Endoreversible Research Articles

Performance at maximum figureof merit for a Brownian Carnot refrigerator.

This paper focuses on the coefficient of performance (COP) at maximum χ^{R} figureof merit for a Brownian Carnot-like refrigerator, within the context of the low-dissipation approach. Our proposal is based on the Langevin equationfor a Brownian particle bounded to a harmonic potential trap, which can perform Carnot-like cycles at finite time. The theoretical approach is related to the equilibrium ensemble average of 〈x^{2}〉_{eq} which plays the role of a statelike equation, x being the Brownian particle position. This statelike equationcomes from the macroscopic version of the corresponding Langevin equationfor a Brownian particle. We show that under quasistatic conditions the COP has the same expression as the macroscopic Carnot refrigerator, while for irreversible cycles at finite time and under symmetric dissipation the optimal COP is the counterpart of Curzon-Ahlborn efficiency as also obtained for irreversible macroscopic refrigerators.

Quantum Dots and Molecular Junctions as Energy Filters for Thermal-to-Electric Energy Conversion

The thermoelectric conversion of heat into electricity requires a form of energy filtering, that allows hot electrons to pass while blocking cold electrons. The energy conversion efficiency is determined by the details of the energy filtering, with an infinitely sharp (delta function) transmission spectrum allowing near ideal (Carnot) efficiency. [1]I will introduce this concept and report on a proof-of-performance experiment based on a quantum dot embedded into a single, semiconductor nanowire. This near-ideal quantum-dot heat engine realized power production at maximum power with Curzon-Ahlborn efficiency as well more than 70% of Carnot efficiency at maximum efficiency settings [2].To further enhance efficiency at maximum power one wishes to shape the transmission function to allow a spectrum of warmer electrons to pass while still completely blocking cold electrons. This may be achieved by exploiting quantum interference phenomena to block low-energy electron transmission by destructive interference, and to thus sculpt the transmission function of quantum dots or molecular junctions.[4] In contrast to quantum dots, molecular junctions have the advantage that they can be used at room temperature and can be produced, in principle, cheaply and scalable. To systematically explore the effect of quantum interferences, we designed and synthesized a new class of porphyrins that are expected to feature destructive quantum interference in single molecule junctions with gold electrodes and may thus show high thermopower, as well as a control that does not. We performed detailed experimental single-molecule break-junction studies of conductance and thermopower on porphyrin molecular junctions. We find a somewhat better thermoelectric performance for the porphyrin where we expect destructive interference. However, the predicted large difference in conductance and thermopower is not supported by our experimental findings.[4]Finally, I will also briefly discuss the potential application of the concepts and systems presented here in hot-carrier photovoltaics, as an approach to increasing photovoltaic energy conversion efficiency beyond current limits.[1] T.E. Humphrey et al., Reversible Quantum Brownian Heat Engines for Electrons. Phys. Rev. Lett., 89, 116801 (2002)[2] M. Josefsson et al. A quantum-dot heat engine operated close to thermodynamic efficiency limits. Nature Nanotechn. 13, 920 (2018)[3] O Karlström et al. Increasing thermoelectric performance using coherent transport. Phys. Rev. B 84, 113415 (2011)[4] H. Xu et al. Electrical Conductance and Thermopower of β-Substituted Porphyrin Molecular Junctions Synthesis and Transport. JACS 145 , 23541 (2023) Figure 1

Maximum power of coupled-qubit Otto engines

We put forward four schemes of a coupled-qubit quantum Otto machine, a generalization of the single-qubit quantum Otto machine, based on work and heat transfer between an internal system consisting of a coupled pair of qubits and an external environment consisting of two heat baths and two work storages. The four schemes of our model are defined by the positions of attaching the heat baths, which play a key role in the power of the coupled-qubit engine. First, for the single-qubit heat engine, we find a maximum-power relation, and the fact that its efficiency at the maximum power is equal to the Otto efficiency, which is greater than the Curzon-Ahlborn efficiency. Second, we compare the coupled-qubit engines to the single-qubit engine from the point of view of achieving the maximum power based on the same energy-level change for work production and find that the coupling between the two qubits can lead to greater powers but the system efficiency at the maximum power is lower than the single-qubit system's efficiency and the Curzon-Ahlborn efficiency. Published by the American Physical Society 2024

Maximum power and the corresponding efficiency for a Carnot-like thermoelectric cycle based on fluctuation theorem.

Here, we investigate the maximum power and efficiency of thermoelectric generators through devising a set of protocols for the isothermal and adiabatic processes of thermoelectricity to build a Carnot-like thermoelectric cycle, with the analysis based on fluctuation theorem. The Carnot efficiency can be readily obtained for the quasistatic thermoelectric cycle with vanishing power. The maximum power-efficiency pair of the finite-time thermoelectric cycle is derived, which is found to have the identical form to that of Brownian motors characterized by the stochastic thermodynamics. However, it is of significant discrepancy compared to the linear-irreversible and endoreversible-thermodynamics based formulations. The distinction with the linear-irreversible-thermodynamics case could result from the difference in the definitions of Peltier and Seebeck coefficients in the thermoelectric cycle. As for the endoreversible thermodynamics, we argue the applicability of endoreversibility could be questionable for analyzing the Carnot-like thermoelectric cycle, due to the incompatibility of the endoreversible hypothesis that attributes the irreversibility to finite heat transfer with thermal reservoirs, though the distinction in the mathematical expressions can vanish with the assumption that the ratio of thermoelectric power factors at the high and low temperatures (γ) is equal to the square root of the temperature ratio, γ=sqrt[T_{L}/T_{H}] (this condition could significantly deviate from the practical case). Last, utilizing our models as a concise tool to evaluate the maximum power-efficiency pairs of realistic thermoelectric material, we present a case study on the n-type silicon.

Enhanced Efficiency at Maximum Power in a Fock-Darwin Model Quantum Dot Engine.

We study the performance of an endoreversible magnetic Otto cycle with a working substance composed of a single quantum dot described using the well-known Fock-Darwin model. We find that tuning the intensity of the parabolic trap (geometrical confinement) impacts the proposed cycle's performance, quantified by the power, work, efficiency, and parameter region where the cycle operates as an engine. We demonstrate that a parameter region exists where the efficiency at maximum output power exceeds the Curzon-Ahlborn efficiency, the efficiency at maximum power achieved by a classical working substance.

Boosting thermodynamic performance by bending space-time

Black holes are arguably the most extreme regions of the universe. Yet, they are also utterly inaccessible to experimentation, and even just indirect observation poses significant technical challenges. The phenomenological approach of thermodynamics is uniquely suited to explore at least some of the physical properties of such scenarios, and this has motivated the study of so-called holographic engines. We show that the efficiency of an endoreversible Brayton cycle is given by the Curzon-Ahlborn efficiency, if the engine is fueled by a 2-dimensional ideal gas; and that the efficiency is higher, if the working medium is a (2 + 1)-dimensional BTZ black hole. These findings may be relevant not only in the quest to unlock the mysteries of black holes, but also for potential technological applications of graphene.

Relation between fluctuations and efficiency at maximum power for small heat engines

We study the ratio between the variances of work output and heat input, $\eta^{(2)}$, for a class of four-stroke heat engines which covers various typical cycles. Recent studies on the upper and lower bounds of $\eta^{(2)}$ are based on the quasistatic limit and the linear response regime, respectively. We extend these relations to the finite-time regime within the endoreversible approximation. We consider the ratio $\eta_{\text{MP}}^{(2)}$ at maximum power and find that the square of the Curzon-Ahlborn efficiency, $\eta_{\text{CA}}^2$, gives a good estimate of $\eta_{\text{MP}}^{(2)}$ for the class of heat engines considered, i.e., $\eta_{\text{MP}}^{(2)} \simeq \eta_{\text{CA}}^2$. This resembles the situation where the Curzon-Ahlborn efficiency gives a good estimate of the efficiency at maximum power for various kinds of finite-time heat engines. Taking an overdamped Brownian particle in a harmonic potential as an example, we can realize such endoreversible small heat engines and give an expression of the cumulants of work output and heat input. The approximate relation $\eta_{\text{MP}}^{(2)} \simeq \eta_{\text{CA}}^2$ is verified by numerical simulations. This relation also suggests a trade-off between the efficiency and the stability of finite-time heat engines at maximum power.

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 equationof 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.

Nonreciprocal photonic management for photovoltaic conversion: design and fundamental efficiency limits

Significant progress in the development of nonreciprocal optical components with broken Kirchhoff symmetry paves the way for increasing the photovoltaic (PV) conversion efficiency beyond the Shockley–Queisser limit due to reuse of emitted photons. Recent papers have analyzed the PV converter with several or an infinite number of multijunction cells, in which the cells are coupled via nonreciprocal filters (optical diodes) in such a way that the light emitted by one cell is absorbed by another cell. We proposed and investigated a single cell converter with nonreciprocal external photon recycling, which provided reabsorption and reuse of the emitting light by the same cell. We considered properties of photons in the sunbeam in terms of ergodicity, disorder, energy availability, information entropy, and coherence, and established fundamental limitations imposed by endoreversible thermodynamics on conversion efficiency at maximal power output. Our results show that the nonreciprocal converter with an ideal multijunction cell can approach the Carnot efficiency, whereas operating exactly at the Carnot limit requires an infinite number of photon recycling processes. This requirement resolves the famous thermodynamic paradox of the optical diode because any small dissipation in the cell or optical system enhanced by infinite recycling will stabilize the converter operation below the Carnot limit. We generalized endoreversible thermodynamics to photonic distributions with nonzero chemical potential and derived the limiting efficiency of the nonreciprocal single-junction PV converter. The performance of this converter with available GaAs solar cells was evaluated.

Thermoelectric generator in endoreversible approximation: The effect of heat-transfer law under finite physical dimensions constraint.

We revisit the optimal performance of a thermoelectric generator within the endoreversible approximation, while imposing a finite physical dimensions constraint in the form of a fixed total area of the heat exchangers. Our analysis is based on the linear-irreversible law for heat transfer between the reservoir and the working medium, in contrast to Newton's law usually assumed in literature. The optimization of power output is performed with respect to the thermoelectric current as well as the fractional area of the heat exchangers. We describe two alternate designs for allocating optimal areas to the heat exchangers. Interestingly, for each design, the use of linear-irreversible law yields the efficiency at maximum power in the well-known form 2η_{C}^{}/(4-η_{C}^{}), earlier obtained for the case of thermoelectric generator under exoreversible approximation, i.e., assuming only the internal irreversibility due to Joule heating. On the other hand, the use of Newton's law yields Curzon-Ahlborn efficiency.

Techno-economic analysis of waste-heat conversion

<h2>Summary</h2> With more than 50% of the primary energy consumed worldwide currently lost as waste heat, waste-heat conversion (WHC) should be considered a promising zero-carbon source of electricity. Despite significant research on WHC, the market penetration of such technologies remains limited, in large part because the R&D community has primarily focused on the development of new WHC heat engines with limited attention given to the techno-economic aspects. As different types of WHC heat engines vary significantly in their physical origins, there is a critical need to develop a system-level techno-economic model that is relatively independent of the detailed physics and design of the engine. In this perspective, we develop a techno-economic model for WHC technologies based on the well-known endoreversible thermodynamics formulations, which results in a fairly universal model. Our results indicate that regardless of the type or efficiency, WHC heat engines are not economically viable below 100°C, which has been the focus of significant research in the literature in recent years. Under highly optimistic assumptions, such as the cost of the WHC heat engines being the same as gas turbines (∼$0.25/W) and the capacity factor of the waste heat source being 0.9, for relative device Carnot efficiency <0.2, which is typical of various WHC heat engines, WHC is economical only for temperatures above 150°C and power output in the range of 100 kW to 1 MW. We conclude this perspective by providing a future outlook on the research needed in the field of WHC heat engines for it to be techno-economically viable. We also propose that along with the temperature and amount of waste heat available from various sources, the capacity factor of the waste-heat sources must be documented.

A Class of Reduced-Order Regenerator Models

We present a novel class of reduced-order regenerator models that is based on Endoreversible Thermodynamics. The models rest upon the idea of an internally reversible (perfect) regenerator, even though they are not limited to the reversible description. In these models, the temperatures of the working gas that alternately streams out on the regenerator’s hot and cold sides are defined as functions of the state of the regenerator matrix. The matrix is assumed to feature a linear spatial temperature distribution. Thus, the matrix has only two degrees of freedom that can, for example, be identified with its energy and entropy content. The dynamics of the regenerator is correspondingly expressed in terms of balance equations for energy and entropy. Internal irreversibilities of the regenerator can be accounted for by introducing source terms to the entropy balance equation. Compared to continuum or nodal regenerator models, the number of degrees of freedom and numerical effort are reduced considerably. As will be shown, instead of the obvious choice of variables energy and entropy, if convenient, a different pair of variables can be used to specify the state of the regenerator matrix and formulate the regenerator’s dynamics. In total, we will discuss three variants of this endoreversible regenerator model, which we will refer to as ES, EE, and EEn-regenerator models.

Quantum Otto engines at relativistic energies

Relativistic quantum systems exhibit unique features not present at lower energies, such as the existence of both particles and antiparticles, and restrictions placed on the system dynamics due to the light cone. In order to understand what impact these relativistic phenomena have on the performance of quantum thermal machines we analyze a quantum Otto engine with a working medium of a relativistic particle in an oscillator potential evolving under Dirac or Klein–Gordon dynamics. We examine both the low-temperature, non-relativistic and high-temperature, relativistic limits of the dynamics and find that the relativistic engine operates with higher work output, but an effectively reduced compression ratio, leading to significantly smaller efficiency than its non-relativistic counterpart. Using the framework of endoreversible thermodynamics we determine the efficiency at maximum power of the relativistic engine, and find it to be equivalent to the Curzon–Ahlborn efficiency.

Multilevel quantum thermodynamic swap engines

We study energetic exchanges and fluctuations in two-stroke quantum thermodynamic engines where the working fluid is represented by two multilevel quantum systems, i.e. qudits, the heat flow is allowed by relaxation with two thermal reservoirs at different temperatures, and the work exchange is operated by a partial-swap unitary interaction. We identify three regimes of operation (heat engine, refrigerator, and thermal accelerator), present the thermodynamic uncertainty relations between the entropy production and the signal-to-noise ratio of work and heat, and derive the full joint probability of the stochastic work and heat. Our results bridge the gap between two-qubit and two-mode bosonic swap engines, and show which properties are maintained (e.g., a non fluctuating Otto efficiency) and which are lost for increasing dimension (e.g., small violations of the standard thermodynamic uncertainty relations or the possibility of beating the Curzon-Ahlborn efficiency).

Thermodynamic Modeling of Heat Engines Including Heat Transfer and Compression–Expansion Irreversibilities

Abstract In this work, a thermodynamic model based on an endoreversible engine approach is developed to analyze the performance of heat engines operating under different thermodynamic cycles. The model considers finite heat transfer rate, variable heat source and sink temperatures, and irreversibilities associated with the expansion and compression. Expressions for the maximum power and efficiency at maximum power output are obtained as a function of hot and cold reservoir temperatures, the equivalent isentropic efficiency of compression and expansion components, and the effective conductance ratio between heat exchangers. In all cases, the Curzon–Ahlborn efficiency is retrieved at constant reservoir temperatures and neglected compression–expansion irreversibilities. The proposed model allows assessing the effect of isentropic efficiencies and heat exchanger design and operation characteristics for different thermodynamic cycles.

Maximal power for heat engines: Role of asymmetric interaction times

The performance of endoreversible thermal machines operating at finite power constitutes one of the main challenges of nonequilibrium classical and quantum thermodynamics, engineering and others. We introduce the idea of adjusting the interaction time asymmetry in order to optimize the engine performance. We consider one of the simplest thermal machines, composed of a quantum dot interacting sequentially with two different reservoirs of heat and particles. Distinct optimization protocols are analyzed in the framework of stochastic thermodynamics. Results reveal that asymmetric interaction times play a fundamental role in enhancing the power output and that maximizations can provide an increase larger than 25\% the symmetric case. As an extra advantage, efficiencies at maximum power are slightly greater than the endoreversible Curzon-Ahlborn efficiency for a broad range of reservoir temperatures.

Optimal Control for a Hydraulic Recuperation System Using Endoreversible Thermodynamics

Energy savings in the traffic sector are of considerable importance for economic and environmental considerations. Recuperation of mechanical energy in commercial vehicles can contribute to this goal. One promising technology rests on hydraulic systems, in particular for trucks which use such system also for other purposes such as lifting cargo or operating a crane. In this work the potential for energy savings is analyzed for commercial vehicles with tipper bodies, as these already have a hydraulic onboard system. The recuperation system is modeled based on endoreversible thermodynamics, thus providing a framework in which realistic driving data can be incorporated. We further used dissipative engine setups for modeling both the hydraulic and combustion engine of the hybrid drive train in order to include realistic efficiency maps. As a result, reduction in fuel consumption of up to 26% as compared to a simple baseline recuperation strategy can be achieved with an optimized recuperation control.

Optimization of an active heat engine.

Optimization of heat engines at the microscale has applications in biological and artificial nanotechnology and stimulates theoretical research in nonequilibrium statistical physics. Here we consider noninteracting overdamped particles confined by an external harmonic potential, in contact with either a thermal reservoir or a stochastic self-propulsion force (active Ornstein-Uhlenbeck model). A cyclical machine is produced by periodic variation of the parameters of the potential and of the noise. An exact mapping between the passive and the active model allows us to define the effective temperature T_{eff}(t), which is meaningful for the thermodynamic performance of the engine. We show that T_{eff}(t) is different from all other known active temperatures, typically used in static situations. The mapping allows us to optimize the active engine, regardless of the values of the persistence time or self-propulsion velocity. In particular, through linear irreversible thermodynamics (small amplitude of the cycle), we give an explicit formula for the optimal cycle period and phase delay (between the two modulated parameters, stiffness and temperature) achieving maximum power with Curzon-Ahlborn efficiency. In the quasistatic limit, the formula for T_{eff}(t) simplifies and coincides with a recently proposed temperature for stochastic thermodynamics, bearing a compact expression for the maximum efficiency. A point, which has been overlooked in recent literature, is made about the difficulty in defining efficiency without a consistent definition of effective temperature.

Irreversible entropy production in low- and high-dissipation heat engines and the problem of the Curzon-Ahlborn efficiency.

Heat engines performing finite time Carnot cycles are described by positive irreversible entropy functions added to the ideal reversible entropy part. The model applies for macroscopic and microscopic (quantum mechanical) engines. The mathematical and physical conditions for the solution of the power maximization problem are discussed. For entropy models which have no reversible limit, the usual "linear response regime" is not mathematically feasible; i.e., the efficiency at maximum power cannot be expanded in powers of the Carnot efficiency. Instead, a physically less intuitive expansion in powers of the ratio of heat-reservoir temperatures holds under conditions that will be inferred. Exact solutions for generalized entropy models are presented, and results are compared. For entropy generation in endoreversible models, it is proved for all heat transfer laws with general temperature-dependent heat resistances, that minimum entropy production is achieved when the temperature of the working substance remains constant in the isothermal processes. For isothermal transition time t, entropy production then is of the form a/[tf(t)±c] and not just equal to a/t for the low-dissipation limit. The cold side endoreversible entropy as a function of transition times inevitably experiences singularities. For Newtonian heat transfer with temperature-independent heat conductances, the Curzon-Ahlborn efficiency is exactly confirmed, which-only in this unique case-shows "universality" in the sense of independence from dissipation ratios of the hot and cold sides with coinciding lower and upper efficiency bounds for opposite dissipation ratios. Extended exact solutions for inclusion of adiabatic transition times are presented.

Power maximization of two-stroke quantum thermal machines

We present a detailed study of quantum thermal machines employing quantum systems as working substances. In particular, we study two different types of two-stroke cycles where two collections of identical quantum systems with evenly spaced energy levels are initially prepared at thermal equilibrium by putting them in contact with a cold and a hot thermal bath, respectively. The two cycles differ in the absence or the presence of a mediator system, while, in both cases, non-resonant exchange Hamiltonians are exploited as particle interactions. We show that the efficiencies of these machines depend only on the energy gaps of the systems composing the collections and are equal to the efficiency of "equivalent" Otto cycles. Focusing on the cases of qubits or harmonic oscillators for both models, we maximize the engine power and analyze, in the model without the mediator, the role of the waiting time between subsequent interactions. It turns out that the case with the mediator can bring performance advantages when the interaction times are comparable with the waiting time of the correspondent cycle without the mediator. We find that in both cycles, the power peaks of qubit systems can surpass the Curzon-Ahlborn efficiency. Finally, we compare our cycle without the mediator with previous schemes of the quantum Otto cycle showing that high coupling is not required to achieve the same maximum power.