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Caking of Granular Materials: An Experimental and Theoretical Study

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PAGE 1

CAKING OF GRANULAR MATERI ALS: AN EXPERIMENTAL AND THEORETICAL STUDY By DAUNTEL WYNETTE SPECHT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Dauntel Wynette Specht

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This document is dedicated to Momma, Daddy, Techia and Danyel.

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iv ACKNOWLEDGMENTS I would like to acknowledge my advisors Dr. Kerry Johanson and Dr. Spyros Svoronos, for their constant guidance and s upport. Dr. Johanson’s extensive experience in the field of particle technology served as a continuous source of knowledge throughout my studies. Dr. Svoronos’s help in all as pects of my graduate studies has been indispensable. I thank my committee members, Dr. Ray Bu cklin and Dr. Oscar Crisalle, and all other faculty who provided useful suggesti ons for the advancement of my studies. I would also like to thank Dr. Brij Moudg il and the National Science Foundation’s Engineering Research Center for Particle Science and Technology and our industrial partners for their financial support. I gratefully acknowledge all of my group members, Brian Scarlett, Jennifer Curtis, Nicolaie Cristescu, Yakov Rabinovich, Olesya Zhupanska, Claudia Genovese, Ali AbdelHadi, Ecivit Bilgili, Mario Hubert, Dim itri Eskin, Nishanth Gopinathan, Madhavan Esayanur, Caner Yurteri, Rhye Hamey, Mari a Palazuelos Jorganes Stephen Tedeschi, Milorad Djomlija, Osama Saada, Julio Castr o, Benjamin James, Mark Pepple, and Bill Ketterhagen, for their help with experiments and modeling. I also acknowledge my friends in Gainesville for ultimately making this experience an enjoyable one. Lastly but certainly not least, I thank my family for thei r constant love and support. To Momma and Daddy, thanks for believing in me and teaching me that I can do anything that I set my mind to. To Techia a nd Danyel, thanks for being the best sisters

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v that a girl could ask for. To Marco, thanks for experiencing the world with me but most of all being my home away from home.

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vi TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES.............................................................................................................ix LIST OF FIGURES.............................................................................................................x ABSTRACT.....................................................................................................................xi ii CHAPTER 1 INTRODUCTION........................................................................................................1 Storage and Handling of Granular Materials................................................................1 Caking in Industrial Processes......................................................................................2 Aim and Outline of Dissertation...................................................................................4 2 CAKING AND CAKING CONDITIONS...................................................................7 Mechanisms of Caking.................................................................................................7 Model of Moisture Migration Caking...........................................................................9 Quantifying the Bulk Cohesive Strength of Caking...................................................10 Direct Shear Testers............................................................................................10 Tensile Testers.....................................................................................................12 Penetration Testing..............................................................................................12 Crushing Test or Uniaxial Compression.............................................................13 Choosing a Tester.......................................................................................................14 Johanson Indicizer...............................................................................................14 Schulze Shear Tester...........................................................................................19 Unconfined Yield Strength Results............................................................................22 Effect of Moisture Content and Consolidation Pressure.....................................22 Effect of Particle Size..........................................................................................24 Effect of the Number of Temperature Cycles.....................................................25 Effect of Relative Humidity................................................................................27 3 ADSORPTION ISOTHERMS AND KINETICS......................................................30 Adsorption Isotherms..................................................................................................30 Mechanisms and Kinetics of Adsorption....................................................................33

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vii Measurement Techniques for Sorption Isotherms......................................................35 Static Vapor Sorption..........................................................................................35 Dynamic Vapor Sorption.....................................................................................35 Symmetrical Gravimetric Analyzer............................................................................36 Experimental Parameters............................................................................................38 Sorption Isotherms for Sodium Carbonate.................................................................39 Effect of Temperature..........................................................................................44 Kinetics of Adsorption........................................................................................46 Sorption Isotherms for Sodium Chloride....................................................................49 4 MOISTURE MIGRATION MODELING..................................................................51 Background.................................................................................................................51 Moisture Migration Model.........................................................................................56 Finite Element Modeling............................................................................................57 About COMSOL Multiphysics............................................................................59 Model Geometry..................................................................................................59 Partial Differential Equations..............................................................................60 Convection-conduction................................................................................61 Convection-diffusion....................................................................................61 Brinkman Equation......................................................................................62 Solids moisture content................................................................................62 Model Parameters................................................................................................63 The Role of Convection in Moisture Migration.........................................................66 Temperature Profiles...........................................................................................66 Solids Moisture Content......................................................................................72 5 EVALUATING THE C AKE STRENGTH OF GR ANULAR MATERIAL.............77 Background.................................................................................................................77 Modified Fracture Mechanics Mode l for Evaluating Cake Strength..........................85 Linear-elastic Fracture Mechanics...............................................................86 Elastic-plastic Fracture Mechanics...............................................................88 Model Parameters................................................................................................91 Calculating the narrowest width of the bridge as a function of the bridge volume....................................................................................................91 Calculating the length of the crack...............................................................95 Stress-Strain parameters F and n ..................................................................95 Geometry of the agglomerate calculating H ..............................................97 Comparison of the Model with Experimental Data....................................................97 Effect of Particle Size..........................................................................................97 Effect of Moisture Content................................................................................100 Effect of Consolidation Stress...........................................................................101 6 CONCLUSIONS AND FUTURE WORK...............................................................111

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viii APPENDIX A NOMENCLATURE.................................................................................................116 B MOISTURE MIGRATION PARAMETERS...........................................................121 LIST OF REFERENCES.................................................................................................123 BIOGRAPHICAL SKETCH...........................................................................................127

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ix LIST OF TABLES Table page 3-1 Kinetics constants for sodium carbonate decahydrate.............................................48 4-1 Parameters for COMSOL simulations.....................................................................64 4-2 Boundary conditions for COMSOL simulations......................................................66 4-3 Subdomain conditions for COMSOL simulations...................................................66

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x LIST OF FIGURES Figure page 1-1 Top down view of a silo wall with a th ick layer of caked material attached.............3 2-1 Representation of the two limiting Mohr circles, showing the major principal stress of the steady state condition 1 and the unconfined yield strength fc.............11 2-2 Johanson Indicizer test cell......................................................................................15 2-3 Schematic of the standard Indicizer test cell..........................................................16 2-4 Consolidation bench used for making the caked samples........................................18 2-5 Temperature profile used for preparing the cakes....................................................19 2-6 Schulze shear tester: (A) image and (B) schematic..................................................20 2-7 Schematic of the permeable Schulze cell.................................................................21 2-8 Unconfined yield strength of sodi um carbonate monohydrate as a function of moisture content in the form of per cent sodium carbonate decahydrate in the mixture.....................................................................................................................23 2-9 The yield strength as a functi on of consolidation pressure......................................24 2-10 The yield strength as a functi on of the mean particle size.......................................25 2-11 Yield locus of sodium carbonate for ai r at 75% RH for 24 hr. and 0 hr with a normal load of 16kPa...............................................................................................26 2-12 The yield strength as a function of the number of temperature cycles.....................27 2-13 Yield locus of sodium carbonate for air as a function of relative humidity with a normal load of 4kPa.................................................................................................28 2-14 Unconfined yield strength as a functi on of relative humidity compared to the isotherm of sodium carbonate..................................................................................29 3-1 The five types of adsorption isothe rms in the classification of Brunauer ...............32 3-2 Mechanisms of mass transfer for absorbent particles............................................. 33

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xi 3-3 Sorption of moisture by a water soluble particle......................................................34 3-4 VTI Symmetrical Gravimet ric Analyzer schematic.................................................36 3-5 User input screen for the VTI Sorption Analyzer....................................................38 3-6 Phase diagram of sodium carbonate.........................................................................40 3-7 Adsorption isotherm of sodium carbonate at 25 C..................................................41 3-8 Sorption isotherm of sodium carbonate at 25 C......................................................43 3-9 Sorption isotherm of sodium carbonate monohydrate at various temperatures.......44 3-10 Sorption isotherm of sodium carbonate decahydrate at various temperatures.........45 3-11 The moisture uptake of sodium carbonate decahydrate as a function of time.........46 3-12 The kinetics of sodium carbonate decahydrate at 50 C...........................................48 3-13 The kinetic rate constant for sodi um carbonate decahydrate adsorption and desorption.................................................................................................................49 3-14 Sorption isotherm of sodium ch loride at various temperatures................................50 4-1 Examples of the elements used in FEM...................................................................58 4-2 Schematic of the geometry used in th e COMSOL solver. The lettered areas label the domains..............................................................................................................60 4-3 Caking cell geometry with mesh elements...............................................................60 4-4 Boundary conditions................................................................................................63 4-5 Temperature profile imposed at the ba se of the cell in the finite element simulations...............................................................................................................65 4-6 Temperature profile along the centerlin e of the cell with c onvection included.......67 4-7 Temperature profile near the insula ted boundary of the cell with convection.........67 4-8 The temperature profile within the ce ll illustrating the convective plumes that develop.....................................................................................................................68 4-9 Temperature profile along the cente rline of the cell w ithout convection.................69 4-10 Temperature profile near the insulate d boundary of the cell without convection....69 4-11 Temperature profile at the center of the cell, with and without convection.............70

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xii 4-12 Temperature profile at the cell wall, with and without convection..........................71 4-13 Solids moisture content profile at th e centerline of the ce ll with convection..........73 4-14 Solids moisture content as a functio n of time at the center of the cell.....................73 4-15 Phase diagram of sodium carbonate with temperature profile imposed..................74 4-16 Equilibrium solids moisture content as various positions along the radius.............76 5-1 Model of contacting spheres with pendul ar water used to calculate the volume and width of the bridge.............................................................................................80 5-2 Mohr circles demonstrating the relati onship between tensile strength and the unconfined yield strength.........................................................................................81 5-3 X-ray tomography slice of caked sodium carbonate................................................90 5-4 Logarithmic plot of the smallest radius of the bridge b vs. the volume of the bridge V ....................................................................................................................92 5-4 Schematic of an agglomerate showing the length of the crack a .............................95 5-5 The stress-strain curve fo r a sodium carbonate tablet..............................................96 5-6 Stress-strain curve for lactose..................................................................................97 5-7 Yield strength of sodium carbonate as a function of particle size...........................99 5-8 The yield strength of sodium carbonate as a function of moisture content............100 5-9 Illustration of He rtz contact model.........................................................................102 5-10 The volume of a bridge usi ng the Hertzian contact model....................................103 5-11 Force chain structure in a DEM simula ted powder bed with a high applied load (right) and a low applied load (left).......................................................................106 5-12 The cumulative distribution of the gra nularity at various co nsolidation loads......107 5-13 Granularity of the major for ce chains in a particle bed..........................................108 5-14 Unconfined yield strength as a functi on of the consolidation stress compared to the model of equation 5-51.....................................................................................110

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xiii Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CAKING OF GRANULAR MATERI ALS: AN EXPERIMENTAL AND THEORETICAL STUDY By Dauntel Wynette Specht May 2006 Chair: Kerry Johanson Cochair: Spyros Svoronos Major Department: Chemical Engineering Many industries such as food service, pharmaceutical, chemical, and agricultural handle and store materials in granular form The processing of granular materials can pose many challenges. The materials can ga in strength during st orage creating flow problems in process equipment. This phenomenon is known as caking and is defined as the process by which free flowing material is transformed into lumps or agglomerates due to changes in environmental conditions. The strength of a cake is a function of various material properties and processing parameters such as temperature, relative humidity, particle size, moisture conten t, and consolidation stress. A fundamental understanding of the entire caking process is needed in order to predict a caking problem. In this research, caking based on the mech anism of moisture migration through the particle bed is studied. The influence of mois ture content, consolidation stress, particle size, and humidity are investigated. An increas e in these variables causes an increase in the unconfined yield strength of the material. Theoretical models exist to describe the

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xiv function of particle size and moisture c ontent on the yield strength. However, the consolidation stress is not included in thes e models. The trend is often qualitatively explained by an increase in the interparticle forces. Applying this theory to the current research yields an incorrect prediction of the unconfined yiel d strength. It is postulated that the increase in unconfined yield strength is attributed to an increase in the number of major force chains in the shear zone. An in crease in the number of major force chains results in the increase of the yield strength of the material. Using the principles of fracture mechanics, a new equation is developed to evaluate the strength of powder cakes. An improvement of existing cake yield stre ngth models is made by including the consolidation stress. Moisture migration through the particle sy stem and moisture uptake by the particles themselves are significant f actors of caking. These processe s are modeled using a finite element partial differential equation solver, COMSOL Multiphysics. The heat and mass transport of the system is described as well as the kinetics of the material. The model is used to predict the areas of caking within a given geometry and approximate the unconfined yield strength.

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1 CHAPTER 1 INTRODUCTION Storage and Handling of Granular Materials Granular materials are common in all facet s of life: from food to baby diapers to geological matter. They encompass the complete range of matter in the form of discrete solid particles. Granular mate rials are often referred to as bulk solids. Many industries must store and handle bulk solid s due to their prevalence in a wide variety of products. In the chemical industry, it is estimated that one-half of the products and at least three quarters of the raw materials are in the form of bulk solids (Nedderman, 1992). With such a vast number of processes involving bulk so lids there is a need for a fundamental understanding of the behavi or of such materials. Granular materials behave differently from any other form of matter—solids, liquids, or gases. They tend to exhibit co mplex behaviors sometimes resembling a solid and other times a liquid or gas (Jenike, 1964) Thus the characterization of granular materials presents a challenge unlike any othe r. It has been estimated that industries processing bulk solids total one trillion dollars a year in gross sales in the United States and they operate at only 63 pe rcent of capacity. By comparison, industries that rely on fluid transport processes opera te at 84 percent of design capacity (Merrow, 1988). Due to the often inadequate and unreliable design of particulate processes, there exists a significant discrepancy in the efficiency of particulate operations compared to fluid operations. Many of the problems in particulate operations deal with th e flow of material through the process. Often times it is the lack of flow that creates additional problems

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2 such as limited capacity of the process e quipment and product degradation. One type of flow problem is caking. A caking event often oc curs when a bulk solid does not flow and is allowed to remain stag nant over a period of time. Caking in Industrial Processes Bulk solids such as food powders, de tergents, pharmaceuticals, feedstocks, fertilizers, and inorganic salts often gain st rength during storage. The increase in strength is caused by particles bonding together at contact points th rough a cementing action. This phenomenon is known as caking and can be defined as the process by which free flowing materials are transformed into lumps and aggl omerates due to changes in atmospheric or process conditions. More precisely, caking is th e increase in bulk c ohesive strength due to changes in interparticle forces. Traditional interparticle forces such as the formation of liquid bridges, Van der Waals forces, or electrostatic forces can cause an increase in bulk cohesive strength. However, none of these phenomena alone cau se caking. Caking occurs when the surface of the particle is modified over time to crea te interparticle bonds due to the formation of solid bridges between similar or dissimila r materials. Thus, the presence traditional interparticle forces may initiate the caking process, but the advent of caking requires some mechanism to change these forces into solid bridges. Caking is the accumulation of smaller partic les held together by solid bridges to produce a cluster of particles. This ph enomenon is related to agglomeration. Agglomeration techniques are often used to improve the sh ape, appearance and handling properties of materials (Schubert, 1981). This can result in better flow characteristics, improved packing density, dust-free operations a nd faster dissolution in liquids (Aguilera et al. 1995). However agglomeration is unwanted and unexpected in the case of caking.

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3 This is attributed to the fact that the pro cess of caking is uncont rolled and the outcome can be useless material. In the food and pha rmaceutical industries, the effect of caking may be detrimental to the process and product. These problems often result in the loss of quality and function of a product (Purutyan et al. 2005). Regardless of the reason for caking, the presence of such agglomerates causes significant problems in process equipment (Johanson and Paul, 1996). Severe caking can result in the solidification of the entire mass within a silo as shown in Figure 1-1. Moreover, consumer products such as de tergent and food powde rs can cake during storage both prior to purchase and after initial use. Hence, the problems of handling such products are often passed on to the consumers. Figure 1-1. Top down view of a silo wall with a thick layer of caked material attached. Currently, there are no predictive tools to forecast the likelihood of a caking event occurring. The industry must rely on empirical knowledge of caking to provide solutions to the problem. This ranges from controlli ng the process environment to maintaining a regular cleaning schedule. A common practice for minimizing the effects of caking is the maintenance of a controlled environment su rrounding the process; i.e., the temperature and relative humidity are maintained at a le vel such that caking may not be induced. The

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4 material response to temperature and relative humidity conditions are determined by measuring sorption isotherms. The isotherms gi ve an indication of the hygroscopic nature of the material and the amount of moisture adsorbed at a sp ecific temperature and relative humidity. In situations where controlling the environmental parameters is not feasible, flow aids are sometimes introduced into the pr oduct mixture. Flow ai ds act as a physical barrier or moisture barrier between particle s that exhibit caking tendencies. However, when controlled environments and flow aids bot h fail, brute force is used to dislodge the cake from process equipment. These actions may result in damage to the equipment and personal injuries. Furthermore, if the caki ng continues to be a persistent problem, a regular cleaning schedule of the pr ocess equipment is established. Aim and Outline of Dissertation A fundamental knowledge of the caking phe nomenon appears to be non-existent. Most of the studies on caking are experiment al, focusing on empirical solutions to the problem. Many of these solutions are not unive rsally applicable. There are few studies which concentrated on predicting the streng th of caking and the moisture diffusion through the bulk. Often times these studies in vestigate ideal situ ations and are not suitable for real materials. As a consequence, when new products are developed it requires a great deal of time, effort, and money to diagnose and remedy a caking problem. Caking is a problem that may be solved in two ways: before the initiation of an event or eliminated after its existence. The goal of this research is to understand the variables that are involved to induce caking. This will enable the prediction as well as the prevention of a caking problem. To accomplish this, a comprehensive knowledge base of the caking phenomenon must be established. W ith this knowledge it is then possible to

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5 develop a model to predict the onset and stre ngth of caking. The following objectives are proposed to achieve the goals of the research: Understand the mechanisms of caking as well as the material properties and process conditions necessary to induce a caking event. Measure the cake strength as a f unction of process variables. Quantify the hygroscopic nature of the bulk material by measuring adsorption isotherms. Develop a model to predict the onset of caking based on the moisture migration through the bulk. Develop a model to predict the strength of caking based on the process variables. The research plan includes both an experiment al and a theoretical investigation to fully describe the caking process. The experi mental work involves measuring the bulk cohesive strength and the adsorption isothe rms. In the theoretical studies various parameters from the experimental work will be used to develop a mathematical model of the caking process. In chapter 2 the reader is further introduc ed to caking and the material properties and process parameters influencing the strength of cakes. Included in this chapter are the possible mech anisms of caking and interp article bond formation. Often times there are several mechanisms involved in a caking event. It is beyond the scope of this dissertation to investigate every mechan ism. Therefore, the focus of this study is moisture migration caking. The various methods for quantifying cake strength are discussed as well as the testing apparatus suit able for this purpose. Experimental data of the bulk cohesive strength are presented and e xplained in regards to observations of other researchers. In order for moisture migration caking to occur, the bulk material must possess a certain affinity for water (in the form of mo isture). Hence the interactions between the

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6 bulk solid and moisture in the system must be understood. This is accomplished by measuring the adsorption isotherms of the material. Chapter 3 provides a brief introduction into adsorption isotherms and pres ents experimental data as a function of temperature. The kinetics of sorption ar e also presented in this chapter. Using the sorption kinetics the moisture migration through the bulk can be modeled. In chapter 4, finite element methods are employed to predict the migration of moisture using an improved model. Many of the existing models assume that the convection in the system is negligible. This research shows that the free convection enhances moisture migration, making it a significant contributor to caking. Chapter 5 begins with a discussion of pr evious models for predicting the cake strength. These models do not adequately pr edict the strength and they are based on single process variables. A new model for predicting the strength is developed which incorporates all the signifi cant parameters of caking. The novelty of the model is the inclusion of the consolidati on stress to predict the stre ngth of the material. The experimental data from chapter 2 are used to verify the model. The final chapter summarizes the findings of this study and provides suggestions for future research in the study of caking.

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7 CHAPTER 2 CAKING AND CAKING CONDITIONS This chapter discusses the various mechan isms of caking along with the methods for quantifying bulk cohesive strength. The ma terial properties and process parameters which influence the initiation of a caking even t as well as the strength of caking are the focus of this chapter. Mechanisms of Caking A powder cake is formed by numerous so lid bridges bonding particles together. The mechanisms of solid bridge formation ar e therefore the key to cake formation. There are several mechanisms for solid bridge formation. Throughout a single caking event more than one mechanism can contribute to th e bulk cohesive strength. An understanding of all the mechanisms of solid bridge forma tion is essential for a thorough explanation of the caking process. Rumpf (1958) was the first to propose seve ral mechanisms for the formation of solid bridges. He named crystallization, si ntering, chemical reaction, partial surface melting, and liquid binder solidification as cau ses for bridge formation. More recently other researchers (Noel et al. 1990 and Farber et al. 2003) have suggested that glass transition may be a potential caking mechanism. The following is a list of mechanisms and a short summary of each phenomenon. Crystallization – Materials that are soluble or slightly soluble are subject to this mechanism. This includes many chemicals and fertilizers. The moisture content of the solid increases as water vapor condens es onto the surface of the particle. A portion of the surface material dissolves and bridges are formed at contact points

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8 between particles. A subsequent change in moisture content evaporates the liquid in the bridge leaving solid crystal bridges between particles. Sintering – Many useful products containing powder metals and ceramics are formed by this mechanism. The molecules or atoms of the material diffuse at the contact points of the particles to form solid bridges. Sintering usually takes place at 1/2 to 2/3 of the melting temperature. Direct reduction of iron or e is one example of a situation where this type of caking is undesired. Chemical reaction or binder hardening – The production of fiberboard from resin impregnated wood chips or flakes ma kes use of this mechanism. A reaction occurs between two different materials to form bridges between the particles much like mortar between bricks. Binder material may be used to form the bridges. Unlike crystallization, the bi nder liquid does not evaporate but is incorporated into the structure of the bridges either by chem ical reaction or bind er solidification. Partial melting – Ice crystals and snow are product s of this mechanism. A pressure induced phase change or a temperature ch ange caused by local friction between the particles at contact points causes the surface material to melt and consecutively solidify, thereby forming bridges. Glass transition – Many pharmaceuticals and food products are subject to this mechanism of caking. An increase in surface moisture causes a lowering of the glass transition temperature Tg of the solid material. As Tg lowers, particles are cemented together by plastic creep. The material changes from a hard crystalline phase into an amorphous plastic phase. A subsequent change in moisture content increases the glass transition temperat ure, solidifying the amorphous mass. The mechanisms of glass transition and crys tallization are domina nt in the cases of undesired caking (Piets ch, 1969; Aguilera et al ., 1995; Johanson et al ., 1996; Hancock et al ., 1998). In these situations, the formation of a cake is initiated from the condensation of moisture onto the surface of individual particles in a bulk assembly. The moisture adsorbs onto the particle su rface and migrates through the bulk at particular process conditions such as temperatur e and relative humidity. A change in these parameters during storage acts as a drivi ng force to induce periodic co ndensation and evaporation of moisture on a particle surface. Once the mois ture is adsorbed it can absorb into the particle creating a layer of amorphous mate rial in the case of glass transition. The

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9 moisture can also remain at the surface causi ng the surface of the part icle to dissolve and crystallize during desorp tion of the moisture. The mechanism of crystallization and associ ated moisture migration are chosen as the focus of this caking research. This type of caking is prevalent in many industries that handle crystalline materials. Although the focus of this research is crystallization caking, the moisture migration analysis is applicable to all mechanisms invol ving the sorption of water vapor. Model of Moisture Migration Caking Solid bridge formation by crystallization is initiated by the presence of moisture in the environment or in the particle. Frequentl y, a change in process conditions such as an increase in temperature or rela tive humidity creates a moistu re concentration gradient in the material. Moisture migrates through the interstitial voids of the bulk adsorbing on the surface of particles at the cont act points to form liquid bri dges. A local increase in the moisture content causes slight dissolution of the particles. A subsequent change in the local temperature or relative humidity cause s the surface moisture to evaporate and the material at the contact points to crystalli ze and form a solid bridge. This process is influenced by process parameters (temperatu re, relative humidity, c onsolidation pressure, storage time) and material prope rties (initial moisture content and particle size). The bulk cohesive strength varies with the magnitude of these parameters. A systematic study of the caking strength with regard to the pr ocess parameters is required for a thorough understanding of this process. This includes the effect of mo isture released or adsorbed by the particle and the number of temperatur e or humidity cycles during storage. From this information a model to predict a nd prevent caking can be developed.

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10 Quantifying the Bulk Cohe sive Strength of Caking Several techniques have been employed to quantify caking and study the effects of properties influencing the bulk cohesive strength of a mate rial. These methods include measuring the unconfined yield strength using direct shear testers, measuring tensile strength using tensile tests, and measuring yield stresses using pe netration testing. In addition, measuring the thickness of a caked laye r of material and classifying the degree of caking are used to characterize cake streng th. A review of the available methods for testing bulk cohesive stre ngth is given below. Direct Shear Testers Direct shear testers are used to charac terize the flow of pow ders and granular materials. One characteristic of flow is th e strength of the material or the unconfined yield strength fc. This property is not directly measur ed. It is extrapolated from shear stress data by the construction of Mohr circ les. The shear stress data are measured using direct shear testers. Direct sh ear testers are construc ted of two parts: a stationary part and a moving part. The moving part is displaced relative to the st ationary part such that the powder shears and a shear zone is created. Prio r to shear the powders are consolidated to a predefined state by applying a load normal to the shear plane. The tester will, after a short transition state, deform the powder with a steady state shear force which is dependent on the applied normal load. The stre ngth of the powder is a function of the applied normal load (or consolidation load ). When the powder is in the steady state condition, the deformation is stopped. The nor mal load is decreas ed and the sample sheared to instant failure. This process is repeated several times with different failure normal loads. The steady state and failure data are used to construct Mohr circles. The Mohr circles define the state of stress of the powder in the plane (shear stress vs.

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11 normal stress). The unconfined yield strength can be determined from the Mohr circles. Figure 2-1 shows an example of Mohr circles. The failure points from the shear tests form the yield locus. The large (limiting) Mohr circle is tangent to this yield locus and includes the steady state conditi on. The Mohr circle intersects the x-axis at the major and minor principle stress, 3 and 1, respectively. The unconfined yield strength, fc, is the major principle stress of the Mohr circle that is tangent to the yield locus and has a minor principle stress of zero (ASTM, 2002). Figure 2-1. Representation of the two limiting M ohr circles, showing the major principal stress of the steady state condition 1 and the unconfined yield strength fc. Shear testers are very useful for designi ng equipment for the storage and handling of bulk materials. The disadvantage of usi ng direct shear testers is that multiple experiments are required to find the unconfined yield strength. Using shear testers can be time consuming for measuring cake strength du e to the time-induced nature of the caking process. An exception to this is the Johanson Indicizer which estimates the unconfined yield strength with a single test. The measur ement principles of th is device are discussed in detail in a later section.

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12 Tensile Testers Tensile testers are used to determine the tens ile strength of materials. From the test various mechanical properties, besides the tensile strength, can be deducted from the stress-strain curve. There are many types of tensile testers, two of which are the horizontal (Leaper et al. 2003) and vertical testers (Pietsch, 1969a, and Schweiger et al. 1999) used for measuring the strength of granul ar material. The horizontal tester consists of a cylindrical split cell with one part rigidl y connected to a load cell and the other part moving. The tensile strength is measured by moving one half of the split cell away from the load cell until the sample breaks. For a vert ical tester the sample is prepared outside of the tester. After the cake is formed, th e sample is glued between two platens. The platens are moved in opposite directions until the sample breaks. For both types of testers, the force required to break the sample divided by the cross se ctional area of the sample defines the tensile strength. The difference between tensile testers a nd shear testers is the mechanism of breakage or shear. With shear testers, a sh earing action is needed to break the cake. Tensile testers use a tensile force to break th e cake. The most active force in the breakage of bonds between particles is often a shear force (Pietsch, 1969b). The tensile strength and the unconfined yield streng th are defined by two different Mohr circles of the same yield locus. The tensile force is consider ed a negative force and the unconfined yield strength is a positive force. The tensile strength t is the x-intercept of the yield locus as shown in Figure 2-1. Penetration Testing The bulk cohesive strength of a cake can be measured by cone penetration. This technique was adapted from soil mechanics where it has been used for years to measure

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13 geotechnical parameters of soil. The measur ement apparatus consists of a cone and a testing machine. Similar to tensile tests the cak ed samples are formed in a cylindrical die. The diameter of the die is large enough such that the walls are at a great enough distance from the failure plane that allows the assump tion that the fail surface is unconfined. After the cakes are formed, a cone indents the su rface to a defined depth and the penetration force is recorded as function of penetration depth. The pene tration force at a given depth is proportional to the unconfined yield strength (Knight, 1988). Similar to shear tests the sample fails in shear but the cone only acts over a limited area of the cake. This technique does not require specialized equipment for ev aluating the cake strength and the tests are not time-consuming. Crushing Test or Uniaxial Compression The crushing test is frequently used in the food industry to study the cohesion of various food powders (Down et al. 1985 and Rennie et al. 1999). The preparation of the caked sample is similar to that of tens ile and penetration tests. The powders are consolidated into a cylindrical mold. After the cake is formed it is removed from the mold and placed between two platens of a co mpression testing machine. The sample is axially loaded until the cake fails. The stress at which the cake breaks is defined as the unconfined yield stress. The cake must be st rong enough to withsta nd the removal from the mold. Thus the caked samples must exhi bit a minimum strength for measurements to be possible. This measuring technique is dire ctly related to the theory of the unconfined yield strength. Unlike the shear testers, the unconfined yield stress of the material is measured directly. For shear cell data, the uncon fined yield strength is extrapolated using Mohr circles.

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14 Choosing a Tester Although there are many testers and numerous methods available to investigate the strength of caking, direct shear testers are used in this research to quantify the bulk cohesive stresses. This choice is partly due to the availability and access to direct shear cells but also because the important inform ation related to caking is obtained in an efficient and accurate manner. The Johanson Hang-Up Indicizer and the Schulze shear cel l are used in this study. The Indicizer is used to measure the strength of the cakes as a function of temperature cycling, initial moisture content, consolidati on load and particle size. The Schulze cell is used to measure the strength of the cakes as a function of temp erature and relative humidity cycling, consolidation load, and time. Johanson Indicizer The Johanson Hang-Up Indicizer is a powder flow tester used to measure material flow properties. The Indicizer is chosen for its ability to estimate the unconfined yield strength from a single test. Many other shea r testers indirectly measure the unconfined yield strength through a series of experiments. This requires the measurement of a yield locus and construction of Mohr circles to de termine the unconfined yield strength. Due to the possible error introduced from inconsistent mixtures (cakes) it is advantageous to minimize this error by using a single e xperiment for the strength estimation. The test cell used to estimate the unconf ined yield strength with the Johanson Indicizer is shown in Figure 2-2 (Johanson, 1992). The test cell consists of an inner piston, outer piston and lower piston. For the ty pical strength test, the sample is first consolidated to a known stress state. The consolidation load and the lower piston are removed after consolidation such that the samp le is in an unconfin ed state. The inner

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15 piston then moved downward to fail the samp le. The diameter of the inner piston is smaller than that of the lower piston. This configur ation allows the creation of an inverted conical shear zone during failu re as shown in Figure 2-3. A force balance on the material displaced from the tester during failure is used to relate the shear stress and normal stress acting at the walls of this conical section to the for ce applied on the inner piston. Figure 2-2. Johanson Indicizer test cell. A) the or iginal test cell and B) modified test cell. The force balance on the conical sect ion is given in equation 2-1 where P is the consolidation load, Vs is the volume of the sample, b is the bulk density of the material and S is the surface area of the shear zone. The equation acco unts for the force of the material within the inverted conical section on the inner piston, the weight of the material in this section, and the normal and shear st resses acting on the inve rted conical section. sin cos S S gV Ps b (2-1) The diverging character of this failure zone (flow channel) makes it possible to estimate the unconfined yield streng th. If it is assumed that the material in the test cell is in an unconfined state of stress and the pi ston is pushed through the material forming a shear plane. The unconfined Mohr circle in Figure 2-3 represents this unique state of

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16 stress. The shear stress acti ng on the failure plane is as shown in Figure 2-3. The normal stress acting on the shear plane is as shown in the same figure. The geometry of the Mohr circle is used to re late the shear and normal stre sses to the unconfined yield strength fc and the internal friction angle These relationships are given in equations 2-2 and 2-3. Figure 2-3. Schematic of the standard Indicizer test cell (A) and M ohr circle used for calculation of the unconfined yield strength (B). sin 1 2 1 cf (2-2) cos 2 1cf (2-3) The volume of the material Vs that is displaced from the tester is given by 2 212 1l l i i sD D D D H V (2-4) where H is the height of the sample and Di and Dl are the inner and lower diameters of the pistons, respectively. The surface area of the shear zone is sin 4 12 2i lD D S (2-5)

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17 The dimensions of the te st cell define the angle of the shear zone. H D Di l2 tan (2-6) The unconfined yield strength is calculated by substituting equations 2-2 through 2-6 into equation 2-1. This yields a value for th e unconfined yield strength based on the dimensions of the test cell, bulk density and the force applied by the inner piston. It is assumed that the shear angle is small. l i l l i i b cD D H F D D D D H f cos 12 3 12 2 (2-7) Failure with the Indicizer is a two step process. Fi rst there is an increase in compaction stress, up to a set maximum va lue. The lower piston is moved downward causing the material to be supported by the li p. Then the inner piston moves at a steady rate to shear the material. After the initial failure, the shear stress and normal stress acting on the displaced material decr ease to the conditions of steady flow. This value is associated with the shape of the flow cha nnel formed after failure (inverted conical section). It is critical that the shear stress on the flow channel walls be less than the shear stress during failure to assure unconfined condi tions. The test is invalid if this condition is not met. The diverging nature of the flow channel ensures that the normal stress is always greater that the c onfining wall stress. The Johanson Indicizer operates in two modes, normal and scientific. In the scientific mode, the sample can be sheared without consolidation and the user manually enters the consolidation load For the cake strength measurem ents the tester is operating in the scientific mode. The consolidation load applied automatically by the tester is not required because the cakes are consolidated before the strength measurements. Pre-

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18 consolidation is performed on a consolidation bench external to the tester and these samples are placed in the Johanson Indicizer for failure. The test cell was modified as shown in Figure 2-2 for the cake strength meas urements. The modified cell consists of two parts and the lower pi ston becomes inoperable. The cakes are formed on a consolidation be nch as shown in Figure 2-4. The bench is housed in an environmental chamber wher e the temperature and relative humidity of the surroundings are controlled. The bench is equipped with a strip heater to control the temperature, position transducers to monitor th e axial strain, and load cells to monitor radial stress on the powder. The powder sample is contained in a 3” diameter cylindrical cell. The cell is constructed from phenolic, an insulating material, to prevent dissipation of heat through the cylinder walls. Figure 2-4. Consolidation bench us ed for making the caked samples. The procedure for making the cakes and measuri ng the strength is given in detail below: Each cake consists of a mixture of s odium carbonate monohydrate and decahydrate. The decahydrate provides a source of mo isture for the caking process and the monohydrate acts as sink for moisture. Th e components are mixed together and a portion of each sample is reserved to meas ure the initial moisture content with an Ohaus MB45 moisture analyzer.

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19 The powder cells are filled using a special feeder to eliminate segregation. The sample is consolidated by weights th at are placed on top of the piston. The temperature profile mimics the changes that occur from day to night within a metallic bin stored outside and exposed to di rect sunlight. It is a ramp and hold, as shown in figure 2-5, varying from 25 C to 50 C and back in a 24 hour cycle. The cakes are held at each temperature for a pproximately 7.5 hours and the temperature is increased or decreased at a rate of 2 per minute. Figure 2-5. Temperature profile used for preparing the cakes After the temperature cycle is complete, the cakes are carefully moved from the consolidation bench and placed in the modified Indicizer test cell. Finally, the strength is measured us ing the Johanson Indicizer. Schulze Shear Tester The Schulze shear cell is a powder tester which is often used to measure the characteristics of powder flow. A schematic of this cell is shown in Figure 2-6. The powder is contained in an annular base whic h has an adjustable annular velocity. A top lid with short vanes sticking into the powder is prevented from rotating by tension bars which are connected to load cells. When the base rotates, the powder will shear somewhere between the bottom of the base and the vanes of the top lid. The load cells measure the force that is needed to keep th e top lid in place. The measured force is equivalent to the shear force. The normal stress on the powder can be adjusted with weigths. The measurement procedure using the Schulze shear cell has been standarized by ASTM (ASTM, 2002).

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20 A) B) Figure 2-6. Schulze shear tester: (A) image and (B) schematic. The rationale for using the Schulze cell was to investigate the effects of relative humidity cycling on the strength of the material. A modified cell is used to enable the humidity cycling in the ce ll. This cell was designed for a previous study which investigated the effects of airflow on mate rial flow properties. A schematic of the modified cell is shown in Figure 2-7. The modified tester can accommodate air flow through the cell. It has a perm eable lid and the air is introduced from the bottom of the cell and leaves through the top lid. A porous me mbrane with a moderate pressure drop is used to ensure that the air is dispersed even ly on the bottom on the cell. The inlet air is conditioned with saturated salt solutions to control the re lative humidity of the sample (Greenspan, 1977 and ASTM, 1996).

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21 Figure 2-7. Schematic of the permeable Schulze cell. The procedure for measuring the cake strength follows the ASTM Standard Shear Test Method for Bulk Solids Using the Sc hulze Ring Shear Tester (ASTM, 2002) with the following modifications. The sample is sheared to steady state and then held under a normal load for a period of 24 hours. Throughout this time, air is bl own through the sample to equilibrate at a specified relative humidity. Humid air is passed through the sample for 12 hours to ensure that the system is in equili brium. The moisture is adsorbed onto the surface of the particles and liquid bridges are formed between adjacent particles. Subsequently, dry air is passed through the sample such that the bridges solidify to form the cake. After drying the sample the air flow is stopped and the sample is sheared to determine the increase in failure stress. A new powder sample must be used for each point on the yield locus due to the formation of cakes in the cell. The points on the yield locus are measured for each relative humidity to create a family of loci The loci are used to construct Mohr circles that determine the unconfined yield strength, fc and the major principal stress, 1. Figure 2-1 is a schematic of data collected from the Schulze cell. The Schulze cell is also used to measure the effects of temper ature cycling on the strength of the material. A flexible heater is attached to the inner and outer circumference of the test cell. The temperat ure of the cell is controlled to the temperature profile as shown in Figure 2-5. Prior to exposure to th e temperature profile the yield locus of the material is measured according to the ASTM procedure. After the temperature cycle the

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22 yield locus of the caked sample is measured. Each point of this yi eld locus requires a new sample. The data from the Schulze cell experime nts are compared to the results from the Johanson Indicizer. Unconfined Yield Strength Results Effect of Moisture Content and Consolidation Pressure The initial moisture content of the powder is important because it gives insight into the material’s tendency to cake and the exte nt of caking. Materials with excess water have the ability to cake depending on the ma terial properties and storage conditions. It has been observed that powders with higher mo isture contents tend to exhibit a stronger propensity for caking than ones w ith lower moisture contents. Sodium carbonate decahydrate with a water content of 61.2 % is used to vary the moisture content of the samples. The water is crystalline water contained in the structure of the material. At 32 C, the material begins to deco mpose and loses water. Therefore adjusting the percentage decahydrate in a mixt ure effectively varies the moisture content of the sample. The unconfined yield st rength results, from the Indicizer, as a function of percentage sodium carbonate de cahydrate in the mixture can be seen in Figure 2-8. The results indicate that the unconfin ed yield strength of the cakes increases as the percentage decahydrate increases. Other researchers ha ve also observed that the cake strength increases with an increase in moisture content (Pietsch, 1969). It can also be seen that only a small pe rcentage of decahydrate is needed to yield strong cakes. If a material is very hygroscopic, such as sodium carbonate monohydrate, the minimum moisture content to induce a caking event is very low. Therefore the storage conditions of these materials are a vita l factor in controlling the unwanted caking. At higher percentages of decahydrate (be yond 4%) the cakes are too strong to be

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23 measured with the Johanson Indicizer. The measurement limit of the Indicizer is stated to be 2000 kPa (Johanson, 1992). 0 20 40 60 80 100 120 140 160 180 012345 % Sodium Carbonate DecahydrateYield Strength (kPa) 17 kPa 250 um 17 kPa 850 um 10 kPa 250 um 10 kPa 850 um Figure 2-8. Unconfined yield strength of sodium carbonate monohydrate as a function of moisture content in the form of per cent sodium carbonate decahydrate in the mixture. It is believed that the consolidation stress has an effect on the bulk cohesive strength of the material. Not all materials exhibit this property but in the case of sodium carbonate, the strength increases with an increas e in consolidation load This is intuitive since larger consolidation loads will result in a denser material and larger interparticle contact areas. Figure 2-8 indicates a nearly linear dependence of consolidation pressure on the yield strength of the cakes. In Fi gure 2-9 the unconfined yield strength of carbonate as a function of the consolidation stre ss is plotted. It can be seen that the strength of sodium carbonate increases as the stress on the material increases.

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24 0 10 20 30 40 50 60 05101520 Consolidation Stress (kPa)Yield Strength (kPa) 250 um particles 850 um particles Figure 2-9. The yield strength as a function of consolidation pressure. The mean particle size of this material is 250 m and the moisture content is 2% decahydrate. Effect of Particle Size The effect of particle size on the strength of powders has been widely researched. Rumpf was the first to theorize that the partic le size influences the strength of a powder. His theory states that the particle size is i nversely proportional to th e tensile strength of the powder assuming that the porosity and the strength of the in terparticle bonds are known. The details of Rumpf’s theory are gi ven in a later chapter. The yield strength measured by the Johanson Indicizer is plotted as a function of the mean particle size d50 of the sodium carbonate. It can be seen in Fi gure 2-10 that the yield strength increases as the particle size decreases. As the particle size decreases the surface area and number of contact points increase such that there are a larger numbers of contact for the water to

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25 migrate and a larger number of solid bridge s formed. This hypothe sis is verified by Figure 2-10 as the strength increases with a decrease in particle size. 0 10 20 30 40 50 60 70 80 0100200300400500600700800900 Particle Size (um)Yield Strength (kPa) 0% Decahydrate 2% Decahydrate 3% Decahydrate Figure 2-10. The yield strength as a function of the mean part icle size. The consolidation pressure is 15 kPa. Effect of the Number of Temperature Cycles It is well known that caking is a time induced phenomenon (Teunou et al. 2000, Purutyan et al ., 2005). Thus the time of storage is also an important factor when considering the strength of caking. The unc onfined yield strength measurements were made with both the Johanson Indicizer and the Schulze shear cell. The yield locus, measured with the Schulze cell, of a sample th at has been exposed to air flow for a period of 24 hours is displayed in Figure 2-11. The tim e consolidation effects can be seen as the yield locus increases with time. The unconfin ed yield strength of the material is a

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26 function of the shear stress as given in equati on 2-3. Thus an increase in the shear stress results in an increase in the unconfined yield strength. 0 1 2 3 4 5 6 7 8 9 0123456 Normal Stress (kPa)Shear Stress (kPa) 24 hr 0 hr Figure 2-11. Yield locus of sodium carbonate fo r air at 75% RH for 24 hr. and 0 hr with a normal load of 16kPa. It has been observed that the caking strength may increase as the number of temperature cycles increases (Johanson et al. ,1996, Cleaver et al. 2004). In other words, the longer a material experien ces the day-to-night temper ature changes, the greater tendency for increased caking. The yield strength of sodium carbonate is investigated as a function of the number of temperature cycl es that the material experiences. The temperature cycles for this set of experiments are 4 hours long. It can be seen in Figure 212 that the unconfined yield strength as measured with the Johanson Indicizer increases as the number of temperature cycles increa ses. It is possible that the duration of temperature cycle is not sufficient for the mate rial to reach an equilibrium state. Thus

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27 subsequent cycles initiate th e release or uptake of moisture causing an increase in the strength until equilibrium. It is shown in Figure 2-12, that for the 2% decahydrate mixture, that the yield after 2 and 4 cycles ar e approximately equal. The same trend is not seen with the 4% decahydrate mixture. This could be due to the fact that the initial moisture content of 4% mixture is higher thus requiring more time for equilibration. It may also be attributed to the migration of dissolved material fro m a non-contact area to the contact zone during the re crystallization process. 0 5 10 15 20 25 30 35 40 45 012345 Number of temperature cyclesYield Strength (kPa) 2% Decahydrate 4% Decahydrate Figure 2-12. The yield strength as a function of the number of temperature cycles. The mean particle size is 250 m and the consolidation pressure is 15 kPa. Effect of Relative Humidity The effect of relative humidity has only re cently received attention in the research of caking (Teunou et al. 1999a, Leaper et al. 2003). It is believ ed that changes in relative humidity could induce a caking event. The effects of relative humidity on the

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28 cake strength of sodium carbonate are investigated using the Schulze shear cell. It can be seen in Figure 2-13 that the sl ope of the yield locus increases with an increase in relative humidity. This indicates that moisture is adsorbed by the particles at higher relative humidities. The moisture sorption causes th e increase in shear stress which in turn increases the unconfined yield strength. 0 1 2 3 4 5 6 0123456 Normal Stress (kPa)Shear Stress (kPa) 0%RH 95%RH Figure 2-13. Yield locus of sodium carbonate for air as a function of relative humidity with a normal load of 4kPa. It can be seen in Figure 2-14 that the unc onfined yield strength increases with the relative humidity. The strength data seem to follow the same trend as the isotherm with the moisture pickup, thus verifying that the adsorbed moisture affects the cake strength. However, a change in relative humidity doe s not seem to produce a cake of the same strength as those created by a temperature change.

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29 0 0.5 1 1.5 2 2.5 3 020406080100 % Relative HumidityUnconfined Yield Strength (kPa)0 5 10 15 20 25 30 35Weight (% change) Figure 2-14. Unconfined yield strength as a f unction of relative humidity compared to the isotherm of sodium carbonate.

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30 CHAPTER 3 ADSORPTION ISOTHERMS AND KINETICS Perry (1997) defines adsorption as the accumulation or depletion of solute molecules at an interface. In the case of moisture migration caking, this entails the transfer and equilibrium dist ribution of moisture between the gas phase (the humid environment) and the particles. In this chapter the adsorption isotherms of sodium carbonate and salt are examined using dynamic vapor sorption techniques. It is shown that the isotherms are a good predictor to the onset of caking. The ki netic sorption curves are also measured to obtain the rates of adsorption and vaporization of water. Adsorption Isotherms During the processing and storage of bulk ma terials, the material is exposed to humid air in the environment. The humid ai r contains water vapor which can have a significant affect on the physical and chemical properties of the material. Water vapor from the environment can be absorbed into the structure of many materials and it may also be adsorbed onto the surface of these ma terials. In the case of amorphous solids, including pharmaceutical excipients and food products, absorbed water vapor is known to lower the glass transition temperature of the material thus promoting a caking event (Zhang et al. 2000). In many crystalline materials, the water vapor is adsorbed on the surface causing dissolution thus also initiati ng a caking event. For food products, water vapor has a critical effect on the dehydrat ion process and storage stability (Iglesias et al. 1976).

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31 At a given temperature and relative humidity, solids exhi bit a maximum capacity to adsorb moisture and this is characterized by the sorption isotherms. Water vapor sorption isotherms are used to describe the interactions between a material and the humid air surrounding it. The isotherms are typically expres sed as the water content (a loss or gain) as a function of the relative humidity (RH) or water activity at a given temperature. The adsorption of water vapor may be either physic al or chemical and the vapor may adsorb in multiple layers. Brunauer et al. (Perry, 1997) described five types of physical adsorption as shown in Figure 3-1. Type I isotherm represent m onomolecular adsorption of a gas and applies to porous materials with small pores. This is the well-known Langmuir isotherm. Type II and III isotherms re present materials with a wider range of pore sizes where gas is adsorbed in eith er a monolayer or multilayer. The sigmoid isotherm of Type II is ty pically obtained from soluble products which show an asymptotic trend as the water activity goes to one (Mathlouthi et al. 2003). The Type III isotherm is known as the Flory-Huggins is otherm. This isotherm represents the adsorption of a solvent gas above the glass transition temperature. Type IV isotherms describe adsorption which causes the formati on of two surface layers. Type V adsorption behavior is found in the adsorption of water vapor on activated carbon. Some crystalline materials will exhibit sorption profiles other than those described by Brunauer et al The adsorption/desorption isotherm of hydrated material sometimes appears as a stair like curve. The hydrates ar e formed at specific relative humidities. The material equilibrates at a low RH and when the RH increases the material becomes saturated increasing the moisture content of the material. This sequence appears as a staircase in the sorption isotherms. Anhydr ous materials only adsorb trace amounts of

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32 water at low RH. At a characteristic RH the adsorption corresponds to saturation equilibrium. A vertical line on the sorp tion isotherm represents the saturation equilibrium. Figure 3-1. The five types of adsorption is otherms in the classification of Brunauer et al. (1940). From the sorption isotherm fundamental material properties and important information about its handling can be derive d. Properties such as hydrate formation, deliquescence, and hygroscopicity can be de termined from the sorption isotherms. Sorption isotherms are also useful for a qualitative prediction of caking. The data obtained from the isotherm are used to de termine the temperature and relative humidity conditions at the onset of caking. This is il lustrated in the isotherms by a sudden increase in the moisture content at a critical relative humidity. Sorption isotherms are also a valuable tool for understanding the moisture relationship of a new product during the formulation stages (Foster et al. 2004).

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33 Mechanisms and Kinetics of Adsorption The adsorption of a solute from a fluid phase is described by mass transport processes such as: interparticl e mass transfer, intraparticle mass transfer and interphase mass transfer. These common transport mechanisms are illustrated in Figure 3-2. Interparticle mass transport is the diffusion of the solute (moisture in the case of caking) in the fluid phase through the particle bed. Th is type of transport is described by heat and mass transfer equations on a continuum scale. Interparticle mass transport is a subject of chapter 4. Intraparticle mass transfer, which includes pore and solid diffusion, describes the transport of the solute through the particle. Interphase mass transpor t is the transfer of a solute at the fluid-particle interface. Fi gure 3-3 is an illustra tion of interphase mass transport showing the sorption of moisture by a sorbent particle. Figure 3-2. Mechanisms of mass transfer fo r absorbent particles (Perry, 1997): 1, pore diffusion; 2, solid diffusion; 3, reac tion kinetics at boundary phase; 4, interphase mass transfer; 5, interparticle mixing. In this research it is assumed that th e particles are nonporous. This assumption is validated by measurements of the specific surface area by gas adsorption. The measured values for the specific surface area correspond the ideal values at a spec ific particle size.

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34 For a packed bed system, the transport of moisture from the bulk of the fluid phase to the surface of the particle is described as interphase transport and the mass transfer rate is given by: q c kA dt dqb sp (3-1) where k is the mass transfer coefficient, Asp is the specific surface area, cb is the moisture concentration in the bu lk of the fluid, and q is the solids moisture content. To simplify the calculations of equation 3-1 it is assumed that the uptake ra te is linearly proportional to the driving force, the so called linear driving force model. The moisture concentration in the bulk is equal to the equilibrium solids mo isture content. Thus the particles in the system are in equilibrium with the fl uid phase concentration of the bulk. q q A k dt dqe sp g (3-2) In equation 3-2 qe is the equilibrium solids moisture content and kg is the Glueckauf factor. The equilibrium concentration is determined from the isotherm of the material. The linear driving force model suggests that the concentra tion gradient is linearly proportional to the average moisture content of the system. Figure 3-3. Sorption of moisture by a water soluble particle.

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35 Measurement Techniques for Sorption Isotherms There are two common methods for determin ing the sorption isot herms of powders or granular materials: static vapor sorption and dynamic vapor sorption. Static Vapor Sorption The method of static vapor sorption is a ve ry time consuming and laborious process for measuring isotherms. A constant envir onment (temperature and relative humidity) must be maintained throughout the duration of the experiment. Several techniques are employed to produce a known constant humidity The use of saturated salt solutions is the most common (Greenspan, 1977, ASTM, 1996). Known constant humidities are created by salts whose affinity for water re gulates the water vapor pressure surrounding the material. Most often the salt solution and the sample are contained in dessicators for the period of time in which the sample is allowed to equilibrate at the specified humidity. The equilibration time is typically several w eeks. The weight of the samples before and after equilibration are recorded and used to determine the moisture content of the sample at a particular relative humid ity. Karl Fischer titration may also be used to determine the moisture content of the sample. Depending on the properties of the powder sample, this method can be time consuming. Dynamic Vapor Sorption An alternative, faster approach to static vapor sorption is dynamic vapor sorption. This method is more time efficient because a smaller sample size is used which decreases the equilibration time. The humid air is also either passed over or through the sample allowing equilibration to occur much faster Dynamic vapor sorption is the chosen method of measurement for the sorption isotherm s and sorption kinetics in this research. The equipment used is the VTI Symmet rical Gravimetric Analyzer (SGA 100).

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36 Symmetrical Gravimetric Analyzer The isotherm and kinetic measurements are made with the VTI Corporation Symmetrical Gravimetric Anal yzer (SGA-100). The SGA-100 is a continuous gas flow adsorption instrument designed to study water vapor sorption isotherms. All measurements are taken at ambient pressu re and the temperature range is from 0 to 80 C. A schematic of the instrument is shown in Figure 3-4. Figure 3-4. VTI Symmetrical Gravimetric Analyzer schematic (VTI website). The instrument is designed with three separate thermal zones such that maximum temperature stability is achieved. Located in zone 1 is the Kahn microbalance. The microbalance has a sensitivity of 0.0001 g. The temperature in this zone is maintained 15 C higher than the experimental temper ature to avoid condensation. Higher temperatures along with a continuous purge st ream of nitrogen in the balance chamber

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37 ensure that there is no vapor condensation on the microbalance which could invalidate the results. Isolating the microbalance also e liminates the weight fluctuations caused by temperature gradients in the environment. Su spended from the microbalance are two thin metal wires, into zone 2, for the samp le holder and reference sample holder. The core of the instrument, an aluminum block containing the sample chamber, is located in zone 2. There ar e two rectangular cutouts (cha mbers); one houses the sample and the other houses a reference sample. Gla ss sample holders are suspended on the thin wires from the microbalance. Both sides ar e subjected to the same temperature and relative humidity conditions. The temperat ure in this zone can range from 0 to 80 C. It is controlled by circulating water from a c onstant temperature bath through the hollow wall aluminum block. The air temperature is measured with an RTD located at the bottom of each chamber. The third zone is for the humidity control. It contains the chilled mirror dew point analyzer and a parallel plate humidifier which are maintained at 40 C. The system is fed with a constant supply of nitrogen. The nitrogen enters the parallel pl ate humidifier and is completely saturated (100 % RH) with resp ect to the experimental temperature. The saturated stream is mixed with a dry stream of nitrogen to obtain the desired relative humidity. The mixed stream passes through the de w point analyzer to be measured before entering the sample chamber. The dew point analyzer continuously measures the relative humidity stream to maintain cont rol of the wet a nd dry mass flows. The VTI Sorption Analyzer is a fully automated system that requires minimal user input. The operator must specify the ope rating conditions such as experimental temperature, relative humidity steps and the equilibrium criterion. Th e user input screen

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38 is displayed in Figure 3-5. The experiment can be set for a drying step before the measurement of the isotherm. The relative hum idity steps are customized by the user or chosen by the computer program if the initial and final step are specified. The data are recorded at intervals specified by the user. Th is is a time or percentage weight change condition. After the experimental parameters ar e set the material is loaded in the sample holder and the experiment begins. Figure 3-5. User input screen for the VTI Sorption Analyzer. Experimental Parameters The sorption isotherms are measured at three temperatures: 25 35 and 50 C. Measuring in this range provi des an indication of how temp erature affects the sorption process. The kinetic sorption curves are also measured to obtain the rates of adsorption and desorption of water. The data from the is otherms can be used to determine if water sorption at a specified relative hum idity is sufficient for caking.

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39 Only the sorption isotherms of single co mponent systems are measured. However, the combined effect of individual adsorpti on isotherms in a mixture on the diffusion of moisture through the bulk and onto the particle surface is important. Consider a system containing two different particle s with different isotherms. One particle can be a moisture sink for the water and the other particle can be a moisture source. Therefore, a caking event can occur due to differences in isothe rms. The differences in isotherms not only affect the diffusion of moisture on the particle surface but also the ki netics of evaporation and crystallization of the bridges within the cake. The standard procedure for measuring ad sorption isotherms suggests drying the material introducing humidity (ASTM, 1996) However, the following isotherms are measured without a prior drying step. The drying step is only necessary to ensure that the starting point is known. Thus the moisture content of the material is taken prior to the isotherm measurements using an Ohaus MB45 moisture analyzer. The moisture contents are consistent within a range of less than 1%. Sorption Isotherms for Sodium Carbonate Sodium carbonate is one of the materials used in this study. This material is widely used in the glass industry as well as in the formulation of powder detergents. Sodium carbonate has a strong propensity to cake if not st ored in a controlled environment. It is a colorless odorless material with three stable hydrate forms, monohydate (SCM), heptahydrate (SCH), and decahydrate (SCD). The phase diagram for sodium carbonate is shown in Figure 3-6 (OCI Chemical website). Heptahydrate is a semi-stable form and likely exist only in solution. It is made up of 57.4% water and is only stable in the temperature range of 32 C to 36 C. Sodium carbonate monoh ydrate granules contain 14.6% water. It loses water on heating (to 100 C), and the solubility increases with

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40 increasing temperature. Sodium carbonate decahydrate contains 61.2% water and the crystals readily effloresce in dry air and fo rm lower hydrates. This is evident from the moisture sorption isotherm, Figure 3-7. Figure 3-6. Phase diagram of sodium carbonate. In Figure 3-7, the expected equilibriu m moisture adsorption for both decahydrate and monohydrate is shown. The decahydrate app ears to lose (47% moisture) at relative humidities lower than 70%. This corresponds to an average molar moisture content

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41 equivalent to 7 moles of crystalline wa ter. However, decahydrate shows a great propensity to deliquesce at relative humiditie s greater than 70% gaining almost 1.5 times its initial weight in moistu re at 98% RH. Comparing this behavior to the monohydrate, shows a very different beha vior. The monohydrate does not lo se or gain moisture at relative humidities lower than 70% RH. This transition humidity, where deliquescence begins, is similar for the decahydrate. Howe ver, the monohydrate does not deliquesce as rapidly as the decahydrate. The deliquescent relative humidity is in the range on 70-80 %. -100 -50 0 50 100 150 020406080100 % Relative HumidityWeight (% Change) Decahydrate Deca Model Monohydrate Monohydrate Model Figure 3-7. Adsorption isotherm of sodium carbonate at 25 C. The points indicate the experimental data and the lines are the adsorption isotherm model predictions. The isotherms of sodium carbonate sugge st that a mixture of decahydrate and monohydrate would exhibit different behaviors in terms of mois ture pick-up at different relative humidities. For example, assume a mixture of decahydrate and monohydrate is in an environment where the humidity was less than 86 % RH but greater than 70% RH.

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42 From the isotherms, it is sufficient to assu me that decahydrate would lose its moisture and that moisture would be picked up by monohydrate. At relative humidities greater than 88% both decahydrate and monohydrate de liquesce. The degree of deliquescence at 90 % RH is the same for both the monohydrat e and decahydrate. However, at relative humidities greater than 90% RH the degree of deliquescence of decahydrate surpasses that observed with monohydrate suggesting th at a mixture of these two hydrates might cause the monohydrate to lose moisture and be picked up by the decahydrate. The pickup of moisture by the monohydrate and the loss of moisture by the decahydrate causes the caking. The additional moisture slightly dissolves the surface of the monohydrate particles. Bridges are formed between part icles and a cake is cr eated. Increasing the amount of decahydrate in the mixture will th erefore increase the strength of the cake. Thus the use of several hydrat es is ideal for controlling the initial water content of the samples. In Figure 3-7, for the monohydrate, a 0 % weight change corresponds to 100% monohydrate. An increase in weight of 87% and 130% correspond to the formation of a heptahydrate and decahydrate, respectively. The endpoint of the monohydrate curve suggests that the material likely converts to a mixture of hept ahydrate and monohydrate. At 25 C, the monohydrate picks up moisture yet not enough to fully transform to the decahydrate or heptahydrate form. For the decahydrate, a decrease in weight of 56% would correspond to a decahydrate transforme d into monohydrate. At a 0% weight change, the decahydrate is transformed to its original state. The endpoint of the decahydrate curve suggests that the material de liquesces with a total moisture gain of 130%.

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43 The sorption of water vapor by sorption carbonate is not sufficiently described by the adsorption models proposed by Brunaeuer in Figure 3-1. It can be seen in Figure 3-7 that the isotherms of sodium carbonate at 25 C are best described by the adsorption model proposed by Miniowitsch (1958). o B RHq Ae q 100 / (3-3) In equation 3-3, A and B are constants, qo is the initial weight lose and q is the moisture content. The constants A and B vary slightly with each fit and qo is zero for monohydrate and -47 for decahydrate. For the decahydrate A =7.97e-7 and B =0.0669. For the monohydrate A =6.38e-7 and B =0.0723. The desorption curves for sodium carbonate are shown in Figure 3-8. -100 -50 0 50 100 150 200 020406080100 % Relative HumidityWeight (% change) Deca Adsorption Deca Desorption Mono Adsorption Mono Desorption Figure 3-8. Sorption isotherm of sodium carbonate at 25 C.

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44 It can be seen that adsorption and de sorption of both the monohydrate and the decahydrate do not follow the same curve. Ther e is a hystersis between the adsorption and desorption which implies that the sorption of sodium carbonate is not isotropic. Thus the desorption must be described by an equati on different from that of the adsorption. Effect of Temperature The effect of temperature on the mois ture sorption of sodium carbonate monohydrate and decahydrate is also investigated It can be seen in Figures 3-9 and 3-10 that increasing the temperature increases the total moisture uptake. -10 10 30 50 70 90 110 130 020406080100 % Relative HumidityWeight (% Change) Monohydrate 25 deg. C Monohydrate 35 deg. C Monohydrate 50 deg. C Heptahydrate Decahydrate Figure 3-9. Sorption isotherm of sodium carbonate monohydrate at various temperatures. The dashed lines correspond to the po int where the monohydrate converts to a heptahydrate or decahydrate. The total moisture uptak e of monohydrate at 35 C is such that heptahydrate is formed. This result is verified by the phase diagram which suggests th at the heptahydrate is stable in the temperature range of 32 -35 C. At 50 C the material shows significant

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45 deliquescence with a gain in weight corresponding to a mi xture of the decahydrate and heptahydrate forms. The same trends are observed with the decahydrate as seen in Figure 3-10. However deliquescence is substantially increased. At 35 C and relative humidity values greater than 80% the material appears to form an unsaturated solution. Likewise, at 50 C and relative humidity values above 85%, the result is an uns aturated solution. -100 -50 0 50 100 150 200 020406080100 % Relative HumidityWeight (% Change) Decahydrate 25 deg. C Decahydrate 35 deg. C Decahydrate 50 deg. C Monohydrate Heptahydrate Decahydrate Figure 3-10. Sorption isotherm of sodium car bonate decahydrate at various temperatures. The dashed lines correspond to the po int where the decahydrate converts to a monohydrate, heptahydrate or decahydrate. The increase in moisture uptake is onl y observed beyond the deliquescence relative humidity. The deliquescence range remains unchanged within the temperatures investigated. This result is contrary to what various isotherm mode ls would predict as a response to a temperature increase (Iglesias et al ., 1976). At higher temperatures on the phase diagram the boundary between the unsat urated solution and the monohydrate plus solution has a negative slope. This suggests th at as temperature increases the percentage

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46 of sodium carbonate decreases. Thus as the temperature of the sample increases the moisture uptake increases. Kinetics of Adsorption The linear driving force model assumes that the order of the reaction of moisture adsorption/desorption is one. The kinetics (moi sture content as a function of time) of sodium carbonate are investigated to validate the use of this model. The curves for the moisture content versus time for sodium car bonate decahydrate are shown in Figure 3-11. 0 10 20 30 40 50 60 70 050100150 Time (min.)Moisture Content (% water)A) 0 20 40 60 80 100 120 140 02004006008001000 Time (min.)Moisture Content ( %water)B) Figure 3-11. The moisture uptake of sodium carbonate decahydrate as a function of time. A)10%RH – represents the kinetics below the deliquescene relative humidity and B) 90%RH – represents the kine tics above the deliquescence relative humidity. These two curves represent the typical ki netics of decahydrate above and below the deliquescence relative humidity. It can be seen that the kinetics at lower relative humidities are significantly faster than the kinetics at higher relative humidities, illustrated by the length of time required to re ach an equilibrium state. This suggests that desorption of water occurs much faster than adsorption. At higher relative humidities there is initia lly a loss of moisture corresponding to 57 moles of water. After desorption the material begins to adsorb moisture. This transient may be attributed to an artifact of the tester Before the material is introduced into the sample chamber, the sorption analyzer operates at a defaul t equilibrium condition of 0%

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47 RH and 25 C. It is possible to set the initial temperature of the chamber to the experimental temperature prior to the test by changing the setpoint of the temperature bath. However, it is not possible to cha nge the default relative humidity. When the material is introduced into the sample chambe r, the system must come to steady state at the specified experimental parameters. This process is not instantaneous. Hence there is an initial desorption of water which is caused by a low relative humidity in the sample chamber. The transient lasts for approximatel y 30 minutes. Thus the first 30 minutes of the curve are disregarded for determin ation of the order of the reaction. To determine the order of the reaction the in tegral method of rate analysis is used. The reaction order is known (or suggested) fr om equation 3-2. If the reaction is first order, integration of equation 3-2 gives t k c q c qg e o e ln (3-4) The slope of the plot of –ln (q-ce/qo-ce) as a function of time t is linear with a slope kg. In Figure 3-11 the concentration da ta is approximated by the line ar driving force model. The slope, intercept and correlation coefficient are given in Table 3-1. The slopes vary with the relative humidity implyi ng that a single rate law may not describe the sorption of water. The rate of moisture sorption changes with a change in relative humidity. The correlation coefficient for the curves of decahydrate at the specified relative humidities is in the range of 0.8453 – 0.9958. A coefficient of 1 would signify a perfect correlation. This result suggest s that the linear driving force model is not perfect. However the kinetics of sodium carbonate may be approximated by this model with a reasonable certainty.

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48 0 1 2 3 4 5 010203040 Time (min)-ln[(q-ce)/(qo-ce)]A) 0 1 2 3 020406080Time (min.)-ln[(q-ce)/(qo-ce)]B) 0 1 2 3 4 0100200300Time (min.)-ln[(q-ce)/(qo-ce)]C) 0 1 2 3 4 5 6 02004006008001000 Time (min.)-ln[(q-ce)/(qo-ce)]D) 0 1 2 3 4 5 05001000 Time (min.)-ln[(q-ce)/(qo-ce)]E) 0 1 2 3 4 05001000150020002500Time (min.)-ln[(q-ce)/(qo-ce)]F) Figure 3-12. The kinetics of s odium carbonate decahydrate at 50 C. A)10%RH, B)70%RH, C)80%RH, D)85%RH E)90%RH, and F)95%RH Table 3-1. Kinetics constants for sodium carbonate decahydrate Relative Humidity (%) Slope Intercept Correlation Coefficient (R2) 10 0.1065 0 0.979 40 0.1049 0 0.9524 70 0.0946 0 0.9142 80 0.0072 1.0059 0.8453 85 0.005 0 0.9056 90 0.005 0 0.9958 95 0.0016 0 0.9077

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49 The rate constants are plotted as a functi on of relative humidity in Figure 3-13. It can be seen that the rate if de sorption is significantly greater than the rates of adsorption. 0 0.02 0.04 0.06 0.08 0.1 0.12 020406080100 Relative HumidityRate Constant (k) Desorption A dsorption Figure 3-13. The kinetic rate constant for sodium carbona te decahydrate adsorption and desorption. Sorption Isotherms for Sodium Chloride Sodium chloride is also used in this study. This is a very complex material due to its deliquescence at hi gh relative humidities. However ther e is an abundance of literature about this material due to its importance in a ll aspects of life. The sorption and kinetics of sodium chloride are well described in literat ure. Sodium chloride is highly soluble in water but only contains small amount of mois ture after dehydration. It varies in color from colorless, when pure, to white, gray or brownish, typical of rock salt. The crystal structure can be modified with a change in temperature. At 20 C, the critical relative humidity is 75%. Above this RH, the salt deliq uesces. This is illustrated by the isotherm

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50 in Figure 3-14. The isotherm for sodium chloride is plotted as function of temperature. It can be seen that the material adsorbs an undetectable amount of water at lower relative humidities. Beyond the deliquescen ce point there is a significan t amount of sorption. This result is consistent with l iterature data (Greenspan, 1977). 0 50 100 150 200 250 300 350 400 450 500 020406080100 % Relative HumidityWeight (% change) Salt 25 deg. C Salt 35 deg. C Salt 50 deg. C Figure 3-14. Sorption isotherm of sodium chloride at various temperatures.

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51 CHAPTER 4 MOISTURE MIGRATION MODELING This chapter explains the modeling of moisture migrat ion through a particle bed. Moisture migration is described by the heat and mass transfer of the system along with the isotherms of the material. In the literatu re, it is assumed that free convection in the system can be neglected. However, it is s hown that free convecti on plays a significant role in the heat and moisture transport of th e system. It is also s hown that the areas of caking within the bulk can be predicted gi ven the proper model for the heat and mass transfer of the system. Background It has been shown in chapters 2 and 3 that heat and moisture play a significant role in the caking process. Thus, in order to bett er understand caking, it is essential that the heat and mass transport (moisture migrati on) is thoroughly explained. An important factor in describing moisture migration caking is the transport of mo isture through the air and onto the surface of the particles. Other res earchers have investigat ed specific parts of the process but few have attempted to describe the entire process of moisture migration. However, all attempts at describing mois ture migration have included simplifying assumptions which render the models inaccurate. Tardos et al (1996a) studied diffusion of atmosp heric moisture into a particulate material inside a container. The authors de veloped a model to describe the amount of moisture that penetrates from a stagnant layer of humid air above a particle bed. The goal of the study was to use the model to calculate the depth of moisture penetration as a

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52 function of time. It was assumed that mois ture migrates through the bulk by diffusion only and is driven by concentration gradients be tween the fluid phase and the particles. In the model, material is exposed to a step chan ge in moisture content due to an increased amount of moisture in the air above the part icle bed. The equations that describe the system are t q z c D t ca 12 2 (4-1) q q k t qe g (4-2) where qe is the equilibrium moisture cont ent given by the sorption isotherm, c is the vapor phase concentration, q is the solid moisture content, D is the diffusion coefficient, is the bed porosity, and a are solid and air densities and z and t are distance and time. It is assumed that the rate of adsorption is pr oportional to the concentration difference such that kg is constant. Kg is the called the Glueckhauf f actor or linear driving force coefficient. The Glueckhauf factor is a function of the particle radius R and the diffusivity De. 215 R D ke g (4-3) Equation 4-1 is based on one-dimensional di ffusion in the z direction. The initial conditions are chosen such that there is a uniform concentration of vapor in the powder bed and the moisture content of the solid is in equilibrium with the vapor phase. Experiments were preformed to verify the model and it was found that the model did not accurately predict the data for all times. At shorter times the model under predicts the measured data. This simple model is only an ap proximate prediction of the data. It fails to

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53 address the effects of temper ature and free convection on th e moisture migration through the particle bed. The results from the moisture migration modeling were applied to caking of fine crystalline powders in a second paper by Ta rdos (1996b). The purpose of this study was to examine the caking behavior of fine bulk powders while exposed to humid atmospheres. The depth of pene tration of moisture into the powder bed was calculated by solving equations 4-1 through 4-3. The caking ratio defined by Ta rdos is the height of the portion of the upper surface of the powder at a critical moisture content divided by the total powder depth. The critical solids moistu re content is the value at which caking of the powder begins. Hence, the tendency of a powder to cake can be determined from this simple model given that the material prope rties of the powder relevant to caking are known. Tardos also suggests that hydrate formation causes swelling in the powder bed. This compression is associated with caking. Due to the adsorption of moisture, various hydrates are formed which cause a swelling of the powder. This increase in bulk volume is on the order of 10% or more and occurs at relative humidities of hydrate formation. Tardos finds a relationship between th e increase in bulk volume and the caking propensity of the powder using dilatometer testing of sodium car bonate. These findings suggest that the powder swelling is primarily responsible for the caking of the powder. Rastikian and Capart (1998) later develope d a model for caking of sugars in a silo during storage. The purpose of the study was to create a model to predict the moisture content profiles, air humidity and temperat ure inside a laborat ory silo. The model includes not only the mass transport but also the heat transport which Tardos fails to

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54 address. The system is described by the following two equations for heat and mass transport as well as the kinetic equation for the drying of sugar. Mass balance equation: c c k z c F z c r c r c r D t ce f a a e 2 2 2 21 (4-4) In equation 4-4 c is the concentration of the air, ce is the equilibrium concentration at the particle surface, De is the effective mass diffusivity of the water vapor, Fa is the flow-rate of the inlet air, a is the density of the air, and r and z are the radial and axial coordinates. Heat balance equation: v e f ps a ps pa a ps sH c c k c z T c c F z T r T r T r c t T 2 2 2 21 (4-5) In equation 4-5 T is the temperature of the air, s is the thermal conduc tivity of the solid, is the density of the solid, cps is the specific he at of the solid, cpa is the specific heat of the air and Hv is the enthalpy of water vaporization. A laboratory silo was constructed in whic h humid air is blown through the bed of sugar. The temperature along the height of th e silo and the relative humidity of the air above the particle bed are measured. These experimental data are compared to the proposed model. It was found that the model approximates the temperature profile within the silo reasonably well. However, Rastikian et al assumed that the mass and heat transfer by diffusion in the radial direction can be neglected. Also, the system is based on forced convection through the pa rticle bed. If material insi de a silo is stored in an uncontrolled environment, it is more realisti c to expect free conv ection in the system. Dehydration during storage and mass transf er likely occur through free convection.

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55 Leaper et al (2002) developed a model of moisture migration through a bulk bag as a function of humidity cycling. The m odel is based on the work of Rastikian et al and Tardos. The purpose of this study was to pred ict the temperature pr ofile and moisture content profile of the material within the bulk bag. It was assumed that moisture migrates by diffusion only due to moisture concentra tion gradients caused by fluctuating local relative humidity. The authors developed a procedure to calculate the profiles as follows: Determine the temperature profile using a simplified one-dimensional finite difference model which does not account for convective heat transfer. 1 1 1 j i j i sp sT T A Q and 1 j i ps solid j iT c m t Q T (4-5) where Ti j is the temperature at node position i and timestep j. Q is the heat flowrate through and specific transfer area Asp, msolid and cps are the mass and heat capacity, respectively. After the temperature profile is determined the RH profile can be calculated for a specific temperature. wH RH H 100 (4-6) where Hw is the saturated humidity and RH is the relative humidity. The saturated humidity is simply the maximum concen tration of water vapor possible at a specific temperature. This variable can be obtained from the partial vapor pressure of water. air water O H T O H wmw mw p p p H2 2 (4-7) where pH2O is the partial pressure of water vapor, pT is the total pressure and mwwater and mwair are the molecular weights of wate r and air, respectively. The solids moisture content q of the sample is calculated using equations 4-6 and 4-7. 100 100q m H RH m qsolid w air tot (4-8) where qtot is the total moisture content, mair is the mass of air and msolid is the mass of solid.

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56 As the temperature changes, the RH profile at each node is adjusted to reflect moisture migration due to diffusion. j i j i j i sp j iH H H DA H1 1 1 1 (4-9) where D is the diffusion coefficient for the moisture migration. Finally, the new equilibrium RHeq and solids moisture content qeq can be calculated. Thus equation 4-8 is adjust ed to include the equilibrium values. 100 100eq solid w eq air totq m H RH m q (4-10) The sorption isotherms are used to derive an empirical relationship between the equilibrium solids moisture content qeq and relative humidity RHeq. To verify the model, the solids moisture content was measured after exposing the material to a temperature cycle. The authors c onclude that they can create a profile of the solid moisture content in the bulk and as a result they can predict cake formation. All of the current models are useful as a first approximation of moisture migration caking. However, they reduce the fundamental transport equations, i.e. the moisture migration, to a simplified case that disregards pertinent details of the caking process. Moisture Migration Model The driving force of moisture migration cak ing is a concentration gradient within the bulk. This gradient causes transport of moisture through the interstitial voids of the bulk solid. Bulk materials are typically stored in an environment where temperature and relative humidity are not controlled. Thus, cha nges in the temperature from day to night can induce a thermal gradient through the ma terial which affects the local relative humidity surrounding the particles. The changes in relative humidity caused by temperature fluctuations initiate the mois ture migration through the bulk. With the additional moisture in the air, particles will adsorb moisture until an equilibrium state is

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57 obtained. The equilibrium cond ition is determined from the sorption isotherms of the material. The isotherm represents the equi librium relationship be tween the bulk solid moisture content and the relativ e humidity of air surrounding th e particles as described in the previous chapter. The process of moisture migration can be described by fundamental transport equations of heat, mass, and energy. It is a ssumed that the air surrounding the material is stagnant; therefore the only velocity gradient is that due to free convection. The partial differential equations will be solved using finite element techniques. The location of the moisture is determined using finite element methods modeling. This result can be used to locate the formati on of solid bridges within a bulk material. The following steps are required to achieve this result: Determine if convection plays a role in the caking process by comparing the temperature profiles from the models with and without a convective term. Analyze the Peclet number in the cell to determine if convection is a dominant mode of transportation for the moisture. Calculate the diffusion through the ce ll from the temperature profile. Determine where the moisture migrates in space and time i.e. the solids moisture content over time. Finite Element Modeling The moisture migration process is modele d using finite element methods utilizing the COMSOL Multiphysics software. This met hod is used to solve partial differential equations (PDEs) which describe and predic t the moisture migration in caking. The behavior of moisture is modele d on a continuum scale. The flow of air, the heat transfer through bulk solids, and the moisture content of the solid within the cell are described. This is used to predict the am ount of material or the thickn ess of the layer of material involved in a caking event and al so the strength of the cake.

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58 The general approach to solving PDEs using the finite element method includes discretization, developing the element equa tions, characterizing the geometry, applying boundary conditions and obtaining a solution. The discretization i nvolves dividing the solution domain into simple shape regions or elements either in one, two, or three dimensions as shown in Figur e 4-1. The points of intersection of the lines are called nodes. Figure 4-1. Examples of the elements used in FEM (A) one-dimensional (B) twodimensional and (C) three-dimensional. Approximate solutions for the PDE’s are developed for each of these elements. The equations for the individual elements must be linked together to characterize the entire system. The total solution is obtained by comb ining the individual solutions. Continuity of the solution must be ensured at the bounda ries of each element. The value of unknown parameters is generated continuously acro ss the entire soluti on domain. After the boundary conditions are applied, the solution is obtained using a variety of numerical

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59 techniques. COMSOL Multiphysics is the software used to solve the partial differential equations. About COMSOL Multiphysics COMSOL Multiphysics is an interactiv e environment for m odeling and solving scientific and engineering problems based on partial differential equations (PDEs). The power of this program lies in its ability to couple several diffe rent physical phenomena into one system and solve the PDEs simultaneously. The finite element method (FEM) is used to solve the PDEs in two dimensions. FEM is a discretization of an original problem using finite elements to de scribe the possible forms of an approximate solution. The geometry of interest is meshed into units of a simple shape. In 2D, the shape of the mesh elements is a triangle. After the mesh is cr eated, approximations of the possible solutions are introduced described by a function with a finite number of parameters, degrees of freedom (DOF). Model Geometry The geometry used in the COMSOL solver is modeled after the cells used to make the cakes as shown in chapter 2, Figure 2-4. A schematic of the geometry is shown in Figure 4-2. The width of the cell is 0.026 meters and the height is 0.02 meters. The bottom plate, labeled subdomain B, is made of aluminum and is the location of the heat source. The walls, labeled subdomain C, are made of phenolic which is a insulating material. The powder is containe d in the center of the cell, labeled subdomain A. The cell is assumed to be axisymmetric for the simplification of the model and reduced computational time. There are 7272 triangular el ements in the mesh as shown in Figure 4-3. The size of the mesh elements along the boundary are smaller than those in the domain.

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60 Figure 4-2. Schematic of the geometry used in the COMSOL solver. The lettered areas label the domains. Figure 4-3. Caking cell geometry with mesh elements. Partial Differential Equations The process of moisture migration through a porous media from the atmosphere is governed by transient differential equati ons of heat (convection-conduction), mass r z z r

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61 transfer (convection-diffusion), adsorption/ desorption, and conti nuity and momentum (Brinkman’s equation). To construct a mathema tical model of the moisture migration, the following assumptions are considered: Powder grains are isothermal and in equi librium with surrounding gas. This can be justified by the fact that grains are small and flow rate of gas is low. Heat generated or absorbed by ad sorption/desorption is neglected. There is no mass (moisture) transfer fr om grain to grain through diffusion or capillary bridges. The partial differential equations used to de scribe the moisture migration are given in detail below. Convection-conduction The convection-conduction equation is us ed to described the temperature distribution within the system. T u C Q N h T k t T Cp i i D i p ts (4-11) where ts is a time scaling coefficient, is the density, Cp is the heat capacity, k is the thermal conductivity, T is the temperature, Q is the heat source, u is the velocity, and hiND,i is a species diffusion term. The temperatur es of the fluid and particles are assumed to be in equilibrium. This can be justified by the fact that, for small particles, heat diffuses by conduction almost instantly. Equa tion 4-11 is applied to every subdomain. However, convection is not include in subdomains B and C. The velocity is determined from the Brinkman equation. The time scaling coefficient is one for all subdomains. The appropriate constants for each subdomain are specified. Convection-diffusion The convection-diffusion equation descri bes the mass transport in the system. c u R c D t cts (4-12)

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62 where D is the diffusion coefficient, R is the reaction rate term and c is the concentration. Since mass transfer only occurs in the bulk, th is equation is only applied to subdomain A. The reaction rate term is a source or sink term for the water va por on the fluid phase which is described by the change in solids mo isture content with time. The time scaling coefficient is one. Brinkman Equation It is believed that free convection plays a role in the moisture migration through the system. Caking is known to be a time induced event. Thus if free convection aids the moisture migration process, the cakes may develop in a shorter period of time. This would indicate that free conv ection can play a major role in inducing cohesive storage time effects. The Brinkman equation coupled with the continuity equation is used to describe free convection in the system. Th e pressure distribution is calculated and consequently the velocity distribution (free convection) within the system and at its boundary. The Brinkman equation is a derivativ e of Darcy’s law. Da rcy’s law describes flow through a porous medium. However, th e Brinkman equation accounts for the viscous forces (Brinkman, 1947). F u u l p u t uT (4-13) where u is the velocity, is the dynamic viscosity, is the permeability, p is the pressure, l is an identity matrix and F is a volume force (g). The equation for the permeability and the density as a function of temperature are given in Appendix B. Solids moisture content The solids moisture content is describe d by the linear driving force model. The equation used by COMSOL is given by:

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63 F t q (4-14) where F is a source term and is a flux vector. The source term F is given in equation 42. However the rate constant kg is determined from the kinetic data in chapter 3. The dissolution of the material and the evaporation of the moisture vapor is described in this model. Both adsorption and desorption must be considered because there is a hystersis in the sorption curve; the adsorption and deso rption are not equal. The equilibrium conditions are not the same. The equations for the isotherm curves used in the model are given in chapter 3. The relative humidity is calculated from the saturation concentration as given in Table 4-1 and Appendix B. Model Parameters The experimental temperature and solid moisture content within the Johanson Indicizer caking test are used as the initial and boundary conditions for the finite element modeling. The boundaries are identif ied in Figures 4-2 and 4-4. The boundary conditions, initial conditions and constant s are listed in Tables 4-1 through 4-3. Figure 4-4. Boundary conditions. Heated z r

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64 Table 4-1. Parameters for COMSOL simulations. Initial Pressure, p_init 101325 Pa Gas Constant (fluid), R_fluid R_fluid Initial Temperature, T_init 298.15 K Linear driving constant (isotherm), k_ads 0.0013 Porosity () 0.5 Linear driving constant (isotherm), k_des 0.0013 Particle Diameter, P_diam 0.001 m Antoine’s coefficient, a_a 16.75667 Viscosity (air), v 1.8e-5 Pa s Antoine’s coefficient, b_b 4087.342 Permeability, permea Antoine’s coefficient, c_c -36.0551 Diffusivity, D 4.2e-6 m2/s Constants (isotherm), A1 0.47 Heat Capacity (solid), Cp 1500 J/kg K Constants (isotherm), A2 0.47 Heat Capacity (fluid), Cf 1005 J/kg K Constants (isotherm), B1 13.5 Conductivity (solid), k 0.2 W/m K Constants (isotherm), B2 13.5 Density (solid), dens_s 2250 kg/m3 Temperature Fluctuation, T_fluct 15 K Density (bulk), 1600.24 kg/m3 Initial Concentration, c_init 0.01 Molecular Weight (water), mol_w 18 Initial Saturated Concentration (moles), c_sat_mol_init c_sat_mol_init Molecular Weight (air), mol_a 29 Initial Saturated Concentration, c_sat_init c_sat_init Gravity, g 9.81 m/s2 Initial Relative Humidity (bulk), RH_init RH_init Gas Constant, R_gas 8.314 Initial Solids moisture content, u_init u_init

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65 The temperature profile imposed on boundary 2 is given in Figure 4-5. The temperature profile is a sine wave and is maintained over a 24 hour time period. The maximum temperature is 55 C and the minimum temperature is 25 C. 295 300 305 310 315 320 325 330 02004006008001000120014001600 Time (min.)Temperature (K) Figure 4-5. Temperature profile imposed at th e base of the cell in the finite element simulations. The initial temperature of the powder is 25 C and the initial temperature of the heat source is 40 C. The temperature surrounding the cell is maintained at 20 C. There is flux at the outer boundaries (5,8,7). Heat is conducted through the material and across the inner boundaries (3,6).

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66 Table 4-2. Boundary conditions for COMSOL simulations. Convectionconduction Convectiondiffusion Brinkman Equation General PDE 1 Axial symmetry Axial symmetry Axial symmetry Axial symmetry 2 TB 3 Insulated No Slip Neumann 4 TB 5 hA(T-T) 6 Insulated No Slip Neumann 7 hA(T-T) Insulated No Slip Neumann 8 hA(T-T) Table 4-3. Subdomain conditions for COMSOL simulations. Convectionconduction Convectiondiffusion Brinkman Equation General PDE A Yes Yes Yes Yes B Yes C Yes The variables listed in Tables 4-1 an d 4-3 are described in Appendix B. The Role of Convection in Moisture Migration Temperature Profiles The current models for moisture migration available in literature assume that free convection does not contribute to the heat transport in the system. Finite element simulations are executed with and without convection to determin e the significance of convection. Temperature profiles along the cen terline and near the insulated boundary (6) are shown in Figures 4-6 and 4-7. The temp erature profiles in these figures include convection.

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67 295 300 305 310 315 320 325 330 00.0050.010.0150.020.025 z coordinateTemperature (K) 0 30 sec 1 min 10 min 30 min 1 hr 4 hr Figure 4-6. Temperature profile along the cente rline of the cell with convection included. 295 300 305 310 315 320 325 330 00.0050.010.0150.020.025 z coordinateTemperature (K) 0 30 sec 1 min 10 min 30 min 1 hr 4 hr Figure 4-7. Temperature profile near the in sulated boundary of the cell with convection.

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68 It can be seen in Figure 4-6 that the te mperature along the cent erline increases until the setpoint temperature is achieved. The set point temperature is obt ained after only five minutes. An oscillation of the te mperature is seen after one minute. This is attributed to the free convection plumes that initially deve lop within in the cell. These plumes are shown in Figure 4-8. The plumes appear from zero to two minutes a nd then disappear. It can be seen in Figure 4-7 that the temperat ure profile near the insulated boundary reaches the setpoint temperature after thirty minutes The oscillations in this figure are also attributed to the formation of plumes. Te mperature profiles for the case of no free convection are shown in Figures 4-9 and 4-10. It can be seen that the setpoint temperature is obtained after four hours. Comparing this result to the free convection included case suggests that the free convection does play a si gnificant role in the heat transport of the system. If free convection is included, the cak es will develop in a s horter period of time. Figure 4-8. The temperature profile within th e cell illustrating the convective plumes that develop.

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69 295 300 305 310 315 320 325 330 00.0050.010.0150.020.025 z coordinateTemperature (K) 0 30 sec. 1 min. 10 min. 30 min. 1 hr. 4 hr. Figure 4-9. Temperature profile along the centerline of the cell without convection. 295 300 305 310 315 320 325 330 00.0050.010.0150.020.025 z coordinateTemperature (K) 0 30 sec. 1 min. 10 min. 30 min. 1 hr. 4 hr. Figure 4-10. Temperature profile near th e insulated boundary of the cell without convection.

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70 The temperature in the center of the cell is plotted in Figure 4-11. The difference between the convection and nonconvective profiles are show n. With the free convection in the system, the response to a change in temperature occurs faster than conduction only. 295 300 305 310 315 320 325 330 0510152025 Time (hr.)Temp (K) without convection with convection Figure 4-11. Temperature profile at the center of the cell, with and without convection. The variation of material temperature as a function of time is most critical in the region where shear takes place i.e. near the cel l wall (insulated boundary). In this region bonds between particles are being broken a nd the force required to break these bonds determine the cohesive properties measured w ith this test cell. Temperature profiles in this region also indicate that free convecti on speeds the caking pr ocess and results in higher cell temperatures during heat-up (see Figure 4-12).

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71 295 300 305 310 315 320 325 330 0510152025 Time (hr.)Temperature (K) without convection with convection Figure 4-12. Temperature profile at the cell wall, with and without convection. An analysis of the cell Peclet number is also used to determine if convection contributes significantly to th e moisture migration through th e cell in caking. The Peclet number is the product of the Reynolds number and Prandtl number. The physical interpretation is the ratio between the heat tr ansfer by convection to the heat transfer by conduction. 2/ ) ( / ) (o o o o o pl T T k l T T v C e P (4-11) The Peclet number gives an idea of the dom inant mode of heat transfer through the cell. A high Peclet number means that the heat transfer from convection can not be neglected and the heat transfer from conduction is not substantial. If the number is low, then heat transfer from convection can be assumed negligible. When convection is added

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72 to the model, the cell Peclet number is in the range of 1 to 5. This suggests that convection is the dominant mode of heat transport. Solids Moisture Content The solids moisture content of the material is important because a change in this quantity determines the extent of caking in the system. Adsorption of moisture and increase in the solids moisture content sp ecifies the amount of dissolved materials available for the creation of solid bridges. Th e desorption of moisture a decrease in the solid moisture content, specifies the amount of crystallized solid material in the bridge. Thus far, it has been determined that th e convective system dominates the moisture migration process. Therefore, the results shown include free convection in the system. The solids moisture content within the cell is given in Figure 4-13. The moisture content is plotted as a function of the position along th e height of the cell at various times. It can be seen that the material in the lower ha lf of the cell does not undergo a change in moisture content. Near the top of the ce ll the moisture content changes in time and increases and decreases with the temperatur e. The moisture content follows the same trend as the temperature profile with time shown in Figure 4-14. This figure shows a cyclic moisture profile in the shear region of the test cell. The frequency of the moisture cycle follows the temperature profile but lags the temperature profile. An increase in the moisture content followed by a subsequent reduction is the mechanism behind solid bridge formation suggests that the cake will develop in the region near the wall and will be most pronounced at the top surface of the test cell. The regions of greatest caking potential are areas with significant adsorpti on and desorption as a function of time.

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73 0.11 0.16 0.21 0.26 0.31 0.36 0.41 0.46 00.0050.010.0150.020.025 z coordinateSolids Moisture Content 0 3 hr 6 hr 9 hr 12 hr 15 hr 18 hr 21 hr 24 hr Figure 4-13. Solids moisture content profile at the centerline of the cell with convection. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0510152025 Time (hr.)Solids Moisture Content Figure 4-14. Solids moisture content as a f unction of time at the center of the cell.

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74 This re-crystallization effect can best be seen by superimposing the temperature profile computed within the shear region on the phase diag ram for the sodium carbonate system. Figure 4-15 shows the operating curve fo r the test cell. Mate rial within the cell initially starts at a solids c oncentration of 14%. The change in temperature and moisture content causes material within the cell to cross a phase boundary. Increasing the temperature causes the sodium carbonate to dissolve forming a solution of monohydrate. Subsequently cooling the material causes th e carbonate to cross the phase boundary and form solids bonds between particles. Figure 4-15. Phase diagram of sodium car bonate with temperature profile imposed.

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75 It is possible to approximate the cake strength if the change in solids moisture content is known. The solids moisture conten t affects the unconfined yield strength by influencing the radius of the crystallized bridge b formed between particles. A detailed description of cake strength predictions and th e factors that influen ce the strength is given in the following chapter. Nonetheless, th e approximate strength of the cake can be determined from the variation in solids moisture content with time. The area of interest is the moisture content near the insulated bounda ry since the shear pl ane in the Johanson Indicizer is located in this region. The shear region is indicated by the hatched area in Figure 4-16. The change in solids moisture cont ent in this region dict ates the strength of the cake. The moisture content distributi on near the insulated boundary is shown in Figure 4-16. It can be seen that the moisture content varies along th e height of the cell. However, it does not change significantly within the radius of the sh ear plane. Because of this insignificance, the unconfined yield strengt h is assumed to be constant in this region. For larger variations in the moisture cont ent within the shear plane, the averaged unconfined yield strength must be considered. Using the equations in chapter 5, it is dete rmined from the moisture content profile within the shear plane that the unconfin ed yield strength increases by 18% at a consolidation stress of 10 kPa over the le ngth of the caking event. Given the same conditions, the yield strength data from chapter 2 indicates th at the strength increases by 40%. Although the unconfined yield strength approximation from the moisture migration analysis and the experimental data are not ex act, these values have the same order of magnitude. This comparison establishes a basis for estimating the unconfined yield

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76 strength as a function of the solids moisture content calculated from an finite element moisture migration analysis. 0.1 0.15 0.2 0.25 0.3 0.35 00.0050.010.0150.02 z coordinateSolids Moisture Content r = 0.024 r = 0.025 r = 0.026 Figure 4-16. Equilibrium solids moisture content as various positions along the radius.

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77 CHAPTER 5 EVALUATING THE CAKE STRENGTH OF GRANULAR MATERIAL In this chapter a model for evaluating th e cake strength of granular material is developed. Although there are models available in literature for estimating cake strength, it is shown that these models are often times inadequate predictors. All of the current models suggest that the streng th is influenced by a single va riable. However, it is well known that several factors affect the stre ngth of cakes. Therefore a new model is developed which includes the dependency on the particle size, moisture content, consolidation stress, and ot her material properties. Background A useful tool for understanding the caki ng of granular materials is a model predicting cake strength. This is not only useful for understanding the effects of the influential properties of caking but it can also be used as a predictive measure for future events. Rumpf (1958) was the first to propos e a theory for the tensile strength of agglomerates and many of the present mode ls are based on his work. He developed expressions for the strength of agglomerates with various types of interparticle bonds. One mechanism for agglomeration is the fo rmation of liquid bridges. Rumpf suggests that the tensile strength of an agglomerate is proportional to the inverse of the particle diameter squared. H d kp o t21 (5-1)

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78 In equation 5-1 is the tensile strength of the agglomerate, is the porosity of the cake, dp is the particle diameter, ko is the coordination number (number of contacts per particle), and H is the strength of the interparticle bond. The fl aw of this equation is based on the unrealistic assumption that the bridges fail simultaneously and that all the particles are of equal size. Rumpf also proposed a theory for agglomeration due to the formation of solid crystalline bridges. This equation is ba sed on moisture content of the material and the concentration of material in the bridge. k k k tq y 1 (5-2) In equation 5-2 yk is the concentration of the dissolved species k q is the moisture content of the particles before caking, is the density of the particle, k is the density of k in the crystal bridge, and k is the strength of a crystal bridge. It is assumed that by random packing the mean volume fraction of particles is equal to the mean cross sectional fraction of these particles. Thus, the dependence of the st rength on particle size is lost and the tensile strength is proportional to the volume fraction of crystalline material in the bridge times the strength of the bridge. This assumption differs from Rumpf’s previous theory that states that th e strength of an agglomerate is inversely proportional to the diameter of the particle. Simi lar to the first equation, it is assumed that all bridges fail simultaneously. Neglecting these critical parameters renders an inadequate predictive model for the cake stre ngth of granular materials. Other researchers have used Rumpf’s model to verify experimental data for various materials. Pietsch (1969a) appl ies Rumpf’s model to investig ate the influence of drying rate on the tensile strength of pellets bound by salt bridges. He measured the agglomerate strength using a vertical tensile tester. The deta ils of this tester are described in chapter 2.

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79 Pietsch altered Rumpf’s equati on to include a mean tensile strength as given in the equation 5-3. k s p s tM M 1 (5-3) where Ms and Mp are the mass of the salt in the dr y agglomerate and the mass of the agglomerate, respectively, s and are the densities of the salt and solid particles, and t is the average tensile strength of the bridges. Since the strength of th e bridges varies with changing crystal structure, an average tensile strength is used. He assumes that a crust, consisting of solid bridges, is formed ar ound the cake during the drying process due to the crystallization of the salt solution. This crust changes the drying rate of the cake, effectively changing the strength of the cake. The crust is removed before measurements are taken for the strength of the material. Piet sch reports that the te nsile strength of the core agglomerate is highly influenced by the drying rate. Tanaka (1978) further developed Rumpf’s model by incorporating the structure of the agglomerate. He included the effects of h eat and mass transfer on the formation of the solid bridges. Tanaka used a model of cont acting spheres with pendular water as shown in figure 5-1. The particles in the agglomerat e are assumed to be monosized with a radius of R. A fictitious sphere of radius r approximates the curv ature of the bridge. The volume of the bridge V and the narrowest width of the bridge b are derived as a function of R and Hence V and b are related by the parameter according to the following equation. 25 0 3/ 82 0 / R V R b (5-4)

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80 Equation 5-4 implies that the cake fails at th e narrowest width of th e bridge (neck). The volume V between two particles is a function of the total volume of a single particle Vt and the number of contact points per particle o tk V V / 2 (5-5) In equation 5-5 ko is the coordination number. This number is approximated by Rumpf (1958) as inversely prop ortional to the porosity (ko /) Figure 5-1. Model of contacting spheres with pendular water used to calculate the volume and width of the bridge. Tanaka used the relationship between the volume and the width of the solid bridge combined with Rumpf’s model to form the following equation for the tensile strength of powders. k X e c tX q C 2 / 1 1 / 1100 / 1 3 8 1 17 0 (5-6) where z is the tensile strength of the recrystallized solid bridge, Ce is the equilibrium concentration, q is the initial moisture content, c is the porosity of the recrystallized bridge, and X is a lumped parameter which is func tion of the temperature and humidity. It is assumed that the solid bridges are formed from dissolved material due to heat and mass

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81 transfer in the system. Equation 5-6 implies that the tensile strengt h is independent of particle size and rather a func tion of the moisture content and equilibrium concentration of the solid and the rate of mass transfer. Thus far the equations for evaluating cak ing have been focused on the tensile strength of the material as a function of particle size and moisture content. However, of greater importance with regards to this rese arch is how the unconfin ed yield strength is affected by these properties. It can be shown that the tensile strength t is proportional to the unconfined yield strength fc from the construction of Mohr circles and a yield locus in figure 5-2. sin 1 sin 1 t cf (5-7) where is the internal angle of friction. Figure 5-2. Mohr circles demonstrating the re lationship between tensile strength and the unconfined yield strength. Tomas et al (1982) formulated a model to invest igate the unconfined yield strength as a function of storage time using Rumpf’s and Tanaka’s model. The unconfined yield

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82 strength is measured using the Jenike shear tester. The authors propose that the unconfined yield strength c varies with the moisture content q of the material as follows: t dq Y dS Ds c 1 (5-8) The kinetics of the materials is assumed to follow the linear driving force model of equation 5-9. This equation is integrated over the storage time to find the moisture content of the material at a particular time: E spq q kA dt dq (5-9) Equation 5-10 relates th e unconfined yield strength to the decrease in moisture content over time through the combinati on of equations 5-8 and 5-9. t kA q q Ysp E O S Ds c exp 1 1 (5-10) In equation 5-10 c is the unconfined yield strength, Ds is the compressive strength of the solid bridge, YS is the solubility, qo and qE are the initial and equilibrium moisture contents, k is the mass transfer coefficient of water, Asp is the specific mass transfer area, and t is the time. This model gave only slight agreement with experimental results. However, the authors state that this mode l can be used to approximate the acceptable moisture content as a function of storag e time to avoid situations of caking. The above models for evaluating cake strength have focused on temperature induced caking. In other words a temperature perturbation is assume d to initiate a caking event. However, it is has been observed that a change in relative humidity can also trigger a caking event (Kun et al., 1998). The effect of humidity cycling has been studied by Leaper et al (2003). The authors make use of Tana ka’s model to develop a relationship

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83 between the cake strength and the number of humidity cycles. It is suggested that if the porosity and particle size remain consta nt the tensile strengt h can be given as N Kh t (5-11) where Kh is a parameter that incorporates the crystal bridge strengt h, particle size, porosity, and humidity swing and N is the number of humidity cycles. This model was compared to experimental results from a simp le compression tester adapted to control the relative humidity of the sample. The authors found that the cake strength does indeed vary with the number of humi dity cycles. The parameter Kh in equation 5-11 was calculated from experimental data to be 26 while the number of cycles N is raised to the power 0.546. Most of the understanding of agglomerate br eakage and all of th e previous models are based on the work of Rumpf. More recently the principles of fracture mechanics have been used to evaluate the strength of agglom erates as an alternative approach to Rumpf’s theory. Kendall (1988) and Adams (1985) have proposed a fracture mechanics description of agglomerate breakage. It is beli eved that internal flaw s or cracks within the material are responsible for the failure of the agglomerate. The authors describe agglomerate strength in terms of fracture mech anics parameters and the size of the crack which depend on the geometry and packing of the agglomerate. Kendall (1988) states that the fracture m echanics approach to agglomerate strength is based on three levels of magnification: pa rticle-particle contacts, an assembly of particles, and a block of material which behave s as an elastic solid. At the particle level the particles are held together, without binder, by an interfacial energy The interfacial

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84 energy can be determined from the size of the contact zone. From fracture mechanics analysis, the diameter d of the contact zone is given by 3 / 1 2 21 2 9 E d dp (5-12) where dp is the diameter of the elastic particles, E is Young’s modulus and is Poisson’s ratio. The Young’s modulus of an assembly of particles differs from that of two particles in contact. This is due to the elastic de formation of the assembly under stress. The effective Young’s modulus E* for an assembly of particles is given by 3 / 1 2 4 *1 1 17 pd E E (5-13) where is the porosity. The assembly of partic les also has an effective cleavage energy Rc *. 3 / 1 2 2 5 4 *1 56 p c cd E R (5-14) In equation 5-14 c is the fracture energy. The agglomerate is tr eated as a continuous medium and the fracture stress f is given by 2 / 1 *893 0 a R Ec f (5-15) where a is the length of the crack. Equation 5-15 applies to clean, smooth elastic spheres and is verified with experimental data on alumina and titania agglomerates. The results prove that this model is a better predictor of agglomerate strength as a function of particle size when compared to the theory of Rumpf. However, Kendall’s model fails to address the inherent plasticity of mo st materials during fracture.

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85 Two different approaches, Rumpf’s theory and fracture mechanics principles, for evaluating the strength of agglomerates have been discussed thus far and the dissimilarities between the models are a pparent. One major difference between the models lies in Rumpf’s derivation of his theo ry. He assumes that the particles are bound together by interparticle forces and the additi on of such forces yields the ultimate strength of the agglomerate. Whereas the fracture mech anics view is that the agglomerate is an elastic body that satisfies the Griffith energy criterion of fracture. Other distinctions between to two theories include the functiona lity of particle si ze and porosity and the assumption of Rumpf that the bridges fail simultaneously. In this research, the principles of fract ure mechanics are applied to evaluate cake strength. The fracture stress f of the material is assessed to determine the unconfined yield strength of the cake. The fracture st ress is defined as the minimum amount of energy needed to fracture the bonds of the cake. An essential factor lacking in the fracture mechanics model as well as the models of Rumpf is the functionality of consolidation stress on the strength of the material. As show n in chapter 2, there is a strong relationship between the unconfined yield strength and th e consolidation stress. Therefore, a new modified fracture mechanics model is proposed for evaluating cake strength which includes the structure of the bridge, particle size, moisture content and consolidation stress. Modified Fracture Mechanics Model for Evaluating Cake Strength The principles of fracture mechanics have been applied to the field of particle technology as an alternative approach to Rump f’s theory in determining the strength of agglomerates. A theory has been developed to explain the failure of solids caused by

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86 flaws or imperfections in the structure of th e solid and the elastic and plastic deformation of the material. The current models proposed by Kendall (1988) and Adams (1985) are based on Linear Elastic Fracture Mechanics (LEFM). This concept is applicable to materials with relatively low fracture resistan ce. Failure below their collapse strength is common therefore these materials can be an alyzed on the basis of LEFM (Broek, 1988). The failure of very brittle materials can be described using LEFM, however most real materials exhibit plastic deformation during failure. For this condition, Elastic-Plastic Fracture Mechanics (EPFM) must be appl ied. The fracture parameters of many crystalline materials, which are prone to cak e, are sufficiently described using LEFM. However the caking process alters the surface characteristics of these materials thus affecting the mechanics of fracture. This ch ange is caused by the creation of ‘soft’ material in the contact zones due to the di ssolution of the particle surface. The bridges formed throughout the process of caking may completely solidify to create a brittle structure. However, it is most probable to assume there is partial solidification of the bridges creating a structure which will deform during fracture. Therefore EPFM will give a better approximation to cake strength. Before introducing EPFM it is useful to review LEFM. The fracture parameters in both areas are directly related and the principles of fractur e mechanics were originally developed for linear elastic materials. Linear-elastic Fracture Mechanics LEFM is based on an energy balance in wh ich the strain energy released at the crack tip provides the driving force to crea te new surfaces (Griffith, 1920). In order for fracture to occur, the rate at which energy is released in the solid must be equal to or greater than the cleavage resistance Rc. Thus the elastic energy release rate of the crack G

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87 must be equal to the crack resistance Rc before crack propagation and fracture can occur. The Griffith criterion fo r fracture is given by: cR E a 2 2 or cR G (5-16) where is a dimensionless factor base d on the geometry of the solid, a is the crack length, and E is Young’s modulus of elasticity. Applying Hooke’s law (=E) equation 5-16 can be written as follows: cR a 2 (5-17) An alternative approach to the Griffith fracture criterion is investigating the stress field at the tip of the crack (Irwin 1957). The stress intensity factor K defines the behavior of the crack under an applied load. This approach is simply the study of stress and strain fields near the tip of a crack in an elastic solid. Since the stresses are elastic, they must be proportional to the stress (or applied load) Irwin proposed the following equation for the stress field near the tip of a crack. 2 / 1a K (5-18) In equation 5-18 is a dimensionless geometry parameter, a is the crack length and is the applied load. The crack will grow when K reaches a critical value Kc. 2 / 1a Kf c (5-19) At the time of fracture the stresses in the crack tip are equal to those in the elastic solid and Kc becomes the toughness of the material. Kc is a measure for the crack resistance of a material and is called the plane strain fracture toughness (Broek, 1977). It can be shown that the elastic energy release rate is equivalent to the st ress intensity factor (Broek, 1988).

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88 E K G2 (5-20) Elastic-plastic Fracture Mechanics EPFM incorporates the plastic deformation at the crack tip during fracture that is neglected in LEFM. The fracture parameter used to describe the crack propagation is the J-integral. The J-integral defines the strain energy release rate for EPFM. Therefore J is equivalent to G for linear-elastic materials. Similarly, the symbol JR is used to denote the fracture energy for non-linear el astic materials. For the case of plastic deformation the geometry factor is termed H and this quantity may vary from the linear elastic case. For LEFM the stress-strain curve is linear and it follows Hooke’s law, however for EPFM the stress-strain curve is non-linear. The stress -strain curve is approximated by the RambergOsgood equation (equation 5-21), where n is the strain ha rdening exponent and F is the plastic modulus. F En (5-21) Using this approximation for the stress-strain relationship, the e quation for the plastic fracture energy becomes: F a H Jn R1 (5-22) where H is the geometry factor, a is the crack size and JR is the fracture energy for nonlinear materials. If n = 1(F = E) equation 5-22 reduces to th e equation 5-16 for linear elastic materials. The total fracture energy is the addition of the elastic and plastic fracture.

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89 Tot RJ J G or (5-23) Tot nJ F a H E a 1 2 2 Fracture will occur when the stress exceeds a critical value th at satisfies equation 5-23. For most real materials, the plastic term is much larger than the elastic term (Mullier et al. 1987). Therefore the elastic term is negligible a nd the fracture stress f can be approximated by 1 / 1 n R fHa FJ (5-24) It must be noted th at the fracture stress f is a tensile stress and must be changed to an unconfined yield strength fc, by equation 5-7, for the purpose of this research. The basis of the new model for evaluating cake strength is given by equation 5-24. The fracture stress is analogous to the tensile stress of the material. This parameter as defined by fracture mechanics is a bulk property. It must be considered that there may not exist a bridge between every ad jacent particle. Thus the tensile stress is defined as the single bridge strength. The bulk tensile strength is determined by multipling the bulk tensile strength by the coordination number of the particles. The probability of a bridge existing is determined from X-ray tomogr aphy images. Using X-ray tomography a three dimensional figure is constructed of the caked sample. The figure is sliced along the x, y, and z planes. The bridges appear as the gray matter between particles, as illustrated in figure 5-3. Each slice is analyzed and the probability Pb of a bridge existing between particles is given by ratio of the number of contacts with a bridge to the total number of contacts.

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90 Figure 5-3. X-ray tomography sl ice of caked sodium carbonate. The existence of a bridge does not mandate th at it will contribute to the unconfined yield strength. It is postulated that only the major forces within the shear zone influence the strength. Thus the probability of shearing across a major force Pf must also be considered. A value of Pf= 1 would represent the breakage of all bridges in the shear zone. Combining the probabilities of bridge exis tence and failure with equations 5-7 and 5-24 the unconfined yield strength fc of a caked material can be written as sin 1 sin 1, tot f cf (5-25) where f b f tot fP P n n 22 1 (5-26)

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91 where f is the average fracture stress of a bridge, Pb is the probability that a solid bridge will form between the particles, n1 is the number of layers in the shear zone and n2 is the total number of possible contacts in one la yer of the shear zone. The average fracture stress is used in equation 5-26 beca use of the distribution particle sizes. Model Parameters The parameters of the model given in equation 5-25 are calculated in the proceeding sections. Calculating the narrowest width of the brid ge as a function of the bridge volume Similar to Tanaka’s derivation, a model of contacting spheres with pendular water is used for the geometrical calculations. Figure 5-1 shows the model of two contacting spheres with the same radius R Circles of radius r represent the boundary of the pendular water with surrounding air. The volume of water at one contact point is V The smallest radius of the bridge is b. The terms b and V are calculated in terms of R and as shown in equations 5-28 and 5-29. sin sin 1 R r (5-27) sin cos 1 1 cos R r r R b (5-28) 0 2 3 3 2cos cos cos cos 2 d R d r r r R V (5-29) It is important to establish a link betw een the smallest radi us of the bridge b and the volume of water at a contact V The smallest radius of the bridge b/R is related to the volume V/R3 by the parameter This relationship is plotted on a logarithmic scale in figure 5-4 and is given by:

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92 24 0 369 0 R V R b (5-30) 0.1 1 0.1110 Vb Figure 5-4. Logarithmic plot of th e smallest radius of the bridge b vs. the volume of the bridge V The volume of water at a contact V is a function of the total volume on the surface of a single particle Vt and the coordination number ko. If it is assumed that the coordination number ko is inversely proportional to the porosity then the volume of a contact is given by equation 5-5 (Rumpf, 195 8). The initial total volume Vto can be expressed in terms of the moisture content of the material q c w w c p o tq R k qV V 3 03 8 (5-31) where Vp is the volume of the particle (33 / 4 R), and w and c are the densities of the cake and water, respectively. Thus th e initial total volume at a contact Vto is given in terms of the particle radius, moisture cont ent, coordination numb er and densities.

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93 The volume calculated in equation 5-31 is for a liquid bridge. For the case of caking, the heat and mass transport in the syst em must be considered. As a bridge forms there will be evaporation of water from the br idge and dissolution of the particle surface. Thus equation 5-30 becomes 24 0 369 0 R V R bsb (5-32) where Vsb is the volume of the solid bridge. The dissolution rate of the solid and vaporization rate of the water must now be included in the calculati on of the volume of a solid bridge. The dissolution rate is given by C C k dt dwe d (5-33) where w is the weight of the dissolved solid, kd is the mass transfer coefficient for dissolution, Ce is the equilibrium concentration, and C is the concentration of the dissolved solid. The vaporization ra te of the water is given by H H k dt dVw v (5-34) where kv is the drying rate, Hw is the saturation humidity of the air and H is the absolute humidity. The concentration may be written as a function of the volume of water at a contact. V w C (5-35) Differentiating equation 5-35 with respect to V gives dV dC V C dV dw (5-36)

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94 From equations 5-33 and 5-34 the relationship between the weight of the dissolved solid and the volume of water at a contact is written as H H k C C k dV dww v e d (5-37) Combining equations 5-36 and 5-37 gives the co ncentration of the dissolved solid as a function of the volume of water at a contact. V C H H k C C k V dV dCw v e d 1 (5-38) Equation 5-38 is solved for the initial conditions of t =0, V = Vto and C =0. ) 1 /( 11 X e e toXC C X XC V V (5-39) where X = kd/kv(Hw-H) and assumed to be constant. If it is assumed that the solids stop dissolving when the equilibrium concentration is reached then C = Ce and V = Ve and equation 5-39 becomes ) 1 /( 1 X to eX V V (5-40) and the volume of the crystal bridge is written as ) 1 /( 1 3 3 3 X b to e b e e sbX R V C R V C R V (5-41) The final form of the relationship between th e smallest bridge radi us and the volume of the bridge is given by substituting equations 5-32 and 5-31 into 5-41. 24 0 ) 1 /( 13 8 69 0 X e b p wX qC R b (5-42)

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95 Calculating the length of the crack The length of the crack is an important parame ter in determining the fracture stress of the material. It is a function of the particle radius R and the smallest radius of the bridge b. Thus the fracture stress dependence on particle size is established in this parameter. The crack size is defined as the void space between the bridges in an assembly of particles as shown in Figure 5-4. From the figure it can be seen that the length of the crack can be expressed as b R a 2 (5-43) Figure 5-4. Schematic of an agglomer ate showing the length of the crack a. Stress-Strain parameters F and n The stress-strain curve is assumed to be nonlinear due to the plastic deformation in the crack zone during fracture and can be estimated by the Ramberg-Osgood model, equation 5-21. The parameters of the curve ar e found by plotting the stress vs. strain on a log scale where the slope of the line is n and the intercept is F. The stress-strain curve for sodium carbonate is shown in figure 5-5. The stress-strain curve is measured using an MTS Alliance RT 30kN. A tablet is made with th e dimensions of 23mm in diameter

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96 and 90mm in height. The load applied to th e tablet and the displacement of the cross beam is recorded as the tablet is compressed at a constant velocity. 0 5 10 15 20 25 00.0050.010.0150.020.025 StrainStress (kPa) Ramberg-Osgood Hooke's law elastic plastic Figure 5-5. The stress-strain curv e for a sodium carbonate tablet. From figure 5-5, the plastic modulus F and strain hardening exponent n are found to be 4604 kPa and 0.834, respectively. It can be seen in figure 5-5 that th e stress-strain curve for sodium carbonate is approximately linear. This implies that there is little plastic deformation during failure. The stress-strain data of sodium carbonate is fitted with Hooke’s law and the Ramberg-Osgood model. Bo th models produce an adequate fit and therefore are equally applicable fo r determining fracture parameters. The stress-strain curve for lactose in the form of infant formula (Nestle Good Start) is shown in figure 5-6. It can be seen the there is plastic deformation and the Ramberg-Osgood equation is a good predic tor of the material response.

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97 0 1 2 3 4 5 6 7 00.0020.0040.0060.0080.01 StrainStress (kPa) Figure 5-6. Stress-strain curve for lactose. Geometry of the agglomerate calculating H The geometry factor H is dependent on the crack size, the width of the sample and the strain hardening exponent n. This factor has been calculated for various geometries and n-values for elastic fractures. It can be assumed that H is a function of the linearelastic geometry factor for small values of n and a (Broek, 1988). 1nH (5-44) Comparison of the Model with Experimental Data Effect of Particle Size The effect of particle size on the strengt h of granular materials has been widely researched. As stated previously, Rumpf was the first to theorize that the particle size influences the strength of a powder. He st ated that the particle size is inversely proportional to the tensile strength of the powder assuming that the porosity and the

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98 strength of the interparticle bonds are known, equation 5-1. This equation implies that the strength increases as the particle size decr eases. Because Rumpf’s model is formulated for the tensile strength of a material it must be modified by equation 5-7 to compare with unconfined yield strength data. Qualitatively, the data for sodium carbonate correlate with this theory as shown in Figure 2-11. From this figure, it can be seen that the unconfined yield strength decreases as a func tion of an increase in the mean particle diameter of the sodium carbonate as measured by the Johanson Indicizer. However, when fitting Rumpf’s model, there is not a good agreem ent with the data. It is shown in figure 5-7 that the model estimates the trend of the data but the magnitude is incorrect. Rumpf’s model fails not only because of the incorrect functionality of pa rticle size but also because of the assumption of simultaneous bridge failure. The fracture mechanics model created by Kendall and Adams is also used to explain the dependency of par ticle size. This model is given in equation 5-15 but must also be modified by equation 5-7 for the comparison of the unconfined yield strength results. In equation 5-15 it is stated that th e strength of the cake is inversely proportional to the square root of the particle diameter Figure 5-7 shows approximations of the yield strength using the models of Rumpf and fract ure mechanics. It can be seen that the fracture mechanics model gives an accurate es timation of the unconfined yield strength as a function of particle size. Th is correlation between experime ntal data and the models is verified by Kendall (1988). He found that the strength of titania powder varied with the particle diameter according to the theory of fracture mechanics.

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99 0 10 20 30 40 50 60 70 80 90 0100200300400500600700800900 Particle Size (um)Yield Strength (kPa) 3% Decahydrate Rumpf model Kendall model Modified model Figure 5-7. Yield strength of sodium carbonate as a function of particle size. The data is fitted with the models of Rumpf and Kendall. The new model proposed in equation 5-25 is also fitted to the data as shown in figure 5-7. It can be seen that the model ag reement with the data only slightly differs from the Kendall model. This can be attributed to the fact that the stress-strain curve for sodium carbonate shown in figure 5-5 shows that there is little plastic deformation during failure. Thus the LEFM model adequately predicts the cake strength. This implies that plastic deformation during failure is not significant to the cake strength of crystalline material. However, for most materials ther e is significant plasti c deformation during failure. This is demonstrated by the stress -strain curve for lactose (Nestle Good Start Infant formula). In this case, the modified fracture mechanics model is a better predictor of the strength.

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100 Effect of Moisture Content In many of the models the moisture content is the determining factor of cake strength. This is first stated by Rumpf in hi s model for solid bridges as he assumes that the mean volume fraction is equal to the mean cross sectional fraction. Thus the strength dependence on particle size becomes a seconda ry affect to the primary influence of moisture content. The unconfined yield stre ngth of sodium carbonate as a function of the moisture content is shown in figure 5-8. The da ta are fitted with the models of Rumpf and Tanaka. The latter suggests that the strength will plateau at a particular moisture content. This implies there is a critical moisture cont ent after which the strength will not further increase. 0 5 10 15 20 25 30 35 40 45 00.010.020.030.040.05 Moisture Content (%)Yield Strength (kPa) Experimental data Rumpf Model Tanaka Model Figure 5-8. The yield strength of sodium carbonate as a func tion of moisture content. This data is fitted with the models of Rumpf and Tanaka.

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101 It can be seen in figure 5-8 that Rumpf’s model is a better predictor than Tanaka’s model. This can be attributed to the varying significance of the moisture content in each model. In equation 5-5 Tanaka suggests that th e strength is proportional to the square root of the moisture content. Rumpf’s model suggest s that the strength is directly proportional to the moisture content of the material. Effect of Consolidation Stress Although the current models are adequate pr edictors of cake stre ngth as a function of particle size and moisture content, ther e exists no model which incorporates all the variables affecting caking. Also lacking in the current models is the effect of the consolidation stress on cake strength. It is shown in chapter 2, th at the unconfined yield strength increases by a factor of three with an increase in the consolidation stress. This observation may be explained with the following theories: an in crease in the contact area, an increase in the true contact area, and an increase in the nu mber of contact forces due to an increase in the consolidation stress. Both Rumpf’s and Tanaka’s models as sume only point contacts between the particles. However, it has been shown by Hertz (Briscoe, 1987) that the size of the contact zone varies with the consolidation stress. 3 / 14 3 E LR A (5-45) where A is the radius of the contact area, L is the stress, R is the radius of the particle and E is Young’s modulus. The change in contact area yields a displacement (or deformation) of the particle within the contact zone as illustrated in figure 5-9.

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102 Figure 5-9. Illustration of Hertz contact model The displacement is defined as the Hertz contact radi us squared divided by the radius of the particle. 3 / 1 2 2 24 3 R E L R A (5-46) According to the Hertz theory the contact area increases as the consolidation stress increases. Consequently the length of the crack a (Figure 5-4) may vary. The crack length is a function of the particle radius R and the smallest radius of the bridge b. While the particle radius remains constant with an incr ease in stress, the bridge radius may change due to a change in volume V. Thus varying the contact area has a direct effect on the relationship between the bridge radius b and the volume of the bridge V. The fracture stress of the material is de pendent on the bridge radius b thus affecting the unconfined yield strength of the material. The curvature of the bridge which is approximated with equation 5-27 is modified to include the Hertz theory. sin 2 sin 1 R r (5-47)

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103 Equation 5-47 is used in equations 5-28 and 5-29 to calculate th e bridge radius and volume using Hertz’s contact model. It can be seen in Figure 5-10 that the bridge radius changes only slightly with the inclusion of Hertzian contac t. This suggests the applied consolidation stress does not significantly eff ect the yield strength w ith a Hertzian contact model. b = 0.7378V0.2012R2 = 0.998 b = 0.6983V0.2366R2 = 0.99960.1 1 0.1110 Vb no load Hertzian load Figure 5-10. The volume of a bridge using the Hertzian contact model. The Hertzian contact model applies to smooth elastic particles. However most materials are not perfectly smooth but have a rough particle surface. Because of the surface roughness, the true contac t area is usually much smalle r than the apparent contact area (Persson, 2000). It has been observed that changes in the true contact area have an effect on the mechanical properties of powders (York et al., 1973). York et al. postulated that the changes occur due to melting at the as perities where the particles actually touch. Under a load, high pressures develop at the asperities, which subsequently lower the

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104 melting point of the material. The Skotni cky thermodynamic equation is used to determine the change in the melting point. At the contact points the material can melt due to the high stress and solid bonds are formed when the material re solidifies. These new bonds cause an increase in the st rength of the material. Conseque ntly, a change in the true contact area due to compression of the materi al can have an effect on the cake strength. However, the applicability of this model is limited to dry materials. During a caking event solid bridges are form ed between the particles. The size of a bridge depends on the volume of water at th e contact. If the water volume fills the space between the asperities, it is likely that the asperities will di ssolve and become a part of a large bridge. This can be seen in the X-ray tomography picture in Figure 5-3. The solid bridges in the contact zone ar e of the same magnitude as th e particle size which indicates that a surface area larger than the asperity size was wetted during caking. In this work a model is proposed for the increase in cake streng th as a function of consolidation stress based on the distribution of contact forces within the particle bed. It is believed that not all bridges in the cake c ontribute to its strength. Instead, it is the distribution of the major contac t forces within the shear zone that dictate the strength. When a consolidation load is applied to a material the stress between the particles not only increase in magnitude but they are also redistributed, creating an increase in the number of major contact forces within the shear zone. The contact forces between the particles are illustrated by the force chains in Figure 5-11. This image is produced from Discrete Element Methods (DEM) modeling as ex ample later in this section. If the major contact forces are redistributed when the c onsolidation stress on the material increases, then the spacing between the force chains will decrease.

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105 Experimentally this concept is difficult to verify. Therefore Discrete Element Methods (DEM) modeling is used to validate this theory. DEM is a numerical technique which solves engineering problems that are modeled as a large system of distinct interacting bodies or particles that are subject to collective motion. This technique is used to investigate the effects of microscopic mechanical properties on the macroscopic response of the body. The stress-strain res ponse of a material under a load can be simulated to produce the details of st ructure throughout the loading period. The Itasca PFC2D software is used to investigate the structure of the force chains within a particle bed. A consolidation load is applied to a bed of pa rticles and the contact forces between the particles are developed. The contact forces between the particles are connected through the contact po ints creating force chains as shown in Figure 5-11. In this figure the force chain structure for a hi gh load and low load can be seen. The high load is a factor of ten greater than the low load It can be seen that an increase in the load applied to a material increases the number and magnitude of the force chains. This implies that the spacing between the forces chains decreases and the number of major force chains increases. The increase in the number of major force chains increases the strength of the material. The term granularity is used to descri be the spacing between the major forces contributing to the cake strength. This para meter gives an indication of the size of the particle clusters between each contact force, i.e. the number of particles not associated with a major contact force. A decrease in the granularity implies that there exist a greater number of major force chains. The force ch ain images, as shown in Figure 5-11, are

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106 analyzed using image analysis to determin e the relative spacing between major force chains within the shear zone. Figure 5-11. Force chain structure in a DE M simulated powder bed with a high applied load (right) and a low applied load (left). A parametric study is conducted which incl udes varying the c onsolidation stress and friction coefficient between the particles. Before determining the granularity each image must be preprocessed to exclude th e magnitude of the contact forces and insignificant contact forces which do not add to the strength. The ma gnitude of a contact force is represented by the thickness of the chains shown in Figure 5-11. The magnitude increases with increasing stress as illustrated by an increase in the thickness of the lines at each load. At higher loads, the increased ch ain thickness reduces the spacing between the chains, thus influencing the gra nularity. To eliminate this artifact each chain is reduced to a unit thickness such that they are consis tent through every image. The major force

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107 chains are those which are continuous through the particles until anot her force chain is contacted. Both ends of the force chain are connected to another fo rce chain. The smaller contact forces not contributing to the streng th are eliminated by introducing a threshold on the data such that these forces are not included. After preprocessing, the images are analy zed using an imaging software package, ImageJ, to determine the granularity. There is a distribution of sizes representing the granularity as shown in Figure 5-12. Thus the median granul arity is considered at each load. 0 0.2 0.4 0.6 0.8 1 1.2 0510152025 Granularity (# of particles in a cluster)Volume distribution 5 kPa 10 kPa 15 kPa 20 kPa 25 kPa Median Granularity Figure 5-12. The cumulative distribution of the gr anularity at various consolidation loads. The dg, median granularity, is the size which splits the distribution into two equal parts. Half of the particle clusters are above this size and half of th e particle clusters are below this size. If the dg decreases with an increase in lo ad this suggests that the spacing between the force chains decreases. It can be seen in Figure 5-13 that the granularity

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108 decreases with a increase in the consolidati on load applied to the material. Thus the granularity is inversely proporti onal to the unconfined yield strength. It can also be seen in Figure 5-13 that the granul arity does change with the coefficient of friction. 2 3 4 5 6 7 8 9 10 11 051015202530 Consolidation Stress (kPa)Granularity 0.15 0.25 0.35 Figure 5-13. Granularity of the major force chains in a particle bed. The granularity as a function of consolida tion stress is related to the unconfined yield strength and incorpor ated into equation 5-25. 3 2) ,% ( og g p cd d O H d A f (5-48) In equation 5-48 dgo is a reference state of granul arity. Because the granularity is defined as a length from a two dimensional imag e, this term must be raised to the third power to represent the volume of the particles. The cube of the ratio of the granularities, dg/dgo, is then proportional to the strength. A approximate comparison of equation 5-48 with the experimental data is given by exam ining the ratio of stre ngth at a high and low

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109 stress. In chapter 2, the data indicate that the unconfined yield st rength increases by a factor of 2.3 when the stress is increased from 10kPa to 17kPa. 3 2, ,low c high cf f (5-49) In Figure 5-12 it is shown th at the mean granularity df decreases by 1.5 when the stress is increased from 10kPa to 15kPa. 5 13 , low f high fd d (5-50) The ratios from the experimental data a nd DEM simulation are the same order of magnitude. This shows that the concept of the granularity is a probable cause of the unconfined yield strength increasing with consolidation stress. Equation 5-50 is incorporated into the unconfin ed yield strength equation 525 and written in the final form as a function of particle size, mo isture content and consolidation stress. 3 ,sin 1 sin 1 of f tot f cd d f (5-51) Equation 5-51 is compared to experimental data shown in chapter 2. In Figure 5-14 it can be seen that the model predicts the trend of the data however the magnitude is not exact. The could be attributed to the use of spherical partic les in the DEM simulations. If non spherical particles are used then the num ber of contacts would increase. Creating a greater number of force chains. Although the exac t magnitude of the data is not predicted, this model provides a good base for the in corporation of strength as function as consolidation stress.

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110 0 10 20 30 40 50 60 05101520 Consolidation Stress (kPa)Unconfined Yield Strength (kPa) 250 um 250 um model 850 um 850 um model Figure 5-14. Unconfined yield strength as a function of the consolid ation stress compared to the model of equation 5-51.

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111 CHAPTER 6 CONCLUSIONS AND FUTURE WORK The prevalence of granular materials is ev ident in many industries. Numerous plant operations involve the storage and handling of such materials. Thus the need to characterize and understand the behavior of gr anular materials is of great importance. The greatest challenge posed with granular materials is the ability to maintain and predict a continuous flow through process equipm ent. One phenomenon jeopardizing powder flow is powder caking. Powder caking is the process in which free flowing powder converts into agglomerates, preventing or reducing the flow of the powder. In this research one particular mechanism of caking, moisture migration caking, is investigated. This type of caking involves ma terials which are soluble or slightly soluble in water. Moisture migration caking can be defined as the process by which free flowing materials are transformed into lumps and aggl omerates due to changes in atmospheric or process conditions. An example of an atmos pheric change is the temperature cycle from day to night. An example of a change in process condition is the introduction of warm material into a cold storage container. Caki ng is a time induced event, i.e., the longer a material is stored or not flowing the greater the likelihood of caking. There are many variables that influen ce a caking event and the bulk cohesive strength of a caked material such as the pa rticle size, consolidation stress, moisture content, relative humidity and time. Other res earchers have studied the influence of some of these variables. However there lacks a complete understanding of the caking process

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112 and the existence of a complete mathematical model which predicts the bulk cohesive strength of caking. Several of the variables influencing the bul k cohesive strength are investigated in this research. The Johanson Indicizer and the Schulze Shear cell are used to measure the unconfined yield strength as a function of particle size, moisture content, relative humidity and consolidation stress. The experi mental work shows that an increase in the investigated variables causes an increase in th e unconfined yield strength of the material. The measured data are used to verify th e current strength models proposed by other researchers. Rumpf (1958) proposed a theory to calculate the tensile stre ngth of a material as a function of the particle size and moisture content. Later Kendall and Adams (1985) disputed the function of particle size a nd proposed a new theory based on fracture mechanics principles. It is shown in this dissertation that the model proposed by Kendall and Adams gives a better prediction of the st rength. However it should be noted that the model by Kendall and Adams was developed for li near elastic materials, an ideal case. The models proposed by Rumpf and Kendall defi ne the state of the art in regards to predicting cake strength. These models do not include an important variable which has a significant influence on the cake st rength, consolidation stress. In this dissertation a new model is pr esented to predict the unconfined yield strength as a function of the caking parameters: particle size, moisture content and consolidation stress. The elastic assumpti on made by Kendall is extended to include materials that exhibit plastic deformati on during failure. Elastic-plastic fracture

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113 mechanics principles are used to include the particle size in the caking model. Therefore this new model should be able to predic t caking in a plastic system as well. The impact of consolidation stress on th e unconfined yield strength is determined using Discrete Element Method (DEM) modeli ng. It is proposed that increasing the consolidation stress increases the major force chains within the particle bed. This is shown by an analysis of the force chain struct ure in a particle bed after a consolidation load is applied. Using the images from th e DEM simulation, the granularity of a powder bed is determined. The granularity is defi ned as the spacing between the major force chains. It is shown that the granularity is a function of the consolidation stress and is inversely proportional to the unconfined yi eld strength of the material. At lower consolidation stresses the granularity is larg er than at higher cons olidation stresses. The granularity function is incorpor ated into the new model for pr edicting the cake strength. It is found that the proposed caking model accurate ly predicts the trend of the unconfined yield strength and its magnitude is w ithin 25% of the experimental data. In order to initiate a moisture migration cak ing event, there must be a change in the water vapor content surrounding th e particles. Thus the heat and mass transport in the system is an important factor in this type of caking. The migration of moisture through a particulate bed is also investigated. COMS OL Multiphysics, a commercial software, is used to describe the moisture migration w ith a finite element analysis. Many of the moisture migration models in literature s uggest that free convection in the system is negligible. However, it is found in this research that the free convection plays a significant role in the heat and mass transport of the system. It is shown that the free convection speeds up the heat transfer process. This suggests that the storage life of a

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114 material prone to caking is shorter due to faster transport processes. The adsorption isotherms and kinetics of moisture uptake of the material are al so included in this analysis. The solids moisture co ntent is determined after th e material is exposed to a change in temperature. This data is used to predict the strength of the material using the model proposed in this res earch. The change in solids moisture content throughout a caking event is used to determine the incr ease in unconfined yield strength. An 18% increase in strength was predicted from the moisture migration analysis compared to a 40% increase observed with the experimental data. From this comparison, it can be concluded that the proposed model for cake st rength combined with a moisture migration analysis can be used to estimate the unconf ined yield strength of a material using the powder material properties a nd the process parameters. Suggestions for Future Work In this study one caking mechanism is i nvestigated. To complete the understanding of caking the other caking mech anisms have to be investigated as well. The moisture migration analysis only included the kineti cs of adsorption and desorption. When the moisture evaporates from the particle crysta l bridges are formed. Thus this analysis can be extended to also include the kinetics of crystallization. Th e amount of moisture adsorbed by the particles are determined from the adsorption isotherm of the material. However this can also be predicted by estimating the amount of water vapor adsorbed by the particles using DEM simulations and a thin film approximation. The DEM simulations in this research are simplified. The simulations are in two dimensions and the particles are spherical. The two dimensional simulations can be extended to three dimensions. The two dimens ional simulations are qualitatively reliable i.e. it is possible to predict trends. Howeve r, the two dimensional simulations may not be

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115 quantitatively reliable in pr edicting the actual values. The three dimensional simulations along with the incorporation of non spherica l particles may provide a more accurate prediction of the granularity a nd unconfined yield strength. The adhesion force model used in this re search is the Hertzian contact model. However the proposed cake strength model shoul d be included in these simulations. The DEM code should also include the rate of ch ange of mass to determine the amount of material that will dissolve and contribute to ca king and also the rate of crystallization to determine the crystal bridge formation. The moisture migration analysis can be incorporated into DEM simulations. Combini ng an improved moisture migration analysis with three dimensional DEM simulations would give a more complete and continuous analysis of cake strength.

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116 APPENDIX A NOMENCLATURE Geometry factor of fracture mechanics Hertzian displacement Interfacial energy c Fracture energy Porosity c Porosity of recrystallized bridge Indicizer cell angle s Thermal conductivity Poisson’s ratio Density of solid a Denisty of gas phase b Denisty of the bulk c Density of cake k Density of dissolved species s Density of salt w Density of water Stress 1 Major principle stress 3 Minor principle stress Ds Compressive strength of solid bridge

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117 f Fracture stress k Tensile strength of crystal bridge k Mean tensile strength of crystal bridge t Tensile strength Shear stress Internal angle of friction a Length of crack (flaw) A Hertzian contact radius Asp Specific transfer area b Narrowest width of bridge cpa Specific heat of air cps Specific heat of solid C Concentration Ce Equilibrium concentration d Diameter of contact zone dp Particle diameter d50 Mean particle diameter D Diffusion coefficient De Effective mass diffusivity Di Inner piston diameter Dl Lower piston diameter E Young’s modulus E* Effective Young’s modulus

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118 F Plastic modulus fc Unconfined yield strength Fa Flow rate of inlet air df Median granularity g Gravity g(L) Granularity as functi on of consolidation stress G Elastic energy release rate Hv Enthalpy of water vaporization Hw Saturation humidity of air J J-integral JR Fracture energy for EPFM JTot Total fracture energy k Heat transfer coefficient kf Mass transfer coefficient kd Dissolution rate kv Drying rate Kh Parameter for humidity cycling model of cake strength K Stress intensity factor Kc Critical stress intensity factor ko Coordination number L Applied load (consolidation stress) msolid Mass of solid Mp Mass of agglomerate

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119 Ms Mass of salt mwair Molecular weight of air mwwater Molecular weight of water n Strain hardening exponent n1 Number of shear layers n2 Number of contacts in a shear layer N Number of humidity cycles pH2O partial pressure of water vapor pT Total pressure P Consolidation load P(b) Probability of a bridge existing between particles P(f) Probability of a bridge c ontributing to the yield strength q Solids moisture content qE Equilibrium moisture content qo Initial moisture content qtot Total moisture content Q Heat flow rate R Radius of sphere Rc Cleavage resistance Rc Effective cleavage energy S Surface area of the shear zone t Time Tg Glass transition temperature

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120 V Volume of water in liquid bridge Ve Volume at equilibrium Vf Stress function of unconfined yield strength Vsb Volume of solid bridge Vs Volume of bulk caked sample Vt Total volume of water possessed by a single particle Vto Initial total volume of water possessed by a single particle w Weight of dissolved solid X Heat and mass transport parameter yk Concentration of dissolved species Ys Solubility

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121 APPENDIX B MOISTURE MIGRATION PARAMETERS The constants and boundary conditions listed in chapter 4 are calculated as follows. 2 3 21 150 _porosity porosity diam P permea a mol init c w mol init c init c gas R fluid R_ 1 1 _ init p c c init T b b a a init mol sat c_ / _ _ exp 1000 _ init mol sat c a mol init mol sat c w mol init mol sat c w mol init sat c_ _ 1 _ _ _ _ _ 100 _ _init sat c init c init RH init sat c init c B A init u _ 1 1 exp 1 Constant Temperature T=0 Axial Symmetry r=0 Insulated (ht) 0 T k n Heat Flux 4 4 infT T const T T h q T k namb o no slip u=0

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122 Normal Flow o Tp n u u l p n n t 0 Insulated (cd) u c c D N N n ; 0 Neumann Boundary G n

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123 LIST OF REFERENCES Adams, M.J. (1985). The strength of particulate solids. Journal of Powder and Bulk Solids Technology, 9(4), 15-20. Aguilera, J., del Valle, J., and Karel, M. (1995). Caking phenome na in amorphous food powders. Trends in Food Science and Technology, 6, 149-155. ASTM (1996). Standard practice for maintain ing constant relative humidity by means of aqueous solutions. in, Annual book of ASTM standards, no E 104-85. ASTM, West Conshoshocken. ASTM (2002). Standard shear test method for bulk solids using the Schulze ring shear tester. in, Annual book of ASTM st andards, 04.09 no D6773-02. ASTM, Philadelphia. Brinkman, H.C (1947). A calculation of the vi scous force exerted by a flowing fluid on a dense swarm of particles. Applied Science Research. Section A1, 27. Briscoe, B. J., Adams, M. J. (1987). Tri bology in Particulate T echnology. Adam Hilger, Bristol, England. Broek, D. (1974). Elementary Engineering Fracture Mechanics. Noordhoff International Publishing, Leyden, The Netherlands. Broek, D. (1988). The Practical Use of Fracture Mechanics. Kluwer Academic Publishers, Dordrecht, The Netherlands. Cleaver, J. A. S., Karatzas, G., Louis, S., Ha yati, I. (2004). Moistu re-induced caking of boric acid powder. Powder Technology, 146(1-2), 93-101. Down, G. R. B., McMullen, J. N. (1985). Th e effect of interparticulate friction and moisture on the crushing strength of sodium chloride compacts. Powder Technology, 42(2), 169-174. Farber, L., Tardos, G. I., Michaels, J. N. (2003). Evolution and structure of drying material bridges of pharmaceutical excipients: Studies on a microscope slide. Chemical Engineering Science, 58(19), 4515-4525.

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124 Greenspan, L. (1977). Humidity fixed points of binary saturated aqueous solution. Journal of Research of the National Bureau of Standards-A. Physics and Chemistry, 81A(1), 89-96. Griffith, A.A., (1921). The phenomena of r upture and flow in solids. Philosophical Transactions of the Royal Society of London A, 221, 163-167. Hancock, B. C., Dalton, Chad R. (1998). The effect of temperature on water vapor sorption by some amorphous pharm aceutical sugars. Pharmaceutical Development and Technology, 4(1), 125-131. Hancock, B. C., Shamblin, S. L. (1998). Water vapour sorption by pharmaceutical sugars. Pharmaceutical Science & Technology Today, 1(8), 345-351. Iglesias, H., Chirife, J. (1976). Prediction of the effect of temper ature on water sorption isotherms of food material. Jour nal of Food Technology, 11, 109-116. Irwin, G.R. (1957). Fracture I. Handbuch der Physik. VI, 558-590. Jenike, A. W. (1964). Storage and flow of solids. Universi ty of Utah, Salt Lake City. Johanson, J. R. (1992). The Johanson Indicizer system vs. Jenike shear tester. Bulk Solids Handling, 12(2). Johanson, J. R., Paul, B. (1996). Eliminati ng caking problems. Chemical Processing: 7175. Kendall, K. (1988). Agglomerate strengt h. Powder Metallurgy, 31(1), 28-31. Knight P., J., S. (1988). Measurement of powde r cohesive strength with penetration test. Powder Technology, 54, 279-283. Kuu, W.-Y., Chilamkurti, R., Chen, C. (1998). Effect of relative humidity and temperature on moisture sorption and st ability of sodium bicarbonate powder. International Journal of Pharmaceutics, 166(2), 167-175. Leaper, M., Bradley, M., Cleaver, J., Bridle I., Reed, A., Abou-Chakra, H., and Tuzun, U. (2002). Constructing an engineering m odel for moisture migration in bulk solids as a prelude to predicting moistu re migration caking. Advanced Powder Technology, 13(4), 411-424. Leaper, M., Berry, M., Bradley, M., Bridle, I ., Reed, A., Abou-Chakra, H., and Tuzun, U. (2003). Measuring the tensile strength of caked sugar produced from humidity cycling. Journal of Process Mech anical Engineering, 217, 41-47.

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125 Mathlouthi, M., Roge, B. (2003). Water vapor sorption isotherms and the caking of food powders. Food Chemistry, 82, 61-71. Merrow, E.W. (1998). Estimating the startup times for solids processing plants. Chemical Engineering, 95(15), 89-92. Mullier, A., Seville, J., and Adams, M. ( 1987). A fracture mechanic s approach to the breakage of particle agglomerates. Chem ical Engineering Science, 42(4), 667677. Nedderman, R.M. (1992). Statics and Kinema tics of Granular Materials. Cambridge University Press, Cambridge, UK. Noel, T.R., Ring, S.G. and Whittam, M.A. Glas s transition in low moisture foods. Trends in Food Science and Technology, 1, 62-67. Perry, R. (1997). Perry’s Chemical Engi neers’ Handbook. McGraw-Hill, New York. Persson, B.N. (2000). Sliding Friction. Springer, New York. Pietsch, W. (1969a). The strength of aggl omerates bound by salt bridges. The Canadian Journal of Chemical Engineering, 47, 403-409. Pietsch, W. (1969b). Adhesion and agglomera tion of solids during storage, flow and handlinga survey. Journal of Engineering for Industry, 435-. Purutyan, H., Pittenger, B., and Tardos, G. (2005). Prevent caking during solids handling. Chemical Engineering Progress, 22-27. Rastikian, K., Capart, R. (1998). Mathem atical model of sugar dehydration during storage in a laboratory silo. Journa l of Food Engineering, 35, 419-431. Rennie, P., Dong Chen, X., Hargreaves, C., and Mackereth, A. (1999). A study of the cohesion of diary powders. Journal of Food Engineering, 39, 277-284. Rumpf, H. (1958). Grundlagen und methoden des granulierens. Chemie-Ing.-Techn., 30(3), 144-158. Schubert, H. (1981). Principles of agglomera tion. International Chemical Engineering, 21(3), 363-377. Schweiger, A., and Zimmermann, I. (1999). A new approach for the measurement of the tensile strength of powders. Powder Technology, 101, 7-15. Tanaka, T. (1978). Evaluating the caking stre ngth of powders. Indus trial & Engineering Chemistry Product Research & Development, 17(3), 241-246.

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126 Tardos, G. I. (1996). Ingress of atmospheric moisture in bulk powders: Application to caking of fine crystalline powders. Po wder Handling and Processing, 8(3), 215220. Tardos, G. I., Gupta, R. (1996). Forces gene rated in solidifying liquid bridges between two small particles. Powder Technology, 87(2), 175-180. Tardos, G. I., Nicolaescu, I.V., and Ahtc hi-Ali, B. (1996). Ingress of atmospheric moisture into packed bulk powders. Powd er Handling and Processing, 8(1), 7-15. Teunou, E., and Fitzpatrick, J.J. (2000). Eff ect of storage time and consolidation on food powder flowability. Journal of Food Engineering, 43, 97-101. Teunou, E., Fitzpatrick, J. J. (1999). Effect of relative humidity and temperature on food powder flowability. Journal of Food Engineering, 42(2), 109-116. Tomas, J. a. S., H. (1982). Modeling of the strength and the flow properties of moist soluble bulk materials. Aufb ereitungs-Technik(9), 507-515. York, P., and Pilpel, N. (1973). The effect of temperature on the mechanical properties of powders: Presence of liquid films. Mate rials Science and Engineering, 12, 295304. Zhang, J., Zografi, G. (2000). The relationship between "BET" and "Free volume" derived parameters for water vapor abso rption into amorphous solids. Journal of Pharmaceutics, 89(8), 1063-1072.

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127 BIOGRAPHICAL SKETCH Dauntel Wynette Specht was born in Houston, Texas, on the 23rd of July 1979. She attended Jesse H. Jones high school. After high school, Dauntel en rolled at Trinity University in San Antonio, Texas, where she received a Bachelor of Science in engineering science. Upon graduation from Trinity, she was admitted into the Chemical Engineering Department at the University of Florida. Dauntel will receive her Doctorate of Philosophy in May of 2006.


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Title: Caking of Granular Materials: An Experimental and Theoretical Study
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Copyright Date: 2008

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Permanent Link: http://ufdc.ufl.edu/UFE0013727/00001

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Title: Caking of Granular Materials: An Experimental and Theoretical Study
Physical Description: Mixed Material
Copyright Date: 2008

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Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
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CAKING OF GRANULAR MATERIALS: AN EXPERIMENTAL AND
THEORETICAL STUDY















By

DAUNTEL WYNETTE SPECHT


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2006

































Copyright 2006

by

Dauntel Wynette Specht

































This document is dedicated to Momma, Daddy, Techia and Danyel.















ACKNOWLEDGMENTS

I would like to acknowledge my advisors, Dr. Kerry Johanson and Dr. Spyros

Svoronos, for their constant guidance and support. Dr. Johanson's extensive experience

in the field of particle technology served as a continuous source of knowledge throughout

my studies. Dr. Svoronos's help in all aspects of my graduate studies has been

indispensable.

I thank my committee members, Dr. Ray Bucklin and Dr. Oscar Crisalle, and all

other faculty who provided useful suggestions for the advancement of my studies. I

would also like to thank Dr. Brij Moudgil and the National Science Foundation's

Engineering Research Center for Particle Science and Technology and our industrial

partners for their financial support.

I gratefully acknowledge all of my group members, Brian Scarlett, Jennifer Curtis,

Nicolaie Cristescu, Yakov Rabinovich, Olesya Zhupanska, Claudia Genovese, Ali Abdel-

Hadi, Ecivit Bilgili, Mario Hubert, Dimitri Eskin, Nishanth Gopinathan, Madhavan

Esayanur, Caner Yurteri, Rhye Hamey, Maria Palazuelos Jorganes, Stephen Tedeschi,

Milorad Djomlija, Osama Saada, Julio Castro, Benjamin James, Mark Pepple, and Bill

Ketterhagen, for their help with experiments and modeling. I also acknowledge my

friends in Gainesville for ultimately making this experience an enjoyable one.

Lastly but certainly not least, I thank my family for their constant love and support.

To Momma and Daddy, thanks for believing in me and teaching me that I can do

anything that I set my mind to. To Techia and Danyel, thanks for being the best sisters









that a girl could ask for. To Marco, thanks for experiencing the world with me but most

of all being my home away from home.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST OF TABLES ....................................................... ............ ....... ....... ix

LIST OF FIGURES ............................... ... ...... ... ................. .x

A B S T R A C T .............................................. ..........................................x iii

CHAPTER

1 IN TR OD U CTION ............................................... .. ......................... ..

Storage and Handling of Granular Materials..................................................
Caking in Industrial Processes......................................................... ..................... 2
A im and O utline of D issertation .............................................................................

2 CAKING AND CAKING CONDITIONS ....................................... ............... 7

M ech anism s of C making ...................................................................... .......... .. .. ... 7
Model of Moisture Migration Caking............................................................9
Quantifying the Bulk Cohesive Strength of Caking .................................................10
D irect Shear T esters ........................ .. .................................. .. .............. 10
T e n sile T e ste rs ............................................................................................... 12
Penetration Testing .................. .................. ............ .. ... ..... .. ........ .... 12
Crushing Test or Uniaxial Compression .................................. ............... 13
C h o o sin g a T e ster ................................................................ ...............................14
Johanson Indicizer ............................................... .. ...... .. ............ 14
Schulze Shear T ester ........................ .................... .. .. .. ...... .......... 19
Unconfined Yield Strength Results .............................. ............. 22
Effect of Moisture Content and Consolidation Pressure ...............................22
E effect of P article S ize ................... ... ..................................................................2 4
Effect of the Number of Temperature Cycles .............................................25
Effect of R elative H um idity ..................................................... ..... .......... 27

3 ADSORPTION ISOTHERMS AND KINETICS ............................................... 30

A dsorption Isotherm s.................. ................ ................................. .............. 30
Mechanisms and Kinetics of Adsorption...... .................. ..............33









Measurement Techniques for Sorption Isotherms.................................................35
Static V apor Sorption ................................................ .............................. 35
D ynam ic V apor Sorption.......................................................... ............... 35
Sym m etrical Gravim etric A nalyzer....................................... ......................... 36
Experim mental Param eters ........................... ......... ................... .. ............... .38
Sorption Isotherms for Sodium Carbonate ...................................... ............... 39
E effect of Tem perature .......................................... .... .. ......... ............... 44
K inetics of A dsorption ......... ................................................. ......... ...............46
Sorption Isotherms for Sodium Chloride.................................................................49

4 MOISTURE MIGRATION MODELING......................................................51

B a c k g ro u n d ........................................................................................................... 5 1
M oisture M igration M odel .............................................................. .....................56
Finite Elem ent M odeling .......................................................................... 57
About COM SOL M ultiphysics....................................... ......................... 59
M odel G eom etry .......... .............................................. .... .... .. .... .. 59
Partial D differential Equations ................................... .......................... .. ......... 60
C onvection-conduction ........................................ .......................... 61
Convection-diffusion ..................................................... ...................61
B rinkm an E qu ation ........................................................... .....................62
Solids m oisture content ................................................... ............... ... 62
M odel Param eters .................................................. ...... ........... .......... .. ..63
The Role of Convection in Moisture Migration ............................... ................ 66
T em p eratu re P rofiles ........................................ ............................................66
Solids M oisture Content ......................................................... .............. 72

5 EVALUATING THE CAKE STRENGTH OF GRANULAR MATERIAL............77

B a ck g ro u n d ...................... ........... ..... .... ..... .......... ..... ........ ............... 7 7
Modified Fracture Mechanics Model for Evaluating Cake Strength.......................85
Linear-elastic Fracture M echanics .................................... ............... 86
Elastic-plastic Fracture M echanics.................................... ............... 88
M odel Param eters .............. ..... ... ............. ...... .. ... ...... ......... 91
Calculating the narrowest width of the bridge as a function of the bridge
volume ................................................... 91
Calculating the length of the crack.....................................................95
Stress-Strain param eters F and n....................................... .....................95
Geometry of the agglomerate calculating H.................. ..................97
Comparison of the Model with Experimental Data..............................................97
E effect of P article Size ........................................................................... ... .. 97
E effect of M oisture C ontent.................................................................... ..... 100
Effect of Consolidation Stress ................................................................ 101

6 CONCLUSIONS AND FUTURE WORK..........................................................111









APPENDIX

A N O M E N C L A T U R E ........................................................................ .................... 116

B MOISTURE MIGRATION PARAMETERS................................121

L IST O F R E FE R E N C E S ......................................................................... ................... 123

BIOGRAPH ICAL SKETCH .............................................................. ............... 127














































viii
















LIST OF TABLES


Table pge

3-1 Kinetics constants for sodium carbonate decahydrate ..........................................48

4-1 Parameters for COMSOL simulations. ....................................... ............... 64

4-2 Boundary conditions for COMSOL simulations.................................................66

4-3 Subdomain conditions for COMSOL simulations. ...............................................66
















LIST OF FIGURES


Figure page

1-1 Top down view of a silo wall with a thick layer of caked material attached ............3

2-1 Representation of the two limiting Mohr circles, showing the major principal
stress of the steady state condition oa and the unconfined yield strength. .............11

2-2 Johanson Indicizer test cell.. ............................. ................................................ 15

2-3 Schematic of the standard Indicizer test cell.......................... ................... 16

2-4 Consolidation bench used for making the caked samples.................. .......... 18

2-5 Temperature profile used for preparing the cakes............................... ...............19

2-6 Schulze shear tester: (A) image and (B) schematic.......................................20

2-7 Schematic of the permeable Schulze cell..............................................................21

2-8 Unconfined yield strength of sodium carbonate monohydrate as a function of
moisture content in the form of percent sodium carbonate decahydrate in the
m mixture. .............................................................................23

2-9 The yield strength as a function of consolidation pressure.. .................................24

2-10 The yield strength as a function of the mean particle size. .....................................25

2-11 Yield locus of sodium carbonate for air at 75% RH for 24 hr. and 0 hr with a
norm al load of 16kP a. ......................... .... ................ ... .... .... ............... 26

2-12 The yield strength as a function of the number of temperature cycles...................27

2-13 Yield locus of sodium carbonate for air as a function of relative humidity with a
norm al load of 4kPa. ............................................... ...............28

2-14 Unconfined yield strength as a function of relative humidity compared to the
isotherm of sodium carbonate. ............................................................................ 29

3-1 The five types of adsorption isotherms in the classification of Brunauer ..............32

3-2 Mechanisms of mass transfer for absorbent particles .......................................... 33









3-3 Sorption of moisture by a water soluble particle................ ...............34

3-4 VTI Symmetrical Gravimetric Analyzer schematic..............................................36

3-5 User input screen for the VTI Sorption Analyzer. ................................................. 38

3-6 Phase diagram of sodium carbonate ................ ....... .............. ... ............... 40

3-7 Adsorption isotherm of sodium carbonate at 250C.. .......................................... 41

3-8 Sorption isotherm of sodium carbonate at 25C. .............................................. 43

3-9 Sorption isotherm of sodium carbonate monohydrate at various temperatures.......44

3-10 Sorption isotherm of sodium carbonate decahydrate at various temperatures.........45

3-11 The moisture uptake of sodium carbonate decahydrate as a function of time. ........46

3-12 The kinetics of sodium carbonate decahydrate at 50C. .....................................48

3-13 The kinetic rate constant for sodium carbonate decahydrate adsorption and
desorption ............................................................................49

3-14 Sorption isotherm of sodium chloride at various temperatures..............................50

4-1 Examples of the elements used in FEM ............................................ .............58

4-2 Schematic of the geometry used in the COMSOL solver. The lettered areas label
th e dom ain s. ....................................................... ................. 60

4-3 Caking cell geometry with mesh elements .......................... ..................60

4-4 B boundary conditions. ...................................................................... ...................63

4-5 Temperature profile imposed at the base of the cell in the finite element
sim ulations. .......................................... ............................ 65

4-6 Temperature profile along the centerline of the cell with convection included.......67

4-7 Temperature profile near the insulated boundary of the cell with convection.........67

4-8 The temperature profile within the cell illustrating the convective plumes that
dev elop .............................................................................68

4-9 Temperature profile along the centerline of the cell without convection ................69

4-10 Temperature profile near the insulated boundary of the cell without convection....69

4-11 Temperature profile at the center of the cell, with and without convection.............70









4-12 Temperature profile at the cell wall, with and without convection.......................71

4-13 Solids moisture content profile at the centerline of the cell with convection. .........73

4-14 Solids moisture content as a function of time at the center of the cell.................73

4-15 Phase diagram of sodium carbonate with temperature profile imposed. ................74

4-16 Equilibrium solids moisture content as various positions along the radius. ............76

5-1 Model of contacting spheres with pendular water used to calculate the volume
and width of the bridge ................. ...... .. ......... ... .. ..............80

5-2 Mohr circles demonstrating the relationship between tensile strength and the
unconfined yield strength. ............................................................. .....................81

5-3 X-ray tomography slice of caked sodium carbonate.............................................90

5-4 Logarithmic plot of the smallest radius of the bridge b vs. the volume of the
b rid g e V ........................................................................................ ..9 2

5-4 Schematic of an agglomerate showing the length of the crack a............................95

5-5 The stress-strain curve for a sodium carbonate tablet...................... ...............96

5-6 Stress-strain curve for lactose. ............................................................................ 97

5-7 Yield strength of sodium carbonate as a function of particle size. ........................99

5-8 The yield strength of sodium carbonate as a function of moisture content ..........100

5-9 Illustration of Hertz contact model............................ ....................... 102

5-10 The volume of a bridge using the Hertzian contact model ............. .....................103

5-11 Force chain structure in a DEM simulated powder bed with a high applied load
(right) and a low applied load (left). ........................................... ............... 106

5-12 The cumulative distribution of the granularity at various consolidation loads......107

5-13 Granularity of the major force chains in a particle bed ............. ...............108

5-14 Unconfined yield strength as a function of the consolidation stress compared to
the model of equation 5-51 ............... .... ...................... ....................... 110















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

CAKING OF GRANULAR MATERIALS: AN EXPERIMENTAL AND
THEORETICAL STUDY

By

Dauntel Wynette Specht

May 2006

Chair: Kerry Johanson
Cochair: Spyros Svoronos
Major Department: Chemical Engineering

Many industries such as food service, pharmaceutical, chemical, and agricultural

handle and store materials in granular form. The processing of granular materials can

pose many challenges. The materials can gain strength during storage creating flow

problems in process equipment. This phenomenon is known as caking and is defined as

the process by which free flowing material is transformed into lumps or agglomerates due

to changes in environmental conditions. The strength of a cake is a function of various

material properties and processing parameters such as temperature, relative humidity,

particle size, moisture content, and consolidation stress. A fundamental understanding of

the entire caking process is needed in order to predict a caking problem.

In this research, caking based on the mechanism of moisture migration through the

particle bed is studied. The influence of moisture content, consolidation stress, particle

size, and humidity are investigated. An increase in these variables causes an increase in

the unconfined yield strength of the material. Theoretical models exist to describe the









function of particle size and moisture content on the yield strength. However, the

consolidation stress is not included in these models. The trend is often qualitatively

explained by an increase in the interparticle forces. Applying this theory to the current

research yields an incorrect prediction of the unconfined yield strength. It is postulated

that the increase in unconfined yield strength is attributed to an increase in the number of

major force chains in the shear zone. An increase in the number of major force chains

results in the increase of the yield strength of the material. Using the principles of fracture

mechanics, a new equation is developed to evaluate the strength of powder cakes. An

improvement of existing cake yield strength models is made by including the

consolidation stress.

Moisture migration through the particle system and moisture uptake by the particles

themselves are significant factors of caking. These processes are modeled using a finite

element partial differential equation solver, COMSOL Multiphysics. The heat and mass

transport of the system is described as well as the kinetics of the material. The model is

used to predict the areas of caking within a given geometry and approximate the

unconfined yield strength.














CHAPTER 1
INTRODUCTION

Storage and Handling of Granular Materials

Granular materials are common in all facets of life: from food to baby diapers to

geological matter. They encompass the complete range of matter in the form of discrete

solid particles. Granular materials are often referred to as bulk solids. Many industries

must store and handle bulk solids due to their prevalence in a wide variety of products. In

the chemical industry, it is estimated that one-half of the products and at least three

quarters of the raw materials are in the form of bulk solids (Nedderman, 1992). With such

a vast number of processes involving bulk solids there is a need for a fundamental

understanding of the behavior of such materials.

Granular materials behave differently from any other form of matter-solids,

liquids, or gases. They tend to exhibit complex behaviors sometimes resembling a solid

and other times a liquid or gas (Jenike, 1964). Thus the characterization of granular

materials presents a challenge unlike any other. It has been estimated that industries

processing bulk solids total one trillion dollars a year in gross sales in the United States

and they operate at only 63 percent of capacity. By comparison, industries that rely on

fluid transport processes operate at 84 percent of design capacity (Merrow, 1988). Due to

the often inadequate and unreliable design of particulate processes, there exists a

significant discrepancy in the efficiency of particulate operations compared to fluid

operations. Many of the problems in particulate operations deal with the flow of material

through the process. Often times it is the lack of flow that creates additional problems









such as limited capacity of the process equipment and product degradation. One type of

flow problem is caking. A caking event often occurs when a bulk solid does not flow and

is allowed to remain stagnant over a period of time.

Caking in Industrial Processes

Bulk solids such as food powders, detergents, pharmaceuticals, feedstocks,

fertilizers, and inorganic salts often gain strength during storage. The increase in strength

is caused by particles bonding together at contact points through a cementing action. This

phenomenon is known as caking and can be defined as the process by which free flowing

materials are transformed into lumps and agglomerates due to changes in atmospheric or

process conditions. More precisely, caking is the increase in bulk cohesive strength due

to changes in interparticle forces.

Traditional interparticle forces such as the formation of liquid bridges, Van der

Waals forces, or electrostatic forces can cause an increase in bulk cohesive strength.

However, none of these phenomena alone cause caking. Caking occurs when the surface

of the particle is modified over time to create interparticle bonds due to the formation of

solid bridges between similar or dissimilar materials. Thus, the presence traditional

interparticle forces may initiate the caking process, but the advent of caking requires

some mechanism to change these forces into solid bridges.

Caking is the accumulation of smaller particles held together by solid bridges to

produce a cluster of particles. This phenomenon is related to agglomeration.

Agglomeration techniques are often used to improve the shape, appearance and handling

properties of materials (Schubert, 1981). This can result in better flow characteristics,

improved packing density, dust-free operations and faster dissolution in liquids (Aguilera

et al., 1995). However agglomeration is unwanted and unexpected in the case of caking.









This is attributed to the fact that the process of caking is uncontrolled and the outcome

can be useless material. In the food and pharmaceutical industries, the effect of caking

may be detrimental to the process and product. These problems often result in the loss of

quality and function of a product (Purutyan et al., 2005).

Regardless of the reason for caking, the presence of such agglomerates causes

significant problems in process equipment (Johanson and Paul, 1996). Severe caking can

result in the solidification of the entire mass within a silo as shown in Figure 1-1.

Moreover, consumer products such as detergent and food powders can cake during

storage both prior to purchase and after initial use. Hence, the problems of handling such

products are often passed on to the consumers.















Figure 1-1. Top down view of a silo wall with a thick layer of caked material attached.

Currently, there are no predictive tools to forecast the likelihood of a caking event

occurring. The industry must rely on empirical knowledge of caking to provide solutions

to the problem. This ranges from controlling the process environment to maintaining a

regular cleaning schedule. A common practice for minimizing the effects of caking is the

maintenance of a controlled environment surrounding the process; i.e., the temperature

and relative humidity are maintained at a level such that caking may not be induced. The









material response to temperature and relative humidity conditions are determined by

measuring sorption isotherms. The isotherms give an indication of the hygroscopic nature

of the material and the amount of moisture adsorbed at a specific temperature and relative

humidity. In situations where controlling the environmental parameters is not feasible,

flow aids are sometimes introduced into the product mixture. Flow aids act as a physical

barrier or moisture barrier between particles that exhibit caking tendencies. However,

when controlled environments and flow aids both fail, brute force is used to dislodge the

cake from process equipment. These actions may result in damage to the equipment and

personal injuries. Furthermore, if the caking continues to be a persistent problem, a

regular cleaning schedule of the process equipment is established.

Aim and Outline of Dissertation

A fundamental knowledge of the caking phenomenon appears to be non-existent.

Most of the studies on caking are experimental, focusing on empirical solutions to the

problem. Many of these solutions are not universally applicable. There are few studies

which concentrated on predicting the strength of caking and the moisture diffusion

through the bulk. Often times these studies investigate ideal situations and are not

suitable for real materials. As a consequence, when new products are developed it

requires a great deal of time, effort, and money to diagnose and remedy a caking

problem.

Caking is a problem that may be solved in two ways: before the initiation of an

event or eliminated after its existence. The goal of this research is to understand the

variables that are involved to induce caking. This will enable the prediction as well as the

prevention of a caking problem. To accomplish this, a comprehensive knowledge base of

the caking phenomenon must be established. With this knowledge it is then possible to









develop a model to predict the onset and strength of caking. The following objectives are

proposed to achieve the goals of the research:

* Understand the mechanisms of caking as well as the material properties and
process conditions necessary to induce a caking event.

* Measure the cake strength as a function of process variables.

* Quantify the hygroscopic nature of the bulk material by measuring adsorption
isotherms.

* Develop a model to predict the onset of caking based on the moisture migration
through the bulk.

* Develop a model to predict the strength of caking based on the process variables.

The research plan includes both an experimental and a theoretical investigation to fully

describe the caking process. The experimental work involves measuring the bulk

cohesive strength and the adsorption isotherms. In the theoretical studies various

parameters from the experimental work will be used to develop a mathematical model of

the caking process. In chapter 2 the reader is further introduced to caking and the material

properties and process parameters influencing the strength of cakes. Included in this

chapter are the possible mechanisms of caking and interparticle bond formation. Often

times there are several mechanisms involved in a caking event. It is beyond the scope of

this dissertation to investigate every mechanism. Therefore, the focus of this study is

moisture migration caking. The various methods for quantifying cake strength are

discussed as well as the testing apparatus suitable for this purpose. Experimental data of

the bulk cohesive strength are presented and explained in regards to observations of other

researchers.

In order for moisture migration caking to occur, the bulk material must possess a

certain affinity for water (in the form of moisture). Hence the interactions between the









bulk solid and moisture in the system must be understood. This is accomplished by

measuring the adsorption isotherms of the material. Chapter 3 provides a brief

introduction into adsorption isotherms and presents experimental data as a function of

temperature. The kinetics of sorption are also presented in this chapter.

Using the sorption kinetics the moisture migration through the bulk can be

modeled. In chapter 4, finite element methods are employed to predict the migration of

moisture using an improved model. Many of the existing models assume that the

convection in the system is negligible. This research shows that the free convection

enhances moisture migration, making it a significant contributor to caking.

Chapter 5 begins with a discussion of previous models for predicting the cake

strength. These models do not adequately predict the strength and they are based on

single process variables. A new model for predicting the strength is developed which

incorporates all the significant parameters of caking. The novelty of the model is the

inclusion of the consolidation stress to predict the strength of the material. The

experimental data from chapter 2 are used to verify the model.

The final chapter summarizes the findings of this study and provides suggestions

for future research in the study of caking.














CHAPTER 2
CAKING AND CAKING CONDITIONS

This chapter discusses the various mechanisms of caking along with the methods

for quantifying bulk cohesive strength. The material properties and process parameters

which influence the initiation of a caking event as well as the strength of caking are the

focus of this chapter.

Mechanisms of Caking

A powder cake is formed by numerous solid bridges bonding particles together.

The mechanisms of solid bridge formation are therefore the key to cake formation. There

are several mechanisms for solid bridge formation. Throughout a single caking event

more than one mechanism can contribute to the bulk cohesive strength. An understanding

of all the mechanisms of solid bridge formation is essential for a thorough explanation of

the caking process.

Rumpf (1958) was the first to propose several mechanisms for the formation of

solid bridges. He named crystallization, sintering, chemical reaction, partial surface

melting, and liquid binder solidification as causes for bridge formation. More recently

other researchers (Noel et al., 1990 and Farber et al., 2003) have suggested that glass

transition may be a potential caking mechanism. The following is a list of mechanisms

and a short summary of each phenomenon.

* Crystallization Materials that are soluble or slightly soluble are subject to this
mechanism. This includes many chemicals and fertilizers. The moisture content of
the solid increases as water vapor condenses onto the surface of the particle. A
portion of the surface material dissolves and bridges are formed at contact points









between particles. A subsequent change in moisture content evaporates the liquid in
the bridge leaving solid crystal bridges between particles.

* Sintering Many useful products containing powder metals and ceramics are
formed by this mechanism. The molecules or atoms of the material diffuse at the
contact points of the particles to form solid bridges. Sintering usually takes place at
1/2 to 2/3 of the melting temperature. Direct reduction of iron ore is one example of
a situation where this type of caking is undesired.

* Chemical reaction or binder hardening The production of fiberboard from
resin impregnated wood chips or flakes makes use of this mechanism. A reaction
occurs between two different materials to form bridges between the particles much
like mortar between bricks. Binder material may be used to form the bridges.
Unlike crystallization, the binder liquid does not evaporate but is incorporated into
the structure of the bridges either by chemical reaction or binder solidification.

* Partial melting Ice crystals and snow are products of this mechanism. A pressure
induced phase change or a temperature change caused by local friction between the
particles at contact points causes the surface material to melt and consecutively
solidify, thereby forming bridges.

* Glass transition Many pharmaceuticals and food products are subject to this
mechanism of caking. An increase in surface moisture causes a lowering of the
glass transition temperature Tg of the solid material. As Tg lowers, particles are
cemented together by plastic creep. The material changes from a hard crystalline
phase into an amorphous plastic phase. A subsequent change in moisture content
increases the glass transition temperature, solidifying the amorphous mass.

The mechanisms of glass transition and crystallization are dominant in the cases of

undesired caking (Pietsch, 1969; Aguilera et al., 1995; Johanson et al., 1996; Hancock et

al., 1998). In these situations, the formation of a cake is initiated from the condensation

of moisture onto the surface of individual particles in a bulk assembly. The moisture

adsorbs onto the particle surface and migrates through the bulk at particular process

conditions such as temperature and relative humidity. A change in these parameters

during storage acts as a driving force to induce periodic condensation and evaporation of

moisture on a particle surface. Once the moisture is adsorbed it can absorb into the

particle creating a layer of amorphous material in the case of glass transition. The









moisture can also remain at the surface causing the surface of the particle to dissolve and

crystallize during desorption of the moisture.

The mechanism of crystallization and associated moisture migration are chosen as

the focus of this caking research. This type of caking is prevalent in many industries that

handle crystalline materials. Although the focus of this research is crystallization caking,

the moisture migration analysis is applicable to all mechanisms involving the sorption of

water vapor.

Model of Moisture Migration Caking

Solid bridge formation by crystallization is initiated by the presence of moisture in

the environment or in the particle. Frequently, a change in process conditions such as an

increase in temperature or relative humidity creates a moisture concentration gradient in

the material. Moisture migrates through the interstitial voids of the bulk adsorbing on the

surface of particles at the contact points to form liquid bridges. A local increase in the

moisture content causes slight dissolution of the particles. A subsequent change in the

local temperature or relative humidity causes the surface moisture to evaporate and the

material at the contact points to crystallize and form a solid bridge. This process is

influenced by process parameters (temperature, relative humidity, consolidation pressure,

storage time) and material properties (initial moisture content and particle size). The bulk

cohesive strength varies with the magnitude of these parameters. A systematic study of

the caking strength with regard to the process parameters is required for a thorough

understanding of this process. This includes the effect of moisture released or adsorbed

by the particle and the number of temperature or humidity cycles during storage. From

this information a model to predict and prevent caking can be developed.









Quantifying the Bulk Cohesive Strength of Caking

Several techniques have been employed to quantify caking and study the effects of

properties influencing the bulk cohesive strength of a material. These methods include

measuring the unconfined yield strength using direct shear testers, measuring tensile

strength using tensile tests, and measuring yield stresses using penetration testing. In

addition, measuring the thickness of a caked layer of material and classifying the degree

of caking are used to characterize cake strength. A review of the available methods for

testing bulk cohesive strength is given below.

Direct Shear Testers

Direct shear testers are used to characterize the flow of powders and granular

materials. One characteristic of flow is the strength of the material or the unconfined

yield strengthfc. This property is not directly measured. It is extrapolated from shear

stress data by the construction of Mohr circles. The shear stress data are measured using

direct shear testers. Direct shear testers are constructed of two parts: a stationary part and

a moving part. The moving part is displaced relative to the stationary part such that the

powder shears and a shear zone is created. Prior to shear the powders are consolidated to

a predefined state by applying a load normal to the shear plane. The tester will, after a

short transition state, deform the powder with a steady state shear force which is

dependent on the applied normal load. The strength of the powder is a function of the

applied normal load (or consolidation load). When the powder is in the steady state

condition, the deformation is stopped. The normal load is decreased and the sample

sheared to instant failure. This process is repeated several times with different failure

normal loads. The steady state and failure data are used to construct Mohr circles. The

Mohr circles define the state of stress of the powder in the Tr- plane (shear stress vs.









normal stress). The unconfined yield strength can be determined from the Mohr circles.

Figure 2-1 shows an example of Mohr circles. The failure points from the shear tests

form the yield locus. The large (limiting) Mohr circle is tangent to this yield locus and

includes the steady state condition. The Mohr circle intersects the x-axis at the major and

minor principle stress, a3 and al, respectively. The unconfined yield strength,f, is the

major principle stress of the Mohr circle that is tangent to the yield locus and has a minor

principle stress of zero (ASTM, 2002).



Shear Stress (kPa)

Steady state point

Yield Locus

Tensile Strength f (Sengt) (Major Principal
X Stress)
Normal Stress (kPa)


Figure 2-1. Representation of the two limiting Mohr circles, showing the major principal
stress of the steady state condition ao and the unconfined yield strengthfc.

Shear testers are very useful for designing equipment for the storage and handling

of bulk materials. The disadvantage of using direct shear testers is that multiple

experiments are required to find the unconfined yield strength. Using shear testers can be

time consuming for measuring cake strength due to the time-induced nature of the caking

process. An exception to this is the Johanson Indicizer which estimates the unconfined

yield strength with a single test. The measurement principles of this device are discussed

in detail in a later section.









Tensile Testers

Tensile testers are used to determine the tensile strength of materials. From the test

various mechanical properties, besides the tensile strength, can be deducted from the

stress-strain curve. There are many types of tensile testers, two of which are the

horizontal (Leaper et al., 2003) and vertical testers (Pietsch, 1969a, and Schweiger et al.,

1999) used for measuring the strength of granular material. The horizontal tester consists

of a cylindrical split cell with one part rigidly connected to a load cell and the other part

moving. The tensile strength is measured by moving one half of the split cell away from

the load cell until the sample breaks. For a vertical tester the sample is prepared outside

of the tester. After the cake is formed, the sample is glued between two platens. The

platens are moved in opposite directions until the sample breaks. For both types of

testers, the force required to break the sample divided by the cross sectional area of the

sample defines the tensile strength.

The difference between tensile testers and shear testers is the mechanism of

breakage or shear. With shear testers, a shearing action is needed to break the cake.

Tensile testers use a tensile force to break the cake. The most active force in the breakage

of bonds between particles is often a shear force (Pietsch, 1969b). The tensile strength

and the unconfined yield strength are defined by two different Mohr circles of the same

yield locus. The tensile force is considered a negative force and the unconfined yield

strength is a positive force. The tensile strength at is the x-intercept of the yield locus as

shown in Figure 2-1.

Penetration Testing

The bulk cohesive strength of a cake can be measured by cone penetration. This

technique was adapted from soil mechanics where it has been used for years to measure









geotechnical parameters of soil. The measurement apparatus consists of a cone and a

testing machine. Similar to tensile tests the caked samples are formed in a cylindrical die.

The diameter of the die is large enough such that the walls are at a great enough distance

from the failure plane that allows the assumption that the fail surface is unconfined. After

the cakes are formed, a cone indents the surface to a defined depth and the penetration

force is recorded as function of penetration depth. The penetration force at a given depth

is proportional to the unconfined yield strength (Knight, 1988). Similar to shear tests the

sample fails in shear but the cone only acts over a limited area of the cake. This technique

does not require specialized equipment for evaluating the cake strength and the tests are

not time-consuming.

Crushing Test or Uniaxial Compression

The crushing test is frequently used in the food industry to study the cohesion of

various food powders (Down et al., 1985 and Rennie et al., 1999). The preparation of the

caked sample is similar to that of tensile and penetration tests. The powders are

consolidated into a cylindrical mold. After the cake is formed it is removed from the

mold and placed between two platens of a compression testing machine. The sample is

axially loaded until the cake fails. The stress at which the cake breaks is defined as the

unconfined yield stress. The cake must be strong enough to withstand the removal from

the mold. Thus the caked samples must exhibit a minimum strength for measurements to

be possible. This measuring technique is directly related to the theory of the unconfined

yield strength. Unlike the shear testers, the unconfined yield stress of the material is

measured directly. For shear cell data, the unconfined yield strength is extrapolated using

Mohr circles.









Choosing a Tester

Although there are many testers and numerous methods available to investigate the

strength of caking, direct shear testers are used in this research to quantify the bulk

cohesive stresses. This choice is partly due to the availability and access to direct shear

cells but also because the important information related to caking is obtained in an

efficient and accurate manner.

The Johanson Hang-Up Indicizer and the Schulze shear cell are used in this study.

The Indicizer is used to measure the strength of the cakes as a function of temperature

cycling, initial moisture content, consolidation load and particle size. The Schulze cell is

used to measure the strength of the cakes as a function of temperature and relative

humidity cycling, consolidation load, and time.

Johanson Indicizer

The Johanson Hang-Up Indicizer is a powder flow tester used to measure material

flow properties. The Indicizer is chosen for its ability to estimate the unconfined yield

strength from a single test. Many other shear testers indirectly measure the unconfined

yield strength through a series of experiments. This requires the measurement of a yield

locus and construction ofMohr circles to determine the unconfined yield strength. Due to

the possible error introduced from inconsistent mixtures (cakes) it is advantageous to

minimize this error by using a single experiment for the strength estimation.

The test cell used to estimate the unconfined yield strength with the Johanson

Indicizer is shown in Figure 2-2 (Johanson, 1992). The test cell consists of an inner

piston, outer piston and lower piston. For the typical strength test, the sample is first

consolidated to a known stress state. The consolidation load and the lower piston are

removed after consolidation such that the sample is in an unconfined state. The inner










piston then moved downward to fail the sample. The diameter of the inner piston is

smaller than that of the lower piston. This configuration allows the creation of an inverted

conical shear zone during failure as shown in Figure 2-3. A force balance on the material

displaced from the tester during failure is used to relate the shear stress and normal stress

acting at the walls of this conical section to the force applied on the inner piston.


Force Measured on Inner Piston
the Inner Piston Only Outer
/ Outer
Piston
I000000000
DO00000? 3 000000000
DO0000OO 000000000
DO0000OO 000000000
DOOOOO0 r 88888 i-
D000000001 00000
DOOOOOCXXX)



Lower
A) Piston B)

Figure 2-2. Johanson Indicizer test cell. A) the original test cell and B) modified test cell.

The force balance on the conical section is given in equation 2-1 where P is the

consolidation load, V, is the volume of the sample, pb is the bulk density of the material

and S is the surface area of the shear zone. The equation accounts for the force of the

material within the inverted conical section on the inner piston, the weight of the material

in this section, and the normal and shear stresses acting on the inverted conical section.

P + pbgV, = r ScosO c SsinO (2-1)

The diverging character of this failure zone (flow channel) makes it possible to

estimate the unconfined yield strength. If it is assumed that the material in the test cell is

in an unconfined state of stress and the piston is pushed through the material forming a

shear plane. The unconfined Mohr circle in Figure 2-3 represents this unique state of









stress. The shear stress acting on the failure plane is r as shown in Figure 2-3. The normal

stress acting on the shear plane is oas shown in the same figure. The geometry of the

Mohr circle is used to relate the shear and normal stresses to the unconfined yield

strengthfc and the internal friction angle q. These relationships are given in equations 2-2

and 2-3.




SF

-Di Tf



-D (A) c (B)
Normal Stress

Figure 2-3. Schematic of the standard Indicizer test cell (A) and Mohr circle used for
calculation of the unconfined yield strength (B).

= f (- sin ) (2-2)
2

1
r= f cos ~ (2-3)
2

The volume of the material V, that is displaced from the tester is given by

V, = 1H(D,2 +D, D +D12) (2-4)
12

where H is the height of the sample and D, and D1 are the inner and lower diameters of

the pistons, respectively.

The surface area of the shear zone is

1 D2 2
S = -- D (2-5)
4 sin 0









The dimensions of the test cell define the angle 0 of the shear zone.


tan0 =D D (2-6)
2H

The unconfined yield strength is calculated by substituting equations 2-2 through 2-6 into

equation 2-1. This yields a value for the unconfined yield strength based on the

dimensions of the test cell, bulk density and the force applied by the inner piston. It is

assumed that the shear angle 0 is small.

1 pb H(D2 +DD + D2)+12F
f= (2-7)
3 ;THcosq~(D, +D)

Failure with the Indicizer is a two step process. First there is an increase in

compaction stress, up to a set maximum value. The lower piston is moved downward

causing the material to be supported by the lip. Then the inner piston moves at a steady

rate to shear the material. After the initial failure, the shear stress and normal stress acting

on the displaced material decrease to the conditions of steady flow. This value is

associated with the shape of the flow channel formed after failure (inverted conical

section). It is critical that the shear stress on the flow channel walls be less than the shear

stress during failure to assure unconfined conditions. The test is invalid if this condition

is not met. The diverging nature of the flow channel ensures that the normal stress is

always greater that the confining wall stress.

The Johanson Indicizer operates in two modes, normal and scientific. In the

scientific mode, the sample can be sheared without consolidation and the user manually

enters the consolidation load. For the cake strength measurements the tester is operating

in the scientific mode. The consolidation load applied automatically by the tester is not

required because the cakes are consolidated before the strength measurements. Pre-









consolidation is performed on a consolidation bench external to the tester and these

samples are placed in the Johanson Indicizer for failure. The test cell was modified as

shown in Figure 2-2 for the cake strength measurements. The modified cell consists of

two parts and the lower piston becomes inoperable.

The cakes are formed on a consolidation bench as shown in Figure 2-4. The bench

is housed in an environmental chamber where the temperature and relative humidity of

the surroundings are controlled. The bench is equipped with a strip heater to control the

temperature, position transducers to monitor the axial strain, and load cells to monitor

radial stress on the powder. The powder sample is contained in a 3" diameter cylindrical

cell. The cell is constructed from phenolic, an insulating material, to prevent dissipation

of heat through the cylinder walls.

Weight

Position
Sii i Transducer


Powder Load Cell


Heater




Figure 2-4. Consolidation bench used for making the caked samples.

The procedure for making the cakes and measuring the strength is given in detail below:

S Each cake consists of a mixture of sodium carbonate monohydrate and decahydrate.
The decahydrate provides a source of moisture for the caking process and the
monohydrate acts as sink for moisture. The components are mixed together and a
portion of each sample is reserved to measure the initial moisture content with an
Ohaus MB45 moisture analyzer.









* The powder cells are filled using a special feeder to eliminate segregation. The
sample is consolidated by weights that are placed on top of the piston.

* The temperature profile mimics the changes that occur from day to night within a
metallic bin stored outside and exposed to direct sunlight. It is a ramp and hold, as
shown in figure 2-5, varying from 250C to 500C and back in a 24 hour cycle. The
cakes are held at each temperature for approximately 7.5 hours and the temperature
is increased or decreased at a rate of 20 per minute.


50C



25C /25C





Figure 2-5. Temperature profile used for preparing the cakes

* After the temperature cycle is complete, the cakes are carefully moved from the
consolidation bench and placed in the modified Indicizer test cell. Finally, the
strength is measured using the Johanson Indicizer.

Schulze Shear Tester

The Schulze shear cell is a powder tester which is often used to measure the

characteristics of powder flow. A schematic of this cell is shown in Figure 2-6. The

powder is contained in an annular base which has an adjustable annular velocity. A top

lid with short vanes sticking into the powder is prevented from rotating by tension bars

which are connected to load cells. When the base rotates, the powder will shear

somewhere between the bottom of the base and the vanes of the top lid. The load cells

measure the force that is needed to keep the top lid in place. The measured force is

equivalent to the shear force. The normal stress on the powder can be adjusted with

weigths. The measurement procedure using the Schulze shear cell has been standarized

by ASTM (ASTM, 2002).























SA) B)

Figure 2-6. Schulze shear tester: (A) image and (B) schematic.

The rationale for using the Schulze cell was to investigate the effects of relative

humidity cycling on the strength of the material. A modified cell is used to enable the

humidity cycling in the cell. This cell was designed for a previous study which

investigated the effects of airflow on material flow properties. A schematic of the

modified cell is shown in Figure 2-7. The modified tester can accommodate air flow

through the cell. It has a permeable lid and the air is introduced from the bottom of the

cell and leaves through the top lid. A porous membrane with a moderate pressure drop is

used to ensure that the air is dispersed evenly on the bottom on the cell. The inlet air is

conditioned with saturated salt solutions to control the relative humidity of the sample

(Greenspan, 1977 and ASTM, 1996).
















1rli, sdulac L n


'ir a i




Figure 2-7. Schematic of the permeable Schulze cell.

The procedure for measuring the cake strength follows the ASTM Standard Shear

Test Method for Bulk Solids Using the Schulze Ring Shear Tester (ASTM, 2002) with

the following modifications.

S The sample is sheared to steady state and then held under a normal load for a period
of 24 hours. Throughout this time, air is blown through the sample to equilibrate at
a specified relative humidity. Humid air is passed through the sample for 12 hours
to ensure that the system is in equilibrium. The moisture is adsorbed onto the
surface of the particles and liquid bridges are formed between adjacent particles.
Subsequently, dry air is passed through the sample such that the bridges solidify to
form the cake. After drying the sample the air flow is stopped and the sample is
sheared to determine the increase in failure stress.

* A new powder sample must be used for each point on the yield locus due to the
formation of cakes in the cell. The points on the yield locus are measured for each
relative humidity to create a family of loci. The loci are used to construct Mohr
circles that determine the unconfined yield strength, fc and the major principal
stress, o-. Figure 2-1 is a schematic of data collected from the Schulze cell.

The Schulze cell is also used to measure the effects of temperature cycling on the

strength of the material. A flexible heater is attached to the inner and outer circumference

of the test cell. The temperature of the cell is controlled to the temperature profile as

shown in Figure 2-5. Prior to exposure to the temperature profile the yield locus of the

material is measured according to the ASTM procedure. After the temperature cycle the









yield locus of the caked sample is measured. Each point of this yield locus requires a new

sample. The data from the Schulze cell experiments are compared to the results from the

Johanson Indicizer.

Unconfined Yield Strength Results

Effect of Moisture Content and Consolidation Pressure

The initial moisture content of the powder is important because it gives insight into

the material's tendency to cake and the extent of caking. Materials with excess water

have the ability to cake depending on the material properties and storage conditions. It

has been observed that powders with higher moisture contents tend to exhibit a stronger

propensity for caking than ones with lower moisture contents.

Sodium carbonate decahydrate with a water content of 61.2 % is used to vary the

moisture content of the samples. The water is crystalline water contained in the structure

of the material. At 32C, the material begins to decompose and loses water. Therefore

adjusting the percentage decahydrate in a mixture effectively varies the moisture content

of the sample. The unconfined yield strength results, from the Indicizer, as a function of

percentage sodium carbonate decahydrate in the mixture can be seen in Figure 2-8. The

results indicate that the unconfined yield strength of the cakes increases as the percentage

decahydrate increases. Other researchers have also observed that the cake strength

increases with an increase in moisture content (Pietsch, 1969).

It can also be seen that only a small percentage of decahydrate is needed to yield

strong cakes. If a material is very hygroscopic, such as sodium carbonate monohydrate,

the minimum moisture content to induce a caking event is very low. Therefore the

storage conditions of these materials are a vital factor in controlling the unwanted caking.

At higher percentages of decahydrate (beyond 4%) the cakes are too strong to be










measured with the Johanson Indicizer. The measurement limit of the Indicizer is stated

to be 2000 kPa (Johanson, 1992).


-- 17 kPa- 250 um
- 17 kPa- 850 um
-A-10 kPa-250 um
-*--10 kPa 850 um


1 2 3 4 5
% Sodium Carbonate Decahydrate


Figure 2-8. Unconfined yield strength of sodium carbonate monohydrate as a function of
moisture content in the form of percent sodium carbonate decahydrate in the
mixture.

It is believed that the consolidation stress has an effect on the bulk cohesive

strength of the material. Not all materials exhibit this property but in the case of sodium

carbonate, the strength increases with an increase in consolidation load. This is intuitive

since larger consolidation loads will result in a denser material and larger interparticle

contact areas. Figure 2-8 indicates a nearly linear dependence of consolidation pressure

on the yield strength of the cakes. In Figure 2-9 the unconfined yield strength of

carbonate as a function of the consolidation stress is plotted. It can be seen that the

strength of sodium carbonate increases as the stress on the material increases.










60
--250 um particles

50 -- 850 um particles


0 40


S30


S20-


10 -


0
0 5 10 15 20
Consolidation Stress (kPa)

Figure 2-9. The yield strength as a function of consolidation pressure. The mean particle
size of this material is 250 |tm and the moisture content is 2% decahydrate.

Effect of Particle Size

The effect of particle size on the strength of powders has been widely researched.

Rumpf was the first to theorize that the particle size influences the strength of a powder.

His theory states that the particle size is inversely proportional to the tensile strength of

the powder assuming that the porosity and the strength of the interparticle bonds are

known. The details ofRumpf's theory are given in a later chapter. The yield strength

measured by the Johanson Indicizer is plotted as a function of the mean particle size dso

of the sodium carbonate. It can be seen in Figure 2-10 that the yield strength increases as

the particle size decreases. As the particle size decreases the surface area and number of

contact points increase such that there are a larger numbers of contact for the water to










migrate and a larger number of solid bridges formed. This hypothesis is verified by

Figure 2-10 as the strength increases with a decrease in particle size.


80
0% Decahydrate
70 A 2% Decahydrate

60 3% Decahydrate


0 50
t---

0)
5 40

30 -

20 -

10 -

10
0 100 200 300 400 500 600 700 800 900
Particle Size (um)

Figure 2-10. The yield strength as a function of the mean particle size. The consolidation
pressure is 15 kPa.

Effect of the Number of Temperature Cycles

It is well known that caking is a time induced phenomenon (Teunou et al., 2000,

Purutyan et al., 2005). Thus the time of storage is also an important factor when

considering the strength of caking. The unconfined yield strength measurements were

made with both the Johanson Indicizer and the Schulze shear cell. The yield locus,

measured with the Schulze cell, of a sample that has been exposed to air flow for a period

of 24 hours is displayed in Figure 2-11. The time consolidation effects can be seen as the

yield locus increases with time. The unconfined yield strength of the material is a










function of the shear stress as given in equation 2-3. Thus an increase in the shear stress

results in an increase in the unconfined yield strength.




9

8 24 hr

7 0 hr

CU 6



u) 4

u 5- 3-

2 -

1 -

0
0 1 2 3 4 5 6
Normal Stress (kPa)


Figure 2-11. Yield locus of sodium carbonate for air at 75% RH for 24 hr. and 0 hr with a
normal load of 16kPa.

It has been observed that the caking strength may increase as the number of

temperature cycles increases (Johanson et al.,1996, Cleaver et al., 2004). In other words,

the longer a material experiences the day-to-night temperature changes, the greater

tendency for increased caking. The yield strength of sodium carbonate is investigated as a

function of the number of temperature cycles that the material experiences. The

temperature cycles for this set of experiments are 4 hours long. It can be seen in Figure 2-

12 that the unconfined yield strength as measured with the Johanson Indicizer increases

as the number of temperature cycles increases. It is possible that the duration of

temperature cycle is not sufficient for the material to reach an equilibrium state. Thus










subsequent cycles initiate the release or uptake of moisture causing an increase in the

strength until equilibrium. It is shown in Figure 2-12, that for the 2% decahydrate

mixture, that the yield after 2 and 4 cycles are approximately equal. The same trend is not

seen with the 4% decahydrate mixture. This could be due to the fact that the initial

moisture content of 4% mixture is higher thus requiring more time for equilibration. It

may also be attributed to the migration of dissolved material from a non-contact area to

the contact zone during the recrystallization process.


45
S 2% Decahydrate
40
S4% Decahydrate
35

c 30-

525

U2 20
-0
> 15

10

5

0
0 1 2 3 4 5
Number of temperature cycles


Figure 2-12. The yield strength as a function of the number of temperature cycles. The
mean particle size is 250tm and the consolidation pressure is 15 kPa.

Effect of Relative Humidity

The effect of relative humidity has only recently received attention in the research

of caking (Teunou et al., 1999a, Leaper et al., 2003). It is believed that changes in

relative humidity could induce a caking event. The effects of relative humidity on the










cake strength of sodium carbonate are investigated using the Schulze shear cell. It can be

seen in Figure 2-13 that the slope of the yield locus increases with an increase in relative

humidity. This indicates that moisture is adsorbed by the particles at higher relative

humidities. The moisture sorption causes the increase in shear stress which in turn

increases the unconfined yield strength.


6
%RH

5 m95%RH


-4



1-4









0
0 1 2 3 4 5 6
Normal Stress (kPa)


Figure 2-13. Yield locus of sodium carbonate for air as a function of relative humidity
with a normal load of 4kPa.

It can be seen in Figure 2-14 that the unconfined yield strength increases with the

relative humidity. The strength data seem to follow the same trend as the isotherm with

the moisture pickup, thus verifying that the adsorbed moisture affects the cake strength.

However, a change in relative humidity does not seem to produce a cake of the same

strength as those created by a temperature change.
-co


























strength as those created by a temperature change.











3 35


2.5 30






0 /
153
.5 25








0 20 40 60 80 100








% Relative Humidity


Figure 2-14. Unconfined yield strength as a function of relative humidity compared to the
isotherm of sodium carbonate.
>- 15




0.5 5




0 20 40 60 80 100

% Relative Humidity


Figure 2-14. Unconfined yield strength as a function of relative humidity compared to the
isotherm of sodium carbonate.














CHAPTER 3
ADSORPTION ISOTHERMS AND KINETICS

Perry (1997) defines adsorption as the accumulation or depletion of solute

molecules at an interface. In the case of moisture migration caking, this entails the

transfer and equilibrium distribution of moisture between the gas phase (the humid

environment) and the particles. In this chapter the adsorption isotherms of sodium

carbonate and salt are examined using dynamic vapor sorption techniques. It is shown

that the isotherms are a good predictor to the onset of caking. The kinetic sorption curves

are also measured to obtain the rates of adsorption and vaporization of water.

Adsorption Isotherms

During the processing and storage of bulk materials, the material is exposed to

humid air in the environment. The humid air contains water vapor which can have a

significant affect on the physical and chemical properties of the material. Water vapor

from the environment can be absorbed into the structure of many materials and it may

also be adsorbed onto the surface of these materials. In the case of amorphous solids,

including pharmaceutical excipients and food products, absorbed water vapor is known to

lower the glass transition temperature of the material thus promoting a caking event

(Zhang et al., 2000). In many crystalline materials, the water vapor is adsorbed on the

surface causing dissolution thus also initiating a caking event. For food products, water

vapor has a critical effect on the dehydration process and storage stability (Iglesias et al.,

1976).









At a given temperature and relative humidity, solids exhibit a maximum capacity to

adsorb moisture and this is characterized by the sorption isotherms. Water vapor sorption

isotherms are used to describe the interactions between a material and the humid air

surrounding it. The isotherms are typically expressed as the water content (a loss or gain)

as a function of the relative humidity (RH) or water activity at a given temperature. The

adsorption of water vapor may be either physical or chemical and the vapor may adsorb

in multiple layers. Brunauer et al. (Perry, 1997) described five types of physical

adsorption as shown in Figure 3-1. Type I isotherm represent monomolecular adsorption

of a gas and applies to porous materials with small pores. This is the well-known

Langmuir isotherm. Type II and III isotherms represent materials with a wider range of

pore sizes where gas is adsorbed in either a monolayer or multilayer. The sigmoid

isotherm of Type II is typically obtained from soluble products which show an

asymptotic trend as the water activity goes to one (Mathlouthi et al., 2003). The Type III

isotherm is known as the Flory-Huggins isotherm. This isotherm represents the

adsorption of a solvent gas above the glass transition temperature. Type IV isotherms

describe adsorption which causes the formation of two surface layers. Type V adsorption

behavior is found in the adsorption of water vapor on activated carbon.

Some crystalline materials will exhibit sorption profiles other than those described

by Brunauer et al. The adsorption/desorption isotherm of hydrated material sometimes

appears as a stair like curve. The hydrates are formed at specific relative humidities. The

material equilibrates at a low RH and when the RH increases the material becomes

saturated increasing the moisture content of the material. This sequence appears as a

staircase in the sorption isotherms. Anhydrous materials only adsorb trace amounts of









water at low RH. At a characteristic RH, the adsorption corresponds to saturation

equilibrium. A vertical line on the sorption isotherm represents the saturation

equilibrium.



Type I Type II Type III




-----
0
-o
Type IV Type V

E




Water Activity


Figure 3-1. The five types of adsorption isotherms in the classification of Brunauer et al.
(1940).

From the sorption isotherm fundamental material properties and important

information about its handling can be derived. Properties such as hydrate formation,

deliquescence, and hygroscopicity can be determined from the sorption isotherms.

Sorption isotherms are also useful for a qualitative prediction of caking. The data

obtained from the isotherm are used to determine the temperature and relative humidity

conditions at the onset of caking. This is illustrated in the isotherms by a sudden increase

in the moisture content at a critical relative humidity. Sorption isotherms are also a

valuable tool for understanding the moisture relationship of a new product during the

formulation stages (Foster et al., 2004).









Mechanisms and Kinetics of Adsorption

The adsorption of a solute from a fluid phase is described by mass transport

processes such as: interparticle mass transfer, intraparticle mass transfer and interphase

mass transfer. These common transport mechanisms are illustrated in Figure 3-2.

Interparticle mass transport is the diffusion of the solute (moisture in the case of caking)

in the fluid phase through the particle bed. This type of transport is described by heat and

mass transfer equations on a continuum scale. Interparticle mass transport is a subject of

chapter 4. Intraparticle mass transfer, which includes pore and solid diffusion, describes

the transport of the solute through the particle. Interphase mass transport is the transfer of

a solute at the fluid-particle interface. Figure 3-3 is an illustration of interphase mass

transport showing the sorption of moisture by a sorbent particle.


2 23



4
I 4






Figure 3-2. Mechanisms of mass transfer for absorbent particles (Perry, 1997): 1, pore
diffusion; 2, solid diffusion; 3, reaction kinetics at boundary phase; 4,
interphase mass transfer; 5, interparticle mixing.

In this research it is assumed that the particles are nonporous. This assumption is

validated by measurements of the specific surface area by gas adsorption. The measured

values for the specific surface area correspond the ideal values at a specific particle size.









For a packed bed system, the transport of moisture from the bulk of the fluid phase

to the surface of the particle is described as interphase transport and the mass transfer rate

is given by:

dq = kA ( q) (3-1)
dt

where k is the mass transfer coefficient, As is the specific surface area, cb is the moisture

concentration in the bulk of the fluid, and q is the solids moisture content. To simplify the

calculations of equation 3-1 it is assumed that the uptake rate is linearly proportional to

the driving force, the so called linear driving force model. The moisture concentration in

the bulk is equal to the equilibrium solids moisture content. Thus the particles in the

system are in equilibrium with the fluid phase concentration of the bulk.


= kA (q q) (3-2)
dt

In equation 3-2 qe is the equilibrium solids moisture content and kg is the Glueckauf

factor. The equilibrium concentration is determined from the isotherm of the material.

The linear driving force model suggests that the concentration gradient is linearly

proportional to the average moisture content of the system.



Water Vapor
Condenses


p -- article -
particle dissolves pticl


Figure 3-3. Sorption of moisture by a water soluble particle.









Measurement Techniques for Sorption Isotherms

There are two common methods for determining the sorption isotherms of powders

or granular materials: static vapor sorption and dynamic vapor sorption.

Static Vapor Sorption

The method of static vapor sorption is a very time consuming and laborious process

for measuring isotherms. A constant environment (temperature and relative humidity)

must be maintained throughout the duration of the experiment. Several techniques are

employed to produce a known constant humidity. The use of saturated salt solutions is

the most common (Greenspan, 1977, ASTM, 1996). Known constant humidities are

created by salts whose affinity for water regulates the water vapor pressure surrounding

the material. Most often the salt solution and the sample are contained in dessicators for

the period of time in which the sample is allowed to equilibrate at the specified humidity.

The equilibration time is typically several weeks. The weight of the samples before and

after equilibration are recorded and used to determine the moisture content of the sample

at a particular relative humidity. Karl Fischer titration may also be used to determine the

moisture content of the sample. Depending on the properties of the powder sample, this

method can be time consuming.

Dynamic Vapor Sorption

An alternative, faster approach to static vapor sorption is dynamic vapor sorption.

This method is more time efficient because a smaller sample size is used which decreases

the equilibration time. The humid air is also either passed over or through the sample

allowing equilibration to occur much faster. Dynamic vapor sorption is the chosen

method of measurement for the sorption isotherms and sorption kinetics in this research.

The equipment used is the VTI Symmetrical Gravimetric Analyzer (SGA 100).










Symmetrical Gravimetric Analyzer

The isotherm and kinetic measurements are made with the VTI Corporation

Symmetrical Gravimetric Analyzer (SGA-100). The SGA-100 is a continuous gas flow

adsorption instrument designed to study water vapor sorption isotherms. All

measurements are taken at ambient pressure and the temperature range is from 0 to

800C. A schematic of the instrument is shown in Figure 3-4.


Balance
Enclosure


L4--:- -- THERMAL ZONE 1
PIIBSEJ i

Sample
Block


THERMAL ZONE 2

To/From
Constant Humidity
Temperature 1EN
Bath DE Control
POINT
I ANALYZER
I. THERMAL ZONE 3

MA55 FLOW MASS FLOW
CONTROLLER CONTROLLER




Figure 3-4. VTI Symmetrical Gravimetric Analyzer schematic (VTI website).

The instrument is designed with three separate thermal zones such that maximum

temperature stability is achieved. Located in zone 1 is the Kahn microbalance. The

microbalance has a sensitivity of 0.0001 jig. The temperature in this zone is maintained

15C higher than the experimental temperature to avoid condensation. Higher

temperatures along with a continuous purge stream of nitrogen in the balance chamber









ensure that there is no vapor condensation on the microbalance which could invalidate

the results. Isolating the microbalance also eliminates the weight fluctuations caused by

temperature gradients in the environment. Suspended from the microbalance are two thin

metal wires, into zone 2, for the sample holder and reference sample holder.

The core of the instrument, an aluminum block containing the sample chamber, is

located in zone 2. There are two rectangular cutouts (chambers); one houses the sample

and the other houses a reference sample. Glass sample holders are suspended on the thin

wires from the microbalance. Both sides are subjected to the same temperature and

relative humidity conditions. The temperature in this zone can range from 0 to 800C. It

is controlled by circulating water from a constant temperature bath through the hollow

wall aluminum block. The air temperature is measured with an RTD located at the

bottom of each chamber.

The third zone is for the humidity control. It contains the chilled mirror dew point

analyzer and a parallel plate humidifier which are maintained at 400C. The system is fed

with a constant supply of nitrogen. The nitrogen enters the parallel plate humidifier and is

completely saturated (100 % RH) with respect to the experimental temperature. The

saturated stream is mixed with a dry stream of nitrogen to obtain the desired relative

humidity. The mixed stream passes through the dew point analyzer to be measured before

entering the sample chamber. The dew point analyzer continuously measures the relative

humidity stream to maintain control of the wet and dry mass flows.

The VTI Sorption Analyzer is a fully automated system that requires minimal user

input. The operator must specify the operating conditions such as experimental

temperature, relative humidity steps and the equilibrium criterion. The user input screen









is displayed in Figure 3-5. The experiment can be set for a drying step before the

measurement of the isotherm. The relative humidity steps are customized by the user or

chosen by the computer program if the initial and final step are specified. The data are

recorded at intervals specified by the user. This is a time or percentage weight change

condition. After the experimental parameters are set the material is loaded in the sample

holder and the experiment begins.


Drying -Relative Humidity Steps
Drying: Off E] On RH, start: 0 %
Temperature: 25'C RH, end: 95 % r Cust.o!m
Heating Rate: 5 'C/min RH, step. __ 5 %
Equilibrium 0.0100 wt % in Des Cutoff: 0 % RH
Criterion: 2.00 min ,
Max Drying Time 5 min .

Adsorption/Desorption :.
Temperature: 25 C .
Bypass Sample. IWaitforRH _
Equilibrium 0.0100 wt % inL OK I
Criteionu 5. rn -Data Logging Interval
Criterion 500 min Cancel
MaxEquilibTime: T min 2 min or 00100



Figure 3-5. User input screen for the VTI Sorption Analyzer.

Experimental Parameters

The sorption isotherms are measured at three temperatures: 250, 350 and 500C.

Measuring in this range provides an indication of how temperature affects the sorption

process. The kinetic sorption curves are also measured to obtain the rates of adsorption

and desorption of water. The data from the isotherms can be used to determine if water

sorption at a specified relative humidity is sufficient for caking.









Only the sorption isotherms of single component systems are measured. However,

the combined effect of individual adsorption isotherms in a mixture on the diffusion of

moisture through the bulk and onto the particle surface is important. Consider a system

containing two different particles with different isotherms. One particle can be a moisture

sink for the water and the other particle can be a moisture source. Therefore, a caking

event can occur due to differences in isotherms. The differences in isotherms not only

affect the diffusion of moisture on the particle surface but also the kinetics of evaporation

and crystallization of the bridges within the cake.

The standard procedure for measuring adsorption isotherms suggests drying the

material introducing humidity (ASTM, 1996). However, the following isotherms are

measured without a prior drying step. The drying step is only necessary to ensure that the

starting point is known. Thus the moisture content of the material is taken prior to the

isotherm measurements using an Ohaus MB45 moisture analyzer. The moisture contents

are consistent within a range of less than 1%.

Sorption Isotherms for Sodium Carbonate

Sodium carbonate is one of the materials used in this study. This material is widely

used in the glass industry as well as in the formulation of powder detergents. Sodium

carbonate has a strong propensity to cake if not stored in a controlled environment. It is a

colorless odorless material with three stable hydrate forms, monohydate (SCM),

heptahydrate (SCH), and decahydrate (SCD). The phase diagram for sodium carbonate is

shown in Figure 3-6 (OCI Chemical website). Heptahydrate is a semi-stable form and

likely exist only in solution. It is made up of 57.4% water and is only stable in the

temperature range of 32C to 36C. Sodium carbonate monohydrate granules contain

14.6% water. It loses water on heating (to 100 C), and the solubility increases with










increasing temperature. Sodium carbonate decahydrate contains 61.2% water and the

crystals readily effloresce in dry air and form lower hydrates. This is evident from the

moisture sorption isotherm, Figure 3-7.


Unsaturated
Solution


Anhydrous + Solution


Monohydrate
+
Solution


Heptahydrate
+
Solution
/^_______


Decahydrate
+
Solution


Ice + Decahydrate


Heptahydrate
+
Monohydrate


Heptahydrate

Decahydrate


0 10 20 30 40 50 60 70 80 90
Percent Sodium Carbonate


Figure 3-6. Phase diagram of sodium carbonate.

In Figure 3-7, the expected equilibrium moisture adsorption for both decahydrate

and monohydrate is shown. The decahydrate appears to lose (47% moisture) at relative

humidities lower than 70%. This corresponds to an average molar moisture content










equivalent to 7 moles of crystalline water. However, decahydrate shows a great

propensity to deliquesce at relative humidities greater than 70% gaining almost 1.5 times

its initial weight in moisture at 98% RH. Comparing this behavior to the monohydrate,

shows a very different behavior. The monohydrate does not lose or gain moisture at

relative humidities lower than 70% RH. This transition humidity, where deliquescence

begins, is similar for the decahydrate. However, the monohydrate does not deliquesce as

rapidly as the decahydrate. The deliquescent relative humidity is in the range on 70-80 %.


150
Decahydrate
Deca Model
100 m Monohydrate
Monohydrate Model


50


P 0



-50
-50 4 ---- *--- -- ( -- --- --



-100
0 20 40 60 80 100
% Relative Huridity


Figure 3-7. Adsorption isotherm of sodium carbonate at 250C. The points indicate the
experimental data and the lines are the adsorption isotherm model predictions.

The isotherms of sodium carbonate suggest that a mixture of decahydrate and

monohydrate would exhibit different behaviors in terms of moisture pick-up at different

relative humidities. For example, assume a mixture of decahydrate and monohydrate is in

an environment where the humidity was less than 86 % RH but greater than 70% RH.









From the isotherms, it is sufficient to assume that decahydrate would lose its moisture

and that moisture would be picked up by monohydrate. At relative humidities greater

than 88% both decahydrate and monohydrate deliquesce. The degree of deliquescence at

90 % RH is the same for both the monohydrate and decahydrate. However, at relative

humidities greater than 90% RH the degree of deliquescence of decahydrate surpasses

that observed with monohydrate suggesting that a mixture of these two hydrates might

cause the monohydrate to lose moisture and be picked up by the decahydrate. The pickup

of moisture by the monohydrate and the loss of moisture by the decahydrate causes the

caking. The additional moisture slightly dissolves the surface of the monohydrate

particles. Bridges are formed between particles and a cake is created. Increasing the

amount of decahydrate in the mixture will therefore increase the strength of the cake.

Thus the use of several hydrates is ideal for controlling the initial water content of the

samples.

In Figure 3-7, for the monohydrate, a 0 % weight change corresponds to 100%

monohydrate. An increase in weight of 87% and 130% correspond to the formation of a

heptahydrate and decahydrate, respectively. The endpoint of the monohydrate curve

suggests that the material likely converts to a mixture of heptahydrate and monohydrate.

At 25C, the monohydrate picks up moisture yet not enough to fully transform to the

decahydrate or heptahydrate form. For the decahydrate, a decrease in weight of 56%

would correspond to a decahydrate transformed into monohydrate. At a 0% weight

change, the decahydrate is transformed to its original state. The endpoint of the

decahydrate curve suggests that the material deliquesces with a total moisture gain of

130%.










The sorption of water vapor by sorption carbonate is not sufficiently described by

the adsorption models proposed by Brunaeuer in Figure 3-1. It can be seen in Figure 3-7

that the isotherms of sodium carbonate at 25 C are best described by the adsorption

model proposed by Miniowitsch (1958).

RH/100
q=Ae B +q, (3-3)


In equation 3-3, A and B are constants, qo is the initial weight lose and q is the moisture

content. The constants A and B vary slightly with each fit and qo is zero for monohydrate

and -47 for decahydrate. For the decahydrate A=7.97e-7 and B=0.0669. For the

monohydrate A=6.3 8e-7 and B=0.0723.

The desorption curves for sodium carbonate are shown in Figure 3-8.



200
SDeca Adsorption

1- *- *Deca Desorption
-A-Mono Adsorption ,* -
*- -Mono Desorption
-Q) 100
-O
50 0






-50 .


-100
0 20 40 60 80 100
% Relative Humidity


Figure 3-8. Sorption isotherm of sodium carbonate at 250C.







44


It can be seen that adsorption and desorption of both the monohydrate and the

decahydrate do not follow the same curve. There is a hystersis between the adsorption

and desorption which implies that the sorption of sodium carbonate is not isotropic. Thus

the desorption must be described by an equation different from that of the adsorption.

Effect of Temperature

The effect of temperature on the moisture sorption of sodium carbonate

monohydrate and decahydrate is also investigated. It can be seen in Figures 3-9 and 3-10

that increasing the temperature increases the total moisture uptake.


130 ......................................
Decahydrate

110


90
Heptahydrate
I 70 7
70

50

Monohydrate 25 deg. C
30
--- Monohydrate 35 deg. C

10 Monohydrate 50 deg. C


-10
-0----------------------------------

0 20 40 60 80 100
% Relative Hurridity


Figure 3-9. Sorption isotherm of sodium carbonate monohydrate at various temperatures.
The dashed lines correspond to the point where the monohydrate converts to a
heptahydrate or decahydrate.

The total moisture uptake of monohydrate at 35C is such that heptahydrate is

formed. This result is verified by the phase diagram which suggests that the heptahydrate

is stable in the temperature range of 32 -35 C. At 50C the material shows significant










deliquescence with a gain in weight corresponding to a mixture of the decahydrate and

heptahydrate forms. The same trends are observed with the decahydrate as seen in Figure

3-10. However deliquescence is substantially increased. At 35C and relative humidity

values greater than 80% the material appears to form an unsaturated solution. Likewise,

at 50C and relative humidity values above 85%, the result is an unsaturated solution.


200
-- Decahydrate 25 deg. C

150 --- Decahydrate 35 deg. C

Decahydrate 50 deg. C
100


50 -
'-c

50 ...

"- Decahydrate

Heptahydrate
-50
Monohydrate

-100
0 20 40 60 80 100
% Relative Hurridity


Figure 3-10. Sorption isotherm of sodium carbonate decahydrate at various temperatures.
The dashed lines correspond to the point where the decahydrate converts to a
monohydrate, heptahydrate or decahydrate.

The increase in moisture uptake is only observed beyond the deliquescence relative

humidity. The deliquescence range remains unchanged within the temperatures

investigated. This result is contrary to what various isotherm models would predict as a

response to a temperature increase (Iglesias et al., 1976). At higher temperatures on the

phase diagram the boundary between the unsaturated solution and the monohydrate plus

solution has a negative slope. This suggests that as temperature increases the percentage










of sodium carbonate decreases. Thus as the temperature of the sample increases the

moisture uptake increases.

Kinetics of Adsorption

The linear driving force model assumes that the order of the reaction of moisture

adsorption/desorption is one. The kinetics (moisture content as a function of time) of

sodium carbonate are investigated to validate the use of this model. The curves for the

moisture content versus time for sodium carbonate decahydrate are shown in Figure 3-11.

70 140
| 60 120-
50 100 -
| 40 80
0 30 60
O O
w 20 40
S10 20
0 0
0 0
0 50 100 150 0 200 400 600 800 1000
Time (min.) A) Time (min.) B)


Figure 3-11. The moisture uptake of sodium carbonate decahydrate as a function of time.
A)10%RH represents the kinetics below the deliquescene relative humidity
and B) 90%RH represents the kinetics above the deliquescence relative
humidity.

These two curves represent the typical kinetics of decahydrate above and below the

deliquescence relative humidity. It can be seen that the kinetics at lower relative

humidities are significantly faster than the kinetics at higher relative humidities,

illustrated by the length of time required to reach an equilibrium state. This suggests that

desorption of water occurs much faster than adsorption.

At higher relative humidities there is initially a loss of moisture corresponding to 5-

7 moles of water. After desorption the material begins to adsorb moisture. This transient

may be attributed to an artifact of the tester. Before the material is introduced into the

sample chamber, the sorption analyzer operates at a default equilibrium condition of 0%









RH and 25 C. It is possible to set the initial temperature of the chamber to the

experimental temperature prior to the test by changing the setpoint of the temperature

bath. However, it is not possible to change the default relative humidity. When the

material is introduced into the sample chamber, the system must come to steady state at

the specified experimental parameters. This process is not instantaneous. Hence there is

an initial desorption of water which is caused by a low relative humidity in the sample

chamber. The transient lasts for approximately 30 minutes. Thus the first 30 minutes of

the curve are disregarded for determination of the order of the reaction.

To determine the order of the reaction the integral method of rate analysis is used.

The reaction order is known (or suggested) from equation 3-2. If the reaction is first

order, integration of equation 3-2 gives


-In q- Ce =kt (3-4)


The slope of the plot of -ln(q-ce/qo-ce) as a function of time t is linear with a slope kg. In

Figure 3-11 the concentration data is approximated by the linear driving force model. The

slope, intercept and correlation coefficient are given in Table 3-1. The slopes vary with

the relative humidity implying that a single rate law may not describe the sorption of

water. The rate of moisture sorption changes with a change in relative humidity.

The correlation coefficient for the curves of decahydrate at the specified relative

humidities is in the range of 0.8453 0.9958. A coefficient of 1 would signify a perfect

correlation. This result suggests that the linear driving force model is not perfect.

However the kinetics of sodium carbonate may be approximated by this model with a

reasonable certainty.




















0 10 20 30 40
Time (min)


100 200
Time (min.)


0 500
Time (min.)


0 20 40
Time (min.)


0 200 400 600
Time (min.)


0 500 1000 1500
Time (min.)


Figure 3-12. The kinetics of sodium carbonate decahydrate at 50C. A)10%RH,
B)70%RH, C)80%RH, D)85%RH, E)90%RH, and F)95%RH

Table 3-1. Kinetics constants for sodium carbonate decahydrate
Relative Humidity (%) Slope Intercept Correlation Coefficient (R2)


0.1065
0.1049
0.0946
0.0072
0.005
0.005
0.0016


0
0
0
1.0059
0
0
0


0.979
0.9524
0.9142
0.8453
0.9056
0.9958
0.9077


W3
o-

0
6-


0



5


-/


60 80


800 1000


2000 2500


r










The rate constants are plotted as a function of relative humidity in Figure 3-13. It

can be seen that the rate if desorption is significantly greater than the rates of adsorption.


0.12


0.1


, 0.08


I 0.06
0
o

S0.04


0.02


Desorption


irption


0 20 40 60 80
Relative Humidity


Figure 3-13. The kinetic rate constant for sodium carbonate decahydrate adsorption and
desorption.

Sorption Isotherms for Sodium Chloride

Sodium chloride is also used in this study. This is a very complex material due to

its deliquescence at high relative humidities. However there is an abundance of literature

about this material due to its importance in all aspects of life. The sorption and kinetics of

sodium chloride are well described in literature. Sodium chloride is highly soluble in

water but only contains small amount of moisture after dehydration. It varies in color

from colorless, when pure, to white, gray or brownish, typical of rock salt. The crystal

structure can be modified with a change in temperature. At 20C, the critical relative

humidity is 75%. Above this RH, the salt deliquesces. This is illustrated by the isotherm










in Figure 3-14. The isotherm for sodium chloride is plotted as function of temperature. It

can be seen that the material adsorbs an undetectable amount of water at lower relative

humidities. Beyond the deliquescence point there is a significant amount of sorption. This

result is consistent with literature data (Greenspan, 1977).


500

450 -A-Salt25deg. C
400 --Salt 35 deg. C
400
--Salt 50 deg. C
S350 -
ci 300
-c
o 250

200

150 -

100

50

0
0 20 40 60 80 100
% Relative Humidity


Figure 3-14. Sorption isotherm of sodium chloride at various temperatures.














CHAPTER 4
MOISTURE MIGRATION MODELING

This chapter explains the modeling of moisture migration through a particle bed.

Moisture migration is described by the heat and mass transfer of the system along with

the isotherms of the material. In the literature, it is assumed that free convection in the

system can be neglected. However, it is shown that free convection plays a significant

role in the heat and moisture transport of the system. It is also shown that the areas of

caking within the bulk can be predicted given the proper model for the heat and mass

transfer of the system.

Background

It has been shown in chapters 2 and 3 that heat and moisture play a significant role

in the caking process. Thus, in order to better understand caking, it is essential that the

heat and mass transport (moisture migration) is thoroughly explained. An important

factor in describing moisture migration caking is the transport of moisture through the air

and onto the surface of the particles. Other researchers have investigated specific parts of

the process but few have attempted to describe the entire process of moisture migration.

However, all attempts at describing moisture migration have included simplifying

assumptions which render the models inaccurate.

Tardos et al. (1996a) studied diffusion of atmospheric moisture into a particulate

material inside a container. The authors developed a model to describe the amount of

moisture that penetrates from a stagnant layer of humid air above a particle bed. The goal

of the study was to use the model to calculate the depth of moisture penetration as a









function of time. It was assumed that moisture migrates through the bulk by diffusion

only and is driven by concentration gradients between the fluid phase and the particles. In

the model, material is exposed to a step change in moisture content due to an increased

amount of moisture in the air above the particle bed. The equations that describe the

system are

c =D 2 (I p ) q (4-1)
at 8z2 E p, at


S= k (q q) (4-2)
at

where qe is the equilibrium moisture content given by the sorption isotherm, c is the

vapor phase concentration, q is the solid moisture content, D is the diffusion coefficient, e

is the bed porosity, p and pa are solid and air densities and z and t are distance and time. It

is assumed that the rate of adsorption is proportional to the concentration difference such

that kg is constant. Kg is the called the Glueckhauf factor or linear driving force

coefficient. The Glueckhauf factor is a function of the particle radius R and the diffusivity

De.
DD,
kg =15 (4-3)
R2

Equation 4-1 is based on one-dimensional diffusion in the z direction. The initial

conditions are chosen such that there is a uniform concentration of vapor in the powder

bed and the moisture content of the solid is in equilibrium with the vapor phase.

Experiments were preformed to verify the model and it was found that the model did not

accurately predict the data for all times. At shorter times the model under predicts the

measured data. This simple model is only an approximate prediction of the data. It fails to









address the effects of temperature and free convection on the moisture migration through

the particle bed.

The results from the moisture migration modeling were applied to caking of fine

crystalline powders in a second paper by Tardos (1996b). The purpose of this study was

to examine the caking behavior of fine, bulk powders while exposed to humid

atmospheres. The depth of penetration of moisture into the powder bed was calculated by

solving equations 4-1 through 4-3. The caking ratio defined by Tardos is the height of the

portion of the upper surface of the powder at a critical moisture content divided by the

total powder depth. The critical solids moisture content is the value at which caking of

the powder begins. Hence, the tendency of a powder to cake can be determined from this

simple model given that the material properties of the powder relevant to caking are

known.

Tardos also suggests that hydrate formation causes swelling in the powder bed.

This compression is associated with caking. Due to the adsorption of moisture, various

hydrates are formed which cause a swelling of the powder. This increase in bulk volume

is on the order of 10% or more and occurs at relative humidities of hydrate formation.

Tardos finds a relationship between the increase in bulk volume and the caking

propensity of the powder using dilatometer testing of sodium carbonate. These findings

suggest that the powder swelling is primarily responsible for the caking of the powder.

Rastikian and Capart (1998) later developed a model for caking of sugars in a silo

during storage. The purpose of the study was to create a model to predict the moisture

content profiles, air humidity and temperature inside a laboratory silo. The model

includes not only the mass transport but also the heat transport which Tardos fails to









address. The system is described by the following two equations for heat and mass

transport as well as the kinetic equation for the drying of sugar.

Mass balance equation:

(c = lc 02c )2c F( ac
SDe +-- + + +kf(c -c) (4-4)
ot r Or r 2 O2) pa oz

In equation 4-4 c is the concentration of the air, ce is the equilibrium concentration at the

particle surface, De is the effective mass diffusivity of the water vapor, Fa is the flow-rate

of the inlet air, pa is the density of the air, and r and z are the radial and axial coordinates.

Heat balance equation:

T +T 2 T+FC T P kf(c c)H (4-5)
at pc9, ^r dr Or 2 OZ 2 pcS OZ pCps

In equation 4-5 Tis the temperature of the air, A, is the thermal conductivity of the solid,

p is the density of the solid, cp is the specific heat of the solid, cpa is the specific heat of

the air and H, is the enthalpy of water vaporization.

A laboratory silo was constructed in which humid air is blown through the bed of

sugar. The temperature along the height of the silo and the relative humidity of the air

above the particle bed are measured. These experimental data are compared to the

proposed model. It was found that the model approximates the temperature profile within

the silo reasonably well. However, Rastikian et al. assumed that the mass and heat

transfer by diffusion in the radial direction can be neglected. Also, the system is based on

forced convection through the particle bed. If material inside a silo is stored in an

uncontrolled environment, it is more realistic to expect free convection in the system.

Dehydration during storage and mass transfer likely occur through free convection.









Leaper et al. (2002) developed a model of moisture migration through a bulk bag as

a function of humidity cycling. The model is based on the work of Rastikian et al. and

Tardos. The purpose of this study was to predict the temperature profile and moisture

content profile of the material within the bulk bag. It was assumed that moisture migrates

by diffusion only due to moisture concentration gradients caused by fluctuating local

relative humidity. The authors developed a procedure to calculate the profiles as follows:

* Determine the temperature profile using a simplified one-dimensional finite
difference model which does not account for convective heat transfer.


Q= AA T +T and T = Qt T, (4-5)
msolld ps

where T/ is the temperature at node position i and timestepj. Q is the heat flowrate
through and specific transfer area Asp, msold and c, are the mass and heat capacity,
respectively.

* After the temperature profile is determined the RH profile can be calculated for a
specific temperature.

RH
H = Hw (4-6)
100

where Hw is the saturated humidity and RH is the relative humidity. The saturated
humidity is simply the maximum concentration of water vapor possible at a
specific temperature. This variable can be obtained from the partial vapor pressure
of water.

H = PH2O water (47)
PT PH2O mWawr

where pH2o is the partial pressure of water vapor, pr is the total pressure and mwwater
and mwair are the molecular weights of water and air, respectively. The solids
moisture content q of the sample is calculated using equations 4-6 and 4-7.

RH m0o/lq
tot = H + sold (4-8)
qtot a 100 w 100

where qtot is the total moisture content, mair is the mass of air and msod is the mass
of solid.









* As the temperature changes, the RH profile at each node is adjusted to reflect
moisture migration due to diffusion.

H = DA H,1-H,' +H, (4-9)

where D is the diffusion coefficient for the moisture migration.

* Finally, the new equilibrium RHeq and solids moisture content qeq can be
calculated. Thus equation 4-8 is adjusted to include the equilibrium values.

RHeq r msohd qeq
qtot = ma eq Hw + (4-10)
100 100

The sorption isotherms are used to derive an empirical relationship between the
equilibrium solids moisture content qeq and relative humidity RHeq.

To verify the model, the solids moisture content was measured after exposing the

material to a temperature cycle. The authors conclude that they can create a profile of the

solid moisture content in the bulk and as a result they can predict cake formation.

All of the current models are useful as a first approximation of moisture migration

caking. However, they reduce the fundamental transport equations, i.e. the moisture

migration, to a simplified case that disregards pertinent details of the caking process.

Moisture Migration Model

The driving force of moisture migration caking is a concentration gradient within

the bulk. This gradient causes transport of moisture through the interstitial voids of the

bulk solid. Bulk materials are typically stored in an environment where temperature and

relative humidity are not controlled. Thus, changes in the temperature from day to night

can induce a thermal gradient through the material which affects the local relative

humidity surrounding the particles. The changes in relative humidity caused by

temperature fluctuations initiate the moisture migration through the bulk. With the

additional moisture in the air, particles will adsorb moisture until an equilibrium state is









obtained. The equilibrium condition is determined from the sorption isotherms of the

material. The isotherm represents the equilibrium relationship between the bulk solid

moisture content and the relative humidity of air surrounding the particles as described in

the previous chapter.

The process of moisture migration can be described by fundamental transport

equations of heat, mass, and energy. It is assumed that the air surrounding the material is

stagnant; therefore the only velocity gradient is that due to free convection. The partial

differential equations will be solved using finite element techniques.

The location of the moisture is determined using finite element methods modeling.

This result can be used to locate the formation of solid bridges within a bulk material.

The following steps are required to achieve this result:

* Determine if convection plays a role in the caking process by comparing the
temperature profiles from the models with and without a convective term. Analyze
the Peclet number in the cell to determine if convection is a dominant mode of
transportation for the moisture.

* Calculate the diffusion through the cell from the temperature profile.

* Determine where the moisture migrates in space and time i.e. the solids moisture
content over time.

Finite Element Modeling

The moisture migration process is modeled using finite element methods utilizing

the COMSOL Multiphysics software. This method is used to solve partial differential

equations (PDEs) which describe and predict the moisture migration in caking. The

behavior of moisture is modeled on a continuum scale. The flow of air, the heat transfer

through bulk solids, and the moisture content of the solid within the cell are described.

This is used to predict the amount of material or the thickness of the layer of material

involved in a caking event and also the strength of the cake.









The general approach to solving PDEs using the finite element method includes

discretization, developing the element equations, characterizing the geometry, applying

boundary conditions and obtaining a solution. The discretization involves dividing the

solution domain into simple shape regions or elements either in one, two, or three

dimensions as shown in Figure 4-1. The points of intersection of the lines are called

nodes.


Line Element

(a) 1-D


Quadrilateral Node
Element
Triangular
(b) 2-D Element


Hexahedron
Element



(c) 3-D


Figure 4-1. Examples of the elements used in FEM (A) one-dimensional (B) two-
dimensional and (C) three-dimensional.

Approximate solutions for the PDE's are developed for each of these elements. The

equations for the individual elements must be linked together to characterize the entire

system. The total solution is obtained by combining the individual solutions. Continuity

of the solution must be ensured at the boundaries of each element. The value of unknown

parameters is generated continuously across the entire solution domain. After the

boundary conditions are applied, the solution is obtained using a variety of numerical









techniques. COMSOL Multiphysics is the software used to solve the partial differential

equations.

About COMSOL Multiphysics

COMSOL Multiphysics is an interactive environment for modeling and solving

scientific and engineering problems based on partial differential equations (PDEs). The

power of this program lies in its ability to couple several different physical phenomena

into one system and solve the PDEs simultaneously. The finite element method (FEM) is

used to solve the PDEs in two dimensions. FEM is a discretization of an original problem

using finite elements to describe the possible forms of an approximate solution. The

geometry of interest is meshed into units of a simple shape. In 2D, the shape of the mesh

elements is a triangle. After the mesh is created, approximations of the possible solutions

are introduced described by a function with a finite number of parameters, degrees of

freedom (DOF).

Model Geometry

The geometry used in the COMSOL solver is modeled after the cells used to make

the cakes as shown in chapter 2, Figure 2-4. A schematic of the geometry is shown in

Figure 4-2. The width of the cell is 0.026 meters and the height is 0.02 meters. The

bottom plate, labeled subdomain B, is made of aluminum and is the location of the heat

source. The walls, labeled subdomain C, are made of phenolic which is a insulating

material. The powder is contained in the center of the cell, labeled subdomain A. The cell

is assumed to be axisymmetric for the simplification of the model and reduced

computational time. There are 7272 triangular elements in the mesh as shown in Figure

4-3. The size of the mesh elements along the boundary are smaller than those in the

domain.















A
C
z


Centerline


Figure 4-2. Schematic of the geometry used in the COMSOL solver. The lettered areas
label the domains.


I

Figure 4-3. Caking cell geometry with mesh elements.
Partial Differential Equations
The process of moisture migration through a porous media from the atmosphere is

governed by transient differential equations of heat (convection-conduction), mass









transfer (convection-diffusion), adsorption/desorption, and continuity and momentum

(Brinkman's equation). To construct a mathematical model of the moisture migration, the

following assumptions are considered:

* Powder grains are isothermal and in equilibrium with surrounding gas. This can be
justified by the fact that grains are small and flow rate of gas is low.
* Heat generated or absorbed by adsorption/desorption is neglected.
* There is no mass (moisture) transfer from grain to grain through diffusion or
capillary bridges.
The partial differential equations used to describe the moisture migration are given in

detail below.

Convection-conduction

The convection-conduction equation is used to described the temperature

distribution within the system.


Pp c +V- -kVT+IhND) Q -pCi VT (4-11)


where ot, is a time scaling coefficient, p is the density, Cp is the heat capacity, k is the

thermal conductivity, Tis the temperature, Q is the heat source, u is the velocity, and

h,ND,, is a species diffusion term. The temperatures of the fluid and particles are assumed

to be in equilibrium. This can be justified by the fact that, for small particles, heat

diffuses by conduction almost instantly. Equation 4-11 is applied to every subdomain.

However, convection is not include in subdomains B and C. The velocity is determined

from the Brinkman equation. The time scaling coefficient is one for all subdomains. The

appropriate constants for each subdomain are specified.

Convection-diffusion

The convection-diffusion equation describes the mass transport in the system.


ac + V -(-DVc)= R -uii Vc (4-12)
at









where D is the diffusion coefficient, R is the reaction rate term and c is the concentration.

Since mass transfer only occurs in the bulk, this equation is only applied to subdomain A.

The reaction rate term is a source or sink term for the water vapor on the fluid phase

which is described by the change in solids moisture content with time. The time scaling

coefficient is one.

Brinkman Equation

It is believed that free convection plays a role in the moisture migration through the

system. Caking is known to be a time induced event. Thus if free convection aids the

moisture migration process, the cakes may develop in a shorter period of time. This

would indicate that free convection can play a major role in inducing cohesive storage

time effects. The Brinkman equation coupled with the continuity equation is used to

describe free convection in the system. The pressure distribution is calculated and

consequently the velocity distribution (free convection) within the system and at its

boundary. The Brinkman equation is a derivative of Darcy's law. Darcy's law describes

flow through a porous medium. However, the Brinkman equation accounts for the

viscous forces (Brinkman, 1947).


pi + = V. [ pl +(Vi +(Vi) )]+F (4-13)
at K

where u is the velocity, r is the dynamic viscosity, K is the permeability, p is the

pressure, I is an identity matrix and F is a volume force (-pg). The equation for the

permeability and the density as a function of temperature are given in Appendix B.

Solids moisture content

The solids moisture content is described by the linear driving force model. The

equation used by COMSOL is given by:









q +V.F=F (4-14)
at

where F is a source term and Fis a flux vector. The source term F is given in equation 4-

2. However the rate constant kg is determined from the kinetic data in chapter 3. The

dissolution of the material and the evaporation of the moisture vapor is described in this

model. Both adsorption and desorption must be considered because there is a hystersis in

the sorption curve; the adsorption and desorption are not equal. The equilibrium

conditions are not the same. The equations for the isotherm curves used in the model are

given in chapter 3. The relative humidity is calculated from the saturation concentration

as given in Table 4-1 and Appendix B.

Model Parameters

The experimental temperature and solid moisture content within the Johanson

Indicizer caking test are used as the initial and boundary conditions for the finite

element modeling. The boundaries are identified in Figures 4-2 and 4-4. The boundary

conditions, initial conditions and constants are listed in Tables 4-1 through 4-3.


7 8



6




Heated
2 4
Centerline
1


Figure 4-4. Boundary conditions.









Table 4-1. Parameters for COMSOL simulations.
Initial Pressure, 101325 Pa Gas Constant R fluid
p init (fluid), R fluid
Initial Temperature, 298.15 K Linear driving 0.0013
T init constant isothermm),
k ads
Porosity (e) 0.5 Linear driving 0.0013
constant isothermm),
k des
Particle Diameter, 0.001 m Antoine's 16.75667
P diam coefficient, a a
Viscosity (air), v 1.8e-5 Pa s Antoine's 4087.342
coefficient, b b
Permeability, K permea Antoine's -36.0551
coefficient, c c
Diffusivity, D 4.2e-6 m2/s Constants 0.47
isothermm), A
Heat Capacity 1500 J/kg K Constants 0.47
(solid), Cp, isothermm), A2
Heat Capacity 1005 J/kg K Constants 13.5
(fluid), Cf isothermm), Bl
Conductivity 0.2 W/m K Constants 13.5
(solid), k isothermm), B2
Density (solid), 2250 kg/m3 Temperature +15 K
dens s Fluctuation, T fluct
Density (bulk), p 1600.24 kg/m3 Initial 0.01
Concentration,
c init
Molecular Weight 18 Initial Saturated c sat mol init
(water), mol w Concentration
(moles),
c sat mol init
Molecular Weight 29 Initial Saturated c sat init
(air), mol a Concentration,
c sat init
Gravity, g 9.81 m/s2 Initial Relative RHinit
Humidity (bulk),
RH init
Gas Constant, 8.314 Initial Solids u init
R gas moisture content,
u init










The temperature profile imposed on boundary 2 is given in Figure 4-5. The

temperature profile is a sine wave and is maintained over a 24 hour time period. The

maximum temperature is 55C and the minimum temperature is 25C.


330


325


320


S315


310
I--
305


300


0 200 400 600 800
Time (min.)


1000 1200 1400 1600


Figure 4-5. Temperature profile imposed at the base of the cell in the finite element
simulations.

The initial temperature of the powder is 25 C and the initial temperature of the heat

source is 40C. The temperature surrounding the cell is maintained at 20C. There is flux

at the outer boundaries (5,8,7). Heat is conducted through the material and across the

inner boundaries (3,6).









Table 4-2. Boundary conditions for COMSOL simulations.
Convection- Convection- Brinkman General PDE
conduction diffusion Equation
1 Axial Axial Axial Axial
symmetry symmetry symmetry symmetry
2 TB
3 -Insulated No Slip Neumann
4 TB
5 hA(T-To)
6 -Insulated No Slip Neumann
7 hA(T-To) Insulated No Slip Neumann
8 hA(T-To)

Table 4-3. Subdomain conditions for COMSOL simulations.
Convection- Convection- Brinkman General PDE
conduction diffusion Equation
A Yes Yes Yes Yes
B Yes
C Yes -

The variables listed in Tables 4-1 and 4-3 are described in Appendix B.

The Role of Convection in Moisture Migration

Temperature Profiles

The current models for moisture migration available in literature assume that free

convection does not contribute to the heat transport in the system. Finite element

simulations are executed with and without convection to determine the significance of

convection. Temperature profiles along the centerline and near the insulated boundary (6)

are shown in Figures 4-6 and 4-7. The temperature profiles in these figures include

convection.












330
-- 0
325 G -- 30 sec
5 -A- 1 min
10 min
320 30 min
-*-1 hr
S-e- 4 hr
a 315


310
I-

305


300


295
0 0.005 0.01 0.015 0.02 0.025
z coordinate


Figure 4-6. Temperature profile along the centerline of the cell with convection included.


330
+0
-^ o
-30 sec
325 -*-1 min
10 min
-)K-30 min
320 -- 1 hr
-G-4 hr

S315


c- 310
I-

305


300


295
0 0.005 0.01 0.015 0.02 0.025
z coordinate


Figure 4-7. Temperature profile near the insulated boundary of the cell with convection.











It can be seen in Figure 4-6 that the temperature along the centerline increases until


the setpoint temperature is achieved. The setpoint temperature is obtained after only five


minutes. An oscillation of the temperature is seen after one minute. This is attributed to


the free convection plumes that initially develop within in the cell. These plumes are


shown in Figure 4-8. The plumes appear from zero to two minutes and then disappear. It


can be seen in Figure 4-7 that the temperature profile near the insulated boundary reaches


the setpoint temperature after thirty minutes. The oscillations in this figure are also


attributed to the formation of plumes. Temperature profiles for the case of no free


convection are shown in Figures 4-9 and 4-10. It can be seen that the setpoint temperature


is obtained after four hours. Comparing this result to the free convection included case


suggests that the free convection does play a significant role in the heat transport of the


system. If free convection is included, the cakes will develop in a shorter period of time.

Max: 313.033

312

310

1308


-306

-304

~302

300


Min 298.15


Figure 4-8. The temperature profile within the cell illustrating the convective plumes that
develop.







69



330
-- 0
-- 30 sec.
325 *- -A--1 min.
o o o o 10 min.

-30 min.
320 -1 hr.
S-e-4 hr.

) 315


c. 310
I-

305


300


295
0 0.005 0.01 0.015 0.02 0.025
z coordinate


Figure 4-9. Temperature profile along the centerline of the cell without convection.


330
-- 0
-- 30 sec.
325 -A- 1 min.
10 min.
-K-30 min.
320 --1 hr.
S-e-4 hr.

315


c. 310
I-

305


300


295
0 0.005 0.01 0.015 0.02 0.025
z coordinate


Figure 4-10. Temperature profile near the insulated boundary of the cell without
convection.










The temperature in the center of the cell is plotted in Figure 4-11. The difference

between the convection and non-convective profiles are shown. With the free convection

in the system, the response to a change in temperature occurs faster than conduction only.


330
-*-without convection
325 --with convection


320


2 315

E
S310


305


300


295
0 5 10 15 20 25
Time (hr.)


Figure 4-11. Temperature profile at the center of the cell, with and without convection.

The variation of material temperature as a function of time is most critical in the

region where shear takes place i.e. near the cell wall (insulated boundary). In this region

bonds between particles are being broken and the force required to break these bonds

determine the cohesive properties measured with this test cell. Temperature profiles in

this region also indicate that free convection speeds the caking process and results in

higher cell temperatures during heat-up (see Figure 4-12).










330


325 -*-without convection
-4-with convection
320


2 315


310
I-
305


300


295
0 5 10 15 20 25
Time (hr.)


Figure 4-12. Temperature profile at the cell wall, with and without convection.

An analysis of the cell Peclet number is also used to determine if convection

contributes significantly to the moisture migration through the cell in caking. The Peclet

number is the product of the Reynolds number and Prandtl number. The physical

interpretation is the ratio between the heat transfer by convection to the heat transfer by

conduction.


P pC, v(T- T)/I
Pe' = Cv(4-11)
k(T-To)/ 1

The Peclet number gives an idea of the dominant mode of heat transfer through the

cell. A high Peclet number means that the heat transfer from convection can not be

neglected and the heat transfer from conduction is not substantial. If the number is low,

then heat transfer from convection can be assumed negligible. When convection is added









to the model, the cell Peclet number is in the range of 1 to 5. This suggests that

convection is the dominant mode of heat transport.

Solids Moisture Content

The solids moisture content of the material is important because a change in this

quantity determines the extent of caking in the system. Adsorption of moisture and

increase in the solids moisture content specifies the amount of dissolved materials

available for the creation of solid bridges. The desorption of moisture, a decrease in the

solid moisture content, specifies the amount of crystallized solid material in the bridge.

Thus far, it has been determined that the convective system dominates the moisture

migration process. Therefore, the results shown include free convection in the system.

The solids moisture content within the cell is given in Figure 4-13. The moisture content

is plotted as a function of the position along the height of the cell at various times. It can

be seen that the material in the lower half of the cell does not undergo a change in

moisture content. Near the top of the cell the moisture content changes in time and

increases and decreases with the temperature. The moisture content follows the same

trend as the temperature profile with time shown in Figure 4-14. This figure shows a

cyclic moisture profile in the shear region of the test cell. The frequency of the moisture

cycle follows the temperature profile but lags the temperature profile. An increase in the

moisture content followed by a subsequent reduction is the mechanism behind solid

bridge formation suggests that the cake will develop in the region near the wall and will

be most pronounced at the top surface of the test cell. The regions of greatest caking

potential are areas with significant adsorption and desorption as a function of time.







73



0.46
-0

0.41 -3 hr
-6 hr
-9 hr
E 0.36 -
S-12 hr
o 15 hr
0.31
a -18 hr
.2 21 hr
S0.26 --24 hr

0
U) 0.21


0.16 -


0.11
0 0.005 0.01 0.015 0.02 0.025
z coordinate


Figure 4-13. Solids moisture content profile at the centerline of the cell with convection.


0.5

0.45

0.4

S0.35
0
o 0.3-

S0.25

0.2 -

UO 0.15

0.1

0.05

0 -
0 5 10 15 20 25
Time (hr.)


Figure 4-14. Solids moisture content as a function of time at the center of the cell.











This re-crystallization effect can best be seen by superimposing the temperature


profile computed within the shear region on the phase diagram for the sodium carbonate


system. Figure 4-15 shows the operating curve for the test cell. Material within the cell


initially starts at a solids concentration of 14%. The change in temperature and moisture


content causes material within the cell to cross a phase boundary. Increasing the


temperature causes the sodium carbonate to dissolve forming a solution of monohydrate.


Subsequently cooling the material causes the carbonate to cross the phase boundary and


form solids bonds between particles.


Unsaturated
Solution









Decahydrate

Solution

Ice + Decahydrate


Anhydrous + Solution


Monohydrate
+
Solution


Heptahydrate
+
Solution
/


Heptahy
+
Monohy


Heptahy
-De +
Decahyd


Temperature
profile





drate Change in

drate Moisture
Content

drate

Irate


0 10 20 30 40 50 60 70
Percent Sodium Carbonate


80 90


Figure 4-15. Phase diagram of sodium carbonate with temperature profile imposed.


' "









It is possible to approximate the cake strength if the change in solids moisture

content is known. The solids moisture content affects the unconfined yield strength by

influencing the radius of the crystallized bridge b formed between particles. A detailed

description of cake strength predictions and the factors that influence the strength is given

in the following chapter. Nonetheless, the approximate strength of the cake can be

determined from the variation in solids moisture content with time. The area of interest is

the moisture content near the insulated boundary since the shear plane in the Johanson

Indicizer is located in this region. The shear region is indicated by the hatched area in

Figure 4-16. The change in solids moisture content in this region dictates the strength of

the cake. The moisture content distribution near the insulated boundary is shown in

Figure 4-16. It can be seen that the moisture content varies along the height of the cell.

However, it does not change significantly within the radius of the shear plane. Because of

this insignificance, the unconfined yield strength is assumed to be constant in this region.

For larger variations in the moisture content within the shear plane, the averaged

unconfined yield strength must be considered.

Using the equations in chapter 5, it is determined from the moisture content profile

within the shear plane that the unconfined yield strength increases by 18% at a

consolidation stress of 10 kPa over the length of the caking event. Given the same

conditions, the yield strength data from chapter 2 indicates that the strength increases by

40%. Although the unconfined yield strength approximation from the moisture migration

analysis and the experimental data are not exact, these values have the same order of

magnitude. This comparison establishes a basis for estimating the unconfined yield











strength as a function of the solids moisture content calculated from an finite element


moisture migration analysis.


0.35
-r= 0.024
-r= 0.025
0.3 -r= 0.026




I 0.25






a)
0
0.2 -
0
CO

0.15




0.1
0 0.005 0.01 0.015 0.02
z coordinate


Figure 4-16. Equilibrium solids moisture content as various positions along the radius.















CHAPTER 5
EVALUATING THE CAKE STRENGTH OF GRANULAR MATERIAL

In this chapter a model for evaluating the cake strength of granular material is

developed. Although there are models available in literature for estimating cake strength,

it is shown that these models are often times inadequate predictors. All of the current

models suggest that the strength is influenced by a single variable. However, it is well

known that several factors affect the strength of cakes. Therefore a new model is

developed which includes the dependency on the particle size, moisture content,

consolidation stress, and other material properties.

Background

A useful tool for understanding the caking of granular materials is a model

predicting cake strength. This is not only useful for understanding the effects of the

influential properties of caking but it can also be used as a predictive measure for future

events. Rumpf (1958) was the first to propose a theory for the tensile strength of

agglomerates and many of the present models are based on his work. He developed

expressions for the strength of agglomerates with various types of interparticle bonds.

One mechanism for agglomeration is the formation of liquid bridges. Rumpf suggests

that the tensile strength of an agglomerate is proportional to the inverse of the particle

diameter squared.


t ( H (5-1)
t 7d2
p









In equation 5-1 a is the tensile strength of the agglomerate, E is the porosity of the cake,

dp is the particle diameter, ko is the coordination number (number of contacts per

particle), and H is the strength of the interparticle bond. The flaw of this equation is based

on the unrealistic assumption that the bridges fail simultaneously and that all the particles

are of equal size. Rumpf also proposed a theory for agglomeration due to the formation of

solid crystalline bridges. This equation is based on moisture content of the material and

the concentration of material in the bridge.


C q k-- (5-2)
Pk

In equation 5-2 yk is the concentration of the dissolved species k, q is the moisture

content of the particles before caking, p is the density of the particle, pk is the density of k

in the crystal bridge, and ok is the strength of a crystal bridge. It is assumed that by

random packing the mean volume fraction of particles is equal to the mean cross

sectional fraction of these particles. Thus, the dependence of the strength on particle size

is lost and the tensile strength is proportional to the volume fraction of crystalline

material in the bridge times the strength of the bridge. This assumption differs from

Rumpf s previous theory that states that the strength of an agglomerate is inversely

proportional to the diameter of the particle. Similar to the first equation, it is assumed that

all bridges fail simultaneously. Neglecting these critical parameters renders an inadequate

predictive model for the cake strength of granular materials.

Other researchers have used Rumpf s model to verify experimental data for various

materials. Pietsch (1969a) applies Rumpf s model to investigate the influence of drying

rate on the tensile strength of pellets bound by salt bridges. He measured the agglomerate

strength using a vertical tensile tester. The details of this tester are described in chapter 2.









Pietsch altered Rumpf s equation to include a mean tensile strength as given in the

equation 5-3.


", (1 )k^ (5-3)
Mppp.

where Ms and Mp are the mass of the salt in the dry agglomerate and the mass of the

agglomerate, respectively, p, and p are the densities of the salt and solid particles, and o,

is the average tensile strength of the bridges. Since the strength of the bridges varies with

changing crystal structure, an average tensile strength is used. He assumes that a crust,

consisting of solid bridges, is formed around the cake during the drying process due to

the crystallization of the salt solution. This crust changes the drying rate of the cake,

effectively changing the strength of the cake. The crust is removed before measurements

are taken for the strength of the material. Pietsch reports that the tensile strength of the

core agglomerate is highly influenced by the drying rate.

Tanaka (1978) further developed Rumpf s model by incorporating the structure of

the agglomerate. He included the effects of heat and mass transfer on the formation of the

solid bridges. Tanaka used a model of contacting spheres with pendular water as shown

in figure 5-1. The particles in the agglomerate are assumed to be monosized with a radius

of R. A fictitious sphere of radius r approximates the curvature of the bridge. The volume

of the bridge V and the narrowest width of the bridge b are derived as a function of R and

0. Hence V and b are related by the parameter 0 according to the following equation.

b/R= O.82(V /R3)25 (5-4)









Equation 5-4 implies that the cake fails at the narrowest width of the bridge (neck). The

volume Vbetween two particles is a function of the total volume of a single particle Vt

and the number of contact points per particle

V = 2V, k, (5-5)

In equation 5-5 ko is the coordination number. This number is approximated by Rumpf

(1958) as inversely proportional to the porosity E (ko ~ 1E).









\ /






Figure 5-1. Model of contacting spheres with pendular water used to calculate the volume
and width of the bridge.

Tanaka used the relationship between the volume and the width of the solid bridge

combined with Rumpf s model to form the following equation for the tensile strength of

powders.


c =0.17 (1 8 Ce(q/100)X1(1 x)} k (5-6)
e [3(1- J E

where o- is the tensile strength of the recrystallized solid bridge, Ce is the equilibrium

concentration, q is the initial moisture content, ec is the porosity of the recrystallized

bridge, and Xis a lumped parameter which is function of the temperature and humidity. It

is assumed that the solid bridges are formed from dissolved material due to heat and mass









transfer in the system. Equation 5-6 implies that the tensile strength is independent of

particle size and rather a function of the moisture content and equilibrium concentration

of the solid and the rate of mass transfer.

Thus far the equations for evaluating caking have been focused on the tensile

strength of the material as a function of particle size and moisture content. However, of

greater importance with regards to this research is how the unconfined yield strength is

affected by these properties. It can be shown that the tensile strength at is proportional to

the unconfined yield strengthfc from the construction of Mohr circles and a yield locus in

figure 5-2.

1 + sin (
f, = r, (5-7)
1 sin o

where q is the internal angle of friction.












tensile unconfined
strength yield strength


Figure 5-2. Mohr circles demonstrating the relationship between tensile strength and the
unconfined yield strength.

Tomas et al. (1982) formulated a model to investigate the unconfined yield strength

as a function of storage time using Rumpf s and Tanaka's model. The unconfined yield









strength is measured using the Jenike shear tester. The authors propose that the

unconfined yield strength o- varies with the moisture content q of the material as follows:

do- = D, [- dq(0)] (5-8)

The kinetics of the materials is assumed to follow the linear driving force model of

equation 5-9. This equation is integrated over the storage time to find the moisture

content of the material at a particular time:

=kA, (q- q) (5-9)
dt

Equation 5-10 relates the unconfined yield strength to the decrease in moisture content

over time through the combination of equations 5-8 and 5-9.

cr = D( )Ys (qo q, )[- exp(- kASt) (5-10)

In equation 5-10 o- is the unconfined yield strength, UDs is the compressive strength of the

solid bridge, Ys is the solubility, qo and qE are the initial and equilibrium moisture

contents, k is the mass transfer coefficient of water, As~ is the specific mass transfer area,

and t is the time. This model gave only slight agreement with experimental results.

However, the authors state that this model can be used to approximate the acceptable

moisture content as a function of storage time to avoid situations of caking.

The above models for evaluating cake strength have focused on temperature

induced caking. In other words a temperature perturbation is assumed to initiate a caking

event. However, it is has been observed that a change in relative humidity can also trigger

a caking event (Kun et al., 1998). The effect of humidity cycling has been studied by

Leaper et al. (2003). The authors make use of Tanaka's model to develop a relationship









between the cake strength and the number of humidity cycles. It is suggested that if the

porosity and particle size remain constant the tensile strength can be given as

= Kh N (5-11)

where Kh is a parameter that incorporates the crystal bridge strength, particle size,

porosity, and humidity swing and N is the number of humidity cycles. This model was

compared to experimental results from a simple compression tester adapted to control the

relative humidity of the sample. The authors found that the cake strength does indeed

vary with the number of humidity cycles. The parameter Kh in equation 5-11 was

calculated from experimental data to be 26 while the number of cycles Nis raised to the

power 0.546.

Most of the understanding of agglomerate breakage and all of the previous models

are based on the work of Rumpf More recently, the principles of fracture mechanics have

been used to evaluate the strength of agglomerates as an alternative approach to Rumpf s

theory. Kendall (1988) and Adams (1985) have proposed a fracture mechanics

description of agglomerate breakage. It is believed that internal flaws or cracks within the

material are responsible for the failure of the agglomerate. The authors describe

agglomerate strength in terms of fracture mechanics parameters and the size of the crack

which depend on the geometry and packing of the agglomerate.

Kendall (1988) states that the fracture mechanics approach to agglomerate strength

is based on three levels of magnification: particle-particle contacts, an assembly of

particles, and a block of material which behaves as an elastic solid. At the particle level

the particles are held together, without binder, by an interfacial energy The interfacial









energy can be determined from the size of the contact zone. From fracture mechanics

analysis, the diameter d of the contact zone is given by

d 2 1/3
d = (i v2) (5-12)
2E

where dp is the diameter of the elastic particles, E is Young's modulus and vis Poisson's

ratio. The Young's modulus of an assembly of particles differs from that of two particles

in contact. This is due to the elastic deformation of the assembly under stress. The

effective Young's modulus E* for an assembly of particles is given by

S 1/3
E*= 17.1(1 -)4 (5-13)
d

where E is the porosity. The assembly of particles also has an effective cleavage energy

R, .

1/3

E 2d

In equation 5-14 FT is the fracture energy. The agglomerate is treated as a continuous

medium and the fracture stress of is given by

ca = 0.893E*"R*('zV)1/2 (5-15)

where a is the length of the crack. Equation 5-15 applies to clean, smooth elastic spheres

and is verified with experimental data on alumina and titania agglomerates. The results

prove that this model is a better predictor of agglomerate strength as a function of particle

size when compared to the theory of Rumpf. However, Kendall's model fails to address

the inherent plasticity of most materials during fracture.









Two different approaches, Rumpf s theory and fracture mechanics principles, for

evaluating the strength of agglomerates have been discussed thus far and the

dissimilarities between the models are apparent. One major difference between the

models lies in Rumpf s derivation of his theory. He assumes that the particles are bound

together by interparticle forces and the addition of such forces yields the ultimate strength

of the agglomerate. Whereas the fracture mechanics view is that the agglomerate is an

elastic body that satisfies the Griffith energy criterion of fracture. Other distinctions

between to two theories include the functionality of particle size and porosity and the

assumption of Rumpf that the bridges fail simultaneously.

In this research, the principles of fracture mechanics are applied to evaluate cake

strength. The fracture stress of of the material is assessed to determine the unconfined

yield strength of the cake. The fracture stress is defined as the minimum amount of

energy needed to fracture the bonds of the cake. An essential factor lacking in the fracture

mechanics model as well as the models of Rumpf is the functionality of consolidation

stress on the strength of the material. As shown in chapter 2, there is a strong relationship

between the unconfined yield strength and the consolidation stress. Therefore, a new

modified fracture mechanics model is proposed for evaluating cake strength which

includes the structure of the bridge, particle size, moisture content and consolidation

stress.

Modified Fracture Mechanics Model for Evaluating Cake Strength

The principles of fracture mechanics have been applied to the field of particle

technology as an alternative approach to Rumpf s theory in determining the strength of

agglomerates. A theory has been developed to explain the failure of solids caused by









flaws or imperfections in the structure of the solid and the elastic and plastic deformation

of the material. The current models proposed by Kendall (1988) and Adams (1985) are

based on Linear Elastic Fracture Mechanics (LEFM). This concept is applicable to

materials with relatively low fracture resistance. Failure below their collapse strength is

common therefore these materials can be analyzed on the basis of LEFM (Broek, 1988).

The failure of very brittle materials can be described using LEFM, however most real

materials exhibit plastic deformation during failure. For this condition, Elastic-Plastic

Fracture Mechanics (EPFM) must be applied. The fracture parameters of many

crystalline materials, which are prone to cake, are sufficiently described using LEFM.

However the caking process alters the surface characteristics of these materials thus

affecting the mechanics of fracture. This change is caused by the creation of 'soft'

material in the contact zones due to the dissolution of the particle surface. The bridges

formed throughout the process of caking may completely solidify to create a brittle

structure. However, it is most probable to assume there is partial solidification of the

bridges creating a structure which will deform during fracture. Therefore EPFM will give

a better approximation to cake strength.

Before introducing EPFM it is useful to review LEFM. The fracture parameters in

both areas are directly related and the principles of fracture mechanics were originally

developed for linear elastic materials.

Linear-elastic Fracture Mechanics

LEFM is based on an energy balance in which the strain energy released at the

crack tip provides the driving force to create new surfaces (Griffith, 1920). In order for

fracture to occur, the rate at which energy is released in the solid must be equal to or

greater than the cleavage resistance Rc. Thus the elastic energy release rate of the crack G