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CFD modeling of wet agglomerate growth in gas-fluidized beds Maturi, Anish 2018

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CFD MODELING OF WET AGGLOMERATE GROWTH IN GAS-FLUIDIZED BEDS  by  Anish Maturi  B.Tech., VIT University, 2015  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  April 2018  © Anish Maturi, 2018 ii  Abstract Oil and gas companies which often face excessive agglomeration when operating Fluidized Bed Reactors (FBRs). In Syncrude Canada’s fluid cokers, agglomeration is highly undesirable because it reduces the product yield and causes significant problems including partial defluidization. In Syncrude Canada’s process, the non-volatiles in bitumen are deposited on the particles and contribute towards agglomeration. There is significant practical knowledge and methods regarding agglomeration. A mathematical model of the agglomeration could contribute significantly to a better understanding and limit the agglomeration problem. This thesis presents a mathematica l model of fluidized beds with liquid injection, with emphasis on the agglomeration process.  Agglomeration takes place via particle coalescence in the presence of a liquid in industrial FBRs. Industrial FBRs are maintained well above the liquid boiling point, and there is significant vaporization. Therefore, the model should account for vaporization. Hence, this work presents a vaporization model followed by an agglomeration model. The model is based on the Kinetic theory of Granular Flows (KTGF) and uses the Population Balance Method (PBM) to solve momentum, heat, and mass transport equations. The simulations were performed using ANSYS Fluent, supplemented by User Defined Functions coded in C Language. Vaporization was modeled, and the influence of temperature and droplet size investigated. The results confirm that boiling is more important than evaporation. Parametric studies on the droplet diameter concluded that, in the absence of agglomeration, vaporization does not vary much with changes in the droplet diameter. They also predicted that agglomeration has a significant impact on the extent of vaporization. iii  The rate of agglomeration calculated by this model was very high, high enough to cause defluidization of the bed. The over-estimation of agglomeration is due to the neglect of agglomerate breakage that is accounted for in this thesis by imposing an artificial limit on the agglomerate diameter. The limit was chosen because less than 1% (wt%) was above 4000 microns . iv  Lay Summary This thesis presents a novel approach to calculating the size distribution of solids in a Fluidized Bed Reactor (FBR). The model presented in Chapter 3 takes into account liquid injection into the reactor, and heat and mass transfer between the various phases in the reactor. Also, the model is mechanistic, and employs few empirical coefficients. Hence, it can be applied to various situations with little or no changes, which makes this model a novel approach. By employing this model with appropriate changes, companies can estimate the size distribution of solids in the reactor at any given time. Because larger diameter solids are undesirable in the oil and gas industry, companies can judge when it is necessary to employ methods which reduce the solid diameter and improve the overall process efficiency. v  Preface I was responsible for developing a mathematical model which could predict the extent of agglomeration in a Fluidized Bed Reactor (FBR) at the University of British Columbia, with the help of my supervisors. The model presented in this thesis is capable of predicting agglomera t ion in FBRs even in the presence of significant vaporization. This model has very few empirica l coefficients, and thus, can be adapted to a wide range of situations when combined with an appropriate breakage model. vi  Table of Contents Table of Contents Abstract...........................................................................................................................................ii Lay Summary ................................................................................................................................iv Preface.............................................................................................................................................v Table of Contents ..........................................................................................................................vi List of Tables ..................................................................................................................................x List of Figures................................................................................................................................xi List of Symbols ............................................................................................................................xiii Acknowledgements  ...................................................................................................................xviii Dedication ....................................................................................................................................xix Chapter 1: Introduction ........................................................................................................... 1 1.1 Background ................................................................................................................. 1 1.2 Motivation ................................................................................................................... 4 1.3 Literature Review........................................................................................................ 4 1.3.1 Experimental Data................................................................................................... 6 1.3.2 Phase Interactions in the Fluidized Bed Reactor .................................................... 7 1.3.2.1 Kinetic Theory of Granular Flows .................................................................. 8 1.3.2.2 Momentum Exchange Coefficient .................................................................. 9 1.3.2.3 Heat Transfer................................................................................................... 9 1.3.2.4 Mass Transfer................................................................................................ 11 1.3.3 Granulation Mechanism ........................................................................................ 13 vii  1.3.4 Solution Methods .................................................................................................. 15 1.3.4.1 Frame of Reference ....................................................................................... 15 1.3.4.2 Population Balance Method .......................................................................... 16 1.4 Objectives.................................................................................................................. 18 1.5 Outline....................................................................................................................... 18 Chapter 2: Theory and Background ..................................................................................... 20 2.1 Background ............................................................................................................... 20 2.2 Conservation Equations ............................................................................................ 21 2.2.1 Conservation of Mass Equations .......................................................................... 21 2.2.2 Conservation of Momentum Equation .................................................................. 22 2.2.3 Conservation of Energy Equations ....................................................................... 24 2.2.4 Kinetic Energy of Collision .................................................................................. 27 2.2.5 Radial Distribution Function................................................................................. 29 2.3 Momentum Exchange ............................................................................................... 30 2.4 Population Balance Method ...................................................................................... 32 2.5 Summary ................................................................................................................... 33 Chapter 3: Model Development............................................................................................. 34 3.1 Model Assumptions .................................................................................................. 34 3.2 Modeling Vaporization ............................................................................................. 37 3.2.1 Model for Heat Transfer ....................................................................................... 38 3.2.1.1 Heat Transfer from Droplets to Gaseous Phase ............................................ 39 3.2.1.2 Heat Transfer from Solid to Gaseous Phase ................................................. 40 3.2.1.3 Heat Transfer from the Solid Phase to Droplets ........................................... 41 viii  3.2.2 Model for Mass Transfer....................................................................................... 41 3.2.2.1 Evaporation from Liquid Phase .................................................................... 42 3.2.2.2 Boiling in the Droplet Phase ......................................................................... 45 3.2.2.3 Mass Transfer from the Droplet to the Film in the Solid Phase  ................... 46 3.2.2.4 Evaporation from the Film ............................................................................ 48 3.2.2.5 Boiling in the film on the Solid particles ...................................................... 50 3.2.2.6 Diffusion of Liquid Film in the Solid Phase ................................................. 50 3.3 Model for Calculating the Rate of Agglomeration ................................................... 51 3.3.1 Physical Probability of Collision .......................................................................... 52 3.3.2 Geometric Probability of Collision ....................................................................... 54 3.3.3 Liquid Distribution in Solid Phase........................................................................ 55 3.4 Summary ................................................................................................................... 56 Chapter 4: Results and Discussion ........................................................................................ 57 4.1 Simulation Parameters .............................................................................................. 57 4.1.1 Material Properties ................................................................................................ 57 4.1.2 Simulation Setup ................................................................................................... 58 4.2 Mesh Selection .......................................................................................................... 61 4.2.1 Mesh Independence Study .................................................................................... 61 4.2.2 Mesh Description .................................................................................................. 63 4.3 Experimental Results ................................................................................................ 65 4.4 Results for Vaporization ........................................................................................... 67 4.4.1 Results for Vaporization at 680C .......................................................................... 67 4.4.2 Effect of Boiling.................................................................................................... 73 ix  4.5 Results for Agglomeration ........................................................................................ 75 4.6 Droplet Diameter Studies.......................................................................................... 80 4.6.1 Effect of Droplet Diameter in the Absence of Agglomeration ............................. 80 4.6.2 Effect of Droplet Diameter on the Extent of Agglomeration................................ 82 4.7 Conclusion ................................................................................................................ 88 Chapter 5: Conclusions and Recommendations for Future Work .................................... 89 5.1 Conclusions ............................................................................................................... 89 5.2 Contributions............................................................................................................. 90 5.3 Limitations and Recommendations for Future Studies............................................. 91 Bibliography ............................................................................................................................ 93 Appendices ....................................................................................................................................98   x  List of Tables Table 4.1 Material Properties........................................................................................................ 57 Table 4.2 Bin Sizes ....................................................................................................................... 61 Table 4.3 Mass Transfer Rates at Different Time Steps ............................................................... 74  xi  List of Figures Figure 1.1 Morales (2013) Setup Side View (left) and Front View (right)  .................................... 7 Figure 1.2 Formation of Liquid Bridge......................................................................................... 14 Figure 1.3 Breakage of Agglomerates .......................................................................................... 15 Figure 3.1 Influence of Contact Angle on the Shape of Liquid Film ........................................... 36 Figure 3.2 Interaction between Phases.......................................................................................... 37 Figure 3.3 Heat Transfer Interactions ........................................................................................... 38 Figure 3.4 Mass Transfer Mechanisms Assumed in the Model.................................................... 42 Figure 4.1 Geometry of the Free Board Section of the FBR ........................................................ 59 Figure 4.2 Pressure Profile along Centre Line of Mid-Plane........................................................ 62 Figure 4.3 Mesh of 64,198 Elements ............................................................................................ 64 Figure 4.4 Morales' Experimental Results Showing the Agglomerate Size Distribution after 45 s (Figure taken from Morales et al. (2016) published in the Canadian Journal of Chemical Engineering and copyright permission obtained from John Wiley and Sons (This image is Copyright © 2016, John Wiley and Sons)) .................................................................................. 66 Figure 4.5 Percentage of Liquid Vaporized for 200 micron Droplets Injected into the FBR....... 68 Figure 4.6 Mean (Time-Averaged) Mass Fraction of Acetone in the Film of the Solid Phase over Time t = 0 s to a) t = 3.5 s b) t = 5.5 s c) t = 12.5 s d) t = 22.5 s at y = 0.075 m .......................... 69 Figure 4.7 Instantaneous Temperature of Solid Phase at Time a) t = 3.25 s b) t = 17.75 s c) t = 29.75 s at y = 0.075 m ................................................................................................................... 71 Figure 4.8 Instantaneous Volume Fraction of Solid Phase at Time a) t = 41 s b) t = 41.25 s c) t = 41.5 s d) t = 41.75 s at y = 0.075 m............................................................................................... 72 Figure 4.9 Comparison of Vaporization Rates for Different FBR Temperatures ......................... 73 xii  Figure 4.10 Effect of Agglomeration on the Extent of Vaporization ........................................... 76 Figure 4.11 Predicted Instantaneous Solid Volume Fraction of Solid Phase at t = 14.125 s at y = 0.075 m.......................................................................................................................................... 78 Figure 4.12 Cumulative Agglomerate Size Distribution at Time t = 14.125 s for 200 micron Droplets ......................................................................................................................................... 79 Figure 4.13 Percentage of Liquid Vaporized as a Function of Time ............................................ 81 Figure 4.14 Effect of Droplet Diameter on the Extent of Vaporization ....................................... 82 Figure 4.15 Cumulative Weight Distribution of Agglomerates at Time t = 3.625 s  .................... 83 Figure 4.16 Cumulative Weight Percentage of Agglomerates at Time t = 4.5 s  .......................... 84 Figure 4.17 Cumulative Weight Percentage of Agglomerates at Time t = 6.5 s  .......................... 85 Figure 4.18 Cumulative Weight Percentage of Agglomerates at Time t = 7 s  ............................. 85 Figure 4.19 Cumulative Weight Percentage of Agglomerates at Time t = 12 s  ........................... 86  xiii  List of Symbols 𝑎(𝑑𝑖 , 𝑑𝑗) Agglomeration kernel between bins “i” and “j”, m3 s-1 𝑎 Radius of film, m 𝑐𝑝 Specific heat of material, J kg-1 K-1 𝑑 Diameter, m 𝑑𝑖𝑗  Equivalent diameter, m 𝑑𝑙_𝑓𝑖𝑙𝑚  Equivalent diameter of liquid film, m 𝑒 Coefficient of restitution, [-] 𝑓 Mass fraction of particles/agglomerates, [-] 𝑔 Acceleration due to gravity, m s-2 𝑔0 Radial distribution function, [-] ℎ Enthalpy, kJ kg-1 ℎ𝑒 Height of film, m ℎ𝑡𝑐 Heat transfer coefficient between the phases, W m-2 K-1 ℎ𝑎 Height of asperities, m 𝑘 Convective mass transfer coefficient, m s-1 𝑘𝑡ℎ𝑒𝑟𝑚𝑎𝑙  Thermal conductivity, W m-1 K-1 𝑚 Mass transfer coefficient, kg m-3 s-1 𝑚0 Equivalent mass of particles, kg 𝑛 Number of phases, [-] 𝑛𝑐𝑜𝑙𝑙̇  Collision frequency, collisions m-3 s-1 xiv  𝑛𝑑 Number density, particles/droplets m-3 𝑞 Heat flux, W m-2 𝑡 Time, s 𝑢 Phase velocity, m s-1 𝑣𝑟𝑒𝑙 Relative velocity, m s-1 𝑦∗ Mole fraction of acetone vapor in gaseous phase, [-] 𝐴𝑖 Area of interface, m2 𝐴𝑖−𝐹𝑜 Dimensionless coefficient, [-] 𝐴𝐿𝐶 Available Liquid Content, [-] 𝐵 Birth of solids, s m3 𝐵𝑑 Dimensionless number, [-] 𝐶𝐷 Drag coefficient, [-] 𝐷 Death of solids, s m-3 𝐷𝑐 Mass diffusivity, m2 s-1 𝐷𝑟 Drag force, N 𝐸𝑓𝑓 Probability, [-] 𝐹 Force, N 𝐾 Momentum exchange coefficient, kg m-3 s-1 𝐿𝑝 Dimensionless number, [-] 𝑁 Number of species, [-] 𝑁𝑢 Nusselt number, [-] P Pressure, Pa xv  𝑃∗ Vapor pressure, Pa 𝑃𝑒𝑟 Perimeter 𝑃𝑟 Prandtl number,[-] 𝑄 Heat transferred by convection, W 𝑅𝑒 Reynolds number, [-] 𝑆𝑐 Schmidt number, [-] 𝑆ℎ Sherwood number, [-] 𝑉𝑏𝑟𝑖𝑑𝑔𝑒  Volume of liquid bridge, m3 𝑉𝐶𝑒𝑙𝑙 Volume of cell, m3 𝑉𝐷  Volume of droplet, m3 𝑉𝑙𝑝 Volume of liquid assigned to particle, m3 𝑊𝑐𝑎𝑝 Capillary force, N 𝑊𝑉𝑖𝑠𝑐  Viscous force, N 𝑌 Species mass fraction, [-] 𝛼 Mass fraction of phase, [-] γ Dilatational viscosity, N s m-2 δ Unit stress tensor, [-] 𝛹 Lump factor, [-] 𝜌 Density, kg m-3 𝜆𝑖−𝐹𝑜  Dimensionless coefficient, [-] 𝜎 Surface tension, N m-1 𝜑 Energy exchanged between phases, J xvi  ɳ Dimensionless number, [-] 𝜃 Contact angle, radians 𝜃𝑔𝑟𝑎𝑛  Granular temperature of mono-disperse system, J kg-1 𝜃𝑚𝑖𝑥  Granular temperature of mixture, J kg-1 𝜏̿ Viscous stress tensor, N s m-2 μ Dynamic viscosity, N s m-2 𝜔 Stickiness coefficient, [-]   Subscripts  acetone_l Acetone in liquid phase Boil Mass transfer by boiling Evap Mass transfer by evaporation G Gas i Bin “i” 𝑗 Bin “j” l Liquid lg Liquid-to-Gas ls Liquid-to-Solid p Phase “p” q Phase “q” pq Phase “p” to “q” qp Phase “q” to “p” xvii  s Solid sg Solid-to-Gas xviii  Acknowledgements I would like to start by thanking my supervisors Dr. Martha Salcudean, Dr. Dana Grecov and Dr. John Grace, for their continued support over the course of my research. Their valuable suggestions and feedback were instrumental in the completion of this thesis. I am extremely grateful for the support and resources which they provided during the course of my studies.  I would like to thank Dr. Amir Motlagh for helping me develop the required knowledge for ANSYS Fluent simulations and his constant feedback on the simulations. Dr. Motlagh’s insight into the problem was invaluable when developing the model. I am grateful to Dr. Konstantin Pougatch for his time and patience in answering my research questions. Dr. Pougatch’s advice helped me select the appropriate formulae when developing the model. I owe special thanks to my parents who supported me financially and emotionally over the years.  Last but not the least I would like to thank NSERC (Natural Sciences and Engineering Research Council of Canada) and Syncrude Canada Ltd. for their financial support. xix  Dedication I dedicate this thesis to my parents. Their support throughout my educational life was invaluab le, without which it would have been infinitely more difficult to become who I am today. 1  Chapter 1: Introduction This chapter describes the phase interactions in Fluidized Bed Reactors (FBRs) and their industr ia l relevance. It also discusses the research methodologies and advancements in FBRs related to momentum exchange, heat and mass transfer, agglomeration and numerical solutions. It concludes with a brief description of the objectives and structure of this thesis. 1.1 Background Fluidized bed reactors (FBRs) are an important scientific advancement. They are commonly adopted by industry due to their advantages over conventional types of reactors such as packed beds. These advantages can be identified by understanding the operating principles behind FBRs. In fluidized bed reactors, physical and chemical changes take place on the surface or within solid particles. These particles, initially at rest, are thrust into complex circulatory motion upon the introduction of a gaseous phase into the reactor. This is termed "fluidization of particles." The process of "fluidization" is a phenomenon wherein the solid particles are lifted by the gas and the mixture’s behavior resembles that of boiling liquid; hence, the term "fluidized bed" is used to describe the system. When the particles are fluidized, there is an increase in volume over which they are spread within the reactor due to the bed expansion and continuous circulation of the particles. In other words, the total volume which the particles occupy remains the same, but the number of particles per unit volume (packing fraction) decreases. This decrease in the packing fraction of the solid particles results in a higher surface area of contact between the solid particles and the remaining phase/phases in the bed. Because the rate of any reaction is dependent on the available surface area, and particles in the reactor offer a larger surface area of contact, favorable rates of reaction are achieved when FBRs are employed (Michael & Michael, 1991b).  2  This aim of this thesis is to model the phenomenon of agglomeration that occurs in fluid cokers used by Syncrude Canada Ltd. These cokers convert bitumen fuel into lighter products by a process known as “thermal cracking.” Products obtained via thermal cracking are easier to transport due to their lower weight and reduced viscosity, thus making the manufacturing process more economical (Pougatch, 2011). The cracking process in fluidized bed reactors occurs when a liquid hydrocarbon is injected into a bed of heated coke particles through steam-assisted nozzles which atomize the fuel. Syncrude Canada Ltd.’s reactors use the hydrocarbon bitumen. The role of the reactor’s nozzles is to distribute the fuel equally among the particles.  In bitumen-wetted coke particles, heat transfer occurs between the solid (hotter phase) and fluid phases (cooler phase) leading to an increase in the temperature of the fluid phase. Once the temperature of the fluid phase is sufficiently high, bitumen undergoes thermal cracking. Over time, due to heat transfer from the solid phase to the fluid phase, the temperature of the coke particles decreases. This leads to a reduction in the heat transferred to the fluid phase as time progresses. A reduction in the magnitude of heat transfer translates to a reduction in product yield. Thus, over time the product yield decreases, and the process ultimately becomes unfeasible. To maintain the efficiency and profitability of the process, i.e., to ensure that the yield of products is adequate at all times, there must be a mechanism to supply heat to the solid phase. Generally, the particle phase is reheated by transferring the particles to a burner that heats the particles to the desired temperature, after which they are reintroduced into the fluid coker reactor (Darabi, 2011).  Agglomeration is another physical phenomenon that affects the product yield. It occurs when bitumen droplets coat the coke particles, causing these particles to stick together, forming 3  “agglomerates." Agglomeration leads to the formation of large dense agglomerates, and it decreases the process efficiency due to two reasons: i. There is a decrease in the total surface area available to the liquid phase. This decrease in surface area, in turn, leads to a decrease in the rate of reaction because the rate of any heterogeneous reaction depends on the available surface area. ii. Heavier agglomerates are harder to lift. Hence, the extent to which the agglomerates in the reactor are "fluidized" decreases. In other words, the behavior of the particles, which make up the agglomerate, stops resembling that of a fluid (defluidization). Defluidization, in turn, leads to lower phase interaction. Very heavy agglomerates can even lead to defluidiza t ion in the FBR.  Liquid content in the FBR also affects the product yield. Liquid content directly affects the rate of agglomeration and, in turn, influences the rate of the reaction. Liquid content is also directly responsible for agglomeration because the number and strength of liquid bridges is dependent on the liquid available to the solids. When solids collide, they agglomerate if, and only if, the liquid bridge formed during collision can dissipate the rebound energy of the solids. Hence, the availability of liquid is an important factor in deciding the success of agglomeration. In addition, the distribution of liquid around the particles, i.e., the film thickness, affects the heat transferred from the solid phase to the liquid phase. Hence, in addition to the liquid content, the film thickness is critical. The thickness of the liquid film is dependent on several factors, such as superficial gas velocity, and size and spatial distribution of the liquid droplets at the nozzle. Factors such as jet penetration, mean droplet diameter and size distribution of droplets can be controlled by choosing an appropriate nozzle. It is to be noted here that in some cases, agglomerates may form in the 4  absence of a liquid binder. The particles in these kind of agglomerates are held together by Van Der Waals forces of attraction. While the aim of this study is to develop a model which is primarily meant for the bitumen refining industry, fluidized bed reactors are also used in other industries, such as the pharmaceutica l industry. However, while agglomeration is undesirable in oil sands refining, it is essential in other industries where the product is an agglomerate, for example, in the pharmaceutical industry. 1.2 Motivation As discussed in the previous section, when liquid is injected into a gas-fluidized bed reactor, wet particles start to agglomerate, leading to a decreased product yield, thereby driving up the production costs and ultimately rendering the process unfeasible. By developing a basic model to predict the product yield at any instant of time, companies such as Syncrude Canada Ltd. can optimize the process of reheating the particles and breaking the agglomerates. This is important for the Canadian economy because oil is one of the country’s most significant exports. Oil sands also provide a large percentage of the country’s own oil consumption. To optimize the refining process, it is essential to have a good understanding of the physics of the process and the interactions between the different phases (solid, liquid and gas) in the reactor. Hence, the objective of this study is to develop a physical model which can simulate the behavior of various phases in a fluidized bed reactor and their role in agglomerate formation and breakage.  1.3 Literature Review In order to develop a model to predict the behavior of fluidized bed reactors, it is essential to understand the physical processes and process variables. Over the years numerous experiments have been performed to determine which variables have the most influence on the product yield. 5  Hence, the best way to start a literature review is to understand the relevant experiments and their conclusions. While many papers present a chronological development of research in fluid ized beds, this thesis chose a paper which focuses on chronological developments in fluidized beds in relation to the pharmaceutical industry (Michael & Michael, 1991b). This paper summarizes the most significant physical parameters in FBRs identified in the literature over the years, experiments that study the feasibility of granulation via fluidized beds and the effect of process variables on the efficiency of granulation beginning in the 1960s. Research in this period did not focus on developing empirical correlations between efficiency and various process variables. It merely observed the trends with respect to changes in process variables. Michael & Michael (1991) noted that in the 1970s researchers started developing mathematical correlations relating to granule growth. During this decade, various papers presented mathematical correlations for the growth of granules with respect to process variables, such as gas and binder flow rate, nozzle geometry and placement, etc. Until 1988, numerous experiments were performed to study the granulation process; however, there was no numerical method which could predict granule growth. This was because the Partial Differential Equations (PDEs) of the Population Balance Method (PBM) had both space and time components. Hence, this equation needed a novel discretization method. Solutions such as Hounslow et al. (1988) paved the way for numerical solutions of the PBM and, in turn, for numerical solutions of granulation in FBRs.  The PBM is an important component of the model presented in this thesis. It is solved along with the conservation and constitutive equations to model several physical phenomena. These 6  phenomena include heat and mass transfer and momentum exchange. The following sub-sections briefly discuss these phenomena. 1.3.1 Experimental Data In order to develop a theoretical model which can predict any physical process with reasonable accuracy, the results must be compared against experimental data. Morales (2013) presented the experimental results chosen for this model for experiments where the experimental conditions are similar to the FBR used by Syncrude Canada Ltd. In addition, these experiments are best suited to model numerically in the Eulerian frame of reference. It is important to note that Morales (2013) used particles and agglomerates which were formed during the injection process. This approach approximates the actual process in an industrial FBR. In Morales’ experiments, the FBR contained particles of silica sand fluidized by nitrogen. A mixture of nitrogen gas and a liquid consisting of a mixture of acetone, pentane, and poly-methyl-methacrylate (PMMA) was injected through a nozzle. Morales (2013) studied evaporation and agglomeration in a FBR and presented a parametric study of different gas-to-liquid (GLR) ratios, fluidization velocity, and acetone content. The experimental setup used by Morales (2013) is shown in Figure 1.1. 7    The heater shown in Figure 1.1 was operated intermittently to maintain the necessary bed temperature. This ensured that the temperature of the silica sand in the FBR is maintained well above the boiling point of the liquid. Taking these operating conditions into consideration, the numerical simulation was set up in this study. 1.3.2 Phase Interactions in the Fluidized Bed Reactor Before discussing the mechanism of agglomeration, it is necessary to understand the phase interactions in the FBR. They include momentum exchange (drag) between different phases, as well as the heat and mass transfer between the phases. These interactions depend largely on the Free Board Section of FBR Heater Expander Section of FBR Nozzle Fluidization Gas (Heated) Figure 1.1 Morales (2013) Setup Side View (left) and Front View (right) 8  distribution and behavior of the solid particles, which is modeled on the basis of the Kinetic Theory of Granular Flows. A comprehensive review of the KTGF and some of the interactions is presented below. 1.3.2.1 Kinetic Theory of Granular Flows Particles in a fluidized bed reactor behave like molecules in a gaseous medium. Hence, to describe and predict their behavior, researchers have developed a theory analogous to the Kinetic Theory of Gases, known as the Kinetic Theory of Granular Flows (KTFG). This theory assumes that the particles are uniformly distributed within a control volume and that the fluctuating component of their velocities obeys a Gaussian distribution around a mean velocity. Working from this assumption, Tan et al. (2004) presented an expression for the fluctuating velocity as a function of the granular temperature. These formulae were tested by simulating various conditions of normal and tangential coefficients of restitutions and the coefficient of friction. In addition to using the KTFG to compute the velocity distribution of particles, researchers applied this model to derive agglomeration kernels and other important factors, such as the spatial distribution of particles in the reactor, and verified them against experimental data (Rajniak et al., 2009).  While these studies receive special mention because they fall within the scope of this research, there are many other correlations which account for physical factors, such as the shear and bulk viscosity of phases in multi-phase flows (Goldschmidt et al., 2001). 9  1.3.2.2 Momentum Exchange Coefficient Wen & Yu (1966) assessed the accuracy of various drag laws available at that time and their applicability over different ranges of Reynolds number. They proceeded to develop a correlation which was applicable over a wide range of the Reynolds number by using correlations for velocity and terminal velocity and verified it against experimental results. Their verification included several parameters such as the Reynolds number, Galileo number, velocity and voidage fraction. They demonstrated that the Wen-Yu drag law was applicable over a wide range of Reynolds numbers, and was a better tool than other drag correlations available at that time (Wen & Yu, 1966). The Wen and Yu drag correlation works best for loosely packed fluidized bed reactors. For FBRs, where the packing fraction varies widely, it is advisable to employ the Gidaspow et al. (1992) drag force equation, which combines the Wen & Yu (1966) force and the Ergun (1952) equation, giving a good fit over a wide range of solid packing fractions. The drag laws used in this work are presented in Chapter 3. 1.3.2.3 Heat Transfer It was noted earlier that robust agglomerates require the presence of a liquid. In the presence of liquid, the solid particles are able to form liquid bridges, facilitating the formation of agglomerates. The mechanism of agglomeration is discussed in the following.  In this research, it is not enough to employ the continuity and momentum conservation equations to predict the liquid distribution. This is because a portion of the acetone vaporizes (i.e., there is mass and heat transfer between the phases involved) due to the heat transferred between the liquid in film on the solid particles and the gaseous phase. Vaporization also takes place from droplets.  In addition to vaporization, mass transfers from the liquid droplets to the film on solid particles 10  due to collisions of particles and droplets. Hence, in order to estimate the extent of agglomerat ion, one should develop a means of predicting the heat and mass transfer to and from the liquid. Because the mass transfer is a function of temperature, it is necessary to start by modeling the heat transfer between the phases.  Heat transfer between phases can occur by conduction, convection, and radiation. In order to simulate convection, it is necessary to define heat transfer coefficients and implement them into the energy equations. Over the years many researchers have developed correlations to predict the heat transfer coefficients, and some of these are discussed below. One of the most widely-used correlations is that of Ranz & Marshall (1952)  𝑁𝑢 = 2 + 0.6 ∗ 𝑅𝑒0.5 ∗ 𝑃𝑟13 (1.1)  It was developed via experimentation, by determining the rate of evaporation of droplets suspended in air. The experimental data were fitted to a modified form of the Froessling equation (mass transfer analogous of the Ranz and Marshall equation) to calculate the Nusselt number, which can be used to calculate the heat transfer coefficient. However, this correlation is valid only when the Reynolds number is 776 or less (Ranz & Marshall, 1952).  In 1969, the Ranz-Marshall correlation was extended to include flows at higher Reynolds numbers, i.e., for Re > 776. The resulting Hughmark (1970) correlation is, 11  𝑁𝑢 = {2 + 0.6𝑅𝑒1/2𝑃𝑟1/3    𝑓𝑜𝑟 0 ≤ 𝑅𝑒 < 776 𝑎𝑛𝑑 0 ≤ 𝑃𝑟 < 250   2 + 0.27𝑅𝑒0.62𝑃𝑟1/3      𝑓𝑜𝑟 776 ≥ 𝑅𝑒 𝑎𝑛𝑑 0 ≤ 𝑃𝑟 < 250 (1.2) Both correlations are used when calculating the heat transfer coefficient between dispersed liquid droplets and gaseous flows; however, it is also necessary to calculate heat transfer between solid and gaseous flows. Gunn (1978) developed a correlation to calculate the heat transfer between suspended solid particles and a gaseous phase. This is applicable in cases where the Reynolds number does not exceed 105 and where the bed voidage ranges between 0.35 and 1.0. Research in this field continued until 1998 when correlations such as (Tomiyama, 1998) were developed to approximate the heat transfer coefficient for bubbles in turbulent flows with relatively low Reynolds Numbers.   Heat transfer by radiation is only important when temperatures are much higher than those encountered in this research (e.g., > 8000C), and therefore radiation is neglected in this study. 1.3.2.4 Mass Transfer When heat is supplied to a droplet, it may change its state, converting from a liquid to a gaseous state. This phase change can take place via evaporation or boiling. When one wishes to develop a model to predict mass transfer, one must understand these separate mechanisms and operational regimes. Evaporation takes place over a wide range of temperatures and is governed by natural and forced convection. On the other hand, boiling is the physical process wherein liquid is transferred to the vapor phase when the temperature of the liquid phase is at the boiling point. For boiling to occur, the temperature of the surrounding media needs to be higher than the boiling point of the liquid. This is because when the liquid reaches the boiling point, it needs additional heat to increase its enthalpy to change its state. The amount of energy required for this phase change is 12  known as the latent heat of evaporation. In the Morales experiments, the temperature of the reactor was well above the boiling point of the liquid (acetone). Hence, both boiling and evaporation occurred in the reactor. Mass transfer by evaporation mostly depends on the vapor pressure of the liquid, which is also a function of the liquid temperature (Kim & Sung, 2003). Hence, in addition to factors such as the surface area of the liquid droplet, the number of droplets must also be calculated. The calculat ion of evaporation from free droplet differs from the calculation of evaporation from a liquid film on a porous structure. The calculation of evaporation from a film is more complex because of factors such as droplet imbibition and non-spherical shape of film. Choi et al. (2017) and Terrazas-Velarde Korina et al. (2011) discuss droplet imbibition and subsequent evaporation from porous particles. While several factors dictate evaporation, boiling is dictated primarily by the boiling point, latent heat of vaporization and the heat available to the droplet phase. In addition, boiling occurs even when liquid imbibes in a solid, unlike evaporation which occurs only when liquid is exposed to air. Hence, modeling boiling is relatively simple when compared to evaporation. In this study, the term vaporization is used to describe the combined effect of boiling and evaporation. The final mechanism of mass transfer calculates mass transfer from the droplet phase to the solid phase during inter-particle collisions. This mechanism ultimately determines the rate of agglomeration because the formation of agglomerates is dependent on the amount of liquid available to the solid particles. 13  1.3.3 Granulation Mechanism As noted previously, the presence of liquid leads to the formation of agglomerates in the reactor, but the process has not been described in detail. This sub-section describes the granulation process (formation of agglomerates). Particles involved in granulation go through the following three stages (Iveson et al., 2001): i. Wetting and nucleation: For two particles to be held together and form agglomera tes, there must be a liquid bridge between them. Hence, the first stage of granulation is particle wetting, i.e., liquid coming into contact with dry solid particles. Ideally, the liquid is distributed evenly throughout the bed; however, this is unlikely. Uneven liquid distribution leads to the formation of agglomeration zones which are called “nuclea t ion zones.” At these nucleation sites, particles start to adhere together via coalescence, forming agglomerates, which is the next stage in granulation. Iveson et al. (2001) note that process conditions and binder properties profoundly influence the type of nucleus formed and, ultimately, the quality of the agglomerates. An excellent example of the process conditions is the diameter of the liquid droplets. If the diameter of the liquid drop is much larger than the diameter of particles, then the resulting nucleus will consist of a droplet with solid particles imbibed on its surface. When the diameter of droplets is comparable to that of the particles, particles do not stick to the liquid droplet (as is the case in this work). Instead, liquid forms a film on the particles, the geometry of which is dictated by the contact angle and volume fraction of the liquid. The geometry of the liquid film is discussed in Chapter 3.   14  ii. Consolidation and Growth: The next stage of granulation occurs after the nuclei start to form granules and increase in size. Granular growth is highly dependent on the collision frequency and the liquid content. The impact of these factors can be understood if one recognizes the underlying mechanism of granulation. When two particles or agglomerates collide in the presence of liquid, they form a liquid bridge  (see Figure 1.2). A variety of factors determines whether or not the liquid bridge holds the particles together, leading to agglomeration. These factors include the viscous and capillary forces of the liquid bridge and the contact angle (Darabi et al., 2009). The liquid bridge can hold the particles together if, and only if, the liquid bridge is able to dissipate the energy of the collision. Because agglomeration happens primarily through collisions, it is important to develop the most suitable model to predict the mechanism of collisions. Depending on factors such as liquid content, capillary, and viscous forces, agglomeration may happen in multiple layers (Kumar et al., 2017).   iii. Breakage and Attrition: When two particles collide, the collision may lead to the formation of agglomerates if the liquid bridge between them can dissipate the rebound energy of the particles. When particles or agglomerates collide, the agglomerate can Figure 1.2 Formation of Liquid Bridge Solid Liquid Liquid Solid 15  undergo coalescence or breakage. Breakage can occur when the energy of the collis ion is more than the strength of the liquid bonds holding the particles together in the agglomerate. Breakage and attrition depend on such factors as agitation intensity, yield stress, and the diameter of particles. The strength of the bond between particles, i.e., the energy required to break the agglomerates, is dependent on the liquid content of the agglomerates. Iveson et al. (2001) summarize the various states of aggloemra tes depending on the liquid content.   1.3.4 Solution Methods Before any model is developed, it is important to understand how the equations can be discretized and in which frame of reference these equations can be solved. This sub-section discusses these aspects of the solution. 1.3.4.1 Frame of Reference Two frames of reference were considered for this study: Eulerian and Lagrangian. They differ by the method in which they treat the particles under consideration. The Lagrangian frame of reference is a more time-consuming method of solving multiphase flows. This is of particular Figure 1.3 Breakage of Agglomerates Solids Fracture 16  concern when there are a large number of particles in the computational space. When using a Lagrangian frame of reference, differential equations are solved for each particle (Zhang & Chen, 2007). In contrast, in the Eulerian frame of reference, the motion and behavior of each particle are not tracked, but the behavior of each phase at particular points is computed. Therefore, one can obtain the representative particle behavior, such as velocity, and temperature. However, it is never as detailed as for the Lagrangian frame of reference, and the behavior of individual particles is not calculated. For this research, information on individual particle behavior is not required. Instead, the purpose is to reduce the computational time and obtain a distribution of particle sizes after collisions, for which the Eulerian framework is more suitable (Loth, 2000).  1.3.4.2 Population Balance Method  The Population Balance Method (PBM) is employed to calculate the growth of agglomerates. It is a set of equations which keep track of the solid diameter and can be used to calculate the size distribution of solids. However, in order to be able to track the size change of solids, this equation must be supplied with appropriate rate terms, called kernels. The agglomeration kernel predicts the coalescence rate, and the breakage kernel predicts the rate at which breakage occurs. The PBM equations are all Partial Differential Equations (PDEs), and, hence, they must be discretized before they can be solved via numerical analysis. Various methods to solve such equations have been developed since Hounslow et al. (1988).  Various discretization schemes include the Discrete Method, Method of Moments and Monte Carlo Methods. The Method of Moments is a term used to describe several similar methods such as the Standard Method of Moments (SMM), Quadrate Method of Moments (QMOM) and the Discrete Quadrature Method of Moments (DQMOM). To understand these methods, one can refer 17  to Randolph & Larson (1971) for the SMM, McGraw (1997), Marchisio et al. (2003) for the QMOM and Fan et al. (2004) for the DQMOM. To gain a general understanding of the Monte Carlo Methods, one can refer to Zhao et al. (2007). The discrete method is adopted in this work because it requires the least solution time, and its focus is not on predicting the exact size of solids (as discussed below). The reader can refer to the above publications for an understanding of the other methods of discretization. The discrete method assumes the presence of various bins which are representative of particle/agglomerate diameters. These bins are in a geometric ratio which can be adjusted according to the problem. Because agglomerates grow or reduce in size due to collisions, newly formed agglomerates are divided between the various bins depending on their diameters. This division is done to keep track of the agglomerate diameter. If the diameter of the new agglomerate matches any of the bins, it is assigned to that specific population balance bin. However, if this is not possible, it is assumed that a certain fraction of the resulting agglomerates belongs to one bin while the remaining fraction is allocated to a different bin. The total volume of the allocated fractions should be the same as that of the resulting agglomerate. Hounslow et al. (1988) describe a relatively straightforward approach for this method of discretization. This will be discussed in Chapter 2. While this method gives a relatively quick solution time, it has a few disadvantages, namely, a. The number of bins heavily influences the final results. In cases where one needs to calculate agglomerate diameters with a high degree of accuracy, one has to use more bins. The higher the number of bins, the more the time required to solve the PBM. 18  b. The overall accuracy of the solution by this method of discretization is not very high. In cases where the focus is not on solution time, but where a high accuracy is needed, this method of discretization becomes unfeasible. 1.4 Objectives The primary objective of this study is to develop a model that can capture the physics of phase interactions that take place in an industrial fluidized bed reactor. In order to achieve this, the following milestones are necessary: i. To develop a mathematical model to monitor the liquid content and position in the fluid ized bed reactor and compare the total liquid content to experimental values. This model must be robust enough to account for variations in temperature and droplet diameter.  ii. To implement a physical model to predict the rate of agglomeration in a fluidized bed reactor with allowances for breakage and attrition probabilities. iii. To investigate  the effect of temperature and droplet diameter on   agglomeration 1.5 Outline The outline of this thesis is as follows:  Chapter 2: The partial differential equations and theories which form the basis of this model are presented in Chapter 2.   Chapter 3: Here the various formulae employed by this thesis are provided and discussed in detail, building on the equations described in Chapter 2.   Chapter 4: Predictions obtained from the simulation of the proposed model are presented and compared with the limited available experimental data.  19   Chapter 5: This chapter summarizes the results obtained by this model and presents the study’s conclusions. It also offers suggestions for future research. 20  Chapter 2: Theory and Background This chapter discusses the theoretical background to solve the multi-phase problem outlined in Chapter 1. It starts by selecting the appropriate frame of reference and the conservation equations. It also discusses the Kinetic Theory of Granular Flows (KTGF) and the Population Balance Method (PBM). The KTGF is employed to solve the distribution of solids in the reactor, whereas the PBM predicts the diameter of agglomerates in the reactor. 2.1 Background The first step to addressing the problem is to choose an appropriate frame of reference, i.e., either the Eulerian-Eulerian or Eulerian-Lagrangian frame of reference. The Literature Review, in Chapter 1, discusses the advantages and disadvantages of these frames of reference. Taking these arguments into account, the Eulerian-Eulerian frame of reference was chosen over the Eulerian-Lagrangian frame of reference because the model suggested and applied in this research project requires significant computational time. This model employs equations derived from both the Kinetic Theory of Granular Flows (KTGF) and the Population Balance Method (PBM) to predict the rate of agglomeration. The KTGF predicts the physical behavior of particles and agglomerates such as velocity and solid pressure, whereas the PBM allows for the calculation of agglomerate diameters. Finally, the PBM equations are discretized using a discrete method. As discussed in the Literature Review, this is the fastest method of solving the PBM. 21  2.2 Conservation Equations In this research, there are three different phases, with each phase consisting of multiple species. To model the behavior of the phases in the reactor, it is necessary to track the behavior of individua l species, rather than the overall phase. The conservation equations used in this research take this into account. 2.2.1 Conservation of Mass Equations This section presents the general equation for the conservation of mass. This equation differs for each species because of the differences in mass transfer terms. The equations for the conservation of mass that are unique to each species are outlined in Appendix A.  Mass transfer is a phenomenon which occurs between the different species, rather than only between phases. Thus, the equation for the conservation of mass must be written for each species and not for the overall phase. Assuming that there are “(n+1)” different phases in a reactor, the general form of the equation of conservation of mass for a species “j” in phase “q” is written (ANSYS Inc., 2013) as: 𝜕𝜕𝑡(𝛼𝑞 𝜌𝑞𝑌𝑞𝑖) + ∇.(𝛼𝑞 𝜌𝑞𝑢𝑞⃗⃗⃗⃗  𝑌𝑞𝑖) = −∇. 𝛼𝑞𝐽 𝑞𝑖+ 𝛼𝑞𝑅𝑞𝑖 + 𝛼𝑞𝑆𝑞𝑖 + ∑(𝑚𝑝𝑖𝑞𝑗̇ − 𝑚𝑞𝑗𝑝𝑖̇ )𝑛𝑝=1 (2.1) In the above equation, the terms 𝛼, 𝜌,𝑢, and 𝑌 represent the phase volume fraction, density, velocity and species mass fraction respectively. The term 𝐽 𝑞𝑖 represents the diffusion flux of species “i” in phase “q”. In addition, 𝑅𝑞𝑖 represents mass addition by chemical reaction, and 𝑆𝑞𝑖 denotes mass addition by an external source. Both of these terms are equal to zero in this research 22  because we do not have any chemical reaction or mass addition. Thus, for the purpose of this research, the equation for conservation of mass becomes: 𝜕𝜕𝑡(𝛼𝑞𝜌𝑞 𝑌𝑞𝑖) + ∇. (𝛼𝑞𝜌𝑞 𝑢𝑞⃗⃗⃗⃗  𝑌𝑞𝑖) = −∇.𝛼𝑞 𝐽 𝑞𝑖+ ∑(𝑚𝑝𝑖𝑞𝑗̇ − 𝑚𝑞𝑗𝑝𝑖̇ )𝑛𝑝=1 (2.2) The summation (Sigma) terms represent the mass transfer between different phases, and 𝑌𝑞𝑖 is the volume fraction of species “i” within phase q. The terms on the left-hand side of the equation represent the rate of increase/decrease in mass (transient term), and the net change in mass due to convection (convection term) (Bird et al.). Assuming that there are (N+1) species in phase q (1,2,3…N+1), then the following equation should be satisfied: ∑ 𝑌𝑞𝑖𝑁+1𝑖=1= 1 (2.3) 2.2.2 Conservation of Momentum Equation The equation for the conservation of momentum works in tandem with the continuity equation to calculate the velocity and pressure of the system. The general form of the momentum conservation equation is written as (ANSYS Inc., 2013): 𝜕𝜕𝑡(𝛼𝑞𝜌𝑞 ?⃗? 𝑞) + ∇.(𝛼𝑞 𝜌𝑞 ?⃗? 𝑞 ?⃗? 𝑞)= −𝛼𝑞∇P+ ∇.𝜏̿𝑞 + 𝛼𝑞 𝜌𝑞𝑔 + ∑(𝐷𝑟⃗⃗⃗⃗  ⃗𝑞𝑝 + ?̇?𝑝𝑞?⃗? 𝑝𝑞 − ?̇?𝑞𝑝?⃗? 𝑞𝑝)𝑛𝑝=1       + (𝐹 𝑒𝑥𝑡 + 𝐹 𝑙𝑖𝑓𝑡,𝑞 + 𝐹 𝑣𝑚,𝑞 ) (2.4) 23  In this equation, the first term represents the rate of increase of momentum per unit volume. The second term designates the rate of momentum addition by convection per unit volume. The first two terms on the right-hand side represent the pressure and stress tensor. Together, these terms calculate the rate of momentum addition by molecular transport per unit volume (Bird et al.). The terms 𝐷𝑟⃗⃗⃗⃗  ⃗𝑝𝑞 , 𝐹 𝑒𝑥𝑡 , 𝐹 𝑙𝑖𝑓𝑡,𝑞 and 𝐹 𝑣𝑚,𝑞  denote the drag force, external forces (such as gravity), lift force and virtual mass, respectively. The vector 𝑔  represents the effect of gravity. The terms ?̇?𝑝𝑞 and ?̇?𝑞𝑝 represent the mass transfer between phases “p” and “q.” The models which calculate these terms are described in Chapter 3. The term ∇.𝜏̿𝑞  accounts for the change in the viscous stress tensor, with: 𝜏?̿? = μ𝑞(∇?⃗? 𝑞 + ∇𝑢𝑞𝑇 ) + (γ − 23μ𝑞 )(∇.𝑢𝑞)δ (2.5) In the above equation, δ is the unit tensor, and γ is the dilatational viscosity. The terms μ and δ  are the viscosity of phase q and unit tensor respectively. The calculation of drag is discussed in Section 2.3. Equation (2.4) represents the general form of the conservation of momentum equation. With the mass transfer between the phases considered, the equations unique to each phase are as follows: i. Equation of Conservation of Momentum for Solid Phase 24  𝜕𝜕𝑡(𝛼𝑠𝜌𝑠?⃗? 𝑠) + ∇.(𝛼𝑠𝜌𝑠?⃗? 𝑠?⃗? 𝑠)= −𝛼𝑠∇P + ∇.𝜏̿𝑠 + 𝛼𝑠𝜌𝑠𝑔 + 𝐷𝑟⃗⃗⃗⃗  ⃗𝑠𝑔 + 𝐷𝑟⃗⃗⃗⃗  ⃗𝑠𝑙 + ?̇? 𝑙𝑠?⃗? 𝑙𝑠 − ?̇?𝑠𝑔_𝑒𝑣𝑎𝑝 ?⃗? 𝑠𝑔− ?̇?𝑠𝑔_𝑏𝑜𝑖𝑙?⃗? 𝑠𝑔 + (𝐹 𝑒𝑥𝑡 + 𝐹 𝑙𝑖𝑓𝑡,𝑠 + 𝐹 𝑣𝑚,𝑠) (2.6) ii. Equation of Conservation of Momentum for Liquid Phase (Droplets) 𝜕𝜕𝑡(𝛼𝑙𝜌𝑙?⃗? 𝑙) + ∇. (𝛼𝑙𝜌𝑙?⃗? 𝑙?⃗? 𝑙)= −𝛼𝑙∇P + ∇.𝜏̿𝑙 + 𝛼𝑙𝜌𝑙𝑔 + 𝐷𝑟⃗⃗⃗⃗  ⃗𝑙𝑔 + 𝐷𝑟⃗⃗⃗⃗  ⃗𝑙𝑠 − ?̇?𝑙𝑠?⃗? 𝑙𝑠 − ?̇?lg _𝑒𝑣𝑎𝑝 ?⃗? 𝑙𝑔− ?̇? lg _𝑏𝑜𝑖𝑙?⃗? 𝑙𝑔 + (𝐹 𝑒𝑥𝑡 + 𝐹 𝑙𝑖𝑓𝑡,𝑠 + 𝐹 𝑣𝑚,𝑠) (2.7) iii. Equation of Conservation of Momentum for Gaseous Phase 𝜕𝜕𝑡(𝛼𝑔𝜌𝑔 ?⃗? 𝑔) + ∇. (𝛼𝑔𝜌𝑔 ?⃗? 𝑔?⃗? 𝑔)= −𝛼𝑔∇P + ∇.𝜏?̿? + 𝛼𝑔𝜌𝑔𝑔 + 𝐷𝑟⃗⃗⃗⃗  ⃗𝑔𝑙 + 𝐷𝑟⃗⃗⃗⃗  ⃗𝑔𝑠 + ?̇?lg _𝑒𝑣𝑎𝑝 ?⃗? 𝑙𝑔+ ?̇? 𝑠𝑔_𝑒𝑣𝑎𝑝?⃗? 𝑠𝑔 + ?̇?lg _𝑏𝑜𝑖𝑙?⃗? 𝑙𝑔 + ?̇?𝑠𝑔_𝑏𝑜𝑖𝑙?⃗? 𝑠𝑔 + (𝐹 𝑒𝑥𝑡 + 𝐹 𝑙𝑖𝑓𝑡,𝑠 + 𝐹 𝑣𝑚,𝑠) (2.8) The terms in these equations which represent drag, virtual mass, and lift forces (𝐷𝑟⃗⃗⃗⃗  ⃗𝑝𝑞 , 𝐹 𝑣𝑚,𝑞 and 𝐹 𝑙𝑖𝑓𝑡,𝑞) are labelled “momentum exchange coefficients.” 2.2.3 Conservation of Energy Equations After writing equations of conservation of the mass and momentum of phases in the system, an equation is required to calculate the transfer of energy (ANSYS Inc., 2013): 25  𝜕𝜕𝑡(𝛼𝑞𝜌𝑞ℎ𝑞) + ∇. (𝛼𝑞𝜌𝑞 ?⃗? 𝑞ℎ𝑞)= 𝛼𝑞𝜕𝑃𝜕𝑡+ 𝜏̿𝑞 :∇?⃗? 𝑞 − ∇𝑞 𝑞 + 𝑆𝑞 + ∑(𝑄𝑝𝑞 + ?̇?pqℎ𝑝𝑞 − ?̇?qpℎ𝑞𝑝)𝑛𝑝=1 (2.9) where 𝑄𝑝𝑞  represents the heat transferred by convection and 𝑞  is the heat flux. The term ℎ represents the change in enthalpy per unit mass between two physical states of a species. For example, the enthalpy difference between the liquid and gaseous state of a substance is calculated by this term. For Eulerian flows ℎ𝑞𝑝 = ℎ𝑝𝑞, where ℎ is the enthalpy change for mass transferred between the phases “p” and “q.” The terms 𝑚pq and 𝑚qp denote mass transfer between phases “p” and “q,” with the models needed to calculate these terms described in Chapter 3. 𝑆𝑞  represents the external heat supply. This is present only at the inlet boundary and only for the gaseous phase entering the reactor. This term represents the effect of the heater, and its calculation is discussed in Appendix 3. Thus, the equation for the conservation of energy for phase “q” becomes: 𝜕𝜕𝑡(𝛼𝑞𝜌𝑞 ℎ𝑞) + ∇. (𝛼𝑞𝜌𝑞 ?⃗? 𝑞ℎ𝑞)= 𝛼𝑞𝜕𝑃𝜕𝑡+ 𝜏̿𝑞 :∇?⃗? 𝑞 − ∇𝑞 𝑞 + ∑(𝑄𝑝𝑞 + ?̇?𝑝𝑞ℎ𝑝𝑞 − ?̇?𝑞𝑝ℎ𝑞𝑝)𝑛𝑝=1 (2.10) Equation (2.10) is the general form of the conservation of energy equation. After considering the mass transfer between the phases, the energy equations for each phase are: i. Conservation of Energy for Liquid Phase (Droplets) 26  𝜕𝜕𝑡(𝛼𝑙𝜌𝑙ℎ𝑙)+ ∇. (𝛼𝑙𝜌𝑙?⃗? 𝑙ℎ𝑙)= 𝛼𝑙𝜕𝑃𝜕𝑡+ 𝜏?̿?: ∇?⃗? 𝑙 − ∇𝑞 𝑙−𝑄𝑙𝑔 − ?̇? 𝑙𝑠ℎ𝑙𝑠 − ?̇? lg _𝑒𝑣𝑎𝑝ℎ𝑙𝑔 − ?̇?lg _𝑏𝑜𝑖𝑙ℎ𝑙𝑔 (2.11) ii. Conservation of Energy for Solid Phase 𝜕𝜕𝑡(𝛼𝑠𝜌𝑠ℎ𝑠) + ∇. (𝛼𝑠𝜌𝑠?⃗? 𝑠ℎ𝑠)= 𝛼𝑠𝜕𝑃𝜕𝑡+ 𝜏?̿?: ∇?⃗? 𝑠 − ∇𝑞 𝑠−𝑄𝑠𝑔 − ?̇?𝑠𝑔_𝑒𝑣𝑎𝑝ℎ𝑠𝑔 − ?̇?𝑠𝑔_𝑏𝑜𝑖𝑙ℎ𝑠𝑔 + ?̇?𝑙𝑠ℎ𝑙𝑠 (2.12) iii. Conservation of Energy for Gaseous Phase 𝜕𝜕𝑡(𝛼𝑔𝜌𝑔ℎ𝑔) + ∇. (𝛼𝑔𝜌𝑔 ?⃗? 𝑔ℎ𝑔)= 𝛼𝑔𝜕𝑃𝜕𝑡+ 𝜏?̿? : ∇?⃗? 𝑔 − ∇𝑞 𝑔+𝑄𝑠𝑔+𝑄𝑙𝑔 + ?̇?𝑠𝑔_𝑒𝑣𝑎𝑝ℎ𝑠𝑔 + ?̇? lg _𝑒𝑣𝑎𝑝ℎ𝑙𝑔+ ?̇?𝑠𝑔_𝑏𝑜𝑖𝑙ℎ𝑠𝑔 + ?̇?lg _𝑏𝑜𝑖𝑙ℎ𝑙𝑔 (2.13) In the above equations, the terms 𝑄𝑙𝑔  and 𝑄𝑠𝑔  represent the heat transferred between the phases by convection and are given by the following equations: 𝑄𝑙𝑔 =6𝛼𝑙 ∗ ℎ𝑡𝑐𝑙𝑔 ∗ (𝑇𝑙 − 𝑇𝑔)𝑑𝑙 (2.14) 𝑄𝑠𝑔 =6𝛼𝑠 ∗ ℎ𝑡𝑐𝑠𝑔 ∗ (𝑇𝑠 − 𝑇𝑔)𝑑𝑠 (2.15) 27  where, ℎ𝑡𝑐𝑠𝑔  and ℎ𝑡𝑐𝑙𝑔 represent the heat transfer coefficients between the phases and are calculated from Nusselt number correlations. The calculation of Nusselt Numbers coefficients is discussed in Chapter 3. These conservation equations form the basis of predicting the behavior of phases present in the reactor. To model the motion of solids, however, formulae derived from the KTGF are employed. These formulae are described in Sections 2.2.4 and 2.2.5. 2.2.4 Kinetic Energy of Collision As outlined in Chapter 1, the KTGF assumes that solid velocities obey Gaussian distributions around a mean velocity. The difference between the time-averaged mean velocity of solids and the instantaneous velocity of the particle in question is termed the “fluctuating velocity of the solids.” It is due to this fluctuating velocity that inter-particle collisions take place. The collision energy is thus determined by the relative velocity between these solids. The kinetic energy of the collis ion is represented by: 𝐾𝑖𝑛𝑒𝑡𝑖𝑐  𝐸𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝐶𝑜𝑙𝑙𝑖𝑠𝑜𝑛 =12∗ 𝑚0 ∗ 𝑣𝑟𝑒𝑙2 (2.16) where, 𝑚0 is the mass of the particles and 𝑣𝑟𝑒𝑙 represents the relative velocity of collision. Substituting the expression for relative velocity, we obtain: 𝐾𝑖𝑛𝑒𝑡𝑖𝑐  𝐸𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝐶𝑜𝑙𝑙𝑖𝑠𝑜𝑛 =12∗ 𝑚0 ∗ (2 ∗ √3 ∗ 𝜃𝑔𝑟𝑎𝑛)2 (2.17) 28  where, 𝜃𝑔𝑟𝑎𝑛  represents the granular temperature of the solid phase. It is important to add that the above equation is valid only in cases where the diameter of the solids is the same. The modified equation for the kinetic energy of collision is presented in Chapter 3. The granular temperature (𝜃𝑔𝑟𝑎𝑛 ), in the above equation is obtained by solving the transport equation of granular energy. The equation for the transport of granular energy for a single phase “p” is given by Gidaspow et al. (1992b) as: 32[𝜕𝜕𝑡(𝛼𝑝𝜌𝑝𝜃𝑔𝑟𝑎𝑛) +  ∇. (𝛼𝑝𝜌𝑝?⃗? 𝑞𝜃𝑔𝑟𝑎𝑛)]= (−𝑃𝛿 + 𝜏𝑝̿̿̿): ?⃗? 𝑞 + ∇.(𝑘𝜃𝑔𝑟𝑎𝑛 ∇𝜃𝑔𝑟𝑎𝑛) − 𝛾𝜃𝑔𝑟𝑎𝑛 + 𝜑𝑝𝑞  (2.18) The first two terms on the right-hand side represent the surface forces, i.e., pressure vector and stress tensor. The next term is the diffusive flux of granular energy, 𝛾𝜃𝑔𝑟𝑎𝑛 , and represents the loss of energy due to inelastic collisions, and 𝜑𝑝𝑞  designates the energy exchanged between phases “p” and “q.” The term 𝛾𝜃𝑔𝑟𝑎𝑛  is calculated based on Lun et al. (1984) as: 𝛾𝜃𝑔𝑟𝑎𝑛 =12(1 − 𝑒)𝑔0𝑑𝑝√𝜋 𝜌𝑝𝛼𝑝2𝜃𝑔𝑟𝑎𝑛32  (2.19) where, 𝑔0 is the radial distribution function, 𝑒 is the coefficient of restitution and 𝑑𝑝 is the particle diameter. The term 𝑘𝜃𝑔𝑟𝑎𝑛  is the diffusion coefficient of granular energy calculated by Syamlal et al. (1993) as follows: 29  𝑘𝜃𝑔𝑟𝑎𝑛 =15 ∗ 𝑑𝑝 ∗ 𝜌𝑝 ∗ 𝛼𝑝  √𝜋𝜃𝑔𝑟𝑎𝑛  4 (41 − 33ɳ)∗ [1 +125ɳ2 (4ɳ − 3)𝛼𝑝𝑔0 +1615 𝜋(41 − 33ɳ) ɳ 𝛼𝑝𝑔0] (2.20) where, ɳ =12 (1 + 𝑒) (2.21) Finally, the term 𝜑𝑝𝑞  was calculated by Gidaspow et al. (1992b): 𝜑𝑝𝑞 = −3𝐾𝜃𝑔𝑟𝑎𝑛  (2.22) where, 𝐾 is the momentum exchange coefficient. With appropriate initial and boundary conditions, Equation (2.18) can be solved to obtain the granular temperature. 2.2.5 Radial Distribution Function The rate of agglomeration is strongly influenced by collisions. The frequency of collisions depends on the number density of the solid particles/agglomerates. The collision frequency is also dependent on the spatial distribution of solids, in addition to the number density. The radial distribution function fulfills this purpose by estimating the particle distribution around the solid under consideration. The model employed in this study employs the expression suggested by Ogawa et al. (1980), i.e.: 𝑔0 = [1 − (𝛼𝑠𝛼𝑠𝑚𝑎𝑥)13]−1 (2.23) 30  2.3 Momentum Exchange When two phases are in contact with each other, there is some resistance to their respective motion due to drag, lift, and virtual mass, factors which must be considered in the equation for the conservation of momentum. The phenomenon of greatest importance in this research is drag, defined by the drag coefficient. The advantages and disadvantages of various drag coeffic ient correlations are discussed in Chapter 1. After considering these issues, the correlation presented by Gidaspow et al. (1992b) was selected for the interaction between the gaseous and solid phases, described below: i. When the fluid volume fraction is greater than 0.8, the Wen & Yu (1966) equation is employed and the solid-gaseous exchange coefficient is defined as: 𝐾𝑠𝑔 =34𝐶𝐷𝛼𝑠𝛼𝑔𝜌𝑔 |𝑢𝑠⃗⃗⃗⃗  − 𝑢𝑔⃗⃗⃗⃗  |𝑑𝑠𝛼𝑔−2.65  (2.24)        with, 𝐶𝐷 =24𝛼𝑔𝑅𝑒𝑠[1 + 0.15(𝛼𝑔𝑅𝑒𝑠)0.687] (2.25) where, 𝐾𝑠𝑔 , 𝐶𝐷, 𝛼, 𝜌, 𝑑𝑠  and ?⃗?  represent the momentum exchange coefficient for the solid and gaseous phase, drag coefficient, volume fraction, density, solid diameter and phase velocity respectively. 𝑅𝑒𝑠 represents the Reynolds number of the solid phase. ii. When, on the other hand, the fluid volume fraction is less than or equal to 0.8, the Ergun equation is employed and the solid-gaseous exchange coefficient is calculated as: 31  𝐾𝑠𝑔 = 150𝛼𝑠(1 − 𝛼𝑔)𝜇𝑔𝛼𝑔𝑑𝑠2+ 1.75𝜌𝑔𝛼𝑠|𝑢𝑠⃗⃗⃗⃗  − 𝑢𝑔⃗⃗⃗⃗  |𝑑𝑠 (2.26) with the coefficient of drag remaining the same.  The Schiller and Neuman equation is employed to calculate the drag for liquid-gaseous interactions. The liquid-gaseous exchange coefficient is calculated by this model as:  𝐾𝑙𝑔 =𝜇𝑞𝐶𝐷𝑅𝑒𝐴𝑖8 𝑑𝑠 (2.27) where, if 𝑅𝑒 ≤ 1000 𝐶𝐷 =24(1 + 0.15𝑅𝑒0.687)𝑅𝑒 (2.28) else, if 𝑅𝑒 > 1000 𝐶𝐷 = 0.44 (2.29) 𝐾𝑙𝑔 is the momentum exchange coefficient between the liquid (droplet) and gaseous phase, and 𝐴𝑖 is the area of interface between the two phases. The third type of momentum exchange involves the solid and liquid (droplet) interactions. The correlation employed for this phenomenon was presented by Li et al. (2010) and is, 𝐾𝑙𝑠 = (1 − 𝜔)𝛹3(1 + 𝑒𝑙𝑠)𝛼𝑠𝛼𝑙𝜌𝑠(𝑑𝑙 + 𝑑𝑠)2|𝑢𝑙⃗⃗  ⃗ − 𝑢𝑠⃗⃗⃗⃗  |4(𝜌𝑙𝑑𝑙3 + 𝜌𝑠𝑑𝑠3) (2.30) In the above equation, 𝜔 represents the “stickiness coefficient” defined in the next section. It is related to the number of liquid droplets collected by the particles during collision. The term 𝛹 32  represents the decrease in momentum exchange due to oblique collisions. This correlation is best suited to calculate the momentum exchange between the solid and liquid phases because it takes into consideration the mass transferred during the collision of droplets and particles. The values of 𝛹 and 𝑒𝑙𝑠 were chosen as 0.4 and 0.5, values used by Li et al. (2010) for a similar case. 𝑒𝑙𝑠 is the coefficient of restitution and 𝛹 is the lump factor taking into account the decrease in momentum due to oblique collisions. In this study, a sensitivity analysis has not been performed, and the values provided by Li et al. (2010) for a similar system have been adopted. The coefficients of drag defined in this section are deployed in the equation for the conservation of momentum presented in Section 2.2.2. 2.4 Population Balance Method In Chapter 1, the granular flow phenomenon is discussed; however, the Population Balance Method (PBM) has not been presented in detail. The PBM is used to calculate the size distribution of solids. It is achieved by distributing the solids among different “bins.” Each “bin” represents a different particle/agglomerate diameter, termed a “bin size.” Solids whose diameter is equal to the bin size belong to that bin. The number of bins is the number of solid diameters allowed in the simulation. The terms “birth” and “death” characterize the transfer of solids between bins, with “birth” referring to the addition of particle/agglomerate to a bin, whereas “death” denotes the loss of a particle/agglomerate from a bin. For example, if two solid particles belonging to bin “i” collide and form an agglomerate belonging to bin “j”, then the two particles are said to have died in bin “i,” while one agglomerate is said to have been born in bin “j.” This method of calculating the diameter of solids is expressed in PDE format as: 33  𝜕𝜕𝑡(𝑓𝑖𝛼𝑠𝜌𝑠) + ∇. (𝑓𝑖𝛼𝑠𝜌𝑠 ?⃗? 𝑠) = 𝛼𝑠𝜌𝑠(𝐵𝑖 − 𝐷𝑖) (2.31) In this equation, 𝑓𝑖 represents the fraction of solids belonging to bin “i,” and 𝐵𝑖 and 𝐷𝑖 represent the “birth” and “death” of solids belonging to bin “i.” If there are “m” bins, then: ∑ 𝑓𝑖𝑚𝑖=1= 1 (2.32) The “birth” and “death” terms are calculated based on the number density of solids and the agglomeration kernel, as discussed in Chapter 3. While the mass is conserved, there may be some momentum loss during the process of coalescence. This loss of momentum is not accounted for in the PBM equations. It is also important to note that the birth and death terms are calculated here based on agglomeration only because, at this point, there is no model for breakage in place. Such a model is currently being developed by others in my research group. The attrition model was developed by Palanisamy (2016). The breakage model is currently in development. 2.5 Summary This chapter presents the basic equations for multiphase flow:  Mass, momentum and energy conservation equations for each phase.  Equations derived from the KTGF for calculating solid distribution.  Equations relating to the PBM for calculating solid diameters. Models for specific terms, such as the mass transfer terms and the agglomeration kernel, are presented and discussed in Chapter 3. 34  Chapter 3: Model Development This chapter discusses the interactions between different phases, specifically with regard to the extent of vaporization and the rate of agglomeration. The model used to calculate evaporation is based on a Sherwood number correlation. The model employed to calculate agglomeration is based on the Kinetic Theory of Granular Flows (KTGF) respectively. This chapter explains the models developed based on these theories. 3.1 Model Assumptions Due to the complexity of the process, the following simplifying assumptions are made:  In the real world, the physical properties of any fluid are temperature-dependent. The relationship between these properties and temperature is determined via experiments. However, the liquid used by Morales (2013) in her experiments was a mixture of three pure substances, acetone, PMMA and pentane (polar, acrylic and non-polar respectively). To obtain a relationship between the physical properties and temperature of the mixture , experiments would need to be performed for the mixture, not for the individual substances. In the absence of these experiments, some properties of the fluid (density, viscosity, specific heat capacity) are assumed to be independent of temperature.   Morales, (2013) used a mixture of three pure substances in her experiments, acetone (80%), pentane (10%) and PMMA (10%) (by weight). Acetone is the major component and influences the mixture’s properties the most. PMMA is the acrylic, and is the binder for agglomerates. Pentane has the least effect on the mixture’s properties and is thus neglected in this research, reducing the model by 1 PDE and 3 mass transfer coefficients. By neglecting pentane, the results will not be significantly different. Neglecting pentane also 35  decreases the simulation time. Hence in the interest of simplicity and solution time, pentane was neglected.  It is not possible to track the exact shape of liquid droplets in Eulerian methods. Due to this  limitation, the droplets are assumed to be spherical, and the diameter of their equivalent spherical shapes is based on the Sauter mean diameter. The diameter of solid particles and agglomerates is also estimated via this method. The diameter of the droplet is assumed to be constant at 200 microns in this work. This approach simplifies the model to a great degree because modeling droplet interaction with the other phases in a FBR is very complex.  In reality the shape of the nozzle influences the diameter of the droplet phase. However, modeling the flow through a nozzle is a very complex process, and it is not the focus of this study. Hence, for the purpose of this study, the effect of nozzle shape on the diameter of droplets are not considered. Models that focus on the flow through the nozzle are available in the literature (for example Pougatch et al., 2012).  When agglomerates in a fluidized bed reactor collide with other agglomerates or particles, they may undergo attrition and break into smaller agglomerates/particles. Here, we assume that attrition and breakage do not occur during these collisions. A suitable breakage/attrition model which estimates the rate of breakage is currently under development in our research group by others.   At each time step, the amount of liquid distributed to the film on each particle/agglomerate is tracked using a volume-based approach. This method is discussed in Section 3.3.3. In this research, the diameter of droplets is similar to the diameter of particles. Hence, the liquid droplets do not collect multiple particles as in the case where the droplet diameter is 36  much larger than the particle diameter. The liquid in film on the solid phase is assumed to form hemispherical caps, whose dimensions are dependent on the contact angle. Figure 3.1 shows the influence of contact angle on the dimensions of the hemispherical cap.   The contact angle in this research is 320 (Morales, 2013). It is important to note that if the liquid content in the solid phase is sufficiently high, then the liquid forms a film instead of a hemishperical cap.  Heat transfer by radiation is assumed to be negligible, because of the moderate temperature of the reactor.  The species model is employed in this research to keep track of the mass transfer. This model assumes that all species in a phase have the same temerature and velocity. Hence, the temperature of each species cannot be different, and there is negligible heat transfer between the species of a phase. However, heat transfer takes place between the various phases. This becomes important when acetone and PMMA are transferred from the droplets to the film in the solid phase. When this type of mass transfer takes place, the temperature of the solid phase (solid and liquid film) is calculated based on the heat balance, and a Contact Angle Contact Angle Figure 3.1 Influence of Contact Angle on the Shape of Liquid Film Solid Collision Liquid Droplet 37  common temperature is assigned to them. In conclusion, at any point of time, there is no heat transfer between the liquid film and the solids in the FBR. 3.2 Modeling Vaporization Heat and mass transfer are two interlinked phenomena which need to be considered when calculating vaporization. Hence, to calculate the extent of vaporization, the heat and mass transfer mechanisms need to be modeled (see Figure 3.2).    ❶ No Heat Transfer  ❶Mass Transfer ❸ Heat     Transfer ❸ Mass    Transfer ❷Heat    Transfer ❷Mass   Transfer Phase 2 (Particle/Agglomerates) Solid + Acetone + PMMA Phase 3 (Liquid) Acetone + PMMA Phase 1 (Gaseous)  Air + Vapor Figure 3.2 Interaction between Phases 38  Models to calculate the heat and mass transfer are described in Sections 3.2.1 and 3.2.2. It is important to emphasize that mass transfer by vaporization can be modeled by considering evaporation and boiling. Hence, models to calculate evaporation and boiling are discussed in Section 3.2.2. 3.2.1 Model for Heat Transfer Heat transfer between any two phases can happen by conduction, convection or radiation. Conduction and radiation are not considered due to the assumptions made in Section 3.1. Only convective heat transfer is modeled in this study as shown in Figure 3.3.   Heat Transfer Neglected   Heat Transfer by Convection Heat Transfer by Convection  Particle/Agglomerate Phase Liquid Phase Gaseous Phase Figure 3.3 Heat Transfer Interactions 39  Heat transfer between droplets and particles/agglomerates is assumed to occur only when these two phases are in contact with each other, i.e., during collisions. To estimate the relevance of this mechanism of heat transfer, the time scales of heat transfer and collision were estimated (by Fourier Number and collision frequency respectively). Analysis showed that the time scale needed for heat transfer was at least an order of magnitude greater than the time scale for collision (refer to Appendix B). For this reason, the heat transfer between the particles and droplets was neglected. There is negligible heat transfer between the liquid film and the solids, for the reasons in the assumptions. 3.2.1.1 Heat Transfer from Droplets to Gaseous Phase Heat transfer from the droplets to gaseous phase can be modeled using the correlation of Ranz and Marshall (1952) obtained by fitting experimental data, with the Nusselt number as a function of both the Reynolds and Prandtl numbers, of the form: 𝑁𝑢 = 2 + 0.6 ∗ 𝑅𝑒𝑎 ∗ 𝑃𝑟𝑏  (3.1) Upon fitting their experimental data, they obtained values of a and b leading to: 𝑁𝑢 = 2 + 0.6 ∗ 𝑅𝑒0.5 ∗ 𝑃𝑟13 (3.2) with, 40  𝑅𝑒 =𝜌𝑔 ∗ |𝑢𝑙 − 𝑢𝑔| ∗ 𝑑𝑙𝜇 (3.3) 𝑃𝑟 =𝑐𝑝 ∗ 𝜇𝑘𝑡ℎ𝑒𝑟𝑚𝑎𝑙 (3.4) where, 𝑁𝑢, 𝑅𝑒, and 𝑃𝑟 are the Nusselt Number, Reynolds Number, and Prandtl Number respectively. 𝑐𝑝, 𝜇, and 𝑘𝑡ℎ𝑒𝑟𝑚𝑎𝑙  heat capacity, viscosity and thermal conductivity of the gas respectively. This heat transfer model works well when considering the interaction between the droplets and gaseous phase. The Gunn Model is chosen to estimate the heat transfer between the solid and gaseous phases as discussed in Chapter 1. This model is presented in Section 3.2.1.2. 3.2.1.2 Heat Transfer from Solid to Gaseous Phase Gunn (1978) took into account the void fraction (volume fraction of gas) when evaluating the heat transfer. He was able to explain the reason for different researchers obtaining different heat transfer coefficients for experiments which had the same Reynolds numbers. Gunn’s correlation is: 𝑁𝑢 = (7 − 10𝛼𝑔 + 5𝛼𝑔2)(1 + 0.7𝑅𝑒𝑠0.2𝑃𝑟13)+ (1.33 − 2.4𝛼𝑔 + 1.2𝛼𝑔2)𝑅𝑒𝑠0.7𝑃𝑟13  (3.5) This model assumes that the solid particles are spherical in diameter, and the diameter of these equivalent solid particles is calculated by the Sauter Mean Diameter. Note that this correlation has two limitations:  1. `0.35 ≤ 𝛼𝑔 < 1.0  2. 𝑅𝑒 ≤ 1𝑥105 41  where, 𝛼𝑔 is the volume fraction of gas. Because the values of the Reynolds number and voidage in this study satisfy these criteria, the Gunn correlation is utilized in our model. 3.2.1.3 Heat Transfer from the Solid Phase to Droplets Heat transfer between the liquid (droplets) and gaseous phase takes place at all times due to the continuous flow of the gaseous medium. However, heat transfer between the solid and liquid phases (droplets) cannot be treated the same because both the solid and liquid phases are discrete in nature. Heat transfer between these two discrete phases is assumed to take place only during collisions. As mentioned before, time-scale analysis showed that the time scale of collision was an order of magnitude smaller than the time scale required for heat transfer (see Appendix B for calculations). Hence, it can be assumed that the magnitude of heat transfer between the solid and liquid phases is small. Hence, heat transfer between the solid and liquid phases is neglected.   3.2.2 Model for Mass Transfer Agglomerates only form in this study if liquid bridges are present between the solids. The number of liquid bridges can be estimated based on the liquid content in this reactor. The liquid content in the reactor can be tracked by understanding the various mass transfer mechanisms between the phases in the reactor, depicted in Figure 3.4. 42   There are two types of mass transfer possible, as shown above: 1. Mass transfer due to vaporization (evaporation and boiling) and,  2. Mass transfer between phases during collisions.  3.2.2.1 Evaporation from Liquid Phase The model for evaporation presented in this section is based on the work of Choi et al. (2017). The main factor which controls the extent of evaporation from the liquid to the gaseous phase is the Acetone and PMMA Transfer by Collisions   Acetone                    State Change (Vaporization) Acetone                 State Change (Vaporization) Phase 2              (Particle/Agglomerate with Liquid Film)                    Solid + Acetone + PMMA Phase 3           (Liquid)         Acetone + PMMA Phase 1 (Gaseous)              Air + Vapor Figure 3.4 Mass Transfer Mechanisms Assumed in the Model 43  amount of acetone vapor in the reactor. This can be formulated by observing the system pressure and vapor pressure of the vapor state. We calculate the number of moles which can change state to vapor at any instant of time by: 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑀𝑜𝑙𝑒𝑠 = (𝑃∗𝑃− 𝑦∗) (3.6) where, 𝑃∗ is the vapor pressure and 𝑦∗ is the mole fraction of vapor in the gaseous phase and P is the pressure in the system. Another factor taken into account is the perimeter of the droplet given by: 𝑃𝑒𝑟𝑙 = 𝜋 ∗ 𝑑𝑙  (3.7) where 𝑑𝑙 is the diameter of the droplets and 𝑃𝑒𝑟𝑙 is the perimeter of the droplets. Next, the number of droplets (𝑛𝑙) in the computational cell needs to be taken into account, as the final expression for the mass transfer coefficient must be in kg/(m3-s) and not kg/droplet. The number of droplets in a computational cell can be calculated by: 𝑛𝑙 =𝛼𝑙 ∗ 𝑉𝐶𝑒𝑙𝑙𝜋 ∗𝑑𝑙36 (3.8) where, 𝑉𝐶𝑒𝑙𝑙 is the volume of the computation space. Next, the Sherwood number (𝑆ℎ) is calculated by the Ranz-Marshall equation as: 44  𝑆ℎ = 2 +  0.6 ∗ 𝑅𝑒0.5𝑆𝑐13 (3.9) where, 𝑆𝑐 is the Schmidt Number. The convective mass transfer coefficient (𝑘) is then derived from the definition of the Sherwood number by:  𝑆ℎ = 𝑘 ∗𝑑𝑙𝐷𝑐 (3.10) where, 𝐷𝑐 is the mass diffusivity. Finally, putting Equations (3.6), (3.7), (3.8) and (3.10) together and changing the units to kg/(m3-s), we obtain the expression for the mass transfer coefficient due to evaporation from droplets (𝑚lg _𝑒𝑣𝑎𝑝) as:  𝑚lg _𝑒𝑣𝑎𝑝 = 6 ∗ 𝑘 ∗ (𝑃∗𝑃− 𝑦∗) ∗𝛼𝑎𝑐𝑒𝑡𝑜𝑛𝑒_𝑙𝑑𝑙∗ 𝜌𝑔  (3.11) In the above equation, 𝛼𝑎𝑐𝑒𝑡𝑜𝑛𝑒_𝑙 is the volume fraction of acetone in the liquid phase. It is important to mention here that the vapor pressure of the liquid phase should be corrected to reflect only the acetone portion of liquid and not the PMMA portion. This correction is required because PMMA is an acrylic, and it does not change from a liquid to vapor state. Thus, its vapor pressure is very close to zero. The vapor pressure of the mixture is corrected using Raoult’s law, which states that the vapor pressure of a mixture is the sum of the products of the mole fraction and vapor pressure of each component. Because PMMA does not change state, its vapor pressure is zero. Thus, the vapor pressure of the liquid is calculated by: 45  𝑃∗ = 𝑃𝑎𝑐𝑒𝑡𝑜𝑛𝑒∗ ∗ (𝑀𝑜𝑙𝑒 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐴𝑐𝑒𝑡𝑜𝑛𝑒) (3.12) Using Equations (3.11) and (3.12), the mass transfer coefficient from the liquid droplets to the gaseous phase can be estimated for each computational cell. This equation is independent of the cell volume. 3.2.2.2 Boiling in the Droplet Phase The mass transfer due to boiling in the Droplet Phase is derived from the Evaporation-Condensation model. This model, unlike the evaporation model, is based on a heat balance: 𝑄𝑖𝑛 − 𝑄𝑜𝑢𝑡 = ?̇?lg _𝑏𝑜𝑖𝑙 ∗ 𝐻𝑙𝑔  (3.13) where, 𝑄and 𝐻 represent the rate of heat transfer and latent heat respectively. Heat transfer from droplets by conduction and radiation is neglected, and it is assumed that heat transfer occurs only by convection. Using this formula, we obtain, ℎ𝑡𝑐𝑙𝑔 ∗ 𝐴𝑖 ∗ 𝑛𝑙 ∗ |𝑇𝑙 − 𝑇𝑔|𝐻𝑙𝑔= ?̇? lg _𝑏𝑜𝑖𝑙 (3.14) where, 𝐴𝑖 is the surface area between the two phases and 𝑛𝑙 is the number density of the droplets. The heat transfer coefficient (ℎ𝑡𝑐) used in the above equation is calculated from the Ranz & Marshall (1952) correlation. The evaporation and boiling mass transfer coefficients described in Sections 3.2.2.2 and 3.2.2.1 together represent the mass transferred from the liquid to the gaseous phase. 46  3.2.2.3 Mass Transfer from the Droplet to the Film in the Solid Phase The next mass transfer coefficient calculates the rate at which acetone and PMMA are transferred from the liquid phase to the liquid film in the solid phase via the collision of droplets with solids. To calculate the mass transfer between the liquid and gaseous phases, the number of droplets, solids (number densities of droplets and solids), and the collision frequency in the computationa l space must be calculated. The number density of droplets (𝑛𝑑𝑙 ) in a computational space is calculated by:  𝑛𝑑𝑙 =𝑉𝑜𝑙𝑢𝑚𝑒 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐷𝑟𝑜𝑝𝑙𝑒𝑡𝑠𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑠𝑖𝑛𝑔𝑙𝑒 𝑑𝑟𝑜𝑝𝑙𝑒𝑡=6 ∗ 𝛼𝑙𝜋 ∗ 𝑑𝑙3 (3.15) where, 𝛼𝑙 and 𝑑𝑙 are the volume fraction and diameter of the droplets in the liquid phase. The number density of the solids (𝑛𝑑𝑠 ) is calculated using a similar formula, by taking the values of volume fraction and diameter of the solids instead of that of the liquid droplets. The collis ion frequency between solids and liquid droplets (number of collisions between droplets and solids per unit volume per unit time) (𝑛𝑐𝑜𝑙𝑙̇ ) can be calculated from the number densities and phase velocities (𝑢𝑙 and 𝑢𝑠) as: 𝑛𝑐𝑜𝑙𝑙̇ = 𝑛𝑑𝑠 ∗ 𝑛𝑑𝑙 ∗𝜋4∗ (𝑑𝑠 + 𝑑𝑙2)2∗ |𝑢𝑙 − 𝑢𝑠| (3.16) The total mass of liquid in contact with solids at any instant of time can be calculated as the product of the collision frequency and mass of a single droplet as shown below (Pougatch, 2011): ?̇?𝑙𝑠 = 𝑛𝑐𝑜𝑙𝑙̇ ∗𝜋 ∗ 𝑑𝑙3 ∗ 𝜌𝑙6=32∗ 𝛼𝑙 ∗ 𝛼𝑠 ∗1𝑑𝑠3∗ 𝜌𝑙 ∗ (𝑑𝑠 + 𝑑𝑙2)2∗ |𝑢𝑙 − 𝑢𝑠| (3.17) 47  This equation only calculates the mass of liquid that is in contact with the solids at any instant of time, but does not represent the mass of liquid which has been permanently transferred to the solids after collisions, i.e., after separation. This quantity is determined by the stickiness ratio which determines the fraction of liquid that remains with the solids after collisions. The stickiness ratio (𝜔) is given (Pougatch, 2011) by: 𝜔 = 1 − 0.35𝐵𝑑 𝑖𝑓 𝐵𝑑 < 2.86, 𝑒𝑙𝑠𝑒 𝜔 = 0 (3.18) In the above equation, 𝐵𝑑 is a dimensionless number derived from the Reynolds Number (𝑅𝑒) (Pougatch, 2011): 𝐵𝑑 =𝑅𝑒0.37𝐿𝑝0.1∗ √𝑑𝑙𝑑𝑝 (3.19) The dimensionless numbers included in Equation (3.19) are calculated by: 𝑅𝑒 =𝜌𝑙 ∗ |𝑢𝑙 − 𝑢𝑝| ∗ 𝑑𝑙𝜇𝑙 (3.20) and,  𝐿𝑝 =𝜎𝜌𝑙𝑑𝑙𝜇𝑙2  (3.21) where, 𝜎 is the surface tension of the liquid and 𝐿𝑝 is a dimensionless number. To complete the model for mass transfer between phases, a mass transfer coefficient between the liquid and gaseous phases must be calculated. 48  3.2.2.4 Evaporation from the Film Once liquid has been transferred from the droplet to the solid phase, forming a film, evaporation from this film leads to mass transfer to the vapor phase. The model for this type of mass transfer is based on the model for evaporation of liquid droplets presented in Section 3.2.2.1. The underlying physical process is similar with subtle differences; namely, evaporation happens from the surface of the liquid film on the solids rather than from individual droplets. Also, droplets are no longer assumed to be spherical. Instead, they are assumed to take the shape of a hemispher ica l cap on the surface of the solids (refer to Section 3.1). Taking these physics into account, the revised formula for mass transfer from the liquid film to the vapor phase is: 𝑚𝑠𝑔_𝑒𝑣𝑎𝑝 = (2𝑎) ∗ 𝑘 ∗ (𝑃∗𝑃− 𝑦∗) ∗ 𝑑𝑙𝑓𝑖𝑙𝑚∗6 ∗ 𝑉𝑜𝑙𝑢𝑚𝑒 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐴𝑐𝑒𝑡𝑜𝑛𝑒 𝑖𝑛 𝐹𝑖𝑙𝑚 ∗ 𝐴𝐿𝐶𝑑𝑠3 ∗ 𝜌𝑔  (3.22) The term 𝑑𝑙_𝑓𝑖𝑙𝑚 represents the equivalent diameter of the film and is obtained by dividing the volume of the film by its area. In other words, 𝑑𝑙_𝑓𝑖𝑙𝑚 =𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑡ℎ𝑒 𝐹𝑖𝑙𝑚𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐹𝑖𝑙𝑚 (3.23) The area of the film is obtained from the radius (𝑎) and height (ℎ𝑒), which are calculated as follows (see Section 3.3.3): 𝑎 = [3𝑉𝑓𝑖𝑙𝑚𝜋𝑠𝑖𝑛3𝜃2 − 3𝑐𝑜𝑠𝜃 + 𝑐𝑜𝑠3𝜃]13⁄ (3.24) 49  ℎ𝑒 = 𝑎1 − 𝑐𝑜𝑠𝜃𝑠𝑖𝑛𝜃 (3.25) where, 𝑉𝑓𝑖𝑙𝑚 = 𝜋 ∗ 𝑑𝑠3 ∗𝑉𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝐴𝑐𝑒𝑡𝑜𝑛𝑒 𝑖𝑛 𝑆𝑜𝑙𝑖𝑑 𝑃ℎ𝑎𝑠𝑒6 ∗ 𝑉𝑜𝑙𝑢𝑚𝑒 𝐹𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝑜𝑙𝑖𝑑𝑠∗ 𝐴𝐿𝐶 (3.26) Hence, substituting these, we obtain, 𝑑𝑙_𝑓𝑖𝑙𝑚 =𝑉𝑓𝑖𝑙𝑚𝜋 ∗ (𝑎2 + ℎ2) (3.27) The term “ALC” stands for available liquid content, representing the percentage of acetone in the solid phase that is available for evaporation. The “ALC” is necessary because some acetone is trapped inside the solid particles/agglomerates and must be accounted for when calculating the extent of evaporation. The value of “ALC” for particles is assumed to be 1.0, i.e., the entire acetone available to the particles is available for evaporation. The “ALC” for the film on an agglomerate is approximated as follows: 𝐴𝐿𝐶 =𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑃𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑃𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝐴𝑔𝑔𝑙𝑜𝑚𝑒𝑟𝑎𝑡𝑒𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐴𝑔𝑔𝑙𝑜𝑚𝑒𝑟𝑎𝑡𝑒=𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑃𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝐷𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝐴𝑔𝑔𝑙𝑜𝑚𝑒𝑟𝑎𝑡𝑒 (3.28) This model for phase interaction completes the mass transfer model, and when combined with the model for heat transfer, is used to estimate the extent of evaporation. 50  3.2.2.5 Boiling in the film on the Solid particles The mass transfer coefficient for boiling for the liquid film present on the solid particles is also modeled based on the evaporation-condensation model. However, it is significantly different from the method presented in Section 3.2.2.2 because the convective heat transfer coefficient used for free droplets is not valid for the film phase. A heat transfer coefficient which is valid for the particle phase must be employed. The Gunn model is chosen for heat transfer to calculate boiling in the film phase. The heat balance equation for this case is similar to the equation described in the boiling for the liquid phase. The final mass transfer coefficient is, 𝑚𝑠𝑔_𝑏𝑜𝑖𝑙 =ℎ𝑡𝑐𝑠𝑔 ∗ 𝐴𝑖 ∗ 𝑛𝑠 ∗ |𝑇𝑠 − 𝑇𝑔|𝐻𝑠𝑔 (3.29) where, 𝐴𝑖 is the interfacial area, ℎ𝑡𝑐 is the heat transfer coefficient, 𝑛𝑠  is the number density of the solids and 𝐻𝑠𝑔  is the latent heat of vaporization of acetone. It is important to note that boiling occurs only when the liquid reaches the saturation temperature. This means that the temperature of the solid and liquid phases in their respective formulae for boiling is the saturation temperature of the liquid. The evaporation and boiling models work simultaneously to calculate the liquid content in the reactor. This leads to the next stage of the project, choosing an appropriate model for agglomeration. The model chosen for agglomeration is described in Section 3.3. 3.2.2.6 Diffusion of Liquid Film in the Solid Phase Section 3.2.2.3 presented a model to calculate mass transfer from liquid droplets to the solid phase during particle-droplet collisions. However, mass transferred to the film in the solid phase does not stay with the solid it was transferred to. It spreads to other particles due to collisions and the random motion of particles. While this study does not consider liquid transfer between the films 51  on the solid particles by solid-solid collision, it considers mass diffusion of liquid due to the random motion of particles. This study uses the formula presented by Hsiau et al. (1992): 𝐷𝑎𝑐𝑒𝑡𝑜𝑛𝑒−𝑓𝑖𝑙𝑚 =𝑑𝑠 ∗ 𝛼𝑠 ∗ √𝜋 ∗ 𝜃𝑔𝑟𝑎𝑛8 ∗ (1 + 𝑒) ∗ 𝑔0 (3.30) where, 𝜃𝑔𝑟𝑎𝑛 , 𝑔0, and 𝑒 are the granular temperature, radial distribution function and coefficient of restitution. 𝐷𝑎𝑐𝑒𝑡𝑜𝑛𝑒−𝑓𝑖𝑙𝑚 is the diffusion of acetone in the film of the solid phase. It is to be noted that this equation is not used to predict the amount of liquid attached to solids. Instead, it is used to calculate the mass of liquid which moves (diffuses) due to the random motion of solids. 3.3 Model for Calculating the Rate of Agglomeration Section 2.4 briefly discussed the method for estimating the diameter of solids. The expression used to solve the rate of agglomeration in this study is based on the work of Rajniak et al. (2009). This model is also built on the foundation provided by Kumar (2015), an internal report at the Univers ity of British Columbia. 𝑎(𝑑𝑖 , 𝑑𝑗) = 𝐸𝑓𝑓𝑖𝑗 𝑔0 (𝜋𝜃𝑔𝑟𝑎𝑛2)0.5. 𝑑431.5. (𝑑𝑖 + 𝑑𝑗)2(1𝑑𝑖3+1𝑑𝑗3)0.5𝛼𝑠2  (3.31) The term 𝐸𝑓𝑓𝑖𝑗 represents the probability of agglomeration for the representative particles/agglomerates “i” and “j.” It is the product of physical and geometrical probabilities (𝐸𝑓𝑓1 and 𝐸𝑓𝑓2 respectively). The term 𝑑43  represents the phase diameter of the solid phase.  52  The physical and geometrical probabilities along with the liquid distribution are discussed in the following sections. 3.3.1 Physical Probability of Collision For a collision to lead to successful agglomeration, it is assumed that the energy of the rebound must be less than the sum of the work done by viscous and capillary forces (Van Der Waals forces are neglected in this study). If this criterion is not satisfied, then the liquid bridge will be unable to dissipate the rebound energy of the collision. This excess energy causes a rupture in the liquid bridge leading to a failure to agglomerate.  The kinetic energy of separation of two colliding particles is first calculated by multiplying the coefficient of restitution by the kinetic energy of the collision. The kinetic energy of separation is calculated from the granular temperature as shown by Hounslow (1998):  𝐾𝑖𝑛𝑒𝑡𝑖𝑐  𝐸𝑛𝑒𝑟𝑔𝑦 𝑜𝑓 𝑆𝑒𝑝𝑎𝑟𝑎𝑡𝑖𝑜𝑛 =  12∗ 𝑒 ∗ 𝑚0 ∗ 𝑣𝑟𝑒𝑙2   =12∗ 𝑒 ∗ 𝑚0 ∗ [16𝜋∗ √3 ∗ 𝜃𝑚𝑖𝑥𝜌∗ (1𝑑𝑖3+1𝑑𝑗3)]2 (3.32) The term 𝜃𝑚𝑖𝑥   (granular temperature of the mixture) is calculated by ( Hounslow, 1998): 𝜃𝑚𝑖𝑥 = 𝜌𝑠𝜋6𝑑433 𝜃𝑔𝑟𝑎𝑛  (3.33) The equations required to calculate the work done by the viscous and capillary forces are based on Pitois et al. (2001) and Girardi et al. (2016): 53  𝑊𝑐𝑎𝑝 = 2 ∗ 𝜋 ∗ cos(𝜃)∗[     ((1 + 0.5 ∗ 𝜃) ∗ (1 − 𝐴𝐴) ∗ (8 ∗ 𝑉𝐵𝑟𝑖𝑑𝑔𝑒𝑑433)13)+   √2 ∗ (8 ∗ 𝑉𝐵𝑟𝑖𝑑𝑔𝑒𝑑433 )𝜋]       (3.34) 𝑊𝑉𝑖𝑠𝑐 = 1.5 ∗ 𝜋 ∗ 𝜇 ∗ 𝑣𝑟𝑒𝑙 ∗𝑑4322∗ [log (𝐴𝐴 ∗ √𝜋(1 + 𝐴𝐴)2) − 𝐹 ′]  (3.35) where, 𝐹′ = 𝑙𝑜𝑔(2 ∗ ℎ𝑎𝑑43) − 2 ∗ 𝑙𝑜𝑔[    2 ∗ ℎ𝑎𝑑43+√(2 ∗ ℎ𝑎𝑑43)2+2 ∗8 ∗ 𝑉𝐵𝑟𝑖𝑑𝑔𝑒𝑑433𝜋]     + 0.5∗ log [(𝜋 ∗ (2 ∗ ℎ𝑎𝑑43)2) + 2 ∗ (8 ∗ 𝑉𝐵𝑟𝑖𝑑𝑔𝑒𝑑433)]   (3.36) where, 54  𝑉𝑏𝑟𝑖𝑑𝑔𝑒 =12(𝑉𝑙𝑝,𝑖 + 𝑉𝑙𝑝,𝑗) ∗ (1 −√32) (3.37) and, 𝐴𝐴 = 1 +(  2 ∗ (8 ∗ 𝑉𝐵𝑟𝑖𝑑𝑔𝑒𝑑433 )13𝜋 ∗ (1 + (𝜃2))2)  12 (3.38) These equations for the work done by the viscous and capillary forces (Equations (3.34) through (3.36)) are employed to calculate the maximum rebound energy which the liquid bridge can dissipate. If the energy of rebound is less than the sum of viscous and capillary forces, then 𝐸𝑓𝑓1 =1. Otherwise 𝐸𝑓𝑓1 is assigned a value of 0.  3.3.2 Geometric Probability of Collision It was noted earlier that the liquid film in the solid phase does not coat the entire solid particle. Instead, it is present in the form of a hemispherical film on a portion of the solid. Hence, when two particles collide, there is a chance that agglomeration might not happen due to the absence of liquid film at the point of collision. This is not considered in the physical probability, and an additiona l probability term (𝐸𝑓𝑓2) is required to take this into account. While the physical probability (𝐸𝑓𝑓1) only considers the kinetic energy of the seperation, the geometric probability (𝐸𝑓𝑓2) is used to predict the probability of liquid film being present at the point of collision. This study assumes that 𝐸𝑓𝑓2 is 1 at this stage, i.e., collisions happen only in the areas where the liquid film is present.  This is not so in reality, and needs further work in the future. Generally a mass based approach is adopted to calculating the value of this term. However, it is proposed that an area based calculat ion 55  be adopted when modeling the geometric probability. This is because in the reactor under consideration, liquid does not wet the particle completely, and an area based approach would be a better fit. It is proposed to adopt the formulae mathematical formulae proposed in the Appendix of Rajniak et al. (2009) by substituting the mass terms with area of the film and solid. The term 𝐸𝑓𝑓 is considered to be equal to:  𝐸𝑓𝑓 = 𝐸𝑓𝑓1 ∗ 𝐸𝑓𝑓2 (3.39) With the efficiencies calculated, in order to complete the agglomeration model, the liquid in each cell must be distributed between the various particles and agglomerates. This model is described in the following section. 3.3.3 Liquid Distribution in Solid Phase The calculation of the physical and geometric efficiencies requires knowledge about how liquid present on film in the solid phase is distributed to the individual particles/agglomerates. The method adopted here is based on the volume of solids because the liquid transferred to the film on the solid phase is stored within the agglomerates instead of just residing on the surface. Hence, it is logical to assume that the liquid is distributed to each particle in proportion to its volume. Using this approach, the liquid is distributed to each bin based on its solid volume fraction. Finally, it is assumed that the liquid is evenly distributed to all particles within the bin. The volume of liquid assigned to each bin is calculated by: 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝐿𝑖𝑞𝑢𝑖𝑑 𝑡𝑜 𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒 =𝑇𝑜𝑡𝑎𝑙 𝐿𝑖𝑞𝑢𝑖𝑑 𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒𝑉𝑜𝑙𝑢𝑚𝑒 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑜𝑙𝑖𝑑𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑖𝑛𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑜𝑙𝑖𝑑𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑏𝑖𝑛 (3.40) 56  This method is an approximation used to estimate the liquid distribution in the solid phase. It is appropriate for this research because the frame of reference cannot track the liquid distribution between particles. In addition, if the exact distribution of liquid were to be calculated, then the pore size and porosity of the agglomerates would be required, information that is not available at this point. 3.4 Summary This chapter defines the equations for calculating the rate of mass transfer and agglomeration. It also proposes models for calculating the heat transfer coefficients. The predictions obtained based on these equations are presented in Chapter 4. Chapter 4 also discusses the properties of materials in the reactor. 57  Chapter 4: Results and Discussion This chapter discusses the material properties and simulation parameters. Following this discussion, the numerical results are presented as well as various graphs and contours. Whenever possible, these results are compared with experimental results. 4.1 Simulation Parameters The physical properties and operating conditions utilized in the numerical simulations performed in this study are based on the experiments performed by Morales (2013). The experimental setup was already discussed in Chapter 1. This section describes aspects not covered in Chapter 1, such as the material properties, geometry, mesh and boundary conditions. 4.1.1 Material Properties Material properties, such as the density, viscosity, specific and latent heat, affect the bed hydrodynamics and heat and mass transfer between the phases. Hence, these must be defined correctly to obtain accurate results from the numerical simulations. The material properties used in this study are provided in Table 4.1.  Table 4.1 Material Properties Physical Property Value Particle Diameter 210 microns Droplet Diameter 150-400 microns Density of Particles 2650 kg/m3 Density of Acetone 791 kg/m3  58  Table 4-1 Material Properties Density of PMMA 1180 kg/m3 Contact Angle of Liquid 320 Boiling Point of Acetone (at 1 atm) 580C Liquid Viscosity 2.2 cP Liquid Surface Tension 22.6 mPa.s Latent Heat of Vaporization 451 kJ/kg Specific Heat of Liquid 1574 J/kg-K Mass of Solids in Bed ~160 kg Molecular Weight of PMMA 100.12 g/mol Molecular Weight of Acetone 58.08 g/mol  These properties were obtained from Morales (2013) and the Dortmund Data Bank (DDB) website (“Dortmund Data Bank - DDBST GmbH,” n.d.). The properties are assumed to not vary with temperature. 4.1.2 Simulation Setup This study employed the use of Phase Coupled SIMPLE scheme in ANSYS Fluent 17.2 with the Second Order Upwind method in space and the First Order Implicit method in time. ANSYS Fluent employs the Finite Volume Method (FVM). The numerical simulations were conducted using a 3-D representation of the Free Board section of the Fluidized Bed Reactor (FBR). The expander section was not modeled because it was observed that very few particles escaped from the Free Board section of the FBR. The geometry used in this study is shown in Figure 4.1. 59   The nozzle is located at a height of 0.38 m above the floor of the bed and is operated at a gas to liquid ratio (GLR) ratio of 3.6%. The nozzle is 2.7 mm in diameter, and it is difficult to pinpoint the nozzle on the figure because of the bed dimensions and the relatively small size of the nozzle. Three-phase flow is considered in this research with the three Eulerian phases as follows:  Gaseous Phase: Comprised of air and acetone vapor  Liquid Phase: Droplets of mixture of acetone and PMMA 1.5 m Figure 4.1 Geometry of the Free Board Section of the FBR Nozzle of 2.7 mm diameter (not to scale) 60   Solid Phase: Silica sand particles that may be covered by a layer of acetone and PMMA mixture The boundary conditions were as follows (Morales, 2013):  The inlet boundary condition at the floor of the bed was air entering uniformly at a superficial velocity of 0.3 m/s at 341.15 K. The inlet condition for all other phases was a velocity of 0 m/s.  The outlet boundary condition was a pressure set to the atmospheric pressure across the top of the bed.  The liquid flow rate at the nozzle was assumed to be 30 g/s (90% acetone and 10% PMMA by weight). The air flow rate was chosen to ensure a 3.6% GLR. The temperature was 294.15 K. No solid phase enters the domain through the nozzle. Air was chosen for simulations instead of pure nitrogen which was used in the experiments. As the properties of air and nitrogen are similar it would not affect the results significantly.  The final aspect of the simulation is the Population Balance Method (PBM). Chapters 1 and 2 note the basic theory regarding the PBM and describe the function of the bins in the PBM and the choice of the discretization method for the equations in the PBM. This study used 7 bins, starting at 210 microns, with the maximum agglomerate diameter of 3,860 microns. Seven bins were chosen because this was the minimum number required to get a bin diameter nearly equal to 355 microns.  This bin was required because the results presented by Morales (2013) started from 355 microns.  The bins were numbered 0-6, and their respective sizes are shown in Table 4.2. 61  Table 4.2 Bin Sizes Bin Number Bin Size 0 0.00386 m 1 0.00237 m 2 0.00146 m 3 0.00090 m 4 0.00055 m 5 0.00034 m 6 0.00021 m  After the simulation was set up as discussed above, the next step was to choose an appropriate mesh to ensure that the discretization error would not significantly affect the results.  4.2 Mesh Selection This section describes the mesh independence study conducted to choose the appropriate mesh. Three different meshes of sizes 38,399, 64,198 and 117,148 elements were chosen for this purpose. This section begins with the mesh independence study and then proceeds to discuss the construction of the chosen mesh. 4.2.1 Mesh Independence Study In this study, the model is complex with many different physical phenomena modeled, includ ing agglomeration, evaporation, boiling and heat transfer. Hence, it is challenging and time-consuming to perform a mesh independence study which considers all these phenomena. Therefore, this mesh-independence study was simplified by considering the bed hydrodynamics in the absence of liquid 62  injection. This choice was made after observing that the behavior of particles was most affected by mesh refinement, focusing only on the behavior of particles, and thus simplifying the complexity of the mesh independence study. The study proceeded as described below.  At time t = 0 s, the solids were patched in the bed at a packing fraction of 0.5, starting at the floor of the bed and extending to a height of 0.68 m. The bed was then fluidized with only the inlet boundary condition for 3 s to reach a quasi-steady state. The state was determined by observing the volume integral of the velocity derivative in the Y-Direction (in other words ∫ (𝑑𝑢𝑠⃗⃗ ⃗⃗ 𝑑𝑦∗ 𝑑𝑦)). Once the quasi-steady state was reached (at time t = 3 s), the mid-plane parallel to the ZX plane was considered. The pressure profile was then plotted along the centre line of this plane (along the Z-Axis) for three mesh sizes, as shown in Figure 4.2.    Figure 4.2 Pressure Profile along Centre Line of Mid-Plane 63  The pressure profile shows that the slope of pressure in the reactor changes abruptly at a certain point. This point approximately demarcates the height of the fully expanded bed. It further shows that the fully expanded height of the bed is the lowest in the case of 38,399 elements at approximately 0.72 m and is approximately 6% greater than the case with 64,198 elements. In addition, the height of the bed with 64,198 and 117,148 elements is approximately the same at approximately 0.68 m. The difference in the height reported in these two cases is approximate ly 0.9%, and agreement is not expected to improve further with mesh refinement because the experimental, fully expanded height of the bed was reported to be 0.68 m. Hence, we can conclude that the pressure profiles do not vary significantly and that the behavior of particles in the bed does not change much with mesh refinement beyond 64,198 elements. Therefore, a mesh of 64,198 elements was chosen for the simulations. 4.2.2 Mesh Description The mesh used in this study is shown in Figure 4.3.  64    It can be observed that there is an area of high mesh density near the lower half of the bed. This refinement is in the vicinity of the nozzle and is necessary because there are significant temperature and pressure differentials near the nozzle caused by liquid injection at a lower temperature. The absence of a finely refined mesh in this region would cause significant errors. The figure also shows that the size of the cells gradually increases away from the refined region. The cells were biased in this manner to avoid divergence issues; the cell size was defined by edge sizing and bias Figure 4.3 Mesh of 64,198 Elements 65  factors (ratio of largest division to the smallest division of the edge in consideration) as shown below:  X-Axis: In this axis, there are 40 divisions without any bias.  Y-Axis: In this axis, there are 25 divisions without any bias.  Z-Axis in the Lower Part of the Bed: In the lower part of the bed, there are 20 divis ions with a bias factor of 3.  Z-Axis in the Upper Part of the Bed: In this part of the bed, there are 40 divisions with a bias factor of 5.  4.3 Experimental Results The primary objective of this study is to develop a suitable model for agglomeration. However, Chapter 3 indicated that the success and extent of agglomeration are based on the physical and geometrical efficiencies. These efficiencies are in turn dependent on the liquid content. Hence, the agglomeration model must be accompanied by a model which can predict with sufficient accuracy the extent of vaporization. Morales (2013) observed that at the end of injection, approximately 75 grams of acetone are left in the reactor at the process conditions described in Section 4.1.2. The extent of vaporization is thus calculated as below: 𝐸𝑥𝑡𝑒𝑛𝑡 𝑜𝑓 𝑉𝑎𝑝𝑜𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 (%)=𝑇𝑜𝑡𝑎𝑙 𝐿𝑖𝑞𝑢𝑖𝑑 𝐼𝑛𝑗𝑒𝑐𝑡𝑒𝑑 − (𝑀𝑎𝑠𝑠 𝑜𝑓 𝐴𝑐𝑒𝑡𝑜𝑛𝑒 𝑎𝑛𝑑 𝑃𝑀𝑀𝐴 𝐿𝑒𝑓𝑡 𝑖𝑛 𝑅𝑒𝑎𝑐𝑡𝑜𝑟)𝑇𝑜𝑡𝑎𝑙 𝐿𝑖𝑞𝑢𝑖𝑑 𝐼𝑛𝑗𝑒𝑐𝑡𝑒𝑑∗ 100% (4.1) 𝐿𝑖𝑞𝑢𝑖𝑑 𝑉𝑎𝑝𝑜𝑟𝑖𝑧𝑒𝑑 =1350 − (75 + 135)1350∗ 100% (4.2) 66  𝐿𝑖𝑞𝑢𝑖𝑑 𝑉𝑎𝑝𝑜𝑟𝑖𝑧𝑒𝑑 =11401350∗ 100% = 84.4% (4.3) Hence, the numerical results shall be near this number. Another important consideration is the extent of agglomeration. The extent of agglomeration in Morales’ experiments is shown in Figure 4.4.    Figure 4.4 Morales' Experimental Results Showing the Agglomerate Size Distribution after 45 s (Figure taken from Morales et al. (2016) published in the Canadian Journal of Chemical Engineering and copyright permission obtained from John Wiley and Sons (This image is Copyright © 2016, John Wiley and Sons)) 67  It can be observed from the above figure that at the end of the experiment, only 5% (wt%) of the total particles exist in the form of agglomerates of diameter ≥ 355 microns. Therefore, while other particles exist in the form agglomerates during intermediate time steps, at the end of 45 seconds only 5% of the total solids are in the form of agglomerates. Simulations show that the rate of agglomeration is proportional the number of agglomerates. The rate of agglomeration when 5% of solids were present as agglomerates, was quite low. Hence, it is logical to assume that the extent of vaporization in the absence of agglomeration must be close to the experimental results (84.4%). The effect of solids which keep agglomerating and breaking is neglected because their number is relatively low. This conclusion was drawn after observing the extent of agglomeration in simulations. Simulation results show that in the absence of breakage, on an average, 3.8% of particles form agglomerates per second (refer to Figure 4.12).  4.4 Results for Vaporization Following the logic described in Section 4.3, the vaporization model was first tested under the conditions described by Morales (2013) by disabling the agglomeration kernel. This section presents an analysis of the results obtained from the stand-alone vaporization model. 4.4.1 Results for Vaporization at 680C For the first test, the temperature of the FBR was initially 680C, and agglomeration was disabled. Liquid was injected through the nozzle at a mass flow rate of 30 g/s and at 294.15 K. The diameter of the liquid droplets was 200 microns. The numerical results obtained for this case are presented in Figure 4.5. The percentage of liquid vaporized was calculated based on Equation (4.1). 68    The above graph indicates that the extent of vaporization changes rapidly in the initial time steps. However, after time t = 25 s the change in the extent of vaporization is very minor. The extent of vaporization at the end of the simulation (t = 48 s) is approximately 88.6%. These results are very close to the experimental results calculated in Equation (4.3), approximately 4.2%. The difference can be attributed to the absence of agglomeration and the assumptions of the numerical model.  The shape of the curve can be explained by observing the contours which represent the Mean (time-averaged) Mass Fraction of Acetone in the Film of the Solid Phase and the Mean (time -averaged) Volume Fraction of the Liquid Droplets. These contours of the mid-plane parallel to the X-Z plane (y = 0.075 m) at different time steps are presented in Figure 4.6.  Figure 4.5 Percentage of Liquid Vaporized for 200 micron Droplets Injected into the FBR 69    a) b) d) c) Figure 4.6 Mean (Time-Averaged) Mass Fraction of Acetone in the Film of the Solid Phase over Time t = 0 s to a) t = 3.5 s b) t = 5.5 s c) t = 12.5 s d) t = 22.5 s at y = 0.075 m 70  It can be observed that initially there is considerable acetone in the vicinity of the nozzle. However, as time progresses, the acetone species (in the film of the Solid Phase) starts to spread throughout the bed, and the acetone content in the vicinity of the nozzle appears to reach a nearly constant value. The contours also show that when this state is achieved, the acetone content in the vicinity of the nozzle is lower than that in the initial time steps. Hence, most of the acetone in the film phase is near the nozzle at any given point in time, and this quantity reaches a nearly steady state over time. The vaporization curve therefore flattens out. It is also to be noted that the above figure is merely the mass fraction contour of the acetone in the film of the solid phase. The numerica l values must be multiplied by the mass of the solid phase to calculate the content of acetone at any location. Figure 4.6 indicates that most of the acetone is in the vicinity of the nozzle. Therefore, it is expected that most evaporation and boiling from this film takes place in this region. Hence, it is expected that the temperature is lower in this region. This temperature difference is observed in the contours shown in Figure 4.7. Figure 4.7 shows the instantaneous temperature of the solid phase at the mid-plane parallel to the X-Z plane (y = 0.075 m). The contours shown in this figure are confined to the region near the nozzle and do not depict the temperature contour of the entire bed. The contours are limited to this region because it is here that most of the acetone is present, and therefore this would be the only region where the temperature is ≤ 331.15 K. Note also that contours in Figure 4.6 depict time-averaged values, whereas the contours in Figure 4.7 depict instantaneous values.  71    The above contours represent the instantaneous temperatures of the solid phase and not time-averaged temperatures. The next contours which were observed were those of the predicted instantaneous solid volume fraction. These contours are shown in Figure 4.8.  a) b) c) Figure 4.7 Instantaneous Temperature of Solid Phase at Time a) t = 3.25 s b) t = 17.75 s c) t = 29.75 s at y = 0.075 m 72   The above figure shows the evolution of the particles in the bed over time. These contours show that there exist bubbles which pass through the bed, and the solids behave in accordance with a) b) c) d) Figure 4.8 Instantaneous Volume Fraction of Solid Phase at Time a) t = 41 s b) t = 41.25 s c) t = 41.5 s d) t = 41.75 s at y = 0.075 m 73  experiments. The fully expanded height of the bed was observed to be 0.7 m and to be within 5% of what was reported for the experiments. There was no escape of solids from the bed at any given time step.  4.4.2 Effect of Boiling In Section 4.4.1 the FBR was operated at 680C, i.e., 100C above the boiling point of 100% pure acetone at 1 atm. The vaporization model for this case returned values close to those reported by experiments, and the next test was designed to test the effect boiling has on the extent of vaporization. For this test, a case was run with the temperature FBR at 580C. At this temperature, there is no latent heat supplied to the acetone species. Therefore, in this case, only evaporation occurs and boiling is not expected. Note that agglomeration was disabled in this case.   Figure 4.9 Comparison of Vaporization Rates for Different FBR Temperatures 74  It was observed that initially, the extent of vaporization was approximately 13% lower when compared to the previous case. However, as time progresses, it appears that the extent of vaporization in this case nearly equals that of the previous case. To analyze the reason behind this difference in trends, the rates of boiling and evaporation are analyzed for these cases. The analysis is shown in Table 4.3.  Table 4.3 Mass Transfer Rates at Different Time Steps  In the above table, one can observe that initially the vaporization rate (sum of the rates of evaporation and boiling) for the acetone in the film on the solid particles when the FBR is at 680C (referred to as Case 1 in this section) is significantly (approximately 57%) greater than the rate of evaporation when the FBR is at 580C (referred to Case 2 in this section). However, as time progresses, this difference drops to 8.6%. Another important observation is that, as expected, the rate of vaporization due to boiling is significantly higher than the rate of evaporation (in the case where the temperature of the FBR was 680C). Further, the results of vaporization for the droplets (liquid phase) are not included because Time Mean (Time-Averaged) Mass Averaged Rate of Film Evaporation with FBR at 680C Mean (Time-Averaged) Mass Averaged Rate of Film Boiling with FBR at 680C Mean (Time-Averaged) Mass Averaged Rate of Film Evaporation with FBR at 580C 4.5 s 0.0194 kg/(m3-s) 0.0931 kg/(m3-s) 0.0714 kg/(m3-s) 20 s 0.0285 kg/(m3-s) 0.0883 kg/(m3-s) 0.1075 kg/(m3-s) 75  it was observed to be at least an order of magnitude smaller than for the results of the liquid film on the solid particles. These observations lead to the following conclusions: i. When liquid is injected into the reactor in Case 1, the higher temperature of the FBR plays a significant role in the rate of vaporization during the initial time steps due to boiling. However, as time progresses, the rate of evaporation in Case 2 increases and becomes roughly equal to the rate of vaporization in Case 1. The reason for this could be that, as time progresses, a higher amount of liquid in Case 2 reaches the boiling point, causing the evaporation rate to also increase.  ii. As time progresses, the extent of vaporization reaches a nearly quasi-steady state. At this stage, the influence of boiling is not as significant compared to the initial time steps. The reduced influence could be because, as time progresses, the temperature of solids near the nozzle marginally decreases. This reduction in temperature in turn leads to a reduced rate of boiling. To conclude, the results indicate that the vaporization model performs well, giving a close fit with experimental results. The model responds to changes in temperature appropriately. Having concluded this, the agglomeration model was enabled. 4.5 Results for Agglomeration After obtaining acceptable results for the vaporization model, the extent of agglomeration and its effect on the extent of vaporization was observed. For this case, droplets of size 200 microns were used (refer to the assumptions listed in Chapter 3), and agglomeration was allowed. The bins in the Population Balance Method (PBM) were defined as described in Table 4.2. Note that the agglomerate size was restricted to 3,860 microns, well below the experimental maximum size of 76  10,000 microns. This limitation was needed because no breakage model was implemented and the agglomerates could grow and cause defluidization of the FBR. In addition, experimental results showed that at the end of 45 s, less than 1% of the solids formed agglomerates of a diameter greater than 4000 microns. It was also observed in the simulations that when there were fewer agglomerates, the rate of agglomeration was slow. Hence, it is logical to assume that agglomerates of diameter greater than 4000 microns have a negligible impact on the phase interactions in the FBR. Before looking at the agglomerate size distribution, the extent of vaporization was observed. The extent of vaporization, in this case, is compared with the results obtained from the case where agglomeration was disabled and is plotted in Figure 4.10.   Figure 4.10 Effect of Agglomeration on the Extent of Vaporization 77  This figure indicates that the initial extent of vaporization is larger when agglomeration is disabled. In these initial time steps, the extent of vaporization in the presence of agglomeration is lower. However, as time progresses, the extent of vaporization in both cases reaches a nearly constant value, and the vaporization rate is nearly the same for both cases. Hence, at this point, it appears that agglomeration reduces the extent of vaporization in the initial period, but soon the effect of agglomeration on the rate of vaporization loses its significance. The initial difference could be because agglomeration reduces the extent of evaporation due to the Available Liquid Content (ALC) which does not allow evaporation to take place from a portion of the acetone, as described in Chapter 3. However, boiling can affect this quantity of acetone, but only when it reaches the boiling point. Hence, initially, the vaporization rate is low due to reduced evaporation, but, as time progresses and more acetone reaches the boiling point, the effect of evaporation on the rate of vaporization is reduced. At the time t = 14.125 s, it was observed that approximately 53% (wt %) of the particles had already formed agglomerates of the largest diameter. This is around 10 times more than expected, and the behavior of the solids in the FBR would significantly change. To confirm that the behavior of solids changes at 14.125 s, the contour depicting the instantaneous solid volume fraction was observed, as shown in Figure 4.11. 78   In the above contour, we observe that there are very few bubbles, and that there is noticeable contraction in the height of the bed (compared with the contours shown in Figure 4.8). This change indicates that the bed has started to defluidize and will continue to do so as time progresses. Hence, the simulation was stopped at this point, and the agglomerate size distribution was observed. The cumulative weight distribution of agglomerates (of diameter greater than or equal to 340 µm) obtained from the numerical simulations at simulation time t = 14.125 s is shown in Figure 4.12. Please note, that in the following figure, the cumulative weight percent of solids at any diameter has been calculated as the percent of solids greater than or equal to the diameter under consideration. These results are not compared with values reported by Morales (2013) because a Figure 4.11 Predicted Instantaneous Solid Volume Fraction of Solid Phase at t = 14.125 s at y = 0.075 m 79  breakage model has not been implemented in the code, and without a breakage model, the agglomerates would grow indefinitely. Therefore, comparison between agglomerate distributions in this work with experimental results would not be meaningful.    The following observations can be made from Figure 4.12: i. The extent of agglomeration is overestimated by an order of magnitude. (It was expected that approximately 5% of solids form agglomerates). ii. Most of the agglomerates in the numerical simulations are in the bin with the largest diameter, with only around 0.25% of agglomerates in the intermediate bins. This is a consequence of not having a breakage model. In its absence, agglomerates can only grow. However, when a breakage model is in place, large agglomerates can break to form multiple smaller agglomerates. Figure 4.12 Cumulative Agglomerate Size Distribution at Time t = 14.125 s for 200 micron Droplets 80  The absence of an appropriate breakage model can explain the above observations, and it is the most important issue. However, other issues could also be important, in particular, the geometrica l probability as described in Chapter 3. It was stated that for now the geometrical probability is assumed to be 1. However, in practical situations, liquid distribution on particles/agglomerates is not homogeneous. By not considering this, the extent of agglomeration may be grossly overestimated. In addition, in larger agglomerates, the geometrical probability becomes very relevant because there is a larger solid surface area, and this may reduce the success of agglomeration. Hence, this may explain why there are almost no agglomerates in the intermed iate bins (agglomerates of diameter greater than 350 microns and less than 4000 microns). 4.6 Droplet Diameter Studies This section presents results of parametric studies where the effect of droplet diameter on the extent of vaporization and agglomeration is studied.  4.6.1 Effect of Droplet Diameter in the Absence of Agglomeration In this section, the effect of droplet diameter on the extent of vaporization in the absence of agglomeration is studied. For this purpose, 4 droplet sizes were chosen (150 µm, 200 µm, 300 µm, and 400 µm). The simulation results are shown in Figure 4.13.  81   One would expect that the rate of vaporization would decrease with increasing diameter of droplets. Although this was observed in the above figure, it is also evident that the extent of vaporization is predicted to be almost independent of the droplet diameter. This can be explained by considering the mass transfer rates of boiling and evaporation for the droplets and the film.  The values for the rate of vaporization shown in Section 4.4.2 indicate that the rate of vaporizat ion from droplets is an order of magnitude smaller than the rate of vaporization from the film on the solid particles. Hence, most of the vaporization takes place from the film on the solid particles. The dimensions of the film dictate this rate of mass transfer. The film dimensions do not vary much with the variation in droplet diameter because of the assumptions made to establish the film dimensions (see Chapter 3 for the assumptions and method). It is to be noted that as droplet dimensions increase, the liquid film may ultimately engulf the particle. However, the droplet diameter and assumptions involved do not allow this to happen (see Section 3.3.3 for liquid Figure 4.13 Percentage of Liquid Vaporized as a Function of Time 82  distribution to the film). Hence, it is to be expected that the rate of vaporization does not vary much with variation of the droplet diameter.  4.6.2 Effect of Droplet Diameter on the Extent of Agglomeration The effect of droplet diameter on the extent of agglomeration is studied in this section. The first step was to observe the percentage of vaporization in the absence of agglomeration as shown in Figure 4.14.   The predicted agglomerate size distribution for different droplet diameters at time t = 3.625 s is observed in Figure 4.15. Figure 4.14 Effect of Droplet Diameter on the Extent of Vaporization 83   At this time, we can clearly observe that the extent of vaporization and the percentage of agglomerates increases with increasing the diameter of droplets. As the simulation progresses, it can be observed that the relative magnitudes of the extent of vaporization change. To understand the reason behind these changes, the agglomerate size distribution at a few critical time steps needs to be observed. Two critical times can be identified where the extent of vaporization changes: i. At time t = 7 s where the extent of vaporization in the case with 200 micron droplets drops below the extent of vaporization for cases where the diameter of the droplets was 300 µm and 400 µm. ii. At time t = 11 s where the extent of vaporization in the case with 300 µm droplets drops below the extent of vaporization for 400 µm droplets. Figure 4.15 Cumulative Weight Distribution of Agglomerates at Time t = 3.625 s 84  Hence, it is essential to observe the extent of agglomeration at these time steps and at time steps leading up to these times. The agglomerate size distribution for different time steps is shown in Figure 4.16 to Figure 4.19.  Figure 4.16 Cumulative Weight Percentage of Agglomerates at Time t = 4.5 s 85     Figure 4.18 Cumulative Weight Percentage of Agglomerates at Time t = 7 s Figure 4.17 Cumulative Weight Percentage of Agglomerates at Time t = 6.5 s 86   The trends shown in the above figures indicate that the extent of agglomeration is strongly related to the diameter of the droplets. Furthermore, when analyzing the vaporization curve, we observe that the presence of larger agglomerates decreases the rate of vaporization. It is particularly interesting to observe the trends at time t = 7 s where the extent of vaporization in the case with 200-micron droplets overtakes the vaporization in the case with 400-micron droplets. The size distribution of agglomerates shows that both cases have nearly the same number of agglomerates, after which, the number of agglomerates in the case with 200-micron droplets increases at a faster pace. This trend is repeated with the 300- and 400-micron droplets at time t = 12 s, after which, we observe that the extent of agglomeration decreases with increasing droplet diameter. Figure 4.19 Cumulative Weight Percentage of Agglomerates at Time t = 12 s 87  However, the above analysis is shown to illustrate the methodology rather than serve for comparison with experiments. The simplifying assumptions needed in the development of this model preclude such comparison because: i. The absence of a suitable breakage model allows agglomerates to grow indefinitely and continuously reduces the number of solid particles. It also does not allow for the presence of a significant quantity of agglomerates with a smaller diameter which may form due to breakage of larger agglomerates. ii. The absence of a geometrical probability term (see Chapter 3) in the agglomeration kernel allows for agglomeration to take place if the Stokes condition is satisfied. However, it does not take into account the non-homogeneous distribution of liquid film on the solid particles. Numerical simulations show that the absence of a geometrical probability does not affect the trends in the short term, but its necessity becomes evident within a few seconds of injection. Its absence changes results in the long term because the droplet behavior changes significantly with changing diameter, namely the mass transfer from droplets to the film in the solid phase. When this mode of mass transfer is influenced by a variation in droplet diameter, the distribution of liquid film on the particles becomes important. Hence, a suitable geometrical probability term is required. Given these deficiencies, one can rely on the trends predicted by this model only when the droplet diameter is within a certain range. To improve the predictions a breakage model and geometric probability must be implemented. The dependence of the vaporization curves on the extent of agglomeration are in line with our expectations. When the FBR has larger agglomerates, they reduce the extent of evaporation (see 88  ALC in Chapter 3). The reduced extent of evaporation leads to a reduction in the overall extent of vaporization. However, agglomeration has a lower impact on the extent of boiling and also does not have any direct effect on the rate of boiling. Because boiling is faster than evaporation, agglomeration has a reduced effect on the extent of vaporization. This explains why the vaporization trends are all close together even when there is a significant difference in the extent of agglomeration.  4.7 Conclusion The results presented in this chapter can be summarized as follows: i. The geometrical probability term is vital for the agglomeration model and must be included to obtain correct results. ii. A breakage model is essential to reproduce agglomerate behavior. iii. The diameter of agglomerates strongly influences the extent of vaporization. That the extent of vaporization was reduced with an increase in the percentage of agglomera tes in the bin with the largest diameter. iv. The extent of vaporization, however, does not vary much with changes in droplet diameter. v. Most of the vaporization takes place from the film in the solid phase and not from the droplets.  The next chapter will discuss these conclusions, as well as recommendations for future studies. 89  Chapter 5: Conclusions and Recommendations for Future Work This chapter presents the conclusions of the study and provides recommendations for future work based on the limitations of the current model. 5.1 Conclusions The conclusions of this study are: i. A model for agglomeration has been developed and applied to an experimental Fluidized Bed Reactor (FBR) comprised of air-fluidized particles of silica sand. Agglomeration was achieved by injecting a liquid mixture of acetone and an acrylic, poly-methyl-methacrylate (PMMA), through a nozzle placed at 0.38 m above the floor of the FBR. ii. A model capable of calculating the extent of agglomeration based on the available liquid content has been developed. It is based on physics with very few empirical parameters and, therefore, allows for the use of a model over a range parameter variation. The computed results were compared with experimental data with reasonable agreement in trends. Quantitative agreement cannot be achieved because of the simplifying assumptions that were necessary within the scope of this work.  iii. Parametric studies were carried out to investigate the effect of bed temperature. It was observed that the impact of boiling was crucial in the short term, but over time the impact of evaporation gains significance, and the overall rate of vaporization does not vary much. This result is caused by the relatively low-temperature difference between the bed and the liquid boiling temperature. iv. The effect of the droplet diameter was investigated in the absence and presence of agglomeration. In the absence of agglomeration, the extent of vaporization did not vary 90  significantly with the change of diameter. Upon further investigation, it was observed that the rate of vaporization was much faster from the liquid film on the solid phase compared with the rate from the droplet phase. The difference in the extent of vaporization exists because most of the acetone is present in the film, and not in the form of droplets. When liquid is transferred to the film, the assumption in this work is that the dimensions of the film do not vary much. Because the film dimensions dictate the vaporization rate, the rate does not vary much with a change in the droplet diameter. v. Agglomeration has a higher influence on the extent of evaporation than boiling. The impact on evaporation is due to the Available Liquid Content (ALC) term. The ALC approximates the quantity of liquid that is shielded from evaporation by agglomerates. Because evaporation is a slower process, a reduction in the rate of evaporation does not change the rate of vaporization to a large extent. Thus, the extent of vaporization is not significantly impacted by agglomeration. 5.2 Contributions This thesis employed existing models to estimate the extent of vaporization and agglomeration in a gas fluidized reactor. The important contributions of this work are: i. A novel model to estimate the extent of vaporization due to boiling and evaporation has been developed. This model considers the mass transfer of liquid from the droplets to the film and subsequent vaporization.  ii. A novel approach has been proposed to calculate the quantity of liquid available for evaporation when transferred to film on the agglomerates. 91  iii. A mathematical model which is based on the Kinetic theory of Granular Flows has been developed to estimate the extent of agglomeration. 5.3 Limitations and Recommendations for Future Studies Despite being a novel approach, this model suffers from a few limitations due to the assumptions made. This section identifies the most significant limitations which future researchers should consider when adopting or adapting this model. The most evident limitation of this model is the absence of agglomerate breakage and attrition. They were accounted for in a simplified way by limiting the growth to a given maximum size (approximately 4000 microns). This limit was chosen after examining the experimental results which showed that less than 1% (wt %) of the solids had diameters greater than 4000 microns. The breakage model is being developed in our group, and will be implemented in the future. In reality, agglomerate breakage and attrition take place to produce a larger percentage of smaller agglomerates. Also, once a breakage and attrition model is implemented, the size limitation on agglomerates can be removed, and there will be larger agglomerates. A significant limitation of this work is the absence of an appropriate geometric probability. This parameter would account for the probability that particles collide in the presence of liquid. This probability is required because the liquid is not distributed homogeneously, and not all portions of the particle are wetted by the liquid. Currently, the model assumes that all collisions take place in the presence of liquid. This assumption affects the rate and extent of agglomeration and, in turn, the size distribution of agglomerates. The inclusion of this term will reduce the extent of agglomeration, and this term is currently being estimated by our research group. 92  The next limitation of this model is the absence of a mechanistic method to calculate the liquid available for evaporation. Future studies can adopt formulae which consider the time required for imbibition of liquid and the porosity of the solids. The fourth limitation is the absence of a method by which the diameter of droplets is tracked. In this model, the diameter of liquid droplets is assumed to be constant at 200 micrometers and does not change with droplet-droplet or droplet-particle interactions. Future researchers can adopt/develop an appropriate model such as  the one proposed by Pougatch (2011) to consider these interactions and improve the accuracy of the overall model. It is to be noted that the effect of the nozzle was not considered in this study. 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Analysis of four Monte Carlo methods for the solution of population balances in dispersed systems. Powder Technology, 173(1), 38–50.  98  Appendices The three Appendices mentioned in the text are presented here. The equations for the Conservation of Mass for the different species are presented in Appendix A. Appendix B contains the calculations for the time scales of collision and heat transfer. Finally Appendix C contains the calculations by which the power of the heater was estimated.             99  Appendix A    The nomenclature used here is the same as described in the List of Symbols and Chapter 2.  i. Equation for the Conservation of Mass for Acetone in the Liquid Phase 𝜕𝜕𝑡(𝛼𝑙𝜌𝑙𝑌𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒 ) + ∇.(𝛼𝑙𝜌𝑙𝑢𝑙⃗⃗  ⃗𝑌𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒 )= −∇.𝛼𝑙𝐽 𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒 − (?̇?𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒 𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒  _ 𝑒𝑣𝑎𝑝 + ?̇? 𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒  _ 𝑏𝑜𝑖𝑙+ ?̇? 𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒  ) (A.1) ii. Equation for the Conservation of Mass for PMMA in the Liquid Phase 𝜕𝜕𝑡(𝛼𝑙𝜌𝑙𝑌𝑙𝑃𝑀𝑀𝐴) + ∇.(𝛼𝑙𝜌𝑙𝑢𝑙⃗⃗  ⃗𝑌𝑙𝑃𝑀𝑀𝐴) = −∇.𝛼𝑙𝐽 𝑙𝑃𝑀𝑀𝐴− ?̇? 𝑙𝑃𝑀𝑀𝐴𝑠𝑃𝑀𝑀𝐴   (A.2) iii. Equation for the Conservation of Mass for Acetone Vapor in the Gaseous Phase 𝜕𝜕𝑡(𝛼𝑔𝜌𝑔𝑌𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒 ) + ∇. (𝛼𝑔𝜌𝑔𝑢𝑔⃗⃗⃗⃗  𝑌𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒 )= −∇. 𝛼𝑔𝐽 𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒 + (?̇?𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒 𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒  _ 𝑒𝑣𝑎𝑝 + ?̇? 𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒  _ 𝑏𝑜𝑖𝑙+ ?̇? 𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒  _ 𝑒𝑣𝑎𝑝 + ?̇?𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒  _ 𝑏𝑜𝑖𝑙 ) (A.3) iv. Equation for the Conservation of Mass for Air in the Gaseous Phase 𝜕𝜕𝑡(𝛼𝑔𝜌𝑔𝑌𝑔𝑎𝑖𝑟 ) + ∇.(𝛼𝑔𝜌𝑔𝑢𝑔⃗⃗⃗⃗  𝑌𝑔𝑎𝑖𝑟 ) = −∇. 𝛼𝑔𝐽 𝑔𝑎𝑖𝑟 (A.4) v. Equation for the Conservation of Mass for Acetone in the Film of the Solid Phase 100  𝜕𝜕𝑡(𝛼𝑠𝜌𝑠𝑌𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒 ) + ∇.(𝛼𝑠𝜌𝑠𝑢𝑠⃗⃗⃗⃗  𝑌𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒)= −∇.𝛼𝑠𝐽 𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒 − (?̇?𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒  _ 𝑒𝑣𝑎𝑝 + ?̇? 𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒𝑔𝑎𝑐𝑒𝑡𝑜𝑛𝑒  _  𝑏𝑜𝑖𝑙− ?̇?𝑙𝑎𝑐𝑒𝑡𝑜𝑛𝑒 𝑠𝑎𝑐𝑒𝑡𝑜𝑛𝑒  ) (A.5) vi. Equation for the Conservation of Mass for PMMA in the Film of the Solid Phase 𝜕𝜕𝑡(𝛼𝑠𝜌𝑠𝑌𝑠𝑃𝑀𝑀𝐴 ) + ∇.(𝛼𝑠𝜌𝑠𝑢𝑠⃗⃗⃗⃗  𝑌𝑠𝑃𝑀𝑀𝐴 ) = −∇. 𝛼𝑠𝐽 𝑠𝑃𝑀𝑀𝐴+ ?̇? 𝑙𝑃𝑀𝑀𝐴𝑠𝑃𝑀𝑀𝐴  (A.6) vii. Equation for the Conservation of Mass for Silica Sand in the Solid Phase 𝜕𝜕𝑡(𝛼𝑠𝜌𝑠𝑌𝑠𝑠𝑖𝑙𝑖𝑐𝑎) + ∇. (𝛼𝑠𝜌𝑠𝑢𝑠⃗⃗⃗⃗  𝑌𝑠𝑠𝑖𝑙𝑖𝑐𝑎 ) = −∇. 𝛼𝑠𝐽 𝑠𝑠𝑖𝑙𝑖𝑐𝑎 (A.7)             101  Appendix B   The collision frequency of particles and droplets is calculated as (Refer to Chapter 2), 𝑛𝑐𝑜𝑙𝑙̇ = 𝑛𝑑𝑠 ∗ 𝑛𝑑𝑙 ∗𝜋4∗ (𝑑𝑠 + 𝑑𝑙2)2∗ |𝑢𝑙 − 𝑢𝑠| (A.8) Substituting 𝑢𝑙𝑥 = 0.03𝑚𝑠, 𝑢𝑙𝑦 = −0.745𝑚𝑠, 𝑢𝑠𝑥 = −3.6𝑚𝑠, 𝑢𝑠𝑦 = −0.078𝑚𝑠 and 𝜀𝑠 = 0.35, the collision frequency is calculated to be 9.39*1013. The time scale of collision was estimated by, 𝐶𝑜𝑙𝑙𝑖𝑠𝑖𝑜𝑛 𝑇𝑖𝑚𝑒 = 𝑛𝑑𝑠 ∗ (1𝑛𝑐𝑜𝑙𝑙̇) = 0.1 𝑚𝑠 (A.9) Time Scale for Heat Transfer is estimated by calculating the Biot Number method. The Biot Number is calculated as, 𝐵𝑖 =((𝑑𝑙6 ) ∗ ℎ𝑡𝑐𝑙𝑠)𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝐴𝑐𝑒𝑡𝑜𝑛𝑒= 0.131 (A.10) where, the heat transfer coefficient is calculated by employing the Ranz-Marshall Correlation and the diameter of droplets as 200 microns. The thermal conductivity of acetone is 0.1642 W/(m-K). Substituting the appropriate values of 𝐴𝑖−𝐹𝑜 and 𝜆𝑖−𝐹𝑜 , we calculate the Fourier Number for heat transfer from 210C to 580C as 102  𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑁𝑢𝑚𝑏𝑒𝑟 = −𝑙𝑜𝑔(  ((𝑇(𝑟,𝑡) − 𝑇∞𝑇𝑙 − 𝑇∞) ∗ 𝜆𝑖−𝐹𝑜)𝐴𝑖−𝐹𝑜 ∗ 𝑠𝑖𝑛(𝜆𝑖−𝐹𝑜))  ∗  1𝜆𝑖−𝐹𝑜2= 5.56 (A.11) The time required for the temperature rise is, 𝑇𝑖𝑚𝑒 = 𝐹𝑜𝑢𝑟𝑖𝑒𝑟 𝑁𝑢𝑚𝑏𝑒𝑟 ∗ 𝑑𝑙2 ∗1𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 𝑜𝑓 𝑑𝑟𝑜𝑝𝑙𝑒𝑡𝑠 (A.12) Substituting the appropriate values (𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑑𝑖𝑓𝑓𝑢𝑠𝑖𝑣𝑖𝑡𝑦 =𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝐶𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦𝜌∗𝑐𝑝𝑙), we obtain the time scale for temperature rise as 46 milliseconds.  Therefore, the timescale required to raise the temperature of liquid droplets is much higher than the time scale of collision. Hence, this mode of heat transfer is neglected.         103  Appendix C   𝑀𝑎𝑠𝑠 𝑜𝑓 𝐿𝑖𝑞𝑢𝑖𝑑 𝐼𝑛𝑗𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑅𝑒𝑎𝑐𝑡𝑜𝑟 = 1350𝑔 (A.13) 𝑀𝑎𝑠𝑠 𝑜𝑓 𝑃𝑀𝑀𝐴 𝐼𝑛𝑗𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑅𝑒𝑎𝑐𝑡𝑜𝑟 = 135𝑔 (A.14) 𝑀𝑎𝑠𝑠 𝑜𝑓 𝐴𝑐𝑒𝑡𝑜𝑛𝑒 𝐼𝑛𝑗𝑒𝑐𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑅𝑒𝑎𝑐𝑡𝑜𝑟 = 1215𝑔 (A.15) 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐  𝐻𝑒𝑎𝑡 𝑜𝑓 𝐿𝑖𝑞𝑢𝑖𝑑 = 1547.74𝑗𝑘𝑔 − 𝐾 (A.16) 𝐿𝑎𝑡𝑒𝑛𝑡 𝐻𝑒𝑎𝑡 𝑜𝑓 𝑉𝑎𝑝𝑜𝑟𝑖𝑧𝑎𝑡𝑖𝑜𝑛 = 451𝑘𝐽𝑘𝑔 (A.17) The temperature of the solids at the end of the simulation is calculated by simple heat balance, (𝑚 ∗ 𝐶𝑝∆𝑇)𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠 = −((𝑚) ∗ 𝐶𝑝∆𝑇)𝑙𝑖𝑞𝑢𝑖𝑑+ ?̇?𝐻𝑣 (A.18) Assuming that 20% of acetone undergoes evaporation and the rest boils (based on the rates of boiling and evaporation), we have, (160 ∗ 800 ∗ (68 − 𝑇))= −(1.215 ∗ 1547.74 ∗ (21 − 58))− (0.135 ∗ 1547.74 ∗ (21 − 68)) + ((0.2 ∗ 1.215) ∗ 451.15 ∗ 1000) (A.19) After the heat balance equation, we obtain, 𝑇 = 66.510𝐶 (A.20) Therefore, the heat required to maintain the solid temperature at 680C is 𝐻𝑒𝑎𝑡 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 = (𝑚∗ 𝐶𝑝∆𝑇) (A.21) 104  𝐻𝑒𝑎𝑡 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 = (160 ∗ 800 ∗ 1.49) = 190.28 𝑘𝐽 (A.22) Therefore, the Power of the heat is calculated by, 𝑃𝑜𝑤𝑒𝑟 =𝐻𝑒𝑎𝑡 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑𝑇𝑖𝑚𝑒=190.2840= 4.76 𝑘𝑊 (A.23) Therefore, the power of the heater is estimated to be 5 kW in this study 

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