Topic: Thermodynamics
Two Carnot heat engines are operating in series such that the heat sink of the first engine serves as the heat source of the second one. If the source temperature of the first engine is $1327^{\circ}C.$ the sink temperature of the second engine is $127^{\circ}C$, and the thermal efficiency of both the engines are the same, the temperature of the intermediate reservoir is:
- A. $627^{\circ}C$
- B. $410^{\circ}C$
- C. $577^{\circ}C$
- D. $527^{\circ}C$
Answer: D. $527^{\circ}C$
Explanation: Given, $T_{1}=1327^{\circ}C+273=1600~K$, $T_{3}=127^{\circ}C+273=400~K$. If two engines are connected in series and their thermal efficiencies are the same, the intermediate temperature is given by the geometric mean: $T_{2}=\sqrt{T_{1}\times T_{3}} = \sqrt{1600\times 400} = 40 \times 20 = 800~K$. Converting to Celsius: $T_{2} = 800 - 273 = 527^{\circ}C$.