Volume 3 solar thermal systems components and applications 3 02 – the solar resource

Volume 3 solar thermal systems components and applications 3 02 – the solar resource Volume 3 solar thermal systems components and applications 3 02 – the solar resource Volume 3 solar thermal systems components and applications 3 02 – the solar resource Volume 3 solar thermal systems components and applications 3 02 – the solar resource Volume 3 solar thermal systems components and applications 3 02 – the solar resource

3.02 The Solar Resource HD Kambezidis, Institute of Environmental Research and Sustainable Development, Athens, Greece © 2012 Elsevier Ltd All rights reserved 3.02.1 3.02.2 3.02.3 3.02.4 3.02.4.1 3.02.4.2 3.02.4.3 3.02.5 3.02.5.1 3.02.5.2 3.02.5.3 3.02.5.4 3.02.6 3.02.6.1 3.02.6.2 3.02.6.3 3.02.6.4 3.02.7 3.02.7.1 3.02.7.2 3.02.7.3 3.02.7.4 3.02.7.5 3.02.7.6 3.02.7.7 3.02.8 3.02.9 3.02.10 3.02.11 3.02.12 3.02.12.1 3.02.12.2 3.02.12.3 3.02.12.4 3.02.12.5 Appendix A: Appendix B: Appendix C: Appendix D: References Introduction Sun–Earth Astronomical Relations Solar Constant Solar Spectrum Planck’s Law Wien’s Displacement Law Stefan–Boltzmann Law Interference of Solar Radiation with the Earth’s Atmosphere The Earth’s Atmosphere Optical Air Mass Attenuation of Solar Direct Radiation Rayleigh and Mie Scattering, Reflection, and Absorption Models of Broadband Solar Radiation on Horizontal and Tilted Surfaces Calculation of Solar Radiation on a Horizontal Plane The Meteorological Radiation Model Calculation of Solar Radiation on a Tilted Surface Quality Control of Solar Radiation Values Evaluation of Models The Standard Deviation The Root Mean Square Error The Mean Bias Error The Mean Absolute Bias Error The t-test The Index of Agreement (d ) The Coefficient of Determination (R 2) Models of Solar Spectral Radiation Net Solar Radiation Networks of Solar Radiation Stations – Solar Atlases Utility Tools for Solar Radiation Calculations Instruments for Measuring Solar Radiation Solar Radiometers The World Radiometric Reference Calibration of Solar Radiometers Uncertainty of Solar Radiometers Correction of Common Solar Radiometer Errors Spectral Distribution of Solar Radiation Radiometric Terminology The Sun as a Blackbody Physical Constants and Conversion Factors Glossary Absorption Transfer of some of the solar radiation power (power of electromagnetic waves) to air molecules during collision of solar radiation with constituents in the atmosphere ACR Active-cavity radiometer (a reference radiometer for calibrating others) Aphelion The longer distance between Earth and Sun (occurring around 3–8 April) Attenuation Depletion of solar radiation (power of electromagnetic waves) due to absorption and scattering Comprehensive Renewable Energy, Volume 28 29 32 33 33 33 34 34 34 35 36 36 39 40 43 45 49 49 50 50 51 51 51 52 52 54 54 58 62 64 64 67 67 67 67 67 78 79 80 80 by the constituents (molecules) of the atmosphere of the Earth DNI Direct normal irradiance DU Dobson Unit (or atm-cm) A measure of the columnar height of ozone in the atmosphere Ecliptic plane The plane on which the Earth orbits around the Sun (heliocentric system) or the Sun around the Earth (geocentric system) Equinox The positions of the Earth around the Sun on the ecliptic with (solar) declination equal to 0; this occurs twice a year, around 20–21 March doi:10.1016/B978-0-08-087872-0.00302-4 27 28 Solar Thermal Systems (vernal equinox) and 22–23 September (autumn equinox) FOV Field-of-view (aperture of pyrheliometer) Mie scattering Scattering of the solar radiation (electromagnetic waves) by molecules comparable in size with the wavelength Net solar radiation The difference between incoming (short-wave) and outgoing (long-wave) radiation Perihelion The shortest distance between Earth and Sun (occurring around 2–4 January) Pyranometer A solar radiation instrument capable of measuring solar radiation in the range 0.29–2.8 μm Pyrheliometer A solar radiation instrument capable of measuring solar radiation at a point (usually the Sun) Radiometer An instrument to measure solar radiation flux (or power) Rayleigh scattering Scattering of the solar radiation (electromagnetic waves) by molecules of bigger dimensions than the wavelength TOA Top-of-the-atmosphere, referring to an altitude of 100 km from the surface of the Earth Scattering Re-distribution of the solar radiation power (power of electromagnetic waves) during collision of solar radiation with constituents in the atmosphere Solar atlas Map of an area showing the distribution of solar radiation (solar energy) over it Solar constant The solar radiation received at TOA on a plane normal to the solar rays at the mean Sun-Earth distance Solar declination The angle formed by the lines joining the centers of the Sun and Earth and the line towards the south of the observer on the Earth along the ecliptic plane Solar geometry The position of the Sun in the sky of any place on earth and any day of the year Solar (radiation) spectrum The electromagnetic waves emitted by the photosphere of the Sun Solstice The apparent position of the Sun in the sky reaching its northernmost or southernmost extremes in the sky; the first is called summer solstice (on 20–21 June) and the second winter solstice (on 21–22 December) Spectrometer A radiometer capable to measure solar radiation at various wavelengths Statistic Statistical estimator 3.02.1 Introduction The Sun emits a tremendous amount of energy, in the form of electromagnetic (EM) radiation, into space Most of the Sun’s energy flows out of our solar system into interstellar space without ever colliding with anything However, a very small fraction of that energy collides with planets, including the Earth, before it can escape into the interstellar void A part of the fraction that the Earth intercepts is sufficient to warm our planet and drive its climate system The Sun emits about 1366 W of power in the form of EM radiation fall normally on an area of m2 at the top of the Earth’s atmosphere (100 km from its surface) Thus, the average surface temperature of the Earth (including the effects of its atmosphere) is about 15 ˚C (http://encarta.msn.com/encyclopedia_761567022/Global_Warming.html) If, though, the Earth were displaced closer to the Sun, where, for example, the planet Mercury is, the number of watts per square meter (W m−2) would be greater, giving an average temperature of about 179 ˚C (http://www.solarviews.com/eng/mercury.htm) If the Earth were further from the Sun, as, for example, the planet Jupiter is, the number of W m−2 would be lesser, giving an average temperature of about −145 ˚C (http://www universetoday.com/guide-to-space/jupiter/temperature-of-jupiter) This is so because the surface area of a sphere varies as the square of the radius of the sphere, so the energy per unit area received varies inversely as the square of the distance from the Sun A planet situated half as far from the Sun as is the Earth would be scorched by four times as much power from the Sun (5472 W m−2) A planet twice as far from the Sun as is the Earth would be warmed by just one-fourth as much radiation (342 W m−2) So our planet’s distance from the Sun is the first key factor influencing the energy we receive, and thus the behavior of our climate Solar radiation refers to the energy coming from the Sun in the wavelength range 0.3–3 μm; it constitutes the principal source of energy for the global Earth–atmosphere system Detailed knowledge of the solar radiation transmission through the atmosphere (under both clear and cloudy conditions) is crucial in determining any possible change in the Earth’s radiation budget in a changing climate For this reason, various solar radiation models have been developed to calculate solar fluxes at the surface either in the whole spectrum (0.3–3 μm) (broadband models) or in a part of it (spectral models) The basis of such models is the so-called radiative transfer models (RTMs), which are complex computer codes taking into account the interaction of solar radiation with the Earth’s atmosphere Each location on the Earth’s surface receives different amounts of solar energy throughout the year This is due to many factors Some of them relate to the geometry of the Earth’s orbit around the Sun and others to the absorption and scattering of solar radiation by its atmosphere In the first set, the eccentricity of the Earth’s orbit, the solar declination, and the geographical coordinates of a location on the surface of the Earth and the position of the Sun in the sky play important roles In the second set, the scattering and absorption of solar energy by the molecules in the atmosphere play important roles In this context, one can distinguish between the direct solar radiation (or beam solar radiation) coming directly from the Sun’s disk and the diffuse solar radiation coming from all parts of the sky (except the Sun’s disk) as a result of scattering (including reflection) of solar rays by the molecules in the atmosphere The sum of direct and diffuse components makes the global (or total) solar radiation Usually, The Solar Resource 29 the measurements of solar radiation refer to the global and diffuse components on a horizontal plane, and so most of the (broadband or spectral) models Nevertheless, due to various applications of solar energy, such as solar thermal and photovoltaic (PV) systems, there is a need for measuring or calculating the incident solar energy on an inclined plane This need is rarely met by measuring equipment (the so-called radiometers) worldwide Therefore, this gap has been the target of various models A solar radiation model is a computer code that tries to simulate the solar radiation received on a (horizontal or inclined) surface with an area of m2 These models simulate solar radiation in either a statistical or a physical way In the first case, a statistical model uses past (or historical) solar radiation data and tries to forecast future values at the same location Such models use the autoregressive and moving average (ARMA) or neuronal technology In the second case, the models take into account the interference between solar radiation and atmospheric molecules; that way they are simple or complicated RTMs Whatever the category of the model is, there is a great need to evaluate simulation results against measurements Several statistical indicators play an important role in this evaluation The need for the knowledge of solar energy received at a place is high, especially at places where no measuring (actinometric) stations exist, and the development of satellites oriented to providing the scientific community with the solar radiation received at regional scale aided this effort Such satellite images are being verified by measurements performed at actinometric stations over Europe and the United States, Canada, and Japan in order for the satellite data to be used with certainty The comparison shows that the satellite data up to now are reliable enough for solar energy engineering purposes; however, they are not ready for use in solar radiation research The outcome of this exercise together with the use of solar radiation models has triggered interest in forming regional maps of solar availability Therefore, such maps (also called solar atlases) exist nowadays covering several regions of the world, for example, the United States, Canada, Japan, the European Union, and India These solar atlases give information about the expected mean levels of solar radiation received on horizontal (and in some cases inclined) surfaces throughout the year as well as seasonally They are used for assessing the available solar energy at the scale of a region in a country or bigger; they are intended for use at specific locations as the satellite image pixels have dimensions of a few kilometers Nevertheless, the most accurate way of knowing the subhourly, hourly, daily, monthly, and annual levels of the various solar radiation components at a location is by performing measurements at the actinometric stations The actinometers are sensors specially designed to measure solar radiation either in the broadband or in the spectral sense The solar instruments that measure the global or the diffuse component in the whole solar spectrum (0.3–3 μm) are called pyranometers If they are used to measure a part of the spectrum, they are called spectroradiometers The instruments that measure the direct solar radiation are called pyrheli­ ometers Various spectral regions are possible with optical filters, for example, 0.525–2.8 µm (OG530 filter, ex OG1), 0.630–2.8 µm (RG630 filter, ex RG2), and 0.695–2.8 µm (RG695 filter, ex RG8) Those radiometers that measure the ultraviolet (UV) spectrum (0.295–0.385 μm) are called UV radiometers Finally, the sensors for measuring the infrared (IR) band (0.750–100 μm) are called pyrgeometers Several radiometer manufacturers exist worldwide A solar radiation terminology is given in Appendix B 3.02.2 Sun–Earth Astronomical Relations The Earth moves around the Sun in an elliptical orbit, making a complete revolution in a year (365.24 days) Figure shows the orbit of the Earth together with the two equinoxes, the two solstices, the aphelion, and the perihelion, and the positions of the smallest distance between the Sun and the Earth Equinox is the position of the Earth on its orbit when the length of the day is equal to that of the night This occurs on 20–21 March (vernal equinox) and 22–23 September (autumnal equinox) each year Solstice is the point on the Earth’s orbit when the day has the longest (summer solstice, 20–21 June) or shortest (winter solstice, 21–22 December) length Vernal equinox 23.5° 3–5 April N Summer solstice 20–21 March 20–21 June ~1.0 AU 23.5° 23.5° N N ~1.017 AU 3–6 July ~0.983 AU 2–4 January Aphelion Perihelion ~1.0 AU 23.5° N 21–22 December Elliptic plane Winter solstice 22–23 September 4–6 October Autumnal equinox Figure The motion of the Earth around the Sun counterclockwise on its ecliptic plane (heliocentric system) The aphelion is approximately on July, the perihelion on about January, while the Earth is at AU distance from the Sun on April and October on average 30 Solar Thermal Systems The aphelion and perihelion are those points of the orbit when the distance of the Earth from the Sun is greatest (152.1 million km) and smallest (147.3 million km) and occurs on 3–6 July and 2–4 January, respectively The mean distance between the two planets occurs on 3–5 April and 4–6 October and is equal to 149.6 million km (more accurately 149.597 890 million km) This distance is called an astronomical unit (AU) and is used in astronomy exclusively The aphelion distance is equal to 1.017 AU and the perihelion distance is equal to 0.983 AU The eccentricity correction factor of the Earth’s orbit, S, is equal to the squared ratio of the mean distance Earth–Sun, r0, to the distance at any instance of the year, r An exact formula giving S according to [1] is S¼  r 2 r ẳ 1:000 110 ỵ 0:034 221cos M ỵ 0:001 280 sin M ỵ 0:000 719 cos 2M ỵ 0:000 077 sin 2M ẵ1 where M (in radians) is called the day angle and is given by [2] Mẳ 2D 365 ẵ2 D is the day number of the year D = on January and 365 on 31 December On leap years, D takes the value of 366 for the last day of December Nevertheless, a more simple formula for S can be employed for engineering and technological applications according to [3]   2D S ẳ ỵ 0:033 cos ẵ3 365 Example 3.02.1 Consider 16 October (D = 289) in a non-leap year Then M = 284.16˚ The eccentricity of the Earth’s orbit is found from eqns [1] and [2] as 1.0064 and 1.0091, respectively In order to better understand the paths of the Sun in the sky, one can imagine a celestial sphere with the Earth at its center and the Sun revolving around it (Figure 2) In the celestial sphere, the celestial poles are the points at which the Earth’s polar axis intercepts with the celestial sphere Similarly, the celestial equator is a projection of the Earth’s equatorial plane on the celestial sphere The plane on which the Earth revolves around the Sun is called the ecliptic plane On the other hand, the Earth spins around its axis (polar axis) The angle between the polar axis and the normal to the ecliptic plane remains unchanged throughout the year However, the angle between the lines joining the centers of the Sun and the Earth to the equatorial plane changes every day (every instant indeed) This angle is called solar declination, δ, and takes values between +23.5° and −23.5° These values are achieved during the summer and winter solstices, respectively Note that when the northern hemisphere experiences summer, the southern hemisphere has winter, and vice versa An accurate formula for calculating δ, in degrees, is given by [1] δ ¼ ð0:006 918 0:399 912  cos M ỵ 0:070 257 sin M 0:006 758 cos 2M ỵ 0:000 907 sin 2M 0:002 697 cos 3M 180 ỵ 0:001 48 sin 3Mị ẵ4a North pole of celestial sphere Plane of celestial equator 23.5° Autumnal equinox Apparent path of sun on the ecliptic plane Polar axis Winter solstice δ Summer solstice 23.5° Earth Sun Vernal equinox South pole of celestial sphere 90° Figure The celestial sphere, the apparent path of the Sun (geocentric system), and the Sun’s declination angle From http://devconsultancygroup blogspot.com/2010/08/will-la-ninas-year-long-cooling-make.html The Solar Resource Simpler formulas but not so accurate are given by   !' & 360 D 82ị ; in degrees ẵ3 ẳ sin − 0:4 sin 365   ! 360 δ ẳ 23:45 sin D ỵ 284ị ; in degrees ẵ4 365 ! D ỵ 284ị ẳ 23:45 sin ; in degrees ½5Š 365 31 ½4bŠ ½4cŠ ½4dŠ Example 3.02.2 Consider 16 October (D = 289) in a non-leap year Then M = 4.975 rad = 284.16° The solar declinations resulting from eqns [4a]–[4d] give the values of −8.67°, −9.42°, −9.97°, and −9.97° The last two equations give identical results To describe the Sun’s path across the sky, one needs to know the angle of the Sun relative to a line perpendicular to the Earth’s surface, the so-called zenith angle, θz, and the Sun’s position relative to the observer’s north–south axis, the azimuthal angle or azimuth, ψ The angle of the position of the Sun on the plane of the Sun’s path in the sky to the observer’s horizon is called solar altitude, γ The hour angle, ω, is easier to use than the azimuthal angle because the hour angle is measured in the plane of the ‘apparent’ orbit of the Sun as it moves across the sky (Figure 3) The position of the Sun in the sky is identified by the values of θz and ψ Since the Earth rotates approximately once every 24 h, the hour angle changes by 15˚ per hour and moves through 360˚ over the day Typically, the hour angle is defined to be zero at solar noon, when the Sun is highest in the sky (Figure 3) cos θz ¼ sin δ sin  ỵ cos cos  cos ẳ sin cos ẳ ẵ5 sin sin  sin δ cos γ cos  ½6Š where φ is the geographical latitude of the observer’s location on the surface of the Earth In the above equations, the refraction of the Earth’s atmosphere has not been taken into account Kambezidis and Papanikolaou [6] give corrections for this effect The trigonometric parameters given above obey the following conditions: γ ¼ 90˚− θz 0 > 0ị ẵ7a ¼ at solar noon; positive in the morning; negative in the afternoon; −90 ≤ω ≤90 ∘ φ > in the northern hemisphere; φ < in the southern hemisphere; −90∘ ≤  ≤90∘ ∘ ∘ ∘ ∘ ∘ ½7bŠ ½7cŠ ∘ ψ ¼ at observe’s south; ψ > to the east; ψ < to the west; ≤ ψ ≤90 with cos ψ ≥0 and 90∘ ≤ ψ ≤180∘ with cos ψ ≤0∘ ½7dŠ At sunrise, θz = 90˚ From eqn [5] the sunrise hour angle, ωs, is found to be ωs ¼ cos − tan  tan ị ẵ8 Zenith Latitude Solar zenith angle Z ϕ Sun’s daily path θz Solar hour angle North celestial axis Autumnal equinox ω West Apparent path of sun on the ecliptic plane Solar altitude angle δ γ Observer’s south Earth ψ Observer’s north East Solar azimuth angle Solar declination angle Vernal equinox South celestial axis Figure The apparent daily path of the Sun in the sky for a place on the Earth (geocentric system) specified by its geographical latitude The coordinates of the Sun are given by the zenith angle and azimuth angle (or equivalently altitude angle) 32 Solar Thermal Systems It must be noted that the sunrise hour angle is equal to the sunset one apart from the difference in sign Then the length of the day, Ldt, is 2ωs:   Ldt ¼ cos − ð− tan  tan δÞ; in hours ½9Š 15 where 15 refers to the arc of 15˚ per hour that the Sun travels in the sky Equations [8] and [9] refer to a flat terrain If obstacles exist at the location of the observer obstructing solar rays during either sunrise or sunset, then other relationships for the hour angles of sunrise and sunset can be given [7] taking into account the height of the obstacle and its distance from the observer 3.02.3 Solar Constant The solar constant, Hex, is the amount of the total solar energy at all wavelengths incident on an area of m2 exposed normally to the rays of the Sun at AU Because of the effects of the Earth’s atmosphere on the transmission of the solar rays through it, the definition of the solar constant is implied at the top of the atmosphere (TOA); TOA is placed at the altitude of 100 km from the surface of the Earth where the density of the atmosphere is null Hex varies along the year due to varying distances of the Earth from the Sun by ∼3.4% of its mean value The first estimated mean value of Hex was 1353 W m−2 [8] This value was updated in 1977 to 1377 W m−2 [9] and later modified to 1367 W m−2 [10, 11] The latest value of the solar constant is 1366.1 W m−2 [12] The spectral distribution of the solar constant at TOA is given in Appendix A The calculation of the solar constant is an arduous process since it involves a series of solar radiation measurements The first measurements were made with ground-based instrumentation These were spectral observations of solar radiation extrapolated to their predicted values at TOA by taking into account the various attenuation effects produced by the molecules in the atmosphere The spectral integration of these values yielded the solar constant However, the ground-based measurements were subject to errors because of the uncertainties involved in estimating the attenuation effects of the atmospheric constituents on solar radiation The second step was to perform these measurements at high-altitude observatories, flying aircraft, balloons, and space probes onboard rockets or satellites lately The solar constant derived from the ground measurements was found to be consistently higher than its estimation at high-altitude platforms Another issue causing uncertainty in the estimation of the solar constant was the intrinsic errors in the radiometers used in such measurements To overcome the problem, the scientists had to intercompare all these devices to ensure that they work within certain limits of uncertainty Furthermore, cavity-type absolute radiometers (see Section 3.02.12.1) started being used giving measurements of the solar irradiance with minimum error For this reason, the World Meteorological Organization (WMO) adopted a new scale, the so-called World Radiometric Reference (WRR), as the basis of all actinometric measurements (see Section 3.02.12.2) Using this reference, Fröhlich and co-workers [10, 11] reexamined all sets of solar constant measurements in the period 1969–80 and recommended the revised value of 1367 W m−2 With the use of satellites equipped with active cavity radiometers (ACRs), the measurement of the solar irradiance at TOA for long periods was possible Such missions were the Nimbus (Earth Radiation Budget) (1978–93), the Solar Maximum Mission (SMM) equipped with the Active Cavity Radiometer Irradiance Monitor I (ACRIM I) (1980–89), the Earth Radiation Budget Satellite (ERBS) Solar Monitor Measurements (1984–2003), and the Upper Atmosphere Research Satellite (UARS) ACRIM II Measurements (1991–97) These missions gave more accurate measurements of the solar constant with the current mean value at 1366.1 W m−2 Figure shows the evolution of all measurements for the solar constant made from airborne sensors onboard satellites Data and further information related to these satellites are available through the NASA Goddard Space Flight Center, Data Archive Center 1995 2000 2005 2010 VIRGO ACRIM II HF 1990 ACRIM I 1985 HF Model 1368 HF ACRIM I PMOD Composite (Wm−2) 1980 1366 1364 Min20/21 Min21/22 Min22/23 Min23/24 Average TSI: 1365.89 Wm−2 Minimum 21/22: 1365.57 Wm−2 Minimum 23/24: 1365.23 Wm−2 1362 1980 1985 1990 1995 Year 2000 2005 2010 Figure Solar irradiance measurements at TOA from airborne sensors onboard satellites in the period 1976–2008 The fluctuation of the measurements is due to the 22-year Sun spot cycle The various sensors are shown at the top of the figure Updated from Fröhlich (2011) The Solar Resource 33 The composite in Figure 4, compiled by the VIRGO team at the Physikalisch Meteorologisches Observatorium, World Radiation Center (PMOD/WRC), Davos, Switzerland, shows the total solar irradiance as daily values plotted in different colors for the various experiments performed The difference between the values at the minima is indicated together with the amplitudes of the three Sun spot cycles 3.02.4 Solar Spectrum The radiant solar energy comes from nuclear fusion happening in the Sun; the Sun has a surface temperature of 5777 K The spectrum of the solar radiation received at TOA (see Figure A.1 in Appendix A) can be well approximated by the spectrum of a blackbody having a surface temperature of 5777 K Thus the Sun may be considered as a blackbody A body is called a blackbody if, at a given temperature, it emits the maximum amount of energy at each wavelength and in all directions and it also absorbs all identical radiation at each wavelength and in all directions The emission from a blackbody obeys the following laws 3.02.4.1 Planck’s Law The power, Ebλ, emitted by a blackbody (or the emissive power) at a given wavelength and temperature is given by the following formula: R 1  ! ẵ10 Eb ẳ R2 λ exp −1 λTK where R1 and R2 are the radiation constants (3.742  108 W μm m−2 and 1.438  104 μm K, respectively), λ the wavelength (in μm), and TK the blackbody temperature (K) Ebλ is plotted for various temperatures in Figure It is seen from the diagram that the maximum of each curve is displaced toward longer wavelengths as the temperature decreases This is known as the Wien’s displacement law The spectral distribution of the solar constant (the most recently measured extraterrestrial radiation at TOA) is given in Appendix A 3.02.4.2 Wien’s Displacement Law By dividing both sides of eqn [10] by TK5, a function of the variable λTK is obtained, that is, Eb ẳ TK5 R1   ! R2 TK ị exp −1 λTK ½11Š From eqn [12] the locus of maximum λTK, called λmaxTK, is 2897.8 μm K The relation max TK ẳ 2897:8 m K ẵ12 is called the Wien’s displacement law Assuming the Sun as a blackbody with a surface temperature of 5777 K, eqn [12] gives λmax = 0.501 μm, which lies in the green region of the visible spectrum 1e+8 T = 300 K T = 500 K T = 1000 K T = 3000 K T = 5777 K Spectral emissive power (W m–2 μm–1) 1e+7 1e+6 1e+5 1e+4 1e+3 1e+2 1e+1 1e+0 10 20 30 40 50 60 Wavelength (μm) 70 80 Figure Spectral emissive power from a blackbody at various temperatures in the wavelength range 0–100 μm 90 100 34 Solar Thermal Systems 3.02.4.3 Stefan–Boltzmann Law By integrating eqn [10] in all wavelengths, one gets the total power emitted from a blackbody at the temperature TK: Eb ¼ σTK4 ½13Š −8 −2 −4 where σ is called the Stefan–Boltzmann constant and is equal to 5.6697  10 W m K This is the theoretical value coming from the integration of eqn [10] Its measured value is 5.6866  10−8 W m−2 K−4 [13] Example 3.02.3 Determine (1) the surface temperature of a blackbody radiating with a total emissive power of 7.25  10−4 W m−2 and (2) the wavelength of maximum emissive power (1) From eqn [13] the temperature TK = (Eb/σ)1/4 = (7.25  104/ 5.6697  10−8) = 1063.4 K (2) From eqn [12], λmax = 2.73 μm 3.02.5 Interference of Solar Radiation with the Earth’s Atmosphere When solar radiation enters the Earth’s atmosphere, a part of the incident energy is attenuated by scattering and another part by absorption from the atmospheric constituents The scattered radiation is called solar diffuse radiation or just diffuse radiation A part of the diffuse radiation goes back to space and a part reaches the ground The radiation that arrives at the surface of the Earth directly from the Sun is called solar direct or solar beam radiation (or just direct or beam radiation) The knowledge of the spectral irradiance (direct and diffuse) arriving at the surface of the Earth is important for the design of certain solar energy applications such as PVs The integration of both diffuse and direct radiation over all wavelengths is called broadband; this is important in calculations concerning heating and cooling loads in architecture, the design of flat-plate collectors (e.g., Reference 14), or the study of radiation climate (e.g., References 15–17) 3.02.5.1 The Earth’s Atmosphere The actual composition of the constituents of the clean dry atmosphere (an atmosphere consisting of its natural chemical elements and no clouds) varies with geographical location, altitude, and season Generally, the vertical structure of the Earth’s atmosphere has been described by the so-called standard atmospheres The standard atmosphere used so far is the US Standard Atmosphere of 1976 (USSA 1976) [18] Figure shows the vertical temperature and pressure profiles indicating the layers of the lower (0–11 km) atmosphere, called the troposphere, the lower-middle (20–50 km) atmosphere, called the stratosphere, the upper-middle (56–80 km) atmosphere, called the mesosphere, and the upper (90–100 km) atmosphere, called the thermosphere The turning points of the temperature profile are formed by intermediate layers, the tropopause (11–20 km), the stratopause (50–56 km), and the mesopause (80–90 km) 200 Pressure (mbar) 400 600 800 1000 1200 100000 Thermosphere 90000 Mesopause 80000 70000 Mesosphere Altitude (m) 60000 Stratopause 50000 40000 Stratosphere 30000 20000 Tropopause 10000 Troposphere –100 –80 –60 –40 –20 Temperature (°C) 20 Figure Air temperature (red curve) and atmospheric pressure (blue curve) profiles from the sea level up to the TOA (100 km) according to the USSA 1976 From http://www.physicalgeography.net/fundamentals/7b.html The Solar Resource 35 Table The main chemical elements comprising the Earth’s clean dry atmosphere (USSA 1976) Name Formula Concentration (% by volume) Nitrogen Oxygen Argon Carbon dioxide Neon Ozone Helium Methane Krypton Hydrogen Nitrous oxide Xenon Water vapor Nitric acid vapor N2 O2 Ar CO2 Ne O3 He CH4 Kr H2 N2O Xe H2O 78.084 20.948 0.934 0.333 0.001 818 0−0.0012 0.000 524 0.000 15 0.000 114 0.000 05 0.000 027 0.000 008 0−0.000 004 Traces Table shows the composition of the Earth’s clean dry atmosphere From this table, it is seen that more than three-quarters of the atmosphere is made up of nitrogen and most of the rest is oxygen However it is the remaining 1%, a mixture of carbon dioxide, water vapor, and ozone, that not only produces important weather features, such as cloud and rain, but also has considerable influence on the overall climate of the Earth, through mechanisms such as the greenhouse effect and global warming Ozone is concentrated in the stratosphere, while water vapor and nitrous oxide in the lower atmosphere The main greenhouse gases are those of carbon dioxide and methane Since methane, carbon dioxide, and ozone are also produced by anthropogenic activities on the surface of the Earth, their concentration is highly variable These gases not exhibit a homogeneous temporal or spatial distribution throughout the atmosphere over a certain location on the surface of the Earth All molecules of air deplete solar radiation by scattering, absorption, and reflection Further details about these mechanisms are gated with the direct irradiance falling on the sensor of the pyranometer under clear skies On the other hand, it was shown that the direct irradiance measurements have higher accuracy than the global ones Therefore, it is wise to calculate global irradiance as the sum of the measured diffuse and direct irradiances Nevertheless, an unshaded pyranometer is still useful This methodology requires proper diffuse irradiance measurements The use of a ventilated instrument is recommended to homogenize temperatures and avoid condensation or frost on the dome Furthermore, the thermal effects have to be minimized (