A roof-cooing system, which operates by maintain a thin film of water on the roof surface, may be used to reduce air-conditioning costs or to maintain a cooler environment in nonconditioned buildings. To determine the effectiveness of such a system, consider a sheet metal roof for which the solar absorptivity is as a₂ 0.5 and the hemispherical emissivity ε 0.3. Representive conditions correspond to a surface heat convection coefficient α 20 W/(m 2.K), a solar irradiation q_s 700 W/m² , a sky temperature of T_s -10^o C , an atmospheric temperature T_{a =} 30^oC, and a relative humidity of φa = 65%. The roof may be assumed to be well insulated from below. Determine the roof surface temperature without the water film. Assuming the film and roof surface temperatures to be equal, determine the surface temperature with the film. The solar absorptivity and hemispherical emissivity of the film-surface combination are a_{sf =} 0.8 and  ε_f = 0.9, respectively.