The performance of a quantum-mechanical heat pump using many non-interacting spin-1/2 systems as the working substance and consisting of two isothermal and two isomagnetic field processes is investigated, based on the quantum master equation and semi-group approach. The inherent regenerative losses in the two isomagnetic field processes are calculated and the influence of non-perfect regeneration on the performance of the cycle is analyzed. Expressions for some important performance parameters, such as the coefficient of performance, heating load, power input, and rate of the entropy production, are derived. Several interesting cases are discussed and, especially, the optimal performance of the cycle at high temperature is discussed in detail. Some important characteristic curves of the cycle, such as the heating load versus coefficient of performance curves, the power input versus coefficient of performance curves, the heating load versus power input curves, and so on, are presented. The maximum heating-load and the corresponding coefficient of performance are calculated. Other optimal performances are also analyzed. The results obtained here are further generalized, so that they may be directly used to describe the performance of the quantum heat-pump using spin-J systems as the working substance.
Heat pumps are important devices for saving energy and have been used widely. Like heat engines and refrigerators, the optimal design of a heat pump is also a major objective of engineering thermodynamics. The new technology and theory concerning heat pumps is being developed continuously. The theory of finite-time thermodynamics has been successfully used to investigate the optimal design of heat pumps, and some novel results, which are more realistic than those of classical thermodynamics, have been obtained [1], [2], [3] and [4]. These results have laid a foundation for analyzing further the optimal performance of a heat-pump.
In recent years, the optimal analyses relative to the performance characteristics of thermodynamic cycles have been extended from classical to quantum cycles [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15] and [16]. Based on the quantum master equation and semi-group approach, the influence of several factors on the performance of quantum heat engines and refrigerators has been investigations and many meaningful conclusions have been obtained. However, the investigations have rarely dealt with the performance of a regenerative quantum heat-pump working with spin systems.
It is well known that when a thermodynamic cycle, whether it is a classical or a quantum cycle, includes regenerative processes [6], [7], [8], [17] and [18], its performance is, in general, closely dependent on the properties of the working substance. For different working substances, such as the spin systems, harmonic oscillator systems, ideal quantum gases, ideal gases, etc., there exist different regenerative losses, so that the performances of cycles are different from each other. Therefore, it is of great significance to study the performance of a regenerative quantum heat-pump using the spin system as the working substance.
In the present paper, a new model of an endoreversible quantum-mechanical heat pump, composed of two isothermal and two isomagnetic field processes, is established. The working substance consists of many non-interacting spin-1/2 systems and the two regenerative processes in the cycle correspond to two isomagnetic processes. The thermodynamic properties of a spin-1/2 system are given in detail, based on the quantum master equation and semi-group approach. The general expressions of several important parameters, such as the coefficient of performance (COP), heating load, power input, and rate of the entropy production, are derived. The general performance characteristics of the cycle are also analyzed. Especially, the maximum heating load and the corresponding parameters of the quantum heat-pump in the high temperature limit are calculated in detail. The optimal region of the heating load and the optimal ranges of the temperatures of the working substance in the two isothermal-processes are determined. The results obtained here are further generalized, so that they are also suitable for the working substance consisting of non-interacting spin-J systems.
We have established the cyclic model of a typical quantum heat-pump cycle working with many non-interacting spin-1/2 systems and consisting of two isothermal and two isomagnetic field processes. On the basis of the statistical mechanics, motion equation, and semi-group formalism, we have analyzed the optimal performance characteristics of the quantum heat-pump cycle and derived concrete expressions for several important parameters such as the COP, heating load, power input, rate of the entropy production, and temperatures of the working substance in the two isothermal processes. Some important conclusions of the cycle in the high-temperature limit are obtained. For example, the maximum heating load and the corresponding parameters are calculated, and the optimal region of the quantum heat-pump is determined. The results obtained here are further generalized and consequently they may be used to describe the optimal performance of the quantum heat-pump working with many non-interacting spin-J systems.
For the different quantum working-substance, the heat transfer law between the working substance and the heat reservoirs is different. For example, in the high-temperature limit, the heat transfer law between the spin-1/2 system and the heat reservoirs can be regarded as the linear law of irreversible thermodynamics, while the heat transfer law between the harmonic oscillator system and the heat reservoirs can be only regarded as the Newtonian law [24]. The difference is because the character of the quantum working-substance depends on the quantum parameters and the coupling action between the different quantum working substance and the heat reservoirs obeys the different laws of quantum statistical mechanics. Consequently, the performances of quantum cycles using different quantum systems as the working substance are different from each other.
