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. How is T related to T1 and T2, if both the engines perform equal amount of work ?

Engineering

Physics

Heat engine Refrigerator Entropy

Question

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. How is T related to T₁ and T₂, if both the engines perform equal amount of work ?

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

$T = \sqrt{T_{1} T_{2}}$

T = 0

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

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Let, Q_H : Heat input to Ist engine

Q_L : Heat rejected from Ist engine

Q_L': Heat rejected from IInd engine

Work done by I^st engine = work done by II^nd engine

Q_H – Q_L = Q_L – Q_L'

2Q_L = Q_H + Q_L'

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

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