A Carnot engine is operated between two heat reservoirs at temperatures of 300K

Question

A Carnot engine is operated between two heat reservoirs at temperatures of 300K and 400K.

a. If the engine recieves 1200kJ from the reservoir at 400K in each cycle, how much heat is transferred to the reservoir at 300K?

b. If the engine is operated as a refrigerator and recieves 1200kJ from the reservoir at 300K, how much heat does it deliver to the reservoir at 400K?

c. How much work is done by the engine in each case?

d. What is the efficiency of the engine and the coefficient of performance of the refrigerator?

e. Sketch a T-S graph for the heat engine.

Explanation / Answer

a) The net entropy change of the engine per cycle equals zero. On the one hand it gains entropy by absorbing heat from hot reservoir and some entropy may be produced within the engine due to irreversibilities . On the other hand it looses entropy by discarding heat to the cold reservoir. ?S_hot + ?S_irr = ?S_cold Since heat transfer occurs at constant temperature: Q_hot/T_hot + ?S_irr = Q_cold/T_cold For an ideal Carnot engine, which operates reversibly - i.e. ?S_irr=0: Q_hot/T_hot = Q_cold/T_cold Hence, Q_cold = (T_cold/T_hot)·Q_hot = (300K / 520K) · 6.45kJ = 3.72kJ b) The net energy change of the heat engine equals zero, too. So the heat transferred to the engine equals the heat discarded plus the work done by the engine: Q_hot = Q_cold + W Hence, W = Q_hot - Q_cold = 6.45kJ - 3.72kJ = 2.73kJ c) Thermal efficiency is defined as the ratio of useful work done to heat absorbed: ? = W/Q_hot = 2.73kJ / 6.45kJ = 0.423 = 42.3%

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