Thermodynamic analysis of a novel multi-target temperature transcritical CO2 ejector-expansion refrigeration cycle with vapor-injection

英文原文与中文译文逐段对照 · English original with Chinese translation

Min-Qiang Zeng ${}^{a}$ , Qiu-Yun Zheng ${}^{c}$ , Xue-Lai Zhang ${}^{a}$ , Fan-Yang Mo ${}^{b}$ , Xin-Rong Zhang ${}^{b, c, * }$

曾敏强${}^{a}$ ，郑秋云${}^{c}$ ，张学来${}^{a}$ ，莫凡洋${}^{b}$ ，张新荣${}^{b, c, * }$

a Merchant Marine College, Shanghai Maritime University, Shanghai, 201306, China

a 上海海事大学商船学院，上海，201306，中国

${}^{\mathrm{b}}$ Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing,100871, China

${}^{\mathrm{b}}$ 北京大学工学院能源与资源工程系，北京，100871，中国

c Beijing Engineering Research Center of City Heat, Beijing, 100871, China

c 北京市城市热工程技术研究中心，北京，100871，中国

A R T I C L E I N F O

文章信息

Keywords:

关键词:

Transcritical ${\mathrm{{CO}}}_{2}$ cycle

跨临界${\mathrm{{CO}}}_{2}$循环

Ejector

喷射器

Vapor-injection

蒸汽喷射

Multi-target temperature

多目标温度

A B S T R A C T

摘要

In the present paper, a multi-target temperature ejector-expansion subcooler vapor-injection refrigeration cycle is proposed in order to improve the performance of natural fluid ${\mathrm{{CO}}}_{2}$ refrigeration cycles. A thermodynamics model to simulate a two-phase ejector cycle integrating a vapor-injection is first established from the energetic and exergetic perspectives. The behavior of the new cycle is then analyzed, and the adjustment direction of the cooling capacity and the evaporating temperature is pointed out. Furthermore, the new cycle is also compared with the subcooler vapor-injection refrigeration cycle and the ejector-expansion refrigeration cycle. The obtained results show that the behavior of the evaporators is affected by the bypass mass ratio and the entrainment ratio, and the two parameter determines the direction of adjusting the cooling capacity distribution and the evaporating temperature. The COP of the new cycle is about 17.9-29.4% and 2.8-7.8% higher than that of subcooler vapor-injection refrigeration cycle and ejector-expansion refrigeration cycle; the exergy efficiency of new cycle is about 14-28.2% and 3.7-11.3% higher than that of subcooler vapor-injection refrigeration cycle and ejector-expansion refrigeration cycle; and it is also found that the performance of the new cycle is improved more significantly under low cooling temperature and high ambient temperature conditions.

本文提出了一种多目标温度喷射器 - 膨胀过冷器蒸汽喷射制冷循环，以提高天然流体${\mathrm{{CO}}}_{2}$制冷循环的性能。首先从能量和㶲的角度建立了一个热力学模型，用于模拟集成蒸汽喷射的两相喷射器循环。然后分析了新循环的性能，并指出了制冷量和蒸发温度的调整方向。此外，还将新循环与过冷器蒸汽喷射制冷循环和喷射器 - 膨胀制冷循环进行了比较。结果表明，蒸发器的性能受旁通质量比和引射比的影响，这两个参数决定了制冷量分布和蒸发温度的调整方向。新循环的性能系数比过冷器蒸汽喷射制冷循环和喷射器 - 膨胀制冷循环分别高出约17.9 - 29.4%和2.8 - 7.8%；新循环的㶲效率比过冷器蒸汽喷射制冷循环和喷射器 - 膨胀制冷循环分别高出约14 - 28.2%和3.7 - 11.3%；并且还发现，在低制冷温度和高环境温度条件下，新循环的性能提升更为显著。

1. Introduction

1. 引言

With the development of cold chain logistics and the increasing demand for food refrigeration and freezing, so far, nearly 20% of the electricity consumed worldwide is used for refrigeration and the demand for refrigerating is expected to grow tenfold by 2050 [1,2]. The critical impact of the refrigeration industry on ozone depletion and global warming has developed into an alarming issue, which is mainly because of the wide application of HCFCs and HFCs refrigerants [3], which are intended to be gradually replaced by more sustainable alternatives. On October 15, 2016, the Kigali Amendment to the Montreal Protocol reached an agreement on the reduction of HFCs and pointed out that HFCs’ production and consumption are to be phased down gradually by between 80 and 85% from 2019 to the late 2040s [4].

随着冷链物流的发展以及食品冷藏冷冻需求的增加，到目前为止，全球近20%的电力用于制冷，预计到2050年制冷需求将增长十倍[1,2]。制冷行业对臭氧层消耗和全球变暖的严重影响已成为一个令人担忧的问题，这主要是由于HCFCs和HFCs制冷剂的广泛应用[3]，而这些制冷剂正逐渐被更具可持续性的替代品所取代。2016年10月15日，《蒙特利尔议定书》基加利修正案就减少HFCs达成协议，并指出从2019年到2040年代末，HFCs的生产和消费将逐步减少80%至85%[4]。

To alleviate the increasingly serious environmental problems around the world, finding and developing green renewable energy sources and improving energy efficiency are urgent issues in the refrigeration field [5], natural refrigerants (such as ${\mathrm{{CO}}}_{2},{\mathrm{{NH}}}_{3},\mathrm{R}{290}$ , etc.) are gradually becoming one of its solutions. Although proper performance can be obtained by using R717, its adoption is not widespread due to its toxicity [6]. HCs (R290, R600a, R1270, etc.) safety is the main bottleneck restricting large-scale applications [7]. ${\mathrm{{CO}}}_{2}$ is considered to be a very potentially environmentally friendly alternative refrigerant due to its good thermophysical properties [8-10]. Because transcritical ${\mathrm{{CO}}}_{2}$ cycles have a higher lift. So their coefficient of performance (COP) is often lower than cycles that run with conventional refrigerants [11,12]. To improve the efficiency of the transcritical ${\mathrm{{CO}}}_{2}$ cycle, a large amount of work such as using ejectors [13-15] have been made by researchers to recover the throttling losses in the systems.

为了缓解全球日益严重的环境问题，寻找和开发绿色可再生能源以及提高能源效率是制冷领域的紧迫问题[5]，天然制冷剂(如${\mathrm{{CO}}}_{2},{\mathrm{{NH}}}_{3},\mathrm{R}{290}$等)正逐渐成为解决方案之一。虽然使用R717可以获得适当的性能，但其毒性导致其应用并不广泛[6]。HCs(R290、R600a、R1270等)的安全性是限制其大规模应用的主要瓶颈[7]。${\mathrm{{CO}}}_{2}$因其良好的热物理性质[8 - 10]被认为是一种极具潜力的环保替代制冷剂。由于跨临界${\mathrm{{CO}}}_{2}$循环的提升更高。因此其性能系数(COP)通常低于使用传统制冷剂的循环[11,12]。为了提高跨临界${\mathrm{{CO}}}_{2}$循环的效率，研究人员做了大量工作，如使用喷射器[(13 - 15)]来回收系统中的节流损失。

Many research results show that the use of ejector as the expansion device of the transcritical ${\mathrm{{CO}}}_{2}$ cycle can improve the performance of the system [16,17]. Li and Groll [18] employed a two-phase ejector assumption for a transcritical air-conditioning system and reported a COP improvement of up to 16%. Ksayer et al. [19] have demonstrated that compared with conventional refrigeration cycles, the performance of cycle using ejectors has increased by more than 15%. Ahammed et al. [20] pointed out that compared with the equivalent conventional ${\mathrm{{CO}}}_{2}$ system, the ejector cycle of COP was improved by 21%. Eskandari Manjili et al. [21] simulated the performance of a new two-stage

许多研究结果表明，使用喷射器作为跨临界${\mathrm{{CO}}}_{2}$循环的膨胀装置可以提高系统性能[16,17]。Li和Groll[18]对跨临界空调系统采用了两相喷射器假设，并报告性能系数提高了16%。Ksayer等人[19]证明，与传统制冷循环相比，使用喷射器的循环性能提高了15%以上。Ahammed等人[20]指出，与等效的传统${\mathrm{{CO}}}_{2}$系统相比，喷射器循环的性能系数提高了21%。Eskandari Manjili等人[21]模拟了一种新型两级

• Corresponding author. Department of Energy and Resources Engineering, College of Engineering, Peking University, Beijing, 100871, China. (X.-R. Zhang).

• 通讯作者。北京大学工学院能源与资源工程系，北京100871，中国。(张XR)。

Nomenclature

符号说明

English letters

英文字母

$C$ Speed of sound (m/s)

$C$ 声速 (米/秒)

${C}_{p}$ Isobaric specific heat capacity $\left( {\mathrm{{kJ}}/\mathrm{{kg}} \cdot \mathrm{K}}\right)$

${C}_{p}$ 定压比热容 $\left( {\mathrm{{kJ}}/\mathrm{{kg}} \cdot \mathrm{K}}\right)$

${C}_{V}$ Isochoric specific heat capacity $\left( {\mathrm{{kJ}}/\mathrm{{kg}} \cdot \mathrm{K}}\right)$

${C}_{V}$ 定容比热容 $\left( {\mathrm{{kJ}}/\mathrm{{kg}} \cdot \mathrm{K}}\right)$

ex Specific exergy $\left( {\mathrm{{kJ}}/\mathrm{{kg}}}\right)$

ex 比㶲 $\left( {\mathrm{{kJ}}/\mathrm{{kg}}}\right)$

$\dot{E}{x}_{D}$ Exergy destruction rate (kW)

$\dot{E}{x}_{D}$ 㶲损失率 (千瓦)

$h$ Specific enthalpy (kJ/kg)

$h$ 比焓 (千焦/千克)

$\dot{m}$ Mass flow rate (kg/s)

$\dot{m}$ 质量流量 (千克/秒)

$M$ Mach number

$M$ 马赫数

$P$ Pressure (MPa)

$P$ 压力 (兆帕)

${\Delta P}$ Pressure drop

${\Delta P}$ 压降

$q$ Specific cooling capacity (kW/kg)

$q$ 比制冷量 (千瓦/千克)

$\dot{Q}$ Rate of net heat (kW)

$\dot{Q}$ 净热率 (千瓦)

$R$ Bypass mass ratio $\left( {\mathrm{{kg}}/\mathrm{{kg}}}\right)$

$R$ 旁通质量比 $\left( {\mathrm{{kg}}/\mathrm{{kg}}}\right)$

$S$ Specific entropy $\left( {\mathrm{{kJ}}/\mathrm{{kg}} \cdot \mathrm{K}}\right)$

$S$ 比熵 $\left( {\mathrm{{kJ}}/\mathrm{{kg}} \cdot \mathrm{K}}\right)$

$T$ Temperature $\left( {\mathrm{K},{}^{ \circ }\mathrm{C}}\right)$

$T$ 温度 $\left( {\mathrm{K},{}^{ \circ }\mathrm{C}}\right)$

${\Delta T}$ Temperature difference $\left( {\mathrm{K},{}^{ \circ }\mathrm{C}}\right)$

${\Delta T}$ 温差 $\left( {\mathrm{K},{}^{ \circ }\mathrm{C}}\right)$

$u$ Velocity $\left( {\mathrm{m}/\mathrm{s}}\right)$

$u$ 速度 $\left( {\mathrm{m}/\mathrm{s}}\right)$

$\dot{W}$ Specific work (kW/kg)

$\dot{W}$ 比功 (kW/kg)

$X$ Quality (kg/kg)

$X$ 干度 (kg/kg)

Acronyms and abbreviations

首字母缩略词和缩写

COP Coefficient of performance

COP 性能系数

${CR}$ Compression ratio

${CR}$ 压缩比

Exp Experiment

Exp 实验

Sim Simulation

Sim 模拟

min Minimum

min 最小值

Greek letters

希腊字母

$\alpha$ Void fraction $\left( {{\mathrm{m}}^{3}/{\mathrm{m}}^{3}}\right)$

$\alpha$ 空隙率 $\left( {{\mathrm{m}}^{3}/{\mathrm{m}}^{3}}\right)$

$\kappa$ Specific heats ratio

$\kappa$ 比热比

$\eta$ Isentropic efficiency

$\eta$ 等熵效率

$\rho$ Density $\left( {\mathrm{{kg}}/{\mathrm{m}}^{3}}\right)$

$\rho$ 密度 $\left( {\mathrm{{kg}}/{\mathrm{m}}^{3}}\right)$

$\omega$ Entrainment ratio of the ejector

$\omega$喷射器的引射比

Subscripts

下标

$c$ Compressor

$c$压缩机

$d$ Outlet of diffuser

$d$扩压器出口

dis Discharge

dis排放

$e$ Evaporator

$e$蒸发器

ev Expansion valve

ev膨胀阀

gc Gas cooler

gc气体冷却器

$G$ Gas

$G$气体

$H$ High

$H$高压

$i$ Isentropic process, intlet

$i$等熵过程，入口

int Intermediate

int中间

$L$ Liquid, low

$L$液体，低压

$m$ Mixing chamber

$m$混合室

0 Outlet

0出口

$p$ Primary

$p$一次

$S$ Second, subcooler

$S$ 第二，过冷器

sh Super heat

sh 过热

$T$ Total

$T$ 总计

0 Reference state

0 参考状态

#1, #2 Component number

#1, #2 组件编号

1,2,3 Cycle locations transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle with two ejectors, and studied the influence of main parameters on cycle performance. Compared with the traditional cycle, the new cycle could increase the COP by more than 40%. Zolcer et al. [10] demonstrated that the combination of ejectors and parallel compression in transcritical systems exhibited suitable efficiencies, with savings ranging between 8 and 10% compared to the basic transcritical system operating in warm climates. The ejector is considered to be a new breakthrough point in the carbon dioxide refrigeration system, which reduces the compression ratio of the compressor and reduces the compressor discharge temperature. With the development of the refrigeration industry, the application of refrigeration systems tends to be multi-evaporation temperature. Although there are many research literatures on ejector systems, the research on multi-evaporator systems is relatively limited. Elakhdar et al. [22] proposed a novel ejector-enhanced vapor compression two-evaporator refrigeration cycle and compared the results with a conventional cycle using pure fluid R134a and azeotrope R290/R600a. The results indicated that the cycle COP and volumetric cooling capacity could be improved by 23% and 62.71%, respectively. Liu et al. [23] proposed three different connection forms of the refrigerator compression-injection hybrid refrigeration cycles, which showed that the use of ejectors can reduce the heat loss during throttling by 7.75%. Bai et al. [16] proposed a modified dual-evaporator ${\mathrm{{CO}}}_{2}$ transcritical refrigeration cycle with two-stage ejector. The simulation results for the modified cycle show that the improvements of the maximum system COP and system exergy efficiency could reach 37.61% and 31.9% over those of the conventional dual-evaporator cycle under the given operating conditions.

1,2,3 循环位置 具有两个喷射器的跨临界${\mathrm{{CO}}}_{2}$制冷循环，并研究了主要参数对循环性能的影响。与传统循环相比，新循环的性能系数可提高40%以上。Zolcer等人[10]表明，跨临界系统中喷射器与并联压缩的组合表现出合适的效率，与在温暖气候下运行的基本跨临界系统相比，节能8%至10%。喷射器被认为是二氧化碳制冷系统的一个新突破点，它降低了压缩机的压缩比并降低了压缩机排气温度。随着制冷行业的发展，制冷系统的应用趋向于多蒸发温度。虽然关于喷射器系统的研究文献很多，但对多蒸发器系统的研究相对有限。Elakhdar等人[22]提出了一种新型喷射器增强型蒸汽压缩双蒸发器制冷循环，并将结果与使用纯流体R134a和共沸物R290/R600a的传统循环进行了比较。结果表明，循环性能系数和容积冷却能力分别可提高23%和62.71%。Liu等人[23]提出了冰箱压缩-喷射混合制冷循环的三种不同连接形式，结果表明使用喷射器可将节流过程中的热损失降低7.75%。Bai等人[16]提出了一种改进的双蒸发器${\mathrm{{CO}}}_{2}$两级喷射器跨临界制冷循环。改进循环的模拟结果表明，在给定运行条件下，最大系统性能系数和系统火用效率比传统双蒸发器循环分别提高了37.61%和31.9%。

In addition to the positive effect of the ejector on the improvement of system efficiency, the vapor-injection is proposed to be an efficient method to improve the performance of the refrigeration systems [24]. Cho et al. [25] presented a theoretical optimization on the two-stage transcritical ${\mathrm{{CO}}}_{2}$ cycle employing three different vapor-injection methods, the experimental results show that the cooling COP of the two-stage gas injection cycle was maximally enhanced by 16.5% over that of the two-stage non-injection cycle. Baek et al. [26] discussed the performance characteristics of jet ${\mathrm{{CO}}}_{2}$ heat pump water heaters under different operating conditions, and found that the COP of the optimized heat pump water heater was 7.6% higher than that of the non-injection heat pump water heater. Peng et al. [27] established a simulation model of a the transcritical ${\mathrm{{CO}}}_{2}$ vapor-injection heat pump system, the heating capacity of the vapor-injection heat pump system is greater than that of the transcritical ${\mathrm{{CO}}}_{2}$ single-stage heat pump system; besides, the COP of vapor-injection heat pump system is 30.7-38.6% greater than that in single-stage heat pump system under the same condition. Bai et al. [28] studied an ejector enhanced vapor injection ${\mathrm{{CO}}}_{2}$ transcritical heat pump cycle with subcooler, the results show that the improvements of the maximum system COP and volumetric heating capacity could reach up to 7.7% and 9.5%, respectively.

除了喷射器对提高系统效率的积极作用外，蒸汽喷射被认为是提高制冷系统性能的一种有效方法[24]。Cho等人[25]对采用三种不同蒸汽喷射方法的两级跨临界${\mathrm{{CO}}}_{2}$循环进行了理论优化，实验结果表明，两级气体喷射循环的制冷性能系数比两级非喷射循环最大提高了16.5%。Baek等人[26]讨论了喷射${\mathrm{{CO}}}_{2}$热泵热水器在不同运行条件下的性能特性，发现优化后的热泵热水器的性能系数比非喷射热泵热水器高7.6%。Peng等人[27]建立了一个跨临界${\mathrm{{CO}}}_{2}$蒸汽喷射热泵系统的模拟模型，蒸汽喷射热泵系统的制热量大于跨临界${\mathrm{{CO}}}_{2}$单级热泵系统；此外，在相同条件下，蒸汽喷射热泵系统的性能系数比单级热泵系统高30.7 - 38.6%。Bai等人[28]研究了一种带有过冷器的喷射增强型蒸汽喷射${\mathrm{{CO}}}_{2}$跨临界热泵循环，结果表明，最大系统性能系数和容积加热能力分别可提高7.7%和9.5%。

After viewing the literatures above, although there are many studies on ejector systems, there are few studies on the combination of vapor-injection in the transcritical ${\mathrm{{CO}}}_{2}$ ejector cycle, especially considering that there are limited conditions for entrainment ratio $\left( \omega \right)$ in the ejector cycle, and there are restrictions on bypass mass ratio $\left( R\right)$ in the vapor-injection cycle (restrictions are discussed in section 4.1). Based on the above discussion, the multi-target temperature ejector-expansion sub-cooler vapor-injection (MTES) transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle is proposed. A numerical model of the two-phase ejector is established based on the constant pressure mixing method. The behavior (including the evaporation temperature of each evaporator, the cooling capacity of each evaporator and the COP of cycle) of the MTES cycle is analyzed, and the performance of MTES cycle is compared to that of the subcooler vapor-injection refrigeration cycle (SRC) as well as to that of the ejector-expansion refrigeration cycle (ERC) at the same operating conditions. The rest of the paper is organized as follows: Section 2 describes the layout of SRC, ERC and MTES cycles. Section 3 describes the thermodynamic model for analysis. Section 4 shows the results of the research and analyzes the discussion. The main conclusions are drawn in Section 5.

在查阅上述文献后，尽管对喷射器系统已有许多研究，但对跨临界${\mathrm{{CO}}}_{2}$喷射器循环中蒸汽喷射的组合研究较少，特别是考虑到喷射器循环中的引射比$\left( \omega \right)$条件有限，且蒸汽喷射循环中的旁通质量比$\left( R\right)$存在限制(限制在4.1节中讨论)。基于上述讨论，提出了多目标温度喷射器-膨胀过冷器蒸汽喷射(MTES)跨临界${\mathrm{{CO}}}_{2}$制冷循环。基于定压混合法建立了两相喷射器的数值模型。分析了MTES循环的性能(包括每个蒸发器的蒸发温度、每个蒸发器的制冷量和循环的COP)，并在相同运行条件下将MTES循环的性能与过冷器蒸汽喷射制冷循环(SRC)以及喷射器-膨胀制冷循环(ERC)的性能进行了比较。本文其余部分的结构如下:第2节描述了SRC、ERC和MTES循环的布局。第3节描述了用于分析的热力学模型。第4节展示了研究结果并进行了分析讨论。主要结论在第5节得出。

Fig. 1. Schematic configurations of the studied system.

图1. 所研究系统的示意图配置。

2. System description

2. 系统描述

Schematic configurations of SRC, ERC and MTES refrigeration cycle are illustrated in Fig. 1. The SRC [24] cycle (Fig. 1(a)) consists of a compressor, a gas cooler, a subcooler, an evaporator and two expansion valves. Its main feature is to bypass a part of the refrigerant through expansion valve#1 for throttling expansion and then enter the sub-cooler, so that the temperature of the refrigerant entering expansion valve#2 is reduced, and the two-phase flow from expansion valve#1 is vaporized. The ERC [18] cycle (Fig. 1(b)) reduces the loss during throttling by adding ejector. The high-pressure supercritical ${\mathrm{{CO}}}_{2}$ fluid at the outlet of the gas cooler is used as the primary flow to drive the ejector. Then, the vapor leaving the ejector is divided into two phases in the separator. The gas-phase fluid enters the compressor, and the liquid-phase fluid enters the ejector as a secondary stream after passing through the evaporator. The proposed MTES cycle is shown in Fig. 1(c). The MTES cycle is a combination of SRC and ERC cycle. At the same time, the value of the $R$ of the SRC cycle and the $\omega$ of the ERC cycle are restricted, therefore, the multi-target temperature refrigeration is achieved by adding evaporator #1 after the subcooler and evaporator#2 after the ejector. Simultaneously, the addition of evaporator#1 and 2 removes the restriction of $R$ and $\omega$ .

图1展示了SRC、ERC和MTES制冷循环的示意图。SRC [24] 循环(图1(a))由压缩机、气体冷却器、过冷器、蒸发器和两个膨胀阀组成。其主要特点是通过膨胀阀#1使一部分制冷剂节流膨胀，然后进入过冷器，从而降低进入膨胀阀#2的制冷剂温度，并使来自膨胀阀#1的两相流汽化。ERC [18] 循环(图1(b))通过添加喷射器减少了节流过程中的损失。气体冷却器出口处的高压超临界${\mathrm{{CO}}}_{2}$流体用作驱动喷射器的主流。然后，离开喷射器的蒸汽在分离器中分为两相。气相流体进入压缩机，液相流体在经过蒸发器后作为二次流进入喷射器。所提出的MTES循环如图1(c)所示。MTES循环是SRC和ERC循环的组合。同时，SRC循环的$R$值和ERC循环的$\omega$值受到限制，因此，通过在过冷器后添加蒸发器#1和在喷射器后添加蒸发器#2来实现多目标温度制冷。同时，蒸发器#1和#2的添加消除了$R$和$\omega$的限制。

3. Modeling for simulation and evaluation

3. 用于模拟和评估的建模

3.1. Assumptions

3.1. 假设

This study will use the 1-D constant pressure model for the ejector calculation [29]. To simplify the theoretical model of the cycle, the following assumptions are made:

本研究将采用一维定压模型进行喷射器计算[29]。为简化循环的理论模型，做出以下假设:

1. The flow through the ejector is regarded as a one-dimensional steady-state flow.

2. 通过喷射器的流动被视为一维稳态流动。

3. The pressure loss at the connection of each component and the speed of the fluid at the inlet and outlet of the ejector are ignored.

4. 忽略各部件连接处的压力损失以及喷射器进出口处流体的速度。

5. The primary flow and the secondary flow reach the same pressure at the inlet of the constant area and are constant pressure mixed in the mixing section.

6. 主流和二次流在等截面入口处达到相同压力，并在混合段进行定压混合。

7. The vapor flow in the gas-liquid separator is saturated vapor, and the liquid flow is saturated liquid.

8. 气液分离器中的蒸汽流为饱和蒸汽，液流为饱和液体。

9. The flow through the expansion valve is ideal isenthalpic expansion.

10. 通过膨胀阀的流动为理想等焓膨胀。

3.2. Two-phase flow ejector model

3.2. 两相流喷射器模型

The use of an expansion energy recovery ejector has been given great concern owing to virtues of no moving elements, cost-effectiveness, high reliability, and comparable efficiency [15]. The first 1-D model describing the flow through the ejector was developed by Keenan et al. [30]. The authors proposed two different approaches to solve the flow. Mixing at constant pressure or constant area. Again, Keenan et al. [31] presented a more advanced method for both approaches mentioned before. In that case, the results reflected that constant pressure mixing was in a better agreement with experimental values. In this study, the 1-D constant pressure model for the ejector calculation was used, in which the state point in the formula is based on the MTES cycle as an example.

由于无运动部件、成本效益高、可靠性高以及效率相当等优点，膨胀能量回收喷射器的应用受到了极大关注[15]。描述通过喷射器流动的第一个一维模型是由Keenan等人[30]开发的。作者提出了两种不同方法来求解流动。在恒定压力或恒定面积下混合。同样，Keenan等人[31]针对上述两种方法提出了一种更先进的方法。在那种情况下，结果表明恒压混合与实验值的一致性更好。在本研究中，使用了用于喷射器计算的一维恒压模型，其中公式中的状态点以MTES循环为例。

3.2.1. Ejector analysis

3.2.1. 喷射器分析

The mass flow rate ratio between the motive and suction fluid determines the entrainment ratio $\omega$ , which is used to evaluate the mass entrainment capacity of the ejector:

引射流体与吸入流体的质量流量比决定了引射系数$\omega$，该系数用于评估喷射器的质量引射能力:

Table 1

表1

Summary of theoretical and experimental values of two-phase ejector efficiency.

两相喷射器效率的理论值和实验值汇总。

In addition, the various losses of the ejector are considered by the actual empirical efficiency, but a one-dimensional theoretical simulation is used in the present study. Therefore, constant coefficients will be used to characterize the theoretical performance of each section of the ejector. The theoretical and experimental values of the efficiency of the two-phase ejector used by previous scholars in the transcritical ${\mathrm{{CO}}}_{2}$ cycle are collected in Table 1. In this study, the smaller relative error between theoretical simulation and experimental data is considered. Motive and suction nozzles efficiencies around ${\eta }_{P} = {\eta }_{s} = {0.75}$ , mixing section efficiency around ${\eta }_{m} = {0.85}$ , and diffuser efficiency distribution around ${\eta }_{d} = {0.8}$ . The comparison of simulation and experimental results using these values is shown in section 3.6.

此外，喷射器的各种损失通过实际经验效率来考虑，但本研究采用一维理论模拟。因此，将使用常数系数来表征喷射器各部分的理论性能。表1收集了先前学者在跨临界${\mathrm{{CO}}}_{2}$循环中使用的两相喷射器效率的理论值和实验值。在本研究中，考虑了理论模拟与实验数据之间较小的相对误差。引射喷嘴和吸入喷嘴效率约为${\eta }_{P} = {\eta }_{s} = {0.75}$，混合段效率约为${\eta }_{m} = {0.85}$，扩压器效率分布约为${\eta }_{d} = {0.8}$。使用这些值的模拟和实验结果比较见3.6节。

3.2.2. Primary and secondary nozzles

3.2.2. 主喷嘴和副喷嘴

Primary flow expands with the given isentropic efficiency and gets accelerated to very high velocity. The conservation of energy and velocity at the exit of the primary nozzle are expressed by Eqs. (2) and (3).

主流在给定的等熵效率下膨胀，并加速到非常高的速度。主喷嘴出口处的能量和速度守恒由式(2)和式(3)表示。

To create a vacuum at the ejector secondary flow nozzle, the pressure drop of the secondary nozzle is defined as:

为了在喷射器副流喷嘴处产生真空，副喷嘴的压降定义为:

The same method is used for the secondary nozzle as the primary nozzle. The conservation of energy and velocity at the exit of the secondary nozzle are expressed by Eqs. (5) and (6).

副喷嘴采用与主喷嘴相同的方法。副喷嘴出口处的能量和速度守恒由式(5)和式(6)表示。

3.2.3. Mixing section

3.2.3. 混合段

The primary flow and secondary flow are mixed under constant pressure, the momentum and energy equations of the mixing process are:

主流和二次流在恒压下混合，混合过程的动量和能量方程为:

(8)

The efficiency of the mixing section adopts a new mixing efficiency formula proposed by Ref. [41].

混合段效率采用文献[41]提出的新混合效率公式。

3.2.4. Shock section

3.2.4. 激波段

Considering that ${\mathrm{{CO}}}_{2}$ in transcritical conditions is entering the primary nozzle, it is likely that the fluid will reach the speed of sound during the mixing process, which will result in a normal shock in the constant area section [42]. presented a simple model for predicting the sonic velocity in two-phase systems. The model has shown to be remarkably successful in predicting the sonic velocity in most of the flow regimes of two-phase gas-liquid flows over the entire range of void and mass fraction. The void fraction and the sound velocity of the two-phase flow are defined as Eqs. (10) and (11).

考虑到跨临界条件下的${\mathrm{{CO}}}_{2}$进入主喷嘴，在混合过程中流体很可能达到声速，这将在等截面段产生正激波[42]。提出了一个预测两相系统中声速的简单模型。该模型在预测气液两相流整个空泡率和质量分数范围内的大多数流态下的声速方面非常成功。两相流的空泡率和声速由式(10)和(11)定义。

If the speed of ${\mathrm{{CO}}}_{2}$ after the mixing process is greater than the speed of sound, Eqs. 12-14 is used to express the impact of the shock wave. Otherwise, all the characteristics of the refrigerants remain unchanged. Conservation of mass, momentum, and energy:

如果混合过程后${\mathrm{{CO}}}_{2}$的速度大于声速，则用式12 - 14来表示激波的影响。否则，制冷剂的所有特性保持不变。质量、动量和能量守恒:

The relationship of the density change during the shock wave is expressed:

激波过程中密度变化的关系表示为:

In which $k$ for two-phase:

其中两相的$k$为:

3.2.5. Diffuser section

3.2.5. 扩压器部分

In the diffusion section, the fluid loses kinetic energy, causing the fluid pressure to increase. The formula is:

在扩压部分，流体动能损失，导致流体压力升高。公式如下:

Table 2

表2

Thermodynamic equations of the studied refrigeration systems.

所研究制冷系统的热力学方程。

3.3. Energetic analysis

3.3. 能量分析

The general energy balance equation can be defined as:

通用能量平衡方程可定义为:

where ${\dot{Q}}_{i}$ means the net heat input rate of each component while ${\dot{W}}_{o}$ means the net work output rate of each component, $\dot{m}$ is the mass flow rate of the working fluid. By using Eq. (19), the thermodynamic equations of each component of the aforementioned systems are obtained as shown in Table 2.

其中${\dot{Q}}_{i}$表示各组件的净热输入率，而${\dot{W}}_{o}$表示各组件的净功输出率，$\dot{m}$是工作流体的质量流量。通过使用式(19)，得到上述系统各组件的热力学方程，如表2所示。

The mass flow rate ratio between the bypass and main cycle fluid determines the bypass mass ratio $R$ .

旁通与主循环流体之间的质量流量比决定了旁通质量比$R$。

The intermediate pressure is obtained from the classical estimated value, given by the geometric mean of discharge pressure and evaporating pressure [43].

中间压力由经典估计值获得，该值由排气压力和蒸发压力的几何平均值给出[43]。

The compression and expansion processes follow the modeling assumptions. In addition, to simplify the calculation, the mass flow through the compressor is the unit mass " ${\dot{m}}^{\prime \prime }\left( {\dot{m} = 1\mathrm{\;{kg}}/\mathrm{s}}\right)$ . The isentropic efficiency expressions of compressor is expressed in the following equations [44]:

压缩和膨胀过程遵循建模假设。此外，为简化计算，通过压缩机的质量流量为单位质量“${\dot{m}}^{\prime \prime }\left( {\dot{m} = 1\mathrm{\;{kg}}/\mathrm{s}}\right)$”。压缩机的等熵效率表达式用以下方程表示[44]:

The evaluation parameter in terms of energetic performance is then given as refrigerating cycle COP.

然后将能量性能方面的评估参数定义为制冷循环COP。

Table 3

表3

The exergy destruction calculation equation for each component.

各组件的火用损失计算方程。

3.4. Exergetic analysis

3.4. 㶲分析

Evaluation of a system also includes the various exergy destructions of various components. Thus, exergetic analysis is applied in this study. The specific exergy of the refrigerant at every state point can be defined as [45]:

对一个系统的评估还包括各个组件的各种㶲损失。因此，本研究应用了㶲分析。制冷剂在每个状态点的比㶲可定义为[45]:

where the subscript 0 represents the reference state which is selected as 300.15K under the ambient atmosphere pressure 101.325 kPa. Based on the second law of thermodynamics, the exergy destruction in each component can be calculated by following equation:

其中下标0表示参考状态，在环境大气压力101.325 kPa下选择为300.15K。基于热力学第二定律，每个组件中的㶲损失可通过以下公式计算:

Based on the above two equations, the exergy destruction in each component of the systems can be obtained and is listed in Table 3. In addition, the cycle total exergy destruction can be obtained as:

基于上述两个公式，可以得到系统中每个组件的㶲损失，并列于表3中。此外，循环总㶲损失可得到为:

The exergy efficiency is defined as the ratio of the effective exergy output to the effective exergy input, where the effective exergy output can be obtained by subtracting the total exergy loss from the effective exergy input. The system exergy efficiency is expressed as follows [46]:

㶲效率定义为有效㶲输出与有效㶲输入之比，其中有效㶲输出可通过从有效㶲输入中减去总㶲损失得到。系统㶲效率表示如下[46]:

3.5. Model procedure

3.5. 模型步骤

The flow chart of the MTES cycle simulation program is shown in Fig. 2. In this study, the performance of the MTES cycle is affected through four design variables. The variables include the gas cooler outlet temperature $\left( {T}_{gc}\right)$ , evaporating temperature of evaporator#3 $\left( {T}_{e3}\right)$ , the entrainment rate of ejector $\left( \omega \right)$ and the bypass mass ratio $\left( R\right)$ . Based on the set working conditions, the range of each variable is as follows:

MTES循环模拟程序的流程图如图2所示。在本研究中，MTES循环的性能受四个设计变量的影响。这些变量包括气体冷却器出口温度$\left( {T}_{gc}\right)$、蒸发器#3的蒸发温度$\left( {T}_{e3}\right)$、喷射器的引射系数$\left( \omega \right)$和旁通质量比$\left( R\right)$。基于设定的工况，各变量的范围如下:

$$

• {30}^{ \circ }C \leq {T}_{e3} \leq {0}^{ \circ }C \tag{29} $$

Fig. 2. The flow chart of the program.

图2. 程序流程图。

Fig. 3. Comparison of simulated and experimental values of ejector model.

图3. 喷射器模型模拟值与实验值的比较。

Fig. 4. The effect of the $\omega$ and $\mathrm{R}$ on the optimal discharge pressure.

图4. $\omega$和$\mathrm{R}$对最佳排出压力的影响。

In this study, the compressor discharge pressure $\left( {P}_{dis}\right)$ , the intermediate pressure $\left( {P}_{\text{ int }}\right)$ and the pressure drop of the secondary nozzle $\left( {\Delta P}\right)$ are taken as the constraints. The optimal value of COP is calculated in MATLAB R2017b based on the Genetic Algorithm Multi-objective simulation model. The thermodynamic data are used REFPROP 9.1 for thermodynamic state and characteristic calculation. In addition to the basic algorithm, it also includes iteration conditions and variable values, the required thermodynamic parameters are calculated.

在本研究中，压缩机排出压力$\left( {P}_{dis}\right)$、中间压力$\left( {P}_{\text{ int }}\right)$和二次喷嘴的压降$\left( {\Delta P}\right)$被作为约束条件。基于遗传算法多目标模拟模型，在MATLAB R2017b中计算COP的最优值。使用REFPROP 9.1进行热力学状态和特性计算的热力学数据。除了基本算法外，它还包括迭代条件和变量值，计算所需的热力学参数。

3.6. Model validation

3.6. 模型验证

To validate the accuracy of the proposed model of two-phase flow ejector and efficiency assumptions, operating data reported in the literature using ${\mathrm{{CO}}}_{2}$ are considered [47-49]. Specifically,36 sets of experimental data are selected, and the results obtained by the model calculation are compared with the experimental data in the literature, which are collected in Table 5. The pressure and temperature of the ejector’s primary and secondary flow are given in the experimental data, and the ejector outlet pressure and entrainment rate are also provided. The ejector outlet pressure is calculated by simulation and compared with the experimental values. The comparison result is shown along with the error bandwidth of the prediction. It can be seen from Fig. 3 that the error is between -8% and +5%. However, considering the constant efficiency assumption is to simplify the model, therefore, the error range can be acceptable, and the mathematical model is considered accurate enough for this investigation.

为验证所提出的两相流喷射器模型和效率假设的准确性，考虑使用${\mathrm{{CO}}}_{2}$的文献中报道的运行数据[47 - 49]。具体而言，选择了36组实验数据，并将模型计算得到的结果与文献中的实验数据进行比较，这些数据收集在表5中。实验数据给出了喷射器一次流和二次流的压力和温度，还提供了喷射器出口压力和引射系数。通过模拟计算喷射器出口压力并与实验值进行比较。比较结果与预测的误差带宽一起显示。从图3可以看出，误差在 - 8%到 + 5%之间。然而，考虑到恒定效率假设是为了简化模型，因此，该误差范围是可以接受的，并且该数学模型对于本研究而言被认为足够准确。

Fig. 5. Variation of $\omega ,\mathrm{{COP}},{\Delta T}$ with $R$ (MTES).

图5. $\omega ,\mathrm{{COP}},{\Delta T}$随$R$的变化(MTES)。

4. Results and discussion

4. 结果与讨论

4.1. MTES cycle behavior

4.1. 多目标温度跨临界喷射器-膨胀制冷循环行为

4.1.1. Effect of discharge pressure

4.1.1. 排气压力的影响

The optimal discharge pressure is one of the indispensable parameters when discussing the transcritical ${\mathrm{{CO}}}_{2}$ cycle. Fig. 4 shows the effect of $\omega$ and $R$ on the optimal exhaust pressure for the MTES cycle. The variation trend of COP with the optimal discharge pressure is also in line with the characteristics of typical transcritical ${\mathrm{{CO}}}_{2}$ systems: the COP increases with the increase of ${\mathrm{P}}_{\mathrm{{gc}}}$ , and COP gradually decreases after the optimal pressure is reached. It can be observed from Fig. 4 that the $\omega$ and $R$ have little effect on the optimal discharge pressure, the optimum discharge pressure is always around ${9.0}\mathrm{{MPa}}$ as the change of $\omega$ and $R$ values.

在讨论跨临界${\mathrm{{CO}}}_{2}$循环时，最佳排气压力是不可或缺的参数之一。图4显示了$\omega$和$R$对MTES循环最佳排气压力的影响。COP随最佳排气压力的变化趋势也符合典型跨临界${\mathrm{{CO}}}_{2}$系统的特性:COP随${\mathrm{P}}_{\mathrm{{gc}}}$的增加而增加，在达到最佳压力后COP逐渐降低。从图4可以看出，$\omega$和$R$对最佳排气压力影响很小，随着$\omega$和$R$值的变化，最佳排气压力始终在${9.0}\mathrm{{MPa}}$左右。

4.1.2. Effect of evaporating temperature, cooling capacity and COP

4.1.2. 蒸发温度、制冷量及性能系数的影响

The MTES cycle includes an ejector and vapor-injection. there are two restrictive conditions in the MTES cycle in order to ensure a constant vapor mass flow at the compressor inlet when the evaporator 1 and 2 are in a non-operating state.

多目标温度跨临界喷射-膨胀制冷循环包括一个喷射器和蒸汽喷射。为了确保在蒸发器1和2处于非运行状态时压缩机入口处的蒸汽质量流量恒定，多目标温度跨临界喷射-膨胀制冷循环中有两个限制条件。

1. The vapor quality at the ejector outlet must meet a condition: $(1 + \; \omega ){X}_{d} \geq 1$ , in other words, the minimum entrainment rate $\left( {\omega }_{\min }\right)$ is limited by this condition.

2. 喷射器出口处的蒸汽干度必须满足一个条件:$(1 + \; \omega ){X}_{d} \geq 1$，换句话说，最小引射系数$\left( {\omega }_{\min }\right)$受此条件限制。

3. The value of $R$ is related to the heat exchange temperature difference $\left( {{\Delta T} = {T}_{4} - {T}_{{5}^{\prime }}}\right)$ of the subcooler, that is to ensure ${\Delta T} \geq {10}^{ \circ }\mathrm{C}$ .

4. $R$的值与过冷器的换热温差$\left( {{\Delta T} = {T}_{4} - {T}_{{5}^{\prime }}}\right)$有关，即确保${\Delta T} \geq {10}^{ \circ }\mathrm{C}$。

Based on the above two restrictions, the variation of $\omega$ , COP and ${\Delta T}$ with $R$ is shown in Fig. 5. The operating conditions are given as follows: the gas cooler outlet temperature ${T}_{gc} = {35}^{ \circ }\mathrm{C}$ , evaporating temperature of evaporator#3 ${T}_{e3} = - {20}^{ \circ }\mathrm{C}$ and the evaporator superheating degree ${T}_{sh} = {5}^{ \circ }\mathrm{C}$ . It can be seen that the COP of the MTES cycle increases with the increase of $R$ , which reflects that vapor-injection is positive for cycle performance, because vapor-injection reduces compressor power consumption. The value of $\omega$ increases as the $R$ is rised, the reason is that the fluid temperature at the inlet of the primary nozzle of the ejector is reduced as the value of $R$ increases, the value of $\omega$ must be increased to ensure sufficient vapor quality at the compressor inlet. In addition, the value of $R$ is restricted by ${\Delta T}$ , it can be seen from Fig. 5 that $R$ reaches the maximum value of 0.35 when ${\Delta T} = {10}^{ \circ }\mathrm{C}$ under this operating condition.

基于上述两个限制条件，图5展示了$\omega$、COP和${\Delta T}$随$R$的变化情况。运行条件如下:气体冷却器出口温度${T}_{gc} = {35}^{ \circ }\mathrm{C}$、蒸发器#3的蒸发温度${T}_{e3} = - {20}^{ \circ }\mathrm{C}$以及蒸发器的过热度${T}_{sh} = {5}^{ \circ }\mathrm{C}$。可以看出，MTES循环的COP随着$R$的增加而增加，这表明蒸汽喷射对循环性能有积极影响，因为蒸汽喷射降低了压缩机功耗。随着$R$升高，$\omega$的值增大，原因是随着$R$值增加，喷射器主喷嘴入口处的流体温度降低，必须增大$\omega$的值以确保压缩机入口处有足够的蒸汽质量。此外，$R$的值受${\Delta T}$限制，从图5可以看出，在此运行条件下，当${\Delta T} = {10}^{ \circ }\mathrm{C}$时，$R$达到最大值0.35。

Fig. 6. The effect of the $\omega$ and $R$ on other parameters. (a) effect on ${Q}_{e1}$ ; (b) effect on ${Q}_{e2}$ ; (c) effect on ${Q}_{e3}$ ; (d) effect on ${Q}_{r2}$ ; (e) effect on ${T}_{e2}$ ; (f) effect on COP.

图6. $\omega$ 和 $R$ 对其他参数的影响。(a) 对 ${Q}_{e1}$ 的影响；(b) 对 ${Q}_{e2}$ 的影响；(c) 对 ${Q}_{e3}$ 的影响；(d) 对 ${Q}_{r2}$ 的影响；(e) 对 ${T}_{e2}$ 的影响；(f) 对COP的影响。

Based on the two restrictions of $\omega$ and $R$ discussed above, evaporator#1 and evaporator#2 were added to break the two restrictions and achieve multi-target temperature refrigeration. The following will discuss the characteristics of the MTES cycle in multi-target temperature refrigeration, where three parameters including the evaporating temperature of each evaporator, the cooling capacity of each evaporator and the COP are studied.

基于上述讨论的$\omega$和$R$两个限制条件，增加了蒸发器#1和蒸发器#2以突破这两个限制条件并实现多目标温度制冷。以下将讨论多目标温度制冷中MTES循环的特性，其中研究了包括每个蒸发器的蒸发温度、每个蒸发器的制冷量和COP在内的三个参数。

Fig. 6(a) depicts the variation of cooling capacity of evaporator#1 on the $R$ and the $\omega$ . It can be seen that the value of ${\dot{Q}}_{e1}$ increases as the $R$ is rised when $R \geq {0.35}$ under this operating condition, and the value of ${\dot{Q}}_{e1}$ is irrelevant to $\omega$ . The reason is that the refrigerant in the bypass cannot completely become saturated vapor when $R \geq {0.35}$ , therefore, the cooling capacity ${\dot{Q}}_{e1}$ of the evaporator#1 will gradually increase to ensure that the intermediate vapor injected into the compressor is in the vapor state.

图6(a)描绘了蒸发器#1在$R$和$\omega$上的制冷量变化。可以看出，在该运行条件下当$R \geq {0.35}$时，随着$R$升高，${\dot{Q}}_{e1}$的值增大，且${\dot{Q}}_{e1}$的值与$\omega$无关。原因是当$R \geq {0.35}$时旁通管中的制冷剂不能完全变为饱和蒸汽，因此，蒸发器#1的制冷量${\dot{Q}}_{e1}$将逐渐增加，以确保注入压缩机的中间蒸汽处于蒸汽状态。

Fig. 7. Variation of COP with evaporating temperature for SRC, ERC and MTES cycles $\left( {{\mathrm{T}}_{\mathrm{{gc}}} = {35}^{ \circ }\mathrm{C}}\right)$ .

图7. SRC、ERC和MTES循环$\left( {{\mathrm{T}}_{\mathrm{{gc}}} = {35}^{ \circ }\mathrm{C}}\right)$的性能系数随蒸发温度的变化。

Fig. 6(b) illustrates the variation of cooling capacity of evaporator#2 on the $R$ and the $\omega$ . The following points can be seen:

图6(b)展示了蒸发器#2在$R$和$\omega$上的制冷量变化。可以看出以下几点:

a) The value of ${\omega }_{\min }$ increases as the $R$ is rised when $R < {0.35}$ ; the value of ${\omega }_{\min }$ remains constant as the $R$ is rised when $R \geq {0.35}$ . Specifically, the fluid temperature at the inlet of the primary nozzle of the ejector is remained constant when $R \geq {0.35}$ , therefore, the value of ${\omega }_{\min }$ does not continue to increase.

a) 当$R < {0.35}$时，随着$R$升高，${\omega }_{\min }$的值增大；当$R \geq {0.35}$时，随着$R$升高，${\omega }_{\min }$的值保持不变。具体而言，当$R \geq {0.35}$时，喷射器主喷嘴入口处的流体温度保持恒定，因此，${\omega }_{\min }$的值不会继续增大。

b) The value of ${\dot{Q}}_{e2}$ increases as the $\omega$ is reduced when the value of $R$ is constant; the value of ${\dot{Q}}_{e2}$ increases as the $R$ is rised when the value of $\omega$ is constant and $R < {0.35}$ . The reason is that with the decrease of $\omega$ and the increase of $R$ , the vapor quality at the ejector outlet continues to decrease, the refrigeration capacity of evaporator#2 is increased.

b) 当$R$的值恒定时，随着$\omega$减小，${\dot{Q}}_{e2}$的值增大；当$\omega$的值恒定且$R < {0.35}$时，随着$R$升高，${\dot{Q}}_{e2}$的值增大。原因是随着$\omega$的减小和$R$的增大，喷射器出口处的蒸汽质量持续减小，蒸发器#2的制冷量增大。

c) The value of ${\dot{Q}}_{e2}$ decreases as the $R$ is rised when the value of $\omega$ is constant and $R \geq {0.35}$ . Specifically, the inlet state of the primary nozzle and the secondary nozzle of the ejector does not change. At this time, when $R$ gradually increases, the mass flow rate of the ejector outlet decreases, which leads to a decrease in ${\dot{Q}}_{e2}$ .

c) 当$\omega$的值恒定且$R \geq {0.35}$时，随着$R$升高，${\dot{Q}}_{e2}$的值减小。具体而言，喷射器主喷嘴和副喷嘴的入口状态不变。此时，当$R$逐渐增大时，喷射器出口的质量流量减小，这导致${\dot{Q}}_{e2}$减小。

Fig. 6(c) represents the variation of cooling capacity of evaporator#3 on the $R$ and the $\omega$ . It can be seen that the value of ${\dot{Q}}_{e3}$ decreases as the $\omega$ is reduced when the value of $R$ is constant, the mass flow through the evaporator#3 is reduced because $\omega$ is reduced, which leads to a reduction in ${\dot{Q}}_{e3}$ .

图6(c)表示蒸发器#3的制冷量随$R$和$\omega$的变化。可以看出，当$R$恒定时，随着$\omega$减小，${\dot{Q}}_{e3}$的值减小，因为$\omega$减小，流经蒸发器#3的质量流量减小，这导致${\dot{Q}}_{e3}$减小。

Fig. 6(d) shows the variation of total cooling capacity of evaporator on the $R$ and the $\omega$ . It can be seen that the $R$ and $\omega$ parameters have little effect on the ${\dot{Q}}_{T}$ .

图6(d)显示蒸发器总制冷量随$R$和$\omega$的变化。可以看出，$R$和$\omega$参数对${\dot{Q}}_{T}$影响很小。

For the evaporating temperature of the evaporator#1, evaporator#2 and evaporator#3:

对于蒸发器#1、蒸发器#2和蒸发器#3的蒸发温度:

a) The evaporating temperature of the evaporator#1 is determined by the pressure on the high and low pressure sides. The problem can be written as follows: ${T}_{e1} = f\left( {{P}_{\text{ int }},{X}_{{7}^{\prime }} = 1}\right)$ . The high and low pressures remain unchanged under the conditions discussed in this section, so ${T}_{e1}$ remains constant.

a) 蒸发器#1的蒸发温度由高低压侧的压力决定。问题可表述如下:${T}_{e1} = f\left( {{P}_{\text{ int }},{X}_{{7}^{\prime }} = 1}\right)$。在本节讨论的条件下，高低压保持不变，因此${T}_{e1}$保持恒定。

b) Fig. 6(e) shows the variation of the evaporating temperature of the evaporator#2 on the $R$ and the $\omega$ . It can be seen that the value of ${T}_{e2}$ increases as the $\omega$ is reduced when the value of $R$ is constant. The value of ${T}_{e2}$ decreases as the $R$ is rised when $R < {0.35}$ ; the value of ${T}_{e2}$ remains constant as the $R$ is rised when $R \geq {0.35}$ .

b) 图6(e)显示蒸发器#2的蒸发温度随$R$和$\omega$的变化。可以看出，当$R$恒定时，随着$\omega$减小，${T}_{e2}$的值增大。当$R < {0.35}$时，随着$R$升高，${T}_{e2}$的值减小；当$R \geq {0.35}$时，随着$R$升高，${T}_{e2}$的值保持不变。

Fig. 8. Variation of COP with gas cooler outlet temperature for SRC, ERC and MTES cycles $\left( {{\mathrm{T}}_{e3} = - {20}^{ \circ }\mathrm{C}}\right)$ .

图8. SRC、ERC和MTES循环中COP随气体冷却器出口温度的变化$\left( {{\mathrm{T}}_{e3} = - {20}^{ \circ }\mathrm{C}}\right)$。

c) The evaporating temperature of the evaporator#3 is set to a constant value $\left( {{T}_{e3} = - {20}^{ \circ }\mathrm{C}}\right)$ .

c) 蒸发器#3的蒸发温度设定为恒定值$\left( {{T}_{e3} = - {20}^{ \circ }\mathrm{C}}\right)$。

Finally, Fig. 6(f) shows the variation of COP of cycle on the $R$ and the $\omega$ . It can be seen that the value of COP increases as the $\omega$ is reduced, and the value of COP increases as the $R$ is rised. As analyzed above, a part of the refrigerant becomes gaseous at the evaporating temperature ${T}_{e1}$ and ${T}_{e2}$ when the evaporator#1 and the evaporator#2 are working, which causes the compressor intake pressure to rise, and then the COP increases.

最后，图6(f)展示了循环的性能系数(COP)在$R$和$\omega$上的变化情况。可以看出，随着$\omega$降低，COP值增大；随着$R$升高，COP值增大。如上文分析，当蒸发器#1和蒸发器#2运行时，一部分制冷剂在蒸发温度${T}_{e1}$和${T}_{e2}$下变为气态，这导致压缩机吸气压力升高，进而COP增大。

4.2. Comparison of SRC, ERC and MTES cycles

4.2. SRC、ERC和MTES循环的比较

To evaluate the thermodynamic performance of the MTES cycle, under the same operating conditions (evaporator#1 and evaporator#2 of the MTES cycle are not operating), the advantages of the MTES cycle are described by comparing the COP and exergy efficiency of SRC, ERC and MTES cycles.

为了评估MTES循环的热力性能，在相同运行条件下(MTES循环的蒸发器#1和蒸发器#2不运行)，通过比较SRC、ERC和MTES循环的性能系数(COP)和火用效率来描述MTES循环的优势。

Fig. 9. Variation of ${\eta }_{Ex}$ with evaporating temperature for SRC, ERC and MTES cycles $\left( {{\mathrm{T}}_{\mathrm{{gc}}} = {35}^{ \circ }\mathrm{C}}\right)$ .

图9. SRC、ERC和MTES循环的${\eta }_{Ex}$随蒸发温度的变化$\left( {{\mathrm{T}}_{\mathrm{{gc}}} = {35}^{ \circ }\mathrm{C}}\right)$。

Fig. 10. Variation of ${\eta }_{Ex}$ with gas cooler temperature for SRC, ERC and MTES cycles $\left( {{T}_{e3} = - {20}^{ \circ }\mathrm{C}}\right)$ .

图10. SRC、ERC和MTES循环的${\eta }_{Ex}$随气体冷却器温度的变化$\left( {{T}_{e3} = - {20}^{ \circ }\mathrm{C}}\right)$。

4.2.1. Parametric analysis on the COP of the SRC, ERC and MTES cycles

4.2.1. 关于SRC、ERC和MTES循环性能系数的参数分析

Fig. 7 shows the COP comparison between SRC, ERC and MTES cycles within the selected evaporating temperature range. As expected, the value of COP decreases as the evaporating temperature is reduced. Compared with SRC and ERC systems, the MTES cycle shows higher COP when applied to the selected temperature range. Specifically, the COP of the MTES cycle in the study temperature range is 17.9-26.7% higher than that of SRC cycle, and 2.8-6% higher than that of ERC cycle, respectively. In addition, the value of ΔCOP increases as the evaporating temperature is reduced, it shows that the MTES cycle performs better under low temperature conditions.

图7展示了在选定蒸发温度范围内SRC、ERC和MTES循环之间的性能系数比较。正如预期的那样，性能系数值随着蒸发温度的降低而减小。与SRC和ERC系统相比，MTES循环在应用于选定温度范围时表现出更高的性能系数。具体而言，在研究温度范围内，MTES循环的性能系数分别比SRC循环高17.9 - 26.7%，比ERC循环高2.8 - 6%。此外，随着蒸发温度的降低，ΔCOP值增大，这表明MTES循环在低温条件下性能更好。

Fig. 8 shows the COP comparison of SRC, ERC and MTES cycles with the change of gas cooler outlet temperature. As expected, the value of COP decreases as the gas cooler outlet temperature is rised. As can be observed, the COP of the MTES cycle in the study temperature range is 23-29.4% higher than that of SRC cycle, and 4.8-7.8% higher than that of ERC cycle, respectively. The value of ΔCOP increases as the gas cooler outlet temperature is rised, which means that the MTES cycle has a more competitive advantage under high ambient temperature conditions.

图8展示了SRC、ERC和MTES循环的性能系数随气体冷却器出口温度变化的比较。正如预期的那样，性能系数值随着气体冷却器出口温度的升高而减小。可以观察到，在研究温度范围内，MTES循环的性能系数分别比SRC循环高23 - 29.4%，比ERC循环高4.8 - 7.8%。随着气体冷却器出口温度的升高，ΔCOP值增大，这意味着MTES循环在高环境温度条件下具有更具竞争力的优势。

4.2.2. Parametric analysis on the exergy efficiency of the SRC, ERC and MTES cycles

4.2.2. SRC、ERC和MTES循环的火用效率参数分析

Fig. 9 illustrates a comparison between the exergy efficiency of the cycles at different evaporating temperatures. The value of ${\eta }_{Ex}$ changes little with the evaporation temperature, the main reason is that the total system exergy loss and the compressor power decrease, and the rate of decrease of the two is similar. In addition, it can be seen that MTES cycle has higher exergy efficiency than SRC and ERC cycles. Specifically, the ${\eta }_{Ex}$ of the MTES cycle in the study temperature range is ${14} - {24}\%$ higher than that of SRC cycle, and 3.7-9.8% higher than that of ERC cycle, respectively. The variation trend of $\Delta {\eta }_{Ex}$ with evaporating temperature is similar to $\Delta \mathrm{{COP}}$ in Fig. 7.

图9展示了不同蒸发温度下各循环的火用效率对比。${\eta }_{Ex}$的值随蒸发温度变化不大，主要原因是系统总火用损失和压缩机功率下降，且二者下降速率相近。此外，可以看出MTES循环的火用效率高于SRC和ERC循环。具体而言，研究温度范围内MTES循环的${\eta }_{Ex}$分别比SRC循环高${14} - {24}\%$，比ERC循环高3.7 - 9.8%。$\Delta {\eta }_{Ex}$随蒸发温度的变化趋势与图7中的$\Delta \mathrm{{COP}}$相似。

Fig. 11. Exergy destruction ratios of different component varieties for SRC, ERC and MTES cycles.

图11. SRC、ERC和MTES循环中不同组件的火用损率。

Table 4

表4

Exergy destruction in cycles $\left( {{\mathrm{T}}_{\mathrm{{gc}}} = {35}^{ \circ }\mathrm{C},{\mathrm{T}}_{\mathrm{e}} = - {20}^{ \circ }\mathrm{C}}\right)$ .

循环$\left( {{\mathrm{T}}_{\mathrm{{gc}}} = {35}^{ \circ }\mathrm{C},{\mathrm{T}}_{\mathrm{e}} = - {20}^{ \circ }\mathrm{C}}\right)$中的火用损。

Table 5

表5

Experimental data and results of the model validation.

模型验证的实验数据和结果。

Fig. 10 illustrates a comparison between the exergy efficiency of the cycles at different gas cooler outlet temperatures. The value of ${\eta }_{Ex}$ decreases as the gas cooler outlet temperature is rised, the main reason is that the total system exergy loss and the compressor power increase, and the former the exergy loss rises faster, resulting in a decrease in exergy efficiency. The ${\eta }_{Ex}$ of the MTES cycle in the study temperature range is 19.2-28.2% higher than that of SRC cycle, and 7-11.3% higher than that of ERC cycle, respectively. The variation trend of $\Delta {\eta }_{Ex}$ with gas cooler outlet temperature is similar to $\Delta \mathrm{{COP}}$ in Fig. 8.

图10展示了不同气体冷却器出口温度下各循环的火用效率对比。随着气体冷却器出口温度升高，${\eta }_{Ex}$的值下降，主要原因是系统总火用损失和压缩机功率增加，且前者火用损失上升更快，导致火用效率降低。研究温度范围内MTES循环的${\eta }_{Ex}$分别比SRC循环高19.2 - 28.2%，比ERC循环高7 - 11.3%。$\Delta {\eta }_{Ex}$随气体冷却器出口温度的变化趋势与图8中的$\Delta \mathrm{{COP}}$相似。

Fig. 11 shows the exergy destruction ratio of each component for the SRC, ERC and MTES cycles. In Table 4, the exergy destruction for each of the components of the SRC, ERC and MTES cycles is presented within the selected temperature conditions. It can be noticed that the compressor, gas cooler and expansion valve are the main exergy destruction in systems, that is because large pressure difference between compressor inlet and outlet, the heat transfer occurs in gas cooler and the large pressure difference during throttling will lead to larger irreversibility. Furthermore, the total exergy destruction of the MTES cycle is minimal, it can be seen from Fig. 11 and Table 4 which is mainly reflected in the small loss at the ejector and expansion valve.

图11展示了SRC、ERC和MTES循环中各组件的火用损率。在表4中，给出了所选温度条件下SRC、ERC和MTES循环各组件的火用损。可以注意到，压缩机、气体冷却器和膨胀阀是系统中火用损的主要部件，这是因为压缩机进出口之间的压差大、气体冷却器中有热传递以及节流过程中的大压差会导致更大的不可逆性。此外，MTES循环的总火用损最小，从图11和表4中可以看出，这主要体现在喷射器和膨胀阀处的损失较小。

5. Conclusions

5. 结论

This paper presents a multi-target temperature ejector-expansion subcooler vapor-injection transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle in which a two-phase ejector and vapor-injection is used to improve the cycle performance. The thermodynamic performance of the proposed MTES cycle is also compared with the SRC and ERC system under the same working conditions. The concluding remarks of the considered MTES cycle are given as follows.

本文提出了一种多目标温度喷射器 - 膨胀过冷器蒸汽喷射跨临界${\mathrm{{CO}}}_{2}$制冷循环，其中使用两相喷射器和蒸汽喷射来提高循环性能。还在相同工况下将所提出的MTES循环的热力性能与SRC和ERC系统进行了比较。所考虑的MTES循环的结论如下。

(1) For the proposed MTES cycle, the changes of $\omega$ and $R$ have no significant effect on the optimal discharge pressure. Then, the factors affecting the evaporating temperature and the cooling capacity of the three evaporators and the change relationship are analyzed in detail, and the direction of cooling capacity distribution and evaporating temperature adjustment is pointed out. The entrainment rate and the bypass mass ratio are the crucial parameters that determine the target evaporating temperature and cooling capacity distribution, providing suggestions on the setting and assignment of the MTES cycle operating conditions.

(1) 对于所提出的MTES循环，$\omega$和$R$的变化对最佳排气压力没有显著影响。然后，详细分析了影响蒸发温度以及三个蒸发器制冷量的因素及其变化关系，并指出了制冷量分配和蒸发温度调节的方向。引射系数和旁通质量比是决定目标蒸发温度和制冷量分配的关键参数，为MTES循环运行工况的设定和分配提供了建议。

(2) Compared with SRC and ERC cycles, the proposed MTES cycle shows higher COP (about 17.9-29.4% higher than that of SRC cycle and about 2.8-7.8% higher than that of ERC cycle), higher exergy efficiency (about 14-28.2% higher than that of SRC cycle and about 3.7-11.3% higher than that of ERC cycle), the MTES cycle is suitable for low temperature refrigeration applications in warm climates. The exergertic analysis also indicates that the compressors, the gas cooler and throttle expansion equipment of the system should be selected and optimized carefully in research and design.

(2) 与SRC和ERC循环相比，所提出的MTES循环具有更高的COP(比SRC循环高约17.9 - 29.4%，比ERC循环高约2.8 - 7.8%)，更高的火用效率(比SRC循环高约14 - 28.2%，比ERC循环高约3.7 - 11.3%)，MTES循环适用于温暖气候下的低温制冷应用。火用分析还表明，在研究和设计中应仔细选择和优化系统的压缩机、气体冷却器和节流膨胀设备。

Based on these conclusions, the MTES cycle has potential practical value in commercial and industrial multi-target temperature refrigeration. It should be mentioned that although the ejector calculation model is a one-dimensional model, it is hoped that the proposed MTES cycle can provide useful guidance for the development of multi-target temperature refrigeration systems. In addition, the parameters $\omega , R$ and compressor frequency also interact with each other during the actual application of the MTES cycle. Therefore, this cycle requires further experimental research to confirm the practical use of the MTES cycle.

基于这些结论，MTES循环在商业和工业多目标温度制冷方面具有潜在的实用价值。应该提到的是，虽然喷射器计算模型是一维模型，但希望所提出的MTES循环能为多目标温度制冷系统的发展提供有益的指导。此外，在MTES循环的实际应用中，参数$\omega , R$和压缩机频率也相互影响。因此，该循环需要进一步的实验研究来确认MTES循环的实际用途。

CRediT authorship contribution statement

CRediT作者贡献声明

Min-Qiang Zeng: Conceptualization, Methodology, Software, Data curation, Validation, Writing - original draft. Qiu-Yun Zheng: Conceptualization, Methodology, Writing - review & editing. Xue-Lai Zhang: Writing - review & editing, Project administration. Fan-Yang Mo: Writing - review & editing, Project administration. Xin-Rong Zhang: Conceptualization, Methodology, Writing - review & editing, Project administration, Funding acquisition.

曾敏强:概念化、方法学、软件、数据管理、验证、撰写原始草案。郑秋云:概念化、方法学、撰写 - 评审与编辑。张学来:撰写 - 评审与编辑、项目管理。莫凡洋:撰写 - 评审与编辑、项目管理。张新荣:概念化、方法学、撰写 - 评审与编辑、项目管理、资金获取。

Declaration of competing interest

利益冲突声明

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

作者声明，他们不存在已知的可能影响本文所报告工作的竞争性财务利益或个人关系。

Data availability

数据可用性

Data will be made available on request.

数据将根据请求提供。

Acknowledgments

致谢

The support from the National Key Research and Development Program (2018YFD0901002) is gratefully acknowledged.

感谢国家重点研发计划(2018YFD0901002)的支持。

References

参考文献

[1] Dupont J-L, Domanski P, Lebrun P, Ziegler F. The role of refrigeration in the global economy-38. Informatory Note on Refrigeration Technologies; 2019.

[2] She X, Cong L, Nie B, Leng G, Peng H, Chen Y, et al. Energy-efficient and -economic technologies for air conditioning with vapor compression refrigeration: a comprehensive review. Appl Energy 2018;232:157-86. https://doi.org/10.1016/j.apenergy.2018.09.067.

[3] Gao E, Cui Q, Jing H, Zhang Z, Zhang X. A review of application status and replacement progress of refrigerants in the Chinese cold chain industry. Int J Refrig 2021;128:104-17. https://doi.org/10.1016/j.ijrefrig.2021.03.025.

[4] Birmpili T. Montreal Protocol at 30: the governance structure, the evolution, and the Kigali amendment. Compt Rendus Geosci 2018;350(7):425-31. https://doi.org/10.1016/j.crte.2018.09.002.

[5] Nawaz K, Ally MR. Options for low-global-warming-potential and natural refrigerants Part 2: performance of refrigerants and systemic irreversibilities. Int J Refrig 2019;106:213-24. https://doi.org/10.1016/j.ijrefrig.2019.05.030.

[6] Jadhav JS, Apte AD. Review of cascade refrigeration system with different refrigerant pairs. Int J Innov Eng Res Technol 2015;2:74-81.

[7] Granryd E. Hydrocarbons as refrigerants-an overview. Int J Refrig 2001;24(1): 15-24.

[8] Lorentzen G. Revival of carbon dioxide as a refrigerant. Int J Refrig 1994;17(5): 292-301.

[9] Abas N, Kalair AR, Khan N, Haider A, Saleem Z, Saleem MS. Natural and synthetic refrigerants, global warming: a review. Renew Sustain Energy Rev 2018;90: 557-69. https://doi.org/10.1016/j.rser.2018.03.099.

[10] Zolcer Skačanová K, Battesti M. Global market and policy trends for ${\mathrm{{CO}}}_{2}$ in refrigeration. Int J Refrig 2019;107:98-104. https://doi.org/10.1016/j.ijrefrig.2019.08.010.

[11] Bellos E, Tzivanidis C. A comparative study of ${\mathrm{{CO}}}_{2}$ refrigeration systems. Energy Convers Manag X 2019;1. https://doi.org/10.1016/j.ecmx.2018.100002.

[12] Ma Y, Liu Z, Tian H. A review of transcritical carbon dioxide heat pump and refrigeration cycles. Energy 2013;55:156-72. https://doi.org/10.1016/j.energy.2013.03.030.

[13] Megdouli K, Sahli H, Tashtoush BM, Nahdi E, Kairouani L. Theoretical research of the performance of a novel enhanced transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle for power and cold generation. Energy Convers Manag 2019;201. https://doi.org/ 10.1016/j.enconman.2019.112139.

[14] Tashtoush BM, Al-Nimr MdA, Khasawneh MA. A comprehensive review of ejector design, performance, and applications. Appl Energy 2019;240:138-72. https://doi.org/10.1016/j.apenergy.2019.01.185.

[15] Tashtoush B, Nayfeh Y. Energy and economic analysis of a variable-geometry ejector in solar cooling systems for residential buildings. J Energy Storage 2020;27. https://doi.org/10.1016/j.est.2019.101061.

[16] Bai T, Yan G, Yu J. Thermodynamics analysis of a modified dual-evaporator ${\mathrm{{CO}}}_{2}$ transcritical refrigeration cycle with two-stage ejector. Energy 2015;84:325-35. https://doi.org/10.1016/j.energy.2015.02.104.

[17] Fangtian S, Yitai M. Thermodynamic analysis of transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle with an ejector. Appl Therm Eng 2011;31(6-7):1184-9. https://doi.org/ 10.1016/j.applthermaleng.2010.12.018.

[18] Li D, Groll EA. Transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle with ejector-expansion device. Int J Refrig 2005;28(5):766-73. https://doi.org/10.1016/j.ijrefrig.2004.10.008.

[19] Ksayer EBL, Clodic D. Enhancement of ${\mathrm{{CO}}}_{2}$ refrigeration cycle using an ejector: 1D analysis. Conference Enhancement of ${\mathrm{{CO}}}_{2}$ refrigeration cycle using an ejector: 1D analysis.

[20] Ahammed ME, Bhattacharyya S, Ramgopal M. Thermodynamic design and simulation of a ${\mathrm{{CO}}}_{2}$ based transcritical vapour compression refrigeration system with an ejector. Int J Refrig 2014;45:177-88. https://doi.org/10.1016/j.ijrefrig.2014.06.010.

[21] Eskandari Manjili F, Cheraghi M. Performance of a new two-stage transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle with two ejectors. Appl Therm Eng 2019;156:402-9. https:// doi.org/10.1016/j.applthermaleng.2019.03.083.

[22] Elakhdar M, Tashtoush BM, Nehdi E, Kairouani L. Thermodynamic analysis of a novel ejector enhanced vapor compression refrigeration (EEVCR) cycle. Energy 2018;163:1217-30. https://doi.org/10.1016/j.energy.2018.09.050.

[23] Liu Y, Xin T, Cao L, Wan C, Zhang M. Compression-injection hybrid refrigeration cycles in household refrigerators. Appl Therm Eng 2010;30(16):2442-7. https:// doi.org/10.1016/j.applthermaleng.2010.06.015.

[24] Xu X, Hwang Y, Radermacher R. Refrigerant injection for heat pumping/air conditioning systems: literature review and challenges discussions. Int J Refrig 2011;34(2):402-15. https://doi.org/10.1016/j.ijrefrig.2010.09.015.

[25] Cho H, Baek C, Park C, Kim Y. Performance evaluation of a two-stage ${\mathrm{{CO}}}_{2}$ cycle with gas injection in the cooling mode operation. Int J Refrig 2009;32(1):40-6. https://doi.org/10.1016/j.ijrefrig.2008.07.008.

[26] Baek C, Heo J, Jung J, Cho H, Kim Y. Performance characteristics of a two-stage ${\mathrm{{CO}}}_{2}$ heat pump water heater adopting a sub-cooler vapor injection cycle at various operating conditions. Energy 2014;77:570-8. https://doi.org/10.1016/j.energy.2014.09.038.

[27] Peng X, Wang D, Wang G, Yang Y, Xiang S. Numerical investigation on the heating performance of a transcritical ${\mathrm{{CO}}}_{2}$ vapor-injection heat pump system. Appl Therm Eng 2020;166:114656.

[28] Bai T, Yan G, Yu J. Thermodynamic analyses on an ejector enhanced ${\mathrm{{CO}}}_{2}$ transcritical heat pump cycle with vapor-injection. Int J Refrig 2015;58:22-34. https://doi.org/10.1016/j.ijrefrig.2015.04.010.

[29] Keenan JH. An investigation of ejector design by analysis and experiment. J Appl Mech 1950;17:299.

[30] Keenan J, Neumann E. A simple air ejector. 1942.

[31] Keenan JH, Neumann EP, Lustwerk F. An investigation of ejector design by analysis and experiment. 1950.

[32] Elbel SW, Hrnjak PS. Effect of internal heat exchanger on performance of transcritical ${\mathrm{{CO}}}_{2}$ systems with ejector. 2004.

[33] Deng J-q, Jiang P-x, Lu T, Lu W. Particular characteristics of transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle with an ejector. Appl Therm Eng 2007;27(2-3):381-8. https:// doi.org/10.1016/j.applthermaleng.2006.07.016.

[34] Sarkar J. Optimization of ejector-expansion transcritical ${\mathrm{{CO}}}_{2}$ heat pump cycle. Energy 2008;33(9):1399-406. https://doi.org/10.1016/j.energy.2008.04.007.

[35] Manjili FE, Yavari MA. Performance of a new two-stage multi-intercooling transcritical ${\mathrm{{CO}}}_{2}$ ejector refrigeration cycle. Appl Therm Eng 2012;40:202-9. https://doi.org/10.1016/j.applthermaleng.2012.02.014.

[36] Zheng L, Deng J, Zhang Z. Dynamic simulation of an improved transcritical ${\mathrm{{CO}}}_{2}$ ejector expansion refrigeration cycle. Energy Convers Manag 2016;114:278-89. https://doi.org/10.1016/j.enconman.2016.01.069.

[37] Zheng L, Deng J. Research on ${\mathrm{{CO}}}_{2}$ ejector component efficiencies by experiment measurement and distributed-parameter modeling. Energy Convers Manag 2017; 142:244-56. https://doi.org/10.1016/j.enconman.2017.03.017.

[38] Taslimi Taleghani S, Sorin M, Poncet S. Modeling of two-phase transcritical ${\mathrm{{CO}}}_{2}$ ejectors for on-design and off-design conditions. Int J Refrig 2018;87:91-105. https://doi.org/10.1016/j.ijrefrig.2017.10.025.

[39] Zare V, Rostamnejad Takleh H. Novel geothermal driven CCHP systems integrating ejector transcritical ${\mathrm{{CO}}}_{2}$ and Rankine cycles: thermodynamic modeling and parametric study. Energy Convers Manag 2020;205. https://doi.org/10.1016/j.enconman.2019.112396.

[40] Peris Pérez B, Expósito Carrillo JA, Sánchez de La Flor FJ, Salmerón Lissén JM, Morillo Navarro A. Thermoeconomic analysis of ${\mathrm{{CO}}}_{2}$ Ejector-Expansion Refrigeration Cycle (EERC) for low-temperature refrigeration in warm climates. Appl Therm Eng 2021;188. https://doi.org/10.1016/j.applthermaleng.2021.116613.

[41] Purjam M, Thu K, Miyazaki T. Thermodynamic modeling of an improved transcritical carbon dioxide cycle with ejector: aiming low-temperature refrigeration. Appl Therm Eng 2021:188. https://doi.org/10.1016/j.applthermaleng.2020.116531.

[42] Nguyen D, Winter E, Greiner M. Sonic velocity in two-phase systems. Int J Multiphas Flow 1981;7(3):311-20.

[43] Yari M. Performance analysis and optimization of a new two-stage ejector-expansion transcritical ${\mathrm{{CO}}}_{2}$ refrigeration cycle. Int J Therm Sci 2009;48(10): 1997-2005. https://doi.org/10.1016/j.ijthermalsci.2009.01.013.

[44] Lee T-S, Liu C-H, Chen T-W. Thermodynamic analysis of optimal condensing temperature of cascade-condenser in ${\mathrm{{CO}}}_{2}/{\mathrm{{NH}}}_{3}$ cascade refrigeration systems. Int J Refrig 2006;29(7):1100-8. https://doi.org/10.1016/j.ijrefrig.2006.03.003.

[45] Ju Ahamed, Saidur R, Masjuki HH. A review on exergy analysis of vapor compression refrigeration system. Renew Sustain Energy Rev 2011;15(3): 1593-600. https://doi.org/10.1016/j.rser.2010.11.039.

[46] Liu J, Lin Z. A novel dual-temperature ejector-compression heat pump cycle - exergetic and economic analyses. Int J Refrig 2021;126:155-67. https://doi.org/ 10.1016/j.ijrefrig.2021.01.005.

[47] Zhu Y, Li C, Zhang F, Jiang P-X. Comprehensive experimental study on a transcritical ${\mathrm{{CO}}}_{2}$ ejector-expansion refrigeration system. Energy Convers Manag 2017;151:98-106. https://doi.org/10.1016/j.enconman.2017.08.061.

[48] Palacz M, Haida M, Smolka J, Nowak AJ, Banasiak K, Hafner A. HEM and HRM accuracy comparison for the simulation of ${\mathrm{{CO}}}_{2}$ expansion in two-phase ejectors for supermarket refrigeration systems. Appl Therm Eng 2017;115:160-9. https://doi.org/10.1016/j.applthermaleng.2016.12.122.

[49] Giacomelli F, Mazzelli F, Banasiak K, Hafner A, Milazzo A. Experimental and computational analysis of a R744 flashing ejector. Int J Refrig 2019;107:326-43. https://doi.org/10.1016/j.ijrefrig.2019.08.007.