Past exam of the physics B course of the University of Cambridge / 2017 / Paper 2 / Section C / C10 / Limit of vanishing heat addition / Solution

For $\rho \to 1^{+}$, Taylor expansion gives ${\rho }^{\gamma }-1=\gamma (\rho -1)+O((\rho -1{)}^2)$. The vanishing-cutoff limit of the Diesel cycle isThe maximum temperature tends to $T_2=T_1{r}^{\gamma -1}$, while the minimum stays at $T_1$. A Carnot engine between those limiting extrema hasso the limiting efficiencies coincide. At exactly $V_3=V_2$, however, both heat input and net work vanish, and the cycle retraces its compression adiabat. The efficiency there is an undefined $0/0$; the boxed value is the continuous limit for neighbouring nonzero cycles, not finite work at zero heat input.

For a finite cutoff ratio $\rho >1$, the Carnot comparison uses $T_{\max }=T_3=T_1{r}^{\gamma -1}\rho$, giving ${\eta }_C=1-1/(r^{\gamma -1}\rho )$. The Diesel cycle has smaller efficiency. For example,which makes its rejected/input heat ratio larger than the Carnot ratio.