All Problems Second law of thermodynamics, entropy
Problem 2.128
Making use of the Clausius inequality, demonstrate that all cycles having the same maximum temperature \(T_{\max }\) and the same minimum temperature \(T_{\min }\) are less efficient compared to the Carnot cycle with the same \(T_{\max }\) and \(T_{\min }\)
2.128. The inequality \(\int \frac{\delta Q_{1}}{T_{1}}-\int \frac{\delta Q_{2}^{\prime}}{T_{2}} \leqslant 0\) becomes even stronger when \(T_{1}\) is replaced by \(T_{\max }\) and \(T_{2}\) by \(T_{\min }\). Then \(Q_{1} / T_{\max }-\) \(-Q_{\mathrm{a}}^{\prime} / T_{\min }<0 .\) Hence \frac{Q_{1}-Q_{2}^{\prime}}{Q_{1}}<\frac{T_{\max }-T_{m \ln }}{T_{\max }}, \text { or } \eta<\eta_{\text {Carnot }}