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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 onvectionconduction ........................................ .......................... 61 Convectiondiffusion ..................................................... ...................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 Linearelastic Fracture M echanics .................................... ............... 86 Elasticplastic 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 StressStrain 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 31 Kinetics constants for sodium carbonate decahydrate ..........................................48 41 Parameters for COMSOL simulations. ....................................... ............... 64 42 Boundary conditions for COMSOL simulations.................................................66 43 Subdomain conditions for COMSOL simulations. ...............................................66 LIST OF FIGURES Figure page 11 Top down view of a silo wall with a thick layer of caked material attached ............3 21 Representation of the two limiting Mohr circles, showing the major principal stress of the steady state condition oa and the unconfined yield strength. .............11 22 Johanson Indicizer test cell.. ............................. ................................................ 15 23 Schematic of the standard Indicizer test cell.......................... ................... 16 24 Consolidation bench used for making the caked samples.................. .......... 18 25 Temperature profile used for preparing the cakes............................... ...............19 26 Schulze shear tester: (A) image and (B) schematic.......................................20 27 Schematic of the permeable Schulze cell..............................................................21 28 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 29 The yield strength as a function of consolidation pressure.. .................................24 210 The yield strength as a function of the mean particle size. .....................................25 211 Yield locus of sodium carbonate for air at 75% RH for 24 hr. and 0 hr with a norm al load of 16kP a. ......................... .... ................ ... .... .... ............... 26 212 The yield strength as a function of the number of temperature cycles...................27 213 Yield locus of sodium carbonate for air as a function of relative humidity with a norm al load of 4kPa. ............................................... ...............28 214 Unconfined yield strength as a function of relative humidity compared to the isotherm of sodium carbonate. ............................................................................ 29 31 The five types of adsorption isotherms in the classification of Brunauer ..............32 32 Mechanisms of mass transfer for absorbent particles .......................................... 33 33 Sorption of moisture by a water soluble particle................ ...............34 34 VTI Symmetrical Gravimetric Analyzer schematic..............................................36 35 User input screen for the VTI Sorption Analyzer. ................................................. 38 36 Phase diagram of sodium carbonate ................ ....... .............. ... ............... 40 37 Adsorption isotherm of sodium carbonate at 250C.. .......................................... 41 38 Sorption isotherm of sodium carbonate at 25C. .............................................. 43 39 Sorption isotherm of sodium carbonate monohydrate at various temperatures.......44 310 Sorption isotherm of sodium carbonate decahydrate at various temperatures.........45 311 The moisture uptake of sodium carbonate decahydrate as a function of time. ........46 312 The kinetics of sodium carbonate decahydrate at 50C. .....................................48 313 The kinetic rate constant for sodium carbonate decahydrate adsorption and desorption ............................................................................49 314 Sorption isotherm of sodium chloride at various temperatures..............................50 41 Examples of the elements used in FEM ............................................ .............58 42 Schematic of the geometry used in the COMSOL solver. The lettered areas label th e dom ain s. ....................................................... ................. 60 43 Caking cell geometry with mesh elements .......................... ..................60 44 B boundary conditions. ...................................................................... ...................63 45 Temperature profile imposed at the base of the cell in the finite element sim ulations. .......................................... ............................ 65 46 Temperature profile along the centerline of the cell with convection included.......67 47 Temperature profile near the insulated boundary of the cell with convection.........67 48 The temperature profile within the cell illustrating the convective plumes that dev elop .............................................................................68 49 Temperature profile along the centerline of the cell without convection ................69 410 Temperature profile near the insulated boundary of the cell without convection....69 411 Temperature profile at the center of the cell, with and without convection.............70 412 Temperature profile at the cell wall, with and without convection.......................71 413 Solids moisture content profile at the centerline of the cell with convection. .........73 414 Solids moisture content as a function of time at the center of the cell.................73 415 Phase diagram of sodium carbonate with temperature profile imposed. ................74 416 Equilibrium solids moisture content as various positions along the radius. ............76 51 Model of contacting spheres with pendular water used to calculate the volume and width of the bridge ................. ...... .. ......... ... .. ..............80 52 Mohr circles demonstrating the relationship between tensile strength and the unconfined yield strength. ............................................................. .....................81 53 Xray tomography slice of caked sodium carbonate.............................................90 54 Logarithmic plot of the smallest radius of the bridge b vs. the volume of the b rid g e V ........................................................................................ ..9 2 54 Schematic of an agglomerate showing the length of the crack a............................95 55 The stressstrain curve for a sodium carbonate tablet...................... ...............96 56 Stressstrain curve for lactose. ............................................................................ 97 57 Yield strength of sodium carbonate as a function of particle size. ........................99 58 The yield strength of sodium carbonate as a function of moisture content ..........100 59 Illustration of Hertz contact model............................ ....................... 102 510 The volume of a bridge using the Hertzian contact model ............. .....................103 511 Force chain structure in a DEM simulated powder bed with a high applied load (right) and a low applied load (left). ........................................... ............... 106 512 The cumulative distribution of the granularity at various consolidation loads......107 513 Granularity of the major force chains in a particle bed ............. ...............108 514 Unconfined yield strength as a function of the consolidation stress compared to the model of equation 551 ............... .... ...................... ....................... 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 onehalf 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 mattersolids, 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, dustfree 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 11. 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 11. 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 nonexistent. 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 21 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 xaxis 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 21. 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 timeinduced 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 stressstrain 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 xintercept of the yield locus as shown in Figure 21. 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 timeconsuming. 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 HangUp 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 HangUp 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 22 (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 23. 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 22. 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 21 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 (21) 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 23 represents this unique state of stress. The shear stress acting on the failure plane is r as shown in Figure 23. 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 22 and 23. SF Di Tf D (A) c (B) Normal Stress Figure 23. Schematic of the standard Indicizer test cell (A) and Mohr circle used for calculation of the unconfined yield strength (B). = f ( sin ) (22) 2 1 r= f cos ~ (23) 2 The volume of the material V, that is displaced from the tester is given by V, = 1H(D,2 +D, D +D12) (24) 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 (25) 4 sin 0 The dimensions of the test cell define the angle 0 of the shear zone. tan0 =D D (26) 2H The unconfined yield strength is calculated by substituting equations 22 through 26 into equation 21. 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= (27) 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 22 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 24. 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 24. 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 25, 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 25. 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 26. 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 26. 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 27. 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 27. 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 21 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 25. 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 28. 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 A10 kPa250 um *10 kPa 850 um 1 2 3 4 5 % Sodium Carbonate Decahydrate Figure 28. 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 28 indicates a nearly linear dependence of consolidation pressure on the yield strength of the cakes. In Figure 29 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 29. 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 210 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 210 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 210. 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 211. 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 23. 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 211. 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 daytonight 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 212, 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 noncontact 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 212. 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 213 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 14 0 0 1 2 3 4 5 6 Normal Stress (kPa) Figure 213. 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 214 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 214. 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 214. 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 31. Type I isotherm represent monomolecular adsorption of a gas and applies to porous materials with small pores. This is the wellknown 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 FloryHuggins 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 31. 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 32. 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 fluidparticle interface. Figure 33 is an illustration of interphase mass transport showing the sorption of moisture by a sorbent particle. 2 23 4 I 4 Figure 32. 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) (31) 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 31 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) (32) dt In equation 32 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 33. 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 (SGA100). The SGA100 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 34. 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 34. 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 35. 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 35. 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 36 (OCI Chemical website). Heptahydrate is a semistable 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 37. 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 36. Phase diagram of sodium carbonate. In Figure 37, 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 7080 %. 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 37. 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 pickup 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 37, 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 31. It can be seen in Figure 37 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, (33) In equation 33, 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=7.97e7 and B=0.0669. For the monohydrate A=6.3 8e7 and B=0.0723. The desorption curves for sodium carbonate are shown in Figure 38. 200 SDeca Adsorption 1 * *Deca Desorption AMono Adsorption ,*  * Mono Desorption Q) 100 O 50 0 50 . 100 0 20 40 60 80 100 % Relative Humidity Figure 38. 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 39 and 310 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 39. 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 310. 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 310. 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 311. 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 311. 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 32. If the reaction is first order, integration of equation 32 gives In q Ce =kt (34) The slope of the plot of ln(qce/qoce) as a function of time t is linear with a slope kg. In Figure 311 the concentration data is approximated by the linear driving force model. The slope, intercept and correlation coefficient are given in Table 31. 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 312. 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 31. 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 313. 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 313. 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 314. 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 ASalt25deg. 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 314. 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 (41) at 8z2 E p, at S= k (q q) (42) 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 (43) R2 Equation 41 is based on onedimensional 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 41 through 43. 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) (44) ot r Or r 2 O2) pa oz In equation 44 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 flowrate 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 (45) at pc9, ^r dr Or 2 OZ 2 pcS OZ pCps In equation 45 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 onedimensional finite difference model which does not account for convective heat transfer. Q= AA T +T and T = Qt T, (45) 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 (46) 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 46 and 47. RH m0o/lq tot = H + sold (48) 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,1H,' +H, (49) 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 48 is adjusted to include the equilibrium values. RHeq r msohd qeq qtot = ma eq Hw + (410) 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 41. The points of intersection of the lines are called nodes. Line Element (a) 1D Quadrilateral Node Element Triangular (b) 2D Element Hexahedron Element (c) 3D Figure 41. Examples of the elements used in FEM (A) onedimensional (B) two dimensional and (C) threedimensional. 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 24. A schematic of the geometry is shown in Figure 42. 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 43. The size of the mesh elements along the boundary are smaller than those in the domain. A C z Centerline Figure 42. Schematic of the geometry used in the COMSOL solver. The lettered areas label the domains. I Figure 43. 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 (convectionconduction), mass transfer (convectiondiffusion), 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. Convectionconduction The convectionconduction equation is used to described the temperature distribution within the system. Pp c +V kVT+IhND) Q pCi VT (411) 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 411 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. Convectiondiffusion The convectiondiffusion equation describes the mass transport in the system. ac + V (DVc)= R uii Vc (412) 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 (413) 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 (414) 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 41 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 42 and 44. The boundary conditions, initial conditions and constants are listed in Tables 41 through 43. 7 8 6 Heated 2 4 Centerline 1 Figure 44. Boundary conditions. Table 41. 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 1.8e5 Pa s Antoine's 4087.342 coefficient, b b Permeability, K permea Antoine's 36.0551 coefficient, c c Diffusivity, D 4.2e6 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 45. 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 45. 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 42. 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(TTo) 6 Insulated No Slip Neumann 7 hA(TTo) Insulated No Slip Neumann 8 hA(TTo) Table 43. 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 41 and 43 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 46 and 47. 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 Se 4 hr a 315 310 I 305 300 295 0 0.005 0.01 0.015 0.02 0.025 z coordinate Figure 46. Temperature profile along the centerline of the cell with convection included. 330 +0 ^ o 30 sec 325 *1 min 10 min )K30 min 320  1 hr G4 hr S315 c 310 I 305 300 295 0 0.005 0.01 0.015 0.02 0.025 z coordinate Figure 47. Temperature profile near the insulated boundary of the cell with convection. It can be seen in Figure 46 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 48. The plumes appear from zero to two minutes and then disappear. It can be seen in Figure 47 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 49 and 410. 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 48. The temperature profile within the cell illustrating the convective plumes that develop. 69 330  0  30 sec. 325 * A1 min. o o o o 10 min. 30 min. 320 1 hr. Se4 hr. ) 315 c. 310 I 305 300 295 0 0.005 0.01 0.015 0.02 0.025 z coordinate Figure 49. Temperature profile along the centerline of the cell without convection. 330  0  30 sec. 325 A 1 min. 10 min. K30 min. 320 1 hr. Se4 hr. 315 c. 310 I 305 300 295 0 0.005 0.01 0.015 0.02 0.025 z coordinate Figure 410. Temperature profile near the insulated boundary of the cell without convection. The temperature in the center of the cell is plotted in Figure 411. The difference between the convection and nonconvective 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 411. 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 heatup (see Figure 412). 330 325 *without convection 4with convection 320 2 315 310 I 305 300 295 0 5 10 15 20 25 Time (hr.) Figure 412. 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(411) k(TTo)/ 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 413. 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 414. 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  S12 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 413. 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 414. Solids moisture content as a function of time at the center of the cell. This recrystallization 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 415 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 415. 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 416. 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 416. 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 416. 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 (51) t 7d2 p In equation 51 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 (52) Pk In equation 52 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 53. ", (1 )k^ (53) 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 51. 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 (54) Equation 54 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, (55) In equation 55 ko is the coordination number. This number is approximated by Rumpf (1958) as inversely proportional to the porosity E (ko ~ 1E). \ / Figure 51. 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 (56) 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 56 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 52. 1 + sin ( f, = r, (57) 1 sin o where q is the internal angle of friction. tensile unconfined strength yield strength Figure 52. 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)] (58) The kinetics of the materials is assumed to follow the linear driving force model of equation 59. This equation is integrated over the storage time to find the moisture content of the material at a particular time: =kA, (q q) (59) dt Equation 510 relates the unconfined yield strength to the decrease in moisture content over time through the combination of equations 58 and 59. cr = D( )Ys (qo q, )[ exp( kASt) (510) In equation 510 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 (511) 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 511 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: particleparticle 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) (512) 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 (513) d where E is the porosity. The assembly of particles also has an effective cleavage energy R, . 1/3 E 2d In equation 514 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 (515) where a is the length of the crack. Equation 515 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, ElasticPlastic 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. Linearelastic 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 