What is the ratio of the efficiencies \(\left( {\frac{{{\eta _I}}}{{{\eta _{II}}}}} \right)\) for the two cycles as shown in the T-s diagrams?

Concept:

The area under the T-S diagram gives the heat interaction by a reversible process, and the ara enclosed by a reversible cycle represents net work or net heat. 

The efficiency of a cycle is given by \(\eta = 1 - \frac{{Heat\;Rejected}}{{Heat\;Supplied}}\)

Calculation:

Case (i)

Heat Supplied \({Q_s} = \frac{1}{2}\left( {{T_1} + {T_2}} \right)\left( {{S_1} - {S_2}} \right)\)

Heat Rejected \(\;{Q_R} = {T_2}\left( {{S_1} - {S_2}} \right)\)

\({\eta _I} = 1 - \frac{{{T_2}\left( {{S_1} - {S_2}} \right)}}{{\frac{1}{2}\left( {{T_1} + {T_2}} \right)\left( {{S_1} - {S_2}} \right)}}\)

\( \Rightarrow {\eta _I} = 1 - \frac{{2{T_2}}}{{{T_1} + {T_2}}}\)

\( \Rightarrow {\eta _I} = \frac{{{T_1} - {T_2}}}{{{T_1} + {T_2}}}\)

Case (ii)

Heat Supplied Q_S = T₁ (S₁ – S₂)

Heat Rejected \({Q_R} = \frac{1}{2}\left( {{T_1} + {T_2}} \right)\left( {{S_1} - {S_2}} \right)\)

\({\eta _{II}} = 1 - \frac{{\frac{1}{2}\left( {{T_1} + {T_2}} \right)\left( {{S_1} - {S_2}} \right)}}{{{T_1}\left( {{S_1} - {S_2}} \right)}}\)

\({\eta _{II}} = 1 - \frac{{{T_1} + {T_2}}}{{2{T_1}}}\)

\( \Rightarrow {\eta _{II}} = \frac{{{T_1} - {T_2}}}{{2{T_1}}}\)

\(\therefore \frac{{{\eta _I}}}{{{\eta _{II}}}} = \frac{{\left( {{T_1} - {T_2}} \right)\;/{T_1} + {T_2}}}{{\left( {{T_1} - {T_2}} \right)/2{T_1}}} = \frac{{2{T_1}}}{{{T_1} + {T_2}}}\)