The preliminary design of a space station calls for a power cycle that at
steady state receives energy by heat transfer at \(T_{\mathrm{H}}=600
\mathrm{~K}\) from a nuclear source and rejects energy to space by thermal
radiation according to Eq. 2.33. For the radiative surface, the temperature is
\(T_{\mathrm{C}}\), the emissivity is \(0.6\), and the surface receives no
radiation from any source. The thermal efficiency of the power cycle is one-
half that of a reversible power cycle operating between reservoirs at
\(T_{\mathrm{H}}\) and \(T_{\mathrm{C}}\) -
(a) For \(T_{\mathrm{C}}=400 \mathrm{~K}\), determine \(\dot{W}_{\text {cycle }}
/ \mathrm{A}\), the net power developed per unit of radiator surface area, in
\(\mathrm{kW} / \mathrm{m}^{2}\), and the thermal efficiency.
(b) Plot \(\dot{W}_{\text {cycle }} / \mathrm{A}\) and the thermal efficiency
versus \(T_{\mathrm{C}}\), and determine the maximum value of \(\dot{W}_{\text
{cycle }} / \mathrm{A}\).
(c) Determine the range of temperatures \(T_{\mathrm{C}}\), in \(\mathrm{K}\), for
which \(\dot{W}_{\text {cycle }} / \mathrm{A}\) is within 2 percent of the
maximum value obtained in part (b).
The Stefan-Boltzmann constant is \(5.67 \times 10^{-8} \mathrm{~W} /
\mathrm{m}^{2} \cdot \mathrm{K}^{4}\).