Numerical Simulation of Fiber Separation in Hydrocyclones

Transcription

1 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 FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 2002 Zheqiong Wang, 2002

2 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

3 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

4 Table of Contents Abstract «List of Tables List of Figures Acknowledgements viii v vi Chapter 1 Introduction Motivation What is a Hydrocyclone? Principle of Operation Some Secondary Flow Patterns Particle Separation Obj ective of the Research Work 6 Chapter 2 Literature Review Overview of the Flow Field Study Overview of the Fiber Motion Study Rigid Fiber Models Flexible Fiber Models i Statie's Wet Fiber Model 13 Chapter 3 Numerical Simulations Modified Turbulence Flow Model Governing Equations Boundary Conditions Flexible Fiber Model Definition of Fiber Flexibility Dynamics Wall Model Random Walk in Fiber Model Coupling the Fiber Model with the Flow Calculation Contributions of this Thesis to the Computer Codes Used in this Research 29 iii

5 Chapter 4 Results and Discussion Results of Flow Model Flow Field in Dabir's Hydrocyclone Flow Field in Bauer's Hydrocyclone Results of Fiber Model Coupled with Flow Calculation Fiber Trajectory in Dabir's Hydrocyclone Separation Performance in Bauer's Hydrocyclone Calculation of Mean Coarseness Prediction of Mean Coarseness Prediction of Arithmetic Average Fiber Length Investigation of Factors Affecting Fiber Separation Influence of Hydrocyclone Geometrical Dimensions Cone Angle Cylindrical Chamber Length Vortex Finder Length Inlet Diameter Downward Diameter Upward Diameter Main Diameter Influence of Fiber Properties Fiber Diameter & Fiber Density Fiber Length Fiber Flexibility Turbulence Effect 63 Chapter 5 Conclusions and Recommendations Conclusions Recommendations for the Future Work 66 Nomenclature 68 References 71 iv

6 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

7 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 Figure 4.3 Dabir's Hydrocyclone: velocity distribution in the (z, r) plane through the middle of the inlet pipe 33 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 Figure 4.6 Bauer's Hydrocyclone 36 Figure 4.7 Figure 4.8 Bauer's Hydrocyclone: velocity distribution in the (z, r) plane through the middle of the inlet pipe 37 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 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

8 Figure Bauer's Hydrocyclone: comparison of mean coarseness in inlet and outlet streams 45 Figure Bauer's Hydrocyclone: comparison of arithmetic average fiber length in outlet streams 45 Figure Influence of cone angle on separation and fractionation 49 Figure Influence of cylindrical chamber length on separation and fractionation 49 Figure Influence of vortex finder length on separation and fractionation 50 Figure Influence of feed diameter on separation and fractionation 51 Figure Influence of downward diameter on separation and fractionation 52 Figure Influence of upward diameter on separation and fractionation 53 Figure Influence of main diameter on separation and fractionation 54 Figure Influence of fiber diameter on separation for a fiber density of 1100kg/m 3 56 Figure Influence of fiber diameter on separation for a fiber density of 1400kg/m 3 57 Figure Influence offiberdiameter and fiber density on separation 57 Figure Influence of fiber density on separation for Fiber A and Fiber B 58 Figure Influence of fiber diameter on separation for Fiber A and Fiber B 59 Figure Influence of fiber length, correlated with fiber diameter, on separation 60 Figure 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

9 Acknowledgements possible. I would like to express my gratitude to all those who have made this project 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

10 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

11 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 INWARD SHORT CIRCUIT BOUNDARY LAYER /' ACROSS THE ROOF FEED INNER HELICAL FLOW ~*«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

12 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

13 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 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 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

14 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

15 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

16 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

17 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

18 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

19 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 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

20 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] 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

21 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

22 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] 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: F D =\pic D AU 2 s (2.1) Fz=(P p -Pi)Vg (2-2) F r =(p/-^- P l^)v (2.3) y y d U F e =-(p p -^- 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

23 Under the assumption that the particle moves at its settling velocity U s, which is determined by the forces acting on the particle, they balance the drag force with the external forces which leads to (2.5) Co is a function of particle Reynolds number which is defined as: Re = M ^ (2.6) 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 U s = Ui -U p. 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 K v, 14

24 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

25 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

26 3.1.1 Governing Equations For many engineering applications in modeling the turbulent flow, the Reynoldsaveraged 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: Vu = 0 (3.1) The Momentum Equation: p u Vu - V {ju eff 'Vu) = -Vp (3.2) The Kinetic Energy Equation: f u ^ puk--^-vk = G-pe (3.3) The Dissipation Equation: V- f U p U ~ Vs A CAG-C 2P^- (3.4) 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 du; G = M, K DX J du. dxj 3U; dx, where (ui, U2, U3) are the Cartesian mean velocity components and i, j = 1, 2, 3. p eff the effective viscosity given by Meff = M, + Mi (3.5) (3.6) fj t is the laminar viscosity H, is the turbulent viscosity evaluated from the relation /j, = pc M k 2 te, 17

27 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 -CI) 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->c i (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 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 (U z =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

28 intersection at various intersecting points are determined analytically. The turbulence energy is calculated from k in =\.5 x (intensity x u in f > where the intensity is The turbulence dissipation is calculated from E 7S { n =C M k. n 5ll in, 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 Sj n 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 (U z = 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, dz ^HJL=Q). dz Axis boundary condition At the axis of the hydrocyclone, by virtue of symmetry, the tangential and radial velocities are zero (U r =0, Ue=0), and zero radial gradient condition is applied to the axial velocity (f^.=o)- dr 19

29 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 ( 3U * =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

30 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 (Nm 2 ) E elastic modulus in bending (Nm") I moment of inertia (m 4 ) 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] 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

31 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

32 N + f i S ia X i Newton's second law: m i r i = i (3.10) a a=\ N The law of conservation of momentum: H; = M, + ^S ia (c ia x X a + Y a ) (3.11) Where mi, Yi, Hz the mass, translational acceleration, and time rate of change of angular momentum of ellipsoid /, respectively; X b, X c the internal constraint forces in joint b and c respectively; Y b, 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; S ia the connectivity matrix which describes how the ellipsoids and joints are connected to one another; c ja 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: (3.12) (3.13) 23

33 where U- 00^ is ambient fluid translational velocity, E^and ft^are the rate of strain tensor and vorticity respectively, and A, Cand 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) =±xab 2 (p p -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 xx 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/* ) +Fp ) +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

34 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" 6 m and lo^m, the relative motion of the particle and the fluid calculated from the code ranges from 10" 6 m/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 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

35 Dong et al. [10] 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, fi wali is a wall friction coefficient. If two or more ellipsoids touch the wall, the reaction forces are added to each of those ellipsoids 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 [11] in their work: "the velocity fluctuation is randomly drawn from a Gaussian probability density distribution of zero mean and a standard deviation V2 / 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 T x is turbulence time scale kls, L t is the approximate size of local size (C^15 k l5 )/s, where (u rd, v rel, w rel ) is the relative velocity of the fiber with respect to the fluid. The fiber will stay in the same eddy until T is expired." 26

36 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