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Based on the thermosize effects of a confined ideal Maxwellian gas, we established a model of micro/nano-scaled heat pump including two isothermal and two isobaric processes, where the Joule-like heating effect was analyzed in detail by using the nonequilibrium thermodynamic theory. Further, in order to improve the performance of a single thermosize device, a model of two-stage micro/nano-scaled heat pump with multiple micro/nano components is for the first time put forward. The general expressions of heating load and coefficient of performance for the single and the two-stage heat pump are derived. The influences of the internal and external irreversibilities of the system, i.e., Fourier heat flow and the Joule-like heating and the finite rate heat transfer with the reservoirs, on the performance characteristic of thermosize devices are studied in detail. Moreover, we also discuss the effect of the allocation the thermosize components on the optimal performance of the two-stage micro/nano-scaled heat pump cycle. It is found that in the presence of the internal and external irreversibilities, the two kinds of the micro/nano-scaled heat pump can achieve specific heating performance. In particular, when the total number M of the thermosize components is fixed, both the maximum of the optimal coefficient of performance Φ opt , max and the maximum of the heating load Π opt , max corresponding to the optimal coefficient of performance appear near n = M / 2 with n being the number of the thermosize elements of the top stage. The results obtained here show several main irreversibility source of the micro/nano-scaled heat pump and have potential applications in the efficient utilization and conversion of energy in the microchannel system using gas.

In this work, we study the dynamic robustness of an endoreversible Carnot cycle working at the maximum per-unit-time performance regime, based on the linearization technique for dynamical systems and the local stability analysis. Our analysis is focused on the endoreversible Carnot refrigerator model, which works in the maximum per-unit-time coefficient of performance. At the steady-state of the maximum performance, the expressions of the relaxation times describing the stability of the system are derived. It is found that the relaxation times in the cycle condition are the function of thermal conductances σh and σc, the temperatures of the heat reservoirs Th and Tc, and the heat capacity C. The influence of the temperature ratio τ = Tc/Th and the thermal conductance ratio σr = σh/σc on the relaxation times is discussed in detail. The results obtained here are useful and provide a potential guidance for the design of an endoreversible Carnot refrigerator working in the maximum performance per cycle time optimization condition.

In this paper, the model of a three-terminal quantum dot heat engine consists of two quantum dots with a single energy level, a cavity and two electron reservoirs is established. According to Landauer formula the expressions for the heat current, the output power and the efficiency are derived analytically. The performance characteristic curves of the output power versus the efficiency are plotted by numerical calculation. Moreover, the optimal performance parameters are determined. Then we optimize the output power and the efficiency respectively, the influence of the width of energy level and the heat leak on performance of the three-terminal thermoelectric heat engine is discussed. Lastly, the variation of the corresponding efficiency at the maximum power output with the Carnot efficiency between two reservoirs is discussed and the corresponding efficiency is compared with Carnot efficiency and CA efficiency. It is shown that this three-terminal heat engine is irreversible due to the existence of the width of energy level and the heat leak. Thus the curve of the power output versus the efficiency is a loop-shaped one. And due to the heat leak, the characteristics of the efficiency and the width of energy level is a non-monotone curve. The efficiency at the maximum power will be greater than the CA efficiency when the heat leak and the width of energy level are not taken into account.

We investigate the finite-power performance of quantum heat engines working in the linear response regime where the temperature gradient is small. The engine cycles with working substances of ideal harmonic systems consist of two heat transfer and two adiabatic processes, such as the Carnot cycle, Otto cycle, and Brayton cycle. By analyzing the optimal protocol under maximum power we derive the explicitly analytic expression for the irreversible entropy production, which becomes the low dissipation form in the long duration limit. Assuming the engine to be endoreversible, we derive the universal expression for the efficiency at maximum power, which agrees well with that obtained from the phenomenological heat transfer laws holding in the classical thermodynamics. Through appropriate identification of the thermodynamic fluxes and forces that a linear relation connects, we find that the quantum engines under consideration are tightly coupled, and the universality of efficiency at maximum power is confirmed at the linear order in the temperature gradient.

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.

We analyze general models of quantum heat engines operating a cycle of two adiabatic and two isothermal processes. We use the quantum master equation for a system to describe heat transfer current during a thermodynamic process in contact with a heat reservoir, with no use of phenomenological thermal conduction. We apply the endoreversibility description to such engine models working in the linear response regime and derive expressions of the efficiency and the power. By analyzing the entropy production rate along a single cycle, we identify the thermodynamic flux and force that a linear relation connects. From maximizing the power output, we find that such heat engines satisfy the tight-coupling condition and the efficiency at maximum power agrees with the Curzon-Ahlborn efficiency known as the upper bound in the linear response regime.

The performance in finite time of a quantum-mechanical Brayton engine cycle is discussed, without introduction of temperature. The engine model consists of two quantum isoenergetic and two quantum isobaric processes, and works with a single particle in a harmonic trap. Directly employing the finite-time thermodynamics, the efficiency at maximum power output is determined. Extending the harmonic trap to a power-law trap, we find that the efficiency at maximum power is independent of any parameter involved in the model, but depends on the confinement of the trapping potential.

We study the performance of minimally nonlinear irreversible refrigerators which works between a hot and cold heat reservoir of constant temperatures [Formula: see text] and [Formula: see text], respectively. Applying optimization criterion [Formula: see text] or [Formula: see text], we analyze the coefficient of performance (COP) at arbitrary value of [Formula: see text] or [Formula: see text] and thus obtain the upper and lower bounds of the COP. Our results show that, for the optimized, tightly-coupled refrigerators, a small loss away from the maximum value of either [Formula: see text] or [Formula: see text] figure of merit means a larger gain in the COP.

The Diesel refrigeration cycle using an ideal quantum gas as the working substance is called quantum Diesel refrigeration cycle, which is different from Carnot, Ericsson, Brayton, Otto and Stirling refrigeration cycles. For ideal quantum gases, a corrected equation of state, which considers the quantum behavior of gas particles, is used instead of the classical one. The purpose of this paper is to investigate the effect of quantum gas as the working substance on the performance of a quantum Diesel refrigeration cycle. It is found that coefficients of performance of the cycle are not affected by the quantum degeneracy of the working substance, which is the same as that of the classical Diesel refrigeration cycle. However, the refrigeration load is different from those of the classical Diesel refrigeration cycle. Lastly, the influence of the quantum degeneracy on the performance characteristics of the quantum Diesel refrigeration cycle operated in different temperature regions is discussed.

The influence of both the quantum degeneracy and the finite rate heat transfer between the working substance and the cylinder wall on the optimal performance of an Otto engine cycle is investigated. Expressions for several important parameters such as the power output and efficiency are derived. By using numerical solutions, the curves of the power output and efficiency varying with the compression ratio of two isochoric processes are presented. It is found that there are optimal values of the compression ratio at which the power output and efficiency attain their maximum. In particular, the optimal performance of the cycle in strong and weak gas degeneracy and the high temperature limit are discussed in detail. The distinctions and connections between the quantum Otto engine and the classical are revealed. Moreover, the maximum power output and efficiency and the corresponding relevant parameters are calculated, and consequently, the optimization criteria of some important parameters such as the power output, efficiency and compression ratio of the working substance are obtained.

The new model of a quantum refrigeration cycle composed of two adiabatic and two isomagnetic field processes is established. The working substance in the cycle consists of many non-interacting spin-1/2 systems. The performance of the cycle is investigated, based on the quantum master equation and semi-group approach. The general expressions of several important performance parameters, such as the coefficient of performance, cooling rate and power input, are given. It is found that the coefficient of performance of this cycle is a close analogue of the classical Carnot-cycle. Some performance characteristic curves relating the cooling rate, the coefficient of performance and power input are plotted. Further, for high temperatures, the optimal relations between the cooling rate and the coefficient of performance are analyzed in detail.

An irreversible cycle model of a quantum heat engine with internal friction is established, which is composed of two adiabatic and two isomagnetic field processes. The working substance of the cycle consists of an ensemble of many non-interacting spin-1/2 systems. Based on a quantum master equation and semi-group approach, the general performance characteristics of the heat engine are investigated. The general expressions for several important parameters, such as the efficiency, power output, and rate of the entropy production, are derived. The performance of the cycle is optimized with respect to the temperatures of the working substance. By numerical solutions, the maximum power output and the corresponding parameters are calculated. The optimal regions of efficiency, temperatures of the working substance, and cycle period are determined. Moreover, the performance of the heat engine in the frictionless case is obtained, which is different from that in the friction case. Finally, the results obtained are generalized to the performance optimization of the heat engine working with spin-J systems.

In this paper, the energy selective electron (ESE) refrigerator with an ideal energy filter based on resonant tunneling is established. It consists of two infinitely large electron reservoirs with different temperatures and chemical potentials, and electrons can be exchanged between the two reservoirs through the ideal energy filter. According to Landauer formula and the assumption of being coupled tightly with the electron current, the expressions for the heat flux, the cooling rate and the coefficient of performance (COP) are derived analytically. The performance characteristic curves such as the cooling rate versus coefficient of performance, the cooling rate and coefficient of performance versus the position of energy level are plotted by numerical calculation. The optimal regions of the cooling rate and the COP are determined. Moreover, we optimize the cooling rate and the COP respectively with respect to the position of energy level. The influence of the width of energy level on performance of the ESE refrigerator is discussed. Finally, based on the optimization criterion for refrigerator, i.e. the product of the COP times the cooling rate, the optimal performance of the ESE refrigerator is discussed in detail.

We study the thermodynamic performance of a nanosized photoelectric refrigerator consisting of three coupled single-level quantum dots embedded between two reservoirs at different temperatures. Based on the quantum master equation, we derive expressions for the cooling power and coefficient of performance (COP) of the refrigerator and plot the characteristic curves between the cooling power and the COP. We analyze the optimal performance parameters under conditions of maximum cooling power and maximum COP, and we discuss the influence of the energy level difference and the temperature ratio on the optimal performance parameters in detail.

A Brownian refrigerator with the cold and hot reservoirs alternating along a space coordinate is established. The heat flux couples with the movement of the Brownian particles due to an external force in the spatially asymmetric but periodic potential. After using the Arrhenius factor to describe the behaviors of the forward and backward jumps of the particles, the expressions for coefficient of performance (COP) and cooling rate are derived analytically. Then, through maximizing the product of conversion efficiency and heat flux flowing out, a new upper bound only depending on the temperature ratio of the cold and hot reservoirs is found numerically in the reversible situation, and it is a little larger than the so-called Curzon and Ahlborn COP ε(CA)=(1/√[1-τ])-1. After considering the irreversible factor owing to the kinetic energy change of the moving particles, we find the optimized COP is smaller than ε(CA) and the external force even does negative work on the Brownian particles when they jump from a cold to hot reservoir.

本文研究了一种由周期性非均匀线性温场所驱动的布朗制冷机性能特征. 利用布朗粒子的动力学方程——Langevin equation, 获得了稳态时布朗制冷机的粒子流、制冷率和制冷系数的解析表达式. 通过数值计算分析了势垒高度、非对称性、外力和温比等参数对制冷机性能的影响. 在最大品质因子条件下, 对制冷机性能参数进行优化, 画出了优化的制冷系数和制冷率随温比的演化曲线. 研究表明: 在准静态极限下, 制冷系数趋于用等效温度给出的卡诺制冷系数, 制冷率趋于零; 制冷系数随制冷率增加而减少; 优化的制冷系数随温比减小而下降, 而对应的制冷率却相反; 相比于在分段均匀温度场中, 非均匀线性温度场将导致制冷系数的提高.

An irreversible model of the air-standard Diesel heat engine cycle is established, in which the primary irreversibilities, such as the heat leak loss through the cylinder wall, friction loss, internal irreversibility and temperature-variable heat capacities, are considered. The analytic expressions of the power output and efficiency are derived. The relations between the power output and the ratio of the highest to the lowest pressure of the working fluid, between the efficiency and the ratio of the highest to the lowest pressure of the working fluid, as well as the optimal relation between the power output and the efficiency of the Diesel cycle are shown by numerical example. Moreover, the influence of these primary irreversible factors on the performance parameters of the Diesel heat engine is discussed. The results obtained here may provide some significant guidelines for the optimal design and operation of the practical Diesel heat engines.

In this paper, we establish a nanothermoelectric engine consisting of two discrete energy levels embedded between two reservoirs at different temperatures and chemical potentials. Based on master equation, the expressions for the power output and efficiency of the nanothermoelectric engine are derived. The characteristic curves between the power output and the efficiency are plotted. Moreover, the optimal performance parameters are obtained by the numerical calculation. The influence of the strength of variations in electron–electron interactions on the optimal performance parameters is analyzed in detail.

We investigate the thermodynamic performance of a quantum-dot refrigerator consisting of a single orbital interacting quantum dot embedded between two electron reservoirs at different temperatures and chemical potentials. Based on the quantum master equation the expressions for the cooling power and the coefficient of performance are derived. The characteristic curves between the cooling power and the coefficient of performance are plotted, and the optimal regions of the performance parameters are determined. Moreover, the optimal performance parameters are calculated numerically. Finally, the influence of the Coulomb interaction and the temperature ratio on optimal performance parameters are discussed in detail. The results obtained here can provide some theoretical guidelines for the design and operation of the practical quantum-dot refrigerator. Our work is not restricted to the linear-response regime.

We setup a three-level heat engine model that works with two noninteracting fermions in a one-dimensional box trap. Besides two quantum adiabatic processes, the quantum heat engine cycle consists of two isoenergetic processes, along which the particles are coupled to energy baths at a high constant energy EH and a low constant energy EC, respectively. Based on the assumption that the potential wall moves at a very slow speed and there exists a heat leakage between two energy baths, we derive the expressions of the power output and the efficiency, and then obtain the optimization region for the heat engine cycle. Finally, we present a brief performance analysis of a Carnot engine between a hot and a cold bath at temperatures TH and TC, respectively. We demonstrate that under the same conditions, the efficiency η=1-(EC/EH) of the engine cycle is bounded from above the Carnot efficiency ηc=1-(TC/TH).

Particle hopping on a one-dimensional lattice driven by an external force in a periodic sawtooth potential and temperature field may act as a micro-Brownian refrigerator. In order to clarify the underlying physical pictures of the refrigerator, heat flows via both the potential energy and the kinetic energy of the particle are considered simultaneously. Based on the master equation describing the jump of the particle among the three states, expressions for the cooling rate and the coefficient of performance of the refrigerator are derived analytically. The general performance characteristic curves are plotted by numerical calculation. It is found that the characteristic curve between the cooling rate and the coefficient of performance is a loop-shaped one; the Brownian refrigerator is irreversible and its coefficient of performance is always less than the Carnot value. The influence of the temperature ratio of the heat reservoirs and the height of the sawtooth potential on the optimal performance characteristic parameters is analyzed. When heat flow via the kinetic energy of the particle is neglected, the characteristic curve between the cooling rate and the coefficient of performance is an open-shaped one. In this case, the Brownian refrigerator is reversible and its coefficient of performance reaches the Carnot value in the quasistatic limit.

An energy selective electron (ESE) refrigerator with heat leaks is established in a one-dimensional system. Based on the theory of electronic transport, the expressions of the heat flux into hot and cold electron reservoirs are derived. When the heat leaks between two electron reservoirs via phonons are taken into account, the cooling rate, coefficient of performance (COP) and input power are obtained. The performance characteristic curves such as the cooling rate versus the COP, the cooling rate and the COP versus the center position of the resonance energy level are plotted by numerical calculation. The influence of the center position and width of the resonance energy level on the performance of the ESE refrigerator is analyzed in detail. Lastly, the influence of heat leaks and average temperature on the performance of the ESE refrigerator is discussed. The results obtained here have theoretical significance for understanding the thermodynamic performance of the practical ESE refrigerator.

An irreversible cycle model of a Bose–Otto engine is established, in which finite time thermodynamic processes and the irreversibility result from the nonisentropic compression and expansion processes are taken into account. Based on the model, expressions for the power output and efficiency of the Bose–Otto engine are derived. On the basis of the thermodynamic properties of ideal Bose gas, the effects of the irreversibility and the compression ratio of the two isochoric processes on the performance of the Bose–Otto engine are revealed and some important performance parameters are optimized. Furthermore, some optimal operating regions including those for the power output, efficiency, and the temperatures of the cyclic working substance at two important state points are determined and evaluated. Finally, several special cases are discussed in detail.