Efficiency at maximum power output of an irreversible Carnot-like cycle with internally dissipative friction

Low-dissipation engines: Microscopic construction via shortcuts to adiabaticity and isothermality, the optimal relation between power and efficiency

We construct a microscopic model of low-dissipation engines by driving a Brownian particle in a time-dependent harmonic potential. Shortcuts to adiabaticity and shortcuts to isothermality are introduced to realize the adiabatic and isothermal branches in a thermodynamic cycle, respectively. We derive an analytical formula of the efficiency at maximum power with explicit expressions of dissipation coefficients under the optimized protocols. When the relative temperature difference between the two baths in the cycle is insignificant, this expression satisfies the universal law of efficiency at maximum power up to the quadratic term of the Carnot efficiency. For large relative temperature differences, the efficiency at maximum power tends to be 1/2. Furthermore, we analyze the issue of power at any given efficiency for general low-dissipation engines and then obtain the supremum of the power in three limiting cases, respectively. These expressions of maximum power at given efficiency provide the optimal relations between power and efficiency which are tighter than the results in previous references.

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.

Cyclic heat engines with nonisentropic adiabats and generalization to steady-state devices including thermoelectric converters

For heat engines (including refrigerators) the separation of total entropy production in reversible parts ΔS and irreversible contributions has proved to be very useful. The ΔS are entropies for ideal lossless processes at the hot- and cold side and are important system parameters. For Carnot-like heat engines performing finite-time cycles, the concern was raised in a preceding paper that the ΔS are not always independent from irreversibilities, if initial and final working fluid temperatures T_{f}(t) differ in the isothermal transitions. It turns out that the ΔS are unchanged and independent, if T_{f} (t) evolution is optimized for entropy minimization and apparent inconsistencies are cleared up. If nonisentropic transitions in the adiabatic cycle branches are taken into account, the difference of cold- and hot-side entropy reversibilities is equal to the entropy production in the adiabats. Maximization of cooling power is studied for various irreversible entropy models. The concepts are extended to noncyclic steady-state engines. Power maximization and efficiency calculations are performed exactly analytically. This serves as prerequisite for the hitherto unsolved problem of an accurate definition of reversible and irreversible entropy parts in thermoelectric (TE) converters in the case of inhomogeneous three-dimensional material distributions. It is revealed that for nonconstant Seebeck coefficients, additional terms to the Joule heat arise that destroy positive generator performance in the limit of heat conductance k→0, in contrast to the traditional constant material properties model. Thus, the concept of improving TE materials by reducing k is in question and an adapted figure of merit Z is presented to deal with the situation.

Unified approach to stochastic thermodynamics: Application to a quantum heat engine

In the present study we have developed an alternative formulation for the quantum stochastic thermodynamics based on the c-number Langevin equation for the system-reservoir model. This is analogous to the classical one. Here we have considered both Markovian and non-Markovian dynamics (NMD). Consideration of the NMD is an important issue at the current state of the stochastic thermodynamics. Applying the present formalism, we have carried out a comparative study on the heat absorbed and the change of entropy with time for a linear quantum system and its classical analog for both Markovian and NMD. Here the strength of the thermal noise and its correlation time for the respective cases are the leading quantities to explain any distinguishable feature which may appear with the equilibration kinetics. For another application, we have proposed a formulation with classical look for a quantum stochastic heat engine. Using it we have presented a comparative study on the efficiency and its value at maximum power for a quantum stochastic heat engine and its classical analog. The engines are Carnot like which are coupled with their respective Markovian thermal baths. Here also the noise strength as well as the diffusion constant are the leading quantities to explain any noticeable feature.

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.

Achieve higher efficiency at maximum power with finite-time quantum Otto cycle

The optimization of heat engines was intensively explored to achieve higher efficiency while maintaining the output power. However, most investigations were limited to a few finite-time cycles, e.g., the Carnot-like cycle, due to the complexity of the finite-time thermodynamics. In this paper, we propose a class of finite-time engine with quantum Otto cycle, and demonstrate a higher achievable efficiency at maximum power. The current model can be widely utilized, benefitting from the general C/τ^{2} scaling of extra work for a finite-time adiabatic process with long control time τ. We apply the adiabatic perturbation method to the quantum piston model and calculate the efficiency at maximum power, which is validated with an exact solution.

Progress in Carnot and Chambadal Modeling of Thermomechanical Engine by Considering Entropy Production and Heat Transfer Entropy

Nowadays the importance of thermomechanical engines is recognized worldwide. Since the industrial revolution, physicists and engineers have sought to maximize the efficiency of these machines, but also the mechanical energy or the power output of the engine, as we have recently found. The optimization procedure applied in many works in the literature focuses on considering new objective functions including economic and environmental criteria (i.e., ECOP ecological coefficient of performance). The debate here is oriented more towards fundamental aspects. It is known that the maximum of the power output is not obtained under the same conditions as the maximum of efficiency. This is shown, among other things, by the so-called nice radical that accounts for efficiency at maximum power, most often for the endoreversible configuration. We propose here to enrich the model and the debate by emphasizing the fundamental role of the heat transfer entropy together with the production of entropy, accounting for the external or internal irreversibilities of the converter. This original modeling to our knowledge, leads to new and more general results that are reported here. The main consequences of the approach are emphasized, and new limits of the efficiency at maximum energy or power output are obtained.

Efficiency and its bounds of minimally nonlinear irreversible heat engines at arbitrary power

The efficiency for minimally nonlinear irreversible heat engines at any arbitrary power has been systematically evaluated, and general lower and upper efficiency bounds under the tight coupling condition for different operating regions have been proposed, which can be seen as the generalization of the bounds [η_{C}/2<η_{maxP}<η_{C}/(2-η_{C})] on efficiency at maximum power (η_{maxP}), where η_{C} means the Carnot efficiency. We have also calculated the universal bounds of the maximum gain in efficiency in different operating regions to give further insight into the efficiency gain with the power away from the maximum power. In the region of higher loads (higher than the load which corresponds to the maximum power), a small power loss away from the maximum power induces a much larger gain in efficiency. As actual heat engines may not work at the maximum power condition, this paper may contribute to operating actual heat engines more efficiently.

Efficiency at maximum power of thermochemical engines with near-independent particles

Two-reservoir thermochemical engines are established by using near-independent particles (including Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein particles) as the working substance. Particle and heat fluxes can be formed based on the temperature and chemical potential gradients between two different reservoirs. A rectangular-type energy filter with width Γ is introduced for each engine to weaken the coupling between the particle and heat fluxes. The efficiency at maximum power of each particle system decreases monotonously from an upper bound η(+) to a lower bound η(-) when Γ increases from 0 to ∞. It is found that the η(+) values for all three systems are bounded by η(C)/2 ≤ η(+) ≤ η(C)/(2-η(C)) due to strong coupling, where η(C) is the Carnot efficiency. For the Bose-Einstein system, it is found that the upper bound is approximated by the Curzon-Ahlborn efficiency: η(CA)=1-sqrt[1-η(C)]. When Γ → ∞, the intrinsic maximum powers are proportional to the square of the temperature difference of the two reservoirs for all three systems, and the corresponding lower bounds of efficiency at maximum power can be simplified in the same form of η(-)=η(C)/[1+a(0)(2-η(C))].

Universality of efficiency at unified trade-off optimization

We calculate the efficiency at the unified trade-off optimization criterion (the so-called maximum Ω criterion) representing a compromise between the useful energy and the lost energy of heat engines operating between two reservoirs at different temperatures and chemical potentials, and demonstrate that the linear coefficient 3/4 and quadratic coefficient 1/32 of the efficiency at maximum Ω are universal for heat engines under strong coupling and symmetry conditions. It is further proved that the conclusions obtained here also apply to the ecological optimization criterion.

Heat devices in nonlinear irreversible thermodynamics

We present results obtained by using nonlinear irreversible models for heat devices. In particular, we focus on the global performance characteristics, the maximum efficiency and the efficiency at maximum power regimes for heat engines, and the maximum coefficient of performance (COP) and the COP at maximum cooling power regimes for refrigerators. We analyze the key role played by the interplay between irreversibilities coming from heat leaks and internal dissipations. We also discuss the relationship between these results and those obtained by different models.

Unified trade-off optimization for general heat devices with nonisothermal processes

An analysis of the efficiency and coefficient of performance (COP) for general heat engines and refrigerators with nonisothermal processes is conducted under the trade-off criterion. The specific heat of the working medium has significant impacts on the optimal configurations of heat devices. For cycles with constant specific heat, the bounds of the efficiency and COP are found to be the same as those obtained through the endoreversible Carnot ones. However, they are independent of the cycle time durations. For cycles with nonconstant specific heat, whose dimensionless contact time approaches infinity, the general alternative upper and lower bounds of the efficiency and COP under the trade-off criteria have been proposed under the asymmetric limits. Furthermore, when the dimensionless contact time approaches zero, the endoreversible Carnot model is recovered. In addition, the efficiency and COP bounds of different kinds of actual heat engines and refrigerators have also been analyzed. This paper may provide practical insight for designing and operating actual heat engines and refrigerators.

Coefficient of performance under maximum χ criterion in a two-level atomic system as a refrigerator

A two-level atomic system as a working substance is used to set up a refrigerator consisting of two quantum adiabatic and two isochoric processes (two constant-frequency processes ω_{a} and ω_{b} with ω_{a}<ω_{b}), during which the two-level system is in contact with two heat reservoirs at temperatures T_{h} and T_{c}(<T_{h}). Considering finite-time operation of two isochoric processes, we derive analytical expressions for cooling rate R and coefficient of performance (COP) ɛ. The COP at maximum χ(=ɛR) figure of merit is numerically determined, and it is proved to be in nice agreement with the so-called Curzon and Ahlborn COP ɛ_{CA}=sqrt[1+ɛ_{C}]-1, where ɛ_{C}=T_{c}/(T_{h}-T_{c}) is the Carnot COP. In the high-temperature limit, the COP at maximum χ figure of merit, ɛ^{*}, can be expressed analytically by ɛ^{*}=ɛ_{+}≡(sqrt[9+8ɛ_{C}]-3)/2, which was derived previously as the upper bound of optimal COP for the low-dissipation or minimally nonlinear irreversible refrigerators. Within the context of irreversible thermodynamics, we prove that the value of ɛ_{+} is also the upper bound of COP at maximum χ figure of merit when we regard our model as a linear irreversible refrigerator.

Performance optimization of minimally nonlinear irreversible heat engines and refrigerators under a trade-off figure of merit

A performance optimization for minimally nonlinear heat engines and refrigerators is conducted under an optimization criterion of Ω. The results show that under tight-coupling conditions, the efficiency and coefficient of performance (COP) bounds in asymmetric dissipation limits are the same as those obtained by de Tomas et al. [Phys. Rev. E 87, 012105 (2013)] for low dissipation heat devices. The efficiency bounds for heat engines under nontight-coupling conditions are also analyzed and the experimental results lie between theoretical results obtained under different coupling strengths. For refrigerators, the theoretical results are also in good agreement with some observed results. The efficiency and COP bounds under the Ω criterion are refined, which are closer to real heat engines and refrigerators.

Stochastic heat engine with the consideration of inertial effects and shortcuts to adiabaticity

When a Brownian particle in contact with a heat bath at a constant temperature is controlled by a time-dependent harmonic potential, its distribution function can be rigorously derived from the Kramers equation with the consideration of the inertial effect of the Brownian particle. Based on this rigorous solution and the concept of shortcuts to adiabaticity, we construct a stochastic heat engine by employing the time-dependent harmonic potential to manipulate the Brownian particle to complete a thermodynamic cycle. We find that the efficiency at maximum power of this stochastic heat engine is equal to 1-sqrt[T(c)/T(h)], where T(c) and T(h) are the temperatures of the cold bath and the hot one in the thermodynamic cycle, respectively.

Weighted reciprocal of temperature, weighted thermal flux, and their applications in finite-time thermodynamics

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. 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.

Coefficient of performance for a low-dissipation Carnot-like refrigerator with nonadiabatic dissipation

We study the coefficient of performance (COP) and its bounds for a Carnot-like refrigerator working between two heat reservoirs at constant temperatures T(h) and T(c), under two optimization criteria χ and Ω. In view of the fact that an "adiabatic" process usually takes finite time and is nonisentropic, the nonadiabatic dissipation and the finite time required for the adiabatic processes are taken into account by assuming low dissipation. For given optimization criteria, we find that the lower and upper bounds of the COP are the same as the corresponding ones obtained from the previous idealized models where any adiabatic process is undergone instantaneously with constant entropy. To describe some particular models with very fast adiabatic transitions, we also consider the influence of the nonadiabatic dissipation on the bounds of the COP, under the assumption that the irreversible entropy production in the adiabatic process is constant and independent of time. Our theoretical predictions match the observed COPs of real refrigerators more closely than the ones derived in the previous models, providing a strong argument in favor of our approach.

From local force-flux relationships to internal dissipations and their impact on heat engine performance: the illustrative case of a thermoelectric generator

We present an in-depth analysis of the sometimes understated role of the principle of energy conservation in linear irreversible thermodynamics. Our case study is that of a thermoelectric generator (TEG), which is a heat engine of choice in irreversible thermodynamics, owing to the coupling between the electrical and heat fluxes. We show why Onsager's reciprocal relations must be considered locally and how internal dissipative processes emerge from the extension of these relations to a global scale: The linear behavior of a heat engine at the local scale is associated with a dissipation process that must partake in the global energy balance. We discuss the consequences of internal dissipations on the so-called efficiency at maximum power, in the light of our comparative analyses of exoreversibility and endoreversibility on the one hand and of two classes of heat engines, autonomous and periodically driven, on the other hand. Finally, basing our analysis on energy conservation, we also discuss recent works which claim the possibility to overcome the traditional boundaries on efficiency imposed by finite-time thermodynamics in thermoelectric systems with broken time-reversal symmetry; this we do by introducing a "thermal" thermopower and an "electrical" thermopower which permits an analysis of the thermoelectric response of the TEG considering a possible dissymmetry between the electrical/thermal and the thermal/electrical couplings.

Efficiency at maximum power of a heat engine working with a two-level atomic system

We consider the finite-time operation of a quantum heat engine whose working substance is composed of a two-level atomic system. The engine cycle, consisting of two quantum adiabatic and two quantum isochoric (constant-frequency) processes and working between two heat reservoirs at temperatures T(h) and T(c)(<T(h)), is a quantum version of the classical Otto cycle. By optimizing the power output with respect to two frequencies, we obtain the efficiency at maximum power output (EMP) and analyze numerically the effects of the times taken for two adiabatic and two isochoric processes on the EMP. In the absence of internally dissipative friction, we find that the EMP is bounded from the upper side by a function of the Carnot efficiency η(C), η(+)=η(C)(2)/[η(C)-(1-η(C))ln(1-η(C))], with η(C)=1-T(c)/T(h). This analytic expression is confirmed by our exact numerical result and is identical to the one derived in an engine model based on a mesoscopic or macroscopic system. If the internal friction is included, we find that the EMP decreases as the friction coefficient increases.

Universal efficiency bounds of weak-dissipative thermodynamic cycles at the maximum power output

Based on the assumption of weak dissipation introduced by Esposito et al. [Phys. Rev. Lett. 105, 150603 (2010)], analytic expressions for the efficiency bounds of several classes of typical thermodynamic cycles at the maximum power output are derived. The results obtained are of universal significance. They can be used to conveniently reveal the general characteristics of not only Carnot heat engines, but also isothermal chemical engines, non-Carnot heat engines, flux flow engines, gravitational engines, quantum Carnot heat engines, and two-level quantum Carnot engines at the maximum power output and to directly draw many important conclusions in the literature.