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Numerical study of a transient bifurcating-flow near slotted apertures Li, Weiyin 2020

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NUMERICAL STUDY OF A TRANSIENT BIFURCATING-FLOW NEAR SLOTTED APERTURES by  Weiyin Li  B.A., Xi’an Jiaotong University, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)   December 2019  © Weiyin Li, 2019   ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, a thesis/dissertation entitled:  Numerical study of a transient bifurcating-flow near slotted apertures  submitted by Weiyin Li in partial fulfillment of the requirements for the degree of Master of Applied Science in Mechanical Engineering  Examining Committee: Dr. James Olson, Mechanical Engineering Supervisor  Dr. Robert Gooding Supervisory Committee Member  Dr. Mark Martinez, Chemical and Mechanical Engineering Supervisory Committee Member  Additional Examiner   Additional Supervisory Committee Members:  Supervisory Committee Member  Supervisory Committee Member iii  Abstract  This thesis numerically studied the transient flow near slotted apertures inside a pulp screen. The capacity of pressure screen is generally defined as the maximum mass throughput before the apertures plug with pulp. Increased capacity follows from factors that either reduce fiber deposition at the aperture or increase the effectiveness with which fibers are removed. This thesis considers this latter effect and in particular the factors that increase the backflush pulse that clears any deposited fibers from the aperture. Three major factors are discussed in this thesis:  (1) higher rotor tip speeds and lower slot velocities support longer and higher reversal flows to backflush and clear the apertures, (2) a foil angle-of-attack of 5-degree generated the longest reversal flow duration and maximum negative pressure pulse, and (3) the maximum reversal velocity was found for an intermediate contour height of 0.9 mm. The reversal flow time increases with the increasing contour height when the slot velocity is reduced.  The study also showed a backflush “flow tunnel” between the vortex and the backside of the wire. When the reversal flow happens, the vortex center would move away from the aperture and the backflush flow tunnel could be found.   iv  Lay Summary  It is industrially proven that pressure screens can improve pulp quality by removing contaminants and separating fibers by length. The rotor inside the screen accelerates the pulp stream and induces turbulence at the surface of the cylinder as well as generating negative pressure pulses at the cylinder surface to backflush the apertures.    The purpose of this study is to develop a detailed computational model of the flow at the surface of a slotted cylinder during the passage of the rotor. Additionally, this study discusses the factors that lead to screen plugging and provides insights into the design of an aperture.  v  Preface  This dissertation is original, unpublished, independent work by the author, Weiyin Li. He is responsible for conducting all the simulations and data analysis in this thesis. Dr. Olson and Dr. Gooding supervised the research and provided valuable feedback.   vi  Table of Contents  Abstract ........................................................................................................................................ iii Lay Summary ................................................................................................................................ iv Preface ............................................................................................................................................ v Table of Contents .......................................................................................................................... vi List of Tables .............................................................................................................................. viii List of Figures ............................................................................................................................... ix List of Symbols ............................................................................................................................ xii List of Abbreviations ................................................................................................................. xiii Acknowledgements ..................................................................................................................... xiv Dedication ..................................................................................................................................... xv Chapter 1: Introduction ................................................................................................................ 1 1.1 Objective .......................................................................................................................... 3 1.2 Outline ............................................................................................................................. 4 Chapter 2: Literature Review ...................................................................................................... 5 Chapter 3: Computational Model ................................................................................................ 9 3.1 Numerical Method ........................................................................................................... 9 3.2 Computational domain and assumptions ....................................................................... 10 3.3 Mesh Generation ........................................................................................................... 12 3.4 Boundary Condition ...................................................................................................... 13 3.5 Validation ...................................................................................................................... 14 3.5.1 Grid Independence ..................................................................................................... 14 vii  3.5.2 Time step convergence .............................................................................................. 16 3.5.3 Periodicity .................................................................................................................. 18 Chapter 4: Results and Discussion ............................................................................................. 19 4.1 Effect of slot velocity and tip speed .............................................................................. 20 4.2 Effect of foil angle of attack .......................................................................................... 35 4.3 Effect of wire geometry ................................................................................................. 42 4.4 Pressure drop Coefficient .............................................................................................. 47 Chapter 5: Results discussion ..................................................................................................... 50 Chapter 6: Conclusions ............................................................................................................... 54 Bibliography ................................................................................................................................. 56 Appendices ................................................................................................................................... 60  viii  List of Tables  Table 3-1 Grid Independence ........................................................................................................ 14 Table 3-2 Time step size ................................................................................................................ 16 Table 4-1 Different wire types ...................................................................................................... 42  ix  List of Figures  Figure 1-1 Schematic of the flow in a modern pulp screen ............................................................. 3 Figure 3-1 Computational domain ................................................................................................. 11 Figure 3-2 Mesh ............................................................................................................................ 12 Figure 3-3 C-mesh around the foil ................................................................................................ 13 Figure 3-4 O-mesh around the wire ............................................................................................... 13 Figure 3-5 Pressure coefficient on the wall versus numbers of mesh ........................................... 15 Figure 3-6 Time step convergence ................................................................................................ 17 Figure 3-7 Instant slot velocity versus flow time .......................................................................... 18 Figure 4-1 Monitor Line for instant y velocity and Monitor point for instant x velocity ............. 19 Figure 4-2 Instant y velocity on the slot entrance versus for Vt = 10m/s ...................................... 21 Figure 4-3 Instant y velocity on the slot entrance versus for Vt = 15m/s ...................................... 21 Figure 4-4 Instant y velocity on the slot entrance versus for Vt = 20m/s ...................................... 22 Figure 4-5 X/chord=0 .................................................................................................................... 22 Figure 4-6  Experimental and numerical results for y velocity at the slot entrance for Vt =10m/s and Vs=1m/s .................................................................................................................................. 23 Figure 4-7  Experimental and numerical results for x velocity at the slot entrance for Vt =10m/s and Vs=1m/s .................................................................................................................................. 23 Figure 4-8 Instant x velocity at the monitor point 	for Vt =15m/s and Vs=4m/s ........................... 24 Figure 4-9 Six rotor positions used in Fig 4.9(a)-Fig4.9 (f) to study the flow field during foil passage over the slotted apertures ................................................................................................. 26 Figure 4-10 Pressure contour and streamlines for Vt =20m/s and Vs=0.5m/s ............................... 27 x  Figure 4-11 Pressure contour and streamlines for Vt =10m/s and Vs=4m/s .................................. 28 Figure 4-12 Downward flow field showing the streamlines and vortex center ............................. 30 Figure 4-13 Reversal flow field showing the streamlines and vortex center ................................ 31 Figure 4-14 Pressure Contour and streamlines at the slot entrance for Vt =10m/s and Vs=1m/s .. 32 Figure 4-15 Pressure Contour and streamlines at the slot entrance for Vt =10m/s and Vs=4m/s .. 33 Figure 4-16 Pressure Contour and streamlines at the slot entrance for Vt =20m/s and Vs=1m/s .. 34 Figure 4-17 Pressure coefficient at the outer wall versus X/chord for different angles-of-attack 35 Figure 4-18 Negative Pressure coefficient at the outer wall on the screen surface for different angles-of-attack and the comparison with experimental result. .................................................... 37 Figure 4-19 Positive Pressure Coefficient at the outer wall for different angles-of-attack ........... 37 Figure 4-20 Instant y velocity versus position for different angles-of-attack (Vs= 0.5 m/s) ......... 38 Figure 4-21 Instant y velocity versus position for different angles-of-attack (Vs= 2 m/s) ............ 38 Figure 4-22 Pressure Contour and streamlines for Vt =15m/s and Vs=2m/s and angle-of-attack = 0-degree ......................................................................................................................................... 40 Figure 4-23 Pressure Contour and streamlines for Vt =15m/s and Vs=2m/s and angle-of-attack = 5-degree ......................................................................................................................................... 41 Figure 4-24  Instant y velocity at the slot entrance versus position for different wire types (Vt =15m/s and Vs=2m/s ) ................................................................................................................... 42 Figure 4-25 Pressure Contour and streamlines at the slot entrance for Vt =15m/s and Vs=2m/s and contour height =0.6 mm .......................................................................................................... 44 Figure 4-26 Pressure Contour and streamlines at the slot entrance for Vt =15m/s and Vs=2m/s and contour height =0.9 mm .......................................................................................................... 45 xi  Figure 4-27 Pressure Contour and streamlines at the slot entrance for Vt =15m/s and Vs=2m/s and contour height =1.2 mm .......................................................................................................... 46 Figure 4-28 Cpd versus Vs for different Vt ..................................................................................... 48 Figure 4-29 Cpd versus Vs for different angles-of-attack ............................................................... 48 Figure 4-30 Cpd versus Vs for different contour heights ................................................................ 49 Figure 5-1  Normalized reversal flow versus Vs for different Vt, angle-of-attack = 5-degree, contour height =0.9mm ................................................................................................................. 51 Figure 5-2  Normalized reversal flow versus Vs for different angles-of-attack, Vt = 15m/s, contour height=0.9mm .................................................................................................................. 52 Figure 5-3 Normalized reversal flow versus Vs for different contour heights, Vt = 15m/, angle-of-attack=5-degree ............................................................................................................................. 52 Figure 5-4  Maximum instant y velocity versus Vs for different Vt, different angles-of-attack, different contour heights ................................................................................................................ 53  xii  List of Symbols  𝑉#:    Rotor tip speed       [m/s] 𝑦% :  Non dimensional distance 𝑘:   Turbulent kinetic energy 𝜀:   Turbulent dissipation rate 𝜌 :   Water density       [kg/m3] 𝐶*+:   Pressure drop coefficient 𝑃-.#/0#: Pressure outlet       [Pa] 𝑃12/0#:  Pressure inlet       [Pa] 𝑃-.#03 :  Pressure at the outer wall     [Pa] 𝐶*45678:  Pressure drop coefficient at the outer wall 𝑉9∗:  Instant slot velocity at the monitor line   [m/s] 𝑉;∗:   Instant x velocity at the monitor point   [m/s]  𝑉9:  Average slot velocity      [m/s] 𝑇30:  Reversal flow Time step in one period 𝑇30∗ :  Normalized reversal time step in one period 𝑉=>?@:  Maximum y velocity in one period at the slot entrance  [m/s] 𝐶*A  :   Negative Pressure coefficient 𝐶*%  :   Positive Pressure coefficient  xiii  List of Abbreviations  CFD:       Computational Fluid Dynamics CSS:       Cross Sectional Screen LDV:       Laser Doppler Velocimetry NACA:       National Advisory Committee for Aeronautics PIV:       Particle Image Velocimetry SIMPLEC:      Semi-implicit Method for Pressure Linked Equations-Consistent   xiv  Acknowledgements  First and foremost, I would like to sincerely thank my supervisor Prof. James Olson for his support and guidance at every stage of researching and writing this thesis. He consistently allowed this paper to be my own work while steered me in the right direction whenever he thought I needed it. It has been an honor to share his scientific knowledge and to be mentored by such an internationally recognized researcher.  I would also like to acknowledge the financial support of Aikawa Fibre Technologies (AFT) and Dr. Robert Gooding from AFT as the second reader of this thesis, and I am indebted to his very valuable and insightful comments on this thesis.  Last but not the least, I must express my very profound gratitude to my parents and to my wife Simin Sun and my daughter Allison Yitong Li for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them. Thank you. xv  Dedication  To my parents, To my wife and daughter, who are always being there for me.  1  Chapter 1: Introduction  A pulp screen is used in the pulp and paper industry to remove oversized contaminants that can degrade the appearance and strength of paper. Figure 1 shows a typical industrial pulp screen. During the screening process, pulp suspensions typically enter the screen tangentially through the feed port and flow between the rotor and the screen cylinder. The accept fibers then pass through the apertures in the cylinder and exit from the accept port while the longer fibers and contaminants are retained by the cylinder and exit from the reject port. The passage ratio was defined by Gooding [5] and Kerekes et al. [21] as the ratio of the fibers in the flow through the aperture to the corresponding concentration upstream of the aperture.   There are two main elements of a pulp screen: a rotor and a screen cylinder with circular or slotted apertures. The rotor accelerates the pulp stream and induces turbulence at the surface of the screen cylinder, as well as, generating negative pressure pulses at the cylinder surface to backflush the apertures. The backflushing action is generated as a result of the flow acceleration between the rotor and the cylinder. According to the Bernoulli Equation, increased velocity leads to reduced pressure, so that the local pressure on the feed side of the cylinder decreases to a point that the flow through the aperture reverses momentarily. This backflush flow passes from the accept side of the screen cylinder to the feed side and releases any plugged fibers in the aperture. This action maintains the flow through the screen cylinder.     2  The capacity of the screen is commonly defined as the maximum mass throughput before the apertures become plugged with pulp, which is the product of the average aperture velocity and the accept pulp consistency, where “consistency” is the mass concentration of the pulp. The maximum aperture velocity is, perhaps, the more critical parameter. It, in turn, is a function of a number of factors, including the feed fiber length [6,14], feed pulp consistency [13], aperture size and type [6,7,8,11,14,15,23,25], and rotor speed and type [2,9,22].   In the past few years, a number of experimental and computational studies have been made to understand pulp screen capacity [2,4,8,9,17]. However, the characteristics of the transient flow field in the pulp screen, which is the flow field that occurs during the backflush pulsation induced by the screen rotor, is still comparatively unknown. In particular, there is no published study of this transient flow field using numerical techniques. 3   Figure 1-1 Schematic of the flow in a modern pulp screen 1.1 Objective  The objectives of this study are: • To develop a two-dimensional, time-varying computational model including a moving rotor foil and slotted cylinder apertures. • To validate the resulting numerical results with Salem et al’s [17] experimental results. • To better visualize and characterize the complex flow phenomena as a function of the rotor and cylinder hardware variables and operating conditions, including backflush flow velocity and volume that is hypothesized to be critical to fiber plug removal and screen capacity.  4  1.2 Outline Chapter 2 starts with the literature review and Chapter 3 introduces the computational fluid dynamics model as well as the boundary conditions and the numerical methods used in this study. Numerical validation, such as, grid independence, time convergence and periodicity check are presented in this chapter. Chapter 4 illustrates how these factors affect the capacity of the pulp screen, as well as how it leads to screen plugging and the backflush flow. Some design insights are discussed in Chapter 5. Finally, a conclusion is provided in Chapter 6. 5  Chapter 2: Literature Review  Karvien et al [18] compared the experimental results with the CFD results and found that CFD could be as an effective tool in the pressure screen research. Gooding [7] used both numerical and experimental methods to examine the flow features and turbulence intensity of several different screen contours. First of all, he applied the pressure drop coefficient K to the pulp screen:                                                            𝐾 = ∆EFG.IJKFL                                                                           (1) Where ∆𝑃9  is the slot pressure loss, Vs is slot velocity, 𝜌  is fluid density, to illustrate the relationship between slot velocity and pressure loss on the surface of the slot.   Wikstrom et al [20] developed a 2D transient CFD model to simulate the hydrodynamics inside the pressure screen. As the flow feature was well captured in the study, the sliding mesh method was proven to be an effective method in the pulp scree transient flow simulation.  Pinon et al. [13] measured the pressure pulse in a cross-sectional pulp screen with a solid core screen rotor. The results suggested that the shape of the pressure pulse as well as the pressure coefficient would not change with the tip speed. Feng et al [9] conducted both numerical and experimental studies of the flow in a pressure screen including different kinds of rotor foil shapes and angle-of-attacks to investigate how the pressure pulse is affected by foil design and operation. The result indicated that the maximum magnitude of the pressure pulse increases linearly with the rotor tip speed squared for all the angles-of-attack. The positive pressure peak along the leading edge of the foil was found to be approximately zero when the angle-of-attack is greater than or 6  equal to 5-degree. It was also found that decreasing the foil clearance could increase the magnitude of the negative pressure pulse. Both the magnitude and range of the negative pressure pulse increased as the foil camber increased.  Delfel [2] developed a CFD model of a pressure screen using a multi-element foil rotor. It was revealed that a multi-element foil rotor was more efficient than the single foil rotor at producing strong negative pressure pulses at low rotor power since the multi-element foil rotor could achieve high angle-of-attack at a lower Reynolds number (thus lower rotational speed and power) without stalling. The design of geometry of the multi-element foil rotor was also studied. Increasing both the angle-of-attack and the flap angle of the multi-element foil rotor could increase the magnitude of the negative pressure pulse and reduce the magnitude of the positive pressure pulse to some extent.  The geometry of the cylinder contour could also affect the capacity and efficiency of pressure screen. Gooding [7] assessed that the flow features at the slot entrance was affected by the contour geometry. As the contour height increases, the vortex would dominate the whole area of the slot entrance. Dong [12] developed a CFD model of a fiber moving in steady flow near a single slot to simulate the flow within a pressure screen. It was shown that slope-shape slot geometry is the most efficient aperture shape for increased fiber passage ratio compared with the smooth slot and the step-step slot. Jokinen et al [16] experimentally studied the effect of screen design parameters on the capacity of the pressure screen. She found that increasing the slot width and the contour height could let more fibers pass the slot as to increase the capacity of the screen. Besides, lowering the wire height could also increase the capacity as well as the backflush time. 7  Mokamati [8] used CFD to simulate the fiber suspension entering a pulp screen aperture. In order to focus on the effect of the wire roughness, he neglected the presence of rotor and the pressure pulse caused by the rotor. The model was validated by comparing the numerical result with Laser Doppler Velocimetry (LDV) experiments and the fluid streamlines are compared to the data obtained from particle image velocimetry (PIV) experiments conducted by Mokamati [15].  It was shown that the exit layer mentioned by Gooding [5] and Olson [6] between the exit line and the dividing streamline created a tunnel for the fibers to redirect and pass through the slot. The size of vortex above the slot entrance increased and stretched as wire width and contour height increased. He speculated that the capacity of pressure screen was related to the size of vortex and larger vortex region would give more chances for the fibers to pass through the slot.  Salem [4] experimentally investigated the turbulence flow at the slot entrance during rotor passing using high speed video in a laboratory cross-sectional screen (CSS). It was stated that the size of vortex above the slot entrance was dependent on the geometry of the contour, slot velocity and rotor speed, which was consistent with the result of Mokamati [8]. More importantly, he studied the time-varying flow near a lab-scale CSS screen. The streamwise flow measured above the wire drops to approximately 30% of the rotor speed when it reached 20% of the foil chord. The backflush flow could be observed in the slot entrance to help clear and unplug the slot. Moreover, he found that the reversal flow was increased by increasing rotor speeds and decreasing average slot velocities. A delayed backflush flow was also examined in the research, which was most likely due to the inertia of the flow passing through the slotted aperture.  8  Olson et al [10] found that the power consumption increased approximately with the rotor speed cubed and rotor diameter squared. Therefore, a small increase of the rotor speed could lead to a significant increase of the power consumption. However, as mentioned before, the growth of the negative pressure generated by the increased rotor speed would increase the capacity of the pressure screen. A balance should be found between the screen capacity and power consumption.   Most of these studies are focused on either steady flow or some idealized component of the flow in a pressure screen. However, the transient effect of rotor foil passage over the screen surface has not been numerically analyzed. A detailed, time-varying CFD model including slotted apertures and a moving foil would provide increased insights on the design of the pressure screen. In this study, we will focus on the effect of rotor foil angle of attack, wire geometry, rotor speed and slot velocity.  9  Chapter 3: Computational Model  3.1 Numerical Method  Fluent 19.0 is used to solve the Navier-Stokes equations for this study. The pressure-based solver is used to solve the continuity, momentum equations. The SIMPLEC algorithm with second-order upwind spatial discretization is employed to couple pressure and velocity in this simulation. The stopping relative criteria for all calculations are below 0.05 for each time step.  Turbulence is modeled by the realizable 𝑘 − 𝜀 turbulence model with enhanced wall treatment. Realizable 𝑘 − 𝜀  turbulence model has been widely used in the numerical pressure screening modeling history and the enhanced wall treatment is a balance between the computation cost and result accuracy [2,8,9,20], which is a near-wall modeling method that combines a two-layer model with enhanced wall functions [1].   Sliding mesh is used to model the unsteady flow field between the foil and wires, which has been proven to be as the most accurate method to simulate rotor-stator interaction in multiple moving reference frames [1,19].     10  3.2 Computational domain and assumptions  The cross-sectional screen is used in this simulation to model the laboratory-based pressure screen PSV 2100 as the variations in the axial flow could be neglected compared with the tangential flow. Furthermore, considering the symmetry of the pulp screen, the flow inside a pulp screen cylinder can be reduced to flow over a rotationally periodic array of wire elements for a 90-degree sector. Figure 3-1 shows a typical computational domain in the simulation. The diameter of the outer cylinder is set to 290 mm. Feng [9] and Delfel [2] stated that the effect of inner wall could be neglected if the inner wall radius is smaller than 0.867 of the radius of the outer wall. As a result, the diameter of inner cylinder wall is set to 105 mm. NACA 0015 foil with different angles-of attack is used in the simulation. The chord length of the foil is 40 mm while the clearance between the rotor foil and the cylinder wall is 3 mm. The downstream tunnel length is set to 200 mm, which gives a fully developed flow at the outlet. Besides, pulp consistency is usually very low (1~2%) during the screening process. Gullichsen and Harkonen [3] found that pulp suspensions had the same flow property with water in a fully turbulent state when pulp consistency was lower than 2%. Therefore, water would be used in the study instead of pulp suspensions that implies an extremely dilute fiber suspension – which is not typical in the industry.  11   Figure 3-1 Computational domain   12  3.3 Mesh Generation  ICEM CFD is used to generate a 2D multi-block mesh. Figure 3-2-3-4 show typical meshes used in this study. As required by the realizable 𝑘 − 𝜀  model with enhanced wall treatment [1], the first mesh cell close to the wires and foil should be in the sublayer region (𝑦%~1). A very fine C-mesh is created around the foil in order to simulate the boundary layer and wake flow while a very fine O-mesh is generated around the wires to obtain a good result across the slot. Besides that, a coarser H-mesh is applied elsewhere to provide good mesh quality as well as saving the computational expense.  Figure 3-2 Mesh 13   Figure 3-3 C-mesh around the foil   Figure 3-4 O-mesh around the wire  3.4 Boundary Condition  Figure 3-1 shows that the entire domain is divided by the interface between the rotor part and wire part. A rotating speed is set up at the center of the rotor part so that the foil is moving together with the whole rotating part. The wire part is set to be stationary in an absolute coordinate system. 14  Periodic boundary conditions are applied at the both sides of the rotating domain so that the flow entering the domain on one side is identical to the flow exiting the domain through the opposite side. Furthermore, periodic boundary conditions are also used at the both sides of the wire domain as shown in Figure 3-1 to make sure the downstream flow is not affected by the side wall effect.  The virtual inner wall is set to a pressure inlet boundary condition, which would generate a specific average slot velocity	𝑉9 in the range of 0.5-4.0 m/s. The pressure outlet boundary condition is set to zero as it is assuming to be a full-developed flow. Besides, a no slip boundary condition is set to the walls of all the wires and the outer cylinder wall.  3.5 Validation 3.5.1 Grid Independence Table 3-1 Grid Independence Case Number Mesh Number 𝐶*45678	 1 220,000 0.07485 2 340,000 0.07588 3 470,000 0.07628 4 760,000 0.07630   The grid independence is conducted by measuring pressure on the outer wall of the cylinder mentioned in Figure 3-1. Four similar meshes from 220,000 to 760,000 control volumes as shown in Table 3-1. Non-dimensional Pressure Coefficient 𝐶*45678 defined as  E45678G.IJK6L , where 𝑃-.#03	is the 15  pressure at the wall, 𝑉#  is the tip speed of the rotor. The case1 mesh is generated to make sure that the 𝑦%on the wall approximately equals to 1 and coarser meshes are used elsewhere. By adding more nodes close to wall and refining the whole domain 1.5 times than case1, the case2 uses 340,000 control volumes. Similarly, the mesh is further refined in the whole domain for case3 and case4. Figure 3-5 shows that pressure coefficient on the wall is independent with the numbers of mesh when the mesh is larger than 470,000 control volumes, which could be considered as grid independence. As a result, case 3 (470,000 control volumes) is the best option that would be used in this study.   Figure 3-5 Pressure coefficient on the wall versus numbers of mesh  16  3.5.2 Time step convergence  Time step convergence, which means the result is independent with time step size, is examined with five different time steps. The validation is conduct by calculating 𝐶*45678with 𝑉# =15m/s and average slot velocity, 𝑉9=2m/s. Table 3.2 shows that the relationships between the time step size and number of time steps. For example, the time step is set that the passing of the foil is divided into 2000 time steps and the rotating part is a 90-degree sector. With a rotational speed of 975 rpm (5850 degree/sec), each time step takes 90/5850/2000=7.69 e-6 sec. Figure 3-6 indicates that the 𝐶*45678 is unchangeable when the time step is smaller than 7.6923e-6 that is 2000 per revolution.  Table 3-2 Time step size Time step size(s) Number of time steps 6.1538e-5 250 3.0769e-5 500 1.5385e-5 1000 7.6923e-6 2000 3.8462e-6 4000   17   Figure 3-6 Time step convergence 18  3.5.3 Periodicity  Figure 3-7 Instant slot velocity versus flow time  As the pressure screen has been simplified to a periodic rotor-stator interaction, time periodicity should also be verified by comparing the instant slot velocity 𝑉9∗ at the slot entrance from one revolution to another. The validation is check with 𝑉# = 10m/s and 𝑉9 = 1m/s. Figure 3.7 indicates that the 𝑉9∗	monitored at the entrance could be considered time-periodic as the change is less than 5% from one period to the next after the third revolution [1].    19  Chapter 4: Results and Discussion  The effects on pressure screen design and operating parameters including slot velocity, tip speed and foil angle-of-attack as well as the wire geometry are studied by simulating transient flow near the screen surface.   Figure 4-1 shows the monitor line and monitor points used in this study. Y-velocity monitor line is chosen at the central slot entrance that is the narrowest distance (0.15 mm) between two wires. The X-velocity measurement point is chosen 1 mm above the wire and is similar to the monitoring line used by Salem [4].  Figure 4-1 Monitor Line for instant y velocity and Monitor point for instant x velocity 20  4.1 Effect of slot velocity and tip speed   Figure 4-2-4-4 shows the spatial average of the y velocity profile at the slot entrance (measured along the y-velocity monitor line) during a single foil passage for different foil tip speeds and different time-averaged slot velocities. These results are illustrated as a function of foil position and X/chord = 0 is set to at the center line of the center wire as shown in figure 4-5.   For 𝑉# = 10 m/s and 𝑉9= 0.5 m/s, the reversal flow (y velocity > 0) starts from X/chord ~0.1 and lasts until X/chord ~1.3. The maximum instant y velocity, 	𝑉9∗ is ~2.5m/s. When 𝑉9 is increased from 0.5m/s to 4 m/s, the reversal flow disappears for the same 𝑉#. At the same time, when 𝑉# is increased from 10 m/s to 20m/s, the reversal flow still exists for high slot velocity (𝑉# =4m/s) and the maximum magnitude of  𝑉9∗ is much bigger (~6m/s). These results are in accordance with Salem [4] experimental result that the reversal flow is increased with increasing foil tip speed and decreasing average slot velocity.   21   Figure 4-2 Instant y velocity on the slot entrance versus for Vt = 10m/s  Figure 4-3 Instant y velocity on the slot entrance versus for Vt = 15m/s 22    Figure 4-4 Instant y velocity on the slot entrance versus for Vt = 20m/s  Figure 4-5 X/chord=0 23   Figure 4-6  Experimental and numerical results for y velocity at the slot entrance for Vt =10m/s and Vs=1m/s  Figure 4-7  Experimental and numerical results for x velocity at the slot entrance for Vt =10m/s and Vs=1m/s 24  Figure 4-6 shows the numerical and the experimental y velocity for the same tip speed (𝑉#= 10m/s) and slot velocity (𝑉9=1m/s) in a same plot. It could be found that the minimum y velocity is ~ -2.2m/s, which matches the Salem’s experimental result within 10%. The maximum y velocity for the experimental result is almost two times bigger than the CFD result. Besides, both results show the different reversal flow times (𝑉9∗ >0). The numerical results states the reversal flow starts at X/chord ~ 0.1 while the experimental reversal flow starts at X/chord~ 0.7, which is a little delay compared with the numerical result. Figure 4-7 states that both results drop to 30% of  𝑉#  at X/chord ~ 0.2. After that, there is acceleration for both methods. The maximum x velocity is seen after one chord for both results.    Figure 4-8 Instant x velocity at the monitor point 	for Vt =15m/s and Vs=4m/s 25  Figure 4-8 indicates a typical curve for 𝑉𝑥∗ at the monitor point versus position during the passage of the foil. It could be found that the flow is deaccelerating as the approaching of leading edge of foil. The minimum 𝑉𝑥∗  (~0.33 of the 𝑉#) occurs at X/chord =0.12. After that, there is a significant increase for 𝑉𝑥∗	and the maximum value could be found at X/chord ~1.2. Then the flow slowly decreases to the average x velocity (~0.65 of the	𝑉#) for approximately 3 chord lengths until the approach of the next foil.                     26                  (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05              (c)X/Chord=0.23                                                   (d) X/Chord=0.5  (e)X/Chord=0.78                                                   (f) X/Chord=1 Figure 4-9 Six rotor positions used in Fig 4.9(a)-Fig4.9 (f) to study the flow field during foil passage over the slotted apertures  In order to better understand the flow feature during the passage of the foil, six different foil positions are selected as shown in Figure 4-9. It could be found that the foil is arriving the first wire at the Figure 4-9(a) and the foil have left the center wire at Figure 4-9(f).      27   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05  (c)X/Chord=0.23                                                   (d) X/Chord=0.5  (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-10 Pressure contour and streamlines for Vt =20m/s and Vs=0.5m/s 28   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05  (c)X/Chord=0.23                                                   (d) X/Chord=0.5  (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-11 Pressure contour and streamlines for Vt =10m/s and Vs=4m/s 29  Figures 4-10-4-11 reveals that the streamlines and pressure contours during the passage of the foil.  Generally speaking, it could be indicated that the reversal flow is in accordance with a larger negative pressure pulse under the leading edge of the foil. For example, the peak negative pressures are in a range of 80 kpa for 𝑉# = 20 m/s and 𝑉9 = 0.5 m/s. When 𝑉# is decreased to 10 m/s and 𝑉9 is increased to 4 m/s, the peak value of the negative pressures are approximately 20 kpa.  Specially, for X/chord= -0.32, there is no reversal flow around the central wire because the foil is far away from the wire in figure 4-10(a). After that, the reversal flow hasn’t happened until X/chord= 0.23 and it would last for the next four plots, which could prove what Salem [4] found that the backflush flow was a little bit delayed. Moreover, another interesting investigation of these plots is that a significantly amount of flow are entering the slot at X/chord = -0.05 to cause the slowing of flow before acceleration shown in Figure 4-8, which is consistent with previous experimental result shown by Salem et al [4]. More results with different tip speeds and slot velocities are shown in Appendix A.   It should be meaningful to zoom in to the flow plots and observe the vortex size and exit layer. Yu and Defoe [24] firstly stated that vortex was an effective method to prevent fibers from blocking the aperture and stapling between two slots. Exit layer is introduced by Gooding [5] and subsequently investigated by Olson [6] and is defined as the area between the dividing streamline and exit streamline. They found that the streamlines inside the exit layer would turn from the upstream flow and then pass through the slot and the vortex was created by the dividing streamline and the wire.   30  Figure 4-12 shows that the center of the vortex is occupying the slot entrance and the exit streamline is allowed to flow around the vortex and led into the slot between the vortex and top face of the wire during the downward flow time. When the reversal flow occurs, as shown in Figure 4-13, the vortex center is moving away from the slot to create a tunnel between the vortex and the backside of the wire to allow the flow to backflush the wire.   Figure 4-12 Downward flow field showing the streamlines and vortex center 31   Figure 4-13 Reversal flow field showing the streamlines and vortex center  Comparing the six plots in Figure 4-14 with those in Figure 4-15, it could be found that the vortex center is closer to the slot entrance while increasing the slot velocity and keeping the tip speed constant. This agrees with Salem [4] and Gooding [5]’s result that high slot velocities would decrease the vortex size. Besides, the exit layer height, which was defined by Mokamati [8] as the vertical distance from the wall crest to the upper exit streamline, would become thicker as the slot velocity increases. While 𝑉# is increased from 10m/s to 20m/s as shown in Figure 4-16, the vortex center is moving to the opposite direction from the slot entrance for the same foil position so that the backflow tunnel is becoming bigger during the reversal flow time that could allow more flows to move upwards through the slot. 32   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-14 Pressure Contour and streamlines at the slot entrance for Vt =10m/s and Vs=1m/s 33     (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5     (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-15 Pressure Contour and streamlines at the slot entrance for Vt =10m/s and Vs=4m/s 34     (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5     (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-16 Pressure Contour and streamlines at the slot entrance for Vt =20m/s and Vs=1m/s 35  4.2 Effect of foil angle of attack  The pressure coefficient 𝐶*45678  measured on the cylinder’s outer wall for four different angles-of-attack are shown in Figure 4-17. All the simulations in this section use the NACA 0015 foil with wire contour height = 0.9 mm. The tip speed of foil 𝑉#	is set to 15m/s and the average slot velocities 𝑉9	are 0.5, 1, 2, 4 m/s. Figure 4-18 shows that 𝐶*45678	has the same form with Gonzalez et al’s [9] experimental results that the 5-degree angle-of-attack has the biggest negative pressure pulse and the value of negative pressure peak for 0-degree angle-of-attack is smaller than the value for a 15-degree angle-of-attack. However, the CFD results predicts the 𝐶*45678 on the outer cylinder wall is smaller than the experimental result because specific boundary conditions are applied in the experimental model.  Figure 4-17 Pressure coefficient at the outer wall versus X/chord for different angles-of-attack 36  More specifically, Figure 4-18 shows that the peak value of the negative pressure coefficient  𝐶*A	versus different angles-of-attack (0-degree, 5-degree, 10-degree and 15-degree) for different average slot velocities. It could be found that the maximum magnitude of 𝐶*A	occurs at 5-degree angle-of-attack with a value of 𝐶*A  ~ -0.4. As the angle-of-attack increases from 5-degree to 15-degree, the negative pressure pulse would decrease.    The maximum value of the positive pressure coefficient  𝐶*% versus different angles-of-attack is shown in Figure 4-19 that 𝐶*%	decreases rapidly when the angle of attack is increased from 0-degree to 5-degree. The pressure changes less as the angle of attack is increased beyond 5-degree. As noted by Delfel [2] and Feng [9], the positive pressure pulse could push contaminants through the screen slot, lead to screening plugging and reduce the efficiency of the pressure screen. Consequently, greater than or equal to 5-degree would be used in the study. 37   Figure 4-18 Negative Pressure coefficient at the outer wall on the screen surface for different angles-of-attack and the comparison with experimental result.  Figure 4-19 Positive Pressure Coefficient at the outer wall for different angles-of-attack 38   Figure 4-20 Instant y velocity versus position for different angles-of-attack (Vs= 0.5 m/s)  Figure 4-21 Instant y velocity versus position for different angles-of-attack (Vs= 2 m/s)  39  Figure 4-20-4-21 shows instant y velocities	𝑉9∗ measured at the slot entrance versus foil positions for different angles-of-attack. It can be noted that 5-degree angle-of-attack generates the longest reversal flow and the maximum 𝑉9∗	for two different average slot velocities. When angle-of-attack is increased to 15-degree as shown in figure 4-21, there is no reversal flow at the slot entrance. Moreover, a delayed reversal flow could be observed on 0-degree angle-of-attack, which is consistent with the pressure coefficient plot in figure 4-17.   Similarly, the pressure contour and streamlines plots are also examined with different angles-of-attack in Figure 4-22-4-23. It could be found that the negative pressure pulse under the leading edge of foil is widen when angle of attack is increased from 0-degree to 5-degree, which is consistent with longer reversal flow time at 5-degree angle-of-attack. Furthermore, the positive pressure at the leading edge and trailing edge of the foil is also becoming smaller as the increase of angle-of-attack because the stagnation points at the leading edge of foil are moving away from the downside of the foil, these result agree with Feng et al [9] result. Another interesting investigation is that the center of vortex and the exit layer height has less change with different angles-of-attack for the same foil position. More results with different angles-of-attack are shown in Appendix B.       40   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5  (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-22 Pressure Contour and streamlines for Vt =15m/s and Vs=2m/s and angle-of-attack = 0-degree 41    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-23 Pressure Contour and streamlines for Vt =15m/s and Vs=2m/s and angle-of-attack = 5-degree  42  4.3 Effect of wire geometry  Table 4-1 Different wire types Wire Type Contour height(mm) Slot width(mm) 0632 0.6 0.15 0932 0.9 0.15 1232 1.2 0.15  Three different wire shapes are simulated to examine the effect of the wire shape on the flow field at the screen surface.  All the simulations use the NACA 0015 foil with a 5-degree angle-of-attack. The tip speed 𝑉#  is set to 15m/s and the average slot velocities 𝑉9  are 0.5, 1, 2, 4 m/s. The information of the wire types is provided in Table 4-1.  Figure 4-24  Instant y velocity at the slot entrance versus position for different wire types (Vt =15m/s and Vs=2m/s ) 43  Figure 4-24 indicates the instant y velocities versus foil position for different wires. It shows that the backflush flows occur at the same foil position for three different contour heights and the reversal flow time would increase as the contour height increases from 0.6 mm to 0.9mm. When the contour height is increased to 1.2 mm, it seems that the reversal flow time has less significant change. Besides, the medium contour height (0.9mm) could generate the highest 𝑉9∗ at the slot entrance.  In order to obtain a much clearer relationship between wire contour height and the flow features, pressure contour plots with streamlines are given in Figure 4-25-4-27.  Firstly, these figures show that as the contour height increases, the vortex size at the slot entrance increases to occupy more areas of the slot entrance and center of vortex is also moving away from the aperture, which are consistent with Mokamati et al [8] steady flow results. Secondly, while the reversal flow happens, for example (X/chord=0.5), a streamline tunnel is generated between the vortex and backside of the wire. The reversal flow would pass through the slot by following this tunnel. As the contour height increases beyond 0.6 mm, the width of the tunnel would increase to some extent. Thirdly, when there is no reversal flow, an exit layer that leading the streamline into the aperture could be discovered above the wire. The exit layer would also increase to allow more streamlines to enter the slot when the contour height decreases. More results with different contour heights could be found in Appendix C.    44    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-25 Pressure Contour and streamlines at the slot entrance for Vt =15m/s and Vs=2m/s and contour height =0.6 mm 45     (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-26 Pressure Contour and streamlines at the slot entrance for Vt =15m/s and Vs=2m/s and contour height =0.9 mm 46     (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure 4-27 Pressure Contour and streamlines at the slot entrance for Vt =15m/s and Vs=2m/s and contour height =1.2 mm 47  4.4 Pressure drop Coefficient   As mentioned in Chapter 3, the average slot velocity at the entrance was controlled by the pressure drop between the pressure inlet and the outlet. The pressure drop coefficient is defined as: 																																																															𝐶*O = E456P76AEQRP76G.IJK6L                                                           (2) Where 𝑉#	is foil tip speed, 𝜌 is the water density, 𝑃-.#/0#  is pressure at the outlet, 𝑃12/0#  is pressure at the inlet.  In this section, the relationship between pressure drop coefficient 𝐶*O and average slot velocity, 𝑉9	is illustrated as follows.  Figure 4-28 shows 𝐶*O as a function of 𝑉9 for varying tip speeds. It shows a linear relationship when 𝑉#	 is 15 m/s. For 𝑉#	 = 10 m/s and 20 m/s, the relationship could be divided into two regimes. At lower average slot velocity ( 𝑉9 <1 m/s), the pressure coefficient reduces slowly when 𝑉9	increases. When average slot velocity is larger than 1 m/s, the pressure coefficient would reduce rapidly. Figure 4-29 shows an approximately linear relationship between pressure drop coefficient and average slot velocity when the angle of attack is smaller than 10 degree. As the angle of attack is bigger than 10 degree, 𝐶*O decreases quickly with a higher slot velocity (𝑉9>2m/s). Figure 4-30 reveals that the contour height has less effect on the 𝐶*O when average slot velocity is lower than and equals to 2m/s. At higher average slot velocity (𝑉9>2m/s), 𝐶*O  with contour height =0.6 mm drops much quickly than the other two contour heights.  48   Figure 4-28 Cpd versus Vs for different Vt   Figure 4-29 Cpd versus Vs for different angles-of-attack 49   Figure 4-30 Cpd versus Vs for different contour heights  50  Chapter 5: Results discussion  It is widely believed that the capacity of screen is defined as the maximum throughput before reversal/backflush flow is incapable of unplugging the screen apertures. Hence, increasing the reversal flow (either peak flow or total flow) may increase the capacity of the screen within a range. In this chapter, we quantify the flow reversal in terms of peak flow reversal velcoity and total reversal flow. Figure 5-1-5-3 shows the quantified reversal flow time for all simulations variables examined in this study.   One perspecitve is the operational parameters of pressure screen. Figure 5-1 indicates the reversal flow time steps 𝑇30	normalised by time steps in one revolution versus different average slot velocities and tip speeds.  																																																		𝑇30∗ = S87S1T0	9#0*9	12	-20	30U-/.#1-2                                   (3)                                    The biggest 𝑇30∗  could be found on 𝑉# = 20 m/s and 𝑉9	= 0.5 m/s. But the difference is really small when 𝑉9 is smaller than 1m/s. As mentioned by Mokamati [8], a high 𝑉9 could casue the contaminants to pass through the slots and decrease the screen efficiency. Furthermore, the target is to design a energy-saving pulp screen. Hence, a low 𝑉9	and relatively low 𝑉#  could make a balance between energy consumption and screen capacity.   Figure 5-2 shows that 5-degree angle-of-attack would provide the biggest 𝑇30∗  for the same 𝑉9. It should be mentioned when the angle-of-attack is greater or equal to 15-degree, there is no reversal flow for a higher 𝑉9. The results in Figure 5-3 suggest increasing contour height could increase the 51  reversal flow time for a smaller 𝑉9	and the reversal flow time difference bewtween contour height = 0.9 mm and 1.2 mm is small when 𝑉9	is lower than 1m/s. Moreover, there is no reversal flow for both contout height = 0.6 mm and 1.2 mm while 𝑉9	is 4m/s.   Another perspective is the 𝑉=>?@ , which is defined as the maximum instant y velocity at the slot entrance in one period. From the Figure 5-4, it could be concluded that 𝑉#	could impact the 𝑉=>?@ more than other factors. Furthermore, as discussed in Chapter 4, the model with contour height =0.9 mm could generate the maximum peak flow, which could be found from Figure 5-4. As described by Olson [11], the target is to find a balance between the energy consumption and capacity of pressure screen. The case that includes 𝑉# = 15m/s , 𝑉9 = 0.5m/s , angle-of-attack= 5-degree and the contour height =0.9mm may be optimal for the pressure screen design.   Figure 5-1  Normalized reversal flow versus Vs for different Vt, angle-of-attack = 5-degree, contour height =0.9mm 52   Figure 5-2  Normalized reversal flow versus Vs for different angles-of-attack, Vt = 15m/s, contour height=0.9mm  Figure 5-3 Normalized reversal flow versus Vs for different contour heights, Vt = 15m/, angle-of-attack=5-degree 53   Figure 5-4  Maximum instant y velocity versus Vs for different Vt, different angles-of-attack, different contour heights  One might be concerned that these simulations and insights are all based water not pulp suspensions. There are many differences in the fluid propeties between water and pulp suspensions. Besides, this model is based on a lab screen (a CSS which is meant to approximate a cross section of an industrial PSV 2100) and not an industrial pressure screen. A 2D 90-degree sector is used instead of 360-degree model that could lead the foil pass over the wires 4 times in one revolution. However, the similar trends and flow features in this study could provide some insights to the industrial pulp screens. 54  Chapter 6: Conclusions   A numerical simulation of the transient flow near the pulp screen apertures with a rotating foil for varying tip speeds, slot velocities, angle-of-attacks and wire types are studied in this thesis.  From this simulation, it was determined that higher 𝑉# and lower 𝑉9	generate reversal flows to backflush the apertures. It was further observed that the exit layer ‘tunnel’ forms during the backflush time between the backside of the wire and the contour vortex. The vortex oscillates between two positions in the contour during foil passage. As 𝑉9 decreases and 𝑉# increases, the exit layer ‘tunnel’ becomes thicker and the vortex center moves away from the aperture.  The simulation results also indicated that a foil with a 5-degree angle-of-attack could generate the longest reversal flow time and maximum negative pressure pulse on the wire surface with a value of 𝐶*A~ -0.4. It was also shown that when the angle-of-attack is greater than 5-degree, positive pressure pulses on the leading edge and the trailing edge of the foil are close to zero. The exit layer size and the vortex size would change less for the different angles-of-attack.  The effect of wire contour height on the y-velocity profile showed complex flow behaviors with increasing contour height.  The maximum 𝑉=>?@ was found for the intermediate contour height of 0.9 mm while the lowest 𝑉=>?@ was found for the 0.6mm contour height. Furthermore, the simulation showed that the effect of wire contour height could extend the reversal flow time to some extent. The vortex size is also increasing with the increasing contour height as well as the vortex center is moving far away from the slot entrance. 55   Pressure drop coefficient 𝐶*O	is introduced to describe the relationship between the average slot velocity and inlet pressure in this study. The pressure drop coefficient would decrease with increasing 𝑉9	and decreasing	𝑉#. The minimum 𝐶*O is accordance with 5-degree angle-of-attack for the same	𝑉9. For a small average slot velocity (𝑉9  < 1m/s), the contour height has the less effect on the pressure drop coefficient.  In summary, the CFD results have stated the character of the reversal flow, which is critical to clear the apertures with any blockages and thus defining the maximum screen capacity. It could provide some insights to industrial pressure screen and increase the screen capacity by optimizing the screen design and operational paramters. In the future, a wide range of screen rotor including the solid core rotor could be simulated by CFD to determine the maxium capacity of pressure screen. The stapling between different slots that leads to plugging could also be tested in CFD.   56  Bibliography  [1] FLUENT, Inc. Fluent 19.0 user guide, 2019  [2] Sean Delfel. A numerical and experimental investigation into pressure screen foil rotor dynamics. PhD thesis, The University of British Columbia, 2009.  [3] J. Gullichsen and E. J. Harkonen. Medium Consistency Technology I: Fundamental Data. Tappi J., 64(6):69–71, 1981.  [4] Hayder J. Salem. Modelling the maximum capacity of a pulp pressure screen. PhD Thesis, The University of British Columbia, Canada, 2013.  [5] Robert W. Gooding. The passage of fibers through slots in pulp screening. MASc thesis, The University of British Columbia, Canada, 1986.  [6] James A. Olson. The effect of fiber length on passage through a single screen aperture. PhD thesis, The University of British Columbia, Canada, 1996.  [7] Robert W. Gooding.  Flow resistance of screen plate apertures. PhD thesis, The University of British Columbia, Canada, 1996.  57  [8] Satya Mokamati, James A. Olson, and Robert W. Gooding. Numerical study of separated cross-flow near a two-dimensional rough wall with narrow apertures and suction. Canadian J. Chem. Eng., 88(1):33-47, 2010.  [9] Mei Feng, Jaime Gonzalez, James A. Olson, Carl Ollivier-Gooch, and Robert W. Gooding. Numerical simulation and experimental measurement of pressure pulses produced by a pulp screen foil rotor. J. Fluids Eng., 127(2):347-357, 2005.  [10] James A. Olson, S. Turcotte, and Robert W. Gooding. Determination of power requirements for solid core pulp screen rotors, Nordic Pulp Pap.Res.J., 19(2), pp .213-217, 2004.  [11] Suqin Dong, Martha Salcudean, and Ian Gartshore. The effect of slot shape on the performance of a pressure screen. Tappi J., 3(5):3-7, 2004.    [12] Albert Yong, Satya Mokamati, Daneil Ouellet, Robert W. Gooding, and James A. Olson.  Experimental measurement of fibre motion at the feed surface of a pulp screen, Appita J., 61(6): 485–489, 2008.  [13] Veronique Pinon, Robert W. Gooding, and James A. Olson. Measurements of pressure pulses from a solid core screen rotor. Tappi J., 2(10):9-12, 2003. [14] Ashok Kumar. The passage of fibres through screen apertures. PhD thesis, The University of British Columbia, Canada, 1991.  58  [15] Satya Mokamati, James A. Olson, Mark D. Martinez, and Robert W. Gooding. Experimental Study of a Turbulent Crossflow near a Rough Wall with Narrow Apertures, AIChE J. 54(10), 2526-2526, 2008.  [16] Hanna Jokinen, Ari Ammala, Jukka A. Virtanen, Kati Lindroos, and Jouko Niinimaki. Pressure screen capacity-current findings on the role of wire width and height. Tappi J., 6(1):3–10, 2007. [17] Hayder J. Salem, Robert W. Gooding, Mark D. Martinez, and James A. Olson. Experimental study of some factors affecting pulp screen capacity. Nordic Pulp Paper Res. J. 64(2): 218-224, 2014. [18] R. Karvien and L. Halonen. The effect of various factors on pressure pulsation of a screen. Paperi ja Puu, 66(7):80-83, 1984.  [19] T. Wikstrom and T. Rasmuson. Transition modelling of pulp suspensions applied to a pressure screen. J. Pulp Paper Sci., 28:374-378, 2002. [20] Stephen Mahon and Xin Zhang. Computational analysis of pressure and wake characteristics of an aerofoil in ground effect, ASME J. Fluids Eng., 127(2), pp. 290–298, 2005. [21] Robert W. Gooding and R. J. Kerekes. Motion of fibres near a screen slot. J. Pulp Paper Sci., 15(2):59–62., 1989. [22] Jaime Gonzalez. Characterization of design parameters for a free foil rotor in a pressure screen. MASc thesis, The University of British Columbia, Canada. 2002. 59  [23] Jouko Niinimaki, O. Dahl, H. Kuopanportti, and Ari Ammala.  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Tappi J., 77(9):125– 131, 1994. 60  Appendices Appendix A  Effect of different tip speeds and slot velocities     (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05    (c)X/Chord=0.23                                                   (d) X/Chord=0.5    (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure A-1 Pressure Contour and streamlines for Vt =10m/s and Vs=1m/s and angle-of-attack = 5-degree, contour height =0.9mm 61    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05     (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure A-2 Pressure Contour and streamlines for Vt =10m/s and Vs=2m/s and angle-of-attack = 5-degree, contour height =0.9mm 62     (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05      (c)X/Chord=0.23                                                   (d) X/Chord=0.5    (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure A-3 Pressure Contour and streamlines for Vt =15m/s and Vs=0.5m/s and angle-of-attack = 5-degree, contour height =0.9mm 63    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05     (c)X/Chord=0.23                                                   (d) X/Chord=0.5    (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure A-4 Pressure Contour and streamlines for Vt =15m/s and Vs=1m/s and angle-of-attack = 5-degree, contour height =0.9mm 64    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05    (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure A-5 Pressure Contour and streamlines for Vt =15m/s and Vs=4m/s and angle-of-attack = 5-degree, contour height =0.9mm 65    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05    (c)X/Chord=0.23                                                   (d) X/Chord=0.5    (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure A-6 Pressure Contour and streamlines for Vt =20m/s and Vs=1m/s and angle-of-attack = 5-degree, contour height =0.9mm 66    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05    (c)X/Chord=0.23                                                   (d) X/Chord=0.5     (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure A-7 Pressure Contour and streamlines for Vt =20m/s and Vs=2m/s and angle-of-attack = 5-degree, contour height =0.9mm 67   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5    (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure A-8 Pressure Contour and streamlines for Vt =20m/s and Vs=4m/s and angle-of-attack = 5-degree, contour height =0.9mm  68  Appendix B  Effect of different angles-of-attack   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-1 Pressure Contour and streamlines for Vt =15m/s and Vs=0.5m/s and angle-of-attack = 10-degree, contour height =0.9mm 69      (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05     (c)X/Chord=0.23                                                   (d) X/Chord=0.5     (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-2 Pressure Contour and streamlines for Vt =15m/s and Vs=1m/s and angle-of-attack = 10-degree, contour height =0.9mm 70   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5      (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-3 Pressure Contour and streamlines for Vt =15m/s and Vs=4m/s and angle-of-attack = 10-degree, contour height =0.9mm 71       (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05      (c)X/Chord=0.23                                                   (d) X/Chord=0.5     (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-4 Pressure Contour and streamlines for Vt =15m/s and Vs=0.5m/s and angle-of-attack = 0-degree, contour height =0.9mm 72       (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05       (c)X/Chord=0.23                                                   (d) X/Chord=0.5        (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-5 Pressure Contour and streamlines for Vt =15m/s and Vs=1m/s and angle-of-attack = 0-degree, contour height =0.9mm 73    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05         (c)X/Chord=0.23                                                   (d) X/Chord=0.5         (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-6 Pressure Contour and streamlines for Vt =15m/s and Vs=4m/s and angle-of-attack = 0-degree, contour height =0.9mm 74    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05    (c)X/Chord=0.23                                                   (d) X/Chord=0.5         (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-7 Pressure Contour and streamlines for Vt =15m/s and Vs=0.5m/s and angle-of-attack = 15-degree, contour height =0.9mm 75    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05     (c)X/Chord=0.23                                                   (d) X/Chord=0.5       (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-8 Pressure Contour and streamlines for Vt =15m/s and Vs=1m/s and angle-of-attack = 15-degree, contour height =0.9mm 76    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05    (c)X/Chord=0.23                                                   (d) X/Chord=0.5     (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-9 Pressure Contour and streamlines for Vt =15m/s and Vs=2m/s and angle-of-attack = 15-degree, contour height =0.9mm 77    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05    (c)X/Chord=0.23                                                   (d) X/Chord=0.5    (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure B-10 Pressure Contour and streamlines for Vt =15m/s and Vs=4m/s and angle-of-attack = 15-degree, contour height =0.9mm  78  Appendix C  Effect of different wire types   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05    (c)X/Chord=0.23                                                   (d) X/Chord=0.5     (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure C-1 Pressure Contour and streamlines for Vt =15m/s and Vs=0.5m/s and angle-of-attack = 5-degree, contour height =0.6mm 79    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5       (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure C-2 Pressure Contour and streamlines for Vt =15m/s and Vs=4m/s and angle-of-attack = 5-degree, contour height =0.6mm 80    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure C-3 Pressure Contour and streamlines for Vt =15m/s and Vs=1m/s and angle-of-attack = 5-degree, contour height =0.6mm 81    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure C-4 Pressure Contour and streamlines for Vt =15m/s and Vs=2m/s and angle-of-attack = 5-degree, contour height =0.6mm 82   (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure C-5 Pressure Contour and streamlines for Vt =15m/s and Vs=0.5m/s and angle-of-attack = 5-degree, contour height =1.2mm 83    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure C-6 Pressure Contour and streamlines for Vt =15m/s and Vs=1m/s and angle-of-attack = 5-degree, contour height =1.2mm 84    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure C-7 Pressure Contour and streamlines for Vt =15m/s and Vs=2m/s and angle-of-attack = 5-degree, contour height =1.2mm 85    (a)X/Chord=-0.32                                                    (b) X/Chord=-0.05   (c)X/Chord=0.23                                                   (d) X/Chord=0.5   (e)X/Chord=0.78                                                 (f) X/Chord=1 Figure C-8 Pressure Contour and streamlines for Vt =15m/s and Vs=4m/s and angle-of-attack = 5-degree, contour height =1.2mm 

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