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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 Schrö dinger 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 Schrö dinger 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 efficiency η and 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 efficiency η 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*-η is parabolic-shaped. There exist a maximum dimensionless power output P* and the corresponding frequency η. 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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Keywords:
- thermoacoustic micro-cycle /
- quantum mechanics /
- power output /
- optimization
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[2] Wu F, Li Q, Guo F Z, Shu A Q 2012 J. Wuhan Inst. Tech. 34 1 (in Chinese) [吴锋, 李青, 郭方中, 舒安庆 2012 武汉工程大学学报 34 1]
[3] Wu F, Shu A Q, Guo F Z, Wang T 2014 Energy 68 370
[4] Yang Z C, Wu F, Guo F Z, Zhang C P 2011 Acta Phys. Sin. 60 084303 (in Chinese) [杨志春, 吴锋, 郭方中, 张春萍 2011 60 084303]
[5] Wang T, Wu F, Li D Y, Chen H, Lin J 2015 Acta Phys. Sin. 64 044301 (in Chinese) [汪拓, 吴锋, 李端勇, 陈浩, 林杰 2015 64 044301]
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[9] Liu X W, Chen L G, Wu F, Sun F R 2014 J. Energy Inst. 87 69
[10] Himangshu P G, Upendra H 2013 Phys. Rev. A 88 013842
[11] Ronnie K 2013 Entropy 15 2100
[12] Wang J H, Xiong S Q, He J Z, Liu J T 2012 Acta Phys. Sin. 61 080509 (in Chinese) [王建辉, 熊双全, 何济州, 刘江涛 2012 61 080509]
[13] Wu F, Wang T, Chen L G, Liu X W 2014 J. Mech. Eng. 50 150 (in Chinese) [吴锋, 汪拓, 陈林根, 刘晓威 2014 机械工程学报 50 150]
[14] Lin B, Chen J 2005 Phys. Scr. 71 12
[15] Wu F, Yang Z, Chen L G, Liu X W, Wu S 2010 J. Therm. Sci. 14 879
[16] Massimiliano E, Maicol A O, Michael G 2015 Phys. Rev. Lett. 114 080602
[17] Wu F, Chen L, Li D 2009 Appl. Energy 86 1119
[18] Tien D, Kieu 2004 Phys. Rev. Lett. 93 140403
[19] Guo F Z, Li Q 2007 Heat Dynamics (Wuhan: Huazhong University of Science and Technology Press) p198 (in Chinese) [郭方中, 李青 2007 热动力学 (武汉: 华中科技大学出版社) 第198页]
[20] Zeng J Y 2000 Quantum Mechanics (Vol. 1) (3th Ed.) (Beijing: Science Press) pp109-113 (in Chinese) [曾谨言 2000 量子力学 (卷I) (第三版)(北京: 科学出版社) 第109-113页]
[21] Xiong H W, Liu S J, Huang G X, Xu Z X 2002 Phys. Rev. A 65 033609
[22] Bender, C M, Brody D C, Meister B K 2000 J. Phys. A: Math. Gen. 33 4427
[23] Wu F, Chen L G, Sun F R, Yu J Y 2008 Study of Finite-time Thermodynamics on Stirling Machines (Beijing: Chemical Industry Press) pp185-188 (in Chinese) [吴锋, 陈林根, 孙丰瑞, 喻九阳 2008 斯特林机的有限时间热力学优化 (北京: 化学工业出版社) 第185-188页]
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[1] Swift G W 1988 J. Acoust. Soc. Am. 84 1145
[2] Wu F, Li Q, Guo F Z, Shu A Q 2012 J. Wuhan Inst. Tech. 34 1 (in Chinese) [吴锋, 李青, 郭方中, 舒安庆 2012 武汉工程大学学报 34 1]
[3] Wu F, Shu A Q, Guo F Z, Wang T 2014 Energy 68 370
[4] Yang Z C, Wu F, Guo F Z, Zhang C P 2011 Acta Phys. Sin. 60 084303 (in Chinese) [杨志春, 吴锋, 郭方中, 张春萍 2011 60 084303]
[5] Wang T, Wu F, Li D Y, Chen H, Lin J 2015 Acta Phys. Sin. 64 044301 (in Chinese) [汪拓, 吴锋, 李端勇, 陈浩, 林杰 2015 64 044301]
[6] Li Q, Wu F, Guo F Z, Wu C, Wu J 2003 Open Syst. Inf. Dyn. 10 391
[7] Wang T, Wu F, Fei J H, Lin J 2013 J. Mech. Eng. 49 183 (in Chinese) [汪拓, 吴锋, 费景华, 林杰 2013 机械工程学报 49 183]
[8] Kan X X, Wu F, Zheng X Q, Shu A Q 2009 J. Wuhan Univ. Tech. 31 130 (in Chinese) [阚绪献, 吴锋, 张晓青, 舒安庆 2009 武汉理工大学学报 31 130]
[9] Liu X W, Chen L G, Wu F, Sun F R 2014 J. Energy Inst. 87 69
[10] Himangshu P G, Upendra H 2013 Phys. Rev. A 88 013842
[11] Ronnie K 2013 Entropy 15 2100
[12] Wang J H, Xiong S Q, He J Z, Liu J T 2012 Acta Phys. Sin. 61 080509 (in Chinese) [王建辉, 熊双全, 何济州, 刘江涛 2012 61 080509]
[13] Wu F, Wang T, Chen L G, Liu X W 2014 J. Mech. Eng. 50 150 (in Chinese) [吴锋, 汪拓, 陈林根, 刘晓威 2014 机械工程学报 50 150]
[14] Lin B, Chen J 2005 Phys. Scr. 71 12
[15] Wu F, Yang Z, Chen L G, Liu X W, Wu S 2010 J. Therm. Sci. 14 879
[16] Massimiliano E, Maicol A O, Michael G 2015 Phys. Rev. Lett. 114 080602
[17] Wu F, Chen L, Li D 2009 Appl. Energy 86 1119
[18] Tien D, Kieu 2004 Phys. Rev. Lett. 93 140403
[19] Guo F Z, Li Q 2007 Heat Dynamics (Wuhan: Huazhong University of Science and Technology Press) p198 (in Chinese) [郭方中, 李青 2007 热动力学 (武汉: 华中科技大学出版社) 第198页]
[20] Zeng J Y 2000 Quantum Mechanics (Vol. 1) (3th Ed.) (Beijing: Science Press) pp109-113 (in Chinese) [曾谨言 2000 量子力学 (卷I) (第三版)(北京: 科学出版社) 第109-113页]
[21] Xiong H W, Liu S J, Huang G X, Xu Z X 2002 Phys. Rev. A 65 033609
[22] Bender, C M, Brody D C, Meister B K 2000 J. Phys. A: Math. Gen. 33 4427
[23] Wu F, Chen L G, Sun F R, Yu J Y 2008 Study of Finite-time Thermodynamics on Stirling Machines (Beijing: Chemical Industry Press) pp185-188 (in Chinese) [吴锋, 陈林根, 孙丰瑞, 喻九阳 2008 斯特林机的有限时间热力学优化 (北京: 化学工业出版社) 第185-188页]
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