Question

42. You can prove in the following way that it is overall more efficient for an

engine to exhaust at the ambient environmental temperature than to exhaust

at a cooler temperature that requires refrigeration. Suppose that the hot

reservoir is at Th, the ambient temperature is Ta, and the refrigerated tem-

perature is Tc. The engine cannot put out more work than would a Carnot

engine operating between Th and Tc, so Weng = eQh = a[(Th − Tc)/Th] Qh,

where a ≤ 1. Likewise the refrigerator cannot require less work input than

a Carnot refrigerator, so Wfrig = bQa[(Ta − Tc)/Ta], where b ≥ 1. The net

work output is that from the engine minus that needed to run the refrigerator,

Wnet = Weng − Wfrig. Show that this quantity is less than the work you would

get out of the engine if it simply ran between Th and Ta without any refrig-

eration, which would be Weng = aQh[(Th − Ta)/Th]. (For the Carnot engine,

Qh/Th = Qa/Ta.)

Answer 1