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 T₁ 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 T₂. For what value of T the efficiences of the two engines are equal

Easy

A

$\large \frac{{{T_1} - {T_2}}}{2}$
B

T₁T₂
C

$\large \sqrt {{T_1}{T_2}}$
D

$\large \frac{{{T_1} + {T_2}}}{2}$

Solution

For the first engine,
$\large {\eta _1}\, = \,1 - \frac{T}{{{T_1}}}$
For the second engine,
$\large {\eta _2}\, = \,1 - \frac{{{T_2}}}{T}$
Since $\large {\eta _1} = {\eta _2}\,$ ,
$\large 1 - \frac{T}{{{T_1}}}\, = \,1 - \frac{{{T_2}}}{T}$
$\large \Rightarrow \,T\, = \,\sqrt {{T_1}{T_2}}$

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