modeling fluidelastic instability of two-phase flow in

Modeling Fluidelastic Instability of Two-Phase

Flow in Tube Bundles

byYasser Selima

A Thesispresented to

The University of Guelph

In partial fulfillment of requirementsfor the degree of

Doctor of Philosophyin

Engineering

Guelph, Ontario, Canadac©Yasser Selima, September, 2018

ABSTRACT

MODELING FLUIDELASTIC INSTABILITY OF TWO-PHASE

FLOW IN TUBE BUNDLES

Yasser Selima Advisors:

University of Guelph, 2018 Dr. Marwan Hassan

Dr. Atef Mohany

Dr. Wael Ahmed

The CANDU steam generator consists of tube bundles where two phase flow across the

bundle takes place. Among many tube vibration mechanisms, the FluidElastic Instability

(FEI) is considered the most destructive one. Many studies focused on FEI of two phase

flow. However, there is no analytical model until now able to predict FEI threshold velocity.

This study presents a novel analytical framework to predict the fluidelastic instability

threshold of two phase flow across normal square bundle. The model focuses on the bubbly

flow (void fractions up to 35%). A single vibrating tube in a fixed bundle was considered.

Flow around the vibrating tube is idealized as one dimensional bubbly flow in two channels.

The fluid in each channel is composed of continuous phase and dispersed phase. The dis-

persed phase is accounted for by spherical bubbles. The motion of each individual bubble

was modeled by accounting for the external forces acting on its surface. Bubble-to-bubble

interaction, bubble break-up, and bubbles coalescence were taken into account. Bubble re-

sponse to the pressure pulsations was integrated in the model by solving Rayleigh-Plesset

equation. By tracking each bubble in the flow channels, it is possible to calculate the change

in the flow density around the tube. The calculated instantaneous local density is used to

solve the unsteady conservation equations to find the fluid forces on the vibrating tube. The

tube is then modeled as a single degree of freedom system to predict its displacement. The

feedback mechanism of the tube on the fluid was then modeled.

Time domain simulations were conducted for air-water mixture and the stability threshold

was predicted. Prediction of the stability threshold showed a very promising results when

compared with the experimental data. A sensitivity study is performed on the model to test

its limitations is presented.

iv

Dedication

To my Mum and Dad,

without you, I would not be able to continue ...

v

Acknowledgements

First of all, thanks to Allah.

Special thanks to Dr. Marwan Hassan for his help, support, and patience during the

most difficult times.

I would like to thank my co-advisors, Dr. Atef Mohany and Dr. Wael Ahmed for the

advise and help provided during this research.

To my wife, Marwa Ragab, thank you for your patience and support during the last few

years.

Thanks to my colleges, Amro Elhelaly, Salim Elbouzaidi, John cloutier and Osama El-

banhawy for the good moments we shared.

vi

Table of Contents

Abstract ii

Dedication iv

Acknowledgements v

Table of Contents viii

List of Tables ix

List of Figures xii

Abbreviations xiii

Symbols xiv

Greek Symbols xv

List of Appendices xvi

1 Introduction 11.1 Flow Induced Vibration (FIV) . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Classification of FIV . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Stability versus Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Cross Flow Heat Exchangers . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Fluidelastic Instability (FEI) . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Ouline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Literature Review 72.1 Tube bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Fluidelastic Instability Models for single phase flow . . . . . . . . . . . . . . 10

2.2.1 Jet Switch Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Quasi-Static Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Unsteady Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

vii

2.2.4 Quasi-Steady Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.5 Flow Cell Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Modeling of two phase flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 Flow patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1.1 Flow patterns in tube bundle . . . . . . . . . . . . . . . . . 212.3.2 Models of two phase flow . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.3.2.1 Two fluid model . . . . . . . . . . . . . . . . . . . . . . . . 242.3.2.2 Homogeneous model . . . . . . . . . . . . . . . . . . . . . . 242.3.2.3 Slip model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.2.4 Void fraction model of Dowlati . . . . . . . . . . . . . . . . 262.3.2.5 Drift flux model . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Two-phase bubbly flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Bubbles Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4.2 Bubbles breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4.2.1 Breakup due to turbulent fluctuation . . . . . . . . . . . . . 322.4.2.2 Breakup due to the viscous shear forces . . . . . . . . . . . 322.4.2.3 Breakup due to shearing-off . . . . . . . . . . . . . . . . . . 322.4.2.4 Breakup due to interfacial instability . . . . . . . . . . . . . 33

2.4.3 CFD modeling of bubbly flow . . . . . . . . . . . . . . . . . . . . . . 332.5 FEI in two phase flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.5.1 Two-phase flow damping . . . . . . . . . . . . . . . . . . . . . . . . . 382.5.2 Modeling FEI in two phase flow . . . . . . . . . . . . . . . . . . . . . 41

2.6 Summary and research needs . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Modelling 443.1 Overview of the analytical framework . . . . . . . . . . . . . . . . . . . . . . 453.2 Structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3 Continuous phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.4 Dispersed phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.4.1 Bubbles generation algorithm . . . . . . . . . . . . . . . . . . . . . . 533.4.2 Forces acting on the bubble . . . . . . . . . . . . . . . . . . . . . . . 55

3.4.2.1 Lift force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.2.2 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4.2.3 Buoyancy force . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.4.3 Bubble kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.4.4 Bubble-to-bubble impact . . . . . . . . . . . . . . . . . . . . . . . . . 623.4.5 Bubble bouncing from the wall . . . . . . . . . . . . . . . . . . . . . 643.4.6 Bubbles coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643.4.7 Bubble breakup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.4.8 Bubble compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5 Fluid structure interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

viii

4 Model Implementation 724.1 Conservation equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.2 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.3 Bubble loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.3.1 Solution of Rayleigh-Plesset equation . . . . . . . . . . . . . . . . . . 764.3.2 Coalescence and breakup procedure . . . . . . . . . . . . . . . . . . . 80

4.4 Steady State Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Results and Discussion 855.1 Homogeneous model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2 Current model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2.1 Stability threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.2.2 FEI Force acting on the tube . . . . . . . . . . . . . . . . . . . . . . 945.2.3 Density distribution in the flow channels . . . . . . . . . . . . . . . . 94

5.3 Sensitivity studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.3.1 Effect of initial bubble size distribution . . . . . . . . . . . . . . . . . 965.3.2 Effect of initial bubbles size . . . . . . . . . . . . . . . . . . . . . . . 1005.3.3 Effect of time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.3.4 Effect of including bubble compressibility . . . . . . . . . . . . . . . . 104

5.4 Effect of Breakup and Coalescence criteria . . . . . . . . . . . . . . . . . . . 1055.4.1 Effect of size breakup criteria . . . . . . . . . . . . . . . . . . . . . . 1055.4.2 Coalescence criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.5 Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.5.1 FEI Threshold (Critical flow Velocity): . . . . . . . . . . . . . . . . . 107

5.6 Slip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.7 Effect of density ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.8 Effect of P/D ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 Conclusion 1196.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.2 Recommendations for future work . . . . . . . . . . . . . . . . . . . . . . . . 121

References 123

A Dimensional Analysis of Rayliegh Plesset equation 130

ix

List of Tables

2.1 Summary of the stability threshold suggested by [82] . . . . . . . . . . . . . 13

3.1 Streamtube parameters for normal square bundle . . . . . . . . . . . . . . . 51

5.1 Simulation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.2 The stability threshold velocities for both beta and normal distributions . . . 995.3 The effect of the minimum bubble radius at the bubble generation stage on

the stability threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.4 The effect of timestep on the stability threshold velocity . . . . . . . . . . . 1035.5 The effect of bubble compressibility on the stability threshold velocity . . . . 1055.6 The effect of forced breakup criteria on the stability threshold velocity . . . . 1075.7 The effect of coalescence criteria on the stability threshold velocity . . . . . . 1105.8 Comparison between the stability threshold velocity obtained by the current

model and the experimental data reported by Pettigrew et al 1989 [61] . . . 1115.9 The initial input pressure used to obtain the density ratio . . . . . . . . . . . 1135.10 The effect of density ratio on the stability threshold velocity . . . . . . . . . 1175.11 The effect of P/D ratio on the stability threshold velocity . . . . . . . . . . . 118

x

List of Figures

1.1 Diagrammatic sketch of CANDU steam generator, Moran 2007 [50] . . . . . 41.2 Relation between the vibration amplitude and the flow velocity, Paıdoussis et

al. [56] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Tube damage caused by FEI, (a), (b) and (c) shows the effects of fretting

wear at the supports (source AECL - currently CNL). (d) damage causedby a broken piece of a tube before hitting another tube (UKAEA Harwell),photos published by Khalifa 2013 [37] . . . . . . . . . . . . . . . . . . . . . . 6

2.1 Tube bundle configurations. (a) Normal square. (b) Normal triangle. (c)Rotated square. and (d) Parallel triangle . . . . . . . . . . . . . . . . . . . . 9

2.2 FEI maps for various array geometry proposed by [82]. (a)Normal square.(b)Rotated square. (c)Normal triangle. and (d)Parallel triangle, [82] . . . . 12

2.3 Cylinder numbering system employed by Tanaka and Takahara 1980 [78] . . 152.4 Flow visualizations showing stream tubes in a cross flow, [43] . . . . . . . . 172.5 Unit cell for fluid elastic model, [28] . . . . . . . . . . . . . . . . . . . . . . 172.6 Comparison between flow cell model and experimental results [42] . . . . . . 182.7 Schematic of flow patterns in vertical upward gas liquid co-current flow, [11] 212.8 Hewitt and Roberts flow-pattern map for vertical upward gas-liquid co-current

flow. G is the mass flux, and ρ is the density. [30] . . . . . . . . . . . . . . . 222.9 Flow-pattern map for vertical flow in tube bundle, [80] . . . . . . . . . . . . 232.10 Flow-pattern map for vertical in tube bundle, [52] . . . . . . . . . . . . . . . 232.11 Sketch of film stretching and bubble surface flattening for two bubbles ap-

proaching each other, [49] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.12 Critical capillary number in simple shear flow [45] . . . . . . . . . . . . . . . 312.13 Map of the three phases used in the coupled model, [83] . . . . . . . . . . . 342.14 Stability map for two phase flow based on the homogeneous model, [50] . . 372.15 Stability map for two phase flow based on the interfacial velocity and measured

RAD density, [50] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.16 Comparison of normalized two phase damping from different studies by [60] 402.17 Stability threshold map obtained from the two-phase flow unsteady forces

simulations and direct flow/structure coupling vs. experimental data fromPettigrew (1989) for airwater flow, [72] . . . . . . . . . . . . . . . . . . . . . 42

xi

3.1 Visual observation of streamtubes flowing through normal square tube bundle,[43] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Streamtubes surrounding the vibrating tube . . . . . . . . . . . . . . . . . . 503.3 A schematic sketch showing the beta distribution function . . . . . . . . . . 533.4 Forces acting on each bubble . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.5 Streamlines of steady flow (from left to right) past a sphere at various Reynolds

numbers, Taneda 1956 (reported by Brennen 1995 [5]) . . . . . . . . . . . . . 573.6 Values of Strouhal at different Reynolds number [73] . . . . . . . . . . . . . 593.7 Elastic Impact between bubbles . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 Screen shot from the simulations showing the vibrating tube (blue) surroundedby the flow channels. The channel oscillating boundary is shown in red . . . 73

4.2 Diagrammatic sketch showing the flowchart of the simulation . . . . . . . . . 774.3 Diagrammatic sketch showing the flowchart of the bubble loop . . . . . . . . 784.4 Response of bubble radius to a sinusoidal pressure wave. Time step = 10−10sec 804.5 Snapshots from the simulation showing bubble breakup . . . . . . . . . . . . 824.6 Snapshots from the simulation showing bubble coalescence . . . . . . . . . . 83

5.1 Tube response for 25% void fraction at various flow velocity using the homo-geneous density model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 Tube response for 25% void fraction at various flow velocity using the currentmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3 Tube response for 68% void fraction, from Moran 2007 [50] . . . . . . . . . . 935.4 Comparison between the stability map obtained by the current model and by

the homogeneous model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.5 Tube displacement and fluid forces on the tube vs. time, Uavg = 1.2m/s and

α = 25% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.6 Tube displacement and fluid forces on the tube vs. time, Uavg = 1m/s and

α = 25% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.7 Tube displacement and fluid forces on the tube vs. time, Uavg = 1m/s and

α = 32% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.8 Average density in the channel vs. the time at different locations. (a)Left

channel s = 0 (tube center), (b) Right channel s = 0 (tube center); (c)Leftchannel s = −so

2; (d)Right channel s = −so

2; (e)Left channel s = −so

4; (f)Right

channel s = −so4

; (g)Left channel s = −so (tube entrance) ; (h)Right channels = −so (tube entrance) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.9 Effect of initial bubble distribution . . . . . . . . . . . . . . . . . . . . . . . 1005.10 Effect of initial bubble size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.11 Effect of time step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.12 Effect of gas compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.13 Effect of using different different forced breakup criteria . . . . . . . . . . . . 1085.14 Effect of using different different coalescence criteria . . . . . . . . . . . . . . 109

xii

5.15 Comparison between the results obtained from the current model with exper-iments Pettigrew et al. 1989a [61] . . . . . . . . . . . . . . . . . . . . . . . . 109

5.16 Comparison between the slip factor obtained by the current model and thecalculated slip using Feenstra’s model [19] applied on the experimental resultsby Pettigrew et al. 1989a[61] . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.17 Effect of the density ratio on the stability threshold . . . . . . . . . . . . . . 1145.18 Effect of P/D ratio on the stability threshold . . . . . . . . . . . . . . . . . . 116

xiii

Abbreviations

FEI FluidElastic Instability

FIV Flow Induced Vibration

MDP Mass Daming Parameter

xiv

Symbols

M Tube total mass per unit length including the added massx Tube displacementζ Damping ratiofn Bundle natural frequency in the two phase mixturek Tube stifnessFFEI Fluidelastic instability forceBo Bond numberCL Coefficient of Lift forceCD Coefficient of Drag forceCa Capillary numberD Tube diameterDb Bubble diameterf FrequencyFr Froude numberFB Buoyancy force.FD Drag forceFL Lift forceG mass fluxJ Volume fluxM Effective tube massm Tube mass per unit lengthma Added massP Center to center pitchRe Reynold’s numberRi Richardson numberS Slip ratioSt Strouhal numberUp Pitch velocityUr Relative VelocityWe Weber numberVf gas Volume fractionX Bubble horizontal location in a vertical channelX Bubble vertical location in a vertical channel

xv

Greek Symbols

ω angular frequencyα Volume fractionδ Damping logarithmic decreamentρl, ρg Density of Liquid, Gasµl Liquid viscousityν Kinamatic viscosityσ Surface tensionφrandom Random phase angleζ Structure damping ratio

xvi

List of Appendices

Appendix A: Dimensional Analysis of Rayliegh Plesset equation

Chapter 1

Introduction

1.1 Flow Induced Vibration (FIV)

Flow Induced Vibration (FIV) is the phenomena when mechanical vibration starts due to

fluid flow. The fluctuations of leaves in a windy day, and the vibration of wires are examples

of FIV. This phonemna occurs in many industries and engineering applications as well. This

obstructs the smooth operation of these applications and cause deviation from the optimal

design conditions. In some cases this could cause disasterous damage. For example, the

collapse of Tacoma bridge in 1940 was caused by vibrations induced by fluid flow or FIV.

Another recent example is the permenant closure of San Onfore Nuclear Generating Station

(SONGS) in 2013. The latter example will be mentioned few times in this thesis.

The physical explanation of FIV could be summerized very briefly as vibration caused by

the fluid forces acting on a certain structure. In some cases, the fluid forces are unsteady in

nature, such as the turbulence forces. These unsteady forces would apply a periodic force on

the structure causing periodic motion; i.e. vibrations. However, even the steady flow would

cause FIV in many cases. This happens because of the existence of a feedback mechanism

from the structure acting back on the fluid. The feedback in this case causes this periodic

1

University of Guelph PhD Thesis - Y. Selima

force of the fluid on the structure pushing it to vibrate. From here, there was a need to

classify the FIV according to its cause and/or mechanism to enable avoiding its damage.

1.1.1 Classification of FIV

Many classification were available in the literature to distinguish the various types of FIV.

Weaver and Fitzpatrick 1988 [82] classified the FIV by the nature of vibration in each case.

1. Forced vibration, in which the excitation fluid force is independant on the structure.

This is the case of turbulent beffeting.

2. Self-controlled vibrations, in which vibration is controlled by the structure. This is the

case of vortex shedding.

3. Self-excited vibration, in which the excitation force that sustains the motion is created

by the motion itself; when the motion stops the excitation force disappears. This is

the case of Fluidelastic Instability (FEI) which is the focus of this thesis.

1.2 Stability versus Instability

A simple definetion of stability will be used in this work, is when the vibration amplitude

reduces to approach zero. If the amplitude doesn’t approach zero, but it is still bounded,

i.e. marginally stable, it will be considered stable. Instability will be considered when the

vibration amplitude keeps increasing with time.

1.3 Cross Flow Heat Exchangers

Heat exchangers are devices used to allow heat transfer from one fluid to another. There are

many types and classifications of heat exchangers. In this thesis, the focus will be on cross

2


flow heat exchangers, specifically the CANDU reactor steam generator. In general, Cross

flow heat exchangers are widely used in many applications such as steam generators and

condensers. It consists of thousands of tubes forming a tube bundle where one fluid flows

inside the tubes while another fluid flows across the bundle. The CANDU reactor’s design

includes a steam generator to transfer the heat from the primary contaminated cooling water

to the secondary cooler. Figure (1.1) shows a diagrammatic sketch of the type CANDU steam

generator. The primary coolant, coming from the reactor, passes inside the U-tubes while

the secondary coolant passes across the tubes. To prevent exceeding the radiation limit, it is

mandatory to monitor and control the leakage from the primary to secondary coolant. This

means that the heat exchanger design should take into account the sources of tube wear and

vibration trying to avoid them.

Increasing the performance of the heat exchangers requires increasing the mass flow rate

and the heat transfer surface area to the maximum possible values. This leads to using

several thousand long tubes (several meters long) with relatively small outer diameters (less

than 2 centimeters). Due to tube thermal expansion at the high operating temperatures,

using tight tube supports becomes a very difficult and expensive task and usually the heat

exchanger ends up with many tubes on loose supports. When the cross flow fluid apply

forces normal to these tubes, they start to vibrate with their natural frequency.

1.4 Fluidelastic Instability (FEI)

Among the vibration excitation mechanisms occurs in heat exchangers, FEI is considered the

most destructive mechanism. Figure 1.2 shows the relation between the vibration amplitude

and the flow velocity for these excitation mechanisms. From the figure we can see that

turbulence buffeting starts at low velocity with an amplitude of vibration proportional to the

flow velocity. This mechanism could not be avoided in the design as the turbulence is needed

3


Figure 1.1: Diagrammatic sketch of CANDU steam generator, Moran 2007 [50]

to enhance the heat transfer. However, the vibration amplitude due to turbulence buffeting

does not reach large destructive values. Second exciting mechanism, vortex shedding, starts

when Strouhal frequency is close to the bundle natural frequency. The amplitude of vibration

resulting from vortex shedding is higher than that from turbulent buffeting. By avoiding

the velocity range that excite the vortex shedding, vibration due to this mechanism could

be avoided.

The third excitation mechanism, Fluidelastic Instability (FEI), starts at a certain flow

velocity called “critical velocity”. When exceeding this critical velocity, vibrations due to

FEI start and the vibration amplitude increases exponentially with increasing flow velocity.

Unless a physical restrain such as impacting another tube, the FEI vibration amplitude

4


Figure 1.2: Relation between the vibration amplitude and the flow velocity, Paıdoussis etal. [56]

never goes down. FEI causes tube-tube clashing and/or tubes impact with supports which

is enough to damage the tubes in few days. Figure 1.3 show some samples of the damage

occured by FEI.

1.5 Ouline of the thesis

The next Chapter 2 will discuss the research published in the area showing the gap to be

covered by this work. Chapters (3) and (4) will explain the current model and sub-models

showing how they were integrated in one code to run the simulations. The Results will be

presented and discussed in chapter (5). The last chapter (6) will summerize the work done,

pointing out the conclusions and recommendations for future work.

5


Figure 1.3: Tube damage caused by FEI, (a), (b) and (c) shows the effects of fretting wearat the supports (source AECL - currently CNL). (d) damage caused by a broken piece of atube before hitting another tube (UKAEA Harwell), photos published by Khalifa 2013 [37]

6

Chapter 2

Literature Review

There are three different flow induced vibration mechanisms that occur in cross flow heat

exchangers; The first is turbulent buffeting which could be classified as forced vibration

mechanism. Second, is the vortex shedding mechanism which could be categorized as self

controlled mechanism. The third is fluidelastic instability (FEI) mechanism which is con-

sidered self excitation mechanism. Among these mechanisms, FEI is considered the most

destructive mechanism which could cause tube failure in a short period of time. Because of

its destructive nature, FEI was the focus of many experimental and analytical research.

There are many theories in the literature trying to explain the physics behind the FEI

mechanism. The most realistic one is that FEI starts by a vibrating tube applying force

on the surrounding fluid which reacts by applying forces back on the tube. Because of the

fluid elasticity a time lag occurs between the tube and fluid forces, Lever and Weaver 1986

[41, 42]. This time lag causes a phase shift between the tube displacement and the fluid

forces. Therefore, the fluid force component acting in the direction of the tube displacement

acts as an excitation force trying to increase the vibration amplitude. When the system

damping is not able to dissipate this vibration energy, the vibration amplitude increases

causing severe damages.

7


Because fluidelastic instability occurs only in tube arrays, the first section 2.1 will present

the basic types of tube bundles used in steam generators. The FEI models for single phase

flow will then be introduced in section 2.2. Section 2.3 will summarize the two phase flow

models in various flow regimes before focusing on the bubbly flow in section 2.4. Section 2.5

will present the work done to model FEI of two phase flow in tube array. A summary of the

research findings will be introduced in section 2.6 focusing on the research gabs need to be

filled.

2.1 Tube bundles

The purpose of heat exchangers, is to transfer the heat between two fluids. One of the

factors affecting the heat transfer positively, is the heat transfer area. Tube bundles provide

large heat transfer area per unit volume enabling more compact design of the heat exchang-

ers. That is why they are widely used in heat exchangers, especially steam generators and

condensers.

The tube bundle arrangement could be inline, or staggered. Usually staggered config-

urations uses the angles 30, 45 or 60 degrees between a tube and its neighbor. The inline

arrangement uses inline straight rows of tubes. The staggered arrays provide higher coef-

ficient of heat transfer because they generate more turbulence in the flow field. The FEI

research identify these various configurations as summarized by Weaver and Fitzpatrick 1988

[82]; normal square, rotated square, normal triangle and rotated triangle. Figure 2.1 shows

the various configurations of tube bundles.

The tube-to-tube distance is identified as the pitch P . The pitch to diameter ratio is

one of the key factors in the heat exchangers. Large P/D will increase the volume of the

heat exchanger for the same heat transfer area. On the other hand, P/D is restricted by a

minimum value of 1.25. Lower values of P/D makes the portion of the material between two

8


Figure 2.1: Tube bundle configurations. (a) Normal square. (b) Normal triangle. (c)Rotated square. and (d) Parallel triangle

adjacent tubes, too weak for proper rolling of the tubes in tube sheet, Thulukkanam 2013

[79]. In most heat exchangers, the pitch to diameter ratio ranges between 1.25 to 1.7. In the

CANDU steam generator, the value of P/D is around 1.5.

As shown earlier in Fig. 1.1, the tubes in the CANDU steam generator take the shape

of inverted U. The cold fluid, uncontaminated water, passes through the shell side of the

CANDU steam generator. It enters from the bottom of the steam generator to be heated by

the pressurized heavy water flowing inside the tubes. As the cold water flows up, it gains

more heat and evaporation occurs. More steam is generated as the flow gains more heat. The

volumetric ratio of the gas phase (steam) to the liquid phase (water) is called void fraction.

At the top of the steam generator, the water mixture void fraction reaches around 95%.

The most critical part in the steam generator is the U bend region at the top of the tubes.

In this U bend region, the flow becomes cross flow. In addition, the mixture velocity reaches

maximum values because of the very high void fraction. While loose supports exist in this

9


region and a tube could be supported on one loose support, the tube acts as a cantilever

supported from one end. For this reason, most of the experimental and theoretical models

uses cantilevers instead of U bend to simplify the setup and/or the model.

2.2 Fluidelastic Instability Models for single phase flow

The recognition of FEI instability started by Roberts 1962 and 1966 [69, 70] when he found

the possibility of self excited vibration of a staggered row of cylinders. Roberts explained

the phenomena by jet switching where a jet pairing between adjacent cylinders, changes its

direction in synchronism with the vibration of the cylinder motions. Roberts presented the

dimensionless reduced velocity parameter Ur as:

Ur =U

fD(2.1)

where U is the pitch velocity, f is the oscillation frequency and D is the tube diameter.

Roberts 1962 [69] suggested that this jet switching mechanism would only occur if Ur is

larger than 12. Roberts found that the instability only happens if the two adjacent cylinders

were free to move in the in-flow direction.

Later, Connors 1970 [14] conducted experiments on a single row of cylinders as well

and proposed a modification for the above instability criteria. Connors proposed a semi-

empirical relation for the instability reduced flow velocity as a function of the dimensionless

mass damping parameter (MDP). MDP was identified as function of the tube mass per unit

length m, the damping δ, fluid density ρ and the tube diameter D. Connor’s semi-empirical

relation is as follows:

UpfD

= K

(mδ

pD2

)n(2.2)

10


where K and n are experimental constants and they were suggested by Connors to be

9.9 and 0.5 respectively. Up is the pitch flow velocity. In a review paper, Paıdoussis 1983 [55]

presented a comparison between Connors 1970 [14] model and Roberts 1962 [69] equation

and other experimental results and showed that the two theories are relatively close for the

studied range.

Blevins 1997 [4] extended Connors model to deal with multiple rows of tubes keeping the

same form of Eq. 2.2. Many experimental studies followed trying to find suitable values for

the constants K and n for different configurations of tube bundles. Chen et al. 1985 [10]

published guidelines for the instability threshold. Weaver and Fitzpatrick 1988 [82] reviewed

the cross flow induced vibrations and attempted to standardize the parameters used to draw

the FEI maps. Weaver and Fitzpatrick 1988 [82] presented flow maps for different patterns

of tube bundles using the pitch flow velocity Up = (P/(P −D))U , and the damping in air δa,

see Fig. 2.2. Weaver and Fitzpatrick 1988 [82] suggested a horizontal line in the flow maps

for values of mδa/ρD2 < 0.3. For higher mass damping parameter, the authors suggested

values for K and n as shown in table 2.1. The authors explained the horizontal lines as

for low flow velocity, the tube velocity is no longer negligible when compared to the fluid

velocity. The drag force, which applies in the same direction of the relative velocity, will

then have a component that opposes the tube motion.

The following few sections will present briefly the leading models to account for the FEI.

2.2.1 Jet Switch Model

Analytical or semi-empirical models of FEI took place as early as it was identified by Roberts

1962 [69]. As discussed earlier, Roberts [69, 70] proposed the first model for self-excitation in

tube bundle, or fluidelastic instability. Roberts showed that instability occurs in the in-flow

direction with adjacent cylinders moving out of phase with each other.

11


Figure 2.2: FEI maps for various array geometry proposed by [82]. (a)Normal square.(b)Rotated square. (c)Normal triangle. and (d)Parallel triangle, [82]

12


Table 2.1: Summary of the stability threshold suggested by [82]

Array geometry mδaρD2 < 0.3 mδa

ρD2 < 0.3

Square Up

fD= 1.4 Up

fD= 2.5(mδa

ρD2 )0.48

Rotated square Up

fD= 2.2 Up

fD= 4.0(mδa

ρD2 )0.48

Normal triangle Up

fD= 2.0 Up

fD= 3.2(mδa

ρD2 )0.40

Parallel triangle Up

fD= 1.0 Up

fD= 4.8(mδa

ρD2 )0.30

Because only in-flow motion was allowed, Roberts suggested that a hypothetical channel

flow involving two half cylinders and imaginary boundary, can represent the flow in tube

bundle, Price 1985 [63]. The FEI was returned to the switching of the fluid jet between two

adjacent tubes. Roberts concluded that one flexible tube in a tightly fixed bundle, can go

unstable.

2.2.2 Quasi-Static Model

This is the most famous model to predict fluidelastic instability threshold and it is being used

currently in the ASME code for heat exchanger design. The model was derived by Connors

1970 [14] and same equation was concluded by Blevins 1997 [4]. Connors considered a

single row of cylinders normal to the flow and measured the fluid forces experimentally.

By Statically displacing two neighboring cylinders in symmetric or antisymmetric directions

while measuring the forces on the central cylinders. Connors reported different cylinder

vibration patterns. The most dominant pattern was a cylinder vibrating in elliptical motion.

Connors concluded that jet switch, reported by Roberts 1962 and 1966 [69, 70], was not

the dominant cause of instability. By Measuring the fluid stiffness, and applying energy

13


balance in the cross flow and parallel flow directions, Connors obtained the famous Eq.

(2.2). Connors found the value of K tends to be 9.9 and n to be 0.5.

Although Eq. (2.2) was derived for a constant pitch to diameter ratio P/D = 1.41, it is

being used until today, with different values of the constants K and n, in the ASME code.

The reason behind using Connors equation until today is its simplicity compared to the

models derived later. This despite the fact that it was proven later by Lever Lever and

Weaver 1982 [43] that one loose tube could be instable in a fixed array which contradicts

with Connors assumptions. Also Connors equation did not take into account the bundle

geometry or the pitch to diameter ratio.

2.2.3 Unsteady Model

This model was presented by Chen 1983 [8, 9] who used the unsteady forces measured by

Tanaka and Takahara 1980 [78]. The later assumed that the fluid forces acting on the tube

are function of its own motion and the motion of the surrounding cylinders. For example,

the forces affecting cylinder O in Fig. 2.3 is function of cylinders R, L, D and U. Then, the

forces acting on cylinder O was expressed by Eq. (2.3)

4CL = CLyo ∗ yo + CLyL ∗ (yL + yR) + CLxL ∗ (xL − xR) + CLyU ∗ yU + CLyD ∗ yD

4CD = CLxo ∗ xo + CDyL ∗ (yL − yR) + CDxL ∗ (xL + xR) + CDxU ∗ xU + CDxD ∗ xD(2.3)

The measured forces coefficients were employed by Chen 1983a [8] as an experimental

input to predict the forces acting on the tube and predict the stability. Chen also showed

that instability could happen by one of two mechanisms, damping controlled or stiffness

controlled insatiability.

Although the unsteady model was able to predict fluidelastic instability successfully, it

14


Figure 2.3: Cylinder numbering system employed by Tanaka and Takahara 1980 [78]

has a major disadvantage which is using many experimental parameters as input to the

model.

2.2.4 Quasi-Steady Model

In the quasi-steady model, the forces acting on a cylinder, in the lift and the drag directions,

are modified according to the cylinder relative velocity. The lift and drag coefficients remain

the same as calculated on stationary cylinders while the fluid relative velocity with respect to

the cylinders changes with its absolute velocity. According to Price 1995 [63], this assumption

is reasonable when the cylinder velocity is small compared to the fluid velocity. However,

when the cylinder vibrational velocity is comparable to the fluid velocity, the assumption

breaks down.

Gross in 1975 (reported by Price 1995 [63] ) was the first to introduce the quasi-static

model for flow across tube bundle, Price 1995 [63]. Gross returned the instability to a

negative damping mechanism. Briefly, this mechanism assumes that a negative damping

15


component resulting from FEI is affecting the total system damping. The magnitude of this

negative component increases when increasing the flow velocity. At certain flow velocity, the

total system damping becomes negative and instability proceeds.

Price and Paıdoussis introduced a series of papers Price and Paıdoussis 1984, 1985 and

1986 [66, 64, 65] on the quasi-steady model. They assumed that the fluid coefficients depend

on the displacements of the neighboring tubes causing a fully coupling between a row of

flexible cylinders. They also considered a constrained mode approach to eliminate the effect

of this coupling. The main problem with using the quasi-steady approach is calculating the

force coefficients which could be done computationally.

2.2.5 Flow Cell Model

The flow cell model was introduced by Lever and Weaver 1982 [43] and adapted by Lever

and Weaver 1986 [41, 42]. Based on flow visualizations and experimental results, the authors

concluded that the flow across tube bundle could be simplified to a flow through stream

tubes, Fig. 2.4. The authors introduced a one dimensional inviscid flow model through

streamtubes in an array with single flexible cylinder, Fig. 2.5. When the flexible cylinder

moves, the streamtube area would change. The fluid neighboring the moving cylinder was

assumed to vibrate with the cylinder (no phase lag between the cylinder and the fluid lying

between attachment and separation). Because of fluid inertia, a time lag between the tube

motion and the resulting change in the streamtube width takes place. This time lag was

thought to cause a phase between the fluid forces and the tube response leading to vibration

instability.

The authors compared the model to experimental results and well agreements were ob-

tained. Khalifa et al. 2013 [38] compared their expeiremntal results with the model showing

well agreement as well, Fig. 2.6. Lever and Weaver 1986 [41, 42] reported the existence of the

16


Figure 2.4: Flow visualizations showing stream tubes in a cross flow, [43]

Figure 2.5: Unit cell for fluid elastic model, [28]

17


Figure 2.6: Comparison between flow cell model and experimental results [42]

multiple stability region at low mass damping parameters. In general, the flow cell model

showed accurate predictions of the stability threshold for low mass damping parameters.

However, as the single tube model does not account for the interaction with the neighboring

tubes, it is less accurate at high values of mass damping parameter. Yetisir and Weaver 1993

[84] attempted to overcome this deficiency by adding the effect of the neighboring tubes.

Later, Hassan and Hayder 2008 [23], Hassan and Hossen 2010 [24] and Hassan and

Mohany [29] performed time domain simulations using the flow cell model to predict the

Fluidelastic instability in tube bundles. Anderson et al. 2014 [3] implemented the boundary

18


layer theory in the flow model for a better prediction of the separation point between the

fluid and the tube. The model was extended later by Hassan and Weaver in a series of

publications 2015, 2016 and 2017 [25, 26, 27] to model the streamwise instability of tube

bundles.

2.3 Modeling of two phase flow

Two phase flow is the type of flow where two phases of the substance are involved. These two

phases could be gas-solid flow like the case in fluidized bed, liquid-solid such as slurry flow

or liquid-gas like the case of boiling and condensation. Also there might be liquid-liquid or

gas-gas in cases of a mixture flow and more generally there might be more than two phases.

The focus here will be liquid-gas flow only as this is the case related to steam generators.

Gasliquid two-phase flow is widely used in many natural and industrial processes. For

this reason, the two phase flow topic has been tackled widely. However, there is a need for

a better insight into the details of this flow (Yan and Che 2010 [83]). In general, adding a

phase to the flow will increase the number of conservation equations by 3 momentum equa-

tions (in case of 3D flow), one continuity equation for the second phase and one or more

equation/equations of state. So, solving the conservation equations for the two phase flow

analytically is almost impossible except for very simple cases. That is why researchers tend

to use different approaches and assumptions to simplify each case. Even numerical tech-

niques use empirical correlations and relations to evaluate different terms of the equations.

These correlations are dependent on the flow pattern. Ishii et al. 2011 [34] summarized the

procedures of solving the conservation equations for each case.

19


2.3.1 Flow patterns

In gas-liquid two phase flow, the flow pattern is a description of how the flow is organized.

There are many flow patterns or regimes that occur in the case of two phase gas-liquid

flow. These patterns are dependent mainly on the void fraction, mass flux, fluid properties,

the flow orientation and the flow direction. Also different flow patterns appear for different

fluid properties such as froth flow for the highly viscous Newtonian flow, Cheng et al. 2008

[11]. The flow pattern has a huge effect on the hydrodynamics and heat transfer. Figure

2.7 shows the different two phase flow patterns in a vertical co-current flow. As shown, at

low void fractions, the flow takes the bubbly flow pattern. In the bubbly flow regime, the

liquid remains as continuous phase while small spherical bubbles appear as discrete phase.

As void fraction increases, the number of bubbles goes up as well as bubbles’ size. With

larger number of bubbles, the number of coalescence between bubbles increases which causes

forming larger bubbles. Larger bubbles or Taylor bubbles deviate from the spherical shape

to the bullet shape. For higher void fractions, the bubbles’ size start to increase forming the

slug flow. In the slug flow, continuous liquid phase could still be found while gas is forming

large Taylor bubbles. Small spherical bubbles are still present as well. Churn flow pattern

could appear as well at higher void fractions. More increase of the void fraction causes the

liquid to flow in annular section around the wall while the gas content flows in the middle

of the tube. This pattern is called annular flow pattern. By increasing the void fractions,

liquid droplets start to appear until droplet or mist flow pattern happens with very high

void fractions.

To identify the flow pattern of each case, flow maps are used. The flow map is a chart

that defines the flow pattern/regime based on the mass flux of each phase and the properties.

Flow maps are created by experiments and visualization of the flow. So, the use of each flow

map is limited by certain conditions. Figure 2.8 shows an example of flow pattern map

20


Figure 2.7: Schematic of flow patterns in vertical upward gas liquid co-current flow, [11]

reported by Hewitt and Roberts 1969 [30] for vertical up-flow. For more readings on the flow

maps, Cheng et al. 2008 [11] published a review on the flow maps and their applications.

2.3.1.1 Flow patterns in tube bundle

For tube bundles, three flow patterns were reported in the literature; Bubbly, intermittent

and dispersed flows. The absence of the slug and annular flow patterns was explained by

Dowlati et al. 1992 [16] and Noghrehkar et al. 1999 [52] as the tube bundle makes an efficient

source of breakup. Ulbrich and Mewes 1994 [80] conducted an experimental study using Air-

Water mixture to predict the flow pattern in tube bundle. The results were presented as

the flow pattern map shown in Fig. 2.9. Later, Noghrehkar et al. 1999 [52] used air-water

mixture as well to present the flow pattern map shown in Fig. 2.10. Although running

experiments using freon, Moran 2007 [50] agreed well with Ulbrich and Mewes 1994 [80] map

(created using air-water).

21


Figure 2.8: Hewitt and Roberts flow-pattern map for vertical upward gas-liquid co-currentflow. G is the mass flux, and ρ is the density. [30]

22


Figure 2.9: Flow-pattern map for vertical flow in tube bundle, [80]

Figure 2.10: Flow-pattern map for vertical in tube bundle, [52]

23


2.3.2 Models of two phase flow

2.3.2.1 Two fluid model

The two fluid model depends basically on solving the coupled conservation equations of each

phase. Thus, for two-phase flow, two different sets of continuity and momentum equations

will be solved. For one dimensional, inviscid flow in the z direction with no phase change,

each phase k will have its continuity equation in the form:

Dk

Dt(αkρk) + αkρk

∂uk∂z

= 0 (2.4)

where αk, ρk and uk are the volumetric fraction, density and velocity of phase k in the

mixture. Thus, for two phase k = 1, 2 and

n∑k=1

αk = 1 (2.5)

The momentum equation for the phase k will be given by

αkρk(∂uk∂t

+ uk∂uk∂z

) = −αk∂pk∂z

+ αkρkgz (2.6)

In addition to the above conservation equations, an empirical equation of state would

be used to describe the flow regime. i.e. annular, bubbly slug .. etc. Although this model

would give the most accurate results, it is very difficult to be applied in tube bundles, Ishii

and Hibiki 2011 [34].

2.3.2.2 Homogeneous model

This model assumes that the velocity of both phases are equal, Ishii and Hibiki 2011 [34].

So, the mixture density could be calculated from:

24


α =

[1 +

ρgρl

(1

x− 1

)]−1

(2.7)

where x is the mixture quality and equal to

x =mg

ml +mg

(2.8)

Because of its simplicity, the homogeneous model was used to report most of the two-

phase, cross flow, FEI data in the literature. However, all experiments showed that the

homogeneous model over-predicts the gas void fraction leading to inaccurate prediction of

the mixture density.

2.3.2.3 Slip model

The slip model was developed by Feenstra et al. 2000 [20] to predict the void fraction more

accurately in tube bundles. It is based on assuming a slip ratio S between the gas and

liquid velocity. By conducting experiments and by using the data available in the literature,

Feenstra et al 2000 [20] was able to correlate the slip as a function of Richardson number

and Capillary number.

S = 1 + 25.7(Ri ∗ Ca)0.5

(P

D

)−1

(2.9)

Then, the void fraction could be calculated from,

α =

[1 + S

ρgρl

(1

x− 1

)]−1

(2.10)

Feenstra et al. 2000 [20] showed superiority of the slip model over the drift flux and

Dowlati’s models. This superiority was confirmed experimentally by Moran 2007 [50].

25


2.3.2.4 Void fraction model of Dowlati

This model was developed by Dowlati et al. 1992 [16]. The authors used the superficial gas

velocity j∗g as the main parameter as follow:

j∗g =xGp√

gDρg(ρl − ρg)(2.11)

Then they calculated the void fraction from Eq. 2.12.

α = 1− (1 + C1j∗g + C2j

∗g

2)−0.5 (2.12)

where C1 and C2 are constants. Dowlati et al. 1992 [16] determined their values for

air-water as

C1 = 30 (2.13)

C2 = 50 (2.14)

2.3.2.5 Drift flux model

The drift flux model was proposed by Zuber and Findlay 1965 [85]. The model was developed

as an intermediate model between the two-fluid model and the homogeneous model. It is

much simple than the two-fluid model and still predict accurate results. It is based on using

a distribution parameter, Co to calculate the weighted average velocity of the gas from:

Ug = Co(jg + jl) + VB (2.15)

where VB was defined as the drift velocity and equal to the rise velocity of a bubble in a

stagnant fluid. jg and jl are superficial velocity of gas and liquid respectively.

26


Many studies were performed to correlate Co and VB in tube bundles. Most of this work

were performed to predict the void fraction in the fuel bundle of nuclear reactors. For more

information on the models and correlations used to predict the void fraction in tube bundle,

please refer to Ozaki et al. 2013 [54], Hibiki et al. 2017 [31] and Mao and Hibiki 2017 [48].

2.4 Two-phase bubbly flow

When modeling two phase bubbly flow, it is a crucial to account for bubble-bubble inter-

action as well as bubble-wall interaction. Bubble-Bubble interactions are mainly: bubble

bounce, coalescence and breakup. This section will focus on the models used to account for

coalescence and breakup.

2.4.1 Bubbles Coalescence

Accounting for bubble coalescence in a two phase bubbly flow model is very important.

Coalescence increases the bubbles’ size and it has a huge effect on the flow pattern. Larger

bubbles shape deforms from the spherical shape and forms Taylor and cap bubbles. Although

the current study assumes a spherical bubble shape regardless of the bubble size, it has to

account for bubble coalescence as this should have an effect on the fluid damping.

Many investigations aimed to explain the coalescence phenomenon such as Marrucci 1968

[49], Chesters 1975 [13], Chesters and Hofman 1982 [12] and Kolev 1993 [40]. Each of these

studies tried to model the coalescence process and whether the coalescence or the bounce

will dominate.

To understand bubble coalescence mechanism, assume having two bubbles approaching

each other. In liquids of low viscosity, both observation and analysis suggest that coalescence

and bouncing processes are in competition. The first step is squeezing the liquid film between

the two bubbles out. This will be accompanied by a bubble surfaces flattening. On one hand,

27


h

i

h

Figure 2.11: Sketch of film stretching and bubble surface flattening for two bubbles ap-proaching each other, [49]

if the liquid film reached a certain length Chesters 1975 [13], van der Waals pressures become

more dominant forming a hole in the liquid film allowing the surface tension forces to expand

the hole forming larger bubble. On the other hand, the deformation of the bubbles increases

the surface energy on the expense of the kinetic energy. The bubbles therefore decelerate

and bounce. Chesters and Hofman 1982 [12]. The two bubbles will coalescence if the liquid

film reached the critical thickness before the bubbles start to bounce. Figure 2.11 shows the

stretching of the liquid film and bubble surface flattening.

Many Experimental studies focused on the parameters dominating coalescence. Kirk-

patrick and Lockett 1974 [39] carried out experiments using high speed camera in which a

cloud of air bubbles has been prevented from rising by down-flowing water in a tube. A

complete absence of bubble coalescence was obtained at large approach velocity. The author

28


concluded that large approach velocity is important to avoid bubble coalescence.

Kolev 1993 [40] assumed that coalescence happens if the contact time between the two

approaching bubbles, exceeds the time required for the bubbles to push the squeezed liquid

film between them. He assumed that coalescence between two equal size bubbles occurs

when the kinetic energy exceeds the surface tension energy. Hence, the author defined a

critical Webber number above which coalescence would occur,

Wecr =D(ρg + αρl)U

2R

σ� 12 (2.16)

where D is the bubbles diameters, UR is the bubbles relative velocities and α is the added

mass ratio.

Senez and Ettienne 2011 [75] reported an experimental study by Duineveld 1998 [17],

where coalescence occurred under a critical value of Webber

Wecr =DρlU

2R

σ≺ 0.36 (2.17)

Senez and Ettienne 2011 [75] used an equivalent radius in Eq. 2.17 to determine if the

bubble will coalesce or not. If the critical Weber number is exceeded, Eq. 2.18 is used to

determine if the coalescence will still happen or not.

Wecr =2Rρl(Ul − Ub)2

σ≺ 6.6 (2.18)

2.4.2 Bubbles breakup

Bubble breakup is one phenomenon that occurs when two phase bubbly flow is found. The

breakup starts by bubble surface deformation resulting from stress balance applied on the

surface. This deformation could result into necking and the breakup. The resulting bubbles

29


after breakup could have the same diameter, in most cases, or they could have different sizes

with different order of magnitude [45].

To understand the mechanism of bubble breakup, it should be known that the surface

tension forces tries, in all cases, to keep the bubble in a spherical shape (lower surface energy).

Due to the pressure, turbulence, or velocity gradients there are some other forces that act

on the bubble surface. These forces could be external and/or internal forces. These forces

act to penetrate the bubble surface. The bubble shape changes to achieve balance between

these forces on one side and the surface tension on the other side. When the applied forces

are large enough, a neck starts to form which ends up by bubble breakup.

Capillary number Ca which represents a ratio between the shear stress and the surface

tension stress plays an important role in determining whether the bubble will breakup or

not. Capillary number could be calculated by

Ca =τvτs

=µfdγ

2σ(2.19)

where τv and τs are the viscous and surface tension stresses, µf is the continuous phase

viscosity, liquid in the case of bubbly flow and γ is the shear rate.

When the capillary number exceeds a critical value Cacr, the bubble will become unstable

and breakup. The region of

Cacr ≤ Ca ≤ kCacr, (2.20)

is dominated by necking, where the droplet breaks up into two equal-sized bubbles. When

the capillary number is suddenly increased to a value well above Cacr, the droplet is rapidly

elongated into a long cylindrical fluid thread, which subsequently breaks into a series of

fragments. In simple shear flow, k = 2.

Many models were introduced in the literature to determine the value of the critical

30


capillary number [22, 6]. It depends on viscosity ratio and the flow type. Figure 2.12 shows

an example of stability maps to determine the bubble stability.

Unstable

Stable

Viscosity ratio

0.001 0.01 0.1 1

0.1

1

10

Ca

pilla

ry n

um

be

r

Figure 2.12: Critical capillary number in simple shear flow [45]

The size of the new resulting bubbles is usually in the same order of magnitude, specially

if they are resulting from spherical bubbles. However, in some cases small bubbles are sheared

off from the original large cap bubble. Liao and Lucas 2009 [45] reviewed bubble breakup

and discussed the breakup mechanisms and models. Liao and Lucas 2009 [45] summarized

the breakup mechanisms as follows,

31


2.4.2.1 Breakup due to turbulent fluctuation

This mechanism is considered the number one cause of bubbles breakup due to the pressure

fluctuations on the bubble surface. Most of the studies in the literature focused on the

breakup due to turbulent fluctuations and turbulent eddy collisions. Liao and Lucas 2009

[45] simplified the mechanism as it is the bubble shape response to the fluctuation of the

surrounding fluid or due to collisions with eddies. When the amplitude of the oscillation is

close to that required to make the particle surface unstable, it starts to deform and stretch

in one direction leading to a neck that contracts further causing breakup.

2.4.2.2 Breakup due to the viscous shear forces

This mechanism happens due to the velocity gradient which causes shear force on the bubble

surface. The viscous shear forces applied on the bubble causes elongation which result in

necking and might lead into breakup.

2.4.2.3 Breakup due to shearing-off

Due to the high velocity gradient on the liquid-gas interface. The shearing-off process is

characterized by a number of small particles sheared-off from a large one, which is also

called erosive breakage. In highly viscous flows, the shearing-off is determined by the balance

between the viscous shear force and the surface tension at skirts of the cap/slug bubble (Liao

and Lucas 2009 [45]). Fu et al. 2002 [21] showed that in the case of air-water mixture, the

shearing off can happen due to the gas boundary layer, inside the bubble, penetrating the

liquid surface (Liao and Lucas 2009 [45]).

32


2.4.2.4 Breakup due to interfacial instability

Breakup due to interfacial instability mechanism can occur in the case of flow characteristics

(could happen in the case of bubbles rising in stagnant fluid. It occurs when a fluid is

accelerated into higher density fluid resulting in interfacial forces able to breakup the bubble.

By looking at the above breakup mechanisms, it could be found that none of them would

exist one dimensional inviscid laminar flow. This could allow bubbles to grow beyond the

limit of being treated as spherical bubble. However, according to Beguin (2010), reported

by Senez and Etienne 2011 [75] in a two-phase internal flow, a bubble subjected to the wake

of other bubbles can be stable up to a radius of 7mm. This is a very large size in a typical

bundle geometry where the distance between tubes is in order of millimeters. This explains

why the tube bundles have an additional breakup mechanism and agrees with Dowlati et al.

1992 [16] and Noghrehkar et al. 1999 [52],

2.4.3 CFD modeling of bubbly flow

Computational fluid dynamics (CFD) is a powerful tool for solving Navier-Stokes equations

numerically combined with turbulence model. It allows simulating wide range of cases while

obtaining accurate results without conducting large number of experiments. Using CFD

for simulating liquid-gas two phase flow has been started in the last decades due to the

improvement of computer technology. One of the main advantage of using the CFD tool is it

can give better understanding of physical mechanisms. However, using CFD to simulate two

phase liquid-gas is faced by many challenges. One of them is that there is a need for different

models to simulate different flow patterns. Another challenge is the large computational time

needed compared to single phase simulation.

Two basic models are being used in the most commercial packages; Eulerian-Eulerian

model and Eulerian-Lagrangian model. The first, Eulerian-Eulerian model considers two

33


continuous fluids. By solving Navier-Stokes equation for the two separate phases, a solution

could be obtained. The second model is Eulerian-Lagrangian model. In this case, one fluid

is treated as continuous phase while the other is treated as dispersed phase.

For the above two models, there is a need to track the interface between the different

phases. A large length scale model is used to track the interface in the case of slug, annular or

churn flow where the interface is usually much larger than the computational grid. In the case

of bubbly or droplet flow, the interface length is small or comparable to the computational

grid. In this case, a small length scale model is used. Yan and Che 2010 [83] developed

a coupled model where it is possible to combine both large-scale and small-scale interfaces

models by considering three phases; liquid, small bubbles and large bubbles, Fig. 2.13.

Figure 2.13: Map of the three phases used in the coupled model, [83]

34


2.5 FEI in two phase flow

The FEI problem is always associated with tube bundles, which is used widely in many heat

exchangers, boilers and condenser, specially in the power generation industry. However, most

of the research performed focused on single phase flow. Surveying the literature, it could be

seen that up to the permanent closure of San Onfore nuclear generating station (SONGS) in

2013, most of the studies on FEI in two phase flow aimed to predict the stability threshold.

Similar to single phase flow, the industry kept using the famous Connors Eq. (2.2 with

different values of K and n. After (SONGS), more fundamental research has been published

trying to analyze the problem specially in the inflow direction where instability occurred.

Experimental studies to predict the stability maps for two phase flow used different

combination of fluids. For example air-water mixture was the most common one such as

Pettigrew 1989 [61] and Tan et al. 2018 [77]. Some experiments used Freon such as Feenstra

2000 [19]. The use of steam-water mixture was very rare in the literature because of the

high cost.

When trying to analyze the applicability of using Connors equation, starting by the left

hand side, the pitch flow velocity U , is the first variable to be determined. In single phase

flow, knowing the mass flux and dimensions, obtaining an estimate value of the mean flow

velocity is a matter of calculations. However, in two phase flow, velocity determination is

much more complex. There are two phases, each has a different velocity. Several approaches

were utilized to estimate the flow velocity. Some of these models are discussed later in this

chapter. Second parameter in Connors equation is the natural frequency f . Again, in single

phase flow it is possible to estimate the natural frequency of the tube in a certain medium

by experimental or analytical technique. However, in two phase flow, natural frequency

changes with the void fraction and flow regime. In the right hand side of the equation, more

unknown parameters will be found such as the mass per unit length m, the system damping

35


δ and the flow density ρ in addition to the constants K and n. The mass per unit length

m is a summation of the tube mass mt and the fluid added mass ma. Obviously, the latter

is a function of the fluid density ρ which is a function of the void fraction α. The system

damping δ is also a function of the void fraction.

In addition to the difficulties discussed above, there are more essential parameters of

two phase flow that were not taken into account in Connors equation, even after ignoring

heat transfer and phase change. An example of these parameters is the surface tension σ

which affects the two phase flow pattern. Also the viscosity µ, which is function of the void

fraction, has a major effect on the flow pattern as well as the pressure drop. Even the flow

pattern was not accounted for in Connors equation. In addition, Connors equation does not

take some bundle essential parameters into consideration such as the pitch to diameter ratio

P/D and the bundle geometry.

It could be concluded from the above that Connors equation is inadequate, even with

different values of K and n, to predict the FEI threshold in two phase flow. However most of

the studies reported experimental data by plotting the flow map between the reduced flow

velocity and the mass damping parameter.

The reported data for FEI instability kept using the homogeneous model to calculate the

void fraction and hence, the mixture density. Figure 2.14 shows an example of two phase

stability maps reported by Moran 2007 [50]. The map was plotted using the homogeneous

model. The figure shows the deviation between the two phase data and the prediction of

Connors equation. Also it is hard to define the different flow patterns on the map.

The main difference between the stability maps of single-phase and two-phase flow is the

way of calculating the density and velocity for the mixture. The most common model used

in the literature was the homogeneous model which is based on the assumption of having

the same flow velocity for both liquid and gas.

Figure 2.15 shows a stability map plotted using the slip density and interfacial flow

36


Figure 2.14: Stability map for two phase flow based on the homogeneous model, [50]

37


Figure 2.15: Stability map for two phase flow based on the interfacial velocity and measuredRAD density, [50]

velocity. As shown in the figure, the accurate prediction of the void fraction by the slip

model enhanced the FEI threshold trend.

recently, [46] conducted experiments using air-water to check the effect of pitch to diam-

eter ration in normal square bundle. The authors confirmed that lower pitch to diameter

becomes unstable at lower velocities.

2.5.1 Two-phase flow damping

To analyze FEI, it is important to study the ability of the system to absorb the fluid energy

which is called “damping”. In tube arrays system, there are many source of damping. These

38


sources act together to form the total damping. [58, 59] presented a review summarizing the

damping mechanism in FEI. Pettigrew et al. 1989 According to Pettigrew et al. 1989a [59],

the most important sources of damping that contributes to the total system damping are:

1. Structure or material damping

2. Squeeze-film between the tube and the supports

3. Viscous damping due to the friction between the fluid and the tubes

4. Flow damping which is dependent on the fluid velocity

5. Friction damping at the supports

For the two phase flow, the damping of the two phases is added to the above sources

of damping. Carlucci 1980 [7] showed experimentally the magnitude of two phase damping

compared to other damping sources . Carlucci showed that the total damping in the case of

two-phase is larger than single phase, whether it is liquid or gas.

In order to compare the damping from different two-phase flow damping, Pettigrew and

Taylor 2004 [57] normalized the damping by the mass ratio and a function of the tube

diameter. Pettigrew and Taylor 2004 [60] compared the normalized two-phase flow damping

published earlier, Fig. 2.16. The figure show a constant trend of the two phase damping

increasing with the void fraction until it hits a maximum at 40 to 60%. Further increase in

void fraction decreases the damping again.

ζnormalized = ζtp

(m

ρlD2

)[(1− (D/De)

2)2

1 + (D/De)3

](2.21)

All the above studies, on the damping, were using the homogeneous model to calculate

the mixture properties. Moran 2007 [50] studied the two phase damping experimentally

and compared between the slip and homogeneous model. Moran 2007 [50] showed that the

39


Figure 2.16: Comparison of normalized two phase damping from different studies by [60]

40


slip model gives better estimation of the maximum damping which occurred at the end

of bubbly flow regime. Through dimensional analysis, Moran showed the importance of

Capillary number on the two phase damping.

2.5.2 Modeling FEI in two phase flow

Several attempts were directed at measuring fluid forces in two phase flow. For example,

Hirota et al. 2002 [32], Inada et al. 2002 [33], Alyaldin and Mureithi 2018 [2] measured the

fluid force coefficients in two-phase steam-water flow in order to utilize them in the unsteady

flow model. Their results showed weak effect of the surrounding tubes which made it difficult

to utilize the full model accounting for a fully flexible tube bundle. Recently, Shahriary et

al. 2007 [76] measured the force coefficients required for the quasi-steady model in the case

of air-water mixture.

Li and Hao 2016 [44] developed a time delay formulation by measuring the unsteady and

quasi-steady forces. The effect of Reynold’s number was taken into consideration. The study

showed an improvement of the predictions of FEI threshold in two phase flow.

Sawadogo and Mureithi 2014 [74] and Olala and Mureithi 2015 [53] used the quasi-steady

model to predict the fluidelastic instability for air-water mixture by measuring the unsteady

fluid forces in the lift direction and comparing the measured forces with the quasi-steady

forces, obtaining an expression for the time delay.

Sadek et al. 2018 [72] used CFD technique to predict the fluid forces acting on a cluster

of tubes subjected to two-phase cross flow. The predicted force coefficients were then used in

an unsteady flow model to predict the stability threshold. The predicted stability threshold

agrees well with the experimental counterparts, Fig. 2.17.

Nai-bin et al. [51] implemented the slip model in the flow cell model, introducing a semi-

analytical model for the two phase flow. By changing the slip ratio, the authors were able to

41


Figure 2.17: Stability threshold map obtained from the two-phase flow unsteady forcessimulations and direct flow/structure coupling vs. experimental data from Pettigrew (1989)for airwater flow, [72]

42


fit the data available in the literature for bubbly and intermittent flows in one single map.

The deviation between the prediction and experimental data exceeds 100% in some cases.

2.6 Summary and research needs

As shown in the literature, FEI is still a major problem in the design of heat exchanger.

There are many successful models for single phase flow able to predict the stability threshold.

Although some of these models need experimental inputs, there are many published studies

that include experimental result to extract these data.

For two phase flow, FEI problem still needs more understandings. Although there were

many attempts to study FEI as well as much experimental studies, very few attempts were

made to model this phenomenon analytically. Among these attempts there is no model that

is able to explain the physical difference between single and two phase flows. This research

tries to shed light on this difference to come up with a mechanistic model that takes various

phenomenas into account.

The objective of this work is to develop an analytical framework, able to predict FEI in

two phase flow across a tube bundle. The model focuses on the bubbly flow regime. Bubble-

to-bubble interction, bubble coalescense and compressibility will be taken into account.

43

Chapter 3

Modelling

As mentioned in section 2.6, this research aims to develop an analytical model to predict

FEI threshold in two phase flow. The model focuses on the bubbly flow regime and takes the

bubble dynamics and interactions into account. Bubble compressibility will be accounted

for in this model. The model is considered a novel work that takes such phenomenas into

account when simulating FEI.

There are many differences between single and two phase flows. In two phase flow, all

properties are function of the void fraction and the flow pattern structure. This means the

density, viscosity, enthalpy and internal energy can have different both; local and average

values in the flow field. Moreover, each fluid velocity can be different due to the slip between

phases. The prediction of the phase velocity in two phase flow can be quite challenging.

This is mainly because it is a function of void fraction, Feenstra et al. 2000 [20], and local

density. Also the orientation of the flow passage affects the flow pattern which in turn has

an effect on the flow behavior as well as the pressure drop. For these reasons, modelling of

two phase flow is such a difficult problem and requires many clouser relations based on the

application.

For the flow across tube bundle, the pressure pulsations in the flow lead to tube vibration.

44


Also, the fluid compressibility plays an important role when studying FEI. Although single

phase flow assumption could be used considering compressible flow at certain flow conditions,

bubbly two phase flow induced pressure waves that could not be ignored, Brennen 1995 [5].

In order to demonstrates how the bubble compressibility can affect the fluidelastic insta-

bility, a single bubble in a flow field that is undergoing rapid pressure change is considered.

In this case, as the pressure increases, the forces acting on the bubble surface is increased.

Consequently, the bubble responds by decreasing its size trying to reach an equilibrium be-

tween the internal and external forces. Theoretically this decrease in bubble diameter would

continue until the internal and external forces are in dynamic equilibrium. However, in a

dynamic system, an overshoot may occur causing the bubble size to go below the equilibrium

size until reaching an equilibrium. The decrease in the bubble size gives a space for the liquid

phase resulting in damping the original pressure signal.

The above explanation intercepted as the bubble always acts as a damper. However, this

is not true. Consider the pressure signal in the above example is an oscillating wave. The

corresponding bubble size will be oscillating with the pressure but with a time lag. Hence,

the bubble would act as a damper in some situations and might accelerate the instability in

other situations due to the time lag between the bubble response and pressure oscillations.

From the above, it is clear that bubble compressibility plays a vital role in determining

the FEI threshold of two phase bubbly flow across tube bundle. This chapter presents the

mail components of the analytical framework used to model two phase flow. Next chapter

(4) will discuss how these components were integrated together into the overall framework.

3.1 Overview of the analytical framework

The current frame work includes modelling the tube bundle dynamics, the continuous phase

flow, the flow of the dispersed phase, and the fluidelastic instability feedback. The two phase

45


flow is modeled utilizing an Eulerian-Lagrangian approach where separate two phases flow

are considered. The interaction between dispersed phase (bubble) and continuous phase

(liquid) is accounted for through two factors: (a)the bubble-liquid momentum exchange,

and (b) void fraction variation within the interstitial flow. The FEI modelling provides the

exchange of pressure from fluid to structure and displacement from the structure to the fluid.

3.2 Structural dynamics

Fluidelastic is a self excitation mechanism where the fluid forces develop as a result of the

tube motion. In other words, the tube has to start vibrating by any other mechanism such

as turbulent buffeting, then, the fluid respond to this vibration. It is expected that the tube

will go unstable at the least stable mode which is the first mode.

In a typical CANDU steam generator, the tubes take the shape of an inverted U-bend.

At the top of this bend, the flow becomes cross flow while the void fraction reaches its

maximum values leading to higher flow velocities. When loose supports exist, the tube acts

as a cantilever fixed from one end and fluidelastic instability forces act as excitation forces.

That is why most of the experimental studies were performed on cantilever tubes trying to

simplify the geometry and lowering the cost.

Based on the above, the current model assumes the following:

1. A single vibrating tube in a tightly fixed bundle. This assumption is based on the

observation by Lever and Weaver 1982 [43] that a single tube would go unstable at

almost the same velocity of a fully flexible tube bundle bundle will. The assumption is

commonly used to determine the FEI threshold experimentally, for example Feenstra

2000 [19] and Moran 2007 [50]

2. The tube will be assumed as a single degree of freedom resembling the first mode of a

46


cantilever tube.

Using these assumptions, the tube vibration was modeled as a forced vibration of single

degree of freedom system where the equation of motion could be simplified as:

Mx+ 4πζMfnx+ kx = FFEI (3.1)

The effective tube mass, M , can be calculated as the sum of the tube mass per unit

length m and the added mass ma. The latter was calculated using Rogers et al. 1984

[71] correlation, Eq. (3.2). Although this correlation was developed for single phase flow,

Pettigrew et al. 1989a [61] reported a good agreement with this correlation when testing

two phase flow, using the homogeneous density, for void fractions up to 80%. Pettigrew et

al. 1989a [61] attributed the discrepancies for higher void fraction to the intermittent flow

regime. In this study, this equation is used as the flow is assumed to be bubbly flow as the

void fraction, calculated using the homogeneous model, does not exceed 40%.

ma =

(ρπd2

4

)(De/d)2 + 1

(De/d)2 − 1(3.2)

In Eq. (3.2), De/d was calculated from Rogers et al. 1984 [71] correlation for square

bundle, Eq. (3.3). The density ρ is calculated using the homogeneous model

De/d = (1.07 + 0.56p/d)p/d (3.3)

In Eq. (3.1), the structure damping ζ could be measured for the vibrating tube. If there

is no available data (design stage for example), it is reasonable ζ between 0.5 to 1%. Other

types of damping discussed in chapter 2 such as the the flow damping, viscous damping and

two phase damping, is considered by the model by considering the gas compressibility.

fn is the tube natural frequency measured in the two phase mixture and k is the stiffness.

47


To calculate k, the tube is assumed as a cantilever fixed at one end and free to vibrate at

the other end.

The Fluidelastic instability force, FFEI , which represents the excitation force of the fluid

on the tube was calculated by integrating the pressure around the tube between the attach-

ment and the separation points. A two phase flow model was developed to calculate the

pressure around the tube and will be discussed in the next section.

3.3 Continuous phase

The liquid was considered as a continuous phase flows across the bundle. Lever and Weaver

1982 [43] observed visually that the flow across the bundle is divided into streamtubes as

shown in Fig. (3.1). Based on this observations, Lever and Weaver 1986a [41] modeled the

flow around a vibrating tube as streamtubes. The current model assumes the following for

the continuous phase:

1. The flow across the tube bundle surrounding the vibrating tube is to be divided into

two streamtubes.

2. The flow in each streamtube is assumed to be one dimensional, inviscid flow.

3. Liquid is always continuous phase and gas can not block the streamtube.

4. The streamtubes boundaries are flexible and vibrates according to the tube under

investigation.

5. No fluid flow across the streamtube boundaries. This applies to both the continuous

and dispersed phases.

Figure (3.2) shows the streamtubes surrounding the tube under investigation. The cross

sectional area of the streamtube area is a function of the position and time A(s, t). This is

48


Figure 3.1: Visual observation of streamtubes flowing through normal square tube bundle,[43]

consistent with the assumption proposed by Lever and Weaver 1986a [41] of presenting the

area in the form of steady state term A and transient term a.

Similar to the flow area, the continuous phase velocity and the pressure are assumed to

have steady state terms U and P as well as transient components u and p.

A = A(S) + a(s, t) (3.4)

U = U(S) + u(s, t) (3.5)

P = P (S) + p(s, t) (3.6)

49


Figure 3.2: Streamtubes surrounding the vibrating tube

In each streamtube, the flow direction is the curvilinear coordinate (s) where the origin

(s = 0) is at the tube center. For the normal square bundle, this curvilinear coordinate is a

straight vertical line except for the distance between the attachment and separation where

if follows the tube curvature. The fluid enters the channel at (s = −so). The entrance area,

Ao is assumed to be constant. At the streamtube entrance, the fluid velocity and pressure

are considered constants and equal to the average two phase velocity Uo and Po, respectively.

Where Po and Uo are independent of the tube displacement. They are function of the bundle

geometry, pitch to diameter ratio P/D and the mass flux across the bundle as well as the

void fraction. The tube contact with the streamtube is between the attachment (s = −sa)

to the separation (s = ss).

The amplitude of streamtube perturbation between the attachment and separation is

similar to the tube displacement. The amplitude of the area perturbations was assumed to

be zero at the entrance of the streamtube (s = −so). A Linear relation was assumed in order

50


Table 3.1: Streamtube parameters for normal square bundle

Parameter Formula

Pitch to Diameter Ratio PD

Ao

DPD− 1

soD

PD

loD

4 soD

βo 20o

xoD

sinβo2

to determine the amplitude of the area perturbation at any location between the entrance to

the attachment. Also, due to fluid elasticity, a time lag between the tube displacement and

the area perturbation is assumed. Another linear relation is assumed to determine this time

lag at any location in the streamtube between the entrance (s = −so) and the attachment

position. Table 3.1 shows the streamtube parameters for the square bundle used in the

current model. In the table βo is the angle between the attachment to the separation. The

conservation equations were solved for the continuous phase to determine the velocity and

pressure. More details are presented in section 4.1.

3.4 Dispersed phase

It should be noted that the current model focuses only on the bubbly flow regime. The

dispersed phase (bubbles) are treated using the Lagrangian approach. By calculating the

forces acting on each bubble, the bubble velocity and acceleration can be determined. Bub-

51


ble bouncing (impact between bubble), coalescence and breakup are taken into account in

the current model. Some assumptions were made to simplify the model which could be

summarized in:

1. The flow in each channel is bubbly flow with one continuous liquid phase and one

dispersed gas phase. This assumption is valid for low and medium void fractions (less

than 40% calculated using the homogeneous model) as observed by Moran 2007 [50].

Moran reported the transition from bubbly to intermittent flow between 40 to 70%.

2. All bubbles take the shape of a sphere at all the time.

3. All the bubbles are small enough to assume uniform properties (pressure and tempera-

ture), velocities inside the bubble. The gas particles inside each bubble have the same

properties and they do not move relative to each other.

4. The pressure gradient, on the bubble outer surface, is too small to affect the bubble

spherical shape.

5. The flow in each streamtube is an isothermal flow where the liquid and gas are in

thermal equilibrium.

6. The time average void fraction does not change in the streamtube and heat transfer

between the tubes and the fluid is considered too small to be accounted in the model.

Although the main purpose of heat exchanger is to transfer heat between two fluids,

the length of the considered streamtube is very short compared to the fluid path in the

heat exchanger. So, the heat added to the fluid along the streamtube could be ignored.

By dividing the time domain into infinitesimal time steps, the forces acting on each bubble

were calculated at each time step. Using the resultant forces, the new bubble velocity was

calculated at each time step by applying Newton’s second law. This velocity, in turn, could

be used to predict the bubble location in the following time step.

52


3.4.1 Bubbles generation algorithm

Bubbles is to be generated randomly taking into account the bubble size distribution to

match the beta distribution function. A schematic sketch showing the beta distribution

function is presented in Fig. (3.3) and given by Eq. 3.7.

Figure 3.3: A schematic sketch showing the beta distribution function

This is the size distribution found by Wallis 1968 [81] and adopted by Beguin, reported

by Senez and Etienne 2011 [75],

P (R) = 10R

Rmax

e−5( RRmax

)2 (3.7)

53


where Rmax represents the bubble radius which have the maximum probability and it

could be calculated from the empirical correlation given by:

Rmax = 0.57σ

ρl

(2fU3

m

Ao

)−25

(3.8)

where σ is the surface tension, ρl is the liquid density, Um is the mean flow velocity,

D is the pipe diameter which was substituted by the width of the the stream tube in this

study (characteristic length), and f is the friction coefficient. It should be mentioned that

the above equation was originally derived for flow in a vertical pipe before replacing the

pipe diameter with the characteristic length Ao for the purpose of the current research. The

friction coefficient was reported by Senez and Etienne 2011 [75] as:

f = 0.046

(UmD

νl

)−0.2

(3.9)

where νl is the liquid kinematic viscosity.

Once Rmax, is calculated, bubble distribution can be generated. After generating each

bubble with radius R, a check for the total volume fraction is performed. If the total volume

fraction does not exceed the input volume fraction, a uniform random function is used to

generate a location for the bubble to place it in the streamtube. An initial vertical velocity

equal to the average flow velocity Um was assumed while a zero initial velocity was assumed

in the horizontal direction.

The pressure inside the bubble is assumed uniform for each bubble and is calculated by

applying force balance on the bubble surface such that:

Pinside − Poutside =2σ

R(3.10)

A normal and uniform distribution functions of bubbles size were also tested as an initial

54


distributions instead of the beta distribution. The results will be presented in chapter 5.

3.4.2 Forces acting on the bubble

There are three main forces acting on each bubble; lift , drag and buoyancy force, as shown

in Fig. 3.4. All these three forces were analyzed into two components; X (perpendicular to

the channel centerline) and Y (parallel to the channel centerline). In addition to these three

forces, the pressure difference between the top and bottom of the bubble adds a force equal

to:

Fpressure = Aprojected ∗ 4p (3.11)

Figure 3.4: Forces acting on each bubble

55


3.4.2.1 Lift force

Many numerical studies on the vortex formation behind a bubble in two phase flow were

presented in the literature, Taneda 1956 —reported by Brennen 1995 [5]—. In the current

research, the bubbles were treated as spheres while the liquid approaches the sphere with

the relative velocity Urel. Each bubble acts as a bluff vortices shed from the body. The

frequency of vortex shedding around a sphere has not been studied as extensively as in the

case of a circular cylinder and seems to vary more with Reynolds number, Brennen 1995 [5].

Brennen 1995 [5] reported experimental results by Taneda 1956, who found that at Re = 30,

a vortex ring develops close to the rear stagnation point, Fig. (3.5). The vortices’s were

found to expand with further increase in the Reynolds number. Defining locations on the

surface by the angle from the front stagnation point, the separation point moves forward

from about 130◦ at Re = 100 to about 115◦ at Re=300. At Re = 130, the wake diameter

reaches comparable value to that of the sphere diameter. At this point the flow becomes

unstable and the ring vortex that makes up the wake begins to oscillate [5]. It should be

mentioned here that the order of magnitude of Reynolds number in the current study is

1000.

For Re = 103 to 3 ∗ 105, laminar boundary layer separation occurs at θ = 84◦ and

a large wake is formed behind the sphere. Laminar flow occurs downstream close to the

sphere while further downstream transition and turbulence occurring in the shear layers

spreads to generate a turbulent “far-wake.” As the Reynolds number increases the shear

layer transition moves forward until, quite abruptly, the turbulent shear layer reattaches to

the body, resulting in a major change in the final position of separation (θ = 120◦) and in

the form of the turbulent wake.

The bubble Reynolds number for ith bubble is calculated by

56


Figure 3.5: Streamlines of steady flow (from left to right) past a sphere at various Reynoldsnumbers, Taneda 1956 (reported by Brennen 1995 [5])

57


Rei =ρlUreliD

µl(3.12)

The vortex induced vibration force acting on the bubble was assumed in the form of

FL = CL(t)ρlU

2r

2

πD2b

4(3.13)

FL is a periodic force where the direction changes with the alternating vortices. The

coefficient of lift force which can be calculated from

CL(t) = CLo sin (2πfV IV t+ φrandom) (3.14)

CLo is the amplitude of the lift force coefficient. The frequency of the alternating vortices’s

fV IV can be determined from Strouhal number St such that

fV IV =St UrD

(3.15)

3.4.2.1.1 Strouhal number

For a spherical bubble, Reynolds number is in the order of 103. Sakamoto and Haniu 1990

[73] conducted an experimental study to find Strouhal number for a sphere. Sakamoto and

Haniu 1990 [73] covered a wide range of Strouhal number showing a good agreement with

the literature, as shown in figure (3.6). The authors showed that vorticity started to shed

behind the sphere at Re = 300. The values of Strouhal number up to Re = 103 were almost

0.2. For Reynold’s between 103 to 104, Sakamoto and Haniu 1990 [73] reported two values

of Strouhal number at the same Reynold’s number. The lower value was found to be around

St1 = 0.2. The values of higher Strouhal number was fitted as in Eq. (3.16) for the purpose

of the current research. For Re > 104, single value of Strouhal was reported once again and

58


Figure 3.6: Values of Strouhal at different Reynolds number [73]

it was around 0.2.

St2 = −10−8Re2 + 0.0003Re+ 0.0138 (3.16)

3.4.2.1.2 Coefficient of lift force CLo

CLo, was obtained from an experimental study to find the coefficient of lift and drag forces

on a bubble conducted by Rastello et al. 2011 [67]. The authors presented the coefficient

as a function of the bubble Reynolds number (calculated using the same parameters as Eq.

(3.12)) as shown in Eq. (3.17). The equation is valid for low Reynolds number Re < 150.

Magnaudet and Legendre 1998 [47] showed that CL approaches 0.5 for higher Re number.

CLo = 0.5 + 4

(1− 6

5Re1/6

)e−Re

1/6

(3.17)

59


In the current model, two random phase lag was generated for each bubble at the bubble

generation stage, then it remains constant during the simulation; one for each value of

Strouhal number. CLo and the second value of Strouhal number (if applicable) St2 are

calculated from Eq. (3.17) and (3.16). St1 is assumed to be 0.2 for all bubbles. By using Eq.

(3.15), the value of the vorticity frequency/frequencies could be determined. Then the lift

coefficient and the lift force are determined. New random phase lag values were generated

for the new bubble/bubbles after coalescence and/or breakup.

3.4.2.2 Drag force

Drag force FD on any body is a function of the velocity components Ux and Uy as well as

the area projected to the flow, Eq. (3.18).

FD = CDρU2

2Aprojected (3.18)

FDx = −CDρUx|Ux|

2πR2 (3.19)

FDy = −CDρUy|Uy|

2πR2 (3.20)

The direction of the drag force is expected to be against the bubble velocity direction,

see Figure (3.4). As the bubble might have a velocity in both x and y directions, the current

model calculates the drag in the x and y directions separately, Eq. (3.19) and(3.20).

The problem in the above equations is calculating the drag coefficient CD. Several empir-

ical formulas has been found in literature to relate the drag coefficient to Reynold’s number.

For example, Ishii and Zuber 1979 [36] related CD as a function of vapor void fraction and

Reynold’s number as:

60


CD =24(1 + 0.1(Re(1− α))0.75)

Re(1− α)(3.21)

For simplification, Senez and Etienne 2011 [75] modified the equation in the numerical

model to be

CD =24

Re(1− α)

Also, Ishii [35] modified Harmanthy, 1960 equation by dividing by√

(1− ε) to be

CD = 1.14R

√g(ρl − ρg)σ(1− α)

(3.22)

Equation (3.22) was used by Senez and Etienne 2011 [75] and others for low Reynold’s

number. In the present research, the coefficient of drag is taken as the larger between the

values calculated by both Eqs. (3.22 and 3.21)

3.4.2.3 Buoyancy force

The buoyancy force FB is is an upward force exerted by the liquid against the bubble weight.

It is equal to the weight of the displaced liquid. By subtracting the bubble weight from the

buoyancy force, the buoyancy can be calculated from:

FB = g4

3πR3(ρl − ρg) (3.23)

The direction of this resultant force will be in the positive y direction.

3.4.3 Bubble kinematics

It should be noted that the bubble location and velocity changes with the time. Therefore, in

order to calculate the change in the bubble location each time frame, the model assumes that

the velocity is constant during each time step. Then by using the two velocity components

61


in the vertical and horizontal direction, it would be possible to find the new location using

the following equations:

Xnew = Xold + Vxdt (3.24)

Ynew = Yold + Vydt (3.25)

Since the bubble velocity changes according to the forces acting on it, this change in the

velocity can be calculated as:

dV =F

mdt (3.26)

3.4.4 Bubble-to-bubble impact

Bubble impact is a difficult problem to model. At low void fraction, when two bubbles

approach each other, they bounce. This could happen before touching each other. However,

some studies modeled the bubble bouncing as an elastic impact such as Delnoij et al. 1997

[15], Senez and Etienne 2011 [75]. This assumption found to be logical when the energy

dissipated in bubble bouncing approaches zero.

For elastic impact between two bubbles, it is assumed that the axis connecting the centers

is aa while the axis bb is perpendicular, Fig. (3.7). The velocity components in the directions

aa and bb could be calculated for each bubble using the horizontal and vertical velocities. For

elastic impact, the velocity component in the perpendicular direction will not be affected. To

calculate the resulting velocities component in the aa-direction, conservation of momentum

and energy equation should be applied. The momentum conservation equation can be written

as:

62


Figure 3.7: Elastic Impact between bubbles

m1Uaa1 +m2Uaa2 = m1U′

aa1+m2U

′

aa2(3.27)

where Uaa1 , Uaa2 are the old velocities of the two bubbles in the aa-direction. U′aa1

and

U′aa2

are the new velocities in the same aa-direction. m1 and m2 are the mass of the first

and second bubbles, respectively.

The conservation of kinetic energy can be expressed by:

m1U2aa1

+m2U2aa2

= m1U′2aa1

+m2U′2aa2

(3.28)

The non-trivial solution for (3.27) and (3.28) obtained by Senez and Etienne 2011 [75],

will be

U′

aa1= −m2 −m1

m1 +m2

Uaa1 +2m2

m1 +m2

Uaa2 (3.29)

63


U′

aa2=

2m1

m1 +m2

Uaa1 +m2 −m1

m1 +m2

Uaa2 (3.30)

3.4.5 Bubble bouncing from the wall

When a bubble impacts the wall, the breakup criteria is checked. If the breakup conditions

does not dominate, elastic impact between the bubble and the wall is assumed. For elastic

impact, the velocity parallel to the wall remains constant while the velocity normal to the

wall is diverted to the opposite direction according to the following:

U′

x1 = −Ux1 (3.31)

3.4.6 Bubbles coalescence

Accounting for bubble coalescence in a two phase bubbly flow model is very important. Since

the coalescence mechanism increases the bubbles’ size and greatly affects the flow pattern.

Larger bubble shape varies from the spherical shape to Taylor and cap bubbles. Although

the current study assumes a spherical bubble shape regardless of the bubble size, it has to

account for bubble coalescence to capture their effect effect on the fluid damping.

Many researches aimed to explain the coalescence Marrucci 1968 [49], Chesters 1975 [13],

Chesters and Hofman 1982 [12] and Kolev 1993 [40]. Each of these studies tried to model

the coalescence process and whether the coalescence or the bounce will dominate.

Bubble coalescence mechanism is affected by two competing processes. The first process

involves squeezing the liquid between the two bubbles. This is accompanied by a bubble

surfaces flattening. If the liquid film reached a minimum thickness, according to Chesters

1975 [13], Van der Waals pressures become dominant forming a hole in the liquid film allowing

the surface tension forces to expand the hole and a larger bubble is formed. In the second

64


process, the deformation of the bubbles increases the surface energy on the expense of the

kinetic energy. The bubbles therefore decelerate and bounce, Chesters and Hofman 1982

[12]. The two bubbles will coalescence if the first process dominates and they will bounce, if

the second process becomes more dominant [12], see Fig. (2.11).

Many experimental studies focused on the parameters dominating coalescence. Kirk-

patrick and Lockett 1974 [39] carried out experiments using high speed camera in which

a cloud of air bubbles has been prevented from rising by downflowing water in a tube. A

complete absence of bubble coalescence was obtained at large approach velocity. The author

concluded that large approach velocities is important to avoid bubble coalescence.

Kolev 1993 [40] assumed that coalescence happens if the contact time bubbles exceeds

the time required for the bubbles to squeeze the liquid film between them. He assumed

that coalescence between two equal size bubbles occurs when the kinetic energy exceeds the

surface tension energy. He defined a critical Weber number above which coalescence would

occur as:

Wecr =d(ρg + αaρl)U

2r

σ� 12 (3.32)

where d is the bubbles diameters, Ur is the bubbles relative velocities and αa is the added

mass ratio.

Senez and Etienne 2011 [75] reported an experimental study by Duineveld, where coa-

lescence occurred under a critical value of Webber calculated as:

Wecr =DρlU

2r

σ≺ 0.36 (3.33)

Senez and Etienne 2011 [75] used an equivalent radius in Eq. 3.33 to determine if the

bubble will coalesce or not. If the critical Weber number is not exceeded, Eq. 3.34 is used

to determine if the coalescence will still happen or not.

65



σ≺ 6.6 (3.34)

3.4.7 Bubble breakup

Bubble breakup is one phenomenon that occur when severe volume deformation takes place.

While the surface tension forces tries to keep the bubble in a spherical shape, the pressure,

turbulence, or velocity gradients act on the bubble surface. These forces tend to penetrate

the bubble surface resulting in changes in the bubble shape to keep balance between the

internal and external forces. When the applied forces are large enough, a neck starts to form

which ends up by bubble breakup.

The breakup starts by bubble deformation resulting from forces balance on the surface.

This deformation could result into necking and the breakup. The resulting bubbles after

breakup could have the same diameter, in most cases, or they could have different sizes with

different order of magnitude, Liao and Lucas 2009 [45].

Capillary number Ca which represents a ratio between the shear stress and the surface

tension stress plays an important role in determining whether the bubble will breakup or

not. Capillary number can be calculated by

Ca =τvτs

=µldγ

2σ(3.35)

where τv and τs are the viscous and surface tension stresses, µl is the continuous phase

viscosity, liquid in the case of bubbly flow and γ is the shear stress rate.

When the capillary number is increased to a critical value Cacr, the bubble will become

unstable and breakup. The region of Cacr ≤ Ca ≤ kCacr , is dominated by necking, where

the droplet breaks up into two equal-sized bubbles. When the capillary number is suddenly

increased to a value well above Cacr, the bubble is rapidly deformed into a long cylindrical

66


shape, which subsequently breaks into a series of fragments. In simple shear flow, k = 2.

Many models were introduced in the literature to determine the value of the critical

capillary number Grace 1982 [22] and Bruijn 1993 [6]. It depends on viscosity ratio and the

flow type.

The size of the new resulting bubbles is usually in the same order of magnitude, specially

if they are resulting from spherical bubble. However, in some cases small bubbles are sheared

of from the original large cap bubble. Liao and Lucas 2009 [45] summarized the breakup

mechanisms and models to be; breakup due to turbulent fluctuation, breakup due to the

viscous shear forces, breakup due to shearing-off and breakup due to interfacial instability.

Richard et al. 2012 [68] define the critical Weber number for a bubble bouncing from the

wall as

Wec = (ρg + αmρl)2RU2

impact

σ= 9, (3.36)

Where αm is the added mass ratio, Uimpact is the impact velocity and σ is the surface

tension. Senez and Etienne 2011 [75] modified this value by adding Bond number Bo to

Webber number. Bond number represents a ratio between the body force, buoyancy, and

the surface tension.

Bo =4g(ρl − ρg)R2

σ(3.37)

Senez and Etienne 2011 [75] considered the bubble to be stable when Eq. (3.38) is

satisfied

Bo+We < 9 (3.38)

67


3.4.8 Bubble compressibility

For a bubbly flow regime suffering pressure perturbations, bubble could act as pressure

dampers. When the pressure increases, the bubble size decreases allowing more room for the

liquid and vice versa. Brennen 1995 [5] showed an extreme increase in the bubble size at low

pressures. Rayleigh-Plesset equation describes the dynamic response of a gas bubble in liquid

subjected to pressure changes, Brennen 1995 [5]. For constant liquid density ρl, viscosity µl

and uniform properties inside the bubble TB(t) and PB(t), Rayleigh-Plesset equation can be

introduced as a non-linear differential equation in the form:

F (t) = R2dR

dt(3.39)

When evaporation or condensation is occurring at the interface, good approximation of

the bubble radius can be obtained by solving Navier Stokes equation for the bubble and

using the normal stress as a function of the bubble pressure PB.

PB(t)− P∞(t)

ρL= R

dR2

dt2+

3

2

(dR

dt

)2

+4νLR

dR

dt+

2σ

ρLR(3.40)

Assuming isothermal compression/expansion for the bubble, the pressure inside the bub-

ble could be calculated from the vapor pressure Pv(TB), the gas pressure and radius at the

beginning of the simulation PGo and Ro, Eq. (3.41). Also, PGo can be obtained from Eq.

(3.42).

PB(t) = Pv(TB) + PGo

(Ro

R

)3

(3.41)

PGo = P∞(0) + Pv(T∞) +2σ

Ro

(3.42)

68


Then Rayleigh-Plesset equation becomes,

Pv(T∞)− P∞(t)

ρL+Pv(TB)− Pv(T∞)

ρL+

PGoρL

(TBT∞

)(Ro

R

)3k

= RdR2

dt2+

3

2

(dR

dt

)2

+4νLR

dR

dt+

2σ

ρLR(3.43)

The first term in Eq. (3.43) is the driving term determined by the liquid conditions far

from the bubble. The second term in Eq. (3.43) is the thermal term which can affect the

bubble dynamics pretty much. The third term accounts for the quantity of contaminant gas

whose partial pressure PGo at a reference radius Ro and temperature T∞. The symbol k in

this term stands for the polytropic index. For isothermal bubble temperature, k = 1 while

k = γ for isentropic relation.

Although the effect of the thermal term in Eq. 3.43 on the bubble dynamic, this term

will be ignored for simplicity and the bubble will be assumed “inertially controlled”. Now,

by assuming that the bubble is initially stable at time t = 0 and liquid pressure P∞(0), the

gas partial pressure PGo will be:

PGo = P∞(0) + Pv(T∞) +2σ

Ro

(3.44)

Substituting from 3.44 into 3.43,

Pv(T∞)− P∞(t)

ρL+PGoρL

(Ro

R

)3k

= RR +3

2(R)2 +

4νLR

R+

2σ

ρLR(3.45)

By applying dimensional analysis, Rayleigh-Plesset equation was decreased to Eq. (3.43).

Appendix A shows the detailed steps of the dimensional analysis.

69


Pv(T∞)− P∞(t)

ρL+PGoρL

(Ro

R

)3k

= RR +3

2(R)2 +

2σ

ρLR(3.46)

Isothermal expansion of the gas is assumed to overcome the difficulty by considering the

change in gas temperature while ignoring the heat transfer and the bubbles generation across

the tube bundle.

By taking this assumption into account, and by substituting of PGo from Eq. (3.42)

1

ρL

[P∞(0)+

2σ

Ro

](Ro

R

)3

−P∞(t)

ρL+

1

ρL

[Pv(T∞)

(1+

(Ro

R

)3)]= RR+

3

2(R)2 +

2σ

ρLR(3.47)

As Eq. 3.47 is still highly nonlinear, a numerical simulation attempt was decided for the

original Rayleigh-Plesset.

3.5 Fluid structure interaction

The coupling between the fluid and the structure can be divided into two main points; the

streamtube movement following the vibrating tube and the fluid forces of the streamtube on

the structure (tube).

The streamtube is assumed to follow the tube vibration with a time lag resulting from the

fluid inertia. Similar to Lever and Weaver 1986a [41], the time lag was assumed to be zero

at the tube itself (between attachment and separation). At the streamtube inlet, the time

is assumed to be maximum and function of the vibration frequency ω, streamtube length lo

and the upstream velocity Uo and related by: equal to

τ(s)−so =ωloUo

(3.48)

70


Between the stream tube entrance to the attachment, a linear relation between τ and s

was assumed to find τ(s).

Also, the fluid forces (FEI) on the vibrating tube were calculated by integrating the

pressure between the attachment to the separation. For each streamtube, this force is:

FFEI = ±∫ so

sa

pdA (3.49)

where the positive/negative sign are used for the two channels, p is the pressure, dA is

the normal area

71

Chapter 4

Model Implementation

The two phase model was implemented in the time domain simulation model developed by

Hassan and Hayder 2008 [23]. The flow is discretized axially into a number of flow cells.

Each flow cell contains two flow channels. Each channel has a length of So and a cross

section area of A. Positions along the channels are represented by the curvilinear coordinate

s, which originates at the moving center tube. For the normal square bundle, the coordinate

s coincides with the vertical coordinate. The flexible tube contacts the channel between the

attachment −sa to the separation ss. Figure 4.1 shows a screen shot of the simulation, where

the vibrating tube appears at the top of the figure. The inner boundaries of the channels

vibrates with the tube. The flow channel boundary along the contact length (−sa < s < ss)

was assumed to instantaneously follow the tube motion. The width at the inlet boundary

at the channel origin s = −so was assumed to be non vibrating (No area perturbation).

Between the inlet boundary and the attachment (−so < s < −sa), the channel width was

assumed to change with time around a mean value Ao. The amplitude of the channel width

oscillation is a and is function of the location. The channel width oscillation a(s, t) follows

the tube oscillation but with a time lag τ(s). The time lag is proportional to the location s.

The maximum time lag is at the channel inlet and could be calculated from

72


Figure 4.1: Screen shot from the simulations showing the vibrating tube (blue) surroundedby the flow channels. The channel oscillating boundary is shown in red

τmax =εsoUo

(4.1)

where ε is the vibration frequency in rad/s. A linear relation was assumed to calculate

the time lag τs at any position.

4.1 Conservation equations

The governing fluid flow equations were solved to determine the flow velocity and the pressure

each time step. As the equivalent density is changing with the bubbles motion, compress-

73


ibility needs to be taken into account The unsteady continuity equation is given by:

∂

∂t(ρA) +

∂

∂sρAU = 0 (4.2)

Substituting with the expression for A and U in the above equation,

∂

∂t(A+ a)ρ+

∂

∂sρ(A+ a)(U + u) = 0 (4.3)

∂

∂sρ(AU + Au+ aU + au) = − ∂

∂tρA (4.4)

integrating both sides w.r.t. s

ρ((A+ a)U + (A+ a)u) = −∫ s

−so

∂

∂tρAds (4.5)

Now the perturbation velocity can be expressed as

u =−1

A+ a[(A+ a)U +

1

ρ

∫ s

−so

∂

∂tρAds (4.6)

The integrals in the above equation are both space and time dependent. The integration

in the above equation was carried out by 5-point Gaussian numerical integrals.

The momentum equation is given by:

∂

∂t(ρU) + U

∂ρU2

∂s= −∂P

∂s+ ρg (4.7)

Again, by using the same analysis as before, the equation becomes

74


∂

∂t(ρ(U + u)) +

∂

∂sρ(U + u)2 = − ∂

∂s(P + p) + ρg (4.8)

∂

∂t(ρu) +

∂

∂sρ(U + u)2 − ρg = − ∂

∂sp (4.9)

and now the perturbation pressure can be expressed as

p =

∫ s

−soρgds− ρ(U + u)2 −

∫ s

−so

∂

∂t(ρu)ds (4.10)

And again, the integration in the above equation was carried out by 5-point Gaussian

numerical integrals to obtain the pressure perturbation p.

4.2 Simulations

A FORTRAN 95 was used as the programming language while Matlab was used for post

processing. The code used parallel processing to decrease the simulation time. The available

processors was divided between the two flow channels according to the number of bubbles.

At the beginning, the code calculates the channel length, the tube natural frequency, and the

actual void fraction in each channel. The bubble generation process starts. After generating

each bubble, a check is done to compare the current void fraction with the required one.

Figure 4.2 shows a diagrammatic sketch of the code organization. As seen from the figure,

first the code read the input file and prepare the channels and parameters. Bubbles were

generated randomly in the flow channels. The size distribution of the bubble follows the

recommendation of Wallis 1968 [81]. Initially each bubble was assumed to move vertically

at a velocity equal to the mean flow and with zero lateral velocity. Non-zero lateral velocity

develops as other effects such as collision, lift, and drag forces develop.

Each time step starts with the bubble loop. More details of the bubble loop will be

75


explained in Section 4.3. Then the conservation equations are solved and the pressure is

integrated to find the tube displacement.

4.3 Bubble loop

Bubble loop starts by calculating the forces and applying Rayleigh-Plesset on each bubble.

Based on the forces, the bubble velocities are updated. Then each bubble is checked for

reaching critical radius for breakup. Bubble coalescence was checked by one processor in

each channel to avoid duplicating the same coalescence. To speed up this check, the bubbles

were sorted by their vertical locations in the channel.

In each channel there are large number of moving bubbles. Detecting the motion of each

bubble is essential for the success of simulation. In general, there are N moving bubbles and

M walls (channel walls and tubes). Each bubble can collide with other moving bubbles as

well as the walls. It is important to keep track of (N2) +NM pairs of objects at every time

step. This can be very time consuming specially for high void fractions as the number of

bubbles gets large. Therefore, the number of possible colliding pairs must be reduced. The

can be carried out by determining the candidate collisions pairs (CCP) which are pairs in

close proximity to each other. A hierarchical bounding box was used. In this scheme, each

bubble is surrounded by a bounding box. CCP are determined if the bounding boxes of the

individual bubbles are overlapping. Once CPP are determined, collision is tested.

At the end of the bubble loop, bubbles were assigned their new locations in the channel.

displacement.

4.3.1 Solution of Rayleigh-Plesset equation

Alehossein and Qin 2007 [1] attempted to solve Rayleigh-Plesset equation using various

numerical techniques. The authors showed that all the tested techniques were unstable

76


Figure 4.2: Diagrammatic sketch showing the flowchart of the simulation

77


Figure 4.3: Diagrammatic sketch showing the flowchart of the bubble loop

78


when the pressure change was relatively high. To solve this instability issue, Alehossein and

Qin 2007 [1] suggested a technique to use variable timesteps which lead to time steps as low

as 10−37s.

In the current model, it is unreasonable to use a similar, computationally expensive,

technique as the expected pressure perturbation should be only due to the change in fluid

velocity. So, it was decided to use Modified Euler Equation. To determine the reasonable

time step, a MATLAB code was used to solve the equation with different time steps, pressure

perturbation function and different pressure frequency. The test showed that dividing the

main time step, used in the two-phase model, into 10,000 to 100,000 resulted in stable

simulation, except for the very small bubbles (R = 10−5m). Decreasing the sub-timestep

further did not change the bubbles size with the time. Figure 4.4 shows the bubble size

response for a sinusoidal pressure wave. The time step selected for this solution was 10−10sec

(equivalent to 110,000

of the time step). To avoid floating error, a safety factor of 10 was applied

and all simulations were conducted with a sub-time step of 10−11sec.

From Fig. 4.4, it could be noticed that the larger bubbles relative response (R/Ro) is

much higher to the pressure wave. Given that the larger bubbles represent higher ratio of

the total gas volume and their absolute response will be order of magnitudes higher than

small bubbles. Another code was written to simulate the flow with pre-determined bubbles’

number and radii. The total change in the gas volume was around ±5%. By ignoring the

compressibility of small bubbles (R = 10−5m), it was shown that change in total gas volume

was affected by less than 0.02%. So, it was decided to avoid applying Rayleigh-Plesset

subroutine for small bubbles while running the simulations.

79


Figure 4.4: Response of bubble radius to a sinusoidal pressure wave. Time step = 10−10sec

4.3.2 Coalescence and breakup procedure

Sections (3.4.6) and (3.4.7) discussed the criteria for bubble-bubble coalescence and bubble

breakup. For each timestep, the proximity of each bubble to another bubble or approaching

the wall is checked. If the bubble was found to be impacting another bubble, coalescence

criteria is checked. The coalescence criteria here was taken similar to Senez and Etienne 2011

[75] by using an equivalent radius in Eq. (3.33) to determine if the bubbles will coalesce

or not. If the critical Weber number is exceeded, Eq. (3.34) is used to determine if the

coalescence will still happen or not. If the coalescence condition was met, a new bubble

is generated which has a mass equal to the original bubbles and had an internal pressure

equal to the initial pressure calculated from the force balance on the bubble surface. If the

coalescence criteria fails, the two bubble bounce as will be discussed later.

When a bubble impacts the wall, breakup criteria is checked. If breakup occurs, the

bubble breaks up into two equivalent bubbles. The location of the new bubbles are 0.1Rnew

80


apart from each other. This distance is assumed to avoid collision between the two new

generated bubbles at the same time step. If there is no breakup, bubble bouncing from the

wall is assumed.

In the current model, it is assumed that the flow is one dimensional inviscid laminar

flow. This means other mechanisms of bubble breakup mentioned are not applicable. The

pressure gradient around the bubble is ignored and the pressure is assumed to be constant

on the bubble outer surface to keep the spherical shape. This resulted in having one source

of breakup which is hitting the channel walls. However, this one source of breakup was not

enough to compensate coalescence and bubbles continued to coalesce and grow until their

radii approach the channel width. To solve this problem, a limit should be put on the bubble

size.

According to Beguin (2010), reported by Senez and Etienne 2011 [75] in a two-phase

internal flow, a bubble subjected to the wake of other bubbles can be stable up to a radius of

7 mm. This bubble size is considered very large when compared to the streamtube dimension

in the current study.

Figure 4.5 shows an example of bubble breakup. The first two frames indicate one bubble

before it is divided in the third frame. The last frame shows the two new bubbles after moving

away from each other.

Figure 4.6 shows an example of bubbles Coalescence. In the figure, the two bubbles in

the red frame approach each other. As the distance between them decreases they coalesce

and appear as one bubble.

4.4 Steady State Simulation

To enable long simulation time while keeping the same mass fraction in the tube, a new

bubble is generated at the flow cell entrance when a bubble leaves the flow cell. The mass,

81


Figure 4.5: Snapshots from the simulation showing bubble breakup

82


Figure 4.6: Snapshots from the simulation showing bubble coalescence

83


volume, velocity and pressure of the two bubbles are kept the same. This enable simulating

large time with the same mass fraction and flow conditions.

The simulation starts by imposing initial displacement to the tube to excite the fluide-

lastic instability. And by dividing the time domain into infinitesimal time steps, the area

of each channel could be obtained for each time step. By solving the continuity and mo-

mentum equations using the average density at each section of the channel, the velocity and

the pressure are obtained. Then the two phase flow model is applied to predict the bubbles

velocities, locations for the next time step. Integrating the pressure around the tube enables

calculating the fluidelastic instability force FFEI . By using this force in equation (3.1), the

displacement of the tube is obtained for the next time step. Running the simulation for a

period of time would allow checking the instability of the tube at the simulation conditions.

Simulation were conducted on SHARCNET. The average simulation time on 4 nodes ranged

between 10 hours to 6 days depending on the number of bubbles.

Average void fraction inside each channel was calculated as the fraction of area occupied

by the air, divided by the whole streamtube area at no perturbation ΣπR2

(Sattachement−So)∗W . At any

section, the local instantaneous void fraction was calculated as the fraction of the channel’s

instantaneous width occupied by the gas αlocal = length occupied by gaschannel width

. This was affected by the

pressure perturbation due to the change of bubbles’ size. Time steps was selected to be 10−5

seconds. Decreasing the time step further showed no effect on the simulation result while

the simulation time was almost doubled. Chapter 5 will show how the time step affected

the simulations. Due to the high nonlinearity of Rayleigh-Plesset equation, each time step

was divided to sub-timesteps when solving the equation. The sub-timestep was selected to

be 10−11 seconds.

84

Chapter 5

Results and Discussion

This chapter presents all the simulations results followed by a discussion. The first section,

5.1, will present the simulations of the homogeneous model to be compared later with the

current model. Simulation procedure of the current model showing the inputs and how the

stability threshold is determined will be presented in section 5.2. A parametric sensitivity

study is presented in sections 5.3 and 5.4 in order to show how much the results is affected

by changing the sub-model parameters. Model validation, using experimental results, is

presented in section 5.5. Section 5.6 will compare the slip ratio between the average gas

velocity to the average liquid velocity obtained from the current model with those obtained

by applying Feenstra’s model [19] on experimental data. Section 5.7 presents the effect of

changing the density ratio between the gas and liquid on the stability threshold. The effect

of the pitch to diameter ratio P/D on the instability is presented in section 5.8.

Early simulations showed the need of a very long simulation time (around 30 days).

For this reason the code was reconstructed to enable parallel processing when possible.

During the parallel simulations, the number of available processors is divided between the

two channels based on the number of bubbles. By dividing the total number of bubbles (in

the two channels) by the number of processors and approximating to the nearest integer.

85


A condition was put to guarantee at least one processor for each channel. During the

simulations in each channel, the bubbles are divided on the dedicated processors, evenly.

All processors (dedicated for one channel) stop after doing the calculations on the assigned

bubbles and wait for the impact loop. Bubble-to-bubble impact loop was done in one thread

for each channel. Solving the continuity and momentum equations is performed in parallel as

well (one thread per channel). After calculating the fluid forces on the tube, the displacement

is calculated by single processor.

Simulations were performed on SHARCNET servers. Each simulation requested between

4 to 6 processors and a total of 2 GB ram. Simulations time varied according to the number

of bubbles. Higher void fractions, in general needed more processing time. Many simu-

lations were not successfully completed because of exceeding the maximum time limit on

SHARCNET (7 days). To overcome this problem, the code was modified to save the results

every given number of timesteps. This number was selected to be 10, 000 time steps which

is equivalent to 0.1 sec.

For some selected simulations, all bubble, channel data were recorded to allow debugging

and/or testing the code. All post processing was performed using Matlab. Because of the

large amount of data being processed, a Matlab code was developed to create simulations

videos using these data. This enabled debugging the code by visual observations in addition

to the regular debugging techniques.

Simulations conditions were selected similar to Pettigrew et al. 1989a [61] to enable

comparison. Air properties at atmospheric conditions is used for the gas properties. Unless

otherwise mentioned, all simulations were performed by the conditions in table 5.1.

Air was treated as an ideal gas to calculate the gas density using the ideal gas relation,

p =1

ρRT (5.1)

86


Table 5.1: Simulation conditions

General

Tube Diameter d [mm] 12.7

P/D 1.47

Time step [s] 10−5

Rayleigh-Plesset time step [s] 10−11

Liquid properties

Density [kg/m3]1000

Kinematic viscosity ν [m2/s] 10−6

Initial displacement [mm] 0.2

Surface tension σ [N/m] 6× 10−2

Gas properties

Temperature [oC] 300

Initial pressure inside the bubbles [Pa] 105

Gas Constant R [J/kg.K] 287

87


The mass damping parameter MDP, in the stability maps, was calculated using the total

mass mt, homogeneous density ρh and the damping in air δa.

MDP =mtotalδaρhd2

(5.2)

The damping was calculated using

δa = 2πζa (5.3)

ζa was assumed to be 0.01 in all simulations. For the experimental data, it was assumed

ζa = 0.01 if no information was reported. This value of damping is considered very when

compared to the damping in the CANDU steam generator. So the assumption is conservative

when compared to the actual applications. And this assumption agrees as well with the

experimental setups used in the literature such as [61, 50, 19, 46] who reported a measured

value of ζ ≤ 1%.

The reduced flow velocity was calculated using the tube diameter d, the tube frequency

in the mixture f , and the average flow velocity of the mixture.

Ur =Uavgfd

(5.4)

The average flow velocity Uavg in the above equation is calculated using the homogeneous

density, mass flow rate of the mixture and the flow area

Uavg =mmixture

ρhA(5.5)

Using the homogeneous velocity and density on the flow maps was selected to enable

comparison with the data published in the literature.

88


5.1 Homogeneous model

First set of simulation was performed using the homogeneous model. This model assumes

homogeneous flow velocity between the gas and liquid. Although it this was proved to

be wrong assumption by [19, 50], the model was used here for comparison purpose. The

homogeneous density is calculated using:

ρhom = αρg + (1− α)ρl (5.6)

And by using the time domain model, developed by Hassan and Hayder 2008 [23] for

single phase flow, the fluidelastic instability threshold of a homogeneous mixture across tube

bundle was obtained. The simulations were performed at various reduced flow velocities,

Ur ranged between 0.4 to 8. For each flow velocity, the vibrating tube was given an initial

displacement of 0.2mm. The closest velocity to the marginally stable oscillation was assumed

to be the instability threshold. Figure (5.1) shows the simulations at four different velocities.

It could be noticed that from Fig. (5.1 a) that the vibration amplitude decreases with the

time until it reaches stability. Similar trend is obtained at Ur = 3.5 in Fig. (5.1 b). At

Ur = 4.0, Fig Fig. (5.1 c), the stability seems to be happening after slightly longer time.

Clear instability is shown in Fig. (5.1 d) where the reduced flow velocity Ur was equal to

4.2. In this case, the instability threshold was taken as Ur = 4.1.

In the case of multiple stability region (low MDP as shown by [41]), the higher stable

velocity was taken as the stability threshold.

89


Figure 5.1: Tube response for 25% void fraction at various flow velocity using the homoge-neous density model

5.2 Current model

The homogeneous model used in the above section aimed to obtain stability maps for single

phase flow. So, the inputs to the model were the reduced flow velocity Ur and the mass

damping parameter MDP in addition to the fluid and tube properties. The fluid velocity

was then calculated by the code to start the simulation. On the other hand, the mechanistic

model developed in this study aimed mainly to predict the stability threshold for a case

by case instead of drawing stability map. The reason is to enable validating the model

by simulating the exact experimental conditions. So, the input to the current model was

90


customized to be the tube mass per unit length mt, damping in the air δa and the average

flow velocity Uavg. Most experimental data report these parameters.

5.2.1 Stability threshold

To obtain stability threshold, simulations were conducted on a range of velocities (usually

10 velocities). The range was selected to start by 0.5m/s and ends by twice the velocity

expected to be the stability threshold. Once the stability is determined to be between two

certain velocities, another set of simulations is conducted with a range of velocities scanning

this range to obtain more precise value.

Figure 5.2 shows the tube displacement for various flow velocities. By comparing figures

5.1 and 5.2, a different trend of the vibration amplitude will be noticed. In Fig. 5.1, corre-

sponding to the homogeneous model, the vibration amplitude decays or increases smoothly

with time. But in Fig. 5.2, corresponding to the current model, the amplitude oscillates

up and down while decaying or growing. This behaviour, obtained from the current model

simulations, is expected for the two phase flow and was shown experimentally by Moran 2007

[50], see Fig. 5.3. Moran 2007 [50] obtained the same trend for all the experiments conducted

at the range of void fractions covered by the current study. Although using different working

fluid (R11), The reason that could explain this behaviour is the instantaneous change of

fluid density around the tube. This density change is caused by the change of local volume

fraction. The stability threshold in Fig. 5.2 was taken at a flow velocity V = 1.31m/s.

Figure 5.4 shows a comparison between the stability map obtained by the current model

and the homogeneous model. The figure shows that the FEI threshold using the homogeneous

model is higher than that obtained by the current model. The difference between the two

models is higher at low void fraction and decreases as the void fraction reaches 30%.

This trend was not expected as the homogeneous equilibrium model was expected to

91


Figure 5.2: Tube response for 25% void fraction at various flow velocity using the currentmodel

have closer prediction at the low void fractions where the effect of the gas phase is still

insignificant. However, this result could be returned to the existence of multiple stability

region at this low void fraction (low mass damping parameter MDP ). Figure (2.6), in

chapter 2, shows a stability map reported by Lever and Weaver 1986b [42] which clarify the

multiple stability region by the dotted lines. The existence of bubbles causes rapid change

of the fluid density, velocity and pressure around the tube. This results in rapid change of

the fluidelastic instability force. With the existence of the multiple stability lines on the

flow map, crossing the stability line, whether up or down is easy. Once the tube crosses the

92


Figure 5.3: Tube response for 68% void fraction, from Moran 2007 [50]

Figure 5.4: Comparison between the stability map obtained by the current model and bythe homogeneous model

93


instability line, it never come back to stability.

5.2.2 FEI Force acting on the tube

In the simulation conducted by the homogeneous equilibrium model, the forces on the tube

was a pure sine wave that have the same frequency of the tube displacement. However, in

the current model, the integration of the forces acting on the tube showed some deviation

from this pure sine wave. Figure (5.5) shows the tube displacement and the fluid forces

versus the time at void fraction 25% and average flow velocity of Uavg = 1.2m/s. From the

figure, an oscillation of the fluid forces with high frequency could be noticed. Also forces

peaks (impulses) that occurs regularly could be noticed. This behaviour of the fluidelastic

instability forces can be returned to the change of of fluid density around the tube.

The force impulses described in figure (5.5) changed in the magnitude between low values

(1 N or less) to very high values (More than 1 kN). Figures (5.6 and (5.7) show an example of

this amplitude change. When comparing the three figures (5.5, 5.6 and (5.7), we can notice

this change in the amplitude.

The change of the FEI forces can be the reason of the non-smooth oscillation obtained

by the current model. Next section (5.2.3) shows an example of the density change in the

flow channel. Also the effect of bubble compressibility is discussed in section (5.3).

5.2.3 Density distribution in the flow channels

Trying to explain the reason of the force peaks, the average density at different positions

in the flow channels was plotted against the time. Figure (5.8) shows the change of the

average density with the time at 4 different locations ((a)Left channel s = 0 (tube center),

(b) Right channel s = 0 (tube center); (c)Left channel s = −so2

; (d)Right channel s = −so2

;

(e)Left channel s = −so4

; (f)Right channel s = −so4

; (g)Left channel s = −so (tube entrance)

94


Figure 5.5: Tube displacement and fluid forces on the tube vs. time, Uavg = 1.2m/s andα = 25%

; (h)Right channel s = −so (tube entrance)) in the two channels. As seen from the figure,

the density change

5.3 Sensitivity studies

As discussed in chapter 3, there are many sub models integrated together to obtain the overall

framework. These sub-models were obtained from the literature where several criterion were

95


Figure 5.6: Tube displacement and fluid forces on the tube vs. time, Uavg = 1m/s andα = 25%

available. This section applies a sensitivity study on the current model to see the effect of

using these various sub-models. This study aimed to check the model limitations and tune

the model parameters if needed.

5.3.1 Effect of initial bubble size distribution

Section (3.4.1) discussed the generation of the bubble algorithm. As mentioned, initial bubble

size distribution was reported in the literature as beta distribution, Senez and Etienne 2011

96


Figure 5.7: Tube displacement and fluid forces on the tube vs. time, Uavg = 1m/s andα = 32%

[75], Wallis 1968 [81] in vertical pipes. Ulbrich and Mewes 1994 [80], Noghrehkar et al. 1999

[52], Feenstra 2000 [19] and Moran 2007 [50] reported difference between the flow regimes

in vertical pipes and tube bundles. So, it is crucial to study the effect of the randomly

generated bubbles’ distribution on the instability threshold.

The Fortran code was customized to have an input distribution parameter to switch

between the beta, normal and uniform distributions. Simulations were conducted using the

same input parameters except the distribution parameter. The simulation time for the beta

97


Figure 5.8: Average density in the channel vs. the time at different locations. (a)Left channels = 0 (tube center), (b) Right channel s = 0 (tube center); (c)Left channel s = −so

2; (d)Right

channel s = −so2

; (e)Left channel s = −so4

; (f)Right channel s = −so4

; (g)Left channel s = −so(tube entrance) ; (h)Right channel s = −so (tube entrance)

distribution was noticeably higher. Table 5.2 shows the difference between the stability

threshold velocities obtained by both the beta and normal distributions.

Figure 5.9 shows the effect of initial beta distribution and normal distribution. As seen

from the figure, the initial size distribution has a minor effect on the stability threshold.

This is explained by that the bubble size reaches a distribution different than the initial one.

Because of having number of coalescence higher than the number of breakups as discussed

in chapters 3 and 4, the bubbles coalesce with each other and a different distribution is

obtained. This was also noticed by following the number of bubbles during simulations. It

should be noted that the beta distribution correlation was obtained in a vertical pipe, not a

98


Table 5.2: The stability threshold velocities for both beta and normal distributions

Void fraction Instability threshold velocity [m/s]

Beta distribution

5% 0.77

10% 0.81

15% 0.95

20% 1.09

25% 1.26

30% 1.30Normal distribution

5% 0.82

10% 0.94

15% 1.04

20% 1.20

25% 1.36

30% 1.29

99


Figure 5.9: Effect of initial bubble distribution

tube bundle.

Based on this result, and by considering the lower simulation time, the normal distribu-

tion was taken as a default for the code. The possibility to change the distribution from the

input file using the distribution parameter was kept though.

5.3.2 Effect of initial bubbles size

The number of bubbles in the channels affects the simulation time significantly. Also as

discussed in chapter 4, the small bubbles, D < 10−2mm need small sub-time-step when

solving Rayleigh-Plesset equation. Hence, in the bubble generation stage, it was reasonable

to avoid going below certain radius. The selection of this radius was done by trial and error.

Table 5.3 shows the stability threshold velocity for three different cases; Rmin = 0.01Rmax,

Rmin = 0.1Rmax, Rmin = 0.2Rmax. Figure 5.10 shows the stability map for the same three

100


cases. Here, Rmin is the minimum radius of the generated bubbles while Rmax is the radius

where the larger number of bubbles exist. This limitation of the bubble radius was applied

only at the bubble generation stage. During simulations, bubbles are allowed to breakup

and reach smaller sizes than Rmin until reaching an absolute value of 0.01mm diameter.

Table 5.3: The effect of the minimum bubble radius at the bubble generation stage on thestability threshold


Rmin = 0.01×Rmax 0.1×Rmax 0.2×Rmax

5% 0.77 0.82 0.75

10% 0.8 0.94 0.84

15% 0.89 1.04 0.91

20% 1.05 1.20 1.08

25% 1.24 1.36 1.26

30% 1.31 1.29 1.31

As shown from Fig. 5.10, the initial bubble size has minor effect on the stability threshold.

Combining this with what has been discussed in section 5.3.1, it could be concluded that the

initial bubble size and distribution have minor effect on the stability threshold. To justify

this, a coalescence and breakup counters were set in each simulation and it was found that

the small bubbles coalesce with each other forming larger bubbles. This was noticed by

the existing large number of coalescences than breakup in the first (2− 3seconds). Then a

balance between the number of coalescence and breakup occur.

101


Figure 5.10: Effect of initial bubble size

5.3.3 Effect of time step

On one hand, decreasing the time step, increase the simulation time exponentially. On the

other hand, large time step would affect the accuracy. That is why selecting a reasonable

time step for the simulations is an important issue. The recommended time step by the

model introduce by [23] was 10−5s. This was done for airflow where the velocities reaches

higher values than the current model. However, the same time step was selected for the

current model. The reason was to be able to capture the bubbles movement relative to

each other. Simulations were performed using various time steps to study the effect of the

time step on the stability threshold. Some of the simulations with time step 10−4s crashed

because of the bubbles intersecting over each other instead of bouncing. Simulations with

time step 5× 10−5 continued without crashing. Table 5.4 presents the stability threshold for

three different timesteps while Fig. 5.11 shows the stability map for these cases. It could

102


be noticed that for the low velocities, the time step between 10−6 to 5× 10−4 has no effect

on the simulations. However, for higher velocities the simulations conducted with time step

5 ∗ 10−5 shows slight difference. From this, it would be recommended to use time step of

10−5 or lower.

Table 5.4: The effect of timestep on the stability threshold velocity


Timestep [sec] 5× 10−4 1× 10−5 1× 10−6

5% 0.78 0.82 0.78

10% 0.84 0.94 0.82

15% 0.97 1.04 0.96

20% 1.05 1.20 1.1

25% 1.2 1.36 1.24

30% 1.31 1.29 1.31

As mentioned in chapter 4, a sub-timestep was used for Rayleigh-Plesset equation and

it was selected to be 1× 10−11sec. An attempt to see the effect of this sub-timestep on the

threshold velocity was started. However, many of the simulations having higher timestep

crashed because of floating point error which indicate that the solution of Rayleigh-Plesset

was unstable. Also, using lower timestep was not feasible as the simulation time increased

by an order of magnitude. So, it was decided to perform this test by solving Rayleigh-

Plesset equation separately using different timesteps, away from the two phase code. This

was presented in chapter 4. Also, studying the effect of including bubble compressibility was

performed and the results is presented in the next section.

103


Figure 5.11: Effect of time step

5.3.4 Effect of including bubble compressibility

The bubble compressibility was implemented in the current model by solving Rayleigh-

Plesset equation. Figure 5.12 shows the effect of disabling Rayleigh-Plesset equation. It could

be seen from the figure that the disabling Rayleigh Plesset equation increased the stability

threshold, specially at high void fraction. Although the difference did not exceed 15%,

disabling Rayleigh-Plesset equation shifted the stability threshold towards the homogeneous

model.

Taking bubbles compressibility into consideration is a novel technique to be applied in

the FEI research. Although the difference in the stability threshold did not exceed 15%, in

the current study, further research is needed to study the effect of fluid compressibility in

two phase flow for higher void fractions and different fluid properties.

104


Table 5.5: The effect of bubble compressibility on the stability threshold velocity


Taking compressibility into effect Ignoring compressibility

5% 0.82 0.78

10% 0.94 0.88

15% 1.04 1.2

20% 1.20 1.3

25% 1.36 1.36

30% 1.29 1.4

5.4 Effect of Breakup and Coalescence criteria

5.4.1 Effect of size breakup criteria

As discussed in chapter 3, the sides of the streamtubes were assumed as flexible walls. No

fluid was allowed to cross these walls. Also this model considered the liquid as a continuous

phase. Because of these reasons, a criteria was used to force the large bubbles to breakup.

The default criteria was to force the bubble to breakup when its diameter is equal to 0.9

of the streamtube width. This criteria was compared by another two criterion where the

forced breakup occurs when the bubble diameter is 0.7 and 0.5 of the channel width. Figure

5.13 shows a comparison between these three cases. As seen from the figure, except for one

point at 25% void fraction, the stability threshold increases with decreasing the maximum

allowed bubble size. In other words, the stability threshold moves towards the single phase.

However, the difference between the stability in the three cases does not exceed 10%.

105


Figure 5.12: Effect of gas compressibility

5.4.2 Coalescence criteria

The coalescence criteria used in this study was similar to that adopted by Senez and Etienne

2011 [75]. They adopted that coalescence will occur if any of these two criterion occur;

Wecr =DρlU

2r

σ≺ 0.36 (5.7)


σ≺ 6.6 (5.8)

Figure 5.14 shows a comparison between the simulations adopted same technique (Default

Criteria), and using equation any of Eq. 5.7 (criteria 2) or Eq. 5.8 (criteria 3) alone. The

exact threshold velocities are presented in table 5.7. The figure shows that stability threshold

when using criteria 2 (Eq. 5.7) is higher less than that of criteria 3 which agrees better with

the criteria adopted by Senez and Etienne 2011 [75]. This could be explained by that criteria

3 precedes criteria 2 most of the time, so it agrees better with the default criteria adopted

in the current model.

106


Table 5.6: The effect of forced breakup criteria on the stability threshold velocity

Void fraction Instability threshold velocity [m/s]Dmax

Wchannel0.5 0.7 0.9

5% 0.8 0.78 0.82

10% 0.89 0.86 0.94

15% 1.01 0.99 1.04

20% 1.15 1.1 1.20

25% 1.28 1.29 1.36

30% 1.34 1.31 1.29

5.5 Model Validation

To validate the model, simulations were conducted for air-water mixture at atmospheric

pressure and room temperature. at the conditions shown in table 5.1 These parameters are

identical to those of Pettigrew et al. 1989 [61, 62] work to facilitate comparison with the

experimental result. In each simulation, the tube was given an initial displacement and no

other force was introduced. Then the tube response was examined for Amplitude decay or

growth.

5.5.1 FEI Threshold (Critical flow Velocity):

Table 5.8 compares the stability threshold velocity obtained from this model with experi-

mental results reported by Pettigrew et al. 1989a [61]. Figure 5.15 compares the stability

map obtained using the current model with the experimental results reported Pettigrew et

al. 1989a [61]. The stability threshold obtained by the current model was the average of

107


Figure 5.13: Effect of using different different forced breakup criteria

three simulations to ensure repeatability. The difference between the three simulations did

not exceed 5% except at 32% void fraction where the difference reached 12% of the average.

The figure also shows the stability threshold obtained by the HEM model. As shown in the

figure, the current model was able to predict the FEI threshold accurately. Although Petti-

grew et al. 1989a [61] published data for higher void fractions, the current model could not

simulate the void fractions larger than 35%. This because of the formation of large Taylor

bubbles which was not implemented in the current model.

5.6 Slip

The slip model, introduced by Feenstra 2000 [19] and Feenstra et al. 2003 [18], provides a

qualitative mean to calculate the slip ratio S (the ratio between the gas velocity and the

liquid velocity). The slip S was found to be dependent on Capillary Ca and Richardson Ri

108


Figure 5.14: Effect of using different different coalescence criteria

Figure 5.15: Comparison between the results obtained from the current model with experi-ments Pettigrew et al. 1989a [61]

109


Table 5.7: The effect of coalescence criteria on the stability threshold velocity


Criteria 2 Criteria 3 Default criteria

5% 0.9 0.75 0.82

10% 0.97 0.81 0.94

15% 1.08 0.95 1.04

20% 1.18 1.03 1.20

25% 1.31 1.25 1.36

30% 1.33 1.29 1.29

numbers. The model showed an excellent prediction of the actual two phase density and

void fraction compared to other two phase models, Feenstra 2000 [19] and Moran 2007 [50].

The slip model equations are given by:

S = 1 + 25.7(Ri ∗ Ca)0.5

(P

D

)(5.9)

α =

[1 + S

ρgρl

(1

x− 1

)]−1

(5.10)

Using the slip model, equations (5.9 and 5.10), a value for the slip S was calculated for

the same experimental conditions reported by Pettigrew et al. 1989a [61] and compared with

the slip factor obtained from the simulations. The slip factor obtained from the simulation

was calculated as a mathematical average of the ratio between the bubbles velocity to the

liquid velocity over a period of time (0.1 second). Figure (5.16) shows a comparison of the

slip factor predicted by the current model and Feenstra’s model [19]. The comparison shows

110


Table 5.8: Comparison between the stability threshold velocity obtained by the currentmodel and the experimental data reported by Pettigrew et al 1989 [61]


Current model Experimental data

5% 0.82 0.79

10% 0.94 N/A

15% 1.04 0.97

20% 1.20 N/A

25% 1.36 1.31

30% 1.29 N/A

32% N/A 1.47

good agreement between the current work and Feenstra’s model.

Ca =µlUGσ

(5.11)

Ri =4ρ2G(P −D)

G2p

(5.12)

where σ is the surface tension

Feenstra 2000 [19] and Moran 2007 [50] compared the results with other two phase models

showing better agreement with the experimental data.

111


Figure 5.16: Comparison between the slip factor obtained by the current model and thecalculated slip using Feenstra’s model [19] applied on the experimental results by Pettigrewet al. 1989a[61]

5.7 Effect of density ratio

A study was performed to see the effect of the density ratio (ρL/ρg) on the stability threshold

using the current model. In this study, air properties were used for the gas for all density

ratios. The initial input pressure was modified to changed the gas density. Six density ratios

were tested; 30, 100, 250, 500, 770 and 1000. The corresponding input pressures are shown

in table 5.9. In the CANDU steam generator, the density ratio between liquid to gas is

around 30.

112


Table 5.9: The initial input pressure used to obtain the density ratio

Density ratio Initial pressure [bar]

30 30

100 10

250 4

500 2

770 1.3

1000 1

Figure 5.17 shows the predicted stability map obtained by the current model at different

density ratios While table 5.10 shows the threshold velocities obtained. As seen from the

figure, the density ratio shows some effect on the stability threshold especially at the higher

mass damping parameter.

Although it was reported in the literature from experimental results, [50, 19] and others,

that the density ratio has a large effect on the stability threshold, this is not reflected in

Fig. 5.17. This discrepancy could be explained by ignoring other factors while reporting

the effect of density ratio. Some parameters such as the surface tension and viscosity might

share part of this effect. Also the phase change between the liquid-vapor combination which

is not accounted in the current model, would make a difference in the two phase damping.

Further investigations are needed to explain this behaviour.

113


Fig

ure

5.17

:E

ffec

tof

the

den

sity

rati

oon

the

stab

ilit

yth

resh

old

114


5.8 Effect of P/D ratio

Although the pitch to diameter ratio P/D is a factor that has a significant effect on the

instability threshold, it was not studied enough in the literature. This section presents a

study of the effect of P/D ratio on the stability threshold. In this study, air properties were

used for the gas while water properties were used for the liquid. The typical P/D ratio in

the CANDU steam generator is between 1.4 to 1.5. It is worth mention here that the ASME

code for the steam generators does not take P/D ratio into effect.

Figure 5.18 shows the predicted stability map obtained by the current model for three

P/D ratios. Table 5.11 shows the threshold velocities obtained. As seen from the figure, the

P/D ratio shows significant effect on the stability threshold. As P/D ratio increases, the

bundle becomes more stable. This result agrees well with the findings of Hassan and Weaver

2018 [28] for single phase flow.

The P/D in the steam generator design is a factor that control the compactness of the

heat exchanger. Lower P/D ratio, means smaller size heat exchanger for the same Heat

transfer. This of course gives an advantage when the space is limited in the nuclear site.

The reason behind this relation between the instability threshold and the P/D ratio can be

explained by the following; At lower P/D ratio, the tube vibration tries to squeeze a thinner

layer of the fluid. This results in a higher pressure which leads to larger fluid forces on the

tube. Hence, the instability threshold occurs earlier, at lower flow velocity.

The steam generators of San Onfore Nuclear Generating Station SONGS, was design

with a lower P/D to diameter. This caused the failure of the steam generator and permanent

closure of the station due to fluidelastic instability. More studies are needed on the effect

of P/D to guarantee safe design of the compact heat exchangers. Special focus is needed on

the instability in the streamwise direction at low P/D ratio.

115


Fig

ure

5.18

:E

ffec

tof

P/D

rati

oon

the

stab

ilit

yth

resh

old

116


Tab

le5.

10:

The

effec

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the

stab

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old

velo

city

Voi

dfr

acti

onIn

stab

ilit

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resh

old

velo

city

[m/s

]

Den

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rati

o30

100

250

500

770

1000

5%0.

770.

780.

770.

780.

790.

82

10%

0.79

0.79

0.8

0.82

0.84

0.94

15%

1.02

1.04

1.04

1.04

1.05

1.04

20%

1.3

1.3

1.3

1.3

1.32

1.20

25%

1.3

1.3

1.3

1.28

1.31

1.36

30%

1.29

1.29

1.3

1.31

1.29

1.29

117


Table 5.11: The effect of P/D ratio on the stability threshold velocity


P/D ratio 1.3 1.47 1.6

5% 0.55 0.77 0.9

10% 0.68 0.79 0.97

15% 0.8 1.02 1.1

20% 1.0 1.3 1.3

25% 1.08 1.3 1.36

30% 1.12 1.29 1.4

118

Chapter 6

Conclusion

6.1 Conclusion

An analytical model to predict FEI threshold of two phase flow was presented in this thesis.

The model expands the flow cell model introduced by [41] to cover two phase flow. Time

domain discretization similar to that adopted by Hassan and Hayder 2008 [23], was performed

to enable predicting the stability threshold. The model focused on the bubbly flow regime

where void fraction, calculated using the homogeneous model, is less than 35%. The flow

across the tube bundle is simplified as two phase bubbly inviscid flow in streamtubes. The

model applied force balance on each bubble to predict its location and velocity. Bubbles

interaction with each other as well as interactions with the streamtube walls were taken

into account. The vibrating tube was simplified as a single degree of freedom cantilever.

Comparison of the results with the data from the literature showed a very well agreement

with an absolute error of 15% in the predicted reduced flow velocity. A parametric study

was presented to show the effect of the sub-models and parameters integrated in this model.

The model is considered the first mechanistic model to simulate fluidelastic instability of

two phase flow. Modeling the movement of each bubble separately while taking the bubbles

119


interaction the FEI threshold is a novel technique. This technique enabled better estimation

of the fluid density and hence the fluid forces. Accounting for the gas compressibility in two

phase flow is being done for the first time in flow induced vibration research area. Bubble

compressibility showed an effect on the stability threshold. Very promising results were

obtained when simulating air-water mixture.

Although the current model is limited to the air-water mixture, the possibility of expand-

ing the model to simulate other mixtures is possible. A sub-model to simulate the phase

change is required for this step to be completed. Then, phase change should be taken into

account in the governing equations.

The model failed to simulate higher void fraction because of the exponential increase

of the number of collisions between the bubbles. This leads to increasing the simulation

time in addition to the formation of large bubbles that were able to block the flow channels.

To simulate higher void fraction, another model is needed that can deal with large bubbles

(Taylor bubbles) which were not implemented in the current model.

More experimental data are needed to allow enhancing the modeling of FEI, specially for

the normal square bundle. Although the normal square geometry is not used widely, but it

is considered the easiest bundle for modeling purpose. After validating the model, this could

be expanded to include more complex geometries.

The closure of San Onofre Nuclear Generating Station (SONGS) very recently confirms

that more analytical models are needed to explain the physics of FEI phenomena in two phase

flow. It should be mentioned that the FEI issues that resulted in the permanent closure of

SONGS was in the streamwise direction. However, it showed some lack of understanding

the physics of FEI.

120


6.2 Recommendations for future work

1. Expand the model to cover the droplet flow regime. This could be done by treating

the gas as compressible continuous phase and treating the liquid droplets as dispersed

phase. The U-bend in the CANDU steam generator lies at the top where void fractions

are very high, (90%) or more. At this high void fraction, the mixture density becomes

lower which leads to higher velocities. In addition, the existence of loose supports at

these high velocities is very common.

2. Expand the model to cover the intermittent flow regime by implementing sub-models

that deal with Taylor bubbles.

3. Implement the various geometries of tube bundles in the model. The most important

geometry to be start with is parallel triangle geometry because of its wide use in

the CANDU steam generator. The current model started with the normal square

geometry to simplify the problem. However, the code have taken the expansion of

other geometries into account.

4. Enable phase change during the expansion/compression of bubbles to be able to model

Steam-Water and Freons accurately. When the bubbles expand because of a low sur-

rounding pressure, a phase change happens from the liquid to vapor and vice versa.

The liquid phase in the current model was selected to be water at atmospheric condi-

tions. With the isothermal assumption, the pressure drop needed to be very large to

reach the saturation temperature. However, in the real CANDU steam generator, the

mixture of steam-water is definitely at saturation conditions. This means any drop of

the pressure will cause evaporation and vice versa. So simulating phase change would

enable simulating steam-water and freons mixtures. Phase change might be the reason

of the high two phase damping reported in many studies such as [7, 50].

121


5. Taking Reynolds, and capillary numbers into account to present the stability maps.

The effect of fluid viscosity plays a crucial rule in the fluid damping and it could have

an effect on the stability threshold for both single and two phase flow. Also, the surface

tension have an essential rule on controlling the gas-liquid interface and could affect

the stability maps. Both of the viscosity and surface tension are not included in the

reduced flow velocity nor the mass damping parameter. The current model was not

able to focus on their effect because it is not able to simulate the phase change yet.

6. Expanding the model to simulate streamwise FEI. Hassan and Weaver have expanded

the flow cell model for single phase flow to the streamwise direction in a series of

publications [25, 26, 27]. This model could be expanded for two phase flow as well.

122

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129

Appendix A

Dimensional Analysis of Rayliegh

Plesset equation

Pv(T∞)− P∞(t)

ρL+Pv(TB)− Pv(T∞)

ρL+ PGo

(TBT∞

)(Ro

R

)3k

= RdR2

dt2+

3

2

(dR

dt

)2

+4νLR

dR

dt+

2σ

ρLR(A.1)

Equation (3.45) is a second order highly nonlinear differential equation which is difficult

to solve, specially that the fluid pressure P∞ is not defined as a continuous function. First

attempt to solve the equation was to apply dimensional analysis to ignore the insignificant

terms.

Assume dimensionless parameters

R∗ =R

Ro

(A.2)

(A.3)

t∗ =tURRo

(A.4)

(A.5)

P ∗x =Px

12ρLU2

R

(A.6)

And

130


dR

dt=R

t=

R∗Ro

t ∗Ro/UR=R∗URt∗

(A.7)

(A.8)

d2R

dt2=R

t2=R∗URt∗

URt∗Ro

=U2RR∗

t∗2Ro

(A.9)

(A.10)

By substituting in equation A.1

1

2U2R[P ∗v (T∞)−P ∗∞(t)] +

1

2U2RP∗Go

(1

R∗

)3k

= R∗RoU2RR∗

t∗2Ro

+3

2

(R∗URt∗

)2

+4νL

R∗UR

t∗

R∗Ro

+2σ

ρLR∗Ro

(A.11)

By dividing the whole equation by 12U2R and rearranging

P ∗v (T∞)− P ∗∞(t) + P ∗Go

(1

R∗

)3k

= 2

(R∗

t∗

)2

+ 3

(R∗

t∗

)2

+νL

URRo

8

t∗+

σ

2ρLU2RRo

8

R∗(A.12)

Substituting Reynold’s number Re = URRo

νLand Webber number We =

2ρLU2RRo

σ,

P ∗v (T∞)− P ∗∞(t) + P ∗Go

(1

R∗

)3k

= 2

(R∗

t∗

)2

+ 3

(R∗

t∗

)2

+1

Re

8

t∗+

1

We

8

R∗(A.13)

To evaluate each term in the right hand side of equation A.13, let’s assume that the bubble

relative velocity UR is in order of [1] m/s and the bubble radius Ro is in order of [10−3] m

and the time step t is in order of [10−4]s−1. By using the air-water mixture properties, the

normalized time t∗ could be found to be in order of [10−1], Reynold’s number in order of

[103] and Webber number We in order of [103].

The evaluation of the normalized radius term is not easy. The radius will change expo-

nentially with the change of the pressure P∞. However, for the normal conditions, the new

bubble radius, R will range between one tenth to ten times the original radius Ro which

gives R∗ to be in order of [10−1 − 10].

By applying dimensional analysis, Rayleigh-Plesset equation was decreased to equation

131


(3.43),

Pv(T∞)− P∞(t)

ρL+PGoρL

(Ro

R

)3k

= RR +3

2(R)2 +

2σ

ρLR(A.14)

And isothermal expansion of the gas will was assumed. This assumption is set due to

the difficulty of considering the change in gas temperature while ignoring the heat transfer

and the bubbles generation across the tube bundle.

By taking this assumption into account, and by substituting by PGo from (3.42), get

1

ρL

[P∞(0)+

2σ

Ro

](Ro

R

)3

−P∞(t)

ρL+

1

ρL

[Pv(T∞)

(1+

(Ro

R

)3)]= RR+

3

2(R)2+

2σ

ρLR(A.15)

132

modeling fluidelastic instability of two-phase flow in

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