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

Optimal figure of merit of low-dissipation quantum refrigerators

The Drazin inverse of the Liouvillian superoperator provides a solution to determine the dynamics of a time-dependent system governed by the Markovian master equation. Under the condition of slow driving, the perturbation expansion of the density operator of the system in powers of time can be derived. As an application, a finite-time cycle model of the quantum refrigerator driven by a time-dependent external field is established. The method of the Lagrange multiplier is adopted as a strategy to find the optimal cooling performance. The figure of merit given by the product of the coefficient of performance and the cooling rate is taken as a new objective function, and, consequently, the optimally operating state of the refrigerator is revealed. The effects of the frequency exponent determining dissipation characteristics on the optimal performance of the refrigerator are discussed systemically. The results obtained show that the adjacent areas of the state of the maximum figure of merit are the best operation region of low-dissipative quantum refrigerators.

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.

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.

Optimal Cycles for Low-Dissipation Heat Engines

We consider the optimization of a finite-time Carnot engine characterized by small dissipations. We bound the power with a simple inequality and show that the optimal strategy is to perform small cycles around a given working point, which can be, thus, chosen optimally. Remarkably, this optimal point is independent of the figure of merit combining power and efficiency that is being maximized. Furthermore, for a general class of dissipative dynamics the maximal power output becomes proportional to the heat capacity of the working substance. Since the heat capacity can scale supraextensively with the number of constituents of the engine, this enables us to design optimal many-body Carnot engines reaching maximum efficiency at finite power per constituent in the thermodynamic limit.

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.

Entropy generation and unified optimization of Carnot-like and low-dissipation refrigerators

The connection between Carnot-like and low-dissipation refrigerators is proposed by means of their entropy generation and the optimization of two unified, compromise-based figures of merit. Their optimization shows that only a limited set of heat transfer laws in the Carnot-like model are compatible with the results stemming from the low-dissipation approximation, even though there is an agreement of the related physical spaces of variables. A comparison between two operation regimes and relations among entropy generation, efficiency, cooling power. and power input are obtained, with emphasis on the role of dissipation symmetries. The results extend previous findings for heat engines at maximum power conditions.

General relations between the power, efficiency, and dissipation for the irreversible heat engines in the nonlinear response regime

We derive the general relations between the maximum power, maximum efficiency, and minimum dissipation for the irreversible heat engine in a nonlinear response regime. In this context, we use the minimally nonlinear irreversible model and obtain the lower and upper bounds of the above relations for the asymmetric dissipation limits. These relations can be simplified further when the system possesses the time-reversal symmetry or antisymmetry. We find that our results are the generalization of various such relations obtained earlier for different heat engines.

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.

Measurement-induced operation of two-ion quantum heat machines

We show how one can implement a quantum heat machine by using two interacting trapped ions, in presence of a thermal bath. The electronic states of the ions act like a working substance, while the vibrational mode is modelled as the cold bath. The heat exchange with the cold bath is mimicked by the projective measurement of the electronic states. We show how such measurement in a suitable basis can lead to either a quantum heat engine or a refrigerator, which undergoes a quantum Otto cycle. The local magnetic field is adiabatically changed during the heat cycle. The performance of the heat machine depends upon the interaction strength between the ions, the magnetic fields, and the measurement cost. In our model, the coupling to the hot and the cold baths is never switched off in an alternative fashion during the heat cycle, unlike other existing proposals of quantum heat engines. This makes our proposal experimentally realizable using current tapped-ion technology.

Efficiency at and near maximum power of low-dissipation heat engines

A universality in optimization of trade-off between power and efficiency for low-dissipation Carnot cycles is presented. It is shown that any trade-off measure expressible in terms of efficiency and the ratio of power to its maximum value can be optimized independently of most details of the dynamics and of the coupling to thermal reservoirs. The result is demonstrated on two specific trade-off measures. The first one is designed for finding optimal efficiency for a given output power and clearly reveals diseconomy of engines working at maximum power. As the second example we derive universal lower and upper bounds on the efficiency at maximum trade-off given by the product of power and efficiency. The results are illustrated on a model of a diffusion-based heat engine. Such engines operate in the low-dissipation regime given that the used driving minimizes the work dissipated during the isothermal branches. The peculiarities of the corresponding optimization procedure are reviewed and thoroughly discussed.

Four-level refrigerator driven by photons

We propose a quantum absorption refrigerator driven by photons. The model uses a four-level system as its working substance and couples simultaneously to hot, cold, and solar heat reservoirs. Explicit expressions for the cooling power Q̇(c) and coefficient of performance (COP) η(COP) are derived, with the purpose of revealing and optimizing the performance of the device. Our model runs most efficiently under the tight coupling condition, and it is consistent with the third law of thermodynamics in the limit T→0.

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.