Theoretical model of transcritical CO2 ejector with non-equilibrium phase change correlation

Accepted Manuscript

Theoretical model of transcritical CO2 ejector with non-equilibrium phase change correlation Yinhai Zhu , Peixue Jiang PII: DOI: Reference:

S0140-7007(17)30435-8 10.1016/j.ijrefrig.2017.10.033 JIJR 3801

To appear in:

International Journal of Refrigeration

Received date: Revised date: Accepted date:

27 July 2017 25 October 2017 26 October 2017

Please cite this article as: Yinhai Zhu , Peixue Jiang , Theoretical model of transcritical CO2 ejector with non-equilibrium phase change correlation, International Journal of Refrigeration (2017), doi: 10.1016/j.ijrefrig.2017.10.033

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ACCEPTED MANUSCRIPT Highlights

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 A theoretical model of two-phase CO2 ejector was developed.  A non-equilibrium correlation in the energy-conservation equation was proposed.  The correlation needs to be adopted when xt > 0.65.  The Mach number in the CO2 ejector is lower than 1.5.  The “double choking” phenomenon will not happen in the CO2 ejector.

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ACCEPTED MANUSCRIPT

Theoretical model of transcritical CO2 ejector with non-equilibrium phase change correlation Yinhai Zhu*, Peixue Jiang

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Key Laboratory for Thermal Science and Power Engineering of Ministry of Education; Beijing Key Laboratory of CO2 Utilization and Reduction Technology,

Department of Thermal Engineering, Tsinghua University, Beijing 100084, China Corresponding author：[email protected]

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*

Abstract. A theoretical model of an ejector was proposed for a transcritical carbon dioxide ejector-expansion refrigeration system capable of predicting the mass flow rates of both primary and secondary flows. A non-equilibrium correlation in the energy-conservation equation was proposed

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and validated using 130 cases obtained from three different ejector configurations. The differences in the predicted primary mass flow rates between the cases with and without the correlation were

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insignificant when the liquid mass fraction at the nozzle throat xt was less than 0.65. However, the

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prediction error increases dramatically when the correlation was not used for xt > 0.65. The Mach number in the CO2 ejector is lower than 1.5 and the “double choking” phenomenon will not happen in

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the CO2 ejector. Finally, a correlation was fitted for the primary flow pressure at the nozzle throat,

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which can be used to replace iterative solving in the model calculation.

Keywords: Ejector; Carbon dioxide; Transcritical cycle; Model; Non-equilibrium

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1. Introduction Heat-pump water heaters using CO2 have a relatively high outlet-water temperature and a superior efficiency, because CO2 has a high-temperature glide during a cooling process (Austin and Sumathy, 2011; Ma et al., 2013; Zhang et al., 2015). The transcritical cycle is preferred when CO2 refrigeration

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systems are employed in warm climate areas because of its lower critical temperature. In a conventional transcritical CO2 refrigeration system, the vapor quality in the expansion valve is high, causing high throttle loss. Ejectors can be used to recover the expansion losses to increase the system efficiency, which has been verified theoretically and experimentally (Jeong et al., 2004; Li and Groll,

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2005; Deng et al., 2007; Sun and Ma 2011; Ahammed et al., 2014).

Elbel and Hrnjak (2008) conducted an experimental study wherein the ejector helped in improving the COP by up to 7% over a conventional system. Lucas and Koehler (2012) experimentally studied

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an ejector-expansion refrigeration cycle. They improved the COP of the ejector cycle to a maximum of 17%. Lee et al. demonstrated that the entrainment ratio of the ejector was a key factor in an

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ejector-expansion system (Lee et al., 2011; Lee et al., 2014). The results showed that the cooling capacity and COP of the ejector-expansion system were approximately in the ranges of 2–5% and

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6–9%, respectively, higher than those of a conventional system for entrainment ratios greater than

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0.76.

Many studies have been conducted on the transcritical CO2 ejector-expansion refrigeration system

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to understand its operating conditions and improve the COP (Elbel, 2011; Xu et al., 2012; Banasiak et al., 2015). Bai et al. (2016) conducted an exergy analysis to investigate an ejector-expansion transcritical CO2 refrigeration system. The results showed that the compressor exhibited the highest avoidable endogenous exergy destruction compared to other components. Two-stage transcritical CO2 heat-pump cycles are advantageous in obtaining dual-refrigeration temperatures and can help in reducing the throttling loss to a greater extent, and thus, enhance the cycle performance (Yari and Mahmoudi, 2011; Xing et al., 2014; Bai et al., 2015). To utilize the waste heat from the exhaust gas of a CO2 compressor to drive the ejector system, hybrid systems were suggested (Chen et al., 2013a; 3

ACCEPTED MANUSCRIPT Chen et al., 2017). Previous studies show that an internal heat exchanger might help in improving the COP of an ejector-expansion transcritical CO2 refrigeration cycle (Nakagawa et al., 2010; Zhang et al., 2013; Goodarzi et al., 2015). The ejector is the key component in the transcritical CO2 ejector-expansion refrigeration system. The flow and mixing process inside the ejector are critically important for the design and performance

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prediction. The study on ejectors has been conducted for decades using experimental, theoretical, empirical, and numerical methods, largely with working fluids such as air, steam, and CFC, HCFC and HFC refrigerants (Chen et al., 2013b; Chen et al., 2015; Besagni et al., 2016).

The ejector in a transcritical CO2 ejector-expansion refrigeration system has a two-phase flow

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inside and a smaller pressure-recovery ratio compared to ejectors using other working fluids. The flow field distribution is very important to explain the complex phenomena occurring in the CO2 ejectors. Numerical investigations were conducted on a CO2 ejector wherein oblique shock waves

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and temperature and pressure distributions were obtained by assuming a homogeneous equilibrium (Lucas et al., 2014; Bodys et al., 2017). Recently, Zhu et al. (2017a) experimentally visualized the

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flow fields in the suction and mixing chambers of a CO2 ejector using a flow-visualization technique. The shock of the primary flow while expanding in the suction chamber was relatively weak compared

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to that in a low-pressure working-fluid ejector (Zhu and Jiang, 2014a; Zhu and Jiang, 2014b).

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The performance of the ejector depends on its geometry and operating conditions. Nakagawa et al. (2011) studied the effect of the mixing length on the ejector in a transcritical CO2 cycle. The

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experimental results showed that an improper size of the mixing length lowered the COP by as much as 10% compared to that in similar conventional systems. Liu et al. (2012a) and Liu et al. (2012b) experimentally studied the effects of different ejector geometries and operating conditions on the performance of a CO2 ejector. They found that the ejector reaches its optimum performance at particular values of the throat diameter, mixing-section constant-area diameter, and diffuser diameter. Smolka et al. (2016) compared the performances of fixed and controllable-geometry ejectors equipped with convergent and convergent–divergent nozzles installed in a CO2 refrigeration system.

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ACCEPTED MANUSCRIPT The results showed that in most cases, the efficiency of the controllable-geometry ejector was 25% higher than that of the fixed-geometry ejector. More recently, Palacz et al. (2017) presented a shape-optimization method for the CO2 ejector considering six geometrical parameters including the primary nozzle and mixing section. Several theoretical and numerical models have been developed for the ejector with a transcritical

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CO2 working fluid. Chen et al. (2010) used “characteristic-curve equations” to evaluate the entrainment ratio of the ejector. The validation results showed that the calculated entrainment ratio was higher than that of the experimental data, because a single-phase flow was assumed. Banasiak and Hafner (2011) presented a one-dimensional mathematical model of the CO2 two-phase ejector

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with a delayed equilibrium model including a homogeneous nucleation theory, which was validated for a typical range of operating conditions. The absolute values of the relative errors between the experimental and simulation results for both the overall pressure lift and the critical primary mass

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flow rate were 2.66% and 1.84% on average, respectively. However, the performance of the model in predicting the mass flow rate of the secondary flow was not reported. Liu et al. (2012a) proposed a

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CO2 ejector model under the assumption of a one-dimensional one-component homogeneous equilibrium two-phase flow. They showed that the simulation model helped in predicting the mass

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flow rate of the primary flow within an error of 2% of the measured data. However, they did not

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compare the mass flow rates of the secondary flow between the calculated and experimental values. Smolka et al. (2013) developed a mathematical model for a compressible transonic two-phase flow of

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a real fluid, wherein the temperature-based energy equation was replaced with an enthalpy-based equation. The main advantage of the developed model is its numerical robustness compared to the Euler–Euler or mixture models. Zheng et al. (2016) developed dynamic models for the transcritical CO2 ejector-expansion refrigeration cycle, which can be used to analyze the dynamic responses of the system performance. The model was validated under two steady conditions and was in good agreement with the experimental data.

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ACCEPTED MANUSCRIPT Predicting the mass flow rate of the secondary flow is important and is more complicated than predicting the primary mass flow rate, particularly when the ejector is operated in the subcritical mode. In this study, a theoretical model of the transcritical CO2 ejector was proposed capable of predicting the mass flow rates of both the primary and secondary flows. We considered the non-equilibrium phase change effect in the flow passing through the nozzle. A non-equilibrium

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correlation for the energy conservation equation was obtained, which was validated using 130 cases obtained from three ejector configurations. The velocity and mass fraction inside the ejector were analyzed under various primary flow pressures, secondary flow pressures, back pressures, and primary flow inlet temperatures. Finally, we developed a correlation for the primary flow pressure at

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the nozzle throat, which can be directly used in modeling the ejector.

2. Experimental setup

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2.1 Experimental system

Two different ejectors were tested on a transcritical CO2 ejector-expansion refrigeration rig as

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shown in Fig. 1. The system comprises a compressor, a gas cooler, an evaporator, an ejector, a vapor–liquid separator, and an oil separator, which was developed by Zhu et al. (2017a).

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The compressor used was a DORIN CD180H with a rated displacement of 1.12 m3 h−1 at 380

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V/50 Hz. A 7.5 kW inverter was used to control the compressor speed. The gas cooler was a counter-flow plate heat exchanger with water as the coolant. The designed capacity of the gas cooler

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was 7.5 kW. The evaporator was a parallel-flow microchannel heat exchanger with a variable-speed fan to control the outflow superheat of the evaporator. The design cooling capacity of the evaporator was 5 kW. In addition, a fan heater with a maximum capacity of 3 kW was placed near the evaporator to keep the room temperature constant. The ejector performance is mainly dependent on the temperature and pressure of the primary flow, the pressure of the secondary flow, and the back pressure of the ejector. We adjusted the rotational speed of the compressor and mass flow rate of the coolant water to control the primary flow

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ACCEPTED MANUSCRIPT pressure of the ejector. The ejector‟s secondary and back flow pressures were controlled by valves V1 and V2. The pressure and temperature transducers with full-scale errors of 0.1% were mounted between the connections of each device as shown in Fig. 1. The mass flow rates of the primary and secondary flows were measured using two Coriolis-type mass flow meters within 0.1% accuracy. Seven PT100

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platinum resistance temperature transducers were used with errors of 0.2oC (-10–50oC), and one PT100 within the range of 0–120oC was used to measure the compressor discharge temperature. Detailed information regarding the accuracy and the range of the sensors used in the test rig is

Compressor

Tw2

Tw1

Water V4

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reported in another study (Zhu et al. (2017b).

1 V5

P4 T4

Cooler

P1 T1

Oil return line

V6

4

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P2 T2 M

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Oil separator

Ejector

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P7 T7 M

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2

V1

P3 T3

Separator

5 Evaporator

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V3

6

P6 T6

V2

Fan heater

Fig. 1 Schematic of the CO2 ejector-expansion refrigeration system

(P: Pressure transducer; T: Temperature transducer; M: Mass flow meter; V1–V4: needle valve; V5–V6: ball valve)

2.2 Test ejectors The primary flow to the ejector is CO2 leaving the gas-cooler outlet at a supercritical pressure. The pressure decreases significantly while flowing through the nozzle and becomes a two-phase flow.

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ACCEPTED MANUSCRIPT The two-phase flow reaches sonic velocity at the nozzle throat and entrains the secondary flow from the outlet of the evaporator. The two flows mix in the mixing chamber and the pressure increases at the end of the mixing chamber or the diffuser. The test ejector can be divided into two parts, as shown in Fig. 3, comprising a convergent nozzle, a suction chamber, a constant-area mixing chamber, and a diffuser. The converging angle of the

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suction chamber was 40°, and the diverging angle of the diffuser was 7°. As suggested by Zhu et al. (2009), the primary-nozzle exit positions (NXP) of the two ejectors were selected as 1.7Dm and

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1.9Dm. Table 1 lists the geometric parameters of the two tested ejectors.

t pm,sm m

Lm

NXP

DP

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θd

θm Dt

Dm

Secondary flow

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Primary flow

DS

Fig. 2 Schematic of the test ejector

1 Geometric parameters of the two tested ejectors EJ1

EJ2

Dt (mm)

0.8

1.0

Dm (mm)

2

2.6

NXP (mm) 3.4

8

B

5

DB

Lm (mm)

18

22

θd (°)

7

7

θm (°)

40

40

DP (mm)

8

8

DS (mm)

8

8

DB (mm)

7

7

s

3. Development of theoretical model

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In this study, the theoretical model of a two-phase flow CO2 ejector was developed based on the

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conservation equations of mass, momentum, and energy. Figure 3 shows the P–h diagram of the two-phase flow inside the CO2 ejector operating in a typical transcritical CO2 ejector-expansion refrigeration cycle.

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The supercritical pressure CO2, termed the primary flow, leaving the gas cooler flows into the ejector. The flow then expands into a subcritical two-phase region through the nozzle. The flow

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reaches sonic velocity at the nozzle throat under the typical operating conditions of the CO2

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ejector-expansion refrigeration system. For an ejector with a convergent nozzle, the flow at the nozzle exit is usually an under-expanded supersonic flow, which implies that the primary flow pressure at

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the nozzle exit Pt is higher than the surrounding pressure PS (Zhu and Jiang, 2014b). The supersonic flow entrains the secondary flow into the ejector. The two flows mix in the mixing chamber and the

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static pressure of the mixed flow increases as the kinetic energy is converted into static pressure. In the present study, we assume that the mixing process of two flows completes at the constant-area mixing chamber entrance. Thus, the mixed flow has a uniform pressure inside the constant-area mixing chamber. The following assumptions were made to develop the model: 1) The flow inside the ejector is steady and one-dimensional. 2) The ejector walls are adiabatic. 9

ACCEPTED MANUSCRIPT 3) The inlet-flow velocity is neglected in the energy-conservation equation. 4) The isentropic relations are used for the flow expect in the mixing process. 5) The mixing process occurs at a constant pressure. 6) The two-phase flow in the suction chamber, mixing chamber and diffuser is homogeneous

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and is in thermodynamic equilibrium.

Primary flow Secondary flow Mixed flow

t pm

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Pressure

P

B

m

S

sm

M

Enthalpy

Fig. 3 P-h diagram of a typical transcritical CO2 ejector. (pm: primary flow at the mixing chamber

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inlet, sm: secondary flow at the mixing chamber inlet, m: mixed flow at the mixing chamber outlet) 3.1 Primary flow from inlet to the mixing-chamber inlet

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Nakagawa et al. (2009) measured the temperature and pressure of supersonic two-phase CO2

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through converging–diverging nozzles. Their results showed that only a few of the obtained data could be predicted by the isentropic homogeneous equilibrium theory. Since the two-phase flow

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through the nozzle is very fast, the fluid may be in subcooled and there is a temperature difference between the liquid and vapor phases. A modified energy-conservation equation for the primary flow from the ejector inlet to the nozzle throat is expressed as follows.

1 hP  ht  Vt 2  QNE 2

(1)

where hP and ht are the enthalpies of the primary flow at the inlet and nozzle throat, respectively, and Vt is the flow velocity at the nozzle throat. QNE is used to correct the error caused by the thermodynamic equilibrium assumption in the energy-conservation equation. The modified 10

ACCEPTED MANUSCRIPT energy-conservation equation was not found in previous CO2-ejector models (Liu et al., 2012a; Chen et al., 2010; Banasiak and Hafner, 2011; Zheng et al., 2016). We found that QNE was related to the liquid mass fraction of the primary flow at the nozzle throat after a lot of data analysis. In addition, the relationship between them satisfies the exponential function. In this study, we used a simple exponential function for QNE, and its parameters were obtained using the Levenberg–Marquardt

QNE  8629e10(1 xt )

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method (Zhu et al., 2008) based on 130 experimental data from three different ejector configurations. (2)

where xt is the liquid mass fraction of the primary flow at the nozzle throat, xt  X ( Pt , ht ) . The

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correlation indicates that QNE is related to xt. A more detailed analysis is given in Section 4.

The primary flow through the nozzle is non-ideal process with the frictional loss. The effect of frictional loss is taken into account by using a nozzle isentropic coefficient, φn. Thus, the primary flow enthalpy at the nozzle throat, ht, can be calculated based on the primary flow inlet pressure, PP,

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enthalpy, hP as follows.

(3)

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ht  hP  n (hP  ht,s )

where ht,s  h( Pt , SP ) . According to ESDU (Pearson, 2005), the common values for φn are between

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0.9 and 0.95, which has been experimentally validated (Zhu et al., 2007; Cardemil and Colle, 2012;

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Zheng et al., 2016). Here, we suggest φn = 0.95 for modeling the CO2 ejector. Note that the primary flow velocity at the nozzle throat, Vt, becomes sonic under the typical

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operating conditions of the ejector in CO2 ejector-expansion refrigeration systems. Therefore, Vt depends only on the thermophysical properties of the primary flow at the nozzle throat. The speed of sound in a two-phase fluid is more complicated than that in a single-phase fluid. Here, we use the model suggested by Credemil and Colle (2012) to calculate the speed of sound in the two-phase CO2 fluid. The detailed derivation process can be found in another study (Lund and Flatten, 2010). 2       C p ,vC p ,l ( l   v )  a 2    v 2  l 2      C p ,v  C p ,l  v av l al  T  

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(4)

ACCEPTED MANUSCRIPT where T and ρ are the temperature and density of the two-phase flow, respectively,  v and  l are the saturated vapor and liquid densities at a given pressure of the two-phase flow, respectively, av and

al are the saturated vapor and liquid speed of sound at the pressure of the two-phase flow, respectively,  v and  l are the void fraction of the vapor and liquid phases in the two-phase flow,

T k  T  k      P s  k cp,k Cp,v   k  k cp,k k  v, l

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respectively, and

(5)

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where  is the thermal expansion coefficient, and cp is the specific heat at constant pressure. By assuming the nozzle-throat pressure Pt, we can determine the enthalpy ht using Eq. (3). Thus, the velocity of the primary flow at the nozzle throat Vt can be determined via Pt and ht using Eq. (4).

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The enthalpy ht can be calculated using Eq. (1). The outlet pressure Pt is updated until the two values of ht converge.

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After determining the velocity and pressure at the nozzle throat, the mass flow rate of the primary

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flow mP through the nozzle can be calculated as follows.

mP  At tVt

(6)

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The expansion process of the primary flow passing from the nozzle throat „t‟ to the mixing chamber inlet “pm” is assumed to be isentropic with the same isentropic efficiency as that of the

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nozzle. Although the pressure at the suction chamber will lower the inlet pressure of the secondary flow to make sure the secondary flow is entrained into the ejector, the difference between the two pressures is small. Therefore, the pressure of the primary flow at the mixing-chamber inlet is assumed equal to that at the secondary-flow inlet (Zhu et al., 2007).

hpm  ht  n (ht  hpm,s )

(7)

where hpm,s  h( PS , St ) . The velocity of the primary flow at the mixing chamber inlet Vpm is then determined using the energy-conservation equation as follows. 12

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hpm 

PS

pm

P 1 1  Vpm 2  ht  t  Vt 2 2 t 2

(8)

where hpm, ρpm, and Vpm are the enthalpy, density, and velocity of the primary flow at the mixing-chamber inlet, respectively. Thus, the flow area occupied by the primary flow at the inlet section Apm of the mixing chamber

Apm 

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can be calculated as follows.

mP pmVpm

Thus, we can obtain Asm as follows.

(10)

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Asm  Am  Apm

(9)

3.2 Mixing process

The primary and secondary flows begin to mix at the mixing chamber inlet section under constant pressure. To simplify the analysis, we assume that the two flows completely mix in the mixing

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chamber.

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The momentum-conservation equation for the mixing process can be expressed as follows. Vm  mP  mS   m  mPVpm  mSVsm 

(11)

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where φm is a coefficient accounting for the frictional loss in the mixing process.  m significantly

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affects the model calculation, which varies with different ejectors and operating conditions. The

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following correlation for  m was obtained after conducting the cases studies.

m  Cm  0.34e3.125m / m S

P

(12)

where Cm is a constant for a given ejector. Note that the larger the ejector efficiency (Elbel and Hrnjak, 2008), the bigger the coefficient φm. The ejector efficiency of EJ1 is in the range of 0.31-0.42 (Zhu et al., 2017b), while the ejector efficiency of the ejector tested in the study by Locus and Koehler (2012) is 0.07-0.22. Through error analysis, Cm is taken as 0.73 for the ejectors EJ1 and EJ2 and 0.58 for the ejector tested in the study by Locus and Koehler (2012). The energy-conservation equation for the mixing process is as follows. 13

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 mP  mS   hm  

1 2 1 1     Vm   mP  hpm  Vpm 2   mS  hsm  Vsm 2  2 2 2     

(13)

Here, the enthalpy of the secondary flow at the mixing-chamber inlet is equal to the enthalpy at the inlet of the secondary flow, i.e., hsm = hS. The mass flow rate of the secondary flow mS is determined using the density and velocity of the secondary flow at the mixing-chamber inlet as follows.

mS  Asm SVsm

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(14)

Because there are four unknown variables, Vsm, mS, Vm, and hm, in the three governing equations Eqs. (11), (13), and (14), an additional equation is required to solve these variables. 3.3 Flow in the diffuser

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In the diffuser, the flow decelerates and the pressure increases. The flow is assumed isentropic as follows.

hB  hm  d (hm  hB,s )

(15)

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where the isentropic efficiency φd is 0.9 in this study and hB,s  h( PB , Sm ) . Based on the energy conservation of the overall ejector, the inlet and outlet enthalpies can be

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determined by neglecting the inlet and outlet kinetic energies as follows. (16)

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 mP  mS  hB  mP hP  mShS

Finally, the five unknown variables Vsm, mS, Vm, hm, and hm can be solved using the five governing

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equations Eqs. (11) and (13)–(16).

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3.4 Solution procedure For a given ejector, the nozzle-throat diameter Dt and the mixing-chamber diameter Dm are

known. The pressures and temperatures at the inlet and outlet of the ejector were measured. The developed two-phase flow ejector model can be used to determine the primary mass flow rate mP and the secondary mass flow rate mS. Because the governing equations are nonlinear and closely coupled, an iterative procedure, given in the flow chart shown in Fig. 4, must be used to solve the set of equations.

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ACCEPTED MANUSCRIPT The ejector model was established based on general mass, momentum and energy governing equations. The speed of sound was calculated for a two-phase fluid. The non-equilibrium correlation QNE was used to correct the error caused by the thermodynamic equilibrium assumption in the energy-conservation equation. Therefore, the developed model and modeling method can be used to any two-phase flow ejector. However, the parameters in Eq. (2) and Eq. (12) such as 8629, -10, -0.34,

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-3.125 need to be changed when the fluid is not CO2. As mentioned before, Cm increases as the ejector efficiency increases, which indicates that Cm depends on the ejector configuration.

Initial Value of Pt

Equations (1)–(5)

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PP, TP

Pt,Vt,ht,xt

PS,TS,Dt,Dm

Equations (6)–(10)

mP,hpm,Vpm,Apm,Asm

Equations (11)–(16)

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PB Initial Value of Vsm

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end

Fig. 4 Calculation flow chart

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PT

mS, hm,Vm, Vsm

4. Results and discussions

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We tested the two ejectors EJ1 and EJ2 with 119 experimental cases to validate the developed

model. In addition, 11 more experimental cases obtained from the study by Lucas and Koehler (2012) were used to validate our model. The ejector model was validated under various conditions: the primary flow pressure of the ejector was varied from 8 to 10.3 MPa, the inlet temperature of the primary flow was in the range of 32–43 °C, and the ejector secondary flow pressure was changed from 2.6 to 4.3 MPa. The

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ACCEPTED MANUSCRIPT temperature at the evaporator outlet was kept at about 22 o. All experiment data of EJ2 are listed in Table 2. The experimental data of EJ1 are reported in Zhu et al. (2017b). Table 2 Details of all experiment data of EJ2 TS (oC) 21.9 22.0 22.2 23.7 24.0 24.1

PB (MPa) 4.62 4.67 4.71 4.76 4.81 4.83

TB (oC) 11.5 12.1 12.2 12.6 13.0 13.3

P1 (MPa) 8.54 8.59 8.56 8.54 8.48 8.45

T1 (oC) 77.5 79.6 78.1 79.0 79.3 79.6

P4 (MPa) 4.45 4.53 4.46 4.45 4.43 4.39

T4 (oC) 16.7 17.6 16.9 17.5 17.7 18.0

P6 (MPa) 4.52 4.55 4.49 4.33 4.30 4.27

T6 (oC) 10.7 10.9 10.6 9.2 9.0 9.0

mS (kg/h) 81.86 81.17 70.09 47.31 36.32 30.00

7.70 7.78 7.93 8.07 8.21 8.38 8.65 8.80 8.96

32.1 32.4 33.3 34.2 35.2 36.5 39.0 41.0 43.4

4.35 4.34 4.36 4.35 4.34 4.32 4.29 4.28 4.28

23.6 23.5 23.2 23.0 22.7 22.7 22.5 22.6 22.5

4.78 4.77 4.80 4.80 4.81 4.80 4.80 4.81 4.83

12.9 12.9 13.0 13.0 13.0 13.0 13.1 13.3 13.6

7.95 8.01 8.17 8.35 8.46 8.65 8.93 9.13 9.28

77.9 78.3 79.6 80.7 81.9 83.0 84.5 84.7 84.5

4.38 4.41 4.47 4.51 4.55 4.58 4.61 4.63 4.65

16.8 16.9 17.5 17.9 18.5 19.2 20.0 20.4 20.5

4.44 4.48 4.52 4.55 4.56 4.57 4.58 4.59 4.61

10.4 10.5 10.9 11.1 11.2 11.3 11.4 11.5 11.6

47.18 98.35 53.82 100.27 60.02 100.09 66.04 99.67 69.33 99.19 74.11 98.26 78.27 95.31 81.13 95.01 83.01 92.91

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PS (MPa) 4.15 4.21 4.22 4.20 4.24 4.25

mp (kg/h) 98.19 98.64 98.41 97.31 95.42 94.69

M

Data in Fig.15

TP (oC) 35.5 35.6 35.7 35.7 35.5 35.5

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Data in Fig.10

PP (MPa) 8.25 8.29 8.30 8.28 8.23 8.20

There are three efficiencies in the ejector model: the isentropic efficiency φn of the nozzle, the

ED

isentropic efficiency φd of the diffuser, and the mixing efficiency φm. In this study, φn = 0.95 and φd =

PT

0.9. Because the calculated mS is very sensitive to φm, φm is not a constant, which is a function of the entrainment ratio mS/mP given in Eq. (12).

CE

The following equations are used in the comparison studies.

AC

The relative error ER is as follows. ER 

Qc ,i  Qe,i Qe,i

 100%

(17)

The root-mean-square of the relative error ERMS is as follows. N

ERMS 

 (E i 1

N

R

)2  100%

(18)

where N is the number of cases, Qc is the value predicted using the model, and Qe is obtained from the experiment. 16

ACCEPTED MANUSCRIPT 4.1 Model validation Unlike in a single-phase ejector, the phase change occurs in a two-phase ejector. Non-equilibrium phenomenon will exist while the two-phase flows through the nozzle wherein the temperature of the liquid and vapor phases are different. The ejector model is simulated with and without the non-equilibrium correlation of QNE to explain the influence of the non-equilibrium phase

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change. The following comparison study is based on three ejector configurations and as many as 130 cases as mentioned previously.

Figure 5 shows the prediction errors of the primary mass flow rate with and without the

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non-equilibrium correlation. The results show that the difference in the prediction errors of the primary mass flow rates between the cases with and without the use of the non-equilibrium correlation of QNE is insignificant when the liquid mass fraction at the nozzle throat xt is less than 0.65. However, the prediction error increases dramatically when the correlation is not used for xt > 0.65,

otherwise, the error increases.

M

implying that the effect of the non-equilibrium phase change cannot be neglected for xt > 0.65;

ED

Based on the case studies, a fitting of QNE for the energy equation is obtained as a function of the

PT

liquid mass fraction at the nozzle throat xt as expressed in Eq. (2). Figure 6 shows QNE for different values of xt calculated using Eq. (2). QNE is close to zero for small xt and the maximum QNE is

CE

approximately 1.4 kJ kg-1 at xt = 0.82. Under this condition, the ratio of QNE to the enthalpy difference between hP and ht (QNE/(hP - ht) is approximately 0.23.

AC

Figure 7(a) shows the comparison between the predicted and measured primary flow mass flow

rates using the model with the non-equilibrium correlation of QNE. The results show that the values predicted using the simulation model are in good agreement with the measured data. The predicted mass flow rate is within ±3.5% of the measured data and the root-mean-square of the relative error ERMS is only 2.2%. The developed model is demonstrated to be capable of predicting the mass flow rate of the secondary flow mS, even when the ejector is operated in subcritical condition of the secondary flow. 17

ACCEPTED MANUSCRIPT Figure 7(b) shows the comparison between the predicted and measured mass flow rates of the secondary flow using the model with the non-equilibrium correlation of QNE. The results show that the errors in predicting mS using the model are largely less than ±15% among the 130 cases, where the root-mean-square of the relative error ERMS is 8.6%.

-4 -8 -12

EJ1 with non-equilibrium correlation without non-equilibrium correlation EJ2

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0

with non-equilibrium correlation without non-equilibrium correlation Experiment data in Lucas and Koehler (2012) with non-equilibrium correlation without non-equilibrium correlation

-16 0.1

0.3

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Relative error of mP (%)

4

0.5

0.7

0.9

Liquid mass fraction at nozzle throat

ED

1600 1200

PT

-1

Non-equilibrium correlation, QNE (J kg )

M

Fig. 5 Prediction errors of the primary mass flow rate with and without non-equilibrium correlation

800

CE AC

EJ1 EJ2 Experiment data from Lucas and Koehler (2012)

(10xt-10)

QED=8629e

400 0

-400 0.1

0.3

0.5

0.7

0.9

Liquid mass fraction at nozzle throat Fig. 6 QNE for different liquid mass fraction at nozzle throat

18

ACCEPTED MANUSCRIPT

EJ1 EJ2 -5% Experiment data from Lucas and Koehler (2012)

100

-1

80

+5%

60 40

CR IP T

Predicted mP (kg h )

120

20 0

0

20

40

60

80

100

-1

120

Measured mP (kg h )

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(a) Primary flow

EJ1 EJ2 Experiment data from Lucas and Koehler (2012)

60

CE

+15%

ED

40 20

0

AC

-15%

M

80

PT

-1

Predicted mS (kg h )

100

0

20

40

60

-1

80

100

Measured mS (kg h ) (b) Secondary flow

Fig. 7 Comparison between predicted and measured mass flow rates

4.2 Effects of the primary flow inlet pressure Figure 8 shows the measured and predicted mass flow rates for different primary flow inlet pressures of EJ1 at PB ≈ 4.3 MPa, PS ≈ 3.6 MPa, TS ≈ 21 °C, and TP ≈ 35 °C. The results show that

19

ACCEPTED MANUSCRIPT both the mass flow rates of the primary and secondary flows increase with the increase in the primary flow inlet pressures. To further explain the phenomena, Figs. 9(a) and (b) show the velocity (Mach number) and the liquid mass fraction distributions along the flow direction in the two cases, marked in Fig. 8, respectively. In Fig. 9(a), the Mach number of the primary flow at the nozzle throat is equal to 1 for

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both cases 1 and 2. Because CO2 is a dense fluid, the Mach number of the primary flow of CO2 at the mixing chamber entrance is only 1.3, which is much lower than that of steam or Freon ejectors (Zhu et al., 2014a). As a result, the Mach number of the mixed flow is below one in the mixing chamber, which indicates that the “double choking” phenomenon will not happen in the CO2 ejector.

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It is interesting that the velocity of the primary flow at the mixing chamber entrance of case 2 is less than that of case 1, though the mass flow rate of the primary flow in case 2 is higher than that in case 1. This is because in case 2, the flow at that location has higher liquid mass fraction, as shown in

M

Fig. 9(b), resulting in a much higher fluid density. The velocity of the secondary flow at the mixing chamber entrance of case 2 is approximately two times that of case 1, resulting in a larger secondary

ED

mass flow rate, as shown in Fig. 8.

The results also show that the liquid-mass fraction gradually decreases along the ejector. The

PT

decrease in the liquid-mass fraction in the nozzle and the suction chamber is due to the flow

CE

expansion process, as shown in Fig. 3. In the mixing chamber, the liquid-mass fraction decreases

AC

because the primary flow mixes with the secondary flow, which is vapor.

20

ACCEPTED MANUSCRIPT

120

Measured mP

100

Measured mS

-1

80

Predictd mS

60 Case 1

40 20 0 7.7

8.2

8.7

CR IP T

ms and mP (kg h )

Case 2

Predictd mP

9.2

PP (MPa)

9.7

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Fig. 8 Measured and predicted mass flow rates for different primary flow inlet pressures (EJ1, PB ≈ 4.3 MPa, PS ≈ 3.6 MPa, TS ≈ 21oC, TP ≈ 35oC)

Ma 1.3 Ma 1.3

120

Mixing chamber outlet

ED

Ma 1

80

Ma 0.6

Case 1 Primary flow Secondary flow Case 2 Primary flow Secondary flow

Nozzle throat

PT CE AC

M

-1

Velocity magnitude (m s )

160

Ejector outlet

Ma 0.54

40

0

0

Mixing chamber inlet

20

40

Position (mm) (a) Velocity magnitude

21

60

80

ACCEPTED MANUSCRIPT

Case 1 Case 2

0.8 0.6 0.4

Nozzle throat

0.2 0.0

Ejector outlet

Mixing chamber outlet

Mixing chamber inlet

0

20

40

Position (mm)

60

80

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(b) Liquid mass fraction

CR IP T

Liquid mass fraction

1.0

Fig. 9 Calculated velocity and liquid mass fraction in the ejector

M

(Cases 1 and 2 were shown in Fig. 8)

4.3 Effects of the outlet pressure of ejector

ED

Figure 10 shows the measured and predicted mass flow rates for different ejector outlet pressures

PT

of EJ2 at PP ≈ 8.3 MPa, PS ≈ 4.2 MPa, TS ≈ 22 °C, and TP ≈ 35 °C. The results show that the mass flow rate of the secondary flow decreases significantly with an increase in the back pressure when the back

CE

pressure is larger than 4.67 MPa. The mass flow rate of the secondary flow decreases because the ejector is operating in the subcritical mode. The results also show that the primary mass flow rate is

AC

independent of the back pressure. Figures 11 (a) and (b) show the velocity (Mach number) and the liquid mass fraction distributions

along the flow direction for the two cases marked in Fig. 10. In Fig. 11(a), the velocity (Mach number) of the primary flow at the nozzle throat and at the mixing-chamber inlet for cases 3 and 4 are largely the same because the primary flow inlet conditions are the same. The behavior of the Mach number is similar to that shown by Banasiak and Hafner (2011) and Palacz et al. (2016).

22

ACCEPTED MANUSCRIPT The velocity of the secondary flow at the entrance of the mixing chamber in case 4 is approximately half that in case 3. Under the same conditions, the secondary mass flow rate in case 4 is approximately half that in case 3 as shown in Fig. 10. The liquid mass fraction of the mixed flow at the mixing-chamber outlet in case 4 is much higher than that in case 3. This is because the mass flow rate of the secondary flow is higher in case 3.

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Figure 12 shows the measured and predicted mass flow rates for different ejector outlet pressures of EJ1 at PP ≈ 8.7 MPa, PS ≈ 3.6 MPa, TS ≈ 21 °C, and TP ≈ 35 °C. The results are similar to that shown in Fig. 10.

120

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Case 3

80 60

Measured mP

20

Measured mS

M

40

Predicted mP Predicted mS

ED

-1

ms and mP (kg h )

100

PT

0 4.5

Case 4

4.6

4.7

4.8

4.9

PB (MPa)

CE

Fig. 10 Measured and predicted mass flow rates for different ejector outlet pressures

AC

(EJ2, PP ≈ 8.3 MPa, PS ≈ 4.2 MPa, TS ≈ 22oC, TP ≈ 35oC)

23

ACCEPTED MANUSCRIPT

Ma 1.2 Ma 1

120

Mixing chamber outlet

Mixing chamber inlet

80

Case 1 Primary flow Secondary flow Case 2 Primary flow Secondary flow

Ma 0.47

Nozzle throat

40 0

0

20

40

Position (mm)

1.0

80

Case 3 Case 4

0.8

M

Mixing chamber inlet

0.6

Ejector outlet

Mixing chamber outlet

Nozzle throat

ED

0.4 0.2

PT

Liquid mass fraction

60

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(a) Velocity magnitude

CE

0.0

AC

Ejector outlet

Ma 0.465

CR IP T

Velocity magnitude (m/s)

160

0

20

40

60

80

Position (mm) (b) Liquid mass fraction

Fig. 11 Calculated velocity and liquid mass fraction in the ejector (Cases 3 and 4 were shown in Fig. 10)

24

ACCEPTED MANUSCRIPT

120 -1

ms and mP (kg h )

100 80 60 Measured mP Predicted mP

20 0 3.9

Measured mS Predicted mS

4.0

4.1

4.2

PB (MPa)

CR IP T

40

4.3

4.4

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Fig. 12 Measured and predicted mass flow rates for different ejector outlet pressures (EJ1, PP ≈ 8.7 MPa, PS ≈ 3.6 MPa, TS ≈ 21oC, TP ≈ 35oC)

M

4.4 Effects of the inlet pressure of secondary flow

Figure 13 shows the measured and predicted mass flow rates for different secondary-flow inlet

ED

pressures of EJ1 at PP ≈ 9.3 MPa, PB ≈ 4 MPa, TS ≈ 21 °C, and TP ≈ 35 °C. Because the primary flow inlet conditions are the same, the primary mass flow rate remains constant in all the cases. The mass

PT

flow rate of the secondary flow increases significantly with the increase in the secondary-flow inlet

CE

pressures.

Figures 14 (a) and (b) present the velocity (Mach number) and the liquid mass fraction

AC

distributions along the flow direction for the two cases marked in Fig. 13. The velocity (Mach number) of the primary flow at the nozzle throat for cases 5 and 6 are the same because the primary flow inlet conditions are the same. However, the velocity (Mach number) at the mixing-chamber inlet in case 5 is higher than that in case 6. This is because the secondary-flow inlet pressure in case 5 is lower than that in case 6, resulting in more further expansion of the primary flow in the suction chamber.

25

ACCEPTED MANUSCRIPT

120 Case 5

Case 6

ms and mP (kg/h)

100 80 60 Measured mP

40

Predicted mP

CR IP T

Measured mS

20

Predicted mS

0 2.9

3.1

3.3

3.5

PS (MPa)

3.7

3.9

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Fig. 13 Measured and estimated mass flow rates for different secondary flow inlet pressures (EJ1, PP ≈ 9.3 MPa, PB ≈ 4 MPa, TS ≈ 21oC, TP ≈ 35oC)

Ma 1.4

120

M

Ma 1.3

-1

Velocity magnitude (m s )

160

Mixing chamber outlet

ED

Ma 1

80

PT

Mixing chamber inlet

AC

CE

40

0

Ma 0.61

Case 1 Primary flow Secondary flow Case 2 Primary flow Secondary flow

Ejector outlet

Ma 0.61

Nozzle throat

0

20

40

Position (mm) (a) Velocity magnitude

26

60

80

ACCEPTED MANUSCRIPT

Case 5 Case 6

0.8

Mixing chamber outlet

Ejector outlet

0.6 Nozzle throat

0.4

Mixing chamber inlet

0.2 0.0

0

20

40

Position (mm)

60

80

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(b) Liquid mass fraction

CR IP T

Liquid mass fraction

1.0

Fig. 14 Calculated velocity and liquid mass fraction in the ejector

M

(Cases 5 and 6 are shown in Fig. 13)

4.5 Effects of the primary flow inlet temperature

ED

Figure 15 shows the measured and predicted mass flow rates for different secondary-flow inlet

PT

pressures of EJ2 at PS ≈ 4.3 MPa, PB ≈ 4.8 MPa and TS ≈ 22 °C. Note that the primary flow inlet pressure changes with the change in the primary flow inlet temperature, the values of which are

CE

shown in Fig. 15. The results show that the primary mass flow rate slightly decreases with an increase in the primary flow inlet temperature. This is largely because the mass fraction of the primary flow

AC

decreases significantly with an increase in the primary flow inlet temperature, as shown in Fig. 16(b). The mass flow rate of the secondary flow increases with an increase in the primary flow inlet temperature. This can be explained using the behavior of the secondary-flow velocity at the mixing-chamber inlet as shown Fig. 16(a).

27

ACCEPTED MANUSCRIPT

120 Case 8 PP= 7.7

8.2

8.4

8.6

8.8

9.0 MPa

7.9 8.0

-1

ms and mP (kg h )

100

7.8

Case 7

80 60

Measured mP Predicted mP

40

CR IP T

Measured mS Predicted mS

20 30

34

38o

42

TP ( C)

46

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Fig. 15 Measured and predicted mass flow rates for different primary flow inlet temperatures (EJ2, PS ≈ 4.3 MPa, PB ≈ 4.8 MPa, TS ≈ 23oC)

Ma 1.27

160

Mixing chamber outlet

Ma 1

Ma 1.07

ED

120

80

Ejector outlet

Nozzle throat

PT 0

Case 7 Primary flow Secondary flow Case 8 Primary flow Secondary flow

Ma 0.45

40

CE AC

M

-1

Velocity magnitude (m s )

200

0

Ma 0.46 Mixing chamber inlet

20

40

Position (mm) (a) Velocity magnitude

28

60

80

ACCEPTED MANUSCRIPT

Case 7 Case 8

Nozzle throat

0.8

Mixing chamber inlet Mixing chamber outlet

0.6

Ejector outlet

0.4 0.2 0.0

0

20

40

Position (mm)

60

80

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(b) Liquid mass fraction

CR IP T

Liquid mass fraction

1.0

Fig. 16 Calculated velocity and liquid mass fraction in the ejector

M

(Cases 7 and 8 were shown in Fig. 15)

4.6 Correlation for primary flow pressure at nozzle throat

ED

The primary flow pressure at the nozzle throat Pt needs to be iteratively solved in the developed

PT

model. After obtaining Pt, the thermal properties and flow rate of the primary flow are then determined. Therefore, the model calculation can be simplified if Pt can be determined using only the

CE

inlet conditions of the primary flow. The following correlation for the pressure Pt is fitted based on

AC

the aforementioned 130 case studies. Pt 

1 0.53  0.121PP 0.5  6 1011 P 3

(19)

7.6  PP  10.3 MPa, 32
Figure 17 shows the pressure at the nozzle throat obtained using the three methods. One of the methods involves iterating Pt using the developed model, and the other two methods are based on the correlations. The result shows that the calculated Pt using the two correlations has a significant difference, particularly at low primary flow inlet pressures. The values of Pt are in the ranges of 4.8–6.3 MPa calculated using Eq. (19) and 3.4–6.8 MPa calculated using Eq. (29), which are reported 29

ACCEPTED MANUSCRIPT in another study (Lucas et al., 2013). The pressure at the nozzle throat should be further studied by conducting an experiment.

7

EJ1 Pt iterated by the develop model

5

EJ2

Pt iterated by the develop model

Determined by Eq. (19) Eq. (29) in Lucas et al. (2013) Experiment Data in Lucas and Koehler (2012) Pt iterated by the develop model

4 3 7.5

Determined by Eq. (19) Eq. (29) in Lucas et al. (2013)

8.0

8.5

9.0

9.5

10.0

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Pt (MPa)

6

CR IP T

Determined by Eq. (19) Determined by Eq. (29) in Lucas et al. (2013)

10.5

PP (MPa)

Fig. 17 Estimated nozzle throat pressure plotted against primary flow inlet pressure by three different

M

methods

ED

5. Conclusions

In this study, we developed a theoretical model of the transcritical CO2 two-phase ejector, to

PT

predict the mass flow rates of both primary and secondary flows. By considering the non-equilibrium

CE

phase change through the primary nozzle, we incorporated a correlation for the non-equilibrium phase change into the energy-conservation equation. The model with and without the

AC

non-equilibrium correlation was simulated under as many as 130 cases, obtained from three different ejector configurations. The results show that the non-equilibrium phase change should be considered when the liquid mass fraction at the nozzle throat is greater than 0.65. The velocity and mass fraction distributions inside the ejector were analyzed under various operating conditions. The Mach number of CO2 in the ejector is lower than 1.5, which is much lower than that of steam or Freon ejectors. Therefore, the “double choking” phenomenon will not happen in the CO2 ejector. The results help in understanding the characteristics of the two-phase flow CO2 ejector. 30

ACCEPTED MANUSCRIPT A correlation for the primary flow pressure at the nozzle throat was fitted, based on the 130 cases. The correlation is a function of only the inlet pressure and the density of the primary flow, which can be used to replace iterative solving of the properties of the primary flow in the nozzle when modeling the ejector.

CR IP T

Acknowledgement This project was supported by the National Natural Science Foundation of China (51576104), the Tsinghua University Initiative Scientific Research Program (2014z21040) and the Science Fund for Creative Research Groups of NSFC (No. 51621062).

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Nomenclature area, m2

D

diameter, mm

h

enthalpy, J kg-1

m

mass flow rate, kg s-1

P

pressure, MPa

QNE

non-equilibrium correlation, J kg-1

s

entropy, J kg-1 K-1

T

temperature, K

V

velocity, m s-1

x

fraction in mass, %

AC

CE

PT

ED

M

A

Greek letters

a

speed of sound, m s-1

ρ

density, kg m-3

φ

efficiency

ω

entrainment ratio, mS/mP

Subscripts B

ejector outlet 31

diffuser

m

mixing chamber

n

nozzle

l

liquid phase

P

primary flow at the nozzle inlet

pm

primary flow at the mixing chamber inlet,

S

secondary flow at the ejector inlet

sm

secondary flow at the mixing chamber inlet

t

nozzle throat

v

vapor phase

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d

CR IP T

ACCEPTED MANUSCRIPT

References

M

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PT

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CE

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PT

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CE

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AC

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CR IP T

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