From maximum power to a trade-off optimization of low-dissipation heat engines: Influence of control parameters and the role of entropy generation

Microscopic theory of the Curzon-Ahlborn heat engine based on a Brownian particle

The Curzon-Ahlborn (CA) efficiency, as the efficiency at the maximum power (EMP) of the endoreversible Carnot engine, has significant impact on finite-time thermodynamics. However, the CA engine is based on many assumptions. In the past few decades, although a lot of efforts have been made, a microscopic theory of the CA engine is still lacking. By adopting the method of the stochastic differential equation of energy, we formulate a microscopic theory of the CA engine realized with a highly underdamped Brownian particle in a class of nonharmonic potentials. This theory gives microscopic interpretation of all assumptions made by Curzon and Ahlborn. In other words, we find a microscopic counterpart of the CA engine in stochastic thermodynamics. Also, based on this theory, we derive the explicit expression of the protocol associated with the maximum power for any given efficiency, and we obtain analytical results of the power and the efficiency statistics for the Brownian CA engine. Our research brings new perspectives to experimental studies of finite-time microscopic heat engines featured with fluctuations.

Maximum efficiency of low-dissipation heat pumps at given heating load

We derive an analytical expression for maximum efficiency at fixed power of heat pumps operating along a finite-time reverse Carnot cycle under the low-dissipation assumption. The result is cumbersome, but it implies simple formulas for tight upper and lower bounds on the maximum efficiency and various analytically tractable approximations. In general, our results qualitatively agree with those obtained earlier for endoreversible heat pumps. In fact, we identify a special parameter regime when the performance of the low-dissipation and endoreversible devices is the same. At maximum power, heat pumps operate as work to heat converters with efficiency 1. Expressions for maximum efficiency at given power can be helpful in the identification of more practical operation regimes.

Applicability of the low-dissipation model: Carnot-like heat engines under Newton's law of cooling

We investigate the validity of using the low-dissipation model (LD model) to describe the maximum power regime of the endoreversible model under Newton's law of cooling. We find it valid only when the temperature difference of heat reservoirs (T_{h}-T_{c}, T_{h}>T_{c}) is small. Thus the efficiency at maximum power derived from the LD model is valid to the first order of Carnot efficiency when describing endoreversible heat engines. We conclude that the LD model produces the Curzon-Ahlborn efficiency (η_{CA}=1-sqrt[T_{h}/T_{c}]) in the maximum power regime with no dependence on dissipation ratios.

Energetic Self-Optimization Induced by Stability in Low-Dissipation Heat Engines

The local stability of a weakly dissipative heat engine is analyzed and linked to an energetic multi-objective optimization perspective. This constitutes a novel issue in the unified study of cyclic energy converters, opening the perspective to the possibility that stability favors self-optimization of thermodynamic quantities including efficiency, power and entropy generation. To this end, a dynamics simulating the restitution forces, which mimics a harmonic potential, bringing the system back to the steady state is analyzed. It is shown that relaxation trajectories are not arbitrary but driven by the improvement of several energetic functions. Insights provided by the statistical behavior of consecutive random perturbations show that the irreversible behavior works as an attractor for the energetics of the system, while the endoreversible limit acts as an upper bound and the Pareto front as a global attractor. Fluctuations around the operation regime reveal a difference between the behavior coming from fast and slow relaxation trajectories: while the former are associated to an energetic self-optimization evolution, the latter are ascribed to better performances. The self-optimization induced by stability and the possible use of instabilities in the operation regime to improve the energetic performance might usher into new useful perspectives in the control of variables for real engines.

Performance optimization of low-dissipation thermal machines revisited

We revisit the optimization of performance of finite-time Carnot machines satisfying the low-dissipation assumption. The standard procedure seeks to optimize an objective function, such as power output of the engine, over the durations of contacts between the working medium and the heat reservoirs. This procedure may lead to unwieldy equations at the optimum of some objective functions. We propose an alternate scheme in which the output or input work is first optimized for a given cycle time, followed by an optimization of another objective function over the cycle time. This optimization problem is solved in a much simplified manner, with closed-form expressions for figures of merit. The approach is demonstrated for various objective functions, both for engines as well as refrigerators.

Optimization induced by stability and the role of limited control near a steady state

A relationship between stability and self-optimization is found for weakly dissipative heat devices. The effect of limited control on operation variables around an steady state is such that, after instabilities, the paths toward relaxation are given by trajectories stemming from restitution forces which improve the system thermodynamic performance (power output, efficiency, and entropy generation). Statistics over random trajectories for many cycles shows this behavior as well. Two types of dynamics are analyzed, one where an stability basin appears and another one where the system is globally stable. Under both dynamics there is an induced trend in the control variables space due to stability. In the energetic space this behavior translates into a preference for better thermodynamic states, and thus stability could favor self-optimization under limited control. This is analyzed from the multiobjective optimization perspective. As a result, the statistical behavior of the system is strongly influenced by the Pareto front (the set of points with the best compromise between several objective functions) and the stability basin. Additionally, endoreversible and irreversible behaviors appear as very relevant limits: The first one is an upper bound in energetic performance, connected with the Pareto front, and the second one represents an attractor for the stochastic trajectories.

Three-level laser heat engine at optimal performance with ecological function

Although classical and quantum heat engines work on entirely different fundamental principles, there is an underlying similarity. For instance, the form of efficiency at optimal performance may be similar for both types of engines. In this work, we study a three-level laser quantum heat engine operating at maximum ecological function (EF) which represents a compromise between the power output and the loss of power due to entropy production. We present numerical as well as analytic results for the global and local optimization of our laser engine in different operational regimes. Particularly, we observe that in low-temperature regimes, the three-level laser heat engine can be mapped to Feynman's ratchet and pawl model, a steady-state classical heat engine. Then we derive analytic expressions for efficiency under the assumptions of strong matter-field coupling and high bath temperatures. Upper and lower bounds on the efficiency exist in case of extreme asymmetric dissipation when the ratio of system-bath coupling constants at the hot and the cold contacts respectively approaches zero or infinity. These bounds have been established previously for various classical models of Carnot-like engines. Further, for weak (or intermediate) matter-field coupling in the high-temperature limit, we derive some new bounds on the efficiency of the engine. We conclude that while the engine produces at least 75% of the power output as compared with the maximum power conditions, the fractional loss of power is appreciably low in case of the engine operating at maximum EF, thus making this objective function relevant from an environmental point of view.

Efficiency Bounds for Minimally Nonlinear Irreversible Heat Engines with Broken Time-Reversal Symmetry

We study the minimally nonlinear irreversible heat engines in which the time-reversal symmetry for the systems may be broken. The expressions for the power and the efficiency are derived, in which the effects of the nonlinear terms due to dissipations are included. We show that, as within the linear responses, the minimally nonlinear irreversible heat engines can enable attainment of Carnot efficiency at positive power. We also find that the Curzon-Ahlborn limit imposed on the efficiency at maximum power can be overcome if the time-reversal symmetry is broken.

Optimization and Stability of Heat Engines: The Role of Entropy Evolution

Local stability of maximum power and maximum compromise (Omega) operation regimes dynamic evolution for a low-dissipation heat engine is analyzed. The thermodynamic behavior of trajectories to the stationary state, after perturbing the operation regime, display a trade-off between stability, entropy production, efficiency and power output. This allows considering stability and optimization as connected pieces of a single phenomenon. Trajectories inside the basin of attraction display the smallest entropy drops. Additionally, it was found that time constraints, related with irreversible and endoreversible behaviors, influence the thermodynamic evolution of relaxation trajectories. The behavior of the evolution in terms of the symmetries of the model and the applied thermal gradients was analyzed.

Ecological efficiency of finite-time thermodynamics: A molecular dynamics study

We present a molecular dynamics simulation of a two-dimensional Carnot engine. The optimization of this engine is achieved through the velocity of the piston, allowing not only the optimization of power output but also some other figures of merit involving entropy production. The maximum power and maximum ecological efficiencies are computed. It is shown that the near ideal gas working substance displays an endoreversible Carnot-like engine behavior. This can be considered as a prove of the validity of the Carnot-like endoreversible model. An effective reversible cycle different than the Carnot one is obtained, in agreement with the endoreversible hypothesis flexibility. We compare the efficiencies stemming from an ideal gas approximation with those of the simulation, and then we propose a suitable approximation to an endoreversible heat engine and to a reversible Joule-Brayton cycle which fits very well to the simulation results. Finally, we show that the maximum ecological efficiency η=1-τ^{3/4}, which is also very close to the upper bound of the low-dissipation heat engine under maximum ecological (and Omega) conditions, is close for describing the dynamics of the simulated cycle under maximum power and maximum ecological conditions in the so-named heat engine operability region.

Diverging, but negligible power at Carnot efficiency: Theory and experiment

We discuss the possibility of reaching the Carnot efficiency by heat engines (HEs) out of quasistatic conditions at nonzero power output. We focus on several models widely used to describe the performance of actual HEs. These models comprise quantum thermoelectric devices, linear irreversible HEs, minimally nonlinear irreversible HEs, HEs working in the regime of low-dissipation, overdamped stochastic HEs and an underdamped stochastic HE. Although some of these HEs can reach the Carnot efficiency at nonzero and even diverging power, the magnitude of this power is always negligible compared to the maximum power attainable in these systems. We provide conditions for attaining the Carnot efficiency in the individual models and explain practical aspects connected with reaching the Carnot efficiency at large power output. Furthermore, we show how our findings can be tested in practice using a standard Brownian HE realizable with available micromanipulation techniques.

The underdamped Brownian duet and stochastic linear irreversible thermodynamics

Building on our earlier work [Proesmans et al., Phys. Rev. X 6, 041010 (2016)], we introduce the underdamped Brownian duet as a prototype model of a dissipative system or of a work-to-work engine. Several recent advances from the theory of stochastic thermodynamics are illustrated with explicit analytic calculations and corresponding Langevin simulations. In particular, we discuss the Onsager-Casimir symmetry, the trade-off relations between power, efficiency and dissipation, and stochastic efficiency.

Local-stability analysis of a low-dissipation heat engine working at maximum power output

In this paper we address the stability of a low-dissipation (LD) heat engine (HE) under maximum power conditions. The LD system dynamics are analyzed in terms of the contact times between the engine and the external heat reservoirs, which determine the amount of heat exchanged by the system. We study two different scenarios that secure the existence of a single stable steady state. In these scenarios, contact times dynamics are governed by restitutive forces that are linear functions of either the heat amounts exchanged per cycle, or the corresponding heat fluxes. In the first case, according to our results, preferably locating the system irreversibility sources at the hot-reservoir coupling improves the system stability and increases its efficiency. On the other hand, reducing the thermal gradient increases the system efficiency but deteriorates its stability properties, because the restitutive forces are smaller. Additionally, it is possible to compare the relaxation times with the total cycle time and obtain some constraints upon the system dynamics. In the second case, where the restitutive forces are assumed to be linear functions of the heat fluxes, we find that although the partial contact time presents a locally stable stationary value, the total cycle time does not; instead, there exists an infinite collection of steady values located in the neighborhood of the fixed point, along a one-dimensional manifold. Finally, the role of dissipation asymmetries on the efficiency, the stability, and the ratio of the total cycle time to the relaxation time is emphasized.

Heat engines at optimal power: Low-dissipation versus endoreversible model

The low-dissipation model and the endoreversible model of heat engines are two of the most commonly studied models of machines in finite-time thermodynamics. In this paper we compare the performance characteristics of these two models under optimal power output. We point out a basic equivalence between them, in the linear response regime.