Unified approach to power-efficiency trade-off relations of generic thermal machines

TL;DR

The paper develops a universal framework connecting power and efficiency for generic thermal machines by tying cycle irreversibility to a scaling $A(\tau)\propto\tau^{-\alpha}$ and describing machines with a sign-function formalism. It derives a saturable $P$–$\eta$ bound and a general EMP formula $\eta_{\rm MP}=\frac{\alpha\eta_{\rm rev}}{1+\alpha-\chi\eta_{\rm rev}}$, showing that increasing $\alpha$ can bring EMP closer to reversible efficiency, while recovering known results for $\alpha=1$ (low-dissipation) and providing the first $\alpha=2$ result for finite-time quantum adiabatic Otto cycles. The framework unifies thermodynamic bounds across engines, refrigerators, and heat pumps and yields concrete, regime-specific forms (e.g., slow-driving isothermal and finite-time quantum adiabatic) to guide practical optimization. The work thus enables consistent optimization of diverse thermal devices in non-equilibrium regimes and predicts enhanced performance under tailored system–reservoir interactions.

Abstract

We present a general framework for determining the power-efficiency trade-off relations across arbitrary thermal machines, addressing the lack of unified optimization results stemming from their diverse functionalities (e.g., heat engines, refrigerators, and heat pumps). For time-dependent cycle irreversibility $A(τ)$ following a $τ^{-α}$ power law, where $α$ is an interaction-dependent parameter, we show that engineering the interactions between thermal machines and reservoirs enables control over the trade-off relations, with the efficiency at maximum power approaching Carnot efficiency as $α$ increases. Setting $α=1$ naturally recovers typical low-dissipation regime results. Additionally, we derive the first power-efficiency trade-off for finite-time quantum adiabatic Otto machines with $τ^{-2}$-scaling. This work establishes a unified constraint for thermodynamic cycles across non-equilibrium regimes, facilitating consistent optimization of diverse thermal devices in practice.

Figures (7)

• Figure 1: (a) schematic of a heat engine operates among multiple heat reservoirs which is represented by red and blue circles. The engine extracts from supplied reservoirs and releases to others. (b) Temperature-entropy ($T-S$) diagram of clockwise and counterclockwise finite-time Carnot cycles. The orientation of the $J$-axis represents the direction of energy flow from the cycles to the heat reservoirs.
• Figure 2: The normalized power-efficiency trade-off relation ($\tilde{\eta}\equiv \eta/\eta_{\mathrm{rev}},~\tilde{P}\equiv P/P_{\mathrm{max}}$) of four thermal machines. The blue dashed (red dash-dotted) curve represents the Eq. \ref{['eq:tradeoff_general']} with $\delta\rightarrow1$($\delta\rightarrow0$), the normalized reversible efficiency $\Tilde{\eta}=1$ is plotted with the green dotted line, and the shadow area marks the available operation region of the machines. In this plot, $\alpha=1$ and $\eta_{\rm{rev}}=0.5$ are used.
• Figure 3: The normalized trade-off relations of heat engines associated with $\delta \to 0$ [(a)] and $\delta \to 1$ [(b)] with different $\alpha$ and $\eta_{\rm{rev}}=0.5$. The blue dotted, red dash-dotted, and purple dashed curve represent $\alpha=1,4,50$, respectively, $\Tilde{\eta}=1$ is plotted with the green dotted line.
• Figure 4: The normalized trade-off relations associated with $\alpha = 1$[(a)], $\alpha = 2$[(b)], $\alpha = 5$[(c)], and $\alpha = 50$[(d)]. The blue dotted, red dash-dotted, and purple dashed curves are plotted with $\delta=0,0.7,1$, respectively. In this plot, $\eta_{\rm{rev}}=0.5$.
• Figure 5: Normalized EMPs of heat engines ($\Tilde{\eta}_{\rm{MP}} \equiv \eta_{\mathrm{MP}}/\eta_{\mathrm{rev}}$) as functions of $\tilde{\eta}_{\rm{rev}}$. (a) The blue dotted, red dash-dotted, yellow solid, and purple dashed curves are plotted with $\alpha=1,2,10,100$, respectively, and $\delta=1$ is fixed. (b) The blue dotted, red dash-dotted, and purple dashed curves are associated with $\delta=0,1/(1+\sqrt{1-\eta_{\mathrm{rev}}}),1$, respectively, and $\alpha=1$ is fixed.