## Calculations

The calculations that were undertaken to determine the solar insolation for each of the point is described below. They are based on the American Society of Heating, Refrigerating, and Air-Conditioning Engineers (ASHRAE) building heat gain calculations.

## Incident Solar Flux on a Surface

## Total Solar Radiation, G

_{t}:G_{t}= G_{D}+ G_{d}+ G_{R}G_{D}is thedirectincidence of solar radiation on a surface, W/m^{2}G_{d}is thediffuseincidence of solar radiation on a surface, W/m^{2}G_{R}is the ground/surroundingsreflectedincidence of solar radiation on a surface, W/m^{2}## Direct Solar Radiation, G

_{D}:G*for values of A, B, and C refer to Table 7, ASHRAE-F-31.14, 2005_{D}= C_{N}G_{ND}cosθ C_{N}is the sky clearness number (C_{N}=1 for clear sky) θ is the solar incident angle (angle between the suns rays direction and normal to the surface) G_{ND}is thenormal directirradiation, W/m^{2}G_{ND}=A/ (e^{(B/sinβ)})Ais the apparent solar irradiation outside the atmosphere, W/m^{2}βis the solar altitude angleBdimensionless atmospheric extinction coefficient## Diffuse Solar Irradiation, G

_{d}:G_{d}= CG_{ND}((1+cos∑) / 2) C is the ratio of diffuse to direct normal irradiation on a horizontal surface ∑ is the surface tilt angle (slope)## Reflected Solar Irradiation, G

_{R}:G_{R}= G_{tH}ρ_{g}((1 − cos∑) / 2) ρ_{g}is the solar reflectance of ground or horizontal surfaces (albedo) G_{tH}is the total solar radiation (direct + diffuse) on horizontal surface: G_{tH}= G_{ND}sin(β+C)

## Solar Angles

∑: Surface tilt angle (vertical ∑=90°; horizontal ∑=0°) ψ: Surface orientation (azimuth) angle (for surface faced South, ψ=0°; faced W, ψ=90°;faced E, ψ= −90°) θ: Solar incident angle (angle between the sun's rays direction and normal to the surface) β: Solar altitude above the horizontal φ: Solar azimuth measured from south (at West, φ=90°; at East, φ= −90°) γ: Surface − Solar azimuth measured from the south; γ=φ − ψ H: Hour angle (angle between the meridian passing the surface and the sun's rays direction); H=15(LST − 12); (Morning H<0 and afternoon H>0) L: Local Latitude; (North, L>0) δ: Solar declination; δ=23.45sin((360(284+n)) / 365); n is the day of the year (n=1 for 1^{st}of Jan.)## Local Solar Time Calculation

LST = (CT - DT) + (1/15)∘(L_{std}− L_{loc})+E LST: Local Solar Time, hr CT: Clock Time, hr DT: Daylight Savings Time Correction (DT=0 if not on daylight savings time, DT equals to the number of hours advanced) L_{std}: Standard Meridian for local time zone (60° Atlantic, 75° Eastern, 90° Central, 105° Mountain, 120° Pacific, and 135° Alaska) L_{loc}: Local longitude, degree E: Equation of Time, hr E=0.165sin(2B) − 0.126cos(B) − 0.025sin(B) B=(360 / 364)∘(n − 81), n is the day of the year## Solar Angle Relations

Solar Altitude: sinβ=cosLcosδcosH + sinLsinδ Solar Azimuth: cosφ=(sinβsinL− sinδ)/cosβcosL Solar Incident Angle: cosθ=cosβcosγsin∑+sinβcos∑