Consider the wintertime heating of a house with a furnace compared to addition of Carnot heat engines/pumps. To compensate for heat losses to the surroundings, the house is maintained at a constant temperature T_house by a constant rate of heat transfer, Q̇_house. The furnace operates at a constant temperature T_F, and with direct heat transfer, the heat required from the furnace, Q̇_F is equal to Q̇_house.

(a) Instead of direct heat transfer, if we utilize the surroundings, at T_S, as an additional heat source and include heat pump technology, Q̇_F may be reduced by generating work from a heat engine operating between T_F and T_S, then applying that work energy to a heat pump operating between T_S and Thouse. Given that T_F = 800 K, T = 293 K, T_S = 265 K, and Q̇_house = 40 kJ/h, determine Q̇_F utilizing heat pump technology. No other sources of energy may be used.

(b) Another option is to run a heat engine between T_F and Thouse and the heat pump between T_S and T_house. Compare this method with part (a).