Self-consistency of optimizing finite-time Carnot engines with the low-dissipation model

The efficiency at the maximum power (EMP) for finite-time Carnot engines established with the low-dissipation model, relies significantly on the assumption of the inverse proportion scaling of the irreversible entropy generation $ΔS^{(\mathrm{ir})}$ on the operation time $τ$, i.e., $ΔS^{(\mathrm{ir})}\propto1/τ$. The optimal operation time of the finite-time isothermal process for EMP has to be within the valid regime of the inverse proportion scaling. Yet, such consistency was not tested due to the unknown coefficient of the $1/τ$-scaling. In this paper, using a two-level atomic heat engine as an illustration, we reveal that the optimization of the finite-time Carnot engines with the low-dissipation model is self-consistent only in the regime of $η_{\mathrm{C}}\ll1$, where $η_{\mathrm{C}}$ is the Carnot efficiency. In the large-$η_{\mathrm{C}}$ regime, the operation time for EMP obtained with the low-dissipation model is not within the valid regime of the $1/τ$-scaling, and the exact EMP is found to surpass the well-known bound $η_{+}=η_{\mathrm{C}}/(2-η_{\mathrm{C}})$

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