NOTICEThis report was prepared as an account of vork sponsored by the United States Government. Neither the United States nor the United States Deyartment of Energy, nor any of their employees, aor a q of their contractors, subcontractors, or their employees, makes any warranty, e-rpress or implied, or assumes any legal liaSility or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disciosed, or represents that its use would not infringe privately owned zights. FOREWORDThis report was prepared under Contract No. SUMMARYThe optical analysis of a solar concentrator is usually carried out by means of computer ray tracing, a microscopic method that provides an enormous amount of detailed information but obscures functional relationships. This paper describes a macroscopic approach that yields all the parameters needed for the optical design of line focus parabolic troughs in closed analytical form, requiring only minimal computation.The goal of the optical analysis developed in this report is to determine the flux a t the receiver as a function of concentrator configuration, receiver size, width of sun, and optical errors (e.g., tracking, reflector contour). All causes of image spreading are quantified as angular standard deviation. Ray tracing with a real reflector and a real sun is shown to be equivalent to convoluting the angular acceptance function of a perfect concentrator with an effective radiation source. This effective source, in turn, is obtained by convoluting the distribution function of optical errors with the angular profile of the sun. The problem is reduced to two dimensions by projecting the threedimensional motion of the sun on the plane normal to the tracking axis. In this frame the apparent width of the sun increases as l/cos 8 with incidence angle 8.A formula is derived for the optimal geometric concentration ratio, maximizing net power output as a function of all relevant variables (all-day average insolation, optical errors, effective transmittance-absorptance, heat loss, and concentrator configuration). Graphical solution of this equation consists of finding the intersection between a universal curve and a straight line representing a critical intensity ratio.In the last section, which is written as a self-contained users guide, the results are summarized and illustrated by specific examples. LIST OF FIGURES NOMENCLATUREThe optimization procedure proposed in this paper is based on typical all-day average values of insolation. All-day averages are designated by angular brackets < >* A subscript tc under the bracket indicates that the average is taken over an operating period from tc hours before until tc hours after solar noon. Subscripts I I and i designate angular variables measured parallel or transverse to the tracking axis.%un ( ' ) Angular profile of sun (w/m2 rad) for line focus geometryEffective source (W/m rad) = convolution of solar profile BSun(8) and distribution

The incidence-angle modifier for parabolic troughs is investigated in order to clarify the connection between collector tests and prediction of long-term energy delivery by collector arrays. The optical efficiency of a parabolic trough collector decreases with incidence angle for several reasons: the decreased transmission of the glazing and the absorption of the absorber; the increased width of the solar image on the receiver; and the spillover of the radiation from troughs of finite length. In order to be able to apply test results from a short collector module to collector arrays of arbitrary length, it is necessary to separate analytically the end loss from the first two effects. This analysis is applied to several collectors that have been tested at Sandia Laboratories and at the Solar Energy Research Institute (SERI). The measurements of the incidence-angle modifier at SERI were, carried out at low temperature with an open water test loop for improved accuracy. The results are presented in two forms: as a polynomial fit to the data; and as a single number, the all-day average optical efficiency for typical operating conditions.

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Improved parabolic trough concentrating collectors will result from better design, improved fabrication techniques, and the development and utilization of improved materials. This analysis quantifies the relative merit of various technological advancements in improving the long-term average performance of parabolic trough concentrating collectors and presents them graphically as a function of operating temperature for north-:south, east-west, and polar mounted parabolic troughs. Substantial annual energy gains (exceeding 50% at 350° C) are shown to be attainable with improved parabolic troughs.

This paper describes a .novel alternative to the lase~ ray trace technique for evaluating the optical quality of focusing solar collectors. The new method does not require any-equipment beyond that which is used for measuring collector efficiency; it could therefore become part of routine collector testing. The total optical errors resulting from imperfect specularity and from inaccuracies in reflector position or slope are characterized by an angular standard deviation O'opticalt the r~ deviation of the reflected rays from the design direction. The method is• based on the fact that the off-axis performance of a concentrator depends on a optical• An angular scan is performed; i.e., the collector output is measured as a function of misalignment angle over the entire range of angles for which there is measurable output (typically a few degrees). This test should be carried out on a very clear day, with the receiver close to ambient temperature (if the ' latter condition cannot be satisfied, appropriate corrections are nec_essary). The parameter a optical is then determined by a leastsquares fit • between the measured and the calculated angular scan. We tested the method on a parabolic trough collector manufactured by Hexcel, but it is suitable for parabolic dishes as well.