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Technical Briefs

Derivation of the Angular Dispersion Error Distribution of Mirror Surfaces for Monte Carlo Ray-Tracing Applications

[+] Author and Article Information
T. Cooper

Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zürich, Switzerlandtcooper@ethz.ch

A. Steinfeld1

Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zürich, Switzerland; Solar Technology Laboratory,  Paul Scherrer Institute, 5232 Villigen PSI, Switzerlandaldo.steinfeld@ethz.ch

1

Corresponding author.

J. Sol. Energy Eng 133(4), 044501 (Oct 13, 2011) (4 pages) doi:10.1115/1.4004035 History: Received October 11, 2010; Revised March 01, 2011; Published October 13, 2011; Online October 13, 2011

Of paramount importance to the optical design of solar concentrators is the accurate characterization of the specular dispersion errors of the reflecting surfaces. An alternative derivation of the distribution of the azimuthal angular dispersion error is analytically derived and shown to be equivalent to the well-known Rayleigh distribution obtained by transforming the bivariate circular Gaussian distribution into polar coordinates. The corresponding inverse cumulative distribution function applied in Monte Carlo ray-tracing simulations, which gives the dispersion angle as a function of a random number sampled from a uniform distribution on the interval (0,1), does not depend on the inverse error function, thus simplifying and expediting Monte Carlo computations. Using a Monte Carlo ray-tracing example, it is verified that the Rayleigh and bivariate circular Gaussian distribution yield the same results. In the given example, the Rayleigh method is found to be ∼40% faster than the Gaussian method.

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Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 4

Right branch of a horizontal slice through the focal plane flux distribution of the HFSS showing equivalence of ray-tracing predictions of the Rayleigh and Gaussian methods. The ray-tracing results are for the best-fit of σerr  = 3.65 mrad from the parameter tuning. Experimental data taken from Ref. [10].

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Figure 3

Probability density function f(θerr ) for the derived distribution showing equivalence to the Rayleigh distribution. The mode occurs at θerr /σerr  = 1.

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Figure 2

Schematic of the different classes of dispersion errors for the facets of a mirror surface. Tracking error is not shown.

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Figure 1

Definition of the azimuthal and circumferential angular dispersion errors, θerr and φerr in terms of: the incident ray r ; specular ray r ′; deviant ray r ″; reflection dispersion error vector uerr , and its Cartesian components ux and uy ; surface normal n ; deviant normal n ′; and normal error vector nerr for a surface element dS

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