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 working substance in each "isothermal" process is a quadratic form of 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 as $\eta_{mP}={\eta_C}/(2-\gamma\eta_C)$ where $\eta_C$ is the Carnot efficiency while $\gamma$ depends on the heat transfer coefficients between the working substance and two reservoirs. The value of $\eta_{mP}$ is bounded between $\eta_{-}\equiv \eta_C/2$ and $\eta_{+}\equiv\eta_C/(2-\eta_C)$. These results are consistent with those obtained by Chen and Yan [J. Chem. Phys. \textbf{90}, 3740 (1989)] based on the endoreversible assumption, those obtained by Esposito \textit{et al.} [Phys. Rev. Lett. \textbf{105}, 150603 (2010)] based on the low-dissipation assumption, and those obtained by Schmiedl and Seifert [EPL \textbf{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 does 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 $\eta_{mP}={\eta_C}/(2-\gamma\eta_C)$ as well as the existence of two bounds $\eta_{-}\equiv \eta_C/2$ and $\eta_{+}\equiv\eta_C/(2-\eta_C)$. 
Yang Wang and Z. C. Tu
cond-mat.stat-mech