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Modeling and Simulation of Particle Agglomeration, Droplet Coalescence and Particle -Wall Adhesion in Turbulent Multiphase Flows
Masterarbeit
Doctoral Thesis
Daniel Jürgens
approved by the
Department of Mechanical Engineering
Modellierung und Simulation
of the der Partikelagglomeration in turbulenten, dispersen Mehrphasenströmungen Helmut-Schmidt-University
University of the German Federal Armed Forces Hamburg
Institut für Mechanik Professur für Strömungsmechanik for obtaining the academic degree of Doktor Ingenieur (Dr.-Ing.)
Betreuer: Michael Alletto
presented by
Naser Almohammed from Deirazor Fakultät für Maschinenbau - Hamburg 2012
Hamburg, February 2018
Referees Univ.-Prof. Dr.-Ing. habil. Michael Breuer Department of Fluid Mechanics Institute of Mechanics Faculty of Mechanical Engineering Helmut-Schmidt-University University of the German Federal Armed Forces Hamburg Holstenhofweg 85 22043 Hamburg Germany
Prof. Dr. J.G.M. Hans Kuerten Computational Multiphase Flow Group Multiphase and Reactive Flows Department of Mechanical Engineering Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands
The day of submission: September 5, 2017 The day of completion of the oral examination: February 9, 2018
Preface This dissertation is the outcome of my efforts during the last challenging and exciting four years as a research fellow at the Department of Fluid Mechanics (Professur f¨ ur Str¨ omungsmechanik) at the Helmut-Schmidt-University (Helmut-Schmidt-Universit¨at / Universit¨at der Bundeswehr Hamburg). In this time frame I worked under the supervision of Univ.-Prof. Dr.-Ing. habil. Michael Breuer, who gave me a wonderful opportunity to research and to teach, which radically changed the way I look at things. It is a great pleasure for me to express my infinite gratitude to all people, who directly or indirectly supported me to accomplish this dissertation. First and foremost, I wish to express my deep felt gratitude towards my supervisor, Univ.-Prof. Dr.-Ing. habil. Michael Breuer, for giving me the opportunity to pursue my Ph.D. at the Department of Fluid Mechanics. I am highly indebted for your guidance and support from the initial to the final level including insights, advices, easy accessibility, sharing ideas on many issues and reviewing my publications and dissertation. Indeed, this helped me to deeply understand the topics and to explore new ideas and to develop my personal skills as well. My heartily special thanks to Prof. Dr. J.G.M. Hans Kuerten from the Eindhoven University of Technology (The Netherlands) for his interest in my research and in becoming the co-reviewer of my dissertation. Again, thanks for your time and kind comments. On the colleagues-side, I am thankful to all fellows of the department for guidance, encouragement and taking care of me during this time. This made me feel like home and made my residence in Hamburg more enjoyable. My special thanks go to my office-mate, Jens Nikolas Wood, for nice discussions and telling great jokes. I would also like to thank Dr.-Ing. Guillaume De Nayer for keeping my computer running during this time. My heartily thanks to Felix Hoppe, Waldemar Stapel and Ali Khalifa for the proofreading of my dissertation. On the friends-side of this list Douha Alfayyad, Mohammad Yehya Aljoneid, Mohammed Berro and Mahmoud Abdelnaby go first, for being a part of my family. I would like to express my sincere gratitude to my parents and parents-in-law, my aunt “Intisar” and my siblings for their blessings and wishes, without which it would have been difficult to complete this work. Your inspiration has always been a very important part of my life and success! My heartily thanks to my twin brother “Fouad” for the invaluable support with scientific articles. I also wish to express my dearest feelings towards my wife “Rawan” to whom I dedicate this doctoral thesis with deepest gratitude to her for her love and countless sacrifices. She would be more proud of this achievement than any other person. Your invaluable support shall always be held in high regards. Last but not least, the Deutsche Forschungsgemeinschaft is gratefully acknowledged for the financial support of the project (contract number BR 1847/13-1). Hamburg, February 2018
Naser Almohammed
i
Contents
Preface
i
Abstract
xi
Kurzfassung
xiii
1
Introduction
1.1
Multiphase Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2
Background on Particle-Laden Turbulent Flows . . . . . . . . . . . . . . . . . . . . . 4
1.3
Motivation for the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4
Summary on Modeling Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4.1 1.4.2 1.4.3
Particle Agglomeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Droplet Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle-Wall Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 12 13
Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.5.1 1.5.2
14 14
1.5
1
Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Analysis and Interpretation. . . . . . . . . . . . . . . . . . . . . . . . .
1.6
Outline of the Thesis
2
Euler-Lagrange Simulation Framework
17
2.1
Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.1.1 2.1.2 2.1.3
Volume Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stokes Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19 20 20
Continuous Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2.1 2.2.2 2.2.3
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21 23 24 24 24 25 25
Disperse Phase (Single Particle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3.1 2.3.2
26 27 28 28 29
2.2
2.3
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Governing Equations of Large-Eddy Simulation Subgrid-Scale Modeling . . . . . . . . . . . . . Boundary Conditions . . . . . . . . . . . . . . 2.2.3.1 No-Slip Boundary Conditions. . . . . . 2.2.3.2 Inflow Conditions . . . . . . . . . . . . 2.2.3.3 Outflow Conditions . . . . . . . . . . . 2.2.3.4 Periodic Boundary Conditions . . . . . Motion of a Single Particle . . Forces Acting on the Particle 2.3.2.1 Gravity Force . . . . 2.3.2.2 Buoyancy Force . . . 2.3.2.3 Drag Force . . . . .
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iii
Contents . . . . . .
30 32 32 33 33 34
Particle-Wall Collision Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.4.1 2.4.2 2.4.3
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37 38 41 41 41 42 43 44
Interphase Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.5.1 2.5.2 2.5.3 2.5.4
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46 46 47 47 48 49
Particle-Particle Collision Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.6.1
51 54 55 57 58
2.3.3 2.3.4 2.3.5
2.4
2.4.4
2.5
2.6
2.6.2
2.3.2.4 Lift Forces . . . . . . . 2.3.2.5 Added Mass Force . . . 2.3.2.6 Pressure Gradient Force Torque Acting on the Particle . . Boundary Conditions . . . . . . Summary of Governing Equations
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Components of the Impulse Vector . . . . . . . . . . . . . . Particle-Wall Collision Type . . . . . . . . . . . . . . . . . . Kinetics of the Particle after the Impact with a Smooth Wall 2.4.3.1 Sticking Collision . . . . . . . . . . . . . . . . . . . 2.4.3.2 Sliding Collision . . . . . . . . . . . . . . . . . . . . Rough Wall (Sandgrain Roughness Model) . . . . . . . . . . 2.4.4.1 Random Normal Unit Vector . . . . . . . . . . . . . 2.4.4.2 Shadow Effect . . . . . . . . . . . . . . . . . . . . Fluid-Particle Interaction (One-Way Coupling) . . Particle-Fluid Interaction (Two-Way Coupling) . . Particle-Particle Interaction (Four-Way Coupling) Subgrid-Scale Models for the Particles . . . . . . 2.5.4.1 Trivial Model . . . . . . . . . . . . . . . 2.5.4.2 Extended Langevin-Type Model . . . . .
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Particle-Particle Collisions with Friction . . . . . . . . . . 2.6.1.1 Components of the Impulse Vector . . . . . . . . 2.6.1.2 Particle-Particle Collision Type . . . . . . . . . . 2.6.1.3 Kinetics of the Collision Partners after the Impact Particle-Particle Collisions without Friction . . . . . . . .
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3
Modeling of Particle Agglomeration
61
3.1
Agglomeration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.1.1
63 64 67 69 73 79 80 81 82 88
3.1.2
iv
Energy-based Agglomeration Models . . . . . . . . . . . . . . . . . . 3.1.1.1 Difference of the van-der-Waals Energy . . . . . . . . . . . . 3.1.1.2 Agglomeration Model by Hiller (1981) . . . . . . . . . . . . . 3.1.1.3 Agglomeration Model by J¨ urgens (2012) . . . . . . . . . . . . 3.1.1.4 Agglomeration Model by Alletto (2014) . . . . . . . . . . . . 3.1.1.5 Present Extension of the Energy-based Agglomeration Model Momentum-based Agglomeration Models . . . . . . . . . . . . . . . . 3.1.2.1 Agglomeration Model by Weber et al. (2004) . . . . . . . . . 3.1.2.2 Agglomeration Model by Kosinski and Hoffmann (2010) . . . 3.1.2.3 Extended Momentum-based Agglomeration Model . . . . . .
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Contents 3.1.3
3.2
Kinetics of the Exact Agglomerate Structure with Multiple Particles . . . . . . . . 3.2.1 3.2.2 3.2.3
3.3
Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Translational Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
104
Position and Translational Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 104 Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Rotational Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Models for the Structure and Kinetics of the Agglomerate 3.4.1 3.4.2 3.4.3 3.4.4
99 99 99
100
Kinetics of the Two-Particle Agglomerate . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 3.3.2 3.3.3
3.4
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3.1 Advantages and Drawbacks of EAM . . . . . . . . . . . . . . . . . . . . 3.1.3.2 Advantages and Drawbacks of MAM . . . . . . . . . . . . . . . . . . . .
Volume-equivalent Sphere Model (VSM) Inertia-equivalent Sphere Model (ISM) . Closely-Packed Sphere Model (CSM) . . Porous Sphere Model (PSM) . . . . . . 3.4.4.1 Limiting Conditions. . . . . . .
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107 . . . . .
107 110 111 112 114
4
Modeling of Droplet Coalescence
117
4.1
Binary Droplet Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
119
4.1.1 4.1.2
4.2
Fundamental Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 Droplet Collision Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Surface-Tension Dominated Droplets . . . . . . . . . . . . . . . . . . . . . . . . . .
122
4.2.1 4.2.2
122 124 125 125 127 129 135 135 135 139 140 141 142 143 144 145 146 151
4.2.3 4.2.4
4.2.5
Collision Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the Boundaries between the Regimes . . . . . . . . . . . . . . . . 4.2.2.1 Boundary between Slow Coalescence (I) and Bouncing (II) . . . . . . . . 4.2.2.2 Boundary between Bouncing (II) and Fast Coalescence (III) . . . . . . . 4.2.2.3 Boundary between Fast Coalescence (III) and Reflexive Separation (IV) . 4.2.2.4 Boundary between Fast Coalescence (III) and Stretching Separation (V) . Description of the satellite droplets . . . . . . . . . . . . . . . . . . . . . . . . . Coalescence Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.1 Energy-based Model by Howarth (1964) . . . . . . . . . . . . . . . . . . 4.2.4.2 Stochastic Model by O’Rourke (1981) . . . . . . . . . . . . . . . . . . . 4.2.4.3 Empirical Model by Podvysotsky and Shraiber (1984). . . . . . . . . . . 4.2.4.4 Film Drainage Model by Chesters (1991) . . . . . . . . . . . . . . . . . Composite Collision Outcome Model . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5.1 Bouncing (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5.2 Fast Coalescence (III) . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5.3 Reflexive Separation (IV) . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5.4 Stretching Separation (V) . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5.5 Overlap of Reflexive (IV) and Stretching (V)Separation . . . . . . . . .
v
Contents 4.2.5.6 4.2.5.7 4.2.5.8
Kinetics of the Coalesced Droplet . . . . . . . . . . . . . . . . . . . . . 152 Calculation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.3
Viscosity Dominated Droplets
5
Modeling of Particle-Wall Adhesion
155
5.1
Overview of Particle Deposition Models . . . . . . . . . . . . . . . . . . . . . . . .
157
5.1.1
5.1.2
5.2
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Empirical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1.1 Correlation by Friedlander and Johnstone (1957) . . . . . . 5.1.1.2 Correlation by Liu and Agarwal (1974) . . . . . . . . . . . 5.1.1.3 Correlation by McCoy and Hanratty (1977) . . . . . . . . . 5.1.1.4 Correlation by Wood (1981) . . . . . . . . . . . . . . . . . 5.1.1.5 Correlation by Kvasnak et al. (1993) . . . . . . . . . . . . . Models based on Physical Relations . . . . . . . . . . . . . . . . . . 5.1.2.1 Wetted-Wall Model . . . . . . . . . . . . . . . . . . . . . 5.1.2.2 Energy-based Model by Dahneke (1971) . . . . . . . . . . . 5.1.2.3 Momentum-based Model by Kosinski and Hoffmann (2009)
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154
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Present Particle-Wall Adhesion Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 5.2.2 5.2.3
Particle-Wall Collision with Adhesion . . . Particle-Wall Collision Type . . . . . . . . Adhesive Impulse Model . . . . . . . . . . 5.2.3.1 Wall-Normal Direction . . . . . . 5.2.3.2 Wall-Tangential Direction. . . . . 5.2.4 Intervals of the Collision Time . . . . . . 5.2.5 Deposition Condition . . . . . . . . . . . . 5.2.6 Kinetics of the Particle without Deposition 5.2.6.1 Sticking Collision . . . . . . . . . 5.2.6.2 Sliding Collision . . . . . . . . . . 5.2.7 Rough Wall Including Adhesion . . . . . . 5.2.8 Calculation Procedure . . . . . . . . . . . 5.2.9 Advantages of the Model . . . . . . . . . 5.2.10 Application and Validity of the Model . .
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157 157 158 158 158 159 159 160 162 163
164 . . . . . . . . . . . . . .
164 165 167 168 169 170 173 174 175 175 175 175 175 176
6
Computational Methodology
177
6.1
CFD Simulation Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179
6.2
Transformation to Curvilinear Coordinate System . . . . . . . . . . . . . . . . . .
179
6.3
Numerical Methods for the Continuous Phase . . . . . . . . . . . . . . . . . . . .
181
6.3.1
vi
Finite-Volume Method . . . . . . . . . . . . . . . 6.3.1.1 Spatial Discretization of Volume Integrals 6.3.1.2 Spatial Discretization of Surface Integrals 6.3.1.3 Convective and Diffusive Fluxes . . . . .
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182 182 183 184
Contents 6.3.2 6.3.3
6.4
Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Pressure-Velocity Coupling (Predictor-Corrector Scheme) . . . . . . . . . . . . . 185
Numerical Methods for the Disperse Phase . . . . . . . . . . . . . . . . . . . . . . 6.4.1 6.4.2
6.4.3 6.4.4 6.4.5 6.4.6 6.4.7
Particle Injection into the Computational Domain Particle Tracking Scheme. . . . . . . . . . . . . . 6.4.2.1 Translational Motion . . . . . . . . . . . 6.4.2.2 Angular Motion . . . . . . . . . . . . . Fluid Velocity at the Particle Position . . . . . . Deterministic Collision Detection Model . . . . . . 6.4.4.1 Particle-Particle Collision Conditions . . Treatment of Particle Agglomeration . . . . . . . Treatment of Droplet Coalescence . . . . . . . . . Treatment of Particle-Wall Adhesion . . . . . . .
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187 . . . . . . . . . .
187 188 188 191 192 193 194 196 198 198
6.5
Determination of the Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199
7
Results for Particle Agglomeration
201
7.1
Validation of the Agglomeration Models . . . . . . . . . . . . . . . . . . . . . . . .
203
7.1.1
7.1.2 7.1.3
7.2
Test Case Description . . . . . . . . . . . . . 7.1.1.1 Simulation Set-up . . . . . . . . . . . 7.1.1.2 Properties of the Particles . . . . . . Theoretical Model for Particle Agglomeration Results and Discussion . . . . . . . . . . . . 7.1.3.1 Agglomeration Rate . . . . . . . . . 7.1.3.2 Particle Number Concentration . . . 7.1.3.3 Summary of Key Findings . . . . . .
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Particle Agglomeration in Turbulent Channel Flow . . . . . . . . . . . . . . . . . 7.2.1
7.2.2
Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.1 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.2 Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1.3 Determination of Agglomeration Statistics . . . . . . . . . . . . . . Results of the Momentum-based Agglomeration Model . . . . . . . . . . . . 7.2.2.1 Effect of the Structure Model of the Agglomerate . . . . . . . . . . . 7.2.2.2 Effect of the Restitution Coefficient . . . . . . . . . . . . . . . . . . 7.2.2.3 Effect of the Friction Coefficient . . . . . . . . . . . . . . . . . . . . 7.2.2.4 Effect of the Two-Way Coupling / Feedback of Particles on the Flow 7.2.2.5 Effect of the Subgrid-Scale Model for the Particles . . . . . . . . . . 7.2.2.6 Effect of the Lift Force . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.7 Cumulative Effect of the Sub-Models . . . . . . . . . . . . . . . . . 7.2.2.8 Effect of the Diameter of the Primary Particles . . . . . . . . . . . . 7.2.2.9 Effect of the Mass Loading . . . . . . . . . . . . . . . . . . . . . . . 7.2.2.10 Effect of the Wall Roughness . . . . . . . . . . . . . . . . . . . . .
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203 204 204 205 206 206 207 209
210 . . . . . . . . . . . . . . .
210 211 211 214 215 215 223 226 227 233 239 243 244 246 250
vii
Contents 7.3
Comparison of Agglomeration Models . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 7.3.2
Effect of the Agglomeration Model without Sub-Models Effect of Different Simulation Parameters . . . . . . . . 7.3.2.1 Cumulative Effect of the Sub-Models . . . . . 7.3.2.2 Effect of the Diameter of the Primary Particles 7.3.2.3 Effect of the Wall Roughness . . . . . . . . .
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257 . . . . .
257 264 264 268 270
8
Results for Droplet Coalescence
275
8.1
Spray Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
277
8.1.1 8.1.2
8.1.3
8.1.4
8.2
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Injected Fuel Mass . . . . . . . . . . . . . . Initial Droplet Position . . . . . . . . . . . Initial Droplet Velocity . . . . . . . . . . . Initial Droplet Diameter (Primary Break-up) 8.2.4.1 Rosin-Rammler Distribution . . . . 8.2.4.2 Log-Normal Distribution . . . . . . 8.2.4.3 Gamma Distribution . . . . . . . . 8.2.4.4 Exponential Distribution . . . . . .
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Test Case Description . . . . . . . Simulation Set-up . . . . . . . . . 8.3.2.1 Properties of the Droplets Model Verification . . . . . . . . . Summary of Key Findings . . . . .
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Validation of the Composite Collision Outcome Model
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8.4.1
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8.4.2
viii
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Test Case Description . . . . . . . . . . . . . . . . 8.4.1.1 Computational Grid . . . . . . . . . . . . 8.4.1.2 Simulation Set-up . . . . . . . . . . . . . . 8.4.1.3 Properties of the Droplets . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . 8.4.2.1 Spray Tip Penetration . . . . . . . . . . . 8.4.2.2 Collision Regimes . . . . . . . . . . . . . 8.4.2.3 Effect of Different Parameters on the Spray
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278 278 279 279 279 279 280 280
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Inter-Impingement Spray System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 8.3.2
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Model for Droplet Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 8.2.2 8.2.3 8.2.4
8.3
Primary Break-up Length . . . . . . . . . . Spray Angle . . . . . . . . . . . . . . . . . 8.1.2.1 Model by Reitz and Bracco (1979) . 8.1.2.2 Model by Arai et al. (1984) . . . . . Spray Tip Penetration . . . . . . . . . . . . 8.1.3.1 Model by Hiroyasu and Arai (1980) 8.1.3.2 Model by Mirza (1991) . . . . . . . Characteristic Mean Diameters . . . . . . .
283 284 285 286 286 288 289 291
292 . . . . .
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296 . . . . . . . .
296 297 298 299 299 299 302 303
Contents 8.4.3
Summary of Key Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
9
Results for Particle-Wall Adhesion
9.1
Validation of the Particle-Adhesion Model 9.1.1
9.2
9.3
311 . . . . . . . . . . . . . . . . . . . . . .
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Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.1.1.1
Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
9.1.1.2
Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
9.1.1.3
Properties of the Particles . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.1.2
Continuous Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.1.3
Two-Phase Flow Including Particle-Wall Adhesion . . . . . . . . . . . . . . . . . 317 9.1.3.1
Dimensionless Deposition Velocity . . . . . . . . . . . . . . . . . . . . . 317
9.1.3.2
Effect of Inter-Particle Collisions on the Deposition Velocity . . . . . . . 319
Particle-Wall Adhesion in Turbulent Channel Flow
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Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
9.2.2
Results of the New Adhesion Model . . . . . . . . . . . . . . . . . . . . . . . . . 321 9.2.2.1
Effect of Adhesion on Particle-Wall Collisions . . . . . . . . . . . . . . . 321
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Effect of Different Parameters on Particle-Wall Deposition . . . . . . . . 324
Particle-Wall Adhesion in Turbulent Flow Past Inclined Airfoil . . . . . . . . . . . 9.3.1
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Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 9.3.1.1
Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
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Properties of the Particles . . . . . . . . . . . . . . . . . . . . . . . . . 333
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 9.3.2.1
Two-Phase Flow Including Particle-Wall Adhesion. . . . . . . . . . . . . 334
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Deposition pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
10 Conclusions and Outlook
339
10.1 Particle Agglomeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10.2 Droplet Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
344
10.3 Particle-Wall Adhesion
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A
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Appendix
349
A.1 Onset of Plastic Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
349
A.2 Impact Time of Fully Elastic Head-on Collision . . . . . . . . . . . . . . . . . . . .
350
A.3 Effect of the Agglomeration Model on the Cohesive Impulse . . . . . . . . . . . .
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A.4 Mechanical Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . .
353
A.5 Algorithm of the Extended Momentum-based Agglomeration Model . . . . . . . .
354
A.6 Packing Fraction of Monodisperse Particles . . . . . . . . . . . . . . . . . . . . . .
355
A.7 Agglomerate Diameter Predicted Using Different Structure Models . . . . . . . .
355
A.8 Effect of the Structure Model on the Cohesive Impulse . . . . . . . . . . . . . . .
355 ix
Contents A.9 Dimensionless Frequencies Predicted by MAM . . . . . . . . . . . . . . . . . . . .
356
A.10 Comparison of the Dimensionless Frequencies Predicted by EAM and MAM . . .
357
B
359
Appendix
B.1 Experimental Studies on Binary Droplet Collisions . . . . . . . . . . . . . . . . . .
359
B.2 Summary of the Composite Collision Outcome Model . . . . . . . . . . . . . . . .
360
B.3 Approximation of the Inverse of the Error Function . . . . . . . . . . . . . . . . .
362
C
365
Appendix
C.1 Model Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
365
C.1.1 Role of the Adhesive Impulse . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 C.1.2 Critical Approach Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 C.1.3 Effect of the Normal Restitution Coefficient . . . . . . . . . . . . . . . . . . . . . 368
C.2 Algorithm of the Particle-Wall Adhesion Model
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C.3 Mechanical Properties of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . .
370
C.4 Effect of the Sub-Models on the Dimensionless Frequencies . . . . . . . . . . . . .
370
D
371
Appendix
D.1 Algorithm for the Four-Way Coupled Euler-Lagrange Simulation . . . . . . . . . .
371
D.2 Trilinear Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
371
D.3 Algorithm of the Deterministic Collision Detection . . . . . . . . . . . . . . . . . .
373
D.4 Determination of the Accumulated Number of Agglomerates . . . . . . . . . . . .
374
References
398
Curriculum Vitae
399
Publications
401
x
Abstract his dissertation is concerned with the development of modern concepts for the modeling, the simulation and the physical analysis of the dynamic process of (i) particle agglomeration, (ii) droplet coalescence and (iii) particle-wall adhesion in turbulent disperse multiphase flows. For this purpose, different models are developed to allow the simulation of these three phenomena within the framework of a four-way coupled Euler-Lagrange approach for disperse, highly-laden two-phase flows in combination with a deterministic collision detection model. The results of the new approaches developed within the context of a hard-sphere model are validated in fully three-dimensional particle-laden turbulent flows using experiments, empirical correlations and other numerical results. In addition to the development of modern, applicable simulation concepts, the second focus of the present thesis is on the physical analysis and interpretation of the simulation results, which are achieved within the framework of LES. Of course, the new models can also be applied in DNS and RANS. In the following the most important contributions of this study structured according to the phenomenon are summarized. As a first topic, two different models are developed to simulate the agglomeration of rigid, dry and electrostatically neutral particles owing to particle-particle collisions with cohesion. These agglomeration approaches rely either on an energy-based model (EAM) or on a momentum-based model (MAM). In this study the energy-based agglomeration model by Alletto (2014) is further improved concerning the post-collision treatment of the collision partners without agglomeration and the agglomeration conditions. Afterwards, the momentum-based model by Kosinski and Hoffmann (2010) is corrected and further extended by considering the dissipative force during the collision for the prediction of the impact time required for the determination of the cohesive impulse. Thus, the new MAM takes into account the influence of the restitution coefficient on the impact time. In contrast to the original model, the cohesive impulse in the normal and tangential direction is distinguished. As a result, the extended momentum-based agglomeration model considers both the normal and the tangential component of the total impulse much more realistically than the original model leading to a more accurate determination of the agglomeration conditions. Besides the classical volume-equivalent sphere model, three new conceptual models are introduced to allow a more realistic description of the structure of the resulting agglomerate. These are (i) the inertia-equivalent sphere model, (ii) the closely-packed sphere model and (iii) the porous sphere model. The enhanced agglomeration approaches are successfully validated in a shear flow using a theoretical model. It is concluded that MAM is superior to EAM due to the reduced necessity of empirical parameters and the slightly more accurate results. Afterwards, MAM and the structure models are used to study in great detail the agglomeration in a particle-laden turbulent vertical channel flow. Here, the effect of various simulation parameters on the agglomeration process is extensively analyzed. Then, a detailed comparison of the results obtained by both agglomeration models is carried out. The comparative study clearly indicates that both approaches predict similar trends of the physical behavior of the agglomeration process for different particle properties, but their results slightly deviate from each other. The focus of the second work package is on the coalescence of surface-tension dominated liquid
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Abstract droplets in a gaseous environment. In this work package a composite collision outcome model is developed to identify the outcome of a binary collision of such droplets. In the framework of this model four regimes of droplet-droplet collisions (i.e., bouncing, fast coalescence, reflexive and stretching separation) are taken into account. Here, the outcome of a binary collision can be identified based on the collision Weber number and a dimensionless impact parameter. The bounding curves between these regimes can be taken from experiments as a function of the droplet size ratio. The implementation of the developed model is first verified in a simplified test case of two crossing water sprays. The predictions of the composite model perfectly agree with the experimental correlations implemented. Afterwards, the composite model is used to simulate the injection process of a solid-cone non-evaporating diesel spray into a quiescent nitrogen environment and validated based on experimental data. In this set-up the primary atomization (or break-up) of the fuel at the nozzle exit is modeled by means of different correlations for the droplet size distribution. The secondary break-up and the evaporation of the droplets are neglected. As one of the most important results, the spray tip penetration is validated based on experimental data. It is found that the predictions of the composite model using the gamma distribution function for modeling the primary break-up are in excellent agreement with the experimental data and an empirical correlation. Furthermore, a detailed parameter study is carried out for different simulation settings and its effect on the penetration depth. The results clearly show that the neglect of coalescence leads to smaller diameters in the drop size distribution. Thus, the calculated penetration depth noticeably deviates from the experimental data, especially during the final phase of the injection process. Taking the adhesion during a particle-wall collision into account, it may lead under certain conditions to the deposition of the particle on bounding walls. Therefore, the aim of the third work package is to develop a suitable methodology for modeling the adhesion between rigid, dry electrostatically neutral particles and smooth or rough walls. The derivation of this model is based on the corresponding momentum-based agglomeration model developed. The new model reveals a significant advantage compared with the state-of-the-art models due to a more reasonable determination of the adhesive impulse. The post-collision translational and angular velocities of the particle are realistically predicted taking the adhesion into account which leads to a more reasonable deposition condition. To examine the effect of the particle-wall adhesion, the developed model is first evaluated using a simple test case. Then, the adhesion model is validated based on a horizontal turbulent channel flow against existing experimental data and numerical results of an energy-based deposition model as well as a common empirical correlation. The predictions of the present model agree very well with the reference data. Afterwards, the effect of different simulation parameters on the particle-wall adhesion in a vertical particle-laden turbulent channel flow is studied in great detail. Finally, the developed adhesion model is employed to study a practically relevant turbulent flow past an inclined airfoil with different diameters of the primary particles. The obtained results are compared with those of the classical wetted-wall model. This test case demonstrates that the application of such an enhanced adhesion model is urgently required for complex geometries.
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Kurzfassung iese Dissertation befasst sich mit der Entwicklung moderner Methoden zur Modellierung, Simulation und Analyse des dynamischen Prozesses (i) der Partikelagglomeration, (ii) der Tropfenkoaleszenz und (iii) der Partikel-Wand-Adh¨asion in turbulenten dispersen Mehrphasenstr¨omungen. Dazu werden unterschiedliche Modelle entwickelt, welche die Simulation dieser Ph¨anomene im Rahmen eines Vierwege-Euler-Lagrange-Ansatzes f¨ ur disperse, hochbeladene Zweiphasenstr¨omungen in Kombination mit einem deterministischen Kollisionsmodell erm¨oglichen. Die Ergebnisse der im Rahmen eines Hard-Sphere-Models entwickelten Ans¨atze werden in dreidimensionalen partikelbeladenen Str¨omungen anhand von Experimenten und empirischen Korrelationen validiert sowie mit anderen numerischen Ergebnissen verglichen. Neben der Entwicklung von modernen, tragf¨ahigen Simulationskonzepten liegt der zweite Schwerpunkt dieser Dissertation auf der physikalischen Analyse und Interpretation der Simulationsergebnisse, welche im Rahmen einer LES erzielt werden. Selbstverst¨andlich k¨onnen die Modelle auch in DNS oder RANS angewandt werden. Im Folgenden werden die wichtigsten Beitr¨age der vorliegenden Arbeit geordnet nach dem betrachteten Ph¨anomen zusammengefasst. Als erstes werden zwei unterschiedliche Modelle zur Simulation der Agglomeration von trockenen, elektrostatisch neutralen Partikeln infolge von Partikel-Partikel-Kollisionen mit Koh¨asion entwickelt. Diese Agglomerationsmodelle beruhen entweder auf einem energiebasierten Modell (EAM) oder einem impulsbasierten Modell (MAM). Im Rahmen dieser Arbeit wird zun¨achst die Beschreibung der Stoßpartner nach der Kollision und die Agglomerationsbedingungen des energiebasierten Modells von Alletto (2014) verbessert. Als n¨achstes wird das impulsbasierte Modell von Kosinski and Hoffmann (2010) korrigiert und weiterentwickelt, indem f¨ ur die Bestimmung der Kollisionszeit die dissipative Kraft w¨ahrend der Kollision in Betracht gezogen wird. Somit ber¨ ucksichtigt das neue MAM den Einfluss des Restitutionskoeffizienten auf die Stoßzeit. Ferner wird im Gegensatz zum urspr¨ unglichen Modell der Koh¨asionsimpuls in der Normal- und Tangentialrichtung unterschieden. Daraus resultierend ber¨ ucksichtigt das erweiterte Modell sowohl die normale als auch die tangentiale Komponente des Gesamtimpulses wesentlich realistischer als das urspr¨ ungliche Modell, was zu einer genaueren Bestimmung der Agglomerationsbedingungen f¨ uhrt. Dar¨ uber hinaus werden neben dem klassischen Volume-equivalent Sphere Model drei weitere Strukturmodelle eingef¨ uhrt, welche eine realistischere Beschreibung der Struktur des resultierenden Agglomerates erm¨oglichen. Diese sind (i) das Inertia-equivalent Sphere Model, (ii) das Closely-packed Sphere Model und (iii) das Porous Sphere Model. Die verbesserten Agglomerationsans¨atze werden erfolgreich in einer Scherstr¨omung anhand eines theoretischen Modells validiert. Dabei wird festgestellt, dass das MAM zu genaueren Berechnungen als das EAM f¨ uhrt. Ferner wird das MAM zusammen mit den Strukturmodellen in einer vertikalen partikelbeladenen, turbulenten Kanalstr¨omung umfangreich untersucht und analysiert. Hierbei wird eine detaillierte Parameterstudie bei verschiedenen Betriebsbedingungen durchgef¨ uhrt. Danach werden die Simulationsergebnisse beider Modelle in dieser Kanalstr¨omung verglichen. Die Vergleichsstudie zeigte an, dass beide Ans¨atze a¨hnliche Tendenzen bez¨ uglich des physikalischen Verhaltens des Agglomerationsprozesses f¨ ur verschiedene Partikeleigenschaften aufweisen, jedoch weichen die Ergebnisse leicht voneinander ab.
D
xiii
Kurzfassung Der Schwerpunkt des zweiten Arbeitspakets liegt auf der Entwicklung eines zusammengesetzten Modells zur Simulation der Koaleszenz von oberfl¨achenspannungsdominierten, fl¨ ussigen Tropfen in einer gasf¨ormigen Umgebung. In Rahmen dieses Modells werden die aufgrund von bin¨aren Tropfen-Tropfen-Kollisionen resultierenden Regime (Abprall, Koaleszenz und reflexive sowie dehnende Trennung) ber¨ ucksichtigt, welche sich anhand der Weber-Zahl und eines dimensionslosen Stoß-Parameters identifiziert lassen. Die Grenzkurven zwischen diesen Regimen k¨onnen aus Experimenten als Funktion des Tropfengr¨oßenverh¨altnisses entnommen werden. Die Implementierung des entwickelten Ansatzes wird zun¨achst mittels eines vereinfachten Testfalls von zwei sich kreuzenden Wasserstrahlen u ¨berpr¨ uft. Anschließend wird das zusammengesetzte Modell zur Simulation des Einspritzvorganges von nicht-verdampfenden Dieseltropfen aus einer solid-cone D¨ use in eine ruhende Stickstoffumgebung eingesetzt und anhand der experimentellen Messungen validiert. In dieser Anordnung wird die Prim¨arzerst¨aubung des Kraftstoffs mittels verschiedener Ans¨atze modelliert. Ferner werden der Sekund¨arzerfall sowie die Verdampfung der Tropfen vernachl¨assigt. Es wird festgestellt, dass die Ergebnisse des zusammengesetzten Modells in Kombination mit der Gamma-Verteilung f¨ ur die Modellierung der Prim¨arzerst¨aubung in sehr ¨ guter Ubereinstimmung mit den Referenzdaten und einer empirischen Korrelation sind. Eine detaillierte Parameterstudie bei verschiedenen Betriebsbedingungen wird durchgef¨ uhrt. Die Ergebnisse zeigen deutlich, dass die Vernachl¨assigung der Koaleszenz zu kleineren Durchmessern in der Tropfengr¨oßenverteilung f¨ uhrt. Folglich weicht auch die berechnete Eindringtiefe von den experimentellen Messungen deutlich ab, insbesondere w¨ahrend der Endphase des Einspritzvorgangs. Die Ber¨ ucksichtigung der Adh¨asion w¨ahrend einer Partikel-Wand-Kollision kann unter bestimmten Bedingungen zur Ablagerung des Partikels an der Begrenzungswand f¨ uhren. Das Ziel des dritten Arbeitspakets ist es daher, eine geeignete Methodik zur Modellierung der Adh¨asion von trockenen, elektrostatisch neutralen Partikeln an glatten sowie rauen W¨anden zu erarbeiten. Die Herleitung dieses neuen Modells basiert auf dem zugeh¨origen impulsbasierten Agglomerationsmodell. Im Vergleich zu anderen modernen Modellen zeichnet sich das neue Modell durch eine realistischere Bestimmung des Adh¨asionsimpulses aus. Somit werden die translatorischen und rotatorischen Geschwindigkeiten des Partikels nach dem Stoß unter Ber¨ ucksichtigung der Adh¨asion realit¨atsnah berechnet und die zugeh¨orige Ablagerungsbedingung bestimmt. Das neue Adh¨asionsmodell wird mit vereinfachten Testf¨allen erprobt, um den Einfluss der Adh¨asionskraft w¨ahrend einer Partikel-Wand-Kollision darzustellen. Anschließend werden Simulationen mit dem neuen Modell in einer horizontalen partikelbeladenen, turbulenten Kanalstr¨omung durchgef¨ uhrt und umfangreich anhand von Experimenten validiert sowie mit numerischen Ergebnissen eines energiebasierten Ablagerungsmodells und einer empirischen Korrelation verglichen. Die Ergebnisse stimmen mit den Messungen und dem empirischen Zusammenhang sehr gut u ¨ berein. Danach wird der Einfluss verschiedener Simulationsparameter auf die Partikelablagerung in einer vertikalen partikelbeladenen turbulenten Kanalstr¨omung ausf¨ uhrlich untersucht. Abschließend wird das neue Adh¨asionsmodell am Beispiel eines angestellten Tragfl¨ ugels mit verschiedenen Partikeldurchmessern getestet und dessen Ergebnisse mit dem klassischen Wetted-Wall-Model verglichen. Dieser Testfall demonstriert anschaulich, dass die Anwendung eines solchen verbesserten Adh¨asionsmodells bei komplexen Geometrien dringend erforderlich ist. xiv
CHAPTER 1
INTRODUCTION
A circulating fluidized bed (see, e.g., Alobaid and Epple, 2011).
1.1
Multiphase Flow Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2
Background on Particle-Laden Turbulent Flows
1.3
Motivation for the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4
Summary on Modeling Concepts
1.5
1.6
. . . . . . . . . . . . . . . . . . . . . . 4
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4.1
Particle Agglomeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4.2
Droplet Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.4.3
Particle-Wall Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
1.5.1
Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
1.5.2
Physical Analysis and Interpretation. . . . . . . . . . . . . . . . . . . . . . . . .
14
Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
1 Introduction In this chapter, first the theoretical background on the relevant topics of the present thesis are presented in Sections 1.1 and 1.2. Then, the motivation related to the goals and scope of the present research study is given in Section 1.3. The subsequent section briefly summarizes the modeling concepts for the agglomeration of cohesive solid particles, the coalescence of liquid droplets and the particle-wall adhesion in turbulent flows. Based on that the objectives of this dissertation are highlighted in Section 1.5. Finally, this chapter is concluded with the outline of this thesis in Section 1.6. 1.1 Multiphase Flow Regimes
The term multiphase flow is commonly used to refer to a fluid flow involving more than one phase of the four states of matter (i.e., solid, liquid, gas and plasma). Thus, a multiphase flow regime is described based on the morphological arrangement of the phases (or components) simultaneously involved in the flow. Note that only solid, liquid and gas phases are relevant for the present study. Relying on these three phases Figure 1.1 shows a schematic representation of the common multiphase flow regimes encountered in a wide range of industrial applications. These are classified into gas-liquid, liquid-liquid, gas-solid or liquid-solid two-phase flows and three-phase flows. Liquid-solid two-phase flow
Gas-solid two-phase flow
Solid
Liquid Liquid Liquid-liquid two-phase flow
Gas
Three-phase flow
Gas-liquid two-phase flow
Figure 1.1: Common regimes of multiphase flows based on solid, liquid and gas phases. The main focus of the present study is on gas-solid and gas-liquid two-phase flows.
According to Brennen (2005), two general topologies of multiphase flows exits, namely disperse and separated flows. The disperse flows consist of finite (or discrete) elements such as solid particles, liquid droplets or gas bubbles distributed in a continuous phase, and hence the disperse phase is not materially connected. On the contrary, separated flows contain two or more continuous phases of different fluids separated by interfaces. The present thesis focuses on the modeling and the simulation of turbulent disperse gas-solid and gas-liquid two-phase flows including important mechanisms observed in these flows as explained next. In the context of this study these flow regimes are termed particle-laden flows, which contain tiny immiscible solid particles or liquid droplets (referred to as the disperse or particle phase) in a continuous gas (referred to as the continuous or carrier phase). 3
1.2 Background on Particle-Laden Turbulent Flows 1.2 Background on Particle-Laden Turbulent Flows
Turbulent disperse multiphase flows are involved in nearly every branch of industry. Particleladen turbulent flow examples include pneumatic conveying systems of granular flows (e.g., the transport of cement, grains and metal powders), combustion technologies (e.g., pulverized coal firing apparatuses), cyclone separators, air classifiers and other items of extremely important process equipments. Turbulent particle-laden flows are also encountered in pharmaceutical and environmental applications including inhalation sprays, dispersion of fine aerosol particles such as atmospheric pollutants in air and dust collectors. Furthermore, they also occur in various natural phenomena such as dust storms and rain droplets in clouds. Figure 1.2 shows examples of turbulent particle-laden flows with applications to a fluidized bed reactor, an inhalation spray and the transport mechanisms of soil particles by wind. It is well known that the optimization of the design and scale-up of various engineering flow systems requires an accurate knowledge about such flows. For these reasons, substantial efforts have been undertaken worldwide over the last decades in the academic and industrial research communities in order to get a clear conceptual understanding of the complex physics of these flows and the phenomena that they exhibit.
(a)
(b)
Figure 3. Soil particles can move through saltation, creep, and suspension.
(c)
deposition may not occur until the particles have traveled thousands of miles. The wind speed at which particle movement is initiated is called the threshold velocity and is dependent on the state of the soil surface. A soil surface that is rough or protected with non-erodible material will require a stronger wind to initiate particle movement than a bare, smooth surface. This means that for a given field, there is no single threshold velocity but rather a range of velocities depending on the soil surface type — aggregation, roughness, crop status, and moisture. Most of these properties also can change during a storm due to the erosive action. There are three ways soil particles are moved by wind: surface creep, saltation, and suspension. Each has its own characteristics and effects. (See Figure 3.) Under surface creep, the force of the wind causes soil particles to roll along the soil surface until the wind slows, they are stopped by other particles, or they are trapped in a sheltered location, such as a furrow or a vegetated area. Surface creep generally involves particles approximately ½ to 1 millimeter in size, small enough to be moved by the wind but too massive to be lifted off the surface. Surface creep contributes to loss and deposition within a localized area. Another mode of transport is saltation, where under the influence of wind, still smaller particles, 1/10 to ½ millimeter in size, bounce or hop along the surface. As they bounce, they strike other particles, causing them to move. The higher the grains jump, the more energy they derive from the wind. Because of this wind-derived energy, the impact of saltating grains initiates movement of larger grains and smaller dust
particles that can be suspended in the air and carried great distances. Saltating grains collide with clods and cause their breakup, reducing roughness. Saltation also damages young plants, threatening their survival and damaging their fruit, which reduces their marketability. Like particles under surface creep, saltating particles continue to move until the wind slows or they are trapped in sheltered areas. Suspension occurs when particles less than 1/10 of a millimeter — smaller than the diameter of a human hair — are lifted far above the surface and carried great distances. Some of these form dust clouds that have been traced across continents, oceans, and around the world. Suspension can cause visibility problems. A small fraction of suspension particles may cause health problems when inhaled. These particles are known as PM10, which are particulate matter smaller than 10 microns in size. The amount of soil that erodes as surface creep, saltation, or suspension depends on the soil type. Soils that are pure sand will move almost completely by surface creep and saltation. However, if the soil is almost pure clay with clods that break down under saltation, a high percentage of soil loss will be by suspension. On an eroding field, the amount of soil movement will tend to increase with distance downwind due to the impact of saltating grains breaking up clods and initiating other particles to move. This increase in erosion across a field is known as the avalanche effect. If the field is large enough, the creep and saltation flow reaches a maximum that a wind of a particular velocity
Figure 1.2: Examples of particle-laden turbulent flows: (a) a circulating fluidized bed (see, e.g., Alobaid and Epple, 2011), (b) an inhalation spray or a powder inhaler (Wort & Bild/Szczesny, 2016) and (c) the transport mechanisms of soil particles by wind (Presley and Tatarko, 2009).
The exploration of the continuous and the disperse phase of particle-laden turbulent flows is commonly achieved by means of experiments, theoretical approaches or computational techniques employing modern computers. It is important to note that the modeling, the characterization and the advanced understanding of multiphase flows in various applications are highly challenging, since the behavior of flow systems is still unknown in many cases and difficult to predict. To investigate particle-laden turbulent flows, experiments using appropriate measuring techniques 4
1. Introduction have provided fundamental insights into the hydrodynamics of these flows. However, they are costly and the usage of full-scale laboratory models is restricted to only a few applications. Thus, in most cases a lab-scale setup of the flow configuration of interest is used. It is worth noting that there also exist cases for which a lab-scale model is impossible due to a wide variety of reasons (see, e.g., Brennen, 2005). On the other hand, due to the complexity of these flows theoretical analysis relying on mathematical approaches and empirical formulations proposed in the past is often limited to strongly simplified models. As a result of the rapid development of high-performance computers and efficient numerical algorithms, computational fluid dynamics (CFD) has become a popular tool in recent years. It has a powerful potential to predict fluid flows bridging the gaps between experiments and theoretical models. Numerical simulations are on the one hand more flexible and less expensive than experiments, especially when parametric studies are carried out for different geometric scales under various operating conditions. On the other hand, they provide extensive data for the entire flow regardless of the complexity of the flow configuration. However, the application of CFD to turbulent disperse multiphase flows is still limited due to the complex physics of these flows. The principal difficulty originates from the variation of the properties of the disperse and the continuous phases and hence a general form of the governing equations can not be established. In addition, suitable closure assumptions are required for these equations. Last and most important, the complexity of these flows strongly increases when increasing the mass loading of the disperse phase. Thus, appropriate coupling closures are necessary to take the feedback effect of the particle phase on the continuous phase into account, the so-called two-way coupling. This issue is of primary importance in case of a moderate or high volume fraction, since it may significantly change the turbulence intensity. It is known that the particle-fluid interaction (see Figure 1.3) should be taken into account for a particle volume fraction of αq & 10−6 (Elghobashi, 1991, 1994; Sommerfeld et al., 2008; Balachandar and Eaton, 2010). It is worth mentioning that in some studies the mass loading is used instead of the particle volume fraction as a criterion to distinguish between the one-way and the two-way coupled regimes. The reason for this choice is that the momentum coupling terms of the fluid phase include the masses of the particles located in the computational cell (see Section 2.2.1). Particle-Particle Interaction
Fluid-Particle Interaction
Figure 1.3: Transport of spherical particles dispersed in a turbulent gas flow by gas-particle interactions and particle-particle collisions (adapted from Bakker, 2015).
5
1.2 Background on Particle-Laden Turbulent Flows Note that it is commonly accepted that in this regime inter-particle collisions still do not play a dominant role (see Section 2.5). If the volume fraction exceeds values of αq & 10−3 , the particleparticle collisions itself play a significant role and hence they have to be taken into account. In such turbulent flow fields the particles are not only transported by the fluid due to the fluid-particle interaction but also by the inter-particle collisions as schematically depicted in Figure 1.3. This leads to the necessity of four-way coupling within the simulation environment. Taking into account all the facts mentioned above, the basic objective of the modeling of the disperse phase in turbulent flows is to reproduce the complex physics of such flows. For this purpose, reliable computational models are required to obtain realistic predictions of the hydrodynamics of particle-laden turbulent flows. Broadly speaking, the computation of particle-laden turbulent flows is commonly achieved by means of two techniques, namely the Euler-Euler and the Euler-Lagrange approach. In both models the continuous phase is considered as a continuum, and the main difference between both concepts is how to treat the disperse phase. The Euler-Euler approach (or more commonly the two-fluid model) is based on the assumption that both phases are modeled as fully inter-penetrating continua by introducing a corresponding viscosity and corresponding stresses of the disperse phase (Anderson and Jackson, 1967). However, the modeling of the motion of the disperse phase including the kinetic and the collisional transport of particles (i.e., the fluid-particle interaction, the particle-particle and the particle-wall collisions) is highly challenging when treating the particles as a continuum phase. For the modeling of the fluid-particle drag correlations and the rheology of the disperse phase many correlations were proposed relying on various assumptions (see, e.g., Wen and Yu, 1966; Syamlal and O’Brien, 1989; Gidaspow et al., 1992; Lun et al., 1984). In most recent Euler-Euler models the interphase coupling is modeled according to the kinetic theory of granular flows (KTGF), in which non-ideal (or inelastic) particle-particle collisions are considered (Lun et al., 1984; Ding and Gidaspow, 1990; Gidaspow, 1994). The Euler-Euler approach was successfully adopted in many studies to predict the hydrodynamics of turbulent gas-solid flows, for example, in spouted fluidized beds (Huilin et al., 2004; Goldschmidt et al., 2004; Du et al., 2006a,b; Gryczka et al., 2009; Lan et al., 2012). Furthermore, a comparative study on the influence of various simulation parameters including drag models, restitution coefficients of particle-particle collisions, granular temperature approaches and wall boundary conditions in terms of the specularity coefficient on the accuracy of the predictions of the two-fluid model in a gas-solid turbulent spouted fluidized bed was published by the author in Almohammed et al. (2014). In the context of the Euler-Lagrange approach (or discrete particle model) applied in this thesis each particle (solid particle or droplet) is individually tracked through the three-dimensional laminar or turbulent flow and its motion is computed relying on first principles (see, e.g, Sommerfeld et al., 2008; Crowe et al., 1998; Van der Hoef et al., 2008; Balachandar and Eaton, 2010). However, the application of such a discrete particle model requires an accurate modeling of the forces and the angular momentum acting on the particle as well as the particle-particle and particle-wall collisions. It is well known that for Euler-Lagrange models an accurate prediction of the particle transport in a turbulent flow strongly depends on the computed velocity field of the continuous phase encountered 6
1. Introduction along particle trajectories. Although the direct numerical simulation (DNS) is the most accurate approach to compute the structure of turbulence and hence the particle transport, it becomes impractical for high Reynolds numbers. On the other hand, traditional statistical turbulence models relying on the Reynolds-averaged Navier-Stokes (RANS) equations do not accurately describe the continuous phase and hence the particle phase in many cases (Sommerfeld et al., 2008). A good compromise between these computational methods is the large-eddy simulation (LES) technique, since it is not restricted to the range of Reynolds numbers as DNS and guarantees much more accurate predictions of the continuous phase than RANS (Breuer, 2002; Sommerfeld et al., 2008). Therefore, one of the principal objectives of the present study is the application of LES as an advanced eddy-resolving approach. In the present study this technique is used for the computation of well-defined turbulent flows given by the experiments of a vertical channel flow by Benson et al. (2005), a direct numerical simulation of a horizontal channel flow by Marchioli and Soldati (2007) and a turbulent flow past an inclined airfoil by Schmidt and Breuer (2014). In the discrete particle models (DPMs) the collisions are detected by either deterministic (Chen et al., 1998; Breuer and Alletto, 2012) or stochastic (Oesterle and Petitjean, 1993; Sommerfeld, 2001) collision detection models. In the deterministic model the probability of inter-particle collisions is defined by purely kinematic conditions (see Section 6.4.4), while it has to be modeled in the stochastic approach based on certain assumptions. The principal advantage of stochastic models is that they can be more easily applied to practical facilities involving disperse multiphase flows due to their relatively low computational effort, whereas deterministic models may be impractical for certain applications, since they require a high computational effort. However, stochastic detection models may lead to unrealistic particle volume fractions in the simulation of dense gas-solid flows (G¨otz, 2006) and hence to low simulation accuracy. This non-physical prediction is attributed to the stochastic nature of the collision detection. Thus, the application of stochastic methods is restricted to gas-particle two-phase flows at low mass loadings (see, e.g., Alobaid, 2013). Based on recent studies by Alobaid (2015a,b) and Stroh et al. (2016), it was found that deterministic models predict much more accurate results in comparison with experiments than stochastic models. In the framework of the Euler-Lagrange approach the treatment of collisions is commonly carried out using two models: the hard-sphere model (Campbell and Brennen, 1985; Hoomans et al., 1996; Crowe et al., 1998) and the soft-sphere model (Cundall and Strack, 1979). If a stochastic method is applied to detect the collisions, the collisions are treated only by the hard-sphere model. On the contrary, the collisions detected by deterministic models can be treated by means of either the hard- or the soft-sphere model (Alobaid, 2013). In the framework of the hard-sphere model only single binary collisions between the collision partners (i.e., particle-particle or particle-wall) are considered as instantaneous processes. In this model the spherical collision partners are assumed to meet at one point in case of collision. Furthermore, the interaction forces between the colliding particles or the particle and the wall are treated as impulses and hence an impulse exchange occurs during the impact (Hoomans et al., 1996; Deen et al., 2007; Breuer et al., 2012). On the contrary, in the soft-sphere model, commonly called the discrete element method (DEM) first introduced by Cundall and Strack (1979), the particles can overlap each other during particle-particle collisions or penetrate into the wall in case of particle-wall collisions. Using a spring-damper-slider system 7
1.3 Motivation for the Thesis the contact force is modeled depending on the penetration depth. Tsuji et al. (1992, 1993), Link et al. (2005), Alobaid and Epple (2013), Alobaid et al. (2013) and Almohammed et al. (2014) successfully employed the DEM to simulate the hydrodynamic behavior of dense gas-solid flows in simplified fluidized bed reactors. According to Deen et al. (2007), for not too dense gas-particle flows systems1 the hard-sphere approach is considerably more efficient than the soft-sphere model. In other words, the application of the soft-sphere model, although offering detailed information on the hydrodynamics of granular flows, becomes impracticable in some cases. The reason for this fact is that this approach accounts for multiple inter-particle collisions at the same instant in time and hence they are substantially depending on the time step chosen which requires special care (Link, 2006; Deen et al., 2007). Taking into account the above considerations, at high mass loadings the hard-sphere model with deterministic collision detection is more efficient than the DEM and hence it is used in the present study. 1.3 Motivation for the Thesis
The motivation for the present thesis is based on the fact that in addition to the phenomena mentioned in the previous section, the transport of particles in disperse turbulent flows involves further challenging phenomena having a wide range of major implications in many process technologies. Therefore, the main focus of the present study is on the modeling and simulation of the following three important mechanisms: 1. The agglomeration of solid particles due to particle-particle collisions 2. The coalescence of liquid droplets due to droplet-droplet collisions 3. The particle-wall adhesion (deposition) The motivations for considering these phenomena are as follows: First, in particle-laden turbulent flows the agglomeration of microscopic particles with diameters up to 10 µm (Ho and Sommerfeld, 2002) is an important phenomenon due to its significant effect on the dynamics of cohesive powders and hence also on the continuous phase. Previous studies on particle agglomeration showed that the enlargement of tiny particles may have either positive effects or negative consequences on the efficiency of many particle technologies. For example, on the one hand Obermair et al. (2005) concluded that the circulation in gas cyclone separators promotes agglomeration of fine particles improving the separation efficiency. On the other hand, Tomas (2007) concluded that in conveyors this mechanism leads to feeding and dosing problems due to avalanching effects (i.e., sudden onset of rapid motion of powders) and oscillating mass flow rates. Figure 1.4 a shows schematic representation of spray agglomeration of powder particles in a fluidized bed. In this granulation process the particles are first sprayed with a binder solution. Afterwards, they collide and begin to stick together (i.e., agglomeration) building up granulates. 1
pp According to Link (2006), a large number of inter-particle collisions (typically 106 ≤ Ncol ≤ 109 ) is observed when simulating gas-solid two-phase flows in fluidized beds.
8
1. Introduction
Spraying
Binder droplets
Powder
Wetting
Solidifying
Finished Granulate
Liquid bridge
Solid bridge
Raspberry structure
Figure 1.4: Spray agglomeration of powder particles in a fluidized bed (Glatt GmbH, 2016; Sommerfeld, 2017).
Second, the phenomenon of droplet coalescence is found in a wide range of industrial and chemical process technologies involving turbulent multiphase flows (e.g., spray processes such as injection systems in combustion engines exemplarily depicted in Figure 1.5 and spray drying techniques). Additionally, it is also observed in many environmental applications such as the growth of rain droplets in clouds. For example, in injection systems of combustion engines droplet-droplet collisions at the exit of a fuel injector are a dominant process leading to significant modifications of the development of the spray and hence the combustion characteristics (see, e.g., Pan and Suga, 2005). In the present study, the principal focus is on the coalescence of liquid droplets in a gaseous environment. In this case two types of droplets are identified, namely surface-tension and viscosity dominated droplet collisions.
Figure 1.5: Fuel injectors in Gasoline Direct Injection (GDI) engines (Chase, 2011; Run-Rite, 2016).
9
1.3 Motivation for the Thesis Third, the particle-wall adhesion plays a significant role in wall-bounded flows, since it influences the rebound behavior of particles colliding with smooth or rough walls and hence the volume fraction in the near-wall region. Classical examples are the deposition of aerosol particles inhaled with the ambient air into the human respiratory system exemplarily depicted in Figure 1.6. Another example is the pore clogging of filters or filtration membranes due to particles deposited on their surfaces. (b)
(a)
Figure 13: Particle distribution at time steps (a) t = 5 · 104 and(b) (b) t = 8 · 104 for simulation S m . The particles are colored by velocity magnitude and their diameter has been scaled by a factor of 50 for visualization purpose. The outer surface has partially removed to show the internal particle distribution. (For the interpretation of the color in this figure, the reader is referred to the theversion deposition pattern of aerosol particles in a human tracheobronchial tract taken of this article.)
Figure 1.6: Examples for from the simulation results by (a) Lin et al. (2013) and (b) Lintermann and Schr¨oder (2017).
h,% h,% bronchus (Pd,s (DR2) = 6.1% and Pd,s (DR7) = tionally, a high deposition load is found at location D h,% (DB1) = 6.91%. The highest load is fo 5.37%) and at DL3 and DL4 in the left main bronchus with Pd,s h,% about these mechanisms h,% knowledge (e.g., particle agglomeration, droplet at DL3, which covers 10.73% of the total amount of (Pd,s (DL3) = 10.73% and Pd,s (DL4) = 5.41%). Addi-
Acquiring advanced coalescence and particle-wall adhesion) is an issue of primary importance, since it allows further insights into the complex hydrodynamics of particle-laden turbulent flows. It is known that the first 15 prerequisite for agglomeration, coalescence and deposition is a particle-particle, droplet-droplet or particle-wall collusion, respectively. As mentioned before, in the framework of a four-way coupled Euler-Lagrange approach using the LES technique, the primary particles are tracked through the computational domain. In each time step, the collision algorithm checks if particle-particle, droplet-droplet or particle-wall collisions occur based on kinematic conditions. Owing to various physical phenomena relevant for the agglomeration and the particle-wall adhesion, the following considerations are restricted to dry, electrostatically neutral solid particles. Thus, the second prerequisite for the occurrence of an agglomeration or a deposition process is a strong enough attractive force between the particles or the particle and the wall generally known as the molecular van-der-Waals force. In case of the coalescence of droplets the surface tension or viscous forces of the liquids also play a significant role. Last but not least, if the Euler-Lagrange approach based on the hard-sphere model and the deterministic collision detection is used, a successful simulation of particle-laden turbulent flows 10
1. Introduction including these phenomena requires three main factors: (i) the large-eddy simulation as an advanced eddy-resolving technique, which guarantees an appropriate prediction of the continuous phase (Breuer, 1998, 2000, 2002), (ii) an efficient particle tracking scheme (Breuer et al., 2006; Sch¨afer and Breuer, 2002) and (iii) an efficient deterministic particle-particle collision detection model (Breuer et al., 2012; Breuer and Alletto, 2012; Alletto and Breuer, 2012, 2013, 2014; Alletto, 2014). Based on this successfully validated four-way coupled Euler-Lagrange approach implemented in the in-house CFD code LESOCC (Large-Eddy Simulation On Curvilinear Coordinates) used in the present study, it can be stated that the time is ripe for these modern algorithms to incorporate and analyze the three additional phenomena mentioned above. 1.4 Summary on Modeling Concepts
In the last decade, significant efforts have been made in the framework of the Euler-Lagrange approach to develop appropriate models to describe the agglomeration of dry particles, the coalescence of liquid droplets and the adhesion (deposition) of dry particles on the wall. A state-of-the-art review on the most important studies relevant for these phenomena including the contributions of this study is provided in Chapters 3, 4 and 5, respectively. To support the motivation of the present thesis, a brief summary on previous works regarding the modeling strategies for these mechanisms in the framework of a hard-sphere model is given next. 1.4.1 Particle Agglomeration
The agglomeration of cohesive solid particles has been investigated in several studies. Basically, the modeling of this phenomenon is achieved by two techniques at diverse levels of complexity and accuracy, namely the energy-based (EAM) and the momentum-based (MAM) agglomeration model. Thus, if a successful particle-particle collision takes place (first prerequisite satisfied), the probability of agglomeration (second prerequisite) is checked based on simplified agglomeration conditions. In the context of the energy-based agglomeration model early ideas were suggested by L¨offler and Muhr (1972) and Hiller (1981). They assumed a frictionless head-on particle-particle collision and determined a critical relative velocity between the collision partners, below which the colliding particles agglomerate building up a new larger particle (agglomerate). In the framework of four-way coupled Euler-Lagrange RANS simulations using the stochastic collision detection model by Sommerfeld (2001), this simple model was adopted in various studies (Ho and Sommerfeld, 2002, 2003; Ho, 2004; Ho and Sommerfeld, 2005; Blei, 2006; Sommerfeld, 2010; St¨ ubing and Sommerfeld, 2010). Later on, this model was improved by J¨ urgens (2012) and extended by Alletto (2014) to include the friction between the particles at the contact point. Note that to the best of the author’s knowledge, only J¨ urgens (2012) and Alletto (2014) used coupled Euler-Lagrange LES predictions with a deterministic collision model and an energy-based agglomeration model to analyze the particle agglomeration in turbulent flows. As a second approach the momentum-based model is the common choice incorporated into the Euler-Lagrange approach using an impulse-based hard-sphere model, since the motion of the particles is described by solving the momentum equations relying on Newton’s second law. This model was proposed in various studies, for example, the square-well potential model (Weber et al., 11
1.4 Summary on Modeling Concepts 2004; Weber and Hrenya, 2006) and the momentum-based agglomeration model (Kosinski and Hoffmann, 2010, 2011). It is worth noting that the EAMs and MAMs mentioned above were not profoundly validated. In the context of the present study the energy-based model by Alletto (2014) and the momentumbased model by Kosinski and Hoffmann (2010) are first further extended to overcome their shortcomings (see Sections 3.1.1 and 3.1.2). In addition, the newly developed models are validated by the application to classical particle-laden turbulent shear flows as will be explained in Chapter 7. Furthermore, in order to take the effect of the porosity in the structure of the resulting agglomerate into account, the volume-equivalent sphere model (VSM) used in many studies (see, e.g., Ho and Sommerfeld, 2002; Blei, 2006; Kosinski and Hoffmann, 2010, 2011; Balakin et al., 2012) is extended towards a more general description by introducing three new concepts for modeling the agglomerate structure (see Section 3.4): (i) the inertia-equivalent sphere model (ISM), (ii) the closely-packed sphere model (CSM) and (iii) the porous sphere model (PSM). Note that some results of the newly developed agglomeration models presented in this thesis were already published in Breuer and Almohammed (2015, 2018) and Almohammed and Breuer (2016a,b). 1.4.2 Droplet Coalescence
Previous studies on the droplet coalescence due to droplet-droplet collisions based on experiments and theoretical models have a long history. Basically, two types of binary droplet collisions are distinguished (i) surface-tension dominated droplets and (ii) viscosity dominated droplets. To describe the boundaries between the outcomes of a binary surface-tension dominated droplet collision (see Section 4.2.1) for various liquid and gas properties in a map of the relevant parameters, many correlations presented in Section 4.2.2 were proposed (see, e.g., Brazier-Smith et al., 1972; Arkhipov et al., 1983; Ashgriz and Poo, 1990; Jiang et al., 1992; Qian and Law, 1997; Estrade et al., 1999; Gotaas et al., 2007; Krishnan and Loth, 2015). In addition, numerical models were introduced to study this phenomenon. Common droplet collision models for surface-tension dominated droplets are (i) the energy-based model by Howarth (1964), (ii) the stochastic model by O’Rourke (1981) employed in many studies (see, e.g., Koll´ar et al., 2005; Ko and Ryou, 2005; Munnannur and Reitz, 2007; Brenn, 2011; Pawar et al., 2012, 2015; Pawar, 2014), (iii) the empirical model by Podvysotsky and Shraiber (1984) and (iv) the film drainage model by, for example, Chesters (1991). However, none of these models can predict all outcomes of a binary droplet-droplet collision. Hence, to allow the predictions of these regimes, a composite collision outcome model was proposed by Post and Abraham (2002) based on the correlations and the models mentioned before. This composite model was applied by many authors to various spray systems (see, e.g., Ko and Ryou, 2005; Koll´ar et al., 2005; Kim et al., 2009; Brenn, 2011). In the present study this approach is improved to allow an adequate prediction of the four possible regimes due to binary collisions of surface-tension dominated droplets. Furthermore, an injection model for the simulation of spray systems is introduced to specify the mass, the initial diameter, the position and the velocity of injected droplets. The present composite model is verified and validated based on the experiment by Gao et al. (2009) as will be explained in Chapter 8. Note that some results of the improved composite collision outcome model presented in this thesis were published in Almohammed and Breuer (2018). 12
1. Introduction On the other hand, viscosity dominated droplets are observed, for example, in spray drying process in which the liquid droplets undergo a significant change of the viscosity and hence the influence of the surface tension can be neglected (see, e.g., Roos, 2002; Sperling, 2005). As a result of the complex physics of the binary collision of viscosity dominated droplets, only a few studies concerning the modeling of the outcomes of such collisions are available (see, e.g., Blei, 2006; Blei and Sommerfeld, 2007). The model by Blei (2006) depends on the fraction of the kinetic energy dissipated during the collision of two viscous droplets and the penetration depth. In this model the collision partners may coalesce forming a new spherical droplet, agglomerate or separate maintaining their pre-collision sizes. However, in the present study the focus is solely on the surface-tension dominated droplets due to the time limitation of the project. 1.4.3 Particle-Wall Adhesion
The deposition of aerosol particles on smooth and rough walls has been the focus of various experimental (see, e.g., Liu and Agarwal, 1974; McCoy and Hanratty, 1977) and theoretical (see, e.g., Wood, 1981) studies. To check if a particle deposits on the wall, common models are (i) the wetted-wall model used in many studies (see, e.g., McLaughlin, 1989; Uijttewaal and Oliemans, 1996; Wang and Squires, 1996; Breuer et al., 2006; Winkler et al., 2006; Koullapis et al., 2016), (ii) the energy-based model by Dahneke (1971) adopted by Li and Ahmadi (1993) and Kvasnak et al. (1993) and (iii) the momentum-based deposition model by Kosinski and Hoffmann (2009). However, these models still have some modeling drawbacks. For example, the simple wetted-wall model is only physically reasonable for the deposition of particles on wetted walls, since the particle-wall collisions are neglected and hence the particle does not bounce back for any value of the restitution coefficient. Based on strongly simplified assumptions the energy-based and momentum-based models predict a critical approach velocity, below which deposition occurs. To avoid these problems (i.e, the neglect of particle rebound and the assumption of critical approach velocity), a new momentum-based particle-wall adhesion model is introduced in the present thesis (see Section 5.2). The newly developed model is validated based on the experiments by Papavergos and Hedley (1984) and Kvasnak et al. (1993) leading to more accurate results in comparison with the wetted-wall and energy-based model. Furthermore, the present model is applied to different particle-laden turbulent flows as will be explained in Chapter 9. Note that parts of the results of the newly developed model were already published in Almohammed and Breuer (2016c) and Breuer and Almohammed (2016). 1.5 Objectives of the Thesis
The present thesis is concerned with the modeling, the simulation and the analysis of the agglomeration of solid particles, the coalescence of liquid droplets and the adhesion (deposition) of solid particles on smooth and rough walls in three-dimensional turbulent disperse flows. It is very important to note that the break-up of particles or droplets induced by any of these three phenomena, among others, is completely excluded in the context of this study. First, the corresponding models for each mechanism are chosen and extended as will be explained in Chapters 3, 4 and 5, respectively. Then, they are implemented in the in-house CFD code 13
1.5 Objectives of the Thesis LESOCC (Breuer, 1998, 2000, 2002; Breuer et al., 2012; Breuer and Alletto, 2012; Alletto and Breuer, 2012, 2013, 2014). Finally, the entire algorithm is validated by applications to different particle-laden turbulent flow configurations. Therefore, the objectives of this study are as follows: 1.5.1 Mathematical Modeling
In the context of the present thesis the mathematical modeling includes the following:
Ê Modeling of the agglomeration of dry, electrostatically neutral particles: This includes two main contributions:
• The energy-based agglomeration model by Alletto (2014) is improved (see Section 3.1.1) • The momentum-based agglomeration model by Kosinski and Hoffmann (2010) is extended (see Section 3.1.2)
Ë Modeling of the structure of the resulting agglomerate: Besides the classical volumeequivalent sphere model, three new concepts are introduced to improve the calculation of the angular velocity of the agglomerate (see Section 3.4): • The inertia-equivalent sphere model • The closely-packed sphere model • The porous sphere model
Ì Modeling of the droplet coalescence: Here, the focus is on the collision outcomes of the surface-tension dominated droplets. The main two contributions are:
• A composite collision outcome model proposed by Post and Abraham (2002) for surfacetension dominated droplets based on the combination of theoretical correlations and numerical models is further improved (see Section 4.2.5). • An injection model is introduced to specify the position, the velocity and the initial diameter of the injected droplets at the break-up length (see Section 8.2).
Í Modeling of the adhesion of dry, electrostatically neutral particles on walls: In this context a new momentum-based particle-wall adhesion model is proposed for smooth and rough walls (see Section 5.2).
It is worth noting that the newly developed models for particle agglomeration, droplet coalescence and particle-wall adhesion are tested using the large-eddy simulation technique, but they can also be applied in the context of other simulation approaches including hybrid LES–RANS, DNS and RANS. 1.5.2 Physical Analysis and Interpretation
The purpose of the analysis and the interpretation of the results is to prove the accuracy of the developed models in comparison with existing experiments or numerical predictions. Furthermore, the effect of various physical parameters on the different phenomena is investigated. 14
1. Introduction
Ê Application and validation of the developed models in turbulent disperse two-phase flows:
The new agglomeration, coalescence and particle-wall adhesion models are applied to fully three-dimensional particle-laden turbulent flows. The following test cases are investigated: • Particle agglomeration in a laminar shear flow and a vertical turbulent channel flows (see Chapter 7). • Droplet coalescence in a inter-impingement spray system and a single non-evaporating diesel spray (see Chapter 8). • Particle-wall adhesion in a horizontal and vertical turbulent channel flow and a turbulent flow past an inclined airfoil (see Chapter 9).
Ë Analysis of the particle agglomeration: In this study the dynamic agglomeration process
of rigid, dry and electrostatically neutral microscopic particles in turbulent shear flows is analyzed using the energy-based and momentum-based agglomeration models. First, both models are validated in a simple shear flow based on an existing theoretical model. Afterwards, a detailed comparative study is carried out in a particle-laden turbulent channel flow to evaluate the performance of both agglomeration models including their advantages and drawbacks.
Ì Characterization of the agglomerate structure: The predictions of the structure models of
the resulting agglomerate are compared against each other and the most appropriate model is used for further investigations.
Í Analysis of the droplet coalescence: The developed composite collision outcome model for
surface-tension dominated droplets is verified using a simple inter-impingement system. Afterwards, the composite model is validated in a single non-evaporation diesel spray based on experimental data.
Î Analysis of the particle-wall adhesion (deposition): The newly developed model is validated
using experiments in a horizontal particle-laden turbulent channel flow and compared to other models based on numerical results and empirical correlations. Then, it is applied to flows in simple and complex geometries including turbulent flows in a vertical channel and past an inclined airfoil.
1.6 Outline of the Thesis
The thesis is organized in the following manner: The next chapter briefly provides some details on the Euler-Lagrange simulation framework applied in the present study. Then, the modeling of the agglomeration of dry, electrostatically neutral particles using the energy-based and the momentum-based agglomeration model is described in Chapter 3. Accordingly, Chapter 4 presents the modeling of the coalescence of surface-tension dominated droplets. The modeling of the adhesion of dry, electrostatically neutral particles on smooth and rough walls based on a momentumbased adhesion (deposition) model is the topic of Chapter 5. Some details of the numerical methods 15
1.6 Outline of the Thesis adopted in this thesis for the computation of the continuous and the disperse phase including agglomeration, coalescence and particle-wall adhesion are explained in Chapter 6. The seventh chapter contains the computational results of the particle agglomeration with application to various test cases. Similarly, the subsequent chapter presents the computational results of the coalescence of surface-tension dominated droplets based on different test cases. Finally, the computational results of the particle-wall adhesion with application to a particle-laden turbulent channel flow and a practically relevant turbulent flow past an inclined airfoil are described in Chapter 9. The last chapter presents conclusions of the entire study. An outlook and recommendations for future work will be also given.
16
CHAPTER 2
EULER–LAGRANGE SIMULATION FRAMEWORK
one-way coupling
two-way coupling
Fluid
Fluid
four-way coupling Fluid
Particles
Particles 1E-9
1E-8
Particles
1E-7
1E-6
1E-5
Dilute systems
1E-4
Particles 1E-3
0.01
0.1
1.0
αq
Dense system
Fluid-particle interphase coupling (adopted from Elghobashi, 1991).
2.1
2.2
2.3
Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.1.1
Volume Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.1.2
Mass Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
2.1.3
Stokes Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
Continuous Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
2.2.1
Governing Equations of Large-Eddy Simulation
. . . . . . . . . . . . . . . . . .
21
2.2.2
Subgrid-Scale Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
2.2.3
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2.3.1
No-Slip Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . .
24
2.2.3.2
Inflow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
2.2.3.3
Outflow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2.2.3.4
Periodic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . .
25
Disperse Phase (Single Particle) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3.1
Motion of a Single Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3.2
Forces Acting on the Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2.3.2.1
Gravity Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.2.2
Buoyancy Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.2.3
Drag Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
2.3.2.4
Lift Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.3.2.4.1 Saffman Force . . . . . . . . . . . . . . . . . . . . . . . . . .
30
2.4
2.3.2.4.2 Magnus Force . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.3.2.5
Added Mass Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.3.2.6
Pressure Gradient Force . . . . . . . . . . . . . . . . . . . . . . . . . .
32
2.3.3
Torque Acting on the Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.3.4
Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.3.5
Summary of Governing Equations of the Particle Motion
. . . . . . . . . . . . .
34
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
2.4.1
Components of the Impulse Vector . . . . . . . . . . . . . . . . . . . . . . . . .
37
2.4.2
Particle-Wall Collision Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.4.3
Kinetics of the Particle after the Impact with a Smooth Wall . . . . . . . . . . .
41
2.4.3.1
Sticking Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.4.3.2
Sliding Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
Rough Wall (Sandgrain Roughness Model) . . . . . . . . . . . . . . . . . . . . .
42
2.4.4.1
Random Normal Unit Vector . . . . . . . . . . . . . . . . . . . . . . . .
43
2.4.4.2
Shadow Effect
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.5.1
Fluid-Particle Interaction (One-Way Coupling) . . . . . . . . . . . . . . . . . . .
46
2.5.2
Particle-Fluid Interaction (Two-Way Coupling) . . . . . . . . . . . . . . . . . . .
46
2.5.3
Particle-Particle Interaction (Four-Way Coupling) . . . . . . . . . . . . . . . . .
47
2.5.4
Subgrid-Scale Models for the Particles . . . . . . . . . . . . . . . . . . . . . . .
47
2.5.4.1
Trivial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
2.5.4.2
Extended Langevin-Type Model . . . . . . . . . . . . . . . . . . . . . .
49
Particle-Particle Collision Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
2.6.1
Particle-Particle Collisions with Friction . . . . . . . . . . . . . . . . . . . . . .
51
2.6.1.1
Components of the Impulse Vector
. . . . . . . . . . . . . . . . . . . .
54
2.6.1.2
Particle-Particle Collision Type . . . . . . . . . . . . . . . . . . . . . .
55
2.6.1.3
Kinetics of the Collision Partners after the Impact . . . . . . . . . . . .
57
2.6.1.3.1 Sticking Collision. . . . . . . . . . . . . . . . . . . . . . . . .
57
2.6.1.3.2 Sliding Collision . . . . . . . . . . . . . . . . . . . . . . . . .
58
Particle-Wall Collision Model
2.4.4
2.5
2.6
Interphase Coupling
2.6.2
Particle-Particle Collisions without Friction
. . . . . . . . . . . . . . . . . . . .
58
2 Euler-Lagrange Simulation Framework This chapter is intended to address the basic principles of the Euler-Lagrange simulation framework used in the present thesis with applications to particle-laden turbulent flows. As mentioned in Section 1.1, the emphasis of this study is only on disperse two-phase flow systems consisting of a continuous phase and a discrete phase (solid particles or liquid droplets). First, fundamental definitions relevant to disperse two-phase flows are given in Section 2.1. Then, the modeling of the fluid (Eulerian approach) and the disperse phase (Lagrangian approach) is presented in Sections 2.2 and 2.3, respectively. 2.1 Basic Definitions
For disperse two-phase flow systems it is well known that the interaction level between the two phases (see Section 2.5) is commonly quantified by two dimensionless parameters (see, e.g., Elghobashi, 1991). These are termed as the volume fraction and the mass loading explained in Sections 2.1.1 and 2.1.2, respectively. In addition, the Stokes number (see Section 2.1.3) is a very important dimensionless parameter when classifying fluid-particle flows. 2.1.1 Volume Fraction
In disperse two-phase flows the volume fraction refers to the volume occupied by each phase. Thus, the volume fraction of the phase q (here q = f for the fluid, q = p for solid particles and q = d for droplets) denoted by αq is given by the following equation: αq =
Vq,tot Vq,tot = , Vtot Vf,tot + Vq,tot
(2.1)
where Vf,tot and Vq,tot stand for the total volume of the fluid and the disperse phase, respectively. Thus, the total volume of the domain is Vtot = Vf,tot + Vq,tot . In this thesis the particles of the disperse phase are assumed to be ideally spherical and hence the total volume of the disperse phase Vq,tot appearing in Eq. (2.1) is given by: Vq,tot =
N π Xq 3 d , 6 i=1 q,i
(2.2)
where Nq and dq,i stand for the total number of the particles and the diameter of the ith particle. Note that for disperse multiphase flows the total volume fraction of all nq phases has to be equal to unity for any volume considered, such that: nq X
αq = 1 .
(2.3)
q=1
Thus, for fluid-particle flows this condition is reduced to αf + αq = 1, where αf is the volume fraction of the fluid phase. It is worth noting that the increase of the volume fraction implies a decrease of the inter-particle spacing (see, e.g., Sommerfeld et al., 2008), which is the mean distance between the centers of the particles regularly arranged in the flow field, leading to a higher probability of particle-particle collisions (see Section 2.5.3). 19
2.1 Basic Definitions 2.1.2 Mass Loading
The second parameter often used to characterize disperse two-phase flows is the mass loading ηq . In such flow systems it is defined as the ratio of the mass flux (or mass flow rate) of the disperse phase m ˙ q to that of the fluid phase m ˙ f and thus reads (see, e.g., Birzer et al., 2004; Sommerfeld et al., 2008): m ˙q 1 mq,tot R = , m ˙f ∆T A uf ρf dA
ηq =
(2.4)
where uf and ρf stand for the bulk velocity and density of the continuous phase passing through the surface A, respectively. mq,tot denotes the total mass of the disperse phase moving through the surface A during the time interval ∆T (i.e., the mass flux of the particle m ˙ q ). Based on the assumption of spherical particles the total mass of the disperse phase is computed as follows: mq,tot
N π Xq = ρq,i d3q,i , 6 i=1
(2.5)
where ρq,i is the density of the ith particle. According to Sommerfeld et al. (2008) the mass loading of the phase q can be expressed in the following form: ηq =
αq ρ q uq αq ρ q uq = . αf ρ f uf (1 − αq ) ρf uf
(2.6)
This relation clearly implies that the mass flux of the two phases is a vectorial quantity and hence it can be determined in each direction of the velocity. In the present study it is assumed that primary particles are injected into the computational domain with the same velocity as the fluid phase in the streamwise direction (i.e., up = uf ) and thus Eq. (2.6) is reduced to: ηq =
ρq ρf
!
αq . (1 − αq )
(2.7)
In disperse two-phase flows high mass loadings lead to a significant feedback effect of the particles on the continuous phase (see Section 2.5.2). 2.1.3 Stokes Number
Generally, in disperse two-phase flows the dimensionless Stokes number St is used to describe the response of a particle (solid particle or droplet) suspended in a fluid flow to the changes in this flow field (Breuer et al., 2006). The Stokes number is defined as the ratio of the relaxation time of the particle τq to the characteristic time scale of the continuous phase τf , such that: St =
τq . τf
(2.8)
Here, the particle relaxation time τq is given by: τq = 20
ρq d2q , 18 µf
(2.9)
2. Euler-Lagrange Simulation Framework where µf is the dynamic viscosity of the fluid. According to Eq. (2.8) the particle relaxation time scales with the square of the particle size (i.e., τq ∝ d2q ). Thus, the Stokes number given by Eq. (2.8) significantly depends on the particle size. Generally, three limiting cases are distinguished in the literature as depicted in Figure 2.1: • St 1: This is observed for very small particle sizes such as tiny particles, which possess a very short relaxation time in comparison with the fluid time scale. Thus, the dynamic behavior of such particles is substantially affected by the small turbulent eddies (see Figure 2.1), since they follow the flow patterns of the continuous phase more or less without delay (Breuer et al., 2006). • St ≈ 1: If the particle relaxation time is of the same order as the characteristic time scale of the eddy, the particle dynamics is mostly affected by the turbulent eddies (see Figure 2.1). • St 1: Due to the effect of inertia large particles require a certain time to adjust themselves to the vortex structure of the flow field. Thus, a high Stokes number (i.e., τq τf ) means that the particle does not respond to the eddies and hence the trajectories of such particles noticeably deviate from the streamlines of the fluid flow as depicted in Figure 2.1. Intermediate particle
St ~ 1 Large turbulent eddy
uf!
Particle Small turbulent eddy
Large particle
St >> 1 Small particle
St 1 µm) such that the Brownian motion of the particles is neglected. It is worth mentioning that the Basset history force is not taken into account in this study due to the high CPU-time requirements and the huge storage capacity, since the integral over the history of the particle during the simulation has to be computed (Kuerten, 2016). For a general formulation of the particle motion, additional forces such as the electrostatic force, the collision force as well as the cohesive and the adhesive force have to be included. Assuming electrostatically neutral particles as done in this study, the electrostatic force is neglected. For tiny particles, the cohesive force is relevant for particle agglomerations due to particle-particle collisions and the adhesive force is important for particle-wall adhesion due to particle-wall collisions. In the framework of the hard-sphere model these forces are taken into account during the impact as will be explained in Chapters 3 and 5, respectively. It is well known (see, e.g., Balachandar and Eaton, 2010) that the principal assumption of the point-particle approach is that the particle diameter dq is much smaller than the Kolomgorov scale ηk (i.e., dq ηk ). This implies only a moderate variation of the fluid velocity at the position of the particle within a range of O(dq ). Thus, it can be assumed that the fluid surrounding the particle remains undisturbed at a distance much smaller than the grid spacing allowing the determination of the fluid forces acting on the particle based on analytical or empirical correlations. The fluid forces on the particle are derived by analyzing the motion of a small rigid sphere in a non-uniform flow (Gatignol, 1983; Maxey and Riley, 1983). Assuming that all forces applied on the moving particle are additive, the resulting fluid force Fq 27
2.3 Disperse Phase (Single Particle) acting on a particle considered in the present study for the one-way coupled regime is given by: Fq =
X
Fq,k = FD + FG + FB + FL + FAM + FP G ,
(2.26)
k
where FD stands for the drag force, FG is the gravitational force, FB is the buoyancy force, FL is the lift force, FAM is the added mass force and FP G is the pressure gradient force. These forces are explained in the following subsections, respectively. Figure 2.2 shows a schematic representation of some forces acting on the particle. g
uf
FB
uf!
high
uf FL
Saf
FL
uq
FD
uq
Mag
uq
uf!
FG low
uf
ωq
Figure 2.2: A schematic representation of the forces acting on a single particle: (left) gravity, buoyancy and drag force, (middle) lift force due to velocity shear (Saffman force) and (right) lift force due to particle rotation (Magnus force).
2.3.2.1 Gravity Force
The gravity force is a volume force acting on a spherical particle (see Figure 2.2) and is given by: FG = mq g =
π ρq d3q g , 6
(2.27)
where g is the gravitational acceleration vector. It is worth noting that the particle motion is significantly affected by this force in particle-fluid flow systems if the gravitational acceleration is not directed in the streamwise direction. In wall-bounded flows, for example, the gravity force noticeably increases the sedimentation velocity of particles towards the wall leading to a higher number of particle-wall collisions and hence deposition processes (see Section 9.1). 2.3.2.2 Buoyancy Force
According to Archimedes’ principle, if a particle is immersed in a carrier fluid, the pressure at the bottom of the particle is higher than at its top part. This pressure difference leads to an upward force exerted on the particle and referred to as buoyancy force depicted in Figure 2.2. For spherical particles the buoyancy force is given by: FB = −mq
ρf ρq
!
π g = − ρf d3q g , 6
(2.28)
where the minus sign means that this force acts parallel to the gravitational acceleration g but in the opposite direction (see Figure 2.2). Note that in the context of this study the density ratio for 28
2. Euler-Lagrange Simulation Framework solid particles is in most cases ρq /ρf 1 and hence the buoyancy force could be neglected in the particle-laden turbulent flows investigated (Sommerfeld et al., 2008). For the droplet case the situation slightly changes. Nevertheless, this force is taken into account in all test cases due to the very low computational effort required. 2.3.2.3 Drag Force
In general, the drag force FD (or resistance force) acts in the direction opposite to the relative motion between the particle and the fluid. In case of a Stokes flow around a spherical particle the drag force (see Figure 2.2) used in this study reads: FD =
ρf π 2 d CD (uf − uq ) |uf − uq | , 2 4 q
(2.29)
where uq and uf are the particle velocity and the fluid velocity at the particle position. Based on Hussainov et al. (1996), among others, the difference between the velocities of the phases is usually termed as the slip velocity uslip = uf − uq . For example, Zhao et al. (2012) found that the slip velocity considerably depends on the particle size and the fluid velocity. The drag coefficient CD appearing in Eq. (2.29) is given by: CD =
h i 24 α with α = 1 + 0.15 Req0.687 , Req
(2.30)
where Req denotes the particle Reynolds number and is defined as: Req =
|uf − uq | dq . νf
(2.31)
Note that the correction factor α on the right-hand side of Eq. (2.30) was proposed by Schiller and Naumann (1933) in order to extend the validity of CD towards higher particle Reynolds numbers Req ≤ 800. In the present study this limiting value of Req is perfectly sufficient for the investigation of particle agglomeration and particle-wall adhesion presented in Chapters 7 and 9, respectively. For example, in case of a particle-laden channel flow (see Sections 7.2 and 9.2) with a Reynolds number Re = UB δ/νf = 11, 900, where UB is the bulk velocity and δ is the channel half-width, the particle Reynolds numbers predicted during the simulation are Req ≤ 15. However, for Req higher than the limiting value of 1000 the drag coefficient is set to a constant value CD = 0.44 (see, e.g., Sommerfeld et al., 2008). In the present thesis this situation is observed, for example, during the investigation of the droplet coalescence in an inter-impingement spray system (see Section 8.3) and a non-evaporating diesel spray (see Section 8.4). Note that if only the drag force is included in the equation of particle motion, reasonable agreement with experiments was found by Alletto and Breuer (2012) and Breuer and Alletto (2012) for a cold flow in a combustion chamber and turbulent channel flows laden with particles of moderate size, respectively. However, for tiny particles the remaining forces in Eq. (2.26) have to be taken into account, since they noticeably affect the agglomeration process, the droplet coalescence and the particle-wall adhesion as will be shown in Chapters 7, 8 and 9, respectively. 29
2.3 Disperse Phase (Single Particle) 2.3.2.4 Lift Forces
In the present study the lift force FL included in Eq. (2.26) has two contributions: (i) FLSaf the lift ag force due to the velocity shear (Saffman force) and (ii) FM the lift force due to particle rotation L (Magnus force), such that: FL = FLSaf + FLM ag .
(2.32)
2.3.2.4.1 Saffman Force
This lift force acts on the particle due to the velocity gradient of the continuous flow field perpendicular to the relative motion between the particle and the fluid. As depicted in Figure 2.2, if the particle moves slower that the fluid (i.e., uq < uf ) the lift due to velocity shear (Saffman force) drives the particle in the direction of a positive velocity gradient and vice versa. In the present thesis two formulations for this lift force acting on the particle due to the velocity shear are used depending on the value of the particle Reynolds number Req and the velocity gradient of the fluid (shear rate). McLaughlin (1991) extended the theoretical formulation of the lift force by Saffman (1964) and stated that the Saffman force decreases when increasing the particle Reynolds number. McLaughlin (1991) assumed that Req and ReG = |G| d2q /νf are smaller than unity, where G = ∇uf stands for the fluid velocity gradient tensor. The derivation of this modified relation of the lift force due to the velocity shear denoted FLM cL is based on the Stokes flow regime. In addition, the velocity disturbance of the particle is assumed to decay at infinity. According to McLaughlin (1991) this relation has the following form: FLM cL
"
|G| 9 µf d2p (uq − uf ) sign(G) = 4π νf
#1/2
J u () ,
(2.33)
where the dimensionless parameter is given in McLaughlin (1991) by: √ ReG [|G| νf ]1/2 = = sign(G) , Req (uq − uf )
(2.34)
where for || 1 and || 1 the dimensionless function J u () is given by J u () = −32π 2 5 ln(1/2 ) and J u () = 2.255 − 0.6463/2 , respectively. If ≈ O(1), the values of the function J u () are tabulated. The formulation by McLaughlin (1991) was used in previous studies using LESOCC (see, e.g., Breuer et al., 2006, 2007; Alletto, 2014) for all particle Reynolds numbers. However, the above relation (2.33) is derived assuming the Stokes flow regime and hence it is in principle only valid for particle Reynolds numbers less than unity, i.e., Req < 1 (McLaughlin, 1991). Thus, in the case of higher particle Reynolds numbers another model for the lift force due to linear shear has to be used. As mentioned before, for the particle-laden channel flow with a Reynolds number Re = UB δ/νf = 11, 900, which is the main test case for particle agglomeration and particle-wall adhesion investigated in this thesis (see Sections 7.2 and 9.2), the predicted particle Reynolds numbers are in the range Req ≤ 15. For this reason the following relation originally 30
2. Euler-Lagrange Simulation Framework proposed by Mei (1992) and modified by Crowe et al. (1988) for three-dimensional flows is applied (see, e.g., Lain et al., 2002; Sommerfeld et al., 2008; Almohammed et al., 2014): ρf π 3 (2.35) FLM ei = d CLS [(uf − uq ) × ωf ] , 2 4 q where ωf denoting the angular velocity (or rotation) vector of the continuous fluid phase is given by: ωf = ∇ × uf .
(2.36)
The empirical factor CLS appearing in Eq. (2.35) proposed by Mei (1992) based on the calculations by Dandy and Dwyer (1990) has the following form:
CLS
q q β exp (−0.1 Re ) + 0.3314 βLS 1 − 0.3314 LS q
4.1126 × = √ Res
q 0.0524 β
LS
Req
for Req ≤ 40 ,
(2.37)
for Req > 40 ,
where the coefficient βLS is given by βLS = 0.5 Res /Req and the shear Reynolds number Res is defined as: Res =
|ωf | d2p . νf
(2.38)
According to Mei (1992) the relation (2.35) is valid for particle Reynolds numbers 0.1 ≤ Req ≤ 100 and 0.005 ≤ βLS ≤ 0.4. Note that based on Eq. (2.37) the lift force decreases with increasing the particle Reynolds number in the range Req ≤ 40, whereas a constant coefficient of CLS = 0.152 is predicted for Req > 40 (Mei, 1992; Lataste et al., 2000). In the present study the Saffman lift force is taken into account, since, for example, in a turbulent particle-laden channel flow this lift force has a significant effect on the agglomeration process (see Section 7.2) and the particle-wall adhesion (see Section 9.2), especially in the near-wall region, where the highest velocity gradient exists. As will be explained in Section 7.2, it is found that if the relation by Mei (1992) is applied to this turbulent particle-laden channel flow (0 < Req ≤ 15) instead of that by McLaughlin (1991), the agglomeration rate is only slightly reduced during the simulation when including the agglomeration model (0 ≤ ∆T ∗ ≤ 200). For this reason the formulation of the lift force by McLaughlin (1991) is used in these simulations (i.e, FLSaf = FLM cL ) as done by Breuer and Almohammed (2015), Almohammed and Breuer (2016b,c) and Breuer and Almohammed (2018). 2.3.2.4.2 Magnus Force
If the particle rotation rate differs from that of the surrounding fluid, an additional force is exerted on the particle. This situation is observed, for example, when particles colliding with the bounding walls acquire high rotation rates which is often the case in the present study. Generally, this force is called the Magnus force (see Figure 2.2) and it is expressed based on Crowe et al. (1998) as follows: ρf π 2 Ωrel × (uf − uq ) ag FM = dq CLR |uf − uq | , (2.39) L 2 4 |Ωrel | 31
2.3 Disperse Phase (Single Particle) where Ωrel stands for the relative rotation of the particle and is given by: Ωrel =
1 ∇ × uf − ωq . 2
(2.40)
In the above relation the first term on the right-hand side refers to the angular velocity (or rotation) vector of the continuous fluid phase and ωq denotes the angular velocity of the particle. The lift coefficient CLR in Eq. (2.39) is determined by means of empirical correlations. For particle Reynolds numbers Req < 140 this coefficient is given by Oesterl´e and Bui Dinh (1998) as: CLR
!
Rer 0.3 − 0.45 exp −0.05684 Re0.4 , = 0.45 + r Req Req
(2.41)
where Rer denotes the Reynolds number of the particle rotation. It is defined as: Rer =
|Ωrel | d2q . νf
(2.42)
2.3.2.5 Added Mass Force
In turbulent particle-laden flows the added mass force1 FAM is caused by the acceleration or deceleration of the particle relative to the continuous phase. In other words, the inertia of the fluid mass encountered by an accelerating or decelerating particle results in an added mass force (or additional drag) acting on this particle (Duarte et al., 2009). The added mass force applied on a spherical particle submerged in an inviscid, incompressible fluid is given by (Crowe et al., 1998): FAM
π = ρf d3q CAM 6
!
duq Duf − , Dt dt
(2.43)
where the added mass coefficient for spherical particles is equal to CAM = 0.5 (Brennen, 1982; Crowe et al., 1998; Kuerten, 2016). Relation (2.43) implies that the added mass force is proportional to the rates of change of the particle velocity and the fluid velocity at the particle position as well as the fluid density and the particle volume. The material derivative for the fluid velocity appearing in Eq. (2.43) is given by the following relation: Duf ∂uf = + uf · ∇uf . Dt ∂t
(2.44)
Obviously, the temporal derivative of the particle velocity appears in the second part of the right-hand side of Eq. (2.43), and thus it can be brought to the left-hand side of the particle equation of motion (2.26) yielding an effective mass of the particle (see Section 2.3.5). 2.3.2.6 Pressure Gradient Force
The pressure gradient force denoted FP G refers to a fluid force applied on the particle caused by the pressure gradient in the fluid surrounding the particle and is given by: π FP G = − d3q ∇p , (2.45) 6 1
32
In some literature it is also called virtual mass force.
2. Euler-Lagrange Simulation Framework where the minus sign means that this force acts in the direction opposite to the pressure gradient ∇p. Assuming a constant pressure gradient across a spherical particle and neglecting the diffusive and the source terms in the momentum equation of the continuous phase (see Section 2.2.1), the pressure gradient force can be written as (Maxey and Riley, 1983): FP G
Duf π = ρf d3q , 6 Dt
(2.46)
where the material derivative for the fluid velocity is given by Eq. (2.44). It is worth mentioning that the pressure gradient force is only significant in case of a large fluid pressure gradient and if ρq ∼ ρf (e.g., neutrally buoyant particles) or ρq ρf (e.g., bubbles). Note that the pressure gradient force and the added mass force are often combined to simplify the numerical implementation in the CFD code (see Section 2.3.5). 2.3.3 Torque Acting on the Particle
The viscous torque Tq acting on a rotating particle due to a local fluid rotation is given by (Crowe et al., 1998; Sommerfeld, 2003): ρf Tq = CR 2
dq 2
!5
|Ωrel | Ωrel ,
(2.47)
where the rotational coefficient CR is estimated based on the analytic solution by Rubinow and Keller (1961) for Rer ≤ 32 and on the numerical simulations by Dennis et al. (1980) as well as the experimental data by Sawatzki (1970) for higher Rer . It is given by:
64π Rer CR = 12.9 128.4 + √ Rer Rer
for Rer ≤ 32 , for 32 < Rer ≤ 1000 .
(2.48)
Note that the first relation of the rotational coefficient, which is only valid for Rer ≤ 32, is perfectly sufficient in the present study. The reason is that although the magnitude of th relative rotation |Ωrel | for microscopic particles dispersed in turbulent flows might be very large, small values of the Reynolds numbers of the particle rotation are predicted, since Rer ∝ d2q based on Eq. (2.42). For example, for a particle-laden channel flow (see Sections 7.2 and 9.2) with a Reynolds number Re = UB δ/νf = 11, 900 the Reynolds number of the particle rotation predicted is Rer ≤ 14. 2.3.4 Boundary Conditions
If periodic boundary conditions (see Section 2.2.3.4) are applied, the particles leaving the computational domain have to be re-injected again into this domain at the inlet. For this purpose, the positions of the particles completely leaving the integration domain in the direction, where the periodic boundary condition is imposed, are shifted back by the length of the computational domain in this direction (e.g., the length Lx in the streamwise direction). Additionally, other properties of the shifted particle (i.e., velocity, density, diameter, etc.) remain unchanged. 33
2.3 Disperse Phase (Single Particle) Assuming that the particle impacts a bounding wall, its translational and angular velocity has to be updated as explained in Section 2.4. Here, two main cases are distinguished. In case of deposition it is assumed that the particle adheres to the wall and is then omitted from the computational domain. Otherwise, the particle is reflected on the wall and the magnitudes of its translation and angular velocity are reduced based on the values of the friction and restitution coefficients. 2.3.5 Summary of Governing Equations of the Particle Motion
In this section a summary of the governing equations of particle motion used in this thesis is presented. By substituting Eqs. (2.27), (2.28), (2.29), (2.43) and (2.46) into Eq. (2.26) and the resulting force into Eq. (2.24a) the equation of the translational motion of the particle can be written as: duq 1 = dt fAM
(
3 1 ρf ρf CD |uf − uq | (uf − uq ) + g 1 − 4 dq ρq ρq
!
)
FL ρf Duf + + (1 + CAM ) . mq ρq Dt (2.49)
Here, the factor fAM results from the inclusion of the added mass force in the particle equation of motion and is given by: fAM = 1 + CAM
ρf ρq
!
(2.50)
.
The drag coefficient CD appearing in relation (2.49) is given by: CD = max
(
!
)
24 α , 0.44 , Req
(2.51)
where the factor α defined in Eq. (2.30) and the particle Reynolds number Req is given by Eq. (2.31). The lift force FL appearing in Eq. (2.49) represents the total lift force (Saffman and Magnus) and is given by Eq. (2.32). Relying on Newton’s second law for the angular momentum the angular velocity of the particle around three Cartesian axes is given by: dωq 1 = Tq dt Iq
with Iq =
π ρq d5q , 60
(2.52)
where Iq is the moment of inertia of the particle. The torque Tq acting on a rotating spherical particle in the presence of the rotation of the surrounding fluid is defined based on Rubinow and Keller (1961) as follows: Tq = µf π d3q Ωrel .
(2.53)
Again, the relation (2.53) is valid up to Rer ≤ 32. Thus, by substituting Eqs. (2.53) and (2.9) into Eq. (2.52), the differential equation of the angular velocity of the particle can be expressed as: dωq 10 10 = Ωrel = dt 3 τq 3 τq 34
1 ∇ × uf − ωq . 2
(2.54)
2. Euler-Lagrange Simulation Framework The numerical integrations of the differential equations of translational and angular motion of the particle given by Eqs. (2.49) and (2.54) will be provided in Section 6.4.2. It is important to note that both the translational and the angular velocity of a particle have to be updated if it collides with bounding walls (see Section 2.4) or with another particle (see Section 2.6). 2.4 Particle-Wall Collision Model
As mentioned in Section 2.3.4, if a particle-wall collision occurs, the translational and angular velocity of the particle impacting a smooth or rough bounding wall have to be updated. In this section the model for particle-wall collisions applied and extended in this thesis is presented. The original model was implemented into the in-house CFD code LESOCC and validated in many studies (see, e.g., Breuer et al., 2012; Alletto, 2014; Alletto and Breuer, 2015). In this original model it is assumed that the adhesion between the particle and the wall is not taken into account (see Section 5.2 for the extension). In addition, the application of the hard-sphere model used in this thesis requires the following assumptions: • The particle is a rigid sphere and hence the deformation of the particle during the impact is neglected (Crowe et al., 1998). This means that the distance between the center of mass of the particle and the contact point located on the wall (see Figure 2.3) remains constant during the collision and is equal to the particle radius denoted rp . • The friction between the particle and the wall obeys Coulomb’s law of dry friction. • The center of the particle displaces linearly within the simulation time step. Particle
ω−p
S
Particle
S
u−p
− − uc,t
rp
mp ol
x p,c
n c
c
Wall
mp xc
− fˆt pw
− fˆ pw − fˆnpw
Impulse components
Figure 2.3: A particle-wall collision including friction without adhesion.
In the framework of the present implementation the condition of a particle-wall collision is given by: ∆ln,p ≤ ∆ncell-center ,
(2.55)
where ∆ln,p stands for the normal distance from the particle center to the bounding wall and ∆ncell-center denotes the distance from the center of the first cell to the wall. In the following the 35
2.4 Particle-Wall Collision Model derivation of this particle-wall model is presented. In the present thesis two types of bounding walls are distinguished to mimic the rebound behavior of the particle on the wall, namely smooth and rough walls. As a starting point for the modeling, it is assumed that the roughness of the wall is neglected. A schematic representation of such a frictional particle-wall collision, during which the adhesion between the particle and the wall is not taken into account, is depicted in Figure 2.3. Here, the contact point between the spherical particle and the wall is denoted c. In the context of this model it is assumed that the collision-normal unit vector n points from the contact point towards the center of mass of the spherical particle denoted S (see Figure 2.3). The conservation of the translational and the angular momentum of the particle impacting the wall reads:
− mp u+ = p − up
Ip ωp+ − ωp− =
Z
Z
f pw dt ,
(2.56a)
rp × f pw dt ,
(2.56b)
+ − + where u− p , up and ωp , ωp are the translational and angular velocities of the particle before (subscript −) and after (subscript +) the particle-wall impact, respectively. The symbol f pw denotes the total impulse vector (or impact force) acting on the particle during the particle-wall collision denoted pw. It is worth mentioning that the terms on the right-hand side of Eq. (2.56) refer to the external compressive and friction forces (Eq. (2.56a)) and the corresponding angular momentum (Eq. (2.56b)) acting on the spherical particle during the impact. The moment of inertia of the particle Ip appearing on the left-hand side of Eq. (2.56a) is given by:
Ip =
1 mp d2p 10
with mp =
π ρp d3p , 6
(2.57)
where mp stands for the mass of the particle with the diameter dp and the density ρp . As visible in Figure 2.3, the radius vector rp appearing in Eq. (2.56b) is the relative vector between the position vectors of the particle-wall contact point xc and the center of mass of the particle at the instant of the collision xp,col and thus: rp = xc − xp,col .
(2.58)
To simplify relation (2.56), the following quantities are introduced. As shown in Figure 2.3, the normal unit vector n can be expressed based on the definition of rp as: n=−
rp |rp |
with
|rp | =
dp , 2
(2.59)
where the minus sign indicates that rp is directed in the opposite direction to the normal unit vector n (see Figure 2.3). Hence, the radius vector can be written in terms of the collision-normal unit vector as follows: rp = − 36
dp n. 2
(2.60)
2. Euler-Lagrange Simulation Framework On the right-hand side of Eq. (2.56) the integral of the forces acting on the particle during the entire collision normalized by the particle mass mp is defined as the specific impulse vector fˆpw (Breuer et al., 2012; Alletto, 2014; Almohammed and Breuer, 2016c): 1 Z pw pw ˆ f = f dt . mp
(2.61)
By substituting Eqs. (2.57), (2.60) and (2.61) into Eq. (2.56), the post-collision translational and angular velocities of the particle read: − ˆpw , u+ p = up + f 5 ωp+ = ωp− − n × fˆpw . dp
(2.62a) (2.62b)
In the above relation the specific impulse vector fˆpw is still unknown and has to be determined as explained next. 2.4.1 Components of the Impulse Vector
To calculate the post-collision translational and angular velocities given by Eqs. (2.62a) and (2.62b), the impulse vector has to be determined. For this purpose, additional relations such as the definition of the restitution coefficient and Coulomb’s law of dry friction are required to close the system of equations. Since friction between the spherical particle and the wall is considered, the total impulse vector has two components (see Figure 2.3), one in the normal (subscript n) and one in the tangential (subscript t) direction: fˆpw = fˆnpw + fˆtpw ,
(2.63)
where fˆnpw and fˆtpw stand for the normal (perpendicular to the wall) and the tangential (parallel to the wall) component of the specific impulse vector. Since the adhesion between the particle and the wall is presently not taken into account, the normal component only consists of one pw pw contribution fˆn,a (i.e., fˆnpw = fˆn,a ). It is important to note that the second index a is introduced here to distinguish the normal impulse component from the case, where the adhesion (index c) is considered as will be discussed in Section 5.2. Thus, the normal impulse vector of a particle-wall collision can be written: pw pw fˆn,a = fˆn,a n,
(2.64)
pw where fˆn,a stands for the magnitude of the normal impulse vector. According to Eq. (2.62a) the total impulse vector can be written in terms of the translational velocities of the particle before and after the impact as follows:
− fˆpw = u+ p − up .
(2.65)
Taking the scalar product of the total impulse vector and the collision-normal unit vector, the pw magnitude of the normal component of the impulse vector fˆn,a reads: pw fˆn,a = fˆpw · n =
h
i
− u+ p − up · n .
(2.66) 37
2.4 Particle-Wall Collision Model However, in the above relation the post-collision velocity of the particle u+ p is still unknown. It is well known that the velocity of the particle after the impact can be written in terms of the velocity before the collision based on the definition of the so-called normal restitution coefficient for particle-wall collisions denoted en,w as follows (see, e.g., Crowe et al., 1988): en,w
u+ u+ p ·n p,n = − − = − up,n u− p ·n
(2.67)
By substituting the above relation into Eq. (2.66), the magnitude of the normal impulse vector reals2 : h
i
pw fˆn,a = − (1 + en,w ) u− p ·n .
(2.68)
As clearly visible in Figure 2.3, a particle-wall collision is only possible if (u− p · n) < 0 and hence pw the magnitude of the normal impulse fˆn,a given by Eq. (2.68) is always positive. In order to determine the total impulse vector, the tangential component of the total impulse fˆtpw appearing in Eq. (2.63) has to be calculated. This impulse component depends on the particle-wall impact type (i.e., sticking or sliding collision) due to the consideration of the friction between the particle and the wall as explained next. 2.4.2 Particle-Wall Collision Type
Two types of the particle-wall impact are distinguished: (i) a sticking (or non-sliding) collision, during which the particle stops sliding and (ii) a sliding collision, during which the particle slides throughout the entire impact time. A sticking particle-wall collision (denoted by the subscript st) can be identified based on the following no-slip condition relying on Coulomb’s law of static friction: ˆpw fst,t
pw ≤ µst,w fˆn,a ,
(2.69)
pw where µst,w stands for the coefficient of static friction for the particle-wall collision. fˆst,t denotes ˆpw the magnitude of the tangential sticking impulse vector, whereas f is equal to fˆpw given by
Eq. (2.68). Hence, relation (2.69) can be written as: ˆpw fst,t
pw ≤ µst,w fˆn,a .
n,a
n,a
(2.70)
The term on the left-hand side is still unknown and can be determined as follows. The so-called tangential restitution coefficient et,w for particle-wall collisions is defined in a similar manner as the normal one as follows: et,w = 2
+ uc,t − uc,t
with et,w ≥ 0 .
(2.71)
In case of a fully elastic particle-wall collision the normal restitution coefficient is equal to unity (en,w = 1) pw and hence fˆn,a = −2 u− p ·n .
38
2. Euler-Lagrange Simulation Framework Thus, the tangential component of the post-collision velocity at the contact point c can be expressed in terms of the corresponding pre-collision velocity, such that: − u+ c,t = −et,w uc,t ,
(2.72)
where the minus sign on the right-hand side means that after the impact the tangential component − of the post-collision velocity u+ c,t points in the direction opposite to uc,t (see Figure 2.3), but its magnitude is reduced by the factor of et,w due to the inelastic deformation during the collision (Langfeldt, 2011; Breuer et al., 2012; Almohammed and Breuer, 2016c). As visible in Figure 2.3, the pre- and post-collision translational velocities of the contact point in the tangential direction with respect to the particle are given by:
− − − u− c,t = up − up · n n + ωp × rp ,
(2.73a)
(2.73b)
− − u− c,t = up − up · n n −
(2.74a)
u+ c,t
(2.74b)
+ + + u+ c,t = up − up · n n + ωp × rp ,
where the second terms on the right-hand side of Eq. (2.73) stand for the normal components of the particle velocity before and after the impact, respectively. By substituting the radius vector rp given by Eq. (2.60) into the above relations, they can be expressed as:
dp − ω × n, 2 p dp + − = u− ω × n, p − up · n n − 2 p
By inserting relation (2.74b) into Eq. (2.72), the latter can be written as: u+ p −
h
i
u+ p ·n n −
dp + ω × n = −et,w u− c,t . 2 p
(2.75)
+ Substituting u+ p and ωp given by Eq. (2.62) into the above equation yields the following relation, which includes the total impulse vector for sticking collisions fˆstpw : + u− p − (up · n) n −
dp − 5 ωp × n + fˆstpw + n × fˆstpw × n = −et,w u− c,t . 2 2
(2.76)
Based on the definition of the normal restitution coefficient given by Eq. (2.67) the scalar product on the left-hand side of the above relation reads:
− u+ p · n = −en,w up · n .
pw n × fˆstpw × n = (n · n) fˆstpw − fˆstpw · n n = fˆstpw − fˆn,a n.
(2.77)
In addition, based on Lagrange’s formula (Lagrange, 1773) the vector triple product on the left-hand side of Eq. (2.76) is related to the scalar product by the following expression:
(2.78)
By substituting Eqs. (2.77) and (2.78) into Eq. (2.76), the resulting relation reads: u− p −
dp − 5 + 7 en,w h − i 7 ˆpw ωp × n + up · n n + fst = −et,w u− c,t . 2 2 2
(2.79) 39
2.4 Particle-Wall Collision Model By solving the above equation with respect to fˆstpw and considering Eq. (2.74), the impulse vector for sticking particle-wall collisions reads:
fˆstpw = − (1 + en,w )
h
i
u− p ·n n +
2 (1 + et,w ) u− c,t . 7
(2.80)
Taking the magnitude of the normal impulse vector given by Eq. (2.68) into account, the above relation can be written as: 2 pw fˆstpw = fˆn,a n − (1 + et,w ) u− c,t . 7
(2.81)
Thus, the tangential component of the impulse vector for sticking particle-wall collisions required for Eq. (2.70) reads: 2 pw fˆst,t = − (1 + et,w ) u− c,t . 7
(2.82)
Substituting the above relation into Eq. (2.70) yields the no-slip condition based on Coulomb’s law of static friction: − uc,t
≤
7 µst,w fˆpw . 2 (1 + et,w ) n,a
(2.83)
If this condition is satisfied, a sticking particle-wall collision occurs implying that the particle stops sliding along the wall during collision. Otherwise, a sliding particle-wall collision has to be taken into account, during which the particle slides along the wall during the entire particle-wall collision. In case of a sliding collision denoted by the subscript sl the normal impulse vector is given by Eq. (2.64), whereas the tangential component is determined based on Coulomb’s law of kinetic friction:
pw pw fˆsl,t = −µkin,w fˆn,a t,
(2.84)
where µkin,w stands for the coefficient of kinetic friction for particle-wall collisions. The collisiontangential unit vector t is defined based on the tangential velocity before the impact at the contact point given by Eq. (2.74a) as follows: u− c,t . t = − uc,t
(2.85)
Thus, the tangential impulse vector of sliding particle-wall collisions reads: − pw pw uc,t fˆsl,t = −µkin,w fˆn,a − . uc,t
(2.86)
pw The minus sign on the right-hand side of Eq. (2.86) means that fˆsl,t points in the direction opposite − to uc,t (see Figure 2.3) and thus reduces this velocity component during the impact.
40
2. Euler-Lagrange Simulation Framework 2.4.3 Kinetics of the Particle after the Impact with a Smooth Wall
If a frictional particle-wall impact occurs, the translational and angular velocities of the particle has to be determined. Thus, for both particle-wall collision types (i.e., sticking and sliding), the total impulse vector fˆpw given by Eq. (2.63) has to be calculated. As discussed before, the normal pw component of the total impulse fˆnpw is equal to fˆn,a given by Eq. (2.64), whereas the tangential pw ˆ component of the total impulse vector ft is calculated according to the collision type. The distinction between both collision types is made based on the no-slip condition (2.83). In the following the post-collision velocities in case of sticking and sliding particle-wall collisions are summarized. 2.4.3.1 Sticking Collision
As mentioned before, if a sticking particle-wall collision occurs and hence the no-slip condition (2.83) is satisfied, the particles stop sliding on the wall during the collision and the total impulse vector given by Eq. (2.63) reads: pw pw fˆpw = fˆnpw + fˆtpw = fˆn,a + fˆst,t .
(2.87)
Thus, for a sticking particle-wall impact the post-collision translational and angular velocities of pw pw the particle are determined by substituting the normal fˆn,a and the tangential fˆst,t components of the total impulse vector of a sticking collision given by Eqs. (2.64) and (2.82) into Eq. (2.87) and the resulting equation into Eq. (2.62):
− ˆpw n − 2 (1 + et,w ) u− t , = u + f u+ p 1 n,a c,t 7 ) ( 10 − + − (1 + et,w ) uc,t (n × t) . ωp = ωp + 7dp
(2.88)
2.4.3.2 Sliding Collision
In this case the no-slip condition (2.83) is not satisfied and thus the particle slides on the wall during the entire collision time. Here, the total impulse vector given by Eq. (2.63) reads: pw pw fˆpw = fˆnpw + fˆtpw = fˆn,a + fˆsl,t .
(2.89)
Thus, for a sliding particle-wall impact the post-collision translational and angular velocities of pw pw the particle are determined by substituting the normal fˆn,a and the tangential fˆsl,t components of the total impulse vector of a sliding collision given by Eqs. (2.64) and (2.86) into Eq. (2.89) and the resulting equation into Eq. (2.62): n
− ˆpw ˆpw u+ p = up + fn,a n − µkin,w fn,a t
ωp+ = ωp− +
(
o
5 pw µkin,w fˆn,a (n × t) dp
)
, .
(2.90)
41
2.4 Particle-Wall Collision Model 2.4.4 Rough Wall (Sandgrain Roughness Model)
In the preceding section it was assumed that the particle collides with a smooth wall. However, this assumption is not valid for surfaces in most practical applications. It is well-known that due to the random nature of three-dimensional rough surfaces y = f (x, z), they can be characterized based on statistical methods. In the present thesis the root-mean-square roughness parameter3 denoted Rrms is used to quantify the quality of rough walls due to its sensitivity to grooves and peaks of rough surfaces (Alletto, 2014): Rrms =
(
M X N 1 X (y(xm , zn ) − hyi)2 N M m=1 n=1
)1/2
,
(2.91)
where N and M denote the number of scanning points to measure the roughness in x and z directions. hyi is the mean height of roughness profile of the surface. The effect of wall roughness on single-phase flows has been investigated for quite a long time (see, e.g., Nikuradse, 1933; Schlichting, 1936; Shockling et al., 2006). Furthermore, in turbulent particle-laden wall-bounded flows it has been found that the wall roughness has a significant influence on the disperse phase and consequently also on the continuous phase due to the two-way coupling (see, e.g., Kussin and Sommerfeld, 2002; Lain and Sommerfeld, 2008; Alletto, 2014; Breuer and Almohammed, 2015). In the present thesis the impact of a particle with a rough wall is modeled based on the sandgrain roughness model proposed by Breuer et al. (2012) based on the idea of Nikuradse (1933). This model was already implemented into the in-house CFD code LESOCC and validated by Breuer et al. (2012) and Alletto (2014). In order to study the effect of the wall roughness on the agglomeration process (see Section 7.2) as well as the rebound behavior of the particles in the presence of particle-wall adhesion (see Section 9.2) this model was incorporated into the new developments. In the framework of this model it is assumed that the wall is covered by a densely packed layer of sand grains, which are idealized by monodisperse spheres of the radius rw (Breuer et al., 2012; Alletto, 2014). The diameter of these spheres is related to the sandgrain roughness ks such that ks = 2 rw . According to the findings by Shockling et al. (2006) the sandgrain roughness ks can be related to the root-mean-square roughness as follows: ks = Csurface Rrms ,
(2.92)
where the surface parameter Csurface ≈ 3. Hence, the radius of the wall spheres rw can be determined as follows: 1 1 rw = ks = Csurface Rrms , (2.93) 2 2 where in this study the surface parameter Csurface required to relate the sandgrain roughness to classical roughness parameters is set equal to Csurface = 3 (Breuer and Almohammed, 2015; Almohammed and Breuer, 2016b,c). In some studies, however, the mean roughness denoted Rz is available instead of the root-mean-square roughness Rrms . Due to the lack of a generalized relation between Rz and Rrms , in this study it is assumed that Rrms ≈ Rz . 3
42
This definition is based on DIN EN ISO 4287:2010-07
2. Euler-Lagrange Simulation Framework 2.4.4.1 Random Normal Unit Vector
To mimic the collision of the particle with the densely-packed layer of spheres, the new random normal unit vector nR seen by the particle at contact with the wall (see Figure 2.4) is determined by inclining the normal unit vector of the smooth wall n by a Gaussian distributed angle αR and then rotating it by the uniformly distributed azimuthal angle φR in the range φR = [0 − 2π] (Breuer et al., 2012; Alletto, 2014). As depicted in Figure 2.4, the random normal unit vector reads: nR = n cos(αR ) + ta sin(αR ) cos(φR ) + tb sin(αR ) sin(φR ) ,
(2.94)
where ta and tb stand for two arbitrarily chosen tangential vectors defined such that they are perpendicular to the normal unit vector of a smooth wall n and to each other. These tangential unit vectors are given by: [nx , nz , −ny ] ta = q n2x + n2y + n2z
and tb = n × ta .
(2.95)
Here, nx , ny and nz are the components of the normal unit vector of the smooth wall n. In this model Breuer et al. (2012) arbitrarily set the component nx to zero (nx = 0) to simplify the description of the tangential unit vector ta (Alletto, 2014). In addition, the components ny and nz are exchanged and partially inverted such that the scalar product of n and ta vanishes (i.e., n · ta = 0). Consequently, they are perpendicular to each other as shown in Figure 2.4. The angles γ and δ − stand for the incidence of the trajectory of the particle in the wall-normal and the wall-tangential direction. n Particle
δ−
γ tb
nR αR φR ta Rough wall
Figure 2.4: Definition of the random normal unit vector nR based on the normal unit vector for a smooth wall n, the tangential unit vectors ta and tb as well as the random angles αR and φR (adopted from Breuer et al., 2012).
The random inclination angle αR is a Gaussian distributed polar angle with zero mean and a standard deviation σw . This random angle is defined by: αR = σw ξ ,
(2.96)
where ξ is a Gaussian distributed random number with a unit standard deviation. Based on the geometrical configuration depicted in Figure 2.5, the standard deviation of the random angle σw 43
2.4 Particle-Wall Collision Model can be defined as follows: !
rw σw = arcsin . rw + dp /2
(2.97)
rw
n
n
dp
+
dp
/2
Particle
σw rw
σw rw
Wall spheres Figure 2.5: Definition of the standard deviation σw (adopted from Breuer et al., 2012).
In this model the standard deviation in restricted to σw,max = 30◦ , since it was found by Breuer et al. (2012) that 99.7% of the computed random numbers based on a Gaussian distribution are in the range of 3σw ≤ 90◦ . The remaining percentage of the calculated random numbers leads to an unrealistic scenario, since the inclination angle becomes αR ≥ 90◦ (Alletto, 2014). Hence, this unphysical scenario is avoided by applying the restriction of σw as mentioned before. It is worth noting that this restriction is only necessary if the wall spheres are larger than the particle (i.e., rw > dp /2). However, in most practical applications rw dp /2 and hence based on Eq. (2.97) the standard deviation becomes σw < 30◦ . In order to take the effect of the wall roughness on the particle rebound into account, the particlewall collision model explained in the preceding sections is used, but the normal unit vector for smooth wall n is replaced by the newly computed random unit vector (i.e., n → nR ). 2.4.4.2 Shadow Effect
As explained before, the random wall-normal unit vector nR lying within a cone predefined by αR implies the local inclination of the wall. It is required to predict the inelastic particle-wall collision. However, if a particle cannot reach a certain region of the wall sphere as depicted in Figure 2.6, this phenomenon has to be separately taken into account. Typically, this phenomenon is called the “shadow effect” (see, e.g., Sommerfeld, 2000), which leads to asymmetric probability density functions of the wall inclination angles (Breuer et al., 2012). As shown in Figure 2.6, the shadow effect generally occurs when the scalar product of the particle velocity before the collision u− p and − the inclined normal vector nR is positive and thus (up · nR ) > 0. 44
2. Euler-Lagrange Simulation Framework
Particle
δ
nR
−
y
nnew R
n Particle
αR
δ−
δ−
Shadow region
nR
S
γ
γ
u− p
x
P
Reachable region
Wall
3σw 3σw
u− p Wall
x
Wall sphere
Shadow region
Figure 2.6: Schematic representation of a particle-wall collision leading to a shadow event (u− p · nR > 0) and the derivation of the new random normal unit vector nnew (adopted from Breuer et al., 2012). R
If the shadow effect occurs, the new normal vector has to be recomputed by calculating the two possible intersection points (denoted P and S) between the straight line x (i.e., the particle trajectory) and the wall sphere of the radius rw (see Figure 2.6). Assuming a three-dimensional space, the straight line x is given by: x = xP + u− p t with xP = rw nR ,
(2.98)
where P refers to the position of the intersection point of the particle at the wall sphere (see Figure 2.6) and t is a free parameter. Substituting Eq. (2.98) into the equation of a spherical wall of the radius rw yields the second intersection point S. As visible in Figure 2.6, the equation of the wall sphere reads:
2
xP + u − pt
2
+ yP + vp− t
2
+ zP + wp− t
The above equation has two solutions:
= rw2 .
(2.99)
• The first solution is the trivial solution obtained by setting t1 = 0 and hence x = xP .
• The second solution gives the parameter t2 used to determine the second intersection point xS (see Figure 2.6): t2 = −2
− − xP u − p + yP vp + zP wp
− 2 up
with
new xS = xP + u− p t2 = rw nR .
= −2
u− p · nR − 2 up
rw ,
(2.100)
(2.101)
By substituting xP = rw nR and t2 based on Eqs. (2.98) and (2.100) into the above relation, the new random normal unit vector can be calculated as follows: nnew = nR − 2 R
− u− p · nR up
− 2 up
.
(2.102)
Again, the treatment of the collision of a particle on a rough wall is the same as explained in the previous section for smooth walls by replacing the normal unit vector n by the newly computed random unit vector nR or nnew in case of a shadow event. R 45
2.5 Interphase Coupling 2.5 Interphase Coupling
In a fluid-particle system the term interphase coupling refers to the effects of the fluid flow on the disperse phase and vice versa. As depicted in Figure 2.7, fluid-particle flows are classified based on the particle volume fraction αq as dilute and dense fluid-particle flow systems (Elghobashi, 1991). In addition, three coupling mechanisms are distinguished: (i) the one-way (ii) the two-way and (iii) the four-way coupling as explained next. one-way coupling
two-way coupling
Fluid
Fluid
four-way coupling Fluid
Particles
Particles
Particles
Particles
αq 1E-9
1E-8
1E-7
1E-6
1E-5
1E-4
Dilute systems
1E-3
0.01
0.1
1.0
Dense system
Figure 2.7: Classification of fluid-particle flows based on the volume fraction αq (Elghobashi, 1991).
2.5.1 Fluid-Particle Interaction (One-Way Coupling)
If the particle volume fraction is αq . 10−6 , the effect of particles on the dynamics of the fluid phase can be ignored (Elghobashi, 1991). As a result, the particles of sufficiently small inertia are transported by the dynamic response to the motion of the continuous phase (see Figure 1.3). As shown in Figure 2.7, this mechanism is commonly referred to as “one-way coupling”. In this case the turbulent carrier flow is treated as a pure fluid and the dynamics of the disperse phase is described by the forces and torques acting on the particles as explained in the preceding section. Note that the particle-particle collisions in this regime are assumed to be irrelevant. 2.5.2 Particle-Fluid Interaction (Two-Way Coupling)
As the particle volume fraction increases up to αq & 10−6 , the fluid-particle flow system is still dilute, but the effect of the particles on the fluid phase becomes important (Elghobashi, 1991, 1994). As mentioned in Section 1.2, instead of the particle volume fraction the mass loading is used in some studies as a criterion to distinguish between the one-way and the two-way coupled regimes. Due to the momentum exchange between the phases the feedback effect of the particles on the continuous fluid alters the dynamics of the continuous phase (i.e., modification of the turbulent structures encountered in the fluid flow) and hence it cannot be neglected (see, e.g., Balachandar and Eaton, 2010). Since the fluid and the disperse phase interact with each other, this regime (see Figure 2.7) is usually known as “two-way coupling” (see, e.g., Elghobashi, 1991, 1994). Furthermore, it is commonly accepted that the particle-particle collisions still do not play a dominant role. 46
2. Euler-Lagrange Simulation Framework In the framework of the Euler-Lagrange approach using LES, the feedback effect of the particle on the fluid phase is considered by adding a source term fjPSIC resulting from the disperse phase on the right-hand side of the momentum equation (2.10b). In the present thesis the particle-source-in-cell (PSI-CELL) method by Crowe et al. (1977) is adopted to consider the influence of the two-way coupling. The source term is given by: fjPSIC
q X ρf π 2 (uf,j − uq,j ) |uf,j − uq,j | FD,j =− =− dq CD , ∆V k=1 2 4 k=1 ∆V
Nq X
N
(2.103)
where Nq stands for the total number of particles contributing to the drag force in the computational cell of the volume ∆V . It is worth to mention that in the formulation of the source term fjPSIC only the drag force predicted based on the filtered fluid and particle velocities is considered. The drag coefficient CD is given by Eq. (2.51). The reason for neglecting other aerodynamic forces is attributed to the fact that the drag force is typically at least one order of magnitude larger than other aerodynamic forces. In the present study the calculation of the source term fjPSIC given by Eq. (2.103) is achieved as follows. First, to determine the drag force, the fluid velocity at the particle position is interpolated by means of a Taylor series expansion scheme by Marchioli et al. (2007) as will be explained in Section 6.4.3. Then, this drag force is distributed to the centers of the eight computational cells surrounding the particle based on a trilinear interpolation (see Appendix D.2) leading to a smooth source term distribution. As a result, the convergence problems due to isolated large source terms can be avoided (Alletto, 2014). 2.5.3 Particle-Particle Interaction (Four-Way Coupling)
For a particle volume fraction αq & 10−3 the fluid-particle flow regime is dense enough, such that the inter-particle collisions and hence the agglomeration or coalescence in case of cohesive particles becomes a dominant momentum exchange mechanism. Thus, the particles are transported by the fluid-particle interaction and particle-wall collisions but additionally by the particle-particle collisions (see Figure 1.3). The regime is commonly referred to as “four–way coupling” (see, e.g., Elghobashi, 1991, 1994). Note that in the test cases investigated in the present thesis the global volume fraction are typically smaller than the limit of the four–way coupling regime (i.e., αq < 10−3 ). However, the local volume fraction, for example, in the near-wall region of a particle-laden turbulent channel flow reaches or exceeds this limit significantly. In addition, as mentioned in Section 1.3 the first prerequisite for an agglomeration or a coalescence process is a particle-particle collision. Thus, the four–way coupling is taken into account in these simulations. A detailed description of these effects will be presented in Chapters 7, 8 and 9. 2.5.4 Subgrid-Scale Models for the Particles
Based on many studies (see, e.g., Armenio et al., 1999; Kuerten and Vreman, 2005; Pozorski and Apte, 2009; Breuer and Alletto, 2012; Alletto, 2014; Breuer and Almohammed, 2015) it has been found that the unresolved scales in a LES have a non-negligible effect on the dynamics 47
2.5 Interphase Coupling of tiny particles. If the relaxation time of the particles is of the same order as the smallest fluid time scale, the unresolved scales in LES become significant and thus their effect has to be considered in the equation of the translational motion of the particle. Broadly speaking, two common kinds of subgrid-scale models for the particles are found in the literature, namely the approximate deconvolution approaches (see, e.g., Shotorban and Mashayek, 2005) and stochastic models (see, e.g., Fede et al., 2006; Pozorski and Apte, 2009; Breuer and Hoppe, 2017). The stochastic subgrid-scale models are usually based on solving a differential equation (also called Langevin equation). In the context of this study two stochastic subgrid-scale models for the particles are adopted, namely a trivial and a Langevin-type model. For both models it is assumed that the instantaneous velocity of the fluid uf is subdivided into the filtered fluid velocity at the particle position uf and the subgrid-scale velocity fluctuation u0s , such that: uf = uf + u0s ,
(2.104)
where the subscript s means that the subgrid-scale velocity “seen” by the particle is considered. These two subgrid-scale models for the particles are briefly summarized in the following. It is also worth noting that in the present thesis the effect of subgrid-scales on the rotational motion (the angular velocity) of the particles is not considered. 2.5.4.1 Trivial Model
Assuming homogeneous isotropic turbulence, a first crude model taking the effect of the unresolved eddies on the translational motion of the particle in a rough manner into account can be suggested. This model is denoted trivial model due to its relatively simple formulation. Thus, in this model the effect of the subgrid scales on the particle motion is not completely ignored, but the use of sophisticated models based on the approximate deconvolution or stochastic differential (Langevintype) equations (see, e.g., Shotorban and Mashayek, 2005; Kuerten, 2006; Fede et al., 2006; Bini and Jones, 2008; Pozorski and Apte, 2009; Jin et al., 2010) is avoided. In the framework of this model the subgrid-scale fluctuation of the fluid u0s required for Eq. (2.104) is given by: u0s = σSGS ξ ,
(2.105)
where ξ is a random number obeying a Gaussian distribution with zero mean and unit variance. The standard deviation σSGS of the fluctuations interpreted as a characteristic velocity of the subgrid scales is computed in terms of the turbulent kinetic energy of the subgrid-scales kSGS as follows: σSGS =
s
2 kSGS , 3
(2.106)
Relying on the scale-similarity approach by Bardina et al. (1980), who suggested a double-filtering ˜ = 2 ∆, the subgrid-scale kinetic energy reads: procedure with a test filter ∆ kSGS = 48
1 2 (uf − uf ) 2
(2.107)
2. Euler-Lagrange Simulation Framework where uf stands for the doubly-filtered fluid velocity at the particle position. Here, the filtering procedure relies on the values of 27 adjacent grid points with weighting factors according to a trilinear interpolation (see Appendix D.2). The above relation implies that the subgrid-scale velocities are assumed to be isotropic. In the equation of the translational motion for the particles the fluid velocity at the particle position is determined as a superposition of the filtered quantity resulting from the solution of the filtered Navier–Stokes equations and the random contribution mimicking the non-resolved part. For more details of the model, the reader is referred to Alletto and Breuer (2012) and Alletto (2014), were this model is applied to different flow configurations. It is important to note that the conditions of this simplified model might not be entirely valid for the simulations carried out in the present study. Hence, it is applied as a first approximation leading to a low computational effort in comparison with other more sophisticated approaches. 2.5.4.2 Extended Langevin-Type Model
The original version of this stochastic Langevin model was first proposed by Pozorski and Apte (2009) to take the effect of unresolved subgrid scales on the particles into account. In this Langevintype model it is assumed that the slip velocity uslip defined as the difference between the filtered velocity of the continuous phase and the particle velocity is aligned with only one of the major axes of the fluid flow. Later on, Breuer and Hoppe (2017) avoided this restriction in their extended model by taking an arbitrary orientation of the slip velocity uslip into account leading to a more general formulation valid in most practically relevant flows. Thus, the cost-efficient subgrid-scale model for the particles by Breuer and Hoppe (2017) considers the effect of subgrid scales on the particles more reliably than the original stochastic model by Pozorski and Apte (2009). The extended version of the equation of the subgrid-scale fluctuations of the fluid is given in index notation by: du0s,i = −Gij du0s,j dt +
q
2 2 σSGS Bij dWj ,
(2.108)
The first term on the right-hand side of this relation represents the drift term, whereas the second term denotes the stochastic diffusion term. The standard deviation σSGS of the fluctuations representing the characteristic velocity of the subgrid scales is computed again in terms of the turbulent kinetic energy of the subgrid scales based on Eq. (2.106), where the turbulent kinetic energy of the subgrid scales kSGS is given by Eq. (2.107). dWj is the Wiener increment obeying a Gaussian distribution with zero mean and a variance equal to the size of the time step. In the above relation the symbols Gij and Bij stand for the drift and diffusion matrices of the stochastic process, respectively. These are defined as follows: 1
1 1 Gij = 0 δij + 0 − 0 ri rj , τL,⊥ τL,k τL,⊥ 1
1 1 Bij = q δij + q 0 − q ri rj , 0 0 τL,k τL,⊥ τL,⊥
(2.109) (2.110)
where ri denotes the component of the unit vector of the slip velocity in the ith direction and hence 0 0 ri = uslip,i / |uslip |. The symbols τL,k and τL,⊥ are the time scales over which the subgrid-scale 49
2.5 Interphase Coupling fluctuations are correlated in the direction parallel and perpendicular to the slip velocity uslip , respectively. In Pozorski and Apte (2009) the components of the time scales are determined by multiplying τSGS by the following factors based on Csanady (1963): τSGS τSGS 0 0 v τL,k =v and τ = , (2.111) L,⊥ u u 2 2 u u t1 + 3 β 2 |uslip | t1 + 6 β 2 |uslip | 2 kSGS kSGS
where the factor β stands for the ratio of the Lagrangian to the Eulerian time scales and is assumed to be equal to unity in the present study as suggested by Pozorski and Apte (2009). The time scale of the subgrid scales τSGS is modeled as: ∆ τSGS = C , (2.112) σSGS where C is a constant set to unity in this model and ∆ is the filter width. An extensive description of two different methods for the numerical implementation of the extended model into the in-house CFD code LESOCC can be found in Breuer and Hoppe (2017). In the following only the solution of the Langevin equation (2.108) applied in the present study is summarized. Minier et al. (2003) and Peirano et al. (2006) suggested that the Langevin equation (2.108) can be solved by means of an analytic integration. The analytically integrated Eq. (2.108) between the instants in time t0 and t according to Kloeden and Platen (1995) and Gardiner (2003) reads: u0s,i (t) = Eij (t, t0 ) u0s,j (t0 ) +
q
2 2 σSGS
Z t t0
Eij (t, t0 ) Bkj dWj (t0 ) ,
(2.113)
where the difference between these time instants is equal to the time step of the simulation ∆t = t − t0 . The integrated solution of the above relation is expressed as follows: 0(n+1)
us,i
= Eij (t, t0 ) us,j + Wij (t, t0 ) ξj ,
(2.114)
0(n)
where Eij (t, t0 ) and Wij (t, t0 ) stand for the time-dependent matrix exponential of Gij and the square-root of the covariance matrix. ξj denotes the j th component of the random vector. The matrices Eij (t, t0 ) and Wij (t, t0 ) are given by: h
i
Eij (t, t0 ) = E⊥ (t, t0 ) δij + Ek (t, t0 ) − E⊥ (t, t0 ) ri rj , h
(2.115)
i
Wij (t, t0 ) = W⊥ (t, t0 ) δij + Wk (t, t0 ) − W⊥ (t, t0 ) ri rj .
(2.116)
The time-dependent coefficients of the matrix Eij (t, t0 ) read:
!
t − t0 Ek (t, t0 ) = exp − 0 τL,k
t − t0 and E⊥ (t, t0 ) = exp − 0 . τL,⊥
(2.117)
In addition, the time-dependent coefficients of the matrix Wij (t, t0 ) read:
1/2
t − t0 1 − exp −2 0 τL,⊥
!)1/2
t − t0 Wk (t, t0 ) = σSGS 1 − exp −2 0 τL,k
W⊥ (t, t0 ) = σSGS 50
(
.
,
(2.118a) (2.118b)
2. Euler-Lagrange Simulation Framework It is worth noting that this Langevin-type model can be reduced to the trivial model for max (Gij ∆t) 1. In this case the coefficients Ek (t, t0 ) and E⊥ (t, t0 ) given by Eq. (2.117) tend to zero, whereas the coefficients Wk (t, t0 ) and W⊥ (t, t0 ) given by Eq. (2.118) converge to 0(n+1) σSGS . Hence, Eq. (2.114) is reduced to us,i = σSGS ξi , which is identical to the formulation of the trivial model given by Eq. (2.105). In the present study the cost-efficient Langevin model by Breuer and Hoppe (2017) is applied as a standard model, since it predicts more reasonable results than the trivial model (see Sections 7.2 and 9.2). 2.6 Particle-Particle Collision Model
As a first step for the modeling of particle agglomeration (see Chapter 3) and droplet coalescence (see Chapter 4) a particle-particle collision model is required. In the framework of this model it is assumed that the cohesion between the collision partners is not taken into account. This model was implemented into the in-house CFD code LESOCC and has been validated in many studies (see, e.g., Alletto and Breuer, 2012, 2013, 2014, 2015; Alletto, 2014; Breuer and Alletto, 2012; Breuer et al., 2012). 2.6.1 Particle-Particle Collisions with Friction
It is well known that if two spherical particles collide with each other (i.e., inter-particle collision) during a very short period of time, impulsive forces are exerted between the collision partners (Crowe et al., 1998). As mentioned before, in the context of a hard-sphere model (Hoomans et al., 1996) used in the present study only binary collisions are taken into account. This means that the simultaneous contact of three or even more collision partners is not allowed, since this case becomes significant only in case of disperse flow systems with very high mass loadings. Figure 2.8 shows a schematic system of two particles during a frictional inter-particle collision, in which the cohesion between the collision partners is not taken into account.
ω−1
Particle 1
ur − el
u2
−
u−1 S1
m1
S1
xr,
col
c
n
− fˆ pp
c fˆ
−
u2
fˆt pp
fˆnpp
r2
pp
,co
l
Particle 1
c
− − uc,t,r
r1
x1
m1
x
S2 2,col
− fˆnpp
S2
u−c,t,r
Particle 2
Impulse components
Particle 2
m2
− fˆt pp
ω−2
m2
Collision is possible
(u−2 - u−1) n Figure 2.8: A frictional binary particle collision without cohesion (Almohammed and Breuer, 2016b).
51
2.6 Particle-Particle Collision Model The application of the hard-sphere model requires the following assumptions: • The colliding particles are rigid spheres and hence the deformation of the collision partners during the impact is neglected (Crowe et al., 1998). This implies that the distance between the centers of mass of the colliding particles remains constant during the collision and is equal to the sum of the radii of the collision partners. • The friction between the collision partners (i.e., spherical particles) is assumed to obey the Coulomb’s law of friction. • The centers of the collision partners displace linearly within the simulation time step and hence the collision is detected relying on purely kinematic conditions (Breuer and Alletto, 2012). The following derivation refers to a model for the collision of two spherical particles with diameter di , mass mi , translational velocity of the center of mass ui and angular velocity about the center of mass ωi (see Figure 2.8). Here, the subscripts i instead of q in Section 2.1 refers to the collision partners (i = 1, 2). The conservation of the translational and angular momentum for this system of two colliding particles reads, respectively: m1
u+ 1
−
u− 1
=−
− m2 u+ = 2 − u2
I1 ω1+ − ω1− = −
I2 ω2+ − ω2− =
Z
f pp dt ,
(2.119a)
f pp dt ,
(2.119b)
Z
r1 × f pp dt ,
(2.119c)
r2 × f pp dt ,
(2.119d)
Z Z
where the superscripts (−) and (+) represent the quantities of the collision partners (ui and ωi ) before and after the impact, respectively. The symbol f pp stands for the total impulse (or impact force) vector during the particle-particle collision denoted pp. Note that the terms on the right-hand side of Eq. (2.119) refer to the external compressive and friction forces (Eqs. (2.119a) and (2.119b)) and angular momentum (Eqs. (2.119c) and (2.119d)) acting on the collision partners during the impact due to the change of their momenta. The moments of inertia of the collision partners (i.e., I1 and I2 ) appearing on the left-hand side of Eqs. (2.119c) and (2.119d) are defined as follows: 1 m1 d21 , 10 1 I2 = m2 d22 . 10
I1 =
(2.120)
To simplify the formulation of relation (2.119), the following quantities are introduced. As visible in Figure 2.8, it is assumed that the unit vector of the collision in the normal direction denoted n points from the center of mass of the first particle S1 (the reference particle) to the center of mass of 52
2. Euler-Lagrange Simulation Framework the second particle S2 . The normal unit vector can be defined based on the relative vector between the centers of mass of the collision partners at the instant of impact xr,col (see Section 6.4.4): xr,col . (2.121) n= |xr,col |
As depicted in Figure 2.8, the relative vector xr,col is defined as follows: xr,col = x2,col − x1,col ,
(2.122)
where x1,col and x2,col are the position vectors of the collision partners at the instant of impact. The magnitude of the relative vector |xr,col | can be written in terms of the diameters of the collision partners d1 and d2 : 1 (d1 + d2 ) . (2.123) 2 The minus sign on the right-hand side of Eq. (2.119a) reveals that the repulsive impulse is directed in the opposite direction to the collision-normal unit vector as depicted in Figure 2.8. The radius vectors r1 and r2 appearing in Eqs. (2.119c) and (2.119d) can be expressed in terms of the collision-normal unit vector as follows: d1 r1 = |r1 | n = n, 2 (2.124) d2 r2 = − |r2 | n = − n . 2 It is well known that the mutual repulsion of the colliding particles is due to the elastic deformation during the impact. In this context the specific vector of the total impulse denoted fˆpp refers to the integral of the impulsive forces f pp acting on the spherical particle during the collision process divided by the effective mass denoted m, ˆ such that: |xr,col | =
1 Z pp pp ˆ f = f dt , m ˆ where the effective (or reduced) mass m ˆ is given by: m1 m2 m ˆ = . m1 + m2
(2.125)
(2.126)
Thus, after the impact the colliding particles bounce back, since the repulsive impulse fˆpp pushes the collision partners apart from each other (see Figure 2.8). By substituting Eqs. (2.120), (2.124) and (2.125) into Eq. (2.119), the post-collision translational and angular velocities of the collision partners after the impact read: m ˆ ˆpp f , m1 m ˆ ˆpp − u+ f , 2 = u2 + m2 m ˆ 5 ω1+ = ω1− − n × fˆpp , m1 d1 m ˆ 5 ω2+ = ω2− − n × fˆpp . m2 d2 − u+ 1 = u1 −
(2.127a) (2.127b) (2.127c) (2.127d) 53
2.6 Particle-Particle Collision Model 2.6.1.1 Components of the Impulse Vector
To determine the post-collision quantities given in Eq. (2.127), the specific impulse vector fˆpp has to be determined. In the present model the friction between the collision partners is taken into account. Thus, the impulse vector appearing in Eq. (2.127) is divided into a normal and a tangential component, respectively, as follows: fˆpp = fˆnpp + fˆtpp .
(2.128)
pp pp If the cohesion is not taken into account, the normal component is denoted fˆn,a (i.e., fˆnpp = fˆn,a ). Note that the second index a is introduced to distinguish the normal impulse component from the case, where the cohesion is taken into account which will be discussed in Section 3.1.2. Thus, the normal impulse vector of a particle-particle collision is given by: pp pp fˆn,a = fˆn,a n,
(2.129)
pp where fˆn,a stands for the magnitude of the normal impulse vector and is derived as follows. By subtracting Eqs. (2.127a) and (2.127b) from each other, the total impulse vector expressed in terms of the translational velocities of the collision partners before and after the impact reads:
+ − − fˆpp = u+ 2 − u1 − u2 − u1 .
(2.130)
Thus, the scalar product of the total impulse vector and the collision-normal unit vector yields the magnitude of the normal component of the impulse vector: pp fˆn,a = fˆpp · n =
h
+ − − u+ 2 − u1 − u2 − u1
i
· n.
(2.131)
However, in the above relation the post-collision velocities are still unknown. Thus, the postcollision relative motion of the particles can be expressed in terms of the velocities before the impact by the so-called normal restitution coefficient for particle-particle collisions denoted en,p as follows: h
en,p = − h
i
+ (u+ 2 − u1 ) · n
i
− (u− 2 − u1 ) · n
(2.132)
.
In general, the normal restitution coefficient en,p refers to the elasticity of the inter-particle collision. Thus, the restitution coefficient is equal to unity (en,p = 1) for a fully elastic collision, whereas en,p = 0 for a plastic (or fully inelastic) collision. By substituting Eq. (2.132) into Eq. (2.131), the magnitude of the normal impulse vector due to the mechanical deformation can be written as (Crowe et al., 1998): pp fˆn,a = − (1 + en,p )
h
i
− u− 2 − u1 · n .
(2.133)
− As depicted in Figure 2.8, a particle-particle collision is only possible if [(u− 2 − u1 ) · n] < 0 pp and hence the magnitude of the normal impulse fˆn,a given by Eq. (2.133) is always positive. To determine the total impulse vector, the tangential component of the total impulse fˆtpp appearing in Eq. (2.128) is still required. However, since the friction between the colliding particles is considered, the tangential component of the total impulse fˆtpp depends on the particle-particle impact type (i.e., sticking or sliding collision) as explained next.
54
2. Euler-Lagrange Simulation Framework 2.6.1.2 Particle-Particle Collision Type
In the framework of the present model two types for the inter-particle collision are distinguished: (i) a sticking (or non-sliding) collision, during which the collision partners stop sliding and (ii) a sliding collision, during which the colliding particles slide throughout the entire collision time. As explained in Section 2.4.1, a sticking collision can be identified by the no-slip condition relying on Coulomb’s law of static friction. Thus, the condition for a sticking collision (denoted by the subscript st) is given by: ˆpp fst,t
pp ≤ µst,p fˆn,a ,
(2.134)
pp where fˆst,t stands for the magnitude of the tangential sticking impulse vector and µst,p is the coefficient of static friction for the inter-particle collisions. Note that fˆpp is equal to fˆpp given by n,a
n,a
Eq. (2.133) and hence the magnitude of the tangential component of the sticking collision reads: ˆpp fst,t
pp ≤ µst,p fˆn,a .
(2.135)
However, in the above relation the term on the left-hand side is still unknown. For this purpose, the total impulse vector for a sticking collision is determined based on the definition of the so-called tangential restitution coefficient et,p for particle-particle collisions. Analog to the normal restitution coefficient given by Eq. (2.132), et,p is defined as the ratio of the tangential component (denoted t) of the relative motion (denoted r) of the particles at the contact point (denoted c) after the collision to that before the impact as follows (see Figure 2.8): et,p =
+ uc,t,r − uc,t,r
with et,p ≥ 0 .
(2.136)
Thus, the tangential component of the relative post-collision velocity at the contact point can be written in terms of the corresponding velocity before the impact: − u+ c,t,r = −et,p uc,t,r ,
(2.137)
where the minus sign on the right-hand side implies that after the impact the tangential component − of the post-collision relative velocity u+ c,t,r points in the direction opposite to uc,t,r . However, due to the inelastic deformation of the contact surfaces of the collision partners the magnitude of the post-collision relative velocity is reduced by the factor of et,p (Langfeldt, 2011; Breuer et al., 2012). According to Figure 2.8, the translational velocities of the contact point with respect to the collision partners read: − − u− c,1 = u1 + ω1 × r1 , − − u− c,2 = u2 + ω2 × r2 ,
(2.138)
where the second term on the right-hand side of each relation stands for the velocity of the contact point with respect to the center of mass of the particle (i.e., rotation due to the angular velocity of the corresponding particle). 55
2.6 Particle-Particle Collision Model By substituting the radius vectors defined by Eq. (2.124), the above equation can be written as: d1 − ω × n, 2 1 d2 − ω × n. = u− 2 − 2 2
− u− c,1 = u1 +
u− c,2
(2.139)
Thus, the tangential components of the velocity of the contact point read:
d1 − − ω1 × n with u− = u · n n, 1,n 1 2 d2 − − − − = u− − u − ω × n with u = u · n n. 2 2,n 2,n 2 2 2
− − u− c,t,1 = u1 − u1,n +
u− c,t,2
(2.140)
The tangential component of the relative velocity at the contact point u− c,t,r results by subtracting both relations (2.140) from each other and thus: − − u− c,t,r = uc,t,2 − uc,t,1 .
(2.141)
By substituting Eq. (2.140) into the above equation, the tangential velocity of the pre-collision relative motion at the contact point u− c,t,r reads: u− c,t,r
=
u− 2
−
u− 1
−
h
u− 2
−
u− 1
i
!
d1 − d2 − ω + ω2 × n . ·n n− 2 1 2
(2.142)
Analog to this analysis, the tangential velocity of the post-collision relative motion at the contact point u+ c,t,r can be expressed as:
+ + u+ c,t,r = u2 − u1 −
h
+ u+ 2 − u1
i
!
d1 + d2 + ·n n− ω + ω2 × n . 2 1 2
(2.143)
− + By substituting Eqs. (2.142) and (2.143) and the corresponding quantities (i.e, u− i , ui and ωi , ωi+ ) given by Eq. (2.127) as well as the definition of en,p given by Eq. (2.132) into Eq. (2.137), the total impulse vector of the sticking particle-particle collision can be written in the following form:
2 pp fˆstpp = fˆn,a n − (1 + et,p )u− c,t,r , 7
(2.144)
where the first term on the right-hand side of the above equation refers to the normal component of the total impulse vector given by Eq. (2.133). The second term of the above relation stands for the tangential component of the sticking impulse vector required for Eq. (2.135): 2 pp = − (1 + et,p ) u− fˆst,t c,t,r . 7
(2.145)
The no-slip condition is required to determine whether a sticking inter-particle collision takes place or not. This is obtained by substituting Eq. (2.145) into Eq. (2.135) yielding the following relation:
56
− uc,t,r
≤
7 µst,p fˆpp . 2 (1 + et,p ) n,a
(2.146)
2. Euler-Lagrange Simulation Framework In case that the condition is not fulfilled a sliding particle-particle collision (denoted by the subscript sl) occurs and the tangential component of the total impulse differs from the case of a pp sticking impact. Relying on Coulomb’s law of kinetic friction the tangential sliding impulse fˆsl,t is given by:
pp pp fˆsl,t = −µkin,p fˆn,a t,
(2.147)
where µkin,p stands for the coefficient of kinetic friction for particle-particle collisions. The symbol t is the collision-tangential unit vector defined based on the tangential relative velocity at the contact point given by Eq. (2.142) as follows: u− c,t,r . t = − uc,t,r
(2.148)
pp It is worth noting that the minus sign on the right-hand side of Eq. (2.147) indicates that fˆsl,t points in the direction opposite to u− c,t,r (see Figure 2.8). By substituting Eqs. (2.133) and (2.148) into Eq. (2.147) the tangential component of the total impulse vector for a sliding collision fˆsl,t
can be expressed as:
− pp pp uc,t,r fˆsl,t = −µkin,p fˆn,a − . uc,t,r
(2.149)
2.6.1.3 Kinetics of the Collision Partners after the Impact
After a successful frictional particle-particle impact without cohesion, the translational and angular velocities of the collision partners have to be determined. Thus, for both particle-particle collision types, the total impulse vector fˆpp given by Eq. (2.128) has to be calculated. As discussed before, pp the normal component of the total impulse fˆnpp is equal to fˆn,a given by Eq. (2.129), whereas the pp tangential component of the total impulse vector fˆt is determined according to the collision type. In the following the post-collision velocities in case of sticking and sliding collisions are summarized, respectively. 2.6.1.3.1 Sticking Collision
As mentioned before, if a sticking particle-particle collision occurs, the particles stop sliding during the collision and the total impulse vector given by Eq. (2.128) reads: pp pp fˆpp = fˆnpp + fˆtpp = fˆn,a + fˆst,t .
(2.150)
Thus, for a sticking particle-particle impact the post-collision translational and angular velocities pp pp of the collision partners are determined by substituting the normal fˆn,a and the tangential fˆst,t components of the total impulse vector of a sticking collision given by Eqs. (2.129) and (2.145) 57
2.6 Particle-Particle Collision Model into Eq. (2.150) and the resulting equation into Eq. (2.127):
m ˆ ˆpp 2 − = fn,a n − (1 + et,p ) u− c,t,r t , m1 7 m ˆ ˆpp 2 − + − u2 = u2 + f n − (1 + et,p ) uc,t,r t , m2 n,a 7 m ˆ 10 ω1+ = ω1− + (n × t) , (1 + et,p ) u− c,t,r m1 7d1 m ˆ 10 − + − ω2 = ω2 + (1 + et,p ) uc,t,r (n × t) . m2 7d2
u+ 1
u− 1
(2.151)
2.6.1.3.2 Sliding Collision
In this case the particles slide during the entire collision period and the total impulse vector given by Eq. (2.128) reads: pp pp fˆpp = fˆnpp + fˆtpp = fˆn,a + fˆsl,t .
(2.152)
Thus, for a sliding particle-particle impact the post-collision translational and angular velocities pp pp of the collision partners are determined by substituting the normal fˆn,a and the tangential fˆsl,t components of the total impulse vector of a sliding collision given by Eqs. (2.129) and (2.147) into Eq. (2.152) and the resulting equation into Eq. (2.127): o m ˆ n ˆpp pp fn,a n − µkin,p fˆn,a t , m1 o m ˆ n ˆpp pp − fn,a n − µkin,p fˆn,a t , u+ 2 = u2 + m2 m ˆ 5 + − pp ˆ ω1 = ω1 + µkin,p fn,a (n × t) , m1 d1 m ˆ 5 + − pp ˆ ω2 = ω2 + µkin,p fn,a (n × t) . m2 d2 − u+ 1 = u1 −
(2.153)
2.6.2 Particle-Particle Collisions without Friction
In this special case the friction between the colliding particles is neglected implying that the tangential impulse vanishes (i.e., fˆtpp = 0) and hence the angular velocities of the collision partners (i = 1, 2) do not change due to the collision (i.e., ωi+ = ωi− ). In general, the translational velocities of the collision partners before and after the impact have two components in the normal and the tangential direction: − − u− i = ui,n + ui,t , + + u+ i = ui,n + ui,t .
(2.154)
In case of a frictionless particle-particle collision solely the translational velocity components in collision-normal direction are changed. Thus, the tangential components remain constant after the impact and are given by: − u+ 1,t = u1,t
58
− and u+ 2,t = u2,t .
(2.155)
2. Euler-Lagrange Simulation Framework In other words, for a frictionless inter-particle collision only the normal component of the impulse vector is considered: pp fˆpp = fˆnpp = fˆn,a .
(2.156)
Thus, based on Eqs. (2.127a) and (2.127b) the normal components of the translational velocities of the colliding particles after collision are given by: m ˆ ˆpp f , m1 n,a m ˆ ˆpp = u− f , 2,n + m2 n,a
− u+ 1,n = u1,n −
u+ 2,n
(2.157)
pp where the normal component of the impulse vector fˆn,a is given by Eq. (2.129). The above equation can also be written in the following form: + u+ 1,n = u1 · n ,
(2.158)
+ u+ 2,n = u2 · n ,
+ where u+ 1,n and u2,n stand for the magnitudes of the normal post-collision velocity components. They are determined by substituting Eq. (2.133) into Eq. (2.129) and the result into Eq. (2.157):
u+ 1,n = u+ 2,n =
− − − m1 u− 1,n + m2 u2,n + m2 u2,n − u1,n en,p
m1 + m2 − − − m1 u− 1,n + m2 u2,n − m1 u2,n − u1,n en,p m1 + m2
,
(2.159)
,
where the magnitudes of the velocities before the collision in the normal direction u− i,n are given by: − u− 1,n = u1 · n , − u− 2,n = u2 · n .
(2.160)
59
CHAPTER 3
MODELING OF PARTICLE AGGLOMERATION
ω−1
ur − el
u2
−
S1
u1
−
fn,cpp Particle 1
n
c
u−2
fn,cpp
x1−
S2
x−2
Particle 2
Collision is possible (u−2 - u−1) n
ω−2
Particle-particle collision including friction and cohesion (Breuer and Almohammed, 2015).
3.1
Agglomeration Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
3.1.1
Energy-based Agglomeration Models . . . . . . . . . . . . . . . . . . . . . . . .
63
3.1.1.1
Difference of the van-der-Waals Energy . . . . . . . . . . . . . . . . . .
64
3.1.1.2
Agglomeration Model by Hiller (1981) . . . . . . . . . . . . . . . . . . .
67
3.1.1.2.1 Critical Relative Velocity and Agglomeration Condition . . . .
67
3.1.1.2.2 Application and Validity of the Model . . . . . . . . . . . . .
69
Agglomeration Model by J¨ urgens (2012) . . . . . . . . . . . . . . . . . .
69
3.1.1.3.1 Agglomeration Condition . . . . . . . . . . . . . . . . . . . .
70
3.1.1.3.2 Kinetics of Collision Partners without Agglomeration . . . . .
70
3.1.1.3.3 Application and Validity of the Model . . . . . . . . . . . . .
73
Agglomeration Model by Alletto (2014) . . . . . . . . . . . . . . . . . .
73
3.1.1.4.1 Agglomeration Condition . . . . . . . . . . . . . . . . . . . .
74
3.1.1.4.2 Kinetics of Collision Partners without Agglomeration . . . . .
75
3.1.1.4.3 Application and Validity of the Model . . . . . . . . . . . . .
79
Present Extension of the Energy-based Agglomeration Model . . . . . .
79
Momentum-based Agglomeration Models . . . . . . . . . . . . . . . . . . . . . .
80
3.1.2.1
Agglomeration Model by Weber et al. (2004) . . . . . . . . . . . . . . .
81
3.1.2.2
Agglomeration Model by Kosinski and Hoffmann (2010) . . . . . . . . .
82
3.1.2.2.1 Particle-Particle Collision Model with Cohesion . . . . . . . .
82
3.1.1.3
3.1.1.4
3.1.1.5 3.1.2
3.1.2.2.2 Model for the Cohesive Impulse . . . . . . . . . . . . . . . . .
84
3.1.2.2.3 Agglomeration Conditions
. . . . . . . . . . . . . . . . . . .
85
3.1.2.2.4 Kinetics of Collision Partners without Agglomeration . . . . .
87
3.1.2.2.5 Application and Validity of the Model . . . . . . . . . . . . .
87
Extended Momentum-based Agglomeration Model . . . . . . . . . . . .
88
3.1.2.3.1 Cohesive Impulse Model
. . . . . . . . . . . . . . . . . . . .
88
3.1.2.3.2 Intervals of the Collision Time . . . . . . . . . . . . . . . . .
91
3.1.2.3.3 Agglomeration Conditions
. . . . . . . . . . . . . . . . . . .
95
3.1.2.3.4 Kinetics of Collision Partners without Agglomeration . . . . .
96
3.1.2.3.5 Calculation Procedure
. . . . . . . . . . . . . . . . . . . . .
98
3.1.2.3.6 Particle-Particle Collisions without Friction. . . . . . . . . . .
98
3.1.2.3.7 Application and Validity of the Model . . . . . . . . . . . . .
98
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
3.1.3.1
Advantages and Drawbacks of EAM . . . . . . . . . . . . . . . . . . . .
99
3.1.3.2
Advantages and Drawbacks of MAM . . . . . . . . . . . . . . . . . . . .
99
3.1.2.3
3.1.3
3.2
3.3
3.4
Kinetics of the Exact Agglomerate Structure with Multiple Particles . . . . . . . . .
100
3.2.1
Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.2.2
Translational Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.2.3
Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Kinetics of the Two-Particle Agglomerate
. . . . . . . . . . . . . . . . . . . . . . .
104
3.3.1
Position and Translational Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.3.2
Angular Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3.3.3
Rotational Kinetic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Models for the Structure and Kinetics of the Agglomerate . . . . . . . . . . . . . .
107
3.4.1
Volume-equivalent Sphere Model (VSM) . . . . . . . . . . . . . . . . . . . . . . 107
3.4.2
Inertia-equivalent Sphere Model (ISM) . . . . . . . . . . . . . . . . . . . . . . . 110
3.4.3
Closely-Packed Sphere Model (CSM) . . . . . . . . . . . . . . . . . . . . . . . . 111
3.4.4
Porous Sphere Model (PSM) 3.4.4.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Limiting Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3 Modeling of Particle Agglomeration This chapter focuses on the modeling of the dynamic process of agglomeration of rigid, dry and electrostatically neutral solid particles due to the van-der-Waals force. It is important to note that in the following the focus is only on the modeling strategies in the framework of a hard-sphere model, since this is the main topic of the present thesis. As a starting point, the background on the model of frictional particle-particle collisions without cohesion was described in Section 2.6. To allow particle agglomeration, the cohesion between the collision partners has to be additionally taken into account. The two techniques (or agglomeration models) adopted to incorporate the cohesion due to the van-der-Waals force into the hard-sphere model are presented in Section 3.1. Then, the modeling of the kinetics of the agglomerate resulting from a successful agglomeration process is explained in Sections 3.2 and 3.3. The present chapter ends up with the concepts for the modeling of the agglomerate structure provided in Section 3.4. 3.1 Agglomeration Models
In the framework of the Eulerian–Lagrangian approach used in the present thesis the focus is on the following two ways to incorporate the cohesion between the collision partners into the hard-sphere model:
Ê The cohesion is considered in the energy conservation equation of the collision process. Ë The cohesive impulse is incorporated into a momentum-based collision model. In the present thesis these two techniques are termed the energy-based (EAM) and the momentumbased (MAM) agglomeration model, respectively. In this section a state-of-the-art review on these modeling strategies for the agglomeration of dry, electrostatically neutral particles due to the van-der-Waals force is given in Sections 3.1.1 and 3.1.2, respectively. The improvement of the energy-based agglomeration model and the extension of the momentum-based agglomeration model are presented in Sections 3.1.1.5 and 3.1.2.3, respectively. A comparison between these two agglomeration models in terms of their advantages and drawbacks is given in Section 3.1.3. The newly developed agglomeration models are implemented in LESOCC (Breuer, 1998, 2000, 2002) and used for the simulations presented in Chapter 7. 3.1.1 Energy-based Agglomeration Models
This section provides an overview on the most important models required to develop the extended energy-based agglomeration model applied in this study. The description starts with Section 3.1.1.1 presenting the derivation of the difference of the van-der-Waals energy required for the energy balance applied in these energy-based models. Then, the extensions of the simplified condition for agglomeration due to head-on collisions by Hiller (1981) carried out by J¨ urgens (2012) and Alletto (2014) are provided in Sections 3.1.1.3 and 3.1.1.4, respectively. At the end, the present improvement of the energy-based model by Alletto (2014) is reported in Section 3.1.1.5. 63
3.1 Agglomeration Models 3.1.1.1 Difference of the van-der-Waals Energy
It is commonly known that the energy between two bodies due to the van-der-Waals force depends on their geometries (see, e.g., Israelachvili, 2011). Generally, this energy is inversely proportional to the free distance δ between these bodies to the power of six (London, 1937; Hamaker, 1937): EvdW ∝ −
1 . δ6
(3.1)
In the present study two geometries are encountered, namely two spherical particles and two flat surfaces. To avoid the singularity (i.e., for δ = 0) in the van-der-Waals energy given by Eq. (3.1), the minimum distance between the bodies (the spheres or flat surfaces) during a particleparticle collision δ is assumed to have a constant value of δ0 = 2 × 10−10 m (see, e.g., Hiller, 1981; Israelachvili, 2011). For these two special cases the van-der-Waals energy EvdW can be determined as follows: EvdW =
Z∞
(3.2)
xx fvdW dδ ,
δ0
where fvdW refers to the cohesive van-der-Waals force, which depends on the geometry of two colliding bodies and is given by (Israelachvili, 2011): fvdW =
H xx f , 12 δ 2 g
(3.3)
where H denotes the material-dependent Hamaker constant, which refers to the strength of the van-der-Waals interactions between the bodies. fgxx is a factor taking the shape (or geometry) of the colliding bodies into account. In the context of the energy-based model it is assumed that the collision partners are two spheres with radii r1 and r2 , which are in contact at a the distance δ. In the case that r1 δ and r2 δ, then the geometry factor for the two spherical particles (denoted xx = pp) reads (Israelachvili, 2011): fgpp
r1 r2 . =2 r1 + r2
(3.4)
After the impact, the collision partners undergo a plastic deformation leading to a flat contact area between the spherical particles as depicted in Figure 3.1. In this case the geometry factor for the two flat surfaces (denoted xx = ww) contacting at a distance δ reads: fgww =
2A , πδ
(3.5)
where A is the cross-sectional area of the contact region. Thus, the van-der-Waals forces before and after the particle-particle collision are obtained by substituting Eqs. (3.4) and (3.5) into Eq. (3.3), respectively: − fvdW =
64
H 6 δ2
r1 r2 r1 + r2
+ and fvdW =
H A . 6 δ3 π
(3.6)
3. Modeling of Particle Agglomeration
h2
u−1
u−2
d1 h − 2 − 1
S1
S2
n d
a
−1 2
Particle 1
Particle 2
h1
Figure 3.1: Contact surface between the collision partners due to a frictionless head-on collision (Almohammed and Breuer, 2016b).
Thus, based on the definition (3.2) the difference between the van-der-Waals energy after and before the impact denoted ∆EvdW reads: ∆EvdW =
+ EvdW
−
− EvdW
=
Z∞
+ fvdW
δ0
dδ −
Z∞
− fvdW dδ .
(3.7)
δ0
By substituting the corresponding terms given by Eq. (3.6) into Eq. (3.7), the difference of the van-der-Waals energy reads: ∆EvdW
H A H = − 2 12π δ0 6 δ0
r1 r2 . r1 + r2
(3.8)
Note that since δ0 is of the order O(nm), it is clearly visible that the van-der-Waals energy before − + the collision is much smaller than that after the impact (EvdW EvdW ). According to Alletto (2014) the second term on the right-hand side of the above relation can be neglected in comparison with the first one and hence Eq. (3.8) is reduced to the following form: ∆EvdW =
H A . 12π δ02
(3.9)
In Eq. (3.9), the circular contact area A of the radius a (i.e., A = π a2 ) can be expressed in terms of the diameter d1 and the depth of the plastic deformation h1 of particle 1 as (see Figure 3.1):
d1 A=π 2
!2
d1 − − h1 2
!2
= π d1 h1 − h21 .
(3.10)
Analogously, based on the diameter d2 and the depth of the plastic deformation h2 of the particle 2 this contact area can be written as:
d2 A = π 2
!2
d2 − − h2 2
!2
= π d2 h2 − h22 .
(3.11) 65
3.1 Agglomeration Models Assuming that the particle diameters are much larger than the depths of the plastic deformation (i.e., d1 h1 and d2 h2 ), the contact area of the collision partners can be approximately determined as follows: A = π d1 h1 = π d2 h2 .
(3.12)
Thus, substituting the contact area given by Eq. (3.12) into Eq. (3.9) yields the difference of the van-der-Waals energy: ∆EvdW =
H d1 h1 . 12 δ02
(3.13)
According to Antonyuk (2006), the work required to plastically deform the collision partners is given by: Epl =
Zh1
p (π d1 h1 ) dh1 +
0
Zh2
p (π d2 h2 ) dh2 =
0
1 π p d1 h21 + d2 h22 , 2
(3.14)
where Epl stands for the plastic deformation energy, which refers to the part of the kinetic energy dissipated during the impact due to an irreversible deformation of the particles (see Figure 3.1). The symbol p denotes the maximum contact pressure, under which the collision partners undergo a plastic deformation. As will be explained in Appendix A.1, this quantity is described as a function of the mechanical properties of the particles. By substituting h2 = h1 d1 /d2 based on Eq. (3.12) into the above relation, the plastic deformation energy reads: !
1 d1 Epl = π p d1 h21 1 + . 2 d2
(3.15)
Assuming no cohesive force between the particles, the energy balance of a frictionless head-on collision in the collision-normal direction reads: − + Ekin,r,n = Ekin,r,n + Epl ,
(3.16)
− + where Ekin,r,n and Ekin,r,n denote the translational kinetic energy of the normal relative motion of the collision partners before and after the impact, respectively. They are given by (see, e.g., J¨ urgens, 2012):
i2 1 h − − · n , Ekin,r,n = m ˆ u2 − u− 1 2 i2 1 h + + Ekin,r,n = m · n , ˆ u2 − u+ 1 2
(3.17a) (3.17b)
where the effective mass m ˆ given by Eq. (2.126) can be written in terms of the densities (ρ1 and ρ2 ) and diameters of the particles as follows : m ˆ = 66
π ρ1 ρ2 d31 d32 . 6 (ρ1 d31 + ρ2 d32 )
(3.18)
3. Modeling of Particle Agglomeration Thus, by inserting Eqs. (3.17a) and (3.17b) as well as the definition of the normal restitution coefficient en,p given by Eq. (2.132) into Eq. (3.16), the plastic deformation energy due to a frictionless head-on collision reads (Hiller, 1981):
− Epl = 1 − e2n,p Ekin,r,n .
(3.19)
By substituting Eq. (3.17a) into Eq. (3.19), the plastic deformation energy can be also expressed as: Epl =
h i2 1 − m ˆ 1 − e2n,p u− 2 − u1 · n 2
(3.20)
Thus, the depth of the plastic deformation h1 is obtained by inserting Eq. (3.18) into Eq. (3.20) and equaling the resulting equation with Eq. (3.15): h1 = d1 d22
h
i
− u− 2 − u1 · n
ρ1 ρ2 1 − e2n,p
6 p¯ (d1
1/2
+ d2 ) (ρ1 d31 + ρ2 d32 )
(3.21)
.
This relation implies that h1 depends on the material properties of the collision partners (the density and the maximum contact pressure) as well as the normal restitution coefficient. Now, the final relation for the difference of the van-der-Waals energy is obtained by substituting Eq. (3.21) into Eq. (3.13): ∆EvdW
1/2
i ρ1 ρ2 1 − e2n,p H 2 2 h − − =− d d u − u · n 2 1 6 p¯ (d1 + d2 ) (ρ1 d31 + ρ2 d32 ) 12 δ02 1 2
,
(3.22)
where the minus sign appears in the above relation to result in a positive value of ∆EvdW , since − [(u− 2 − u1 ) · n] < 0 as mentioned in Section 2.6.1.1. It is worth mentioning that for a fully elastic collision (en,p = 1), where no kinetic energy is dissipated during the impact, the difference of the van-der-Waals energy given by Eq. (3.22) vanishes. This result is consistent with the extended momentum-based agglomeration model, which will be presented in Section 3.1.2.3. 3.1.1.2 Agglomeration Model by Hiller (1981)
As mentioned in Section 1.4, the first energy-based agglomeration model due to a frictionless head-on particle-particle collision was proposed by Hiller (1981). In the framework of this model the collision partners agglomerate if the normal relative velocity between the particles does not exceed a critical value derived next. 3.1.1.2.1 Critical Relative Velocity and Agglomeration Condition
It is well known that the van-der-Waals force acts only in the collision-normal direction. Thus, the derivation of the critical velocity was based on the following energy balance of the collision process in the normal direction taking the cohesion into account: − − + + E˜kin,r,n + EvdW = E˜kin,r,n + EvdW + ∆Edis ,
(3.23) 67
3.1 Agglomeration Models − + where E˜kin,r,n and E˜kin,r,n stand for the normal relative translational kinetic energy before and after the collision, in which the cohesion between the colliding particles is considered. The symbol ∆Edis denotes the kinetic energy dissipated during the impact. Hiller (1981) determined ∆Edis by assuming that the whole dissipated energy is transformed into plastic deformation. Hence, it is equal to the plastic deformation energy given by Eq. (3.19) (i.e., ∆Edis = Epl ). In addition, since before the impact the collision partners are only marginally affected by the cohesive force, it is assumed that the pre-collision kinetic energies appearing in Eq. (3.23) are equal to − − the corresponding ones without taking the cohesion into account (i.e., E˜kin,r,n = Ekin,r,n ). Thus, relation Eq. (3.23) can be written as:
− − + Ekin,r,n − E˜kin,r,n − 1 − e2n,p Ekin,r,n = ∆EvdW ,
(3.24)
− + where the difference of the van-der-Waals energy ∆EvdW = EvdW − EvdW is given by Eq. (3.22). If the cohesion between the collision partners is strong enough, they agglomerate and hence the + post-collision relative kinetic energy vanishes (i.e., E˜kin,r,n = 0). The reason is that the particles stick together and the newly formed two-particle system (or agglomerate) moves with the velocity of its center of mass uag (see Section 3.3). Thus, the above relation can be written for the limiting case as:
− − Ekin,r,n − 1 − e2n,p Ekin,r,n = ∆EvdW .
(3.25)
− By eliminating Ekin,r,n from the left-hand side of the above equation, it reduces to the following form: − e2n,p Ekin,r,n = ∆EvdW .
(3.26)
By substituting Eqs. (3.17a) and (3.22) into Eq. (3.26), the critical (or limiting) relative velocity, at which agglomeration occurs, is determined as follows: h
u− 2
−
u− 1
i
·n
crit
H 1 =− 2 π δ0 d1 d2
(
(ρ1 d31 + ρ2 d32 ) 6 p¯ ρ1 ρ2 (d1 + d2 )
1 − e2n,p en,p
!)1/2
.
(3.27)
If the normal velocity between the collision partners is smaller than this limiting value, an agglomerate is formed. Otherwise, the colliding particles separate after the collision according to the description in Section 2.6. Expressed as an energy balance, the agglomeration condition reads: − ∆EvdW ≥ e2n,p Ekin,r,n ,
(3.28)
where the difference of the van-der-Waals energy is given by Eq. (3.22). If the above criterion is satisfied, the velocity and the position of the newly formed agglomerate are determined as explained in Section 3.3.1. In addition, the structure of the agglomerate is modeled based on the volume-equivalent sphere model presented in Section 3.4.1. 68
3. Modeling of Particle Agglomeration 3.1.1.2.2 Application and Validity of the Model
This agglomeration model was applied by Hiller (1981) to investigate the particle deposition in fiber filters. Furthermore, in the context of four-way coupled Euler-Lagrange RANS simulations using the stochastic collision detection model proposed by Sommerfeld (2001), this simple agglomeration model was adopted by Ho and Sommerfeld (2002), Ho (2004), Blei (2006), Sommerfeld (2010), St¨ ubing and Sommerfeld (2010) and Lipowsky (2013) to study the agglomeration of solid particles. In addition, Ho and Sommerfeld (2003, 2005) applied this model to investigate the influence of the particle agglomeration on the separation efficiency of a gas cyclone separator using a very low number of particles (Np = 1000). They found that the application of the agglomeration model leads to a higher separation efficiency of fine particles (smaller than 2 µm). On the other hand, Ho (2004) applied this model to a vertical turbulent mixing layer laden with a huge number of particles of various classes at different mass loadings and inflow velocity ratios. He concluded that the predictions of this model are in reasonable agreement with the experiments for a suitable value of the restitution coefficient for particle-particle collisions. However, the assumption of frictionless head-on particle-particle collisions in the above mentioned studies implies that the particle rotation is neglected, which is a rather crude constriction not justified by “real-world” conditions. Furthermore, if the agglomeration condition is not satisfied, the kinetics of the collision partners after the impact has to be treated taking the cohesion into account. To avoid these drawbacks, the model by Hiller (1981) was extended as presented next. 3.1.1.3 Agglomeration Model by Jurgens ¨ (2012)
J¨ urgens (2012) first extended the energy-based model by Hiller (1981) towards oblique collisions allowing relative tangential velocities at the contact point. For this purpose, J¨ urgens (2012) slightly extended the agglomeration condition of the original model by Hiller (1981) and focused on the treatment of the collision partners without agglomeration taking the cohesion into account. Note that the friction between particles during the collision process is still neglected. In this model the energy balance of the collision process is given by: − − + + E˜kin,r + EvdW = E˜kin,r + EvdW + ∆Edis ,
(3.29)
− + where E˜kin,r and E˜kin,r stand for the relative translational kinetic energy before and after the collision taking the cohesion into account. Analog to the assumption made in Section 3.1.1.2, the − pre-collision relative kinetic energy E˜kin,r is assumed to be identical to that of the collision without − − cohesion (i.e., E˜kin,r = Ekin,r ). In addition, the dissipated kinetic energy ∆Edis is again set equal to the plastic deformation energy given by Eq. (3.19) (i.e., ∆Edis = Epl ). Thus, the energy balance given by Eq. (3.29) can be written as:
− − + Ekin,r − E˜kin,r − 1 − e2n,p Ekin,r,n = ∆EvdW ,
(3.30)
− + where the relative kinetic energies Ekin,r and E˜kin,r are defined by:
2 1 − m ˆ u2 − u− , 1 2 2 1 + ˜2 − u ˜+ , ˆ u = m 1 2
− Ekin,r =
(3.31)
+ E˜kin,r
(3.32) 69
3.1 Agglomeration Models ˜+ ˜+ where u 1 and u 2 denote the post-collision velocities of the colliding particles when the cohesion is taken into account. These are still unknown. 3.1.1.3.1 Agglomeration Condition
Assuming that the collision partners agglomerate, the post-collision relative motion between the + particles vanishes and hence E˜kin,r = 0. Thus, the agglomeration condition can be expressed based on Eq. (3.30) as follows (J¨ urgens, 2012): − − ∆EvdW ≥ Ekin,r − (1 − e2n,p ) Ekin,r,n ,
(3.33)
where the difference of the van-der-Waals energy is given by Eq. (3.22). If the above criterion is satisfied, the velocity and the position of the newly formed agglomerate are determined. In addition, the structure of the agglomerate is modeled based on the volume-equivalent sphere model. Note that the main difference between the agglomeration conditions based on J¨ urgens (2012) and Hiller (1981) is that J¨ urgens (2012) considered both the normal and the tangential component of − − − the relative kinetic energy. Thus, by substituting Ekin,r = Ekin,r,n + Ekin,r,t into the agglomeration condition (3.33) by J¨ urgens (2012), it can be expressed as: − − ∆EvdW ≥ e2n,p Ekin,r,n + Ekin,r,t ,
(3.34)
− where Ekin,r,t is the tangential relative kinetic energy before the impact and is defined as follows: − Ekin,r,t =
i2 1 h − m ˆ u2 − u− . 1 ·t 2
(3.35)
When comparing Eqs. (3.34) and (3.28), the agglomeration condition by J¨ urgens (2012) reduces to − that by Hiller (1981) if Ekin,r,t in Eq. (3.34) is set equal to zero (i.e., the tangential relative motion vanishes). 3.1.1.3.2 Kinetics of Collision Partners without Agglomeration
If the agglomeration condition given by Eq. (3.33) is not satisfied, the collision partners bounce + back, since the relative kinetic energy after the impact is no longer zero (E˜kin,r 6= 0). Thus, the ˜+ ˜+ post-collision velocities of the particles taking the cohesion into account (i.e., u 1 and u 2 ) are determined as follows: m ˆ ˆpp f , m1 tot m ˆ ˆpp − ˜+ f , u 2 = u2 + m2 tot − ˜+ u 1 = u1 −
(3.36a) (3.36b)
pp pp pp where fˆtot = fˆtot,n + fˆtot,t is the total impulse vector including the cohesion. The normal and the tangential component of the total impulse vector are given by: pp pp pp pp fˆtot,n = fˆtot,n n and fˆtot,t = fˆtot,t t,
70
(3.37)
3. Modeling of Particle Agglomeration pp pp where fˆtot,n and fˆtot,t are the magnitudes of the normal and the tangential component of the total impulse vector, respectively. By subtracting Eqs. (3.36b) and (3.36a) from each other, the total pp impulse vector fˆtot reads:
pp − − ˜+ ˜+ fˆtot = u 2 −u 1 − u2 − u1 .
(3.38)
pp pp Hence, the magnitudes fˆtot,n and fˆtot,n can be written as: pp pp fˆtot,n = fˆtot ·n= pp pp = fˆtot ·t = fˆtot,t
h
h
− − ˜+ ˜+ u 2 −u 1 − u2 − u1
− − ˜+ ˜+ u 2 −u 1 − u2 − u1
i
i
· n, · t,
(3.39)
where the tangential unit vector t is given by Eq. (2.148), but without rotation of the collision partners when calculating u− c,t,r . As a result of the reduction of the normal and the tangential impulse vector due to the inclusion of the van-der-Waals forces, the restitution coefficients along these directions have to be modified. In this model the modified restitution coefficient in the normal direction e˜n,p has the following form:
e˜n,p = −
˜+ ˜+ u 2 −u 1 ·n
− u− 2 − u1 · n
(3.40)
.
In addition, the modified restitution coefficient in the tangential direction e˜t,p is defined as:
e˜t,p =
˜+ ˜+ u 2 −u 1 ·t
− u− 2 − u1 · t
(3.41)
.
Note that the definition of e˜t,p implies that the magnitude of the tangential relative velocity after the collision is reduced, but its sign is not inverted. By substituting Eqs. (3.40) and (3.41) into Eq. (3.39) the magnitudes of the components of the total impulse can be expressed as: pp fˆtot,n = − (1 + e˜n,p ) pp fˆtot,t = − (1 − e˜t,p )
h
i
− u− 2 − u1 · n ,
h
− u− 2 − u1 · t
i
.
(3.42a) (3.42b)
Thus, the post-collision velocities can be obtained by substituting Eq. (3.37) into Eq. (3.36): o m ˆ n ˆpp pp ftot,n n + fˆtot,t t , m1 o m ˆ n ˆpp pp − ˜+ u ftot,n n + fˆtot,t t , 2 = u2 + m2
− ˜+ u 1 = u1 −
(3.43)
pp pp where fˆtot,n and fˆtot,n are determined based on Eqs. (3.42a) and (3.42b), respectively. Note that only one equation for the energy balance (3.29) is available, whereas Eqs. (3.42a) and (3.42b) have two unknowns (i.e., e˜n,p and e˜t,p ). Thus, further assumptions are required to determine the modified restitution coefficients as explained in the following.
71
3.1 Agglomeration Models As mentioned before, in model by J¨ urgens (2012) the relative translational kinetic energies given by Eqs. (3.31) and (3.32) are split into normal and tangential components. Thus, the energy balance (3.30) reads:
− − − + + Ekin,r,n + Ekin,r,t = ∆EvdW + E˜kin,r,n + E˜kin,r,t + 1 − e2n,p Ekin,r,n ,
(3.44)
− − + + Ekin,r,t = ∆EvdW + E˜kin,r,n + E˜kin,r,t − e2n,p Ekin,r,n ,
(3.46)
− + where Ekin,r,t is given by Eq. (3.35). E˜kin,r,t stands for the tangential relative kinetic energy after the impact taking the cohesion into account and is defined as follows: i2 1 h + + ˜2 − u ˜+ E˜kin,r,t = m ˆ u . (3.45) 1 ·t 2 − The relation (3.44) can be simplified to the following form by eliminating Ekin,r,n from both sides:
+ + Obviously, this energy balance includes two unknowns (E˜kin,r,n and E˜kin,r,t ). To close this equation, the following two cases are distinguished depending on the direction, along which the colliding particles are assumed to rebound.
Rebound in the Collision-Normal Direction
In this case it is assumed that the collision partners bounce back solely in the collision-normal direction. This assumption implies that only the normal component of the relative motion between the collision partners is affected by the cohesive force. On the other hand, it means that the − + tangential component of the relative velocity remain unchanged (Ekin,r,t = E˜kin,r,t ) and hence the energy balance (3.46) can be rearranged as follows: − + E˜kin,r,n = e2n,p Ekin,r,n − ∆EvdW .
(3.47)
Note that based on the definition (3.40), the square of the modified restitution coefficient in the normal direction can be written as: h
i2
+ E˜kin,r,n e˜2n,p = h = . i2 − − − E kin,r,n u2 − u1 · n
˜+ ˜+ u 2 −u 1 ·n
(3.48)
Substituting relation (3.47) into Eq. (3.48), the square of the modified restitution coefficient in the normal direction is determined as follows: E˜ + ∆EvdW e˜2n,p = kin,r,n = e2n,p − − . (3.49) − Ekin,r,n Ekin,r,n J¨ urgens (2012) assumed that if e˜2n,p > 0, the modified restitution coefficient in the normal direction required for Eq. (3.42a) is calculated based on the above relation, such that: e˜n,p =
(
e2n,p
∆EvdW − − Ekin,r,n
)1/2
.
(3.50)
It is worth mentioning that this relation implies that the normal restitution coefficient is reduced due to the cohesion. This is a reasonable result for the role of the cohesive force, since it decreases the relative motion during the impact. In addition, the modified restitution coefficient in the tangential direction required for Eq. (3.42b) is set to unity (i.e., e˜t,p = 1), which means that the pp magnitude of the tangential component of the total impulse vector vanishes (i.e., fˆtot,t = 0). 72
3. Modeling of Particle Agglomeration Rebound in the Collision-Tangential Direction + If e˜2n,p ≤ 0 results from Eq. (3.49), it is assumed that E˜kin,r,n = 0 implying that the attractive van-der-Waals force is strong enough to overcome the relative kinetic energy in the collision-normal direction. Hence, based on the definition (3.48) the normal restitution coefficient required for Eq. (3.42a) is set equal to zero (i.e., e˜n,p = 0). Thus, the collision partners bounce back only in the collision-tangential direction and hence the energy balance (3.46) reads: − − + Ekin,r,t = ∆EvdW + E˜kin,r,t − e2n,p Ekin,r,n .
(3.51)
− By dividing this relation by Ekin,r,t , the square of the modified restitution coefficient in the tangential direction given by Eq. (3.41) reads:
e˜2t,p
( ) − + E˜kin,r,t ∆EvdW − e2n,p Ekin,r,n = − =1− . − Ekin,r,t Ekin,r,t
(3.52)
Thus, the modified restitution coefficient e˜t,p required for Eq. (3.42a) is given by: e˜t,p
(
− ∆EvdW − e2n,p Ekin,r,n = 1− − Ekin,r,t
!)1/2
.
(3.53)
3.1.1.3.3 Application and Validity of the Model
J¨ urgens (2012) applied his extended model to investigate the agglomeration process in particleladen turbulent channel and pipe flows at different mass loadings. Unfortunately, the predictions of the extended model were not validated or even compared to the original model by Hiller (1981) to evaluate the impact of the model improvement on the results. Later on, this model was further extended by Alletto (2014) to include the friction between the particles as explained next. 3.1.1.4 Agglomeration Model by Alletto (2014)
In the recent study of Alletto (2014) the energy-based model by Hiller (1981) and J¨ urgens (2012) was extended by taking the friction between the colliding particles into account. For this purpose, the rotational kinetic energies of the collision partners and the resulting agglomerate are considered in the energy balance leading to a “generalized” agglomeration condition. In this model the energy balance of the collision process reads: − − + + − + E˜kin,r + E˜rot + EvdW = E˜kin,r + E˜rot + EvdW + ∆Edis ,
(3.54)
− + where E˜rot and E˜rot stand for the rotational kinetic energy before and after the collision if the cohesion is considered. As assumed by Hiller (1981) and J¨ urgens (2012), the kinetic energy dissipated during the collision ∆Edis is determined in this model by neglecting the cohesion between the collision partners and hence the colliding particles rebound after the impact (Alletto, 2014), such that:
− + + − ∆Edis = Ekin,r + Erot , − Ekin,r + Erot
|
{z
− Ekin,tot
}
|
{z
+ Ekin,tot
}
(3.55)
73
3.1 Agglomeration Models where the terms on the right-hand side refer to total kinetic energy before and after the impact − + denoted Ekin,tot and Ekin,tot , which is the sum of the translational and rotational kinetic energies when the cohesion is not taken into account. In analogy to the assumptions made in the previous − model regarding the translational kinetic energy, the pre-collision rotational energy E˜rot appearing − − in Eq. (3.54) is assumed to be equal to the energy without considering the cohesion (i.e., E˜rot = Erot ). Since the particles are only marginally influenced by the cohesive forces before the impact, this − + assumption is valid. The sum of the rotational kinetic energies of both particles Erot , Erot and + E˜rot remaining in Eq. (3.54) are given by: 1 − 2 I1 ω1 + 2 1 2 = I1 ω1+ + 2 1 + 2 ˜1 + = I1 ω 2
− Erot = + Erot + E˜rot
1 − 2 , I2 ω2 2 1 + 2 I2 ω2 , 2 1 + 2 ˜2 . I2 ω 2
(3.56a) (3.56b) (3.56c)
− − By substituting Eq. (3.55) into Eq. (3.54) and eliminating E˜kin,r and E˜rot from the resulting relation, the final energy balance reads: + + + + ∆EvdW = Ekin,r − E˜kin,r + Erot − E˜rot .
(3.57)
3.1.1.4.1 Agglomeration Condition
If the collision partners agglomerate (or stick together), the relative motion vanishes and hence the relative translational kinetic energy after the impact appearing in Eq. (3.57) is set equal to + + zero (i.e., E˜kin,r = 0). In addition, the rotational kinetic energy E˜rot is set equal to the rotational kinetic energy of the two-particle system (i.e., the resulting agglomerate) Eag,rot , which will be derived in Section 3.3.3. Thus, the agglomeration criterion for the case including the friction between the colliding particles is given by (Alletto, 2014): + + ∆EvdW ≥ Ekin,r + Erot − Eag,rot ,
(3.58)
+ + where ∆EvdW , Erot and Eag,rot are given by Eqs. (3.22), (3.56b) and (3.176), respectively. Ekin,r stands for the translational kinetic energy without cohesion and is defined by: + Ekin,r =
2 1 + . m ˆ u2 − u+ 1 2
(3.59)
Physically, the agglomeration criterion (3.58) implies that the collision partners only agglomerate if the cohesion due to the van-der-Waals force between the colliding particles is strong enough. If the agglomeration criterion given by Eq. (3.58) is satisfied, the kinetics of the newly formed agglomerate is determined as explained in Section 3.3. In addition, the structure of the agglomerate is modeled based on the volume-equivalent sphere model presented in Section 3.4.1. It is worth to mention that to determine the kinetic energies after the impact in Eq. (3.58) (i.e., + + + + + Ekin,r and Erot ), the post-collision translational (i.e., u+ 1 , u2 ) and angular (i.e., ω1 , ω2 ) velocities can be calculated without considering the cohesion between the collision partners. As explained 74
3. Modeling of Particle Agglomeration in Section 2.6.1.3, these quantities are calculated using Eq. (2.151) for a sticking collision and Eq. (2.153) for a sliding collision. Note that the difference of the van-der-Waals energy ∆EvdW in Eq. (3.58) is determined as done before (see Section 3.1.1.1). 3.1.1.4.2 Kinetics of Collision Partners without Agglomeration
Assuming that the agglomeration condition (3.58) is not satisfied, the colliding particles are pushed apart from each other due to resulting restitution impulse, which has the same magnitude but is oppositely directed. In this case the translational and angular velocities of the collision partners have to be determined taking the cohesive force into account, since it affects the resulting impulse pp significantly. For this purpose, a cohesive impulse denoted fˆag was introduced by Alletto (2014) to take the van-der-Waals interaction into account. In this model the post-collision translational velocities of the collision partners are calculated based on the conservation of the translational momentum as follows (Alletto, 2014): m ˆ ˆpp f , m1 ag m ˆ ˆpp + ˜+ u f . 2 = u2 + m2 ag + ˜+ u 1 = u1 −
(3.60a) (3.60b)
In addition, Alletto (2014) assumed that the post-collision angular velocities are proportional to those without taking the collision into account with the factors kω1 and kω2 , respectively: ˜ 1+ = kω1 ω1+ , ω
˜ 2+ = kω2 ω2+ . ω
(3.61a) (3.61b)
Thus, the calculation procedure of this model including the friction and the cohesion between the collision partners is divided into two steps:
Ê The cohesion is not taken into account and the post-collision translational (i.e., u+1 and u+2 )
and angular (i.e., ω1+ and ω2+ ) velocities required for Eqs. (3.60) and (3.61) are determined based on the collision type as presented in Section 2.6.1.3. In other words, for a sticking and sliding collision these intermediate quantities are given by Eqs. (2.151) and (2.153), respectively.
Ë The final post-collision quantities with cohesion (i.e., u˜ +1 , u˜ +2 , ω˜ 1+ and ω˜ 2+ ) are determined
based on Eqs. (3.60) and (3.61). It is worth noting that in this step the modeling is achieved analogously to the model by J¨ urgens (2012), but taking the friction between the collision pp partners into account as explained next. The calculation of the cohesive impulse vector fˆag as well as the factors kω1 and kω2 required for Eqs. (3.60) and (3.61) is explained next.
Model for the Cohesive Impulse
The cohesive impulse vector appearing in Eq. (3.60) is split into two components in the normal and tangential direction: pp pp pp fˆag = fˆag,n + fˆag,t .
(3.62) 75
3.1 Agglomeration Models The components of the cohesive impulse vector are given by: pp pp pp pp fˆag,n = fˆag,n n and fˆag,t = fˆag,t t,
(3.63)
pp pp where fˆag,n and fˆag,n are the magnitudes of the normal and the tangential cohesive impulse vector, respectively. It is important to note that if the cohesion is taken into account during the impact, pp refers to a tangential impulse due to the increasing friction at the contact point. Thus, by fˆag,t inserting Eq. (3.63) into Eq. (3.60), it can be expressed as:
o m ˆ n ˆpp pp fag,n n + fˆag,t t , m1 o m ˆ n ˆpp pp + ˜+ u fag,n n + fˆag,t t . 2 = u2 + m2 + ˜+ u 1 = u1 −
(3.64)
pp The total cohesive impulse fˆag can be determined by subtracting Eqs. (3.60b) and (3.60a) from each other, such that:
pp + + ˜+ ˜+ fˆag = u 2 −u 1 − u2 − u1
(3.65)
pp pp Thus, the magnitudes of the cohesive impulse vector fˆag,n and fˆag,n can be written as: pp pp fˆag,n = fˆag ·n= pp pp fˆag,t = fˆag ·t =
h
h
− − ˜+ ˜+ u 2 −u 1 − u2 − u1
− − ˜+ ˜+ u 1 − u2 − u1 2 −u
i
i
· n,
(3.66)
· t,
where the tangential unit vector t is given by Eq. (2.148). In this model the reduction of the relative velocities of the collision partners in both normal and tangential direction is accounted for by introducing the following new factors kn,p and kt,p defined by Alletto (2014) as:
˜+ ˜+ u 2 −u 1 ·n
kn,p = + u2 − u+ 1 ·n
and kt,p =
h
+ u+ 2 − u1 · n ,
˜+ ˜+ u 2 −u 1 ·t
+ u+ 2 − u1 · t
.
(3.67)
pp pp By substituting kn,p and kt,p into Eq. (3.66), the magnitudes fˆag,n and fˆag,n can be expressed as follows: pp fˆag,n = − (1 − kn,p ) pp fˆag,t = − (1 − kt,p )
h
+ u+ 2 − u1 · t
i
i
.
(3.68a) (3.68b)
Determination of the Final Post-Collision Quantities
˜+ ˜+ ˜ 1+ and ω ˜ 2+ ) given by Eqs. (3.64) and To calculate the final velocities after the impact (u 1,u 2,ω (3.61), four factors (kn,p , kt,p , kω1 and kω2 ) have to be first determined. However, only one equation for the energy balance given by Eq. (3.57) is available. Thus, to determine these factors, the following model was proposed by Alletto (2014). 76
3. Modeling of Particle Agglomeration At first, the translational kinetic energies in the energy balance of the collision process (3.57) are written in terms of their normal and tangential components, such that: + + + + + + ∆EvdW = Ekin,r,n − E˜kin,r,n + Ekin,r,t − E˜kin,r,t + Erot − E˜rot .
(3.69)
+ + + Note that the above relation includes three unknowns, namely E˜kin,r,n , E˜kin,r,t and E˜rot . To simplify this problem, the following three different cases were distinguished to allow the determination of the four factors (kn,p , kt,p , kω1 and kω2 ) required for Eqs. (3.64) and (3.61) based only on the energy balance (3.69).
Ê It is assumed that the collision partners bounce back only in the normal direction. In other
words, this assumption suggests that only the normal component of the relative motion and hence the kinetic energy in this direction is affected by the van-der-Waals force. Consequently, the tangential translation and the rotational relative energies appearing in Eq. (3.69) remain + + + + unchanged (i.e., E˜kin,r,t = Ekin,r,t and E˜rot = Erot ). Thus, based on these assumptions the energy balance (3.69) is reduced to the following form: + + ∆EvdW = Ekin,r,n − E˜kin,r,n .
(3.70)
According to Eq. (3.67) the square of the factor kn,p can be written in terms of the kinetic energy as follows: 2 kn,p
h
i2
+ E˜kin,r,n = h . i2 = + + Ekin,r,n u+ 2 − u1 · n
˜+ ˜+ u 2 −u 1 ·n
(3.71)
In addition, the the square of the factor kt,p reads: 2 kt,p
h
i2
+ E˜kin,r,t . = h i2 = + + Ekin,r,t u+ − u · t 2 1
˜+ ˜+ u 2 −u 1 ·t
(3.72)
+ + Resolving Eq. (3.70) for Ekin,r,n and dividing the result by E˜kin,r,n yields based on Eq. (3.71) the 2 square of the factor kn,p : 2 kn,p
+ E˜kin,r,n ∆EvdW = + =1− + , Ekin,r,n Ekin,r,n
(3.73)
2 If based on Eq. (3.73) kn,p > 0, the factor kn,p is set to this value. According to Eq. (3.72) the pp + + factor kt,p is equal to unity if E˜kin,r,t = Ekin,r,t and hence based on Eq. (3.68b) fˆag,t = 0. In addition, the coefficients kω1 and kω2 in Eq. (3.61) are also set to unity, since in this case the + + rotational kinetic energy is not influenced by the cohesive force (i.e., E˜rot = Erot ). In summary, for these assumptions the collision partners rebound only in the collision-normal direction and the cohesive force does not enhance the friction at the contact point. Thus, the tangential component of the cohesive impulse vanishes and hence the angular velocities remain the same as those calculated using the standard hard-sphere model for particle-particle collision presented ˜ i+ = ωi+ ). in Section 2.6.1 (i.e., ω
77
3.1 Agglomeration Models 2 Ë If kn,p < 0 results from Eq. (3.71), it is assumed that collision partners bounce back only in the
collision-tangential direction. In this case the normal translational kinetic energy vanishes (i.e., + E˜kin,r,n = 0) and the rotational kinetic energy is not affected by the cohesive force and hence + + Erot = E˜rot . These assumptions mean that the energy balance (3.69) can be written as follows: + + + ∆EvdW = Ekin,r,n + Ekin,r,t − E˜kin,r,t .
(3.74)
+ + Thus, resolving Eq. (3.74) for E˜kin,r,t and dividing the result by Ekin,r,t yields based on Eq. (3.72) 2 the square of the factor kn,t :
2 kt,p =
+ + E˜kin,r,t ∆EvdW − Ekin,r,n = 1 − . + + Ekin,r,t Ekin,r,t
(3.75)
2 If based on Eq. (3.75) kt,p > 0, the factor kt,p is set accordingly. Based on Eq. (3.71) the + ˜ assumption of Ekin,r,n = 0 leads to kn,p = 0. This means that the normal cohesive impulse is strong enough to hold the collision partners in contact, but they rebound in the tangential pp direction, since in this case 0 < kt,p < 1 and hence fˆag,t 6= 0. In addition, analog to the first case, the rotational kinetic energy and hence the angular velocities predicted by the hard-sphere ˜ i+ = ωi+ . model remain unchanged implying that kω1 = kω2 = 1 and hence ω 2 2 Ì If based on the above equations kn,p < 0 and kt,p < 0, it is assumed that the components of the
+ + relative kinetic energy after the impact vanish and hence E˜kin,r,n = E˜kin,r,t = 0. In other words, this implies that the collision partners bounce back neither in the normal nor in the tangential direction. Thus, they roll or slide over each other during the collision in case of a sticking of sliding collision, respectively. Accordingly, the energy balance of the collision process (3.69) reads: + + + + ∆EvdW = Ekin,r,n + Ekin,r,t + Erot − E˜rot .
(3.76)
Rearranging the above equation with respect to the difference of the rotational kinetic energy + + (Erot − E˜rot ) and substituting the corresponding terms given by Eqs. (3.56b) and (3.56c) into the resulting equation yields: 1 1 + + + 2 + 2 + 2 + 2 ˜1 ˜2 I1 ω1 − ω + I2 ω2 − ω = ∆EvdW − Ekin,r,n − Ekin,r,t . (3.77) 2 2
˜ 1+ and ω ˜ 2+ ). To avoid this However, the above relation still has two unknown quantities (ω problem, a dimensionless factor denoted β was introduced by Alletto (2014). It is defined as the ratio of the rotational energy of particle 1 to the total rotational energy of both particles after the collision: 1 + 2 I1 ω1 β = 1 2 2 1 2 . I1 ω1+ + I2 ω2+ 2 2 78
(3.78)
3. Modeling of Particle Agglomeration By substituting Eq. (3.78) into Eq. (3.77), the difference of the rotational kinetic energy of both collision partners without and with the consideration of the cohesive force reads:
n o 2 2 1 + + ˜ 1+ = β ∆EvdW − Ekin,r,n − Ekin,r,t , I1 ω1+ − ω 2 n o 2 2 1 + + ˜ 2+ = (1 − β) ∆EvdW − Ekin,r,n − Ekin,r,t . I2 ω2+ − ω 2
(3.79a) (3.79b)
By dividing Eqs. (3.79a) and (3.79a) by the rotational kinetic energy of the corresponding particle without taking the cohesion into account, the square of the factors kω2 and kω2 in Eq. (3.61) can be written as follows: kω2 1 = 1 − kω2 2
2β
I1 ω1+
2
n
o
+ + ∆EvdW − Ekin,r,n − Ekin,r,t ,
o 2(1 − β) n + + = 1 − 2 ∆EvdW − Ekin,r,n − Ekin,r,t . I2 ω2+
(3.80)
According to the above assumptions of this case, the factors kn,p and kt,p are set to zero based on Eqs. (3.71) and (3.72), respectively. In addition, the angular velocities of the collision partners are reduced by the factors kω1 and kω2 determined by relation (3.80). 3.1.1.4.3 Application and Validity of the Model
To the best of the authors knowledge, only J¨ urgens (2012) and Alletto (2014) used coupled EulerLagrange LES predictions with a deterministic collision model and an energy-based agglomeration model to analyze the particle agglomeration in turbulent flows. However, a profound validation of this agglomeration model is missing, since Alletto (2014) only carried out an a-priori analysis and then an a-posteriori evaluation with application to a downward pipe flow at low Reynolds numbers. To overcome the modeling shortcomings (see Section 3.1.1.5) of the energy-based agglomeration model by Alletto (2014), this model is further improved in the present study as explained next. 3.1.1.5 Present Extension of the Energy-based Agglomeration Model
Although Alletto (2014) extended the energy-based agglomeration model to a general case, it still has the following modeling drawbacks. As explained in Section 3.1.1.4.2, three different cases for the treatment of the kinetics of the collision partners without agglomeration taking the cohesion into account were discussed. However, this model has two main shortcomings: 2 2 • As explained before, the third case occurs if kn,p < 0 and kt,p < 0. These conditions indicate that the cohesion between the collision partners is strong enough to keep the two particles in contact. Consequently, they do not rebound after the collision as assumed by Alletto (2014), but rather roll or slide over each other during a sticking or a sliding collision. For this reason, this case refers to an agglomeration process for both collision types and the particle-pair forms an agglomerate. Thus, in the present thesis a new condition is added to Eq. (3.58),
79
3.1 Agglomeration Models such that the modified agglomeration conditions read: + + ∆EvdW ≥ Ekin,r + Erot − Erot,ag
or
2 2 kn,p < 0 and kt,p Bcrit (Wec ) .
(4.92)
It was explained in Section 4.2.2.4 that different correlations for the bounding curve between the fast coalescence (III) and the stretching separation (V) are available in the literature. The focus of this section is on the most appropriate criterion for this boundary. As visible in Figure 4.7, for ∆ = 1.0 the experiments by Kuschel and Sommerfeld (2013) showed that for stretching separation the predictions (red curve) of the correlation by Ashgriz and Poo (1990) agree more closely with the experiments than the correlation by Brazier-Smith et al. (1972) (orange curve) used in the stochastic model by O’Rourke (1981). Nevertheless, the results of Brazier-Smith et al. (1972) are still in very good agreement with the experiments of Kuschel and Sommerfeld (2013) and Estrade et al. (1999) as displayed in Figures 4.7 and 4.8, respectively. However, the correlation by Ashgriz and Poo (1990) leads to unphysical predictions for small size ratios, especially at high Weber numbers7 . To confirm this statement, this correlation is analyzed for various size ratios and compared with the corresponding results by Brazier-Smith et al. (1972). It is important to note that the main test cases investigated in the present study (see Chapter 8) are spray systems, where high collision Weber numbers and various size ratios are encountered, especially in case of polydisperse size distribution of the droplets. As evidenced in Figure 4.9, for ∆ = 0.5 the correlation by Ashgriz and Poo (1990) yields unrealistic results if Wec & 195 and this problem becomes even worse for ∆ = 0.25, while the predictions of Brazier-Smith et al. (1972) are realistic leading to a larger area of the fast coalescence when reducing the size ratio as observed in most experiments. Another reason for the application of the correlation of Brazier-Smith et al. (1972) is that due to the complex formulation of the correlation (4.54) by Ashgriz and Poo (1990) it is very difficult 7
146
Ashgriz and Poo (1990) conducted experiments with Wec ≤ 100 and 0.5 ≤ B ≤ 1.0 (see Table B.1).
4. Modeling of Droplet Coalescence to estimate the value of Bcrit (Wec ) required for the calculation of the dissipation factor fBss for stretching separation explained next. 1.0
Brazier-Smith et al. (1972) Ashgriz and Poo (1990)
Bouncing (II)
0.8
0.6
B
Stretching Separation (V)
0.4
Fast Coalescence (III) 0.2
Reflexive Separation (IV) 0.0 0
25
50
75
100
125
150
Wec Figure 4.7: Comparison of the models for the critical impact parameter Bcrit and the experimental results of stretching separation by Kuschel and Sommerfeld (2013) (blue symbols) using water droplets with ∆ = 1.0. Dashed black lines represent the boundaries of the bouncing regime by Estrade et al. (1999) and the reflexive separation by Ashgriz and Poo (1990). 1.0
Brazier-Smith et al. (1972) Ashgriz and Poo (1990)
Bouncing (II)
0.8
0.6
B
Stretching Separation (V)
0.4
Fast Coalescence (III) 0.2
Reflexive Separation (IV) 0.0 0
25
50
75
100
125
150
Wec Figure 4.8: Comparison of the models for the critical impact parameter Bcrit and the experimental results of stretching separation by Estrade et al. (1999) (blue symbols) using ethanol droplets with ∆ = 1.0. Dashed black lines represent the boundaries of the bouncing regime by Estrade et al. (1999) and the reflexive separation by Ashgriz and Poo (1990).
Taking these considerations into account, in the present composite model the critical impact parameter Bcrit (Wec ) is determined based on Eq. (4.69) proposed by O’Rourke (1981) relying on the theoretical correlation of Brazier-Smith et al. (1972), which was applied in may studies (see, 147
4.2 Surface-Tension Dominated Droplets e.g., Amsden et al., 1989; Koll´ar et al., 2005; Blei, 2006; Munnannur and Reitz, 2007; Kim et al., 2009). 1.0
Bouncing (II) Brazier-Smith et al. (1972) Ashgriz and Poo (1990) Ashgriz and Poo (1990)
0.8
∆
B
0.6
=
0.5
∆
=
0.5
Stretching Separation (V)
0.2 5
0.4
Fast Coalescence (III)
∆=
0.2
Reflexive Separation (IV)
0.0 0
30
60
90
120
150
180
210
Wec Figure 4.9: Boundaries between the four regimes: Dashed curves represent the bouncing regime by Estrade et al. (1999) and reflexive separation based on Ashgriz and Poo (1990) for the size ratio ∆ = 0.5. Solid curves are the boundary between stretching separation and fast coalescence according to Brazier-Smith et al. (1972) (orange curve ∆ = 0.5) and Ashgriz and Poo (1990) (red curve ∆ = 0.5; blue curve ∆ = 0.25).
Although in reality the stretching separation leads to a mass exchange during the collision (Jiang et al., 1992; Kim et al., 2009), in the present model it is assumed that the droplets conserve their pre-collision sizes (O’Rourke, 1981). Despite the extensive literature on the model by O’Rourke (1981), the treatment of the kinetics of the collision partners after a grazing separation schematically depicted in Figure 4.10 is not clearly explained. Therefore, in the present study the post-collision velocities of the collision partners are calculated based on the extended model by Kim et al. (2009) using a deterministic collision detection similar to the one applied in this thesis. Here, it is assumed that the momentum exchange between the droplets occurs only in the direction − − of the relative velocity between the collision partners u− rel = us − ul (see Figure 4.10). For this purpose, the velocity of each droplet is divided into two components in the directions parallel (denoted k) and perpendicular (denoted ⊥) to the relative velocity as shown in Figure 4.10, such that: − − u− k = uk,k + uk,⊥ , + + u+ k = uk,k + uk,⊥ .
(4.93)
Here, the index k refers to either the small or the large droplet, respectively. The unit vector of the relative velocity is given by: eˆk = 148
u−
rel
−
urel
.
(4.94)
4. Modeling of Droplet Coalescence Droplet s
e^||
Droplet l
Ss
e^!
u−rel
u−s
Sl
u−l Figure 4.10: Sketch of a grazing collision leading to a stretching separation.
Thus, the velocity components of the collision partners in the directions parallel and perpendicular to the relative velocity before and after the impact can be written as: − ˆk u− k,k = uk,k e + ˆk u+ k,k = uk,k e
− − and u− k,⊥ = uk − uk,k , + + and u+ k,⊥ = uk − uk,k ,
(4.95)
+ where u− k,k and uk,k denote the magnitudes of the droplet velocity in the parallel direction before and after the collision and read: − ˆk u− k,k = uk · e
+ ˆk . and u+ k,k = uk · e
(4.96)
Note that during the collision the momentum is conserved, whereas a part of the kinetic energy is lost. It is important to stress that in this model the friction between the collision partners is not taken into account. Thus, for a stretching separation using the model by Kim et al. (2009) the momentum exchange occurs solely in eˆk direction. That means that (i) the relative velocity in eˆk direction is reduced by the factor fBss and (ii) the droplet velocity components in eˆ⊥ direction remain unchanged (no momentum exchange) and thus: + u− k,⊥ = uk,⊥ .
(4.97)
The fraction of the kinetic energy in eˆk direction dissipated during this collision is given by:
fBss =
+ Ekin,r,k − Ekin,r,k
1/2
(4.98)
,
− + where Ekin,r,k and Ekin,r,k stand for the relative kinetic energies in eˆk direction before and after the impact, respectively. These quantities are defined as: i2 1 h − − ˆ ˆ us − u− · e , Ekin,r,k = m k l 2 (4.99) i2 1 h + + + Ekin,r,k = m ˆ us − ul · eˆk . 2 Thus, the dissipation factor fBss can be expressed as the ratio of the relative velocity between the droplets in eˆk direction after the collision to that before the impact and hence it reads:
fBss
=
h
h
+ u+ · eˆk s − ul
− u− · eˆk s − ul
i
i
=
+ u+ s,k − ul,k − u− s,k − ul,k
,
(4.100) 149
4.2 Surface-Tension Dominated Droplets + The post-collision velocities in the direction parallel to the relative velocity (i.e., u+ s,k and ul,k ) are determined relying on the definition of fBss and the conservation law of the linear momentum in eˆk direction given by: − + + ms u− s,k + ml ul,k = ms us,k + ml ul,k .
(4.101)
+ Thus, solving the momentum conservation equation (4.101) for u+ s,k and ul,k and incorporating the dissipation factor fBss given by Eq. (4.103) into the resulting relations yields the components of the post-collision velocities in the direction parallel to the relative velocity:
u+ s,k = u+ l,k =
− − − ss ms u− s,k + ml ul,k + ml us,k − ul,k fB
ms + ml − − − ss ms us,k + ml ul,k − ms u− s,k − ul,k fB ms + ml
,
(4.102)
.
However, the dissipation factor fBss given by Eq. (4.103) required for the determination of the post-collision velocities of the droplets is still unknown. In this model fBss is calculated based on the assumption of a grazing collision by O’Rourke (1981) as follows: fBss =
B − Bcrit (Wec ) 1 − Bcrit (Wec )
with 0 ≤ fBss ≤ 1 .
(4.103)
Thus, B = 1 (grazing collision) leads to fBss = 1 meaning that the relative velocity between the − + ˆk ] = [(u− ˆk ]. According to droplets in eˆk direction is conserved, i.e., [(u+ s,k − ul,k ) · e s,k − ul,k ) · e Eq. (4.97) the velocity components in eˆ⊥ direction remain also unchanged. Consequently, the droplets move in the same direction and with the same pre-collision velocities as before the collision − + − ss (i.e., u+ s = us and ul = ul ). On the other hand, B = Bcrit (Wec ) leads to fB = 0 implying that ˆk = u+ ˆk ), since the droplets coalesce the relative velocity in eˆk direction vanishes (i.e., u+ s,k · e l,k · e forming a new larger droplet (droplet-pair) moving with the common velocity as will be explained in Section 4.2.5.6. To simplify the implementation of relation (4.102), it is written in a similar manner as for the case of bouncing regime presented in Section 4.2.5.1: m ˆ ˆdd f eˆk , ml k m ˆ ˆdd − f eˆk . u+ s = us + ms k − u+ l = ul −
(4.104)
Here, fˆkdd stands for the magnitude of the impulse vector parallel to the relative velocity and is defined by: fˆkdd = −(1 − fBss )
h
i
− u− · eˆk , s − ul
(4.105)
where the dissipation factor fBss for the stretching separation is given by Eq. (4.103). Thus, the second term on the right-hand side of relation (4.104) is responsible for the reduction of the magnitudes of the velocity components in the direction of the relative velocity. 150
4. Modeling of Droplet Coalescence 4.2.5.5 Overlap of Reflexive (IV) and Stretching (V)Separation
As depicted in Figure 4.6, for a constant size ratio ∆ and high Weber numbers the stretching and reflexive separation regimes overlap at an intersection point. Ashgriz and Poo (1990) distinguished these outcomes based on the effective kinetic energy of the stretching and reflexive separation. Munnannur (2007) assumed that the collision outcome in the overlapping region is treated as a rs ss reflexive separation if Ekin ≥ Ekin . Otherwise, it is treated as a stretching separation. Note that the post-collision velocities for these regimes are determined as explained in Sections 4.2.5.3 and 4.2.5.4. However, the above condition by Munnannur (2007) might be satisfied for small Weber numbers outside the overlap region and hence an additional condition is required. Since the overlap of these outcomes occurs only at high Weber numbers, a limiting value of Weov at the intersection of the corresponding curves (see Figures 4.7 and 4.8) has to be determined depending on the size ratio ∆ and hence a new condition can be introduced. Thus, for B ≥ Bcrit (Wec ) the improved conditions for a reflexive separation inside the overlap region are: Wec ≥ Wef c/rs
and Wec ≥ Weov ,
(4.106)
where the Weber number at the intersection Weov is determined by analyzing the corresponding boundaries of the reflexive separation based on Ashgriz and Poo (1990) and the stretching separation by Brazier-Smith et al. (1972) for all impact parameters (i.e., 0 ≤ B ≤ 1) and size ratios within the range 0.01 ≤ ∆ ≤ 1.0. Examples for the obtained values of the impact parameter Bov and Weber number Weov at the intersection as a function of the size ratio (∆ = ds /dl ) are listed in Table 4.2. ∆
Bov
Weov c
1.0 0.8 0.6 0.4 0.2
0.2293 0.2233 0.2012 0.1585 0.0930
119.5436 133.2520 269.4171 1407.745 43035.42
Table 4.2: Examples for the impact parameter Bov and Weber number Weov at the overlapping point of the reflexive separation by Ashgriz and Poo (1990) and stretching separation by Brazier-Smith et al. (1972) for different size ratios.
To minimize the computational effort, the impact parameter at the intersection Bov is fitted as a function of the size ratio as exemplarily depicted in Figure 4.11 by the following polynomial relation: Bov = 0.0023669 + 0.5152400 ∆ − 0.3143200 ∆2 − 0.0071091 ∆3 + 0.0330610 ∆4 . (4.107)
Thus, for B = Bov the overlapping Weber number Weov can be determined based on either Eq. (4.24) for reflexive separation by Ashgriz and Poo (1990) or Eq. (4.32) for stretching separation by Brazier-Smith et al. (1972) such that Weov = Wef c/rs = Wef c/ss . 151
4.2 Surface-Tension Dominated Droplets In the present study due to the complex formulation of relation (4.24) Weov is calculated based on Brazier-Smith et al. (1972) for B = Bov . Thus, the overlapping Weber number reads: Weov =
4.8 f (γ) , 2 Bov
(4.108)
where the dimensionless function f (γ) is given by Eq. (4.81). 0.30 0.25
Calculation Fitting using Eq. (4.105)
Bov
0.20 0.15 0.10 0.05 0.2
0.4
0.6
0.8
1.0
∆ Figure 4.11: Curve fitting of the calculated impact parameters at the intersection point Bov of the boundaries of the reflexive separation by Ashgriz and Poo (1990) and the stretching separation based on Brazier-Smith et al. (1972) for different size ratios ∆.
4.2.5.6 Kinetics of the Coalesced Droplet
If a coalescence process occurs, the newly formed droplet (referred to as agglomerate) is assumed to have a spherical shape. Hence, the diameter of the agglomerate dag is calculated based on the mass conservation using the volume-equivalent sphere model presented in Section 3.4.1. Assuming this agglomerate structure, the density of the droplets is maintained (i.e., ρag = ρs = ρl = ρd ) and hence the diameter of the coalesced droplet is determined based on the mass conservation:
dag = d3s + d3l
1/3
.
(4.109)
The position of the resulting coalesced droplet (or agglomerate) is given by: x+ ag =
− ms x− s + ml xl . ms + ml
(4.110)
The translational velocity of the agglomerate is determined based on the momentum conservation and hence it reads: u+ ag = 152
− ms u− s + ml u l . ms + ml
(4.111)
4. Modeling of Droplet Coalescence Although the droplet-droplet collision is assumed to be frictionless, the droplet rotation based on Eq. (2.54) is caused by the viscous torque acting on the droplet if the local fluid rotation does not + vanish (i.e., ∇ × uf 6= 0). Thus, the components of the angular velocity of the agglomerate ωag can be determined based on the conservation of the angular momentum: + ωag =
Lag . Iag
(4.112)
Here, Lag stands for the total angular momentum about the center of mass of the agglomerate (see Section 3.3.2) and is given by: Lag = Is ωs− + Il ωl− +
m ˆ − . (ds + dl ) n × u− s − ul 2
(4.113)
The moment of inertia of the resulting agglomerate (or the volume-equivalent sphere) Iag appearing in Eq. (4.112) reads: Iag =
π 1 mag d2ag = ρag d5ag . 10 60
(4.114)
As discussed before, in case of no coalescence the collision partners bounce back maintaining their angular velocities before the impact due to the assumption of frictionless collision (see Section 2.6.2). 4.2.5.7 Calculation Procedure
The algorithm of the improved composite collision outcome model for surface-tension dominated droplets implemented in LESOCC is presented in Appendix B.2. In addition, a detailed description of the calculation steps is provided there. 4.2.5.8 Concluding Remarks
As mentioned before, in spray injection systems high Weber numbers and a wide range of the size ratio ∆ are encountered. It was experimentally observed that if the size ratio decreases the boundaries of bouncing (see, e.g., Estrade et al., 1999) as well as reflexive and stretching separations (see, e.g., Brazier-Smith et al., 1972; Ashgriz and Poo, 1990) are shifted leading to a larger region of the fast coalescence as schematically depicted in Figure 4.12. Thus, taking the entire spectrum of size ratios, for example, in the simulation of spray systems, in the regime map the occurrence (or overlap) of the points representing the fast coalescence in other three regimes is no more surprising (see Chapter 8).
153
4.3 Viscosity Dominated Droplets 1.0 Bouncing (II)
0.8 Stretching Separation (V)
B
0.6
0.4 Fast Coalescence (III)
0.2
Reflexive Separation (IV)
0.0 0
30
0.7
60
90
120
150
180
Wec
Figure 4.12: Enlargment of the region of fast coalescence when decreasing the size ratio ∆. Boundaries between the four regimes based on (bouncing) Estrade et al. (1999), (reflexive separation) Ashgriz and Poo (1990) and (stretching separation) Brazier-Smith et al. (1972) for different size ratios: ∆ = 1.0 (solid black curves), ∆ = 0.75 (dashed red curves) and ∆ = 0.5 (dashed orange curves).
4.3 Viscosity Dominated Droplets
If the colliding droplets are partially dried (i.e., high solids content) during, for example, a spray drying process, the viscosity of the droplets significantly increases. For such droplets with a very high droplet viscosity (typically, µd > 1 Pa s) the surface-tension forces do not play a significant role compared to the viscous and inertia forces. Hence, the collisions of these droplets are dominated by viscous forces (Oh2c > 1). Blei and Sommerfeld (2007) stated that the dynamic viscosity of the droplet is the most decisive property strongly varying during the drying process, whereas the surface tension is only slightly changed. Thus, for these viscosity-dominated fluids the collision partners may not coalesce, since they roughly maintain their shape leading to a partial or full penetration or even a full passage of the collision partners, which may yield structured agglomerates (Blei and Sommerfeld, 2007). As a result of the complex physics of this phenomenon, a few studies are available in the literature concerning the modeling of the outcomes of such binary collision of two viscous droplets. For example, Blei (2006) modeled the collision of viscosity dominated droplets by assuming that high low the high viscosity droplet (dlow ) penetrates into the low viscosity droplet (dlow d , µd d , µd ). In the context of this model three outcomes are distinguished based on a critical relative approach velocity and the penetration depth: (i) agglomeration, (ii) coalescence and (iii) separation. However, in the present study the focus is only on surface-tension dominated droplets explained in Section 4.2 due to the time limitation of the current project. Therefore, the inclusion of the viscosity dominated droplets in the in-house CFD code LESOCC shall be done in the future.
154
CHAPTER 5
MODELING OF PARTICLE-WALL ADHESION
Particle
ω−p
S
fˆn,cpw
mp
ol
x p,c
u−p
n c
Wall
xc
Particle-wall collision including adhesion (Almohammed and Breuer, 2016b)
5.1
Overview of Particle Deposition Models 5.1.1
5.1.2
5.2
. . . . . . . . . . . . . . . . . . . . . . . .
157
Empirical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.1.1.1
Correlation by Friedlander and Johnstone (1957) . . . . . . . . . . . . . 157
5.1.1.2
Correlation by Liu and Agarwal (1974) . . . . . . . . . . . . . . . . . . 158
5.1.1.3
Correlation by McCoy and Hanratty (1977) . . . . . . . . . . . . . . . . 158
5.1.1.4
Correlation by Wood (1981) . . . . . . . . . . . . . . . . . . . . . . . . 158
5.1.1.5
Correlation by Kvasnak et al. (1993) . . . . . . . . . . . . . . . . . . . . 159
Models based on Physical Relations . . . . . . . . . . . . . . . . . . . . . . . . . 159 5.1.2.1
Wetted-Wall Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.1.2.2
Energy-based Model by Dahneke (1971) . . . . . . . . . . . . . . . . . . 162
5.1.2.3
Momentum-based Model by Kosinski and Hoffmann (2009)
Present Particle-Wall Adhesion Model
. . . . . . . 163
. . . . . . . . . . . . . . . . . . . . . . . . .
164
5.2.1
Particle-Wall Collision with Adhesion . . . . . . . . . . . . . . . . . . . . . . . . 164
5.2.2
Particle-Wall Collision Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
5.2.3
Adhesive Impulse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.2.3.1
Wall-Normal Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
5.2.3.2
Wall-Tangential Direction. . . . . . . . . . . . . . . . . . . . . . . . . . 169
5.2.4
Intervals of the Collision Time
. . . . . . . . . . . . . . . . . . . . . . . . . . . 170
5.2.5
Deposition Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.2.6
Kinetics of the Particle without Deposition . . . . . . . . . . . . . . . . . . . . . 174 5.2.6.1
Sticking Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.6.2
Sliding Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.7
Rough Wall Including Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.8
Calculation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.9
Advantages of the Model
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.10 Application and Validity of the Model . . . . . . . . . . . . . . . . . . . . . . . 176
5 Modeling of Particle-Wall Adhesion To consider the particle-wall adhesion, the adhesive force between the particle and the wall has to be taken into account during the collision process. Under specific conditions this effect leads to the deposition of the particle on the bounding wall. As a starting point, the background on the model of frictional particle-wall collisions without adhesion was described in Section 2.4. The present chapter is concerned with the modeling of particle-wall adhesion and is organized in the following manner. A state-of-the-art review on the particle deposition models applied in most studies is given in Section 5.1. Then, the newly developed momentum-based particle-wall adhesion model is presented in Section 5.2. 5.1 Overview of Particle Deposition Models
Basically, if the adhesion between the particle and the wall is strong enough, the particle adheres (or sticks) to the wall. This phenomenon is denoted “particle deposition” and has been commonly characterized in terms of the particle deposition velocity vd = Jd /C0 . It refers to a ratio of the flux of the particles Jd = Ndpp /(Ad td ) to the deposition surface per unit time and the mean particle concentration C0 = N0 /Vtot (i.e., the number of particles divided by the occupied volume Vtot ). The symbol N0 stands for the total number of primary particles released into the computational domain. Ndpp is the total number of primary particles deposited on a deposition surface of the area Ad within the time interval td . Typically, in most studies the deposition velocity is made dimensionless with respect to the friction velocity uτ and thus: vd vd+ = . (5.1) uτ Often, the dimensionless deposition velocity1 denoted vd+ is plotted versus the dimensionless particle relaxation time defined by: τp+ =
τp u2τ , νf
(5.2)
where τp denotes the particle relaxation time given by Eq. (2.9) and νf stands for the kinematic viscosity of the continuous phase. In the following, the most common methods used to determine the dimensionless deposition velocity are presented. 5.1.1 Empirical Models
In the last decade, many studies focused on the investigation of the deposition of aerosol particles on smooth and rough walls. The mostly cited empirical correlations of the dimensionless deposition velocity are listed next, which are of particular importance for the present thesis. 5.1.1.1 Correlation by Friedlander and Johnstone (1957)
Friedlander and Johnstone (1957) proposed a theoretical “diffusion free-flight” model to study the particle deposition. In this simplified model the effect of gravity on the deposition velocity 1
In some studies it is also termed as the deposition coefficient denoted Kd+ .
157
5.1 Overview of Particle Deposition Models was eliminated using vertical channels. Brownian diffusion was neglected by using particles with dp > 0.5 µm. Thus, the suspended particles are transported towards the wall solely by turbulent diffusion until they reach a certain distance from the wall called “stopping distance”. Beyond this distance the particles are transported to the wall by the free-flight mechanism. That means that a particle with a given initial velocity detaches from the turbulent eddies due to sufficient inertia, then penetrates the viscous sublayer and eventually deposits on the wall. The stopping distance is denoted ls and is given by ls = τp vp− , where vp− is the particle velocity before the impact. Friedlander and Johnstone (1957) and Beal (1970) found that the stopping distance is proportional to the root-mean-square of the fluctuating gas velocity in the wall-normal direction. 5.1.1.2 Correlation by Liu and Agarwal (1974)
Liu and Agarwal (1974), among others, conducted careful experiments to explore the deposition of fine particles with application to a turbulent vertical pipe flow. They used olive oil droplets with a dimensionless particle relaxation time τp+ ranging from 0.27 to 774. Here, the droplets were assumed to be spherical and electrostatically neutral particles. Based on their observations Liu and Agarwal (1974) suggested the following empirical correlation between the dimensionless deposition velocity and the relaxation time: 2
vd+ = 6 × 10−4 τp+ .
(5.3)
However, this linear increase of vd+ with the square of the dimensionless relaxation time holds true only for τp+ < 10. In addition, Liu and Agarwal (1974) observed an increase of the dimensionless deposition velocity up to τp+ ≈ 30, whereas it slightly decreases beyond this value (i.e., τp+ > 30). These experimental observations were found to agree quite well with the theoretical “diffusion free-flight” model by Friedlander and Johnstone (1957). 5.1.1.3 Correlation by McCoy and Hanratty (1977)
Later on, McCoy and Hanratty (1977) experimentally investigated the deposition rate of droplets in a turbulent vertical pipe flow. Based on a large number of experiments, they suggested that for 0.2 < τp+ < 22.9 the dimensionless deposition velocity can be expressed as: 2
vd+ = 3.25 × 10−4 τp+ .
(5.4)
Obviously, in comparison with Liu and Agarwal (1974) this empirical correlation differs only by the constant of proportionality. Additionally, McCoy and Hanratty (1977) found that for τp+ > 22.9 the dimensionless deposition velocity reaches a relatively constant value vp+ ≈ 0.17 implying its insensitivity to the particle size or to the velocity of the continuous phase. 5.1.1.4 Correlation by Wood (1981)
Wood (1981) proposed an analytical relation between the dimensionless particle deposition velocity and the relaxation time for smooth and rough walls. In case of smooth walls this relation has the following form: 2
vd+ = 0.057 Sc−2/3 + 4.5 × 10−4 τp+ , 158
(5.5)
5. Modeling of Particle-Wall Adhesion where Sc = νf /Dp stands for the Schmidt number and Dp is the mass diffusivity of the particle. Note that either the first or the second term of this expression is dominant for small and large particles, respectively. In case of rough walls, Wood (1981) derived different empirical expressions for the dimensionless deposition velocity of the form vd+ = f (τp+ , k + , Sc), where k + = ks uτ /ν stands for the dimensionless wall roughness and ks is the equivalent sandgrain roughness (see Section 2.4.4). It was concluded that the predictions of this model agree well with experimental data using smooth walls. However, in case of rough walls the predictions were not reliable, since the deposition rates were extremely sensitive to the wall roughness (Wood, 1981). 5.1.1.5 Correlation by Kvasnak et al. (1993)
Kvasnak et al. (1993) experimentally investigated the wall deposition rate of particles on a flat gold plate in a horizontal turbulent channel flow. They used spherical glass particles and five dust components with size ranges of dp = 5 − 45 µm and 1 − 10 µm, respectively. To take the influence of the restitution coefficient on the particle-wall collisions into account, the flat gold plate was covered by a thin film coating. Kvasnak et al. (1993) measured the deposition velocity for different particle types and sizes. They concluded that in case of spherical particles the increase of the particle diameter leads to a higher deposition velocity. For horizontal flow configurations Kvasnak et al. (1993) extended the analytical relation (5.5) of Wood (1981) by including an additional term to take the significant role of the gravity force on the deposition into account, especially in the case of horizontal channel flows, such that: 2
vd+ = 0.057 Sc−2/3 + 4.5 × 10−4 τp+ + g + τp+ ,
(5.6)
where g + = g νf /u3τ stands for the dimensionless gravitational acceleration in wall units. Kvasnak et al. (1993) found that the measured values of the deposition velocity were in good agreement with the experiments carried out by Papavergos and Hedley (1984) and the empirical model by Wood (1981). This correlation is used in the present study to validate the newly developed particle-wall adhesion model (see Section 5.2) in a horizontal particle-laden turbulent channel flow (see Section 9.1), since it also takes the effect of the gravity (or sedimentation) into account. 5.1.2 Models based on Physical Relations
Besides the empirical expressions for the dimensionless deposition velocity vd+ mentioned before, the particle deposition was also investigated using Euler-Lagrange simulations. Here, it is assumed that N0 primary particles are uniformly distributed in a certain region within a dimensionless distance of h+ 0 = h0 uτ /νf from the wall. If the number of deposited primary particles Ndpp 2 during the dimensionless time t+ d = t uτ /νf approaches a constant rate (i.e., dNdpp /dt = 0), the dimensionless deposition velocity is commonly defined as: vd+
Ndpp = N0
!
h+ 0 . t+ d
(5.7)
For this purpose, deposition models of different levels of complexity were applied. Typically, three models are commonly used to check if a particle deposits on the wall, namely (i) the wetted-wall, 159
5.1 Overview of Particle Deposition Models (ii) the energy-based and (iii) the momentum-based deposition models. These deposition models are briefly summarized next, whereas in the context of the present thesis the emphasis will mainly be on the newly developed momentum-based deposition model (see Section 5.2). 5.1.2.1 Wetted-Wall Model
In the framework of this model a relatively simple criterion is applied to check the deposition of a particle colliding with the wall. In this model it is assumed that the particle deposits on the wall, if the particle reaches a certain distance to the closest wall. This distance is usually one radius of the spherical particle (i.e., dp /2), i.e., the particle touches the wall. The main drawback of this model is that all particle-wall collisions are neglected and hence the particle does not bounce back for any value of the restitution coefficient for particle-wall collisions en,w . In other words, the particle touching the (absorbing) wall adheres directly to it, which implies that the particle-wall adhesion is strong enough to maintain the particle sticking to the wall. This model has been applied in many numerical studies. For instance, McLaughlin (1989) studied the particle deposition in a vertical fully-developed turbulent channel flow using direct numerical simulations (DNS). Hence, in this flow configuration the effect of gravity (or sedimentation) does not directly cause deposition of the particles on the channel walls. As a deposition criterion, McLaughlin (1989) assumed that if the gap between a particle and the closest wall is smaller than the particle size, the particle deposits on this wall. In comparison with Liu and Agarwal (1974) lower deposition rates were predicted for τp+ < 2 and higher values outside this range of the dimensionless relaxation time. Uijttewaal and Oliemans (1996) carried out direct numerical and large-eddy simulations to study the deposition of particles in vertical pipe flows for 5 ≤ τp+ ≤ 104 at various shear Reynolds numbers (Reτ = 360, 1000 and 2100) under different conditions of gravity and lift forces. They assumed dilute flow and hence particle-particle collisions were not considered. They neglected the gravity and the lift forces and found that for a vertical pipe flow the number of active particles Np (t) decayed exponentially with time according to the following expression: !
Np (t) 4 vd+ t+ = exp − . N0 Reτ
(5.8)
Uijttewaal and Oliemans (1996) also found that for τp+ < 100 variations of turbulence properties noticeably affect the particle deposition. Contrary to other studies, vd+ depends on the Reynolds number of the flow. For Reτ = 360 they observed over the used range of τp+ excellent agreement with McCoy and Hanratty (1977) as well as reasonable agreement with the DNS of McLaughlin (1989) and the experiments by Liu and Agarwal (1974). In addition, Uijttewaal and Oliemans (1996) studied the effect of the gravity force and compared the results with predictions without this force. They found that the gravity slightly increased the deposition velocity in the case of upward flow for τp+ < 200 and then sharply decreased beyond this value of τp+ . In the case of downward flow the deposition velocity decreased for τp+ > 50. They also observed that the inclusion of the Saffman lift force (see Section 2.3.2.4.1) increased the deposition rate due to the fact that in the direct vicinity of the wall the particles are faster than the fluid, which results in a lift force towards the wall. The predictions with lift force were in close agreement with the results of McLaughlin 160
5. Modeling of Particle-Wall Adhesion (1989) for small values of τp+ . In many studies DNS was also carried out to investigate the particle deposition based on the wetted-wall model, for example, by van Haarlem et al. (1998) for a free-slip and no-slip channel walls and Zhang and Ahmadi (2000) for different directions of the gravity in vertical and horizontal turbulent duct flows. To reduce the high computational effort of DNS, many studies were carried out using the large-eddy simulation (LES) technique. For example, Wang and Squires (1996) used LES to investigate the deposition of solid particles with dimensionless particle relaxation times in the range of 2 ≤ τp+ ≤ 6 in a vertical turbulent channel flow at two different Reynolds numbers Re =11,160 and 79,400. As a deposition criterion, they applied the wetted-wall model. They found a similar dependency of the dimensionless deposition velocity on τp+ and the density ratio ρp /ρf as observed in the DNS by McLaughlin (1989). Although their LES predictions tend to predict lower values of vd+ than DNS, the results still reasonably agree with the empirical relation (5.3) by Liu and Agarwal (1974). Furthermore, Wang and Squires (1996) found that the Reynolds number of the flow only slightly affects the dimensionless deposition velocity for considered range of τp+ . They also examined the effect of the subgrid-scale velocity fluctuations on the deposition velocity by incorporating them into the equation of the particle motion. They noticed that for small particles the inclusion of the subgrid-scale velocities increased the dimensionless deposition velocity by about 30%, whereas no effect was observed on large particles (τp+ = 6). Wang and Squires (1996) suggested that the predictions can be further improved in comparison with the experiments by taking the particle-particle collisions into account. It is worth noting that due to the relatively low volume fractions used in the above mentioned studies, among others, the two-way coupling and the particle-particle collisions were not considered. Furthermore, Wang et al. (1997) proposed an accurate formulation of the lift force acting on a particle in wall-bounded flows and studied the deposition of particles in vertical turbulent channel flows using LES. They found good agreement of the deposition velocity with experimental measurements. Later on, Breuer et al. (2006) applied the Euler-Lagrange approach using LES and a wetted-wall model to study the deposition in a 90◦ bend flow laden with 250,000 monodisperse particles. They used various particle sizes and two different Reynolds numbers based on the bend diameter and mean flow velocity, i.e., Re = 1000 (laminar flow) and 10,000 (turbulent flow) and validated their predictions with the experiments by Pui et al. (1987). In their study the deposition efficiency is defined as the ratio of the number deposited particles Ndpp to the total number of released particles N0 and is determined according to Tsai and Pui (1990). Excellent agreement of the prediction was found in comparison with the experimental results of Pui et al. (1987). Winkler et al. (2006) also employed LES to study the influence of the two- and four-way coupling on the dimensionless deposition velocity in turbulent square duct flow with Reτ = 360 using different particle sizes and densities. They applied the wetted-wall model and compared their predictions with the experiments by McCoy and Hanratty (1977) in pipe flows. For a volume fraction of αp < 10−4 , they found that the deposition velocity is not noticeably affected by including the two-way coupling, and hence in this case a one-way coupled approach is sufficient. They also observed a significant increase of the dimensionless deposition velocity if the subgrid-scale fluctuations are considered. In addition, the inclusion of the two- and four-way coupling led to 161
5.1 Overview of Particle Deposition Models higher deposition rates. Overall, Winkler et al. (2006) found that for large particles the predicted rates of particle deposition in this square duct agree well with the experimental results by McCoy and Hanratty (1977), while these are two orders of magnitude higher for small particles. Koullapis et al. (2016) employed large-eddy simulation with the wetted-wall model to investigate the particle deposition in complex geometries with application to human airways under various conditions. In their study the inhaled aerosol particles are injected into the computational domain with various diameters ranging from dp = 1 to 10 µm. However, their predictions were not validated or even compared to other numerical simulations. It is important to note that the wetted-wall model is only physically reasonable if it is applied to study the deposition of particles on wetted walls, for example, inhaled particles on the wetted airway walls (see e.g., Breuer et al., 2006; Koullapis et al., 2016). However, in most engineering applications this is typically not the case because 0 < en,w < 1, and hence improved models are urgently required to allow the particles to bounce back from the wall after the impact. 5.1.2.2 Energy-based Model by Dahneke (1971)
In the context of the energy-based deposition model Dahneke (1971) proposed a simple model relying on an energy balance before and after the particle-wall impact. In this model it is assumed that the particle impacting the wall adheres to it if its wall-normal velocity component before the − ∗,D impact vp,n is less than or equal to a critical value denoted vp,n . Hence, the deposition condition in the energy-based model by Dahneke (1971) can be expressed as: (5.9)
− ∗,D vp,n ≤ vp,n ,
∗,D where the magnitude of the critical approach velocity vp,n , up to which the particle sticks to the wall, is given by:
∗,D vp,n
"
2Es = mp
1 e2n,w
!#1/2
−1
,
(5.10)
where mp is the particle mass and en,w is the normal restitution coefficient for the particle-wall collisions given by Eq. (2.67). The symbol Es = H dp /(12 δ0 ) denotes the surface potential energy. Here, H is the Hamaker constant and δ0 stands for the minimum separation distance between the particle and the wall (typically, δ0 = 2 × 10−10 m). Li and Ahmadi (1993) applied this energy-based model to investigate the particle deposition in horizontal and vertical turbulent channel flows at a Reynolds number of Re = 6657. They used silicon, quartz and gold particles, whose diameters were in the range between 0.01 µm and 10 µm, whereas gold channel walls were assumed. Furthermore, different values of the restitution coefficient of particle-wall collisions were used: en,w = 0.5, 0.85 and 0.96. Their predictions were in qualitative agreement with the analytic relation of Wood (1981) and the DNS results by McLaughlin (1989). Nevertheless, these predictions agreed well with the experiments by Papavergos and Hedley (1984) and Kvasnak (1991). Additionally, Li and Ahmadi (1993) stated that for a particle diameter dp > 2 µm the gravity plays a significant role on the deposition velocity. Furthermore, the effect of particle rebound becomes more pronounced for particles with 162
5. Modeling of Particle-Wall Adhesion dp > 10 µm. This leads to a noticeable deviation in the predicted dimensionless deposition velocity when increasing the normal restitution coefficient. Li and Ahmadi (1993) also concluded that for dp > 10 µm the deposition velocity decreases when increasing the restitution coefficient en,w . 5.1.2.3 Momentum-based Model by Kosinski and Hoffmann (2009)
It is well known that the standard hard-sphere model applied in the present thesis originally does not take the particle-wall adhesion into account (Hoomans et al., 1996; Crowe et al., 1998). Thus, an extension of this model is required to include the adhesive force during the impact. In this context Kosinski and Hoffmann (2009) incorporated the adhesive impulse into the hard-sphere approach and proposed a deposition criterion. To determine this impulse, they assumed that a constant van-der-Waals force acts over the surface separations between δ1 and δ0 during the compression phase of the particle-wall collision and vice versa during the restitution phase. This leads to a spatially averaged mean value of the adhesive force over these distances. Here, δ1 was determined based on Weber et al. (2004) by assuming that at this surface separation the adhesive force becomes small and comparable to the gravity force. Thus, by setting the van-der-Waals force equal to the gravity force, the separation distance reads (Weber et al., 2004): H 2πρp g d2p
δ1 =
!1/2
(5.11)
.
As a deposition criterion, Kosinski and Hoffmann (2009) estimated the magnitude of the critical normal velocity of the particle2 , below which deposition occurs: ∗,KH vp,n
"
2 H(δ1 − δ0 ) = dp 4πρp δ1 δ0
!#1/2
1
−1
e2n,w
,
(5.12)
Kosinski and Hoffmann (2009) applied their momentum-based model to investigate the influence of the adhesive force on a single particle impacting with a smooth wall. They found that for − ∗,KH vp,n > vp,n the inclusion of the adhesive force reduces the normal component of the particle + velocity after the impact vp,n . However, these results were not validated at all or even compared to numerical results using other deposition models or empirical correlations. It is worth noting that for a given material of a microscopic particle it can be easily proved that both the energy-based model by Dahneke (1971) and the momentum-based model by Kosinski ∗,KH ∗,D and Hoffmann (2009) lead to the same value of the critical approach velocity (i.e., vp,n = vp,n ). In the case of tiny particles it is reasonable to assume that δ1 δ0 . Hence, the above relation reduces to: ∗,KH vp,n
"
2 H = dp 4πρp δ0
1 e2n,w
!#1/2
−1
.
(5.13)
By substituting the surface potential energy Es = H dp /(12 δ0 ) and the particle mass mp = π/6 ρp d3p into Eq. (5.10), the limiting approach velocity by Dahneke (1971) reads: ∗,D vp,n 2
"
2 H = dp 4πρp δ0
1 e2n,w
!#1/2
−1
.
(5.14)
Note that Kosinski and Hoffmann (2009) made a typo in the corresponding equation.
163
5.2 Present Particle-Wall Adhesion Model ∗,D ∗,KH Thus, for tiny particles the above relation implies that vp,n by Dahneke (1971) is identical to vp,n given by Eq. (5.13). However, the momentum-based model by Kosinski and Hoffmann (2009) is superior to the energy-based model by Dahneke (1971), since the former takes the effect of the adhesive impulse into account if the deposition condition is not fulfilled leading to a lower wall-normal particle velocity after the particle-wall collision.
5.2 Present Particle-Wall Adhesion Model
To avoid the drawbacks of the deposition models mentioned before, the goal is to develop a methodology for modeling the particle-wall adhesion in the context of a hard-sphere model. The modeling strategy of the newly developed momentum-based adhesion model is based on the corresponding agglomeration model (see Section 3.1.2.3) by assuming an infinitely large size of one of the collision partners (i.e., d2 → ∞ implying a flat surface). The key difference between the new approach and the model by Kosinski and Hoffmann (2009) is the modeling of the adhesive impulse and the deposition condition as explained next. Note that the derivation and the applications (see Section 5.2.10) of the newly developed particle-wall adhesion model presented here were already published in Almohammed and Breuer (2016c) and Breuer and Almohammed (2016). 5.2.1 Particle-Wall Collision with Adhesion
In this section the particle-wall collision model including dry friction explained in Section 2.4 is extended to consider the adhesive force between the particle and the wall during the impact. Thus, due to the assumption of a hard-sphere model the particle deformation is neglected during the entire particle-wall impact and the friction between the particle and the wall is based on Coulomb’s law. As assumed for the agglomeration model (see Chapter 3), the particle impacting the wall is considered to be dry and electrostatically neutral and hence only the adhesive van-der-Waals force is taken into account. A schematic particle-wall collision including the adhesive impulse fˆpw n,c
is depicted in Figure 5.1.
Particle
ω−p
S
fˆn,cpw
mp
ol
x p,c
u−p
n c
Wall
xc
Figure 5.1: Particle-wall collision with friction and adhesion (Almohammed and Breuer, 2016c).
As explained in Section 2.4, the post-collision translational and angular velocity of the particle are 164
5. Modeling of Particle-Wall Adhesion given by: − ˆpw , u+ p = up + f 5 ωp+ = ωp− − n × fˆpw . dp
(5.15a) (5.15b)
Again, the total impulse vector fˆpw appearing on the right-hand side of Eq. (5.15) has two components pointing in the collision-normal and tangential directions: fˆpw = fˆnpw + fˆtpw .
(5.16)
The inclusion of the adhesion during the impact implies that an additional force acting in the collision-normal direction has to be taken into account. Thus, the normal component of the total impulse vector fˆnpw is split up into two contributions: pw pw fˆnpw = fˆn,a + fˆn,c ,
(5.17)
pw where fˆn,a denotes the repulsive impulse due to the mechanical deformation (see Section 2.4.1) and is given by: pw pw fˆn,a = fˆn,a n=
h
i
− u+ p − up · n n .
(5.18)
pw Here, fˆn,a stands for the magnitude of normal component of the repulsive impulse and is given by:
pw fˆn,a = −(1 + en,w ) u− p ·n .
(5.19)
where en,w is the normal restitution coefficient for a particle-wall collisions defined by Eq. (2.67). As mentioned in Section 2.4.1, the first condition for a particle-wall collision is that the particle is approaching the wall and hence the scalar product (u− p · n) < 0. Based on Eq. (5.19) this implies pw pw pw ˆ that the magnitude fn,a is always positive. The symbol fˆn,c = fˆn,c n appearing in Eq. (5.17) stands for the adhesive impulse between the particle and the wall due to the van-der-Waals force. The pw determination of the magnitude of the adhesive impulse fˆn,c for the newly developed particle-wall adhesion model will be explained in Section 5.2.3. On the other hand, the tangential component of the total impulse fˆtpw appearing in Eq. (5.16) depends on the collision type (sliding or sticking) as explained in the following. 5.2.2 Particle-Wall Collision Type
As mentioned in Section 2.4.2, two types of the particle-wall impact are distinguished, namely a sticking and a sliding collision. The impact type is determined based on the no-slip condition by applying Coulomb’s law of static friction, which for a sticking particle-wall impact is defined by: ˆpw fst,t
≤ µst,w fˆnpw ,
(5.20)
where µst,w is the static friction coefficient of the particle-wall impact. Thus, by substituting the magnitude of the normal component of the total impulse vector given by Eq. (5.17) into Eq. (5.20), 165
5.2 Present Particle-Wall Adhesion Model the magnitude of the tangential component of a sticking particle-wall collision can be expressed as: ˆpw fst,t
pw pw ≤ µst,w fˆn,a + fˆn,c .
(5.21)
pw It is worth mentioning that the adhesive impulse fˆn,c is always positive as will be explained in Section 5.2.3. Thus, relation (5.21) clearly implies that the tangential component of the total impulse is increased due to the inclusion of the adhesion, since it enhances the friction between the particle and the wall during the impact. pw The derivation of the tangential component of the impulse vector fˆst,t for a sticking collision required for Eq. (5.21) was given in detail in Section 2.4.2. It describes the contribution of the impulse exerted by the wall on the particle due to the static friction. This tangential impulse acts parallel to the wall and is given by:
2 pw (5.22) fˆst,t = − (1 + et,w ) u− c,t , 7 where et,w stands for the tangential restitution coefficient for a particle-wall impact given by Eq. (2.71). It is worth noting that the minus sign on the right-hand side of Eq. (5.22) means that pw the tangential impulse vector fˆst,t points in the direction opposite to the tangential component of the velocity at the contact point before the impact u− c,t . As explained in Section 2.4.2, the pre-collision tangential velocity components at the contact point u− c,t reads:
dp − ω × n. (5.23) 2 p By substituting Eq. (5.22) into Eq. (5.21), the impact type is identified by the following no-slip condition relying on Coulomb’s law of static friction: − − u− c,t = up − up · n n −
− uc,t
≤
7 µst,w ˆpw ˆpw f + fn,c . 2 (1 + et,w ) n,a
(5.24)
Thus, if relation Eq. (5.24) is satisfied, a sticking particle-wall collision occurs, during which the particle stops sliding on the wall. Otherwise, a sliding particle-wall collision takes place, during which the particle continues sliding throughout the entire impact time. In case of a sliding particle-wall collision the tangential component of the total impulse is expressed relying on Coulomb’s law of kinetic friction as follows:
pw fˆsl,t = −µkin,w fˆnpw t ,
(5.25)
where µkin,w is the kinetic friction coefficient for a particle-wall impact and the tangential unit vector t is defined as: u− . t = c,t u− c,t
(5.26)
By substituting Eqs. (5.17) and (5.26) into Eq. (5.25) the tangential component of the sliding impulse vector can be written as:
166
−
pw pw pw uc,t fˆsl,t = −µkin,w fˆn,a + fˆn,c − . uc,t
(5.27)
5. Modeling of Particle-Wall Adhesion As visible in Eq. (5.27), the inclusion of the particle-wall adhesion increases the tangential component of the total sliding impulse due to the enhancement of the friction during the impact. 5.2.3 Adhesive Impulse Model
In analogy to the momentum-based agglomeration model (Section 3.1.2.3), the collision process is divided into a compression and a restitution phase. It is known that the adhesive (van-der-Waals) pw force acts in the collision-normal direction. Therefore, the effect of the repulsive fn,a and the pw adhesive fn,c forces during a frictional head-on particle-wall impact with en,w ≤ 1 is analyzed. The time history of these two forces during such a particle-wall collision is schematically sketched in pw Figure 5.2. The repulsive force fn,a increases as the particle is approaching the wall (compression phase), whereas it decreases during the restitution phase. Thus, the integral of the repulsive force pw over the collision time yields the impulse fˆn,a , whose magnitude given by Eq. (5.19) is always pw positive. In the following the focus is only on the influence of the adhesive impulse fn,c during the particle-wall collision.
Particle
Particle
Particle
S
S
S
fn,cpw
fn,cpw
fn,cpw
Wall
Compression phase
Restitution phase
fnpw Fully elastic collision (en,w= 1)
fn,apw
Inelastic collision (en,w< 1)
fn,cpw t pw
Δtcpw* pw Δtcom
Δtrespw Δtcpw
pw pw Figure 5.2: Schematic representation of the time history of the repulsive fn,a and adhesive fn,c forces during a frictional head-on particle-wall collision with adhesion.
167
5.2 Present Particle-Wall Adhesion Model As visible in Figure 5.2, while the particle is approaching the wall, the adhesive force accelerates the motion of the particle towards the wall in the compression phase. In other words, since the adhesive force acts in the direction of motion, the center of mass of the particle is pushed towards the contact point at the wall. Thus, the normal component of the total impulse vector is increased leading to an enhanced friction between the particle and the wall. On the contrary, the motion of the particle is decelerated during the restitution phase, since the adhesive force still pulls the particle towards the wall as it rebounds from the wall due to the resulting impulse. As a result, the normal component of the total impulse vector is reduced when the particle bounces back, since the adhesive force acts in the opposite direction to the particle motion. It is worth noting that based on Figure 5.2 an asymmetric distribution of the repulsive impulse is observed for an inelastic head-on collision (en,w < 1), whereas a symmetric distribution is found for a fully elastic particle-wall collision (en,w = 1). This behavior is attributed to the dissipation of a part of the kinetic energy of the particle during an inelastic particle-wall impact (see Section 5.2.4). Thus, it can be concluded that the normal restitution coefficient en,w significantly affects the time intervals of both collision phases (see Figure 5.3). Thus, its effect has to be considered in the newly developed particle-wall adhesion model. Note that the adhesive force has a different effect on the tangential motion of the particle than on the normal direction, since it enhances the friction between the particle and the wall during the entire collision (Almohammed and Breuer, 2016c). Therefore, the role of the adhesive force is separately analyzed depending on the collision direction as explained next. 5.2.3.1 Wall-Normal Direction
As discussed before, in the wall-normal direction a different effect of the adhesive force on the motion of the particle is observed during the compression and the restitution phase as displayed in Figure 5.2. In the present particle-wall adhesion model it is again assumed that the van-derpw Waals force acting between the particle and the wall fn,c is constant during the entire impact pw∗ (Almohammed and Breuer, 2016c). Thus, the adhesive impulse in the wall-normal direction fˆn,c can be expressed by the following relation: pw pw pw 1 Z ∆tcom +∆tres pw 1 Z ∆tcom pw pw∗ ˆ fn,c = fn,c dt + −fn,c dt, mp 0 mp ∆tpw com
|
{z
Compression phase
}
|
{z
Restitution phase
(5.28)
}
pw where ∆tpw com and ∆tres stand for the time intervals of the compression and restitution phase of the particle-wall impact (Figure 5.3), respectively. The adhesive force between the particle and the pw wall fn,c is given by: pw fn,c =
H rp , 6δ02
(5.29)
where H denotes the Hamaker constant and δ0 stands for the minimum separation distance between the particle and the wall (typically, δ0 = 2 × 10−10 m according to Israelachvili (2011)). Note that the different signs of the adhesive impulse appearing in Eq. (5.28) are due to the acceleration and the deceleration of the particle motion towards the wall during the compression and the pw∗ restitution phase, respectively. It is worth mentioning that the new notation fˆn,c is used here to 168
5. Modeling of Particle-Wall Adhesion distinguish the role of the adhesive force in the wall-normal and the wall-tangential direction of the impact. Assuming a constant adhesive force between the particle and the wall, the integrals on the right-hand side of Eq. (5.28) can be easily determined: 1 pw pw 1 pw pw pw∗ fˆn,c = fn,c ∆tcom − f ∆t . mp mp n,c res
(5.30)
Assuming an inelastic particle-wall collision (en,w ≤ 1), the time interval of the restitution phase pw is longer than that of the compression phase ∆tpw res ≥ ∆tcom . Thus, the above expression for the adhesive impulse in the wall-normal direction is reduced to the following form: 1 pw pw∗ pw∗ fˆn,c =− f ∆t , mp n,c c
(5.31)
where ∆tpw∗ denotes the difference between the time intervals of the restitution and the compression c phase and is therefore given by: pw ∆tpw∗ = ∆tpw c res − ∆tcom ≥ 0 ,
(5.32)
Thus, the normal component of the total impulse vector can now be expressed as follows:
pw pw∗ + fˆn,c n. fˆnpw∗ = fˆn,a
(5.33)
It is important to note that in case of an inelastic particle-wall impact (en,w < 1) the adhesive pw∗ impulse in the wall-normal direction fˆn,c given by Eq. (5.31) is always negative. This means that based on Eq. (5.33) the adhesion between the particle and the wall during the impact reduces the total impulse in the collision-normal direction. Consequently, the post-collision velocity of the particle in the wall-normal direction is lower than for the case without adhesion. Assuming a fully elastic particle-wall collision (en,w = 1), the time intervals of the compression and pw the restitution phases are equal (i.e., ∆tpw res = ∆tcom ) due to the symmetric time history as depicted in Figure 5.3. Based on Eq. (5.32) this leads to ∆tpw∗ = 0. In other words, for a fully elastic c particle-wall impact the adhesive impulse in the wall-normal direction vanishes (i.e., fˆnpw∗ = 0). This realistic result is consistent with the energy-based and the momentum-based deposition models explained in Sections 5.1.2.2 and 5.1.2.3, since no kinetic energy is dissipated during the impact. However, the effect of the adhesion has to be also considered in the wall-tangential direction. Contrary to the simplified deposition models by Dahneke (1971) and Kosinski and Hoffmann (2009), the later issue is considered in the newly developed particle-wall adhesion model as explained next. 5.2.3.2 Wall-Tangential Direction
The role of the adhesive force in the wall-tangential direction is an increase of the contact force between the particle and the wall during the compression and the restitution phase. That means that for any value of the normal restitution coefficient en,w the tangential component of the total impulse is increased due to the enhanced friction between the particle and the wall (see pw Section 5.2.2). Thus, the magnitude of the adhesive impulse fˆn,c required for the determination 169
5.2 Present Particle-Wall Adhesion Model of the collision type (Eq. (5.24)) and the tangential component of a sliding particle-wall collision (Eq. (5.27)) is computed based on the total impact time ∆tpw c (the sum of the time intervals for the compression and the restitution phase) as follows: pw 1 Z ∆tc pw pw ˆ fn,c dt . fn,c = mp 0
(5.34)
pw Again, the van-der-Waals force fn,c is assumed to be constant during the entire particle-wall pw ˆ impact and hence fn,c reads:
1 pw pw pw f ∆t , fˆn,c = mp n,c c
(5.35)
where ∆tpw c stands for the total particle-wall impact time and is given by: pw pw ∆tpw c = ∆tcom + ∆tres .
(5.36)
pw∗ pw Thus, the adhesive impulse in the wall-normal fˆn,c and wall-tangential fˆn,c direction are determined pw∗ pw by calculating either the time difference of the impact periods ∆tc = ∆tpw res − ∆tcom or the total pw pw pw pw impact time ∆tpw c = ∆tcom + ∆tres . For the determination of the time intervals ∆tcom and ∆tres during a frictional head-on particle-wall collision the same idea as used in the momentum-based agglomeration model is applied.
5.2.4 Intervals of the Collision Time pw The time intervals ∆tpw com and ∆tres of a particle-wall collision are estimated based on a simplified model relying on a head-on collision and the corresponding equation of motion. The derivative of the normal overlap between the particle and the wall δnpw with respect to time is defined as:
dδnpw − = upw n = up · n . dt
(5.37)
Based on the fundamental principle of dynamics, the equation of motion of the particle in the normal direction reads: mp
d u− p ·n dt
= fnpw (t) .
(5.38)
where fnpw (t) stands for the normal repulsive force at the instant t of the particle-wall collision and mp is the particle mass. Substituting Eq. (5.37) into the above relation yields: d2 δnpw 1 pw = f (t) . 2 dt mp n
(5.39)
Neglecting the adhesive force during a frictional particle-wall impact, the repulsive force in pw the normal direction has two components due to (i) the elastic deformation fn,e (t) and (ii) the pw dissipation of the kinetic energy during a head-on impact fn,d (t): pw pw fnpw (t) = fn,e (t) + fn,d (t) .
170
(5.40)
5. Modeling of Particle-Wall Adhesion Thus, the ordinary differential equation for the particle-wall overlap in the normal direction δnpw can be obtained by inserting Eq. (5.40) into Eq. (5.39): mp
o d2 δnpw n pw pw − f (t) + f (t) = 0. n,e n,d dt2
(5.41)
According to Hertz (1882) the elastic response of the particle reads: pw fn,e = −K pw (δnpw )3/2
q 4 with: K pw = Eˆpw rˆpw , 3
(5.42)
where K pw stands for the stiffness coefficient. The symbols Eˆpw represents the reduced Young’s modulus and is defined as follows: Eˆpw
"
1 − νp2 1 − νw2 + = Ep Ew
#−1
(5.43)
,
where Ep , Ew and νp , νw denote the Young’s modulus and the Poisson’s ratio of the particle and the wall, respectively. Thus, the mechanical properties of the particle and the wall are taken into account in this model. The symbol rˆpw appearing in Eq. (5.42) stands for the reduced radius and is given by: rˆpw =
rp rw , rp + rw
(5.44)
However, for the wall it is assumed that rw → ∞ and hence rˆpw can be reduced to the radius of the pw particle, i.e., rˆpw = rp (see, e.g., Antonyuk et al., 2010). The dissipation force fn,d (t) is modeled by Tsuji et al. (1992) as follows:
pw pw pw fn,d (t) = −ηnpw u− p · n = −ηn un ,
(5.45)
where ηnpw is the damping parameter and was determined by the following expression (Tsuji et al., 1992): ηnpw = αpw (mp knpw )1/2
with knpw = K pw (δnpw )1/2 ,
(5.46)
where knpw is the elastic stiffness. The friction coefficient αpw is expressed in terms of the normal restitution coefficient en,w as follows (see, e.g., Almohammed et al., 2014):
αpw =
2 ln en,w −q for en,w 6= 0 , π 2 + ln2 en,w
2
(5.47)
for en,w = 0 .
Thus, the dissipation force is obtained by inserting Eqs. (5.37) and (5.46) into Eq. (5.45): pw fn,d (t) = − αpw (mp K pw )1/2 (δnpw )1/4
dδnpw . dt
(5.48) 171
5.2 Present Particle-Wall Adhesion Model By substituting Eqs. (5.42) and (5.48) into Eq. (5.41), the differential equation of the overlap in the normal direction δnpw can be written as: d2 δnpw + αpw dt2
K pw mp
!1/2
(δnpw )1/4
!
dδnpw K pw + (δnpw )3/2 = 0 . dt mp
(5.49)
The differential equation of the normal overlap δnpw is made dimensionless as follows (Almohammed and Breuer, 2016c): ˆpw d2 δˆnpw pw ˆpw 1/4 dδn + α ( δ ) + (δˆnpw )3/2 = 0 , n pw 2 pw ˆ ˆ (dt ) dt
(5.50)
where tˆpw and δˆnpw are the dimensionless time and the dimensionless normal particle-wall overlap, respectively. These quantities are given by:
tˆpw = t C −1/2 rp1/2 K pw /mp
1/2
and δˆnpw = C δnpw /rp .
(5.51)
Here, the constant C is defined as follows: C = rp
(
K pw 2 mp (upw n )
)2/5
(5.52)
.
ˆpw ˆpw and hence: Thus, the dimensionless normal velocity can be defined as uˆpw n = dδn /dt
pw uˆpw C −5/2 rp5/2 K pw /mp n = un
−1/2
(5.53)
.
In this model the ordinary differential equation (5.50) is numerically solved using Matlab with a Runge-Kutta method and a small time step of ∆tˆpw = 10−4 . The initial conditions required for solving this system are δˆnpw (0) = 0 and uˆpw n (0) = 1. Figure 5.3 shows the dimensionless normal pw displacement δˆn as a function of the dimensionless time tˆpw for different restitution coefficients of the particle-wall collision en,w .
1.2 1.0
Compression
Restitution
∆tˆpw com
∆tˆpw res en,w en,w en,w en,w en,w
δˆpw
0.8 0.6 0.4
= 1.0 = 0.8 = 0.6 = 0.4 = 0.2
0.2 0.0 0.0
1.0
2.0
3.0
ˆpw
4.0
5.0
t
Figure 5.3: Dimensionless particle-wall overlap in the normal direction δˆnpw as a function of the dimensionless time tˆpw for different values of the normal restitution coefficient en,w .
172
5. Modeling of Particle-Wall Adhesion Analog to the strategy applied to the corresponding momentum-based agglomeration model ˆpw presented in Section 3.1.2.3.2, the results concerning the dimensionless times ∆tˆpw∗ = ∆tˆpw c res −∆tcom ˆpw ˆpw and ∆tˆpw c = ∆tres + ∆tcom are pre-computed as a function of the normal restitution coefficient en,w and stored in a look-up table, which is used once at the beginning of the simulation. That reduces the computational effort significantly. As depicted in Figure 5.3, a symmetrical distribution of the time history of the overlap is observed in case of a fully elastic particle-wall collision (i.e., en,w = 1) and hence both time intervals are ˆpw equal, i.e., ∆tˆpw res = ∆tcom . However, in case of an inelastic particle-wall impact (i.e., en,w < 1), the time interval of the restitution phase is longer than that of the compression phase and hence ˆpw ∆tˆpw∗ = ∆tˆpw c res − ∆tcom > 0. Combining Eqs. (5.42), (5.51) and (5.52) yields an expression for the time t as a function of the dimensionless time tˆpw : 2/5
3 t= 4
1/5 −m2p rp−1
E ˆ2
− pw (up
· n)
tˆpw .
(5.54)
As mentioned before, a particle-wall collision only occurs if (u− p · n) < 0 and hence the time t pw∗ in the above equation is always positive. Since the adhesive impulse fˆn,c depends only on the pw∗ pw time difference between the compression and the restitution phase ∆tc = ∆tpw res − ∆tcom , the corresponding dimensionless time ∆tˆpw∗ has to be inserted into Eq. (5.54). By doing that, the c normal adhesive impulse given by Eq. (5.31) reads: pw∗ fˆn,c
2/5
1 3 =− 6 4
E ˆ2
1/5
−m−3 p
pw
(u− p · n)
H 4/5 ˆpw∗ r ∆tc . δ02 p
(5.55)
On the other hand, it was stressed before that the adhesive (van-der-Waals) force increases the impact force in the tangential direction of the particle motion during the entire collision time (compression and restitution phase). Furthermore, the magnitude of the adhesive impulse during pw the entire collision time fˆn,c occurs in the no-slip condition given by Eq. (5.24). Based on the pw same consideration as applied to Eq. (5.55), the magnitude of the adhesive impulse fˆn,c can be pw expressed depending on the total dimensionless time of the collision ∆tˆc . Therefore, Eq. (5.35) can be written as: pw fˆn,c
2/5
1 3 = 6 4
E ˆ2
−m−3 p
pw
5.2.5 Deposition Condition
1/5
(u− p · n)
H 4/5 ˆpw r ∆tc . δ02 p
(5.56)
Assuming that the adhesion between the particle and the wall is strong enough, the particle sticks to the bounding wall during a collision. In other words, the particle impacting the wall stays at rest after the impact and hence its post-collision velocity vanishes (i.e., u+ p = 0). Thus, in case of deposition the total impulse vector given by Eq. (5.15a) reduces to: fˆlpw = fˆpw = − u− p .
(5.57) 173
5.2 Present Particle-Wall Adhesion Model Here, fˆlpw denotes the limiting impulse, up to which the adhesive impulse exceeds the repulsive one leading to particle deposition. Thus, based on Eq. (5.57) the normal and the tangential component of this limiting impulse read:
pw fˆl,n = fˆlpw · n = − u− p ·n , pw fˆl,t = fˆlpw · t
= − u− p · t .
(5.58)
In the framework of the momentum-based agglomeration model the agglomeration conditions (see Section 3.1.2.3.3) were expressed based on the particle-particle collision type by Eq. (3.133). It was stated that a sticking particle-particle collision leads to an agglomeration process if only the first condition in Eq. (3.133) is satisfied, whereas both conditions (normal and tangential) of Eq. (3.133) have to be fulfilled in case of a sliding particle-particle collision. Thus, in the context of the momentum-based agglomeration model the second condition has to be taken into account, since in case of a sliding collision the impacting particles may separate from each other in the collision-tangential direction even if the first condition of Eq. (3.133) is satisfied. However, in case of a sliding particle-wall impact including adhesion the latter behavior is not relevant. The reason is attributed to the fact that if the first condition is satisfied the particle cannot escape from the wall and hence adheres to the wall. Based on these arguments the common deposition condition for a sticking and a sliding particle-wall impact can be expressed as: pw pw pw∗ fˆl,n > fˆn,a + fˆn,c .
(5.59)
By substituting Eqs. (5.58) and (5.19) into Eq. (5.59), the deposition condition for a sticking and a sliding particle-wall impact reads:
pw∗ −fˆn,c > −en,w u− p ·n .
(5.60)
If the above condition is not fulfilled, the particle deposits on the wall and is then directly omitted from the computational domain. Otherwise, the particle rebounds from the wall and the postcollision translational and angular velocities of the particle are determined by taking the effect of the adhesion into account. The main advantages of the present particle-wall adhesion model are summarized in Section 5.2.9. 5.2.6 Kinetics of the Particle without Deposition
Assuming that the deposition condition (5.60) is not satisfied, the particle bounces back. In this case the extended hard-sphere model including friction and adhesion still works correctly. Thus, the post-collision translational and angular velocities can be determined based on Eq. (5.15) taking the adhesive impulse into account. For this purpose, the newly developed formulation of the adhesive impulse is applied. Note that the normal component of the total impulse fˆnpw∗ for both sticking and sliding particle-wall collisions is given by Eq. (5.33). Contrarily, the tangential component is calculated depending on the impact type distinguished by the no-slip condition (5.24). In the following the kinetics of the particle is presented for both sticking and sliding collisions, respectively. 174
5. Modeling of Particle-Wall Adhesion 5.2.6.1 Sticking Collision
If a sticking particle-wall impact occurs, the post-collision translational and angular velocity of the pw pw pw∗ particle are calculated by substituting the normal fˆn,a + fˆn,c and the tangential fˆst,t component of the total impulse vector of a sticking collision given by Eqs. (5.33) and (5.22) into Eq. (5.15):
2 pw pw∗ = + fˆn,a + fˆn,c n − (1 + et,w ) u− c,t t , 7 10 1 ωp+ = ωp− + (1 + et,w ) u− c,t (n × t) . 7 dp
u+ p
u− p
(5.61a) (5.61b)
5.2.6.2 Sliding Collision pw pw∗ For a sliding collision the velocities are calculated by substituting the normal fˆn,a + fˆn,c and the pw tangential fˆsl,t component of the total impulse vector of a sliding collision given by Eqs. (5.33) and (5.27) into Eq. (5.15): − u+ p = up +
ωp+ = ωp− +
n
o
pw pw∗ pw pw fˆn,a + fˆn,c n − µkin,w fˆn,a + fˆn,c t ,
5 pw pw + fˆn,c µkin,w fˆn,a (n × t) . dp
(5.62a) (5.62b)
5.2.7 Rough Wall Including Adhesion
Assuming that the particle collides with a rough wall, the new particle-wall adhesion model is applied, but the normal unit vector is replaced by the random one calculated using the sandgrain roughness model by Breuer et al. (2012) explained in Section 2.4.4. This step is included in the calculation algorithm as shown in Appendix C.2. 5.2.8 Calculation Procedure
The algorithm of the newly developed momentum-based particle-wall adhesion model implemented in LESOCC is presented in Appendix C.2. 5.2.9 Advantages of the Model
The newly developed particle-wall adhesion model is superior to the energy-based and momentumbased deposition approaches presented in Sections 5.1.2.2 and 5.1.2.3 due to the following main advantages:
Ê As explained in Section 5.1.2, the compression and the restitution phase are not distinguished
in the simplified energy-based model and the momentum-based deposition model of Kosinski and Hoffmann (2009). In the present model the adhesive impulse is modeled more realistically relying on the time intervals of the compression and the restitution phase of the particle-wall impact, since the calculation of these time intervals takes the normal restitution coefficient en,w into account. 175
5.2 Present Particle-Wall Adhesion Model
Ë The role of the adhesive force in the wall-normal and the wall-tangential direction during the
impact is distinguished. In case of a fully elastic particle-wall collision (en,w = 1), the time intervals of the compression and the restitution phases are equal and hence the adhesive impulse in the wall-normal direction vanishes (i.e., fˆnpw∗ = 0) yielding a realistic result. In addition, the present model takes the effect of in the wall-tangential direction during the entire impact into account (i.e., fˆnpw 6= 0), whereas this is not the case in the formulation of the simplified deposition models by Dahneke (1971) and Kosinski and Hoffmann (2009).
Ì If the deposition condition is not satisfied, the particle-wall adhesion is also taken into
account for the determination of the kinetics of the post-collision translational and angular motion of the particle for both impact types (sticking and sliding collision). Thus, the adhesive impulses in the wall-normal fˆnpw∗ and the wall-tangential fˆnpw direction occur in the corresponding equations as explained in Section 5.2.6. This issue was completely ignored in the energy-based model by Dahneke (1971) and the momentum-based model by Kosinski and Hoffmann (2009)
5.2.10 Application and Validity of the Model
To examine the performance of the newly developed particle-wall adhesion model, simple test cases are carried out as presented in Appendix C.1 to investigate the effect of the adhesive impulse on the particle rebound. In addition, the predictions of the new model are compared with those of the deposition models by Dahneke (1971) and Kosinski and Hoffmann (2009). As will be explained in Section 9.1, the entire algorithm is validated based on the experimental test case by Kvasnak et al. (1993) and compared to the numerical results using the wetted-wall model (see Section 5.1.2.1) and the data by Fan and Ahmadi (1993) using of the energy-based model of Dahneke (1971) as well as the analytic relation by Wood (1981). Then, the particle-wall adhesion model is applied to a vertical particle-laden turbulent channel flow with different particle diameters. As a practical application of the new particle-wall adhesion model, the particle deposition in a turbulent flow past an airfoil is investigated in Section 9.3. In addition, the predictions of this particle-wall adhesion model are compared with those by the wetted-wall model. Note that parts of the investigations using the developed particle-wall adhesion model presented here were already published in Almohammed and Breuer (2016c) and Breuer and Almohammed (2016).
176
CHAPTER 6
COMPUTATIONAL METHODOLOGY
y t + ∆tmin
t + ∆tcol
t
t + ∆t
u− rel x− r,col x− r,min
x− r
Particle 2
x
(0, 0) Particle 1
Relative motion of the collision partners (adopted from Breuer and Alletto, 2012).
6.1
CFD Simulation Software
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
179
6.2
Transformation to Curvilinear Coordinate System . . . . . . . . . . . . . . . . . . .
179
6.3
Numerical Methods for the Continuous Phase . . . . . . . . . . . . . . . . . . . . .
181
6.3.1
6.4
Finite-Volume Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.3.1.1
Spatial Discretization of Volume Integrals . . . . . . . . . . . . . . . . . 182
6.3.1.2
Spatial Discretization of Surface Integrals . . . . . . . . . . . . . . . . . 183
6.3.1.3
Convective and Diffusive Fluxes . . . . . . . . . . . . . . . . . . . . . . 184
6.3.2
Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.3.3
Pressure-Velocity Coupling (Predictor-Corrector Scheme) . . . . . . . . . . . . . 185
Numerical Methods for the Disperse Phase
. . . . . . . . . . . . . . . . . . . . . .
187
6.4.1
Particle Injection into the Computational Domain . . . . . . . . . . . . . . . . . 187
6.4.2
Particle Tracking Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.4.2.1
Translational Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 6.4.2.1.1 Particle Velocity (First Integration)
. . . . . . . . . . . . . . 188
6.4.2.1.2 Particle Position (Second Integration). . . . . . . . . . . . . . 190 6.4.2.2
Angular Motion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.4.3
Fluid Velocity at the Particle Position . . . . . . . . . . . . . . . . . . . . . . . 192
6.4.4
Deterministic Collision Detection Model . . . . . . . . . . . . . . . . . . . . . . . 193 6.4.4.1
6.4.5
Particle-Particle Collision Conditions . . . . . . . . . . . . . . . . . . . 194
Treatment of Particle Agglomeration . . . . . . . . . . . . . . . . . . . . . . . . 196
6.5
6.4.6
Treatment of Droplet Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.4.7
Treatment of Particle-Wall Adhesion . . . . . . . . . . . . . . . . . . . . . . . . 198
Determination of the Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199
6 Computational Methodology This chapter gives a general overview on the computational methodology of the four-way coupled Euler-Lagrange simulation framework employed in the present thesis. Here, the reader is briefly introduced into the main numerical methods implemented in the in-house CFD code LESOCC (Breuer, 1998, 2000, 2002) for the fluid and the disperse phase. A flow chart of the four-way coupled Euler-Lagrange simulation adopted in this study is provided in Appendix D.1. To understand the main steps of this simulation approach, the following points are explained. At first, a summary on the development of the CFD simulation software employed in this study is given in Section 6.1. Then, the transformation of the coordinate system used for the computation of the continuous and the disperse phase is shortly described in Section 6.2. As mentioned before, the focus of the present study is on the disperse phase. Therefore, the numerical methods for the continuous phase are only briefly summarized in Section 6.3. Afterwards, the computational methodology for the disperse phase including the treatment of particle agglomeration, droplet coalescence and particle-wall adhesion is presented in Section 6.4. Lastly, the determination of the statistics of both phases is described in Section 6.5. 6.1 CFD Simulation Software
In the present thesis the CFD code LESOCC (Large-Eddy Simulation On Curvilinear Coordinates) is employed. This computer simulation tool is based on a three-dimensional finite-volume method (see Section 6.3.1) for arbitrary non-orthogonal and block-structured grids. This CFD code is highly vectorized and additionally parallelized by domain decomposition using MPI (Breuer and Rodi, 1994, 1996; Breuer et al., 1996a,b; Breuer, 1997, 1998, 2000, 2002). This means that efficient computations on vector-parallel machines and Symmetric Multi-Processing (SMP) clusters can be carried out using this CFD software. The code was developed to predict complex turbulent flows based on the large-eddy simulation (LES) technique. It was also extended to compute particleladen turbulent flows and validated based on various test cases of different complexity (see, e.g., Breuer et al., 2006; Alletto and Breuer, 2012, 2013, 2014; Breuer et al., 2012; Breuer and Alletto, 2012; Breuer et al., 2012; Breuer and Alletto, 2013; Breuer and Hoppe, 2017). In the context of the present study the CFD code is further extended to include the particle agglomeration, droplet coalescence and particle-wall adhesion (see, e.g., Breuer and Almohammed, 2015, 2016, 2018; Almohammed and Breuer, 2016a,b,c, 2018). In the next sections the computational methodology related to the present study is presented. 6.2 Transformation to Curvilinear Coordinate System
It is known that in most practical particle-laden flows a complex computational grid is required to carry out numerical simulations. However, the question is how to describe the computational domain of such complex geometries using a curvilinear coordinate system. This automatically implies a transformation of the coordinate system from the physical space (x, y, z) to the computational space (ξ, η, ζ) as exemplarily depicted in Figure 6.1 for a three-dimensional case. In this 179
6.2 Transformation to Curvilinear Coordinate System section this coordinate transformation [(x, y, z) (ξ, η, ζ)] is only briefly discussed. For more details about this topic the interested reader is referred, for example, to Ferziger and Peri´c (2002) and Breuer (2013, 2014). η
j
ξ
i k
ζ
k
j+3
j+3
Transformation
j
i+2
i+1
i
y
k
j+1
k+2
j+1
∆ξ
j+2
k+1
j+2
∆η
∆ζ
k+1 k+2
j
i+3
i
η
Physical Space
x
i+1
i+2
i+3
Computational Space
ξ
z
zet1a
ζ
Figure 6.1: Schematic transformation of three-dimensional coordinate system from the physical space to the computational space or vice versa.
In this description the coordinates of the computational and the physical space can be expressed in terms of each other as follows: ξ = ξ(x, y, z) ,
η = η(x, y, z) and ζ = ζ(x, y, z) .
(6.1)
Accordingly, the changes along each direction of the coordinate system (ξ, η, ζ) can be determined by: ∂ξ dx + ∂x ∂η dη = dx + ∂x ∂ζ dζ = dx + ∂x dξ =
∂ξ dy + ∂y ∂η dy + ∂y ∂ζ dy + ∂y
∂ξ dz , ∂z ∂η dz , ∂z ∂ζ dz . ∂z
(6.2a) (6.2b) (6.2c)
The matrix form of the above relation reads:
∂ξ ∂x dξ ∂η dη = ∂x dζ ∂ζ ∂x 180
|
∂ξ ∂y ∂η ∂y ∂ζ ∂y {z J
∂ξ ∂z dx ∂η , dy ∂z dz ∂ζ ∂z }
(6.3)
6. Computational Methodology where ∂ξj /∂xi denote the metric coefficients of the Jacobian matrix J . Note that the volume of the cell (control volume) in the computational space is equal to unity ∆Vc = 1, since it is assumed that ∆ξ = ∆η = ∆ζ = 1 (see Figure 6.1). On the other hand, the back-transformation from the p-space to the c-space can be achieved based on the inverse of the Jacobian matrix J −1 (see, e.g., Ferziger and Peri´c, 2002; Breuer, 2013). Thus, the cell volume in the physical space and the computational space reads: ∆V = J −1 (∆ξ ∆η ∆ζ)
and ∆Vc = J (∆x ∆y ∆z) ,
(6.4)
where J −1 is the determinant of the inversed Jacobian matrix and thus J −1 = ∆V and J = 1/∆V . In the following the most important differential operators required for the transformation of the governing equations are summarized1 . For example, the gradient of a scalar quantity φ (e.g., in the momentum equations φ = p) is given by: ∂φ = ∂xj
!
∂φ βij J , ∂ξi
(6.5)
where ξi = (ξ, η, ζ) and the metric coefficient βij is defined as βij = J −1 (∂ξi /∂xj ). The divergence of the velocity ui reads (e.g., mass conservation): ∂ui ∂ Ui J −1 J , = ∂xi ∂ξi
(6.6)
where the vector ui = (u, v, w) and the contravariant velocity is defined by Ui = (U, V, W ) = J (uk βik ). The Laplace operator of the scalar quantity φ is given by: ∂ ∂xi
∂φ ∂xi
!
∂ = ∂ξm
!
∂φ J Bkm J , ∂ξk
(6.7)
where the matrix Bkm is given by Bkm = βki βmi . Based on these transformations the entire system of governing equations can be transformed to the computational space. Finally, it is worth noting that for a complex geometry the computational domain is spit up into several blocks. As noted before, the CFD code LESOCC used in this study works on curvilinear body-fitted coordinates. Hence, each block has a block-structured body-fitted grid and an individual curvilinear coordinate system (Alletto, 2014). 6.3 Numerical Methods for the Continuous Phase
In this section the numerical methods implemented in the in-house code LESOCC to compute the continuous phase are presented. The finite-volume method (FVM) used for the spatial discretization of the fluxes is briefly introduced in Section 6.3.1. Then, the temporal discretization based on a low-storage multi-stage Runge-Kutta method is explained in Section 6.3.2. Afterwards, the predictor-corrector scheme for the pressure-velocity coupling is presented in Section 6.3.3. At the end of this section, the numerical solution of the linear system of equations is shortly described. 1
More details about the transformation of the coordinates can be found in Breuer (2013).
181
6.3 Numerical Methods for the Continuous Phase If the transformation explained in the previous section is applied to the mass and the momentum conservation equations given by Eq. (2.10), they have the following form in a curvilinear coordinate system (see, e.g., Ferziger and Peri´c, 2002; Breuer, 2013):
J
−1
∂uj ∂t
!
∂(Ui J −1 ) = 0, ∂ξi
(6.8a) !
∂ ∂p + βij + uj Ui J −1 = − ∂ξi ∂ξi " !# ∂ ∂uj 1 J Bkm + J −1 fj PSIC , µT + ∂ξm Re ∂ξk
(6.8b)
where the source term fj PSIC due to the momentum exchange between the particles and the fluid (two-way coupling) is given by Eq. (2.103). The conservation equations are discretized by means of the finite-volume method explained next. 6.3.1 Finite-Volume Method
Assuming an incompressible fluid, the integral form of the generic conservation equation for the transport quantity φ in a curvilinear coordinate system is given by (see, e.g., Breuer, 2013): J
−1
! ! ∂ ∂φ ∂φ ∂ −1 + Γφ J Bki + qφ J −1 , φ Ui J = ∂t ∂ξi ∂ξi ∂ξk
(6.9)
where Γφ stands for the diffusion coefficient and qφ J −1 is a source term. By integrating the above relation over the volume of the computational cell and transforming the volume integrals of the convection and the diffusion terms into surface integrals by applying the Gauss’s theorem, it can be written as: Z
|
V
J
−1
!
!
Z Z Z ∂φ ∂φ dV + Γφ J Bki · ni dA + qφ J −1 dV , (6.10) φ Ui J −1 · ni dA = ∂t ∂ξk A A V {z } {z } | {z } | {z } |
Local change
Convection term
Diffusion term
Source term
where ni denotes the normal unit vector with respect to the area of the cell face A (see Figure 6.2). 6.3.1.1 Spatial Discretization of Volume Integrals
The spatial discretization of the volume integrals in the transport equation (6.10) is achieved by applying the simple second-order accurate midpoint rule (Ferziger and Peri´c, 2002): Z
V
qφ J −1 dV = q φ J −1 ∆Vc ≈ qφ J −1 ,
(6.11)
where q φ is the mean value of the integrand approximated at the cell center denoted P as schematically depicted in Figure 6.2. 182
6. Computational Methodology T
N
t
∆ζ
n w
E
e P
W s
b
S
ζ
η
∆η
B
ξ
∆ξ
Figure 6.2: Notation used in the framework of the finite-volume method for a three-dimensional orthogonal computational cell with a cell-centered arrangement of the variables (adopted from Ferziger and Peri´c, 2002).
It is worth noting that Eq. (6.11) implies that no interpolation is required, since in the context of the finite-volume method combined with a cell-centered variable arrangement all variables are available at the cell center (node P). 6.3.1.2 Spatial Discretization of Surface Integrals
In this section the spatial discretization of the surface integrals (convective and diffusive terms) included in the transport equation (6.10) is presented. In general, the net flux through the CV boundary is the sum of integrals over the six faces of the computational cell and is given by (see, e.g., Breuer, 2002; Ferziger and Peri´c, 2002): Z
A
fi · ni dA =
X
c={e,w,s,n,t,b}
Z
Ac
fi,c · ni,c dAc =
X
Fi,c ,
(6.12)
c={e,w,s,n,t,b}
where fi is the component of the convective (φ Ui J −1 ) or the diffusive (Γφ ∂φ/∂ξk J Bki ) flux vector in Eq. (6.10). The index c = {e, w, s, n, t, b} stands for the six sides of the hexahedral control volume and the lower-case letters refer to the corresponding direction of each cell face with respect to the cell center P as depicted in Figure 6.2. It is worth noting that the surfaces of the cell faces in the computational space Ac = 1, since ∆ξ = ∆η = ∆ζ = 1. In the present study the surface integral is approximated by means of the second-order accurate midpoint rule. Thus, the integral is approximated as the product of the flux at the center of the cell face (i.e., the mean value over the surface) and the area of the cell face (Ferziger and Peri´c, 2002): Fi,c =
Z
Ac
fi,c · ni,c dAc = f i,c · ni,c Ac ≈ fi,c · ni,c Ac = Fi,c .
(6.13)
However, the value of flux fi,c is not available at the center of the cell face c and thus it has to be interpolated to this point. For this purpose, a second-order accurate linear interpolation is used in this study, which preserves the accuracy of the midpoint rule applied for the approximation of the surface integrals. Note that this linear interpolation corresponds to the central differencing scheme (CDS) in the context of the finite-difference method (FDM). 183
6.3 Numerical Methods for the Continuous Phase 6.3.1.3 Convective and Diffusive Fluxes
This section presents a simplified example for the spatial discretization of the convective and the diffusive fluxes on the west side of the computational cell (Cartesian grid) depicted in Figure 6.3 using the finite-volume method. Here, the transported quantity φ in relation (6.10) is set equal to φ = uj,w and the diffusion coefficient Γφ = (µT + 1/Re) leading to the momentum equations (6.8). As explained in the previous section, the convective flux (superscript conv) through the west cell face is approximated at the center of the west side based on the midpoint rule given by Eq. (6.12) as follows: Fwconv =
Z
Aw
uj Ui J −1
w
where the contravariant velocity Uw is given by: Uw = uj,w
∂ξ ∂xj
!
(6.14)
· ni,w dAw = uj,w Uw Jw−1 Aw ,
(6.15)
.
w
Here, the velocity uj,w is linearly interpolated to the corresponding cell face due to the collocated arrangement of the variables in LESOCC and thus reads: uj,w = λw uj,P + (1 − λw ) uj,W ,
(6.16)
where the interpolation factor λw for the example displayed in Figure 6.3 is given by: λw =
|xw − xW | . |xP − xW |
(6.17)
N n ww
W
P
w
e
E
ee
s
η
S
ξ Figure 6.3: Example for the spatial discretization of fluxes on the west side of the computational cell.
The diffusive flux (superscript diff) is approximated based on the midpoint rule given by Eq. (6.12) as follows: Fwdiff 184
=
Z
Aw
Γ
∂uj J Bki ∂ξk
!
w
· ni,w dAw = Γw
∂uj ∂ξ1
!
w
∂ξ1 ∂ξj ∂xj ∂x1
!
w
Jw−1 Aw ,
(6.18)
6. Computational Methodology where the diffusion coefficient Γw = (µT + 1/Re) is also linearly interpolated to the cell face. In this example, the gradient of the velocity uj is approximated at the center of the west side of the computational cell: ∂uj ∂ξ1
!
!
uP,w − uj,W . ξP − ξW
= w
(6.19)
6.3.2 Temporal Discretization
The spatial discretization of Eq. (6.8) based on the finite-volume method leads to a non-linear differential equation of the following form: J −1
∂uj ∂t
!
(n)
= F uj
−R p
∂ξi ∂xj
!(n)
(6.20)
where the superscript (n) denotes the values at the previous time step. By integrating both sides of the resulting relation in the time interval between t(n) and t(n+1) , the following relation is obtained (Breuer, 2013; Alletto, 2014): (n+1) uj,P
=
(n) uj,P
!(n) ∆t (n) ∂ξi + −1 F uj − R p , J ∂xj
(6.21)
where ∆t is the time step and the superscript (n + 1) stands for the values at the new time step. In the present study a low-storage multi-stage Runge-Kutta method of second-order accuracy is applied for the time-marching of the momentum equations (Binninger, 1989). The three sub-steps of the numerical integration of Eq. (6.21) are given by: (1)
uj,P (2)
uj,P (n+1)
uj,P
(∗)
with α1 =
(6.22a)
1 3
with α2 =
(6.22b)
1 2
with α3 = 1
(6.22c)
!(n) ∆t (n) ∂ξi (n) = uj,P + α1 −1 F uj − R p J ∂xj !(n) ∆t (1) ∂ξi (n) = uj,P + α2 −1 F uj − R p J ∂xj
!(n) ∂ξi ∆t (2) (n) = uj,P + α3 −1 F uj − R p J ∂xj (n+1)
uj,P = uj,P
(6.22d)
where the superscript (∗) refers to the uncorrected (not divergence-free) velocity field and hence the time integration performed applying this low-storage multi-stage Runge-Kutta method is called the predictor step. 6.3.3 Pressure-Velocity Coupling (Predictor-Corrector Scheme)
As mentioned in the previous section, the velocity field predicted by the time integration (6.22) is not divergence-free and hence it has to be corrected at the new time step to ensure that the mass conservation is achieved. In this section the predictor-corrector scheme adopted in LESOCC is 185
6.3 Numerical Methods for the Continuous Phase summarized. For the sake of simplicity this scheme is first described in the physical space and the final relation is transformed into the computational space. To derive an equation for the pressure, which is then used to correct the velocity field, the momentum equations (2.10b) are written in a semi-discrete form (Breuer, 2013): (∗)
"
(n)
uj,P − uj,P ∂(ui uj ) = ∆t ∂xi
#(n) P
1 + µT + Re
"
∂ 2 uj ∂xi ∂xi
#(n) P
(∗)
−
∂pP , ∂xj
(6.23)
(∗)
where the uncorrected velocity filed ui,P is given by Eq. (6.22d). The intermediate pressure field p(∗) is still unknown and therefore it is approximated in LESOCC by the pressure at the previous time step p(n) . Assuming that the pressure in the next time step p(n+1) leading to a divergence-free (n+1) velocity uj,P is known, the momentum equations can be expressed as: (n+1)
uj,P
(n)
"
− uj,P ∂(ui uj ) = ∆t ∂xi
#(n) P
1 + µT + Re
"
∂ 2 uj ∂xi ∂xi
#(n) P
(n+1)
∂p − P ∂xj
(6.24)
.
Subtracting Eq. (6.23) from Eq. (6.24) leads to the following expression: (n+1)
uj,P
(∗)
(n+1) (∗) − uj,P ∂p ∂p = − P − P . ∆t ∂xj ∂xj
(6.25)
(n+1)
To eliminate the unknown velocity field uj,P , the divergence operator is applied to relation (6.25) yielding the Poisson equation for the pressure correction: ∂p0P ∂xj
∂ ∂xj
!
(∗)
1 ∂uj,P = , ∆t ∂xj
(6.26)
where p0 = p(n+1) − p(∗) denotes pressure correction and hence the pressure field at the new time step reads p(n+1) = p(∗) + p0 . By transforming Eq. (6.26) to a curvilinear coordinate system based on Eqs. (6.5) and (6.7), it has the following form: ∂ ∂ξj
"
∂p0 J Bkj ∂ξk
!#
P
"
1 ∂ (∗) −1 = Uj J ∆t ∂ξj
#
(6.27)
. P
The Poisson equation (6.27) is solved numerically by the iterative incomplete LU decomposition2 by Stone (1968). Details about this procedure are documented in Stone (1968) and can be also found, for example, in Ferziger and Peri´c (2002), Sch¨afer (2006) and Breuer (2013). It is important to note that the improved pressure field is substituted into Eq. (6.25) yielding a new guess of the (∗) velocity field uj,P and thus: (n+1) uj,P
=
(∗) uj,P
∂p0P − ∆t ∂xj
!
(6.28)
, (n+1)
It is accepted that the numerical solution is converged if the velocity field uj,P satisfies a predefined convergence criterion (i.e., a nearly divergence-free velocity field). Otherwise, the Poisson 2
186
It is also called the strongly-implicit procedure (SIP).
6. Computational Methodology equation has to be solved again leading to further corrections of the pressure and the velocity field. Five to ten iterations are usually required (see, e.g., Schmidt, 2016). Note that in the context of the finite-volume method the collocated arrangement of the variables used in the CFD code LESOCC (Alletto, 2014; Schmidt, 2016) leads to pressure-velocity decoupling, which causes unphysical pressure oscillations. To avoid this behavior, the interpolation technique by Rhie and Chow (1983) is applied to the corrected velocity ensuring a proper coupling of the pressure and velocity fields. 6.4 Numerical Methods for the Disperse Phase
In this section the numerical methods used in the in-house CFD code LESOCC to compute the disperse phase in turbulent two-phase flows is presented. The first step of the calculation is the injection of the particles into the computational domain as explained in Section 6.4.1. Then, relying on a Lagrangian frame of reference each particle is tracked individually through the fluid field. The algorithm of particle tracking, which includes the numerical integration of the governing equations of the particle (i.e., solid particle or liquid droplet), is presented in Section 6.4.2. Afterwards, the numerical scheme used to interpolate the velocity of the continuous phase at the particle position required for the solution of the equations of particle motion is briefly explained in Section 6.4.3. The deterministic detection of particle-particle collisions within the framework of the hard-sphere model applied for the four-way coupled simulations is briefly summarized in Section 6.4.4. More details on these methods employed in LESOCC can be found, for example, in Breuer et al. (2012), Breuer and Alletto (2012), Alletto and Breuer (2012, 2013) and Alletto (2014). In addition, a detailed description of the post-collision treatment of the collision partners in case of particle agglomeration or droplet coalescence and in case of particle-wall adhesion of particles is given in Sections 6.4.5, 6.4.6 and 6.4.7, respectively. 6.4.1 Particle Injection into the Computational Domain
As a first step for the simulation of the disperse phase, the properties of particles injected into the computational domain have to be defined. In LESOCC these quantities are the diameter dq , the density ρq , the position xq , the translational uq and the angular ωq velocity of each particle. In addition, the block (or processor), into which the particle is released, has to be specified. Here, the subscript q = p for solid particles and q = d for droplets. In LESOCC the position of the particle is defined in the computational space (ξ, η, ζ) and each primary particle iq is given a positive source number and stored in a linear array denoted NUMQPA(iq ) as will be described in Section 6.4.5. Furthermore, the injection time of the primary particle into the computational domain is stored in a linear array denoted NUMTPA(iq ). In the standard set-up it is assumed that the injected particle has the same velocity as the fluid at the position of the particle implying that uq = uf |particle . For this purpose, a trilinear interpolation scheme (see Appendix D.2) is applied to approximate the fluid velocity at the particle position, for which the fluid velocities at the eight cell centers surrounding the particle are required. Regarding the angular velocity, it is generally assumed that the particle does not rotate at the time instant of injection and hence its angular velocity is set to zero (i.e., ωq = 0). 187
6.4 Numerical Methods for the Disperse Phase 6.4.2 Particle Tracking Scheme
Assuming that the volume fraction of the disperse phase is small enough (typically αq . 10−6 ), a one-way coupling is sufficient (see Section 2.5). In this regime the particle motion is treated such that the particles have no effect on the fluid phase and the particle-particle collisions are neglected. As concluded in Section 2.3.5, six ordinary differential equations for the translational and angular motion of the disperse phase have to be numerically solved. After computing the continuous phase, the velocity components and the pressure of the fluid, which are stored in the cell centers (i.e., colocated arrangement), are available in the physical space.
η
η
j+3 j+2
ξ
j+2
∆ξ
j+1
∆η j
j+1
y
y j
ξ i
i+1
i+2
i+3
x (a) Physical space
i−1
i
i+1
i+2
i+3
x (b) Computational space
Figure 6.4: Particle tracking in (a) the physical and (b) the computational space (Breuer et al., 2007).
A two-dimensional representation of the particle tracking (for the sake of simplicity) is depicted in Figure 6.4. As explained next, the first numerical integration of the equation of the translational and the angular motion of the particle is done in physical space. However, the second integration yielding the particle position is performed in the computational space. The reason for this procedure is that the computation of the particle position in the physical space is not trivial in case of curvilinear grids as depicted in Figure 6.4. 6.4.2.1 Translational Motion 6.4.2.1.1 Particle Velocity (First Integration)
The ordinary differential equation of the translational motion of the particle given by Eq. (2.49) can be expressed in the following form (see Section 2.3.5): duq = fD (uf − uq ) + fG + fL + fM . dt
(6.29)
Here, the following factors are introduced in order to simplify the formulation of relation (2.49). 188
6. Computational Methodology The factor of the drag force fD is given by: fD =
1 fAM
(
)
3 1 ρf CD |uf − uq | 4 dq ρ q
,
(6.30)
where the added mass factor fAM is given by Eq. (2.50). The factor of the sum of the gravity and the buoyancy force fG reads: fG =
1
ρf 1− ρq
fAM
!
(6.31)
g.
The factor of the lift forces fL appearing in Eq. (6.29) represents the total lift force (Saffman and Magnus) and is given by: fL =
1 fAM
(
FLSaf + FLM ag mq
)
(6.32)
.
The last term on the right-hand side of Eq. (6.29) fM considers the material derivative included in the added mass and the pressure gradient forces and is given by: fM =
1 fAM
(
ρf Duf (1 + CAM ) ρq Dt
)
.
(6.33)
In the present study the material derivative appearing in Eq. (6.33) is trilinearly interpolated to the particle position as explained in Appendix D.2. The fluid velocity at the particle position uf appearing in relation (6.29) is interpolated relying on a Taylor series expansion by Marchioli et al. (2007) as will be explained in Section 6.4.3. In the context of the present study the ordinary differential equation Eq. (6.29) is solved either numerically or analytically. In order to distinguish between the numerical and the analytical integration of Eq. (6.29), the following condition is used: β = min {(fD ∆t) , 128} ,
(6.34)
where ∆t is the time step of the simulation (i.e., for the continuous and the disperse phase). It has been shown that the classical fourth-order Runge-Kutta scheme is absolutely stable for 0 ≤ β ≤ 2.78 (Antia, 2002; Breuer et al., 2006; Alletto, 2014). In the CFD code LESOCC this Runge-Kutta scheme (see, e.g., Faires and Burden, 1994) is applied under the condition β < 2.0. To calculate the translational velocity of the particle, relation (6.29) is integrated in the physical space by this classical Runge-Kutta scheme (see, e.g., Breuer et al., 2006, 2012; Alletto and Breuer, 2013; Alletto, 2014; Breuer and Almohammed, 2015; Almohammed and Breuer, 2016b,c). Thus, the particle velocity at the new time step u(n+1) reads: q u(n+1) = u(n) q q +
1 (k1 + 2k2 + 2k3 + k4 ) , 6
(6.35) 189
6.4 Numerical Methods for the Disperse Phase where the coefficients of the Runge-Kutta scheme including the quantities from the previous time step denoted by the superscript (n) are given by: n
(n)
(n)
(n)
(n)
o
uf − u(n) + fG + fL + fM , q 1 (n) (n) (n) (n) (n) k2 =∆t fD uf − uq + k1 + fG + fL + fM , 2 1 (n) (n) (n) (n) (n) k3 =∆t fD uf − uq + k2 + fG + fL + fM , 2 k1 =∆t fD
(n)
k4 =∆t fD
uf − u(n) q + k3
(n)
(n)
(n)
+ fG + fL + fM
.
(6.36a) (6.36b) (6.36c) (6.36d)
Since the lift, the added mass and the pressure gradient forces are included in Eq. (6.36), they should in general be updated for each sub-step of the Runge-Kutta scheme. However, to increase the efficiency of the calculation, the corresponding terms are computed only once. This is motivated by the fact that the lift, the added mass and the pressure gradient forces are smaller than the drag force by at least one order of magnitude (see, e.g., Alletto, 2014; Kuerten, 2016). If the condition mentioned above is not satisfied (i.e., β ≥ 2), the particle velocity is analytically calculated by the following relation:
u(n+1) = uf + u(n) q q − uf exp (−β) +
o 1 n (n) (n) (n) + f [1 − exp (−β)] , + f f M L G (n)
fD
(6.37)
It is worth noting that relations (6.35) and (6.37) differ from the corresponding equations in Alletto (2014) due to the inclusion of the general form of the forces acting on the particle in the present description (see Section 2.3.5). 6.4.2.1.2 Particle Position (Second Integration)
Based on Eq. (2.25) the second integration of Eq. (6.29) yields the particle position. In LESOCC this step is performed in the computational space. By doing that, time-consuming search algorithms can be avoided, which leads to a highly efficient particle tracking scheme and hence the paths of millions of particles can be predicted (see, e.g., Breuer et al., 2012; Alletto and Breuer, 2013). This choice is motivated by the following explicit relations between the particle location and the indices of the cell containing it, which are required to determine the fluid forces displacing the particle: iq = int(ξq ) ,
jq = int(ηq ) and kq = int(ζq ) .
(6.38)
where int(ξq ) stands for the integer value of particle coordinate ξq . As exemplarily depicted in Figure 6.4(a), it is not trivial to identify the new computational cell containing the particle in the physical space (Sch¨afer and Breuer, 2002) and hence this step is done in the computational space: Using the notion ξq = (ξq , ηq , ζq ) for the particle position in the computational space, it can be determined based on Eq. (2.25) transformed to the c-space which reads: dξq = Uq . dt 190
(6.39)
6. Computational Methodology The components of the contravariant velocity Uq = (Uq , Vq , Wq ) are calculated based on Eq. (6.2). Thus, they can be written as follows:
∂ξ ∂ξ ∂ξ dξq uq + vq + wq = Uq , = dt ∂x q ∂y q ∂z q
dηq ∂η ∂η ∂η uq + vq + wq = V q , = dt ∂x q ∂y q ∂z q
(6.40)
dζq ∂ζ ∂ζ ∂ζ uq + vq + wq = Wq , = dt ∂x q ∂y q ∂z q
where the components of the particle velocity uq = (uq , vq , wq ) are known from the first integration of relation (6.29). The metric coefficients (e.g., ∂ξ/∂x|q ) are trilinearly interpolated (see Appendix D.2) to the particle position. Finally, the coordinates of the particle position at the new time step are determined by integrating Eq. (6.39) by means of the Runge-Kutta scheme which reads: 1 (6.41) ξq(n+1) = ξq(n) + (k1 + 2k2 + 2k3 + k4 ) . 6 The coefficients for the Runge-Kutta scheme appearing in Eq. (6.41) are given by: k1 = ∆t Uq ,
(6.42a)
k2 = ∆t (Uq + 1/2 k1 ) ,
(6.42b)
k3 = ∆t (Uq + 1/2 k2 ) ,
(6.42c)
k4 = ∆t (Uq + k3 ) .
(6.42d)
It is important to note that in the entire procedure described above a back-transformation of these particle coordinates to the physical space is not required. However, for visualization purposes the coordinates of the particle position are transformed back using the trilinear interpolation presented in Appendix D.2 leading to xq = (xq , yq , zq ). 6.4.2.2 Angular Motion
In the present study the components of the angular velocity of the particle ωq = (ωq , ωq , ωq ) are determined by analytically solving the set of ordinary differential equations (2.54). Thus, the angular velocity in the new time step ωq(n+1) reads: ωq(n+1) = ωq(n) exp (−β) + where the factor β is given by: β = min
(
!
)
1 ∇ × uf [1 − exp (−β)] , 2
10 ∆t , 128 . 3τq
(6.43)
(6.44)
The solution of relation (6.43) requires the calculation of the fluid vorticity at the particle position at each time step (details in Breuer (2013)). For this purpose, the six partial derivatives of the fluid velocities are computed for each control volume and then trilinearly interpolated to the particle position (see Appendix D.2). 191
6.4 Numerical Methods for the Disperse Phase 6.4.3 Fluid Velocity at the Particle Position
The determination of the slip velocity between the disperse and the continuous phase requires the interpolation of the fluid velocity at the particle position uf |particle appearing in the governing equations of the particle motion. For this purpose, Breuer et al. (2006, 2007) adopted the trilinear interpolation scheme (see Appendix D.2). However, it was found by Alletto (2014) that this approximation leads to a pronounced filtering effect on the velocity of the continuous phase and hence the second-order statistics of a turbulent channel flow can not be correctly reproduced in comparison with DNS results by Kim et al. (1987). Therefore, a more accurate interpolation scheme based on a Taylor series expansion proposed by Marchioli et al. (2007) is used in the present thesis. This interpolation scheme was implemented in the in-house code LESOCC and applied in many studies (see, e.g., Alletto and Breuer, 2013; Alletto, 2014). The Taylor series expansion about the cell center next to the particle, whose position has to be computed, is expressed in Cartesian coordinates as follows: ∂uf |N ∂uf |N ∂uf |N ∆x + ∆y + ∆z uf |particle = uf |N + ∂x ∂y ∂z (6.45)
+ O ∆x2 , ∆y 2 , ∆z 2 , ∆x∆y, ∆x∆z, ∆y∆z ,
where N refers to the nearest cell center (xN , yN , zN ) with respect to the particle position. ∆xi = (∆x, ∆y, ∆z) represents the relative distance between the coordinates of the particle and those of the nearest cell center and hence ∆xi = (xq,i − xN,i ). Based on Marchioli et al. (2007) only the first derivatives appearing in Eq. (6.45) are taken into account and thus this second-order accurate interpolation scheme has the truncation error of the order O[(xq,i − xN,i )2 ]. As mentioned before, the in-house CFD code works with curvilinear body-fitted coordinates and hence the derivatives appearing on the right-hand side of Eq. (6.45) have to be determined as follows (Alletto, 2014; Breuer, 2014):
∂uf |N ∂uf ∂ξ ∂uf ∂η ∂uf ∂ζ = + + , ∂x ∂ξ ∂x N ∂η ∂x N ∂ζ ∂x N
∂uf |N ∂uf ∂ξ ∂uf ∂η ∂uf ∂ζ = + + , ∂y ∂ξ ∂y N ∂η ∂y N ∂ζ ∂y N
(6.46)
∂uf |N ∂uf ∂ξ ∂uf ∂η ∂uf ∂ζ = + + . ∂z ∂ξ ∂z N ∂η ∂z N ∂ζ ∂z N
where ξ, η and ζ denote the three coordinates of the computational space. If the above operators are applied to Eq. (6.45), the derivatives of the velocity of the continuous phase with respect to the coordinates of the computational domain (ξ, η, ζ) have to be determined. This step is carried out applying the second-order accurate central difference scheme: o ∂uf 1n = uf |(i+1,j,k) − uf |(i−1,j,k) + O ∆ξ 2 , ∂ξ 2 o ∂uf 1n = uf |(i,j+1,k) − uf |(i,j−1,k) + O ∆η 2 , ∂η 2 o ∂uf 1n = uf |(i,j,k+1) − uf |(i,j,k−1) + O ∆ζ 2 . ∂ζ 2
192
(6.47)
6. Computational Methodology For the metric coefficients (e.g., ∂ξ/∂x|N ) the already available and stored quantities can be used. This interpolation scheme is adopted for all simulations, where most of the results were already published in Breuer and Almohammed (2015, 2016, 2018) and Almohammed and Breuer (2016a,b,c, 2018) . 6.4.4 Deterministic Collision Detection Model
In the framework of a four-way coupled Euler-Lagrange simulation particle-particle collisions have to be taken into account. In the present study the particle-particle collisions are detected deterministically by applying the recently developed model by Breuer and Alletto (2012). This collision detection model relies on the technique of uncoupling developed by Bird (1976), in which the computation of particle trajectories is carried out in two stages (Breuer and Alletto, 2012): 1. In the first stage the particles are moved based on the equation of motion without taking the particle-particle collisions into account. 2. In the second stage the occurrence of collisions during the first stage is examined for all particles. If a particle-particle collision is found, the velocities of the collision pair are replaced by the post-collision velocities without changing their position, which is also advantageous for parallelization (Breuer and Alletto, 2012). The main reason why the displacements are preferred to be neglected after the collisions is the prevention of an otherwise appearing inherent risk that an overlap with a third particle is generated which could disarrange the entire procedure (Wunsch et al., 2008). Note that nevertheless a re-collision of the same two collision partners is impossible as will be explained below. The collision treatment itself is a critical issue from the computational burden evolving from this task for a large number of particles Nq at high mass loadings. By using small time steps, it is accepted that only collisions between neighboring particles are likely. Thus, the computational effort can be substantially reduced by dividing the domain into virtual cells as schematically depicted in Figure 6.5(a). This concept was applied in many studies (see, e.g., Viccione et al., 2008; Hopkins and Louge, 1991; Wunsch et al., 2008), in which the collision detection was restricted to neighboring particles in the virtual cell. By using this method, the cost of the detection of collisions can be reduced from the usual order O(Nq2 ) to O(Nq ). To achieve a nearly optimal virtual cell size, it is dynamically adjusted during the simulation based on the number of particles found in the virtual cell. Furthermore, to avoid overlapping cells or the necessity to take the 26 surrounding cells into account, the search and collision detection procedure is carried out a second time with slightly different cell sizes as depicted in Figure 6.5(b). For this purpose the ratio of edge lengths is chosen according to the quotient of two prime numbers, here 17/13 ≈ 1.3. This measure reduces the possibility that the boundaries of the virtual cells of the first and second sweep match and thus guarantees that the maximum number of possible collisions is found. If likely collision partners are identified by this concept, the further inspections about possible collisions can be restricted to particles in one virtual cell. More details about the formulation of this concept of collision detection can be found in Breuer and Alletto (2012) and Alletto (2014). 193
6.4 Numerical Methods for the Disperse Phase
Virtual grid for the first search
Virtual grid for the first search
Virtual grid for the second search
✓
?
(a) First search
(b) Second search
Figure 6.5: Treatment of deterministic collisions by virtual cells (taken from Breuer and Alletto, 2012).
6.4.4.1 Particle-Particle Collision Conditions
In this model it is assumed that the centers of mass of the collision partners move with a constant velocities implying that they displace linearly within the time step (Tanaka and Tsuji, 1991; Chen et al., 1998; Yamamoto et al., 2001). Thus, a binary collision can be detected based on purely kinematic conditions. In this model a particle-particle collision occurs if two conditions are satisfied (Breuer and Alletto, 2012):
Ê The collision partners are approaching each other, which is only possible if the following condition is fulfilled as schematically depicted in Figure 6.6.
− x− r · urel < 0 ,
(6.48)
− − − − − where x− r = x2 − x1 and urel = u2 − u1 stand for the relative position vector and the relative velocity between the two particles, respectively. It is worth noting that only for those potentially colliding particles located in a virtual cell, which satisfy the first condition given by Eq. (6.48), the second condition has to be checked.
Ë The minimum separation between the collision partners xr,min within a time step has to be less than the sum of their radii l12 (see Figure 6.6) and thus: |xr,min | ≤ l12
with l12 = r1 + r2 =
1 (d1 + d2 ) . 2
(6.49)
Assuming a linear displacement of the centers of mass of the collision partners, the minimum separation between the particles required for the second collision condition (6.48) is determined by: − xr,min = x− r + urel ∆tmin ,
194
(6.50)
6. Computational Methodology where the minimum time ∆tmin , at which the separation distance between the particles becomes minimal, is computed as follows: ∆tmin = −
− x− r · urel
− 2 urel
(6.51)
.
As mentioned in Section 2.6, the relative velocity between the two particles changes its sign after the impact and thus a re-collision of the collision partners is automatically excluded (Breuer and Almohammed, 2015). y t + ∆tmin
t + ∆tcol
t
t + ∆t
u− rel x− r,col x− r,min
x− r
Particle 2
(0, 0)
x
Particle 1
Figure 6.6: Schematic representation of the relative motion between collision partners (adopted from Tanaka and Tsuji, 1991; Breuer and Alletto, 2012).
Based on these considerations the conditions of a particle-particle collision can be expressed as follows (see Figure 6.6): (∆tmin ≤ ∆t and
|xr,min | ≤ l12 )
or
− xr
≤ l12 .
(6.52)
It is worth noting that the second condition in relation (6.52) implies that the relative distance between the colliding particles at the beginning of the time step is less than the distance between the centers of mass of the particles l12 . This case indicates that the collision partners interpenetrate into each other (overlap) leading automatically to a particle-particle collision. If the collision conditions are satisfied, the two particles collide and hence the collision-normal unit vector n required for the particle-particle collision model has to be determined. As explained in Section 2.6, this quantity can be computed based on the relative position vector between the centers of mass of the collision partners at the instant of impact xr,col : n=
xr,col , |xr,col |
(6.53)
where the relative vector xr,col and its magnitude |xr,col | reads: − xr,col = x− r + urel ∆tcol
and
|xr,col | = l12 .
(6.54) 195
6.4 Numerical Methods for the Disperse Phase However, the collision time ∆tcol appearing in Eq. (6.53) is still unknown. If the collision conditions are satisfied, this time period is computed by equating the relative distance between the centers of mass of the particles at that time instant to l12 and thus: − xr
2
2 + ∆tcol u− rel = l12 .
(6.55)
The solution of the above equation is given by Chen et al. (1998):
∆tcol = ∆tmin 1 −
q
1 − K1 K2 ,
(6.56)
where the coefficients K1 and K2 are defined by: 2
K1 =
2
− |x− r | urel
− 2 (x− r · urel )
and K2 = 1 −
2 l12 2 . |x− r |
(6.57)
If a collision is detected, the kinetics of the collision partners have to be updated based on the hard-sphere model with en,p ≤ 1. If the friction between the particles is taken into account, it is modeled by Coulomb’s law (see, e.g., Crowe et al., 1998; Sommerfeld et al., 2008). Thus, the translational and angular velocities of the collision partners are updated depending on the normal and the tangential restitution coefficients (en,p and et,p ) as well as the static and kinetic coefficients of friction (µst,p and µkin,p ). In this study these issues are directly related to modeling of the particle agglomeration and droplet coalescence (more details in Sections 3.1 and 4.2.5). The algorithm of the deterministic collision detection model for binary particle and droplet collisions implemented in LESOCC is presented in Appendix D.3. 6.4.5 Treatment of Particle Agglomeration
If the agglomeration of rigid, dry and electrostatically neutral solid particles is taken into account, a particle-particle collision leads to either an agglomeration or the collision partners move apart from each other. The post-collision treatment of the collision partners and the resulting agglomerates depends on the agglomeration model applied, namely the energy-based model and the momentumbased agglomeration model (see Section 3.1). Note that in the in-house code LESOCC the properties of the particles, namely the position in c- and p-space, the translational and angular velocities in p-space, the density, the diameter, the injection time and the source number, are stored in linear arrays. If the collision partners bounce back, the source numbers of the particles maintain their positive values. However, if the agglomeration conditions are satisfied, the colliding particles agglomerate and the kinetics and the structure of the resulting agglomerate are modeled as explained in Sections 3.3 and 3.4, respectively. A negative source number is given to this agglomerate in order to distinguish it from the two colliding primary particles. For this purpose, two steps are carried out: 1. The first collision partner denoted ip (i.e., reference particle), whose source number is stored in the array NUMQPA(ip ), is replaced by the formed agglomerate and hence iag = ip . The size of the agglomerate defined as the number of primary particles included in the this 196
6. Computational Methodology agglomerate Napp (i.e., agglomerated primary particles) is now stored in the array defining the source number as follows: NUMQPA(iag ) = −Napp .
(6.58)
2. The second collision partner is no longer required and thus it is directly deleted from the computational domain. The source number of the formed agglomerate is determined based on the source numbers of the collision partners. Here, four combinations are possible for the collision partners:
Ê Two primary particles. Hence, both source numbers are positive, i.e., NUMQPA(1) > 0 and NUMQPA(2) > 0. In this case the source number of the agglomerate is defined by: NUMQPA(iag ) = −2 .
(6.59)
Thus, the agglomeration of two primary particles yields a two-particle agglomerate (i.e., Napp = 2) based on Eq. (6.58), which is the smallest possible size of an agglomerate.
Ë A primary particle (positive source number) and an agglomerate (negative source number). This means that NUMQPA(1) > 0 and NUMQPA(2) < 0 and hence the source number of the agglomerate reads: NUMQPA(iag ) = (NUMQPA(2) − 1) .
(6.60)
For example, a primary particle collides with a two-particle agglomerate leading to a threeparticle agglomerate (i.e., NUMQPA(iag ) = −2 − 1 = −3) and hence Napp = 3 based on Eq. (6.58).
Ì An agglomerate (negative source number) and a primary particle (positive source number). Thus, NUMQPA(1) < 0 and NUMQPA(2) > 0. Here, the source number of the agglomerate is given by: NUMQPA(iag ) = (NUMQPA(1) − 1) .
(6.61)
For example, the agglomeration of a three-particle agglomerate and a primary particle yields a four-particle agglomerate (i.e., NUMQPA(iag ) = −3 − 1 = −4) and hence Napp = 4.
Í Two agglomerates. Hence, both source numbers are negative. In this case the source number of the agglomerate reads:
NUMQPA(iag ) = NUMQPA(1) + NUMQPA(2) .
(6.62)
For example, a two-particle agglomerate collides with a three-particle agglomerate resulting in a five-particle agglomerate (i.e., NUMQPA(iag ) = −2 − 3 = −5) and hence Napp = 5. 197
6.4 Numerical Methods for the Disperse Phase At the end of each time step all removed particles reduce the sizes of the linear arrays for the properties of the particles (position, velocity, etc.). Thus, the number of particles has to be readjusted. Here, the number of active particles is the sum of the remaining primary particles and the total number of agglomerates. Note that for each time step the total number of injected particles pp(n) N0 , active particles Np(n) , agglomerated primary particles n(n) app , particle-particle collisions ncol th and agglomeration processes n(n) agp at the n time step are stored in external files (agglomeration statistics) and then used for the analysis of the results. Thus, the accumulated number of the pp(n) (n) (n) particle-particle collisions Ncol , agglomeration processes Nagp and formed agglomerates Nag (see Appendix D.4) as a function of time required for the analysis of the results can be determined during the post-processing based on these variables. 6.4.6 Treatment of Droplet Coalescence
In Chapter 4 two types of liquid droplets are distinguished, namely surface-tension and viscosity dominated droplets. In this context, it is worth stressing that the focus is solely on the droplet coalescence due to the binary collision of surface-tension dominated droplets. For this purpose, the composite collision outcome model presented in Section 4.2.5 is applied in this thesis. The treatment of the kinetics of the droplets after the impact is achieved depending on the collision outcome, namely bouncing, stretching separation and reflexive separation. Additionally, in case of a coalescence the position and the velocity of the coalesced droplet (i.e, agglomerate) are determined. The entire calculation procedure for the composite collision outcome model is summarized in Appendix B.2. The treatment of the coalesced droplets (i.e., agglomerates) in terms of the source numbers of the collision partners is treated based on the same procedure as explained in the preceding section. In addition, since the identification of the collision outcome is done based on the dimensionless parameter B and the collision Weber number Wec , these parameters are stored at the end of each time step in external files and then used for plotting the regime map (Wec , B). 6.4.7 Treatment of Particle-Wall Adhesion
As done in most numerical studies, if the particle deposits on the wall, it is directly removed from the computational domain (see, e.g., McLaughlin, 1989; Kvasnak et al., 1993; van Haarlem et al., 1998; Zhang and Ahmadi, 2000; Breuer et al., 2006; Winkler et al., 2006; Koullapis et al., 2016). The motivation for this assumption is that (i) the cover rate of the deposited particles on the wall (i.e., the number of deposited particles on the bounding surface as a function of the simulation time) is in general very small and (ii) the treatment of these deposited particles on the wall generating a certain roughness is not trivial. In the present study this approach is applied as done in Almohammed and Breuer (2016c) and Breuer and Almohammed (2016). In other words, using the newly developed momentum-based particle-wall adhesion model (see Section 5.2), the deposited particles are deleted from the domain if the deposition condition (5.60) is satisfied. Basically, within a time step the particle-wall collision including adhesion is done as follows: • First, it is checked whether a particle-wall collision occurs. In the framework of this particlewall collision model two conditions have to be satisfied: (i) the particle is approaching 198
6. Computational Methodology the wall when (u− p · n) < 0 and (ii) the normal distance of the particle center to the wall is less or equal to the distance between the center of the first fluid cell and the wall (i.e., ∆ln,p ≤ ∆ncell-center ).
• If a particle-wall collision occurs, the deposition condition (5.60) is checked. If this relation is satisfied, the particle is given a zero source number. The remaining particles maintain their source numbers and are treated as explained in Section 5.2.6 taking the effect of the particle-wall adhesion into account.
• At the end of the time step the deposited particles with zero source numbers are removed from the domain and the sizes of the arrays for the properties of the particles (e.g., position and velocity, etc.) are readjusted based on the number of the remaining active particles. 6.5 Determination of the Statistics
The statistics of the continuous phase are calculated on-the-fly as follows: hφf (x, y, z)i(n) =
1 1 φf (n) (x, y, z) + 1 − hφf (x, y, z)i(n−1) , n n
(6.63)
where φf (x, y, z) stands for any characteristic fluid quantity of interest (e.g., mean velocity or velocity fluctuations) and n = 1, ..., Nav denotes the time-averaging period with the total number of averaging time steps Nav . The brackets h· · · i stand for the mean value of the fluid quantity and hence relation (6.63) yields an arithmetic mean value of φf . Thus, for n → ∞ the series (6.63) converges to the true mean value of the quantity φf (x, y, z). It is worth noting that in case of a (n) homogeneous direction, the instantaneous fluid quantity φf (x, y, z) is substituted by its spatially averaged value in this direction (Alletto, 2014). Similarly, the statistics of the disperse phase are computed on-the-fly as done by Alletto and Breuer (2013). For this purpose, the following series is updated if a particle is found in a cell whose center is located at the point (x, y, z). This situation is denoted as event (Breuer and Almohammed, 2015; Almohammed and Breuer, 2016b,c): hφp (x, y, z)i
Nk
=
k φN p (x, y, z)
!
1 1 + 1− hφp (x, y, z)i(Nk −1) , Nk (x, y, z) Nk (x, y, z)
(6.64)
where φp (x, y, z) can be any characteristic particle quantity of interest (e.g., mean velocity, velocity fluctuations or particle diameter). The superscript Nk = Nk (x, y, z) denotes the event number defining the accumulated number of particles found in a specific control volume located at the position (x, y, z). As for the fluid statistics, the expression (6.64) leads to an arithmetic mean value of the quantity φp . Note that if required, the three-dimensional statistics predicted by Eq. (6.64) needed for the analysis can be further spatially averaged in the homogeneous direction during the post-processing. It is important to mention that this on-the-fly prediction for the statistics of the fluid and the disperse phase is carried out in order to significantly reduce the amount of stored data, which is urgently required in case of fine grid resolutions or huge numbers of particles in the computational domain. 199
CHAPTER 7
RESULTS FOR PARTICLE AGGLOMERATION
Snapshot of agglomerates in a particle-laden turbulent channel flow (Almohammed and Breuer, 2016b).
7.1
Validation of the Agglomeration Models. . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1
7.2
Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 7.1.1.1
Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.1.1.2
Properties of the Particles . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.1.2
Theoretical Model for Particle Agglomeration
7.1.3
Results and Discussion
. . . . . . . . . . . . . . . . . . . 205
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.1.3.1
Agglomeration Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
7.1.3.2
Particle Number Concentration . . . . . . . . . . . . . . . . . . . . . . 207
7.1.3.3
Summary of Key Findings . . . . . . . . . . . . . . . . . . . . . . . . . 209
Particle Agglomeration in Turbulent Channel Flow . . . . . . . . . . . . . . . . . . . 7.2.1
203
210
Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.2.1.1
Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.2.1.2
Simulation Set-up
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7.2.1.2.1 Properties of the Particles
. . . . . . . . . . . . . . . . . . . 212
7.2.1.2.2 Coefficients of the Collision Models . . . . . . . . . . . . . . . 213 7.2.1.3 7.2.2
Determination of Agglomeration Statistics
. . . . . . . . . . . . . . . . 214
Results of the Momentum-based Agglomeration Model 7.2.2.1
. . . . . . . . . . . . . . 215
Effect of the Structure Model of the Agglomerate . . . . . . . . . . . . . 215 7.2.2.1.1 Summary of Key Findings . . . . . . . . . . . . . . . . . . . . 222 7.2.2.1.2 Comparison with the Model by Kosinski and Hoffmann . . . . 223
7.2.2.2
Effect of the Restitution Coefficient . . . . . . . . . . . . . . . . . . . . 223
7.2.2.3
Effect of the Friction Coefficient . . . . . . . . . . . . . . . . . . . . . . 226
7.2.2.4
Effect of the Two-Way Coupling / Feedback of Particles on the Flow . . 227
7.2.2.5
Effect of the Subgrid-Scale Model for the Particles . . . . . . . . . . . . 233 7.2.2.5.1 Trivial Model . . . . . . . . . . . . . . . . . . . . . . . . . . 234 7.2.2.5.2 Extended Langevin Model . . . . . . . . . . . . . . . . . . . . 237
7.2.2.6
Effect of the Lift Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.2.2.7
Cumulative Effect of the Sub-Models . . . . . . . . . . . . . . . . . . . 243
7.2.2.8
Effect of the Diameter of the Primary Particles . . . . . . . . . . . . . . 244
7.2.2.9
Effect of the Mass Loading . . . . . . . . . . . . . . . . . . . . . . . . . 246
7.2.2.10 Effect of the Wall Roughness
. . . . . . . . . . . . . . . . . . . . . . . 250
7.2.2.10.1 Different Wall Roughnesses Considering Large Particles . . . . 253 7.2.2.10.2 Different Wall Roughnesses Considering Small Particles . . . . 255 7.3
Comparison of Agglomeration Models
. . . . . . . . . . . . . . . . . . . . . . . . .
257
7.3.1
Effect of the Agglomeration Model without Sub-Models . . . . . . . . . . . . . . 257
7.3.2
Effect of Different Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 264 7.3.2.1
Cumulative Effect of the Sub-Models . . . . . . . . . . . . . . . . . . . 264
7.3.2.2
Effect of the Diameter of the Primary Particles . . . . . . . . . . . . . . 268
7.3.2.3
Effect of the Wall Roughness
. . . . . . . . . . . . . . . . . . . . . . . 270
7 Results for Particle Agglomeration This chapter is concerned with the numerical results of the agglomeration of dry, electrostatically neutral particles in turbulent flows using the newly developed agglomeration models implemented in LESOCC. The content of this chapter is organized as follows. In Section 7.1 the energy-based and the momentum-based agglomeration models are validated in a shear flow against a theoretical model. Then, a comparative study using the momentum-based model is carried out in Section 7.2 to investigate the effect of the agglomerate structure models and various simulation parameters on the dynamic agglomeration process in a particle-laden turbulent channel flow. Afterwards, the predictions of both agglomeration models are compared using this particle-laden turbulent channel flow in Section 7.3. Note that most of the results presented in this chapter were already published1 in Breuer and Almohammed (2015), Almohammed and Breuer (2016a), Almohammed and Breuer (2016b) and Breuer and Almohammed (2018). 7.1 Validation of the Agglomeration Models
This section aims at validating the capability of the newly developed agglomeration models to adequately predict the agglomeration process in disperse two-phase flows. Although a variety of experimental studies can be found in the literature regarding fluid and particle statistics of particle-laden flows including preferential concentration, to the best of the author’s knowledge no such data are available for the agglomeration of solid particles. However, for a simple shear layer theoretical results including the agglomeration rate and the rate of change of the particle number concentration can be found under certain simplifications (Almohammed and Breuer, 2016b). 7.1.1 Test Case Description
The simple shear layer considered is schematically depicted in Figure 7.1. The size of the rectangular computational domain in streamwise, wall-normal and spanwise direction is 2δ × 2δ × 0.2δ, respectively. Here, δ = 0.02 m is the half-width of the channel. Moving Wall
Uw
y 2δ
x u(y) = −γ˙ y Moving Wall
Uw
2δ Figure 7.1: Set-up for the simulation of the shear layer used in the validation study. 1
Only full-paper contributions in peer-reviewed journals and conference proceedings are mentioned.
203
7.1 Validation of the Agglomeration Models An equidistant mesh consisting of 64 × 64 × 10 grid points is used. The distribution of the fixed fluid velocity in streamwise direction is given by u(y) = −γ˙ y, where γ˙ stands for a constant shear rate and y is the distance from the wall (see Figure 7.1). It is assumed that the shear rate γ˙ = 71 s−1 is established by moving the upper and lower walls in opposite directions with a constant velocity Uw = γ˙ δ = 1.42 m/s. The density and the kinematic viscosity of the continuous phase (water) are ρf = 1000 kg/m3 and νf = 1 × 10−6 m2 /s, respectively. Hence, the Reynolds number based on the wall velocity and δ reads Re = Uw δ/νf = 28, 400. Note that the required values for the simulation of the continuous and the disperse phase are made dimensionless using the wall velocity Uw , the half-width δ and the density of the continuous phase ρf (water). 7.1.1.1 Simulation Set-up
In this numerical experiment periodic boundary conditions are used in streamwise and spanwise directions and no-slip conditions hold on the walls. In the simulation the velocity components v(x, y, z) and w(x, y, z) as well as the pressure p(x, y, z) are set to zero. It is assumed that the flow filed is frozen and thus steady. In the present study, N0 = 75, 000 monodisperse primary particles with a dimensionless diameter of d∗p = dp /δ = 1.25 × 10−3 are randomly released into the domain (see Figure 7.4(a)) with a velocity equal to the fluid velocity at the corresponding position (i.e., up = uf |particle ). This corresponds to a volume fraction of αp = 9.58 × 10−5 leading to a mass loading of ηp = 10.05% based on Eq. (2.7). The mechanical properties of polystyrene particles dispersed in the fluid flow are listed in Table A.2. The start of the particle release into the computational domain is denoted t∗ = 0, at which the entire algorithm including the energy-based and the momentum-based agglomeration model is applied for a dimensionless time interval of ∆T ∗ = ∆T Uw /δ = 1400 (corresponding to 20 s). 7.1.1.2 Properties of the Particles
For both agglomeration models the dimensionless Hamaker constant is H ∗ = H/(ρf Uw2 δ 3 ) = 1.41 × 10−19 and the dimensionless separation between two particles during the contact is δ0∗ = δ0 /δ = 1 × 10−8 . The dimensionless maximum contact pressure and Young’s modulus are E ∗ = E/(ρf Uw2 ) = 1.49 × 106 and p∗ = p/(ρf UB2 ) = 7.40 × 104 , respectively. Note that for polystyrene particles with ν = 0.34 the maximum contact pressure is p = 1.492 σyield (see Appendix A.1), where the value of the compressive strength σyield is included in Table A.2. The walls are assumed to be smooth (i.e., the roughness coefficient Rz is set to zero) and made of the same material as the particles. The restitution and friction coefficients for particle-particle (x = p) and particle-wall (x = w) collisions are en,x = 0.90, et,x = 0, µst,x = 1.0 and µkin,x = 0.15 (Balakin et al., 2012). In this test case, the gravity, buoyancy, drag, lift, pressure gradient and the added mass forces are assumed to be relevant for the particle dynamics (see Section 2.3.2), since the particle-to-fluid density ratio ρ∗p = ρp /ρf = 1.05 (i.e., neutrally buoyant particles). The gravitational acceleration points in the wall-normal direction and its dimensionless value is gy∗ = −g δ/Uw2 = −9.73 × 10−2 . The subgrid-scale model for the particles is switched off due to the assumption of a steady flow. The feedback effect of the particles on the continuous flow (i.e., two-way coupling) is not taken 204
7. Results for Particle Agglomeration into account, since the fluid field is assumed to be frozen. For simplicity, the volume-equivalent sphere model is applied for modeling the structure of the agglomerate. 7.1.2 Theoretical Model for Particle Agglomeration
It is well known (see, e.g., Smoluchowski, 1917; Wang et al., 1998; Mumtaz and Hounslow, 2000; Dogon and Golombok, 2015) that for a monodisperse system consisting of Np primary particles in pp a volume Vtot , the specific rate of particle-particle (superscript pp) collisions per unit volume r˙col is given by: pp r˙col =
pp N˙ col 1 = K n2p , Vtot 2
(7.1)
pp where N˙ col is the number of collision events per unit time. K stands for the size-independent collision kernel and np = Np /Vtot is the particle number concentration (the total number of particles per unit volume of the suspension). Since not all inter-particle collisions lead to agglomeration, the agglomeration rate2 β (dimensionless quantity) is defined as the number of particle-particle collisions leading to agglomeration Nagp divided by the total number of particle-particle collisions pp pp Ncol and hence β = Nagp /Ncol . Thus, the specific rate of agglomeration processes per unit volume r˙agp can be expressed as (Mumtaz and Hounslow, 2000):
r˙agp =
N˙ agp 1 = β K n2p , Vtot 2
(7.2)
According to Hounslow et al. (1988) the rate of decrease of the particle number concentration for a size-independent agglomeration kernel β0 = β K is given by: 1 1 dnp (t) = − β0 n2p (t) = − β K n2p (t) . dt 2 2
(7.3)
The integration of Eq. (7.3) yields: np (t) Np (t) 1 ≡ = . np (0) Np (0) 1 + 0.5 β K np (0) t
(7.4)
This relation implies that the number of primary particles reduces with time based on the occurrence of agglomeration processes. For a suspension of neutrally buoyant spheres with a diameter dp subjected to a simple shear flow with a constant shear rate γ, ˙ the following relation for the theoretical agglomeration rate βth was proposed by van de Ven and Mason (1977): 8H βth = f (λ) 36 π µf γ˙ d3p
!0.18
,
(7.5)
where µf stands for the dynamic viscosity of the fluid. The parameter λ is equal to λ/πdp with the assumption that the characteristic wavelength of the dispersion interaction is λ = 10−7 m. 2
It is also denoted collision efficiency in the literature.
205
7.1 Validation of the Agglomeration Models According to van de Ven and Mason (1977) the dimensionless value of f (λ) is equal to 0.95, 0.87 and 0.79 for primary particles with a diameter of 1, 2 and 4 µm, respectively. Balakin et al. (2012) carried out two-dimensional simulations to compare the performance of the agglomeration model proposed by Kosinski and Hoffmann (2010) with the above mentioned theoretical model. To determine the time history of the particle number concentration given by Eq. (7.4), they used the agglomeration rate from their simulation and applied the collision kernel proposed by Saffman and Turner (1956): K=
γ˙ 1/3 1/3 3 V1 + V2 , π
(7.6)
where V1 and V2 are the volumes of the collision partners. However, their predictions do not agree with the theoretical results neither qualitatively nor quantitatively. Possible reasons for these deviations are the assumption of a two-dimensional simulation and the application of an inaccurate agglomeration model leading to a much higher agglomeration rate than given by Eq. (7.5). Furthermore, they employed the size-dependent collision kernel defined by Eq. (7.6) for the model given by Eq. (7.4), which is only valid for a size-independent collision kernel (see, e.g., Kumar, 2013). It is also important to note that Balakin et al. (2012) did not use common material properties, but estimated the values required for the agglomeration model. The reason is that the value of the compressive strength can not be found in the literature prohibiting the application of the energy-based agglomeration model. Thus, owing to the above mentioned drawbacks of the test case by Balakin et al. (2012) in the present study the agglomeration process of polystyrene particles dispersed in a three-dimensional shear flow (water) is validated against the theory. Note that the comparison between both agglomeration models carried out in Section 7.3 for a particle-laden turbulent channel flow is based on fused quartz particles in air. In addition to the full availability of the material properties of polystyrene particles, when dispersed in water flow, they satisfy the main assumption of neutrally buoyant spheres made by van de Ven and Mason (1977). 7.1.3 Results and Discussion 7.1.3.1 Agglomeration Rate
The first step in the present validation study is to evaluate the agglomeration rate predicted by both models against the empirical correlation of van de Ven and Mason (1977) given by Eq. (7.5). Figure 7.2(b) shows the time history of the agglomeration rate predicted by the energy-based (EAM) and the momentum-based (MAM) agglomeration model in comparison with the theoretical value within a dimensionless time interval of ∆T ∗ = 1400. Using a quadratic fitting of the values of f (λ) provided by van de Ven and Mason (1977), the corresponding value of f (λ) for the diameter used in this study is about 0.59 and hence based on Eq. (7.5) the theoretical agglomeration rate reads βth = 3.4%. It is clear that the momentum-based agglomeration model yields slightly more accurate predictions than EAM in comparison with the theoretical model. To analyze this behavior, the time history of the accumulated number of collisions and agglomeration processes predicted by both models is depicted in Figure 7.2(a). It is visible that both models predict almost the same number of 206
7. Results for Particle Agglomeration agglomeration processes, while the energy-based model results in a higher number of particleparticle collisions than MAM. The main reason for this observation can be attributed to the assumptions made in EAM for the treatment of the kinetics of the collision partners without agglomeration. As explained in Section 3.1.1.4, it is assumed that the collision partners separate only in either the normal or the tangential direction and the van-der-Waals force is neglected when calculating the angular velocities after the collision.
6.0
×10+5
EAM MAM
4.0 3.0
Collision
2.0
×10−2
7.5 pp Nagp /Ncol
pp Nagp × 20, Ncol
5.0
9.0
Agglomeration
1.0 0.0
6.0
βth βth (dp = dp ) EAM MAM
4.5 3.0 1.5 0.0
0
200
400
600
800
1000 1200 1400
t Uw /δ (a)
0
200
400
600
800
1000 1200 1400
t Uw /δ (b)
pp Figure 7.2: Time history of (a) the accumulated number of particle-particle collisions Ncol and agglomerapp tion processes Nagp and (b) the agglomeration rate Nagp /Ncol predicted by the energy-based (EAM) and the momentum-based (MAM) agglomeration model against the theoretical value given by Eq. (7.5) based on the diameter of either the primary particle dp or the mean diameter of the particles dp .
It is important to note that the assumption of a constant agglomeration rate as used in the theoretical model (Figure 7.2(b)) is not realistic due to the fact that the agglomeration process leads to an enlargement of the mean particle size and hence to a weaker cohesive impulse reducing the probability of agglomeration. This behavior is reasonably reproduced by the present agglomeration models. Furthermore, when setting the diameter dp in the theoretical correlation (7.5) equal to the mean diameter of the particles dp , an excellent agreement with the momentum-based model is observed. It is worth mentioning that the agglomeration rate predicted by the improved momentum-based agglomeration model by Breuer and Almohammed (2015) is in much closer agreement with the theory than the original model of Kosinski and Hoffmann (2010) applied by Balakin et al. (2012). 7.1.3.2 Particle Number Concentration
In a second step, the decrease of the particle number concentration predicted by EAM and MAM as a function of time is validated against the theory. Since the particle agglomeration process leads to larger particles, a size-dependent collision kernel has to be used. Wang et al. (1998) studied the effect of the post-collision treatment and collision kernel on the particle number concentration. They distinguished three counting schemes, namely the agglomerating particles are (i) not removed from the domain after collision, (ii) marked and excluded from the collision detection and (iii) 207
7.1 Validation of the Agglomeration Models removed immediately from the computational domain. Wang et al. (1998) concluded that if the collision kernel by Saffman and Turner (1956) is applied to real agglomeration processes (i.e, the particle number concentration changes in time), corrections have to be made, since this model assumes a uniform, time-independent concentration field. In the present study, if the collision partners collide and agglomerate, they form a particle of larger size. That means that the agglomerating primary particles are removed immediately from the computational domain and are replaced by an agglomerate, which corresponds to the third post-treatment model by Wang et al. (1998). This implies that if an agglomeration process occurs due to a successful particle-particle collision, the number of active particles decreases by one. Hence, the total number of particles participating in the collision detection decreases with time. In analogy to Eq. (7.1) the collision kernel at the time t(n) is defined by Wang et al. (1998) as: K
(n)
=
pp(n) → (n+1)
2 Vtot Ncol h
i (n) 2
∆t
Np
(7.7)
, pp(n) → (n+1)
where ∆t is the simulation time step, Ncol stands for the collision count in the time step (n) (n+1) (n) t 1 and the fluid turbulence in a fully developed downwards directed channel flow. Kulick et al. (1994) also stated that the degree of attenuation of the fluid turbulence increases with the particle Stokes number, the particle mass loading and the distance from the wall. The reduction of the wall-normal and spanwise velocity fluctuations of the fluid due to the two-way coupling was also found in numerical investigations by Kuerten et al. (2011) for a plane channel flow without taking particle-particle collisions into account. Furthermore, Vreman et al. (2009) used a volume fraction of about 1.3% and performed two- and four-way coupled simulations of a vertical turbulent particle-laden channel flow. They also showed that due to particle-fluid interactions the wall-normal and spanwise turbulence intensities of the fluid are reduced, whereas the streamwise turbulent intensity is amplified by the presence of large particles. In addition, Vreman et al. (2009) compared the four-way coupled simulations with two-way coupled simulations and found that the inclusion of the inter-particle collisions has a large influence on the main statistics of both phases. 228
7. Results for Particle Agglomeration
3.5
×10−2
TWC = OFF TWC = ON
×10−2
3.0
u′p u′p /UB2
2.5 2.0
TWC = OFF TWC = ON
2.5 2.0
E
E
u′f u′f /UB2
3.0
3.5
1.5
D
D
1.5 1.0
1.0
0.5
0.5
0.0
0.0 -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2
y/δ
3.5
2.5
2.5
vp′ vp′ /UB2
3.0
2.0
0.2 0.4 0.6 0.8
1
0.2 0.4 0.6 0.8
1
−3
×10
2.0
E
E
vf′ vf′ /UB2
×10
1.5
D
D
1.5 1.0
1.0
0.5
0.5
0.0
0.0 -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2
y/δ
0
y/δ
(c)
(d)
−3
×10
5.0
−3
×10
4.0
wp′ wp′ /UB2
4.0 3.0
3.0
E
E
wf′ wf′ /UB2
1
(b)
−3
3.0
5.0
0.2 0.4 0.6 0.8
y/δ
(a) 3.5
0
2.0
D
D
2.0 1.0
1.0
0.0
0.0 -1 -0.8 -0.6 -0.4 -0.2
0
y/δ (e)
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2
0
y/δ (f )
Figure 7.16: Effect of the two-way coupling (TWC) on the dimensionless averaged fluctuations of the fluid (left) and the particles (right) along the dimensionless channel width y/δ in (a) & (b) the streamwise, (c) & (d) the wall-normal and (e) & (f) the spanwise directions. Dimensionless averaging time for the continuous and the disperse phase ∆T ∗ = 200 (closely-packed sphere model).
229
7.2 Particle Agglomeration in Turbulent Channel Flow As visible in Figure 7.16, due to the reduction of the fluctuations of the continuous phase by the two-way coupling also the fluctuations of the particles are reduced. However, as underlined in Figures 7.16(a) and (b), it is obvious that the particle fluctuations in streamwise direction are almost the same as the carrier phase, except in the region near the walls, where the particle fluctuations are stronger than those of the continuous phase. It is worth noting that similar observations were made for downward particle-laden flows at different mass loadings by Breuer and Alletto (2012) and Yamamoto et al. (2001). The reason for the differences between the streamwise fluctuations of the particles and the fluid in the near-wall region is the effect of particle-wall collisions. Yamamoto et al. (2001) stated that due to the presence of the walls additional streamwise particle fluctuations are produced by the mixing of the particles reflected on the walls in the wall-normal direction. In contrast to the observations concerning the streamwise direction, Figures 7.16(d) and (f) indicate that the particle fluctuations in wall-normal and spanwise directions are weaker than those of the continuous phase. This result is attributed to the relatively high inertia of the particles used in the present simulations. Furthermore, due to the two-way coupling the wall-normal and spanwise velocity fluctuations are reduced by the presence of the particles (see Figures 7.16(d) and (f)). It is worth noting that for TWC = ON/OFF the differences between the particle fluctuations of both cases are smaller than for the fluid, since it is an indirect effect resulting from the alteration of the continuous phase. Figure 7.17 compares the results with and without the effect of the two-way coupling on the agglomeration process. Figure 7.17(a) shows that the accumulated number of agglomerated primary particles Napp increases if the two-way coupling is switched on, especially after a dimensionless time of ∆T ∗ ≈ 80. This leads to a slightly higher number of arising agglomerates Nag of all classes for the case with two-way coupling compared to the case without two-way coupling as displayed in Figure 7.17(b). Thus, the overall number of active particles in the computational domain (i.e., the sum of the non-agglomerated primary particles and the agglomerates) is slightly smaller in the case of TWC (not depicted here for brevity). On the other hand, the averaged number of primary particles included in the agglomerate Napp /Nag is almost the same until ∆T ∗ ≈ 120, while it increases during the remaining time of the simulation due to the higher number of Napp when TWC is switched on (not depicted here for brevity). As explained in the preceding sections, the increase of the number of agglomerates or the size of the agglomerates in terms of the number of included primary particles should in principle enhance the inter-particle collisions. Nevertheless, the accumulated number of particle-particle collisions decreases when the two-way coupling is taken into account as shown in Figure 7.17(c). The reason for this behavior is the reduction of the particle velocity fluctuations in the wall-normal and spanwise directions along the entire channel width when the two-way coupling is taken into account as demonstrated in Figures 7.16(d) and (f). Although the accumulated number of particle-particle collisions is reduced by the two-way coupling as depicted in Figure 7.17(c), a marginally higher number of agglomeration processes is predicted when the two-way coupling is taken into account. To understand this behavior, the distribution of the averaged concentration of inter-particle collisions and those leading to agglomeration are shown in Figure 7.18. Additionally, the agglomeration rate is included. 230
7. Results for Particle Agglomeration
2.5
×10+5
10
1.5
Nag
Napp
2.0
10+5
TWC = OFF TWC = ON
10+3
1.0
10+2
0.5
10+1 10+0
0.0 0
50
100
150
TWC = OFF TWC = ON
+4
200
2
3
4 5 6 7 8 Agglomerate Type
t UB /δ (a)
×10+7
10
(b) 2.0
TWC = OFF TWC = ON
1.2
×10−2
TWC = OFF TWC = ON
1.5 pp Nagp /Ncol
pp Nagp × 20, Ncol
1.6
9
0.8
Collision
0.4
1.0
0.5
Agglomeration
0.0
0.0 0
50
100
t UB /δ (c)
150
200
0
50
100
150
200
t UB /δ (d)
Figure 7.17: Effect of the two-way coupling (TWC) on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : (a) time history of the accumulated number of the agglomerated primary particles Napp , (b) number of agglomerates of the same type (2 = two-particle agglomerate, 3 = three-particle agglomerate, etc.) at a dimensionless time of 200, (c) time history of the accumulated number of particle-particle collisions pp pp Ncol and agglomeration processes Nagp and (d) time history of the agglomeration rate Nagp /Ncol (closely-packed sphere model).
Regarding the reduction of the wall-normal and spanwise particle fluctuations along the channel width and the resulting influence on the inter-particle collisions, two regions have to be distinguished: 1. Region far away from the wall (−0.8 ≤ y/δ ≤ 0.8): Due to the reduction of the particle velocity fluctuations visible in Figures 7.16a(c) and (e) the concentration of particle-particle collisions is decreased when the two-way coupling is taken into account, whereas the concentration of the agglomeration processes is hardly affected. As a result, the local agglomeration rate in this region slightly increases as shown in Figure 7.18(c). 2. Near-wall region (−1.0 ≤ y/δ ≤ −0.9 and 0.9 ≤ y/δ ≤ 1.0): The slight reduction of the particle fluctuations due to the two-way coupling marginally influences the number of 231
7.2 Particle Agglomeration in Turbulent Channel Flow particle-particle collisions, while it enhances the agglomeration processes (see Figure 7.18(d)). The reason for this observation is that the reduction of the particle velocity fluctuations pp weakens the repulsive impulse fˆn,a separating the collision partners and enhances the cohesive pp∗ ˆ impulse fn,c between the collision partners due to the lower relative velocities. Thus, the local agglomeration rate also increases in this region (see Figure 7.18(c)) implying a slightly larger number of agglomerates than for TWC = OFF.
9.00 6.00
×10−4
10+8
MAM = OFF TWC = OFF TWC = ON
10+6
hαp i
3.00
TWC = OFF TWC = ON
10+7
pp hNcol i
10+5 10+4 10+3 0.18 0.12
10 1
10
y+
100
hNagp i
+2
1000
-1 -0.8 -0.6 -0.4 -0.2
(a)
pp hNagp i / hNcol i
0
0.2 0.4 0.6 0.8
1
0.96
1
y/δ (b)
0.40 0.20 0.10 0.05
10+8
0.01
10+5
TWC = OFF TWC = ON
10+7 pp hNcol i
10+6
10+4
hNagp i
10+3 -1 -0.8 -0.6 -0.4 -0.2 0
y/δ (c)
0.2 0.4 0.6 0.8
1
10+2 0.9
0.92
0.94
0.98
y/δ (d)
Figure 7.18: Effect of the two-way coupling (TWC) on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : (a) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dots represent the distribution of the mean volume fraction without taking the agglomeration model into account (MAM = OFF) and the dashed line denotes the global volume fraction of αp = 0.18 × 10−4 ), (b) pp averaged concentration of the number of collision events hNcol i and the agglomeration processes hNagp i along pp pp y/δ, (c) averaged agglomeration rate hNagp i / hNcol i along y/δ and (d) averaged concentration hNcol i and hNagp i close to the walls. Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
It is commonly known that in wall-bounded turbulent flows turbophoresis refers to a preferential accumulation (or migration) of inertial particles toward the wall due to the gradient of the turbulent 232
7. Results for Particle Agglomeration velocity fluctuations in the wall-normal direction (see, e.g., Nowbahar et al., 2013). Based on the numerical study by Kuerten et al. (2011), who performed one-way and two-way coupled simulations without inter-particle collisions, it was concluded that compared to the one-way coupled case the two-way coupling reduces the turbophoresis effect, since the wall-normal velocity fluctuations of the continuous phase are considerably decreased leading to a lower particle concentration close to the walls. Vreman et al. (2009) observed a relatively small turbophoresis effect in the four-way coupled case due to the large particles and the high volume fraction used in their work. In addition, they did not observe an appreciable turbophoresis effect in their two-way coupled simulation. However, both studies can not be directly compared with the present study, since Kuerten et al. (2011) neglected the inter-particle collisions in the two-way coupled case and Vreman et al. (2009) used a much larger volume fraction than in the present simulations, where the process is strongly dominated by inter-particle collisions. As evidenced in Figure 7.16(c), the wall-normal turbulent fluctuations of the fluid are reduced due to the inclusion of the two-way coupling in the present study implying a weaker effect of turbophoresis. Hence, a lower particle concentration (or volume fraction) is expected in the near-wall region based on the observation by Kuerten et al. (2011). However, Figure 7.18(a) shows that in the present study the particle volume fraction slightly increases near the walls if the two-way coupling is taken into account. This observation can be attributed to the inclusion of the agglomeration applying the closely-packed sphere model leading to particle enlargement in the near-wall region, where the highest number of particle-particle collisions and hence agglomeration processes occurs and most of the agglomerates are located. Note that Kuerten et al. (2011) neglected the inter-particle collisions, whereas they are a prerequisite for the agglomerations in the present analysis. Based on the above observations, the agglomeration rate at the end of the simulation increases from about 0.73% (without two-way coupling) to about 0.79% when the two-way coupling is switched on as depicted in Figure 7.17(d). The corresponding dimensionless frequency of the agglomeration process f˜agp increases from about 5.39 × 102 to about 5.62 × 102 when the two-way coupling is taken into account as listed in Table A.6. In summary, it can be concluded that the effect of the two-way coupling is not negligible and thus should be taken into account 7.2.2.5 Effect of the Subgrid-Scale Model for the Particles
As explained in Section 2.5.4, the unresolved scales within a LES may have an appreciable influence on the particle motion and thus this effect should be investigated. In the previous analysis this effect was not taken into account to reduce the number of influencing parameters. This section aims at extending these investigations towards the effect of the subgrid-scale models for the particles on the particle motion and the subsequent agglomeration process. For this purpose, two stochastic subgrid-scale models for the particles are applied, namely (i) the trivial and (ii) the extended Langevin-type model. The simulation set-up is equal to the standard case applying the closely-packed sphere model and smooth channel walls. Furthermore, the two-way coupling is again not taken into account. The inclusion of the subgrid-scale model for the particles means that the subgrid-scale velocity fluctuations u0f are added to the filtered fluid velocity at the particle position uf . Kuerten (2006) stated that the incorporation of the subgrid-scale contribution results 233
7.2 Particle Agglomeration in Turbulent Channel Flow in stronger turbophoresis enhancing the accumulation of particles near the channel walls. However, in his study the effect of the gravity force and even more important the inter-particle collisions were not considered. 7.2.2.5.1 Trivial Model
As a first step, the trivial subgrid-scale model presented in Section 2.5.4.1 is used. Figure 7.19 depicts the influence of the subgrid-scale model on the agglomeration process. As visible in Figure 7.19(a), the accumulated number of agglomerated primary particles Napp is significantly reduced if the trivial subgrid-scale model is considered. Figure 7.19(b) depicts a statistic on the number of existing agglomerates at the end of the simulation (i.e., ∆T ∗ = 200) depending on the number of primary particles included in the arising agglomerate. 2.5
×10+5
10
1.5
Nag
Napp
2.0
10+5
SGS = OFF SGS = ON
10+3
1.0
10+2
0.5
10+1 10+0
0.0 0
50
100
150
SGS = OFF SGS = ON
+4
200
2
3
4 5 6 7 8 Agglomerate Type
t UB /δ (a)
×10+7
2.0
SGS = OFF SGS = ON
1.2 0.8
Collision
0.4
10
(b)
pp Nagp /Ncol
pp Nagp × 20, Ncol
1.6
9
×10−2
SGS = OFF SGS = ON
1.5 1.0
0.5
Agglomeration
0.0
0.0 0
50
100
t UB /δ (c)
150
200
0
50
100
150
200
t UB /δ (d)
Figure 7.19: Effect of the trivial subgrid-scale model (SGS) for the particles on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : (a) time history of the total number of agglomerated primary particles Napp , (b) number of agglomerates of the same type (2 = two-particle agglomerate, 3 = threeparticle agglomerate, etc.) at a dimensionless time of 200, (c) time history of the accumulated number of pp particle-particle collisions Ncol and agglomeration processes Nagp and (d) time history of the agglomeration pp rate Nagp /Ncol (closely-packed sphere model).
234
7. Results for Particle Agglomeration Note that the diameters of the agglomerates of the same type (e.g., two-particle agglomerates) are the same, since the same structure model (CSM) is applied, but the predicted number of agglomerates of the same size is different. It is clearly visible that the number of all classes of agglomerates are reduced in case with the subgrid-scale model for the particles. As a result, less and smaller agglomerates are predicted in case of the application of the subgrid-scale model. Consequently, at the end of the simulation the average number of primary particles included in an agglomerate Napp /Nag decreases, i.e., Napp /Nag ≈ 2.04 for SGS switched off and Napp /Nag ≈ 2.01 for SGS switched on (time history not shown here for brevity). As evidenced in Figure 7.19(c), the accumulated number of particle-particle collisions decreases when the trivial subgrid-scale model is taken into account. The reduction of the inter-particle collisions due to the subgrid-scale model for particles may be indeed attributed to the following reasons: 1. The lower number and the smaller sizes of the formed agglomerates (see Figure 7.19(b)) indicate that the probability of collisions between the primary particles and the agglomerates as well as the agglomerates themselves is reduced. 2. The consideration of the subgrid-scale model leads to higher particle velocity fluctuations (not shown here for brevity). As a result, in principle a higher number of inter-particle collisions is expected. However, the subgrid-scale contribution also influences the particlewall collisions, which have an effect on the distribution of the particles in the channel as will be explained below. To discuss the combined effect of the above mentioned two reasons on the number of particleparticle collisions, the mean particle volume fraction along the dimensionless wall coordinate y + = y uτ /νf is depicted in Figure 7.20(a). It shows that in the direct vicinity of the wall the particle volume fraction is reduced when the subgrid-scale model is taken into account. However, for a wall distance y + ≥ 2 the volume fraction slightly increases by the subgrid-scale model due to the increasing turbophoretic drift pumping the particles from the bulk towards the channel walls as stated in Kuerten (2006). The reason for the unexpected behavior observed in the direct vicinity of the wall is the higher particle fluctuations in the wall-normal direction due to the inclusion of the subgrid-scale model also leading to a higher number of particle-wall collisions. Thus, the particles colliding with the channel walls are more often reflected away from the walls than without the subgrid-scale model. This observation is obvious in the present simulation, since the number of particle-wall collisions increases by a factor of about 1.6 when taking the subgrid-scale model into account. A further reason for the reduction of the particle volume fraction very close to the wall is the lower number and smaller sizes of the arising agglomerates if the subgrid-scale model is taken into account (see Figure 7.19(b)). As visible in Figure 7.19(c), the accumulated number of agglomeration processes also becomes smaller when including the subgrid-scale model. This behavior is attributed to the following two reasons: • The lower number of inter-particle collisions as explained before. 235
7.2 Particle Agglomeration in Turbulent Channel Flow • The subgrid-scale velocity fluctuations leading to larger relative velocities between the pp particles and hence to a stronger repulsive impulse fˆn,a separating the collision partners and pp∗ ˆ a weaker cohesive impulse fn,c attracting the collision partners. As a result, the global agglomeration rate at the end of the simulation is reduced from about 0.73% without the subgrid-scale model to about 0.20% with the model as depicted in Figure 7.19(d). The corresponding dimensionless frequency of the agglomeration processes f˜agp decreases from about 5.39 × 102 to about 1.33 × 102 when the subgrid-scale model is taken into account as listed in Table A.6.
9.00 6.00
×10−4
10+8
MAM = OFF SGS = OFF SGS = ON
3.00
SGS = OFF SGS = ON
10+7 10+6
pp hNcol i
hαp i
10+5 10+4 10+3
hNagp i
10+2
0.18 0.12
10+1 1
10
y
+
100
1000
-1 -0.8 -0.6 -0.4 -0.2
0.2 0.4 0.6 0.8
1
0.96
1
y/δ
(a)
pp hNagp i / hNcol i
0
(b)
0.40 0.20 0.10 0.05
10+8
0.01
10+5
SGS = OFF SGS = ON
10+7 10+6
10+4 10+3
pp hNcol i
hNagp i
10+2 -1 -0.8 -0.6 -0.4 -0.2 0
y/δ (c)
0.2 0.4 0.6 0.8
1
10+1 0.9
0.92
0.94
0.98
y/δ (d)
Figure 7.20: Effect of the trivial subgrid-scale model (SGS) for the particles on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : (a) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dots represent the distribution of the mean volume fraction without taking the agglomeration model into account (MAM = OFF) and the dashed line denotes the global volume fraction pp of αp = 0.18 × 10−4 ), (b) averaged concentration of the number of collision events hNcol i and agglomeration pp processes hNagp i along y/δ, (c) averaged agglomeration rate hNagp i / hNcol i along y/δ and (d) averaged concenpp tration hNcol i and hNagp i close to the walls. Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
236
7. Results for Particle Agglomeration Figure 7.20(b) shoes the local distribution of the averaged concentration of the number of collision pp events hNcol i and agglomeration processes hNagp i along the dimensionless channel width y/δ. As explained before, the inclusion of the subgrid-scale model enhances the inter-particle collisions, except in the direct vicinity of the wall due to the increased number of particle-wall collisions. As evidenced in Figure 7.20(d), the number of particle-particle collisions in the near-wall region, where the highest number of inter-particle collisions takes place, is marginally reduced, whereas the number of agglomeration processes significantly reduces due to the higher particle velocity fluctuations leading to stronger repulsive and weaker cohesive impulses when the subgrid-scale model is switched on. As a result, the distribution of the local agglomeration rate displayed in Figure 7.20(c) is noticeably reduced along the channel width, especially near the walls. Based on the above discussion, it can be concluded that the subgrid-scale model for the particles should be taken into account, since it affects the agglomeration rate significantly. 7.2.2.5.2 Extended Langevin Model
To study the effect of different subgrid-scale models for the particles on the agglomeration process, the extended Langevin model (see Section 2.5.4.2) is used in the present simulation. This analysis is motivated by the fact that the Langevin subgrid-scale model predicts more realistic results and better correlated particle fluctuations than the trivial model (Breuer and Hoppe, 2017). As mentioned before, the key difference between the subgrid-scale models is the velocity fluctuations of the particle. As evidenced in Figure D7.21,Ethe Langevin modelD predicts noticeably higher E 0 0 2 0 0 particle fluctuations in the wall-normal vp vp /UB and spanwise wp wp /UB2 directions than predicted by the trivial model. Note that this difference is more pronounced in the regions −0.95 ≤ y/ δ ≤ −0.45 and 0.45 ≤ y/ δ ≤ 0.95 than outside theses regions. ×10−2
SGS = OFF SGS = Trivial SGS = Langevin
2.5
×10−3
3.0 2.5 2.0
E
2.0
E
vp′ vp′ /UB2
3.5
wp′ wp′ /UB2
3.0
1.5
1.5 1.0
D
D
1.0 0.5
0.5
0.0
0.0 -1 -0.8 -0.6 -0.4 -0.2
0
y/δ (a)
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
y/δ (b)
Figure 7.21: Effect of the subgrid-scale model on the dimensionless averaged fluctuations of the particles in the
(a) wall-normal vp0 vp0 /UB2 and (b) spanwise wp0 wp0 /UB2 directions. Dimensionless averaging time ∆T ∗ = 200.
Figure 7.22 shows that the application of the trivial and the Langevin subgrid-scale models yields similar trends of the physical behavior of the agglomeration processes, but with strongly different rates. 237
7.2 Particle Agglomeration in Turbulent Channel Flow 1.8
×10+7
SGS = OFF SGS = Trivial SGS = Langevin
1.2 0.9
Collision
0.6
SGS = OFF SGS = Trivial SGS = Langevin
0.9 0.6 0.3
Agglomeration
0.3
×10−2
1.2 pp Nagp /Ncol
pp Nagp × 20, Ncol
1.5
1.5
0.0
0.0 0
50
100
150
200
0
50
t UB /δ ×10−4
200
(b) 10+5
MAM = OFF SGS = OFF SGS = Trivial SGS = Langevin
10
Nag
hαp i
3.00
150
t UB /δ
(a) 9.00 6.00
100
SGS = OFF SGS = Trivial SGS = Langevin
+4
10+3 10+2 10+1
0.18 0.12
10+0 1
10
y+ (c)
100
1000
2
3
4 5 6 7 8 Agglomerate Type
9
10
(d)
Figure 7.22: Effect of the subgrid-scale model for the particles on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : (a) time history of the accumulated number of particle-particle collisions pp pp Ncol and agglomeration processes Nagp , (b) time history of the agglomeration rate Nagp /Ncol , (c) mean particle + volume fraction hαp i along the dimensionless wall coordinate y (the dots represent the distribution of the mean volume fraction without taking the agglomeration model into account (MAM = OFF) and the dashed line denotes the global volume fraction of αp = 0.18 × 10−4 ) and (d) number of agglomerates of the same type (2 = two-particle agglomerate, 3 = three-particle agglomerate, etc.) at a dimensionless time of 200. Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
As clearly obvious in Figure 7.22(a), compared to the case without a subgrid-scale model (SGS = pp OFF) a significantly higher number of particle-particle collisions Ncol is predicted when applying the Langevin model, whereas less collision events are computed by the trivial model. This outcome is attributed to the appreciably higher particle fluctuations predicted by the Langevin model in comparison with the trivial model (see Figure 7.21). However, the number of agglomeration processes Nagp noticeably decreases for both subgrid-scale models, where the effect is even stronger for the Langevin model. The reason for this behavior is that the increase of the particle fluctuations leads to higher relative velocities between the collision partners and hence to a stronger repulsive pp pp∗ impulse fˆn,a and a weaker cohesive impulse fˆn,c . As a result, a lower number of agglomerates is predicted by the extended Langevin model than for the trivial model as shown in Figure 7.22(d). 238
7. Results for Particle Agglomeration Of course, the highest number of agglomerates (i.e., black histograms) corresponds to the case without a subgrid-scale model for the particles. Another key difference between the predictions of both subgrid-scale models can be easily noticed in Figure 7.22(c). It shows that the Langevin subgrid-scale model leads to a higher mean volume fraction near the walls implying stronger turbophoresis enhancing the accumulation of particles in the near-wall region as concluded by Kuerten (2006), which is not the case using the trivial model. As displayed in Figure 7.22(b), if the Langevin subgrid-scale model is applied, the global agglomerapp tion rate Nagp /Ncol at the end of the simulation decreases from about 0.73% (without subgrid-scale model) and about 0.20% for the trivial model to about 0.04%. The corresponding dimensionless frequency of the agglomeration process f˜agp drastically decreases from about 5.39 × 102 (without subgrid-scale model) to about 0.34 × 102 when the Langevin subgrid-scale model is taken into account as listed in Table A.6. Based on the above discussion, it can be concluded that the Langevin subgrid-scale model for the particles predicts significantly different results than the trivial model, which are assumed to be more reasonable and realistic. Hence, it should be used in the next simulations. 7.2.2.6 Effect of the Lift Force
It is well known that particles moving at a constant velocity relative to the fluid in a shear flow experience a transverse lift force, the so-called Saffman force. This force originates from the pressure difference across the particle resulting from a non-uniform velocity distribution over the particle. The lift force acts perpendicular to the relative velocity. It is directed towards or against a positive velocity gradient depending on uslip = (uf − up ) being greater or smaller than zero, respectively. On the other hand, a rotating particle in a flow field may also experience a lift force due to rotation, the so-called Magnus force. In order to investigate the influence of the lift forces due to velocity shear (i.e., Saffman force) and particle rotation (i.e., Magnus force), these forces are taken into account in the subsequent analysis. The same computational set-up as presented in Section 7.2.1 is used applying the closely-packed sphere model and smooth channel walls. Furthermore, the two-way coupling and the subgrid-scale model are switched off. Thus, the present case including both lift forces (see Section 2.3.2.4) can be directly compared with the data presented in Section 7.2.2.1 without lift forces. It is well known that heavy particles in a channel with downward flow direction lead to a Saffman lift force towards the walls. Furthermore, in wall-bounded turbulent flows, particle-wall collisions may result in high particle rotation rates leading to a transverse Magnus lift force due to the modification of the flow field around the particle (see, e.g., Sommerfeld et al., 2008). In the present shear flow, the high velocity gradients in the near-wall region and the fast rotation due to the particle-wall collisions lead to a lift force resulting in a cross-stream migration of the particles towards the walls. Figure 7.23(a) shows the effect of the consideration of the lift forces on the mean particle volume fraction as a function of the dimensionless wall coordinate y + . It is clearly visible that the consideration of the lift forces noticeably reduces the particle volume fraction in the direct vicinity of the wall, while it is slightly enhanced in the remaining parts of the channel. The reason for this unexpected behavior can be attributed to the higher number of particle-wall collisions, which 239
7.2 Particle Agglomeration in Turbulent Channel Flow increases by a factor of about 1.8 when the lift forces are taken into account. In other words, the lift forces push the particles towards the walls. Subsequently, the particles collide with the wall and are more strongly reflected than in the case without the lift forces. Thus, the overall effect of the lift forces is an enhancement of the migration of the particles from the near-wall region.
9.00 6.00
×10−4
10+8
MAM = OFF LFs = OFF LFs = ON
10
10+6
hαp i
3.00
LFs = OFF LFs = ON
+7
pp hNcol i
10+5 10+4 10+3 0.18 0.12
hNagp i
10+2 1
10
y
+
100
1000
-1 -0.8 -0.6 -0.4 -0.2
0.2 0.4 0.6 0.8
1
0.96
1
y/δ
(a)
pp hNagp i / hNcol i
0
(b) 10+8
0.40 0.20 0.10 0.05
LFs = OFF LFs = ON
10+7 10+6
0.01
pp hNcol i
10+5 10+4 10+3 -1 -0.8 -0.6 -0.4 -0.2 0
y/δ (c)
0.2 0.4 0.6 0.8
1
hNagp i
10+2 0.9
0.92
0.94
0.98
y/δ (d)
Figure 7.23: Effect of the lift forces (LFs) on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : (a) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dots represent the distribution of the mean volume fraction without taking the agglomeration model into account (MAM = OFF) and the dashed line denotes the global volume fraction of αp = 0.18 × 10−4 ), (b) averaged pp concentration of the number of collision events hNcol i and agglomeration processes hNagp i along y/δ, (c) averaged pp pp agglomeration rate hNagp i / hNcol i along y/δ and (d) averaged concentration hNcol i and hNagp i close to the walls. ∗ Dimensionless averaging time ∆T = 200 (closely-packed sphere model).
Note that the increase of the number of particle-wall collisions due to the lift forces is higher than for the case when the subgrid-scale model is switched on (see Section 7.2.2.5), and hence it is obvious that the particle volume fraction in the direct vicinity of the wall is smaller for the present case (see Figure 7.23(a)). As a result of the lower particle concentration in the immediate near-wall region, where the highest number of inter-particle collisions takes place, the number of 240
7. Results for Particle Agglomeration particle-particle collisions in this region is reduced as visible in Figure 7.23(d). Furthermore, the averaged concentration of agglomeration processes is significantly reduced in the region close to the wall. The following reasons are responsible for this reduction when the lift forces are taken into account: 1. The overall reduction of the number of inter-particle collisions as shown in Figure 7.24(a). 2. Due to the increased number of particle-wall collisions when the lift forces are taken into account, on average the reflected primary particles or agglomerates possess a stronger impulse in the collision-normal direction reducing the probability of a successful agglomeration. On the other hand, the migrated particles slightly enhance the local number of inter-particle collisions outside the region near the wall as displayed in Figure 7.23(a). In accordance with the pp information mentioned above, the local agglomeration rate hNagp i / hNcol i depicted in Figure 7.23(c) is more or less the same in the central region, while it is reduced in the near-wall region. As evidenced in Figure 7.24(a), the accumulated number of inter-particle collisions and hence also the agglomeration processes is reduced when the lift forces are taken into account. The resulting agglomeration rate shown in Figure 7.24(b) is reduced from about 0.73% to about 0.58% at the end of the simulation (i.e., ∆T ∗ = 200) when the lift forces are considered. The corresponding dimensionless frequency of the agglomeration process f˜agp decreases from about 5.39 × 102 to about 3.39 × 102 when the lift forces are taken into account as listed in Table A.6. ×10+7
2.0
LFs = OFF LFs = ON
1.2
×10−2
LFs = OFF LFs = ON
1.5 pp Nagp /Ncol
pp Nagp × 20, Ncol
1.6
0.8
Collision
0.4
1.0
0.5
Agglomeration
0.0
0.0 0
50
100
t UB /δ (a)
150
200
0
50
100
150
200
t UB /δ (b)
Figure 7.24: Effect of the lift forces (LFs) on the agglomeration process predicted by MAM for large par pp and ticles d∗p = 6 × 10−4 : Time history of (a) the accumulated number of particle-particle collisions Ncol pp agglomeration processes Nagp and (b) the agglomeration rate Nagp /Ncol (closely-packed sphere model).
Furthermore, the average number of primary particles included in an agglomerate is lower when the lift forces are considered implying a lower number and smaller sizes of the arising agglomerates. Note that the lower number and the smaller sizes of the agglomerates reduce the probability of collisions between the primary particles and the agglomerates as well as the agglomerates 241
7.2 Particle Agglomeration in Turbulent Channel Flow themselves. This influence has to be added to the issues mentioned above as a further reason for the lower number of inter-particle collisions when considering the lift forces. Finally, it should be clarified which lift force has the most dominant effect on the collision and agglomeration processes for the present simulations. Therefore, the lift forces either due to velocity shear (Saffman) or due to particle rotation (Magnus) are separately taken into account in the next simulations. Figure 7.25 shows that both forces noticeably reduce the global agglomeration rate, in particular the Magnus force indicating the influence of the high rotation rate of the particles colliding with the walls. ×10+7
2.0
LFs = OFF SLF = ON pp Nagp × 20, Ncol
pp Nagp × 20, Ncol
1.6 1.2 0.8
Collision
0.4
Agglomeration
0.0
×10−2
LFs = OFF SLF = ON
1.5 1.0
0.5 0.0
0
50
100
150
200
0
50
t UB /δ ×10+7
200
150
200
(b) 2.0
LFs = OFF
×10−2
MLF = ON
LFs = OFF
MLF = ON
1.2
1.5 pp Nagp /Ncol
pp Nagp × 20, Ncol
150
t UB /δ
(a) 1.6
100
0.8
Collision
0.4
1.0
0.5
Agglomeration
0.0
0.0 0
50
100
t UB /δ (c)
150
200
0
50
100
t UB /δ (d)
Figure 7.25: Effect of the Saffman lift force (SLF) and the Magnus lift force (MLF) on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : Time history of (a) & (c) the accumulated pp number of particle-particle collisions Ncol and agglomeration processes Nagp and (b) & (d) the agglomeration rate Napp /Nag (closely-packed sphere model).
As mentioned before, in the direct vicinity of the walls the lift forces push the particles towards the walls, and hence they are more strongly reflected from the walls. As a result, an overall enhancement of the migration of the particles towards the channel center is expected. A direct 242
7. Results for Particle Agglomeration comparison between Figure 7.24 and Figures 7.25(a) and (b) indicates that the Saffman force is more dominant than the Magnus force, since only marginal differences between the time histories of the predicted variables can be noticed between the combined effect of both forces and the Saffman force alone. Nevertheless, when looking at Figures 7.25(b) and (d), it is astonishing that the agglomeration rate predicted for the case when only the Magnus force is taken into account is obviously lower than that for the pure Saffman force. This difference can be directly related to the higher number of particle-particle collisions for the case with the Magnus force compared to that of the Saffman force as visible in Figures 7.25(a) and (c). The reduction of the number of inter-particle collisions for the case with the Saffman force implies that this force is stronger than the Magnus force, and hence the particles colliding with the walls have stronger impulses while they rebound. Thus, the particle concentration or the number of particles (not shown for brevity) in the direct vicinity of the wall is smaller for the case with the Saffman force than that with the Magnus force. Note that the combination of both forces increases the impulse of the reflected particles and thus reduces the number of inter-particle collisions and agglomeration processes in the region near the walls as shown in Figure 7.23(d). In conclusion, the deviations observed in Figure 7.24 between the cases with and without lift forces can be attributed to both forces owing to (i) the high velocity gradient in the region close to the wall and (ii) the high rotation rates of the particles impacting the walls. The analysis depicted in Figure 7.24 suggests that both lift forces should be taken into account, since they alter the main predictions perceptibly. Therefore, the combined effect of both lift forces will be considered in the next simulations. 7.2.2.7 Cumulative Effect of the Sub-Models
The assessment of the numerical results discussed in Sections 7.2.2.4, 7.2.2.5 and 7.2.2.6 shows that the inclusion of the two-way coupling, the subgrid-scale model for the particles and the lift forces alters the agglomeration rate significantly. Nevertheless, their cumulative effect may be stronger and thus their overall influence within the computational set-up presented in Section 7.2.1 has to be evaluated. Note that in this analysis both subgrid-scale models for the particles (trivial and the Langevin model) are applied to show their impact on the predictions. Figure 7.26 shows the cumulative effect of the three issues on the agglomeration process as a function of time. As expected from the conclusions given in Sections 7.2.2.4, 7.2.2.5 and 7.2.2.6, the accumulated number of inter-particle collisions and agglomeration processes depicted in Figure 7.26(a) is reduced due to the combination of the three effects, since all three sub-models except the Langevin subgrid-scale model for the particles reduce the number of particle-particle collisions. Furthermore, the two-way coupling marginally enhances the accumulated number of agglomeration processes as explained in Section 7.2.2.4, while the consideration of the subgridscale model and the lift forces leads to a lower number of successful agglomerations. As a result, the cumulative effect leads to a reduced number of agglomeration processes. Regarding the agglomeration rate shown in Figure 7.26(b), the effect of the three sub-models is again superimposed. That means that the slight increase of the agglomeration rate due to the two-way coupling is overwhelmed by the reduction due to the subgrid-scale model and the lift forces. 243
7.2 Particle Agglomeration in Turbulent Channel Flow
1.8
×10+7
SMs = OFF SMs = ON (Trivial SGS) SMs = ON (Langevin SGS)
1.2 0.9
Collision
0.6
SMs = OFF SMs = ON (Trivial SGS) SMs = ON (Langevin SGS)
0.9 0.6 0.3
Agglomeration
0.3
×10−2
1.2 pp Nagp /Ncol
pp Nagp × 20, Ncol
1.5
1.5
0.0
0.0 0
50
100
150
200
t UB /δ (a)
0
50
100
150
200
t UB /δ (b)
Figure 7.26: Cumulative effect of the three sub-models (SMs = two-way coupling, subgrid-scale model for the particles and lift forces) on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : pp Time history of (a) the accumulated particle-particle collisions Ncol and agglomeration processes Nagp and (b) pp the agglomeration rate Nagp /Ncol (closely-packed sphere model).
The present analysis shows that the global agglomeration rate at the end of the simulation is reduced from about 0.73% to about 0.23% (trivial model) and about 0.019% (Langevin model) when taking the three sub-models into account. The corresponding dimensionless frequency of the agglomeration process f˜agp decreases from about 5.39 × 102 to about 1.29 × 102 (trivial model) and about 0.15 × 102 (Langevin model). These results are also included in Table A.6 for comparison purposes. Based on these observations, it can be concluded that the cumulative effect of the three sub-models (i.e., the feedback effect of the particles on the fluid, the subgrid-scale model for the particles and the lift forces) is not negligible and has to be considered in the next simulations, whereas the subgrid-scale model plays the most important role. 7.2.2.8 Effect of the Diameter of the Primary Particles
The motivation for this investigation is the fact that practical applications of turbulent particleladen flows generally involve primary particles of different sizes. Hence, it is meaningful to investigate the influence of the diameter of the primary particles and the mass loading on the agglomeration process. For this purpose, the diameter of the particles is reduced from dp = 12 µm to 4 µm keeping the number of particles constant (i.e., N0 = 6, 000, 000). Thus, the mass loading decreases from ηp = 3.32% to 0.12%, respectively. In other words, in this analysis the same computational set-up as presented in Section 7.2.1 is used, except for the diameter of the primary particles. For these simulations, only the closely-packed sphere model is applied for modeling the structure of the agglomerate and the walls are assumed to be smooth. Furthermore, the two-way coupling, the subgrid-scale model (trivial or Langevin) and the lift forces are taken into account due to their significant effect as concluded in Section 7.2.2.7. Figure 7.27 shows the effect of the diameter of the primary particles on the global agglomeration 244
7. Results for Particle Agglomeration process. The reduction of the diameter of the particles implies that the inter-particle distances become larger, and thus it is expected that the number of collisions is significantly reduced compared with the case of large particles. As expected, comparing Figure 7.27(a) for the small particles with Figure 7.26(a) for the large ones approves that the accumulated number of interpp particle collisions Ncol is extremely influenced by the diameter of the primary particles and the resulting mass loading. The total number of collisions drops about two orders of magnitude for the small particles in relation to the large particles. As a result, also the total number of agglomeration processes Nagp has to differ significantly. However, the ratio of the agglomeration processes between the small and the large particles is much smaller than for the collision events. Thus, the number of collisions leading to agglomeration increases for the smaller particles. ×10+5
2.0
SMs = ON (Trivial SGS) SMs = ON (Langevin SGS)
4.5
×10−2
SMs = ON (Trivial SGS) SMs = ON (Langevin SGS)
1.5 pp Nagp /Ncol
pp Nagp × 20, Ncol
6.0
3.0
Collision 1.5
1.0
0.5
Agglomeration
0.0
0.0 0
50
100
t UB /δ (a)
150
200
0
50
100
150
200
t UB /δ (b)
Figure 7.27: Effect of the particle diameter on the agglomeration process predicted by MAM for small par pp ticles d∗p = 2 × 10−4 : Time history of (a) the accumulated particle-particle collisions Ncol and agglomeration pp processes Nagp and (b) the agglomeration rate Nagp /Ncol . Note that the three sub-models (SMs = the two-way coupling, the trivial or the Langevin subgrid-scale model for the particles and the lift forces) are taken into account (closely-packed sphere model).
Figure 7.27(b) confirms this statement demonstrating that using the trivial subgrid-scale model the pp agglomeration rate Nagp /Ncol increases by a factor of about 4.8 when the diameter of the particles is reduced by a factor of three (about 1.09% for dp = 4 µm and about 0.23% for dp = 12 µm). On the other hand, if the extended Langevin subgrid-scale model is applied, the corresponding agglomeration rate even increases by a factor of about 34 (about 0.68% for dp = 4 µm and about 0.02% for dp = 12 µm). The reason for this behavior is clear, since based on Eqs. (3.131) and (3.132) the cohesive impulse is inversely proportional to the diameter of the agglomerating pp∗ ∝ 1/dp . A stronger cohesive impulse can more often overcome the repulsive impulse particles fˆn,c of the collision and hence the probability of agglomeration increases significantly by reducing the diameter of the primary particle. It is worth mentioning that this analysis also confirms the increasing importance of considering the agglomeration in technical applications when the particles encountered are getting smaller. 245
7.2 Particle Agglomeration in Turbulent Channel Flow 7.2.2.9 Effect of the Mass Loading
In this section the effect of the particle mass loading on the agglomeration process is studied. For this purpose, three cases are distinguished: (i) case Large denotes 6, 000, 000 primary particles with dp = 12 µm (St+ = 15.04) leading to a mass loading of ηp = 3.22%. (ii) Small denotes the same number of particles with a smaller diameter of dp = 4 µm (St+ = 1.67) yielding a mass loading of ηp = 0.12%. (iii) case Many stands for the small particles with dp = 4 µm but increasing the number of particles to N0 = 162, 000, 000 in order to reach the same mass loading as in the first case. For these simulations, only the closely-packed sphere model is applied for modeling the structure of the agglomerate and the walls are assumed to be smooth. Furthermore, the two-way coupling, the trivial subgrid-scale model3 and the lift forces are taken into account due to their significant effect. Figures 7.28(a), (b) and (c) depict the time history of the accumulated number of inter-particle pp collisions Ncol and agglomeration processes Nagp for the three set-ups, respectively. As expected, the number of collisions does not only depend on the mass loading, but also on the diameter of the primary particles. Hence, the largest number of collisions is observed for the case Many including 162 million small particles. For the case Large with the same mass loading but 27 times less particles of dp = 12 µm, the number of particle-particle collisions is more than two orders of magnitude smaller. As expected, for the case Small a drastically lower number of particle-particle collisions is observed. This behavior also influences the number of successful agglomeration processes, since inter-particle collisions are the first prerequisite for agglomeration. However, for the case Large an additional effect is visible in Figure 7.28, namely the number of agglomerations significantly drops in comparison with the number of particle-particle collisions. This phenomenon is even more clearly obvious based on the global agglomeration rate also displayed in Figure 7.28. At the end of the simulation (∆T ∗ = 200) the values of the agglomeration rate pp Nagp /Ncol are 0.23%, 1.09% and 1.22% for the cases Large, Small and Many, respectively. Thus, for the two cases with the same particle diameter (i.e., dp = 4 µm) the agglomeration rate is nearly identical although the total number of collisions and agglomeration processes differs by four orders of magnitude. However, for the case with the large particles the agglomeration rate is a factor of about five smaller. The reason for this behavior can be attributed to the cohesive impulse, which pp∗ is inversely proportional to the diameter of the agglomerating particles fˆn,c ∝ 1/dp . As a result, stronger cohesive impulses found for the smaller particles can more often overcome the repulsive impulses of the particle-particle collisions and hence the probability of agglomeration increases significantly as visible in Figure 7.28. To get insight into the local distribution of the collision events and agglomeration processes, these quantities are accumulated during the simulation and then averaged in streamwise and spanwise directions. Figures 7.29(a), (b) and (c) depict the concentration of the averaged number of inter-particle collisions and agglomeration processes. Obviously, for all three cases most of the collisions and agglomeration processes take place in the near-wall region. 3
Note that the Langevin subgrid-scale model was not available in LESOCC when the simulation of the third case was carried out.
246
7. Results for Particle Agglomeration ×10+7
0.5
×10−2
0.4
0.9 pp Nagp /Ncol
pp Nagp × 20, Ncol
1.2
0.6
Collision 0.3
0.3 0.2 0.1
Agglomeration 0.0
0.0 0
50
100
150
200
0
50
t UB /δ
100
150
200
150
200
150
200
t UB /δ (a) Case: Large
×10+5
2.5
2.0
×10−2
2.0 pp Nagp /Ncol
pp Nagp × 20, Ncol
2.5
1.5
Collision
1.0
Agglomeration
0.5
1.5 1.0 0.5
0.0
0.0 0
50
100
150
200
0
50
t UB /δ
100
t UB /δ (b) Case: Small
1.8
×10+9
2.5 2.0
1.2
pp Nagp /Ncol
pp Nagp × 20, Ncol
1.5
×10−2
0.9
Collision
0.6
Agglomeration
1.5 1.0 0.5
0.3 0.0
0.0 0
50
100
150
200
t UB /δ
0
50
100
t UB /δ (c) Case: Many
Figure 7.28: Effect of the mass loading on the agglomeration process predicted by MAM: Time history of pp the accumulated number of particle-particle collisions Ncol and agglomeration processes Nagp as well as the pp agglomeration rate Nagp /Ncol for three cases (a) Large [N0 = 6, 000, 000 and d∗p = 6 × 10−4 ], (b) Small [(N0 = 6, 000, 000 and d∗p = 2 × 10−4 ] and (c) Many [N0 = 162, 000, 000 and d∗p = 2 × 10−4 ] (closely-packed sphere model).
247
7.2 Particle Agglomeration in Turbulent Channel Flow 10+8 pp hNagp i / hNcol i
10+7 10+6 10+5 10+4 10+3 10+2
pp hNcol i
hNagp i
0.40 0.20 0.10 0.05
×10−2
0.01
10+1 -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ
0.2 0.4 0.6 0.8
1
0.2 0.4 0.6 0.8
1
0.2 0.4 0.6 0.8
1
y/δ (a) Case: Large
10+4 10+3
pp hNagp i / hNcol i
10+5
pp hNcol i
10+2
10
+1
0.40 0.20 0.10 0.05
×10−2
0.01
hNagp i -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ
y/δ (b) Case: Small
10+9
10+7 10+6
pp hNagp i / hNcol i
10+8
pp hNcol i
10+5 10+4
0.40 0.20 0.10 0.05
×10−2
0.01
hNagp i -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
y/δ
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (c) Case: Many
Figure 7.29: Effect of the particle mass loading on the agglomeration process predicted by MAM: The conpp centration of the averaged number of collision events hNcol i and agglomeration processes hNagp i as well as pp the averaged agglomeration rate hNagp i / hNcol i along y/δ for three cases (a) Large [N0 = 6, 000, 000 and d∗p = 6 × 10−4 ], (b) Small [N0 = 6, 000, 000 and d∗p = 2 × 10−4 ] and (c) Many [N0 = 162, 000, 000 and d∗p = 2 × 10−4 ]. Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
248
7. Results for Particle Agglomeration
Nag
As expected, the high turbulence level of the fluid and especially of the particles in this region leads to higher numbers of collisions than at other locations. Additionally, due to turbophoresis the highest particle volume fractions (not shown for brevity) are found in the region closest to the channel walls. As depicted in Figure 7.29, the number of collisions is low at the channel center due to the low particle velocity fluctuations. pp Furthermore, Figure 7.29 shows the distribution of the averaged agglomeration rate hNagp i / hNcol i along the channel width. Although the largest numbers of collisions and agglomeration processes are found close to the walls, the agglomeration rate is largest in the center of the channel. That is first of all astonishing but can be explained as follows: The velocity gradient and thus the velocity difference between two particles moving in a similar distance to the wall is smallest in the center of the channel. Additionally, the velocity fluctuations are low in the center. As a result, the repulsive impulse of two colliding particles is small and can be easily outplayed by the cohesive impulse leading to agglomeration. A comparison of the three cases reveals that the distributions of the agglomeration rates of the large and the small particles differ over the entire channel width, where the deviations are smallest in the center and increase towards the walls. Overall, that explains the significant difference between the global agglomeration rates found for the small and the large particles as mentioned before. On the contrary, the distribution of the local agglomeration rate for the case Small and Many are qualitatively resembling, which explains the similar global agglomeration rates mentioned above. Figure 7.30 depicts a statistic on the number of existing agglomerates of the same type at the end of the simulations (∆T ∗ = 200). The sizes of the agglomerates are given in terms of the number of primary particles included in the agglomerate. It is visible that for the case Small solely a few two-particle agglomerates exist. Thus, although the agglomeration rate is relatively high, the probability that these agglomerates collide with primary particles or even other agglomerates is low. Hence, larger agglomerates are unlikely to appear. 10
+7
10
+6
10
+5
10
+4
10
+3
10
+2
10
+1
10
+0
dp = 12 µm, 6M dp = 4 µm, 6M dp = 4 µm, 162M
2
3
4
5
6
7
Agglomerate Type Figure 7.30: Effect of the particle mass loading on the agglomeration process predicted by MAM for the three cases Large, Small and Many investigated: Total number of agglomerates of the same type in terms of included primary particles (2 = two-particle agglomerate, 3 = three-particle agglomerate, etc.) at a dimensionless time of 200 (closely-packed sphere model).
249
7.2 Particle Agglomeration in Turbulent Channel Flow The situation is completely different for the case Many. Here, the agglomeration rate is comparable to the previous case, but due to 27 times more particles of the same size the total number of agglomerates is much higher. This circumstance also increases the probability that existing agglomerates collide and agglomerate with primary particles or even other agglomerates building up larger agglomerates consisting of up to six primary particles. Contrarily, for the case Large a much lower number of two-, three- and four-particle agglomerates is observed in comparison with the case Many. The reason is the decrease of the cohesive impulse when increasing the particle diameter from dp = 4 µm to 12 µm. That also hinders the formation of larger agglomerates, since the effect is strongly augmented in this case. 7.2.2.10 Effect of the Wall Roughness
In this section the effect of the wall roughness on the global agglomeration process is investigated. This analysis is motivated by the fact that the walls of most industrial apparatuses are not ideally smooth (i.e., Rz /δ 6= 0) as assumed in the previous simulations. For this purpose, typical values of the restitution and friction coefficients of rough steel walls (typical mean roughness Rz = 10 µm) encountered in practical applications are listed in Table 7.1. Note that in comparison to the previous simulations the properties of the particles are not changed, and hence solely the wall model leads to differences in the results. It is worth mentioning that these adjusted restitution and friction coefficients are also used during the dispersion period of the particles (∆T ∗ = 50) in order to allow the particles and the continuous phase to adjust to the new situation before the agglomeration model is taken into account. The diameter of the primary particles is again set to the standard value dp = 12 µm. Additionally, the three sub-models (i.e., the two-way coupling, the subgrid-scale model for the particles and the lift forces) are considered due to their significant effect as concluded in Section 7.2.2.7. In the present simulations only the Langevin subgrid-scale model for the particles is applied for both the smooth and rough walls. Again, solely the closely-packed sphere model is adopted for modeling the agglomerate structure. The effect of the wall roughness on the agglomeration process is depicted in Figure 7.31. pp As clearly visible in Figure 7.31(a), the accumulated number of inter-particle collisions Ncol and agglomeration processes Nagp is significantly reduced when the wall roughness is taken into account, but with different rates. It is known that at rough walls the normal unit vector at the contact point strongly varies, since its orientation is adjusted leading to a Gaussian distributed random wall-normal unit vector (see, e.g., Breuer et al., 2012; Alletto, 2014). Thus, a large spreading of the particle trajectories is expected after the collision with the rough wall. Furthermore, the restitution and friction coefficients for particle-wall collisions provided in Table 7.1 are different for smooth and rough walls. Hence, the reduction of the accumulated number of particle-particle collisions can be attributed to the fact that the wall roughness considerably alters the rebound behavior of the particles at the channel walls, which is visible by looking at the distribution of the mean volume fraction displayed in Figure 7.32(a). As observed in the previous simulations, the highest numbers of inter-particle collisions and those leading to agglomeration occur in the region close to the wall (see Figure 7.32(b)). Figures 7.33(a) and (b) show that the inclusion of the wall roughness enhances the particle fluctuations in the streamwise and wall-normal directions, especially close to the walls. Breuer et al. (2012) found 250
7. Results for Particle Agglomeration that the increase of the particle fluctuations can be attributed to two reasons: 1. Additional momentum loss of the particles hitting the rough wall decelerates the particles coming from the bulk flow. Thus, due to its inertia a particle needs a certain time to adjust to the mean flow and hence the particle fluctuations are increased throughout the entire channel, especially close to the walls. 2. Due to the shadow effect (i.e., a particle can not hit a section of a wall which has a negative inclination angle with respect to the particle trajectory) the cone, in which the possible trajectories of the reflected particles lie, points slightly towards the bulk flow leading to a large spreading and hence dispersion of the particles throughout the channel. As a result, the fluctuations of the particle are further increased. ×10+7
1.2
0.05
Rz /δ = 0 Rz /δ = 5 × 10−4
0.9 0.6
Collision
0.3
×10−2
0.04 pp Nagp /Ncol
pp Nagp × 400, Ncol
1.5
Rz /δ = 0 Rz /δ = 5 × 10−4
0.03 0.02 0.01
Agglomeration
0.0
0.00 0
50
100
t UB /δ (a)
150
200
0
50
100
150
200
t UB /δ (b)
Figure 7.31: Effect of the wall roughness model on the agglomeration process predicted by MAM for large pp particles d∗p = 6 × 10−4 : Time history of (a) the accumulated number of particle-particle collisions Ncol and pp agglomeration processes Nagp and (b) the agglomeration rate Nagp /Ncol (closely-packed sphere model).
These differences can be more clearly explained by looking at Figure 7.32(b), which shows the local pp distribution of the concentration of inter-particle collisions hNcol i and agglomeration processes hNagp i along the channel width averaged over a dimensionless time interval of ∆T ∗ = 200. Obviously, although it is expected that a higher intensity of the particle fluctuations in the region close to the wall (∆y/δ < 0.02) compared to the smooth wall enhances the inter-particle collisions as concluded earlier, the concentrations of the collision events and hence also the agglomeration processes are appreciably reduced there for rough walls as visible in Figures 7.32(c). The reduction of the number of inter-particle collisions in the near-wall region (y + ≤ 10) is attributed to the migration of the particles away from the walls as mentioned before (see the distribution of the mean volume fraction in Figure 7.32(a)). On the other hand, the migration of the particles increases the volume fraction in the central region as shown in Figure 7.32(a). This leads to a slightly higher number of particle-particle collisions and agglomeration processes outside the near-wall region for the rough wall compared to the smooth wall as visible in Figures 7.32(b). Hence, the agglomeration processes outside the region near the wall are slightly augmented. 251
7.2 Particle Agglomeration in Turbulent Channel Flow Nevertheless, as explained before a significantly lower number of accumulated inter-particle pp collisions Ncol shown in Figure 7.31(a) is observed for the rough wall in comparison with the smooth wall. That implies that the reduction of the number of inter-particle collisions in the region near the walls overwhelms the slight increase in the central region of the channel leading to pp a noticeably lower number of collisions Ncol and agglomeration processes Nagp for the case with rough walls. As a result, the number of formed agglomerates is noticeably reduced.
9.00 6.00
×10−4
10+8
Rz /δ = 0 Rz /δ = 5 × 10−4
3.00
10+7 10+6
hαp i
10
pp hNcol i
+5
10+4 10+3 10+2 10+1
0.18 0.12
hNagp i
10+0 1
10
100
y+
1000
-1 -0.8 -0.6 -0.4 -0.2
(a)
10+8 10+7 10+6 10+5 10+4 10
+3
10+2
pp hNcol i
0.2 0.4 0.6 0.8
1
0.2 0.4 0.6 0.8
1
(b)
Rz /δ = 0 Rz /δ = 5 × 10−4
pp hNagp i / hNcol i
10+9
0
y/δ
hNagp i
0.40 0.20 0.10 0.05
×10−2
0.01
10+1 10+0 0.9
0.92
0.94
0.96
y/δ (c)
0.98
1
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (d)
Figure 7.32: Effect of the wall roughness model on the agglomeration process predicted by MAM for large particles d∗p = 6 × 10−4 : (a) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed line denotes the global volume fraction of αp = 0.18×10−4 ), (b) averaged concentration of the number pp pp of collision events hNcol i and agglomeration processes hNagp i along y/δ, (c) averaged concentration hNcol i and pp hNagp i close to the walls and (d) averaged agglomeration rate hNagp i / hNcol i along y/δ. Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
Figure 7.31(b) shows that the global agglomeration rate at the end of the time interval considered decreases from about 0.019% to about 0.012% when the wall roughness model and the coefficients of the particle-wall model listed in Table 7.1 are taken into account. The corresponding dimensionless frequency of the agglomeration processes f˜agp decreases from about 0.15 × 102 to about 0.05 × 102 252
7. Results for Particle Agglomeration when taking the wall roughness into account as listed in Table A.6. The reduction of the global agglomeration rate can be explained by the relatively higher particle velocity fluctuations in the near-wall region (see Figure 7.33) leading to stronger repulsive and weaker cohesive impulses. Figure 7.32(d) shows the distribution of the local agglomeration rate (i.e., the ratio of the pp concentration of the inter-particle collisions and agglomeration processes hNagp i / hNcol i) as a function of the dimensionless wall-normal coordinate y/δ. Similar to the previous simulations with smooth walls, it is obvious for rough walls that the local agglomeration rate strongly increases with increasing distance to the wall. This phenomenon was already explained in Section 7.2.2.1 by the smaller velocity gradients and the weaker velocity fluctuations observed for larger wall distances. Consequently, the local agglomeration rate is largest at the channel center.
3.5
×10−2
1.2 1.0
vp′ vp′ /UB2
2.5 2.0
0.6 0.4
D
D
1.5 1.0
Rz /δ = 0 Rz /δ = 5 × 10−4
0.5 0.0 0.94
0.8
E
E
u′p u′p /UB2
3.0
×10−3
0.95
0.96
0.97
0.2 0.98
0.99
1
0.0 0.94
0.95
0.96
y/δ (a)
0.97
0.98
0.99
1
y/δ (b)
Figure 7.33: Effect of the wall roughness model on the dimensionless averaged fluctuations of the particles near the wall in the (a) streamwise and (b) wall-normal directions. Dimensionless averaging time ∆T ∗ = 200.
Lastly, since it is interesting to know whether this behavior changes in case of lower values of the wall roughness, the effect of different wall roughnesses on the agglomeration process for large and small particles is investigated next. 7.2.2.10.1 Different Wall Roughnesses Considering Large Particles
In this section the effect of different wall roughnesses on the agglomeration processes is analyzed. In the present simulations the same numerical set-up for rough walls is used but with two different values of the wall roughness, namely a low roughness Rz = 1 µm (i.e., Rz /δ = 5 × 10−5 ) and the standard roughness 10 µm (i.e., Rz /δ = 5 × 10−4 ). Figure 7.34 shows the effect of different values of the wall roughness on the agglomeration process of the large particles in comparison with the case with smooth walls (i.e., Rz /δ = 0). It is clearly visible that similar trends of the physical behavior of the agglomeration process are predicted when increasing the wall roughness from zero to either Rz = 1 µm or 10 µm. This includes two main observations: (i) As shown in Figure 7.34(a), the number of particle-particle pp collisions Ncol significantly decreases due to the wall roughness, which appreciably alters the 253
7.2 Particle Agglomeration in Turbulent Channel Flow rebound behavior of the particles colliding with the rough walls. As a result, the mean volume fraction hαp i is significantly reduced in the direct vicinity of the wall (y + ≤ 10) as displayed in Figure 7.34(c). (ii) The so-called shadow effect enhances the wall-normal velocity fluctuations as visible in Figure 7.34(d), which leads to a appreciably lower number of agglomeration processes Nagp displayed in Figure 7.34(a). The reason for this observation was discussed in the previous investigation on the subgrid-scale model for the particles. ×10+7
1.2
0.05
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
0.9 0.6
Collision
0.3
×10−2
0.04 pp Nagp /Ncol
pp Nagp × 400, Ncol
1.5
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
0.03 0.02 0.01
Agglomeration
0.0
0.00 0
50
100
150
200
0
50
t UB /δ (a)
150
200
(b)
×10−4
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
×10−3
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
1.0 0.8
E
hαp i
3.00
1.2
vp′ vp′ /UB2
9.00 6.00
100
t UB /δ
0.6
D
0.4 0.2
0.18 0.12 1
10
y (c)
+
100
1000
0.0 0.94
0.95
0.96
0.97
0.98
0.99
1
y/δ (d)
Figure 7.34: Effect of different values of the wall roughness on the agglomeration process predicted by MAM for the large particles d∗p = 6 × 10−4 : (a) time history of the accumulated number of particle-particle collisions pp pp Ncol and agglomeration processes Nagp , (b) time history of the agglomeration rate Nagp /Ncol , (c) mean particle + volume fraction hαp i along the dimensionless wall coordinate y (the dashed line denotes the global volume fraction of αp = 0.18 × 10−4 ) and (d) dimensionless averaged fluctuations of the particles in the wall-normal
direction vp0 vp0 /UB2 . Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
However, a further inspection of Figure 7.34(a) shows that if the wall roughness is increased from Rz = 1 µm to 10 µm, the number of particle-particle collisions is only very slightly increased, whereas the agglomeration processes are hardly affected. As a result, the global agglomeration rate for Rz = 1 µm is slightly lower than for Rz = 10 µm. Note that a noticeable difference between 254
7. Results for Particle Agglomeration the agglomeration rates is observed within the dimensionless time interval of ∆T ∗ ≤ 80, whereas an almost constant temporal behavior is predicted during the remaining time. To discuss this unexpected behavior, it is worth recalling that in the framework of the sandgrain roughness model applied in this study (see Section 2.4.4) the random inclination angle αR required for the determination of the normal unit vector nR of rough walls is defined by: αR = σw ξ ,
(7.14)
where ξ is a Gaussian distributed random number with a unit standard deviation. The angle σw reads: rw σw = arcsin rw + dp /2
!
with rw =
1 Csurface Rz , 2
(7.15)
where the radius of the wall spheres rw is expressed in terms of the wall roughness Rz and the constant Csurface = 3 as mentioned in Section 7.2.1. Note that in the context of the sandgrain roughness model by Breuer et al. (2012) the angle is restricted to σw,max = 30◦ to avoid unrealistic scenario for αR ≥ 90◦ . Table 7.5 shows the values of σw for large particles determined based on Eq. (7.15) with different wall roughnesses. Obviously, for Rz /δ = 5 × 10−4 the value exceeds the maximum value set in the corresponding routine of the in-house code LESOCC, since rw > dp /2 and thus for this case σw = σw,max . Rz /δ
dp /δ
rw /δ
rw /(dp /2)
σw
5 × 10−5 5 × 10−4
6 × 10−4 6 × 10−4
7.5 × 10−5 7.5 × 10−4
0.25 2.50
11.54◦ < σw,max 45.58◦ > σw,max
Table 7.5: The angle σw required for the determination of the random inclination angle αR = σw ξ for large particles with different wall roughnesses (σw,max = 30◦ ).
It is known than in most practical applications rw dp /2 and hence σw < 30◦ based on Eq. (7.15). In the present simulations the primary particles satisfy the condition rw < dp /2 only in the case of the low roughness (i.e., Rz /δ = 5 × 10−5 ) and thus the random inclination angle is restricted for the large roughness. Obviously, the present particles are less affected by the wall roughness Rz = 10 µm than for Rz = 1 µm. A possible reason for this unexpected behavior is the restriction made for the inclination angle σw as explained before. 7.2.2.10.2 Different Wall Roughnesses Considering Small Particles
To study the effect of the wall roughness on the agglomeration process for the small particles, the previous set-up is applied using the same three roughness values. The corresponding values for the inclination angle σw are listed in Table 7.6. Note that the condition rw < dp /2 is again satisfied only for the value of the low wall roughness (i.e., Rz /δ = 5 × 10−5 ). Figure 7.35 shows the effect of different values of the wall roughness on the agglomeration process of the small particles in comparison with the case with smooth walls (i.e., Rz /δ = 0). 255
7.2 Particle Agglomeration in Turbulent Channel Flow Rz /δ
dp /δ
rw /δ
rw /(dp /2)
σw
5 × 10−5 5 × 10−4
2 × 10−4 2 × 10−4
7.5 × 10−5 7.5 × 10−4
0.75 7.50
25.38◦ < σw,max 61.93◦ > σw,max
Table 7.6: The angle σw required for the determination of the random inclination angle αR = σw ξ for small particles with different wall roughnesses (σw,max = 30◦ ).
×10+5
4.5
2.0
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
3.0
Collision 1.5
×10−2
1.5 pp Nagp /Ncol
pp Nagp × 20, Ncol
6.0
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
1.0
0.5
Agglomeration
0.0
0.0 0
50
100
150
200
0
50
100
t UB /δ
(b)
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
×10−3
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
2.5 2.0
E
6.00
3.0
vp′ vp′ /UB2
×10−6
12.00
3.00
1.5 1.0
D
hαp i
200
t UB /δ
(a) 24.00
150
0.5 0.67 0.50 1
10
y (c)
+
100
1000
0.0 0.94
0.95
0.96
0.97
0.98
0.99
1
y/δ (d)
Figure 7.35: Effect of different values of the wall roughness on the agglomeration process predicted by MAM for the large particles d∗p = 2 × 10−4 : (a) time history of the accumulated number of particle-particle collisions pp pp Ncol and agglomeration processes Nagp , (b) time history of the agglomeration rate Nagp /Ncol , (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed line denotes the global volume fraction of αp = 0.67 × 10−6 ) and (d) dimensionless averaged fluctuations of the particles in the wall-normal
direction vp0 vp0 /UB2 . Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
Figures 7.35(a) clearly indicates similar observations for the accumulated number of particleparticle collisions as for the large particles. However, compared with the case with smooth walls the effect of the wall roughness on the small particles is less pronounced than for the large particles. This observation is supported by comparing Figure 7.34(d) with 7.35(d), which indicates a less 256
7. Results for Particle Agglomeration pronounced modification of the particle fluctuations in comparison with the case of the large particles. Obviously, despite the noticeably higher wall-normal particle fluctuations for small particles in comparison with the large ones, a lower reduction of the volume fraction is observed in the vicinity of the wall for small particles as visible in Figure 7.35(c). That can be attributed to the 27 times lower mass loading for the small particles in comparison with the large ones. Although the wall-normal particle fluctuations are only marginally affected by the wall roughness, pp a slightly higher number of inter-particle collisions Ncol and agglomeration processes Nagp is predicted for Rz = 1 µm than for Rz = 10 µm as visible in Figure 7.35(a). Furthermore, the simulation with a wall roughness of Rz = 1 µm leads to a slightly higher agglomeration rate pp Nagp /Ncol than for Rz = 10 µm as shown in Figure 7.35(b). Contrary to the case with large particles, a more reasonable time history of the agglomeration rate is predicted for small particles when increasing the wall roughness from zero to either Rz = 1 µm or 10 µm. That means that an increase of the wall roughness reduces the global agglomeration rate as expected. pp∗ It is worth noting that since fˆn,c ∝ 1/dp , the agglomeration rates for the small particles shown in Figure 7.35(b) using the same three roughness values are more than one order of magnitude higher than the agglomeration rates of the large particles (see Figure 7.34(b)). As concluded in Section 7.2.2.8, this observation is attributed to the fact that in the context of MAM the cohesive pp∗ impulse is inversely proportional to the particle diameter as discussed before (i.e., fˆn,c ∝ 1/dp ). This implies a higher probability of satisfying the agglomeration conditions for small particles than for large ones. 7.3 Comparison of Agglomeration Models
The objective of this section is to compare the predictions of the energy-based (EAM) and the momentum-based (MAM) agglomeration model against each other using the test case of the particle-laden turbulent channel flow. In the following the numerical results are organized as follows. In Section 7.3.1 the results of both agglomeration models are analyzed without taking into account the cumulative effect of the three sub-models (the feedback effect of the particles on the continuous flow, the subgrid-scale model for the particles and the lift forces). Afterwards, the performance of both agglomeration models is studied for different simulation parameters in Section 7.3.2. For all steps the most important reasons for the deviations observed between the results of both models are discussed in detail. 7.3.1 Effect of the Agglomeration Model without Sub-Models
In this section the standard computational set-up for the large particles (dp = 12 µm) explained in Section 7.2.1 is applied assuming smooth channel walls. As a first step, the three sub-models mentioned above are not taken into account to minimize the influencing parameters. Figure 7.36(a) shows the accumulated number of the agglomerated primary particles Napp as a function of the dimensionless time. It is clearly visible that the energy-based model predicts a higher number of agglomerated primary particles than the momentum-based agglomeration model. The simulation data also reveal that the energy-based agglomeration model predicts a higher number of agglomerates Nag than the momentum-based model (not shown for the sake of brevity). 257
7.3 Comparison of Agglomeration Models The number of agglomerates of the same class are different for both models. Figure 7.36(b) depicts the total number of agglomerates Nag at the end of the simulation (i.e., ∆T ∗ = 200) as a function of the number of primary particles included in an agglomerate. This diagram shows that the number of two-particle agglomerates is noticeably higher for EAM than for MAM, but a lower number of larger agglomerates is observed for EAM. In other words, for the energy-based agglomeration model the tendency that an existing agglomerate agglomerates with a primary particle or another agglomerate building up a larger agglomerate decreases in comparison with the momentum-based model. However, for primary particles the tendency to agglomerate is slightly higher for EAM compared with MAM. The reason for these observations will become clear in the next paragraph.
3.0
×10+5
2.5
EAM MAM
10+5
EAM MAM
10+4
Nag
Napp
2.0 1.5
10+3
1.0
10+2
0.5
10+1
0.0
10+0
0
50
100
150
200
2
3
4 5 6 7 8 Agglomerate Type
t UB /δ (a)
×10+7
2.0
EAM MAM
1.2
×10−2
1.5
0.8
Collision 0.4
10
(b)
pp Nagp /Ncol
pp Nagp × 20, Ncol
1.6
9
EAM MAM Alletto (2014)
1.0
0.5
Agglomeration
0.0
0.0 0
50
100
t UB /δ (c)
150
200
0
50
100
150
200
t UB /δ (d)
Figure 7.36: Comparison between of the energy-based (EAM) and the momentum-based (MAM) agglom eration models for large particles d∗p = 6 × 10−4 : (a) time history of the total number of the agglomerated primary particles Napp , (b) number of agglomerates of the same type (2 = two-particle agglomerate, 3 = threeparticle agglomerate, etc.) at a dimensionless time of 200, (c) time history of the accumulated number of pp particle-particle collisions Ncol and agglomeration processes Nagp and (d) time history of the agglomeration pp rate Nagp /Ncol (closely-packed sphere model). The agglomeration model by Alletto (2014) is used here for the purpose of comparison.
258
7. Results for Particle Agglomeration Since the agglomeration conditions of both models given by Eqs. (3.58) and (3.133) cannot be directly compared to each other, some assumptions are made to simplify the relations in order to explain the observations. As found in the previous investigations using MAM (see Section 7.2.2.3), the dominant particle-particle collisions leading to agglomeration are sticking events. Thus, an agglomeration process occurs if only the first condition in Eq. (3.133) is satisfied. On the other hand, this implies that in a first guess the rotation of the collision partners and the arising agglomerates can be neglected. Hence, for MAM the cohesive impulse responsible for a successful agglomeration can be written based on Eq. (3.131) in terms of the diameter dp and the density ρp of the agglomerating primary particles as follows: pp∗ fˆn,c ∝
1 3/5 ρp
dp
(7.16)
.
Applying the same assumptions to the energy-based agglomeration model, the last two terms on the right-hand side of Eq. (3.58) cancel out and hence for a head-on collision the agglomeration + condition reduces to ∆EvdW ≥ Ekin,r,n , where these terms can be expressed as a function of ρp and dp as follows: ∆EvdW ∝ ρp1/2 d2p
+ and Ekin,r,n ∝ ρp d3p .
(7.17)
Eqs. (7.16) and (7.17) show that both agglomeration conditions lead to a lower agglomeration probability when the diameter of the agglomerated particles increases, which is the case for further agglomeration processes between agglomerates and primary particles. This indicates that both techniques (i.e., EAM and MAM) reproduce the physical behavior of the agglomeration process in a similar manner, but with different rates. The reason for the higher number of two-particle agglomerates predicted by EAM (see Figure 7.36(b)) is attributed to the different formulations of the agglomeration conditions. Although the trends regarding a variation of the diameter are qualitatively reproduced by both models in a similar manner, that does not automatically mean that the agglomeration rates are identical. Obviously, the condition regrading the difference of + the van-der-Waals energy ∆EvdW in relation to the relative kinetic energy Ekin,r,n is more often satisfied for two-particle agglomerates than the corresponding agglomeration condition relying on the cohesive impulse. If the diameter of the agglomerate reaches a certain size, the trend is reversed. Thus, the energy-based agglomeration model predicts lower numbers of large agglomerates than MAM, which is the case when a primary particle collides with an existing agglomerate. A more detailed analysis suggests that the increase or reduction of the number of agglomerates of the same size is similar along the channel width (not shown for brevity). In other words, the number of two-particle agglomerates predicted by EAM is higher than for MAM along the entire channel width, whereas the number of larger agglomerates is lower. pp Figure 7.36(c) depicts the accumulated number of particle-particle collisions Ncol and agglomeration processes Nagp computed by both agglomeration models as a function of time. It shows that the energy-based agglomeration model predicts a higher number of inter-particle collisions than the momentum-based model. To further analyze this behavior, the local distribution of the number of pp collision events hNcol i and agglomeration processes hNagp i using both agglomeration models is considered. Figure 7.37(a) depicts the distribution of the concentration of the averaged number of 259
7.3 Comparison of Agglomeration Models inter-particle collisions and agglomeration processes along the channel width at the end of the simulation. It is worth noting that the total values are not of interest, but solely the distribution over the channel width is of relevance. Obviously, the highest numbers of particle-particle collisions and agglomeration processes occur in the near-wall region. As mentioned before, this behavior can be attributed to: (i) the high level of turbulence (particle velocity fluctuations are higher than at other locations as displayed in Figure 7.38 leading to more inter-particle collisions) and (ii) the high particle volume fraction in the region closest to the channel wall driven by turbophoresis (see Figure 7.37(c)). Furthermore, it is obvious that both agglomeration models predict almost the same number of inter-particle collisions and agglomeration processes in the direct vicinity of the walls, whereas a significant difference is observed in other regions, especially in the central area. 10+8 10+7
pp hNagp i / hNcol i
EAM MAM
10+6 pp hNcol i
10+5 10
+4
10
+3
0.40 0.20 0.10
0.01 EAM MAM
hNagp i
10+2
-1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (a) 9.00 6.00
0.2 0.4 0.6 0.8
1
0.2 0.4 0.6 0.8
1
y/δ (b)
×10−4
6.09
EAM MAM
6.08
hαp i
hdp /δi
3.00
×10−4
EAM MAM
6.07 6.06 6.05
0.18 0.12
6.04 1
10
y (c)
+
100
1000
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (d)
Figure 7.37: Comparison between the energy-based (EAM) and the momentum-based (MAM) agglomeration pp models for large particles d∗p = 6 × 10−4 : (a) averaged concentration of the number of collision events hNcol i and agglomeration processes hNagp i along y/δ, (b) averaged agglomeration rate along y/δ, (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dots represent the distribution of the mean volume fraction without taking the agglomeration model into account and the dashed line denotes global volume fraction of αp = 0.18 × 10−4 ) and (d) dimensionless mean diameter of the active particles hdp /δi along y/δ. Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
260
7. Results for Particle Agglomeration To explore this observation, the distribution of the dimensionless mean diameter hdp /δi of all active particles in the channel (i.e., the remaining primary particles and the existing agglomerates) is calculated during the simulation time. As visible in Figure 7.37(d), EAM results in a larger mean diameter of the particles along the channel width than MAM. Thus, the higher number of particle-particle collisions predicted by EAM is directly related to the larger mean diameter (i.e., the overall higher number of agglomerates), since the probability of inter-particle collisions between primary particles and agglomerates as well as the agglomerates themselves increases. Figure 7.37(c) displays the distribution of the mean particle volume fraction as a function of the dimensionless wall coordinate y + = y uτ /ν using both agglomeration models. Here, the distance y to the wall is made dimensionless with the friction velocity uτ /UB = 0.053 of the one-way coupled case and the kinematic viscosity of the fluid. The highest volume fraction is observed in the region close to the channel walls. As expected, EAM predicts a marginally higher volume fraction due to the larger mean diameters of the active particles (Figure 7.37(d)) along the channel width compared with MAM. Furthermore, Figure 7.36(c) shows that the energy-based model predicts a higher number of agglomeration processes than observed for the momentum-based agglomeration model. This observation can be attributed to the stronger tendency of EAM to predict two-particle agglomerates as explained before (see also Table 7.7). Hence, the global agglomeration rate defined as the total number of particle-particle collisions leading to agglomeration to the total number of collisions pp (i.e., Nagp /Ncol ) is higher for EAM than for MAM as depicted in Figure 7.36(d). At the end of the simulation (i.e., ∆T ∗ = 200), the agglomeration rate is about 0.87% for EAM and about 0.73% for MAM. As mentioned before, the momentum-based agglomeration model leads to a higher number of larger agglomerates including more than two particles. However, the number of these large agglomerates is very small compared to the two-particle agglomerates. Figure 7.37(a) shows that both models predict almost the same number of agglomeration processes in the near-wall region, while EAM leads to a higher number of agglomeration processes in the range −0.9 < y/δ < 0.9. For the purpose of direct comparison Figure 7.36(d) also includes the agglomeration rate predicted by the original energy-based model by Alletto (2014). At the end of the simulation the agglomeration rate is about 0.49% and thus about a factor of two smaller than the value for EAM. The reason for this deviation (defective third case in the agglomeration model) was already discussed in Section 3.1.1.5 and has led to the corrections of this approach by modifying the agglomeration conditions resulting in the EAM employed in the present study. Figure 7.37(b) depicts the distribution of the averaged agglomeration rate along the channel width at the end of the simulation time. It is obvious that the agglomeration rate predicted by EAM and MAM are nearly identical in the region near the walls. On the other hand, this behavior is different in the central area of the channel within the range −0.4 < y/δ < 0.4. Note also that the highest agglomeration rate of both models is located in the center of the channel although the largest number of collisions and agglomeration processes are found close to the walls. This observation can be attributed to the fact that the velocity gradient as well as the velocity fluctuations and thus the velocity difference between two particles moving in a similar distance to the wall is smallest in the center of the channel leading to a weaker repulsive impulse between the collision partners. As a result, this impulse can be more often overcome by the cohesive impulse between two 261
7.3 Comparison of Agglomeration Models colliding particles leading to agglomeration. The different values of the agglomeration rates in the central region can be explained in a similar manner as before based on the different agglomeration conditions of EAM and MAM. These rely on either the difference of the van-der-Waals energy pp∗ ∆EvdW or the cohesive impulse fˆn,c given by Eqs. (3.22) and (3.131), respectively. In general, + the post-collision relative kinetic energy Ekin,r,n (normal component) can be written in terms of − + that before the collision and the normal restitution coefficient as Ekin,r,n = e2n,p Ekin,r,n . Thus, the + terms included in the simplified agglomeration condition for EAM (i.e., ∆EvdWh ≥ Ekin,r,n )scalei − with the normal component of the relative velocity between the collision partners u− 1 − u2 · n as follows: ∆EvdW ∝
h
i
− u− 1 − u2 · n
+ and Ekin,r,n ∝
h
i2
− u− 1 − u2 · n
.
(7.18)
Hence, for small relative velocities the relative kinetic energy reduces more drastically than the difference of the van-der-Waals energy leading to a higher probability of agglomeration and vice pp pp∗ versa. On the other hand, for MAM the repulsive fˆn,a and cohesive fˆn,c impulses scale with the normal relative velocity as follows: pp fˆn,a ∝
h
i
− u− 1 − u2 · n
pp∗ and fˆn,c ∝
h
i−1/5
− u− 1 − u2 · n
.
(7.19)
pp Thus, relation (7.19) implies that fˆn,a decreases with smaller relative velocities as observed in the pp∗ channel center, whereas fˆn,c increases. Hence, the tendency of an increasing agglomeration rate for smaller differences between the velocities of the collision partners in the central region is correctly reproduced by both agglomeration models. Nevertheless, due to the complex relationships in both agglomeration models a slightly higher probability of satisfying the agglomeration conditions is found for MAM compared with EAM. pp Figure 7.36(c) shows that the accumulated numbers of inter-particle collisions Ncol and agglomeration processes Nagp vary almost linearly after ∆T UB /δ = 100 allowing the definition of the dimensionless frequencies given by Eq. (7.12). Table 7.7 provides the dimensionless frequencies pp of the particle-particle collisions f˜col and the agglomeration processes f˜agp . These frequencies allow a direct evaluation of the influence of different sub-models and are therefore also provided in Table A.7 for all cases investigated in the following subsections. In the present analysis a comparison between both agglomeration models yields a difference of about 4.30% for the collision frequency and about 19.20% for the agglomeration frequency. That is a clear evidence that the increase of the collision frequency for EAM is a secondary effect due to the increased mean diameter of the particles as already shown in Figure 7.37(d).
Agglomeration model
pp f˜col
f˜agp
EAM MAM Difference
8.14 × 104 7.79 × 104 4.3%
6.67 × 102 5.39 × 102 19.2%
pp Table 7.7: Dimensionless frequencies of the particle-particle collisions f˜col and the agglomeration processes f˜agp for the large particles predicted by EAM and MAM after a dimensionless time ∆T ∗ = 100.
262
7. Results for Particle Agglomeration Figure 7.38 shows the dimensionless mean particle velocity and the streamwise and wall-normal fluctuations as well as the Reynolds shear stress determined during the dimensionless time interval of ∆T ∗ = 200 and additionally averaged in both homogeneous directions. Obviously, the particle fluctuations predicted by the energy-based agglomeration model are marginally higher than those computed by the momentum-based model. Furthermore, for both agglomeration models the statistics are nearly identical to the case without agglomeration (not shown here). The main reason why these particle statistics are not noticeably influenced by the agglomeration process is the fact that in the present work the agglomeration rates are quite low and the overall number of arising agglomerates is still below 2.2% of the total number of particles for EAM and below 2% for MAM at the end of the simulation. 3.5
1.0
3.0
u′p u′p /UB2
1.2
hup i /UB
0.8
2.5 2.0
E
EAM
0.6
×10−2
MAM
1.5
D
0.4
1.0
0.2
0.5
0.0
0.0 -1 -0.8 -0.6 -0.4 -0.2
0
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2
y/δ
0.2 0.4 0.6 0.8
1
0.2 0.4 0.6 0.8
1
y/δ
(a) 2.5
0
(b)
×10−3
4.0
×10−3
3.0
u′p vp′ /UB2
2.0 1.0
E
1.5
E
vp′ vp′ /UB2
2.0
0.0
-1.0
D
D
1.0
-2.0
0.5
-3.0 0.0
-4.0 -1 -0.8 -0.6 -0.4 -0.2
0
y/δ (c)
0.2 0.4 0.6 0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (d)
Figure 7.38: Comparison between of the energy-based (EAM) and the momentum-based (MAM) agglomera tion models for large particles d∗p = 6 × 10−4 : The dimensionless averaged (a) particle velocity in streamwise
direction hup i /UB , (b) streamwise fluctuations u0p u0p /UB2 , (c) wall-normal fluctuations vp0 vp0 /UB2 and (d)
0 0 2 Reynolds shear stress of the particles up vp /UB . Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
Finally, a note concerning the fluid statistics is given. As already shown in Section 7.2.2.1 using 263
7.3 Comparison of Agglomeration Models MAM, the mean velocities and Reynolds stresses of the continuous phase are hardly influenced by the disperse phase. Solely the wall-normal and spanwise stresses are slightly attenuated by the particles. However, the question whether an agglomeration model (i.e., EAM or MAM) is taken into account or not does not play a role. The reason for this observation is the same as mentioned for the particle statistics, i.e., the low number of arising agglomerates. 7.3.2 Effect of Different Simulation Parameters
In this section the predictions of the energy-based and momentum-based model are compared against each other for different simulation parameters. These are the effect the three sub-models, the diameter of the primary particles and the wall roughness. 7.3.2.1 Cumulative Effect of the Sub-Models
To reduce the number of influencing parameters, the investigations in the previous sections were carried out without the two-way coupling, the subgrid-scale model for the particles and the lift forces. In the context of the investigations on particle agglomeration using MAM presented in Section 7.2.2, it was concluded that the cumulative effect of these sub-models significantly reduces the agglomeration rate due to the following reasons: • The feedback of the particles on the continuous phase reduces the number of inter-particle collisions, whereas the number of agglomeration processes is hardly affected yielding a slightly higher agglomeration rate. • Taking the subgrid-scale model for the particles into account, the particle velocity fluctuations increase leading to a lower number of agglomeration processes. Additionally, a lower particle concentration is predicted in the direct vicinity of the wall resulting in a lower number of inter-particle collisions. As a result, the global agglomeration rate is perceptibly reduced. • The inclusion of the lift forces reduces the global agglomeration rate noticeably, since they enhance the migration of primary particles and agglomerates away from the region close to the walls, where the highest number of inter-particle collisions and agglomerations occurs. Based on these findings, in this section the investigations are extended towards the energy-based model and the comparison of both agglomeration models when the sub-models are taken into account. Again, the standard computational set-up for the large particles (dp = 12 µm) explained in Section 7.2.1 is applied assuming smooth channel walls. At a first glance, Figure 7.39 shows that both techniques predict similar trends of the agglomeration process as observed in Section 7.3.1, but with different rates when the sub-models are included. As revealed by Figure 7.39(a), the cumulative effect of the three sub-models reduces the number of pp accumulated particle-particle collisions Ncol and agglomeration possesses Nagp compared with the case without the sub-models depicted in Figure 7.36(c). However, different rates of reduction are found4 . This behavior can be more clearly explained by looking at Figure 7.40(a). It shows that if the sub-models are considered, the volume fraction is noticeably reduced in the direct vicinity 4
264
Note the different scaling of the axes used in Figures 7.36 and 7.39 for the number of agglomeration processes.
7. Results for Particle Agglomeration of the channel walls (y + ≤ 2) due to the migration of the particles out of the near-wall region leading to a higher volume fraction in the region 2 ≤ y + ≤ 15. The reason for the reduction of the particle volume fraction very close to the wall is the higher particle fluctuations in the wall-normal direction (see Figure 7.40(b)) due to the inclusion of the subgrid-scale model also leading to a higher number of particle-wall collisions. Thus, the particles colliding with the channel walls are more often reflected away from the walls than without the sub-models. Moreover, a lower number and smaller sizes of the arising agglomerates in the direct vicinity of the wall is predicted if the sub-models are switched on (compare Figure 7.39(d) with Figure 7.36(b)). ×10+7
0.05
EAM MAM
×10−2
EAM MAM
0.04
1.2 pp Nagp /Ncol
pp Nagp × 400, Ncol
1.6
0.8
Collision 0.4
0.03 0.02 0.01
Agglomeration 0.0
0.00 0
50
100
150
200
0
50
t UB /δ (a) 0.6
×10+5
0.5
100
150
200
t UB /δ (b) 10+4
EAM MAM
EAM MAM
10+3
Nag
Napp
0.4 0.3 0.2
10+2 10+1
0.1 0.0 0
50
100
t UB /δ (c)
150
200
10+0 2
3
4 5 6 7 8 Agglomerate Type
9
10
(d)
Figure 7.39: Results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for pp large particles d∗p = 6 × 10−4 : (a) time history of the accumulated number of particle-particle collisions Ncol pp and agglomeration processes Nagp , (b) time history of the agglomeration rate Nagp /Ncol , (c) time history of the total number of the agglomerated primary particles Napp and (d) number of agglomerates of the same type (2 = two-particle agglomerate, 3 = three-particle agglomerate, etc.) at a dimensionless time of 200 (closely-packed sphere model).
Figure 7.40(c) depicts the distribution of the concentration of the averaged number of inter-particle pp collisions hNcol i and agglomeration processes hNagp i along the channel width at the end of the 265
7.3 Comparison of Agglomeration Models simulations. Obviously, the cumulative effect of the sub-models noticeably enhances the number of inter-particle collisions in the central region due to the particle migration into this region, wheres the collision events are only slightly reduced in the near-wall region (see Figure 7.40(d)), where the highest numbers of inter-particle collisions and agglomeration processes occur. As visible in Figure 7.40(d), the agglomeration processes are significantly reduced in the near-wall region due to the higher particle fluctuations, especially in the wall-normal direction. Figure 7.40(c) also reveals that the number of agglomeration processes is drastically reduced along the entire channel width in comparison with the case without sub-models. That holds true for both agglomeration pp models leading to a significant reduction of the local agglomeration rate hNagp i / hNcol i along the channel width (not shown here for brevity).
9.00 6.00
×10−4
1.2
EAM MAM
×10−3
1.0 0.8
E
hαp i
vp′ vp′ /UB2
3.00
0.6
D
0.4 0.2
0.18 0.12 1
10
y+
100
1000
0.0 0.94
0.95
0.96
(a) 10+8 10+7 10+6 10+5
10+9
EAM MAM
10
+1
10+0
0.99
1
EAM MAM
10+8 10+7
pp hNcol i
10+6 10+5 10+4
10+3 10
0.98
(b)
10+4 +2
0.97
y/δ
10+3
pp hNcol i
hNagp i
10+2 10+1
hNagp i -1 -0.8 -0.6 -0.4 -0.2
0
y/δ (c)
10+0 0.2 0.4 0.6 0.8
1
0.9
0.92
0.94
0.96
0.98
1
y/δ (d)
Figure 7.40: Results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for large particles d∗p = 6 × 10−4 : (a) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed blue line denotes the global volume fraction of αp = 0.18 × 10−4 ), (b) dimensionless averaged
fluctuations of the particles in the wall-normal direction vp0 vp0 /UB2 close to the wall, (c) averaged concentrapp tion of the number of collision events hNcol i and agglomeration processes hNagp i along y/δ and (d) averaged pp concentration hNcol i and hNagp i close to the wall. The dashed red and black lines stand for the corresponding cases without the three sub-models. Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
266
7. Results for Particle Agglomeration This observation can be clarified based on the discussion presented in Section 7.3.1 regarding the relationships the magnitude of the normal relative velocity between the collision partners h between i − − + u1 − u2 · n and the normal component of the relative kinetic energy Ekin,r,n , the difference pp∗ ˆ of the van-der-Waals energy ∆EvdW (EAM) or the cohesive impulse f (MAM). The effect n,c
of higher particle fluctuations and hence the increased relative velocities between the collision partners on the predictions of both agglomeration models can be explained as follows: (i) Based + + on Eq. (7.18) the energies Ekin,r,n and ∆EvdW increase. However, Ekin,r,n increases more quickly pp∗ than ∆EvdW . (ii) According to Eq. (7.19) the cohesive impulse fˆn,c reduces. As a result, for both models the increase of the particle fluctuations reduces the probability of satisfying the agglomeration conditions leading to a lower number of agglomeration processes. It is worth noting that in the present simulations the level of the increase of the particle fluctuations is not constant along the channel width, but rather a higher level of increase is observed in the near-wall region than in the central region (not shown for brevity). Hence, two regions have to be distinguished. (i) In the near-wall region (∆y/δ < 0.02), a lower number of agglomeration processes is predicted by both models due to the stronger particle fluctuations. Note that although the energy-based model leads to slightly lower wall-normal particle fluctuations close to the walls than MAM (see Figure 7.40(b)), a higher number of agglomeration processes is predicted by MAM in comparison with EAM as visible in Figure 7.40(c). In other words, a lower probability of satisfying the agglomeration conditions is observed for EAM than for MAM due to the reasons mentioned before. Thus, near the channel walls the momentum-based model predicts a higher concentration of agglomeration processes than the energy-based model (see Figure 7.40(d)). (ii) In the central region, the energy-based model predicts slightly higher wall-normal particle fluctuations than MAM. However, this slight difference between the particle fluctuations leads to almost the same probability of satisfying the agglomeration conditions for both EAM and MAM. Thus, both the energy-based and the momentum-based model predict a similar concentration of agglomeration processes in the central region as visible in Figure 7.40(c). In summary, if the three sub-models are taken into account, both agglomeration models predict much lower numbers of inter-particle collisions and agglomeration processes, but with different rates. Nevertheless, Figure 7.40(c) shows that both models predict almost the same concentration of the inter-particle collisions. Contrarily, based on Figure 7.40(d) the energy-based model predicts a lower concentration of agglomeration processes in the near-wall region than MAM, whereas a similar numbers of agglomeration processes are observed in the central region. A possible reason for this behavior is the different level of particle fluctuations observed along the channel width. However, a general statement why the changes are different for both agglomeration models cannot be easily made due to the complex formulation of the agglomeration conditions. Compared to the case without sub-models, the lower number of agglomeration processes predicted by both models in the present simulations leads to a lower number of agglomerated primary particles Napp (compare Figure 7.36(a) with Figure 7.39(c)). As a result, also the number of agglomerates Nag predicted by both models decreases when the sub-models are taken into account, which is visible by comparing Figure 7.39(d) and Figure 7.36(b). In contrast to the results presented in the previous section, for the predictions including the sub-models the energy-based 267
7.3 Comparison of Agglomeration Models model delivers a lower number of agglomerates of the same class (see Figure 7.39(d)) than observed for MAM. This is directly related to the lower number of agglomeration processes predicted by EAM in comparison with MAM. Along the channel width the number of two-particle agglomerates predicted by the momentumbased model is higher than for the energy-based model (not shown here). Furthermore, the number of three-particle agglomerates computed by MAM is higher in the central region than near the walls, whereas no three-particle or larger agglomerates are predicted by EAM within ∆T ∗ = 200. The reason for this observation is the higher number of two-particle agglomerates predicted by MAM accompanied with weaker particle fluctuations than for EAM leading to a higher probability of satisfying the agglomeration conditions of MAM in the central region, if a two-particle agglomerate collides with a primary particle building up a three-particle agglomerate. At the end of the simulation including the three sub-models, the agglomeration rates are about 0.007% for the energy-based model and about 0.019% for the momentum-based model as depicted in Figure 7.39(b). The corresponding dimensionless frequency of the agglomeration process f˜agp decreases significantly from about 6.67×102 to about 0.05×102 for EAM and from about 5.39×102 to about 0.15 × 102 for MAM as listed in Table A.7. Thus, it can be concluded that the cumulative effect of the three sub-models is not negligible and should be considered for reliable predictions of the agglomeration process. Hence, it will be taken into account in the next simulations. 7.3.2.2 Effect of the Diameter of the Primary Particles
Previous results show a noticeable difference between the agglomeration rates predicted by both agglomeration models for dp = 12 µm. In this section the performance of the agglomeration models is analyzed for different diameters of the primary particles. For this purpose, the same number of primary particles as in the previous simulations is used (i.e., N0 = 6, 000, 000), but the diameter is reduced from 12 µm to 4 µm. Thus, the corresponding mass loading decreases from 3.32% to 0.12%, respectively. Here, the standard computational set-up is applied assuming smooth channel walls. Additionally, the sub-models (i.e., the feedback effect of the particles on the fluid, the subgrid-scale model for the particles and the lift forces) are taken into account due to their significant effect as concluded in Section 7.3.2.1. Note that the Langevin subgrid-scale model for the particles is applied in the present simulations. Compared to Figure 7.39 for large particles (dp = 12 µm), Figure 7.41 shows a similar trend concerning the number of collisions predicted by both models for small particles. However, the time history of the number of agglomeration processes and the agglomeration rate calculated by both models shows a different behavior when the diameter of the particles is reduced from 12 µm to 4 µm, i.e., EAM now predicts a higher agglomeration rate. Figure 7.41(c) displays the number of the agglomerated primary particles Napp as a function of time. This diagram also shows that the energy-based model predicts a higher number of agglomerated primary particles than the momentum-based model, and hence a higher number of agglomerates Nag (Figure 7.41(d)). Obviously, at the end of the simulation the number of two-particle agglomerates predicted by EAM is noticeably higher than for MAM, where the situation is vice versa for the three-particle agglomerates. In summary, this means that due to the different formulations of the agglomeration conditions of both models a successful agglomeration process using EAM more often occurs for 268
7. Results for Particle Agglomeration two-particle agglomerates than for MAM. If the diameter of the agglomerate exceeds a certain size (e.g., a primary particle collides with an existing agglomerate), the trend is reversed. ×10+5
2.0
EAM MAM
4.5
×10−2
EAM MAM
1.5 pp Nagp /Ncol
pp Nagp × 20, Ncol
6.0
3.0
Collision 1.5
1.0
0.5
Agglomeration
0.0
0.0 0
50
100
150
200
0
50
t UB /δ
1.2
200
(b) 10+4
EAM MAM
EAM MAM
10+3
Nag
0.9
Napp
150
t UB /δ
(a)
×10+4
100
0.6
10+2 10+1
0.3 0.0 0
50
100
t UB /δ (c)
150
200
10+0 2
3
4 5 6 7 8 Aagglomerate Type
9
10
(d)
Figure 7.41: Results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for pp small particles d∗p = 2 × 10−4 : (a) time history of the accumulated number of particle-particle collisions Ncol pp and agglomeration processes Nagp , (b) time history of the agglomeration rate Nagp /Ncol , (c) time history of the total number of the agglomerated primary particles Napp and (d) number of agglomerates of the same type (2 = two-particle agglomerate, 3 = three-particle agglomerate, etc.) at a dimensionless time of 200 (closely-packed sphere model). pp Figure 7.41(a) depicts the accumulated number of particle-particle collisions Ncol and those leading to agglomeration Nagp computed by both agglomeration models as a function of time. It is visible that although a higher number of two-particle agglomerates is computed by EAM (Figure 7.41(d)), both models predict almost the same number of inter-particle collisions. It is noteworthy that the reduction of the diameter of the primary particles implies larger inter-particle distances. Consequently, when comparing Figure 7.41(a) and Figure 7.39(a), the total number of the inter-particle collisions is about two orders of magnitude lower for the small particles in relation to the large particles. Correspondingly, the total number of agglomeration processes is
269
7.3 Comparison of Agglomeration Models significantly decreased. Furthermore, Figure 7.41(a) shows that although the number of accumulated inter-particle collisions only marginally deviates, the number of agglomeration processes predicted by EAM is noticeably higher than for MAM. This leads to a higher agglomeration rate computed by the energy-based model than observed for the momentum-based model as depicted in Figure 7.41(b). At the end of the simulation, the agglomeration rate is about 1.02% for the energy-based model and about 0.68% for the momentum-based model. Note that the corresponding values for the large particles (dp = 12 µm) are 0.007% and 0.019% for EAM and MAM, respectively. Hence, as expected, the agglomeration rates predicted by both agglomeration models increase when the diameter is reduced. In other words, the reduction of the diameter of the primary particles leads to a higher probability of satisfying the agglomeration conditions of both models. This observation can be explained based on Eq. (7.17) for the energy-based model and based on Eq. (7.16) for the momentum-based model. As mentioned before, for EAM the kinetic energy + Ekin,r,n decreases more quickly with a decreasing diameter of the particle than the difference of the van-der-Waals energy ∆EvdW . Furthermore, in the momentum-based model the cohesive pp∗ impulse fˆn,c is inversely proportional to the diameter of the primary particles and thus increases for dp = 4 µm in comparison with dp = 12 µm. The corresponding dimensionless frequency of the agglomeration process f˜agp for the small particles (dp = 4 µm) is about 0.32 × 102 for EAM and about 0.21 × 102 for MAM, whereas for the large particles (dp = 12 µm) it is about 0.05 × 102 for EAM and about 0.15 × 102 for MAM (see Table A.7). Finally, it is worth noting that the increase of the agglomeration rate and the decrease of the agglomeration frequency observed for the reduction of the particle size from dp = 12 µm to 4 µm is not a contradiction, but instead complementary. 7.3.2.3 Effect of the Wall Roughness
In the previous simulations the channel walls were assumed to be ideally smooth (i.e., Rz /δ = 0). As motivated before, most walls encountered in practical applications are rough and thus the effect of the wall roughness on the agglomeration process is studied in this section using both models. Here, rough walls with a mean roughness Rz = 10 µm (Rz /δ = 5 × 10−4 ) found in practice are assumed as reported in Section 7.2.1. The corresponding restitution and friction coefficients are listed in Table 7.1. The diameter of the primary particles is again set to dp = 12 µm. Note that in comparison to Section 7.3.2.1 the properties of the particles are not changed, and hence solely the wall model leads to differences in the results. In order to allow the particles and the fluid phase to adjust to the new situation before considering the agglomeration model, the coefficients listed in Table 7.1 are used during the dispersion period of the particles. Additionally, the three sub-models are taken into account relying on the Langevin subgrid-scale model for the particles. According to the wall roughness model by Breuer et al. (2012), the wall-normal unit vector at the contact point of a particle colliding with the wall strongly varies, since its orientation is adjusted leading to a random Gaussian distributed wall-normal unit vector. Thus, after the collision the particle trajectories spread largely. pp Figure 7.42(a) shows that the accumulated number of particle-particle collisions Ncol predicted by both models is significantly reduced when the wall roughness is considered. In order to explain 270
7. Results for Particle Agglomeration the reasons for the reduction of the number of inter-particle collisions, it is worth noting that the simulation data reveal that for both models the inclusion of the wall roughness enhances the particle fluctuations near the walls (not shown for brevity). Furthermore, the particle fluctuations predicted by the energy based-model are slightly higher than for the momentum-based model, especially close to the walls. For a detailed explanation for the reasons of the increase of the particle fluctuations the reader is referred to Breuer et al. (2012). ×10+7
0.05
EAM MAM
×10−2
0.04
1.2 pp Nagp /Ncol
pp Nagp × 400, Ncol
1.6
0.8
Collision 0.4
EAM MAM
0.03 0.02 0.01
Agglomeration 0.0
0.00 0
50
100
150
200
0
50
100
t UB /δ (a) 10+9
10+6 10+5
9.00 6.00
10+2 10
+1
EAM MAM
3.00
pp hNcol i
10+4 10+3
×10−2
hαp i
10+7
200
(b)
EAM MAM
10+8
150
t UB /δ
hNagp i 0.18 0.12
10+0 0.9
0.92
0.94
0.96
0.98
1
1
10
y/δ (c)
y
+
100
1000
(d)
Figure 7.42: Effect of the wall roughness on the results of the energy-based (EAM) and the momentum-based (MAM) agglomeration models for large particles d∗p = 6 × 10−4 : (a) time history of the accumulated number pp of particle-particle collisions Ncol and agglomeration processes Nagp , (b) time history of the agglomeration rate pp pp Nagp /Ncol , (c) averaged concentration of the number of collision events hNcol i and agglomeration processes hNagp i along y/δ close to the walls, (d) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed blue line denotes the global volume fraction of αp = 0.18 × 10−4 ), The dashed red and the black lines stand for the corresponding cases with smooth walls. Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
Since the highest number of inter-particle collisions and those leading to agglomeration occurs in the region close to the wall (y + ≤ 10), it is expected that higher particle fluctuations due to the 271
7.3 Comparison of Agglomeration Models wall roughness enhance the total number of inter-particle collisions in this region. However, a lower number of inter-particle collisions is observed near the walls as visible in Figure 7.42(c). This behavior is a direct consequence of the wall roughness, since it considerably alters the rebound behavior of the particles at the wall. Thus, the particles migrate away from the wall leading to a lower volume fraction in the direct vicinity of the wall as clearly visible in Figure 7.42(d). Hence, a lower number of inter-particle collisions is found there. A second but less important reason for the reduction of the inter-particle collisions is given by the different values of the restitution and friction coefficients for the particle-wall collisions for smooth and rough walls as listed in pp Table 7.1. The corresponding dimensionless frequency of the inter-particle collisions f˜col decreases 4 4 appreciably from about 7.72 × 10 (smooth wall) to about 4.13 × 10 (rough wall) for EAM and from about 7.67 × 104 (smooth wall) to about 4.23 × 104 (rough wall) for MAM as listed in Table A.7. This again confirms the significant reduction of the number of inter-particle collisions when the wall roughness is taken into account. Note that on the other hand, a very slight deviation between the dimensionless frequencies of the particle-particle collisions predicted by both models is observed. Furthermore, Figure 7.42(a) shows that the accumulated number of agglomeration processes Nagp predicted by both models is reduced in comparison with smooth walls. Note that although the rate of reduction of the agglomeration processes predicted by MAM is higher than for EAM in comparison with smooth walls, the number of agglomeration processes predicted by the momentumbased model is still higher than for EAM. Accordingly, if the wall roughness model is considered, the dimensionless frequency of the agglomeration processes f˜agp decreases from about 0.05 × 102 to about 0.03 × 102 for EAM and from about 0.15 × 102 to about 0.05 × 102 for MAM (see Table A.7). In summary, the inclusion of the wall roughness results in a significant reduction of the number of particle-particle collisions and a noticeably lower number of agglomeration processes leading to lower agglomeration rates predicted by both models. However, for EAM the trend is similar but less pronounced than for MAM. As displayed in Figure 7.42(b), for the case with rough walls the momentum-based model predicts a higher agglomeration rate than the energy-based model. At the end of the simulations, the agglomeration rate is about 0.008% for the energy-based model and about 0.012% for the momentum-based model. Note that the corresponding values for the smooth wall are 0.007% and 0.019%, respectively. As expected, the inclusion of the wall roughness noticeably decreases the agglomeration rate predicted by MAM, whereas surprisingly it is slightly increased for EAM in comparison with smooth walls. A possible reason for this unexpected behavior is that for the energy-based model the higher number of inter-particle collisions in the central region of the channel due to the roughness effect enhances the agglomeration rate more significantly than the reduction of the number of agglomeration processes in the near-wall region. Figure 7.42(c) depicts the local distribution of the concentration of inter-particle collisions and agglomeration processes along the channel width in the near-wall region averaged over the time when the agglomeration model is taken account. Obviously, in comparison to the case with smooth walls the concentrations of the inter-particle collisions and hence also the agglomeration processes are reduced near the walls due to the migration of the particles away from the walls as mentioned before. On the other hand, the migration of the particles increases the volume fraction in the 272
7. Results for Particle Agglomeration central region as shown in Figure 7.42(d). This leads to a slightly higher number of particle-particle collisions and agglomeration processes outside the near-wall region for the rough wall compared to the smooth wall (not shown for brevity). However, as explained before, a lower number of accumulated inter-particle collisions is observed for the rough wall in comparison with the smooth wall. That implies that the reduction of the number of inter-particle collisions in the region near the walls overwhelms the slight increase in the central region of the channel. Figure 7.42(c) also shows that in the near-wall region the momentum-based model predicts a sightly higher concentration of the agglomeration processes than the energy-based model, while this behavior is reversed outside this area (not shown for brevity). To explain this difference, it should be again recalled that the energy-based model predicts slightly higher particle fluctuations near the walls than MAM. Thus, the observation concerning the different distributions of the agglomeration processes for both models can be explained based on the discussion presented in Section 7.3.2.1, where a similar behavior was observed for the case with smooth walls, but of course with different rates owing to the roughness effect. On average, the number of agglomeration processes predicted by both models for rough walls is lower than for smooth walls. For both types of walls the energy-based model predicts a lower number of agglomeration processes than MAM as stated before.
273
CHAPTER 8
RESULTS FOR DROPLET COALESCENCE
Primary Break-up
Nozzle
Needle
Liquid film
Db
Collision and Coalescence
φ
Evaporation
Nozzle
Lb Seco
ψ 2
ndar
y Br eak-
Sp (t )
Y
up
X Z
Characterization of a solid-cone fuel spray injected through a single-hole nozzle.
8.1
Spray Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1
Primary Break-up Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.1.2
Spray Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
8.1.3
8.1.4 8.2
277
8.1.2.1
Model by Reitz and Bracco (1979) . . . . . . . . . . . . . . . . . . . . . 279
8.1.2.2
Model by Arai et al. (1984) . . . . . . . . . . . . . . . . . . . . . . . . . 279
Spray Tip Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 8.1.3.1
Model by Hiroyasu and Arai (1980) . . . . . . . . . . . . . . . . . . . . 279
8.1.3.2
Model by Mirza (1991) . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Characteristic Mean Diameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Model for Droplet Injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
282
8.2.1
Injected Fuel Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
8.2.2
Initial Droplet Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.2.3
Initial Droplet Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8.2.4
Initial Droplet Diameter (Primary Break-up) . . . . . . . . . . . . . . . . . . . . 286 8.2.4.1
Rosin-Rammler Distribution . . . . . . . . . . . . . . . . . . . . . . . . 286
8.2.4.2
Log-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 288
8.2.4.3
Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
8.2.4.4 8.3
Inter-Impingement Spray System . . . . . . . . . . . . . . . . . . . . . . . . . . . .
292
8.3.1
Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
8.3.2
Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 8.3.2.1
8.4
Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Properties of the Droplets . . . . . . . . . . . . . . . . . . . . . . . . . 294
8.3.3
Model Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294
8.3.4
Summary of Key Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
Validation of the Composite Collision Outcome Model . . . . . . . . . . . . . . . . 8.4.1
8.4.2
296
Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 8.4.1.1
Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
8.4.1.2
Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
8.4.1.3
Properties of the Droplets . . . . . . . . . . . . . . . . . . . . . . . . . 299
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299 8.4.2.1
Spray Tip Penetration . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
8.4.2.2
Collision Regimes
8.4.2.3
Effect of Different Parameters on the Spray Tip Penetration . . . . . . . 303
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
8.4.2.3.1 Initial Diameters of the Injected Droplets . . . . . . . . . . . 304 8.4.2.3.2 Break-up Length
. . . . . . . . . . . . . . . . . . . . . . . . 305
8.4.2.3.3 Interphase Coupling . . . . . . . . . . . . . . . . . . . . . . . 306 8.4.2.3.4 Pressure Gradient and Added Mass Forces . . . . . . . . . . . 307 8.4.2.3.5 Subgrid-Scale Model for the Continuous Phase
. . . . . . . . 308
8.4.2.3.6 Effect of Coalescence . . . . . . . . . . . . . . . . . . . . . . 309 8.4.3
Summary of Key Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
8 Results for Droplet Coalescence This chapter is concerned with numerical results related to the droplet coalescence in the framework of the hard-sphere model. According to the classification made in Section 4.1.2, the focus is only on the collision outcome of surface-tension dominated droplets. In this context a detailed verification and validation study of the improved composite collision outcome model (see Section 4.2.5) is carried out. Here, the four-way coupled Euler-Lagrange simulation framework is applied to different test cases related to spray systems. For this purpose, this chapter is organized in the following manner. First, an overview of the most important models for the characterization of single sprays related to the present thesis is given in Section 8.1. Then, a model for the droplet injection of solid-cone spays used in the present simulations is introduced in Section 8.2. Afterwards, the performance of the composite model is verified in an inter-impingement spray system (two crossing water jets) in Section 8.3. Lastly, the composite model is validated against experimental data of a non-evaporating diesel spray with high injection pressure in Section 8.4. Note that most of the results presented in this chapter were published in Almohammed and Breuer (2018). 8.1 Spray Characterization
In the present study the focus is on conical sprays, since they are encountered in a wide range of applications. The fuel injection system used in this study is schematically depicted in Figure 8.1. Here, a high pressure-driven liquid jet is injected through a single-hole nozzle into a quiescent gas environment (or chamber). The most important parameters characterizing this spray are briefly summarized next. More details can be found, for example, in Lefebvre (1988) and Brenn (2011). Note that these spray characteristics are required for the droplet injection model presented next. Primary Break-up
Nozzle
Needle
Liquid film
Db
Collision and Coalescence
φ
Evaporation
Nozzle
Lb Seco
ψ 2
ndar
y Br eak-
Sp (t )
Y
up
X Z
Figure 8.1: Characterization parameters of an oblique (angle φ) solid-cone fuel spray of the angle ψ injected through a single-hole nozzle into a quiescent gas environment. Here, the global coordinate system is (x,y,z).
277
8.1 Spray Characterization 8.1.1 Primary Break-up Length
At the nozzle exit (i.e., pressure atomizer), the liquid ligament atomizes into fragments of different sizes and shapes and eventually into droplets. This fragmentation process of the injected liquid ligament into fragments is commonly referred to as primary break-up. As depicted in Figure 8.1, it is typically assumed that the jet is fully atomized up to the break-up length Lb . Since the break-up process is excluded in the present thesis, a droplet atomization model is required. In the present study, the primary break-up length is modeled using the empirical correlation by Levich (1962): Lb = kb
s
ρd DN , ρf
(8.1)
where the constant kb depends on the nozzle design and DN is the inner diameter of the nozzle. ρd and ρf stand for the density of the liquid droplets and the surrounding gas environment (fluid phase), respectively. This correlation implies that the increase of the density of the continuous phase ρf (e.g., high back-pressure in the combustion chamber) reduces the distance of the primary break-up. In the framework of the droplet injection model used for the simulation of spray systems in the present thesis (see Section 8.2), the break-up length is required to determine the diameter of the injection area Db depicted in Figure 8.1. Here, the primary break-up is modeled by assuming that the droplets injected into the computational domain are fully atomized and have a spherical shape. Based on this assumption, the droplets are released at the break-up length. Assuming that Db DN , the diameter of the injection area Db can be written as: !
ψ , Db = 2 Lb tan 2
(8.2)
where the break-up length Lb is given by Eq. (8.1) and ψ denotes the conical spray angle determined as explained next. 8.1.2 Spray Angle
As depicted in Figure 8.1, for a solid-cone fuel spray the angle ψ is defined as the angle between two straight lines, which start from the nozzle exit and are tangents to the spray morphology (see, e.g., Mart´ınez-Mart´ınez et al., 2010). Typically, the angle of a solid-cone spray is measured from the nozzle axis at a penetration length of about 60% of the spray tip penetration (see, e.g., Pastor et al., 2001). As mentioned before, in addition to the break-up length the spray angle is required to compute the diameter of the injection area. For the simulation of spray systems, in the present study the mean spray angle is taken from the corresponding experimental data by Gao et al. (2009). However, if the mean value of this conical angle is not known, it can be determined based on empirical correlations. In the following two common correlations are presented. A review on empirical correlations describing the spray angle can be found, for example, in Santos and Moyne (2011). 278
8. Results for Droplet Coalescence 8.1.2.1 Model by Reitz and Bracco (1979)
Reitz and Bracco (1979) proposed the following empirical correlation for the spray angle: ψ tan 2
!
4π = A
ρf ρd
!1/2
√ 3 . 6
(8.3)
Here, the constant A is given by: lN A = 3 + 0.28 DN
!
(8.4)
,
where lN is the length of the nozzle hole. As visible in relation (8.3), this model depends only on the nozzle design (i.e., lN and DN ) and density ratio ρf /ρd , but not on the injection parameters. 8.1.2.2 Model by Arai et al. (1984)
Taking the injection conditions into account, Arai et al. (1984) determined the spray angle as follows: 2 ψ ρf ∆p DN = 0.025 2 µ2f
!1/4
,
(8.5)
where ∆p refers to the mean pressure difference across the nozzle and is given by:
∆p = pinj − pback ,
(8.6)
where pinj stands for the mean injection pressure and pback is the constant chamber pressure (commonly called the back-pressure). µf is the dynamic viscosity of the chamber gas at a constant temperature Tf . 8.1.3 Spray Tip Penetration
In the literature there is no general definition of the spray tip penetration Sp (t) displayed in Figure 8.1 (also termed penetration depth) in either computational or experimental studies. In the present study the definition of the spray tip penetration is the point behind which 99% of the mass of the solid-cone spray is located (see, e.g., Beck and Watkins, 2004). The tip penetration of fuel sprays is described based on fittings of experimental data, which are used to validate numerical simulations. In the following two common correlations for the spray tip penetration are presented. A review on empirical correlations describing the spray tip penetration can be found, for example, in Santos and Moyne (2011). 8.1.3.1 Model by Hiroyasu and Arai (1980)
Hiroyasu and Arai (1980) divided the injection process of fuel sprays into two zones. In the first zone it is assumed that the droplets move with a constant velocity and hence Sp (t) ∝ t, whereas √ in the second zone Sp (t) ∝ t. This “classical” two-line fit for the penetration depth has the 279
8.1 Spray Characterization following form:
Sp (t) =
kd
2∆p ρd
2.95
!1/2
2∆p ρf
t
!1/4 q
for 0 ≤ t < tb ,
(8.7)
DN t for t ≥ tb ,
where the pressure difference ∆p is given by Eq. (8.6) and t is the time measured from the start of the fuel injection. The break-up time tb is defined as the time, at which the spray penetration Sp (tb ) is equal to the break-up length Lb . Beyond this time, it is assumed that Lb has an approximately constant value, while the spray continues to penetrate into the surrounding gas environment (Mirza, 1991; Yule and Filipovi´c, 1992). In the formulation of the correlation by Hiroyasu and Arai (1980) the break-up time is given by:
4.351 ρd DN q tb = , kd2 ρf ∆p
(8.8)
where kd stands for the discharge coefficient of the nozzle. In the experiment by Hiroyasu and Arai (1980) a discharge coefficient of kd = 0.39 was used and hence the pre-factor in relation (8.8) is about 28.7. Note that kd is an empirical parameter depending on the nozzle design and the injection conditions. 8.1.3.2 Model by Mirza (1991)
The “classical” correlation for the spray jet penetration of Hiroyasu and Arai (1980) was modified by Mirza (1991) yielding an excellent single-line fit to the entire spray length. In this model the modification is based on the introduction of an additional term including an adjustable constant, which should be proportional to, but not necessarily equal to, the break-up time (Mirza, 1991; Yule et al., 1991; Yule and Filipovi´c, 1992). For non-evaporation fuel sprays the hyperbolic function proposed by Mirza (1991) is given by: Sp (t) = k1
∆p ρf
!1/4
h
tanh (k2 t)0.6
iq
DN t ,
(8.9)
where the adjustable coefficients were experimentally determined by Mirza (1991) and have the values k1 = 3.8 and k2 = 4.1 × 103 . The pressure difference ∆p is given by Eq. (8.6). Note that the correlation by Mirza (1991) is more attractive than the common two-line fit correlation (e.g., by Hiroyasu and Arai (1980)), since the single equation (8.9) smoothly blends the full spray (i.e., the first and the second zone in the classical model). 8.1.4 Characteristic Mean Diameters
After the primary break-up, most sprays consist of fully atomized droplets of various sizes (i.e., poly-disperse distribution). Assuming spherical liquid droplets, different mean diameters are used to characterize the change of droplet sizes during the injection process. The two principal 280
8. Results for Droplet Coalescence mechanisms causing changes in the droplet diameters are the fragmentation and the enlargement of the primary droplets due to break-up and coalescence, respectively. The poly-disperse droplet sizes of spray systems are described using the distribution functions fr (dd ) and the cumulative distribution functions Fr (dd ), where the subscript r refers to the distribution type: r = 0 for the number, 1 for the length, 2 for the surface and 3 for the volume or mass distribution. In the present study only the number distribution function f0 (dd ) is relevant. Assuming a constant density of the droplets, the common definition for the characteristic mean diameter Dnm is expressed in terms of the number distribution function f0 (D) as follows (see, e.g., Loth et al., 2004): Dnm =
"R∞ R 0∞ 0
f0 (D) Dn dD f0 (D) Dm dD
#1/(n−m)
(8.10)
,
where f0 (D) stands for the number fraction of droplets in the diameter interval (D, D + dD). R Here, f0 (D) ≥ 0 has to satisfy the following condition 0∞ f0 (D) dD = 1. The exponents n and m appearing in Eq. (8.10) are typically positive integers chosen based on the characteristic mean diameter Dnm determined. In a fully atomized spray system (i.e., disperse phase) the characteristic mean diameters can be determined by dividing the droplets into Nc classes as depicted in Figure 8.2. If each droplet class (superscript k) has a mean diameter Dk , the discrete version of the characteristic mean diameter Dnm given by Eq. (8.10) reads: Dnm =
N Pc
k=1 N Pc
k=1
n
f0,k Dk
m f0,k Dk
(8.11)
,
where f0,k = Nd,k /Nd is the number fraction of the k th droplet class defined as the number of droplets counted in this class Nd,k to the total number of droplets Nd . The symbol Nc stands for the total number of droplet classes considered. f0,k
f0 (dd )
f0 (dd,i )
N1
Nk dd,i
Nc
dd
Figure 8.2: Schematic representation of the continuous number distribution function f0 (dd ) and its discrete version obtained by dividing the spectrum of the droplet diameters into Nc classes.
281
8.2 Model for Droplet Injection In the present study two different mean diameters are used to characterize the droplet sizes:
Ê Sauter Mean Diameter The Sauter mean diameter (SMD) was introduced by Sauter (1926). It is defined as the diameter of a droplet having the same volume-to-surface ratio as that of the entire spray. In a fully atomized spray system the SMD denoted D32 is obtained based on the discrete version of the characteristic mean diameter given by Eq. (8.11) by setting the exponents n = 3 and m = 2. Thus, the mathematical expression of the Sauter mean diameter reads: D32 =
N Pc
k=1 N Pc k=1
3
f0,k Dk 2
(8.12)
.
f0,k Dk
By substituting f0,k = Nd,k /Nd into Eq. (8.12), the Sauter mean diameter D32 can be defined by the following expression: D32 =
N Pc
k=1 N Pc
k=1
3
Nd,k Dk
(8.13)
.
2 Nd,k Dk
Ë Arithmetic Mean Diameter The arithmetic mean diameter of the droplets denoted D10 is obtained from Eq. (8.11) by setting the exponents n = 1 and m = 0 and thus it is expressed as: D10 =
N Pc
f0,k Dk
k=1 N Pc
k=1
(8.14)
. f0,k
where again f0,k is the number fraction of the class k of the mean diameter Dk and Nc is the total number of classes considered. By substituting f0,k = Nd,k /Nd and Eq. (8.14), the arithmetic mean diameter D10 is computed as follows: D10 =
Nc 1 X Nd,k Dk , Nd k=1
N Pc
k=1
f0,k = 1 into
(8.15)
where again Nd,k is the number of droplets counted in the k th class and Nd is the total number of droplets. 8.2 Model for Droplet Injection
As mentioned previously, only solid-cone sprays are considered in the present thesis. The numerical set-up for the droplet injection used in the present study is schematically depicted in Figure 8.3. 282
8. Results for Droplet Coalescence In the following the injected mass, the position, the velocity and the diameter of the injected droplets are modeled. y y
Ub
Spray cone
ϕi yd,i
Injection area
ri b
N
z
zd,i
x
b
D
Nozzle exit
(at break-up length)
Db
Lb
Y
Db
ψ 2
z
ud,i Ub
eˆi ψi
X Z
x
Figure 8.3: Position and velocity of the injected spherical droplets at the primary break-up length in a solidcone spray. Here, the global coordinate system is (x,y,z), whereas the local one at the origin of the injection area is denoted (x, y, z).
8.2.1 Injected Fuel Mass
Analog to many previous studies on non-evaporating sprays, it is assumed that the droplets are injected into a constant-volume chamber at a constant back-pressure and ambient temperature (quiescent gas). The total fuel mass injected at the nozzle exit during the time of the injection process ∆tinj is determined by: minj =
Z ∆tinj 0
m ˙ inj (t) dt ,
(8.16)
where m ˙ inj (t) is the transient mass flow rate of the fuel injected at the nozzle exit. In most studies it is assumed that the needle of the nozzle is fully opened and hence the mass flow rate m ˙ inj is constant during the injection process. Thus, the total injected fuel mass at the nozzle exit reads: minj = m ˙ inj ∆tinj ,
(8.17)
where ∆tinj stands for the total injection time. The fuel injection flow rate m ˙ inj is given by: m ˙ inj =
π 2 ρd Uinj DN , 4
(8.18)
where Uinj is the mean velocity of the liquid jet at the nozzle exit. Note that the assumption of a fully opened needle implies that the mean injection pressure pinj and the back-pressure of the 283
8.2 Model for Droplet Injection chamber pback are constant during the injection period. Hence, the jet velocity can be determined relying on the orifice relation (see, e.g., Madsen, 2006; Kim et al., 2009; Pawar et al., 2012, 2015): Uinj = kd
s
2 ∆p ρd
with ∆p = pinj − pback ,
(8.19)
where kd denotes the discharge coefficient of the nozzle and typically has a value between 0.6 and 0.8 (see, e.g., Madsen, 2006; Jung and Assanis, 2001). In the simulations of non-evaporating sprays the total fuel mass ∆minj injected at the nozzle exit within the time interval (t, t + ∆t) has to be specified. Assuming steady-state injection conditions, the total fuel mass at each time step is determined as follows: ∆minj = m ˙ inj ∆t ,
(8.20)
where ∆t is the time step used in the simulation. By substituting Eq. (8.18) into Eq. (8.20), the expression for the total mass injected at the primary break-up position within the time interval (t, t + ∆t) has the following form: ∆minj =
π 2 ρd Uinj DN ∆t . 4
(8.21)
8.2.2 Initial Droplet Position
As depicted in Figure 8.3, in the present droplet injection model it is assumed that fully atomized spherical droplets are injected into the computational domain with random initial positions located inside the injection area of the diameter Db . As mentioned in Section 8.1.1, the droplets are released at the break-up length and hence for Db DN the diameter Db of the injection area can be expressed by substituting Eq. (8.1) into Eq. (8.2) as follows: Db = 2 kb
s
!
ψ ρd tan DN . ρf 2
(8.22)
As depicted in Figure 8.3, besides the global coordinate system (x,y,z) a local coordinate system (x, y, z) is introduced at the center of the injection area. Thus, with respect to the global coordinate system the initial position of the injected droplet denoted by the subscript i reads: xd,i = ax , yd,i = ay + (ri cos ϕi ) ,
(8.23)
zd,i = az + (ri sin ϕi ) , where ax , ay and az are the components of the relative vector between the origin of the global coordinate system (x,y,z) and the center of the injection area. The injection radius ri (i.e., droplet position relative to the local coordinate system) lies within the range 0 ≤ ri ≤ Db /2 (see Figure 8.3) and is randomly chosen by the following relation: ri = 284
q 1 Db ξi , 2
(8.24)
8. Results for Droplet Coalescence where the injection diameter is given by Eq. (8.22) and ξi is a uniformly distributed random number with a zero mean and a unit standard deviation. It is worth noting that taking the square √ root of a random number ξi yields a uniform distribution of the generated droplet positions, whereas using the random number instead leads to random positions concentrated in the central region of the nozzle (see, e.g., Weisstein, 2017). The random angle ϕi appearing in Eq. (8.23) has to be chosen within the range 0 ≤ ϕi ≤ 2π as follows ϕi = 2π ξi . If the spray is additionally rotated about the z-axis in counterclockwise direction by the angle φ (i.e., oblique spray), the location of the ith injected droplet is expressed as follows: xd,i = ax − (ri cos ϕi ) sin φ ,
yd,i = ay + (ri cos ϕi ) cos φ ,
(8.25)
zd,i = az + (ri sin ϕi ) . 8.2.3 Initial Droplet Velocity
It is commonly known that the penetration velocity in the break-up zone denoted Ub is considerably less than the jet velocity at the nozzle exit Uinj due to the drag force acting on the liquid ligament injected from the nozzle (see, e.g., Mirza, 1991; Yule et al., 1991; Yule and Filipovi´c, 1992). At the break-up length, where the entire jet is assumed to be fully atomized and hence no liquid ligament is present, the initial velocity of the injected droplet ud,i is given by the following expression: ud,i = Ub eˆi ,
(8.26)
where Ub and eˆi are the velocity magnitude and the unit vector of the injected droplets as depicted in Figure 8.3. In the present injection model it is assumed that the injected droplets have the magnitude Ub of the characteristic break-up zone velocity. Mirza (1991) expressed the magnitude of the break-up zone velocity in terms of the jet velocity at the nozzle exit Uinj as follows: Ub = kv Uinj ,
(8.27)
where the coefficient kv takes the reduction of the droplet velocity within the primary break-up zone into account and has a value of kv ≤ 1 depending on the injection conditions. For example, Mirza (1991) found that for pinj = 21 MPa and DN = 0.25 × 10−3 m the coefficient kv ≈ 0.6 led to the best agreement with the experimental data. The direction of the injected droplet is randomly sampled inside a three-dimensional cone, whose spreading angle is equal to the spray angle ψ. Thus, this direction is defined based on the random unit vector eˆi appearing in Eq. (8.26). To ensure that the released droplet i lies with in the solid-cone spray, the corresponding random angle ψi is defined as follows: ψ (2 ξi − 1) with − ψ/2 ≤ ψi ≤ ψ/2 . 2 Thus, the components of the unit vector eˆi of the injected droplet read: ψi =
(8.28)
eˆx,i = cos ψi , eˆy,i = eˆz,i =
q
1 − cos2 ψi cos ϕi = sin ψi cos ϕi ,
q
(8.29)
1 − cos2 ψi sin ϕi = sin ψi sin ϕi .
285
8.2 Model for Droplet Injection If the spray is additionally rotated in counterclockwise direction by the angle φ about the z-axis (i.e., oblique spray), the components of the unit vector of the injected droplet can be expressed as follows: eˆx,i = cos ψi cos φ − (sin ψi cos ϕi ) sin φ , eˆy,i = cos ψi sin φ + (sin ψi cos ϕi ) cos φ ,
(8.30)
eˆz,i = sin ψi sin ϕi . In summary, by multiplying the unit vector eˆi with Ub , the three Cartesian velocity components required for the initialization of the injected droplet i are obtained. 8.2.4 Initial Droplet Diameter (Primary Break-up)
As mentioned before, no atomization model is used in the present study. Instead, the diameters of injected droplets are determined using the number distribution functions, which are mostly based on curve fittings of experimental data. Assuming that the jet is fully atomized at the primary break-up length, the basic idea of the present injection model is that at each time step the initial diameters of the released droplets are randomly generated based on a prescribed distribution P function until the sum of the droplet masses md,i is equal to the mass ∆minj injected at one time step. In other words, this procedure ensures that for each time step different droplet diameters are calculated. As explained in Section 8.2.2, for steady-state injection conditions the total fuel mass injected ∆minj (see Eq. (8.21)) is constant at each time step. Since the droplet mass md,i can be expressed in terms of the droplet density ρd and the randomly generated diameters dd,i , the condition for the random generation of the initial droplet diameters reads: ∆minj
nd π X = ρd d3d,i , 6 i=1
(8.31)
where nd refers to the total number of droplets injected into the computational domain within the current time step until the fuel mass ∆minj is reached. By substituting Eq. (8.21) into Eq. (8.31), the criterion required for the random generation of the initial diameters of the injected droplets using the number distribution functions listed below has the following form: nd X i=1
d3d,i =
3 2 D Uinj ∆t . 2 N
(8.32)
It is worth noting that randomly generated droplet diameters can be divided into Nc classes to ensure desired distribution functions and thus also characteristic mean diameters (e.g., SMD) as described in Section 8.1.4. In the following, four droplet distribution functions used in the present study to determine the diameters of the injected droplets are briefly summarized. 8.2.4.1 Rosin-Rammler Distribution
The classical distribution proposed by Rosin and Rammler (1933) is often expressed by the cumulative volume (or mass) distribution for the droplet sizes, which has the following form: (
dd F3 (dd ) = 1 − exp − D 286
!q )
,
(8.33)
8. Results for Droplet Coalescence where q is the spreading parameter indicating the distribution width. F3 (dd ) is the cumulative volume fraction of the droplets, whose diameters are less than dd . D stands for the Rosin-Rammler mean diameter, which by definition is the diameter at a cumulative mass fraction of F3 (D) = 63.2%. This cumulative volume distribution function was widely used for spray simulations applying the parcel approach due to its mathematical simplicity (see, e.g., Yoon et al., 2007; Montazeri et al., 2015; Pawar et al., 2015). However, in the present study the fully atomized jet is described using primary droplets (i.e., no parcels) and thus only the number distribution function f0 (dd ) by Rosin and Rammler (1933) is relevant, which has the following form (see, e.g., Crowe et al., 1998; Loth et al., 2004): 1 q f0 (dd ) = Γ (1 − 3/q) D
dd D
!q−4
(
dd exp − D
!q )
(8.34)
, R
where Γ(α) stands for the gamma function defined for any value α as Γ(α) = 0∞ xα−1 exp(−x) dx. Note that for a positive integer α the gamma function is given by Γ(α) = (α − 1)!. In general, the characteristic mean diameters for the distribution by Rosin and Rammler (1933) can be expressed as (Loth et al., 2004): !
n−3 D Γ 1+ q ! = m−3 m D Γ 1+ q n
Dnm
(8.35)
Using relation (8.35), the Rosin-Rammler mean diameter D is defined in terms of the spreading parameter q and the Sauter mean diameter D32 by the expression: D = Γ (1 − 1/q) D32 ,
(8.36)
In addition, the Rosin-Rammler mean diameter can also be expressed based on Eq. (8.35) in terms of the arithmetic mean diameter D10 and the spreading parameter q as follows: D=
(
)
Γ (1 − 3/q) D10 . Γ (1 − 2/q)
(8.37)
By substituting Eq. (8.36) into Eq. (8.37), the arithmetic mean diameter D10 can be written as a function of D32 and the spreading parameter q as follows: D10 =
(
)
Γ (1 − 1/q) Γ (1 − 2/q) D32 . Γ (1 − 3/q)
(8.38)
According to the number distribution function given by Eq. (8.34), the droplet diameter dd can not be expressed in terms of the corresponding number fraction f0 (dd ). Thus, a classical inversion method can not be applied. However, it is possible to randomly generate the droplet diameters dd,i (see Figure 8.2) required for Eq. (8.31) using a rejection sampling procedure (von Neumann, 1951) also applied in the CFD-code OpenFOAM, which includes the following steps: 287
8.2 Model for Droplet Injection
Ê A random droplet diameter is generated in the range dd,min ≤ dd,i ≤ dd,max by the following relation:
dd,i = dd,min + ξi (dd,max − dd,min ) ,
(8.39)
where ξi ∈ [0, 1] is a uniformly distributed random number with a zero mean and a unit standard deviation. Here, dd,min and dd,max are the minimum and maximum droplet diameter in the number distribution, respectively.
Ë The corresponding number fraction f0 (dd,i ) is determined based on Eq. (8.34). Ì An additional uniformly distributed random number ξj ∈ [0, f0,max ] is generated. Í If the condition ξj ≤ f0 (dd,i ) is satisfied, this droplet diameter is considered. Otherwise, it is rejected and the previous steps are repeated unit this criterion is fulfilled.
Figure 8.4 shows the distribution function (8.34) for q = 3.2 in comparison with the distribution functions used for the validation study presented later on in Section 8.4 for D32 = 55 µm taken from the experimental data by Gao et al. (2009).
Number Distribution Function
0.07
Villermaux and Bossa (2009) Rosin and Rammler (1933) Amsden et al. (1989) Log-Normal
0.06 0.05 0.04 0.03 0.02 0.01 0 0
20
40
60
80
100
120
140
160
Droplet Diameter (µm) Figure 8.4: Distribution function of droplets with a Sauter mean diameter of D32 = 55 µm based on (red line) gamma distribution by Villermaux and Bossa (2009) (green line) Rosin and Rammler (1933), (black line) exponential distribution by Amsden et al. (1989) and (blue line) log-normal distribution.
8.2.4.2 Log-Normal Distribution
The log-normal distribution function is given by (see, e.g., Feingold and Levin, 1986; Paloposki, 1994; Loth et al., 2004):
288
1 1 ln f0 (dd ) = √ exp − 2 2π σln dd
2 dd /Dln σln
,
(8.40)
8. Results for Droplet Coalescence where σln = ln(σ0 ). Here, σ0 is a variance parameter referring to the distribution width and has to be always positive. The symbol Dln stands for the logarithmic mean diameter. The characteristic mean diameter for the log-normal distribution reads (Loth et al., 2004): n
o
2 Dnm = Dln exp (n + m) σln /2 .
(8.41)
Using relation (8.41), the logarithmic mean diameter can be expressed based on the Sauter mean diameter (n = 3 and m = 2) or the arithmetic mean diameter (n = 1 and m = 0) as follows: Dln =
D10 D32 = . 2 2 /2) exp (5 σln /2) exp (σln
(8.42)
Thus, for this distribution the arithmetic mean diameter in terms of the SMD reads:
2 D10 = D32 exp −2 σln .
(8.43)
Figure 8.4 shows the log-normal distribution function (8.40) for σln = ln(1.8) in comparison with other distribution functions used in this study. As visible, the log-normal distribution tends to predict a higher number of larger droplets than the Rosin-Rammler distribution. According to Yoon et al. (2007) the droplet diameters dd,i based on the log-normal distribution function given by Eq. (8.40) can be randomly generated as follows: h√ i (8.44) dd,i = Dln exp 2 σln erf−1 (2 ξi − 1) ,
where ξi a uniformly distributed random number with a zero mean and an unit standard deviation. The error function erf(x) for x > 0 is given by: x 2 Z erf(x) = √ exp(−t2 ) dt . π
(8.45)
0
The inverse of the error function erf −1 (x) appearing in Eq. (8.44) is determined based on the approximation by Giles (2011) presented in Appendix B.3. 8.2.4.3 Gamma Distribution
The probability density function (PDF) of the gamma distribution for a variable x ≥ 0 has the following form: f (x) =
β α α−1 x exp (−β x) , Γ(α)
(8.46)
where α and β stand for the shape and the rate parameters. The general formulation of the gamma distribution given by Eq. (8.46) is required to formulate the distribution functions presented next. Thus, it is useful to explain the general procedure used to randomly generate the PDF of the gamma distribution. It is clearly visible that relation (8.46) can not be analytically solved for the variable x (i.e., x = f (x, α, β)). For this reason, Opfer et al. (2012) proposed a numerical method to randomly generate the gamma distribution using a set of uniformly distributed random 289
8.2 Model for Droplet Injection numbers. As depicted in Figure 8.5, the cumulative distribution function (CDF) for x ≤ xi reads R F (xi ) = 0xi f (t) dt. If the shape parameter α is a positive integer, F (xi ) has the following series expansion (Papoulis and Pillai, 1985): F (xi ) = 1 −
(β xi )n exp(−β xi ) . n! n=0
α−1 X
(8.47)
Since the cumulative distribution function F (xi ) varies between 0 and 1, the value xi can be determined based on the following relation using a uniformly distributed random number ξi ∈ [0, 1] (Opfer et al., 2012): ξi = 1 − F (xi ) .
(8.48)
By substituting Eq. (8.47) into Eq. (8.48), it can be written as: ξi =
(β xi )n exp(−β xi ) or n! n=0
α−1 X
(β xi )n exp(−β xi ) − ξi = 0 . n! n=0
α−1 X
(8.49)
This relation means that for a given random number ξi the value xi is a root of Eq. (8.49). In other words, a random number ξi is first chosen and then the value xi is varied in the range xi ∈ [0, ∞] until a pre-defined convergence criterion is reached. Thus, the numerical solution of Eq. (8.49) proposed by Opfer et al. (2012) can be expressed as follows: (
)
β2 2 β3 3 β α−1 1 + β xi + xi + xi + · · · + xα−1 exp(−β xi ) − ξi = , 2 6 (α − 1)! i
(8.50)
where the error is set in this study to = 1 × 10−4 . Note that more efficient methods to find the root xi such as the secant method are not used here for the sake of simplicity.
f (x)
f (xi )
F (xi )
xi
x
Figure 8.5: Schematic representation of the probability density function f (x) of the gamma distribution.
The gamma distribution given by Eq. (8.46) was used by Villermaux and Bossa (2009) to describe the primary break-up of rain droplets. For this purpose, they set the shape and rate parameters to α = β = 4 and replaced the variable xi by the ratio of the droplet diameter to the arithmetic 290
8. Results for Droplet Coalescence mean diameter, i.e., xi = dd,i /D10 . Here, the arithmetic mean diameter D10 for this distribution is defined in terms of the Sauter mean diameter D32 by the following expression (Villermaux and Bossa, 2009): D10 =
2 D32 . 3
(8.51)
By substituting these quantities into relation (8.46) with D = D32 /6 and Γ(4) = 6, the distribution function by Villermaux and Bossa (2009) has the following form: 2 f (x) = 3
dd D
!3
!
dd exp − . D
(8.52)
Based on this PDF the number distribution function by Villermaux and Bossa (2009) has the following form f0 (dd ) = f (x)/D10 and thus: !
1
dd 3 f0 (dd ) = . 4 dd exp − D 6D
(8.53)
Figure 8.4 shows this number distribution function in comparison with other distributions used in the present study. Applying the method by Opfer et al. (2012), the diameters dd,i of the injected droplets can be randomly generated by iteratively solving the following equation, which originates from Eq. (8.50) for α = β = 4:
dd,i 1+ D
!
1 + 2
dd,i D
!2
1 + 6
dd,i D
8.2.4.4 Exponential Distribution
!3
dd,i exp − D
!
− ξi = .
(8.54)
Amsden et al. (1989) proposed an exponential number distribution function to describe the sizes of the injected droplets. This distribution is a special case of the gamma function f (x) given by Eq. (8.46) for α = 1 and β = 1. The exponential number distribution by Amsden et al. (1989) reads f0 (dd ) = f (x)/D, where the variable x = dd /D. Hence, it has the following form: !
1 dd f0 (dd ) = exp − , D D
(8.55)
where for this distribution the mean diameter reads D = D32 /3. In Figure 8.4 this exponential distribution function is compared to other distributions used in the present study. It is worth noting that the method by Opfer et al. (2012) is not applied, since relation (8.55) can be solved analytically. Here, the diameters dd,i of the injected droplets required for Eq. (8.31) are randomly generated based on the exponential distribution as follows:
dd,i = −D ln D f0,i ,
(8.56)
where the random number fraction is given by: f0,i = f0,min + (f0,max − f0,min ) ξi .
(8.57) 291
8.3 Inter-Impingement Spray System Here, the minimum and maximum number fractions (i.e., f0,min and f0,max ) computed based on Eq. (8.55) read: f0,min f0,max
!
1 dd,max = exp − , D D ! dd,min 1 exp − , = D D
(8.58)
where dd,min and dd,max stand for the minimum and maximum droplet diameters considered in the simulation. 8.3 Inter-Impingement Spray System
In this section the implementation of the improved composite collision outcome model (Section 4.2.5) into the in-house code LESOCC is verified. For this purpose, the dynamic process of binary collisions of surface-tension dominated droplets in inter-impingement water sprays (see Figure 8.6) injected into a quiescent air environment is studied. A detailed description of the numerical set-up and the predictions of the present test case is presented next. NOZZLE 1
NOZZLE 2
Z Y
X
φ1 ψ1 Sp,1
l1
φ2 θ
ψ2
c Sp,2
l2
Figure 8.6: Schematic representation of the inter-impingement spray system with an impingement angle θ and the rotation angles φ1 and φ2 .
8.3.1 Test Case Description
As schematically depicted in Figure 8.6, the inter-impingement spray system investigated in this analysis consists of two sprays crossing each other with an impingement angle θ. In this set-up the two sprays are rotated about the z-axis of the global coordinate system by the angles φ1 = −45◦ and φ2 = 45◦ , such that the impingement angle θ = 180◦ − (|φ1 | + φ2 ) = 90◦ . It is assumed that the water droplets are injected from two single-hole nozzles and the spray angle is set to 292
8. Results for Droplet Coalescence ψ = ψ1 = ψ2 = 10◦ . The distance from each nozzle exit to the impingement point c is set equal to √ l1 = 0.05 m. Thus, the distance between the two nozzles reads l1 2 and hence it is equal to about 0.07 m. 8.3.2 Simulation Set-up
The simulation for this simplified test case of an inter-impingement spray system is carried out in a three-dimensional vertical channel. As depicted in Figure 8.7, the size of the computational domain is 8 δ × 2 δ × 2 δ in the streamwise (x), wall-normal directions (y) and (z), respectively. Here, δ = 0.25 m stands for the channel half-width. In this set-up an equidistant mesh consisting of 256 × 64 × 64 grid points is used. Boundary Conditions (BC)
z
Y
Y
Y
x
x
z
8δ
Nozzle 2
Nozzle 1
Z
2δ
x
2δ Figure 8.7: Computational set-up for the simulation of the inter-impingement spray system.
At the beginning of the simulation, the velocity and the pressure field is initialized to zero, since the surrounding air (i.e., the continuous phase) is assumed to be quiescent. The air density is ρf = 1.25 kg/m3 at atmospheric pressure. Boundary conditions for the continuous flow are not required, since due to the switched-off two-way coupling (see below) the continuous fluid stays at rest. The dimensionless time step is set to ∆t∗ = ∆t Uinj /δ = 1 × 10−4 , where Uinj is the magnitude of the injection velocity. The parameters used in this simulation are listed in Table 8.1. Assuming that the injected water droplets are fully atomized at the nozzle exit, the primary break-up zone is not taken into account. Thus, the droplets are directly injected at the nozzles. In this numerical experiment one droplet is injected from each nozzle into the computational domain at each time step. This assumption means for the injection model that: • There in no need to determine the primary break-up length Lb and hence the diameter of the injection area Db . • The initial position of the released droplet is set equal to the coordinate of the nozzle centers. 293
8.3 Inter-Impingement Spray System • The initial droplet velocity of the released droplets is computed as follows: ud,i = Uinj eˆi ,
(8.59)
where the magnitude of the velocity at the nozzle exit is set to Uinj = 10 m/s. The components of the unit vector eˆi are determined based on Eq. (8.30) for φ1 = −45◦ and φ2 = 45◦ . Continuous Phase Dynamic viscosity, µf Density ρf
1.8 × 10−5 1.25
kg/(m · s) kg/m3
Disperse Phase Dynamic viscosity, µd Density, ρd Surface tension, σ Diameter (monodisperse), dd Injection velocity magnitude, Uinj
1.0 × 10−3 998.8 0.072 175 × 10−6 10
kg/(m · s) kg/m3 N/m m m/s
Table 8.1: Input parameters used in the present simulation for the continuous and the disperse phase.
In order to minimize the influencing parameters, only the drag force is taken into account. In other words, the remaining fluid forces (i.e., lift, pressure gradient, added mass, etc.) and the gravity force acting on the injected droplets are set to zero. In addition, the feedback effect of the droplets on the continuous phase (i.e., two-way coupling) and the subgrid-scale model for the droplets are switched off. The composite collision outcome model for surface-tension droplets is applied to predict the regimes of binary droplet-droplet collisions occurring during the simulation time. 8.3.2.1 Properties of the Droplets
In this simplified test case monodisperse primary droplets with dd = 175 µm are used and hence the dimensionless droplet diameter reads d∗d = dd /δ = 7 × 10−4 . Thus, the distribution functions presented in Section 8.2.4 are not required. These droplets are injected within a dimensionless time interval of about ∆T ∗ = ∆T Uinj /δ = 0.5, which is the total simulation time. The total number of droplets injected from both nozzles into the computational domain during the entire simulation is N0 = 10, 000 (5000 droplets from each nozzle). The dimensionless density and surface tension 2 of the water droplets are ρ∗d = ρd /ρf = 800 and σ ∗ = σ/(ρf Uinj δ) = 23.04 × 10−4 , respectively. It is worth noting that the use of a short interval of droplet injection is motivated by the fact that in “real” processes (e.g., combustion engines) the injection time is typically ∆tinj = O(10−3 ) s. 8.3.3 Model Verification
As depicted in Figure 8.8, the inclined water sprays are injected from the single-hole nozzles and cross each other in the impingement zone, where the highest number of droplet-droplet collisions occurs, which eventually lead to coalescence resulting in an enlargement of the droplet sizes. At 294
8. Results for Droplet Coalescence the end of the simulation, it is observed that the largest coalesced droplet (or agglomerate) is about four times larger in diameter than the injected primary droplets (dd = 175 µm). As assumed in the framework of the composite model, the droplets maintain their sizes in the cases of bouncing, stretching and reflexive separations. Note that these regimes can not be distinguished based on the qualitative representation displayed in Figure 8.8. Therefore, the predictions are examined in more detail as explained next. Y
x
SIZE 175 250 300 350 400
Figure 8.8: Snapshot of the distribution of droplet sizes at the end of the simulation of two crossing water sprays with an impingement angle θ = 90◦ and rotation angles φ1 = −45◦ and φ2 = 45◦ . Here, blue dots represent the remaining primary droplets (dd = 175 µm).
To evaluate different outcomes (i.e., bouncing, fast coalescence, reflexive and stretching separation) occurring within the present simulation quantitatively, the collision regime map predicted by the composite model at the end of the simulation is depicted in Figure 8.9. It is visible that the predicted collision outcomes perfectly agree with the empirical boundaries (dashed curves) between the regimes for a droplet size ratio ∆ = 1.0. As explained in the context of the composite model, these correlations were proposed by Estrade et al. (1999) for bouncing, Ashgriz and Poo (1990) for reflexive separation and Brazier-Smith et al. (1972) for stretching separation. In this description the rate of the outcome j is defined as the number of droplet-droplet collisions dd,(j) dd (superscript dd) leading to this outcome Ncol divided by the total number of collisions Ncol . dd,(j) dd Thus, the rate of the outcome j reads Ncol /Ncol for j = II, III, IV and V. As evidenced in Figure 8.9, most likely droplet-droplet collisions are stretching separations (V) with a rate of about 50.15%. This observation can be attributed to the inclination of the crossing sprays leading to the impingement angle θ = 90◦ shown in Figure 8.6. The reflexive separations (IV) have the lowest rate implying that the number of nearly head-on droplet-droplet collisions is the smallest 295
8.4 Validation of the Composite Collision Outcome Model one in this set-up. The coalescence rate predicted in this simulation is about 27%, which based on the description in Section 4.2.2.4.1 perfectly agrees with the range of 10% to 40% as suggested by Brazier-Smith et al. (1972) for monodisperse water droplets. 1.0
1.0 Bouncing
Bouncing (II) 0.8
0.6
dd /Ncol
0.8
dd,(j)
Ncol
B
Stretching Separation (V)
0.4
Fast Coalescence (III)
0.2
Fast Coalescence Reflexive Separation Stretching Separation
0.6
0.4
0.2
Reflexive Separation (IV) 0.0
0.0 0
30
60
90
120
150
180
Wec
(II)
(III)
(IV)
(V)
Collision Outcome
Figure 8.9: Results for the two crossing water sprays in a quiescent air at atmospheric pressure using equalsized primary droplets with d∗d = 7 × 10−4 : (left) collision outcomes • bouncing (II), • fast coalescence (III), • reflexive separation (IV) and • stretching separation (V). Dashed lines represent the boundaries of the bouncing regime by Estrade et al. (1999), the reflexive separation by Ashgriz and Poo (1990) and the stretching separation by Brazier-Smith et al. (1972) for a droplet size ratio ∆ = 1.0; (right) ratio of the number of collisions leading dd,(j) dd to a certain outcome to the total number of droplet-droplet collisions Ncol /Ncol for j = II, III, IV and V.
8.3.4 Summary of Key Findings
Based on the results of the present analysis it can be concluded that the improved composite collision outcome model is correctly implemented into the in-house code LESOCC and can be used for further numerical investigations on droplet coalescence of surface-tension dominated droplets in real applications as explained next. 8.4 Validation of the Composite Collision Outcome Model
In this section a four-way coupled Euler-Lagrange large-eddy simulation using the composite collision outcome model is adopted to simulate the injection process of a non-evaporating single diesel spray into a quiescent nitrogen environment. Particularly, a validation study of the composite model is carried out in terms of the predicted spray tip penetration against the experimental data by Gao et al. (2009) and the empirical correlation by Mirza (1991). 8.4.1 Test Case Description
The simulation set-up is based on the experimental study by Gao et al. (2009), who investigated the main characteristics of a non-evaporating diesel spray. The experiment was conducted in a constant-volume vessel charged with a nitrogen gas at a back-pressure of pback = 1.10 MPa. Based 296
8. Results for Droplet Coalescence on the experimental data the mean injection pressure is pinj = 32.42 MPa and the total injection time is about ∆tinj = 1.65 × 10−3 s. The mean conical angle for this diesel spray is about ψ = 14◦ . The physical properties of the continuous and the disperse phase are listed in Table 8.2. Continuous Phase Dynamic viscosity, µf Density ρf
17.8 × 10−6 12.7
kg/(m · s) kg/m3
Disperse Phase Dynamic viscosity, µd Density, ρd Surface tension, σ
2.902 × 10−3 853.8 0.025
kg/(m · s) kg/m3 N/m
Table 8.2: Physical properties of the continuous and the disperse phase used in the present simulation.
8.4.1.1 Computational Grid
As schematically depicted in Figure 8.10, the simulations are performed in a three-dimensional domain of the size 2400 DN × 800 DN × 800 DN in the streamwise (x) and both wall-normal directions (y) and (z), respectively. Here, the nozzle diameter is DN = 0.18 × 10−3 m. It is worth noting that using the present domain dimensions, it is ensured that no droplet-wall collisions occur during the entire simulation. Boundary Conditions (BC) x
Z
7
6
5
4
2
3
1
82 DN
200 DN
800 DN
Wall
Inflow
Injection block
Db
8
Wall
200 DN
800 DN
Wall
Wall
9
gx
Y
Wall
Back Pressure pback = 1.10 Mpa ρf = 12.7 kg/m3 Quiescent nitrogen (uf = 0 at t = 0)
Outlow
Y
Injection area
Wall
2400 DN
Figure 8.10: Computational set-up for the simulation of a single non-evaporating diesel spray.
In this set-up the computational domain is split up into nine blocks as shown in Figure 8.10. The computational grid consists of 300 × 100 × 100 control volumes (CVs). For all blocks an equidistant grid with a dimensionless grid spacing of ∆x/DN = 8 is used in the streamwise direction. Figure 8.11 shows the cross-section of the computational grid in the yz-plane. Here, 297
8.4 Validation of the Composite Collision Outcome Model the injection block (block 5 in Figure 8.10) is located in the middle of the domain and consists of 300 × 40 × 40 CVs. For this block an equidistant grid with a dimensionless grid spacing of ∆y/DN = ∆z/DN = 5 in y- and z-directions is used, whereas for the remaining blocks the grid is geometrically stretched in outwards direction with a stretching factor of 1.04.
Figure 8.11: A cross-section of the computational grid in the injection plane (yz).
8.4.1.2 Simulation Set-up
A four-way coupled LES prediction is carried out using the Smagorinsky subgrid-scale model with Cs = 0.1. In the present simulations inflow and outflow boundary conditions are applied at the inlet and outlet. For the former the inflow velocity is set to zero. Furthermore, the no-slip condition is used at the walls as depicted in Figure 8.10. At the beginning of the simulation, the velocity and the pressure field is initialized to zero, since the surrounding nitrogen (continuous phase) is assumed to be at rest (Gao et al., 2009). The droplets are released at each time step with random positions and diameters using the injection model presented in Section 8.2. The primary break-up length is modeled using the empirical correlation by Levich (1962) and its dimensionless value based on Eq. (8.1) is Lb /DN = 102.5, where the best predictions are found for the constant kb = 12.5. Thus, the origin of the injection area is set at the center of the computational domain with a streamwise distance of xb /DN = Lb /DN = 102.5. The dimensionless diameter of the injection area computed based on Eq. (8.22) is Db /DN = 25.17 (see Figure 8.10). The discharge coefficient of the nozzle is assumed to be kd = 0.70 yielding an injection velocity of Uinj = 189.6 m/s based on relation (8.19). Using relation (8.27) and assuming a coefficient for the break-up zone velocity of kv = 0.75, the velocity magnitude is Ub = kv Uinj = 142.2 m/s leading to a Reynolds number of Re = Ub DN /νf = 18, 350. Note that both velocities Uinj and Ub are assumed to be constant during the entire injection. As assumed by Mirza (1991) and Yule and Filipovi´c (1992), the primary break-up time is approximated as tb ≈ Lb /Ub and hence its dimensionless value reads t∗b = tb Ub /DN ≈ 102.5. The primary droplets are injected within a time interval of tb ≤ t ≤ ∆tinj and the dimensionless time step is set to ∆t∗ = ∆t Ub /DN = 0.36. 298
8. Results for Droplet Coalescence In this test case the gravity, the buoyancy, the drag, the pressure gradient and the added mass forces are assumed to be relevant for the droplet dynamics (see Section 2.3.2), since the droplet-to-fluid density ratio is ρd /ρf < 100. The gravitational acceleration points in the main flow direction and its dimensionless value is gx∗ = gx DN /Ub2 = 8.73 × 10−8 . Since a fluid flow is induced due to the injection of the droplets with high velocities, the feedback effect of the particles on the continuous phase (i.e., two-way coupling) is taken into account. The lift force is not considered and the subgrid-scale model for the particles is switched off, since at the beginning of the simulation the flow is assumed to be quiescent. As mentioned before, in the framework of the composite model the volume-equivalent sphere model is applied for modeling the structure of the coalesced droplets. In addition, frictionless, fully elastic droplet-droplet collisions are assumed. 8.4.1.3 Properties of the Droplets
The dimensionless values of the density and the surface tension of the injected diesel droplets are ρ∗d = ρd /ρf = 67.23 and σ ∗ = σ/(ρf Ub2 DN ) = 5.52 × 10−4 , respectively. Assuming that the injected diesel jet is fully atomized at the break-up length, the diameters of the droplets are determined based on various size distributions introduced in Section 8.2.4. In the standard set-up the gamma distribution function by Villermaux and Bossa (2009) is used. In the experiment by Gao et al. (2009) the Sauter mean diameter (SMD) is D32 = 55 µm at the instant in time of the primary break-up. Hence, its dimensionless value is D32 /DN = 0.305. The constant mass flow rate is m ˙ inj = 4.12 × 10−3 kg/s. The number and the diameters of the injected droplets at each time step varies, since the droplet sizes are randomly chosen until the fuel mass ∆minj = m ˙ inj ∆t = 1.88 × 10−9 kg is reached. Thus, the criterion (8.32) required for the random generation of the dimensionless initial diameters d∗d,i = dd,i /DN of the injected droplets using a prescribed number distribution function has the following form: n X i=1
d∗d,i
3
=
3 ∗ U ∆t∗ , 2 inj
(8.60)
∗ where the dimensionless velocity at the nozzle exit reads Uinj = Uinj /Ub = 1/kv .
8.4.2 Results and Discussion
The objective of this section is to analyze the performance of the composite collision outcome model concerning the prediction of the most important characteristics of the non-evaporating diesel spray. The section is organized in the following manner. In Section 8.4.2.1 the predictions of the composite model are validated in terms of the spray tip penetration based on the experiment by Gao et al. (2009) and the empirical correlation by Mirza (1991). Then, the collision outcomes occurring during of the simulation are presented in Section 8.4.2.2. At the end, the effect of various simulation parameters on the predicted penetration depth is discussed in Section 8.4.2.3. 8.4.2.1 Spray Tip Penetration
In most numerical studies the spray tip penetration is used to validate the predictions against the experimental data. Figure 8.12 shows the time history of the dimensionless spray tip penetration 299
8.4 Validation of the Composite Collision Outcome Model Sp (t)/DN predicted by the present composite model in comparison with the experiment by Gao et al. (2009) and the single-line empirical correlation of Mirza (1991) given by Eq. (8.9). As mentioned before, the gamma number distribution by Villermaux and Bossa (2009) is applied here to describe the initial diameters of the droplets injected at the break-up length. 6.0
×10+2
Sp (t)/DN
5.0
Gao et al. (2009) Mirza (1991) Present Simulation
4.0 3.0 2.0 Primary break-up
1.0 0.0 0
260
520
780
1040
1300
t Ub /DN Figure 8.12: Penetration depth of a solid-cone non-evaporating diesel spray in quiescent nitrogen environment at 1.1 MPa: • Experimental data by Gao et al. (2009), (blue line) empirical correlation by Mirza (1991) and (red line) present results predicted by the composite model. The droplet diameters at the instant of the primary break-up time are determined based on the gamma number distribution by Villermaux and Bossa (2009).
Obviously, the prediction of the developed composite model agrees very well with the references mentioned above. Nevertheless, the penetration depth is slightly overestimated in the dimensionless time interval 200 ≤ t Ub /DN ≤ 450. This behavior can be attributed to the neglect of the secondary break-up yielding smaller droplets and hence a lower penetration depth. This issue shall be considered in the future. Note that the predictions (red curve) start at the primary break-up defined in the above diagram by the coordinates (t∗b , Sp∗ (tb )). The primary break-up time is approximated as explained before and the corresponding dimensionless penetration length is Sp∗ (t∗b ) = Sp (tb )/DN = Lb /DN = 102.5. Figure 8.13 shows the distribution of the primary and coalesced droplets in the computational domain at the end of the simulation. It is obvious that a higher probability of fast coalescence is observed for low values of the droplet velocity (i.e., lower collision Weber numbers observed further downstream) which is consistent with the theory. In this test case the ratio of the number of agglomerates to the total number of injected droplets is about 15.87%, which indicates the importance of coalescence. The largest agglomerate predicted during the simulation includes 67 primary droplets. To quantitatively show the effect of the coalescence on droplet sizes, the number distribution function f0 (dd ) is determined at the end of the simulation. Here, Nc = 50 classes are considered for the determination of the number size distribution. As visible in Figure 8.14, if the droplet coalescence is taken into account, lower number fractions are observed for small droplets with 300
8. Results for Droplet Coalescence dd . 35 µm. However, since coalescence leads to larger droplets, increased number fractions are observed for dd > 35 µm. Y
ud Ub
x 80
yd /DN
40 0 -40 -80 50
100
150
200
250
300
350
400
450
500
300
350
400
450
500
350
400
450
500
xd /DN (a) Y
uag Ub
x
yag /DN
80 40 0 -40 -80 50
100
150
200
250
xag /DN (b) Y
dag DN
x
yag /DN
80 40 0 -40 -80 50
100
150
200
250
300
xag /DN (c) Figure 8.13: Snapshot of the distribution of (a) all droplets (i.e., primary and coalesced droplets); only the coalesced droplets colored by (b) the dimensionless streamwise velocity uag /Ub and (c) the dimensionless diameter dag /DN at the end of the simulation.
301
8.4 Validation of the Composite Collision Outcome Model
Number Distribution Function
0.12
Coalescence = OFF Coalescence = ON
0.10 0.08 0.06 0.04 0.02 0.00 0
20
40
60
80
100
Droplet Diameter (µm)
120
140
160
Figure 8.14: Number distribution function of the droplets at the end of the simulation for the case without (gray histogram) and with (red histogram) coalescence using the composite collision outcome model.
As a result of the high injection velocity of the droplets into the stagnant nitrogen environment, a fluid flow is induced. Note that this effect is taken into account by the two-way coupling. Figure 8.15 shows a snapshot of the contour of the induced fluid field due to the droplet injection colored by the dimensionless fluid velocity in streamwise direction. Here, the expected physical behavior of the flow field is predicted, since the fluid flow moves slower than the injected droplets. Unfortunately, the corresponding experimental data were not provided in Gao et al. (2009) and hence a direct comparison between the predictions and the experiment is not possible. Y
uf Ub
x
0.02
0.1
0.2
0.3
0.4
0.5
80
y/DN
40 0 -40 -80 50
100
150
200
250
300
350
400
450
500
x/DN Figure 8.15: Snapshot of the contour of the induced fluid field due to the droplet injection at the end of the simulation colored by the dimensionless fluid velocity in streamwise direction uf /Ub .
8.4.2.2 Collision Regimes
In this section the outcomes of droplet-droplet collisions occurring during the simulation are quantitatively evaluated based on the regime map predicted by the composite model. It is worth noting that this regime map depends on the droplet size ratio of the collision partners defined 302
8. Results for Droplet Coalescence as ∆ = ds /dl , where ds and dl are the diameters of the small and large droplets, respectively. Figure 8.16 shows the regime map exemplarily for the predicted droplet size ratio in the range 0.95 ≤ ∆ ≤ 1.0 in comparison with the theoretical boundaries between the regimes for ∆ = 1.0. Furthermore, the rate of each outcome predicted by the composite model during the entire simulation is shown for all size ratios (i.e., 0 < ∆ ≤ 1.0).
As expected, the predicted collision outcomes for size ratios 0.95 ≤ ∆ ≤ 1.0 lie perfectly within the empirical boundaries (dashed curves) between the regimes by Estrade et al. (1999), Ashgriz and Poo (1990) and Brazier-Smith et al. (1972). In the present simulation the most likely dropletdroplet collisions lead to stretching separations (V) with a rate of about 47.57%. The reflexive separations (IV) resulting from nearly head-on collisions have the lowest rate of about 1.20%. The coalescence rate predicted in this simulation is about 10.76%.
1.0
1.0 Bouncing
Bouncing (II) 0.8
0.6
dd /Ncol
0.8
dd,(j)
Ncol
B
Stretching Separation (V)
0.4
Fast Coalescence (III)
0.2
Fast Coalescence Reflexive Separation Stretching Separation
0.6
0.4
0.2
Reflexive Separation (IV) 0.0
0.0 0
25
50
75
Wec
100
125
150
(II)
(III)
(IV)
(V)
Collision Outcome
Figure 8.16: Result for a solid-cone non-evaporating diesel spray in a quiescent nitrogen environment at a backpressure of 1.1 MPa: (left) collision outcomes occurring during the entire simulation time for 0.95 ≤ ∆ ≤ 1.0: • bouncing (II), • fast coalescence (III), • reflexive separation (IV) and • stretching separation (V). Dashed lines represent the boundaries of the bouncing regime by Estrade et al. (1999), the reflexive separation by Ashgriz and Poo (1990) and the stretching regime by Brazier-Smith et al. (1972) for a droplet size ratio ∆ = 1.0; (right) dd,(j) dd the outcome rate Ncol /Ncol with j = II, III, IV and V predicted for all size ratios (i.e., 0 < ∆ ≤ 1.0).
8.4.2.3 Effect of Different Parameters on the Spray Tip Penetration
In this section the effect of various simulation parameters on the spray tip penetration is studied. First of all, the effect of the droplet size distribution mimicking the primary break-up process is investigated. Afterwards, the effect of the break-up length, the interphase coupling (i.e., one-, two-, three- and four-way coupled simulations), the pressure gradient force and the added mass force as well as the subgrid-scale model for the continuous flow is analyzed. The section ends up with the effect of the droplet-droplet collision model (i.e., the standard versus the composite model) on the spray tip penetration. 303
8.4 Validation of the Composite Collision Outcome Model 8.4.2.3.1 Initial Diameters of the Injected Droplets
To investigate the effect of the droplet size distribution representing the outcome of the primary break-up, the standard set-up is applied using the four number distribution functions presented in Section 8.2.4 for a fixed Sauter mean diameter D32 = 55 µm. Note that for the distribution by Rosin and Rammler (1933) given by Eq. (8.34) a spreading parameter of q = 3.2 is found by try-and-error to give the best results compared to those in the range 3 ≤ q ≤ 4 typically used in the simulations of spray systems. The width of the log-normal distribution given by Eq. (8.44) is also determined by try-and-error to be σln = ln(1.8). Figure 8.17 shows the effect of these four number distribution functions on the spray tip penetration. It is obvious that the model by Villermaux and Bossa (2009) leads to the best agreement with the experiment. The number distribution functions by Rosin and Rammler (1933), Amsden et al. (1989) and the log-normal distribution predict noticeably lower spray tip penetrations than the experiment for a dimensionless time of about t∗ > 520. The reason for the different results due to the four number distributions considered for D32 = 55 µm can be explained by again looking at Figure 8.4. It is visible that the probability of generating large droplets is higher for the gamma distribution by Villermaux and Bossa (2009) than for the other models. As will be shown in Section 8.4.2.3.3, large droplets penetrate faster into the ambient gas environment than small ones and thus the distribution of Villermaux and Bossa (2009) predicts a higher penetration depth than the other number distribution functions. 6.0
×10+2
Villermaux and Bossa (2009) Rosin and Rammler (1933) Amsden et al. (1989) Log-Normal
5.0
Sp (t)/DN
Gao et al. (2009)
4.0 3.0 2.0
Primary break-up
1.0 0.0 0
260
520
780
1040
1300
t Ub /DN Figure 8.17: Effect of the initial diameter distribution on the penetration depth of a solid-cone non-evaporating diesel spray in quiescent nitrogen environment at 1.1 MPa: • Experimental data by Gao et al. (2009); present results predicted by the composite model using the number distribution functions by (red line) Villermaux and Bossa (2009), (green line) Rosin and Rammler (1933), (black line) Amsden et al. (1989) and (blue line) the log-normal distribution.
In summary, based on these findings the gamma number distribution function by Villermaux and Bossa (2009) used for modeling the primary break-up is most appropriate and thus used for further investigations. 304
8. Results for Droplet Coalescence 8.4.2.3.2 Break-up Length
The length of the primary break-up is the most sensitive parameter for the initialization of the present simulation, since it is used to calculate the location and the diameter of the injection area. Therefore, the effect of the primary break-up length on the penetration depth is analyzed in this section. As explained in Section 8.1.1, the primary break-up length Lb depends on the empirical constant kb (Levich, 1962), which in the standard set-up is set to kb = 12.5 leading to Lb /DN = 102.5. In this analysis two additional values are considered, namely kb = 9.5 and 15.5 yielding a dimensionless primary break-up length of Lb /DN = 77.9 and Lb /DN = 127.1, respectively. Note that the magnitude of the break-up velocity Ub is set to the same value for the three cases and hence the curves of the predicted penetration depth are plotted such that they start at the primary break-up time and length of the standard set-up. Figure 8.18 shows that the increase of the constant kb and hence the primary break-up length leads to significantly lower spray tip penetrations. The reason for this observation is the increase of the diameter of the injection area (see Eq. (8.22)), which can be explained as follows. The total fuel mass injected at each time step is identical for the three cases considered. Thus, the diameters of the released droplets are determined using the gamma number distribution function by Villermaux and Bossa (2009) yielding N0 = 163, 000 primary droplets for all cases. The increase of the diameter of the injection area implies larger inter-droplet distances, which leads to a lower number of droplet-droplet collisions and hence to a different dispersion behavior within the computational dd dd domain. For the three cases the ratio Ncol /N0 of the number of inter-droplet collisions Ncol to the total number of injected primary droplets N0 is about 3.46, 3.18 and 2.92 for Lb /DN = 77.9, 102.5 and 127.1, respectively. 6.0
×10+2
5.0
Sp (t)/DN
Gao et al. (2009)
Lb /DN = 77.90 Lb /DN = 102.5 Lb /DN = 127.1
4.0 3.0 2.0
Primary break-up
1.0 0.0 0
260
520
780
1040
1300
t Ub /DN Figure 8.18: Effect of the primary break-up length on the penetration depth of a solid-cone non-evaporating diesel spray in quiescent nitrogen environment at 1.1 MPa predicted by the composite model: • Experimental data by Gao et al. (2009); (black line) Lb /DN = 77.9, (red line) Lb /DN = 102.5 and (blue line) Lb /DN = 127.1.
Note that using a dimensionless primary break-up length of Lb /DN = 77.9 the minor kink in the 305
8.4 Validation of the Composite Collision Outcome Model curve of the predicted penetration depth for t∗ > 1000 means that due to the higher collision rate compared to the other two cases the leading droplets quickly coalesce and thus penetrate faster into the ambient environment (i.e., higher penetration depth). In summary, for the present set-up the best result is obtained for Lb /DN = 102.5 (i.e., kb = 12.5) and hence this primary break-up length will be used in the next simulations. 8.4.2.3.3 Interphase Coupling
To analyze the effect of the interphase coupling, in addition to the standard set-up relying on a four-way coupled simulation, predictions based on one-, two- and three-way coupling are carried out using the composite collision model. Here, three-way coupling means that compared to the four-way coupling the feedback effect of the droplets on the continuous flow is neglected. As visible in Figure 8.19, a noticeable influence of the interphase coupling on the predicted spray tip penetration is observed. If the droplet-droplet collisions and the feedback effect of the droplets on the continuous phase are not taken into account, the one-way coupled simulation noticeably underpredicts the spray tip penetration for t∗ > 500. This observation can be explained based on the drag force FD , which is the most dominant force among other fluid forces taken into account in the present set-up. 6.0
×10+2
Sp (t)/DN
5.0 4.0
Gao et al. (2009) 4-Way Coupling (Composite Model) 3-Way Coupling (Composite Model) 2-Way Coupling 1-Way Coupling
3.0 2.0 Primary break-up
1.0 0.0 0
260
520
780
1040
1300
t Ub /DN Figure 8.19: Effect of the interphase coupling on the penetration depth of a solid-cone non-evaporating diesel spray in quiescent nitrogen environment at 1.1 MPa: • Experimental data by Gao et al. (2009); (red line) present results predicted by four- and (green line) three-way coupling using the composite model; (black line) two- and (blue line) one-way coupled simulations (i.e., without coalescence).
According to Eq. (2.29), the drag force depends on the slip velocity uslip = (uf − uq ) and the droplet diameter dd by the following relation: FD ∝ d2d (uf − uq ) |uf − uq | .
(8.61)
Assuming an quiescent gas means that uf vanishes, since for one-way coupling a fluid flow is not induced by the droplets. As a result, in the case of one-way coupling the droplets undergo higher 306
8. Results for Droplet Coalescence drag forces than in the case of two- and four-way coupled simulations. Thus, the droplets move slower than in the case of two- and four-way coupling leading to an appreciably lower penetration depth as shown in Figure 8.19. Looking again at Eq. (8.61), the drag force FD ∝ d2d and hence the enlargement of droplets due to coalescence increases the drag force and thus affects the penetration depth. However, the inertia of the droplets scales with d3d implying that for an identical initial velocity large droplets penetrate quicker than small ones. To study the effect of the droplet diameter on the penetration depth a three-way coupled simulation using the composite model (i.e., feedback effect = OFF) is carried out. Figure 8.19 shows that the penetration depth predicted by the three-way coupling including coalescence is higher than for the one-way coupled simulation, since larger droplets are generated due to coalescence. Note that the predictions of the one- and two-way coupled simulations differ only due to the different slip velocities, since in both cases the droplet coalescence is neglected and hence the initial diameter distribution does not change. Figure 8.19 also shows that the two-way coupled simulation underpredicts the experimental observations for t∗ < 1000, since droplet coalescence is not taken into account in this simulation. However, for the four-way coupled simulations two effects have to be distinguished. (i) Taking the two-way coupling into account implies lower slip velocities and thus weaker drag forces. This is also the case for two-way coupling. However, the droplet enlargement increases the drag force according to Eq. (8.61) which is not the case for two-way coupling. (ii) Larger diameters due to coalescence lead to higher inertia, which is proportional to d3d . Figure 8.19 shows that the four-way coupled simulation predicts slightly higher spray tip penetrations than the case with two-way coupling. This observation implies that the increase of the drag forces (∝ d2d ) due to droplet enlargement (coalescence) is overwhelmed by the higher inertia (∝ d3d ). Note that for a dimensionless time t∗ > 1000 the two-way coupled simulation overpredicts the experimental observation and leads to a higher penetration depth than the four-way coupling. The reason for this observation is the neglect of inter-droplet collisions in this simulation (see Section 8.4.2.3.6). This unexpected behavior confirms again the necessity of using the four-way coupling. In summary, the four-way coupling using the composite model taking the coalescence into account leads to the best agreement with the experimental data by Gao et al. (2009) and hence is solely considered for the following investigations. 8.4.2.3.4 Pressure Gradient and Added Mass Forces
As mentioned before, for a droplet-to-fluid density ratio of ρd /ρf < 100 the added mass force FAM and the pressure gradient force FP G become relevant and hence they should be taken into account (see, e.g., Elghobashi and Truesdell, 1992; Kuerten, 2016). To examine the effect of these two forces on the predicted spray tip penetration, both forces are switched off in the present simulation (ρd /ρf = 67.23) and the predictions are compared with those using the standard set-up (FAM + FP G = ON). Figure 8.20 shows the effect of the inclusion of these forces on the penetration depth. Obviously, the neglect of the added mass and pressure gradient forces leads to a slightly lower tip penetration for a dimensionless time of t∗ > 800. The reason for this observation can be explained 307
8.4 Validation of the Composite Collision Outcome Model by looking at the general equation for the translational motion of the droplet given by Eq. (2.49). The acceleration of the droplet scales with the factor fAM as follows: dud ∝ 1/fAM dt
with fAM = 1 + CAM
ρf ρq
!
(8.62)
,
where the added mass coefficient for spherical particles is equal to CAM = 0.5 (Brennen, 1982; Crowe et al., 1998; Kuerten, 2016). In the present set-up ρd /ρf = 67.23 and hence the factor fAM , by which the right-hand side of Eq. (2.49) is divided, is about fAM = 1.007. This confirms again the marginal effect of the pressure gradient and added mass forces in the present test case. Nevertheless, the added mass and pressure gradient forces are taken into account in the standard set-up, since they are computationally not expensive. 6.0
×10+2
Sp (t)/DN
5.0
Gao et al. (2009) FAM + FP G = ON FAM + FP G = OFF
4.0 3.0 2.0 Primary break-up
1.0 0.0 0
260
520
780
1040
1300
t Ub /DN Figure 8.20: Effect of the inclusion of the pressure gradient force and the added mass force on the penetration depth of a solid-cone non-evaporating diesel spray in quiescent nitrogen environment at 1.1 MPa: • Experimental data by Gao et al. (2009); present results predicted by the composite model with (red line) and without (black line) the inclusion of the pressure gradient force and the added mass force.
8.4.2.3.5 Subgrid-Scale Model for the Continuous Phase
In the standard set-up a four-way coupled LES prediction is carried out using the Smagorinsky subgrid-scale model with Cs = 0.1. In this section the effect of the subgrid-scale model on the spray tip penetration is investigated. For this purpose, the simulation is repeated but without taking the Smagorinsky model into account (SGS = OFF). Figure 8.21 shows that there is almost no effect of the subgrid-scale model on the predictions. This observation can be explained as follows. In the present simulation the fluid flow is purely induced by the injection of the droplets with a high velocity into the quiescent nitrogen environment. Hence, no turbulence exists at the beginning of the simulation in the chamber. That is confirmed by Figure 8.15 showing a smooth jet of the continuous fluid induced by the droplets. Owing to the short injection interval of only 1.65 × 10−3 s turbulent structures have not yet developed although 308
8. Results for Droplet Coalescence strong velocity gradients appear in the shear layers of the jet. Note that for a longer simulation interval the situation might change. 6.0
×10+2
Sp (t)/DN
5.0
Gao et al. (2009) SGS = ON SGS = OFF
4.0 3.0 2.0 Primary break-up
1.0 0.0 0
260
520
780
1040
1300
t Ub /DN Figure 8.21: Effect of the subgrid-scale model (SGS) for the continuous phase on the penetration depth of a solid-cone non-evaporating diesel spray in quiescent nitrogen environment at 1.1 MPa: • Experimental data by Gao et al. (2009); present results predicted by the composite collision model with (red line) and without (black line) the SGS model.
8.4.2.3.6 Effect of Coalescence
In the present analysis the effect of the droplet-droplet collision model on the predicted penetration depth is studied. As depicted in Figure 8.22, the inter-droplet collision model without friction underpredicts the experimental results in comparison with the composite model. The reason for this observation becomes clear in the next paragraph. As mentioned before, in the framework of the classical collision model and the composite collision outcome model frictionless, fully elastic collisions are assumed. Additionally, for the composite model the treatment of the kinetics of the collision partners after the impact depends on the collision outcome (i.e., bouncing, fast coalescence, reflexive and stretching separations). On the contrary, in the framework of the classical collision model the colliding droplets solely bounce back after the impact. In other words, in the classical droplet-droplet collision model only the bouncing outcome is taken into account and thus the droplet size distribution does not change during the simulation. Contrarily, due to coalescence larger droplets are generated by the composite collision outcome model. As a result of these principal differences between the two collision models, a deviation in the results is expected. Figure 8.22 shows that the classical inter-droplet collision model predicts a lower penetration depth than the composite model. This observation is due to the droplet enlargement by coalescence using the composite model, since larger droplets more quickly penetrate into the surrounding nitrogen environment than primary droplets as concluded before. In summary, it can be stated that the composite model predicts binary collisions of surface-tension dominated droplets more realistically than the classical inter-droplet collision model. 309
8.4 Validation of the Composite Collision Outcome Model 6.0
×10+2
Sp (t)/DN
5.0
Gao et al. (2009) Composite Model Classical Collision Model
4.0 3.0 2.0 Primary break-up
1.0 0.0 0
260
520
780
1040
1300
t Ub /DN Figure 8.22: Effect of the collision model on the penetration depth of a solid-cone non-evaporating diesel spray in quiescent nitrogen environment at 1.1 MPa: • Experimental data by Gao et al. (2009); present results predicted by (red line) the composite collision outcome model and (black line) the droplet-droplet collision model without friction.
8.4.3 Summary of Key Findings
Taking the above findings into account, it can be concluded that the improved composite model using the proposed injection model can realistically predict the penetration of a non-evaporating diesel spray into a quiescent gas environment. The results of droplet coalescence provide the following key findings: • The initial diameters of the droplets injected at the primary break-up length can be best described by the gamma number distribution function of Villermaux and Bossa (2009). • The increase of the length of the primary break-up leads to a lower spray tip penetration. • A four-way coupled simulation leads to the best agreement with the experimental data. • The application of the two-way coupling allows to predict the induced fluid flow, which leads to a realistic determination of the drag forces acting on the primary and coalesced droplets. • The inclusion of the added mass and pressure gradient forces marginally affects the predicted spray tip penetration. • The subgrid-scale model for the continuous flow can be neglected in the present test case due to the short injection time. • The classical droplet-droplet collision model without friction underpredicts the penetration depth, since this model only considers one regime (i.e., bouncing) of the four possible outcomes of binary droplet-droplet collisions. 310
CHAPTER 9
RESULTS FOR PARTICLE-WALL ADHESION
Particle-wall collision past an airfoil (Breuer and Almohammed, 2016).
9.1
Validation of the Particle-Adhesion Model . . . . . . . . . . . . . . . . . . . . . . . 9.1.1
9.2
313
Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 9.1.1.1
Computational Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
9.1.1.2
Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
9.1.1.3
Properties of the Particles . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.1.2
Continuous Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
9.1.3
Two-Phase Flow Including Particle-Wall Adhesion . . . . . . . . . . . . . . . . . 317 9.1.3.1
Dimensionless Deposition Velocity . . . . . . . . . . . . . . . . . . . . . 317
9.1.3.2
Effect of Inter-Particle Collisions on the Deposition Velocity . . . . . . . 319
Particle-Wall Adhesion in Turbulent Channel Flow . . . . . . . . . . . . . . . . . . .
321
9.2.1
Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
9.2.2
Results of the New Adhesion Model . . . . . . . . . . . . . . . . . . . . . . . . . 321 9.2.2.1
Effect of Adhesion on Particle-Wall Collisions . . . . . . . . . . . . . . . 321
9.2.2.2
Effect of Different Parameters on Particle-Wall Deposition . . . . . . . . 324 9.2.2.2.1 Particle-Wall Restitution Coefficient . . . . . . . . . . . . . . 324 9.2.2.2.2 Diameter of the Primary Particles . . . . . . . . . . . . . . . 325 9.2.2.2.3 Particle Agglomeration . . . . . . . . . . . . . . . . . . . . . 327 9.2.2.2.4 Wall Roughness . . . . . . . . . . . . . . . . . . . . . . . . . 329 Different Wall Roughnesses Considering Large Particles . . . . 330 Different Wall Roughnesses Considering Small Particles . . . . 331
9.3
Particle-Wall Adhesion in Turbulent Flow Past Inclined Airfoil . . . . . . . . . . . . 9.3.1
332
Test Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 9.3.1.1
Simulation Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
9.3.1.2 9.3.2
Properties of the Particles . . . . . . . . . . . . . . . . . . . . . . . . . 333
Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 9.3.2.1
Two-Phase Flow Including Particle-Wall Adhesion. . . . . . . . . . . . . 334
9.3.2.2
Deposition pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
9 Results for Particle-Wall Adhesion This chapter covers the numerical simulation results using the newly developed particle-wall adhesion model (see Section 5.2) in turbulent particle-laden flows. It is organized as follows. In Section 9.1 the adhesion model is validated in a horizontal particle-laden turbulent channel flow against existing experimental and numerical results. Afterwards, the developed model is adopted to study the effect of different simulation parameters on the particle-wall adhesion (deposition) again in a particle-laden turbulent channel flow but with other fluid and particle parameters. Furthermore, the effect of particle-wall adhesion on the agglomeration process and vice versa is evaluated by simultaneously applying the corresponding models. Finally, in Section 9.3 the newly developed adhesion model is applied to study a practically relevant turbulent flow past an inclined airfoil with separation, transition and reattachment. Note that most of the results presented in this chapter were already published1 in Almohammed and Breuer (2016c) and Breuer and Almohammed (2016, 2017). 9.1 Validation of the Particle-Adhesion Model
In this section the newly developed particle-wall adhesion model is validated based on the experimental study by Kvasnak et al. (1993) and Papavergos and Hedley (1984). Furthermore, the present predictions are compared with the numerical results by Fan and Ahmadi (1993) using the energy-based deposition model by Dahneke (1971) and the empirical relation by Wood (1981). In the following the test case used in this study is described in Section 9.1.1. Afterwards, the present LES predictions of the continuous phase are validated against existing DNS data in Section 9.1.2. Finally, the validation study for the disperse phase is done in Section 9.1.3 in terms of the dimensionless deposition velocity. 9.1.1 Test Case Description
The simulation set-up is based on the experiments conducted by Kvasnak et al. (1993), who investigated the deposition velocity of particles of different materials and sizes on a flat gold plate covered with a thin coating film in a horizontal turbulent channel flow. In this experiment the Reynolds number of the fluid flow is Re = UB dh /νf = 15, 000 based on the hydraulic diameter dh of the channel and the bulk velocity UB . The corresponding Reynolds number based on the bulk velocity and the channel half-width δ is Re = UB δ/νf = 4230. In the present validation study only the deposition of glass particles is considered, since their material properties required for the momentum-based adhesion model are available in the literature. The first step is the validation of the continuous phase predicted by the large-eddy simulation technique. However, since experimental data of the fluid flow are not available for the reference case, a direct numerical simulation (DNS) of a horizontal turbulent channel flow by Marchioli and Soldati (2007) at a comparable Reynolds number of Re = 4200 based on the bulk velocity UB and the channel half-width δ is used. 1
Only full-paper contributions in peer-reviewed journals and conference proceedings are mentioned.
313
9.1 Validation of the Particle-Adhesion Model 9.1.1.1 Computational Grid
As schematically depicted in Figure 9.1, the simulations are performed in a three-dimensional domain of the size 4 πδ × 2 δ × 2π δ in streamwise, wall-normal and spanwise direction, respectively. The computational domain is discretized by 256 × 256 × 256 grid points. In streamwise (x) and spanwise (z) direction an equidistant grid is used, while in wall-normal (y) direction the grid is geometrically stretched with a stretching factor of 1.023 and the first cell center is located at + ∆y1st = 0.63. Boundary Conditions (BC)
y
z
δ 2π
z
Wall
∆T ∗= 60
1
Disperse Phase Continuous Phase
Averaging
x
Periodic BC Periodic BC
2
∆T ∗= 780
Wall
z
Periodic BC
4πδ
Wall
h0 = δ
Flow Direction
x
y
Wall
Periodic BC
2δ
gy y
x
Periodic BC Simulation Time
Figure 9.1: Set-up for the simulation of the horizontal particle-laden channel flow.
9.1.1.2 Simulation Set-up
In the present simulations the bulk velocity is UB = 3.3 m/s and the channel half-width is δ = 0.02 m. The fluid is assumed to be air with the kinematic viscosity νf = 15.7 × 10−6 m2 /s and the density ρf = 1.3 kg/m3 . The shear Reynolds number is Reτ = uτ δ/νf = 300. Periodic boundary conditions are applied on the fluid velocity field in the streamwise and spanwise direction, and no-slip conditions are imposed at the walls of the channel. For the subgrid-scale modeling the Smagorinsky model with a constant of Cs = 0.065 and van Driest damping near the walls is used. To drive the flow a dynamically adjusted pressure gradient is applied to assure a constant mass flux through the channel in the streamwise direction. The dimensionless time step is set to ∆t∗ = ∆t UB /δ = 0.003. After a fully developed unladen flow has established within a dimensionless time interval of ∆T ∗ = ∆T UB /δ = 780 (about 62 flow-through times), N0 = 400, 000 primary particles are released into the computational domain. Hereafter, this instant in time is denoted t∗ = 0, at which the entire algorithm including the adhesion model is applied for a dimensionless time interval of ∆T ∗ = ∆T UB /δ = 60 (see Figure 9.1). Analog to many previous studies, the primary particles are uniformly distributed in the lower half of the channel, i.e., 0 ≤ y ≤ h0 = δ as depicted in Figure 9.1 leading to a constant volume fraction in this region. The velocity of the released particles is set to that of the fluid at the corresponding position (i.e., up = uf |particle ). For each run the same 314
9. Results for Particle-Wall Adhesion number of primary particles with different sizes within the particle size range by Kvasnak et al. (1993) is considered. The values of the particle diameters and the corresponding relaxation times are listed in Table 9.1. Case dp [µm] 1 2 3 4 5 6 7 8 9
5 10 15 20 25 30 35 40 45
dp /δ
τp [s]
τp+ = St+
αp
ηp [%]
2.50 × 10−4 5.00 × 10−4 12.5 × 10−4 10.0 × 10−4 12.5 × 10−4 15.0 × 10−4 17.5 × 10−4 20.0 × 10−4 22.5 × 10−4
0.017 × 10−2 0.068 × 10−2 0.153 × 10−2 0.272 × 10−2 0.425 × 10−2 0.612 × 10−2 0.833 × 10−2 1.088 × 10−2 1.378 × 10−2
0.425 1.699 3.823 6.797 10.62 15.29 20.82 27.19 34.41
2.072 × 10−8 1.658 × 10−7 5.595 × 10−7 1.326 × 10−6 2.590 × 10−6 4.476 × 10−6 7.108 × 10−6 1.061 × 10−5 1.511 × 10−5
0.004 0.032 0.108 0.255 0.498 0.861 1.367 2.040 2.905
Table 9.1: Particle diameters used in the simulations and the corresponding relaxation times, global volume fractions αp and mass loadings ηp . Note that the dimensionless relaxation time τp+ is equal to the Stokes number St+ = τp u2τ /νf .
In the standard set-up the three sub-models (i.e., the feedback effect of the particle on the continuous phase, the lift forces and the subgrid-scale model for the particles) are taken into account. As discussed in Section 7.2, the motivation for the consideration of the three sub-models is that the total number of particle-wall collisions increases due to the following observations: (i) The feedback effect of the particles on the continuous phase noticeably reduces the particle fluctuations in the wall-normal and spanwise directions in the region outside the direct vicinity of the wall. Hence, the particle-wall collisions are slightly affected by the inclusion of the two-way coupling. (ii) The velocity gradient near the wall and the high angular velocities of the particles due to particle-wall collisions lead to shear-induced and rotation-induced lift forces pushing the particles towards the wall. (iii) The subgrid-scale contribution of the particles results in higher fluctuations of the particle velocities in the direct vicinity of the wall. As a result, the combined effect of the sub-models is on the one hand a higher wall-normal particle velocity yielding a stronger repulsive force and a higher number of particle-wall collisions. On the other hand, the increase of the particle fluctuations augments the relative velocity between the particles and the wall pw∗ 1/5 leading to a weaker adhesive impulse, since fˆn,c according to Eq. (5.55). Thus, ∝ 1/(u− p · n) the cumulative effect of these sub-models significantly influences the deposition rate and hence should be taken into account for a realistic simulation. In the present study it is assumed that heavy particles are dispersed in a light fluid and hence the density ratio ρp /ρf 1. Thus, the significant forces are only the drag, gravity, buoyancy and lift forces due to rotation and velocity shear (Sommerfeld et al., 2008). Note that the lift force on a particle due to the velocity shear (Saffman force) is calculated using the correlation by McLaughlin (1991). The effect of the subgrid scales of the continuous phase on the particle motion is considered by applying the Langevin model originally proposed by Pozorski and Apte 315
9.1 Validation of the Particle-Adhesion Model (2009) and adapted by Breuer and Hoppe (2017). 9.1.1.3 Properties of the Particles
In the present test case, the particles disperse inside the channel due to the fluid and gravitational forces. The dimensionless gravitational acceleration is directed against the wall-normal direction and its dimensionless value is gy∗ = −gy δ/UB2 = −1.80 × 10−2 . The walls are assumed to be made of gold. Typical values of the mechanical properties of the materials2 are listed in Table C.3. The dimensionless Hamaker constant is H ∗ = H/(ρf UB2 δ 3 ) = 2.79 × 10−15 and the dimensionless separation between two particles during the contact is δ0∗ = δ0 /δ = 1 × 10−8 . The dimensionless Young’s moduli of the particles and the walls are Ep∗ = Ep /(ρf UB2 ) = 4.94 × 109 and Ew∗ = Ew /(ρf UB2 ) = 5.58 × 109 , respectively. For particle-wall collisions the following restitution and friction coefficients are used: en,w = 0.96 (Kvasnak et al., 1993; Almohammed et al., 2014), et,w = 0.44, µst,w = 0.94 and µkin,w = 0.092 (see, e.g., Vreman, 2007; Lain and Sommerfeld, 2008). In the first step particle-particle collisions are not taken into account. 9.1.2 Continuous Phase
Figure 9.2 shows the distribution of the dimensionless meanDfluid velocity in streamwise direction E 0 0 2 huf i/uτ , the dimensionless mean fluid velocity fluctuations ui,f ui,f /uτ and the Reynolds shear D
E
stress u0f vf0 /u2τ along the channel width. It is clearly visible that the present LES prediction agrees very well with the DNS data of Marchioli and Soldati (2007). Minor deviations are visible for the wall-normal and spanwise fluctuations in the buffer layer. Since only the resolved parts of the fluctuations are shown here, such minor differences to the DNS data have to be expected. 20.0
8.0
D
6.0
15.0
huf i/uτ
D
10.0
E
vf′ vf′ /u2τ
D
2.0
E
wf′ wf′ /u2τ
D
4.0
E
u′f u′f /u2τ
E
u′f vf′ /u2τ
5.0 0.0 0.0 0.001
-1.0 0.01
0.1
y/δ (a)
1
0
0.2
0.4
0.6
0.8
1
y/δ (b)
Figure 9.2: (a) Dimensionless mean streamwise fluid velocity huf i/uτ and (b) dimensionless mean fluid velocity fluctuations in the streamwise hu0f u0f i/u2τ , wall-normal hvf0 vf0 i/u2τ and spanwise hwf0 wf0 i/u2τ directions as well as the dimensionless Reynolds shear stress of the continuous phase hu0f vf0 i/u2τ . Symbols stand for the DNS results (Marchioli and Soldati, 2007) and the solid lines are the present LES results. Here, the friction velocity uτ used or the normalization of the LES data is taken from the LES itself (here uτ /UB = 0.06). 2
316
The Hamaker constant for a glass-gold interface is H = 31.6 × 10−20 J (Kvasnak et al., 1993).
9. Results for Particle-Wall Adhesion 9.1.3 Two-Phase Flow Including Particle-Wall Adhesion 9.1.3.1 Dimensionless Deposition Velocity
As mentioned in Section 5.1, the process of particle deposition is commonly expressed in terms of the dimensionless deposition velocity vd+ against the dimensionless relaxation time τp+ . The dimensionless deposition velocity is given by (see Section 5.1.2):
Ndpp vd+ = N0
!
h+ 0 , + td
(9.1)
where N0 stands for the total number of primary particles released into the computational domain. 2 Ndpp is the total number of primary particles deposited during the dimensionless time t+ d = t uτ /νf . For the present simulation the dimensionless distance is h+ 0 = h0 uτ /νf = 300 with h0 = δ, since the primary particles are injected in the region 0 ≤ y ≤ δ as depicted in Figure 9.1. Furthermore, in the numerical experiments the dimensionless time range t+ d should be selected such that a quasi-equilibrium condition is realized, which means that Ndpp /t+ d becomes a constant as stated by Li and Ahmadi (1993). Figures 9.3(a) and (b) show the time history of the accumulated number of particle-wall collisions pw Ncol and deposited primary particles Ndpp (i.e., deposition processes) for different particle sizes. Note that in the present case Ndpp is equal to the number of deposition processes Ndep . Obviously, the number of particle-wall collisions drastically increases when increasing the particle diameter. As visible in Figures 9.3(b), the number of deposited particles first non-linearly increases, where the duration of this phase depends on dp and is attributed to an increasing number of particle-wall collisions until a particle deposits on the wall. Afterwards, a quasi-equilibrium condition is reached (approximately linear increase with time) for a certain time interval. Since the sedimentation velocity of a particle in a fluid at rest is proportional to d2p , larger particles reach the lower wall much faster. That also holds true for the horizontal channel flow. Thus, the drastic increase of particle-wall collisions when increasing the particle diameter leads to a higher number of deposition processes, whereby the increase with dp remains smaller than that of the collisions due to a decreasing tendency of deposition as explained below. Figure 9.3(c) depicts the time history of the deposition rate defined as the number of deposition pw processes Ndep divided by the total number of particle-wall collisions Ncol . Although the number of particle-wall collisions significantly increases for larger particles, a lower deposition rate is observed for increasing dp . This physically realistic behavior is attributed to the lower deposition pw∗ probability due to a weaker adhesive impulse, since based on Eq. (5.55) fˆn,c ∝ 1/dp . It is worth noting that for the present case the particles deposit only on the lower channel wall. To determine the dimensionless deposition velocity, the dimensionless time in wall units can be ∗ ∗ expressed by t+ d = ∆Td (uτ /UB ) Reτ , where ∆Td stands for the dimensionless time interval during which the quasi-equilibrium condition is satisfied. Based on Figure 9.3(b) this time interval is chosen for the present set-up between t∗d = 15 and 30 leading to ∆Td∗ = ∆T UB /δ = 15. Figure 9.4 depicts the predicted dimensionless deposition velocity as a function of the dimensionless relaxation time in comparison with the experimental data of Kvasnak et al. (1993) and Papavergos and Hedley (1984), the numerical results by Fan and Ahmadi (1993) using the energy-based model 317
9.1 Validation of the Particle-Adhesion Model and the empirical relation of Wood (1981). As evidenced in Figure 9.4 the predictions of the new adhesion model are in reasonable agreement with the experimental results as well as the empirical model.
6.0
×10+6
10 µm 20 µm 30 µm 40 µm
×10+5
10 µm 20 µm 30 µm 40 µm
3.0
Ndpp
4.5 pw Ncol
4.0
3.0
1.5
2.0
1.0
0.0
0.0 0
10
20
30
40
50
60
0
10
20
t UB /δ (a)
40
50
60
(b) 90.0
×10−2
10 µm 20 µm 30 µm 40 µm
75.0 pw Ndep /Ncol
30
t UB /δ
60.0 45.0 30.0 15.0 0.0 0
10
20
30
40
50
60
t UB /δ (c) pw Figure 9.3: Time history of the accumulated number of (a) particle-wall collisions Ncol , (b) deposition processes pw Ndep = Ndpp and (c) the deposition rate Ndep /Ncol predicted by the present adhesion model for particle sizes from dp = 10 µm to 40 µm.
As observed in many studies on particle deposition in turbulent channel flows (see, e.g., Liu and Agarwal, 1974; Li and Ahmadi, 1993; Fan and Ahmadi, 1993), the dimensionless deposition velocity vd+ rapidly increases with an increasing particle relaxation time τp+ until it reaches a peak, beyond which vd+ decreases for higher values of τp+ . Li and Ahmadi (1993) also stated that the effect of particle rebound (how many times a particle rebounds before depositing on the wall) becomes considerable when increasing the particle size. They found that for Re = UB δ/νf = 3330 particle rebound from the wall noticeably reduces the dimensionless deposition velocity for τp+ > 5. A further inspection of Figure 9.4 reveals that the scattered values in the experimental results Kvasnak et al. (1993) show a drop of the dimensionless deposition velocity for τp+ > 10. Obviously, 318
9. Results for Particle-Wall Adhesion this phenomenon is correctly reproduced by the new adhesion model, since the effect of particle rebound after particle-wall collisions is considered. Furthermore, for τp+ > 4 the present results agree with the experiments more reasonably than the predictions of the energy-based model by Fan and Ahmadi (1993). The main reasons for this observation are (i) the realistic determination of the adhesive impulse and (ii) the consideration of the effect of the adhesion if the deposition condition is not satisfied, since it has a significant influence on the number of particle-wall collisions as will be shown in Section 9.2.2.1. To compare the present model with the wetted-wall model, the same simulations are carried out using the latter model. Figure 9.4 shows that the wetted-wall model tends to predict higher values of vd+ for τp+ > 10, since it ignores the rebound effect. Thus, the present deposition model is superior to the wetted-wall model for τp+ > 10. 10+1 +0
Papavergos and Hedley (1984) Kvasnak et al. (1993)
−1
Present: PPC = OFF Present: PPC = ON
10
vd+
10
10−2 10−3 Fan and Ahmadi (1993) Wood (1981) Wetted-wall model
10−4 10−5 10−2
10−1
10+0
τp+
10+1
10+2
Figure 9.4: Dimensionless deposition velocity of spherical glass particles with dp = 5 µm to 45 µm predicted by the new adhesion model without and with considering particle-particle collisions (PPC = OFF/ON) in comparison with the wetted-wall model, experimental data by Papavergos and Hedley (1984) and Kvasnak et al. (1993) and the numerical results by Fan and Ahmadi (1993) using the energy-based model and the modified empirical relation of Wood (1981).
9.1.3.2 Effect of Inter-Particle Collisions on the Deposition Velocity
It is well known that high local volume fractions (αp & 10−3 ) affect the transport of particles due to inter-particle collisions. Since for the cases with high global volume fractions (see Table 9.1) the highest local volume fractions exceed the above mentioned limit, the particle-particle collisions have to be taken into account. To investigate the effect of collisions on the deposition velocity, four-way coupled simulations are carried out. The restitution and friction coefficients for particle-particle collisions are en,p = 0.96, et,p = 0.44, µst,p = 0.94 and µkin,p = 0.092. Otherwise, everything is identical to the previous case. The analysis shows only a slight influence of the particle-particle collisions on the particle-wall adhesion for τp+ < 15, while a significant effect is observed for higher values which according to Table 9.1 correspond to a mass loading ηp > 1%. Exemplarily, Figure 9.5 depicts the results 319
9.1 Validation of the Particle-Adhesion Model for dp = 35 µm (τp+ = 20.82 and ηp = 1.37%). Obviously, the consideration of inter-particle collisions reduces the number of deposition processes and the deposition rate. Figure 9.5(a) shows pw that the number of particle-wall collisions Ncol is reduced within the dimensionless time interval 0 ≤ t UB /δ ≤ 32 and augmented during the remaining time, whereas the number of deposition processes Ndep is attenuated during the entire simulation time. The reason for these observations can be explained as follows. The results reveal that if the particle-particle collisions are taken into account, a lower particle velocity is observed in the streamwise and wall-normal directions. The latter corresponds to a lower sedimentation velocity of the particles and hence to a lower number of particle-wall collisions. However, this effect is observed only for t∗ ≤ 32. Afterwards, the number of particle-wall collisions is significantly affected by the particle-particle collisions in the direct vicinity of the wall, since the particles reflected from the wall and collide with other particles located there. Consequently, they are pushed back again towards the wall leading to a higher number of particle-wall collisions. Taking the case for dp = 35 µm into account, Figure 9.5(b) shows that the same deposition rates are predicted for the cases without and with inter-particle collisions during the first stage (t∗ ≤ 7) implying the same rate of reduction of the particle-wall collisions and deposition processes. Afterwards, a lower deposition rate is found for PPC = ON than for PPC = OFF. As evidenced in Figure 9.5(a), the inclusion of the particle-particle collision reduces the number of particle-wall collisions within 7 ≤ t∗ ≤ 32 leading to a lower number of deposition processes than for PPC = OFF. Contrarily, an opposite trend is observed during the remaining simulation time. It can be concluded that at the end of the simulation the inclusion of the inter-particle collisions for ηp > 1% increases the number of particle-wall collisions but slightly reduces the number of deposition processes yielding a lower deposition rate. ×10+6
20.0
PPC = OFF PPC = ON
4.5
×10−2
PPC = OFF PPC = ON
15.0
3.0
Collisions
pw Ndep /Ncol
pw Ndep × 5, Ncol
6.0
Depositions
1.5
10.0
5.0
0.0
0.0 0
10
20
30
t UB /δ (a)
40
50
60
0
10
20
30
40
50
60
t UB /δ (b)
Figure 9.5: Effect of the inclusion of particle-particle collisions (PPC) on: (a) the time history of the accupw mulated number of particle-wall collisions Ncol and deposition processes Ndep and (b) the time history of the pw deposition rate Ndep /Ncol for dp = 35 µm (dp /δ = 17.5 × 10−4 ).
To evaluate the effect of particle-particle collisions on the dimensionless deposition velocity as a function of τp+ , the predictions for PPC = ON are included in Figure 9.4 and compared 320
9. Results for Particle-Wall Adhesion with the reference results mentioned above. Comparing both cases (i.e., PPC = ON/OFF), a noticeable effect of the inter-particle collisions on the deposition velocity is observed for τp+ > 10 (dp > 25 µm). It is visible that the predictions of the newly developed particle-wall adhesion model are in better agreement with the experiments than for the case without inter-particle collisions. Furthermore, compared with the experiments the present model predicts more accurate results than the wetted-wall model and the energy-based model by Fan and Ahmadi (1993). 9.2 Particle-Wall Adhesion in Turbulent Channel Flow
In this section the newly developed particle-wall adhesion model is applied to a particle-laden turbulent vertical channel flow presented in Section 7.2.1. This set-up has the advantage that no sedimentation of the particles takes place, since the gravitational acceleration points in the streamwise direction. Hence, a statistically quasi-steady state of the disperse phase is reached. 9.2.1 Test Case Description
The test case employed was already used to study the particle agglomeration. It includes the three sub-models but without considering particle deposition. In the standard set-up the effect of the subgrid scales of the continuous phase on the particle motion is considered by applying the Langevin model originally proposed by Pozorski and Apte (2009) and adapted by Breuer and Hoppe (2017). The small particles (dp /δ = 2 × 10−4 ) are used and the walls are assumed to be smooth (Rz /δ = 0). As a first step, the agglomeration model is switched off and the focus is on the particle-wall adhesion within the dimensionless time interval ∆T ∗ = 200. Nevertheless, the particle-particle collisions are considered. As will be shown in Section 9.2.2, an almost linear behavior of the accumulated number of pw particle-wall collisions Ncol and deposition processes Ndep is observed for t∗ ≥ 100. Thus, the temporal behavior of these characteristic quantities within the time interval 100 ≤ t UB /δ ≤ 200 is pw characterized in terms of the dimensionless frequency of particle-wall collisions f˜col and depositions ˜ ˜ fdep , respectively. The dimensionless frequency fφ of the quantity Nφ is defined as: Nφ (t∗ = 200) − Nφ (t∗ = 100) . f˜φ = (t∗ = 200) − (t∗ = 100)
(9.2)
For example, the dimensionless deposition frequency f˜φ = f˜dep expresses how many deposition processes occur within a dimensionless time unit. 9.2.2 Results of the New Adhesion Model
In this section two main issues are investigated. First, the effect of the adhesion on the particle-wall collisions is analyzed without taking agglomeration into account. Second, the effect of several simulation settings on the particle-wall adhesion (deposition) is studied. 9.2.2.1 Effect of Adhesion on Particle-Wall Collisions pw Figure 9.6(a) shows the time history of the accumulated number of particle-wall collisions Ncol and deposition processes Ndep . Obviously, a linear growth in time is observed for both quantities
321
9.2 Particle-Wall Adhesion in Turbulent Channel Flow pw yielding a nearly constant deposition rate Ndep /Ncol of about 2.82% as depicted in Figure 9.6(b). ˜ The dimensionless deposition frequency fdep for the present case is about 2.91 × 103 . It is clear that the inclusion of particle-wall adhesion significantly decreases the number of particle-wall collisions. The main reason for this observation is the deposition of particles on the wall, since it leads to a lower number of active particles in the near-wall region and hence to a lower probability of further particle-wall collisions. The explanation for this observation will become clear in the next paragraph.
×10+7
3.0 2.0
4.0
Adhesion = OFF Adhesion = ON Wetted-Wall Model
Collisions
1.0
×10−2
2.0
1.0
Depositions
0.0
0.0 0
50
100
150
200
0
50
t UB /δ
24.00
×10−6
1.8 1.2
hvp i /UB
6.00
150
200
0.2 0.4 0.6 0.8
1
(b)
Adhesion = OFF Adhesion = ON Wetted-Wall Model
12.00
100
t UB /δ
(a)
hαp i
Adhesion = ON
3.0 pw Ndep /Ncol
pw Ndep × 10, Ncol
4.0
3.00
0.67
×10−3
Adhesion = OFF Adhesion = ON Wetted-Wall Model
0.6 0.0 -0.6 -1.2
0.30
-1.8 1
10
y (c)
+
100
1000
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (d)
Figure 9.6: Effect of the adhesion model on the particle-wall collisions for the small particles d∗p = 2 × 10−4 : pw (a) time history of the accumulated number of particle-wall collisions Ncol and deposition processes Ndep , (b) pw time history of the deposition rate Ndep /Ncol , (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed line denotes the global volume fraction of αp = 0.67×10−6 ) and (d) dimensionless averaged particle velocity in the wall-normal direction hvp i /UB . The dots represent the corresponding results using the wetted-wall model. Dimensionless averaging time ∆T ∗ = 200.
Figure 9.6(c) depicts the distribution of the averaged volume fraction hαp i along the channel width at the end of the simulation (i.e., ∆T ∗ = 200). Note that the global volume fraction is αp = 0.67 × 10−6 . It is visible that if the particle-wall adhesion is taken into account, the mean 322
9. Results for Particle-Wall Adhesion volume fraction is noticeably reduced along the channel width due to the deposition of primary particles on the walls. The ratio of the total number of deposited particles to the number of released particles Ndpp /N0 is about 10% at the end of the simulation (Note: Ndpp = Ndep ). However, the particles participating in the particle-wall collisions are only those located in the direct vicinity of the wall. It is observed that at the end of the simulation the number of particles in the two blocks of the computational domain adjacent to the walls (∆y/δ ≈ 0.056) is a factor of about 1.4 higher for the case without particle-wall adhesion than for the case with adhesion. As depicted in Figure 9.6(a), this observation leads to a factor of about 1.8 higher number of particle-wall collisions for the case without adhesion in comparison with the case taking particle-wall adhesion into pw account. The corresponding dimensionless frequency of the particle-wall collisions f˜col significantly 5 5 decreases from about 1.90 × 10 for the case without adhesion to about 1.03 × 10 for the case with adhesion. To allow a direct evaluation of the influence of different parameters on the results, the pw values of f˜col and f˜dep for all cases investigated in the following subsections are listed in Table C.4. pw A second reason for the significant decrease of Ncol is the reduction of the wall-normal velocity after a particle-wall collision with adhesion but without deposition as presented in Appendix C.1. This effect is explained as follows. If a particle collides with the wall, its post-collision velocity becomes smaller than for the case without considering the adhesive impulse. If such a particle is reflected from the wall and collides with other particles in the vicinity of the wall, the probability that the particle reaches the wall again is reduced for the case with adhesion. In order to evaluate whether this phenomenon plays a significant role or not, an additional simulation with adhesion but without inter-particle collisions was carried out and compared with the present case. It is pw found that the number of particle-wall collisions Ncol and the deposition rate are only slightly affected by the inclusion of the particle-particle collisions. Thus, the first reason mentioned above is much more significant for the reduction of the number of particle-wall collisions than the second one. Figure 9.6(d) shows that the mean wall-normal particle velocity hvp i along the channel width is directed towards the walls and significantly increases when including the particle-wall adhesion. That is first of all astonishing but can be explained as follows. The reason for this observation is the fact that if a particle collides with the wall, the absolute value of the post-collision velocity component in the wall-normal direction is reduced for a normal restitution coefficient en,w < 1 (see pw∗ Appendix C.1). This reduction is augmented by the additional adhesive impulse fˆn,c . Additionally, some particles stick to the wall and thus possess a post-collision wall-normal velocity of zero. Consequently, the averaged magnitude of the wall-normal velocity of the particles colliding with the wall is larger for the case with adhesion. It is worth noting that the case including adhesion the most dominant particle-wall collisions are of sticking type (about 95% of the total number of particle-wall collisions). According to Eq. (5.61) it is clear that for sticking collisions the adhesive impulse only influences the wall-normal component of the particle velocity. Hence, the average particle velocities in the streamwise direction (not shown here for brevity) are only very slightly affected by the adhesive impulse. Lastly, to evaluate the effect of the deposition model on the results, the present adhesion model is replaced by the simple wetted-wall model. That means that each particle-wall collision immediately pw leads to a deposition and hence to a deposition rate of unity (i.e., Ndep /Ncol = 100%). Figure 9.6(a) 323
9.2 Particle-Wall Adhesion in Turbulent Channel Flow shows that at the very early stage of the simulation (t∗ < 4) the number of particle-wall collisions is similar to the case without considering the particle-wall adhesion. Due to the significant reduction of the volume fraction in the near-wall region depicted in Figure 9.6(c) owing to deposition, the accumulated number of particle-wall collisions considerably reduces within the remaining time. pw The corresponding dimensionless frequency of particle-wall collisions is f˜col = 6.53 × 103 and thus more than one order of magnitude smaller than for the momentum-based adhesion model (see pw Table C.4). However, since for the wetted-wall model f˜col = f˜dep , its deposition frequency f˜dep is still a factor of about 2.25 higher than observed for the present adhesion model. In summary, this analysis confirms that the wetted-wall model is a very crude assumption leading to an unrealistic deposition behavior which has direct repercussions on the entire disperse phase and for high mass loadings also on the continuous phase. 9.2.2.2 Effect of Different Parameters on Particle-Wall Deposition
In this section the effect of various simulation parameters on the deposition process is studied. In Section 9.2.2.2.1 the effect of the normal restitution coefficient for particle-wall collisions is analyzed. Then, the effect of the diameter of the primary particles is studied in Section 9.2.2.2.2 by using three times larger particles leading to a 27 times higher particle mass loading. Afterwards, the agglomeration model is considered in Section 9.2.2.2.3 and the effect of agglomeration on the particle-wall adhesion and vice versa is investigated. At the end, the effect of wall roughness on the deposition process is examined in Section 9.2.2.2.4 using different values of the mean wall roughness. 9.2.2.2.1 Particle-Wall Restitution Coefficient
The present investigation aims at studying the effect of the normal restitution coefficient for particle-wall collisions en,w on the deposition process, since different values of en,w can be found in the literature. Here, the particle-wall adhesion model is applied for en,w = 0.97 and 0.80, respectively. As expected, Figure 9.7(b) confirms the significant increase of the number of deposition processes Ndep by reducing en,w from en,w = 0.97 to 0.80. Based on the discussion presented in the previous section, this observation can be explained by the reduction of the number of particles located in the direct vicinity of the wall and participating in the particle-wall collisions. At the end of the simulations it is found that the number of particles in the two blocks adjacent to the walls (∆y/δ ≈ 0.056) is a factor of about 1.8 higher for the case with en,w = 0.97 than for en,w = 0.80, pw which correspondingly leads to an about five times higher number of particle-wall collisions Ncol as visible in Figure 9.7(a). Based on the explanation presented in Section 5.2.4, the reason for these observations is clear, since the reduction of en,w increases the difference of the time intervals between the compression and the restitution phases ∆tˆpw∗ leading to a stronger adhesive c pw impulse. Furthermore, the magnitude of the repulsive impulse fˆn,a given by Eq. (5.19) decreases with decreasing en,w . Consequently, the compiled effect of a decreasing repulsive impulse and an increasing adhesive impulse for lower en,w leads to a higher probability of satisfying the deposition condition and hence to a higher number of deposited particles. 324
9. Results for Particle-Wall Adhesion ×10+7
pw Ncol
2.0
2.0
en,w = 0.97 en,w = 0.80
×10+6
en,w = 0.97 en,w = 0.80
1.5
1.5
Ndep
2.5
Collisions
1.0
Depositions
1.0
0.5
0.5 0.0
0.0 0
50
100
150
200
0
50
100
t UB /δ
200
t UB /δ
(a)
(b) 0.6
pw Ndep /Ncol
150
en,w = 0.97 en,w = 0.80
0.4
0.2
0.0 0
50
100
150
200
t UB /δ (c) Figure 9.7: Effect of the normal restitution coefficient en,w on the particle-wall adhesion for small particles pw d∗p = 2 × 10−4 : Time history of (a) the accumulated number of particle-wall collisions Ncol , (b) the accumupw lated number of deposition processes Ndep and (c) the deposition rate Ndep /Ncol .
As clearly visible in Table C.4, the corresponding dimensionless frequency of the particle-wall pw collisions f˜col decreases appreciably from about 1.03 × 105 for en,w = 0.97 to about 0.20 × 105 for en,w = 0.80 and the dimensionless frequency of the deposition processes increases from about 2.91 × 103 for en,w = 0.97 to about 6.10 × 105 for en,w = 0.80. On the basis of these observations, the reduction of the number of particle-wall collisions and the increase of the probability to satisfy the deposition condition due to the altered impulses yield an about eleven times higher deposition pw rate (i.e., Ndep /Ncol ≈ 30.8%) when the normal restitution coefficient for particle-wall collisions is reduced from en,w = 0.97 to 0.80 as depicted in Figure 9.7(c). 9.2.2.2.2 Diameter of the Primary Particles
In practical applications of turbulent particle-laden flows a wide range of diameters of the primary particles can be found. Thus, it is important to study the effect of the particle-wall adhesion for different particle sizes and hence mass loadings. In the present analysis the same number of 325
9.2 Particle-Wall Adhesion in Turbulent Channel Flow particles is used, but the diameter is increased from dp = 4 µm to 12 µm. The corresponding mass loading increases from ηp = 0.12% to 3.32%, respectively. If the particle-wall adhesion is not taken into account, Figure 9.8(a) shows that despite the higher pw mean volume fraction for the large particles a slightly lower number of particle-wall collisions Ncol is predicted in comparison with the small particles (see Figure 9.6(a)). This observation can be attributed to the stronger effect of the subgrid-scale model for the particles on the small particles than on the large ones. ×10+7
4.0
Adhesion = OFF Adhesion = ON
3.0
×10−2
Adhesion = ON
3.0
2.0
pw Ndep /Ncol
pw Ndep × 10, Ncol
4.0
Collisions
1.0
2.0
1.0
Depositions
0.0
0.0 0
50
100
150
200
0
50
t UB /δ
150
200
0.2 0.4 0.6 0.8
1
t UB /δ
(a) 9.00 6.00
100
(b)
×10−4
1.0
Adhesion = OFF Adhesion = ON
×10−3
Adhesion = OFF Adhesion = ON
0.5
hαp i
hvp i /UB
3.00
0.0
-0.5 0.18 0.12
-1.0 1
10
y+ (c)
100
1000
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (d)
Figure 9.8: Effect of the particle diameter on the particle-wall adhesion for the large particles d∗p = 6 × 10−4 : pw (a) time history of the accumulated number of particle-wall collisions Ncol and deposition processes Ndep , (b) pw time history of the deposition rate Ndep /Ncol , (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed line denotes the global volume fraction of αp = 0.18×10−4 ) and (d) dimensionless averaged particle velocity in the wall-normal direction hvp i /UB . Dimensionless averaging time ∆T ∗ = 200.
As depicted in Figure 9.8(a), the number of particle-wall collisions for the large particles is only slightly reduced due to adhesion, whereas the effect is much more pronounced for the small particles. For the large particles, the corresponding dimensionless frequency of the particle-wall pw collisions f˜col decreases from about 1.68 × 105 for the case without adhesion to about 1.52 × 105 326
9. Results for Particle-Wall Adhesion for the case with adhesion (see Table C.4). This outcome can be explained by the lower number of deposited particles (about 6% at the end of the simulation) compared to the case of the small particles (about 10%), since the adhesive impulse is inversely proportional to the diameter of pw∗ ∝ 1/dp . In other words, the adhesive impulse influences the particle-wall the particle, i.e., fˆn,c collision process of the large particles less than for the small particles which on the one hand explains the much smaller effect on the number of particle-wall collisions and on the other hand is the reason for a lower probability of satisfying the deposition condition. The dimensionless deposition frequency f˜dep is about 1.81 × 103 . Thus, the deposition rate displayed in Figure 9.8(b) reduces by a factor of about 2.4 (about 2.82% for dp = 4 µm and 1.19% for dp = 12 µm). Figure 9.8(c) shows that in the present case the mean volume fraction hαp i is only slightly reduced along almost the entire channel width, since the number of deposited particles is relatively low. Furthermore, analog to the corresponding case with the small particles, the mean wall-normal particle velocity is noticeably increased by the inclusion of the adhesive impulse as visible in Figure 9.8(d). The reason for this effect of adhesion was already explained above (i.e., the reduction of the post-collision wall-normal velocity after the particle-wall collisions and the deposition processes). 9.2.2.2.3 Particle Agglomeration
In this section the effect of particle agglomeration on the particle-wall adhesion is investigated. The simulations are carried out including the momentum-based agglomeration model (MAM) with the closely-packed sphere model for modeling the agglomerate structure. Since the adhesive impulse acts on all particles colliding with the walls, primary particles (dp = 4 µm) and arising agglomerates may deposit on the wall if the adhesive impulse is strong enough. Figure 9.9 shows the effect of the agglomeration on the deposition process. ×10+7
2.0
6.0
MAM = OFF MAM = ON
×10−2
MAM = OFF MAM = ON
4.5 pw Ndep /Ncol
pw Ndep × 10, Ncol
2.5
1.5
Collisions
1.0
Depositions
0.5
3.0
1.5
0.0
0.0 0
50
100
t UB /δ (a)
150
200
0
50
100
150
200
t UB /δ (b)
Figure 9.9: Effect of the particle agglomeration predicted by the momentum-based model (MAM) on the particle-wall adhesion for small particles d∗p = 2 × 10−4 : Time history of (a) the accumulated number of pw pw particle-wall collisions Ncol and deposition processes Ndep and (b) the deposition rate Ndep /Ncol .
327
9.2 Particle-Wall Adhesion in Turbulent Channel Flow The agglomeration model only marginally affects the number of particle-wall collisions and deposition processes and consequently the deposition rate. The main reason for this observation is that the number of deposited agglomerates is very small compared to the number of the deposited primary particles (1:1621) as shown in Table 9.2. As mentioned before, the adhesive impulse decreases when the diameter of the particle increases due to agglomeration. Thus, the deposition condition is less likely satisfied when an agglomerate collides with the wall, and hence the overall effect of agglomeration on the deposition process is marginal. This issue is also clearly visible pw based on the dimensionless frequencies of the particle-wall collisions f˜col and deposition processes f˜dep for both cases (Agglomeration = OFF/ON) listed in Table C.4. Primary Particles
N0
Ndpp
active Npp
6,000,000
612,708
5,383,607
Ndag
active Nag
376 2
1459 3
Agglomerates Two-particle Agglomerate Three-particle Agglomerate
0 0
Table 9.2: Number of deposited primary particles Ndpp (without agglomerates) and the number of deposited active and active agglomerates agglomerates Ndag as well as the total number of active primary particles Npp active at the end of the simulation. Nag
Figure 9.10 shows the effect in the other direction, i.e., how the particle-wall adhesion affects the agglomeration process. Despite the marginal effect of agglomeration on the particle-wall adhesion, the latter noticeably affects the dynamic process of particle agglomeration. As evidenced pp in Figure 9.10(a), the adhesion leads to a lower number of particle-particle collisions Ncol and hence to a lower number of agglomeration processes Nagp . The reason for the reduction of the inter-particle collisions is explained as follows. The inclusion of the adhesion reduces the number of primary particles and arising agglomerates in the direct vicinity of the walls, where the highest number of particle-particle collisions and hence agglomeration processes takes place. As a result, a lower mean volume fraction is observed along the channel width, especially in the region near walls as visible in Figure 9.10(c). Note that for the case with adhesion the reduction of the number of inter-particle collisions and agglomeration processes does not automatically lead to a lower agglomeration rate as depicted in Figure 9.10(b). To explain this observation, the distribution of the dimensionless mean wall-normal particle velocity hvp i /UB along the channel width is depicted in Figure 9.11(d). It is clearly visible that for the case with adhesion a significant increase of the absolute value of the post-collision wall-normal velocity is observed. The latter results from the pre- and post-collision wall-normal velocities of the particles colliding with the wall, where the velocity prior to the wall collision is not directly affected by adhesion. The increase of the post-collision wall-normal velocity leads to a pp pp∗ stronger repulsive impulse fˆn,a and a weaker cohesive impulse fˆn,c for subsequent particle-particle collisions. As a result, a significantly lower number of agglomeration processes is observed for the case with adhesion as obvious in Figure 9.10(a). 328
9. Results for Particle-Wall Adhesion In summary, the agglomeration rate at the end of the simulation is reduced form about 0.68% to about 0.45% for the case with adhesion. ×10+5
1.2
Adhesion = OFF Adhesion = ON
4.5
×10−2
Collisions
3.0
1.5
0.6
0.3
Agglomerations
0.0
0.0 0
50
100
150
200
0
50
t UB /δ
24.00
×10−6
150
200
0.2 0.4 0.6 0.8
1
(b) 1.2
Adhesion = OFF Adhesion = ON
12.00
100
t UB /δ
(a)
0.8
hvp i /UB
6.00
hαp i
Adhesion = OFF Adhesion = ON
0.9 pp Nagp /Ncol
pp Nagp × 20, Ncol
6.0
3.00
0.67
×10−3
Adhesion = OFF Adhesion = ON
0.4 0.0 -0.4 -0.8
0.30
-1.2 1
10
y (c)
+
100
1000
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (d)
Figure 9.10: Effect of the particle-wall adhesion on the agglomeration process predicted by MAM for small pp and particles d∗p = 2 × 10−4 : (a) time history of the accumulated number of particle-particle collisions Ncol pp agglomeration processes Nagp , (b) time history of the agglomeration rate Nagp /Ncol , (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed line denotes the global volume fraction of αp = 0.67 × 10−6 ) and (d) dimensionless averaged particle velocity in the wall-normal direction hvp i /UB . Dimensionless averaging time ∆T ∗ = 200 (closely-packed sphere model).
9.2.2.2.4 Wall Roughness
In this section the investigations are extended towards rough walls applying the sandgrain roughness model by Breuer et al. (2012) with the constant Csurface = 3 modeling the surface finishing. In the present analysis the mean roughness is varied: Rz = 0 (smooth wall), 1 µm (low roughness, Rz /δ = 5 × 10−5 ) and 10 µm (standard roughness, Rz /δ = 5 × 10−4 ). This effect is first studied on the large particles (i.e., dp /δ = 6 × 10−4 ) due to its more significant impact than on small particles as shown below. 329
9.2 Particle-Wall Adhesion in Turbulent Channel Flow Different Wall Roughnesses Considering Large Particles pw Figure 9.11(a) shows the time history of the accumulated number of particle-wall collisions Ncol and deposition processes Ndep for the large particles and the three roughness values. Obviously, a linear growth in time is found for both quantities in all three cases yielding nearly constant pw deposition rates Ndep /Ncol as depicted in Figure 9.11(b). It is obvious that if the wall roughness is increased, the number of particle-wall collisions and deposition processes decreases significantly. The deposition rate decreases from about 1.18% (smooth wall) to 0.38% (rough wall).
×10+7
2.5 2.0 1.5
4.0
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
Collisions
1.0
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
2.0
1.0
Depositions
0.5
×10−2
3.0 pw Ndep /Ncol
pw Ndep × 10, Ncol
3.0
0.0
0.0 0
50
100
150
200
0
50
t UB /δ
150
200
0.2 0.4 0.6 0.8
1
t UB /δ
(a)
(b)
×10−4
0.9
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
0.6
hαp i
3.00
hvp i /UB
9.00 6.00
100
0.3
×10−3
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
0.0 -0.3 -0.6
0.18 0.12
-0.9 1
10
y+ (c)
100
1000
-1 -0.8 -0.6 -0.4 -0.2 0
y/δ (d)
Figure 9.11: Effect of different values of the wall roughness on the particle-wall adhesion for large particles pw d∗p = 6 × 10−4 : (a) time history of the accumulated number of particle-wall collisions Ncol and deposition pw processes Ndep , (b) time history of the deposition rate Ndep /Ncol , (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed line denotes the global volume fraction of αp = 0.18 × 10−4 ) and (d) dimensionless averaged particle velocity in the wall-normal direction hvp i /UB . Dimensionless averaging time ∆T ∗ = 200.
330
9. Results for Particle-Wall Adhesion To understand the effect of the wall roughness, the distribution of the mean volume fraction hαp i along the channel width is depicted in Figure 9.11(c) at the end of the simulation. Obviously, the wall roughness strongly reduces the mean volume fraction of the particles in the direct vicinity of the wall. To further analyze this behavior, Figure 9.11(d) shows the dimensionless mean wall-normal particle velocity hvp i /UB along the channel width. It is clearly visible that an increasing roughness leads to a significant decrease of the magnitude of the wall-normal velocity, which is directed towards the wall. Since the wall-normal velocity is the result of the pre- and post-collision wall-normal velocities of the particles colliding with the wall and the velocity prior to the wall collision is not directly affected by the wall roughness, it is obvious that the reason for this observation is an increase of the absolute value of the post-collision wall-normal velocity component due to the wall roughness. The above observation can be explained by the so-called shadow effect (i.e., a particle can not hit a roughness structure which has a negative inclination angle with respect to the particle trajectory). This shadow effect leads to asymmetric probability density functions of the wall inclination angles, where the mean normal vector is turned towards the incoming particle trajectory. Thus, momentum is transferred from the streamwise direction towards the direction normal to the original wall. As a result, the particles migrate away from the wall leading to a lower mean volume fraction in the direct vicinity of the wall as obvious in Figure 9.11(c). A clear evidence for this behavior is the fact that in the present simulation for the rough walls with Rz /δ = 5 × 10−5 about one third of the particle-wall collisions of the large particles are shadow events, whereas for Rz /δ = 5 × 10−4 the ratio increases to about 41%. Note that also for the small particles the ratio of shadow events is on the same level. Different Wall Roughnesses Considering Small Particles
Figure 9.12(a) depicts the time history of the accumulated number of particle-wall collisions pw Ncol and deposition processes Ndep for the small particles taking the same three roughness values into account. Obviously, similar observations can be made as for the large particles. However, the effects are not so pronounced as for the large particles, for example, concerning the effect of the wall roughness on the volume fraction in the vicinity of the wall (see Figure 9.12(c)). The pw most prominent difference between both particle sizes is the deposition rate Ndep /Ncol depicted in Figure 9.12(b), which is between about 2.82% (smooth wall) and 2.04% (rough wall) and thus considerably higher than for the large particles. That is consistent with the theory, since the pw∗ adhesive impulse is inversely proportional to the diameter of the primary particle (fˆn,c ∝ 1/dp ). As a result, the probability of satisfying the deposition condition increases with decreasing the particle size yielding a higher deposition rate. In summary, the wall roughness considerably alters the rebound behavior of the particles at the wall. Due to the increased momentum in wall-normal direction by the shadow effect the probability that the adhesion impulse is overcome is reduced leading to lower deposition rates. The only issue which could not be clarified is the question why the increased roughness has a stronger effect on the large particles than on the small ones. According to theory of the sandgrain roughness model by Breuer et al. (2012), the opposite is expected. However, since the roughness model is not a main topic of the present thesis, this issue is not further analyzed. 331
9.3 Particle-Wall Adhesion in Turbulent Flow Past Inclined Airfoil ×10+7
2.0
6.0
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
1.5 1.0
Collisions
×10−2
4.5 pw Ndep /Ncol
pw Ndep × 10, Ncol
2.5
Depositions
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
3.0
1.5
0.5 0.0
0.0 0
50
100
150
200
0
50
t UB /δ
200
0.2 0.4 0.6 0.8
1
(b)
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
4.00
0.9 0.6
hvp i /UB
×10−6
8.00
hαp i
150
t UB /δ
(a) 16.00
100
0.3
×10−3
Rz /δ = 0 Rz /δ = 5 × 10−5 Rz /δ = 5 × 10−4
0.0 -0.3 -0.6
0.67 0.50
-0.9 1
10
y
+
100
1000
-1 -0.8 -0.6 -0.4 -0.2 0
(c)
y/δ (d)
Figure 9.12: Effect of different values of the wall roughness on the particle-wall adhesion for small particles pw d∗p = 2 × 10−4 : (a) time history of the accumulated number of particle-wall collisions Ncol and deposition pw processes Ndep , (b) time history of the deposition rate Ndep /Ncol , (c) mean particle volume fraction hαp i along the dimensionless wall coordinate y + (the dashed line denotes the global volume fraction of αp = 0.18 × 10−4 ) and (d) dimensionless averaged particle velocity in the wall-normal direction hvp i /UB . Dimensionless averaging time ∆T ∗ = 200.
9.3 Particle-Wall Adhesion in Turbulent Flow Past Inclined Airfoil
In this section the newly developed adhesion model is applied to study a practically relevant turbulent flow past an inclined airfoil with separation, transition and reattachment. This test case mimics in a simplified manner the icing problem and shows the necessity of the new adhesion model. The main outcome was already published in Breuer and Almohammed (2016). 9.3.1 Test Case Description
The test case is taken from Schmidt and Breuer (2014) and Schmidt (2016), who simulated the external flow around a SD7003 airfoil. In the present set-up the Reynolds number is Re = 332
9. Results for Particle-Wall Adhesion U∞ c/νf = 60, 000 based on the chord length c and the free-stream velocity U∞ . An angle of attack of α = 4◦ is applied. The relative thickness and the relative camber of the airfoil are 8.51% and 1.46%, respectively. The flow is characterized by a pressure-induced separation at the suction side of the wing. Due to the adverse pressure gradient a stable laminar separation bubble (LSB) is observed in measurements starting at about 0.25 c. Inside the LSB transition to turbulence starts with the amplification of two-dimensional Tollmien-Schlichting waves. In the separated shear layer Kelvin-Helmholtz instabilities are observed. Due to the onset of turbulence a rapid pressure drop leads to a highly instantaneous reattachment at about 0.7 c. 9.3.1.1 Simulation Set-up
A wall-resolved LES is carried out using the Smagorinsky subgrid-scale model with Cs = 0.1. A block-structured curvilinear C-grid consisting of about 5,800,000 control volumes (CVs) is applied. The width of the computational domain is set to z/c = 0.25 (50 CVs). A constant inflow profile is employed and a convective boundary condition is applied at the outlet. At the lateral edges symmetry boundary conditions and in spanwise direction periodic boundary conditions are used. A no-slip boundary condition is employed at the surface of the airfoil, since the viscous sublayer is + resolved with a wall-normal distance of the center of the first control volume of y1st ≤ 1.5 In the present set-up 100 new monodisperse particles are released at each dimensionless time step ∆t∗ = ∆t U∞ /c = 2.75 × 10−4 randomly distributed in a y-z-plane located at about 1.2 c upstream of the nose. The particles are tracked through the flow field until they either deposit on the walls or leave the domain at the outlet. After a while a total of about 2,300,000 particles are permanently found in the computational domain. Due to the low mass loading considered a one-way coupled simulation is sufficient and hence also no particle-particle collisions and no particle agglomeration are taken into account. The drag, lift and buoyancy forces are considered and the subgrid-scale model for the particles (Langevin) is switched on.
During the initialization phase the injected particles are assumed to collide with an elastic wall of the airfoil taking friction into account. The surface of the airfoil is assumed to be smooth. If the continuous and disperse phase are fully developed, the conditions for the particle-wall collisions are modified, i.e., either the new particle-wall adhesion model described above is considered or for comparison purposes, a wetted-wall assumption is used. In this trivial model it is assumed that a particle deposits on the wall if the center of the particle is within a distance of one radius of the spherical particle to the closest wall, i.e., the particle touches the wall and adheres to the wall. Thus, particle-wall collisions and hence particle rebounds are not considered in the wetted-wall model. The evaluation phase of the particle-wall adhesion model covers a dimensionless time interval of ∆T ∗ = ∆T U∞ /c = 55. 9.3.1.2 Properties of the Particles
The restitution and friction coefficients for the particle-wall collisions as well as the dimensionless values of the density, Hamaker constant, Young’s modulus and Poisson’s ratio are adopted from the test case of the particle-laden turbulent channel flow presented in Section 7.2, whereas the gravitational acceleration is set to zero. Two different particle diameters are taken into account, 333
9.3 Particle-Wall Adhesion in Turbulent Flow Past Inclined Airfoil namely small particles with dp /c = 1 × 10−5 and large particles with dp /c = 5 × 10−5 . Assuming a chord length of c = 1 m, that corresponds to particles with dp = 10 and 50 µm. 9.3.2 Results and Discussion
In this section the results of the new particle-wall adhesion model are compared with the outcome of the wetted-wall model. 9.3.2.1 Two-Phase Flow Including Particle-Wall Adhesion
A first impression on the flow field and the paths of the particles can be gained by looking at Figure 9.13, which depicts a side view of the stream of particles around the airfoil. It is important to note that this is not a contour plot but the particles are shown as scatter points colored according to their instantaneous velocity in mean flow direction (x).
(a) dp /c = 1 × 10−5
(b) dp /c = 5 × 10−5 Figure 9.13: Stream of particles around the airfoil for two different particle diameters colored by the streamwise velocity (snapshot).
Obviously, a significant difference between both cases is observed. As visible in Figure 9.13(a), the small particles (dp /c = 1 × 10−5 ) closely follow the continuous fluid phase. As a result, a nearly perfect coincidence with the contour plot of the instantaneous streamwise fluid velocity depicted in Figure 9.14 is observed. In contrast, Figure 9.13(b) reveals that the large particles (dp /c = 5 × 10−5 ) can not follow the curved surface of the airfoil at the suction side due to their higher inertia. Thus, a limiting particle path exists, which separates the stream of particles from the upper surface of the airfoil. That starts closely behind the nose and ends at about 75% of the chord length. However, this bound is not so sharp. 334
9. Results for Particle-Wall Adhesion
Figure 9.14: Contour levels of the dimensionless instantaneous streamwise fluid velocity uf /U∞ around the airfoil. The predictions of the continuous phase were validated by Schmidt (2016).
9.3.2.2 Deposition pattern
Figure 9.15(a) and (b) depicts the deposition pattern of the small particles at the end of the evaluation phase. It is visible that these particles deposit in the nose region and in the reattachment region on the suction side (x/c > 0.6).
(a) Wetted-Wall Model
(b) Present Adhesion Model Figure 9.15: Deposition pattern at the airfoil surface for dp /c = 1 × 10−5 using (a) the wetted-wall model and (b) the present momentum-based particle-wall adhesion model.
As shown in Table 9.3, the total number of depositions is about 21% higher in the case of the trivial wetted-wall model than for the new adhesion model. Surprisingly, the total number of particle-wall collisions is about a factor of 5 higher for the new adhesion model, which can be explained by multiple repeated collisions of rebounding particles on the airfoil surface. This situation is impossible for the trivial model, since all particles coming into contact with the wall are assumed to directly stick to the surface (i.e., they are omitted from the computational domain). 335
9.3 Particle-Wall Adhesion in Turbulent Flow Past Inclined Airfoil dp /c
Adhesion Model
pw Ncol
Ndep
pw Ndep /Ncol
f˜dep
1 × 10−5
Wetted-Wall Model Present Adhesion Model
68,430 342,798
68,430 56,522
100% 16.48%
1244 1028
5 × 10−5
Wetted-Wall Model Present Adhesion Model
209,859 303,637
209,859 13,808
100% 4.55%
3816 251
pw Table 9.3: Comparison between the accumulated number of particle-wall collisions Ncol , depositions Ndep , the pw ˜ deposition rate Ndep /Ncol and the dimensionless deposition frequency fdep for small (dp /c = 1 × 10−5 ) and large (dp /c = 5 × 10−5 ) particles at the end of the simulation.
An even more significant deviation between the results of both models is observed for the large particles according to Table 9.3. The wetted-wall model predicts an about 15 times larger number of depositions than the new wall adhesion model. That is also directly visible in the deposition patterns in Figures 9.16(a) and (b).
(a) Wetted-Wall Model
(b) Present Adhesion Model Figure 9.16: Deposition pattern at the airfoil surface for dp /c = 5 × 10−5 using (a) the wetted-wall model and (b) the present momentum-based particle-wall adhesion model.
As mentioned before, the dimensionless deposition frequency f˜dep expresses how many depositions occur within a dimensionless time unit. Using this quantity, the differences between the different sizes of the particles and the two deposition models applied are most clearly visible. In case of the small particles the trivial wetted-wall model still seems to do a reasonable job overestimating f˜dep
by only about 21%. In this case the adhesion impulse taken into account in the new particle-wall adhesion model is relatively strong such that about 16.48% of the primary particles colliding with the airfoil surface deposit, whereas in the context of the wetted-wall assumption all particles pw touching the surface for the first time stick to the wall yielding a deposition rate Ndep /Ncol = 100% 336
9. Results for Particle-Wall Adhesion pw∗ (see Table 9.3). For the large particles, however, according to fˆn,c ∝ 1/dp the adhesive impulse is much weaker than for small particles and thus most particles colliding with the wall bounce back from the wall leading to a deposition rate of about 4.55%. Obviously, the wetted-wall model does not allow to reproduce these results, since the deposition rate is always 100% and thus the deposition frequency is more than one order of magnitude too high (see Table 9.3). In summary, with increasing particle size the assumption made for the wetted-wall model no longer holds true and thus a “real” particle-wall adhesion model as suggested in the context of the present thesis is urgently required.
337
CHAPTER 10
CONCLUSIONS AND OUTLOOK
(Magoosh.com, 2017)
10.1 Particle Agglomeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
341
10.2 Droplet Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
344
10.3 Particle-Wall Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
346
10 Conclusions and Outlook The focus of this dissertation is on the modeling, the simulation and the physical analysis of three challenging phenomena having a wide range of major implications in many technologies involving turbulent disperse multiphase flows:
Ê Agglomeration of rigid, dry and electrostatically neutral particles due to particle-particle collisions with cohesion,
Ë Coalescence of surface-tension dominated liquid droplets due to binary droplet collisions and Ì Deposition of rigid, dry and electrostatically neutral particles on smooth and rough walls. Acquiring advanced knowledge about these mechanisms and developing enhanced models based on first principles allows further insights into the complex hydrodynamics of particle-laden turbulent flows. Thus, in the present dissertation the corresponding models for each mechanism are extended and implemented into the in-house CFD code LESOCC (Breuer, 1998, 2000, 2002; Breuer et al., 2012; Breuer and Alletto, 2012; Alletto and Breuer, 2012, 2013). The newly developed models for agglomeration, coalescence and particle-wall adhesion are applied in the framework of a four-way coupled Euler-Lagrange approach using the large-eddy simulation technique and a deterministic collision detection. They are validated based on various turbulent disperse flow configurations. This chapter contains conclusions on key findings obtained from the numerical results of the three phenomena investigated. An outlook is given for future developments of the numerical methods proposed in the thesis. 10.1 Particle Agglomeration
Two different models are extended allowing the simulation of the agglomeration of rigid, dry and electrostatically neutral particles due to particle-particle collisions including cohesion, namely the energy-based (EAM) and the momentum-based agglomeration model (MAM). The original contributions related to this topic include the following key points (Breuer and Almohammed, 2015, 2016; Almohammed and Breuer, 2016a,b):
Ê The energy-based agglomeration model by Alletto (2014) is corrected regarding the treatment of post-collision kinetics of the collision partners without agglomeration taking the cohesion into account. The original agglomeration condition is improved by including an additional condition representing the case of non-separating particles after the collision leading to more realistic predictions than the model by Alletto (2014). Furthermore, in contrast to the original model, the maximum contact pressure required for the determination of the difference in the van-der-Waals energy is realistically modeled.
Ë The momentum-based agglomeration model originally introduced by Kosinski and Hoffmann
(2010) is further extended. In contrast to the original model, where the collision time of a fully elastic, frictionless impact is used to determine the cohesive impulse, in the extended 341
10.1 Particle Agglomeration model only the difference between the times of the restitution and the compression phase is considered depending on the restitution coefficient. Thus, the effect of the cohesive impulse is much weaker and vanishes in the case of a fully elastic collision. In addition, the new model distinguishes the cohesive impulse in the normal and tangential directions. Hence, both components of the total impulse and the agglomeration conditions are more realistically modeled than in the original model yielding more accurate predictions of the agglomeration rate.
Ì Besides the classical volume-equivalent sphere model, three structure models of the arising
agglomerate are introduced and analyzed, namely (i) the inertia-equivalent sphere model, (ii) the closely-packed sphere model and (iii) the porous sphere model.
Í The application area of both agglomeration models is extended towards three-dimensional particle-laden turbulent flows.
As a first step, both agglomeration models are successfully validated in a simple shear flow against a theoretical model. This validation study is carried out in terms of the agglomeration rate and the particle number concentration. It is found that both the energy-based and the momentum-based agglomeration model realistically reproduce the physical behavior of the agglomeration process in disperse shear flows. Taking the advantages and drawbacks of each model and the results of the validation study into account, it can be stated that MAM is superior to EAM due to the reduced necessity of empirical parameters and the noticeably more accurate predictions. Then, the new momentum-based agglomeration model and the newly introduced structure models are applied to a vertical particle-laden turbulent channel flow based on the experiment by Benson et al. (2005). The effect of various simulation parameters on the agglomeration process is studied providing the following key findings: • Although the total numbers of the inter-particle collisions and agglomeration processes predicted by the four agglomerate structure models vary, the global agglomeration rates differ only slightly. Among the structure models proposed, the closely-packed sphere model is used in the parameter study, since this model (i) satisfies the conservation of mass and translational and angular momentum, (ii) reasonably considers the interstitial space between the primary particles included in the agglomerate and (iii) is more efficient than the porous sphere model, since no system of equations has to be solved. • The increase of the normal restitution coefficient for the particle-particle collisions leads to a higher repulsive impulse separating the collision partners from each other and a weaker cohesive impulse attracting the colliding particles. As result, a lower probability of satisfying the first agglomeration condition is found leading to a lower agglomeration rate. • The coefficient of kinetic friction for particle-particle collisions marginally affects the agglomeration process, since it is included only in the second agglomeration condition. Thus, this friction coefficient solely plays an important role for a high number of sliding collisions leading to agglomeration in comparison with sticking ones. However, since the 342
10. Conclusions and Outlook dominant collisions leading to agglomeration are sticking events, higher values of the kinetic friction coefficient only marginally alters the agglomeration rate. • The feedback of the particles on the continuous phase leads to weaker particle fluctuations along the channel width. Hence, a lower number of inter-particle collisions is observed, but the agglomeration processes is enhanced, especially in the near-wall region. As a result, a slightly higher agglomeration rate is predicted. • If the trivial or the extended Langevin subgrid-scale model for the particles is switched on, the fluctuations of the particle velocity are enhanced leading to an increased number of particle-wall collisions. Hence, a lower volume fraction is observed in the direct vicinity of the walls leading to a lower number of inter-particle collisions. Furthermore, the probability of agglomeration reduces due to the stronger fluctuations caused by the subgrid-scale model, especially close to the walls. As a result, the global agglomeration rate is perceptibly reduced. • The lift forces due to velocity shear and particle rotation significantly affect the agglomeration process, since they enhance the migration of primary particles and agglomerates away from the region close to the walls. Thus, lower numbers of inter-particle collisions and agglomeration processes occur in this region. Additionally, the probability of agglomeration reduces due to a stronger impulse of the reflected particles, especially close to the walls. Hence, the global agglomeration rate is noticeably reduced. It is also found that the Saffman lift force is mostly responsible for the different results with and without lift forces. • Keeping the number of primary particles constant and reducing their diameters, a realistic behavior of the predicted agglomeration rate is obtained. Here, the number of inter-particle collisions drastically reduces due to larger mean distances between the particles. However, the number of agglomeration processes enhances, since the cohesive impulse is inversely proportional to the particle diameter leading to a significantly higher agglomeration rate. • For the same mass loading but different diameters of the primary particles, the number of collisions does not only depend on the mass loading, but also on the diameter of the primary particles. A significant difference between the global agglomeration rates is found for the small and the large particles owing to the dependency of the cohesive impulse on the particle diameter as mentioned before. • Assuming rough walls, the particle fluctuations near the wall increase, which enhances the migration of the particles from the region close to the wall leading to a lower volume fraction there. In addition, the particles colliding with the wall lose additional momentum and the reflected ones are largely spread across the channel. The inclusion of the wall roughness leads to a lower global agglomeration rate. It is also found that the agglomeration rate reduces by increasing the value of the mean wall roughness. Lastly, a comparative study of the two different agglomeration models (i.e., EAM and MAM) is carried out. The results reveal that both agglomeration models predict similar trends concerning the physical behavior of the agglomeration process, but the predictions slightly deviate from each 343
10.2 Droplet Coalescence other. The main reason for the different results is attributed to the different formulations of the agglomeration conditions and the treatment of the post-collision kinetics of the collision partners. It is also found that almost the same computational time is required for both agglomeration models. In this test case the agglomeration routine including collision handling requires only about 6% of the total computational time. In summary, it can be concluded that the new energy-based and momentum-based agglomeration models and the newly introduced structure models realistically predict the physical behavior of the agglomeration process in particle-laden turbulent flows. However, MAM is superior to EAM due to the reasons mentioned above. As an outlook for future studies applying the energy-based and momentum-based agglomeration models, the break-up of the formed agglomerates due to aerodynamic forces and inter-particle or particle-wall collisions has to be taken into account. This consideration allows the application of the agglomeration models to disperse multiphase flow systems of industrial interest. In addition, detailed experiments are required to validate the predictions of both agglomeration techniques. 10.2 Droplet Coalescence
A composite collision outcome model is developed enabling the simulation of the coalescence of surface-tension dominated droplets in a gaseous environment. In this composite model, four possible regimes of binary droplet-droplet collisions are taken into account, namely bouncing, fast coalescence, reflexive and stretching separations. An outcome is identified by the collision Weber number and the impact parameter representing the collision angle. The empirical correlations of the bounding curves between the regimes are based on experimental studies and defined as a function of the droplet size ratio. In the present thesis the focus is on conical sprays, since they are encountered in a wide range of practical applications. The main contributions are:
Ê A composite collision outcome model is improved in the framework of the hard-sphere model
to identify the four possible collision regimes of surface-tension dominated droplets without applying the parcel approach widely used for spray simulations.
Ë Assuming that the jet is atomized at the primary break-up length and the droplets are injected
into a constant-volume chamber at a constant back-pressure and ambient temperature (quiescent gas), a model is suggested for the injection of the droplets in simulations of spray systems. This approach includes the determination of the injected mass at each time step and the position and velocity of these droplets.
Ì The initial diameters of the droplets injected at the primary break-up length mimicking the atomization process are modeled based on various number distribution functions.
As a first step, the implementation of the composite collision outcome model is verified using a simplified test case of an inter-impingement spray system (two crossing conical water sprays) with an impingement angle θ = 90◦ . Assuming monodisperse droplets injected at the nozzle exit, it is found that predictions of the composite model perfectly agree with the experimental correlations 344
10. Conclusions and Outlook implemented. In addition, the predicted coalescence rate lies within the range experimentally observed by Brazier-Smith et al. (1972) for monodisperse water droplets. Afterwards, the entire algorithm is used to simulate the injection process of a solid-cone nonevaporating diesel spray into a stagnant nitrogen. The validation study based on the experiment by Gao et al. (2009) is carried out in terms of the spray tip penetration. The evaporation of the droplets and the secondary break-up are not taken into account. The results of the composite model are in excellent agreement with the experimental data and the empirical correlation by Mirza (1991). A parameter study is carried to investigate the effect of different simulation settings on the penetration depth yielding the following scientific insights: • The initial diameters of the droplets injected at the primary break-up length can be described based on different number distribution functions. It is found that the gamma distribution function by Villermaux and Bossa (2009) leads to best predictions of the spray tip penetration against the experimental data and the empirical correlation. • The increase of the length of the primary break-up leads to a lower spray tip penetration. The reason is the increased injection area leading to a different dispersion behavior of the droplets. • The application of the two-way coupling allows the prediction of the induced fluid flow guaranteeing a realistic determination of the drag forces acting on the primary and coalesced droplets. • A four-way coupled simulation using the composite model leads to the best agreement with the experimental data. As expected, the drag force is reduced due the induced fluid flow by taking the feedback effect of the droplets on the fluid phase into account. Additionally, due to coalescence larger droplets are formed, which penetrate into the gaseous environment more quickly than the primary droplets leading to a higher penetration depth than for the case of one-way coupling. Although droplet enlargement due to coalescence leads to higher drag forces, this effect is overcome by the inertia of the droplets. Thus, higher penetration depths are observed for the case of four-way coupling than for two-way coupling. • The inclusion of the added mass and pressure gradient forces only slightly affects the predicted spray tip penetration due to the moderate droplet-to-fluid density ratio in the present set-up (ρd /ρf = 67.23). • The subgrid-scale model for the fluid phase can be neglected in the present set-up due to the short time interval of the injection. Although strong velocity gradients appear in the shear layer of the jet, turbulent structures have not yet developed due to the short injection time. • The classical droplet-droplet collision model without friction leads to a lower penetration depth than the composite model, since this model only considers one regime (i.e., bouncing). The neglect of coalescence means that the initial droplet diameters do not change during the simulation. Thus, the penetration depth predicted by the classical collision model significantly deviates from the experimental results due to the neglect of coalescence. 345
10.3 Particle-Wall Adhesion To improve this composite model, the following key extensions should be considered in future studies. Presently, the primary break-up is modeled using appropriate number distribution functions. However, this assumption can be avoided by incorporating a break-up model to describe this atomization process. The secondary break-up of the liquid droplets due to the aerodynamic forces or inter-droplet collisions has to be also taken into account. Furthermore, in the case of a high temperature of the surrounding gas the evaporation of the injected droplets has to be modeled. 10.3 Particle-Wall Adhesion
A new momentum-based particle-wall adhesion model for rigid, dry and electrostatically neutral particles is suggested. This model takes the physics of particle-wall adhesion more realistically into account than previous deposition models and is thoroughly validated based on experimental data and a common empirical correlation. Compared to other deposition models such as the trivial wetted-wall approach or the energy-based model by Dahneke (1971), the adhesive impulse is not only considered for the deposition condition but also taken into account for all concerns of a particle-wall collision (Almohammed and Breuer, 2016c; Breuer and Almohammed, 2016):
Ê In the criterion for the determination of deposition. Ë In the distinction between sticking and sliding particle-wall collisions. Ì For the determination of the post-collision velocity of the particle if the deposition condition is not satisfied.
In contrast to the model by Kosinski and Hoffmann (2009), the time intervals of the compression and the restitution phases are clearly distinguished. Furthermore, the different effects of the adhesive force in the normal and the tangential direction are separately treated. All these issues lead to a more realistic description of the particle-wall collision process including adhesion. As a first step, the particle-wall adhesion model is successfully validated in a horizontal turbulent channel flow using different diameters of the primary particles. The results of the present adhesion model are found to be in excellent agreement with the experimental data by Kvasnak et al. (1993) and Papavergos and Hedley (1984) and the empirical relation of Wood (1981). Additionally, the new adhesion model predicts more accurate results than the energy-based deposition model by Fan and Ahmadi (1993). The results of the new deposition model are also compared with the wetted-wall model. It is concluded that the present model is clearly superior to the wetted-wall model, which tends to predict too high values of the dimensionless deposition velocity due to the neglect of the rebound effect. Taking the inter-particle collisions into account, a lower deposition velocity is observed leading to a lower deposition rate. Compared with the experiments the predictions of the new particle-wall adhesion model are in better agreement than for the case without inter-particle collisions and more accurate than the results of the wetted-wall and the energy-based model. The adhesion model is then applied to the test case of a vertical particle-laden turbulent channel flow. Here, the effect of various simulation parameters on the particle deposition is studied. A 346
10. Conclusions and Outlook significant effect of the inclusion of the particle-wall adhesion on tiny particles is observed near the walls. The main outcomes of this parameter study are: • Taking the particle-wall adhesion into account, a lower absolute value of the wall-normal velocity of the particle after the impact is observed. If particles deposit on the walls, the post-collision wall-normal velocities of these particles vanish. As a result, higher averaged absolute values of the post-collision wall-normal velocities of the particles impacting the wall are observed in comparison with the case without adhesion. • The deposition of particles on the channel walls leads to a lower number of active particles in the direct vicinity of the walls and thus to significantly lower particle-wall collisions. As a result, a lower volume fraction is observed, especially in the near-wall region. • The reduction of the normal restitution coefficient for particle-wall collisions significantly reduces the number of particle-wall collisions and hence a lower number of deposition processes occurs. However, the deposition rate noticeably increases. This effect is explained by the reduction of the repulsive impulse and the increase of the difference of the time intervals between the compression and the restitution phases leading to a stronger adhesive impulse. Thus, this combined effect leads to a higher probability of satisfying the deposition condition. • Keeping the number of primary particles constant and increasing their diameters, the adhesive impulse decreases, since it is inversely proportional to the diameter of the primary particle. As a result, a lower probability of satisfying the deposition condition is found. • In the present set-up the agglomeration process influences the particle deposition only very slightly, since the total number of deposited agglomerates is very low in comparison with the number of deposited primary particles. However, this effect might change for a longer simulation time, other flow configurations or boundary conditions. • If the particle-wall adhesion is taken into account, the numbers of particle-particle collisions and agglomeration processes are noticeably reduced, but with different rates. This leads to a lower agglomeration rate in comparison with the case without adhesion. The reason for the reduction of the number of particle-particle collisions and agglomeration processes with particle-wall adhesion is a non-negligible number of deposited particles reducing the volume fraction in the near-wall region, where the highest numbers of inter-particle collisions and agglomerations take place. • Assuming rough walls, the rebound behavior of the particles at the wall is significantly altered. The increase of the wall roughness leads to a significant reduction of the number of particle-wall collisions and deposition processes. The reason is attributed to the shadow effect leading to a momentum transfer from the streamwise direction towards the wall-normal direction. As a result, the absolute value of the post-collision wall-normal velocity increases and thus the particles migrate away from the wall leading to a strongly reduced mean volume fraction in the direct vicinity of the wall. Due to the increased momentum in wall-normal 347
10.3 Particle-Wall Adhesion direction by the shadow effect, the probability that the adhesive impulse is overcome is reduced leading to a lower deposition rate. Lastly, the developed adhesion model is used to investigate the particle deposition on an inclined airfoil as a practically relevant turbulent flow. Here, different diameters of the primary particles are used and the results are compared with those of the classical wetted-wall model. It is concluded that if the particle size is increased, the assumption made for the wetted-wall model no longer holds true. Hence, the application of a “real-world” particle-wall adhesion model as proposed in the present thesis is urgently required for complex geometries. As mentioned before, due to a low cover rate of the depositing particles on the wall the deposited particles are presently omitted from the domain. However, this assumption is no longer realistic in the case of high deposition rates. As an outlook for future studies, the present adhesion model has to be accordingly extended. As a simplified approach, it is suggested that the deposited particles are considered as a part of the wall roughness, i.e., deposited particles are transferred into a kind of wall roughness. In other words, the wall roughness has to be locally modified at each time step based on the number of deposited particles in a region of the wall. A further issue to be taken into account is the fact that if a particle collides with the wall, it may break up into smaller particles, which have to be reintroduced into the computational domain as new active particles. Therefore, this break-up process due to particle-wall collisions should also be considered in future studies.
348
Appendix A PARTICLE AGGLOMERATION A.1 Onset of Plastic Yield
It is well known that if the normal or tangential stresses acting on any solid material exceed the maximum limit of elastic deformation, this body undergoes plastic deformation. Based on, for example, Johnson (1985) the distribution of the “Hertzian” pressure p(r) at the radius r within the contact area of the radius a can be expressed as: r2 p(r) = p 1 − 2 a
!1/2
(A.1)
,
where p denotes the maximum contact pressure. Assuming that the colliding spherical particles approach each other along the z-axis, the components of the shear stress vanish for all points along the direction of movement (Popov, 2010). In other words, this means that the principal axes coincide with the axes of the coordinate system, i.e., σxx = σ1 , σyy = σ2 and σzz = σ3 . According to Popov (2010) the components of the stress tensors are given by:
σ2 z σ1 a = = −(1 + ν) 1 − arctan p p a z z2 σ3 =− 1+ 2 p a
!−1
1 z2 + 1+ 2 2 a
!−1
, (A.2)
.
This relation clearly implies that the penetration depth z depends on the Poisson’s ratio of the particle material ν. The maximum shear stress, at which the plastic yield starts, based on the criterion by Tresca (1869) reads as follows: 1 |σ1 − σ3 | (A.3) 2 By substituting Eq. (A.2) into Eq. (A.3), for example, with ν = 0.33 the maximum shear stress τy,max occurs at a dimensionless ratio z/a ≈ 0.49 (Johnson, 1985). In the present study primary particles made of fused quartz with ν = 0.17 are used. Plotting Eqs. (A.2) and (A.3) as a function of the ratio z/a, the maximum shear stress occurs at z/a ≈ 0.44 as depicted in Figure A.1. The maximum contact pressure p required for Eq. (3.22) in the context of the energy-based agglomeration model is estimated as a function of the compressive strength σyield of the particle’s material by the criterion of Mises (1913): τy =
σyield =
r
n
o
0.5 (σ1 − σ2 )2 + (σ1 − σ3 )2 + (σ3 − σ2 )2 = σ1 − σ3 ,
(A.4)
Hence, the onset of the plastic deformation of the colliding particles is obtained by substituting Eq. (A.2) into Eq. (A.4), such that:
σyield z a = −(1 + ν) 1 − arctan p a z
3 z2 + 1+ 2 2 a
!−1
.
(A.5) 349
A.2 Impact Time of Fully Elastic Head-on Collision In case of fused quartz particles with ν = 0.17, for z/a ≈ 0.44 the maximum contact pressure is p = 1.466 σyield . 1.0
−σ3 /p −σ1 /p τy /p
0.8 0.6
τy,max
0.4 0.2 0.0 0.0
0.5
1.0
1.5
2.0
2.5
z/a Figure A.1: Distribution of the principal stresses and the maximum shear stress normalized by the maximum contact pressure p for fused quartz particles with ν = 0.17 (Almohammed and Breuer, 2016b).
A.2 Impact Time of Fully Elastic Head-on Collision
The derivation of the duration of a frictionless, fully elastic head-on impact between two spheres is based on the classical theory of impact by Hertz (1882). It is assumed that two spheres of mass m1 − and m2 are moving with the velocities u− 1 and u2 along the line between their centers as depicted in Figure A.2. Thus, the centers of the spheres approach each other by a displacement δnpp in the collision-normal direction due to the elastic deformation during the impact (Johnson, 1985). According to Fan and Zhu (2005) the first derivative of the normal overlap δnpp with respect to the time t can be expressed in terms of the relative normal velocity between the particles upp rel,n : dδnpp = − upp (A.6) rel,n . dt The normal component of the relative velocity is defined in the context of the present thesis as follows (see Section 2.6): h
i
− − upp rel,n = (u2 − u1 ) · n .
(A.7) h
i
− As mentioned before, the collision occurs only if (u− 2 − u1 ) · n < 0 (see Figure A.2) and hence upp rel,n is always negative. By inserting Eq. (A.7) into Eq. (A.6), the first derivative of the normal overlap δnpp with respect to the time reads:
i dδnpp h − = (u1 − u− ) · n , 2 dt
(A.8)
Commonly (see, e.g., Fan and Zhu, 2005; Marshall, 2009), the term on the right-hand side of the pp above relation is referred to upp 12,n ≡ − urel,n and hence: h
i
− − upp 12,n = (u1 − u2 ) · n .
350
(A.9)
A. Appendix
Particle 1
Collision is possible (u−2 - u−1) n Particle 2
n
u−1
u−2 S2
S1
m2
m1
Figure A.2: Relative motion of a fully elastic, frictionless head-on collision without rotation.
The repulsive force between the particles at any instant during this collision reads: du− du− 1 pp m1 = −f (t) and m2 2 = f pp (t) . dt dt
(A.10)
Thus, the normal component of the impulse can be written as: i d h − 1 pp (u1 − u− ) · n = f (t) , 2 dt m ˆ n
(A.11)
i d h − d2 δnpp − = (u − u ) · n . 1 2 dt2 dt
(A.12)
1 pp d2 δnpp = f (t) . 2 dt m ˆ n
(A.13)
where the effective mass m ˆ is defined in Eq. (2.126). The term on the left-hand side of Eq. (A.11) can be determined by taking the derivative of Eq. (A.8) with respect to the time leading to:
Substituting the above relation into Eq. (A.11) yields:
It is well known (see, e.g., Hertz, 1882; Johnson, 1985) that the normal repulsive force for an elastic contact as a function of the normal overlap δnpp can be expressed as: fnpp (t) = −K pp (δnpp )3/2
with K pp =
4 ˆ 1/2 Epp rˆpp , 3
(A.14)
where Eˆpp and rˆpp are the effective Young’s modulus and radius of the collision partners given by Eqs. (3.94) and (3.92), respectively. By inserting the above relation into Eq. (A.13), the differential equation of δnpp has the form: d2 δnpp K pp pp 3/2 = − (δn ) . dt2 m ˆ
(A.15)
The initial conditions required for the solution of the above relation read (Fan and Zhu, 2005): dδnpp dt
!
= u12,n
and
(δnpp )t=0 = 0 .
(A.16)
t=0
351
A.3 Effect of the Agglomeration Model on the Cohesive Impulse Integration of Eq. (A.15) with respect to δnpp gives: (
dδnpp 1 pp 2 u12,n − 2 dt
!)
2 K pp pp 5/2 (δn ) . 5 m ˆ
=
(A.17)
Thus, the maximum compression is approached at dδnpp /dt = 0, which yields: pp = δn,max
5
m ˆ upp 12,n
4
K pp
2 2/5
15
2 2/5
m ˆ upp 12,n = 1/2 ˆ 16 Epp rˆpp
.
(A.18)
The time of the compression period of the impact is found by a second integration, thus: ∆tpp com
Z1 pp pp pp δn,max d(δn,max /δnpp ) δn,max = 1.47 , = pp pp pp 5/2 1/2 u12,n upp {1 − (δ /δ ) } 12,n n,max n 0
(A.19)
where the integral in the above relation was evaluated numerically by Deresiewicz (1968). After the instant of the maximum compression at ∆tpp com , the spheres rebound during the restitution phase. Since this particle-particle collision is fully elastic and frictionless, the deformation process is perfectly reversible. Therefore, the periods of the restitution and the compression phase are pp equal and hence the total time of impact ∆tpp c = 2 ∆tcom . By substituting Eq. (A.18) into this equation, the impact duration for a fully elastic (en,p = 1), frictionless head-on collision reads: ∆tpp c = 2.87
E ˆ2
pp
1/5
−1 m ˆ 2 rˆpp
[(u1 − u2 ) · n]
.
(A.20)
pp Note that in case of a fully elastic collision (en,p = 1 and hence fn,d = 0 based on Eq. (3.121)) the time interval of the compression phase based on the above relation is identical to that computed in the framework of the extended momentum-based agglomeration model (see Section 3.1.2.3.2).
A.3 Effect of the Agglomeration Model on the Cohesive Impulse
In this section the effect of the normal restitution coefficient en,p on the magnitude of the cohesive pp impulse in the normal direction fˆn,c modeled by Kosinski and Hoffmann (2010) and the extended pp∗ ˆ momentum-based model fn,c is discussed. For this purpose, it is assumed that the values of the Hamaker constant H, the minimum distance δ0 , the effective Young’s modulus Eˆpp, thei h pp pp − effective radius rˆpp and the normal relative particle velocity u12,n = − urel,n = u− 1 − u2 · n are constant. As explained in Section 3.1.2.2.2, in the model by Kosinski and Hoffmann (2010) the pp magnitude of the normal cohesive impulse denoted here fˆn,c can be written based on Eq. (3.95) as follows: pp fˆn,c = 0.478 A ,
(A.21)
where the constant A based on the above assumptions reads: A= 352
E ˆ2
pp
h
m ˆ −3 u− 1
−
u− 2
i
·n
1/5
H 4/5 rˆ = const . δ02 pp
(A.22)
A. Appendix Eq. (A.21) clearly shows no effect of the restitution coefficient on the normal cohesive impulse (see Table A.1). On the other hand, in the extended model the absolute value of the normal cohesive impulse given by Eq. (3.131) can be written as: ˆpp∗ fn,c
1/5
1 9 = 6 4
∆tˆpp∗ c A,
(A.23)
where the value of the dimensionless time difference ∆tˆpp∗ is calculated as a function of the normal c restitution coefficient en,p as presented in Section 3.1.2.3.2. Examples for the ratio of the normal cohesive impulse predicted by Kosinski and Hoffmann (2010) and the extended model are listed in Table A.1. It is worth mentioning that in the present study the standard normal restitution coefficient for particle-particle collisions is en,p = 0.97 leading to a significant difference between the predicted values of the cohesive impulses as clearly visible in Table A.1. en,p
∆tˆpp∗ c
pp /A fˆn,c
0.97 0.80 0.60
0.0215 0.1617 0.3880
0.478 0.478 0.478
ˆpp∗ fn,c /A
4.214 × 10−3 3.169 × 10−2 7.605 × 10−2
pp ˆpp∗ / fn,c fˆn,c
113.4 15.08 6.285
Table A.1: Cohesive impulse in the collision-normal direction based on the model by Kosinski and Hoffmann pp∗ pp for different values of en,p . and on the extended momentum-based agglomeration model fˆn,c (2010) fˆn,c
Note that in comparison with the extended momentum-based agglomeration model the high value of the normal cohesive impulse predicted by the original model of Kosinski and Hoffmann (2010) leads to a much higher number of agglomeration processes Nagp due to particle-particle collisions and hence to a much higher agglomeration rate (see Section 7.2). A.4 Mechanical Properties of Materials
Table A.2 shows typical values of the mechanical properties of materials of the primary particles and the bounding walls used in the present thesis for the simulations of the particle agglomeration. Property
Symbol
Unit
Fused quartz
Polystyrene
Density Hamaker constant Young’s modulus Compressive Strength Poisson’s ratio
ρp H E σyield ν
kg/m3 J N/m2 N/m2 −
2200 6.3 × 10−20 7.2 × 1010 1.1 × 109 0.17
1050 2.28 × 10−21 0.3 × 1010 1.0 × 108 0.34
Table A.2: Mechanical properties of the materials of particles and walls used in the simulations of particle agglomeration (Momentive, 2014; Saint-Gobain, 2015; AZoMATERIALS.com, 2015).
353
A.5 Algorithm of the Extended Momentum-based Agglomeration Model A.5 Algorithm of the Extended Momentum-based Agglomeration Model
START Yes
Every Possible Pair checked ?
No
Select a Pair of Particles
No
END
Particle-Particle Collision ?
No Collision !
Yes
Compute Cohesive Impulses pp pp∗ fˆn,c & fˆn,c
No
No-Slip Condition satisfied ?
Sliding Collision
Yes
Sticking Collision
I. & II. Agglomeration Conditions satisfied ?
No
No
I. Agglomeration Condition satisfied ?
Yes
Yes
Compute Impulse Vector
fˆpp = fˆnpp + fˆtpp
Compute Post-Collision Properties of the Collision Partners + + + u+ 1 u2 & ω 1 ω 2
Momentum-based Agglomeration Model VSM ISM CSM PSM
Compute Kinetics & Structure of the Agglomerate + + x+ ag uag ωag (ρag & dag )
Structure Models of the Agglomerate
Figure A.3: Flowchart of the extended momentum-based agglomeration model implemented in the in-house CFD code LESOCC (adopted from Breuer and Almohammed, 2015). In the present thesis, four models for the structure of the resulting agglomerate are considered: VSM = Volume-equivalent Sphere Model (see Section 3.4.1), ISM = Inertia-equivalent Sphere Model (see Section 3.4.2), CSM= Closely-packed Sphere Model (see Section 3.4.3) and PSM= Porous Sphere Model (see Section 3.4.4).
354
A. Appendix A.6 Packing Fraction of Monodisperse Particles
Napp
fpack
2 3 4 5 6 7 8
0.266835854698718 0.315745968789368 0.420994625052491 0.385065731603426 0.428409926205119 0.458897881424660 0.523598775598298
Table A.3: Examples for the packing fraction fpack of the agglomerate made up of Napp agglomerated monodisperse primary particles based on Packomania (2013). (2 = two-particle agglomerate, 3 = three-particle agglom√ erate, etc.). Note that the maximum packing fraction (or the asymptotic value) is equal to π/(3 2) ≈ 0.7405.
A.7 Agglomerate Diameter Predicted Using Different Structure Models
Table A.4 shows a comparison between the diameters of the agglomerates of the same size predicted by different structure models.
Napp
Combination of agglomerated particles
2 3
1+1 1+2
4 5 6 7 8
1+3 1+4 1+5 1+6 1+7
2+2 2+3 2+4 2+5 2+6
3+3 3+4 3+5
4+4
dag /dp VSM
ISM
CSM
1.259 1.442
1.527 1.904
1.957 2.118
1.587 2.148 − 2.333 1.710 2.288 − 2.604 1.817 2.350 − 2.908 1.912 2.354 − 3.231 2.000 2.317 − 3.564
2.118 2.350 2.410 2.480 2.481
Table A.4: Examples for the spectrum of the diameters of the agglomerates with the same number of agglomerated primary particles Napp predicted by the volume-equivalent (VSM), inertia-equivalent (ISM) and the closely-packed (CSM) sphere model (1= primary particle, 2 = two-particle agglomerate, 3 = three-particle agglomerate, etc.).
A.8 Effect of the Structure Model on the Cohesive Impulse
In this section the influence of the structure model on the cohesive impulse is discussed. Assuming the same value of the Hamaker constant H, the distance between the spheres during the collision δ0 , the Young’s modulus E and the relative particle velocity between the collision partners, Eq. (3.95) 355
A.9 Dimensionless Frequencies Predicted by MAM can be written as: 4 rˆpp m ˆ3
pp =A fˆn,c
!1/5
(A.24)
,
where the constant A is given by:
1/5 1 H h i A = 0.478 2 = const . − 2 δ0 Eˆpp u− 1 − u2 · n
(A.25)
Assuming that a primary particle with a density ρp and a diameter dp agglomerates with an agglomerate including Napp agglomerated primary particles, the effective mass m ˆ and radius rˆ appearing in Eq. (A.24) can be expressed in terms of the density and the diameter of the primary particle as follows: m ˆ =
m1 m2 ∝ ρp d3p m1 + m2
and rˆ =
r1 r2 ∝ dp . r1 + r2
(A.26)
By substituting Eq. (A.26) into relation (A.24), it can be rewritten in the following form: pp ∝ fˆn,c
χ 3/5 ρp
dp
(A.27)
,
where χ is a variable depending on the agglomerate size and the structure model. Table A.5 demonstrates an example for the calculation of the variable χ for the volume-equivalent sphere model and the closely-packed sphere model. Based on Sections 3.4.1 and 3.4.3, in the case of VSM the density of the arising agglomerate is equal to that of the primary particle, while it is reduced when CSM is adapted. Table A.5 shows that, for the same number of primary particles involved in the agglomerate, the value of χ using CSM is larger than that computed by VSM implying a stronger cohesive impulse. Furthermore, χ decreases with the increase of the diameter of the agglomerated particles resulting in a lower probability for a further agglomeration process as expected.
Napp
ρag /ρp
VSM dag /dp
3=1+2 4=1+3
1.00 1.00
1.44 1.54
χ 0.68 0.66
ρag /ρp
CSM dag /dp
χ
0.32 0.42
2.10 2.12
0.78 0.74
3/5 pp Table A.5: Example for the calculation of the cohesive impulse fˆn,c ∝ χ/(ρp dp ) as a function of the density ρp and diameter dp of the primary particle. Here, the volume-equivalent sphere model and the closely-packed sphere model are used for a three-particle and a four-particle agglomerate.
A.9 Dimensionless Frequencies Predicted by MAM pp Table A.6 shows the dimensionless frequencies of the particle-particle collisions f˜col and the agglomeration processes f˜agp determined based on Eq. (7.12) as well as the agglomeration rate
356
A. Appendix pp f˜agp /f˜col for the simulations of particle agglomeration in a particle-laden turbulent channel flow predicted by MAM.
Case
SGS model
No sub-model Two-way coupling
− − Trivial Langevin − Trivial Langevin Trivial
12 µm 12 µm 12 µm 12 µm 12 µm 12 µm 12 µm 12 µm
Langevin
12 µm
− Trivial Langevin
4 µm 4 µm 4 µm
Subgrid-scale model Lift forces Cumulative effect Wall roughness model No sub-model Cumulative effect
dp
pp f˜col
f˜agp
pp f˜agp /f˜col
7.79 × 104 7.52 × 104 6.93 × 104 8.83 × 104 6.03 × 104 5.83 × 104 7.67 × 104 3.88 × 104
5.39 × 102 5.62 × 102 1.33 × 102 0.34 × 102 3.39 × 102 1.29 × 102 0.15 × 102 1.39 × 102
0.69 × 10−2 0.74 × 10−2 0.19 × 10−2 0.04 × 10−2 0.56 × 10−2 0.22 × 10−2 0.02 × 10−2 0.36 × 10−2
0.07 × 104 0.12 × 104 0.30 × 104
1.25 × 102 0.14 × 102 0.21 × 102
18.5 × 10−2 1.11 × 10−2 0.71 × 10−2
4.23 × 104
0.05 × 102
0.01 × 10−2
pp Table A.6: Effect of the sub-models on the dimensionless frequencies of the particle-particle collisions f˜col , pp ∗ the agglomeration processes f˜agp and the agglomeration rate f˜agp /f˜col after a dimensionless time ∆T = 100 predicted by MAM using the closely-packed sphere model.
A.10 Comparison of the Dimensionless Frequencies Predicted by EAM and MAM pp Table A.7 shows a comparison of the dimensionless frequencies of the inter-particle collisions f˜col and the agglomeration processes f˜agp determined based on Eq. (7.12) as well as the agglomeration pp rate f˜agp /f˜col for the simulations of particle agglomeration in a particle-laden turbulent channel flow predicted by EAM and MAM.
Case
Wall
dp
pp f˜col
MAM
EAM
MAM
No sub-models smooth 12 µm 8.14 × 104 Cumulative effect smooth 12 µm 7.72 × 104 Cumulative effect rough 12 µm 4.13 × 104
7.79 × 104 7.67 × 104 4.23 × 104
6.67 × 102 0.05 × 102 0.03 × 102
5.39 × 102 0.15 × 102 0.05 × 102
Cumulative effect
0.30 × 104
0.32 × 102
0.21 × 102
smooth
4 µm
EAM
f˜agp
0.30 × 104
pp Table A.7: Effect of the sub-models on the dimensionless frequencies of the particle-particle collisions f˜col and the agglomeration processes f˜agp after a dimensionless time ∆T ∗ = 100 predicted by EAM and MAM using the closely-packed sphere model. Note that for the simulations with the cumulative effect the Langevin subgrid-scale model is applied.
357
Water Water Water & n-alkane Water Ethanol Water & tetradecane Water, saccharose & PVP
1 2 3 4 5 6 7
Air Air Air Air Air N & He Air
Gas [µm]
[bar] 1.0 300 − 1500 1.0 300 − 1200 1.0 150 1.0 100 − 500 1.0 80 − 300 0.6 − 2.4 200 − 400 1.0 380
dd
pg [m/s]
urel [−]
Wec
1.0 − 2.5 0.3 − 3.0 0.0 − 80 1.15 − 2.6 0.1 − 120 1.0 0.4 − 4.0 0.0 − 60 1.0 & 2.0 5.0 − 100 1.0 & 2.0 3.0 − 12 5.0 − 200 1.0 0.4 − 5.0 0.0 − 80 1.0 0.5 − 4.0 2.0 − 100
[−]
γ = 1/∆
Table B.1: Review on the common experimental studies on binary droplet collisions: (1) Brazier-Smith et al. (1972), (2) Arkhipov et al. (1983), (3) Jiang et al. (1992), (4) Ashgriz and Poo (1990), (5) Estrade et al. (1999), (6) Qian and Law (1997) and (7) Kuschel and Sommerfeld (2013).
Droplet
Study #
Appendix B
DROPLET COALESCENCE
B.1 Experimental Studies on Binary Droplet Collisions
359
B.2 Summary of the Composite Collision Outcome Model B.2 Summary of the Composite Collision Outcome Model
Figure B.1 shows how the improved composite collision model implemented in LESOCC works including the kinetics and structure model of the coalesced droplet (or agglomerate) presented in Section 4.2.5.6. START Yes
Every possible pair checked ?
No
Select a Pair of Liquid Droplets
No
END
Droplet-Droplet Yes? Collision
No collision !
Yes
Compute Dimensionless Quantities
Wec < Webo/f c
Yes
Bouncing (II)
1
2
&
B ≥ Bhelp No No
&
Yes
B ≤ Bcrit
5
Wec ≥ Weov
3
4
No
Yes
rs Ekin ≥ 0.75 Eσrs
rs Ekin < 0.75 Eσrs
Yes No
5b 4b
Stretching Separation (V)
5a 4b
4b
Reflexive Separation (IV)
Compute Post-Collision Properties of the Collision Partners + u+ 1 u2
VSM
Particle-Particle Collision Model
4a Fast Coalescence (III)
Compute Kinetics & Structure of the Newly Formed Droplet (Agglomerate) + + x+ ag uag ωag (ρag & dag )
Structure Model of the Agglomerate
Figure B.1: Flowchart of the Composite Collision Outcome Model (CCOM) for surface-tension dominated droplets (see Section 4.2.5) implemented in the in-house CFD code LESOCC. The agglomerate structure model: VSM = Volume-equivalent Sphere Model (see Section 3.4.1).
360
B. Appendix To identify the outcome of a binary droplet collision using the composite model, Bhelp is first specified as described in Section 4.2.5.1 based on a new integer variable ibofcmod. In CFD code LESOCC this variable is set equal to unity (ibofcmod = 1) for the general case (e.g., hydrocarbon droplets in nitrogen) or to zero (ibofcmod = 0) for the special case (e.g., water droplets in air). As depicted in Figure B.1), the steps below are then followed:
Ê Determine the dimensionless quantities: X Wec =
ρd u2rel
ds
σ
X Bcrit (Wec ) =
, B = 1 −
v ( u u tmin 1.0,
2 1/2 − u · n rel − urel
4.8 f (γ) Wec
!)
and ∆ =
ds dl
,
where f (γ) is given by Eq. (4.81) with γ = 1/∆ X Webo/f c =
∆ (1 + ∆2 ) (4 λ − 12) φl (1 − B 2 )
with φl =
1 − τ2
1 (2 − τ )2 (1 + τ ) for τ > 1.0 , 4
(3 − τ ) for τ ≤ 1.0 , 4 where τ = (1 − B) (1 + ∆) and the shape factor λ = 3.351 (Estrade et al., 1999).
Ë Check bouncing condition: If Wec < Webo/f c and B ≥ Bhelp −→ Bouncing (II) Bhelp =
0
for the general case ,
B
crit
for the special case .
X Keep the diameters of the droplets the same as before the impact X Determine the post-collision velocities of the droplets using Eqs. (4.83) and (4.85)
Ì Otherwise, check the second coalescence condition: If B ≤ Bcrit then X Calculate the effective reflexive kinetic energy rs Ekin
= σπ
d2l
"
1+∆
2
− 1+∆
where ηs = 2 (1 − ξ)2 1 − ξ 2
3 2/3
1/2
ηl = 2 (∆ − ξ)2 ∆2 − ξ 2
Wec 6 ∆ η + η + s l 12 ∆ (1 + ∆3 )2
#
−1
1/2
− ∆3
B (1 + ∆) 2 X Calculate the surface energy of the collision partners ξ=
Eσrs = σ π d2l 1 + ∆3
2/3
361
B.3 Approximation of the Inverse of the Error Function
Í Check the third coalescence condition: 4a
rs If Ekin < 0.75 Eσrs −→ Fast Coalescence (III)
X Calculate the diameter and the density of the newly formed droplet (Section 4.2.5.6):
dag = d3s + d3l
1/3
and ρag = ρd
X Determine the position and velocity of the newly formed droplet (agglomerate) using Eqs. (4.110) and (4.111) X Determine the angular velocity of the arising agglomerate using Eq. (4.112) 4b
rs Otherwise (i.e., Ekin ≥ 0.75 Eσrs ) −→ Reflexive Separation (IV)
X Keep the diameters of the droplets the same as before the impact X Calculate the critical Weber number for reflexive separation:
2
∆ (1 + ∆3 ) −4 1+∆ , Wef c/rs = 3 7 1 + ∆ ∆6 ηs + ηl rs where ηs and ηl are determined as in the previous step for Ekin . X Calculate the dissipation factor fBrs based on Eq. (4.91) X Determine the post-collision velocities of the droplets for reflexive separation using Eqs. (4.83) and (4.90) 3 2/3
2
Î Otherwise (i.e., B > Bcrit (Wec )): Check overlap condition X Determine the overlapping impact parameter based on the polynomial (4.107) Bov = 0.0023669 + 0.5152400 ∆ − 0.3143200 ∆2 − 0.0071091 ∆3 + 0.0330610 ∆4
X Determine the critical overlapping Weber number Weov Weov = 5a
4.8 f (γ) 2 Bov
rs If Wec ≥ Weov and Ekin ≥ 0.75 Eσrs −→ Reflexive Separation (IV)
X Repeat the steps listed in 5b
4b
Otherwise: −→ Stretching Separation (V)
X Keep the diameters of the droplets the same as before the impact X Calculate the factor fBss based on Eq. (4.103) X Determine the post-collision velocities of the droplets for reflexive separation by means of Eqs. (4.104) and (4.105) B.3 Approximation of the Inverse of the Error Function
Giles (2011) approximated the inverse of the error function by the following relation: erf −1 (x) = x ∗ p ,
where the coefficient p is determined as follows: 362
(B.1)
B. Appendix w = - log ((1. d0 - x ) * (1. d0 if ( w . lt .5. d0 ) then w = w - 2.5 d0 p = 2.81022636 d -08 p = 3.43273939 d -07 + p p = -3.5233877 d -06 + p p = -4.39150654 d -06 + p p = 0.00021858087 d0 + p p = -0.00125372503 d0 + p p = -0.00417768164 d0 + p p = 0.246640727 d0 + p p = 1.50140941 d0 + p else w = sqrt ( w ) - 3. d0 p = -0.00020021426 d0 p = 0.000100950558 d0 + p p = 0.00134934322 d0 + p p = -0.00367342844 d0 + p p = 0.00573950773 d0 + p p = -0.0076224613 d0 + p p = 0.00943887047 d0 + p p = 1.00167406 d0 + p p = 2.83297682 d0 + p endif
+ x ))
* * * * * * * *
w w w w w w w w
* * * * * * * *
w w w w w w w w
363
Appendix C PARTICLE-WALL ADHESION C.1 Model Testing C.1.1 Role of the Adhesive Impulse
In order to understand the effect of the adhesive impulse during a particle-wall impact, a simple test case of an oblique collision including friction and adhesion is carried out in a three-dimensional computational domain. In order to simplify the analytic calculation, a two-dimensional motion of the particle is assumed as schematically sketched in Figure C.1. ω−p Particle
SS
u−p
ψ
−
ω−p
−
mp
u−p S
Particle
ψ n
Wall
Figure C.1: A simplified test case for an oblique frictional particle-wall collision including adhesion.
Assuming a smooth wall, the wall-normal unit vector points in y-direction as depicted in Figure C.1. Thus, the post-collision translational and angular velocities of the particle can be simplified. For a sticking particle-wall collision the components of the translational velocity of the particle can be expressed based on relation (5.61a) as: !
dp − 2 − u+ (1 + et,w ) u− ω , p + p = up − 7 2 z pw pw∗ vp+ = vp− + fˆn,a + fˆn,c ,
(C.1)
pw∗ pw where the magnitudes of the adhesive impulse in the wall-normal fˆn,c and wall-tangential fˆn,c direction are given by Eqs. (5.31) and (5.35), respectively. In this setup the angular velocity of the particle has only one component in the z-direction based on Eq. (5.61b):
!
10 dp − ωz+ = ωz− − (1 + et,w ) u− ω . p + 7dp 2 z
(C.2) 365
C.1 Model Testing If the no-slip condition given by Eq. (5.24) is not satisfied, a sliding collision occurs. Thus, based on Eq. (5.62a) the components of the translational velocity of the particle reduce to:
− ˆpw ˆpw u+ p = up − µkin,w fn,a + fn,c ,
pw pw∗ vp+ = vp− + fˆn,a + fˆn,c .
(C.3)
In addition, based on Eq. (5.62b) the component of the angular velocity of the particle in the z-direction reads: 5 pw pw + fˆn,c . (C.4) ωz+ = ωz− − µkin,w fˆn,a dp
In the following the influence of the adhesive impulse on the post-collision velocity of the particle is investigated. In order to minimize the influencing parameters, the fluid (e.g., drag, lift, etc.) and the gravity forces acting on the particle are neglected. A smooth wall and a particle with a diameter of dp = 4 µm is assumed to be made of fused-quartz. Typical values of the mechanical properties of fused quartz required for the momentum-based particle-wall adhesion model are listed in Table C.3. The normal and the tangential restitution coefficients are assumed to be en,w = 0.9 and et,w = 0.44, respectively. In addition, the static and kinetic friction coefficients are µst,w = 0.94 and µkin,w = 0.092, respectively (Breuer and Almohammed, 2015; Almohammed and Breuer, 2016c). To show the influence of the adhesion on the translational and the angular velocities for both collision − − types, the incident angle ψ in Figure C.1 defined as ψ = arctan up /vp is set to 30◦ . The velocity component of the particle before the impact is assumed to be vp− = −0.4 m/s. Furthermore, three different values of the angular velocity are used leading to both collision types, namely ωz− = 0 and 2 × 104 1/s for a sticking collision as well as ωz− = 2 × 106 1/s for a sliding collision. It is worth noting that the values of the translational and angular velocities are chosen such that the particle does not deposit on the wall (i.e., the deposition condition given by Eq. (5.60) is not satisfied). The corresponding results relying on Eqs. (C.1), (C.2), (C.3) and (C.4) are listed in Table C.1. ωz− [1/s]
Collision type
Adhesion
u+ p [m/s]
vp+ [m/s]
ωz+ [1/s]
0.0
sticking
OFF ON
0.1359 0.1359
0.3600 0.3576
2 × 104
sticking
OFF ON
0.1195 0.1195
0.3600 0.3576
−1.1877 × 105 −1.1877 × 105
2 × 106
sliding
OFF ON
0.1610 0.1506
0.3600 0.3576
−1.1934 × 105 −1.1934 × 105 +1.9126 × 106 +1.8996 × 106
Table C.1: Influence of the particle-wall adhesion on a sticking and sliding two-dimensional oblique collision without and with rotation in the z-direction.
It is clear that the inclusion of the particle-wall adhesion reduces the wall-normal velocity component vp+ for both collision types. This effect can also be easily noticed in Eq. (C.1) and 366
C. Appendix + Eq. (C.3). Furthermore, the adhesion influences the tangential component u+ p and the angular ωz velocity after the impact only in case of a sliding particle-wall collision (see Table C.1). In case of a sticking particle-wall collision the particle velocity component u+ p is not affected by the adhesive impulse as visible from Eq. (C.1). In addition, the reduction of the angular velocity in case of a sliding particle-wall collision is due to the increase of the friction between the particle and the wall when taking the adhesive impulse into account (see Eq. (C.4)).
C.1.2 Critical Approach Velocity
As mentioned before, the particle deposits on the wall if its wall-normal velocity component − before the impact vp,n is less than or equal to a critical value (see Sections 5.1.2.2 and 5.1.2.3). To ∗ determine this limiting velocity vp,n the same values of the restitution and friction coefficients and the incident angle ψ as applied to the previous step are used while varying the wall-normal velocity within the range 0.0 ≤ vp− ≤ 0.10 m/s. It is important to stress that the increase of the
pre-collision angular velocity ωz− and hence u− c,t changes the impact type from a sticking to a sliding particle-wall collision (see Table C.1). However, it does not influence the post-collision velocity of the particle in wall-normal direction vp+ . Thus, the value of ωz− is not relevant here. The results for all numerical experiments are shown in Figure C.2. 0.10
[m/s]
Adhesion = OFF 0.08
Adhesion = ON
vp+
0.06 0.04 0.02 0.00 0.00
∗,D ∗,KH vp,n = vp,n
0.02
0.04 0.06 0.08 − vp [m/s]
0.10
Figure C.2: Effect of the particle-wall adhesion on the post-collision velocity of a fused-quartz particle in the wall-normal direction for the oblique collision depicted in Figure C.1 for a normal restitution coefficient en,w = 0.9 (Almohammed and Breuer, 2016c).
As expected, the predictions confirm that the wall-normal velocity after the impact vp+ is reduced when including the adhesion. In the present setup the particle deposits on the wall (i.e., vp+ = − 0) if the pre-collision particle velocity in the wall-normal direction vn,p ≤ 0.62 × 10−2 m/s. However, the magnitude of the critical approach velocity predicted by the energy-based model of Dahneke (1971) and the momentum-based deposition model by Kosinski and Hoffmann (2009) is ∗,D ∗,KH vp,n = vp,n = 2.58 × 10−2 m/s. Thus, the deposition models by Dahneke (1971) and Kosinski and Hoffmann (2009) overpredict the limiting velocity computed by the new momentum-based adhesion model. For en,w = 0.9 the critical approach velocity predicted by the energy-based 367
C.1 Model Testing model by Dahneke (1971) and the momentum-based model by Kosinski and Hoffmann (2009) is about four times higher than that computed by the new momentum-based adhesion model. In other words, this result implies that the application of the deposition models by Dahneke (1971) and Kosinski and Hoffmann (2009) leads to a much higher probability of deposition in comparison with the newly developed adhesion model. As stated by Almohammed and Breuer (2016c), this difference is explained by the more realistic determination of the adhesive impulses pw∗ pw in the wall-normal fˆn,c and wall-tangential direction fˆn,c based on the time intervals of the two different collision periods (i.e., compression and restitution) leading to a weaker effect of the adhesive impulse. C.1.3 Effect of the Normal Restitution Coefficient
In this section the influence of the normal restitution coefficient en,w for particle-wall collisions on ∗ the critical approach velocity vp,n is investigated. For this purpose, the same setup as used in the previous section with different values of en,w is applied to compute the limiting velocity using the newly developed particle-wall adhesion model as well as the deposition models by Dahneke (1971) and Kosinski and Hoffmann (2009). A comparison between the predicted results is displayed in Table C.2. en,w 0.999 0.9 0.8 0.7 0.6 0.5
∗ vp,n [m/s]
∗,D ∗,KH vp,n = vp,n [m/s]
0.14 × 10−3 0.62 × 10−2 1.29 × 10−2 2.15 × 10−2 3.34 × 10−2 5.09 × 10−2
2.38 × 10−3 2.58 × 10−2 4.00 × 10−2 5.44 × 10−2 7.12 × 10−2 9.24 × 10−2
Table C.2: Effect of the normal restitution coefficient for particle-wall collisions en,w on the critical approach ∗ computed by the momentum-based adhesion model in comparison with the deposition models by velocity vp,n Dahneke (1971) and Kosinski and Hoffmann (2009).
Obviously, the critical approach velocity increases and hence the probability of deposition rises when the normal restitution coefficient en,w decreases. In addition, it is clear that for all values of en,w the present momentum-based approach predicts lower critical velocities than the deposition models by Dahneke (1971) and Kosinski and Hoffmann (2009). The newly developed adhesion model is superior to these deposition models and the empirical correlations listed in Section 5.1.1 as shown in Section 9.1.
368
C. Appendix C.2 Algorithm of the Particle-Wall Adhesion Model
Figure B.1 shows how the present particle-wall adhesion model implemented in LESOCC works for smooth and rough walls. START Yes
Every Particle checked ?
No
Select the Particle
No
END
Particle-Wall Collision ?
No Collision !
Sandgrain Roughness Model
Yes
Yes
Rough Wall ? No
Compute Adhesive Impulses pw pw∗ fˆn,c & fˆn,c
Deposition condition Satisfied ?
No
Yes Delete Particle from Domain
Compute the Post-Collision Properties of the Particle + u+ p & ωp
Sticking Collision Sliding Collision
Compute Impulse Vector
fˆpw = fˆnpw + fˆtpw
Momentum-based Particle-Wall Adhesion Model
Figure C.3: Flowchart of the newly developed momentum-based particle-wall adhesion model implemented in the in-house CFD code LESOCC.
369
C.3 Mechanical Properties of Materials C.3 Mechanical Properties of Materials
Table C.3 shows typical values of the mechanical properties of materials of the primary particles and the bounding walls used in the present thesis for the simulations of the particle-wall adhesion. Property
Symbol
Unit
Fused quartz
Glass
Gold
Density Compressive Strength Hamaker constant Young’s modulus Poisson’s ratio
ρp σyield H E ν
kg/m3 N/m2 J N/m2 −
2200 1.1 × 109 6.3 × 10−20 7.2 × 1010 0.17
2500
19,300
11.5 × 10−20 7.0 × 1010 0.22
40 × 10−20 7.9 × 1010 0.42
Table C.3: Mechanical properties of the materials of particles and walls used in the simulations of particle-wall adhesion (Momentive, 2014; Dahneke, 1971; Saint-Gobain, 2015; AZoMATERIALS.com, 2015).
C.4 Effect of the Sub-Models on the Dimensionless Frequencies pw Table C.4 shows the dimensionless frequencies of the particle-wall collisions f˜col and the deposition processes f˜dep determined based on Eq. (9.2) for the simulations of particle-wall adhesion in a particle-laden turbulent channel flow.
Case
dp /δ
en,w
Adhesion
Agglomeration
pw f˜col
f˜dep
1 2 3 4
2 × 10−4 2 × 10−4 2 × 10−4 2 × 10−4
0.97 0.97 0.80 0.97
OFF ON ON ON
OFF OFF OFF ON
1.90 × 105 1.03 × 105 0.20 × 105 1.04 × 105
− 2.91 × 103 6.10 × 103 2.92 × 103
4 5
6 × 10−4 6 × 10−4
0.97 0.97
OFF ON
OFF OFF
1.68 × 105 1.52 × 105
− 1.81 × 103
pw Table C.4: Dimensionless frequencies of the particle-wall collisions f˜col and the deposition processes f˜dep after ∗ a dimensionless time ∆T = 100.
370
Appendix D COMPUTATIONAL METHODOLOGY D.1 Algorithm for the Four-Way Coupled Euler-Lagrange Simulation
Computational Mesh Boundary Conditions Initial Conditions Fluid Properties
START
Solve the Momentum Equations
Define Particle Properties Particle Injection
Solve the Pressure Correction Equation
Particle Tracking Algorithm (One-Way Coupling)
Disperse Phase
Correct the Pressure and the Velocity Field
Outer Iteration
(LAGRANGE)
Particle-Particle Collision?
Yes
Agglomeration Models Coalescence Model
Yes
Particle-Wall Adhesion Model
No
Convergence?
No
Particle-Wall Collision?
Yes
Continuous Phase
No
(EULER) Include the Source Term in the Momentum Equations
Next Time Step
Momentum Exchanage ? (Two-Way Coupling)
No
Time to Stop?
Yes
END
Figure D.1: Flowchart of the four-way coupled Euler-Lagrange simulation framework adopted in the present study. It includes the particle agglomeration (Chapter 3), the droplet coalescence (Chapter 4) and the particlewall adhesion (Chapter 5), which are the main topics of this thesis.
D.2 Trilinear Interpolation
The trilinear interpolation scheme is applied to approximate the fluid quantity φf (e.g., velocity) at the particle position in the computational space. As schematically depicted in Figure D.2, the particle is located within a cube, whose vertices are the centers of the fluid cells surrounding the 371
D.2 Trilinear Interpolation particle. Thus, the cube is partitioned into eight volumes. To trilinearly interpolate the quantity φf at the particle position, the volumes of the partitions of the cube have to be calculated first. The distances between the particle position (ξq , ηq , ζq ) and the node of the cell with the triple index (i, j, k) in c-space are defined by (see Figure D.2): ∆ξq+ = ξq − int (ξq ) ,
∆ηq+ = ηq − int (ηq ) , ∆ζq+
(D.1)
= ξq − int (ζq ) ,
where the indices of the cell containing the particle are i = int(ξq ), j = int(ηq ) and k = int(ζq ). (i+1,j+1,k+1)
(i,j+1,k+1)
Particle path (i,j,k+1)
(i+1,j,k+1) (i+1,j+1,k)
(i,j+1,k)
∆ζq−
∆ηq−
Particle
∆ζq+
(i,j,k)
∆ηq+
(i+1,j,k)
∆ξq+
∆ξq−
Figure D.2: Trilinear interpolation of a fluid quantity at the particle position in the computational space.
As mentioned before, the control volume in the computational space has a dimension of unity in the three directions and hence the distances to the neighboring node are given by: ∆ξq− = 1 − ∆ξq+ ,
∆ηq− = 1 − ∆ηq+ , ∆ζq−
=1−
∆ζq+
(D.2)
.
Based on this arrangement the fluid quantity can be trilinearly interpolated to the particle position φf |particle as follows: φf |particle = φf |(i,j,k) φf |(i,j+1,k) φf |(i+1,j+1,k) φf |(i,j+1,k+1)
h
h
i
h
∆ξq+ ∆ηq− ∆ζq− +
i
h
∆ξq+ ∆ηq− ∆ζq+ +
∆ξq− ∆ηq− ∆ζq− + φf |(i+1,j,k) i
∆ξq− ∆ηq+ ∆ζq− + φf |(i,j,k+1)
h
∆ξq+ ∆ηq+ ∆ζq− + φf |(i+1,j,k+1)
h
i
∆ξq− ∆ηq+ ∆ζq+ + φf |(i+1,j+1,k+1)
h
i i
∆ξq− ∆ηq− ∆ζq+ +
h
i i
(D.3)
∆ξq+ ∆ηq+ ∆ζq+ ,
where the second part of each term on the right-hand side refers to the volume of the corresponding partition of the cube (i.e., weighting factors). Note that this trilinear interpolation is not applied to the fluid velocity at the particle position as described in Section 6.4.3. 372
D. Appendix D.3 Algorithm of the Deterministic Collision Detection
START Yes
No
Every Possible Pair checked ?
Select a Pair of Particles
− − u− rel = u2 − u1 − − x− r = x2 − x1
END
No Collision !
No
− x− r · urel < 0
1
Yes
Compute minimum separation
No
∆tmin & xr,min
2
Collision Conditions satisfied ?
3
Yes
Compute Contact Time ∆tcol
Post-Collision Treatment 6
4
Particle-Particle Collision Model
EAM & MAM
Particle Agglomeration Models
CCOM
Droplet Coalescence Model
Compute Normal Unit Vector
n=
xr,col |xr,col |
5
Figure D.3: Flowchart of the deterministic collision detection including the post-collision treatment: (i) Particle-particle collision model (see Section 2.6), (ii) particle agglomeration models or (iii) droplet coalescence model. Here, EAM = Energy-based Agglomeration Model (see Sections 3.1.1.4 and 3.1.1.5), MAM = Momentum-based Agglomeration Model (see Section 3.1.2.3), CCOM = Composite Collision Outcome Model (see Section 4.2.5).
To identify the collision partners, the steps below are followed:
Ê Check for every possible particle pair whether they are approaching each other within the time
step by calculating the scalar product of their relative position vector x− r and their relative − velocity vector urel : − − − − x− r · urel < 0 with xr = x2 − x1
− − and u− rel = u2 − u1
373
D.4 Determination of the Accumulated Number of Agglomerates
Ë If the above condition is satisfied, calculate the minimum time ∆tmin and the minimum particle separation distance between the collision partners xr,min : x− · u− ∆tmin = − r rel − 2 urel
− and xr,min = x− r + urel ∆tmin
Ì Check the collision conditions: (∆tmin ≤ ∆t) where l12 =
and
(|xr,min | ≤ l12 )
or
1 (d1 + d2 ) 2
− xr
≤ l12 ,
Í If these conditions are fulfilled, determine the collision time ∆tcol :
∆tcol = ∆tmin 1 − 2
where K1 =
q
1 − K1 K2
2
− |x− r | urel
− 2 (x− r · urel )
and K2 = 1 −
2 l12 2 |x− r |
Î Determine the collision-normal unit vector n: n=
xr,col |xr,col |
− with xr,col = x− r + urel ∆tcol
Ï Compute the post-collision properties based on the model applied. In the present study three options are available:
• Particle-particle collision (Section 2.6),
• Particle agglomeration: Energy-based and momentum-based agglomeration model (see Section 3.1), • Droplet coalescence: Composite collision outcome model (see Section 4.2.5). D.4 Determination of the Accumulated Number of Agglomerates
As explained before, a two-particle agglomerate results from one agglomeration process of two primary particles. However, if the total number of primary particles included in this agglomerate Napp > 2, more than one agglomeration process is required to form this agglomerate (see Figure D.4). In general, the number of agglomerates n(n) ag arising during the time step n is not equal to the (n) number of agglomeration processes nagp occurring within this time step, since many primary particles might agglomerate building up a multi-particle agglomerate. In the analysis of the results for the particle agglomeration presented in Chapter 7, the determination of the accumulated number of agglomerates is required. 374
D. Appendix Assuming that the total number of primary particles released into the computational domain before switching on the agglomeration model is N0 (i.e., at the time t = 0), the accumulated (n) number of agglomerates Nag up to the time step n reads: n X
(n) Nag =
n=0
n
o
(n) (n) n(n) ag = Np − N0 − Napp ,
(D.4)
where Np(n) is the total number of active particles in the domain at the nth time step (i.e., the sum of (n) the total number of the remaining primary particles and formed agglomerates). The symbol Napp stands for the accumulated number of agglomerated primary particles included in all agglomerates (i.e., from the instant in time of switching on the agglomeration model) and thus it is given by: (n) Napp =
n X
n(n) app ,
(D.5)
n=0
where n(n) app stands for the number of agglomerated primary particles due to the agglomeration processes occurring within the nth time step. As mentioned in Section 6.4.5, in the in-house code (n) LESOCC the variables N0 , Np(n) and Napp are stored for each time step n in an external file. Thus, the determination of the accumulated number of agglomerates based on Eq. (D.4) is done during the post-processing, since all required quantities are available. The application of this algorithm can be illustrated based on the simplified example depicted in Figure D.4. Here, it is assumed that the number of injected primary particles into the computational domain is equal to N0 = 12. (0)
Np
(0) napp
= 12
Np
(1)
=9
=0
(1) napp
=6
(0)
Napp = 6
(0)
Nag = 3
(1)
Napp = 0
(1)
Nag = 0 (Agglomeration = OFF)
(2)
Np
(2) napp (2) Napp
(n = 1)
=8
Np
(3)
=7
=1
(3) napp
=2
=7
(3) Napp
=9
(3)
(2)
Nag = 3
Nag = 4 (n = 2) Primary particle
(n = 3) Agglomerated primary particle
Agglomerate
Figure D.4: Example for the determination of the total number of agglomerates for different time steps.
375
D.4 Determination of the Accumulated Number of Agglomerates In this example the accumulated number of the formed agglomerates as a function of time is determined by Eqs. (D.4) and (D.5). At the first time step (i.e., n = 1) the accumulated number of (1) (1) primary particles included in all agglomerates based on Eq. (D.5) is equal to Napp = n(0) app + napp = 0 + 6 = 6. The total number of active particles Np(1) = 9. Thus, the accumulated number of (1) agglomerates in this time step based on Eq. (D.4) is equal to Nag = 9 − (12 − 6) = 3. In the following the number of agglomerates predicted in this example is summarized in Table D.1. n
Active particles Np(n)
0 1 2 3
12 9 8 7
Agglomerated primary particles (1) n(n) Napp app 0 6 1 2
0 0+6=6 6+1=7 7+2=9
Agglomerates (n) Nag 0 9 − (12 − 6) = 3 8 − (12 − 7) = 3 7 − (12 − 9) = 4
Table D.1: Total number of agglomerates as a function of time for the example depicted in Figure D.4 with N0 = 12.
376
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Curriculum Vitae PERSONAL DATA
First & Last Name Date of Birth Place of Birth Nationality Marital Status
Naser Almohammed January 20, 1984 Deirazor German Married, One child
ACADEMIC QUALIFICATIONS
04/2013 – 05/2017
PhD Candidate
Helmut-Schmidt-University Hamburg Faculty of Mechanical Engineering, Department of Fluid Mechanics
10/2010 – 02/2013
Master of Science
Technical University of Darmstadt Major: Maschinenbau – Mechanical and Process Engineering
09/2001 – 06/2006
Bachelor of Science
Aleppo University Major: Mechanical and Energy Engineering
09/1998 – 06/2001
High-School Diploma
Technical Gymnasium Muheeb-Alwan
WORK EXPERIENCE
10/2017 – Present
CFD/FEA Simulation Engineer
Continental AG, Division Powertrain, BU Hybrid Electric Vehicles
04/2013 – 05/2017
Research Associate
Helmut-Schmidt-University Hamburg Faculty of Mechanical Engineering, Department of Fluid Mechanics
09/2012 – 03/2013
Graduate Research Assistant
Technical University of Darmstadt Faculty of Mechanical Engineering, Institute of Fluid Systems
08/2008 – 04/2010
Teaching Assistant
Al-Furat University Faculty of Petrochemical Engineering
06/2007 – 07/2008
Mechanical Field Engineer
CONOCOPhillips (Syrian Gas Company)
399
Publications Articles in Peer-Reviewed International Journals
[1] Almohammed, N., Breuer, M., A deterministic composite collision outcome model for surface-tension dominated droplets: A verification and validation study, submitted (2018). [2] Almohammed, N., Breuer, M., Modeling and simulation of particle-wall adhesion of aerosol particles in particle-laden turbulent flows. International Journal of Multiphase Flow, 85: 142-15, (2016). [3] Almohammed, N., Breuer, M., Modeling and simulation of agglomeration in turbulent particle-laden flows: A comparison between momentum-based and energy-based agglomeration models. Powder Technology, 294: 373-402, (2016). [4] Breuer, M., Almohammed, N., Modeling and simulation of particle agglomeration in turbulent flows using a hard-sphere model with deterministic collision detection and enhanced structure models. International Journal of Multiphase Flow, 73: 171-206, (2015). [5] Almohammed, N., Alobaid, F., Breuer, M., Epple, B., A comparative study on the influence of the gas flow rate on the hydrodynamics of a gas-solid spouted fluidized bed using Euler-Euler and Euler-Lagrange/DEM models. Powder Technology, 264: 343-364, (2014). Conference Proceedings
[2] Breuer, M., Almohammed, N., Particle Agglomeration in Turbulent Flows: A LES investigation based on a deterministic collision and agglomeration model. In: Grigoriadis, D.G.E., Geurts, B.J., Kuerten, H., Fr¨ohlich, J., Armenio, V. (Eds.), ERCOFTAC Series 24, Direct and Large-Eddy Simulation X, 10th International ERCOFTAC Workshop on Direct and Large-Eddy Simulation, Limassol, Cyprus, May 27-29, 2015, Springer International Publishing AG, 24: 119-124 (2018). [1] Breuer, M., Almohammed, N., Turbulent particle-laden and droplet-laden flows: An advanced eddy-resolving simulation methodology with deterministic collision, agglomeration and coalescence models, International Symposium on Transport Phenomena and Dynamics of Rotating Machinery, Maui, Hawaii, December 16-21, (2017). [3] Almohammed, N., Breuer, M., Deterministic agglomeration models for the Euler-Lagrange approach using large-eddy simulation, 9th International Conference on Multiphase Flow, Firenze, Italy, May 22-27, (2016). [4] Breuer, M., Almohammed, N., Particle-wall adhesion model for turbulent disperse multiphase flows within an Euler-Lagrange LES approach, 9th International Conference on Multiphase Flow, Italy, May 22-27, (2016). 401
Publications [5] Almohammed, N., Breuer, M., Comparison of an energy-based and a momentum-based agglomeration model within an Euler-Lagrange LES approach, 14th Workshop on Two-Phase Flow Predictions, Halle (Saale), Germany, September 7-10, (2015). Colloquia:
[1] Almohammed, N., Breuer, M., Numerical simulation of droplet-laden turbulent flows including coalescence, Energietechnisches Seminar 2017, Helmut-Schmidt-University, Hamburg, Germany, May 4, (2017).
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