In order to calculate the expected energy output, the expected solar irradiance—solar power per unit area—of the roof was calculated based on solar energy data for Chapel Hill, compiled by Solar Energy Local, and a study on absorbed solar radiation of vaulted roofs compared to flat roofs.
Solar Energy Local collects data for solar radiation measured in three ways: direct normal irradiance (DNI), average tilt at latitude (ATaL), and global horizontal irradiance (GHI). Each method measures the total amount of solar radiation received per unit area, but with a different orientation of the receiving surface to the sun. DNI has a surface always positioned
angle that is equal to the current latitude, often producing optimal energy output [51]. For GHI, the surface is always positioned in a horizontal manner—much like a flat roof. The different average monthly values for each type of measurement are displayed in Figure 24. As shown, GHI has the largest seasonality within the data, with higher amounts of irradiance in the spring and summer than in the fall and winter. In this analysis, the monthly GHI values for Chapel Hill were used because they represent irradiance of a flat roof, which can be related to radiation absorption of a vaulted roof.
Fig. 24. Various Monthly Solar Energy Values in Chapel Hill [51]. This chart depicts the monthly trend for solar radiation in kilowatt hours per square meter per day (kWh/m2/day) for DNI, ATaL, and GHI surface orientations.
After collecting the global horizontal irradiance data for Chapel Hill, this flat roof irradiation was converted to a vaulted roof irradiation based on conversion ratios calculated by the Desert Architecture Unit of the J. Blaustein Institute for Desert Studies at the Ben-Gurion
account angular dependence of absorption and solar geometry. The GHI irradiance in CH was converted to vaulted roof irradiance by using the ratios found in the study, shown inTable 1.
First, it is important to define the parameters and terms used in the vaulted roof analysis and how they relate to this study. A “south-north facing vaulted roof” describes a roof that curves from south to north and has flat planes on the east and west ends. Figure 25 shows a diagram of a south-north facing vaulted roof on a coordinate axis. Figure 26 depicts the spatial orientation of the IPF. As shown, south-north facing best describes the orientation of the roof, so that assumption will be carried out throughout the study.
Fig. 25. South-North Facing Vaulted Roof [53]. Illustration of the orientation of a south-north facing vaulted roof, where R is the radius of the circular segment and ϴ0 is the half angle of the central angle in degrees.
Fig. 26. Aerial View of IPF Spatially [52]. This map shows the roof orientation coincides most closely with a south-north classification.
Similarly to the circular segment, R represents the radius of the circle. In contrast, ϴ0 represents the half angle of the central angle in degrees, rather than in radians. Given that the central angle of the roof is 82 degrees, the half angle is 41 degrees. ƛ represents the latitude of the building geographically. The latitude of Chapel Hill is 35.9 N. KT represents the average clearness index from 0 being the least clear to 1 being the most clear. Qband Qdrepresent monthly average daily absorbed beam radiation and diffuse radiation, respectively. Qtotis the sum of Qb and Qd. Adding the subscript “flat,” for example, Qd, flat, indicates that the radiation is received by a flat roof. Kb, vault and Ktot, vault are defined as Qb/Qb, flat and (Qb + Qd)/(Qb, flat + Qd, flat), respectively. Using a computing program, the researchers computed ratios Kb, vault and Ktot, vault for various values of ϴ0, ƛ, and KT. The results for KT = 0.65 and ƛ = 32 N are shown in Table 1.
Table 1. Irradiance Conversion Ratios. Ratio Kb, vault and Ktot, vault varied with half angle ϴ0 (KT = 0.65 and ƛ = 32 N) for a south-north oriented vaulted roof[53].
As shown, the Ktot, vault ratio decreases as the half angle decreases. The parameters that most accurately resemble the IPF’s parameters (ƛ = 35 N, ϴ0 = 41) from the study are ƛ = 32 N and ϴ0 = 45. Because ratios are only given for the months June, July, and August, ratios for the remaining months of the year were extrapolated from a full set of monthly data of Ktot, vault for a south-north facing vaulted roof at ϴ0 = 90, shown in Table 2. Total annual solar irradiance for Chapel Hill was calculated to be 1770 kWh/m2/year by multiplying monthly GHI by monthly Ktot, vault ratios for ϴ0 = 45, KT = 0.65, and ƛ = 32 N and summing the values.
Table 2. Annual Irradiance Conversion Ratios. Ktot, vault ratios for ϴ0 = 90 and ϴ0 = 45 at ƛ = 32 N for a south-north facing vaulted roof. Extrapolated values are shaded red.
Month Ktot, vault, ϴ0 = 90 Ktot, vault, ϴ0 = 45
Jan 1.35 1.28 Feb 1.29 1.22 Mar 1.18 1.12 Apr 1.10 1.02 May 1.09 1.02 Jun 1.10 1.02 Jul 1.09 1.02 Aug 1.09 1.02 Sep 1.12 1.05 Oct 1.22 1.15 Nov 1.40 1.33 Dec 1.51 1.44