A receiver runs at under concentration

PHYSICS · Multi-part · Fermi University Physics · Earth, Climate & Energy Physics · Energy resources and conversion · Energy resources and conversion

Energy from Physics, Ch The solar resource resolved: geometry, spectrum and the limits of concentration

Problem

A receiver runs at

T = 500\,\mathrm{K}

under concentration

C = 10

, with

F_{\rm c} = 900\,\mathrm{W\,m^{-2}}

on its aperture. Its surface is an ideal step selective absorber: black below a cut-off wavelength

\lambda_c

and a perfect mirror above it. Per unit wavelength, the concentrated sunlight it receives is

CF_{\rm c}\,M_\lambda(T_\odot)/(\sigma T_\odot^4)

and its own emission is

M_\lambda(T)

, where

M_\lambda(T) = \dfrac{2\pi hc^2}{\lambda^5}\,\dfrac{1}{e^{hc/\lambda k_BT}-1}

is the blackbody emissive power per unit wavelength and

T_\odot = 5772\,\mathrm{K}

. ((a)) Evaluate both spectral curves on a grid of wavelengths between

2

and

5\,\mu\mathrm{m}

, and find the best cut-off

\lambda_c

to the nearest

0.1\,\mu\mathrm{m}

. ((b)) At that cut-off,

0.982

of the

5772\,\mathrm{K}

spectrum and

0.020

of the

500\,\mathrm{K}

spectrum lie below

\lambda_c

. Find the net gain

g(\lambda_c)

per unit area, and compare it with the net gain of a plain black absorber at the same temperature.