Efficiency at maximum power output of linear irreversible Carnot-like heat engines

The efficiency at maximum power output of linear irreversible Carnot-like heat engines is investigated based on the assumption that the rate of irreversible entropy production of the working substance in each “isothermal” process is a quadratic form of the heat exchange rate between the working substance and the reservoir. It is found that the maximum power output corresponds to minimizing the irreversible entropy production in two isothermal processes of the Carnot-like cycle, and that the efficiency at maximum power output has the form η_mP=η_C/(2-γη_C), where η_C is the Carnot efficiency, while γ depends on the heat transfer coefficients between the working substance and two reservoirs. The value of η_mP is bounded between η_-≡η_C/2 and η₊≡η_C/(2-η_C). These results are consistent with those obtained by Chen and Yan [J. Chem. Phys.JCPSA60021-960610.1063/1.455832 90, 3740 (1989)] based on the endoreversible assumption, those obtained by Esposito [Phys. Rev. Lett.PRLTAO0031-900710.1103/PhysRevLett.105.150603 105, 150603 (2010)] based on the low-dissipation assumption, and those obtained by Schmiedl and Seifert [Europhys. Lett.EULEEJ0295-507510.1209/0295-5075/81/20003 81, 20003 (2008)] for stochastic heat engines which in fact also satisfy the low-dissipation assumption. Additionally, we find that the endoreversible assumption happens to hold for Carnot-like heat engines operating at the maximum power output based on our fundamental assumption, and that the Carnot-like heat engines that we focused on do not strictly satisfy the low-dissipation assumption, which implies that the low-dissipation assumption or our fundamental assumption is a sufficient but non-necessary condition for the validity of η_mP=η_C/(2-γη_C) as well as the existence of two bounds, η_-≡η_C/2 and η₊≡η_C/(2-η_C).