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UBC Theses and Dissertations

Electro-hydrodynamics of gas-solid fluidized beds Jalalinejad, Farzaneh 2013

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 ELECTRO-HYDRODYNAMICS OF GAS-SOLID FLUIDIZED BEDS by Farzaneh Jalalinejad B.Sc., Sharif University of Technology, 2004 M.A.Sc., University of Tehran, 2007   A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in   THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES  (Chemical and Biological Engineering)  The University of British Columbia (Vancouver) September 2013  ? Farzaneh Jalalinejad, 2013  - ii - Abstract The generation of electrical charges, reported in gas-solid fluidized beds for over sixty years, can cause serious problems like wall sheeting in polyolefin reactors, leading to costly shutdown, electrical shock hazards and even explosions. Understanding the associated phenomena plays an important role to avoid these problems. In this study an attempt has been made to broaden the understanding of electrostatics in fluidized beds by adopting computational fluid dynamics (CFD), using the Two-Fluid-Model in MFIX (an open-source code originated by the U.S. Department of Energy). The Maxwell equations were incorporated in the MFIX code. The resulting model is then used to investigate how electrostatics modify bubble shape, size, velocity and interaction for three cases: (a) single bubbles, (b) bubble pairs in vertical and horizontal alignment, and (c) a freely-bubbling bed. In each of these cases, a two-dimensional column, partially filled with mono-sized particles, is simulated for both uncharged and charged particles.  In case (a), it is predicted that single bubbles elongate and rise more quickly in charged particles than in uncharged ones. For case (b), electrostatics cause asymmetry of coalescence for a pair of vertically-aligned bubbles, while leading to the migration of a side bubble towards the axis of the column and changing the leading-trailing role for a pair of horizontally-aligned bubbles. Finally in case (c), the simulation predicts that electrostatics decrease bubble size and frequency in the free bubbling regime, accompanied by a change in the spatial distribution of bubbles, causing them to rise more towards the axis of the column.  An attempt was also made to test experimentally the single bubble simulations. To reach this goal, a two-dimensional fluidization column was built with a central jet to inject single bubbles. The setup is equipped with a novel Faraday-cup device to measure the charge density accurately. The experimental results indicates a small decrease in bubble size and an increase in bubble height-to-width ratio with increasing charge density, accompanied by an increase in particles raining from the bubble roof. The assumption of uniform charge density on the particles is identified as a significant reason for differences between observed and predicted behaviour.  - iii - Preface The main chapters of this thesis have already been, or will be, submitted to journals for publication. A paper incorporating very similar materials as Chapters 1 to 3 has been published as Jalalinejad, F., Bi, X. T., Grace, J. R., Int. J. Multiphase Flow 44, 15-28, 2012.   - iv - Table of Contents Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iii Table of Contents ......................................................................................................................... iv List of Tables ............................................................................................................................... vii List of Figures ............................................................................................................................. viii Nomenclature .............................................................................................................................. xii Acknowledgements ..................................................................................................................... xv Dedication ................................................................................................................................... xvi Chapter 1. Introduction ........................................................................................................... 1 1.1. Electrostatics in Gas-Solids Fluidized Beds ................................................................... 1 1.2. Thesis Objectives, Approach and Layout ....................................................................... 5 Chapter 2. Model Development .............................................................................................. 8 2.1. Introduction ..................................................................................................................... 8 2.2. Two Fluid Model (TFM) .............................................................................................. 10 2.3. Boundary Conditions .................................................................................................... 14 2.4. Electric Force Density................................................................................................... 15 2.5. MFIX Numerical Method ............................................................................................. 17 2.6. Numerical Discretization .............................................................................................. 18 2.7. Verification of Electric Potential Solver ....................................................................... 20 Chapter 3. Effect of Electrostatics on Single Bubbles ........................................................ 23 3.1. Introduction ................................................................................................................... 23 3.2. Simulations ................................................................................................................... 23 3.3. Comparison of Numerical and Experimental Results for Uncharged Particles ............ 26 3.3.1. Bubble Shape ........................................................................................................ 26 3.3.2. Solid Pressure........................................................................................................ 27 3.4. Effect of Electrostatic Charges on Isolated Bubbles ..................................................... 32 3.4.1. Bubble Shape for Charged Particles ..................................................................... 32 3.4.2. Bubble Rise Velocity for Charged Particles ......................................................... 35 3.5. Main Cause of Bubble Shape Change .......................................................................... 37  - v - 3.6. Conclusions ................................................................................................................... 43 Chapter 4. Influence of Electrostatics on Bubble Pair Interaction ................................... 44 4.1. Introduction ................................................................................................................... 44 4.2. Simulations ................................................................................................................... 45 4.3. Interaction of Bubbles in Vertical Alignment ............................................................... 46 4.4. Interaction of Bubbles in Horizontal Alignment .......................................................... 50 4.5. Conclusions ................................................................................................................... 51 Chapter 5. Effect of Electrostatics on Freely-Bubbling Beds ............................................ 53 5.1. Introduction ................................................................................................................... 53 5.2. Simulations ................................................................................................................... 54 5.3. Frictional Model............................................................................................................ 56 5.3.1. Bubble Size ........................................................................................................... 57 5.3.2. Time-Average Vertical Solid Velocity ................................................................. 59 5.3.3. Comparison of Frictional Models and Final Choice ............................................. 61 5.4. Effect of Electrostatics on Freely-Bubbling Beds ........................................................ 62 5.4.1. Bubble Size ........................................................................................................... 62 5.4.2. Bubble Spatial Distribution................................................................................... 63 5.4.3. Time-Average Solid Velocity ............................................................................... 66 5.5. Connection between Results ......................................................................................... 68 5.6. Conclusions ................................................................................................................... 70 Chapter 6. Experimental Tests for Single Bubbles ............................................................. 71 6.1. Introduction ................................................................................................................... 71 6.2. Experimental Setup ....................................................................................................... 71 6.3. Gas and Particles ........................................................................................................... 74 6.4. Experimental Method.................................................................................................... 75 6.5. Experimental Results .................................................................................................... 78 6.6. Comparison of Experimental Results with Simulation Predictions .............................. 82 6.7. Conclusions ................................................................................................................... 85 Chapter 7. Summary, Contribution and Future work ....................................................... 87 7.1. Summary and Contribution ........................................................................................... 87 7.2. Recommendation for Future Work ............................................................................... 89  - vi - References .................................................................................................................................... 90 Appendix A. Challenges Faced in this Project ......................................................................... 97 Appendix B. Initialization Effect ............................................................................................. 100 Appendix C. Influence of Specularity Coefficient on Solid Phase Circulation ................... 101 Appendix D. Mesh Study .......................................................................................................... 103 Appendix E. Difference between Voidage of 0.8 and 0.85 as Bubble Boundary Criterion 105 Appendix F. Electric Field around a Circle............................................................................ 106 Appendix G. Error Bars for Bubble Diameter ...................................................................... 107 Appendix H. Design Sketches of Faraday-Cup Device.......................................................... 109  - vii - List of Tables Table 2-1: Stress terms.................................................................................................................. 11  Table 2-2: Granular temperature relations .................................................................................... 14  Table 3-1: Simulation parameters for different case studies ........................................................ 25  Table 3-2: Bubble rise velocity from available correlations for Db= 0.16 m ............................... 37  Table 4-1: Simulation parameters ................................................................................................. 46  Table 4-2: Comparison of sum of areas of original bubbles with area of resultant bubble for coalescence of bubbles in vertical alignment................................................................................ 50  Table 5-1: Simulation parameters (two-dimensional column) ..................................................... 56  Table 6-1: Comparison of average bubble diameter and aspect ratio for beds previously fluidized at different superficial gas velocities (Data are reported in the form of the average value ? 90% confidence interval.) ..................................................................................................................... 79  Table 6-2: The charge density distribution in beds of low- and high-charge system (dp= 518 ?m, ?s= 2500 kg/m3). Data points are reported as average value ? 90% confidence interval. ............ 83   - viii - List of Figures Figure 2-1: (a) Schematic of key variables around control volume cell in MFIX code; (b) Schematic of scalar, x-momentum and y-momentum control volumes based on MFIX discretization. ................................................................................................................................ 18  Figure 2-2: Contour plot of electric potential (V) for packed bed with L=1 m, bed height= 0.5 m, ?g= 0.5, qm= ?1.0 ?C/kg and ?s= 2500 kg/m3. .............................................................................. 22  Figure 2-3: Comparison of electric potential (V) where L=1 m, bed height= 0.5 m, ?g= 0.5, qm= ?1.0 ?C/kg and ?s= 2500 kg/m3: (a) along x- direction for Y= 0.5 m, (b) along y- direction for X= 0.5 m. ...................................................................................................................................... 22  Figure 3-1: Comparison of predicted and experimental bubble shapes at different heights. (Solid lines are simulation results obtained at different times, whereas dashed lines are experimental results from Gidaspow et al., 1983). ............................................................................................. 27  Figure 3-2: Comparison of bubble diameters predicted by simulation and Rahman and Campbell (2002) experimental results for 500 ?m glass beads fluidized with background velocity of 0.23 m/s based on S-S frictional model. The bubble was injected with Vjet = 18 m/s for a duration of 0.2 s. .............................................................................................................................................. 28  Figure 3-3: Solid pressure contours during bubble rise for simulation of the Rahman and Campbell (2002) experiment (pressures in Pa) for 500 ?m glass beads fluidized with 0.23 m/s background velocity and S-S frictional model. Bubble is injected with Vjet = 18 m/s for a duration of 0.2 s. Points in Fig. 3.4 (a): (a) A (b) B (c) C. .......................................................................... 30  Figure 3-4: Comparison of solid pressure between simulation with S-S frictional model and the Rahman and Campbell (2002) experimental results for 500 ?m glass beads with Vsup= 0.23 m/s and Vjet= 18 m/s: (a) probe 1; (b) probe 2; (c) probe 3; (d) probe 4. For probe locations, see Figure 3.3. ..................................................................................................................................... 31  Figure 3-5: Bubble shapes for 500 ?m glass beads at different times, for S-R-O frictional model with Vsup= 0.2 m/s and Vjet= 15 m/s.: (a) Uncharged particles; (b) Charged particles (qm= ?0.36 ?C/kg). .......................................................................................................................................... 33  Figure 3-6: Bubble shapes for 500 ?m glass beads at different times predicted based on S-S frictional model with Vsup= 0.23 m/s, Vjet= 14 m/s.: (a) Uncharged particles; (b) Charged particles (qm= ?0.36 ?C/kg). ....................................................................................................................... 34  Figure 3-7: Effect of charge density on bubble shape for 500 ?m glass beads based on S-S frictional model. Vsup= 0.23 m/s and Vjet= 14 m/s. ........................................................................ 35  Figure 3-8: Comparison of effect of boundary conditions on predicted bubble rise velocity through bed of 500 ?m glass beads, with S-S frictional model. Vsup= 0.23 m/s and Vjet= 14 m/s. 36  - ix - Figure 3-9: Electric field vectors shown with electric field magnitude (V/m) contours for charged bed (qm= ?0.36 ?C/kg of 500 ?m glass beads, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model). .......................................................................................................................................... 38  Figure 3-10: Predicted change in bubble shape, when electric field is turned on, after bubble enters uniform electric field region of 500 ?m charged glass beads (qm= ?0.36 ?C/kg, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model). Vectors show electric field force where voidage = 0.5.................................................................................................................................................. 39  Figure 3-11: Force components along contour where ?g= 0.5 around bubble 1 versus angle from nose in bed of 500 ?m charged glass beads (qm= ?0.36 ?C/kg, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model): (a) x-direction; (b) y-direction. ................................................................. 41  Figure 3-12: Force components along contour where ?g= 0.5 around bubble 4 versus angle from nose in bed of 500 ?m charged glass beads (qm= ?0.36 ?C/kg, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model): (a) x-direction; (b) y-direction. ................................................................. 42  Figure 4-1: Simulated coalescence of two bubbles in vertical alignment in: (a) uncharged; (b) charged particles (dp= 300 ?m, ?s= 2450 kg/m3, Vj,v= 3.4 m/s, Vsup= 0.084 m/s). ........................ 47  Figure 4-2: Snapshot of coalescence of two bubbles in vertical alignment (Clift and Grace, 1970) in a two-dimensional column. Grid lines shown have a 25.4 mm spacing. From right to left: t= 0, 0.08, 0.095, 0.11, 0.125 s. ............................................................................................................. 48  Figure 4-3: Nose coordinates of bubbles in vertical alignment versus time (dp= 300 ?m, ?s= 2450 kg/m3, Vj,v= 3.4 m/s, Vsup= 0.084 m/s). ......................................................................................... 49  Figure 4-4: Coalescence of two bubbles in horizontal alignment in: (a) uncharged; (b) charged particles ......................................................................................................................................... 51  Figure 5-1: Comparison of bubble diameter predicted by S-R-O and S-S frictional models with experiemental data (Laverman et al., 2008) (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S) and 0.45 m/s (2.5Umf, S-R-O and S-S). ........................ 58  Figure 5-2: Sensitivity of S-S model to mins? (minimum threshold of solid volume fraction to account for the frictional stress) (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.29 m/s). ........................ 58  Figure 5-3: Comparison of horizontal profiles of vertical solid velocity predicted based on S-R-O and S-S frictional models for particles of diameter 485 ?m and density 2500 kg/m3 at y= 100 mm for: (a) Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S); (b) Vsup= 0.45 m/s (2.5Umf, S-R-O and S-S). .................................................................................................................................... 60  Figure 5-4: Comparison of bubble diameter in uncharged and charged particles with experimental data of Laverman et al. (2008) at both velocities (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S) and 0.45 m/s (2.5Umf, S-R-O and S-S). 63   - x - Figure 5-5: Cumulative bubble number fraction percentage in zone 2 (y= 200-300 mm) predicted for uncharged and charged particles of dp= 485 ?m, ?s= 2500 kg/m3 at: (a) Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S); (b) 0.45 m/s (2.5Umf, S-R-O and S-S). ........................ 65  Figure 5-6: Comparison of vertical solid velocity in uncharged and charged particles with experimental data of Laverman et al. (2008) for particles of diameter 485 ?m and density 2500 kg/m3 at y= 100 mm: (a) Vsup= 0.27 m/s (1.5Umf); (b) Vsup= 0.45 m/s (2.5Umf). .......................... 67  Figure 5-7: Comparison of bubble height/bubble width versus x in bed with uncharged and charged particles at height 150 mm (Vsup= 0.27 m/s (1.5Umf), dp= 485 ?m, ?s= 2500 kg/m3). .... 69  Figure 6-1: Schematic of experimental setup ............................................................................... 72  Figure 6-2: Windbox with inner reservoir (box) for bubble injection .......................................... 73  Figure 6-3: Faraday cup device mounted at back of column: (a) outer cup is closed, (b) outer cup is open. .......................................................................................................................................... 74  Figure 6-4: Particle size distribution for glass beads particles ..................................................... 75  Figure 6-5: Bubble rise in bed previously fluidized at 1.3Umf (dp= 518 ?m, ?s= 2500 kg/m3). Successive frames were taken at 39 frames/s. Shadow corresponds to Faraday cup sampling device. ........................................................................................................................................... 80  Figure 6-6: Two patterns of bubble rise in bed previously fluidized at 1.8Umf (dp= 518 ?m, ?s= 2500 kg/m3). Successive frames taken at 39 frames/s. (a) pattern 1, (b) pattern 2. Shadow corresponds to Faraday cup sampling device. .............................................................................. 81  Figure 6-7: Effect of charge density distribution on single bubbles for: (a) uncharged particles, (b) uniformly charged particles with charge density of ?1 ?C/kg, (c), (d) three horizontal layers of particles with charge densities defined in Table 6-3. ............................................................... 85  Figure C-1: Time-averaged solid velocity for (a) specularity coefficient = 0, (b) specularity coefficient = 0.05 (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.45 m/s (2.5Umf, S-R-O)). .................. 101  Figure C-2: Experimental results of Laverman et al. (2008) for superficial gas velocity of 2.5Umf. (dp= 485 ?m, ?s= 2500 kg/m3). ................................................................................................... 102  Figure D-1: Mesh refinement for single bubble case study in section 3.4.1 (qm= ?0.36 ?C/kg, dp= 500 ?m glass beads, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model). ............................ 103  Figure D-2: Mesh study for freely-bubbling regime in section 5.3.1 (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.27 m/s). ............................................................................................................... 104   - xi - Figure E-1: Influence of bubble boundary voidage on single bubble for typical voidages of 0.8 and 0.85 (qm= ?0.36 ?C/kg, dp= 500 ?m glass beads, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model).......................................................................................................................... 105  Figure F-1: Electric field around circle in a bed of 500 ?m particles with a charge density of ? 0.36 ?C/kg .................................................................................................................................. 106  Figure G-1: Mean and standard deviation of predicted bubble diameters (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S). Experiemental data reported here are based on Laverman et al. (2008) studies. .............................................................................. 107  Figure G-2: Mean and standard deviation of predicted bubble diameters (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.45 m/s (2.5Umf, S-R-O and S-S). Experiemental data reported here are based on Laverman et al.  (2008) studies. .................................................................................................. 108  Figure H-1: Front and side views of Faraday-cup device ........................................................... 109  Figure H-2: Separate parts of Faraday-cup device ..................................................................... 109                              - xii - Nomenclature c  instantaneous velocity, (m/s) B  magnetic field, (N/m.A) CD  drag coefficient, (?) dp  particle diameter, (m) D  electric displacement field, (C/m2) E  electric field, (N/C) H  magnetising field, (C/m.s) ep  particle-particle restitution coefficient, (?) ew  particle-wall restitution coefficient, (?) fe  electrical force density, (N/m3) Fe  electrical force, (N) g0  radial distribution, (?) Gb  total gas flow rate (2D), (m2/s) Gb_th gas flow rate in bubble phase (2D), (m2/s) Ge  gas flow rate in emulsion phase (2D), (m2/s) I  unit vector, (?) I2D  second invariant of deviator of strain rate, (s?2) J  free current density, (C/m2.s) Jcoll  loss term due to particle-particle collision, (kg/m.s2) Jvis  loss term due to interaction of gas and particles, (kg/m.s2) n  unit normal vector, (?) P  pressure, (N/m2) Pc  critical pressure, (N/m2) qm  charge density based on mass of solids, (C/kg) qv  charge density based on volume of solids, (C/m3) Q  point charge density, (C) Re  Reynolds number, (?) S  strain rate tensor, (s-1) t  time, (s) U  velocity vector U=(U,V)T, (m/s)  - xiii - V  vertical component of velocity vector, (m/s) W  energy of system of charged particles, (N.m) X (x) Cartesian coordinates in horizontal direction, (m)  Y(y)  Cartesian coordinates in vertical direction, (m)   Greek Letters ?  constant, (?) ?i  variable property, (?) ?gs  gas-solid momentum exchange coefficient, (N.s/m4) ?  volume fraction, (?) ?   electrical permittivity of medium, (F/m) 0?   electrical permittivity of free space, (F/m) ?smin  minimum threshold to account frictional stress in S-S frictional model, (?) ?smax max solid packing volume fraction, (?) ?e  electric susceptibility, (?) ?  granular conductivity, (W/m.k) ?  electric potential, (V) ?   angle of internal friction, degree ?  angle from bubble nose, degree ?  specularity coefficient, (?) ?  density, (kg/m3) ?  viscosity, (kg/m.s) ?b  bulk viscosity, (kg/m.s) ?  constant, (?) ?  stress tensor, (N/m2) ?   granular temperature, (m2/s)  Subscripts ave  average g  gas mf  minimum fluidization  - xiv - s  solids sup  superficial sl  slip  Superscripts k  kinetic part of stress f  frictional part of stress max  maximum   - xv - Acknowledgements I would like to express my sincere gratitude to my Supervisors: Dr. Xiaotao Bi and Dr. John Grace for their advice, support, encouragement and patience, which made my PhD experience not only an opportunity to build problem solving skills, but more importantly a unique experience to grow. I consider myself fortunate and honoured to have such role models in my life.  I also want to thank my committee members Dr. Carl Ollivier-Gooch, from whom I was fortunate to learn the basics of CFD in a beautiful style, and Dr. James Feng for their valuable suggestions and support through the course of my PhD project.  I like to thank Dr. Bud Homsy for his patience in reviewing part of my numerical method and his curious questions that shed light on my path. I would like to express appreciation to Doug Yuen and his team members in the workshop for their valuable advice and patience in constructing the experimental setup.  I?m also grateful for my friends and group members for their valuable suggestions and support in this path, in particular: Dr. Lifeng Zhang, Dr. Yolung Ding, Dr. Tingwen Li, Turki Alsmari and Chuan He.  Finally, I would like to express my everlasting gratitude to my parents for their love, support, and tireless deduction. My father, Masoud, is not among us anymore, but his memories and persistence are and will be alive in my heart forever.   - xvi - Dedication  To my parents for their endless love and dedication     - 1 - Chapter 1. Introduction 1.1. Electrostatics in Gas-Solids Fluidized Beds Despite the fact that the physics of electrostatic charging is well understood at a fundamental level, prediction and control are difficult due to the role of key parameters such as humidity, purity and surface properties. When two bodies contact each other, electrical double layers of opposite sign form close to each surface due to the transfer of electrons from the material with lower work function to equalize the Fermi energy levels at the interface. The distance between these layers is only a few molecular diameters. If these bodies are suddenly separated, the original electronic equilibrium cannot be re-established, with the result that one surface gains electrons at the expense of the other (Guardiola et al., 1996, Yang, 1998). Since fluidization involves continuous particle-particle and particle-wall collisions, as well as rubbing of particles together, generation of electrostatic charges is inevitable for insulating materials. On the other hand, electrostatic charges cannot escape from the vessel easily in these systems, although some charges may leave the system through grounded walls. Overall, significant charge accumulates inside the vessel, and if the magnitude of charge inside the bed reaches a critical value, particles tend to adhere to the reactor wall and form sheets whose thickness can reach 5-15 mm. In severe cases these can cause plugging and loss of fluidization, even leading to costly reactor shutdowns (Hendrickson, 2006).  Therefore understanding electrostatic charging and its influence on hydrodynamics of fluidized beds is important to reduce and control the charge build-up to avoid sparks, explosions, wall sheeting and plugging of discharge lines in processes such as drying and polyolefin production. Since the 1940s, there have been a number of experimental and numerical studies to understand this phenomenon and reduce its effects, e.g. by grounding the walls, increasing the relative humidity (RH) or adding antistatic powders. Some of those works are reviewed here; those which are more relevant to the problem definition of this thesis.  Boland and Geldart (1971) conducted one of the first studies to understand the mechanism of charge generation in fluidized beds. They investigated the voltage signal during passage of a  - 2 - single bubble in 200-300 ?m glass beads, applying an induction probe located centrally on one face of a two-dimensional column. This voltage signal was found to have a positive peak (associated with the bubble nose passing from the probe location) followed by a negative peak (associated with passage of the wake). Since the voltage did not return to its initial state after bubble eruption at the surface, the authors suggested that the measured charge was on the walls of the equipment, rather than on the particles.   Years later, Chen et al. (2006) used a multiple induction probe, to reconstruct the magnitude of charges on particles around a single bubble for 560 ?m glass beads, both close to minimum fluidization and in free bubbling. The induced charge signal for the first case only showed a negative peak, while two patterns were seen for the second case, one with a negative peak and the other with a negative peak followed by a positive peak. Reconstruction of the signal for those two patterns revealed that the particles were negatively charged, with the charge density gradually increasing towards the bubble interface; while, it reached zero inside the bubble, a region in which there were no particles. On the other hand, a high charge density in the bubble wake region was detected for the pattern with two peaks. Bubble splitting and coalescence close to the probe were reported to be the possible cause of the two-peak pattern. Thus charge distribution was detected around bubbles in this system. These results also imply that the two-peak pattern reported by Boland and Geldart (1971) may have been due to changes in bubble shape close to the probe.   Non-uniformity of charging was also reported for a dense fluidized bed by Napier (1994), for 425 to 500 ?m glass beads, the range of primary interest in this study. In that study charge generation was investigated for a dense fluidized bed, passage of particles from a valve, pneumatic conveying and unloading of different particulate materials. It was shown that the magnitude of charge density varied during all of these processes. A field mill and an electrostatic probe were utilized in their work to measure charge density.  The work reviewed above involved studies with narrow-sized particles. However, most particles have wide size distributions in industrial reactors. Thus studies on poly-dispersed systems are of great interest. Mehrani et al. (2005) studied the difference in charge generation of mono-sized  - 3 - particles and a binary mixture of glass beads by developing an on-line Faraday cup fluidized bed, fabricated from copper. It was found that the magnitude of the charge density was negligible for mono-sized particles compared to a binary system, and the existence of a small amount of fines led to significant negative charges on large particles, and positive charges on fines. Bipolar charging was detected in their system, related to fines entrainment, due to the fact that the Faraday cup measured only the net charge on particles in the bed.  Giffin and Mehrani (2010) studied the influence of fluidization time on charge build-up for polyethylene particles of wide size distribution in the bubbling and slugging flow regimes. When they measured the charge density of particles in a dense bed, particles adhered to the wall and fines collected at the top of the column. The charge densities in different sections were found to reach a constant range (i.e., called saturation state), after 1 hour of fluidization. The reported charge densities differed by several orders of magnitude for three sections, and a change in charge polarity from negative in the dense part of the bed to positive in the freeboard was also reported, indicating that larger polyethylene particles tend to become negatively charged, and fines to become positively charged. This phenomenon was also detected by collision probes by Fang et al. (2008) and Zhou et al. (2013) who measured the electric potential distribution in fluidized beds of polyethylene particles. Negative and positive potentials were detected in the lower and upper section of the column, respectively, implying negative and positive charge density of particles in the lower and upper parts of the bed, respectively.   From a theoretical perspective, extracting an electron is harder for smaller particles. This indicates that smaller particles should be negatively charged and larger ones positively charged, but this is not always the case, e.g., Gallo and Lama (1976) for polyethylene and glass beads. This discrepancy is usually explained by the ionic nature of the charge carriers. Therefore, bipolar charging occurs naturally in poly-dispersed systems of dielectric particles. Different polarities can cause particle agglomeration, resulting in particle clusters which may be permanent or temporary due to interaction with other particles or gas (Cocco et al., 2010, McMillan et al., 2013). On the other hand, for a mono-sized uniformly charged system, either positive or negative, repulsion between particles can prevent the formation of clusters and change the collision rate between particles (Scheffler and Wolf, 2002).   - 4 - Clusters can also form as a result of collisional cooling, hydrodynamics, cohesive bridging or van der Waals forces. Different causes usually combine in initiating and stabilizing the clusters (Cocco et al., 2010, Chew et al., 2012, McMillan et al., 2013). As dielectric particles become smaller and have a wider particle size distribution, the role played by electrostatics increases in enhancing particle agglomeration. On the other hand, clusters can result in deviation from the predictions of fundamental models, due to the lack of considering cohesive forces in those models, which leads to the over-prediction of drag in some cases, as observed in Li et al. (2008) in simulating Geldart Group A particles.   Cluster formation in risers, even without considering cohesive forces, can be captured numerically by continuum models, in particular the Two Fluid Model, in which both gas and solids are considered as interpenetrating continua. However, these clusters can only be fully detected using a fine mesh, which is computationally expensive, which gives rise to the need to use sub-grid models to simulate large scale industrial units (Agrawal et al., 2001, Igci et al., 2008).   While there have been a number of experimental studies related to electrostatic fields in gas-fluidized beds and related systems, few numerical investigations have included the effects of electrostatic charges. An exception is Al-Adel et al. (2002 a,b) where electrostatic charges were studied in a vertical riser for a fully-developed steady gas-solid flow. The model predicted that the electrical body force introduced in the radial solid momentum balance could force the particles towards the wall, influencing particle segregation in the riser.  In another numerical study, Rokkam et al. (2010) investigated the effect of electrostatics on overall hydrodynamics of gas-solid fluidized beds for particles with a non-uniform size distribution by applying Eulerian multiphase flow. Different particle sizes were assumed to carry different constant charge densities, producing an electric field that exerts a force on the solid particles. The magnitude of this force was calculated based on the Lorentz force equation as the product of charge and electric field. The model was used to simulate a pilot plant of polymer particles. It was predicted that the negatively charged medium and large polymer particles mostly resided in the fluidized zone (lower part of the column), while small positively charged particles  - 5 - predominated in the expansion region (upper part of the column), in agreement with the experimental results reported above.   To summarize, experimental studies have shown that there is a distribution of charges inside gas-fluidized beds, even for mono-sized particles. For binary or poly-dispersed systems, both positive and negative charges are involved, and the results indicate that there are regions with positive and negative charges in different regions of the column. In addition, numerical studies have predicted that electrostatic charges influence the spatial distribution of particles in risers, dense beds and freeboards.  While previous work has provided an indication of the role of electrostatics in gas-fluidized beds, how these charges influence bubbles has not been investigated yet. Bubbles play a very important role in determining the overall properties of many gas-fluidized beds. Therefore this study addresses the question of, ?How do electrostatics modify bubbles?? Answering this question can assist in understanding the nature and extent of the effect of these forces.  Finally it is important to mention that available Computational Fluid Dynamics (CFD) models for prediction of fluidization behaviour are still under development, hampered by the complex natures of gas-solids interactions. Inter-particle forces like electrostatic, van der Waals and other cohesive forces can play important roles in determining the behaviour of the system. However, quantifying these forces remains a major challenge. In this study, an attempt has been made to shed light on quantifying the impact of electrostatic forces by implicitly studying their influence on bubbles, as well as providing visual observations to directly compare, and thus test, the theory with experiments for the first time. Overall, this work is a step toward improving two-phase CFD models and the understanding of the role of electrostatics for gas-fluidized beds.  1.2. Thesis Objectives, Approach and Layout The objective of this study is to understand how electrostatic charges modify bubble shape, size, velocity and interaction. To answer this question, the Two Fluid Eulerian-Eulerian Model (TFM) is used, considering both gas and solids as continuous phases. It is assumed that particles are  - 6 - mono-sized and uniformly charged and therefore the complex nature of polydispersed systems is greatly simplified. It is also assumed that the charge density inside the system reaches a constant value (i.e., a saturation state) after one hour of fluidization, as explained above. Furthermore, the charge on the particles produces a continuous electric field in the system which exerts a force on the particles, with this force calculated as proposed by Melchers (1981).  In other words, the mass and momentum transport equations are solved in this thesis for gas and solids, with the electric field calculated from Gauss? law, considering all walls as grounded. MFIX (an open-source Multiphase Flow with Interphase eXchange code originated by the U.S. Department of Energy) is adopted to solve the transport equations, and the electrical parts are implemented to that code in this work. The hydrodynamic part of the model is then compared with experimental results of Gidaspow et al. (1983) and Rahman and Campbell (2002) for uncharged particles. The electrical part of the model is verified by comparing with the theoretical solution for a two-dimensional column partially filled with charged particles at time zero. The model is then used to investigate the effect of electrostatics on bubble properties by simulating three different cases: (a) single bubbles, (b) bubble pairs in vertical and horizontal alignments, and (c) freely bubbling beds. In all of these cases, a two-dimensional bed is partially filled with mono-sized particles, and the results are compared for cases with both uncharged and charged particles.   Although experimental validation is difficult (Grace and Taghipour, 2004), an attempt is made here to test the single bubble simulation result. To reach this goal, an experimental two-dimensional fluidization column was built, equipped with a central jet to inject single bubbles. The setup also includes a novel Faraday cup device mounted on the back face of the column, to measure the charge density of particles accurately, while minimizing particle handling (For more information about challenges faced in this work see Appendix A).  This thesis is divided into seven chapters. The story begins by explaining the basic idea of modeling and code verification in Chapter 2. Key questions are then answered in Chapters 3 through 5 as follows:   - 7 - Chapter 3: ?How do electrostatics influence a single bubble?? Chapter 4: ?How do electrostatics affect interaction of pairs of bubbles in vertical and horizontal alignment??  Chapter 5: ?How do electrostatics change the properties and behaviour of bubbles in a freely-bubbling bed??  In Chapter 6, the simulation predictions are compared with experimental results from the two-dimensional column for single bubbles. Finally Chapter 7 summarizes the key conclusions drawn from this work, together with recommendations for future works.    - 8 - Chapter 2. Model Development 2.1. Introduction In the last few decades, considerable progress has been made to predict the hydrodynamics of gas-solid fluidized beds by using CFD models to enhance the design and scale up of industrial reactors. This goal has not been fully achieved yet, due to the lack of understanding different interactions between gas, particles and wall, as well as complex cohesive forces blamed for agglomerating particles such as electrostatics, van der Waals and capillary forces. Another obstacle in achieving that goal is the big difference between flow structures scale from ?m (particle scale) to meters (industrial reactor scale).  These challenges have led to the generation of different models, from small to large scale such as lattice Boltzmann models, discrete particle models, the two (multi)-fluid models and discrete bubble models. In recent years, combinations of these models have been proposed, with the basic idea of using smaller scale models to develop closures for larger-scale ones. (See van der Hoef et al. (2006, 2008) for more details on these models).   In this thesis, a Two-Fluid model (TFM) is chosen as the starting point to incorporate electrostatic forces. This model, which considers both gas and solids as interpenetrating continua, is widely used in capturing the hydrodynamics of gas-solid flows and is still one of the most feasible approaches for simulating industrial systems. In this model, the equations employed are a generalization of the Navier-Stokes equations for interacting continua and locally-averaged quantities such as the volume fractions, velocities, species concentrations, and temperature for each phase appear as dependent variables. These variables are averaged over a region that is large compared with the particle dimensions and spacing, but much smaller than the flow domain, and the volume fractions are introduced to track the fraction of the averaging volume occupied by the phases (Syamlal et al., 1993). Since solids are considered as a continuum phase in this model, closures are required to describe the solid stress tensor and the fluid-particle momentum transfer (i.e. drag). This model has been further developed to consider multiple solid phases in recent years and then renamed to as the Multi-Fluid model, giving rise to  - 9 - the need to characterize particle-particle momentum transfer closure, as well as the closures mentioned above.  In the last decade, researchers have tried to improve Two (Multi)-fluid model in different aspects such as developing closures for drag (van der Hoef et al., 2005; Benyaha et al., 2006) and frictional stress (Sirivastava and Sundaresun, 2003), and developing sub-grid models to enhance the large scale simulation results. The sub-grid models are obtained by performing finer-grid simulation and then statistical averaging (Agrawal et al., 2001, Holloway and Sundaresan, 2012). Progress has also been made in incorporating particle size distributions (PSD) (Fan, 2006; Fan et al., 2007) by combining quadrature method of moments (QMOM) with chemical reaction engineering (CRE) model to study the effect of catalyst PSD on the overheating and final PSD of polymer particles in fluidized bed polymerization. Others have tried to incorporate van der Waals forces (van Wachem and Sasic, 2008) and electrostatic forces (Rokkam et al., 2010).  One of the important challenges in modeling electrostatics phenomena is to predict the electrical forces which arise due to bulk charges and polarization. Only for a very dilute medium subject to an imposed field can these forces be calculated from first principles. For dense gases and liquids, recourse to thermodynamics is necessary (Castellanos, 1998). In 1981, Melchers derived the electrical force density from the concept of virtual work for systems in which there was a combined effect of free charge and polarization for both compressible and incompressible fluids.   In previous incorporation of electrostatic force to fluidized beds, explained in more detail in Section 1.1, Rokkam et al. (2010) calculated this force based on the Lorentz force equation as the product of charge and electric field. As a result, the polarization effect in dense particle systems was neglected. In this work, TFM is developed further by incorporating Melcher?s electrostatic force density, and therefore the polarization effect is included. Other differences between this work and this earlier study lie in the number of solid phases and the approach taken. While they focused on polydisperse polymer particles and tried to understand the influence of electrostatic forces on overall behavior of a pilot scale bubbling bed qualitatively, this thesis starts with mono-sized glass beads particles and focuses on the quantitative influence of electrical forces on bubbles shape, size, velocity and interaction in a smaller scale system.  - 10 - 2.2. Two Fluid Model (TFM) The Two-Fluid Model treats both the gas and solid phase as interpenetrating continua. The conservation equations for each phase take the form: Continuity equation: ? ?. 0m m m mt? ? ? ?? ?? ?? mU (2-1) where ? , ?  and U  are the voidage, density and velocity, and subscript m refers to either the gas or solid phase. The Momentum equation for gas phase is: ( . ) . ( )g g g g g sU U U g U Ug g g g g gst? ? ? ? ? ?? ? ? ? ? ? ? ?? ? (2-2) Here gs? denotes the gas-solid momentum exchange coefficient; g ?? , the gas phase stress tensor, is defined as 2 .3Tg g g gU U U Ig g gP? ?? ?? ?? ? ?? ? ? ?? ? ? ?? ?? (2-3) g? and gP  in the above equation are gas viscosity and gas pressure, respectively. The Momentum equation for the solid phase is: ? ?( . ) .s s s s g sU U U g U U fs s s g s s gs ePt? ? ? ? ? ?? ? ? ?? ? ? ? ? ? ?? ? (2-4) where ef  is the electric force density for charged particles, and the solid stress tensor, s? , is defined by T 2 .3s s sU U Is s b sP? ?? ?? ?? ?? ?? ? ?? ? ? ? ? ?? ?? ?? ? ? ?? ?sU? (2-5) wheres? and sP denote solid viscosity and solid pressure, and b? refers to the bulk viscosity for perfectly elastic particles; its multiplication by a constant ? , which is a function of the restitution coefficient (ep) as defined in Table 2-1, results in the bulk viscosity for inelastic particles. The constitutive equation proposed by Savage (1998) is used to define the solid phase shear stress tensor, as a sum of kinetic and frictional terms, denoted by subscripts k and f, as presented in Table 2-1. The kinetic part of the stress in MFIX is based on the model of Lun et al. (1984), modified to account for the effect of interstitial gas on particle phase viscosity through terms with an asterisk superscript (Agrawal et al., 2001), such as *?  in Eq. 1.7 in Table 2-1. Replacing  - 11 - *?  by ?  recovers the Lun et al. (1984) model, where ? is the shear viscosity for perfectly elastic particles with dilute concentrations.  Table 2-1: Stress terms  (1.1)? ?? ? ?k fs s s  ? ? 2 . (1.2)3Ts s s s s b s sP? ?? ?? ?? ?? ? ?? ? ? ? ? ?? ?? ?? ?? ?? U U U I ? ?* 0 002 8 8 3? 1 1 3 23 (2 ) 5 5 1 3)5 ( .kk f fs s s s s b sg g ?g? ?? ? ? ? ? ? ? ? ? ?? ?? ??? ? ? ? ? ?? ? ? ? ? ? ? ?? ?? ? ? ? ? ??? ? ? ? ? ?? ? ? ?1.6?,?? 1 / 2? (1.4 5)pe? ?? ? ? ? ? ?0( ? 1 4 ) (1.6)k f k fs s s s s s s sP P P g P? ? ??? ? ? ? ? ? ?? ?1 22* 0205 ? 256?,?? ,  (1.7 9)96 51 2 / ?s p sbgs s sd gg? ? ???? ? ? ?? ? ? ?? ? ? ??  Carnaham-Starling (radial distribution) for mono-sized particles: ? ? ? ?s0 s 3s1 0.5?g ? (1.10)1 ??? ? S-R-O frictional model: ? ? ? ?2( ) / 4 , * (1.11)f max maxs c D s s smin P sin I? ? ? ? ???  ? ?24 1010 ? (1.12)0?max maxs s s sc maxs sP ? ? ? ?? ?? ? ??? ? ??? S-S frictional model: ? ?? ?? ? ? ?1/( 1)2( ) 1* (1.13)2 ?//:f f cf mins s ss pnsin P n n P Pd?? ? ??? ?? ???S S ? ?T1 1 . (1.14)2 3ss sS U U U I? ? ?? ? ? 3 / 2? ( )???? . 0?? (1.15)1.03??????????????? . 0??sin dilationn compaction?? ? ??? ? ? ???ssUU 2( 1) (1.1.(1 6)/ ) ?2 ( ) :fs csnpP Pn sin d???? ? ??sUS S ? ?10242510( )0.05 ???? ? (1.17)( )0max maxs s s sminmin maxs sc s s smaxs smins sP? ? ? ?? ?? ? ?? ?? ?? ? ??? ?? ????? ????   - 12 - The frictional part of the stress is implemented in MFIX by two methods, namely: the Syamlal et al. (S-R-O) and Srivastava and Sundaresan (S-S) models. The S-R-O model is based on plastic flow of a granular material, and it allows for compressibility near the packing limit. The frictional stress from this model is only non-zero when the solid volume fraction exceeds (maxs?). The critical pressure is a power law function of voidage (Syamlal et al., 1993), and the frictional viscosity is proportional to the critical solid pressure, as proposed by Schaeffer (1987). As shown in Eq. 1.11 in Table 2-1. ? , 2DI  and maxs?  in this equation denote the angle of internal friction, the second invariant of the deviator of the strain rate tensor for solid phase. The maximum solid viscosity limit, set to 100 kg/m.s by default in MFIX. ? ?maxs s? ??  is a Boolean expression that is equal to 1 or 0 when the expression is true or false, respectively.  The S-S model is based on a compressible granular flow, with the frictional stress starting to play a role at lower solid volume fraction?( mins?) than for the S-R-O model. The S-S model includes the fluctuation in the strain rate associated with the formation of shear layers, proportional to the root of granular temperature over particle diameter, 0.5s p? d  , preventing numerical singularity in regions where the magnitude of strain tensor, denoted by (:S S), is zero as shown in equation 1.13 in Table 2-1. In this equation, S  is the strain rate tensor and ? ?mins s? ??  is a Boolean expression that acts as explained above. The critical solid pressure in this model is calculated in the same way as in the S-R-O model when the solid volume fraction exceeds? maxs?, but it is based on the Johnson et al. (1990) empirical correlation between mins?  and maxs? , and equals zero when the solid volume fraction falls below the minimum solid volume fraction, mins? , needed to account for the frictional effect.   The solids and gas momentum equations are coupled by the drag term, which plays an important role in the overall force balance. In the present study, we use the Gidaspow (1994) drag relation which combines the Ergun (1952) equation for voidage < 0.8 and the Wen &Yu (1966) correlation for voidage ? 0.8:  - 13 - 2.6523??????????????????? 0.8??41.75?150 (1 ) ?? ?? 0.8?g g sD gpgsg ss g ggpg pCddd? ? ? ???? ?? ? ??????? ???? ???????g sg sU UU U (2-6) ? ?0.68724 ???????? 10001 0.15??????????0.44?????????????????????????????? 1000DReRe ReCRe? ?? ?? ?? ?? (2-7) g g pgdRe ? ? ??? g sU U (2-8) Heregs? , DC and Re are a momentum exchange coefficient, drag coefficient and particle Reynolds number, respectively. The Eulerian multiphase model considers the solid phase as a continuum, called a ?granular gas?, whose properties are defined by analogy with the kinetic theory of gases, but unlike gases, which have elastic collisions, particles lose part of their energy because the collisions are inelastic. This model provides closures for solid viscosity and solid pressure as functions of granular temperature for the solid phase. The granular temperature, analogous to gas temperature, is proportional to the kinetic energy of the random motion of particles, defined as 2 / ? 3? ? c  (2-9) where c is the instantaneous particle velocity. The granular temperature conservation equation takes the form ? ?3 ? . ?  : . ?2 ks s s s coll visJ Jt? ? ??? ?? ? ? ? ? ? ? ? ?? ?? ? ?? s sU U? (2-10) The granular temperature is generated by the kinetic portion of shear stress between particles (first term on the right-hand side of Eq. 2-10), and it can be conducted between particles (second term in RHS of Eq. 2-10). Some portion of it can be lost through particle-particle collisions ? ??collJ, and some through interaction between gas and particles ? ?visJ. The granular conductivity and the loss terms are defined in Table 2-2; whereas, the redial distribution, 0g  and parameter, ? , are defined in Table 2-1. ? , *?  and s? in Table 2-2 refer to granular conductivity for inelastic particles in dilute concentration, proposed by Lun et al. (1984), the modified solid conductivity needed to account for the interstitial gas effect and the granular conductivity of inelastic particles for high dense concentrations, respectively.  - 14 - Table 2-2: Granular temperature relations ? ?? ?? ?1/2*2075 ?? ?,  648 41 33 15 ?s pgss sdg? ? ?? ? ? ?? ?? ?? ?? ? ? ? ? ?? ?* 220 0 0012 12 641 1 4 3 41 335 5 25s s s sg g gg?? ? ? ? ? ? ? ???? ?? ? ? ?? ? ? ? ? ?? ?? ? ? ?? ? ? ?? ? 2230813 s g g svis gsp sJ g d? ?? ? ??? ?? ?U U ? ? 2 3/2048 1 s scollpJ gd? ?? ??? ? ?  2.3. Boundary Conditions Atmospheric constant pressure is assumed as the top boundary condition, while uniform gas velocity is adopted at the distributor, except over the central orifice, where a transient jet boundary is applied. The no-slip boundary condition for the gas phase, and free-slip as well as a partial-slip for the solid phase are employed at the side walls. The partial slip boundary conditions based on Johnson and Jackson (1987) take the form ? ? 0max. . ( . . ) tan 0| | 2 3s s ssl k f sl fsl sg??? ? ???? ? ? ?U n U n n? ? ? (2-7) 2 20 3/20maxmax3 (1 ). 42 3sl s s s s s ws s sssg g e?? ? ? ?? ?? ??? ?? ?? ? ? ?Un (2-8) Here n and ew are the unit vector normal to the wall and particle-wall restitution coefficient, respectively.slU in these equations denotes s wall?U U (slip velocity); ?  is the specularity coefficient. The electrical boundary conditions for all surrounding boundaries are set to zero potential, corresponding to being fully grounded.  It is important to note that electrostatics do not influence particle-wall boundary conditions, since these forces are considered as a body force in our model like gravitational force. In other words, electrostatics do not affect particle-wall or particle-particle collisions directly.   - 15 - 2.4. Electric Force Density The charges on particles produce an electric field that exerts forces on particles. This field is described based on electromagnetic governing equations, Maxwell equations, for charged particles: .D vq? ?  Gauss law of electricity (2-9) BE t??? ? ? ? Faraday?s law  (2-10) . 0? ?B Gauss law of  magnetism (2-11) t??? ? ? ?DH J Ampere?s Law with  Maxwell?s displacement current correction (2-12) where  v s s mq q? ??  (2-13) D, qv, qm, E, B, H, J and t refer to the electric displacement field, charge density based on bulk volume of solids, charge density based on mass of solids, electric field, magnetic field, magnetizing field, free current density and time, respectively. We neglect the magnetic field, so that Faraday?s law reduces to  0?? ?E (2-14) Likewise, Gauss? law of magnetism and Ampere?s law are not considered in this study due to the neglect of the magnetic field. In view of equation (2-14), E can be written as the gradient of a scalar potential field????E. The electric force (eF ) for a point charge Q, moving at velocity pV, is given by the Lorentz force: e EF V BpQ Q? ? ? (2-15) For a dielectric continuum, the electric force is more complicated. Based on thermodynamics, Melchers (1981) exploited the principle of virtual work to estimate the electrical force density  ( fe ) of a dielectric material where there is a combined effect of free charge and polarization, leading to 1 1 1?f E Em m me v i v i ii i ii i iW W Wq q? ? ?? ? ?? ? ?? ? ?? ? ? ? ? ? ??? ? ?? ? ? (2-16)  - 16 - W refers to energy of the system of charged particles, and ? is a variable property of a dielectric continuum. The electric displacement field takes the form 0 ?(1 )e??? ? ??D ? E E (2-17) where 0???  and ? denote the electric permittivity of free space and the medium, respectively, while?e? is the susceptibility of the material. We use the Maxwell-Garnett relation (Robinson and Friedman, 2005) to define the average relative permittivity of mixtures of mono-sized particles in the background fluid as ? ?? ?? ?? ?2 110 1 2 2 13 11 2 1gaveg??? ?? ?? ?? ? ?? ? ?? ?? ? ?? ? ? ?? ?? ? (2-18) where 1? and 2?  denote the permittivity (a measure of the resistance encountered when forming an electric field in a medium) of the background fluid and solids, respectively. In the current study, the relative permittivity (ratio of permittivity of medium to permittivity of vacuum) of the gas phase (air) and solid phase (glass beads) are 1 and 7.6, respectively (Robinson and Friedman, 2005). By considering E in Eq. 2-17 as a negative gradient of scalar potential and then inserting the resultant expression back into Eq. 2-9, a variable coefficient Poisson equation is derived. This equation is then solved as explained in Section 2.4 to calculate the electric potential.  For a compressible material, the susceptibility is assumed to be a function of mass density, .?  The energy of a system of charged particles, equivalent to the energy required to bring charge from infinity to its current location, can be derived (Reitz et al., 1993) as: . / 2W ? D E  (2-19) Replacing E by /?D  in the above equation results in ? ?? ?20 ?2 1 eW ? ?? ? ?D (2-20) Finally, by inserting W from Eq. 2-20 into Eq. 2-16 and replacing ?  by mass density ( ),?  in Eq. 2-16, the electric force density for compressible materials can be obtained: 2 21 12 2f E E Ee vdq d? ?? ??? ? ????? ?? ? (2-21)  - 17 - This equation is also known as the Korteweg-Helmholtz force density equation. (See Melchers, 1981; Zahn and Rhee, 1984; Rietz et al., 1993 for more details). The gas-solid mixture in fluidized beds can be assumed to be compressible. Equation 2-21 can then be applied to define the electrical force density by replacing the mass density ( ? ) by ? ?1s g? ?? . The electric force density inside the fluidized bed is then defined as  ? ?2 21 1 12 2f E E Ee v g gdq d? ?? ??? ? ???? ?? ?? ? (2-22) This equation is central to this study; thus, ef  is added to the left side of the solid momentum equation. In this study we assume that the fluidized particles have reached a charge saturation level, typically requiring about 1 h of fluidization. At this state, the total amount of charges in the system is constant, implying that the magnitudes of charge generation and dissipation are equal. We also assume that all particles carry the same magnitude of charges, regardless of particle-particle and particle-wall collisions. Therefore, the electric field is a function of particle volume fraction and collisions do not change the electric field, except by changing the particle volume fraction.  2.5. MFIX Numerical Method  The numerical methods employed in MFIX for the discretization of temporal and convective terms are the implicit backward Euler method and superbee metheod, respectively, which are both of second order accuracy. A modified version of the SIMPLE algorithm using void fraction and gas pressure correction equations is used to solve a set of linearized equations using the BiCGStab method (Barrett et al., 2006). Details are available in the MFIX numerical guide (Syamlal et al., 1998).           - 18 - 2.6. Numerical Discretization As explained above, Gauss? law of electricity and Faraday?s law of electromagnetic induction are combined to form a Poisson equation with variable coefficients. This equation is solved numerically at each time step to find the electric potential. Figure 2-1(a) describes different variables in the MFIX code. Points P, E, W, N and S in this figure represent the center of the scalar cell P and its east, west, north and south neighbouring cell centers. Ug and Us denote the x-components of the gas and solid phase velocity, respectively, while Vg and Vs represent their y-components; ?g and ? refer to the voidage and electrical potential.     Figure 2-1: (a) Schematic of key variables around control volume cell in MFIX code; (b) Schematic of scalar, x-momentum and y-momentum control volumes based on MFIX discretization.   The x- and y-momentum equations are discretized around the east and north faces, respectively, due to the staggered discretization in MFIX. Scalar quantities like pressure, voidage and electrical potential are defined at cell centers, whereas the x-component of velocity is defined at the east and west faces, and the y-component at the north and south faces. The variable-coefficient Poisson equation is discretized around each scalar cell as: ? ?0. .ave av ve qdV dA dV? ?? ? ? ? ? ? ? ??? ? ? (2-23)   - 19 - so that , , , ,0? ? ? ?P W N P P S vE Pave e e ave w w ave n n ave s s qA A A A Vx x y y? ? ? ? ? ?? ? ? ? ???? ?? ?? ?? ? ?? ? ? ? ? (2-24) Subscripts e, w, n and s in this equation refer to the east, west, north and south face of scalar cell P, while A and V? refer to its facial area and volume. For a uniform Cartesian two-dimensional mesh weA A y? ? ? , snA A x? ? ? , and V x y? ? ? ? (with unit depth). Inserting these values leads to  ? ? ? ?, ? , ? , ? , ? , ? , ? , ? , ?2 20ave w W ave e ave w P ave e E ave s S ave n ave s P ave n N vqx y? ? ? ? ? ??? ? ? ?? ?? ?? ? ? ?? ??? ? ?? ? (2-25) The relative permittivity of each scalar cell is calculated at its centre, while this property at each face is calculated based on the harmonic average of adjacent cells. For example, the average relative permittivity for the south face is calculated by , ,,, ,2 ave P ave Save save P ave S? ?? ?? ?? (2-26) The discretized Poisson equation with variable coefficient with second order accuracy, is then solved by applying Successive Line Over-Relaxation (SLOR) (Jaluria and Torrance, 2003). Here the electrical force is considered to be a body force, and the discretized formula for calculating the x- component of the electric field at the center of the x-momentum control volume is ? ?? ? ? ?22,21121)2  (   ?2EEEggv xe E Pv E Pe xE Pddddq E dVdx dxq Pst Pstx xFVx??? ?? ???? ?? ?? ? ?? ?? ?? ??? ? ?? ?? ? ? ?? ?? ? ?? ?? ? (2-27) where  ? ? 2,1 1 |2E g E E EgdPst d? ??? ? ? E (2-28) ? ? 2,1 1 |2P g P P PgdPst d? ??? ? ? E (2-29) The x- and y-momentum control volumes, as well as other notation required for this equation, are shown in Figure 2-1(b). This force is then added to the RHS of the x-momentum equation.  - 20 - The magnitude of the electric field at the east face of a scalar cell (i.e., centre of the x-momentum control volume) is calculated as the average of its adjacent cells? electric fields. ( ) / 2E E Ee E P? ? (2-30) where the magnitude of the electric field is calculated at the scalar cell centre as  222 2EE W N SP x y? ? ? ?? ?? ?? ?? ? ? ?? ?? ?? ? ? ? (2-31) The same approach is used to calculate the electric field at the east scalar cell. ?|Egdd?? can be obtained by substitution of voidage at the east face of the x-momentum control volume (i.e. point) into  ? ?? ?? ?? ?? ?1 2 1 1 221 2 2 13 22 1g gdd? ?? ? ? ?? ? ??? ?? ?? ? ? ? ?? (2-32) This quantity is calculated for the west side by an approach similar to that for the east side. The same approach is taken to calculate,? e yFat the center of the y-momentum control volume.   2.7. Verification of Electric Potential Solver The numerical solver for the variable coefficient Poisson equation was verified by comparing the results with the analytical solution for a rectangular column of 1 m width and L (m) height, filled with uniformly charged solid particles to a height of L/2. This problem consists of solving the Poisson equation in the bed and the Laplace equation in the freeboard. The electric potential at the walls is assumed to be zero, which holds for grounded metal walls, but is not an accurate assumption for grounded dielectric walls. Separation of variables was used to solve each section separately, and the coefficients in derived solutions were then found by assuming equal potential at the bed surface (originating from Faraday?s law of induction) with the following relation for a dielectric-dielectric boundary, derived from Gauss? Law of electricity (Reitz et al, 1993): avge sy y? ?? ???? ? (2-33)  - 21 - Here g?  and s?  denote the electric potential in the freeboard and dense bed, respectively, whereas ave? is the average relative permittivity of the dense phase. The analytical solution for each part is as follows, with qv being the charge density based on the bulk volume of solids, as defined previously in Eq. 2-13.   ? ? ? ?? ? ? ?1,   ,      =oddgnA sinh y B cosh y sin x n? ? ? ? ? ?? ?? (2-34) ? ? ? ? ? ?3031, 04 4 odd ave avvsevnq qC sinh y cosh y sin x? ? ? ?? ?? ?? ? ?? ?? ? ? ?? ?? (2-35)  ? ? ? ? /A B cosh y sinh L? ?? ?  (2-36) ? ?? ?? ?0 38 cosh / 2 11vaveq LB ???? ? ?? (2-37) ? ? ? ?? ?? ? ? ?? ?23028 / 2 / 2 ( / 2)1 ? ?vaavea eve vq sinh L cosh L cosh LCsinh L? ? ?? ?? ? ??? ? ?? (2-38)  Figure 2-2 presents a sample solution for the above system, representing the packed bed electric potential at its initial state for a case where L=1 m, ?g=0.5, ?s=2500 kg/m3 and qm= ?1 ?C/kg. The relative permittivities of the gas phase (air) and solid phase (glass beads) are 1 and 7.6, respectively. In Figure 2-3, the magnitudes of the analytical and numerical electric field along X=0.5 m and Y=0.5 m are seen to be almost identical.   - 22 -  Figure 2-2: Contour plot of electric potential (V) for packed bed with L=1 m, bed height= 0.5 m, ?g= 0.5, qm= ?1.0 ?C/kg and ?s= 2500 kg/m3.   Figure 2-3: Comparison of electric potential (V) where L=1 m, bed height= 0.5 m, ?g= 0.5, qm= ?1.0 ?C/kg and ?s= 2500 kg/m3: (a) along x- direction for Y= 0.5 m, (b) along y- direction for X= 0.5 m.    - 23 - Chapter 3. Effect of Electrostatics on Single Bubbles 3.1. Introduction In this chapter, the effect of electrostatic charges on single bubbles in a mono-sized system is investigated by solving the electrical governing equation and including the resultant force in the solid momentum equation. In this manner, the electrical field can change the flow and vice versa. Two cases are simulated, based on experiments reported by Gidaspow et al. (1983) and Rahman and Campbell (2002). The effect of charge on bubble shape is then studied for the latter setup. After sensitivity analysis to assist in choosing the frictional model and charge density, the bubble rise velocity is investigated for different solid phase boundary conditions. Finally, different forces on the bubble periphery are evaluated to provide a better understanding of the electrical field effect.   3.2. Simulations As outlined in the previous chapter, the Two-Fluid Model was employed to investigate the effect of electrostatic charge on fluidized bed hydrodynamics. The ability of this model to predict bubble shape and solid pressure is addressed, by simulating two previous experimental studies. The first case study is the single-jet-fluidized-bed of Gidaspow et al. (1983). The set-up dimensions and simulation parameters are provided in Table 3-1.   In the initial second of the simulation, the central jet velocity is equal to the uniform velocity at the distributor (background velocity), after which the orifice velocity is increased to 5.77 m/s. By impulsively increasing the jet velocity after 1 s, the effect of the initialization on the first bubble is minimized (See Appendix B for details.).  The constitutive relations for solid stress were not fully developed when the Gidaspow et al. (1983) paper was published. Due to the need to add a normal component of solid stress to prevent excessive compaction of solid, the modulus of elasticity was utilized with fitted experimental parameters. In our study, the S-S frictional model is employed to compare with  - 24 - their experimental results. This model has been demonstrated to have good promise (Patil et al., 2005; Passalacque and Marmo, 2009; Benyahia, 2008).   As discussed below, the magnitude of electrical force due to charged particles can be of the same order as the solid pressure force. Thus, the second case study is based on the system studied by Rahman and Campbell (2002), where the magnitude of solid pressure on the face of a two-dimensional bed of glass beads fluidized by air was measured for both a single bubble and a freely bubbling bed. Their first case is simulated here, with slight modification in bubble injection. In that study, a bubble was injected through a tube located 0.23 m above the distributor, but here only part of their column is simulated from the injection level to the top. Therefore, the distributor is shifted 0.23 m upward in the simulation, and it has a 12.7 mm (arbitrary size) central orifice added. The goal is to produce a single bubble with the same frontal diameter as in the physical experiment. The injection method is not important after the bubble has formed and rises. The modified set-up dimensions and simulation parameters for this case are listed in Table 3-1. As in the other case study, the first second of simulation is used to diminish the initialization effect, and after which a single bubble is injected.  The simulation parameters (see Table 3-1) used in this section and in later sections of this thesis are the same as, or close to, values typically used in CFD literature (e.g. Gidapow et al, 1983; Sirivastava and Sundaresun, 2003; van Wachen, 2004; Li et al., 2010). Some of these parameters like particle-particle restitution coefficient and particle-wall restitution coefficient in the range of 0.8 to 0.99 have been found to exert negligible influence on the final results (Reuge et al., 2008). However, the specularity coefficient, with a default value of 0.5, can influence the solids velocity distribution and circulation pattern significantly, (See Li et al., 2010 and Appendix C for more details) especially in bubbling fluidized beds. When the bed is close to Umf, which is the case in this chapter, using the default value only increase the computational time without significantly changing the overall results. Therefore, in this study, the smaller value of 0.05 was chosen to reduce the simulation time.  The values of the threshold solid fraction for friction, ?smin, and maximum solid packing volume fraction, ?smax, (S-S frictional model parameters) were chosen by trial and error to produce a  - 25 - stable single bubble at superficial gas velocity close to the experimental value reported in the literature. These values are then used in other chapters of this thesis. As explained further in Chapter 5, they play an important role in determining the average voidage of the bed and bubble size in freely-bubbling beds. Finally see Appendix D for details about grid size.    Table 3-1: Simulation parameters for different case studies              Gidaspow.  Rahman and               et al. (1983)  Campbell (2002) L  Column height (m)        0.584   1.3 LB  Bed height (m)        0.292   0.87 W  Column width (m)        0.394   0.46 do  Orifice diameter (mm)       12.7    12.7 ?t  Initial time step (s)        0.0001   0.0001 tstop  Simulation time (s)        4.0    3.0 ?x, ?y Grid size (mm)        5.0     5.0 dp  Particle diameter (?m)       500    500 ?s  Particle density (kg/m3)       2610   2500 MW  Gas molecular weight (kg/kmol)     29    29 Tg  Gas temperature (C)       24    24 ?g  Gas viscosity (kg/m.s)       1.8e-5   1.8e-5 Vjet  Jet velocity (m/s)        5.77  18b & 14c (S-S),15c(S-R-O) ?tjet  Jet duration (s)        3.0    0.2 Vsup  Superficial velocity (m/s)      0.284 0.23b,c (S-S), 0.2c(S-R-O) Pinlet  Inlet pressure (Pa)        1.06e5   1.12e5 ?   Angle of internal  friction (degrees)    30    30  w?   Angle of internal friction at walls (degrees)   N.A.a   11.3 ep  Particle-particle  restitution coefficient (-)   0.8    0.9 ?   Specularity coefficient (-)      N.A.a   0.05 ?smin  Threshold solid fraction for friction (-)    0.55    0.55   ?smax  Maximum solid packing volume fraction (-)  0.6    0.6 a N.A.= not applicable  b Section 3.3.2 velocity  c Section 3.4 and 3.5 velocity  - 26 - 3.3. Comparison of Numerical and Experimental Results for Uncharged Particles 3.3.1. Bubble Shape Simulation predictions are compared in Figure 3.1 with experimental results of Gidaspow et al. (1983). Unlike Gidaspow et al. (1983), who only compared the second numerical bubble with experiment, different bubbles passing each height after the eruption of the first bubble at the surface, are compared in this work. In the literature, the bubble boundary is defined as the locus of where the voidage is 0.8 or 0.85 (Gidaspow et al., 1983; Kuipers et al., 1991). In the current study, 0.8, the most commonly chosen values, is employed (See Appendix E). In Figure 3.1, the predicted bubbles are smaller than the experimental ones, consistent with earlier simulations (Patil et al., 2005). The difference between theoretical and experimental results is more significant for bubbles not yet detached from the distributor and for bubbles close to the bed surface (Figures 3-1 a and d). The predictions are closer to experimental results for bubbles with centroids passing heights 0.09 and 0.17 m above the distributor (Figures 3-1 b and c, respectively). Although predicted sizes are smaller than the experimental ones, the shapes are in reasonable agreement. As can be seen from Figure 3-1(d), some bubbles have deformed bases due to recent coalescence with a trailing bubble. Note also that the experimental column thickness was 38.1 mm, so that the assumption of two-dimensional flow (neglecting the third dimension) may have influenced the results (see Li et al., 2010).  - 27 -  Figure 3-1: Comparison of predicted and experimental bubble shapes at different heights. (Solid lines are simulation results obtained at different times, whereas dashed lines are experimental results from Gidaspow et al., 1983).  3.3.2. Solid Pressure  Rahman and Campbell (2002) indicated that most particle pressure is generated below and to the sides of a bubble, with its magnitude reaching a maximum when the bubble erupts at the bed surface. It was concluded that this pattern is mainly due to defluidization, resulting from drawing fluidizing gas into the bottom of a bubble and away from the surrounding solids. Details of the  - 28 - injection velocity and bubble shape were not reported in their work; only the area-equivalent bubble diameter was provided. Hence, the jet velocity and background velocity were varied here to obtain a similar area-equivalent bubble diameter ( 4 /bD A ?? ), as shown in Figure 3-2.   The experimental bubble was injected at time 0 for a duration of 0.2 s; its nose reached the centerline probe at 0.75 s, its base passed the probe at 1.2 s, and it finally erupted at 1.8 s, whereas the simulated bubble was injected between 1 and 1.2 s, its nose reached the central probe at 1.64 s, the base reached the probe at 1.86 s and the bubble erupted at 2.54 s. Thus, the experimental bubble rose more slowly than the simulated one. The two cases are synchronized in Figures 3-3 and 3-4, so that t=0 corresponds to the beginning of bubble injection.   Figure 3-2: Comparison of bubble diameters predicted by simulation and Rahman and Campbell (2002) experimental results for 500 ?m glass beads fluidized with background velocity of 0.23 m/s based on S-S frictional model. The bubble was injected with Vjet = 18 m/s for a duration of 0.2 s.  Figure 3-3, presents solid pressure contours during bubble rise inside the bed at different times. The white lines indicate where the voidage equals 0.8, whereas the black dots, numbered from 1 to 4, show the centre of the solid pressure probes, located 0.46 m above the injection port and 50  - 29 - mm apart, similar to the Rahman and Campbell (2002) experiment. The chosen time steps in this figure correspond to points A to D on the central numerical probe signals in Figure 3-4 (a).   At the start of bubble injection, the solid pressure increases in the lower part of the bed and then decreases to a negligible value (0.08 Pa) at 0.24 s. As the bubble starts to rise, the solid pressure increases around the bubble, most noticeably in the region below the bubble. The surge in solid pressure precedes the bubble and reaches probe 1 at 0.36 s, as depicted in Figure 3-3 (a). The bubble envelopes the three innermost probes at 0.78 s, while its diameter is ~ 190 mm. (experimental bubble diameter at this point was 180 mm.)   The solid pressure drops to almost zero and then increases quickly after the bubble moves away, as depicted in Figure 3-3 (b) at 0.88 s. The solid pressure continues to increase until the bubble reaches the bed surface (see Figure 3-3 (c)) at time 1.5 s, and then decreases as the bubble erupts. This behaviour is consistent with that reported by Rahman and Campbell (2002). However, although the simulated bubble diameter is close to theirs, the bubble shapes apparently differ since the experimental bubble only enveloped the two innermost probes.  The magnitudes of solid pressure for each probe are compared with experimental results in Figure 3-4. The reported surge seen in Figure 3-3 (a) is labeled A here; it produces a hump around 0.36 s, as depicted in Figure 3-4 for all probes. The solid pressure decreases to zero as the bubble envelopes the probes, and the increase of solid pressure after bubble passage (Point B) is also seen here as a larger hump that increases continuously until bubble eruption (Point C). As seen in this figure, the simulated magnitude of solid pressure is larger than the experimental value, but they are of the same order of magnitude and follow similar trends.   It is worth mentioning that the dimension of the experimental probe diaphragm was 25.4 mm, so it measured the average solid pressure over a significant area, while the simulation values refer to a single point (center of the probe); this could have affected the results to some extent. In addition, the experimental column was 25 mm thick, and the assumption of two-dimensional flow may have again influenced the simulation results.    - 30 -  Figure 3-3: Solid pressure contours during bubble rise for simulation of the Rahman and Campbell (2002) experiment (pressures in Pa) for 500 ?m glass beads fluidized with 0.23 m/s background velocity and S-S frictional model. Bubble is injected with Vjet = 18 m/s for a duration of 0.2 s. Points in Fig. 3.4 (a): (a) A (b) B (c) C.  - 31 -  Figure 3-4: Comparison of solid pressure between simulation with S-S frictional model and the Rahman and Campbell (2002) experimental results for 500 ?m glass beads with Vsup= 0.23 m/s and Vjet= 18 m/s: (a) probe 1; (b) probe 2; (c) probe 3; (d) probe 4. For probe locations, see Figure 3.3.    - 32 - 3.4. Effect of Electrostatic Charges on Isolated Bubbles In the previous section, the hydrodynamic model predictions were compared with the earlier experimental studies for the uncharged particles, and the results were in good agreement. In this section, the modified experimental set-up of Rahman and Campbell (2002) is reinvestigated for uniformly charged particles.  The electrostatic charge generation on particles is a function of many parameters; for example, the superficial gas velocity, particle size, particulate material, humidity, temperature, pressure, surface roughness and wall material of construction. Due to weather changes, electrostatic charge on particles can vary from day to day. Here we consider typical charges for 500 ?m mono-sized particles, which, from the literature (Chen et al., 2007; Sowinski et al., 2009), lie in the range of ?0.22 E-7 to ?3.6 E-7 C/kg.  The maximum effect of electric field can be estimated from Crowley (1995) by calculating the electrical pressure as 2?eff E?. This is of the order 175 Pa, assuming a maximum electric field of 3 MV/m. If 0.1s? gDb is taken as a representative solid pressure (Rahman and Campbell, 2002) for a 0.2 m bubble, the solid pressure is of order 490 Pa. Note that the electrical and solid pressures are of similar magnitude. Figure 3-4 indicates that there are times during which the magnitude of solid pressure falls below the representative solid pressure, indicating that the electric field can significantly affect bubble behaviour, as confirmed below.   3.4.1. Bubble Shape for Charged Particles 3.4.1.1. Frictional Model The particulate pressure plays a role in maintaining the force balance in a thin boundary layer adjacent to the bubble surface, while being insignificant elsewhere (Buyevich et al., 1995). Hence, the frictional stress choice may influence the overall effect of the electric field on the particle motion. As a result, two alternate frictional models are compared below to determine the possible effect of electrical charges. The effect of the frictional model on bubble shape for charged particles is examined by considering the maximum charge. The central jet and  - 33 - background velocity were adjusted to obtain a bubble which did not split for each case. The velocities and simulation parameters for this section are presented in Table 3-1.  Figures 3-5 and 3-6 display predicted bubble shapes with and without an electric field at different times for the two frictional models. While both models predict elongation in the flow direction due to the electrical field, the extent of bubble splitting and details of bubble shape differ. As seen in Figures 3-5 (b) and 3-6 (b), small voids are generated close to the bed surface during the bubble rise. Wittman and Ademoyega (1987) reported elongation of bubbles in the electric field direction in a study of the effect of external electric field on hydrodynamics of a fluidized bed of 177-210 ?m silica gel particles with O2 as the fluidizing gas. Deformation of an injected bubble was observed along the field line, with the bubble adopting an elliptical shape and rising more slowly in the presence of a horizontal electrical field.    Figure 3-5: Bubble shapes for 500 ?m glass beads at different times, for S-R-O frictional model with Vsup= 0.2 m/s and Vjet= 15 m/s.: (a) Uncharged particles; (b) Charged particles (qm= ?0.36 ?C/kg).  - 34 -  Figure 3-6: Bubble shapes for 500 ?m glass beads at different times predicted based on S-S frictional model with Vsup= 0.23 m/s, Vjet= 14 m/s.: (a) Uncharged particles; (b) Charged particles (qm= ?0.36 ?C/kg).   3.4.1.2. Charge Density The charge density based on signal reconstruction from an array of induction probes for closely-sized glass beads of mean diameter ~ 500 ?m was found (Chen et al., 2007) to be of order ? 0.36 ?C/kg, while experiments by Sowinski et al. (2009) for similar particles fluidized at velocities of 0.27 and 0.37 m/s led to charge densities of ? 0.022 to ? 0.026 ?C/kg. To assess the sensitivity to charge density, three simulations were carried out for a column of the same dimensions as the experimental column of Rahman and Campbell (2002), with different charge densities and applying the S-S frictional model, with the same jet and background velocity as explained in section 3.4.1.1. Figure 3-7 portrays the bubble shape for different magnitudes of charge density based on the minimum, average and maximum of the above values. It is seen that as the magnitude of the charge increases, the bubble is predicted to become more elongated.  - 35 -  Figure 3-7: Effect of charge density on bubble shape for 500 ?m glass beads based on S-S frictional model. Vsup= 0.23 m/s and Vjet= 14 m/s.  3.4.2. Bubble Rise Velocity for Charged Particles As seen in the previous section, a bubble subject to an electrical field takes a prolate shape, independent of frictional model, with the extent of deformation depending on the charge density. Bubble shape affects its rise velocity, with more streamlined bubbles (elongated in flow direction) tending to rise more quickly (Grace and Harrison, 1967). In this part of the study, the bubble rise velocity was calculated for an elongated bubble in charged particles for the maximum charge density (? 0.36 ?C/kg) with different solid phase boundary conditions (free slip and partial slip), based on the S-S frictional model, with the same jet and background velocities as explained in Section 3.4.1.1. Figure 3-8 shows that the bubble rise velocity in the charged particles for both boundary conditions increases significantly, never reaching a constant value for the current geometry. The bubble rise velocity is lower for the partial slip condition than for the free slip boundary condition, though both cases give similar results for the uncharged system.    - 36 - The bubble rise velocity for uncharged particles is around 0.6 m/s, whereas it is 0.65 to 0.95 m/s for charged particles. During its rise, the bubble diameter varies in the range of 0.15 to 0.16 m and 0.16 to 0.17 m for the uncharged and charged systems, respectively. Thus the bubble diameter remains almost constant for both cases (see Figure 3-6). The bubble reaches a spherical-cap shape for uncharged particles, but for charged particles the bubble is elongated vertically. The choice of boundary conditions does not affect the bubble shape over the ranges investigated.  Figure 3-8: Comparison of effect of boundary conditions on predicted bubble rise velocity through bed of 500 ?m glass beads, with S-S frictional model. Vsup= 0.23 m/s and Vjet= 14 m/s.  The bubble rise velocities are calculated based on the displacement of the nose position versus time for an interval of 0.02 s. There are many correlations in the literature to predict isolated bubble velocities (Karimipour and Pugsley, 2011). Three of those correlations are used here to calculate the bubble rise velocity for Db = 0.16 m, as shown in Table 3.2. The simulated rise velocity for uncharged particles is closer to the Wallis (1969) correlation, where the wall effect is considered, though those of Rowe and Partridge (1965) and Rowe (1971) are closer for the charged particle case. However, the experimental bubble rise velocity data are widely scattered.   - 37 - Table 3-2: Bubble rise velocity from available correlations for Db= 0.16 m Correlations    Particles      Rise velocity (m/s) Author(s) 0.84 24 /b bU gD?    460-550 ?m glass beads   0.75    Rowe (1971) 1.02 bbU gD?    Geldart group B (52-550 ?m) 1.28    Rowe & Partridge (1965) 0.711 1.2?exp( 1.49 )/b b b cU gD D D? ?      0.64    Wallis (1969)  3.5. Main Cause of Bubble Shape Change In this section, different forces are evaluated at the periphery of a bubble to provide a better understanding of the causes of the predicted bubble deformation. Since the resultant interpolated curves are smoother for a voidage of 0.5 than for a voidage of 0.8 (somewhat higher than the predicted minimum fluidization voidage of 0.44), 0.5 is chosen for the path along which forces are evaluated and compared. This is referred to as the force path in this study.  To simplify reasoning, a further simulation was conducted with a bubble injected for a duration of 0.2 s for the same column width (0.46 m) as in the Rahman and Campbell (2002) experiment, but with a deeper bed (1.35 m) and taller column (1.75 m). The taller column provides a region in the mid-section of the bed where the electric field varies only horizontally, not vertically, helping to explain the bubble deformation. The magnitude of charge was set to a maximum value (qm= ? 0.36 ?C/kg of solids), and the S-S frictional model was utilized, with the free-slip boundary condition for the solid phase to minimize wall effects and increase the computational speed. The simulation parameters are summarized in Table 3-1.   Figure 3-9 presents the electric field vectors and their magnitudes for t=0. It can be seen that the electric field is stronger close to the walls and the distributor, with reduced magnitude towards the centre of the bed. The electric field only varies appreciably in the horizontal direction between heights 0.5-1.1 m (mid-section), while it varies in both the horizontal and vertical directions close to the distributor. On the other hand, the injected bubble at the distributor experiences a two-directional electric field region before reaching the mid-section of the column. As a result, the bubble deforms prior to entering the mid-section of the bed. To resolve this, the code was modified to change the charge magnitude stepwise from zero to a maximum charge density after a specified time (TE), as in the previous simulations. This was performed for 3 s,  - 38 - with the bed initially uncharged for the first second to diminish initialization effects, and then a bubble was injected over an interval of 0.2 s, initially rising in uncharged particles. At t = 1.98 s, the time at which the charge on particles was subjected to a step change from 0 to a maximum value, the bubble was completely inside the mid-section, so that after this time, the bubble rose in a region of charged particles. It immediately began to deform.  Figure 3-9: Electric field vectors shown with electric field magnitude (V/m) contours for charged bed (qm= ?0.36 ?C/kg of 500 ?m glass beads, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model).    Figure 3-10, shows how the bubble is predicted to deform due to the presence of the electric field at different times. The bubble (red line) is defined where the voidage is equal to 0.8, whereas the vectors represent the electric field forces where the voidage is 0.5. As shown in Figure 3-9, the electric force vectors have similar spatial distribution on the force path, justifying the constant magnitude of electric field in the y-direction for this region. These forces tend to push the particles slightly downward and outward towards the wall (The direction of electrical forces is also visualized around a circle in the bed. See Appendix F for details.). Other applicable forces  - 39 - were calculated on the periphery of the bubble using a MATLAB code based on the MFIX output. The resultant forces are then estimated by interpolation. Different force components in the x- and y- directions are plotted for bubbles 1 and 4 in Figures 3-11 (b) and 3-12 (b).    Figure 3-10: Predicted change in bubble shape, when electric field is turned on, after bubble enters uniform electric field region of 500 ?m charged glass beads (qm= ?0.36 ?C/kg, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model). Vectors show electric field force where voidage = 0.5.  As expected, the dominant forces in the x-direction are the drag and gas pressure forces, opposed by solid pressure and shear forces. In the y-direction, the dominant forces are drag and gas pressure, mainly opposed by gravity in the first stage of shape change. As the bubble deforms further, it rises more quickly and the magnitude of the solid pressure force, which is always larger than the solid shear force component, and the solid shear force increase with increasing distance from the axis of symmetry, while remaining almost constant in the y- direction.    - 40 - The maximum magnitude of the electric force, ~ 0.5 N/kg, is in the x-direction, compared to the maximum drag force of order 1.8 N/kg for the first bubble. On the other hand, the magnitude of the electric force in the y-direction is almost negligible, less than 0.1 N/kg, as shown in Figures 3-11 and 3-12. The net force in the y-direction for bubble 1 shows that for ? < 55o, particles tend to rise since they experience an upward force; beyond that, they tend to descend. Which way they move depends on their previous history, i.e., on their velocity from the previous time step. Similar patterns are seen for bubble 4, but particles move downward sooner compared to bubble 1, at an angle of ~ 35o to the vertical direction, indicating bubble elongation.   Let us return to the main question: why does the bubble shape change? The charge on particles after TE= 1.98 s in the simulation adds a force to the system that must be balanced by changes in the other forces. As seen in Figures 3-11 and 3-12, rebalancing the forces alters all forces in such a way that the net vertical force on the leading edge of the bubble increases, whereas that on the sides decreases, resulting in an elongated bubble that rises more quickly.   The magnitude of throughflow velocity increases as a bubble elongates (Grace and Harrison, 1969; Gera and Gautam, 1993). Therefore, for this case, the throughflow velocities were computed for bubbles 1 and 4. They turned out to be as 0.402 and 0.414 m/s, respectively, values which are 1.7 and 1.8 times the simulation background velocity (0.23 m/s), and 2.0 to 2.1 times the experimental minimum fluidization velocity of ~ 0.2 m/s.   The voidage and granular temperature are also compared around the bubble, when the bubble base is at a height of 0.6 m above the distributor for the uncharged and charged systems. Both quantities decrease at the sides of the bubble, mainly due to the gradual increase in electric field strength towards the walls, which pushes the particles slightly outwards towards the wall, leading to a denser bed in these regions. The increase in solid volume fraction near the walls causes the gas to migrate inwards towards the centre, resulting in a decrease in granular temperature and an increase in solid pressure at the sides of the bubble. This leads to a rebalancing of all forces in such a way that the bubble width decreases and its throughflow increases.   - 41 -  Figure 3-11: Force components along contour where ?g= 0.5 around bubble 1 versus angle from nose in bed of 500 ?m charged glass beads (qm= ?0.36 ?C/kg, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model): (a) x-direction; (b) y-direction.  - 42 -  Figure 3-12: Force components along contour where ?g= 0.5 around bubble 4 versus angle from nose in bed of 500 ?m charged glass beads (qm= ?0.36 ?C/kg, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model): (a) x-direction; (b) y-direction.  - 43 - 3.6. Conclusions Coupled with the Srivastava and Sundaresan (2003) frictional model, the two-fluid CFD model gives predictions which are in reasonable agreement with experimental results of Gidaspow et al. (1983) and Rahman and Campbell (2002) for bubbles rising in fluidized beds of uncharged particles.  The electric force density equation proposed by Melchers (1981) was used for the first time to simulate single bubble rise in a bed of charged particles. The electrostatic charges are predicted to cause bubbles to elongate and rise more quickly, with the extent of elongation dependent on the charge density. These findings apply regardless of which of the two frictional models investigated is assumed. The main reason for bubble elongation was identified by comparing different forces for a case in which the electrical field varies only in the horizontal direction. It is found that the magnitude of electrical force is comparable to other forces, requiring rebalancing of forces in such a way that the bubble width decreases, causing increased throughflow. The electric field causes the voidage of the bed to be lower towards the outer walls, leading to more vertical gas flow near the axis of column.     - 44 - Chapter 4. Influence of Electrostatics on Bubble Pair Interaction 4.1. Introduction The interaction of bubbles plays an important role in determining the distribution of bubble size. It also strongly influences how much gas bypasses as bubbles and through bubbles in the bubbling regime of gas-fluidized beds. In this regime, many bubbles can be interacting/coalescing at any time. However, pair-wise interaction of bubbles is usually the main focus of studies in the literature to simplify the analysis. Based on the observations of previous study (Clift and Grace, 1970), coalescence of obliquely-aligned pairs of bubbles proceeds via the following steps: (1) relative motion to give near vertical alignment of both bubbles along the same vertical axis; (2) acceleration and elongation of the trailing bubble; (3) trailing bubble overtakes the leading one; (4) rupture of the thin film of particles separating the two bubbles.  The interaction of pairs of bubbles in vertical alignment was investigated by Clift and Grace (1970) experimentally and theoretically by approximating the solids flow around the bubbles as potential flow. It was shown that the velocity of a bubble could be approximated by adding the bubble velocity in isolation to the velocity the continuous (dense) phase would have at the position of the nose if the bubble were absent. This postulate predicted the acceleration of the trailing bubble, and the results were in good agreement with experiments, even though the flow around the bubble was approximated with a flow around a cylinder (two dimensions) or sphere (three dimensions). Our simulation results are compared with their experiment in this chapter.   The model proposed by Clift and Grace (1970) was extended to predict the multiple pair-wise interactions between leading and tailing bubbles (Clift and Grace, 1971; 1985). Afterwards, Farrokhalaee (1979) showed that the trailing bubble did not affect the leading bubble significantly and she proposed a simpler model and verified it experimentally. It was found that the simplified model could predict the bubble behaviour with a slight deviation from the more complicated model proposed by Clift and Grace (1971; 1985). The Farrokhalaee (1979) model has been later adopted as one of the closure methods in the Discrete Bubble Model (DBM) to predict the interaction of bubbles (Laverman et al., 2007; Movahedirad et al., 2012).   - 45 - In this chapter, the effect of electrostatics is studied on the interaction of pairs of bubbles in vertical and horizontal alignment. The predictions are compared with those for uncharged and charged particles, with a maximum charge density (qm= ? 0.36 ?C/kg) for mono-sized particles, as reported in the literature.  4.2. Simulations The interaction of bubbles is simulated in a two-dimensional fluidized column filled with 300 ?m glass beads, in both horizontal and vertical alignment. The simulations are based on the previous work of Clift and Grace (1970), who studied the interaction of pairs of bubbles in vertical alignment. The geometry of the system and simulation parameters are presented in Table 4-1 (See Chapter 3 for more information about how these parameters were chosen). The S-S frictional model is used for all cases, since it leads to better predictions when the superficial gas velocity is close to Umf as shown in Chapter 3.  To study the vertical interaction of bubbles, two bubbles are injected at the distributor from a central orifice, 12.8 mm wide, separated by a time interval of 0.14 s and each over a duration of 0.1 s, to mimic the size of bubbles in the experimental work of Clift and Grace (1970). The jet velocity, interval between the injections and the duration of each injection were chosen by trial and error, and the orifice size was arbitrarily selected. Note that the experimental bubbles were injected simultaneously at two locations in the bed, but here they are injected one after the other from the distributor to mimic the experimental bubble sizes and locations.  To study the horizontal interaction of bubbles, two bubbles are injected simultaneously from the central and side orifices at the distributor, separated by 175 mm. The orifice size is the same as the vertical case and the duration of injection is 0.2 s. The jet velocities are arbitrarily chosen. The no slip and partial slip (Johnson and Jackson, 1987) boundary conditions are considered for gas and solids respectively, and the walls are assumed to be at zero electric potential (grounded) for the case with charged particles.    - 46 -  4.3. Interaction of Bubbles in Vertical Alignment In the uncharged system, the initial simulated bubble diameters were 72 and 77 mm, close to experimental values of 71 and 78 mm (Clift and Grace, 1970). Afterwards, the same case was simulated for charged particles. Snapshots of interaction of two simulated bubbles in vertical alignment are presented in Figure 4-1 for uncharged and charged particles. Both cases follow the same pattern, with the trailing bubble elongating as it enters the wake of the leading bubble, and Table 4-1: Simulation parameters L  Column height (m)             1.22 W  Column width (m)             0.54 ?t  Initial time step (s)             0.0001 tstop  Simulation time (s)             3 ?0  Initial voidage              0.44 LB  Initial bed height (m)            0.86 ?x, ?y Grid size (mm)             4.0 dp  Particle diameter (?m)            300 ?s  Particle density (kg/m3)            2450 MW  Gas molecular weight (kg/kmol)          29 Tg  Gas temperature (C)            24 ?g  Gas viscosity (kg/m.s)            1.8e-5 Vsup  Superficial velocity (m/s)            0.084 Vj,v  Vertical jet velocities (m/s)           3.4  Vj,h  Horizontal jet velocities (m/s)          4.5 Pinlet  Inlet pressure (Pa)             112880 ?   Angle of internal  friction (degrees)         30  w?   Angle of internal friction at walls (degrees)        11.3 ep  Particle-particle restitution coefficient (-)        0.9 ew  Particle-wall restitution coefficient (-)         0.8 ?   Specularity coefficient (-)           0.05 ?smin  Threshold solid fraction for friction (-)         0.55   ?smax  Maximum solid packing volume fraction (-)       0.6   - 47 - then it expands and the bubbles start to coalesce. During the coalescence, the trailing bubble splits into smaller bubbles, which coalesce with the leading bubble and eventually form a spherical-cap bubble.  (a)  (b) Figure 4-1: Simulated coalescence of two bubbles in vertical alignment in: (a) uncharged; (b) charged particles (dp= 300 ?m, ?s= 2450 kg/m3, Vj,v= 3.4 m/s, Vsup= 0.084 m/s).  The main difference between the uncharged and charged cases lies in the symmetry of bubbles. The bubbles showed greater symmetry for the uncharged particles than for the charged particles. Moreover, the leading bubble in the charged system wobbled, probably due to the change in its wake shape and particles raining was predicted from the roof of the trailing bubble, but this did not lead to splitting, with the curtain of particles moving to one side, as shown in the first snapshot of Figure 4-1(b). Another difference between the two cases is a different pattern of splitting. For the uncharged particles, the trailing bubble splits from the base and the resulting bubbles split again, forming four trailing bubbles. On the other hand, for the charged system, the trailing bubble splits only once, forming two trailing bubbles. This can be due to two- - 48 - dimensional simulation of three-dimensional column even with small width, where the influence of front and back walls are neglected as shown in Li et al. (2010).  What happened experimentally is shown in Figure 4-2 (from Clift and Grace, 1970). The coalescence pattern is similar in the figure to that in the simulation, except for the predicted splitting of the trailing bubble. The final shape of the experimental bubble was not reported by Clift and Grace (1970). Therefore, no qualitative comparison can be made between simulation and experiment at that point.   Figure 4-2: Snapshot of coalescence of two bubbles in vertical alignment (Clift and Grace, 1970) in a two-dimensional column. Grid lines shown have a 25.4 mm spacing. From right to left: t= 0, 0.08, 0.095, 0.11, 0.125 s.   When two bubbles interact in vertical alignment, the velocity of the leading bubble is not significantly affected by the trailing bubble, whereas the trailing bubble accelerates and its velocity increases. The relative motion can be visualized by mapping the coordinates of the bubble nose as in Figure 4.3 for uncharged and charged particles, together with the experimental data of Clift and Grace (1970). The simulation data are synchronized with experiment by assuming t=0, after injection of both bubbles from an orifice. N1 and N2 in this figure refer to the leading and trailing bubble noses respectively, and the slopes of the corresponding lines indicate the bubble velocities. The initial bubble heights differ due to the different method of bubble injection in the simulation and experiment.    - 49 -  Figure 4-3: Nose coordinates of bubbles in vertical alignment versus time (dp= 300 ?m, ?s= 2450 kg/m3, Vj,v= 3.4 m/s, Vsup= 0.084 m/s).  The predicted rising velocity of the leading bubble is close to the experimental velocity, but the predicted velocity of the trailing bubble is greater than experimental values for uncharged particles. Bubbles are predicted to rise more quickly in the bed of charged particles, and the trailing bubble acceleration relative to the leading bubble is predicted to be larger than for the uncharged case. Simulations also indicate that the velocity of the leading bubble is almost unaffected by the presence of the trailing bubble. The acceleration of the trailing bubble is predicted to be larger in the charged system than in the uncharged one. The increase in velocity of bubbles in the charged system is also in line with what was predicted for a single bubble in Chapter 3.   - 50 - The experiments also show that when two bubbles coalesce, the volume of the resultant bubble becomes larger than the sum of the volumes of the initial bubbles before coalescence (Grace and Venta, 1973), since part of the cloud merges into the resultant bubble. This phenomenon is examined here based on our two-dimensional simulations for the uncharged and charged particles. Instead of volumes, areas are compared in the 2-D bed, as presented in Table 4-2.  Table 4-2: Comparison of sum of areas of original bubbles with area of resultant bubble for coalescence of bubbles in vertical alignment            No Charge (m2)    Charged (m2) Initial area of leading bubble       0.0041      0.0052 Initial area of trailing bubble       0.0047      0.0049 Sum of initial areas        0.0088      0.0101 Resultant bubble area from      0.0102 to 0.098    0.0125 to 0.014 formation to near eruption  The simulation results predict that the area of resultant bubble is larger than the sum of the areas of the initial bubbles for both the uncharged and charged cases. Table 4-2 also shows that the bubbles are predicted to be larger in the charged system compared to the uncharged system, implying more bypassing of gas via bubbles. Note that the area of the resultant bubble decreases during its rise in the bed for the uncharged case, while it increases in the charged particle system.  4.4. Interaction of Bubbles in Horizontal Alignment Figure 4-4 shows the trajectory of bubbles injected in horizontal alignment in uncharged and charged particles, when bubbles start in horizontal alignment. In the uncharged case, the two bubbles move side by side, but the side bubble becomes slightly larger, rises more quickly and becomes a leading bubble. The central bubble takes the trailing role and they coalesce. The resultant bubble splits soon afterwards near the upper bed surface.   In the charged case, the electrostatics cause the side bubble to move towards the axis of the column, and it takes the trailing role, while the central bubble becomes the leading bubble. In this case, the coalescence is completed at a lower height than for the uncharged case. Therefore,  - 51 - the leading-trailing role appears to differ, due to electrostatics, for bubbles in horizontal alignment.  Figure 4-4: Coalescence of two bubbles in horizontal alignment in: (a) uncharged; (b) charged particles  (dp= 300 ?m, ?s= 2450 kg/m3, Vj,h= 4.5 m/s, Vsup= 0.084 m/s)  4.5. Conclusions The interaction and coalescence of vertically-aligned pairs of bubbles in charged and uncharged particles are simulated and compared with experimental results of Clift and Grace (1970) for uncharged particles, when bubbles are vertically-aligned. Both cases follow similar patterns, and the predicted rise velocity of bubbles is in good agreement with experimental results. However, the predicted splitting of the trailing bubble in the simulations was not observed in the experiments.   For two bubbles in vertical alignment in a bed containing charged particles, bubble coalescence is predicted to be asymmetric, with the leading bubble wobbling, and particles raining from the roof of the trailing bubble. The resultant bubble is predicted to be larger in the charged particle case than for uncharged particles. On the other hand, for bubbles in horizontal alignment in the  - 52 - charged particles, the side bubble is predicted to migrate towards the axis of the column, with coalescence being completed at a lower height than for uncharged particles.      - 53 - Chapter 5. Effect of Electrostatics on Freely-Bubbling Beds 5.1. Introduction Freely-bubbling is the fluidization flow regime widely used, where electrostatics often cause major operating problems, in gas phase polymerization reactors. In this regime, the bubble size and spatial distribution determine the internal solids circulation and consequently the efficiency of gas and solid contact. Key properties such as heat transfer, mass transfer and reactor efficiency are related to bubble motion and the distribution of gas flow between the bubble and emulsion phases.  The ?two phase theory? assumes that enough gas goes to emulsion phase to keep the particles at the minimum fluidization superficial velocity, and any excess gas beyond that needed for minimum fluidization forms bubbles. Experimental results have shown that this assumption is an oversimplification of the real case, resulting in over-prediction of visible bubble flow. As explained by Grace and Clift (1974), some researchers have attributed the discrepancy to an interstitial dense phase velocity greater than Umf/?mf, while others have attributed it to increased throughflow inside bubbles. Grace and Harrison (1969) showed that the velocity of a bubble in a swarm of bubbles is higher than for isolated bubbles of the same size. Since bubbles tend to enter the wake of others, causing them to elongate, this is expected to increase the throughflow across the trailing bubbles. The true distribution of gas flow between the phases remains unclear. The advent of CFD should shed more light on the flow distribution, but CFD models need extensive validation for their predictions to be quantitatively reliable.  One of the experimental studies used later in this chapter to compare with CFD predictions is that of Laverman et al. (2008), who investigated the hydrodynamics of a two-dimensional fluidized bed by combining Particle Image Velocimetry (PIV) with Digital Image Analysis (DIA) to measure bubble size and solids velocity at different superficial gas velocities. They reported that the combination of the two methods, PIV and DIA, resulted in more accurate measurements of particle velocities.  - 54 - Li et al. (2010) simulated the experiments of Laverman et al. (2008) based on MFIX software and the S-R-O frictional model to investigate the effect of the wall boundary condition on freely-bubbling fluidization. It was found that the specularity coefficient, ,? a measure of wall roughness, influences the expanded bed height, solid velocity profile and bubble diameter. This parameter varies between 0 and 1, with the lower limit corresponding to free slip. The specularity coefficient was found to affect the predictions significantly, while the influences of the particle-wall restitution coefficient on the expanded bed height and solid velocity were small. The bed thickness in the Laverman et al. (2008) experiments was only 0.015 m. Li et al. (2010) in comparing 2D and 3D cases showed that the 3D results were in better agreement with experimental results in terms of bubble diameter and solid velocities.   Previous chapters of this thesis have shown that bubble shape and interaction can be influenced by electrostatic charges. In this chapter, we go one step further and investigate the effect of electrostatic charges on freely-bubbling bed of mono-sized particles, by comparing simulation results for uncharged and charged fluidized beds of 500 ?m glass beads. To compare with experimental results, the geometry of the system is matched with that of Laverman et al. (2008). The S-S frictional model was previously used to see the effect of electrostatics on single bubbles in fluidized bed, and showed a better agreement with experimental results than the S-R-O model (see Chapter 3). Here, both frictional models are employed, and the one which works better is retained to determine how the inclusion of electrostatics modifies the bubble diameter, bubble spatial distribution and solids velocities.  5.2. Simulations A two-dimensional column is simulated in this study as in the experiments performed by Laverman et al. (2008) at superficial gas velocities of 1.5Umf and 2.5Umf. The particle Reynolds number at mentioned superficial gas velocities are about 9 and 15, respectively, and therefore the flow regime is close to laminar. The dimensions and simulation parameters are listed in Table 5-1 (See Chapter 3 for more information about how these parameters were chosen). Simulations were performed for 30 s except where otherwise stated, with the results for the first 2 s discarded in all cases to minimize initialization effects.   - 55 - Mesh independency is usually achieved at a grid size of 10 times the particle diameter (Andrews et al., 2005). To find a mesh-independent result, 5dp, 10dp and 15dp mesh sizes were tested, and time-averaged voidages were plotted against bed height and compared for a superficial gas velocity of 1.5Umf (See Appendix D for more details). The results were similar, with 10dp then being chosen as the mesh size for the remaining simulations. In the CFD literature, the bubble boundary is commonly defined as the locus of where the voidage is 0.8 or 0.85 (e.g. Gidaspow et al., 1983; Kuipers et al., 1991). In the current study, 0.8 is chosen as the boundary as in previous chapters. The areas and centroids of bubbles were analyzed by generating threshold images of voidage > 0.8 using Paraview software (www.paraview.org) and converting images to binary with Image J software (http://rsbweb.nih.gov/ij/). Area-equivalent bubble diameters were calculated as4 /bD A ?? and averaged over 30 mm height intervals.                                - 56 - 5.3. Frictional Model As discussed in Chapter 2, two frictional models (S-R-O and S-S) are available in MFIX. There are several studies in the literature that compare these two models in predicting bed expansion and the relative height of fluctuations in the freely-bubbling and slug flow regimes (e.g. Reuge et al., 2008), bin discharge flow rate and granular flow in a shear cell (Benyahia, 2008) and single bubbles in uncharged and charged systems (Chapter 3). In each of these studies, the S-S model works better than the S-R-O model. However, there have been no published studies comparing Table 5-1: Simulation parameters (two-dimensional column) L  Column height (m)            0.7 W  Column width (m)            0.3 ?t  Initial time step (s)            0.00005 tstop  Simulation time (s)            30 ?0  Initial voidage             0.4 LB  Initial bed height (m)           0.3 ?x, ?y Grid size (mm)            5.0 dp  Particle diameter (?m)           485 ?s  Particle density (kg/m3)           2500 MW  Gas molecular weight (kg/kmol)         29 Tg  Gas temperature (C)           24 ?g  Gas viscosity (kg/m.s)           1.8E-5 Umf  Minimum fluidization velocity (m/s)        0.18 Vsup  Superficial gas velocity (m/s)         0.29, 0.45 (S-S),                  0.27, 0.45 (S-R-O) Pinlet  Inlet pressure (Pa)            1.06E5 ?   Angle of internal  friction (degrees)        30  w?   Angle of internal friction at walls (degrees)       11.3 ep  Particle-particle restitution coefficient (-)       0.95 ew  Particle-wall restitution coefficient (-)        0.8 ?   Specularity coefficient (-)          0.05 ?smin  Threshold solid fraction for friction (-)        0.55   ?smax  Maximum solid packing volume fraction (-)      0.6   - 57 - these frictional models in terms of bubble size and time-averaged solid velocity in freely-bubbling fluidized beds. Hence, both models are compared with experiments in this chapter, with the one which works better chosen for comparing the results of uncharged and charged systems in the later part of the chapter.   5.3.1. Bubble Size Figure 5-1 compares the predicted mean bubble diameter in the bed with uncharged particles as a function of height for the S-R-O and S-S frictional models and superficial gas velocities of 1.5 and 2.5Umf (0.27 and 0.45 m/s) with the experimental results of Laverman et al. (2008) (Check Appendix G to see the spread of bubble diameters at each height). The S-S model is seen to under-predict the bubble diameter, while the S-R-O model under- and over-predicts the bubble size at lower and upper sections of the bed, respectively. These patterns are seen for both superficial gas velocities. Note that the S-S model requires a higher superficial gas velocity of 1.6Umf  (0.29 m/s) than the experiment to match the bubble size determined experimentally.  While the bubble diameters predicted by the S-R-O model are closer to the experimental results at both superficial gas velocities, this may only be due to the choice of parameters in the S-S model. The sensitivity of the S-S model to mins? , the minimum threshold voidage to account for the solid frictional stress, was tested for a superficial gas velocity of 1.6Umf. Three mins? values, 0.55, 0.56 and 0.58, were tested. The results in Figure 5-2 show an increase in predicted bubble size with increasingmins? . These results lead to two questions: ?Why does the S-S model predict a smaller mean bubble diameter than the S-R-O model?? and ?Why does an increase in the minimum threshold voidage in the S-S model lead to larger bubbles?? The answers to the two questions are revealed in Section 5.3.3 below.  - 58 -  Figure 5-1: Comparison of bubble diameter predicted by S-R-O and S-S frictional models with experiemental data (Laverman et al., 2008) (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S) and 0.45 m/s (2.5Umf, S-R-O and S-S).   Figure 5-2: Sensitivity of S-S model to mins? (minimum threshold of solid volume fraction to account for the frictional stress) (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.29 m/s).  - 59 - 5.3.2. Time-Average Vertical Solid Velocity  The predicted time-average vertical solid velocities with the two frictional models are compared with the experimental results of Laverman et al. (2008) for superficial gas velocities of 1.5 and 2.5Umf at a height of 100 mm in Figure 5-3. Figure 5-3 (a) shows that at Vsup= 1.5Umf, the S-R-O frictional model overpredicts the vertical solid velocity compared to S-S model. On the other hand, Figure 5-3 (b) presents results for Vsup= 2.5Umf. The results from both models are approximately of the same order in the middle section of the column, with the S-R-O model overpredicting the solids downflow close to the walls more than the S-S model.   Overall, the S-S model predicts a velocity range closer to the experimental results. However, the velocity profiles of S-R-O are in better agreement with experiemental observation. Overprediction of solid velocities in a 2D simulation was also reported by Li et al. (2010), who simulated the same experimental setup of Laverman et al. (2008), based on the S-R-O model. In their work, it was shown that 3D simulation results were closer to experimental values, suggesting that the assumption of two-dimensional flow for a column of thickness 15 mm (Laverman et al., 2008) may not be accurate.     - 60 -    Figure 5-3: Comparison of horizontal profiles of vertical solid velocity predicted based on S-R-O and S-S frictional models for particles of diameter 485 ?m and density 2500 kg/m3 at y= 100 mm for: (a) Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S); (b) Vsup= 0.45 m/s (2.5Umf, S-R-O and S-S).  - 61 - 5.3.3. Comparison of Frictional Models and Final Choice  The results above show that the predicted bubble diameter is smaller for the S-S model than for the S-R-O frictional model. In addition, the time-average vertical solid velocity is lower for the S-S model. What is the main difference between these two models that leads to these differences?  The answer to this question lies in the non-zero stress in the S-S model compared with the zero stress in the S-R-O model, when the solid volume fraction falls between the minimum threshold to account the frictional stress, mins? , and the maximum solid packing voidage, maxs? . In other words, the frictional forces are larger for the S-S model. On the other hand the solid pressure, which is the combination of frictional and collisional terms, dictates how closely particles can be packed. In dense beds, the frictional stress dominates the collisional part and since the solid pressure does not allow particles to pack beyond a certain critical value, larger solid pressures result in larger voidage. As a consequence, the emulsion phase voidage predicted by the S-S model always exceeds that from the S-R-O model. The higher voidage implies easier passage of gas through the emulsion phase, explaining why the bubbles predicted by the S-S model are smaller than those predicted by the S-R-O model at the same superficial gas velocity.   As seen in section 5.3.1, increasing mins? , the parameter in the S-S model, increases the bubble size. This phenomenon again goes back to the change in solid pressure and consequent change in the emulsion phase voidage. Increasing mins?  from 0.55 to 0.58 decreases the solid pressure, causing the emulsion phase voidage to decrease from 0.45 to 0.42, close to 1-mins? , implying the bed becomes denser. A denser bed means greater resistance to interstitial gas flow and a consequent decrease in gas exchange from bubbles to the dense phase, leading to larger bubbles. Note that the mins? = 0.5 and maxs?  = 0.65, values chosen arbitrarily by Johnson et al. (1990), were also tried in this study, but failed to predict the formation of bubbles inside the bed.   At this point, one of the two frictional models needs to be selected for the rest of this work. The S-R-O model predicts the bubble diameter closer to experimental data and requires less computational time than the S-S model. In addition, the S-S model needs a higher superficial gas  - 62 - velocity, 1.6Umf instead of 1.5Umf, to reach a comparable level of bubbling. Hence the S-R-O frictional model was chosen for the rest of this work.  5.4. Effect of Electrostatics on Freely-Bubbling Beds The case studies examined above are now conducted for particles with a charge density of ? 0.36 ?C/kg, a value reported in the literature for 500 ?m glass beads (Chen et al., 2007), to investigate the effect of electrostatics on average bubbles size, bubble spatial distribution and time-average solid velocities. The simulation parameters are the same as reported in Table 5-1.  5.4.1. Bubble Size Figure 5-4 plots the time-average bubble diameter against height for uncharged and charged particles for a superficial gas velocity of 1.5 and 2.5Umf. This figure shows that bubble size is predicted to decrease when charges are introduced at both superficial gas velocities over most of the bed height. The number of bubbles counted also decreases by 33% and by 0.25% at 1.5 and 2.5Umf, respectively.  Although no experimental results are currently available for direct comparison with the predictions, results with an external electric field have been reported to lead to a decrease in the number of bubbles and in bubble size (Colver, 1977, Wittmann and Ademoyyenga, 1987, Kleijn van Willigen et al., 2005). However, the external electric field in those studies differed substantially from the system simulated here, where the electric field is created internally by charged particles. Yao et al. (2002) reported a decrease in the standard deviation of pressure fluctuation signals with increasing relative humidity for polyethylene particles, presumably corresponding to a lower charge density. They ascribed this decrease to a decrease in bubble size, although no direct measurement on bubble size was performed. Thus the decrease in charge density of particles likely resulted in a decrease in bubble size in their studies, contrary to our predictions. This discrepancy could be due to the irregular shape and surface roughness of polyethylene resin powders and the wide particle size distribution in their experiments. The increase in relative humidity in their study could also introduce capillary forces, making the particles stickier and harder to fluidize.  - 63 -  Figure 5-4: Comparison of bubble diameter in uncharged and charged particles with experimental data of Laverman et al. (2008) at both velocities (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S) and 0.45 m/s (2.5Umf, S-R-O and S-S).  5.4.2. Bubble Spatial Distribution The bubble population can be characterized by the Cumulative Bubble Number fraction (CBN). The bed width is divided into n columns and the bed height into m rows. Bubble centroids are located at one of these rows and columns at each time step. The cumulative bubble number fraction percentage for each zone is then defined as  CBN fraction percentage for zone (m, n) = (Number of bubbles in that zone for t s) / (Total number of bubbles in the bed for same interval)*100 %  A larger CBN indicates more passing bubbles. Here the bed is arbitrarily divided into 20 columns and 3 rows (y=0-100 mm (Zone 1), y=100-200 mm (Zone 2), y=200-300 mm (Zone 3), and bubbles are counted every 0.02 s between 2 and 30 s. CBN is plotted for the uncharged and  - 64 - charged particle cases at two superficial gas velocities, with the results for zone 2 shown in Figure 5-5.   Figure 5-5 (a) presents CBN profile over the column width for uncharged and charged particles, at a superficial gas velocity of 1.5Umf. As shown in this figure, both cases have two peaks, implying that more bubbles pass through those areas. However, the peaks are closer together for the charged particles. The change in spatial bubble distribution in the charged system is due to more compact regions near the walls because of the outward electrical force acting on the particles, making the gas choose the less resistant path in the middle of the column. The predicted bubble distribution with two peaks, which approach each other as height increases, has also been observed experimentally (e.g. Grace and Harrison, 1968; Werther and Molerus, 1973).  Figure 5-5 (b) shows the predicted CBN at a superficial gas velocity of 2.5Umf. It can be seen that electrostatics barely influenced the rather uniform time-average bubble distribution at this relatively high velocity. The reason for the predicted different trend at two gas velocities is discussed below.         - 65 -    Figure 5-5: Cumulative bubble number fraction percentage in zone 2 (y= 200-300 mm) predicted for uncharged and charged particles of dp= 485 ?m, ?s= 2500 kg/m3 at: (a) Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S); (b) 0.45 m/s (2.5Umf, S-R-O and S-S).  - 66 - 5.4.3. Time-Average Solid Velocity In this section, the effect of electrostatic charges on time-averaged vertical component of solid velocity is investigated for two superficial gas velocities at a height of 100 mm. Figure 5-6 (a) presents the time-averaged vertical solid velocity for Vsup= 1.5Umf. In the uncharged system, the solids are predicted to move primarily upward close to the wall and downward in the middle of the bed, as well as adjacent to the walls. However, in the charged system, solids are predicted to rise moves upward in the middle of the bed and travel downward close to the outer walls. The difference between the maximum and minimum solid velocities is predicted to be greater in the charged system than for the uncharged case.  On the other hand, as presented in Figure 5-6 (b) at a superficial gas velocity of 2.5Umf, particles in the uncharged system move upward from x ~ 15 mm to 240 mm and downward close to the walls. The profile is asymmetric with a maximum and minimum on the left and right sides of the bed, respectively. In the charged system, the solids velocity starts from zero, and particles move downward on the left side until x= 275 mm and then upward. Moreover, the average solids velocity is predicted to be lower in the charged system compared to the uncharged one at this height.   Overall, comparison of time-average predicted vertical solid velocities with experiments shows that at Vsup= 1.5Umf , the solid velocities are over-predicted in both the uncharged and charged systems, while at Vsup= 2.5Umf, the solid velocities are in reasonable agreement with experiments. The asymmetry of results at both superficial gas velocities can be due to the limited duration of time-averaging and/or to numerical round-off errors. However, it is important to note that a much longer duration of time-averaging, up to 120 s instead of 30 s, was also applied for the uncharged particles, with no significant improvement in symmetry.        - 67 -     Figure 5-6: Comparison of vertical solid velocity in uncharged and charged particles with experimental data of Laverman et al. (2008) for particles of diameter 485 ?m and density 2500 kg/m3 at y= 100 mm: (a) Vsup= 0.27 m/s (1.5Umf); (b) Vsup= 0.45 m/s (2.5Umf).  - 68 - 5.5. Connection between Results Some of the above findings warrant further investigation, such as why bubbles become smaller in the charged system, and why the spatial distribution of bubbles is not affected significantly by electrostatics for Vsup= 2.5Umf.  The first question can be answered by considering how gas is divided between bubbles and the emulsion phase. Ge and Gb_th are defined as the gas flowrates in the emulsion and bubble phases, respectively. Gb_th consists of two parts: the gas which is displaced with bubbles (Gb) and the throughflow relative to the bubbles (Gth). The total dilute phase flow rate is then Gt = Gb_th + Ge. The gas flow rate is calculated as ?g Vg dx in each computational cell, and if the voidage in that cell is greater or equal to 0.8, the cell is inside a bubble; otherwise it is assigned to the emulsion phase. The sums of flow rates in the emulsion and bubble cells at each cross section give Ge and Gb_th, respectively.   The time-averaged Ge /Gt ratio is calculated at a height of 150 mm (i.e., initial bed height/2) for a superficial gas velocity of 1.5Umf, resulting in 0.82 and 0.85 for the uncharged and charged cases. These values indicate that somewhat more gas goes through the emulsion phase in the charged system, consistent with the decrease in bubble size and frequency predicted in the previous simulations (see Section 5.4.1). The extent of bubble diameter decrease of approximately 12%, however, is more than the 3% increase in the emulsion phase flow. Therefore, the distribution of gas between the bubble and the throughflow should be further analyzed to find a more convincing answer to the question. Finding Gb and Gth is time-consuming, since the velocity of bubbles passing the height in question is required.   However, there is an easier path which compares the bubble height-to-width ratio for uncharged and charged particles as presented in Figure 5-7. This figure shows bubble elongation in the central part of the bed at a superficial gas velocity of 1.5Umf, indicating an increase in throughflow. An increase in throughflow suggests that more gas passes vertically through bubbles, not carried as bubbles, and this can be another cause for the decrease in bubble size.  - 69 - Moreover, it is worth noting that the vertical component of bubble rise velocity passing between x= 100 and 200 mm, the domain in which bubble elongation is observed in Figure 5-7, is found to be 0.47 and 0.61 m/s in the uncharged and charged systems, respectively. These values are averaged over 20 bubbles, and they indicate an increase in bubble rise velocity in the charged system compared to the uncharged case. Note that the average bubble size decreases in the charged system, but bubble elongation leads to a higher bubble rise velocity in this part of the bed. This is in line with previous work of Grace and Harrison (1967), who found that elongated bubbles rise at higher velocity.  Figure 5-7: Comparison of bubble height/bubble width versus x in bed with uncharged and charged particles at height 150 mm (Vsup= 0.27 m/s (1.5Umf), dp= 485 ?m, ?s= 2500 kg/m3).  The second question was about the impact of electrostatic charges on the spatial distribution of bubbles in the charged system at a superficial gas velocity of 2.5Umf. For the lower superficial gas velocity of 1.5Umf, the average bubble size is predicted to decrease along with a change in spatial bubble distribution, raising the question of why the trend is not the same for both  - 70 - superficial gas velocities. The answer to these questions is related to the strength of electric field for both cases.  Note that the bed voidage is lower at a lower superficial gas velocity, and therefore if the charge density on particles remains constant, the charge density per volume of bed would be higher for lower superficial velocity, indicating stronger electric field for this case. Stronger electric field and lower vertical momentum of particles at lower gas superficial velocity push particles toward the walls, leading to the generation of a larger compact zone close to the walls at this velocity. Thus, the gas prefers to go more towards the centre of the bed at lower superficial gas velocity due to less resistance in that area, resulting in the more prominent change in bubble spatial distribution.  5.6. Conclusions The two frictional models, S-R-O and S-S, are compared in this chapter in terms of bubble diameter and time-average solid velocity with experimental data of Laverman et al. (2008) at superficial gas velocities of 1.5 and 2.5Umf. The S-R-O model gives better overall agreement with experimental data in the bubbling regime, and therefore, is used for further simulations. The effects of electrostatics are then investigated based on bubble diameter and spatial distribution, as well as time-average solid velocity, in the same system at the same superficial gas velocities, comparing the results for uncharged and charged particles.   The simulations at a superficial velocity of 1.5Umf, indicate that with charged particles, bubble size and bubble frequency decrease and bubbles tend to rise more towards the axis of the column relative to uncharged particles. The change in bubble size at 1.5Umf in the charged particles is possibly caused by more gas passing through the emulsion phase and an increase in throughflow across bubbles, leading to elongation of bubbles in the central part of the bed and consequently resulting in an increase in bubble rise velocity. At a superficial gas velocity of 2.5Umf, a decrease in average bubble size is predicted, but bubble frequency and distribution are not influenced significantly by electrostatics because of the decrease in intensity of the electric field for deeper beds, with the same particle charge density assumed.   - 71 - Chapter 6. Experimental Tests for Single Bubbles 6.1. Introduction The simulation results predicted that single bubbles elongate in bed of uniformly charged particles compared to uncharged particles, with the degree of elongation increasing with increasing charge density. In this chapter, the simulation predictions are tested by comparing the single bubble shape in particles with different magnitudes of charge densities, achieved by fluidizing the bed at different superficial gas velocities. This is the first time that the electrostatics theory has been tested, and therefore it could play an important role in building confidence for applying electrical governing equations to the simulation of fluidized beds containing charged particles.   6.2. Experimental Setup The setup built for this work consists of a ?two-dimensional? Plexiglas column, a pressure vessel and a solenoid valve for injection of single bubbles, and a novel Faraday cup device. The setup is illustrated schematically in Figure 6-1.  The two-dimensional column, with cross section of 280 mm ? 12.7 mm and 965 mm height, is equipped with a perforated copper distributor, designed for operation close to minimum fluidization. This distributor has a central 12.7 mm square orifice, as well as 18 holes, each of diameter 1 mm, distributed uniformly in a row symmetrically on either side of the central orifice. Copper mesh was also deployed on top of the distributor plate to prevent the particles from falling into the windbox. The inner part of the side walls of the column is covered with copper tape, electrically connected to the distributor. By grounding the distributor, side walls and distributor all have zero potential, matching the electrical boundary conditions applied in the simulations carried out in the earlier chapters of this thesis.   - 72 -  Figure 6-1: Schematic of experimental setup            N2  tank Rotameter Electrometer Pressure vessel PT Faraday cups  Grounding line from distributor PC Fluidized bed column Solenoid valve Light Camera PT  - 73 - The windbox of the column was designed to have two chambers, as illustrated in Figure 6-2. The inner reservoir (box) is connected to a pressure vessel of volume 1 L via a solenoid valve, and it is used for bubble injection. The outer chamber (main windbox) is connected to a nitrogen tank, which supplies the fluidizing gas.   Figure 6-2: Windbox with inner reservoir (box) for bubble injection The custom-made novel Faraday cup device is mounted on the back wall of the column, as shown in Figure 6-3. This device is used to measure the charge density of particles, increasing the measurement accuracy by reducing the handling effect by allowing particles to flow directly into the inner cup. It consists of inner and outer copper cups separated by a Teflon spacer, which acts as an insulator. The inner cup is placed at 45o to the vertical, and it is made of two separable pieces screwed together. This cup is connected to the electrometer via a triax cable. Particles are collected in this cup by raising the gate shaped handle, placed on the outside end of the channel that passes through the column?s back wall. The outer cup is made of two copper pieces, secured with latches, which can be opened to gain access to the inner cup. Design sketches of this device are provided in Appendix H.  - 74 -  Figure 6-3: Faraday cup device mounted at back of column: (a) outer cup is closed, (b) outer cup is open.  A Keithley Model 6514 electrometer is used to measure the electrical charge on the collected particles. This device can measure charges in the range of 10-13 to 20?10-6 C. The pressures inside the pressure vessel and across the bed are measured with pressure transducers, model PX139-030A 4V and 142 PC 01D, respectively. Labview software is used to read/record the pressure signals at a frequency of 100 Hz and to control the solenoid valve.   6.3. Gas and Particles Nitrogen (99.998% Purity) is used as the fluidizing gas to avoid the effect of humidity. Glass beads are employed as an ideal particulate system, due to their smooth surface and nearly spherical shape. The batch used here was provided by Potters Industries, and particles were sieved to obtain a narrow size range between 500 to 600 ?m. The sieved particles were further analyzed by Mastersizer 2000 and their size distribution is shown in Figure 6-4. The particle surface-area weighted mean diameter and volume weighted mean diameter are 518 and 530 ?m, respectively.    - 75 -  Figure 6-4: Particle size distribution for glass beads particles  The density of the particles, measured with water, two measuring cylinders and a scale, is 2250 kg/m3, by assuming non-porous glass bead particles. The minimum fluidization velocity of particles Umf was experimentally determined to be ~ 0.3 m/s by plotting the bed pressure drop versus superficial gas velocity.  6.4. Experimental Method The initial experimental objective was to inject a single bubble into beds of uncharged and charged particles to compare with the simulation predictions. However, the experiments showed that particles always carried a minimum charge, depending on how they were handled previously. We tried to reduce the magnitude of this minimum charge by heating the particles in the oven at 100oC for a day. Although the charge density on the particles was reduced, it was found that the particles recharged as a result of being poured into the column, making it impossible to obtain truly ?uncharged? particles. However, fluidizing the bed at different superficial gas velocities produced different levels of charging of particles inside the bed. Therefore the objective of the experiments was revised to comparing the shape of single bubbles injected in beds of particles of different charge levels, obtained by fluidizing at a specific gas velocity and then lowering the gas velocity quickly to Umf. In this manner, the bubble shape in particles with different charge densities could be investigated.  - 76 - The static bed height was 0.42 m for all experiments. Superficial gas velocities of 1.3Umf and 1.8Umf were used to charge the bed. Single bubbles were injected into the bed through the square orifice in the distributor, by opening the solenoid valve for 1 s, with the pressure in the pressure vessel set at 35.8 kPa. More charges are generated when the bed is fluidized at higher velocity, and therefore the bed with a history of fluidization at superficial gas velocities of 1.3Umf and 1.8Umf are referred to below as ?low-charge? and ?high-charge? systems, respectively.   At the beginning of each set of runs, the bed was fluidized for an hour at a pre-set superficial gas velocity to produce low- or high-charge system of particles, which reaches a steady-state. Then the superficial gas velocity was reduced quickly to Umf, and the first bubble was injected. After each injection, the superficial gas velocity was increased to the previous value for 10 minutes and then reduced to Umf for the next injection. Thereafter, the cycle of 10 minutes between injections was repeated, and each bubble rise was recorded with EO-1312c CMOS color USB camera at a rate of 39 frame/s. To obtain excellent contrast, i.e., helping to distinguish between bubbles and the surrounding emulsion phase, the lamp was placed behind the column, as indicated in Figure 6.1.  After the last bubble injection of each day, the bed was fluidized for 10-20 minutes at a pre-set superficial velocity, and the charge density was measured with the Faraday cup device (see above) mounted on the back wall of the column, while the bed was kept fluidized. Particles residing at a height of ~ 210 mm were drained from the bed into the inner-cup of this device, and their net charge was measured by the electrometer and recorded manually. Next, the outer-cup was removed and the collected particles inside the inner-cup were weighted on a sensitive scale. The charge density of particles was calculated, by dividing the net charge by the weight of the collected sample. This procedure was repeated two more times to obtain three charge densities in succession, and their average was calculated. After each measurement, the inner- and outer-cups were cleaned by a stream of air to make sure that no particles were left adhering to the walls.   - 77 - To obtain a rough idea of the distribution of charge density in the area between the Faraday cup device and the bed surface, while the bed was still at a pre-set superficial gas velocity, particles were drained from the inner-cup of Faraday cup device, to reduce the bed height by 50 mm. After removing these particles, the particles remaining above the sampling port (which was at y ~ 210 mm) were on average closer to the sampling port. For this new bed height, three samples were taken in succession, with their charge density again measured, as described above. The volume of samples was small enough, that the withdrawal of each sample did not influence the bed height appreciably. This procedure, removing particles and taking samples, was repeated four more times, and three samples were collected for each new bed height.  The process of collecting samples and recording the data took 3-4 minutes, whereas particle draining took 1-2 minutes. The experimental results showed that the charge density of particles dissipated quickly, after reducing superficial gas velocity. Therefore, the bed was kept fluidized at the pre-set superficial gas velocity, to make sure that the measured charge densities remained close to their values during the bubble injections. When the superficial velocity was impulsively dropped in order to inject a bubble, the bubble injection process required 1-2 minutes, this time being needed to adjust the gas velocity to Umf and to allow the bed surface to become horizontal at its new level. After injection, the bubble residence time in the bed was only of the order of a few seconds.              - 78 - 6.5. Experimental Results Figures 6-4 and 6-5 show typical snapshots of bubbles rising in the bed previously fluidized at superficial gas velocities of 1.3Umf and 1.8Umf. Frames where a bubble is located in the lower part of the bed are not included in these figures, because of the Faraday cup device, which looks like a circle from the front, prevents bubbles from being seen clearly with the backlighting. However, those frames in which part of bubbles were obscured by this device (First 8, 4, 7 frames of Figs. 6-4, 6-5a and 6-5b, respectively) are included to show the elongation of bubbles in the lower part of the bed.  Figure 6-5 for the low-charge system shows that the bubble elongated in the lower half of the bed and becomes more circular in the middle part, afterwards flattening close to the bed surface, with particles raining from the roof then being observed. Two patterns for bubble rise were observed for the high-charge system as presented in Figure 6-6. In Figure 6-6 (a) pattern 1 (observed 60% of the time) the bubble adopts a kidney shape in the middle part of the bed along with particles raining from the roof that sometimes led to complete bubble splitting, although not in the case shown here. On the other hand, in Figure 6-6 (b) pattern 2 (40% of the time), bubble elongation followed by bubble flattening was observed in the lower and middle parts of the bed, and close to the bed surface, respectively.  Similar patterns of bubble elongation and flattening were observed in both the low- and high-charge systems, with the main difference between them being the extent of particle raining, as well as the bubble size and shape. As shown in these figures, bubbles deformed in different regions of the bed. Thus, to compare bubbles quantitatively, the bubble diameter and bubble height-to-width ratios are compared for the low- and high-charge systems, above the area obscured by blockage of backlighting by Faraday cup device. For each case, 3 to 5 frames which show the bubble in the region above the cup were analyzed and averaged. 15 bubbles were analyzed in this manner for both the low-charge and high-charge systems, with the results presented in Table 6-1. Note that the 30 bubbles analyzed here were collected after many unsuccessful injections, in which bubbles either rose toward the walls or became unstable due to  - 79 - coalescence with smaller bubbles or voids that occasionally appeared in vicinity of walls or distributor due to lack of perfect uniformity of gas distribution. The bubble diameter and aspect ratio for those cases in which particle raining seemed to split bubbles are, based on the average frontal areas of the two split parts and the ratio of average bubble height of the two parts to the sum of widths of the two parts, respectively.  Table 6-1: Comparison of average bubble diameter and aspect ratio for beds previously fluidized at different superficial gas velocities (Data are reported in the form of the average value ? 90% confidence interval.) Superficial gas velocitiy     1.3Umf      1.8Umf Charge density (?C/kg)     ?0.98 ? 0.21     ?2.03?0.66 Bubble diameter (mm)     170 ? 3.5      149 ? 5.7 Bubble height to bubble width ratio  0.71 ? 0.06     0.79 ? 0.06   This table shows that the bubbles were found to have a smaller size and to be elongated slightly more in the high-charge system than in the low-charge one.   - 80 -  Figure 6-5: Bubble rise in bed previously fluidized at 1.3Umf (dp= 518 ?m, ?s= 2500 kg/m3). Successive frames were taken at 39 frames/s. Shadow corresponds to Faraday cup sampling device.  - 81 -  (a)  (b) Figure 6-6: Two patterns of bubble rise in bed previously fluidized at 1.8Umf (dp= 518 ?m, ?s= 2500 kg/m3). Successive frames taken at 39 frames/s. (a) pattern 1, (b) pattern 2. Shadow corresponds to Faraday cup sampling device.  - 82 - 6.6. Comparison of Experimental Results with Simulation Predictions The experimental observations in this chapter for single bubbles in particles previously fluidized at two different superficial gas velocities, show the same pattern of bubble elongation/flattening as bubbles rise towards the bed surface in low-charge particles, and two patterns of behaviour: bubble elongation/flattening (40% of the time) and kidney-shaped bubbles with particles raining from the roof (60% of the time) in high-charge system. On the other hand in Chapter 3, it was predicted that the single bubbles elongate in a bed with uniformly charged particles compared to uncharged particles. Both simulation and experimental results indicate that electrostatics influence bubble shape, but they predict different deformation patterns. Moreover, the sensitivity analysis on charge density in Section 3.4.1 showed an increase in bubble elongation with increasing charge density. Therefore, if experimental low- and high-charge densities are used in the simulation, the elongation/flattening pattern will not be predicted.  One of the sources of this discrepancy may come from the assumption of uniform charge density in the simulation. To test this, charge densities were measured for particles between the Faraday cup device and bed surface, as described in section 6.4. The results are shown in Table 6-2. The data reported under ?qday 1? to ?qday 3?columns are averages of three measurements in succession, for each case. Moreover, the data under ?qaverage? column present the final average of the data collected over the three days at each height. As seen in this table, the final average charge densities of approximately ?1 and ?2 ?C/kg were measured at the Faraday cup height (i.e., y=210 mm) for low- and high-charge particles, respectively. In both cases, the absolute magnitude of the final average charge densities decreases as bed height decreases, until it reaches a nearly neutral region and finally positive charges are detected in the lowest bed height. In the bed with low-charge particles, the charge density sign switches from negative to positive at height of 0.35 m, while in high-charge system, the sign switches at height of 0.27 m, corresponding to particles closer to the bed surface.      - 83 - Table 6-2: The charge density distribution in beds of low- and high-charge system (dp= 518 ?m, ?s= 2500 kg/m3). Data points are reported as average value ? 90% confidence interval. Bed height (m)  qday 1(?C/kg)  qday 2(?C/kg)  qday 3(?C/kg)  qaverage(?C/kg)          Low-charge system 0.42     ?0.83   ?1.23   ?0.88   ?0.98 ? 0.21 0.37     ?0.95   ?1.17   ?0.60    ?0.91? 0.27 0.32     0.48    ?1.0    0.70    0.06 ? 0.87 0.27     2.18    ?1.02   ?0.19   0.32 ? 1.6 0.23     1.79    0.28    0.26    0.78? 0.83          High-charge system 0.42     ?1.50   ?2.50   ?2.52   ?2.17 ? 0.55 0.37     ?0.70   ?1.50   ?1.88    ?1.36 ? 0.57 0.32     ?0.65   ?2.14    ?2.58   ?1.79 ? 0.96 0.27     ?0.10    ?1.65   ?1.97   ?1.24 ? 0.95 0.22     0.02    0.12    0.15    0.097 ? 0.063  Although these results are approximate, they indicate that there is a distribution of charge density in the system in the vertical direction, accompanied by a change in the sign of the charge. The distribution of charge density differs in low- and high-charge systems, indicating that the saturation charge density is a function of the superficial gas velocity. Thus, the axial distribution of charge density is likely major source of the lack of agreement between the experiments and the CFD simulations. It is interesting to mention that this is the first time that bipolar distribution of charge has been detected in mono-sized particles. Non-uniform charge distributions have been reported for polymer particles with a particle size distribution (e.g. Moughrabiah et al., 2009; 2012). However, for glass beads no one has reported this phenomenon so far, and, even more surprisingly in a system of glass beads fluidized with air in a steel column, the particles were all negatively charged (Moughrabiah et al., 2009).  The different charging behaviour in our system was probably due to the effect of Plexiglas walls and the velocity of collision between wall and particles, since particle-wall collisions are likely to be the major source of charge generation in our system.   - 84 - As a result of this factor to compare simulation and experimental results, another approach is taken. Four cases are simulated with the same experimental geometry, but with different magnitude and distribution of charge densities as follows: (a) uncharged particles; (b) uniformly charged particles with charge density of ?1 ?C/kg; (c) three horizontal layers of particles with charge densities defined in Table 6-2; and (d) three horizontal layers of particles, with charge densities as defined in Table 6-2. In these simulations, the same model parameters were used as reported in Chapter 3 (see Table 3-1), with the S-S frictional model. The jet velocity was 7 m/s and charge densities were chosen arbitrarily, within the order of magnitude of experimentally measured charge densities, in order to have an identical average charge density in the bed for all charged cases.  Table 6-3: Charge densities at different parts of the bed        qdown (?C/kg)   qmiddle(?C/kg)   qup (?C/kg)         (y=0-0.14 m)   (y=0.14-0.28 m)  (y=0.28-0.47 m) Case (c)       ? 1.7    ? 1.0    ? 0.3 Case (d)       ? 2.3    ? 1.0    + 0.3  Figure 6-7 shows the simulation results for these four cases. For uncharged particles (Case a), the bubble is stable and adopts a spherical-cap shape. When particles are uniformly charged with qm= ?1 ?C/kg (Case b), the bubble elongates and, takes a bullet shape. For Case (c) with three horizontal layers of different charge density, the bubble elongates in the lower part of the bed, while the area above the bubble nose becomes unstable as it passes through zones of different charge density, and, as a result, a void appears above the bubble nose. At higher elevations the bubble and void gradually merge, and a spherical-cap shape bubble finally resulted close to the upper bed surface which is flat. On the other hand, for case d, the bubble becomes unstable and particles raining from an unstable roof tend to split the bubble into several parts. However, as the bubble rises, different parts tend to coalesce again into a stable spherical-cap bubble which gradually flattens in the upper part of the bed.   - 85 -  Figure 6-7: Effect of charge density distribution on single bubbles for: (a) uncharged particles, (b) uniformly charged particles with charge density of ?1 ?C/kg, (c), (d) three horizontal layers of particles with charge densities defined in Table 6-3.  Overall, these cases show that both the magnitude and spatial distribution of charge density play important roles in determining the bubble shape. As a result, comparison between simulation and experiment (the main objective of this chapter) requires accurate experimental charge spatial distribution data and more experimental runs to derive statistically-meaningful average bubble size and shape data. This is left for further studies. However, the basic question of ?whether electrostatics can modify bubble shape?? is answered positively in this study, although the ?How?? requires further elucidation.  6.7. Conclusions In this chapter, an attempt has been made to compare the single bubble simulation results of Chapter 3 with experimental results obtained by injecting bubbles into the two-dimensional fluidized bed, charged by having been previously fluidized at superficial gas velocities of 1.3Umf  - 86 - and 1.8Umf. Comparison of these cases shows a decrease in bubble size and a small increase in bubble height- to-width ratio in a system with higher charge density, accompanied by an increase in particles raining from the bubble roof.  A new series of model simulations based on the experimental geometry was then performed for three cases with different charge density distributions but identical average charge density in the bed. These simulations predict that bubble shape is function of spatial distribution of charge density. Unstable bubbles and particle raining from the bubble roofs are also predicted for cases with charge distribution. Accurate spatial charge density distribution is required to draw firm conclusions about the agreement between simulations and experimental results.                   - 87 - Chapter 7. Summary, Contribution and Future work 7.1. Summary and Contribution In this thesis, the influence of electrostatics on bubble shape, size, velocity and interaction has been investigated for the first time. Three different cases are considered: single bubbles, pairs of bubbles initially in horizontal and vertical alignment, and a freely bubbling bed. The investigation was performed by using a computational fluid dynamics (CFD) approach and adopting the Two-Fluid Model in MFIX (an open source code from the Department of Energy in the U.S.). The electrical governing equations are implemented in that code based on previous work of Melchers (1981). Simulation results are compared for uncharged and charged particles. It is shown that, even for the system of mono-sized glass beads with a mean diameter of 500 ?m where the charge density is believed to be low, bubbles can be influenced significantly by electrostatic charges.  Comparison of results for single bubbles in uncharged and charged systems indicates that single bubbles tend to be elongated in a bed containing charged particles, leading to an increase in throughflow. These changes originate from the generation of more compact regions close to the containing walls, caused by the outward electrical forces on the charged particles. The change in bubble shape also leads to an increase in rise velocity of bubbles in the charged system. Sensitivity analysis predicts that bubbles elongate more when the charge density of the particles is increased.  The interaction of bubbles in vertical and horizontal alignment is found to differ in uncharged and charged particles. In charged particles, bubbles are predicted to rise faster than for uncharged particles with the leading bubble wobbling and particles raining from the roof of the trailing bubble. The trailing bubble can split during coalescence in both uncharged and charged particles. On the other hand, when bubbles are in horizontal alignment in the charged particle, the side bubble is predicted to migrate towards the axis of the column, and coalescence is completed at a  - 88 - lower height than in the uncharged system. The resultant bubble is also predicted to be larger in the charged system than for the uncharged system for vertically- aligned bubbles.  For a freely-bubbling bed, it is predicted that the bubble size is reduced and bubbles tend to migrate more towards the center of the bed in charged particles compared to the uncharged particle system.   The sensitivity of results to two different frictional models, the Syamlal et al. (S-R-O) and Srivastava and Sundaresan (S-S) models, were studied for single bubbles and freely-bubbling beds. For the cases investigated, it was found that the S-S model works better at superficial gas velocities close to Umf, while the S-R-O model is preferred in the freely-bubbling flow regime.  An experimental setup was built to test the model predictions. A novel Faraday cup device was mounted on the back of the column to directly measure the charge from the bed, minimizing the effect of handling the particles on the charge density. A series of experiments was performed to test the effect of electrostatic charges on single bubble shape and size by injecting single bubbles into the bed of particles given different charge densities, achieved by fluidizing the bed at two superficial velocities equal to 1.3Umf and 1.8Umf. The bubble size and bubble shape were then compared for the low- and high-charge system, in the middle part of the bed and the results showed a decrease in bubble size and an increase in bubble-height-to-bubble-width aspect ratio in the system with a higher charge density, accompanied by increased particle raining from the bubble roof.   Approximate charge densities were also measured roughly above the sampling Faraday cup device across the column?s axis, for the bed fluidized at superficial gas velocities of 1.3Umf and 1.8Umf. The results show that for both superficial gas velocities, the magnitude of the charge density decreased over the height until it reaches a neutral region, and finally positive charges were detected towards the bed surface. The change in sign of charge density occurred at a lower height in the low-charge system compared to the high-charge one. The influence of charge  - 89 - density distribution was also investigated by simulating three cases, and it was shown that non-uniformity from charge density can play an important role in determining bubble shape.   7.2. Recommendation for Future Work  Although an attempt has been made here to test the electro-hydrodynamics model of gas-fluidized beds, more extensive experimental work is required on the measurement of charge density spatial distributions, together with simulations which take into consideration the spatial charge distributions to reach a firm conclusion on the validity of the proposed CFD model.  The electrostatic charge accumulation in a bed of dielectric particles is a function of charge generation and dissipation at each superficial gas velocity. These charges can be transferred between particles, depending on their electrical properties. As a result, the dynamics of charge generation, charge accumulation and charge transfer should be incorporated when undertaking further development of the proposed model.  Mono-sized particles were investigated in this study. However, real fluidized beds almost always contain a wide range of particle sizes. Hence then is a need to extend the work to binary and polydisperse particle systems. Moreover, the model in this thesis was limited to two-dimensional flow; future work should extend the model to three-dimensional flow.       - 90 - References Al-Adel, M., Saville, D., Sundaresan S., 2002a. The effect of static electrification on gas-solid  flows in a vertical riser. Ind. Eng. Chem. Res. 41, 6224-6234. Al-Adel., M. F., 2002b. Role of electrostatics on gas-particle flows in vertical ducts. M.S. Thesis,  Princeton University, Princeton, NJ., USA. Agrawal, K., Loezos, P., Syamlal, M., Sundaresan, S., 2001. The role of meso-scale structures in  rapid gas-solid flows. J. Fluid Mech. 445, 151-185. Andrews, A.T., Loezos, P.N., Sundaresan S., 2005. Course-grid simulation of gas-particle flows  in vertical risers. Ind. Eng. Chem. Res. 44, 6022-6037. Barrett, R., Berry, M., Chan, T., Demmel, J., Donato, J., Dongarra, J., Eijkhout, V., Pozo, R.,  Romine, C., vander Vorst, H., (2006). Templates for the solution of linear systems: building  blocks for iterative methods. SIAM, Philadelphia, PA. Available at http:  //www.netlib.org/templates/templates.pdf. Benyahia, S., 2008. Validation study of two continuum granular frictional flow theories. Ind.  Eng. Chem. Res. 47, 8926-8932. Benyahia, S., Syamlal, M., O?Brien, T.J., 2006. Extension of Hill-Koch-Ladd drag correlation over  all ranges of Reynolds number and solids volume fraction. Powder Technol.162, 166?174. Boland, D., Geldart, D., 1971. Electrostatic charging in gas fluidized beds. Powder Technol 5,  289-297. Buyevich, Y., Yates, J., Cheeseman, D., Wu, K., 1995. 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Frictional-collisional equations of motion for particulate  flows and their application to chutes. J. Fluid Mech. 210, 501-535. Karimipour, S., Pugsley, T., 2011. A critical evaluation of literature correlations for predicting  bubble size and velocity in gas-solid fluidized beds. Powder Technol 205, 1-14. Kleijn van Willigen, F., Van Ommen, J.R., Van Turnhout, J., Van den Bleek, C.M., 2005.  Bubble size reduction in electric-field-enhanced fluidized beds. J. Electrostatics 63, 943- 948. Kuipers, J.A.M., Prins, W., Swaaij van, W.P.M., 1991. Theoretical and experimental bubble  formation at a single orifice in a two-dimensional gas-fluidized bed. Chem. Eng. Sci. 46,  2881?2894. Laverman, A., Roghair I., Annaland, M.V., Kuipers, H., 2008. Investigation into the  hydrodynamics of gas-solid fluidized beds using particle image velocimetry coupled with  digital image analysis. Can. J. Chem. Eng. 86 (3), 523-535. Laverman, J.A., Van sent annaland, M., Kuipers, J.A.M., 2007. Influence of bubble-bubble  interactions on the micro scale circulation patterns in a bubbling gas-solid fluidized bed.  12th. Int. Conf. Fluidization, Vancouver, Canada. Li, T., Grace, J.R., Bi, X.T., 2010. Study of wall boundary condition in numerical simulations of   bubbling fluidized beds. Power Technol., 203, 447-457. Li, T.W., Pougatch K., Salcudean, M., Grecov, D., 2008. Numerical simulation of horizontal jet  penetration in a three-dimensional fluidized bed. Power Technol. 184, 89-99. Lun, C., Savage, S., Jeffrey, D., Chepurniy, N., 1984. Kinetic theories for granular flow: inelastic  particles in Couette flow and slightly inelastic particles in a general flow field. J. Fluid  Mech. 140, 223-256. McMillan, J., Shaffer, F., Gopalan, B., Chew, J., Hrenya, C., Hays, R., Reddy Karri, S. B., Cocco  R., 2013. Particle cluster dynamics during fluidization. Chem. Eng. Sci., In press. Mehrani, P., Bi, H.T., Grace, J.R., 2005. Electrostatic charge generation in gas-solid fluidized  beds. J. Electrostatics 63, 165-173. Melchers, J.R., 1981. Continuum electromechanics. MIT Press, Cambridge, Mass., USA. Moughrabiah W.O., Grace, J.R., Bi, X.T., Effects of pressure, temperature, and gas velocity on  electrostatics in gas-solid fluidized beds, 2009. Ind. Eng. Chem. Res. 48, 320?325.  - 94 - Moughrabiah, W.O., Grace, J.R., Bi, X.T., 2012. Electrostatics in gas?solid fluidized beds for  different particle properties. Chem. Eng. Sci. 75, 198?208. Movahedirad, S., Molaei dehkordi, A., Manaei, N., Deen, N.G., Van sint Annaland, M., Kuipers,  J.A.M, 2012. Bubble size distribution in two dimensional gas-solid fluidized beds. Ind. Eng.  Chem. Res. 51, 6571-6579. Napier, D., 1994. Generation of static electricity in a fluidized bed and in powder conveying.  Proc. 2nd World Congress Particle Technology, Nuremberg, Germany. Passalacque, A. and Marmo, L., 2009. A critical comparison of frictional stress model applied to  the simulation of bubbling fluidized beds. Chem. Eng. Sci. 160, 2795-2806. Patil, D., Van Sint Annaland, M., Kuipers J., 2005. Critical comparison of hydrodynamic models  for gas-solid fluidized beds ? part 1: Bubbling gas-solid fluidized beds operated with a jet.  Chem. Eng. Science 60, 57-72. Rahman, K., Campbell, C., 2002. Particle pressures generated around bubbles in gas fluidized  beds. J. Fluid Mech. 455, 103-127. Reitz, J., Milford, F., Christy, R., 1993. Foundation of electromagnetic theory. Addison-Wesley,  USA. Reuge, N., Cadoret, L., Coufort-Saudejaud, C., Pannala, S., Syamlal, M., Caussat, B., 2008.  Multifluid Eulerian modeling if dense gas-solids fluidized bed hydrodynamics: Influence of  the dissipation parameters. Chem. Eng. Sci. 63, 5540-5551. Robinson D.A., Friedman S.P., 2005. Electrical conductivity and dielectric permittivity of  sphere packings: measurements and modeling of cubic lattices, randomly packed monosize  spheres and multi-size mixtures. Physica A 358, 447-465. Rokkam, R., Fox, R., Muhle, M., 2010. Computational fluid dynamics and electrostatic  modeling of  polymerization fluidized-bed reactors. Powder Technol. 203, 109?124. Rowe, P.N., 1971. Experimental properties of bubbles: Chap. 4 in Davidson, J.F., Harrison, D.,  in Fluidization. Academic Press, New York, U.S.A. Rowe, P.N., Partridge, B.A., 1965. An X-ray study of bubbles in fluidized beds. Chem. Eng.  Res. Des. 43, 157-175. Savage, S., 1998. Analyses of slow high-concentration flows of granular materials. J. Fluid  Mech. 377, 1-26.  - 95 - Schaeffer, D.G.,1987. Instability in the evolution equation describing incompressible granular  flow. J. Diff. Eq.,66, 19-50. Scheffler, T., Wolf, D.E., 2002. Collision rates in charged granular gases. Granular Matter 4,  103-113. Sowinski, A., Salama, F., Mehrani, P., 2009. New technique for electrostatic charge  measurement in gas?solid fluidized beds. J. Electrostatics 67, 568?573. Srivastava, A., Sundaresan, S., 2003. Analysis of a frictional?kinetic model for gas?particle  flow. Powder Technol. 129, 72?85. Syamlal, M., Rogers, W., O?Brien, T., 1993. MFIX Documentation: Theory Guide;  DOE/METC-94/1004 (DE94000087). U.S. Department of Energy: Morgantown, West  Virginia. Syamlal, M., Rogers, W., O?Brien, T., 1998. MFIX Documentation: Numerical Technique;  DOE/MC31346-5824 (DE98002029). U.S. Department of Energy: Morgantown, West  Virginia. van der Hoef, M.A., Boetstra, R., Kuipers, J.A.M, 2005. Lattice Boltzmann simulations of low  Reynolds number flow past mono- and bi-disperse arrays of spheres: results for the  permeability and drag forces. J. Fluid Mech. 528, 233?254. van der Hoef, M.A., van Sint Annaland, M., Deen, N.G., Kuipers, J.A.M., 2008. Numerical  simulation of dense gas-solid fluidized beds: a multiscale modeling strategy. Annu. Rev.  Fluid Mech. 40, 47?70. van der Hoef, M.A., Ye, M., van sint Annaland, M., Andrews, A.T., Sundaresan, S., Kuipers, J.A.M, 2006. Multiscale Modeling of Gas-Fluidized Beds. Adv. Chem. Eng. 31, 65-149.  van Wachem, B.G.M., Schouten, J.C., Van den Bleek, C.M., Krishna, R., Sinclair, J.L., 2004. Comparative analysis of CFD models of dense gas?solid systems. AIChE 47, 1035-1051. van Wachem, B., Sasic, S., 2008. Derivation, simulation and validation of a cohesive particle  flow CFD model. AIChE J. 54, 21?32. Wallis, G., 1969. One dimensional two-phase flow. McGraw-Hill, New York. Wen, C. Y., Yu, Y. H., 1966. Mechanics of Fluidization. Chem. Eng. Prog. Symp. Ser. 62, 100- 111. Werther, J., Molerus, O., 1973. The local structure of gas fluidized beds ? II. The spatial  distribution of bubbles. J. Multiphase flow 1, 123-138.  - 96 - Wittmann, C.V., Ademoyega B.O., 1987. Hydrodynamic changes and chemical reaction in a  transparent two-dimensional cross-flow electrofluidized bed. 1. Experimental results. Ind.  Eng. Chem. Res. 26, 1586-1593. Yang, W., 1998. Fluidization, solids handling, and processing ? industrial applications. Noyes  Publications, Westwood, New Jersey. U.S.A. Yao, L., Bi H.T., Park, A., 2002. Characterization of electrostatic charges in freely bubbling  fluidized beds with dielectric particles. J. Electrostatics 56, 183-197. Zahn, M., Rhee, S., 1984. Electric field effects on the equilibrium and small signal stabilization  of electrofluized beds. IEEE Trans. Ind. Appl.  IA-20, 137-147. Zhou, Y., Ren, C., Wang, J., Yang, Y., Dong, K., 2013. Effect of hydrodynamics behaviour on  electrostatic potential distribution in gas-solid fluidized bed. Powder Technol. 235, 9-17.   - 97 - Appendix A. Challenges Faced in this Project The nature of this project was challenging from the beginning, since there was no previous study in the literature that combines the effect of electrostatics and hydrodynamics through CFD models. However, after two years we found out that there was another group working in parallel with us on the same topic, but they took another more qualitative approach, as discussed in more detail in Chapters 1 and 2.  The model to capture the physics of the problem had to be chosen carefully. DEM was the first suggested approach, but this changed to TFM after one year, since it was felt that DEM itself was not sufficiently developed, and it requires high computational time as well as parameter tuning. Thus TFM was chosen as the most feasible model. The next step was the choice of using black box codes like Fluent, or an open source one like MFIX. The latter was finally chosen, since it is the most developed code for fluidization in the literature, and once learns how it works, everything can be modified without restriction.  MFIX consists of at least 370 inter-connected procedures and modules. The general idea about the code itself and the numerical procedure are well-documented, and the code has a support network which is helpful. The code comes with several tutorials for beginners and those who simply want to make different runs without changing the code. However code modification requires more details about data structure, and the connection between those procedures. It is always the duty of developer to learn by exploring, something that is quite challenging. In this study, code modification was required to incorporate electrical forces to the solid momentum equation.   Another challenge was how to define the electrical forces in granular matter acting as a continuum. The electrical force for two point charge points without existence of any other object can be determined from the Coulomb equation of physics. However, when we have many charged moving dielectric particles of mm size, electrical force definition is required to consider  - 98 - polarization, as well as the Coulomb force. That was another unknown or challenge at the beginning of this work. Exploration revealed that electrical force for continuum as defined by Melchers (1981) (see Chapter 2 for more details) could be adopted for our case study.   Another challenge was the computational time. The simulations without considering electrical forces required long times, and as a result, using computational grid became necessary. Westgrid (Western Canada Research Grid) was used to run the simulations in the Linux environment. Generally speaking, the computational time for 3 s of a single bubble case study with a mesh size of 10dp, when the whole granular temperature PDE and frictional model were considered, required around 1 day. Considering electrical forces increased the computational time to around 3 days. The computational time for 30 s of freely-bubbling bed simulation without considering electrostatics, took about a week. As a result, the computational time was a burden, especially when parameters like bubble size had to be computed to compare with experimental results in Chapters 3 and 4, or when different frictional models were analyzed in Chapters 3 and 5.  Generating a clear single bubble when the bed was fluidized closed to experimental Umf was another challenge, especially when S-S frictional model was employed. mins? and maxs? play important roles in that model in determining the bed voidage, and consequently, the stability of single bubbles. At the beginning of this work, values of 0.5 and 0.6 were tried for mins? and maxs?respectively, leading to an average bed voidage of nearly 0.48. This then required the higher background velocity of around 0.3 m/s compared to experimental Umf (~ 0.2 m/s) to obtain a stable bubble.  The challenges above were faced in the numerical part of this work. The experimental part of the work itself, from building a setup to running experiments, also provided challenges, as well as good experience for someone like me who spent most of her life behind computers. Building a setup took around 1.5 year, and getting the experimental results reported in Chapter 6 took 6 months. The biggest challenge of the experimental work was to generate bubbles which rose vertically. This depends mainly on uniform distribution of gas, which is easily set in models, but  - 99 - not in experiments. Even though the distributor was designed to provide a uniform distribution of gas by producing enough pressure drop across the distributor, and the bed was leveled with great precision, uniform gas distribution was still hard to achieve. The electrostatic charges caused particles to stick to the walls and resulted in the generation of voids close to the distributor and side walls. This led to non-uniform gas distribution and migration of bubbles towards the walls in some cases. In other cases, the voids became bigger during bubble injection and then coalesced with the injected bubble making the bubble unstable. After sometime, it was noticed that a sudden reduction of superficial gas velocity from the preset high value to slightly below Umf and then slightly above Umf could be used as a remedy. It was also noticed that bubbles tended to move towards the walls more in beds, which might be caused by the non-uniform distribution of charge in the bed, when the bed was not fluidized for a long time before bubble injection.   Overall, many unsuccessful bubbles were injected into the bed (i.e. order of 100 to 200) to identify the sweet set of operating parameters to generate stable single bubbles. After that one could generate a good bubble for every 2 or 3 injections, if that was a lucky day. So the 30 bubbles reported in Chapter 6 resulted from 6 months of experiments.    - 100 - Appendix B. Initialization Effect In Chapter 3, a single bubble was injected into the bed of particles fluidized at a velocity close to Umf and the bubble was injected after 1 s of the simulation. However, in Chapter 5, the first 2 s of simulation were discarded for the freely-bubbling regime. The reason for discarding the data before 1 s or 2 s for each case is related to the initialization effect.  At the beginning of simulations, the bed is usually initialized with approximate values of voidage, velocities, pressures and granular temperatures. After several iterations, those field variables are updated based on governing equations. As an example for single bubble simulation in Chapter 3, the bed initial voidage was usually set to 0.5 for smooth convergence at the beginning of the simulation and after several iterations the bed voidage reached 0.4 to 0.43 as a result of the balance of different forces. These changes can influence the injected bubble. The existence of charges on the particles can also introduce some initial variations in voidage, which disappear after a certain time, usually around 1 s, and that explains why a single bubble was injected after 1 s of simulation.  At the beginning of simulation of the freely-bubbling flow regime, different layers of particles became unstable and afterwards bubbles formed. At this stage, the bubble formation was symmetrical, but after approximately 2 s, the bed became unsymmetrical. This is the reason for discarding the first 2 s of simulation in different analysis of Chapter 5 as an initialization effect, since the formation of those unstable layers mainly depends on how the bed is initialized in terms of voidage and gas velocity. For the case studied in Chapter 5, the voidages of 0.4 and 1, and the initial vertical gas velocities of 8 and 4 m/s were used for the dense bed and freeboard initialization, respectively, values which are similar to those used previously by Li et al. (2010).          - 101 - Appendix C. Influence of Specularity Coefficient on Solid Phase Circulation  The effect of specularity coefficient has been studied by Li et al. (2010) previously, and it was found that this parameter can influence the vertical solid velocity significantly. However, the sensitivity of the solid circulation to this parameter was not studied. Here values of 0 and 0.05 are tested for this parameter for uncharged particles fluidized at a superficial gas velocity of 2.5Umf. Time-averaged solid velocities (averaged between time 2 and 30 s) are visualized and compared with the experimental results of Laverman et al. (2008) in Figure C-1.   As presented in Figure C-1 (a) when the specularity coefficient is zero, particles rise in the center of the bed and come down close to the walls. As a result two circulation regions exist on each half of the bed. This pattern is observed, since bubbles rise more in the axis of the bed and therefore push the particles upward in that area.  Figure C-1: Time-averaged solid velocity for (a) specularity coefficient = 0, (b) specularity coefficient = 0.05 (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.45 m/s (2.5Umf, S-R-O)).  - 102 - By increasing the specularity coefficient to 0.05 (the value that used in this work), the two continuous circulating regions become asymmetrical and break into two circulating regions on each half of the bed as shown in Figure C-1 (b). On the right half of the bed, the upper region rotates clockwise and the lower one rotates counter clockwise. The rotation directions are reversed on the left side.   Comparison of results for two cases with experimental results of Laverman et al. (2008) (presented in Figure C-2) shows that the results with specularity coefficient of 0.05 are in better agreement with experimental data. However the predicted circulation pattern is unsymmetrical, whereas the experimental one is symmetrical. Note that the asymmetry of the predicted circulation was also detected, when the solid velocity was averaged between 2 and 120 s.   Figure C-2: Experimental results of Laverman et al. (2008) for superficial gas velocity of 2.5Umf. (dp= 485 ?m, ?s= 2500 kg/m3).   - 103 - Appendix D. Mesh Study The influence of mesh refinements on a single bubble is presented in Figure D-1 for the case studied in section 3.4.1, when particles are uniformly charged, with a charge density of ? 0.36 ?C/kg and when the S-S frictional model was employed.   Figure D-1: Mesh refinement for single bubble case study in section 3.4.1 (qm= ?0.36 ?C/kg, dp= 500 ?m glass beads, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model).   The mesh size in both x- and y- directions are equal. Different values of 0.75, 0.6, 0.5, 0.48, 0.43 mm are considered for mesh size. As can be seen, refining the mesh causes bubble size reduction and accordingly a decrease in the expanded bed height. The results became closer as the mesh was refined further, and the computational time increased accordingly. Finally a mesh size of 5  - 104 - mm = 10dp (i.e. the typical mesh size used in the literature) was chosen as the optimum mesh size, as a tradeoff between accuracy and computational cost.  The effect of mesh refinement was also investigated for the freely-bubbling bed of uncharged particles fluidized at a superficial gas velocity of 1.5Umf in section 5.3.1, with the S-R-O frictional model employed, as presented in Figure D-2. In this figure, the time-average voidage is plotted over bed height for mesh sizes of 2.5, 5, 7.5 mm. It can be seen that the results are close. Moreover, the effect of mesh refinement on bubble diameter was performed previously by Li et al. (2010) for this case study using exactly the same parameters, and a mesh size of 5 mm, i.e. ~ 10dp, was found to be adequate to get a converged solution. As a result, the same mesh size was used here.  Figure D-2: Mesh study for freely-bubbling regime in section 5.3.1 (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.27 m/s).     - 105 - Appendix E. Difference between Voidage of 0.8 and 0.85 as Bubble Boundary Criterion In the literature, a voidage of 0.8 or 0.85 is usually used to define the bubble boundary (e.g. Gidaspow et al., 1983; Kuipers et al., 1991). In this work a voidage of 0.8 was chosen arbitrarily. Figure E-1 presents the bubble defined by both criteria for the case studied in section 3.4.1, when the bed charge density is ? 0.36 ?C/kg. As can be seen in this Figure, the results are close for both cases especially on the bubble side and base, and differ slightly close to the bubble nose.   Figure E-1: Influence of bubble boundary voidage on single bubble for typical voidages of 0.8 and 0.85 (qm= ?0.36 ?C/kg, dp= 500 ?m glass beads, Vsup= 0.23 m/s and Vjet= 14 m/s, S-S frictional model).                  - 106 - Appendix F. Electric Field around a Circle In Chapter 3 electric forces were visualized around a bubble during its deformation as presented in Figure 3-10. Those forces push the particles down around the bubble nose and toward the walls on the bubble side. To have a better understanding about directions of electrical forces, a geometry similar to that used in section 3.5 is considered. Like a bubble, a circle was placed in the middle of the bed width at height 0.8 m, with a diameter of 90 mm (i.e. the maximum width of simulated bubble in section 3.5). The voidage inside the circle was set to 1 (as if there are no particles), and the bed was considered to have a uniform voidage of 0.43. It was assumed that particles in the bed were uniformly charged with a charge density of ? 0.36 ?C/kg. Figure F-1 presents the calculated electric field around this circle.  Figure F-1: Electric field around circle in a bed of 500 ?m particles with a charge density of ? 0.36 ?C/kg  As can be seen, the electric field is mainly horizontal and towards the axis of the column on the sides of the circle, and then it deforms around the circle until it becomes upward and downward on the top and bottom of the circle, respectively. The direction of Coulomb force (qE) is in reverse direction of electric field since particles are negatively charged, consistent with what was presented previously in Figure 3-10. From another perspective, it looks like that electric field is compressing the circle on its sides and pulling it on the top and bottom.     - 107 - Appendix G. Error Bars for Bubble Diameter Bubble diameter has a wide distribution in the freely-bubbling flow regime of fluidization. In Figure 5-1 only the mean values of simulation results were reported. Both the mean value and standard deviation of bubble diameters are reported as a function of bed height for two superficial gas velocities of 1.5Umf and 2.5Umf in Figures G-1 and G-2, respectively. These figures show that the spread of bubble sizes distribution increases with height.   Figure G-1: Mean and standard deviation of predicted bubble diameters (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.27 m/s (1.5Umf, S-R-O), 0.29 m/s (1.6Umf, S-S). Experiemental data reported here are based on Laverman et al. (2008) studies.  - 108 -  Figure G-2: Mean and standard deviation of predicted bubble diameters (dp= 485 ?m, ?s= 2500 kg/m3, Vsup= 0.45 m/s (2.5Umf, S-R-O and S-S). Experiemental data reported here are based on Laverman et al.  (2008) studies.                       - 109 - Appendix H. Design Sketches of Faraday-Cup Device                     Figure H-1: Front and side views of Faraday-cup device   - 110 - Figure H-2: Separate parts of Faraday-cup device 

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