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Machines

Three Carnot ...

Question

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 $T_{1} > T_{2} > T_{3} > T_{4}$ . The three engines are equally efficient if

Diagram

T_{2} = (T_{1}^{3} T_{4})^{1 / 4}; T_{3} = (T_{1} T_{4}^{3})^{1 / 4}

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T_{2} = (T_{1}^{2} T_{4})^{1 / 3}; T_{3} = (T_{1} T_{4}^{2})^{1 / 3}

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T_{2} = (T_{1} T_{4})^{1 / 2}; T_{3} = (T_{1}^{2} T_{4})^{1 / 3}

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T_{2} = (T_{1} T_{1}^{2})^{1 / 3}; T_{3} = (T_{1}^{2} T_{4})^{1 / 3}

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Solution

The correct option is B $T_{2} = (T_{1}^{2} T_{4})^{1 / 3}; T_{3} = (T_{1} T_{4}^{2})^{1 / 3}$
$η_{1} = 1 - \frac{T_{2}}{T_{1}}, η_{2} = 1 - \frac{T_{3}}{T_{2}}, η_{3} = 1 - \frac{T_{4}}{T_{3}}$

Given, $η_{1} = η_{2} = η_{3}$

$1 - \frac{T_{2}}{T_{1}} = 1 - \frac{T_{3}}{T_{2}} = 1 - \frac{T_{4}}{T_{3}}$

$\Rightarrow \frac{T_{2}}{T_{1}} = \frac{T_{3}}{T_{2}} = \frac{T_{4}}{T_{3}}$

So, $T_{2} = \sqrt{(T_{1} T_{3})}$ and $T_{3} = \sqrt{(T_{2} T_{4})}$

$\Rightarrow T_{2} = \sqrt{T_{1} \sqrt{T_{2} T_{4}}}$

$T_{2}^{3 / 4} = T_{1}^{1 / 2} T_{4}^{1 / 4}$

$T_{2} = T_{1}^{2 / 3} T_{4}^{1 / 3}$

Final answer: (a)

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