The Path of the Sun Larry Caretto Mechanical Engineering 83 Alternative Energy Engineering II March, 2 Outline Review radiation properties for solar collectors Orientation of earth and sun Earth-based solar path Definition of angles Calculation of path Optimum tilt angles for solar collectors Solar air mass 2 From last lecture From Çengel, Heat and Mass Transfer 3 Emissivity and Absorptivity Data on previous chart are due to difference in radiation properties associated with radiation temperature For solar collector incoming radiation has source temperature about 58 K Outgoing radiation from collector has source temperature just over 3K Kirchoff s Law says, but this is true only at individual wavelengths What is average over all wavelengths? Average Radiation Properties Integrated average properties over all wavelengths E b d' E b d' Look at simple example where for < and 2 for > E ' ' + b d E b d 2E b d' 5 Average Radiation Properties II Rearrange to get f, the fraction of black body radiation between and Eb d 2 ' + ' + 2 σ Eb d f T ( f ) Similar equation for absorptivity ( ) Eb d 2 ' + ' + 2 σ Eb d f T ( f ) 6 ME 83 Alternative Energy Engineering II
Example Data:.9 for < 3 μm and.2 for > 3 Solar T 58 K, T 7, μm K, f (7, μm K).9855, find Use f + f.9.98 +.2.98 ( ) ( ) ( ). 886 58 K 2 Earth T 3 K, T 9 μm K, f (7, μm K)., find ( ).9(.) +.2(.). 2 3 K f + 2 f 7 8 Declination Angle (degrees 3 2 - -2 Declination Angle & Relative Earth-Sun Distance Solar Declination Angle Declination angle Relative distance.3.2...99.98 Relative Earth-sun Distanc Solar Angles Azimuth is daily variation around horizon Declination is annual altitude variation -3.97 8 2 6 2 2 28 32 36 Julian Date 9 Solar Angles II Solar Angles III Sometimes called altitude angle from the sun center 2 φ s ME 83 Alternative Energy Engineering II 2
Angles for Tilted Collector Tangent plane to earth s surface at 3 given location Equation of Time Solar Time Standard Time + Equation of Time + ( min/ o ) * (Standard Longitude Local Longitude) Standard Time DST hour 5 Computing the Sun Path Input data: Latitude, L, date, hour h Find declination from serial date, n δ o 36 π ( 23.5 ) sin ( 28 + n) ( δ in degrees) 365 8 Two angles: altitude () and azimuth (φ) sin() sin(l) sin(δ) + cos(l) cos(δ) cos(h) sin( s ) sin(φ) cos(δ) sin(h) / cos() Sun path is plot of vs. φ s for one day Plot is symmetric about solar noon Typically plot data for 2 st of month 6 Path Calculation Problem Angles given as sin(angle) x require arcsin function calculation Typical arcsine function returns angle between 9 o and 9 o Limits correspond to range for sine between and + Special calculation for hour angle limit h limit ±tan(δ) / tan(l) φ ±[π arcsin(sin φ)] for h > h limit 7 8 ME 83 Alternative Energy Engineering II 3
Optimum Fixed Collector Tilt Solar 35 o Insolation by collector orientation and month at o N latitude 9 2 Solar Irradiance Cycles 2 22 23 2 ME 83 Alternative Energy Engineering II
25 26 July Solar Energy 983-93 April Insolation 98-993 27 28 January Insolation Solar Irradiation by Month in Los Angeles (LAX) Average of Monthly 96-99 NREL Data for different collectors 9 Irradiation (kwh/m 2 /day) 8 7 6 5 3 Fixed, tilt Fixed, tiltl-5 Fixed, tiltl Fixed,tiltL+5 Fixed,tilt9 -axis,track,ew horizontal -axis,track,ns horizontal -axis,track,tiltl axis,tiltl+5 2-axis,track 2 Notes: All fixed collectors are facing south The L in tilt L means the local latitude (33.93 o N) Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Month 29 3 ME 83 Alternative Energy Engineering II 5