Solar Energy Vol. 35, No. 5, pp. 393-399, 1985 0038-092X/85 $3.00 + .00 Printed in the U.S.A. © 1985 Pergamon Press Ltd. TRUNCATION OF CPC SOLAR COLLECTORS AND ITS EFFECT ON ENERGY COLLECTION M. J. CARVALHO and M. COLLARES-PEREIRA L.N.E.T.I., Departamento de Energias Renovaveis, Estrada do Paco do Lumiar 22, Lisboa, Portugal J. M. GORDON'[" Applied Solar Calculations Unit, Blaustein Institute for Desert Research, Ben-Gurion University of the Negev, Sede Boqer Campus, Israel and A. RABL Center for Energy and Environmental Studies, Princeton University, Princeton, NJ 08544, U.S.A. IReceived 8 May 1984; revision received 22 February 1985; accepted 22 February. 1985) Abstract--Analytic expressions are derived for the angular acceptance function of two-dimensional compound parabolic concentrator solar collectors (CPC's) of arbitrary degree of truncation. Taking into account the effect of truncation on both optical and thermal losses in real collectors, we also evaluate the increase in monthly and yearly collectible energy. Prior analyses that have ignored the correct behavior of the angular acceptance function at large angles for truncated collectors are shown to be in error by 0-2% in calculations of yearly collectible energy for stationary collectors. l. INTRODUCTION Compound parabolic concentrators (CPC's) are of interest for the design of nontracking solar collec- tors because they achieve the highest possible con- centration ratio evaluate the long-term average energy collection (monthly and yearly), taking into account the ef- fects of truncation in both optical and thermal losses in real collectors. The results can serve as a basis for optimizing the design of CPC collectors. C = Cma× = l/sin 0,, for a specified acceptance half-angle 0a, C being defined as the ratio of aperture area and absorber surface area[l, 2]. There are many different em- bodiments of the CPC, depending on the applica- tion. Of particular importance are the CPC with flat one-sided absorber in Fig. 1 and the CPC with tubular absorber in Fig. 2. The top portion of a CPC reflector is nearly nor- mal to the aperture and contributes therefore little concentration. In practice one usually truncates a CPC to about half of its full height to save reflector material and to reduce the depth of the collector container. For example in Fig. 1 the concentration ratio decreases only from 1.74 to 1.53 if the reflector is truncated to 0.4 of its full height. Truncation modifies the field of view of a CPC, allowing some rays beyond the nominal acceptance half angle 0o to reach the absorber. The resulting change in the collection of diffuse radiation is easy to estimate because a collector of concentration ratio C accepts 1/C of isotropic radiation[2]. But the increased collection of beam insolation outside the nominal acceptance angle has not been analyzed before, and forms one subject of this paper. We also ~"Present address: Dept. of Civil, Environmental & Ar- chitectural Engineering, University of Colorado, Boulder, CO 80309 2. ANGULAR ACCEPTANCEFUNCTION TO discuss the effect of truncation on the field of view, it is convenient to define an angular ac- ceptance function F(O) as that fraction of the rays incident on the aperture at angle 0 that reaches the absorber. It is a purely geometric property and does not include absorption in the reflector. For trough- like concentrators the angular acceptance function depends only on the projected incidence angle 0p, defined as the angle between the optical axis and the projection of the incident ray onto the plane perpendicular to the trough axis[2]. In this section, we designate the incidence angle by 0, omitting the subscript p for simplicity. For an untruncated CPC the angular acceptance function is given by F(O) = 1 101_< 0o F<0) = 0 101> 0° regardless of absorber shape. On the other hand, the effect of truncation does depend on the shape of the absorber. Let us begin with the case of the flat one-sided absorber (Fig. 1). To understand the following arguments the reader may note that the left portion of the reflector is a parabola whose axis makes an angle 0, with the optical axis and whose focus is at the right edge B of the absorber. The reflector extends from the absorber to the point E, 393