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Optical Analysis of Point Focus Parabolic Radiation Concentrators

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Optical Analysis of Point Focus Parabolic Radiation Concentrators SERI/TR-631 -336 April 1980 Optical Analysis of Point Focus Parabolic Radiation Concentrators Paul Bendt Ari Rabi DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. Printed in the United States of America Available from: National Technical Information Service U.S. Department of Commerce 5285 Port Royal Road Springfield, VA 22161 Price: Microfiche $3.00 Printed Copy $4. 50 NOTICE This report was prepared as an account of work sponsored by the United States Govern­ ment. Neither the United States nor the United States Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owne? rights. r SERI/TR-631-336. UC CATEGORYs UC-62 OPTICAL ANALYSIS OF POINT FOCUS PARABOLIC RADIATION CONCENTRATORS PAUL BENDT ARI RABL APRIL 1980 ~-------DISCLAIMER---------, This book was prepare<! asan account of WO!k sponsored by an agency of the United S1a1es Government. Neither the Uniled States Government nor any agency thereof, nor any ol their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any informa1ion, apparatus, producl. or process disclosed, or represents thal its uso would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or Otherwise, does not necessarily cons1i1ute or imply its endorsement, recommendation. or favoring by the United States Government or· any agency thereof, The views and opinions of authors expressed herein do no1 necessarily state or rellect those of tho United States Government or any agency thereof. i .. PREPARED UNDER TASK No. 3527 Solar Energy Research Institute 1536 Cole Boulevard Golden, Colorado 80401 A Division of Midwest Research Institute Prepared for the U.S. Department of Energy Contract No. EG · 77·C·01·4042 DiGTR!BUT!llfl OF THIS OOCIJME•T I~ UNLlfilT1~. THIS PAGE WAS INTENTIONALLY LEFT BLANK S:~1,·-~, __________________________TR_-_3_36 FOREWORD This report documents work done on Task 3527 in the Solar Thermal Research Branch of the Solar Energy Re·search Institute. Frank Kreith, Branch Chief Solar Thermal Research Branch Approved for: SOLAR ENERGY RESEARCH INSTITUTE Division ' iii ·.r·.,. THIS PAGE WAS INTENTIONALLY LEFT BLANK S:~I f"'-1, ____________________________T_R_-_3_3_6_ SUMMARY A simple formalis~ is developed for analyzing the optical performance of point focus parabolic radiation concentrators. To account for off-axis aberrations of the parabola, ari angular acceptance function is defined as that fraction of a beam of parallel ~adiation incident on the aperture which would reach the receiver if the . optics· were perfect. The radiation intercepted by the re­ ceiver of a real concentrator is obtained as a convolution of angular accep­ tance function, of .optical error distribution, and of angular brightness dis­ tribution of the radiation source. Losses resulting from ab~orption in the reflector or reflection at the receiver are treated by a multiplicative factor pa where. p = reflectance -of the reflector and a = absorptance of the re­ ceiver. For· numerical calculations, ·this method is more accurate and less time-consuming than the ray-tracing method. In many cases, there are accept­ able approximations whereby the results can be obtained by reading a graph or evaluating a simple c~rve fit. V • ,ttJr"• T THIS PAGE · WAS INTENTIONALLY LEFT BLANK 55~1,fl, --------------------------TR-336 TABLE OF CONTENTS 1.0 Introduction •••••••••••••••••••••••••••••••••••••••••••••••••••••••• 1 2.0 Angular Acceptance Function••••••••••••••••••••••••••••••••••••••••• 3 2.1 Symmetry Considerations •••••••••••••••••••••••••••••••••••••••• 3 2.2 Spherical, Receiver••••••••••••••••••••••••••••••••••••••••••••• 6 2.3 Flat Receiver•••••••••••••••••••••••••••••••••••••••••••••••••• 9 3.0 Calculation of Flux at Receiver••••••••••••••••••••••••••••••••••••• 19 3.1 Effective. Source••••••••••••••••••••••••••••••••••••••••••••••• 19 3.2 Flux at Receiv~r ••• e••••••••••••••••••••••••••••••••••••••••••• 22 3.3 RMS Width for 2-D Distributions •••••••••••••••••••••••••••••••• 23 4.0 Example: Circular .Gaussian Source ••••••••••••••••••••••••••••••• : •• 25 s.o Referenc~s •••••••••••••••••••••••••••••••••••••••••••••••••••••••••• 29 vii THIS PAGE W.AS INTENTIONALLY LEFT BLANK sa,1 TR-336 LIST OF FIGURES 2-1 Parabolic dish reflector••••••••••••••••••••••••••••••••••••••••••••• 4 2-2 Geometrical relations for the calculation of the angular acceptance function of parabolic dish wit1'. ·,·.,\._(~rfr·.;:,·; .receiver................... 7 (~ 2-3 Geometrical relations for the calculation of ·the angular acceptance function of parabolic dish with flat receiver, showing elliptical boundary of rays emitted by receiver and reflected at P ••••••••••••••••••••••••••••••••••••••.• .- • • • • • • • • • • • • • • • • • • • • • • • 11 2-4 Angular acceptance function for parabolic dish with flat receiver ••• 17 3-1 Equivalence between (a) imperfect reflector with point source and (b) perfect reflector with smeared source••••••••••••••••••••••• 20 4-1 Intercept factory for parabolic dish with flat receiver, geometric concentration C, and rim angle~, if source is gaussian with width a••••••••••••••••••••••••••••••••••••••••••••••••••••••••· 27 LIST OF TABLES 2-1 Coefficients of polynomial fit Eq. 2-44 to angular acceptance function, with v1 = 81 -yG and v2 = 82 ~; (a) three-term expan- sion, and (b) four-t.erm expansion • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 18 l.x. :r· ..... THIS PAGE WAS INTENTIONALLY LEFT BLANK· S:fl ,----- _______________________r_R_-3_3_6 11 BOMKNCLATUU a = ratio of absorber radius over focal length b = ratio of reflector radius over focal length 2 8source ( S) = angular profile of source (W/m sterad) = effective source (W/m2 sterad) • convolution+ of source profile Bsource(0) and distribution function E(e) of optical errors C = geometric concentration ratio = ratio of aperture area over receiver surface area Eoptical(S) = distribution function of optical errors (sterad-l) f(e) = angular acceptance function = fraction of rays incident on aperture at incidence angle e from optical axis, which reach receiver = beam component of solar irradiance (W/m2), as measured by pyrheliometer = that portion of lb which reaches the receiver (in W per m2 of aperture area) a = absorptance of receiver y = intercept factor Iin/Ib = rim angle 0 optical = angular spread caused by all optical errors 0 source = angular width of source a = total beam spread p = reflectance of reflector xi . S:~I t'"·,, --------'-----.,..-------------------TR_-_3_'3_6 SECTION 1.0 INTRODUCTION Parabolic reflectors are employed to concentrate incoming radiation onto a smaller receiver. · For example, in solar energy applications, concentrators can reduce heat losses in thermal collectors (Rabl 1976); and in radiation de­ tectors, the signal-to-noise ratio may be improved if the incident radiation is concentrated onto a smaller receiver (Harper et al. 1976). In these situa­ tions' the incident rays come from a range of directions (e.g.' from the en­ tire solar disk) and their angular spread is further increased by optical er­ rors. Optical errors may be caused by deviations of the reflector surface from the specified parabolic contour, by lack of perfect specularity of the reflector material (Pettit 1977; Butler and Pettit 1977), by displacement of the receiver from its. design position, .etc. In these applications, the imag­ ing properties of the parabola are not directly relevant--one needs to . know how much of the incident radiation is intercepted by the receiver and, per­ haps, the spatial or angular intensity distribution, but the exact correspon­ dence between points of the radiation source and impact points on the receiver is immaterial. Traditionally, the optical analysis of radiation concentrators has been carried out by means of computer ray-trace programs* (Biggs and Vittoe; Schrenk 1963). Ray tracing is a microscopic method that can provide an enormous amount of numerical information but obscures functional relationships. Recognition of functional relationships is extremely useful for design optimization. Recently, an interesting analytical solution for the
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