Renewable Energy ResourcesRenewable Energy Resources is a numerate and quantitative text covering subjectsof proven technical and economic importance worldwide. Energy supplies fromrenewables (such as solar, thermal, photovoltaic, wind, hydro, biofuels, wave, tidal,ocean and geothermal sources) are essential components of every nations energystrategy, not least because of concerns for the environment and for sustainability.In the years between the first and this second edition, renewable energy has comeof age: it makes good sense, good government and good business.This second edition maintains the books basis on fundamentals, whilst includ-ing experience gained from the rapid growth of renewable energy technologies assecure national resources and for climate change mitigation, more extensively illus-trated with case studies and worked problems. The presentation has been improvedthroughout, along with a new chapter on economics and institutional factors. Eachchapter begins with fundamental theory from a scientific perspective, then considersapplied engineering examples and developments, and includes a set of problems andsolutions and a bibliography of printed and web-based material for further study.Common symbols and cross referencing apply throughout, essential data are tabu-lated in appendices. Sections on social and environmental aspects have been addedto each technology chapter.Renewable Energy Resources supports multi-disciplinary master degrees in sci-ence and engineering, and specialist modules in first degrees. Practising scientistsand engineers who have not had a comprehensive training in renewable energy willfind this book a useful introductory text and a reference book.John Twidell has considerable experience in renewable energy as an academic pro-fessor, a board member of wind and solar professional associations, a journal editorand contractor with the European Commission. As well as holding posts in the UK,he has worked in Sudan and Fiji.Tony Weir is a policy adviser to the Australian government, specialising in theinterface between technology and policy, covering subjects such as energy supplyand demand, climate change and innovation in business. He was formerly SeniorEnergy Officer at the South Pacific Forum Secretariat in Fiji, and has lectured andresearched in physics and policy studies at universities of the UK, Australia and thePacific.Also available from Taylor & FrancisEvaluation of the Built Environment forSustainabilityV. Bentivegna, P.S. Brandon and P. Lombardi Hb: 0-419-21990-0Spon PressGeothermal Energy for Developing CountriesD. Chandrasekharam and J. BundschuhHb: 9058095223Spon PressBuilding Energy Management Systems, 2nd edG. LevermoreHb: 0-419-26140-0Pb: 0-419-22590-0Spon PressCutting the Cost of Cold: Affordable Warmthfor Healthier HomesF. Nicol and J. RudgePb: 0-419-25050-6Spon PressInformation and ordering detailsFor price availability and ordering visit our website www.sponpress.comAlternatively our books are available from all good bookshops.Renewable EnergyResourcesSecond editionJohn Twidell and Tony WeirFirst published 1986by E&FN Spon LtdSecond edition published 2006by Taylor & Francis2 Park Square, Milton Park, Abingdon, Oxon OX14 4RNSimultaneously published in the USA and Canadaby Taylor & Francis270 Madison Ave, New York, NY 10016, USATaylor & Francis is an imprint of the Taylor & Francis Group 1986, 2006 John W. Twidell and Anthony D. WeirAll rights reserved. No part of this book may be reprinted orreproduced or utilised in any form or by any electronic, mechanical, orother means, now known or hereafter invented, including photocopyingand recording, or in any information storage or retrieval system, withoutpermission in writing from the publishers.The publisher makes no representation, express or implied, with regardto the accuracy of the information contained in this book and cannotaccept any legal responsibility or liability for any errors oromissions that may be made.British Library Cataloguing in Publication DataA catalogue record for this book is availablefrom the British LibraryLibrary of Congress Cataloging in Publication DataTwidell, John.Renewable energy resources / John Twidell andAnthony Weir. 2nd ed.p. cm.Includes bibliographical references and index.ISBN 0419253203 (hardback) ISBN 0419253300 (pbk.)1. Renewable energy sources. I. Weir, Anthony D. II. Title.TJ808.T95 2005621.042dc222005015300ISBN10: 0419253203 ISBN13: 9780419253204 HardbackISBN10: 0419253300 ISBN13: 9780419253303 PaperbackThis edition published in the Taylor & Francis e-Library, 2006.To purchase your own copy of this or any of Taylor & Francis or Routledgescollection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.ContentsPreface xiList of symbols xvii1 Principles of renewable energy 11.1 Introduction 11.2 Energy and sustainable development 21.3 Fundamentals 71.4 Scientific principles of renewable energy 121.5 Technical implications 161.6 Social implications 22Problems 24Bibliography 252 Essentials of fluid dynamics 292.1 Introduction 292.2 Conservation of energy: Bernoullis equation 302.3 Conservation of momentum 322.4 Viscosity 332.5 Turbulence 342.6 Friction in pipe flow 352.7 Lift and drag forces: fluid and turbine machinery 39Problems 41Bibliography 443 Heat transfer 453.1 Introduction 453.2 Heat circuit analysis and terminology 463.3 Conduction 49vi Contents3.4 Convection 513.5 Radiative heat transfer 613.6 Properties of transparent materials 733.7 Heat transfer by mass transport 743.8 Multimode transfer and circuit analysis 77Problems 80Bibliography 824 Solar radiation 854.1 Introduction 854.2 Extraterrestrial solar radiation 864.3 Components of radiation 874.4 Geometry of the Earth and Sun 894.5 Geometry of collector and the solar beam 934.6 Effects of the Earths atmosphere 984.7 Measurements of solar radiation 1044.8 Estimation of solar radiation 107Problems 110Bibliography 1125 Solar water heating 1155.1 Introduction 1155.2 Calculation of heat balance: general remarks 1185.3 Uncovered solar water heaters progressive analysis 1195.4 Improved solar water heaters 1235.5 Systems with separate storage 1295.6 Selective surfaces 1345.7 Evacuated collectors 1375.8 Social and environmental aspects 140Problems 141Bibliography 1456 Buildings and other solar thermal applications 1466.1 Introduction 1466.2 Air heaters 1476.3 Energy-efficient buildings 1496.4 Crop driers 1576.5 Space cooling 1616.6 Water desalination 162Contents vii6.7 Solar ponds 1646.8 Solar concentrators 1666.9 Solar thermal electric power systems 1706.10 Social and environmental aspects 173Problems 175Bibliography 1797 Photovoltaic generation 1827.1 Introduction 1827.2 The silicon pn junction 1847.3 Photon absorption at the junction 1937.4 Solar radiation absorption 1977.5 Maximising cell efficiency 2007.6 Solar cell construction 2087.7 Types and adaptations of photovoltaics 2107.8 Photovoltaic circuit properties 2207.9 Applications and systems 2247.10 Social and environmental aspects 229Problems 233Bibliography 2348 Hydro-power 2378.1 Introduction 2378.2 Principles 2408.3 Assessing the resource for small installations 2408.4 An impulse turbine 2448.5 Reaction turbines 2498.6 Hydroelectric systems 2528.7 The hydraulic ram pump 2558.8 Social and environmental aspects 257Problems 258Bibliography 2619 Power from the wind 2639.1 Introduction 2639.2 Turbine types and terms 2689.3 Linear momentum and basic theory 2739.4 Dynamic matching 2839.5 Blade element theory 288viii Contents9.6 Characteristics of the wind 2909.7 Power extraction by a turbine 3059.8 Electricity generation 3079.9 Mechanical power 3169.10 Social and environmental considerations 318Problems 319Bibliography 32210 The photosynthetic process 32410.1 Introduction 32410.2 Trophic level photosynthesis 32610.3 Photosynthesis at the plant level 33010.4 Thermodynamic considerations 33610.5 Photophysics 33810.6 Molecular level photosynthesis 34310.7 Applied photosynthesis 348Problems 349Bibliography 35011 Biomass and biofuels 35111.1 Introduction 35111.2 Biofuel classification 35411.3 Biomass production for energy farming 35711.4 Direct combustion for heat 36511.5 Pyrolysis (destructive distillation) 37011.6 Further thermochemical processes 37411.7 Alcoholic fermentation 37511.8 Anaerobic digestion for biogas 37911.9 Wastes and residues 38711.10 Vegetable oils and biodiesel 38811.11 Social and environmental aspects 389Problems 395Bibliography 39712 Wave power 40012.1 Introduction 40012.2 Wave motion 40212.3 Wave energy and power 40612.4 Wave patterns 41212.5 Devices 418Contents ix12.6 Social and environmental aspects 422Problems 424Bibliography 42713 Tidal power 42913.1 Introduction 42913.2 The cause of tides 43113.3 Enhancement of tides 43813.4 Tidal current/stream power 44213.5 Tidal range power 44313.6 World range power sites 44713.7 Social and environmental aspects of tidal range power 449Problems 450Bibliography 45114 Ocean thermal energy conversion (OTEC) 45314.1 Introduction 45314.2 Principles 45414.3 Heat exchangers 45814.4 Pumping requirements 46414.5 Other practical considerations 46514.6 Environmental impact 468Problems 469Bibliography 46915 Geothermal energy 47115.1 Introduction 47115.2 Geophysics 47215.3 Dry rock and hot aquifer analysis 47515.4 Harnessing Geothermal Resources 48115.5 Social and environmental aspects 483Problems 487Bibliography 48716 Energy systems, storage and transmission 48916.1 The importance of energy storage and distribution 48916.2 Biological storage 49016.3 Chemical storage 49016.4 Heat storage 49516.5 Electrical storage: batteries and accumulators 49916.6 Fuel cells 506x Contents16.7 Mechanical storage 50716.8 Distribution of energy 50916.9 Electrical power 51316.10 Social and environmental aspects 520Problems 521Bibliography 52417 Institutional and economic factors 52617.1 Introduction 52617.2 Socio-political factors 52617.3 Economics 53017.4 Some policy tools 53417.5 Quantifying choice 53617.6 The way ahead 545Problems 550Bibliography 550Appendix A Units and conversions 553Appendix B Data 558Appendix C Some heat transfer formulas 564Solution guide to problems 568Index 581PrefaceOur aimRenewable Energy Resources is a numerate and quantitative text coveringsubjects of proven technical and economic importance worldwide. Energysupply from renewables is an essential component of every nations strat-egy, especially when there is responsibility for the environment and forsustainability.This book considers the timeless principles of renewable energy tech-nologies, yet seeks to demonstrate modern application and case studies.Renewable Energy Resources supports multi-disciplinary master degrees inscience and engineering, and also specialist modules in science and engineer-ing first degrees. Moreover, since many practising scientists and engineerswill not have had a general training in renewable energy, the book has wideruse beyond colleges and universities. Each chapter begins with fundamentaltheory from a physical science perspective, then considers applied exam-ples and developments, and finally concludes with a set of problems andsolutions. The whole book is structured to share common material and torelate aspects together. After each chapter, reading and web-based materialis indicated for further study. Therefore the book is intended both for basicstudy and for application. Throughout the book and in the appendices, weinclude essential and useful reference material.The subjectRenewable energy supplies are of ever increasing environmental and eco-nomic importance in all countries. A wide range of renewable energy tech-nologies are established commercially and recognised as growth industriesby most governments. World agencies, such as the United Nations, havelarge programmes to encourage the technology. In this book we stress thescientific understanding and analysis of renewable energy, since we believethese are distinctive and require specialist attention. The subject is not easy,mainly because of the spread of disciplines involved, which is why we aimto unify the approach within one book.xii PrefaceThis book bridges the gap between descriptive reviews and specialisedengineering treatises on particular aspects. It centres on demonstrating howfundamental physical processes govern renewable energy resources and theirapplication. Although the applications are being updated continually, thefundamental principles remain the same and we are confident that this newedition will continue to provide a useful platform for those advancing thesubject and its industries. We have been encouraged in this approach by theever increasing commercial importance of renewable energy technologies.Why a second edition?In the relatively few years between the first edition, with five reprintedrevisions, and this second edition, renewable energy has come of age; itsuse makes good sense, good government and good business. From being(apart from hydro-power) small-scale curiosities promoted by idealists,renewables have become mainstream technologies, produced and operatedby companies competing in an increasingly open market where consumersand politicians are very conscious of sustainability issues.In recognition of the social, political and institutional factors which con-tinue to drive this change, this new edition includes a new final chapteron institutional and economic factors. The new chapter also discusses anddemonstrates some tools for evaluating the increasingly favourable eco-nomics of renewable energy systems. There is also a substantial new sectionin Chapter 1 showing how renewable energy is a key component of sus-tainable development, an ideal which has become much more explicit sincethe first edition. Each technology chapter now includes a brief concludingsection on its social and environmental impacts.The book maintains the same general format as the first edition, butmany improvements and updates have been made. In particular we wishto relate to the vibrant developments in the individual renewable energytechnologies, and to the related commercial growth. We have improved thepresentation of the fundamentals throughout, in the light of our teachingexperience. Although the book continues to focus on fundamental physi-cal principles, which have not changed, we have updated the technologicalapplications and their relative emphases to reflect market experience. Forelectricity generation, wind-power and photovoltaics have had dramaticgrowth over the last two decades, both in terms of installed capacity andin sophistication of the industries. In all aspects of renewable energy, com-posite materials and microelectronic control have transformed traditionaltechnologies, including hydro-power and the use of biomass.Extra problems have been added at the end of each chapter, with hintsand guidance for all solutions as an appendix. We continue to emphasisesimplified, order-of-magnitude, calculations of the potential outputs of thevarious technologies. Such calculations are especially useful in indicatingPreface xiiithe potential applicability of a technology for a particular site. However weappreciate that specialists increasingly use computer modelling of whole,complex systems; in our view such modelling is essential but only afterinitial calculation as presented here.ReadershipWe expect our readers to have a basic understanding of science and tech-nology, especially of physical science and mathematics. It is not necessaryto read or refer to chapters consecutively, as each aspect of the subject istreated, in the main, as independent of the other aspects. However, somecommon elements, especially heat transfer, will have to be studied seriouslyif the reader is to progress to any depth of understanding in solar energy.The disciplines behind a proper understanding and application of renew-able energy also include environmental science, chemistry and engineering,with social science vital for dissemination. We are aware that readers witha physical science background will usually be unfamiliar with life scienceand agricultural science, but we stress the importance of these subjects withobvious application for biofuels and for developments akin to photosynthe-sis. We ourselves see renewable energy as within human-inclusive ecology,both now and for a sustainable future.OurselvesWe would like our readers to enjoy the subject of renewable energy, aswe do, and to be stimulated to apply the energy sources for the benefitof their societies. Our own interest and commitment has evolved from thework in both hemispheres and in a range of countries. We first taught,and therefore learnt, renewable energy at the University of Strathclyde inGlasgow (JWT) and the University of the South Pacific in Fiji (ADW andJWT). So teaching, together with research and application in Scotland andthe South Pacific, has been a strong influence for this book. Since the firstedition we have made separate careers in universities and in governmentservice, whilst experiencing the remarkable, but predicable, growth in rele-vance of renewable energy. One of us (JWT) became Director of the EnergyStudies Unit, in the Faculty of Engineering at the University of Strathclydein Glasgow, Scotland, and then accepted the Chair in Renewable Energyat the AMSET Centre, De Montfort University, Leicester, England. He iseditor of the academic journal Wind Engineering, has been a Council andBoard member of the British Wind Energy Association and the UK SolarEnergy Society, and has supervised many postgraduates for their disserta-tions. The AMSET Centre is nowa private company, for research, educationand training in renewables; support is given to MSc courses at ReadingUniversity, Oxford University and City University, and there are Europeanxiv PrefaceUnionfunded research programmes. TW was for several years the SeniorEnergy Officer of the South Pacific Forum Secretariat, where he manageda substantial program of renewable energy pilot projects. He then workedfor the Australian Government as an adviser on climate change, and lateron new economy issues.We do not see the world as divided sharply between developed industri-alised countries and developing countries of the Third World. Renewablesare essential for both, and indeed provide one way for the separating con-cepts to become irrelevant. This is meaningful to us personally, since wewish our own energies to be directed for a just and sustainable society,increasingly free of poverty and the threat of cataclysmic war. We sincerelybelieve the development and application of renewable energy technologywill favour these aspirations. Our readers may not share these views, andthis fortunately does not affect the content of the book. One thing they willhave to share, however, is contact with the outdoors. Renewable energy isdrawn from the environment, and practitioners must put on their rubberboots or their sun hat and move from the closed environment of buildingsto the outside. This is no great hardship however; the natural environmentis the joy and fulfilment of renewables.Suggestions for using the book in teachingHow a book is used in teaching depends mainly on how much time isdevoted to its subject. For example, the book originated from short andone-semester courses to senior undergraduates in Physics at the Universityof the South Pacific and the University of Strathclyde, namely EnergyResources and Distribution, Renewable Energy and Physics and Ecology.When completed and with regular revisions, the book has been mostly usedworldwide for MSc degrees in engineering and science, including those onrenewable energy and on energy and the environment. We have alsotaught other lecture and laboratory courses, and have found many of thesubjects and technologies in renewable energy can be incorporated withgreat benefit into conventional teaching.This book deliberately contains more material than could be covered inone specialist course. This enables the instructor and readers to concentrateon those particular energy technologies appropriate in their situation. Toassist in this selection, each chapter starts with a preliminary outline andestimate of each technologys resource and geographical variation, and endswith a discussion of its social and environmental aspects.The chapters are broadly grouped into similar areas. Chapter 1 (Principlesof Renewable Energy) introduces renewable energy supplies in general, andin particular the characteristics that distinguish their application from thatfor fossil or nuclear fuels. Chapter 2 (Fluid Mechanics) and Chapter 3 (HeatTransfer) are background material for later chapters. They contain nothingPreface xvthat a senior student in mechanical engineering will not already know.Chapters 47 deal with various aspects of direct solar energy. Readersinterested in this area are advised to start with the early sections of Chapter 5(Solar Water Heating) or Chapter 7 (Photovoltaics), and review Chapters 3and 4 as required. Chapters 8 (Hydro), 9 (Wind), 12 (Waves) and 13 (Tides)present applications of fluid mechanics. Again the reader is advised to startwith an applications chapter, and review the elements from Chapter 2 asrequired. Chapters 10 and 11 deal with biomass as an energy source andhow the energy is stored and may be used. Chapters 14 (OTEC) and 15(Geothermal) treat sources that are, like those in Chapters 12 (wave) and 13(tidal), important only in fairly limited geographical areas. Chapter 16, likeChapter 1, treats matters of importance to all renewable energy sources,namely the storage and distribution of energy and the integration of energysources into energy systems. Chapter 17, on institutional and economicfactors bearing on renewable energy, recognises that science and engineeringare not the only factors for implementing technologies and developments.Appendices A (units), B (data) and C (heat transfer formulas) are referred toeither implicitly or explicitly throughout the book. We keep to a commonset of symbols throughout, as listed in the front. Bibliographies include bothspecific and general references of conventional publications and of websites;the internet is particularly valuable for seeking applications. Suggestionsfor further reading and problems (mostly numerical in nature) are includedwith most chapters. Answer guidance is provided at the end of the bookfor most of the problems.AcknowledgementsAs authors we bear responsibility for all interpretations, opinions and errorsin this work. However, many have helped us, and we express our gratitudeto them. The first edition acknowledged the many students, colleagues andcontacts that had helped and encouraged us at that stage. For this secondedition, enormously more information and experience has been available,especially from major international and national R&D and from commer-cial experience, with significant information available on the internet. Weacknowledge the help and information we have gained from many suchsources, with specific acknowledgement indicated by conventional referenc-ing and listing in the bibliographies. We welcome communications from ourreaders, especially when they point out mistakes and possible improvement.Much of TWs work on this second edition was done while he wason leave at the International Global Change Institute of the Universityof Waikato, New Zealand, in 2004. He gratefully acknowledges the aca-demic hospitality of Neil Ericksen and colleagues, and the continuing sup-port of the [Australian Government] Department of Industry Tourism andxvi PrefaceResources. JWT is especially grateful for the comments and ideas fromstudents of his courses.And last, but not least, we have to thank a succession of editors at SponPress and Taylor & Francis and our families for their patience and encour-agement. Our children were young at the first edition, but had nearly all lefthome at the second; the third edition will be for their future generations.John Twidell MA DPhil A.D. (Tony) Weir BSc PhDAMSET Centre, Horninghold CanberraLeicestershire, LE16 8DH, UK AustraliaandVisiting Professor in Renewable EnergyUniversity of Reading, UKemail -info.renewable@tandf.co.uk>see -www.amset.com>List of symbolsSymbol Main use Other use or commentCapitalsA Area (m2) Acceptor; ideality factorAM Air-mass-ratioC Thermal capacitance ( J K1) Electrical capacitance (l); constantCP Power coefficientCr Concentration ratioC! Torque coefficientu Distance (m) Diameter (of pipe or blade)L Energy ( J)LF Fermi levelLg Band gap (eV)LK Kinetic energy ( J)EMF Electromotive force (V)l Force (N) Faraday constant (Cmol1)l/|j Radiation exchange factor (| to j)C Solar irradiance (Wm2) Gravitational constant (Nm2kg2);Temperature gradient (Km1);Gibbs energyCb, Cd, Ch Irradiance (beam, diffuse, onhorizontal)H Enthalpy (J) Head (pressure height) of fluid (m);wave crest height (m); insolation( J m2day1); heat of reaction (AH)l Electric current (A) Moment of inertia (kg m2)j Current density (Am2)K Extinction coefficient (m1) Clearness index (KT); constantL Distance, length (m) Diffusion length (m); litre (103m3)N Mass (kg) Molecular weightN Concentration (m3) Hours of daylightN0 Avogadro numberP Power (W)P/ Power per unit length(Wm1)PS Photosystem(Continued)Symbol Main use Other use or comment0 Volume flow rate (m3s1) Thermal resistance (KW1) Radius (m); electrical resistance (H);reduction level; tidal range (m); gasconstant (0);m Thermal resistance (masstransfer)n Thermal resistance(conduction)r Thermal resistance (radiation)v Thermal resistance(convection)RFD Radiant flux density (Wm2)S Surface area (m2) entropySv Surface recombinationvelocity (ms1)STP Standard temperature andpressure7 Temperature (K) Period (s1)U Potential energy ( J) Heat loss coefficient (Wm2K1)\ Volume (m3) Electrical potential (V)W Width (m) Energy density (Jm3)X Characteristic dimension (m) Concentration ratioScript capitals (Non-dimensional numberscharacterising fluid flow) Rayleigh number Grashof number Nusselt number Prandtl number Reynolds number Shape number (of turbine)Lower casea Amplitude (m) Wind interference factor; radius (m)h Wind profile exponent Width (m)c Specific heat capacity( J kg1K1)Speed of light (ms1); phase velocityof wave (ms1); chord length (m);Weibull speed factor (ms1)d Distance (m) Diameter (m); depth (m); zero planedisplacement (wind) (m)e Electron charge (C) Base of natural logarithms (2.718)[ Frequency of cycles(Hz =s1)Pipe friction coefficient; fraction;force per unit length (Nm1)g Acceleration due to gravity(ms2)h Heat transfer coefficient(Wm2K1)Vertical displacement (m); Planckconstant ( Js)Symbol Main use Other use or comment| 1k Thermal conductivity(Wm1K1)Wave vector (=2=\); Boltzmannconstant (=1.381023J K1)| Distance (m)m Mass (kg) Air-mass-ration Number Number of nozzles, of hours ofbright sunshine, of wind-turbineblades; electron concentration(m3)p Pressure (Nm2=Pa) Hole concentration (m3)o Power per unit area (Wm2)r Thermal resistivity of unitarea (R-value = RA)(m2KW1)Radius (m); distance (m)s Angle of slope (degrees)t Time (s) Thickness (m)u Velocity along stream (ms1) Group velocity (ms1)v Velocity (not along stream)(ms1)w Distance (m) Moisture content (dry basis, %);moisture content (wet basis, %)(w/)x Co-ordinate (along stream)(m), Co-ordinate (across stream)(m)z Co-ordinate (vertical) (m)Greek capitals! (gamma) Torque (Nm) Gamma functionA (delta) Increment of . . . (othersymbol)A (lambda) Latent heat ( J kg1)l (sigma) Summation sign4 (phi) Radiant flux (W) Probability function4u Probability distribution ofwind speed ((ms1)1)H (omega) Solid angle (steradian) Phonon frequency (s1); angularvelocity of blade (rads1)Greek lower caseo (alpha) Absorptance Angle of attack (deg)o\ Monochromatic absorptance8 (beta) Angle (deg) Volumetric Expansion coefficient(K1) (gamma) Angle (deg) Blade setting angle (deg)o (delta) Boundary layer thickness (m) Angle of declination (deg): epsilon Emittance Wave spectral width; permittivity;dielectric constant:\ Monochromatic emittance (eta) Efficiency(Continued)Symbol Main use Other use or comment0 (theta) Angle of incidence (deg) Temperature difference (oC) (kappa) Thermal diffusivity (m2s1)\ (lambda) Wavelength (m) Tip speed ratio of wind-turbine (mu) Dynamic viscosity (Nm2s) (nu) Kinematic viscosity (m2s1) (xi) Electrode potential (V) Roughness height (m)= (pi) 3.1416 (rho) Density (kg m3) Reflectance; electrical resistivity(H m)\ Monochromatic reflectanceu (sigma) StefanBoltzmann constantt (tau) Transmittance Relaxation time (s); duration (s);shear stress (Nm2)t\ Monochromatic transmittancem (phi) Radiant flux density (RFD)(Wm2)Wind-blade angle (deg); potentialdifference (\); latitude (deg)m\ Spectral distribution of RFD(Wm3)+ (chi) Absolute humidity (kg m3) (psi) Longitude (deg) Angle (deg)w (omega) Angular frequency (=2=[ )(rads1)Hour angle (deg); solid angle(steradian)SubscriptsB Black body BandD Drag DarkE EarthF ForceG GeneratorL LiftM MoonP PowerR RatedS SunT Tangential Turbinea Ambient Aperture; available (head); aquiferabs Absorbedb Beam Blade; bottom; base; biogasc Collector Coldci Cut-inco Cut-outcov Coverd Diffuse Dopant; digestere Electrical Equilibrium; energyf Fluid Forced; friction; flowg Glass Generation current; band gaph Horizontal HotSymbol Main use Other use or commenti Integer Intrinsicin Incident (incoming)int Internalj Integerm mass transfer Mean (average); methanemax Maximumn conductionnet Heat flow across surfaceo (read as numeral zero)oc Open circuitp Plate Peak; positive charge carriers(holes)r radiation Relative; recombination;room; resonant; rockrad Radiatedrefl Reflectedrms Root mean squares Surface Significant; saturated; Sunsc Short circuitt Tip Totalth Thermaltrans Transmittedu Usefulv convection Vapourw Wind Waterz Zenith\ Monochromatic, e.g. o\0 Distant approach Ambient; extra-terrestrial;dry matter; saturated;ground-level1 Entry to device First2 Exit from device Second3 Output ThirdSuperscriptm or max Maximum Measured perpendicular to directionof propagation (e.g. Cb).(dot) Rate of, e.g. mOther symbolsBold face Vector, e.g. F= Mathematical equality Approximate equality (within afew %) Equality in order of magnitude(within a factor of 210) Mathematical identity (or definition),equivalentChapter 1Principles of renewable energy1.1 IntroductionThe aim of this text is to analyse the full range of renewable energy sup-plies available for modern economies. Such renewables are recognised asvital inputs for sustainability and so encouraging their growth is signifi-cant. Subjects will include power from wind, water, biomass, sunshine andother such continuing sources, including wastes. Although the scale of localapplication ranges from tens to many millions of watts, and the totality isa global resource, four questions are asked for practical application:1 How much energy is available in the immediate environment what isthe resource?2 For what purposes can this energy be used what is the end-use?3 What is the environmental impact of the technology is it sustainable?4 What is the cost of the energy is it cost-effective?The first two are technical questions considered in the central chapters bythe type of renewables technology. The third question relates to broad issuesof planning, social responsibility and sustainable development; these areconsidered in this chapter and in Chapter 17. The environmental impactsof specific renewable energy technologies are summarised in the last sectionof each technology chapter. The fourth question, considered with otherinstitutional factors in the last chapter, may dominate for consumers andusually becomes the major criterion for commercial installations. However,cost-effectiveness depends significantly on:a Appreciating the distinctive scientific principles of renewable energy(Section 1.4).b Making each stage of the energy supply process efficient in terms ofboth minimising losses and maximising economic, social and environ-mental benefits.c Like-for-like comparisons, including externalities, with fossil fuel andnuclear power.2 Principles of renewable energyWhen these conditions have been met, it is possible to calculate the costsand benefits of a particular scheme and compare these with alternatives foran economic and environmental assessment.Failure to understand the distinctive scientific principles for harnessingrenewable energy will almost certainly lead to poor engineering and uneco-nomic operation. Frequently there will be a marked contrast between themethods developed for renewable supplies and those used for the non-renewable fossil fuel and nuclear supplies.1.2 Energy and sustainable development1.2.1 Principles and major issuesSustainable development can be broadly defined as living, producing andconsuming in a manner that meets the needs of the present without com-promising the ability of future generations to meet their own needs. It hasbecome a key guiding principle for policy in the 21st century. Worldwide,politicians, industrialists, environmentalists, economists and theologiansaffirm that the principle must be applied at international, national and locallevel. Actually applying it in practice and in detail is of course much harder!In the international context, the word development refers to improve-ment in quality of life, and, especially, standard of living in the less devel-oped countries of the world. The aim of sustainable development is for theimprovement to be achieved whilst maintaining the ecological processes onwhich life depends. At a local level, progressive businesses aim to report apositive triple bottomline, i.e. a positive contribution to the economic, socialand environmental well-being of the community in which they operate.The concept of sustainable development became widely accepted fol-lowing the seminal report of the World Commission on Environment andDevelopment (1987). The commission was set up by the United Nationsbecause the scale and unevenness of economic development and populationgrowth were, and still are, placing unprecedented pressures on our planetslands, waters and other natural resources. Some of these pressures are severeenough to threaten the very survival of some regional populations and, inthe longer term, to lead to global catastrophes. Changes in lifestyle, espe-cially regarding production and consumption, will eventually be forced onpopulations by ecological and economic pressures. Nevertheless, the eco-nomic and social pain of such changes can be eased by foresight, planningand political (i.e. community) will.Energy resources exemplify these issues. Reliable energy supply is essentialin all economies for lighting, heating, communications, computers, indus-trial equipment, transport, etc. Purchases of energy account for 510% ofgross national product in developed economies. However, in some devel-oping countries, energy imports may have cost over half the value of total1.2 Energy and sustainable development 3exports; such economies are unsustainable and an economic challenge forsustainable development. World energy use increased more than tenfoldover the 20th century, predominantly from fossil fuels (i.e. coal, oil andgas) and with the addition of electricity from nuclear power. In the 21stcentury, further increases in world energy consumption can be expected,much for rising industrialisation and demand in previously less developedcountries, aggravated by gross inefficiencies in all countries. Whatever theenergy source, there is an overriding need for efficient generation and useof energy.Fossil fuels are not being newly formed at any significant rate, and thuspresent stocks are ultimately finite. The location and the amount of suchstocks depend on the latest surveys. Clearly the dominant fossil fuel type bymass is coal, with oil and gas much less. The reserve lifetime of a resourcemay be defined as the known accessible amount divided by the rate ofpresent use. By this definition, the lifetime of oil and gas resources is usuallyonly a few decades; whereas lifetime for coal is a few centuries. Economicspredicts that as the lifetime of a fuel reserve shortens, so the fuel priceincreases; consequently demand for that fuel reduces and previously moreexpensive sources and alternatives enter the market. This process tends tomake the original source last longer than an immediate calculation indi-cates. In practice, many other factors are involved, especially governmentalpolicy and international relations. Nevertheless, the basic geological factremains: fossil fuel reserves are limited and so the present patterns of energyconsumption and growth are not sustainable in the longer term.Moreover, it is the emissions from fossil fuel use (and indeed nuclearpower) that increasingly determine the fundamental limitations. Increasingconcentration of CO2 in the Atmosphere is such an example. Indeed, froman ecological understanding of our Earths long-term history over billions ofyears, carbon was in excess in the Atmosphere originally and needed to besequestered below ground to provide our present oxygen-rich atmosphere.Therefore from arguments of: (i) the finite nature of fossil and nuclear fuelmaterials, (ii) the harm of emissions and (iii) ecological sustainability, itis essential to expand renewable energy supplies and to use energy moreefficiently. Such conclusions are supported in economics if the full externalcosts of both obtaining the fuels and paying for the damage from emissionsare internalised in the price. Such fundamental analyses may conclude thatrenewable energy and the efficient use of energy are cheaper for societythan the traditional use of fossil and nuclear fuels.The detrimental environmental effects of burning the fossil fuels likewiseimply that current patterns of use are unsustainable in the longer term. Inparticular, CO2 emissions from the combustion of fossil fuels have signifi-cantly raised the concentration of CO2 in the Atmosphere. The balance ofscientific opinion is that if this continues, it will enhance the greenhouse4 Principles of renewable energyeffect1and lead to significant climate change within a century or less, whichcould have major adverse impact on food production, water supply andhuman, e.g. through floods and cyclones (IPCC). Recognising that this isa global problem, which no single country can avert on its own, over 150national governments signed the UN Framework Convention on ClimateChange, which set up a framework for concerted action on the issue. Sadly,concrete action is slow, not least because of the reluctance of governmentsin industrialised countries to disturb the lifestyle of their voters. However,potential climate change, and related sustainability issues, is nowestablishedas one of the major drivers of energy policy.In short, renewable energy supplies are much more compatible with sus-tainable development than are fossil and nuclear fuels, in regard to bothresource limitations and environmental impacts (see Table 1.1).Consequently almost all national energy plans include four vital factorsfor improving or maintaining social benefit from energy:1 increased harnessing of renewable supplies2 increased efficiency of supply and end-use3 reduction in pollution4 consideration of lifestyle.1.2.2 A simple numerical modelConsider the following simple model describing the need for commercialand non-commercial energy resources:B =IN (1.1)Here B is the total yearly energy requirement for a population of N people.I is the per capita energy-use averaged over one year, related closely toprovision of food and manufactured goods. The unit of I is energy perunit time, i.e. power. On a world scale, the dominant supply of energy isfrom commercial sources, especially fossil fuels; however, significant use ofnon-commercial energy may occur (e.g. fuel wood, passive solar heating),which is often absent from most official and company statistics. In terms oftotal commercial energy use, the average per capita value of I worldwideis about 2 kW; however, regional average values range widely, with NorthAmerica 9 kW, Europe as a whole 4 kW, and several regions of CentralAfrica as small as 0.1 kW. The inclusion of non-commercial energy increases1 As described in Chapter 4, the presence of CO2 (and certain other gases) in the atmospherekeeps the Earth some 30 degrees warmer than it would otherwise be. By analogy withhorticultural greenhouses, this is called the greenhouse effect.Table1.1ComparisonofrenewableandconventionalenergysystemsRenewableenergysupplies(green)Conventionalenergysupplies(brown)ExamplesWind,solar,biomass,tidalCoal,oil,gas,radioactiveoreSourceNaturallocalenvironmentConcentratedstockNormalstateAcurrentorflowofenergy.AnincomeStaticstoreofenergy.CapitalInitialaverageintensityLowintensity,dispersed:300Wm2Releasedat100kWm2LifetimeofsupplyInfiniteFiniteCostatsourceFreeIncreasinglyexpensive.EquipmentcapitalcostperkWcapacityExpensive,commonlyUS$1000kW1Moderate,perhaps$500kW1withoutemissionscontrol;yetexpensive>US$1000kW1withemissionsreductionVariationandcontrolFluctuating;bestcontrolledbychangeofloadusingpositivefeedforwardcontrolSteady,bestcontrolledbyadjustingsourcewithnegativefeedbackcontrolLocationforuseSite-andsociety-specificGeneralandinvariantuseScaleSmallandmoderatescaleofteneconomic,largescalemaypresentdifficultiesIncreasedscaleoftenimprovessupplycosts,largescalefrequentlyfavouredSkillsInterdisciplinaryandvaried.Widerangeofskills.ImportanceofbioscienceandagricultureStronglinkswithelectricalandmechanicalengineering.NarrowrangeofpersonalskillsContextBiastorural,decentralisedindustryBiastourban,centralisedindustryDependenceSelf-sufficientandislandedsystemssupportedSystemsdependentonoutsideinputsSafetyLocalhazardspossibleinoperation:usuallysafewhenoutofactionMaybeshieldedandenclosedtolessengreatpotentialdangers;mostdangerouswhenfaultyPollutionandenvironmentaldamageUsuallylittleenvironmentalharm,especiallyatmoderatescaleEnvironmentalpollutionintrinsicandcommon,especiallyofairandwaterHazardsfromexcessbiomassburningSoilerosionfromexcessivebiofueluseLargehydroreservoirsdisruptiveCompatiblewithnaturalecologyPermanentdamagecommonfromminingandradioactiveelementsenteringwatertable.DeforestationandecologicalsterilisationfromexcessiveairpollutionClimatechangeemissionsAesthetics,visualimpactLocalperturbationsmaybeunpopular,butusuallyacceptableiflocalneedperceivedUsuallyutilitarian,withcentralisationandeconomyoflargescale6 Principles of renewable energyall these figures and has the major proportional benefit in countries wherethe value of I is small.Standard of living relates in a complex and an ill-defined way to I. Thusper capita gross national product S (a crude measure of standard of living)may be related to I by:S =f I (1.2)Here f is a complex and non-linear coefficient that is itself a function ofmany factors. It may be considered an efficiency for transforming energyinto wealth and, by traditional economics, is expected to be as large aspossible. However, S does not increase uniformly as I increases. IndeedS may even decrease for large E (e.g. because of pollution or technicalinefficiency). Obviously unnecessary waste of energy leads to a lower valueof f than would otherwise be possible. Substituting for I in (1.1), thenational requirement for energy becomes:B = (SNif (1.3)soABB = ASS ANN Aff (1.4)Now consider substituting global values for the parameters in (1.4). In50 years the world population N increased from 2500 million in 1950 toover 6000 million in 2000. It is now increasing at approximately 23% peryear so as to double every 2030 years. Tragically, high infant mortality andlow life expectancy tend to hide the intrinsic pressures of population growthin many countries. Conventional economists seek exponential growth of Sat 25% per year. Thus in (1.4), at constant efficiency f , the growth oftotal world energy supply is effectively the sum of population and economicgrowth, i.e. 48% per year. Without new supplies such growth cannotbe maintained. Yet at the same time as more energy is required, fossiland nuclear fuels are being depleted and debilitating pollution and climatechange increase; so an obvious conclusion to overcome such constraints isto increase renewable energy supplies. Moreover, from (1.3) and (1.4), it ismost beneficial to increase the parameter f , i.e. to have a positive value off . Consequently there is a growth rate in energy efficiency, so that S canincrease, while B decreases.1.2.3 Global resourcesConsidering these aims, and with the most energy-efficient modern equip-ment, buildings and transportation, a justifiable target for energy use in a1.3 Fundamentals 7modern society with an appropriate lifestyle is I =2kW per person. Sucha target is consistent with an energy policy of contract and converge forglobal equity, since worldwide energy supply would total approximatelythe present global average usage, but would be consumed for a far higherstandard of living. Is this possible, even in principle, from renewable energy?Each square metre of the earths habitable surface is crossed by, or accessibleto, an average energy flux from all renewable sources of about 500 W (seeProblem 1.1). This includes solar, wind or other renewable energy formsin an overall estimate. If this flux is harnessed at just 4% efficiency, 2 kWof power can be drawn from an area of 10m10m, assuming suitablemethods. Suburban areas of residential towns have population densitiesof about 500 people per square kilometre. At 2 kW per person, the totalenergy demand of 1000kWkm2could be obtained in principle by usingjust 5% of the local land area for energy production. Thus renewable energysupplies can provide a satisfactory standard of living, but only if the tech-nical methods and institutional frameworks exist to extract, use and storethe energy in an appropriate form at realistic costs. This book considersboth the technical background of a great variety of possible methods anda summary of the institutional factors involved. Implementation is theneveryones responsibility.1.3 Fundamentals1.3.1 DefinitionsFor all practical purposes energy supplies can be divided into two classes:1 Renewable energy. Energy obtained from natural and persistent flowsof energy occurring in the immediate environment. An obvious exampleis solar (sunshine) energy, where repetitive refers to the 24-hour majorperiod. Note that the energy is already passing through the environmentas a current or flow, irrespective of there being a device to interceptand harness this power. Such energy may also be called Green Energyor Sustainable Energy.2 Non-renewable energy. Energy obtained from static stores of energythat remain underground unless released by human interaction. Exam-ples are nuclear fuels and fossil fuels of coal, oil and natural gas. Notethat the energy is initially an isolated energy potential, and externalaction is required to initiate the supply of energy for practical pur-poses. To avoid using the ungainly word non-renewable, such energysupplies are called finite supplies or Brown Energy.These two definitions are portrayed in Figure 1.1. Table 1.1 provides acomparison of renewable and conventional energy systems.8 Principles of renewable energyNatural Environment:green Mined resource: brownCurrent source of continuousenergy flowACDEFBDeviceUseDEFDeviceUseEnvironment Sink Environment SinkFinite source ofenergy potentialRenewable energy Finite energyFigure 1.1 Contrast between renewable (green) and finite (brown) energy supplies.Environmental energy flow ABC, harnessed energy flow DEF.1.3.2 Energy sourcesThere are five ultimate primary sources of useful energy:1 The Sun.2 The motion and gravitational potential of the Sun, Moon and Earth.3 Geothermal energy from cooling, chemical reactions and radioactivedecay in the Earth.4 Human-induced nuclear reactions.5 Chemical reactions from mineral sources.Renewable energy derives continuously fromsources 1, 2 and 3 (aquifers).Finite energy derives from sources 1 (fossil fuels), 3 (hot rocks), 4 and 5.The sources of most significance for global energy supplies are 1 and 4. Thefifth category is relatively minor, but useful for primary batteries, e.g. drycells.1.3.3 Environmental energyThe flows of energy passing continuously as renewable energy through theEarth are shown in Figure 1.2. For instance, total solar flux absorbed atsea level is about 1.2 1017W. Thus the solar flux reaching the Earthssurface is 20MW per person; 20 MW is the power of ten very large1.3 Fundamentals 9Reflectedto space50 000SolarradiationFromSunFromEarthFromplanetarymotion120 000Absorbed onEarth40 00080 000SensibleheatingLatent heatand potentialenergy300Kinetic energyPhotonprocessesGeothermal30100HeatGravitation,orbital motion Tidal motion3InfraredradiationSolar water heatersSolar buildingsSolar dryersOcean thermal energyHydropowerWind and wave turbinesBiomass and biofuelsPhotovoltaicsGeothermal heatGeothermal powerTidal range powerTidal current powerFigure 1.2 Natural energy currents on earth, showing renewable energy system. Notethe great range of energy flux (1: 105) and the dominance of solar radiationand heat. Units terawatts (1012W).diesel electric generators, enough to supply all the energy needs of a townof about 50 000 people. The maximum solar flux density (irradiance)perpendicular to the solar beam is about 1kWm2; a very useful andeasy number to remember. In general terms, a human being is able tointercept such an energy flux without harm, but any increase begins tocause stress and difficulty. Interestingly, power flux densities of 1kWm2begin to cause physical difficulty to an adult in wind, water currents orwaves.However, the global data of Figure 1.2 are of little value for practicalengineering applications, since particular sites can have remarkably differentenvironments and possibilities for harnessing renewable energy. Obviouslyflat regions, such as Denmark, have little opportunity for hydro-power butmay have wind power. Yet neighbouring regions, for example Norway, mayhave vast hydro potential. Tropical rain forests may have biomass energysources, but deserts at the same latitude have none (moreover, forests mustnot be destroyed so making more deserts). Thus practical renewable energysystems have to be matched to particular local environmental energy flowsoccurring in a particular region.10 Principles of renewable energy1.3.4 Primary supply to end-useAll energy systems can be visualised as a series of pipes or circuits throughwhich the energy currents are channelled and transformed to become use-ful in domestic, industrial and agricultural circumstances. Figure 1.3(a)is a Sankey diagram of energy supply, which shows the energy flowsthrough a national energy system (sometimes called a spaghetti diagrambecause of its appearance). Sections across such a diagram can be drawnas pie charts showing primary energy supply and energy supply to end-useThermalelectricitygenerationRefiningCrude oilPRIMARYENERGYSUPPLIESCoalFossil gasBiomassHydroOil productsNon-energy use ENERGYEND-USETransportIndustryResidentialand otherDistrictheatingWaste heatElectricity(a)300 PJFigure 1.3 Energy flow diagrams for Austria in 2000, with a population of 8.1 million.(a) Sankey (spaghetti) diagram, with flows involving thermal electricity showndashed. (b)(c) Pie diagrams. The contribution of hydropower and biomass(wood and waste) is greater than in most industrialised countries, as is theuse of heat produced from thermal generation of electricity (combined heatand power). Energy use for transport is substantial and very dependent on(imported) oil and oil products, therefore the Austrian government encouragesincreased use of biofuels. Austrias energy use has grown by over 50% since1970, although the population has grown by less than 10%, indicating the needfor greater efficiency of energy use. [Data source: simplified from InternationalEnergy Agency, Energy Balances of OECD countries 20002001.]1.3 Fundamentals 11(b)Energy End-Use(total: 970 PJ)Industry30%Transport30%Residential28%Other12%(c)Figure 1.3 (Continued).(Figure 1.3(b)). Note how the total energy end-use is less than the pri-mary supply because of losses in the transformation processes, notably thegeneration of electricity from fossil fuels.1.3.5 Energy planning1 Complete energy systems must be analysed, and supply should not beconsidered separately from end-use. Unfortunately precise needs forenergy are too frequently forgotten, and supplies are not well matchedto end-use. Energy losses and uneconomic operation therefore fre-quently result. For instance, if a dominant domestic energy require-ment is heat for warmth and hot water, it is irresponsible to generategrid quality electricity from a fuel, waste the majority of the energyas thermal emission from the boiler and turbine, distribute the elec-tricity in lossy cables and then dissipate this electricity as heat. Sadly12 Principles of renewable energysuch inefficiency and disregard for resources often occurs. Heatingwould be more efficient and cost-effective from direct heat productionwith local distribution. Even better is to combine electricity genera-tion with the heat production using CHP combined heat and power(electricity).2 System efficiency calculations can be most revealing and can pinpointunnecessary losses. Here we define efficiency as the ratio of the usefulenergy output from a process to the total energy input to that pro-cess. Consider electric lighting produced from conventional thermallygenerated electricity and lamps. Successive energy efficiencies are: elec-tricity generation 30%, distribution 90% and incandescent lighting(energy in visible radiation, usually with a light-shade) 45%. The totalefficiency is 11.5%. Contrast this with cogeneration of useful heatand electricity (efficiency 85%), distribution (90%i and lighting inmodern low consumption compact fluorescent lamps (CFL) (22%i.The total efficiency is now 1418%; a more than tenfold improvement!The total life cycle cost of the more efficient system will be much lessthan for the conventional, despite higher per unit capital costs, because(i) less generating capacity and fuel are needed, (ii) less per unit emissioncosts are charged, and (iii) equipment (especially lamps) lasts longer(see Problems 1.2 and 1.3).3 Energy management is always important to improve overall efficiencyand reduce economic losses. No energy supply is free, and renewablesupplies are usually more expensive in practice than might be assumed.Thus there is no excuse for wasting energy of any form unnecessarily.Efficiency with finite fuels reduces pollution; efficiency with renewablesreduces capital costs.1.4 Scientific principles of renewable energyThe definitions of renewable (green) and finite (brown) energy supplies(Section 1.3.1) indicate the fundamental differences between the two formsof supply. As a consequence the efficient use of renewable energy requiresthe correct application of certain principles.1.4.1 Energy currentsIt is essential that a sufficient renewable current is already present in thelocal environment. It is not good practice to try to create this energy currentespecially for a particular system. Renewable energy was once ridiculedby calculating the number of pigs required to produce dung for sufficientmethane generation to power a whole city. It is obvious, however, thatbiogas (methane) production should only be contemplated as a by-productof an animal industry already established, and not vice versa. Likewise1.4 Scientific principles of renewable energy 13for a biomass energy station, the biomass resource must exist locally toavoid large inefficiencies in transportation. The practical implication of thisprinciple is that the local environment has to be monitored and analysedover a long period to establish precisely what energy flows are present. InFigure 1.1 the energy current ABC must be assessed before the divertedflow through DEF is established.1.4.2 Dynamic characteristicsEnd-use requirements for energy vary with time. For example, electricitydemand on a power network often peaks in the morning and evening,and reaches a minimum through the night. If power is provided from afinite source, such as oil, the input can be adjusted in response to demand.Unused energy is not wasted, but remains with the source fuel. However,with renewable energy systems, not only does end-use vary uncontrollablywith time but so too does the natural supply in the environment. Thus arenewable energy device must be matched dynamically at both D and Eof Figure 1.1; the characteristics will probably be quite different at bothinterfaces. Examples of these dynamic effects will appear in most of thefollowing chapters.The major periodic variations of renewable sources are listed in Table 1.2,but precise dynamic behaviour may well be greatly affected by irregularities.Systems range from the very variable (e.g. wind power) to the accuratelypredictable (e.g. tidal power). Solar energy may be very predicable in someregions (e.g. Khartoum) but somewhat random in others (e.g. Glasgow).1.4.3 Quality of supplyThe quality of an energy supply or store is often discussed, but usuallyremains undefined. We define quality as the proportion of an energy sourcethat can be converted to mechanical work. Thus electricity has high qualitybecause when consumed in an electric motor >95% of the input energymay be converted to mechanical work, say to lift a weight; the heat lossesare correspondingly small, -5%. The quality of nuclear, fossil or biomassfuel in a single stage thermal power station is moderately low, becauseonly about 33% of the calorific value of the fuel can be made to appearas mechanical work and about 67% is lost as heat to the environment.If the fuel is used in a combined cycle power station (e.g. methane gasturbine stage followed by steam turbine), then the quality is increased to50%. It is possible to analyse such factors in terms of the thermody-namic variable energy, defined here as the theoretical maximum amount ofwork obtainable, at a particular environmental temperature, from an energysource.Table1.2IntensityandperiodicalpropertiesofrenewablesourcesSystemMajorperiodsMajorvariablesPowerrelationshipCommentTextreference(equation)Directsunshine24h,1ySolarbeamirradianceC b(Wm2);PC bcos0zDaytimeonly(4.2)Angleofbeamfromvertical0zPmax=1kWm2Diffusesunshine24h,1yCloudcover,perhapsairpollutionP 71).In free convection (sometimes called natural convection) the movementis caused by the heat flow itself. Consider the fluid in contact with thehot surfaces of Figure 3.3; for example, water against the inside surfacesof a boiler or a solar collector. Initially the fluid absorbs energy by con-duction from the hot surface, and so the fluid density decreases by vol-ume expansion. The heated portion then rises through the unheated fluid,thereby transporting heat physically upwards, but down the temperaturegradient.In forced convection the fluid is moved across a surface by an externalagency such as a pump (e.g. water in a solar collector to a storage tankbelow) or wind (e.g. heat loss from the outside surfaces of a solar collector).The movement occurs independently of the heat transfer (i.e. is not a func-tion of the local temperature gradients). Obviously convection is usuallypartly forced and partly free, yet usually one process dominates.3.4.2 Nusselt number The analysis of convection proceeds from a gross simplification of theprocesses. We imagine the fluid near the surface to be stationary. We thenconsider the heat flowing across an idealised boundary layer of stationaryfluid of thickness o and cross-sectional area A(Figure 3.4). The temperaturesFigure 3.4 Idealised thermal boundary layer in free convection. (a) Hot surfacehorizontal. (b) Hot surface vertical.3.4 Convection 53across the fictitious boundary are Tf, the fluid temperature away from thesurface, and Ts, the surface temperature. This being so, the heat transfer byconduction across unit area of the stationary fluid iso = IA = l(TsTfio (3.16)where l is the thermal conductivity of the fluid.As described here, o is fictitious and cannot be measured. We can, how-ever, measure X, a characteristic dimension specified rather arbitrarily foreach particular surface (see Figure 3.4 and Appendix C).From (3.13),o = IA = l(TsTfio = Xol(TsTfiX =l(TsTfiX (3.17) is the Nusselt number for the particular circumstance. It is a dimension-less scaling factor, useful for all bodies of the same shape in equivalentconditions of fluid flow. Tables of values of are available for spec-ified conditions, with the appropriate characteristic dimension identified(Appendix C).From Section 3.2 it follows that:thermal resistance of convection Bv= XlA (3.18)convective thermal resistivity of unit area rv=BvA = Xl (3.19)convective heat transfer coefficient lv= 1rv= lX (3.20)The amount of heat transferred by convection will depend on three factors:1 The properties of the fluid2 The speed of the fluid flow and its characteristics, i.e. laminar or tur-bulent3 The shape and size of the surface.The Nusselt number is a dimensionless measure of the heat transfer.Therefore it can depend only on dimensionless measures of the three factorslisted. In choosing these measures, it is convenient to separate the cases offorced and free convection.54 Heat transfer3.4.3 Forced convectionFor a given shape of surface, a non-dimensional measure of the speed ofthe flow is the Reynolds number:= uX: (3.21)We saw in Section 2.5 that determines the pattern of the flow, andin particular whether it is laminar or turbulent. In flow over a flat plate(Figure 3.5), turbulence occurs when 3105, with subsequent increasein the heat transfer because of the perpendicular motions involved.The flowof heat into or froma fluid depends on (a) the thermal diffusivity of the fluid, and (b) the kinematic viscosity :, (2.10), which may beconsidered the diffusivity of momentum, since it affects the Reynoldsnumber and thus the character of the flow. These are the only two propertiesof the fluid that influence the Nusselt number in forced convection, sincethe separate effects of l, j, and c are combined in (see (3.15)).A non-dimensional measure of the properties of the fluid is the Prandtlnumber: =: (3.22)If is large, changes in momentum diffuse more quickly through the fluidthan do changes in temperature. Many common fluids have 1, e.g.7.0 for water at 20oC. For air at environmental temperatures =0.7 (seeAppendix B).Thus, for each shape of surface, the heat transfer by forced convectioncan be expressed in the form =(, i (3.23)That is, for each shape, the Nusselt number is a function only of theReynolds number and the Prandtl number. These relationships may beFigure 3.5 Fluid flow over a hot plate. General view of pathlines, showing regions:(A) well away from the surface; (B) laminar flow near the leading edge;(C) turbulent flow in the downstream region.3.4 Convection 55expressed with other closely related dimensionless parameters, e.g. the Stan-ton number (i and the Pclet number , but neither are used inthis book.Numerical values of are determined from experiment, with theirmethod of use explained in Appendix C. An example is given inSection 3.4.5. It is important to realise that these formulas are mostly onlyaccurate to 10%, partly because they are approximations to the exper-imental conditions, and partly because the experimental data themselvesusually contain both random and systematic errors.3.4.4 Free convectionIn free convection, often called natural convection, the fluid speed dependson the heat transfer (whereas in forced convection, heat transfer depends onthe fluid speed). Analysis still depends on determining the Nusselt number,but as a function of other dimensionless numbers. We replace (3.23) by =(, i (3.24)where the Rayleigh number = g8X3AT: (3.25)These formulae are sometimes expressed in terms of the Grashof number =

=g8X3AT2 (3.26)Both the Grashof and Rayleigh numbers are dimensionless measures of thedriving temperature difference AT; g is the acceleration due to gravity; 8 isthe coefficient of thermal expansion and the other symbols are as before.In this book we prefer to use the Rayleigh number because it moredirectly relates to the physical processes indicated in Figure 3.6. Heated fluidis forced upwards by a buoyancy force proportional to g8AT, and retardedby a viscous force proportional to :. However, the excess temperature (andtherefore the buoyancy) is lost at a rate proportional to thermal diffusivity .Therefore the vigour of convection increases with g8AT(:i, i.e. with .The factor X3is inserted to make this ratio dimensionless. It is found exper-imentally that free convection is non-existent if Rayleigh number -103and is turbulent if 105.This argument shows that the Nusselt number in free convection dependsmainly on the Rayleigh number. The dimensional argument used inSection 3.4.3 suggests that may also depend on the Prandtl number, asindicated in (3.24). Formulas to calculate these Nusselt numbers are given56 Heat transferFigure 3.6 Schematic diagram of a blob of fluid moving upward in free convection.It is subject to an upward buoyancy force, a retarding viscous force, anda sideways temperature loss.in Appendix C for various geometries. These cannot be expected to givebetter than 10% accuracy. Note that the Nusselt number (and thus thethermal resistance) in free convection depends on AT, through the depen-dence on . This is because a larger temperature difference drives a strongerflow, which transfers heat more efficiently. By contrast, in forced convec-tion, the Nusselt number and thermal resistance are virtually independentof AT.3.4.5 Calculation of convective heat transferBecause of the complexity of fluid flow, there is no fundamental theory forcalculating convective heat transfer. Instead we use experiments on geomet-rically similar objects. By expressing the results in non-dimensional form,they can be applied to different sizes of the objects and for different fluids.For this, working formulas are given in the tables of Appendix C for shapescommon in renewable energy applications; more extensive collections aregiven in textbooks of heat transfer.All this seems very confusing. However, when used in earnest for cal-culating convection, confusion lessens by using the following systematicprocedure:1 Open the tables of heat transfer processes and equations (e.g.Appendix C)2 Draw a diagram of the heated object.3 Section the diagram into standard geometries (i.e. parts correspondingto the illustrations in the tables)3.4 Convection 574 For each such section:a Identify the characteristic dimension (X)b As required in the tables, calculate and/or for each section ofthe object.c Choose the formula for from tables appropriate to that range of or . (The different formulas usually correspond to laminar orturbulent flow.)d Calculate the Nusselt number and hence the heat flow across thesection I =oA.5 Add the heat flows from each section to obtain the total heat flow fromthe object.Example 3.2 Free convection between parallel platesTwo flat plates each 1.0m1.0m are separated by 3.0 cm of air. Thelower is at 70oC and the upper at 45oC. The edges are sealed togetherby thermal insulating material acting also as walls to prevent air move-ment beyond the plates. Calculate the convective thermal resistivity ofunit area, r, and the heat flux, I, between the top and the bottom plate.SolutionFigure 3.7 corresponds to the standard geometry given for (C.7) inAppendix C. Since the edges are sealed, no outside air can enterbetween the plates and only free convection occurs. Using (3.25) andTable B.1 in Appendix B, for mean temperature 57oC(=330Ki= g8X3AT: =

g8:

X3AT

= (9.8ms1i(1330Ki(2.6105m2s1i(1.8105m2s1i(0.03mi3(25Ki=4.1104Using (C.7) a reasonable value for can be obtained, although isslightly less than 105: =0.0620.33=2.06(From (3.17), this implies the boundary layer is about half way acrossthe gap.)From (3.19)rv= Xl = 0.03m(2.06i(0.028Wm1K1i =0.52KW1m258 Heat transferFigure 3.7 Diagrams for worked examples on convection. (a) Parallel plates,as in Example 3.2 (b) Cooking pot with lid, as in Example 3.3.From (3.17),I = AATr = (1m2i(25Ki0.52KW1m2 =48WNote the following:1 The factor (g8:i =(X3ATi is tabulated in Appendix B for airand water.2 The fluid properties are evaluated at the mean temperature (57oCin this case).3 It is essential to use consistent units (e.g. SI) in evaluating dimen-sionless parameters like .Example 3.3 Convective cooling of a cooking potA metal cooking pot with a shiny outside surface, of the dimensionsshown in Figure 3.7(b), is filled with food and water and placed on acooking stove. What is the minimum energy required to maintain it at3.4 Convection 59boiling temperature for one hour, (1) if it is sheltered from the wind(2) if it is exposed to a breeze of 3.0ms1?SolutionWe assume that the lid is tight, so that there is no heat loss by evapora-tion. We also neglect heat loss by radiation, as justified in Problem 3.4.Since the conductive resistance of the pot wall is negligible, the prob-lem is then reduced to calculating the convective heat loss from thetop and sides of a cylinder with a surface temperature of 100oC. Weshall consider the ambient (air and surrounding walls) temperature tobe 20oC. Therefore heat transfer properties of the air are evaluated atthe mean temperature T =60oC.1 Free convection alone For the top (using Table B.1 and (3.25)), =(5.8107m3K1i(0.22mi3(80Ki =4.9107and (from (C.2)) =0.140.33=48.4andItop=AlATX=(r4i(0.22mi2(0.027Wm1K1i(48.4i (80Ki0.22m =18WFor the sides, X =0.11m:

side=top(0.11m0.22mi3=6.1106and (from (C.5)) =560.25=27.8soIside=r(0.22mi(0.11mi(0.027Wm1K1i(27.8i (80Ki0.11m =41WHenceIfree=ItopIside=59W=(0.059kWi(3600s h1i 0.2MJ h1.60 Heat transfer2 Forced plus free convection Here we calculate the forced con-vective power losses separately, and add them to those alreadycalculated for free convection, in order to obtain an estimate ofthe total convective heat loss Itotal.For the top,= (3ms1i(0.22mi1.9105m2s1 =3.5104which suggests the use of (C.8): =0.664 0.5

0.33=110SoItop=Al(ATX i =42WFor the sides, as for the top,=3.5104which suggests the use of (C.11): =0.26 0.6

0.3=124andIside=r(0.22mi(0.11mi(0.027Wm1K1i(124i (80Ki0.22m=93WHenceIforced=9342 =135WThe total estimate isItotal=IforcedIfree=194W=(194Wi(3600s h1i 0.7MJ h1i.e. about 3 times the energy per unit time of the shelteredcooking.3.5 Radiative heat transfer 61The overall accuracy of calculations like those in Example 3.3 may be nobetter than 50%, although the individual formulas are better than this.This is because forced and free convection may both be significant, buttheir separate contributions do not simply add because the flow induced byfree convection may oppose or reinforce the pre-existing flow. Similarly theflows around the separate sections of the object interact with each other.In Example 3.3 there is an additional confusion about whether the flow islaminar or turbulent. For example, on the top of the pot > 105, suggestingturbulence, but using the external flow speed to calculate (as earlier)gives - 105, suggesting laminar flow. In practice such a flow acrossthe top would be turbulent, since it is difficult to smooth out streamlineswhich have become tangled by turbulence. The only safe way to accuratelyevaluate a convective heat transfer, allowing for all these interactions, isby experiment! Some formulas, such as (C.15), based on such specialisedexperiments are available, but have a correspondingly narrow range ofapplicability. Nevertheless, calculation of convection is essential to giveorder-of-magnitude understanding of the processes involved.3.5 Radiative heat transfer3.5.1 IntroductionSurfaces emit energy by electromagnetic radiation according to fundamentallaws of physics. Absorption of radiation is a closely related process. Sadly,the literature and terminology concerning radiative heat transfer are confus-ing; symbols and names for the same quantities vary, and the same symboland name may be given for totally different quantities. Here, we have triedto follow the recommendations of the International Solar Energy Society(ISES), whilst maintaining unique symbols throughout the whole book, ason the opening list. In this chapter we consider radiative heat transfer ingeneral. In Chapter 4 we shall consider solar radiation in particular, and inChapters 5 and 6 heating devices using solar energy.3.5.2 Radiant flux density (RFD)Radiation is energy transported by electromagnetic propagation throughspace or transparent media. Its properties relate to its wavelength \. Thenamed regions of the spectrum are shown in Figure 3.8. The flux of energyper unit area is the radiant flux density (abbreviation RFD, unit Wm2,symbol m). The variation of RFD with wavelength is described by thespectral RFD (symbol m\, unit (Wm2i m1or more usually Wm2m1),which is simply the derivative dmd\. Thus m\A\ gives the power per unitarea in a (narrow) wavelength range A\, and integration of m\ with respect62 Heat transferFigure 3.8 Some of the named portions of the electromagnetic spectrum. (The spectrumextends both longwards and shortwards from that shown.)Figure 3.9 Measurements of various radiation parameters using a small totally absorbingplane. (a) Absorbs all directions. (b) Absorbs from hemisphere above one sideonly. (c) Absorbs from one direction only. (d) Absorbs from one solid angleonly.to wavelength gives the total RFD, i.e. m=

m\d\. Radiation coming ontoa surface is usually called irradiance.It is obvious that radiation has directional properties, and that these needto be specified. Understanding of this is always helped by:1 Drawing pictures of the radiant fluxes and the methods of measurement2 Clarifying the units of the parameters.Consider a small test instrument for measuring radiation parameters in anideal manner. This could consist of a small, totally absorbing, black plane(Figure 3.9) that can be adapted to (a) absorb on both sides, (b) absorb onone side only, (c) absorb from one direction only and (d) absorb from onethree-dimensional solid angle only.3.5 Radiative heat transfer 63The energy AI absorbed in time Ai could be measured from the tempera-ture rise of the plane of area of one side, AA, knowing its thermal capacity.From Figure 3.9(a) the radiant flux density from all directions would bem=(AIAii(2AAi. In Figure 3.9(b) the radiation is incident from the hemi-sphere above one side of the test plane (which may be labelled or ), som = AI(AAAii (3.27)In Figure 3.9(c) a vector quantity is now measured, with the direction ofthe radiation flux perpendicular to the receiving plane. In Figure 3.9(d) theradiation flux is measured within a solid angle Aw, centred perpendicularto the plane of measurement and with the unit of W(m2sri1.The wavelength(s) of the received radiation need not be specified, sincethe absorbing surface is assumed to be totally black. However, if a dispersingdevice is placed in front of the instrument which passes only a small rangeof wavelength from \(A\2i to \(A\2i, then the spectral radiant fluxdensity may be measured asm\= AIAAAiA\ |unit Wm2m1] (3.28)This quantity can also be given directional properties per steradian (sr) aswith m. Difficulty may sometimes arise, especially regarding certain measur-ing instruments. There are two systems of units relating to the measurementof radiation quantities photometric and radiometric units (see Kaye andLaby 1995). Photometric units have been established to quantify responsesas recorded by the human eye, and relate to the SI unit of the candela. Radio-metric units quantify total energy effects irrespective of visual response, andrelate to the basic energy units of the joule and watt. For our purposes, onlyradiometric units need be used. As an aside, we note that a similar pair ofunits exists for noise.3.5.3 Absorption, reflection and transmission of radiationRadiation incident on matter may be reflected, absorbed or transmitted(Figure 3.10). These interactions will depend on the type of material, thesurface properties, the wavelength of the radiation and the angle of incidence0. Normal incidence (0 = 0i may be inferred if not otherwise mentioned,but at grazing incidence (90o >0 70oi there are significant changes in theproperties.At wavelength \, within wavelength interval A\, the monochromaticabsorptance o\ is the fraction absorbed of the incident flux density m\A\.Note that o\ is a property of the surface alone, depending, for example, onthe energy levels of the atoms in the surface . It specifies what proportion of64 Heat transferFigure 3.10 Reflection, absorption and transmission of radiation (m is the incidentradiation flux density).radiation at a particular wavelength would be absorbed if that wavelengthwas present in the incident radiation. The subscript on o\, unlike that onm\, does not indicate differentiation.Similarly, we define the monochromatic reflectance \ and the monochro-matic transmittance t\.Conservation of energy implies thato\\t\=1 (3.29)and that 0 o\, \, t\1. All of these properties are almost independent ofthe angle of incidence 0, unless 0 is near grazing incidence. In practice, theradiation incident on a surface contains a wide spectrum of wavelengths,and not just one small interval. We define the absorptance o to be theabsorbed proportion of the total incident radiant flux density:o =mabsmin (3.30)It follows thato =

~\=0o\m\,ind\

~\=0m\,ind\(3.31)Equation (3.31) describes how the total absorptance o, unlike o\, doesdepend on the spectral distribution of the incident radiation. For example,a surface appearing blue in white daylight is black in orange sodium-light.This is because the surface absorbs the photons of orange colour of bothlights, and so the remaining reflected daylight appears blue.3.5 Radiative heat transfer 6510(a)(b)2 4 6 80.80.2 m12 4 6 8 m1200010000 / Wm2m1IIIIIIFigure 3.11 Data for Example 3.4. The maxima of curves I, II, III in (b) are at (0.5,2000), (3.0, 1000) and (6.0, 400) respectively.The total reflectance =mreflmin and the total transmittance t =ttranstinare similarly defined, and againot =1 (3.32)Example 3.4 Calculation of absorbed radiationA certain surface has o\ varying with wavelength as shown inFigure 3.11(a) (this is a typical variation for a selective surface, asused on solar collectors, Section 5.6). Calculate the power absorbedby 1.0m2of this surface from each of the following incident spectraldistributions of RFD:1 m\ given by curve I of Figure 3.11(b) (this approximates a sourceat 6000 K).2 m\ given by curve II (approximating a source at 1000 K).3 m\ given by curve III (approximating a source at 500 K).66 Heat transferSolution1 Over the entire range of \, o\=0.8. Therefore from (3.31) o=0.8also, and the absorbed power isI =o(1m2i

m\,ind\=(0.8i(1m2i12

2000Wm2m1

(2mi=1600W(The integral is the area under curve I.)2 Here we have to explicitly calculate the integral o\m\,ind\ of(3.31). Tabulate as follows (the interval of \ is chosen to matchthe accuracy of the data; here the spectra are obviously linearised,so an interval A\ 1m is adequate):\(m) A\(m) o\ m\(Wm2m1) o\m\A\(Wm2)2.5 1 0.62 500 3103.5 1 0.33 750 2504.5 1 0.2 200 40Total 600Therefore the power absorbed is approximately 600 W.3 In a manner similar to part 1 of this solution, o\= 0.2 over therelevant wavelength interval. Thus the power absorbed isI =(0.2i(1m2i12

400Wm2m1

(5mi=200W.Note: The accuracy of calculations of radiative energy transfer is gen-erally better than for convection. This is because the theory of thephysical processes is exactly understood.3.5.4 Black bodies, emittance and Kirchhoffs lawsAn idealised surface absorbing all incident irradiation, visible and invisible,is named a black body. The name is because surfaces having the colourblack absorb all visible radiation; note, however, that black bodies absorbat all wavelengths, i.e. both visible and invisible radiation. Therefore, ablack body has o\= 1 for all \, and therefore also has total absorptanceo = 1. Nothing can absorb more radiation than a similarly dimensionedblack body placed in the same incident irradiation.3.5 Radiative heat transfer 67Kirchhoff proved also that no body can emit more radiation than asimilarly dimensioned black body at the same temperature.The emittance a of a surface is the ratio of the RFD emitted by the surfaceto the RFD emitted by a black body at the same temperature:a = mfrom surface(Timfrom blackbody(Ti (3.33)The monochromatic emittance, a\, of any real surface is similarly definedby comparison with the ideal black body, as the corresponding ratio of RFDin the wavelength range A\ (from \(A\2i to (\A\2i. It follows that0 a, a\1 (3.34)Note that the emittance a of a real surface may vary with temperature.Kirchhoff extended his theoretical argument to prove Kirchhoffs law: forany surface at a specified temperature, and for the same wavelength, themonochromatic emittance and monochromatic absorptance are identical,o\=a\ (3.35)Note that both o\ and a\ are characteristics of the surface itself, and notof the surroundings.For solar energy devices, the incoming radiation is expected from theSuns surface at a temperature of 5800 K, emitting with peak intensityat \ 0.5m. However, the receiving surface may be at about 350 K,emitting with peak intensity at about \ 10m. The dominant monochro-matic absorptance is therefore o\=0.5m and the dominant monochromaticemittance is a\ = 10m. These two coefficients need not be equal, seeSection 5.6. Nevertheless, Kirchhoffs Law is important for the determina-tion of such parameters, e.g. at the same wavelength of 10m, a\=10m=o\=10m3.5.5 Radiation emitted by a bodyThe monochromatic RFD emitted by a black body of absolute temperatureT, mB\, is derived from quantum mechanics as Plancks radiation law:mB\= C1\5|exp(C2\Ti 1] (3.36)where C1=lc2, and C2=lcl (c, the speed of light in vacuum; l, Planckconstant; and l, Boltzmann constant). Hence C1= 3.741016Wm2andC2= 0.0144mK are also fundamental constants. Figure 3.12 shows howthis spectral distribution mB\ varies with wavelength \ and temperature T.68 Heat transfer1081061041024(a)8 12 16 20Locus ofmaxima B/Wm2m1 m1T=6000K1000K400K(b)0 2000 4000 6000 8000 10 00010.80.60.40.20T/(mK)DFigure 3.12 (a) Spectral distribution of black body radiation. After Duffie andBeckman (1991). (b) Cumulative version of (a), in dimensionless form,as in eqn (3.40). Note that u 1 as \7 ~.Note that the wavelength \m, at which mB\ is most intense, increases asT decreases. Indeed, as we know also from experience, when any surfacetemperature increases above T 700K(430oCi significant radiation isemitted in the visible region and the surface does not appear black, butprogresses from red heat to white heat.By differentiating (3.36) and setting d(mB\id\ =0, we find that\mT =2898mK (3.37)This is Wiens displacement law. Knowing T, it is extremely easy to deter-mine \m, and thence to sketch the form of the spectral distribution mB\.3.5 Radiative heat transfer 69From (3.36) the total RFD emitted by a black body ismB=

~0mB\d\Standard methods (e.g. see Joos and Freeman, Theoretical Physics, p. 616)give the result for this integration asmB=

~0mB\d\ =uT4(3.38)where u = 5.67 108Wm2K4is the StefanBoltzmann constant,another fundamental constant.It follows from (3.33) that the heat flow from a real body of emittancea(a -1i, area A and absolute (surface) temperature T isIr=auAT4(3.39)Note:a in using radiation formulae, it is essential to convert surface tempera-tures in, say, degrees Celsius to absolute temperature, Kelvin; i.e. xoC=(x273i K,b the radiant flux dependence on the 4th power of absolute temperatureis highly non-linear and causes radiant heat loss to become a dominantheat transfer mode as surface temperatures increase more than 100oC.The StefanBoltzmann equation (3.39) gives the radiation emitted by thebody. The net radiative flux away from the body may be much less(e.g. (3.44)). More convenient for calculation than (3.36) is the dimension-less function D, whereD=

\0mB\d\uT4 (3.40)which turns out to be a function of the single variable \T. This function isgraphed in Figure 3.12(b).3.5.6 Radiative exchange between black surfacesAll material bodies, including the sky, emit radiation. However, we do notneed to calculate how much radiation