Universality of energy conversion efficiency for optimal tight-coupling heat engines and refrigerators

A unified $\chi$-criterion for heat devices (including heat engines and refrigerators) which is defined as the product of the energy conversion efficiency and the heat absorbed per unit time by the working substance [de Tom\'{a}s \emph{et al} 2012 \textit{Phys. Rev. E} \textbf{85} 010104(R)] is optimized for tight-coupling heat engines and refrigerators operating between two heat baths at temperatures $T_c$ and $T_h(>T_c)$. By taking a new convention on the thermodynamic flux related to the heat transfer between two baths, we find that for a refrigerator tightly and symmetrically coupled with two heat baths, the coefficient of performance (i.e., the energy conversion efficiency of refrigerators) at maximum $\chi$ asymptotically approaches to $\sqrt{\varepsilon_C}$ when the relative temperature difference between two heat baths $\varepsilon_C^{-1}\equiv (T_h-T_c)/T_c$ is sufficiently small. Correspondingly, the efficiency at maximum $\chi$ (equivalent to maximum power) for a heat engine tightly and symmetrically coupled with two heat baths is proved to be $\eta_C/2+\eta_C^2/8$ up to the second order term of $\eta_C\equiv (T_h-T_c)/T_h$, which reverts to the universal efficiency at maximum power for tight-coupling heat engines operating between two heat baths at small temperature difference in the presence of left-right symmetry [Esposito \emph{et al} 2009 \textit{Phys. Rev. Lett.} \textbf{102} 130602].