B. Eng. Northwestern Polytechnical University, China, Eng. Beijing University of Aeronautics and Astronautics, China, 1988

Transcription

1 FIBER ORIENTATION IN A H E A D B O X by Xun Zhang B. Eng. Northwestern Polytechnical University, China, 1985 Eng. Beijing University of Aeronautics and Astronautics, China, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF MASTER OF APPLIED SCIENCE in THE F A C U L T Y OF G R A D U A T E STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 2001 X u n Zhang, 2001

2 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia 2324 Main Mall Vancouver, BC V6T 1Z4 Date: January 2001

3 ABSTRACT The prediction of fiber orientation is a critical parameter for papermakers who wish to control the quality of their paper products. The wet end processes, especially the headbox and the drainage stage on the forming wire, play important roles in determining the fiber orientation characteristics. The current thesis is focused on the headbox flow effect on fiber orientation. It summarizes a mathematical method, which has been developed by other researchers, for predicting the orientation of rigid fibers in dilute suspensions. This method, composed of a turbulent flow simulation model and a fiber motion model, has been applied to predict fiber motion in a headbox. To validate the method, experiments have been conducted by measuring the orientation of dyed nylon fibers moving in a pilot plexiglass headbox. Comparison of experiments and the present numerical simulations of the fiber orientation shows that the simulation method proposed can predict the trend of the statistical orientation distribution of dilute suspensions in headboxes, although there exists obvious deviations between the simulations and experiments. The fibers are seen to be more strongly oriented by the predictions than is observed in the experiments. The anisotropy of the fiber orientation in the headbox flow is caused not only by the mean flow field characteristics, but also by the turbulence characteristics, and the explicit effects of the turbulence are not yet included in the predictions. The simulation method is applied to predict fiber orientations for different headbox geometry, fiber aspect ratio and flow rate. From the prediction method, using only the mean flow effects, a larger contraction ratio is found to produce more concentrated fiber orientation in the flow direction at the exit of the headbox. The channel length, the flow velocity and the fiber aspect ratio within the range of study have little influence on the fiber orientation properties. ii

4 TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES ACKNOWLEDGEMENTS ii iii v vi viii 1. INTRODUCTION 1 2. LITERATURE REVIEW Fiber Orientation and Paper Quality The Definition of Fiber Orientation Factors Affecting Fiber Orientation Headbox Jet to Wire Speed Difference Forming Wire Fiber Suspension Consistency Headbox Flow Simulations to Investigate Fiber Orientation Fiber Suspension Simulation The Scope of This Thesis Work EXPERIMENTAL ARRANGEMENTS Objectives of the Experimental Work Fiber Suspensions Flow Loop Image Analysis System 19 iii

5 3.5 Measurement COMPUTER SIMULATION OF FLOW A N D FIBER ORIENTATION The Headbox Flow Model Fiber Model RESULTS A N D DISCUSSION Analysis of the Headbox Flow Field Comparison of Simulation and Experimental Results Factors Affecting Fiber Orientation The Effect of Contraction Ratio on Fiber Orientation The Effect of Flow Rate on Fiber Orientation The Effect of Channel Length on Fiber Orientation The Effect of Fiber Aspect Ratio on Fiber Orientation The Effect of Flow Elongation Symmetric Channel Statistical Error Estimation S U M M A R Y A N D CONCLUSIONS RECOMMENDATIONS FOR FUTURE WORK N O M E N C L A T U R E REFERENCES 66 iv

6 LIST OF TABLES Table 3.1. The Geometry of the Headbox Converging Section 18 Table 3.2. The Sign of the Orientation Angles 21 Table 3.3. The Number of Fibers at Each Measurement Point 22 Table 5.1. Orientation Parameters Obtained from Experiments and Simulations: 43 Table 5.2. Fiber Orientation Parameters for Different Rc (Uo = 0.24 m/s, L c = m) 45 Table 5.3. The Orientation Parameters for Different Uo (Rc= 10, L c = m) 46 Table 5.4. Fiber Orientation Parameters for Different L c (Rc= 10, Uo-0.24 m/s) 46 Table 5.5. The Orientation Parameters for Different A r...47 Table 5.6. The Elongation of Flow at the Channel Exit for Different Rc 48 Table 5.7. Orientation Parameters at Exit of A Symmetric Headbox for Different U 0: 48 v

7 LIST OF FIGURES Figure 2.1. Fiber orientation distribution pattern in a piece of paper 16 Figure 3.1. The length distribution of nylon fibers 23 Figure 3.2. Images of fibers: (a) dry dyed nylon fibers, (b) fiber suspension 23 Figure 3.3. The flow loop in the experiment 24 Figure 3.4. The scaled plexiglass headbox used in the experiment 24 Figure 3.5. Cross sectional view of the scaled headbox (dimensions in cm) 25 Figure 3.6. The photographic arrangement for (a) side view and (b) bottom view 25 Figure 3.7. Typical picture of fibers in the flow: (a) before analysis; (b) after analysis.. 26 Figure 3.8. The measurement points along the headbox channel 27 Figure 4.1. A fiber in three-dimensional coordinates 35 Figure 4.2. The initial random distribution of 1000 fibers 35 Figure 5.1. The physical mesh of the asymmetric converging section 52 Figure 5.2. The streamlines of the flow in the headbox convergent channel 52 Figure 5.3. The pressure and u-velocity changes along the central streamline 53 Figure 5.4. The u-velocity contours on the central symmetry plane 53 Figure 5.5. The v-velocity contours on the central symmetry plane 53 Figure 5.6. The elongation of the flow changes along the central streamline 54 Figure 5.7. The fiber orientation distribution at x = 4.5 cm, 54 Figure 5.8. The fiber orientation distribution at x = 12.2 cm, 55 Figure 5.9. The fiber orientation distribution at x = 15.7 cm, 55 Figure The fiber orientation distribution at x = 19.2 cm, 56 Figure The fiber orientation distribution at x = 22.7 cm, 56 Figure The fiber orientation distribution at x = 26.2 cm, 57 Figure The fiber orientation distribution at the channel exit, 57 Figure The orientation parameters along the central streamline, 58 Figure Fiber orientation distributions at the channel exit for various Contraction ratios, (a) in x-y plane, (b) in x-z plane 58 vi

8 Figure Cross sectional view of the symmetric headbox (dimensions in mm) 59 Figure The physical mesh of the symmetric headbox 59

9 A C K N O W L E D G E M E N T S I express sincere gratitude to my supervisors, Dr. Martha Salcudean and Dr. Ian Gartshore, for their helpful advice and suggestions. I would also like to thank my colleagues, Mohammad R. Shariati and Suqin Dong. Their effort and help has made things much easier both in my experimental work and my simulation work. I am grateful for the financial support provided by FRBC Research Award. Finally, I wish to thank my wife, Yinghui, for her constant support and encouragement during the past two years. viii

10 1. INTRODUCTION Paper is a heterogeneous three-dimensional composite of fibers and other materials. Its mechanical properties are highly dependent on the microstructure characteristics such as fiber properties, and the formation and orientation distribution of the fibers. The demand for high quality paper and paperboard has focussed the attention of papermakers on how to control these critical characteristics in the papermaking processes. The fiber orientation distribution in a piece of paper determines the distribution of strength, permeability and absorbency, and affects the dimensional stability, runability and printability of the paper. The fiber orientation in paper is determined by the processing conditions in the wet-end stage of the headbox and in the forming process. Experimental evidence has shown that fibers have some preferred orientation direction depending on the specific flow field. The headbox has a significant effect on the orientation of fibers leaving the slice. The elongation and shear in the flow leading to the slice tend to orient fibers in the machine direction. If the fiber orientation can be predicted for a given set of processing conditions, manufacturing paper with optimum mechanical properties will become much easier. The general objective of this thesis is to investigate, both numerically and experimentally, the three-dimensional fiber orientation produced by a dilute headbox flow. In the numerical simulations, both symmetric and asymmetric headboxes are studied. The numerical simulation method introduced here provides a quantitative methodology for the prediction of the fiber orientation resulting from the fluid kinematics. It can be used to predict fluid-fiber interactions and provide paper manufacturers a better knowledge of fiber orientation distribution and sheet properties. In this research work, several elements which affect the fiber orientation in a headbox, such as the headbox 1

11 geometry, flow conditions and fiber properties, are investigated with the predictive ability of this simulation method and the results are analyzed. Following this chapter, the relevant literature is reviewed in Chapter 2. The detailed experimental conditions and methods of measuring the orientation distribution of fibers in a headbox flow are presented in Chapter 3. Chapter 4 describes the numerical simulation of the fiber orientation distribution by a combination of a flow model and a fiber motion model. Chapter 5 presents the comparison of measured and numerically simulated results. Parametric studies, obtained using the numerical method, show the influence of headbox geometry, flow velocity and fiber property. Chapters 6 and 7 summarize the major conclusions of this thesis and give recommendations for future research, respectively. 2

12 2. LITERATURE REVIEW 2.1 Fiber Orientation and Paper Quality A piece of paper is composed of numerous fibers which are located within the paper plane and oriented in different directions. Statistically, however, most of the fibers may be aligned in one direction. This anisotropy of fiber orientation is produced by the paper manufacture process and is closely related to several critical paper properties. The orientation pattern retained in the final paper controls the mechanical properties of the sheet. Nordstrom and Norman [1] indicated that depending on the grade, a certain degree of fiber orientation anisotropy in the paper is desired. For newsprint, a rather high anisotropy is required for good runnability in the paper machine and the printing press. But for wood-free sheet grades, on the other hand, a lower anisotropy is desired to ensure isotropic dimensional changes with variations in moisture and temperature. Nordstrom and Norman also pointed out that the strength in the paper thickness direction is affected positively by the degree of fiber-to-fiber bonding and also by the degree of fiber orientation in that direction. The strength in the paper thickness direction must be sufficient to avoid delamination in coldset offset printing, or blistering in heatset offset printing. A sheet is stronger and stiffer in the direction in which most fibers are oriented, and weaker and more compliant in the direction of least orientation. We can understand this by studying the properties of the principal constituents of paper, the wood fibers, because the fiber properties and paper properties are closely correlated [2, 3]. Fiber dimensions, flexibility and coarseness are connected with the mechanical properties, structural variables and formation of paper, such as tensile strength, tearing strength, bursting and bonding strength, porosity and sheet density. 3

13 A wood fiber has quite different properties along its axis compared to those across it. For example, the strength of a fiber is much greater along the fiber axis than across it, whereas the wet-expansivity is greater across the fiber axis than along it [4]. If the major direction is defined as the direction in the paper surface toward which most fibers are aligned and the minor direction as the direction normal to the major direction, the following conclusions about paper property can be inferred from the fiber-paper relationship. The tensile stiffness, tensile energy absorption, bending stiffness and crush strength are higher in the major direction, but the tear strength and wet expanding tendency are higher in the minor direction. Directional differences in mechanical properties have been experimentally correlated with fiber orientation [5, 6, 7]. Significant fiber misalignment may cause serious defects, leading to poor dimensional stability [8] and reduced strength. Loewen [6] summarized the paper quality problems that are related to poor fiber orientation as follows: Twist, warp, curl and stack-lean. Web wandering, misregistering in multi-pass printing and colour printing. Paper feed path j amming. Multi-part forms debonding. Low tensile strength, low tear strength and weak stiffness. Wrinkles on long lead presses and dryer wrinkles. 2.2 The Definition of Fiber Orientation Fiber orientation refers to the angular distribution of fibers relative to the paper-machine direction (MD). This can be visualized in the polar diagram of Fig The distance from the origin at a given angle is proportional to the number of fibers oriented in that direction. The polar diagram describes two commonly used fiber orientation terms: the fiber orientation angle and fiber orientation index. The fiber orientation angle, 9 as shown in the diagram, is the angle from the machine direction in which most of the fibers are

14 oriented. The fiber orientation index is the ratio of the fibers oriented in the M D over those oriented in the cross machine direction (CD), which is often defined as the ratio of M D to CD strength based on the knowledge that the fiber orientation distribution corresponds to the distribution of strength. The fiber orientation index of Fig. 2.1 is equal to the ratio of lengths a / b. 2.3 Factors Affecting Fiber Orientation Paper is made in a continuous process. The suspension of fibers and fillers is discharged from the slice of a headbox and distributed at high speed onto the forming wire. On the wire a sheet is formed through de-watering. The sheet thus formed is wet and weak and needs to be further processed in presses and dryers. The primary mechanism of orienting fibers in the sheet is the hydrodynamic shear flows in the early forming section or wet end operations of the paper machine, i.e., headbox discharge and formation process. The headbox design can have an effect on the orientation of fibers leaving the slice. The elongation and shear leading to the slice tend to orient the fibers in the machine direction. Many researchers [4, 9, 10] have analyzed the factors that affect fiber orientation and have agreed that the primary mechanism in the sheet is the hydrodynamic process in the headbox discharge and formation operations of the paper machine. Wrist [11] studied the fiber orientation in the jet and on the forming wire and concluded that the relative spatial arrangement of the fibers in a machine-made sheet of paper is very largely determined between the headbox and the end of the forming table. Within this space, the orientation of the fibers, the degree of flocculation, the relative distribution of materials through the thickness of the sheet and the macro- and micro-mass distribution in the plane of the sheet are all laid down. Subsequent processes, like pressing and drying, have minor effects on fiber orientation with only some micro-rearrangement and consolidation of the web. N e x t w e will summarize the major factors in the wet end processes that lead to n o n- 5

15 uniformity of fiber orientation. Because this work is focused on the headbox, we will start first with the headbox effect on the fiber orientation, then consider relative wire speed, wire types and fiber consistency Headbox A fundamental function of any headbox is to ensure the machine and cross machine directional uniformity. Two areas that are facing increasingly stringent quality demands are uniformity of basis weight profiles on finer scales and the controllability of fiber orientation profiles. A headbox can be divided into three sections by the principal flow patterns involved [12]: fluid distribution, flow rectification and jet development. In the first section, a tapered header is used to achieve ideally uniform flow into the distributor. Then the flow from the distributor is improved through the rectification processes. In a hydraulic headbox, a tube bank is often used in these processes. The wall friction in the tubes dampens flow disturbances originating in the stock approach system and creates turbulence which is needed to prevent fiber flocculation in the paper-machine forming zone. The processes in the tube bank may include mixing and blending of separate flows from a distributor, eliminating undesirable cross-flow and eddies, improving the velocity profile and developing turbulence of desired scale and intensity [13]. The jet development process can be described as delivering the stock to the sheet forming section. An ideal headbox should produce a uniform and stable jet over the width of the machine, without lateral velocities and machine-direction perturbations. In brief, the headbox spreads the flow of pulp out of the stock approach piping along the width of the paper machine, provides turbulence "blending" and delivers the furnish to the machine forming section. Kyosti et al. [14] pointed out that in a headbox, fiber orientation can be influenced by the recirculation rate, header pressure distribution, flow distribution units and headbox tube patterns. Many researchers [4, 15, 16, 17] have agreed that the adjustment of the slice lip 6

16 profile not only dominates the basis weight profile of paper in the CD direction, but also significantly affects the fiber orientation distribution profile. In a conventional headbox, the slice lip shape is governed by the basis weight profile controller, which keeps the basis weight at the reel as flat as possible. However, this demand for a uniform basis weight competes with the demand for uniformity of fiber orientation profiles, because a change in the shape of the slice lip may result in significant cross flow, which leads to a variation in fiber orientation in the cross machine direction. For a conventional headbox, it is impossible to adjust basis weight and cross-machine fiber orientation profiles independently of each other. To solve the problem, a revolutionary headbox design, which is called the consistency profiled headbox or dilution control headbox, has been introduced [9, 18, 19, 20, 21, 22, 23, 24]. This headbox enables independent control of CD basis weight and fiber orientation profiles. The basis weight profile is controlled by varying the stock consistency profile in the headbox and the slice lip is then used in the control of fiber orientation. Nordstrom and Norman [1,7] found that a high headbox nozzle contraction ratio, which is the ratio between the inlet area and the outlet area, can not only generate a high degree of anisotropy of fiber orientation, but can also improve the formation. They attributed this effect to the enhanced strength of the elongational strain field in the nozzle and the changes in turbulence intensity. The amount of eddy deformation is dependent on the degree of contraction. Ullmar and Norman [25] indicated that the contraction ratio of the jet developing section plays an important role in fiber orientation at the nozzle exit. Their results indicated that the effect of contraction ratio is more significant on the fiber orientation than that of the flow velocity. The fibers have been found to be more strongly oriented in the machine direction for higher contraction ratio. Bandhakavi and Aidun [26] reported that the accelerating flow in the converging section tends to orient the fibers in the machine direction, and stretch and rupture the floes. The turbulence in the flow may decrease fiber orientation but may also improve the suspension dispersions. 7

17 Lee and Pantaleo [17] indicated that besides the headbox, the forming process also contributes to the resultant fiber orientation depending on the type of former, and operating conditions such as jet to wire speed difference, wire tension and drainage rate. Several of these effects are summarized in the following sections Jet to Wire Speed Difference The most significant factor determining the fiber orientation is usually the speed difference between the jet and the forming wire. Ideally, the jet is assumed to be in the machine direction, but in practice, there exist small transverse flows. The magnitude of the cross flows varies from layer to layer within the jet, and also varies across the width of the jet. The difference between the jet speed and wire speed is usually small. But even a small cross flow may cause a significant change in fiber orientation angle when the suspension is delivered onto the forming wire. This is the reason why, in the industry, fiber orientation is primarily controlled by changing the jet to wire speed difference. As the difference between the M D component of jet velocity and the wire speed is increased, the average fiber orientation angle is reduced, and the fiber orientation index is increased [4] Forming Wire Because the jet discharged from the slice may have a non-uniform velocity profile due to boundary effects and wake effects, and also has an impingement angle when the stock is spread on the wire, it is impossible to eliminate the difference in the velocity between the jet and the wire. The velocity difference may cause shear forces in the region of the stock-wire interface, and produce further variations in the fiber orientation. Erikkila et al. [27] pointed out that the fiber orientation for each individual layer of the sheet is finally settled down in the drainage process and is affected by the shear, de-watering velocity of the suspension, consistency and the turbulence during the process. The difference in the 8

18 manner of de-watering, in one direction such as in the Fourdrinier case or in two directions such as in gap forming, produces different orientation two-sidedness in the fiber orientation. In addition to the hydrodynamic effect on the fiber orientation, the turbulence effect should also be considered. Turbulence is generated in the headbox and maintained during drainage by the drainage elements. In addition, turbulence is induced by the speed difference between the suspension and the wire. As the turbulent energy is increased, the average fiber orientation angle is not changed, but the in-plane fiber orientation anisotropy is decreased, and the fiber orientation index is reduced [1] Fiber Suspension Consistency The orientation produced at the slice is found to be consistency sensitive, and to be a function of the fiber network strength. Fiber-fiber interactions determine how many individual fibers can be rotated with the oriented shear field. At higher concentrations, fibers are less aligned in the flow direction, presumably as a result of flocculation. Kerekes and Schell [28] defined a crowding factor N, which is based on the volume concentration C v, the fiber length L and diameter d, to represent the degree of flocculation: N = -Cv 3 (2.1) As the consistency increases, the crowding factor will increase. Ullmar [29] did experiments which showed that the fiber alignment decreases as the crowding factor increases. Curly fibers were also found to be less aligned than straight fibers [29, 30]. According to their concentrations, fiber suspensions are usually classified into three regimes: dilute, semi-concentrated and highly concentrated. If fibers are considered to be 9

19 rigid cylinders with length L and diameter d, and occupy a fraction C v of the total volume of the suspension, Dinh [31] shows that the dilute regime is defined when C v < (d/l) 2, the semi-concentrated regime is defined as (d/l) 2 < C v < (d/l), and the highly concentrated regime is defined as C v > (d/l). In the dilute regime, the distance between a fiber and its nearest neighbor is greater than L, so the fibers are free to rotate, and interactions between fibers are rare. In the semi-concentrated regime, the spacing between fibers is less than L but greater than d, and interaction between fibers are frequent. When the suspension falls into the highly concentrated regime, the spacing between fibers is on the order of fiber diameter d. Three regimes can also be defined in terms of fiber volume fraction C v and fiber aspect ratio A r, which equals L/d [32]: The dilute regime is when: C A, 2 «1 (2.2) the semi-concentrated regime is given by: A; 2 < Cv < V (2.3) and the concentrated regime is defined as: CvAr» 1 (2.4) In headboxes of conventional paper machines, the fiber weight consistencies vary between 0.1 and 1.5%, fiber lengths vary between 1 and 5 mm and aspect ratios vary between 30 and 200 [33]. For example, if the volume concentration of the fiber suspension C v is 1%, fibers have a uniform length of 3 mm and a uniform diameter of 40 um, then the aspect ratio A r is 75. The suspension is then in the semi-concentrated regime, because A r" 2 < C v < A r _ 1 (that is 0.018% < 1% < 1.3%). There would then exist 10

20 frequent fiber-fiber interactions in the headbox flow. The dilute suspension assumption in the current study is therefore a simplification of the actual problem. 2.4 Headbox Flow Simulations to Investigate Fiber Orientation Computer simulation has been widely used in the study of processes that occur in engineering equipment. The simulation investigations not only meet the need for understanding and prediction, but also have large economic benefits. Several researchers have conducted headbox flow simulations in order to investigate the flow induced fiber orientation. Aidun [34, 35] studied the secondary flows in the headbox and their effects on nonuniform fiber orientation and mass formation by using a non-linear k-s turbulence model to investigate the characteristics of turbulent flow in a low consistency headbox. The author indicated that the cause of non-uniformity in fiber orientation in the cross machine direction is the secondary flows that are generated inside the headbox induced either by the geometric effects and the kinematics, or by the anisotropy of turbulent flows. Lee and Pantaleo [17] used a standard k-s turbulence model to analyze headbox flow when different flow control devices were employed, such as slice profiling, edge valve control, bleed controls, tube inserts and header re-circulation valves. They examined the relationship between the headbox flow characteristics and the fiber orientation, and correlated the headbox flow characteristics in terms of flow angle obtained from CFD solutions with the measured fiber orientation. They tried to use the flow angle [3, which is defined by the M D velocity, u, and CD velocity, v, to represent the average fiber orientation angle: P = tan" ( -1 (2.5) 11

21 Shimizu and Wada [36] applied the k- s turbulence model and a finite difference method to study the influence which elements of an imaginary headbox, such as a tapered header, side wall, contracting part and slice lip, have on paper quality, especially the uneven basis weight profiles and fiber orientation. The researchers mentioned above have the common problem that they tried to study fiber orientation in the flow without a specific simulation of fiber behavior. 2.5 Fiber Suspension Simulation The first fundamental study of the orientation of a rigid ellipsoidal particle in a dilute viscous Newtonian liquid was conducted by Jeffery [37]. He solved the flow field around a rotating ellipsoid by solving Stokes equations, using a no-slip boundary condition at the surface of the particle. The angular velocity vector of the particle was then found from the requirement that the total torque acting on the particle be zero. The following Jeffrey's equation describes a simplified case of a fiber lying in a two-dimensional flow field [38].. du _. 2 du 2 dv dv_ -smtpcostp sin tp + cos (p + sin^cos^ dx dy dx dy,., du i, du. i,dv.,, 5v ( 1 ^ -sin^cos^ + cos <p dx dy dx sin <p -fsin^cost? (2.6) The angle ^, which is the angle between fiber axis and x-axis, describes the orientation of the fiber. When the aspect ratio is greater than unity, the orientation of the fiber changes mainly in response to deformation or rotation of the fluid. Besides rotation, fibers also translate with the velocity that the unperturbed fluid would have at the centroid of the fiber. Jeffery's theory has been verified in the experimental work by Mason and Bartok [39]. 12

22 Since Jeffery's work, several constitutive models have been developed from which flow induced orientation can be predicted for the dilute or semi-concentrated or concentrated suspensions. Rao et al. [40] indicated that in a complex flow, if the spatial stress gradients due to fibers are very small compared to spatial viscous stress gradients, then the fluid behavior is Newtonian, i.e. the presence of fibers does not alter the flow kinematics. Consequently, the implementation of an anisotropic model is not needed and the sole use of Jeffrey's equation is sufficient to characterize the orientation field. On the other hand, if stress gradient contributions from the particles are comparable or larger than the suspending fluid contributions, the suspension exhibits non-newtonian characteristics with directionally dependent properties. This necessitates the simultaneous solution of the flow and orientation fields by using a proper anisotropic constitutive model, or by accounting for particle/particle interaction in some other way. The behavior of an individual fiber in a dilute suspension is only a function of its orientation and of the flow field, since the fiber's orientation will not be affected by other fibers. Givler et al. [10] have developed a numerical scheme to solve for the fiber orientation in dilute suspensions and in confined geometries by integrating Jeffery's equation along the streamlines. The velocity field used in order to determine the streamlines was obtained by assuming that the fibers do not disturb the flow. It was shown in their work that shear flows always induce a periodic rotation of the particles, and particles with large aspect ratios spend most of the time in a period aligned with the streamlines of the flow, although they are subject to a cyclic tumble. For expansion flow, the fibers will assume a transverse orientation with respect to the streamlines of the flow. Conversely, flow in a convergent geometry will orient fibers closer to the fluid streamline direction. Stable equilibrium orientation exists for elongational flow and not for shear flow. In an elongation flow, it is well known that stretching flows align fibers in the direction of stretching. A fiber oriented in the principal stretching direction, orientation angle ^ = 0, is in stable equilibrium, and a fiber at = ± n/2 is in unstable equilibrium. All other fibers rotate toward the stable equilibrium position with ^ changing monotonically. The eventual orientation distribution is perfectly aligned in the stretching direction. 13

23 Akbar and Altan [41] use a combination of analytical solutions and statistical methods to study fiber orientation behavior in arbitrary two-dimensional homogeneous flows. They use an orientation distribution function, which is generated statistically by considering the frequency distribution curve of the orientation of a large number of fibers, and they found that the accuracy of the orientation distribution function is dependent on the number of fibers used in the analytical solution. As the suspension concentration increases towards the semi-concentrated regime, the behavior of fibers changes because of interactions between fibers. The interactions cause changes in the angles of both interacting fibers. In a concentrated suspension each fiber interacts with many other fibers simultaneously, so a mechanistic model would be very difficult to create. There are some studies directly relevant to the development of a mathematical model to predict the orientation distribution of rigid fibers in semiconcentrated and concentrated suspensions. Rao [40] provides an approach for the simultaneous solution of the flow and orientation fields. In his research, the orientation of the fibers is first computed by assuming that the stresses generated due to the presence of fibers are zero. Then, the orientation field computed from the Newtonian solution is coupled back to the governing equations of flow to solve the anisotropic flow of fiber suspensions. Folgar and Tucker [38] developed a model for concentrated fiber suspensions, where fiber-fiber interactions are taken into account by adding a diffusion term to the Jeffery's equations. They used a statistical approach and introduced an orientation function to describe the fibers' orientational state. Advani et al. [42] proposed a more efficient approach for numerical simulation of fiber orientation which uses a set of orientation tensors. Altan et al. [41, 43, 44] investigated the two- and three-dimensional description of fiber orientation in semi-concentrated homogeneous flow fields by using Dinh- Armstrong rheological model [45] to describe fiber motion in non-dilute solutions. Folgar and Tucker [38] pointed out that all of these researchers who studied semiconcentrated or concentrated suspensions have observed fiber orientation behavior which 14

24 is qualitatively similar to the dilute suspension models. While most of the studies have focused on suspensions of rigid fibers, flexible fibers were also investigated by Ross and Klingenberg [46], Wherrett et al. [47], and Dong et al. [48]. The present fiber model uses the method of Ross and Klingenberg [46] and the numerical scheme of Dong [48], both of which are based on Jeffery's original assumptions [37]. 2.6 The Scope of This Thesis Work The current research is part of an effort at the University of British Columbia to develop computational methods to simulate the motion of fibers in a way that can be directly applicable and beneficial to the pulp and paper industry. This thesis is limited to the study of dilute fiber suspensions in a headbox converging section. Two models have been developed in the UBC research group: a turbulent flow model is used to calculate the headbox flow field, and a fiber model is used to simulate fiber motion. These two models have been combined together and applied to the study of fiber orientation in a headbox in this thesis research. By applying the simulation method, reasons for the fiber orientation can be identified and the headbox or flow conditions needed to enhance paper quality and increase production efficiency can be recommended. A scaled plexiglass headbox, built at the University of British Columbia, is used in the present experiments [49]. Photos of dyed nylon fibers in the flow are taken at several locations along the central streamline of the channel from side and bottom direction. An image analysis method is applied to measure the fiber orientation angles. Direct comparison between the statistical results of experiments and predictions is performed. The numerical simulation method is further used to predict the fiber orientation for different flow rates, headbox geometries and fiber aspect ratios. 15

25 Figure 2.1. Fiber orientation distribution pattern in a piece of paper. 16

26 3. EXPERIMENTAL ARRANGEMENTS 3.1 Objectives of the Experimental Work The objective of the experimental work is to obtain data to validate the simulation model by comparison between the experimental results and the simulation results. 3.2 Fiber Suspensions The fibers are made of nylon and have a nominal length of 3 mm and coarseness of 15 denier (1 denier = 1 g/9000m). A simple calculation gives the width of the fiber as 44^m (the density of nylon fibers is 1,140 kg/m 3 ). As a result, the fiber aspect ratio, which is the ratio between fiber length and fiber width, is 68. Nylon fibers were chosen for the experiment because they can be colored, have a density close to water, can be cut to specific lengths and can be considered as rigid rods. The fibers were dyed with Rit blue marine 30 by soaking them overnight. In the experiment, water was used as the working fluid. Between 2000 and 3000 fibers were placed in each liter of water for tracking single fibers by means of photos. As the consistency is no more than 0.001%, the suspension was well within the dilute regime, which means there was little interaction between fibers. The lengths of dry nylon fibers were tested with the image analysis system. In a sample of 200 fibers, 95% of the fibers lay in the range of 2.4 mm to 3.2 mm (the mean fiber length was 2.8 mm). The distribution of the fiber length is shown in Fig Most of the fibers were straight or nearly straight when the fibers were in the dry condition or in the suspension as shown in Fig

27 3.3 Flow Loop The experimental set-up [49] used a closed flow system diagrammatically shown in Fig Experiments were conducted in a transparent plexiglass headbox, shown in Fig. 3.4, to allow for visual inspection of the flow. This headbox is a scaled model of a typical headbox with the size reduced by a factor of 5. In the flow loop, the dilute fiber suspension is pumped from the reservoir tank, which can contain a total volume of 3 m 3 of fluid, to the headbox through the pipes and rectifier tubes. The rectifier tubes are round at the inlet and rectangular at the outlet with slowly increasing cross sectional areas and are typically used both to provide the turbulence energy needed for fiber dispersion and to generate a fairly uniform velocity profile at the converging section inlet. At the outlet of these tubes, it is assumed (and verified by observation) that the fibers are oriented randomly because of the turbulence effects on the fibers. Altogether there are 40 rectifier tubes, two rows in the headbox height direction and 20 in the span direction. The flow through each tube is metered and adjusted individually, so that the flow rate at each tube exit is 0.34 liter per second. As a result, the average velocity at the inlet of the converging section is 0.24 m/s. After travelling through the asymmetric converging section, the flow is finally discharged at the nozzle or "slice". The converging section starts with a rectangular channel which remains constant in cross sectional area until the channel length reaches m (the origin is at the entrance). Downstream of this point, the channel converges to the nozzle with a contraction ratio of 10. Details of the headbox geometry are given in Table 3.1 and Fig. 3.5 presents the cross sectional view. Table 3.1. The Geometry of the Headbox Converging Section. parameters values width inlet height slice height contraction ratio length 0.76 m (constant) m m m 18

28 It is important to avoid entrapment of air bubbles in the suspension when taking photographs, because the bubbles may deteriorate the quality of pictures. Air bubbles are avoided by circulating the suspension flow through the headbox for at least 2 hours before taking pictures. 3.4 Image Analysis System An image analysis method was used in this study of fiber orientation. To detect the fibers in the flow, an Optikon MotionScope CCD (charge-coupled device) video system with Cosmicar/Pentax T V lens (3.7mm, 1:1.6) and a Sony DCR-TRV320 digital video camera were mounted and connected together. The Optikon system captures up to 500 frames per second with a resolution of 336 x 243 pixels for each picture. The Sony camera has a shutter speed of 1/4 to 1/4000 second. While the Optikon camera was used to capture pictures of fibers in the high-speed flow zone, i.e. very close to the exit, the Sony camera was used where the flow speed was not too high (channel length < 26 cm). In the experiment, the cameras were mounted at several locations either below or beside the headbox (as shown in Fig. 3.6) in order to obtain views from beneath or from the side of the headbox. Fig. 3.6 also defines the three dimensional coordinates: the x-axis is in the machine direction, the y-axis represents the paper thickness direction, and the z-axis represents the headbox span direction (cross-machine direction). Lighting is extremely important for obtaining good pictures. Back lighting was provided with a 150 w Sylvania floodlight bulb. A sheet of fine ground glass, 4 mm in thickness, was used to scatter the light for better photograph qualities. In the experiment, video pictures were taken of fibers in motion in the plexiglass channel. To establish the field of focus, a tube was temporarily inserted into the channel. Cameras were focused at the center of the channel while taking pictures from the bottom and in a plane 6 cm next to the side channel wall when taking pictures from side views. It was difficult to control the camera to ensure a very shallow depth of field, so in each picture 19

29 the fibers may have been located at a different distance from the camera, from the boundary near the wall to the far inside of the flow, about 20 cm away from the wall. The Sony PictureGear 4.1 Lite software was used to download the video pictures from the recorder to the PC. Matrox Inspector software was then used to evaluate fiber orientations from the recorded images. Matrox Inspector has the power to automatically recognize a target and measure the required parameters, such as length, width, angle, area, etc. When this software is used to deal with a picture of fibers, automatic measurement becomes difficult. The prime problem is contrast. Fibers have a large length-to-width ratio and the width of a fiber is only a very small fraction of the field of view, so that the contrast of the fiber in the picture is very poor. There were also some fine scratches on the plexiglass plates, and the channel width was too large to have clear fiber pictures from the side views. The resolution of the cameras also needs to be improved. All of these made it difficult to distinguish fibers from their background automatically. If the automatic function of the software is used, a single fiber is often viewed as several segments and treated as several separate fibers by the computer, or some fibers are not recognized at all. Therefore, the measurements were conducted with the software but the recognition of the fibers was completed manually to ensure the quality of measurement. The quantitative results were obtained by further processing the measured data. 3.5 Measurement In order to collect enough data for statistical analysis, hundreds of pictures were taken at each experimental point. There are between 5 and 30 fibers in each picture. The resulting sample size of 1600 to 2400 fibers at each measurement point represents a compromise, from a statistical viewpoint, between accuracy and effort. The orientation of fibers was evaluated by measuring the angle between the line connecting the two ends of a fiber and the machine direction (x-axis). This definition is reasonable when almost all the fibers are straight or nearly straight. Fiber orientation angles vary between -90 and +90 with 0 corresponding to the paper machine direction. The determination of the sign of an 20

30 orientation angle depends upon the location of the fiber-end in the Cartesian coordinate system with the origin located at the mid-point of the fiber, as shown in Table 3.2. During the measurement, blurry fibers, fibers located outside the border of interest, and highly curved fibers were ignored. A typical picture of fibers in the flow is shown in Figure 3.7. Table 3.2. The Sign of the Orientation Angles view or plane the quadrant sign of angles side view or 1 - on x-y plane IV + bottom view or 1 + on x-z plane IV - In order to compare the experimental results with the simulation results, for the side view case, these studies were restricted to measurements close to the central streamline of the converging section, because fiber orientations on the central streamline were simulated in the computational study. It is important not to bring the effect of the tubes upstream of the headbox on fiber orientation into the measurement. This can not be avoided entirely for the side view case, because the central streamline is located in the wake of a tube wall. One can assume that the wake effect on fiber orientation is quickly lost as the fibers enter the converging channel and are subject to strong stretching of the flow [50]. While taking pictures from the bottom view, the measurements are made along both the centerline of one rectifier tube and the line extended downstream from a tube wall. The results from these two categories of measurements are then mixed together to produce the final averaged results. The measurements were taken at several points along the headbox channel as shown in Figure 3.8. The first point is selected very close to the inlet of the channel where x = 4.5 cm. This point is still within the flat section so that the fiber orientation situation should not change as the fiber moves downstream in the constant area section. The second point, x = 12.2 cm, is located near the beginning of the converging section. The following points are further downstream in the converging section where x = 15.7 cm, 19.2 cm, 22.7 cm, 26.2 cm and 31cm. The point x = 31 cm is very close to the exit of the channel 21

31 and therefore only bottom view measurements were conducted. Clear pictures from the side could not be obtained at this point. During measurements, only the fibers in the specific area are counted, i.e. the fibers in a square area of 2 x 2 cm 2 when the measurement is close to the inlet (x < 20 cm) and lxl cm 2 when the measurement is close to the exit. The number of fibers counted at each measurement point is listed in Table 3.3. Table 3.3. The Number of Fibers at Each Measurement Point x-positions number of fibers in x-y plane in x-z plane N/A

32 fiber length (mm) Figure 3.1. The length distribution of nylon fibers. Figure 3.2. Images of fibers: (a) dry dyed nylon fibers, (b) fiber suspension. 23

33 Y A Fluid Tank Figure 3.3. The flow loop in the experiment.

34 Figure 3.5. Cross sectional view of the scaled headbox (dimensions in cm). Headbox z 4 "I *x Headbox [k Lighting Lighting Camera Camera (a) (b) Figure 3.6. The photographic arrangement for (a) side view and (b) bottom view. 25

35 (a) (b) F i g u r e 3.7. T y p i c a l p i c t u r e o f f i b e r s i n t h e f l o w : (a) b e f o r e a n a l y s i s ; ( b ) a f t e r a n a l y s i s. ( X d i r e c t i o n is the m a c h i n e d i r e c t i o n. ) 26

36 central streamline 4> focus point ,, i,,,, i, 1 1,,,, 1, X Figure 3.8. The measurement points along the headbox channel. 27

37 4. COMPUTER SIMULATION OF FLOW AND FIBER ORIENTATION The fiber orientation in a piece of paper is determined by the papermaking process, especially in the headbox and on the forming wire. Efforts must be made to derive quantitative relationships between processing conditions and fiber orientations. The intent is to learn how to design and control manufacturing processes to generate favorable orientation states, so as to obtain the best possible paper products. To perform the prediction of fiber orientation that is required for the design and process control, must have an accurate quantitative model of the way fibers change orientation as one they move in the flow. To simulate fiber motion, two models need to be developed and effectively combined together. The first model is used to describe the fluid motion in a 3-dimensional domain, as constrained by the specific boundary conditions. The second model describes fiber motion and orientation in the flow field. The important part of the method is to combine these two models. The simulation method is used here for the solution of fiber orientation in a flow field of a Newtonian fluid where the fibers do not alter the flow. For suspensions with higher volume fractions, these solution techniques can be utilized with some modifications. Currently, the above two models are uncoupled in that the fiber orientation state will not alter the governing equations for the flow. Fiber orientations are calculated subsequent to the velocity field determination, hence, the two models may be solved consecutively. The detailed approach can be described as follows. Firstly, the flow field is predicted by the solution of the Reynolds averaged Navier-Stokes equations. Then the translation and rotation of a rigid fiber is described based on Newton's Second Law and the law of angular momentum. The angular velocity of a fiber depends upon the local flow conditions, such as vorticity and the components of the rate of deformation tensor. 28