Characterization of Choking Flow Behaviors inside Steam Ejectors Based on the Ejector Refrigeration System

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Characterization of Choking Flow Behaviors inside Steam Ejectors Based on the Ejector Refrigeration System Yu Han , Xiaodong Wang , Anthony Chun Yin Yuen , Ao Li , Lixin Guo , Guan Heng Yeoh , Jiyuan Tu PII: DOI: Reference:

S0140-7007(20)30056-6 https://doi.org/10.1016/j.ijrefrig.2020.02.003 JIJR 4662

To appear in:

International Journal of Refrigeration

Received date: Revised date: Accepted date:

13 September 2019 30 December 2019 2 February 2020

Please cite this article as: Yu Han , Xiaodong Wang , Anthony Chun Yin Yuen , Ao Li , Lixin Guo , Guan Heng Yeoh , Jiyuan Tu , Characterization of Choking Flow Behaviors inside Steam Ejectors Based on the Ejector Refrigeration System, International Journal of Refrigeration (2020), doi: https://doi.org/10.1016/j.ijrefrig.2020.02.003

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Characterization of Choking Flow Behaviors inside Steam Ejectors Based on the Ejector Refrigeration System Yu Han a,b,∗, Xiaodong Wang a,∗, Anthony Chun Yin Yuen b, Ao Li b, Lixin Guo a, Guan Heng Yeoh b,c and Jiyuan Tu d a

School of Mechanical Engineering and Automation, Northeastern University, Shenyang

110819, China b

School of Mechanical and Manufacturing Engineering, University of New South Wales,

Sydney, NSW 2052, Australia c

Australian Nuclear Science and Technology Organisation (ANSTO), Locked Bag 2001,

Kirrawee DC, NSW 2232, Australia d

School of Engineering, RMIT University, Victoria 3083, Australia

Corresponding Author 1 :

Professor Xiaodong Wang School of Mechanical Engineering and Automation Northeastern University Shenyang 110819, China

Email

:

[email protected]

Phone number

:

+86-24-8368 7618

Corresponding Author 2 :

Yu Han School of Mechanical Engineering and Automation Northeastern University & The University of New South Wales Shenyang 110819, China & Sydney NSW 2052, Australia

Email

:

[email protected]

Phone number

:

+61 403684760

Abstract In order to maximize the working potential of a steam ejector, aside from experiment testing of various design parameters, it is also important to understand the involved thermal fluid mixing behaviors. In this study, an ejector refrigeration experimental system was established, and a numerical model was developed to comprehend the complex and non-linear flow characteristic using an ideal gas model which is more consistent with the experimental data compared with the wet steam model. Herein, it was discovered that the choking flow and the occurrence locations of this phenomenon played an essential role in the system efficiency. For the first time, three choke behaviors modes were characterized by means of numerical investigations: ―fit-choked flow mode‖, ―sub-choked flow mode‖ and ―super-choked flow mode‖. The relationship between the choke and the normal shock wave was revealed to analyse the two mixing fluids flow conditions. The primary fluid pressure and the back pressure under three choke flow modes were comprehensively discussed. The simulation results indicated that the choke was a critical factor in affecting the performance of the steam ejector, and the ejector performance optimizes at ―fit-choked flow mode‖ when the primary fluid pressure was 0.36 MPa with a remarkable entrainment ratio of 0.525.

Keywords: refrigeration, steam ejector, choking flow, turbulent flow, shock wave

Nomenclature ms

mass flow rate of primary fluid

mp

mass flow rate of secondary fluid

he,i

evaporator inlet enthalpy

he,o

evaporator outlet enthalpy

hg,i

generator inlet enthalpy

hg,o

generator outlet enthalpy density ( kg / m3 ) mass fraction

Vd

average droplet volume (m3) mass generation rate (kg/s) velocity ( m / s )

̅

droplet average radius (m)

∗

critical droplet radius (m) thermal conductivity ( W / m  K ) temperature ( K ) non-isothermal correction factor specific heat capacities ratio isobaric heat capacity ( J / kg  K )

T0

droplet temperature (K) viscous stress ( N / m2 )

hv

vapor specific enthalpy (J/kg)

η

droplet number density (1/m3)

ρv

vapor density (kg/m3)

I

nucleation rate (1/s)

qc

evaporation coefficient

σ

droplet surface tension (kg/m)

M

molecular mass (kg)

R

super saturation ratio

Vd

average droplet volume (m3)

B, C

Virial coefficients (m3/kg, m6/kg2)

Cp0

isobaric specific heat at zero pressure

h0

standard state enthalpy

s0

standard state entropy

GCI

grid convergence index

f

21 [exact]

exact extrapolation solution between the mesh 1 and mesh 2

1. Introduction Ejector refrigeration system, which utilizes steam droplets has recently received wide research attention, benefits by its capability of using recyclable working fluid (i.e. water vapour) to generate a secondary flow via steam entrainment. The second flow will take away the heat in the evaporator and cause heat exchange between the evaporator and the environment, thereby generating a cooling effect (Besagni et al., 2016a). The ejector refrigeration system is mainly composed of four parts: generator, evaporator, ejector and condenser. The working fluid and the secondary flow are generated in the generator and evaporator, respectively. The working fluid and the secondary flow are mixed inside the ejector, and the mixing fluid is discharged into the condenser. The ejector is the core component of the ejector refrigeration system and governs the efficiency of the entire refrigeration system (Sun, 1997). The ejector itself is a piece of important vacuum equipment which is widely applied in metallurgy, automobile, chemical, pharmaceutical and other industries to achieve various vacuuming conditions (Riffat et al., 2005) (Tashtoush et al., 2019). Although the steam ejector has many advantages, such as no moving parts, simple installation, easy to maintain and operate, no pollution to the environment and the structural design of the steam ejector is relatively simple, the efficiency is comparatively low. Two of the most important parameters to assess the efficiency of a steam ejector are entrainment ratio (ER) and the compression ratio (Riffat et al., 2005). ER can be expressed as (Besagni et al., 2016b):

(1) The performance evaluation factor of the refrigeration system is the coefficient of performance (COP) which is given in terms of ER (Chen et al., 2015): (

)

(

)

(2)

Moreover, COP is also written by applying Eqs. (1) as follows: (

)

(

)

(3)

ER can evaluate the efficiency of the refrigeration cycle, which could be obtained by Eq. (3). The compression ratio is defined as the ratio of back pressure and the secondary fluid pressure which is a measure of the operating range of the ejector (Besagni et al., 2016b). As well as the internal flow is transonic and complicated, and the performance of the ejector drastically varies depending on the internal flow mechanisms (Sriveerakul et al., 2007). Thence, it is essential to develop a fundamental and comprehensive understanding of the flow behaviours of steam ejectors. In the past few decades, researchers have carried out extensive studies on steam ejectors to understand the thermal interacting flow. Results in numerous studies have indicated that the viscous loss caused by heat transfer, irreversible oblique shock wave, as well as the choking flow were the main causes of the poor efficiency of the steam ejector (Haghparast et al., 2018). Since the 1920s, the ejector has already been incorporated in the refrigeration and air conditioning system. Despite the effect of the geometry structure (Bartosiewicz et al., 2005) and the operating parameters (Zhu et al., 2009) on the ejector were previously proposed, the internal complex flow phenomena such as the shock wave (Allouche et al., 2014) and the boundary layer separation of the ejector (Han et al., 2019b) yet to be studied. Choking flow is defined as a flow at a certain pressure passes through a restriction into a lower pressure, and the velocity reaches to sound speed under specific conditions, in which the

mass flow is constant even the downstream pressure further decreases (Huang et al., 2009) (Eames et al., 1995) (Chen et al., 2015). Practically, two choking position exist in the ejector. One occurs in the primary flow through the nozzle, the other was in the throat where the acceleration of the secondary flow was from a stagnant state to a sonic state (Monday and Bagster, 1977) (Huang et al., 1999). Huang (Huang et al., 2009) shown that the choking flow behaviour was associated with the effective area of the secondary fluid when the ejector was operating at a back pressure below the critical value. Sun (Sun, 1997) pointed out that when choking flow occurred, the condensation pressure would have a smaller effect on system efficiency. However, there was no further analysis examining the relationship between choking flow and shock wave for the entrainment ratio of the steam ejector. The choking of the secondary flow in the mixing chamber was important for the system performance was proposed through the experimental method by Eamas et al. (Eames et al., 1995). Selvaraju (Selvaraju and Mani, 2004) stated that choking flow played an influential role in the steam ejector effectiveness. It was agreed from many research studies that two chokes were required in the steam ejector system in order to create a quality vacuuming effect, where one occurred within the nozzle, while the other one occurred at the ejector out of the nozzle (Monday and Bagster, 1977) (García Del Valle et al., 2012). Aphornratana (Aphornratana et al., 2001) made a conclusion contrary to Huang et al. (Huang et al., 1999) and showed that the choke might occur only in the particular nozzle structure and no generality. Chou (Chou et al., 2001) divided the choke into three parts, but no specific judgment basis was given for the specific location of each part of the choke. Nevertheless, the choking flow was still lack of full understanding and no overall research on judgment and determination. With the application of Computational Fluid Dynamics (CFD) simulations on the steam ejector system (Varga et al., 2009), it enabled the in-depth analysis of the internal flow and the mixing process of the ejector, as well as providing valuable insights for the geometric and parametric optimizations of the system (Wang and Dong, 2010) (Wang et al., 2014) (Fu et al.,

2018). Lots of studies forced on the nozzle part only, the study on the flow behaviour of the whole steam ejector is not enough (Dai and Huo, 2011) (Fu et al., 2016). CFD simulation could provide additional detailed information for thermally reacting flows, in which the complex heat and fluidic interactions could be thoroughly characterised, which would be extremely difficult to obtain via experimentation (Yuen et al., 2017) (Yuen and Yeoh, 2013). In this study, a comprehensive understanding of the choking flow and its effects on the ejector performance will be analysed. An overall reference for the study of the internal flow characteristics of the ejector will be provided. The key objectives of this study are: 1) to validate the numerical model by comparing the ideal gas model and the wet steam model with the experimental data; 2) to identify the occurrence location of the choking effect by analysing the variation in Mach number across the steam ejector chamber; 3) to study the influence of the choked behaviour with regards to the ejector entrainment ratio and efficiency of the ejector refrigeration system; 4) to categorize the choking flow phenomenon to three different modes: ―fit-choked flow mode‖, ―sub-choked flow mode‖ and ―super-choked flow mode‖; 5) to investigate the importance of incorporating the double choking flow behaviour for the prevention of backward flows occurring in the outlet; 6) to optimize the working condition for the primary and secondary flows (i.e. pressure and temperature) in order to make use of the choking effect.

2. Experimental system setup An experimental setup for a ejector refrigeration was carried out to validate the CFD model and investigated the performance of the ejector. The saturated steam produced by the generator heating is referred as the primary fluid. After the primary fluid stream is compressed and expanded by the nozzle, a low-pressure area is created at the exit of the

nozzle. The liquid in the evaporator was evaporated to absorb external heat to produce a cooling effect, and the evaporated gas is referred as the secondary fluid. The secondary fluid is pumped to the mixing chamber of the ejector under the pressure difference to exchange energy and momentum with the primary fluid. The mixed fluid is decelerated and pressurized by the ejector and discharged into the condenser. After being condensed into a liquid in the condenser, it is sent to the generator and the evaporator via the feed-water pump and the gas storage tank for the next refrigeration cycle. The distribution of the experimental system is shown in Fig. 1(a), and this refrigeration system consists of seven components which are listed as follows: 1) steam ejector, 2) condenser, 3) gas storage tank, 4) evaporative condenser, 5) boiler, 6) evaporator and 7) water-ring pump. The specific operation process of the entire ejector refrigeration system can be summarized as the following procedures. The steam boiler produces saturated steam at a specific pressure and temperature, which is then transferred to the steam storage tank through the steam pressure regulating valve. The water ring pump is installed at the outlet of the condenser as a replenishment pump to provide a stable pressure to the steam ejector. The different back pressure is regulated by a back-pressure valve that is mounted at the outlet. The desired water vapour pressure can be obtained by using it in combination with a pressure transmitter and a PID hand-held controller. Liquid state water is evaporated into a gaseous state by absorbing external heat in the evaporator. At the same time, it produces refrigeration effect when it exchanges heat with the outside environment. The mixed steam coming out of the ejector is decelerated and pressurized by the ejector and discharged into the condenser. Herein, it is condensed and converted from gaseous to the liquid state. The liquid water from the condenser enters the next refrigeration cycle. The condenser is used to cool the mixed steam, while the evaporative condenser is used to provide circulating water for condenser and water-ring pump. The pressure transmitter is used to monitor the primary fluid pressure in the gas storage tank in real-time. It can directly convert the digital signal into an electrical signal and send it

to the control valve switch to adjust the pressure in the steam storage tank. To ensure that the primary fluid pressure output from the storage tank is maintained at a relatively stable value. At the same time, a pressure regulating device and a temperature measuring instrument is installed on the steam storage to measure the pressure and temperature, which can be used to control and evaluate the thermodynamic state of the working fluid at the inlet to the nozzle. The secondary fluid is mainly formed by the low-temperature evaporation in the evaporator, so the flow rate of the monitored suction fluid can be measured by the vortex flowmeter. The vortex flowmeter has many advantages with a simple and firm structure, no moving parts, and high reliability. The primary fluid pressure and the secondary flow rate are measured by a pressure transmitter with a full-scale error of 0.5% and a vortex flowmeter with the uncertainty less than 0.33%, respectively. The temperature of the water in the evaporator was measured by a T-type thermocouple temperature controller. The T-type thermocouple is continuously calibrated with the preset temperature to control the water temperature near the set temperature. The temperature is measured by a T-type thermocouple temperature control meter which the accuracy is

0.5℃ and the resolution is 0.1℃. As can be seen from Fig.

1(b), according to the present numerical simulation results, 15 small holes are uniformly applied on the wall surface of the ejector to measure the pressure at a certain distance, so that the position of the shock wave can be accurately captured. Each small hole through a vacuum silicon tube is linked to a valve. The pressure information is conveyed to the vacuum gauge through the air guide valve. The pressure is detected by a film vacuum gauge to sequentially display the values on the gauge through these small holes. 3. Mathematical model The total energy equation, including viscous dissipation, was considered and coupled with the gas law. The governing equations are mathematical statements of the physical and chemical phenomena of the considered fluid mixture, and they are resolved in the form of

computational codes, obtaining numerical solutions for fluid problems. Based on the finite volume method, Fluent ANSYS 19.2 was used to solve the governing equations of continuity, momentum and energy. The wet steam transport equation needs to be considered for the wet steam simulation. Owing to describe the mass and heat transfer between the vapour phase and the liquid phase during the condensation process, the liquid-phase mass-fraction transport equation and the droplet number density transport equation should be solved. Several assumptions were preset in the present model: (1) the velocity slip between the droplets and gaseous-phase is negligible; (2) the condensation is non-equilibrium; (3) the interactions between droplets are neglected as the mass fraction of the condensed phase is small; (4) due to droplet sizes are very small, the volume of the condensed phase and the heat capacity are negligible. The first addition transport equation governing the mass fraction of the condensed liquid phase (K. Ishazaki, T. Ikohagi, 1995) can be written as below: (

(

)

(

⃑)

4) where Г is the mass generation rate due to condensation and evaporation, which is correlated to the nucleation rate I and the growth/demise of these droplets in the classical nucleation theory (K. Ishazaki, T. Ikohagi, 1995):

∗

( ̄

̄

5) The nucleation rate (Young, 1988) can be written in this form: (

∗

(

)

)√

(

(

)

6)

Here, θ is given by (Young, 2015): ( (

) ( )

)(

)

(7)

The wet steam density can be determined by the vapour density ρv and the liquid phase mass fraction β: ( (

)

8)

The Kelvin-Helmholtz critical droplet radius r*, above which the droplet will grow and below which the droplet will evaporate (Young, 2015), is given by: ( ∗

9) The size of droplets is affected by two mechanisms: the transfer of mass from the vapor to the droplets, and the transfer of heat from the droplets to the vapor in the form of latent heat (K. Ishazaki, T. Ikohagi, 1995). It can be written as below: ̄

(

√

)

(10)

The second addition transport equation governing the droplet number density η that is defined as the number of droplets per unit volume can be expressed as: ( (

(

)

)

11)

The number density of the droplets transport equation obtained from Ref (K. Ishazaki, T. Ikohagi, 1995) is described as: (

(

)

(

⃑)

12) In addition, the wet steam equation of state which relates the pressure to vapor density and temperature (Young, 1988) is given by: ( (

)

13) Here, B and C are the functions of temperature.

The wet steam isobaric specific heat capacity Cp, specific enthalpy h and specific entropy s are given by (Young, 2015): ( )

((

(((

)(

)

)

)

)

(1 4)

)

(1 ( )

((

)

(

)

)

5) ( )

(

(

)

(

)

)

(1

)

6) 4. Numerical method 4.1 Algorithms The structure of the ejector refrigeration system except the ejector is determined by the experimental device as described in the previous study (Han et al., 2019c). The dimensions of the geometric parameters of the ejector are shown in Table 1. Based on the pioneering numerical work reported by Pianthong and Seehanam (Pianthong et al., 2007), the simulation results yielded from 2D axisymmetric and the three-dimensional models were almost identical. Accordingly, to enhance the prediction accuracy at the near-wall regions (i.e. using a strict refined meshing system), and to avoid the false scaling numerical error led by the third-dimensional model, the 2D axisymmetric model is adopted for this numerical assessment. The operating conditions are shown in Table 2. The density-based model was employed for the simulation. The turbulent closure was modelled by k-ω SST (Besagni and Inzoli, 2017) (Besagni et al., 2015), accounting for the transport of the turbulent shear stress to properly predict the onset and amount of flow separation from smooth surfaces. It has a more accurate and reliable for a wide range of flows such as adverse pressure gradient flows, aerofoils, transonic shock waves (Menter, 1994). This

model modified turbulent viscosity formulation to account for the transport effects of the principal turbulent shear stress, which delivers an enhanced performance at the near-wall regions of the steam ejector. The two inlets and outlet were prescribed as pressure-inlets and pressure-outlet, respectively. The wall-boundary was assumed to be no-slip and adiabatic. Second-order upwind was implemented as the discretization scheme. The mass flow rate tends to be stable when the iteration is converged and terminated. The quadrilateral structure grid was applied as depicted in Fig. 2. Based on the configuration of the steam ejector, the four main parts of a typical steam ejector were modelled, including (i) primary nozzle, (ii) mixing chamber, (iii) throat, and (iv) subsonic diffuser. Due to the flow feature inside the whole steam ejector, the quadrilateral structure grid with adaptive grid technique was employed to have a better understanding of the flow behaviour in the steam ejector. The convergence criterion for residual of all dependent variables and the mass imbalance value was set to 1×10−5. The mass flow difference between the two inlets and outlet flows of the ejector is less than 10-7 kg/s. 4.2 Mesh independency A grid sensitivity test was performed to evaluate the mesh independency towards the computational results. Since the focal point of this study was the choking flow behaviour and it can be studied by the change of the Mach number, similar to other numerical studies, the axial Mach number profile can be applied the functional for the sensitivity test (Yuen et al., 2018) (Chen et al., 2018). The GCI method which was a measure of the relative discrete error of the computed solution which is based on generalized Richardson Extrapolation involving a comparison of discretization solutions at different grid spacings was applied to analyze the mesh sensitive (Roache PJ, 1998). Accordingly, four structured mesh systems have been constructed at increasing refinement levels with 34361, 55742, 76895 and 122132 elements, respectively. The results of the GCI method were shown in Table 3. It is shown that both GCI values are small and the discrete errors on the three grids are also small, which can ensure

certain accuracy of the simulations. The fine mesh and the medium mesh have good precision as numerically simulated meshes. Since a large number of simulation calculations are required later, a medium mesh of 55742 cells is selected for all simulation carried out in this study to save the computation time and costs. 4.3 The comparison of the ideal gas model and wet steam model In the numerical simulation, spontaneous condensation occurs within the steam ejector (Li et al., 2019) (Sharifi et al., 2013). It has been demonstrated through previous studies that the wet steam simulation is capable of replicating the condensation effect, thus enhancing the prediction accuracy against experimental measurements (Sharifi et al., 2012). Nonetheless, Javier (García Del Valle et al., 2012) and Aphornratana (Aphornratana and Eames, 1997) have pointed out that although the ideal gas model has some variations against the actual real gas mixture behavior, this difference is insignificant for a superheated fluid mixture for low operating pressure conditions (i.e. boiler temperature from 120°C-140°C), and it has also been identified by experiments. The ideal gas model is then the option for the real gas due to its computational efficiency. Sriveerakul et al. (Sriveerakul et al., 2007) showed that the simulation results compare well against the experimental results with the assumption of an ideal gas. In comparisons of the experimental values with the ideal gas and wet steam simulation values, Ariafar et al. (Ariafar et al., 2015) concluded that the ideal gas model provided more consistent results versus the experimental data, where the primary fluid temperature, secondary fluid temperature and the back pressure are 403K, 287K and 4200Pa, respectively. Fig. 3 shows the comparison of the Mach number calculated by the ideal gas model and the wet steam model within the operating condition of the primary fluid pressure, the secondary fluid pressure and the back pressure of 0.36 MPa, 1710 Pa and 3500 Pa, respectively. It can be seen in Fig. 3(a) that the internal flow of the ejector has similar fluid characteristics such as

shock train, choking and normal shock wave using both ideal gas and wet steam models. Typically, the shock train occurs at the mixing section and the choke effect occurs at the throat section. Although the shock train mainly happens in the mixing section, the length of this shock train behaviour is generally shorter when using the wet-steam model. Owing to the presence of condensation, the intensity of the normal shock wave is greater for the wet steam model at the throat exit, which from the occurrence location is different when using the ideal gas model. It indicates that the higher critical back pressure can be obtained, and the ejector flow is more stable when the wet steam model is applied. As indicated by the Mach number distribution along the axis, as shown in Fig. 3(b), the Mach number remains the same within the entire nozzle. The increase in humidity and temperature of the fluid mixture is attributed to the phase change effect led by condensation of vapours, thus creating a lower Mach number flow against the ideal gas model simulation. The Mach number is 4.3 at x=0.12m in the wet steam model, whereas it is 4.4 at x=0.13m for the ideal gas model. Nevertheless, it should be noted that the Mach number in the wet steam model simulation can reach the local maximum of 1.8 at x=0.55 m while it is 1.65 at x=0.52 m for an ideal gas model. It further reflects the intensity of the normal shock wave under the wet steam model is much stronger, which can sufficiently withstand the disturbance from propagating upstream caused by back pressure. Through numerical simulation using wet steam model, it shows the ejector could function for a wide working range. The results of the wall static pressure for different working properties under the same operating condition as Fig 3 are depicted in Fig. 4. It further illustrates there is no significant difference in the choke occurrence locations for the two models, where it is located at the throat section and happens when the minimum pressure is reached. Simultaneously, the simulation results under the ideal gas model are more consistent when compared with the experimental results. The reason is that since the speed of the working fluid has reached the supersonic speed after exiting from the nozzle, the droplets in the smaller-sized ejector quickly condense and

grow into larger droplets. As shown by the experimental results of Tang et al. (Tang et al., 2019), when the condensated droplets contact with the ejector wall in a limited space will attach to the wall surface. This will result in a decrease in the mass flow rate of the mixed fluid as well as a decrease in energy and momentum. There is a certain difference from the situation where the energy will increase after the steam condenses into liquid under the wet steam model. Although the ideal gas model does not consider the condensation of droplets, the relatively low energy and momentum of the gas itself coincides with the experimental results. Therefore, the ideal gas model can reflect the flow structure inside the ejector and the properties of the working fluid have little effect on the choking flow in this working. Although the operation of the ejector is more stable using the wet steam model, the ideal gas model is capable of reflecting the fluid characteristics of the ejector.

5. Results and Discussion 5.1 Numerical model validation It is owing to the time and expense cost of the ejector refrigeration system experiments, while the efficiency is not good enough. The numerical simulation is widely used with the advantages of its high precision, good visibility and good predictive. The validation of the simulation results under the ideal gas model was demonstrated against the experiment data. Fig. 5 shows the entrainment ratio is numerically investigated under the wet steam model and the ideal gas model within different back pressure when the primary pressure is 0.36 MPa, and the secondary pressure is 1710 Pa. The simulated results of the two models are close to the experimental values, with the error of 5.8% in the ideal gas model and 16.8% for the wet steam flow, which are acceptable in engineering. However, the simulation results using the ideal gas model is more consistent with the experimental value. Additionally, as displayed in Fig. 6, the trend of the mass flow rate of the primary fluid between CFD values and the experimental values is well consistent. Though the experimental value is higher than the

simulated value, the error between them is less than 9%, which can also accept in engineering. As can be seen in Fig. 7, the static pressure distribution along the wall using the ideal gas model is compared with the experimental values. The CFD simulation and the experimental results are recorded under the condition of the primary fluid pressure at 0.36 MPa and the secondary fluid pressure at 1710 Pa, as well as the back-pressure ranges from 4000 Pa to 5500 Pa, respectively. The experimental and simulated values in the diffusion section are in good agreement with each other. The deviation is reasonable, and the error is less than 6% which is an ideal result in engineering. Therefore, the CFD model was validated and verified by the experimental results. In general, this numerical model can be reliably applied in the following simulations. 5.2 The influence of the primary fluid pressure on the ejector performance The performance of the steam ejector is affected by both operating conditions and geometric parameters. The choking flow, the effective area of the secondary fluid and the shock wave are critical factors in dominating the pumping efficiency (Wang et al., 2014) (Han et al., 2019c) (Yang et al., 2012). As mentioned above, the key factor in assessing the performance of the steam ejector is the entrainment ratio (ER), which is defined as the ratio of the mass flow rate of the secondary fluid and the primary fluid. The maximum entrainment ratio of the ejector will be obtained as the ejector operated in the double-choked flow mode (Fu et al., 2016). However, the research of the definition and characteristics of the choking flow is still limited by the present existing research methods for the complex flow condition in the steam ejector. Only the flow field with the Mach number range below one is displayed in Fig 8. The secondary fluid pressure and the back pressure of the steam ejector are 1710 Pa and 4000 Pa, respectively. As can be seen in Fig. 8, the primary fluid jet core has a similar pattern, the

maximum expansion core occurs inside the throat (the constant-area section) and the choking position is gradually moving downstream as the primary fluid pressure increases when the pressure is less than 0.36 MPa or more than 0.35 MPa. There is no significant velocity difference inside the convergent and diffuser sections as the pressure changes. As the primary fluid pressure further increases after 0.36 MPa, the effective area of the secondary fluid decreases, as shown in Fig. 9. Meanwhile, the effective area of secondary fluid flow decreases as the primary pressure decreases except for 0.33 MPa when the primary fluid pressure is less than 0.36 MPa. Even if the length of the jet core remains almost constant, the size of the maximum expansion core of the primary fluid and the choking position differ considerably under different primary fluid pressure. Owing to the reduction in the effective area, the flow passage of the secondary fluid is narrowed, resulting in a decrease in the mass flow rate of the secondary fluid. As the mass flow rate of the primary fluid is increased, the ratio between the two inlet flows is reduced so that the entrainment ratio is decreased as shown in Fig. 5. Although the size of the effective area is the largest at the pressure of 0.33 MPa, the energy of the primary fluid is not enough for the two fluids to mix, and the secondary fluid has not reached the choking flow completely at the choking position which pumps less mass flow rate of the secondary fluid in this pressure (in Fig. 8). Meanwhile, the thermal and molecular exchange between the primary fluid and secondary fluid is insufficient. Therefore, its entrainment ratio remains small, and it is still less than the pressure at 0.36 MPa. This indicates that the effective area will affect the performance of the ejector, but it is not the directly decisive factor in a certain primary fluid pressure range. However, when the primary fluid pressure exceeds 0.36 MPa, the jet core enlarges obviously with the increasing of primary pressure resulting in a larger expansion core and smaller effective area. The choking position at various primary pressures only changes at the inlet of the throat section (in Fig. 8), the local peak value of the ER is consistent with the distance from the throat entrance to the choking position, as shown in Fig. 8 and Fig. 10. The value of the ER mainly

corresponds to the choking position and is affected to some extent by the size of the effective area. As the primary pressure is less than 0.36 MPa, the chocking position gradually approaches the throat inlet and the ER is improved with the primary fluid pressure increasing (in Fig. 8 and Fig. 10). It indicates that the mass flow rate of the secondary fluid is still increasing at this pressure range (in Fig. 6). We claim the steam ejector operates at ―sub-choked flow mode‖. When the primary fluid pressure is at 0.36 MPa, the ER of the secondary fluid reaches the maximum value while the choking position near to the throat entrance (in Fig. 8). Meanwhile, the mass flow rate reaches the maximum value and stay unchanged (in Fig. 6) which illustrate the choking position is closest to the inlet of throat section (in Fig. 8), and the ER reaches its maximum value (in Fig. 10). Thus, in this situation, it is in line with the definition of the choking mentioned in previous studies (Huang et al., 2009). We call the steam ejector operates at ―fit-choked flow mode‖. The choking position moves downstream with the primary fluid pressure increasing; the maximum expansion core of the primary fluid flow has significantly enlarged when the primary pressure exceeds 0.36 MPa. The effective area of secondary fluid flow in the throat section is significantly reduced. That makes the mass flow of secondary fluid decreased, and ER in the choking position is lower than the pressure at 0.36 MPa. At this moment, we call the steam ejector is operated under a ―super-choked flow mode‖. The comparison of Mach number distribution along the wall under different primary pressure in the part of the throat is shown in Fig. 11 with the condition of the secondary pressure and back pressure are 1710 Pa and 3500 Pa, respectively. From this figure, it can be seen that the peak values of the Mach number located at the front part of the throat are the choking positions. According to our previous research (Han et al., 2019a), the flow passage of the secondary pressure is a contraction channel between the ejector wall and the shock mixing layer. The locations of the peak value move downstream with the increase of the primary fluid pressure. Meanwhile, the peak values increase firstly and then decrease. The velocity and the

mass flow of the secondary fluid constantly change. To be specific, the velocity and the mass flow rate of the secondary fluid increase when the primary pressure is less than 0.36 MPa. When the primary pressure is more than 0.36 MPa, the velocity and the mass flow rate value of the secondary fluid declines. This further indicates that the ejector is in fit-choking flow mode only when the primary fluid pressure is 0.36MPa. With the decrease of primary fluid pressure, the fluctuation in Mach number is significantly less. At back area of the throat, the Mach number curve is relatively stable with little change. There is no change in the ability to overcome the disturbance caused by back pressure and the ejector is in stable working condition under different choking flow modes. It means the influence of the three choking modes on the operation stability of an ejector, as well as the capability to overcome the disturbance can be neglected. 5.3 The influence of the back pressure on the ejector performance The comparison of the secondary fluid mass flow rate between experiment values and simulated values at different back pressures, as shown in Fig. 12. The combined analysis of Fig. 12 and Fig. 5 shows that the experimental results exhibit a higher ability to overcome the resistance produced by the back pressure than the simulation results. When the back pressure is less than 5000 Pa, the mass flow rate of the secondary fluid is a constant value. When the back pressure is above 5000 Pa, the mass flow rate of the secondary fluid gradually decreases with the increasing back pressure. From the experimental values, the mass flow rate of the secondary fluid is 0 when the back pressure is 6500Pa. However, for the CFD simulation values, the mass flow rate of the secondary fluid is 0 when the back pressure is between 6000Pa and 6500Pa. This shows that the back pressure range of the experiment is larger than the simulated value. This may be attributed to the fact that the primary fluid in the experiment is the actual fluid, and there is a condensation phenomenon of the droplets. Since the water vapour condenses into small water droplets, the momentum of the fluid increases while the

temperature remains unchanged. In this regard, condensation facilitates the expansion of the back pressure range of the ejector, thereby increasing the ability of the ejector against back pressure disturbance. At the same time, 5000Pa is the critical back pressure of the ejector in the distribution of experimental and simulation values. From Fig. 13(a), the choking position of the primary fluid moves from the throat to the mixing chamber with the increasing of the back pressure while the primary and secondary fluid pressures remain constant at 0.36 MPa and 1710 Pa. When the back pressure is less than 5000 Pa, the normal shock wave occurs in the diffuser section and the choking position is in the throat and almost no change. For the choking position at the entrance of the throat, the normal shock wave can be generated and block the impact of back pressure spreading upstream. Simultaneously, the flow passage of the secondary fluid not only satisfies the convergent-divergent channel requirement which can pump more fluid but also has a fairly long throat length, which can make the two fluids exchange energy adequately and steadily. The ejector can obtain higher pumping efficiency and can be worked at more stable working condition. Meanwhile, in Fig. 13(b), the same trend has been demonstrated in the temperature contours with different back pressure. The temperature is small at the location of the choking flow, the normal shock wave and the shock train. Due to the working fluid in the simulations is an ideal gas, the region with a higher fluid flow velocity is in a lower temperature region. As the back pressure increases, the high temperature region increases and gradually expands upstream, and the low temperature region decreases. Until the high temperature area is filled in the ejector throat and the diffuser sections. When the back pressure is too high, it is easy to find the secondary fluid was pushed back, which means the steam ejector lost its entraining performance. The choking position occurs inside the throat section and remains constant as the back pressure is less than 5000 Pa, the effective area, and the Mach number are both unchanged as illustrated in Fig. 14. With increasing back pressure, the number of vortices between the fluid

and the wall increases. As a result, the boundary layer separation condition is increased, and the flow of the mixed fluid is reduced. As the degree of boundary layer separation increases, the thickness of the boundary layer gradually is reduced. The range of the boundary layer is gradually narrowing. The shock wave is gradually weakened, and the back pressure resistance of the ejector decreases. In the case of double-choking flow, the small degree of boundary layer separation has less effect on the performance of the ejector. As the fluid flows from the double choking to the single choking, the degree of separation of the boundary layer gradually increases, and the shock wave gradually moves upstream as it is demonstrated in Fig. 12. When the back pressure exceeds the 5000 Pa, there will be no normal shock wave inside the steam ejector. Then the disturbance caused by the change of the back pressure will propagate to the upstream, the steam ejector operates at the ―no-choking flow mode‖. In this case, the choking flow cannot be formulated, and the mass flow rate of the secondary fluid will sharply drop until zero, which means the ejector loses its ability to pump fluid. The boundary layer separation zone, the choking zone and the shock zone can be clearly seen from Fig. 15. In this figure, it is demonstrated that reverse flow will happen when the back pressure is too high more than 5000 Pa and result in the larger boundary separation and the lower mass flow rate. The higher pressure in the boundary layer separation zone is caused by the upward propagation of the back pressure disturbance. The lower pressure in the normal shock region is due to the greater velocity in this region. It indicates that the shock wave is not only the cause of the transonic flow of the fluid inside the ejector but also one of the reasons for the gradient change of the pressure. Fig. 16 shows that the entrainment ratio follows the change of secondary fluid mass flow rate when the primary fluid pressure is a constant value at 0.36 MPa and the secondary pressure is 1710 Pa. The above discussion is a quantitative description of the changes in ejector performance when the entrainment ratio varies based on experimental research. However, the causes of a sharp drop in the secondary fluid mass flow and the entrainment ratio are not given, especially when the external pressure changes. With the help of the powerful visualization and

post-processing functions of the CFD method, this study conducted more comprehensive and detailed research of the internal flow of the steam ejector. The effective area remains constant and the steam ejector is operating at the double-choked flow mode (in Fig. 16). When the back pressure is higher than the critical value, the primary fluid is forced to move further upstream. The back pressure is too high for the primary jet to flow into the diffuser section, and the jet core moves into the mixing section. The energy and the momentum exchange between the primary fluid and the secondary fluid are too short of constituting the convergent-divergent structure for pumping the secondary fluid. The secondary fluid cannot be further accelerated; then the flow rate is rapidly reduced. The steam ejector lost its function because of no choking formation in the secondary fluid. There would be no normal shock wave, and choke occurred when the pressure is more than 5500 Pa, the pumping capacity of steam ejector decreases or even lost (the pressure is 6000 Pa). From the above analysis, it can be concluded that the choke only occurs at the entrance of the steam ejector throat that can ensure the chokes can be established in a stable state. Moreover, the choking position determines the performance of the ejector which means that the closer the choke position is to the throat, the greater the performance of the ejector.

6. Conclusions In this work, numerical simulations have been performed to comprehensively analyse the choking flow behaviour of an ejector refrigeration system. Through the application of CFD approach, the occurrence of the choke location and the influence of the choking behaviour towards the flow was thoroughly analysed. The relationship between the choking flow and the operating parameters was established. Simulation case studies with various primary fluid pressure have been examined. Based on the numerical findings, it was discovered that there were three major choked flow behaviours including the ―sub-choked flow mode‖, ―fit-choked flow mode‖ and ―super-choked flow mode‖. When the primary fluid pressure was working in

the ―sub-choked flow mode‖, the mass flow rate proportionally increased versus the primary fluid pressure. The steam ejector was identified to be operating at ―fit-choked flow mode‖ when the entrainment ratio reached its maximum point, in which the highest pumping efficiency was achieved. With further increased of primary fluid pressure after the ―fit-choked flow mode‖, there would only be a negative influence on the entrainment ratio, due to the enlarged motive fluid jet core and the effective area, which was numerically represented via the prediction of velocity vectors. Additionally, the typical shock wave at the diffuser prevented the disturbance caused by back pressure, maintaining the ejector to be operating in fit-choked flow condition. As the back pressure exceeded the critical pressure, the normal shock wave disappeared and the choking condition was exterminated, leading to a dramatic drop of the pumping efficiency. When the back pressure was higher than the critical value, backflow might occur causing the steam ejector to lose its pumping capacity completely. For the first time, the choked flow behaviour has been characterized from a numerical perspective, which will provide additional insights for the optimization of the steam ejector and the ejector refrigeration system, idealising the running conditions for the primary flow (i.e. 0.36 MPa).

Acknowledgement This work was supported by the Australian Research Council Industrial Transformation Training Centre in the University of New South Wales under Grant [number: IC170100032]; National Natural Science Foundation of China under Grant [number: 51775098]; and Australian Research Council under Grant [number: DP160101953]. All financial and technical supports are deeply appreciated by the authors.

Conflict of Interest Declaration

The Authors declare that there is no conflict of interest.

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List of Figure Captions Fig. 1

The ejector refrigeration experimental system；(a) The main components: 1 steam ejector, 2 condenser, 3 water-ring pump, 4 evaporator, 5 boiler, 6 evaporative condenser, 7 gas storage tank, 8 pressure measurement apparatus; (b) the steam ejector and the testing holes along the wall of the steam ejector.

Fig. 2

The structural mesh results of the computational domain.

Fig. 3

(a) Contours of Mach number under the ideal gas model and wet steam model; (b) Temperature contours of different back pressures.

Fig. 4

Comparison of the static pressure distribution along the wall between the ideal gas model and wet steam model.

Fig. 5

Comparison of entrainment ratio between experiments and simulations.

Fig. 6

Comparison of the mass flow rate varies with different primary fluid pressure between experiments and simulations.

Fig. 7

Comparison of the static pressure distribution along the wall between the experimental model and the CFD model under different back pressure.

Fig. 8

Contours of Mach number under different primary fluid pressure.

Fig. 9

Contours of Mach number for the various effective area.

Fig. 10

The effect of the primary fluid pressure on the ejector performance.

Fig. 11

Comparison of Mach number distribution along the wall of the ejector.

Fig. 12

Comparison of the mass flow rate with different back pressure between experiments and simulations.

Fig. 13

(a) Contours of Mach number for various back pressure; (b) Temperature contours of different back pressures.

Fig. 14

The path lines of the boundary layer separation under different back pressure.

Fig. 15

Comparison of the static pressure distribution along the axis under different back pressure.

Fig. 16

The effect of back pressure on the ejector performance.

List of Table Captions Table 1

Geometrical parameters of the steam ejector.

Table 2

Working conditions of the steam ejector.

Table 3

GCI analysis Table 1 Geometrical Parameters Diameter of nozzle inlet section Diameter of nozzle outlet section Diameter of nozzle throat Distance of the nozzle from inlet to throat Distance of the nozzle from throat to outlet Expand angle of nozzle Nozzle exit position Diameter of mixing chamber inlet section Diameter of throat Length of mixing chamber Length of throat Length of diffuser

Value 12 mm 11 mm 2.5 mm 40 mm 40 mm 10° -10 mm 70 mm 28 mm 160 mm 180 mm 260 mm

Table 2 Boundary conditions Primary fluid pressure Secondary fluid pressure Back pressure

Value 310,000~390,000 Pa 1,710 Pa 3,500~5,500 Pa

Table 3 Para meter f 21 [exact] GCI Nume rical Interval

1(Grid 1-2, 1-fine,2-medium)

2(Grid 2-3, 2-medium,3-coarse)

2.1235 1.63%

2.2541 1.98%

[2.1058，2.2877]

[2.2762，2.3146]

(a)

(b) Fig.1

Fig. 2

(a)

(b) Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

Fig. 15

Fig. 16

48