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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.

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.

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.

A local stability analysis of an endoreversible Carnot refrigerator, working at the maximum objective function of the product of the cooling rate R and the coefficient of performance e, is presented. The endoreversible Carnot refrigerator consists of a reversible Carnot refrigerator that exchanges heat with the heat reservoirs TH through the thermal conductance α and with the cold reservoirs TL through the thermal conductance β. In addition, the working fluid has the same heat capacity C in the two isothermal branches of the cycle. By linearization and stability analysis, we find that the relaxation times are a function of α, β, the heat capacity C and τ=TL /TH; that the endoreversible Carnot refrigerator is stable for every value of α, β, C and τ; that after a perturbation, the system state exponentially decays to the steady state with either of two different relaxation times; that both relaxation times are proportional to α/2C; and that one of them is a monotonically increasing function τ and the other is almost independent of τ. Finally, the phase portraits for the trajectories after a small perturbation over the steady-state values of internal temperatures are presented.

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.