Q. Three Carnot engines operate in series between a heat source at a temperature $T_1$ and a heat sink at temperature $T_4$ (see figure). There are two other reservoirs at temperature $T_2$, and $T_3$, as shown, with $T_2 > T_2 > T_3 > T_4$ . The three engines are equally efficient if:

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Solution:

$t_{1} = 1- \frac{T_{2}}{T_{1}} = 1- \frac{T_{2}}{T_{2}} = 1 - \frac{T_{4}}{T_{3}}$
$ \Rightarrow \frac{T_{2} }{T_{1}} = \frac{T_{3}}{T_{4}}= \frac{T_{4}}{T_{3}} $
$ \Rightarrow T_{2} = \sqrt{T_{1}T_{2}} = \sqrt{T_{1} \sqrt{T_{2}T_{4}}}$
$ T_{3} = \sqrt{T_{2}T_{4}} $
$ T_{2}^{3/4} = \sqrt{T_{1}^{1/2}} T^{1/4}_{4} $
$ T_{2} =T_{1}^{2/3} T_{4}^{1/3} $

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