Luận văn thạc sĩ Kỹ thuật hàng không vũ trụ: Two-phase simulation considering phase change due to depressurization

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY

HO CHI MINH CITY UNIVERSITY OF TECHNOLOGY

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LUONG HUYNH DANG KHOA

TWO-PHASE SIMULATION CONSIDERING PHASE CHANGE DUE TO DEPRESSURIZATION

Major: Aerospace Engineering Major ID: 8520120

MASTER’S THESIS

HO CHI MINH CITY, February 2023

THIS RESEARCH IS COMPLETED AT:

HO CHI MINH UNIVERSITY OF TECHNOLOGY – VNU HCM Instructor: PhD Dang Le Quang

Examiner 1: Assoc Prof PhD Le Tuan Phuong Nam Examiner 2: PhD Tran Tien Anh

Master’s Thesis is defended at HCMC University of Technology, VNU-HCM on February 04, 2023

The Board of The Master’s Thesis Defense Council includes:

1 Chairman: Assoc Prof PhD Ngo Khanh Hieu 2 Secretary: PhD Le Thi Hong Hieu

3 Counter-Argument Member: Assoc Prof PhD Le Tuan Phuong Nam 4 Counter-Argument Member: PhD Tran Tien Anh

5 Council Member: PhD Pham Minh Vuong

Verification of the chairman of the Master’s Thesis Defense Council and the Dean of the Faculty of Transportation Engineering after the defense is correct (if any)

CHAIRMAN OF THE COUNCIL

(Full name and signature)

DEAN OF FACULTY OF TRANSPORTATION

ENGINEERING

(Full name and signature)

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VIETNAM NATIONAL UNIVERSITY HCMC

VNUHCM UNIVERSITY OF TECHNOLOGY _

SOCIALIST REPUBLIC OF VIETNAM Independent – Liberty – Happiness

_

MASTER’S THESIS ASSIGNMENTS

Full name: LUONG HUYNH DANG KHOA Learner ID: 2170727

Date of birth: January 13, 1998 Place of birth: Ho Chi Minh City

I TITLE: TWO-PHASE SIMULATION CONSIDERING PHASE CHANGE

DUE TO DEPRESSURIZATION / MÔ PHỎNG DÒNG HAI PHA CÓ XEM XÉT SỰ CHUYỂN PHA DO GIẢM ÁP

II ASSIGNMENTS AND CONTENTS:

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III ASSIGNMENT DELIVERING DATE (based on the Decision on

Assignments Delivering): September 05, 2022

IV ASSIGNMENT COMPLETING DATE (based on the Decision on

Assignments Delivering): December 18, 2022

V INSTRUCTOR: PhD Dang Le Quang

Ho Chi Minh City, December 16, 2022

INSTRUCTOR

(Full name and signature)

HEAD OF DEPARTMENT

(Full name and signature)

DEAN OF FACULTY OF TRANSPORTATION ENGINEERING

(Full name and signature)

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Acknowledgements

First of all, I would like to express my deep gratitude to my family, who have supported me throughout my studies During the thesis process, my family has always been by my side, supported and motivated me, which gives me more positive energy to successfully complete this master’s thesis

Next, I would like to thank Mr Dang Le Quang (Dr Dang Le Quang) for his enthusiastic guidance, provided helpful documents and tools to help me complete my master’s thesis well He was the one who has guided enthusiastically, spent time on supporting and helping me in disorientation times

Finally, I want to thank myself for constantly working hard every day to gain the right knowledge to solve this problem and complete the master’s thesis

Student

Luong Huynh Dang Khoa

iv Abstract

The master’s thesis aims to simulate the flashing flow phenomena inside a nozzle called “Super Moby-Dick” nozzle in high temperature and pressure difference between inlet and outlet conditions applying the proposed phase change model in Ref [1]

Based on the most optimal chosen mesh with slip model approach, and the combination of mixture model and mixed phase change model UDF code, the simulation provides interested information about the two-phase flashing flow behavior by results of pressure along nozzle, global and detailed local void fraction, as well as, the effect of bubble number density to pressure and void fraction

A good agreement of different average mass flow rate among inlet, outlet and experiment below 2.6% by using the standard K-ω turbulence model with Low-Re correction, and the closest pressure along nozzle profile compared to the experiment in almost points are achieved Besides, the discussion of the effect of bubble number density parameter to pressure and global void fraction are presented Finally, the detailed explanation of two-phase flashing flow behavior based on the simulation results of the global average void fraction and the local average void fraction in 8 sections along nozzle are discussed carefully in the thesis

v Tóm tắt luận văn

Mục tiêu của đề tài luận văn thạc sĩ là mô phỏng ứng xử hiện tượng “flashing flow” của dòng chảy hai pha bên trong vòi phun “Super Moby-Dick” trong điều kiện nhiệt độ cao và chênh lệch áp suất cao giữa đầu vào và đầu ra áp dụng mô hình chuyển pha được đề xuất trong bài báo [1]

Dựa trên mô hình lưới được chọn tối ưu nhất với cách tiếp cận mô hình trượt và sự kết hợp giữa mô hình hỗn hợp và bộ mã UDF về mô hình chuyển pha hỗn hợp, kết quả mô phỏng đã cung cấp các thông tin quan trọng về ứng xử của hiện tượng “flashing flow” của dòng chảy hai pha bằng kết quả của áp suất dọc theo vòi, tỉ lệ hơi toàn cục và tỉ lệ cục bộ tại nhiều vị trí dọc trục, cũng như ảnh hưởng của mật độ bong bóng đến áp suất và tỉ lệ hơi

Mô phỏng cung cấp giá trị sai số lưu lượng khối lượng trung bình giữa đầu vào, đầu ra và thí nghiệm dưới 2,6% bằng cách sử dụng mô hình rối K-ω tiêu chuẩn với chức năng hiệu chỉnh Low-Re, và đường áp suất dọc theo trục khớp với giá trị thực nghiệm ở hầu hết các điểm Ngoài ra, phần thảo luận về ảnh hưởng của tham số mật độ bong bóng đến áp suất và tỷ lệ hơi toàn cục cũng được trình bày trong luận văn Cuối cùng, phần giải thích chi tiết về ứng xử của hiện tượng “flashing flow” của dòng chảy hai pha dựa trên kết quả mô phỏng của tỉ lệ hơi trung bình toàn cầu và tỉ lệ hơi cục bộ thông qua 8 mặt cắt dọc theo trục được trình bày cẩn thận trong luận án

vi Commitment

I confirm that:

- This is my master’s thesis

- The data and results stated in the thesis are honest and have never been published in any other researches

- The quotations and results used for comparison in the thesis are all cited and have the highest accuracy to the extent of my knowledge

Student

Luong Huynh Dang Khoa

1.4.2 Numerical study reviews 5

1.4.2.1 Homogeneous equilibrium (HEM) models 6

1.4.2.2 Non-homogeneous equilibrium (NHEM) models 7

1.4.2.3 Homogeneous non-equilibrium (HNEM) models 8

1.4.2.4 Non-homogeneous non-equilibrium (NHNEM) models 9

1.4.3 Computational fluid dynamics (CFD) models 11

1.4.3.1 Thermal phase-change models 12

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1.4.3.2 Pressure phase-change models 13

1.4.3.3 Mixed phase change model 14

1.5 Outline of the thesis 15

2.3.2 Axial momentum conservation equation 24

2.3.3 Radial momentum conservation equation 25

3.2.1 Mass flow rate 31

3.2.2 Pressure along nozzle 32

3.2.3 Average vapor fraction along nozzle 32

4.2.1 Solution convergence type identification 38

4.2.2 Mesh convergence evaluation 40

4.2.2.1 Y plus comparison 40

4.2.2.2 Grid convergence index test 43

4.2.3 Turbulence modeling 44

4.2.4 Pressure profile analysis 48

4.2.5 The effect of bubble number density on pressure analysis 49

4.2.6 Inlet turbulent intensity analysis 51

4.2.7 Average void fraction analysis 52

4.2.7.1 Global results 53

4.2.7.1.1 Global average void fraction profile 53

4.2.7.1.2 The effect of bubble number density on void fraction 56

4.2.7.2 Local results 57

4.3 Discussion 59

61

CONCLUSION AND FURTHER DEVELOPMENT 61

5.1 Numerical results conclusion 61

A Mass Transfer UDF Code 70

B Experimental Pressure Data 72

C Experimental Void Fraction Data 73

D Y Plus Comparison Data 74

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List of Figures

Figure 2.1: Super Moby-Dick experiment setup [48] 16

Figure 2.2: The Super Moby-Dick nozzle geometry [48] 17

Figure 2.3: Moby-Dick nozzle geometry [48] 18

Figure 2.4: Measured pressure chart [48] 19

Figure 2.5: Cross-sectional average void fraction 20

Figure 2.6: Measured void fraction chart [48] 20

Figure 2.7: Flashing flow behavior: (a) thermodynamic diagram of phase change of water with equilibrium assumption and (b) start of vaporization [1] 21

Figure 3.1: Vertical circular convergent-divergent nozzle geometry [3] 29

Figure 3.2: Absolute pressure at nozzle axis comparison 32

Figure 3.3: Average vapor fraction comparison 32

Figure 4.1: Nozzle geometry separated sections 34

Figure 4.2: Moby-Dick meshing 35

Figure 4.3: Water properties 36

Figure 4.4: Water and vapor properties at saturation point 36

Figure 4.5: Scaled residuals chart 39

Figure 4.6: Mass flow rate trend line chart 39

Figure 4.7: Pressure at throat 40

Figure 4.8: Velocity at throat 40

Figure 4.9: Velocity section positions 42

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Figure 4.10: Velocity profile comparison at section 2 of mesh 4 42

Figure 4.11: Pressure Profile comparison 46

Figure 4.12: Pressure comparison among K-ω models 46

Figure 4.13: Velocity contour with the turbulence Standard K- ω (Low-Re correction) model 47

Figure 4.14: Pressure profile with the turbulence Standard K-ω (Low-Re correction) model 49

Figure 4.15: Pressure comparison of bubble number density values 50

Figure 4.16: Average void fraction profile with the turbulence Standard K-ω(Low-Re correction) model 53

Figure 4.17: Void fraction contour 54

Figure 4.18: Mixture fluid density contour 54

Figure 4.19: Mixture velocity contour 55

Figure 4.20: Average void fraction at nozzle axis 55

Figure 4.21: Average void fraction of bubble number density values 56

Figure 4.22: Local average void fraction sections 57

Figure 4.23: Turbulent kinetic energy contour 57

Figure 4.24: Average void fraction section profiles 57

Figure 5.1: Velocity profile comparison at section 1 of mesh 1 74

Figure 5.2: Velocity profile comparison at section 2 of mesh 1 74

Figure 5.3: Velocity profile comparison at section 3 of mesh 1 75

Figure 5.4: Velocity profile comparison at section 1 of mesh 2 75

Figure 5.5: Velocity profile comparison at section 2 of mesh 2 76

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Figure 5.6: Velocity profile comparison at section 3 of mesh 2 76

Figure 5.7: Velocity profile comparison at section 1 of mesh 3 77

Figure 5.8: Velocity profile comparison at section 2 of mesh 3 77

Figure 5.9: Velocity profile comparison at section 3 of mesh 3 78

Figure 5.10: Velocity profile comparison at section 1 of mesh 4 78

Figure 5.11: Velocity profile comparison at section 2 of mesh 4 79

Figure 5.12: Velocity profile comparison at section 3 of mesh 4 79

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List of Tables

Table 1: Numerical models overview [8] 5

Table 2: Experiment conditions 17

Table 3: Moby-Dick geometry parameters 18

Table 4: BNL working condition 29

Table 5: Boundary conditions 30

Table 6: BNL Ansys Fluent method 30

Table 7: Numerical approach list 30

Table 8: Mass flow rate comparison 31

Table 9: Nozzle section dimension 34

Table 10: Meshing information 35

Table 11: Meshing quality information 36

Table 12: Material properties 37

Table 13: Boundary conditions 37

Table 14: Method setup information 38

Table 15: First cell height of mesh information 41

Table 16: Grid convergence index information 43

Table 17: Turbulence K-ε models’ comparison 44

Table 18: Turbulence K-ω models’ comparison 45

Table 19: Mass flow rate comparison of bubble number density values 50

Table 20: Turbulent intensity comparison 51

When the water flows under high velocity and pressure working conditions, the erosion of pipe walls trends to accelerate generally due to the increased friction rate between dissolved solids in the water and the inner pipe walls Flashing flow is one of the most interested phenomena related the high-pressure factor The “flashing flow” term means a phase change from liquid to vapor due to the depressurization related to the nonequilibrium phase change effects If the vaporization happens because local pressure drops below the vapor pressure of fluid, as well as, the and the downstream pressure keeps below the vapor

1.3 Objectives and scope of study

The thesis is focused on simulations of “flashing flow” phenomena in entrance high pressure and temperature condition based on the modified thermal phase change model proposed in Ref [1] There are 5 objectives covered in the thesis including:

• Evaluate the suitable turbulence model for the flashing flow phenomena • Compare and discuss simulation results of pressure and void fraction to

Abuaf et al The experimental design concept was to measure net vapor

generation rates under nonequilibrium conditions by using a steady water loop with the well-controlled flow and thermodynamic The simple geometry of the nozzle comprised approximately 0.0512 meters of the inlet and outlet diameter and 0.0256 meters of throat diameter With different conditions of the constant temperature of subcooled water entrance combined with pressure inlet and outlet, various test cases were performed and effects of various parameters were reported and very useful for any simulated approaches [3]

An experimental program using the Moby–Dick nozzle was conducted at the Centre d’Etude Nucleaire (CEA) de Grenoble, whose purpose is to study the flashing flow phenomena of two-phase critical flow in different conditions for the hypothetical Loss of Coolant Accidents (LOCA) study in Pressurized Light Water reactors The Moby-Dick nozzle consisted of a smooth convergent-divergent section with an approximate outlet angle of 7 degrees, and a relatively long cylindrical throat One of the popular nozzle dimensions included a total length of one meter with an inlet diameter of 66.7 millimeters and an outlet diameter of 73.7 millimeters Subcooled liquid materials with various subcooling degrees to saturation ones, as well as, with different vapor mass fractions were taken into account as the nozzle inlet conditions The continuous reduce of pressure downstream was found, in which a maximum flow rate was

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obtained through the nozzle in the experiment The behavior of nozzle flow according to several results of critical mass flow rate, pressure, vapor fraction, etc was analyzed by different charts during the tests [4]

Investigation of a water and steam mixture through a naval nozzle was presented

by Deich et al, in which a broad spectrum of gas and various outlet pressure

conditions are applied The wet steam whose wetness ranged from 0.0 in a superheated vapor state to a maximum value of 0.83 was applied as the material for experiments The geometry of the nozzle consisted of a circular inlet section with a radius of 28 mm, connected to a cone having the length of 122 mm and an approximate angle of 3.3 degrees The fixed value of pressure mixture entrance at 1.2 bar was used, as well as, the outlet pressure was reduced down to various values from 0.1 bar to 0.95 bar As a result, the flashing flow occurred due to the high depressurization in the nozzle The static pressure along the nozzle, as well as, liquid and vapor mass flow rates were collected and be preferably considered as qualitative measures for the flow behavior [5]

Besides, the flashing flow was also considered to be one of the preferred topics in pipe blowdown experiments One of the famous pipe blowdown transients in the 1970s was presented by Edwards In the experiment, an initial subcooled liquid at a high pressure of about 7.0 MPa and a temperature of 502K corresponding to a subcooling of 56.8K was fulfilled in a long one-closed end pipe The outlet was controlled by a rupture disk allowing the rapid depressurization to the environment at atmospheric pressure The values of pipe diameter and length were 0.076 m and 4.096 m respectively The area of flow at the rupture was reduced by about 13 percent The pressure along the pipe and void fraction at different sections of the pipe were found These results were used to compare to estimated values in many research in the safety nucleation major [6]

An experiment called “Super-Canon” test program was performed at the Centre d’Etude Nucleaire (CEA) de Grenoble, which was similar to Edward’s pipe

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blowdown experiment A horizontal pipe of 4.39 meters in length and an internal diameter of 0.1 m with a one-closed end was also used However, the outlet of the pipe was opened totally to the atmosphere environment An initial pressure of 15.0 MPa and a temperature of 507K which was equivalent to a subcooling of 42K were chosen as the material of the experiment The temperature, pressure, and void fraction at different positions of the pipe were found [7]

1.4.2 Numerical study reviews

Various numerical studies related to flashing scenarios were conducted with application to nuclear safety analysis In general, it could be divided into 4 types of numerical models for flashing flow including the homogeneous equilibrium model (HEM), non-homogeneous equilibrium model (NHEM), homogeneous non-equilibrium model (HNEM), and non-homogenous non-equilibrium model

(NHNEM) approaches that are summarized in Table 1 below

Table 1: Numerical models overview [8] Models

and references Features Remarks HEM

[9, 10, 11, 12] Equal velocity Equal temperature Acceptable agreement for low metability, low void fraction, and long pipes

NHEM

[13, 14, 15] Unequal velocity Equal temperature Mechanical considered with a two-fluid interaction is model, but reliable closure models are needed

HNEM [16, 17, 18]

Equal velocity Unequal temperature

Modeling of thermal equilibrium by nucleation and interphase heat transfer models is recommended, and further model improvement and development need take more efforts

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1.4.2.1 Homogeneous equilibrium (HEM) models

The two-phase flows are simplified to a pseudo or an equivalent single phase one by HEM models The equivalent one flows have average velocities and mean thermodynamic properties, which are derived by interpolation process between the saturated liquid and vapor ones, in which the equilibrium quality equation is applied

A generalized correlation for a simple homogeneous-equilibrium choked flow was presented by Lueng While experiencing an isentropic expansion with identical phasic velocities and temperature, the flashing two-phase mixture was considered as a single-phase compressible fluid Good agreements in normalized critical mass flux and critical pressure ratio for water and ten common fluids such as ammonia, nitrogen, propane at different stagnation pressure conditions were provided It was suitable for long pipes, where the time was enough for equilibration between the phases to happen [21] In contrast, for short pipe, where the time was not enough to reach the equilibration, there were differences between the empirical and numerical results The two-phase critical flow of steam-water mixtures in a short pipe had been investigated in various

pressure conditions by Isbin et al The result showed that the obtained critical

flows are greater than the preceding reported ones by others [10]

A study of the critical flow of steam-water mixture in blowdown pipe was

performed by Ardron et al, which provided some limits of the HEM model in

thermal equilibrium There was a good comparison with all available critical flow rate of steam-water mixtures evaluation in a restricted range of thermodynamic and geometry conditions However, some worse cases were found with low quality of flows, in which an influence of depressurization transient in the mixture was reported as one of the most important parameters that should be considered [22]

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A steady one-dimensional separated two-phase flow model applied hydrodynamic as well as thermal non-equilibrium effects under quickly depressurizing were present was publicized by Richter In the first stage of flashing, when small bubbles’ dispersion was finely occurred in the liquid, in which hydrodynamic momentum equilibrium and thermal non-equilibrium assumptions were considered due to the limit of the interfacial area for heat transfer Whenever the bubbles grew and the void faction had a value more than 0.3, thermal equilibrium might be used for one of assumptions but the slip between the phases became critical and it would cause inaccuracies if the slip model between two phases was ignored [23]

1.4.2.2 Non-homogeneous equilibrium (NHEM) models

In the non-homogeneous equilibrium (NHEM) approach, the velocity of two phases became separated and interfacial slip or drift-flux models were applied to define the relative velocity between the two phases, however, the thermodynamic equilibrium condition was still assumed

The interfacial slip effect on bubbly two-phase flow through a sudden extension

was researched by Attou et al The global mass, momentum, and energy

conservation equations, as well as, the thermal equilibrium of the phases were considered Especially, the kinematic non-equilibrium effect had to be taken into account because of the different mechanical inertia of the phases To determine the role of mechanical non-equilibrium, two assumptions were considered An infinite momentum transfer coefficient and no momentum transfer between phases were assumed, in which ideal gas was applied With the first assumption, a higher-pressure recovery than experiments was found because of the faster liquid deceleration than in reality due to the lower gas inertia, which was opposite to the second assumption In the MFM assumption, the slower liquid deceleration leads to the lower pressure recovery than the experimental data [24]

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To develop empirical or theoretical correlations for the slip ratio between two phases, there were several researched topics related to the NHEM approach An analytical model for predicting the highest flow rate of a two-phase mixture based on annular flow, steady linear velocities of each phase, and liquid-vapor equilibrium was conducted by Moody The result showed that the highest flow rate value was obtained when concerning the local slip ratio and static pressure for known stagnation conditions [13]

A two-phase critical discharge of a one-component mixture through a convergent nozzle model considering the interphase heat, mass, and momentum

transfer rates was investigated by Henry et al The results showed a good

agreement in critical flow rate with the data over broad stagnation conditions based on the comparison between the theoretical and experimental results in different materials including water, nitrogen, potassium and carbon dioxide [14] The mechanical interaction between the phases considering a separated flow model or two-fluid model should be applied in the flashing flow simulation [24]

1.4.2.3 Homogeneous non-equilibrium (HNEM) models

For the homogeneous non-equilibrium (HNEM) approach, a similar assumption to the homogeneous equilibrium model in velocity was taken into account However, the HNEM models are concerned with the non-equilibrium thermal effect on flows

A one-dimensional non-equilibrium relaxation model of flashing phenomena in

liquid was presented by Downar-Zapolski et al According to them, the most

critical character of flashing flows was thermal non-equilibrium due to nucleation delay and rate of vapor generation process as pressure drops, which greatly influenced the void fraction as well as the pressure and velocity distribution along a flow Besides, a correlation for the relaxation time which was a closure law for the homogeneous relaxation model (HRM) taken into

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account to the non-equilibrium evaporation leading to the metastable liquid conditions was also provided The HRM prediction of the critical mass-flow rates and the pressure distributions had been validated by available experimental data [25]

The thermodynamic effects in cavitation bubble growth for hot water, liquid

hydrogen, and nitrogen were proved by a numerical study of Kato et al It was

shown that the model had to capture accurately the non-equilibrium nature of the flow by solving the combination of energy equation and the Raleigh’s equation [26]

A new model called Delay Equilibrium Model (DEM) was not yet widespread due to its complex assumption of three phases, in which the third phase was in thermal non-equilibrium with the saturated ones The purpose of the DEM model was to describe the flashing flow using the three-phase mixture including saturated liquid, saturated vapor, and metastable liquid A flashing model of a supersaturated liquid using the DEM approach had been developed by Lackme It was shown that vaporization is not complete, which leads to approximate 90% of the liquid in a supersaturated state When vaporization was choked, the vaporized products were blown out at the local sonic velocity [27]

A flashing liquid nitrogen flow in a venturi using a one-dimensional HNEM, in which the heterogeneous nucleation was taken into account by making a similarity to the classical homogeneous nucleation theory with the dominant

bulk nucleation assumption was simulated by Rohatgi et al An overall good

agreement to the experiment was shown by using two adjustable parameters including nucleation site density and heterogeneity factor In addition, the author confirmed that bubble coalescence in further studies was needed [28]

1.4.2.4 Non-homogeneous non-equilibrium (NHNEM) models

Besides the above homogeneous models, the non-equilibrium effects considering the nucleation and interphase heat transfer are preferred in the non-

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homogeneous non-equilibrium models (NHNEM), in which the drift-flux or two-fluid model is used to present the different velocities between the gas and liquid phases

A non-equilibrium drift flux model applied to the discharge of two-phase mixtures under choked flow conditions, including equilibrium as well as non-equilibrium vapor generation models were presented by Kroeger Good agreements in void fraction and pressure along pipe length were shown after being validated by Edwards’ blowdown experiment [16]

The effort of taking both the effect of unsteady flow and concentration profiles as well as the effect of the local relative velocity between the phases for getting a general expression which could be applied to the average volumetric concentration prediction was presented by Zuber and Findlay Good agreement with experimental data was shown by comparisons between the analysis and experiment obtained for various two-phase flow regimes, with various liquid-gas mixtures in adiabatic, vertical flow over a wide pressure range [29] A one-dimensional mechanistic non-equilibrium model for two-phase critical flow using bubble formation, growth, and convection, as well as, thermal non-equilibrium vapor drift velocity and non-uniform bubble distribution was

investigated by Elias et al It was shown that the theoretical critical mass fluxes

were acceptable by comparison between numerical results and the horizontal and vertical discharge pipes and converging-diverging nozzles data [30] A new improved numerical model based on the wall nucleation theory, bubble

growth, and drift-flux bubble transport model was performed by Riznic et al

The bubble number transport equation was solved with a source term of wall nucleation distribution Satisfactory agreements were proved by the comparison of the void fraction to BNL [3] and Moby Dick [4] experimental data The model was calculated without any floating parameter to be adjusted with data [31]

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A non-equilibrium vapor generation model for flashing flows consisting of a flashing inception point, a bubbly flow regime followed by a bubbly-slug regime, an annular or annular-mist regime, and a dispersed droplet regime was

investigated by Saha et al The two-fluid model was suggested to be superior to

the drift-flux model in calculating the relative velocity in the model Good agreements of area-averaged void fraction were acquired when compared with experimental data The optimum bubble number density at the inception point was also found at a temperature of liquid superheat reached 3 degrees [32] The critical mass flux of subcooled water discharging from a pipe was estimated by Ardron’s one-dimensional two-fluid equations with high accuracy compared to the experiment The thermal non-equilibrium between the liquid phase and vapor phase was applied in the model [33]

A two-fluid model applied thermal non-equilibrium effect due to rapid depressurization was developed by Richter to the calculation of critical flow rates for steam-water mixtures from nozzles In the model, the mass transfer, evaporation, or condensation rate was accompanied with the heat transfer between the two phases With these assumptions, the results were shown good agreements of critical flux with different experiments [23]

The critical flashing flow in a converging-diverging nozzle used a steady-state

two-fluid six-equation model in RELAP5 was simulated by Tiselj et al The

BNL nozzle data was used to compare with the simulation results [3] The estimated results were shown that the virtual mass term should be taken into account when simulating the flashing flow phenomena of a two-phase flow entrance [34]

1.4.3 Computational fluid dynamics (CFD) models

In general, two important aspects must be considered in the CFD approach to model flashing flows including the phase change during the flashing process and the liquid-vapor interaction during the vaporization process Numerical

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models can be divided into two main categories based on the change of phase during the flashing flow, comprising models accounting for the nucleation process and models neglecting the one and assuming constant bubble radius or number density, with a special quantity called artificial coefficients to control the nonequilibrium process Concerning the interaction between two phases during the vaporization process, the mixture approach, and the Eulerian multi-fluid approach, which is more general than the first one, are usually applied in the CFD approach The CFD works on flashing nozzle flow are also divided into two categories comprising thermal phase change models (TPCMs) and pressure phase change models (PPCMs) based on phase change models In the thermal phase-change model, the main considered force factor for phase change is the temperature difference across the interface between two phases In contrast, for the pressure phase change model approach, the pressure difference is applied instead of the previous one

1.4.3.1 Thermal phase-change models

In thermal phase change models, temperature is the main parameter used to define the fluid state If the temperature of the mixture is below the saturation temperature at a certain pressure value, the process will be considered an evaporation process, otherwise, a condensation

The transient flashing flow calculation of a three-dimensional CFD model using

the commercial CFD-code CFX4.2 was presented by Maksic et al The

simplified two-phase model considered two phase flow, as well as, the thermal non-equilibrium between the phases To verify the model ability, calculations of the flow through the BNL 237 nozzle [3] were carried out, in which the turbulence K-Ɛ model and the neglect of the influence of the dispersed vapor phase were applied In addition, the vapor generation and propagation were modelled by the bubble number density equations in the Jones’s model [35] and vapor mass conservation equations for the vapor phase The model results

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showed a good agreement on mass flow rate with the experiment data Moreover, a pressure fluctuation profile and under-prediction of void fraction in the divergent section were also found [36]

The flashing flow process in the BNL nozzles [3] was modelled by Marsh and O’Mahony, in which the commercial CFD code, with a User-Defined Scalar (UDS) transport equation used to solve bubble nucleation and transport were used The two fluid model with separate mass, momentum and enthalpy conservation equations, inter-phase mass and momentum as well as energy transfer derived result from nucleation process and phase change process were considered However, the non-drag forces effect on the momentum exchange and heat transfer between phases were ignored The results show good agreements for almost cases except the Run 309 which ran under a high entrance pressure and a low outlet pressure [37] There was a main difference between the simulated results when comparing with simulation of Maksic’s research Specifically, the Blander and Katz model [38] used in Marsh and O’Mahony study was deduced from the classical nucleation theory for bulk nucleation and affected on both the walls and the entire remaining domain depending on local liquid superheat, which is more different than the nature of the Jones wall cavity model [35] applied in the preceding work

1.4.3.2 Pressure phase-change models

In pressure phase change models, pressure is the main parameter used to define the fluid state If the pressure of the mixture is below the saturation temperature at a certain pressure value, the process will be considered an evaporation process, otherwise, a condensation

The flashing flow phenomena simulation of steady flow through BNL nozzles

[3] using the of Singhal et al’ cavitation model [39] available in Ansys Fluent 6.1 was performed by Palau-Salvador et al The authors found that pre-existing

nuclei growth increased when the local pressure dropped below the saturation

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pressure, and collapse in the reverse case The results show good agreements between simulation and experiment data in all cases With a difference less than 4% between the measured and calculated mass flow rates was achieved, the simple model was proposed to apply in steady flow case The authors also applied the model to the unsteady flow, and the simulation results obtained from the numerical model did not match with experimental data [40]

The two-phase critical flow simulation in the nozzles and breaks using another cavitation model called Schnerr and Sauer model [41] in the Ansys Fluent code,

was presented by Ishigaki et al The Super Moby Dick (SMD) experiment [42]

and the SGTR experiment [43] were used for the model validation The numerical results showed good agreements of the mass flow rates to the experimental results Besides, the authors also emphasized about the possibility of CFD code Ansys Fluent to simulate two-phase critical flows referred to the nuclear safety analysis [44]

1.4.3.3 Mixed phase change model

A new model for flashing flow prediction using two-phase mixture with slip model approach was performed in Ref [1] In the simulation, the thermal phase change Lee model [45] was modify by combination of Clapeyron-Clausius equation and the formula of vaporization pressure [46] The Clapeyron-Clausius equation was used to convert the thermal phase model to pressure phase change model The model used pressure instead of thermal to define the flow states A new parameter called “accommodation” coefficient was taken into account for thermal non-equilibrium effects The model was validated against BNL experimental data [3] Simulated results showed good agreement of mass flow rate with the highest relative error below 6%, average void fraction and static pressure along nozzle to experiment [1]

A numerical study on heat transfer effects of cavitating and flashing flows was

conducted by Jin et al The combination of a homogeneous mixture model in an

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in-house CFD code, and the effects of both pressure and thermal difference on the interfacial mass transfer was applied in the model The results showed good agreement of pressure profiles and void fraction in two BNL nozzle cases including the Run 309 and the Run 268 [47]

Based on author’s knowledge, most of previous numerical solutions on flashing flow are focus on low Mach below 0.3, incompressible and mildly compressible flows However, the numerical researches for high compressible flow are still limited up to now The thesis aims to investigate the flashing flow behavior in high compressible flow based on homogeneous mixture model and validate with a Super Moby Dick nozzle experimental case [48] After considering the results of several cases applied thermal and pressure phase changes in the BNL nozzle test case 309 [3], which is described detailed in Chapter 3, the modified thermal phase change model provided in Ref [1] shows higher precise values in mass flow rate and average vapor fraction than the others Therefore, the model proposed in Ref [1] is applied in the thesis for simulating the flashing flow phenomena in high compressible flow

1.5 Outline of the thesis

The content of the thesis consists of 5 chapters with the following structure: Chapter 1: Introduction and literature review

Chapter 2: Flashing flow phenomena and mathematical model Chapter 3: Chosen phase change model

Chapter 4: Simulation of flashing phenomena in the Super Moby-Dick nozzle Chapter 5: Conclusion and further development

Figure 2.1: Super Moby-Dick experiment setup [48]

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The primary loop is composed of a centrifugal pump with nominal point of 20 kg/s and 5 MPa, a preheater, a test section and a condenser The preheater has a maximum power of 3.5 MW and the 1m3 condenser can support a maximum pressure of 1.3 MPa The pressure at the inlet of the test section and the inlet temperature are regulated by a by-pass and two valves, and the electrical power of the preheater (0.1oC maximum oscillation) [48]

2.1.2 Super Moby-Dick nozzle test section

Figure 2.2: The Super Moby-Dick nozzle geometry [48]

The Moby Dick nozzle components shown in Figure 2.2 above include a convergent section, a long cylindrical throat and a divergent section The divergent section has an approximate angle of aperture of 7 degrees

The nozzle operating condition is presented in the Table 2 below with the entrance temperature of liquid phase being approximate 9K lower than the saturation temperature based on inlet pressure

Table 2: Experiment conditions

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Figure 2.3: Moby-Dick nozzle geometry [48] Table 3: Moby-Dick geometry parameters

Sections Inlet Extension

Inlet Convergent

Throat Outlet Divergent

Outlet Extension Diameter

(mm) 65.5 65.5 20 73.5 73.5 Length

Gc s

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Figure 2.5: Cross-sectional average void fraction

The experimental void fraction chart is presented in the Figure 2.6 below:

Figure 2.6: Measured void fraction chart [48]

The process measuring void fraction is very difficult to evaluate and the averaging process in the cross section certainly led to a loss of accuracy, which is reported in the experiment paper From simulation results including periodical trend lines of residuals, and mass flow rates of inlet and outlet in section 4.2.1 below, the obtained simulated flow behaviors are not constant In addition, a remarkable disadvantage of the “Cross-sectional average void fraction” method is that the method should not be applied in the inconstant fluid behavior, which usually makes the void fraction measurement is incorrect Therefore, the

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average void fraction will not be used as a critical parameter of validation in the thesis

2.2 Flashing flow phenomena

The thermodynamic diagram of water phase change in the boiling process

is shown in Figure 2.7 below

Figure 2.7: Flashing flow behavior: (a) thermodynamic diagram of phase change of water with equilibrium assumption and (b) start of

vaporization [1]

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At a constant pressure, the phase change process between the liquid and vapor phases occurs in a saturation region under thermal equilibrium conditions However, the non-equilibrium effects are important factors affected to the phase change of high-speed flows inside nozzles, for instance, ejector-based systems Therefore, it is impossible to use this diagram above for fluid dynamics prediction in the phase change region Specifically, sudden depressurization occurs when a subcooled water flow enters the nozzle, which leads to superheated water, and the phase change appears The rapid depressurization leads the liquids to be in metastable state until the phase change happens [50] Consequently, the nonequilibrium effect will critically affect to the bubble nucleation process [51] In general, the phase change mechanism driven by an abrupt pressure drop is divided into in three stages including formation of nucleation droplets, bubble growth, and the bubble boiling explosion The flashing phenomena of two-phase flows is usually found in industrial energy systems such as trilateral cycle power systems, supercritical CO2 power systems and ejector heat pumps [52]

In the first stage, small nuclei appearing in non-wettable rough wall cavities and in the bulk the formation of start growing after the pressure decreases below saturation pressure [1] Another explanation is that the nucleation takes place due to the difference temperature between the liquid phase and vaporization state, specifically, the saturated temperature at a given ambient pressure is significantly lower than the temperature of liquid phase In this stage, the bubble nucleation is critically affected by the superheated degree of the droplets [51] The surface tension forces play dominated factors that directly influence These forces limit the nuclei growth rate, therefore, this stage is also called as delay period [53] In the initiating nucleation process, the liquid superheat degree required for the nucleation is dependent on the depressurization rate, surface conditions and the flow characteristics The liquid superheat degree at the begin of flashing inception is noticeably smaller than the kinetic limit of nucleation in

The heat transfer around the bubble interface is the most important factor which dominate and control the final stage The heat transfer mechanism at water-bubble interface is critically influenced by the turbulence fluctuations and relative motion between the dispersed phase and the continuous phase [56] The bubble boiling explosion is broadly known as the main dominant mechanism for the fuel atomization after flashing happens [51] The flashing increases the liquid jets atomization performance, and the higher-quality liquid fuel atomization is an important keyword to acquire the better combustion efficiencies and lower the pollutant emissions [57]

2.3 Mathematical Model

The flashing flow mathematical model of 2D axisymmetric geometry with the appearance of time differential terms was reviewed in Ref [1] and the Ansys

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Fluent’s tutorials, in which all vectors are presented in axial, x, and radial, r, components The model is a modified version of the Lee thermal phase change model, in which a combination of Clapeyron-Clausius and vaporization pressure formula [46] is applied to solve the problem with pressure values instead of thermal values

of x ,r, m are axial component, radial component, and mixture

2.3.2 Axial momentum conservation equation

mkk dr x k dr x kkk dr r k dr x kk

axial component of drift velocity, radial component of drift velocity, and kth phase