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Critical Angle We wish to know the time (or hour angle !) when light is cut off from the white, reflective backplane because of blockage of the light by the tubes. To do this, we need to know something about the angle of the incoming sunlight and the geometry of the array of tubes (or cylinders). The angle of incidence, !, of light on the collector plane is given by the well-known equation [1,2]

!"#! ! !"# ! ! ! !"# ! !"#! ! !"# ! ! ! !!"# ! (1) From previously published studies regarding the geometry of tubular solar collectors, the projected angle, !, in the PE (or transversal) plane, is given by [2,3]

!"# ! ! !"#! !"# !!"#! !!

If the tube axis is oriented N-S, then the transverse plane is perpendicular to the tube axis. Combining these two equations and setting ! ! !! for ! ! !!, one obtains

!"#!! !!"# ! ! !!"#!!

! !"# ! ! ! !!"#!!!"# !

Furthermore, we have for the array of tubes

!"#!! !!! !!!

where D is the tube diameter and d is the center to center spacing between the tubes. For example, if the tubes are spaced out so that every other tube is missing, d = 2D and!!!= 60 degrees. The use of these equations is straightforward and yields some fundamental insights. If the light was always in the transversal plane, the rays reaching the array of tubes would never be skewed and !! ! !!! This is not the condition for most cases. For zero tilt (s = 0) and a latitude, L, of 37! N (approximately that of San José or Santa Clara, CA), we may use the equations above to obtain the plots shown in Figs. A-C from which we obtain the critical angle during the following times of year:

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Summer Solstice (! ! !!!!!"#! !! ! !!"# or 4 hours 37 minutes from solar noon Equinoxes (! ! !"! !! ! !!"!!"# or 3 hours 37 minutes from solar noon Winter Solstice (! ! !!!!!"#! !! ! !!"!!"# or 2 hours 37 minutes from solar noon

This closely matches what is observed experimentally for the horizontally-mounted Solyndra PV module. In the early morning, the incoming light is blocked from the backplane until the critical angle is reached. In addition, a tube is partially shaded by its neighboring tube. For ! ! !! , light reaches the backplane and can be diffusely reflected onto the underside of the tubes. Later in the afternoon, the critical angle is again reached and, for the rest of the day, light does not reach the backplane (and tubes are partially shaded). This helps to explain the shape of the short circuit current, or alternatively the power, output vs. time during the day. This exhibits a nearly linear portion in the morning and afternoon, and a more rounded profile in between that is centered on solar noon. During the two linear portions of the daily curve, the array of tubes behaves like a flat plate collector, albeit one with crests and troughs. One can see this when looking at the array transverse to the cylindrical axis, but at a shallow (nearly horizontal) angle. A full theoretical description of the behavior of arrays of tubular solar collectors can be obtained by using the previously published study by Beekley and Mather [4]. In that DOE/NASA report, they assumed !! ! !!, which is the case only when the tilt is at latitude and the system is considered at the Equinoxes. Although formulated to describe evacuated tube thermal collectors, they first considered the amount of light incident on the surface of the tubes as a function of the geometry of the system. This is applicable to photovoltaic systems that utilize the same arrays of cylinders, but with photovoltaic (PV) absorber materials. References [1] Duffie, J. A.; Beckman, W. A. Solar Engineering of Thermal Processes; Wiley-Interscience: New York, 1991. [2] Mather, Jr., G. R. ASHRAE 93-77 Instantaneous and All-Day Tests of the Sunpak Evacuated-Tube Collector. J. Sol. Energy Eng. 1980, 102, 294. [3] Theunissen, Ph. –H.; Beckman, W. A. Solar Transmittance Characteristics of Evacuated Tubular Collectors with diffuse Back Reflectors. Solar Energy. 1985, 35, 311. [4] Analysis and experimental tests of a high-performance evacuated tubular collector, D. C. Beekley and G. R. Mather, Jr., Publication Date: 12/1978. A PDF is available on line via the U.S. Department of Energy - NASA Technical Report Server (NTRS), DOE/NASA contractor report; DOE/NASA CR-150874", http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19790008199_1979008199.pdf