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DISCRETE PHASE SIMULATION OF BUBBLE AND PARTICLE DYNAMICS IN GAS-LIQUID-SOLID FLUIDIZATION SYSTEMS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the
Graduate School of The Ohio State University
By
Jianping Zhang, M.S.
*****
The Ohio State University 1999
Dissertation Committee:
Professor Liang-Shih Fan, Adviser
Professor Ken Cox
Professor Bhavik Bakshi Department of Chemical Engineering ÜMI Number*: 9931708
UMI Microform 9931708 Copyright 1999, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized copying under Title 17, United States Code.
UMI 300 North Zeeb Road Ann Arbor, MI 48103 Copyright by Jianping Zhang 1999 ABSTRACT
The dynamic behavior of the colhsion of two elastic spheres is studied in stagnant viscous fluids for particle Reynolds numbers ranging from 5 to 300. The interactive behavior of these particles is examined both experimentally and theoretically.
The lattice-Boltzmann (LB) simulation is conducted to obtain the detailed three- dimensional flow field and the forces around the particles during the course of collision.
Furthermore, a mechanistic model is developed which accounts for four stages of collision processes. The LB simulation and experimental results lead to an empirical expression for the drag force on the particle during the close-range particle-particle interaction.
A computational scheme for discrete-phase simulation of a gas-liquid-solid fluidization system and a two-dimensional code based on it are developed. In this scheme, the volume-averaged method, the dispersed particle method, and the volume-of- fluid (VOF) method are used for simulating the flow of liquid, solid particles, and gas bubbles, respectively. The close-distance interaction model is included which illustrates the motion of the particle prior to its collision; upon collision, the hard sphere model is employed. The particle-bubble interaction is formulated by incorporating the surface
11 tension force in the particle motion equation. The particle-liquid interaction is brought
into the liquid phase Navier-Stokes equations through the use of the Newton’s third law
of motion.
The simulation results using this scheme are verified for bed expansion and pressure drop in liquid-solid fluidized beds. Simulations of a single bubble rising in a
liquid-solid suspension and the particle entrainment in the freeboard by an emerging bubble are also in agreement with the experimental data. Numerical studies are performed on the bubble and particle dynamics in bubble columns and three-phase fluidized beds at high pressures. Simultaneous simulations for both liquid and gas phases are performed with a volume fraction weighted averaged method to calculate the properties on the interface. The wake interactions and the bubble coalescence and breakup are also simulated and agree with the experimental observations. The effects of pressure and solids holdup on the bubble rise characteristics such as the bubble rise velocity, bubble shape and trajectory are examined. The simulation results of particle- bubble interactions and the maximum stable bubble size agree well with the experimental data.
Ill Dedicated to my family
IV ACKNOWLEDGMENTS
I wish to express my profound gratitude to my adviser. Professor Liang-Shih Fan, for his intellectual support, encouragement, guidance, and enthusiasm which made this dissertation possible. I would like to thank Dr. Ken Cox and Dr. Bhavik Bakshi for serving on my Dissertation Committee and for their valuable comments on my research.
I am also grateflil to Dr. Dewei Qi, Dr. Chao Zhu, and Dr. Yong Li for their assistance during the research included in this dissertation.
Thanks also go to Dr. Peijun Jiang for his help and usefiil discussions during the whole period of my graduate study. The assistance from Mr. D.J. Lee and Mr. Wildon
Pan in part of image analysis process is gratefully acknowledged. To other former and current members in the Particulate and Multiphase Reaction Engineering Lab, Dr.
Katsumi Tsuchiya, Dr. Tao Hong, Dr. Su-Chien Liang, Dr. Jack Reese, Dr. Suhas
Mahuli, Dr. Rajeev Agnihotri, Dr. Xukun Luo, Dr. Shriniwas Chauk, Mr. Guoqiang
Yang, Mr. Raymond Lau, I offer sincere thanks for their support and friendship.
Special thanks go to my psnents, my wife and son, for their understanding, support, encouragement, and endless love. The financial support in part from the National Science Foundation and the Ohio
Supercomputer Center for the research included in this dissertation is gratefully acknowledged.
VI VITA
March 26, 1963 ...... Bom —Ningbo, China
1983...... B.S., Engineering Thermophysics, University of Science and Technology of China
1983 — 1988...... Research Engineer, Research Institute of Jinling Petrochemical Corporation, SINOPEC, China
1988 — 1991...... M.S., Research Associate, Thermoenergy Engineering Institute, Southeast University, China
1991 — 1993...... Research Engineer, Research Institute of Jinling Petrochemical Corporation, SINOPEC, China
1993 — 1994...... Visiting Scholar, Department of Chemical Engineering, The Ohio State University
1994 — 1996...... M.S., Research Associate, Department of Chemical Engineering, The Ohio State University
1996 — present ...... Research Associate, Department of Chemical Engineering, The Ohio State University
PUBLICATIONS
1. Jianping Zhang, Liang-Shih Fan, Chao Zhu, and Dewei Qi, Dynamic behavior of collision of elastic spheres in viscous fluids, Powder Technology, (in print), 1999.
2. Yong Li, Jianping Zhang, and Liang-Shih Fan, Numerical simulation of gas-
vii liquid-solid fluidization systems using a combined CFD-VOF-DPM method: bubble wake behavior, Chem. Eng. Set, (in print), 1999.
3. Jianping Zhang,Yong Li, and Liang-Shih Fan, Discrete phase simulation of gas- liquid-solid fluidization systems: single bubble rising behavior. Powder Technology, (in review), 1999.
4. L.S. Fan, P. Jiang, R. Agnihotri, S.K. Mahuli, J. Zhang, S.Chauk, and A. Ghosh- Dastidar, Dispersion & ultra-fast reaction of calcium-based sorbent powders for 802 and air toxics removal in coal combustion, Chem. Eng. Set, (in print), 1999.
5. Jianping Zhang,Peijun Jiang, and Liang-Shih Fan, Dynamics behavior in transients of solids flow rates in a circulating fluidized bed system, Fluidization IX, ed. by L.-S. Fan and T.M. Knowlton, Engineering Foundation, New York, 1998.
6. Jianping Zhang,Peijun Jiang and Liang-Shih Fan, Flow characteristics of coal ash in a circulating fluidized bed, Ind. Eng. Chem. Res. 37,1998.
7. Peijun Jiang, Jianping Zhangand Liang-Shih Fan, Electrostatic charge effects on the local solids distribution in the upper dilute region of circulatmg fluidized beds. Circulating Fluidized Bed-V, ed. by M. Kwauk and J. Li, Science Press, Beijing, 1997.
8. Katsumi Tsuchiya, Akihiko Furumoto, Liang-Shih Fan and Jianping Zhang, Suspension viscosity of and bubble rise velocity in liquid-solid fluidized beds, Chem. Eng. Set 52, 1997.
9. Shu-Chien Liang, Jianping Zhangand Liang-Shih Fan, Electrostatic characteristics of hydrated lime powder during transport, Ind. Eng. Chem. Res. 35, 1996.
10. Xiangyong Chen, Jianping Zhangand Xianglin Shen, Investigation into control characteristics of fly ash flow in an L-valve, J. Southeast Univ. 22 (in Chinese), 1992.
11. Jianping Zhang, Calculation on separation of diisopropanalamine. Petrochemical Technology (in Chinese), March, 1988.
V lll FIELDS OF STUDY
Major Field: Chemical Engineering
IX TABLE OF CONTENTS
Page ABSTRACT...... ii
DEDICATION...... iv
ACKNOWLEDGMENTS...... v
VITA...... vii
LIST OF TABLES...... xiii
LIST OF FIGURES...... xiv
CHAPTERS:
1. INTRODUCTION...... 1
2. BUBBLE AND PARTICLE DYNAMICS IN GAS-LIQUID-SOLID FLUIDIZATION SYSTEMS - LITERATURE REVIEW
2.1 Introduction ...... 7 2.2 Experimental Studies ...... 8 2.2.1 Single bubble rise characteristics ...... 8 2.2.1.1 Bubble rise velocity ...... 8 2.2.1.2 Bubble shape and wake structure ...... 10 2.2.2 Pressure effects on the bubble rise behavior ...... 12 2.2.3 Particle-particle interactions ...... 14 2.2.4 Particle-bubble interactions ...... 16 2.3 Numerical Studies ...... 18 2.3.1 Eulerian continuum method ...... 18 2.3.2 Dispersed particle/bubble tracking method ...... 19 2.3.3 Surface tracking method ...... 21 2.3.4 Direct simulation and lattice-Boltzmann simulation ...... 23 2.4 Summary...... 24 3. DYNAMIC BEHAVIOR OF COLLISIONS OF ELASTIC SPHERES IN VISCOUS FLUIDS
3.1 Introduction ...... 27 3.2 Lattice-Boltzmann Simulation ...... 28 3.3 Mechanistic Model ...... 31 3.3.1 Contact velocity ...... 31 3.3.2 Compression in a collision ...... 34 3.3.3 Rebound in a collision ...... 36 3.3.4 Velocity after rebound ...... 36 3.4 Experimental...... 37 3.5 Results and Discussion ...... 38 3.6 Concluding Remarks ...... 43 3.7 Nomenclature ...... 44
4. DISCRETE PHASE SIMULATION OF SINGLE BUBBLE BEHAVIOR IN GAS-LIQUID-SOLID FLUIDIZATION SYSTEMS 4.1 Introduction ...... 65 4.2 Governing Equations for Individual Phases ...... 66 4.2.1 Liquid-phase model ...... 66 4.2.2 Gas-phase model ...... 67 4.2.3 Discrete particle model ...... 68 4.3 Particle-Particle Collision Dynamics ...... 70 4.3.1 Liquid shear effect...... 70 4.3.2 Particle collision analysis ...... 72 4.4 Interphase Coupling ...... 73 4.4.1 Coupling between gas and liquid phases ...... 73 4.4.2 Coupling between particle and liquid phases ...... 74 4.4.3 Coupling between particle and gas phases ...... 76 4.5 Numerical Methods ...... 76 4.5.1 Two-step projection method ...... 76 4.5.2 Particle movement ...... 78 4.6 Results and Discussion ...... 79 4.6.1 CVD-2 code ...... 79 4.6.2 Verification of liquid-solid fluidization ...... 80 4.6.3 Bubble rising in liquids-solid fluidized media ...... 81 4.6.4 Bubble wake structure and wake instability ...... 84 4.6.5 Particle entrainment ...... 85 4.7 Concluding Remarks ...... 87 4.8 Nomenclature ...... 88
XI 5. DISCRETE PHASE SIMULATION OF BUBBLE DYNAMICS IN BUBBLE COLUMNS AT ELEVATE PRESSURES
5.1 Introduction ...... 113 5.2 Theoretical and Numerical Models ...... 113 5.2.1 Governing equations ...... 114 5.2.2 Interface tracking ...... 115 5.2.3 Solution technique ...... 116 5.2.3.1 Continuous transition treatment ...... 116 5.2.3.2 Two-step projection method ...... 117 5.3 Simulation Conditions ...... 118 5.3.1 CVD-2 code ...... 118 5.3.2 Bovmdary and initial conditions...... 119 5.3.3 Selection of parameters ...... 119 5.4 Results and Discussion ...... 120 5.4.1 Pressure effect on the bubble rise velocity ...... 120 5.4.2 Pressure effect on the bubble shape...... 121 5.4.3 Pressure effect on the bubble size ...... 122 5.4.4 Experimental validation ...... 124 5.4.5 Bubble-bubble interactions ...... 125 5.5 Concluding Remarks ...... 127 5.6 Nomenclature ...... 127
6. NUMERICAL STUDIES OF BUBBLE AND PARTICLE DYNAMICS IN A THREE-PHASE FLUIDIZED BED AT ELEVATED PRESSURES 6.1 Introduction ...... 167 6.2 Simulation Scheme ...... 168 6.3 Results and Discussion ...... 169 6.3.1 Simulation conditions ...... 169 6.3.2 Bubble rise velocity...... 170 6.3.3 Bubble shape and trajectory ...... 171 6.3.4 Bubble-particle interactions ...... 173 6.3.5 Bubble breakage and stability ...... 174 6.4 Concluding Remarks ...... 175 6.5 Nomenclature ...... 176
7. RECOMMENDATIONS FOR FUTURE RESEARCH ...... 189
7.1 Parallel Computing ...... 189 7.2 Turbulence M odel ...... 190 7.3 Heat, Mass Transfer and Reaction Kinetics ...... 191
BIBLIOGRAPHY...... 192
XII LIST OF TABLES
Table Page
3.1 Velocity vector for cubic lattice in 3-D ...... 47
5.1 Physical properties of the gas and liquid used in the experiments ...... 130
6.1 Physical properties of the gas and liquid used in simulations ...... 178
6.2 Comparison of the simulation results with the experimental data for the rise velocity of a bubble 7.5 mm in diameter ...... 179
X lll LIST OF FIGURES
Figure Page
3.1 The schematic diagram of the experimental setup...... 48
3.2 An example of the spatial resolution of two consecutive images ...... 49
3.3 SEM of the surface of a spherical Teflon particle ...... 50
3.4 Distance change with time for a Teflon ball approaching to another ...... 51
3.5 Velocity change with time for a Teflon ball approaching to another ...... 52
3.6 Settling of spherical particles in aqueous glycerin (in collision with another particle), = 0.053 kg/m-s, rip = 1.27 cm ...... 53
3.7 Settling of spherical particles in aqueous glycerin (in collision
with another particle), // = 0.135 kg/m-s, r ip — 1.27 cm ...... 54
3.8 Settling of a glass bead in water (in collision with wall), rip — 2.0 mm ...... 55
3.9 Comparison of the lattice-Boltzmann simulation and the analytical model with the experimental data for the settling of spherical particles in aqueous glycerin (in collision with another particle), // = 0.053 kg/m-s, rip = 1.27 cm ...... 56
3.10 Comparison of the lattice-Boltzmann simulation and the analytical model with the experimental data for the settling of spherical particles in aqueous glycerin (in collision with another particle), // = 0.135 kg/m-s, rip = 1.27 cm ...... 57
3.11 Comparison of the lattice-Boltzmann simulation and the analytical model with the experimental data for the settling of a glass bead in water (in collision with wall), rip = 2.0 mm ...... 58
XIV 3.12 Velocity vector field of liquid around two particles when the dimensionless separation distance is 0.7 ...... 59
3.13 Velocity vector field of liquid around two particles when the separation distance is of the size of one lattice unit ...... 60
3.14 The particle deformation and velocity change in the compression and rebound process during the elastic collision of two spheres in viscous fluid (symmetric for low viscosity and high elasticity modulus) ...... 61
3.15 The particle deformation and velocity change in the compression and rebound process during the elastic collision of two spheres in viscous fluid (asymmetric for high viscosity and low elasticity modulus) ...... 62
3.16 The motion of particle after contact, = 1.27 cm, /Cp = 2,180 kg/m^, // = 0.053 kg/m-s ...... 63
3.17 Experimental and simulation results of trajectories of a particle colliding with another ...... 64
4.1 Flowchart of the particle-phase simulation...... 92
4.2 Flowchart of the main program for discrete phase simulation ...... 93
4.3 Simulation results of sedimentation of 500 particles in water (dp = 0.8 mm, pp = 2,500 kg/m^, domain size: 2x6 cm^) ...... 94
4.4 Simulation results of fluidization of 500 particles in water (dp = 0.8 mm, pp = 2,500 kg/m^, domain size: 2x6 cm^) ...... 95
4.5 Comparison of the pressure gradient with the weight per unit volume in the liquid-solid fluidized bed ...... 96
4.6 Simulation and experimental results of a bubble rising in a liquid- solid fluidized bed (de = 1.0 cm, 1,000 particles, dp = 1.0 mm, pp = 2,500 kg/m^, domain size: 3x8 cm^, pi = 1,206 kg/m^, pi - 5.29x10'^ kg/m-s, ai = 6.29x10'^ N/m) ...... 97
4.7 Comparison of the simulation and experimental results of the bubble rise velocity ...... 98
XV 4.8 Comparison of the simulation and experimental results of the bubble aspect ratio...... 99
4.9 The simulated velocity vector field of liquid flow surrounding the rising bubble in the liquid-solid fluidized bed (a) t = to, (b) t = to + 0.1 s ...... 100 (c) t = to + 0.2 s, (d) to + 0.3 s ...... 101
4.10 The simulated velocity vector field of liquid flow surrounding the rising bubble in the liquid-solid fluidized bed based on a moving coordinate system. (a) t = to, (b) t = to + 0.1 s ...... 102 (c) t = to + 0.2 s, (d) to + 0.3 s ...... 103
4.11 The simulated velocity vector field of particles following the liquid motion induced by the bubble flow. (a) t = to, (b) t = to + 0.1 s ...... 104 (c) t = to + 0.2 s, (d) to + 0.3 s ...... 105
4.12 S imulation results o f bubble wake structure. (a) t = to, (b) t = to + 0.1 s, (c) t = to + 0.2 s, (d) to + 0.3 s ...... 106
4.13 Simulation of a bubble emerging from a liquid-solid fluidized bed (de = 0.8 cm, 1,000 particles, dp = 0.7 mm, pp = 2,500 kg/m'’, domain size; 6x12 cm^, pi = 1,000 kg/m^, pi = 1.0x10"^ kg/m s, (71 = 7.19x10'^ N/m). (a) t = to, (b) t = to + 0.03 s ...... 107 (c) t = to + 0.06 s, (d) to + 0.09 .s...... 108
4.14 The simulated velocity vector field of liquid flow surrounding a rising bubble in a Liquid-solid fluidized bed. (a) t = to, (b) t = to + 0.03 s ...... 109 (c) t = to + 0.06 s, (d) to + 0.09 .s...... 110
4.15 The simulated velocity vector field of particles following the liquid motion induced by the bubble flow. (a) t = to, (b) t = to + 0.03 s ...... 111 (c) t = to + 0.06 s, (d) to + 0.09 .s...... 112
5.1 Flowchart of discrete phase simulation program including CTT ...... 131
5.2 Simulation results of a rising bubble in cylindrical coordinate system (Dg= 7.5 mm, P = 0.1 MPa). (a) t = to, (b) t = to + 0.08 s ...... 132
xvi (c) t = to + 0.16 s, (d) to + 0.24 s ...... 133
5.3 Simulation results of a rising bubble in cylindrical coordinate system (De = 7.5 mm, P = 19.4 MPa). (a) t = to, (b) t = to + 0.08 .s...... 134 (c) t = to + 0.16 s, (d) to + 0.24 s ...... 135
5.4 Comparison of the rising velocities under different pressures ...... 136
5.5 Simulation results of a rising bubble in Cartesian coordinate system (Dg = 7.5 mm, P = 0.1 MPa). (a) t = to, ...... 137 (b) t = to + 0.08 s, ...... 138 (c) t = to + 0.16 s, ...... 139 (d) to + 0.24 s ...... 140
5.6 Simulation results of a rising bubble in Cartesian coordinate system (Dg = 7.5mm, P = 19.4 MPa). (a) t = to, ...... 141 0?) t = to + 0.08 s, ...... 142 (c) t = to + 0.16 s, ...... 143 (d) to + 0.24 s ...... 144
5.7 Simulation results of a rising bubble in Cartesian coordinate system (Dg =11.0 mm, P = 0.1 MPa). (a) t = to, ...... 145 O?) t = to + 0.06 s, ...... 146 (c) t = to + 0.12 s, ...... 147 (d) to + 0.18 s ...... 148
5.8 Simulation results of a rising bubble in Cartesian coordinate system (Dg =11.0 mm, P = 19.4 MPa). (a) t = to, ...... 149 Ô?) t = to + 0.06 s, ...... 150 (c) t = to + 0.12 s, ...... 151 (d) to+ 0.18 s ...... 152
5.9 Schematic diagram of high-pressure fluidized bed ...... 153
5.10 Bubbles under various pressures (Dg « 0.7 mm) ...... 154
5.11 Bubble aspect ratios under various pressures ...... 155
5.12 Pressure effect on bubble rise velocity ...... 156
xvii 5.13 Bubble trajectories and liquid velocity vector field when two bubbles rising side-by-side at a pressure of 19.4 MPa. (a) t = to, ...... 157 Ô?) t = to + 0.15 s, ...... 158 (c) t = to + 0.3 s, ...... 159 (d) t = to + 0.45 s, ...... 160 (e) t = to + 0.6 s...... 161
5.14 Bubble trajectories and liquid velocity vector field when two bubbles rising in line at a pressure o f 19.4 MPa. (a) t = to, ...... 162 Ô?) t = to + 0.1 s, ...... 163 (c) t = to+ 0.15 s, ...... 164 (d) t = to + 0.2 s, ...... 165 (e) t = to + 0.25 s ...... 166
6.1 The simulated results of the bubble aspect ratio changing with time at various solids holdups and pressures ...... 180
6.2 Bubble rising trajectory at different pressures and solids holdups. (a) p = 0.1 MPa, Ss = 0.384, dy = 7.5 mm ...... 181 (b) p = 17.3 MPa, Es = 0.384, db = 7.5 mm ...... 182 (c) p = 17.3 MPa, Es = 0.545, dy = 7.5 mm ...... 183
6.3 The simulation results of bubble-particle interactions (p = 17.3 MPa, Es = 0.384, db = 7.5 mm). (a) t = to, (b) t = to + 0.15 sec ...... 184 (c) t = to + 0.30 sec, (d) t = to + 0.45 sec ...... 185
6.4 The simulated sequence of bubble shape change and the bubble breakage in a liquid-solid medium (p = 17.3 MPa, Ss = 0.384, db = 10.0 mm). (a) t = to, (b) t = to + 0.05 s, (c) t = to + 0.10 s, (d) t = to + 0.15 s, (e) t = to + 0.20 s ...... 186
6.5 The simulated sequence of bubble shape change and the bubble breakage in a liquid medium (p = 17.3 MPa, Ss = 0., db = 10.0 mm). (a) t = to, (b) t = to + 0.05 s, (c) t = to + 0.10 s, (d) t = to + 0.15 s, (e) t = to + 0.20 s ...... 187
6.6 Velocity vector field of gas and liquid phases before the bubble breakup (p = 17.3 MPa, Es = 0.384, db = 10.0 mm) ...... 188
X V lll CHAPTER 1
INTRODUCTION
Gas-liquid-solid fluidization systems are widely used in physical, chemical, petrochemical, and biochemical processes (e.g., Deckwer, 1985; Fan, 1989).
Applications of three-phase fluidization systems in petrochemical and chemical industries include the hydrotreating of resid oil (H-Oil)/(T-Star), productions of methanol/polyethylene/polyolefins, coal gasification, and coal liquifaction. Sand filter cleaning, crystallization, and drying are some of applications of three-phase fluidization systems in physical processes. Three-phase fluidization systems are found in boichemical applications such as productions of ethanol/enzyme/penicillin, hydrogenation of glucose, and bioleaching of metals. In a gas-liquid-solid fluidized bed, solid particles are suspended by the gas bubbles or the flow of liquid mixed with gas bubbles. The strong interactions among the individual phases provide an intensive mixing which is desirable for effective heat and mass transfer and for chemical reactions.
The design, scale-up, and optimization of three-phase reactor rely heavily on the knowledge of the hydrodynamics as well as heat and mass transfer in the system. There have been extensive investigations on the fluid dynamic characteristics of the gas-liquid-
1 solid fluidization systems in the last three decades (Fan, 1989). In early studies, the macroscopic flow properties such as pressure drop (Ostergaard, 1976; Epstein, 1981), incipient fluidization (Song et al., 1987; Jean, 1988), and hed expansion and regime transition (Fan et al., 1987; Tzeng et al., 1993; Chen et al., 1994) are exploited under various operating conditions. For better predictions of the macroscopic flow properties, empirical correlations and phenomenological models in the mesoscale flow range have been developed to account for single bubble rising characteristics (Darton, 1985;
Tsuchiya and Furumoto, 1995; Tsuchiya et al., 1997), bubble wake dynamics (Miyahara et al., 1989; Tsuchiya and Fan, 1989), bubble-particle interactions (Chen and Fan, 1989), and particle-particle interactions (Tsuji et al. 1981; Zhu et al., 1994; Liang et al., 1996).
However, due to the complex nature of the non-linear interactions of the hydrodynamics among individual phases, it is very difficult to obtain general analytic or empirical models that can satisfy the need in a wide range of applications of the three-phase fluidization systems. It is the recent advancement of the computer power that grants the approaches based the computational fluid dynamic models becoming feasible in providing a generalized solution in predicting the hydrodynamics behavior of the three- phase fluidization systems.
This dissertation presented for the first time a systematic study on the modeling and numerical results of the discrete phase simulation of gas-liquid-solid fluidization systems. The simulation scheme and the computer code developed are able to predict the local and dynamic flow behavior of gas-hquid-solid fluidization including the motion of individual bubbles and particles and interactions among the three-phases. In Chapter 2, the experimental and numerical studies related to the bubble and particle dynamics in three-phase fluidization systems are briefly reviewed.
Chapter 3 studies the dynamic behavior of the collision of two elastic spheres in a stagnant viscous fluid for particle Reynolds numbers ranging firom 5 to 300. The interactive behavior of these particles is examined both experimentally and theoretically.
Specifically, the trajectory and velocity of a moving particle in collinear and oblique collisions with a fixed particle are measured using a high-speed video system and an
Infinity lens. The lattice-Boltzmann (LB) simulation is conducted to obtain the detailed three-dimensional flow field and the forces around the particles during the course of collision. Furthermore, a mechanistic model is developed which accounts for four stages of collision processes, including (1) immediately before the collision, (2) compression during the collision, (3) rebound during the collision, and (4) immediately after the collision. The LB simulation and experimental results lead to an empirical expression for the drag force on the particle during the close-range particle-particle interaction. This close-range interaction between two approaching particles is taken into account in the equation of motion of the particle. The pressure force and added mass force are derived, based on collisions in inviscid fluids, as a function of separation distance. Results of the
LB simulation and prediction by the mechanistic model are in good agreement with the experimental results. The viscous effects on the compression and rebound processes of colliding particles with regard to the elasticity properties of the particle are examined.
The studies are also conducted for simulation based on a hard sphere model, which is commonly used in accounting for the particle collision behavior in gas. The study concludes that the key to proper quantification of the particle collision characteristics in liquid is the ability to accurately predict the particle velocity upon contact.
In Chapter 4, a computational scheme for discrete-phase simulation of a gas- liquid-solid fluidization system and a two-dimensional code based on it are developed. In this scheme, the volume-averaged method, the dispersed particle method, and the volume-of-fluid (VOF) method are used to account for the fiow of liquid, solid particles, and gas bubbles respectively. The gas-liquid interfacial mass, momentum, and energy transfer is described by a continuum surface force (CSF) model. A close-distance interaction (GDI) model is introduced which illustrates the motion of the particle prior to its collision; upon collision, the hard sphere model is employed. The particle-bubble interaction is formulated by incorporating the surface tension force in the equation of motion of particles. The particle-liquid interaction is brought into the liquid phase
Navier-Stokes (N-S) equations through the use of the Newton’s third law of motion. The volume-averaged liquid phase N-S equations are solved using the time-split two-step projection method. The simulation results using this scheme are verified for bed expansion and pressure drop in liquid-solid fluidized beds. Simulations of a single bubble rising in a liquid-solid suspension and the particle entrainment in the freeboard by an emerging bubble are also in qualitative agreement with the experimental data. From the simulation results, it is seen that particles follow the liquid motion induced by the bubble flow. Particles immediately behind the bubble base move at high velocity due to the high liquid velocity. In the case of a low bubble Reynolds number, a closed wake structure with two symmetric vortices is observed in the velocity vector field of the liquid based on the coordinates moving with the bubble. In the case of a high bubble Reynolds number, unstable and periodic asymmetrically shed wakes are obtained in the simulation.
The simulated wake shedding frequency is comparable to the experimental findings.
In Chapter 5, numerical studies are performed on the flow behaviors in high pressure bubble columns. A computational fluid dynamics (CFD) method is used to account for the continuous phases, and the volume tracking represented by the volume- of-fluid (VOF) method is applied to track the free surface of gas bubble, as well as to describe the bubble shapes. Simultaneous simulations for both liquid and gas phases are performed with a volume fraction weighted averaged method to calculate the properties on the interface. In this study, a single gas bubble rising in Paratherm NF heat transfer liquid in a two-dimensional column (height 106 mm, vridth 80 mm) at pressures of 0.1
MPa and 19.4 MPa is investigated. Simulation results indicate that the present numerical approach can not only capture the features of external liquid flow such as bubble wake structure, but also predict the internal flow circulation inside the gas bubble that is hard to be observed in the measurements. It is found that the elevated pressure reduces the bubble rising velocity and maximum stable bubble size, which agrees well with the experimental findings quantitatively. The simulation results reveal that the combined effect of the centrifugal force of internal gas flow circulation and the upward force of external liquid wake flow, plays an important role in breaking up the bubble at elevated pressure, which is different from the mechanism of bubble breakage at ambient pressure.
The wake interactions and the bubble coalescence and breakup are also simulated and agree with the experimental observations. In Chapter 6, the bubhle and particle dynamics in a three-phase fluidized bed at high pressures are numerically studied using the discrete phase simulation. The effects of pressure and solids holdup on the bubble rise characteristics such as the bubble rise velocity, bubble shape and trajectory are examined. The particle-bubble interactions and the maximum stable bubble size are also simulated. The simulations of the bubble rise velocity at various solids holdups and elevated pressures agree well with the experimental data. The simulation results indicate that the bubble aspect ratio increases with the increasing of solids holdup. For the same solids holdup, the bubble trajectory is more tortuous at high pressure than that at low pressure. The trajectory of a rising bubble is more stable at high solids holdup than that at low solids holdup. For the bubble-particle interactions, it is found that most of the particles in front of a bubble passing around the bubble surface while only very few particles break into the rising bubble. The results of the simulated bubble breakage and the maximum stable bubble size agree well with the experimental data and predictions from the mechanistic model.
Finally, some recommendations for further research are described in Chapter 7. CHAPTER 2
BUBBLE AND PARTICLE DYNAMICS IN GAS-LIQUID-SOLID FLUIDIZATION SYSTEMS - LITERATURE REVIEW
2.1 Introduction
The hydrodynamics of a multiphase system is dictated by the motions of individual phases and interactions among them. A fundamental understanding of the bubble and particle dynamics in gas-liquid-solid fluidization systems is essential to a comprehensive description of the overall system performance. This chapter provides a brief review of the experimental and numerical studies on the bubble and particle dynamics in three-phase fluidization systems. The experimental studies being reviewed in this chapter cover the single bubble rise characteristics including the bubble rise velocity, bubble shape and wake structure, pressure effects on the bubble rise characteristics; particle-particle interactions; and particle-bubble interactions. Numerical studies on the simulations of the bubble and particle flows in multiphase systems are reviewed with respect to the two-fluid method, dispersed particle method, surface tracking method, direct numerical simulation, and the lattice-Boltzmann method. Finally, a summary ou the previous studies on the particle and bubble dynamics is given and the objectives of the present study are introduced.
2.2 Experimental Studies
2.2.1 Single bubble rise characteristics
2.2.1.1 Bubble rise velocity
The characteristics of a rising bubble can be described in terms of the rise velocity, shape and motion of the bubble. These rise characteristics are closely associated with the flow and physical properties of the surrounding medium as well as the interfacial properties. The bubble rise velocity is the single most critical parameter in characterizing the hydrodynamics and transport phenomena in liquids and liquid-solid suspensions (Fan and Tsuchiya, 1990). Understanding of the behavior of bubble rise velocity in a liquid-solid medium is important as it is closely associated with the hydrodynamics of gas-liquid-solid fluidization systems. The rise velocity of bubbles in liquid and liquid-solid media is affected by a number of variables including bubble size and shape, surface tension and viscosity of the medium, and densities of the liquid and particles.
Experimental and theoretical work of the single bubble rise velocity in liquids has been reported extensively in the literature. Three approaches have been adopted in the development of predictive equations for the terminal rise velocity of a single bubble in liquids: (1) largely theoretical, accounting for bubble size effect based on a prevailing physical mechanism over a specific bubble size range (e.g., Levich, 1962; Wallis, 1969);
8 (2) largely empirical, covering a wide range of bubble sizes (e.g., Grace et al., 1976); (3) semi-empirical, taking into account prevailing physical mechanisms covering the entire bubble size range (Fan and Tsuchiya, 1990; Tomiyama et al., 1995).
The rise velocities of single bubbles in liquid-solid fluidized beds were found, under certain conditions, similar to those in highly viscous liquids (Massimilla et al.,
1961). Using the Newtonian or non-Newtonian fluid analogy, the liquid-solid suspension was treated as a homogeneous medium, depending on solids holdup and bubble size
(Darton and Harrison, 1974; Verbitskii and Vakhrushev, 1975; Darton, 1985; Tsuchiya et al., 1997). However, when applying to the rise velocity of smaller bubbles (e/y < 1 2 -1 7 mm), the effective viscosity of the liquid-solid medium deviates from the viscosity of the corresponding Newtonian liquid (Darton, 1985; Fan and Tsuchiya, 1990; Tsuchiya et al.,
1997). The deviations which mark the reduction in the bubble rise velocity reflect a significant close-range interactions of the bubble with the liquid-solid mediums or with individual particles.
To account for the heterogeneous characteristics of liquid-solid suspensions with respect to the rising bubble, Jean and Fan (1990) developed a mechanistic model based on a force balance on a rising bubble involving the net gravity, liquid drag, and particle-bubble collision forces. The model can predict the bubble rise velocity in liquid-solid media for small particles {dp < 500 pm), low-to-intermediate solids holdups
{Ss < 0.45) and large spherical bubbles (<7y > 15 mm). Luo et al. (1997b), in studying the bubble rise velocity in liquid-solid suspensions at elevated pressure and temperature, extended Jean and Fan’s model to cover the range of small bubble size ((Jy < 15 mm). However, their results have shown that the model overestimates the bubble rise velocity at low temperatures while better agreement with the experimental data is obtained at higher temperatures.
2.2.1.2 Bubble shape and wake structure
Interactions between a rising gas bubble and the surrounding liquid or liquid-solid medium determine the shape of bubble and the extent of disturbance in the surrounding flow field. Spherical, oblate ellipsoidal, and spherical/ellipsoidal cap are the most frequently observed bubble shapes for small, intermediate, and large bubbles, respectively. Quantitative correlations of the bubble shape pertaining to the bubble size, bubble rise velocity, and the physical properties of the surrounding media are obtained from experimental measurements (e.g., Tadaki and Maeda, 1961; Kojima et al., 1968;
Bhaga and Weber, 1981). However, applicability of the obtained correlations is limited to the experimental conditions. A shape-regime map of single rising bubble was given in terms of three dimensionless groups: the bubble Reynolds number (Reb =yOiûfbt/l///i); the
Eotvos number (Eo = gp\d^/(f)\ and the Morton number (Mo = gii\^/p\c^) (Grace, 1973;
Bhaga and Weber, 1981). This regime diagram shows that bubbles are spherical at either low Reb or low Eo; ellipsoidal at relatively high Rcb and intermediate Eo; and spherical cap at high Rey and Eo. A correlation of bubble aspect ratio, h/b, as a function of Tadaki number (Ta = RebMo*^"^), was developed by Vakhrushev and Efremov (1970), which covers a wide range of physical properties.
10 Information of the bubble shape in three-dimensional liquid-solid suspensions, is very limited due to the experimental difficulties involved in the measurement of bubble shape in the liquid-solid suspensions. It is found that at low solids holdup (Ss < O.I) the bubble shape in liquid-solid fluidized bed can be reasonably represented by the
Vakhrushev-Efremov correlation when the bubble Reynolds number is moderately high
(> 2000) (Miyahara et al., 1988). At lower Rey (< 2000), bubbles in the liquid-solid fluidized bed are flatter than those in pure liquids. At high solids holdups, experimental studies are performed in two-dimensional systems (e.g., Henriksen and Ostergaard, 1974;
Tsuchiya and Fan, 1988; Song, 1989; Tsuchiya et al., 1990). It was found that for small particles (0.1 mm ~ 0.5 mm), the variation in solids holdup has little effect on the bubble aspect ratio. For large particles (0.5 mm ~ 1 mm) at various solids holdups, the bubble shape is flatter than that in pure liquid provided the bubble Reynolds number is not too large (< 2000).
In addition to the bubble rise velocity and bubble shape, the wake behind a rising bubble also plays an important role in the hydrodynamics of the gas-liquid, and gas- liquid-solid multiphase systems. It has been noted that the flow in the wake region controls the diffusion and dispersion of gas from bubbles into liquid media (e.g., Levich,
1962; Brignell, 1974; Yabe and Kunii, 1978). The wake is a key factor responsible for solids mixing in gas-liquid-solid fluidized beds (Fan, 1989). The bubble wake has been identified as the primary factor for the bed contraction phenomenon in three-phase fluidized beds (Massimilla et al., 1959; Stewart and Davidson, 1964; Ostergaard, 1965;
Rigby and Capes, 1970). The wake behind gas bubbles exhibits a variety of structures
11 depending on the shape and size of bubble and the physical properties of surrounding
medium. The wake structures are classified into five categories depending on the
Reynolds number (Fan and Tsuchiya, 1990). These categories are: (1) a steady wake
with a negligible circulation region; (2) a steady wake with a weU-developed circulation
region followed by a laminar stream wise tail; (3) an unsteady wake with large-scale
vortical structures; (4) an unsteady wake with a high degree of turbulence; and (5) a
highly turbulent wake. Quantitative information and empirical correlations of the onset
of the unstable wake and the wake shedding frequency of a rising bubble are reported for
bubbles in the liquids by Lindt (1971, 1972), Hill (1975), Yabe and Kunii (1978); and in
liquid-solid fluidized beds by Tsuchiya and Fan (1988), Miyahara et al. (1988), Song
(1989), and Tsuchya et al. (1990).
2.2.2 Pressure effects on the bubble rise behavior
Many bubble column and three-phase reactors are operated at high pressure
conditions such as in methanol synthesis (at P = 5.5 MPa and T = 0°C), resid hydrotreating (at P = 5.5 to 21 MPa and T= 300 to 425°C), Fischer-Tropsch synthesis (at
P = 1.5 to 5.0 MPa and T = 250°C) and benzene hydrogenation (at P = 5.0 MPa and T =
180°C) (Fox, 1990; Jager and Espinoza, 1995; Saxena, 1995; Mill et al., 1996; Peng et al., 1998). Proper understanding of the pressure effect on the hydrodynamics as well as heat and mass transfer is essential for the development of optimum design and operation criteria for the commercial applications. Studies in the literature have indicated significant eSects of pressure on the hydrodynamics and transport phenomena in bubble
12 columns (Idogawa et al., 1986; Wilkinson, 1991), slurry bubble columns (Tarmy et al.,
1984) and three-phase fluidized bed (Jiang et al., 1992; Luo et al., 1997a). It has been found that an increase in operating pressure leads to an increase in the gas holdup and a decrease in the average bubble size (Chiba et al., 1989; Jiang et al., 1995). Experimental studies have found that the increased gas holdup can be related to the smaller bubble size
(Jiang et al., 1992; Jiang et al., 1997), slower bubble rise velocity (Luo et al., 1997b), reduced bubble coalescence rate and increased bubble breakup rate (Luo et al., 1999) at high, pressures.
The variation in physical properties of the gas and liquid with pressure is considered to be the most fundamental reason for the bubble size reduction. Slowinski et al. (1957) and Massoudi and King (1974) found that the gas-liquid interfacial surface tension decreases approximately linearly with increasing gas pressure. The liquid viscosity below the normal boiling point is not particularly affected by pressure at low to moderate pressures; but under very high pressures, a large increase in viscosity has been noted (Reid et al., 1977). At a higher pressure, the larger gas density leads to a smaller initial bubble size from the gas distributor; the lower surface tension and larger liquid viscosity are contributed to the reduced bubble coalescence rate; and the larger gas density and smaller surface tension are contributed to the increased bubble breakup rate
(Lin et al., 1998; Luo et al., 1999).
13 2.2.3 Particle-particle interactions
To accurately account for the microscopic and macroscopic behavior of particulate flows and three-phase fluidization systems, fundamental knowledge of the mechanism of particle-particle interactions is required. The Stokes equation is suitable in predicting the steady state motion of a single particle settling in an infinite large container with Reynolds number much less than 1. For particles moving in a finite Reynolds number, a drag factor is often used to correct the Stokes equation (e.g., Schiller and
Naumann, 1933). The presence of other particles will significantly affect the particle motion in liquids. In the liquid-solid suspension, the drag force depends strongly on the local liquid holdup in the vicinity of the particle under consideration. The effective drag coefficient can be obtained by the product of the drag coefficient for an isolated particle and a correction factor (Wen and Yu, 1966). When two particles moving in-line in some direction, the drag force of the trailing particle is reduced due to the wake effect of the leading particle. Experimental studies have been conducted to measure the drag force reduction due to the wake effect for different ranges of Reynolds number (Rowe and
Henwood, 1961; Lee, 1979; Tsuji, et al., 1982, Zhu et al, 1994).
Studies on the two-particle collision in liquid media are mainly conducted in creeping fiow conditions. For the steady-state creeping fiow, the hydrodynamic interactions between two spheres can be illustrated by a set of linear equations that can be analytically solved. From the classical lubrication theory (Reynolds, 1886), the lubrication force approaches infinity when the distance between two smooth spheres approaches zero and hence prevents the spheres from direct contact with each other.
14 Solutions to the creeping flow around two approaching particles were obtained by using bi-spherical coordinates (Stimson and Jeffirey, 1926; Goldman et al., 1966; Lin et al.,
1970), method of reflections (Happel and Brenner, 1965), lubrication theory (O’Neill and
Majumdar, 1970), multipole expansions (Jeffirey and Onishi, 1984), and boundary collocation technique (Kim and Mifflin, 1985). All solutions indicate an infinitely high drag at the contact point. However, experimental and theoretical studies have found that physical contact occurs at low Reynolds number when the particle surface is rough. The particle-particle contact would significantly affect the relative motion of spheres (Arp and
Mason, 1977). Smart and Leighton (1989) reported that the lubrication force between a rough surface and an opposing surface is small compared to that between the nominal surfaces even when the separation distance is within the molecular range. Gelbard et al.
(1991) demonstrated the particle contact in a viscous fluid using an acoustic detection technique. Davis (1992) developed a model combining the hydrodynamic interactions with the contact fiction for a sphere settling through a dilute suspension of neutrally buoyant spheres. Davis’s roll/slip model was further experimentally verified by Zeng et al. (1996). Ekiel-Jezewska et al. (1998) presented a simple model of contact mechanics, with pure rolling and rolling with slip, and compared the results with the experimental data.
Studies of particle collision in viscous fluids beyond the creeping flow condition are very limited. Zenit and Hunt (1997) presented an analytical model which accounts for the resistance of the fluid on a spherical particle when the particle closely approaches a flat solid wall. In their model, the pressure imposed on the particle was obtained by the
15 mass and momentum balance of the fluid flow between the solid wall and the particle.
The deceleration of the particle was calculated from the equation of motion by incorporating the net force obtained from integration of the pressure variation over the surface. Their model, however, was developed based on the one-dimensional flow, and thus the complex three-dimensional field of the liquid flow between the colliding particle and the wall was not considered.
2.2.4 Particle-bubble interactions
In a three-phase fluidization system, particle-bubble interactions and their momentum transfer are key factors contributing to the hydrodynamic behavior of the system. Experimental, analytical, and numerical studies of the bubble-particle interactions have been carried out by Henriksen and Ostergaard (1974), Chen and Fan
(1989a, 1989b), and Hong et al. (1999). The basic mechanism of the bubble breakup due to a particle-bubble collision was described by analyzing the force balance for a spherical particle colliding with a spherical-cap bubble (Chen and Fan, 1989a, 1989b). In their analysis, the bubble being penetrated is assumed to deform into a doughnut shape when the particle penetrates the bubble. Using the Boys’ (1890) instability criteria, they illustrated that the particle penetration may not necessarily result in bubble disintegration.
The bubble will break only if the penetrating particle has a diameter greater than the height of the doughnut-shaped bubble. Hong et al. (1999) reported the force variation on a particle during a bubble-particle collision in two different liquid phases: (1) distilled water and (2) 80 wt% glycerin in water solution. In their study, an analytical model was
16 developed to account for the pressure force on a particle induced by bubble-particle contact. They also conducted the numerical simulation for the bubble-particle collision process and found the agreement of the numerical results with the experimental data.
In addition to the particle penetration and the bubble breakage, the particle-bubble interaction also affects other bubble rising behavior such as bubble rise velocity and bubble wake structure. Jean and Fan (1990) developed a mechanistic model based on a force balance on a rising bubble involving the net gravity, liquid drag, and particle-bubble collision forces. Their model is able to predict the bubble rise velocity in liquid-solid media for small particles (Jp < 500 fj.m), low-to-intermediate solids holdups (£^ < 0.45) and large spherical bubbles {d\y> 15 mm). Luo et al. (1997b), in studying the bubble rise velocity in liquid-solid suspensions at elevated pressure and temperature, extended Jean and Fan’s model to cover a smaller bubble size range.
However, their results have shown that the model overestimates the bubble rise velocity at low temperatures while better agreement with the experimental data is obtained at higher temperatures. Due to the complex nature of the hydrodynamics of the bubble flow in liquid-solid media, it is difficult to obtain a mechanistic model that can be used to calculate the bubble rise velocity in various physical properties and system parameters.
For a wider range prediction of the bubble rising characteristics in liquid-solid suspensions and the dynamic behavior of gas-liquid-solid fluidization systems including the interactions of individual bubbles and particles, numerical simulations based on the computational fluid dynamics are required.
17 2.3 Numerical Simulations
2.3.1 Eulerian continuum method
The Eulerian continuum method is an extension of the single-phase fluid dynamic to multiphase flows. In the continuum approach, the individual phases are treated as pseudo-continuous fluids, each being governed by conservation laws in the averaging forms (e.g., Anderson and Jackson, 1967; Ishii, 1975; Drew, 1971; Nigmatulin, 1979;
Zhang and Prosperetti, 1994; and Jackson, 1997). The volume/time- or ensemble- averaged governing equations are solved in the Eulerian coordinates. The averaging procedure leads to a number of undetermined terms that need to be expressed in terms of averaged variables themselves. Rigorous closures of the constimtive relationships are only found in very limited conditions. For example, the closed forms of the motion equations, entirely in terms of the averaged variables, are obtained in the limiting case of
Stokesian particles at very low concentration for a mixture of identical spherical particles and a Newtonian fluid (Jackson, 1997). In practice, in a multiphase system where the interparticle collision dominating, the kinetic theory, which is an analogous of the gas kinetic theory to the multiphase flow, is the most frequently used method to formulate the constitutive relationships reflecting the pseudo-properties of the discrete phases and the interactions between individual phases (Savage, 1983; Gidaspow, 1993).
A numerical technique, called the implicit, multiIfield (IMF) solution method, based on the Eulerian continuum approach for multiphase fluid flow was first developed by Harlow and Amsden (1975). The basic numerical procedure of IMF is an extension of the implicit, continuous fluid, Eulerian (ICE) technique (Harlow and Amsden, 1971).
18 The Euleriaa continuum approach has been widely used in fluidization applications.
Numerical simulations using the continuum approach range from the gas-solid (Sinclair and Jackson, 1989; Ding and Gidaspow, 1990; Pita and Sundaresan, 1993; Dasgupta et al., 1994), gas-liquid (Torvik and Svendsen, 1991; Svendsen, et al., 1992; Sokohchin and
Eigenberger, 1994; Boisson and Malin, 1996), to gas-liquid-solid fluidization systems
(Gidaspow et al., 1994; Grevskott et al., 1996; Mitra-Majumdar et al., 1997).
2.3.2 Dispersed particle/bubble tracking method
While the continuum approach can predict the macroscopic flow behavior of a multiphase fluidization system in the conditions at or not far from the homogeneous regimes, it has difficulties to model the discrete flow characteristics, for example, the particle size effect, the bubble formation, coalescence and breakage in those systems.
Furthermore, the validity of the closure laws is an area requiring constant refinement.
In the Lagrangian approach, the volume-averaged equations are used only for the continuous phase and motions of the discrete particles are modeled individually. The discrete particles/bubbles/droplets are treated as a group of “point” masses with their position, velocity, and other quantities being tracked based on the motion equation of indi\ddual particles/bubbles/droplets. The Lagrangian approach is often called the dispersed particle method when applied to the granular and fluidization systems. In the study of soil behavior, Cundall and Strack (1979) calculated the particle interaction forces using simple mechanistic models such as a spring, a dash-pot and a fiiction slider.
A similar model is formulated to simulate the dense-phase pneumatic conveying flow by
19 Tsuji et al. (1992) where a simple Ergun equation, is used for the continuous phase flow.
Using the similar treatment for the particle contact, Tsuji et al. (1993) simulated a two-
dimensional gas-solid fluidized bed with the gas phase solved by the volume-averaged
equations. However, in their simulation, the gas phase is treated as an inviscid fluid and
the particle stiffiiess is assumed to be small in order to save the computation time. These
simplified assumptions lead to an unrealistically large displacement or overlap between
the contacting particles and hence result in false fluid drag and inter-particle forces.
Recently, Hoomans et al. (1996) have simulated the bubble and slug formation in a two-
dimensional gas-fiuidized bed using a hard sphere model which is commonly used in the
molecular dynamic simulations (Walton, 1984; Allen and Tildesley, 1987). In their
model, the motion of the particles is directly calculated from the forces acting on them,
accounting for the interactions between the particles and the interstitial gas phase. On the
other hand, Xu and Yu (1997) have developed a model for the gas-solid flow using the
soft sphere approach. The coupling between the continuous phase and the discrete phase
is achieved by applying the principle of Newton’s third law of motion to the individual
phases that are modeled at different length and time scales.
The Lagrangian approach is also used in simulating the flow of small bubbles in the gas-liquid bubble columns. This approach is often called the bubble tracking method.
Webb et al. (1992) simulated the turbulent-induced gas bubble dispersion and the
buoyancy-induced liquid flow by incorporating the time-averaged, mean velocity flow
field with the individual bubble motion resulting from the drag, buoyancy and pressure
forces. Lapin and Lubbert (1994) proposed a model that tracks the movement of bubble
20 clusters based on a spatial density function. In these models, the interaction of bubbles to the liquid is only accounted for by the effective density of the gas-liquid two-phase mixtures and no coupling of the momentum transfer from the gas phase to the liquid phase is included. The hubble-bubble collision and bubble breakage are either treated in a simple manner (Webb et al., 1992) or neglected (Lapin and Lubbert, 1994). Recently,
Delnoij et al. (1997) developed a dispersed gas-liquid two-phase flow model that incorporates all relevant forces acting on a bubble and the bubble-bubble interactions.
Forces on a bubble include the pressure gradient, drag, virtual mass, liquid-phase vorticity and gravity. Their simulation results show the importance of the added mass and lift force to the dynamic behavior of the bubble flow. However, it should be noted that the discrete particle model can only simulate the flow of small bubbles. In a bubble column operating in the chum turbulent or heterogeneous regime which is frequently encountered in industrial bubble columns, it is important to model the behavior of large, deformable bubbles.
2.3.3 Surface tracking method
It is necessarily to discretize and track the movement of the surface of particle when the length scale of the discrete particles in consideration, i.e., solid particles, bubbles, and droplets, is larger than the computational cell. Early studies have been limited to the motion of a single bubble or a drop. Using the boundary integral technique,
Youngren and Acrivos (1976) calculate a gas bubble in viscous extensional flow, and
Rallison (1981) studied the time-dependent deformation of a non-axisymmetric droplet
21 with a viscosity equal to the surrounding fluid. Ryskin and Leal (1984) examined the
steady-state shape of a rising axisymmetric bubble using a finite difference technique and body-fitted coordinate system.
A firont tracking/finite difference method is introduced by Unverdi and
Tryggvason (1992) for computing the unsteady motion of drops and bubbles. In their study, the flow fields of individual phases are solved on a fixed uniform grid using the finite difference scheme. The interface between two fluids is tracked by a set of points that are moved by interpolating their velocity firom the fixed grid. In their formulation, the interface is given a finite thickness of the order of the mesh size. To provide the stability and a continuous change of density and viscosity across the interface, a smooth function is introduced and the surface points are reconstructed at each step.
The volume of fluid (VOF) method (Hirt and Nichols, 1981) is another firont tracking algorithm that has proven to be a useful and robust tool in calculating the flow of bubbles and droplets. In the VOF method, the interface is constructed firom the local volume fraction which is solved firom a standard convection equation. Using this method, Tomiyama et al. (1993) examined the single bubble rise at varying flow conditions. Hong et al. (1996) simulated the formation of single bubble chain and the bubble breakage due to the collision with particle in liquid. A good agreement between the simulation with experiments is shown by Lin et al. (1996) for the multi-bubble flow in a two-dimensional gas-liquid bubble column with up to 12 bubbles.
22 2.3.4 Direct simulation and Lattice-Boltzmann simulation
When applied to the liquid-solid flow, the approach based on the surface tracking/moving boundary technique is often noted as the direct numerical simulation of multiphase flow (Joseph, 1994; Shih, 1998), since the flow surrounding individual particles is resolved by using the Navier-Stokes equations. Tezduyar et al. (1992a, b) developed a deforming-spatial-domain/space-time procedure for finite element computations involving moving boundaries and interfaces. The flows with drafting cylinders were tested in their work. Using the same method, Johnson and Tezduyar
(1995) simulated sedimentation of up to five solid spheres in a tube at a Reynolds number of 100. Using a finite element technique, Hu et al. (1992) simulated two-dimensional motions of up to four sedimenting circular and elliptic cylinders in a channel; Feng et al.
(1994a, b) studied the motion and interaction of circular and elliptical particles in sedimenting, Couette and Poiseuille flows of a Newtonian fluid at particle Reynolds numbers in the himdreds. Recently, Hu (1996) simulated the flow of solid-liquid mixtures up to 400 particles using a finite element technique based on moving unstructured grids. The hydrodynamic forces and moments acting on the solid particles are formulated into a single variational equation incorporating both the fluid and particle equations using a generalized Galerkin finite element formulation. An arbitrary
Lagrangian-Eulerian (ALE) technique is adopted to deal with the motion of the particles.
Direct numerical simulations of Uquid-soUd flow are often based on the finite element or finite difference method. On the other hand, an approach, called lattice-
Boltzmann (LB) simulation has been developed to simulate liquid-solid suspensions
23 including 3-D spherical (Ladd, 1994a, b), nonspherical particles (Aidun and Lu, 1995;
Qi, 1997) and deformable membrane (Aidun and Qi, 1998). The LB method simulates fluid motion at a microscopic level similar to a molecular dynamic simulation. It has been proven that the Navier-Stokes equations are fully recovered at the macroscopic scale through a Chapman-Enskog like expansion (Chen et al., 1992). In the LB method, the fluid particles reside at a given lattice and move to neighboring nodes. The fluid particle movement represents real fluid flows through a distribution function of fluid particle density. There are two speeds of moving fluid particles in addition to fluid particles at rest. The particles of speed cr = 1 move along links of the lattice in axial directions and the particles of speed c = 2 move along the diagonal links of the lattice. The solid particles are discretized and move over on the lattice. The hydrodynamic forces and torques acting on the solid particle can be determined from a summation of the momentum of all the fluid particles hitting the solid particle boundaries. Then the motion of the solid particles is determined at each time step from the forces and the torques by using the Newton’s second law similar to the molecular dynamical simulations.
2.4 Summary
The experimental and numerical studies on the bubble and particle dynamics related to three-phase fluidization systems are reviewed in this chapter. Studies on the single bubble characteristics, particle-particle, and bubble-particle interactions provide useful information in modeling the hydrodynamics of large-scale reactors. However, due to the complex nature of multiphase interactions, the applicable range of analytical and
24 empirical models is limited. Numerical simulation is therefore a powerful tool in predicting the hydrodynamics of multiphase systems. Although the Eulerian approach can be applicable to practical problems, the accuracy of this approach strongly depends on empirical constitutive equations. The dispersed particle/bubble tracking method uses less empirical equations with the compensation of larger computing time for tracking the motion of each particle/bubble comparing with the Eulerian approach. Furthermore, both
Eulerian and Lagrangian approaches have difSculties to simulate the flow of large bubbles because the length scale of large bubbles is usually larger than the computational grid size. On the other hand, the interface tracking, direct simulation, and lattice-
Boltzmann simulation require no empirical constitutive equations and can yield detailed information on the flow field around bubbles and particles; but these methods are applicable to small scale systems due to the confinement in the computational power and time.
Studies presented in this dissertation focus on the development of a discrete phase simulation for the gas-liquid-solid fluidization systems. In order to account for the liquid interstitial effect on the particle-particle collision, the flow field and the motion of the approaching particle are examined experimentally and theoretically. Simulations based on the lattice-Boltzmann method is conducted to obtain the detailed three-dimensional flow field and the forces around the particles during the course of collision. The discrete phase simulation uses the dispersed particle method, and volume-of-fluid method to simulate the solid and liquid phases, respectively. A bubble induced force model, a continuum surface force model, and Newton’s third law are applied to account for the
25 couplings of particle-bubble, gas-liquid, and particle-liquid interactions, respectively.
The close distance interaction model developed in this study is included in the particle- particle collision analysis, which considers the liquid interstitial effects between colliding particles. The hydrodynamic characteristics of bubble and particle in liquid-solid fluidized beds at ambient conditions, and in liquids and liquid-solid fluidized beds at elevated pressures are numerically studied using the discrete phase simulation code developed in this study.
26 CHAPTERS
DYNAMIC BEHAVIOR OF COLLISIONS OF ELASTIC SPHERES IN VISCOUS FLUIDS
3.1 Introduction
For two approaching particles with finite Reynolds numbers flowing in viscous fluid, there is no theoretical solution available which can account for the hydrodynamic interactions between these particles. Therefore, in this study, the flow field and the motion of the approaching particle are examined experimentally and theoretically. The trajectory and velocity of a moving particle in collinear and oblique collisions with a fixed particle are measured using a high speed video system and an Infinity lens. The lattice-Boltzmann (LB) simulation is used to obtain the detailed three-dimensional flow field and the forces around the particles during the course of collision. Furthermore, a mechanistic model is developed which accounts for various stages of collision processes.
Predictions by the simulation and the mechanistic model are also compared with the experimental results.
27 3.2 Lattice-Boltzmann Simulation
In this study, a two-speed model with 15 bit variations for the motion of the “fluid particle” is used for the LB simulation. The lattice-Boltzmann (LB) equation with a single relaxation time is given by
/ (x + e + (x,0=-- f - f \x,t) (3.1) O I O I O I 'Ç a I where f ;;,''(x,f) = 4 (3.2) where cr = 1 corresponds to the fluid particles moving to their first-near neighbors along axial directions; a = 2 corresponds to the fluid particles moving to their second-near neighbors along diagonal directions; cr = 0 and / = 0 correspond to the fluid particles at rest; e^i is the vector of fluid particle velocity; and u is the mean velocity of fluid particles at a node, which can be calculated from p(x, t)u = Yufoi O^oi (3-3) a t The suitable coefficients in fluid density distribution function are given in Qian (1990). The complete lists of the velocities of fluid particles, ed, are given in Table 3.1 for 3-D cubic lattice. When solid particles are suspended in the fluid, the fluid elements collide with the surface boundary of the solid particles. In order to match the fluid velocity with the 28 velocity of the solid in the solid-fluid interface, Ladd (1994a, 1994b) proposed a collision rule which is given by (x,r +1) = (x ,L ) - - U^) (3.4) wherex is the position of the node adjacent to the solid-surface with velocity Z7b; t+ is the post-collision time, which is the same as defined by Ladd (1994a); /' and / denote the reflected and incident directions, respectively. The above rule is applied to the boundary nodes on both sides of the sohd-surface. As a result, a no-slip boundary condition for moving solid particles is imposed correctly by the collision rule in such a way that the fluid mass is conserved at each time step by allowing exchange of population of fluid elements at the boundary nodes adjacent to both sides of the solid surface. The hydrodynamic force exerted on the solid particle at the boundary node is f (JC + ) = 2e^ (/^ ix ,t, ) - (U^ ■ )) (3.5) where U\,= Uq + Q x. jCb; Uq is the velocity of the center of mass; is the velocity of solid-fluid interface at the node; D is the angular velocity of the solid particle; and x^ = x + ‘/ 2 e^\ — R, whereR is the mass center of the solid particle. The total force F t and torque Tt on the solid particles are obtained as f r = g f ( % + L ^ ) (3.6) + -.R)xf(A: + L.) (3.7) where the summation is over all the boundary nodes in the fluid region associated with a particular solid particle. 29 In the lattice-BoItzmann method, the nodes are fixed and the solid particles move over the (nodes) grids. Whenever a node crosses the fluid-solid interface and enters the solid region, the momentum of the flow at the boundary node may exert a force on the solid particle. The force F\ at the node is (Aidun et al., 1998, Qi, 1999) F^= pix,t)u{x,t) (3.8) where p is the density of the fluid at the node. Similarly, whenever a node crosses the solid-fluid interface and leaves the solid region, the flow in the node would add a force Fq on the solid particle, Fq = t)u(x, t) (3.9) It is important to note that the present approach allows fluid to enter the solid phase to conserve total mass of the fluid at each time step. Conservation of the fluid mass guarantees the recovery of the Navier-Stokes equations from the LB method. The motion of rotation is governed by the Euler equations. If using the Euler angles directly, there is a singularity in the equations of motion of Euler angles themselves whenever 6 approaches 0 or tt. Therefore, Euler angles, (f>, 9, yr, are not appropriate for solving the equation directly, and four quaternion parameters have to be used as generalized coordinates (Evans, 1977; Qi, 1997) to avoid the singularity. The translations of the mass center of each particle are updated at each Newtonian dynamic time step by using a so-called half-step 'leap-frog' scheme which is popular in molecular dynamic simulations (Allen and Tildesley, 1987). 30 3.3 Mechanistic Model 3.3.1 Contact velocity The dynamic equation of the translational motion of a sphere in a viscous fluid under gravity field can be expressed as = (3.10) df where m is the mass of particle, Fd the drag force, Fp the pressure force, Fg the buoyancy force, Fba the Basset force, and Fam the added mass force. The drag force on a single sphere moving in the Stokes flow regime is given by Fus-^^f^U a (3.11) where Fps is the drag force, and a is the radius of the sphere. When the particle Reynolds number is larger than 1, it is assumed that the drag force can be obtained by simply multiplying Eq. (3.11) by a drag factor,/ The drag factor that is applicable for particle Reynolds numbers up to 800 is given by (Schiller and Naumann, 1933) / = l + 0.15Re°^'^ (3.12) When two spheres are moving towards each other, the interaction between the two will affect the drag force on the particle. A correction factor, / can be introduced to account for the deviation of the drag force from the non-interactive single particle situation. This correction factor is a function of particle Reynolds number, particle and fluid density ratio, and the separation distance between two particles. Quantification of the correction factor is given in section 3.5. 31 To account for the added mass of sphere and fluid pressure force in this collision process, the results of Milne-Thomson (1968), who considered a sphere moving normally towards a wall in an inviscid fluid, can be used. Accordingly, the total kinetic energy of the spheres and the inviscid liquid is conserved during the normal collision process (Milne-Thomson, 1968); hence. = constant (3.13) "I? where m is the mass of sphere, m ' is the mass of liquid displaced by the sphere, and h is the distance between the center of a sphere and the symmetrical plane of the two spheres. Noting that (3.14) dr dr éh the equation of motion of the sphere moving towards the wall can thus be obtained based on Eqs. (3.13) and (3.14) as dU 9 m' a m + —m ’ U~ (3.15) 2 dr 32 A" with the added mass force given by ^am= Y (3.16) dr The right-hand side of Eq. (3.15) represents the resistance force from the fluid due to the sphere approaching the wall. More precisely, the resistance force can be denoted as the pressure force given by 32 Fp- (3.17) 32 A" The buoyancy force is given as (3.18) For a particle moving in a liquid with a finite Reynolds number, the modified Basset force is given as (Mei and Adrian, 1992) ^BA = K (r- r ) ^ d r (3.19) K(r-r) in Eq. (3.19) is given as -2 -il/4 -,1/2 n:(t — t ) v K (f-r) = a' 2 a i / ’(Re) (3.20) /„(R e) =0.75+ 0.105 Re; Re = lU ^ a / V. where v is the kinematic viscosity of the fluid; Um is the mean stream velocity and equals zero in the current case. Substituting Eqs. (3.11), (3.12), and (3.16) through (3.19) into Eq. (3.10), the dynamic motion equation of the sphere is obtained as 3 dU dt (3.21) —dK/jXJcif(f)—-^;^— — K(r — r)dr + /ng'(l — — ) Substituting Eq. (3.14) into Eq. (3.21) yields 33 2 Pp 16 àh (3.22) 2aV 32 p, A- 2aV„C/ !/ /?/ i.i.2 Compression in a collision During the compression process of the collision, the forces acting on each individual sphere include the viscous drag force, the pressure force, the compression force due to the elastic deformation of the sphere, and the gravity force. It is assumed that the particle deformation is small so that the viscous drag force and the pressure force still take the same form as for a sphere, while the compression force takes the same form as if the collision were in a vacuum. Moreover, during collision the added mass is considered constant. Denoting s' as the approaching distance from the center of sphere to the symmetrical plane of the two spheres, the equation of motion of the sphere can be expressed as + = m g -Füc~ Fvz~Fc~Fq (3.23) d r where Foc, and represent the added mass, drag force, pressure resistance, and collision force in the compression process, respectively. The added mass in this case is obtained from Eq. (3.16) as /Mac ^ — a (3.24) 34 The viscous drag force and pressure force for the compression process are obtained from Eqs. (3.11) and (3.17), respectively, as Foc = 6 ; r //a ^ (3.25) at and = (3.26) The compression force can be expressed by (Hertz, 1881) whereE is Young’s modulus of the sphere and vis Poisson’s ratio. Substituting Eqs. (3.18), (3.24) through (3.27), Eq. (3.23) can be solved fds^ numerically with the initial conditions given by — =Ucl^o = ^ • The compression Vdtjo time, tc, and the maximum approach distance, 5m, can be obtained, respectively, by Ut^tc " 0 and . The maximum collision force is thus obtained from Eq. (3.25) as Fc.max= -, (3.28) 3( 1 - V) 35 3.3.3 Rebound in a collision During the rebound process of the collision, the pressure force and the compression force aid the reboimd while the viscous drag and gravity forces resist the rebound. Hence, the equation of motion of the sphere in rebound can be expressed by (w+mac)~^ = ~^g^“ FDc + Fpc + Fc + FB (3.29) Since dy/dr is negative during the rebound, the rebound process in the collision can also r dy ^ be solved numerically with the initial conditions of — =0 and 5 o = s'm • After V dr solving Eq. (3.29), the rebound time, and the rebound velocity, Ur, can be determined, respectively, by 5 t=,, = 0 and jJr = Ut=i,- The total collision time, % is thus calculated byrT = rc+ri. 3.3.4 Velocity after Rebound During the after-rebound process, the motion of the sphere is again under the action of the pressure force, the viscous drag force, the gravity force, the buoyancy force, and the Basset force. The pressure force aids the motion of the sphere away from the wall, while the viscous drag force and gravity force work against the motion. Thus, motion of the sphere during after- rebound process is governed by (/77 + /Ma ) = + —F ba + F b ( 3 .3 0 ) at with the initial conditions o f Uo~Uv\ h^ — ci. Note that 36 Substituting Eqs. (3.12), (3.17) through (3.19), and (3.31) into Eq. (3.30) yields i + l£ - * ± £ - S LdU 2 /7p 16 Pp dh (3.32) dU 9f/é9 p a \, . + — —(1-—) 2 a ' / ? 3 2 p p A " p ,p 3.4 Experimental A schematic diagram of the experimental setup is shown in Fig. 3.1. A spherical particle is fixed to three stainless steel rods that are separated by 120° in the horizontal plane of the center of the particle. The falling sphere is released firom the center of a guider which is center-aligned with the fixed sphere for collinear collision. The guider can be moved in the vertical direction. The velocity of the falling sphere is measured by visualization. A high speed CCD camera (HSC 250x2, JB Labs) is used to obtain the images at a rate of 480 firames per second with a resolution of 765x246 pixels per firame. Images from the CCD camera are captured by a frame grabber board (Epix 4 MEG video Model 12). An Infinity lens is used which can provide a full frame of image for a 2 mm sphere when the object distance is 1 cm. Therefore, the imaging system gives a spatial resolution of 2.6 pm/pixel at the object distance of 1 cm. The velocity of the moving ball is calculated by counting the number of pixels that the sphere moves in consecutive frames. The resolution of the image is calibrated on-line when the measurement is 37 performed. The diameters of the particles are measured by an Optomax V image analyzer cormected with a Nikon Optiphot biological microscope which gives a 2.8 pm resolving power for a 2.0 cm object. Figure 3.2 shows an example of the spatial resolution of two consecutive images. In the figure, the measured distance between two particles is 125 pm for the left frame and zero (in contact) for the right firame. Therefore, the contact velocity obtained in the current study is defined within a distance of the order of 10'^ particle diameter. The experimental data are averaged over at least five runs. The fluid viscosity is measured by a rotational viscometer (Fann series 35 viscometer). The surface roughness of the particle is observed using the scanning electron microscope (SEM). An SEM picture of a 1.27 cm Teflon particle is shown in Fig. 3.3. As can be seen firom the figure, the surface roughness of the particle is about 5 micron. The spatial resolution for the same ball is about 25 micron per pixel. Distilled water and aqueous glycerin solutions with viscosities ranging firom 0.001 kg/m s to 0.135 kg/m-s are used in the experiments. The diameters of the spherical particles range fi-om 2 mm to 1.27 cm. The densities of the particles vary firom 1,386 to 8,451 kg/m^. With the measurable resolution, the particle Reynolds numbers vary firom 5 to 300, which cover most of the operating conditions involved in slurry bubble column and three-phase fluidized bed systems. 3.5 Results and Discussion Figure 3.4 shows the measured distance variation with time for a Teflon ball of 1.27 cm in diameter with the initial dimensionless separation distance, x/d, of unity. The 38 corresponding velocity variation with time is shown in Fig. 3.5. In the figure, the positive value of the velocity corresponds to the downward movement. When measuring the velocity, the fixed sphere is attached to the bottom of the container. As shown in the figure, the moving particle is bounced back in the first contact. After the second contact, the particle slides down along the surface of the fixed sphere. Figure 3.6 shows the change of the velocity with distance for spherical particles colliding with another particle in aqueous glycerin solution. The particle diameter is 1.27 cm and the liquid viscosity is 0.053 kg/m-s. Results for particles with densities of 1,386 kg/m^ and 2,180 kg/m^ are shown in the figure. As can be seen firom the figure, the velocity of the approaching particle starts to decrease around 0.05 x/d. The magnitude of the velocity reduction is higher for the particle with higher density, and hence with a higher Reynolds number. A similar trend of the velocity reduction is also observed in Fig. 3.7, where the particle of 1.27 cm diameter and with densities of 2,180 kg/m^ and 8,451 kg/m^ collides with another particle in aqueous glycerin solution of viscosity 0.135 kg/m-s. In Fig. 3.8, the results of the settling of a glass bead with a diameter o f 2 mm in water and those of collision with a wall are shown. It can be seen from the figure that, again, the velocity reduces more for the high Reynolds numbers than for low Reynolds numbers. The physical parameters controlling the collision process are: (1) particle density, (2) particle radius, a; (3) fluid density, p\ (4) fluid viscosity, (5) particle velocity, U\ and (6) separation distance, h. From the dimensional analysis, three dimensionless groups are obtained to correlate the experimental results. The correlation 39 for the drag correction factor in the current experimental ranges (Rep: 5 ~ 300, p^p\ 1.2 8, (fp: 2 -1 2 .7 mm) is given as ^ Pp _p Comparisons of the results from the lattice-Boltzmann simulation and from the mechanistic model, i.e., Eq. (3.22), incorporating Eq. (3.33) are shown in Figs. 3.9 through 3.11. For the lattice-Boltzmann simulation, a 65x65x65 lattice system is used. One diameter of the solid sphere is represented by 20 lattice units. The center of the fixed particle is located at the 11th lattice from the bottom of the computational domain. The center of the moving particle is initially located at the 51st lattice from the bottom, which gives the initial separation distance of one diameter of the sphere. The first order differential equation of the mechanistic model is solved by the Runge-Kutta method. As can be seen from the figures, both the lattice-Boltzmann simulation and the mechanistic model predict the particle settling process reasonably well. The lattice-Boltzmann simulation can also provide information on the velocity field as shown in Figs. 3.12 and 3.13. These figures show a cross section of the three-dimensional velocity vector field at the X-Y plane with the Z-direction located at the center of the simulation domain. The contours of the velocity vector field are color-coded in the figures. Figure 3.12 shows the velocity vector field when the dimensionless separation distance is about 0.7. As can be seen from the figure, at this separation distance, the flow field of the approaching particle is not affected by the presence of the other particle. No interactions between two particles are present. At this point, the correction factor from Eq. (3.33) is equal to 1. 40 When the approaching particle moves close to the fixed one, the fiow field of the approaching particle begins to be afîected by the other particle. Figure 3.13 shows the flow field around the two particles when the separation distance is the size of one lattice unit. As can be seen firom the figure, significant liquid outward flow is present in the interstice between the two. The liquid in the gap undergoes forward and outward flow simultaneously, exhibiting a highly complex three-dimensional flow structure. When two particles are in contact, both particles undergo the compression and rebound process. During compression, the kinetic energy of the particles converts to elastic energy which is retained in the particle in the form of material deformation. The elastic energy of the particle is converted back to kinetic energy during the rebound process. For the elastic colhsion of two spheres without the presence of the viscous fluid, the rebound velocity in normal direction will remain the same as its contact velocity, i.e., no energy loss is incurred during the compression and rebound process for the ideal elastic condition. However, when the particle collision takes place in the viscous fluid, the kinetic and the elastic energy of the particle can also be dissipated by the fluid drag. Therefore, the rebound velocity of the particle is less than the contact velocity even for an ideal elastic collision. The deformation and velocity change of the contact particle in the compression and rebound process can be calculated from Eqs. (3.23) and (3.29). Figures 14 and 15 show the particle deformation and the velocity change during the compression and rebound process of the moving particle. As shown in Fig. 3.14, for the Young’s modulus of 4.0x10*° N/m^ and the liquid viscosity of 0.053 kg/m-s, the fluid drag force is negligibly small compared to the elastic force of the particle, which results in a 41 symmetric compression and rebound process. As a result, the rebound velocity of the particle is very close to its contact velocity. When the fluid viscosity is high and the Young’s modulus of the particle is small, the fluid drag force will be comparable to the particle’s elastic force. As shown in Fig. 3.15, the rebound velocity of the particle is smaller than the contact velocity in the high viscosity and low elastic modulus condition. It should be noted that for the low elastic modulus, an inelastic collision will take place. The current study only demonstrates the effect of the fluid drag on the compression and rebound process of the collision as described in Eqs. (3.23) and (3.29). The effect of inelastic compression and rebound can be taken into account in these equations by replacing with the equation of compression force for an inelastic collision. The motion of the particle after the contact is governed by Eq. (3.32). A comparison of the experimental results with the lattice-Boltzmann simulation and the mechanistic model is shown in Fig. 3.16. As shown in the figure, both the simulation and the mechanistic model results agree with the experimental data. The agreement also demonstrates the importance of the drag correction factor for the hydrodynamic interaction in the interstice of the two particles. The mechanistic model developed for particle collision given in this study is incorporated into a computational fluid dynamic code for two-dimensional gas-liquid, gas-solid, and gas-liquid-solid fluidized systems developed by Zhang et al. (1998). In this code, the particle movement in the viscous fluid is simulated using a Lagrangian approach, while the liquid phase is simulated in Eulerian coordinates using the finite difference method. Three mechanistic models are used to simulate the particle motion 42 and particle-particle collision: (1) the hard sphere model without including the drag correction factor for the close distance interaction; (2) the hard sphere model including the drag correction factor, and (3) the complete collision model including four sub processes described in this study. For the tangential velocities after the collision, a slip/stick model (Li et al., 1999) is employed. The simulation of the entire process of the oblique collision of two glass beads is performed, and the results based on different mechanistic models are shown in Fig. 3.17 along with the experimental data. The diameter of the glass beads is 2 mm. Water is used as the liquid phase. The approaching particle is released from four particle diameters above the particle resting on the bottom of the vessel. The motion of particles is recorded by a high speed CCD camera with 480 frames per second. The trajectories of the centers of particles are measured from the recorded frames. As can be seen from the figure, results from the complete collision model yield the best agreement with the experimental data. The hard sphere model including the drag correction factor also gives a good prediction. The prediction based on the hard sphere model without including the drag correction factor gives rise to significant deviations from the experimental data. These comparisons reveal the importance of liquid interstitial effects on the particle collision in viscous fluids. 3.6 Concluding Remarks The dynamic behavior of the collision of two elastic spheres in a stagnant viscous fluid is investigated. The lattice-Boltzmann simulation reveals the complex three- dimensional flow field of the liquid in the gap between two colliding particles. The 43 mechanistic model is presented which takes into consideration the equations of particle motion in the entire process of collision, which includes the particle motion before the collision, compression in contact, rebound in contact, and motion after the collision. Results of the lattice-Boltzmaim simulation and the mechanistic model compare favorably with the experimental data for the particle motion before and after the collision. It is shown that the viscous effect on the compression and rebound processes of contacting particles is significant for a single collision only when the liquid is highly viscous and the elastic modulus of the particle is very small. The mechanistic model is also incorporated into a computational fluid dynamic code developed for multi-particle flow simulation. The simulation results for oblique collision of two spherical particles compare well with the experimental data. It is shown that the key to proper quantification of the particle collision characteristics is the ability to accurately predict the particle velocity upon contact. The particle drag equation developed in this study for particle Reynolds numbers ranging from 5 to 300 provides the required information for such prediction for most slurry bubble column and fluidized bed operating systems. 3.7 Nomenclature a particle radius A, B, C, D coefficients for the equilibrium distribution function d particle diameter ggj vector of fluid particle velocity E Young’s modulus 44 F force on a solid particle at the boundary nodes / correction factor Fam added mass force Fb buoyancy force Fba Basset force Fd drag force F d s Stokes drag force F\ force due to the fluid entering the solid nodes Fq force due to the fluid leaving the solid nodes Fp pressure force F t total force on a solid particle fax fluid particle distribution function equilibrium fluid particle distribution function g gravitational acceleration h separation distance from the center of the approaching particle to the symmetric plane of two colliding particles m mass of a particle wî’ mass of liquid displaced by a particle R mass center of the solid particle Rep particle Reynolds number s center approaching distance t time 45 7r torque on a solid particle U particle velocity u mean velocity vector o f fluid particles i7b velocity vector of solid-surface X position vector X separation distance Greek letters correction factor H dynamic viscosity V kinematic viscosity, Poisson’s ratio r single relaxation time Q angular velocity of solid particle 46 Table 3.1 Velocity vector for cubic lattice in 3-D 1 1 1 0 0 1 1 2 -1 0 0 1 1 3 0 1 0 1 1 4 0 -1 0 1 1 5 0 0 1 1 1 6 0 0 1 2 1 1 1 1 3 3 1/2 2 2 -1 -1 2 3 -1 1 1 3 2 4 1 -1 3 2 5 -1 -I 1 3 2 6 1 1 3 ^ U2 2 7 1 -I 1 2 8 -I 1 3 47 1 1. Container, 2. Fixed ball, 3. Falling ball, 4. Guider, 5. Infinity video microscope, 6. CCD camera, 7. Image grabber board, 8. PC Figure 3.1 The schematic diagram of the experimental setup. 48 Figure 3.2 An example of the spatial resolution of two consecutive images. 49 % m Figure 3.3 SEM of the surface of a spherical Teflon particle. 50 0.8 0.6 0.4 0.2 00.01 0.02 0.03 0.04 0.05 0.06 t(s) Figure 3.4 Distance change with time for a Teflon ball approaching to another. 51 0.8 0.6 0.4 0.2 - 0.2 0 0.01 0.02 0.03 0.04 0.05 0.06 t(s) Figure 3.5 Velocity change with time for a Teflon ball approaching to another. 52 80.0 60.0 Pp = 2,180 kg/m 20.0 Pp = 1,386 kg/m 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x/d Figure 3.6 Settling of spherical particles in aqueous glycerin (in collision with another particle), fi = 0.053 kg/m-s, = 1.27 cm. 53 40.0 Pp = 8,451 kg/m i 20.0 Pp = 2,180 kg/m 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x/d Figure 3.7 Settling of spherical particles in aqueous glycerin (in collision with another particle), // = 0.135 kg/m-s, dp = 1.27 cm. 54 350 250 - 0.5 1.5 2.5 x/d Figure 3.8 Settling of a glass bead in water (in collision with wall), i/p = 2.0 nun. 55 80.0 ^ Experiments Anafytfcal model 60.0 - X LB simulation Pp =2,180 kg^rn^ of 40.0 04 20.0 pp = 1,386 kg/nf Figure 3.9 Comparison of the lattice-Boltzmann simulation and the analytical model with the experimental data for the settling of spherical particles in aqueous glycerin (in collision with another particle), jj. = 0.053 kg/m-s, = 1.27 cm. 56 50.0 ^ Experiments AnafytKal model 40.0 V LB simulation 30.0 Pp = 8,451 kg/m 20.0 Pp = 2,180 kg/m 10.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 x/d Figure 3.10 Comparison of the lattice-Boltzmann simulation and the analytical model with the experimental data for the settling of spherical particles in aqueous glycerin (in collision with another particle), // = 0.135 kg/m-s, = 1.27 cm. 57 250 û Experiments _x Anafytical model 200 LB simulation 150 100 50 0 0 0.51 1.5 2 x/d Figure 3.11 Comparison of the lattice-Boltzmann simulation and the analytical model with the experimental data for the settling of a glass bead in water (in collision with wall), dç = 2.0 mm. 58 HI <-w/ ' .' Figure 3.12 Velocity vector field of liquid around two particles when the dimensionless separation distance is 0.7. 59 0 (cm /s) Figure 3.13 Velocity vector field of liquid around two particles when the separation distance is of the size of one lattice unit. 60 4.E-04 -, E = 4.0e+10 N/m^, v = 0.35, p. = 0.053 kg/m*s 3.E-04 - f 2.E-04 - U > 0 : conpression 1.E-04 - U < 0 : rebound O.E+00 - 20.0 - 10.0 0.0 10.0 20.0 U (cm/s) Figure 3.14 The particle deformation and velocity change in the compression and rebound process during the elastic collision of two spheres in viscous fluid (symmetric for low viscosity and high elasticity modulus). 61 E = 1.0e+6 N/m , v= 0.35, 2.E-02 - \j. = 2.0 kg/m*s S l.E-02 - U > 0 : conçression U < 0 : rebound O.E+00 - 20.0 • 10.0 0.0 10.0 20.0 U (cm/s) Figure 3.15 The particle deformation and velocity change in the compression and rebound process during the elastic collision of two spheres in viscous fluid (asymmetric for high viscosity and low elasticity modulus). 62 40.0 ^ Experiments Analytical model 30.0 LB simulation. i 20.0 10.0 \ A 0.0 0.00 0.04 0.08 0.12 x/d Figure 3.16 The motion of particle after contact, dp = 1.27 cm, pp = 2,180 kg/m^, // = 0.053 kg/m-s. 63 12.0 ♦ Ejqjerimental Data 10.0 - . Hard sphere model without approaching interaction . Hard sphere model with approaching interaction 4.0 Complete collision model 2.0 with compression and rebound 0.0 (3 ~ 0.0 2.0 4.0 6.0 8.0 x(mm) Figure 3.17 Experimental and simulation results of trajectories of a particle colliding with another. 64 CHAPTER 4 DISCRETE PHASE SIMULATION OF SINGLE BUBBLE BEHAVIOR IN GAS-LIQUm-SOLH) FLUTDIZATION SYSTEMS 4.1 Introduction In this study, gas-liquid-solid flow in a fluidized bed is simulated using the Eulerian fluid dynamic model in combination with the discrete particle method (DPM) and the VOF front-tracking method. The liquid phase hydrodynamics is described using the volume-averaged, time dependent Navier-Stokes equations. The trajectories of the individual solid particles are computed using the dynamic motion equation of particles. The dynamic interactions between particle-particle and particle-bubble in the liquid medium are also considered. Coupling of the particle-liquid interactions is considered by applying Newton's third law of motion. The gas-liquid interface is obtained by using the VOF front-tracking method. The gas-liquid mterfacial mass, momentum, and energy transfer is modeled by a continuum surface force (CSF) model (Brackbill et al., 1992). A two-dimensional code is developed in this study which incorporates a program for incompressible flows with free surfaces, titled Ripple (Kothe, et al. 1991). The present code implements various steps of calculation to 65 obtain flow field properties for the volume-averaged liquid phase and the discrete particle phase while considering the coupling effects among individual phases. The simulation results are also compared to the experimental results on the expansion of the liquid-solid fluidized bed and the single bubble rising behavior in liquid-solid fluidized beds. 4.2 Governing Equations for Individual Phases 4.2.1 Liquid-phase model In multiphase simulation, the expression of the liquid phase governing equations is related to the expressions of the governing equations for the other two phases. In this study, the motion of particles is described by the discrete particle method while the motion of gas bubbles is described by the VOF front-tracking method. For the discrete particle method, the interaction forces between the fluid and particles are considered. For the VOF front-tracking method, the interface dynamics between the liquid and gas bubbles are considered. The governing equations for the liquid phase are derived for liquid-solid suspension flow based on the Navier-Stokes equations by considering the presence of dispersed particles in the liquid phase flow. The resulting volume-averaged mass and momentum equations are given as the continuity equation = 0 (/1.1) ot and the momentum equation 66 P i + P ,^ ' C^i^) = + £■! V • r + £T,p,g- + /(, (4.2) where v is the liquid velocity vector; fi is the liquid holdup in the liquid-soUd mixture (£i + £j = 1); yO[ is the liquid density;p is the scalar pressure; r is the viscous stress tensor; g is the acceleration of gravity; and/b is the total volumetric body force which includes the forces acting on the liquid from the particles and bubbles,/i, = / p f + ^bf- The particle-liquid interaction force,/pf, and the bubble-liquid interaction force,/ÿf, are obtained by using a continuum surface force (CSF) model and Newton’s third law, respectively, as will be discussed in Sections 4.4.1 and 4.4.2. The Newtonian viscous stress tensor is used which is given as r = 2/iS = //[(Vv) + (Vv)^] (4.3) whereS is the rate-of-strain tensor and p is the coefficient of dynamic viscosity. 4.2.2 Gas phase model The gas phase in the system is present in the form of bubbles. The flow inside the gas bubbles is governed by the single phase Navier-Stokes equations. Due to the significant density difference between the gas and the liquid-solid mixture, the momentum transfer from the flow inside the bubble to the mixture is negligible except in the gas-liquid interface where the surface tension force across the interface acts on the liquid from the gas phase. Therefore, the gas bubble can be treated as a void with the motion and the topological change of its surface governed by the liquid-solid flow and interface dynamics. The free surface of the gas void is reconstructed by a scalar 67 field aix,t), where a(r,0 = 1 in the liquid or liquid-solid mixture, 0 < cc{x,t) < I at the free surface, and a(x,t) = 0 in the void. The advection equation for a{x,t) is + (v • V )a = 0 (4.4) dt 4.2.3 Discrete particle model The motion of a particle in a flow field can be described in the Lagrangian coordinate system with its origin set at the center of the moving particle. The particle movement in a non-uniform flow field includes acceleration and rotation. As the particle size is smaller than the grid size in the present method, the particle rotational motion can be neglected in the dynamic motion equation. The change of particle rotation due to collision, however, is considered as will be discussed in Section 4.3.2. The translational motion of a particle in the liquid is governed by Newton’s second law of motion dv '«p-^ = ^.ouü (4.5) Forces acting on a particle include: interface forces between fluid and particle; and forces imposed by external fields. Interactions between particles are accounted for by collision mechanics and therefore the collision forces are not included in the total force. The total force acting on a particle is composed of all applicable forces, including drag ( F d ) , added mass (F a m ), gravity ( F g ) , buoyancy ( F b ) , Basset history force (F b a ), and other forces (Z F j), 68 ^total --^D +-^AM +-^G ^-^BA The drag force acting on a suspended particle is proportional to the relative velocity between the phases as follows - Vp |(v - Vp ) (4.7) where A is the exposed frontal area of the particle to the direction of the incoming flow, Cd is the drag coefficient, which is a function of the particle Reynolds number. Rep. For rigid spherical particles the drag coefficient Cd can be estimated by the following equations (Rowe and Henwood, 1961) ^"^■(l + 0.15Rep°®®"), Rep <1000 Cd = (4.8) 0.44, Rep >1000 In the liquid-solid suspension, the drag force depends strongly on the local liquid holdup in the vicinity of the particle under consideration. The effective drag coefficient can be obtained by the product of the drag coefficient for an isolated particle and a correction factor as given by (Wen and Yu, 1966) C D = C D g r' (4.9) The added mass force accounts for the resistance of the fluid mass that is moving at the same acceleration as the particle. For a spherical particle, the volume of the added mass is equal to one-half of the particle volume, so that (4.10) 69 The Basset force accounts for the effect of past acceleration. The original formulation of the Basset force is derived based on the creeping flow condition. For a particle moving in a liquid with, finite Reynolds number, the modified Basset force is given as (Mei and Adrian, 1992) ft à(v — v„) ’ ' dr (4.U) K{t - r) in Eq. (4.11) is given as 1/4 - i l / 2 7t{t—v)V I (C/ + Vp-v)^ K { t-v ) = 2 + — TV L '■p J 2 /fj (Re) = 0.75 + 0.105 Re, (4.12) Re = Ud^ / V where v is the kinematic viscosity of the fluid; U is the mean stream velocity. The sum of the gravity and buoyancy forces has the form Fq +Fb ={Pf,-P\W.g (4.13) 4.3 Particle-Particle Collision Dynamics A hard sphere approach is used for the particle-particle collision analysis. In this approach, collisions between spherical particles are assumed to be binary and quasi-instantaneous, and further, that there is a sequence of collisions during each time step. The equations, which are similar to the equation o f molecular dynamic simulation (Allen and Tidesley, 1987), are used to locate the minimum flight time of particles before any collision. 70 4.3.1 Liquid shear effect While the shear force can be neglected in gas-solid flow systems, in liquid- solid systems this is not the case. The liquid shear effect between particles becomes important when two particles move close to each other in liquid-solid systems, especially when the distance between two particles is less than 0. Wp (Zhang et al., 1998). Thus, the close-distance interaction (CDI) model is used to locate the particle contact velocity just before collision, which considers the strong damping effect due to the liquid film before particle contact. The particle normal contact velocity can be described by (Zhang et al., 1998) 9 fff(j> (Wp-M) 9 p < (u^-u)\u^-u\ 2/?p 16 p. A' u.. 32 p . A' (4.14) g dt 1-- p y where h is the distance from the center of the approaching particle to the midpoint between the two particles; is the radius of particle; and f (Schiller and Naumann,1933) and 0 (Zhang et al., 1998) are the correction functions and can expressed as .0 47 \ .0 .4 4 0.19 Re Re. 0 .687 (4.15) ^ =exp , / = l + 0.15Re P ,1.7, .P. Using the Runge-Kutta method, Eq. (4.14) can be solved to locate the particle normal contact velocity just before the collision. 71 4.3.2 Particle collision analysis When two particles are in contact, collision analysis can be conducted to obtain the velocities of the particles after collision. It is assumed that tangential traction and the resulting displacements have no effect on normal collision. For the collision between particles a and b, the normal components after collision can be obtained by solving the equations for the restitution coefficient and the conservation of momentum - U ^ (4.16) where jJ^ is the normal velocity of the particle (a or b) at the contact point before or after (with superscript ' ) the collision. From Mindlin’s contact theory, there are three kinds of frictional contact during the collision: 1. sliding contact, 2. non-sliding or sticking contact, 3. torsion of elastic particles in contact. By neglecting the effect of particle torsion during collision, the simplified Mindlin's contact theory is applied to obtain the tangential components after the collision. If the incident angle, defined as the ratio of the particle-particle relative velocity in the tangential direction to velocity in the normal direction, is less than the critical angle (acr = tan'^(%), where fk is the friction coefficient), the sticking collision occurs (4.17) Otherwise, the sliding collision occurs, in which (Fan and Zhu, 1998) (C/J-(/J)-([/7'-[/b')=2A(:/r-^r) (4.18) 72 where i f is the tangential particle velocity (a or b) at the contact point. The conservation of momentum is given as 77Z.C/J (4.19) The tangential velocities after the collision can be obtained by solving Eqs. (4.17) or (4.18) together with Eq. (4.19). As mentioned in Section 4.2.3, the collision induces a change in particle rotation. The angular velocities after the collision are determined by 4 -co,) = m^iUl -U l) - ^^ 20) where a is the angular velocity of the particle (a or 6), and I is the moment of inertia defined by 7= USm^r^. The tangential velocities of the particle center are given as Ul=Ul-o>,r^ (4.21) K = U l , 4.4, Interphase Couplings 4.4.1 Coupling between gas and liquid phases In the gas-liquid free surfaces, the stress boundary condition follows the Laplace equation as p^=p-p^= interface. The continuum surface force (CSF) model (Brackbill et al., 1992) converts 73 the surface force into a volume force within free surfaces. The volume force at the free surfaces is given by the CSF model as /bf =œc{x,t)^a{x,t) (4.23) where the free surface curvature xris given as K = —(V • rt) = — A-v [/il-(V-n) (4.24) /I; \n\ y The unit normal h , n (4.25) \n\ is derived from the normal vector n, n = V a (4.26) The volume force given in Eq. (4.23) is the interface force between the gas and liquid phases, which is added to the volumetric body force term in the momentum equation, Eq. (4.2), at the free surfaces. 4.4.2 Coupling between particle and liquid phases Based on Newton’s third law of motion, the forces acting on the particles from liquid yield a reaction force on the liquid. Therefore, the momentum transfer from particles to liquid is taken into account by adding the particle-fluid interaction force, /pf, to the body force term of Eq. (4.2) f - i n •'pf A T^k ’ -^P (4.27) 74 where subscript ij defines the location of a computational cell; and AV are the domain and volume of this cell, respectively; jCp*^ is the location vector of particle h. Ffp is the fluid-particle interaction force acting on any individual particles, which includes the drag, added mass, and Basset force. At the liquid-bubble interface area, the fluid-particle interaction force also includes the bubble-induced force as will be discussed in Section 4.4.3. Liquid properties on a particle are obtained by an area-weighted averaging method based on the properties at the four grid points of the computational cell containing the particle. The liquid holdup, Si, is obtained by subtracting the volume fraction of the particles from that of the liquid-solid suspension in a computational cell. However, if the grid size is less than five particle diameters, the liquid holdup is obtained based on an averaged value from the adjacent cells. Furthermore, it is found that the cell averaged liquid holdup cannot represent well the phase holdups surrounding a particle when the particle is within an area that has significant solids concentration variations. Therefore, the cell-averaged liquid holdup is restricted in solving the volume-averaged equations for the liquid phase, Eqs. (4.1) and (4.2). A particle-centered area averaging method is used to obtain the liquid holdup, S[, in solving the particle phase equation, Eq. (4.9). A correlation based on the comparison between a hexagonal lattice and an FCC unit cube (Hoomans et al., 1996) is used to obtain the psuedo-three-dimensional liquid holdup, S|, from the two-dimensional data. 75 4.4.3 Coupling between particle and gas phases In the VOF surface-tracking method, the gas-liquid interface is assumed to have a finite thickness. When particles move into the gas-liquid interface where the fraction of the fluid, a, is less than I, the volumetric surface tension force acts on the particles through the liquid film. Since the size of the computational cell is larger than the thickness of the gas-liquid interface, a bubble induced force model (BIF) is applied to the particle Fbp =Fp07c(A:,r)Va(jc,r) (4.28) If the particle overcomes this bubble-induced force, the particle would penetrate the bubble surface. The penetrating particle breaks the bubble surface momentarily upon contact. If the penetrating particle is small, the bubble may recover its original shape upon particle penetration (Chen and Fan, 1989). However, if the penetrating particle is large, then bubble breakage may take place. 4.5 Numerical Methods 4.5.1 Two-step projection method The liquid N-S equations, Eqs. (4.1) and (4.2), are solved using the time-split two-step projection method. Unlike the case of single-phase incompressible flow, the volume averaged equations for the liquid velocity field, Eqs. (4.1) and (4.2), do not retain the zero-divergence vector field. Therefore, the original method used in the Ripple program (Kothe et al., 1991) is modified. 76 The first order time difference of Eq. (4.1) can be written as n + I _ n = -V-(^,v)" (4.29) in the explicit form; or n + l _ n = -V-(£-,v)""‘ (4.30) At in the implicit form. By applying the continuity equation, the acceleration term in the momentum equation, Eq. (4.2), can be written as Substituting Eq. (4.31) into Eq. (4.2), we can write the difference equation as (4.32) On the right side of the above equation, only the pressure term is taken at the advanced time Applying the two-step projection method, we have >j,£,“^ ^ = Piv’V.fev)"-A>'"V(£,..)"+s,"V.r" +£,V,g + /," (4.33) and _ Z- ^ = (4.34) At Eq. (4.34) can be expressed in terms of a Poisson equation for the pressure, by taking V • (£■," v""^* ) = V • (£•, v)". Then, we have 77 At The velocity at is then obtained as v""*=v ΗVp""'Ar (4.36) A 4 The difference equations are obtained in a staggered grid system. The finite difference approximation of the pressure Poisson equation, Eq. (4.35), leads to a set of linear equations. The resulting matrix equation is solved by the incomplete Cholesky- conjugate gradient (ICCG) method (Kershaw, 1978). The numerical methods for the VOF advection and the CSF method are described in Kothe et al. (1991). 4.5.2 Particle movement Within a time step of advance. At, particles are moved to a new position according to -^p =-^p.o -Ar (4.37) if no collision is encountered. The particle velocity is updated using a simple explicit integration formula (4.38) where the acceleration of the particle is obtained from Fq. (4.5). The equations, which are similar to those in the molecular dynamic simulation (Allen and Tidesley, 1987), are used to determine the minimum flight time of particles 78 before any collision. If the collisions occur within time step A/, this time step is split into several flight time steps, A/ci, and Af = g A /d (4.39) i The flight time of a particle is determined as the minimum flight time of this particle to its neighboring particles and/or walls. The flight time for two particles a and h is obtained by ■'"ab '^ab "V (^ b '^ab)^ “ '’^ab^ /lb "C ^a + ^ b ) " _ ab = 2 (4.40) Vab where i?a and are radii of particles a and h, respectively, and Tab — ra - ry and Vab = Va - Vb. For the particle-wall collision, the flight time is given as (4.41) ^x,a The velocity of the particle before each collision is updated within the time duration using Eq. (4.38). The detailed procedure of calculating the particle movement is shown in Fig. 4.1. 4.6 Results and Discussion 4.6.1 CVD-2 code Based on the theoretical models and the numerical methods presented in Sections 4.2-4.5, a two-dimensional code, named CVD-2 (combined ÇFD-VOF-DPM for 2-D three-phase flows), is developed by incorporating a program on incompressible flows with free surface, titled Ripple (Kothe et al., 1991). The present 79 code implements various steps of programming to address flow properties of the volume-averaged liquid phase, the Lagrangian formulation of the particle phase flow, and all the couplings among the three individual phases. The flowchart of the main program of the discrete phase simulation of gas-liquid-solid fluidization systems is shown in Fig. 4.2. Simulations are performed on the Cray T-90 supercomputer at the Ohio Supercomputer Center. 3.6.2 Verification o f liquid-solidfluidization Simulations of liquid-solid sedimentation and fluidization are performed in order to verify the prediction of the simulation code for the limiting case of three- phase flow. The snapshots of the predicted settling of 500 glass beads in water are shown in Fig. 4.3. The diameter and density of the particles used are 0.8 mm and 2,500 kg/m^, respectively. The simulation domain is 2 cm and 6 cm in the horizontal and vertical directions, respectively. In the computation for sedimentation, initially, the 500 particles are randomly placed in the liquid in the computational domain. Then they fall under gravity. The movement of the particles is determined by the dynamic motion equation and the collision mechanisms of the simulation model. Within about 6 seconds, a stable packing configuration with a bed height of 1.5 cm and an equivalent three-dimensional solids holdup of 65.7% is obtained. For fluidization, the simulation starts from the sedimented packing configuration. A liquid velocity of 6.0 cm/s is introduced evenly to the inlet boundary of the computational domain. In about the same time as the sedimentation simulation (6 seconds), a dynamically stable 80 fluidized condition is reached. The equilibrium bed height is 3.0 cm and the equivalent three-dimensional solids holdup is 23.2%. For the same fluidization condition, the solids holdup calculated from the Richardson-Zaki equation (Richardson and Zaki, 1954) gives rise to a solids holdup of 20.6%. With a ±20% deviation in the Richardson-Zaki equation, the simulation can be considered to be reasonably good. Snapshots of the bed expansion in simulated liquid-solid fluidization are given in Fig. 4.4. In liquid-solid fluidization, the pressure drop across the bed is known to be equal to the weight per cross-sectional area of the bed. Figure 4.5 shows the pressure gradient in the bed along different bed heights. The weight per unit volume of the bed is also shown in the figure. A good agreement between the pressure gradient and the weight per unit volume is seen in the figure. For liquid-solid fluidization, the microstructure of the horizontal particle alignment observed in the experiments (Fortes et al., 1987) and in direct simulation of particles in Newtonian fluids (Hu, 1996) is directly reproduced in the current simulations of sedimentation and fluidization as shown, respectively, in Figs. 4.3 and 4.4. This matching suggests that the discrete particle model proposed in this study is capable of capturing microscopic flow behavior of multiphase flow systems. 4.6.3 Bubble rising in liquid-solidfluidized medium The shape and rise velocity of a bubble in the liquid-solid medium is affected by a number of variables including the surface tension and viscosity of the liquid, the 81 densities of liquid and particles, the solids holdup, and the bubble size. Figure 4.6 shows the simulation and the experimental results of a single bubble rising in a liquid- solid fluidized bed. The simulation domain is 3x8 cm^. One thousand particles with a density of 2,500 kg/m^ and a diameter of 1.0 mm are used as the solid phase. An aqueous glycerin solution (80 wt%) with a density of 1,206 kg/m"*, a viscosity of 5.29x10*^ kg/m-s and a surface tension coefficient of 6.29x10'^ N/m is used as the liquid phase. A spherical bubble with a diameter of 1.0 cm is initially positioned in the computational domain with its center located 1.5 cm above the bottom. Initially, the particles are randomly positioned in a 3x24 cm^ area. Particles are then settle against a liquid inlet velocity of 0.5 cm/s. At this stage of the simulation, the spherical bubble is treated as a stationary obstacle. When an equilibrium bed height is reached, a three-dimensional equivalent solids holdup of 44% is achieved. With the particles at the equilibrium bed height, the simulation is then restarted. This time, the bubble is released, which yields subsequent bubble induced motion of liquid and particles. The time step of simulation for the liquid and solid phases is 5x10^ second. Experiments are performed in a two-dimensional column with a thickness of 7.0 mm. The comparison of the rise velocity based on the simulation and the experimental results is shown in Fig. 4.7. The relative deviations between the simulated and experimental results are within 7%. It is noted that a similar magnitude of the relative deviations was reported by Grace and Harrison (1967) when comparing the rise velocity of a two-dimensional bubble evaluated theoretically and that obtained experimentally from a two-dimensional column. The comparison of the bubble shape 82 from the simulation and the experiments is shown in Fig. 4.8. The bubble shape is described in terms of the aspect ratio, h/b, defined as the ratio of the m in or axis (vertical) over the major axis (horizontal) of the bubble. As shown in the figure, the simulation and the experimental results generally agree well. Figure 4.9 shows the velocity vector field of the liquid flow in the simulation domain relative to the four frames given in Fig. 4.6. As shown in the figure, the liquid flow is induced by the upward motion of the rising bubble. The liquid flows around the rising bubble with the highest velocity right behind the bubble. A symmetric flow pattern is observed initially as shown in Fig. 4.9(a). In Figs. 4.9(b)-4.9(d), two symmetric vortex cells behind the rising bubble are seen. In the figure, the bubble that appears in the four frames undergoes acceleration with an increase in the bubble Reynolds number from 0 to 28. Bubbles in this Reynolds number range exhibit a stable closed wake consisting of a pair of stationary vortices (Coppus et al., 1977). The liquid flow field around the bubble is often obtained in a moving coordinate system (Collins, 1966; Miyahara et al., 1989; and Tsuchiya et al., 1992). The velocity vector field of the liquid based on the moving coordinate system of the present simulation is given in Fig. 4.10 which is obtained by subtracting the rise velocity of the bubble from the velocity vectors shown in Fig. 4.9. A closed wake structure with two symmetric vortices is clearly evident in Figs. 10(b)—10(d). Figure 4.11 shows the velocity vector field of the particles. As shown in the figure, particles follow the liquid motion induced by the bubble flow. Particles immediately behind the bubble base move at a high velocity due to the high liquid 83 velocity. A significant downward flow of particles is observed at both sides of the bubble. As can be seen from the velocity vectors of the liquid in Fig. 4.9, the liquid flow further upstream than approximately two minor axes of bubble is not affected by the bubble motion. From Fig. 4.11, it is also seen that the particles in this flow region are almost in stationary. These particles are at the equilibrium state, suspended by the upward flow of the liquid. 4.6.4 Bubble wake structure and wake instability In general, the observed variation of the wake structure can be described based on the dual-wake-structure concept, that is, the wake consists of a primary wake and a secondary wake. In this study, the bubble wake structure and the wake instability are numerically studied. The solid phase used in the simulation is one thousand particles of glass beads with a density of 2,500 kg/m^ and a diameter of 0.7 mm. The computational domain is 6x12 cm". Water is used as the liquid phase and the liquid inlet velocity is set at 7.5 cm/s. The equilibrium bed height is 6.5 cm, yielding a solids holdup in the fluidized bed of 0.13. A spherical bubble with a diameter of 0.8 cm is initially imposed in the liquid-solid suspension at 1.5 cm above the bottom. As shown in the Fig. 4.12, the simulation results capture a primary wake moving in close association with the bubble, and a secondary wake extending far downstream. In this case, the bubble Reynolds number is about 1600, which falls into the range between the Second Critical Reynolds Number Recz (about 100-400) and the Third Critical Reynolds Number Rec 4 (above 6000). Considerable experimental results indicate that 84 the bubble wake in this range should be unsteady with a large-scale vortical structure (Fan & Tsuchiya, 1990). This behavior is reflected in the simulation results in Fig. 3.12. The figure reveals periodic asymmetric wakes about the vertical axis of the bubble movement, indicating the existence of the helical vortex wake typically observed in the range of Rec 2 The wake instability is closely associated with the wake interaction with the external flow. The wake instability is manifested by cyclic phenomena of vortex formation and shedding. In present study, with the bubble Reynolds number of about 1600, the wake oscillation of the bubble and the periodic asymmetric vortex shedding are observed as shown in the simulation results given in Fig.l. For the simulated case, the Menton number (Mo) is about 3.56x10'^* and the Eotvos Number (Eo) is 9.65. Thus, from empirical correlations of Fan & Tsuchiya (1990), the Strouhal Number (Sr) can be calculated as 0.14, and vortex shedding firequency as 3.58. The simulation results shown in Fig. 4.12 yield the wake vortex shedding frequency of 3.71, which is in reasonable agreement with experimental finding. 4.6.5 Particle entrainment When bubbles disengage firom the bed surface, particle entrainment takes place. Experimental observations of the evolution of particle flow around a single bubble have elucidated the mechanisms of particle entrainment in a three-phase fluidized bed (Miyahara et al., 1989; Fan and Tsuchiya, 1990; and Tsuchiya et al., 1992). The reported mechanisms indicate that particles are drawn by the wake behind 85 the bubble from the upper surface of the fluidized bed into the freeboard, and vortices containing particles are shed from the wake in the freeboard. The simulation conditions are the same as described in section 4.6.4. Figure 4.13 shows the bubble emerging from the bed surface in four frames of the simulation with a time interval between two frames of 0.03 second. Frame 1 of Fig. 4.13 shows the bubble emerging from the upper free surface of the fluidized bed. A group of particles are dragged by the bubble wake in the subsequent frames. The simulation results agree, in bubble- particle entrainment, and the related tim in g and displacement variations, with the experimental results given in photographs by Miyahara et al. (1989) and in sketches by Tsuchiya et al. (1992). Figure 4.14 shows the velocity vector field of the liquid for the four frames given in Fig. 4.13. An asymmetric vortex street is observed with the rising bubble. In this case, the bubble Reynolds number is about 1600, which falls into the range between the second critical Reynolds number Recz (about 100—300) and the third critical Reynolds number Recs (above 6000); bubbles in this Reynolds number range have unstable wakes and experience periodic asymmetric sheddings (Fan & Tsuchiya, 1990). The development of the shed vortices is observed in Fig. 4.14. The velocity vector field of the particles for the four frames given in Fig. 4.13 is shown in Fig. 4.15. As shown in the figure, most of the particles are suspended by the liquid flow and the velocities of these particles are very small. Particles with large velocities are found in the vortex areas and behind the rising bubble. The movement and the velocity 86 variation of the particles are closely related to the development of vortices and the motion of bubble flow. 4.7 Concluding Remarks A computational model for a gas-liquid-solid three-phase fluidization system and a two-dimensional code are developed in this study. The volume-averaging for the liquid phase flow, the Lagrangian simulation for particle, and the volume-of-fluid approach for the bubble flow are employed for the computation. The model takes into account the dynamic and discrete flow behavior of the gas-liquid-solid flow such as hubble-bubble, bubble-particle, and particle-particle interactions. The simulation results are verified with experimental results on the bed expansion and pressure drop in the liquid-solid fluidization. Simulations of a single bubble rising in a liquid-solid suspension and the particle entrainment in the fi-eeboard by an emerging bubble are also in qualitative agreement with the experimental data. From the simulation results, it is seen that particles follow the liquid motion induced by the bubble flow. Particles immediately behind the bubble base move at high velocity due to the high liquid velocity. In the case of a low bubble Reynolds number, a closed wake structure with two symmetric vortices is observed in the velocity vector field of the liquid based on the coordinates moving with the bubble. In the case of a high bubble Reynolds number, unstable and periodic asymmetrically shed wakes are obtained in the simulation. The simulated wake shedding firequency is comparable with the experimental findings. The discrete phase simulation scheme developed in this study 87 can effectively simulate the dynamic flow behavior of bubbles and particles in three- phase fluidized systems. 4.8 Nomenclature A Area Co Drag coefficient d Diameter e Restitution coefficient F Force / Volumetric body force / Correction function Friction coefficient g Gravity h Separation distance, minor axis of bubble / Moment of inertia n Normal vector of free surface p Scalar pressure Ps Surface pressure Pm Gas phase pressure inside bubbles r Radius of particle Re Reynolds number S Rate-of-strain tensor 88 t Time U Mean stream velocity, particle velocity Ub Rise velocity of a bubble u velocity V Volume V Velocity vector X Coordinate vector Greek letters a Volume fraction of fluid cccz Critical angle s Holdup <1> Correction function K Free surface curvature fi Dynamic viscosity V Kinematic viscosity p Density cr Surface tension r Viscous stress tensor, time CO Angular velocity 89 Subscripts G Initial condition AM Added mass a Particle index ac Center of particle a b Particle index be Center of particle b B Basset, bubble bf Bubble-fluid interaction bp Bubble-particle interaction D Drag fb Fluid-bubble interaction G/B Gravity/Buoyancy g Gas phase ij Cell indices I Liquid phase M Magnus P Pressure P Particle pb Particle-bubble interaction SV Surface tension w Wall 90 X X-component y Y-component Superscripts k Particle index N Normal direction T Tangential direction T Tangential direction 91 locate the nei^bor list for each particle calculate mininium fli^ t time o f each particle, tmin move particle during tmin, increment by tmin perform contact dynamics, update Vp, update the nei^bor list calculate newt^m move particle during ^l~Ctflig}it~lmin) _____ Figure 4.1 Flowchart of the particle-phase simulation. 92 start read input, mesh generation intialize fields ^ —— liquid-phase simulation solve N-S equtions bubble free surface simulation solve VOF equation calculate the effects of liquid on particle phase particle phase simulation calculate the effects of particle on liquid phase output Figure 4.2 Flowchart of the main program for discrete phase simulation of gas-liquid-solid fluidization systems. 93 liî »- w .. ## m i i \.:W:CL Figure 4.3 Simulation results of sedimentation o f 500 particles in water (dp = 0.8 mm, pp = 2,500 kg/m^, domain size: 2x6 cm^). 94 + ^ 4 4 4 4 4 4 4 4 4 4 Figure 4.4 Simulation results of fluidization of 500 particles in water (dp = 0.8 mm, pp = 2,500 kg/m^, domain size: 2x6 cm^). 95 2000 eu 1500 - !§> ■a jet) ' od , 1000 dP/dH C/3i c/5 X Unit weight of bed i . -o 500 1 2 3 Bed height (cm) Figure 4.5 Comparison of the pressure gradient with the weight per unit volume in the liquid-solid fluidized bed. 96 (a) t = 0.06 sec (b) t = 0.16 sec m - Æ m . R ^ ' I S' (c) t = 0.26 sec (d) t = 0.36 sec Figure 4.6 Simulation and experimental results of a bubble rising in a liquid-solid fluidized bed (ds = 1.0 cm, 1,000 particles, dp = 1.0 mm, pp = 2,500 kg/m\ domain size: 3x8 cm^, pi = 1,206 kg/m^, p.i = 5.29x10'^ kg/m-s, ai = 6.29x10 ^ N/m). 97 25 20 15 o 10 X experimental simulation 0 0.1 0.2 0.3 0.4 t(s) Figure 4.7 Comparison of the simulation and experimental results of the bubble rise velocity. 98 1.2 0.8 o experimental simulation I 0.6 0.4 0.2 0 0.1 0.2 0.3 0.4 t(s) Figure 4.8 Comparison of the simulation and experimental results of the bubble aspect ratio. 99 ^iuiA/ I uTfi i, a , I, I ( A/ i) U f. U/<. uw (a) t = to (b) t = to + 0.1 s (con’t.) Figure 4.9 The simulated velocity vector field of liquid flow surrounding the rising bubble in the liquid-solid fluidized bed. 100 Figure 4.9 (con’t.) S \ \ I K \ t 1 t / ^ ^ .^AAAAA. M; JU lUM, I ( . 1 J,/ ihUnA, 1,; f j ::::::: I::* ■ . m ] 1 a/ui, ' n/./. - ,a a J i i ,j t f I t \ \ N (c) t = to + 0.2 s (d) to + 0.3 s 101 (a) t = to (b) t = to + 0.1 s (con’t.) Figure 4.10 The simulated velocity vector field of liquid flow surrounding the rising bubble in the liquid-solid fluidized bed based on a moving coordinate system. 102 Figure 4.10 (con’t.) nr f l i l u-r r ' ' ' / / / n [i U.l \ ‘AV S \ 1 > > i U l w m (c) to + 0.2 s (d) t = to + 0.3 s 103 (a) t = to (con’t.) Figure 4.11 The simulated velocity vector field of particles following the liquid motion induced by the bubble flow. 104 Figure 4.11 (con’t.) (c) to + 0.2 s (d) t = to + 0.3 s 105 : T: lîxîxx: : : : : : :::::: : : : : ::: :::: : ; : : ; : : : ' y % ^ '*• m m « S S i -r -m 2. (a) t = to (b)t = to + 0.1s (b)t = to + 0.2s (b)t = to+0.3s Figure 4.12 Simulation results of bubble wake structure. 106 /• • • > >- (a) t = to (b) t = to -f- 0.03 (con’t.) Figure 4.13 Simulation of a bubble emerging from a liquid-solid fluidized bed (de = 0.8 cm, 1,000 particles, dp = 0.7 mm, Pp = 2,500 kg/m^, domain size: 6x12 cm^, pi = 1,000 kg/m^. Pi = 1.0x10'^ kg/m-s, ai = 7.19x10'^ N/m). 107 Figure 4.13 (con’t.) vr i* • f • * n / #, (c) t = to + 0.06 (d) t = te + 0.09 108 # 1 iimmi /f/11 ililll (a) t = to (b) t = to + 0.03 s (con’t.) Figure 4.14 The simulated velocity vector field of liquid flow surrounding a rising bubble in a liquid-solid fluidized bed. 109 Figure 4.14 (con’t.) I I I I r ( I I I r I t I I i I I t I f trrirtitti I f I t t r I I I f I t t t • I t I I r t I f I I r t I r r t t I I I r I I / r I 1 I t I ( r r r r t t t t I r f r / r WWW\\\\\\\ W \ \ I I (!!!!! ! t t t t I ! I t t %#Ëtt////ft f f/f r r 11 X// ////I ^ / / / f f t iii/ \ V I i 1 Nü^üÉ"- .\l\\\WVv\ IJJiUU'v \ \ \ i \ \ \ \ M l I \ \ » » L^///f»»»ll!!!!1%lt »! t t t t » t % ///Mrtitititlll / / //r/ifrtiiiiitt /ffrttiritittit rffrtifrtiiiiit ffifrrirtitii ffrfrritiiiii ; f I I I * I ; I t 1 « 1 ffrriitiitiii (c) t = to + 0.06 s (d) t = to + 0.09 s 110 •✓/ t (a) t = to (b) t = to + 0.03 s (con’t.) Figure 4.15 The simulated velocity vector field of particles following the liquid motion induced by the bubble flow. Ill Figure 4.15 (cou’t.) '/* / f (c) t = to + 0.06 s (d)t 0.09 112 CHAPTER 5 DISCRETE PHASE SIMULATION OF BUBBLE DYNAMICS IN BUBBLE COLUMNS AT ELEVATE PRESSURES 5.1 Introduction In the present study, numerical simulations on the single bubble dynamics at the elevated pressure are performed by using an improved VOF method including a continuous transition approach or treating the bubble-liquid interface. In order to verify the numerical studies, experiments are performed to study the effect of pressure on bubble shape and bubble rise velocity. The computational results provide intrinsic information describing the effect of pressure on such bubble flow properties as velocity, shape, size of a single rising bubble, and maximum stable bubble size. Comparisons of computational results with experimental data are also made. 5.2 Theoretical and Numerical Models The conventional VOF method for simulating gas bubble treats a gas bubble as a void with an assumption that the momentum transfer from the flow inside the bubble to its surroimding liquid is negligible due to the significant density difference between the 113 gas and liquid (Kothe et aL, 1991; Harper et al., 1991). Under high-pressure conditions, however, the effect of gas density and viscosity inside the bubble plays a significant role on the gas-liquid flow behavior. Thus, under high pressure conditions, the flow fields of both the liquid phase outside bubble and the gas phase inside bubble need to be considered. 5.2.1 Governing equations For both liquid flow outside the bubble and gas flow inside the bubble, the governing equations can be written based on the Navier-Stokes equations as given by the continuity equation, ^^V-(pU ) = Q (5.1) at and the momentum equation, 4- V • {pUU) = -Vp + V-{r) + pg + F^^ (5.2) where U is the velocity vector; p is the density; p is the scalar pressure; Fbf is the interface force including the surface tension force; and r is the Newtonian viscous stress tensor which is given as: r = 2pS = p[(VU) + (V U f ] (5.3) whereS is the rate-of-strain tensor and p is the coefficient of dynamic viscosity. The properties of liquid or gas are used in the Eqs. (5.1) to (5.3) when the computational cell is in the liquid phase or the gas phase respectively. When it is in the 114 interface of the gas and liquid phases, the properties of mixture of the gas and liquid phases based on the volume fraction weighted average are used, which will be discussed in Section 5.2.3.1. 5.2.2 Interface tracking The movement of the gas-liquid interface is tracked based on the distributions of F(x,t), the volume fraction of liquid in a computational cell, i.e., F(x,t)=\ in the liquid phase and F(x,t)=Q in the gas phase (Hirt and Nichols, 1981). Therefore, the gas-liquid interface exists in the cells whereF(x,t) lies between 0 and 1. From the values of F(x,t) in neighboring cells, the size and shape of gas bubble can be reconstructed. The advection equation for F(x,t) is given by: ^ ^ ^ ^ ^ ^ + (U-V)F(x,t)=0 (5.4) dt The piecewise linear interface calculation (PLIC) method (Youngs, 1982) is applied to reconstruct the bubble free surfaces with a multidimensional algorithm to determine the slope of each reconstructed line. The curvature of the interface is applied to obtain the surface tension force. The effect of the surface tension force is incorporated in the computation of the velocity and pressure fields via an equivalent surface pressure. In the gas-liquid free surfaces, the stress boundary condition follows the Laplace equation as: p,= p-p^=OK (5.5) 115 whereps is the surface tension induced pressure difference across the interface; and cr is surface tension, and /cis the free surface curvature as r \ K = -{V -h )= ^ «.V M — (V •«) (5.6) \n\ vi«l ; where /i=VF (5.7) rl The surface force can be converted into the volume force by the continuum surface force model (Brackbill et al., 1992) as given by Fbf(x,0 = a7c(x,f)VF(x,0 (5.8) The volume force is the interface force between the gas and liquid phases, which is applied to the governing equation, Eq. (5.2), for the both phases. 5.2.S Solution technique 5.2.3.1 Continuous transition treatment In this smdy, the numerical simulation is performed on the entire flow domain including the flow field inside the gas bubble. In order to circumvent the divergence problem resulting from a discontinuity in fluid properties across the interface, a continuous transition treatment (CTT) is applied to the evaluation of the fluid properties across the interface. In the CTT method, the property discontinuity is approximated by a smooth variation of mixture properties from one phase to the other within a finite 116 interface thickness. The mixture properties are then calculated using a volume fraction weighted averaged method using the scalar fraction function, F(x,t), as: P = P\- ^ (x ,0 + Pg • (1 - /y = //,-F(x,0+Pg-(l-FCx,/)) (5.9) where the subscribes / and g represent the liquid and gas phases respectively. 5.2.3.2 Two-step projection method Equations (5.1) and (5.2) are solved using a two-step projection method. The momentum equation can be written in the form of time discretization as ( pEI ~(P^) =-S/.(pUUf-^p'^^^ +V-r"+r/:g^/+F^ / (5.10) A t where only pressure term is taken at f , while all the right hand side terms are taken at f. In the two-step projection, Eq. (5.10) is decomposed to two equations: P - R j z i Æ Ï - = -sj.( pUU f ^(pgf + F . / (5.11) At -p'^Ü (5.12) At The variations of advection, viscosity, gravity force and interface force are considered to obtain a temporary velocity field U in Eq. (5.11), and new velocity field is then calculated in Eq. (5.12) with the pressure information at time 117 The time discretization of the continuity equation can be written as = 0 (5.13) Thus, the velocity field is projected onto a zero-divergence vector field, and Eq. (5.12) can be expressed in a Poisson equation for the pressure as V-Cp Û) n+\^ (5.14) At With the pressure at calculated firom Eq. (5.12), the value for can be evaluated as = U (5.15) The difference equations are obtained in a staggered grid system, and the finite difference approximation of the pressure Poisson equation leads to a system of linear equations. The resulting matrix equation is solved with the incomplete Cholesky- Conjugate gradient (ICCG) method (Kershaw, 1978). 5.3 Simulation Conditions 5.3.1 CVD-2-HP code The flow advection inside a gas bubble and the treatment of smooth variation of fluid properties across the interface are included in the two-dimensional code CVD-2 as discussed in Chapter 2. The flowchart of the code CVD-2-HP is given in Fig. 5.1. The simulations are carried out using the Cray T-90 supercomputer at the Ohio State Supercomputer Center; it takes about 1.2 hours of CPU time to compute for 1 second real 118 time. The simulation, results in Cartesian coordinates are presented in this study to reveal the pressure effects on the bubble size and shape. Furthermore, to compare the simulation results with the 3-D experimental data, the bubble rise velocity is simulated in cylindrical coordinate system. 5.3.2 Boundary and initial conditions For the simulation in Cartesian coordinates, a 101.6 mm (diameter) x 80.0 mm (height) domain is used with three walls and an exit on the top as boundary conditions. For the simulation in cylindrical coordinates, a half domain is used with a symmetric boundary. A spherical gas bubble is artificially imposed at 15 mm above the bottom of the computational domain. In this study, two bubble size are studied, 7.5 mm and 11 mm in diameter. The numerical studies indicate that the effect of the side walls on the bubble behavior can be neglected under present simulation conditions, due to the fact of small ratio of bubble diameter to the width of computation domain (less than about 0.1). 5.3.3 Selection o f parameters In the simulation, Paratherm NF heat transfer fluid is used as the liquid phase and nitrogen constitutes the gas phase as those used in the experiments discussed in Section 5.4.4. The physical properties of the fluids under the simulation conditions are given in Table 5.1. The bubble behavior is numerically studied for two pressure conditions: 0.1 MPa and 19.4 MPa with the computation time step of 5x10"^ second. 119 5.4 Results and Discussion 5.4.1 Pressure effect on the bubble rise velocity The rise velocity of a single bubble of 7.5 mm in spherical equivalent diameter is numerically simulated at 27°C for two pressures of 0.1 MPa and 19.4 MPa. The numerical simulation in this case is performed in a cylindrical coordinate system with 54 X 80 grids. As shown in Fig. 5.4, the bubble in the column rises from rest quickly accelerates to its terminal rise velocity. For a given size bubble, it rises faster at the lower pressure of 0.1 MPa (Fig. 5.2) than that at the pressure of 19.4 MPa (Fig. 5.3). The pressure induces the physical property variation of the fluids; at the pressure of 19.4 MPa, the gas phase density increases by about two orders of magnitude over that at the pressure of 0.1 MPa. The numerical simulated terminal rise velocity of the bubble in the diameter of 7.5 mm is 23.21 cm/s at 0.1 MPa or 19.16 cm/s at 19.4 MPa. The comparison of the simulation results with the experimental results described in Section 5.4.4 shown in Fig. 5.4 reveals the accurate numerical simulation of the pressure effect on the bubble rise velocity. The comparison of the simulated bubble rise velocity is also proved by the Fan- Tsuchiya equation (1990) for high-pressure systems. The Fan-Tsuchiya equation in dimensionless form can be given by og P\ Pi 2 (5.16) 120 where Mo is Morton Number, C/ÿ is the bubble rise velocity and De' is the dimensionless bubble diameter given by (5.17) where Dg is the bubble equilibrium diameter. The empirical parameters, n, c and Æy, are three specific factors governing the rate of bubble rise, which have the values for this case: n = 0.8, c = 1.2, and ATyo = 14.7 for ,12) (5.18) The calculated value for the bubble rise velocity in this case, based on the Fan- Tsuchiya correlation are 22.6 cm/s at 0.1 MPa and 18.9 cm/s at 19.4 MPa. The simulation results are also in good agreement with the calculated values. 5.4.2 Pressure effect on the bubble shape To examine the effects of pressure on the bubble shape, the above case is simulated in Cartesian coordinate. The simulated results are plotted in Figs. 5.5 and 5.6 with the origin of the coordinate system fixed on the gas bubble and rising with it. It is seen in the figures that when the bubble rises at the pressure of 19.4 MPa, the bubble shape changes from the initial imposed spherical (Fig. 5.6a) to ellipsoidal cap (Fig. 5.6b), the oblate (Fig. 5.6c) and then flatter oblate (Fig. 5.6d). At the beginning, two symmetric laminar vortexes can be observed closely behind the rising bubble (Fig. 5.6b) due to the low rise velocity. As the rise velocity increases, asymmetric bubble wake dynamics becomes evident with a primary wake moving in close association with 121 the bubble and a secondary wake extending to the downstream (Figs. 5.6c and 5.6d). The alternate vortex shedding is observed in Fig. 5.6d. The internal gas flow patterns are shown in Figs. 5.5 and 5.6. The internal gas circulation is symmetric (Figs. 5.5a, 5.5b and Figs. 5.6a, 5.6b) when bubble rise velocity is small occurring at the initial rising stage; and asymmetric when the bubble rise velocity further increases to its terminal rise velocity (Figs. 5.5c, 5.5d and Figs. 5.6c, 5.6d). The variation in the internal gas flow pattern is in concert with that in the bubble shape and the instability of external wake structure. Several differences of bubble behavior are observed from the simulation results. The bubble shape is flatter at 19.4 MPa (Fig. 5.6d) than at 0.1 MPa (Fig. 5.5d) based on the simulation. This is caused, in part, by the increased centrifugal force of the internal gas circulation at higher pressure, which stretches the bubble. Differing from the bubble shape variation at 19.4 MPa described above, the ellipsoidal cap bubble is not observed at 0.1 MPa in the shape transition. More wake sheddings are observed in the flow field at the low pressure (Fig. 5.5) than that at the high pressure due to the higher rise velocity at lower pressure. 5.4.3 Pressure effect on the bubble size Luo et al. (1999) measured bubble sizes under various pressures, and found that the maximum stable bubble size decreases with increasing pressure. A theoretical equation is developed by them to account for the maximum stable bubble size. Demax, as given below 122 ^emax*C — ( C = 2.53 for = 0.21, C = 3.27 for ûT = 0.3) (5.19) \SP, where a , the aspect ratio of bubble, is defined as the ratio of bubble height to bubble breadth. From Eq. (5.19), the maximum stable diameter of bubble is predicted as 15.2 mm at 0.1 MPa; and 9.5 mm at 19.4 MPa. Figures 5.5 and 5.6 show that a bubble of the equivalent spherical diameter of 7.5 mm can rise up stably, as predicted, under pressures of 0.1 MPa and 19.4 MPa. In contrast, a bubble of equivalent spherical diameter of 11 mm would be stable for 0.1 MPa but unstable for 19.4 MPa from Eq. (5.19). Figures 5.7 and 5.8 show the simulation study on stable bubble size at 0.1 MPa and 19.4 MPa. In Fig. 5.7, it can be seen that the bubble changes its shape from spherical to oblate cap (Fig. 5.7b) and oblate (Figs. 5.7c and 7d), and rises up without breakage at 0.1 IvIPa. However, at 19.4 MPa (Fig. 5.8), the bubble shape becomes concave-up at first (Fig. 5.8b) due to the stronger upward liquid inertia in the bubble wake and the strong internal circulation at the edge o f the bubble. The centrifugal force of the internal gas circulation together with the upward force of the external liquid flow in the wake, further flatten the up and down curved bubble (Fig. 5.8c), and finally break the bubble into three small bubbles (Fig. 5.8d). These three small bubbles then recover into stable shapes (oblate and cap) and rise stably. 123 5.4.4 Experimental validation The numerical simulation results are validated by experiments using apparatus shown in Fig. 5.9. Experiments are performed in a high pressure stainless-steel column of 101.6mm in diameter and 1.58m in height with three pairs of quartz windows for flow visualization. Each window has the dimensions of 12.7mm in width and 92 mm in height. A high-resolution (800 x 490 pixels) CCD camera with an infinity lens is used to quantify the bubble flow behavior. Paratherm NF heat transfer fluid is used as the liquid phase and nitrogen is used as the gas phase. The physical properties of the fluids are measured in situ with the surface tension obtained by the emerging bubble method (Lin et al., 1996); the liquid density and viscosity are obtained by the hydrostatic weighing and falling ball techniques (Lin and Fan, 1997). A two-stage pressure regulator and a flow control valve are used to provide control of the prescribed gas flow rate. Gas enters the column through a multiorifice sparger in a ring arrangement with an orifice diameter of 3 mm. Figure 5.10 shows the pressure effect on the bubble shape. The photographs of the bubbles in the equivalent spherical diameter of 7.5 mm at various pressures indicate that at elevated pressure the bubble shape becomes flatter. Specifically, the bubble shape varies from the oblate (Fig. 5.10a) to spherical cap (Fig. 5.10d). The simulated results at O.lMPa (Fig. 5.5d) and 19.4MPa (Fig. 5.6d) exhibit similar bubble shape variations to this experimental findings. As shown in Fig. 5.11, for a given bubble volume, the aspect ratio ((z) of the single bubble is reduced with an increase in pressure. That is, for a bubble of volume 0.22 ml, (equivalent diameter of 7.5 mm), a changes from 0.34 to 0.23 124 with increasing pressure from 0.1 MPa to 19.4 MPa. The comparison between the simulation and experimental results in Fig. 5.11 shows a good agreement. The rise velocities of single bubbles of a given size are measured and simulated at pressures of 0.1 MPa and 19.4 MPa, as shown in Fig. 5.12. It is seen that a single bubble of a given size has a higher rise velocity at 0.1 MPa than that at 19.4 MPa and a large bubble rises faster than small one. Fig. 5.12 shows the agreement between the measured and the simulated results. A bubble of an equivalent spherical diameter of 11 mm disintegrates at 19.4 MPa, and thus only the simulated rise velocity of an equivalent spherical diameter of 7.5 mm at 19.4 MPa is shown in the figure. 5.4.5 Bubble-bubble interactions Bubble-bubble interactions when two bubbles rising side-by-side and in-line at a pressure of 19.4 MPa are simulated. Figure 5.13 shows a series of simulated bubble trajectory and liquid velocity vector field. The simulation domain is 8 x 9 cm^ with the liquid filled from bottom up to 8 cm in height. The area above 8 cm is filled with gas. Initially, two bubbles of 0.8 cm in diameter are placed with centers at 1.5 cm above the bottom and 1 cm from the both sides of the vertical symmetric line of the simulation domain as shown in Fig. 5.13(a). Significant interactions of bubble wakes are observed from Figs 5.13(b) ~ 5.13(e). At first, two bubbles rise rectilinearly with little interactions. As the wakes behind the bubbles grow, bubbles start to separate as shown in Fig. 5.13(b). The bubbles separate further as the result of the wake interactions. However, when the separation distance is large enough, interactions between two bubbles 125 become less significant. In the current simulation, bubbles are seen flowing towards each other after they separate to a certain distance as shown in Figs. 5.13(d)&5.13(e). It is seen that the separation distance is larger than the bubble-wall distance when two bubbles change the flow direction. This shows the wall effect on the multi-bubble rising trajectories. Experimental studies of two bubbles rising side by side were reviewed by Fan and Tsuchiya (1990). It is reported that bubble-bubble interactions are less significant when separation distance between the bubbles is large enough. Dynamic bubble-bubble interactions are seen when the separation distance is not sufficiently large. If two bubbles are close enough, bubble coalescence wül take place. The initial separation distance is selected in this simulation to show the significant interactions between two bubbles and bubbles and walls. The bubble and wake interactions are further studied for two bubbles rising in line. Figure 5.14 shows the rise pattern o f successive bubbles and the resulting bubble coalescence and breakup. The simulation domain is 6 x 9 cm^. Two bubbles of 0.8 cm in diameter are initially placed in the center of the simulation domain with the vertical distance 1.5 and 2.8 cm above the bottom, respectively. When two bubbles rise, the trailing bubble accelerates due to the wake effect of the leading bubble as shown in Fig. 5.14(b)&5.14(c). The trailing bubble eventually catches up with the leading one and two bubbles merge to a single large bubble shown in 5.14(d). In the current simulation, the merged bubble is much larger than the stable bubble size. Therefore, the merged bubble breaks into two bubbles shortly after the coalescence as shown in Fig. 5.14(e). The simulation results agree with the mechanism revealed from the experimental observations 126 of the two bubble coalescence and breakup reported by many researchers (Crabtree and Bridgwater, 1971; de Nevers and Wu, 1971; Narayanan et al., 1974; Otake et al., 1974; Bbaga and Weber, 1980; Komasawa et al., 1980; Tsuchiya et al., 1988). 5.5 Concluding Remarks Single bubble rising characteristics in a bubble column under the high pressure (19.4 MPa) condition are numerically simulated and compared with those under the ambient (0.1 MPa) condition in this study. The flow field of the gas phase inside the bubble and the liquid phase outside the bubble is obtained, which reveals significant asymmetric gas circulation patterns inside the bubble and liquid vortices associated with wake formation and shedding. The results of computation demonstrate the transient bubble rise velocity variation to its terminal value and the transient bubble shape variation to its stable shape. An experimental validation of the computed results on the bubble rise velocity and bubble shape is conducted which shows a good agreement. The simulation also confirms with the phenomena of higher pressure yielding a smaller maximum stable bubble size in a bubble column. The wake interactions and the bubble coalescence and breakup are also simulated and agree with the experimental observations. 5.6 Nomenclature c Empirical parameter governing the bubble rising velocity C Empirical parameter calculating the stable maximum bubble size 127 De Equilibrium diameter of gas bubble F Volume fraction o f fluid fbf Gas-liquid interaction force g Gravity acceleration Ko Empirical parameter governing the bubble rising velocity Mo Morton number n Empirical parameter governing the bubble rising velocity p Scalar pressure Ps Surface pressure /7v Gas phase pressure inside bubble O Fluid property S Rate-of-strain tensor t Time U Velocity vector C/b Bubble rising velocity X Position Greek letters a Aspect ratio of bubble K Curvature of free surface jJL Dynamic viscosity p Density 128 cr Surface tension coefficient r Viscous stress tensor Subscripts 0 Initial condition e Equilibrium value g Gas phase max Maximum value 1 Liquid phase Superscripts ' Dimensionless value n Time step 129 Table 5.1 Physical properties of the gas and liquid used in the experiments P = 0.1 MPa P = 19.4 MPa Gas density 1.1 kg/m^ Gas density 220.15 kg/m^ Liquid density 868 kg/m^ Liquid density 896.32 kg/m^ Liquid viscosity 0.0289 Pas Liquid viscosity 0.0431 Pas Surface tension coefficient 0.0296 N/m Surface tension coefficient 0.02295 N/m 130 start read input, mesh generation intialize fields solve N-S equtions for the liquid and gas phases solve VOF equation for tracking the bubble surface calculate the gas-liquid interaction by CSF model update the field properties by CTT method output end Figure 5.1 Flowchart of discrete phase simulation program including CTT. 131 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 0 0 (a) to (b) to+0.08s (con’t.) Figure 5.2 Simulation results of a rising bubble in cylindrical coordinate system (Dg = 7.5 mm, f = 0.1 MPa). 132 Figure 5.2 (con’t.) 8 8 7 7 6 6 \\ 5 5 4 4 3 3 2 2 1 1 0 0 0 1 2 0 1 2 (c) to+0.16s (d) to+0.24s 133 8 7 6 6 - 5 4 4 - 3 # 2 u'!:: 1 1 - 0 1 2 (a) to (b) to+0.08s (con’t.) Figure 5.3 Simulation results of a rising bubble in cylindrical coordinate system (£>e = 7.5 mm, P = 19.4 MPa). 134 Figure 5.3 (con’t.) 8 8 7 7 6 6 5 5 r 1 4 4 3 3 2 2 1 1 0 0 (c) to+0.16s (d)to+0.24s 135 30 20 CM OS D ■simulated results at P = 0.1 MPa 10 - ■simulated results at P = 19.4 MPa • experimental data at P = 0.1 MPa X experimental data at P = 19.4 MPa 0.05 0.1 0.15 0.2 0.25 0.3 Time (second) Figure 5.4 Comparison of the rising velocities under different pressures. 136 8 7 6 5 4 3 2 1 0 (a) t = to (con’t.) Figure 5.5 Simulation results of a rising bubble in Cartesian coordinate system (De = 7.5 mm, f = 0.1 MPa). 137 Figure 5.5 (con’t.) 8 if!!;! p " ' (b) t = to+0.08s (con’t.) 138 Figure 5.5 (con’t.) 8 7 6 5 4 3 2 1 0 (con’t.) 139 Figure 5.5 (con’t.) 8 7 6 5 4 3 2 1 0 (d) t = t o+0.24s 140 8 7 6 5 4 3 2 1 0 (a) t = to (con’t.) Figure 5.6 Simulation results of a rising bubble in Cartesian coordinate system (De= 7.5mm, f = 19.4 MPa). 141 Figure 5.6 (con’t.) 8 7 6 5 4 3 2 1 0 (b) t = to+0.08s (con’t.) 142 Figure 5.6 (con’t.) 8 7 6 5 V \ 4 3 2 1 0 l , (c)t = to+0.16s (con’t.) 143 Figure 5.6 (con’t.) 8 7 6 5 4 3 2 1 0 (d) t = to+0.24s 144 8 7 6 5 4 3 2 1 0 (a) t = to (con’t.) Figure 5.7 Simulation results of a rising bubble in Cartesian coordinate system {De =11.0 mm, P = 0.1 MPa). 145 Figure 5.7 (con’t.) ' " P i r s e t 3 4 (b) t = to+0.06s (con’t.) 146 Figure 5.7 (con t.) !! i (c) t = to+0.12s (con’t.) 147 Figure 5.7 (con’t.) 8 7 6 5 4 3 ill 2 1 0 (d) t = to+0.18s 148 8 7 6 5 4 3 2 m 1 O. 0 (a) t = to (con’t.) Figure 5.8 Simulation results of a rising bubble in Cartesian coordinate system (Dg= 11.0 mm, P = 19.4 MPa). 149 Figure 5.8 (con’t.) 8 7 M!1 6 5 4 3 2 1 0 (b) t = to+0.06s (con’t.) 150 Figure 5.8 (con’t.) S N. SN \ (c) t = to+0.12s (con’t.) 151 Figure 5.8 (con’t.) 8 llll 7 6 5 V ^ V \ 4 3 2 1 0 (d) t =to+0.18s 152 — 4 I. Gas inlet 2. Liquid inlet 3. Perforated plate distributor 4. Quartz windows 5. Heat transfer probe 6. Support of the probe 7. Copper screen 8. Sealing and signal cables 9. Gas and liquid outlet 10. Lighting II. Differential pressure transducer 12. Thermocouple 13. TV and VCR 14. Video camera 15. DC source 16. Signal amplifier 17. Data acquisition system 18. Computer Figure 5.9 Schematic diagram of high-pressure fluidized bed. 153 P = 0.1 MPa P = 3.45MPa P = 10.34MPa P = 19.3MPa a = 0.34 a = 0.31 a — 0.265 a - 0.23 (a) (b) (c) (d) Figure 5.10 Bubbles under various pressures (De « 0.7 mm). 154 0.35 0.3 0 .2 5 od ü (U 0.2 & ♦ — Experimental data < 0 .1 5 ♦ Simulation results 0.1 010 20 Pressure (MPa) Figure 5.11 Bubble aspect ratios under various pressures. 155 0.4 Experimental data at 0.1 MPa Experimental data at 19.4 MPa 0.3 Simulation results at 0.1 MPa Simulation results at 19.4 MPa 0.2 0.1 10 100 Dg (m m ) Figure 5.12 Pressure effect on bubble rise velocity. 156 (a) t = to (con’t.) Figure 5.13 Bubble trajectories and liquid velocity vector field when two bubbles rising side-by-side at a pressure of 19.4 MPa. 157 Figure 5.13 (con’t.) ii 9 (b)t = to + O.I5 s (con’t.) 158 Figure 5.13 (con’t.) mmi « r >■ / > / \ y ' f (c) t = to + 0.3 s (con’t.) 159 Figure 5.13 (con’t.) (d) t = to + 0.45 s (con’t.) 160 Figure 5.13 (con’t.) ;i I mi mil (e) t = to + 0.6 s 161 (a) t = to (con’t.) Figure 5.14 Bubble trajectories and liquid velocity vector field when two bubbles rising in-line at a pressure of 19.4 MPa. 162 Figure 5.14 (con’t.) ilil il IM (b) t = to + 0.1 s (con’t.) 163 Figure 5.14 (con’t.) (c) t = to + 0.15 s (con’t.) 164 Figure 5.14 (con’t.) (d) t = to + 0.2 s (con’t.) 165 Figure 5.14 (con’t.) » (e) t = to + 0.25 s 166 CHAPTER 6 NUMERICAL STUDIES OF BUBBLE AND PARTICLE DYNAMICS IN A THREE-PHASE FLUIDIZED BED AT ELEVATED PRESSURES 6.1 Introduction In this study, the bubble and particle dynamics in a gas-liquid-solid fluidized bed at elevated pressures is numerically studied using a discrete phase simulation. The Eulerian volume-averaged method, the Lagrangian dispersed particle method and the volume-of-fluid (VOF) method are employed for simulations of liquid, solid particles, and gas bubbles, respectively. A bubble induced force model, a continuum surface force model, and Newton’s third law are applied to account for the couplings of particle- bubble, gas-liquid, and particle-liquid interactions, respectively. A close distance interaction model is included in the particle-particle collision analysis, which considers the liquid interstitial effects between colliding particles. The flow characteristics studied include single-bubble rise velocity at various solids holdups, the stable maximum bubble size and bubble breakup, as well as the bubble-particle interactions. Comparisons of computational results with experimental data are also made. 167 6.2 Simulation Scheme In this study, the volume-averaged Navier-Stokes equations are used to solve the liquid phase flow in the presence of dispersed particles in the liquid phase. The motion of particles is described by the dispersed particle method and the motion of gas bubbles is described by the VOF firont-tracking method. The gas-liquid interfacial mass, momentum, and energy transfer is described by a continuum surface force (CSF) model. A close-distance interaction (GDI) model, which considers the strong damping effect due to the liquid fihn before particle contact, is used to determine the particle contact velocity just before collision. Upon contact of two particles, a hard sphere approach is used to calculate the velocities of the particles after collision. The particle-bubble interaction is formulated by incorporating the surface tension force in the equation of motion of particles. The particle-liquid interaction is brought into the liquid phase Navier-Stokes equations through the use of Newton’s third law of motion. Under high-pressure conditions, the effect of gas density and viscosity inside the bubble plays a significant role on the gas-liquid flow behavior. Thus, the flow field of both the liquid phase outside bubble and the gas phase inside bubble is simulated simultaneously in this study. A continuous transition treatment (CTT) is applied to evaluate the fluid properties across the interface. Details of the simulation scheme and numerical methods are given in Chapters 4 and 5. 168 6.3 Results and Discussion 6.3.1 Simulation conditions Simulations are performed using the two-dimensional code, CVD-2, which has been described in previous chapters, Chapters 4 and 5. In the simulation, Paratherm NF heat transfer fluid, nitrogen, and glass beads are used as liquid, gas, and solid phase, respectively. The hydrodynamics of bubble rising in a liquid-solid fluidized bed are numerically studied under an elevated pressure of 17.3 MPa. To study the pressure effects, simulations for bubble rising in similar conditions but atmospheric pressure are also conducted. The physical properties of the gas and liquid phases under the simulation conditions are given in Table 6.1. Solid particles are spherical with a density of 2,500 kg/m^ and diameter of 0.088 cm. The solids holdups studied are 0.384 and 0.545. To simulate the bubble rising in a liquid-solid fluidized bed, initially, a spherical bubble is positioned in the computational domain with its center located 1.5 cm above the bottom; the particles are randomly positioned in a 3x24 cm^ area. Particles are then settle under gravity in the liquid medium with an evenly distributed inlet velocity. The spherical bubble is treated as a stationary obstacle at this stage of the simulation. At a certain liquid inlet velocity, the desired solids holdup is reached in a 3x8 cm^ area. The simulation is then restarted in this 3x8 cm^ domain with the particles at their equilibrium positions and the bubble released. The time step of simulation for the liquid and solid phases is 5x10"^ second. Th inlet liquid velocity is 1.0 cm/s and 0.45 cm/s for the solids holdups of 0.384 and 0.545, respectively 169 6.3.2 Bubble rise velocity In this study the simulation is performed in a two-dimensional column. In order to compare with the experimental data for the bubble rising in liquid-solid media at high pressures obtained in a three-dimensional column, a conversion for the bubble rise velocity of two-dimensional case to that of three-dimensional case is required. It is noted that for large bubbles, the theoretical Davies—Taylor equation (Davies and Taylor, 1950) gives the two-dimensional bubble rise velocity C/b 2 = H2{gR) and the three-dimensional bubble rise velocity U\,t, = 2l3(gR). Therefore, the conversion factor of the bubble rise velocity from two-dimensional to that of three-dimensional is % = 4/3 Ubz- Although it is not appropriate to use the Davies—Taylor equation to predict the rise velocities of bubbles of small diameter (d^ < 2 cm), this relationship between the two-dimensional and three-dimensional rise velocities is used in the current study as an approximation. This conversion factor is used to obtain the three-dimensional rise velocity from the simulation results. Additionally, the following equation is used to correct the wall effect (Chft et al., 1978) on the bubble rise velocity (6 .1) 1 -M{ d "J Eq. (6.1) is recommended for bubbles and drops for Eo < 40, Re > 200, and db/D < 0.6. The comparison of the simulation and the experimental results on the bubble rise velocity is given in Table 6.2. As shown in the table, the simulation results agree well with the experimental data for the rise velocity of a gas bubble in the liquid-solid medium under various solids holdups and system pressures. The reduction of the bubble rise 170 velocity with an increase in pressure is one of the major reasons for the significant increase in the gas holdup of three-phase fluidized beds at elevated pressures. When the pressure is increased from 0.1 to 15.6 MPa, a 100% increase of gas holdups was reported at all gas velocities (Luo et al., 1997a). 6.3.3 Bubble shape and trajectory Similar to the rise velocity of a bubble, the shape of a bubble is affected by the physical properties and system parameters, including the surface tension and viscosity of the liquid, densities of liquid and particles, solids holdup, bubble size, and system pressure and temperature. The aspect ratio of bubble, h/b, defined as the ratio of the minor axis over the major axis of the bubble, is the parameter used to characterize the bubble shape. The simulation results of the bubble aspect ratio changing with the time during the bubble rising in the liquid-solid fluidized bed with solids holdup of 0.384 at pressures of 0.1 MPa and 17.3 MPa, and a solids holdup of 0.545 and pressure of 17.3 MPa are shown in Fig. 6.1. As shown in the figure, the bubble aspect ratio is similar at different pressures in the same sohds holdup 0.384, although the bubble rise velocity is reduced from 20.6 cm/s to 14.0 cm/s with the increasing pressure as shown in Table 2. In the high solids holdup 0.545, the bubble aspect ratio is larger than that in low solids holdup. When rising in pure liquids, the bubble aspect ratio increases with the liquid viscosity. The presence of the solid particles has a similar effect of increase liquid viscosity regarding to the bubble rise characteristics. The simulation results indicated that the pressure effect is insignificant to the bubble aspect ratio. Meanwhile, the 171 increase of solids holdup has an appreciable effect to the bubble aspect ratio. Usually, the particle effect is small at low solids holdup (Ss < 0.4) and is significant at high solids holdup as found from experimental studies of bubble rise characteristics in liquid-solid fluidized systems under various pressures and temperatures (Fan and Tsuchiya, 1989; Tsuchiya et al., 1997; Luo et al., 1997b). The change of physical properties and system parameters can also affect the bubble rising trajectory in addition to affect the rise velocity and shape of the bubble. The bubble rising trajectory, in turn, can also affect the bubble rise velocity. The simulated trajectories a bubble rising in the liquid-solid fluidized bed with solids holdup of 0.384 at pressures of 0.1 MPa and 17.3 MPa, and a solids holdup of 0.545 and pressure of 17.3 MPa are shown in Fig. 6.2. The time step between two bubbles in Fig. 6.2 is 0.05 second. As can be seen from the figure, for the same solids holdup, the bubble trajectory is more tortuous at high pressure than that at low pressure. Noted that the liquid viscosity increases and the surface tension decreases with the increasing pressure. The difference of the bubble trajectory at various pressures in the same solids holdup may be resulted from the competing effects of the liquid viscosity and surface tension. As the higher liquid viscosity favors a more stable, rectilinear trajectory of a rising bubble, the smaller surface tension would lead to a more tortuous trajectory. For bubbles rising at the same pressure but different solids holdups, it is seen that the trajectory of the rising bubble is more stable at high solids holdup than that at low solids holdup as shown in Figs 6.2(b)—6.2(c). This is in consistent to the viscosity effect on the bubble rising trajectory since increasing the solids holdup would give the similar effect as increasing liquid 172 viscosity. However, at a high solids holdup, more interactions between particles and the bubbles are encountered. Therefore, more bubbles with irregular shapes are observed at a high solids holdup as shown in Fig. 6.2(c). 6.3.4 Bubble-particle interactions When the bubble rising in the liquid-solid fluidized bed, the bubble will contact with a number of particles. In this study, the bubble-particle interaction is accounted for by adding a surface tension induced force to the particle motion equation. In turn, this force is also added to the source term of the liquid momentum equation for the liquid elements in the interfacial area to account for the particle effect on the interface. The particle movement is determined based on the resulting total force acting on the particle. From the simulation results, it is seen that most of particles contacting with the bubble do not fall into the bubble. Instead, they follow the liquid flow passing around the bubble surface. Occasionally, there is one or several particles breaking into the bubble. When the particles moves into the bubble, the particles fall quickly to the bubble base because of the low viscosity and density of the gas phase. Figure 6.3 illustrated the particle- bubble interactions in four snapshots of the simulated results. As shown in the figure, most of tlie particles on the bubble surface do not fall into the bubble while only one or two particles is seen inside the rising bubble. 173 6.3.5 Bubble breakage and stability In addition to the bubble rise velocity, the bubble size is another important parameter determining the overall gas holdup in a three-phase fluidized system. Luo et al. (1999) measured bubble sizes under various pressures, and found that the maximum stable bubble size decreases with increasing pressure. They found that the centrifugal force induced by internal circulation of gas inside a bubble can disintegrate the bubble at high pressures. A mechanistic model is developed to account for the maximum stable bubble size, afymax, as given below ^bmax ~ C — ( C = 2.53 for or = 0.21, C = 3.27 for a: =0.3) (6.2) \SP, where a is the aspect ratio of bubble. This model reveals that the gas inertia and gas- liquid surface tension dictate the maximum stable bubble size at high pressures. From Eq. (2), the maximum stable diameter of bubble is predicted as 9.0 mm at 17.3 MPa. The rise trajectory of a bubble with the equivalent diameter of 7.5 mm is shown in Fig. 6.2, which shows the bubble rising without significant breakage. The breakup of the bubble is simulated when the bubble diameter is larger than the maximum stable bubble at the given condition. Figure 6.4 shows a series of bubble shape change for a bubble 10.0 mm in diameter rising from its initial position in a liquid-solid fluidized bed with solids holdup of 0.384 at 17.3 MPa. As shown in the figure, the rise bubble changes its shape drastically and eventually breaks into three parts. The largest part of the breaking bubble has an equivalent diameter about 9.0 mm and rises without further breakage. Similar results are obtained for the bubble rising in pure liquid as shown in Fig. 6.5, which shows 174 the relatively small effects of solids concentration on the maximum bubble size and on the gas holdup as observed by Luo et al. (1999). The results of the simulated bubble breakage agree well with the experimental data and predictions from the mechanistic model. Figure 6.6 shows the velocity field of the gas and liquid phases before the bubble breakup in a coordinate moving with the bubble. The bubble rises in a liquid-solid fluidized bed with solids holdup of 0.384 and pressure of 17.3 MPa. As shown in the figure, a significant internal circulation in the bubble is observed. Luo et al. (1999) assumes that the centrifugal force induced by this internal circulation is the main driving force for the bubble breakup at elevated pressures. The simulation results of this study agree with the predictions from the theoretical model developed based on this assumption. 6.4 Concluding Remarks The bubble and particle dynamics in a three-phase fluidized bed at elevated pressures are numerically studied using a discrete phase simulation. The Eulerian volume-averaged method, the Lagrangian dispersed particle method and the volume-of- fluid (VOF) method are employed for the simulations of liquid, solid particles and gas bubbles, respectively. The simulations of the bubble rise velocity at various solids holdups and elevated pressures agree well with the experimental data. The simulation results indicate that the bubble aspect ratio increases with the increasing of solids holdup. For the same solids holdup, the bubble trajectory is more tortuous at high pressure than 175 that at low pressure. The trajectory of a rising bubble is more stable at high solids holdup than that at low solids holdup. For the bubble-particle interactions, it is found that most of the particles in front of a bubble passing around the bubble surface while only very few particles break into the rising bubble. The results of the simulated bubble breakage and the maximum stable bubble size agree well with the experimental data and predictions from the mechanistic model. 6.5 Nomenclature C Constant D Column diameter db Volume equivalent bubble diameter dbmax Maximum stable bubble diameter g Gravity acceleration p pressure R Volume equivalent bubble radius Ub Bubble rise velocity Ub 2 Two-dimensional bubble rise velocity Ub 3 Three-dimensional bubble rise velocity f/boo Bubble rise velocity in an infinite container 176 Greek letters a Constant Es Solids holdup Ps Gas density a Surface tension 177 Table 6.1 Physical properties of the gas and liquid used in simulations p = QA MPa p = 17.3 MPa Gas density Gas density 0.94 kg/'m^ 154 kg/m^ Liquid density Liquid density 843 kg/m^ 872 kg/m^ Liquid viscosity Liquid viscosity 0.00379 Pa s 0.00417 Pa s Surface tension coefficient Surface tension coefficient 0.0253 N/m 0.0192 N/m 178 Table 6.2 Comparison of the simulation results with the experimental data for the rise velocity of a bubble 7.5 mm in diameter. Simulation & Pressure: Pressure: Pressure: experimental 0.1 MPa, 17.3 MPa, 17.3 MPa, conditions Solids holdup: Solids holdup: Solids holdup: 0.384 0.384 0.545 Simulated velocity 20.6 15.4 11.9 (cm/s) Velocity from 20.8 15.7 12.3 experiments (cm/s) Relative error (%) 1.08 1.85 3.2 179 Solids holdup. Pressure 1 # A 0.384, 0.1 MPa X 0.384, 17.3 MPa 0.8 o 0.545, 17.3 MPa ^ o 2 0.6 ■4o—» a, 0 0 0.2 0.4 0.6 time (sec) Figure 6.1 The simulated results of the bubble aspect ratio changing with time at various solids holdups and pressures. 180 8 7 6 5 4 3 2 1 0 0 12 3 (a) p = 0.1 MPa, Ss = 0.384, dy = 7.5 mm (con’t.) Figure 6.2 Bubble rising trajectory at different pressures and solids holdups. 181 Figure 6.2 (con’t.) 8 7 6 5 4 3 2 1 0 0 1 2 3 (b) p = 17.3 MPa, 8s = 0.384, db = 7.5 mm (con’t.) 182 Figure 6.2 (con’t.) 8 7 G 5 4 3 2 1 0 0 1 2 3 (c) p = 17.3 MPa, Es = 0.545, dy = 7.5 mm 183 • • **. • • • • i (a) t = to (b) t = to+ 0.15 sec (con’t.) Figure 6.3 The simulation results of bubble-particle interactions (p = 17.3 MPa, Ss = 0.384, db = 7.5 mm). 184 Figure 6.3 (con’t.) mmS-.:%-g-M mmm> ♦. f (c) t = to + 0.30 sec (d) t = te + 0.45 sec 185 8 7 6 S 4 3 2 1 0 3 01 2 3 (a)t = to (b) t = to + 0.05 s (c) t = to + 0.10 s (d) t = to+0.15 s (e) t = to + 0-2G s Figure 6.4 The simulated sequence of bubble shape change and the bubble breakage in a liquid-solid medium (p = 17.3 MPa, Ss = 0.384, db = 10.0 mm). 186 4 - 3 3 2 2 3 0 1 3 3 (a)t = to (b) t = to + 0.05 s (c) t = to + 0.10 s (d) t = to + 0.15 s (e) t = to + 0.20 s Figure 6.5 The simulated sequence of bubble shape change and the bubble breakage in a liquid medium (p = 17.3 MPa, Ss = 0., db = 10.0 mm). 187 mm Èi Iff/// Figure 6.6 Velocity vector field of gas and liquid phases before the bubble breakup (p = 17.3 MPa, Ss = 0.384, dy = 10.0 mm). 188 CHAPTER 7 RECOMMENDATIONS FOR FUTURE RESEARCH This dissertation presented, for the first time, a systematic study on the modeling and numerical results of the discrete phase simulation for gas-liquid-solid fiuidization systems. The simulation scheme and the computer code developed are able to predict the local and dynamic flow behavior of gas-liquid-soHd fiuidization including the motion of individual bubbles and particles and interactions among the three-phases. For more general applications of this code, recommendations for further research are given below. 7.1 Parallel Computing In the current discrete phase simulation, tracking the movement of individual particles is a time consuming procedure, especially when a large number of particles need to be simulated. When applied to large/practical systems, parallel computing is recommended for simulating a large number of particles and bubbles in three-phase fiuidization systems. There are two major developments of parallel computing system: 1. massively parallel processors (MPPs), such as Cray T3D and IBM SP2, and 2. the widespread use of distributed computing. MPPs combine hundreds to thousands CPUs in 189 a single cabinet shared same large memory. They offer enormous computational power and are used to solve computational grand challenge problems. Distributed computing is a process whereby a group of homogeneous or heterogeneous computers connected by a network is used collectively to solve a single large problem. Parallel Virtual Machine (PVM) software system developed at University of Tennessee and Oak Ridge National Laboratory is one of the well known software packages that have been developed for distributed computing. For parallel computing in the discrete phase simulation, domain decomposition, exchange of zonal boundary conditions, and data communication are some major areas for further research. 7.2 Turbulence Model As a simulation program for dynamic flow, the current code is able to simulate the temporal fluctuation in three-phase fiuidization systems since very small time step, ~ 10"^ s is often used in simulations. To account for the spatial fluctuations which will be present in high Reynolds number flows, a liquid phase turbulence model can be included. K -s model, Reynolds stress model, and large eddy simulation (LES) are some o f methods that are commonly used in predicting the turbulence in liquid and multiphase flows. However, when applied to three-phase fiuidization, very limited information is available for the turbulence interactions among individual phases. Therefore, experimental and analytical studies are also recommended for providing constitutive relationships and verifications of the simulation models. 190 7.3 Heat. Mass transfer, and Reaction Kinetics In practice, most of the gas-liquid-solid fiuidization systems are involved in heat, mass transfer and reactions. Therefore, the inclusion of simulations of heat, mass transfer, and reaction kinetics is recommended in future research. 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