Effect of Back Pressure on Performances and Key Geometries of the Second Stage in a Highly Coupled Two-Stage Ejector

In this paper, for a highly coupled two-stage ejector-based cooling cycle, the optimization of primary nozzle length and angle of the second-stage ejector under varied primary nozzle diameters of the second stage was conducted first. Next, the evaluation for the influence of variable back pressure on ER of the two-stage ejector was performed. Last, the identification of the effect of the variable back pressure on the key geometries of the two-stage ejector was carried out. The results revealed that: (1) with the increase of the nozzle diameter at the second stage, the ER of both stages decreased with the increases of the length and angle of the converging section of the second-stage primary nozzle; (2) the pressure lift ratio range of the second-stage ejector in the critical mode gradually increased with the increase of the nozzle diameter of the second-stage; (3) when the pressure lift ratio increased from 102% to 106%, the peak ER of the second-stage decreased, and the influence of the area ratio and nozzle exit position of the second-stage ejector on its ER was reduced; (4) with the increase of nozzle diameter of the second-stage, the influence of area ratio and nozzle exit position of the second-stage on the second-stage performance decreased; and (5) the optimal AR of the second stage decreased but the optimal nozzle exit position of the second stage kept constant with the pressure lift ratio of the two-stage ejector.


Introduction
Nowadays, we are facing global warming and resource shortage [1]. Therefore, energysaving and environmentally friendly refrigeration technologies have become a topic of widespread interest to refrigeration researchers and practitioners [2]. Ejector-based refrigeration systems have some advantages, such as no moving parts, waste heat driving and low operating costs [3,4]. Undoubtedly, ejector-based refrigeration systems are a promising industry [5,6].
In recent years, scholars have engaged in the research and development of two-stage ejector-based refrigeration systems. Kong et al. [7] investigated a supersonic two-stage ejector-diffuser system with numerical methods, and the system had four times better performance than a single ejector system. In addition, Kong et al. [8] predicted the flow phenomenon inside the two-stage ejector-diffuser. The results showed that the entrainment effects of the system greatly increased. Liu et al. [9] proposed a modified transcritical CO 2 ejector enhanced two-stage compression cycle. The results showed that the heating coefficient of performance (COP) of the cycle, and so on, outperformed others. Wang et al. [10] presented a gas-fired air-to-water ejector heat pump. The system performance was improved with a high entrainment ratio. Liu et al. [11] proposed a novel two-stage compression transcritical CO 2 refrigeration system with an ejector. The results indicated that the performance of the novel system was better than those of conventional systems. Yan et al. [12] presented a dual-ejector refrigeration system, and the area ratio (AR) had the most significant influences on the performance of the two-stage ejector. Ierin et al. [13] optimized a hybrid two-stage CO 2 ejector-based cooling system, and the efficiency of the system increased by up to 32.7%. Ghorbani et al. [14] investigated a twostage ejector cooling system, and the consumed power of the system decreased by 12.37%. Cao et al. [15] proposed a two-stage evaporation cycle and their numerical results disclosed that the COP of the cycle was improved, and the exergy was reduced. Sun et al. [16] claimed that the influence of phase transition in the ejector contributed to the ejector optimization. Chen et al. [17] studied the effect of the second-stage geometrical factors on the system performance and the optimized length to diameter ratio was 5. Similarly, Yadva et al. [18] numerically analyzed the performance of a two-stage ejector. Yang et al. [19] evaluated the exergy destruction characteristics inside a transcritical CO 2 two-stage refrigeration system and showed that the system exergetic performance can be improved by enhancing the efficiency of the ejector. Yang et al. [20] also found that the gas cooler temperature had the greatest influence on ejector performance. Surendran et al. [21] explored a novel transcritical ejector regenerative refrigeration system and identified the system performance. Asfahan et al. [22] presented a system with two-stage ejectors and investigated them numerically. The results showed that the system had good performance when using the two-stage ejectors. Ding et al. [23] performed numerical studies using computational fluid dynamics (CFD) to predict two-stage ejector performance for subzero applications. Manjili et al. [24] used Engineering Equation Solver (EES) software to investigate a twostage transcritical CO 2 refrigeration cycle. It was found that the COP of new cycle was improved from 20% to 80% compared to the conventional cycle. Xue et al. [25] proposed and studied a two-stage vacuum ejector by comparing seven different ejector models. The results showed the two-stage ejector could provide superior suction pressure. Exposito-Carrillo et al. [26] optimized a two-stage CO 2 refrigeration system, and the COP improved up to 13%. Wang et al. [27] developed a CFD model to investigate the performance of a proposed two-stage ejector. Viscito et al. [28] proposed a seasonal performance analysis of a hybrid ejector cooling system, they claimed that system required three or four ejectors for any reference climate, and they obtained an increase of the seasonal energy efficiency ratio up to 107%. Lillo et al. [29] presented a thermo-economic analysis of a waste heat recovery hybrid ejector cycle with a cooling load of 20 kW, the thermo-economic performance of this cycle has evident advantages over other waste heat driven system. Li et al. [30] carried out a numerical analysis of the influence of nozzle geometries on steam condensation and irreversibility in the ejector nozzle. The results indicated that the condensation of the steam makes a large amount of irreversible energy. Wen et al. [31] presented a two-stage ejector-based refrigeration system and optimized the two-stage ejector (TSE) geometries and system performance. Yan et al. [32] proposed another type of a highly coupled TSEbased system as shown in Figure 1, and they also optimized the key geometries such as area ratio (AR) and nozzle exit position (NXP) of the two-stage ejector, as illustrated in Figure 2, in which the mixture coming from the outlet of the first stage enters the primary nozzle of the second stage and entrains its secondary flow refrigerant.
However, no studies in the literature have mentioned the optimization of the key geometries, such as AR and NXP, of the highly coupled second-stage ejector under different back pressures and varied primary nozzles of the second stage (PNTD 2 ). To bridge the gap, and based on our previous studies [33], further works in this study included: • CFD modelling and model validation of the highly coupled TSE; • Optimization of primary nozzle geometry of the second-stage ejector under varied primary nozzle of the second stage; • Evaluation the influence of variable back pressure on entrainment ratio (ER) of the TSE; • Identification of the effect of the variable back pressure on the key geometries of the second-stage ejector.  However, no studies in the literature have mentioned the optimization of the k geometries, such as AR and NXP, of the highly coupled second-stage ejector under d ferent back pressures and varied primary nozzles of the second stage (PNTD2). To brid the gap, and based on our previous studies [33], further works in this study included: • CFD modelling and model validation of the highly coupled TSE; • Optimization of primary nozzle geometry of the second-stage ejector under var primary nozzle of the second stage;  However, no studies in the literature have mentioned the optimization of the key geometries, such as AR and NXP, of the highly coupled second-stage ejector under different back pressures and varied primary nozzles of the second stage (PNTD2). To bridge the gap, and based on our previous studies [33], further works in this study included: • CFD modelling and model validation of the highly coupled TSE; • Optimization of primary nozzle geometry of the second-stage ejector under varied primary nozzle of the second stage;

System and Initial TSE Geometries
The schematic of the highly coupled TSE-based refrigeration system is shown in Figure 1, and the initial geometrical parameters of the TSE are presented in Figure 3.

System and Initial TSE Geometries
The schematic of the highly coupled TSE-based refrigeration system is shown Figure 1, and the initial geometrical parameters of the TSE are presented in Figure 3.

CFD Modelling
The flow inside the TSE is calculated by using governing equations [33,34], an Gambit 2.4 and Ansys 19.0 [35] are used in this simulation. Grids with 103,000 quad lateral elements are created as shown in Figure 4. R134a is the working fluid with parameters from NIST [36], the RNG k-ε turbulen model was selected in this study. The standard wall function was chosen, and the ran of the first grid cell is in the region of 30 < y+ < 300. The residual convergence limit f each equation is below 10 −5 , except that for energy equation is set to less than 10 −6 . In a dition, to ensure that the refrigerant liquid completely evaporates into refrigerant ga three inlet streams are set as 10 K superheat. The primary fluid inlet and the seconda fluid inlet are both set as the pressure inlet, while the outlet is set as the pressure out [32], and the boundary conditions of the TSE are illustrated in Table 1.

CFD Modelling
The flow inside the TSE is calculated by using governing equations [33,34], and Gambit 2.4 and Ansys 19.0 [35] are used in this simulation. Grids with 103,000 quadrilateral elements are created as shown in Figure 4.

System and Initial TSE Geometries
The schematic of the highly coupled TSE-based refrigeration system is shown Figure 1, and the initial geometrical parameters of the TSE are presented in Figure 3.

CFD Modelling
The flow inside the TSE is calculated by using governing equations [33,34], a Gambit 2.4 and Ansys 19.0 [35] are used in this simulation. Grids with 103,000 quad lateral elements are created as shown in Figure 4. R134a is the working fluid with parameters from NIST [36], the RNG k-ε turbulen model was selected in this study. The standard wall function was chosen, and the ran of the first grid cell is in the region of 30 < y+ < 300. The residual convergence limit each equation is below 10 −5 , except that for energy equation is set to less than 10 −6 . In a dition, to ensure that the refrigerant liquid completely evaporates into refrigerant g three inlet streams are set as 10 K superheat. The primary fluid inlet and the seconda fluid inlet are both set as the pressure inlet, while the outlet is set as the pressure out [32], and the boundary conditions of the TSE are illustrated in Table 1.  R134a is the working fluid with parameters from NIST [36], the RNG k-ε turbulence model was selected in this study. The standard wall function was chosen, and the range of the first grid cell is in the region of 30 < y+ < 300. The residual convergence limit for each equation is below 10 −5 , except that for energy equation is set to less than 10 −6 . In addition, to ensure that the refrigerant liquid completely evaporates into refrigerant gas, three inlet streams are set as 10 K superheat. The primary fluid inlet and the secondary fluid inlet are both set as the pressure inlet, while the outlet is set as the pressure outlet [32], and the boundary conditions of the TSE are illustrated in Table 1. Three levels of grids (71,000, 103,000 and 138,000) are used to validate the grid independence as illustrated in Figure 5. Since the three grid levels are quite close with each other, the medium one is finally used in the following simulation. Three levels of grids (71,000, 103,000 and 138,000) are used to validate the grid independence as illustrated in Figure 5. Since the three grid levels are quite close with each other, the medium one is finally used in the following simulation.

Experimental Setup
The experimental setup is presented in Figure 6, in the setup, Evaporator 1 is simulated as an air conditioner, and its evaporating temperature is set as 7 °C; Evaporator 2 is simulated as a refrigerator, and its evaporating temperature is arranged as −5 °C; whist, Evaporator 3 is simulated as a freezer, and its evaporating temperature is specified as −30 °C. The ambient temperature is valued at 36 °C. Based on the thermodynamic calculation, the individual required cooling loads for three evaporators are 1566.2 W, 609.4 W and 997.2 W, respectively. Other details can refer to our previous study [32]. The range and accuracy of sensors are presented in Table 2.

Experimental Setup
The experimental setup is presented in Figure 6, in the setup, Evaporator 1 is simulated as an air conditioner, and its evaporating temperature is set as 7 • C; Evaporator 2 is simulated as a refrigerator, and its evaporating temperature is arranged as −5 • C; whist, Evaporator 3 is simulated as a freezer, and its evaporating temperature is specified as −30 • C. The ambient temperature is valued at 36 • C. Based on the thermodynamic calculation, the individual required cooling loads for three evaporators are 1566.2 W, 609.4 W and 997.2 W, respectively. Other details can refer to our previous study [32]. The range and accuracy of sensors are presented in Table 2.
Three levels of grids (71,000, 103,000 and 138,000) are used to validate the grid independence as illustrated in Figure 5. Since the three grid levels are quite close with each other, the medium one is finally used in the following simulation.

Experimental Setup
The experimental setup is presented in Figure 6, in the setup, Evaporator 1 is simulated as an air conditioner, and its evaporating temperature is set as 7 °C; Evaporator 2 is simulated as a refrigerator, and its evaporating temperature is arranged as −5 °C; whist, Evaporator 3 is simulated as a freezer, and its evaporating temperature is specified as −30 °C. The ambient temperature is valued at 36 °C. Based on the thermodynamic calculation, the individual required cooling loads for three evaporators are 1566.2 W, 609.4 W and 997.2 W, respectively. Other details can refer to our previous study [32]. The range and accuracy of sensors are presented in Table 2.

Sensors
Position Unit Range Accuracy Volume flow rate

Validation of the CFD Model
Fifteen CFD simulation results, as illustrated in Table 3, were validated by the experimental data. The average and maximum discrepancy of ER 1 were 7.2% and 11.9%, and those for ER 2 are 5.9% and 10.6%, respectively; therefore, the models can be used in the following simulations. Table 3. Operating conditions of the TSE for CFD model validation.  Figure 7 shows the results of ER 1 and ER 2 with the second-stage ejector nozzle length (LC 2 ) when PNTD 2 is 4.1 mm. When LC 2 changes from 5 mm to 40 mm, ER 1 first rises from 0.595 to the maximum value of 0.646 (LC 2 = 25 mm), and then slowly decreases to 0.636 (LC 2 = 40 mm). ER 2 rises slowly from 2.096 to 2.153 (LC 2 = 20 mm) and then decreases rapidly until it reaches a minimum of 2.022. When LC 2 changes from 5 mm to 40 mm, the maximum deviations of ER 1 and ER 2 reach 8.571% and 6.479%, respectively, which means Entropy 2022, 24, 1847 7 of 17 that the change of LC 2 has an impact on the performance of both the first stage and the second-stage, but the impact on the first-stage is more obvious. At the same time, it can be seen that when LC 2 = 15-25 mm, the values of ER 1 and ER 2 are relatively large.  Figure 8 displays the results of ER1 and ER2 with the second-stage ejector nozzle angle (AC2) when PNTD2 is 4.1 mm. With the increase of AC2, ER1 first increases and then decreases, and the maximum value of ER1 is 0.653 (AC2 = 16°). As AC2 increases from 6° to 10°, ER2 increases from 2.146 to 2.171, and ER2 gradually decreases to 2.014 as AC2 continues to rise to 22°. Compared with the initial values of ER1 and ER2 (0.632 and 2.153), the maximum values of ER1 and ER2 increase by 0.021 and 0.019, respectively, and the maximum deviations of ER1 and ER2 are 11.054% and 7.795%, respectively. This means that AC2 has an impact on the performance of both stages, but ER1 is more sensitive to the changes of AC2. Moreover, it can be seen that AC2 has a greater impact on the two-stage performance than LC2. In addition, the AC2 range for which ER1 achieves large values is 10°-22°, while the AC2 range for which ER2 achieves large values is 6°-14°, indicating that the AC2 range for which both ER1 and ER2 obtain large values is 10°-14°.   Figure 7. Changes of ER 1 and ER 2 with LC 2 when PNTD 2 is 4.1 mm. Figure 8 displays the results of ER 1 and ER 2 with the second-stage ejector nozzle angle (AC 2 ) when PNTD 2 is 4.1 mm. With the increase of AC 2 , ER 1 first increases and then decreases, and the maximum value of ER 1 is 0.653 (AC 2 = 16 • ). As AC 2 increases from 6 • to 10 • , ER 2 increases from 2.146 to 2.171, and ER 2 gradually decreases to 2.014 as AC 2 continues to rise to 22 • . Compared with the initial values of ER 1 and ER 2 (0.632 and 2.153), the maximum values of ER 1 and ER 2 increase by 0.021 and 0.019, respectively, and the maximum deviations of ER 1 and ER 2 are 11.054% and 7.795%, respectively. This means that AC 2 has an impact on the performance of both stages, but ER 1 is more sensitive to the changes of AC 2 . Moreover, it can be seen that AC 2 has a greater impact on the two-stage performance than LC 2 . In addition, the AC 2 range for which ER 1 achieves large values is 10 • -22 • , while the AC 2 range for which ER 2 achieves large values is 6 • -14 • , indicating that the AC 2 range for which both ER 1 and ER 2 obtain large values is 10 • -14 • .  Figure 8 displays the results of ER1 and ER2 with the second-stage ejector nozzle angle (AC2) when PNTD2 is 4.1 mm. With the increase of AC2, ER1 first increases and then decreases, and the maximum value of ER1 is 0.653 (AC2 = 16°). As AC2 increases from 6° to 10°, ER2 increases from 2.146 to 2.171, and ER2 gradually decreases to 2.014 as AC2 continues to rise to 22°. Compared with the initial values of ER1 and ER2 (0.632 and 2.153), the maximum values of ER1 and ER2 increase by 0.021 and 0.019, respectively, and the maximum deviations of ER1 and ER2 are 11.054% and 7.795%, respectively. This means that AC2 has an impact on the performance of both stages, but ER1 is more sensitive to the changes of AC2. Moreover, it can be seen that AC2 has a greater impact on the two-stage performance than LC2. In addition, the AC2 range for which ER1 achieves large values is 10°-22°, while the AC2 range for which ER2 achieves large values is 6°-14°, indicating that the AC2 range for which both ER1 and ER2 obtain large values is 10°-14°.   Figure 9 reveals the results of ER 1 and ER 2 with LC 2 when PNTD 2 is 4.7 mm. It can be seen that when LC 2 changes from 5 mm to 45 mm, ER 1 first rises from 1.031 to the maximum value of 1.098 (LC 2 = 35 mm), and then rapidly decreases to 1.074 (LC 2 = 45 mm). However, ER 2 slowly increases to the maximum value of 1.861 (LC 2 = 15 mm), follows by a rapid decline in ER 2 until it decreases to the lowest value of 1.799. When LC 2 changes from 5 mm to 45 mm, the maximum deviations of ER 1 and ER 2 reach 6.499% and 3.446%, respectively, which reflects that the performance of the first-stage and the second-stage are affected by the change of LC 2 , but the first-stage can be affected more obviously. Furthermore, it can be seen that when LC 2 = 15-25 mm, the values of ER 1 and ER 2 are relatively large.

First-Stage Primary Flow First-Stage Secondary Flow Second-Stage Secondary Flow Outflow
Entropy 2022, 24, 1847 8 of 18 a rapid decline in ER2 until it decreases to the lowest value of 1.799. When LC2 changes from 5 mm to 45 mm, the maximum deviations of ER1 and ER2 reach 6.499% and 3.446%, respectively, which reflects that the performance of the first-stage and the second-stage are affected by the change of LC2, but the first-stage can be affected more obviously. Furthermore, it can be seen that when LC2 = 15-25 mm, the values of ER1 and ER2 are relatively large. Figure 9. Changes of ER1 and ER2 with LC2 when PNTD2 is 4.7 mm. Figure 10 indicates the results of ER1 and ER2 with AC2 when PNTD2 is 4.7 mm. As shown in the figure, with the increase of AC2, ER1 increases first and then decreases, and the highest value of ER1 is 1.091 (AC2 = 14°). With the increase of AC2 from 6° to 8°, ER2 increases from 1.849 to 1.856; and with the increase of AC2 to 22°, ER2 decreases to 1.811. The maximum deviations of ER1 and ER2 are 5.106% and 2.485%, respectively, which means that AC2 has an impact on the performance of both stages, but ER1 is more sensitive to the change of AC2. In addition, the range of AC2 where both ER1 and ER2 are at large values is still within 10°-14°.   Figure 9. Changes of ER 1 and ER 2 with LC 2 when PNTD 2 is 4.7 mm. Figure 10 indicates the results of ER 1 and ER 2 with AC 2 when PNTD 2 is 4.7 mm. As shown in the figure, with the increase of AC 2 , ER 1 increases first and then decreases, and the highest value of ER 1 is 1.091 (AC 2 = 14 • ). With the increase of AC 2 from 6 • to 8 • , ER 2 increases from 1.849 to 1.856; and with the increase of AC 2 to 22 • , ER 2 decreases to 1.811. The maximum deviations of ER 1 and ER 2 are 5.106% and 2.485%, respectively, which means that AC 2 has an impact on the performance of both stages, but ER 1 is more sensitive to the change of AC 2 . In addition, the range of AC 2 where both ER 1 and ER 2 are at large values is still within 10 • -14 • .
Entropy 2022, 24, 1847 8 of 18 a rapid decline in ER2 until it decreases to the lowest value of 1.799. When LC2 changes from 5 mm to 45 mm, the maximum deviations of ER1 and ER2 reach 6.499% and 3.446%, respectively, which reflects that the performance of the first-stage and the second-stage are affected by the change of LC2, but the first-stage can be affected more obviously. Furthermore, it can be seen that when LC2 = 15-25 mm, the values of ER1 and ER2 are relatively large. Figure 9. Changes of ER1 and ER2 with LC2 when PNTD2 is 4.7 mm. Figure 10 indicates the results of ER1 and ER2 with AC2 when PNTD2 is 4.7 mm. As shown in the figure, with the increase of AC2, ER1 increases first and then decreases, and the highest value of ER1 is 1.091 (AC2 = 14°). With the increase of AC2 from 6° to 8°, ER2 increases from 1.849 to 1.856; and with the increase of AC2 to 22°, ER2 decreases to 1.811. The maximum deviations of ER1 and ER2 are 5.106% and 2.485%, respectively, which means that AC2 has an impact on the performance of both stages, but ER1 is more sensitive to the change of AC2. In addition, the range of AC2 where both ER1 and ER2 are at large values is still within 10°-14°.   Figure 11 presents the results of ER 1 and ER 2 with LC 2 when PNTD 2 is 5.3 mm. As illustrated in the figure, when LC 2 changes from 5 mm to 40 mm, ER 1 increases from 1.502 to the maximum value 1.585. ER 2 decreases from 1.645 to a minimum of 1.598. Compared with the initial values of ER 1 and ER 2 (1.573 and 1.627), the maximum values of ER 1 and ER 2 increase by 0.012 and 0.018. In addition, when LC 2 changes from 5 mm to 40 mm, the maximum deviations of ER 1 and ER 2 are 5.526% and 2.941%, respectively, which means that the performance of both stages is affected by the change of LC 2 , and the performance of the first-stage is slightly more affected. In contrast, when LC 2 is 15-25 mm, the values of ER 1 and ER 2 are at large values. Figure 11 presents the results of ER1 and ER2 with LC2 when PNTD2 is 5.3 mm. As illustrated in the figure, when LC2 changes from 5 mm to 40 mm, ER1 increases from 1.502 to the maximum value 1.585. ER2 decreases from 1.645 to a minimum of 1.598. Compared with the initial values of ER1 and ER2 (1.573 and 1.627), the maximum values of ER1 and ER2 increase by 0.012 and 0.018. In addition, when LC2 changes from 5 mm to 40 mm, the maximum deviations of ER1 and ER2 are 5.526% and 2.941%, respectively, which means that the performance of both stages is affected by the change of LC2, and the performance of the first-stage is slightly more affected. In contrast, when LC2 is 15 Figure 11. Changes of ER1 and ER2 with LC2 when PNTD2 is 5.3 mm. Figure 12 shows the changes of ER1 and ER2 with AC2 when PNTD2 is 5.3 mm. It can be seen that with the increase of AC2, ER1 first increases and then decreases, and the maximum value of ER1 is 1.573 (AC2 = 14°). As AC2 increases from 6° to 10°, ER2 increases from 1.629 to 1.632, when AC2 continues to increase to 18°, ER2 gradually decreases to 1.621. The maximum deviations of ER1 and ER2 are 3.897% and 1.527%, respectively, which indicates that AC2 has an impact on the performance of both stages, but ER1 is slightly more sensitive to AC2. In addition, the range of AC2 is still within 10°-14° when both ER1 and ER2 are at large values.   Figure 12. Changes of ER1 and ER2 with AC2 when PNTD2 is 5.3 mm. Figure 11. Changes of ER 1 and ER 2 with LC 2 when PNTD 2 is 5.3 mm. Figure 12 shows the changes of ER 1 and ER 2 with AC 2 when PNTD 2 is 5.3 mm. It can be seen that with the increase of AC 2 , ER 1 first increases and then decreases, and the maximum value of ER 1 is 1.573 (AC 2 = 14 • ). As AC 2 increases from 6 • to 10 • , ER 2 increases from 1.629 to 1.632, when AC 2 continues to increase to 18 • , ER 2 gradually decreases to 1.621. The maximum deviations of ER 1 and ER 2 are 3.897% and 1.527%, respectively, which indicates that AC 2 has an impact on the performance of both stages, but ER 1 is slightly more sensitive to AC 2 . In addition, the range of AC 2 is still within 10 • -14 • when both ER 1 and ER 2 are at large values. illustrated in the figure, when LC2 changes from 5 mm to 40 mm, ER1 increases from 1.502 to the maximum value 1.585. ER2 decreases from 1.645 to a minimum of 1.598. Compared with the initial values of ER1 and ER2 (1.573 and 1.627), the maximum values of ER1 and ER2 increase by 0.012 and 0.018. In addition, when LC2 changes from 5 mm to 40 mm, the maximum deviations of ER1 and ER2 are 5.526% and 2.941%, respectively, which means that the performance of both stages is affected by the change of LC2, and the performance of the first-stage is slightly more affected. In contrast, when LC2 is 15 Figure 11. Changes of ER1 and ER2 with LC2 when PNTD2 is 5.3 mm. Figure 12 shows the changes of ER1 and ER2 with AC2 when PNTD2 is 5.3 mm. It can be seen that with the increase of AC2, ER1 first increases and then decreases, and the maximum value of ER1 is 1.573 (AC2 = 14°). As AC2 increases from 6° to 10°, ER2 increases from 1.629 to 1.632, when AC2 continues to increase to 18°, ER2 gradually decreases to 1.621. The maximum deviations of ER1 and ER2 are 3.897% and 1.527%, respectively, which indicates that AC2 has an impact on the performance of both stages, but ER1 is slightly more sensitive to AC2. In addition, the range of AC2 is still within 10°-14° when both ER1 and ER2 are at large values. 6 Figure 12. Changes of ER1 and ER2 with AC2 when PNTD2 is 5.3 mm. Figure 12. Changes of ER 1 and ER 2 with AC 2 when PNTD 2 is 5.3 mm.

mm
In conclusion, both LC 2 and AC 2 have certain effects on the performance of the TSE. With the increase of PNTD 2 , the influence of LC 2 and AC 2 on the performance of the two stages is gradually weakened. With a comprehensive consideration, LC 2 = 25 mm and AC 2 = 12 • are selected as the optimized geometries of the second-stage ejector converging nozzle to carry out the following study on the influence of variable back pressure on the ER of TSE.

Influence of Variable Back Pressure on ERs of the TSE
Boundary conditions of the TSE except the back pressure are kept constant, and the change of the back pressure of the TSE is expressed as the percentage of pressure lift, namely the change of PLR (the ratio of the back pressure to the secondary inlet pressure of the second-stage ejector). The initial PLR is 108%. When PNTD 2 is 4.1 mm and PLR changes in the range of 102-118%, the influence of the changed PLR on ER 1 and ER 2 is shown in Figure 13. It can be seen that when PLR increases from 102% to 118%, ER 1 gradually decreases, but its maximum value and minimum value are 0.644 and 0.643, respectively. Therefore, ER 1 almost does not change. ER 2 decreases almost linearly from 2.442 to 1.105, and the maximum deviation of ER 2 is 121.0%. Therefore, changes in PLR have a significant impact on ER 2 . nozzle to carry out the following study on the influence of variable back pressure on the ER of TSE.

Influence of Variable Back Pressure on ERs of the TSE
Boundary conditions of the TSE except the back pressure are kept constant, and the change of the back pressure of the TSE is expressed as the percentage of pressure lift, namely the change of PLR (the ratio of the back pressure to the secondary inlet pressure of the second-stage ejector). The initial PLR is 108%. When PNTD2 is 4.1 mm and PLR changes in the range of 102-118%, the influence of the changed PLR on ER1 and ER2 is shown in Figure 13. It can be seen that when PLR increases from 102% to 118%, ER1 gradually decreases, but its maximum value and minimum value are 0.644 and 0.643, respectively. Therefore, ER1 almost does not change. ER2 decreases almost linearly from 2.442 to 1.105, and the maximum deviation of ER2 is 121.0%. Therefore, changes in PLR have a significant impact on ER2. When PNTD2 = 4.7 mm, the influence of changing PLR on ER1 and ER2 is shown in Figure 14. ER1 is almost unaffected by PLR, with the highest and lowest values of 1.092 and 1.090, respectively. In addition, when PLR increases from 102% to 104%, ER2 remains at 1.938; when PLR increases from 104% to 118%, ER2 decreases to the minimum value of 1.253, and the maximum deviation of ER2 is 54.7%. Therefore, the change of PLR has a relatively obvious impact on ER2. Moreover, it can be seen that compared with the ER value of PNTD2 = 4.1 mm, ER1 increases a lot, while ER2 decreases a little.  Figure 13. Effect of varied PLR on ER 1 and ER 2 (PNTD 2 = 4.1 mm).
When PNTD 2 = 4.7 mm, the influence of changing PLR on ER 1 and ER 2 is shown in Figure 14. ER 1 is almost unaffected by PLR, with the highest and lowest values of 1.092 and 1.090, respectively. In addition, when PLR increases from 102% to 104%, ER 2 remains at 1.938; when PLR increases from 104% to 118%, ER 2 decreases to the minimum value of 1.253, and the maximum deviation of ER 2 is 54.7%. Therefore, the change of PLR has a relatively obvious impact on ER 2 . Moreover, it can be seen that compared with the ER value of PNTD 2 = 4.1 mm, ER 1 increases a lot, while ER 2 decreases a little. The effect of the changing PLR on ER1 and ER2 at PNTD2 = 5.3 mm is shown in Figure  15. With the increase of PLR, ER1 is still almost unaffected. However, compared with PNTD2 = 4.7 mm, ER1 increases largely. The maximum value of ER2 is lower than that of PNTD2 = 4.7 mm. When PLR increases from 102% to 106%, ER2 remains at 1.654 and it still shows a downward trend while PLR increases from 106% to 118%, and its maximum and minimum values are 1.654 and 1.213, respectively. The maximum deviation of ER2 is 36.4%. It is noted that at PNTD2 = 5.3 mm, the PLR affects ER2 to a smaller extent than at PNTD2 = 4.7 mm.
The effect of the changing PLR on ER 1 and ER 2 at PNTD 2 = 5.3 mm is shown in Figure 15. With the increase of PLR, ER 1 is still almost unaffected. However, compared with PNTD 2 = 4.7 mm, ER 1 increases largely. The maximum value of ER 2 is lower than that of PNTD 2 = 4.7 mm. When PLR increases from 102% to 106%, ER 2 remains at 1.654 and it still shows a downward trend while PLR increases from 106% to 118%, and its maximum and minimum values are 1.654 and 1.213, respectively. The maximum deviation of ER 2 is 36.4%. It is noted that at PNTD 2 = 5.3 mm, the PLR affects ER 2 to a smaller extent than at PNTD 2 = 4.7 mm. The effect of the changing PLR on ER1 and ER2 at PNTD2 = 5.3 mm is shown in Figure  15. With the increase of PLR, ER1 is still almost unaffected. However, compared with PNTD2 = 4.7 mm, ER1 increases largely. The maximum value of ER2 is lower than that of PNTD2 = 4.7 mm. When PLR increases from 102% to 106%, ER2 remains at 1.654 and it still shows a downward trend while PLR increases from 106% to 118%, and its maximum and minimum values are 1.654 and 1.213, respectively. The maximum deviation of ER2 is 36.4%. It is noted that at PNTD2 = 5.3 mm, the PLR affects ER2 to a smaller extent than at PNTD2 = 4.7 mm. In summary, the change of PLR basically has no effect on the performance of the first stage and a significant effect on the performance of the second stage. Furthermore, the effect of PLR on the second-stage performance gradually diminishes with increasing PNTD2. It can also be seen that when PNTD2 = 4.1 mm, the ejector with a PLR of 102% is already in subcritical mode. When PNTD2 = 4.7 mm and PLR is 102-104%, the second-stage ejector is in critical mode, and when PLR is greater than 104%, it is in subcritical mode. When PNTD2 = 5.3 mm, the PLR is in the range of 102-106%, and the ejector is in critical mode. Therefore, the critical back pressure increases with the increase of 102% 104% 106% 108% 110% 112% 114% 116% 118%  Figure 15. Effect of varied PLR on ER 1 and ER 2 (PNTD 2 = 5.3 mm).
In summary, the change of PLR basically has no effect on the performance of the first stage and a significant effect on the performance of the second stage. Furthermore, the effect of PLR on the second-stage performance gradually diminishes with increasing PNTD 2 . It can also be seen that when PNTD 2 = 4.1 mm, the ejector with a PLR of 102% is already in subcritical mode. When PNTD 2 = 4.7 mm and PLR is 102-104%, the second-stage ejector is in critical mode, and when PLR is greater than 104%, it is in subcritical mode. When PNTD 2 = 5.3 mm, the PLR is in the range of 102-106%, and the ejector is in critical mode. Therefore, the critical back pressure increases with the increase of PNTD 2 . The reason for this phenomenon is that, when the PNTD 2 increases, which means the area ratio of the secondary stage reduces, normally the entrainment ratio increases with the area ratio when the back pressure keeps unchanged; as a result, the critical pressure increases with the increase of PNTD 2 . For different PNTD 2 , the next study will be carried out for the optimization of the key geometries of the second-stage ejector under different PLR, such as AR 2 and NXP 2 , to identify the influence of back pressure and PNTD 2 on the best AR 2 and NXP 2 .

Optimized AR 2
The effect of AR 2 at PNTD 2 = 4.1 mm on ER 2 at different PLR is shown in Figure 16. It can be seen that the change trend of ER 2 affected by AR 2 under the three PLR is consistent, that is, ER 2 first increases, and then decreases as AR 2 increases. At PLR = 102%, ER 2 reaches its maximum value of 3.354 at AR 2 = 23.5, so the best value for AR 2 is 23.5, which is 13.0 times more than the optimal value of AR 2 at PLR = 108%, and ER 2 increases by 37.3% compared to the optimal value of 2.442 at PLR = 108%. When PLR is 104%, ER 2 reaches a maximum value of 2.682 at AR 2 = 17.5, then the optimal value of AR 2 is 17.5, which is 7.0 times greater than the optimal value of AR 2 at PLR = 108%; the optimal value of ER 2 increases by 13.1% compared to the optimal value of 2.372 for ER 2 at PLR = 108%. At PLR = 106%, the optimal value of 2.480 for ER 2 is obtained at AR 2 = 14.5, so that the optimal value for AR 2 is 14.5, which is 4.0 times greater than the optimal value for AR 2 at PLR = 108%, and the optimal ER 2 increases by 9.4% over the optimal value for ER 2 (2.266) at PLR = 108%. In summary, with the PLR increasing from 102% to 108%, the maximum ER 2 decreases from 3.354 to 2.266, and the corresponding optimal AR 2 decreases from 23.5 to 10.5.
At PLR = 106%, the optimal value of 2.480 for ER2 is obtained at AR2 = 14.5, so that the optimal value for AR2 is 14.5, which is 4.0 times greater than the optimal value for AR2 at PLR = 108%, and the optimal ER2 increases by 9.4% over the optimal value for ER2 (2.266) at PLR = 108%. In summary, with the PLR increasing from 102% to 108%, the maximum ER2 decreases from 3.354 to 2.266, and the corresponding optimal AR2 decreases from 23.5 to 10.5.  Figure 17 shows the effect of AR2 at PNTD2 = 4.7 mm on ER2 under the three PLR. When PLR is 102%, ER2 increases first and then decreases with the change of AR2, and its maximum value of 3.022 is obtained at AR2 = 20.1, which is 55.9% higher than the optimal value of ER2 (1.938) at PLR of 108%; and the optimal AR2 is 12.0 times larger than that at PLR of 108%. With a PLR of 104%, when AR2 changes, the optimal value for AR2 is 16.1, and its corresponding maximum ER2 of 2.598, and ER2 increases by 0.662 over the optimal value of ER2 (1.936) at a PLR of 108%. When the PLR is 106%, and when AR2 changes, ER2 rises first and then decreases; its optimal value of 2.283 is obtained at AR2 = 13.1; and the optimal value of ER2 is increased by 0.368 compared to the optimal value of ER2 of 1.915  Figure 17 shows the effect of AR 2 at PNTD 2 = 4.7 mm on ER 2 under the three PLR. When PLR is 102%, ER 2 increases first and then decreases with the change of AR 2 , and its maximum value of 3.022 is obtained at AR 2 = 20.1, which is 55.9% higher than the optimal value of ER 2 (1.938) at PLR of 108%; and the optimal AR 2 is 12.0 times larger than that at PLR of 108%. With a PLR of 104%, when AR 2 changes, the optimal value for AR 2 is 16.1, and its corresponding maximum ER 2 of 2.598, and ER 2 increases by 0.662 over the optimal value of ER 2 (1.936) at a PLR of 108%. When the PLR is 106%, and when AR 2 changes, ER 2 rises first and then decreases; its optimal value of 2.283 is obtained at AR 2 = 13.1; and the optimal value of ER 2 is increased by 0.368 compared to the optimal value of ER 2 of 1.915 (PLR of 108%). Similar to PNTD 2 = 4.1 mm, the maximum value of ER 2 decreases and the corresponding optimal AR 2 decreases as the PLR increases from 102% to 108%. (PLR of 108%). Similar to PNTD2 = 4.1 mm, the maximum value of ER2 decreases and the corresponding optimal AR2 decreases as the PLR increases from 102% to 108%.  Figure 18 shows the change of ER2 at PNTD2 = 5.3 mm with AR2 under the different PLR conditions. At PLR = 102%, as AR2 changes from 2.9 to 19.9, ER2 increases from 0.336 to a maximum of 2.663 at AR2 = 16.9, thus, the best value for AR2 is 16.9. Furthermore, the optimal value of ER2 is increased by 1.008 when compared to the optimal ER2 (1.655) at PLR = 108%. At PLR = 104%, ER2 achieves a maximum value of 2.299 at AR2 = 12.9. Therefore, the optimal AR2 is 12.9. In addition, the optimal value of ER2 is increased by 0.645 when compared to the best ER2 (1.654) at PLR = 108%. When PLR = 106%, ER2 shows a trend of increasing first and then decreasing with the increase of AR2. Its maximum value 2.050 is obtained at AR2 = 10.9, hence, the optimum of AR2 is 10.9. Compared  Figure 18 shows the change of ER 2 at PNTD 2 = 5.3 mm with AR 2 under the different PLR conditions. At PLR = 102%, as AR 2 changes from 2.9 to 19.9, ER 2 increases from 0.336 to a maximum of 2.663 at AR 2 = 16.9, thus, the best value for AR 2 is 16.9. Furthermore, the optimal value of ER 2 is increased by 1.008 when compared to the optimal ER 2 (1.655) at PLR = 108%. At PLR = 104%, ER 2 achieves a maximum value of 2.299 at AR 2 = 12.9. Therefore, the optimal AR 2 is 12.9. In addition, the optimal value of ER 2 is increased by 0.645 when compared to the best ER 2 (1.654) at PLR = 108%. When PLR = 106%, ER 2 shows a trend of increasing first and then decreasing with the increase of AR 2 . Its maximum value 2.050 is obtained at AR 2 = 10.9, hence, the optimum of AR 2 is 10.9. Compared with the optimal ER 2 (1.653) when PLR is 108%, the maximum ER 2 increases by 0.397. It can be seen that when PNTD 2 is 5.3 mm and PLR changes from 102% to 108%, the maximum value of ER 2 and the corresponding optimal AR 2 also show a decreasing trend. This phenomenon can be probably explained as follows: when the PLR increases, the pressure difference between the outlet and the secondary flow inlet pressure increases, in order to maintain the increased pressure lift; thus, the ER 2 usually drops and the suitable AR 2 reduces accordingly.  Figure 19 shows the influence of NXP2 on ER2 under different PLR when PNTD2 is 4.1 mm. With given PLR = 102%, as NXP2 increases from 18 mm to 30 mm, the change trend of ER2 rises first and then decreases. Its maximum value of 2.486 is obtained when NXP2 = 26 mm, and the maximum deviation of ER2 is 1.944%. With given PLR = 104%, as NXP2 increases from 18 mm to 26 mm, ER2 increases from 2.341 to 2.382 and then decreases to 2.369, so the maximum deviation of ER2 is 1.756%. With given PLR = 106%, ER2 also shows a trend of first increasing and then decreasing, and its peak value of 2.269 is obtained at NXP2 = 26 mm, and the maximum deviation of ER2 is 1.584%. Therefore, it can be seen that when PNTD2 = 4.1 mm, the optimal value of NXP2 does not change with the change of PLR, and thus the influence of NXP2 on ER2 is far less than that of AR2.   Figure 19 shows the influence of NXP 2 on ER 2 under different PLR when PNTD 2 is 4.1 mm. With given PLR = 102%, as NXP 2 increases from 18 mm to 30 mm, the change trend of ER 2 rises first and then decreases. Its maximum value of 2.486 is obtained when NXP 2 = 26 mm, and the maximum deviation of ER 2 is 1.944%. With given PLR = 104%, as NXP 2 increases from 18 mm to 26 mm, ER 2 increases from 2.341 to 2.382 and then decreases to 2.369, so the maximum deviation of ER 2 is 1.756%. With given PLR = 106%, ER 2 also shows a trend of first increasing and then decreasing, and its peak value of 2.269 is obtained at NXP 2 = 26 mm, and the maximum deviation of ER 2 is 1.584%. Therefore, it can be seen that when PNTD 2 = 4.1 mm, the optimal value of NXP 2 does not change with the change of PLR, and thus the influence of NXP 2 on ER 2 is far less than that of AR 2 . Figure 20 demonstrates the influence of NXP 2 on ER 2 under different PLR when PNTD 2 is 4.7 mm. When PLR is 102%, as NXP 2 changes from 18 mm to 24 mm, ER 2 rises from 1.932 to 1.950, and then decreases to 1.929 when NXP 2 is 30 mm, and the maximum deviation of ER 2 is 1.057%. When PLR is 104%, the change trend of ER 2 also increases first and then decreases, and its peak value of 1.936 appears at the value of 24 mm of NXP 2 , while the maximum deviation of ER 2 is 0.868%. When PLR is 106%, the variation trend of ER 2 is similar to the previous two, and its maximum value of 1.921 is obtained at NXP 2 = 24 mm, and the maximum deviation of ER 2 is 0.744%. Therefore, when PNTD 2 is 4.7 mm, the optimal value of NXP 2 is 24 mm, which does not change with the change of PLR.

Optimized NXP2
NXP2 increases from 18 mm to 26 mm, ER2 increases from 2.341 to 2.382 and then decreases to 2.369, so the maximum deviation of ER2 is 1.756%. With given PLR = 106%, ER2 also shows a trend of first increasing and then decreasing, and its peak value of 2.269 is obtained at NXP2 = 26 mm, and the maximum deviation of ER2 is 1.584%. Therefore, it can be seen that when PNTD2 = 4.1 mm, the optimal value of NXP2 does not change with the change of PLR, and thus the influence of NXP2 on ER2 is far less than that of AR2.  and then decreases, and its peak value of 1.936 appears at the value of 24 mm of NXP2, while the maximum deviation of ER2 is 0.868%. When PLR is 106%, the variation trend of ER2 is similar to the previous two, and its maximum value of 1.921 is obtained at NXP2 = 24 mm, and the maximum deviation of ER2 is 0.744%. Therefore, when PNTD2 is 4.7 mm, the optimal value of NXP2 is 24 mm, which does not change with the change of PLR.  Figure 21 shows the influence of NXP2 on ER2 under different PLR when PNTD2 is 5.3 mm. The changing trend of ER2 is the same under the three PLR conditions, that is, ER2 increases first and then decreases, and all the maximum value of ER2 appear at NXP2 = 22 mm. Therefore, when PNTD2 is 5.3 mm, the best value of NXP2 is 22 mm, that is, it is not affected by the change of PLR.

Conclusions
In this paper, the CFD simulation method was first used to optimize the nozzle 18 Figure 21 shows the influence of NXP 2 on ER 2 under different PLR when PNTD 2 is 5.3 mm. The changing trend of ER 2 is the same under the three PLR conditions, that is, ER 2 increases first and then decreases, and all the maximum value of ER 2 appear at NXP 2 = 22 mm. Therefore, when PNTD 2 is 5.3 mm, the best value of NXP 2 is 22 mm, that is, it is not affected by the change of PLR. and then decreases, and its peak value of 1.936 appears at the value of 24 mm of NXP2, while the maximum deviation of ER2 is 0.868%. When PLR is 106%, the variation trend of ER2 is similar to the previous two, and its maximum value of 1.921 is obtained at NXP2 = 24 mm, and the maximum deviation of ER2 is 0.744%. Therefore, when PNTD2 is 4.7 mm, the optimal value of NXP2 is 24 mm, which does not change with the change of PLR.  Figure 21 shows the influence of NXP2 on ER2 under different PLR when PNTD2 is 5.3 mm. The changing trend of ER2 is the same under the three PLR conditions, that is, ER2 increases first and then decreases, and all the maximum value of ER2 appear at NXP2 = 22 mm. Therefore, when PNTD2 is 5.3 mm, the best value of NXP2 is 22 mm, that is, it is not affected by the change of PLR.

Conclusions
In this paper, the CFD simulation method was first used to optimize the nozzle geometry of the second-stage ejector under different PNTD2 with the operating condi-

Conclusions
In this paper, the CFD simulation method was first used to optimize the nozzle geometry of the second-stage ejector under different PNTD 2 with the operating conditions given in Table 1. Then, the effect of variable back pressure on the ejector performance was studied. Finally, three PLRs that place the ejector in critical or near critical mode were selected to study the influence of AR 2 and NXP 2 on the performance of the second-stage ejector. The main findings obtained are as follows: (1) When LC 2 and AC 2 change, the maximum values of ER 1 and ER 2 do not appear at the same length or angle. LC 2 = 25 mm and AC 2 = 12 • are the relative optimal combination of values for the second-stage ejector nozzle; (2) With the increase of PNTD 2 , ER 1 and ER 2 decrease with the increase of LC 2 and AC 2 , and the PLR range of the ejector in the critical mode gradually increases; (3) The change of PLR has no effect on the performance of the first-stage ejector, but has a significant effect on the performance of the second-stage ejector; with the increase of PNTD 2 , the influence of PLR on the performance of the second stage is gradually weakened; (4) When PNTD 2 is 4.1 mm, the optimal value of AR 2 decreases from 23.5 to 14.5 with the increase of PLR, and the peak value of ER 2 decreases from 3.354 to 2.480. When PNTD 2 is 4.7 mm, the optimal value of AR 2 decreases from 20.1 to 13.1 with the increase of PLR, and the maximum value of ER 2 decreases from 3.022 to 2.383. When PNTD 2 is 5.3 mm, with the change of PLR from 102% to 106%, the optimal value of AR 2 is from 16.9 to 10.9, and the peak value of ER 2 is reduced from 3.354 to 2.382; (5) The optimal value of NXP 2 is not affected by the change of PLR. When PNTD 2 is 4.1 mm, 4.7 mm and 5.3 mm, the corresponding optimal value of NXP 2 is 26 mm, 24 mm and 22 mm, respectively.