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, Gt:

Gt = GD + Gd + GR
		
GD is the direct incidence of solar radiation on a surface, W/m2
Gd is the diffuse incidence of solar radiation on a surface, W/m2
GR is the ground/surroundings reflected incidence of solar radiation on a surface, W/m2
		
solar flux diagram

Direct Solar Radiation, GD:

GD = CNGNDcosθ
	
CN is the sky clearness number (CN=1 for clear sky)
θ is the solar incident angle (angle between the suns rays direction and normal to the surface)
GND is the normal direct irradiation, W/m2
	
GND = A / (e(B/sinβ))
	
A is the apparent solar irradiation outside the atmosphere, W/m2
β is the solar altitude angle
B dimensionless atmospheric extinction coefficient
	
*for values of A, B, and C refer to Table 7, ASHRAE-F-31.14, 2005

Diffuse Solar Irradiation, Gd:

Gd = CGND((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, GR:

GR = GtHρg((1 − cos∑) / 2)

ρg is the solar reflectance of ground or horizontal surfaces (albedo)
GtH is the total solar radiation (direct + diffuse) on horizontal surface:
	GtH = GNDsin(β +C)
	
		

Solar Angles

solar
∑: 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 1st of Jan.)
		

Local Solar Time Calculation

LST = (CT - DT) + (1/15)∘(Lstd − Lloc)+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)
Lstd: Standard Meridian for local time zone 
	(60° Atlantic, 75° Eastern, 90° Central, 105° Mountain, 120° Pacific, and 135° Alaska)
Lloc: 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∑