First slide

Heat engine

Question

Two Carnot engines A and B are operated in series. The engine A receives heat from the source at temperature $T_{1}$ and rejects the heat to the sink at temperature T. The second engine B receives the heat at temperature T and rejects to its sink at temperature $T_{2}$ . For what value of T the efficiencies of the two engines are equal

Easy

A

$\frac{T_{1} - T_{2}}{2}$
B

$T_{1} T_{2}$
C

$\sqrt{T_{1} T_{2}}$
D

$\frac{T_{1} + T_{2}}{2}$

Solution

$Efficiency of a Carnot engine η = 1 - \frac{T_{2}}{T_{1}} where T_{1} is the temperature of source and T_{2} is the temperature of sink respectively For engine A, η_{A} = 1 - \frac{T}{T_{1}} For engine B, η_{B} = 1 - \frac{T_{2}}{T} As per question, η_{A} = η_{B} ∴ 1 - \frac{T}{T_{1}} = 1 - \frac{T_{2}}{T} \Rightarrow \frac{T}{T_{1}} = \frac{T_{2}}{T} or T^{2} = T_{1} T_{2} or T = \sqrt{T_{1} T_{2}}$

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