A parabolic solar collector of clear aperture diameter m focuses sunlight onto a…

PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Geometrical optics and imaging · Geometrical optics and imaging

Optics and Photonics, Ch Radiometry and photometry

Problem

A parabolic solar collector of clear aperture diameter

D = 1.0

m focuses sunlight onto a small target at its focus, working at

f/1

: its image-side cone is described by the f-number through

\sin\theta' = 1/(2f_\#)

, with

f_\#=1

. The Sun subtends a half-angle of

4.65\times10^{-3}

rad as seen from the Earth, and the solar constant (the irradiance on a surface facing the Sun) is

1361

W m

^{-2}

. Treat the Sun's disc as a small, uniform (Lambertian) source and the collector as a lossless imaging system. ((a)) Find the radiance

L_\odot

of the Sun's disc. ((b)) Find the irradiance delivered to the target at the focus, using

I_{\rm image}\approx \pi L_\odot/(4f_\#^2)

. ((c)) Find the irradiance (radiant exitance) leaving the Sun's own surface. Compare it with your answer to (b), and say why, whatever the collector's design, the target's irradiance can never exceed this value. ((d)) Find the largest concentration ratio (image irradiance divided by the solar constant) that any lossless collector could deliver, and the image-side cone at which it would be reached, given as

\sin\theta'

and as the f-number

1/(2\sin\theta')

that describes it.

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