The concepts of weighted reciprocal of temperature and weighted thermal flux are proposed for a heat engine operating between two heat baths and outputting mechanical work. With the aid of these two concepts, the generalized thermodynamic fluxes and
forces can be expressed in a consistent way within the framework of irreversible thermodynamics. Then the efficiency at maximum power output for a heat engine, one of key topics in finite-time thermodynamics, is investigated on the basis of a generic model under the tight-coupling condition. The corresponding results have the same forms as those of low-dissipation heat engines [M. Esposito, R. Kawai, K. Lindenberg, and C. Van den Broeck, {Phys. Rev. Lett.} textbf{105}, 150603 (2010)]. The mappings from two kinds of typical heat engines, such as the low-dissipation heat engine and the Feynman ratchet, into the present generic model are constructed. The universal efficiency at maximum power output up to the quadratic order is found to be valid for a heat engine coupled symmetrically and tightly with two baths. The concepts of weighted reciprocal of temperature and weighted thermal flux are also transplanted to the optimization of 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 $eta_Cequiv (T_h-T_c)/T_h$, 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].
