热声技术以无运动部件和采用与环境友好的工质这两个突出特点，催生着动力和机械装置的重大变革。量子力学是揭示微观世界本质规律的有力工具，为了揭示热声微循环的本质规律，根据量子力学基本原理对量子热声微循环的优化性能进行了较深入的研究。把热声微团看作是许多服从量子力学规律的热声子，建立了热声微循环的量子力学理论模型。借助于二能级谐振子系统薛定谔方程的能量解以及Gibbs热平衡概率分布导出了量子热声微循环输出功率、热效率以及临界温度梯度的解析表达式，得到了无量纲输出功率和热效率的优化关系。量子热声微循环输出功率关于热效率、高温端温度和低温端温度都存在极大值。所得结果不但为热声理论提供了一个新的研究方法，而且拓宽了量子热力学的应用领域。%The purpose of this paper is to optimize the performance of a quantum thermoacoustic micro-cycle. Thermoacoustic devices, such as thermoacoustic engines, thermoacoustic refrigerators, and thermoacoustic heat pumps are a new class of mechanical equipments without moving part and pollution. The thermoacoustic technology associated with these devices will hasten significant revolution in power engineering and mechanical devices. The work substance of a thermoacoustic device is composed of a number of parcels of fluid. Each parcel consists of a lot of molecules or atoms. The thermodynamic cycle is realized by the heat exchange between the parcel and the solid wall of the channel. The thermodynamic cycle of the parcel of fluid is called the thermoacoustic micro-cycle. The thermodynamic behavior of a thermoacoustic system may be described by studying that of the thermoacoustic micro-cycle. It is necessary to study the model and performance of the thermoacoustic micro-cycle in order to promote the development of thermoacoustic technology. The quantum mechanics, which was one of the great achievements in the 20 th century, can reveal the secret of the micro particle world. Quantum thermodynamics is an inter-discipline that combines quantum dynamics and thermodynamics. It provides a useful tool for analyzing the quantum cycles and devices. In this paper, the method of the quantum thermodynamics is employed to analyze the performance of a quantum thermoacoustic micro-cycle. The thermoacoustic parcel is modeled as a gas composed of many micro particles, which abide by the quantum mechanics. These particles are referred to as thermal phonons. Thermal phonons are bosons. The evolution of each thermal phonon must satisfy the Schrödinger equation in quantum mechanics. The quantum mechanics model of the thermoacoustic micro-cycle, which is called the quantum thermoacoustic micro-cycle, is established in this paper. The quantum thermoacoustic micro-cycle consists of two constant force processes and two quantum adiabatic processes. The quantum thermodynamical behavior and evolution of the thermal phonon in a one-dimensional harmonic trap are investigated based on the Schrödinger equation and the two-eigenstates system. The energy eigenvalue of the thermal phonon are employed. The analytical expressions of the optimal dimensionless power output P∗, the thermal efficiencyηand the critical temperature gradient (dT/dx)ex for the quantum thermoacoustic micro-cycle are derived by considering Gibbs probability distribution. The optimal relationship between dimensionless power output P∗ and thermal efficiencyη is obtained. The analysis shows that both the power output and the thermal efficiency decrease with the increase of width of the harmonic trap L1. One can find that the characteristic curve of P∗-ηis parabolic-shaped. There exist a maximum dimensionless power output P∗and the corresponding frequencyη. It is noteworthy that there is a critical temperature gradient for the quantum thermoacoustic micro-cycle. The critical temperature gradient is important because it is the boundary between the heat engine and the heat pump. The optimal design and these operating conditions for the quantum thermoacoustic micro-cycle are determined in this paper. The results provide a new method for studying the thermoacoustics by means of the quantum thermodynamics, thereby broadening the application range of the quantum thermodynamic.
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