Pareto fronts and trade-off relations from exact multi-objective optimization of thermal machines

Abstract

Thermal machines are physical systems that, when fueled by input energy, perform output tasks such as heat pumping or the production of work. Their performance is characterized with several, often competing quantities, such as power, efficiency, energy waste, and resilience to environmental noise. Multi-objective optimization provides a key tool to investigate the characterization of the best thermal machines operating in the irreversible linear-response regime. Here, we derive exact analytical parameterizations for the optimal (Pareto) fronts associated with any given choice of relative weights assigned to their mean extracted power $P$, efficiency $η$, entropy production $Σ$ and the amplitude of power fluctuations $σ^2_P$. The geometry of the front of endoreversible machines is universal: two-, three-, and four-objective trade-offs follow analytical formulae that do not depend on the value of any physical parameter of the machine. We show that such universal thermodynamic Pareto fronts also set quantitative fundamental limits for the performance of non-endoreversible machines. Furthermore, we demonstrate that our results apply to existing experimental data from different physical systems also beyond the linear regime, ranging from atomic to macroscopic scales, including single-atom engines, colloidal systems, macroscopic engines and power plants.

Figures (3)

• Figure 1: (a) Sketch of a multi-objective optimization for the case of two objectives. The blue region represents the permitted values for the two objectives while the white region represents unattainable configurations. The red solid line at the frontier between the two regions corresponds to optimal configurations (the so-called Pareto front). The Pareto front quantifies a trade-off relation inasmuch if one wants to increase one objective then the other objective's value has to decrease. (b) Sketch of an example multi-objective optimization in a windmill (in the linear-response regime), where we keep the quantities $A_2$ (the height of the waterfall) and $L_{22}$ (the rate of water flow of the river atop the waterfall) fixed, as they represent the thermodynamic resources available to the windmill. The quantities varied are $A_1$ (the weight of the load), $L_{11}$ (the effectivity of the pulley mechanism) and $L_{12}$ (the effectivity of the windmill blades). (c) Onsager affinities $A_1$, $A_2$ and their conjugate fluxes $J_1$, $J_2$ for different thermal machines. Here $F$ is an external force, $\dot{x}$ the linear velocity, $e$ the elementary charge, $\Delta V$ the electrical potential difference, $I$ the electric current, $\Delta \mu_{\mathrm{ATP}}$ the chemical potential difference per ATP, and $\dot{n}_\mathrm{ATP}$ the ATP consumption rate. The symbols $T_{\mathrm{h}}$, and $T_{\mathrm{m}}$ denote hot and intermediate reservoir temperatures, while $\dot{Q}_{\mathrm{h}}$ and $\dot{Q}_{\mathrm{c}}$ are the heat fluxes from the respective reservoirs.
• Figure 2: Pareto fronts (red solid lines) derived in this work in two (a-c) and three (d-f) dimensional spaces (see Sec. 3.1 for more details). The blue points are configurations in the linear regime with parameters: $A_2=1$, $L_{22}=1$.
• Figure 3: (a–c) Pareto fronts (red solid lines) derived in this work for the two-dimensional trade-offs considered. Symbols correspond to experimental data from different systems, as specified in the legend. (d) Bar plot of the ratio of the achieved efficiency to the optimal efficiency for different generations of nuclear power plants. For each generation, values are obtained by averaging over individual plants, with error bars indicating the corresponding standard deviations.