Question: Two Carnot engines 'A' and 'B' are operated in succession. The first one, A receives heat from a sou...

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

Solution

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, ${W_1} = {W_2}$

\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.