Serway & Jewett — Heat Engines, Entropy, and the Second Law of Thermodynamics: Problem 16

Suppose you build a two-engine device with the exhaust energy output from one heat engine supplying the input energy for a second heat engine. We say that the two engines are running in series. Let \(e_1\) and \(e_2\) represent the efficiencies of the two engines. (a) The overall efficiency of the two-engine device is defined as the total work output divided by the energy put into the first engine by heat. Show that the overall efficiency \(e\) is given by \(e = e_1 + e_2 - e_1 e_2\). What If? For parts (b) through (e) that follow, assume the two engines are Carnot engines. Engine 1 operates between temperatures \(T_h\) and \(T_i\). The gas in engine 2 varies in temperature between \(T_i\) and \(T_c\). In terms of the temperatures, (b) what is the efficiency of the combination engine? (c) Does an improvement in net efficiency result from the use of two engines instead of one? (d) What value of the intermediate temperature \(T_i\) results in equal work being done by each of the two engines in series? (e) What value of \(T_i\) results in each of the two engines in series having the same efficiency?