constant: c2 5 hc=k 5 1.439 3 10
2 m ⋅ K Planck’s function (spectral energy density): uðl; T Þ 5

8phc l ðe 5

hc=lkT


1Þ

Point-focus concentrator concentration area: A 5 pr1 r2 5 r1 5 f tanðus þ sÞ

pf 2 tanðus þ sÞ cosðuÞ r2 5 f

tanðus þ sÞ cosðuÞ

Power: P l;rec 5 M l Arec Projected area: Ap 5 A cosðuÞ Ray reflection:

!

! ˆ nˆ P
F 5 FS
2ðFS ⋅ nÞ where P is the ray intersection point on the target surface, F is the ray intersection on the reflector surface, and S is the source point. Reflectivity: rs 5

n1 cosðu1 Þ
n2 cosðu2 Þ n1 cosðu1 Þ þ n2 cosðu2 Þ

rp 5

n2 cosðu1 Þ
n1 cosðu2 Þ n2 cosðu1 Þ þ n1 cosðu2 Þ

Field Guide to Solar Optics

105

Equation Summary Reflectance: rt 5

r2s þ r2p 2

Retinal irradiance: 

rðui ÞE DNI Er 5 b2



d2p t f2



Shading and blocking efficiencies: hshade 5

AT
As AT

hblock 5

AT
Ab AT

Snell’s law: n1 sin ui 5 n2 sin ut Solar multiple: SM 5

SM 5

aperture of reflective area exact aperture of reflective area

daily solar hours þ thermal storage hours daily solar hours

Solid angle: v5

A ½ster r2

Solar spectral irradiance on Earth: Z E sun 5 0

þ∞


E l;sun dl 5

1000 W=m2 ! global tilt 900 W=m2 ! direct þ circumsolar

Field Guide to Solar Optics

106

Equation Summary Sun angular extent: 2us 5 2 tan

r

sun



d

Sun position:
þf for north of equator Latitude: 5
f for south of equator Hour angle: vs 5 15ðts
12Þ   284 þ n Declination: d 5 23.45 sin 360 365 Solar altitude angle: sin a 5 cos f cos d cos vs þ sin f sin d  cos a sin f
sin d   Solar azimuth: g 5 signðvs Þcos
1  sin a cos f Solar unit vector: Sˆ 5 ½ˆx; yˆ ; zˆ , where xˆ 5 sin g cos a yˆ 5 sin a zˆ 5 cos g cos a Sunshape (Buie): 8 < cosð0.326uÞ fðuÞ 5 cosð0.308uÞ : ek ug

for 0 ≤ u ≤ 4.65 mrad for

where k 5 0.9 lnð13.5xÞx
0.3 , and x 5 FFcsi .

u . 4.65 mrad

g 5 2.2 lnð0.52xÞx
0.43
1,

Transmissivity: ts 5

2n1 cosðu1 Þ n1 cosðu1 Þ þ n2 cosðu2 Þ

tp 5

2n1 cosðu1 Þ n2 cosðu1 Þ þ n1 cosðu2 Þ

Field Guide to Solar Optics

107

Equation Summary Transmittance (total): n2 cosðu2 Þ t2s þ t2p ttotal 5 2 n1 cosðu1 Þ 

Wien’s displacement law: lpeak T 5 2898 mm ⋅ K

Field Guide to Solar Optics

108

Cited References 1. J. A. Duffie and W. A. Beckman, Solar Engineering of Thermal Processes, 4th ed., Wiley, Hoboken (2013). 2. C. Honsberg and S. Bowden, explanation of air mass: https://www.pveducation.org/pvcdrom/propertiesof-sunlight/air-mass 3. E. J. Smith and C. K. Ho, “Field demonstration of an automated heliostat tracking correction method,” Energy Procedia 49, 2201–2210 (2014). 4. D. Buie, A. G. Monger, and C. J. Dey, “Sunshape distributions for terrestrial solar simulations, Solar Energy 74, 113–122 (2003). 5. A. B. Zavoico, “Solar Power Tower Design Basis Document, Revision O,” Sandia National Laboratories, SAND2001-2100 (2001). 6. M. Schmitz, P. Schwarzbözl, R. Buck, and R. Pitz-Paal, “Assessment of the potential improvement due to multiple apertures in central receiver systems with secondary concentrators,” Solar Energy 80, 111–120 (2006). 7. B. L. Kistler, “A User’s Manual for DELSOL3: A Computer Code for Calculating the Optical Performance and Optimal System Design for Solar Thermal Central Receiver Plants,” Sandia National Laboratories, SAND86-8018 (1986). 8. T. J. Wendelin and J. W. Grossman, “Comparison of Three Methods for Optical Characterization of PointFocus Concentrators,” National Renewable Energy Laboratory, NREL/TP-471-7296 (1994). 9. N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011). 10. C. K. Ho, C. M. Ghanbari, and R. B. Diver, “Methodology to assess potential glint and glare hazards from concentrating solar power plants: analytical models and experimental validation,” ASME J. Solar Energy Eng. 133, 031021 (2011). Field Guide to Solar Optics

109

Bibliography of Further Reading Andraka, C. E., S. Sadlon, B. Myer, K. Trapeznikov, and C. Liebner, “Rapid reflective facet characterization using fringe reflection techniques,” J. Solar Energy Eng. 136(1), 011002 (2013). Benítez-Gimenez, P., J. C. Miñano, J. Blen, R. M. Arryo, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004) [doi: 10.1117/1.1752918]. Biggs, F. and C. N. Vittitoe, “HELIOS: A Computational Model for Solar Concentrators,” Sandia National Laboratories, SAND-77-1185C (1977). Blair, N., N. DiOrio, J. Freeman, P. Gilman, S. Janzou, T. Neises, and M. Wagner, “System Advisor Model (SAM) General Description, Version 2017.9.5,” National Renewable Energy Laboratory, NREL/TP-6A20-70414, (2018). Boubault, A., J. Yellowhair, and C. K. Ho, “Design and characterization of a 7.2 kW solar simulator,” J. Solar Energy Eng. 139(3), 031012 (2017). Buie, D., A. G. Monger, and C. J. Dey, “Sunshape distributions for terrestrial solar simulations,” Solar Energy 74(2), 113–122 (2003). Cassarly, W., “Nonimaging Optics: Concentration and Illumination” in Handbook of Optics, 3rd ed., Vol. II, Bass, M., Ed., Chapter 39, McGraw Hill, New York (2010). Chavez, K., E. Sproul, and J. Yellowhair, “Development and analysis of the heliostat focusing and canting enhancement technique for full heliostat alignments,” Proc. ASME ES2012-91039, 237–246 (2012). de Winter, F., Ed., Solar Collectors, Energy Storage, and Materials, MIT Press, Cambridge (1990). Duffie, J. A. and W. A. Beckman, Solar Engineering of Thermal Processes, 4th ed., Wiley, Hoboken (2013).

Field Guide to Solar Optics

110

Bibliography of Further Reading Garcia, P., A. Ferriere, and J.-J. Bezian, “Codes for solar flux calculation dedicated to central receiver system applications: a comparative review,” Solar Eng. 82(3), 189– 197 (2008). Genevet, P., F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4(1), 139–152 (2017). Grant, B., Field Guide to Radiometry, SPIE Press, Bellingham, Washington (2011) [doi: 10.1117/3.903926]. Hecht, E., Optics, 5th ed., Pearson, London (2017). Ho, C. K., C. M. Ghanbari, and R. B. Diver, “Methodology to assess potential glint and glare hazards from concentrating solar power plants: analytical models and experimental validation,” ASME J. Solar Energy Eng. 133, 031021 (2011). Ho, C. K. and S. S. Khalsa, “A photographic flux mapping method for concentrating solar collectors and receivers,” J. Solar Energy Eng. 134(4), 041004 (2012). Imenes, A. G. and D. R. Mills, “Spectral beam splitting technology for increased conversion efficiency in solar concentrating systems: a review,” Solar Eng. Mat. and Solar Cells 84(1–4), 19–69 (2004). Kistler, B. L., “A User’s Manual for DELSOL3: A Computer Code for Calculating the Optical Performance and Optimal System Design for Solar Thermal Central Receiver Plants,” Sandia National Laboratories, SAND86-8018 (1986). Malacara, D., Ed., Optical Shop Testing, 2nd ed., WileyInterscience, New York (1992). McDaniels, D. K., The Sun: Our Future Energy Source, 2nd ed., Krieger Publishing, Malabar, Florida (1984). O’Gallagher, J. J., Nonimaging Optics in Solar Energy, Morgan and Claypool Publishers, San Rafael, California (2008). Field Guide to Solar Optics

111

Bibliography of Further Reading Palz, W. and J. Greif, Commission of the European Communities (CEC), European Solar Radiation Atlas, Springer Science & Business Media (2012). Schmitz, M., P. Schwarzbözl, R. Buck, and R. Pitz-Paal, “Assessment of the potential improvement due to multiple apertures in central receiver systems with secondary concentrators,” Solar Energy 80(1), 111–120 (2006). Smith, E. J. and C. K. Ho, “Field demonstration of an automated heliostat tracking correction method,” Energy Procedia 49, 2201–2210 (2014). Stine, W. B. and R. W. Harrigan, Solar Energy Fundamentals and Design, Wiley-Interscience, New York (1985). Welford, W. T. and R. Winston, The Optics of Nonimaging Concentrators: Light and Solar Energy, Academic Press, New York (1978). Wendelin, T. J. and J. W. Grossman, “Comparison of Three Methods for Optical Characterization of Point-Focus Concentrators,” National Renewable Energy Laboratory, NREL/TP-471-7296 (1994). Wendelin, T. and A. Dobos, “SolTrace: A Ray-Tracing Code for Complex Solar Optical Systems,” National Renewable Energy Laboratory, NREL/TP-5500-59163 (2013). Winston, R., J. C. Minano, and P. Benitez, Nonimaging Optics, Elsevier/Academic Press, Boston (2005). Winter, C. J., R. L. Sizmann, and L. L. Vant-Hull, Eds. Solar Power Plants: Fundamentals, Technology, Systems, Economics, Springer-Verlag, New York (1991). Wood, M., “Etendue,” ‘Out of the Wood’ column in Protocol Winter issue (2012). Ye, J., L. Chen, X. Li, Q. Yuan, and Z. Gao, “Review of optical freeform surface representation technique and its application,” Opt. Eng. 56(11), 110901 (2017) [doi: 10.1117/1.OE.56.11.110901]. Field Guide to Solar Optics

112

Bibliography of Further Reading Yellowhair, J. and C. E. Andraka, “Evaluation of advanced heliostat reflective facets on cost and performance,” Energy Procedia 49, 265–274 (2014). Yellowhair, J. and C. K. Ho, “Heliostat canting and focusing methods: an overview and comparison, Proc. ASME ES2010-90356, 609–615 (2010).

Field Guide to Solar Optics

113

Online Resources National Oceanic and Atmospheric Administration, SURFRAD Aerosol Optical Depth, https://www.esrl.noaa.gov/ gmd/grad/surfrad/aod/ National Renewable Energy Laboratory, SMARTS: Simple Model of the Atmospheric Radiative Transfer of Sunshine, https://www.nrel.gov/grid/solar-resource/smarts.html National Solar Radiation Database, https://nsrdb.nrel.gov/ Solargis: Weather data and software for solar power investments, https://solargis.com World Radiation Data Centre, http://wrdc.mgo.rssi.ru/ U.S. Department of Energy, SunShot Vision Study, February 2012, https://www.energy.gov/sites/prod/files/ SunShot%20Vision%20Study.pdf U.S. Energy Information Administration (EIA), International Energy Outlook 2019 (with Projections to 2050); https://www.eia.gov/outlooks/ieo/

Field Guide to Solar Optics

114

Index absorptance, 33 accelerated aging, 57 acceptance angle, 93 aerosol optical depth, 20 aimpoint, 52 aluminum, 42 angle of incidence, 31 annual optical efficiency, 77 astigmatism, 38 ASTM G-173-03, 17 astronomical unit, 9 atmospheric attenuation, 76 attenuator, 57 autumnal equinox, 10 axial tilt, 9 axis of revolution, 49 back focal distance, 59 beam characterization system, 87 beamsplitter, 96 blackbody source, 16 blocking, 72 Boltzmann’s constant, 16 boresight, 35 Brewster’s angle, 32 camera look-back, 86 canting, 65 canting strategy, 66 Cassegrain, 59 CCD camera, 85 celestial equator, 10 celestial sphere, 10 center of curvature, 35 centroid, 87 circumsolar irradiance, 18 circumsolar radiation, 69 Field Guide to Solar Optics

circumsolar ratio, 69 clear aperture, 35 collector field optimization, 81 compound parabolic concentrator, 91 concentrating photovoltaic, 56 concentration ratio, 43 cone optics, 78 conservation of energy, 28 contaminants, 75 convergence test, 80 cosine efficiency, 73 cosine loss, 73 critical angle, 93 declination, 13 deflectometry, 82 diffuse radiation, 20 diffuse reflection, 63 direct beam radiation, 20 direct normal irradiance, 18 directional emittance, 90 directional reflectance, 89 dish concentrator, 53 dish-engine system, 53 eccentricity, 9 ecliptic, 10 effective sunshape, 78 electromagnetic spectrum, 15 energy balance, 44 energy consumption, 1 energy cost, 81 equatorial plane, 11 etendue, 28

115

Index extraterrestrial solar spectrum, 18 Fermat’s principle, 95 first surface reflector, 40 float glass, 41 flux, 23, 79 flux density, 23 flux gauge, 88 fluxmapper, 87 fluxmapping, 57 focal length/distance, 35 f-number, f/#, 35 frequency, 15 Fresnel equations, 32 fringe reflection, 82 geometric concentration ratio, 43 glint/glare, 97 global coordinate system, 13 global horizontal irradiance, 22 global solar radiation, 20 global tilt, 18 global tilted irradiance, 22 gold, 42 gravity force vector, 70 gravity impacts, 70 ground cover ratio, 81 ground-reflected radiation, 20 Hartmann test, 82 heliostat, 50 hemispherical emittance, 90 Hermite polynomial expansion, 79

hour angle, 13 hour-angle equation, 14 insolation, 43 instantaneous efficiency, 77 intercept factor, 74 irradiance, 24 isotropic source, 29 jitter, 70 Lambertian source, 29 laser, 85 latitude, 13 law of reflection, 30, 80 law of refraction, 30 levelized cost of energy, 81 limb darkening, 69 linear Fresnel concentrator, 46 longitude, 13 low-iron glass, 41 maximum concentration, 44 measurement sensitivity, 86 metal coating, 42 metal halide, 58 metamaterial/surface, 95 mirror facet, 64 mirror soiling, 75 Monte Carlo sampling, 80 nameplate power output, 55 NOAA, 21 nonimaging optics, 40 north celestial pole, 10 NSRDB, 21 ocular parameters, 97 Field Guide to Solar Optics

116

Index off-axis canting, 67 off-axis parabola, 92 on-axis canting, 66 on-sun canting, 67 on-sun tracking, 68 optical efficiency, 55 optical error, 71 optical fiber, 93 orbit, 9 orbital prograde motion, 9 ozone, 19 parabolic concentrator, 49 parabolic mirror, 34 parabolic trough concentrator, 46 phase unwrapping, 84 photovoltaic modules, 45 photovoltaic technologies, 5 Planck’s constant, 16 Planck’s function, 16 plant efficiency, 55 point source, 29 pointing error, 68 power block, 4 power block efficiency, 55 power tower, 55 projected area, 26 pyranometer, 22 pyrheliometer, 22 radiance, 25 radiant energy, 23 radiant exitance, 24 radiant intensity, 25 radiant power, 23 radiation, 15 radiation constants, 16 radiometer, 87 Field Guide to Solar Optics

radius of curvature, 35 ray tracing, 80 receiver efficiency, 55 reflectance, 32 reflection coefficient, 32 reflective film, 40 reflectivity, 32 reflectometer, 89 right ascension, 10 roughness, 97 second law of thermodynamics, 44 second surface reflector, 40 secondary concentrator, 47 secondary reflector, 59 shading, 72 shape error, 62 silver, 42 sinusoidal fringe, 82 sky radiation, 20 slab waveguide, 93 slant range, 79 slope error, 61 Snell’s law, 30 solar altitude angle, 13 solar azimuth angle, 13 solar collector, 39 solar furnace, 57 solar multiple, 60 solar radiation, 20 solar resource data, 21 solar simulator, 58 solar spectrum, 42 solar system, 8 solar thermal power, 4 solar unit vector, 13 solar-weighted reflectance, 42

117

Index solid angle, 25 south celestial pole, 10 spectral irradiance, 19 spherical aberration, 37 spherical mirror, 35 spillage, 74 Stefan–Boltzmann constant, 44 subtended angle, 12 summer solstice, 11 sun angle subtense, 12 sun angular extent, 12 sun diameter, 8 sun half angle, 49 sun mean distance, 8 sun position, 13 sun surface temperature, 8 sun vector, 14 sunshape, 69 surface irregularity, 61 systems performance modeling, 81 target imaging metrology, 86 temporal drill-down, 84 terrestrial solar spectra, 19 theoretical image overlay, 86

thermal energy storage, 60 throughput, 28 total internal reflection, 30 tracking error, 68 transmission coefficient, 33 transmissivity, 33 transmittance, 33 truncated elliptical reflector, 58 truncated parabolic reflector, 92 typical meteorological year, 21 vernal equinox, 10 visibility, 76 visible region, 15 waveguide, 93 wavelength, 15 Wien’s displacement law, 16 wind force vector, 70 wind impacts, 70 winter solstice, 11 xenon, 58

Field Guide to Solar Optics

Julius Yellowhair is an optical engineer and has worked in the area of concentrating solar technologies while a technical staff member at Sandia National Laboratories in Albuquerque, New Mexico between 2010 and 2020. While with the concentrating solar technologies group at Sandia, he worked to develop opticalbased metrology tools, analyze collector cost and performance, and develop novel solar collector concepts. His interests are in optical metrology development, optical system modeling and analysis for performance predictions, and evaluating novel optical concepts. Julius received his B.S. in optical engineering from the University of Arizona. He received his M.S. in electrical engineering with a photonics concentration from the University of New Mexico in 2002. He received his M.S. (2005) and a Ph.D. (2007) in optical sciences from the University of Arizona College of Optical Sciences. His dissertation topic was manufacturing large (.1 m diameter) flat mirrors under the direction of Prof. James Burge (retired).

Solar Optics Julius Yellowhair Optics related to solar technologies is a wide field of study. The topics covered in this Field Guide are frequently encountered in solar engineering and research for energy harvesting, particularly for electricity generation. The book includes: background on energy needs and usage, including where solar technologies fit into the energy mix; properties of the sun and foundations for solar energy collection; optical properties, concepts, and basic components; various optical systems used in solar engineering and solar energy collection; concepts for characterizing optical components/ systems and analysis approaches; and measurement tools commonly used in solar engineering and research. The presentation is slanted toward solar thermal power, or as it is commonly called, concentrating solar power (CSP) technologies.

SPIE Field Guides The aim of each SPIE Field Guide is to distill a major field of optical science or technology into a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena. Written for you—the practicing engineer or scientist— each field guide includes the key definitions, equations, illustrations, application examples, design considerations, methods, and tips that you need in the lab and in the field.

J. Scott Tyo Series Editor P.O. Box 10 Bellingham, WA 98227-0010 ISBN: 9781510636972 SPIE Vol. No.: FG47

www.spie.org/press/fieldguides