A very simple solar engine absorbs heat through a collector. The collector loses some of the heat it absorbs by convection, and the remainder is passed through a heat engine to produce electricity. The heat engine operates with one-half the Carnot efficiency with its low temperature side at ambient temperature T_amb and its high-temperature side at the steady-state temperature of the collector, T_c. The expression for the heat loss from the collector is

Rate of heat loss from the collector = k (T_c − T_amb)

Where T₁ is the temperature at which heat enters the solar collector, and k is the overall heat transfer coefficient; all temperatures are absolute. What collector temperature produces the maximum rate at which work is produced for a given heat flux to the collector?