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Understanding the dynamics of pulp fibre suspension dewatering Paterson, Daniel Thomas 2020

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Understanding the Dynamics of Pulp Fibre SuspensionDewateringbyDaniel Thomas PatersonM.A.Sc., The University of British Columbia, 2016B.A.Sc., The University of British Columbia, 2013A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Chemical and Biological Engineering)The University of British Columbia(Vancouver)July 2020© Daniel Thomas Paterson, 2020The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Understanding the Dynamics of Pulp Fibre Suspension Dewateringsubmitted by Daniel Thomas Paterson in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Chemical and Biological Engineering.Examining Committee:Mark Martinez, Chemical and Biological EngineeringCo-SupervisorNeil Balmforth, MathematicsCo-SupervisorDaan Maijer, Materials EngineeringUniversity ExaminerBrian Wetton, MathematicsUniversity ExaminerAnders Dahlkild, Engineering Mechanics, KTHExternal ExaminerAdditional Supervisory Committee Members:John Frostad, Chemical and Biological EngineeringSupervisory Committee MemberRobert Gooding, VP of Aikawa Fiber Technologies, Mechanical EngineeringSupervisory Committee MemberiiAbstractDewatering of pulp fibre suspensions is a fundamental process in many unit op-erations in the production of pulp and paper. A theoretical understanding of thedynamics of these networking fibre suspensions can prove valuable in the designof industrial equipment and supplement the general field of compressive rheol-ogy. This project aims to provide a general understanding of the consolidation, inparticular, how the network of fibres responds to the stresses experienced duringdewatering events. This begins with assessing the robustness of our previous mod-elling effort, which extended the traditional multi-phase, deformable porous mediaframeworks, through the accommodation of a rate-dependent constitutive modelfor the solid effective stress (bulk viscosity). Robustness studies include: testinga large variety of pulp types, investigating low concentration dynamics, and usingindustrial and lab equipment. Results from these studies motivated extending stud-ies of the solid network’s response during high speed dewatering and shear stressapplication.For each study, a combined theoretical and experimental approach was un-dertaken to formulate appropriate model equations, independently calibrate therequired material parameters, and collect experimental dynamic dewatering re-sults. Model robustness for varying pulp suspensions at intermediate concentra-tions utilized a Darcian flow cell to calibrate permeability, a uni-axial dewateringexperiment to determine their compressive yield stress, and dynamic dewateringexperiments at modest rates to characterize the suspension’s bulk viscosity. Thelow concentration investigation introduced experiments for calibrating permeabil-ity and compressive yield stress around the suspension’s gel point and utilizedgravitational drainage experiments to gauge bulk viscosity’s importance. In bothiiiinvestigations, the inclusion of a sizable bulk viscosity was necessary to effectivelyrepresent the dewatering behaviour. Dewatering dynamics in the Twin Roll press,collected at a pilot-scale facility, primarily highlighted the limitations of our pre-vious modelling efforts. Rebuilding of the uni-axial experiment and constitutivemodel for the solid effective stress was undertaken to capture the solid network’selastic response evident at elevated dewatering rates. Additionally, a unique ap-paratus was developed to experimentally calibrate a pulp suspension’s significantshear yield stress at concentrations above traditional rheometer approaches.ivLay SummaryThe removal of water from pulp fibre mixtures is a fundamental step in the pro-duction of pulp and paper. To optimize the design of the industrial machines, astrong understanding of the forces required to squeeze the water out of the mixtureis needed. Generally, these compressive loads are governed by the difficulty ofexpelling the liquid, and the forces required to slide and deform the fibres closertogether. The understanding of these factors constitute the key goals of this thesis.Although this may seem trivial, the forces become highly dependent on the typeof pulp fibres used, the concentration of fibres, and, most importantly, how fast thefibres are slid together. The contribution of this study is a mathematical model thatdescribes the dynamic forces experienced by various pulp fibre suspensions acrossa variety of experiments performed.vPrefaceThis thesis represents my vision, understanding, and contribution to an importantfundamental and industrial question. That said, the thesis also reflects my col-laboration with various individuals across the chapters. For each chapter, I wasinvolved in all aspects of the work, including hypothesis and experimental design,building of the equipment, measurement of the data, and reporting of the results,with mentorship and guidance provided by the research group, constituting MarkMartinez, Neil Balmforth, Duncan Hewitt, and Tom Eaves.A version of Chapter 4 has been published in Physical Review Fluids as D.Paterson, T. Eaves, D. Hewitt, N. Balmforth, and D. Martinez. Flow-driven com-paction of a fibrous porous medium. Phys. Rev. Fluids, 4 (7):074306, 2019.The writing of the manuscript was primarily provided by Tom, which I supportedthrough writing select sections (the Freeness tester), generating various figures,and providing editing support. Mark and Neil provided guidance and the remain-ing supplemental discussions throughout.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Background Summary and Gap . . . . . . . . . . . . . . . . . . . . . 112.2 Project Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Model Robustness: Representation of Varying Suspensions . . . . . . 163.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.1 Hardwoods and Softwoods . . . . . . . . . . . . . . . . . . . 193.1.2 Chemical and Mechanical Pulping . . . . . . . . . . . . . . . 193.1.3 Mechanical Low Consistency (LC) Refining . . . . . . . . . 213.1.4 Chemical Additives . . . . . . . . . . . . . . . . . . . . . . . 22vii3.2 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.1 Fitting Bulk Viscosity: η∗ . . . . . . . . . . . . . . . . . . . 253.3 Results and Discussion: Material Parameters . . . . . . . . . . . . . 273.3.1 Permeability . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3.2 Compressive Yield Stress . . . . . . . . . . . . . . . . . . . . 313.4 Results and Discussion: Dewatering . . . . . . . . . . . . . . . . . . 363.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 454 Model Robustness: Representation of Low Concentration . . . . . . . 474.1 Model Geometry and Equations . . . . . . . . . . . . . . . . . . . . . 494.1.1 Unnetworked and Clear Zones . . . . . . . . . . . . . . . . . 544.1.2 Networked Zone . . . . . . . . . . . . . . . . . . . . . . . . . 554.1.3 Capillary Stresses for Drainage and Freeness . . . . . . . . . 564.2 Material and Experimental Details . . . . . . . . . . . . . . . . . . . 574.3 Calibration of Py(φ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.4 Calibration of k(φ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4.1 Crude Calibration of Py(φ) from Pump-Out Experiment . . 664.5 Results: Model Representation of Suspension Dynamics . . . . . . 684.5.1 Fixed-Volume Compaction Dynamics . . . . . . . . . . . . . 684.5.2 Drainage Dynamics . . . . . . . . . . . . . . . . . . . . . . . 734.6 Understanding Freeness . . . . . . . . . . . . . . . . . . . . . . . . . 764.6.1 Freeness Scores . . . . . . . . . . . . . . . . . . . . . . . . . 764.6.2 The Dynamics of Freeness . . . . . . . . . . . . . . . . . . . 784.6.3 Parameter Sensitivity . . . . . . . . . . . . . . . . . . . . . . 834.6.4 Varying Pulp Suspensions . . . . . . . . . . . . . . . . . . . 854.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.8 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 875 Model Robustness: Representation of Industrial Twin Roll Press . . . 895.1 Model Geometry and Equations . . . . . . . . . . . . . . . . . . . . . 915.1.1 2D Stress State and Constitutive Model . . . . . . . . . . . . 945.1.2 Non-Dimensionalized, Leading Order Model . . . . . . . . 985.1.3 Reduction: A Uni-Axial-Like Model . . . . . . . . . . . . . 101viii5.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2.1 Finding φstart . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.3.1 Uncertainty in φstart . . . . . . . . . . . . . . . . . . . . . . 1145.3.2 Temperature Correction: Λ(φ) . . . . . . . . . . . . . . . . . 1155.3.3 Wash Section . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.4 Discussion: Reflection and Moving Forward . . . . . . . . . . . . . 1205.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1236 Model Extension: High Speed Uni-Axial Dewatering . . . . . . . . . . 1246.1 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.2 Solid Network Rheology: An Elastic-Viscoplastic-Like Constitu-tive Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.2.1 Loading/Unloading Experiments . . . . . . . . . . . . . . . 1286.2.2 Constitutive Model Development . . . . . . . . . . . . . . . 1316.3 Model Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1437 Model Extension: 2D Stress State and Shear Yield Stress . . . . . . . . 1447.1 Proposed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 1467.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1497.3 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.4 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 1568 Summary, Conclusions, and Future Work . . . . . . . . . . . . . . . . . 1588.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163A Experimental Details and Equipment . . . . . . . . . . . . . . . . . . . 173A.1 Suspension Preparation . . . . . . . . . . . . . . . . . . . . . . . . . 173A.2 Uni-Axial Dewatering Experiment . . . . . . . . . . . . . . . . . . . 174A.2.1 Protocol: Dewatering . . . . . . . . . . . . . . . . . . . . . . 176ixA.2.2 Protocol: Calibration of Compressive Yield Stress . . . . . . 177A.2.3 Retrofit: Spring 2018 . . . . . . . . . . . . . . . . . . . . . . 179A.3 Permeability Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 185A.3.1 Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187A.3.2 Retrofit: Spring 2018 . . . . . . . . . . . . . . . . . . . . . . 192B Varying Suspensions’ Details and Calibrated Parameters . . . . . . . 196C Freeness Cone Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200D Twin Roll Press Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203xList of TablesTable 3.1 Details of Various Suspensions . . . . . . . . . . . . . . . . . . . 18Table B.1 CSF and FQA Results of Various Suspensions . . . . . . . . . . . 197Table B.2 Material Parameter and Dewatering Results of Various Suspen-sions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199Table D.1 Roll Press Experimental Results . . . . . . . . . . . . . . . . . . . 204Table D.2 Roll Press Model Results . . . . . . . . . . . . . . . . . . . . . . . 205xiList of FiguresFigure 1.1 Roll Press Schematics . . . . . . . . . . . . . . . . . . . . . . . . 2Figure 2.1 Uni-Axial Dewatering and Permeability Models . . . . . . . . . 8Figure 2.2 Viscoplastic Constitutive Model Free Body Diagram . . . . . . 12Figure 3.1 Schematics of Fibre Details . . . . . . . . . . . . . . . . . . . . . 21Figure 3.2 Details of Bulk Viscosity Fitting . . . . . . . . . . . . . . . . . . 26Figure 3.3 Permeability Results . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 3.4 Collapsed Permeability Results . . . . . . . . . . . . . . . . . . . 31Figure 3.5 Compressive Yield Stress Results . . . . . . . . . . . . . . . . . 33Figure 3.6 Dewatering Experimental Results . . . . . . . . . . . . . . . . . 37Figure 3.7 Good Dewatering Model Representation . . . . . . . . . . . . . 39Figure 3.8 Model Error Measurements . . . . . . . . . . . . . . . . . . . . . 41Figure 3.9 Average Model Representation . . . . . . . . . . . . . . . . . . . 42Figure 3.10 Fitted η∗ to k∗ and p∗ . . . . . . . . . . . . . . . . . . . . . . . . 43Figure 4.1 Canadian Standard Freeness Tester . . . . . . . . . . . . . . . . 49Figure 4.2 Model Geometries and Zones of Suspension . . . . . . . . . . . 51Figure 4.3 Results of Sedimentation Experiments . . . . . . . . . . . . . . . 60Figure 4.4 Bridged Py(φ) Results . . . . . . . . . . . . . . . . . . . . . . . 62Figure 4.5 Flow Through Experimental Results . . . . . . . . . . . . . . . . 63Figure 4.6 Bridged k(φ) Results . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 4.7 Sedimentation Dynamics: Experiments and Model . . . . . . . 69Figure 4.8 Flow Through Dynamics: Experiments and Model . . . . . . . . 70Figure 4.9 Error of Interface Postion for Sedimentation and Flow Through 72xiiFigure 4.10 All Drainage Dynamics: Experiments and Model . . . . . . . . 74Figure 4.11 Select Drainage Dynamics: Experiment and Model . . . . . . . 75Figure 4.12 Error of hˆ(tˆ) and Countour Distance . . . . . . . . . . . . . . . 77Figure 4.13 Experimental and Model Predictions of Freeness Scores UsingCSF Tester . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79Figure 4.14 Detailed Model Solutions for Freeness Test: Varying η∗ Values 80Figure 4.15 Freeness Score Material Parameter Sensitivity & Model Pre-dictions for Varying Suspensions of Chapter 3 . . . . . . . . . . 85Figure 5.1 Schematic of the Twin Roll Press . . . . . . . . . . . . . . . . . 90Figure 5.2 Model Geometry: Roll Press Nip . . . . . . . . . . . . . . . . . . 92Figure 5.3 Permeability and Compressive Yield Stress of Series 17 . . . . 105Figure 5.4 Picture and Flowchart of the Pilot Plant Twin Roll Press . . . . 107Figure 5.5 Approximations for φstart . . . . . . . . . . . . . . . . . . . . . . 108Figure 5.6 Roll Press Mass Balance . . . . . . . . . . . . . . . . . . . . . . 109Figure 5.7 Model Results for Trial 103 . . . . . . . . . . . . . . . . . . . . . 111Figure 5.8 Model vs. Experimental Line Loads . . . . . . . . . . . . . . . . 113Figure 5.9 Strength and Viscous Components of Load for Trial 103 withUniform Compaction . . . . . . . . . . . . . . . . . . . . . . . . 114Figure 5.10 Investigation of φstart Uncertainty . . . . . . . . . . . . . . . . . 115Figure 5.11 Model vs. Experimental Line Loads: Temperature CorrectedBulk Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Figure 5.12 Model Results for Trial 601: η∗ and ηTC∗ . . . . . . . . . . . . . . 118Figure 5.13 Wash-Section Schematic and Corresponding φstart,wash Ap-proximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Figure 5.14 Model Results for Trial 103: With and Without a Wash-Section 121Figure 6.1 Viscoplastic Constitutive Model Representation: High SpeedUni-Axial Dewatering . . . . . . . . . . . . . . . . . . . . . . . . 125Figure 6.2 Sample Loading and Unloading Experiment . . . . . . . . . . . 127Figure 6.3 Elastic Experimental Rheology . . . . . . . . . . . . . . . . . . . 129Figure 6.4 Elastic-Viscoplastic Constitutive Model Free Body Diagram . . 132Figure 6.5 Elasticity and Compressive Yield Stress . . . . . . . . . . . . . . 135xiiiFigure 6.6 Model Unloading/Reloading Curves . . . . . . . . . . . . . . . . 137Figure 6.7 Parabolic Compression Dewatering Experiments . . . . . . . . . 138Figure 6.8 Model Representation: Select Parabolic Compression Dewa-tering Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 139Figure 6.9 Experimental and Model Solid Phase Velocity Contours: SlowSpeed Dewatering . . . . . . . . . . . . . . . . . . . . . . . . . . 141Figure 6.10 Experimental and Model Solid Phase Velocity Contours: HighSpeed Dewatering . . . . . . . . . . . . . . . . . . . . . . . . . . 142Figure 7.1 Traditional Vane and Cup Rheometer . . . . . . . . . . . . . . . 146Figure 7.2 Shear Yield Stress Experimental Determination . . . . . . . . . 147Figure 7.3 Experimental Shear Yield Stress Tester . . . . . . . . . . . . . . 151Figure 7.4 Experimental Shear Yield Stress Results . . . . . . . . . . . . . 153Figure 7.5 Yield Stress Results . . . . . . . . . . . . . . . . . . . . . . . . . 155Figure A.1 Uni-Axial Dewatering Experimental Apparatus (UDEA) . . . . 175Figure A.2 UDEA Suspension Chamber and Permeable Piston . . . . . . . 176Figure A.3 UDEA Overall Layout . . . . . . . . . . . . . . . . . . . . . . . . 177Figure A.4 UDEA Retrofit Images 1 . . . . . . . . . . . . . . . . . . . . . . 180Figure A.5 UDEA Retrofit Images 2 . . . . . . . . . . . . . . . . . . . . . . 181Figure A.6 Compression Profiles . . . . . . . . . . . . . . . . . . . . . . . . 183Figure A.7 D-Ash Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184Figure A.8 Permeability Experimental Apparatus (PEA), Section and Ex-ploded View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186Figure A.9 PEA Permeable Piston and Screen Spacer . . . . . . . . . . . . . 187Figure A.10 PEA Retrofit Images 1 . . . . . . . . . . . . . . . . . . . . . . . . 193Figure A.11 PEA Retrofit Images 2 . . . . . . . . . . . . . . . . . . . . . . . . 194Figure A.12 PEA Retrofit Images 3 . . . . . . . . . . . . . . . . . . . . . . . . 195Figure A.13 Comparsion of Permeability Before and After Retrofit . . . . . 195Figure B.1 Roll Press Schematics . . . . . . . . . . . . . . . . . . . . . . . . 198Figure C.1 Funnel Geometry of the Freeness Device . . . . . . . . . . . . . 201xivAcknowledgmentsThank you to Valmet Corp., NSERC, and UBC for the financial support providedfor this project. Thank you to Tomas Vikstrom and Patrik Pettersson of Valmetfor their guidance throughout. Within UBC, many individuals have supported andassisted through this project. In particular, the excellent efforts of the various sum-mer students including Devon, John, Ash, and Mark, and the support of the PPC’sGeorge Soong. A great level of appreciation must be acknowledged to my supervi-sors, Mark and Neil, and the postdoctoral members in the group, Tom and Duncan,for the countless hours of support, guidance, and mentorship.To my friends and family: thank you Jordan, William, and Samim for yourhelp, and for taking me out of it for a short time. To the Figueiras and my extendedfamily, thank you for all the visits that helped smooth the choppy times. Thank youto Emily and Graeme for the letters, support, and visits throughout which alwaysseemed to be so timely with needing a good break. Thank you Devon for yoursupport throughout, and I will always cherish our adventure in the big city!To my parents. Thank you for the completely unwavering and endless quantityof heart warming support. From the momentary visits to the city, to the near dailycalls and emails, I cant begin to articulate how much I appreciate the efforts you twoput forth to support and encourage me to continue forward through this enduring,challenging, humbling process.Finally, to Alicia. I am so fortunate to have such a strong and understandingpartner throughout this endeavour, my entire educational pursuit, and in my life.Thank you for all your love, your patience, and support. On to the next adventure!xvChapter 1IntroductionThe production of pulp and paper is a major economic provider for the BritishColumbian economy. According to the BC Forestry Innovation Investment 1,the industry provided $4.4 billion in exports in 2017, constituting 10% of theprovince’s total. Pulp and paper operations can be found across the province, em-ploying thousands, and provide an excellent example of an industry that producesa value-added product that, in turn, is sold around the world.During the production of either pulp or paper, ligno-cellulose-based pulp fibresare processed in suspension. These fibres exist as a complex mixture of parti-cles, which vary in length, aspect ratio and cross-sectional shape. They may havekinks, splits and various other features, as discussed in Appendix B. Generally,these pulp fibres have been considered as high aspect-ratio cylindrical particles,with average lengths of 1 − 3mm and widths on the order of 15 − 30µm [19].The fibres are longitudinally hollow in structure, with walls constructed from var-ious layers comprised of cellulose, hemicellulose, and lignin. The fibre walls trapvarying amounts of water within the structure, constituting intra-fibre water of thesuspension. At a threshold concentration, these fibres in suspension begin to en-tangle and contact one another, which in turn develops a continuous solid-phasenetwork which can sustain certain levels of stress. When a stress on the networkexceeds this concentration-dependent strength, consolidation of the network of fi-bres occurs through either rearrangement or deformation of the fibres. At elevated1 www.bcfii.ca16.5 mFigure 1.1: Schematics of a Twin Roll press. Significant dewatering of in-terest occurs in the nip region, highlighted by the green box.concentrations, this network of pulp fibres can begin to withstand high levels ofstress without further deformation. As well, the porous nature of the solid networkcreates resistance to the flow of water outside the fibres, or the inter-fibre water.This resistance will also depend on the concentration of the fibre network.The consolidation of the network of pulp fibres is at the heart of several unit op-erations in the production of pulp and paper, which aim at removing large volumesof water that are added to the suspension to both rinse the suspension and to aidin forming a homogeneous sheet [79]. In these operations, mechanical means areused to squeeze water from the suspension. A particular example of a machine thatperforms dewatering is a Twin Roll press, shown in Figure 1.1. This machine isfound in the wash section of a pulp mill and has two functions: the first is to rinsethe suspension of undesired residuals in the product, and the second is to dewa-ter the suspension to a high solid volume fraction. The action of dewatering occurswhere the two counter-rotating permeable rolls entrain and squeeze the suspension,identified with a green box in Figure 1.1.Although this is a particular machine of interest, the focus of this study is aimedat a general understanding of the basic phenomenon of dewatering of pulp suspen-sions. This includes understanding how the water is expelled from the network,how the network responds to stress, what measurable physical parameters gov-ern the problem, and finally, develop a model for representing the fibre network’s2experimental behaviour. A study of this sort also proves relevant to a variety ofindustrial settings where consolidation of suspensions occurs.Due to its prevalence in the production of pulp and paper, a large numberof studies investigating the dewatering behaviour of these fibre networks can befound in the literature. Select examples of experimental studies include investi-gations of the spatial compaction of the fibre network passing through a papermachine’s pressing section [7, 8], investigations on water removal from fibre walls[13], compression levels at which the intra- and inter-fibre water drain from thesuspension [50], and measurements of compression loads during dewatering [97].Many modelling approaches have also been suggested over the years, with severalexamples provided that use empirically-driven approaches to capture compressionmeasurements of either lab-scale experiments or industrial equipment [31, 44, 62,86, 103, 105]. While these studies prove successful for their particular application,their translation into a more general application is rather limited. We thereforeare drawn to a subset of the available modelling studies which are based upona more complete theoretical basis (examples [6, 25, 41, 42, 66, 78, 81]). Thesemodelling efforts represent the dewatering of the pulp fibre suspension througha multi-phase, fluid mechanical approach. The fibre network is represented by adeformable porous media, which impedes the fluid escaping the suspension (rep-resented by the network’s permeability), and provides resistance to consolidationdue to the network’s inherent strength (provided by an effective stress of the solidnetwork). Both the network’s permeability and strength are material dependentand will vary with concentration. The experimental calibration of these materialparameters is therefore required.The concepts behind these pulp-suspension specific models have been extended,adapted, or rediscovered in a wide range of contexts, from geological compactionto waste water treatment [2, 16, 17, 75, 87]. Highly cited examples, who often areattributed as pioneers of this approach, include the works of Landman, Buscall,and White [11, 53]. These authors were interested in the compression dewater-ing of flocculated mineral suspensions, and defined this deformable porous mediamodelling approach as compressive rheology. The pulp-specific examples above,therefore, are supplemented by a much broader selection of studies found in thegeneral flocculated suspension literature.3A recent publication from our group has contributed to these pulp-specificmodelling studies, and showed improved representation of the network of pulpfibres during a uni-axial compression dewatering experiment, which resembled aFrench coffee press [32]. The notable distinction of the modelling in this studycame from the constitutive function for the solid effective stress. Traditionally,the solid effective stress of the network during yielding (i.e. during dewatering)is assumed to be fully prescribed by the concentration of the particles in suspen-sion. What was demonstrated in this study was that this simple constitutive modelwas insufficient for properly representing the dynamic response of the pulp fibresuspensions during dewatering. Instead, a viscoplastic-like constitutive functionwas required, which defined the solid effective stress as the sum of a concentrationdependent term and a compression rate dependent term, to properly represent thenetwork’s dewatering response. This extension of the constitutive model, whichwas interpreted as a bulk viscosity of the solid network, was speculated to be re-quired due to the drainage of water trapped within the walls of the pulp fibres (theintra-fibre water).The main purpose of this thesis is to further evaluate the suitability of thisviscoplastic-like constitutive model across a larger variety of pulp suspensions (ma-terials) and dewatering experiments. A second underlining purpose is to explore thegoverning material parameters to investigate if insight into their micro-structuralorigins can be obtained. We begin with Chapter 2, a background which presentsthe general model framework for the coming chapters, and an in-depth descriptionof the constitutive model under scruity. The chapter highlights the need for furtherevaluation, and motivates the work of the following chapters. Chapter 3 starts thisevaluation by investigating the model’s suitability over a large variety of pulp fi-bre suspensions. In Chapter 4, the range of concentrations evaluated is expandedto a low-concentration regime. In Chapter 5, we advance from simple geometriesto investigate the constitutive model’s suitability in representing the dewateringseen in the Twin Roll press introduced earlier. Results of Chapter 5 motivate fur-ther scrutiny of the constitutive model at elevated compression rates, provided inChapter 6, and the determination of a pulp suspension’s shear yield stress at ele-vated concentrations, found in Chapter 7. A general summary, conclusions, andsuggested future work is then provided in Chapter 8.4Chapter 2BackgroundWe begin with a general description of the dewatering model equations that wereused in our previous study [32] and constitute the backbone of the various modelsto come. This general framework is then simplified for the respective experimentsand geometries of the various chapters. The modelling approach represents thepulp fibre suspension as a two phase medium, whereby the solid phase representsthe structure of the pulp fibres and the liquid phase includes both the intra- andinter-fibre water. Above the suspension’s gel point concentration, the fibres contactand entangle, which develops a continuous network of fibres (a solid network).This network can sustain certain levels of stress without consolidating to higherconcentrations. The concentration of the pulp fibre suspension is expressed by the(solid) volume fractionφ = VsVl + Vs ≡ msρs (VTotal) , (2.1)with Vs being the solid volume, Vl being the liquid volume, and VTotal being theirsum. Both the solid and liquid phases are assumed to be independently incompress-ible, with densities of 1.5 g/cm3 (taken as the density of cellulose) and 1.0 g/cm3respectively. φ, therefore, can be defined with respect to a suspension’s oven-drysolid mass, ms, and the solid phase density, ρs. Dewatering, or consolidation ofthe pulp suspension, occurs through the collapse of the solid network of fibres.5Continuity expressions for both phases are∂φ∂tˆ+ ∇ˆ ⋅ (φuˆs) = 0, (2.2)−∂φ∂tˆ+ ∇ˆ ⋅ ((1 − φ)uˆl) = 0, (2.3)where uˆs is the solid phase velocity and uˆl is the liquid phase velocity. The hatson select variables are used throughout to indicate dimensional quantities, and boldvariables indicate vectors.As is typical in various compressible porous media studies [32], phase inertiais assumed negligible, and viscous effects are with respect to momentum transferbetween the phases, which is accommodated through Darcy’s law(1 − φ)(uˆs − uˆl) = k(φ)µ[∇ˆpˆ − ρlg] , (2.4)where pˆ is the liquid pore pressure, µ is the liquid viscosity, ρl is the liquid density,and g is the acceleration of gravity. k(φ) is the solid network’s volume fraction-dependent permeability, which is material specific and therefore requires experi-mental calibration. Particular to fibre suspensions, the permeability generally de-pends on the size of the fibres, the volume fraction, and the arrangement of thefibres [37]. Additionally, due to the fibres’ inherent elongated shape, anisotropicpermeabilities arise for these suspensions, depending upon the fibre orientation.This has been demonstrated for pulp suspensions by Lindsay et al. [60, 61], andfor other fibrous materials such as textiles and felts (examples [1, 46]). In addi-tion to Darcy’s law, a bulk conservation of stress statement, following Terzaghi’sprinciple [93] is defined as∇ˆ ⋅ [−pˆI + Σˆ] + [ρl + φ(ρs − ρl)]g = 0, (2.5)where the gradients in liquid pore pressure and the solid network’s total effectivestress, Σˆ, balance the body forces of the two phases. The total effective stress rep-resents a structural stress within the network of fibres, which needs an appropriateconstitutive law. We note that Equations 2.4 and 2.5 can be derived from individ-ual momentum expressions for both phases with an appropriate inter-phase transfer6term.This general modelling framework, comprising of Equations 2.2-2.5, is notunique to this study. Many examples implementing this approach can be foundboth in pulp and paper literature (examples, [6, 25, 41, 42, 66, 78, 81]) and themore general compressive rheology literature, which spans a variety of contextsfrom geological compaction, to mining tailings, to waste treatment (examples,[2, 11, 16, 17, 74, 75, 87, 99]). The differences between these studies come fromtheir respective constitutive models for the solid network’s effective stress, the cal-ibration of the material dependent parameters, or the geometry to which the equa-tions are applied. In our previous study [32], which summarized the content of myMASc. thesis [76], we used a simple uni-axial dewatering geometry, similar to aFrench coffee press, shown schematically in Figure 2.1a, to develop an appropri-ate constitutive model for capturing the dewatering dynamics of fibre suspensions.The experimental equipment, detailed in Appendix A, had a permeable piston atzˆ = hˆ(tˆ), which compressed into a suspension of fibres at varying compressionrates. The solid phase was retained in the volume between the permeable pistonand the closed base at zˆ = 0, and so∫ hˆ(tˆ)0φ(zˆ, tˆ)dzˆ = φ0h0, (2.6)where φ0 and h0 are the suspension’s initial volume fraction and height. Typicalvalues chosen were φ0 ≈ 0.02−0.03 and h0 ≈ 50mm, and the maximum compres-sion rate collected was 10mm/s. The height of the piston and compressing stressσˆ, provided by a hydraulic material tester were recorded through the experiment.Using a simplistic dewatering geometry allowed a simplified set of governingequations. With an assumption that the solid phase slips on the cup walls [53],this geometry lent well to a simplified total effective stress whereby only isotropicstresses, Pˆ , are experienced by the solid networkΣˆ = −PˆI, (2.7)and a one-dimensional set of Equations 2.2-2.5 can be applied to represent theexperiment.7(a)ො𝒛ෝ𝝈 (ො𝒕)ො𝒛 = ෡𝒉(ො𝒕)Suspensionො𝒛 = 𝟎(b)ෝ𝝈ො𝒛 = ෡𝒉ො𝒛 = 𝟎෡𝑸෡𝑸Suspension∆ෝ𝒑Figure 2.1: Experimental geometries used in [32, 76]. In (a), a uni-axialdewatering experiment schematic is shown. In (b), a transverse perme-ability experiment schematic is shown.A one-dimensional framework also extinguishes concern of the permeability’sanisotropic nature, provided that it is calibrated with the fibres oriented equiva-lently. A transverse-permeability cell was developed to do so, as shown schemati-cally in Figure 2.1b, and detailed in Appendix A. The equipment retained the fibresbetween a permeable surface at zˆ = 0 and a permeable piston at zˆ = hˆ, which isdriven by a hydraulic actuator. A constant flow rate Qˆ is permeated through thesuspension held at an average volume fraction value of φ¯ = φ0h0/hˆ, allowing thepermeability to be defined by the expressionk(φ¯) = QˆµhˆA∆pˆ. (2.8)Inherent in this protocol is an assumption that φ(zˆ) ≈ φ¯ throughout, and thereforeminimal variation in volume fraction is seen across the sample. This is experimen-tally controlled by ensuring the pressure drop ∆pˆ is maintained sufficiently lowwith respect to the network’s strength, which eliminates the complications of dif-ferential compaction. This protocol was found suitable in a range of φ ≈ 0.05−0.40,8with further details found in Appendix A.3.1. The experimental permeability re-sults collected were well fit by the functional formk(φ) = aφln( 1φ) e−bφ, (2.9)which is motivated by the dilute limit of an ideal suspension of solid rods [37].The constants a and b were fitted to a particular suspension’s results. The resultscollected using an ideal nylon fibre suspension agreed well with a large variety offibre suspensions found in the literature [37]. The pulp suspensions’ results werefound to agree with simliar pulps in literature (examples, [60, 79, 96]), howeverwere found not to scale similarly to the general fibrous suspensions of [37], withrespect to mean fibre widths.The constitutive model for the (isotropic) solid effective stress explored in thisstudy was equivalent to that suggested in [11]. It can be expressed as the followingevolution expression of the solid volume fraction,DφDtˆ≡ ∂φ∂tˆ+ uˆs ⋅ ∇ˆφ =max⎛⎝0, φ (Pˆ − Py(φ))Λ(φ) ⎞⎠ . (2.10)The max-condition toggles between a yielded condition (when Pˆ > Py(φ)), andan un-yielded condition (when Pˆ < Py(φ)). In the yielded condition, the solideffective stress, Pˆ , can be defined fromPˆ − Py(φ) = Λ(φ)φDφDtˆif Pˆ > Py(φ), (2.11)whereas in the un-yielded condition, the solid network withstands the imposedstress without deformation and soDφDtˆ= 0 if Pˆ ≤ Py(φ), (2.12)and the solid effective stress is left undetermined (as for any material with a yieldcondition, or a friction law with a threshold). Py(φ) represents the compressiveyield stress of the fibre network. This is defined as the maximum stress, for a givenvolume fraction, that the network can withstand without irreversible plastic consol-9idation. As the network stiffens with increasing value of φ, we would expect Py(φ)to increase with φ, with related studies typically suggesting a simple power law forits functional form (examples, [94, 95]). A functional form found to represent theexperimental results, and that has been used previously [32, 76, 78], isPy(φ) = aφb(1 − φ)c , (2.13)where a, b, and c are fitted to collected results. The experimental results collectedagreed with others found in the literature (examples, [78, 98]). Calibration of thisparameter utilized our uni-axial dewatering equipment, outlined in Appendix A.The protocol involves a slow speed compression, which utilizes a uniformly com-pacting asymptotic limit of the model equations. This allows direct calibration ofcompressive yield stress from the load on the permeable piston, i.e. Py(φ) = σˆ(tˆ)where φ = φ0h0/hˆ(tˆ). This follows from Equation 2.5, with assumptions of negli-gible body forces and fluid pore pressure drop across the piston, and from Equation2.11, with an assumption that the overpressure is small due to the low speeds (i.e.Pˆ ≈ Py(φ)). This protocol was found suitable in a range of φ ≈ 0.05 − 0.45, withfurther details provided in Appendix A.2.2.The function Λ(φ) in Equation 2.10 behaves as a bulk viscosity, and representsthe distinction of our study. Typical implementations of this constitutive modelassume that the bulk viscosity is small, suggesting its origins are due to the motionof the particles through the viscous fluid of the suspension. Therefore, its sizing issuggested to be of the order of the liquid viscosity [11, 21], and therefore is neg-ligibly small with respect to the compressive yield stress. If this sizing of Λ(φ) isassumed, negligible overpressure in the solid effective stress above the compres-sive yield stress occurs at all rates, providing the simplified yielding constitutivemodel Pˆ − Py(φ) ≈ 0 if Pˆ > Py(φ). (2.14)With its prevalence in the literature, we began [32] with this simplified constitu-tive model which closed the set of equations. After calibration of the compressiveyield stress and permeability, we were able to demonstrate this simplified consti-tutive model was effective in representing the dewatering dynamics of the ideal10nylon fibre and glycerin suspensions, however was not able to represent the dewa-tering dynamics of the pulp fibre suspensions at elevated compression rates aboveO(1)mm/s.In order to recover effective representation of the pulp suspensions’ dewateringdynamics, the full constitutive model shown in Equation 2.11 was used, with achosen simple bulk viscosity form ofΛ(φ) = η∗φ2. (2.15)The constant η∗ was set to a value of O(107)Pa ⋅ s, which optimally pulled themodel’s predictions for the compression loads and the motion of the solid phaseinto agreement with those seen experimentally. The specific source of this ratedependency in the effective stress of the network was not clear, however viscousbehaviour of the fibre network has been noted in the pulp and paper literature andoften is attributed to intra-fibre water leaving the porous fibre walls [62, 79, 97]. Weprovided a similar suggestion for its source, however, with no evidence in support.Outside the specific dewatering literature, further precedence of pulp suspensions’viscous behaviour can be found in rheological studies (with the following reviewsciting various examples [19, 43]).A mechanistic illustration of the constitutive model of Equation 2.10 for a givenvolume element is shown in Figure 2.2, where the compressive yield stress is rep-resented as an increasing resistance ratchet, and the bulk viscosity is representedas a dashpot. Compaction of the volume element occurs only when the effectivestress exceeds the compressive yield stress, and when yielding occurs, it does soviscously. Analogies to viscoplastic fluid flows can be seen with this mechanisticillustration, and therefore we refer to it as a viscoplastic-like constitutive model.Formulation of the constitutive model from this schematic is provided in Chapter6.2.1 Background Summary and GapA general model framework for describing the dewatering behaviour of a pulpfibre suspension has been presented. In a previous study of our own [32], wefound this approach successful in representing a limited set of uni-axial dewatering11Λ(𝜙)𝑃𝑦(𝜙)𝑒𝑃𝑦𝑃𝑦−Λ 𝜙 ⋅ ሶ𝑒−Λ 𝜙 ⋅ ሶ𝑒Ƹ𝑧Simple SaramitoModel(unyielded, acts as a simple Spring system)෠𝒫𝑅𝑎𝑡𝑐ℎ𝑒𝑡 𝐷𝑎𝑠ℎ𝑝𝑜𝑡Figure 2.2: Pictorial representation of a viscoplastic-like constitutive modelshown in Equation 2.11, which was utilized in [32]. The continuedsuitability of this constitutive model across a larger range of pulp sus-pension experiments constitutes the main purpose of this thesis.experiments. The key for successful representation came from a viscoplastic-likeconstitutive model for the solid effective stress. This previous study presented apromising advancement of the pulp and paper literature, however left several gapsfor this thesis to explore.The first opportunity is with respect to a continued exploration with the exper-iments and protocols at hand. Varying pulp suspensions can have a large variationin dewatering behaviour due to many biological and process factors [86]. Froman industrial perspective, this variability leads to a complex production process tooptimize and control, resulting in broad operating conditions which can make thedesign of industrial equipment difficult. Although acknowledgment of various fac-tors can be found in the literature (examples being degree of mechanical refining,fines content, or dewatering chemicals), these factors’ clear impact on a fibre sus-pension’s dewatering behaviour is often not reached. This is because these factors’impact on the suspension dewatering are quite complex to understand, which iscompounded by the polydisperse, multi-scale porosity of the pulp fibres of the sus-pension. The true understanding of these select factors’ impact on the dewatering12properties of pulp suspensions would demand extensive micro-structure investiga-tions, which is not investigated in this thesis. Instead, the model framework andexperimental protocols developed could be applied to these varying suspensions inorder to observe continued suitability of the viscoplastic-like constitutive model,and to observe the variation in the governing material parameters. Interesting ques-tions with respect to the material parameters are left after [32], including the needof bulk viscosity across all suspensions, or the cause for permeability not scalingas they do in other fibre suspensions. Providing further evaluation of the modellingapproach would also improve its industrial usability.A second opportunity is the application of the model framework to industriallyrelevant applications. As mentioned in Chapter 1, a variety of unit operations inthe productions of pulp and paper implement dewatering in some way. In mostexamples, either a two or three dimensional dewatering model would be necessaryto capture the dynamics of the compression loads experienced by the equipment,however, there are examples where simple assumptions based on the geometry ofthe dewatering can lead to a simpler model frame work that resembles the uni-axialformulation. Examples in the literature can be found where a similiar multi-phaseapproach to that taken in [32] have been applied to pulp and paper relevant indus-trial examples, including in the forming and pressing section of a paper machine(example [27, 66, 69]). In these examples, either due to low compression loadsor simply through an assumption, a viscous solid effective stress was not imple-mented. Additionally, only limited model validation was provided. Aside fromthe added complexity of the industrial geometry, these machines also operate atelevated compression rates compared to those evaluated in [32]. Having demon-strated the poor representation of traditional effective stress modelling approaches(example [51]) on the representation of pulp fibre suspensions at increasing com-pression rates, it is tempting to investigate if the constitutive model in Equation2.10 continues its representation at industrially relevant compression rates.A final identified opportunity would be to expand the range of volume fractionin which the viscoplastic-like constitutive model has been evaluated, and the mate-rial parameters have been calibrated. Both at concentrations above and below thoseof [32] would warrant further investigation. At higher concentrations, the materialparameters’ maximum packing behaviour could be investigated, which as shown13in [32], can impact the dewatering dynamics. The maximum volume fractionsreached in the experiments thus far have only just reached values where industrialequipment can be found operating (dewatering upwards of φ > 0.30− 0.40), whichleaves little overhead for certainty in the material parameters. It should be notedhowever, this upper limit posses some difficult experimental challenges. Equallyinsightful would be an investigation at the lower end of the volume fraction, nearthe suspension’s gel point, φg. The gel point is defined as the minimum volumefraction that the suspension fibres form a continuous network. Consolidation atthese concentrations occurs predominantly due to the reduction of inter-fibre porespace, with minimal deformation of the fibres’ structure. Relevant studies in thisconcentration regime can be found, applying the model framework of Chapter 2 toslow speed sedimentation experiments both for pulp suspensions [104] and in themore general dewatering literature [36]. Although these studies did not require arate-dependent constitutive model, it is unclear if this would continue to be the casewith increased dewatering rate experiments. This study also may help inform theorigins of this bulk viscosity, depending upon its necessity in an order of magnitudelower volume fraction regime.2.2 Project ObjectivesBased upon the background provided and the gaps that have been identified, thisthesis began with the following project objectives:• Collect the material parameters and uni-axial dewatering experiments for alarger variety of pulp suspensions. The intention of this objective is to in-crease confidence in the viscoplastic-like constitutive model shown in Equa-tion 2.11, and to investigate the variation of the material parameters with thegoal of gaining insight into their micro-structural origins.• Perform a dewatering study near a pulp suspension’s gel point and apply themodelling framework to the industrially relevant Canadian Standard Free-ness drainage test. The intention of this objective is to further evaluate thesuitability of the viscoplastic-like constitutive model in this concentrationlimit, and provide an industrial example to which we can apply the model.14This objective will require the calibration of the material parameters in thelow concentration regime, which may provide insight into their origins aswell.• Apply the modelling framework of the uni-axial experiment to the dewa-tering occurring in a Twin Roll press. This objective aims at utilizing theslender geometry of the region between the two counter rotating permeablerolls to evaluate the viscoplastic-like constitutive model at elevated compres-sion rates. Additionally, this provides a valuable industrial application of thismodelling approach.Upon completing these objectives, two further objectives became clear:• Investigate the representation of the viscoplastic-like constitutive model atelevated compression rates in the uni-axial dewatering device. This objec-tive utilizes a redesigned uni-axial experiment for elevated compression rateexperiments.• Determine a pulp suspension’s shear yield stress at elevated concentrations.This objective evaluates a crucial assumption in the reduction of the TwinRoll press dewatering model and additionally provides insight into a pulpsuspension’s two-dimensional stress state.Each research objective corresponds to a coming chapter where details, results,discussions, and section summaries and conclusions are found. An overall projectsummary and conclusions, as well as suggested future work, is found in Chapter 8.15Chapter 3Model Robustness:Representation of VaryingSuspensionsThis chapter implements the existing equipment and protocol to investigate theviscoplastic-like constitutive model across a large variety of pulp fibre suspensions.For each suspension, the material parameters are first calibrated, followed by uni-axial dewatering experiments for model evaluation. The intention of this chapteris to increase the confidence in the proposed constitutive model shown in Equation2.11, and investigate the variations of the material parameters.In terms of measuring the dewatering or select material parameters across avariety of pulp suspensions, this chapter is not unique. Particular examples in thepulp and paper literature can be found for small selections of pulp suspensions(examples [14, 60, 78, 79, 96, 98]) which each implement various unique equip-ment and protocols. Although these various results can be compared and compliedacross these various studies, significant variation in results can be observed due tovarious experimental factors, including appropriate height calibration, suspensionthickness, and protocols implemented. A self-contained study for a large varietyof pulp suspensions, whereby the material parameters are calibrated systemati-cally, followed with experimental dewatering results, would prove very valuablefor the further evaluation of the viscoplastic-like constitutive model. Additionally,16it would provide a library of material parameters, which can be used to begin un-derstanding their origins.This chapter begins with a materials section, which introduces the various sus-pensions investigated. Brief sub-sections which distinguish the various pulps arefound, which are meant to serve as an introduction to the reader of some of thefactors of a particular pulp suspension, and the complexity allotted. As discussedin Section 2.1, clear translation of these various factors into modifications of eitherthe material parameters or the overall dewatering behaviour is not the intention ofthis data collection, and therefore these various sub-sections should be taken withthis under consideration. Following the material section, the formal reduction ofthe general equations of Chapter 2 for the uni-axial geometry, shown in Figure 2.1a,is provided, along with details of a fitting protocol to determine the values of η∗for a particular suspension. A discussion of the results begin with first focusing onthe collection of permeability and then compressive yield stress, with discussionsof the results provided throughout. Finally, we turn to the dewatering results: firstfocusing on select experimental results and finally onto the model representation.3.1 MaterialsTwenty-seven pulp fibre suspensions have been investigated to provide varying de-watering behaviour. Additionally, nylon fibre results are borrowed from [32] to addto various discussions. Each suspension corresponds to a specific series number, asymbol used throughout the results of this chapter, and a line style for the fits of thematerial parameters, as shown in Table 3.1. Each pulp suspension is described withrespect to the wood type, the pulping process used, and whether chemical additivesare used or if low consistency (LC) refining has occurred. Suspension preparationdetails can be found in Appendix A.1.In addition to the coming material parameters and dewatering results, Cana-dian Standard Freeness (CSF) tests were performed for each series. The detailsof this standardized drainage experiment are provided in Chapter 4. Also, for allthe suspensions without chemical additives, trials using an Optest Fibre QualityAnalyser (FQA) machine were also performed, which provides various statisticalsizing information of the fibres in each suspension. Both the FQA and CSF results17Table 3.1: Description and details for the various suspensions investigated.Series Symbol Fit DescriptionMaterial Pulping Additives a Beating1b △ ––– Softwood, LP+WS d Chemical2 + ––– Softwood, LP+WS Chemical EKA FIX 413 b × − − − Softwood, LP+WS Chemical EKA PL 15104 b ◻ ⋯⋯ Softwood, LP+WS Chemical EKA PL 1510 +EKA NP 3205 ◇ – ⋅ – ⋅ – Softwood, LP+WS Chemical Fennobond 3300E6 ◯ –– –– –– Softwood, LP+WS Chemical Fennosil ES3257 ∎ − − − Softwood, LP+WS Chemical LCR, 47 kWh/T i8 ⧫ ⋯⋯ Softwood, LP+WS Chemical LCR, 95 kWh/T9 ▲ – ⋅ – ⋅ – Softwood, LP+WS Chemical LCR, 151 kWh/T10 b ● ––– Hardwood, Aspen Chemical11 ▲ ––– Hardwood, Euc e Chemical12 × ––– Hardwood, Euc 80% + ChemicalSoftwood, (LP+WS) 20%13 ● − − − Hardwood, Euc 60% + ChemicalSoftwood, LP+WS 40%14 ▲ ⋯⋯ Hardwood, Euc 40% + ChemicalSoftwood, LP+WS 60%15 ∎ – ⋅ – ⋅ – Hardwood, Euc 20% + ChemicalSoftwood, LP+WS 80%17 ∎ ––– Softwood,SP+NS f Chemical19 ∎ ––– Hardwood, Aspen Chem.-Mech.20 ⧫ − − − Hardwood, Aspen Chem.-Mech. LCR, 13 kWh/T21 ▲ ⋯⋯ Hardwood, Aspen Chem.-Mech. LCR, 30 kWh/T22 ∎ ––– Softwood, LP+WS Thermo.-Mech.23 ▲ ⋯⋯ Softwood, LP+WS Thermo.-Mech. LCR, 56 kWh/T24 ⧫ − − − Softwood, LP+WS Thermo.-Mech. LCR, 112 kWh/T25 ● – ⋅ – ⋅ – Softwood, LP+WS Thermo.-Mech. LCR, 229 kWh/T26 ◻ ⋯⋯ Softwood, BS+ WS+BF g Thermo.-Mech. LCR 37 kWh/T27 ◇ – ⋅ – ⋅ – Softwood, BS+ WS+BF Thermo.-Mech. LCR, 73 kWh/T28 ◯ –– –– –– Softwood, BS+ WS+BF Thermo.-Mech. LCR, 110 kWh/T29 ◻ ––– Softwood, SP h ChemicalNF c ☀ ––– Nylon, 1.5 Deniera Chemical additive concentrations are 0.1 (wt/wt %).b Suspensions presented in [32].c Nylon fibre suspension presented in [32]. 1.5 denier corresponds to a 0.167mg/m coarseness. Fibres are 3.05mm inlength, and a mean width of 13.58µm. Density is assumed 1.15 g/cm3.d LP+WS: mixture of lodgepole pine and white spruce [71].e Euc: Eucalyptus.f SP+NS: mixture of scots pine and norway spruce [71].g BS+ WS+BF: Mixture of black spruce, white spruce, and balsam fir.h SP: Mixture of southern pines (slash, longleaf, shortleaf, lobolly, and virginia pines)[71].i LCR: Low consistency refining.18are included in Table B.1 of Appendix B.In the following subsections, guiding discussions on the differences betweenthese suspensions are provided.3.1.1 Hardwoods and SoftwoodsThe pulp fibre suspensions of interest to this study consist of wood-sourced fibres.A typical wood fibre is a hollow, elongated particle that is constructed from mul-tiple layers comprised of varying amounts of cellulose, hemicellulose, and lignin[47]. An interpretation of a crossection of a wood fibre is shown schematically inFigure 3.1a. The tertiary wall (T), being the innermost layer, surrounds the cen-tral void of the fibre which is referred to as the lumen. The main body (S2) andthe windings (S1) of the secondary wall contain the majority of the cellulose inthe fibre. In the main body, micro-fibrils of cellulose are arranged in bundles calledmacro-fibrils that are bound together and to one another with a lignin-hemicellulosematrix, forming an interrupted lamella structure [26, 45, 88]. The macro-fibrils arehelically wound around the fibres longitudal axis [40] and constitute the majority ofthe fibre wall structure. The primary wall (P), composed of cellulose and hemicel-lulose, provides the thin outer layer of the wood fibre. Arrays of these wood fibresare aligned vertically and are bound together by a lignin rich layer, referred to asthe middle lamella (ML). This bounded array of fibres forms the wood structure,shown in Figure 3.1b. Depending on the species of tree and where in the wood itresided, the wood fibres can vary in length, width, and wall thickness. Typically,fibres sourced from softwood trees (conifers) are longer, wider, and have greaterwall thickness than those from hardwoods (deciduous)[85].3.1.2 Chemical and Mechanical PulpingThe process of separating these fibres from the wood matrix is referred to as pulp-ing. Two philosophies exist for doing this. The first, a chemically pulped process,separates the fibres by dissolving the lignin-rich middle lamella binding the fibrestogether. A schematic of a chemically pulped fibre is shown in Figure 3.1c. Duringthe pulping process, the lignin and a certain amount of the hemicellulose in themain body of the fibre is also dissolved, leaving a considerable amount of porosity19in the fibre walls as the spacing between the macro and micro-fibrils [88]. Theexposed surfaces of the cellulose fibrils have hydroxyl sites causing an anionic na-ture of the fibre wall which attract water molecules, filling the porous space. Thesehighly swollen fibres, referred to as never-dried fibres, are flexible and have a highsurface area which are ideal properties for bonding between fibres. However, dueto the significant amount of water stored in the pores of the fibre wall, these fibresare difficult to dewater [50]. In subsequent steps in the pulping process, the suspen-sion of fibres are dried, either through pressing or thermal means. With the waterevacuated between the fibrils, hydrogen cross-linking occurs between the surfacesof some of the pores, which reduces the fibre’s ability to retain water in furtherreslushing processes [82]. These fibres, refered to as dried pulp fibres, are stifferbut dewater more readily. This reduction of water holding capacity is referred to ashornification [39], and is a well studied distinction between never-dried and driedpulp fibres. Water retention values (WRV), a metric of the water stored in the fibrewalls, can be expected to drop by 20-30% when comparing never-dried and driedfibres [13]. The chemically pulped suspensions of this study are all reslushed fromdried pulp fibre sheets. In terms of relative dewatering performance, dried chemi-cal pulp fibres constitute some of the best dewatering performance achievable witha pulp suspension. Chemical pulping is suitable for both hardwoods and softwoods,and provides a long, flexible, and strong fibre, but due to the significant amount ofthe wood structure being dissolved, this pulping process has a relatively low yieldof 40-55% [85].The second philosophy, a mechanical pulping process, takes a cruder approachof bashing and cracking the matrix of wood fibres apart. A schematic of a mechan-ically pulped fibre is shown in Figure 3.1d. These processes provide much higheryields of 85-95%, however result in fracturing the fibres and the binding middlelamella, producing smaller fibres and a high content of fines (small debris-like ma-terial) [85] which negatively impact the dewatering. Often the mechanically pulpedprocess is aided through either thermal means (hot steam used to soften woodchips), chemical means (chemicals used to begin dissolving the lignin) or both. Thedelicacy of hardwood fibres somewhat limits these pulping processes to softwoodspecies, though with the help of chemicals in the pulping process can be extendedto hardwoods as well [85]. Due to the majority of the lignin-hemicellulose ma-20TPS1S2(a)     (b)    (c) (d)ML~15 𝜇𝑚 ~25 𝜇𝑚 ~30 𝜇𝑚~50 𝜇𝑚Figure 3.1: In (a), a simplified cross-sectional view of a wood fibre is shown,illustrating the various constituting layers. In (b), a wood structure, con-sisting of varying sized wood fibres is shown. In (c), a toy chemicallypulped fibre is illustrated, with the exposed helical macro-fibrils of theS2 layer. In (d), a toy mechanically pulped fibre is shown, with the ma-jority of the fibre structure still encapsulated by the lignin-rich MiddleLamella (ML).trix surrounding the micro and macro-fibrils of the fibre still remaining intact, theWRV of mechanically pulped fibres is lower, and the fibres are stiffer than chemi-cally pulped fibres [49]. However, due to the limited number of exposed cellulosefibrils, the pore structure the mechanically pulped fibres do possess is less prone tohornification [13, 82].3.1.3 Mechanical Low Consistency (LC) RefiningIn addition to the wood type and pulping process, there are various other factorsin the production that can alter a pulp suspension’s dewatering behaviour. A firstexample is with respect to whether a suspension has been mechanically workedor beaten, and to what degree. In some circumstances, a modification of the fi-bre structure is sought after. An example is the low consistency (LC) refining ofreslushed, chemically pulped fibre suspensions, which improves the bonding be-tween fibres at the expense of poorer dewatering behaviour [100]. While the equip-ment used for this process is rather complex, the goal of the process is to providenormal and shear forces to the fibres of the suspension [30]. The three effects to thefibre suspension are increased levels of fibre fibrillation (both internal and exter-21nal), production of fines, and straightening of the fibres [100]. Internal fibrillation,coming predominantly from the normal forces experienced by the fibres, refers tothe breaking of the cross-linking of the fibrils in the fibre wall, which develops fur-ther porosity and thereby increases the fibres WRV while making it more flexible[101]. This is suggested to occur in the early stages of refining [89, 101, 102].External fibrillation, coming from the shear forces and the abrasive action betweenfibres, refers to some of the fibrils peeling away from the fibre walls, making thefibre appear hairy. These external fibrils increase the surface area and thereforeprovide more locations for fibres to bond [102]. Fibre straightening allows bond-ing between an increased number of fibres, whereas fines generation is a byproductfrom fibres being cut. To some extent, low consistency refining is also performedon mechanically pulped fibres in industry to further develop the fibre structure [68],however due to their greater stiffness, refining predominantly results in cutting ofthe fibres [85]. The degree of refining can be quantified with the specific refiningenergy (SRE) value, which simply is the net power transferred to the suspensionper ton. SRE values therefore have units of kWh/T .3.1.4 Chemical AdditivesA second example of a factor found in production that alters the suspension’s dewa-tering behaviour is the use of chemical additives [86]. Various additives are appliedin the production of pulp products for many reasons, which can include improvingthe dewatering or drainability of the suspension. These alterations can come in theform of improving retention of fines, or changing the flocculation state of the fibres[85]. Typically, these additives utilize the negatively charged fibrils of the fibresin order to alter the suspension. Several chemical additives are observed for theirimpact on dewatering behaviours in this study, however the details underlying thesurface chemistry are beyond the scope of this thesis.3.2 Model EquationsAs described in Chapter 2, a one-dimensional form of the general equations can beused for the uni-axial experiment shown in Figure 2.1a. Additionally, in this par-ticular experiment, we can neglect the body forces as they are small in comparison22to the pressure and solid stresses. Therefore we start with the equations∂φ∂tˆ+ ∂∂zˆ(φvˆs) = 0, (3.1)−∂φ∂tˆ+ ∂∂zˆ((1 − φ) vˆl) = 0, (3.2)(1 − φ)(vˆs − vˆl) = k(φ)µ∂pˆ∂zˆ, (3.3)∂∂zˆ[−pˆ − Pˆ] = 0, (3.4)where vˆs and vˆl are the solid and liquid velocities in the zˆ-direction. Reductionof the equations is done by combining the two continuities expressions, and inte-grating using the closed base boundary condition of the suspension chamber (i.e.vˆs = vˆf = 0 at zˆ = 0). We arrive at the following expression for the solid phasevelocityvˆs = (1 − φ)(vˆs − vˆl). (3.5)This expression allows us to eliminate the fluid velocity in Darcy’s law shown inEquation 3.3, and with the conservation of stress Equation 3.4, we can express thesolid velocity in terms of the gradient of the solid effective stress asvˆs = −k(φ)µ∂Pˆ∂zˆ. (3.6)This manipulated Darcy’s law, the solid phase continuity of Equation 3.1, and theconstitutive model shown in Equation 2.11 can be used to solve for the volumefraction and solid phase velocity during dewatering, if the permeability, compres-sive yield stress and bulk viscosity are specified. The model prediction of the loadon the permeable piston is simplyσˆ(tˆ) = (p + Pˆ)∣zˆ=hˆ ≡ Py(φ) −Λ(φ) ∂vˆs∂zˆ ∣zˆ=hˆ(tˆ) , (3.7)which is arrived at by integrating the Equation 3.4 and assuming no pore pressuredrop across the permeable piston.To non-dimensionalize the equations, the following scalings of the primary23parameters are usedzˆ = h0z, hˆ = h0h, vˆs = V v, tˆ = h0Vt, σˆ = p∗σ and Pˆ = p∗P, (3.8)where h0 represents the initial height of the suspension, V represents the permeablepiston speed, and p∗ represents a pressure scaling (to be defined). The materialparameters are scaled ask(φ) = k∗K(φ) and Py(φ) = p∗Πy(φ), (3.9)where k∗ and p∗ are defined as the fitted functions of k(φ) and Py(φ) at φ = 0.10,respectively. The scaling of the bulk viscosity simply is the fitted value of η∗ inEquation 2.15. We arrive at the following dimensionless equations describing thecompaction as∂φ∂t= − ∂∂z(vφ) where v = −γ ∂Πy(φ)∂φK(φ)∂φ∂z+ K(φ) ∂∂z[φ2∂v∂z] ,(3.10)and two dimensionless parameters defined asγ = p∗k∗µh0Vand  = k∗η∗µh20. (3.11)The γ term represents the ratio between the compressive strength of the networkto the viscous drag force of the escaping fluid, and can be used to compare relativecompression rate severity between suspensions. The  term carries the bulk viscos-ity scaling, and represents the ratio of the timescale for relaxation of the networkto that of the pore pressure diffusion, as defined in [32]. Finally, the initial and24boundary conditions of the problem areφ(z,0) = φ0, (3.12)v(0, t) = 0, and v(h, t) = −1, (3.13)σ(t) = Πy(φ) − γφ2∂v∂z∣z=h(t) . (3.14)To solve the model Equations 3.10 throughout a dewatering event, we utilizea finite difference scheme on a fixed domain coordinate system with respect toh(t). From a known volume fraction profile at time step n, we determine the solidphase velocity profile by discretizing the velocity expression in Equation 3.10 andsolving with respect to the boundary conditions provided in Equation 3.13. Tosolve the volume fraction profile at timestep n + 1, an iterative Crank-Nicholsonmethod for Equations 3.10 is implemented, where an explicit solid phase velocityis used in the first iteration. From this first approximation of the solid volumefraction profile at n + 1, we determine a corresponding solid phase velocity profileat n + 1, and iterate until the determined volume fraction profile converges. Asimilar iterative approach is adopted to accommodate the non-linearity in φ fromour chosen form of Λ(φ). In implementing the scheme, a variable time step wasutilized and staggered volume fraction and solid phase velocity grids were chosen.3.2.1 Fitting Bulk Viscosity: η∗To fit the scaling of the bulk viscosity, an optimization, using the solver for thescheme described above, was developed, with η∗ left as a single free parameter.The optimization determines the value of η∗ that minimizes the discrepancy be-tween the model predictions and the various experimental dewatering results. Theparticular trends that are utilized in the optimization are a suspension’s compressiveload versus average volume fraction trends (σˆ versus φ¯ ≡ φ0h0/hˆ(tˆ)). An illustra-tion of this experimental trend for a given compression rate is shown in Figure 3.2aas the blue solid line. We note that increasing values of φ¯ are representative of timethroughout the dewatering experiment (due to the decreasing hˆ(tˆ)). Additionallyshown in Figure 3.2a as a blue dashed line is a toy model result. The error between25𝑀𝑜𝑑𝑒𝑙𝐸𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡(b)(a)Figure 3.2: Images clarifying the fitting of bulk viscosity, η∗. In (a), toymodel and experimental compressive load versus average volume frac-tion trends are shown. In these trends, φ ≡ φ0h0/hˆ(tˆ) and thereforecan be considered as time. These trends can be found throughout thethesis for representing dewatering results, and are used in fitting η∗ asdescribed. In (b), a pictorial representation of the minimization is pro-vided. An area is formed under the four discrete values of E , which isused in the minimization.these two trends is defined as,E(η∗) = ∫ ∣σModel(η∗) − σExp.∣dφ¯∫ σExp.dφ¯ , (3.15)or equivalently, the grey shaded area divided by the yellow shaded area. Eachcompression rate value, corresponding to a value of γ, that was investigated for agiven pulp suspension will correspond to a value of E . The optimizer, therefore,aims to minimize the discrepancy across all the various rates simultaneously. Todo this, minimization of the following expression is performedF(η∗) = ∫ E(η∗, γ)dγγmax − γmin (3.16)which represents some averaged error across the various rates collected. Pictorially,this equates to minimizing the height of the shaded area formed under the variousvalues of E for each value of γ, as shown in Figure 3.2b. An optimal value of η∗ isdetermined for each suspension in this way, as well as a range, η∗,range, bounding26where F(η∗) ≤ 1.05F(η∗,opt). These values are reported in Table B.2.3.3 Results and Discussion: Material Parameters3.3.1 PermeabilityWe begin our discussion of permeability with a representative selection of the se-ries of pulp suspensions, as well as the nylon fibre suspension shown in Figure 3.3a.For each suspension, the symbols represent sample measurements, and the line isthe fitted functional form of Equation 2.9. Varying fit quality is seen, however weconsider the representation in all cases acceptable. The representative results showsignificant variation across the pulp suspensions, through considerable differencesin magnitude, and subtle variations in trend steepness. The general results withrespect to the different pulp suspensions are consistent with focused pulp suspen-sion permeability studies (examples [14, 58, 60, 61, 79, 96]). Examples include thechemically pulped fibre suspensions which are typically higher when compared tomechanically pulped, and where LC refining is found to drastically reduce the net-work permeability. Details of the respective fits for the complete list of series arefound in Table B.2. A final observation with this representative selection is withrespect to the permeability of all of the pulps being significantly lower, and havinga steeper trend than the nylon fibres. This is an important contrast to a more ideal-ized fibre suspension, and is explored further for investigating the pulp suspensionresults as a class of materials.To begin understanding the high variation observed across the pulp suspen-sions, and the lower measurements compared to the nylon fibres, we turn to arelated study in the literature. The study of Jackson and James [37] is a well citedpublication which collected permeability data for a large selection of fibre sus-pensions found in the literature. Example materials included nylon fibres, wools,wire crimps, and filter pads. The study focused on approximately circular shaped,uniform sized fibres that were sufficiently long such that the aspect ratio of the fi-bre was not a parameter. Permeability for a fibre suspension was suggested to be afunction of the radius of the particles (a proxy for the pore size within the network),the arrangement of the fibres, and the solid volume fraction. Scaling the variety of27(a)0 0.1 0.2 0.3 0.410-1510-10(b)10-2 10-1 10010-5100Figure 3.3: In (a), select results are shown for Series 1 (△), 4 (◻), 8 (⧫), 10(●), 11 (▲), 26 (◻), 27 (◇), 29 (◻) and nylon fibres (☀). In (b), allseries’ results (excluding Series 2-6) are scaled by their respective fibrehalf mean width squared, and are compared to results from Jackson andJames [37] (shown as the cloud of varying gray symbols).28data by their fibres’ respective radius squared, provided convincing collapse ofthe various magnitudes of permeabilities, and draws attention to the similarity intrends. These scaled permeabilities found in Jackson and James are shown as thevarious gray-scaled symbols in Figure 3.3b. The remaining scatter of the Jacksonand James data was attributed to predominantly three factors: the orientation of thefibres in the network, the cross-sectional shape of the particles (if not truly circu-lar), and the homogeneity of the porous network. The permeability of our nylonfibres, scaled with respect to their fibres’ radius squared is shown as well in Fig-ure 3.3b, and agrees well with the Jackson and James points. When consideringthe criteria defining the fibres used in the study, it might not be surprising whypulp suspensions were excluded. As mentioned in Chapter 1, pulp suspensions arecomprised of a variety of particle sizes, and their cross-sectional area is difficultto define. All pulp fibres have varying hollow structures, which will be prone toeither partial or full collapse as the suspension is consolidated [67]. Therefore, inthe two extremes, the pulp fibres may resemble a tube-like particle, or flattenedribbon structure.Irrespective of this, we scale our data in a similar way using the fibres’ halfmean width squared, found in Table B.1. If the fibres are truely cylindrical, this isequivalent to the radius squared. The scaled data is shown in Figure 3.3b. Whatis evident in this figure is the scaling that proved successful for a variety of otherfibrous suspensions has not been successful in collapsing the spread between serieswith respect to themselves, and with respect to other materials. Two rough group-ings of the series do seem to be evident. When considering the factors Jackson andJames attributed to the spread in their scaled data, we find it difficult to envisionsufficient variation in orientation between the various series to accommodate thespread shown. Homogeneity of the pulp suspensions does play an important fac-tor when measuring permeability of thin pulp fibre suspensions [61], however, oursuspensions are far beyond the thickness where this factor is relevant, suggestingthat any preferential pore structure in the z-direction averages between trials. Fi-nally of course the non-circular cross-section should provide a discrepancy whencomparing to circular fibres (the nylon fibres, for example), though again it seemsunlikely there is sufficient variation between the series to accommodate four ordersof magnitude in the scaled permeability.29Having seen little success in collapsing the data with respect to mean fibrewidth, we speculate on the difference between our pulp suspensions and the moreideal fibre suspensions of Jackson and James. We would expect a characteristicpore size to exist for the various pulp suspensions, however it appears, at leastacross the whole selection of pulps, that this may not be strongly proportional to themean fibre width. Further, when consideration to a pulp fibre’s more complicatedstructure (hollow, with anionic fibre walls that attract water), it does seems possiblethat only a fraction of the non-solid volume fraction contributes to the flow pathsthrough the solid network, as suggested and studied by [59, 80]. Perhaps then,a redefined volume fraction is in order, which may explain the variation in trendsteepness seen across the pulps. Motivated by this, we performed a two parameterfit to the experimentally collected pulp results to force a collapse of the data ontothe well-behaved, dimensionless nylon fibre functional form,k(φ)R2= 0.147φln( 1φ) e−5.4φ. (3.17)We seek the factors F1 and F2 which are applied to the solid volume fraction andpermeability asΦ = F1φ ≡ (1 +WRV ρsρl)φ and K = kF 22(3.18)and are interpreted as immobile water in the suspension, and a characteristic poresize. Notice we can define F1 from a water retention value (WRV ), howeverthese were not collected for the various pulps. The fitted functional form for thismodified permeability, K(Φ), and solid volume fraction, Φ, therefore will beK(Φ) = 0.147Φln( 1Φ) e−5.4Φ. (3.19)The collapsed trends are shown in the main plot of Figure 3.4. The determinedfactors F1 and F2 are tabulated in Table B.2, and plotted in the insert of Figure 3.4,defined as a WRV , and with respect to the suspensions’ respective half mean fibrewidth. What is seen in the insert is the two families of suspensions again, with thechemically pulped softwood and hardwood fibres having a rather consistent ratio3010-1 10010-51000.5 1 1.5 200.10.2Figure 3.4: Collapsed permeability results. In the insert, the required factorsF1 (interpreted as a WRV ) and F2 divided by the suspensions’ respec-tive half mean width are plotted.(0.15-0.22) of characteristic pore scale F2 to half fibre width. The ratio is verylow for the mechanically pulped suspensions, along with the refined chemicallypulped fibres (0.02 to 0.05). Not having the WRV s makes it difficult to judge itsinterpretation as the correction factor to φ, however roughly speaking theWRV ofthe chemically pulped fibres are in the correct range (on the order of 1 g/g), and theaction of low consistency refining monotonically increases the value to reasonablelevels as well. Little was found on WRV of mechanical pulps, however the valuessuggested here are suspected to be too high. This could suggest this factor cannotbe solely interpreted as immobile water, trapped in the fibre walls.3.3.2 Compressive Yield StressWe begin our discussion of compressive yield stress with a representative selectionof the series of pulp suspensions, as well as the nylon fibre suspension shown inFigure 3.5a. For each suspension, the symbols represent select points along the av-eraged continuous trend, the error bars represent two standard deviations, and theline represents the fitted functional form of Equation 2.13. Good representation forall suspensions is found with this functional form. The representative results show31minimal variation between the pulp suspensions, particularly when compared tothe variation observed with permeability. The data is replotted with logarithmicaxes in the insert, in which subtle curvature in the various trends can be seen. Thissuggests more structure in the data than a simple power law, and supports the func-tional form chosen. The only notable observation made between the various pulpsis the mechanical versus chemical pulps. The mechanical pulp suspensions havea higher compressive yield stress at low concentrations, but also a smaller volumefraction dependency at higher concentrations. The experimental equipment wasvalidated against data found in the literature (example [78]), however, unlike per-meability, focused studies investigating the variation in this parameter for multiplepulp suspensions have not been found. Perhaps this is due to the minimal variationsobserved. Details of the respective fits for the complete list of series are found inTable B.2. Turning now to the nylon fibres, we see a close agreement in magnitudewith respect to the various pulps. However, in the insert of Figure 3.5a, it is ap-parent the nylon fibres have a stronger volume fraction dependency. We again willcontrast our results to more idealized suspensions like nylon fibres, to investigatethe pulp suspensions as a class of materials.To begin this investigation, we will consider the study provided by Toll [94] onthe compaction of fibre networks. We would expect the mechanisms in this highlyidealized, uniform fibre study to only apply in low concentrations, and therefore donot expect insight into the entire range of volume fraction investigated. Irrespectiveof this, it provides a model approach to a compressive yield stress that may proveinsightful. In the study, the compaction strength of a fibre network is modelled.For slender fibres of length L and typical cross-section width W , the expectednumber density of contacts nc and the typical distance between each contact alongthe fibres λ scale asnc ∼ φ2W 3and λ ∼ Wφ. (3.20)If each contact point sustains a force f , the bulk stress supported by the micro-structure isPy(φ) ∼ nchf (3.21)where h represents the typical length scale for microscopic deformation. Toll [94],32(a)0 0.1 0.2 0.3 0.4 0.5051015 1050.1 0.2 0.3 0.4104106(b)0 5 10 15 20 25 301051061070 5 10 15 20 25 301230 5 10 15 20 25 3005Figure 3.5: In (a), select results are shown for Series 1 (△), 5 (◇), 8 (⧫),10 (●), 11 (▲), 22 (∎), 24 (⧫), 29 (◻) and nylon fibers (☀). In (b), theresulting values of a, b, and c from the Py(φ) form in Equation 2.13 areshown for the various suspensions and the nylon fibres, with the meanof the pulp fibres’ values shown as the dashed lines.33following Van Wyck [95], assumes that the contact force is due to elastic bendingof roughly cylindrical, randomly oriented, and relatively dilute fibres, leading toh ∼ λ andf ∼ ωEW 4λ2, (3.22)where E is the fibre wall’s Young’s modulus, and ω is a parameter that incorporatesthe true geometry of the fibres and their contact points for a given suspension.Hence,Py(φ) ∼ ωEφ3, (3.23)which has been shown to agree with experimental results for suspensions of wool[95].Despite not fitting our experimental data to a simple power law, the chosenform conveniently approaches one in the low φ limit, i.e.Py(φ) = aφb(1 − φ)c ≈ aφb whenφ≪ 1. (3.24)The fitted values of b for all the pulps and the nylon fibres are shown in Figure 3.5b.We see the φ3 dependence agrees closely with the experimental nylon fibre results,however the various pulp series hover around a value of b = 2. A φ2 dependence ofPy(φ) may be suggestive of a local contact force f : for example, if deformationtakes place over the scale of the fibre cross-section (either collapse or buckling ofthe fibre wall). We can speculate that this variation in predominant deformationmechanism is due to the hollow structure of the pulp fibres, which may be moreprone to local collapse than sustained bending moments. In this case, h ∼ W andf ∼ ωEW 2, givingPy(φ) ∼ ωEφ2. (3.25)Turning next to the values of a, with taking the Young’s modulus for the ny-lon and cellulose as E = O(109) and (1010)Pa respectively, we find values ofω = O(10−4 − 10−3) across the various suspensions. Van Wyk [95] speculates thisgeometrical corrector would be O(1), though it is unclear the motivation behindthis suggestion, with his experimental wool results finding a value of O(10−2).Van Wyk also suggests the value of ω would vary little from sample to sample,34however with the varying structural properties of pulp fibres, its seems reasonableto have varying values of ω (and therefore a) across the suspensions. Qualitatively,the variations in a follow broad expected results. Examples include the softwoodchemically pulped results are somewhat equivalent, mechanically refining the soft-wood chemically pulped fibres reduces the fibre stiffness (and therefore a), thechemical additives’ impact is minimal, the mechanically pulped fibres are stiffer,and the nylon fibres, being solid particles, are stiffer yet.An inconvenience, however, with these model efforts of the compressive yieldstress is that they are elastic, and therefore deformations should be recoverable.Experimentally however, the deformations of the suspensions showed very littlerecovery during unloading, consistent with the notion that deformations are pri-marily plastic. To account for this, we follow Servais, Manson & Toll [84] andargue that frictional rearrangements under elastic formal forces dominate the com-pressive yield stress. That is,Py(φ) ∼ νωEφ3 or ∼ νωEφ2, (3.26)where, ν is a friction coefficient at the contact points.In an effort to understand Py(φ) at higher volume fractions, we turn to the finalfitted parameters c. This power is responsible for the steepening of the trends athigh values of φ and therefore is capturing the network’s response upon approachto maximum packing. We see for most of the fibres, including the nylon, a ratherconsistent value of c around the mean value of 3. The two notable exceptions arethe refined chemically pulped softwood fibres, with values greater than the mean,and the mechanically pulped fibres, which are significantly less. Toll [94] consid-ered higher volume fraction behaviour as well and suggested two high φ behaviourswhich may explain these outlying pulp groups. If the fibres are highly ductile, asare the refined chemically pulped softwoods, they may nearly indefinitely deform,eventually resulting in a stronger trend of contact points and thus the strength ofthe network grows exponentially. On the other hand, if the fibres are brittle, suchas the mechanically pulped fibres, they may reach a point of deformation wherethey begin breaking, relieving internal stresses, resulting in a weaker trend.353.4 Results and Discussion: DewateringDewatering experiments at varying rates were performed for the various series.The compression rates, corresponding to ranges of γ values, are listed in TableB.2 of Appendix B. Three representative suspensions’ results will be highlighted.We start with the dewatering results for Series 4, shown in Figure 3.6a. In thisplot, four averaged dewatering trends at varying compression rates are shown asthe red lines. The black arrow illustrates the shifting of the trends with increasingcompression rates. The error bars at the location of the symbols represent twostandard deviations of the averaged trend. The suspensions’ fitted compressiveyield stress is re-shown as the gray dotted line. Illustrated in this figure is theincreasing difficulty of dewatering the pulp suspension at higher rates (lower valuesof γ). The severity of this will be dictated by the material parameters of the pulpsuspension. Identically depicted are the results for Series 13 and 22, in Figure3.6b and 3.6c respectively, which qualitatively demonstrate the same behaviour.Coincidently, all three series used the same compression speeds of 0.25, 1.5, 5,and 10mm/s, which adds to the comparison of the experimental results betweenthe series.We first compare Series 4 to 13 (Figure 3.6a to 3.6b). Comparing equivalentrates, we see increased difficulty (higher compressive loads) in dewatering Series13. Inspecting the material parameters found in Table B.2, we find a lower per-meability for Series 13, represented by a smaller k∗, and a similar compressiveyield stress, represented by equivalent p∗ values. This difference in material pa-rameters certainly contributes to the difference in dewatering difficulty, howeveris neglecting any contribution from the suspensions’ bulk viscosity. To investigatefurther, we turn to the model load expression, shown in Equation 3.14. From thedefinitions of the dimensionless groups, the first difference between the two seriesat a given compression rate V will be a smaller value of γ for Series 13 due to k∗,which would suggest a higher model load. Conveniently, due to the differences inmaterial parameters, virtually equivalent values of γ are found between Series 4 at1.5mm/s, and Series 13 at 0.25mm/s (γ = 0.36 and 0.37, respectively). Thesetrends are identified in Figure 3.6a as the second red trend in from the right, and inFigure 3.6b as the rightmost green trend, respectively. Interestingly, these two ex-36(a)0 0.1 0.2 0.3 0.4 0.5051015 105(b)0 0.1 0.2 0.3 0.4 0.5051015 105(c)0 0.1 0.2 0.3 0.4 0.5051015 105Figure 3.6: In (a), experimental compressive load vs. average solid fractionfor Series 4 is shown. Four dewatering trends are shown as the colouredlines, corresponding to 0.25, 1.5, 5, and 10mm/s. Increasing dewa-tering rates (decreasing values of γ) are identified by the black arrow.Compressive yield stress is re-shown as the gray dotted line. Equivalentdepiction for Series 13 and Series 22, at the same dewatering rates, isshown in (b) and (c), respectively.37perimental load curves look virtually equivalent, with respect to the σˆ at a given φ¯.Returning to Equation 3.14, this would suggest the second term, constituting thebulk viscosity contribution through , is somehow equivalent between these twopulp suspensions at the separate compression rates. Not having the bulk viscosityvalues yet, we will return to this discussion later in this section.Next, when comparing Series 22 to Series 4 (Figure 3.6c to 3.6a), looking atits material parameters illustrates that something wrong is occurring with these ex-perimental results. Despite Series 22 having a considerably lower permeability, therelative dewatering difficulty appears minimal (equivalent compressive loads). Se-ries 22 is presented in this section to highlight an experimental challenge found fora subset of the mechanically pulped suspensions (Series 22 - 25). With the higherfines content, retention of the solid phase underneath the permeable piston was achallenge at elevated compression rates. This was evident visually during the ex-periment as cloudy water above the piston after dewatering. Ineffective retentionof the solid phase manifests in an artificially low compression load measurementduring dewatering. Retention is a common problem in industry as well when deal-ing with mechanically pulped suspensions. For the remaining mechanical pulps(Series 26-28), experimental retention was effectively managed.For each series, the model equations were used to fit an optimal value of thescaling of the bulk viscosity, η∗. The model was also solved with a zero bulkviscosity to demonstrate its importance of inclusion. The range and optimal valuesof bulk viscosity, as well as the minimized error values are shown in Table B.2.Due to the retention challenges, Series 22-25 should be considered with caution.Demonstration of the model suitability begins with Series 4, a particularly wellrepresented suspension. The four averaged dewatering trends from Figure 3.6a areindividually presented in Figures 3.7a-d as the solid red trends in order of increas-ing dewatering rates (or decreasing values of γ). The compressive yield stress isshown as the gray dotted trend in the various figures. For each compression rate,the model trends, with a fitted bulk viscosity and zero bulk viscosity, are shown asthe short and long dashed red lines, respectively. For all but the highest dewater-ing rate, we see equivalent representation in both the fitted and zero bulk viscositymodel trends (Figure 3.7a-3.7c). Representation by the fitted bulk viscosity trendis greatly improved over the zero bulk viscosity for the highest compression rate38(a)0 0.1 0.2 0.3 0.4104106(b)0 0.1 0.2 0.3 0.4104106(c)0 0.1 0.2 0.3 0.4104106(d)0 0.1 0.2 0.3 0.4104106Figure 3.7: Series 4 model representation presented for the dewatering rates0.25, 1.5, 5, and 10mm/s (values of γ are 2.2, 0.36, 0.11, and 0.05) aredepicted in (a) - (d) respectively. In each figure, the averaged experi-mental trend is shown as the coloured solid line, the model result with afitted bulk viscosity is shown as the coloured short dashed line, and thezero bulk viscosity model trend is shown as the coloured long dashedline. The compressive yield stress is re-shown as the grey dashed line.(Figure 3.7d). This is repeating the observation in Hewitt et al. [32] where exper-iments with small values of γ (high dewatering rates) required the bulk viscosity’sinclusion.The fitted and zero bulk viscosity model trends’ quantitative representation ofthe experiments is presented in Figure 3.8a and 3.8b respectively, for all of theseries investigated. In these two figures, the majority of series results are anony-39mously plotted as gray symbols. A significant number of points in Figure 3.8b arebeyond the error values shown in the axis. Interestingly, this graph appears to showa range of γ at which point the majority of the suspensions’ representation for zerobulk viscosity diverges. This suggests that if values of γ are kept above O(1), thenthe bulk viscosity is not necessary to model the dewatering behaviour. Series 4 re-sults are indicated by the red open faced squares, appearing as a flat error versus γtrend for the fitted bulk viscosity. For zero bulk viscosity, a diverging trend for thelowest γ experiment is seen. In comparison to the other materials, we recognizehow strongly the viscoplastic-like model fits to this particular suspension.Typical model representation is seen with Series 13, whose model results areshown in Figure 3.9, and are depicted identically to those of Series 4 in Figure 3.7.In these figures, the zero bulk viscosity trends are seen to struggle at lower com-pression rates when compared to Series 4. The fitted bulk viscosity trends are stillrepresentative of the experimental results, however the discrepancy is more notice-able for some of the rates. Despite this, when compared to the zero bulk viscositytrends, it is clear the fitted bulk viscosity is effective in reducing the discrepancy ofrepresentation of the dewatering experiments.We now turn to the fitted values of bulk viscosity for the various series, plottingη∗ versus their k∗ and p∗ values, in Figure 3.10a and 3.10b, respectively. We notethat due to the experimental retention challenges for Series 22-25, some uncertaintyarrives with the fitted η∗ values. These values are highlighted with green in thefigure for this reason, however their relative locations are encouraging, despitetheir poor representation (see Figure 3.8). Observed across all of the series is aconvincing power inverse proportionality of η∗ with respect to k∗, and appearsrather immune to the varying values of p∗. The simple power law, η∗ ∼ k−1.04∗ isfit to the data, with a 95% confidence interval of the power ranging from -0.92 to-1.16. Due to its closeness to unity, we opt for a power of −1 and arrive with thefollowing fitted functionη∗[MPa ⋅ s] = 2.2(10−11)k∗[m2] , (3.27)shown as the dashed line in Figure 3.10a.If the bulk viscosity of a suspension that has been found is a true material pa-40(a)10-2 100 102100105Error(b)10-2 100 102100105ErrorFigure 3.8: Quantitative model representation for the various series. In bothplots, error is provided by Equation 3.15. In (a), the error vs. γ areshown for all of the series with their fitted bulk viscosity. In (b), theerror vs. γ are shown for all of the series with zero bulk viscosity.Highlighted results of Series 4 (◻), 13 (●), and 22 (∎) are shown in bothfigures, however in (b), Series 22 error values exceed 106.41(a)0 0.1 0.2 0.3 0.4104106(b)0 0.1 0.2 0.3 0.4104106(c)0 0.1 0.2 0.3 0.4104106(d)0 0.1 0.2 0.3 0.4104106Figure 3.9: Series 13 model representation presented for the compressionrates 0.25, 1.5, 5, and 10mm/s (values of γ are 0.37, 0.061, 0.018, and0.009) are depicted in (a) - (d) respectively. In each figure, the averagedexperimental trend is shown as the coloured solid line, the model resultwith a fitted bulk viscosity is shown as the coloured short dashed line,and the zero bulk viscosity model trend is shown as the coloured longdashed line. The compressive yield stress is re-shown as the grey dashedline.42(a)10-15 10-14 10-13 10-12 10-11100102104(b)102 103 104 105 106100102104Figure 3.10: The various series’ fitted bulk viscosity versus k∗ in (a) andversus p∗ shown in (b). The error bars represent the range of η∗, asreported in Table B.2 in Appendix B.43rameter, one implication of this relation is that , defined in Equation 3.11, becomesindependent of the solid network’s material parameters, as = 2.2(10−11)µh20(106), (3.28)being solely defined by the fluid viscosity and the initial height. In the variousexperiments of this study, µ was maintained as a constant and h0 varied onlymarginally, and therefore  ∼ O(100 − 101) range. First, this consistent valueof  is insightful for the Py(φ) calibration (see Section A.2.2). Second, this  valuesets the threshold observed in Figure 3.8b, as the transition point of representationfor the zero bulk viscosity model. As long as γ remains large in comparison to ,i.e. γ ≫ O(1), the bulk viscosity inclusion is negligible. Third, this seems to agreewith the experimental load discussion between Series 4 and 13 at the start of thissection.To end this section, an effort to rationalize a bulk viscosity is investigated.Djalili-Moghaddam, and Toll [20] observe that viscous flow around the solid fibrescan provide a bulk viscosity. Following a similar approach to the Toll discussionsin Section 3.3.2, we aim to develop a model for the bulk viscosity term. If theentire fibre is in motion, then we define the deformation length, h, as h = L. Thecompression rate gradient ∂vˆ/∂zˆ sets a velocity difference between fibres of V =L∂vˆ/∂zˆ at the micro-scale. Considering the sliding of fibres past one another,separated by a distance of the order ofW , the viscous shear stress is µV /W , whereµ is the solvent viscosity. Thus, f ∼ µVW , and so the additional rate-dependentsolid stress is expected to bePˆ − Py(φ) ∼ nchf ∼ µ( LW)2 φ2∂vˆ∂zˆ. (3.29)This would suggest a bulk viscosity scaling ofO(10−1−101)Pa ⋅s with the aspectratio range of 20 − 100 measured experimentally. This is much smaller than thecalibrated values required to match the model with compression experiments. Therelatively strong dependence on the aspect ratio is also a little unsettling, eventhough the φ2 dependence is consistent with model results, and there is no clearconnection to the permeability scale k∗.44With the encouraging φ dependency, we reconsider the dimensional ratio inthis problem. With the correlation with respect to k∗, it seems the dissipative lengthscale would be appropriately set by√k∗, providingPˆ − Py(φ) ∼ µh2k∗φ2∂vˆ∂zˆ. (3.30)This arrives at an expression of the bulk viscosity scaling as η∗ = µh2/k∗, whichcan be related to the fit in Equation 3.27. However, issues arise when solving forthe appropriate deformation length h, which is found to be large, beingO(10−1)m.In summary, a clear viscous argument for the required bulk viscosity is notreached, however, we have demonstrated the bulk viscosity far exceeds simplythe motion of the fibres through the viscous fluid, as suggested by various studies[11, 20].3.5 Summary and ConclusionsA large variety of pulp suspensions were investigated in this chapter to increasethe confidence in the viscoplastic-like constitutive model, and provide insight intothe variation of the governing material parameters. Beginning with permeability,this material parameter was found to vary significantly between the various pulps,and does not scale similarity to more ideal fibre suspensions found in the literature.This could be due to the complex fibre structure which creates a smaller pore scalefor the permeating fluid and immobilizes a fraction of the water (trapped in thefibre walls), thereby only allowing a fraction of the non-solid volume to contributeto the flow paths. Compressive yield stress is found to vary minimally across thepulps and demonstrates a smaller φ-dependency at low concentrations compared toa suspension of nylon fibres. We investigated, through an exploration of a relatedstudy, that this lower dependency on volume fraction may also be attributed to thecomplex fibre structure, whereby the fibres preferentially collapse at the networkcontact points rather than sustain bending moments. At high volume fractions,compressive yield stresses are equivalent across the majority of the pulps, how-ever, two outlying subsets appear to demonstrate the two expected high packingbehaviours: continuous rearrangement of nearly infinitely flexible fibres, or the45breaking of stiff fibres.Varying dewatering behaviour were observed across the pulp suspensions. Typ-ically, their difficulty in dewatering is qualitatively proportional to their collectedmaterial parameters. Varying quality of representation by the viscoplastic-like con-stitutive model was observed across the pulps, however in all cases the inclusion ofbulk viscosity improved its representation. The fitted values of η∗ were determinedto be inversely proportional to the fibre suspensions characteristic permeabilities,k∗, and not strongly correlated to their characteristic compressive yield stresses,p∗. Finally, the magnitude of bulk viscosity was demonstrated to far exceed simplythe motion of the fibres through the viscous fluid. The particular source of the highbulk viscosity, however, remains undetermined.46Chapter 4Model Robustness:Representation of LowConcentrationThis chapter details a dewatering study near a pulp suspension’s gel point. Includedin this study is a reduction of the general equations presented in Chapter 2, cali-bration of the necessary material parameters in the low concentration regime, anddewatering experiments to evaluate the continued suitability of the viscoplastic-like constitutive model. In addition, the modelling framework is applied to theindustrially-relevant Canadian Standard Freeness (CSF) drainage test. The objec-tives of this study were to investigate the continued necessity of the bulk viscosity,to gain insight into the material parameters at concentrations an order of magni-tude lower than originality investigated, and to provide an industrial context for theapplication of this modelling approach.A suspension of fibres can establish a connected network that supports stressand resists deformation even at relatively low solid volume fractions. As the con-centration continues to decrease, eventually the fibres will be sufficiently dispersedsuch that they do not contact one another. At these concentrations, the fibres cannotprovide a continuous network, and therefore cannot support a solid effective stress,(i.e. Pˆ = 0). The lowest solid volume fraction, at which the suspension networks,is referred to as the suspension’s gel-point, φg. For pulp fibre suspensions, this47is typically a fraction of a percent [34]. If φ < φg, the fibres are assumed to bedispersed, non-Brownian particles. Once φ > φg, the characteristics of a structurednetwork that resists collapse and provides resistance to the flow of inter-fibre waterare recovered. Based on the trends and intuition of the material parameters, we canenvision this network of pulp fibres having a very small compressive yield stressand relatively high permeability, both due to the low degree of contact between fi-bres (large pore space, and weak contact points). Due to the low compressive yieldstresses of the network, body forces become significant in this regime and requireinclusion. It is unclear if the viscoplastic-like constitutive model will still be nec-essary at these concentrations. A previous study of Young [104] would suggest aplastic-only constitutive model is appropriate at these concentrations, however thisalso only investigated sedimentation and not increasing severities of dewatering, aswe aim to do.An industrially relevant dewatering experiment that is performed around φg isthe Canadian Standard Freeness Test (CSF). This is a widely used, gravity-drivendrainage test for pulp (TAPPI Standard T221). The test uses a carefully calibratedexperimental device, shown in Figure 4.1. A specified amount of a pulp sample(1000 g) is released from the upper portion of the device. As the water drainsfrom the sample, a simple funnel geometry is employed to divert a fraction ofthe water into a collection chamber. The volume of the diverted water, or the“Freeness Score”, is set by the rate at which water leaves the pulp sample, whichis controlled by the degree to which the fibre network compacts and impedes flowas it drains from the upper potion of the device. This test is further explained inSection 4.6.1. Although the test is simple, the precise relationship between theFreeness Score and the different material properties of the fibrous medium is nottransparent. Moreover, the only existing models of freeness are semi-empiricaland based on the average filter resistance of the compacting pulp [24, 48, 91].We therefore apply our two-phase model to the Freeness test, to explore whetherits predictions match the observed values of Freeness, to dissect the underlyingdynamics and to identify the controlling material properties. A more general aimis to determine whether the Freeness test provides a useful device to interrogatethe behaviour of a deformable porous media under relatively rapid and substantialcompaction.48Figure 4.1: Schematic of a Canadian Standard Freeness (CSF) tester. Thediverted volume out the side channel collected constitutes a FreenessScore.We start this chapter with the formulation of the model equations for thislow concentration regime and the experimental geometries used. With the modelsorted, we provide material and overall experimental details before a discussionof the calibration of permeability and compressive yield stress. We then discussthe model’s representation of the pulp suspension dynamics, focusing first on lowspeed experiments (example, sedimentation) before moving onto high rate, signif-icant compaction results. Finally, we turn to the discussion of the CSF test, beforeending with a general discussion.4.1 Model Geometry and EquationsSeveral geometries, aside from the CSF tester, were used in this low concentrationstudy for either calibration of the material parameters, or for experiments to evalu-ate the dewatering model. The first is a fixed volume chamber through which water49can be recirculated to assist the gravitational sedimentation of fibres, shown in Fig-ure 4.2a. In this experiment, the suspension is retained between the permeable baseat zˆ = 0, and a permeable top at zˆ = hˆ. The second experimental geometry, shownin Figure 4.2b, is a drainage tank that allows water out of the permeable surface atzˆ = 0, which is driven by the head difference between the pipe exit, hˆexit, and thesuspension height hˆ, which falls as the water leaves.In all the experiments, the initial concentrations and heights are defined as φ0and h0, and the pulp fibres are retained by the permeable surfaces, and therefore∫ hˆ0φ(zˆ, tˆ)dzˆ = φ0h0. (4.1)Similar to the uni-axial dewatering experiment shown in Figure 2.1a, the geome-tries of this chapter lend well to a one-dimensional framework. Different, however,is the necessity of inclusion of body forces to drive the dynamics of the problems.Due to this, and the potential that the initial volume fraction φ0 < φg, various zoneswithin the consolidating pulp suspension can occur, that are illustrated in 4.2c. Weput off the discussion of these zones for now, and introduce the simplified versionsof the equations from Chapter 2 which pertain to the geometries of this chapter as∂φ∂tˆ+ ∂∂zˆ(φvˆs) = 0, (4.2)−∂φ∂tˆ+ ∂∂zˆ((1 − φ) vˆl) = 0, (4.3)(1 − φ)(vˆs − vˆl) = k(φ)µ[∂pˆ∂zˆ+ ρlg] , (4.4)∂∂zˆ[−pˆ − Pˆ] = (ρl + φ(ρs − ρl)) g, (4.5)where g = 9.81m/s2 (we note its negative zˆ direction is accommodated for informulations). Inspired again by the φ ≪ 1 asymptotic limit of an array of rigidrods e.g. ([33, 37]), we define a permeability function,k(φ) = Aφln(αφ) (0 < φ ≲ 0.006), (4.6)where A and α are experimentally calibrated parameters. Results presented by50Figure 4.2: Schematics of the first two flow configurations. The recirculationloop of the rectangular tank in the flow-through experiments is sketchedin (a), and the tank for the drainage experiments in (b). Panel (c) il-lustrates the geometry of the three possible phase arrangements: (i) and(ii) illustrate the clear, freely falling and networked layers in a closedcontainer, in the manner of the sedimentation experiments. (iii) showsthe final drainage stage of a configuration in which water is withdrawnand the top surface of the clear water layer meets that of the solid.51Higdon & Ford [33] suggest that this form is adequate over the calibrated range ofsolid fraction. Away from this low-φ limit, we extend (4.6) by bridging to our ex-perimental results of Chapter 3, using a simple interpolation based on the variableln(φ), which will be detailed in Section 4.4.As we did for the uni-axial experiment in Chapter 3, we can combine the con-tinuities, integrate, and define with respect to a net bulk flow velocity, V(tˆ),vˆs − V(tˆ) = (1 − φ) (vˆs − vˆl) , (4.7)where V(tˆ) equals the fluid velocity through the permeable surface at zˆ = 0, andis therefore defined for a given geometry. For the fixed volume chamber, V(tˆ) =−Q/A (as Q is drawn) where Q is a volumetric flux driven through the suspension.For the drainage tank and later for the CSF chamber, V(tˆ) = dhˆ/dtˆ. Equations 4.7and 4.5 can be substituted into Equation 4.4, arriving atvˆs − V(tˆ) = −k(φ)µ[∂Pˆ∂zˆ+ (ρs − ρl) gφ] . (4.8)We make note of this equation’s resemblance to Equation 3.6, with the variationcoming from drainage out of the suspension’s base and inclusion of body forces.As discussed, if the solid fraction of the suspension lies below the network’sgel fraction φg, then Pˆ = 0, as the suspension cannot sustain compressive stress.However, for regions where the suspension is networked (i.e. φ > φg), we are keenon investigating if the constitutive model for the solid effective stressPˆ = Py(φ) −Λ(φ)∂vˆs∂zˆif Pˆ > Py(φ), (4.9)still holds its improved representation seen previously. As was the case in Chapter3, when Pˆ < Py(φ), the solid is assumed to withstand the imposed stress withoutdeforming, and so ∂vˆs/∂zˆ = 0, with the solid stress remaining otherwise undeter-mined. For compressive yield stress, the assumed form isPy(φ) = ⎧⎪⎪⎪⎨⎪⎪⎪⎩0, φ < φg,m(φ − φg)n, φ > φg, (4.10)52where n, m, and φg are calibrated experimentally. Again, extension to higher solidfraction results shown in Chapter 3 is performed, which will be detailed in Section4.3. For the bulk viscosity, we continue with the formΛ(φ) = ⎧⎪⎪⎨⎪⎪⎩ 0, φ < φg,η∗φ2, φ > φg, (4.11)assuming it only applies in the networked regions. We again do not have a directcalibration of the scaling η∗, and will leave it as a free parameter.The yield condition Pˆ > Py(φ) can be awkward to deal with in situations inwhich unyielded (but networked, i.e. φ > φg) plugs of solid appear in the suspen-sion. Although one does not expect such plugs to appear in sedimentation tests,they can, in fact, appear in the drainage problems we consider. In principle, oneis then forced to track moving yield surfaces and ensure the plugs remain rigid.Rather than deal with such complications, we adopt a “regularization” of the con-stitutive model in Equation 4.9, similar to to that used for viscoplastic fluid models[29]: Pˆ = −⎡⎢⎢⎢⎣ Py(φ)∣∂vˆs∂zˆ ∣ + δ +Λ(φ)⎤⎥⎥⎥⎦ ∂vˆs∂zˆ , (4.12)where δ ≪ 1 is a regularization parameter. Equation 4.12 applies everywherethat φ > φg; the unyielded regions are taken care of approximately in that, when∣∂vˆs/∂zˆ∣ ≪ δ, the regularization renders the first term into a relatively large viscous-like stress, supplementing Λ(φ)∂vˆs/∂zˆ. But where ∣∂vˆs/∂zˆ∣ ≫ δ, the regularizedlaw in Equation 4.12 coincides with the original model in Equation 4.9. In practice,we take δ = 10−7s−1, the precise value having been verified as irrelevant.The regularization of the constitutive law permits a relatively straightforwardimplementation scheme to solve the model equations numerically. However, thechoice η∗ = 0 is inaccessible in this scheme. To access and compare with the rate-independent limit η∗ → 0, and reduce the model to more conventional formulations[11, 52], we therefore select a relatively small value for η∗ to minimize the effectof the solid viscosity.534.1.1 Unnetworked and Clear ZonesIf φ < φg, Pˆ = 0 and it then follows from Equation 4.5 that the pore pressure ishydrostatic∂pˆ∂zˆ= − (ρl + φ(ρs − ρl)) g, (4.13)and from Equation 4.4, the solid sediments at the local free fall velocity Vsed.:Vsed.(φ, t) = vˆs = V(tˆ) − φk(φ)gµ(ρs − ρl). (4.14)The solid phase continuity in this zone is therefore∂φ∂tˆ+ ∂∂zˆ(φVsed.) = 0. (4.15)This hyperbolic problem implies that an initially uniform state with φ = φ0 < φgfalls uniformly and a shock forms underneath where the solid consolidates to thegel point at height zˆ = zˆg(t), as shown in Figure 4.2c(i). The unnetworked solidalso falls away from the top surface, leaving behind an overlying clear fluid layeroccupying zˆf(tˆ) < zˆ < hˆ, also shown in Figure 4.2c(i). In the constant-volumeexperiments (Figure 4.2a), the top surface is fixed, so that hˆ = h0 and V is set bya constant imposed background flow. For the drainage problems (Figure 4.2b), thesurface falls as water leaks out, and V(tˆ) = ∂hˆ∂tˆ. We see in this case, V(tˆ) is not aconstant in time, however, it is spatially independent and therefore the uniformlyfalling unnetworked zone still remains. The top of the free-fall zone is given by∂zˆf∂tˆ= Vsed.(φ0, t), (4.16)orzˆf(tˆ) = ∫ tˆ0V(tˆ)dtˆ − [φ0k(φ0)gµ(ρs − ρl)] tˆ + h0. (4.17)Note that there is only a free-fall zone when zˆf > zˆg; eventually, the freely fallingunnetworked solid completely falls into the compacting layer, leaving clear fluidabove and zˆf = zˆg, as shown in Figure 4.2c(ii).544.1.2 Networked ZoneIn the networked layer, i.e. φ > φg, we use Equations 4.2 and 4.8 with the consti-tutive model Equation 4.12 substituted in to arrive at the following equations thatdescribe the consolidation∂φ∂tˆ+ ∂∂zˆ(φvˆs) = 0, where, (4.18)vˆs + k(φ)µ∂∂zˆ⎡⎢⎢⎢⎢⎣−⎛⎝ Py(φ)∣∂vˆs∂zˆ ∣ + δ +Λ(φ)⎞⎠ ∂vˆs∂zˆ⎤⎥⎥⎥⎥⎦ = V(tˆ) − k(φ)µ (ρs − ρl)gφ. (4.19)The spatial boundaries of this zone are 0 ≤ zˆ ≤ zˆg. At the shock at the top of theconsolidated layer, zˆ = zˆg(tˆ), the effective stress vanishes, and soφ(zˆ−g , tˆ) = φg, (4.20)∂∂zˆvˆs(zˆ−g , tˆ) = 0, (4.21)where the boundary growth is defined based upon mass conservation across theshock as∂zˆg∂tˆ= φ(zˆ+g , tˆ)Vsed.(φ(zˆ+g , tˆ), tˆ) − φgvˆs(zˆ−g , tˆ)φ(zˆ+g , tˆ) − φg , (4.22)where the ± superscripts indicate the limits from above and below, respectively.Also, φ(zˆ+g , tˆ) = φ0 if zˆf > zˆg and there is an overlying free-fall zone, or φ(zˆ+g , tˆ) =0 if zˆf = zˆg and clear fluid overlies the compacting layer.For fixed-volume compaction, as in Figure 4.2a, V is a prescribed parameter,and so the boundary condition at the bottom of the network zonevˆs(0, tˆ) = 0, (4.23)is sufficient for closing the model equations for sedimentation with or without flow-through.For drainage experiments, an additional expression for the net bulk flow V(tˆ)is required, which is set self-consistently by the stress conditions at the permeablelower boundary. In particular, there is an outflow condition that dictates the pore55pressure at zˆ = 0:pˆ(0, tˆ) = ρl(ghˆexit + cV2), (4.24)where ρlghˆexit denotes the hydrostatic head at the outflow and ρlcV2 is the resis-tance to the outflow velocity V , modelled using a friction coefficient c. The stressconservation of Equation 4.5 now demands that the fluid and solid stresses at zˆ = 0balance the overlying weight of the suspension:Pˆ(0, tˆ) + pˆ(0, tˆ) = ρlghˆ + φ0h0(ρs − ρl)g. (4.25)As a consequence,Pˆ(0, tˆ) + ρlcV2 = ρlg(hˆ − hˆexit) + φ0h0(ρs − ρl)g, (4.26)which completes the model for drainage. A similar scheme to that described inSection 3.2 is used to solve the model.4.1.3 Capillary Stresses for Drainage and FreenessIf the water drains from the chamber, the model outlined above applies to the net-worked zone only as long as hˆ > zˆg and there is an unnetworked or clear-fluid zoneabove the consolidating region. However, the fluid surface eventually descendsto zˆ = zˆg(tˆ). Any subsequent drainage must either expose a dry fibre network, orprompt capillary stresses to appear that maintain contact between the solid and fluidsurfaces. In the experiments, the top surface always remains wet, with no evidenceof an overlying unsaturated solid. Therefore, we assume that capillary stresses actto hold the fluid and solid surfaces together once they meet, and zˆg = hˆ (see Fig-ure 4.2c(iii)). This demands that we abandon the upper boundary conditions inEquations 4.20 and 4.21, and instead applyvˆs(zˆ−g , tˆ) = ∂zˆg∂tˆ= ∂hˆ∂tˆ. (4.27)A similar situation may arises when the initial suspension is already networked,φ0 > φg, and the water and solid surfaces coincide at tˆ = 0. Such initial statesturn out to characterize the freeness test for the pulp suspension we employ in our56experiments, which begin from a well-mixed initial condition that is slightly abovethe gel point. In this situation, and without any interfacial interaction, the phasesseparate at the top under sedimentation. But, by including capillary effects, wemay again prevent the water surface from detaching from the top of the solid.4.2 Material and Experimental DetailsThe pulp suspension used for this study is the same bleached, northern softwoodKraft (NBSK) pulp used for Series 1 from Chapter 3. Details of the suspension canbe found in Table B.1.To calibrate the compressive yield stress Py(φ), permeability k(φ) and thengauge the importance of any rate-dependent solid effective stress, we perform avariety of laboratory experiments. In a first series of tests, we conducted sedimen-tation experiments in a selection of three cylinders with interior diameters of 10.0,13.6 and 19.0 cm. The initial depth of the suspension h0 was varied from 3.1 to88.3 cm. From these tests, we measured the final sedimented height, i.e. the levelof the top surface of solid. As will be described in Section 4.3, this height can beused to calibrate Py(φ).The second set of tests used a tank comprised of a large rectangular chamberand a water reservoir as sketched in Figure 4.2a. The suspension was placed in thechamber and was confined between screen meshes. Water could be recirculatedbetween the reservoir and test chamber by a pump, providing a closed flow loop.In this second arrangement, we first performed more sedimentation tests withoutflow-through (Q = 0) to supplement the final height data from the cylinders andfurther constrain Py(φ). We then activated the pumps and measured the effectof the additional flow-induced compaction on the final solid height, which allowsus to calibrate the permeability (see Section 4.4). The rectangular chambers hada cross-section of 15.2 cm by 14.0 cm, and the depth of the pulp compartmentwas h0 = 28.2 cm. Flow speeds in the range 2.26 × 10−5m/s to 2.07 × 10−4m/swere attained. A variant of the flow-through experiments, in which water wasdrained from the arrangement at a fixed rate, also allowed a cruder estimation ofthe compressive yield stress at higher volume fractions, which will be detailed inSection 4.4.1.57For the flow-through tank, we also recorded high resolution camera images ofthe compacting suspension through the side wall during the experiments. Fromthese images we measured the instantaneous height of the top surface of the solid,zˆf(tˆ). We also seeded the pulp with small pieces of black paper (of a few mil-limeters in size) to act as tracers for one-dimensional particle image velocimetry(PIV). The PIV extracts the horizontally averaged vertical displacements betweenconsecutive frames, which were spaced by 10 seconds in these tests with relativelyslow flow speeds. Both measures of the time-dependent dynamics allowed us toquantify the transient adjustments that took place after the suspension was mixedand left to settle (with or without flow-through), and when the pump was suddenlyswitched on or changed flow rate.A third suite of experiments focussed on the time-dependent dynamics ofdrainage tests in a large, open top rectangular tank within which the pulp wassuspended above a permeable screen; see Figure 4.2b. The bottom of the tankwas connected to an outlet pipe ending in a valve fixed at an adjustable verticalposition that was below the initial top surface of the suspension. The height ofthe valve sets a net hydrostatic pressure drop across the system that forces waterwithin the pulp chamber to drain to a given level, thereby compacting the solidagainst the screen. The initial height of the suspension in the drainage tank wash0 = 41.9 cm; the tank’s cross-sectional area was approximately 390 cm2. The exitpipe was one metre long and 2.54 cm in diameter and its end was held at a rangeof heights from 28.5 cm down to 3.5 cm. The resistance coefficient was measuredto be c = 3.44× 104 from steady flow-through tests with pure water. Again, we usethe black paper tracers to perform PIV for the arrangement; this time, the imageswere spaced by 1 second in view of the faster flow speeds that were achieved. Boththe drainage tests and the transient adjustments in the flow-through tank are usedto gauge the importance of any solid viscosity Λ(φ) in Section 4.5.4.3 Calibration of Py(φ)Calibration of compressive yield stress was performed through various sedimen-tation tests. For a vertical tube filled with a suspension of volume fraction φ0to a height h0, sedimentation proceeds until the total weight of solid material58(ρs − ρl)gφ0h0 is balanced by the compressive yield stress at the bottom, Py(φB).Moreover, in the steady state, the gravitational stratification of the solid dictatesthe final height hˆf , establishing a connection with the base solid fraction φB thatprovides the means to calibrate the compressive yield stress. For the task, we con-ducted sedimentation tests in the three cylinders and the rectangular tank of theflow-loop arrangement, beginning with initial concentrations φ0 < φg. For equiva-lent selections of φ0 and h0, the final settled height was consistent across all fourcontainers, as expected. However, the time taken to reach the final settled heightwas significantly longer in the smallest diameter cylinder than for the other con-tainers, for which the sedimentation rates were similar.Figure 4.3a shows the average final solid fraction, φ¯ = φ0h0/hˆf , against thetotal gravitational stress on the solid, (ρs − ρl)gφ0h0. When the final sedimentedstate is almost uniform, the overburden is balanced by Py(φ¯). Thus, aside fromany gravitational stratification, the plot in Figure 4.3a reflects the compressive yieldstress function. Although there is some spread, the data are suggestive of a lineardependence of Py(φ) on solid fraction, which guides us to take n = 1 in Equation4.10 for a deeper analysis of the experimental results. Further evidence for thischoice is provided later.For Py(φ) = m(φ − φg) and no flow-through (V = 0), the steady-state form ofEquations 4.8 and 4.9 reduce todPˆdz=mdφdz= −(ρs − ρl)gφ, (4.28)which, given that φ = φg at zˆ = hˆf , has solutionφ = φg exp [ gm(ρs − ρl)(hˆf − zˆ)] . (4.29)φB is thereforeφB = φge(ρs−ρl)ghˆf /m. (4.30)The integral of Equation 4.28 across the height of the solid also provides the globalforce balance condition,Py(φB) =m(φB − φg) = (ρs − ρl)gφ0h0, (4.31)592 2.5 310-302462 3 4 510-30246Figure 4.3: Results of the sedimentation experiments, plotting in (a) the totalgravitational stress on the solid, (ρs−ρl)gφ0h0, against the average finalsolid fraction, φ¯ = φ0h0/hˆf . This data is replotted in (b) as Py(φB)against φB , using (4.30) and (4.31), having fitted the parameters m =1.756 × 103 Pa and φg = 0.00178 according to the procedure outlinedin Section 4.3; the fit itself is shown by the dashed line. The differentsymbols refer to the two batches of pulp with different φ0 (blue forφ0 = 0.00165 and red for φ0 = 0.00167), and the different containers(circles for the cylinders, with size corresponding to radii; squares forthe rectangular flow-through tank).with the help of Equation 4.1. Equations 4.30 and 4.31 can then be combined toarrive at a prediction for the final settled height,hˆf = mg(ρs − ρl) ln [1 + (ρs − ρl)gφ0h0mφg ] . (4.32)The constant (ρs − ρl)g ≈ 5 × 103 kg/m2s2, and each experiment furnishes a pair(φ0h0, hˆf). We therefore determine the values form and φg that best approximatesthis equation in a least squares sense over all 49 experiments. We find the fittedfunctional form to bePy(φ) = 1.756 × 103(φ − 0.00178) [Pa]. (4.33)The fitted linear compressive yield stress function is plotted in Figure 4.3balong with the experimental data for Py(φB) and φB , implied by Equation 4.31 or604.30, given the fitted values of m and φg. The agreement between the model fitand the experimental data is satisfactory, leading us to conclude that Py(φ) is wellrepresented by a linear function near the gel point. This conclusion is reinforced byrepeating the analysis, but including the exponent n as a further free parameter: thenon-linear least squares fit to the generalization of Equation 4.32 predicts that n ≈0.93. This fitted exponent is not significantly different from unity given that it leadsto no qualitative change to Figure 4.3. In particular, the root mean square (RMS)error in Equation 4.32 normalised by the observed hˆf is reduced from 4.55% to4.52% upon changing n from 1 to 0.93.The linear behaviour near the gel point is somewhat surprising, with extrap-olations from higher solid fraction suggesting a low φ limit Py(φ) ∼ φ2, as wasseen in Chapter 3. For other flocculated dispersions, an even stronger dependencePy ∝ (φ − φg)n, with n between 2 and 4 is seen [11, 92].The linear calibrated Py(φ) is replotted in Figure 4.4 as the blue trend, alongwith the lower end of the results of Series 1 in Chapter 3, which is shown as thered trend. In some of the dynamic experiments, we venture into the φ domainnot covered by either experimental results. We therefore introduce the bridgedfunctional formPy(φ) = exp{ln [1.756 × 103(φ − 0.00178)]S(φ) +[1 − S(φ)] ln [6.70 × 105 φ1.89(1 − φ)2.98 ]} [Pa], (4.34)with an interpolant functionS(φ) = 12[1 − tanh Υ(1 − lnφln Φ)] , (4.35)where the parameters are Φ = 0.01 and Υ = 10. This too is shown in the figure asthe thin purple line. The parameters of the interpolant were found to align the formwith rough measurements of Py(φ) found through a variant of the flow throughexperiment, which will be addressed in the next section and are shown as the yellowcircles.610 0.1 0.2 0.3 0.410-21001021041060.021001030 0.1 0.2 0.3 0.4 0.510-1610-1410-1210-1010-80 0.0210-1010-8Figure 4.4: Plot of the bridged Py(φ) function, shown in Equation 4.34, asthe thin purple line. The thicker blue line shows the results from sed-imentation experiments, and the red line shows the fit for Series 1 inChapter 3. The yellow dots show the cruder estimate described in Sec-tion 4.4.1.4.4 Calibration of k(φ)The permeability function can be calibrated by measuring the steady-state solidheights for given bulk flow velocities V in the flow-through tank. To build sucha data set, we first mixed the suspension, then let the solid settle to steady statewithout any flow-through (V = 0). The pumps were then turned on and the rate in-creased sequentially, waiting at each pump setting for the steady state to be reached.The settled heights obtained in this way are plotted against V in Figure 4.5a as thefilled squares and circles.In steady state flow-through, Equation 4.8 becomesmdφdz= µVk(φ) − (ρs − ρl)gφ, (4.36)62-2 -1 010-40.140.160.180.22.6 2.8 3 3.2 3.410-35678910-9Figure 4.5: In (a), settled heights hˆf against flow velocity V , showing boththe experimental results (filled squares and circles) and the theoreticalpredictions (solid lines), given the fits established in this section. In (b),this implied mean permeability for a nearly uniform suspension givenby Equation 4.37, plotted against φ¯ = φ0h0/hˆf ; the black line showsEquation 4.6 with the calibrated A and α. The (blue) circles have φ0 =0.0015 and are used in the fits for A and α; the (red) squares show anindependent data set with φ0 = 0.0017.which, for a nearly uniform suspension, indicates thatk(φ¯) ∼ µVhˆfφ0h0⎡⎢⎢⎢⎢⎣2mhˆf⎛⎝φghˆfφ0h0 − 1⎞⎠ + (ρs − ρl)g⎤⎥⎥⎥⎥⎦−1. (4.37)Figure 4.5b plots the implied k(φ¯) against φ¯ ≡ φ0h0/hˆf for our steady-state flow-through experiments. This data suggests that the main dependence of the perme-ability on the solid fraction is through a factor φ−1, as in the fit shown in Equation4.6 and found by previous studies [104].With the adopted form k(φ) = Aφ−1 ln(α/φ), the implicit solution to Equation4.36 for φ(zˆ) ishˆf − zˆ = −mAΓµV ln⎡⎢⎢⎢⎢⎣ φφg ( Γ − ln(φ/α)Γ − ln(φg/α))Γ⎤⎥⎥⎥⎥⎦ , (4.38)where Γ = −µV/[(ρs − ρl)gA]. The global stress balance obtained by integrating63Equation 4.36 now givese−Γ (φg − φBΓα− µVmAΓ2αφ0h0) = E1 [Γ − ln(φgα)] − E1 [Γ − ln(φBα)] , (4.39)whereE1(x) = ∫ ∞xe−t/t dt, (4.40)is an exponential integral. φB follows from Equation 4.38 with zˆ = 0. We use thisexpression to fit A and α in a least squares sense to the collected hˆf versus V data,given the previously determined values of m and φg. This procedure leads to thefitted functional formk(φ) = 3 × 10−12φln(0.84φ) [m2]. (4.41)The fitted value of α is not far from the unity used in Equation 2.9. The effective-ness of the fit can be judged by Figure 4.5a, which includes the theoretical pre-dictions for the final height (given the fitted values of the parameters). The figurecontains two sets of data with slightly different φ0; only one of these sets (shownwith blue circles) was used to fit A and α. Given those parameter settings, thesecond set of data is reproduced with an RMS error (normalised by the observedhˆf ) of 1.23%; the first set has an error of 0.32%.The fidelity of the fit can be further justified by using a more general permeabil-ity function of the form Aφ−` ln(α/φ) and varying `. We find that ` = 1 providesa superior fit than any other power larger than 1.1 or less than 0.9; the RMS errorwhen ` = 1.1 or 0.9 is larger than that for ` = 1 by approximately 30%.Note that the pump rate was only increased in steps in this series of experi-ments. With a reduction of the flow rate, the suspension shows highly hystereticbehaviour, with very little recovery and expansion to a less consolidated structure.In the most extreme case of turning off a test with the highest pump rate, the topsurface of the solid rebounds upwards by at most 1− 2mm. By comparison, whenthat pump rate is switched on to compress the gravitationally sedimented state,the top surface is pushed down by around 7 cm. Thus, at least for the degree ofconsolidation experienced in the current tests, the fibre matrix must deform almostentirely plastically on compression, as assumed in the model.640 0.1 0.2 0.3 0.410-21001021041060.021001030 0.1 0.2 0.3 0.4 0.510-1610-1410-1210-1010-80 0.0210-1010-8Figure 4.6: Plot of the bridged k(φ) function, shown in Equation 4.42, asthe thin purple line. The thicker blue line shows the results from theflow through experiments, and the red line shows the fit for Series 1in Chapter 3. A representative fit of permeability from Jackson andJames [37] is plotted as the black line, and estimates of an effective bulkpermeability obtained via a Pulmac tester are shown with red triangles.The calibrated permeability is replotted in Figure 4.6, again as the blue trend.Similarly to Py(φ), we wish to bridge the permeability with the higher concentra-tion results reported for Series 1 in Chapter 3, shown as the red trend. The bridgedfunction, plotted as the purple trend, isk(φ) = exp{ln [3 × 10−12φln(0.84φ)]S(φ) +[1 − S(φ)] ln [3.60 × 10−13φln( 1φ) e−18.52φ]} [m2]. (4.42)and utilizes the same interpolant function as Py(φ).Support of the interpolant is provided by an independent set of permeabilitymeasurements of the same pulp using a commercial device (a Pulmac tester), whichis also included in Figure 4.6 as the orange symbols. This device places the pulpunder a given compression and a given pore pressure drop to measure the mean65permeability of a sample. Differential compaction at low φ implies that these mea-surements become inaccurate, and too low in comparison to the actual permeabilityfor φ→ φg.The accumulated results of Jackson and James [37] are shown as the black linein Figure 4.6, being a fit of their data and using the W /2 (where W is the meanfibre width) for this particular pulp, being Series 1 found in Table B.1. Despitethe order of magnitude jump of the blended trend, the permeability is still low andsteeper in comparison.Another perspective on this anomaly is provided by the fit of the constant A,which in the dilute limit should equal (W /2)2, times a constant between 0.125 and0.25 dependent on fibre orientation [33]. The typical fibre width of this chemicallypulped softwood is ≈ 30µm, implying thatA should be of order (3−6)×10−11m2,which is an order of magnitude larger than the fitted value shown in Equation4.41. This may arise because the dilute theory assumes that the fibres are solidand straight, rigid rods, whereas in reality the fibres are hollow and deformable.Moreover, the conduits in the fibre walls through which the water inside may leakout are small, suggesting that at low solid fractions, it may be more appropriate toconsider the fluid contents of the fibres as part of the solid matrix (as suggestedin [80]). Thus, the effective solid fraction is higher than expected by a factor cor-responding to the volume ratio of the fibre to its interior. Therefore, a correctionsimilar to F1 in Chapter 3 may effectively pull the pulp suspension’s results in linewith the ideal materials presented in Jackson and James.4.4.1 Crude Calibration of Py(φ) from Pump-Out ExperimentTo shore up the interpolation of the compressive yield stress, we conducted a sim-ple experiment in the flow-through arrangement, designed to extract a crude esti-mate of Py(φ) over a wider range of φ. More specifically, for an equilibrated testin which the solid had been compressed at the maximum pump rate, we removedthe return pipe of the pump from above the pulp chamber, and instead withdrewwater from the back tank at the same fixed rate. Although the pump rate was atits maximum, this withdrawal experiment is relatively slow, taking about an hourto remove the 12 litres of water from the two tanks. This leads us to assume that66the solid adjusts quasi-statically during the withdrawal, allowing us to estimate thecompressive yield stress as outlined below.To begin with, the slowly descending top surface of the water remains above thesolid. In this state, the draining of the fluid from the arrangement is not expectedto affect the pulp, and no changes in the solid were observed during this phase ofthe experiment. Once the water level in the pulp chamber meets the top of thecompacted solid layer, however, capillary stresses prevent drainage through thatmatrix. The top surface of the suspension then begins to fall less quickly than in thereservoir connected to it because the solid matrix supports a hydrostatic pressuredrop across the two. If 2Vw denotes the flux at which water is withdrawn, per unitcross sectional area of the two tanks, thenhˆ + hb = 2(h0 − Vwt), (4.43)where hb is the water level in the back reservoir.At this stage, the pulp is also relatively compacted, with an elevated solid stresscountering the capillary pressure that is exerted at zˆ = hˆ to maintain the coincidenceof the top surfaces of the water and pulp. This leads us to suppose that the solidbecomes relatively uniform, φ ≈ φ(t) ≈ φ0h0/hˆ, in which case vˆs ≈ zˆhˆ dhˆdtˆ . Ignoringthe rate-dependent stress in view of the relatively slow withdrawal, Equation 4.5then reduces to − ∂∂zˆ[Py(φ) + pˆ] = (ρl + φ(ρs − ρl)) g. (4.44)Given that Py(φ)+ pˆ = 0 at zˆ = hˆ, pˆ(0, tˆ) = ρlghb, and φ(ρs−ρl) ≪ ρl, the integralof Equation 4.44 now impliesPy (φ0h0hˆ) ≈ 2ρlg(hˆ − h0 + Vwt). (4.45)This allows us to extend the compressive yield stress function to higher φ using theobserved hˆ(tˆ) from the withdrawal experiment, as shown in Figure 4.4.674.5 Results: Model Representation of SuspensionDynamics4.5.1 Fixed-Volume Compaction DynamicsWe begin our assessment of the time-dependent dynamics predicted by the modelwith the transient dynamics observed in the flow-through tank. Figures 4.7a and4.8a show the solid velocity measured by PIV during fixed-volume sedimentationexperiments either without flow-through (V = 0; Figure 4.7a) or assisted with V =−2.07 × 10−4 m/s (Figure 4.8a). The figures compare these results with modelpredictions, computed using the bridged calibrations of Py(φ) and k(φ), describedin Section 4.3 and 4.4. For the rate-dependent solid stress, we again leave η∗ as afree parameter, which is varied to observe representation. Its magnitude can alsobe compared to the value found in Chapter 3 for Series 1, being redefined here asηPF∗ , to denote it is the value obtained through pressure filtration. The value, asreported in Table B.2, is ηPF∗ = 10MPa ⋅ s.The model solutions are weakly sensitive to the value of η∗ in these calcu-lations, as illustrated by the time series of zˆf(t) and zˆg(t) for different choicesof the bulk viscosity in Figures 4.7d and 4.8d; only for relatively large values ofη∗ (in comparison to ηPF∗ ) is there any suggestion that the rate-dependent solidstress participates in the settling dynamics. Moreover, the comparison with theexperimental PIV, which is otherwise qualitatively successful, implies that suchcases are not realistic. The insignificance of the bulk solid viscosity in these testsfor η∗ = O(ηPF∗ ) can be established more directly by dimensional analysis withthe model equations: in comparison to the relative velocity vˆs − V , the viscousterm on the left of Equation 4.19 is of order ηPF∗ φ2gk(φg)/(µh20) = O(10−3), ifwe use the initial height h0 and gel fraction φg as characteristic scales for lengthand solid fraction. Our fixed-volume compaction tests can therefore be adequatelyreproduced by the two-phase model with just a compressive yield stress for theconstitutive model, as found by previous studies both in the pulp (example [104])and the general literature (example [16]).A different comparison of theory and experiment is given by the φ−distributionof the final steady state. This distribution is, however, difficult to extract from the68(a) (b)0 0.1 0.2 0.300.050.10.150.2(c) (d)0 500 1000 150000.10.20.3* Figure 4.7: Dynamics of a sedimentation experiment with φ0 = 0.00154and h0 = 0.272m. Contour plots of solid phase velocity, vˆs [m/s],are shown from (a) experimental PIV and (c) a model solution withη∗ = ηPF∗ = 107 Pa ⋅ s. In (b), the PIV displacements are integrated todetermine the final positions of tracers that were uniformly distributedthrough the column (dots), with solid line showing the correspondingtheoretical prediction. Panel (d) shows the time series of the interfacezˆf (solid) and networked height zˆg (dashed) for further theoretical so-lutions with η∗/ηPF∗ = 10−3, 10−2, 0.1, 1, 10 and 100; the dotted blackline shows the experimental observation of zˆf .69(a) (b)0 0.1 0.2 0.300.050.10.15(c) (d)0 500 1000 150000.10.20.3* Figure 4.8: A similar set of plots as in Figure 4.7, but for a flow-assistedsedimentation experiment with V = −2.07 × 10−4m/s, φ0 = 0.00166and h0 = 0.282m.PIV measurements because of the need to differentiate the particle displacementsin zˆ: in our one-dimensional approximation of the compaction problem, conserva-tion of mass demands that the solid fraction at any time tˆ be related to the initialdistribution bydzˆdz0= φ(z0,0)φ(zˆ, tˆ) = φ0φ(zˆ, tˆ) , (4.46)where z0 is the initial position of an element of solid. The mapping from initialto final positions of tracers encoded in zˆ(z0; t) provides an alternative means ofcomparing the theory and experiments. For example, for sedimentation without70flow-through (V = 0),zˆ = m(ρs − ρl)g ln [ mφg + (ρs − ρl)gφ0h0mφg + (ρs − ρl)gφ0(h0 − z0)] , (4.47)corresponding to Equation 4.29. The mapping zˆ = zˆ(z0; tˆ) is conveniently ex-tracted from the experiments using the cumulative displacements between con-secutive images found by PIV. Figures 4.7b and 4.8b include the experimentallymeasured and theoretically predicted final profiles of zˆ(z0) for those two partic-ular experiments. Overall, the experiment and theory are in agreement (with theexample in Figure 4.8 being one of the more demanding cases), providing furtherconfidence in our fits of the constitutive functions Py(φ) and k(φ).Despite the qualitative agreement of theory and experiment in Figure 4.7, thereis a notable difference between the two in the free-fall zone zˆg(tˆ) < zˆ < zˆf(tˆ)which, according to the model, should contain solid with the initial fraction φ0falling at the constant free-fall velocity Vsed.(φ0, tˆ). By contrast, the experimentalPIV data indicate a variable fall velocity over this region, which was also noticeablein the descent of the solid surface hˆ(tˆ) (see Figure 4.7d), and evident in all ourexperiments. A variable fall speed may arise because a well-mixed initial conditionis difficult to establish in the experiments, resulting in a non-uniform initial solidfraction. Indeed, assuming that the final sedimented state is given by Equation 4.29,one can use Equation 4.46 and the PIV data to trace φ back to where the solid wasunnetworked and in free fall. This procedure implies an initial state characterizedby irregular spatial structure with φ0 = 0.0017 ± 0.0008.It is also possible that a non-monotonic sedimentation flux could generatea non-uniform free-fall zone, as suggested previously for other suspensions [3,4]. However, one still expects a uniform region embedded within that zone (ifφ(zˆ,0) = φ0), unlike what is seen in Figure 4.7a. Moreover, although the spatialdifferentiation of the PIV measurements introduces significant noise in the localestimates of φ, the same data also suggests that the sedimentation flux is not asimple function of the local solid fraction. The discrepancy may well thereforepoint to some other behaviour in the free-fall zone that is not accounted for in themodel (such as spatial inhomogeneity induced by the flocculation of fibres, or the7110-2 100 10210-210-1Figure 4.9: Root-mean-square error in the interface position zˆf(tˆ), normal-ized by its mean value for a number of sedimentation and flow-throughexperiments. These experiments include an example of pure sedimen-tation (red squares), a test in which sedimentation was assisted with thelargest bulk flow rate (V = −2.07 × 10−4m/s; black stars), and tests inwhich the solid was allowed to sediment before turning the pumps on.For the latter, two cases are shown: one in which the pumps were againturned to the maximum (green filled circles), and a second in which thepump rate was increased in five steps up to that maximum (blue opencircles; this test corresponds the squares in Figure 4.5).additional drag experienced by the markers nearer the side walls). Awkwardly,this feature also precludes us from exploiting the PIV data to directly infer reliablepermeability data from the sedimentation experiments.A measure of the goodness-of-fit of the theoretical model over all of the sedi-mentation and flow-through experiments is shown in Figure 4.9. This figure plotsthe root-mean-square error in the interface position zˆf(tˆ) for the model using aspread of values of η∗. The error is normalized by the mean interface position and,unless the solid viscosity is chosen to be excessively high, is of the order of a fewpercent.724.5.2 Drainage DynamicsThe higher fall speeds achieved during the drainage experiments provide a moredemanding test of the model and an indication as to whether or not a bulk viscosityis needed: in the most extreme case with hˆexit = 0.035m, the mean solid fractionreaches approximately 0.02 and the solid speed peaks at around 1 cm/s (about fourtimes denser and fifty times faster than in the constant-volume tests). Importantly,in the drainage tests, the relatively high resistance of the output pipe has the effectof throttling the fall speed of the water. Consequently, the water height hˆ(tˆ) islargely set by the “pipe law” c (dhˆdtˆ)2 ≈ g(hˆ − hˆexit), in all but the most extremecases of hˆexit where the densification of the solid can offset the pipe resistance.Moreover, in most cases, the interface zˆf(t) does not have sufficient time to sedi-ment below the free surface to create an observable clear layer at the top. Thus, thetime series of both hˆ(tˆ) and zˆf(tˆ) are not the best statistics to gauge the bulk vis-cosity. Instead, we use the growth of the networked zone at the base of the columnas a clearer diagnostic.Figure 4.10a shows the solid phase velocities obtained by PIV for a num-ber of drainage experiments with varying hˆexit starting from h0 = 0.419m withφ0 = 0.0016. Except for the cases with small hˆexit, the water heights follow theparabolas predicted by the pipe law, and the bulk of the solid falls at roughly uni-form velocity given by dhˆdtˆ. More significant is the sharp decline of the fall speedinto the networked layer underneath, which is highlighted by plotting the fall ve-locity normalized by the free surface velocity, vˆs(zˆ, tˆ)/dhˆdtˆ (Figure 4.10b); althoughthe division introduces some noise, the free-fall zone and compacted layer be-come more apparent. Figure 4.10c adds complementary model computations ofvˆs(zˆ, tˆ)/dhˆdtˆ using a bulk viscosity scaling of η∗ = 12ηPF∗ , a choice that is motivatedbelow.Figure 4.11 shows further details of the theoretical solutions for one of the tests.Computations with different bulk viscosities η∗ are shown, illustrating how the po-sition of the top surface is not particularly sensitive to this parameter. The degreeof compaction at the base of the column is, however, controlled by η∗, with lowbulk viscosities generating a compacted layer bordered from the overlying fallingzone by a relatively sharp interface with pronounced velocity gradients. Raising73(a)(b)(c)Figure 4.10: (a) Solid phase velocity obtained from PIV of eight drainagetests with varying hˆexit values, plotted in series. The experimentallyobserved hˆ(tˆ) and the solution to the pipe law c (dhˆdtˆ)2 = ρlg(hˆ−hˆexit)are shown as black and dashed red lines, respectively. In (b) we re-plotthe data using the scaling vˆs(zˆ, tˆ)/dhˆdtˆ . In (c), we show the theoreticalcounterpart to (b) using the solid bulk viscosity η∗ = 12ηPF∗ .74(a) (b)(c) (d)Figure 4.11: A drainage test with hˆexit = 10.3 cm. Panels (a)-(c) show plotsof the scaled solid phase velocity vˆs/dhˆdtˆ for (a) the experimental PIV,and model solutions with (b) η∗ = 10−3ηPF∗ and (c) η∗ = ηPF∗ . Theblack lines indicate the contours of vˆs/dhˆdtˆ = 0.5 and 0.8. Panel (d)plots time series of the interfaces hˆ ≈ zˆf and zˆg for solutions withη∗/ηPF∗ = 10−3, 10−2, 0.1, 0.2, 0.333, 0.5, 1, 10 and 100; the dashedline shows the free surface of the experiment.the bulk viscosity smooths these gradients to furnish a more gradual transition.Furthermore, bulk viscosities close to the value suggested by pressure filtration,η∗ = ηPF∗ , lead to velocity gradients that are more consistent with the PIV obser-vations. Thus, the qualitative comparison of the PIV and theoretical velocity plots,and in particular the transition above the compacted layer, indicates that the modelperforms better with a bulk viscosity scaling η∗ = O(ηPF∗ ) than without one.A more quantitative evaluation of this conclusion is given in Figure 4.12, which75reports, for all the drainage tests, the root-mean-square error in the free surfaceheight hˆ(tˆ) normalized by h0, and the average distance between the contours forwhich vˆs(zˆ, tˆ)/dhˆdtˆ = 0.5 and 0.8; see Figure 4.11. The latter is a direct measureof the sharpness of the transition in solid velocity above the compacted layer, andhas the curious feature of being roughly independent of time and hˆexit for boththe experiments and model solutions. The plots of this diagnostic in Figure 4.12bsuggest that that the model fits the experiments best for bulk viscosities close toηPF∗ ; a near-optimal choice indicated by Figure 4.12b is η∗ = 12ηPF∗ , as used inFigure 4.10c.Note that the PIV detects some degree of rebound of the solid once the waterdrainage terminates; although there is no change in the position of the top surfaceof the solid, fibres near the base of the column deform back upwards over distancesof up to a few millimetres over times of order tens of seconds. We interpret thisrebound to be the signature of recovery from a small amount of elastic stress su-perimposed on the plastic compression (which will be addressed in Chapter 6). Asthe model incorporates no such dynamics, we have avoided any comparisons withthe PIV after the termination of the drainage.4.6 Understanding FreenessEquipped with a calibrated two-phase model for a specific suspension of cellulosefibres, we now turn to an exploration of the Canadian Standard Freeness test. Inparticular, in addition to verifying that the calibrated model reproduces the freenessscore for the pulp suspension, we explore the dynamics of the test in order to gaugewhat material metric the freeness score provides for a two-phase medium.4.6.1 Freeness ScoresThe arrangement of the Canadian Standard Freeness (CSF) test is sketched in Fig-ure 4.1: a suspension of pulp held in a cup drains through a thin permeable screeninto a funnel; a side channel diverts part of the discharge into a collection cup toregister the “Freeness Score” (in mL). Because the build up of solids above thescreen limits the drainage, the freeness score represents a non-linear integral mea-sure of the drainage dynamics. Under standard conditions, the tester is initialised76(a)10-2 10-1 100 101 10200.020.040.060.080.0350.0850.1350.1850.2350.285(b)10-2 10-1 100 101 10200.020.040.060.080.10.12Figure 4.12: (a) Root-mean-square error in hˆ(tˆ) and (b) the average distancebetween the contours along which vˆs(zˆ, tˆ)/dhˆdtˆ = 0.5 and 0.8; see Fig-ure 4.11. In (a), the error is normalized by h0. In (b), the dots showmodel predictions; the grey shaded region indicates the range of ex-perimental measurement as defined from the cumulative distributionof observed values (the distribution is strongly skewed; we use thelimits within which 68% of the data lie). The colour bar in (a) mapsthe colour of the dots in both panels to hˆexit.77with 1L of the suspension (leading to h0 ≈ 0.122m) with a consistency of 0.3%-by-weight (corresponding to φ0 ≈ 0.002, if ρs = 1.5 g/cm3 and ρl = 1.0 g/cm3),and a temperature of 20oC. Clear water has a freeness score in the range 880 to890mL.To compute theoretical predictions for the freeness score, we supplement themodelling posed in Section 4.1 with a model of the funnel that determines howmuch of the discharge from the cup enters the side channel. We relegate the detailsof that extension to Appendix C, its sole purpose being to convert the flow historyto the freeness score.Because the permeable screen of the cup presents less resistance to outflow thanthe pipe of our drainage tests (the resistance coefficient of the screen is estimatedto be c = 180, in comparison to the pipe coefficient c = 3.4 × 104), the dynamics ofthe freeness score are richer than the experiments in Section 4.5. Thus, as well asperforming a further verification of the model and its calibration, the freeness testserves a potentially informative application of the model. However, a single scorefor a given pulp suspension is limiting, leading us to perform a series of tests withthe freeness device in which we departed from standard procedure and varied theamount of pulp in the cup and its initial solid fraction.The results of these tests are displayed in Figure 4.13; the freeness score showsa remarkably linear dependence on the initial weight of pulp, over a range of initialconsistencies. Both the values of freeness and the trend with initial weight andconsistency are recovered by the model when the bulk viscosity is included witha value close to ηPF∗ . By contrast, solutions in the limit η∗ ≪ ηPF∗ of traditionalcompaction theory [11, 52] significantly underestimate the freeness score and fur-nish a trend with initial weight that is noticeably non-linear, except at the lowestsolid concentrations.4.6.2 The Dynamics of FreenessFor a more complete analysis of the freeness dynamics, we interrogate the modelsolutions. Figure 4.14 presents solutions for the parameters of the CSF test withη∗ = 0.01ηPF∗ , ηPF∗ and 100ηPF∗ . Shown are the predictions for hˆ(tˆ) and a se-lection of snapshots of φ and solid velocity during the period before the discharge780 200 400 600 800 100002004006008001000C = 0.1%C = 0.2%C = 0.3%C = 0.4%C = 0.5%(C,*) = (0.1%,*PF)(0.3%,*PF)(0.5%,*PF)(0.1%,10-3*PF)(0.3%,10-3*PF)(0.5%,10-3*PF)Figure 4.13: Freeness score measured experimentally and predicted by themodel against initial volume for the initial concentrations C (by mass)indicated. Solid lines show the model results for η∗ = ηPF∗ , and dottedlines for η∗ = 10−3ηPF∗ .from the side channel in the cone switches off. For a small bulk viscosity (Fig-ure 4.14b,e), the bulk of the material falls as an unyielded plug against the screen,rapidly crushing the material there so that the solid stress and hydrostatic pressurecome into balance. A boundary layer forms near the screen in which the solid issignificantly compacted, and this layer subsequently chokes the falling plug flowbecause of its low permeability. Consequently, the initial fall of the suspension israpid, but soon decelerates. With a large bulk viscosity (Figure 4.14d,g), the rate-dependent stress significantly supports the suspension, reducing the hydrostatichead and the drainage rate; the material remains roughly uniform and the velocityprofiles are linear. Bulk viscosities with scalings of η∗ = O(ηPF∗ ) lead to solutionswith characteristics between these two extremes (Figure 4.14c,f). The different dy-namics of the three cases is reflected by the resulting freeness scores (which are490mL for panels (b,e), 736mL for panels (c,f) and 243mL for panels (d,g)).7910-3 10-2 10-1 100 10100.050.10 0.0200.51-0.04 -0.02 000.510 0.02-0.1 -0.05 00 0.02-0.02 -0.01 0Figure 4.14: Model solutions for the freeness test for η∗ = 0.01ηPF∗ , ηPF∗and 100ηPF∗ . Panel (a) plots the free surface position hˆ(tˆ) against time.Also shown are the asymptotic predictions for η∗ ≪ 1 (Section 4.6.2)and η∗ ≫ 1 (from Equation 4.53 with η∗ = 100ηPF∗ ). Panels (b-d) and(e-g) show snapshots of φ and vˆs at the times indicated in (a) by thinvertical lines. Thick red lines in (e-g) show unyielded zones, definedfor our regularized constitutive law by the condition ∣Pˆ ∣ < Py(φ).80The Limit of Zero Bulk ViscosityIn the limit η∗ → 0 described by the traditional rate-independent models [11, 52],the equations for the freeness problem can be simplified owing to the plastic formof the solid stress: when φ0 < φg, the release of the flow at the base of the containerimmediately causes the bulk of the solid to move downwards with the speed of thetop surface and to compact into a much thinner consolidated layer above the screen.If the screen resistance and the contribution of the solid to the weight are relativelysmall, the force balance at the base, Equation 4.26, implies thatPˆ(0, tˆ) = ρlghˆ. (4.48)Thus, at the moment of release, the solid phase must immediately compact atthe base to a solid fraction given by Py(φ) = ρlgh0. Thereafter, however, theoverlying weight declines due to the falling height of the suspension, leaving thestress Pˆ(0, tˆ) below the compressive yield stress Py(φ(0, tˆ)) and the solid over-consolidated at the base. Immediately above, the downward flow continues to pushsolid into the compacted layer, thickening it and forcing the solid to consolidatelocally up to a yield stress Py(φ) that balances the instantaneous overlying weight.In other words, an upward-migrating compaction front forms at zˆ = Y (tˆ) wherePˆ(Y, tˆ) = Py(φ(Y, t)) = ρlg[hˆ(tˆ) − Y (tˆ)]. (4.49)Above the front, the bulk of the solid still moves down with φ = φ0, whereaswithin the compacted layer, the φ−distribution is a frozen record of the evolvingyield condition. For φ0 > φg, this simple structure is complicated by the factthat the solid is networked above the compaction front and the sharp jump abovezˆ = Y (tˆ) begins to diffuse upwards into the bulk of the falling layer. Provided thejump remains sharp, we may approximate it as a discontinuity at the compactionfront and apply the mass conservation constraint,dYdtˆ= dhˆdtˆφ0φ0 − φ(Y, t) , (4.50)given that φ0 dhˆdtˆ is the flux into the front from above, and the unyielded layer be-81low is stationary. Last, Darcy’s law of Equation 4.4 integrated across the over-consolidated layer implies thatp(Y, tˆ) ≡ ρlg(hˆ − Y ) − Py(φ0) = −µdhˆdtˆ∫ Y0dzˆk(φ) − ρlgY (4.51)(since pˆ(0, tˆ) = 0), where we have used the hydrostatic pressure pˆ = ρlg(hˆ − zˆ) −Py(φ0) in Y < zˆ < hˆ, which includes the contribution of the capillary pressure atthe top if φ0 > φg. Equations 4.50, 4.51, and 4.49 constitute two coupled ODEs forthe yield surface and suspension height; the solution of hˆ(tˆ) for the conditions ofthe CSF is included in Figure 4.14a. Both the compressive yield stress and perme-ability feature in this reduced model: Py(φ) determines the solid distribution at thebottom via force balance, whereas the permeability controls the water flow acrossthe solidified cake above the screen. The main limitation of the approximation inEquations 4.50–4.51 is the diffusive spread of the jump in φ above the compactionfront.The Limit of Large Bulk ViscosityIn the opposite limit of a large bulk viscosity, the solid becomes almost uniformthroughout the compacting column. This implies that φ = φ0h0/hˆ(tˆ) and vˆs =zˆ/hˆdhˆdtˆ. The bottom boundary condition, shown in Equation 4.26, now demandsthatdhˆdtˆ∼ −ρlgΛh2 ≡ − ρlgη∗φ20h20h4, (4.52)if the compressive yield stress, screen resistance ρlc (dhˆdtˆ )2 and solid buoyancyφ0h0(ρs − ρl)g are all small in comparison to the bulk viscous stress. Thus,hˆ ∼ h0 (1 + 3ρlgh0η∗φ20 tˆ)−1/3, (4.53)which is also plotted in Figure 4.14a for η∗ = 100ηPF∗ .In this limit, the freeness score is therefore completely determined by the bulkviscosity; the permeability and compressive yield stress play no role. To under-stand how those additional effects come into play, we estimate the corrections to82Equation 4.52 as follows: from Equation 4.19 we observe that the permeabilityenters when the Darcy drag is no longer negligible in comparison to the rate-dependent stress, implyingk∂∂zˆ(Λµ∂vˆs∂zˆ) ∼ dhˆdtˆ− vˆs ∼ (1 − zˆhˆ) dhˆdtˆ. (4.54)The bulk viscous stress at the bottom is then,Λ∂vˆs∂zˆ(0, tˆ) ∼ η∗φ20h20 1hˆ3dhˆdtˆ− µhˆ2k(φ0h0/hˆ) dhˆdtˆ . (4.55)Hence, Equation 4.52 (being a representation of Equation 4.26) can be generalizedtoPy(φ) − η∗φ2 1hˆdhˆdtˆ+ µhˆ2k(φ) dhˆdtˆ + ρlc(dhˆdtˆ )2 = ρlghˆ + (ρs − ρl)gφ0h0, (4.56)with φ ∼ φ0h0/hˆ. Equation 4.56 can be attacked with dimensional analysis togauge when effects other than the bulk viscosity come into play. In fact, Equa-tion 4.56 offers a convenient setting in which to assess parameter sensitivity moregenerally, as we detail next.4.6.3 Parameter SensitivityTo understand in more detail how the freeness score relates to the underlying ma-terial behaviour of a fibrous suspension, we vary parameters in the model solutionsand perform a dimensional analysis based on the simplifications afforded by theη∗ ≫ 1 limit exposed above: first, by varying the resistance parameter in the com-putations, we find that the screen resistance is important in limiting fall speeds atearly times. Thus, the initial velocity scale is provided by balancing the screenresistance against the hydrostatic head in Equation 4.56, giving∣dhˆdtˆ∣ ∼ √gh0c∼ 0.07 m/s. (4.57)83The rate-dependent stress, of order η∗φ2 dhˆdtˆ /hˆ ∼ η∗φ3 dhˆdtˆ /(φ0h0), only enters inthe main balance of Equation 4.56 when the compaction at the bottom becomessufficiently high, which suggests a typical solid fraction there:φ(0, t) = φB ∼ ⎛⎜⎝ρlgh20φ20η∗ dhˆdtˆ⎞⎟⎠1/4 ∼ 5 × 10−3. (4.58)Both estimates in Equations 4.57 and 4.58 compare well with the model solutionsin Figure 4.14(c,f).Given these estimates, we now gauge the importance of the remaining termsin Equation 4.56. From our calibrations of the compressive yield stress, and withη∗ = ηPF∗ , we find that φBPy(φB)/(ρlgφ0h0) ∼ 0.03 (since Py(φB) ∼ 10Pa).Thus, the compressive yield stress is likely to be a relatively small contributor tothe Freeness Score unless it is made larger by a factor of order ten or more. Onthe other hand, the relative size of the term originating from the Darcy drag isµ√h0/[2ρlk(φB)√gc] ∼ 1.8 (using k(φB) ∼ 2 × 10−9m2). The permeability istherefore expected to affect the freeness score except when made larger by a factorof ten or so. Last, the overlying solid weight (ρs − ρl)gφ0h0 is negligible in viewof the relatively small initial solid fraction.Since the screen resistance is fixed, these scalings suggest that the bulk viscos-ity and permeability have the most immediate effect on the freeness score, whereasa material requires substantially larger values of Py(φ) for the compressive yieldstress to become important. These predictions are confirmed in Figure 4.15a, whichdisplays Freeness Scores obtained from model solutions in which relevant param-eters are varied.More specifically, when we vary η∗, holding all the other parameters fixed, wesee that the Freeness Score does indeed depend sensitively on the bulk viscosity.Interestingly, the Freeness is maximized for η∗ = O(ηPF∗ ), and the observed val-ues in the actual Freeness tester lie close to the maximum. Evidently, Freeness iseffectively reduced by the choking of the drainage rate by excessive compaction atlow bulk viscosity, or the viscous support of the suspension at high values of η∗.Notably, neither reduction is compatible with the observed Freeness Score.Likewise, Figure 4.15a also highlights how reducing the permeability by a con-84(a)10-2 100 102200400600800(b)10-2 100 102 1040200400600800Figure 4.15: (a) Freeness Score against constant multiplier of η∗ (blue cir-cles), k(φ) (red diamonds) and Py(φ) (yellow triangles). The exper-imental Freeness of Series 1 pulp (714 mL) is shown by dashed line.(b) Freeness Score against bulk viscosity for the varying suspensionsof Chapter 3. Additionally, the model predictions for Series 1, varyingη∗ whilst keeping the product η∗k(φ) constant (as per Figure 3.10a) isshown by the solid red line connecting the open circles.stant factor has a dramatic effect on the Freeness Score, an effect we attribute to theelevated Darcy drag enhancing the compaction of the solid above the screen. Bycontrast, increasing the permeability by a constant factor has little effect on Free-ness, because the Darcy drag is then made unimportant, precisely as anticipated bythe scaling analysis. Similarly, changing the compressive yield stress by a constantfactor has no effect on the predicted Freeness except when Py(φ) is increased bymore than a factor of ten, to enable the solid yield stress to contribute to the supportof the hydrostatic load above the screen.4.6.4 Varying Pulp SuspensionsFrom the preceding discussion, we conclude that the model is able to reproducethe Freeness Score of the pulp suspension evaluated, being Series 1 of Chapter 3.A rate-dependent stress is certainly needed, and the material parameters that moststrongly affect the Freeness Score are the bulk viscosity and the permeability of thepulp suspension. Moreover, the best fitting choice of the bulk viscosity parameterη∗ is consistent with that found in uni-axial tests of Chapter 3.85As seen in Table B.1, we do have the Freeness Scores for the remaining sus-pensions investigated in Chapter 3, and therefore could evaluate the model for theremaining suspensions. To formally do this however, the lengthy process of cali-brating the material parameters for the various suspensions would need to be un-dertaken. Instead, we shall utilize the consistent bulk viscosity from Chapters 3and 4 found for Series 1, the relation between η∗ and k∗ shown in Equation 3.27,and the insensitivity to compressive yield stress to solve the model approximatelyacross the range of η∗ found in Chapter 3. More specific, we perform further com-putations of the model and vary η∗ and the k(φ) trend in the solver, keeping theterm η∗k(φ) fixed. The freeness scores of the varying suspensions are shown asthe symbols in Figure 4.15b, which are identified in Table 3.1. The model solutionswith the term η∗k(φ) fixed are shown as the connected red open circles. We finda similar trend for the Freeness Score with η∗, although none of the pulps testedhave substantially lower values of bulk viscosity.4.7 DiscussionThrough this study, we have demonstrated a continued need for the viscoplastic-like constitutive model for effective representation of high rate dewatering dynam-ics of pulp suspensions. Although bulk viscous effects are to be expected in two-phase media, they are normally ignored because dimensional analysis suggests theviscosity to be relatively small. The core of the argument is that the rate-dependentsolid stress originates from the viscous flow of the solvent around the solid particlesduring compaction. Thus, in the low φ limit, the solid viscosity scales with that ofthe solvent, as also predicted by general two-phase flow theory [21]. For our fibresuspension, this suggests that Λ ∼ µ = 10−3 Pa ⋅ s. The bulk viscosity adopted inour model, Λ = η∗φ2, is four orders of magnitude larger with the calibrated valuefor η∗ and φ = O(10−3). This anomaly mirrors two other results: the compres-sive yield stress depends linearly on φ − φg near the gel point, whereas a strongerdependence is found for other suspensions (Section 4.3), and pulp permeability isunusually low in comparison to other fibres (see Section 4.4).A rationalization of all these observations requires a micromechanical modelof the pulp fibre suspension, which is beyond the scope of this thesis. Never-86theless, we speculate that the unexpected material behaviour originates from thestructure of the hollow, deformable fibres themselves. For example, the abnor-mally low permeability may arise because the solid fraction, again, is misidentifiedin the suspension due to water trapped inside the fibre walls. Similar to Chapter3, perhaps the high bulk viscosity originates not from the larger-scale viscous flowaround fibres, but from flow within the much narrower regions where the fibres arein sliding contact. The reduced scale of those regions could, in principle, enhancethe viscous dissipation, and therefore the solid viscosity.Overall, pulp suspensions constitute an interesting two-phase material withsomewhat poorly understood micro-structural properties, despite widespread us-age. Our efforts here have highlighted the macroscopic rheology that this micro-structure must dictate. Whether many other materials share similar properties re-mains to be seen, although our speculations about the micro-structure dynamicsare not particularly specific to cellulose.4.8 Summary and ConclusionsIn this chapter, we explored the viscoplastic-like constitutive model’s representa-tion and the material parameters for a pulp suspension in the low concentrationlimit. Equipment and protocols were developed for calibrating the permeability,the compressive yield stress, and to collect dewatering behaviour. A variant of thegeneral model of Chapter 2 was used which accommodates body forces, varyinggeometries, and the multiple zones of the suspension. Additionally, the model wasadapted to an industrially relevant low concentration drainage test called a Cana-dian Standard Freeness test.Permeability was found to be low with respect to other fibre suspensions. Thewater trapped inside the fibre walls may significantly contribute to this difference.Compressive yield stress was found to be linear at low concentrations, which iscontrary to other materials. The viscoplastic-like constitutive model is found to benecessary for increased severity dewatering (drainage) experiments. The necessaryscaling of the bulk viscosity, η∗, was found to be equivalent to the value found inthe uni-axial experiments of Chapter 3. This continuity in the bulk viscosity acrossa large range of volume fractions is interesting, however, is equally perplexing as87to its source, with its magnitude substantially greater than that of traditional twophase theory would suggest at low concentrations. As is the case in Chapter 3, asthe dewatering rates diminish, as does the necessity of the bulk viscosity (i.e. notnecessary for sedimentation or slow flow-through experiments).Evaluation of the Canadian Standard Freeness test shows that the bulk viscos-ity is required to provide effective representation. Through a parameter sensitiv-ity study, the critical parameters that govern the Freeness test are found to be thepermeability and the bulk viscosity, whereas the test is insensitive to the suspen-sion’s compressive yield stress. This suggests, therefore, a Freeness test may bea valuable experiment for an independent calibration of a pulp suspension’s bulkviscosity.88Chapter 5Model Robustness:Representation of Industrial TwinRoll PressThis chapter aims at applying a variant of the calibrated modelling framework de-veloped for the uni-axial experiment to the operations of a pilot-scale Twin Rollpress. This will involve a two-dimensional model geometry and therefore a two-dimensional variant of the equations introduced in Chapter 2. Experimental resultsfor the roll press are collected using Valmet’s facility (in Sundsvall Sweden), andcalibration of the particular suspension’s material parameters follows the detailsof Chapter 3. The objective of this study was to evaluate the constitutive modelin a two-dimensional context and at elevated compression rates. Additionally, thisstudy provides an industrial context for the application of this modelling approach.The Twin Roll press, as introduced in Chapter 1, has two primary functions,which are illustrated in Figure 5.1. Its first function is to facilitate displacementwashing of the pulp and liquid suspension. This removes residuals, lignin, andcontaminants from previous operational steps. This washing action occurs throughthe injection of clean wash fluid into the suspension at several locations around thevat wall, along the length of the machine. As the wash fluid is added to the suspen-sion, it displaces the liquid of the suspension which is driven into the permeabledrums. Considerable amounts of wash liquor can be added to the suspension, re-89Figure 5.1: Detailed schematic of a Twin Roll press, illustrating the flow of apulp suspension, wash nozzles used in the vat region, location of a pres-sure transducer (at the bottom of the vat), and the region of significantdewatering referred to as the nip region.placing the amount of liquid in the suspension several times over [85]. The inflowof suspension and the rotation of the permeable roll transports the pulp suspensionaround the vat, into the region referred to as the nip. The nip, highlighted in greenin Figure 5.1, represents the zone of the machine where the second major operationof the machine occurs: the dewatering of the fibre suspension to high volume frac-tions. The two streams of the suspension meet in the nip and are driven through thegeometrical restriction, imposing considerable loads onto the suspension. Throughthis step, water is evacuated from the suspension into the permeable drums andaway from the suspension.Roll presses come in a variety of designs and sizes, depending on the man-ufacturer and their operational requirements, however generally are constructedwith roll radii and lengths of O(1)m, nip thickness O(10−2)m, and vat heightsof O(10−2)m. Roll surface speeds typically are O(1)m/s, which correspond tocompression rates normal to the roller surfaces of O(10−1)m/s. To fully cap-ture all the dynamics of dewatering in the nip region, a full three-dimensionalevaluation would be necessary. However, the thin-gap geometry of the machine90(with respect to the roll radii) would suggest variations in the machine and cross-machine directions are small in comparison to variations in the through-plane di-rection, which implies the steady-state dewatering should be suitability representedby some reduced order model. Similar geometrical arguments are used in papermachine dewatering studies, which we follow similar forumulations to (examples,[27, 66, 69]). Further, with the synchronous rotation speed of the two rolls, andtherefore equal surface traction provided to the suspension on either side, we couldimagine a fairly low level of shear in the nip region as well. For these two reasons,we speculate that a model can be devised that will resemble the uni-axial modelsand experiments presented thus far, which allows this machine to provide an in-dustrially relevant geometry to evaluate the continued success of our constitutivemodel at dewatering rates approximately an order of magnitude higher than havebeen achieved.The chapter begins with a lengthy, but necessary, description of the model ge-ometry and the pertaining equations, including a two-dimensional stress descrip-tion and the necessary, equivalent formations of the constitutive laws. Followingthis is a section on the materials and methods used. Next, the results of the modeland experiments are provided, along with discussions of reconciliatory attemptsto regain representation. Finally, a discussion reflecting on the steps taken andmotivation for moving forward in the next two chapters is provided.5.1 Model Geometry and EquationsHaving introduced the region of significant dewatering in the roll press, we presentan idealized geometry of the dewatering seen in the nip region, shown in Figure5.2. The dewatering zone occurs from xˆstart ≤ xˆ ≤ 0. Immediately apparent isthe absence of one of the rolls in the geometry. This is possible by acknowledgingthe symmetry of the problem, which dictates that no mass or momentum transportcan occur across zˆ = 0. h0 in this geometry does not set the initial height as in theprevious chapters, rather it is used to define the distance between the rolls and thesymmetric line. Therefore the minimum nip thickness is equivalent to 2 × h0. Theplanar geometry follows from an underlying assumption that no variation along thelength of the roll press is seen. The validity of this is difficult to say at this time.91x^z^x^startz = h(x)^ ^ ^tnRh0^^Figure 5.2: Idealized model geometry for the dewatering seen in the TwinRoll press nip region. The symmetry line is shown at zˆ = 0.The pulp suspension is fed into the geometry from the left. It is assumed it entersthe dewatering region as a uniform plug (both in velocity and concentration), that istranslating purely in the xˆ−direction, at a rate equivalent to the xˆstart−componentof the roll’s surface velocity. Non-slip conditions of both phases is assumed atthe roll’s permeable surface throughout the dewatering zone, which is located atzˆ = hˆ(xˆ). The permeable surface is assumed to retain all of the solid phase, andprovide negligible resistance to the escaping liquid. The pulp suspension exits thedewatering geometry at xˆ = 0, and no re-wetting is assumed. Finally, we assumethe operation of this machine is steady with time, and that body forces are small incomparison to the large compression forces.With the model description and geometry, we present the variants of the gov-erning equations introduced in Chapter 2, which pertain to the dewatering modelledin this chapter. We start with continuities of the two phases as∇ˆ ⋅ (φuˆs) = 0, (5.1)∇ˆ ⋅ ((1 − φ)uˆl) = 0, (5.2)where uˆs,f = (uˆs,f , vˆs,f), being the solid and liquid phase velocities respectively,92each have components in the xˆ−direction and zˆ−direction. For Darcy’s law,(1 − φ)(uˆs − uˆl) = k(φ)µ∇ˆpˆ, (5.3)where k(φ) = (kxx(φ), kzz(φ)). Finally, the stress conservation,∇ˆ ⋅ [−pˆI + Σˆ] = 0, (5.4)where we need a description of the total solid effective stress Σˆ. The boundaryconditions leading from the model description at the symmetric line arevˆs,f = 0 at zˆ = 0, (5.5)∂uˆs,f∂zˆ= 0 at zˆ = 0, (5.6)whereas at the permeable surface arepˆ = 0 at zˆ = hˆ(xˆ), (5.7)uˆs ⋅ nˆ = 0 at zˆ = hˆ(xˆ), (5.8)uˆs,f ⋅ tˆ = Uˆ ⋅ tˆ at zˆ = hˆ(xˆ), (5.9)where tˆ and nˆ are tangential and normal unit vectors that follow the roll surface,and Uˆ is the surface velocity of the spinning roll. hˆ(xˆ) can be found from Figure5.2 ashˆ(xˆ) = h0 +R −√R2 − xˆ2, (5.10)which can be used to define tˆ, nˆ, and Uˆ in the xˆ − zˆ coordinate system astˆ ≡ (tˆx, tˆz) = ⎛⎝1 + (dhˆdxˆ)2⎞⎠−1/2 (1, dhˆdxˆ) , (5.11)nˆ ≡ (nˆx, nˆz) = ⎛⎝1 + (dhˆdxˆ)2⎞⎠−1/2 (−dhˆdxˆ,1) , (5.12)Uˆ ≡ (Uˆ , Vˆ ) = (Ω (R + h0 − hˆ(xˆ)) ,Ωxˆ) . (5.13)93Finally, the initial conditions are defined asφ = φstart at xˆ = xˆstart, (5.14)uˆs,f = (Ω (R + h0 − hˆ(xˆstart)) ,0) at xˆ = xˆstart. (5.15)5.1.1 2D Stress State and Constitutive ModelTo implement the modelling framework, we need a definition of the total effectivestress and appropriate constitutive expressions. As discussed, the geometry of theroll press dewatering is not expected to accommodate a significant amount of shear.However, in an effort to provide a more general formulation, we do entertain thepossibility that the pulp suspension may experience a more general solid effectivestress state than the isotropic only state investigated in the previous chapters. Wetherefore define the total solid effective stress as,Σˆ = −Pˆ I + τˆ , (5.16)where Pˆ and τˆ represent isotropic and deviatoric solid effective stresses, respec-tively. We note that although Pˆ is similar to Pˆ in the previous chapters, it is notprecisely equal, as will be presented shortly.Constitutive models for the components of the total effective stress are re-quired. Our intent is to formulate viscoplastic representations for both the isotropicand deviatoric solid effective stresses, having seen the importance of the bulk vis-cosities in previous chapters. We start, however, with the plastic components only.As in previous chapters, we assume there are no elastic components to the stress,and in both compression and shear that a maximum yield stress exists that mustbe overcome for yielding to occur. The most basic yield criterion relating thesetwo yield stresses, as seen in other two-dimensional frameworks [35, 87], is thefollowing elliptic yield expressionτˆ2τy(φ)2 + Pˆ 2Ny(φ)2 = 1, (5.17)where τˆ2 = ∑i∑jτˆij τˆij/2 is the second invariant of the deviatoric stress tensor, and94τy(φ) andNy(φ) represent the deviatoric and isotropic yield stresses, respectively.We start by rearranging this criteria to define a pressure dependent shear yield stressTy(Pˆ , φ), asTy(Pˆ , φ) ≡ τˆ = τy(φ)¿ÁÁÀ1 − Pˆ 2Ny(φ)2 . (5.18)With this, we can define a yield surfaceFy = τˆ − Ty(Pˆ , φ) = 0. (5.19)In the plastic deformation literature (example [35]), this yield surface will be usedin conjunction with a specified flow rule to define the constitutive model, such asγ˙ij = λ˙ ∂Fy∂Σˆij≡ λ˙( τˆij2τˆ+ ∂Ty∂Pˆδij2) , where λ˙ > 0, (5.20)where λ˙ is a Lagrange multiplier that enforces the constraint Fy = 0, δij is theKronecker delta, and γ˙ is the solid phase strain rate tensor. This strain rate tensor,in turn, can be decomposed into the sum of a deviatoric and isotropic strain-ratetensor asγ˙ = Dˆ + (∇ˆ ⋅ uˆs)I = [∇ˆuˆs + ∇ˆuˆTs − (∇ˆ ⋅ uˆs)I] + (∇ˆ ⋅ uˆs)I. (5.21)A more convenient form of the constitutive model can be arrived at throughfirst defining the second invariant of Equation 5.20 asγ˙2 = λ˙2 (14+ (∂Ty∂Pˆ)2 14) , (5.22)where γ˙2 = ∑i∑jγ˙ij γ˙ij/2. This expression, along with Equation 5.19, can be usedto eliminate λ˙ in 5.20, arriving atτˆijTy= γ˙ijγ˙¿ÁÁÀ1 + (∂Ty∂Pˆ)2 − ∂Ty∂Pˆδij . (5.23)An expression for the isotropic solid effective stress can lead from this equation as95well, by defining the trace of Equation 5.23 to zero. Through various manipula-tions, one then can arrive at the following compact forms of the plastic componentsof the isotropic and deviatoric stresses asPˆ = −Ny(φ)2(∇ˆ ⋅ uˆs)√τy(φ)2Dˆ2 +Ny(φ)2(∇ˆ ⋅ uˆs)2 , (5.24)andτˆ = τy(φ)2Dˆ√τy(φ)2Dˆ2 +Ny(φ)2(∇ˆ ⋅ uˆs)2 , (5.25)where Dˆ2 = ∑i∑jDˆijDˆij/2. To include the viscous contributions to each, we sim-ply append contributions of an extensional bulk viscosity, µˆe, and a shear bulkviscosity µˆs arriving at the following viscoplastic constitutive modelsPˆ = −Ny(φ)2(∇ˆ ⋅ uˆs)√τy(φ)2Dˆ2 +Ny(φ)2(∇ˆ ⋅ uˆs)2 − µˆe(φ)(∇ ⋅ uˆs), (5.26)andτˆ = τy(φ)2Dˆ√τy(φ)2Dˆ2 +Ny(φ)2(∇ˆ ⋅ uˆs)2 + µˆs(φ)Dˆ. (5.27)Applied in Uni-Axial GeometryIn order to relate the stresses and material parameters from these constitutive mod-els to those of the previous chapters, we will apply these multi-dimensional con-stitutive expressions to the uni-axial dewatering experiment. With the simple one-dimensional velocity, uˆs = (0, vˆs(zˆ)), we find the deviatoric strain rate tensor asDˆ = ⎡⎢⎢⎢⎢⎢⎢⎣−∂vˆs∂zˆ 00 ∂vˆs∂zˆ⎤⎥⎥⎥⎥⎥⎥⎦ , (5.28)leading to Dˆ2 = (∂vˆs∂zˆ )2 and ∇ˆ ⋅ uˆs = ∂vˆs∂zˆ. (5.29)96With that, we write out the xˆ− and zˆ− components of Equation 5.4, with no bodyforces as∂∂xˆ(−pˆ − Pˆ + τˆxx) + ∂∂zˆ(τˆxz) = 0, (5.30)∂∂xˆ(τˆzx) + ∂∂zˆ(−pˆ − Pˆ + τˆzz) = 0, (5.31)where τˆxz = τˆzx = 0. Defined for compression (where ∂vˆs∂zˆ < 0), we haveτˆxx = τy(φ)2√τy(φ)2 +Ny(φ)2 − µˆs(φ)∂vˆs∂zˆ , (5.32)τˆzz = −τy(φ)2√τy(φ)2 +Ny(φ)2 + µˆs(φ)∂vˆs∂zˆ , (5.33)andPˆ = Ny(φ)2√τy(φ)2 +Ny(φ)2 − µˆe(φ)∂vˆs∂zˆ . (5.34)These definitions can be substituted into Equation 5.31, arriving at∂∂zˆ(−pˆ − (√τy(φ)2 +Ny(φ)2 − (µˆe(φ) + µˆs(φ)) ∂vˆs∂zˆ)) = 0. (5.35)This expression provides an equivalence to the solid effective stress Pˆ of the pre-vious chapters asPˆ ≡ √τy(φ)2 +Ny(φ)2 − (µˆe(φ) + µˆs(φ)) ∂vˆs∂zˆ. (5.36)This also provides the bridges for Py(φ) and Λ(φ) with respect to the materialparameters of this 2D stress state:Py(φ) ≡ √τy(φ)2 +Ny(φ)2, (5.37)Λ(φ) ≡ µˆe(φ) + µˆs(φ). (5.38)In this chapter, we will continue the investigate the suitability of the suggested formof Λ(φ) = η∗φ2.975.1.2 Non-Dimensionalized, Leading Order ModelTo exploit the favourable geometry of the nip dewatering, we start by non-dimensionalizing the problem. The following scalings are implemented:xˆ = √h0Rx, zˆ = h0z, hˆ = h0h, Dˆ = ΩRh0D, (5.39)uˆs,f = (ΩRus,f ,Ω√h0Rvs,f) , Uˆ = (ΩRU,Ω√h0RV ) , (5.40)pˆ = p∗p, Pˆ = p∗P, τˆ = p∗τ , σˆ = p∗σ, (5.41)k(φ) = k∗K(φ), Ny(φ) = p∗Ny(φ), τy(φ) = p∗Sy(φ), (5.42)µˆs = p∗h0ΩRµs, and µˆe = p∗h0ΩRµe, (5.43)where R and Ω are the roll radii and rotational rates, respectively. As in previ-ous chapters, the stresses are scaled by p∗, being the value of the fitted Py(φ) ≡√Ny(φ)2 + τy(φ)2 function defined at φ = 0.10, and the permeability is scaledby the permeability fitted function, which would correspond to kzz(φ), again atφ = 0.10. With these scalings, Equations 5.1 - 5.4 are∂∂x(φus) + ∂∂z(φvs) = 0, (5.44)∂∂x[(1 − φ)uf ] + ∂∂z[(1 − φ)vf ] = 0, (5.45)(1 − φ)(us − uf) = γδ2Kxx(φ)∂p∂x, (5.46)(1 − φ)(vs − vf) = γKzz(φ)∂p∂z, (5.47)δ∂∂x(−p − P + τxx) + ∂τzx∂z= 0, (5.48)δ∂τxz∂x+ ∂∂z(−p − P + τzz) = 0, (5.49)98in which two dimensionless constants appear, which are defined asγ = p∗k∗µΩ√h0Rh0, δ = √h0R≪ 1. (5.50)As in Chapter 3, γ is the ratio of compressive strength of the solid network to theviscous drag force from the fluid. δ is a small parameter that represents the slen-derness of the geometry. The simplification of the equations with respect the theslender geometry comes from the consideration of these scaled equations at leadingorder, with respect to δ. This means cancellations in the x−direction of the Darcyexpression in Equation 5.46, suggesting no relative motion between the phases inthis direction, and simplifications in the two stress conservations in Equations 5.48and 5.49 where δ is seen. Interestingly, we still see shear at leading order in thex−direction.We next return to the boundary conditions, beginning with the symmetric lineconditions asvs,f = 0 at z = 0, (5.51)∂us,f∂z= 0 at z = 0, (5.52)and at the permeable roll surface, beingp = 0 at z = h(x), (5.53)usnx + δvsnz = 0 at z = h(x), (5.54)us,f tx + δvs,f tz = Utx + δV tz at z = h(x), (5.55)whereh(x) = 1 + δ−2 − δ−2√1 − δ2x2, (5.56)99and the tangent, normal vectors, and the roller surface velocity can be expressed ast ≡ (tx, tz) = (1 + δ2 (dhdx)2)−1/2 (1, δ dhdx) , (5.57)n ≡ (nx, nz) = (1 + δ2 (dhdx)2)−1/2 (−δ dhdx,1) , (5.58)U = (1 + δ2 − δ2h(x), x) . (5.59)Equations 5.53 - 5.59 are sufficient for defining the boundary conditions atz = h(x), however, a further simplification can be made through Taylor seriesexpansions in δ of h(x), t, and n. This provides the following approximations forthe terms ash(x) = 1 + x22+O(δ2), (5.60)t = (1 − δ2x22+O(δ3), δx +O(δ3)) , (5.61)n = (−δx +O(δ3),1 − δ2x22+O(δ3)) , (5.62)which can then be used to write the boundary conditions at z = h(x) asδ(−xus + vs) = 0 at z = h, (5.63)us,f + δ2(−x2us,f2+ xvs,f) = 1 at z = h. (5.64)These expressions, at leading order, simplify tous,f = 1 at z = h(x), (5.65)vs = x at z = h(x). (5.66)100To close the set of equations, we need our scaled constitutive modelsP = −δNy(φ)2 (∇ ⋅ us)√Sy(φ)2D2 + δ2Ny(φ)2 (∇ ⋅ us)2 − δµe(φ) (∇ ⋅ us) , (5.67)τij = S2yDij√Sy(φ)2D2 + δ2Ny(φ)2 (∇ ⋅ us)2 + µs(φ)Dij , (5.68)where the dimensionless divergence of the velocity and the deviatoric stress tensorare given by ∇ ⋅ us = ∂us∂x+ ∂vs∂z, (5.69)D = ⎡⎢⎢⎢⎢⎢⎢⎣δ (∂us∂x − ∂vs∂z ) ∂us∂z + δ2 ∂vs∂x∂us∂z + δ2 ∂vs∂x δ (∂vs∂z − ∂us∂x )⎤⎥⎥⎥⎥⎥⎥⎦ . (5.70)We leave simplification of Equations 5.67-5.70 with respect to δ for now.5.1.3 Reduction: A Uni-Axial-Like ModelAs mentioned earlier in the chapter, we are keen to recover a model resemblingthat of the uni-axial experiments in the previous chapters and as found in relevantpaper machine dewatering studies [27, 66, 69]. At leading order with respect to δ,the equations are certainly getting close, however Equation 5.48 and the definitionof τzx are causing complications. At leading order, we have τzx defined asτzx = Sy(φ) + µs(φ) (∂us∂z) , (5.71)which can be substituted into Equation 5.48 as∂∂z(Sy(φ) + µs(φ) (∂us∂z)) = 0. (5.72)As discussed earlier, we expect that τzx is small in the nip due to the synchro-nized motion of the permeable rolls, which should amount to a plug flow in themachine direction (i.e. ∂us∂z ≈ 0). One way to impose this, with our definition ofτzx, is to assume Sy(φ) is O(δ), which amounts to an assumption that the suspen-101sion does not have significant strength in shear (in comparison to compression).The validity of this assumption is unclear at this point, having found no results inthe literature for pulp suspensions at the volume fractions of interest. For otherflocculated suspensions, with examples including latex and silica [12], titania andagregated aluminua [15, 106], this assumption is appropriate with at least an orderof magnitude difference between the shear and compressive yield stresses, acrossthe volume fractions measured. To demonstrate this assumption’s effect, we holdoff imposing it at first, and integrate Equation 5.72 with respect to the boundarycondition shown in Equation 5.52 at z = 0, arriving atµs∂us∂z= [Sy(φ(z = 0)) − Sy(φ)] . (5.73)If Sy(φ) = O(δ), then we see that us has no z dependency, and so us = B(x).From the boundary condition shown in Equation 5.65 at z = h at leading order wefind B(x) = 1, thereforeus = 1. (5.74)Finally from Equation 5.46, at leading order, we find uf = 1 as well, and so wearrive at our desired plug flow of the suspension in the machine direction.With the assumption of shear yield stress being small, we can also simplifyour constitutive expressions. Starting with P , we cancel out the Sy(φ) and thegradients of us, and for compression (∂vs∂z < 0) we arrive atP = Ny(φ) − δµe(φ) (∂vs∂z) . (5.75)In spite of the δ factor, we retain the viscous terms for the time being. From thisexpression, and from Equation 5.37 with Sy(φ) being negligible, we findNy(φ) orNy(φ) being equivalent to Py(φ) or Πy(φ) of previous chapters respectively. Thisconveniently allows direct insertion of the calibrations of compressive yield stressprovided in Chapter 3. We make this substitution for the remaining equations.The final constitutive expression, τzz , is simplified by cancelling the Sy(φ) andgradients of us,τzz = δµs(φ) (∂vs∂z) , (5.76)102again retaining the viscous term despite the δ factor.We next turn back to our governing equations. Since us,f = 1, the continuityexpressions are simplified to∂φ∂x+ ∂∂z(φvs) = 0, (5.77)−∂φ∂x+ ∂∂z[(1 − φ)vf ] = 0. (5.78)The remaining reductions follow the approach performed in the previous chapters.We combine and integrate the continuity expressions, and with the help of the vs,fboundary condition at z = 0, we arrive at(1 − φ)(vs − vf) = vs. (5.79)This expression can be plugged into the z−direction Darcy expression shown inEquation 5.47, along with Equations 5.75 and 5.76 to furnishvs = −γ ∂Πy∂φK(φ)∂φ∂z+ γK(φ) ∂∂z[δ(µe + µs)∂vs∂z] , (5.80)where K(φ) ≡ Kzz(φ). From Equation 5.49, we also can obtain the load ex-pression on the roll surface, σ(x), at leading order for which we can compare toexperimental results, asσ(x) = Πy(φ) − δ (µe(φ) + µs(φ)) ∂vs∂z∣z=h(x) . (5.81)Finally, we also find that with the scaling and Equation 5.38, we can showδ (µe + µs) = δη∗ΩRp∗h0 φ2 ≡ γ φ2, where  = η∗k∗µh20 . (5.82)In summary, a two-dimensional framework has been applied to the geometry ofthe nip of the roll press at leading order δ. With an assumption that the suspensionhas negligible strength in shear Sy(φ) = O(δ), we can recover a set of equations103equivalent to the uni-axial experiments of the previous chapters, as∂φ∂x+ ∂∂z(φvs) = 0, (5.83)vs = −γ ∂Πy∂φK(φ)∂φ∂z+ K(φ) ∂∂z[φ2∂vs∂z] , (5.84)where the spatial variable x takes the place of the temporal term of previous chap-ters. The boundary and initial conditions arevs = x at z = h, (5.85)vs = 0 at z = 0, (5.86)φ(xstart, z) = φstart, (5.87)and the load on the permeable roll is found asσ(x) = Πy(φ) − γφ2∂vs∂z∣z=h(x) . (5.88)A similar scheme to that described in Section 3.2 is used to solve the model.5.2 Materials and MethodsA Scandinavian softwood Kraft pulp was used for the evaluation of representationof the presented model. This is the same pulp as Series 17 in Chapter 3. Its perme-ability, compressive yield stress, and bulk viscosity therefore have been calibratedthrough the experiments described in Chapter 3, with the fitted functional formsk(φ) = 2.67−13φln( 1φ) e−20.38φ [m2], (5.89)Py(φ) = 6.20 × 105 φ1.87(1 − φ)3.83 [Pa], (5.90)Λ(φ) = 2.86 × 107 φ2 [Pa ⋅ s]. (5.91)The experimental results, along with their corresponding fits, are shown in Figure5.3.Experimental roll press results for comparison to the model were collected at104(a)0 0.1 0.2 0.3 0.410-1510-10(b)0 0.1 0.2 0.3 0.4 0.5051015 105Figure 5.3: Material parameter results for Series 17 of Chapter 3. In (a), se-lect permeability points are shown along with the fitted functional formof Equation 5.89. In (b), the symbols represent select points along theaveraged continuous trend, the error bars represent two standard devia-tions, and the blue line represents the fitted functional form in Equation5.90.105Valmet’s pilot-plant Twin Roll press facility, in Sundsvall Sweden, shown in Fig-ure 5.4a. A suspension of pulp is prepared in a mixing chest to a given consistency(solid mass fraction), which is pumped to the roll press, and distributed to bothrolls at controllable volume fractions. Upon exiting the roll press, the fibre suspen-sion is re-diluted (with the filtrate from the roll press) and returned to the mixingchest. The wash fluid volumetric flow rates are controllable for either side of thevat. The rolls themselves measure 0.75m in radii, and 0.80m in length, which arefull size in radius and the shortest length of rolls Valmet provides to the industry.The rolls are housed in a vat with a height of 40mm, and their rotation rate iscontrollable. For this geometry, the location of xˆstart = −0.233m, which corre-sponds to the merging of the two vats. The pilot plant has one fixed location roll,and the other can move radially towards the second roll (in the model geometry, itwould be zˆ−direction) to vary the nip thickness from 7 to 20mm. The translationof only one roll does introduce slight asymmetry, however the effect is assumed tobe negligible. Measurements from the instrumented roll press include consistencyor solid mass fraction (Cin) and flow rates of input pulp suspension (Qa and Qb),flow rates of wash fluid (Wa andWb) and fluid pressure at the bottom of the vat (Paand Pb), solid in filtrate (FF ), outlet consistency (Cout), suspension temperature(T ), and torque and line load experienced by the rolls (Torquea, Torqueb, and LLrespectively). A schematic of the facility’s layout is shown in Figure 5.4b, and theexperimentally measured results can be found in Table D.1.5.2.1 Finding φstartDespite knowing the pulp suspension’s consistency and volumetric flow rates intothe roll press, and therefore able to solve the volume fraction entering, this is notnecessarily equivalent to φstart. This is due to the pulp suspension’s unknownliquid content at the beginning of the nip, being first impinged upon the permeablesurface of the rolls at the inlet to the vat, and second, having large amounts of liquidadded in the vat to rinse the suspension. Making it more complicated still, samplingor even viewing of the suspension at the entrance of the nip is not possible due tothe construction of the machine. We therefore need to approximate φstart. Tworeasonable methods of approximating this volume fraction have been implemented,106(a) (b)𝑊𝑎𝑠ℎ𝐹𝑙𝑢𝑖𝑑𝐹𝑖𝑙𝑡𝑟𝑎𝑡𝑒𝐼𝑛𝑙𝑒𝑡 𝑃𝑢𝑙𝑝𝑂𝑢𝑡𝑙𝑒𝑡 𝑃𝑢𝑙𝑝Figure 5.4: A picture of the pilot plant Twin Roll press in Valmet’s Sundsvallfacility is shown in (a). In (b), a basic layout of the facility is provided.which are shown schematically in Figure 5.5. Both approaches are based uponmass balances of the steady state operation of the roll press.We start with the first approach, shown schematically in Figure 5.5a, whichutilizes the geometric change seen through the nip and the volume fraction exitingthe machine asφstart,geo. = φexit hniphinlet. (5.92)To define φexit from the exit consistency, an overall mass conservation of the solidand liquid phases is required. A simplistic representation of the two zones of theroll press, along with the inlets and outlets, is represented by the schematic in Fig-ure 5.6. The relevant control volume used is shown as the the outer grey boundary.The resulting balances arem˙l,1 + m˙l,4 = m˙l,2 + m˙l,3, (5.93)m˙s,1 = m˙s,2 + m˙s,3, (5.94)107(a)Ω𝑅Ω𝑅ℎ𝑖𝑛𝑙𝑒𝑡 ℎ𝑛𝑖𝑝(b)Ω𝑅Ω𝑅𝜙𝑠𝑡𝑎𝑟𝑡Figure 5.5: Two approaches for approximating φstart. In (a) we use the ge-ometry change and φexit determined to find initial solid volume fraction.In (b), we assume a volumetric flow rate into the nip such that a plugflow occurs spanning the width of the vat, moving at the roll surfacespeed. Using mass balances we find the necessary solid to liquid ratioto make this happen.where m˙l,1 and m˙s,1 are the mass flow rates into the Twin Roll press (defined fromfeed consistency, Cin, and total suspension flow rate, Qin,a + Qin,b), m˙l,4 is themass flow rate of wash fluid in (defined from Qwash,a +Qwash,b), m˙l,2 and m˙s,2are the mass flow rates of the filtrate stream, and finally m˙l,3 and m˙s,3 are the massflow rates out of the Twin Roll press. Using the exit consistency, Cout, and inthe trials for which we have the fibre-in-filtrate amount, FF , we can arrange thetwo equations to solve for the two unknown values m˙l,2 and m˙l,3, which then canbe used to define m˙s,2 and m˙s,3. For trials without a fibre-in-filtrate amount, weassume FF = 0. With that, we can define φexit asφexit = m˙s,3/ρsm˙s,3/ρs + m˙l,3/ρf . (5.95)The resulting values of φstart,geo. can be found in Table D.2. This approach hasan important assumption that the volume fraction remains unchanged from hnip(or xˆ = 0) to outside the machine where the consistency is measured. Althoughthis may seem reasonable, it assumes no rewetting of the suspension, a knownconcern with roll presses and other roller-type compression machines. Due to thegeometrical expansion on the back-side of the nip (small positive xˆ values), and108ሶ𝑚𝑙,1ሶ𝑚𝑙,2ሶ𝑚𝑙,4ሶ𝑚𝑙,3ሶ𝑚𝑠,1ሶ𝑚𝑠,2ሶ𝑚𝑠,3𝑴𝒂𝒄𝒉𝒊𝒏𝒆 𝑰𝒏𝒍𝒆𝒕 𝑴𝒂𝒄𝒉𝒊𝒏𝒆 𝑶𝒖𝒕𝒍𝒆𝒕𝑭𝒊𝒍𝒕𝒓𝒂𝒕𝒆𝑾𝒂𝒔𝒉 𝑭𝒍𝒖𝒊𝒅ሶ𝑚𝑙,5 ሶ𝑚𝑠,5ሶ𝑚𝑙,6ሶ𝑚𝑠,6𝑫𝒓𝒂𝒊𝒏𝒊𝒏𝒈𝑷𝒍𝒖𝒈 𝑭𝒍𝒐𝒘,𝑵𝒊𝒑 𝑰𝒏𝒍𝒆𝒕Figure 5.6: Simplified schematic of the Twin Roll press from a mass balanceperspective. The dashed line represents the permeable boundary of rolls.the abundance of water in the vicinity (being on the other side of the permeablesurface), inevitably a small amount of water will re-enter the suspension, and re-duce its solid volume fraction. This approximation therefore represents a low-endvalue for φstart.A second approximation, which builds upon the solutions of the first massbalance, uses an assumption that the suspension in the vat, just prior to the nip, isflowing as a plug at the speed of the roll surface. This requires an additional massbalance of the inner control volume, shown in red in Figure 5.6, to define φstartsuch that the phases’ fluxes at the beginning of the nip are sufficient to create theplug flow. This approach is shown schematically in Figure 5.5b. The inner massbalances, and the additional plug flow constraint are as followsm˙l,1 + m˙l,4 = m˙l,5 + m˙l,6, (5.96)m˙s,1 = m˙s,5 + m˙s,6, (5.97)Q6 = m˙l,6ρl+ m˙s,6ρs= ΩR(2A), (5.98)where A is the area between the roll and the trough, doubled for the two streams.109This inner control volume, which ends at the beginning of the nip, represents thewash section of the roll press. We assume all of the solids lost in the filtrate are lostfrom this inner control volume, i.e. m˙s,5 = m˙s,2. With this final assumption, we cansolve the inner control volume’s mass balances and get our second approximationfor the volume fraction at the start of the nip asφstart,plug = m˙s,6/ρsQ6. (5.99)The resulting values can also be found in Table D.2. This approach ends up rep-resenting an upper end approximation for φstart and therefore can accommodaterewetting.5.3 Results and DiscussionFifty seven data points were collected using the Twin Roll press pilot facility overthree days of trials. All of the experimental results are provided in Table D.1. Thecalculated φstart values, as described in the previous section, are provided in TableD.2. We begin with the model results for a particular sample Trial 103, shown inFigure 5.7. In Figure 5.7a, we show the model’s compressive load as a function ofxˆ for φstart,geo. = 0.020, shown as the red curve, and for φstart,plug = 0.026, shownas the blue curve. The hˆ(xˆ) trend is also shown in the figure, with its axis on theright. It is quite apparent that the two model solutions look equivalent if scaled bythe maximum compressive load. The similarities between the two results continueinto Figure 5.7b and 5.7c as well, with a modest structure in the volume fractionprofiles, and nearly linear velocity profiles throughout the dewatering.Comparing the compressive loads again, we see significant compression loadsbeing suggested by the model, exceeding 5MPa. This is much higher than ex-pected, and corresponds to model line loads of 571N/mm and 1010N/mm, re-spectively for the two values of φstart, which can be found by the following ex-110-0.25 -0.2 -0.15 -0.1 -0.05 0051015 10600.020.040.060 0.1 0.2 0.300.010.020.030.04-0.4 -0.3 -0.2 -0.1 000.010.020.030.04Figure 5.7: Model results are shown for Trial 103, with the results usingφstart,geo. shown as the red trends, and φstart,plug shown as the bluetrends. In (a), the model load curves through the compression are shownalong with the position of the permeable roll, hˆ(xˆ), shown as the greencurve. In (b) and (c), select volume fraction and solid phase velocityprofiles are shown through the compaction, with the black arrows indi-cating increasing xˆ or ‘time’.111pressionLLModel [N/mm] = ∫ 0xˆstartσˆ(xˆ)dxˆ≡ p∗√h0R∫ 0xstart[Πy(φ) − γφ2∂vs∂z]∣z=h(x) dx.(5.100)Unfortunately, this integrated load is the only compression metric available fromthe experimental data, and these model predictions far exceed the experimentallycollected value of 78N/mm. Further, this is not isolated to this particular trial.Shown in Figure 5.8 are all the line load predictions from the model for the datacollection performed, with φstart,geo. shown as the red squares and φstart,plug asblue squares. The targeted 1:1 ratio is shown in black. Trial 103 results are high-lighted in green. Regardless of the chosen φstart, the model is consistently overpredicting the line load measured on the roll press. In addition, there is significantvariation in line load predicted from the model, with the rather modest changes ofφstart predictions.The model’s poor performance motivated several investigations to improve rep-resentation. The details of those are to follow, however, first we acknowledge theimportance of the inclusion of the bulk viscosity, as without it, the dewateringmodel is unable to be solved, with φ(z) exceeding 1. We therefore have obtainedimproved representation with the inclusion of Λ(φ), however, we would like toimprove it further.To help guide our investigation, we return to the compressive load shown inEquation 5.88. Having seen the model provide near uniform velocity profiles, andonly marginal gradients in the solid volume fraction, we can approximate the vol-ume fraction profile and gradient in the solid phase velocity asφ(z) ≈ φstarthstarthˆ(xˆ) and ∂vs∂z ∣z=h(x) ≈ xh(x) . (5.101)With these solutions, we can straight forwardly solve the two components of Equa-tion 5.88 for a uniformly dewatering suspension. Re-shown in Figure 5.9 is thefull model load curve of Figure 5.7a, using φstart,plug (shown in blue). Addition-ally shown are the uniformly compacting first and second terms of Equation 5.88,112100 150 200 250102103104105Figure 5.8: Model versus experimental line loads for two approximations ofφstart. The blue squares represent the results with φstart,plug and thered squares represent the results with φstart,geo.. Trial 103 points arehighlighted with green, and Trial 601 are highlighted with cyan. Theblack line shows the targeted 1:1 ratio.shown as the black dot-dashed and black dashed lines, respectively. Two valuableinsights are provided by this plot. First, it is clear that a uniformly compactingmodel provides nearly an equivalent representation to that of the full model (withthe summation of the two black trends nearly equal to the blue). This indicates thatthe differential compaction that is provided by the full model is negligibly small(with respect to the suggested compressive load on the surface), and therefore abulk behaviour of the suspension must be responsible for the high loads predicted.Second, it clearly indicates the viscous contribution to the load on the roll surfacefar out weighs the strength component. Both of these observations are useful tohave in mind moving forward.113-0.25 -0.2 -0.15 -0.1 -0.05 0051015 106Figure 5.9: Trial 103 compression load result with φstart,plug re-shown asthe blue trend. Additionally shown are the two components of Equation5.88, assuming uniform compaction as per Equations 5.101. The firstcomponent, corresponding to the strength is shown as the black dot-dashline, and the viscous contribution is shown as the black dashed line.5.3.1 Uncertainty in φstartAs mentioned previously, the pulp suspension immediately before the nip dewater-ing region is inaccessible to sample, and therefore approximations of φstart werenecessary. Having seen substantial differences in the predicted line load for themodest variation in φstart,plug and φstart,geo., we first investigated the values ofφstart that provided a model prediction equal to the experimental measurements.Representative trial points’ results are shown in Figure 5.10. Shown in this plotare φstart,geo. and φstart,plug, shown as the bounds of the red error bars, and thedetermined value that provides the experimental value of line load.Unfortunately, these results suggest the sole culprit of the significant over pre-diction of the line load cannot be simply a misidentified φstart. As mentionedpreviously, φstart,geo. bounds the lowest possible entrance φ while satisfying themass balance by assuming zero rewetting. Perhaps if the required φstart to ob-tain experimental line load was sufficiently close to φstart,geo., we could entertaina consistent experimental error with the data collection or measurement of outletconsistency, however the span demonstrated here is not that case.114Figure 5.10: Investigation into uncertainty in φstart. Seven representativetrial results are shown. The bounds on the right show φstart,geo. −φstart,plug. The green filled squares on the left show the required φstartto hit the experimentally determined line load.5.3.2 Temperature Correction: Λ(φ)Presently, the scaling of Λ(φ), being η∗, has been directly implemented into thisproblem having found it as the ideal value in Chapter 3. As has been demonstrated,a substantial proportion of the load on the permeable drum is coming from the bulkviscosity term. As suggested in the previous chapters, the source of the pulp sus-pension’s bulk viscosity is attributed to flow of the liquid through small restrictionsin the pulp suspension, and therefore proportional to the fluid viscosity. Perhapsthen, due to the elevated suspension temperatures that occur in the roll press (seeTable D.1), we should be correcting η∗ for the difference in fluid viscosity, i.e.ηTC∗ = µµ20○C η∗, (5.102)which may help counteract the largeness of the calculated line load. The modelline load results with ηTC∗ are shown for our two estimates for φstart in Figure 5.11.First comparing the results for Trial 103, which are again highlighted in green, weobserve a lowering of the model predicted line loads for both initial φ estimates.115100 150 200 250102103104105Figure 5.11: Model versus experimental line loads for two approximationsfor φstart, with a temperature corrected bulk viscosity, ηTC∗ . Blue andred open squares represent the mass balanced and geometrically deter-mined φstart values, respectively. Trials 103 and 601 are highlightedwith green and cyan, respectively, and the ideal 1:1 ratio is shown asthe black line.Particularly with φstart,geo. (red symbols), we do seem to have several trial resultsthat are approaching the 1:1 ratio. However, there is an increase in scatter betweenboth groups of points. A representative example of this is Trial 601, highlightedwith cyan, which has a drastically larger spread in the two model estimates whencompared to the results with η∗ found from Chapter 3 shown in Figure 5.8.To get a better look of the source of this increased instability in the model so-lutions with respect to φstart, we investigate the results for Trial 601 with bothbulk viscosity scalings. The results are shown in Figure 5.12. In 5.12a, the redtrends correspond to φstart,geo., with the solid and dashed lines representing η∗and ηTC∗ respectively. It is clear in this figure the temperature corrected bulk vis-cosity will result in a lower line load, and therefore a lower bulk viscosity can beeffective in reducing the model predicted line load. However, as has been shown116in the previous chapters, the effect of the bulk viscosity is to reduce gradients inφ during dewatering events and so caution must be taken to not reduce the scalingexcessively. The implication of an insufficient bulk viscosity to suppress large gra-dients in φ can be observed in Figure 5.12a, with the blue trends that correspondto φstart,plug. We see with the dashed line trend, again representing ηTC∗ , that theload at first is lower than the solid line trend, representing η∗. However, as thedewatering gets into its late stages, the smallness of the bulk viscosity is unable toprevent a large gradient in φ near the permeable roll, which results in a divergenceof the load. This large gradient is evident when comparing the blue trends in Figure5.12b and Figure 5.12c, corresponding to η∗ and ηTC∗ respectively.Temperature correction to the bulk viscosity seems plausible, and in some casesreduces the model predicted line load, however it comes with the large drawbackof making the model far less robust with variations in φstart.5.3.3 Wash SectionNot having seen significant success with reducing the viscous contribution term inEquation 5.100, we consider if the neglect of the wash function of the Twin Rollpress is a culprit in the high model line load predictions. Again, due to the con-struction of the machine, it is difficult to know how the suspension responds in thisregion. We hypothesize, then, that the wash fluid evacuating through the perme-able surface of the roll as the suspension travels around the vat is in fact inducingcompaction of the pulp suspension against the permeable roll, and a clear layerdevelops on the lower wall of the vat. We speculate that this impacts the line loaddue to the compact plugs from both streams not coming into contact until furtherinto the nip, and thereby delaying the network compaction and thus reducing themodel’s line load. A schematic of the speculated behaviour is shown in Figure5.13a. Due to the speculative nature, a formal model of the wash section was notundertaken at this time, however a crude representation is proposed as follows.The goal is a simplistic representation that can provide insight into whether this isa worthy direction to investigate.The crude wash press representation is based upon the pressure measured atthe bottom of the vat. Assuming body and inertial forces are negligible, we can117-0.25 -0.2 -0.15 -0.1 -0.05 00123 1070 0.2 0.400.020.040.060 0.2 0.4 0.600.020.040.06Figure 5.12: Model results for Trial 601 used for investigating a temperature-corrected value of η∗, being ηTC∗ . The various blue and red trendscorrespond to φstart,geo. and φstart,plug, respectively. In (a), the solidand dashed lines correspond to a bulk viscosity scalings of η∗ and ηTC∗ .In (b), the volume fraction profiles for η∗ are provided, and in (c) thevolume fraction profiles for ηTC∗ are shown.118(a)(b)Ω𝑅𝜙𝑠𝑡𝑎𝑟𝑡ො𝑥𝑐𝑙𝑒𝑎𝑟Figure 5.13: Schematics corresponding to a basic wash section model. In(a), a wash section behaviour is suggested whereby a compacted cakebuilds upon the permeable roll due to the permeating wash fluid exitinginwards into the roll, and a clear layer forms on the wall of the vat.In (b), the assumed wash section’s corresponding initial φstart,washprofile is shown, which does not meet and begin dewatering until afterxˆclear.apply a total stress conservation in the normal direction, nˆ, as∂∂nˆ(−pˆ − Pˆ) = 0, (5.103)where we have adopted the stress notation of previous chapters. Integrating thisand assuming the pulp suspension has compacted against the roller surface, we getthe total stress is equal to the vat fluid pressure, and therefore if negligible pressuredrop across the permeable surface occurs, then we find the solid effective stress atthe permeable surface must balance the vat fluid pressurePˆ ∣hˆ(xˆ) = pˆvat. (5.104)Assuming the compaction in the vat has sufficient time to equilibrate, the viscouscontributions to Pˆ disappear, and we can find the φ at the roll surface simply fromour Py(φ) function. In reality, our study in Chapter 4 would suggest a structuredprofile of φ would occur, varying from φ at the inlet of the machine to φ definedat the surface. However for simplicity we assume the entire plug is at the roller119surface volume fraction. We therefore determine φstart,wash asPy(φstart,wash) = pˆvat, (5.105)using Equation 5.90. Since the pulp has packed against the roller surface, as shownin Figure 5.13b, the wash model also assumes a plug flow moving at the roll speed.Using φstart,wash, and φstart,plug, we can define the new height of the plug, andthus determine the xˆclear required before the two compact plugs meet. Thereforethe boundary conditions remain the same for the model, however the dewateringmodel begins at xˆ = xˆstart − xˆclear.Sample results for Trial 103 are shown in Figure 5.14, using η∗ from Chapter3. Results with and without the wash section approximation are shown as themagenta and blue trends, respectively. Beginning with Figure 5.14a, we observethe impact of the wash section approximation as offsetting the compression untilaround xˆ ≈ −0.1m. Once compaction begins, the load instantaneously goes toessentially an identical magnitude of the result without the wash section. Thisimmediate growth in load is not physical, and exists due to the omission of elasticeffects in the constitutive model (which will be address in Chapter 6). Trivially, themodel results with the inclusion of the wash section are going to equal a lower lineload, however the manner in which this occurs is not particularly satisfying withthe substantial peak loads still present. This result suggests that if the wash sectiondoes, in fact, offset the compression further into the nip, then investigation of thecurrent model’s suggested instantaneous load growth needs to be examined further.5.4 Discussion: Reflection and Moving ForwardSeveral investigations aimed at improving the representation of the roll press modelhave proven unsuccessful, leaving significant over-estimations of line load with thecurrent model across the selection of data. Early it was detected that the majority ofthe line load was a result of the bulk viscosity term, however through reducing η∗,either by accommodating the temperature or by tuning, the model becomes volatilefor several of the trial points with respect to variations in φstart. Tuning of theφstart can obtain experimentally determined line loads, but does so in conflict withthe mass balance of the experimental machine. Finally, although the wash section120-0.25 -0.2 -0.15 -0.1 -0.05 0051015 1060 0.2 0.400.010.020.030.04-0.4 -0.3 -0.2 -0.1 000.010.020.030.04Figure 5.14: Results of the model for Trial 103 with φstart defined from themass balance, with and without a wash-section (i.e. φstart,plug andφstart,wash, respectively). In the various figures, blue trends representwithout a wash-section, and the magenta are with a wash-section.121can reduce the measured line load, it does not do so in a particularly inspiringmanner.Further investigations were undertaken as well, including an attempt to accom-modate reasonable levels of roll compliance due to the high compression loads inthe nip. Due to the short length and large radii of the rolls, very little deflection ofthe permeable surface would be expected. Based upon discussions with the indus-trial sponsor, a compliance function added as much as 1 − 2mm of height to hˆ(xˆ)trends, depending upon the magnitude of the line load measured experimentally.This was deemed the maximum reasonable deflection that could occur on the ma-chine, and still the model results far exceeded the experimental values. Tuning ofthe other material parameters was also undertaken, where an arbitrary permeabil-ity trend increase was found to provide a more stable model with respect to φstartand the various trial points. This potential motivated a rebuild of the experimentalequipment for determining permeability, as detailed in Appendix A.3.2. However,after a more robust design providing a higher assurance of the height calibrationand implementation of the permeable surface of the rolls, equivalent permeabilitycalibrations were found.With marginal success seen with the modelling approach at this point, we re-evaluated if a larger problem existed with the modelling approach. The first majordifference between the roll press dewatering and the experiments done in the pre-vious chapter are the compression rates found in the equipment. With compressiondewatering rates in the roll press V = O(0.1)m/s, a first concern could be iner-tial terms coming into the problem. We can judge inertial effects by defining apore-scale Reynolds number asRe = ρfV Dµ= 1.0 ∗ 103kg/m3 ⋅ 0.1m/s ⋅ 2 ∗ 10−6m0.001Pa.s= O(10−1), (5.106)where we use our pore scale size, F2, from Chapter 3. Degradation of modelrepresentation then is not likely due to inertial effects.Despite inertia not being relevant, the suitability of the viscoplastic-like con-stitutive model at these elevated compression rates should be considered. It is un-clear whether representation would continue to be sufficient at an order magnitudehigher compression rates. An additional consideration comes from the simplifying122assumption in the reduction that Sy(φ) = O(δ). The validity of this is open, andtherefore should be explored. We therefore arrive at the motivation for the final twoproject objectives, being the evaluation of the model effectiveness at elevated uni-axial compression rates, and to determine the pulp suspension’s shear yield stressbehaviour. These studies are detailed in the following two chapters.5.5 Summary and ConclusionsIn this chapter we have taken the general equations of Chapter 2 and appliedthem to the steady, slender geometry dewatering occurring in the nip of a TwinRoll press. This has provided an analogous set of equations to that of the uni-axial experiments found in Chapter 3, and therefore allows an evaluation of theviscoplastic-like constitutive model in an industrially relevant application. Similarmodel reductions can be found for other pulp-specific machinery as well. Experi-mental results for model validation were collected on a pilot-scale Twin Roll press,and constitute results collected at an order of magnitude higher compression ratesthan previously investigated.Using the calibrated permeability, compressive yield stress, and bulk viscos-ity scaling determined in Chapter 3, we evaluated the model’s effectiveness inreproducing the experimental line loads measured. What was determined is theviscoplastic-like constitutive model was ineffective at representing the integratedcompressive load experienced by the permeable rolls by far exceeding the experi-mental values. Particular uncertainties, from the roll press or the material parame-ters, were investigated to determine if proper model representation could be recov-ered, with no avail. This study therefore constitutes a thorough invalidation of amodelling approach that has been applied to pulp and paper equipment in the past,and provided two paths to explore. First, we hypothesized that the viscoplastic-like constitutive model is no longer valid at an order of magnitude higher compres-sion rates. Second, we reconsider a rheological assumption in the model reductionwhereby the pulp suspension has negligible strength in shear when compared tocompression. These paths are evaluated in the coming two chapters.123Chapter 6Model Extension: High SpeedUni-Axial DewateringThis chapter details a second uni-axial dewatering study at elevated dewateringrates. Included in this study is a reworking of the constitutive model, an explorationof a pulp suspension’s unloading elastic behaviour, development of a new materialparameter protocol, and model comparisons to experiments collected with reviseddewatering equipment capable of capturing higher speed dynamics, as detailed inAppendix A. The intention of this study was to evaluate the continued success ofthe viscoplastic-like constitutive model shown in Equation 2.11, to clarify if the rollpress model’s poor performance was due to an inappropriate constitutive model.We begin this chapter with an evaluation of the model’s suitability to describethese new experiments collected at elevated compression rates. As will be detailedfurther in the coming sections, the high speed uni-axial dewatering experiment nolonger uses a constant compression rate experiment, as was seen in Chapter 3,rather it uses a parabolic hˆ(tˆ) trend which has multiple advantages. An averagedexperimental compression load trend is shown in Figure 6.1, with a hˆ(tˆ) trend suchthat the initial compression rate was 40mm/s. The black line shows the average offour repeated experiments, with the two standard deviations of the average shownas the dotted bounds. The blue dashed line is the representation provided by theviscoplastic constitutive model presented thus far. As can be seen, the model isbeginning to struggle with representation at these elevated compression rates. Al-1240 0.05 0.1 0.15 0.2104106Figure 6.1: Representation of the viscoplastic constitutive model of Equation2.11 for higher speed uni-axial experiments, with an initial compressionspeed of 40mm/s. The averaged experimental curve is shown in black,and the model’s representation is shown as the blue dashed line.though the magnitude does not match, the more notable observation is the differentshape of the load curve, with a maximum seen late in the compression that is notobserved experimentally. Knowing the constitutive model struggled with the rollpress results in Chapter 5, in particular by suggesting a much higher line load thanwas experimentally measured, this preliminary result certainly was suggestive thatsomething was missing from the current constitutive model.Moving forward we aimed to improve the model’s representation of these el-evated compression rate experiments in a way that complements the success ofthe current constitutive model thus far. The approach taken up to this point hasbeen to develop a constitutive model for the network’s solid effective stress thatresembles a complex fluid (a viscoplastic), in that it has a yield stress that it canwithstand before consolidating, and when it does collapse, it does so viscously andplastically. In reality, albeit small, elastic behaviour in a pulp solid network is ev-ident, as discussed and shown in several studies (example [23, 63, 90, 98]). Wealso observed evidence of a small bounce back in the flow through experiments inChapter 4. Qualitatively, the experimental elastic behaviour of pulp suspensionsappears similar to the response seen in soils or woll (example [2, 22]). Previousmodelling efforts of pulp suspensions with the inclusion of elastic and plastic de-125formations can even be found in the literature (examples [54–57, 64, 65]), alongwith further general visco-elasticity evidence in fibre suspensions [72]. The prob-lem with elasticity in complex fluids, is it begins to show with small deflections(where the material acts elastically), and at higher speeds where the time-scale ofthe experiment begins approaching the elastic time-scale of the material. Up untilnow, we have ignored the pulp suspension’s elastic response that is superimposedon top of the much more pronounced plastic deformation response, however atthese elevated rates, the experimental time-scales may be sufficiently small thatelastic effects are becoming important.We therefore undertake experiments to observe the elastic behaviour, build itinto our constitutive modelling for the solid effective stress, and see if our modelprovides better representation to the experimental results at increasing rates. Ourhypothesis is that elastic effects added into the constitutive model will fix the issuesseen with representation at elevated compression rates.This chapter begins with a description of the materials and methods that wereused, followed by the development of an elastic-viscoplastic-like model for thesolid effective stress. The model equations are then provided, and the results, usingthe modified equipment, are used to evaluate the model’s suitability.6.1 Materials and MethodsThe same Scandinavian softwood Kraft pulp in Series 17 from Chapter 3, andthat was used in Chapter 5 was used here. Its permeability, compressive yieldstress, and bulk viscosity therefore have been calibrated through the experimentsdescribed in Chapter 3, with the fitted functional forms,k(φ) = 2.67−13φln( 1φ) e−20.38φ [m2], (6.1)Py(φ) = 6.20 × 105 φ1.87(1 − φ)3.83 [Pa], (6.2)Λ(φ) = 2.86 × 107 φ2 [Pa ⋅ s]. (6.3)Like Chapter 3, the suspensions are made from dried pulp sheets, and prepared toan initial concentration of φ0 = 0.02 − 0.03.126(a)0 1 2 3 4 51040510152025(b)0 1 2 3 4 51040510105Figure 6.2: Sample unloading and reloading experiments used for exploringthe solid network’s rheology. In both figures, the black indicates com-pression, blue indicates unloading, and the green indicates hold periods.The main experimental apparatus used in this study is the uni-axial dewateringequipment detailed in Appendix A.2. Experiments exploring the pulp suspension’selastic response and its high dewatering rate behaviour are used throughout thischapter. For both of these experiments, modifications were needed to be madeto the uni-axial dewatering equipment. In particular, two new experimental testswere developed, and retrofitting of the components and the LabVIEW interfacewas performed. Details of the changes to the equipment can be found in AppendixA.2.3.The elastic response was explored using the new loading/unloading test. Thisnew experiment performs various loading and then unloading experiments to char-acterize the suspension’s elastic response with network stresses belowPy(φ). Vary-ing configurations of this experiments are possible with the equipment. A sampleexperiment is shown in Figure 6.2 with the colors illustrating piston direction.The high dewatering rate experiments utilized a parabolic piston hˆ(tˆ) curvein lieu of the ramp compression profile, as seen in Chapter 3. Two benefits areallotted from this change. First, the parabolic compression curve has the benefitof approximating the compression profile between the permeable rolls of the TwinRoll press (as seen in Equation 5.60). Second, the new experiment has a smoothstop condition, which allows higher speed dynamics to be obtained safely within127the load limits of the equipment. The experimental protocol measures the initialheight of the suspension, h0, and specifies the initial compression rate, dhˆdtˆ (tˆ = 0),and the final height, hend, from which the parabolic hˆ(tˆ) profile is constructed.Further details of these new tests can be found in Appendix A.2.3.6.2 Solid Network Rheology: AnElastic-Viscoplastic-Like Constitutive Model6.2.1 Loading/Unloading ExperimentsTo start exploring the pulp suspension’s complete solid network compression rhe-ology, we turn to the experiment illustrated in Figure 6.2. In this experiment, mul-tiple loading and unloading cycles were performed on the pulp sample. Both theloading and unloading speeds were chosen as 1µm/s, being a rate sufficiently slowsuch that the suspension compacts uniformly [32]. The unloading of the suspen-sion ceased when the σ(t) ≈ 0Pa, though a finite load was always maintainedto not lift the permeable piston off the compressed suspension. What is evident,particularily at the higher levels of compression, is rather significant elastic strainrecovered during unloading. The data is plotted in Figure 6.3a, where φ = φ0h0/hˆ.Again the blue and black distinguish the direction of the piston, and the greenrepresents holding periods where the piston remained approximately fixed. Alsoshown in this plot is the Py(φ) experimental trend from Chapter 3, shown in redwith the error bars representing two standard deviations. Plotted in this way, weget another appreciation of significance of elastic strain, observing considerablehysteritic loops of unloading and loading. The reloading of the pulp smoothly con-verges to the compressive yield stress trend and follows it once the compressivestress surpasses the point where unloading initiated. Comparing from loop to loop,we see a similar shape, and the intermediate unloading loops do not seem to im-pact later behaviour. These looping trends have been shown in the literature forpulp suspensions [56, 65], soils [2, 28], and wools [22]. In the insert, the data isreplotted on a log-log scale. What is notable in this plot is the consistent trend forthe unloading curves and the substantially steeper slope with respect to the com-pressive yield stress.128(a) (b)(c) (d)Figure 6.3: Unloading experiments collected to explore the elastic rheologyof the solid network, along with the suspension’s Py(φ) trend shown inred. In (a), near full unloading (blue) before reloading (black) is per-formed multiple times. The data is replotted on a log-log scale in theinsert. In (b), near full unloading and reloading was repeated multipletimes at the same initial height, and again is replotted on a log-log scalein the insert. In (c), partial unloading (66% of take-off load) was per-formed before reloading. The insert shows a single loop, along witha linear approximation used for fitting values of E(φ). In (d), partialunloading (66% of take-off load) was performed, at elevated unload-ing rates before reloading. The insert shows the results overlaid on theresult from (c), which is re-shown in cyan.129Observing these pronounced unloading and loading trends, we next investi-gated their repeatability. Shown in Figure 6.3b is an experiment where the unload-ing and loading loop was repeated at the same take-off height multiple times, againat loading and unloading rates of 1µm/s. What can be seen in the main figure,and in the insert, is slight variation between each loop as it approaches a stablestate. Slight reduction in the hysteresis is seen in the stable state, however it is stillsignificant. A possible source of the hysteresis being viscous dissipation, as sug-gested by Vomhoff [98], seems unlikely due to the very slow rate of compressionand unload. The rate dependency in the viscoplastic-like constitutive model, whichis quite large, has negligible impacts at these rates. Therefore, if dissipation wasthe culprit to the hysteritic elastic nature, it would need to be orders of magnitudelarger than previously calibrated which is not physical. This leaves the mechanis-tic explanation of the loops undetermined at this point, however motivates furtherstudies. El-Hosseiny [23] suggested friction between the fibres as the source of thehysteresis. Upon exiting the final loop we see the trend rejoins the compressiveyields stress seamlessly.To approximate the unloading and reloading behaviour, we undertook severalexperiments where we only partially unloaded the suspension to 66% the load atwhich the piston reversal started. Both the loading and unloading rates were keptat 1µm/s. This reduced the elastic strain recovered, and returned a much tighterhysteresis loop, as shown in Figure 6.3c. Although these unloading trends do notshare the whole picture of the networks elasticity, they provide a good experimentto calibrate approximate elastic properties of the suspension as linear, which provemuch easier to implement into the numerical solver. Like the previous experiments,the reloading trends converge to compressive yield stress.A final confirmatory experiment was undertaken to verify the hysteresis wasnot dissipative. An experiment where the unloading rate was 10 times that of theloading rate was done to observe if the loops grew substantially (due to the higherdissipation). The loops from this experiment are shown in Figure 6.3d, whichappear unchanged from the lower rate unloading experiment, as highlighted in theinsert, with the cyan loop showing the result of Figure 6.3c. We feel confidentthat under this partial unloading, any rate dependency in the network’s elasticityappears negligible. Higher speed full unloading curves were also performed, with130only marginal changes observed as well.6.2.2 Constitutive Model DevelopmentHaving explored the network’s response in both the loading and unloading states,we begin constructing a constitutive model that can replicate the behaviour. Thedesired constitutive model will represent the solid network as a near linear elasticmaterial below the yield stress and a viscoplastic above the yield stress. This issimilar to the modellings of some non-Newtonian fluid flows, such as the workof Saramito et al. [83]. Schematically, the solid network’s stress model is shownin Figure 6.4 for an element of the solid network. The elasticity, compressiveyield stress, and bulk viscosity are represented as a spring, a ratchet, and dashpot,respectively. A similar schematic is found in Lobosco and Kaul [65], who imposea single phase, spatially uniform approach for modelling the compressive load onthe pulp suspension.Similarly, we begin translating this model into usable equations by first consid-ering a spatially uniform solid effective stress. Total strain of this network elemente = ep + ee, (6.4)is defined as the sum of the plastic and elastic strain components. With only con-sidering compression of the network, we note ep,e ≤ 0, with respect to the originalsuspension. The irreversibility of the plastic strain and the inability of the perme-able piston to impose tension on the network ensure this. The elastic and plasticstrains, or their respective rates, can be represented with respect to the elements ofthe model, with the spring following Hooke’s law asPˆ = −E(e) ⋅ ee or dPˆdtˆ≈ −E(e) ⋅ deedtˆ, (6.5)where E(e) represents some strain dependent spring constant, analogous to the131Λ(𝑒)𝑃𝑦(𝑒)𝜀(𝑒)෠𝒫𝑒p𝑒e𝑒෠𝒫෠𝒫𝑃𝑦𝑃𝑦−Λ 𝑒 ⋅ ሶ𝑒𝑝−Λ 𝑒 ⋅ ሶ𝑒𝑝Ƹ𝑧Simple SaramitoModel(unyielded, acts as a simple Spring system)𝑅𝑎𝑡𝑐ℎ𝑒𝑡 𝐷𝑎𝑠ℎ𝑝𝑜𝑡𝑆𝑝𝑟𝑖𝑛𝑔Figure 6.4: Free body diagram of a proposed elastic-viscoplastic constitutivemodel for the solid effective stress.network’s elasticity, and the dashpot as−Λ(e)depdtˆ= Pˆ − Py(e) if Pˆ > Py(e),−Λ(e)depdtˆ= 0 if Pˆ ≤ Py(e), (6.6)where Λ(e) is some strain-dependent damping coefficient, analogous to the bulkviscosity. We therefore arrive at the rate form of the total strain equation asdedtˆ= −max(0, Pˆ − Py(e)Λ(e) ) − 1E(e) dPˆdtˆ . (6.7)132It is more convenient to represent this expression in terms of solid volume fractionchanges, rather than strain changes. The relation between the two isdedtˆ= − 1φdφdtˆ, (6.8)and so we can eliminate the total strain rate from the expression above, and haveour material parameters calibrated in terms of solid volume fraction and not interms of strain. In order to accommodate spatial variation in the solid effectivestress, we redefine the constitutive model in a Eulerian framework, asDφDtˆ=max⎛⎝0, φ (Pˆ − Py(φ))Λ(φ) ⎞⎠ + φE(φ)DPˆDtˆ . (6.9)We assume the bulk viscosity function remains the previously presented formΛ(φ) = η∗φ2. In this formulation, the previous constitutive model presented inChapter 2 and used throughout Chapters 3-5 can be recovered, if we consider theelastic deformations negligible, through an infinite elastic stiffness to the network.We see if E(φ)→∞, we recover Equation 2.10.An additional convenience with this functional form is a clear method of ap-proximating the elasticity of the network from the unloading/loading loops. Whenthe unloading and initial reloading occurs, Pˆ < Py(φ) and the ratchet locks theplastic deformation. Therefore the first term on the right side of the equality inEquation 6.9 equals zero. If the elastic response is collected sufficiently slow, nogradient in the solid volume fraction will occur and therefore the convective termsequal zero in both material derivatives. We therefore are left withdφdtˆ= φE(φ) dPˆdtˆ . (6.10)Using the partial unload loops, we can fit a value of the network elasticity asE(φ¯) = φ¯ (Pˆ2 − Pˆ1)(φ2 − φ1) where φ¯ = φ1 + φ22 , (6.11)as illustrated in the insert of Figure 6.3c. Performing these unloading loops at133varying values of φ¯ is used to calibrate elasticity, E(φ¯).6.3 Model EquationsSimplification of the general model equations presented in Chapter 2 follows whatwas implemented in Chapter 3. Again we are interested in a one-dimensional setof the equations, and assume body forces are negligible. Reduction and non-dimensionalization of the model equations also follows Chapter 3, but of courseinstead uses our new elastic viscoplastic-like constitutive model shown in Equa-tion 6.9. The additional scaling of the elasticity isE(φ) = e∗E(φ), (6.12)where e∗ is defined as the fitted functional form E(φ) at φ = 0.10. The scalingof the solid phase velocity, V , is chosen as the initial compression speed of thepermeable piston with the parabolic hˆ(tˆ) trend discussed, i.e.V = ∣ ddtˆhˆ(tˆ = 0)∣ . (6.13)The reduced non-dimensional model equations arrived at areDφDt= −φ∂v∂z, (6.14)v = −γK(φ)∂P∂z, (6.15)DPDt= −αE(φ)∂v∂z−max(0, αγ(E(φ) [P −Πy(φ)]φ2)) , (6.16)with the following dimensionless groupsγ = p∗k∗µh0V,  = k∗η∗µh20, and α = e∗p∗ . (6.17)We again recognize the recovery of our viscoplastic modelling efforts can beachieved by making the solid network infinitely stiff, which can be done with α →∞. The boundary conditions of the solid phase velocity, and the initial condition1340.1 0.2 0.3 0.4 0.5104106108Figure 6.5: E(φ) results shown as the orange symbols, with the fitted func-tional form as the solid orange line. Also shown is Py(φ) as the redtrend, with the error bars representing two standard deviations of theaverage of four trends.arev(0, t) = 0, (6.18)v(h, t) = 1Vdhˆdtˆ, (6.19)φ(z,0) = φ0. (6.20)As was the case in Chapter 3, the load on the permeable piston is found as the solideffective stress at the top of the suspension, i.e.σ(t) = Πy(φ) − γφ2∂v∂z− γ αφ2E(φ)DPDt ∣z=h(t) . (6.21)A similar scheme to that described in Section 3.2 is used to solve the model.1356.4 Results and DiscussionMultiple partial unloading experiments (unloaded to 66% of take-off load) wereconducted to calibrate values of E(φ), and to capture experimental variability. Theresults are shown as the orange symbols in Figure 6.5. The closed squares areperformed on the MTS, whereas the open circles were determined on a low-loadexperimental apparatus using the same protocol (slow compaction, and partial un-loading). A fitted functionE(φ) = 1.08(108) φ2.71(1 − φ)0.688 [Pa], (6.22)is shown as the solid orange line. The compressive yield stress is also shown in thefigure as the red line. The two measured parameters do seem to be proportional toone another, by a weakly φ dependent factor. This result supports the discussion ofSection 3.3.2, whereby the source of Py(φ) was suggested to be due to frictionalre-arrangements under elastic formal stresses. The weakly φ-dependent propor-tionality perhaps therefore may be best interpreted as a weakly varying frictionfactor at the fibre contacts (ν in Equation 3.26).With the necessary materials calibrated, we start with observing the model’sreproduction of some unloading loops, shown in Figure 6.6. In this model run, thepiston velocity was reversed temporarily three times. We see with our constitutivemodel having non-dissipative response in the unyielded/elastic regime, that theunloading and reloading falling on the same line (the unloading curves shown asblue dashed lines). We also see the reloading rejoins the trend of Py(φ) oncereaching the load of take-off. The model result roughly captures the main detailsof the partial unloading loops, and therefore we turn our focus to the higher speeddewatering experiments.Experimental parabolic compression curves were collected for a variety of ini-tial velocities, V , and are presented in Figure 6.7, plotted as load versus averagevolume fraction (φ¯ = φ0h0hˆ). Each initial velocity was repeated four times, withthe solid line of each color showing the average of the four trends, and the dottedlines banding a standard deviation of the average. The same values of h0 and hendwere used for each initial velocity. The shape of the load trend is indicative of1360.1 0.2 0.3 0.4024681012 105Figure 6.6: Model results imposing three unloading and reloading cycles.The unloading is shown as the blue dashed line (virtually indistinguish-able from the reloading), and the loading are shown as the black trend.Both directions’ speeds were 1µm/s.the negative acceleration of the piston due to the chosen hˆ(tˆ) profile. It is notedthat as the initial compression rate increases, a steeper initial linear growth in theload is occurring at small times. It can be seen in the experiments with this linearload growth that highly differential compaction, with a building boundary layer ofpulp packing under the permeable piston is occuring. This is similar to the spatialcompaction observed by studies such as those of Burton and Burns [7–9]. Even-tually during the experiment, this packing boundary layer stops building. We nowevaluate our new constitutive model with these experimental trends.To begin with, we evaluate a slow initial rate (V = 5mm/s) dewatering experi-ment, replotted in Figure 6.8a. Two model trends are plotted in this figure. The redtrend is our new constitutive model (Equation 6.9), whereas the blue dashed lineis our previous constitutive model (Equation 2.10). Compared to the experimen-tal results, we see only a marginal difference between the two model trends. Atstartup, we see the elastic model has a gentler onset of load, and over the range ofexperiment, is slightly closer to the experimental results, however the difference isnot substantial. This reflects the conclusions of [32] and Chapter 3, having found1370 0.05 0.1 0.15 0.2104106Figure 6.7: Multiple parabolic compression dewatering experiments areshown with initial speeds, V , equalling 5, 15, 25, 35, 45, 50, 55, and60mm/s, respectively. Increasing rates are indicated by the arrow.Each initial speed had four experiments collected, with the averageshown as the solid line and its standard deviation shown as the dot-ted lines. The initial and final heights for all experiments were heldconstant.a bulk viscosity alone was effective in representing the experimental dewateringbehaviours at these rates.Moving onto a higher compression rate, we show the experimental results andthe two model representations at V = 35mm/s in Figure 6.8b. At this rate, asignificant difference in the model trends is becoming apparent. We notice againthe elastic model has a gradual onset of load at the beginning of the experiment,which is representing the experimental trend much better than the previous model.We see a flatter trend with the elastic model, which looks closer to the stable loadsseen experimentally, whereas the old model shows a maximum at approximatelyφ = 0.15, which is not evident in the experiment.Finally, we move onto the highest compression rate experimentally collected,shown in Figure 6.8c. With V = 60mm/s, we see the elastic model is significantlycloser both in magnitude and shape to the experimental trend, when compared tothe previous constitutive model. A small saddle is evident in the new model trend,138(a)0 0.05 0.1 0.15 0.2104106(b)0 0.05 0.1 0.15 0.2104106(c)0 0.05 0.1 0.15 0.2104106Figure 6.8: Compressive loads with initial rates of V = 5, 35, and 60mm/sare shown in (a)-(c). In each figure, the average of four experimentaltrends is shown in black, with one standard deviation of the averageshown as the dashed lines. The red curve shows the elastic model, andthe blue curve shows the inelastic model results.139however, it provides a flatter load curve, which is not evident with the previousmodel.Better representation of the load on the permeable piston has been demon-strated with the elastic constitutive model. With an objective to further illustratean improved representation, we performed a particle tracking velocimetry (PTV)study to compare the solid phase velocity measured experimentally to the modelpredictions. This was performed by repeated dewatering experiments with smalltorn pieces of black paper speckled throughout the suspension that followed themotion of the solid phase (similar to experiments described in Chapter 4). Videowas taken of the experiment from the side, from which the solid phase velocity wasdetermined. The results are shown in Figures 6.9 and 6.10.Starting with a slow dewatering rate (with V = 1.5mm/s), shown in Figure6.9a, we see after start-up a rather linear solid phase velocity with respect to ele-vation and the piston velocity, with the majority of the solid phase in motion. Thiswould result in an approximately uniform compaction. Very little difference can beseen between the two model predictions of this velocity contour, comparing Fig-ures 6.9b and 6.9c. Again, at slower rates we see the elastic and the non-elasticmodels are indistinguishable. Next, turning to the higher speed dewatering exper-iment (with V = 55mm/s) we see a substantially different experimental result,shown in Figure 6.10a. What is shown is the majority of the suspension has lit-tle motion except for a building region near the piston. This solid phase motioninduces high spatial variation in φ, and is illustrating the packing boundary layerunder the permeable piston that was discussed at the beginning of this section.Comparing the models for this experiment, we see a significant distinction betweenthe elastic and the non-elastic constitutive models, in Figures 6.10b and 6.10c re-spectively. Much better representation of the experimental dewatering event isobserved with the inclusion of elasticity, leading to confidence in the calibratedmaterial parameters.Before ending the chapter, we wish to make note of a sensitivity of the modelresults that was observed with respect to the fitted material parameters. Small vari-ations in the fitted functional form of permeability was found to eliminate both theimproved shape in the compression load trend and the effective representation ofthe solid phase motion gained with the inclusion of elasticity that was presented140(a)0 10 20 3001020304050 -1.5-1-0.50(b)0 10 20 3001020304050 -1.5-1-0.50(c)0 10 20 3001020304050 -1.5-1-0.50Figure 6.9: Experimental PTV results showing the solid phase velocity forV = 1.5mm/s dewatering rate are shown in (a). The piston locationis shown as the red line. The corresponding contours from the modelwith the elastic-viscoplastic, and the viscoplastic constitutive model areshown in (b) and (c), respectively.141(a)0 0.2 0.4 0.6 0.8 101020304050-50-40-30-20-100(b)0 0.2 0.4 0.6 0.8 101020304050-50-40-30-20-100(c)0 0.2 0.4 0.6 0.8 101020304050-50-40-30-20-100Figure 6.10: Experimental PTV results showing the solid phase velocity forV = 55mm/s dewatering rate are shown in (a). The piston locationis shown as the red line. The corresponding contours from the modelwith the elastic-viscoplastic, and the viscoplastic constitutive modelare shown in (b) and (c), respectively.142above. When we consider the complexity of the constitutive model, with both se-ries and parallel components which are separately calibrated or specified, and theunderlying flow-induced compaction due to the network’s permeability, we beginto gain an appreciation of the significant interplay between several sensitive mate-rial parameters. Although the model has been found to be effective, the sensitivityof the model results is important to be aware of if applying this elastic viscoplastic-like constitutive model to pulp dewatering experiments.6.5 Summary and ConclusionsThe objective of this chapter was to evaluate the continued success of theviscoplastic-like constitutive model of Equation 2.11 at elevated compression rates.This stemmed from a hypothesis that the poor representation seen of the modelin Chapter 5 was due to the viscoplastic-like constitutive model no longer beingappropriate at the elevated dewatering rates experienced. Modification to the ex-perimental equipment enabled higher speed dewatering dynamics to re-evaluatethe modelling framework. The upper limit of suitability for the viscoplastic-likeconstitutive model was quickly established. In efforts to regain appropriate rep-resentation, an extension of the constitutive model was undertaken to include thenetwork’s elastic experimental response. A protocol to independently calibratethe network elasticity was developed. The results collected suggest a weakly φ-dependent proportional factor of the network’s compressive yield stress. This sup-ports the discussion of Section 3.3.2, whereby the source of Py(φ) was suggestedto be due to frictional re-arrangements under elastic formal stresses.With this evolved calibrated elastic viscoplastic-like constitutive model, repre-sentation of higher speed dynamics by the model framework was recovered, boththrough representation of the observed compressive loads and in the motion ofthe solid phase. It is therefore concluded that the inclusion of elasticity in theconstitutive model is critical for properly representing high speed pulp suspensiondewatering dynamics. Therefore, this provides meaningful insight into the poorrepresentation of the model in Chapter 5, in which the viscoplastic-like constitu-tive function was utilized.143Chapter 7Model Extension: 2D Stress Stateand Shear Yield StressThis chapter details the final experimental study aimed at determining a pulp sus-pension’s shear yield stress, τy(φ), at relevant volume fractions to the dewateringseen in the Twin Roll press. Included in this study is the development of a uniquetesting protocol that aims to reconcile typical challenges using standard rheologi-cal equipment for fibre suspensions, details of a unique testing apparatus developedand constructed, and the results obtained. The intention of this study is to evalu-ate the assumption, put forth in Chapter 5, that the pulp suspension has negligiblestrength in shear in comparison to compression, which reduced the complexity ofthe model equations and constitutive expressions.Measurements of τy(φ) and exploration of the rheological behaviours of pulpsuspensions in general is an area that has been well explored (with several examplesfound in the review Derakhshandeh et al. [19]). The motivation of many of thesestudies is to understand the complex flow behaviour of these suspensions to provideinsight into its transport and processing at unit operations in the production ofpulp and paper [43]. Unfortunately, the studies are hampered by several attributesof the suspension that make rheological studies quite complicated. These factorsinclude the distribution of fibre sizes, their hollow construction and porous walls,and their tendency to first flocculate and then network at increasing concentrations.These factors limit both the tools and methods available for calibrating rheological144parameters like τy(φ), and result in only a small range of concentrations that canbe evaluated.A particular challenge when performing rheology experiments for suspensionsis preventing slip between the suspension and the walls of the tester, which whenpresent, skews the measurement artificially low [5]. An extensive investigation ofthis challenge, specifically for pulp suspensions, is found in Mosse et al. [70].Typical approaches for combating this is to either roughen or create protrusions onthe walls of the tester that aid in immobilizing the suspension. Generally, the rule ofthumb for suspensions is roughness on the order of the particle size [10], however,in the context of pulp suspensions, particle size needs to refer to the average flocsize rather than the individual fibre size. Floc sizes in a suspension depend ona variety of factors, in some cases being as large as 1 − 2 cm in diameter [38].A particular testing approach that has been successful in limiting wall slip is thevane-in-cup technique, in which τy(φ) can be found from a torque measurementcollected initiating a shear flow at a fix rotation rate. As was shown by Nguyen andBoger [73], the peak torque found during start up can be used to determine τy(φ)asτy(φ) = Tm [piD32(HD+ 13)]−1 , (7.1)where Tm, H and D are the maximum torque, height and diameter of the vane, asshown in Figure 7.1.Several studies implementing this approach for pulp suspensions can be foundin the literature (examples, [18, 34, 70]). However, the significant limitation ofthis approach for our purposes is the low volume fractions to which the techniqueis confined. Similar to the challenge realized with elevated φ0 values in the otherexperiments performed in this project, obtaining a uniformly distributed sample ina cup is difficult due to the suspension behaving more solid-like at increasing con-centrations. Therefore, the suspension does not readily pour into a rheometer’s cupand therefore large air pockets in the sample will occur. Further challenges withhigher concentrations include increased torque beyond what typical rheometers canprovide, and the immobilization of the suspension on the cup walls. Due to thesefactors, it is not practical to use traditional rheometry techniques to obtain τy(φ)at volume fractions relevant to the Twin Roll press, and so we look to define a new145𝐻𝐷𝑇𝑇𝑡𝑖𝑚𝑒𝑇𝑚Figure 7.1: Schematic of a traditional vane and cup rheometer used for de-termining τy(φ) for pulp suspensions. The expected torque versus timetrend of an initialized steady rate shear flow is shown as well.methodology that accommodates elements of experiments and theory developed inthe previous chapters.This chapter begins with a development of a proposed methodology based uponthe two-dimensional yielding state discussed in Section 5.1.1. Following this is amaterial and methods section and results. Finally, a discussion of the implicationsand section conclusions.7.1 Proposed MethodologyThe experimental approach to approximate τy(φ) utilizes an ideal two-dimensionalgeometry shown in Figure 7.2a. A permeable surface of infinite length in thexˆ−direction can dewater the uniform suspension beneath by applying a compres-sive load σˆc, as well as impose a shear stress to the suspension σˆs. The experi-mental device is assumed to strain the suspension at sufficiently slow speeds suchthat the bulk viscosity components are negligible, and that the suspension dewa-ters uniformly, i.e. φ(x, z) = φ0h0/hˆ. The experiment also assumes body forcesand pressure drop across the permeable surface are negligible. The most gen-eral velocity this equipment can impose on the suspension’s phases is uˆs,f(z) =(uˆs,f(z), vˆs,f(z)), which therefore defines the divergence of the solid phase ve-146(a)ො𝒛 = ෡𝒉ො𝒛 = 𝟎ො𝒛ෝ𝝈𝒄Suspensionෝ𝒙ෝ𝝈𝒔(b)ෝ𝝈𝒄෠𝒫Ƹ𝜏𝑥𝑧ෝ𝝈𝒔𝜏𝑦(𝜙2)𝑃𝑦(𝜙2)𝜙1 𝜙2 𝜙3𝑃𝑦(𝜙3)𝑃𝑦(𝜙1)𝜏𝑦(𝜙3)𝜏𝑦(𝜙1)Figure 7.2: Schematics for illustrating the shear yield stress experimental de-termination. In (a), an ideal two-dimensional geometry is shown. In (b),the two step procedure is illustrated.locity, and the solid phase deviatoric strain-rate tensor as∇ˆ ⋅ uˆs = ∂vˆs∂zˆ, (7.2)Dˆ = ⎡⎢⎢⎢⎢⎢⎢⎣−∂vˆs∂zˆ ∂uˆs∂zˆ∂uˆs∂zˆ∂vˆs∂zˆ⎤⎥⎥⎥⎥⎥⎥⎦ . (7.3)Starting again from the general equations of Chapter 2, we are interested in definingthe stress conservations in the two dimensions of this geometry. We also will followthe same definition of the total solid effective stress and the developed constitutivemodels put forth in Chapter 5 (in particular the details provided in Section 5.1.1).Therefore, the stress balances in the xˆ and zˆ− directions can be expressed as∂∂xˆ(−pˆ − Pˆ + τˆxx) + ∂∂zˆ(τˆxz) = 0, (7.4)∂∂xˆ(τˆzx) + ∂∂zˆ(−pˆ − Pˆ + τˆzz) = 0, (7.5)respectively. Despite them equalling zero, the xˆ−gradients are retained for now.147The specific protocol utilizing this experimental geometry for approximatelycalibrating τy(φ) aims at tracing out the suspension’s yield surface at a fixed valueof φ. The simple elliptical yield surface, as first introduced in Equation 5.17, isτˆ2τy(φ)2 + Pˆ 2Ny(φ)2 = 1 (reshown (5.17)), (7.6)where τˆ2 is the second invariant of the deviatoric stress tensor as defined in Equa-tion 5.25. With the form of Dˆ, we can express τˆ2 simply asτˆ2 =∑i∑jτˆij τˆij/2 ≡ τˆ2zz + τˆ2xz, (7.7)given that τˆzz = −τˆxx and τˆxz = τˆzx. This definition of τˆ2 can be substituted intoEquation 7.6, which arrives atτˆ2xzτy(φ)2 + τˆ2zzNy(φ)2 + Pˆ 2 τy(φ)2Ny(φ)2 τy(φ)2 = 1, (7.8)which provides a more convenient form of the yield surface, that will become evi-dent shortly.The protocol for tracing out the yield surface is a two-step process. Step oneinvolves dewatering the suspension uni-axially, i.e. uˆs,f(z) = (0, vˆs,f(z)), at aquasi-steady state rate to a specified hˆ, which sets the suspension’s φ(x, z). In thisstep, the formulations of Section 5.1.1 are relevant, which provide expressions forPˆ and τˆzz in compression (where ∂vˆs∂zˆ < 0) asPˆ = Ny(φ)2√τy(φ)2 +Ny(φ)2 , and τˆzz = −τy(φ)2√τy(φ)2 +Ny(φ)2 . (7.9)These expressions lead to the definitions of Py(φ), and σˆc with help of the zˆ−direction force balance and negligible pressure drop across the permeable surface,asσˆc = Py(φ) ≡ √τy(φ)2 +Ny(φ)2. (7.10)148These definitions can be used to simplify Equation 7.8 asτˆ2xzτy(φ)2 + σˆ2cPy(φ)2 = 1, (7.11)where τˆ2xz = 0 for this first step of the protocol. This form of the volume fractionspecific yield surface is schematically illustrated in Figure 7.2b for three differentφ values as the three elliptic dotted curves. The first step of the calibration proto-col represents translation up the Pˆ axis to the specified location corresponding todesired value of φ (shown with the red arrow).Once the desired φ has been achieved, the dewatering stops, i.e. vˆs = 0. Thecompressive load is maintained, and since the dewatering was performed quasi-statically, we assume Pˆ is defined and remains at its yield point in compressionfor the time being. The second step in the protocol now begins, which beginsintroducing a slow, ramping shear σˆs while maintaining the hˆ and the compressionof the suspension. From the xˆ−direction force balance, we find τˆxz = σˆs, whichcan be plugged into Equation 7.11 arriving atσˆ2sτy(φ)2 + σˆ2cPy(φ)2 = 1. (7.12)Since φ is maintained, this expression remains true, and as σˆs builds, σˆc will reduceto compensate. Assuming this experimentally determined interplay between σˆs andσˆc tracks along the yield surface (shown as the blue arrow in Figure 7.2b), we canuse Equation 7.12 to fit the τy(φ) at the specified φ, which is equivalent to the τxzintercept in Figure 7.2b.7.2 Materials and MethodsThe same Scandinavian softwood Kraft pulp in Series 17 from Chapter 3, and thatwas used in Chapters 5 and 6, was used here. Its permeability, compressive yieldstress, and bulk viscosity therefore have been calibrated through the experimentsdescribed in Chapter 3. The fitted functional form of the compressive yield stress,149which is used in the coming discussions, isPy(φ) = 6.20 × 105 φ1.87(1 − φ)3.83 [Pa]. (7.13)The ideal geometry used in the protocol development cannot be realized ex-perimentally. Instead, a proxy cylindrical geometry is chosen for the experimentalapparatus which is mechanically actuated by hand. A section view of the devicedevised is shown schematically in Figure 7.3. The main components of the appa-ratus include: 1) a frame with a bearing/lead-nut assembly pressed into its top, 2)a lead-screw that threads through the lead-nut, which has a hex feature machinedinto its top and is constrained to the permeable piston at its base, and 3) an annulusshaped cup, constrained in the θ−direction, with a clear outer wall. In addition tothese main components, an instrumented torque wrench is attached to the top ofthe lead screw which can either restrain or induce θ−direction motion of the per-meable surface, and a load cell at the base of the cup allows measurement of thecompressive loads. A coarse 10 x 10 mesh wire is attached to the underside of thepermeable surface and on the base of the annulus cup to facilitate shear transfer tothe pulp suspension. Discussion of this surface’s roughness is returned to at the endof this section. Pure vertical translation of the permeable piston or pure shear un-der fixed piston position can be realized by this apparatus, which will be describedshortly. The device outputs measurements of σˆc and σˆs, and uses a camera andan encoder strip mounted on the outer face of the permeable piston to measure theheight and angular position.To invoke the protocol described in the previous section, the first step is todewater the pulp suspension uni-axially to a target hˆ, corresponding to a desiredφ = φ0h0/hˆ. This requires pure translation of the permeable surface. This is per-formed by constraining the piston in the θ−direction via stabilizing the wrenchfit to the lead screw, and twisting the lead nut which winds the lead screw down-wards. This in turn compresses the permeable piston into the suspension and causesdewatering. As described, this dewatering should be performed at quasi-steadystate compression rates, which are O(10−3)mm/s. These slow of rates were notachieved experimentally, with typical dewatering rates of 0.5mm/s used. Instead,a minimum 20 minute hold period after translation was provided to allow the dif-150T.C.L.C.CAMERADAQDAQDAQWrenchLead NutPistonCupƸ𝑧𝜃Figure 7.3: Schematic section view of the shear yield stress tester.fusion of any gradients in the volume fraction that may have occurred, which weassume are small. The stabilized load is a measurement of Py(φ).Once the suspension has stabilized, we can perform the data collecting steptwo, for which a ramping shear stress will be applied to the compressed pulp sus-pension that is constrained at the target height. To do this, first a thumb set-screwhoused in the lead-nut is wound inwards which binds against the flights of the lead-screw. This set-screw inhibits relative motion between the lead-nut and lead-screw,and therefore locks the current piston height hˆ. With that, the torque wrench canbe used to increase the torque applied to the suspension, at a rate of 0.5kPa/s, andthe interplay of σˆc to σˆs can be measured for the given value of φ.The schematic shown in Figure 7.3 of the shear yield stress tester illustratesapproximately the correct proportion between the inner and outer radius of the an-nulus cup, r and R respectively. With this large variation, significant curvatureeffects are expected. Therefore, we require a small operating θ sweep. Curva-ture effects were a strong consideration in the design of the apparatus, with thepermeable piston and annulus cup having features that can utilize a varying inner151wall radius insert. However, with reduced curvature effects comes reduction ofsuspension area for which shear can be transferred. This becomes detrimental ifthe suspension cannot withstand considerable shear, as the torque meter will notdetect a sufficient force imposed. A maximum angular deflection of 3 degrees wasdetermined experimentally. Beyond this angular deflection, irregularities begin toappear, including an increasing compression load with increasing shear and radi-ally aligned tube-like structures on the suspension surface (i.e. the top layer of thesuspension seems to tumble under the piston surface, forming tight cylinders ofpulp fibres).An initial 1000 g of φ0 = 0.02 − 0.03 pulp suspension was found to work bestin the equipment. The pulp samples were found to accommodate as many as fiveascending φ targets so long as the angular deflection at any given test did not exceedthe threshold specified.To end this section, we return to a discussion of the coarse wire mesh that pro-vides the roughness to immobilize the constrained suspension. As was discussed inthe introduction of this chapter, studies using traditional rheology techniques havesuggested that surface structures of the order of floc sizes are necessary to ensure noslip between the suspension and the walls of the tester. The wire mesh used in thisdevice is made from 1.2mm diameter wire with opening size of 1.35mm, whichcloser represents the order of magnitude of the fibres themselves rather than theflocs. The justification provided for this choice is two-fold. First, we assume thesignificant normal force provided by the compression is helping facilitate propertraction between the suspension and the tester walls. Second, for increasing volumefractions, the individual flocs are driven together and eventually the suspension isstructurally better thought of as a continuous network rather than flocs surroundedby interstitial fluid. At these higher volume fractions, the significance of flocs islost, and therefore we assume the significant length scale to immobilize the suspen-sion returns down to the individual particle size. We assume the volume fractionstested in this study are sufficiently high such that the chosen wire mesh is sufficientfor providing traction to the suspension. Visually, this assumption appears valid.1520 100 200 30002040608010012000.511.522.5Figure 7.4: Select results from the shear yield stress tester developed. Sam-ple experimental trends are shown with their corresponding fitted ellip-tic yield surfaces for initial motion detection and 3 degrees of angulardeflection, shown as the blue and red dashed lines, respectively.7.3 Results and DiscussionSample experimental results, which show the tracing of the yield curves at fivediscrete φ values, are shown in Figure 7.4 as the scatters of coloured circles, whosecolor correspond to the angular deflection. In all cases, with increasing σˆs, the σˆcdecreases. If the data is assumed to follow the yield surface, this would suggestan elliptic yield surface was an appropriate assumption to have been made. Anadditional observation is the respective data trends’ angular sweep from the σˆc axisis linearly proportional to the magnitude of σˆc. This is illustrated with the greydiagonal line, which bounds the experimental trends up to an angular deflection of3 degrees. The grey diagonal line’s effectiveness in bounding the experimental datais not limited to these five sample results either, with all 26 trials collected showinga similar experimental sweep. The cause or implication of this observation is notunderstood at this time.We next turn to interpreting a shear yield stress from the experimental trends.If we assume the entire data trends follow their respective yield surfaces, then wesimply fit the functional form in Equation 7.12 to the data sets and determine the153values of τy(φ). The fitted ellipses in this manner are illustrated as the red dashedlines, with the resulting τy(φ) values being the x-axis intercepts in Figure 7.4.However, there is a possibility that at some point during the experiment, the col-lected trends diverged from their true yield surfaces. This could be due to curvatureeffects, or other errors in the measurements. To try to bound this uncertainty, wefit a second set of elliptic yield curves to the experimental trends that only use thedata up to initial motion detection via the camera and encoder (approximately 0.1degrees). These respective fits are shown as the blue dashed lines in Figure 7.4. Weassume the true τy(φ) for a given φ, will lie between these two bounds described.The collection of τy(φ) bounds found for all the experiments performed areshown in Figure 7.5 as the connected blue crosses. The majority of the exper-imental points have a sizeable range of approximated shear yield stress. A fewpoints scattered throughout the data have a very tight spread in τy(φ), which seemto align closer to the higher approximation of the data points with larger spread.This may suggest the true measurement lies closer to the upper end of the discretepoints’ ranges, however not necessarily. Included as well in the main figure arethe Py(φ) measurements collected using the shear tester, which are taken as thestabilized σˆc before shear was induced and are shown as red dots. To comple-ment these points, the fitted Py(φ) function in Equation 7.13 is also shown as thesolid red line. Beginning at large φ, we see that the τy bounds are observed to beequivalent to that of Py(φ), and as we travel down the lower values of φ, τy(φ)begins dropping slightly below the measured and fitted Py(φ) measurements. It isunclear at this point whether this observation is truly representative of the suspen-sion’s behaviour, or whether this is merely an experimental artefact. As the volumefraction reduces, so does the traction between the suspension and the rough wallsof the tester. Additionally, the assumption of sufficient roughness made earlier isweakened as the volume fraction is reduced. Both these points would suggest anincreased likelihood of slip occurring between the surfaces of the tester and thesuspension. Inherently, this would produce an artificially low measurement of thesuspension’s shear yield stress at lower values of φ.To further investigate the lower end of the results, we can turn to the rheologicalstudies in the literature. The insert of Figure 7.5 replots the experimental data ofthe main figure along with select, lower concentration results found. Beginning1540 0.1 0.2 0.310010210-2 10-110-2100102Figure 7.5: Results from the shear yield stress tester, and comparison to se-lect results. In the main figure, the corresponding bounds for τy(φ)and the Py(φ) measurements from the shear tester are shown as theblue crosses and red circles, respectively. The fitted trend for Py(φ)for Series 17 in Chapter 3 is shown as the solid red line. In the insert,the results of the main figure are compared to select results at lowerconcentrations of similiar pulps. τy(φ) fits found in Derakhshandeh etal. [18], Hofgen et al. [34], and Mosse et al. [70] are shown as thedashed, dotted, and dash-dotted blue lines, respectively. Py(φ) fit from[34] is shown as the red dashed line. The red dotted line and the redopen circles show Py(φ) measurements from sedimentation and fromthe pump-out experiments of Chapter 4, respectively.with τy(φ), we have included the results collected by Derakhshandeh et al.[18],Hofgen et al. [34] and Mosse et al. [70] as the dashed, dotted, and dash-dottedblue lines respectively, which all used the vane in cup technique introduced earlierfor similar bleached softwood Kraft suspensions. In general, we see our boundedshear yield stresses at higher values of φ agree with the lower concentration results,both in trend and magnitude. A slight inconsistency at the transition between theirsand our results can be seen, however it should be considered that this is a φ regionwhere both methodologies are struggling, being either at their upper or lower endof their suitability ranges. This improves our confidence in the collected τy(φ)155results, even at the lower end of the volume fractions investigated.Turning next to Py(φ) at low concentrations, we re-plot the results from Chap-ter 4 as the dotted red line (sedimentation experiments), and the open face redcircles (pump-out experiments). Although Series 1 and Series 17 pulps are notequal, they are similiar enough to suppliment the discussion here. Additionally, weinclude the compressive yield stress results of Hofgen et al. [34] as the red dashedline. Interestingly, Hofgen et al.’s fitted data aligns well with the pump-out experi-ments collected in Chapter 4, and seem to even suggest the same kink in the trendthat we observed. At these lower concentrations, there does appear to be a moresignificant difference between the two yield stresses (with compressive strengthgreater than shear strength). The literature results also do not seem to suggest thekink in the τy(φ) results. Overall, the experiments collected on the shear testerseem to agree quite well, both in its τy(φ) and its Py(φ) measurements.We return now to data in the main plot of Figure 7.5. Despite τy(φ) report-ing marginally less than the Py(φ) at low volume fractions, this graph has a majorimplication for the modelling of the roll press in Chapter 5, as these results convinc-ingly suggest that over the range of volume fractions seen in the Twin Roll press,that the shear yield stress, τy(φ) is within the same order as the compressive yieldstress, Py(φ). This is a significant difference compared to other flocculated partic-ulate suspensions that are found in the literature, where at least an order magnitudedifference was observed between the suspension’s strength in shear and compres-sion (examples [12, 15, 106]). Further, with respect to the discussion earlier of slipbetween the suspension and the surfaces, if any slip occurred in the experimentalresults in Figure 7.5, then they would be an underestimate of the true shear yieldstress, and therefore even more significant.7.4 Summary and ConclusionsThe objective of this chapter was to investigate if a pulp suspension at elevatedvolume fractions has significant strength in shear with respect to compression.The conclusion of this investigation provides insight into an important assump-tion made in the model reduction found in Chapter 5 for the dewatering in the nipof the Twin Roll press. A unique protocol and experimental apparatus was de-156veloped for determining a pulp suspension’s shear yield stress, τy(φ), at elevatedconcentrations, which reconciled the challenges found using traditional rheologicaltechniques.The results show that the suspension’s shear yield stress is equivalent in mag-nitude to that of the compressive yield stress at volume fractions relevant to the rollpress. This study refutes the assumption made in Chapter 5, and therefore providesmeaningful insight into the failure of the reduction.157Chapter 8Summary, Conclusions, andFuture WorkThis thesis began with presenting a promising modelling approach for capturingthe dewatering behaviour of pulp fibre suspensions [32]. The advancement in thatstudy is a proposed viscoplastic-like constitutive model for the solid effective stressdeveloped by the network of fibres. The primary purpose of this thesis was to inves-tigate the continued suitability of this viscoplastic-like constitutive model acrossa larger variety of pulp suspensions and dewatering experiments. To fullfill thispurpose, five research objectives were undertaken, with intermediate summariesand conclusions found at the ends of Chapters 3-7. Starting with Chapter 3, theviscoplastic-like constitutive model was evaluated across twenty-seven pulp fibresuspensions, constituting a broad range of dewatering behaviours. Calibration ofthe material parameters was performed and simple dewatering experiments, up tointermediate compression rates (≤ 10mm/s), were used to critique the modellingapproach. Across the various suspensions, varying representation was observed,however in all cases, the inclusion of a bulk viscosity improved representationover more traditional, concentration only constitutive model approaches. Chap-ter 4 continued model exploration into the low concentration limit around a pulpsuspension’s gel point. Various experiments were performed both to calibrate therequired material parameters in the new concentration range and to provide increas-ingly severe dewatering experiments, analogous to what was provided in Chapter1583. Concluded from this study is the continued necessity of the viscoplastic-likeconstitutive model for effective representation of the dewatering dynamics of pulpsuspensions in this concentration limit at similar intermediate compression rates(≲ 10mm/s). Indeed, an industrially relevant drainage test (the Canadian StandardFreeness test) that occurs in this concentration limit finds the inclusion of a bulkviscosity paramount in capturing the experimental behaviour. Moving to Chapter5, a model reduction was provided that allowed the evaluation of the viscoplastic-like constitutive model in the steady, two-dimensional dewatering that occurs in anindustrially relevant, Twin Roll press. Difficulty in representation was observedwith the viscoplastic-like constitutive model in this dewatering geometry. This washypothesized to be either due to the elevated compression rates experienced bythe pulp suspension (> 100mm/s), or due to an assumption of the suspension’sstrength in shear, which was made through the reduction. Chapter 6 investigatedthis first hypothesis by returning to the simple uni-axial dewatering geometry usedin Chapter 3. The experimental equipment was modified in order to obtain de-watering dynamics above 10mm/s. Exploring with this new equipment demon-strated an onset of deteriorating representation provided by the viscoplastic-likeconstitutive model at compression rates ≳ 30mm/s. The cause of deteriorationwas attributed to the purely plastic constitutive model and the neglect of the net-work’s minimal elastic response. An evolved, independently calibrated, elasticviscoplastic-like constitutive model was purposed for elevated rates, which demon-strated effective representation of the experimental compressive loads and the mo-tion of the solid phase throughout dewatering experiments with rates ≲ 80mm/s.Finally, arriving at Chapter 7 we explored the pulp suspension’s shear yield stressat increased concentrations to provide clarity on the suitability of reduction used inChapter 5. A new experimental protocol and apparatus was developed for findingthe pulp suspension’s shear yield stress above concentrations accessible by stan-dard rheology techniques. What was determined was that in the concentrationsrelevant to the dewatering of the Twin Roll press, a pulp suspension’s shear yieldstress is equivalent in magnitude to its compressive yield stress. This, therefore,suggests an alternative reduction is necessary to appropriately represent the pulpsuspension in the dewatering action of the Twin Roll press, and therefore excludesa simple re-evaluation with the extended elastic viscoplastic-like constitutive model159for now.Throughout the various chapters, additional discussions, investigations andspeculations, aside from simply the suitability of the constitutive model, were pro-vided. This extended content aimed at fullfilling a second underlying purpose ofthe thesis: an exploration of the governing material parameters to investigate if in-sight into their micro-structural origins can be obtained. While one would expectminimal conclusions reached with respect to the micro-structure through investi-gating these macro-scale parameters, we consider this underlying purpose impor-tant to add context with respect to other materials and to motivate future studies.As described, pulp fibre suspensions constitute a complex material, comprised ofhighly varying mixtures of fibres with a complicated construction. Above the gelpoint threshold, these fibres in suspension network and form a solid, deformable,porous structure with unique material properties when compared to other fibre net-works. To start, the evaluation of the viscoplastic-like constitutive model acrossvarying pulp suspensions and into a low concentration range has illustrated a sub-stantial bulk viscosity effect of the network of fibres. As discussed, a suspensionwill exhibit a bulk viscous effect due to the motion of the particles through thefluid. However, in the case of pulp suspensions, the required bulk viscosity foundwas shown to far exceed this simplistic mechanism. The scaling of the bulk vis-cosity was also shown to be inversely proportional to each suspension’s charac-teristic permeability, and insensitive to its compressive yield stress. The variouspermeability measurements made through Chapters 3 and 4 displayed different be-haviours with respect to scaling than more ideal fibre networks (such as nylon),which effectively scale with respect to their fibre radii. Compressive yield stress ofthe various pulp suspensions was found to vary minimally, and displays differentvolume fraction dependency when compared to other fibre suspensions, both in theintermediate and low concentration limits. Finally, a pulp suspension’s completenetwork elastic response was found to be quite complex, however, when simplis-ticly represented, was found to be nearly proportional to the suspension’s compres-sive yield stress. The fibres’ complex structure was speculated to be the source ofthese various findings throughout the chapters. Rationalization of these behaviours,however, clearly demand micro-structural investigations and further future work,which are presented in the next section.1608.1 Future WorkAs is the case with any study, addressing the project’s objectives has provided addi-tional questions to investigate. Beginning in Chapter 3 and 4, it is clear that furtherexperimental efforts in permeability should be undertaken. This could include trialswith very select suspensions. For example, this could include systematic investiga-tions controlling individual factors such as: particle size distribution, varying levelsof refining, fines content, or varying chemical additives. In addition, collections ofthe suspensions’ water retention values (WRV ) would be insightful to investigatethe merit of the speculations that the definition of φ is incorrect due to the watertrapped in the fibre walls. Additionally, some investigation to understand the porescale in which the permeating fluid passes through would be valuable. Future workinto the compressive yield stress would include evaluation of the micro-structuralmodel, comprising of frictional rearrangements under elastic formal forces at thecontact points. To investigate the frictional component, perhaps compression be-haviour could be determined in a different fluid with lower lubricating properties(i.e. a suspension of fibres in air). To evaluate the proposed fibre-scale deforma-tions, microscopic compaction trials should be undertaken, which may also pro-vide insight into the linear φ dependency near the suspension’s gel point. Throughdynamic microscopic experiments of pulp suspension dewatering, insight into theorigin of bulk viscosity may also become evident. Finally, returning to the uni-axialtester, dewatering experiments at varying initial height should be undertaken to en-sure the fitted bulk viscosity scaling is determined to be equivalent, and, therefore,represents a true material parameter and not simply a tuning parameter.Moving on to Chapter 5, the first suggested future work is a reduction ofthe equations with the new knowledge of the elastic viscoplastic-like constitutivemodel and the significant shear yield stress. Second, in order to support expandedmodelling efforts, richer dynamic compression load information of the Twin Rollpress is needed. As was shown in Chapter 6, a significant amount of informationwith respect to the constitutive modelling can be inferred simply from the shape ofthe compressive load trend. If, for example, a force or pressure transducer could bebuilt into the permeable surface of the roll, or passed through the nip, buried in thesuspension, the corresponding compressive load pulse could be obtained, similar161to Martinez [69]. Practically, this experiment certainly would cause some compli-cations, however the value of this data would be significant for future model com-parisons. Additionally, perhaps trials with the Twin Roll press could be performedwith significantly slower rotation rates. Should sufficiently slow compressions oc-cur, perhaps the suspension’s elastic response can be neglected, and therefore theinterplay between compression and shear, without the added complexity of elastic-ity, could be investigated.Future work from the progress of Chapters 6 and 7 would start with an investi-gation into the full unloading behaviour of the pulp suspension. Highly hystereticbehaviour was observed between the unloading and loading curves, which remainsunexplained. 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Large torn sections of thepulp sheet (approximately 25 g) were soaked in reverse osmosis (RO) water (ap-proximately 2L) for a minimum of 30 minutes. After soaking, we tore the largesections of pulp in the soaking water down to “chunks” of approximately 5 cm x5 cm. The sample was next poured into a bench top re-pulper. We ran the re-pulperfor 15000 revolutions to obtain our suspension. This was repeated multiple times toobtain the appropriate amount of suspension necessary for the various experiments.The re-pulping process resulted in a 0.0125 (wt/wt) consistency suspension. Formost applications, φ0 was chosen to be 0.02 − 0.03, and so we concentrated thesuspensions before storing. This was done using a tap aspirator, vacuum flask witha filtration funnel, and a nylon mesh screen. In order to avoid pulling fibres throughthe nylon mesh, we sparingly provide the vacuum and pour substantial amounts ofsuspension onto the nylon mesh. A thick filter cake develops, which traps the smallparticles from passing through the nylon mesh. Once prepared, the suspension iskept in a cold room to slow degradation. Vigorous mixing of the suspension isperformed before sampling for trials.173The mechanically pulped suspensions used in this project came in the form ofmoist, shredded stock. A similar approach used for the chemically pulped suspen-sions was undertaken. Approximately 25 g was soaked in reverse osmosis (RO)water (approximately 2L), however the water was microwaved to obtain a tem-perature of approximately 60○C. Again the sample was put into the re-pulper for15000 revolutions, and concentrated to the target φ0 = 0.02− 0.03. Minor attemptswere made with Series 22 through 25 to minimize fines, by rinsing the concentratedsuspensions over a mesh with 0.149mm hole size. After which, the retained solidwas recovered and diluted back to φ0 = 0.02 − 0.03.The distinction in suspension preparation for chemical additive trials (Series 2through 6) begins after the re-pulping step. Instead of concentrating the obtained0.0125 (wt/wt) consistency suspension, it is further diluted to 0.005 (wt/wt) con-sistency in preparation for the additives. This low consistency is necessary to allowthe chemicals to attach to the fibre surfaces effectively. Chemical additives usedwere at a 0.1 (wt/wt %) concentration. The appropriate amount of additive wasintroduced slowly to a vigorously stirred sample of suspension. With all the chem-ical added, the suspension was mixed for a further 30 seconds. At this point, thesuspension was concentrated to the target φ0 = 0.02 − 0.03.Finally, the nylon fibres came in the form of dry, chips of fibres. They weremixed into suspension using a table top mixer and RO water, to our target φ0 =0.02 − 0.03.A.2 Uni-Axial Dewatering ExperimentThe benchtop uni-axial dewatering experimental apparatus developed for [32] and[76] is implemented again in this study. The dewatering apparatus utilises an MTSSystems Corporation 858 Table Top System (www.mts.com) hydraulic materialtester which drives a permeable piston into a closed based suspension chamber. Aload cell measuring the compression force is positioned underneath the suspensionchamber, as can be seen in Figure A.1.At the beginning of the study, a pre-existing permeable piston was imple-mented, which was constructed from a solid cylinder of stainless steel measuring76.2mm in diameter, and 19.05mm tall. Eight 12.7mm holes drilled through the174Permeable PistonSuspension ChamberLoad CellLoad Cell PlatformFigure A.1: Uni-axial dewatering experimental apparatus utilizing an MTS858 Table Top System that drives a permeable piston into a suspensionchamber. Figure taken from [76].cylinder allowed water flow through, and an O-ring groove was machined aroundits circumference. The piston attached to the MTS hydraulic actuator rod with acentrally located tapped hole. To promote flow distribution, a coarse wire meshwas attached to its bottom face, and a fine plastic mesh with hole sizes of 0.33mmwas attached upon the coarse weave. The permeable piston’s assembly is shownin Figure A.2. The complementary suspension chamber, also shown in FigureA.2, is constructed from a clear PVC pipe, with an inner diameter of 76.2mm,sandwiched between a base plate and top collar. Four long fasteners are used toassemble the cup, and a rubber gasket is used to seal the base plate. The inner lip ofthe cup was chamfered to help with aligning the the suspension chamber with theinserting piston. The chamber sits unconstrained on a flat platform that is threadedinto the load cell.The load transducer and the MTS frame itself have a maximum compressionforce of 11kN , however we limited the maximum compression force to 6kN .The maximum stroke achievable with the piston is 100mm, and the position of175Top CollarRubber GasketBase PlateRemovable Side WallFine MeshCoarse MeshPermeable PistonFigure A.2: Details of the suspension chamber (left) and the permeable pis-ton (right) of the uni-axial dewatering experimental apparatus. Figuretaken from [76].the hydraulic actuator is measured with an integrated LVDT. The movement ofthe hydraulic actuator is provided by the stand-alone controller unit that can con-trol the piston position and perform basic motion functions (ramps, sine waves,square waves, etc.). The ramp function was implemented, with max compres-sion rates limited to 10mm/s. Measurements of compression load and pistonposition are displayed on the controller unit as well. Remote connection to thecontroller via LabVIEW was implemented and an interface for specifying exper-imental inputs and measuring and recording data was developed. Recording ofexperimental data was performed through a National Instruments 6009 USB DAQ(www.nationalinstruments.com). The overall experimental apparatus is shown inFigure A.3.A.2.1 Protocol: DewateringThe experimental protocol developed uses a 250 − 275 g suspension with φ0 =0.02 − 0.03. The piston first is manually lowered to the top of the suspension us-ing the stand-alone controller and the initial height h0 is recorded. The piston isthen raised approximately 20mm above the top of the suspension at which pointautomation of the trial begins. The piston first moves downwards to the top of thesuspension at a rate of 0.5mm/s, at which point it pauses temporarily. Data collec-tion begins, and the dewatering experiment commences at the specified compres-176Figure A.3: Complete uni-axial dewatering experimental apparatus with allcomponents shown and identified. Figure taken from [76].sion rate until the maximum compression load is achieved. Once the experimentaltrend has been collected, compliance of the equipment is accommodated for bycorrecting the height of the piston with respect to the magnitude of load experi-enced. Piston friction of 6500Pa is also subtracted from the measured load. Thedata is saved to a text file. The experimental sample is recovered, remixed, filtered,and dried to obtain the dry mass of solids ms for φ determination as per Equation2.1. Typically, four experiments (fresh samples each time) are performed for agiven compression rate for a particular suspension, to accommodate experimentalvariability. The average of the four trends is reported, along with two standarddeviations of the average.A.2.2 Protocol: Calibration of Compressive Yield StressThe protocol for calibrating compressive yield stress using the uni-axial dewateringequipment is to perform a very slow dewatering experiment, such that any differ-ential compaction that may occur has sufficient time to diffuse back to uniformity.This protocol was first proposed by Pettersson et al. [78], and has been imple-mented in our previous studies [32, 76]. To demonstrate the limit under which this177experiment is performed, we start with Equations 3.10 combined as∂φ∂t= ∂∂z(γ ∂Πy(φ)∂φφK(φ)∂φ∂z− φK(φ) ∂∂z[φ2∂v∂z]) , (A.1)When dewatering is very slow, γ ≫ 1, both ∂φ∂t and ∂v∂z are small, thusO(1). For theequation to then balance, ∂φ∂z is O(1/γ). With this conclusion, we can approximatethe solid phase continuity of Equation 3.10 as∂φ∂t= − ∂∂z(vφ) = −φ∂v∂z− v∂φ∂z∂φ∂t≈ −φ∂v∂z(A.2)which leads toφ = φ0hand v = − zh. (A.3)Therefore, at sufficiently slow dewatering rates, the volume fraction within thesuspension remains approximately uniform, and the compressive yield stress canbe calibrated directly from the compressing load of the permeable piston as shownwith Equation 3.14:Py(φ(tˆ)) ≈ σ(tˆ). (A.4)The appropriate dewatering rate for this slow speed limit was found by successivelyslower experiments, until the trends collapse onto one another. A compression rateof 0.001mm/s was determined sufficient.The experimental protocol is the same as a typical dewatering experiment. Theonly variation is with respect to the starting condition. Instead of the piston movingto the top of the suspension at 0.5mm/s, it compresses into the suspension to atarget height such that φ = 0.04, as approximated from the initial consistency andthe h0. This lower start point was chosen to save time during the experiment, andsaw no implication on the measurement due to negligible compressive loads at thisvolume fraction. Four experiments (fresh samples each time) are performed fora particular suspension to accommodate experimental variability. The average ofthe four trends is reported, along with two standard deviations of the average. Theaverage trend is fitted to the functional form shown in Equation 2.13 by linearizing178the expressionln [Py(φ)] = ln [a] + b ln [φ] − c ln [1 − φ] (A.5)and using linear regression to find our empirical constants a, b, and c.A.2.3 Retrofit: Spring 2018Part way through the project, the decision was made to modify the existing uni-axial dewatering equipment to expand its capability, and to improve certain aspectsof its design. The motivation behind this came from a hypothesis that the poorfitting model results seen of the experimental roll press results in Chapter 5 wasdue to having calibrated the bulk viscosity at insufficiently high speeds. In orderto obtain higher speed dynamics, a reworking of the experimental equipment andthe LabVIEW interface was necessary to reduce delay and improve precision ofthe compression profile, improve certainty in the height calibration and surfaceparallelism, and finally reduce drag of the fluid passing through permeable piston.Additional aspects of the equipment that were improved through the rebuild in-clude an improved experimental start condition for better detection of h0, reducedcompliance of the suspension cup, and improved viewing of the suspension and itsbase through the clear cup walls. Finally, the rebuild provided an opportunity tofulfill an industrially encouraged modification of changing the permeable surfaceof the piston to the surface found on the rolls of the Twin Roll press.We start detailing the rebuild with the hardware. The new cup’s two-piece de-sign has an aluminium machined base that is faced on both surfaces, as seen inFigure A.4a. This drastically reduced the compliance of the cup (and improvedparallelism with the piston). A groove in the face houses an o-ring that seals thecup wall when assembled. The stepped cylinder in its center inserts into the cupwall and is viewable from the side when the wall is fastened down to the base, al-lowing clear detection of the lower bound of the suspension during particle trackingstudies. Further clarity for imaging is provided by the removal of the long fastenersused for assembling the cup as found on the previous design (see Figure A.2). Theconstruction of the cup wall is very basic, with a 92mm section of a clear PVCpipe, with an inner diameter of 79.5mm and a flange cemented to its base, and aninner chamfer added to the top to aid in piston alignment. The assembly of the cup179(a) (b)Figure A.4: In (a), the new base plate of the cup assembly is shown. In (b),the cup wall is assembled onto its base. Notice the stepped surface ofthe base plate viewable through the wall of the cup.is shown in Figure A.4b.The new piston shares its construction to the piston developed for the perme-ability tester, which is detailed in the following section. The construction beginswith an aluminium body with various through holes for the evacuating fluid toflow through, as can be seen in Figure A.5a. A groove is machined into its cir-cumference for a PTFE spit ring used to locate in the suspension cup. On thebottom side of the piston, square posts are machined to aid in flow convergenceto the discrete passages through the body of the piston and to support the perme-able surface attached with fasteners, as can be seen in Figure A.5b. The permeablesurface is constructed from stainless steel with repeating arrays of 0.66mm di-ameter through-holes spaced 1.5 − 2mm apart, resulting in approximately 11%open area. Additionally shown in this view is a 3D-printed component that slipsover the square posts and sits on the flat plane of the piston body. This componentaims at further guiding the evacuating water to the discrete passes of the pistonbody. It has a complex topography on the flow side from exaggerated fillets aroundholes aligned with those in the body of the permeable piston. The purpose of thiscomponent is to reduce the drag of the evacuating fluid.180(a) (b)Figure A.5: Two views of the new permeable piston for the uni-axial dewa-tering experimentExtensive modifications were also made to the LabVIEW interface used forcontrolling the uni-axial dewatering equipment. A new interface was created, basedupon a superior state-machine architecture, which significantly improves modular-ity for expansion of experiments, decreases delay and redundancies, is safer forboth the operator and the equipment, and allows far more effective troubleshoot-ing. With the new LabVIEW program, two additional experiments were developedfor exploring the elastic response of the suspension, and suspension dynamics atelevated rates. Details of these new tests are provided in the following two sub-sections.Loading/Unloading ExperimentsFor exploring the network’s elastic response, a new test that utilizes the ramp func-tion generator of the MTS stand-alone controller is used to perform various load-ing, and then unloading experiments through piston motion reversals. Flexibilityof the test is provided by a number of factors, which include 1) the number ofloading/unloading cycles performed at target volume fractions, 2) the stop condi-tion (being either a load condition or an unloaded target volume fraction), and 3)181independent specifications of loading and unloading piston speeds. All of thesefactors are inputs into the new LabVIEW interface at the commencement of an ex-periment. At moments during the test where a direction change is to occur, a 30second hold is maintained first to stabilize the suspension. Upon completing thefinal load-reversal loop, the experiment proceeds until the maximum compressionload of 6kN is achieved.The experimental protocol from the operator’s perspective is the same as theother uni-axial experiments performed. An initial sample of 250 − 275 g of φ0 =0.02 − 0.03 is put into the suspension cup and placed underneath the permeablepiston. The various details of the experiment are entered into the input boxes ofthe LabVIEW interface. The piston is first manually lowered to the top of thesuspension and the initial height h0 is recorded. The piston is then raised approx-imately 20mm above the top of the suspension at which point automation of thetrial begins. The piston first moves downwards to the top of the suspension at arate of 0.5mm/s, at which point it pauses temporarily. Data collection begins, andthe dewatering experiment commences. Once the experimental trend has been col-lected, compliance of the equipment is accommodated for by correcting the heightof the piston with respect to the magnitude of load experienced. The data is savedto a text file. The experimental sample is recovered, remixed, filtered, and dried toobtain the dry mass of solids ms for φ determination as per Equation 2.1.One difference with these reversal experiments is the treatment of piston fric-tion. Addressing piston friction with reversals of direction becomes quite compli-cated due to experimental realities such as the hydraulic ram’s dither and tolerancesbetween the groove and the piston seal. We therefore do not attempt to subtract theload contribution due to friction. The impact of this should be quite minimal.High Speed Uni-Axial ExperimentsFor exploring the pulp suspension’s high speed dynamic response, a new test uti-lizing the external waveform input function of the MTS stand-alone controller hasbeen developed. This function of the controller reads a supplied input voltage be-tween −10V and 10V , which corresponds to the location of the piston between−50mm and 50mm of its central location. Calibration with respect to the base182ℎ0(𝑇𝑦𝑝. 40 𝑚𝑚)ℎ𝑒𝑛𝑑(𝑇𝑦𝑝. 5 𝑚𝑚)𝑑෠ℎ𝑑 Ƹ𝑡 𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑇𝑖𝑚𝑒𝑃𝑖𝑠𝑡𝑜𝑛𝐻𝑒𝑖𝑔ℎ𝑡,෠ ℎ(Ƹ 𝑡)ParabolicExponentialRampFigure A.6: Sample compression profiles implemented with the retro-fitteduni-axial dewatering experiment.of the suspension cup can use this range of position to convert the voltage to an hˆrange. A voltage trend in time therefore can be used to induce an hˆ(tˆ) profile andtherefore a variety of functions could be realized. Sample profiles implemented todate are shown in in Figure A.6, with the parabolic hˆ(tˆ) profile being the desirablefunctional form for the high speed uni-axial experiments.The experimental implementation of this begins in the LabVIEW interface.First, parameters of dhˆdtˆ initialand hend are specified, and h0 is collected experi-mentally. The volume of the suspension and the hend are typically chosen to beclose to those of the compression seen in the Twin Roll press. With these threeparameters, the constants defining a parabolic hˆ(tˆ) profile can be solved for andhold periods at h0 and hend are appended at the start and end of the experimentrespectively (as can be seen in Figure A.6). This represents our target experimentaltrend, in which we now turn to converting to an analog voltage trend to send to thecontroller. This starts with the analog output from the National Instruments 6009USB DAQ. Unfortunately, the output range of the DAQ is 0 − 5V . This requiredbuilding a voltage scaling device, affectionately named D-Ash box, that takes theDAQ 0 − 5V input, and outputs the required −10 to 10V for inputting to the MTScontroller. The circuit diagram and the completed construction of D-Ash box isshown in Figure A.7. With the communication and scaling between LabVIEW and183(a)𝐿 𝑁𝑊𝑎𝑙𝑙 𝑆𝑜𝑐𝑘𝑒𝑡𝐿 𝑁 −𝑉 +𝑉12 𝑉 𝑃𝑜𝑤𝑒𝑟 𝑆𝑢𝑝𝑝𝑙𝑦−𝑉 +𝑉𝐿 𝑁12 𝑉 𝑃𝑜𝑤𝑒𝑟 𝑆𝑢𝑝𝑝𝑙𝑦PC + LabVIEW−+22           Digital           171                  Analog               16NI USB 6009USB~~𝑅1𝑅2𝑅3𝑅4(b)Figure A.7: Circuit and assembled view of D-ash box.the MTS piston position complete, calibration for proper translation of hˆ(tˆ) fromLabVIEW to an outputted 0 − 5V was completed.The experimental protocol remains quite similar to the original dewatering ex-periment. An initial sample of 250 − 275 g of φ0 = 0.02 − 0.03 is put into thesuspension cup and placed underneath the permeable piston. The various detailsof the experiment are entered into the input boxes of the LabVIEW interface. Thepiston is first manually lowered to the top of the suspension and the initial height h0is recorded. The piston is then raised approximately 20mm above the top of thesuspension at which point automation of the trial begins. The piston first movesdownwards to the top of the suspension at a rate of 0.5mm/s, at which point itpauses temporarily. Data collection, and hˆ(tˆ) specification via analog voltage out-put begins, and the dewatering experiment commences. Once the specified hˆ(tˆ)trend has completed, the piston reverses away from the suspension. An additionalmax load condition is monitored during the experiment, should the dynamics pro-duce a load above our 6kN limit. If the load exceeds this limit, the piston is rapidlystopped, regardless of location along the hˆ(tˆ) trend, and the experiment is aborted.This emergency stop of the experiment is one example where minimal delay iscrucial, for which the new LabVIEW interface is far superior.Should the experiment run successfully, the compliance of the equipment isaccommodated for by correcting the height of the piston with respect to the mag-184nitude of load experienced. Friction of 6500Pa between the permeable piston andthe cup walls is subtracted off from the load, and the data is saved to a text file.The experimental sample is recovered, remixed, filtered, and dried to obtain thedry mass of solids ms for φ determination as per Equation 2.1. Typically, fourexperiments (fresh samples each time) are performed for a given hˆ(tˆ) profile fora particular suspension, to accommodate experimental variability. The average ofthe four trends is reported, along with two standard deviations of the average.A.3 Permeability ExperimentPermeability data collection is performed on a Darcian flow cell experimental ap-paratus that was developed for [32] and [76]. The machine consists of a cylindricalsuspension chamber with a permeable top surface, and a permeable piston lowersurface. The piston is capable of compressing the suspension to varying loadsand a flow loop permeates de-aerated water through the suspension at controllablerates. The suspension chamber is built from a steel pipe with inner diameter of101.6mm, and height of 305mm. Flanges are welded onto either end of the pipe,and the chamber is oriented vertically. The bottom flange is mated to a flange cap,which is assembled and bolted down to a supporting frame. A centrally locatedhole in the flange cap provides passage of a hydraulic actuator rod, which connectsto the bottom of the permeable piston. Additionally, two flow ports are drilled inthe flange cap for allowing permeating fluid in and for measuring fluid pressure atthe bottom of the suspension. A section view of the permeable tester is shown inFigure A.8.The permeable piston within the suspension chamber has a range of 160mm,and can provide compressive loads up to approximately 1MPa. The permeablepiston is constructed from aluminium and has two PTFE seals around its circumfer-ence. Twelve through holes of varying size are machined into the piston allowingpermeable water to flow up to a flow distribution plateau constructed from ma-chined square posts on the top of the piston that support an assembled permeablesurface made from a fine metal mesh with hole sizes of 0.28mm. The top perme-able surface, referred to as the screen spacer, has a similar construction, both ofwhich can seen in Figure A.9. The assembly of the top of the experimental tester185Flange CapFlangeScreen SpacerFlangeFlange CapSuspension ChamberPermeablePistonHydraulic Actuator Pressure/FlowPortsPressure/Flow PortsFlange CapWashers & O-Ring SealWashers & O-Ring SealScreen SpacerFigure A.8: Section and exploded view of the permeability experimental ap-paratus. Figure taken from [76].is shown in Figure A.8. The top flange cap, like the bottom, has two flow pas-sages; one for the permeating fluid to leave and the second for measuring the fluidpressure. The removal of these components is necessary to add a pulp suspensionfor testing. O-ring seals and bolt washers were used between components to en-sure a seal was achieved, and to assure assembly height was consistent betweenexperiments. Flange bolt torque was also repeatedly set to 80ft ⋅ lbs.A 0.56kW Flint and Walling booster pump (www.acklandsgrainger.com), con-trolled by an Eaton M-Max (www.mcmaster.com) VFD is used to permeate de-aerated RO water through the suspension. Temperature of the water is maintainedusing a plate heat exchanger, and the fluid reservoir is kept in a 0.95MPa vac-uum. Temperature and dissolved oxygen are measured by a Eutech InstrumentsDO 500 meter (www.coleparmer.ca). The compressive load and movement ofthe permeable piston is provided by a hydraulic linear actuator, which is pow-ered by a 1.5kW hydraulic power supply provided by Monarch Hydraulics Inc.186Fine MeshFine Mesh Support DiscPermeablePistonFine MeshScreen SpacerFine Mesh Support DiscFigure A.9: Permeable piston and screen spacer of the permeability experi-mental apparatus. Figure taken from [76].(www.mcmaster.com). Pressure drop across the compressed suspension is mea-sured using an OMEGA 1.03MPa (www.omega.ca) differential pressure trans-ducer. The height of the piston is measured by an OMEGA 0.305m stroke linearpotentiometer. Finally, the compressive load imposed on the suspension is mea-sured by an OMEGA 13.79MPa pressure transducer, measuring the hydraulicfluid pressure powering the linear actuator.A LabVIEW interface was developed for data collection. Analog signals fromthe various sensors are read by a National Instruments 6009 USB DAQ(www.nationalinstruments.com). Controls of the apparatus (permeating flow ratesand suspension height) are manually recorded.A.3.1 ProtocolThe protocol for calibrating permeability using the tester involves low flow rate,steady state permeation experiments, where the suspension can be approximated asuniform (negligible differential compaction due to the permeating flow). Schemat-ically, the protocol is shown in Figure 2.1b. This protocol was implemented inother studies found in the literature (eg. [77, 79]). Due to the one-dimensionalnature of the experiment and negligible body forces, we can demonstrate the limitunder which this experiment is performed with equations from Chapter 3. We start187with the steady, stationary solid phase forms of Equations 3.2-3.4 as∂∂zˆ((1 − φ) vˆl) = 0, (A.6)(1 − φ)(vˆl) = −k(φ)µ∂pˆ∂zˆ, (A.7)∂∂zˆ[−pˆ − Pˆ] = 0. (A.8)The experimental method involves permeating a volumetric flow rate Qˆ through thesuspension of area A, held at a fixed compression load of σ between the permeablepiston at zˆ = hˆ and the permeable top boundary at zˆ = 0. The permeating pres-sure drop ∆pˆ is maintained. The average volume fraction within the suspension isdefined asφ¯ = 1hˆ∫ hˆ0φdzˆ. (A.9)We start by integrating Equation A.6, with the condition that vˆl = − QˆA if φ = 0.This expression, along with Equation A.8 and the constitutive model Equation 2.11with stationary solid phase (vˆs = 0) are substituted into Equation A.7, arriving atk(φ)∂Py(φ)∂zˆ= −QˆµA. (A.10)Boundary conditions at the permeable surfaces can be found asPy(φ) = σ +∆pˆ, zˆ = 0,Py(φ) = σ, zˆ = hˆ. (A.11)With ∂Py(φ)∂zˆ = P ′y ∂φ∂zˆ , a pressure scaling of P ′y(φ) is introduced along with thefollowing dimensionless parametersz = zˆhˆ, K = k(φ)A∆pˆQˆµhˆ, P = Py − σP ′y(φ¯) , δ = ∆pˆP ′y(φ¯) , (A.12)which when substituted into Equation A.10 and the boundary conditions giveK∂P∂z= −δ (A.13)188P = δ, z = 0,P = 0, z = 1. (A.14)If δ ≪ 1, then ∂φ∂z ≪ 1, which is the desired limit.The following expansions and balances were provided in [76], and reproducedas follows. We start by expanding asφ(z) = φ¯ + δφ1(z) + δ2φ2(z) +O(δ3),K =K0(φ) + δK1(φ) + δ2K2(φ) +O(δ3),P = P0(φ) + δP1(φ) + δ2P2(φ) +O(δ3), (A.15)whereφ¯ = ∫ 10φdz, (A.16)∫ 10φi dz = 0 for all i ≥ 1. (A.17)The resulting leading order balance and its boundary conditions can be foundto beK0(φ¯)∂P0(φ¯)∂z= 0, (A.18)P0(φ¯) = 0, z = 0,P0(φ¯) = 0, z = 1, (A.19)which when solved givesP0(φ¯) = 0. (A.20)We next determine the O(δ) balance and its boundary conditions asK0(φ¯) ∂∂z[φ1P ′0(φ¯) + P1(φ¯)] = −1, (A.21)φ1P′0(φ¯) + P1(φ¯) = 1, z = 0,φ1P′0(φ¯) + P1(φ¯) = 0, z = 1, (A.22)with prime indicating partial derivatives with respect to φ, thus P ′0(φ¯) = ∂P0(φ¯)∂φ .The results of Equation A.21 with the boundary conditions shown in Equation A.22189areK0(φ¯) = 1, (A.23)P1(φ¯) = 12, (A.24)φ1 = 1P ′0(φ¯) [12 − z] . (A.25)Finally, we turn to the O(δ2) balance and its boundary conditions[φ1K ′0(φ¯) +K1(φ¯)] = ∂∂z [φ2P ′0(φ¯) + φ21P ′′0 (φ¯)2 + φ1P ′1(φ¯) + P2(φ¯)] , (A.26)φ2P′0(φ¯) + φ21P ′′0 (φ¯)2 + φ1P ′1(φ¯) + P2(φ¯) = 0, z = 0,φ2P′0(φ¯) + φ21P ′′0 (φ¯)2 + φ1P ′1(φ¯) + P2(φ¯) = 0, z = 1,(A.27)which give the resultK1(φ¯) = 0. (A.28)Inserting these solutions and reintroducing the dimensional variables, we arrive atthe following expressionφ(zˆ) = φ¯ + ∆pˆP ′y(φ¯) [12 − zˆhˆ] +O(δ2), (A.29)given that P ′0(φ¯) = P ′(φ¯) + O(δ) = 1 + O(δ) from our definition of P shownin Equation A.12. Finally, expressions for the compressive yield stress and thepermeability are found asPy = ∆pˆ2+ σ + P ′y(φ¯) [(φ − φ¯) +O(δ2)] , (A.30)k = QˆµhˆA∆pˆ[1 + (φ − φ¯)K ′0(φ¯) +O(δ2)] . (A.31)We now consider the expression for k and φ(zˆ) that have been found. If we190evaluate these expressions for zˆ = hˆ/2, we haveφ = φ¯ +O(δ2),k(φ¯) = QˆµhˆA∆pˆ+O(δ2). (A.32)The conclusion of this analysis suggests that if evaluated at zˆ = hˆ/2, we can neglectthe impact of flow induced compaction, and use the leading order term of perme-ability as an approximation with error O(δ2). Accuracy of this approach involveskeeping the experimentally defined error term δ small, which for the data collectedwas always maintainedδ = ∆pˆP ′y(φ¯) < 0.05. (A.33)The experimental protocol developed used a 500 or 1000 g suspension withφ0 = 0.02 − 0.03. With the suspension added into the chamber, and the systemclosed, a low permeation began with no compression of the suspension at first tocirculate water through the system. This state is maintained for approximately60mins to allow stability of dissolved oxygen (always below 3ppm) and temper-ature. After this, the piston is moved slowly to its first of multiple heights that willbe collected for the particular sample. The suspension is allowed to stabilize forapproximately 20mins and the flow rate is controlled to maintain δ < 0.05. Oncethe suspension has stabilized, a flow rate measurement (taken via the filling of astandpipe and timer), a pressure drop, height, compression load, and temperaturemeasurements (all averaged over the duration of the flow collection) are taken andrecorded in the LabVIEW interface. These experimental values constitute a dis-crete permeability measurement (as per Equation A.32) at a given average volumefraction. Once collected, the permeable piston is moved to the next descendingvalue of hˆ and the protocol repeats. Once the final compression height has beencollected, the suspension chamber is opened, the sample is collected, remixed, fil-tered, and dried to obtain the dry mass of solids ms for φ determination as perEquation 2.1. Typically, four experiments (fresh samples each time) are performedfor a particular suspension, with each experiment contributing 5 − 10 data points.The discrete points are fitted to the functional form shown in Equation 2.9 by lin-191earizing the expressionln (k(φ¯)) − ln (ln(1/φ¯)/φ¯) = ln (a) − bφ¯, (A.34)and using linear regression to find the empirical constants a and b.A.3.2 Retrofit: Spring 2018Part way through the project, the decision was made to modify the existing perme-ability equipment to improve certain aspects of its design. The motivation behindthis came from a hypothesis that the poor fitting model results seen of the exper-imental roll press results in Chapter 5 could be traced back to an incorrectly cal-ibrated permeability, because of primarily two factors, being: 1) the open area ofthe permeable screens did not span all the way to the suspension chamber’s wall,and 2) a lack of confidence in height calibration due to poor surface parallelismbetween the permeable piston and the top permeable surface. Additionally, the re-build provided an opportunity to fulfill an industrially encouraged modification ofchanging the permeable surface to the surface found on the rolls of the Twin Rollpress.To start the redesign, the inside of the cylinder of the suspension chamber wasturned to improve concentricity down its length. The faces of the welded flangeswere also machined to improve mating with the flange cap and screen spacer. Anew lower flange cap (which also was faced) was then assembled after which twolocation holes were machined through the flange and cap, in which high-precisionlocation pins (−0.0005”) were inserted. With the cap centrally located to the sus-pension cylinder, a central hole was drilled through the cap for the insertion of abrass bushing that would align the shaft extending up to the permeable piston. Onthe inside face of the flange, a step was machined for the insertion of a lip seal.The inserted brass bushing and inner face lip seal are shown in Figures A.10a andA.10b respectively.Centric, parallel alignment of the rod extending up to the permeable piston isachieved at this point. Next, focus turned to the permeable piston itself. As canbe seen in Figure A.8, the original design implemented a rod-end bolt, and a cle-vis rod-end bolt to reduce constraint of the piston in the original cylinder. Since192(a) (b)Figure A.10: Pictures of the rebuilt lower flange cap. In (a), we can seethe brass sleeve inserted into the cap for aligning the rod attached tothe permeable piston. In (b), the inner face of the cap shows a smallcover plate over the shaft’s lip seal. Two additional holes for flow andpressure measurement can be seen in both figures.the original cylinder was not machined, this had some merit, however the piston’sparallelism with the top surface then was solely left to the the two seals of the pis-ton, which had a certain degree of compliance. Additionally damning, with respectto parallelism of the piston, was an absence of an alignment bushing through thebase flange cap in the original design. To eliminate this in the new design, the rodthreaded directly into the permeable piston, as is shown in Figure A.11a, and theseals on the piston were replaced with stiffer and taller PTFE split rings which weremachined to a close fit inside cylinder. To accommodate the desired permeable sur-face, an adapter that fit into the permeable piston’s existing construction provided asecondary square post construction that a disc of the permeable surface could boltdown to. The permeable surface is constructed from stainless steel with repeatingarrays of 0.66mm diameter through-holes spaced 1.5-2mm apart, resulting in ap-proximately 11% open area. The assembled permeable piston is shown in FiguresA.11a and A.11b.A similar modification to that of the permeable piston was performed to the193(a) (b)Figure A.11: Pictures of the rebuilt permeable piston. In (a), the seals, theassembled permeable surface, and the rod threaded directly into thepiston assembly can be seen. In (b), we see a view of the assembledpiston, showing the through holes for the permeating fluid.screen spacer to accommodate the new permeable surface. Its final assembly isshown in Figure A.12a. The installed permeable piston into the suspension cham-ber is shown in Figure A.12b. Finally, calibration of height was further assured bythe machining of two pucks of known heights, which were used during setup.Upon completion of the retrofit, experimental results of a particular pulp (Se-ries 17 of Chapter 3) were collected and compared to results collected just beforedismantling the equipment for the rebuild. The before and after results are shownas the red and blue results respectively in Figure A.13. As can be seen, despite theimprovements to the equipment, the calibration of permeability remained for allintensive purposes, unchanged. This disproved the hypothesis for the poor TwinRoll press representation being incorrect permeability calibration, however did alsoprovide assurances that the results collected to date were correct. The robustnessof the equipment has also been increased, which has proven valuable for new usersoperating this interference-prone device.194(a) (b)Figure A.12: In (a), a picture of the retrofitted screen spacer is shown. In(b), a picture of the rebuilt permeable piston inside the suspensionchamber is shown.0 0.1 0.2 0.3 0.410-1610-1410-1210-10Figure A.13: Experimental results for Series 17 of Chapter 3 collected beforeand after the retrofit detailed in this section. The red points and linerepresent the results and fit respectively before the rebuild, and theblue points and line represent results after.195Appendix BVarying Suspensions’ Details andCalibrated ParametersSupplementary information for the various suspensions investigated throughout thethesis is provided in Table B.1. This includes average Canadian Standard Freeness(CSF) scores, and the statistically determined sizing information from the FibreQuality Analyser (FQA). Suspensions with chemical additives were not tested inthe FQA.While it is convenient to represent the size distribution of the various pulp sus-pensions by a single, average value, as is done throughout this thesis, this approachrisks obscuring some important aspects of the pulp fibre suspensions. Pulp suspen-sions exist as a complex mixture of particles, as shown in Figure B.1. Generallyspeaking, and as viewed in most of this thesis, these pulp fibres can be consid-ered as high aspect-ratio cylindrical particles with average lengths of 1−3mm andwidths in the order of 15− 30µm. While the majority of the pulp mass can be rep-resented by such fibres, one should first recognize that these larger fibres are oftennot strictly cylindrical, but may be more ribbon-like, and may have kinks, splits,and surface fibrillation that affect their dewatering performance. The presence ofa considerable number of small, fines particles, is also very evident in Figure B.1.The smallest of these particles will have an aspect ratio of 1 and a size that is onthe order of the pore size in the networks described in Chapter 3. An arithmeticdistribution shows that the small fibre particles are predominant, and represent the196Table B.1: Canadian Standard Freeness test and FQA results for various sus-pensions investigated in Chapter 3.Series CSF Fine % LW Length LW Mean Width, W Coarseness, ω[mL] [mm] [µm] [mg/m]1 703.40 2.95 2.57 26.67 0.1432 703.303 743.634 759.075 710.316 720.097 502.02 2.78 2.60 26.87 0.1588 384.51 2.88 2.59 27.03 0.1539 288.10 3.36 2.53 26.97 0.15210 611.17 5.46 0.79 20.13 0.09711 416.96 2.85 0.76 16.20 0.06512 480.43 2.79 0.95 16.60 0.07713 531.94 2.87 1.16 17.27 0.09214 575.70 2.77 1.46 18.23 0.10615 636.49 2.92 1.83 20.30 0.13117 679.88 4.42 2.15 25.53 0.18319 250.45 10.90 0.82 23.37 0.16820 200.46 11.84 0.77 23.53 0.15921 127.73 13.44 0.72 23.73 0.14822 718.70 6.21 1.64 36.67 0.36623 680.77 6.45 1.63 36.63 0.33524 580.60 7.24 1.59 36.43 0.33225 389.83 9.11 1.50 35.60 0.31626 216.2 11.58 1.00 31.73 0.23827 159.63 13.81 0.87 30.93 0.24928 125.73 15.02 0.82 30.93 0.22529 729.63 5.67 2.45 26.83 0.229NF 3.05 13.58 0.167largest number of particles in suspension, outnumbering the long, cylindrical fi-bres. Moreover the prevalence of small particles is under-represented by the FQA(which has particle detection threshold of 0.070mm) and thus cannot measure thepresence of “sub-fine” particles that are in the order of the pore size in the net-work. Typically for industrial assessments of fibre length, and as done here, massdistribution rather than arithmetic distribution of particle size is preferred, whichsuppresses the contribution of fines in the mean measurements, and emphasizes thetrue particles of interest (being the larger fibres). A length-weighted approach is1971 𝑚𝑚Figure B.1: A sample image of LC refined softwood chemically pulped fi-bres. These fibres would be similar to those found for Series 7 in Chap-ter 3, for example. Illustrated in the image are the high-aspect cylindri-cal fibres, and various fine particles.applied here that yields the data in Table B.1. Therefore, with respect to an arith-metic distribution, higher value of average fibre length and lower estimate of Fine% will be reported.What this discussion is meant to provide is insight into the significant simplicityallotted by the values reported in the Table B.1. Although potentially flawed incertain perspectives, they do provide a means to capture a measurement of fibre sizefor the various suspensions, and therefore do find value throughout the discussionsof this thesis. The true impact of the distribution of particle sizes and the impactdue to the presence of fines is not understood, and left for future studies.Details of the investigation in Chapter 3 are provided in Table B.2. This in-cludes fitted parameters, characteristic values and the range of volume fractionsinvestigated for permeability and compressive yield stress experiments, as well asdetermined η∗ values and ranges, the fit score, and the range of γ values investi-gated.198Table B.2: Fitted material parameters and dewatering details collected in Chapter 3.Permeability Compressive Yield Stress DewateringParameters Parameters ParameterSeries a b R2 k∗ φ range F1 F2 a b c R2 p∗ φ range γPy η∗, range η∗ F(η∗) γrange[m2] [m2] [m] [MPa] [kPa] [MPa ⋅ s] [MPa ⋅ s]1 3.60e-13 18.52 0.985 1.30e-12 0.03 - 0.38 2.41 2.21e-06 0.67 1.89 2.98 0.999 11.82 0.09 - 0.48 362.69 7.83 - 13.3 10.0 0.531 33.7 - 0.0342 1.89e-13 14.21 0.894 1.05e-12 0.09 - 0.40 2.01 1.52e-06 0.70 1.99 3.09 0.999 9.92 0.10 - 0.48 304.90 6.73 - 9.42 7.92 0.459 0.790 - 0.0203 3.72e-13 13.72 0.908 2.17e-12 0.11 - 0.40 1.96 2.11e-06 0.84 2.13 2.86 0.999 8.42 0.11 - 0.49 554.10 3.00 - 4.45 3.53 0.135 1.41 - 0.0354 5.28e-13 14.25 0.954 2.92e-12 0.11 - 0.39 2.02 2.57e-06 0.69 1.98 3.05 0.999 9.96 0.10 - 0.49 763.89 7.08 - 10.5 8.84 0.141 2.23 - 0.0565 2.43e-13 17.77 0.824 9.46e-13 0.05 - 0.37 2.39 1.90e-06 0.43 1.72 3.77 0.999 12.19 0.09 - 0.47 374.28 17.3 - 25.6 20.6 1.23 0.919 - 0.0226 2.69e-13 16.37 0.795 1.21e-12 0.05 - 0.38 2.27 2.02e-06 0.78 2.16 3.12 0.999 7.50 0.12 - 0.48 283.45 10.6 - 15.1 12.5 0.641 0.699 - 0.0177 1.19e-14 22.99 0.652 2.75e-14 0.05 - 0.34 2.84 4.18e-07 0.39 1.82 4.42 0.999 9.40 0.11 - 0.45 8.54 582 - 774 661 0.611 0.101 - 0.0208 8.54e-15 25.74 0.680 1.50e-14 0.05 - 0.26 3.41 4.55e-07 0.31 1.70 4.76 0.998 10.21 0.10 - 0.45 4.97 4544 - 6163 5224 2.57 0.307 - 0.0129 6.02e-15 25.72 0.824 1.06e-14 0.05 - 0.20 3.61 4.24e-07 0.24 1.61 5.16 0.998 10.15 0.09 - 0.44 3.46 3955 - 5214 4495 1.28 0.211 - 0.02110 2.76e-13 14.06 0.883 1.56e-12 0.05 - 0.40 2.02 1.93e-06 1.31 2.23 2.13 0.999 9.65 0.10 - 0.50 395.76 7.40 - 11.23 8.71 0.364 30.4 - 0.03011 8.12e-14 19.74 0.855 2.60e-13 0.05 - 0.32 2.67 1.21e-06 1.05 2.11 3.02 0.999 11.20 0.10 - 0.45 104.96 61.9 - 89.8 73.6 1.58 0.214 - 0.00512 9.43e-14 19.42 0.921 3.11e-13 0.05 - 0.37 2.55 1.22e-06 0.87 2.03 3.19 0.999 11.36 0.10 - 0.46 127.77 53.4 - 79.4 63.5 2.27 0.256 - 0.00613 1.21e-13 18.89 0.889 4.21e-13 0.05 - 0.34 2.56 1.45e-06 0.70 1.95 3.36 0.999 11.19 0.10 - 0.46 170.15 28.3 - 40.3 32.5 0.690 0.348 - 0.00914 1.63e-13 19.11 0.805 5.55e-13 0.05 - 0.39 2.47 1.54e-06 0.64 1.91 3.54 0.999 11.43 0.09 - 0.46 229.25 22.2 - 32.2 26.4 0.463 0.474 - 0.01215 2.12e-13 19.06 0.837 7.26e-13 0.05 - 0.40 2.40 1.66e-06 0.51 1.81 3.69 0.999 11.65 0.09 - 0.47 305.51 18.3 - 24.7 21.5 0.355 0.626 - 0.01617 2.67e-13 20.38 0.683 8.02e-13 0.05 - 0.36 2.62 2.05e-06 0.62 1.87 3.83 0.999 12.52 0.09 - 0.44 335.37 24.0 - 34.6 28.6 1.17 0.728 - 0.01819 1.62e-14 22.91 0.536 3.77e-14 0.04 - 0.30 3.02 5.75e-07 1.62 2.12 2.87 0.999 16.63 0.08 - 0.41 19.65 278 - 368 323 0.839 0.043 - 0.00520 1.21e-14 24.99 0.498 2.29e-14 0.04 - 0.30 3.16 4.76e-07 1.34 2.04 3.45 0.999 17.58 0.08 - 0.39 12.61 292 - 420 338 0.322 0.069 - 0.00721 1.65e-14 26.14 0.554 2.78e-14 0.07 - 0.27 3.35 5.89e-07 1.30 2.03 3.30 0.999 17.18 0.08 - 0.40 15.00 856 - 1401 1112 0.378 0.165 - 0.01622 2.41e-14 21.22 0.399 6.65e-14 0.04 - 0.31 2.84 6.77e-07 3.93 2.20 1.09 0.998 27.81 0.07 - 0.44 62.69 276 - 382 329 17.72 0.133 - 0.00323 1.28e-14 15.27 0.669 6.40e-14 0.05 - 0.33 2.23 4.61e-07 3.89 2.25 1.01 0.999 24.33 0.07 - 0.46 51.27 149 - 214 174 3.98 0.107 - 0.00324 3.56e-14 21.22 0.416 9.82e-14 0.05 - 0.34 2.71 7.28e-07 3.16 2.18 1.34 0.999 24.04 0.07 - 0.45 67.15 180 - 251 211 2.07 0.175 - 0.00425 1.35e-14 19.08 0.326 4.61e-14 0.05 - 0.34 2.58 4.78e-07 2.83 2.20 1.31 0.999 20.50 0.07 - 0.46 30.58 244 - 360 284 1.22 0.069 - 0.00226 9.71e-15 18.19 0.676 3.63e-14 0.07 - 0.29 2.58 4.27e-07 2.68 2.29 2.06 0.999 17.08 0.08 - 0.41 21.36 103 - 118 109 0.207 0.23 - 0.01227 2.36e-15 16.55 0.392 1.04e-14 0.07 - 0.28 2.41 2.06e-07 1.40 1.99 2.76 0.999 19.16 0.07 - 0.41 7.46 602 - 816 692 0.724 0.076 - 0.00928 1.47e-15 23.10 0.426 3.36e-15 0.05 - 0.25 3.19 1.90e-07 1.66 2.09 2.48 0.999 17.52 0.08 - 0.41 2.23 4880 - 6955 5609 8.92 0.044 - 0.00429 5.67e-13 21.19 0.785 1.57e-12 0.05 - 0.35 2.70 2.97e-06 0.77 2.04 3.62 0.999 10.28 0.10 - 0.39 644.72 9.71 - 14.8 11.6 1.58 1.36 - 0.034199Appendix CFreeness Cone ModelA sketch of the cone in the freeness device is shown in Figure C.1. Water is fed inat the top with a flux Q(t) and fills the main chamber of the freeness device to adepth of Z(t); just below that interface, the instantaneous (downward) flow speedis v. At the bottom of the chamber, water enters an exit channel with flow speedvE. A side channel with an entrance at a height ZS= 6.6 cm above the bottom ofthe chamber removes water with speed vSto register the freeness score.Taking atmospheric pressure to be zero and assuming that the flow inside thecone is quasi-steady, we apply Bernoulli’s law to determine the pressures at thebottom of the vessel and the entrance to the side channel,pE+ 12ρv2E= ρgZ + 12ρv2 (C.1)andpS+ ρgZS+ 12ρv2S= ρgZ + 12ρv2, (C.2)respectively. We also adopt the friction factors fE≈ 0.14 and fS≈ 0.36 tomodel the resistance of the two outflow channels (as estimated by circulating waterthrough the cone at various constant flow rates), so thatpE= fEρv2E− ρghE& pS= fSρv2S− ρghS, (C.3)where hE= 1 cm and hS= 7 cm denote the vertical lengths of the exit and side200Figure C.1: Sketch of the Funnel Geometry of the Freeness Device.channels. Thus,vE= ¿ÁÁÀv2 + 2g(Z + hE)(1 + 2fE) (C.4)andvS= ¿ÁÁÀv2 + 2g(Z −ZS + hS)(1 + 2fS) (C.5)provided Z > ZSthe side-channel discharge switches off if Z < ZS.Mass conservation within the vessel demands thatV˙ = AZ˙ = Q −AEvE−ASvS, (C.6)whereA(Z),AE= 0.086 cm2 andAS= 1.29 cm2 are the horizontal cross-sectionalareas at the top free surface within the cone and of the exit and side channels re-spectively, and the volume of water in the main compartment V can be related toZ through an empirical relation accounting for the geometry; a simple fit suggeststhat V (Z) = 442Z9−0.639Z6+0.060Z3+1.7×10−5m3 (measuring Z in metres).We simplify further by observing that in the freeness test the main compartmentfills up to a depth on the order of tens of centimetres in a few seconds. Thus201Z˙ = O(0.1)m/s. By contrast, (vE, vS) ∼ √gZ = O(1)m/s. This disparityimplies that the flow within the cone is quasi-steady, and suggests that we reduce(C.6) toZ˙ = QA− AEA¿ÁÁÀ2g(Z + hE)1 + 2fE−ΘASA¿ÁÁÀ2g(Z −ZS + hS)1 + 2fS, (C.7)where Θ = 1 if Z > ZSand Θ = 0 otherwise. We solve equation (C.7), startingfrom Z(0) = 0 and given the input flux Q(t), up until the height Z(t) falls backbelow ZS. The integral of the discharge through the side channel provides thefreeness score.As a check of the applicability of the model, we perform two tests: first, werecord the freeness scores (i.e. the total discharge through the side channel) whendraining a fixed amount of water from the cone. For 1L of water in the cone, themodel predicts a score of 902mL, in comparison to a measurement of 899mL; for500mL of water, the theoretical score is 440mL, whereas we observe 436mL.Second, conducting the freeness test itself with pure water (and using the calibratedvalue for the screen friction factor quoted earlier, c = 180) we calculate a score of903mL, in comparison to the operating range of 880 − 890mL stated in the CSFdocumentation.Earlier models [24, 48, 91] adopt a rather simpler description of the cone, tak-ing the flux through the bottom to be a fixed discharge (i.e. AEvE≈ 8.833mL/s)and assuming the main compartment fills instantaneously. This approximation al-lows one to relate the influx Q directly to the side-channel discharge ASvS. If weintroduce the same assumption into (C.6), we find that the freeness score is higherby roughly 20 points.202Appendix DTwin Roll Press ResultsExperimental trials were performed using Valmet’s pilot-scale Twin Roll press fa-cility in Sundsvall Sweden. Two sets of data are used in this study, providing atotal of fifty seven trials. Catalogued results provided from several years back con-stitute Trials 4040− 4084, and results collected in person in March 2018 constituteTrials 601 − 507. The raw experimental data collected for each trial is provided inTable D.1. Processed results, including results from the mass balance and modelare found in Table D.2.203Table D.1: Results provided from the pilot-scale Twin Roll press facility.Trial Cin Qin,a Qin,b Qwash,a Qwash,b FF Pa Pb Cout T Ω hnip Torquea Torqueb LL[%] [m3/h] [m3/h] [m3/h] [m3/h] [mg/L] [kPa] [kPa] [%] [oC] [rpm] [mm] [Nm] [Nm] [N/mm]4040 4.6 120 121 24 24 128 21 20 36 70 10 11 13071 11745 1404041 4.7 138 140 28 28 129 19 19 28.6 71 14 11 10331 10361 1184042 4.7 140 140 28 29 154 26 25 26.6 72 13 11 12285 11792 1364043 4.7 140 140 28 28 238 35 35 32.3 72 11 11 15402 13679 1594044 4.7 160 159 32 32 139 21 20 21.9 72 18 11 9338 9572 1104045 4.8 159 159 32 32 180 31 30 23.7 73 15 11 11029 11190 1194046 4.7 159 160 32 31 231 41 40 25.7 73 14 11 13335 12949 1374047 4.8 161 159 32 31 520 51 50 28.8 73 13 11 15276 14356 1544050 10.4 66 65 29 29 200.2 9 9 40.1 72 11 11 16016 14082 1774051 10.3 86 85 29 29 374.1 15 12 32.2 75 17 11 13612 13145 1704060 10.1 73 72 33 32 284.6 15 17 35 72 12 13 14192 12620 1534061 9.4 98 96 29 30 607.1 28 31 37.1 72 11 13 17544 17440 1584070 9.7 90 90 35 32 175.9 20 20 26.5 71 17 16 9908 9804 1044071 10 92 91 35 33 404.2 24 27 32.8 72 13 16 15068 13726 1464072 9.91 100 102 35 33 422 27 31 31.8 72 15 16 14491 13488 1464080 10.3 90 88 35 34 150.1 19 20 24.6 72 16 19 9231 9057 874081 10.4 90 92 36 35 438.8 28 34 31.8 73 12 19 15656 14015 1304082 10.2 99 100 36 35 332.4 25 29 27.6 73 15 19 11859 11128 1204083 10.3 98 103 36 35 421.8 31 36 30.8 73 14 19 14848 13608 1324084 9.86 110 112 36 35 587.5 36 42 29.7 74 15 19 14989 13938 139601 8.8 70 69 29 29 0 21 22 33.4 65 11.15 11.05 12075 12172 128602 9.1 69 67 29 29 0 24 26 37.1 65 9.68 11.08 14629 14958 169603 9.3 77 79 33 33 0 18 19 28.6 65 14.39 11.04 9226 9099 98604 9.2 81 81 33 33 0 32 34 30.4 66 12.3 11.1 14050 14171 161605 9.2 91 89 37 37 0 32 32 25.4 66 16.75 11.05 10745 10583 131606 9.3 91 89 37 37 0 32 32 25.7 66 16.75 11.05 10877 10786 133607 9.3 90 89 37 37 350 39 42 32.4 66 13.95 11.1 13970 13951 163608 9.2 90 90 39 39 0 49 51 34.2 67 13.35 11.1 15932 15785 183609 9.4 97 93 39 39 0 41 44 27.8 67 16.15 11.1 13338 13138 166610 9.5 94 96 39 39 0 46 50 31.6 67 14.58 11.1 15115 15022 180611 9.9 103 103 41 41 0 56 56 29.7 67 16.05 11.1 15672 15604 196612 10.4 85 85 37 37 0 34 36 25.1 67 17.55 11.05 11279 10640 142613 10.7 85 90 37 37 0 40 40 28.3 67 16.55 11.1 13157 12842 164614 10.6 90 90 37 37 0 44 44 31.5 67 15.56 11.1 14726 14708 181701 10.5 95 93 37 37 0 41 42 27 67 17.48 11.1 13115 12746 168702 10.3 89 90 3 3 0 14 11 29.4 67 17.56 11.05 11074 11066 133703 10.2 92 90 3 3 0 18 12 33.5 67 14.7 11.1 14915 15345 169101 4.3 80 80 15 15 0 15 15 39.9 65 6.19 8.1 13250 13269 181102 4.6 80 80 15 15 0 10 10 27 65 10.78 8.1 6617 6303 69103 4.6 95 94 17 17 0 13 13 27 65 12.69 8.1 7124 6867 78104 4.5 95 95 17 17 0 16 17 34.2 65 9.73 8.1 9541 9428 125105 4.5 94 95 17 17 0 20 21 40.5 65 7.76 8.1 13784 13689 185106 4.7 109 110 20 20 140 21 21 29.2 65 12.22 8.1 9801 9576 120107 4.8 110 109 20 20 0 30 30 37.8 65 9.58 8.1 13379 13251 177108 4.8 119 120 22 22 0 21 20 24.2 65 14.88 8.1 8676 8352 105109 4.7 119 119 22 22 0 30 30 30.3 66 11.95 8.1 11722 11513 146110 4.6 119 120 22 22 530 39 40 34.9 66 10.96 8.1 13533 13398 172111 4.7 119 119 22 22 0 44 46 36.6 66 10.35 8.1 14279 14190 181112 4.8 119 120 22 22 320 33 35 31.1 66 12.15 7.35 12905 12793 182113 4.8 120 119 22 22 0 58 61 39.2 66 9.77 9 14940 14913 150114 4.9 119 121 22 22 0 46 49 36.6 66 10.95 8.15 13726 13704 178501 4.6 119 119 3 3 0 21 21 31 66 13.13 8.1 10842 10806 109502 4.8 120 120 3 3 390 29 30 39 66 10.54 8.1 14215 14242 171503 5 109 111 3 3 0 20 19 31.5 66 12.55 8.1 10242 10199 110504 4.8 109 110 3 3 500 28 29 39.6 66 9.32 8.1 15034 15166 182505 4.6 95 94 3 3 710 20 21 39.3 66 7.94 8.1 13817 13938 165506 5 130 130 3 3 390 37 41 37.2 66 11.75 9 12939 12713 111507 4.8 130 130 3 3 0 41 43 31 66 11.83 9 13834 13667 136204Table D.2: Various results output from the mass balance and model equationsfor the experimental trials performed and provided.η∗ ηTC∗ Wash Section Model, η∗Trial φstart, geo. φstart, plug LLgeo. LLplug LLGeo. LLplug φstart,wash Mix Length LLwash[N/mm] [N/mm] [N/mm] [N/mm] [m] [N/mm]4040 0.036 0.040 1053 1471 715 14902 0.121 0.122 12464041 0.028 0.034 769 1194 368 672 0.117 0.135 9494042 0.026 0.036 598 1385 275 953 0.132 0.138 10814043 0.032 0.043 829 2308 434 10016 0.150 0.135 18214044 0.021 0.030 528 1157 232 570 0.121 0.148 8344045 0.023 0.037 527 1616 232 1129 0.142 0.144 12124046 0.025 0.038 593 1769 266 2021 0.159 0.150 12744047 0.028 0.042 725 2426 341 - 0.172 0.149 17554050 0.041 0.045 1782 4377 - - 0.085 0.075 37794051 0.032 0.038 1260 2039 635 1858 0.102 0.110 18014060 0.040 0.044 1241 1637 1083 74085 0.109 0.107 14064061 0.043 0.059 1436 - 31043 7412 0.141 0.102 -4070 0.035 0.035 834 848 458 469 0.120 0.146 5344071 0.045 0.049 1212 1664 1461 10897 0.133 0.122 12524072 0.043 0.046 1252 1552 1075 33846 0.139 0.134 10904080 0.037 0.038 679 727 390 431 0.119 0.144 4294081 0.049 0.053 1133 1459 37922 23110 0.143 0.128 9724082 0.042 0.045 873 1053 602 920 0.136 0.140 6474083 0.048 0.049 1167 1309 2015 84496 0.148 0.140 8014084 0.046 0.049 1094 1314 1038 106279 0.157 0.149 726601 0.033 0.040 943 1725 562 12702 0.123 0.123 1445602 0.038 0.047 1167 15073 1022 - 0.131 0.115 25401603 0.028 0.037 798 1637 413 1289 0.116 0.125 1363604 0.030 0.044 799 3661 425 10759 0.147 0.130 2747605 0.025 0.036 695 1776 340 1265 0.145 0.147 1296606 0.025 0.037 715 1839 351 1371 0.145 0.146 1353607 0.032 0.044 1077 3526 609 7341 0.159 0.139 2608608 0.034 0.045 1206 6127 737 8372 0.172 0.142 3456609 0.027 0.040 832 2527 415 11952 0.162 0.147 1847610 0.031 0.045 1047 6495 569 8340 0.169 0.141 3774611 0.029 0.047 977 82525 504 9628 0.179 0.144 7394612 0.024 0.037 708 1990 340 1562 0.150 0.148 1441613 0.028 0.042 891 2954 448 10527 0.158 0.143 2222614 0.031 0.045 1107 5922 600 7986 0.164 0.139 3830701 0.026 0.041 839 3079 413 15974 0.160 0.144 2292702 0.029 0.039 1040 2290 532 2968 0.098 0.105 2039703 0.034 0.046 1246 91901 732 - 0.106 0.095 94423101 0.031 0.041 827 8343 551 - 0.106 0.101 7952102 0.020 0.025 501 855 238 436 0.089 0.127 752103 0.020 0.026 590 1014 279 517 0.100 0.136 861104 0.026 0.033 809 1569 418 1644 0.110 0.124 1401105 0.032 0.041 1092 9397 783 - 0.121 0.113 12298106 0.022 0.031 683 1699 329 1211 0.122 0.135 1456107 0.029 0.041 1062 20455 614 - 0.141 0.126 14933108 0.018 0.029 538 1595 251 906 0.121 0.142 1321109 0.023 0.035 729 2549 350 11245 0.141 0.139 2162110 0.027 0.037 961 4560 497 5635 0.157 0.142 3469111 0.028 0.040 1038 25588 564 5868 0.165 0.140 37237112 0.021 0.036 781 7389 370 6506 0.148 0.139 5116113 0.033 0.043 1278 25244 936 - 0.183 0.145 14377114 0.028 0.040 1099 14983 599 - 0.169 0.142 37352501 0.023 0.031 847 1757 409 1184 0.122 0.136 1500502 0.030 0.041 1288 12988 789 - 0.140 0.126 13420503 0.024 0.033 842 2003 410 2081 0.118 0.130 1757504 0.031 0.042 1202 12223 772 - 0.138 0.122 16890505 0.030 0.041 999 11031 626 - 0.121 0.113 21309506 0.031 0.041 1280 27620 774 - 0.156 0.137 7131507 0.026 0.039 775 3396 386 10897 0.161 0.143 2654205

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