广义一维势中热声制冷微循环的性能分析

鄂青，吴锋



广义一维势中热声制冷微循环的性能分析

鄂青1, 2，吴锋2, 3

(1. 华中科技大学 能源与动力工程学院，湖北 武汉，430074；2. 武汉工程大学 理学院，湖北 武汉，430205；3. 海军工程大学 动力工程学院，湖北 武汉，430032)

从工质粒子在不同声波势场条件下的量子力学行为入手，建立一套适用于各种一维势场条件的广义量子热声制冷微循环分析模型并推导出广义量子热声制冷微循环的性能参数表达式。以几个典型的一维势场为例，计算分析工质粒子在不同势场中运动时的循环性能。通过比较，确定当工质粒子工作于一维无限深势阱或谐振势阱条件下时，循环的性能系数和制冷率的综合性能比其他势场条件时的优。研究结果表明：要使热声制冷机性能达到最优，必须对声场进行控制，使其能够在回热器中建立起一维无限深势阱或谐振势阱。

有限时间热力学；广义一维势；量子热声制冷机微循环；量子热力学

1 广义量子热声制冷微循环模型的建立

1.1 系统的量子力学基础

在经典热声热力过程中，外界通过振动膜片的运动对系统作功。与振动膜片的经典运动类似，可以假设广义一维势阱的壁面在有限的速度下运动。这样，当系统消耗外界功量或对外界输出功量时，其施加于势阱壁面的合力F就可以写为

式中：F为处于状态时级本征态的粒子施加于势阱壁面的力；L为处于状态时的势阱宽度。

将式(1)代入式(4)可得

表1 不同势阱的比较

1.2 理想广义量子热声制冷微循环

(a) F−V图；(b) 微循环示意图

本文从量子力学的角度分析，可将上述气体微团视为1种被限制在广义一维势阱中的粒子。为了简单起见，在分析量子热声制冷微循环的过程中，只考虑由出现概率较高的2个特征态粒子构成的二能级系统。1台真正的制冷机中的工作介质是由无数这样的粒子组成的。从粒子的量子行为角度分析，每个微循环都可归纳为由2个量子绝热过程和2个量子等压过程环绕而成的。由此得到本文的主要研究对象：理想的广义量子热声制冷微循环(generalizedquantum thermoacoustic refrigeration cycle, GQTARC)，如图2所示。

图2 理想广义量子热声制冷微循环示意图

式中：为玻尔兹曼常数；T为系统平衡温度。

由式(6)可求得系统处于激发态的概率为

由式(7)可得

联立式(1)，(3)和(5)可得出在此过程中系统耗功12的计算式为

由于系统始终处于热平衡状态，所以向高温热源放热过程中的换热量12为

将式(9)和式(11)代入式(10)可得

＜0 (13)

在过程3—4中，系统在等作用力状态下从1个低温热源(温度为L)吸取热量，随着自身体积的膨胀向外界输出功。采用类似于过程1—2的分析方法可得到此过程中交换的功34和热量34分别为：

＞0 (15)

根据热力学第一定律，在理想广义量子热声制冷微循环过程中，系统的制冷量c为

＞0 (17)

系统的净耗功net为

1.3 循环周期

1.4 循环性能参数

由式(5)可知：1个二能级系统施加在势阱壁上的力可写为

由式(7)可知：当系统处在宏观状态2和状态4时，其内部激发态粒子的概率分别为：

在理想情况下，可取状态点4和状态点2的温度分别为冷、热端温度(图2)，即

将式(22)和式(23)代入式(17)和式(18)可将循环制冷量及耗功量计算式改写为：

可得出广义量子热声制冷微循环的性能系数p为

循环的制冷率为

(27)

2 广义量子热声制冷微循环性能分析

式(26)和式(28)指出了广义量子热声制冷微循环的性能与系统参数间的关系，在给定了一部分参数的情况下，可用图线的形式对其性能与重要参数间的关系进行研究。

对于工作介质气体微团被束缚于各种一维势阱(如一维无限深势阱、包含相对论粒子的一维势阱、谐振势阱、四次势等)中的量子热声制冷微循环，可统称为一维量子热声制冷微循环(1D quantum thermo-acoustic refrigeration cycle, 1DQTARC)，即一维量子热声制冷微循环包括工作于一维无限深势阱的量子热声制冷微循环(1D infinite potential quantum thermo-acoustic refrigeration cycle, 1DIQTARC)、相对论粒子系统量子热声制冷微循环(relativistic particles quantum thermo-acoustic refrigeration cycle, RQTARC)、谐振系统量子热声制冷微循环(harmonic potential quantum thermo-acoustic refrigeration cycle, HQTARC)和四次势系统量子热声制冷微循环(quartic potentialquantum thermo-acoustic refrigeration cycle, QQTARC)。

(a) RQTARC，相对粒子在一维势中运动；(b) HQTARC，粒子在谐振势或一维无限势中运动；(c) QQTARC，粒子在四次势中运动

(a) RQTARC，相对粒子在一维势中运动；(b) HQTARC，粒子在谐振势或一维无限势中运动；(c) QQTARC，粒子在四次势中运动

(a) RQTARC，相对论粒子在一维势中运动；(b) HQTARC，粒子在谐振势或一维无限势中运动；(c) QQTARC，粒子在四次势中运动

3 结论

2) 当2e给定时，3种势场条件下循环的p都会随的增加呈单调递减；而当给定时，3种势场条件下循环的p都会随2e的取值变化而分别达到最大值pmax和最小值pmin。通常，pmax出现在2e＞0.5的区间中，而pmin出现在2e＜0.5的区间。在相同参数条件下，处于3种不同势场的量子热声制冷微循环满足pR＞pH＞pQ。

4) 在相同参数条件下，当粒子工作于一维无限深势阱或谐振势场时，循环的制冷系数p和制冷率*的综合效果比其他2种势场系统的综合效果好。因此，可通过调节声场，使其能够在热声回热器中建立一维无限深的或谐振的势阱条件，来优化热声热机性能。

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Performance analysis for quantum thermoacoustic refrigeration micro-cycle working in generalized 1D potential

E Qing1, 2, WU Feng2, 3

(1. School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China; 2. School of Science, Wuhan Institute of Technology, Wuhan 430205, China;3. Institute of Thermal Science and Power Engineering, Naval University of Engineering, Wuhan 430032, China)

A set of analysis model for generalized quantum thermoacoustic refrigeration micro-cycle working in various 1D potential wells was established started with the quantum mechanical behavior of working medium particles under different acoustic potential field conditions. And the performance parameter expressions of the generalized quantum thermoacoustic refrigeration micro-cycle were derived. Taking several typical 1D potentials as examples, the cyclic properties of working medium particles moving in different potential fields were calculated and analyzed. By comparison, it was show that when particle works in 1D infinite deep potential well or resonance potential well, the comprehensive performance of micro-cycle was better than that of other potential well conditions. The results show that, to achieve the optimal performance of the thermoacoustic refrigerator, the sound field must be controlled so that it can establish a 1D infinite deep potential well or resonant potential well in the regenerator.

finite time thermodynamics; generalized potential well; quantum thermoacoustic refrigeration cycle; quantum thermodynamics

TB65

A

1672−7207(2019)03−0726−08

10.11817/j.issn.1672-7207.2019.03.028

2018−03−01；

2018−05−20

湖北省教育厅科学研究基金资助项目(Q20141506)；武汉工程大学教学研究基金资助项目(X2016036) (Project (Q20141506) supported by the Science Research of Hubei Provincial Department of Education; Project(X2016036) supported by the Teaching Research Funding of Wuhan Institute of Technology)

吴锋，博士，教授，从事有限时间热力学及热声热机系统研究；E-mail: wufeng@wit.edu.cn

(编辑 刘锦伟)