Question

Two Carnot engines 'A' and 'B' are operated in succession. The first one, A receives heat from a source at \[{{\text{T}}_{\text{1}}}{\text{ = 800K}}\] and rejects to a sink at \[{{\text{T}}_{\text{1}}}{\text{ K}}\]. The second engine B receives heat rejected by the first engine and rejects to another sink at \[{{\text{T}}_3}{\text{ = 300K}}\]. If the work outputs of two engines are equal, then the value of \[{{\text{T}}_2}\] is
A.100 K
B.300 K
C.550 K
D.700 K

Answer 1

Hint: In this question, Carnot engine is a theoretical thermodynamic cycle proposed by Leonard Carnot. It provides the estimation of the potential efficiency that a heat engine will attain during the heat-to-work transfer cycle, and conversely, operating between two reservoirs.

Complete answer:
Let, \[{Q_1}\]= heat absorbed in first engine, \[{Q_2}\]= heat released in first engine = heat absorbed in second engine, \[{Q_3}\]= heat released in second engine,
\[{T_1}\]= source temperature of first engine, \[{T_2}\]= sink temperature of first engine = source temperature of second engine, \[{T_3}\]= sink temperature of second engine
For the 2 Carnot Engines, \[\dfrac{{{Q_2}}}{{{Q_1}}} = \dfrac{{{T_2}}}{{{T_1}}}\]
\[\dfrac{{{Q_3}}}{{{Q_2}}} = \dfrac{{{T_3}}}{{{T_2}}}\]
The condition given is, \[{W_1} = {W_2}\]
\[\begin{gathered}
\Rightarrow {Q_2} - {Q_1} = {Q_3} - {Q_2} \\
\Rightarrow 2{Q_2} = {Q_1} + {Q_3} \\
 \Rightarrow 2{T_2} = {T_1} + {T_3} \\
 \Rightarrow 2{T_2} = 300 + 800 = 1100 \\
\end{gathered} \]
Hence the required temperature is 550 K .

Note:
The Carnot principles are for cyclic devices such as heat engines only, which state:
• The output of an irreversible heat engine is often smaller than that of a reversible engine working between the two reservoirs themselves.
• Efficiencies are equivalent to other operating reversible heat engines between the two reservoirs.
It is essential to increase the temperature of the combustion room to increase the thermal efficiency of a gas-power turbine. For example, turbine blades are unable to hold the gas at high temperatures and eventually lead to early fatigue.