鄂青，吳鋒

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廣義一維勢(shì)中熱聲制冷微循環(huán)的性能分析

鄂青1, 2，吳鋒2, 3

(1. 華中科技大學(xué) 能源與動(dòng)力工程學(xué)院，湖北 武漢，430074；2. 武漢工程大學(xué) 理學(xué)院，湖北 武漢，430205；3. 海軍工程大學(xué) 動(dòng)力工程學(xué)院，湖北 武漢，430032)

從工質(zhì)粒子在不同聲波勢(shì)場條件下的量子力學(xué)行為入手，建立一套適用于各種一維勢(shì)場條件的廣義量子熱聲制冷微循環(huán)分析模型并推導(dǎo)出廣義量子熱聲制冷微循環(huán)的性能參數(shù)表達(dá)式。以幾個(gè)典型的一維勢(shì)場為例，計(jì)算分析工質(zhì)粒子在不同勢(shì)場中運(yùn)動(dòng)時(shí)的循環(huán)性能。通過比較，確定當(dāng)工質(zhì)粒子工作于一維無限深勢(shì)阱或諧振勢(shì)阱條件下時(shí)，循環(huán)的性能系數(shù)和制冷率的綜合性能比其他勢(shì)場條件時(shí)的優(yōu)。研究結(jié)果表明：要使熱聲制冷機(jī)性能達(dá)到最優(yōu)，必須對(duì)聲場進(jìn)行控制，使其能夠在回?zé)崞髦薪⑵鹨痪S無限深勢(shì)阱或諧振勢(shì)阱。

有限時(shí)間熱力學(xué)；廣義一維勢(shì)；量子熱聲制冷機(jī)微循環(huán)；量子熱力學(xué)

1 廣義量子熱聲制冷微循環(huán)模型的建立

1.1 系統(tǒng)的量子力學(xué)基礎(chǔ)

在經(jīng)典熱聲熱力過程中，外界通過振動(dòng)膜片的運(yùn)動(dòng)對(duì)系統(tǒng)作功。與振動(dòng)膜片的經(jīng)典運(yùn)動(dòng)類似，可以假設(shè)廣義一維勢(shì)阱的壁面在有限的速度下運(yùn)動(dòng)。這樣，當(dāng)系統(tǒng)消耗外界功量或?qū)ν饨巛敵龉α繒r(shí)，其施加于勢(shì)阱壁面的合力F就可以寫為

式中：F為處于狀態(tài)時(shí)級(jí)本征態(tài)的粒子施加于勢(shì)阱壁面的力；L為處于狀態(tài)時(shí)的勢(shì)阱寬度。

將式(1)代入式(4)可得

表1 不同勢(shì)阱的比較

1.2 理想廣義量子熱聲制冷微循環(huán)

(a) F?V圖；(b) 微循環(huán)示意圖

本文從量子力學(xué)的角度分析，可將上述氣體微團(tuán)視為1種被限制在廣義一維勢(shì)阱中的粒子。為了簡單起見，在分析量子熱聲制冷微循環(huán)的過程中，只考慮由出現(xiàn)概率較高的2個(gè)特征態(tài)粒子構(gòu)成的二能級(jí)系統(tǒng)。1臺(tái)真正的制冷機(jī)中的工作介質(zhì)是由無數(shù)這樣的粒子組成的。從粒子的量子行為角度分析，每個(gè)微循環(huán)都可歸納為由2個(gè)量子絕熱過程和2個(gè)量子等壓過程環(huán)繞而成的。由此得到本文的主要研究對(duì)象：理想的廣義量子熱聲制冷微循環(huán)(generalizedquantum thermoacoustic refrigeration cycle, GQTARC)，如圖2所示。

圖2 理想廣義量子熱聲制冷微循環(huán)示意圖

式中：為玻爾茲曼常數(shù)；T為系統(tǒng)平衡溫度。

由式(6)可求得系統(tǒng)處于激發(fā)態(tài)的概率為

由式(7)可得

聯(lián)立式(1)，(3)和(5)可得出在此過程中系統(tǒng)耗功12的計(jì)算式為

由于系統(tǒng)始終處于熱平衡狀態(tài)，所以向高溫?zé)嵩捶艧徇^程中的換熱量12為

將式(9)和式(11)代入式(10)可得

＜0 (13)

在過程3—4中，系統(tǒng)在等作用力狀態(tài)下從1個(gè)低溫?zé)嵩?溫度為L)吸取熱量，隨著自身體積的膨脹向外界輸出功。采用類似于過程1—2的分析方法可得到此過程中交換的功34和熱量34分別為：

＞0 (15)

根據(jù)熱力學(xué)第一定律，在理想廣義量子熱聲制冷微循環(huán)過程中，系統(tǒng)的制冷量c為

＞0 (17)

系統(tǒng)的凈耗功net為

1.3 循環(huán)周期

1.4 循環(huán)性能參數(shù)

由式(5)可知：1個(gè)二能級(jí)系統(tǒng)施加在勢(shì)阱壁上的力可寫為

由式(7)可知：當(dāng)系統(tǒng)處在宏觀狀態(tài)2和狀態(tài)4時(shí)，其內(nèi)部激發(fā)態(tài)粒子的概率分別為：

在理想情況下，可取狀態(tài)點(diǎn)4和狀態(tài)點(diǎn)2的溫度分別為冷、熱端溫度(圖2)，即

將式(22)和式(23)代入式(17)和式(18)可將循環(huán)制冷量及耗功量計(jì)算式改寫為：

可得出廣義量子熱聲制冷微循環(huán)的性能系數(shù)p為

循環(huán)的制冷率為

(27)

2 廣義量子熱聲制冷微循環(huán)性能分析

式(26)和式(28)指出了廣義量子熱聲制冷微循環(huán)的性能與系統(tǒng)參數(shù)間的關(guān)系，在給定了一部分參數(shù)的情況下，可用圖線的形式對(duì)其性能與重要參數(shù)間的關(guān)系進(jìn)行研究。

對(duì)于工作介質(zhì)氣體微團(tuán)被束縛于各種一維勢(shì)阱(如一維無限深勢(shì)阱、包含相對(duì)論粒子的一維勢(shì)阱、諧振勢(shì)阱、四次勢(shì)等)中的量子熱聲制冷微循環(huán)，可統(tǒng)稱為一維量子熱聲制冷微循環(huán)(1D quantum thermo-acoustic refrigeration cycle, 1DQTARC)，即一維量子熱聲制冷微循環(huán)包括工作于一維無限深勢(shì)阱的量子熱聲制冷微循環(huán)(1D infinite potential quantum thermo-acoustic refrigeration cycle, 1DIQTARC)、相對(duì)論粒子系統(tǒng)量子熱聲制冷微循環(huán)(relativistic particles quantum thermo-acoustic refrigeration cycle, RQTARC)、諧振系統(tǒng)量子熱聲制冷微循環(huán)(harmonic potential quantum thermo-acoustic refrigeration cycle, HQTARC)和四次勢(shì)系統(tǒng)量子熱聲制冷微循環(huán)(quartic potentialquantum thermo-acoustic refrigeration cycle, QQTARC)。

(a) RQTARC，相對(duì)粒子在一維勢(shì)中運(yùn)動(dòng)；(b) HQTARC，粒子在諧振勢(shì)或一維無限勢(shì)中運(yùn)動(dòng)；(c) QQTARC，粒子在四次勢(shì)中運(yùn)動(dòng)

(a) RQTARC，相對(duì)粒子在一維勢(shì)中運(yùn)動(dòng)；(b) HQTARC，粒子在諧振勢(shì)或一維無限勢(shì)中運(yùn)動(dòng)；(c) QQTARC，粒子在四次勢(shì)中運(yùn)動(dòng)

(a) RQTARC，相對(duì)論粒子在一維勢(shì)中運(yùn)動(dòng)；(b) HQTARC，粒子在諧振勢(shì)或一維無限勢(shì)中運(yùn)動(dòng)；(c) QQTARC，粒子在四次勢(shì)中運(yùn)動(dòng)

3 結(jié)論

2) 當(dāng)2e給定時(shí)，3種勢(shì)場條件下循環(huán)的p都會(huì)隨的增加呈單調(diào)遞減；而當(dāng)給定時(shí)，3種勢(shì)場條件下循環(huán)的p都會(huì)隨2e的取值變化而分別達(dá)到最大值pmax和最小值pmin。通常，pmax出現(xiàn)在2e＞0.5的區(qū)間中，而pmin出現(xiàn)在2e＜0.5的區(qū)間。在相同參數(shù)條件下，處于3種不同勢(shì)場的量子熱聲制冷微循環(huán)滿足pR＞pH＞pQ。

4) 在相同參數(shù)條件下，當(dāng)粒子工作于一維無限深勢(shì)阱或諧振勢(shì)場時(shí)，循環(huán)的制冷系數(shù)p和制冷率*的綜合效果比其他2種勢(shì)場系統(tǒng)的綜合效果好。因此，可通過調(diào)節(jié)聲場，使其能夠在熱聲回?zé)崞髦薪⒁痪S無限深的或諧振的勢(shì)阱條件，來優(yōu)化熱聲熱機(jī)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

湖北省教育廳科學(xué)研究基金資助項(xiàng)目(Q20141506)；武漢工程大學(xué)教學(xué)研究基金資助項(xiàng)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)

吳鋒，博士，教授，從事有限時(shí)間熱力學(xué)及熱聲熱機(jī)系統(tǒng)研究；E-mail: wufeng@wit.edu.cn

(編輯 劉錦偉)