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Fiber orientation in a headbox Zhang, Xun 2001

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FIBER ORIENTATION IN A HEADBOX 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 FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 2001 ©Xun Zhang, 2001 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 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 TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iiLIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS vii1. INTRODUCTION 1 2. LITERATURE REVIEW 3 2.1 Fiber Orientation and Paper Quality 3 2.2 The Definition of Fiber Orientation 4 2.3 Factors Affecting Fiber Orientation 5 2.3.1 Headbox 6 2.3.2 Jet to Wire Speed Difference 8 2.3.3 Forming Wire 8 2.3.4 Fiber Suspension Consistency 9 2.4 Headbox Flow Simulations to Investigate Fiber Orientation 11 2.5 Fiber Suspension Simulation 12 2.6 The Scope of This Thesis Work 5 3. EXPERIMENTAL ARRANGEMENTS 17 3.1 Objectives of the Experimental Work3.2 Fiber Suspensions 17 3.3 Flow Loop 8 3.4 Image Analysis System 19 iii 3.5 Measurement 20 4. COMPUTER SIMULATION OF FLOW AND FIBER ORIENTATION 28 4.1 The Headbox Flow Model 29 4.2 Fiber Model 32 5. RESULTS AND DISCUSSION 36 5.1 Analysis of the Headbox Flow Field 35.2 Comparison of Simulation and Experimental Results 39 5.3 Factors Affecting Fiber Orientation 44 5.3.1 The Effect of Contraction Ratio on Fiber Orientation 44 5.3.2 The Effect of Flow Rate on Fiber Orientation 45 5.3.3 The Effect of Channel Length on Fiber Orientation 46 5.3.4 The Effect of Fiber Aspect Ratio on Fiber Orientation 47 5.3.5 The Effect of Flow Elongation 45.4 Symmetric Channel 48 5.5 Statistical Error Estimation 9 6. SUMMARY AND CONCLUSIONS 60 7. RECOMMENDATIONS FOR FUTURE WORK 62 8. NOMENCLATURE 64 9. REFERENCES 6 iv 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, Lc = 0.225 m) 45 Table 5.3. The Orientation Parameters for Different Uo (Rc= 10, Lc = 0.225 m) 46 Table 5.4. Fiber Orientation Parameters for Different Lc (Rc= 10, Uo-0.24 m/s) 46 Table 5.5. The Orientation Parameters for Different Ar ....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 U0: 48 v 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 fibersFigure 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, 5Figure 5.8. The fiber orientation distribution at x = 12.2 cm, 5 Figure 5.9. The fiber orientation distribution at x = 15.7 cm, 5Figure 5.10. The fiber orientation distribution at x = 19.2 cm, 6 Figure 5.11. The fiber orientation distribution at x = 22.7 cm, 5Figure 5.12. The fiber orientation distribution at x = 26.2 cm, 7 Figure 5.13. The fiber orientation distribution at the channel exit, 57 Figure 5.14. The orientation parameters along the central streamline, 58 Figure 5.15. Fiber orientation distributions at the channel exit for various Contraction ratios, (a) in x-y plane, (b) in x-z plane 58 vi Figure 5.16. Cross sectional view of the symmetric headbox (dimensions in mm) 59 Figure 5.17. The physical mesh of the symmetric headbox 5ACKNOWLEDGEMENTS 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 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 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 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 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. 2.1. 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 oriented. The fiber orientation index is the ratio of the fibers oriented in the MD over those oriented in the cross machine direction (CD), which is often defined as the ratio of MD 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. Next we will summarize the major factors in the wet end processes that lead to non-5 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. 2.3.1 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 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 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. 2.3.2 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 MD 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]. 2.3.3 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 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]. 2.3.4 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 Cv, the fiber length L and diameter d, to represent the degree of flocculation: 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 N = -Cv 3 (2.1) 9 rigid cylinders with length L and diameter d, and occupy a fraction Cv of the total volume of the suspension, Dinh [31] shows that the dilute regime is defined when Cv < (d/L)2, the semi-concentrated regime is defined as (d/L)2 < Cv < (d/L), and the highly concentrated regime is defined as Cv > (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 Cv and fiber aspect ratio Ar, 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 Cv is 1%, fibers have a uniform length of 3 mm and a uniform diameter of 40 um, then the aspect ratio Ar is 75. The suspension is then in the semi-concentrated regime, because Ar"2 < Cv < Ar_1 (that is 0.018% < 1% < 1.3%). There would then exist 10 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 non uniform 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 MD velocity, u, and CD velocity, v, to represent the average fiber orientation angle: P = tan" ( -1 (2.5) 11 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]. . _ . 2 du 2 dv -smtpcostp sin tp— + cos (p— + sin^cos^ du dx dy dx dv_ dy, ( 1 ^ (2.6) . , i , du . i ,dv ., , -sin^cos^— + cos <p sin <p — -fsin^cost? du dx dy dx 5v 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 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 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 semi-concentrated 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 semi-concentrated or concentrated suspensions have observed fiber orientation behavior which 14 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 Figure 2.1. Fiber orientation distribution pattern in a piece of paper. 16 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/m3). 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. 3.1. 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. 3.2. 17 3.3 Flow Loop The experimental set-up [49] used a closed flow system diagrammatically shown in Fig. 3.3. 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 m3 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 0.0922 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) 0.075 m 0.0075 m 10 0.3172 m 18 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 TV 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 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 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 on x-y plane 1 -IV + bottom view or on x-z plane 1 + 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 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 cm2 when the measurement is close to the inlet (x < 20 cm) and lxl cm2 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 1 1325 1638 2 2031 1793 3 2066 2885 4 2056 2325 5 1554 2333 6 1823 1356 7 N/A 1586 22 1.4-1.6 1.6-1.8 1.8-2 2-2.2 2.2-2.4 2.4-2.6 2.6-2.8 2.8-3 3-3.2 3.2-3.4 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 Y A Fluid Tank Figure 3.3. The flow loop in the experiment. Figure 3.5. Cross sectional view of the scaled headbox (dimensions in cm). z 4 Headbox "I *x Headbox [k Lighting Lighting (a) Camera Camera (b) Figure 3.6. The photographic arrangement for (a) side view and (b) bottom view. 25 (a) (b) Figure 3.7. Typical picture of fibers in the flow: (a) before analysis; (b) after analysis. (X direction is the machine direction.) 26 0.1 - — central streamline 0.08 0.06 4> focus point 0.04 0.02 - --0 -0.02 , , i , , , , i , 1 1 , , , , 1 , 0 0.05 0.1 0.15 0.2 0.25 0.3 X Figure 3.8. The measurement points along the headbox channel. 27 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, one must have an accurate quantitative model of the way fibers change orientation as 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 4.1 The Headbox Flow Model In the simulation study, the liquid is assumed to be pure water or a dilute fiber suspension. The consistency of the suspension is very low, so there is no interaction between fibers, and the fibers do not affect the flow field. The fibers are uniform and long enough so that the Brownian motion can be ignored [51]. The suspension can therefore be viewed as a uniform incompressible Newtonian fluid. Based on the above assumptions, the available continuum theories can be used for the dilute suspension [32]. The numerical simulation of the flow has been carried out by solving the three-dimensional incompressible Reynolds averaged Navier-Stokes equations. Turbulence closure is obtained by the use of the standard k-s model with the wall function treatment. We can write the continuity equation in the form of: where: u = instantaneous fluid velocity vector p = modified pressure including the gravitational forces p = density x = fluid stress tensor. For a Newtonian fluid with constant viscosity, V u = 0 (4.1) The equation of conservation of momentum is: u-Vu = -Vpl p + V -r I p (4.2) V-T = //V2u (4.3) where: 29 ^ = dynamic viscosity of the fluid. In turbulent flow, the velocity and pressure can be expressed as a mean and a fluctuating part: u = u + u' (4-4) p = p + p' (4-5) In the equations, the over-bar denotes the mean, and the prime indicates the fluctuating component. After substituting Equations (4.3), (4.4) and (4.5) into Equations (4.1) and (4.2), and taking a time average, the governing equations of the turbulent flow become: V-u = 0 (4-6) uVu = -Vp/p + V-r/p (4-7) Now the stress tensor x includes both the viscous and turbulent Reynolds stress tensors: VT = /J V2u + V-(-p u\u'j ) (4-8) where: r, = uTV; (i,j = l,2,3) (4.9) can be used to simplify Equation (4.8). The Reynolds stress tensor T.. introduces additional unknowns for the turbulent flow. In order to describe the mean velocity and pressure fields, a closure formulation is necessary to relate the components of the Reynolds stress tensor to the mean flow velocity or velocity gradients. The standard linear k-s model [52] is employed to solve the closure problem. 30 The Reynolds stress tensor can then be expressed as: T,J = vtSu--kSv where: du, dui + dx, dx, (4.10) (4.11) k = — u'u' 2 ' ' (4.12) k is the turbulent kinetic energy, 5.. is the Kronecker Delta, and Vt is the turbulent kinematic viscosity which, unlike its laminar counterpart, varies spatially and is not a property of the fluid. The transport equation for k is: where: and 5 lr- 9 ku, = dx dk + G-E G = -u] Uj Sa (4.13) (4.14) E = VS'iS'g (4.15) G and E are the rates of kinetic energy production and dissipation per unit mass, respectively. The transport equation for the kinetic energy dissipation E is given in the form of: 5 zr- 8 EU: = dx, 1 dx v + -^ + (ClG-c2E)^ (4-16) dx, K 31 A final correlation, based on an isotropic viscosity assumption for the turbulent viscosity in terms of k and E, as given by V{ = k2/E, closes the system of equations. The usual values of the constants are: = 0.09, Cj = 1.44, C2 = 1.92, ak = 1.0, aE = K2/[(C2-C,) Cm], where K = 0.41 is the Von Karman constant. A finite volume method in conjunction with general curvilinear grids is used in the computational code, which was developed in our research group by Nowak [53]. Use and validation of this code for headbox flows has been reported in the work of Shariati et al. [49] andHuaetal. [50]. 4.2 Fiber Model The main concern of this work is the prediction of fiber orientation in the headbox flow field. In this work, the fiber orientation is calculated for a given velocity field, decoupling the flow and fiber orientation calculations. Theoretically, the solutions are only valid for zero fiber consistency, because the fibers influence the flow. However, for dilute solutions, this model should not give significant errors. At fiber locations, the velocity and velocity gradients are calculated by interpolating the values at neighboring nodes and are used to determine the translation and rotation of the fiber. The calculated fiber position and orientation are used as initial conditions for the calculation of the fiber's new location and orientation at the next time step (t + At). The fiber motion model used in this work was first developed by Ross and Klingenberg [46, 54] and adapted by Dong [48]. Jeffrey's equation [37] is used for the verification of this model. Dong showed the identity of the results from both approaches. Although flexible fibers can be simulated with this model, only rigid fibers are simulated in this thesis. As described by Dong, a fiber is represented by one or more prolate spheroids. In this thesis research, a rigid fiber is represented by one prolate spheroid with uniform length L and diameter d. The density of the fiber is the same as the fluid. The motion of a 32 fiber is determined by solving the translation and rotation equations which are based on Jeffery's original work [37]. The orientation of a fiber is three-dimensional and can be determined by two angles as expressed in Figure 4.1. The azimuthal angle <j, is the angle between the projection of the fiber axis on the x-y plane and the y-axis with 0 < <j, < n (one end of the fiber is not distinguishable from the other). The polar angle 0 is the angle between the fiber axis and the z-axis with 0 < Q < N. The motion of the fiber relative to the surrounding fluid is so small that the inertial force is negligible. The center of the fiber translates with the local fluid velocity, and the fiber orientation is governed by the components of the rate of deformation tensor, the vorticity of the flow and the previous fiber orientation. In order to predict a fiber motion, the velocity, strain tensor, and vorticity vector of the flow at the center of a fiber must be known. The approach developed is firstly to determine in which cell the fiber lies. Then accurate values of the velocity, strain and vorticity at the fiber center are interpolated from the surrounding nodal points. In each time step, the calculation uses the fiber position and orientation obtained from the preceding time step and the local flow kinematics information. Fiber particles suspended in a turbulent flow experience a mean velocity associated with the mean flow field and a random velocity due to the fluctuating component of the turbulent flow. The orientation of fibers depends on a combination of orienting effects of the mean velocity gradients and the randomizing effects of turbulence. Only mean velocity gradients are considered in the present simulation. In this study, rigid fibers are considered with uniform length L and diameter d, and aspect ratio L/d. The position and orientation of a single fiber can be described by the coordinates (x, y, z) and angles (Q, §). The orientation field that is expressed in the form of fiber orientation angles, is specified from the solution of the orientation equations along the particle trajectory. A large number of fibers is considered by using statistical 33 expressions developed to describe the orientation state. In brief, the numerical scheme deals with each individual fiber separately. Then the behavior of large numbers of fibers is characterized with the statistical methods. The fiber orientation distribution and the fiber orientation parameter f are based on the statistical data. The fibers are initially injected at the center of the inlet of the converging section with random orientations, although it is possible to set up a non-random distribution for the initial fiber state. At the inlet, random fiber orientation is implemented by choosing the fiber angles ^ and 9 for a large number of fibers. The angles Q and ^ are not selected randomly from uniform distributions Q E [0, n] and <j) G [0, n], since on the surface of a unit sphere, the area element dQ = singd^dQ is a function of Q. TO distribute fibers equally in all possible directions, the correct selection method should be followed [55]: choose aj and a2 to be random variables on [0, 1], then = nax (4-17) 9 = cos-1(2a2-l) (4-18) The initial orientation of 1000 fibers is shown in Figure 4.2. Obviously, as the number of fibers is increased, a more accurate representation of the orientation distribution will be obtained. From such an orientation distribution, the preferred orientation angle and the degree of alignment for that angle can be obtained. Although the accuracy of the statistical solution is dependent on the number of fibers considered, one should consider the available computational resources and the desired accuracy level. Hence, the number of fibers utilized in the statistical solution can be based on the desired accuracy and the available computational resources. After trying different numbers of fibers in the simulation, from 500 to 100,000, a sample size of 3000 fibers was chosen to represent the bulk fiber behavior as this gives a reasonable statistical expression. 34 Figure 4.1. A fiber in three-dimensional coordinates. Figure 4.2. The initial random distribution of 1000 fibers. 35 5. RESULTS AND DISCUSSION 5.1 Analysis of the Headbox Flow Field The quality of any CFD analysis depends, among other factors, on the quality of the chosen grid. In the flow simulation, the geometric dimensions are the same as the asymmetric converging section of the headbox in the experiment, only the size in the headbox span direction is reduced from 0.76 m to 0.8 cm. The 80x32x8 mesh as shown in Fig. 5.1 is generated over the converging section. By mapping the shape from the physical domain to a computational domain, the governing equations can be transformed and solved in the latter domain, using the finite-volume technique, and then the solutions at every node are mapped back onto the physical domain. The Cartesian coordinates x, y and z represent the machine direction, paper thickness direction and cross machine direction respectively. As we have noted already, the headbox span is reduced in the z direction in the computation. The simulation only models a spanwise section of the laboratory headbox, because it is assumed that there is no significant variation in velocity and velocity gradients in the z direction. The tubes are not modeled. The mesh is refined in the region close to the walls and the nozzle because the velocity gradients are larger there than in other place. The mesh in the machine direction (x-axis) is gradually refined since the velocity and pressure change rapidly as the flow approaches the exit. By refining the mesh, we can obtain comparatively accurate solutions on the corresponding nodes near the walls and the nozzle. The maximum mesh sizes at the inlet in x and y directions are around 8 mm and 2.8 mm respectively; the minimum mesh sizes at the nozzle in x and y directions are around 2.4 mm and 0.2 mm respectively. The mesh size in z direction is 1 mm. The boundary conditions are uniform velocity profile at the inlet (with initial k = 4 x 10"3 m2/s2, E = 3 x 10"4 m2/s3 for all cases), zero velocity gradient at the slice exit for all 36 velocities in the x direction, and the wall conditions at the upper and lower channel walls. Usually, zero velocity gradient condition is applied to fully developed flow. However, we find in this study that if the mesh grids close to the channel exit are small enough, zero velocity gradient can be approximately accepted as the boundary condition for the outflow of the converging channel case. The more physically correct exit boundary condition of constant pressure has been tried elsewhere. This condition leads to numerical simulations which converge with great difficulty, and has not been used here for that reason. The symmetry condition (zero gradients normal to the boundary of all velocity components, and zero value of the velocity component normal to the boundary) is imposed on both sides of the channel because only a section of the channel is modelled and there is no wall effect. Appropriate velocity gradients are set to zero and a no penetration boundary condition is imposed at each symmetry plane. The fluid used in this simulation is water. The uniform velocity at the inlet is 0.24 m/s, which is the same as the average inflow velocity in the experiment. There are no y- and z-direction velocity components at the inlet. Typical streamlines in this converging section are shown in Figure 5.2. The fiber orientation study is focused on the fibers travelling on the central streamline only. The variations of the x-direction velocity and pressure along the central streamline are shown in Figure 5.3. It can be seen from the plots that a large pressure drop occurs near the nozzle exit whereas the fluid velocity increases rapidly near the exit of the channel and reaches its maximum at the exit. Figure 5.4 gives u-velocity (x-direction) contours on the central symmetry plane. Figure 5.5 gives the v-velocity (y-direction) contours on the same symmetry plane. The flow, beginning with a uniform velocity profile at the inlet, gradually increases its speed as the channel becomes narrower, and the velocity gradients increase and reach a maximum at the headbox nozzle. The fiber orientation will be strongly affected by the increased elongation and shear stresses. 37 As some researchers have stated [25], fiber orientation is governed by the elongation effect in a convergent channel, and both the elongation of the flow and the fiber alignment in the flow direction reach their maximum at the channel exit. For this headbox channel, the flow elongation along the central streamline can be expressed as: s = \*±dt (5-D J ds • where us is the flow velocity on the central streamline and s is the distance along the central streamline. Because ds dt = — the flow elongation can also be calculated as: f -1 ds (5-2) 1 dus us ds The velocity us and distance s on the central streamline are related to the x direction velocity u and x, in the starting flat section, by: u = u„ x = s and in the consequent contraction section by: u = us coscp x = xt + (s - xx) cos q> where ^ is the angle between the central streamline and x-axis in the contraction section, Xj is the length of the flat section. Substituting the above relations into (5.2), the flow elongation becomes: s -r 1 du , J —— dx u dx 38 (5.3) If u is dependent on x only, equation (5.3) can be further simplified as: du (5.4) £ U or £ (x) = In u(x) (5.4') At the channel exit, the flow elongation is therefore: (5.5) where U0 and Ue are the velocity component in the x direction at the channel inlet and outlet. Because the ratio of Ue/U0 is approximately equal to the contraction ratio (R.) of the channel, then the flow elongation at the exit is obtained as: One can find from (5.6) that the flow elongation s at the channel exit only depends on the contraction ratio of a convergent channel. From (5.4) the flow elongation can be calculated varying with channel length as shown in Fig. 5.6. The flow elongation can also be calculated by equation (5.4'), which is essentially identical to the numerical plot of Fig. 5'.6. 5.2 Comparison of Simulation and Experimental Results A convenient graphical representation is needed to present the three-dimensional fiber orientation results. The projections of the orientation of large numbers of fibers can be (5.6) 39 obtained on three different planes. The projections on two of the planes, x-y and x-z plane, are used here, because the results can be compared directly with the experimental results for the corresponding planes. The fiber orientation angle a, either on the x-y or on the x-z projection plane, is defined to be the angle between the projection of the fiber axis on that plane and the machine direction (x-axis). The x-y plane corresponds to the side view in the measurements and the x-z plane corresponds to the bottom view in the measurements. The fiber orientation projection on the y-z plane is not considered because it is difficult to obtain the fiber images on that projection plane in the experiments. It would be possible to obtain the fiber orientation results on the y-z plane from the simulation, but this analysis has not been done. When all the orientation angle data at each specific point on the projection planes are available, the results of the fiber orientation distributions from experiments and from simulations can be compared. An angular interval must be chosen to provide a reasonable picture of fiber orientation distribution. For the fiber orientation distribution diagram used here, the horizontal axis represents the orientation angle a. It is separated into 18 zones of 10 degrees each, from -90° to +90°, with 0° indicating the machine direction (x-axis). The vertical axis represents the statistical probability density p(a)> such that: fj p(a)da = 1 (5-7) Fiber orientation distributions close to the inlet (x = 4.5 cm) as seen from the side view and the bottom view of the headbox are shown in Figure 5.7. One can see that the fibers are almost randomly distributed at this location. If more fibers had been used in either the measurements or the numerical simulations, p(a) would have become more constant, approaching a value equal to l/n (~ 0.318) for a very large number of fibers. As fibers enter the converging section of the headbox, they gradually exhibit the tendency to align in the flow direction. Figure 5.8 and 5.9 show the fiber orientation distributions at x = 12.2 cm and 15.7 cm. An interesting phenomenon was found from these two diagrams: the fiber orientation distributions from numerical simulations change faster than that from 40 experiments. When fibers move further towards the nozzle, this phenomenon becomes more significant as shown in Figure 5.10 (x = 19 cm), Fig. 5.11 (x = 22 cm) and Fig. 5.12 (x = 26 cm). At the exit where x = 31 cm, the fibers are highly aligned in the flow direction as shown in Fig. 5.13. We do not have the experimental data of fiber orientation in the x-y plane, because the channel is too narrow and the flow speed is too high at that location to obtain clear fiber images. These diagrams show that the simulation can predict the trend of the fiber orientation in a dilute headbox flow, but there exist obvious differences between the numerical data and the experimental data. Another phenomenon shown from these diagrams was that the fiber orientation distributions or the alignments in the x-y plane are stronger than that in the x-z plane. In other words, the orientation state is changed more in the shear and extensional plane (x-y plane) than in the neutral plane (x-z plane). This is caused by different velocity gradient effect. In this headbox structure, the velocity gradients du/dx, du/dy, dv/dx and dv/dy all affect the fiber orientation in the x-y and y-z planes, but only du/dx and du/dy affect the fiber orientation in the x-z plane. That is why the fiber alignment in x-y plane is stronger than that in the x-z plane. In general, the simulated fiber orientations tend to align more with the flow compared to the experimental results. The major reason for this phenomenon is almost certainly that the turbulence effect is not considered in our fiber simulation. In the experiment, the flow enters the headbox channel through the rectifier tubes. As we have mentioned earlier, the rectifier tubes have two fundamental functions on the incoming flow. First, they provide turbulence of desired intensity and scale to breakup fiber floes. Secondly, they produce a fairly uniform inflow at the channel inlet. The turbulence created in the rectifier tubes will affect the fiber behavior downstream. Turbulence tends to randomize the fiber orientation, which makes the fiber orientation distribution have a comparatively flat appearance. The current numerical fiber model simulates only the fiber motion in a mean flow field, so the predicted fiber orientation distribution is more organized than in the observed situation, where turbulence plays an important role. 41 The fiber orientation distribution in the side view (x-y projection plane) is not symmetrical about zero because of the asymmetric structure of the headbox in the x-y plane. The channel is not symmetric so that zero degree orientation would indicate a fiber being parallel to the x-axis or the lower channel wall. The angle between the upper and lower channel wall is 16.7°. As a result, the preferred orientation in the side view should be between Oo and 16.7°. On the other hand, the simulation results give a symmetrical distribution on x-z projection plane. This is because there is no restriction in this plane, and fibers are allowed to freely move in all possible directions. Another concise way to describe fiber orientation is to use a fiber "orientation parameter". The plane orientation parameter f used by McCullough [56] is defined by fp = 2 p(a) cos2 (a-a0) da -1 (5-8) 2 where ao is the mean of the distributed angles or the preferred alignment angle, and p(a) represents the probability density function. For the case of a finite number of fiber orientations, n, the above equation has been approximated by York [57] as: 2 n fP = -Scos2 (ai~ao) n ,=i (5.9) The parameter fp provides a convenient method to describe a particular state of in-plane fiber orientation. For perfect alignment, f = 1, and for a completely random orientation distribution, f = 0. The mean direction ao can be calculated with the following equations [58]: -i C an = cos — 0 R (5.10) or 42 a0 = sin"1- (5.H) 0 R where: 1 " c = 1S cos a, 1 " S = — V sin a, n M R = (C2 +S2 Table 5.1 presents the observed and predicted orientation parameters at several measurement points along the central streamline (see Fig. 3.8). Table 5.1. Orientation Parameters Obtained from Experiments and Simulations: x-positions Experiments Simulations x-y plane x-z plane x-y plane x-z plane 1 0.04 -0.01 0.01 -0.00 2 0.10 0.02 0.17 0.07 3 0.14 0.05 0.33 0.15 4 0.18 0.05 0.50 0.26 5 0.22 0.15 0.68 0.38 6 0.33 0.20 0.82 0.51 7 N/A 0.53 0.96 0.76 Fig. 5.14(a) and Fig. 5.14(b) present the fiber orientation parameters along the central streamline in the x-y and x-z projection planes respectively. The results show that there are differences between the measurement data and the simulation data, although the trend is the same. For both measurements and simulations, the fiber orientation distribution develops from a random initial condition to a much more aligned status at the exit, although the distribution at the exit is not fully aligned. As will be discussed in the next 43 section, the degree of alignment depends on the headbox geometry, or more specifically, the contraction ratio of the channel. 5.3 Factors Affecting Fiber Orientation When subjected to the plane rate of strain, QU/QX, in the convergent channel, the fibers have the tendency to align in the direction of the flow. For different headbox geometries and flow conditions, the degree of alignment of fibers in the flow direction is different and is determined by the rates of strain in the flow, and the time that the fibers are exposed to the forces. High rate of strain and long time duration would be likely to produce highly concentrated fiber orientations. In order to detect what factors may influence the fiber orientation characteristics, the headbox geometry, the flow velocity and the fiber aspect ratio have been changed and their effects on fiber orientation have been studied. When one parameter is changed, the others are kept the same. This exploration of the effects of selected parameters is much easier to do in the simulation studies than in the experimental work. In the simulation, we investigate the final fiber orientation distribution for different values of the contraction ratio (R.), which is the ratio between inlet area and exit area, the channel length (Lc), the inflow rate U0 and the fiber aspect ratio (Ar). 5.3.1 The Effect of Contraction Ratio on Fiber Orientation In this part of the present study, only the channel's contraction ratio is changed by adjusting the exit area, while the channel length and the flow velocity remain the same. The fiber orientation parameters at the channel exit for different Rc are shown in Table 5.2. 44 Table 5.2. Fiber Orientation Parameters for Different R (U = 0.24 m/s, L = 0.225 m) Rc Orientation parameter x-y plane x-z plane 6.7 0.96 0.73 10 0.98 0.81 15 0.98 0.86 The fiber orientation parameter (which corresponds to the fiber orientation distribution) in the x-y and x-z projection planes at the channel exit are increased with increasing contraction ratio, although the increment in the x-y plane is slowed down as it approaches its maximum value of 1. The conclusion is that higher Rc corresponds to more aligned fiber orientation in the flow direction. 5.3.2 The Effect of Flow Rate on Fiber Orientation Keeping other parameters the same, only the flow rate at the inlet of the channel is changed. In every case the contraction ratio is 10. The fiber orientation parameters at the channel exit for different flow rates are shown in Table 5;. 3. The Reynolds number Re is calculated by the following equation: Re = U° H° (5.12) v where U0 is the velocity at the channel inlet, HQ is the channel height at inlet, v is the kinematic viscosity of water. 45 Table 5.3. The Orientation Parameters for Different IL (R = 10, L = 0.225 m) Uo (m/s) Reynolds number Orientation parameter x-y plane x-z plane 0.16 12,000 0.98 0.81 0.24 18,000 0.98 0.81 0.36 27,000 0.98 0.80 As already noted, the experimental results were obtained for a value of UQ of 0.24 m/s. From the simulation results, it appears that the fiber orientation at the channel exit is not affected by the flow velocity. Increasing the flow rate of course increases the normal and shear rates of strain in the flow, but at the same time, it also shortens the fiber residence time in the flow. The present results of fiber orientation for different flow velocities needs further investigation, because in our simulation the turbulence effect is not specifically included, and the flow velocity has close relationship with the turbulence. 5.3.3 The Effect of Channel Length on Fiber Orientation Besides the effect of Rc and UQ, it is important to determine the effect of channel length on fiber orientation. The fiber orientation parameters at the channel exit for different channel lengths are shown in Table 5.4. The initial uniform cross section channel length is the same. L is the length of the convergent section. Table 5.4. Fiber Orientation Parameters for Different L (R = 10, U = 0.24 m/s) Orientation parameter (m) x-y plane x-z plane 0.1500 0.98 0.82 0.2250 0.98 0.81 0.3375 0.99 0.82 46 Although a shorter contraction corresponds to higher rates of strain in the flow field, the fiber residence duration in the flow is also reduced. The total effect of changing channel length on fiber orientation is therefore eliminated. Here again, only the mean flow has been considered in the simulation. For further investigation, turbulence effect should be included. 5.3.4 The Effect of Fiber Aspect Ratio on Fiber Orientation Finally, the effect of changes to the fiber aspect ratio is investigated by changing the fiber length and diameter. The prediction of fiber orientation parameters at the channel exit is shown in Table 5.5. Table 5.5. The Orientation Parameters for Different A r (Rc = 10, UQ = 0.24 m/s, Lc = 0.225 m) Ar (L/d) ( L and d in mm ) orientation parameter x-y plane x-z plane 133.3 (4/0.030) 0.98 0.81 68.2 (3/0.044) 0.98 0.81 33.3 (2/0.060) 0.98 0.81 From the table it appears that, within the range of interest, the change in Ar does not affect the fiber orientation distribution. The turbulence length scale could be of importance in a more complete simulation which includes turbulent dispersion. 5.3.5 The Effect of Flow Elongation The flow elongation is also examined for its relations with the contraction ratio of the channel. Table 5.6 shows the effect of R on the elongation of flow at the exit of the c ° channel. 8 is calculated from Equation (5.6). Because the change of flow rate and channel 47 length does not affect s, it is clear that the elongation of flow can be used for the determination of the fiber orientation status in a convergent channel, if only the mean velocities are considered. However, turbulence dispersion almost certainly a significant factor which is not covered by the elongation of the mean flow. Table 5.6. The Elongation of Flow at the Channel Exit for Different R (U =0.24 m/s, L = 0.225 m): Conditions Rc=15 Rc = 10 Rc = 6.7 s 2.7 2.3 1.9 5.4 Symmetric Channel The previous prediction results are for an asymmetric convergent channel. A symmetric channel is now explored using the numerical simulations. The symmetric headbox used in Ullmar's work [25] with a contraction ratio of 16.7 is adapted for the current simulation. Fig. 5.16 gives the cross sectional dimensions of this headbox, and Fig. 5.17 shows the computational mesh of this design. The channel width is constant. During the simulation, four inflow velocities are tested. The fiber orientation parameters on the central streamline at the channel exit for various inflow rates are shown in Table 5.7. Table 5.7. Orientation Parameters at Exit of A Symmetric Headbox for Different U • Inflow rate (m/s) Reynolds number Orientation parameter x-y plane x-z plane 0.10 25,000 0.98 0.88 0.30 75,000 0.98 0.88 0.43 107,000 0.98 0.88 0.56 140,000 0.98 0.88 From this study of a symmetric channel flow, the following conclusions are obtained from the numerical simulations: 48 a. Fiber orientation changes from random at the channel inlet to the highly preferred orientation at exit as fibers move along the central streamline; b. As in the asymmetric converging section, an increase of inflow velocity has no effect on the fiber orientation; c. The difference in fiber orientation for two projection planes exists both in the asymmetric and symmetric convergent channel. 5.5 Statistical Error Estimation An estimate can be made of the error which will be present in the probability distributions measured or calculated, due to the limited sample size which is being used. In general, the standard deviation within repeated samples consisting of n objects, of which a fraction r has a particular characteristic, is given approximately by [59] where b is the fraction of n not contained in r and is therefore equal to (1 - r). This estimate is accurate when r and b are approximately equal for small n and also for r and b unequal if n is large. Since n is larger than 1,000 in all the cases of our experiments, we assume this expression can be used as an error estimate when applied to the probability density function p(a) for which In the equation, & is the interval of orientation angle a for which the value of p(a) is to be evaluated, sometimes called the "bin width" of the angle a. Although A. can vary with (5.13) p(a) = r (5.14) A 49 a if the bin widths are not constant, the assumption is made here that A is a constant, independent of a. The standard deviation of the probability density value p(a) will be (5.15) and therefore equal to r(l-r) 1 n A (5.16) Substituting (5.14) into (5.16), we have \E^-p(af (5-17) In general, if the orientation is initially completely random, p(a) can be expressed as p(a) = - (5-18) . n In the present evaluation of p(a), a constant "bin width" A of nl\% has been used. For a sample size n of 1325 used in the experimental evaluations for the x-y plane at x = 4.5 cm, the standard deviation of every point in the probability density would be ~Ar - 0.036 For a sample size n of 1638 used in the experimental evaluations for the x-z plane at x 4.5 cm, the standard deviation is 50 The error can be expected to be within a band of + 3a about the mean value (equal to 1/jr), which implies a value of the probability density in x-y plane in the range 0.318 + 0.108 and a value of the probability density in x-z plane in the range 0.318 + 0.096. The observed values at the first point (x = 4.5) for the x-y plane lie between 0.221 and 0.458 (see Fig.5.7), a little bit outside of the upper limit of the predicted scatter. Presumably, the orientations of fibers at this point are not completely random because of the upstream flow effect. The observed values at the same point for the x-z plane lie between 0.245 and 0.392, falling within the predicted band. In summary, the present evaluations show the effect of sample size on the probable scatter in measured values and give a fair estimate of the observed distribution of data in both the x-y and x-z planes. The simulations, shown in Fig. 5.7, have an initial scatter which lies well within the predicted statistical error estimates, as expected. 51 Figure 5.2. The streamlines of the flow in the headbox convergent channel. 52 2.5 Ol 1 1 1 1 1 1 1 i i I I i i i I I oi 1 1—•• ' 1 i i i i 1 i i 1 1-0 0.1 0.2 0.3 0 °1 0.2 0.3 channel length (m) channel length (m) Figure 5.3. The pressure and u-velocity changes along the central streamline. Figure 5.4. The u-velocity contours on the central symmetry plane. 003 Figure 5.5. The v-velocity contours on the central symmetry plane. 53 channel length (m) Figure 5.6. The elongation of the flow changes along the central streamline. Figure 5.7. The fiber orientation distribution at x = 4.5 cm, (a) in x-y plane, (b) in x-z plane 54 £ 0.4 JJj 0.3 cn 3 e °- 0.21-SIMULATION • EXPERIMENT -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 angle (radians) (a) (b) Figure 5.8. The fiber orientation distribution at x = 12.2 cm, (a) in x-y plane, (b) in x-z plane. 0.5 angle (radians) •0.5 0 0.5 angle (radians) (a) (b) Figure 5.9. The fiber orientation distribution at x = 15.7 cm, (a) in x-y plane, (b) in x-z plane. 55 56 57 .2 i E .» o E g 0.4 o | 02 C i SIMULATION • EXPERIMENT • • channel length (cm) channel length (cm) (a) (b) Figure 5.14. The orientation parameters along the central streamline, (a) in the x-y plane, (b) in the x-z plane. -0.5 0 0.5 angle (radians) angle (radians) (a) (b) Figure 5.15. Fiber orientation distributions at the channel exit for various Contraction ratios, (a) in x-y plane, (b) in x-z plane. 58 Figure 5.17. The physical mesh of the symmetric headbox. 59 6. SUMMARY AND CONCLUSIONS This thesis is concerned with fiber orientation in the converging section of a headbox. A summary of the work and some findings based on the experiments and computational simulations are reported as follows. A mathematical flow model and a fiber motion model have been combined for predicting the orientation of rigid fibers in dilute suspensions. There is no parameter in the model to be determined by experiment. Rigid fibers flowing in a symmetric and asymmetric converging section are numerically simulated. Random initial fiber orientations are specified at the inlet of the channel. The statistical expressions of the orientation of a large number of fibers can be evaluated by computing the orientation of each single fiber along the central streamline. Experiments were performed in a scaled headbox model to validate the numerical model. The orientation distributions of dyed nylon fibers with uniform length and width were studied using an image analysis method. An asymmetric headbox with a contraction ratio of 10 is used in this study as the basis for experimental and simulation work. The comparison shows that the simulation method can predict the trend of fiber orientation in a dilute headbox flow, although the difference between the experimental data and the numerical prediction is significant. The predicted orientation distributions, being based only on the mean flow in the headbox, show greater organization than is observed, in every case. The turbulence dispersion effect must be added to the current fiber model before it is capable of accurately predicting the fiber orientation in a given flow field. The simulation method has further been used to predict fiber orientations for different headbox geometry, flow velocity and fiber aspect ratio. Changes in contraction ratio causes changes in velocity gradients, and rates of strain, and therefore affect the alignment of fibers in the flow direction. Changes in channel length and flow velocity can also change the flow field, but their effects on the fiber orientation are eliminated in this 60 simulation by the fact that the residence time is changed. The simulation results also show that for dilute suspensions, the effect of fiber aspect ratio and fiber length in the range of our interest (i.e. Af = 33-133, L = 2~4 mm) on fiber orientation is small. Both experimental and simulation results showed differences in the fiber orientation in different projection planes. This phenomenon is caused by the flow characteristics in a convergent channel, i.e., the contributions of velocity gradients to fiber orientation are not the same in all directions. 61 7. RECOMMENDATIONS FOR FUTURE WORK The current work has established a method to investigate fiber orientation in a headbox flow and has found important effects on the fiber orientation state resulting from different headbox geometries and flow conditions. However, there are problems to be solved in future work before reaching a complete understanding of fiber orientation in headboxes. 1. Currently, only the effect of mean velocities and their gradients have been considered in the fiber model. There exist significant differences between the experimental data and the numerical predictions. Clearly, turbulence plays an important role in the determination of fiber orientation in the flow. If the influence of the fluctuating velocities on the fiber is added, the simulation could produce results closer to the experiments. It is therefore important to know the characteristics of turbulence in the convergent section which are inherited from the upstream flow characteristics in the tube bank and header, and how these characteristics are modified by the headbox contraction. 2. Flexible fibers should be used in the simulation instead of the rigid fibers used in this research. Fortunately the flexible fiber model is available in the research group at UBC. To quantify such results, one must provide a reasonable definition of orientation for flexible fibers. The curl or kink of fibers should be studied as well. 3. The current research has focussed on fiber motion along the central streamline. A detailed study of the fiber orientation profiles in the machine direction, cross machine direction and paper thickness direction at the headbox exit is required. Then one may find the answer to questions such as what causes fiber orientation non-uniformity in the cross machine direction, or find fiber orientation in the paper thickness direction. 62 4. In order to have an overall view of the fiber orientation in a headbox, the fiber-wall interactions must be included. As shown, fiber orientation in a headbox is mainly determined during the short distance that fibers move very close to the exit. Near the exit, the channel opening becomes narrower and fiber-wall interactions may not be negligible. 5. During this work only the convergent slice section of a headbox is simulated with the assumption of initial uniform velocity and random fiber orientation distribution at the inlet. A more complete work should involve the flow of the suspension in the diffuser tubes or even through the manifold. The paper manufacturers are concerned about the fiber orientation profiles in the cross-machine direction. That requires a thorough understanding of fluid-fiber interactions and fiber orientation characteristics within the entire headbox. To give a total view of the fiber orientation in the paper making processes, the fiber orientation in the jet and on the forming wire should also be studied. All these studies can then be combined together to give an overall view of the fiber orientation in the paper. 6. In our research, the fiber-fiber interaction is neglected, because the suspensions used in the simulations and experiments are very dilute. At commercially used papermaking consistencies, i.e., in the range of 0.1 to 1.5%, fibers do not exist independently in suspension, but are present in a network form. There are seldom fibers freely floating about. Each fiber is in contact with many other fibers, and is bent out of its natural relaxed shape. The bending and pushing of fibers against each other gives fiber networks a certain amount of mechanical strength, which in turn affects the flow pattern. As a result, fiber-fiber interaction and its effect on the flow should also be investigated in the future. When considering fiber-fiber and fluid/fiber interactions, it is important to solve simultaneously for the velocity field and the fiber orientation in order to obtain an accurate description of the flow and the fiber motion. 63 8. NOMENCLATURE Ar fiber aspect ratio av a2 random variables b fraction of fibers not in the r CD cross machine direction C volume concentration V d fiber diameter E rate of kinetic energy dissipation per unit mass f orientation parameter G rate of kinetic energy production per unit mass HQ channel inlet height k turbulent kinetic energy L fiber length L channel length c ° MD machine direction N crowding factor n number of fibers p pressure p(a) the statistical probability density function r fraction of fibers R contraction ratio c Re Reynolds number s distance along the central streamline S.. rate of strain tensor u t time u hydrodynamic velocity component in the x-direction UQ u-velocity at the inlet U u-velocity at the outlet 64 u instantaneous fluid velocity vector us flow velocity along the central streamline v hydrodynamic velocity component in the y-direction w hydrodynamic velocity component in the z-direction x the x-direction in a Cartesian system Xj the length of the flat section y the y-direction in a Cartesian system z the z-direction in a Cartesian system a fiber orientation angle in a projection plane aQ the mean of distributed angles p flow angle Kronecker Delta A interval of orientation angle s elongation of the flow 8e elongation of the flow at the channel exit (j, fiber orientation angle <p the angle between the central streamline and x-axis 1^ dynamic viscosity of the fluid v kinematic viscosity of the fluid v turbulent kinematic viscosity 0 fiber orientation angle p fluid density CTp standard deviation of the probability density p (a) CTr standard deviation of the fraction r x fluid stress tensor Xij Reynolds stress tensor Q solid angle 65 9. 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