In 2009, a group of environmentalists approached the Boulder City Council with a request to deny an application to continue operating the local coal-fired \(200 \mathrm{MW}\) power plant in order to reduce \(\mathrm{CO}_{2}\) emissions. The group suggested that the coalfired power plant be replaced by a nonpolluting solar plant.

To evaluate the feasibility of their request, a steel plate is placed in sunlight in Boulder at an angle equal to the latitude. The plate is treated with a coating that has an absorptivity for solar insolation of 0.9 and an emissivity in the long-range spectral range of 0.2.

a. If the average insolation between 9 a.m. and 5 p.m. is equal to \(400 \mathrm{~W} / \mathrm{m}^{2}\) and the convective heat-transfer coefficient from the top surface of the plate is \(8 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}\), estimate the equilibrium temperature of the plate assuming the bottom is insulated and the effective sky temperature is \(50 \mathrm{~K}\).

b. If you could operate a heat engine between the plate temperature calculated in part (a) and the environment at \(10^{\circ} \mathrm{C}\), estimate the plate surface area required to generate power equal to the coal-fired power plant. Then comment on the result and compare it with a PV system of \(12 \%\) efficiency.