Isobaric-isochoric-adiabatic heat engine

Consider an ideal-gas cycle with constant heat-capacity ratio $\gamma >1$: expand isobarically from $V_1$ to $r{V}_1$, cool isochorically, then compress reversibly and adiabatically to the initial state. With $P_{\ast }=p_1{V}_1=nR{T}_1$, closure gives $T_2=r{T}_1$ and $T_3=r^{1-\gamma }{T}_1$. The heat input is $Q_h=\gamma P_{\ast }(r-1)/(\gamma -1)$ and the rejected heat magnitude is $Q_c=P_{\ast }(r-r^{1-\gamma })/(\gamma -1)$. Thus $\eta =1-(r-r^{1-\gamma })/[\gamma (r-1)]$. Reservoir temperatures determine feasibility and entropy generation but need not coincide with the gas temperatures at heat-exchange endpoints.