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Numerical simulation of the pressure pulses produced by a pressure screen foil rotor Feng, Monica Mei 2003

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NUMERICAL SIMULATION OF THE PRESSURE PULSES PRODUCED BY A PRESSURE SCREEN FOIL ROTOR  By Monica Mei Feng B. Eng., Southeast University, 1990 M . Eng., Southeast University, 1993  A thesis submitted in partial fulfillment of the requirements for 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 March 2003  © Monica Mei Feng, 2003  In  presenting  degree freely  this  at the available  copying  of  department publication  of  in  partial  fulfilment  University  of  British  Columbia,  for reference  this or  thesis  thesis by  this  and study.  for scholarly  his  or  thesis  her  of  DE-6  e  VWoU»»'c«*  (2/88)  Ap^l/«3 3,  <^UO 3  I further  purposes  for financial  The University of British C o l u m b i a Vancouver, Canada  D a t  I  gain  the agree agree  may be  representatives.  permission.  Department  of  shall  It  is  requirements that that  the  for  an  advanced  Library shall  make  it  permission for extensive  granted  by the  understood  not be allowed  head  that  without  of my  copying  or  my written  At our request, the commencement of the period for which the partial license shall operate shall be delayed from April, 17 , 2003 for a period of six months; such operation may be delayed for an additional period with good cause, as determined by the undersigned. th  Candidate: Monica Mei Feng Thesis Supervisors: Df James'X^Oison Dr. Carl Ollivier-Gooch r-V*  •  Department Head: Di?Nimal kajapakse /Dean of Graduate Studies: Frieda Granot  ABSTRACT Pressure screening is the most industrially efficient and effective means of removing contaminants that degrade the appearance and strength of paper and of fractionating fibres for selective treatments and use in specialty products. A critical component of a screen is the rotor which induces a tangential velocity in the suspension and produces pressure pulses on the screen cylinder surface to keep the screen apertures clear. To understand the effect of the key design and operating variables for a NACA foil rotor, a computational fluid dynamic (CFD) simulation was developed using FLUENT, and the numerical results were compared with experimental measurements. All the experimental measurements for negative pressure peak were 50% larger than the numerical results over a wide range of foil tip-speeds, clearances, angles of attack and foil cambers. In addition, it was shown that the magnitude of the pressure pulse peak increases linearly with the square of tip-speed for all the angles of attack studied. The maximum negative pressure pulse occurred for NACA 0012 and 4312 foils at 5 degrees angle of attack and NACA 8312 at 0 degrees. The stall angle of attack was found to be approximately 5 degrees for NACA 8312, 10 degrees for NACA 4312 and 15 degrees for NACA 0012. The positive pressure peak at the screen cylinder surface near the leading edge of the foil was eliminated for foils operating at a positive angle of attack. The magnitude of the negative pressure peak increased as clearance decreased. Increased camber increases both the magnitude and the width of the pressure pulse. In addition to, and more important than, these specific results, we have shown that CFD is a viable tool for the optimal design and operation of rotors in industrial pressure screens.  ii  TABLE OF CONTENTS  ABSTRACT  ii  TABLE OF CONTENTS  iii  LIST O F F I G U R E S  v  NOMENCLATURE  viii  ACKNOWLEDGEMENTS  ix  CHAPTER 1 - INTRODUCTION  1  CHAPTER 2 - BACKGROUND  4  2.1  Airfoil Nomenclature and Airfoil Study Development  4  2.2  Previous Experimental Studies of Pressure Pulse  6  2.3  Previous CFD Studies of Pressure Pulse  9  CHAPTER 3 - E X P E R I M E N T A L W O R K ON PRESSURE PULSE  3.1  Experimental Apparatus  11  11  CHAPTER 4 - CFD MODEL  16  4.1  Numerical method  16  4.2  Mesh Generation  17  4.3  Boundary Conditions for Fluid Flow  18  4.4  Model Validation  19  C H A P T E R 5 - R E S U L T S A N D DISCUSSION  22  5.1  Tip Speed  23  5.2  Angle of Attack  27  5.3  Foil Clearance  32  iii  5.4  Foil Camber  35  CHAPTER 6 - DISCUSSION OF PRESSURE PULSE AND SCREENING PERFORMANCE  46  CHAPTER 7 - CONCLUSIONS  48  CHAPTER 8 - RECOMMENDATIONS FOR FUTURE WORK  50  REFERENCES  51  APPENDIX 1 - MULTIBLOCK DESIGN FOR STRUCTURED GRIDS  53  APPENDIX 2 - USING TECPLOT TO CREATE CONTOUR PLOTS AND STREAMLINES  59  IV  LIST OF FIGURE Figure 1: Airfoil Nomenclature  5  Figure 2: Foils Used in Experimental and Numerical Studies of Pressure Pulses  5  Figure 3: Test Section of Cross Section Screen  12  Figure 4: Schematic diagram of cross sectional screen  13  Figure 5: Foil type of rotor used in the experiment  15  Figure 6: Representative computational mesh  17  Figure 7: Calculation Domain and Boundary Condition  18  Figure 8: Flow Domain for Study of Airfoil in Straight Channel  20  Figure 9: Computed skin friction coefficient distributions over an NACA 0012 airfoil...21 Figure 10: Effect of rotating speed on pressure pulse (0 degree angle of attack)  23  Figure 11: Numerical pressure coefficient vs. x/chord (0 degree angle of attack)  24  Figure 12: Experiment pressure coefficient vs. x/chord (0 degree angle of attack)  25  Figure 13: Numerical pressure coefficient vs. x/chord (20 degree angle of attack)  26  Figure 14: Experimental pressure coefficient vs. x/chord (20 degree angle of attack)  27  Figure 15: Pressure contour and particle path lines; degree of attack (a) = 0, (b) =5, (c) = 10, (d) = 15,(e) = 20  ,  29  Figure 16: Numerical pressure coefficient vs. x/chord for 5 different angles of attack. ...30 Figure 17: Experimental pressure coefficient vs. x/chord for 5 different angles of attack.31 Figure 18: Numerical pressure coefficient vs. x/chord for 4 different clearances (0 degree angle of attack)  33  Figure 19: Numerical pressure coefficient vs. x/chord for 4 different clearances (5 degree angle of attack)  33  Figure 20: Numerical pressure coefficient vs. x/chord for 4 different clearances (20 degree angle of attack)  34  Figure 21: Numerical and experimental suction pulse peak vs. clearance for 3 different angles of attack  35  Figure 22: Numerical Pressure Coefficient vs. Position (x/chord) for 0 degree angle of attack  '.  37  Figure 23: Experimental Pressure Coefficient vs. Position (x/chord) for 0 degree angle of attack  \  37  Figure 24: Pressure Contour and Flow Pattern for NACA 4312 (5 angles of attack)  38  Figure 25: Pressure Contour and Flow Pattern for NACA 8410 (5 angles of attack)  39  Figure 26: Numerical Pressure Coefficient vs. Position (x/chord), NACA4312 (5 angles of attack)  41  Figure 27: Experimental Pressure Coefficient vs. Position (x/chord), NACA4312 (5 angles of attack)  42  Figure 28: Numerical Pressure Coefficient vs. Position (x/chord), NACA8312 (5 angles of attack)  42  Figure 29: Numerical and Experimental Minimum Pressure Coefficient vs. Angle of Attack (Degree)  43  Figure 30: Pressure Pulse Width vs. Angle of Attack  44  Figure 31: Pressure pulse strength vs. Angle of attack  45  Figure A. 1.1: Block Design for NACA 0012 (0 degree angle of attack)  54  Figure A.1.2: Computational Mesh for NACA 0012 (0 degree angle of attack)  55  Figure A. 1.3: Block Design for NACA 0012 (20 degree angle of attack)  57  Figure A. 1.4: Computational Mesh for NACA 0012 (20 degree angle of attack)  57  Figure A.1.5: Convergence Curve for NACA 0012 (20 degree angle of attack)  58  Vll  NOMENCLATURE Cr  skin friction coefficient  C  Pressure coefficient  p  Re  Reynolds number  x  distance along the screen cylinder  X, Y  dimension co-ordinates, Cartesian co-ordinate system  y  dimensionless distance to wall  k  kinetic energy of turbulence  s  rate of dissipation  +  Vlll  ACKNOWLEDGEMENTS I would like to express my sincere gratitude to my supervisors, Dr. James A . Olson and Dr. Carl Ollivier-Gooch, for their great guidance, support and patience throughout my years of study in U B C .  I am grateful to my student colleagues in the research group for their assistance on countless occasions. The experimental work done by Jaime Gonzalez served as the basis of this study.  I am grateful to Brenda Dutka for helping with the format of papers and Judy Mackenzie for assisting with the references.  I would like to pay special tribute to my parents for their unconditional love and constant support.  The financial support from the Advanced Fiber Technologies (AFT), Pulp and Paper Research Institute of Canada (Paprican) and Natural Sciences Engineering Research Council (NSERC) is gratefully acknowledged.  ix  CHAPTER 1 INTRODUCTION  Pressure screening is the most industrially efficient and effective means of removing contaminants that degrade the appearance and strength of paper. In addition, pressure screens are increasingly used to fractionate fibres by length so that the long or short fibre streams can be processed separately or used in high-value paper grades. For these reasons, pressure screens are increasingly important in the production of mechanical and recycled pulp, as well as the manufacture of high-quality mechanical printing papers.  In general, pressure screens divide a contaminated feed flow into an accept stream of clean pulp and a reject stream laden with contaminant. During screening, the pulp and contaminants enter the screening zone tangentially. The pulp suspension flows between a rotor and the inlet side of a cylindrical screen with narrow slot or hole apertures. The fibres flow through the apertures in the screen cylinder and exit through the accept port. The oversized particles and a fraction of the long fibres are retained by the small apertures and travel down the annulus, leaving the screen through the reject port. To prevent the apertures from becoming plugged by the fibres, airfoils are passed over the feed side of the screen surface to create a negative pressure pulse that backflushes the apertures. Along with their support structure, the airfoils and other hydrodynamic elements that serve to unplug the apertures make up the screen rotor.  1  The rotor plays a critical role in screen operation. Despite the differences in rotor design, all rotors serve two main functions. One function is to accelerate the pulp suspension on the feed side of the screen to a high tangential velocity, and to induce turbulence at the surface of the screen plate. The turbulence mixes the suspension and keeps the pulp fluidized. The other more important function is to create a negative pressure pulse that backflushes the screen apertures, clearingfibreaccumulations and preventing plugging of the apertures.  The principal measures of screen performance are: 1) Contaminant removal efficiency, defined as the mass percentage of contaminants leaving the screen through the reject port to that entering the screen; 2) Capacity, defined as the maximum mass flow rate of pulp in the accept stream; 3) Power consumption, defined as the power required by the rotor; and 4) Reject rate, defined as the mass flow of fibres rejected with the contaminants. Achieving high capacity and high efficiency with reduced energy demand at a low reject rate is the goal of an optimal rotor design.  Although rotor design is critical in determining pulp screen performance, the mechanism of pressure pulse generation and the factors that affect the pressure pulse are poorly understood.  To better understand the critical rotor design factors affecting screen  performance, we use computational fluid dynamics (CFD) to analyze the complex flow created by the rotor. In particular, this paper discusses how the key design and operating variables, i.e., rotating speed, clearances, angles of attack and camber of a NACA airfoil rotor influence the pressure pulse at the screen plate surface. The numerical results are  2  also compared with corresponding experimental measurements. This information provides a basic physical understanding of rotor performance experimentally validated computational tool for optimal rotor design.  3  and provides an  CHAPTER 2 BACKGROUND  2.1 Airfoil Nomenclature and Airfoil Study Development Anderson (2001) gave an introduction to fundamentals of aerodynamics and the development of airfoils. In the early 1930s, the National Advisory Committee for Aeronautic (NACA) - the forerunner of NASA - embarked on a series of definitive airfoil experiments using airfoil shapes that were constructed rationally and systematically. Many of these NACA airfoils are in common use today.  Consider the airfoil sketched in Figure 1. The mean camber line is the locus of points halfway between the upper and lower surfaces as measured perpendicular to the mean camber line itself. The most forward and rearward points of the mean camber line are the leading and trailing edges, respectively. The straight line connecting the leading and trailing edges is the chord line of the airfoil, and the precise distance from the leading to the trailing edge measured along the chord line is simply designated the chord (c) of the airfoil. The camber is the maximum distance between the mean camber line and the chord line, measured perpendicular to the chord line. The thickness is the distance between the upper and lower surfaces. The angle of attack a is defined as the angle between the chord and U . The shapes of all standard NACA airfoils are generated by x  specifying the shape of the mean camber line and then wrapping a specified symmetrical thickness distribution around the mean camber line.  4  Figure 1: Airfoil Nomenclature  The N A C A identified different airfoil shapes with a logical numbering system. The first family of N A C A airfoils was the four digit series, such as the N A C A 4312 airfoil. Here, the first digit is the maximum camber in hundredths of chord, the second digit is the location of maximum camber along the chord from the leading edge in tenths of chord, and the last two digits give the maximum thickness in hundredths of chord. For the N A C A 4312 airfoil, the maximum camber is 0.04c located at 0.3c from the leading edge, and the maximum thickness is 0.12c. A n airfoil with no camber is called a symmetric airfoil. For example, the N A C A 0012 airfoil is a symmetrical airfoil with a maximum  NACA 0012 ( 0% Camber)  NACA 4312 (4% Camber)  NACA 8312 (8% C a m b e r )  Figure 2: Foils Used in Experimental and Numerical Studies of Pressure Pulses  5  thickness of 12 percent. Figure 2 shows the foils used in experimental and numerical studies of pressure pulses.  During the 1930s and 1940s, the NACA carried out numerous measurements of the lift, drag, and moment coefficients on the standard NACA airfoils. Even today, the NACA airfoils are sometimes the most expeditious choice of the airplane designer. However, today the power of computational fluid dynamics (CFD) is revolutionizing airfoil design and analysis. The overall automated procedure that results in a completely optimized airfoil shape for a given design point is being made tractable by CFD. CFD solutions of the continuity, momentum, and energy equations for viscous flow (the Navier-Stokes equations) are carried out for the purpose of airfoil design.  2.2 Previous Experimental Studies of Pressure Pulse  Rotor design and operation are critically important for screen performance. There are two classes of rotor design in widespread use: solid-core and foil rotors. The solid-core rotors have a solid cylindrical core with various shaped hydrodynamic elements on the outer surface. The key feature of these rotors is that the flow is only on one side of the element. In foil rotor designs, there is central rotor shaft and vertical foils are held by radial supports. This arrangement allows flow to pass over both sides of the foil, similar to an airplane wing. One important advantage of the foil design is the ability to optimize the angle of attack and the gap between the foil and the screen cylinder for a given type of pulp. This study focuses on the design and operation of foil rotors.  6  The pressure pulse generated by a foil rotor has not been the subject of much published research. However, a few studies have given quantitative information about the effect of rotor design parameters. The literature review given here represents the development of experimental and computational studies on rotor design in the pulp and paper industry.  Some experimental studies have been conducted to explain how screening performance is affected by the pressure pulse generated by the rotor. According to Niinimaki (1998), pressure pulses influence screening in two ways. First, the suction pressure pulses keep the screen apertures unplugged by disrupting the fibre mat and lifting the lodged particles away from apertures into the turbulent zone above the screen plate. Also, the suction pulses move a considerable amount of water and fines from the accept chamber back into the screen basket, and therefore influence screening by diluting pulp on the inner surface of the screen basket. According to Cox and Fredriksson (2000), dewatering or thickening occurs during the phase of positive flow through the screen plate. This loss of water is compensated by filtrate recovered from the accept side of the screen basket by a long suction pulse. The amount of filtrate recovered can be controlled by the intensity and duration of the suction pulse.  Further experimental studies have been done on the factors that influence the shape and magnitude of the rotor pressure pulse and the effect of these pulse parameters on screen performance. Levis (1991) studied the pressure pulse created by a foil type rotor for screening of secondary fibres. He stated that the magnitude and shape of the pressure  7  pulse is dependent upon the foil shape, tip-speed and clearance between the foil and the screen cylinder. He suggested that as rotor speed is increased, the magnitude of the pressure pulse increases. Furthermore, he hypothesized that higher pulse magnitudes decrease the contaminant removal efficiency but increase the maximum capacity of the screen. In addition, he showed that a critical tip-speed exists after which no further increase in capacity is achieved by increasing the rotor speed. Repo and Sundholm (1996) investigated the effect of rotor speed on separation of coarse fibres in a pressure screen. Their experimental results indicated that decreasing rotor speed reduces the mass reject rate and improves separation efficiency compared with a higher speed. Gooding (1996) measured the pressure pulse in an industrial Hooper PSV 2100 pulp screen. His results showed that increasing rotor speed strongly increases pulse strength. While increased rotor speed shortens the duration of the pulse, the shape of the pulse is relatively unchanged. Increasing the speed of the rotor can consume significantly more power. Levis (1991) found that the consumed horsepower increases as the 2.5 power of rotor tipspeed. Niinimaki (1999) presented the power consumption measured at different rotor speeds and pulp consistencies. He showed that power consumption increases as rotating speed increases and consistency has only a small effect on power consumption.  Typical experimental studies on the effects of clearance and foil angle of attack were reported by Y u (1993) and Niinimaki (1999). Y u (1993) studied the pressure pulse signatures for a foil rotor and contoured drum rotor. Yu's results showed a significant decrease in pulse magnitude for foil type rotors when clearance increases. Niinimaki (1999) reported the effects of clearance between the foil and screen surface on screen  8  performance. As the clearance is reduced, the magnitude of the pulse increases, so screening efficiency decreases and capacity increases. He found that changes in foil angle of attack have a much greater effect on screen performance. Furthermore, he showed that a greater angle of attack reduces the screening efficiency markedly with a corresponding increase in screen capacity. In addition, he suggested that screen performance can be optimized by adjusting the foil angle of attack, especially if screening at higher consistencies.  2.3 Previous CFD Studies of Pressure Pulse  A significant number of CFD studies examine the flow around a 2-D airfoil. Although there are still challenges for CFD prediction of turbulent shear and separation, computations can generally predict surface pressures, velocity profiles, skin friction, lift and drag with reasonably good accuracy at angles of attack below stall. Standard CFD methods provide reasonable prediction of flow variations with Reynolds number below stall, but predictions of when stall occurs are inconsistent and the skin friction predictions are often too high. Even with these limitations, CFD is a powerful technique for industrial equipment design and optimization.  Despite a rapid and substantial increase in the use of computer simulation in the pulp and paper industry in recent years, little CFD research has been done to study the pressure pulse generated by foil-type rotors. One useful study in this area was made by Karvinen and Halonen (1984) who assessed rotor pressure pulsations using experimental and  9  computational techniques. They found that the backflushing action of the pressure pulse is created by the acceleration of the flow through the gap between the moving rotor tip and stationary screen plate. This acceleration causes the local pressure on the feed side of the screen plate to decrease to the point that the flow through the aperture reverses. The flow then passes from the accept side of the screen plate to the feed side, and releases any plugged fibres. They used numerical methods to calculate the turbulent velocity field in a clearance between the rotating foil and screen basket and to simulate the pressure pulse generated in a screen. Their results showed that the foil shape greatly affects the form of pressure pulse. In addition, they found that the peak-to-peak pressure difference (maximum pressure minus minimum pressure) increased rapidly with increased rotation speeds.  Wikstrom (2002) investigated the hydrodynamics inside a pulp screen using commercial CFD simulation software. While the general flow behaviour is captured well, pressure pulse data showed a deviation in absolute values when comparing simulations and experiments for water. CFD model overestimated the magnitude of pressure pulses and simulation showed a delay in pressure drop at the beginning of the pulse.  None of the C F D studies on pressure pulse generation included the necessary detailed information required to evaluate the quality of the CFD analysis. Moreover, while several studies have compared foil shapes, none has examined a generic foil shape and considered how parameters affect flow patterns and the pressure pulse.  10  CHAPTER 3 EXPERIMENTAL WORK ON PRESSURE PULSE  The experimental study of pressure pulse presented in this thesis was done by Jaime Gonzalez (Gonzalez, 2002).  3.1 Experimental Apparatus  A comprehensive experimental program to measure the pressure pulses caused by foil rotors for a wide range of design and operating variables was conducted to validate the CFD solutions.  An experimental Cross Sectional Screen (CSS) was designed to simulate a slice of an industrial PSV 2100 pressure screen and has been used in several experimental studies (e.g., Gonzalez, 2002; Pinon, Gooding and Olson, 2002; Olson and Wherrett, 1996). Figure 3 shows the test section of the cross section screen. The CSS test section is 30 cm in diameter and 5 cm in depth. A 6 cm wide removable screen cylinder coupon is located in the bottom wall with the remaining circumference a solid wall. Figure 4 shows schematic of the flow loop and the principal components of this apparatus. The flow loop consists of a 150-litre reservoir tank, a mixer, a pump, two flow meters and two pressure transducers and P V C piping. A flow meter and pressure sensor are installed in both the feed line and rejects line to monitor the flow rate and pressure. The fibre suspension  11  flows through a 25mm diameter round pipe, flow meter and then enters the CSS through the feed port at the upper part of CSS. Some suspension flows out of the device through the openings of the slot coupon (i.e. the accept flow). The rest (i.e. the rejects) leaves the test section through the reject port and flows back to the reservoir tank. The flow rate at feed line and reject line is negligible.  Pressure Transducer  Feed Port  Rotor Reject Port Diameter (30 cm) Accept Port Screen Plate  Figure 3: Test Section of Cross Section Screen  A high-speed data acquisition system, a strain-gauge pressure transducer and an optical encoder that measures rotor position were installed in the CSS to record the form of the pressure pulsations. To measure the resulting pressure pulses a high frequency 4mm  12  diameter pressure transducer is installed flush with the inside (feed) surface on the top wall of the CSS. The optical encoder installed on the shaft sends 2048 signals per revolution to record the precise position of the rotor in the test section.  Pump Figure 4: Schematic diagram of cross sectional screen.  The experimental CSS has been designed to experimentally measure the effect of the key rotor operating and design variables on screen performance and pulse generation. Figure 5 shows the bracket used to hold the foil and vary the rotor foil, clearance and angle of attack in the CSS. The rotor is driven by a 10 hp electric motor, controlled by a variable frequency drive that enables the rotating speed to be varied. The effect of different  13  rotating speeds, clearances, angles of attack and cambers on pressure pulse shape and magnitude were tested on the experimental CSS. The technical characteristics of the CSS and the range of variables tested are summarized as the following:  •  •  Screen cylinder diameter: 300 mm  •  Foils: NACA 0012, 4312 and 8312, all with a chord length of 40 mm  •  Rotor rotating speed: 600, 800, 1000, 1200 and 1400 rpm  •  Clearance between the foil and screen plate: 2, 3, 4 and 5 mm  •  Foil angle of attack: 0, 5, 10, 15 and 20 degrees  The objectives of this study are to numerically calculate the pressure pulse on the screen cylinder wall generated by a rotating foil rotor using commercial CFD software and to compare these predictions with experimentally measured pressure pulse data from the CSS. In addition, the experimentally validated CFD methodology is used to provide a more complete understanding of how the key parameters affect the flow structures around the foil and the resulting pressure pulse profile. Numerical results will be used to establish the relationship between the pressure pulse shape and rotor design and operating variable, e.g., rotating speed, angle of attack, foil clearance and camber. The CFD model developed can be used to optimize these parameters in high performance rotors design.  14  Setting clearance  Setting angle of attack  Changing rotor foil  Figure 5: Foil type of rotor used in the experiment.  15  CHAPTER 4 CFD MODEL  4.1 Numerical Method  The commercial code F L U E N T 5.4 has been used for the numerical solution of the Navier-Stokes equations. The numerical method is based on a finite volume formulation applicable to structured grids. A l l variables, including velocity components and pressure, are the control volume averages. A second-order spatial interpolation method was employed to obtain the velocity components and pressure on the control volume faces from those at the control volume centres. • The control volume face values of the dependent variables are used to evaluate the convective fluxes.  F L U E N T uses a segregated solution method which solves the momentum and continuity equations sequentially. The SIMPLEC algorithm is used to couple pressure and velocity. The pressure-correction equation is solved to update the pressure and face mass flow rate.  The standard k-s  turbulence model is used for all calculations. Transport equations for  k, the kinetic energy of turbulence, and s, the rate of dissipation, are solved together with the continuity and momentum equations.  16  4.2 Mesh Generation  G a m b i t , the m e s h i n g tool p a c k a g e d w i t h 2D  multiblock  experimental  FLUENT C F D  m e s h based o n the surface information  software, w a s used to generate a obtained  from models  C S S apparatus. C a r e must b e t a k e n i n m e s h g e n e r a t i o n to o b t a i n  numerical solutions for pressure pulse simulation.  17  o f the reliable  Figure 6 shows a structured multi-block grid typical of those used in this study. A Cmesh is used around the foil to provide good resolution around the leading edge of the foil, in the wake, and in the boundary layer. Because a logarithmic wall law is used to compute the skin friction coefficient, the first grid cell near the airfoil should be in the logarithmic region; that is, y for the first cell center should be between 30 and 60. Near+  wall mesh spacing was checked as a post processing step to ensure that this requirement was met. A n H mesh is used upstream and downstream of the foil to provide good mesh quality elsewhere.  4.3 Boundary Conditions for Fluid Flow  Outer Wall  Periodic 1  Periodic 2  Figure 7: Calculation Domain and Boundary Condition  18  A rotating reference frame is used so that the flow is modeled in a coordinate system moving at the same speed as the rotating foil. In this case, the flow is steady relative to the rotating frame. For the wall boundary conditions as shown in Figure 7, the relative angular velocity was set to zero for the foil and inner wall which moves at the speed of the rotating frame (and hence are stationary in the rotating frame). Fluid within the inner wall was treated as solid body rotation. To study the effect of inner wall radius on pressure pulse magnitude, we run the simulations for 0 degree angle of attack with inner wall radius of 0.1m, 0.11m, 0.12m and 0.13m. The difference of the negative pressure peak values for 4 different inner wall radiuses is negligible comparing with pressure pulse magnitude. For the convenience of result analysis, the inner wall radius was set as 0.1 lm for all the calculations except for NACA 0012 at 0 and 5 degree angles of attack. The outer wall was stationary in the non-rotating frame of reference, so the velocity was set to zero in the absolute reference frame. If the wall velocities are specified in this manner, only the rotating speed of the reference frame needs to be changed when there is a change in rotating speed of the foil. Periodic boundary conditions are used, because the flows across two opposite planes in the computational model are identical. The flow entering the computational model through one periodic plane is identical to the flow exiting the domain through the opposite periodic plane.  4.4 Model Validation  19  05 m  Figure 8: Flow Domain for Study of Airfoil in Straight Channel  The pressure pulsations are affected by the turbulence model used, especially for separated flows. Therefore, it is worth considering the errors in the k-s  model for CFD  study of pressure pulsation. Model validation was conducted by calculating the skin friction on a NACA 0012 airfoil in a straight channel at zero angle of attack for Reynolds Uc  number (Re = — , U is inlet velocity, c is chord length, y is kinematic viscosity of water 7  ) of 3xl0 . The channel shown in Figure 8 is 0.5m long and 0.2m high. The airfoil is in 6  the middle of the channel with 0.5m chord length. The velocity at the inlet boundary was 59.7m/s and pressure at outlet boundary was 0 Pascal. In Figure 9, the local friction drag coefficient distribution obtained by the k-s  model shows good agreement with the  numerical results calculated using a boundary-layer code by Lombardi, Salvetti and Pinelli (2000). Airfoil skin friction coefficient rapidly increases from a value of zero at the stagnation point to a peak value shortly downstream of the leading edge. This rapid increase is due to the rapidly increasing velocity as the flow external to the boundary  20  layer rapidly expands around the leading edge. Beyond the peak, C f monotonically decreases in the same qualitative manner as for a flat plate. The results for friction drag from the boundary-layer code of Lombardi, Salvetti and Pinelli had been validated with experiment, and were considered the baseline for accuracy. Based on this validation case of an airfoil, we have reasonable confidence in the predicted drag coefficient for attached flows. The study of the effects of different parameters on pressure pulse then followed.  0.010 H •  Fluent Numerical Result  -HK-  B o u n d a r y Layer C o d e , L o m b a r d i , Salvetti a n d Pinelli  0.008 A  o o  0.006 - |  CD O  o  g  0.004 -I  %  0.002 4  o.ooo A —i  0.0  1  1  0.2  1  1  1  0.4  1  0.6  1  1—  0.8  1.0  x/Chord  Figure 9: Computed skin friction coefficient distributions over an N A C A 0012 airfoil.  21  CHAPTER 5 RESULTS AND DISCUSSION  The static pressure at location (x = 0.135 and y = 0.0075) was set to zero for all the calculations. The reference pressure is the average pressure value of the first 18 points along the outer wall starting from the point (x = -0.15, y = 0). The reference pressure was then subtracted from the pressure values at the outer wall. The dimensionless quantity of pressure coefficient is defined as:  P is the pressure values (already subtracted the reference pressure) at the outer wall, p is the density of water and U is rotor tip speed. ti  All the experimental measurements for negative pressure peak are 50% larger than the numerical results over a wide range of foil tip-speeds, clearances, angles of attack and foil cambers. This is influenced by the presence of the side walls at the front and back of the test section. These side walls slow down the flow near them. The fluid velocity in the mid section where the pressure sensor was installed increases. This increased velocity causes the local pressure to drop more, giving higher magnitude of the negative pressure peak. In 2D numerical simulations, we did not count the effects of the side walls which slow down the fluids in the test section.  22  5.1 Tip Speed  Figure 10: Numerical Estimation of Effect of rotating speed on pressure pulse (0 degree angle of attack).  Figure 10 shows numerical results of pressure pulses calculated at 5 different rotating speeds for 0 degree angle of attack. The negative suction pressure peaks are 60 kPa at 1400 rpm (tip speed = 21.55 m/s) and 10 kPa at 600 rpm (tip speed = 9.24 m/s) for a NACA 0012 airfoil. Magnitudes of both positive and negative pressure pulse increase with increased rotating speeds for all the angles of attack studied. Pressure coefficient C„  23  was obtained by normalizing the pressure with the dynamic pressure associated with the foil tip speed and fluid density. All the pressure curves in Figure 10 collapsed into a single C curve, as shown in Figure 11. p  Figure 11: Numerical pressure coefficient vs. x/chord (0 degree angle of attack).  24  0.1 H  1  -4  •  1  -2  •  1  0  1  1  2  <  1  4  Position, x/Chord  Figure 12: Experiment pressure coefficient vs. x/chord (0 degree angle of attack).  Comparing the numerical C curves for 0 degree angle of attack in Figure 11 with p  experimental C curves for the same range of tip speeds in Figure 12, the pressure pulse p  profiles are similar, although the magnitudes of positive and negative pulse are smaller for the numerical cases. Likewise, C F D calculation (Figure 13) captured the shape of pressure pulse for 20 degree angle of attack comparing with the experimental measurements (Figure 14). But, simulation showed a delay in pressure recovery right after the negative pressure peak. This might due to the spatial filtration of 4mm diameter of the pressure sensor and average of the 1000 pressure pulse traces in the experimental measurements. Also the numerical result of pressure pulse for 20 degree is offset about a half chord length ahead of experimental result.  The collapsed C curves in Figure 11 and Figure 13 further prove that magnitudes of p  positive and negative suction pressure pulse peak are proportional to the square of foil tip velocity.  0.05-,  Position, x/Chord  Figure 13: Numerical pressure coefficient vs. x/chord (20 degree angle of attack).  26  Position, x/Chord  Figure 14: Experimental pressure coefficient vs. x/chord (20 degree angle of attack).  5.2 Angle of Attack The foil's angle of attack greatly affects the flow around the airfoil and the resulting pressure pulse.  For all the pressure contour and streamline plots, the x, y coordinates are in metres and foil chord length is 4 cm. The typical variation of pressure distribution and flow pattern with angle of attack for NACA 0012 is shown in Figure 15. As seen from the streamlines for low angle of attack (under 10 degrees), the flow is smooth over the foil and is attached over most of the surface. However, as the angle of attack increases to 10 degrees or larger the flow tends to separate from the top surface of the foil, creating a vortex that  27  extends beyond the trailing edge of the foil, as shown by the streamlines in the figure. One consequence of this separated flow at high angle of attack is a large increase in pressure drag on the foil. Under these conditions, the increase in drag will increase the amount of power required to operate the rotor.  28  Figure 15: Pressure contour and particle path lines: degree of attack (a) = 0, (b) (c) = 10, (d) = 15, (e) = 20.  29  0.1  1 -4  1  1  1  1  -2  0  •  1  2  '  1 4  Position, x/Chord  Figure 16: Numerical pressure coefficient vs. x/chord for 5 different angles of attack.  Figure 16 shows the calculated pressure pulse profiles on the screen cylinder surface for the NACA 0012 foil with 5 different angles of attack (0, 5, 10, 15 and 20 degree). The corresponding experimental results for the effect of angle of attack are shown in Figure 17.  In general there is an excellent correspondence between the experimental  measurements and numerical predictions.  Both numerical results (Figure 16) and  experimental results (Figure 17) show that 5 degree angle of attack has the highest negative suction pressure pulse, which is expected to give the highest capacity. Both experimentally and computationally, the value of negative suction pulse peak for 0 degrees angle of attack falls between the values for 15 and 20 degree angle of attack. As  30  angle of attack increases beyond 5 degrees, the magnitude of the negative suction pulse decreases with increased angle of attack. Again, this trend holds true for both the numerical and experimental cases.  -0.8  |  -4  i  |  -2  1  1 0  1  1 2  1  1 4  Position, x/Chord  Figure 17: Experimental pressure coefficient vs. x/chord for 5 different angles of attack.  When the foil is at a non-zero angle of attack, near 5 degrees, the positive pressure pulse at the leading and trailing edges of the foil are approximately zero. The positive pressure pulse is reduced for positive angles of attack because the stagnation point at the leading edge of the foil is located on the surface of the foil opposite to the screen cylinder and the foil effectively blocks the high pressure region around the stagnation point from the cylinder. The resulting pressure pulse is effectively flat with a sharp negative peak only in the vicinity of the foil. Reduction of the positive pressure pulse while maintaining a  31  high negative pulse may significantly increase contaminant and fractionation efficiency of the screen while increasing the maximum achievable capacity of the screen.  In both Figure 16 for numerical C results and Figure 17 for experimental C results there p  p  is a pressure plateau after the negative pressure peak. This phenomenon is observed for angles of attack higher than 15 degrees. This plateau is a result of flow separation from the upper surface of the foil. Thus flow separation on the foil surface has a significant effect on pressure pulse shape along the screen plate.  Certainly, we see that foil angle of attack strongly affects the shape and magnitude of the pressure pulse generated on the screen cylinder and that angle of attack is an important design parameter that needs to be optimized for a specific application, especially in higher consistency applications where tip-speed is constrained.  5.3 Foil Clearance  The clearance between the foil and screen cylinder also affects the magnitude of the pressure pulse on the screen cylinder surface. Figures 18, 19 and 20 show the pressure pulses on the screen cylinder surface for 3 different angles of attack (0, 5 and 20 degree) with a range of clearances (2, 3, 4 and 5 mm). Although increasing the clearance decreases the magnitude of both the positive and negative pressure pulse peaks, increasing the clearance appears to have a stronger effect on the suction pressure peak.  32  0.1 - i  0.0 J  a U  «01 0  -0.1 -I  1  -Clearance 5% •Clearance 7.5% •Clearance 10% • Clearance 12.5%  -0.2  o 01 3  o f chord o f chord of chord o f chord  -0.4.  Position, x/Chord  Figure 18: Numerical pressure coefficient vs. x/chord for 4 different clearances (0 degree angle of attack).  0.0  5" -  02  c  Clearance 5% Clearance 7.5% Clearance 10% Clearance 12.5%  OI  o £  -0.4 4  8  oSi 01  -0.6  of of of of  chord chord chord chord  4  Q.  -0.8  0 Position, x / C h o r d  Figure 19: Numerical pressure coefficient vs. x/chord for 4 different clearances (5 degree angle of attack).  33  Figure 18 shows the magnitude of the computed negative suction pressure peaks as a function of clearance for the range of angle of attacks along with experimental results for the zero angle of attack foil. The experimental and computed peaks demonstrate the same trend: reducing the clearance increases the magnitude of the negative pressure pulse. We also see that the foil at 5 degrees angle of attack, which creates the greatest negative peak magnitude, is the most sensitive to variations in clearance.  0.05  -i  Position, x/Chord  Figure 20: Numerical pressure coefficient vs. x/chord for 4 different clearances (20 degree angle of attack).  34  —•—Angle —•—Angle — A — Angle --*--Angle  i  5  1  1  6  1  1  7  1  1  8  1  of Attack 0 Degree of Attack 5 Degree of Attack 15 Degree of Attack 0 Degree  1  1  9  1  10  1  (Numerical) (Numerical) (Numerical) (Experimental)  1  11  1  1  ' 12  1 13  C l e a r a n c e (% o f c h o r d )  Figure 21: Numerical and experimental suction pulse peak vs. clearance for 3 different angles of attack.  Decreasing the clearance increases the pressure pulse magnitude and the rotor's ability to clear plugged apertures; however, in practice, decreasing the clearance increases the probability of large contaminants becoming wedged between the cylinder and the rotor, damaging the cylinder. For this reason, foil clearances are seldom below 2 mm.  5.4 Foil Camber  To determine the effect of foil camber on the magnitude and shape of the rotor pressure pulse, numerical and experimental studies of NACA 0012 (no camber), NACA 4312  35  (4% camber) and NACA 8312 (8% camber) airfoils were conducted. Classical aerodynamics predicts that a cambered airfoil will produce more lift than a non-cambered airfoil at a given angle of attack. This is because the flow moves faster over the top of the curved surface than the lower surface which results in a lower pressure on the top surface, hence creating lift. We expect that the same mechanism will result in a higher negative pressure pulse on the screen cylinder.  Figure 22 shows the numerically determined pressure pulses for the three airfoils with 0%, 4% and 8% camber at 0 degree angle of attack. The increased camber increases both the magnitude and the width of the pressure pulse. Figure 23 shows the experimentally determined pressure pulses for the same three airfoils as Figure 22 (0%, 4% and 8%). The experimentally measured pressure pulses show the same increase in magnitude and width as shown in the numerical results, further validating the numerical predictions.  36  Figure 22: Numerical Pressure Coefficient vs. Position (x/chord) for 0 degree angl of attack  Position, x/Chord  Figure 23: Experimental Pressure Coefficient vs. Position (x/chord) for 0 degree angle of attack  37  Figure 24: Pressure Contour and Flow Pattern for NACA 4312: degree of attack (a) = 0 , (b) =5, (c) = 10, (d) = 15, (e) = 20.  38  016 r  Static Pressure f 2.08E+04 j 1.77E+04 1.46E+04 1.15E+04 8.40E+03 5.3DE+03 2.19E+03 -9.17E+02 -4.02E+03 -7.13E+03 -1.02E+04 -1.33E+04 -1.64E+04 -1.96E+04 -2.27E+04  Figure 25: Pressure Contour and Flow Pattern for NACA 8410: degree of attack (a) = 0 , (b) =5, (c) = 10, (d) = 15, (e) = 20.  39  Figures 24 and 25 show the flow streamlines and pressure distribution for the NACA 4312 and NACA 8312 at 5 angles of attack equal to 0, 5, 10, 15 and 20 degrees. The filled pressure contours show that the stagnation point progressively moves downstream of the leading edge over the bottom surface of the airfoil as the angle of attack is increased. This results in a decrease of the positive pressure pulse at the screen cylinder surface near the leading edge of the foil.  As foil camber is increased, the adverse pressure gradient on the foil surface facing the screen cylinder increases, and the flow separates at lower angles of attack. The stall angle of attack is approximately 5 degrees for NACA 8312, 10 degrees for NACA 4312 and 15 degrees for NACA 0012. More precise determination of the stall angle would require computation at intermediate angles of attack. The stalling phenomenon shown in Figure 24 & 25 is trailing-edge stall. We see a progressive and gradual movement of separation from the trailing edge toward the leading edge, as angle of attack is increased. The larger low pressure region for cambered foils in the gap between the rotating foil and screen cylinder means that the width (or duration) of the negative pressure pulse is larger for cambered foils than for non-cambered foils.  Figures 26 and 27 show the numerical and experimental pressure pulse profiles on the screen cylinder wall for a NACA 4312 foil with 5 different angles of attack (0, 5, 10, 15 and 20 degree). Both results show that 5 degree angle of attack has the highest negative suction pressure pulse, which is expected to give the highest capacity. As the angle of attack increases beyond 5 degrees, the magnitude of the negative suction pulse decreases  40  with increasing angle of attack. Again, the positive pressure peak near the leading edge of the foil is completely eliminated for foils operating at a positive angle of attack.  0.1 -. 0.0Q.  o  i r C  CD  -0.1 -0.2-  -  effi  O  -0.3-  o  O  CD  -0.4-  3 CO CO  -0.5-  |  £  CL  -  -0.6-0.7  -1  V  NACA 4312 (Angle of Attack 0 Degree) (Angle of Attack 5 Degree) (Angle of Attack 10 Degree) (Angle of Attack 15 Degree) (Angle of Attack 20 Degree)  0  1  2  Position, x/Chord  Figure 26: Numerical Pressure Coefficient vs. Position (x/chord), NACA4312 (5 angles of attack)  41  Figure 27: Experimental Pressure Coefficient vs. Position (x/chord), NACA4312 (5 angles of attack)  i  -  •  1  0  i  '  i  1  1  1  2  Position, x/Chord  Figure 28: Numerical Pressure Coefficient vs. Position (x/chord), NACA8312 (5 angles of attack)  42  Figure 28 shows the computed pressure pulses at the screen cylinder for the N A C A 8312 foil. This figure indicates that the magnitude of the negative suction pulse decreases as the angle of attack increases due to the onset of separation. Thus for a N A C A 8312 foil the maximum pressure pulse occurs at 0 degree angle of attack.  Figure 29 shows the minimum of the pressure pulse for the three cambers over all angles of attack for both the experimental and numerical results. The numerical and experimental peak magnitudes have the same trends.  However, the experimentally  measured negative pulse peak is significantly less than the numerical peaks for the N A C A 4312 at 15 and 20 degree angle of attack. The reason for the discrepancy between the experimental and numerical pressure peaks for only these two points in unknown.  - • - N u m e r i c a l NACA0012 - • - N u m e r i c a l NACA4312 - A - Numerical N A C A 8312  c jg  -°-  75  4  1 0  <  1  1  5  1 10  1  1 15  1  r 20  Angle of Attack (Degree)  Figure 29: Numerical and Experimental Minimum Pressure Coefficient vs. Angle of Attack (Degree)  43  For the 3 foils and 5 different angles of attack we studied, the NACA 4312 (4% camber) at 5 degree angle of attack has the highest negative suction pulse peak. As angle of attack increases beyond 5 degrees, the magnitude of the negative suction pulse decreases with increased angle of attack because of flow separation. The largest negative peak for the NACA 8312 foil occurs at 0 degree angle of attack. This is because flow separation occurs at less than 5 degree angle of attack.  It is still relatively unknown what is required for improved screen performance, either a stronger negative pulse or a wider negative pulse. For this reason we examine the pulse width in addition to the pulse magnitude. Figure 30 shows that the negative pressure pulse width at half magnitude of negative pressure pulse decreases as angle of attack increases for NACA 0012, 4312 and 8312 foils. The pulse widths for cambered foils are wider than that for non-cambered foil.  0.50-1 0.480.46-  - • -  0.44 0.42-1  N A C A 0012 (0% Camber)  •  N A C A 4312 (4% Camber)  - A -  N A C A 8312 (8% Camber)  O 0-40-1 O  0.38-1  =5 0.36 A 5 0.34 <H 0.32 °-  0.30-1 0.28  j  0.26 0.24 -r 10 -  15  20  Angle of Attack (Degree)  Figure 30: Pressure Pulse Width vs. Angle of Attack  44  -0.04 -0.06 -0.08 -0.10£  -0.12-  g  -0.14-  v.  W  o  - • -  N A C A 0 0 1 2 (0% C a m b e r ) N A C A 4312 (4% C a m b e r ) - A - N A C A 8312 (8% C a m b e r )  -0.16-  2> -0.183  °"  -0.20-0.22 -0.24 -0.26 0  5  10  15  20  Angle of Attack (Degree)  Figure 31: Pressure pulse strength vs. Angle of attack The pulse strength (defined as pulse width normalized by chord length multiplied by minimum pressure coefficient) is hypothesized to be critical to screen operation. Figure 31 shows that the pulse strengths for N A C A 4312 at 5 degrees and N A C A 8312 at 0 degrees have the highest value with almost the same magnitude.  45  CHAPTER 6 DISCUSSION OF PESSURE PULSE AND SCREENING PERFORMANCE  Although it is clear that the performance of pressure screens is strongly affected by the shape and magnitude of the pressure pulse induced by the rotor, and this study shows how the key operating and design variables affect the pulse, the exact relation between screen performance and pulse shape is still unclear. The positive pressure pulse is hypothesized to decrease the contaminant removal efficiency of the screen. A high positive pressure pulse (found at the leading and trailing edge of the foil at zero degrees angle of attack) forces contaminants through narrow apertures, decreasing efficiency. This is especially true for flexible fibres and deformable contaminant particles, such as stickies, which are adhesives found in recycled pulp. The small to non-existent positive pressure pulse generated by foils at non-zero angle of attack would be ideal for removing deformable contaminants as they will not be pushed through the apertures by the positive pressure pulse.  The maximum achievable capacity is believed to be strongly correlated with the magnitude of the negative pressure pulse. The negative pulse reverses the flow through the apertures, backflushing the apertures and clearing particles lodged near the slot inlet and lifting them into the turbulent zone. As the peak strength increases, the suction pulse  46  is more effective at clearing the screening apertures. Increasing the rotation speed will increase the pulse strength and the frequency of pulses.  Using an airfoil that produces a strong negative pressure peak provides the opportunity to reduce the total energy consumed by the rotor by providing a sufficient negative pulse at a low rotational speed. Ideally, the rotor speed would be set just high enough to meet the required capacity and runnability but no higher to minimize energy usage. Optimization of the rotation speed, clearance, angle of attack and foil camber makes it possible to achieve increased efficiency at a reduced power consumption for each pulp type.  As discussed previously, the ability of the foil to remove pulp accumulation may be due to a combination of pulse height and width.  For this reason, we defined the pulse  strength as a combination of both width and height. The pulse strength defined above may indicate how much suspension flows back into the screening zone from the accept chamber, and in turn, may indicate the ability of the screen to reduce thickening of the rejects. If the negative suction pulse is extended, such as with a cambered foil at low angles of attack, pulp thickening at the reject port may be reduced.  47  CHAPTER 7 CONCLUSION Using both numerical and experimental approaches, the key design and operating variables affecting the pressure pulse on the surface of a pressure screen cylinder were investigated. A l l the experimental measurements for negative pressure peak are 50% larger than the numerical results over a wide range of foil tip-speeds, clearances, angles of attack and foil cambers due to the side wall effect.  In this study, the tip speed, angle of attack, clearance and camber were shown to greatly affect the pressure pulse generated by the foil. In particular, we have been able to show that:  1. The magnitude of the pressure pulse peak increases linearly with the square of rotor tip speed for all angles of attack studied. 2. The maximum negative pressure pulse occurs for the N A C A 0012 and 4312 foils at 5 degree angle of attack and N A C A 8312 at 0 degree. The positive pressure peak near the leading edge of the foil is completely eliminated for foils operating at greater than or equal to 5 degree angle of attack. 3. The magnitude of the negative pressure peak increases as clearance decreases. 4. The stall angle of attack is approximately 5 degrees for N A C A 8312, 10 degrees for N A C A 4312 and 15 degrees for N A C A 0012. As the foil camber is increased, the flow separates at lower angles of attack because of the higher adverse pressure gradient on the foil surface near the screen cylinder.  48  5. The magnitude and the width of the negative suction pulse increase with increased camber. 6. The optimal operating foil angles of attack are around 5 degrees for NACA 0012 and 4312 and 0 degrees for NACA 8312.  Most importantly, we have shown that CFD is an important tool for the optimal design and operation of rotors in industrial pressure screens.  49  CHAPTER 8 RECOMMENDATIONS F O R F U T U R E W O R K This thesis has shown how CFD and CSS experimental studies can be combined to elucidate the key design and operating variables affecting the pressure pulse on the surface of a pressure screen cylinder. Some studies which would extend this understanding would examine the following topics:  •  Improve the C F D model to be able to predict the pressure pulses under different consistencies of pulp suspension.  •  Investigate pulse interaction with screen slots.  •  Investigate the relationship between screen performance, such as capacity, efficiency and thickening with the pressure pulse magnitude and width.  •  Examine different foil shapes to increase the negative pulse peak.  •  Experimentally confirm pulse strength correlation with screen performance.  •  Optimize rotor designs for different industrial applications.  •  Compare competitive foil shapes.  •  Conduct industrial trials for the best foils under different operating conditions.  50  REFERENCES  1. Anderson, J. D., "Fundamentals of Aerodynamics", 3 ed., McGraw-Hill Book rd  Company, 2001. 2. Cox, M., Fredriksson, B., and J. Koikkalainen, "Mill Experience with Highconsistency, Screening Technology for Recycled Fibre", Paper Technology, 41(9), 31-36 (2000). 3. Gonzalez, J.A., "Characterization of Design Parameters for a Free Foil Rotor in a Pressure Screen", M.A.Sc. thesis, The University of British Columbia (2002). 4. Gooding, R.W., "Flow Resistance of Screen Plate Apertures", Ph.D. thesis, The University of British Columbia (1996). 5. Karvinen, R., and L. Halonen, "The Effect of Various Factors on Pressure Pulsation of a Screen", Paperija Puu, 66(2), 80-83 (1984). 6. Levis, S.H., "Screening of Secondary Fibers", Progress in Paper Recycling, 5(1), 31-45 (1991). 7. Niinimaki, J., and O. Dahl, "A Comparison of Pressure Screen Baskets with Different Slot Widths and Profile Heights - Selection of the Right Surface for a Groundwood Application", Paperi ja Puu - Paper and Timber, 80(8), 31-45 (1998).  8. Niinimaki, J., "Phenomena Affecting the Efficiency of a Pressure Screen", Tappi Pulping Conf, 957-966 (1999). 9. Olson, J.A., and G.W. Wherrett, " A Model of Fibre Fractionation by Slotted Screen Apertures", J. Pulp and Paper Sci., 24 (12), 398-403 (1998). 10. Pinon, V., Gooding, R.W., and J.A. Olson, "Measurement of Pressure Pulses from a Solid Core Rotor", Tappi Pulping/Engineering Conf, San Diego (2002). 11. Repo, K . , and J. Sundholm, "The Effect of Rotor Speed on the Separation of Coarse Fibres in Pressure Screening with Narrow Slots", Pulp & Paper Canada, 97(7), T253-257 (1996).  12. Lombardi, G., Salvetti, M . V . , and D. Pinelli, "Numerical Evaluation of Airfoil Friction Drag", J. Aircraft, 37(2), 354-356 (2000).  13. Wikstrom, T., "Transition Modelling of Pulp Suspensions Applied to a Pressure Screen", J. Pulp and Paper Sci., 28(11), 374-378 (2002).  14. Yu, C.J., and R.J. DeFoe, "Fundamental Study of Screening Hydraulics Part 3: Model for Calculating Effective Open Area", Tappi J., 77(9), 125-131 (1994).  52  APPENDIX - 1 B L O C K DESIGN FOR S T R U C T U R E D GRIDS  This appendix explains the block designs that were used to generate the mesh for two dimensional N A C A 0012 foil geometries and provides some basic information about the grid distribution required to predict the flow behaviour and pressure pulse reasonably well.  Grid generation is an essential aspect of all numerical methods for the solution of partial differential equations (PDE). Grid generation consists in sub-division of the flow domain into a number of smaller sub-domains: a grid (or mesh) of cells (or control volumes). A good grid can accelerate the convergence of the solution, while a bad grid can lead to a divergent iteration history. Adequate resolution of important flow phenomena, such as vortices, boundary layers and wakes, and separation regions are crucial to obtaining physically representative flow solutions. This implies that in areas with large gradients of a physical parameter (such as velocity and pressure), the mesh should not be too coarse, otherwise the desired physical phenomenon can not be resolved properly. Also, to maintain the accuracy of difference approximations of the derivatives in flow solvers, structured grids have to be smooth and orthogonal. To address these issues for domains with complex geometries, a large number of solvers today use structured multiblock approach. With Fluent CFD software it is possible to have several blocks which have a regular structured grid combined together. Fluent uses structured grids in an attempt to gain the speed advantage that comes from using a regular mesh while retaining the  53  flexibility to model complex geometry. This combination of meshes is called a multiblock grid as it can be seen as a series of blocks built together. In a multiblock structured grid, there is a two level subdivision of solution domain. The solution domain is first subdivided into several subdomains in such a way that each subdomain can be fitted with a structured grid with good properties (not too non-orthogonal, individual control volume aspect ratios not too large). Since the accuracy of the solution depends as much on the grid quality as on the approximations used for discretization of the equations, block design involves constructing the blocks to fill the irregular domain so that each block can be filled with a high quality structured grid.  Figure A . l . l : Block Design for NACA 0012 (0 degree angle of attack).  54  The block design is based on the geometrical features and the physical problem to be solved. Figure A . 1.1 shows the multiblock design used for a N A C A 0012 at 0 degree angle of attack. The computational domain was divided into 30 blocks and total of 10,000 cells. Figure A. 1.2 shows the correspondent structured grid around a N A C A 0012 foil. The outer grid in Figure A. 1.2 is of H-type and the block grid around the foil is of C-type. The C type of grid is often used for bodies with sharp edges to have good quality grid. This approach enhanced grid quality near the foil and leaded to both more accurate solutions and better convergence of numerical solution method. The ratio of the sizes of adjacent cells was kept under control, as accuracy is adversely affected i f it is large.  0 . 1 5  5-  0 . 1 3 5  0 . 1 2  0  - 0 . 0 2  0 . 0 2  X  Figure A.1.2: Computational Mesh for NACA 0012 (0 degree angle of attack).  55  As the angle of attack increases, the gap distance between the foil and screen cylinder increases rapidly from the leading edge to the trailing edge, the block design showed in Figure A. 1.1 can no longer give high quality grid in the gap region along the whole foil length. In Figure A. 1.3 a multiblock design of structured grid for the calculation of 2D flow around NACA 0012 at 20 degree angle of attack is shown. The computational domain contained 51 blocks. The grid used for the study of 20 degree angle of attack had a total of 30,400 cells. In Figure A. 1.4 block-structured grid matches at the interfaces and the cells are of nearly equal size near block interface. The domain mesh presented in Figure A. 1.4 for a rotor foil with tip clearance gap shows the advantages of such an approach: A C - mesh was wrapped around the foil and H meshes were used upstream of the leading edge and downstream of the trailing edge. This kind of grid is flexible as local refinement is possible blockwise (i.e., the grid may be refined in some blocks). The locations of grid points on the boundaries were determined prior to the generation of the internal grid. The mesh was refined near the foil to provide good resolution around the leading edge of the foil, in the wake, and in the boundary layer. Grid lines intersected the solid boundaries of foil and screen cylinder at right angles. The grids with skew angles were kept away from the foil and the screen cylinder (outer wall). The quality of the mesh was checked to avoid very skewed cells and high aspect ratio cells especially around the foil and in the gap between the foil and screen cylinder.  56  Figure A.1.3: Block Design for NACA 0012 (20 degree angle of attack).  Figure A.1.4: Computational Mesh for NACA 0012 (20 degree angle of attack).  57  Because a logarithmic wall law is used to compute the velocity near the wall, the first grid cell near the airfoil should be in the logarithmic region, that is y for the first cell +  center should be between 30 and 60. Near-wall mesh spacing was checked as a post processing step to ensure that this requirement was met. The residual is related to the physical complexity, the numerical algorithm chosen and also to the quality of the mesh. The mesh used had a stable convergence curve and all calculations were run until the mass, x-velocity, y-velocity, k and epsilon residuals had leveled out. The convergence of mass, x-velocity, y-velocity, k and epsilon residuals for a typical case are shown in Figure A. 1.5. For the converged result, the calculation was taken as being correct when the mass residual dropped below 10" and x-velocity, y-velocity, k and epsilon residuals 5  dropped below 10" . 7  Residuals —continuity  1e-08 4—-—,—.—,—.—,—.—,—,—,—,—,—.—,—.—,—,—,—,—, 0  2000 4000 6000 8000 1 0 0 0 0 1 2 0 0 0 1 4 0 0 0 1 6 0 0 0 1 8 0 0 0 20000  Iterations  Scaled Residuals  Mar 24, 2003 F L U E N T 5.4 (2d, segregated, ke)  Figure A.1.5: Convergence Curve for NACA 0012 (20 degree angle of attack).  58  APPENDIX - 2 USING TECPLOT TO CREATE CONTOUR AND STREAMLINE PLOTS  To visualize a wide range of numerical data, Tecplot was used to create pressure contours and streamlines. In Figure A . 2 . 1 , the flooded pressure contour plot gives an immediate visual impression of how the pressure is changing. The streamline profile reveals the flow separation on the foil and the flow recirculation zone.  A . l Create pressure contour plots A pressure contour plot needs three variables (X and Y coordinates and pressure) to show the variation of static pressure across the datafield.The basic procedures for creating a pressure contour plot are as following: 1. Read in Fluent datafilein Tecplot format. 2. Assign the contour variable to be static pressure. 3. Deselect the Mesh check box to turn off the mesh layer. 4. Create contour legend.  The pressure contour legend relates the displayed colors to the actual values of the static pressure. Creating the pressure contour legend involves following steps: 1. From the Field menu, select Contour Legend. The Contour Legend dialog appears.  59  2. Select the check box labeled Show Contour Legend. 3. Select the check box labeled Show Header to include the name of the contour variable -static pressure. 4. Enter the number of levels between each entry in the legend in the Level Skip text field, and enter the line spacing between entries in the Line Spacing text field. Together, these two text fields control the overall size of the legend. 5. In the Tecplot sidebar, click Redraw.  A.2 Generating streamlines  The streamlines were placed on top of pressure contour plotting layer. Because streamlines are dependent upon velocity field, the vector components of velocity are defined as X-relative velocity and Y-relative velocity before creating the streamlines.  Streamlines in Figure A.2.1 were added to the plot in a rake, which is a set of streamlines with starting positions along a defined line. The streamlines are placed using the Streamtrace Placement dialog which gives the precise control over the starting points. Define the line by entering the starting and end position by specifying the X and Y coordinates. By default, Tecplot draws ten streamlines per rake.  To capture all the information on the plot, a layout file was saved for later use with a different set of data. When opening the Layout file with different data files, the new data file overrides the data files that were referenced in the layout file, yet the X , Y value ranges, the position of the rake, the number and range of contour levels, and the contour  60  color map are all preserved. Use of a layout file makes all the plots consistent and saves data post-processing time.  61  

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