Advancement of closure relations for annular flow modeling in CFD

Advancement of Closure Relations

for Annular Flow Modeling in CFD

by

Giulia Agostinelli

B.S., Engineering, Universitá degli Studi di Pisa, 2009 M.S., Engineering, Universitá degli Studi di Pisa, 2013

SUBMITTED TO THE

DEPARTMENT OF NUCLEAR SCIENCE AND ENGINEERING

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN NUCLEAR SCIENCE AND ENGINEERING

AT THE

MASSACHUSETTS INSTITUTE OF TECHNOLOGY SEPTEMBER 2020

©2020 Massachusetts Institute of Technology All rights reserved.

Author:

Giulia Agostinelli Department of Nuclear Science and Engineering August, 21 2020 Certified by:

Emilio Baglietto Associate Professor of Nuclear Science and Engineering Thesis Supervisor Certified by:

Jacopo Buongiorno TEPCO Professor of Nuclear Science and Engineering Thesis Reader Certified by:

Jean-Marie Le Corre Affiliated Faculty of Nuclear Engineering, KTH Thesis Reader Certified by:

Eissa Al-Safran Professor of Petroleum Engineering, Kuwait University Thesis Reader Accepted by:

Ju Li Battelle Energy Alliance Professor of Nuclear Science and Engineering and Professor of Materials Science and Engineering Department Committee on Graduate Students

Advancement of Closure Relations for Annular Flow Modeling in CFD

Submitted to the Department of Nuclear Science and Engineering on August, 21 2020

in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Nuclear Science and Engineering

Abstract

In Boiling Water Reactors (BWRs), the presence of a liquid film in contact with the heated rod surface is crucial to ensure an efficient heat removal and prevent the threatening occurrence of dryout. The accurate prediction of the complex multidimensional liquid film behavior in advanced BWR fuel assemblies is critical to guarantee improved reactors performance and safety. Multiphase-CFD (M-Multiphase-CFD) brings the ability to model the complex three-dimensional flow structures in annular flow regime [1], while physics-based constitutive equations are needed to accurately represent the phase interactions, particularly at the liquid film interface. The development of closure relations for droplet deposition and entrainment as well as wave-induced interfacial shear, is a major priority for the modeling of annular flow in M-CFD.

In annular flow conditions, liquid is continuously exchanged at the interface between the bulk steam and the film on the walls. While liquid droplets deposit onto the film driven by turbulent diffusion, new ones are entrained from the waves appearing on the film surface. A modeling approach is proposed and assessed to represent the local subgrid-scale deposition in CFD, showing comparable results with existing integral correlations, and an average error of 30%. Available closures are also evaluated for their ability to represent entrainment in the CFD implementation. Finally, in order to drive the advancement of the representation of interfacial shear, as well as physics-based droplet entrainment, the work focuses on the analysis and modeling of disturbance waves. The recent high resolution film measurements collected by Robers [2] are analyzed and leveraged to propose a physical representation of disturbance waves, which can be implemented into a complete model. The proposed model is successfully assessed against the experimental measurements of Sawant [3], while a large disagreement is found in comparison with the high pressure data evaluated at the RISO facility [4]. The new model predictions are consistent with existing integral correlations, demonstrating the need for further advancement of high pressure experiments with high resolution, necessary to drive more general representations.

The complete set of closures is implemented in a commercial CFD software, and demonstrated adopting data from the Robers experiments. While the lingering limitations of the CFD implemen-tation to transport thick films lead to overprediction of the local film thickness, the formulation shows promising performance towards more fundamental modeling of annular flow in M-CFD.

Thesis Supervisor: Emilio Baglietto

This dissertation is the result of a team effort and I would like to express my gratitude to all those who played a fundamental part in the completion of my doctoral work.

First, I would like to sincerely thank my supervisor, Professor Emilio Baglietto, for his excep-tional support and technical expertise during these years. He encouraged me to apply to MIT that afternoon at the NURETH-15 conference in Pisa, and welcomed me as part of his CFD group. He provided financial assistance and fostered precious research collaborations, allowing me to focus on my studies and produce meaningful work. Above all, he challenged me and pushed me through a deep technical and personal growth, while committing to my success and helping me achieve my goals. I am truly grateful.

Thank you to my Ph.D. committee members, Professors Jacopo Buongiorno, Eissa Al-Safran, and Dr. Jean-Marie Le Corre for providing invaluable perspectives, feedback and guidance in my doctoral journey. Particularly, I’m especially thankful to Professor Buongiorno for being an inspiring teacher and making me appreciate the reactor thermal-hydraulics, to Professor Al-Safran for encouraging me when I needed the most, with long chats and tasty figs, and to Dr. Le Corre for motivating me to think outside the box and never stop questioning my results. It has been a pleasure and an honor to be mentored by you.

My doctoral research would not have been possible without the collaboration with dedicated experimentalists. I would like to extend my gratitude to Professor Horst-Michael Prasser and Lukas Robers at ETH and Professor Michitsugu Mori for generously offering me access to incredibly valuable databases and kindly providing all the answers to my many questions.

I would like to thank Dr. Neroorkar and Dr. Tandon for their exceptional support in the use of the Fluid Film framework in STAR-CCM+ and helping me in the implementation of my closure models. I hope my results will serve in their efforts to develop robust modeling tools for annular flow at Siemens.

My research was funded by the Consortium for Advanced Simulation of Light Water Reactors (CASL) and I am thankful for this opportunity. I am proud that I have contributed to the improve-ment of advanced nuclear reactor designs. In particular, I would like to thank Professor Richard Lahey and Dr. Jens Andersen for their knowledgeable insights and suggestions.

Thank you to Dr. Guanuy Su for the good discussions on liquid films and disturbance waves. They were extremely helpful in finding solutions to my annular flow challenges.

I would like to thank also the NSE administrative staff for providing endless care and support while navigating the graduate student life and doctoral requirements, in particular, Brandy Baker, Lisa Magnano-Bleheen, Carolyn Carrington, Rachel Batista and Peter Brenton. Their contagious smiles are part of my best memories in the NSE department, from the moment I arrived at MIT, to the day I took my qualifying exams, to my virtual defense.

Acknowledgements

I am extremely lucky that since prior to my arrival in Cambridge until today, I could rely on the precious assistance and advice of my CFD teammates, in particular, Giancarlo Lenci, Melanie Tetreault-Friend, Ravikishore Kommajosyula, Mike Acton, Ben Magolan, Etienne Demarly, Rose-mary Sugrue, Nazar Lubchenko and Carolyn Coyle. Thank you for their constant help, meaningful feedback and friendship. Thank you to the bubble bros for welcoming me as part of their family despite my rather liquid droplet sis nature. Thank you to the thesis support group for lightening the thesis writing process and making the final steps more enjoyable.

I would like to thank my classmates, in particular, Karen Dawson, Mike Acton, Pierre Guenoun, Daniel Stack and Nestor Sepulveda, for helping with homework assignments and finals and keep-ing me company those early years’ late nights. I would have not made it through quals without them.

I am very fortunate to have extraordinary and caring friends. Thank you to my friends back home, Flavia, Giovanna and Elena for understanding me and supporting me, beyond time and distance, through the rigors and evolution of these years. Thank you to Melanie, Giancarlo and Karen, for their sincere friendship and unrelenting motivation. Thank you to Juan, Rami and Elena for creating a family and a place that I could call home. Thank you to MITaly for the fun times and making me feel close to my beloved country. Thank you to the MIT Tango Club and my wonderful tango community, especially Wendy and Stefano, for nurturing my love for dance and constantly recharging me. Finally, thank you to Steve for making me laugh and believing in me. I know he would be proud today.

I would like to give special thanks to my parents, Maria Rosa and Giovanni, and my brother Lorenzo, for encouraging me and being my number one fan in the pursuit of my Ph.D., and to my aunt Carmelina and uncle Turi for welcoming me in the US and offering me a safe and comforting nest as if I were their daughter. My family’s support has been fundamental for the accomplishment of my doctoral degree.

No words are enough to express my gratitude to my husband Jeremiah for his unconditional love, patience and faith. He has been a solid rock and an infinite source of energy and motivation. I look forward to our future adventures together.

Abstract 3 Acknowledgements 4 Contents 6 List of Figures 9 List of Tables 13 Nomenclature 14 1 Introduction 17 1.1 Motivation . . . 17 1.2 Thesis Objectives . . . 19 1.3 Structure of Thesis . . . 19

2 Modeling Annular Flow with M-CFD 20 2.1 Annular Flow Regime . . . 20

2.2 CFD Modeling Strategies . . . 22

2.2.1 The Eulerian-Eulerian Approach . . . 22

2.2.2 The Computational Multi-Fields Dynamics (CMFD) method . . . 23

2.2.3 The Extended Boiling Framework (EBF). . . 24

2.2.4 KTH model . . . 26

2.3 Selected Modeling Strategy and Development Opportunity. . . 27

2.4 Governing Equations . . . 27

2.4.1 Gas-Droplet Core . . . 27

2.4.2 Liquid Film . . . 29

3 Deposition 31 3.1 Separation of deposition effect. . . 31

3.2 Classic definition of deposition . . . 32

3.3 Mechanics of droplet deposition . . . 33

3.3.1 Turbulent diffusion deposition . . . 33

3.3.2 Deposition rate model . . . 35

3.3.3 Droplet velocity fluctuation . . . 36

3.4 Validation of deposition rate closure in M-CFD . . . 38

3.4.1 Jepson and Govan Test Cases . . . 38

3.4.2 M-CFD Setup . . . 38

3.4.3 Jepson Test Case Analysis . . . 40

3.4.4 Govan Test Case Analysis . . . 42

3.4.5 Local vs. Averaged Implementation . . . 43

Contents

4 Entrainment 46

4.1 Mechanics of droplet entrainment . . . 46

4.2 Entrainment rate correlations . . . 48

4.2.1 Ishii - Mishima . . . 48

4.2.2 Govan-Hewitt . . . 49

4.2.3 Okawa . . . 49

4.3 Assessment of Entrainment Correlations in M-CFD . . . 50

4.3.1 Sawant Test Case . . . 50

4.3.2 M-CFD Setup . . . 51

4.3.3 Results . . . 52

4.4 Future Reasearch Directions on Entrainment Modeling . . . 55

5 Film Interface Characterization using ETH Experimental Database 56 5.1 Disturbance Wave Characterization. . . 56

5.2 ETH Experimental Database . . . 58

5.3 Data Analysis . . . 60

5.3.1 Film Thickness . . . 61

5.3.2 Disturbance Wave Identification . . . 62

5.3.3 Base Film Thickness . . . 63

5.3.4 Wave Frequency . . . 64

5.3.5 Wave Velocity . . . 64

5.3.6 Wavelength and Wave Spacing . . . 64

5.3.7 Wave Induced Roughness . . . 64

5.3.8 Film Velocity . . . 66

5.4 Disturbance Wave Analysis . . . 69

5.5 Note on different post-processing techniques . . . 71

6 Disturbance Wave Modeling 76 6.1 Modeling Proposal . . . 76

6.1.1 Base Film Thickness . . . 77

6.1.2 Interfacial Shear Stress . . . 79

6.1.3 Wave Velocity . . . 81

6.1.4 Wave Geometry. . . 82

6.2 Results and Discussion . . . 84

7 Disturbance Wave Model Assessment with Experimental Data 92 7.1 Sawant Database . . . 92

7.1.1 Sawant Experimental Conditions and Measurements . . . 92

7.1.2 Results for Sawant Test Case . . . 93

7.2 RISØ Database. . . 99

7.2.1 RISØ Experimental Conditions and Measurements . . . 99

7.2.2 Results for RISØ Test Case. . . 100

8 Implementation of Disturbance Wave Model in M-CFD and Challenges 102 8.1 M-CFD . . . 102

8.1.1 Models . . . 102

8.1.2 Simulation Setup . . . 102

8.3 Results of the Implementation in M-CFD . . . 107 8.4 Challenges in the Wavy Interface Effect Predictions in M-CFD . . . 111

9 Conclusions and Future Work 113

9.1 Summary . . . 113 9.2 Recommended Future Work . . . 114

List of Figures

1.1 Westinghouse BWR Fuel Design Evolution. . . 17

1.2 Demonstration of M-CFD modeling capabilities [1] [9]. . . 18

2.1 Two-phase flow regimes in vertical flow heated channel. . . 21

2.2 Key characteristic mechanisms of annular flow regime. . . 22

2.3 Different interface topologies in annular flow regime. . . 23

2.4 CMFD four-field two-fluid model schematic representation [15]. . . 24

2.5 EBF flow topology map [20].. . . 25

2.6 EBF master-cell topology as combination of basic regime interface configurations [20]. 25 2.7 Schematic representation of the liquid film mass, momentum and energy balances [24]. 26 3.1 Schematic representation of experimental techniques: double extraction technique (left) and localized source droplet introduction (right).. . . 32

3.2 Results of the test of Okawa correlation against M-CFD and experiments [42]. . . 33

3.3 Characterization of deposition regimes [43]. . . 34

3.4 Comparison between gas velocity fluctuation [54], particle velocity fluctuation com-puted with available relations and experimental measurements of particle velocity fluctuation [39], for different droplet size (50µm and 150µm). . . . 37

3.5 Comparison between gas velocity fluctuation [54], droplet velocity fluctuation com-puted with original and corrected relation and experimental measurements of droplet velocity fluctuation [39], for different droplet size (50µm and 150µm). . . . 38

3.6 Computational meshes used in simulations performed for the Jepson (left) and Govan (right) test cases. . . 39

3.7 Test of deposition rate correlation, comparison between M-CFD and experiments [38]. 40 3.8 Comparison between available droplet size correlations: deposited liquid mass flux GDvs. entrained liquid mass flux Gle(left), relative error vs. entrained liquid mass flux Gle(right). . . 42

3.9 Test of deposition rate correlation, comparison between M-CFD and experiments [37]. 42 3.10 Test of deposition rate correlation, comparison between M-CFD computed locally and section-averaged and experiments, in Govan (left) and Jepson (right) test cases. 43 3.11 Comparison of measured and computed deposited liquid mass flux GD with the Okawa correlation (left) and with equation 3.13 in M-CFD (right) for both Govan and Jepson test cases. . . 44

3.12 Schematic of potential ballistic droplet formation. . . 45

4.1 Mechanisms leading to entrainment [10]. . . 47

4.2 Schematics of the limits for entrainment inception identified by Ishii & Grolmes [58]. 47 4.3 Comparison between experiments and predicted entrained fraction by Ishi & Mishima [61]. . . 49

4.4 Comparison between experiments and predicted entrainment by Okawa [67]. . . 50

4.5 Schematic of the experiment and test facility [64]. . . 51

4.7 Comparison between entrainment rate correlations (Okawa, Ishii-Mishima, Govan-Hewitt) and experiments at p=1.2 bar. . . 53 4.8 Comparison between entrainment rate correlations (Okawa, Ishii-Mishima,

Govan-Hewitt) and experiments at p=6.0 bar. . . 53 4.9 Comparison between implementing both deposition and entrainment and

entrain-ment only in the liquid film model and experientrain-ments, at p = 6bar and with Ishii-Mishima entrainment rate correlation. . . 54 5.1 Layout of a section (3x3) of the used sensor. The blue area represents a measuring

lo-cation when the transmitter line (in red) and the receiver line (in green) are activated [86]. . . 59 5.2 Film thickness signal in time at a centered location of the upper sensor for jg =50

m/s and j_l =0.15 m/s. . . 59 5.3 Test section geometry and sensors layout (left), cross-section with sensor location

and orientation (center) and schematic representation of the lower sensor and the end of the PLRs (right) in the experiments performed at ETH [2]. . . 60 5.4 Time-averaged film thickness spacial distribution in lower (left) and upper (right)

sensor at jg =50 m/s and jl =0.15 m/s. . . 61

5.5 Schematic representation of the disturbance waves geometric approximation. . . 61 5.6 Identification of local peaks and computation of each amplitude and width. . . 62 5.7 Histogram of local peaks heights of the film thickness signal at the center of the

upper sensor at jg =50 m/s and jl =0.15 m/s. . . 63

5.8 Modified film thickness signal for the computation of the base film thickness at the center of the upper sensor at jg=50 m/s and jl =0.15 m/s. . . 63

5.9 Disturbance wave-only modified signal at the center of the upper sensor at jg =50

m/s and jl =0.15 m/s. . . 64

5.10 Effect of the distance between measuring points on velocity computation. The choice of l =0.08m is highlighted. . . 65 5.11 Comparison between turbulent and laminar velocity profile in the time-averaged

liquid film at the center of the upper sensor for jg =40 m/s and jl =0.24 m/s.. . . . 68

5.12 Percent bias in case of turbulent and laminar velocity profiles. . . 68 5.13 Plotted film and wave characteristic quantities as a function of liquid Reynolds number. 69 5.14 Plotted film and wave characteristic quantities as a function of film velocity. . . 71 5.15 Plotted film and wave characteristic quantities as a function of film thickness. . . 72 5.16 Results of the no-filtering post-processing method. Plotted film and wave

character-istic quantities as a function of liquid Reynolds number.. . . 73 5.17 Results of the no-filtering post-processing method. Plotted film and wave

character-istic quantities as a function of film velocity. . . 74 5.18 Results of the no-filtering post-processing method. Plotted film and wave

character-istic quantities as a function of film thickness. . . 75 6.1 Schematics of liquid droplet formation through instabilities from film interface. . . . 77 6.2 Experimental measurements and predictions of base film thickness (left) and percent

error (right). . . 79 6.3 Comparison between film thickness experimental measurements and predictions

using Haaland correlation for computing interfacial friction factor. . . 80 6.4 Comparison between interfacial friction factor predictions obtained by

List of Figures

6.5 Experimental evaluations of ratio between wave velocity and average film velocity.. 82 6.6 Experimental ratio of wave amplitude to wave width. . . 83 6.7 Experimental ratio of wave spacing to wave width. . . 83 6.8 Proposed model predictions of wave amplitude, wave width and wave spacing with

film Reynolds number and comparison with experiments. . . 84 6.9 Proposed model predictions of wave amplitude, wave width and wave spacing with

film velocity and comparison with experiments. . . 85 6.10 Proposed model predictions of wave amplitude, wave width and wave spacing with

film thickness and comparison with experiments. . . 85 6.11 Comparison between proposed model and available correlations predictions of wave

amplitude for film Reynolds number variations. . . 86 6.12 Comparison of proposed model with available correlations predictions of wave

amplitude for film velocity variations. . . 87 6.13 Comparison of proposed model with available correlations predictions of wave

amplitude for film thickness variations. . . 88 6.14 Mass transported in base film and disturbance waves in experiments (left) and

according to model predictions (right). . . 89 6.15 Comparison between transported mass in average liquid film and realistic liquid

film, using experimental and modeled quantities (left). and error (right). . . 90 6.16 Comparison between computed wave velocity and maximum velocity in the base film. 91 7.1 Experimental conditions of Sawant tests [76]. . . 92 7.2 Conductance probes layout for the the film thickness measurements performed in

Sawant tests [76]. . . 93 7.3 Comparison of proposed model and measured predictions of wave amplitude at

p=1.2bar (left) p=4.0bar (center) and p=5.8bar (right) in the Sawant test case. . . 94 7.4 Proposed model predictions of wave amplitude, wave width and wave spacing with

film Reynolds number and comparison with experiments in the Sawant test case for p=1.2bar . . . 94 7.5 Model predictions of wave amplitude, wave width and wave spacing with film

Reynolds number and comparison with experiments in the Sawant test case for p=4.0bar . . . 95 7.6 Model predictions of wave amplitude, wave width and wave spacing with film

Reynolds number and comparison with experiments in the Sawant test case for p=5.8bar. . . 95 7.7 Comparison between proposed model and available correlations predictions of wave

amplitude for film Reynolds number variations in the Sawant test case for p =1.2bar. 96 7.8 Comparison between proposed model and available correlations predictions of wave

amplitude for film Reynolds number variations in the Sawant test case for p =4.0bar. 97 7.9 Comparison between proposed model and available correlations predictions of wave

amplitude for film Reynolds number variations in the Sawant test case for p =5.8bar. 98 7.10 Comparison between proposed model and available correlations predictions of wave

amplitude . . . 99 7.11 Example of the needle contact time curve obtained with the RISØ tests.. . . 99 7.12 Model predictions of wave amplitude, wave width and wave spacing with film

Reynolds number in the RISØ test case. . . 100 7.13 Model predictions of wave amplitude, wave width and wave spacing with film

7.14 Comparison between proposed model and available correlations predictions of wave

amplitude for film Reynolds number variations in the RISØ test case. . . 101

8.1 Computational mesh used in simulations performed for the ETH Test Case. . . 103

8.2 Computational mesh used in simulations performed for the ETH Test Case. . . 103

8.3 Velocity profile near the wall. . . 105

8.4 M-CFD predictions of film thickness (left) film velocity (center) and interfacial shear stress (right) on upper sensor obtained at each step of the disturbance wave model implementation. . . 107

8.5 Film thickness distribution obtained from the simulation of the ETH test at ˙Vg=112 l/h and ˙Vl =896 l/h. . . 108

8.6 Film velocity distribution obtained from the simulation of the ETH test at ˙Vg =112 l/h and ˙Vl =896 l/h. . . 109

8.7 Interfacial shear stress distribution obtained from the simulation of the ETH test at ˙ Vg=112 l/h and ˙Vl =896 l/h. . . 109

8.8 Equivalent sand-grain roughness distribution obtained from the simulation of the ETH test at ˙Vg=112 l/h and ˙Vl =896 l/h. . . 110

8.9 M-CFD predictions of equivalent sand-grain roughness on upper sensor obtained at the second and third step of the disturbance wave model implementation. . . 110

8.10 Disturbance wave profile representative of the corrected equivalent sand-grain roughness. . . 111

8.11 M-CFD predictions of equivalent sand-gran roughness on upper sensor obtained with the proposed correction on the evaluation of the equivalent sand-grain roughness.112 8.12 M-CFD predictions of film thickness (left) film velocity (center) and interfacial shear stress (right) on upper sensor obtained with the proposed correction on the evaluation of the equivalent sand-grain roughness. . . 112

List of Tables

3.1 Experimental setup and conditions for [42] and Jepson [38] test cases. . . 39

4.1 Ranges of experimental conditions [3]. . . 51

5.1 Available models for predicting wave amplitude.. . . 57

5.2 Experimental conditions of tests performed at ETH. . . 58

6.1 Coefficients for interfacial friction model. . . 81

8.1 Summary of the disturbance wave model. . . 105

Latin Symbols

Notation Description Unit

a Wave amplitude [m]

A Area [m2]

C Concentration [kg m−3]

C0 Kolmogorov constant [-]

Cµ 0.09 [-]

Ct Turbulence response constant [-]

CD Drag coefficient [-]

d Droplet diameter [m]

D_h Hydraulic diameter [m]

˙

D Deposition rate [kg m−2_s−1_]

˙E Entrainment rate [kg m−2s−1]

f Frequency [s−1]

f Friction factor [-]

F Force per unit volume [kg m−2s−2]

g Standard gravity [9.81 m s−2_]

G Dimensionless number [-]

˙

G Mass flux [kg m−2_s−1_]

ha Sinusoidal wave amplitude [m]

j Superficial velocity [m s−1_]

k Turbulent kinetic energy []

kD Deposition coefficient [m s−1]

Lp Particle characteristic length [m]

M Momentum exchange term [kg m−2s−2]

˙

m Mass flow rate [kg s−1_]

Nµ Viscosity number [-] Pw Wetted perimeter [m] P Pressure [Pa] rd Droplet radius [m] rc Cylinder radius [m] R Radius [m] Re Reynolds number [-]

Ra Equivalent sand-grain roughness [m]

s Wave spacing [m]

Sδ Film mass source/sink term [kg m−2s−1]

Svf Film momentum source/sink term [kg m

−1_s−2_]

t Time [s]

TL Lagrangian integral time scale [s]

T Temperature [°C]

uτ Friction velocity [m s−1]

v Velocity [m s−1]

vin Initial velocity [m s−1]

Nomenclature

Nomenclature – Continued from previous page

v0 Velocity fluctuation [m s−1]

V Volume [m3_]

˙

V Volumetric flow rate [m3s−1]

w Wave width [m]

We Weber number [-]

y+ Non-dimensional distance from wall [-]

zD Deposition length [m]

Greek Symbols

Notation Description Unit

α Volume fraction [-]

α Inverse of integral time scale [s−1]

β Inverse of relaxation time [s−1]

δ Film thickness [m]

δb Base film thickness [m]

e Turbulent dissipation rate []

ep Particle diffusivity [m2s−1]

em Momentum diffusivity [m2s−1]

κ Wave number [-]

κ Von Karman constant [-]

λ Wavelength [m]

µ Dynamic viscosity [Pa s]

µt Turbulent eddy viscosity [Pa s]

π 3.14159265359... [-]

πE Entrainment dimensionless number [-]

ρ Density [kg m−3]

σ Surface tension [Pa]

σTD Turbulent Prandtl number [-]

τ Shear stress [Pa]

τ Laminar stress tensor [Pa]

τRe Turbulent Reynolds stress tensor [Pa]

τp Particle relaxation time [s]

τc Contact time [s] ω Growth rate [m s−1] Subscripts Subscript Description avg Average b Base c Critical co Core dw Disturbance wave dep Deposition ent Entrainment f Film f Fluid

Nomenclature – Continued from previous page

g Gas phase

k Phase

i Interfacial

i, j Cartesian coordinate components

l Liquid phase le Entrained liquid n Normal direction p Particle p Peak r Relative s Tangential direction sin Sinusoidal wave

tot Total

w Wall

Superscripts Superscript Description

dri f t With drift nodri f t Without drift

Acronyms Acronym Description

BWRs Boiling Water Reactors

CASL Consortium for Advanced Simulation of Light Water Reactors CFD Computational Fluid Dynamics

CMFD Computational Multi-Fields Dynamics DNS Direct Numerical Simulation

EBF Extended Boiling Framework LWRs Light Water Reactors

M-CFD Multiphase Computational Fluid Dynamics NPPs Nuclear Power Plants

Chapter 1

Introduction

1.1

Motivation

Boiling Water Reactors (BWRs) are the second most common nuclear technology, representing 18% of the nuclear power generation fleet [5]. As for other Light Water Reactors (LWRs), water is used both as the coolant and moderator. While going through the hot fuel rods in the core of BWRs, the water boils under a pressure of 7 MPa, causing the generation of a high content of vapor. The phase change leads the transition from single-phase subcooled liquid flow through multiple two-phase flow regimes to the final annular flow regime, which is expected to occupy most of the height on a BWR fuel assembly.

As the costs associated with Nuclear Power Plants (NPPs) nowadays represents a major barrier in making nuclear energy competitive with other energy generation options [6], nuclear power plants manufacturers have been intensifying the efforts to improve reactor economy while ensuring safety. New BWR fuel designs are being developed aiming at reducing fuel cycle costs and increasing operational flexibility while maintaining reliability. Enhanced burnups and the capability to work at high efficiency with flexible cycle lengths are crucial requirements for the success of a new fuel design. Fig.1.1offers a schematic of the evolution of BWR fuel designs proposed by Westinghouse Electric Company over the last decades.

Figure 1.1:Westinghouse BWR Fuel Design Evolution.

The assessment of new advanced BWR fuel assembly designs is difficult to carry out experimen-tally, due to the associated time and costs. This has driven the development of numerical analysis

that can replace the use of experiments. Sub-channel codes have been representing a valid method for the prediction of two-phase flow inside BWR fuel bundles but have some limitations. Since they predict sub-channel averaged values, the focus is mainly directed on the modeling of the coolant in the flow direction, obtaining a quasi-three-dimensional resolution with the representation of the exchanges between neighboring sub-channels. In addition, a sub-channel code analysis is based on empirical correlations that are highly accurate in specific conditions similar to the ones used for their development but unreliable for the prediction of two-phase flow in complex geometries because of the lack of physical modeling [7] [8]. The advancement of BWR fuel assembly design is character-ized by an increasing geometric complexity that limits the one-dimensional case-specific modeling capabilities of sub-channel codes and instead requires a higher multidimensional physics-based modeling representation. With Computational Fluid Dynamics (CFD) codes, a three-dimensional detailed resolution of the flow is achieved. As CFD predictions are the solution of a fundamental set of fluid-dynamics equations, CFD simulations are able to predict the effect of complex flow or geometrical variations on three-dimensional flow structures. Unfortunately, while the nuclear industry relies on CFD for a wide variety of single-phase flow studies, CFD is not yet fully mature to carry out accurate two-phase flow analysis.

Multiphase CFD (M-CFD) has demonstrated the capability to generally represent the two-phase flow typical of BWRs. The M-CFD modeling of the coolant inside a prototypical BWR fuel assembly is presented in Fig. 1.2[9]. The distribution of vapor volume fraction αvshows that the content

of vapor phase is maximum in the center of sub-channels and minimum next to the wall of fuel rods. This configuration is representative of annular flow regime, which is characterized by a core mixture of vapor and liquid droplets and a liquid film in contact with the fuel rods. When M-CFD predictions are compared to the experimental data the weaknesses of the modeling can be observed. While the average volume fraction is well captured, large discrepancies are shown locally close to the fuel rods walls where the liquid film is located [1]. M-CFD results are not able to accurately predict the presence and behavior of the liquid film in annular flow.

Figure 1.2:Demonstration of M-CFD modeling capabilities [1] [9].

The characteristic physics of annular flow regime directly influences the heat exchange between fuel rods and the coolant. The presence of a liquid film in contact with the heated rod surface allows the transfer of latent heat due to evaporation, ensuring an efficient heat removal and hence, a good

CHAPTER 1. INTRODUCTION

reactor performance. The complete evaporation of the liquid film, also called dryout, results in a rapid increase of temperature at the hot rod surface and can potentially lead to fuel failure. For this reasons, the correct modeling of the liquid film is of fundamental importance in the representation of the coolant two-phase flow and prevention of the threatening occurrence of dryout. Therefore, the need to accurately predict the liquid film and its behavior in annular flow represents the central focus when improving the modeling of the coolant two-phase flow in M-CFD for the assessment of new BWR fuel assembly designs.

1.2

Thesis Objectives

The main purpose of this work is to advance the development of CFD suitable closure relations for the accurate modeling of liquid film and annular flow regime in M-CFD. The objectives of this thesis are listed below:

– Identify the key physical mechanisms of annular flow regime and review the existing frame-works to modeling annular flow in CFD.

– Investigate the mechanics of droplet motion and deposition and propose a CFD suitable model for deposition rate.

– Test existing correlations for modeling droplet entrainment and identify new research direc-tions for the development of a CFD closure for entrainment rate.

– Develop consistent post-processing techniques to obtain indirect measurements of film and wave quantities from film thickness time traces.

– Leverage experimental measurements to characterize the wavy liquid film interface and develop a close disturbance wave model.

– Assess the potential of the disturbance wave model for its applicability in annular flow test cases.

– Provide an implementation of the complete set of closure relations in a commercial CFD software and demonstrate the predictive capabilities of the proposed closure relations for the modeling of annular flow in M-CFD.

1.3

Structure of Thesis

An overview on annular flow regime and two-phase modeling frameworks in CFD is offered in Chapter 2. The physical mechanisms driving the liquid droplet deposition is studied and a model for deposition rate is proposed in Chapter 3. Existing entrainment models are tested and a model for entrainment rate is selected in Chapter 4. Chapter 5 presents the performed analysis of the ETH experimental database for the characterization of the liquid film interface. A close model for the prediction of disturbance waves is proposed in Chapter 6 and assessed with experimental data in Chapter 7. The implementation of the complete set of closure relations in a commercial CFD software (STAR-CCM+) and the results are presented in Chapter 8. Finally, the conclusions of this thesis and the scope for future work are listed in Chapter 9.

Modeling Annular Flow with M-CFD

Accurate numerical analysis of two-phase flows allows a time- and cost-effective assessment of new advanced BWR fuel assembly designs. In particular, M-CFD has shown the potential to be a valid tool for the modeling of the three-dimensional complex structures of the two-phase flow typical of the coolant in BWRs [1] [9]. Special attention must be reserved to the prediction of the behavior of liquid film in annular flow, to be able to enhance the reactor performance while ensuring safety. This chapter wants to provide an overview on annular flow and two-phase flow modeling in M-CFD. After introducing the annular flow regime in Sec. 2.1, the existing M-CFD modeling strategies are explored in Sec.2.2. The selected approach and the governing equations for the M-CFD modeling of annular flow are presented respectively in Sec.2.3and Sec.2.4.

2.1

Annular Flow Regime

In multiphase flow, flow regimes (or flow patterns) are representative of a specific flow topology for each phase. If sub-cooled water is injected upward into a vertical heated channel different flow regimes can be observed as the volume fraction of the vapor increases because of the phase change: single phase flow, bubbly flow, slug/churn flow, annular flow and mist flow. Bubbles start forming on the heated walls of the channel and increase in size due to evaporation of the water surrounding them. Because of the interactions among themselves and with the liquid flow, bubbles become deformed and assume non-spherical shapes, such as those in slug and churn flow regimes. The further increase of the volume fraction of the vapor drives large bubbles to merge and form the core of annular flow regime, in which the liquid phase appears in the shape of a film and dispersed droplets. Finally, the complete evaporation of the liquid film leads to mist flow, in which the liquid phase is present only in the shape of droplets. A schematic of the transition through multiple flow regimes is illustrated in Fig.2.1.

As just described, the annular flow topology is characterized by a liquid layer or film on the channel walls and a core occupied mostly by vapor and by liquid droplets. The presence of the liquid film in annular flow allows an efficient heat removal from the walls of the channel while not limiting the content of vapor. This makes annular flow regime suitable for a large range of industrial applications, particularly in heat transfer systems, playing a central role in the operation, safety and cost of such processes. For this reason, some consider it the most important two-phase flow regime [10]. However, the complete evaporation of the liquid film, also called dryout, is to be avoided in annular flow because it results in the rapid degradation of the heat transfer between the wall and the flow. In a BWR fuel assembly, dryout represents a threatening occurrence because it can potentially lead to fuel damage and release of radioactive fission products into the reactor coolant. Therefore, the understanding and the accurate modeling of annular flow are crucial to predict the heat transfer through the liquid film and prevent the dryout occurrance, hence for the assessment of the reactor performance and safety.

The complex physics of annular flow has been the focus of numerous research studies. The main challenge in describing the annular flow regime is the interdependency between different mechanisms. While a fraction of the liquid droplets that are flowing in the core deposits onto the

CHAPTER 2. MODELING ANNULAR FLOW WITH M-CFD

Figure 2.1:Two-phase flow regimes in vertical flow heated channel.

wall and participates in the formation of the liquid film, part of the liquid in the film detaches and forms entrained droplets. Additionally, the interface between the liquid film and the vapor in the core is characterized by waves and an interfacial shear stress. Fig.2.2provides an overview of the key mechanisms happening in annular flow regime.

The liquid film surface and its mass, momentum and energy are continuously changing because of droplet deposition, entrainment and wavy interface. These mechanisms are rarely independent, but rather have a reciprocal effect on each other. For example, the size and momentum of the entrained droplets affect their deposition, while depositing droplets may perturb the liquid film interface. Waves are formed because of the shear stress due to the velocity difference between the two phases but simultaneously have a friction effect on the flow in the core of annular flow and drive the entrainment. As a result, the study and understanding of all these mechanisms, separately whenever this is possible, are crucial for the accurate modeling of annular flow regime.

Figure 2.2:Key characteristic mechanisms of annular flow regime.

2.2

CFD Modeling Strategies

A common agreement on best modeling practices is still missing when studying two-phase flows in CFD codes, especially in case of more complex flow regimes, such as annular flow regime. M-CFD includes different modeling approaches that have been introduced and applied on multiphase flows based on specific research goals and available computational resources. The three-dimensional detailed resolution of flow structures can make CFD codes too computationally expensive. This is the case of Direct Numerical Simulations (DNS), where the interfaces between different phases are resolved at a sufficiently fine scale. Since DNS simulations are not computationally affordable for practical applications they serve to generate numerical results that can support the development of closure laws for lower resolution M-CFD models [11] [12]. Instead, for industrial applications, the focus has been on strategies that achieve the necessary model resolution while reducing computational cost. In the following sections the existing M-CFD modeling strategies are discussed.

2.2.1 The Eulerian-Eulerian Approach

The Eulerian-Eulerian approach is a frequently adopted strategy to model multiphase flow in CFD because it represents a good compromise between the need for detailed prediction of the flow structures and the need to limit computational costs. This strategy uses an interpenetrating continua formulation for which each phase is treated as a continuum that fills up the entire volume and which is described by its own system of conservation equations. The local instantaneous continuity, momentum, and energy equations are separately averaged for each phase using a double averaging technique [13] or an ensemble average approach [14]. The resulting equations are weighed by their respective volume fraction, which represents the probability of occurrence of each phase at a certain point in time and space. The averaging process also introduces new terms to the conservation equations that represent the exchange of mass, momentum and energy at the interface between two phases. These terms depend on the small-scale phase interface structure, reintroducing the information that was lost with the averaging process. Constitutive relations for these additional terms are required in order to obtain a closed model. These relations must fully describe the interaction between the different phases and incorporate all relevant phenomena that are not resolved as they occur at physical scales that are smaller than that of the numerical mesh.

When only two phases are involved, the Eulerian-Eulerian approach is referred to as two-fluid model. In annular flow regime the vapor/gas interacts with the liquid phase through two different

CHAPTER 2. MODELING ANNULAR FLOW WITH M-CFD

interface structures: the liquid droplets and the liquid film (Figure2.3). As constitutive relations are required to describe the physical interactions between the two phases at the interface, they are expected to be representative of the specific interface configuration. Therefore, while modeling a two-phase flow in M-CFD, it becomes important to characterize all the expected interface topologies and to identify the specific interface configuration in each computational cell of the numerical domain.

Figure 2.3:Different interface topologies in annular flow regime.

A few research groups further developed the concept of the Eulerian-Eulerian two-fluid ap-proach and proposed distinctive strategies to model two-phase flows in BWR fuel assemblies. As of today, the existing complete frameworks are the Computational Multi-Fields Dynamics method, the model presented at Argonne National Laboratory and the model advanced at KTH.

2.2.2 The Computational Multi-Fields Dynamics (CMFD) method

Lahey [15] first advanced a viable solution to model the different interface structures in two-phase flow. Starting from the two-fluid approach, he proposed the Computational Multi-Fields Dynamics (CMFD) method, which consists in modeling each configuration that a fluid assumes in relation with the other fluid as a different field. Four fields result from two-phase flow in a heated channel, specifically continuous vapor, continuous liquid, dispersed vapor and dispersed liquid. Each field is described with a separate system of conservation equations and characterized by a volume fraction, which determines the presence of the specific field in each computational cell of the domain. Fig. 2.4schematically shows the expected distributions of the four different fields throughout a vertical channel where the typical transition between different regimes is observed. While bubbly flow regime is represented by the dispersed vapor and continuous liquid fields, in annular flow regime all four different fields are interacting.

The CMFD method shows good prediction capabilities in a wide variety of adiabatic and diabatic bubbly flow conditions, using different geometries and different gravity conditions as well as adopting different CFD solvers (such as NPHASE, PHOENICS and CFX) [15] [16] [17]. While an

Figure 2.4:CMFD four-field two-fluid model schematic representation [15].

extensive selection of closure models is available for bubbly flow regime, the lack of mature closure relations for the continuous vapor-continuous liquid interface topology makes the modeling of high volume fraction regimes still challenging [17]. However, the CMFD method shows a promising performance when capturing the primary physical features of bubbly flow, transitional and annular flow regimes in the modeling of a full transition two-phase flow [18].

Nevertheless, the computational effort required in solving a different set of conservation equa-tions for multiple fields can represent a restriction for practical applicaequa-tions, leading researchers to explore two-phase modeling methods that allow to reduce the number of fields to model.

2.2.3 The Extended Boiling Framework (EBF)

The research group at Argonne National Laboratory (ANL) proposed an alternative strategy, the Extended Boiling Framework (EBF), to model the wide spectrum of interface topologies in the two-phase flow typical of BWR fuel assemblies. The EBF is based on the idea of using only two separate fields, one for each fluid, while locally identifying the specific interface configuration and providing appropriate closure models for each flow topology [19].

Consider for example the two different cells of the computational domain in Figure2.3. These can be occupied by the same content of liquid phase, while representing different interfaces. In the EBF, a flow topology map is employed to determine the local interface configuration as a function of local flow conditions, such as volume fraction and volume fraction gradient (Fig.2.5). Once the

CHAPTER 2. MODELING ANNULAR FLOW WITH M-CFD

interface topology is identified, the specific constitutive relations are prescribed.

Figure 2.5:EBF flow topology map [20].

While the interaction at the interface typical of bubbles in bubbly flow regime and droplets in annular flow regime has been well investigated, validated closures for the exchange terms in case of a sharp interface and any other interface forming in transitional regimes are lacking. Therefore, while validated constitutive equations are used to model the phase interactions in bubbly and mist flow regimes, the exchange terms in case of other interface configurations are obtained by interpolating between those for the known regimes [21]. This is based on the assumption that any interface configuration can be represented by an appropriate combination of bubbly and mist flow topologies (Fig.2.6).

Figure 2.6:EBF master-cell topology as combination of basic regime interface configurations [20].

The EBF presents good results when modeling bubbly flows in simple geometries [19] [21] as well as for a wider range of flow conditions [20] and in BWR channel representative geometries [22] [23], also using different CFD codes (such as STAR-CD and NEK-2P). While the framework shows potential for the modeling of two-phase flow in a BWR fuel assembly, its major limitation is represented by the artificial determination of the exchange terms for all those interface topologies that are not typical of bubbly or mist flow. The use of interpolating functions that impose the combination of the basic flow regimes in the representation of unknown interface topologies, as well as the arbitrary choice of regime transition points in the flow topology map, constitute a source of inaccuracy and affects the final two-phase flow predictions. In conclusion, the need of validated constitutive relations for the interface configurations typical of a liquid film and transitional regimes

is further demonstrated.

2.2.4 KTH model

With the objective of predicting the dryout occurrance in BWRs, Li & Anglart [24] at KTH proposed to model only annular flow regime using a three-field approach. The focus is directed on the phase interactions occurring at the interface of the liquid film, which is hence modeled as a separate field. Instead, the vapor and liquid droplets in the core of annular flow are described as two fields using the two-fluid Eulerian-Eulerian or Eulerian-Lagrangian technique.

The liquid film model is based on two assumptions, which state that the flow is neglected in the wall normal direction and the spatial gradients in the tangential direction are negligible compared to the ones in the normal direction [25]. Under these assumptions, films are considered thin and modeled with a two-dimensional system of equations.

In order to couple the liquid film model with the two-fluid model in the core, the KTH three-field approach requires to define constitutive equations for the mass, momentum and energy exchanges at the interface between liquid film and vapor-droplets core. Fig. 2.7schematically summarizes the proposed contributions to mass, momentum and energy balances. Specifically, the interactions between liquid film and liquid droplets as well as those between liquid film and vapor flow have to be described. The rates of deposition and entrainment at which droplets are respectively depositing and being entrained, represent key terms for the characterization of the interaction between the film and liquid droplets, and they are described adopting available one-dimensional empirical correlations [24].

Overall, the KTH three-field approach demonstrates good modeling capabilities, showing favorable agreement with the experimental predictions of film thickness and dryout occurrence [26] as well as temperature in post-dryout conditions [27], in simple geometries and also with the introduction of obstacles [28].

CHAPTER 2. MODELING ANNULAR FLOW WITH M-CFD

2.3

Selected Modeling Strategy and Development Opportunity

As presented in Sec.2.2, different approaches can be used to model two-phase annular flow. For generality, the two-field two-fluid EBF and four-field two-fluid CMFD method are preferable in applications involving the different regimes of two-phase flow. Instead, the KTH three-field model is limited to annular flow regime only. In this work, the objective is representing the physical phenomena from a general point of view, which could be then leveraged in different frameworks. As annular flow and liquid film interactions are the focus of this research, the three-field approach is selected because it is the most applicable to the study and modeling of annular flow regime.

Closure relations describing the rates of deposition and entrainment are key to the liquid film model. While multiple correlations are available in literature to model droplet deposition and entrainment, multidimensional closures are still lacking. Existing correlations were generally designed and developed for lumped parameter method applications and as a result present two main issues for their adoption in M-CFD. First, they make use of integral quantities, such as geometric or averaged flow conditions, so they are capable of modeling only one-dimensional flow structures. Second, one-dimensional correlations were developed based on experiments and as a consequence, their applicability is often limited to those specific conditions used for their development. In order to be suitable for M-CFD applications, closures should instead represent the local subgrid-scale physics of interaction at the liquid film interface and be applicable to any flow condition. Multidimensional physics-based constitutive models are expected to predict complex flow and geometrical variations as well as to improve the accuracy of flow predictions and contribute in making M-CFD a reliable tool for industrial use. In this research, the focus is on governing models, so new closure relations for deposition and entrainment rates could ideally be leveraged to model the exchange terms at the liquid film interface in other modeling approaches, as well. The research performed on the modeling of deposition and entrainment mechanisms is presented respectively in Chapter3and Chapter4.

2.4

Governing Equations

In order to reduce the complexity of the problem when analyzing the phase interactions at the liquid film interface, flow is assumed adiabatic in this work.

2.4.1 Gas-Droplet Core

The gas-droplet core region of annular flow is modeled using the two-fluid Eulerian-Eulerian approach. The gas is considered the continuous phase, while the liquid droplets are represented as dispersed phase. Assuming adiabatic, incompressible air-water flow, the averaged equations for mass and momentum conservation become [29]:

∂(α_kρ_k) ∂t + ∂ ∂xi (α_kρ_kv_i,k) =0 (2.1) ∂(αkρkvi,k) ∂t + ∂ ∂xi (αkρkvi,kvi,k) = − ∂ ∂xi (αkPk) + ∂ ∂xj

[αk(τij,k+τ_ij,kRe)] +αkρkgi+Mi,k (2.2)

where subscripts i and j denote Cartesian coordinate components, αkρkvkare the volume fraction,

density and velocity of phase k, P is the pressure, τij,k is the laminar stress tensor, τij,kRe is the

forces.

Turbulence

The standard k−eturbulence model is used to model turbulence in the gas (g) flow. The

turbu-lent Reynolds stress tensor τ_ij,gRe in equation2.2is defined adopting the Boussinesq approximation:

τ_ij,gRe =µt,g  ∂vi,g ∂xj + ∂vj,g ∂xi  (2.3)

The turbulent eddy viscosity µt,gis defined as a function of the turbulent kinetic energy k and

the turbulent dissipation rate e:

µt,g=Cµρg

k2_g

eg

(2.4)

where Cµ =0.9. The transport equations for turbulent kinetic energy and turbulent dissipation rate

can be found in [30].

As liquid droplets are transported by the gas turbulent eddies, the turbulence of the liquid phase (l) is correlated to that of the gas phase through the function Ct:

Ct=

|v0_l|

|v0_g| (2.5)

where v0is the velocity fluctuation. Ctis defined with equation3.11.

Interfacial Forces

Liquid droplets can be treated similarly to solid particles if their shape is spherical and does not change significantly because of flow conditions. However, the droplet shape can be affected by the interaction with other droplets and the gas flow. The dimensionless Weber number represents the ratio between the liquid inertia and surface tension and provides a measure of the droplet deformation.

We= ρlv

2 rd

σ (2.6)

In this work, low Weber numbers (We<10) are observed and for this reason, liquid droplets are assumed to remain spherical [31].

The forces acting on the interface between the gas and the liquid droplets and contributing in the momentum exchange are the drag force and the turbulent dispersion:

M =FD+FTD (2.7)

Specifically, drag force is computed as in equation2.8with CDbeing the drag coefficient:

FD =

3 4

CD

CHAPTER 2. MODELING ANNULAR FLOW WITH M-CFD

The model proposed by the Schiller-Naumann [32] for solid particles is adopted to determine the drag coefficient:

CD = ( 24 Red(1+0.15Re 0.687 d ) 0< Red61000 0.44 Re_d>1000 (2.9)

where Ddis the droplet diameter and Redis the droplet Reynolds number:

Re_d = ρld|vg−vl|

µl

(2.10) The formulation by Burns (the Favre Averaged Drag Model) [33], with σTD = 1.0, is used to

model the turbulent dispersion force:

FT D = 3 4 CD d αl|vg−vl| µt,g σTD  1 αl + 1 1−αl  ∇αl (2.11)

The droplet diameter D_dis modeled using the correlation provided by Azzopardi et al. [34], selected after performing a comparative analysis of existing correlations for the mean droplet size (Sec. 3.4.3).

2.4.2 Liquid Film

Thin Film Assumption:Similarly to the KTH model, in order to simplify the liquid film model and save computational cost in M-CFD, the thin-film assumption is adopted. This assumption al-lows to neglect the flow in the wall normal direction and describe the liquid film two-dimensionally, only in the wall tangential directions. Also, the film is considered thin enough for the laminar boundary layer approximation to apply. Hence, the film velocity profile can be easily computed without modeling turbulence.

Hence, the liquid film is modeled as a two-dimensional field separate from the gas-droplet core region. Assuming adiabatic, incompressible liquid flow, the mass and momentum conservation equations are shown below [35]:

∂(ρlδ) ∂t + ∂ ∂xs (ρlδvf) =Sδ (2.12) ∂(ρlδvf) ∂t + ∂ ∂xs (ρlδvfvf) = − ∂ ∂xs (δPf) + ∂ ∂xs (δτf) +Svf (2.13)

where δ is the film thickness, vf is the mean film velocity, τf is the viscous stress tensor within the

film and subscript s indicates the tangential direction to the surface.

The velocities and momentum exchanges due to the local pressure and the viscous shear stress are imposed to be the same for the liquid film and the gas phase at the interface:

v_f =vg (2.14) − ∂ ∂xs (δPf) + ∂ ∂xs (δτf) = − ∂ ∂xs (αgPg) + ∂ ∂xs [αg(τsj,g+τ_sj,kRe )] (2.15)

The source terms Sδ and Svf represent the mass and momentum exchanges at the interface

between the liquid film and the gas-droplet core. Under the assumption of adiabatic flow, the mass exchange at the liquid film interface occurs only due to droplets depositing and liquid entraining, therefore:

Sδ = D˙ − ˙E (2.16)

where ˙D and ˙E are respectively the deposition and entrainment rates per unit area.

In order to represent the transfer of mass between film and liquid phase in the core of annular flow, a sink term of equivalent absolute value and opposite sign of the film source term (S_l = −Sδ) is

added to the RHS of the equation for the mass conservation of liquid droplets in the bulk (equation

2.16).

The contributions to the momentum source term are wall shear stress, gravity, deposition and entrainment. The momentum source terms deriving from the wall shear stress and gravity are computed using equations2.17and2.18.

Svf,τw = −µl  ∂vf ∂y     y=0 (2.17) Svf,g = −ρlδn·g (2.18)

where y is the wall normal direction and n is the film surface normal vector.

The momentum exchanges resulting from droplet deposition and entrainment can be defined as a function of the velocity at which the droplet is depositing or entraining in the tangential direction to the surface:

S_v

f, ˙D =Dv˙ s,dep (2.19)

S_{vf, ˙E} = ˙Evs,ent (2.20)

In this work, in order to simplify the model and reduce the source of inaccuracies deriving from a required assumption on the depositing and entraining droplet tangential velocity, the momentum exchanges due to deposition and entrainment are neglected.

Chapter 3

Deposition

The interdependency between different physical mechanisms represents the main challenge in investigating and modeling the complex physics of annular flow. Liquid droplets are continuously entrained and deposited, so the liquid film is the result of a continuous exchange of liquid mass departing from and arriving to the film interface. The independent study of a specific mechanism is possible only if the reciprocal effects can be separated. For example, in order to study the droplet deposition, droplets cannot be entrained and the structure of the liquid film cannot affect the way and the amount with which droplets are depositing. The separation of the effects of different mechanisms in annular flow is challenging when carrying out experimental work. However, through dedicated experimental designs it is possible to mostly separate the droplet deposition. For this reason, in this work, the deposition of liquid droplets on the film of annular flow is studied first.

In this chapter, the droplet deposition is investigated to gain a detailed physical understanding as well as to improve its modeling in M-CFD. The experimental approaches that allow the separation of this key mechanism are described in Sec.3.1. The droplet deposition is commonly defined by the classic definition presented in Sec.3.2. The fundamental mechanics leading droplets to deposit is explored and discussed in Sec.3.3. Finally, a new approach in defining the droplet deposition rate is introduced and tested against two test cases in Sec.3.4.

3.1

Separation of deposition effect

Deposition is investigated leveraging a separate effects approach. Cleverly designed tests can allow the separation of the effect of deposition. The removal of the liquid film and the introduction of droplets from a controlled source permit the study of the deposition of the dispersed droplets without these being affected by the process of entrainment.

The double extraction technique was introduced by Cousins [36] and adopted by many re-searchers later, such as Govan [37] and Jepson [38]. First, the liquid film is extracted a first time after reaching annular flow equilibrium conditions, then, the liquid film starts reforming from the deposition of droplets migrating from the core of the annular flow towards the wall. A uni-directional deposition region is created, in which conditions below the onset of entrainment are maintained and droplets are only allowed to deposit. In this way, it is possible to measure through a second extraction the liquid film mass flux, that is equivalent to the mass flux of the liquid phase that was deposited along the deposition length zD.

Another way to separate the deposition is by introducing droplets of a certain size from a localized source. Similarly to the case of the double extraction technique, no liquid film is initially attached at the wall and droplets can only deposit and start generating a liquid film. As no entrainment occurs, deposition is studied independently and measured through a liquid film extraction. The experiments performed by Lee [39] and Binder [40] follow this technique. Fig.3.1

Figure 3.1:Schematic representation of experimental techniques: double extraction technique (left) and localized source droplet introduction (right).

3.2

Classic definition of deposition

In classic literature, the deposition rate ˙D at which droplets are depositing at the wall per unit area is described as the product of droplet concentration C and a deposition coefficient kD:

˙

D=kDC (3.1)

where the concentration is defined as a function on mass fluxes ˙G, densities ρ and velocities v of entrained liquid droplets (le) and gas (g) phases:

C= _˙ G˙le Gg ρg vle vg + ˙ Gle ρl (3.2)

Many one-dimensional correlations offer to define the deposition coefficient kD. In the absence

of multidimensional closures, a first attempt has been to evaluate the adoption of existing integral empirical correlations. As an example, the formulation of the well-known Okawa correlation [41] for the deposition coefficient kDis given by the following equation:

kD = r σ ρgDh 0.0632 C ρg −0.5 (3.3)

where σ is the surface tension and Dhis the hydraulic diameter. The droplet concentration C is

defined in M-CFD as the product of the volume fraction and density of the liquid phase.

For the sake of consistency with the one-dimensional nature of the Okawa correlation, the section-averaged value for the volume fraction of liquid phase is used. The Okawa correlation for the deposition coefficient and the classic formulation for deposition rate are tested in M-CFD against the experimental database collected by Govan [42]. The experimental results are compared with the prediction from the lumped formulation of the Okawa correlation and the M-CFD results in Fig.3.2.

CHAPTER 3. DEPOSITION

Figure 3.2:Results of the test of Okawa correlation against M-CFD and experiments [42].

The comparison is useful to evidence both the applicability and limitation of this simple approach. When deposition rate is evaluated in M-CFD from averaged quantities across the channel the results can be made consistent with the one-dimensional lumped approach; the remaining discrepancy is due to the approximated expression used for the droplet concentration in the one-dimensional approach under the assumption that ρg <<ρl. Interestingly, the Okawa correlation

still underpredicts the deposition, further demonstrating the challenge of reducing deposition to a simple integral phenomenon. The results serve to prove the need to develop appropriate multidimensional closures for M-CFD application, where the rate of deposition is a function of local quantities, and that represent the subgrid-scale physics of deposition.

3.3

Mechanics of droplet deposition

In the interest of representing the subgrid-scale physics of droplet deposition in future closures for M-CFD applications, the mechanics that lead droplets to move and eventually to deposit on the wall are investigated.

3.3.1 Turbulent diffusion deposition

Young and Leeming [43] proposed to identify the different mechanisms that induce droplets to deposit, by collecting empirical data and plotting a dimensionless deposition velocity versus a dimensionless relaxation time. Three different trends representing three different regimes leading to deposition can be clearly recognized. The differentiation between a diffusion driven deposition, that is predominant for small particles, and an inertial deposition, characteristic of bigger particles, is established (Fig.3.3).

Brownian diffusion can be neglected when the droplet size is larger than 0.5µm [44]. As the size of the droplet in BWR conditions is much larger than the above limit [45], Brownian diffusion is not considered. The motion of liquid droplets flowing in the gas core of annular flow is deeply influenced by the turbulence of the continuous phase. With a dependence on their size and density, droplets are transported by the turbulent eddies and acquire sufficient inertia to reach the wall and

Figure 3.3:Characterization of deposition regimes [43].

deposit. According to Binder [40], a turbulent diffusion model can properly work within a range for the length scale characterizing the motion of a particle moving into a turbulent flow. This length scale is defined as:

Lp = ep q v02 p =τp q v02 p (3.4)

where epis the particle diffusivity, which is defined as the product of the mean square of the particle

velocity fluctuation v02

p and the particle relaxation time τp, and

q v02

p is the root mean square velocity

of the particle. When the length scale Lp is larger than the viscous wall boundary layer (or the

dimensionless relaxation time is larger than 20) and smaller than the radius of the pipe, deposition can be assumed to be driven by turbulent diffusion.

Earlier, James [46] observed that the interaction between droplets and eddies in the turbulent gas flow changed depending on the droplet size. Specifically, droplets with a diameter smaller than 80µm were clearly affected by the turbulent eddies and engaged in a random-walk type of motion. Instead, droplets with a diameter above 120µm were not affected by turbulence and followed clear trajectories. The dimensionless number G is derived from the ratio between the droplet momentum and its drag force:

G= ρlvind CDρguτDh

(3.5) where vin is the droplet initial velocity, uτ is the friction velocity and the drag coefficient CD is

CHAPTER 3. DEPOSITION

Shepherd [47]:

CD = 18.5

Re0.6_d (3.6)

The droplet Reynolds number is defined as:

Red=

ρgduτ µg

(3.7)

The gas friction velocity uτ is assumed to be equal to the relative velocity between droplet and

eddy, while the initial velocity of the droplet is determined by the following correlation:

vin =12

r ρg

ρl

uτ (3.8)

Andreussi and Azzopardi [48] proposed a critical value of G below which the momentum of the droplet is not large enough to overcome the turbulent eddy. It is found that when G<0.7 droplets are deposited as a result of their interaction with the turbulent eddies.

Two experiments based on the double extraction technique performed by Govan [42] and Jepson [38], and the experiments using a localized droplet injection performed by Lee [39] are used to evaluate the applicability of different modeling approaches. For all the test cases both criteria for turbulent diffusion-driven deposition, proposed by Binder (δv < Lp < R) and by Andreussi

and Azzopardi (G<0.7) are met, therefore the focus of this study is on the deposition driven by turbulent diffusion.

3.3.2 Deposition rate model

Many researchers explored the possibility of solving the diffusion equation for the droplet concentration to study the distribution of the droplets in time and space. Diffusion is neglected in the axial direction of the flow and a pipe is chosen as geometry. The diffusion equation describing the particles concentration in the radial direction r is:

∂C ∂t =ep  ∂2C ∂r2 + 1 r ∂C ∂r  (3.9)

where C is the number of particles per unit volume. Here t is the time that particles at a certain distance z from the source have been in the system.

When using a diffusion model a few challenges have been commonly encountered. The first challenge is the choice of the boundary condition at the wall. The definition of the particle diffusivity is another point of discussion, since a relation between particle diffusivity and fluid diffusivity is not yet available.

A zero concentration at the wall condition was selected by some [44] [49], while others preferred the use of a radiation boundary condition [50] [40]. The definition of the proposed radiation boundary condition comes from the assumption that the length scale Lp is large enough that

the hypothesis of a perfect absorber wall is not valid [39]. The depositing droplets travel at a characteristic velocity that is equal to the conditionally averaged radial velocity given that this