Chapter 6. Solar Geometry Contents 6.1 Introduction 6.2 The Sun 6.3 Elliptical Orbit 6.4 Tilt of the Earth s Axis 6.5 Consequences of the Altitude Angle 6.6 Winter 6.7 The Sun Revolves Around the Earth! 6.8 Sky Dome 6.9 Determining Altitude and Azimuth Angles 6.10 Solar Time 6.11 Sun-Path Diagrams 6.12 Solar Site-Evaluation Tools 6.13 Sun Machines (Heliodon) 6.14 Sundials for Model Testing 6.15 Conclusion Appendix SOLAR RADIATION AND GEOMETRY
6.1 INTRODUCTION In sustainable or green design in architecture, the buildings are designed to let the sun shine in during the winter and are shaded from the sun in the summer. This approach to architecture requires that the designer have a good understanding of the relationship of the sun to the earth. 6.2 THE SUN The sun is a huge fusion reactor in which light atoms are fused into heavier atoms, and in the process energy is released. The reaction can occur only in the interior of the sun where the necessary temperature of 14,000,000 exists. The solar radiation reaching earth is emitted from the sun s surface, which is much cooler (Fig. 6.2a).
The amount and composition of solar radiation reaching the outer edge of the earth s atmosphere are quite varying and are called the solar constant. The amount and composition of the radiation reaching the earth s surface, however, vary widely with sun angles, elevation, and the composition of the atmosphere (Fig. 6.2b). 6.3 ELLIPTICAL ORBIT The orbit of the earth is not a circle but an ellipse, so that the distance between the earth and the sun varies as the earth revolves around the sun (Fig. 6.3). The distance varies about 3.3 percent, and this results in a small annual variation in the intensity of solar radiation. Since the sun is very far away and since it lies in the plane of the earth s orbit, solar radiation striking the earth is always parallel to this plane.
6.4 TILT OF THE EARTH S AXIS Because the tilt of the earth s axis is fixed, the Nothern Hemisphere faces the sun in June and the Southern Hemisphere faces the sun in December (Fig. 6.4a). On June 21, the sun s rays are perpendicular to the earth s surface along the Tropic of Cancer, which is at latitude 23.5 N (Fig. 6.4b). It is also the longest day in the Northern Hemisphere, and is called the summer solstice ( 夏至 ).
On December 21, at the opposite end of the earth s orbit around the sun, the North Pole points so far away from the sun. In the Northen Hemisphere this is the day with the longest night and is also known as the winter solstice ( 冬至 ). On this day, the sun is perpendicular to the Southern Hemisphere along the Tropic of Capricorn, which is at latitude 23.5 S (Fig. 6.4c). 6.5 CONSEQUENCES OF THE ALTITUDE ANGLE The first effect of the altitude angle is illustrated by Fig. 6.5b, which indicates that at low altitude angles the sun s rays pass through more of the atmosphere. Consequently, the radiation reaching the earth surface will be weaker and more modified in composition (spectrum).
The second effect of the altitude angle is illustrated in the diagram of the cosine law (Fig. 6.5c). This law says that a given beam of sunlight will illuminate a larger area as the sun gets lower in the sky. As the given sun beam is spread over larger areas, the sunlight on each square unit of land naturally gets weaker. The amount of sunlight that a surface receives changes with the cosine of the angle between the sun s rays and the normal to the surface. 6.6 WINTER Now we can understand what causes winter. The temperature of the air, as well as that of land, is mainly a result of the amount of solar radiation absorbed by the land. The air is mainly heated or cooled by its contact with the earth. The reason for less radiation falling on the ground in the winter are the following. 1) Most important is the fact that there are fewer hours of daylight in the winter, i.e. shorter daytime hours. 2) The second reason for reduced heating of the earth is the cosine law due to the low solar altitude angles. 3) Lastly, the lower sun angles increase the amount of atmosphere the sun must pass through, therefore, there is again less radiation reaching each square meter of land, i.e. low value of W/m 2.
6.7 THE SUN REVOLVES AROUND THE EARTH! Even though actually the earth revolves around the sun, the assumption that the sun revolves around the earth makes it more convenient to understand sun angles. 6.8 SKY DOME An imaginary sky dome can be placed over the building site (Fig. 6.8a). The position of the sun on the sky at different hours are marked. When all the points for one day are connected, it becomes a line on the sky dome called the sun path for that day. Figure 6.8a shows the highest sun path of the year on June 21 (summer solstice 夏至 ), the lowest sun path on December 21 (winter solstice 冬至 ), and the middle sun paths on March 21 and September 21(equinoxes 春分秋分 ).
Since the solar radiation is quite weak in the early and late hours of the day, the part of the sky dome through which the most powerful sun rays enter is called the solar window. Figure 6.8b shows the conventional solar window, which is assumed to begin at 9 A.M. and end at 3 P.M. Ideally, no trees, buildings, or other obstacles should block the sun rays entering through the solar window during those months when solar energy is desired. 6.9 DETERMINING ALTITUDE AND AZIMUTH ANGLES The most useful sun angles related to solar radiation are the altitude angle, which is measured vertically, and the azimuth angle, which is measured horizontally.
6.10 SOLAR TIME At 12 noon solar time, the sun is always due south. However, the sun is not due south at 12 noon clock time because solar time varies from clock time. The first reason is the deviation in longitude of the building site from the standard longitude of the time zone. The second reason is a consequence of the fact that the earth s rotation speed in its orbit around the sun changes during the year. However, changing solar time to clock time or vice versa is almost never necessary. In most cases, the sun angles are calculated for solar times. 6.11 SUN-PATH DIAGRAMS Altitude and azimuth angles can be readily from sun-path diagrams. In Fig. 6.11a we see the sky dome, but this time it has a grid of altitude and azimuth lines drawn on it, just as a globe of the earth has latitude and longitude lines. Just as there are maps of the world that are usually either cylindrical or polar projections, there are vertical or horizontal projections of the sky dome.
1) Horizontal Sun-Path Diagram When the sun paths are plotted on a horizontal projection of the sky dome, we get a sun-path diagram such as the one shown in Figure 6.11c. 2) Vertical Sun-Path Diagram
6.12 SOLAR SITE-EVALUATION TOOLS
6.13 SUN MACHINES (HELIODON) A heliodon simulates the relationship between the sun and a building. The three variables that affect this relationship are latitude, time of year, and time of day. Every heliodon has a light source, an artificial ground plane, and three adjustments so that the light will strike the ground plane at the proper angle corresponding to the latitude, time of year, and time of day desired.
6.14 SUNDIALS FOR MODEL TESTING The least expensive way to test models for shading, solar access, and daylighting is to use a sundial (Fig. 6.16). A sundial would be mounted on a model so that its south and that of the model align. The model along with the sundial is then rotated and tilted until the shadow of the gnomon (peg) points to the time and day to be analyzed.
6.15 CONCLUSION The concepts on the relationship between the sun and the earth are fundamental for an understanding passive solar energy, shading, passive cooling, and daylighting for buildings. Solar radiation reaching the earth s surface consists of about 47 % visible (light), 48 % shortwave infrared (heat), and about 5 % ultraviolet radiation. The solar window is that part of the sky dome through which the sun shines throughout the year. Sun path diagrams present both the pattern of the sun s motion across the sky and specific sun angle data. APPENDIX SOLAR RADIATION AND GEOMETRY Fusion Energy Generated by the Sun Einstein s Mass-Energy Equivalence Equation: E = MC 2 [J] [kg/(m/s) 2 ] Where, M=Mass, C=Speed of Light (Approx. 300,000 km/s) J = Nm = (Mass x Acceleration) m= (kg x m/s 2 )m = kg / (m/s) 2
Spectrum of Solar Radiation 1) ultra violet ( 자외선 ): 5% - Wave length: 20-380 nm - 315-380 nm (UV-A): Black light. Causes tanning and premature aging of skin. Not absorbed by the ozone layer - 280-315 nm (UV-B): Dangerous UV. Produces Vitamin D in human skin. Causes skin cancer. Mostly absorbed by the ozone layer. - 200-208 nm (UV-C): Germicidal UV (kills microorganisms such as bacteria, viruses, molds). Mostly absorbed by the ozone layer 2) visible light ( 가시광선 ): 46% - Wave length: 380-780 nm - Light: Sensed by human eyes. Used by plants for photosynthesis. 3) Infrared ( 적외선 ): 49% - Wave length: 780-3000 nm - Thermal radiation (heat) Absorption and Scattering of Solar Radiation in the Atmosphere 1) Absorption ( 흡수 ) - Affects spectral power intensity (W/m 2 λ) and change the color of visible light Ozone (O 3 ): Absorbs ultra violet radiation (UV rays shorter than 300 nm are almost absorbed Oxygen (O 2 ): Absorbs visible light Moisture (H 2 O): Absorbs short-wave infrared radiation (780-2000 nm) Carbon Dioxide (CO 2 ): Absorbs long-wave infrared radiation (2000 nm or longer)
2) Scattering ( 산란 ) - Affects the direction of radiation and the color of light (1) Reyleigh scattering (blue sky): by air molecules (very small) (2) Mie scattering (overcast sky and white clouds): by airborne water droplets or dusts (relatively large) Solar Geometry ( 태양기하학 ) Sunrise and Sunset hours Sun angles (Altitude, Azimuth, Profile, Incident) To calculate solar irradiation To calculate outdoor illuminance To design sun shades 1. Earth-to-Sun Distance(R), Declination angle ( 적위 ), and Day angle ( 일각 ) 1) Distance between the sun and the earth - Elliptical Orbiting of the Earth around the sun: The extraterrestrial solar constant (W/m 2 ) changes according to the changing distanct. The speed of earths revolve also changes.
Average Distance R O = 1 A.U.(Astronomical Unit)= approximately 150,000,000km Extratrestrial solar constants: Irradiation= 1353 W/m 2, Illuminance= 127,500 lx Sun Diameter 1.3910 6 km θ θ=32 Earth Diameter 12,700km R 8 O 1.49610 km R MIN = 0.983 A.U.(on January 3), Solar Constant 1,418 W/m 2 R MAX = 1.017 A.U.(on July 4), Solar Constant 1,325 W/m 2 Equation: R O R 2 2J r 1 0.033cos 365 [km] R R O r J: Julian date, 1 J 365 (e.g. Feb. 1=32, Dec. 31=365) 2) Declination Angle () - The angle subtended by the equator and the line connecting the centers of the sun and the earth Summer solstice (6/21): 23.45 Winter solstice (12/21): -23.45 23.45 Equator Equinoxes (3/21 and 9/21): 0 δ Sun Earth s axis Earth 360( J 284) 23.45 sin [] 365
3) Day Angle (d) Jan. 2 =0.986 365 days Sun d Jan. 1 =0 Earth Earth d 360( J 1) 365 [ ] Sun 2. Solar Angles (Sun location) Zenith angle Azimuth angle Altitude angle Profile angle Incident angle The zenith angle ( 천정각 ) or altitude angle ( 고도각 ) is calculated first and the other angles are calculated for different purposes. Azimuth angle ( 방위각 ): Used to design vertical fins of sunshades Profile angle ( 일영각 ): Used to design horizontal overhangs of sunshades Incident angle ( 입사각 ): Used to calculate external irradiation [W/m 2 ] or illuminances [lx]
1) Zenith angle (Z) and Altitude angle cos Z sin sin sin cos cos cos sin 1 (sin sin cos cos cos[] ) ψ: site latitude (north +, south -) [] : declination angle [] : solar hour angle (24 hours=360, 1 hour=15, Noon =0), = (12 - T) x 15 [] T: Solar time T T T T s S L L 15 s e 60 4( L LS ) e 60 T S : Local standard time(clock time) L S : Longitude of Standard Meridian. L S of Korea = 135E L: Longitude of Site e: equation of time in minute(min) (discrepancy in solar time and local standard time) Latitude of Seoul= 3734 37.5 N, Longitude of Seoul= 12658 127E Equation of Time ( 균시차 ) The equation of time describes the discrepancy between the solar time (also called apparent solar time) and the local standard time (also called mean solar time). It is caused by the elliptical orbiting of the earth around the sun. 4 2 e 0.170 sin ( J 80) 0.129 sin ( J 8) 60 373 365 [min] Clock with auxiliary dial displaying the equation of time. Piazza Dante, Naples (1853). 2) Solar azimuth angle sin cos sin cos cos sin sin 1 cos a.m.= (+), p.m.=( -)
3. Profile Angle and Incident Angle ( 일영각, 입사각 ) 1) Profile angle (P) tan p tan cos ws P tan 1 tan ( ) cos ws [] N W E S w Φ ws Φ ws : Wall-Solar azimuth angle = w - where,: w azimuth angle of wall 2) Incident Angle (i) cosi cos cos ws i cos 1 (cos cos ws ) [] i normal line
4. Sunrise and Sunset Hours ( 일출및일몰시각 ) 1) Sunrise hour angle ( s ) and Sunrise hour (SRH) The sun altitude at the moment of sunrise is assumed to be 0. = sin 1 (sin sin cos cos cos cos cos 1 ( tan tan ) s S s sin sin (tan tan ) cos cos [] SRH= 12 - s /15 [in solar time] ) 0 2) Sunset hour (SSH) In solar time, the duration of morning and afternoon hours are the same. SSH = 12+ s /15 [in solar time]