1. SOLAR GEOMETRY, EXTRATERRESTRIAL IRRADIANCE & INCIDENCE ANGLES

Transcription

1 1. SOLAR GEOMETRY, EXTRATERRESTRIAL IRRADIANCE & INCIDENCE ANGLES

2 The Sun A blackbody with T ~ 6000 K Blackbody radiation with the same amount of energy per unit of area T ~ 5762 K Blackbody radiating with the same wavelength of the maximum T ~ 6300 K

3 The Sun Blackbody: a body that absorbs everything, therefore, all radiation coming from it is due to itself It follows the Planck s law (spectral emissive power) 2π hc Eλ (T ) = (W.m.m ) hc " % 5 λ $ e λkt 1' # &

4 The Sun Stefan-Boltzmann equation E(T ) = σ T 4 Wien s displacement law B λmax = T

5 The Sun As seen from the earth the sun can be considered a point source at infinity (parallel sun rays); only for high concentration purposes the exact geometry has to be taken into account

6 The Sun The intensity of the solar radiation is nearly constant outside the earth s atmosphere. It has a variation of 3.3% due to the eccentricity of the earth s orbit (aphelion and perihelion) G sc, the solar constant, is the average solar irradiance at normal incidence outside the earth s atmosphere (extraterrestrial irradiance)

7 The Sun The intensity of the solar Irradiance: radiation is instantaneous nearly constant power per unit of area (W.m outside the earth s atmosphere. -2 ) Irradiation (insolation): total It has a variation of 3.3% radiated due to the energy eccentricity during a time of the earth s orbit (aphelion and interval perihelion) unit of area ( J.m -2 ) G sc, the solar constant, is the average solar irradiance at normal incidence outside the earth s atmosphere (extraterrestrial irradiance)

8 The Sun A value of G sc =1367 W.m -2 will be used (World Radiation Center) but other values are usually found in literature. The extraterrestrial irradiance on the n th day of the year is given with a good approximation by '! G on = G sc cos# 360º " 365 n $ * ) &, ( % + G on,max = 1412 W.m -2 G on,min = 1322 W.m -2

9 The Sun A more precise value of G on can be evaluated with the equation G on = G sc ( cosB sin B cos2b sin 2B) B = (n 1)

10 The Sun Problem 1.1: when does the maximum of G on occur according to the simpler equation? Calculate that value and compare it with the values obtained for ± 4 days away from the date you found. R: 31 th December G on (31th Dec.)= W.m -2 G on (27 th Dec.)=G on (4 th Jan.)= W.m -2 Difference = 0.008%

11 The Sun Problem 1.2: when does the maximum of G on occur according to the more precise equation? R: 3 rd January 2015: 4 th January at 6h : 2 nd January at 22h49

12 Sun-Earth Geometry The interaction of the solar radiation with the earth produces several components of radiation that may reach a given surface. The next goal is to describe the angle between the beam radiation and a surface.

13 Sun-Earth Geometry Solar time is different from local standard time Solar time-standard time=e + 4( L st L loc ) 60ST (min) E = 2.292( cosB sin B cos2B 4.089sin 2B) B = ( n 1) 360! # ST = " $# summer +1h is in use 0 summer +1h is not in use

14 Sun-Earth Geometry Solar time: 12h or 0h p.m. at solar Solar time is different from local noon standard time Solar time-standard time=e + 4( L st L loc ) 60ST (min) E = 2.292( cosB sin B cos2B 4.089sin 2B) B = ( n 1) 360! # ST = " $# summer +1h is in use 0 summer +1h is not in use Solar noon: sun points towards south (northern hemisphere)

15 Sun-Earth Geometry Problem 1.3: What is the solar time in Lisbon, on 23 rd September, at 12h? (Lisbon: 9º W, +1h Summer time) Solar time-standard time=e + 4( L st L loc ) R: 10h32min E = 2.292( cosB sin B cos2B 4.089sin 2B) B = n 1 ( )

16 Sun-Earth Geometry The earth orbit plane and the earth s rotation axis

17 Sun-Earth Geometry Declination, δ: the angle between the earth orbit and the equatorial plane It can be calculated with a good approximation by! δ = 23, 45ºsin 360º n $ # & " 365 %

18 Sun-Earth Geometry For a more precise calculation of the declination the following equation can be used δ = 180º ( cosB sin B + π cos2B sin2B cos3B sin 3B) B = (n 1)

19 Sun-Earth Geometry Problem 1.4: find out when δ=0 both with the simple and the more precise equation.! δ = 23, 45ºsin 360º n $ δ # = 180 ( & cosB sin B + " π 365 % cos2B sin2B cos3B sin 3B) R: simple, 22 nd March, 21 st September precise, 21 B st = March, (n 1) rd September 365 Correct values for 2015: 20 th March, 22h45 23 rd September, 8h21

20 Incidence Angle! n s = cosδ cosω, cosδ sinω,sinδ ( ) ω = 360 τ day t t = 0pm at noon ω = 360 τ day t -180º t [0h, 24h] Incidence angle, θ: cosθ = n s n

21 Incidence Angle on Fixed Surfaces Case 1: horizontal surface at latitude ϕ (θ will be θ z, the zenithal angle) cosθ z = cosδ cosω cosφ + sinδ sinφ

22 Incidence Angle on Fixed Surfaces Case2: sunset angle and time, ω s and t s (θ z =90º) cosω s = tanδ tanφ t s = ω s τ day 360º

23 Incidence Angle on Fixed Surfaces Problem 1.5: at what solar hour will sunset occur in Lisbon, on 23 rd September? (ϕ=38.7º N) R: 5h57min

24 Incidence Angle on Fixed Surfaces Problem 1.6: Show that the zenith angle, θ z, can be expressed using the sunset angle, ω s, as follows: cosθ z = cosδ cosφ ( cosω cosω s )