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Investigation of fibre interaction with a forming fabric Li, Jingmei 2013

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Investigation of Fibre Interaction with a Forming Fabric by  Jingmei Li  B.Sc., Qingdao Agricultural University, China, 2005 M.Sc., Tsinghua University, China, 2008  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy  in The Faculty of Graduate Studies (Mechanical Engineering)  The University of British Columbia (Vancouver) January, 2013 © Jingmei Li, 2013  Abstract The forming stage of the papermaking process plays an important role in shaping the quality of final paper sheets. This thesis focuses on studies of fibre motion in the forming section. Wire mark was investigated both numerically and experimentally. Initial sheet forming was simulated with hundreds of fibres of random initial distribution placed into the flow above the fabric and advected onto the fabric. The surface roughness of the resulting fibre mat was calculated. The results show that during initial formation, topographic wire mark is caused in part by fibre bending and in part by the geometry of the fabric. For the specific fibres and sinusoidal forming fabric examined, more than 50% of topographic wire mark was caused by fabric geometry, with the remainder caused by fibre bending. In the experiments, the surface roughness of paper sheets made from different fibre properties was studied using an X-ray tomography device. Light-weight fibre mats were made in a handsheet former machine. A surface map of the wire side of the paper was produced via image analysis. The results reveal that increasing fibre coarseness decreases the surface smoothness of paper. As fibre length increases, surface roughness decreases slightly. Both fine and coarse forming fabrics were used in the sheet forming section. The surfaces of fibre mats made from finer forming fabrics were found to be smoother. The fibre orientation distribution of final paper sheets is closely dependent on the physical properties of the sheets. Fibre orientation in the forming section was studied numerically. In the simulations, one end of each fibre was held by the wire/fibre mat, with the other end carried in the flow. In the uniform flow, analyzed solution from the analysis was obtained. The fibre angle after deposition was only found to be a function of flow direction and initial fibre position. In the shear flow, a dimensionless group of was defined. As the value of  increases, fibre mats increase in anisotropy.  Fibre properties such as flexural rigidity and aspect ratio were found to have a insignificant effect on fibre orientation.  ii  Preface Versions of the data presented in this dissertation have been published or submitted for publication: 1. Li, J., & Green, S. I. (2012). Fibre interaction with a forming fabric. Tappi Journal, 11(8), 39-46 2. Li, J., & Green, S. I. (2012). Investigation of fibre interaction with a forming fabric. In proceedings of the Annual Conference of the Canadian Pulp and Paper Industry, PaperWeek Canada 2012, Montreal. 3. Li, J., & Green, S. I. (2012). X-ray Microtomography Measurements of Paper Surface Roughness. Nordic Pulp Paper Research Journal, 27(5), 952-957. The Chapters in this dissertation were created based on these publications. Chapter two includes the data in article one and two in the list. I am responsible for all aspects of work which includes, but not limited to, developing the code, conducting the simulations, analyzing data. Dr. Green put insight during the work. With his direction, I performed the data analysis and summarized the study. I wrote the manuscript and it was revised and edited by Dr. Green. Chapter 3 includes the data in article three in the list above. The authors of chapter 3 are Jingmei Li and Sheldon Green. Dr. Green initiated the experimental study of measuring fibre mat surface roughness. I am responsible for producing all the samples, conducting the microtomography measurements and analyzing the image data. With Dr. Green’s suggestions, I performed the data analysis. I wrote the manuscript and Dr. Green revised and edited it. The authors of chapter 4 are Jingmei Li and Sheldon Green. I am responsible for developing the code and conducting all the simulations. Data analysis was performed  iii  with direction from Dr. Green. I wrote the manuscript with revisions and suggestions from Dr. Green.  iv  Table of Contents Abstract ............................................................................................................................... ii Preface ............................................................................................................................... iii Table of Contents ................................................................................................................ v List of Tables ................................................................................................................... viii List of Figures .................................................................................................................... ix Lists of Symbols and Abbreviations ................................................................................. xii Acknowledgements.......................................................................................................... xiii Chapter 1 Introduction ........................................................................................................ 1 1.1  Background .......................................................................................................... 1  1.2  Assumptions ......................................................................................................... 2  1.2.1  Fibre suspensions.......................................................................................... 2  1.2.2  Hydrodynamics of fibres .............................................................................. 3  1.3  Literature Review ................................................................................................. 4  1.3.1  Wire mark and paper structure (Chapter 2&3) ............................................. 4  1.3.2  Particle motion in a flow field (Chapter 2&3&4) ........................................ 7  1.3.3  Particle orientation in a flow field (Chapter 2&3&4) ................................ 10  1.3.4  Flow through a forming fabric (Chapter 2&4) ........................................... 12  1.3.5  Particle motion in the bounded flow (Chapter 2&4) .................................. 13  1.3.6  Evolution of fibre orientation in the headbox and sheet forming (Chapter 4) 15  1.4  Objectives .......................................................................................................... 18  1.5  Structure of the Thesis ....................................................................................... 18  Chapter 2 Fibre Interaction with a Forming Fabric .......................................................... 20 2.1  Introduction ........................................................................................................ 20  2.2  Methods.............................................................................................................. 21  2.3  Results and Discussion ...................................................................................... 24  2.3.1  Code validation ........................................................................................... 24  2.3.2  Sine-wave forming fabric ........................................................................... 26  v  2.3.3 2.4  Monoshape forming fabric ......................................................................... 34  Conclusion ......................................................................................................... 39  Chapter 3 3D X-Ray Microtomography Measurement of Paper Surface Roughness ...... 41 3.1  Introduction ........................................................................................................ 41  3.2  Methods.............................................................................................................. 42  3.2.1  Paper sample preparation ........................................................................... 42  3.2.2  CT scanning ................................................................................................ 45  3.2.3  Image analysis ............................................................................................ 45  3.2.4  Repeatability ............................................................................................... 48  3.3  Results ................................................................................................................ 50  3.3.1  Basis weight................................................................................................ 50  3.3.2  Fibre coarseness.......................................................................................... 51  3.3.3  Fibre length ................................................................................................. 52  3.3.4  Drainage velocity........................................................................................ 53  3.3.5  Forming fabric structure ............................................................................. 54  3.3.6  Comparison of experimental results with simulations ............................... 55  3.4  Conclusion ......................................................................................................... 56  Chapter 4 Evoluation of Fibre Orientation in the Forming Section ................................. 57 4.1  Introduction ........................................................................................................ 57  4.2  Numerical Model ............................................................................................... 59  4.2.1  Problem simplifications .............................................................................. 59  4.2.2  Hydrodynamic force on the fibre ............................................................... 60  4.2.3  Equations of motion ................................................................................... 62  4.3  Results ................................................................................................................ 63  4.3.1  Fibre orientation in a uniform flow ............................................................ 63  4.3.2  Fibre orientation distribution ...................................................................... 65  4.3.3  Fibre orientation in a simple shear flow ..................................................... 67  4.3.4  The effect of fibre properties ...................................................................... 70  4.4  Conclusion ......................................................................................................... 73  Chapter 5 Summary and Recommendations for Future Work ......................................... 75  vi  5.1  Summary ............................................................................................................ 75  5.1.1  Investigation of fibre interaction with a forming fabric ............................. 75  5.1.2  Microtomography measurements of paper surface roughness ................... 76  5.1.3  Evolution of fibre orientation in the forming section ................................. 77  5.2  Study Applications ............................................................................................. 78  5.3  Limitations and Discussions ............................................................................. 78  5.3.1  Investigation of fibre interaction with a forming fabric ............................. 78  5.3.2  Microtomography measurements of paper surface roughness ................... 79  5.3.3  Evolution of fibre orientation in the forming section ................................. 80  5.4  Future Work ....................................................................................................... 80  Bibliography ..................................................................................................................... 82 Appendix A Numberical model of a Fibre ....................................................................... 89 A.1  Equations of Motions ......................................................................................... 89  A.2  Hydrodynamic Force ......................................................................................... 91  A.3  Numerical Model ............................................................................................... 93  A.3.1  Fibre in the flow and contact with a surface ............................................... 93  A.3.2  Fibre with one end fixed ............................................................................. 95  A.4  Solution to the Group of Equations ................................................................... 96  A.5  Numerical Stability ............................................................................................ 96  Appendix B Validations ................................................................................................... 98 B.1  Jeffery’s Theory ................................................................................................. 98  B.2  Small-Deflection Beam Theory ......................................................................... 99  B.3  Capstan Equation ............................................................................................. 101  B.4  Rigid Beam ...................................................................................................... 103  B.5  Hydrodynamic force ........................................................................................ 105  vii  List of Tables Table 2.1: Standard deviation of the surface height of the fibre mat with different fibre concentrations ................................................................................................................... 36 Table 2.2: Standard deviation of the surface heights of fibre mats with different fibre orientations ....................................................................................................................... 38 Table 3.1: The effect of pixel size on the standard deviation of surface height ............... 48 Table 3.2: Standard deviation of mat surface height over six zones ................................ 49 Table 3.3: Characteristics of fibres used in coarseness study ........................................... 51 Table 3.4: Types of fibres of similar coarseness but different lengths ............................. 53 Table 3.5: Standard deviation of mat surface height made using different drainage velocities ........................................................................................................................... 54 Table 3.6: Four different forming fabrics used in the experiments .................................. 54 Table 4.1: Hydrodynamic force on cylinder at 45˚ (see text for details) .......................... 62  viii  List of Figures Figure 1.1: The rotation orbit at different value C............................................................ 11 Figure 1.2: The configuration of a fibre in the proximity of a cylinder ........................... 15 Figure 2.1: Schematic of a discretized fibre and neighbouring segments i and i+1 ......... 21 Figure 2.2: Schematic of a fibre in contact with a forming fabric .................................... 23 Figure 2.3: Schematic of a cantilever beam under uniform load ...................................... 25 Figure 2.4: Comparison of maximum deflection of beam with one free end ................... 26 Figure 2.5: Geometry of the sine-wave forming fabric .................................................... 27 Figure 2.6: Computational domain and boundary conditions .......................................... 27 Figure 2.7: Z-velocity contour in a plane d/4 above the forming fabric........................... 28 Figure 2.8: Schematic of the fibre position ...................................................................... 29 Figure 2.9: Fibre average height and its standard deviation versus initial offset at different EI ........................................................................................................................ 29 Figure 2.10: Fibre average height and its standard deviation versus initial angle θ at different EI ........................................................................................................................ 30 Figure 2.11: Standard deviation of the surface height of the fibre mat made from fibres of different lengths ................................................................................................................ 31 Figure 2.12: Standard deviation of the surface height of the fibre mat made from fibres of differing EI........................................................................................................................ 32 Figure 2.13: Standard deviation of the surface height of the fibre mat made at different drainage velocities ............................................................................................................ 33 Figure 2.14: For varying levels of forming fabric coarseness .......................................... 33 Figure 2.15: Geometry of the forming fabric ................................................................... 35 Figure 2.16: Dependence of mean velocity through fabric on mesh size ......................... 35 Figure 2.17: Z-velocity contour in a plane d/4 from the forming fabric with bulk velocity 0.1m/s ............................................................................................................................... 36 Figure 2.18: Standard deviation of the surface height of the fibre mat made from fibres of different lengths ................................................................................................................ 37 Figure 2.19: Fibres aligned in MD ................................................................................... 38 Figure 2.20: Fibres aligned in CMD ................................................................................. 38 Figure 2.21: Comparison of surface roughness for fibre mats made from fibres with differing EI........................................................................................................................ 39 Figure 3.1: Schematic of a modified handsheet former (Montgomery 2010) .................. 43 Figure 3.2: Geometry of the forming fabric ..................................................................... 44 Figure 3.3: Four cross sections of one sample .................................................................. 45 Figure 3.4: Surface smoothing with different Gaussian blur radii at one cross section ... 46 Figure 3.5: Surface map over one fabric repeat at different Gaussian blur radii (The radii are respectively 0, 27 μm, 54 μm, 109 μm, 163 μm from top-left to bottom-right) ......... 47  ix  Figure 3.6: Standard deviation of paper surface height as a function of Gaussian blur radius................................................................................................................................. 47 Figure 3.7: Surface map in six fabric weave repeats ........................................................ 49 Figure 3.8: Standard deviation of surface height for four fibre mat samples made under the same conditions........................................................................................................... 50 Figure 3.9: Standard deviation of fibre mat surface height for different basis weights ... 50 Figure 3.10: Standard deviation of fibre mat surface height with difference coarseness levels ................................................................................................................................. 52 Figure 3.11: Standard deviation of surface height for fibre mats with different fibre lengths ............................................................................................................................... 53 Figure 3.12: Comparison of surface roughness of fibre mats made from different forming fabrics ............................................................................................................................... 54 Figure 3.13: Comparison of numerical and experimental results of standard deviation .. 55 Figure 4.1: Fibre orientation distribution pattern in paper ............................................... 58 Figure 4.2: Schematic of the initial fibre orientation........................................................ 60 Figure 4.3: Schematics of computational domain ............................................................ 61 Figure 4.4: Schematic of one fibre rotation plane ............................................................ 64 Figure 4.5: Fibre orientation angle β as a function of initial orientation β0 (α=1.19) ...... 65 Figure 4.6: Fibre orientation distribution after headbox ................................................... 66 Figure 4.7: Anisotropy of a fibre mat with different fibre initial orientation distributions .......................................................................................................................................... 66 Figure 4.8: Simulated fibre mats at different anisotropy (from top left to bottom right: 1, 2, 4, and 9) ........................................................................................................................ 67 Figure 4.9: Fibre orientation angle β as a function of initial orientation β0 at .................................................................................................................................... 68 Figure 4.10: Fibre orientation angle β as a function of initial orientation β0 (ϴ0=0.94) in a shear flow.......................................................................................................................... 69 Figure 4.11: Anisotropy of fibre mats plotted with the velocity angle at different velocity gradients ............................................................................................................................ 69 Figure 4.12: Fibre angle β after deposition as a function of initial value β0 at different aspect ratio L/d ................................................................................................................. 71 Figure 4.13: Fibre configuration after deposition at θ0=0.942, L/d=100, (β0=0.471(L), β0=0.972(R)) ..................................................................................................................... 71 Figure 4.14: Fibre configuration after deposition at θ0=0.942, L/d=100, (β0=1.414) ...... 72 Figure 4.15: Fibre configuration after deposition at θ0=0.942, L/d=50 (β0=0.471(L), β0=0.972(R)) ..................................................................................................................... 72 Figure 4.16: Fibre configuration after deposition at θ0=0.942, L/d=50, (β0=1.414) ........ 73 Figure A.1: Schematic of a discretized fibre and neighbouring segments i and i+1. ....... 89 Figure B.1: The geometry of the particle.......................................................................... 99 Figure B.2: Comparisons of rotation orbit between numerical results and Jeffery’ theory .......................................................................................................................................... 99  x  Figure B.3: A beam with two ends simply supported .................................................... 100 Figure B.4: Change of ω with number of segments ....................................................... 101 Figure B.5: Comparison of maximum deflection ω between beam theory results and numerical results ............................................................................................................. 101 Figure B.6: Discretized rope wrapping around the cylinder.......................................... 102 Figure B.7: The ratio of tension calculated in the simulations based on different numbers of segments ..................................................................................................................... 103 Figure B.8: A rigid beam in touch with a cylinder ......................................................... 104 Figure B.9: The effect of the number of segments on the critical value of the initial offset . ................................................................................................................................ 105 Figure B.10: Position of the fibre above the sine-wave forming fabric (1) .................... 106 Figure B.11: Position of the fibre above the sine-wave forming fabric (2) .................... 106 Figure B.12: Velocity profile along the fibre at different distance from the forming fabric ........................................................................................................................................ 107 Figure B.13: Comparison of hydrodynamic force on the fibre calculated with two different methods ............................................................................................................ 108  xi  Lists of Symbols and Abbreviations MD  Machine Direction  CMD  Cross Machine Direction  GSM  Gram per Squre Meter  2D  Two Dimensional  3D  Three Dimensional  CT  Computed Tomography  μ  Dynamic viscosity[Pa.s] Hydrodynamic force[N] Drag coefficient[] Lift coefficient[] Particle velocity[m/s] Flow velocity[m/s] Fibre length[m] Fibre diameter[m] Aspect ratio of particle[] Flexural rigidity[Nm2] Root-mean-square deviation of the surface height[μm]  xii  Acknowledgements I would like to express my sincere gratitude to my supervisor Dr. Sheldon Green for his guidance on my work. He put insight into my research and helped me understanding the basics of the research problems. I could not have finished my thesis without his support and encouragement. I would like to thank AstenJohnson Inc. for their financial support, and thank Dr. John Xu and others there for providing their expertise and insight in the process of my study. I extend my thanks to Prof. Markku Kataja, Jarno Alaraudanjoki and the graduate students from the Department of Physics at the University of Jyvaskyla, Finland. Without their help with the microtomography experiments, I could not have finished the experiments. I would like to thank Dr. Schajer and Dr. Martinez at my supervisor committee. Thank you for putting thoughts in to my research. I am very grateful to Dr. Kerekes from the Pulp and Paper Center at UBC. Thanks for helping obtain the pulp samples and for giving me comments on my research. Finally, I extend my thanks to Dr. Ali Vakil a senior student from my office. I’ve learned a lot about the research problems from him, especially at the beginning of my project. I also want to thank other graduate students in my office for being so nice to me and sharing things with me.  xiii  Chapter 1  Introduction 1.1  Background  Paper making involves three main processes: forming, pressing and drying. The forming fabric is responsible for sheet forming. When a dilute suspension of fibres in water is forced through a woven forming fabric, it is dewatered into a wet fibre sheet. In the process, fibres are filtered out from the suspension as water flows through the fabric (Adanur 1997). The relative spatial arrangement of fibres in completed paper sheets is largely determined in the space between the headbox and the end of the forming section. Within this space, the final orientation of the fibres, degree of agglomeration, and relative distribution of material (e.g. fillers through the thickness of the paper and macro- and micro-mass distribution in the plane of the sheet) are determined (Wrist 1962). The forming section thus plays an important role in shaping the final properties of paper sheets. This thesis focuses on hydrodynamic processes that occur in the forming section. Wire mark, which is present on the wire side of the paper web, replicates, to some degree, the geometry of the forming fabric. It is especially noticeable within the surface topography and light transmittance characteristics of paper (Helle 1988a). In a finished paper sheet, wire mark may be either topographic (a three-dimensional paper surface produced by knuckles and holes in the fabric) or hydrodynamic (non-uniform mass distribution in the sheet caused by non-uniform mass flux through the fabric). Wire mark can affect the printing quality of paper (Danby 1994). In this thesis, fibre interaction with the forming fabric in the forming section was studied to understand the mechanism of topographic wire marking in the forming process. Fibre interactions with forming fabrics  1  were studied numerically to understand the surface roughness of fibre mats as a function of fibre length, fibre flexibility and drainage velocity (Chapter 2). Experimental research using Microtomography and image analysis was also conducted to measure the surface roughness of handsheets (Chapter 3). The numerical and experimental results were compared qualitatively. The physical properties of paper sheets are closely related to fibre orientation distribution in finished paper (Loewen 1997). The directionality and curl of paper may be partially determined by the degree of fibre orientation developed in the forming section (Parker 1972). This thesis presents a numerical study of fibre orientation in the forming section and statistically measures fibre orientation distribution in fibre mats (Chapter 4). Single fibres of differing initial orientations were simulated to calculate differences in resultant orientation after deposition. The anisotropy of fibre mats was calculated as a function of fibre initial orientation, drainage velocity and oriented shear.  1.2  Assumptions  1.2.1 Fibre suspensions Fibre suspensions can range widely in consistency. Fibre behavior depends greatly on fibre concentration. Several regimes have been defined based on collisions induced by fibre asymmetry and suspension concentration. Mason (1980) created a dilute regime in which there was less than one fibre in a volume swept out by the length of a single fibre. Kerekes et al. (1985) introduced the ―crowding number‖ to generalize Mason’s criterion. The dimensionless number is a function of suspension volumetric concentration Cv, fibre length L and diameter d, as shown below: (1.1)  Three regimes based on fibre behavior -- dilute, semi concentrated and concentrated -were defined by Soszynski (1987). For dilute suspensions, the hydrodynamic interaction  2  between fibres may be ignored. In this thesis, the fibre suspension was assumed to be dilute. Since the distance between fibres in dilute regimes is greater than fibre length, the fibres are free to rotate, and thus interaction between fibres was neglected. In pulp suspensions, fibres can vary in length, diameter, wall thickness, curvature stiffness and so on. In this thesis, fibres were assumed to be uniform in length diameter and stiffness. A particle-level method was used to model flexible fibres, crafted as chains of particles connected by flexible joints. 1.2.2 Hydrodynamics of fibres Three principle hydrodynamic processes occur in the forming section: drainage, orientated shear and turbulence (Parker 1972). Simplifications were made to calculations of flow through the forming fabric. Only drainage was considered in the simulations of fibre interaction with the forming fabric. Vakil et al. (2009) simulated flow through a forming fabric with an impingement angle, and found that the shear stress does not have a significant effect on the flow velocity perpendicular to the forming fabric. For fibre-fluid interactions, one-way coupling was considered and the flow field was prescribed. This assumption was made to simplify the simulation given the complexity of the flow field. Lindstrom and Uesaka (2007) took two-way coupling into account in their simulations and compared the results with simulations conducted with one-way coupling. The results show that the difference increases as the fibre aspect ratio increases. The dimensionless period of rotation for particles in a shear flow is roughly 25% different when the aspect ratio is 30. In my simulations, the fibre aspect ratio was lower than 30 in most cases. Krochak et al. (2008) studied the orientation of rigid fibres in a flow. They justified the one-way coupling assumption between fibre and flow. Their results show that the effect of two-way coupling on the orientation distribution function is very small relative to flow geometry (Krochak et al. 2008).  3  1.3  Literature Review  1.3.1 Wire mark and paper structure (Chapter 2&3) Paper surface topography plays an important role in shaping ink transfer to paper, an important aspect of paper printability (Borch et al. 2001). Wire mark is present on the wire side of the paper web and replicates, to some degree, the geometry of the forming fabric. The existence of wire mark is known to affect the printing quality of paper (Danby 1994). There are two main types of wire mark: topographic wire mark (a threedimensional paper surface created by knuckles and holes in the fabric) and hydrodynamic wire mark (non-uniform mass distribution in the sheet caused by non-uniform mass flux through the fabric). Hydrodynamic wire mark has been studied by Vakil et al. (2009, 2010). They simulated the flow through forming fabrics, and averaged the velocity field over areas comparable to the projected area of a fibre; the resulting average is related to the amount of fines that should accumulate in different regions of the fabric during the initial stages of dewatering. Most research on wire mark and paper structure has been carried out through image analysis. Several image acquisition techniques have been used to assess printing paper sheet smoothness, including desktop scanning, profilometry, X-ray microtomography (Samuelsen et al. 1999; Chinga et al. 2007) and scanning electron microscopy (Reme et al. 2002; Ashori et al. 2008). Helle characterized the wire mark as a very weak systematic variation in the distribution and orientation of structural elements in paper (Helle 1988a). Du Roscoat et al. (2005) investigated the three-dimensional structure of paper using synchrotron radiation microtomography. They concluded that, compared to classical laboratory equipment, X-ray synchrotron microtomography provides higher quality signal to noise ratio data and high spatial resolution. Wire mark is closely related to forming fabric geometry. Danby (1994) studied the impact of forming fabric structure on sheet formation and wire mark. He concluded that  4  micro density differences on the paper surface, resulting from the knuckles of the yarns in the forming fabrics, are the main cause of poor printing quality. He shows that knuckles in the fabrics create light areas in the sheet, while holes in the fabric create heavier areas in the sheet. Bennis and Benslimane (2010) studied pore size distribution in the z-direction of paper via synchrotron X-ray microtomography, finding that fewer pores on the paper surface correspond with smoother paper surfaces. Helle (1988b) theoretically studied initial paper web formation, using beam theory to calculate the optimum geometrical shape of ―frames‖ formed between crossing strands in the forming surface of the wire. The structure of paper as a fibrous network determines its mechanical properties. The properties of paper are very much a product of the fibres from which it is made. Much research has been done to study the effect of fibre properties such as fibre length and fibre coarseness on paper strength. Few studies have been found that investigate the effect of fibre properties on the paper surface topography. El-Hosseiny and Anderson (1999) studied the effect of fibre length and coarseness on the burst strength of paper. They concluded that while increasing fibre length increases the burst strength, increasing fibre coarseness decreases burst strength. Broderick et al. (1996) investigated the relationship between fibre characteristics and the handsheet properties of high-yield pulp. They showed that increasing fibre flexibility through chemical treatment increases the sheet tensile strength by increasing the bonded area. They also found that handsheet strength increases as average fibre length increases, but only to a certain point; the presence of fibres longer than 2mm does not further increase tensile strength. Forseth et al. (1997) studied the surface roughening mechanisms for printing paper containing mechanical pulp. They showed that increasing coarse fibre content decreases paper surface smoothness, and that the smoothest handsheets prior to calendaring were those that contained the most fines.  5  Several parameters have been developed to describe surface roughness (Chinga et al. 2007; Sacerdotti et al. 2000). Sacerdotti et al. (2000) studied various measures of surface roughness, concluding that the average deviation from the mean surface height, Sa, and the root-mean-square deviation of the surface height, Sq, were the most robust measurements of surface fluctuation. In Chinga’s (2007) study, the surface of the paper is represented by subtracting a regression plane from the surface. He included several surface roughness measures, including arithmetic mean deviation Sa (shown in Eq.(1.2)), root-mean-square deviation (standard deviation) Sq (shown in Eq.(1.3)), skewness Ss (shown in Eq.(1.4)), and kurotsis Sku (shown in Eq.(1.5)). Skewness expresses the ―skewed‖ deviations from a symmetrical distribution of peak and valley deviations in the profile. Kurtosis expresses the abundance of larger deviations relative to the Gaussian normal distribution. Vernhes et al.(2008) selected seven amplitude parameters and several others to study the surface roughness of different paper sheets. In this thesis, for the purpose of comparing different paper surfaces, Sq is used to represent the paper surface roughness, as recommended by Sacerdotti et al. (2000).  (1.2)  (1.3)  (1.4)  (1.5)  6  Where ha - average height of the paper surface relative to a datum plane parallel to the surface. h(m,n) - surface height at the point (m,n). M,N - The number of data in the x and y direction. There is no previous work that explains paper wire marks through the interaction between fibres and the forming fabric. This thesis numerically studies fibre interaction with a forming fabric and interprets the mechanism of topographic wire mark. Given its ability to provide high quality volume images of paper, the microtomography method was used to obtain 3D image data of the fibre mat and forming fabric. Using this image analysis, surface roughness was calculated. 1.3.2 Particle motion in a flow field (Chapter 2&3&4) Both theoretical and experimental approaches have been applied to study the motion of particles in a flow field. Due to the inherent complexity of flexible particles, numerical simulations have been used widely in this area. Numerical simulations are also a very useful complement to theoretical and experimental research, as theoretical research has primarily focused on the motion of rigid bodies. Relevant work is reviewed as followed. Jeffery (1922) theoretically studied the equations of motion of a non-sedimenting ellipsoid suspended in a flow field, neglecting the inertia force. According to Jeffery's theory, the axis of an isolated, rigid and neutrally buoyant ellipsoid in an uniform simple shear flow at a low Reynolds number moves on one of a family of closed periodic orbits, the center of the particle moving with the velocity of the undisturbed fluid at that point. The dimensionless period of ellipsoid rotation in a simple shear flow depends merely on the aspect ratio of the particle as shown in the equation below (Jeffery 1922).  7  (1.6)  Here, T is the period,  is the shear rate and  is the aspect ratio of the ellipsoid.  Mason and his coworkers developed a series of experiments that identified a broad range of fibre behaviour. They verified the Jeffery’s orbit in their experiments and defined five regimes of flexible fibre motion in a creeping flow: rigid motion, spring motion, snakelike motion, coiled motion and coil motion with self-entanglement (Forgacs & Mason 1959). Due to the inherent complexity of flexible particles, numerical simulations have been applied to study the motion of such particles in a flow field. One such numerical approach includes particle-level simulation (Yamamoto & Matsuoka 1993; Yamamoto & Matsuoka 1995; Ross & Klingenberg, 1997; Schmid & Klingenberg 2000; Lindstrom & Uesaka 2007; Lindstrom & Uesaka 2008). Flexible particles are modeled as a chain of rigid, massless particles connected by flexible links. Yamamoto and Matsuoka (1993) (modeled the flexible particles as chains of interconnected spherical beads linked by springs; interacting with a prescribed fluid flow through viscous drag forces (Yamamoto & Matsuoka 1993; Yamamoto & Matsuoka 1995). Ross and Klingenberg (1997) modeled fibres as a chain of spheroids connected by ball and socket joints, which reduced the computational time by eliminating the iterative constraints for the connectivity of segments. Schimd and Klingenberg (2000) employed chains of elongated bodies (rigid rods) connected by ball and socket joints, enabling the simulation of higher aspect ratio (rp=50-280) fibres using relatively few bodies per chain. In those works, fibre motions in a free flow field were simulated and the creeping flow assumption was used to determine the hydrodynamic force on the fibre. Lindstrom and Uesaka (2007) proposed a new model based on Schmid and Kingenberg’s model. The model takes into account two-way coupling between fibres and the fluid. The model applies the immerse boundary method and considers a finite particle Reynolds number. The model was used  8  to simulate semi-dilute suspension of fibres in a shear flow, and fibre network formation in papermaking (Lindstrom & Uesaka 2008a, 2008b). Fibre-fibre interaction, and interactions between fibres and the solid walls were examined in Lindstrom and Uesaka’s (2008) research. The interaction forces between fibres include the normal force, frictional force and lubrication force. The forces were prescribed. The normal force was taken as a repulsive force, which increases exponentially with fibre surface overlap. Lubrication force was applied only when the distance between two fibres was less than the fibre diameter. Frictional force was estimated based on the normal force (Lindstrom & Uesaka 2007). Stockie and Green (1998) used immersed boundary methods to simulate the motion of a fibre immersed in a shear flow. Two-way coupling between the fluid and fibres was considered, and the flow field, including disturbance caused by the fibre motion, was observed. Qi (2006) simulated the motion of an isolated fibre using the Lattice-Boltzman method. His simulation examined two-way interaction between the fibres and flow. The method can be applied for both low Reynolds number flow and finite Reynolds number flow (Qi 2006). Lawryshyn (1997) studied the statics and dynamics of pulp fibres. He applied the equations of motion of a flexible beam in vibration, and discretized PDEs using the finite element method. A flexible fibre moving in a channel flow with a slot was simulated. Previous work has shown that particle level-simulation is effective at representing fibres and replicating single fibre motions. This method was used in our simulations, with oneway coupling between the fibre and fluid, and with interactions between fibres neglected. One way coupling means that the fibres are affected by the flow, but the flows isn’t affected by the fibres. Physical contact between fibres and the forming fabric was included, with normal and frictional contact forces calculated through equations of motions.  9  1.3.3 Particle orientation in a flow field (Chapter 2&3&4) The fundamental theoretical framework that describes the evolution of rigid ellipsoidal particle orientation in a viscous flow was conducted by Jeffery in 1922. He showed that the motion of a single ellipsoidal particle suspended in a Newtonian fluid within a Stokes flow field will rotate around the vorticity axis. Other particles, such as rigid rods, follow similar orbits that one can predict by defining an equivalent aspect ratio using Jeffery’s equations. The time rate of change of the fibre’s orientation p is given by: (1.7)  Where E is the rate of strain tensor, and  is the vorticity tensor.  The orientation angles θ and φ are given by the following equations. (1.8)  (1.9)  C and k are parameters decided by the initial conditions of the particle. rp is the particle aspect ratio. Figure 1.1 shows the rotation orbits of one end of a spheroid at a different value of C in a shear flow (u=γy).  10  Figure 1.1: The rotation orbit at different value C  Jeffery’s theory can be applied to the dilute regime of suspensions, where particles are hydrodynamically independent. Givler et al. (1983) developed a numerical scheme for the determination of the fibre orientation state in dilute suspensions. Fibre orientation is calculated through the numerical integration of Jeffery’s orientation along the streamlines, obtained via the finite element method. The orientation distribution and spatial configuration of fibres are determined by the number of fibres per unit value, n, fibre aspect ratio (L/d) and the hydrodynamics of the processing flow (Sundarajajakumar & Koch 1997). Rahnama, Koch and Shaqfeh (1995) studied the effect of hydrodynamic interactions on the orientation distribution of dilute (nL3<<1) and semi-dilute fibre suspensions (nL3≥O(1), nL2d<<1). It was found that hydrodynamic interactions among fibres are rather weak, causing minor orientational displacement. Sundarajajakumar and Koch (1997) studied fibre orientation in semi-concentrated suspensions (nL2d≥O(1)). In this case, direct mechanical contracts among fibres play an important role. They showed that hydrodynamic lubrication cannot prevent physical contact between fibres, and thus non-hydrodynamic contact forces will become important. Mechanical contacts are the primary mechanism for increasing the frequency of fibre rotation in a shear flow as  11  (nL2d) becomes (1) (Sundarajajakumar & Koch 1997). Folgar and Tucker (1984) developed a mathematical model to predict the orientation distribution function of rigid fibres in more concentrated suspensions, which is an empirically modified form of Jeffery’s equations. With measured phenomenological interaction coefficient for the suspension, the orientation distribution of rigid fibres in the highly viscous fluid can be predicted. Holm and Soderberg (2007) experimentally studied shear influence on fibre orientation for dilute suspension in the near wall region. The fibre angle relative to the wall was measured. The results indicate that in the near wall region, fibres are oriented perpendicular to the streamwise direction. This differs from fibres in a free shear flow, which remain largely parallel to the streamwise direction. Fibre orientation distribution and spatial configuration are closely related to suspension concentration. Researchers have studied fibre orientation in dilute, semi-concentrated and concentrated suspensions. This thesis focuses on dilute suspensions with given boundary conditions. 1.3.4 Flow through a forming fabric (Chapter 2&4) Forming fabrics in the forming section of paper machines have three main functions: to allow water to pass through the structure, to support and retain fibres (enabling the sheet to form on the top of it), and to act as a conveyor belt that transports the sheet to the press section (Adanur 1997). Forming fabrics may be single- or multi-layered, although both are complex structures. For purposes of experimentation, complexities in the forming fabric geometry have been simplified in most previous works. Huang et al. (2006) simplified fabrics as rows of dissimilar cylinders, and studied the effect of the second row on the first row. His work was validated by the experimental research of Gilchrist (2006). Green et al. (2008) displaced individual filaments in simulations of flow through single layer fabrics, and presented the effect of filament displacement on the flow field.  12  Vakil et al. (2008) presented a novel method for building an accurate, three-dimensional CAD model of forming fabrics based on cross-sectional images of forming fabrics taken at varying depths of CMD and MD. The models of forming fabrics were generated and meshed in Gambit, and then exported into Fluent to conduct the simulation. Numerical results of air permeability through the fabric were compared with experimental results, and the difference in most cases was less than 2%. Using the precise model of a forming fabric, the researchers simulated the flow through the forming fabric, and averaged the velocity field over areas that were comparable to the projected area of a fibre; the resulting average is related to the amount of fines that should accumulate in different regions of the fabric during the initial stages of dewatering. Vakil et al. (2009) also conducted simulations of flow through a forming fabric at different impingement angles, and showed that the pressure drop through an overall forming fabric is nearly equal to the sum of the pressure drop through each layer. Simplified sine-wave and single layer forming fabrics used in industry have also been studied in simulations. The flow field through these forming fabrics has been simulated applying the methods used in the experiments explained above. 1.3.5 Particle motion in the bounded flow (Chapter 2&4) Many previous research studies have examined the characteristics of a particle moving toward a surface. Berner (1961) studied the movement of a sphere through a viscous fluid towards a plane surface. He modified the Stokes’ resistance formula, which applies only to the unbounded flow field. Goren and O’Neill (1971) studied hydrodynamic resistance to a particle that occurs when a particle is close to a large obstacle. They concluded that the deviation of particles from the streamlines of the undisturbed flow within the collector (cylinder) is insignificant until the particle collector separation is of the order of 2 or 3 particle radii.  13  Gradon et al. (1988) analyzed the motion and deposition of fibrous particles on a single filter element. Stokes flow through an infinite cylinder was prescribed. The equations of motion of the particle were solved using the average flow velocity and velocity gradient around the particle. Any local contact between the particle and fibre surface was assumed to be sufficient to cause particle deposition. The coefficients of deposition efficiency were defined and calculated. The results were believed to supplement data on the deposition of oblong aerosols particles in the lungs (Gardon et al. 1988). Bellani et al. (2008) studied experimentally the fluid-fibre interaction using Particle Image Velocimetry (PIV). They measured the amplitude of the velocity fluctuations above a forming fabric with fibres present in the flow, and concluded that the fluctuations are higher with shorter fibres. Vakil and Green (2011) studied fibre interaction with an infinite cylinder using both the particle level simulation method and a fibre model similar to the model described by Schmid and Klingenberg (2000). A fibre was placed in a prescribed 2D flow field around a cylinder, with the fibre and cross section of the cylinder in the same plane. The only external forces exerted onto the fibre as it moved through the free stream were hydrodynamic. While the fibre was in contact with the cylinder, contact forces were calculated, and the contact behaviour was determined using the normal and frictional force values. Seven dimensionless groups involving the initial position of the fibre relative to the cylinder, fibre properties and fluid properties were defined as shown in Eq. (1.1). Dimensionless numbers include the Reynolds number and the ratio of bending force to viscous force. The fibre and cylinder configurations are shown in Figure 1.2. Vakil and Green (2011) determined the region of fibre ―hang up‖ on the cylinder as a function of several dimensionless numbers as shown in Eq. (1.1) (1.10)  14  Figure 1.2: The configuration of a fibre in the proximity of a cylinder  Following Vakil and Green’s(2011) work, this thesis studies fibre interaction with two cylinders in a 2D flow, and fibre interaction with one cylinder in a 3D flow, presenting new findings on the nature of fibre interactions with real forming fabrics. This thesis focuses on real forming fabrics, with the flow field prescribed in simulations. 1.3.6 Evolution of fibre orientation in the headbox and sheet forming (Chapter 4) The primary mechanism used to orient fibres in the sheet involves the manipulation of hydrodynamic shear flows in the head box and forming process. Fibre orientation distribution in the paper can be determined by many variables, such as fibre properties, headbox geometry and forming conditions. Headbox design can have an effect on preorientation (Loewen 1997). In the headbox, the contraction ratio of the headbox nozzle strongly affects the fibre orientation distribution in the nozzle of the headbox (Nordstrom & Norman, 1994).  15  The fibre orientation distribution in the headbox has been observed both experimentally and numerically. Akbar and Altan (1992) studied the orientation behaviour of short fibres in an arbitrary, two-dimensional homogeneous flow. The results showed that a combination of analytical solutions and statistical methods can be used to provide a convenient description of fibre orientation behaviour. The expression for orientation distribution function for finite aspect ratio particles for planar elongational flow is given by (1.11)  Here, ø is the orientation angle, λ is a function of the fibre aspect ratio rp, , and ε is the total elongation,  (c is the velocity gradient, t is time).  Bellani et al Wrist (1961) measured fibre orientation in the jet immediately following the slice. The results show that the orientation produced at the slice is very sensitive to the consistency. The slice design and machine speed are also important, but to a lesser degree (Wrist 1961). Three principal hydrodynamic processes occur during the sheet forming process: drainage, oriented shear and turbulence. Oriented shear occurs between the fibre suspension and the wire/fibre mat, and can be characterized by a mean velocity gradient or less precisely velocity difference and by a mode of orientation. Paker (1972) found that oriented shear in conjunction with the consolidating and immobilizing effects of drainage may orient the fibre to a preferred direction. Finger and Majewski (1954) studied the mechanism of formation and structure of the sheet of the Fourdrinier machine. In their work, the layered structure is attributed to the discrete stages of the drainage process. They demonstrate that the fibres in any of the layers reflect the relative motion of the free slurry at the moment the layer is deposited. It  16  is assumed that the fibre becomes vertically-oriented over the suction zone; then the leading end becomes anchored while the other end is carried over with the flow. Wrist (1972) pointed that ―the boundary between the deposited fibres and the free slurry is very diffuse‖. Therefore, it is likely that one end of the fibre is of a higher consistency, and thus located in a stronger part of the network structure, while the other still occupies the weaker network of the free suspension. When relative motions between the free suspension and fabric/fibre mat occur, the fibre is preferentially oriented in the direction of relative motion, held at one end by the network of the deposited mat (Wrist 1982). This interpretation of fibre orientation in the fibre mat was applied in the following simulations, in that the fibre is assumed to be held at one end by the fibre network, with the other end carried in the flow. For a roll twin wire former, the speed of mix after jet deceleration by the dewatering pressure T/R (T is the tension of wire, R is the radius of the roller) between the wires was obtained using Bernoulli equations (Nordstrom &Norman 1994).  1 1 u m 2  pm  u j 2  p0 2 2  (1.12)  Nordstrom &Norman (1994) showed that in a roll former, minimum in-plane anisotropy is theoretically obtained when the mix speed equals the speed of the wires after jet deceleration by the dewatering pressure T/R. 2T u jw  uw ( 1   1) Ru w 2  (1.13)  Image analysis has been applied to study the layered structure of a sheet (Erikkila et al. 1998, Rosen et al. 2008, Axelsson 2008). For example, Erikkila et al. (1998) studied sheet forming through layered orientation analysis, in which the sheet was split and an optical method determined the orientation for each layer. This kind of analysis provides insight into the sheet forming mechanism.  17  Fibre orientation in the forming section has been interpreted in previous studies. However, limited literature examines the effect of fibre initial condition, drainage velocity and shear on fibre orientation during the forming stage. This thesis conducts a numerical study to simulate fibre deposition in the forming section, and aims to understand the mechanism of fibre orientation in the forming section of the Forurdiner former.  1.4  Objectives  This thesis studies fibre interaction with a forming fabric both numerically and experimentally. The main objectives include 1) Proposing a numerical procedure for simulating the initial sheet forming process. 2) Understanding the mechanism of topographical wire mark and the effect of fibre properties and forming fabric geometry on paper surface roughness. 3) Presenting an experimental method for obtaining the wire-side surface roughness of fibre mats using microtomography and image analysis. 4) Understanding the mechanism of fibre orientation in the forming section through numerical simulations. 5) Studying the anisotropy of paper as a function of forming conditions (initial conditions, flow direction and orientation shear).  1.5  Structure of the Thesis  Chapter 1 introduces the background of the thesis and reviews relevant literature. The thesis objectives and assumptions made in the thesis calculations are also listed. Chapter 2 includes the numerical study of fibre interaction with a forming fabric. A numerical model was built to simulate single fibres moving in the flow through the  18  forming fabric and interacting with the forming fabric. The model was validated by comparing the results of the simulation with the theoretical results. Surface roughness is calculated as a function of fibre length, fibre stiffness, drainage velocity and so on. A simplified sine-wave forming fabric and real single layer fabric were used in the simulations. Single fibres were studied to understand the mechanism of tomographical wire mark. Initial sheet forming was simulated by releasing hundreds of fibres over top of the forming fabric, which later deposited to create a formed mat. A standard deviation of the surface height of the formed mat was calculated to characterize the surface roughness. An experimental study of the surface roughness of light-weight fibre mats is presented in Chapter 3. Several fibre mats created under varying conditions in a handsheet former were produced. 3D X-ray microtomography was used to obtain high resolution, 3Dvolume images of both the fibre mats and the forming fabrics. Image analysis of the tomographic images was carried out to find the surface topography of the wire side of the fibre mats. The experiments were conducted to complement and validate the numerical research results presented in Chapter 2. Chapter 4 presents the evolution of fibre orientation in the forming section. The effect of forming conditions on fibre orientation distribution was studied to understand the mechanism of fibre orientation in the forming section. Numerical simulations were conducted, and the anisotropy of paper was calculated as a function of fibre initial orientation, shear stress and flow direction. Chapter 5 presents the thesis conclusions.  19  Chapter 2  Fibre Interaction with a Forming Fabric 2.1  Introduction  Wire mark, which is present on the wire side of the paper web replicates, to some degree, the geometry of the forming fabric. It is especially noticeable in the surface topography and light transmittance characteristics of paper (Helle 1988a). In the finished paper, wire mark may be either topographic (a three-dimensional paper surface caused by knuckles and holes in the fabric) or hydrodynamic (non-uniform mass distribution in the sheet caused by non-uniform mass flux through the fabric). Wire mark can change the printing quality of paper. This thesis focuses primarily on the mechanism of topographic wire marking. Topographic wire mark was studied through an investigation of fibre interaction with a forming fabric. Particle-level simulations were used to describe the movement of flexible fibres through a forming fabric. Unlike previous studies, the creeping flow assumption is not made here. Rather, the hydrodynamic force was calculated based on the lift and drag coefficient of cylindrical sections in a flow field. The flow is steady and one-way coupling of fibre and flow is considered; the flow field is assumed to be the single phase flow through the fabric, and fibres do not interact with one another. A sine-wave (2×2 shed) single layer forming fabric was used in the simulations. Simulations of flow through the fabric were conducted using FLUENT software. Single fibres of different orientations and positions were released into the flow upstream from the forming fabric. These fibres then deposited onto the fabric surface, creating a simulated lightweight fibre  20  mat. The surface roughness of the simulated fibre mat, which was related to the fibre mat wire mark, was then calculated as a function of fibre rigidity and fibre length.  2.2  Methods  The particle-level simulation method involves modeling a fibre as a chain of particles connected by flexible joints, as shown in Figure 2.1. For segment i, ri is the vector pointing to its center, and ci is the vector along its axis.  Figure 2.1: Schematic of a discretized fibre and neighbouring segments i and i+1  The governing equations of motion for the fibre include kinematic equations and dynamic equations. For each segment of the fibre, there are connectivity equations, linear momentum equations and angular momentum equations. The details of the governing equations are shown in the Appendix A. With some manipulation of the governing equations, the translational velocity and angular velocity for each segment can be solved directly using the following equations: (2.1)  21  (2.2)  Where,  is the angular velocity of segment i, and (c)  segment i. F  is the contact force and T  (c)  is the translational velocity of  is the contact torque. N1, N2, N3, M1, M2, M3  are matrices associated with the position and properties of the particles, and with the velocity and properties of the flow field. These matrices are listed in the Appendix A. When the fibre moves in a non-constrained flow, the contact force is zero. In this case, the equations can be simplified as: (2.3)  (2.4)  As shown in Figure 2.2, when a fibre is in contact with a forming fabric, there are normally multiple support points over the length of the fibre. Due to the hydrodynamic force, the contact force normal to the contact surface is much higher compared to the tangential force. Hence, the tangential force cannot overcome the frictional force or cause the fibre to slide. Therefore, while the fibre is in contact with the forming fabric, the segments in contact with the surface are assumed to be non-sliding. In this case, the contact forces are added to the equations as unknown variables, while the segments in contact with surface at zero velocity are added as boundary conditions.  22  Figure 2.2: Schematic of a fibre in contact with a forming fabric  The hydrodynamic force of a spheroid in a creeping flow is a linear function of the relative velocity of the particle. In our simulations, the Reynolds number of the flow based on the dimension of fibre diameter was much higher than that for creeping flow. For example, a typical fibre diameter is 20-40μm, and a typical flow velocity is 0.1-1m/s, therefore a typical Re is 2-40. The equations are therefore not suitable in these cases. Vakil and Green (2009) simulated the flow field around a cylinder and calculated the hydrodynamic force on the cylinder at different angles and aspect ratios. A series of formulas for drag and lift coefficients under different conditions were obtained by curve fitting the simulation results. The hydrodynamic forces at a moderate Reynolds number (1-40) were calculated using the following equations based on the drag coefficient CD and lift coefficient CL. (2.5)  (2.6) (2.7)  (2.8)  23  d is the unit directional vector along the axis of the cylinder. δ is an identity matrix of size 3. U∞ and U are the velocity of the flow and the particle, respectively. Ur is the magnitude of relative velocity of the particle. L and D are the length and diameter of cylinder, respectively. θ is the angle between the axis of the cylinder and its relative velocity vector.  2.3  Results and Discussion  2.3.1 Code validation C++ code was written to simulate the motions of a fibre and its interaction with the forming fabric. The code was verified through a comparison with two analytical results as shown below. More validation cases were shown in Appendix B. Jeffery (1922) theoretically studied the equations of motion of a neutrally buoyant ellipsoid suspended in a flow field, where the inertia force was neglected. According to Jeffery’s theory, the axis of an isolated, rigid and neutrally buoyant ellipsoid in a uniform simple shear flow at a low Reynolds number moves on one of a family of closed periodic orbits. The center of the particle at that point moves with the velocity of the undisturbed fluid. The dimensionless period of rotation of the ellipsoid in a simple shear flow depends merely on the aspect ratio of the particle, as shown in Eq.(2.9) (2.9)  Here, T is the period of rotation;  is the shear rate and rp is the aspect ratio of the  ellipsoid. Numerical simulations of spheroid motion in a shear flow were conducted to compare with Jeffery’s equation. The hydrodynamic force for creeping flow conditions was used in the simulations. The period of rotation calculated based on simulations and Eq.(2.9)  24  were compared. The numerical results match the theoretical solution well with a difference of less than 0.03%.  Figure 2.3: Schematic of a cantilever beam under uniform load  Small deflection beam theory describes the deflection of beams under a small force. For a cantilever beam with uniform load q (load per length) as shown in Figure 2.3 , the maximum deflection ω is a function of the beam flexural rigidity EI, beam length L and q. (2.10)  The deflection of beams under uniform force was calculated in the simulations. The beam was modeled as a number of rigid particles connected to flexible joints. The force was applied to the center of each segment. The results in Figure 2.4 show that as the number of segments increases, the numerical results draw closer to the theoretical results. The numerically-computed beam deflection is within 3% of the small deflection beam theory result when the number of segments is 20 or more. In the following simulations, fibre with small deflection was assumed to calculate the bending moment.  25  Figure 2.4: Comparison of maximum deflection of beam with one free end  2.3.2 Sine-wave forming fabric Flow through the forming fabric The sine-wave single layer forming fabric geometry shown in Figure 2.5 was studied. The diameter of the filaments d was 150μm, and the gap between two filaments was 200μm. A second sine-wave fabric, with twice the gap (400µm), was also studied. The 3D flow field through the forming fabric was simulated using FLUENT. The computational domain and boundary conditions are shown in Figure 2.6. The CFD mesh was generated in Gambit and imported into FLUENT. The z velocity contour in a plane d/4 above the forming fabric in the upstream is shown in Figure 2.7. In this figure, the bulk velocity through the fabric is 0.5m/s. The flow field data was imported into written code to simulate the motion of fibres in the vicinity of the fabric.  26  Figure 2.5: Geometry of the sine-wave forming fabric  Figure 2.6: Computational domain and boundary conditions  27  (m/s)  Figure 2.7: Z-velocity contour in a plane d/4 above the forming fabric  Individual fibres interacting with the fabric A single fibre with a given initial position was released into the flow upstream from the forming fabric. The fibre was assumed to parallel to the fabric. Different initial positions X0 and orientations θ were considered in the simulations, as shown in Figure 2.8. Unless otherwise stated, fibres used in the simulations were 1mm in length and 40μm in diameter. Previous researchers have measured the bending stiffness, EI, of pulp fibres to be between 10-10 and 10-13 Nm2 (Tam Doo& Kerekes 1981; Yan&Li 2008a; Yan&Li 2008b). For this reason, bending stiffnesses in this range (and as small as 10-14 Nm2) were considered in the simulations.  28  Figure 2.8: Schematic of the fibre position  Figure 2.9: Fibre average height (left) and its standard deviation (right) versus initial offset at different EI  After the fibre was released, it rested on top of the forming fabric. From the fibre geometry, one may compute the average height of the fibre, measured relative a reference plane taken flush with the top of the fabric. As shown in Figure 2.9, the average height of the fibre did not vary significantly with changes in EI; the three curves for different EI show similar trends. Therefore, the geometry of the forming fabric is primarily responsible for fibre mat topography. The standard deviation of fibre height decreases as the fibre grows more rigid, as Figure 2.9 shows.  29  Figure 2.10: Fibre average height (left) and its standard deviation (right) versus initial angle θ at different EI  For fibres centered on one fabric knuckle (i.e., Xo=0), but with variable θ, because the height in the center is fixed, the average height of fibres varies substantially with both EI and θ, except for very rigid fibres (see Figure 2.10), expect for very rigid fibres. The two cases shown in Figure 2.9 and Figure 2.10 respectively are situations where forming fabric geometry and fibre bending contributes most to changes in the fibre average height. Therefore, in general, both fibre bending and the geometry of the forming fabric are responsible for topographical wire mark. Lightweight fibre mat To better understand topographical wire mark in paper making, the initial stage of sheet forming was studied. Simulations were conducted to simulate a number of fibres deposited on top of the forming fabric. Fibres of initially randomly distributed positions and orientations were simulated. The fibres moved independently of other fibres. The surface roughness of the fire mat is represented by the standard deviation of the surface height, given by Eq.(1.3). The standard deviations of the surface heights of fibre mats made from differing numbers of fibres were compared. The standard deviation changes only very slightly (2.5%  30  difference) from 100 fibres to 200 fibres, and therefore, in the simulations, one hundred fibres gave statistically robust values of mat surface roughness. Fibres of different lengths were simulated to study the effect of fibre length on surface roughness. The EI of each fibre was fixed at 10-12 Nm2. Figure 2.11 shows that as fibre length increases, the value of standard deviation decreases. However, in real paper making, paper made from short fibres is smoother than paper made from longer fibres. The reason for this difference might be that in the simulations, only the initial formed fibre mat was calculated and the pressing and drying sections were not accounted for. It is thought that pressing may change the topography of short fibre mats more than that of long fibre mats due to the fact that short fibres may be less constrained by other fibres in the mat. Furthermore, it is very difficult to vary one pulp property without affecting other properties. As fibre length is correlated with fibre stiffness, for instance, the smoothness of paper made from short fibres may also be a function of stiffness.  Figure 2.11: Standard deviation of the surface height of the fibre mat made from fibres of different lengths  Fibre mats made from fibres with different initial orientations (random and machinedirection oriented) were compared. The standard deviation does not change significantly between the two orientations (30.3μm for randomly oriented and 33.0μm for machine-  31  direction oriented). For the sine-wave forming fabric, the fibre orientation did not affect the surface smoothness significantly. It is believed that this result would not apply for non-symmetric forming fabrics. Figure 2.12 shows the surface roughness of the fibre mat for randomly distributed fibres as a function of the fibre bending stiffness. When EI is 10-10 Nm2 or larger, the fibre is very rigid and deflection of the fibre is negligible. Therefore, a standard deviation of surface height 18 μm is solely a function of the forming fabric geometry. Fibres used in paper making possess EI in the range of 10-10 Nm2 to 10-13 Nm2, for which the corresponding standard deviation of surface height varies from 18 μm to 37 μm. Thus, for the simple sine-wave forming fabric and realistic fibres, 50% or more of the surface roughness is caused by the geometry of the forming fabric.  Figure 2.12: Standard deviation of the surface height of the fibre mat made from fibres of differing EI  Fibre mats were also formed at different drainage velocities. In these simulations, the stiffness of fibres was fixed at 10-12 Nm2. As shown in Figure 2.13, when the velocity increases from 0.1 to 1 m/s, the standard deviation of the mat surface height also increases. The increased deviations are a consequence of greater fibre bending caused by the increased hydrodynamic force acting on fibres at higher flow velocities.  32  Figure 2.13: Standard deviation of the surface height of the fibre mat made at different drainage velocities  A coarser sine-wave fabric with twice the gap spacing compared to the sine-wave fabric shown in Figure 2.14 was also studied. Figure 2.14 shows two curves for the two fabrics. Fibres in the larger gap size forming fabric tend to bend more. Due to its smaller filament pitch, the fibre mat surface produced by the fine sine-wave forming fabric is much smoother.  Figure 2.14: For varying levels of forming fabric coarseness  33  2.3.3 Monoshape forming fabric The single layer MonoshapeTM forming fabric shown in Figure 2.15 was also simulated. The diameter of the yarns in the MD (Machine direction) and CMD (Cross machine direction) was 400μm. The yarn count (per inch) was 36 (MD) and 32 (CMD). The geometry was created and meshed using Gambit and the flow field was simulated using FLUENT software. To ensure that the simulations are meshing independent, the flow field through the forming fabric was simulated for an air pressure drop by 10 Pa (Vakil&Green 2009). As shown in Figure 2.16, the inlet average velocity varies by only one percent as the number of mesh elements increases from 4.2 million to 8 million. Therefore, with 4.2 million mesh elements, the results are considered to be sufficiently accurate. Simulations were done over one repeated pattern of the forming fabric. The computational domain was similar to the one shown in Figure 2.6, though the boundary conditions were different. In the fabric volume, the sides were set as non-slip walls instead of slip walls. This showed that the effect of boundary conditions involves less than one filament. Therefore, in simulations, one more filament on each side is involved. The inlet velocity was set to 0.1m/s, with a Reynolds number of 40. The z velocity contour in a plane a quarter of a yarn diameter upstream from the forming fabric is shown in Figure 2.17. The specific flow field was used in the following simulations. The flow field data were imported into the written code to simulate the motion of a fibre in the fabric flow field.  34  Figure 2.15: Geometry of the forming fabric  Figure 2.16: Dependence of mean velocity through fabric on mesh size  35  (m/s)  Figure 2.17: Z-velocity contour in a plane d/4 from the forming fabric with bulk velocity 0.1m/s  To understand topographical wire mark in paper making, the initial stage of sheet forming was examined. Simulations of fibres deposited on the top of the forming fabric were conducted. Fibres with initially randomly distributed positions and orientations were simulated. The fibres moved independently of other fibres. The surface roughness of the fire mat is represented by the standard deviation of the surface height, given by Eq.(1.3). Table 2.1 shows the standard deviation of the surface height of fibre mats made from different fibre concentrations. The standard deviation changes very slightly from 200 fibres to 300 fibres (in one repeated pattern) (at EI=10-12 Nm2, L=2mm). Therefore, in the simulations, two hundred fibres give statistically robust values of mat surface roughness. Table 2.1: Standard deviation of the surface height of the fibre mat with different fibre concentrations Number of fibres  Standard deviation (μm)  100  99.0  200  106.4  300  109.1  36  Topographic variations in the sheet surface were found to be much larger than those found through the experimental measurements of Ashori et al. (2008). Ashori et al. studied the surface topography of paper using a non-contact profilometer. The surface roughness of the sheet they studied was less than 10µm. It is believed that the pressing, drying and calendaring sections dramatically reduce topographic variations generated in the forming section. Fibres of different lengths were simulated to study the effect of fibre length on surface roughness. The EI of each fibre is fixed at 10-12 Nm2. Figure 2.18 shows that as the fibre length increases, the value of standard deviation decreases. The results are consistent with the results for sine-wave forming fabrics.  Figure 2.18: Standard deviation of the surface height of the fibre mat made from fibres of different lengths  Fibre mats made from fibres of different initial orientations are compared in Table 2.2 (at EI=10-12 Nm2, L=2mm). The standard deviation increased by about 60% for the MDorientated fibre mat (shown in Figure 2.19) compared to the CMD-orientated fibre mat (shown in Figure 2.20). The fibre orientation affected the surface smoothness significantly in non-symmetric forming fabrics. Fibres aligned with the machine direction became embedded in the forming fabric more easily, while fibres aligned with  37  the cross machine direction were better supported by the yarns. For symmetric forming fabrics such as the sine-wave forming fabric, fibre orientation does not significantly change the surface smoothness of the fibre mat.  Figure 2.19: Fibres aligned in MD  Figure 2.20: Fibres aligned in CMD Table 2.2: Standard deviation of the surface heights of fibre mats with different fibre orientations Orientation of fibres  Standard deviation (μm)  Randomly distributed  106.8  MD  136.4  CMD  84.5  Figure 2.21 shows the surface roughness of the fibre mat for randomly distributed fibres as a function of fibre bending stiffness. The flexural rigidity ranges from 10-10 Nm2 to 1014  Nm2. When EI is 10-10 Nm2 or larger, the fibre is very rigid and the deflection of the  fibre is negligible. Therefore, at the highest bending stiffness, the standard deviation of the surface height is solely a result of the geometry of the forming fabric. Fibres used in papermaking have EI in the range of 10-10 Nm2 to 10-13 Nm2 (Tam Doo& Kerekes 1981;  38  Yan&Li 2008a; Yan&Li 2008b). Thus, for the MonoshapeTM forming fabric and realistic fibres, about 50% or more of the surface roughness is shaped by the geometry of the forming fabric.  Figure 2.21: Comparison of surface roughness for fibre mats made from fibres with differing EI  2.4  Conclusion  Topographical wire mark of the fibre mat has been studied numerically. The physical interaction between the fibres and the forming fabric was simulated. The numerical methods used were verified through a comparative study of Jeffery’s equations and beam theory. Two single layer forming fabrics, a simplified sine-wave forming fabric and a real forming fabric were used in the simulations. A lightweight fibre mat was formed in the simulations from a few hundred fibres, which were initially randomly distributed on top of the forming fabric. The surface roughness of the fibre mat was calculated and compared as a function of fibre EI and length. For fixed EI, increasing the fibre length decreased pulp mat surface roughness owing to the presence of multiple support points over the length of each fibre. For fixed fibre length, increasing the EI decreased the surface roughness owing to the reduced bending of fibres into fabric surface holes. The  39  surface roughness predicted numerically is much larger than the experimental results presented in the existing literature. It is believed that the pressing, drying and calendaring sections dramatically reduce topographic variations generated in the forming section. Different fibre alignments were also studied. It was shown that for non-symmetric forming fabrics, the surface roughness of the fibre mat changes substantially with changes in fibre orientation. Surface roughness was also calculated as a function of drainage velocity. As drainage velocity increased fibre bending, and therefore mat surface roughness, also increased. Based on these simulations of surface roughness at different EI levels, it is concluded that topographical wire mark is caused partially by fibre bending and partially by the geometry of the fabric. With small deflection beam theory assumption, fibre bending moment was calculated. For the simple sine-wave single layer fabric and monoshape forming fabric, more than 50% of topographical wire mark was caused by the geometry of the forming fabric, with the remainder caused by fibre bending.  40  Chapter 3  3D X-ray Microtomography Measurements of Paper Surface Roughness Owing to the complexity of the fibre interaction with the forming fabric, numerous simplifications were made in the simulations – the use of uniform fibres and simple fabric geometry, negligible interaction between fibres, etc. In designing a study that closely reflected real conditions, measurements of the topography of fibre mats were carried out to follow up with numerical studies, and to compare the surface roughness of the formed mat with numerical results.  3.1  Introduction  Paper surface topography plays an important role in ink transfer to paper, which in turn is an important feature of paper printability (Mark et al. 2001). Wire mark is present on the wire side of the paper web and replicates, to some degree, the geometry of the forming fabric. The existence of wire mark is known to affect the printing quality of paper (Danby 1994). There are two main types of wire mark: topographic wire mark (a threedimensional paper surface caused by knuckles and holes in the fabric) and hydrodynamic wire mark (non-uniform mass distribution in the sheet caused by non-uniform mass flux through the fabric). Hydrodynamic wire mark has been studied by Vakil et al. (2009, 2010). They simulated the flow through forming fabrics, and averaged the velocity field over areas comparable to the projected area of a fibre; the resulting average is related to the amount of fines that should accumulate in different regions of the fabric during the  41  initial stages of dewatering. This paper focuses on topographic wire mark during initial sheeting forming. We have found no previous work related to the influence of the forming section and pulp furnish variables (drainage velocity, jet rush, fibre coarseness, fibre length, etc.) on sheet topography in the forming section. To fill this gap in knowledge we have formed many different fibre mats under varying conditions in a handsheet former. 3D X-ray microtomography was used to obtain high resolution 3D volume images of both the fibre mats and forming fabrics, and image analysis of the tomographic images was conducted to find the surface topography of the wire side of the fibre mats.  3.2  Methods  3.2.1 Paper sample preparation The paper samples were prepared using the specialized handsheet forming device shown in Figure 3.1(Montgomery 2010). The apparatus consisted of a 3‖ (75mm) diameter clear acrylic circular cylinder above a forming fabric, and a vacuum chamber mounted below the fabric. With this device, it’s possible to vary the speed of dewatering. A 5cm depth of pulp slurry at 0.03% consistency was poured into the cylinder. At time t=0, a valve separating the vacuum chamber from the underside of the forming fabric was opened, which initiated dewatering and generated a fibre mat of 15 gsm. The rate of dewatering is a function of the set vacuum chamber pressure. The typical dewatering velocity through the fabric and fibre was about 4 cm/s with gravity drainage (no vacuum applied), and about 12 cm/s with 50 kPa of vacuum. Following sheet formation, the handsheet was taken from the forming device and air dried at room temperature on the fabric. One handsheet was imaged both before and after wetting the fibre mat. Fibre swelling caused by wetting was shown to have a negligible impact on the wire-side mat surface topography. Transporting the fabric and fibre mat  42  involved holding the fibre mat and fabric between two rigid plastic sheets. Though the plastic sheets deformed the top side of the fibre mat, this had a negligible impact on the topography of the wire-side fibre mat. Both chemical pulp and TMP have been used to make fibre mats in the handsheet former, although for the majority of tests a hemlock kraft pulp was used. The other kraft pulps tested were made from radiata pine, SPF (spruce, pine and fir), and abaca. All pulps were supplied by Canfor. To study fibre mats made from fibres with different lengths, fibres were fractionated by length using a Bauer-McNett fibre classifier following TAPPI Test Method standard T233. Five different length cuts were separated and in most tests the R28 fraction was studied. A Fibre Quality Analyzer (FQA) measured the fibre length and fibre coarseness of each of the different length cuts. The FQA was calibrated to meet TAPPI Test Method standard T271.  Figure 3.1: Schematic of a modified handsheet former (Montgomery 2010)  A coarse weave, single layer forming fabric (AstenJohnson Monoshape M36 AJ-170), was used in sheet forming (refer to Figure 3.2). The diameter of the yarns in both the Machine direction (MD) and Cross machine direction (CMD) was 400μm. The fabric  43  had 14.2 yarns/cm in MD and 12.6 yarns/cm in CMD. This simple fabric was chosen for the study to simplify the interpretation of the results.  Figure 3.2: Geometry of the forming fabric  Unless otherwise stated, experiments were conducted with the following variables held constant:   Pulp: Hemlock (Bauer-McNeet 28 fraction)    Forming fabric: Monoshape M36 AJ-170    Fibre length: 2.63mm    Fibre coarseness: 0.174mg/mm    Drainage velocity: 4cm/s    Basis weight of fibre mat: 15g/m2  44  3.2.2 CT scanning A micro-CT scanner (Skyscan1172 1 ,Bruker microCT) was used to measure the 3D structure of the forming fabric and fibre mat. The reconstruction process is handled by NreconTM software. As discussed below, for most imaging the image voxel size was selected to be 5.44 μm, which is fine enough to capture individual pulp fibres, but coarse enough to image multiple repeats of the fabric structure. Fabric samples were cut to 2 3 mm to fit the area of view of the scanner. Figure 3.3 shows four cross-sectional images of one sample. After reconstruction of the tomographic images, the forming fabric image is digitally removed to create an image of just the paper.  Figure 3.3: Four cross sections of one sample  3.2.3 Image analysis Image analysis of the paper was performed using ImageJ software. The wire-side surface of the fibre mat was detected based on the ―flying carpet‖ method. This method was implemented as a plugin to ImageJ (Mettänen et al. 2011). In brief, the plugin uses a discrete version of the Edwards-Wilkinson equation. In the plugin a dynamic interface is forced to approach the sample surface, and the interface is repelled by an opposing force which monotonically increases with the local grayscale value of the voxels (Voutilaninen  1  The scanner was made available by Prof. Kataja, from University of Jyvaskyla.  45  et al. 2012). The Gaussian blur algorithm (Nixon & Aguado 2008) was used to smooth the surface. Figure 3.4 shows the evaluated paper surface in one cross-section as a function of the Gaussian blur radius. Figure 3.5 shows the surface map over one fabric repeat smoothing with different Gaussian blur radii. The grayscale value in the image represents the height of the surface relative to the reference surface, which parallels to the interface between the plastic sheet and the fibre mat. For small Gaussian blur radii topographic variations caused by individual pulp fibres are apparent, whereas for larger radii primarily topographic variations caused by the fabric structure are visible. The standard deviation of surface height versus Gaussian blur radius is plotted in Figure 3.6. With a voxel size of 5.44 μm, six fabric repeats were included in the microtomography measurement. The standard deviation of each repeat was calculated, and the variance of the six standard deviations was set to be the standard error and plotted as the error bars in Figure 3.6. Table 3.2 lists the standard deviations and their variance over six repeat for one particular paper sample. A Gaussian blur radius of 109 μm was used throughout the following data analysis. This radius is substantially larger than an individual pulp fibre diameter, but significantly smaller than the fabric filament diameter. The use of this radius permit one to resolve features on the scale of the forming fabric topographic wire mark while minimizing the noise associated with individual fibres. The trends described below are unaffected by the blur radius (for any reasonable radius), although the choice of blur radius affects the magnitude of the trends and the level of noise in the results.  Figure 3.4: Surface smoothing with different Gaussian blur radii at one cross section (The radii are 27 μm, 54 μm, 109 μm, 163 μm, respectively from top to bottom)  46  Figure 3.5: Surface map over one fabric repeat at different Gaussian blur radii (The radii are respectively 0, 27 μm, 54 μm, 109 μm, 163 μm from top-left to bottom-right)  80  Standard deviation(µm)  70 60 50 40 30 20 10 0 0  100  200  300  400  500  600  Guassian blur radius(µm) Figure 3.6: Standard deviation of paper surface height as a function of Gaussian blur radius  In imaging there is often a trade-off between image resolution and image size. To understand this trade-off in the context of sheet topography, the same sample was measured at two different resolutions – 2.97 μm and 5.44 μm. With the higher resolution,  47  only one fabric weave repeat could be imaged. When the pixel size is increased by 2.47 μm (from 2.97 μm to 5.44 μm) the standard deviation increases by almost the same amount, 2.86 μm as shown in Table 3.1. The pixel size was set to 5.44 μm as that size represents a reasonable balance between maximizing the sample area (to be able to see several fabric repeats, over which there will be differences owing to the random arrangement of fibres in the paper) while maintaining reasonable resolution of the measurements. It is not the case that the effective experimental error is greater than or equal to 5.44 μm; much smaller differences between experiments can be distinguished statistically because the pixilation noise is averaged out over about 400,000 pixels per image. Table 3.1: The effect of pixel size on the standard deviation of surface height Pixel size(μm)  Standard Deviation(μm)  2.97  26.76  5.44  29.62  3.2.4 Repeatability To assess the repeatability of the measurements, a surface map of the wire-side fibre mat over multiple fabric repeats was created, as shown in Figure 3.7. Six fabric weave repeats are included in the image. The standard deviation of the fibre mat height was calculated separately in the six zones. As shown in Table 3.2, the standard deviation of surface height varies by about 5% from zone to zone. The difference between zones is believed to be caused primarily by the random distribution of fibres and to a lesser extent by slight geometric inconsistencies in the forming fabric. The standard deviation of surface height for one sample is an average of its value in six fabric repeats. The variance of the six values is set to be the error for each calculation of standard deviation of surface height.  48  Figure 3.7: Surface map in six fabric weave repeats Table 3.2: Standard deviation of mat surface height over six zones  1  Standard Deviation (μm) 31.07  2  27.79  3  28.10  4  29.28  5  29.21  Zone  6  26.34  Average  28.63  Variance  1.47  To further assess experimental repeatability, four paper samples were made from the same pulp suspension under the same forming conditions. The fibre mat wire-side topography was measured and the surface roughness of the samples was calculated. Figure 3.8 shows that the variation from sample to sample is comparable to the variation between multiple weave repeats on a single sample.  49  35.00 30.83  Standard deviation (μm)  29.52  28.63  30.00  28.78  25.00 20.00 15.00 10.00 5.00 0.00  Sample  Figure 3.8: Standard deviation of surface height for four fibre mat samples made under the same conditions  3.3  Results  3.3.1 Basis weight  Standard deviation (μm)  35.00 30.00  29.48  29.43  10  15  28.94  25.00 20.00 15.00 10.00 5.00 0.00  Basis weight  20  (g/m2)  Figure 3.9: Standard deviation of fibre mat surface height for different basis weights  Fibre mats of three different basis weights were made under otherwise identical forming conditions. As seen in Figure 3.9, all fibre mats possessed identical surface roughness  50  values within experimental error. This finding implies that the surface roughness of a fibre mat is determined by the first 10 g/m2 of fibre, and is unaffected by subsequent fibre deposition. 3.3.2 Fibre coarseness As explained in the methods section, four different types of fibres were fractionated in a Bauer-McNett classifier. As shown in Table 3.3, the four pulps possess similar mean lengths but dramatically different coarsenesses. The ratio of fibre flexible stiffness EI and fibre coarseness is proportional to the second order of fibre diameter approximately; therefore the approximated EI was calculated and listed in Table 3.3. As seen in Figure 3.10, there is a very strong correlation between fibre coarseness and sheet surface roughness; coarser fibres produce substantially rougher sheets. This finding is consistent with the work of Forseth et al (1997). In their work, handsheets were made from coarse fibres mixed with varying amounts of mechanical pulp fines under a wet pressing pressure of 60kpa. The paper was moistened by humid air of 97% RH for 24h. The results show that the handsheets containing most the fines are the smoothest. Because the coarseness increases with increasing EI (see Table 3.3), the positive dependence of surface roughness on coarseness implies an equivalent dependence on EI. Table 3.3: Characteristics of fibres used in coarseness study Type of pulp  Mean length(mm)  Coarseness(mg/m)  Fibre diameter(μm)  Abaca kraft(28 2.79 0.10 19.0 fraction) SPF kraft(28 2.89 0.18 28.3 fraction) Pine kraft(28 2.77 0.23 30.7 fraction) TMP(14 2.94 0.34 32.0 fraction) * The value was estimated based on the data from Kerekes and Tam Doo(1985)  51  Approximated EI(N/m2) 1e-12 4e-12 7e-12* 1e-10*  Figure 3.10: Standard deviation of fibre mat surface height with difference coarseness levels  3.3.3 Fibre length To study the influence of fibre length on sheet surface roughness, different BauerMcNett fractions produced from different pulps, but sharing the same coarseness, were compared. These pulp fractions are of the same coarseness but possess significantly different mean lengths. The relevant characteristics of the pulp fractions are given in Table 3.4. Fibre mats were made from these pulp fractions. As Figure 3.11 shows, there is a clear correlation between fibre length and fibre mat surface roughness; increasing fibre length is associated with a decrease in surface roughness. A linear least-squares fit to the results (Reed, 1989) shows the best fit line has a slope (in units of µm per mm) in the range (-3.46, -2.38). In other words, longer fibres are statistically correlated with a decrease in surface roughness. The correlation is consistent with the results of numerical simulations (Li & Green 2012). The effect of fibre length on surface roughness is small in this case. In the study, the mean length of fibres was over 1.87mm. As short fibres were not included, the conclusion might not apply for short fibres.  52  Table 3.4: Types of fibres of similar coarseness but different lengths Type of pulps  Mean length(mm)  Coarseness(mg/m)  Fibre diameter(μm)  Pine kraft(48 1.87 0.18 27.8 fraction) SPF kraft (28 2.89 0.18 29.3 fraction) SPF kraft (14 3.32 0.18 29.0 fraction) * The value was estimated based on the data from Kerekes and Tam Doo(1985)  Approximated EI(N/m2) 4e-12 4e-12* 4e-12*  Figure 3.11: Standard deviation of surface height for fibre mats with different fibre lengths  3.3.4 Drainage velocity Two different drainage conditions were studied through experimentation. One is drainage by gravity, with a drainage velocity of 0.04m/s; the other is drainage by suction with a drainage velocity 0.12m/s. Fibre mats made under these two conditions were compared. As shown in Table 3.5, the standard deviation increases slightly as drainage velocity increases from 0.04 to 0.12m/s. This increase is statistically significant and is consistent with the physical insight that more forceful drainage should embed fibres more deeply in the fabric.  53  Table 3.5: Standard deviation of mat surface height made using different drainage velocities Drainage Velocity(m/s)  Standard Deviation(μm)  Error(μm)  0.04  29.52  1.40  0.12  32.60  0.67  3.3.5 Forming fabric structure Forming fabric geometry is known to change the appearance of paper produced overtop of it. Measurements have been conducted to compare fibre mats made from forming fabrics of different coarsenesses. All fabrics used were single layer forming fabrics supplied by AstenJohnson, and geometric details of the fabrics are given in Table 3.6. As seen in Figure 3.12, high mesh count fabrics produce paper that is a factor of two smoother than paper produced by low mesh count fabrics. Table 3.6: Four different forming fabrics used in the experiments Name  Mesh(No. per cm) Knock(No. per cm)  Monoshape M36 AJ-170 MonoFlex AJ-100 MonoFlex AJ-102 MonoFlex AJ-105  14.2 21.6 34.6 35.0  Diameter(μm)  12.6 21.2 28.7 40.9  400 270 170 170  35.00  Standard deviation (μm)  30.00 25.00 20.00 15.00 10.00 5.00 0.00 14.2*12.6  21.6*21.2  36.4*28.7  35.0*40.9  Forming fabric Figure 3.12: Comparison of surface roughness of fibre mats made from different forming fabrics  54  3.3.6 Comparison of experimental results with simulations In a companion paper (Li and Green 2012), we report on numerical simulations of the initial stages of sheet forming. In general, there are the following points of qualitative agreement between the simulations and the experiments:   Increased fibre length is associated with decreased surface roughness    Increased drainage velocity results in increased surface roughness    Coarser fabrics produce paper of greater surface roughness  Experiments show that surface roughness has a positive dependence on EI, which is opposite to the simulation results. One reason might be that in simulations, the fibres are uniform and the diameter of fibres at different EI is set to be the same. To compare quantitatively with simulation results, in experiments, only the positions of fibres were taken into account to calculate the standard deviation of the fibre mats as in simulations. After calculations of the standard deviation of the sample used in 3.2.4, the comparisons was plotted in Figure 3.13. The EI value of fibres used in the experiment was given approximately.  Figure 3.13: Comparison of numerical and experimental results of standard deviation  55  Although the qualitative agreement between the simulations and experiments is good, the quantitative agreement is less robust. There are many factors that cause agreement between simulations and experiments to be only qualitative. First, real pulp fibres are polydisperse; not monodisperse as assumed in the simulations. Furthermore, real pulp fibres interact with one another both hydrodynamically and through direct contact. Such interactions were not considered in the simulations.  3.4  Conclusion  The surface roughness of fibre mats made from different fibres and under different forming conditions was studied experimentally using an X-ray microtomography device. Light-weighted fibre mats were produced in a handsheet former device. Using 3D microtomography, high resolution volume images were obtained and the 3D structure of paper was reconstructed. A surface map of the fabric-side surface of the fibre mat was created using image analysis techniques. The fabric-side surface roughness of fibre mats was shown to increase with increasing fibre coarseness, drainage velocity and fabric weave coarseness, and to decrease slightly in fibre length.  56  Chapter 4  Evolution of Fibre Orientation in the Forming Section 4.1  Introduction  Understanding fibre orientation in a suspension flow is important for the production of several industrial materials, including fibre-reinforced composites and paper, and paper board products made from cellulose fibres. Fibre orientation refers to the angular distribution of fibres relative to the paper-machine direction. The mechanical properties of the sheet are closely related to the fibre orientation distribution in the final paper (Loewen 1997). Directionality and curl of paper may be determined in part by the degree of fibre orientation developed in the forming process (Parker 1972). As shown in Figure 4.1, the ellipse in the polar diagram represents the fibre orientation distribution. Two main features are used to describe fibre orientation: the fibre orientation angle (angle from MD) ϴ, and the fibre orientation ratio (the ratio of major to minor axes of the orientation distribution ellipse), also known as anisotropy.  57  Figure 4.1: Fibre orientation distribution pattern in paper  The primary mechanism of orienting fibres in the sheet is the hydrodynamic shear flow in the head box and forming process. Fibre orientation distribution in the paper can be determined using many variables, such as fibre properties, headbox geometry and forming conditions. In this paper, we studied the effect of forming conditions on the fibre orientation distribution and aimed to understand the mechanism of fibre orientation in the forming section. Numerical simulations were conducted, and paper anisotropy was calculated as a function of fibre initial orientation, shear stress and flow direction. Finger and Majewski (1954) studied the mechanism of formation and the structure of the sheet on the Fourdrinier machine. In their work, the layered structure is attributed to the discrete steps of the drainage process. They demonstrated that the fibres in any layer reflect the relative motion of the free suspension at the instant the layer is deposited. It is assumed that the fibre becomes vertically oriented over the suction zone; then the leading end becomes anchored and the other is carried over with the flow. Wrist (1972) pointed that ―the boundary between the deposited fibres and free slurry is very diffuse‖. Therefore, it is likely that one end of the fibre is of a higher consistency, and thus located in a stronger part of the network structure, while the other still occupies the weaker network of the free suspension. When relative motions between the free suspension and  58  wire/fibre mat occur, the fibre is preferentially oriented in the direction of relative motion, held at one end by the network of the deposited mat (Wrist 1972). This interpretation of fibre orientation in the fibre mat was applied in the following simulations, in that the fibre is assumed to be held at one end by the fibre network, while the other end is carried in the flow.  4.2  Numerical Model  4.2.1 Problem simplifications To study fibre orientation in the forming process, a few simplifications were made. Three main hydrodynamic processes occur at the forming stage: drainage, oriented shear and turbulence. Turbulence was neglected. The flow was simplified to uniform flow and simple shear flow. The shear was characterized by a mean velocity gradient. In pulp suspensions, fibres are of various lengths, diameters, wall thicknesses, curvature stiffnesses and so on. In this paper, all fibres were assumed to be uniform in diameter. Rigid fibres were assumed to consist of one spheroid, and flexible fibres were modeled as chains of spheroids connected by flexible joints. In the calculations, the fibre suspension was assumed to be very dilute. Interactions between fibres were neglected. As fibres approached the fabric/fibre mat, one end of each fibre became attached to the network of the deposited fibre mat and the other end was mobilized in the flow. The fibre mat was assumed to be a porous media in parallel with the MD-CMD plane. The fibre was modeled as a spheroid with one end pinned to the fibre mat as shown in Figure 4.2. Angle β0 and θ0 defines the initial orientation of the fibre, β0 ϵ(0, π) and θ0 ϵ(0, π/2).  59  Figure 4.2: Schematic of the initial fibre orientation  4.2.2 Hydrodynamic force on the fibre For a particle in an unbounded Stokes flow, the hydrodynamic force depends linearly on the relative velocity between the fibre and the fluid. In the simulations, the fibre was assumed to be in a free stream. Hydrodynamic force was calculated based on the formula shown in following equations (Kim & Karrila 2005). (4.1) (4.2)  i and j is the index of direction.  are the velocity, vorticity and rate of strain  of the ambient flow, respectively. The resistance tensors are defined by (4.3) (4.4) (4.5)  XA, YA, XC, YC, YH are a function of the particle eccentricity (Kim & Karrila 2005). Their expression is shown in Appendix A. To test our assumption that the hydrodynamic force acting on a fibre could be well represented by that acting on a fibre in the local freestream flow conditions, the  60  hydrodynamic force acting on a cylinder above a porous media (with porosity 0.5) was computed for two cylinder configurations as shown in Figure 4.3. The free stream velocity is 0.5m/s. In one configuration, a cylinder at an angle of 45˚ to the flow was placed above the porous medium perpendicular to the flow. The cylinder diameter was 50 µm and its length was 500 µm. The computational domain is shown in Figure 4.3. The computed hydrodynamic force is compared with that of the same cylinder in a freestream flow in Table 4.1. The two components of force acting on the cylinder differ by less than 10% between when it is located above the porous medium and when it is in an unbounded flow. In another configuration, the cylinder was positioned parallel to, and one diameter above, the porous media. The force on the cylinder was -5.60e-6N above the porous media compared to -5.51e-6N in a free stream. In both cases, the hydrodynamic force on the fibre did not vary significantly when it was placed upstream of a porous media. This agreement implies that our assumptions about the hydrodynamic force acting on a fibre, in the presence of a fibre mat, are plausible.  Figure 4.3: Schematics of computational domain  61  Table 4.1: Hydrodynamic force on cylinder at 45˚ (see text for details)  Hydrodynamic force Above the porous media In the free stream  Fx (N)  Fz (N)  1.28e-6  -4.01e-6  1.25e-6  -3.76e-6  4.2.3 Equations of motion The governing equations of motion for the fibre include kinematic equations and dynamic equations. For each segment in the fibre, there are connectivity equations, linear momentum equations and angular momentum equations. Since the fibre is inextensible, system connectivity for the fibre can be described as (4.6)  r is the vector to the center of the spheroids, and c is the vector along the major axis of the spheroids. The equation of momentum for particle i is given by (Ross & Klingenberg 1997)  (4.7)  Where, m is the mass of the segment; F(h) is the hydrodynamic force; F(c) is the contact force; Xa is the internal force at the joint a; S is connectivity matrix. N is the number of segments. The equation of angular momentum is given by (Ross & Klingenberg 1997)  (4.8)  Where, T(h) is the hydrodynamic torque; T(c) is the contact torque; c is the connectivity vector. Ya is the internal torque at joint a, which includes the twisting torque and  62  bending moment.  is the time derivative of angular momentum. A detailed description  of the equations can be found in Li and Green (2012). The boundary condition for the fibre is that one end of the fibre is fixed. Therefore, and there is a following relationship in the calculations.  (4.9)  The Nth segment is at one end that is in contact with the fabric/fibre mat. By some manipulation of the governing equations, the translational velocity and angular velocity for each segment can be solved directly with the following equations: (4.10)  (4.11)  N1, N2, N3, N4, M1, M2 and M3 are matrices associated with the position and properties of the particles, and with the velocity and properties of the flow field. These matrices are given in the Appendix A. C++ code was written to simulate the motion of the fibre. The code was validated by comparing the simulation results with theoretical solutions. Jeffery’s theory and small beam theory were used to verify the code. Details of validation were shown in a previous paper (Li & Green 2012) and were also included in Appendix B.  4.3  Results  4.3.1 Fibre orientation in a uniform flow In uniform flow, fibres move in a constrained plane defined by two vectors, the initial long axis of the fibre and the flow velocity (Figure 4.4). As shown in Eq.(4.14), the fibre  63  orientation P after deposition is a function of its initial orientation P0 (Eq. (4.13) and flow direction U (Eq.(4.12)). Figure 4.5 shows the curve of the azimuth angle after deposition as a function of the initial azimuth angle at a given flow direction and different initial elevation angle.  Z  Figure 4.4: Schematic of one fibre rotation plane (4.12) (4.13) (4.14)  β is the azimuth angle and θ is elevation angle. α is the angle of the freestream velocity vector (both MD and Z-direction components) makes relative to the Machine direction.  64  Figure 4.5: Fibre orientation angle β as a function of initial orientation β0 (α=1.19)  4.3.2 Fibre orientation distribution Using the formula above, the fibre orientation distribution was studied statistically. An orientation distribution of 40000 fibres of initially randomly distributed orientation was calculated based on Eq. (1.11). The anisotropy of the deposited fibre mat was calculated based on the distribution ellipse, and was plotted against the flow velocity direction as shown in Figure 4.7. Fibres with initial distribution given according to the Fokker-Planck orientation distribution function (  ) as shown in Figure 4.6 are also  considered and plotted in Figure 4.7.  65  Figure 4.6: Fibre orientation distribution after headbox  Figure 4.7: Anisotropy of a fibre mat with different fibre initial orientation distributions  Figure 4.8 shows that the initial orientation distribution of fibres has a strong effect on the orientation distribution of paper. Anisotropy of fibre mats increases when the velocity angle α decreases. During the forming process, drainage velocity and machine direction velocity both decreases, based on the change of their ratio, anisotropy of the fibre mat can be estimated. When the jet to wire speed ratio increases, the relative velocity between  66  the stock and the forming fabric also increases, which in turn increases fibre mat anisotropy. Figure 4.8 shows computer-generated fibre mats of varying anisotropy.  Figure 4.8: Simulated fibre mats at different anisotropy (from top left to bottom right: 1, 2, 4, and 9)  4.3.3 Fibre orientation in a simple shear flow The flow is simplified as simple shear flow, when the jet to wire velocity ratio is not equal 1. The shear rate is simplified to be in linear relationship with the jet to wire velocity difference. In a simple shear flow, the fibre does not move in a constrained plane as it did in the uniform flow because the direction of hydrodynamic torque is different from the direction of rotation plane. Simulations were conducted to obtain the curve of fibre orientation angle β as a function of fibre initial orientation angle β0, θ0, and ( is the velocity gradient).  is a dimensionless velocity gradient, and can be  viewed as the ratio of two time scales:  is the time taken to move one fibre length,  67  and  is proportional to one period of fibre rotation. Compared to Figure 4.5, Figure  4.9 shows one group of curves at  . Due to the shear stress, the fibre tends to  move toward the plane of MD and Z. Figure 4.10 shows a given initial elevation angle θ0=0.942 with a curve of angle β after deposition versus the initial angle β0 at different shear stress.  Figure 4.9: Fibre orientation angle β as a function of initial orientation β0 at  68  Figure 4.10: Fibre orientation angle β as a function of initial orientation β0 (ϴ0=0.94) in a shear flow  In Figure 4.11, a different value of  is compared. Fibre mats were formed under the  same initial conditions. We can conclude that with an increasing velocity gradient or decreasing drainage velocity, the anisotropy of paper will increase.  Figure 4.11: Anisotropy of fibre mats plotted with the velocity angle at different velocity gradients  69  4.3.4 The effect of fibre properties Fibre properties, including fibre length and fibre stiffness, were considered in the simulations. Their effects on fibre orientation after deposition were calculated and plotted. The shear rate was fixed to be 150s-1, and  in the following  simulations. The results are shown below. In Figure 4.12, the curves represent the angle β as a function of β0 at ϴ0=0.942 for different aspect ratios of rigid fibres. When the value fibre aspect ratio L/d increases, it does not have a significant effect on the results. When the aspect ratio is quite small (less than 10), the difference becomes more significant. For realistic fibres, with a diameter of 20-40μm and length of 0.3-3mm, the aspect ratio is much higher than 10; we can conclude that the aspect ratio in this range does not greatly affect fibre deposition. Fibres of varying flexural rigidity were used in the simulations. Figure 4.13 shows that the fibre configuration after deposition varies with flexural rigidity EI at a given , initial elevation angle ϴ0=0.942 and given initial angle β0. The EI value of the fibre affects the orientation of the fibre after deposition due to fibre bending. For longer and more flexible fibres, fibre bending is more obvious. Two fibre lengths were used: L=4mm (L/d=100) (shown in Figure 4.13 and Figure 4.14) and 2mm (L/d=50) (shown in Figure 4.15 and Figure 4.16). Compared to the two groups of figures at different L, the EI value has less of an effect on fibre orientation for shorter fibres. The dimensionless parameter  (μ is dynamic viscosity, uz is drainage velocity and L is fibre length)  has been defined in Vakil’s work (Vakil & Green 2011), which is the ratio of the resistive force to bending force. In this case, with shorter fibres, the dimensionless flexural rigidity is higher; therefore shorter fibres tend to bend less. In reality, the fibres from the headbox are pre-oriented to some degree in the flow direction and typical EI values of fibres are between 10-11 and 10-12 Nm2. For these typical fibres, fibre flexibility does not have a significant effect on fibre orientation.  70  Figure 4.12: Fibre angle β after deposition as a function of initial value β 0 at different aspect ratio L/d  Figure 4.13: Fibre configuration after deposition at θ0=0.942, L/d=100, (β0=0.471(L), β0=0.972(R))  71  Figure 4.14: Fibre configuration after deposition at θ0=0.942, L/d=100, (β0=1.414)  Figure 4.15: Fibre configuration after deposition at θ0=0.942, L/d=50 (β0=0.471(L), β0=0.972(R))  72  Figure 4.16: Fibre configuration after deposition at θ0=0.942, L/d=50, (β0=1.414)  4.4  Conclusion  The mechanical properties of the sheets are closely related to the fibre orientation distribution in final paper. This paper focuses on the evolution of fibre orientation in the forming section. Fibre orientation in the forming section was studied numerically. The fibres in the simulations were positioned with one end held by the fabric/fibre mat, and the other end carried in the flow. An analytical solution was obtained for the case of a pinned fibre carried by a uniform flow. In this case, the fibre angle after deposition was found to be only a function of flow direction and fibre initial position. Fibre mat anisotropy was calculated based on a statistical study of a large number of fibres. The results show that with a decreasing drainage velocity to MD velocity ratio, anisotropy in the formed paper increases. These results can be used to interpret the anisotropy of paper made using different jet-to-wire ratios and layered structures of paper formed in the forming section. Two different initial fibre orientation distributions were considered. The  73  results show that the initial orientation distribution of the fibres has a strong effect on the anisotropy of paper. For fibres in a shear flow, another dimensionless group, deposition on the fabric. With a higher value of  , plays a role in fibre , fibre mats develop higher  anisotropy. Increased oriented shear or decreased drainage velocity results in increased anisotropy. Fibre properties such as flexural rigidity (  ) and the fibre aspect ratio  did not play important roles in the fibre orientation distribution of the paper.  74  Chapter 5  Summary and Recommendations for Future Work 5.1  Summary  This thesis studied fibre interactions with a forming fabric in the forming section of a paper machine. Initial sheet forming and the evolution of fibre orientation in the forming section were studied numerically. Experiments were carried out to study the surface roughness of light-weighted fibre mats created under different conditions. Conclusions for each study are reviewed below. 5.1.1 Investigation of fibre interaction with a forming fabric Topographical wire mark of the fibre mat was studied numerically in this section. The physical interaction between the fibres and the forming fabric were simulated. The numerical methods used were verified through a comparison with Jeffery’s equations and beam theory. Two single layer forming fabrics, a simplified sine-wave forming fabric and a real forming fabric were used in the simulations. A light-weighted fibre mat was formed in the simulations from a few hundred fibres which were initially randomly distributed on top of the forming fabric. The surface roughness of the fibre mat was calculated and compared as a function of fibre EI and length. For fixed EI, increasing the fibre length decreased the pulp mat surface roughness owing to the multiple support points over the length of each fibre. For fixed fibre length, increasing the EI decreased the surface roughness due to reduced fibre bending in the fabric surface holes. The surface roughness  75  predicted through the numerical experiments is much higher than the results of experimental studies from literature (Ashori et al., 2008). It is believed that the pressing and drying sections dramatically reduce topographic variations generated in the forming section. Different fibre alignments were also studied. For non-symmetric forming fabrics, the surface roughness of the fibre mat changed substantially with fibre orientation. However for symmetric forming fabrics such as sine-wave forming fabrics, the effect of fibre alignment was not significant. Surface roughness was also calculated as a function of drainage velocity. As drainage velocity increased, so too did fibre bending and therefore mat surface roughness. Based on the surface roughness results found at different EI levels, it is concluded that topographical wire mark is caused partially by fibre bending, and partially by the fabric geometry. For the simple, sine-wave single layer fabric and monoshape forming fabric, more than 50% of topographical wire mark was caused by the geometry of the forming fabric, with the remainder caused by fibre bending. 5.1.2 Microtomography measurements of paper surface roughness Due to the complexity of the interaction between the fibre mat and forming fabric, numerous simplifications were made in the simulations (i.e. the use of uniform fibres and simple fabric geometry, negligible interaction between fibres, etc.). To design a study that reflected real conditions, measurements of the topography of the fibre mats were conducted to correspond with the numerical studies. These measurements were also used to compare the surface roughness of the formed mat with the numerical results. The surface roughness of fibre mats made from different fibres and under different forming conditions were studied experimentally using an X-ray tomography device. Light-weighted fibre mats were made in a handsheet former device. Through 3D microtomography, high resolution volume images were obtained and the 3D structure of  76  paper was reconstructed. A surface map of the fabric-side surface of the fibre mat was created using image analysis techniques. The fabric-side surface roughness of the fibre mats was shown to increase with increasing fibre coarseness, drainage velocity and fabric weave coarseness, and to decrease slightly with fibre length. Handsheets made from finer forming fabrics possessed smoother surfaces. This relationship between fibre mat surface roughness and fibre length is consistent with the numerical results. 5.1.3 Evolution of fibre orientation in the forming section Fibre orientation distribution in final paper sheets is closely related to the mechanical properties of the sheets. This chapter numerically studied the evolution of fibre orientation in the forming section. The fibres in the simulations were positioned with one end held by the fabric/fibre mat, and the other end carried in the flow. Using a uniform flow, an analytical solution was obtained. The fibre angle after deposition was only a function of flow direction and fibre initial position. Anisotropy of fibre mats was calculated based on a statistical study of a large number of fibres. The results show that with a decreasing drainage velocity to MD velocity ratio, anisotropy of the formed paper increases. In the forming section, as the jet to speed ratio increases, the ratio of machine velocity to drainage velocity increases, which in turn increases the anisotropy of the paper. Two different initial fibre orientation distributions were considered. The results show that the initial orientation distribution of fibres has a strong effect on the anisotropy of paper. In a shear flow, the dimensionless group of  was defined. Higher values of  corresponded with high fibre mat anisotropy. Increased oriented shear also caused increased anisotropy. Fibre properties such as flexural rigidity (dimensionless group ) and aspect ratio were found to not play important roles in the fibre orientation distribution of the paper.  77  5.2  Study Applications  Chapter 2 elaborates on the numerical model that was built to simulate the initial sheet forming process using a real forming fabric. This simulation of initial sheet forming can be extended to other studies on paper structure, since the fibre network is closely related to the mechanical properties of paper, including tensile and tear strength. The model of fibre interaction with a forming fabric can be applied to other studies of particle-structure interaction in a flow, such as studies of air particles trapped by filters. The model can also be applied to biomedical studies, with potential applications for studies of particle deposition in lungs. Although several previous studies have used microtomography devices to study paper structures, this study is distinct in that both the fibre mat and the forming fabric were measured in order to maintain the structure of the wire side of the paper after leaving the forming section. This method can be used to study other fabric-related properties, such as the effect of forming fabric structures on filler/additive distributions. The method illustrated in Chapter 4 can be used to study the layered structure of paper. Anisotropy of paper under varying initial conditions and forming conditions may be calculated and compared. Both the numerical method and the conclusion of this chapter can be used to study other fibre-reinforced composites and the evolution of particle orientation in a flow near a boundary.  5.3  Limitations and Discussions  5.3.1 Investigation of fibre interaction with a forming fabric In the simulations, fibre suspension was assumed to be dilute, and the fibre-fibre interaction was neglected. Therefore, in simulating the initial sheet forming, the fibres were assumed to be independent from one another. This simulation diverged from real  78  conditions by neglecting fibre interaction, producing surface roughness results that may differ slightly from real paper sheets. In the simulations, the flow field was assumed to be prescribed while the fibre mat was formed on top of the forming fabric. Given this assumption, the model cannot be understood as a simulation of the whole paper forming process. As fibres accumulate onto the forming fabric, the hydrodynamic resistance of the fibre mat increases, causing the flow field to grow progressively more uniform. The method described can thus only be applied to the initial forming process. Transient flow must also be considered when simulating the paper forming process. Finally, the surface roughness levels calculated in the simulations apply only to fibre mats located in the forming section. Following the pressing and drying stages, surface roughness characteristics change significantly. Hence, the value of surface roughness obtained in the simulations is not comparable to the results of studies that have measured the surface roughness of final paper sheets. 5.3.2 Microtomography measurements of paper surface roughness Using 3D microtomography, high resolution volume images were obtained and the 3D structure of paper was reconstructed with good resolution. After the 3D structure of the fibre mat and forming fabric was obtained, image analysis was processed to create a surface map of the fabric-side of the fibre mat, given the fact that the forming fabric was attached to the fibre mat in the measurements. In the process of conducting the image analysis to remove the forming fabric, some peak value of surface may have been cut, which may have decreased the surface roughness. When the surface roughness was calculated, the standard deviation of the surface height was averaged to remove variations caused by single fibres, and to highlight the wire mark. Thus, the absolute value of surface roughness calculated in the experiments cannot be compared directly to the simulation results.  79  5.3.3 Evolution of fibre orientation in the forming section In order to conduct focused studies, the simulations in this chapter were simplified significantly. In reality, the flow field of the paper machine during drainage is more complicated, and the boundary conditions of fibres during formation are more complex than particles with one fixed end. Although the forming section simulated in this study closely replicates the forming section of a Fourdriner machine, forming processes in roll twin wire formers are quite different. There are thus limitations in applying the results of this study to twin wire formers.  5.4  Future Work  (1) Future work could extend the study of fibre mat formation simulations that consider fibre and fibre interaction. When considering particle interactions, additional simulations of sheet forming could be conducted. Research relevant to the fibre concentration could be continued, and the effect of fibre concentration on paper properties could be studied. (2) The forming process is transient; a fibre mat built on the forming fabric alters the flow field. Future work could couple fibre mat formation and the flow field. By taking into account the fibre interaction and by coupling the fibre mat and the flow field, the paper forming process could be simulated. A more precise paper structure could be obtained through such simulations. (3) The anisotropy of fibre mats under different flow velocities calculated in the simulations could be validated by experiments. Fibre mats made under different flow conditions could be measured using microtomography. Anisotropy of the fibre mat could be calculated in accordance with algorithms using image analysis. The layered structure of the fibre mat could also be studied through image analysis.  80  (4) The hydrodynamic process in the headbox and forming section determines the fibre orientation distribution of paper. The simulations conducted in the forming section showed the effect of flow conditions and fibre properties. This work can be extended by simulating fibre orientation in the headbox. 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Dynamic simulation of fibre suspensions in shear flow. Journal of Chemical Physics, 102(5), 2254. Yan, D., & Li, K. (2008a). Measurement of wet fibre flexibility by confocal laser scanning microscopy. Journal of Materials Science, 43(8), 2869-2878. Yan, D., & Li, K. (2008b). Wet fibre shear flexibility and its contribution to the overall transverse deformation of fibres. Journal of Materials Science, 43(8), 7210-7218.  88  Appendix A  Numerical Model of a Fibre A.1  Equations of Motions  In the particle-level simulation method, the fibre is modeled as a chain of particles connected by flexible joints, as shown in Figure A.1. For segment i, ri is the vector to its center, and ci is the vector along its axis.  Figure A.1: Schematic of a discretized fibre and neighbouring segments i and i+1.  The governing equations of motion for the fibre include kinematic equations and dynamic equations. For each segment in the fibre, there are connectivity equations, linear momentum equations and angular momentum equations. As shown in Figure B.1, if the fibre is inextensible, system connectivity for the fibre can be described as (A.1)  r is the vector to the center of spheroids, and c is the vector along the major axis of spheroids.  89  The equation of momentum for particle i is given by (Ross & Klingenberg 1997)  (A.2)  Where, m is the mass of the segment; F(h) is the hydrodynamic force; F(c) is the contact force; Xa is the internal force at the joint a; S is the connectivity matrix; N is the number of segments. The equation of angular momentum is given by (Ross & Klingenberg 1997)  (A.3)  Where, T(h) is the hydrodynamic torque; T(c) is the contact torque; c is connectivity vector. Ya is the internal torque at joint a, which includes the twisting torque and bending moment.  is the time derivative of angular momentum, and can be extended as  (Lindstrom & Uesaka 2007) (A.4)  Where, I is the tensor of inertia, and  is the angular velocity.  Y represents the internal resistance torque at the joints, which includes torsional torque Yt and bending moment Yb. According to Bernoulli-Euler Law, the bending moments can be given by (A.5)  Where, θ is the angle of bending in radiant and θe is the equilibrium bending angle. EI is the flexural rigidity. nb is the unit vector normal to the plane of bending. For a uniform fibre under torsion, the torsional torque Yt can be written as (Ning & Melrose 1999)  90  (A.6)  Where, φ is the angle of torsion in radiant and φ e is the equilibrium torsion angle. GJ is the torsional rigidity. nt is the unit vector normal to the plane of torsion.  A.2  Hydrodynamic Force  The hydrodynamic force and hydrodynamic torque of a prolate spheroid in an unbounded creeping flow can be expressed by (Kim & Karrila 2005): (A.7) (A.8)  i and j is the index of direction.  are the velocity, vorticity and rate of strain  of the ambient flow, respectively. The resistance tensors are defined by (A.9) (A.10) (A.11)  Where XA, YA, XC, YC, YH are only functions of eccentricity e. d is the unit directional vector along the major axis of the spheroid. (A.12) (A.13) (A.14) (A.15)  (A.16)  91  (A.17)  (A.18)  a and b are the semi-major axis and semi-minor axis of the spheroid, respectively. When the hydrodynamic force on a cylinder in a creeping flow was calculated, the equivalent aspect ratio as shown below was used in the equations (Cox 1971). (A.19)  The hydrodynamic force of a spheroid in a creeping flow is a linear function of the relative velocity of the particle as shown in Eq. (A.7). In our simulations, the Reynolds number of the flow was much higher than that for creeping flow. Therefore, the equations are not suitable in these cases. Vakil simulated the flow field around a cylinder and calculated the hydrodynamic force on the cylinder at different angles and aspect ratios (Vakil & Green 2009). A series of formulas for drag and lift coefficients at different conditions were concluded by curve fitting. The hydrodynamic forces at a moderate a Reynolds number were calculated by the following equations based on the drag coefficient CD and lift coefficient CL. (A.20)  (A.21)  (A.22)  (A.23)  d is the unit directional vector along the axis of the cylinder. δ is an identity matrix of size 3. U∞ and U are the velocity of the flow and the particle, respectively. Ur is the  92  magnitude of relative velocity of the particle. L and D are the length and the diameter of the cylinder. θ is the angle between the axis of the cylinder and its relative velocity.  A.3  Numerical Model  The time derivative term in Eqs.(A.2) and (A.3) can be substituted by a forward finite different formula using the Euler method as shown below.  (A.24)  (A.25)  Where, Δt is the time step of discretization; n is the number of time steps. A.3.1 Fibre in the flow and contact with a surface Contact torque was neglected in the calculations. Through some manipulation of the governing equations, the translational velocity and angular velocity for each segment can be solved directly with the following equations: (A.26)  (A.27)  Where, ω is the angular velocity of segments, and  is the translational velocity of  segments. F(c) is the contact force, and T(c) is the contact torque. N1, N2, N3, M1, M2, M3 are matrices associated with the position and properties of the particles, and with the velocity and properties of the flow field.  93  Here A′ i = Ai + miI3×3Δt and  .  and  are anti-symmetrical matrices of  vector di and bi. The details are given by Ross and Klingenberg (1997). When the fibre moves in a non-constrained flow, the contact force is zero. In this case, the equations can be simplified as: (A.28)  (A.29)  94  A.3.2 Fibre with one end fixed In the simulations of the evolution of fibre orientation in the forming section, the fibre was fixed at one end. Therefore  and the following relationship applies:  (A.30)  Nth is the segment at one end that in contact with fabric-wire mat. By some manipulation of the governing equations, the translational velocity and angular velocity for each segment can be solved directly using the following equations:  (A.31)  (A.32)  N1, N2, N3, N4, M1, M2, M3 are matrices associated with the positions and properties of the particles, and with the velocity and properties of the flow field.  95  Here A′ i = Ai + miI3×3Δt and  A.4  .  is the Skew-symmetric matrix of  .  Solution to the Group of Equations  C++ code was written to simulate the motion of the fibre. The linear algebraic equations were solved through Gaussian elimination with back-substitution (Press et al. 1986). The typical step is  (A.33)  x is the unknown; a is the left-side matrix; b is the right-side vector. The Gaussian elimination and back-substitution yields a solution to the set of equations. After the translational velocity and angular velocity are obtained, the displacement and rotation of the segments are calculated with the following equations.  (A.34)  (A.35)  A.5  Numerical Stability  In the simulations, the continuity of the fibre was calculated at each step and was required to satisfy  96  (A.36)  ε is set to be 10-4. Time step size was chosen to meet the requirement. In calculations, when the velocity of the fibre is less than 10-7m/s, the fibre is said to be in steady state.  97  Appendix B  Validations B.1  Jeffery’s Theory  C++ code was written to simulate the motions of the fibre and its interaction with the forming fabric. Jeffery’s theory, beam theory and capstan’s equation were applied to validate the code. The fundamental theoretical framework that describes the evolution of the orientation of a rigid ellipsoidal particle in a viscous flow was conducted by Jeffery in 1922. He showed that the motion of a single ellipsoidal particle suspended in a Newtonian fluid in a Stokes flow field will rotate around the vorticity axis. The orientation angles θ and φ as shown in Figure B.1 are given by the following equations. (B. 1)  (B.2)  C and k are parameters decided by the initial conditions of the particle. Figure B.2 shows the rotation orbit obtained from simulations and equations at two different initial orientations. The simulation results are very consistent with the theoretical results based on both initial orientations.  98  Figure B.1: The geometry of the particle  Figure B.2: Comparisons of rotation orbit between numerical results and Jeffery’ theory  B.2  Small-Deflection Beam Theory  Small deflection beam theory describes the deflection of beams at given conditions. For a beam with two ends simply supported, given uniform distributed load as shown in Figure B.3, the maximum deflection ω is a function of rigidity EI, distributed load q and beam length L.  99  Figure B.3: A beam with two ends simply supported (B.3)  EI is the flexural rigidity of the beam, in which E is the Young’s modulus, and I is the second moment of inertia. The deflection of beams under uniform force was calculated in the simulations. The boundary conditions in the simulations are that two segments in the ends are simply supported by two balls. The beam was modeled as a number of rigid particles connected by flexible joints. The force was applied to the center of each segment. In Figure B.4, simulation results of ω based on different numbers of segments are compared. As the number of segments increases, the simulation results draw closer to the theoretical results. Figure B.5 shows the results of maximum deflection calculated based on simulations (18 segments) and beam theory. Compared to small deflection beam theory, the numerical results of the deflection are about 2% different.  100  Figure B.4: Change of ω with number of segments  Figure B.5: Comparison of maximum deflection ω between beam theory results and numerical results  B.3  Capstan Equation  The capstan equation was applied in the simulations to verify the determination of fibre beheavior after it comes into contact with the cylinder. Based on capstan equations as shown in Eq. (B.4), the ratio of tension on the tauter side to the tension on the slacker  101  side of rope wrapping a cylinder is an exponential function of the coefficient of friction μ and angle of contact φ.  (B.4)  T1 is the tension on the tauter side, and T2 is the tension on the slacker side. In the capstan equation, the rope is assumed to be perfectly flexible. In the simulations, the rope was modelled as a number of rigid particles connected by perfectly flexible joints as shown in Figure B.5. The threshold of the ratio of T1 and T2 which makes the rope slide was calculated in the simulations. Different numbers of segments were used to model the fibre. Figure B.6 shows the ratio of tension calculated by the simulations for different numbers of segments. As the number of segments increases, the tension ratio tends to reach a constant value. In Table B.1 , the result of the tension ratio is based on 20 segments.  Figure B.6: Discretized rope wrapping around the cylinder  102  Figure B.7: The ratio of tension calculated in the simulations based on different numbers of segments Table B.1: The ratio of tension calculated from the capstan equation and simulations  B.4  Method  Capstan equation  Simulation results  Ratio of Tension  1.874  1.870  Rigid Beam  For a rigid beam with a uniform load as shown in Figure B.7, we can calculate the initial offset of fibre χ theoretically when the beam begins to slide off the cylinders. When the initial offset should satisfy (B.5)  The beams will slide off the cyinder. Where μ denotes the frictional coefficient. D is the diameter of the cylinder. α is the inclination angle of the beam,  when the beam slides.  103  Figure B.8: A rigid beam in touch with a cylinder  In the simulations, a very short beam with a length of 0.25mm was used and discretized into segments connected by joints. With satisfies results of  , based on equations, if the initial offset  , the beam must slide off the cylinder. Figure B.9 shows the numerical . As the number of segments increases to 20 (segment size=0.0125mm), the  results are quite close to 0.1 (5% difference). Therefore, the numerical method to determine the contact behaviour of the fibre after it is in contact with the cylinder is appropriate when the size of segments is very small. However when we simulated the motion of realistic fibres, the length of the fibre was approximately 1-3mm. It will take a large computational time to calculate a large number of segments. Hence, it is not practical to use this method to calculate contact between a flexible fibre and a cylinder.  104  Figure B.9: The effect of the number of segments on the critical value of the initial offset  .  The simulations of the fibre moving in a free flow are verified by Jeffery’s equations. The results of the maximum deflection of the beam with both ends simply supported based on the simulations is consistent with beam theory. Simulations of fibre sliding on the surface of the cylinder are also verified by capstan equations. Therefore, the numerical method and code work properly for the investigation of fibre interaction with cylinders.  B.5  Hydrodynamic force  In the simulations, the hydrodynamic force acting on each segment was calculated based on the flow velocity in the position of the segment center, and the flow through the forming fabric was prescribed without considering the existence of fibres. The formulas of hydrodynamic force on a cylinder were drawn in a freestream. To test our assumption that the hydrodynamic force acting on a fibre could be well represented by that acting on a fibre in the local freestream flow conditions, simulations were conducted to calculate the hydrodynamic force on a cylinder placed in the upstream of the flow through a  105  forming fabric. The results were compared with those calculated by formulas based on local flow velocity.  Figure B.10: Position of the fibre above the sine-wave forming fabric (1)  Figure B.11: Position of the fibre above the sine-wave forming fabric (2)  A fibre placed in the upstream of the flow through a sine-wave forming fabric was simulated as shown in Figure B.11. The fibre was represented as one cylinder of a diameter 50um and length 1mm. The average velocity of the flow is 0.5m/s. The fibre was placed at different distances (D) from the forming fabric as shown in Figure B.11. Hydrodynamic force on the fibre with was calculated. Simulations show that the forming fabric has pretty small effect on the force acting on the fibre, especially when the fibre is  106  1d away from the fabric. Hydrodynamic force  on the fibre at given position was  normalized by that on the fibre in the upstream 9d away from the fabric  and plotted in  Figure B.13 . The flow through the sine-wave forming fabric without the existence of fibres was simulated. Figure B.12 shows the velocity profile at different positions above the forming fabric, which are the same as the positions of the fibre in the upstream as shown in Figure B.11. Hydrodynamic force  was calculated based on the local flow velocity, and the  results are normalized by the hydrodynamic force on the fibre 9d from the forming fabric . The results were plotted in Figure B.13 and compared with the above case in which two- way coupling between fibre and water was considered. Results show that as the fibre moves close to the forming fabric, the difference of hydrodynamic force between the two cases increase. As shown in Figure B.13, when the fibre is 1d away from the forming fabric, the difference between the two cases is about 17%. When the fibre is further than 3d away from the forming fabric, the difference is less than 5%.  Figure B.12: Velocity profile along the fibre at different distance from the forming fabric  107  Figure B.13: Comparison of hydrodynamic force on the fibre calculated with two different methods  We can conclude that the assumption of one way coupling doesn’t change the hydrodynamic force on the fibre significantly, especially when the particle is more than one diameter away from the forming fabric. That applies to most cases during fibre motion in the flow through the forming fabric. This agreement implies that our assumptions about the hydrodynamic force acting on a fibre, in the presence of a forming fabric, are plausible. In these calculations, the force is normalized to exclude the effect of boundaries of the computational domain and focus on the effect of the forming fabric.  108  

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