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Numerical simulation of fiber separation in hydrocyclones Wang, Zheqiong 2002

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Numerical Simulation of Fiber Separation in Hydrocyclones By Zheqiong Wang B. Eng., Huazhong University of Science & Technology, China, 1996 M . Eng., Huazhong University of Science & Technology, China, 1999 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T OF THE REQUIREMENTS OF THE DEGREE OF M A S T E R OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES DEPARTMENT OF M E C H A N I C A L ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH C O L U M B I A May 2002 © Zheqiong Wang, 2002 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia 2324 Main Mall Vancouver, BC Canada V6T 1Z4 Date: May, 2002 Abstract Hydrocyclones are used in the pulp and paper industry to eliminate undesirable particles as well as for fiber fractionation. This current thesis is focused on modeling the performance of hydrocyclone, which can be used to predict and optimize the hydrocyclone design. The computational model developed in this study consists of two models. The flow model is a three-dimensional k-e turbulence model. The Navier-Stokes equations are solved in a curvilinear coordinate system. The Launder correction is used to model the turbulence in the highly swirling flow. Then the flow model is coupled with a Lagrangian tracking of solid particles representing the fibers. The fiber model allows for the motion in three dimensions. Fibers are constituted of ellipsoids and allow for the representation of flexible behavior. Interaction with the wall is implemented. Separation characteristics are investigated for different fiber properties and hydrocyclone design parameters. The predictions of the proposed model are compared with elaborate published experimental data sets. Good agreement is obtained between the model predictions and the experimental data. ii Table of Contents Abstract « List of Tables v List of Figures vi Acknowledgements • viii Chapter 1 Introduction 1 1.1 Motivation 1 1.2 What is a Hydrocyclone? 2 1.3 Principle of Operation 4 1.4 Some Secondary Flow Patterns 4 1.5 Particle Separation 5 1.6 Obj ective of the Research Work 6 Chapter 2 Literature Review 7 2.1 Overview of the Flow Field Study 7 2.2 Overview of the Fiber Motion Study 10 2.2.1 Rigid Fiber Models 10 2.2.2 Flexible Fiber Models i 11 2.2.3 Statie's Wet Fiber Model 13 Chapter 3 Numerical Simulations 16 3.1 Modified Turbulence Flow Model 16 3.1.1 Governing Equations 17 3.1.2 Boundary Conditions 18 3.2 Flexible Fiber Model 20 3.2.1 Definition of Fiber Flexibility 21 3.2.2 Dynamics 21 3.2.3 Wall Model 25 3.2.4 Random Walk in Fiber Model 26 3.3 Coupling the Fiber Model with the Flow Calculation 27 3.4 Contributions of this Thesis to the Computer Codes Used in this Research 29 iii Chapter 4 Results and Discussion 30 4.1 Results of Flow Model 30 4.1.1 Flow Field in Dabir's Hydrocyclone 30 4.1.2 Flow Field in Bauer's Hydrocyclone 35 4.2 Results of Fiber Model Coupled with Flow Calculation 40 4.2.1 Fiber Trajectory in Dabir's Hydrocyclone 40 4.2.2 Separation Performance in Bauer's Hydrocyclone 42 4.2.2.1 Calculation of Mean Coarseness 43 4.2.2.2 Prediction of Mean Coarseness 44 4.2.2.3 Prediction of Arithmetic Average Fiber Length 46 4.3 Investigation of Factors Affecting Fiber Separation 47 4.3.1 Influence of Hydrocyclone Geometrical Dimensions 47 4.3.1.1 Cone Angle 48 4.3.1.2 Cylindrical Chamber Length 48 4.3.1.3 Vortex Finder Length 50 4.3.1.4 Inlet Diameter 51 4.3.1.5 Downward Diameter 52 4.3.1.6 Upward Diameter 53 4.3.1.7 Main Diameter 54 4.3.2 Influence of Fiber Properties 55 4.3.2.1 Fiber Diameter & Fiber Density 55 4.3.2.2 Fiber Length 59 4.3.2.3 Fiber Flexibility 61 4.3.3 Turbulence Effect 63 Chapter 5 Conclusions and Recommendations 64 5.1 Conclusions 64 5.2 Recommendations for the Future Work 66 Nomenclature 68 References 71 iv List of Tables Table 4.1 Dabir's hydrocyclone dimensions and parameters 31 Table 4.2 Three sets of flow rates in Bauer's hydrocyclone 36 Table 4.3 Bauer's hydrocyclone dimensions and parameters 36 Table 4.4 Properties of nylon fibers used in the experiments 42 Table 4.5 Properties of fibers used in this investigation 48 Table 4.6 Dabir's hydrocyclone: turbulence effect for fibers with length 3.5mm and diameter 50.7pm 63 v List of Figures Figure 1.1 Diagram of a hydrocyclone : 2 Figure 1.2 Different types of hydrocyclones ...3 Figure 1.3 Minor flow patterns in a hydrocyclone 5 Figure 3.1 Boundary conditions in a hydrocyclone 19 Figure 3.2 Representation of fiber using linked rigid ellipsoids 22 Figure 3.3 Free-body diagram for spheroid i in a fiber 22 Figure 3.4 Diagram of fiber interaction with the wall 25 Figure 4.1 Dabir's Hydrocyclone 31 Figure 4.2 Dabir's Hydrocyclone: velocity distribution in the (z, r) plane through the middle of the inlet pipe 33 Figure 4.3 Dabir's Hydrocyclone: pressure contours in the (z, r) plane through the middle of the inlet pipe 33 Figure 4.4 Dabir's Hydrocyclone: axial velocity at different axial locations 34 Figure 4.5 Dabir's Hydrocyclone: tangential velocity at different axial locations.... 34 Figure 4.6 Bauer's Hydrocyclone 36 Figure 4.7 Bauer's Hydrocyclone: velocity distribution in the (z, r) plane through the middle of the inlet pipe 37 Figure 4.8 Bauer's Hydrocyclone: pressure contours in the (z, r) plane through the middle of the inlet pipe 37 Figure 4.9 Bauer's Hydrocyclone: axial velocity at different axial locations 38 Figure 4.10 Bauer's Hydrocyclone: tangential velocity at different axial locations.... 38 Figure 4.11 Flow reversal observed by Nuttal for swirling flow in a circular pipe 39 Figure 4.12 Typical fiber trajectories in Dabir's Hydrocyclone 41 vi Figure 4.13.1 Bauer's Hydrocyclone: comparison of mean coarseness in inlet and outlet streams 45 Figure 4.13.2 Bauer's Hydrocyclone: comparison of arithmetic average fiber length in outlet streams 45 Figure 4.14.1 Influence of cone angle on separation and fractionation 49 Figure 4.14.2 Influence of cylindrical chamber length on separation and fractionation 49 Figure 4.14.3 Influence of vortex finder length on separation and fractionation 50 Figure 4.14.4 Influence of feed diameter on separation and fractionation 51 Figure 4.14.5 Influence of downward diameter on separation and fractionation 52 Figure 4.14.6 Influence of upward diameter on separation and fractionation 53 Figure 4.14.7 Influence of main diameter on separation and fractionation 54 Figure 4.15.1 Influence of fiber diameter on separation for a fiber density of 1100kg/m3 56 Figure 4.15.2 Influence of fiber diameter on separation for a fiber density of 1400kg/m3 57 Figure 4.15.3 Influence of fiber diameter and fiber density on separation 57 Figure 4.16.1 Influence of fiber density on separation for Fiber A and Fiber B 58 Figure 4.16.2 Influence of fiber diameter on separation for Fiber A and Fiber B 59 Figure 4.17.1 Influence of fiber length, correlated with fiber diameter, on separation 60 Figure 4.17.2 Influence of fiber length on separation for Fiber A and Fiber B 60 Figure 4.18 Influence of fiber flexibility on separation for fibers with length 3.5mm and diameter 50.7pm 62 vii Acknowledgements I would like to express my gratitude to all those who have made this project possible. First of all, I would like to thank my supervisors, Dr. Martha Salcudean and Dr. Ian Gartshore, who have kindly offered me the opportunity to pursue this research project. They have given me not only their valuable suggestions and encouragement for my course study and research work, but also their concern and help for my living. I am also grateful to Dr. Paul Nowak and Dr. Jerry Yuan for their advice on flow model, my colleague Suqin Dong for her advice on fiber model and Dr. Emil Statie for discussions on the study of hydrocyclones. Many thanks to my colleagues, Xiaosi Feng, Jason Xun Zhang and Yaoguo Fan, who have been very friendly and supportive throughout the course of the project and especially in times of technical difficulties. Besides, I wish to acknowledge the financial assistance from FRBC Research Award and the useful experimental data provided by Dr. Branion. Most important is the consideration and support from my husband, Zhengbing Bian, without which I would have never been able to overcome the challenges and to get through this. viii Chapter 1 Introduction 1.1 Motivation A wide variety of paper is used in the world. Many properties of a paper product are determined by the characteristics of the fibers from which it is manufactured. The separation of pulp fibers with different properties is presently being achieved in industry using fiber fractionation — "the mechanical separation of fibers from a mixture to produce at least two fractions that have higher percentages of fibers with certain properties" [44] in each fraction. In pulp mills, pressure screens and centrifugal cleaners are the common devices for fiber fractionation. They can remove impurities such as dirt or plastic as well as shives from the slurry. By understanding the mechanisms of their operation, the separation of pulp fibers based on the differences in physical properties can be improved in these devices. Hydrocyclones were originally designed for use in the pulp and paper industry to remove high specific gravity debris from paper stock. Later, they were applied to remove undesired particles or classify different properties of materials in many other fields of industry. They emerge as an economical and effective alternative for classification. They are inexpensive, small relative to other separators, simple in design, easy to run, and have low maintenance cost [42]. They can be widely used to "clarify liquids, concentrate slurries, classify solids, wash solids, separate two immiscible liquids, degas liquids or sort solids according to density or shape" [42]. Each application of hydrocyclones has its particular requirements and goals, and changes in design and operation are needed to optimize each application. In particular, the separation of particles from the liquid in each hydrocyclone depends heavily on 1 particle properties. The strong dependence of their separation performance on particle properties and body geometry makes the design of hydrocyclones different for each application. Because of this, a numerical method needs to be developed to predict and optimize hydrocyclones for each application. The numerical method developed here is directly beneficial to the pulp and paper industry since it can be used to improve the quality of pulp and paper products. 1.2 What is a Hydrocyclone? Hydrocyclones are usually referred to as the centrifugal cleaners in the pulp and paper industry. A hydrocyclone is a device having no moving parts and the centrifugal forces generated by swirling fluid motion can separate solid particles from the suspending fluid [1,5,42]. VORTEX FINDER FEED INNER HELICAL FLOW ~ * « INWARD SHORT CIRCUIT BOUNDARY LAYER / ' ACROSS THE ROOF OUTER HELICAL FLOW AND BOUNDARY LAYER UNDERFLOW Figure 1.1 Diagram of a hydrocyclone [19] A hydrocyclone usually consists of a cylindrical section followed by a conical section, as shown in Figure 1.1. It also can consist of only a conical section. This conical 2 or cylindrical-conical section is usually called the vessel. A vortex finder, also called an overflow nozzle, is located on the central line of the vessel. It is "the pipe protruding through the top lid some length into the hydrocyclone body" [42]. There are two exits of a hydrocyclone: an upper exit from the vortex finder, which is called an overflow opening; and a bottom exit, which is called an underflow orifice or apex opening [1]. Accepts Accepts Accepts Accepts Rejects (b) Reverse hydrocyclones (c) Through-flow hydrocyclones Figure 1.2 Different types of hydrocyclones [1] 3 Three types of cleaners are currently used in the industry. Forward cleaners are used for the rejection of higher density particles. In this kind of cleaners, the overflow is called the accepts-flow while the underflow exit is called the rejects-flow. The opposite terminology is the case for a reverse cleaner which is used for the rejection of lower density particles such as plastics suspended in water [1, 19]. Reverse cleaners are slightly modified forward cleaners, i.e. with oversize apex opening, or slightly undersize vortex finders [1]. Besides the physical modifications, they are operated with much higher flow rates at the apex end, typically 40% -60% of the feed flow [1]. For a through-flow cleaner, both exits are from the apex. They "typically discharge their accept-flow coaxially with the light reject-flow at the apex end of the cleaner" [1]. These three types of cleaner are shown in Figure 1.2. 1.3 Principle of Operation As noted by Bliss [1] whose description is followed here, the operation of a hydrocyclone can be described as follows: the suspension is injected tangentially through a feed opening located near the top of the hydrocyclone. While the flow moves away from the inlet and toward the underflow exit along the inside wall of the vessel, it rotates around the central axis and its velocity increases as the cone diameter decreases. When the flow approaches the underflow orifice, the small outlet diameter prevents the discharge of some of the flow, so some flow rotates in a smaller-diameter inner vortex. These flows move away from the apex opening and finally leave the main body from overflow opening. The major flow pattern is shown in Figure 1.1. 1.4 Some Secondary Flow Patterns In addition to the major flow pattern which consists of "a spiral within another spiral moving in the same circular direction" [38], the shear conditions inside the hydrocyclone also produce secondary flow patterns, as shown in Figure 1.3. These flow patterns are significantly dependent on the shape and operating conditions of the hydrocyclone [1]. 4 Locus of zero ^ vertical velocity Air core Figure 1.3 Minor flow patterns in a hydrocyclone [5] Short circuit flow on the top of the hydrocyclone, some flow goes directly from the inlet, passes across the cyclone roof, moves down along the outside wall of the vortex finder and joins the upflow stream leaving out from the overflow opening. Its existence is due to the lower pressure regions near the cyclone walls. Eddy flow near the vortex finder, there are eddy flows. They exist in the form of recirculating eddies. The locus of zero vertical velocity "The existence of an outer region of downward flow and an inner region of upward flow necessitates a position at which there is no vertical velocity"[5]. The major eddy flow center is around this locus. The air core an area of low pressure in the center created by the rotation of the fluid often results in a rotating free liquid surface. If either or both outlets are connected to the atmosphere, the core is filled with air. 1.5 Particle Separation Under the effect of the centrifugal force field developed by the swirling fluid within the cyclone body, movement of solid particles relative to the fluid is created [19]. This relative motion between a particle and the suspending fluid depends on the particle 5 properties, the viscosity and the flow pattern of the suspending fluid. Therefore, in a suspension containing particles with different properties, some of the particles move towards the outer wall and leave through the underflow orifice, while others move to the center and go out through vortex finder. Thus hydrocyclones can be used to separate pulp suspensions into fractions having different properties. 1.6 Objective of the Research Work The first objective of the research work is to validate the proposed numerical prediction method against measured values to ensure that it is suitable for studying the performance and fiber separation of a hydrocyclone. Then the integrated model, which includes flow model and fiber model, can be used to study the influence of different fiber properties and geometrical hydrocyclone parameters on separation characteristics. The effects of fiber properties such as fiber density, diameter, length and flexibility are then investigated. In addition, the influence of different geometrical dimensions on separation and fractionation are considered. The developed model will allow the performance prediction for a given geometry and operating condition. Also, it will permit the design of alternative geometries under similar conditions for optimization purposes. This modeling tool will benefit users to maximize the value of pulp resources. 6 Chapter 2 Literature Review 2.1 Overview of the Flow Field Study Earlier work on determining the fluid flow patterns used various experimental techniques. Photographic or optical methods, Pitot tubes and Laser Doppler Anemometry (LDA) [8] were the common techniques for flow field measurements within hydrocyclones. LDA has been the preferred method in the last two decades. Its capability of high-speed data acquisition and no flow disturbances gain an obvious advantage over the other techniques [8]. Theoretical studies on different models were also carried out. The models can be divided into empirical and semi-empirical simulations, analytical solutions and numerical modeling [8]. Empirical models are based on correlations of the key parameters and fitting formulas to experimental data. Particle separation efficiency is estimated by equations relying on empirical formulas. The semi-empirical approach is focused on the prediction of the velocity field in the main flow using existing data. However, these models can only be used within the range of the experimental data from which the model parameters were determined. Because of this shortcoming, mathematical models, which are based on fluid mechanics and apply some version of the non-linear Navier-Stokes equations, are highly desirable [8,31]. The analytical model is a mathematical solution with various simplifying assumptions. Bloor et al. have pursued it for many years [2,3,4,]. They at first used spherical polar coordinates (r, 0, a) with the origin at the vertex of the cone, but later they recast their equations in cylindrical coordinates. No matter which kind of coordinates they use, they cleverly used stream function concept to mathematically solve the 7 conservation equations for mass and momentum under the assumption that the flow is incompressible, axisymmetric and fully inviscid. Their model successfully predicted the experimental data of Kelsall [23] and gave a result reasonably consistent with the measurements of Knowles et al. [26], but "some of their assumptions are rather simplistic" [8] and the contribution of some terms in the expression of the solution is open to discussion [8]. With the fast development of numerical methods and computer technology in recent years, computational fluid dynamics (CFD) becomes an efficient means to study the dynamics of many physical systems. Thus, numerical models using the power of CFD to predict turbulent flows in hydrocyclones have emerged in recent years. As Svarovsky [42] comments, it seems that the analytical flow models are being abandoned in favor of numerical simulations. For a hydrocyclone, the presence of high swirl and hence very large curvature of the streamlines make the conventional turbulence models unsuitable for modeling the fluid flow [18]. A large swirl in a flow makes the turbulence anisotropic, with effective viscosities different in the axial and radial directions [12,20,35,36,40]. For this reason, several models other than the conventional k-e two-equation turbulence model have been proposed. A commercial computer code, PHOENICS, was used by Rhodes et al. [35] to solve the required partial differential equations. The authors used a modified Prandtl mixing-length model with an axisymmetry assumption to account for the viscous momentum transfer effect. Hsieh and Rajamani [20] used a modified Prandtl mixing-length model with a stream function-vorticity form of the Navier-Stokes equations. For wide variations in hydrocyclone dimensions and operating conditions, the velocity field can be predicted quite accurately. But due to the inherent limitation of the axisymmetric assumption, the 8 separation efficiency curve can only be predicted for those hydrocyclones fitted with an axisymmetric tangential inlet tube which is not common in industrial hydrocyclones [31]. Dyakowski and Williams [12] used the conventional k-e model combined with appropriate equations for the normal components of Reynolds stresses to overcome the anisotropy of turbulent viscosity and the non-linear interaction between mean vorticity and mean strain rate. Malhotra et al. [29] have developed a new formulation of the turbulence dissipation equation based on the turbulence length scale, which they implemented in the TEACH code to predict the flow field in the hydrocyclone. Hsieh and Rajamani [20] mentioned that the key to success is choosing the appropriate turbulence model and numerical solution scheme. In the above models, as He et al [18] said, two points need to be noticed. For commercial hydrocyclones, the inlet conditions are clearly not axisymmetric. Besides, errors near the wall are produced unavoidably and the computation becomes more inaccurate when a three-dimensional treatment is applied by using a step-wise rectangular grid to represent the inclined sidewalls. In the last few years, efficient CFD codes have been developed at UBC [17, 32] to model the turbulent flow in an arbitrary complex geometry including the effects of large swirling components of the velocity. This code can be applied to the investigation of the flow field in hydrocyclones. It includes a fully three-dimensional modified k-e turbulence model with a cylindrical coordinate system and curvilinear grid for the calculation of flow fields. The grids can be made to represent exactly the hydrocyclone geometry and preserve the advantage of the cylindrical coordinates for rotational flow in hydrocyclones. 9 2.2 Overview of the Fiber Motion Study Understanding the motion of fibers in suspension plays a crucial role in many fields "ranging from reinforced composites to biotechnology" [44]. This includes the pulp and paper industry, where all fiber processing and papermaking is performed at high speeds in turbulent fluids. Determining the motion of a particle or particles in bounded and unbounded flows is a central problem in micro-hydrodynamics [25]. When a suspension of fibers is subjected to a turbulent flow field, the fibers rotate, translate, and deform. As Kim and Karrilla noted in their book [25], fiber motion dictates the evolution of the suspension microstructure and the microstructure in rum shapes the forces acting on the particles which induce further motion. These forces include the viscous resistance of the fluid, usually referred to as hydrodynamic drag. It is shape-dependent, so the particle trajectories are no longer described by a lumped parameter like the mass [25]. In the past years, the literature on the hydrodynamics of suspended particles and its applications has grown enormously. Only the previous theories that are relevant to the model being developed in this work will be reviewed here. 2.2.1 Rigid Fiber Models The earliest investigation into the behavior of fiber suspensions in a flow field is that of Jeffery [21]. He calculated the total force and moment exerted on a rigid, neutrally buoyant, ellipsoidal particle moving in a homogeneous Stokes flow in a Newtonian fluid. In a homogeneous Stokes flow, the drag force on the particle has a linear relationship with the relative velocity between the fluid and the particle. Jeffery's model showed that an isolated particle in a simple shear flow rotates in a periodic orbit and the center of the particle follows the fluid streamline. Classical expressions for the ellipsoidal particle's orbital motion were given in his model, in which the period is a function of aspect ratio and shear rate, and the orbit depends on the initial orientation of the ellipsoid relative to the shear plane. 10 Based on Jeffery's work, Bretherton [6] subsequently used the concept of an equivalent ellipsoidal aspect ratio (the ratio of the length of the ellipsoid to its maximum diameter), which related the shape of the particle to an ellipsoid, to describe the motion of any axisymmetric particle. Trevelyan et al [43] also showed that Jeffery's equation could be used to-describe the motion of a cylindrical particle by substituting an equivalent ellipsoidal aspect ratio for that appearing in Jeffery's equation. Consequently pulp fibers have been modeled as rigid spherical or cylindrical particles [15,19,41,45]. 2.2.2 Flexible Fiber Models Sometimes shear induces deformation of the fiber particle, which makes the behavior of the particle and the flow pattern around the particle complicated. The motion of a flexible pulp fiber could not be modeled by Jeffery's equation due to the fiber flexibility [30]. Forgacs and Mason [13,14,15] observed that a fiber could undergo four different complex rotational motions depending on its flexibility. With lower fiber flexibility, the fiber will undergo rigid rotation. The period and rate of the rotation can be described by Jeffery's equation. Springy rotation will occur when a compressive force acts on the fiber due to the fluid, and the fiber responds by bending if the force continues to increase. If the flexibility of the fiber is higher, the fiber will bend into an S-shape or coil up as it rotates. Forgacs and Mason [13,14,15] developed a theory for the onset of deformation in a cylindrical particle rotating in a shear flow, but they did not model it. In recent years, with the fast development of computers, computer simulations have been carried out which allow the dynamics of flexible particles to be modeled. Most models simulate the flexible fibers as chains of rigid bodies. Yamamoto et al [45,46] proposed a method to simulate the motion of arbitrarily shaped, deformable fibers by modeling the fiber as a chain of osculating rigid spheres connected through springs. By altering bond distance, bond angle, and torsion angle between spheres, the fiber model can be changed from rigid fiber model to flexible fiber model. 11 Based on the work of Yamamoto et al [45,46], Wherrett [44] modeled a fiber as a series of spherical elements. Translational and rotational equations of each element are developed according to the hydrodynamic force and torque exerted on the element. By solving these equations, the motion of the fiber can be determined. It was concluded that the rigid fiber motion in the model followed the theoretical results of Jeffery for rigid fibers. However, compared to Jeffery's theory for rigid fibers, the model overestimated the hydrodynamic force at lower aspect ratios and underestimated it at higher ratios [47]. A major problem was identified that the computation time was too long, and the model was limited to a two-dimensional motion which is not practical for industrial applications like hydrocyclones [47]. Similar to Yamamoto's method, Ross et al [37] proposed a particle-level simulation method for the structural evolution of flexible fiber suspensions. By modeling the fiber as a chain of elongated spheroids connected through ball and socket joints and by introducing the resistance in the joints, the dynamic behavior of both rigid and flexible fibers can be simulated. In this model, the need for iterative constraints to maintain fiber connectivity is eliminated and large aspect ratio fibers can be represented with relatively few bodies. These features help to reduce the computation time. When a fiber is located at finite distance from a solid wall, the boundary can have significant effects on the fiber motion. Burget [7] experimentally investigated fiber-wall interactions. In his investigation, several analytical and computational models were explored, which includes a cell model, a thin rod-circular cylinder model, and a translation model. From his experimental investigations, Jeffrey's equations were verified when the center of the fiber was located at a distance greater than a fiber length from the wall. In regions less than a fiber length and greater than a fiber diameter, the motion of the fiber can be described by Jeffery's equations if an effective shear rate is used. The effective shear rate increased logarithmically with decreasing separation distance. The wall effect was higher for longer aspect ratio fibers and was also a function of orientation. 12 For real industry applications, a fiber model which includes fiber-wall interaction is essential to represent the wall effects. Based on Ross and Klingenberg's model, Dong et al. [10] in the UBC research group developed an efficient fiber model to simulate fiber motion in screens. A wall model was implemented in their fiber model to account for the fiber-wall interaction. Higher order methods, Runge-Kutta and Hamming, were used in the model so that the time step can be adjusted and thus the computational time can be significantly reduced [47]. 2.2.3 Statie's Wet Fiber Model Statie et al. [41] in the research group at UBC developed a wet fiber model for hydrocyclones with certain assumptions. Since the calculations to be presented in this thesis will use some of their results as a basis of comparison, this model will be described briefly. In Statie's model, there are four forces acting on a particle in the rotational flow field, the drag force, the gravitational force, the centrifugal force and the Coriolis force. They are expressed as follows: FD=\piCDAU2s (2.1) Fz=(Pp-Pi)Vg (2-2) F r = ( p / - ^ - P l ^ ) V (2.3) y y d F e = - ( p p U - ^ - P l ^ ) V ' (2.4) r r where the subscripts p and 1 represent values for particle and liquid, respectively, V is the volume of the particle, g is the gravitational acceleration, A is an appropriate projected area of the particle, CD is the drag force coefficient, r is the distance of particle's center to the axis. 13 Under the assumption that the particle moves at its settling velocity Us, which is determined by the forces acting on the particle, they balance the drag force with the external forces which leads to where d is the volume equivalent diameter to a sphere. They calculate the slip velocity, which is the relative velocity between fluid and particle, as Us = Ui -Up. The particle is assumed to move at the same tangential velocity as the fluid and always keeps its long axis tangential to the flow direction. The particle is injected at the speed of the fluid at random positions inside the inflow area. The motion is then iteratively calculated until the particle moves out of the flow field [47]. In their study, both spherical and cylindrical particles were considered and the results have been compared with available experimental data. When they developed mathematical models of particles, CD, A , and V were different for the spherical and cylindrical particles, and the expressions for the drag coefficient were from equations available in published work. The computation of spherical particles can be simply accomplished from the above equations, once the size and density of the particle are known. For cylindrical particles, properties in the above equations are for a wet fiber, which must be correlated to the 'dry' properties measured through experiments. The authors assume that for the wet particle, the water completely replaces the air, so the particle has a liquid part. They also assume that some of the water is absorbed by the solid part and causes swelling, so that the particle has a solid-wet part. With these assumptions and an experimentally determined value of the volume-swelling factor Kv, (2.5) Co is a function of particle Reynolds number which is defined as: Re = M ^ (2.6) 14 they could develop the model by introducing shape factors. A volume equivalent diameter and surface area of the particle instead of the actual cylinder surface area and volume were used in their model. Their results showed that the accuracy of the model was improved by using drag expressions for cylindrical particles with smooth and complex surface shape. 15 Chapter 3 Numerical Simulations In this simulation study, the liquid is assumed to be pure water. It is assumed that the consistency of the suspension is very low, so there is no interaction between fibers. The presence of the fibers does not change the velocity field so that the velocities, once calculated, are unaffected by the fibers. This restricts the problem to low fiber concentrations. Under the assumption that the effect of the particle's motion on the flow is neglected, the modeling can be separated into two parts: the flow model and the fiber model. The flow model is used to study the liquid phase flow field to predict liquid velocities. The fiber model is used to study the particle motion within the predicted fluid flow. 3.1 Modified Turbulence Flow Model The flow model used in this study is a modified k-e model with the wall function treatment. Turbulence closure can be obtained in this model. The water is treated as an incompressible Newtonian fluid. The flow field can be simulated numerically by solving the three-dimensional incompressible Reynolds averaged Navier-Stokes equations. Nowak [32] in the UBC research group originally wrote a three-dimensional orthogonal coordinate-based CFD code for the standard k-e model using a finite volume method. Based on Nowak's work, He [16,17] in the UBC group developed a modified k-8 model proposed by Launder et al. [27] using curvilinear grids. After that, Nowak developed his own curvilinear code and modified his code to be capable of solving high swirl problem in the turbulence flow field. His current code with local segmentation capabilities can be used to handle the complex calculations for the flow field in many commercial industrial applications [47]. For the present hydrocyclone application, the code developed by Nowak was used to compute the flow field. 16 3.1.1 Governing Equations For many engineering applications in modeling the turbulent flow, the Reynolds-averaged Navier-Stokes equations together with a turbulence model are appropriate. The most commonly used turbulence, model is the conventional k-e model which has the following equations: The Continuity Equation: V u = 0 (3.1) The Momentum Equation: p u • Vu - V • {jueff'Vu) = - V p (3.2) The Kinetic Energy Equation: f u ^ puk--^-Vk = G-pe (3.3) The Dissipation Equation: V -f U A p U £ ~ — Vs CAG-C2P^-where: u — instantaneous fluid velocity vector p — modified pressure including the gravitational forces p — the flow density (assumed constant) k — kinetic energy of turbulence e — dissipation rate of turbulence kinetic energy G — the turbulence energy generation rate given by G = M, du; du. KDXJ dxj 3U; dx, (3.4) (3.5) where (ui, U2, U3) are the Cartesian mean velocity components and i , j = 1, 2, 3. peff —• the effective viscosity given by Meff = M, + Mi fjt is the laminar viscosity H, is the turbulent viscosity evaluated from the relation /j, = pCMk2 te, (3.6) 17 The usual values of the constants are: Ci = 1.44, C2= 1.92, C u= 0.09, a^= 1.0 and a E = K 2 / [ ( C 2 - C I ) C u 1 / 2], where K = 0.41 is the Von Karman constant. For strongly swirling flows such as the flow in a hydrocyclone, this standard k-s model is not appropriate. Launder et al. [27] proposed a modified k-e model for the prediction of anisotropic wall bounded turbulent flow with streamline curvature, in which the effect of the curvature on turbulence is controlled by a single empirical coefficient C c through the Richardson number Ri described as follows: C]/>y->Ci(l-C„RI,)/>y (3.7) s r or He et al [18] applied this modified model for the numerical simulation of hydrocyclones. Their numerical study indicated that by giving the value of 0.2 to the constant C c , most of the velocity distributions in a typical hydrocyclone flow field could be predicted accurately. 3.1.2 Boundary Conditions In this simulation study, we model the intersection between the inlet pipe and the cyclone as the flow inlet instead of simulating the inlet flow pipe itself, a practice which helps to decrease the computational time. There are five types of boundary conditions used, which include the inlet, outlet, axis, wall, and periodic condition, as illustrated in Figure 3.1. Inlet boundary conditions It is assumed that the flow in the inlet pipe is uniform and is parallel to the pipe axis, so uniform velocity boundary conditions are imposed at the feed opening, and the inlet velocity, which is perpendicular to the hydrocyclone axial direction, has zero axial velocity (Uz=0). The tangential and radial components of the inlet velocity can be calculated based on the inlet mass flow rate and the angle of the flow once the angles of 18 intersection at various intersecting points are determined analytically. The turbulence energy is calculated from kin=\.5 x (intensity x uinf > where the intensity is 0.05. The turbulence dissipation is calculated from E { n =C°M7Sk.n5llin, where li n is the turbulence length scale and is estimated to be half of the inlet pipe diameter. It is assumed that the values of ki n and Sjn are not important because the turbulence production inside the hydrocyclone is large enough to make the initial conditions unimportant. Outlet BC Inlet BC Q Outlet BC Figure 3.1 Boundary conditions in a hydrocyclone [ 18] Outlet boundary conditions At the top vortex finder and the bottom orifice exits, the axial velocity is prescribed to be uniform (Uz= Constant Value). Its value is determined from the outlet mass flow rates measured from the experiments; zero axial gradient conditions are applied for the tangential and radial velocity components at both exits (?H±=Q, ^HJL=Q). dz dz Axis boundary condition At the axis of the hydrocyclone, by virtue of symmetry, the tangential and radial velocities are zero (Ur=0, Ue=0), and zero radial gradient condition is applied to the axial velocity (f^.=o)-dr 19 If an air-core is observed by experiments, then at the air-core surface, the air/water interface is assumed impermeable and stress free, so impermeable and free-slip conditions are applied in this interface. The shape of the air-core is typically not modeled, so the radius of air-core is specified by an experimentally measured air-core size which is of constant radius, independent of axial position in the hydrocyclone. At the air/water interface, the following conditions are typically applied: the radial velocity is zero (U r=0), and the radial gradients of tangential and axial velocity are also set to zero ( 3 U* =p, dr ^ = o ) . dr . Wall boundary condition The wall of the hydrocyclone is impermeable and the no-slip condition applies. There is no flow through the solid boundary, so all the velocity components there are set to zero (Ur=U z =Ue=0) . Due to the characteristics of standard k-s model, a wall function treatment is applied near the wall. The wall shear stress, the turbulence kinetic energy k and its dissipation rate 8 can be calculated based on the wall function which is a function of the dimensionless distance y + from the wall [18]. Instead of using the no-slip wall condition, the velocity boundary at the wall is implemented by appropriately modifying the flux transport terms at the cell surfaces adjoining the boundary and taking the wall shear stress as an auxiliary force in the momentum equations [16]. Periodic boundary condition The axial-radial plane is chosen to partition the circumferential domain into a finite number of cells. Periodic boundary condition is applied to this plane, in which we assume that the flow at one end is connected with the other end to form an interior domain. 3.2 Flexible Fiber Model 20 3.2.1 Definition of Fiber Flexibility Wet fiber flexibility determines the ability of fibers to deform and entangle during the consolidation stage and is recognized as an important fundamental fiber property. It influences flocculation, drainage and retention characteristics, wet web strength and paper structure [28]. These properties affect the strength, surface and optical properties of the paper. Fiber flocculation plays a significant role in the behavior of pulp suspensions and paper formation. Experiments [28] showed that as the average fiber flexibility increases, fiber flocculation decreases because fibers will conform more easily to one another. Wet fiber flexibility is governed by the modulus of elasticity (material property) and its moment of inertia (geometric property). It is defined as: WFF = - = — (3.9) S EI 1 2 where WFF — wet fiber flexibility (N~ m") S — stiffness (Nm2) E — elastic modulus in bending (Nm") I — moment of inertia (m4) Since papermaking fibers are often damaged, the Effective Fiber Flexibility (EFF), defined as "the flexibility of a perfect fiber which deforms to the same extent as an imperfect fiber under the same loading condition", is often used as a measure of wood fiber flexibility [28]. 3.2.2 Dynamics The flexible model used in this thesis was based on a model proposed by Ross and Klingenberg [37]. Dong et al. [10] in the UBC research group adapted their model and wrote a code to predict the fiber motion in a predetermined flow field including the 21 effect of walls. The code written by Dong et al. [10] was modified for the present study of fiber fractionation in hydrocyclones. Figure 3.2 Representation of fiber using linked rigid ellipsoids [10, 37] According to the model of Ross et al. [37] which was further developed by Dong et al. [10], each fiber in the suspension is modeled by N rigid ellipsoids connected through N - l ball and socket joints, as showed in Figure 3.2. The three degrees of rotational freedom in each joint enable the model to bend and twist much like a real fiber. The configuration of the fiber in a fixed reference frame is determined by defining the positions and orientations of each ellipsoid. Cartesian vectors r, (i = 1,2, • • •, N) in a fixed reference frame define the positions. Euler parameters, which are a set of generalized orientation coordinates derived from Euler's theorem [37], define the orientations. Any orientation of body-fixed frame can be achieved by a rotation from the fixed reference frame about some unit vector. The motion of the fiber is determined by solving the translation and rotational equations of motion for each ellipsoid. They are derived from Newton's second law and the law of conservation of momentum. F,. • M ; Figure 3.3 Free-body diagram for spheroid i in a fiber [10, 37] From the free-body diagram Figure 3.3, for ellipsoid i in one fiber, we can get: 22 miri=¥i + fiSiaXi N Newton's second law: a (3.10) a=\ N The law of conservation of momentum: H ; = M , + ^Sia(cia x X a + Y a ) (3.11) Where mi, Yi, Hz — the mass, translational acceleration, and time rate of change of angular Xb, X c — the internal constraint forces in joint b and c respectively; Yb, Y c — the resultant internal torques in joint b and c respectively; Fj — the resultant external force acting through the center of mass. In a hydrocyclone, the centrifugal force is a dominant phenomenon. So F,- includes hydrodynamic force Fp^, interparticle force FJ-P^, body force F^ g^, and centrifugal force F ^ . Here we don't consider the interparticle forces. M , — the resultant external torque, which includes hydrodynamic torque, and torques produced by external moments of interparticle forces; Sia —the connectivity matrix which describes how the ellipsoids and joints are connected to one another; cja — a set of body-fixed connectivity vectors which is introduced to establish the relationship between the ellipsoid positions; If hydrodynamic interactions and fluid inertia are neglected, then the hydrodynamic forces and torques can be written as follows: momentum of ellipsoid /, respectively; (3.12) (3.13) 23 where U- 0 0 ^ is ambient fluid translational velocity, E ^ a n d ft^are the rate of strain tensor and vorticity respectively, and A , C a n d H-^ are resistance tensors. This model has been developed by Dong, whose description has been followed here. Modifications were added in the present work to account for the large centrifugal forces, which are present in the hydrocyclone flow field. Body force and centrifugal force can be written as follows: $g)=±xab2(pp-p,)g (3.14) F / , = ( ^ - , , ^ ^ ) - 4 w ( 3 , 5 ) In hydrocyclones, it is usually assumed that the fiber accelerates rapidly to its terminal velocity at which the forces acting on the fiber are balanced. It is also assumed in this study that the centrifugal force acts at the center of the fiber, no matter how many ellipsoids are linked to model the fiber. Based on these assumptions, Equation (3.10) and (3.11) can be reduced to F(*) + F/s> + F / c > + f ] $ f l X f l = 0 • (3.16) a=l MJ*> + f ; 5 t o ( c < 8 x X f l + Y f l ) = 0 (3.17) a=l Summing the Equation (3.16) for i=\ to N , the constraint forces cancel, so f ] F / * ) + F p ) + F / c ) = 0 (3.18) a=\ After mathematical manipulation of the above equations, the translational and rotational equations of motion for an ellipsoid can be deduced. They are similar to'those reported by Ross and Klingenberg [37]. The difference is that the centrifugal force is added here in the same way as the gravitational force. The expressions of some terms in the above equations, such as A ^ , CJ^ , etc., are provided by Kim et al. [25]. 24 This model is based on the theory of microhydrodynamics in which the Stokes equations are used to get the expression of the above resistance tensors. In the vicinity of the particle boundary, by considering the fixed density and viscosity of the fluid and the relative speed of the fluid, the assumption of Stokes flow is approximately valid. In the present study, the investigated particle size is between 10"6m and lO^m, the relative motion of the particle and the fluid calculated from the code ranges from 10"6m/s to 10"1 m/s. By using the properties of the water, the Reynolds number typically ranges from 10"4 to IO"1 and occasionally as high as 10 so that Stokes flow assumptions can apply. In general, it is assumed that the Reynolds number is low enough to be a good approximation when calculating the hydrodynamic force on the ellipsoid. This model does not use the concept of a drag coefficient. In this sense it is different from other models that use drag coefficient for the study of particle separations. A direct comparison of the present results with a model which does use the concept of a drag coefficient is made in a later chapter and from this comparison it is concluded that the present model produces results which are very similar to those which use drag coefficients. 3.2.3 Wall Model As indicated in Chapter 2, a fiber model without considering fiber-wall interactions would not be appropriate for most industrial applications. In the present work, the wall model of Dong is included. Figure 3.4 shows a diagram of fiber interaction with the wall. Figure 3.4 Diagram of fiber interaction with the wall [10] 25 Dong et al. [ 1 0 ] stated clearly in their developed fiber model how the wall model is implemented; this can be described as follows: when one ellipsoid in the fiber chain is close to the wall, the equation of the wall surface grid line predetermined from the flow model and the ellipsoid equation which describes the ellipsoid position and orientation can be solved to judge whether the ellipsoid touches the wall; if it does, a normal force (F) and a tangential friction force (T), which are created from the translation and rotation equation, are added to this ellipsoid to stop it going through the wall, where T = P w a l l F, fiwali is a wall friction coefficient. If two or more ellipsoids touch the wall, the reaction forces are added to each of those ellipsoids. 3.2.4 Random Walk in Fiber Model In a turbulent flow, the velocity of a fiber consists of a mean velocity associated with the mean flow field and a random velocity due to the fluctuating component of the turbulent flow. In this fiber model, besides the mean component of velocity and angular velocity, we are able to consider the fluctuating component as well. By assuming that it is homogeneous in the fluctuating components of turbulence flow, the random walk in fiber model is implemented as described by Dong et al [ 1 1 ] in their work: "the velocity fluctuation is randomly drawn from a Gaussian probability density distribution of zero mean and a standard deviation V 2 £ / 3 ; The angular velocity fluctuation is randomly drawn from a Gaussian probability density of zero mean and a standard deviation /(0.09k). The residence time of the fiber in the present eddy is determined by T = min(7i,L t IVel), where Tx is turbulence time scale kls , Lt is the (urd, vrel, wrel) is the relative velocity of the fiber with respect to the fluid. The fiber will stay in the same eddy until T is expired." approximate size of local size (C^15kl5)/s , where 26 3.3 Coupling the Fiber Model with the Flow Calculation When the fiber model is coupled with the calculated flow field, to track the motion of the flexible fiber, we assume that the flow imposes forces and moments which act through the center of mass of each ellipsoid, which permits different parts of the fiber subject to experience different forces. During the coupling procedure, the information on the flow field used by the flow model, such as computational grid points, boundary conditions, the velocity, strain tensor and vorticity vector of the flow at each grid point, will be stored as inputs to the fiber model. Each fiber's initial position, initial orientation, and its properties such as fiber length, fiber diameter, fiber flexibility will also be given before running the fiber model code. How many ellipsoids are used to model the fiber should also be prescribed before running the fiber model. The three-dimensional orientation of a fiber is determined by two angles: the azimuthal angle (j), which is the angle between the projection of the fiber axis on the x-y plane and the y-axis; and the polar angle 9, which is the angle between the fiber axis and the z-axis. They are limited to the range 0 to 180 degrees. In the fiber model, the translational and rotational equations of the fiber will be solved at time step t to determine in which cell each ellipsoid lies. At each ellipsoid's location, the accurate values of the flow variables, such as the velocity, strain and vorticity, are obtained by interpolation of the flow variable from the surrounding cells. The calculation continues at the next time step t + At , using the local kinematics information obtained from the preceding time step. In this way, the fiber's path can be tracked. The numerical method originally used by Dong [47] for the fiber motion was a first order Euler method using a constant time step. This resulted in a relatively long computational time, which is not suitable for real applications. Later Dong [47] 27 implemented higher order methods — Runge-Kutta and Hamming. These methods provide the ability to adjust the time step according to the local truncation error, and thus significantly reduce the computational time. For hydrocyclones, the expression of the centrifugal force includes the square of the fiber velocity. When the translational and rotational equations at time step t are derived for each ellipsoid of a fiber, as in the Ross and Klingenberg's model [37], the mathematical manipulation can be very complicated and the computational time will be long, sacrificing the advantage of the proposed model. If a very small time interval is used, the centrifugal force of the preceding time step can be added into the derived equations at the current time step given by Ross and Klingenberg. The error is assumed to be negligibly small using this procedure. When the coupling method is used to study the performance of hydrocyclones, the separation efficiency is often computed. In this thesis, it is assumed that a particle is either carried out through the vortex finder or through the apex opening, although experiments and predictions both show that particles can orbit for lengthy periods in a hydrocyclone without being accepted or rejected [19]. Based on this assumption, the separation efficiency of the hydrocyclone is evaluated in the present work by the carried-down percentage, which is the ratio of the number of fibers that move out of the field from the underflow orifice to the total number of fibers injected into the hydrocyclone. Held-up fibers, which orbit in the hydrocyclone without showing any tendency to move out, are treated either as accepted or as rejected. Between the carried-over number (number of fibers moving out of the overflow) and the carried-down number (the number of fibers moving out through the underflow), held-up number (the number of held-up fibers) is added to the smaller of the two. If the smaller number is then larger than the other one because of this addition, then this held-up number is important to both sides, and cannot be simply added to the smaller number as described. In this case, this held-up number is split in the ratio of the carried-over number to the carried-down number and 28 the two split values are added to their corresponding outflow numbers. This treatment is used only for convenience; there is no theory to support this procedure. In this way, the coupling of the fiber model and the flow field is implemented, and the separation performance in hydrocyclones can be predicted. 3.4 Contributions of this Thesis to the Computer Codes Used in this Research Paul Nowak in the UBC group wrote the overall CFD code for the flow model [32]. Suqin Dong in the UBC group wrote the code for the fiber model and the coupling procedure [10,11]. These codes can be used for many industrial applications. For the hydrocyclone application, the work of the present thesis included the following modifications to these existing codes: input the appropriate boundary data and generate appropriate cylindrical grids in the physical domain for hydrocyclones in the flow code; for the needed interpolation, transform the grid in physical space into the grid in the computational space for hydrocyclones in the flow code; add the centrifugal force into the flow code and write appropriate codes necessary for generating initial conditions for fibers entering hydrocyclones. 29 Chapter 4 Results and Discussion In the present study, two different hydrocyclone geometries are simulated to demonstrate the capabilities of the developed numerical method. One is Dabir's Hydrocyclone [9,18], the other one is Bauer's Hydrocyclone [19,49]. These two hydrocyclones are chosen because detailed experimental flow field data is reported for Dabir's Hydrocyclone, while experimental data of fiber fractionation is available for Bauer's Hydrocyclone. Throughout the whole study of hydrocyclone performance, one hundred particles are distributed at the entrance using a random function generator provided by the Fortran compiler. The inlet orientation of these particles is also generated a by a second random function generator. The numbers of fibers above one hundred were also tried and the predicted results showed variations of the carried-down percentage from 2% to 4%. This difference was considered acceptably low. Because the use of more particles requires more computational time to run the code, the conclusions reached in this thesis are all based on results obtained using one hundred particles. 4.1 Results of Flow Model 4.1.1 Flow Field in Dabir's Hydrocyclone The first hydrocyclone, studied by Dabir [9, 18], was operated with water only and no particles or air core were present in the original experiments. Since the method proposed here predicts the flow and fiber motion separately, the experimental flow field data of Dabir's hydrocyclone is suitable for the validation of the velocity prediction model. 30 An illustration of the Dabir's hydrocyclone is shown in Figure 4.1 with dimensions. The parameters and dimensions are listed in Table 4.1. Table 4.1 Dabir's hydrocyclone [9,18] dimensions and parameters Feed" Composition Value D (mm) 76.0 D 0 (mm) 21.3 Di (mm) 25.8 D2 (mm) 12.0 D3 (mm) 28.0 L (mm) 380.0 Li (mm) 50.9 L2(mm) 30.4 Feed (kg/min) 24.42 Overflow (kg/min) 19.54 Underflow (kg/min) 4.88 Rejects Figure 4.1 Dabir's Hydrocyclone [ 18] For present computational purposes, the entire domain is divided into three segments, i.e., segment 1 for the inside of the vortex finder, segment 2 for the outside of the vortex finder, and segment 3 for the main body of the hydrocyclone below the vortex finder. The grid contains 13*25*11, 12*25*16, 26*25*40 cells in segment 1, 2 and 3 respectively in (r, 0, z) coordinates. The grid in the circumferential direction has uniformly distributed nodes. In the other two directions, to smooth the procedure and to obtain more accurate values at corresponding nodes near the region, the mesh is gradually refined in the region close to the boundary, such as the walls, inlet boundary, axis etc., and also in the region close to the inter-segment. 31 Five types of boundary conditions are used: inlet, outlet, walls, axis, and periodic conditions. They are described in Chapter 3. The predicted velocity distribution for the Dabir's hydrocyclone in the (z, r) plane (axial-radial plane) through the center of the inlet pipe is shown in Figure 4.2 and the pressure contours in that plane is presented in Figure 4.3. Figure 4.4 and Figure 4.5 show the axial velocities and the tangential velocity predicted by the model at different axial locations. In figure 4.4, positive values of axial velocity show downward flow near the wall, which is the direction of the main flow, while negative values represent upward or reverse flow near the axis. The values are only shown from the center to the wall. The experimental values of the axial and tangential velocity at Z=0.18m are represented as small circles in figures 4.4 and 4.5. Z represents the distance downward from top wall of the hydrocyclone. From Figure 4.2 to Figure 4.5, it is apparent that: • The major and minor flow patterns described in Chapter 1 can be clearly seen, and the model predictions for the axial and tangential velocities are close to the experimental data. • The velocity distribution near the top section of the hydrocyclone is non-symmetrical. At a section below the vortex finder, the flow approaches axisymmetry. • In most sections of the hydrocyclone, the velocity distribution has a three-cell structure (two flow reversals). This agrees well with the experimental observations. One axial flow reversal takes place in the region not far from the wall so that some flow moving down along the wall is reversed and moves upward. Another reversal takes place near the axis of the hydrocyclone so that some flow near the axis moving upward may be reversed and moves down along the axis and out from the underflow. • Near the inlet region and the bottom of the vortex finder, there are some re-circulating flows. 32 Figure 4.2 Dabir's Hydrocyclone: velocity distribution in the (z, r) plane through the middle of the inlet pipe 7 98418.8 6 97787.5 5 97156.3 4 96525 3 95893 8 2 95262.5 1 94631.3 Figure 4.3 Dabir's Hydrocyclone: pressure contours in the (z, r) plane through the middle of the inlet pipe 33 34 • Near the top cover there are strong inward radial velocities directed towards the root of the vortex finder, so that some flow from the inlet moves radially to the overflow and exits. • The flow in the outer vortex moves down and that in the inner vortex moves up. There is a well-defined locus of zero vertical velocity between the two vortices which roughly follows the profile of the cyclone. • A strong pressure variation exists at the exits of the cyclone with the lowest pressure at the center. The maximum pressure occurs at the feed inlet and the minimum pressure occurs at the center of the underflow orifice. • As the radial distance from the axis increased, the tangential velocity gradually increases to its maximum value close to the axis and then decreases toward to the wall. The radius of the peak in the tangential velocity is smaller than the inside radius of the vortex finder. • For each curve in Figure 4.4 and 4.5, the upper limit of the radial distance from axis is not at the wall because the inherent characteristics of the k-e model implies that the value of the velocity at the wall is not given by the code to be zero, although it can be imposed in a subroutine for,the sake of plotting. The proposed method showed that the value of dimensionless distance y + locating the point nearest the wall falls into the range 10<y+<100. Given finer grids, the predicted results did not change appreciably so the solution obtained by this grid is believed to be essentially grid independent. 4.1.2 Flow Field in Bauer's Hydrocyclone A 3-in (76 mm) commercial Bauer Centri-Cleaner used for fiber fractionation is shown in Figure 4.6. Three sets of flow rates are listed in Table 4.2. The parameters and dimensions are listed in Table 4.3. Fiber separations have been reported at each of these three flow rates as will be described later. The measurements have been made and provided by Dr. Branion's group at UBC [49]. This Bauer's hydrocyclone is different from Dabir's hydrocyclone because the main body of the hydrocyclone is only conical, 35 not like the commercial cylindrical-conical hydrocyclone. One peculiar structural part of Bauer's hydrocyclone is the flared end of the vortex finder. Unfortunately, there is no flow field data for this hydrocyclone, so that the radius of air-core (if it existed) could not be specified in the flow model. It was reported by Sevilla et al. [19, 38] that there was no clear evidence that the air-core influenced the axial and radial velocity, and the presence of the air core only slightly influenced the pressure drop. It was therefore assumed that the separation efficiency would not be affected significantly by the absence of the air-core and in this study, the air-core was not included. Table 4.2 Three sets of flow rates in Bauer's hydrocyclone [19] Feed flow rate Rejects flow rate Accepts flow rate (Kg/min) (Kg/min) (Kg/min) Case 1 36.5 4.8 31.7 Case 2 49.5 5.8 43.7 Case 3 72.4 7.4 65.0 Table 4.3 Bauer's hydrocyclone [19, 49] dimensions and parameters Composition Value D (mm) 76.0 D 0 (mm) 12.0 D] (mm) 16.0 D2 (mm) 5.0 D 3 (mm) 24.0 D4(mm) . 31.0 L (mm) 810.0 Li (mm) 9.0 L2 (mm) 50.0 Feed Figure 4.6 Rejects Bauer's Hydrocyclone [49] 36 Figure 4.7 Bauer's Hydrocyclone: velocity distribution in the (z, r) plane through the middle of the inlet pipe 5 216650 4 206100 3 195550 2 185000 1 174450 Figure 4.8 Bauer's Hydrocyclone: pressure contours in the (z, r) plane through the middle of the inlet pipe 37 7 h 6 h (fi I r 5f- '/-• o o > 9> 3 t-c 2 1 h 0.01 - r -0.02 -4-0.03 Z=0.1m — -2-— Z=0.2m — -3—- Z=0.4m 4 Z=0.6m 5 - Z=0.7m i i i I i I I i I I I I I I 0 01 0.02 R(m) 0.03 8 - 7 6 5 - ] 4 - 3 - 2 - 1 0 Figure 4.10 Bauer's Hydrocyclone: tangential velocity at different axial locations 38 For the present CFD predictions, the entire domain of Bauer's hydrocyclone is divided into three segments, the same as for Dabir's hydrocyclone. The grid contains 8*30*24, 12*30*41, 24*30*42 cells in segment 1, 2 and 3 respectively in (r, 9, z) coordinates. The predicted velocity distribution and pressure for Case 1 are shown in the same manner as for Dabir's hydrocyclone in Figure 4.7 and Figure 4.8. Figures 4.9 and 4.10 show the comparison of the axial and tangential velocities at several locations for Case 1. There is no reported measurement of flow patterns for Bauer's hydrocyclone, but from the predicted velocity plot, Figure 4.7, the common major and minor flow patterns can still be clearly seen. Different from the velocity distribution in Dabir's hydrocyclone, the flare end of the vortex finder induces some recirculating eddies near it. In addition, the three-cell structure prevailing in Dabir's hydrocyclone does not prevail in Bauer's hydrocyclone. Away from the vortex finder, the structure of the velocity pattern in Bauer's hydrocyclone is two-cellular. In the lower sections of Bauer's hydrocyclone, the two-cell structure transits to a three-cell structure, which is due to the increase of swirl component of the flow. It can be seen in Figure 4.10 that the tangential velocity at Z=0.7m is higher than that at Z=0.1m and Z=0.2m. Nuttal [33] observed flow reversal for swirling flow in a circular pipe. He concluded that the single-cell structure usually occurs in a low swirl flow field, the two-cell structure in a medium swirl flow and the three-cell structure in a high swirl flow, as shown in Figure 4.11. One cell, Low swirl Two cell, Medium swirl Three cell, High swirl Figure 4.11 Flow reversal observed by Nuttal [9,33] for swirling flow in a circular pipe 39 4.2 Results of Fiber Model Coupled with Flow Calculation In this section, firstly we will report particle trajectories in Dabir's hydrocyclone to validate the coupling procedure used. Then we will use the experimental data of Bauer's hydrocyclone to see whether the centrifugal forces are correctly implemented in the code and whether the proposed numerical solution is suitable for studying the separation performance of hydrocyclones. 4.2.1 Fiber Trajectory in Dabir's Hydrocyclone For Dabir's hydrocyclone, a particle with a density of 1.1 times the fluid (water) density is represented by only a single rigid ellipsoid. Two representative trajectories (1) and (2) are shown in Figure 4.12.1 and Figure 4.12.2. Figure 4.12.0 is used to show the outline of the hydrocyclone. One representative trajectory of the held-up fiber is also shown in Figure 4.12.3. We can see that in Figure 4.12.2, the particle moves along the cyclone wall and proceeds toward the underflow first, then it reverses its direction and moves up to the vortex finder further away from the wall, and finally it moves out of the cyclone through the overflow exit. In Figure 4.12.1, the particle moves along the wall and toward the underflow exit until it moves out of the apex opening without reversing its direction. For the held-up fiber, we can see in Figure 4.12.3, that the fiber circles around the central axis up and down within a certain range in the vertical (axial) direction, the center of which is not far away from the bottom of the vortex finder. In any of the trajectories, fibers can rotate at all times, which is not shown in the figures. These different trajectories depend on the different initial positions and orientations of the fiber. The method shows that fibers with the same inlet position but different inlet orientation can have these different trajectories. Fibers with the same inlet orientation but different inlet position can also have different trajectories. This means that 40 the trajectory is a function of inlet position and inlet orientation and verifies that the inlet pipe has an important role in the design and performance of hydrocyclones. Figure 4.12 Typical fiber trajectories in Dabir's Hydrocyclone 41 4.2.2 Separation Performance in Bauer's Hydrocyclone A study of wood pulp fiber fractionation is complicated by the presence of fines and defects in the wood fibers caused by refining or other fiber processing operations [34]. To avoid these complexities and to simplify the description of fiber geometry, nylon fibers of known length and coarseness were used by Dr. Branion's group at UBC [19] in their experimental study of Bauer's hydrocyclone. There were no fines in these nylon fibers. Their properties are listed in Table 4.4. Table 4.4 Properties of nylon fibers used in the experiments [19] Density (kg/m3) Fiber Length (mm) Fiber Diameter (nm) Fiber Coarseness (mg/m) Crowding Factor* (N) Fiber 1 1145 1 13.75 0.17 9 Fiber 2 1145 1 27.50 0.68 2 Fiber 3 1145 3 13.75 0.17 79 *Based on feed suspension consistency, mean fiber length and mean coarseness. [19,24] Two cases are chosen to compare the predicted results to Ho's experimental data. The feed consistency suitable for the comparison of the model and the experimental data is chosen as the lower consistency of 0.3%. Case 1: Feed suspension consisted of a 50:50 (mass basis) mixture of Fiber 1 and Fiber 2. Case 2: Feed suspension consisted of a 50:50 (mass basis) mixture of Fiber 1 and Fiber 3. In the fiber model, for shorter fibers (L=lmm), one ellipsoid is used to model each fiber, while for the longer fibers (L=3mm), two ellipsoids are used to model the fiber, and the flexibility is chosen as 1012(N~'m"2). The reason for this difference is that modeling the fiber as several linked ellipsoids will take a longer computational time than when using one ellipsoid, and the wall model in the fiber model implies that using several ellipsoids instead of one would be better for longer fibers. The fiber model can only predict the hydrocyclone's separation efficiency by calculating the percentage of fibers exiting from each orifice, while the experimental 42 results of Ho only provide the mean coarseness and arithmetic average fiber length at the exits. It is therefore necessary to transform the predicted results to the same form as the experimental results. In the following section the definitions and the formulae used in Casel to transform the separation percentage to coarseness is described. The percentage efficiency can be easily transformed to an arithmetic average fiber length in a similar way. 4.2.2.1 Calculation of Mean Coarseness • Definition: Coarseness= total Mass/ total length • Concepts: In a hydrocyclone, the separation process does not change the coarseness and mean length of fibers because, unlike refining, there is no fiber development such as fiber breakage, delamination or peeling-off. • Formula: Summary: Fiber A: 0.17mg/m 1mm Fiber B: 0.68mg/m 1mm Mass of feed: M , I »~ Mass of Fiber A in feed: Ma=0.5M Mass Basis: 50:50, 1 Mass of Fiber B in feed: Mb=0.5M Assume: for Fiber A, the percentage of fibers moving out from underflow orifice is X , for Fiber B, the percentage of fibers moving put from underflow orifice is Y , At the rejects. Total mass of mixture = total mass of A + total mass of B : Ma * X + Mb * Y Total length of mixture = total length of A + total length of B, „ . . . , . Total mass of A Total length of A = -Coarseness of A Fiber A's coarseness will not change during the separation in hydrocyclone, so Ma*X Total length of A = 0.17 This can be explained in another way: 43 Total length of A = number of A x mean length of A Number of A at the rejects = number of A in feed * X , . . , . Total mass of A in feed Number of A in feed= Coarseness of A x mean length of A So: Ma Ma * X Total length of A = — * X x 1 = 0.17x1 0.17 „ . . . , . Ma*X Mb* Y Total length of mixture -^The coarseness of mixture at the rejects 0.17 0.68 Ma*X + Mb*Y Ma*X Mb*Y - + 0.17 0.68 Put Ma=0.5M and Mb=0.5M into the above equation, we can get: X +Y X Y - + -0.17 0.68 Similarly, we can find the coarseness of the mixture at the accepts to be: (i-X) + (i-Y) l -X l - Y 0.17 0.68 In this way, we can use the measured experimental coarseness values to validate the model. 4.2.2.2 Prediction of Mean Coarseness Figure 4.13.1 is a comparison of the predicted values to the experimental values for the feed, accepts and rejects coarseness at various feed flow rates for Casel. Held-up fibers are treated in the way described in section 3.3. 44 30 40 50 60 70 80. 1 i—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—11 0.75 E E 8 0.5 o> c 0) _ re o o 0.25 -a rejects - experimental -b rejects - model -ft accepts - experimental — —6—- — accepts - model Rejects Feed Accepts J I I I I I _ I 30 40 50 60 flow rate (kg/min) 70 0.75 0.5 0.25 80 Figure 4.13.1 Bauer's Hydrocyclone: comparison of mean coarseness in inlet and outlet streams „30 40 50 60 70 2.75 ~ 2.5 E E B 2.25 cn B 0) > < I 1.75 < 1.5 1.25 T— i—i—i— I—r T — i — I I I 3 0 I 1 1 1 1 I — A accepts - experimental • —B accepts - model —a rejects - experimental -b rejects - model Accepts • -B A - -*" "-Rejects _ ta • a _____ A to a 40 50 60 flow rate (kg/min) I I I I I I I l _ l L 80„ 2.75 2.5 2.25 2 1.75 1.5 1.25 70 80 Figure 4.13.2 Bauer's Hydrocyclone: comparison of arithmetic average fiber length in outlet streams 45 It can be seen in the figure that in all cases the rejects coarseness was greater than the feed coarseness and the accepts coarseness was less than the feed coarseness. In addition, as the flow rate increased, the coarseness value of both the rejects and accepts decreased. But accepts coarseness looks very nearly constant because experiments and predictions both show that almost all high coarseness fibers (C=0.68mg/m) are rejected. The numerical fiber model predicts that the coarser fibers are mostly in the rejects and the finer fibers are in the accepts. It also predicts that at high flow rates the accepts stream contained only low coarseness fibers (C=0.17mg/m) while the rejects stream contained both low (C=0.17mg/m) and high (C=0.68mg/m) coarseness fibers. This is consistent with experimental observations. The fiber model prediction agrees well with experimental values for the accepts coarseness, but at lower flowrates, there is a little difference for the rejects coarseness. That can be explained by flocculation. Flocculation is more likely to happen at lower flow rates. Higher flow rate introduces higher shear forces. Higher shear force is an advantage for classification because it breaks the agglomerates easily [42]. In the fiber model, no fiber-fiber interaction is considered. Thus, at lower flow rates, this assumption may cause some error. 4.2.2.3 Prediction of Arithmetic Average Fiber Length Figure 4.13.2 is the comparison of the predicted values to the experimental values for the arithmetic average fiber length at the feed, accepts and rejects at various feed flow rates for Case2. The arithmetic average fiber length is the mean length, that is, the total length of fibers divided by the number of fibers. It can be seen in the figure that in all cases the average fiber length in the rejects was lower than the average fiber length in the accepts. As the feed flow rate increased, the arithmetic average fiber length of the rejects and the accepts increased. 46 The fiber model predicts that the shorter fibers tend to go to the rejects while the longer fibers to the accepts. This agrees with the observations in the experimental studies. The fiber model prediction shows some differences compared to the experimental values. This is mainly due to the existence of the longer fiber (L=3mm) in the mixture, which increases the possibility of fiber-fiber interactions, an effect that is neglected in the fiber model. The effect of these longer fibers can be seen from a consideration of the crowding factor. The crowding factor is the number of fibers in a spherical volume of a diameter equal to the length of a fiber. It is a parameter that can be related to fiber flocculation [24]. The crowding factors for the three kinds of fibers used in this work were given in Table 4.4. According to the criterion of Kerekes and Schell [24], when the crowding factor is less than 30, flocculation between fibers can be negligible. We can see that in Table 4.4, the 1mm fibers (Fiber 1 and Fiber 2) have lower crowding factors, so for Casel, the error is smaller. But for Case 2, the 3 mm fibers (Fiber 3) have higher crowding factors and hence flocculation could be expected, producing a larger error. 4.3 Investigation of Factors Affecting Fiber Separation 4.3.1 Influence of Hydrocyclone Geometrical Dimensions In this section, the effects of several geometrical dimensions on separation and fiber fractionation of a hydrocyclone are investigated using the numerical method already described. The fiber is modeled as one ellipsoid. The base case is Dabir's hydrocyclone. When we change the geometrical dimensions, the flow rates at inlet and outlet exits are kept the same. From Figure 4.14.2 to Figure 4.14.7, the word 'relative' is used which means the ratio of the value in the current case to the value in the base case. 47 The separation occurring in the hydrocyclone is evaluated by the carried-down percentage, which is the ratio of the number of fibers carried downward to the total number of inlet fibers. The fractionation effect for two different fibers is evaluated by the difference between their carry-down percentages. The properties of these two fibers used in this comparison are presented in Table 4.5. These have been chosen in order to make a comparison with the results of Statie et al. [41] who had used a different fiber model (see section 2.2.3). As will be seen in later paragraphs, all the general conclusions of the results using the present model are in agreement with the corresponding conclusions reached by Statie et al [41]. Table 4.5 Properties of fibers used in this investigation [41] — - - - - - — T Density (kg/m ) Diameter (um) Length (mm) Fiber A 1050 48 3.1 Fiber B 1100 39 3.5 4.3.1.1 Cone Angle A l l geometrical dimensions of the hydrocyclone, other than the cone angle, were kept constant. The influence of cone angle on separation and fiber fractionation is shown in Figure 4.14.1. A decrease in the cone angle corresponds to an increase in the length of the conical chamber, therefore, increasing the effective length of the vortex. So as a general trend for separation, we can see from the figure that a decrease of cone angle increases the. number of particles carried downward, and separation efficiency is improved. Figure 4.14.1 also shows that cone angle influences separation for these two kinds of fibers to the same extent, so the angle of the cone has no significant influence on fiber fractionation which is evaluated by the difference of their separation efficiency. 4.3.1.2 Cylindrical Chamber Length A l l geometrical dimensions of the hydrocyclone were kept constant, except the length of the cylindrical chamber. As presented in Figure 4.14.2, the length of the 48 cylindrical chamber does not have an important influence either on separation or on fiber fractionation. e o •c E ra a. a co 100 90 F-80 -70 -60 fc-50 40 30 20 10 0 12 1 1 1 Fractionation Fiber A FiberB 17 100 -390 - 80 - 70 -160 c o o •s 40 E _i i i i_ •3 30 - 20 10 0 12 17 Cone Angle Figure 4.14.1 Influence of cone angle on separation and fractionation 100° 90 80 70 =S 60 c o 1 50 co I 40 30 20 10 1.5 -i 1 1 1 1 1 Fractionation Fiber A ~i 1 1 1 1 1 r Fiber B _L 100 90 80 70 60 c o 50 1 o tj 40 E 30 _= 20 - 10 .0 1 1.5 relative length of cylindrical chamber Figure 4.14.2 Influence of cylindrical chamber length on separation and fractionation 49 4.3.1.3 Vortex Finder Length A l l geometrical dimensions of the hydrocyclone were kept constant, except the length of the vortex finder. The influence of the length of the vortex finder on separation and fiber fractionation can be seen in Figure 4.14.3. io6°i 90 80 70 g 60 c o 1 50 ra I 4 0 .6 30 20 10 Ve 0.8 R T _ 1 1.2 1.4 1-§ Fractionation — — — FiberA Fiber B _L _L 100 90 80 70 60 _ o 50 g o o 40 E u. 0.8 1 1.2 1.4 relative length of vortex finder 30 20 10 Figure 4.14.3 Influence of vortex finder length on separation and fractionation The earliest hydrocyclones had no vortex finders and the overflow discharged through a simple hole in the cyclone cover. To reduce the bypassing of particles to the overflow via the short circuit flow under the top cover and to allow an opportunity for the re-entrainment of the particles in the short circuit flow, the vortex finder was introduced [5,42]. In order to avoid turbulence, the shortest of vortex finder analyzed in the study could not be less than the lowest end of the inlet opening (L 2 couldn't be less than D 0), which is the requirement of the design [5,42]. For the two chosen kinds of fibers, from the Figure 4.14.3, we see the carried-down percentage for base case is below 50%, so most of them reach the overflow from 50 the underflow by flow reversal. An extension of the vortex finder shortens the forced vortex in the cyclone body and allows less time for the re-entrainment of fibers and therefore decreases separation efficiency [5,42]. An increase in the length of the vortex finder slightly affects the fractionation efficiency. 4.3.1.4 Inlet Diameter A l l geometrical dimensions of the hydrocyclone were kept constant, except the inlet diameter. The influence of the inlet diameter on separation and fiber fractionation is presented in Figure 4.14.4. The size of the inlet opening plays an important role in the separation efficiency because it controls the inlet velocity and therefore to a great extent the tangential velocity inside the cyclone. The largest value of the inlet diameter analyzed in the study could not be more than the lowest end of the vortex finder (D 0 couldn't be more than L2) because an inlet opening located below the vortex finder would lead to higher turbulence. 51 A strong effect on separation can be observed in the figure. Even a small change in the inlet diameter, produces a large change in separation. According to the figure, smaller inlets lead to higher efficiency. The inlet diameter also has an important influence on fiber fractionation: a large inlet causes less fractionation, but the effect is not as strong as on separation. There may be a greater risk of flocculation occurring for small feed diameters and this should be considered in new designs. 4.3.1.5 Downward Diameter A l l other diameters and axial lengths of the hydrocyclone were kept constant, except the diameter of the underflow orifice. The change of this diameter corresponds to a small change of the cone angle. The influence of the change of the downward diameter on separation and fiber fractionation is presented in Figure 4.14.5. relative downward diameter Figure 4.14.5 Influence of downward diameter on separation and fractionation Generally, separation efficiency goes up with increasing size of the downward diameter. This is reasonable because the bigger the diameter, the greater the probability 52 that the fibers will move out. However, a very small underflow orifice may become plugged because of an increasing risk of flocculation. The underflow orifice size affects the separation efficiency continuously, but does not significantly influences the fractionation. 4.3.1.6 Upward Diameter A l l geometrical dimensions of the hydrocyclone were kept constant, except the diameter of the overflow exit. The influence of the size of the upward diameter on separation and fiber fractionation is shown in Figure 4.14.6. The overflow diameter is another important dimension affecting the hydrocyclone's performance. A decreasing vortex finder diameter leads to increasing separation efficiency, but it does not significantly influence the fractionation efficiency. According to Bradley [5], a small size of the vortex finder should be avoided because if the diameter is smaller than the radius of maximum tangential velocity, the particles in 53 the short circuit flow under the roof of the cyclone do not encounter strong centrifugal forces as they pass the bottom edge of the vortex finder so that they cannot be re-entrained into the main flow. Also the vortex finder diameter should not be greater than the locus of zero vertical velocity because this causes collapse of the normal patterns of inward radial flow [42]. 4.3.1.7 Main Diameter The cyclone main diameter is the diameter of the base of the cone. If a cylindrical section is used, the diameter is also the diameter of this cylinder. A l l geometrical dimensions of the hydrocydone were kept constant, except the diameter of the cylindrical section. The influence of the main diameter on separation and fiber fractionation is demonstrated in Figure 4.14.7. 10875 relative main diameter Figure 4.14.7 Influence of main diameter on separation and fractionation 54 The main diameter of the cylindrical part of the cyclone body is the primary design variable and all the other dimensions are usually related to it. Consistent with what was reported by other researchers, the separation efficiency increases with decreasing cyclone diameter. This is why in industry, multi-cyclone units are built consisting of hundreds of little cyclones in parallel manifolds. From the figure, for this conventional cylindrical-conical hydrocyclone and for these two fiber kinds, the effect of the cyclone diameter is somewhat less pronounced, probably because this cyclone is already small. Also, the main diameter affects fiber fractionation, but not remarkably. The increasing risks of fiber flocculation must be investigated for a large main diameter cyclone because of the associated low levels of radial and axial velocities in it. 4.3.2 Influence of Fiber Properties Fiber properties have profound effects on the performance of a hydrocyclone. This section will discuss the effects of fiber properties on separation and fiber fractionation in Dabir's hydrocyclone. The fiber is modeled as one ellipsoid unless otherwise specified. Two kinds of fibers (Fiber A and Fiber B) are again used in this section. Their properties are listed in Table 4.5. 4.3.2.1 Fiber Diameter & Fiber Density The diameter and density of particles have a very important influence on separation and fractionation in hydrocyclones. The density difference is necessary for separation to take place because effective buoyancy occurs in the strong centrifugal fields much more than occurs under normal gravity. The centrifugal force is also dependent on the particle volume which is based on the shape and size of the particle. The influence of particle density on separation, correlated with the shape and the particle diameter, is presented in Figure 4.15.1, 4.15.2, and Figure 4.15.3. The length of all particles is 1mm in these figures. Because particles have the same length and diameter, the volume of a particle in Statie's cylindrical fiber model is 1.5 times the 55 volume of the ellipsoid in the present rigid fiber model. The volume is therefore adjusted to make them consistent, which makes the elliptical shaped fiber model results directly comparable to Statie's model results. This adjustment is applied in the figures. Figure 4.15.1 is for particles with density 1.1 times the flow (water) density. Figure 4.15.2 is for particles with density 1.4 times the flow (water) density. Figure 4.15.3 combines Figure 4.15.1 and Figure 4.15.2 into one graph to make the influence of the particle density more evident. By applying the volume adjustment, it is clear that the present model results are very close to Statie's cylindrical fiber model results. diameter (m) Figure 4.15.1 Influence of fiber diameter on separation for a fiber density of 1100kg/m3 56 100 10"* 10" c o SB 2 ro a . s> CO 90 80 70 60 50 40 30 h 20 P 10 -i 1 1 — i — i— i— r r ( density=1400 kg/m3 ) • -A — cylindrical •HE— ellipsoidal (not adjusted) // / • -B — ellipsoidal (adjusted) / / / A / / / / i T ^ T ^ T — i — i t ' / / // / * / J. / / /// 7 / / J i i i_ 10" if? diameter (m) rH$ 100 90 SC 70 60 50 40 30 20 10 ,0 10 Figure 4.15.2 Influence of fiber diameter on separation for a fiber density of 1400kg/m3 100 90 80 70 60 o S ro I 4 0 30 20 10 0 10"6 -A-•-&• •-B-density is 1100 kg/m3 / / // i I I I ellipsoidal (not adjusted) 11400 k'g/m3^  / / ellipsoidal (adjusted) < 11 I I I T 1 1 1—i—i—i—r-f spherical — cylindrical ellipsoidal (not adjusted) ellipsoidal (adjusted) — cylindrical 1 d e n s i t y i s /// (density=14^ kcj7rn3 ) IDT 10"4L 10"' diameter (m) Figure 4.15.3 Influence of fiber diameter and fiber density on separation 57 From these three figures, we see that at the same diameter, in the order of cylindrical particle, ellipsoidal particle and spherical particle, the downward separation for them is decreasing. At the same diameter, for the same shape, the separation efficiency for particles with higher density is always greater than that for lower density. The number of downward separated particles increases dramatically with particle diameter. For particles with very small diameter, particle density and diameter have little influence on separation. The effects of fiber density and diameter for Fiber A and Fiber B are shown in Figure 4.16.1 and 4.16.2. The effects are evaluated by the carried-over percentage, which is the ratio of the number of fibers carried upward to the total inlet fibers. For separation, the two fibers are considered separately and for fractionation, two fibers are considered together. 1100 1200 1300 140-Poo 90 80 70 60 50 40 30 20 10 1100 1200 1300 140% fiber density (kg/m3) Figure 4.16.1 Influence of fiber density on separation for Fiber A and Fiber B 58 7 ° , ™ 100 90 80 70 60 50 40 30 20 10 40 50 60 70° fiber diameter (microns) Figure 4.16.2 Influence of fiber diameter on separation for Fiber A and Fiber B For values less than 1200 kg/m 3, particle density is one of the most important factors on separation and fractionation. For values greater than 1200 kg/m 3, the influence is small. Separation and fractionation are sensitive to particle diameter for the range of values investigated. The separation is more strongly affected by diameter in a small range of diameters for denser particles (Fiber B). 4.3.2.2 Fiber Length The influence of particle length on separation, correlated with particle diameter, is presented in Figure 4.17.1. The solid line is for particles with length 1mm, while the dashed line is for particles with length 6 mm. As we see in the figure, separation is not sensitive to particle length as much as to particle density and diameter. Figure 4.17.2 shows the results of this investigation for Fiber A and Fiber B . The solid line in Figure 4.17.2 shows results for the fiber modeled as two ellipsoids with very p — i — | — i — i — i — i — | — i — i — i i — | — i — i — i — i — q 59 small flexibility 109. The figure also shows separation is not sensitive to particle length as much as to particle density and diameter. diameter (m) Figure 4.17.1 Influence of fiber length, correlated with fiber diameter, on separation 100 90 80 70 60 50 40 30 20 10 0 i 1 r Fiber A Fiber B 1 -ellipsoid (FiberA) 2-ellipsoids (Fiber A) 1 -ellipsoid (Fiber B) 2-ellipsoids (Fiber B) J I I I I I i_ J I L 100 90 80 70 60 50 40 30 20 10 0 2 3 Fiber Length (mm) Figure 4.17.2 Influence of fiber length on separation for Fiber A and Fiber B 6 0 In Figure 4.17.2, for Fiber B, which has a higher aspect ratio than Fiber A, a greater sensitivity to the number of ellipsoids used becomes apparent. When Bradley [5] studied the influence of particle size and shape on hydrocyclone performance using theoretical efficiency relationships, he said "it is believed that particles which have one dimension very much greater than the other, such as needles, show a greater departure from theoretical performance in a cyclone due to the ease of misclassification". He explained that in a small volume of liquid in a hydrocyclone, there are liquid streams which flow in many directions, and the existence of shear in the liquid 'confuses' the long particle, one end of which is being propelled at a different velocity from the other. Therefore, long particles can be readily entrained in the wrong stream. As has already been described, fibers in a shear flow can have four distinct types of rotations [15,44], rigid rotation, springy rotation, flexible rotation, and complex rotation. For a longer fiber, a rigid fiber model may give incorrect results because the wall model implemented in the present fiber model can provide a difference when used with the one or two ellipsoid models. In the fiber model, the body-fixed reference frame is at the center of each ellipsoid. The distance of the body-fixed frame to the position where the fiber touches the wall is different between the one-ellipsoid model and the two-ellipsoids model, so the reaction force acting on each ellipsoid in the two models is different, causing the displacement of each ellipsoid and the relative change of centrifugal force to be different. The two-ellipsoids model result agrees with experimental observations in Bauer's hydrocylone. Unlike what is observed in Bauer's hydrocyclone [19], the tendency that longer fibers are likely to move upward is not evident here, probably because of the different hydrocyclones and fibers studied here. Ho himself noted in his paper [19] that some other workers reported that their hydrocyclones rejected long fibers because they used different hydrocyclones than the one used in his work for nylon fibers. 4.3.2.3 Fiber Flexibility The influence of fiber flexibility on separation, correlated with the number of ellipsoids used in the model, is presented in Figure 4.18. The fiber with length 3.5mm and diameter 50.7pm is used in this investigation. It is known that particles can orbit for 61 lengthy periods in a hydrocyclone without being accepted or rejected, so in this figure, the held-up percentage, which is the percentage of fibers exhibiting this held-up tendency, is included. By the definition of fiber flexibility described in Chapter 3, the range of fiber flexibility used in this investigation is from 109 to 1012 (N"'m"2) which are the lower and upper limit that the flexibility of most fibers would reach [28,48]. When we compare one-ellipsoid model results with two-ellipsoids model results, another adjustment needs to be made. For a fiber with only one ellipsoid having length (long axis) 2a and diameter (short axis) 2b, the volume is ^mb2; For a fiber with n ellipsoids, to keep the aspect ratio unchanged, each ellipsoid has length 2aIn and diameter 2b I n , and the volume of each ellipsoid is ^mb1 ir?, so that the volume of the fiber is n^nab2/^ , n2 times less than that with one ellipsoid. To make the result comparable, the volume of each ellipsoid has been adjusted by multiplying n . 109 Q. _ J 0) £ o -o 100 90 80 70 60 50 40 30 20 10 10" 1 1 — l l l l l 11 10" I I I 1 1 1 1 — 10s A down, 1-ellipsoid down, 2-ellipsoids — -E — down, 4-ellipsoids —a— heldup, 1-ellipsoid — to heldup, 2-ellipsoids — -_ — heldup, 4-ellipsoids down _. % held-up 101' -f 100 - \ 90 80 70 60 50 40 30 20 -4 1 0 1 1 1 1 1 1 1 J _ J I I 10 1 0 10" flexibility (1/NM**2) 101 Figure 4.18 Influence of fiber flexibility on separation for fibers with length 3.5mm and diameter 50.7pm 6 2 From the figure, we see that: • For several ellipsoids, the held-up percentage decreases as the flexibility increases. This can be explained that as the fiber flexibility increases, the fiber will conform more easily to the flow stream so that it will more easily move out either as accepts or as rejects. • The carried-down percentage in the 2-ellipsoids model is very close to that in 1-ellipsoid model, but at higher flexibility, held-up percentage is less in 2-ellipsoids model than that in 1-ellipsoid model, while at lower flexibility, they are very close. • For 4-ellipsoids model, as flexibility increases, carried-down percentage increases a little. When flexibility changes from 1011 to 1012, held-up percentage does not change. 4.3.3 Turbulence Effect The fiber with length 3.5mm and diameter 50.7pm is used this investigation. The results of the model with random walk and without random walk are presented in Table 4.6 for comparison in the form "carried-over (%): carried-down (%): held-up (%)". Table 4.6 Dabir's hydrocyclone: turbulence effect for fibers with length 3.5mm and diameter 50.7pm Without Random Walk With Random Walk One-ellipsoid model 43:45:12 54:22:24 Two-ellipsoids model with flexibility 109 42:45:13 62:15:23 Two-ellipsoids model with flexibility 1010 43:45:12 65:18:17 Two-ellipsoids model with flexibility 1011 45:45:10 68:15:17 Two-ellipsoids model with flexibility 1012 47:45:8 66:24:10 We see that for all cases, there are more held-up fibers, less carried-down fibers in the model with random walk. This is an unexpected result since the turbulence was thought to carry more fibers out of the flow, leaving fewer held-up in the hydrocyclone. The effect of turbulence in this model needs further investigation. 63 Chapter 5 Conclusions and Recommendations 5.1 Conclusions A numerical method has been developed for the three-dimensional modeling of hydrocyclones. A modified k-s turbulence flow model capable of predicting the hydrodynamic flow field has been used. A flexible fiber model coupled with the flow model has been developed to simulate fiber motion and to predict the separation performance in any hydrocyclone. The prediction results of the proposed model are compared with published experimental data sets. Good agreement is obtained between the model predictions and the experimental data with respect to both flow field and separation performance. The influences of the fiber properties and geometrical dimensions on separation and fiber fractionation of hydrocyclones are investigated. A summary of the most important conclusions from this investigation is described in the following paragraphs. In both Dabir's and Bauer's hydrocyclone, an outer vortex and an inner vortex, which is the common major flow patterns in hydrocyclones, and the common minor flow patterns in hydrocyclones, such as short circuit flow, recirculating eddies, and the locus of zero vertical velocity, can be clearly seen. The velocity near the top section of the hydrocyclones is not symmetrical, but at a section below the vortex finder, the flow approaches axisymmetry. As the radial distance from the axis decreases, the axial and tangential velocity gradually increases to maximum values close to the axis and then decreases again toward to the wall. The maximum pressure occurs at the feed inlet and the minimum pressure occurs at the center of the underflow orifice. In most sections of Dabir's hydrocyclone, the velocity vectors have a three-cell structure, while in Bauer's hydrocyclone, the structure of the velocity vectors is two-cellular. 64 In Bauer's hydrocyclone, from predicted results, it can be concluded that for a nylon fiber mixture of the same fiber length, but different coarseness, the rejects fibers were coarser than the accepts. As the flowrate increased, the coarseness values of the rejects decreased. For a nylon fiber mixture of the same coarseness, but different fiber lengths, the accepts fibers were significantly longer than the rejects fibers. Prediction also showed that short coarse fibers were preferentially rejected and long fine fibers were preferentially accepted. All of these are consistent with experimental observations. Based on the case for- Dabir's Hydrocyclone, from the investigation of factors affecting fiber separation, it can be concluded that: • For conventional cylindrical-conical hydrocyclones, a smaller angle of the conical section can improve the separation and fractionation performance of a hydrocyclone. The length of the cylindrical section has little influence on separation and fractionation. Longer vortex finders decrease the separation and fractionation efficiency slightly. Smaller inlets lead to higher separation and fractionation efficiency, but separation efficiency is much more affected by the inlet diameter than is the fractionation. An increase in downward diameter and a decrease in upward diameter can improve the separation performance and slightly improve fractionation efficiency. The smaller the diameter of the hydrocyclone, the higher the separation and fractionation efficiency, but this effect is not remarkable in the range of geometries investigated here. • The diameter and density of particles have a very important influence on separation and fractionation by hydrocyclones. For particles with length 1mm, the prediction shows that at the same diameter, for particles with higher density, the separation efficiency is always higher than that with lower density. The separation percentage increases dramatically with particle diameter. When the diameter decreases to very small value, particle density and diameter have little influence on separation. Separation is less sensitive to particle length than the particle density and diameter. Fiber flexibility is an important fundamental fiber property. For two-ellipsoids and four-ellipsoids models, as the flexibility increases, the held-up percentage decreases 65 and the carried-down percentage increases a little. When the flexibility is small, the separation percentage for all cases is very close to the rigid fiber model results, as expected. • When we consider the effect of turbulence on separation, the model predicts more held-up fibers and less carried-down fibers in the model with random walk. This effect needs further investigation, as held-up fibers were expected to decrease in number when the random walk model was added to the simulations. 5.2 Recommendations for the Future Work The current work has established a method to model the fiber separation performance in a hydrocyclone and to investigate important effects on separation resulting from changes in geometrical dimensions and fiber properties. However, the model still needs to be developed in order to increase its applicability as a design tool. Some recommendations that may improve the applicability of the model are as follows. Most of the cases of the fiber motion and separation were simulated in turbulent mean flow but without fluctuation effects. Although the turbulence effect in the model was briefly investigated, its effect is still not clear. The assumption that it is homogeneous in the fluctuating components of turbulence flow needs to be verified, perhaps by adding experimental parameters to the model. Currently, all fiber-fiber interactions are neglected, so the model is limited to very dilute suspension applications. At higher feed consistency, the effect of fiber-fiber interactions must be included because fibers seldom exist independently in suspension. Fibers contact other fibers and their hydrodynamic and other physical interaction cause the formation of fiber networks, which affects the flow field in return. To provide a realistic range of applicability of the model, we should consider fiber-fiber and fluid-fiber interactions. Solving simultaneously for the velocity field and the fiber orientation would provide a more accurate description of the flow and the fiber motion. 66 In this thesis the performance of only Dabir's and Bauer's hydrocyclone are investigated. Further comparisons with experimental data from different hydrocyclones would be very useful. 67 Nomenclature A a A . ( h ) j c . ( h ) ) g(A) b C C i , C2, C D Cia d E E F F F F D Fi F.(0 F i ( g ) . F L ( H ) F.(P) g G I k K v L L t M appropriate projected area of the particle short axis of ellipsoid resistance tensors long axis of ellipsoid coarseness constant values drag force coefficient body-fixed connectivity vector volume equivalent diameter to a sphere rate of ambient fluid strain tensor elastic modulus in bending (Nm"2) effective fiber flexibility normal force drag force external force acting at spheroid i centrifugal force acting at spheroid i body force acting at spheroid i hydrodynamic force acting at spheroid i interparticle force acting at spheroid i gravitational acceleration turbulence energy generation rate moment of inertia (m4) kinetic energy of turbulence volume-swelling factor fiber length approximate size of local size of eddy mass of feed 68 M A mass of Fiber A in feed M B mass of Fiber A in feed M ; external torque acting at spheroid i mi mass of spheroid i M i ( h ) hydrodynamic torque acting at spheroid i n number of ellipsoids p modified pressure including the gravitational forces qn, qii, q i i , q u Euler parameters r distance of particle's center to the axis. R c Reynolds number , ri position of spheroid i , measured from a fixed reference frame R i t Richardson number S stiffness (Nm2) S i a connectivity matrix T residence time of the fiber T tangential friction force t time step Ti turbulence time scale u instantaneous fluid velocity vector ue tangential velocity uei liquid's tangential velocity uep particle's tangential velocity Ui ( o o ) ambient fluid translational velocity U mean velocity Uri liquid's radial velocity Urp particle's radial velocity V volume of the particle V r e i relative velocity of the fiber with respect to the fluid WFF wet fiber flexibility (N" V 2 ) X b , X c internal constraint forces in joint b and c y + local Reynolds number 69 YB, YC Z internal torques in joint b and c distance from top wall of the hydrocyclone Greek Letters K Von Karman constant Q ( c o ) rate of ambient fluid vorticity P density e dissipation rate of turbulence dynamic viscosity 00i absolute angular velocity of spheroid i 0"k, d e constant Values P w a l l wall friction coefficient Subscripts 6 tangential direction A, B index of two kinds of fibers b,c index of joint D drag eff effective i index of spheroid in inlet 1 liquid P particle r radial direction rel relative s slip t turbulence z axial direction 70 References 1. Bliss, T., "Stock Cleaning Technology, A Literature Review", Pira International, Leatherhead, U.K., 1997. 2. Bloor, M . and Ingham, D., "The Flow in Industrial Cyclones". Journal of Fluid Mechanics, Vol. 178, pp. 507-519, 1987. 3. Bloor, M . and Ingham, D., "The Influence of Vorticity on the Efficiency of the Hydrocyclone". In 2nd International Conference on Hydrocyclones, BHRA, Bath, England, paper B2, pp. 19-21, 1984. 4. Bloor, M . and Ingham, D., "Theoretical Investigation of the Flow in a Conical Hydrocyclone", Transactions of the Institution of Chemical Engineers, Vol. 51, pp. 36-41, 1973. 5. Bradley, D., "The Hydrocyclone", Pergamon Press, London, 1965. 6. Bretherton, F.P., "The Motion of Rigid Particles in a Shear Flow at Low Reynolds Number", Journal of Fluid Mechanics, Vol. 14, pp. 284-304, 1962. 7. Burget, K . 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Statie, E . , Salcudean, M . , and Gartshore, I., "The Influence of Hydrocyclone Geometry on Separation and Fiber Classification", Filtration and Separation, pp. 36-41,2001. 42. Svarousky, L . , "Hydrocyclones", Technomic Publishing Company Inc., 1984 43. Trevelyan, B.J . , and Mason, S.G., "Particle Motions in Sheared Suspensions I. Rotations", Journal of Colloid Science, Vol.6, pp.354-367, 1954. 44. Wherrett, G . , Gartshore, I., Salcudean, M . and Olson, J . , " A Numerical Model of Fiber Motion in Shear", The 1997 ASME Fluids Engineering Division Summer Meeting, FEDSM'97, June 22-26, 1997. 74 45. Yamamoto, S., and Matsuoka, T., "A Method for Dynamic Simulation of Rigid and Flexible fibers in a Flow Field", Journal of Chemical Physics, Vol. 98, No. 1, pp. 644-650, 1993. 46. Yamamoto, S., and Matsuoka, T., "Viscosity of Dilute Suspensions of Rodlike Particles: A Numerical Simulation Method", Journal of Chemical Physics, Vol. 100, No. 4, pp. 3317-3324, 1994. 47. FRBC Year Progress Report I, II, HI, FRBC Research Award PA97422-IRE, submitted by Dr. Salcudean, M . , and Dr. Gartshore, I., Department of Mecahnical Engineering, University of British Columbia, Vancouver, B.C., 1999, 2000, 2001. 48. http://www.kcl.fi/scico/pakka.html 49. Private Communication with Dr. Branion, R.M.R, Department of Chemical Engineering, University of British Columbia, Vancouver, B.C., 2001. 75 

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