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Modelling of jet impingement and early roll forming Dalpke, Barbara 2002

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MODELLING OF JET IMPINGEMENT AND EARLY ROLL FORMING By Barbara Dalpke Dipl. Ing. (Mechanical and Process Engineering, Paper Science and Technology) Darmstadt University of Technology, 1 9 9 7 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 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 2002 © Barbara Dalpke, 2002 In presenting this thesis in partial fulfilment 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. 1 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 Jjfcir\&.i~>ica\ I ^ g C rtnc^ The University of British Columbia Vancouver, Canada Date /^or>7 ML , 9002, DE-6 (2/88) Abstract Twin-wire paper machines have gained great importance in the production of printing paper and other paper grades. The two major former types used are roll and blade formers, with each having fluid dynamics that significantly influence the quality of the final paper. While many aspects of these forming hydrodynamics are well understood, the first part of the drainage section, where the flow impinges between the two fabrics, has been largely neglected, although it is known that certain paper properties are strongly influenced in this zone. The objective of this study was to model the hydrodynamic details of jet impingement and drainage in the early part of roll forming in twin-wire paper machines. A theoretical approach based on computational fluid dynamics was chosen to model the free jet and impingement zone. For the first time, a two-dimensional, viscous model was employed using a Volume of Fluid method. The first step modelled jet impingement on a single fabric. This case represented jet impingement on the outer fabric in twin-wire formers. In addition, with slight modifications, it applies to Fourdrinier papermaking as well. Computations were carried out for cases with or without fibre mat build-up for different machine settings (jet velocity, impingement angle and jet rush or drag). It was shown that both the inertial and viscous component of resistance are important. It was further shown that forces from jet impingement influence flow over only a short distance around the impingement point, causing most of the drainage to occur over a distance of about one or two jet thicknesses. This force on the fabric is mainly influenced by the jet velocity and the impingement angle, while drainage depends mainly on the fibre mat resistance. Rush or drag affects the shear stress at the fabric, but has little influence on pressure and drainage. To validate the computations, drainage velocity profiles were measured for impingement on a stationary single fabric. Agreement with computations was reasonable over the first centimetre after impingement, but was less good further downstream. Probable causes are inaccurately measured fabric resistances at low flow velocities and a dependence of the re-Ill sistance on the flow angle. The fabric roughness also might have an important effect on drainage in single-fabric-impingement. The model was then extended to include the wedge of roll formers. The outer fabric cur-vature was calculated based on a force balance. Findings by other researchers regarding the dependence of the forming zone length on machine variables were confirmed. The pressure distribution in early roll forming takes on a more complex form than that given by P = T/R (pressure = fabric tension/roll radius). It is lower than T/R at the wedge entry, and then increases. In some specific cases, it increases to local pressures exceeding T/R. The increase depends on the jet velocity, wrap angle, and fibre mat resistance. Higher fabric tensions cause overall higher pressures. The forming zone geometry, which is partly influenced by the impingement forces, affects the pressure distribution, but the impingement position does not affect the flow in the wedge. Contents Abstract ii List of Figures viii List of Tables xviii Acknowledgements xx 1 Introduction 1 1.1 History of twin-wire formers 2 1.2 Twin-wire forming principles 3 1.2.1 Blade forming 3 1.2.2 Roll forming 5 1.2.2.1 Jet-to-wire speed ratio 7 1.2.2.2 Experimental work 8 1.2.2.3 Theoretical modelling 10 1.2.3 The jet impingement region in twin-wire formers 13 2 Drainage resistance of the fabric and fibre mat 17 2.1 Background 17 2.1.1 Previous experimental work on fibre mat resistance 19 2.1.2 Theoretical models to describe fibre mat resistance 27 2.1.3 Fabric resistance and fabric/mat interaction 29 2.2 Choosing resistance descriptions for the model 31 iv CONTENTS v 2.2.1 Experimental determination of fabric resistance: flow perpendicular to the fabric 32 2.2.1.1 Experimental set-up 33 2.2.1.2 Test conditions 33 2.2.1.3 Results 35 2.2.2 Experimental determination of fabric resistance: flow at varying angles to the fabric 39 2.2.2.1 Modified experimental set-up 40 2.2.2.2 Test conditions 41 2.2.2.3 Results 41 3 Jet impingement on a single fabric 48 3.1 Computational work 48 3.1.1 Analysis 48 3.1.2 Method of solution 51 3.1.2.1 Boundary conditions 52 3.1.2.2 Modelling the fabric 52 3.1.2.3 Modelling fibre mat build-up 53 3.1.3 Validation of the V O F model with potential flow theory 57 3.1.4 Grid independence 59 3.1.5 Results 61 3.1.5.1 Computational results neglecting fibre mat build-up 61 3.1.5.2 Computational results including the fibre mat build-up . . . 72 3.1.6 Discussion of assumptions 86 3.1.7 Summary -. . 90 3.2 Experimental work 91 3.2.1 Drainage profile measurements at impingement on a stationary fabric 91 3.2.1.1 Experimental set-up 92 3.2.1.2 Test conditions 95 3.2.1.3 Experimental results 96 CONTENTS vi • 3.2.2 Comparison of experimental and computational results 104 4 The initial flow in roll forming 118 4.1 The computational model 118 4.1.1 Analysis 118 4.1.2 Method of solution 119 4.1.2.1 Boundary conditions 122 4.1.3 Grid independence 124 4.1.4 Convergence issues 124 4.1.5 Centrifugal forces 126 4.1.6 Forming conditions in the computations 128 4.1.6.1 Jet velocity 128 4.1.6.2 Impingement angle 128 4.1.6.3 Fabric tension 128 4.1.6.4 Fabric and fibre mat resistance 129 4.1.6.5 Forming roll radius 129 4.1.6.6 Wrap angle 129 4.1.6.7 Jet thickness 130 4.1.6.8 Impingement position 130 4.1.7 Results 132 4.1.7.1 Comparison with experimental data 132 4.1.7.2 Influence of jet velocity 137 4.1.7.3 Influence of fabric tension 141 4.1.7.4 Influence of filtration resistance 143 4.1.7.5 Influence of wrap angle 147 4.1.7.6 Influence of impingement position 150 4.1.8 Summary 154 5 Summary and conclusions 158 5.1 Impingement on a single fabric 158 5.1.1 Computational model 158 CONTENTS vii 5.1.2 Experimental measurements 159 5.2 Initial drainage zone in roll formers 160 6 Recommendations for future work 162 Bibliography 165 Appendicies 173 A Results of the drainage experiments 173 A . l Completed drainage as a function of distance from the impingement point. . 173 A. 2 Drainage velocity profile 176 B Comparison experiments/computations 180 B. l Grids for the computations 180 B.2 Results: Fabric C 181 B.2.1 Varying jet velocity 181 B.2.2 Varying impingement angle 181 B.3 Results: Fabric D 186 B.3.1 Varying jet velocity 186 B.3.2 Varying impingement angle 189 B.4 Results: Fabric E 195 B.4.1 Varying jet velocity 195 B.4.2 Varying impingement angle 198 List of Figures 1.1 Webster's roll former [1] 2 1.2 Gap blade former (Black Clawson) 4 1.3 Pressure pulses in a blade gap former [20] 4 1.4 Gap roll former (Voith) 5 1.5 Principle of two-sided dewatering in roll forming [25] 6 1.6 Pressure measurement in the wedge of a gap roll former [30] 7 1.7 Typical drainage profiles for roll, spray and total drainage around a forming roll 9 1.8 Impingement zone of a blade former. 15 2.1 Filtration versus thickening process [59] 21 2.2 Typical oscilloscope trace of the force on the fabric at impingement during simulated jet impingement [46] 31 2.3 Schematic of flow loop to determine fabric characteristics 34 2.4 Pressure drop data for different fabrics 36 2.5 Dimensionless pressure coefficient as a function of flow velocity for different fabrics 37 2.6 Definition of flow loop variables in the modified set-up 40 2.7 Pressure drop data versus flow velocity for different approach angles (fabric C). 42 2.8 Pressure drop data versus flow velocity for different approach angles (fabric D). 42 2.9 Pressure drop data versus flow velocity for different approach angles (fabric E). 43 2.10 Dimensionless pressure coefficient as a function of the flow velocity (fabric C). 43 2.11 Pressure drop data versus superficial velocity through the fabric for different approach angles (fabric C) ' 44 viii LIST OF FIGURES ix 2.12 Pressure drop data versus superficial velocity through the fabric for different approach angles (fabric D) 45 2.13 Pressure drop data versus superficial velocity through the fabric for different approach angles (fabric E) 45 2.14 Calculated resistance factor as a function of the approach flow angle for purely inertial resistance 47 3.1 Jet impingement on a single fabric with the important parameters 49 3.2 Approximation of mat resistance profile with a step function 55 3.3 Fibre mat build-up: Schematic approach 56 3.4 Convergence of the drainage velocity profile during fibre mat build-up calcu-lation 57 3.5 Comparison between computation and potential flow theory 58 3.6 Grid convergence of drainage for different fabrics (meshes are 200 x 100, 400 x 200, 800 x 200 and 800 x 400 cells) 60 3.7 Grid convergence of pressure for fabric lb) 61 3.8 Influence of viscous resistance on pressure profiles along the fabric 63 3.9 Influence of jet velocity on the computed pressure profiles at the fabric. . . . 64 3.10 Computed integrated pressure at the fabric as a function of jet velocity and comparison with a simple one-dimensional analysis 65 3.11 Computed drainage as a function of jet velocity (given in absolute value per metre of machine width and as a fraction of incoming flow volume) 66 3.12 Influence of jet velocity on the drainage velocity through the fabric 66 3.13 Influence of impingement angle on the computed pressure profiles at the fabric. 68 3.14 Computed integrated pressure at the fabric as a function of impingement angle and comparison with a simple one-dimensional analysis 68 3.15 Computed drainage as a function of impingement angle (given in absolute value per metre of machine width and as a fraction of incoming flow volume). 69 3.16 Influence of impingement angle on the drainage velocity through the fabric. . 70 3.17 Computed integrated pressure at the fabric and drainage as a function of rush/drag 71 LIST OF FIGURES x 3.18 Shear stress along the fabric for various cases of rush/drag. From top to bottom, the different curves represent decreasing amounts of drag (jet velocity was 15 m/s) 73 3.19 Computed pressure profiles at the fabric for jet impingement of different fur-nishes 74 3.20 Integrated pressure at the fabric and relative drainage for different furnishes (SFR=0 represents impingement without fibre mat build-up) 75 3.21 Build-up of basis weight as a function of the furnish 76 3.22 Integrated pressure and relative drainage for different furnishes (SFR values) in conjunction with different fabrics. SFR=0 represents impingement without fibre mat build-up 78 3.23 Computed pressure profiles at the fabric for different jet velocities, G W newsprint furnish with fabric 1 at 10° impingement angle 80 3.24 Computed integrated pressure at the fabric as a function of jet velocity and comparison with a simple one-dimensional analysis, G W newsprint furnish with fabric 1 at 10° impingement angle 81 3.25 Computed drainage as a function of jet velocity (given in absolute value per metre of machine width and as a fraction of incoming flow volume), G W newsprint furnish with fabric 1 at 10° impingement angle 82 3.26 Computed pressure profiles at the fabric for different impingement angles. G W newsprint furnish with fabric 1 at 15 m/sjet velocity. 84 3.27 Computed integrated pressure at the fabric as a function of impingement angle and comparison with a simple one-dimensional analysis. G W newsprint furnish with fabric 1 at 15 m/s jet velocity 84 3.28 Computed drainage as a function of impingement angle (given in absolute value per metre of machine width and as a fraction of incoming flow volume). G W newsprint furnish with fabric 1 at 15 m/s jet velocity 85 3.29 Force balance for the fabric 88 3.30 Principle of drainage profile measurements 92 3.31 Front view of the mobile flow loop 93 LIST OF FIGURES xi 3.32 The Drainage Fractionator 94 3.33 Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric C, 7.0 m/s jet velocity) 97 3.34 Fraction of completed drainage as a function of the distance from impingement point for different jet velocities (fabric C, 20° impingement angle) 98 3.35 Fraction of completed drainage as a function of the distance from impingement point for different jet velocities (fabric E, 20° impingement angle) 99 3.36 Overall relative drainage as a function of jet velocity (20° impingement angle). 99 3.37 Overall relative drainage as a function of impingement angle 100 3.38 Drainage velocity as a function of distance from impingement point for differ-ent jet velocities (fabric C, 20° impingement angle) 101 3.39 Drainage velocity as a function of distance from impingement point for differ-ent impingement angles (fabric C, 7.0 m/s jet velocity) 102 3.40 Repeatability of measurements: fraction of completed drainage as a function of distance from impingement point (jet velocity 7.0 m/s, fabric C) 103 3.41 Repeatability of measurements: drainage velocity profiles (jet velocity 7.0 m/s, fabric C) 103 3.42 Computed drainage velocity profiles for different jet velocities (impingement angle 20°, fabric C) 105 3.43 Experimental and computed drainage velocity profiles for different jet veloci-ties (impingement angle 20°, fabric C) 106 3.44 Measured and computed drainage velocities at 0.5 and 1.5 cm downstream from impingement as a function of jet velocity (impingement angle 20°, fabric C). Drainage velocities are averaged over 1 cm length (from 0 to 1 cm and 1 to 2 cm) 107 3.45 Computed fraction of completed drainage as a function of distance from im-pingement point for different jet velocities (impingement angle 20°, fabric C). 108 3.46 Experimental and computed fraction of completed drainage as a function of distance from impingement point (impingement angle 20°, fabric C) 108 LIST OF FIGURES xii 3.47 Computed and experimental overall relative drainage as a function of jet ve-locity (impingement angle 20°) 109 3.48 Computed drainage velocity profiles for different impingement angles (jet ve-locity 7.0 m/s, fabric C) 110 3.49 Experimental and computed drainage velocity profiles (jet velocity 7.0 m/s, fabric C) I l l 3.50 Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 7.0 m/s, fabric C) 112 3.51 Computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 7.0 m/s, fabric C) 113 3.52 Experimental and computed fraction of completed drainage as a function of distance from impingement point (jet velocity 7.0 m/s, fabric C) 113 4.1 Schematic of the determination of the forming zone geometry in roll forming. 121 4.2 Forming zone with fixed points P i , P 2 , a) without fibre suspension, b) with fibre suspension. C i and C 2 are corner points of the modelled domain 123 4.3 Grid convergence in roll forming computation (case 1 of Table 4.1) 125 4.4 Comparison of theoretical and computed pressure difference between outer and inner fabric due to centrifugal force (case 1 of Table 4.1) 127 4.5 Gap width as a function of distance from nip: case 1 and comparison with Gooding et al. [32]. 0 cm in the nip is defined in Figure 4.2 134 4.6 Computed drainage velocity through inner and outer fabric as a function of distance from nip: case 1 and comparison with Gooding et al. [32] 136 4.7 Pressure in the wedge as a function of distance from nip: case 1 and compar-ison with Gooding et al. [32] 137 4.8 Pressure in the wedge for different jet velocities 138 4.9 Drainage velocity through inner and outer fabric for different jet velocities. . 139 4.10 Gap width for different jet velocities 140 4.11 Machine direction velocity in the wedge for different jet velocities 140 LIST OF FIGURES xii i 4.12 Pressure in the wedge for different fabric tensions 142 4.13 Drainage velocity through inner and outer fabric for different fabric tensions. 143 4.14 Gap width for different fabric tensions 144 4.15 Machine direction velocity in the wedge for different fabric tensions 144 4.16 Pressure in the wedge for different SFR values 145 4.17 Drainage velocity through inner and outer fabric for different SFR values. . . 146 4.18 Gap width for different SFR values 147 4.19 Machine direction velocity in the wedge for different SFR values 148 4.20 Pressure in the wedge for different wrap angles 149 4.21 Drainage velocity through inner and outer fabric for different wrap angles. . 149 4.22 Gap width for different wrap angles 150 4.23 Machine direction velocity in the wedge for different wrap angles 151 4.24 Pressure in the wedge for different impingement positions 152 4.25 Drainage velocity through inner and outer fabric for different impingement positions 152 4.26 Gap width for different impingement positions 153 4.27 Machine direction velocity in the wedge for different impingement positions. 153 4.28 Pressure and drainage velocity in the wedge for case 8 (no fibre mat build-up). 155 A . l Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric C, 3.7m/sjet velocity) 173 A.2 Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric D, 7.0 m/s jet velocity) 174 A.3 Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric D, 3.7 m/s jet velocity) 174 A.4 Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric E, 7.0 m/s jet velocity) 175 A.5 Fraction of completed drainage as a function of the distance from impingement point for different jet velocities (fabric D, 20° impingement angle) 175 A.6 Drainage velocity as a function of distance from impingement point for differ-ent impingement angles (fabric C, 3.7 m/sjet velocity) 176 LIST OF FIGURES xiv A.7 Drainage velocity as a function of distance from impingement point for differ-ent impingement angles (fabric D, 7.0 m/s jet velocity) 177 A.8 Drainage velocity as a function of distance from impingement point for differ-ent impingement angles (fabric D, 3.7m/sjet velocity) 177 A.9 Drainage velocity as a function of distance from impingement point for differ-ent impingement angles (fabric E, 7.0 m/s jet velocity) 178 A.10 Drainage velocity as a function of distance from impingement point for differ-ent jet velocities (fabric D, 20° impingement angle) 178 A. 11 Drainage velocity as a function of distance from impingement point for differ-ent jet velocities (fabric E, 20° impingement angle) 179 B. l Experimental and computed drainage velocity profiles for jet velocities of 4.8 m/s and 6.1 m/s (impingement angle 20°, fabric C) 181 B.2 Experimental and computed fraction of completed drainage as a function of distance from impingement point for jet velocities of 4.8 m/s and 6.1 m/s (impingement angle 20°, fabric C) 182 B.3 Experimental and computed drainage velocity profiles for 15°, 30° and 45° impingement angle (jet velocity 7.0 m/s, fabric C) 182 B.4 Experimental and computed fraction of completed drainage as a function of distance from impingement point for 15°, 30° and 45° impingement angle (jet velocity 7.0 m/s, fabric C) 183 B.5 Computed drainage velocity profiles for different impingement angles (jet ve-locity 3.7 m/s, fabric C) 183 B.6 Experimental and computed drainage velocity profiles for 20° and 90° im-pingement angle (jet velocity 3.7 m/s, fabric C) 184 B.7 Experimental and computed drainage velocity profiles for 30° and 45° im-pingement angle (jet velocity 3.7 m/s, fabric C) 184 B.8 Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 3.7 m/s, fabric C) . 185 LIST OF FIGURES xv B.9 Experimental and computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 3.7 m/s, fabric C) 185 B.10 Computed drainage velocity profiles for different jet velocities (impingement angle 20°, fabric D) 186 B . l l Experimental and computed drainage velocity profiles for jet velocities of 3.7 m/s and 7.0 m/s (impingement angle 20°, fabric D) 187 B.12 Experimental and computed drainage velocity profiles for jet velocities of 4.8 m/s and 6.1 m/s (impingement angle 20°, fabric D) 187 B.13 Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of jet velocity (impingement angle 20°, fabric D) 188 B.14 Experimental and computed fraction of completed drainage as a function of distance from impingement point for different jet velocities (impingement an-gle 20°, fabric D) 188 B.15 Computed drainage velocity profiles for different impingement angles (jet ve-locity 7.0 m/s, fabric D) 189 B.16 Experimental and computed drainage velocity profiles for 20° and 90° im-pingement angle (jet velocity 7.0 m/s, fabric D) 190 B.17 Experimental and computed drainage velocity profiles for 15°, 30° and 45° impingement angle (jet velocity 7.0 m/s, fabric D) 190 B.18 Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 7.0 m/s, fabric D) 191 B.19 Experimental and computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 7.0 m/s, fabric D) 191 B.20 Computed drainage velocity profiles for different impingement angles (jet ve-locity 3.7 m/s, fabric D) 192 LIST OF FIGURES xvi B.21 Experimental and computed drainage velocity profiles for 20° and 90° im-pingement angle (jet velocity 3.7 m/s, fabric D) 192 B.22 Experimental and computed drainage velocity profiles for 30° and 45° im-pingement angle (jet velocity 3.7 m/s, fabric D). . 193 B.23 Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 3.7 m/s, fabric D) 193 B.24 Experimental and computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 3.7 m/s, fabric D) 194 B.25 Computed drainage velocity profiles for different jet velocities (impingement angle 20°, fabric E) 195 B.26 Experimental and computed drainage velocity profiles for jet velocities of 3.7 m/s and 7.0 m/s (Impingement angle 20°, fabric E) 196 B.27 Experimental and computed drainage velocity profiles for jet velocities of 4.8 m/s and 6.1 m/s (impingement angle 20°, fabric E) 196 B.28 Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of jet velocity (impingement angle 20°, fabric E) 197 B.29 Experimental and computed fraction of completed drainage as a function of distance from impingement point for different jet velocities (impingement an-gle 20°, fabric E) 197 B.30 Computed drainage velocity profiles for different impingement angles (jet ve-locity 7.0 m/s, fabric E) 198 B.31 Experimental and computed drainage velocity profiles for 20° and 90° im-pingement angle (jet velocity 7.0 m/s, fabric E) 199 B.32 Experimental and computed drainage velocity profiles for 15°, 30° and 45° impingement angle (jet velocity 7.0 m/s, fabric E) 199 LIST OF FIGURES xvii B . 3 3 Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 7.0 m/s, fabric E) 200 B . 3 4 Experimental and computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 7.0 m/s, fabric E) 200 List of Tables 2.1 Fabric characteristics for the three fabric samples (courtesy of AstenJohnson). 35 2.2 Polynomial coefficients to describe pressure drop data for different fabrics and the respective resistance coefficients to be used in F L U E N T 38 2.3 Error estimate from a Monte Carlo simulation for the curve fit parameters and the resulting fabric resistance numbers 39 2.4 Polynomial coefficients to describe pressure drop data for different fabrics at angles of 90°, 45° and 22.5° flow angle 46 3.1 Fabric characteristics for input in F L U E N T 60 3.2 Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different jet velocities 67 3.3 Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different impingement angles 70 3.4 Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different furnishes 77 3.5 Basis weight after jet impingement for different furnishes with different fabrics. 78 3.6 Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different furnishes with fabric 2. The numbers in brackets are for fabric 1. . . 79 3.7 Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different jet velocities (GW newsprint furnish with fabric 1 at 10° impingement angle) 82 3.8 Basis weight at the end of the jet impingement region for different jet velocities (GW newsprint furnish with fabric 1 at 10° impingement angle) 83 xviii LIST OF TABLES xix 3.9 Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different impingement angles (GW newsprint furnish with fabric 1 at 15 m/s jet velocity) 85 3.10 Basis weight at the end of the jet impingement region for different impinge-ment angles (GW newsprint furnish with fabric 1 at 15 m/s jet velocity). . . 86 3.11 Test conditions for determination of drainage profiles 96 4.1 Variable selection for roll forming computations 131 4.2 Variable settings for comparison with experimental results 133 4.3 Drainage length, integrated pressure, relative drainage and resulting impinge-ment angle for the computed roll forming cases (i.f. = inner fabric, o.f. = outer fabric) 141 B . l Domain sizes and grids used in the computations for the drainage experiments. 180 Acknowledgement s I would like to thank all those who offered me support and assistance throughout the duration of this work. Special thanks goes to my research supervisors, Dr. Sheldon I. Green and Dr. Richard J . Kerekes, for their thoughtful advice and encouragement. The time they invested and the ideas they contributed over many fruitful discussions were greatly appreciated. I would also like to thank my research committee members, Dr. Carl F. Ollivier-Gooch, Dr. Chad P.J. Bennington, Dr. Ian S. Gartshore and Dr. Greg A. Lawrence, for their conti-nous assistance and support. The staff at the Pulp and Paper Centre deserves special thanks for their frequent and valuable help, and their ever present friendliness. Thanks to Peter Taylor for his help in the design of the experiments and for carrying out all work promptly, even on short notice. My fellow students at the Pulp and Paper Centre and at the Department of Mechanical Engineering have provided friendship and helpful discussions throughout all my time at the University of British Columbia. Peter Ostafichuk and Jens Heymer were a great help in conducting the experiments. Most of all I would like to thank my partner Jorg and my family for their unconditional support throughout this work. xx Chapter 1 Introduction Since its invention in China more than 2000 years ago, the art of papermaking has undergone dramatic changes. For a long time, paper was a luxury commodity available only to the wealthy. Only in 1804 did paper become widely available through the invention of the first continuous paper machine, named after the Fourdrinier brothers. Since then, papermakers have constantly striven to build larger and faster paper machines for a range of different applications. In the original Fourdrinier machine, a jet containing a suspension of water, fibres, mineral fillers and other chemical additives impinges on a moving permeable fabric. Water is drained through the fabric while the fibres and other particles are retained by the fabric in a filtration process that forms a paper sheet. At the end of this "forming" section of the paper machine, the paper passes through a press section, where further drainage is caused by mechanical pressing, and finally into the drying section where the remaining water is evaporated. While the Fourdrinier concept worked and still works well in paper machines running at modest speeds, new concepts had to be found to produce paper at high speeds. One breakthrough here was the invention of the first twin-wire former by David Webster. He proposed a former in which the fibre suspension would be constrained between two fabrics, so that drainage would take place from both sides. Since then several types of twin-wire formers have been built, and efforts to improve their functionality and extend their range of application are still ongoing. 1 CHAPTER 1. INTRODUCTION 2 1.1 History of twin-wire formers Webster filed his first patent on a roll former in 1953. The patent describes the pulp sus-pension fed into a wedge between a rotating roll and a wire wrapping the roll, with the wire free to adjust in the radial direction. His roll former of a later patent is shown in Figure 1.1. This roll former was the base for further developments. Although Webster can be seen as the inventor of modern roll forming, the credits for the first commercial twin-wire machine go to a group headed by Brian Attwood, then with St. Anne's Board Mills in Bristol, England. The early patents for the former resulting from the work, the Inverform, were assigned to R. J . Thomas, then manager at St. Anne's [2]. The first commercial unit of the Inverform, built by Beloit, started up in 1958. Webster's first former concept was subsequently developed into the Papriformer [3] while the Inverform was developed into the Vertiforma by Black Clawson and the Twinverform by Beloit. Other companies followed with their own twin-wire concepts. A good summary of the historical development of early twin-wire formers and develop-ments in forming in general is given in [4] and [5]. A more recent look at developments in twin-wire forming and a summary of the function of drainage elements in different formers is given in [6]. Oct. 2, 19S2 JZa we /?. iVea$T£ft Figure 1.1: Webster's roll former [1]. CHAPTER 1. INTRODUCTION 3 1.2 Twin-wire forming principles Different principles of twin-wire forming exist. Their common feature is a fibre suspension that is trapped between two fabrics. This avoids free surface disturbances that limit machine speeds. Except in tissue forming, where the forming roll is solid, drainage takes place to both sides, increasing the drainage capacity and leaving a less two-sided sheet than in Fourdrinier forming. With the higher drainage capacity and no restriction on the orientation of the fabric as in Fourdrinier forming, shorter and compacter forming zone designs are possible. The two basic concepts in twin-wire forming are roll forming and blade forming. The two have their different advantages and disadvantages ([6],[7]). Roll forming gives a high retention (a measure of how much of the solids material is retained on the paper machine fabric and how much is washed out) combined with high drainage capacities and low drive requirements. A disadvantage of roll forming is a very grainy sheet formation. Blade forming on the other hand offers good formation, but at the cost of low retention and higher drive requirements. The reason for this behaviour lies in the characteristic of the drainage pressure, which will be described later on. In recent years, twin-wire formers have sought to gain the advantages of both roll and blade formers by combining them in sequence. These combined roll-blade formers offer good retention while maintaining the potential to improve formation in the blade forming zone [8]. The performance of combined roll-blade formers depends on the proportion of roll to blade forming [9]. Finding the optimal configuration is not easy and still achieved mainly by empirical trials. A good summary of the roll, blade and combined roll-blade forming principle is given in [10]. The possibilities of controlling important variables such as filler distribution, internal bond strength, layered orientation of fibres and formation with different former configurations are discussed in that reference. 1.2.1 Blade forming In a pure blade former, the headbox jet is fed between two converging fabrics, which pass over fixed blades. The blades are either arranged on a large radius only along one of the fabrics, or more often alternating between the opposite sides of both fabrics, causing a zig-zag path CHAPTER 1. INTRODUCTION 4 of the fabrics. An example of a blade gap former with blades on opposite sides is given in Figure 1.2. The blades introduce a sharp pressure pulse which causes drainage (Figure 1.3). The pressure pulses are strong enough to disrupt fibre floes leading to improved formation. Outer/Backing Wire Figure 1.2: Gap blade former (Black Clawson). Standard Forming Zone 40H -20 I I I l I I I I I 0 3 6 9 12 15 18 21 24 Inches Figure 1.3: Pressure pulses .in a blade gap former [20]. Much attention has been placed on blade forming, resulting in extensive theoretical mod-elling (see [11]-[19] for the newest developments) as well as some experimental investigations ([20]-[24]). Many aspects of the blade forming process can be considered well understood. CHAPTER 1. INTRODUCTION 5 1.2.2 Roll forming In roll forming, the headbox jet is delivered into a nip in which one fabric lies on a forming roll and the other is supported only in tension by a breast roll. A modern roll former developed by Voith is shown in Figure 1.4. The wrap angle of the fabrics around the forming roll in pure roll forming is large. Except in tissue forming, the forming roll surface is usually partly open with a small vacuum applied so that dewatering takes place into the roll through the inner fabric, as well as through the outer fabric. Even dewatering to both sides is possible. The principle of two-sided dewatering is shown in Figure 1.5. The water drained into the roll is kept in there by a small vacuum and released once the inner fabric parts from the roll. The main dewatering force in roll formers is given by the curved, tensioned outer fab-ric wrapping the roll. As noted by Norman [25] a common misunderstanding was that the pressure increase at the entrance of the suspension jet in the nip is generated by the jet de-celeration from the headbox. In fact it is the other way around with the jet being decelerated from the pressure generated by the wire as an outer force. Another incorrect impression was that the dewatering pressure in roll formers is mainly due to centrifugal forces. It can be shown that centrifugal forces have little effect on drainage [26]. However they can lead to instabilities under certain conditions. The topic of centrifugal forces will be revisited in greater detail later in the thesis. Inner/Backing Wire Wire /~c 7-v Figure 1.4: Gap roll former (Voith). CHAPTER 1. INTRODUCTION 6 Figure 1.5: Principle of two-sided dewatering in roll forming [25]. Assuming a zero bending stiffness of the fabric and neglecting centrifugal forces, the dewatering pressure in roll formers is commonly described by Equation 1.1, where R is the roll radius and T is the fabric tension. The equation is not exactly correct, as the local radius of the outer fabric should be used in the equation. However, it is assumed that, because the gap width, which is usually less than 1 cm, is small compared to the roll radius, the roll radius can be used as a good approximation. This is not true at the beginning of the nip where the outer fabric path changes gradually from a straight path to the roll radius. Variations in the pressure therefore may occur at the entrance of the nip as discussed by Ingemarsson [27], with the exact geometry of the impingement zone affecting the pressure. Local variations from the supposedly constant pressure in the forming wedge are also possible because real wires, especially multi-layer wires, might not have a negligible bending stiffness. Figure 1.6 shows the pressure curve in one-sided dewatering measured with a pressure transducer embedded in the surface of the solid forming roll. T (1.1) CHAPTER 1. INTRODUCTION 7 J Jet Impact on Roll Forming Pressure, 0 -kPa -40. 404 +f*— Forming Length Jet Impact on Wire 0 120 240 360 Position on Forming Roll, Degrees Figure 1.6: Pressure measurement in the wedge of a gap roll former [30]. 1.2.2.1 Jet-to-wire speed ratio In papermaking, the jet-to-wire speed ratio is used to influence fibre orientation. Minimum fibre anisotropy is produced if the suspension velocity on the wire equals the wire velocity. In roll forming, this can be achieved only with a jet-to-wire speed ratio higher than unity due to the initial deceleration of the jet. Equal velocity of suspension and wire speed should be reached only after this initial deceleration. An estimate of the necessary jet-to-wire speed ratio for minimal anisotropy can be given by using the Bernoulli equation and the equation for the pressure difference in a roll former to get: In [28] the influence of shear and secondary flow on fibre orientation is discussed. Wahren notes that although it is possible to match suspension to wire speed for any machine speed, this can be done for only one layer of the wedge. Above and below this layer, shear flow influences fibre orientation, leading to the difficulties in producing a non-oriented sheet in roll formers for thick jets (higher basis weights or very low headbox consistency). Roll forming since its invention has received much attention, resulting in many suggestions on how to model the process. Some of the proposed models as well as experimental work are discussed in the following section. (1.2) CHAPTER 1. INTRODUCTION 8 1.2.2.2 Experimental work Some of the early experimental work reported for roll forming was done during the devel-opment of new roll formers. Initial trial runs during the development of the Papriformer are described in [3]. In these trials the outer wire position, and thus the gap size, along the forming zone was measured with a probe. These measurements were used to evaluate the influence of machine variables like jet velocity and fabric tension as well as jet consistency. The trials also evaluated the influence of these variables on the resulting paper quality. Some of the later experimental work focused on determining the fluid pressure in the forming wedge in an attempt to relate pressure to paper quality variables. Wahren et al. [29] first reported the recording of pressure traces in the forming wedge along the forming roll surface for one-sided dewatering by embedding a pressure transducer in the roll surface. The pressure traces were utilized for capacity determinations by using them to calculate the outer fabric shape. They observed the pressure rising to approximately T/R after the initial impingement. At the end of a distance which will be referred to as the forming length, the pressure drops to atmospheric pressure. They concluded that at this point all the free water is drained and the fibre network then supports the outer fabric. Hergert and Sanford [30] in 1984 measured the pressure in a twin-wire tissue machine (one-sided dewatering), also with a pressure transducer in the roll surface. They observed the dependence of the length of the forming zone on the basis weight, machine speed, wire tension and jet-to-wire speed ratio. As would be expected, the forming length increases with increasing basis weight and machine speed, but decreases with wire tension. An interesting finding is that the forming length also varies with the jet-to-wire speed ratio. A plot of forming length as a function of jet-to-wire speed ratio is similar to a plot of tensile ratio against the same abscissa. This dependence of forming length on jet-to-wire speed ratio suggests that fibre alignment influences the drainage resistance of the fibre mat. Martinez [31] measured the pressure in the forming wedge of a roll former with two-sided dewatering by using a pressure probe fixed to a long flexible wire. The wire was fed through the headbox into the forming wedge. Keeping the wrap angle constant at 40°, he measured pressure profiles for various machine variables. He also measured the drainage amounts through each fabric. The pressure traces he measured displayed the same behaviour as CHAPTER 1. INTRODUCTION 9 reported by Hergert and Sanford, rising to a value of T/R towards the end of the forming zone (here the whole gap length). The measured pressure profiles were also used as input to a numerical model, which predicts drainage. For validation, the results from the model were then compared to the drainage measurements. Good agreement was found. In 2001 Gooding et al. [32] undertook a more detailed pilot paper machine study of drainage around a forming roll. They characterized the drainage profile around the roll using different techniques. Photographs were used for a qualitative assessment of the spray drainage through the outer fabric. As reported by commercial paper machine operators, they were able to distinguish between a momentum-driven-drainage spray at impingement and a tension-driven spray further around the roll. Drainage profiles through the outer fabric were quantified measuring the flow with a scoop of small opening area. By traversing the scoop along the outer fabric and assuming that the drainage spray leaves the fabric tangentially to the roll, it is possible to obtain drainage profiles. Finally, the gap size was measured using a strain gauge mounted on a flexible paddle which again was traversed around the outer fabric (see also [3]). From the information about the gap width and the spray drainage, overall and roll drainage profiles were calculated. A typical result for measured drainage profiles is shown in Figure 1.7. 40 -10 0 5 10 15 20 Distance from nip (cm) Figure 1.7: Typical drainage profiles for roll, spray and total drainage around a forming roll. CHAPTER 1. INTRODUCTION 10 Pressure in the gap was also recorded by feeding a capillary tube into the nip through the headbox. It was found that the pressure traces showed a more complex behaviour than the simple T/R with the average pressure being lower than that value. In these studies, fabric type, fabric tension, machine speed, jet-to-wire speed ratio and impingement position were all varied. They found little opportunity to influence roll former performance with the latter three variables, except some possible surface improvement by optimizing the impingement location. 1.2.2.3 Theoretical modelling Besides the experimental work, scientists also have always attempted to model the roll former behaviour. Although the models vary in their complexity, even the simpler ones in general have to be solved numerically. One of the first models was presented by Baines in 1967 [33]. He tried to describe the free jet zone, wedge zone and press zone with simplified equations, assuming quasi one-dimensional flow, where the machine direction (MD) component of velocity is constant over the width of the gap. He further assumed the validity of Bernoulli's equation in the wedge zone and used Darcy's law to describe fibre mat resistance. The resistance factor is described by fibre and suspension properties. Even these simplified equations present a system of nonlinear differential equations that have to be solved numerically. No solution was given by Baines. However, he does give solutions for some simplifying approximations for the pressure profile. With this he estimated the influence of machine variables on forming zone length and undesired backflow. As noted by Baines, many of the difficulties in modelling roll former behaviour result from the problem of finding a sensible description of fibre mat properties. In his simplified approach he considers the fibre mat resistance as constant per unit thickness and therefore dependent only on the fibre mat growth. Fibre mat compressibility is neglected. Different models have been used to describe fibre mat resistance, as will be discussed later. Meyer in 1971 [34] published a detailed review of the hydrodynamics of forming. The work also includes different approaches for one-dimensional modelling of roll forming. A streamline approach for a symmetrical former is outlined in some detail. A second approach CHAPTER 1. INTRODUCTION 11 based on a momentum balance is briefly introduced for an asymmetrical former. No solutions for either approach are given. Koskimies et al. [35] also proposed a one-dimensional model, based on the momentum equation, continuity equation, a force balance relating fabric shape to fluid pressure and a drainage resistance equation. The drainage resistance equation is based on experimental work by Meyer [36]. Numerical results for the model looking at the influence of different machine variables are presented. Unlike earlier models ([33]), no assumption regarding the pressure profile is necessary. Hauptmann and Mardon [37] considered a force balance, Darcy's law, a relationship be-tween drainage resistance and wire gap and the continuity equation in their one-dimensional model. Compared to earlier models, they included the possibility of external suction as well as the effect of centrifugal forces. They were mainly interested in the application of the model to predict operational conditions that would lead to backflow in the beginning of the nip, which causes unsatisfactory operation of the paper machine. Wahren et al. [29] developed a flow model for one-sided dewatering in Webster-type formers, for comparison with their pressure measurements mentioned earlier. They chose an inviscid description of the flow, as did all other models to that date. However, they did note that there might be some zones of the former where viscous effects are significant. Their model is based on a control volume approach, with the control volume being infinitesimal in the machine direction (MD), but using real dimensions in the radial direction, which is possible due to the very small thickness of the wedge. The drainage resistance of the fibre mat is described by an equation fitted to experimental values. This one-dimensional approach leads to ordinary differential equations that can be solved. They found reasonable agreement between their model and the experiments. In 1978, Wahren extended his model to two-sided dewatering [38]. The results show the pressure building up almost instantaneously at the outer fabric, but only gradually at the inner fabric. In Ingemarsson's model [27] the wedge around the forming roll is divided into several small segments, and going iteratively from the beginning of the forming zone to the end, the drainage equations and a force balance for the wire curvature are solved. He uses CHAPTER 1. INTRODUCTION 12 empirical equations for the retention and consistency derived from measurements on the K M W drainage tester. He found that the curvature of the outer fabric is not constant, as is often assumed. Rather the curvature in the beginning is larger and therefore the pressure in the beginning of the forming section is lower than later in the forming section. He predicts backflow in the forming zone relative to the wire due to the very fast drainage in the beginning where mat resistance is low. He concludes that drainage in the initial region should be kept as low as possible, but acknowledges, that because his model does not include inertial effects, it may inaccurately predict drainage for basis weights lower than 25 g /m 2 . Thorp and Baratsch in [39] challenge the common belief that the main drainage force is given by the wire tension. They propose that drainage is mainly caused by the change in momentum, which in other models is often neglected. However, their attempt to apply the momentum equation contains some mistakes, as noted in [26]. Turnbull et al. [40] were mostly interested in MD basis weight variations when they developed a one-dimensional unsteady model for a crescent former (one-sided dewatering). The inviscid model is based on the conservation laws, with the wire being represented as an axially moving medium subjected to fluid loading. When an initial disturbance is introduced, the model shows heavy dampening in the response of the system, however near resonant frequencies significant amplification occurs. This can lead to substantial variations in the M D basis weight profile. In [41], the model of Turnbull et al. is extended to two dimensions: the machine direction and the cross direction. They also included viscosity in the model. With this model, it is possible to examine variations of basis weight in both MD and CD due to different initial disturbances. Jong [42] also used a one-dimensional inviscid model predicting flow characteristics for cases where the wedge shape is either known from paper machine trials or not known but calculated. In this model the fibre mat resistance is based on the specific filtration resistance with values taken from drainage experiments. Jong assumes a linear variation for the MD-velocity and pressure in the wedge from the beginning of the wedge zone to the end. Martinez' model for predicting drainage in roll forming [31] is based on a series of force and mass balances for the forming web. The fibre mat is characterized by fibre properties CHAPTER 1. INTRODUCTION 13 and process conditions, including the effect of web compressibility. The drainage rate is determined from Darcy's law and the web thickness and solidity are also output from the model. The model needs the applied pressure as input, which is determined experimentally. The work is extended by Zahrai et al. [43] to two dimensions. The web growth is expressed using a similarity solution, which is based on the assumption that the pressure on the fabrics is nearly constant and the pressure drop over the fabric is small compared to T/R. In 2000, Boxer et al. [44] published an analytical solution to the Martinez model. They revealed that the Martinez model is in fact based on an implicit assumption of constant pressure and therefore does not even need the experimental pressure profiles as input. With the constant pressure assumption, an analytical solution to the model for the flow rate, the average solidity of the web and the sheet thickness can be found. Particularly the solution for flow rate is useful as a predictive tool. However, the model is still based on compressibility constants which need to be determined experimentally. Using the machine variables from the trials which Martinez conducted as input, the analytical solution shows as good agreement with the experiments as the numerical solution for the flow rate through the outer fabric, and even better agreement for the flow rate through the inner fabric. 1.2.3 The jet impingement region in twin-wire formers Although extensive studies have been done on roll and blade forming, very few models include a detailed view of the very first region of the former, where the free jet coming from the headbox impinges on the fabrics. In fact, most models do not include inertial forces, which due to the high jet velocities in modern twin-wire formers (and a low mat resistance in the beginning) should be dominant in the impingement region. As noted in some of the studies discussed earlier, the impingement region might have a substantial influence on paper qualities such as formation, wire marks and z-profile of fines and fillers, as well as on retention. The aim of this thesis is therefore to provide a more detailed study of the impingement region, focused on impingement in combined roll-blade formers. The earlier mentioned study by Baines [33] was probably the first that considered the impingement zone in roll forming. Baines developed an expression for the contour of the expanding jet before impingement. However, as he considers the free jet zone separately CHAPTER 1. INTRODUCTION 14 from the wedge zone, the expansion at the jet impingement point needs to be known as a boundary condition. Another early study specifically including the jet impingement region was done by Haupt-mann et al. [45] in 1985. They investigated jet impingement into narrow channels formed by two counter rotating rolls, which represents a simplified case of jet impingement in a twin-wire former. They were able to observe flow reversal as the jet loses its momentum while encountering an adverse pressure gradient under certain conditions. This resulted in the development of a pond in the nip, which would be strongly detrimental to formation. Osterberg [46] calculates the pressure pulse at impingement for Fourdrinier and twin-wire formers, assuming that initial drainage is brought about by inertial forces. He applies a simplified momentum analysis using an inertial resistance factor for the fabric and initial fibre mat. The determination of this resistance number will be discussed in Chapter 2. The length of the initial forming zone for which his equations are valid is defined from geometrical considerations as the thickness of the jet divided by the impingement angle. He compares his model with initial pressure pulses measured for different machine and stock conditions. The impingement zone of blade formers is very different from roll formers. However, impingement on the conveying fabric resembles single fabric impingement, which is also the case in roll formers when impingement is preferentially on the outer fabric. For this reason it is instructive to consider the literature related to impingement in blade formers. Johnson in 1992 [47] did an experimental study of the impingement zone of Bel Baie machines, which are pure blade formers. He looked at the influence of impingement conditions on retention, wire marks and fines distribution. Impingement was on the conveying fabric at different distances ahead of the first deflector blade (see Figure 1.8). Johnson measured pressure and drainage profiles, observing impingement pressure peaks proportional to V?et sin f3, where (3 is the impingement angle. His study showed a strong influence of impingement forces on wire marks, fines distribution and two-sidedness. Condon in his review of forming [48] discusses the differences between the impingement zones of single wire formers, roll formers and blade formers. He estimates the impingement angles, which are restricted by the geometry of formers. Assuming tangential impingement on the roll, impingement angles in roll formers are estimated to be between 7 and 12° for CHAPTER 1. INTRODUCTION 15 .375" -rrTTTTTT I " ' 111 « 11.5* from headbox 5.6" Breast roll « 10' to fabric convergence Figure 1.8: Impingement zone of a blade former. typical former configurations, which are higher than is the case in blade impingement. High impingement angles also imply high initial drainage forces. Jong in his one-dimensional study of roll forming [42] included the possibility of the jet impinging preferentially on one fabric and presented an equation that can be used to find the position of the free surface of this flow. He neglects the jet expansion taking place before impingement and therefore uses the original jet height as a boundary condition. As a second boundary condition, he also assumes that the drainage velocity at the impingement point equals the vertical component of the jet velocity. To further simplify he assumes that the velocity in fabric direction is constant and equals the jet velocity. He shows results for an impinging water jet as well as a suspension jet, and also includes the effect of fabric tension. The calculated free surface positions can be used for input in his wedge model, which was discussed earlier. Audenis [49] looked at analytical and numerical solutions for an inviscid jet impinging on a single paper machine wire. He included the flexibility and porosity of the wire, but only in separate cases. Impingement on a flexible, but impermeable wire was solved analytically for different jet angles, using the pressure profiles obtained for impingement on a rigid imper-meable wire. In the case of a permeable, but rigid wire, his numerical analysis is based on the Baiocchi transformation. Only for jet impingement at an angle of 90° does he consider the low permeabilities of paper machine fabrics, where most of the fluid is not drained but leaves the modelled domain flowing on top of the wire. For impingement at different angles he solves for the flow for high permeabilities and suction. These constraints ensure that all fluid is drained through the fabric in the modelled impingement region and no fluid has to CHAPTER 1. INTRODUCTION 16 leave the domain on top of the fabric. An attempt to investigate the influence of jet impingement on fines retention and drainage was done by Mitsui [50]. He used the laboratory twin-wire former developed by Hammock and Gamier [51] to study the impingement region under different conditions, varying jet velocity and impingement angle. Due to geometrical restrictions of the former, impingement is always on the outer wire at higher angles than used in industry. Therefore drainage velocities are kept lower to match the component perpendicular to the fabric to values used in the industry, based on the assumption that the magnitude of drainage velocity is the major influence on drainage. However, he does not specify the distance ahead of the nip where the jet impinges and the distance over which drainage measurements were taken, which makes an evaluation of the results difficult. He found that varying the impingement angle did not significantly influence fines retention, while varying the jet velocity did. He concludes that the impingement pressure (which is much more influenced by the jet velocities than the impingement angles) and the difference between the jet velocity component in the machine direction and the wire velocity most influence fines retention. His approach of matching only the normal velocity to industrial values thus seems questionable. Chapter 2 Drainage resistance of the fabric and fibre mat A model to describe the early forming process needs a means of representing the drainage resistance offered by the fibre mat and fabric. Describing the fibre mat resistance is par-ticularly difficult. Different methods to measure or model resistance have been developed, but a consensus on the best approach has not been reached. In this chapter, some aspects of modelling of fibre mat and fabric resistance are described and some of the work done by other researchers is reviewed. The approach chosen for this work is then presented. Some experimental measurements of fabric resistance are also presented. 2.1 B ackgr ound Finding a suitable description for the drainage resistance of the fabric and fibre mat is one of the most fundamental problems in modelling the wet end of a paper machine. Deter-mining the fabric drainage characteristics should be possible with suitable experiments, or even theoretical models if the fabric structure is known. However, the complexity of the fabric structure makes such direct modelling difficult. Experimental correlations should be more easily obtainable. The drainage resistance of the fibre mat is much more difficult to determine, as the fibre mat resistance depends on the properties of the formed web. The web in turn is a complex heterogeneous structure, which is not only defined by the geometry 17 CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 18 of the individual fibres, but also by the process conditions. The deposition of the fibres rel-ative to each other will depend on drainage characteristics and shear forces, and the density of the mat will depend on the applied pressure. Even the single fibre properties are not constant, but defined by a distribution of fibre length, diameter, coarseness, etc. Fines and filler retention and deposition add to the complexity of the problem. Thus one can see that a direct description of fibre mat resistance based on fibre properties and web structure is not possible, as the latter is not known. However, mat resistance to flow can be described by some simple approaches. The most common is to express fibre mat resistance by considering it as a homogeneous, incompressible material. Darcy's law relates the volumetric flow rate Q through a porous medium to the pressure drop A P , the thickness of the medium L, the cross sectional area A, and a factor called the hydraulic conductivity K: AP Q = KA~— (2.1) E This can also be written as an equation for the pressure drop, utilizing the superficial velocity v defined as Q/A: AP = ^Lv (2.2) The hydraulic conductivity depends on the fluid properties in the form of the viscosity and on the pore structure of the web, expressed as the permeability k: K = - (2.3) A4 Using this expression leads to Darcy's law in a form well known in papermaking appli-cations: A P = (2.4) k In most papermaking applications, the permeability k is assumed to be constant over the thickness of the web, although at least for higher basis weights, when more than one layer CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 19 of fibres have been deposited under different conditions, this is probably not true. However k can be taken as an average over the mat thickness. Strictly speaking, Darcy's law is only valid for creeping flow with a Reynolds number (based on a characteristic diameter of the fibre and the superficial velocity) of less than about one. However, it seems to work reasonably well for forming conditions in actual papermaking, where the Reynolds number might be somewhat higher. For high Reynolds number flow, the equation must include an inertial term, leading to the expression A P = ^av + pbv2 (2.5) which is also known as Forcheimer's equation [52]. The permeability from Darcy's law is here written as a viscous resistance factor a; b represents the inertial resistance factor. Forcheimer's equation is widely used to describe fabric resistance, as will be discussed later on. Researchers generally agree that the inertial resistance is of little influence as soon as a noticeable fibre mat builds up, and therefore it is usually not used in describing the fibre mat resistance. The difficulty in using Darcy's law to explain drainage through a fibre mat lies in finding a suitable expression for the permeability k. Attempts have been made to measure it directly, where it is often expressed as the specific filtration resistance (SFR). Other approaches are to develop empirical models for the fibre mat permeability, or to describe it by theoretical models based on fibre properties and process conditions. 2.1.1 Previous experimental work on fibre mat resistance Traditionally, the Canadian Standard Freeness (CSF) test or the Schopper-Riegler (SR) test were used to describe drainage properties of pulp suspensions by measuring the drainage of water through a pulp pad. The results of such tests are only useful for the comparison of different pulp suspensions, but cannot really be used to describe drainage in a paper machine, as the results depend strongly on the test apparatus geometry. Furthermore, the test conditions (low flow velocity, low pressure and long drainage time) do not represent the drainage conditions in the forming section of a paper machine'. It is known that the fibre mat properties depend on the process conditions, which is why newly developed tests CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 20 try to simulate the drainage conditions, especially the higher drainage velocities, of forming sections. Instead of measuring only qualitative numbers, many such tests aim at providing resistance numbers which can be used in theoretical models such as Darcy's law. A good review of some of the newer approaches to measure drainage characteristics is given in [53]. Often measurement results are expressed as the aforementioned SFR. The SFR can be defined as "the pressure differential in metres of water to produce a flow rate of 1 m 3 /s per m 2 through a filter cake weighing 1 kg per m 2 , and porosity equivalent to that as deposited under the actual filtration process" [54]. The SFR can be substituted for the permeability factor in Darcy's law by applying a mass balance to the solids as shown by Ingmanson and coworkers ([55]-[57]). The thickness of the porous medium can be expressed as the basis weight BW [kg/m2] divided by the mass concentration cmat [kg/m3] of the mat. Darcy's law becomes: BW AP = vvf^— (2.6) The factor l/kcmat represents the SFR, so AP = nvSFRBW (2.7) The SFR can then be determined by recording pressure and drainage velocity (or drainage rate) at known basis weights. When trying to determine the SFR one has to keep in mind though that it is not only a property of the pulp which is used, but also of the process conditions. A good overview of how process variables influence the SFR can be found in Pires et al. [58]. They used a modified Britt's jar to measure the drained volume of a pulp slurry as a function of time. Expressing Darcy's law in terms of inverse drainage rate and drained volume leads to At ^slurry VSFR AV ~ A2AP { ' } with inverse drainage rate [s/m3] A P pressure drop across the fibre mat [Pa] A cross sectional area of the fibre mat [m2] CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 21 csiurry fibre concentration of the slurry [kg/m3] V accumulated volume [m3] fj, fluid viscosity [Pa-s] SFR specific filtration resistance [m/kg] Plotting the inverse drainage rate against the drained volume should yield a linear curve if the pressure during the test is kept constant. The slope then represents the SFR. This is based on the following assumptions: • The flow is laminar. • The pressure gradient across the mat is constant (and therefore the SFR is also constant across the mat). • The process is a filtration process rather than a thickening process, so that the concen-tration of fibres in the slurry remains constant during the test (see Figure 2.1). This assumption is quite common for describing the forming process in paper machines and was discussed by Parker [59]. TVTfTT Figure 2.1: Filtration versus thickening process [59]. Pires et al. measured the SFR for different pulps at different pressure drops, different consistencies, different viscosities (changed by varying the temperature) and different tur-bulence levels (varied by stirring the suspension at different rotation speeds). They found CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 22 that the viscosity, at least in the range tested, had no influence on the SFR, but all other variables did. The SFR increased with increasing pressure drop and increasing consistency. They developed an empirical relationship for the SFR as a function of pressure drop and con-sistency for a given pulp. They did not try to quantify the effect of turbulence on the SFR, but observed that increased turbulence decreased the SFR. They found the pulp properties had the biggest influence on the SFR. In particular, refining had a large effect on SFR, which is reasonable as refining changes the surface area and geometry of the fibres. For different refining states represented by freeness values between 700 and 100 (CSF), the SFR increases from about 5 • 107 to 3 • 10 1 0 m/kg. Although the relationship found for the SFR seems to be useful for the theoretical mod-elling of drainage in a forming section, one should remember that it is empirical and not based on theoretical reasoning. For example, the influence of pressure may be due to the compressibility of the mat, which is important only at higher basis weights. Therefore one should be careful when using the equation outside the range of process conditions for which it was determined. Springer and Kuchibhotla [65] based their study, in which they look at the influence of fillers on the SFR, on the work of Pires et al. They looked at the influence of filler grade, filler fraction and retention on the SFR at constant pressure drops. Besides the influence of pressure drop and slurry consistency, they also found a strong influence of the filler concentration and freeness of the stock (where the freeness represents the effect of fines as well as of the used filler type). They extended the empirical model by Pires to also include the filler concentration and freeness. However, the new model included 10 constants which all have to be determined from experiments for every different stock. The model therefore seems to have limited use in predictive models for drainage in a paper machine. Also, it suffers from the same limitations as the one by Pires et al. Recognizing the superiority of the SFR to other parameters such as CSF or SR, Springer et al. [66] proposed an on-line method to monitor SFR. Another study that examines the influences on the SFR experimentally was done by Mantar et al. [53]. They used a modified Dynamic Drainage Jar (DDJ) to estimate the SFR based on the same theory as above, measuring the filtrate volume versus time at a constant CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 23 pressure drop. They looked at the influence of consistency and fines content on the SFR, using different pulps. In addition to the influence of slurry consistency observed before by Pires et al., they found an influence of the basis weight on the SFR for beaten chemical pulps and mechanical pulps, but not for unbeaten chemical pulps. They attributed the increasing SFR with increasing basis weight for these pulps to the presence of fines, which will experience an increase in retention at higher basis weights, which causes a higher SFR. Herzig and Johnson [60] measured the filtration resistance in a closed flow loop with pulp injection at constant drainage rates (thus constant flow velocities). High flow velocities up to 2 m/s are possible in this device. Different basis weights can be simulated by injecting different amounts of pulp. Plotting the measured pressure drop versus basis weight for different pulps at a constant approach velocity results in a linear curve, which implies a constant SFR for the basis weight range studied (0-20 g/m 2). This suggests that the pressure drop at low basis weights has little influence on the SFR. However, some discrepancy can be found in tests with different approach velocities (done for only one furnish), where the linear increase of pressure drop against basis weight is not obvious. Herzig and Johnson attributed this to the slow response of the flow loop to sudden changes in basis weight. Jong et al. [61] also studied fibre mat resistance in a flow loop with pulp injection. However their flow loop is neither at constant pressure drop nor at constant drainage velocity. Using an LWC furnish, they found three distinct regions of behaviour as a function of basis weight. In the first region they found that the pressure drop increased linearly with basis weight. The velocity in the loop decreased. The second region showed a constant pressure drop while in the third region it slightly diminished. A plot of the SFR revealed the same behaviour with a linearly increasing SFR in the first region up to 20 g /m 2 , followed by a constant region up to 40 g /m 2 and then a slightly decreasing SFR. The study was extended to other furnishes [42]. While a newsprint furnish also showed an increasing SFR value with higher basis weights in the first region, the SFR of an unbleached softwood furnish remains constant over the whole basis weight region. This agrees with observations by Mantar et al. [53]. Wildfong et al. [62] developed a rapid drainage tester to determine fibre mat resistance at forming conditions and validated it with data from a pilot paper machine [63]. A certain CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 24 volume of fibre suspension at headbox consistency is drawn through a fabric by a vacuum pump. The drainage time is on the order of less than two seconds. The pressure differential across the fabric, the elapsed time and the drained volume are recorded at high frequency. They calculate a viscous resistance coefficient based on Darcy's law in the following form: APmat = a{t) • L(t) • v(t) (2.9) with a(t) viscous resistance coefficient [kg/m3s] L[t) mat thickness [m] L(t) is determined using a mass balance, i. e. , it is calculated using the drained volume, consistency of the suspension, fibre density, cross sectional area of the test device, and the retention coefficient and porosity of the fibre mat. The last two variables were assumed constant during the drainage process and typical numbers were used. The authors found the viscous resistance coefficient increased exponentially with basis weight. If expressed in the form of a SFR, the resistance would still increase with basis weight, as was suggested before. To explain the increasing resistance, they speculated that for the low basis weights they investigated (10 - 55 g/m 2), inertial effects might be important and therefore the transitional model by Forcheimer should be used rather than Darcy's law. However, calculations of the inertial resistance coefficient for the fibre mat resulted in values near or less than zero. They concluded that the increasing resistance is not due to inertial forces at high velocities, but rather needs to be explained from the dynamics of the drainage process. They also note that the behaviour is not an effect of choosing a certain retention coefficient or permeability, as changing these values would shift the resistance versus basis weight curve left or right, without changing the slope. On studying different furnishes, they found a larger increase with basis weight of the resistance factor for newspaper furnishes than for fine paper furnishes. The viscous resistance coefficient values of different furnishes relative to each other were not consistent with CSF values, which emphasizes the point that the CSF values are not suitable to describe the drainage behaviour of pulp suspensions in the forming section. In a later paper, Wildfong et al. [64] discuss the possibility that the measured varying resistance is due to varying retention coefficients or porosities with increasing basis weight. CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 25 Such variations could be due to fines washing and compression. Using one furnish with different fines fractions, they found that at higher fines contents the increase in resistance with basis weight is much more pronounced than at lower fines content. This becomes especially obvious at basis weights higher than 10 to 20 g/m 2 , where the fines retention becomes important. At even higher basis weights (>40 g/m 2) the viscous resistance coefficient appears to level out for furnishes with high fines content. They also looked at a furnish where all the fines were removed. The viscous resistance coefficient still increased with basis weight, but only marginally. They attributed the latter increase to the compressibility of the mat and concluded that pore clogging by fines retention was by far the more influential factor than mat compressibility. They also looked at the relative importance of compression and fines retention by varying the drainage velocity (vacuum level). At higher drainage velocities, compression should be higher but retention lower due to the higher drag forces. Therefore if compression is the main factor, resistance values should be higher with higher velocities, while if fines retention is more important, resistance should be lower at higher velocities. Although they used a furnish with very low fines content, thereby already minimizing the effect of fines retention, they found lower resistance values at higher drainage velocities, indicating that fines retention is indeed more important than compressibility, at least at these low basis weights. Ingmanson [67] carried out an earlier study showing the importance of keeping all drainage conditions similar to machine conditions when determining the SFR. A comparison of SFR values measured in a laboratory sheet former were much higher than those on a Fourdrinier machine at the same pressure drop. He concluded that the duration of the pressure applica-tion (drainage time) seems to be of importance as well. Andrews and White [68] also noted the difficulties in using filtration resistance data measured in laboratory devices for the prediction of drainage on Fourdrinier machines. They suggest that the viscous resistance might not be sufficient to describe drainage at high velocities. They developed a constant drainage rate tester and measured pressure drop as a function of basis weight for velocities up to 1 m/s. In their device non-linear behaviour was apparent only at basis weights >20 g/m 2 . As they were using a stock free from fines or fillers, the non-linear behaviour should be due to compression, which is therefore likely only CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 26 important at higher velocities and basis weights. Their study and an earlier one [69] that looked into the possibility of an additional inertial fibre mat resistance at high velocities (up to 1.6 m/s), found that inertial effects should be considered for drainage velocities higher than 1 m/s. One should bear in mind that in most applications these high velocities occur only over a very short distance in the beginning of the forming zone. The fibre mat builds up quite quickly, which reduces drainage velocities to below this value. Therefore the common assumption that inertial effects are negligible when modelling the fibre mat resistance seems justified. The latest interest in measuring fibre mat resistance involves introducing defined shear to simulate closely real forming conditions. As discussed above, it is known from earlier work which involved drainage testers with stirring of the suspension that shear influences resistance, but it was not possible to quantify the influence. Such quantification would be possible if shear were introduced in a defined way. Attwood and Jopson [70] tried this by moving a long fabric in one direction while measuring drainage. However, their apparatus limited wire speeds to only around 35 m/min, which are much lower than in real forming. Another promising approach was presented by Green et al. [71]. They developed a drainage tester where the fabric covers only a thin outer annulus of a cylindrical device. By rotating the fabric it is possible to introduce a defined amount of shear. Work on this device is ongoing. First measurements of drainage resistance under known shear conditions were presented by Paradis et al. [72]. Wildfong's rapid drainage tester was modified by adding a rotating inverted cone to generate known shear rates on the wire. They found increasing resistance values with increasing shear rate up to a certain point. At even higher shear rates the drainage resistance drops sharply. They attributed the increasing drainage resistance at low shear rates to the ability of the shear helping to form a compacter mat. At high shear rates however, the shear stress might become too large and therefore disrupt the fibre mat or lift the fibre mat from the fabric, leading to a steep drop in drainage resistance. Trying to approximate shear conditions in commercial top-wire units, they chose a shear rate of 0.3 Pa for the remaining experiments, based on a numerical model by Franzen [73]. Using different headbox samples, they found further proof for their earlier conclusion that the fibre mat CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 27 resistance is strongly dependent on the fines and filler content of a given fibre suspension. 2.1.2 Theoretical models to describe fibre mat resistance Besides the extensive experimental work done on fibre mat resistance, different theoretical approaches to relate the fibre mat resistance to properties of a porous medium exist as well. The most popular of these is probably the use of the Kozeny-Carman equation relating the permeability factor of Equation 2.4 to the porosity of the medium and the specific surface area of the fibres. To get to the Kozeny-Carman form of Darcy's law, it is assumed that the medium consists of a series of continuous channels. By using fibre matrix theory and using the concept of superficial velocity [74] one can express the permeability as k = 7 (2.10) /CS 2 (1 - e)2 K ' with e porosity of the medium SV specific surface per unit volume of fibres [m 2/m 3] K, Kozeny factor [1/m] The Kozeny factor is commonly assumed to be a function of the porosity. Ingmanson and coworkers ([69], [57]) found the empirical equation 3.5e3 K = 1 + 57(1 - ef (2.11) /-, \0.5 ( 1 - e ) for water flow through uniform synthetic fibre pads. One difficulty in applying the Kozeny-Carman equation to pulp mats lies in estimating the specific surface area SV. Due to the variability of the surface geometry of individual fibres, it is not possible to use simple geometrical considerations to find SV. SV usually needs to be evaluated with separate flow experiments. Some of the theoretical approaches include also a description of the inertial resistance factor b of Forcheimer's Equation 2.5. One suggestion to relate b to fibre and mat properties is given in [75], where the following equation is proposed: CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 28 b = V(l-e)^ (2.12) In this equation the factor b' still needs to be evaluated. For uniform synthetic fibre mats with a basis weight of > 20 g /m 2 the following correlation was found by Ingmanson and Andrews [69]. b' = 0A^JC{e) (2.13) A discussion of the application of their theories to wood pulps is also given in that work. Sayegh and Gonzalez [76] tried to include the compressibility of fibre mats in a theoretical approach using the Kozeny-Carman theory. Using the Kozeny expression, the SFR becomes: They stated that only the bed height L and the void fraction are influenced by compres-sion and can be related to the pad strain 7 = ^ (V= pad volume). They arrived at the relationship SFR = SFR0-^—% (2-15) where the zero-subscript refers to zero-compression and /CS 2 L0 (1 - eof BW el They used their theory to model the SFR with a Maxwell element (spring and dashpot in series), which describes the relation between pad strain, pressure and time. They were able to break the SFR down into a zero-pressure term, a pressure history term and an instantaneous pressure term. Using constant flow drainage tests to obtain model parameters, they related the SFR to the CSF and were able to predict freeness reasonably well. A second approach to describe the mat permeability is by using drag theory for flow around solid objects, in this case the fibres which make up the porous medium. The fibres dissipate energy as a result of viscous drag, which can be estimated from the Navier-Stokes equations. The sum of the drag equals the flow resistance K CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 29 One of the important investigations using drag theory is Happel's work [77]. He solved the Navier-Stokes equations for creeping flow parallel and perpendicular to circular cylinders. As in the Kozeny-Carman equation, the resulting equations are functions of the porosity and specific surface. Meyer [78] compares the theoretical results from Happel with experimental results by other researchers and concludes that for common values of initial consistency and compres-sion the multi-body drag model seems to be more useful than the capillary approach. He uses Happel's equations together with a power law for compression, an empirical retention equation, and conservation of mass to develop general equations for filtration and discusses their application to fibre mats at low basis weights. 2.1.3 Fabric resistance and fabric/mat interaction The pressure drop in the forming section is not only caused by the fibre mat resistance, but also by the resistance of the fabric. Once a substantial fibre mat has built up on the fabric, the fabric resistance is usually very low compared to the fibre mat resistance, but at lower basis weights the fabric resistance can be of considerable influence and therefore should be included in any approach to model the forming section. In general the pressure drop caused by fabric and fibre mat are considered separately and the contributions are assumed to be additive: APtotal = Pfabric + APmat (2-17) If this is true, the fabric resistance can be determined separately from the fibre mat resistance. As the drainage velocities in the beginning of the forming zone, where the fabric resistance is of importance, are high, inertial forces need to be considered in a description of the fabric resistance. Therefore Forcheimer's equation is commonly used with constants specific to the fabric: APfabric — ^fabricV + pbfabricV (2-18) The inertial and viscous resistance coefficients a,fabric and 6/a&™c can be determined from CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 30 flow measurements through a fabric with pure water. Open or closed flow loops are most often used to measure the pressure drop over the fabric as a function of flow velocity. The data can be fitted to the above equation (see for example [60] and [62]). In [60] it was additionally attempted to relate the resistance coefficients to the air permeability of the fabrics which were included in the study, which resulted in a power law relationship for both coefficients. The constants in these empirical functions were determined from the experiments, but could be related to the geometry and fabric design. Sometimes the viscous resistance is neglected, on the grounds that inertial forces are dominant at the start of forming [61]. For the fabric, it should also be easier to express the resistance based on theoretical con-siderations, as the structure of the fabric is known and can be assumed to be incompressible. One theoretical approach was developed by Ingmanson and coworkers [79], where the fabric resistance is related to the specific surface, the porosity and dimensionless constants which should be common to all fabrics. They then converted the relationship to dimensionless numbers, using a modified Reynolds number. Some researchers considered the possibility of interaction between the forming fabric and the fibre mat, by adding a third term to the total pressure drop, representing an additional pressure loss by fibre/fabric interaction: ^-Ptotal — 'fabric + &Pmat + ^Pinter (2.19) Correcting for this interaction seems reasonable, although equation 2.19 is purely em-pirical. Andrews and White [68] studied the flow through thin fibre mats and found that fibre/wire interactions seemed much more pronounced for wood fibres than for the synthetic fibres studied earlier and therefore the mat/fabric interaction issue should be addressed further. Meyer [36] compares experimental studies to a theoretical total pressure drop and finds that the experimental values are usually higher than estimated from Equation 2.17. This might be an effect of fibres getting trapped within the wire structure and thereby changing the fabric characteristics. The effect of fibre/wire interaction is obvious in particular for thin fibre mats. He formulates interaction factors to account for the interdependence. These factors appear to vary with the fabric type. CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 31 A very different approach to describe the fabric and fibre mat resistance was used by Osterberg [46]. He determined an impulse resistance number ip of the fabric together with the initial fibre mat. He used this number in his model to describe the initial impingement mentioned earlier. The impulse resistance number is defined as the integral over time of the perpendicular force F acting on the fabric divided by the impulse of the jet perpendicular to the fabric. The resistance number was determined by dropping a cylinder fitted with a fabric sample into a tank filled with either water or fibre suspension. The force and impulse were recorded as a function of time. A typical trace of the force is shown in Figure 2.2. Falling time Point of impact-^ ^ Reactive force ^ Figure 2.2: Typical oscilloscope trace of the force on the fabric at impingement during simulated jet impingement [46]. The impulse was evaluated as a function of the penetration depth of the fabric into the suspension, which would correspond to the jet thickness in impingement. The approach avoids any problems with fibre/fabric interactions by modelling both resistances as one. However, the resistance measurements are applicable only to this specific model. 2.2 Choosing resistance descriptions for the model As can be seen from the discussion above, there is no agreement yet about what is the correct description of fibre mat and fabric resistance. In the specific application investigated in this thesis, the following descriptions were chosen: • The pressure drops caused by the fabric and the fibre mat are purely additive. Any influence of the fibre mat on the fabric resistance or vice versa is neglected. Although CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 32 some evidence can be found in the literature that the interaction between fibre mat and fabric, especially at lower basis weights, is significant, there seems to be no widely accepted formulation to quantify this influence yet. • The pressure drop caused by the fabric includes inertial as well as viscous contributions. Different descriptions of the fabric resistance were tried early in the project and it was shown that neither of these contributions should be neglected. The chosen fabric resistances vary throughout this work and will be mentioned in the respective sections where they were used. The fabric resistances used in the computations are taken from the literature. However, for the validation experiments it was necessary to measure the fabric resistance for the specific sample fabrics which were used. This is described below. • The pressure drop through the fibre mat was assumed to be caused by only viscous forces, as is widely accepted. Darcy's law using a specific filtration resistance was used (Equation 2.7). This description has the advantage that the resistance depends on the basis weight and not on the thickness of the mat. The basis weight can be calculated from drainage data, using certain assumptions regarding retention and the mode of fibre deposition. A constant SFR was chosen for the model. Although the SFR is known to depend on different variables, as discussed above, no general description is presently available. If a variable SFR were to be used, it would have to be based on empirical relations specific to certain systems and fibre suspensions. Also this work is primarily focussed on fibre mat build-up at jet impingement, where the basis weight is relatively low. Under such conditions certain variables like the compression of the fibre mat should only slightly influence resistance, although fines retention, which is not considered in this model, might be a significant issue. 2.2.1 Experimental determination of fabric resistance: flow per-pendicular to the fabric In Chapter 3.2.2, a comparison of experimentally determined drainage curves to computa-tional ones will be described. To do the comparative computations, the specifications of the CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 33 fabrics tested needed to be known. The manufacturer provides only air permeability data, which are not useful in theoretical drainage models. Therefore it was necessary to determine the fabric characteristics experimentally. Fabric resistance was determined by measuring the pressure drop over the fabric in an experimental flow loop similar to the ones used by other researchers. The loop was filled with water and flow velocities typically found at impingement were used. The pressure drop versus velocity data is fitted to Forcheimer's equation to determine the inertial and viscous resistance of the fabric. 2.2.1.1 Experimental set-up t ' An existing open flow loop was used for the measurements. The flow loop (Figure 2.3) consists of a holding tank, a centrifugal pump and 3/4" diameter piping. The flow loop also includes a pitot tube installation connected to a manometer that can be used to read the flow velocity. The reading from the pitot tube was taken as the average flow velocity. This is not entirely correct, but as the flow in the loop is turbulent (the Reynolds number ranges from 4 • 103 to 4 • 104 for flow velocities from 0.2 m/s and 2 m/s), the error introduced by this approximation is small, in particular compared to the error in reading the manometer. To determine the pressure drop through a fabric an additional section was installed downstream of the pitot tube that included the fabric and differential pressure transducers upstream and downstream of the fabric. The fabric was mounted between two steel rings and held between two flanges (inner diameter also 3/4") which were sealed to the outside to prevent any water leakage. If water is pumped through the flow loop, the fabric causes a pressure loss which can be determined with the two differential pressure transducers. The flow loop can be operated at different flow velocities with the maximum flow velocity given by the maximum pump head and the resistance of the fabric. 2.2.1.2 Test conditions Measurements were done for three different fabrics at flow velocities ranging from 0.2 to 1.3 m/s (the maximum flow velocity is given by the fabric resistance), which is in the range CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT Figure 2.3: Schematic of flow loop to determine fabric characteristics. CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 35 of drainage velocities found in the impingement region. The fluid was filtered pure water. The air permeabilities of the three fabrics ranged from 415 to 585 f t 3 /min. See Table 2.1 for the fabric specifications. The water was at a temperature of 20.0°C. For every fabric the pressure drop was measured at 10 or 11 different velocities. Table 2.1: Fabric characteristics for the three fabric samples (courtesy of AstenJohnson). Fabric Product code Air permeability [ft3/min] at 0.5"H 2 0 Mesh [1/inch] Number of layers C D717 525 85 3 D D493 415 85 2 E D061 585 87 1 2.2.1.3 Results The averaged pressure drop versus velocity data for all three fabrics are shown in Figure 2.4. The standard deviation of the pressure data was low, varying from around 50 Pa for lower flow velocities up to 150 Pa for the high velocities. As expected, the single-layer fabric with its high air permeability causes the lowest pressure drop. The double-layer fabric causes a higher pressure drop than the three-layer fabric, which agrees with the lower air permeability of the double-layer fabric. The pressure loss at 1 m/s reaches between 10 and 25 kPa, which is in the same range as found by other researchers. At low flow velocities, the pressure drop is approximately linear, indicating that only viscous forces are important. At higher velocities where inertial forces are also important the pressure loss becomes non-linear. The pressure loss data can also be represented in the form of a dimensionless pressure coefficient: C> = T^ <2-2°) The pressure coefficient versus velocity data is shown in Figure 2.5. Again it can be seen that at lower flow velocities viscous forces are more important which causes the pressure CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 36 35 30 25 20 15 10 5 fabric C measured fabric C curve-fit fabric D measured fabric D curve-fit fabric E measured fabric E curve-fit R =0.9995 R 2 -09?J51 J * ^ * - - - - - , R - 0.9977 ^.lili''i''.L-B--'-—1-4 PS - ' -H 0.2 0.4 0.6 0.8 1 Velocity [m/s] 1.2 1.4 Figure 2.4: Pressure drop data for different fabrics. coefficient to drop with increasing velocity. At higher velocities inertial forces are dominant, which causes a nearly constant pressure coefficient. From the pressure drop data, the resistance coefficients of Forcheimer's equation can be determined. A second degree polynomial was fitted to the pressure versus velocity data: AP = K0 + Kxv + K2v2 (2.21) The constant K0 was forced to be zero, as for zero velocity the pressure drop also has to be zero. The model which will be developed in the next chapter is based on the commercial C F D code F L U E N T , which uses the following form of Forcheimer's equation to describe porous media: A P = yvAn + \p C2 v2An (2.22) k 2 k viscous permeability coefficient [m2] C2 inertial resistance coefficient [m_ 1] . A n thickness of porous medium [m] CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 37 80 70 p. 60 r-50 u a ' o o e 40 ^ 30 20 10 fabric C —>— fabric D — - x -fabric E o-0.2 0.4 0.6 0.8 1 Velocity [m/s] 1.2 1.4 Figure 2.5: Dimensionless pressure coefficient as a function of flow velocity for different fabrics. By comparing the formulations 2.21 and 2.22, the viscous permeability and inertial re-sistance used in F L U E N T can be expressed as: k = - ^ A n (2.23) C72 = ™j± (2.24) p A n where the fabrics will be modelled with a thickness of A n = l mm. Table 2.2 shows the results of the curve fit. Figure 2.4 shows both the measured data and the respective curve fits, which agree very well with the measured data. It is interesting that fabrics C and E, although very different in air permeability, have approximately the same inertial resistance. Fabric D with the lowest air permeability also has the highest inertial resistance. Fabric E with the highest air permeability also has the highest viscous permeability, but it is not much higher than the one of fabric D. Also, the viscous permeability CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 38 of fabric D is higher than the one of fabric C, although fabric D has a lower air permeability than fabric C. Therefore both the inertial resistance and viscous permeability coefficients show that air permeability is not sufficient to describe the drainage characteristics of a fabric. Fabric manufacturers note that this is in particularly true for multilayer fabrics, where the bottom layers can have a big influence on the air permeability but only a small influence on the drainage rate. Air permeability is therefore now mainly used as a quality control tool by fabric manufacturers. Table 2.2: Polynomial coefficients to describe pressure drop data for different fabrics and the respective resistance coefficients to be used in F L U E N T . Fabric K2 [103-kg/m3] K, [103- kg/(m2s)] c2 k C 7.79 3.77 15600 2.65 • 10" 1 0 D 14.75 2.01 29500 4.98 • 10" 1 0 E 7.83 1.91 15700 5.24 • 10~ 1 0 Error discussion The main source for error in the measurements is the reading of the velocity. While the digital pressure reading is quite exact, the velocity reading off the manometer scale can introduce bigger errors. Although an estimate of the reading error is mainly subjective ([80]), it is often given as a standard deviation calculated by the smallest scale unit divided by \ / l 2 , which seemed a good estimate for this specific case. For the 1 mm scale used with the manometer this corresponds to a standard deviation of 0.288 mm which equals approximately ± 0.075 m/s. At low flow velocities this deviation can be significant and therefore the possible influence on the resistance numbers should be discussed. Typical error bars, based on this analysis, are also shown in Figure 2.4. As the error in reading the velocity cannot be studied directly, the Monte Carlo method [81] was used to simulate the error from the known standard deviation. Random numbers are created and used to determine x-values (flow velocity) and y-values (pressure drop) from the respective cumulative probability distribution functions of each measurement point. From CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 39 the x/y-data series, a curve is then created to which the polynomial (Equation 2.21) is fitted. More curves are then created (in this case 30) with new random numbers. The variance in the curve fit parameters can be interpreted as the experimental error in these parameters, and can be translated into an error for the resistance numbers used in F L U E N T . Table 2.3 gives the averaged curve fit parameters and their standard deviations estimated with the Monte Carlo simulation, and the resulting fabric resistance numbers. Table 2.3: Error estimate from a Monte Carlo simulation for the curve fit parameters and the resulting fabric resistance numbers. Fabric K2 Kx c2 k [10-1 0- m2] [103-kg/m3] [103- kg/(m2s)] av max min C 7.91 ± 1.39 3.57 ± 1.23 15800 ± 2800 2.80 4.28 2.08 D 14.51 ± 2.50 2.67 ± 1.51 29000 ± 5000 3.74 8.59 2.39 E 7.61 ± 1.25 2.08 ± 1.06 15200 ± 2500 4.80 9.72 3.18 Clearly it can be seen that the resistance number errors associated with reading errors can be significant, especially for the viscous permeability, which is the dominant factor at low velocities. The uncertainties in the resistance values have to be kept in mind when they are used for computations aimed at a comparison with experiments. Another source of error was the pump, which at low flow velocities (up to 0.4 m/s) delivered a somewhat unsteady flow that caused further fluctuations in both velocity and pressure readings. 2.2.2 Experimental determination of fabric resistance: flow at varying angles to the fabric The comparison between computed and measured drainage profiles, which will be shown in the next chapter, showed a possible dependence of the fabric resistance on the approach flow angle. To examine such a dependence, the fabric resistance measurements were repeated at different flow angles. CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 40 2.2.2.1 Modified experimental set-up To measure the fabric resistance at different flow angles, the fabric test section in the flow loop was modified as shown in Figure 2.6. In the modified test section, the fabric was mounted at different angles to the flow direction. Three different angles were chosen for the trials, and for each angle a separate pipe section that can be inserted into the flow loop was manufactured. The same 3/4" piping as before was used. The fabric area in the pipe was now an ellipse. Flow normal to the fabric surface defines the superficial flow velocity. Placing the fabric at an angle to the oncoming flow changes the fabric area as well as potentially the fabric resistance. The details of the test section are shown in Figure 2.6. A i o o p = Flow loop area Vj = Flow velocity in pump loop a = Approach flow angle A f a b r i c = F a b r i c a r e a V g u p e r = Normal velocity through fabric Steel plate Flange Figure 2.6: Definition of flow loop variables in the modified set-up. The fabrics were mounted between two steel rings held between two flanges, into which the piping is glued at the desired angle (Figure 2.6). (The ends of the pipes were cut at the respective angles to produce the elliptical area.) The pipe ends were flush with the fabric so that no water can escape between the pipe end and fabric. CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 41 Some modifications were also made on the flow loop to achieve a steadier flow. A new power supply was used, which at the same time also allowed for higher flow velocities. However, the velocity was still determined by manometer reading, so that significant errors at low velocities are still present. 2.2.2.2 Test conditions Tests were carried out for all three fabrics at three different angles: 90° (to confirm the repeatability of the earlier measurements), 45° and 22.5°. Measurements were taken at different velocities, ranging from about 0.2 m/s to 2.3 m/s. Between 15 and 25 different velocities were used for each series. The pressure transducer readings were recorded by a data acquisition system instead of the multimeter used in the earlier trials. 2.2.2.3 Results Pressure drop as a function of flow velocity The raw data from the measurements are shown in Figures 2.7, 2.8 and 2.9. They show the measured pressure drop as a function of the flow velocity. The effect of the fabric area on the flow velocity through the fabric is not included. The figures also show the error bars, based on a slightly different error analysis than in the earlier trials. For the velocity measurements, the reading error of the manometer scale was assumed to linearly vary from 0.15 cm at zero velocity to 1.15 cm at full velocity. The error in the pressure data is based on the manufacturer's specifications for the pressure transducers. The error bars were then calculated using a first order Taylor expansion for pressure and velocity. As seen earlier, the errors are larger at small velocities. Also, for the curves at 90° flow angle, the pressure at low velocities is even lower than determined in the first trials. This confirms that the results for low velocities are unreliable. The large errors at low velocities become even more apparent if the pressure coefficient (also based on the flow velocity in the pump loop) is plotted against flow velocity with the error bars included. This is shown in Figure 2.10 for fabric C. Pressure drop as a function of superficial flow velocity To compare the results for different angles, the flow velocity is converted into a normal superficial velocity through the CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 42 0.5 1 1.5 Flow loop velocity V i o o p [m/s] 2.5 Figure 2.7: Pressure drop data versus flow velocity for different approach angles (fabric C). 60 50 40 r 30 _o S g 20 <D t-« PH 10 0 -10 90° 45° r-SE-i r * - i HH rSH H >-£»-; :--B--: HEH HJH H ;-G-; t-SH . .}£3i na-i na-i na-: HI H£H an i-s .*rEai'B' 0.5 1 1.5 Flow loop velocity V l o o p [m/s] 2.5 Figure 2.8: Pressure drop data versus flow velocity for different approach angles (fabric D). CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 43 45 40 35 30 & 25 I-C3 Sa •2 20 15 10 5 0 -5 90° 45° 22.5° 0.5 1 1.5 Flow loop velocity V ) o 0 p [m/s] 2.5 Figure 2.9: Pressure drop data versus flow velocity for different approach angles (fabric E). u a .2 ' o o o ID 40 35 30 25 20 15 10 5 90° '—•-45 0 !---x-22.5° •=> i-y-i ^ • •x.-J- x r{9-; USH 0.5 1 1.5 Flow loop velocity V l o o p [m/s] 2.5 Figure 2.10: Dimensionless pressure coefficient as a function of the flow velocity (fabric C). CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 44 fabric by accounting for the ratio of the fabric area to the cross sectional area of the pipe. Vsuper = Vtoop-—^- = Vioop sin a (2.25) •^•fabric Figures 2.11, 2.12 and 2.13 show the measured pressure drop data versus superficial velocity and the polynomial curve-fit. Error bars are not shown. PH 80 60 B3 & 40 h 20 h 0 f 3 -20 i 1— 90° measured 90° curve-fit 45° measured 45° curve-fit 22.5° measured 22.5° curve-fit 0.5 1 1.5 Superficial velocity V s u p e r [m/s] 2.5 Figure 2.11: Pressure drop data versus superficial velocity through the fabric for different approach angles (fabric C). For all three fabrics, the pressure drop is higher at comparable superficial velocities for lower angles. Second degree polynomials with the intercept at zero pressure for zero velocity can be fitted to the curves. The resulting constants Kx and K2 are given in Table 2.4. K2, which represents the inertial resistance factor, is similar to what was measured in the earlier trials at 90° for fabrics C and D, and somewhat greater for fabric E. In all cases, the coefficients increase with decreasing angle. The results also show that the data for K\, which represent the viscous resistance, are considerably less for the 90° case than measured earlier, and in some cases are even negative. Negative resistances do not exist. This suggests that the values for K\ are unreliable. This is likely due to inaccurate measurements at low flow CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 45 80 60 h ^ 40 o u co co Oh 20 -20 90° measured 90° curve-fit 45° measured 45° curve-fit 22.5° measured 22.5° curve-fit 0 ,^.^ssm§^f^^: • .X' 0.2 0.4 0.6 O i 1.2 1.4 1.6 Superficial velocity V s u p e r [m/s] 1.8 Figure 2.12: Pressure drop data versus superficial velocity through the fabric for different approach angles (fabric D). 80 60 40 20 -20 - i 1 1— 90° measured 90° curve-fit 45° measured 45° curve-fit 22.5° measured 22.5° curve-fit -? 1 r J"3- ' ...X V- CA— — 0 0.2 0.4 0.6 O i Superficial velocity V s u p e r [m/s] 1.2 1.4 1.6 1. Figure 2.13: Pressure drop data versus superficial velocity through the fabric for different approach angles (fabric E). CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 46 velocities and to the fact that inertial resistance dominates over most of the range tested (see Figure 2.10) and therefore weights the curve-fit to the inertial term. Table 2.4: Polynomial coefficients to describe pressure drop data for different fabrics at angles of 90°, 45° and 22.5° flow angle. Fabric Angle K2 [103-kg/m3] Kx [103- kg/(m2s)] 90 9.73 0.46 C 45 15.53 -3.92 22.5 25.38 0.49 90 15.91 0.86 D 45 18.94 -1.17 22.5 37.53 1.65 90 11.33 -0.51 E 45 17.08 -0.94 22.5 26.74 -0.03 If reliable resistance coefficients for both inertial and viscous resistance at different angles were known, it would be possible to find a function that describes both the viscous and inertial resistance as a function of the flow angle. Such a function can also be found for the inertial resistance only, as in our case most of the measured values lie within a range where inertia dominates. Thus the pressure data could be described with a polynomial with only a quadratic term. A P = F{a)Vl0V (2.26) F(a) includes the resistance factor which should be a function of a and also the density and thickness of the permeable medium. The quadratic equation is fitted to the pressure versus flow velocity data (where the actual flow velocity in the pump loop is used), using only the data for velocities higher than 0.5 m/s, where the viscous resistance should be negligible. F(a) is then plotted against the flow angle (Figure 2.14). As only three different angles were investigated, it is difficult to describe the resulting curve reliably, but for fabrics C and D, CHAPTER 2. DRAINAGE RESISTANCE OF THE FABRIC AND FIBRE MAT 47 F{a) seems to depend linearly on a. Therefore a linear fit was chosen. The resulting factors and R 2 values are given in the plot and could be used to extrapolate the curves to find the resistance also for other angles. However, for a reliable fit, more measurements at different angles should be done. 00 a, 18000 16000 14000 12000 10000 8000 6000 h 4000 2000 p 0 fabric C • fabric D X fabric E * y=159x+2110 R2=0.9985 y-99x-;-2485 R2=b.9105 0 10 20 30 40 50 60 Approach flow angle [°] ..-tl yj=90x+1920 R 2-0.9965 70 80 90 Figure 2.14: Calculated resistance factor as a function of the approach flow angle for purely inertial resistance. This new study of the fabric resistance yields two main results. Firstly, the fabric is not a homogeneous medium. If it were, the pressure loss versus normal velocity curves should collapse to one curve for different angles. Secondly, the tests carried out yield unreliable values for the viscous permeability coefficient, due in part to measurement errors and in part to the fact that inertial resistance dominates over most of the range over which data have been fitted. If the measured fabric resistance values are used in the computational model, this can lead to poor predictions of drainage at low drainage velocities. At this point it is not possible to quantify the influence of flow angle for both the inertial and viscous resistance. Chapter 3 Jet impingement on a single fabric In this chapter, a two-dimensional viscous model for jet impingement on a single fabric is developed. The influences of different machine variable settings on jet impingement are then computed. To validate the model, drainage profiles were experimentally determined and compared to computed drainage profiles. 3.1 Computational work To avoid some of the difficulties in modelling jet impingement in twin-wire machines, im-pingement on a single fabric is modelled as a first step. To simplify further the physical situation, fibre mat build-up is also initially neglected. A two-dimensional viscous model is developed for this simplified case. The results can be compared to potential flow theory for validation of the model. Then a model for fibre mat build-up is added. Modelling jet impingement on a single fabric is not only a good starting point for mod-elling the twin-wire geometry, but it can also be used with only slight modifications to predict jet impingement on a Fourdrinier. It is also relevant to jet impingement in twin-wire machines if the jet impinges preferentially on the backing wire, as is often the case. 3.1.1 Analysis Figure 3.1 shows schematically jet impingement on a single fabric and the important param-eters. The following assumptions are made to simplify the problem: 48 CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 49 • The fibre suspension is very dilute and therefore can be modelled as pure water (New-tonian fluid). • The consistency of the suspension remains unchanged during drainage (fibre mat build-up as a filtration process rather than a thickening process). • Laminar flow is assumed. • The velocity profile of the incoming jet is constant over the height of the jet (negligible boundary layer thickness exiting the headbox). • The fabric remains straight in the impingement zone (wire tension —>• oo). • The fabric can be described as a homogeneous porous medium with a smooth surface. • The influence of gravity is neglected. • Retention is assumed to be 100% when fibre mat build-up is computed, and filtration is the prevalent drainage mode. • The SFR of the fibre mat is constant. surrounding air (p,p) Figure 3.1: Jet impingement on a single fabric with the important parameters. The flow is described by the two-dimensional Navier-Stokes equations. The free surface is accounted for by a Volume of Fluid (VOF) method ([82]). This approach determines CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 50 fluid properties according to the volume fraction of water and air at each position (in each computational cell). The interface between the air-water (which here represents the jet surface) is tracked through a continuity equation for the volume fraction. The equations in detail are: 1) Volume fraction equation (shown for the k-th phase, which will be solved for N- l phases, with N=2 in this case): where is the volume fraction of the k-th phase. 2) Continuity equation: dx + dy ~ U {6-Z} 3) Momentum equations in x and y direction: / du du\ _ dp (d2u d2u\ . . ^ \ dx Vdy) dx ^ \dx2 dy2) ( dv dv\ dp (d2v d2v\ P{Ud-x+%)=-^ + »{w + dV2) (3>4) 4) Equations to calculate the fluid properties p and p: P = Y.ekPk (3-5) p = tkHk (3.6) 5) The volume fractions of different fluids in each computational cell add up to one: with k=l,2 (air and water) Forcheimer's equation was used to describe the flow through the fabric and therefore replaces the momentum equations within the porous medium : CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 51 dp dx JJLOL\U + pb\U2 (3.8) dp dy p,a2v + pb2v2 (3.9) With the above equations it is possible to determine the five unknowns: • fluid velocity in x-direction u(x,y) • fluid velocity in y-direction u(x,y) • pressure in the fluid p(x,y) • volume fraction of fluid efc(x,y) 3.1.2 Method of solution The F L U E N T code was used to solve the above equations for a jet impinging on a single fabric. F L U E N T uses a control volume formulation [83] to convert the differential equations into algebraic ones which are then solved numerically for each cell of the computational domain. The modelled domain in this case is a simple rectangle with the fabric along one side of the domain. The solution algorithm is based on a SIMPLE scheme (Semi-Implicit Method for Pressure-Linked Equations). A blended second-order-upwind/central-difference interpolation scheme was chosen for all variables except the volume fraction. As the numerical diffusion at the air-water interface proved to be a main source of error, the QUICK (Quadratic Upwind Interpolation) scheme was chosen for the volume fraction. The QUICK scheme offers the highest numerical accuracy in F L U E N T . Some problems were experienced with convergence which by default in this code is nor-mally assumed to be reached when the sum of the normalized residuals falls below 10~3. (The residuals for the conservation equations in F L U E N T are taken as sum of the imbalance in each equation for all cells in the domain. Only the pressure residual is defined differently, as the imbalance in the continuity equation.) The default value was usually not reached as the residual of pressure in this problem did not always fall below 1 0 - 3 . Therefore, the convergence criterion was not based on the residual sum. Rather the solution was assumed CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 52 to be converged when there was a levelling out of the residuals when plotted as a function of the number of iterations. Convergence then was verified by checking mass continuity for the problem. If the difference between incoming and outgoing mass fluxes was 0.5% or smaller, the solution was assumed to be converged. The problems with convergence of pressure could be linked to the air included in the domain and therefore should not affect the solution for the water jet. 3.1.2.1 Boundary conditions The necessary boundary conditions to solve the flow can be set up from the following con-siderations. The angle and velocity of the jet coming from the headbox are known. We know that the volume fraction of water in the jet is one. The pressure on the outer surface of the fabric as well as in the jet far downstream from the impingement point is zero. The boundary conditions therefore are: • Velocity and angle of the flow as well as ewater = 1 where the jet enters the domain. • Zero static pressure boundaries at all other boundaries. These boundaries allow flow to enter or exit the domain and the volume fraction is calculated from upstream conditions for flow exiting the domain. Two of the boundaries (above the incoming jet and the top of the domain, opposite to the fabric) were later defined as slip-walls for the final computations. This was done to improve convergence. The solution for the impinging jet was not influenced by this modification. 3.1.2.2 Modelling the fabric The fabric is modelled as a 1 mm thick porous medium adjacent to the pressure boundary at the lower side of the domain. The porous medium in F L U E N T [84] is assumed to be homogeneous, with 100% open area. A paper machine fabric is not homogeneous, as it consists of interwoven strands, and also the open area is not 100%. However, the open area does not influence the pressure drop as long as the fabric characteristics are determined appropriately (i. e. based on the superficial velocity). Also, as the dimensions of the jet are CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 53 more than an order of magnitude larger than the single fabric strands, assuming the fabric to be a homogeneous material should be reasonable. To represent the velocity of the fabric, the velocity in the machine direction M D (x-velocity) in the fabric was set to a constant value equalling the machine speed. At the same time, the resistance of the fabric in machine direction, was assumed to be infinite, so that water could pass through the fabric only in the y-direction. It is transported along with the fabric in M D while passing through the fabric. The infinite resistance can be modelled by setting the inertial resistance coefficient in M D to a very large value and the viscous permeability factor to a very low value. The coefficients in the y-direction (perpendicular to MD) are set to the appropriate values representing the fabric resistance. In the previous chapter, it was shown that the resistance in reality is a function of the flow angle, but as this relationship was not quantified, the dependence on the flow angle in the simulations was neglected. 3.1.2.3 Modelling fibre mat build-up The deposited fibre mat adds to the drainage resistance. As the fibre mat gradually builds up, the drainage resistance will be a function of the x-direction distance from the jet impingement point downstream. The approach taken here to model the fibre mat build-up was to calculate the drainage resistance as a function of the deposited basis weight and then combine the drainage resistance from fabric and fibre mat, representing it as a single porous medium with varying drainage resistance in the machine direction. The thickness of the deposited fibre mat, which is very low even at the end of the impingement region, is neglected. (In [60], S E M images of fibre mats up to 10 g /m 2 were published and clearly show that the fibre mat thickness is small compared to that of the fabric.) The inertial resistance of the fibre mat is assumed to be negligible so that Darcy's law can be used. A constant specific filtration resistance is used to describe the viscous resistance of the fibre mat. By comparing Darcy's law for porous media when based on the SFR A P = fivSFRBW with the viscous part of the porous medium description used in Fluent (3.10) CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 54 APviscous = ^ » A n (3.11) leads to an expression of the viscous permeability of the fibre mat in terms of SFR as k m a t = SFR • BW ( 3 ' 1 2 ) The calculated mat resistance was then applied in the model in the following iterative procedure: A first calculation (base case) was done assuming an average fibre mat basis weight of 5 g /m 2 in the computed domain. (The computations were done assuming a target basis weight of 50 g/m 2.) The combined viscous permeability coefficient of fabric and mat for use in the model can be calculated as: k = 1 SFR-BW1 1 L k fabric An (3.13) with BW = 5 g/m 2 Using the combined viscous permeability and the additional inertial resistance of the fabric, drainage was computed and the drainage velocity data as a function of distance from the impingement point was saved. From this data the drained volume (per metre of machine width) up to a certain point can be calculated: VS+i = te+i - xt) + Vi (3.14) with V drained volume [m2/s] v drainage velocity [m/s] x distance from impingement point [m] and subscript i denotes the cell number, which increases in the machine direction. Assuming a filtration process rather than a thickening process, and also assuming 100% retention, the basis weight as a function of distance can be established: BWi = ^BWend (3.15) 'in CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 55 with Vin = volume flow from the headbox [m2/s] BWend = 50 g /m 2 Now it is possible to describe the mat resistance as a function of distance in the form of an array and use this data as input in the code: 1 SFR • BWj - l k' + j-,'yi ^2 _k fabric A 71 In the code it is only possible to input constant mat resistance coefficients to describe porous media. Instead of fitting the ki data to a function k = f(x), the data was therefore approximated by a step function. The porous medium was then divided into different zones, with k = constant in each zone. See Figure 3.2 for an example of the approximation of the mat resistance profile with a step function. I 2.4e-10 2.2e-10 2e-10 1.8e-10 1.6e-10 1.4e-10 1.2e-10 le-10 8e- l l 6e - l l 1 ! 1 ! j I I 1 V | I I 5 6 7 Distance [cm] 10 Figure 3.2: Approximation of mat resistance profile with a step function. With the new values for the viscous permeabilities, the computation was repeated. The resulting drainage data as well as the drainage velocity profile were compared to the results from before. If the results did not agree, the drainage data were used to calculate a new CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 56 mat resistance profile for k as before and another computation was done. The procedure was repeated until convergence of the resulting data was reached. Convergence was assumed for an L2-error norm of < 10~2 m/s for the change in drainage velocity profiles for two consecutive iteration steps. Figure 3.3 shows the schematic approach to compute fibre mat build-up. Figure 3.4 shows an example of the drainage velocity profiles during the iteration process until convergence is reached. (Impingement is at about 3 cm with the jet coming from the left. This will be the same in the following plots showing pressure and drainage profiles.) assume 5 gsm mat calculate a (constant) input in FLUENT run computation I get v-velocity profile calculate BW, a = f(x) run new computation get v-velocity profile if not converged compare to last v-profile for convergence if converged end iteration Figure 3.3: Fibre mat build-up: Schematic approach. In nearly all cases as few as two iterations were sufficient to reach convergence. One concern was the use of a piecewise continuous function to describe A; as a function of x. However the resulting pressure profiles and drainage velocity profiles in the "step-function" approach were smooth so the approximation was considered sufficiently accurate. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 57 4 base case after 1. iteration after 2. iteration 0 3 4 5 6 7 8 9 10 Distance [cm] Figure 3.4: Convergence of the drainage velocity profile during fibre mat build-up calculation. 3.1.3 Validation of the V O F model with potential flow theory To verify the applicability of the V O F model to this specific problem, computations were done for jet impingement on a straight, impermeable wall. The Reynolds number of this problem is 1.5 • 105 at a velocity of 15 m/s, which shows that inertial forces will be high. The use of potential flow theory therefore is reasonable and the computational results should show good agreement with that theory. To represent the assumptions of potential flow theory more closely in the model, a slip-wall was used for the impingement wall, where the no-slip condition at the wall is turned off. Therefore no shear forces occur along the wall as is the case in potential flow theory. The potential flow theory for this free surface flow is based on Prandtl's hodograph method, shown in detail in [85]. Figure 3.5 shows the computed and theoretical pressure distributions along the impinge-ment wall for impingement angles of 30° and 10° at 15 m/s jet velocity. One can see that the overall pressure distributions match very well. However, the peak pressure is predicted poorly, especially at lower impingement angles. This poor prediction CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 58 Figure 3.5: Comparison between computation and potential flow theory. of peak pressure is caused by numerical diffusion, which occurs at the interface between the air and water. The prediction of peak pressure can be improved by using finer meshes and higher order solution procedures, but computation time goes up. At lower angles, it was not possible to obtain mesh-converged solutions for peak pressure, due to limitations in the maximum mesh size caused by the available computational power. Therefore, at least at lower angles, the prediction of peak pressure will not be accurate. On the other hand, the integrated pressure along the wall agrees very well between computation and theory. For different meshes and solution methods, the integrated pressure was overpredicted in the computation compared to the theoretical values by 3 to 8% at an impingement angle of 10°. Keeping in mind that we are comparing a viscous code with an inviscid theory, these predictions are fairly accurate. The good agreement, despite the poor prediction of peak pressure, can be explained by the fact that the peak pressure occurs only over a very small distance (in the order of 1 mm for a = 10°) and the pressure gradient at the impingement point is very high. The peak pressure therefore has only little influence on the overall integrated pressure. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 59 Another consideration is that if we look at phenomena occurring over such small distances, the assumption of the fabric being homogeneous ceases to be reasonable. If the peak pressure on a permeable fabric also occurs over a distance of about 1 mm, the fabric structure will definitely influence the peak pressure. Even if it were possible to accurately reproduce the peak pressure for impingement on a solid wall, it would not be possible to conclude that the same would be the case for impingement on the fabric. Rather it makes sense to consider integrated values and look carefully at any phenomena that occur over distances which are the same order of magnitude as the fabric strands and voids. 3.1.4 Grid independence When looking at jet impingement on a moving permeable wire, another factor affecting the accuracy of computed results has to be considered. Depending on the fabric resistance, the interface between water and air needs some time to penetrate the fabric after impingement. In practice, this can be seen as a separation between the location of impingement and where water first comes through the wire. In the computations this penetration time causes some complications, as the question of numerical diffusion, which is closely related to mesh convergence becomes more important than in the case of jet impingement on an impermeable wall. As long as the interface between air and water is located in the fabric, the values for fluid properties will be influenced by the remaining air, which in turn influences the calculation of pressure drop over the thickness of the fabric. Computations for different fabrics, listed in Table 3.1, were carried out with different mesh sizes. Jet impingement angle in all cases is 10° and jet velocity is 15 m/s. The fabric characteristics were determined from work done by [60], [61] and [62]. If both inertial resistance and viscous permeability were given (sets b)), computations were also done using only the inertial resistance (sets a)) to see how much influence the viscous permeability has at impingement. Figure 3.6 shows the calculated relative drainage amounts (where the relative drainage is the volume of water drained through the fabric divided by the incoming jet volume) as a function of mesh size. For most sets the drainage amounts when using different mesh sizes differ only within the range of uncertainty in the calculations. However, for set lb) mesh CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 60 Table 3.1: Fabric characteristics for input in F L U E N T . Set la) Set lb) Set 2a) Set 3a) Set 3b) 3.37 • 104 3.37 • 104 6-10 4 1.9 • 105 1.9 • 105 kfabric [m2] oo 1.27 • 10" 1 0 oo oo 8- 10~ n convergence was reached only for the finer meshes. Looking at the pressure distributions for this set for different meshes (Figure 3.7), only slight differences in the pressure distribution can be detected. Equal differences were found in the plots for those sets where mesh conver-gence was given for all three meshes. Therefore, one has to be careful when judging mesh convergence. Also, the pressure distributions in the early part of jet impingement, when the interface between the two phases has not yet penetrated through the fabric, have to be viewed with some scepticism. set 1 a) — ' — set 1 b) —-x—-set 2 a) * set 3 a) a set3b) - - - - - - -25 h £ 20 ao > P i 15 10 10000 100000 Number of cells le+06 Figure 3.6: Grid convergence of drainage for different fabrics (meshes are 200 x 100, 400 x 200, 800 x 200 and 800 x 400 cells). The proposed mesh size of 400 x 200 cells seems sufficient for the given conditions and CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 61 35 30 25 'to & 20 B 3 3 15 OH 10 5 0 200x100 400x200 -son v Ann B 1 \ I \ -\ i I i i 2 3 4 5 6 7 8 9 10 Distance [cm] Figure 3.7: Grid convergence of pressure for fabric lb). therefore was used in the computations. However, when certain variables are changed mesh convergence was checked again. Lower impingement angles, lower jet velocities and higher fabric resistances are particularly problematic. 3.1.5 Results 3.1.5.1 Computational results neglecting fibre mat build-up Computations for jet impingement neglecting the fibre mat build-up were done for the fol-lowing conditions: Jet height H Fabric thickness A n Domain size Mesh density Density of suspension p 10 mm 1 mm 10 cm x 3 cm 400 x 200 cells 1000 kg/m 3 Viscosity of suspension fi 5 -10 4 Ns /m 2 CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 62 The viscosity is that of water at approximately 60°C, which corresponds to the headbox temperature. For air, properties at 18° were used. The mesh was uniform in x-direction; in y-direction a successive ratio of 1.01 was used to achieve better resolution at the impingement point. Jet speed, impingement angle, fabric resistance and fabric speed were varied throughout the computations and therefore are described in the respective sections. Influence of fabric resistance It is interesting to examine the influence of inertial and viscous resistance of the fabric at jet impingement, knowing that in many studies the viscous resistance of the fabric is assumed to be negligible. Therefore, a comparison of the pressure distribution at the fabric was done for the fabric sets 1 and 3 of Table 3.1, which gave a distinction between inertial and viscous resistance. For the computations further conditions were: Jet velocity Vjet 15 m/s Fabric velocity Vfabric 15 m/s Impingement angle a 10° The values found in the literature for drainage resistance of fabrics vary considerably. Figure 3.6 showed that drainage through the fabric depends very much on the resistance of the fabric. Adding a viscous resistance changes the drainage amount noticeably, especially if the inertial resistance is in the lower range (set la), b)). However, comparison of the pres-sure distribution at the fabric (Figure 3.8) shows little difference between the computations for inertial resistance only and the ones for both inertial resistance and viscous resistance. Furthermore, when comparing the pressure profile of set la) or b) with set 3a) or b), the curves differ only close to the impingement point, where a prediction is difficult due to the problems with numerical diffusion. Farther from the impingement point, where the results are not influenced by the interface and therefore are assumed to be accurate, the pressure profiles of sets 1 and 3 nearly overlap. This is unexpected, as drainage for these sets was CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 63 very different. An explanation might be that different fabric resistances will influence the pressure distribution only close to the impingement point, but further downstream the free surface accommodates for different fabric resistances. An alternative explanation is that for all cases 75% or more of the water is not drained, and the small amount that is drained does not substantially affect the pressure distribution. C3 3 5 6 7 Distance [cm] Figure 3.8: Influence of viscous resistance on pressure profiles along the fabric. Influence of jet velocity In order to investigate the influence of jet velocity, computations for jet velocities ranging from 15 m/s (900 m/min) to 25 m/s (1500 m/min) were done. The impingement angle was set at 10° for these computations. The velocity of the fabric in each case was the same as the jet velocity. The fabric resistance of set lb) was used for these computations. This resistance includes inertial and viscous terms and gave reasonably high drainage amounts. Figure 3.9 shows a comparison of the pressure profiles along the fabric. As expected, the pressure is higher and the curve also somewhat broader for higher jet velocities. The pressure pulse is significant only over the first 2 to 3 cm after impingement, but some effect extends up to 6 or 7 cm downstream of the impingement point. This seems sensible, as the CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 64 projected length of the jet width on the fabric equals the jet width (1 cm) divided by the sine of the impingement angle (10°), which is approximately 6 cm. OH 5 6 7 Distance [cm] 15 m/s 17.5 m/s 20 m/s 22.5 m/s 25 m/s Figure 3.9: Influence of jet velocity on the computed pressure profiles at the fabric. Figure 3.10 shows the integrated pressure (pressure integrated over the distance in MD) as a function of jet velocity. The computed integrated pressure was compared to a simple one-dimensional analysis. Neglecting drainage, the momentum equation in this case can be used to calculate the force on the fabric (for one metre machine width) as F = Jpdx = pHVfet sin a (3.17) The comparison shows that the computed values as a function of jet velocity follow the same trend as the results from the one-dimensional analysis, but are about 10% lower. This seems reasonable, as with drainage the force on the fabric should be lower than without, as a result of the momentum carried by the water passing through the fabric. Figure 3.11 gives the drainage amounts (absolute and as a fraction of inflow), also as a function of jet velocity. The drainage amounts include the water that actually drains from the fabric plus the water remaining in the fabric. The absolute drainage increases approximately linearly CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 65 1200 Computed 1-D-theory 200 0 14 16 18 20 22 24 26 Jet velocity [m/s] Figure 3.10: Computed integrated pressure at the fabric as a function of jet velocity and comparison with a simple one-dimensional analysis. with the drainage velocity, and if the curve would be extended to the left, the intercept with the x-axis would be at approximately 0 cm. Correspondingly the relative drainage amount as a fraction of incoming flow remains constant even at different jet velocities. For the given variables the relative drainage amount is about 20%. Another interesting aspect is to look at the distance over which the majority of drainage occurs at impingement. The drainage velocity profiles along the fabric (y-component of the velocity) for different jet velocities are shown in Figure 3.12. From the drainage velocity profile it is possible to calculate the fraction of drainage completed over a certain distance (as the drained volume up to position x of the fabric divided by the volume drained over the whole domain). Table 3.2 shows the fraction of the overall drainage achieved during impingement at certain distances after jet impingement. Independent of jet velocity, nearly 50% of the drainage is completed after only one centimetre. After that the drainage rate falls rapidly so that an additional 20% is drained in the second centimetre and after 4 cm drainage is between 85 and 90% complete. It can be concluded CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 66 U 60 es .9 eS •s •*-» P < 0.06 0.05 0.04 h 0.03 0.02 0.01 h 1 I I I ] Absolute drainage — 1 — Relative drainage — -*— , --x X - J Jfr - — u — i i i i 14 16 18 20 22 Jet velocity [m/s] 24 26 100 80 60 40 20 0 s -u 00 es a "8 T 3 <D .> °-4-> 43 Figure 3.11: Computed drainage as a function of jet velocity (given in absolute value per metre of machine width and as a fraction of incoming flow volume). £ 3 I ; f 1 ! ! 1 > 15 m/s ! ! 17 5 m/s j i 20 m/s : i l.i ! 22.5 m/s . ! j i\ \ i 25 m/s H% : j . 1 J \ \' \; i i : i i : T « V' \ i i ; \V" \ ^ ^ ^ ^ i i i j : ^ ; - - - _ . . . i | -2 3 4 5 6 7 8 9 10 Distance [cm] Figure 3.12: Influence of jet velocity on the drainage velocity through the fabric. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 67 that the influence of jet impingement is significant only over a very small distance; one-half of the drainage occurs within a distance comparable to the jet height. Table 3.2: Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different jet velocities. Jet velocity [m/s] 1 cm [%] 2 cm [%] 4 cm [%] 15 49 70 90 17.5 48 69 88 20 47 68 87 22.5 48 67 88 25 • 44 64 85 Influence of impingement angle Computations were done for various impingement angles while keeping the other variables constant. Impingement angles of 4°, 6°, 8° and 10° were used. The jet velocity and fabric velocity were 15 m/s and the fabric resistance was described by set lb) from Table 3.1. Figure 3.13 shows the pressure profiles along the fabric for different impingement angles. The pressure profiles have similar shapes with a steep gradient at the impingement point and the maximum pressure at about the same location, but are lower in magnitude for lower impingement angles. Figure 3.14 shows the computed integrated pressure along the fabric and Figure 3.15 the drainage amounts (absolute and as a fraction of inflow), both as function of impinge-ment angle. As expected the integrated pressure is higher with higher impingement angles and so is the drainage (absolute and relative). In both cases the dependence seems to be approximately linear. The integrated pressure was again compared to the one-dimensional analysis. According to the analysis, the integrated pressure should depend on the sine of the impingement angle, which at these low angles is approximately linear. The computed values are again on average around 10% lower, consistent with the fact that the forming fabric is not the impermeable medium assumed in the one-dimensional analysis. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 68 Figure 3.13: Influence of impingement angle on the computed pressure profiles at the fabric. Figure 3.14: Computed integrated pressure at the fabric as a function of impingement angle and comparison with a simple one-dimensional analysis. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 69 0 0 C3 e 1 +-» J 3 "3 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 6 7 8 9 Impingement angle [°] Absolu Relath te draina re drains ige —<• ige — - x ..... * J-} ~ — 3 10 100 80 60 & a T3 OJ 40 •£ "5 1 20 11 Figure 3.15: Computed drainage as a function of impingement angle (given in absolute value per metre of machine width and as a fraction of incoming flow volume). Drainage velocity profiles were computed (shown in Figure 3.16) and the drainage com-pleted after certain distances as a function of overall drainage was calculated (see Table 3.3). The drainage velocity profiles again look very similar in shape; they nearly overlap further away from the jet impingement point and differ only in magnitude over the first centimetre after impingement. From the values for completed fraction of drainage after the first and second centimetre, it can be concluded that with a lower angle of impingement, the frac-tion of completed drainage over a certain distance is slightly diminished. The fraction of drainage after one centimetre drops from approximately 50% for 10° to 42% at 4° and after two centimetres 70% of drainage are completed at 10° compared to 65% at 4°. At a distance further downstream the fraction of completed drainage is about the same for all angles with approximately 90% at 4 cm. Even at lower angles, therefore, drainage occurs mainly over the first 2 cm after impingement. Influence of rush/drag The influence of rush/drag on jet impingement was investigated. Computations were done for jet velocities of 15 m/s (900 m/min) combined with different CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 70 Figure 3.16: Influence of impingement angle on the drainage velocity through the fabric. Table 3.3: Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different impingement angles. Impingement angle [°] 1 cm [%} 2 cm [%] 4 cm [%] 10 49 70 90 8 48 69 89 6 45 68 87 4 42 65 88 CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 71 fabric velocities. Drag (Vfabric — Vjet) was set to 20 m/min, 40 m/min and 60 m/min and rush (Vjet — Vfabric) to 10 m/min. The impingement angle was 10° and set 1 b) was used to describe the fabric resistance. The pressure profiles along the fabric for all cases were very similar, with only slight differences close to the impingement point. As the jet has no information about the fabric velocity except through a very thin boundary layer, this result was expected. The integrated pressure and drainage amounts, given in Figure 3.17, consequently also remain approximately constant for different cases of rush/drag. The small differences lie within a range that can be attributed to inaccuracies in the computations, e. g. from different convergence behaviour. 400 U 3 co CO a S 6 0 CD 350 h 300 250 200 150 100 50 0 -20 •10 I I i ) e 3 sjc Integrated pres Relative draii sure — 1 — lage —-x---10 20 30 40 vfabric-vjet [m/min] 50 60 30 25 20 15 10 70 CD 6 0 ca a '3 T3 CD > CD Figure 3.17: Computed integrated pressure at the fabric and drainage as a function of rush/drag. When looking at the influence of rush/drag, it is also important to look at the velocity profiles over the height of the jet or at the shear stress at the fabric. The x-component of velocity as a function of y-direction fell in all cases rapidly from the velocity of the fabric to the velocity of the jet (over the distance of the first two cells, which equals a distance of 0.2 mm) and was then relatively constant over the remaining height of the jet. It is CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 72 meaningless to attempt to resolve the boundary layer more accurately because its dimension is clearly in the range of the fabric roughness height, which itself is not considered in the simulations. The velocity of the fibre suspension equalled the velocity of the fabric at the end of the computed domain only without rush or drag. For the three computed cases of drag the jet was slowly accelerated by the fabric, but a difference in jet and fabric velocity remains at the end of the domain. In the case of 10 m/min rush, the jet velocity was lower than the fabric velocity around the impingement point, but further downstream the jet velocity was higher, which can be seen in the negative values for shear stress along the fabric in Figure 3.18. An estimate of the shear stress can be made based on the velocity difference of the suspension and fabric: r Ufabrrc ~ U j e t Ay v ' where the difference in u-velocity was taken over the two cells next to the fabric and thus Ay = 0.2 mm. This distance was in all cases the approximate thickness of the boundary layer; further inwards the average jet velocity was reached. It has to be noted that with the used mesh resolution the boundary layer itself is not properly resolved and the calculation of the shear stress is therefore only an approximation. Figure 3.18 shows shear stress as a function of distance from the impingement point along the fabric. At the jet impingement point the shear stress is relatively high due to the deceleration of the jet, but diminishes quickly and remains relatively constant further downstream. One should be aware however, that this calculation is based on the assumption of a smooth surface of the permeable medium. As mentioned above, the computed thickness of the boundary layer is in the range of the fabric roughness and therefore the boundary layer flow and with it the shear stress will in reality be influenced by the roughness of the fabric. Resolving the boundary layer properly would be important. 3.1.5.2 Computational results including the fibre mat build-up Fibre mat build-up was computed for different furnishes as well as different fabrics. The influences of jet velocity and impingement angle were also investigated. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 73 14.8 m/s' 15.0 m/s 15.3 m/s 15.7 m/s 16.0 m/s 3 4 5 6 7 8 9 10 Distance [cm] Figure 3.18: Shear stress along the fabric for various cases of rush/drag. From top to bottom, the different curves represent decreasing amounts of drag (jet velocity was 15 m/s). Fibre mat build-up for different furnishes Fibre mat build-up was computed for different furnishes to investigate the difference between fast and slow draining furnishes. The following furnishes were included in the study: Eucalyptus SFR = 3.0 • 109 m/kg Groundwood directory grade SFR = 2.2 • 109 m/kg Groundwood newsprint grade SFR = 0.9 • 109 m/kg Kraft O C C SFR = 0.5 • 109 m/kg Fine paper furnish SFR = 1.67 • 109 m/kg The fine paper furnish was a sample from a pulp and paper mill, including fillers and chemical agents, while the other furnishes were pure pulps. A l l data are taken from [86]. Jet impingement for these calculations was at a 10° impingement angle and the jet and fabric velocity were 15 m/s. The characteristics for the fabric (also from [86]) used in the computations were: CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 74 Inertial resistance C2 = 1.36 • IO4 m 1 Viscous permeability kfabric = 2.43 • I O - 1 0 m 2 In all cases only two iterations were needed to reach convergence of the resistance profile. In addition to computing fibre mat build-up, jet impingement on the fabric with only water (no fibre mat build-up) was computed for comparison. Figure 3.19 shows the pressure profiles along the fabric for different furnishes. Differ-ences between the pressure profiles are not significant: easy-to-drain furnishes like the kraft pulp show a somewhat lower pressure around the peak region, and because more fibres are deposited in the beginning, the pressure drop gradient further downstream is less steep than for furnishes which do not drain that easily, like eucalyptus. The pressure pulse is important only over the first 2 to 3 cm of drainage. ! ! ! 1 1 Eucalyptus GW directory <A 1 i Fine paper ' \\ • : • GW newsprint \ \ \ Kraft Fabric only —• — \ \ I \ | 1 i -i i i rsii-.i... i ' I 2 3 4 5 6 7 8 9 10 Distance [cm] Figure 3.19: Computed pressure profiles at the fabric for jet impingement of different fur-nishes. Figure 3.20 shows the integrated pressure and relative drainage for different furnishes. The difference in integrated pressure for the different furnishes is not very significant, even CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 75 if compared to the case where fibre mat build-up was neglected. As expected the trend shows higher integrated pressures for furnishes with higher drainage resistance (higher SFR value). But even for the eucalyptus pulp (highest drainage resistance) the integrated pressure ( « 350 N) is only 10% higher than for the plain fabric ( « 320 N). to 3 CD (-. ft -a CD 6 0 CD 350 345 h 340 335 330 325 -~~-x I ntegrated Relative pressure drainage X 30 25 20 15 10 CD 6 0 CO a 2 T3 CD > OH 320 1 1 1 1 1 1 ' 1 0 0 5e+08 le+09 1.5e+09 2e+09 2.5e+09 3e+09 3.5e+09 SFR [m/kg] Figure 3.20: Integrated pressure at the fabric and relative drainage for different furnishes (SFR=0 represents impingement without fibre mat build-up). Looking at the relative drainage, the influence of fibre mat build-up is much more pro-nounced, with a relative drainage of w 14% for eucalyptus compared to 28% for the plain fabric, a drop in relative drainage by 50% based on the plain fabric. Relative drainage for the different furnishes range from « 22% for the kraft pulp to « 14% for the eucalyptus pulp. The basis weight as a function of distance from jet impingement is shown in Figure 3.21. As implied by the pressure profiles, drainage is important only over the first centimetres downstream of impingement and therefore fibre deposition takes place in the same area. Further downstream the basis weight remains nearly constant. The slope of the curves in the beginning is the same for all furnishes. For higher resistance pulps the fibre deposition CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 76 is finished within a shorter distance than for lower resistance pulps, as also illustrated in Table 3.4, which shows how much of the overall drainage takes place over certain distances. If fibre mat build-up is included, even more drainage (which is proportional to the amount of fibre deposition), relatively, takes place over the first 1 to 2 cm than for the plain fabric. It can also be seen that the higher the drainage resistance of the pulp, the shorter will be the distance over which the majority of drainage takes place. This finding is reasonable because in the early stages of fibre deposition the overall drainage resistance increases faster for the higher resistance pulps. The basis weight reached at the end of the jet impingement region (in conjunction with this specific fabric) ranges from « 7 g /m 2 for the eucalyptus pulp to « 11 g /m 2 for the kraft pulp. 12 10 ~ J 8 +-> JS .2? 6 a A m 4 Eucalyptus G W directory Fine paper G W newsprint Kraft 6 7 Distance [cm] 10 Figure 3.21: Build-up of basis weight as a function of the furnish. Influence of fabric resistance on fibre mat build-up To investigate how fibre mat build-up is influenced by the drainage characteristics of the fabric used in the computation, some of the computations from above were repeated using a different (higher resistance) fabric. The characteristics for this fabric (referred to as fabric 2) were: CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 77 Table 3.4: Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different furnishes. Furnish (SFR in [m/kg]) 1 cm [%] 2 cm [%] 4 cm [%] Eucalyptus SFR=3.0 • 109 74 89 95 G W directory SFR=2.2 • 109 71 88 96 Fine paper SFR=1.67-10 9 65.5 85 95 G W newsprint SFR=0.9 • 109 59 80.5 94.5 Kraft SFR=0.5 • 109 54.5 76 93 Fabric only 45 66.5 87.5 Inertial resistance C2 = 3.37 • 104 m 1 Viscous permeability kfabriC = 1-27 • 10~ 1 0 m 2 Figure 3.22 shows the integrated pressure and relative drainage for different furnishes when this new fabric description is used, as well as for the fabric from before (referred to as fabric 1). The figure also shows these values for both fabrics without fibre mat build-up. With the higher drainage resistance of the fabric, the influence of fibre mat build-up is less significant than before. While with the lower resistance fabric the integrated pressure was around 10% higher for the eucalyptus pulp compared to the fabric without fibre mat, it is only 3% higher when the higher resistance fabric is used in the model. The relative drainage with the high resistance fabric drops from m 17.5% for the plain fabric to « 12.5% for the eucalyptus pulp. This is a drop of about 28%, half of the drop which occurred when the lower resistance fabric was used. The low resistance kraft pulp, with 16% relative drainage with fabric 2, drains only slightly less than if no fibre mat at all were present (17.5%). Also, the difference in relative drainage between the pulps with high and low resistance is less for the higher resistance fabric (3.5%) than for the lower resistance fabric (7.5%). The difference in relative drainage with different fabrics is also reflected in Table 3.5, which shows a comparison of the basis weights achieved during jet impingement for the different furnishes in conjunction with different fabrics. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 78 380 360 340 £ 320 S. 300 -a CD 60 CD 280 t 260 240 220 Integrated pressure (fabric 1) Integrated pressure (fabric 2) Relative drainage (fabric 1) Relative drainage (fabric 2) 50 45 40 35 ^ 30 & C 25 1 20 15 10 5 0 CD _ > *•*-* 13 Pi 5e+08 le+09 1.5e+09 2e+09 2.5e+09 3e+09 3.5e+09 SFR [m/kg] Figure 3.22: Integrated pressure and relative drainage for different furnishes (SFR values) in conjunction with different fabrics. SFR=0 represents impingement without fibre mat build-up. Table 3.5: Basis weight after jet impingement for different furnishes with different fabrics. Furnish Basis weight [g/m2] Fabric 1 Fabric 2 Eucalyptus 7.05 6.3 G W directory 7.7 not computed Fine paper 8.85 6.95 G W newsprint 10.15 not computed Kraft 11.3 8.0 CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 79 Although the overall influence of the fibre mat build-up is less significant with the higher resistance fabric, there is still an influence on how fast drainage takes place. Table 3.6 shows again the distance over which a certain fraction of the drainage is completed. Once again a higher fraction of drainage takes place over the first 1 or 2 cm for the high resistance eucalyptus pulp than for the lower resistance pulps or the plain fabric, but the difference for high and low resistance pulps is less than with the lower resistance fabric. Table 3.6: Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different furnishes with fabric 2. The numbers in brackets are for fabric 1. Furnish 1 cm [%] 2 cm [%] 4 cm [%] Eucalyptus SFR=3.0 • 109 71 (74) 87.5 (89) 95 (95) Fine paper SFR=1.67-10 9 76 (65.5) 85.5 (85) 95 (95) Kraft SFR=0.5-10 9 60.5 (54.5) 80.5 (76) 94 (93) Fabric only 55 (45) 75 (66.5) 90.5 (87.5) Influence of jet velocity For one of the furnishes (GW newsprint), some computations were done to investigate the influence of jet velocity on fibre mat build-up. Five different computations with jet velocities ranging from 10 m/s to 22.5 m/s were carried out. The fabric velocity equalled the jet velocity and the impingement angle was kept constant at 10°. The fabric characteristics were those of the lower resistance fabric (fabric 1). In all cases two iterations were enough to reach convergence. Figure 3.23 shows the computed pressure profiles along the fabric. As expected, the pres-sure at impingement is higher and the curves are somewhat broader at higher jet velocities. As before, the pressure pulse at all velocities seems important only over the first 2 to 3 cm after impingement. Figure 3.24 shows the integrated pressure as a function of jet velocity, and the com-puted values are compared to the simple one-dimensional theory where drainage is neglected (Equation 3.17). The comparison shows that the computed values are somewhat lower than the theoretical values, owing to the dewatering that occurs. The computed values show the CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 50 80 es £ £ 45 40 35 30 25 20 15 10 5 0 5 6 7 Distance [cm] 10 m/s -15 m/s -17.5 m/s -20 m/s 22.5 m/s — 10 Figure 3.23: Computed pressure profiles at the fabric for different jet velocities, G W newsprint furnish with fabric 1 at 10° impingement angle. same trend as the theoretical values, although the two curves diverge a little more with higher jet velocities (at 10 m/s, the computed value is « 8% lower than the theoretical one, at 22.5 m/s the difference is w 14%). This can be explained with Figure 3.25, which shows the absolute and relative drainage as a function of jet velocity. Slightly higher relative drainage values at higher jet velocities imply a larger difference between the computed integrated pressure and the theoretical one from the simplified theory. The finding of an increasing relative drainage with higher jet velocity is different from what was found for impingement on a fabric when fibre mat build-up was neglected. For that case the relative drainage was independent of jet velocity. There are two possible reasons for the increasing relative drainage with jet velocity. Firstly, the resistance of the porous medium is not constant as would be the case without considering fibre mat build-up. The average fibre mat resistance can be different for different velocity cases, therefore the relative drainage can also differ with the jet velocity. Secondly, the viscous permeability is more important when fibre mat build-up is included, as it is relatively higher than in cases without fibre mat build-up. This might also change the drainage behaviour as CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 81 1000 Computed 1-D-theory 800 0 8 10 12 14 16 18 20 22 24 Jet velocity [m/s] Figure 3.24: Computed integrated pressure at the fabric as a function of jet velocity and comparison with a simple one-dimensional analysis, G W newsprint furnish with fabric 1 at 10° impingement angle. a function of jet velocity. Table 3.7, which gives the fraction of the overall drainage completed after certain dis-tances from the jet impingement point, shows that for a jet velocity of 10 m/s 65% of the overall drainage is completed after only 1 cm, while for a jet velocity of 22.5 m/s it is only 54%. Therefore, relatively speaking, the fibre mat builds up over a shorter distance at lower jet velocities, which leads to an earlier increase in mat resistance, which might also cause the slightly lower relative drainage amounts. After 2 cm, about 85% of the drainage is completed for a jet velocity of 10 m/s compared to 75% at 22.5 m/s, after 4 cm it is about 96% for 10 m/s compared to 92% for 22.5 m/s. Again it becomes clear from these numbers that most of the drainage takes place over a very short distance of a few centimetres. Table 3.8 gives the basis weight at the end of the jet impingement region for different jet velocities. In like fashion to the relative drainage, the basis weight varies from « 9 g /m 2 for a 10 m/s jet velocity to « 11.6 g /m 2 for a 22.5 m/s jet velocity. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 82 00 cd a "8 •a <u 2^ CA XI 0.06 0.05 0.04 0.03 0.02 h 0.01 h 1 1 1 1 Absolute drainage — 1 — Relative drainage —-x—-1 --j - x — ^ •*-X : • * i i i i i 10 12 14 16 18 20 Jet velocity [m/s] 22 H 60 100 80 H 40 20 24 u 00 C3 a 1 CD > Pi Figure 3.25: Computed drainage as a function of jet velocity (given in absolute value per metre of machine width and as a fraction of incoming flow volume), G W newsprint furnish with fabric 1 at 10° impingement angle. Table 3.7: Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different jet velocities (GW newsprint furnish with fabric 1 at 10° impingement angle). Jet velocity [m/s] 1 cm [%] 2 cm [%] 4 cm [%} 10 65 85 96 15 59 80.5 94.5 17.5 57.5 79 94 20 56 77.5 93.5 22.5 54 75.5 92 CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 83 Table 3.8: Basis weight at the end of the jet impingement region for different jet velocities (GW newsprint furnish with fabric 1 at 10° impingement angle). Jet velocity [m/s] Basis weight [g/m2] 10 9.05 15 10.15 17.5 10.7 20 11.05 22.5 11.6 Influence of impingement angle For the G W newsprint furnish in conjunction with fabric 1, computations were also done for different impingement angles. Figure 3.26 shows the pressure profiles along the fabric for jet impingement at different impingement angles. Again, the curves differ mostly around the impingement point, with higher pressures at higher impingement angles. At lower angles the gradient leading to the maximum pressure is less steep. Looking at the integrated pressure along the fabric (Figure 3.27), again compared to the simple one-dimensional theory, we see that the computed integrated pressures follow the same trend as the theoretical values (Equation 3.17) at a somewhat lower level. The relative difference between the two curves is approximately constant, with the computed values being 10% less than the theoretical ones. Figure 3.28 shows the absolute and relative drainage amounts. There is a slight increase in relative drainage at higher impingement angles. This increase could also be observed when studying jet impingement without fibre mat build-up, although the increase in that case was higher. Referring to Table 3.9 more drainage takes place over the first centimetre at higher impingement angles than at lower angles (60% at 12° compared to 48.5% at 6°). This implies that the fibre mat builds up faster at higher angles and therefore the mat resistance will increase faster. Using the same argumentation as before for the influence of the jet velocity, the overall relative drainage with a fibre mat present should therefore not increase as much with the impingement angle as it would if fibre mat build-up was neglected. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 84 ! ! ! ! ! ! ! i 12° 10° 8° A \ I 1 1 I I 6 ° "~ j i- i i i f i l l I 1 i /••• i i i i —i i i i 2 3 4 5 6 7 8 9 10 11 12 Distance [cm] Figure 3.26: Computed pressure profiles at the fabric for different impingement angles. G W newsprint furnish with fabric 1 at 15 m/sjet velocity. 9 10 11 12 13 Impingement angle [°] Figure 3.27: Computed integrated pressure at the fabric as a function of impingement angle and comparison with a simple one-dimensional analysis. G W newsprint furnish with fabric 1 at 15 m/sjet velocity. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 85 0.036 0.034 _ 0.032 CO B 00 C3 .3 CS u T 3 U 0.03 0.028 0.026 0.024 0.022 0.02 Absolute draim Relative draiiu ige — i — ige — e 1 <- -| j -5 8 9 10 Impingement angle [°] 11 12 100 80 60 20 13 £1 tu 00 es a 40 •£ Pi Figure 3.28: Computed drainage as a function of impingement angle (given in absolute value per metre of machine width and as a fraction of incoming flow volume). G W newsprint furnish with fabric 1 at 15 m/s jet velocity. Table 3.9: Fraction of drainage completed after distances of 1 cm, 2 cm and 4 cm for different impingement angles (GW newsprint furnish with fabric 1 at 15 m/s jet velocity). Impingement angle [°] 1 cm [%] 2 cm [%] 4 cm [%] 6 48.5 74.5 94 8 56 78 93 10 59 80.5 94.5 12 60 81 94.5 CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 86 Table 3.10 gives the basis weight at the end of the impingement region for different impingement angles. It ranges from « 8 g /m 2 at an impingement angle of 6° to « 11 g /m 2 at an impingement angle of 12°. Table 3.10: Basis weight at the end of the jet impingement region for different impingement angles (GW newsprint furnish with fabric 1 at 15 m/s jet velocity). Impingement angle [°] Basis weight [g/m2] 6 8.3 8 9.45 10 10.15 12 11.15 3.1.6 Discussion of assumptions Some of the assumptions made in the beginning of the chapter will now be discussed in more detail to show their validity. Lamina r flow The jet leaves the headbox in turbulent flow, but the turbulence decays very fast once the jet enters the ambient air. Therefore it can be assumed that the jet is laminar at impingement and afterwards. Some turbulent phenomena might occur directly at the impingement point and in the flow through the fabric. However, this would influence the flow only locally, but would not influence the overall flow. Also, the important length scales would be influenced by the length scales of the single fabric strands and voids. As the fabric is modelled as a homogeneous medium, it would not be possible to correctly compute influences from turbulence. If the fabric is considered a flat plate, the transition Reynolds number for the boundary flow is 5 • 105. The Reynolds number for the jet has to be based on the velocity difference between the jet and the fabric and the distance from the impingement point x. An average value for the velocity difference could be 0.5 m/s. The distance from the impingement point where transition would occur can then be estimated. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 87 Re = {Vfabric ~~ Vjet) X 0.5 s 1 -X s = 5-10 5 =• x = 0.5 m (3.19) v 0.5-10-4 This is larger than the modelled domain length of 0.07 m, so transition to turbulence will not occur within the modelled domain. Accounting for the roughness of the fabric would lead to earlier transition, although it would likely still occur outside of the modelled domain. Neglecting the influence of gravity In modern paper machines jet velocities are very high and the influence of gravity therefore should be negligible. The error introduced by this assumption can be estimated. Assuming that the forming fabric is horizontal (the worst case), gravity would add to the drainage force. Also assuming that no drainage takes place (so the jet thickness remains at 1 cm), the additional force on the fabric per metre machine width would be (The length L downstream of the impingement point in the modelled domain is approx-imately 7 cm.) Depending on impingement angle and jet velocity, the value of Fgrav is between one and two orders of magnitude lower than the impingement force and therefore can be neglected. The fabric remains straight in the impingement zone As mentioned, this would only be the case for an infinite fabric tension. With the fabric tensions used in practice the fabric will deform, which will change the effective impingement angle. Approximating the impingement pressure with a point force, a force balance can be done for the fabric according to Figure 3.29 to estimate how much the angle would change: From Figure 3.29 the following relations can be established: Fgrav = mg = pHLg « 7N (3.20) T s i n 0 i + T s i n 0 2 - - F i m p = O , „ Ay , „ Ay CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC F, 88 imp undeformed fabric A" deformed fabric A y 4-F imp T sin 0,1, T T 1 ^ , T s i n 0 2 Figure 3.29: Force balance for the fabric. Now, for small angles sint9 w tan# « 0, and the two equations can be simplified and combined: Ay Aj/ _ T ~ r J j- — r imp &Av = ^ For an estimate, assume L i = 30 cm L2 = 5 cm T = 7 kN F i m p - 500 N then Ay « 3 mm and 8i « 0.6° At higher impingement angles and high velocities (=high impingement forces), the in-fluence of the change in impingement angle therefore would be noticeable. However it would be easy to adjust for the change in angle if this would be desired. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 89 The fabric can be represented as a homogeneous porous medium This assumption was already discussed. It is valid as long as the important length scale (here the jet thickness) in the model is much larger than the length scale of single fabric strands and voids. Any phenomena occurring over smaller distances, like the maximum pressure peak, have to be treated carefully. Also, the model cannot predict the flow field in the fabric itself, but this was not intended and is not necessary to predict the jet flow properties and drainage. The dependence of the fibre mat resistance on the flow angle, which was found later on, is neglected. None of the computations were repeated including this dependence, as no exact information is available. Assuming constant SFR and 100% retention The reasons and consequences for choos-ing a constant SFR value to describe fibre mat resistance were already discussed in the previous chapter. The pressure dependence (due to the mat compressibility) of the SFR should introduce only a small error in the computations, as it becomes important only at higher basis weights, where a significant number of fibre layers have been deposited. A more significant error will be related to the fines and filler content of the fibre suspension and is interwoven with the assumption of 100% retention. The assumption of 100% retention influences two aspects of the calculations. Firstly, the actual basis weight will be less if real retention values would be used in the calculations. Secondly, and even more important, the varying retention of fines and fillers throughout the drainage process does influence the SFR of the fibre mat noticeably. As discussed earlier, at least at low basis weights, this is in fact the most important factor leading to a varying SFR value throughout the forming region. Despite these limitations, the assumption of 100% retention was made for simplifi-cation and because no reasonable, generally applicable model for retention values in forming is available. Retention values are unique to each fibre suspension and paper machine and therefore in general have to be determined experimentally. In addition, to be useful for the model, not only would separate retention values for fibres, fines and fillers need to be known, but also the dependence of the SFR on the retention values would need to be determined experimentally. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 90 3.1.7 Summary In this section, a two-dimensional viscous model was developed to compute jet impingement on a single fabric, including fibre mat build-up. Computations were first limited to cases without fibre mat build-up. A comparison with potential flow theory for impingement on a solid wall showed good agreement except in the near vicinity of the impingement point. The area over which results do not agree is negligible. Investigating the influence of different fabric resistance numbers revealed that, contrary to common opinion, not only the inertial resistance but also the viscous resistance of the fabric are important. Although the influence of the viscous resistance on the impingement pressure is small, it has significant influence on the amount of drainage at impingement. Neglecting the fibre mat build-up, reasonable results for drainage, pressure profiles and velocity profiles were then predicted for different machine variable settings. The majority of drainage always occurs over a distance on the order of one or two times the jet thickness. Further downstream, pressure profiles along the fabric are not significantly influenced by different variable settings. The model shows that drainage is higher with higher jet velocity and higher impingement angle, as would be expected. The integrated pressure follows the predictions of a simplified one-dimensional analysis. The shear stress along the fabric for different rush/drag of the jet was computed and found to be large at impingement and nearly constant (at values according to the jet/fabric speed difference) some distance downstream from jet impingement. Due to the drainage through the fabric, the boundary layer cannot grow and remains negligible. Fibre mat build-up was then included in the computations with the fibre mat resistance determined based on the basis weight and the specific filtration resistance. Even more so than without a fibre mat, the major part of drainage occurs over small distances comparable to the jet thickness. The model predicts higher relative drainage for furnishes with lower resistance (lower SFR value) as expected. It also predicts lower values of integrated impingement pressure for lower resistance furnishes, but the difference is very small for different furnishes. At higher SFR values, more drainage takes place over the first one to two centimetres. Therefore the fibre mat build-up for high resistance pulps relatively seen occurs over smaller distances than for low SFR pulps. The influence of the fibre mat diminishes with higher CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 91 resistance fabrics because the fabric characteristics become more dominant. With fibre mat build-up included, drainage and integrated pressure are again higher with higher jet velocities and higher impingement angles. The integrated pressure accords with what is predicted from the simple one-dimensional analysis, although in the case of different jet velocities the computed values at higher velocities deviate somewhat more than at lower velocities due to the influence of the fibre mat build-up. The influence of different velocities and angles on the integrated pressure is much more pronounced than the influence of different furnishes. This is not the case for their influence on relative drainage. 3.2 Experimental work There is little experimental data available regarding jet impingement. No drainage or pres-sure data could be found for impingement on a single fabric which could have been used for a comparison with the computations. It was therefore decided to perform some experiments as part of this thesis. A device was built to measure drainage profiles for jet impingement on a single stationary fabric. A stationary fabric was used for the experimental set-up as a moving fabric would be much more complicated. To compare predictions from the computations to these experimental data, the fabric characteristics are required. The determination of fabric characteristics of three sample fabrics was described earlier (Chapter 2.2.1). 3.2.1 Drainage profile measurements at impingement on a station-ary fabric Drainage profiles for impingement of a two-dimensional jet on a stationary fabric were de-termined by capturing the drainage flows in compartments under the fabric and measuring these flows (Figure 3.30). The compartments were formed by placing blades beneath the fabric at certain positions downstream of the impingement point to capture the drainage from each respective section. The blades were designed to be close to the fabric but not touching it, to avoid creating local "blade" pressures. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 92 2 cm 4 cm Figure 3.30: Principle of drainage profile measurements. The following requirements had to be taken into account in the design of this drainage experiment: • The impinging jet should have an approximately constant velocity profile. • The fabric should be under a known tension comparable to values used in real appli-cations. • The drainage collection under the fabric should not influence the drainage process itself, but still give an accurate drainage profile. This means that the blades used to divide the drainage zone should be as close as possible to the fabric, but without touching it. Also, no back pressure must develop under the fabric that influences drainage. • The jet should impinge always at the same position, independent of impingement angle and velocity. 3.2.1.1 Experimental set-up A device was designed, combining a frame to hold the fabric, a nozzle creating a plane jet and an apparatus which divides and catches the drained water. For convenience, this device is called the Drainage Fractionator (DF). The DF was used together with an existing mobile flow loop that was modified to accommodate the installation of the DF. The mobile flow loop, once modified for this application, is composed of four major parts, which are all mounted on CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 93 a wheeled cart and so are portable (see Figure 3.31). The four components are the holding tank with piping, the pump with motor, the control panel and the Drainage Fractionator. Figure 3.31: Front view of the mobile flow loop. Tank and p ip ing The tank used in the mobile flow loop is a 270 L tank, connected to the pump by a 1.5" diameter P V C pipe. On the pressure side the pump is connected to the nozzle of the Drainage Fractionator via the same 1.5" piping followed by a flexible tube. A magnetic flowmeter is mounted vertically inline with the pressure pipe of the pump. P V C ball valves are installed before and after the pump. P u m p and motor The flow loop is equipped with a centrifugal pump powered by a 7.5 hp electric motor. The pump and motor are mounted at the bottom of the cart and protected against splashing with a Plexiglas water shield. Con t ro l panel The variable frequency drive (VFD) for controlling the motor, the magnetic flowmeter transmitter, and the magnetic contactor are located on the control panel fixed to the side of the cart. During operation of the flow loop the valves were always fully open and flow velocity was adjusted with the V F D . The control panel was also shielded with a Plexiglas panel. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 94 Drainage Fract ionator The DF itself is composed of three principle parts, which are the flow nozzle, the fabric with frame, and the chambers to divide and capture the drained water (see Figure 3.32). It is portable and therefore could also be used in a different flow loop if, for example, higher velocities are required. The device was mounted horizontally on the flow loop cart. The nozzle has the dimensions of 5 mm height by 50 mm width and therefore, with its aspect ratio of 10:1, should deliver a jet that can be considered two-dimensional. The nozzle was fed by three smaller tubes to deliver an approximately constant velocity profile over the nozzle width. A constant velocity profile was then assumed but not verified experimentally. The angle of the nozzle with respect to the fabric as well as the distance from the impingement point can be adjusted. The fabric was stretched in a tensile tester at a specified tension of approximately 7 kN /m and then fixed in a 100 mm wide and 250 mm long aluminum frame. The drained water is captured in four chambers over an area of 40 mm width by 100 mm length. The water draining outside of this area is collected in a tank. The drainage collecting device is placed beneath the fabric with the four side walls, which form the main frame of the 40 by 100 mm measurement area, touching the fabric. The side walls are bevelled so that they have a small width where they touch the fabric, to avoid a significant influence on drainage. Impingement takes place 20 mm downstream of the front end of the drainage area, with the impingement point being defined as the point where the lower surface of the Figure 3.32: The Drainage Fractionator. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 95 jet touches the fabric. A blade is fixed beneath the fabric at the impingement point, defining the actual beginning of the measurement area. At 10 mm, 20 mm and 40 mm downstream of the impingement point additional blades are mounted, dividing the area over which drainage is measured into 10 mm, 10 mm, 20 mm and 40 mm long chambers. The drainage from the first 20 mm, before the impingement point, is discarded. The blades are very thin and vertically adjustable, so that they do not touch the fabric, which ensures that drainage is uninfluenced. The distance between blades and fabric was 0.25 mm with no flow. This ensured that even when the fabric was bent by the impingement force, the blades did not touch the fabric. From the chambers, the water was drained through hoses into different buckets. 3.2.1.2 Test conditions Drainage profiles were determined for all three fabrics. Different angles and jet velocities were used. For some angle/velocity combinations, the drainage profiles were determined on different days to determine the repeatability of measurements. Measurements with different velocities (at a constant impingement angle) were carried out in one series, followed by a series of measurements at different angles from 20° to 90° at constant jet velocity. This was done for all fabrics. The distance of the nozzle from the impingement point was always 20 mm. Only the measurements for 15° were done separately in the end, as the distance of the nozzle from the impingement point had to be adjusted (due to geometrical restrictions). This may also have changed the impingement point. Preliminary trials showed that drainage profiles can change significantly with even the smallest changes in impingement point, presumably because of the position of the first blade right at the impingement point. Because the drainage velocities are highest at the impingement point, the drainage in the first chamber will change noticeably if impingement takes place at a small distance before or after the presumed impingement point. See Table 3.11 for a complete list of test conditions. Clean water at a temperature of 17°C was used in the tests. During start up of the pump the fabric was covered with a thin plastic foil to avoid drainage. The plastic foil was removed and the measurement started once the water level in the tank reached a certain mark. The measurement was stopped by putting the plastic foil back on once the water level CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 96 Table 3.11: Test conditions for determination of drainage profiles. Fabric Jet velocity Vjet [m/s] Impingement angle a [°] C, D, E 7.0 15, 20, 30 (2x), 45, 90 3.7, 4.8, 6.1, 7.0 (2x) 20 C, D 3.7 20, 30, 45, 90 reached a second mark. In doing so it was ensured that measurements were always taken for the same total amount of flow (0.12 m 3). During the measurement, the drained water from the different chambers was collected in buckets and the volume was determined. The time during which the measurement took place was noted. For each combination of variables (jet velocity, impingement angle and fabric), 10 mea-surements were taken and averaged. In the following, this average will be referred to as a "measurement". The standard deviation in the measured drainage velocities for each series of 10 measurements was typically in the range of 1 to 2%. 3.2.1.3 Experimental results Drainage as a function of distance from the impingement point Drainage profiles were determined by assuming that the drainage from impingement was fully completed within the test section, and therefore the total collected water was defined as 100% drainage. With this, the fraction of completed drainage can be calculated and is represented in a drainage curve as a function of distance from the impingement point. Figure 3.33 shows the drainage curves for differing impingement angles for jet impingement at 7.0 m/s on fabric C. With increasing impingement angle, the relative amount of water drained within the first 10 mm increases. This is consistent for all fabrics. (Plots with the results of all fabrics are given in Appendix A.) Figure 3.34 shows the drainage curves for different jet velocities. The difference between the curves is very small and becomes negligible at higher velocities. (The variance between the curves for velocities higher than 4.8 m/s is less than 5%.) The small differences might show the influence of gravity on drainage, which at the lower velocities may still be of CHAPTER 3 JET IMPINGEMENT ON A SINGLE FABRIC 97 120 100 0 2 3 4 5 6 7 8 Distance [cm] Figure 3.33: Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric C, 7.0 m/s jet velocity). influence, but at higher velocities is not. If this is the case, the results of the experiments at the lowest velocity of 3.7 m/s should be looked at with caution. It is interesting to note that for fabric C and D the completed drainage is a little higher at respective points for higher velocities, but for fabric E drainage at respective points seems to be lower with higher velocities (see Figure 3.35). This suggests a very different drainage behaviour for this single-layer fabric. As the impingement pressure is lower for this low resistance fabric than for the higher resistance fabric, gravity might have a bigger influence not only at the low jet velocity but at all jet velocities, therefore changing the behaviour compared to the other fabrics. However, the differences between the curves are very small and therefore could also just lie in the range of measurement errors. Overall relative drainage Overall relative drainage can be determined by dividing the sum of the collected drainage water by the flow from the nozzle. Results for the overall relative drainage as a function of jet velocity of all three fabrics are plotted in Figure 3.36. It is interesting that for fabric C and D drainage is basically constant for different velocities, CHAPTER 3, JET IMPINGEMENT ON A SINGLE FABRIC 98 120 100 0 2 3 4 5 6 7 8 Distance [cm] Figure 3.34: Fraction of completed drainage as a function of the distance from impingement point for different jet velocities (fabric C, 20° impingement angle). while for fabric E the relative drainage decreases significantly with higher jet velocities. Again this shows a very different drainage behaviour of fabric E (the single-layer fabric). The overall relative drainage as a function of impingement angle is shown in Figure 3.37. It did not follow a clear trend, and the curves for 7.0 m/s and 3.7 m/s show very different behaviour. One reason for the inconsistent behaviour might be that at higher impingement angles, flow in the direction opposite to the impingement direction on the fabric is significant, but any drainage which takes place upstream of the earlier defined impingement point is not captured. This would lead to lower than expected relative drainage amounts for higher impingement angles. Another reason for the inconsistent behaviour might be the fact that the experimental set-up reacted very sensitively to any changes in impingement position as mentioned earlier. Although the experiment was designed to keep the impingement position as constant as possible, even for different impingement angles, it is possible that during adjustment to a new angle the impingement position might have changed a little. This makes a comparison of the overall relative drainage at different angles questionable. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 99 CD J D "E. S o o CD 6 0 ca g ?3 120 100 80 60 h 40 h 20 h J i l l 7.0 m/s ^ — i — i 6.1 m/s x- • 4.8 m/s x 3.7 m/s * a >• 1 1 1 I 0 1 2 3 4 5 6 7 8 Distance [cm] Figure 3.35: Fraction of completed drainage as a function of the distance from impingement point for different jet velocities (fabric E, 20° impingement angle). 100 80 60 CD 6 0 CS a "8 T3 CD •B 40 JS T3 Pi 20 3.5 4.5 5 5.5 6 6.5 Jet velocity [m/s] Si.... i i 1 1 fabric C fabric D !•-fabric E - X ! • • * "•*-..[ T ^ ^ T 1 1 i ' t i 7.5 Figure 3.36: Overall relative drainage as a function of jet velocity (20° impingement angle). CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 100 0 s CD 00 c3 C "S -o > Pi fabric C, 7.0 m/s fabric D, 7.0 m/s fabric E, 7.0 m/s fabric C, 3.7 m/s fabric D, 3.7 m/s 30 40 50 60 Impingement angle [°] 90 Figure 3.37: Overall relative drainage as a function of impingement angle. Dra inage veloci ty profi le From the collected volume of water, the area from which the drainage is collected and the duration of the measurement, the average drainage velocity for each compartment of the device can be calculated. Figure 3.38 shows the drainage velocity profile for different jet velocities for fabric C. The drainage velocity is higher with higher jet velocities as expected. For all curves the drainage velocity is highest close to the impingement point but then diminishes, as also expected. The computations have shown that the pressure pulse at impingement, which causes drainage, also diminishes quickly downstream from the impingement point. It is interesting that fabric E again shows a very different behaviour than the other fabrics, with the drainage velocity being lower for higher jet velocities (see appendix, Figure A.11). This would imply a lower pressure pulse for higher jet velocities, which seems unreasonable. The drainage behaviour of fabric E therefore can't be readily explained. Figure 3.39 shows drainage velocity profiles for different impingement angles at constant jet velocity. For higher impingement angles, the drainage velocity is also higher close to the impingement point, but drops more quickly than for lower angles. This implies that the CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 101 0.7 7.0 m/s '— '—' 6.1 m/s -x—-! 0 0 2 3 4 5 6 Distance [cm] Figure 3.38: Drainage velocity as a function of distance from impingement point for different jet velocities (fabric C, 20° impingement angle). pressure pulse for higher impingement angles is higher but less wide (and with a steeper drop-off) than at lower angles, which again agrees with the trend of the computations. The behaviour of drainage velocity as a function of impingement angle is similar for all three fabrics. Repeatability of the measurements Two of the measurements were repeated on a different day. The measurement at 20° impingement angle for fabric C at 7.0 m/s jet velocity was measured on one day when a series of different jet velocities were measured for fabric C, and then again in the following series of measurements for fabric C on the next day (different impingement angles). The impingement angle between these two measurements was not changed. Both measurements agreed very well: the overall drainage was 45.5% for one test and 45.0% for the other and the drainage profiles (Figure 3.40) and drainage velocity profiles (Figure 3.41) are similar. The measurement for a 30° impingement angle for fabric C at 7.0 m/s jet velocity was also repeated, but this time there was a week between the measurements and the impingement CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 102 1.2 20° x— -30° • * • 45° > e < 90° 0 0 2 3 4 5 6 Distance [cm] Figure 3.39: Drainage velocity as a function of distance from impingement point for different impingement angles (fabric C, 7.0 m/s jet velocity). angle had been changed numerous times during this week. The agreement between these two measurements is poor. The drainage profiles still look quite similar (Figure 3.40), but overall drainage (44.4% and 52.3%) and therefore also the drainage velocity profiles (Figure 3.41) do not agree as well as before. This finding suggests that the accuracy of the drainage measurements is probably ±10%. However, as long as the nozzle angle is not changed between measurements, the repeata-bility of the measurements is very good and therefore each series of measurements at different jet velocities (where impingement angle was not changed) should be very consistent. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 103 120 100 h I 80 o o CD 00 CS a 2 Q 60 h 40 20 0 j--^^^ i 1 >0°, test 1 (23.8.) + >0°, test 2 (24.8.) SO", test 1 (23.8.) SO", test 2 (30.8.) • 0 1 2 3 4 5 6 7 8 Distance [cm] Figure 3.40: Repeatability of measurements: fraction of completed drainage as a function of distance from impingement point (jet velocity 7.0 m/s, fabric C). 1 CJ o > CD 00 es g 'ed 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 1 Q 20°, test 1 (23.8.) + 20°, test 2 (24.8.) * 30°, test 1 (23.8.) * 30°, test 2 (30.8.) a ! \ \ S i ; i i Distance [cm] Figure 3.41: Repeatability of measurements: drainage velocity profiles (jet velocity 7.0 m/s, fabric C). CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 104 3.2.2 Comparison of experimental and computational results The measurements described in the previous section were modelled in F L U E N T . Computa-tional and experimental results were then compared. The computations should resemble the real situation as closely as possible. The following set-up was therefore chosen. To model the fabrics, the average fabric resistance values determined in the first fabric resistance trials were used, as outlined in Chapter 2.2.1. (The measurements for fabric resistance at different approach flow angles were carried out later.) It must be noted that the error in the resistance numbers can be substantial, especially for the viscous permeability. It was assumed that the resistance in fabric direction is high so that no flow parallel to the fabric plane (in x-direction) takes place within the fabric. In reality this is probably not true, but the resistance in fabric direction is not known. However, using a fabric resistance in the x-direction similar to the value for the y-direction (which might be closer to reality) did not alter the results. Due to better convergence, the high resistance was therefore used throughout the computations. As the jet velocities used in the measurements are rather low, gravity may be significant and was therefore included in the computations. The porosity of the fabric must then be given to correctly compute the influence from body forces in the porous medium. The porosity most likely varies for the different fabrics; therefore an average value of 60% porosity was assumed because a precise value for this variable is unknown. The fabric is stationary and therefore the x-velocity in the fabric is not fixed. Using a stationary instead of a moving fabric altered the convergence behaviour. Con-vergence for the stationary fabric is more difficult to achieve if a long distance after the impingement point is modelled, presumably due to air eventually reentering the fabric if drainage velocities approach zero. Therefore the modelled domain, in general, had to be shorter than that used in the experiments. Also, different meshes had to be used for differ-ent cases. A list of the meshes used and their dimensions is given in Table B. l in Appendix B. Comparisons of different jet velocities First, a comparison was carried out for fabric C. Figure 3.42 shows the computed drainage velocity profiles as a function of distance from the impingement point (which is at 0 cm) for the jet velocities that were used in the experiment CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 105 for this particular fabric. As in the experiment, the impingement point is defined as the theoretical impingement position of the lower surface of the jet if the influence of gravity and jet spread are neglected. The drainage velocities are higher over the whole drainage length for increasing jet velocities. The experimentally determined drainage velocities (Figure 3.38) showed the same be-haviour. However it becomes apparent that the computed drainage velocities, when averaged over the respective distances that were used in the experiment to collect water, seem to be noticeably lower than the measured ones. 7.0 m/s 6.1 m/s 4.8 m/s 3.7 m/s 0 1 2 3 4 5 Distance [cm] Figure 3.42: Computed drainage velocity profiles for different jet velocities (impingement angle 20°, fabric C). This can be seen clearly in Figure 3.43, which shows the comparison between experiment and computation for the highest and lowest velocity case. For the low velocity, the difference between computation and experiment is about 0.1 m/s and is approximately constant over the whole drainage length. At the high velocity, the difference is also about 0.1 m/s in the beginning, but even larger at the second and third measurement point. As in these computations the average determined drainage resistance of the fabric was used (Table 2.2), CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 106 and the error in this average can be relatively large, the computation for the high velocity was repeated using the minimum estimated fabric resistance numbers (Table 2.3). In this case, the average drainage velocity over the first centimetre matches the measured drainage velocity very well. Over the remaining length, the computed drainage velocities are still lower than the measured velocities. 1 + \ ' ' . 1 7.0 m/s, 7.0 n 7.0 m/ 3.7 n 3.7 m/ computed, rr l/s, compute s, experimer i/s, compute s, experimer lin. resistanc d + It x d * it • e • \ \ ! \ 0 1 2 3 4 5 6 Distance [cm] Figure 3.43: Experimental and computed drainage velocity profiles for different jet velocities (impingement angle 20°, fabric C). It was found that the difference between the average measured and computed drainage velocities over the first centimetre from impingement is constant for all jet velocities tested. Figure 3.44 shows the average drainage velocities over the first centimetre for the compu-tations and experiments as a function of jet velocity. It also shows the drainage velocities over the second centimetre, where the difference is not constant but increases with higher jet velocities. The drainage velocity behaviour is reflected in the drainage profiles. Again, the behaviour of the computed drainage profiles (Figure 3.45) for different jet velocities is similar to what was seen in the experiments (Figure 3.34), with the drainage profiles being relatively inde-CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 107 0.7 0.6 0.5 0.4 0.3 Q 0.2 0.1 0 o o CD 00 CO .3 ca 3.5 4.5 5 5.5 Jet velocity [m/s] 1 "I exp. at 0.5 cm comp. at 0.5 cm exp. at 1.5 cm comp. at 1.5 cm I x X-B , . x Mr''' x " ~ ' * * : • ' 1 i i ! n i i v) ."' • 6.5 Figure 3.44: Measured and computed drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of jet velocity (impingement angle 20°, fabric C). Drainage velocities are averaged over 1 cm length (from 0 to 1 cm and 1 to 2 cm). pendent of the jet velocity. However, a direct comparison, as done in Figure 3.46 for the highest and lowest jet velocity cases, shows the difference between the computations and the experiments. The fraction of completed drainage in the experiments is significantly less at comparable distances, based on the higher drainage velocities especially further away from the impingement point, where the computed drainage velocities are nearly negligible and, therefore, do not significantly contribute to the drainage amounts. The comparisons were then also done for fabrics D and E (see plots in Appendix B). Fabric D shows the same trends in the comparison as fabric C. For fabric E, the behaviour of the predicted drainage velocities was similar to fabrics C and D, however, due to the very different behaviour of fabric E in the experiments, the discrepancies between experiments and computations are even larger and also show a different trend. But, as discussed previously, the reliability of the measured values for fabric E is questionable. As shown in Figure 3.47, the difference in computed and measured drainage velocities can CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 108 120 Distance [cm] Figure 3.45: Computed fraction of completed drainage as a function of distance from im-pingement point for different jet velocities (impingement angle 20°, fabric C). Distance [cm] Figure 3.46: Experimental and computed fraction of completed drainage as a function of distance from impingement point (impingement angle 20°, fabric C). CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 109 also be seen in the overall relative drainage numbers for different jet velocities. For fabrics C and D, the computed values show a very subtle increase with higher jet velocity, and are only about half of the measured relative drainage numbers, which were approximately constant over the covered jet velocity range. For fabric E, the computed values show the same trend as for fabric C and D, but the experimental values show a completely different behaviour, as discussed before. Of interest is the fact that the computations for fabric D show about the same relative drainage as fabric C, while in the experiments it is about 5% higher for fabric C. This might be attributed to errors in the measured fabric resistance numbers and shows that this can introduce noticeable errors in the computational results. <0 so g 2 > 100 80 60 •a 40 Pi 20 1 1 I 1 I • -C, comp. — 1 — C, exp. — - x -D, comp. *—-D. exp. a ... E, comp. - - - • — oj - : m m i »— | . [ "1 L *L i i i i i 3.5 4 4.5 5 5.5 6 6.5 Jet velocity [m/s] 7.5 Figure 3.47: Computed and experimental overall relative drainage as a function of jet velocity (impingement angle 20°). Comparisons of different impingement angles A comparison of the drainage velocities as a function of the impingement angle between computation and experiment (again done here for fabric C) shows the same agreement in basic behaviour. At higher impingement angles the drainage velocity is higher in the beginning, but drops faster (the computational results are shown in Figure 3.48 and the experimental results in Figure 3.39). This trend is CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 110 a little less consistent for the experimental results, which suggests inaccuracies introduced when the impingement angle was changed. If actual values of the drainage velocities are compared (as in Figure 3.49), they do not agree very well as in the case of differing jet velocities. 1.6 i 1 1 1 1 15° / \ j 20° 1-4 : \ [ \ i 30° -\ 4 5 ° 1 - - V 0.8 r \ 0.6 h 0.4 0.2 0 1 ' ,.- --—r-~~ , , i I 0 1 2 . 3 4 5 Distance [cm] Figure 3.48: Computed drainage velocity profiles for different impingement angles (jet ve-locity 7.0 m/s, fabric C). 00 C3 a 2 Q Unlike for different jet velocities, the drainage velocities however are not always un-derpredicted in the F L U E N T model. Over the first centimetre, the drainage velocity is overpredicted at high impingement angles, but underpredicted at low impingement angles, as can be seen in Figure 3.50, which shows the average drainage velocity as a function of impingement angle over the first and second centimetre after impingement. Over the second centimetre (and also subsequently), the drainage velocities are again underpredicted at all impingement angles. The inconsistent behaviour of the experimental values over the second centimetre shows that the measurements at different impingement angles, at least further downstream from the impingement point, might not be that reliable. Also, if the same plot is done for a jet velocity of 3.7 m/s (see Appendix B), the behaviour seems different. Drainage CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 111 1.2 V 0.8 o > SO a 0.6 0.4 Q 0.2 -0.2 20 °, computed + 20 °, experiment * \ 90 ' experimen t H X '• i \ > — i Distance [cm] Figure 3.49: Experimental and computed drainage velocity profiles (jet velocity 7.0 m/s, fabric C). velocities are predicted to be too small at all angles, even over the first centimetre. At this low velocity, a deviation of the jet from a straight path before impingement was observed, possibly due to gravity, so the actual impingement point and impingement angle are differ-ent than assumed, which influences the drainage results and makes them unreliable. This deviation from a straight path is not apparent when looking at the computed jet shape, and therefore the influence of gravity is probably not correctly computed. As in the case of different impingement angles, the drainage velocity behaviour is mirrored in the drainage curves. Relative to each other, the drainage curves show similar behaviour for the computations (Figure 3.51) and experiment (Figure 3.33). The higher impingement angle gives a higher fraction of drainage over the first and second centimetre. However when absolute values are compared (as in Figure 3.52 for 20° and 90° impingement angle), less drainage is completed over respective distances in the experiment. As before, this can be attributed to the measured drainage velocities which are substantially higher than the computed ones for positions farther away from the impingement point. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 112 1.2 exp. at 0.5 cm comp. at 0.5 cm exp. at 1.5 cm "g 0.4 -Q x ' 0.2 k } a i 0 10 20 30 40 50 60 70 80 90 Impingement angle [°] Figure 3.50: Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 7.0 m/s, fabric C). For the case of varying impingement angle, both other fabrics, D and E, show the same trends when predicted values are compared to measured ones. Discussion of the comparison Different possible causes of the discrepancies between computed and measured drainage velocities can be found from the above observations: • The fabric resistance depends on the flow angle. • The measured fabric resistance, particularly the viscous permeability, is inaccurate. • The experimental set-up to determine drainage profiles might introduce errors. • Fabric roughness, neglected in the computations, may affect drainage. • The influence of gravity is not accounted for correctly. A possible influence of the flow angle on the fabric resistance is suggested by the observa-tion that the difference between measured and computed drainage velocity seems to depend CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 113 120 100 2? - 80 to *9-o o 0> 00 ca a 2 Q 60 40 20 ! ! ! ! ! til : & 15° — — 9 0 ° i i i i 30° 45° 90 o 0 1 2 3 4 5 6 7 8 Distance [cm] Figure 3.51: Computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 7.0 m/s, fabric C). 100 80 •a CD -4-» JJ "E. 3 o o D .3 2 Q 60 V 40 20 ! "i I— •• -> . 1 — i : -| '•' // / - //"/ _f" 20 °, computed 20.°, experiment + 90 °, computed 90 °, experiment x 3 4 5 Distance [cm] Figure 3.52: Experimental and computed fraction of completed drainage as a function of distance from impingement point (jet velocity 7.0 m/s, fabric C). CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 114 on the impingement angle. The drainage velocity over the first centimetre is predicted to be smaller than measured at small impingement angles for all jet velocities. However, as the impingement angle is increased, the prediction improves. (This is not the case if a jet velocity of 3.7 m/s is used, but the measurements at this velocity seem to be unreliable). In addition, the flow model predicts drainage more poorly at points further from the impinge-ment point (from about the second measurement point on), where the flow is nearly parallel to the fabric. One can therefore conclude that the resistance of the fabric may depend on the flow angle. If this is the case, the computational model cannot correctly predict the flow, because all resistance values used in the computations were determined for a flow angle of 90°, as is common practice. The dependence of the fabric resistance on the jet angle is a potentially important factor and therefore it was decided to repeat the fabric resistance measurements at different flow angles. These measurements are described in Chapter 2.2.2. A dependence of the fabric resistance on the flow angle was indeed observed for the inertial resistance. Resistance increases at smaller impingement angles for a common superficial velocity through the fabric. The new measurements showed again that results are unreliable at small pressure drops and small drainage velocities, and therefore the viscous permeability is difficult to determine. Thus, an effect of jet angle on the drainage velocities is likely, but is not possible to specify in detail. Computations therefore were not repeated with the newly determined resistance values. As discrepancies were always observed at low flow velocities, which are mainly influenced by the viscous permeability, and where inertia is of less influence, the error in the viscous permeability itself is also likely an important factor causing discrepancies between experiment and computation. Errors may also originate during measuring drainage profiles, but are probably not sig-nificant. If water does not drain perpendicularly through the fabric as assumed, some water carry-over at the leading edge of the Drainage Fractionator and then from chamber to cham-ber might occur due to water travelling in x-direction within the fabric before it is drained. The amount should be negligible compared to the overall drainage from one chamber. Also, the side walls of the Drainage Fractionator, which touch the fabric, might introduce a pres-CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 115 sure pulse, which should be very small. A pressure pulse from this source can originate only where the force on the fabric is large enough to cause some wrap of the fabric around the side walls. The fabric experiences high pressures only at the impingement point. Thus, even if a pressure pulse occurs here, it cannot explain the differences between experiment and computation further downstream. A more likely source of error in the measurements is the nozzle. The design of the nozzle is very simple and might deliver a jet with an uneven velocity profile. The part of the jet from which drainage is measured therefore might have a higher velocity than the assumed average. This could explain why measured drainage velocities are higher than the predicted ones, but it cannot explain why the prediction is poor further away from the impingement point. Neglecting the fabric roughness in the computations might cause lower predicted drainage velocities than measured ones. A typical value for the fabric roughness, based on the diameter of the yarn used in top layers of forming fabrics, is a minimum of 0.2 mm. The ratio of fabric roughness to jet thickness, based on the original jet thickness of 5 mm, is then 0.04 or 4%. The yarn is woven into a weave, which likely introduces an even larger scale of roughness. Towards the end of the modelled region, part of the jet is drained and the ratio would also be higher. Therefore it is likely that the fabric roughness affects the pressure on the fabric surface, leading to slightly higher drainage velocities than predicted in the computations, for which a smooth surface is assumed. It is difficult to quantify the effect from fabric roughness, but low flow velocities combined with low flow angles may introduce differences between the experimental measurement and what has been modelled that account for the differences that have been observed. As mentioned earlier, it was suspected that the influence of gravity is not correctly represented in the computations. If this is a factor, measured drainage velocities would be higher than predicted ones. The computed pressure towards the end of the modelled domain (where the pressure is caused only by gravity) was therefore compared to the theoretical pressure caused by gravity (Equation 3.20). Based on the remaining jet thickness at the end of the domain, it agreed very well with the computed ones. Therefore, an incorrectly computed influence from gravity can be ruled out. CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 116 In summary the following can be concluded from the comparison between measured and computed drainage profiles. The most likely sources for the discrepancies between computation and experiment are inaccurately determined viscous permeabilities of the fabric and fabric roughness neglected in the computations. A dependence of the fabric resistance on the flow angle may also influence the accuracy of the computations,.but the effect cannot be quantified at the moment. It is possible that errors are introduced during measuring the drainage profiles, but these appear to be insignificant. The possible impact of these factors on the accuracy of the computations carried out in the first part of this chapter and in the next chapter should be small for the following reasons. When fibre mat build-up is included in the computations, the fabric resistance is important only in the vicinity of the impingement point. Drainage velocities here are high, and thus the inertial resistance, which has been more accurately determined than the viscous permeability, is of more importance. Even in the absence of mat build-up, errors should be small as the jet velocities used in those computations are much higher than in the experiment, which again means that the inertial resistance will be of more influence. The errors introduced by neglecting a dependence of the fabric resistance on the flow angle become smaller in the case of a moving fabric, as the flow angles relative to the fabric become larger. The flow angle that the fabric sees depends on the ratio of the x-velocity (of the flow relative to the fabric: VjettX — Vfabric) to the drainage velocity. In the experiment (and the computations for comparison), the fabric velocity is zero, therefore this ratio is large, giving a small relative angle between the flow and fabric. If the fabric is moving, Vjet,x ~ yfabric becomes smaller, and therefore the relative flow angle is much larger. In addition, the jet thickness in the experiment (5 mm) was small (typical values in twin-wire paper machines are around 10 mm). If the jet thickness is larger, as in all other computations, small drainage velocities have less influence on the relative drainage. Overall it can be said that the qualitative agreement of predicted and measured drainage velocity profiles is reasonable. The velocities over the first centimetre after impingement are predicted with a reasonable accuracy compared to the measured ones if all possible sources for errors are considered. To improve the agreement further downstream from the impingement point, the fabric description would need to be explored in a more thorough CHAPTER 3. JET IMPINGEMENT ON A SINGLE FABRIC 117 study of flow through a fabric. The computational model for impingement on a single fabric therefore might have some errors in exact drainage values, but the overall trends and order of magnitudes should be predicted correctly. Chapter 4 The initial flow in roll forming In the previous chapter a two-dimensional model for jet impingement on a single fabric was developed. This chapter extends this model to the wedge of a twin-wire former, including the jet impingement zone and the early wedge flow zone of roll forming. Computations were carried out for different machine settings, including different impingement positions. 4.1 The computational model Sub-models for computing the free surface flow, modelling permeable moving fabrics and computing fibre mat build-up during forming were introduced in previous chapters. They were combined to model impingement on a single fabric. If this model is to be extended to impingement and flow in the wedge of a roll former, an additional problem arises as the geometry of the forming zone is not known. The inner boundary of the forming wedge is given by the radius of the forming roll. However the outer boundary is determined by the flexible fabric deformed by flow in the wedge. An additional sub-model therefore needs to be developed which enables the calculation of the outer fabric shape. 4.1.1 Analysis The governing equations for the flow with free surface and flow through a porous medium are the same as shown in Chapter 3.1.1. Therefore, only an analysis to determine the correct shape of the outer fabric will be described here. The assumptions regarding the model are 118 CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 119 also similar, except the following: • The fabric now is under a finite tension T and can deform. It has no bending stiffness and no mass and is extensible. The tension remains constant even under deformation. • The forming roll is open and does not offer a drainage resistance additional to that of the fabric. According to membrane theory, for a membrane of zero stiffness, its radius of curvature is a function of the pressure differential across the membrane and the membrane tension. This theory can be used to describe the radius of curvature of the outer fabric with the pressure distribution in the nip and the fabric tension: J^local The radius of curvature is simply the reciprocal value of the curvature Rlocal = — (4.2) X The curvature of a function in a polar coordinate system g = £>(</>) is described by the radius and the first two derivatives of the radius with respect to the angle: x = S + 2 l t V - * . ' ( 4 . 3 ) (e2 + (e')2)' If the pressure and tension are known, the radius of curvature can be calculated. From Equation 4.3 the radius g of the fabric that is consistent with the assumed pressure distri-bution can be determined. 4.1.2 Method of solution The exact shape and position of the outer fabric in roll forming is determined iteratively. Using an initial guess of the geometry of the wedge, the flow field is solved in F L U E N T . The resulting pressure profile along the outer fabric is used to determine a new position for the CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 120 outer fabric with Equations 4.1 to 4.3. Solving for the fluid flow and updating the geometry is repeated until convergence is obtained. Equation 4.3 cannot be solved analytically. Therefore a 4th order Runge-Kutta method was used to solve for the radius, with the solution array for the pressure from F L U E N T used directly as input. The step-size of the angle in the Runge-Kutta calculation is then automatically fixed, as it is given by the mesh used in F L U E N T . The step-size is therefore not necessarily constant. The change in angle between two cells da can be calculated from the x and y positions of those cell centres which represent the inner edge of the fabric. To prove that with the F L U E N T mesh size the step-size is small enough to give a grid converged solution, a Runge-Kutta calculation was also done using half the step-size (and interpolating values for the pressure for the new grid points). The resulting fabric shape was unchanged. The overall method of solution for computations of impingement in the nip combines the two iteration procedures for fibre mat build-up and location of the outer fabric as follows. An initial location for the fabric is calculated from a one-dimensional analytical model. An initial deposited constant basis weight of 5 g /m 2 on each fabric is assumed and used to de-termine the combined fabric and fibre mat resistance. Then, the fluid flow is computed with the F L U E N T code, including the porous medium model for the fabric and the fibre mat. (The convergence criterion for the F L U E N T code was chosen as in single fabric impingement, although convergence was in general better with the residual sum often falling below 10 - 3 . ) Keeping the geometry constant, the correct fibre mat as a function of distance from impinge-ment is then determined with the iterative procedure described in Chapter 3.1.2.2. Once the fibre mat profile is converged for this specific geometry, the resulting pressure distribution along the outer fabric is used to determine the new location of the fabric as described above. The new geometry is imported into F L U E N T and the flow field is solved using the fibre mat resistance profile calculated in the last previous computation. The correct fibre mat profile for the new geometry is then determined. The calculation of a new geometry is repeated until the pressure distribution in the nip satisfies the force balance equations for the fabric. The overall procedure is summarized in Figure 4.1. One difference between the roll former model and the single fabric model lies in the description of the moving fabric. To model the movement of the fabric, the MD-velocity of CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 121 update geometry use last a = f(x) I initial geometry from 1-d-model assume 5 gsm mat input in FLUENT run computation 1 get v-velocity profile I calculate BW, oc=f(x) run new computation if not converged 1 get v-velocity profile compare to last v-profile for convergence get p-profile if converged if not converged 1 compare to last p-profile for convergence 11 if converged end iteration Figure 4.1: Schematic of the determination of the forming zone geometry in roll forming. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 122 the flow within the fabric would need to be fixed to the fabric velocity. In the single fabric model the M D coincides with the x-direction of the model, so that the x-component of the velocity is fixed. In the roll former model however, the fabrics follow a curved path and therefore the MD-velocity consists of x- and y-velocity component in the coordinate system used in F L U E N T . Both velocity components would have to be fixed, which would also fix the CD-velocity (drainage velocity) and make the calculation of this velocity impossible. Instead of using a fixed velocity, the fabric was represented with a zero resistance to water in the MD. The drained water can then move freely in the M D within the fabric. Although this is not the same as fixing the velocity, a comparison between the two ways of describing the moving fabric in single fabric impingement resulted in virtually the same results for pressure and drainage. The only drawback to that approach is that rush or drag cannot be modelled. However, in single fabric impingement it was shown that rush or drag does not affect the pressure field or drainage amount. The only influence of rush or drag was on shear, which was shown to be dependent on the velocity difference between the jet and the fabric, which could be estimated for roll forming in the same way if desired. 4.1.2.1 Boundary conditions The modelled domain in F L U E N T includes the inner and outer fabric. In machine direction, the domain starts some distance before the wedge and ends at that point where the fabrics part from the forming roll, therefore modelling the whole wrap length. Fluid flow boundary conditions The boundary conditions for the fluid flow are similar to the earlier ones. When the jet enters the domain, a velocity inflow condition is used, specifying the angle and magnitude of the entering jet velocity by the u and v velocity components. The volume fraction of water at the jet inflow is specified as one. At the outflow, a zero static pressure boundary is imposed, as the pressure drops to zero when the fabrics leave the roll. A l l remaining boundaries are also zero static pressure boundaries. The zero pressure boundary along the forming roll side of the inner fabric is based on the assumption that the forming roll is open so that drainage into the roll is possible. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 123 Fabric boundary conditions To solve the fabric equation, some boundary conditions have to be specified for the outer fabric. An analysis of the problem reveals that there are two fixed points which can be used as boundary conditions upstream and downstream. One is the fixed point at which the fabric leaves the breast roll. The second is where the fabric touches the first blade of the blade section. Typical tangential distances between the breast roll and the forming roll are 25 to 40 cm and between the forming roll and the first blade are in the range of 10 cm. Figure 4.2 a) shows schematically the geometry of the forming section when the machine is at rest with no fluid flow. The coordinates for the two fixed points can be calculated. a) without fibre suspension R , U l l l l l l l g 1 U I 1 1. blade breast roll 25 cm Distance from nip 0 cm b) with fibre suspension inner fabric outer fabric modelled domain ^ drainage Figure 4.2: Forming zone with fixed points P i , P 2 , a) without fibre suspension, b) with fibre suspension. C i and C 2 are corner points of the modelled domain. Figure 4.2 b) shows the wedge geometry with the fibre suspension. The approach for the fabric calculation is as follows. Once the fibre suspension impinges in the wedge, the outer fabric is deformed, but P i and P 2 remain fixed. As P i and P 2 do not coincide with the corner points C i and C 2 of the modelled domain on the outer fabric, the fabric shape is linearly interpolated between P i and C i , and P 2 and C 2 . The new points of this interpolation CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 124 are added to the data array of the fabric position from F L U E N T . The pressure along the additional fabric lengths is zero. For this extended domain, the calculation of fabric position can be done with the first and last point fixed as boundary conditions. From the resulting data array, only the x and y positions which describe the fabric for the domain modelled in F L U E N T are saved and can be imported back into F L U E N T to describe the new geometry. 4.1.3 Grid independence To verify grid independence, one case was computed with consecutively finer meshes of 150 x 112, 300 x 224 and 600 x 448 cells. The cells were spaced uniformly in the machine direction. In the cross direction, they were spaced consecutively closer towards the fabrics. Figure 4.3 shows the pressure distributions along the inner and outer fabric for these meshes. The overall relative drainage through the outer fabric was 40.9% for the coarse mesh, 41.2% for the medium mesh and 41.8% for the fine mesh. For the inner fabric, overall relative drainage was 39.7% (coarse mesh), 39.5% (medium mesh) and 40.2% (fine mesh). The differences in relative drainage are in the range of the numerical error. The pressure profiles also are very close, and although there is still a small difference between the medium and fine mesh, this is also in the range of inaccuracies and therefore grid independence can be assumed for the 300 x 224 mesh. The case shown here is for a 10° wrap angle around the forming roll. If higher wrap angles were chosen (longer forming zone), the number of cells in the M D was increased accordingly. 4.1.4 Convergence issues As in single fabric impingement, the fibre mat resistance as a function of distance from impingement point was described by a step function. However, from Figure 4.3 it is evident that in the case of roll forming the assumption is not as good as before. While in single fabric impingement all resulting profiles (pressure, velocity and resistance) showed a smooth curve, the pressure profile in roll forming contains small tips which have to be attributed to the use of a non-smooth mat resistance. Although this definitely introduces an error in the results, it should still be possible to obtain good results by fitting a smooth curve to CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 125 16 4 6 8 10 12 14 16 18 Distance [cm] Figure 4.3: Grid convergence in roll forming computation (case 1 of Table 4.1). these profiles. The plots shown on the following pages will generally contain the smoothed profiles. Good convergence results could be achieved despite the problems with using a step-function approach. With only two iteration steps the L2-error norm for the change in drainage velocity dropped in all cases to 5 • 10~3 m/s or lower. The average drainage velocities over the wedge are around 0.5 m/s. Another problem was the necessary computation time. The combination of the two itera-tion procedures is time consuming. The fabric shape calculation in particular often converges slowly. Even small deviations in the pressure distribution along the fabric (compared to the previous pressure distribution) can cause the fabric path to change considerably. The change in fabric position from one iteration to the next often seemed to be overpredicted. The solu-tion was therefore dampened by adjusting the fabric position by only half of the theoretically calculated value. Although this improved the convergence, especially in the early stages of the iteration, computation times for one case were still significant. As the convergence be-haviour was different for each computed case, the convergence was not judged by setting a certain convergence criterion. Rather it was judged from the convergence history. If a CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 126 couple of consecutive iterations would show only small differences in the fabric position, the case was considered to be converged. In other cases, convergence was more easily achieved and no change in fabric position was noticeable from iteration to iteration. In these cases, the L2-error norm of change in fabric position was in the range of 0.05 mm. In cases where convergence was not as good, the L2-norm was still below 0.15 mm. 4.1.5 Centrifugal forces As mentioned previously, centrifugal forces have very little influence on the drainage in roll forming and therefore are often neglected in analytical models. They do not enhance drainage through the outer fabric as was often assumed in earlier studies. The pressure at the outer fabric which determines drainage through that fabric is solely given by the fabric tension and the local radius of curvature (Equation 4.1). However, the pressure at the inner fabric is reduced by centrifugal forces. Therefore centrifugal forces have a negative, although very small, effect on drainage. If equal drainage to both sides is desired, the centrifugal force can always be countered with a small vacuum in the forming roll. This is often done in practice but not considered in this study. The only important consideration regarding the centrifugal force for a papermaker is concerned with instabilities, which would develop if the centrifugal force exceeds the outer fabric pressure. A maximum safe machine speed can be calculated by comparing the rela-tionships for the centrifugal pressure and the pressure at the outer fabric ([26]), leading to an expression for the critical speed of: The critical speed therefore depends on the fabric tension T, fluid density p and suspen-sion radial thickness H, but is independent of the local radius. For typical modern twin-wire machine running conditions the critical velocity due to centrifugal force is generally not a concern; only at very high suspension thicknesses may problems arise. In F L U E N T , centrifugal effects are automatically included by using the full Navier-Stokes-equations. A difference between the pressure distributions along the inner and outer (4.4) CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING- 127 fabric should therefore be present. Assuming that the difference is mainly based on centrifu-gal effects, and not on other phenomena, the pressure results can be checked for reasonable-ness by calculating the theoretical pressure difference due to centrifugal forces (based on the gap width, the roll radius and the computed MD-velocity in the wedge, which was averaged over the gap width) and comparing it to the computed one. This check was done for all computations. Theoretical and computed pressure differences always agreed well with only small differences. Figure 4.4 shows the results for one example calculation. The pressure profiles along the inner and outer fabric, the difference between these two curves and the centrifugal pressure calculated from the MD-velocity are shown. 18 r 16 -14 -12 -la & 10 -CD u I-* 8 -CD & 6 -4 -2 -0 -10 12 14 Distance [cm] Figure 4.4: Comparison of theoretical and computed pressure difference between outer and inner fabric due to centrifugal force (case 1 of Table 4.1). CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 128 4.1.6 Forming conditions in the computations As mentioned earlier, the computations are very time consuming and convergence in some cases is difficult to achieve. Only a few cases were thus computed. A decision of what cases should be computed was based on the following considerations regarding the significance of different variables. 4.1.6.1 Jet velocity The pressure and therefore also the drainage velocity in the forming wedge (but not at the impingement point) are determined by the fabric tension and the local radius of curvature of the outer fabric. However the relative drainage will depend on the jet velocity. If the pressure is indeed approximately constant in the forming zone as is often assumed, then higher jet velocities should lead to less relative drainage and the nip width at the end of the modelled forming zone should be larger. The nip width is one measure that was experimentally determined by other researchers and could therefore be used for a validation of the model. Different jet velocities thus were included in the study. 4.1.6.2 Impingement angle The effect of different impingement angles should be only noticeable in the vicinity of the impingement point. The flow in the wedge should not be influenced. Also, it is difficult to control the impingement angle on the outer fabric in the computations, as the actual impingement angle depends on the deformation of the fabric. To obtain a certain angle, the jet angle would need to be adjusted throughout the computation, which would make convergence nearly impossible. Although the change in impingement angle due to the fabric deformation turned out to be insignificant, it was decided to use only one impingement angle. This angle was set relative to the undeformed fabric. 4.1.6.3 Fabric tension The fabric tension is one of the factors that determine the pressure at the outer fabric and therefore also the drainage velocity. With higher fabric tensions more drainage should occur CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 129 over the same forming length (keeping all other variables constant). Cases for two different fabric tensions were thus computed. 4.1.6.4 Fabric and fibre mat resistance The fabric and fibre mat resistance will influence the drainage velocities through the fabrics. The pressure is still given by the fabric tension and geometry. However if the drainage velocity changes, the local radius of the outer fabric might also change. Looking at different resistance values for the porous medium therefore might show interesting results. Based on the high drainage in roll forming, the resistance of the fibre mat quickly dominates over the fabric resistance and it was decided to look at different SFR values rather than at different fabric resistances. 4.1.6.5 Forming roll radius The roll radius influences the local radius of the fabric, which in turn determines the pressure along the outer fabric. Therefore the roll radius should have some influence on the flow in the forming wedge. However it can be assumed that the influence will be similar to the one of different fabric tension, changing the magnitude of the pressure pulse in the wedge, leading to higher drainage velocities for a smaller roll radius. As the influences are probably similar, the roll radius was not changed throughout the computations. 4.1.6.6 Wrap angle An increased wrap angle results in increased available drainage time and therefore the drainage should increase. However the flow field itself should be independent of the wrap an-gle, if it is assumed that the conditions at the wedge end do not influence the flow upstream. It needs to be verified if this influence of the wrap angle is indeed negligible. One case with an increased wrap angle was therefore computed. If the influence is negligible, then this model should accurately predict the flow in the beginning of the wedge independently of the wrap angle, demonstrating that it is hot necessary to model the full length of the wedge. If the influence is not negligible, the model results are valid only for the specific wrap angle used in the computation. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 130 4.1.6.7 Jet thickness The jet thickness (if the target basis weight is unchanged) influences only the relative drainage, but the absolute drainage and the flow field itself should be independent. There-fore a constant jet thickness was chosen so that a comparison with experimental results from other researchers was possible. 4.1.6.8 Impingement position Like the impingement angle, the impingement position is also hard to control exactly in the computations, as it changes every time that a new fabric position is calculated throughout the iteration process. In most computations it was attempted, successfully, to have the jet impinge right into the nip. In only a few cases was the resulting impingement position slightly before the nip, preferentially on the outer fabric. One case with the impingement position clearly ahead of the nip was also computed. Roll impingement was avoided as this seems to lead to unstable flow at impingement (the computations, in which steady flow is assumed, did not converge for such cases). Presumably, a different impingement position will change the results only in the vicinity of the impingement point. If impingement is ahead of the nip, a first pressure pulse should be seen from the impact with a form similar to the pressure pulse in single fabric impingement. A second pressure pulse should build when the jet enters the wedge. If the jet impinges right into the nip, the pressure pulse from impingement (which depends on the impingement angle and jet velocity) should only be visible if it is higher than the initial pressure in the nip. In one case convergence was reached first with a resulting impingement position on the outer fabric. This case was then repeated with the impingement position changed to nip impingement to verify the above hypotheses. A list of the variable settings used in the computations for most cases (labelled cases # 1 to 7) is given in Table 4.1. Common variables in all computations are: Impingement angle (undeformed fabric) 4° Forming roll radius 0.4 m Jet thickness 9 mm CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 131 Fabric thickness (inner and outer fabric) Fabric resistance Density of suspension Viscosity of suspension 1 mm C 2 = 2.96 • 104 m - \ kfabric = 2.43 • 1CT1 0 m 2 1000 kg/m 3 5 -IO- 4 Ns /m 2 Table 4.1: Variable selection for roll forming computations. Case # Jet velocity Fabric tension Wrap angle SFR Impingement position [m/s] [N/m] [°] [m/kg] 1 10 7000 10° 0.6-10 1 1 nip 2 15 7000 10° 0.6- 10 1 1 nip 3 15 7000 10° 0.9-10 1 1 outer fabric 4 15 7000 10° 0.3-10 1 1 nip 5 15 8000 10° 0.6-10 1 1 nip 6 15 7000 15° 0.6-10 1 1 nip 7 15 7000 10° 0.9-10 1 1 nip The aforementioned case (labelled case # 8) where impingement was clearly ahead of the nip was computed as a preliminary case and therefore has different settings than the other cases. This case does not include fibre mat build-up (and therefore the permeable medium has a constant resistance throughout the wedge), and was computed with a rather high fabric resistance to avoid draining the water too quickly. Also, the boundary conditions for the fabric geometry calculation were slightly different. The fixed point P i from Figure 4.2 is only 9 cm ahead of the nip. The settings for the computations for this case which are different from the other cases are: Forming roll radius 0.5 m Jet thickness 10 mm Fabric resistance C 2 = 5 • 104 m _ 1 , hf^^c = 2 • 1 0 - 1 1 m 2 Jet velocity 15 m/s CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 132 Fabric tension 7000 N/m Wrap angle 20° Impingement position 3 cm ahead of nip on outer fabric 4.1.7 Results 4.1.7.1 Comparison with experimental data Before studying the behaviour of the flow in the forming wedge as a function of different variables, an attempt was made to validate the model by comparison with experimental results. This proved to be difficult, as measurements in the forming wedge are subject to substantial uncertainty, have rarely been done, and the exact forming conditions are often not known or difficult to express in numbers which are suitable for numerical modelling. Carrying out experiments in the roll forming zone is a major, time consuming task and involves equipment which is not easily available. Therefore it was beyond the scope of this thesis to experimentally determine validation data for the wedge zone, and data measured by others had to be used. However, as mentioned above, not all the variables that are needed as input for the computational model are given in these studies, or they are given in a form which is difficult to translate into the model. Therefore only order-of-magnitude qualitative comparisons can be expected. The study by Gooding et al. [32] was chosen for a comparison with the computational model. This is the most recent experimental study of roll forming. Most of the necessary variables are given in the model, and drainage profiles as well as pressure profiles are given. There are two major difficulties in a comparison. Although it is known that reslushed newsprint was used in the experimental study, the SFR of this stock and whether the SFR changes as a function of basis weight is not known. Also, the wrap angle in the experiments was 65°. Thus, the wedge includes the forming zone where free fibre suspension is still available in the center of the wedge, but also a "press zone" in which no free suspension exists. The fibre mats on the two fabrics meet and therefore the fibre network supports the load. In the computational model, only the free forming zone can be modelled. The modelled wedge length is therefore much shorter. Only a few data points were measured CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 133 over this limited length in the experiment, which makes a comparison of profiles difficult. However, a qualitative comparison of flow behaviour can be done, and average values can also be compared. Results of the experimental work were shown in detail for the "base-case" Gooding et al. looked at. Table 4.2 shows the machine settings used for this base case and the respective settings chosen for the computation (case 1 from Table 4.1). Table 4.2: Variable settings for comparison with experimental results. Variable Gooding et al. Computation (case 1) Headbox consistency 0.73% 0.56% Slice opening (jet width) 9 mm 9 mm Nominal B W 46 g /m 2 50 g /m 2 Radius of forming roll 0.4 m 0.4 m Wrap angle 65° 10° Machine speed 10 m/s 10 m/s Jet rush 40 m/min 0 (not possible to model) Fabric tension 7 kN/m 7 kN /m Impingement position nip nip Impingement angle not known 4° (without fibre suspension) Furnish reslushed newsprint SFR not known SFR = 0.6-10 1 0 m/kg Fabric 163 x 118 double-layer C 2 = 2.96 • 104 m r 1 (resistance not known) kfabric = 2A3 • lO"" 1 0 m 2 The setting of the nominal basis weight chosen for the computation has been rounded off for easier use in later computations. The headbox consistency is then given by the slice width, the target basis weight and the assumption of 100% retention. As long as it is not possible to match the SFR to the experiment, an exact agreement of the consistency and basis weight settings is of little importance. The SFR was chosen slightly higher than the ones used in single-fabric-impingement. Due to filler content in paper machine stock and CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 134 increasing fines and filler retention with distance from the impingement point, it is likely that the average SFR in the forming region will be higher than the ones of pure pulps. As discussed earlier, the computational model does not include jet rush or drag, but rush or drag should not influence drainage and pressure values. The impingement angle of the experiments was not given, but again this should only influence the impingement pressure and not the pressure which builds up in the wedge. A first comparison can be done for the gap width, which was measured experimentally by a strain gauge mounted on a flexible paddle. The paddle was moved around the forming roll to track the fabric path. The computed gap width as well as the data points measured in the experimental study are shown in Figure 4.5. When comparing the gap width, one has to be aware that the gap width is governed by the drainage which has taken place, which in turn depends strongly on the SFR. Therefore exact agreement cannot be expected. 60 •4—» d 10 9 8 7 6 5 4 3 2 1 I I I I computed, case 1 Gooding et al. —--<— j i -4 -2 0 2 4 Distance from nip [cm] Figure 4.5: Gap width as a function of distance from nip: case 1 and comparison with Gooding et al. [32]. 0 cm in the nip is defined in Figure 4.2. Gooding et al. measured a gap width of about 4 mm at the nip (the nip being defined as the merging point of the outer fabric and forming roll without flow from the headbox). CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 135 56% of the fibre suspension therefore is removed at impingement even before the nip. (They estimate the average drainage velocity based on this number as 5.3 m/s, which seems very high, however they do not mention over what length this number was averaged). The computed gap width at the nip is 5.5 mm. Any difference here can be based on using a different SFR, as well as different impingement conditions (exact position and angle), which influences the initial drainage. Over the next 4 cm from the nip, the measured gap width decreases by about 1 mm, while the computed value decreases by 2.5 mm. (The results over the last 3 cm of the computational domain are influenced by the zero static pressure outlet condition, as can be clearly seen from Figure 4.7. Due to the different wrap angles used in the computation and experiment these data are therefore not used in the comparison.) The gap width decreases faster in the computation (but this again could be owing to the different SFR value). Also, with only three data points measured over the 4 cm, a comparison of the profiles has to be judged critically. What can be said safely from the comparison is that the computed gap width as well as the decrease in gap width seems to be in a reasonable range. Closely related to the gap width is the drainage velocity through the fabrics. Gooding et al. measured the drainage through the outer fabric by collecting the spray water with a small scoop which was moved along the outer fabric. The drainage velocity was calculated from the measured drainage volume over time. The drainage velocity at the inner fabric (drainage into the forming roll) was calculated from the drainage velocity through the outer fabric and the gap width measurements (which can be translated into total drainage). The computed and measured drainage velocity profiles are shown in Figure 4.6. With the same limitations regarding the choice of SFR discussed earlier, the agreement between measured and computed drainage velocities through the outer fabric seems to be reasonable. For the inner fabric, the computed drainage velocity shows nearly the same profile as the drainage velocity through the outer fabric. In the experiment, however, the drainage into the roll increased from 0 to 0.12 m/s over the first 4 cm after the nip and only then started to decrease. This seems odd, especially as the measured pressure trace showed that the pressure has already built up before the nip and therefore drainage at the beginning of the roll should be present. The disagreement of inner fabric drainage velocities between computation and experiment thus could also be based on the way the experiments were done. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 136 1.6 1.4 h outer fabric, case 1 inner fabric, case 1 outer fabric, Gooding et al. inner fabric, Gooding et al. X 1.2 h 0 -6 -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.6: Computed drainage velocity through inner and outer fabric as a function of distance from nip: case 1 and comparison with Gooding et al. [32]. Figure 4.7 shows the computed pressure profiles along inner and outer fabric and the measured profile. The pressure was determined with a capillary tube of 0.8 mm outer diameter, which was fed into the wedge through the headbox. The exact position of the tube in respect to the gap width is not known, but the measured values should represent an average pressure in the wedge. However, due to the thickness of the tube it is possible that at lower gap width further along the forming wedge, the pressure might actually be influenced by the presence of the tube. The experimental pressure trace further downstream than shown in Figure 4.7 experiences another sharp increase to about 18 kPa, before it drops to a value of about 6 kPa. This could be the point where free fibre suspension is no longer present. The computed pressure drops to zero at the end of the modelled domain due to the imposed zero pressure boundary condition when the fabrics part from the roll. Comparing the experimental and computed pressure up to about 4 cm downstream of the nip shows good agreement. In both cases, the pressure is below the commonly assumed value oiT/R, as also reported in other studies. It makes sense that the pressure profiles agree better than CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 137 the drainage profiles, as the pressure depends on the fabric tension (which is the same in computation and experiment) and the local radius of the fabric. The local radius depends slightly on the dewatering that occurs, which in turn is a function of the furnish SFR, but the influence on pressure should be relatively small. 3 <L> OH -2 0 2 Distance from nip [cm] Figure 4.7: Pressure in the wedge as a function of distance from nip: case 1 and comparison with Gooding et al. [32]. In view of the shortcomings of the experiment and the computational model, and also the problems in matching the computation to the experiment, the agreement between the computation and experiment is good. The model therefore can be used to predict and compare flow conditions in the early wedge of roll formers, although one needs to bear in mind the limitations of the model. 4.1.7.2 Influence of jet velocity Figure 4.8 shows the pressure profiles for jet velocities of 10 m/s and 15 m/s. As already seen in the previous section, the pressure distribution at 10 m/s increases only slightly throughout the nip and remains below the value of the fabric tension divided by the roll radius (17.5 kPa). CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 138 The pressure difference between inner and outer fabric is due to centrifugal forces. At 15 m/s, the impingement position is later than for 10 m/s, presumably because for the faster jet the initial impingement force is larger and the fabric is pushed out further. The pressure distribution shows a different form than at 10 m/s jet velocity, with the pressure increasing noticeably throughout the wedge to about 18 kPa (outer fabric). The maximum pressure therefore even exceeds the value of 17.5 kPa given by T/R. 20 18 16 14 I 12 a 3 o< 10 8 6 h 4 2 0 T/R outer fabric, V : e t = 10 m/s inner fabric, V - e t = 10 m/s outer fabric, vL = 15 m/s inner fabric, V j e t = 15 m/s -2 0 2 4 Distance from nip [cm] Figure 4.8: Pressure in the wedge for different jet velocities. At 15 m/s, the drainage velocities (Figure 4.9) are more than 0.1 m/s or on average about 30% higher than at 10 m/s. In the beginning of the wedge, this difference in drainage velocity is based on the different impingement position. (When drainage starts for the higher velocity case, a fibre mat has already built up for the lower jet velocity case, thereby increasing the resistance and thus reducing the drainage velocity.) As over the remaining wedge length less fibres are deposited (due to the lower relative drainage, even though the drainage velocities are higher), the resistance for the higher jet velocity case remains lower throughout the wedge. Therefore the drainage velocities are different even if the pressure is the same. In both cases, the drainage velocities decrease throughout the modelled domain. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 139 -6 -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.9: Drainage velocity through inner and outer fabric for different jet velocities. Even though the drainage velocities are higher for higher jet velocities, the relative drainage, reflected in the gap width (Figure 4.10), is lower, due to the higher overall flow. This agrees with observations by other researchers (e. g. de Montigny et al. [3]) that the forming length increases with higher machine velocities. At the respective wedge entrance positions (the beginning of the curve) the wedge is slightly wider for the 10 m/s case, based on less drainage at initial impingement, the gap width then decreases much faster at the lower jet velocity case to a difference in gap width of about 1.5 mm for the two computed cases. The velocities in the machine direction in the wedge are in both cases lower than the incoming jet velocities (see Figure 4.11). The velocities decrease slightly throughout the wedge and only at the end of the domain, as a result of the imposed zero pressure outlet condition, does the velocity increase again to the original jet velocity. The differences between MD-velocity at the inner and outer fabrics are not significant. Table 4.3 gives the integrated pressure at the fabrics and the relative drainage to each side (based on the incoming volume flow) for all cases. The drainage length at each fabric, CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 1 1 1 V j e t = 10 m/s vL = 15 m/s i i -6 -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.10: Gap width for different jet velocities. I 1 1 j ^ ^ ^ ^ outer fabric, V j e t = 10 m/s inner fabric, V - e t = 10 m/s — ... outer fabric, V J e t =15 m/s inner fabric, V j ? t = 15 m/s -6 -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.11: Machine direction velocity in the wedge for different jet velocities. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 141 denned as the length over which drainage takes place, is also given. In all cases, the integrated pressure along the outer fabric is higher, which agrees with the pressure profiles. The relative drainage is also higher through the outer fabric, according to the drainage velocity profiles. A comparison of the drainage length at inner and outer fabric shows how well centered the flow impinges into the nip. For the above discussed case 2, the impingement takes place a few millimetres before the nip. The drainage length in this case is about 1 cm shorter than in case 1 with the lower jet velocity. Table 4.3: Drainage length, integrated pressure, relative drainage and resulting impingement angle for the computed roll forming cases (i.f. = inner fabric, o.f. = outer fabric). Case # Drainage length [cm] / P [ N ] Rel. drainage [%} Imp. angle [°] i.f. o.f. i.f. o.f. i.f. o.f. o.f. 1 11.3 11.2 1220 1330 39.6 41.2 4.1 2 10.4 10.8 1160 1470 29.9 . 35.3 4.0 3 9.1 11.3 1200 1550 26.1 30.8 3.8 4 11.2 11.3 1090 1350 36.9 43.1 4.1 5 10.5 11.0 1340 1640 31.6 37.7 4.0 6 13.8 14.3 1630 1990 36.5 41.4 4.0 7 10.1 10.4 1220 1560 25.3 31.5 3.9 8 14.7 17.6 1590 2050 35.0 45.4 4.1 4.1.7.3 Influence of fabric tension The influence of fabric tension on early roll forming was studied in cases 2 and 5 from Table 4.1 (fabric tensions of 7 kN/m and 8 kN/m with otherwise similar settings). The impingement position was approximately centered into the nip for both cases. With higher fabric tension it would be expected that the pressure in the wedge is also higher by a constant value. This was confirmed (Figure 4.12). In both cases, the pressure rises linearly throughout the wedge to about the value of T/R (outer fabric). The average pressure is lower than the theoretical value of T/R. The impingement position for the case with the higher fabric CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 142 tension is about 1 cm earlier, as the fabric deforms less at the initial impingement point. The integrated pressure (Table 4.3) is about 200 N higher with the higher tension. -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.12: Pressure in the wedge for different fabric tensions. Interestingly the drainage velocities, shown in Figure 4.13 are not that different for high and low fabric tensions, except for a small difference between the drainage velocities through the outer fabric. Presumably the fibre mat builds up faster at higher fabric tensions and therefore the drainage velocities, which depend on the pressure as well as the mat resistance, are about the same even though the pressure is higher at comparable positions. Even though the drainage velocities are not that different, the difference is sufficient to cause the gap width in the case of the higher fabric tension to decrease faster than in the case of the lower fabric tension (Figure 4.14). The gap width of the two cases differs by about 0.4 mm at the end of the modelled domain. A small difference in the beginning can be attributed to the slightly earlier impingement at higher fabric tension. Although the difference in gap width is not that large (as can be also seen from the relative drainage values, which are only about 2% higher with the higher fabric tension), it is noticeable. The decrease of gap width with higher tension (and therefore decrease in necessary forming CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 143 0 I i i i i i I -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.13: Drainage velocity through inner and outer fabric for different fabric tensions. length) was also observed by other researchers. De Montigny et al. [3] measured the gap width for different fabric tensions. The roll radius in their study was 0.38 m, the machine velocity was 7.6 m/s and the slice height was 11.4 mm. They measured a drop in gap width of about 4 mm over a length of 8 cm for a fabric tension of 5.3 kN/m. At a fabric tension of 2.6 kN/m, this drop was about 3 mm. Due to the different settings in this study compared to the computations, an exact comparison is not possible, but the computed difference in gap width for the different tensions lies in a reasonable range compared to the measured values. The velocity in machine direction in the wedge is shown in Figure 4.15. Only small differences along inner and outer fabric and for high or low fabric tensions are apparent. 4.1.7.4 Influence of nitration resistance Cases 2, 4 and 7 predict the flow conditions for different filtration resistances of the pulp mat. Figure 4.16 shows that a different filtration resistance will result in a different pressure distribution in the wedge. This shows that the local radius of curvature of the outer fabric CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 144 60 O •c 0 2 4 Distance from nip [cm] Figure 4.14: Gap width for different fabric tensions. 1 outer fabric, T = 7 kN/i inner fabric, T = 7 kN/i outer fabric, T = 8 kN/i inner fabric, T = 8 kN/i n n n -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.15: Machine direction velocity in the wedge for different fabric tensions. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 145 must be influenced by the filtration resistance, presumably due to different drainage velocity distributions. From the pressure distributions it can be seen that with all other variables held constant and the jet centered into the nip, the pressure starts to build up earlier at lower SFR values, meaning that the fabric deforms less in the beginning at lower SFR values so that the contact point between fibre suspension and fabrics is earlier. Although again in all three cases the pressure increases throughout the nip, it does so only very slightly for the lowest SFR value, where the maximum pressure remains below T/R, but much more so for the higher SFR values, where the pressure increases to about the value of T/R respectively to a value about 25% higher than T/R. A pressure higher than T/R implies that the local radius can be smaller than the roll radius, at least locally. The pressure differences between the inner and outer fabrics can be attributed to the centrifugal forces as shown before. A higher filtration resistance gives a higher integrated pressure. 25 20 h S 15 3 £ 10 T/R /outer fabric, SFR = 0.3-10 " m/kg / inner fabric, SFR =0.3-10 "m/kg :> V * / outer fabric, SFR = 0.6-10 "m/kg u inner fabric, SFR - 0.6-10 m/kg \* outer fabric, SFR = 0.9-10 " m/kg inner fabric, SFR = 0.9-101 0 m/kg -2 0 2 4 Distance from nip [cm] Figure 4.16: Pressure in the wedge for different SFR values. With greater SFR values, the drainage velocity (Figure 4.17) is smaller throughout most of the gap, even though the pressure is higher for higher SFR values. The relative drainage therefore also diminishes with higher resistances. The drainage velocity profiles agree with CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 146 the pressure profiles. For the lowest SFR, where the pressure is approximately constant throughout the nip, the drainage velocity decreases due to the increasing resistance from fibre mat build-up. At the highest SFR, the drainage velocity can remain constant in the wedge even though the resistance increases, as the pressure also increases significantly in the machine direction. -6 -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.17: Drainage velocity through inner and outer fabric for different SFR values. The drainage velocity distributions are also reflected in the gap width in Figure 4.18. Due to the higher drainage velocities, the gap width decreases faster with lower SFR values, so the difference in gap width grows. Some of the difference in drainage values is due to the earlier beginning of drainage at lower SFR values, but even if the curves would be moved horizontally to adjust for the different impingement position, there would still be a difference in the gap width at the end of the domain. The difference in the gap width then would be about 0.5 mm between the high and medium SFR value, and about 1 mm between the medium and the low SFR value. Figure 4.19 shows the velocity in M D in the wedge. As seen before, it is lower than the incoming jet velocity and decreases slightly throughout the wedge until it increases again at CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 147 -6 -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.18: Gap width for different SFR values. the end. The decrease is more pronounced for higher SFR values, due to the higher pressures towards the end of the wedge. 4.1.7.5 Influence of wrap angle The pressure distributions for different jet velocities and different SFR values show some unexpected behaviour, increasing noticeably throughout the wedge and rising to values even larger than T/R towards the end of the wedge. The question arises whether this is caused by the short wrap angles in conjunction with the chosen boundary condition of a fixed point used in the fabric position calculation. Therefore one of the cases (case 2) was repeated with the wrap angle increased to 15° (case 6). The pressure distributions of these computations (Figure 4.20) show the same profile in the beginning of the nip, but from about 2 cm after the nip on, the pressure increases less for the case with larger wrap angle. The pressure distributions confirm that the flow in the beginning of the nip is not influenced by the wrap angle (and therefore by the boundary condition at the outlet), but the distance before the wedge end over which some influence CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 148 16 14 12 o o > Q 6 4 2 0 outer fabric, SFR = inner fabric, SFR = outer fabric, S F R : inner fabric, SFR = outer fabric, SFR = inner fabric, SFR = i Li -4 -2 0 2 4 Distance from nip [cm] 0.3-10° m/kg -•0.3-10 "m/kg -0.6-10 "m/kg • 0.610 "m/kg 0.9-10 "m/kg -O ^ l O ^ m / k g , A £L Figure 4.19: Machine direction velocity in the wedge for different SFR values. can be seen is longer than expected (in this case about 2 to 3 cm for the smaller wrap angle, which excludes the last 2 cm where the pressure obviously drops because of the zero pressure boundary condition). Based on this result it can be assumed that the high pressures towards the end of the nip in all cases of higher jet velocity and higher filtration resistance are caused by the short wrap angle. At a larger wrap angle, the pressure would probably still increase throughout the nip, but at a lower rate. This also implies that the wrap angle in conjunction with the boundary condition (e. g. the position of the first blade) must influence the local radius towards the end of the nip. As with the pressure distributions, the drainage velocities (Figure 4.21) through the inner and outer fabric of the two cases overlap in the beginning, but diverge somewhat after about 2 cm from the nip, although only slightly. As a consequence, the gap width, shown in Figure 4.22, also does not differ much. Only from about 4 cm after the nip on does the difference become obvious, which is where the pressure for the lower wrap angle case starts dropping due to the outlet boundary condition. The MD-velocity distributions (Figure 4.23) at the start of the roll completely overlap. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 149 CO 2 3 CA cn t> i-> PH 2 4 6 Distance from nip [cm] Figure 4.20: Pressure in the wedge for different wrap angles. 1.4 1.2 v 1 t 0.8 h o "33 > CD ao CO 0.6 0.4 0.2 outer fabric, wrap = 10° inner fabric, wrap = 10° outer fabric, wrap =15° inner fabric, wrap =15° 2 4 6 Distance from nip [cm] 10 Figure 4.21: Drainage velocity through inner and outer fabric for different wrap angles. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 150 Figure 4.22: Gap width for different wrap angles. From about 3 cm along the roll on, a slight difference becomes noticeable with increasing total wrap. 4.1.7.6 Influence of impingement position Cases 3 and 7 use the same machine settings and the same pulp resistance (jet velocity = 15 m/s, fabric tension = 7 kN/m, SFR = 0.9-10 1 0 m/kg), but slightly different impingement positions. In case 3 the jet impinges approximately 2 cm ahead of the nip on the outer fabric. In case 7 the jet impinges centered into the nip. As discussed above, this presumably should change the flow only in the very beginning of the nip, which was confirmed. Figure 4.24 shows the pressure distributions along the fabrics for both cases. When the impingement takes place on the outer fabric, a first pressure pulse builds up only on this fabric, according to the impingement force. The pressure then decreases, as it would in single fabric impingement, and only when the fluid enters the forming wedge does the pressure start to rise again. The impingement angle on the outer fabric is low (3.8°), so the first pressure pulse is lower than the pressure which builds up in the wedge. If impingement takes place into the nip, the CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 151 o o > 16 14 h 12 1 10 1 1 i outer 1 inner i outer 1 inner 1 abric, wrap = 10° abric, wrap =10° abric, wrap =15° abric, wrap = 15° • 1 2 4 6 Distance from nip [cm] 10 12 Figure 4.23: Machine direction velocity in the wedge for different wrap angles. pressure rises constantly throughout the wedge from the beginning on. While the pressure along the outer fabric is different for the different impingement positions in the beginning of the wedge (due to the different deformation of the fabric), the pressure distributions start overlapping only a short distance downstream of the impingement positions. For the inner fabric, the pressure distributions overlap even earlier with only a very small difference in the beginning. The integrated pressures are also nearly the same. The drainage velocities in Figure 4.25 show the same trend, with noticeable differences at the start of drainage, but further downstream the drainage velocities are nearly the same (for both the inner and outer fabrics), with only insignificant differences. Consistent with these nearly similar drainage velocities, the inter-fabric gap (Figure 4.26) is also independent of the impingement position, as can be seen from the almost perfectly overlapping curves. The velocities in the machine direction show the same behaviour as seen before, with the velocities being the same at inner and outer fabric. For fabric impingement, the MD-velocities are somewhat, but not significantly higher than for nip impingement (about 0.3 m/s). The build-up of a first separate pressure pulse for fabric impingement can be seen even CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 152 -4 -2 0 2 4 6 8 Distance from nip [cm] Figure 4.24: Pressure in the wedge for different impingement positions. Figure 4.25: Drainage velocity through inner and outer fabric for different impingement positions. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 153 10 9 h & 7 60 o •c § 6 nip impingement fabric impingement -4 -2 0 2 4 6 Distance from nip [cm] Figure 4.26: Gap width for different impingement positions. o o "3 > 16 14 12 10 4 h 2 0 -6 outer fabric, nip impingement inner fabric, nip impingement outer fabric, fabric impingement inner fabric, fabric impingement | -2 0 2 Distance from nip [cm] Figure 4.27: Machine direction velocity in the wedge for different impingement positions. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 154 more clearly for case 8 (Figure 4.28), where impingement takes place noticeably before the nip (about 3 cm). As one of the preliminary computations this case was computed without including the fibre mat build-up (constant drainage resistance) and for slightly different settings than the other cases. Figure 4.28 shows the pressure and drainage velocity distributions in the wedge. At the outer fabric, the pressure at the initial impingement peaks at about 10 kPa, and then drops to about 5 kPa before the flow actually enters the nip. In the wedge, the pressure at the outer fabric is approximately constant; at the inner fabric the pressure increases slightly throughout the nip. The drainage velocity at the outer fabric is high at the impingement location, then decreases before it increases again when the jet enters the wedge, in parallel with the pressure changes at the outer fabric. In the wedge the drainage velocities are approximately constant. With the constant fabric resistance, the velocity in this case is higher at the outer fabric due to the higher pressure at the outer fabric. It is interesting that although the pressure is higher in the wedge, the drainage velocity is still higher at the impingement position. As this case was computed with a constant fabric resistance, the initially high drainage velocities must be due to the initial deceleration of the jet in machine direction. 4.1.8 Summary In this chapter, the flow model for impingement on a single fabric was extended to the roll former geometry, including impingement and early roll forming. The flow geometry is determined by the inner and outer fabrics wrapping the forming roll. The position of the outer fabric is flexible and can adjust to the flow conditions. A model to compute the correct shape of the outer fabric was developed and combined with the fluid flow model. Preliminary computations showed that the step function approach which was used to describe fibre mat resistance introduced larger errors in the roll forming computations than in the single fabric impingement calculations, presumably because the resistance varies more in a roll former. The main implication of the larger resistance errors was slow convergence. The resulting profiles of velocity etc. need to be smoothed. The slow convergence was also caused by the fact that the fabric position calculation often needed many iteration steps to converge, as even small changes in the pressure distribution (compared to the previous one) CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 155 Figure 4.28: Pressure and drainage velocity in the wedge for case 8 (no fibre mat build-up). can cause the fabric shape to change significantly. Due to the slow convergence it was only possible to compute a few cases to show the influence of certain variables. A more thorough study would need improved iteration procedures. A first computation was done for comparison to available experimental data. The pressure distributions showed very good agreement. A comparison of the drainage velocities and the gap width did not show as good agreement, but numbers were in the same range. The poorer agreement for drainage velocity can be attributed to differences in the drainage resistance of fabric and fibre mat of the computation and experiment. As the drainage resistance of the fabric and pulp in the experiment is not exactly known, it was not possible to match it exactly. The drainage resistance has a much bigger influence on the drainage velocities than on the pressure distribution. The comparison between the computation and experiment gives credence to the model so it could be confidently used in the following computations. The model was used to compute the wedge flow for different conditions. The precise values computed should be treated with caution, as they depend very much on the fibre mat resistance model used. CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 156 The difference between the pressure distribution along the inner and outer fabric was shown to equal the effect from centrifugal forces, as would be expected. This gives further confidence in the results. It was possible to confirm certain trends found in experimental studies of previous workers ([3], [30]). With higher jet velocity, higher pulp mat resistance and lower fabric tension the forming length increases. The impingement position has no influence on the flow in the wedge. If the jet impinges ahead of the nip, the pressure distribution resembles single fabric impingement until the suspension enters the wedge. If the jet impinges right into the nip, no separate influence from the impingement force (in the form of a pressure peak) could be observed. However, differences in the impingement force can bend the fabric further outwards in the beginning, causing the point where the suspension enters the wedge to move downstream. This is particularly evident at higher jet velocities and higher resistances (which lead to higher impingement forces), for which the drainage starts only later in the wedge. It is possible that this dependence of the initial forming geometry on the impingement force can also be seen in the pressure distributions. The integrated pressure in the wedge behaves as expected. It increases with higher jet velocity, higher fabric tension, and higher mat resistance. However in the cases of varying jet velocity and mat resistance, the form of the pressure pulse also changes noticeably. This is not the case for varying fabric tension. At a higher jet velocity, and also at higher SFR values, the pressure seems to increase noticeably throughout the nip. In particular the pressure distributions for different SFR values show this clearly: At the lowest SFR value, the pressure is nearly constant throughout the nip (and lower than T/R). Increasing the SFR leads to a pressure that increases slightly through the nip to about T/R. This increase is even more pronounced when the SFR is increased again, with the maximum pressure higher, than'T/R. However the additional increase occurs mostly over the last couple of centimetres in the wedge. If a larger wrap angle is used, the results are independent of the wrap angle in the first part of the wedge. Over the last part of the wedge, the pressure distribution is affected by the downstream conditions. Therefore the wrap angle is shown to be of influence over the last few centimetres of the wedge, which most likely implies also a dependence on the outlet condition (e. g. the position of the first CHAPTER 4. THE INITIAL FLOW IN ROLL FORMING 157 blade after the roll forming part). This study confirms that the simple T/R assumption for the pressure in the wedge is not sufficient. A non constant pressure throughout the nip was also reported by other researchers ([30], [31], [32]). The computed pressure is lower than T/R throughout most of the wedge in all cases. In some cases it can rise to a peak of T/R or even higher towards the end of the wedge. The combination of different influences, including machine variables, pulp suspension properties and the geometry of the initial forming zone is complex, but could be investigated by the methodology used for this thesis. Chapter 5 Summary and conclusions In this thesis a two-dimensional viscous model to describe jet impingement in a roll former was developed. The model was used to investigate the influence of different variables on the hydrodynamics of jet impingement, in particular the influence on pressure distribution and drainage velocities. Pressure and drainage velocity may have a significant influence on the quality of the finished paper product. The major findings of the thesis are summarized below. 5.1 Impingement on a single fabric 5.1.1 Computational model A computational model was used to describe impingement on a single fabric. This is the case in Fourdrinier machines, as well as in twin-wire machines when the jet impinges pref-erentially on the outer fabric. A volume of fluid method was used to model the free surface flow. Comparison with potential flow theory for impingement on a solid wall showed good agreement. • Computations for jet impingement in paper machines were first carried out for cases without fibre mat build-up, using different machine variable settings (jet velocity, impinge-ment angle and jet rush or drag). It was shown that, contrary to a common assumption, both inertial and viscous flow resistance caused by the fabric are important. The computed forces from jet impingement have an influence only a short distance downstream from the 158 CHAPTER 5. SUMMARY AND CONCLUSIONS 159 impingement point. Most of the drainage occurs over this distance, which is about one or two jet thicknesses. The force on the fabric was found to agree with the prediction from a simple one-dimensional theory. It increased with greater jet velocities and impingement angles. Accordingly, drainage is also greater in these cases, although the relative drainage (fraction of incoming flow volume which is drained within the modelled domain) remains constant for different jet velocities. Jet rush or drag influences the shear stress along the fabric, but has negligible influence on the pressure and other flow properties due to negligible boundary layer growth. Fibre mat build-up was then included in the model. The basis weight profile as a function of distance from the impingement point was calculated by an iterative procedure, based on the drainage velocities and a constant specific filtration resistance of the fibre mat. Compu-tations were carried out for different machine settings (jet velocity and impingement angle) and different fibre mat and fabric resistance values. To a greater degree than was found without mat build-up, drainage was important only over a short distance, approximately one jet thickness. The fibre mat resistance influences the relative drainage considerably, although this influence diminishes if fabrics with a higher resistance are used. The influence of different fibre mat resistances on the integrated pressure along the fabric is small. If the jet velocity or impingement angle are changed, the opposite behaviour is observed, with a large influence on the integrated pressure but only small influence on the relative drainage. 5.1.2 Experimental measurements To validate some of the computations, drainage velocity profiles were measured for im-pingement on a single stationary fabric and compared to computational predictions. Fabric resistance was determined in a flow loop. It was found to depend on the flow angle, but the precise influence of angle could not be quantified accurately. This is partly due to inaccura-cies in reading low flow velocities, to unsteady flow caused by the pump at low velocities, and to curve-fitting a single equation over a regime which is dominated by the inertial resistance. The agreement between measured drainage velocity profiles and the computations was reasonable given the shortcomings in measuring the fabric resistance. The drainage velocity profiles exhibited the same form, and the computed drainage velocities over the first centime-CHAPTER 5. SUMMARY AND CONCLUSIONS 160 tre are within a reasonable range of the measured ones. Further away from the impingement point, discrepancies were greater. The predicted drainage velocities were lower than the measured ones. Various explanations for the discrepancies were postulated, in particular inaccurate knowledge of the viscous permeability, which can lead to large errors in the pre-diction of low drainage velocities. Fabric roughness also might have an important effect on drainage. The uncertainty regarding fabric resistance values had little influence on impingement once the fibre mat build-up was included. In this case, the fabric resistance is important only in the beginning, where drainage velocities are high, so the inertial resistance, which could be determined more accurately, is of more consequence. The dependence of the inertial resistance on the jet angle can have only a small influence at the high relative flow angles seen by the moving fabric. Overall, the fibre mat resistance is much more important, so the accuracy of the model predictions will depend mainly on the quality of the fibre mat resistance model. 5.2 Initial drainage zone in roll formers The single-fabric-impingement model was extended to describe the flow in the wedge of a roll former. The modelled domain includes the impingement zone and the wedge zone for small wrap angles. The shape of the outer fabric, which is flexible and therefore deforms according to the pressure in the wedge, was computed in a second iteration step with the help of a force balance. Drainage conditions used in an earlier experimental study were modelled. The computed pressure distribution showed very good agreement with the measured one. The results for drainage velocities were of the same order of magnitude. .Exact agreement was not expected as the drainage velocities depend strongly on the fibre mat resistance, which was not known with precision. The agreement between computation and experiment validated the predictions of the computational model. The model was then used to evaluate the influence of machine variables and fibre mat properties on the wedge hydrodynamics. It was shown that the impingement position has CHAPTER 5. SUMMARY AND CONCLUSIONS 161 no effect on the flow in the wedge. It was confirmed that for jet impingement on the outer fabric, the model for single fabric impingement predicts the size of the first pressure pulse well. Changing jet velocity, fibre mat resistance, or fabric tension influenced the pressure distribution in the wedge and the relative drainage (forming length). The forming length increased with higher jet velocities, higher fabric resistance and lower fabric tension, in agreement with observations from experimental studies reported in the literature. The pressure distribution generally increased but did not always reach the value T/R, which is often assumed. This accords with recent experimental findings in which a non-constant pressure distribution was also observed. The pressure distribution was often less than T/R at the entry of the wedge and in some cases locally rises to values higher than T/R towards the end of the wedge. Some evidence was found that the initial forming zone geometry (which in turn is influenced by the forces at impingement and by the fabric boundary conditions) influences the form of the pressure distribution. A change in wrap angle was found to influence flow within the later part of the wedge, but not at the wedge entry. In conclusion, this work enables us to predict the impingement pressure and the flow in the wedge of a roll former. The work showed clearly that the impingement pressure on a single fabric is important only over a distance of one to two jet thicknesses from the impingement point. It was further found that fabric roughness played a more important role in drainage than was previously apparent. Regarding the roll forming model, it was shown that the jet impingement position had little influence on the flow in the wedge, but that the impingement force influenced the fabric shape, and thereby the pressure in the wedge. In some cases this led to a local computed pressure exceeding the value T/R. The hydrodynamics model developed in this thesis can be used by researchers to predict pressure and drainage in the wedge of a roll former. With additional experimental informa-tion in the future, it may be possible to link the computed forming hydrodynamics to paper properties, and thereby optimize forming conditions to attain desired paper quality. Chapter 6 Recommendations for future work This thesis contains the first known two-dimensional viscous model for jet impingement, on a single fabric and in roll forming. The influence of some of the major variables were studied. For an extended study, modifications to the model would be necessary, especially to overcome the slow convergence in the roll forming computations. In addition, a more rigorous description of the fibre mat resistance is required. Specific recommendations for future work follow. • To improve convergence, the step-function approach to describe fibre mat resistance should be improved. A continuous function based on the distance from the impinge-ment point should be used. In F L U E N T , this would be possible only with a user-defined subroutine that should be relatively straightforward to develop. The subroutine would be programmed to allow for input of a function for the permeable medium resistance as a function of i-direction distance. A drainage resistance profile would be calculated the same way as in this thesis, but instead of approximating it with a step-function, a continuous function would be fitted to it and the function thus determined would be used for input in F L U E N T . • To shorten the convergence time, the iteration used to find the correct fibre mat profile could be included in the F L U E N T calculation. This would be accomplished with a user-defined subroutine as above, where the resistance would be defined by given constants and the drainage velocity in each cell. From the drainage velocity, the drained volume and subsequently the basis weight would be calculated as is now done 162 CHAPTER 6. RECOMMENDATIONS FOR FUTURE WORK 163 outside of F L U E N T . The function used for input in F L U E N T would include the basis weight as a variable. Each iteration step in F L U E N T then also updates the resistance in each cell, until convergence is reached. • With a more accurate and faster converging model, a more thorough study of the roll forming region could be done, varying the variables over a wider range than done in this thesis, including the variables which were left out here. Studying the influence of the former geometry might also show interesting results. • An experimental study of jet impingement on a stationary single fabric was carried out, and in roll forming, the results were compared to previous experimental work. However, further experimental validation of the roll forming model should be done. This must include using a moving fabric. Drainage profile measurements on experimental roll formers might be possible with a device based on the concept used here to determine drainage profiles. It would need to be modified for the roll former geometry. • The experimental study for single fabric impingement showed discrepancies with the computations. These are most probably caused by the fabric resistance description. A dependence of the fabric resistance on the flow angle was found. Further work is needed to accurately describe the flow through a fabric and to provide a model for the fabric resistance which includes dependence on flow angle. • The fibre mat description used in this thesis is also simplified. More knowledge of the fibre mat resistance would be needed if it is intended to compute accurate absolute values for drainage and pressure for certain drainage conditions. A remaining issue therefore is to develop a widely applicable expression for the fibre mat resistance as a function of pressure and basis weight. To model realistic cases, fines and filler retention and their influence on fibre mat resistance would have to be included as well. • Another downside of the F L U E N T model for roll forming is that rush or drag cannot be modelled, as the fabric velocity cannot be defined separately for the machine direction. Different possibilities to model the moving fabric could be explored. However, if a more CHAPTER 6. RECOMMENDATIONS FOR FUTURE WORK 164 detailed study of the influence of rush or drag was to be done, it eventually would also require the modelling of turbulence close to the fabric or fibre mat. • The given model is based on the assumption of filtration dewatering, where the consis-tency of the remaining fibre suspension is unchanged. However, this is probably true only for the very initial drainage. Later in the forming wedge, a mix between filtration and thickening might take place. Therefore models to describe the fibre mat and fibre suspension consistency would be needed. When the suspension thickens, it no longer behaves like a Newtonian fluid. Fibre-fibre interactions would become important and would need to be included in the model. • At the moment it is only possible to model short wrap angles. This might be satis-factory for roll-blade formers, where the wrap angles indeed are small. However, in pure roll forming, larger wrap angles are used. 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BIBLIOGRAPHY 171 [65] SPRINGER, A. and K U C H I B H O T L A , S., "An Investigation into the Influence of Filler Components on Specific Filtration Resistance", TAPPI Papermakers Conference Pro-ceedings Vol. 2: 533-547(1992). [66] SPRINGER, A., P E N N I M A N , J . G., and PIRES, E. C , "Innovative Procedure for Automatic Measurement of Specific Filtration Resistance and Electrostatic Charge", TAPPI Journal 77(8): 121-127(1994). [67] INGMANSON, W. L., "Filtration resistance on the Fourdrinier Table Roll Section", TAPPI Journal 40(12): 936-943(1957). [68] A N D R E W S , B. D. and WHITE, L. R., "A Constant-Rate Rapid Drainage Tester", TAPPI Journal 52(6): 1171-1175(1969). [69] I N G M A N S O N , W. L. and A N D R E W S , B. D., "High Velocity Flow Through Fibre Mats", TAPPI Journal 46(3): 150-155(1963). [70] A T T W O O D , B. W. and JOPSON, R. N., "Dynamic Drainage Simulation", Paper Tech-nology 39(4): 53-56(1998). [71] H U N G , L., L E U N G , W. K., and G R E E N , S. I., "Pulp Fibre Mat Permeability Mea-surement Under Simulated Forming Conditions", TAPPI Engineering Conference Pro-ceedings: CD-Rom(2000). [72] PARADIS, M. A., G E N C O , J . M., BOUSFIELD, D. W., HASSLER, J . C , and W I L D F O N G , V. J . , "Determination of Drainage Resistance Coefficients Under Con-ditions of Known Shear Rate", TAPPI Engineering/Finishing & Converting Conference Proceedings: CD-Rom(2001). [73] F R A N Z E N , M. F., "Simulation and Optimization of the Papermaking Sheet Forming Process", A S M E Meeting, Boston, Mass. (2000). [74] H A N , S. T. and INGMANSON, W. L., "A Simplified Theory of Filtration", TAPPI Journal 50(4): 176-180(1967). BIBLIOGRAPHY 172 [75] E R G U N , S., "Fluid Flow through Packed Columns", Chemical Engineering Progress 48(2): 89-94(1952). [76] S A Y E G H , N. N. and G O N Z A L E Z , T. O., "Compressibility of Fibre Mats During Drainage", Journal of Pulp and Paper Science 21(7): J255-J261(1995). [77] H A P P E L , J . , "Viscous Flow Relative to Arrays of Cylinders", A.I.Ch.E. Journal 5(2): 174-177(1959). [78] M E Y E R , H., "A Filtration Theory for Compressible Fibrous Beds Formed from Dilute Suspensions", TAPPI Journal 45(4): 296-310(1962). [79] INGMANSON, W. L., H A N , S. T., WILDER, H. D., and M Y E R S , W. T. JR., "Resis-tance of Wire Screens to Flow of Water", TAPPI Journal 44(1): 47-54(1961). [80] T A Y L O R , J . R., "Estimating Uncertainties when Reading Scales" in An Introduction to Error Analysis: The Studies of Uncertainties in Physical Measurement, , Mi l l Valley, Calif., University Science Books: 9-11(1982). [81] BRONSHTEIN , I. N. and S E M E N D Y A Y E V , K. A., "Simulation and Statistical Plan-ning and Optimization of Experiments", in Handbook of Mathematics, 3rd edition, Berlin Heidelberg New York, Springer: 936-946(1997). [82] F L U E N T User's Guide Vol. 2, Lebanon, NH, Fluent, 10:1-72(1996). [83] F L U E N T User's Guide Vol. 4, Lebanon, NH, Fluent, 19:1-154(1996). [84] F L U E N T User's Guide Vol. 1, Lebanon, NH, Fluent, 6:159-186(1996). [85] S C H A C H , W., "Umlenkung eines Freien Fliissigkeitsstrahles an einer Ebenen Platte", Ingenieur-Archiv 5: 245-265(1934). [86] Dale Johnson, AstenJohnson Company, Private communication Appendix A Results of the drainage experiments This appendix shows the measured drainage profiles and drainage velocity profiles at jet impingement on a stationary fabric which were not shown earlier. A . l Completed drainage as a function of distance from the impingement point T3 2 t o o ID 60 C3 d 120 100 80 60 3 40 20 1 1 " \ . . . . . .1. /"" ? x ^x / /// i r " / •''/ \ \ 20° ^ 1 i i 30" ! • — X - — ! 45° — 90° ' a < 0 1 2 3 4 5 6 7 8 Distance [cm] Figure A . l : Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric C, 3.7 m/s jet velocity). 173 APPENDIX A. RESULTS OF THE DRAINAGE EXPERIMENTS 174 120 100 - 80 "a o o 00 ca c 60 a 40 20 0 I I | • - i i i I \--'-::'::\ .^^ 4^ .. | ! ff / I i -;;// iff/ \ \ \ 15° ^ '— i 20° H—x~~• . f 1 ! * i i i i 30° » - : 45° ' a < 90° 0 1 2 3 4 5 6 7 8 Distance [cm] Figure A.2: Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric D, 7.0 m/s jet velocity). 100 h I 80 B B "a o o <D 60 a a 60 a 40 20 I t / . - * ' • 20° > — " — i . 1*1 30° . 45° • * : 90° • e t Distance [cm] Figure A.3: Fraction of completed drainage as a function of the distance from impingement point for different impingement angles (fabric D, 3.7 m/s jet velocity). APPENDIX A. RESULTS OF THE DRAINAGE EXPERIMENTS 175 120 Distance [cm] Figure A.5: Fraction of completed drainage as a function of the distance from impingement point for different jet velocities (fabric D, 20° impingement angle). APPENDIX A. RESULTS OF THE DRAINAGE EXPERIMENTS A.2 Drainage velocity profile 176 Figure A.6: Drainage velocity as a function of distance from impingement point for different impingement angles (fabric C, 3.7 m/s jet velocity). APPENDIX A. RESULTS OF THE DRAINAGE EXPERIMENTS 177 0 1 2 3 4 5 6 Distance [cm] Figure A.7: Drainage velocity as a function of distance from impingement point for different impingement angles (fabric D, 7.0 m/s jet velocity). 0.8 I 1 1 1 1 r o h -0.1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 Distance [cm] Figure A.8: Drainage velocity as a function of distance from impingement point for different impingement angles (fabric D, 3.7 m/sjet velocity). APPENDIX A. RESULTS OF THE DRAINAGE EXPERIMENTS 178 0.7 0 0 1 2 3 4 5 6 Distance [cm] Figure A.10: Drainage velocity as a function of distance from impingement point for different jet velocities (fabric D, 20° impingement angle). APPENDIX A. RESULTS OF THE DRAINAGE EXPERIMENTS 179 Figure A . l l : Drainage velocity as a function of distance from impingement point for different jet velocities (fabric E, 20° impingement angle). Appendix B Comparison between experiments and computations This appendix shows plots comparing the predicted and measured drainage values which were not shown earlier. The computations were done with the average fabric resistance numbers determined in the flow loop with the flow perpendicular to the fabrics. B . l Grids for the computations Table B . l : Domain sizes and grids used in the computations for the drainage experiments. Domain size length x height Grid i x j Modelled length from impingement point Used for cases fabric/angle [°]/velocity [m/s] 7 cm x 1.7 cm 352 x 214 5 cm C/20/al l velocities D+E/20/3.7+4.8+6.1 C+D+E/15/7.0 6.4 cm x 2 cm 322 x 294 5 cm C+D/30/3.7+7.0 E/30/7.0 8 cm x 2 cm 402 x 214 6 cm D+E/20/7.0 5 cm x 1.1 cm 302 x 294 4 cm C+D+E/90/7.0 6.1 cm x 1.1 cm 301 x 294 5 cm C+D/45/3.7 ~ C+D+E/45/7.0 4 cm x 1.1 cm 242 x 294 3 cm C+D/90/7.0 180 APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 181 B.2 Results: Fabric C B.2.1 Varying jet velocity B.2.2 Varying impingement angle 0.6 0.5 "§ 0.4 o o <a ao 1 0.2 Q 0.1 6.1 m/s, computed 6.1 m/s, experiment 4.8 m/s, computed 4.8 m/s, experiment Distance [cm] Figure B . l : Experimental and computed drainage velocity profiles for jet velocities of 4.8 m/s and 6.1 m/s (impingement angle 20°, fabric C). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 100 182 2? 8 60 h a o o CD SO cd c 2 Q 40 h 20 h [.....//'., 1 : ..-j . 1 X'-.'. 1 j / 3 T ! 6.1 m/s, experiment 4.8 m/s, computed 4.8 m/s, experiment 0 1 2 3 4 5 6 7 8 Distance [cm] Figure B.2: Experimental and computed fraction of completed drainage as a function of distance from impingement point for jet velocities of 4.8 m/s and 6.1 m/s (impingement angle 20°, fabric C). 0.9 0.8 0.7 co | 0.5 u 0.4 60 C3 | 0.3 Q 0.2 0.1 0 a *. 1 i i i 15°, computed + 15 °, experiment * 30 °, computed * 30 °, experiment 0 45 °, computed • 45 °, experiment ° \ \ + ---•lis- - -• — \ i ^ ' f-—* \ 0 1 2 3 4 5 6 Distance [cm] Figure B.3: Experimental and computed drainage velocity profiles for 15°, 30° and 45° impingement angle (jet velocity 7.0 m/s, fabric C). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 183 120 Distance [cm] Figure B.4: Experimental and computed fraction of completed drainage as a function of dis-tance from impingement point for 15°, 30° and 45° impingement angle (jet velocity 7.0 m/s, fabric C). 0.8 | 1 1 1 1 1 20° 30° 0.7 r \ | ! | 45° " ; 90° o.6 - \ \ i i I I -0 I i i - ••' i 0 1 2 3 4 5 Distance [cm] Figure B.5: Computed drainage velocity profiles for different impingement angles (jet velocity 3.7 m/s, fabric C). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 184 o > 60 c '3 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 ' ' X . ' 20 °, computed 20 °, experiment 90 °, computed 90 °, experiment a— 0 1 2 3 4 5 6 Distance [cm] Figure B.6: Experimental and computed drainage velocity profiles for 20° and 90° impinge-ment angle (jet velocity 3.7 m/s, fabric C). o o "5 > 60 CS a cs 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 1 1 1 30 °, computed + 30 °, experiment x 45 °, computed * 45 °, experiment Q x \j ; | 1 * \ \ \ \ —-::.-;:4-.,,, - . . -_,„..^ ..n- .^r-.-r?^*" I i i Distance [cm] Figure B.7: Experimental and computed drainage velocity profiles for 30° and 45° impinge-ment angle (jet velocity 3.7 m/s, fabric C). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 185 o o C3 .3 S3 0.2 exp. at 0.5 cm comp. at 0.5 cm exp. at 1.5 cm comp. at 1.5 cm 20 30 40 50 60 Impingement angle [°] 70 80 90 Figure B.8: Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 3.7 m/s, fabric C). N T O -•73 O O <D ao C3 .3 2 Q 3 4 5 Distance [cm] Figure B.9: Experimental and computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 3.7 m/s, fabric C). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 186 B.3 Results: Fabric D B.3.1 Varying jet velocity 1 7.0 m/s 6.1 m/s H.H mi 3.7 m/ s s \ \ \ \ \ \ \\ -i 0 1 2 3 4 5 6 Distance [cm] Figure B.10: Computed drainage velocity profiles for different jet velocities (impingement angle 20°, fabric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 187 * 1 1 7.0 m/s, computed + 7.0 m/s, experiment * 3.7 m/s, computed * 3.7 m/s, experiment Q + \ i \ \ \ * \ ! \ *-i I * ! '-'m h i i i 0 1 2 3 4 5 6 Distance [cm] Figure B . l l : Experimental and computed drainage velocity profiles for jet velocities of 3.7 m/s and 7.0 m/s (impingement angle 20°, fabric D). 0.5 0.45 0.4 ^ 0.35 'o -§ 0.25 u so ca a 0.2 0.15 0.1 0.05 0 1 1— 6.1 m/s, computed 6.1 m/s, experiment 4.8 m/s, computed 4.8 m/s, experiment .......... B . ... Distance [cm] Figure B.12: Experimental and computed drainage velocity profiles for jet velocities of 4.8 m/s and 6.1 m/s (impingement angle 20°, fabric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 188 0.6 0.5 J L 0 4 o o f 0 3 oo CS 1 0.2 0.1 h 3.5 4.5 5 5.5 Jet velocity [m/s] 1 1 1 1 | • • • - , , T 1 " exp. at 0.5 cm — 1 — comp. at 0.5 cm - - x — exp. at 1.5 cm *—- x ^ comp. at 1.5 cm a x-""" I i i ; i * i _ *-"*[ 1 * - i ; •••-B i } i Q T i i i i i • i 6.5 Figure B.13: Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of jet velocity (impingement angle 20°, fabric D). T3 D •*-» CD I O o a> 00 CS .3 2 Q 7.0 m/s, computed 7.0 m/s, experiment + 6.1 m/s, computed 6.1 m/s, experiment * 4.8 m/s, computed 4.8 m/s, experiment * 3.7 m/s, computed 3.7 m/s, experiment D 3 4 5 6 7 Distance [cm] Figure B.14: Experimental and computed fraction of completed drainage as a function of distance from impingement point for different jet velocities (impingement angle 20°, fab-ric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS B.3.2 Varying impingement angle 189 Figure B.15: Computed drainage velocity profiles for different impingement angles (jet ve-locity 7.0 m/s, fabric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 190 £ o > 00 CO .3 C3 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 • i i 20 °, computed + 20 °, experiment x 90 °, computed * 90 °, experiment Q - \V \ H \ \ i "\ 'k i i'" — - a * 0 1 2 3 4 5 6 Distance [cm] Figure B.16: Experimental and computed drainage velocity profiles for 20° and 90° impinge-ment angle (jet velocity 7.0 m/s, fabric D). 1 .£> o o 13 > <D 00 C3 .3 «5 Q 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 1 O 1 : i i i 15 °, computed 15 °, experiment 30 °, computed 30 °, experiment 45 °, computed 45 °, experiment + X • • o X-. \ VsJ \ i i 1 +• V V \ \ i i i : \ '"iV'Vx I I I ! " !"<x• i '^"^....-i i Distance [cm] Figure B.17: Experimental and computed drainage velocity profiles for 15°, 30° and 45° impingement angle (jet velocity 7.0 m/s, fabric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 191 1 .£ 3 o > <L> 00 CO .3 CO a 40 50 60 Impingement angle [°] Figure B.18: Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 7.0 m/s, fabric D). T3 ca 3 o o <0 00 C3 .3 2 Q 120 100 3 4 5 Distance [cm] Figure B.19: Experimental and computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 7.0 m/s, fabric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 192 > u 60 cd a 2 Q 2 3 Distance [cm] Figure B.20: Computed drainage velocity profiles for different impingement angles (jet ve-locity 3.7 m/s, fabric D). o o "3 > u 00 03 .9 03 Q 0.8 0.7 0.6 0.5 0.4 0.3 h 0.2 0.1 0 -0.1 0 20 °, computed 20 °, experiment 90 °, computed 90 °, experiment 5r>. . ts-Distance [cm] Figure B.21: Experimental and computed drainage velocity profiles for 20° and 90° impinge-ment angle (jet velocity 3.7 m/s, fabric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 193 0.7 W 0.5 • 30 °, computed + 30 °, experiment * 45 °, computed * 45 °, experiment D * \ x ia - j Vj, '*% """ r - i -• • ~ i 1 — 0 1 2 3 4 Distance [cm] Figure B.22: Experimental and computed drainage velocity profiles for 30° and 45° impinge-ment angle (jet velocity 3.7 m/s, fabric D). o "5 > 00 CS d '§ Q 0.8 h 0.6 0.4 0.2 1 1 1 exp. at 0.5 cm — 1 — comp. at 0.5 cm -— x — ~ exp. at 1.5 cm *—-comp. at 1.5 cm i —' —r i i i 20 30 40 50 60 Impingement angle [°] 70 80 90 Figure B.23: Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 3.7 m/s, fabric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 194 S o o <u ao CO C 120 100 I 80 60 40 20 20 °, computed 20 °, experiment + 30 °, computed 30 °, experiment x 45 °, computed 45 °, experiment * 90 °, computed 90 °, experiment H 3 4 5 6 7 Distance [cm] Figure B.24: Experimental and computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 3.7 m/s, fabric D). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 195 B.4 Results: Fabric E B.4.1 Varying jet velocity 7.0 m/s 6.1 m/s — 4.8 m/s 3.7 m/ s \\ \ \ \ '\. \ \ -0 1 2 3 4 5 6 Distance [cm] Figure B.25: Computed drainage velocity profiles for different jet velocities (impingement angle 20°, fabric E). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 196 1 o _ o 1 3 > 00 CS g '3 Q 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Q, 1 1 I -1 7.0 m/s, computed + - \ 7.0 m/s, experiment x ... 3.7 m/s, computed * 3.7 m/s, experiment Q ... X \ * i j " " " i i - . . 1 i * " - " • ' ] t Distance [cm] Figure B.26: Experimental and computed drainage velocity profiles for jet velocities of 3.7 m/s and 7.0 m/s (Impingement angle 20°, fabric E). 3 o > o 00 CS c '3 u Q 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 i 1 1 1 6.1 m/s, computed + - a 6.1 m/s, experiment x 4.8 m/s, computed * 4.8 m/s, experiment Q , ' t i : \ \ \ -« ; s v \ K 1 '•• *. \ X . : ~ -4-i Distance [cm] Figure B.27: Experimental and computed drainage velocity profiles for jet velocities of 4.8 m/s and 6.1 m/s (impingement angle 20°, fabric E). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 197 o 1 1 ' 1 ' 1 1 1 3.5 4 4.5 5 5.5 6 6.5 7 Jet velocity [m/s] Figure B.28: Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of jet velocity (impingement angle 20°, fabric E). 120 CD OO ca S3 2 Q 7.0 m/s, computed 7.0 m/s, experiment + 6.1 m/s, computed 6.1 m/s, experiment x 4.8 m/s, computed 4.8 m/s, experiment * 3.7 m/s, computed 3.7 m/s, experiment ° 3 4 5 6 7 8 Distance [cm] Figure B.29: Experimental and computed fraction of completed drainage as a function of distance from impingement point for different jet velocities (impingement angle 20°, fabric E). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 198 B.4.2 Varying impingement angle 0 1 2 . 3 4 5 Distance [cm] Figure B.30: Computed drainage velocity profiles for different impingement angles (jet ve-locity 7.0 m/s, fabric E). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 199 1 .£ 3 o *3 > ao C3 a 2 Q 1.4 1.2 1 0.8 0.6 0.4 0.2 0 * 4 > 20 °, computed + 20 °, experiment x \ 90 °, experimer It • x . \ 1 'v. " a 1-Distance [cm] Figure B.31: Experimental and computed drainage velocity profiles for 20° and 90° impinge-ment angle (jet velocity 7.0 m/s, fabric E). 1.2 1 3 o "3 > ao ca a 'ca t-i Q 0.8 0.6 0.4 0.2 15 °, computed 15 °, experiment 30 °, computed 30 °, experiment 45 °, computed 45 °, experiment _ i Distance [cm] Figure B.32: Experimental and computed drainage velocity profiles for 15°, 30° and 45° impingement angle (jet velocity 7.0 m/s, fabric E). APPENDIX B. COMPARISON EXPERIMENTS/COMPUTATIONS 200 o > bo C3 .a C3 40 50 60 Impingement angle [°] Figure B.33: Measured and computed average drainage velocities at 0.5 and 1.5 cm down-stream from impingement as a function of impingement angle (jet velocity 7.0 m/s, fabric E). 100 T3 B I o o <u ao a .3 e Q 15 °, computed 15 °, experiment + 20 °, computed 20 °, experiment * 30 °, computed 30 °, experiment * 45 °, computed 45 °, experiment D 90 °, computed 90 °, experiment • 3 4 5 6 7 8 Distance [cm] Figure B.34: Experimental and computed fraction of completed drainage as a function of distance from impingement point for different impingement angles (jet velocity 7.0 m/s, fabric E). 

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