Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, T1 and T2. The temperature of the hot reservoir of the first engine is T1 and the temperature of the cold reservoir of the second engine is T2 . T is temperature of the sink of first engine which is also the source for the second engine. If T is related to T1 and T2 as . T=T1+T2K Where K = ?(Assume both the engines perform equal amount of work)

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Two ideal Carnot engines operate in cascade (all heat given up by one engine is used by the other engine to produce work) between temperatures, T₁ and T₂. The temperature of the hot reservoir of the first engine is T₁ and the temperature of the cold reservoir of the second engine is T₂ . T is temperature of the sink of first engine which is also the source for the second engine. If T is related to T₁ and T₂ as . $T = \frac{T_{1} + T_{2}}{\sqrt{K}}$ Where $K$ = ?
(Assume both the engines perform equal amount of work)

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answer is 4.

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Detailed Solution

Efficiency of first engine, $η_{1} = 1 - \frac{T}{T_{1}}$

Efficiency of second engine, $η_{2} = 1 - \frac{T_{2}}{T}$

$Q_{1} = Heat absorbed by 1 st engine$

$Q_{2} = Heat rejected by 1 st engine = Heat absorbed by 2 nd engine$

$Q_{3} = Heat rejected by 2 nd engine$

$W_{1} = Q_{1} - Q_{2}$

$W_{2} = Q_{2} - Q_{3}$

Given : $W_{1} = W_{2}$

$\Rightarrow Q_{1} - Q_{2} = Q_{2} - Q_{3}$

$\Rightarrow Q_{2} = \frac{Q_{1} + Q_{3}}{2}$

For ideal heat engine, $Q_{1} \propto T_{1}, Q_{2} \propto T, Q_{3} \propto T_{3}$

$\Rightarrow T = \frac{T_{1} + T_{2}}{2}$

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