An engineer proposes a two-stage solar furnace to beat the Sun's own surface temperature

PHYSICS · Subjective · Fermi University Physics · Optics & Photonics · Earth, Climate & Energy Physics · Geometrical optics and imaging

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

Problem

An engineer proposes a two-stage solar furnace to beat the Sun's own surface temperature, and argues as follows: "A primary parabolic mirror gives concentration

C_1 = 200

. I focus its output beam onto a second, smaller concentrator that itself achieves

C_2 = 250

when fed with parallel light. Feeding the output of the first stage into the second multiplies the ratios, so the overall system concentration is

C = C_1 \times C_2 = 50\,000

, above the quoted air limit of

C_{\max} = 46\,200

. Using the stagnation-temperature formula

T_{\rm abs} = T_\odot(C/C_{\max})^{1/4}

with

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

, this gives

T_{\rm abs} = 5772\,\mathrm{K}\times(50\,000/46\,200)^{1/4} = 5887\,\mathrm{K},

already above

T_\odot

. Adding a third stage with

C_3 = 100

gives a total

C = 5\,000\,000

and

T_{\rm abs} = 5772\,\mathrm{K}\times(5\,000\,000/46\,200)^{1/4} = 18\,600\,\mathrm{K},

so with enough stages a solar furnace can comfortably exceed

6000\,\mathrm{K}

, whatever materials allow it to survive." Find the error in this reasoning, and state the highest stagnation temperature that any passive arrangement of mirrors, however many stages, can in principle deliver from unconcentrated sunlight outside the atmosphere.

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