13 Due 11/2

1. You want to keep your house at 25$\POW9,{\circ}$C. What is the maximum coefficient of performance of a heat pump if it is 10$\POW9,{\circ}$C outside? What is it if it is -20$\POW9,{\circ}$C outside? Assume that if it is -20$\POW9,{\circ}$C outside, the amount of heat needed is four times the amount of heat needed when it is 10$\POW9,{\circ}$C outside. How much more electricity would an ideal heat pump then require if it is -20$\POW9,{\circ}$C outside instead of 10$\POW9,{\circ}$C? Also, if it is 10$\POW9,{\circ}$C outside, then how much more power would a resistance heater require compared to an ideal heat pump?

2. Suppose that you have a basin of 35$\POW9,{\circ}$C warm water that you use to run a heat engine that cools its substance with air at 20$\POW9,{\circ}$C. What is the maximum efficiency of the heat engine? What is the work you get out of each kJ of heat extracted from the basin? Repeat for the case that the water is at 100$\POW9,{\circ}$C. Comment on why efficient car engines run so hot.

3. Assume that your car engine burns 2 kg of fuel per hour in its cylinders at a temperature of 1500 K. The heating value of the fuel is 50 MJ/kg. Assume that the combustion heat is the equivalent of the heat $\dot Q_H$ in a heat engine. Assume that the engine rejects heat at an average temperature of 750 K through its exhaust, radiator, etcetera. Find the maximum amount of horsepower the engine can provide. But don't think you are going to get it from your car.

4. A Carnot refrigerator uses helium as refrigerant. It is 5$\POW9,{\circ}$C inside the fridge and 20$\POW9,{\circ}$C in the kitchen. The helium pressure decreases from 100 kPa to 90 kPa during the heat extraction from the fridge. How much heat is extracted from the fridge per kg of helium in each cycle, and what is the work required?