Optimization of the primary nozzle based on a modified condensation model in a steam ejector

Journal Pre-proofs Optimization of the primary nozzle based on a modified condensation model in a steam ejector Guojie Zhang, Sławomir Dykas, Sichong Yang, Xinzhe Zhang, Hang Li, Junlei Wang PII: DOI: Reference:

S1359-4311(19)37276-X https://doi.org/10.1016/j.applthermaleng.2020.115090 ATE 115090

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

22 October 2019 12 February 2020 14 February 2020

Please cite this article as: G. Zhang, S. Dykas, S. Yang, X. Zhang, H. Li, J. Wang, Optimization of the primary nozzle based on a modified condensation model in a steam ejector, Applied Thermal Engineering (2020), doi: https://doi.org/10.1016/j.applthermaleng.2020.115090

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2020 Published by Elsevier Ltd.

Optimization of the primary nozzle based on a modified condensation model in a steam ejector Guojie Zhanga,*, Sławomir Dykasb, Sichong Yangc, Xinzhe Zhangd, Hang Lia,1, Junlei Wanga,1 a

b

School of Mechanical and Power Engineering, Zhengzhou University, Zhengzhou, 450001, China

Department of Power Engineering and Turbomachinery, The Silesian University of Technology, Gliwice, Poland c

d

School of Chemical Engineering, Zhengzhou University, Zhengzhou, 450001, China

School of Aeronautical Engineering, Zhengzhou University of Aeronautics, Zhengzhou, 450046, China

Abstract: The beauty and complexity of steam ejectors have fascinated scientists for decade. A modified condensation model is presented to evaluate the steam ejector performance and to optimize the steam ejector geometry. Four models consisting of different nucleation models and droplet growth models are employed to predict the nonequilibrium condensation phenomenon in a nozzle and in a steam ejector. The results show that the modified condensation model is more appropriate than the other three models. The internal flow structures as well as p-h diagram are applied in the steam ejector performance analysis. The results show that the steam ejector entrainment ratio predicted by the modified condensation model is smaller than the value predicted by the dry gas model. Given that the nucleation process mainly occurs in the steam ejector primary nozzle, only

* First author. Tel.: +18336362536: E-mail address: [email protected], 1

Corresponding author. Hang Li, Junlei Wang. E-main address: [email protected]; [email protected] 1 / 50

the primary nozzle is optimized using the Multi-Objective Genetic Algorithm method based on the modified condensation model. The optimal results indicate that the entrainment ratio of the optimized steam ejector increases by about 27.5% against the original value. The findings are of fundamental and practical relevance. It hopes that experimenters will be motivated to check our predictions. Keywords: Steam ejector; Modified model; Flow structures; Primary nozzle optimization 1. Introduction Steam ejectors play a prominent part in power generation [1-2], fuel cell system [3], chemical processing industries [4], distillation [5-6], vacuum evaporation and drying [7], to name just a few areas of their common application. A typical steam ejector is made of a primary nozzle coupled with a secondary one [8] (cf. Fig. 1). High-pressure steam is accelerated to supersonic velocity in the primary nozzle, which will produce a low-pressure zone in the vicinity of the primary nozzle outlet. Then the secondary flow is induced since the pressure difference so as to attain the purpose of enhancing low-grade thermal energy [9]. It is also justified to add that, compared to a conventional pump or compressor, the steam ejector is characterized by excellent operating reliability and stability. Fig. 2 is presented for a better understanding of three segments of a typical performance curve, namely critical mode, subcritical mode and back-flow mode, respectively [10]. Region I, the condenser pressure (pc) is lower than the critical condenser pressure and the entrainment ratio (ER) keeps constant. Under such condition, the steam 2 / 50

ejector is in the critical mode. In Region II (the subcritical mode) the ER is rapidly reduced with an increase in pc. It is in Region III that the backflow mode will occur in the secondary flow, except that pc exceeds the condenser limiting pressure.

Fig. 1. Schematic diagram of a typical steam ejector

Fig. 2. Performance curve

The steam ejector performance can be significantly affected by its overall structure, the relative position of components [11], working circumstances [12], working fluids [13] etc. Dong et al. [14] present experimental investigations of the effects of operating conditions and geometry on the cooling capacity. Ramesh and Sekhar [15] discuss the influence of the suction chamber on the ejector performance. Considering how powerful they are, CFD (computational fluid dynamics) tools are 3 / 50

being gradually adopted by many researchers to predict the flow in the steam ejector. The works of Sriveerakul et al. [16-17] enable a clear understanding of the application of CFD techniques. Yang et al. [18] present numerical studies on the effects of the nozzle structure and they also suggest that the most crucial parameters which account for the phenomenon are vortex distribution and internal energy variation. Tang et al. [19-21] evaluate the way to improve the performance of steam ejectors under designed parameters with a novel auxiliary entrainment structure, focusing on the low-pressure potential and structure optimization. Han et al. [22] validate the numerical model by refrigeration experimental system and estimate the degree of boundary layer separation. The study is concluded emphasizing the importance of the nozzle exit position (NXP), which will create much more severe boundary-layer separation if selected wrongly. Ariafar et al. [23] provide a simulation to describe entrainment due to the mixing layer and pressure-driven effects in specific working conditions. Moreover, the CFD method is extensively used in many other studies on optimization based on the single-phase flow and the ideal gas model, such as the works on: 1.the primary nozzle structure and the NXP location [24-25]. 2. the diameter adjustment [26]. However, the single-phase flow and the ideal gas model are scarce to cope with the elaborate circumstances in steam ejector. There are two main reasons for that. Firstly, an excessive velocity gradient will lead to an excessive time difference between inertial restoration and temperature restoration, and the non-equilibrium condensation (NEC) will 4 / 50

follow. Secondly, the superheating degree of the primary and the secondary flow is relatively low, which will make it easy for static temperature to reach the critical point of the beginning of NEC. Previous theoretical and numerical studies focus on the following issues: the NEC impact on steam ejector performance [27], the influence of area ratio of primary nozzle outlet [28] and mixing section inlet, and surface roughness [29], comparison of different models [30-32]. Nevertheless, the wet steam model embedded in FLUENT is not competent to predict the NEC [33] because the solver for wet steam model lacks robustness. Moreover, a high-quality mesh is required. The result obtained by Ariafar et al. [34] is contrary to that obtained by Wang et al. Yang et al. [35], who develop a new wet steam model to capture characteristic parameters of steam ejector and obtain a lower result of ER, which is reasonable compared to the ideal gas model. The research results would have been more convincing, if the steam ejector parameters, especially the critical pc and the degradation rate in the subcritical mode, had been studied. It is the phase change accompanied by the heat and mass transfer and other complicated factors that makes the NEC process difficult to predict. The point is that CFD makes it much easier to capture parameters. It also creates a number of possibilities that were previously very difficult or not practical to implement, such as key parameters in the supersonic nozzle [36-37] and in the stator blade [38]. Based on current rationale, NEC is regarded as two processes: the droplet nucleation process and the droplet growth process. The former one can be calculated using a classical or a modified nucleation model. The 5 / 50

latter is mainly in the form proposed by Gyarmathy [41], Young [42] and Hill [43]. Another modified model, which is more accurate and reliable, is also presented and tested [44-45]. This paper presents a novel nucleation model to investigate and enhance the steam ejector performance. It makes use of the Benson surface tension model. Four NEC models are built, which are composed of the novel nucleation model and the classical nucleation model with several droplet growth models. The accuracy of the four NEC models is tested against experimental data obtained in a typical supersonic nozzle. At the same time, the steam ejector is investigated using the modified model and the ideal gas model to check the modified model rationality. On that basis, the steam ejector designed according to a one-dimension theoretical model is studied using the dry gas model and the modified model. The steam ejector internal flow structures are obtained and the differences between flow structures predicted by different models are found and discussed. At the same time, the ph diagram of the steam ejector operating under different modes is obtained using different models. Finally, the steam ejector primary nozzle, where the nucleation process mainly occurs, is optimized to enhance the steam ejector entrainment capability using the MultiObjective Genetic Algorithm (MOGA) method based on the modified condensation model. 2. Numerical models 2.1 Wet steam flow equations When the steam expands rapidly in steam ejector due to lower superheating, the bigger velocity gradient, the larger time difference between restoration of parameters, and a 6 / 50

certain value of the steam supercooling trigger the NEC process, which involves formation of tiny droplets. The two-phase flow is chosen to tackle the internal flow of steam ejector because the flow cannot be treated as a single-phase one in that point. To cope with the NEC process using numerical methods, the following assumptions are needed [46]: (a) The droplet diameter is small enough to omit the velocity slip between the droplets and vapor and the condensed liquid phase volume. (b) The interactions between droplets, including mass, heat and momentum transfer, are neglected. (c) The liquid mass fraction (𝛽) is less than 0.2. (d) The temperature and pressure of mixture flow is equal to that of the vapor phase. Then the relationship between the mixture density (𝜌) and the vapor density (𝜌𝑣) is described by Eq. (1): 𝜌𝑣 = 𝜌(1 ― 𝛽)

(1)

The flow governing equation that defines the mixture flow in the vector form is expressed as: ∂ ∂𝑡

∫ 𝑊𝑑𝑉 + ∮[𝐹 ― 𝐺] ∙ 𝑑𝐴 = 0 V

(2)

𝐴

[] [ ] [ ]

𝜌 𝜌𝑢 Where 𝑊 = 𝜌𝑣 , 𝐹 = 𝜌𝑤 𝜌𝐸

𝜌𝑣 0 𝜏𝑥𝑖 𝜌𝑣𝑢 + 𝑝i 𝜏𝑦𝑖 𝜌𝑣𝑣 + 𝑝𝑗 ,𝐺 = 𝜏𝑧𝑖 𝜌𝑣𝑤 + 𝑝𝑘 𝜏𝑖𝑗𝑣𝑗 + 𝑞 𝜌𝑣𝐸 + 𝑝𝑣

In addition, the liquid phase is controlled by the mass conservation equation (3): ∂𝜌𝛽 + ∇ ∙ (𝜌𝑣𝛽) = 𝑚 ∗ ∂𝑡

The droplet number per unit volume per second is defined by Eq. (4) below: 7 / 50

(3)

∂𝜌𝑁 + ∇ ∙ (𝜌𝑣𝑁) = 𝜌𝐽𝑑 ∂𝑡

(4)

Where 𝐽𝑑 is the droplets nucleation rate [47]. Eq. (1) and the droplet volume 𝑉𝑑 determine the number of droplets per unit volume, which is shown as Eq. (5): 𝑁=

𝛽 (1 ― 𝛽)𝑉𝑑(𝜌𝑙/𝜌𝑣)

(5)

The droplet volume 𝑉𝑑 is given by Eq. (6): 4 𝑉𝑑 = 𝜋𝑟3 3

(6)

Where 𝑟 is the droplet radius. 2.2 Phase change model Using the nucleation and droplet growth model to simulate the NEC process, the following hypotheses are proposed: (a) The condensation is homogeneous, namely, there are no impurities present to form nuclei. (b) The droplet is treated as spherical. (c) The droplet is tiny enough (about from approximately 0.1 microns to approximately 100 microns), that is, the droplet is surrounded by infinite vapor space. (d) The heat capacity of droplet is negligible against the latent heat released from the droplet condensation process. Mass production rate 𝑚 ∗ is determined by the droplet nucleation process and the droplets growth process throughout the NEC flow, and can be written as Eq. (7): 𝑑𝑟 4 𝑚 ∗ = 𝜋𝜌𝑙𝐽𝑑𝑟3∗ + 4𝜋𝜌𝑙𝑁𝑟2 3 𝑑𝑡

(7)

Where 𝑟 ∗ is the Kelvin-Helmholtz critical radius [48-49], which is given by Eq. (8): r∗ =

2𝜎 𝜌𝑙𝑅𝑇𝑣ln(𝑝/𝑝𝑠𝑎𝑡) 8 / 50

(8)

2.2.1 The modified nucleation model When the steam expands rapidly, tiny liquid droplets called “nuclei” are formed in the vapour and the phenomenon is referred to as “nucleation” [50]. Nowadays, the nucleation model corrected by Kantrowitz [51] has been widely employed. It can be written as follows:

𝐽𝑑 =

𝜌2v

()

𝑞𝑐

1 + 𝐶 𝜌𝑙

― 2𝜎(𝑇𝑑) exp 𝑀3𝑚𝜋

4𝜋𝑟2∗ 𝜎(𝑇𝑑)

(

3𝑘𝑇𝑣

)

(9)

Where 𝑞𝑐, 𝑘, 𝑀𝑚 and C are the condensation coefficient (generally 𝑞𝑐 = 1), and the other quantities are the Boltzmann constant, the individual water molecular mass and the non-isothermal correction coefficient, respectively. In addition, C can be written as Eq. (10): 𝐶 =

2(γ ― 1) ℎ𝑣𝑙 ℎ𝑣𝑙 1 ― (𝛾 + 1) 𝑅𝑇𝑣 𝑅𝑇 2

(

)

(10)

The surface tension 𝜎 is given by Eq. (11) [52]:

(

𝜎 = 𝜎𝑝(𝑇𝑑) = 235.8 1 ―

𝑇𝑑

1.256

) [

647.3

(

1 ― 0.625 1 ―

𝑇𝑑

)]

647.3

(11)

The droplet surface tension model has a direct influence on the nucleation process, as Eq. (9) shown. From Eq. (11), it is clear that the surface tension 𝜎 is only a function of temperature, which is not reasonable. When the liquid-plane tension model is applied in the nucleation model, a high error will be introduced. For this reason, the Benson model [53], which takes the nonlinear effect of both the droplet radius and the droplet temperature on the surface tension into consideration, is employed. It is described as follows:

(

σ(𝑇𝑑,𝑟) = 𝜎𝐵(𝑇𝑑,𝑟) = 𝜎𝑝 ∙ 1 ―

9 / 50

3

𝜌𝑙 𝑚

)

4.836𝑟

(12)

When the Benson surface tension model is employed, the nucleation model can be written as (namely the modified nucleation model): 𝑞𝑐 𝜌2 v 𝐽𝑑 = 𝐶 𝜌𝑙

()

― 2𝜎(𝑇𝑑,𝑟) exp 𝑀3𝑚𝜋

4𝜋𝑟2∗ 𝜎(𝑇𝑑,𝑟)

(

3𝑘𝑇

)

(13)

2.2.2 The droplet growth model A comprehensive review on the droplet growth model was given by Young [54]. There are three droplet growth models widely employed by researches, including Gyarmathy’s droplet growth model, Young’s droplet growth model and Hill’s droplet growth model. The Gyarmathy’s droplet growth model just solves the droplet energy equation, which leads the model is limited by the heat transfer rate from droplet to its surrounding vapor. The Gyarmathy’s droplet growth model is defined as: 𝑑𝑟 𝑑𝑡

=

ℎ𝑡

(𝑇𝑑 ― 𝑇𝑣) =

𝜌𝑙ℎ𝑣𝑙

𝜆𝑣(𝑇𝑑 ― 𝑇𝑣) ℎ𝑣𝑙𝜌𝑙𝑟(1 + 3.18Kn)

(14)

Where ℎ𝑡 is the heat transfer coefficient between the droplet and its surrounding vapor, which is determined by the Knudsen number (Kn = 𝑙/2𝑟). The expression for the mean free path is defined as follows: 𝑙 = 1.5𝜇𝑣 𝑅𝑇𝑣/𝑝

(15)

According to the reference [48], when the radius of the droplet is smaller than 100 μm, Eq. (16) can be employed to evaluate the droplet temperature. The Eq. (14) eliminates the heat and mass balances for the droplet, making the droplet growth model much easier to be solved. 10 / 50

𝑇𝑑 = 𝑇𝑠𝑎𝑡(𝑝) ― ∆𝑇

𝑟∗

( )

(16)

𝑟

Where ∆𝑇 reflects the vapor supercooling degree: (17)

𝛥𝑇 = 𝑇𝑠𝑎𝑡(𝑝) ― 𝑇v(𝑝)

Young’s droplet growth model is proposed on the basis of the Gyarmathy’s droplet growth model. To be better agreement with the experimental data, Young modified the Eq. (14) [42] based on the Gyarmathy’s assumptions, and the final droplet growth model is expressed as: 𝑑𝑟 𝜆𝑣(𝑇𝑑 ― 𝑇𝑣) = ℎ𝑣𝑙𝜌𝑙𝑟 𝑑𝑡

(1 ― 𝑟 ∗ /𝑟) 1 Kn + 3.78(1 ― 𝜓) ) 1 + 2𝜙Kn Pr

(

(18)

Where 𝜆𝑣 is the vapor thermal conductivity, Pr is the Prandtl number. 𝜓 is the correction factor and can be written as: 𝜓=

[

𝑅𝑇𝑠𝑎𝑡 ℎ𝑣𝑙

𝛼 ― 0.5 ―

( )(

)( )]

𝛾 + 1 𝐶𝑝𝑣𝑇𝑠𝑎𝑡 2 ― 𝑞𝑐 2𝛾 ℎ𝑣𝑙 2𝑞𝑐

(19)

Parameter 𝜙 and 𝛼 are introduced into Young’s droplet growth model. Many researchers set the 𝜙 = 0, but a few researchers set the 𝜙 = 2 to obtain better results. In this paper, 𝜙 = 0.75 shows a better agreement with experimental data. Parameter 𝛼 represents the relationship between the condensation and evaporation coefficients (𝑞𝑐 and 𝑞𝑣) based on the Taylor expansion, and 𝛼 = 9 is recommended by Young. Hill’s droplet growth model is based on the energy equation in the free molecular regime [48], which indicates that the model is applicable for large Knudsen numbers only. The expression is as follows: 11 / 50

𝑝 γ+1 𝑑𝑟 = 𝐶 (𝑇 ― 𝑇𝑣) 𝑑𝑡 ℎ𝑣𝑙𝜌𝑙 2𝜋𝑅𝑇𝑣 2γ 𝑝𝑣 𝑑

(20)

3. Solution schemes In this paper, the context of calculation is the ANSYS FLUENT 2019R1 and the UDF (User-Defined-Function). As for the turbulence model, the impact of the turbulence model on the numerical results has been sufficiently discussed by Besagni G [55] in supersonic flow. The 𝑘 ― 𝜔 SST with non-equilibrium wall function performs the best ability compared with the experimental data. Besides, the results calculated by using the secondorder upwind discretization scheme serve as the initial value, the third-order MUSCL scheme is adopted to obtain more accurate results. The double precision solver is recommended to solve the multiphase flow for a better result. The absolute convergence criterion for all equations are 10-6. 4. Model test The modified model reliability and accuracy are tested using the Moses-Stein [37] nozzle with detailed experimental data. The nozzle geometry and mesh are shown in Fig. 3 and Fig. 4, respectively. The nozzle throat is at 𝑋 = 0.0822m. The mesh around the nozzle throat is refined to cope with the large gradient of the physical parameters to ensure the calculation results accuracy.

Fig. 3. The Moses-Stein nozzle geometry [37] 12 / 50

Fig. 4. The nozzle mesh

It is indispensable to verify the mesh independence before the calculation. In this paper, four different mesh densities are selected to check the calculation result sensitivity to the mesh number. The inlet total pressure and total temperature are 43.02kPa and 366K, respectively. Supersonic flow occurs at the outlet. The pressure distribution calculated by different mesh densities and the values measured experimentally along the nozzle centerline is shown in Fig. 5. The pressure distribution is in good agreement with experimental data when the mesh density reaches 201 thousand elements. 0.60 P0=43.02kPa T0=366.0K

P/P0[-]

0.50

0.40

0.30

0.20

0.10 0.08

54 thousand 110 thousand 201 thousand 225 thousand Experiment [37] 0.09

0.10

0.11 0.12 X[m]

0.13

0.14

0.15

Fig. 5. Nozzle centerline pressure distribution

In order to facilitate analysis, a summary of the model is presented in Table 1. Table 1 The models summary

13 / 50

Name

Nucleation model

Droplet growth model

M-Y model

Modified nucleation model

Young’s droplet model (𝜙 = 0.75,𝛼 = 9)

M-G model

Modified nucleation model

Gyarmathy’s droplet growth model

K-Y model

Kantrowitz nucleation model

Young’s droplet model (𝜙 = 0.75,𝛼 = 9)

M-H model

Modified nucleation model

Hill’s droplet growth model

The calculation results obtained using different models and the experimental data are compared in Fig. 6. The results calculated by all four models can accurately map the pressure distribution along the nozzle centre line, except the rapid condensation zone. Except for the result predicted by the M-H model, the results predicted by the other three models are higher than the experimental data. The M-Y model shows the best ability to predict the pressure distribution in the NEC flow. The main reason is that the modified nucleation model takes account of the effect of the droplet temperature and the droplet radius on the droplet surface tension, and parameters 𝜙 and 𝛼 are adjusted. The M-H model shows the most unsatisfactory ability to calculate the pressure distribution. This is mainly due to the fact that Hill’s droplet growth model is based on the energy equation in the free molecular regime, which decides that the model is applicable for large Knudsen numbers only.

14 / 50

Experimental Data [37]

0.6

The M-Y model The M-G model The K-Y model

P/P0[-]

0.5

The M-H model

0.4 0.4

P/P0[-]

0.3 0.2

0.3 0.11

0.1 0.08

0.12 X[m]

0.09

0.10

0.11

0.12

0.13

0.14

0.15

X[m]

Fig. 6. Nozzle centerline pressure ratio distribution calculated by different models

To test the modified model accuracy in predicting the steam ejector performance, a steam ejector [56] with experimental data is used. The total pressure and total temperature of the primary nozzle inlet are 270kPa and 130℃, respectively. The total pressure and total temperature of the secondary nozzle inlet are 1.6kPa and 14℃. Moreover, the condenser pressure is 4.2kPa. The static pressure distribution along the steam ejector wall predicted by different models and the experimental data is shown in Fig. 7. The static pressure distribution predicted by the modified model and the ideal gas model, in general, shows little difference compared to the experimental data. However, the static pressure predicted by the modified model is still more in line with the experimental data, especially before the flow reaches the diffuser. This indicates that the modified model is more suitable to predict the flow in the steam ejector than the ideal gas model. Besides, the flow is handled as a single phase and the working fluid is considered as ideal gas in steam ejector, which are 15 / 50

not reasonable.

Static pressure[kPa]

8

6

Experimental data [56] Modified model Ideal gas model

4

2

0 0.0

0.1

0.2

0.3

0.4

0.5

Axial position[m]

Fig. 7. Static pressure distribution predicted by different models along the steam ejector wall

To check the modified model accuracy further, the steam ejector performance is predicted using different models. It can be concluded that under double-choking mode, the entrainment ratio predicted by the modified model shows a smaller relative error compared to the ideal gas model (about 14% and about 35%, respectively), and compared to the value obtained from the experiment. This indicates that the modified model is more suitable to predict the steam ejector performance and the flow structures. Moreover, the condenser critical pressure calculated by the modified model shows more satisfactory agreement.

16 / 50

Entrainment ratio [-]

0.5 0.4 0.3 0.2 Experimental data [56] Modified model Ideal gas model

0.1 0.0

4

5

6

7

8

Condenser pressure pc[kPa]

Fig. 8. Steam ejector performance predicted by different models

In this section, the modified model excellent performance is given by the comparison presented in Fig. 6, Fig. 7 and Fig. 8, which establishes a firm foundation for the following evaluation of NEC in the steam ejector. 5. Steam ejector analysis A steam ejector was designed according to a one-dimension theoretical model [57] by Ural Federal University [58]. The device had been checked for dry gas. The steam ejector geometry with detailed dimensions is shown in Fig. 9. The primary nozzle and the secondary nozzle angle are about 12° and 10°, respectively. The boundary conditions are listed in Table 2. The p-h diagram presented in Fig. 10 for the steam ejector tested operating conditions is made taking no account of the NEC process at pc=5.0 kPa.

17 / 50

Fig. 9. Steam ejector geometry with detailed dimensions Table 2 Boundary conditions for the steam ejector Boundary condition Primary Inlet Secondary Inlet

𝑝0,1 = 47.5𝑘𝑃𝑎, 𝑇0,1 = 140℃ 𝑝0,2 = 1.23𝑘𝑃𝑎, 𝑇0,2 = 10℃

Outlet

𝑝𝑐 = 5.0~8.0𝑘𝑃𝑎

Steam ejector wall

Smooth adiabatic wall

60 47.8

p =47.5kPa 0,1

47.6

p1=47.31kPa

47.2

40

h1=2663.76kJ/kg

47.0

1.3

20

2760

2762

2764

h[kJ/kg]

p2=1.224.28kPa h2=2517.49kJ/kg

1.2

p0,2=1.23kPa 1.1

10

2758

1.4

30 p[kPa]

p[kPa]

2756

apor line

47.4

Saturate dv

p[kPa]

50

h0,1=2759.31kJ/kg

1.0 2515

p0,out=8.228kPa

h0,2=2517.99kJ/kg 2516

2517

2518

2519

h0,out=2722.98kJ/kg

2520

h[kJ/kg]

pout=5kPa

0 -10

hout=2663.76kJ/kg

2200

2300

2400

2500

2600

2700

2800

h[kJ/kg] Fig. 10. Thermodynamic transformations of fluids in the steam ejection ignoring the NEC

As a significant parameter, the ER reflects the steam ejector performance. It is defined as: 𝐸𝑅 =

𝑚𝑠 𝑚𝑝

(21)

Where 𝑚𝑠 and 𝑚𝑝 are the inlet mass flow rate of the secondary and the primary 18 / 50

nozzle, respectively. Taking ER as criterion, the mesh independence test is carried out using the modified model. Four sets of meshes (a: 340065 quadrilateral cells, b: 314971 quadrilateral cells, c: 296145 quadrilateral cells, d: 276543 quadrilateral cells) are selected and calculated in this paper under critical mode. The Grid Convergence Index (GCI) method is employed to obtain an optimal mesh density. The GCI for the fine mesh can be defined as follows [46]: 𝐺𝐶𝐼 =

F𝑠|𝜀| 𝑟𝑝 ― 1

(22)

× 100

Where 𝐹𝑠 is a safety factor being an empirical parameter, 𝑝 is the formal order of the algorithm accuracy, 𝜀 is the relative error. The GCI test results are shown in Table 3. It can be seen that the GCI values of Grid a-b are satisfactory, and the mesh b is selected. Table 3 The Grid Convergence Index (GCI) test results

ER

𝐹𝑠

𝑝

3

3

Grid a-b 𝜀𝑎,𝑏(%) 𝐺𝐶𝐼𝑎,𝑏(%) 0.11

0.41

Grid b-c 𝜀𝑏,𝑐(%) 𝐺𝐶𝐼𝑎,𝑏(%) 2.63

9.48

Grid c-d 𝜀𝑐,𝑑(%) 𝐺𝐶𝐼𝑎,𝑏(%) 4.50

16.56

The refine mesh is shown in Fig. 11 (a). The mesh quality is checked and shown in Table 4. The three-dimensional model and periodic boundary are adopted to tackle the details, as shown in Fig. 11 (b). The period is 7.5°.

(a) Fig. 11. Steam ejector mesh with enlarged zones 19 / 50

(b)

Table 4 Mesh quality parameters Cells number

/ Worst value > 0.95 Worst value > 4.8 Worst value > 0.95 Worst value > 4.5 Worst value > 81°

Eriksson Skewness Aspect ratio Quality Taper Angle

314971 0.81 98.67% 8.6 4.6% 0.846 99.89% 18 1.1% 55° 96%

5.1 Steam ejector performance evaluation The ER is one of the most critical parameters to characterize the steam ejector performance. The modified model, the dry gas model and the ideal gas model are adopted to evaluate the steam ejector performance. The aim is to determine differences between the results of calculations performed using the three models mentioned above. Fig. 12 presents a comparison of results calculated by different models. It can be concluded that: (a) The ER predicted by the ideal gas model and the modified model is smaller compared to the dry gas model in the double-choking mode. This is mainly due to the fact that the ideal gas model treats steam as ideal gas, while the dry gas model adopts steam as a real gas, which results in a difference in the steam property. The ER predicted by the modified model is much smaller than the value predicted by the dry gas model and the ideal gas model. This is mainly due to the fact that too much loss caused by NEC degrades the steam ejector performance excessively. Moreover, it also indicates that the reduction in the ER is mainly due to NEC and not to the adopted property of steam. (b) 20 / 50

The condenser critical pressure predicted by the modified model is the highest, which indicates that the critical mode zone taking account of NEC is wider than that predicted by the dry gas model and the ideal gas model. The onset of NEC can delay the occurrence of the single-choking mode. (c) In the single-choking mode, the steam ejector performance deterioration rate predicted by the modified model is sharp, which shows that the ER is very sensitive to the condenser pressure in the single-choking mode when NEC is taken into account. (d) The condenser critical pressure calculated by the dry gas model is higher compared to the modified model and the ideal gas model, which indicates that steam treated as dry gas and the single-phase-flow approach will increase the single-choking zone.

Entrainment ratio [-]

0.2

0.0

-0.2 Ideal gas model Dry gas model Modified model

-0.4

-0.6

5

6

7

8

Condenser pressure pc[kPa]

Fig. 12. Steam ejector ER predicted by different models

5.2 Steam ejector internal flow structures investigation using different models In this section, the steam ejector internal flow structures are investigated in different operating regions, i.e. in double-choking (𝑝𝑐 = 5𝑘𝑃𝑎), single-choking (𝑝𝑐 = 7.6𝑘𝑃𝑎) and 21 / 50

backflow (𝑝𝑐 = 8𝑘𝑃𝑎), to establish the NEC impact using the dry gas model and the modified model.

𝑝𝑐 = 5.0𝑘𝑃𝑎 (Doublechoking)

𝑝𝑐 = 7.6𝑘𝑃𝑎 (Singlechoking)

𝑝𝑐 = 8.0𝑘𝑃𝑎 (Backflow)

Fig. 13. Static pressure distribution

Fig. 13 presents a comparison of the static pressure distribution in different operating regions. In general, a series of oblique shock and expansion waves can be observed in the secondary nozzle. It follows from the pressure distribution of the primary nozzle outlet that the primary nozzle flow is under expansion. When the steam ejector is in the doublechoking mode, i.e. when 𝑝𝑐 = 5𝑘𝑃𝑎, the location of the shock wave moves far away from the steam ejector outlet. However, when the steam ejector is in the single-choking and in the backflow mode, the location of the shock wave is shifted back forward. Moreover, the influence moves gradually upstream with a rise in the condenser pressure. This indicates that the static pressure distribution is more sensitive to the condenser pressure when the 22 / 50

double-choking mode is broken down. When the steam ejector is in the backflow mode, backflow occurs in the secondary nozzle inlet, and the location of the shock wave moves closer to the secondary nozzle throat. The shock waves emerge near the primary nozzle outlet in the backflow mode when the modified model is used. This is mainly due to the fact that during the condensation process, the heat and mass transfer between the droplet and the surrounding vapour, and the static pressure jump, will be diminished.

𝑝𝑐 = 5.0𝑘𝑃𝑎 (Double-choking)

𝑝𝑐 = 7.6𝑘𝑃𝑎 (Single-choking)

𝑝𝑐 = 8.0𝑘𝑃𝑎 (Backflow)

Fig. 14. Mach number distribution

Fig. 14 presents the steam ejector Mach number distribution predicted by the two models in different operating regions. Obviously, when the condenser pressure is 5kPa, the flow in the throat of the primary and the secondary nozzle enters the double-choking mode. With the condenser pressure increasing, the choking state is broken down in the second nozzle and the steam ejector becomes gradually off-design. When the condenser pressure 23 / 50

reaches 8kPa, backflow occurs in the secondary nozzle inlet, and clear backflow emerges near the secondary nozzle wall in the mixing section. Mach number predicted by the modified model is smaller. This is due to the fact that the heat and mass transfer in rapid condensation reduces the steam velocity, which results in an energy loss. Fig. 15 presents the static pressure distribution along the steam ejector centre line determined using the two models under different condenser pressures. The static pressure distribution predicted by both models is almost the same before rapid condensation occurs in the primary nozzle. When rapid condensation occurs, the droplet releases latent heat to the surrounding vapour, causing vapour re-compression and a pressure jump, which results in a smaller ER predicted by the modified model compared to the dry gas model. Besides, no matter how the condenser pressure changes, the location of the pressure jump keeps almost constant, which indicates that the condenser pressure has little influence on NEC in the primary nozzle. When the steam ejector is in the double-choking state, a clear shock wave can be observed in the mixing section and in the diffuser section. In the mixing section, only a clear shock wave emerges, and the pressure fluctuation predicted by the modified model is bigger compared to the prediction of the dry gas model in the mixing section, and the pressure crest location predicted by the modified model moves forward. The other shock waves arise in the diffuser section, and the pressure fluctuation calculated by the modified model is smaller compared to the dry gas model, but the predicted pressure crest location is almost the same in both models. As the condenser pressure increases, the 24 / 50

shock wave in the diffuser section moves upstream, but the pressure crest location predicted by the modified model moves forward compared to the dry gas model prediction. Meanwhile, the pressure fluctuation frequency increases with a rise in the condenser pressure, but the pressure fluctuation amplitude decreases. When the condenser pressure reaches 8kPa, the pressure fluctuation predicted by the modified model takes place in the mixing section, while the pressure fluctuation calculated by the dry gas model emerges in the diffuser section, which indicates that NEC diminishes the pressure fluctuation and the shock intensity in the backflow mode substantially. 1.0 Dry gas model Modified model

𝑝𝑐 = 5.0𝑘𝑃𝑎 (Double-choking)

P/P0,1[-]

0.8 0.6 0.4 0.2 0.0 0.0

0.1

0.2

0.4

0.3

0.5

0.6

0.7

X[m] 1.0 Dry gas model Modified model

𝑝𝑐 = 7.6𝑘𝑃𝑎 (Single-choking)

P/P0,1[-]

0.8 0.6 0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4 X[m]

25 / 50

0.5

0.6

0.7

1.0 Dry gas model Modified model

𝑝𝑐 = 8.0𝑘𝑃𝑎 (Backflow)

P/P0,1[-]

0.8 0.6 0.4 0.2 0.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X[m]

Fig. 15. Pressure distribution along the steam ejector centerline

The static temperature distribution predicted by different models under different modes is shown in Fig. 16 along the steam ejector centerline. The conclusions can be received that: (a) In most region of the steam ejector, the temperature predicted by the dry gas model is low the triple point temperature, and with the condenser pressure increasing, the region where the temperature is low the triple point temperature gradually gets smaller. Meanwhile, with the condenser pressure increasing, the temperature fluctuation region reduces. (b) The temperature predicted by the modified model is much higher than that calculated by the dry gas model, the main reason is that the NEC emerging leads the droplet releasing latent heat into its surrounding vapor, increasing the vapor temperature. Notably, the temperature distribution under backflow mode, the higher temperature is mainly caused by the high temperature steam back flowed from steam ejector outlet. (c) The temperature predicted by the modified model is low the triple point temperature in some regions, however, due to the limited region involved and the subsequent abrupt temperature increase caused by the appearance of a shock, it is reasonable to believe that no ice form.

26 / 50

(Double-choking)

Temperature[K]

𝑝𝑐 = 5.0𝑘𝑃𝑎

Dry gas model Modified model

400 300 200

Triple point temperature

100 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

𝑝𝑐 = 7.6𝑘𝑃𝑎 (Single-choking)

Temperature[K]

X[m]

Dry gas model Modified model

400 300 200 100 0.0

Triple point temperature

0.1

0.2

0.4

0.3

0.6

0.5

0.7

𝑝𝑐 = 8.0𝑘𝑃𝑎 (Backflow)

Temperature[K]

X[m]

400 Triple point temperature

300 200 100 0.0

Dry gas model Modified model 0.1

0.2

0.3

0.4

0.5

0.6

0.7

X[m]

Fig. 16. Temperature distribution along the steam ejector centerline

Fig. 17 presents the Mach number distribution along the steam ejector centre line in different operating modes using different models. It can be concluded that: (a) The flow is always supersonic in the secondary nozzle when the steam ejector is in the critical mode. As the condenser pressure increases, the flow in the secondary nozzle becomes subsonic. (b) Compared to the Mach number distribution predicted by the dry gas model, a sudden 27 / 50

decrease in the Mach number occurs in the primary nozzle, which is caused by NEC. Besides, the sudden decrease has nothing to do with the condenser pressure, which indicates that the condenser pressure has no effect on the nucleation process and rapid condensation. (c) In the backflow mode, the reduction in the Mach number in X=0.25m is caused by the backflow.

𝑝𝑐 = 5.0𝑘𝑃𝑎 (Double-choking)

Mach Number[-]

5 Dry gas model Modified model

4 3 2 1 0 0.0

0.1

0.2

0.4

0.3

0.6

0.5

0.7

X[m]

𝑝𝑐 = 7.6𝑘𝑃𝑎 (Single-choking)

Mach Number[-]

5 Dry gas model Modified model

4 3 2 1 0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X[m]

𝑝𝑐 = 8.0𝑘𝑃𝑎 (Backflow)

Mach Number[-]

5 Dry gas model Modified model

4 3 2 1 0 0.0

0.1

0.2

0.3

0.4

0.5

X[m]

Fig. 17. Mach number distribution along the steam ejector centerline 28 / 50

0.6

0.7

The nucleation distribution is shown in Fig. 18 under different steam ejector operation modes. The nucleation process mainly takes place in the primary nozzle divergence section, and a small nucleation zone emerges near the primary nozzle outlet. Besides, no matter how the condenser pressure changes, the nucleation zone almost keeps constant, which indicates that the condenser pressure has nothing to do with the nucleation process, and the nucleation process is done before the primary steam begins mixing with the secondary steam. Emerging NEC reduces the primary steam quality in the primary nozzle, which involves deterioration in the entrainment capacity.

𝑝𝑐 = 5.0𝑘𝑃𝑎 (Double-choking)

𝑝𝑐 = 7.6𝑘𝑃𝑎 (Single-choking)

𝑝𝑐 = 8.0𝑘𝑃𝑎 (Backflow)

Fig. 18. Droplets nucleation rate distribution under different modes

5.3 p-h diagram of the steam ejector under different modes predicted by different models This section presents the steam ejector p-h diagram investigations using different 29 / 50

numerical models in different modes. The characteristics of the steam ejector thermal dynamics are found and the work process difference predicted by different models is established. Fig. 19 shows the steam ejector p-h diagram for pc=5.0kPa, the parameters distribution is listed in Table 5. It can be concluded that: (a) The expansion line predicted by both models crosses the saturation line, but steam reaches lower pressure and enthalpy values compared to the dry gas model. This is probably due to the excessive energy loss caused by NEC. (b) The expansion line predicted by the modified model has a rising region (cf. enlargement in Fig. 19), which is different from the dry gas model. The main reason is that the droplet latent heat released into the surrounding vapour involves a jump in the vapour pressure and enthalpy. (c) The difference between pressure or enthalpy and total pressure or total enthalpy in the steam ejector outlet is about 3.1kPa and 58.6 kJ/kg, respectively, which indicates that the steam ejector outlet velocity is relatively high. It also reflects that in the critical mode the steam ejector has a strong entrainment capability.

30 / 50

Dry gas model Modified model

60

3.5

50

3.0

30

Saturated va por line

2.5

p[kPa]

p[kPa]

40

2.0 1.5 1.0 0.5

20

0.0 2240 2260 2280 2300 2320 2340 h[kJ/kg]

10

0 2100

2200

2300

2400

2500

2600

2700

2800

h[kJ/kg]

Fig. 19. Steam ejector p-h diagram obtained from different models for Pc=5.0kPa Table 5 Parameters distribution predicted by different models under critical mode

Primary inlet Secondary inlet Outlet

Dry gas model Modified model Dry gas model Modified model Dry gas model Modified model

p0 (kPa)

h0 (kJ/kg)

p (kPa)

h (kJ/kg)

47.50 47.50 1.23 1.23 8.229 8.197

2759.31 2759.31 2517.99 2517.99 2722.99 2731.95

47.31 47.31 1.224 1.228 5.011 5.007

2758.70 2758.70 2517.49 2517.80 2663.76 2673.36

The p-h diagram of the steam ejector with pc=7.6kPa is charted in Fig. 20, and the parameters distribution is listed in Table 6. Compared with the p-h diagram of the steam ejector under pc =5.0kPa, the results can be received that: (a) The total enthalpy and total temperature in steam ejector outlet under pc =7.6kPa are lower than that under pc =5.0kPa, which indicates that the entrainment ratio decreases. (b) The steam expansion process is 31 / 50

the same in the primary nozzle, which indicates that the condenser pressure has nothing to do with the expansion process of the primary nozzle. (c) The difference between static enthalpy or static pressure and the total enthalpy or total pressure is little under singlechoking mode in steam ejector outlet, which reflects the outlet velocity is small. And the entrainment capability diminishes. Dry gas model Modified model

60

3.5

50

3.0

30

Saturated va por line

2.5

p[kPa]

p[kPa]

40

2.0 1.5 1.0 0.5

20

0.0 2240 2260 2280 2300 2320 2340 h[kJ/kg]

10

0 2100

2200

2300

2400

2500

2600

2700

2800

h[kJ/kg]

Fig. 20. Steam ejector p-h diagram obtained from different models for Pc=7.6kPa Table 6 Parameters distribution predicted by different models under single-choking mode

Primary inlet Secondary inlet Outlet

Dry gas model Modified model Dry gas model Modified model Dry gas model Modified model

p0 (kPa)

h0 (kJ/kg)

p (kPa)

h (kJ/kg)

47.50 47.50 1.230 1.230 7.874 7.926

2759.31 2759.31 2517.99 2517.99 2748.83 2740.49

47.31 47.31 1.229 1.228 7.60 7.60

2758.70 2758.70 2517.94 2517.82 2743.53 2734.29

Fig. 21 and Table 7 illustrate the steam ejector thermodynamic process in the 32 / 50

backflow mode obtained from different models. Compared to the process in the critical mode and in the single-choking mode, it can be concluded that the steam ejector secondary inlet enthalpy or pressure increases substantially. This is mainly due to the fact that the high temperature of steam flowing back from the steam ejector outlet causes an increase in the secondary flow enthalpy. Dry gas model Modified model

60

Saturated va por line

2.5 50

p[kPa]

40

30

20

p[kPa]

2.0 1.5 1.0 0.5 2240 2260 2280 2300 2320 2340 h[kJ/kg]

10

0 2200

2300

2400

2500

2600

2700

2800

h[kJ/kg]

Fig. 21. Steam ejector p-h diagram obtained from different models for Pc=8.0kPa Table 7 Parameters distribution predicted by different models under backflow mode

Primary inlet Secondary inlet Outlet

Dry gas model Modified model Dry gas model Modified model Dry gas model Modified model

p0 (kPa)

h0 (kJ/kg)

p (kPa)

h (kJ/kg)

47.50 47.50 1.2311 1.354 8.186 8.045

2759.31 2759.31 2732.40 2767.65 2760.61 2750.12

47.31 47.31 1.23 1.23 8.0 8.0

2758.70 2758.70 2732.27 2752.99 2757.12 2749.27

6. Steam ejector structure optimization based on the modified model 33 / 50

Based on above conclusions, the structure optimization, considering the NEC impact on the improvement in the steam ejector entrainment performance, should focus on the primary nozzle. The primary nozzle geometry, with five arcs, is shown in Fig. 22. During the optimization process, the size of the primary nozzle inlet and outlet is constrained, and parameters R1-R5 are optimized.

Fig. 22. Primary nozzle geometry

The Genetic algorithm (GA) is known for its reliability, gradient-free and varied solutions [59]. As a hybrid variant of the popular NSGA-Ⅱ (Non-dominated Sorted Genetic Algorithm-II), the MOGA (Multi-Objective Genetic Algorithm) is selected based on the controlled elitism concept. The process of MOGA workflow is shown in Fig. 23. The optimization algorithm detail can be found in [60].

34 / 50

Fig. 23. The MOGA workflow

The upper bound and lower bound of the optimized parameters are listed in Table 8. In this section, the steam ejector is optimized, which is under double-choking operating region, namely, the pc is 5kPa. Table 8 The constraint of optimized parameter Parameters

Setup condition

R1 R2 R3 R4 R5

Variable: 52.164-63.756mm Variable: 49.5-60.5mm Variable: 126.9-155.1mm Variable: 261.57-319.69mm Variable: 234-286 mm

2000 design points are calculated in this analysis to achieve the best three candidate points. The three points under optimization are shown in Table 9. After the primary nozzle is optimized, the ER increases remarkably by about 27.5%, which means that NEC occurred in the primary nozzle and that the primary nozzle geometry has a great impact on the steam ejector performance. Table 9 Three candidate points for optimum ER 35 / 50

Parameters

Candidate point 1

Candidate point 2

Candidate point 3

R1 R2 R3 R4 R5 ER

60.351mm 56.612 mm 150.650 mm 319.150 mm 285.540 mm 0.1347

60.185 mm 56.404 mm 137.570 mm 319.160 mm 285.390 mm 0.1327

60.182 mm 60.421 mm 135.400 mm 319.240 mm 284.690 mm 0.1312

The geometry of the optimized primary nozzle is shown in Fig. 24. It is obvious that the converging part is substantially changed for optimization, which indicates that the converging part of the primary nozzle has a greater impact on the steam ejector ER. This is mainly due to the fact that the optimized primary nozzle decreases the strength of NEC, reducing the flow energy loss.

25 Candidate 1

Y [mm]

20

Candidate 2

Candidate 3

Original

15 10 5 0

0

15

30

45

60

75

90

105

120

135

150

165

180

X [mm]

Fig. 24. The optimized primary nozzle geometry

To check the optimization influence on the steam ejector condenser critical pressure, the steam ejector optimized performance curve (i.e. candidate 1) is calculated and compared with the original steam ejector performance curve using the modified model, as shown in Fig. 25. The following can be observed: (a) The optimized steam ejector ER is increased by about 27.5% compared to the original steam ejector ER, mainly due to the fact that the NEC strength is weakened in the optimized steam ejector, which decreases the energy loss and enhances the steam ejector ER. (b) The optimized steam ejector critical 36 / 50

pressure gets slightly higher compared to the original steam ejector: an increase from 7.596kPa to 7.62kPa. Besides, the optimized steam ejector limiting pressure is increased. 0.2

Entrainment ratio [-]

0.1 0.0 -0.1 -0.2 -0.3 Original steam ejector Optimized steam ejector

-0.4 -0.5 -0.6 4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

Condenser pressure pc[kPa] Fig. 25. Performance curves of the optimized steam ejector and the original steam ejector

7. Conclusions This paper presents a modified condensation model to investigate the steam ejector internal flow structure and to optimize the ejector primary nozzle. The model accuracy is tested in a supersonic nozzle and a steam ejector to estimate the model feasibility. Different models are compared to prove the modified model advantage and the novel model is used to determine the optimization program. The conclusions are as follows: (a) Coupled with the radius and temperature of the droplet, a novel nucleation model is presented. Several NEC models are checked in the nozzle and in the steam ejector. The results show that the modified model demonstrates the best ability to predict the pressure 37 / 50

distribution in a supersonic nozzle, achieving good agreement with experimental data. At the same time, the modified model is more suitable for the steam ejector investigations than the ideal gas model. (b) The steam ejector is investigated numerically. The device performance is checked using different models and the results show that emerging NEC will decrease the steam ejector entrainment capability. Compared to the steam ejector ER predicted by the ideal gas model in the critical mode, the ER calculated by the dry gas model is higher, which is due to the steam property. It turns out, however, that the steam ejector ER reduction is mainly caused by NEC. (c) The steam ejector internal flow structures are studied using the modified model and the dry gas model. The results show that the pressure fluctuation predicted by the modified model moves far away from the steam ejector outlet and the temperature distribution taking account of NEC is much higher compared to the dry gas model prediction. Although in some regions the temperature falls below the triple point temperature, no ice is formed due to the limited region involved and the subsequent abrupt temperature increase caused by the appearance of a shock. (d) This paper is focused on optimization of the primary nozzle, where the nucleation process mainly arises. The optimal result obtained using the MOGA method based on the modified model indicates a profound influence of the converging part of the primary nozzle. The optimized steam ejector entrainment ratio is by 27.5% higher compared to the original. 38 / 50

Besides, the condenser critical pressure of the optimized steam ejector is slightly higher compared to the original steam ejector. In the future, the modified model should be used to optimize the geometry of the steam ejector secondary nozzle. Acknowledgements Thanks are due to the support by the National Natural Science Foundation of China (21676257, 51606171), the Postdoctoral Foundation of Henan Province (19030014, 1902008) and China Postdoctoral Science Foundation (2019M652575, 2019M652565). Nomenclature 𝑢,𝑣,𝑤 𝐸 𝑞 𝑁 𝐽𝑑 𝑉 𝑟 𝑟∗ 𝑝 𝑇 ℎ𝑣𝑙 𝐾𝑛 𝑙 𝑘 𝑀𝑚

velocity component ( m s ―1) total energy per unit mass ( J kg ―1) heat flux ( W m ―2) droplet number per unit volume (m ―3) nucleation rate (m ―3s ―1) volume (m3) droplet radius (m) droplet critical radius (m) pressure (N m ―2) temperature (K) latent heat (J kg ―2) Knudsen number (𝑙/2𝑟) mean free path of vapor molecules (m) Boltzmann’s constant (1.3807 × 10 ―23 J K ―1) the individual water molecular mass (kg) mass flow rate (kg 𝑠 ―1)

𝑚 𝑚∗ 𝐶𝑝 𝑞𝑐 𝛥𝑇 𝑡 NEC

―1

mass production rate (kg 𝑚 ―3 s ) specific heat at constant pressure (J kg ―1 K ―1) condensation coefficient supercooling degree (K) time (s) Non-equilibrium condensation 39 / 50

GCI MOGA NSGA-Ⅱ GA 𝐸𝑅 𝑃𝑟 𝑅 ℎ𝑡

Grid Convergence Index Multi-Objective Genetic Algorithm Non-dominated Sorted Genetic Algorithm-II Genetic algorithm entrainment ratio Prandtl number gas constant (J kg ―1 K ―1) heat transfer coefficient (𝑊 𝑚 ―2 𝐾 ―1)

Greek symbols specific heat ratio tuning parameter tuning parameter

𝛾 𝛼 𝜙 𝜆

thermal conductivity (W m ―1 K ―1) relative error density (𝑘𝑔 𝑚 ―3) viscous stress tensor liquid mass fraction surface tension (N m ―1) kinetic viscosity (Pa s)

𝜀 𝜌

𝜏 𝛽 𝜎 𝜇 Subscript 𝑙 B p 𝑣 𝑑 𝑠𝑎𝑡 0 𝑖,𝑗,𝑘

liquid phase Benson surface tension model Liquid-plant surface tension model vapor phase droplet saturation stagnation condition cartesian tensor notation

References [1] Liu X, Chen Z, et al. Reducing the unburned combustible in the fly ash from a 45,000 Nm3/h Ende Pulverized-Coal Gasifier by applying steam-solid ejector[J]. Applied Thermal Engineering, 2019, 149:34-40. [2] Chen W , Li B , Zhang S , et al. Simulation investigation on the design and operation strategy of a

40 / 50

660 MW coal-fired power plant coupled with a steam ejector to ensure NOx reduction ability[J]. Applied Thermal Engineering, 2017, 124:1103-1111. [3] Besagni G , Mereu R , Inzoli F , et al. Application of an Integrated Lumped Parameter-CFD approach to evaluate the ejector-driven anode recirculation in a PEM fuel cell system[J]. Applied Thermal Engineering, 2017, 121:628-651. [4] Besagni G , Mereu R , Inzoli F . CFD study of ejector flow behavior in a blast furnace gas galvanizing plant[J]. Journal of Thermal Science, 2015, 24(1):58-66. [5] Liu J, Wang L, et al. Thermodynamic analysis of the steam ejector for desalination applications[J]. Applied Thermal Engineering, 2019, 159:113883. [6] Gao S, Zhao H, et al. Study on the performance of a steam ejector with auxiliary entrainment inlet and its application in MED-TVC desalination system[J]. Applied Thermal Engineering, 2019, 159:113925. [7] Zhang G, Zhang X, et al. Performance evaluation and operation optimization of the steam ejector based on modified model[J]. Applied Thermal Engineering, 2019, 163:114388. [8] Besagni G , Mereu R , Chiesa P , et al. An Integrated Lumped Parameter-CFD approach for offdesign ejector performance evaluation[J]. Energy Conversion and Management, 2015, 105:697715. [9] Besagni G . Ejectors on the cutting edge: The past, the present and the perspective[J]. Energy, 2019, 170:998-1003. [10] Besagni G , Mereu R , Inzoli F . Ejector refrigeration: A comprehensive review[J]. Renewable and

41 / 50

Sustainable Energy Reviews, 2016, 53:373-407. [11] Dong J , Yu M , Wang W , et al. Experimental Investigation on Low-temperature Thermal Energy Driven Steam Ejector Refrigeration System for Cooling Application[J]. Applied Thermal Engineering, 2017: 123:167-176. [12] Yongzhi Tang, Zhongliang Liu, Yanxia Li, Can shi, Chen Lv. A combined pressure regulation technology with multi-optimization of the entrainment passage for performance improvement of the steam ejector in MED-TVC desalination system [J]. Energy, 2019, 175:46-57. [13] Besagni G , Mereu R , Di Leo G , et al. A study of working fluids for heat driven ejector refrigeration using lumped parameter models[J]. International Journal of Refrigeration, 2015, 58:154-171. [14] Dong J, Chen X, Wang W, Kang C, Ma H. An experimental investigation of steam ejector refrigeration system powered by extra low temperature heat source [J]. Int Communications Heat Mass Transfer 2017; 81:250-256. [15] Ramesh A, Sekhar SJ. Experimental studies on the effect of suction chamber angle on the entrainment of passive fluid in a steam ejector [J]. J Fluids Eng 2018; 140:011106. [16] Sriveerakul T , Aphornratana S , Chunnanond K . Performance prediction of steam ejector using computational fluid dynamics: Part 1. Validation of the CFD results [J]. International Journal of Thermal Sciences, 2007, 46(8):812-822. [17] Sriveerakul T , Aphornratana S , Chunnanond K . Performance prediction of steam ejector using computational fluid dynamics: Part 2. Flow structure of a steam ejector influenced by operating

42 / 50

pressures and geometries [J]. International Journal of Thermal Sciences, 2007, 46(8):823-833. [18] Yang X , Long X , Yao X . Numerical investigation on the mixing process in a steam ejector with different nozzle structures [J]. International Journal of Thermal Sciences, 2012, 56:95-106. [19] Tang YZ, Liu ZL, Li YX, Wu HQ, Zhang XP, Yang N. Visualization experimental study of the condensing flow regime in the transonic mixing process of desalination-oriented steam ejector. Energy Conversion and Management 2019; 197:111849. [20] Tang YZ, Liu ZL, Shi C, Li YX. A novel steam ejector with pressure regulation to optimize the entrained flow passage for performance improvement in MED-TVC desalination system. Energy 2018; 158:305-316. [21] Tang YZ, Liu ZL, Li YX, Shi C, Lv C. A combined pressure regulation technology with multioptimization of the entrainment passage for performance improvement of the steam ejector in MED-TVC desalination system. Energy 2019; 175:46-57. [22] Han Y, Wang X, Sun H, Zhang G, Guo L, Tu J. CFD simulation on the boundary layer separation in the steam ejector and its influence on the pumping performance [J]. Energy, 2019; 167:469-483. [23] Ariafar K , Buttsworth D , Al-Doori G , et al. Mixing layer effects on the entrainment ratio in steam ejectors through ideal gas computational simulations [J]. Energy, 2016, 95:380-392. [24] Varga S , Oliveira A C , Diaconu B . Influence of geometrical factors on steam ejector performance - A numerical assessment [J]. International Journal of Refrigeration, 2009, 32(7):1694-1701. [25] Wu Y, Zhao H, Zhang C, Wang L, Han J. Optimization analysis of structure parameters of steam ejector based on CFD and orthogonal test [J]. Energy 2018; 151:79-93.

43 / 50

[26] Ruangtrakoon N, Thongtip T, Aphornratana S, Sriveerakul T. CFD simulation on the effect of primary nozzle geometries for a steam ejector in refrigeration cycle[J]. Int J Thermal Sciences 2013; 63:133-45. [27] Sharifi , Boroomand M , Sharifi M . Numerical assessment of steam nucleation on thermodynamic performance of steam ejectors[J]. Applied Thermal Engineering, 2013, 52(2):449-459. [28] Wang C, Wang L, Zou T, Zhang H. Influences of area ratio and surface roughness on homogeneous condensation in ejector primary nozzle [J]. Energy Conversion & Management, 2017; 149:168174. [29] Ariafar K, Buttsworth D, Sharifi N, et al. Ejector primary nozzle steam condensation: Area ratio effects and mixing layer development [J]. Applied Thermal Engineering, 2014, 71(1):519-527. [30] Wang X D , Lei H J , Dong J L , et al. The spontaneously condensing phenomena in a steam-jet pump and its influence on the numerical simulation accuracy [J]. International Journal of Heat and Mass Transfer, 2012, 55(17-18):4682-4687. [31] Wang X , Dong J , Li A , et al. Numerical study of primary steam superheating effects on steam ejector flow and its pumping performance [J]. Energy, 2014, 78:205-211. [32] Wang X, Dong J, Zhang G, Fu Q, Li H, Han Y, et al. The primary pseudo-shock pattern of steam ejector and its influence on pumping efficiency based on CFD approach [J]. Energy, 2019;167:224234. [33] Zhang G, Zhang S, Zhou Z, et al. Numerical Study of Condensing Flow Based on the Modified Model [J]. Applied Thermal Engineering, 2017; 127:1206-1214.

44 / 50

[34] Ariafar K , Buttsworth D , Al-Doori G , et al. Effect of mixing on the performance of wet steam ejectors [J]. Energy, 2015, 93:2030-2041. [35] Yan Yang, Xiaowei Zhu, Yuying Yan, Hongbing Ding, Chuang Wen. Performance of supersonic steam ejectors considering the nonequilibrium condensation phenomenon for efficient energy utilization [J]. Appl Energy 2019; 242:157-167. [36] Moore MJ, Walters PT, Crane RI, Davidson BJ. Predicting the fog-drop size in wet steam turbines [C]. In: Proc of IMECHE Conf Wet Steam 4, Coventry, UK, 1973. [37] Moses, C. A., & Stein, G. D. (1978). On the Growth of Steam Droplets Formed in a Laval Nozzle Using Both Static Pressure and Light Scattering Measurements [J]. Journal of Fluids Engineering, 100(3), 311. [38] Han X, Zeng W, Han Z. Numerical investigation of the condensation flow characteristics and modification optimization of a condensing steam turbine cascade[J]. International Journal of Numerical Methods for Heat & Fluid Flow,2019,1-19. [39] J. Bian, X.W. Cao, W. Yang, X.D. Song, C.C. Xiang, S. Gao, Condensation characteristics of natural gas in the supersonic liquefaction process [J], Energy, 168 (2019) 99-110. [40] Zhang G, Wang L, Zhang S, et al. Numerical study of the dehumidification structure optimization based on the modified model [J]. Energy Conversion & Management, 2019, 181:159-177. [41] Gyarmathy G. The spherical droplet in gaseous carrier streams: Review and synthesis [J]. Multiphase Science & Technology, 1982, 1(1):99-279. [42] Young JB. The spontaneous condensation of steam in supersonic nozzles [J]. Physico Chem

45 / 50

Hydrodyn 1982; 3: 57-82. [43] Hill PG. Condensation of water vapor during supersonic expansion in nozzles[J]. J Fluid Mech 1966; 25:593-620. [44] Guojie Zhang, Xinzhe Zhang, et al. The relationship between the nucleation process and boundary conditions on non-equilibrium condensing flow based on the modified model [J], International Journal of Multiphase Flow,2019(114)180-191. [45] Guojie Zhang, Xinzhe Zhang, et al. Design and optimization of novel dehumidification strategies based

on

modified

nucleation

model

in

three-dimensional

cascade

[J],

Energy,

DOI:10.1016/j.energy.2019.115982. [46] Mohammad Ali Faghih Aliabadi, Esmail Lakzian, et al. Numerical investigation of effects polydispersed droplets on the erosion rate and condensation loss in the wet steam flow in the turbine blade cascade[J]. Applied Thermal Engineering, 2020; 164:114478. [47] Dykas S, Wróblewski W. Single- and two-fluid models for steam condensing flow modeling [J]. International Journal of Multiphase Flow, 2011, 37(9):1245-1253. [48] Young J B. Two-Dimensional Non-equilibrium Wet-Steam Calculations for Nozzles and Turbine Cascades [J]. Journal of Turbomachinery.1992, 114(3):569-579. [49] Sharifi N, Boroomand M, Kouhikamali R. Wet steam flow energy analysis within thermocompressors [J]. Energy, 2012, 47(1):609-619. [50] Han Xu, Zeng Wei, Han Zhonghe. Investigating the dehumidification characteristics of the lowpressure stage with blade surface heating[J]. Applied Thermal Engineering, 2020,164:114538.

46 / 50

[51] Kantrowitz A. Nucleation in Very Rapid Vapor Expansions [J]. Journal of Chemical Physics, 1951, 19(9):1097-1100. [52] Guojie Zhang, Xinzhe Zhang, et al. Numerical investigation of novel dehumidification strategies in nuclear plant steam turbine based on the modified nucleation model [J], International Journal of Multiphase Flow,2019(120)1-13. [53] Benson, G.C.; Shuttleworth, R. The surface energy of small nuclei [J]. Chem. Phys. 1951, 19, 130131. [54] J. Bian, X.W. Cao, W. Yang, D. Guo, C.C. Xiang, Prediction of supersonic condensation process of methane gas considering real gas effects[J], Appl. Therm. Eng. 164 (2020) 114508. [55] Besagni G, Inzoli F. Computational fluid-dynamics modeling of supersonic ejectors: Screening of turbulence modeling approaches[J]. Applied Thermal Engineering, 2017, 117:122-144. [56] Al-Doori GFL. Investigation of refrigeration system steam ejector performance through experiments and computational simulations [PhD thesis]. University of Southern Queensland; 2013. [57] Sokolov E.Y., Singer N.M., The jet devices, 3rd ed., Rev. M., Energoatomizdat, 1989. [58] F. Mazzelli, D. Brezgin, I. Murmanskii, N. Zhelonkin e A. Milazzo, Condensation in supersonic steam ejectors: comparison of theoretical and numerical models, in Proceedings of 9th International Conference on Multiphase Flow, Florence, 2016. [59] K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multi-objective genetic algorithm, NSGA-II [J], IEEE Trans. Evol. Comput. 6 (2002) 182–197.

47 / 50

[60] Mitchell M . An introduction to genetic algorithms [M]. MIT Press, 1996.

48 / 50

Highlights:    

The modified nucleation model is built. The modified model is more suitable to describe the non-equilibrium condensation. It is the first time to optimize steam ejector geometry based on modified model. The optimized steam ejector ER increases by about 27.5%.

49 / 50

We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted.

50 / 50