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On low consistency refining of mechanical pulps Rubiano Berna, Jorge Enrique 2018

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On Low Consistency Refining of Mechanical PulpsbyJorge Enrique Rubiano BernaM.Sc. Royal Institute of Technology, Stockholm, Sweden, 2013B.Sc. Universidad del Valle, Cali, Colombia 2007a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoral studies(Chemical and Biological Engineering)The University of British Columbia(Vancouver)November 2018c© Jorge Enrique Rubiano Berna, 2018The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:  On Low Consistency Refining of Mechanical Pulps  submitted by Jorge Enrique Rubiano Berna  in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemical and Biological Engineering  Examining Committee: James Olson, Department of Mechanical Engineering Co-supervisor Mark Martinez, Department of Chemical and Biological Engineering Co-supervisor   Supervisory Committee Member Bern Klein, School of Mining Engineering University Examiner Steven Rogak, Department of Mechanical Engineering University Examiner   Additional Supervisory Committee Members: Heather Trajano, Department of Chemical and Biological Engineering Supervisory Committee Member Rodger Beatson, British Columbia Institute of Technology Supervisory Committee Member  iiAbstractThe aim of this thesis is to develop comprehensive knowledge to fill the gaps in the understandingof three key aspects of low consistency refining of mechanical pulps.Firstly, the fibre shortening mechanisms are formally studied by using a comminution model.Fibre length distribution data from before and after refining with a variety of pulp types,net-powers, feed flow rates, angular velocities and plate geometries was analyzed. Fibres’ cuttingrate and cutting location were found to be highly correlated with refiner gap. Plate geometrywas also demonstrated to have a role in the fibre cutting location.Secondly, the relationship between net-power and gap was described using a correlation builtentirely from pilot-scale refining data. Results showed that a properly defined dimensionlessnet-power number is crucial to compare different refiner sizes under the same grounds. Thedeveloped correlation was compared to industrial-scale data showing that the correlation is wellsuited for predictions. Key assumptions of the correlation were validated using bar-force sensormeasurements data.Finally, the framework developed in the first two parts of this thesis were used togetherwith pressure screening models available in literature to theoretically analyze refining systemstypically found in TMP lines. Fibre length was used to assess each system performance in termsof refiner gap, reject ratio and refiner power. Moreover, the impact of some design aspects suchas refiner size, recirculation and split-ratios was also described.iiiLay SummaryThe core unit operation in the production of thermomechanical pulp is refining as it is neededto convert wood chips into pulp suitable for paper making. However, refining demands largeamounts of electrical energy, thus energy savings in refining directly benefit the sector.The current understanding of refining mechanisms is limited. In order to achieve importantenergy savings, it is imperative to fully understand the process and its effects on pulp.This research addressed key components of the mechanisms of low consistency refining. First,fibre shortening which is the most important aspect in pulp quality, was described using astatistical model. Second, the refining power consumption was related to machine and processvariables. Lastly, the two frameworks were put together to analyze and assess refining systemscommonly found in industry.ivPrefaceAll the work presented herein is part of the Energy Reduction Mechanical Pulp researchprogram which is a consortium between several industries, research institutes, universities andgovernmental entities.• A version of Ch. 3 was presented as J.E. Rubiano Berna, D.M. Martinez and J.A.Olson. “Analysis of fibre shortening during low consistency refining of mechanical pulpsusing a comminution model” at the 10th Fundamental Mechanical Pulp Research Seminar.Jyva¨skyla¨, Finland. June 13–14 2017.• Ch. 3 has been published as: J.E. Rubiano Berna, D.M. Martinez and J.A. Olson. “AComminution Model Parametrization for Low Consistency Refining”. Powder Technology,328. pp. 288–299, 2018. This paper proposes a parametrization of a comminution modelto mechanistically describe fibre shortening due to low consistency refining. The author ofthis thesis was the principal contributor to this publication. Professors James Olson andMark Martinez supervised the research.• A version of Ch. 4 and 5 was presented as J.E. Rubiano Berna, C. Sandberg, D.M.Martinez and J.A. Olson. “Theoretical study of systems composed of Low ConsistencyRefining and Pressure Screening” at the 31st International Mechanical Pulping Conference.Trondheim, Norway. May 27–30 2018.• Ch. 4 has been accepted for publication to Nordic Pulp&Paper Research Journal as:J.E. Rubiano Berna, M. Martinez and J. Olson. “Power–Gap Relationships in LowConsistency Refining”. In this publication, a correlation to describe power–gap relationshipsin low consistency refiners is developed. The author of this thesis was the principalcontributor to this publication. Professors James Olson and Mark Martinez supervisedthe research.• Ch. 5 has been accepted for publication to Nordic Pulp&Paper Research Journal as:J.E. Rubiano Berna, C. Sandberg, M. Martinez and J. Olson. “Theoretical Analysisof Refining Systems in TMP”. This publication used the frameworks developed in theprevious two publications together with screening models available in literature to analyserefining systems from a system-design point of view and compare simulation resultsvwith industrial-scale data. The author of this thesis was the principal contributor to thispublication. Christer Sandberg of Holmen Paper designed and gathered the industrial-scaledata. Professors James Olson and Mark Martinez supervised the research.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAbbreviations and Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Mechanical Pulping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Low Consistency Refining Overview . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Fibre Structural Changes during LC . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 LC Refining Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Summary of Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.6 Objectives of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Fibre Shortening Analysis Using a Comminution Model . . . . . . . . . . . 143.1 Comminution model equation for LC refining . . . . . . . . . . . . . . . . . . . . 143.2 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32vii4 Power-Gap Relationships in LC Refining . . . . . . . . . . . . . . . . . . . . . 344.1 Normal Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Refining Net-Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 Refining Systems Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1 Screening Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.2 Comminution Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Refining Size Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.4 Refining Power Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.5 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Appendix A Grinding equation for continuous systems . . . . . . . . . . . . . . 74A.1 Specific Refining Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74A.2 Refiner radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75Appendix B Experimental and predicted fibre length distributions for se-lected trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Appendix C Comminution in Pulp Mixtures . . . . . . . . . . . . . . . . . . . . 80Appendix D Affinity Laws in Low Consistency Refining . . . . . . . . . . . . . 81Appendix E Mathematical Analysis of a Refining System . . . . . . . . . . . . 83E.1 Flows fibre length distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83E.2 Calculation of E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84viiiList of TablesTable 3.1 Summary of refining trials performed at Andritz R&D facility . . . . . . . . . 21Table 3.2 Summary of refining trials performed at the UBC Pulp and Paper Centre . . . 22Table 3.3 Comparison of K values for trials (1-9) and (A-F) in two scenarios. . . . . . . 27Table 3.4 Model comparisons using likelihood ratio test . . . . . . . . . . . . . . . . . . 31Table 4.1 Fitting results and statistics of experimental data on the power-gap correlation 44Table 5.1 Summary of the experimental data and simulation results . . . . . . . . . . . 60ixList of FiguresFigure 2.1 Symbols and block diagram of TMP lines . . . . . . . . . . . . . . . . . . . . 4Figure 2.2 Schme of LC refining operation. . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 2.3 Lateral-view and Cross-view of a disc refiner . . . . . . . . . . . . . . . . . . 6Figure 3.1 Diagram showing how a fibre is shortened depending on the cutting position 17Figure 3.2 Example of shape for the Bi,3 matrix. . . . . . . . . . . . . . . . . . . . . . . 18Figure 3.3 Diagram showing refining in stages . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 3.4 Evolution of fibre length distribution during refining in stages . . . . . . . . . 19Figure 3.5 Configuration for batch refining . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 3.6 K vs gap for the trials (1-9) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Figure 3.7 K vs gap for trials (1-9) and (A-F) . . . . . . . . . . . . . . . . . . . . . . . . 25Figure 3.8 K¯ parameters vs gap for trials (1-9) and (A-F) . . . . . . . . . . . . . . . . . 26Figure 3.9 m vs gap for trials (1-9) and (A-F). . . . . . . . . . . . . . . . . . . . . . . . 27Figure 3.10 n vs gap for trials (1-9) and (A-F) . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 3.11 Four examples of Kq vs gapq . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 3.12 Kp values obtained considering a non-constant matrix A . . . . . . . . . . . . 29Figure 3.13 mp values obtained considering a non-constant matrix A . . . . . . . . . . . 30Figure 3.14 np values obtained considering a non-constant matrix A . . . . . . . . . . . . 30Figure 4.1 Normal forces at different gaps and RPM . . . . . . . . . . . . . . . . . . . . 41Figure 4.2 Experimental data of net-power Pnet vs. gap andPd/α2 versus gap . . . . . . 42Figure 4.3 Parity plot between predicted and measured gap . . . . . . . . . . . . . . . . 44Figure 4.4 Plot of dimensionless quantity Pd/α2 as a function of gap for different lw . . . 45Figure 5.1 Block diagrams of the three refining systems studied . . . . . . . . . . . . . . 48Figure 5.2 Block diagram of pressure screening . . . . . . . . . . . . . . . . . . . . . . . 49Figure 5.3 Block diagram of a single LC refining stage . . . . . . . . . . . . . . . . . . . 49Figure 5.4 Twin-Flow disc refiner recommended flow rates and rotational speeds . . . . 50Figure 5.5 Block diagram showing an arbitrary system . . . . . . . . . . . . . . . . . . . 51Figure 5.6 Curves of lw vs. gap for five different refiner sizes . . . . . . . . . . . . . . . . 52Figure 5.7 Scheme of Reject-refining . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53xFigure 5.8 Ratio E as a function of gap at different values of Rv . . . . . . . . . . . . . 55Figure 5.9 Performance curves in terms of gap . . . . . . . . . . . . . . . . . . . . . . . . 61Figure 5.10 Performance curves in terms of dimensionless quantity Pdα . . . . . . . . . . . 62Figure 5.11 Performance curves in terms of dimensionless quantity E Pdα . . . . . . . . . . 63xiAbbreviations and NomenclatureAbbreviationsBEL bar edge lengthBIL bar interaction lengthCSF Canadian standard freenessFLD fibre length distributionHC high consistencyLC low consistencyLRT likelihood ratio testLVDT linear variable differential transformerMEL modified edge loadMLE maximum likelihood estimationNLP no-load-powerSEL specific edge lengthSPF Spruce-Pine-FirSEC specific energy consumptionSRE specific refining energySR Schopper-RieglerSSL specific surface loadTMP Thermomechanical pulpxiiRoman SymbolsA Comminution model matrixAc Area of a bar-bar crossing [m2]Acs Cross section area of flow of a refiner [m2]B Breakage function matrixBij Breakage functionBw Bar width [mm]C Pulp consistencyCo Pulp consistency at an uncompressed stateC C-factorD Likelihood ratio test statisticDo Outer refiner diameter [in]Dw Groove depth [mm]e Flow energy content [kWh/t]E Ratio: mass flow rate flowing through the refiner over mass flow rate leaving the systemf¯N Normal force [N]f¯s Shear force [N]G Dimensionless number related to gapgap Distance between refiner plates [mm]gapo Distance between refiner plates at no-load-power regime [mm]Gw Groove width [mm]I Intensity of impacts per unit mass [J/kg]I Identity matrixK¯ Comminution model parameter [(kWh/t)−1 ]K Comminution model parameter related to the selection functionli Bin size [mm]xiiilo Arbitrary constant [mm]lw Length-weighed mean fibre length [mm]m Comminution model parameter related to the breakage functionm˙ Mass flow rate [kg/s]N Number of crossings per unit mass [kg−1]Nc Number of crossingsni Number of fibres in the ith binn Comminution model parameter related to the selection functionP Pressure [Pa]Pc Power consumed by a bar-bar crossing [kW]Pd Dimensionless powerPnet Refiner net-power [kW]p Number of binsP (li) Passage ratio fuctionq Number of refining stagesQ Volumetric flow rate [L/min, gpm]r Refiner radius [m]Ri Refiner inner radius [m]Ro Refiner outer radius [m]Rv Volumetric reject ratioSi Selection functionS¯i Selection function [(kWh/t)−1]S Selection function matrixT Reject thickening ratiot Time [s, h]VG Radial pulp flow speed at the outer radius [m/s]xivVT Rotor tip tangential speed [m/s]yi Proportion of fibres in the ith bin (relative frequency)Y Fibre length distribution vectorGreek Symbolsα Plate geometry constant related to the effective refining areaβ Plate geometry constant related to the open area of flow [mm]η Dimensionless number associated to power consumptionΘ Residence time [s]λ Ratio: outer refiner radius over inner refiner radiusµf Friction coefficientρ Density [kg/m3]τ Torque [N m]χd Chi distribution function with d degrees of freedomΨ Multinomial distribution probabilityω Refiner rotational speed or angular velocity [rad/ sec, RPM]xvAcknowledgmentsI would like to express my gratitude to my supervisor James Olson for his enthusiasm andguidance and my co-supervisor Mark Martinez for his support and scientific rigorosity. Also, Iwould like to extend my gratitude to the members of my supervisory committee Heather Trajanoand Rodger Beatson for their contribution.I gratefully acknowledge the financial support provided by the Natural Sciences and Engineer-ing Research Council of Canada (NSERC) through the Collaborative Research and Development(CRD) grant CRDPJ 437223-12 and the support of AB Enzymes, Alberta Newsprint Company,Andritz, BC Hydro, Canfor, Catalyst Paper, FPInnovations, Holmen, Meadow Lake Pulp (PaperExcellence), Millar Western, NORPAC, West Fraser, Westcan Engineering and Winstone PulpInternational.I would like to thank all the good friends I made during my doctoral studies for theirfriendship. In particular, my dear padawan Nick McIntosh for his absolute presence andCanadian touch and Joaquin Esquivel for being a such a great adult-childhood friend andanimate my after work life.I would also like to thank my beloved parents Gustavo and Matilde and my beloved sisterDiana for being the greatest source of love and support.Finally, I would like to express my deepest gratitude to Maddalena for her love, care andpatience during this stage of my life.xviDedicationA mis colegas favoritos, Poncio & Poncia.xviiChapter 1IntroductionEvery new beginning comes from some other beginning’s end. — Seneca (Romanphilosopher, 4 BC – 65 AD)Thermomechanical pulp (TMP) is produced by converting wood chips into fibres usingmultiple refining stages. Refiners are used to initially grind wood chips into fibres and later onto develop pulp properties needed for paper making and other related applications. This processprovides high yields because between 94-98% of wood chips are converted to pulp but requireslarge amounts of electrical energy to defibrate chips and develop fibres.There has been a substantial number of studies in chip pre-treatment, high consistency (HC)refining and low consistency (LC) refining of mechanical pulps whose ultimate goal is to reduceenergy consumption. Gradually, TMP mills have implemented these strategies in an attempt toreduce production prices.In light of the interests of this dissertation, the focus will now be on LC refining . The LCrefining of mechanical pulps has proven to have advantages in terms of energy savings. However,compared to HC refined pulps, LC refined pulps generally produces a slightly inferior pulp. Thus,its use is oftentimes limited by the trade off between pulp quality and energy savings.Although LC refining has been extensively studied through the years, there is still the needfor a more fundamental understanding of the refining mechanisms and effects of pulp to takefull advantage of the energy savings potential.This dissertation provides comprehensive knowledge that improves the current understandingof following key aspects of LC refining: (1) the fibre shortening mechanisms, (2) the relationshipbetween refining power, gap and pulp and (3) the impact of LC refining on a TMP system. Thisthesis is composed of 6 chapters and is structured as follows:The motivation of this work is presented in Ch. 1. The background of TMP and LC refiningis presented in Ch. 2 along with the missing gaps in the understanding of LC refining that arein the scope of this dissertation. Ch. 3 uses a comminution model to describe how fibres areshortened due to LC refining. Ch. 4 analyzes the refining power consumption and develops acorrelation that describes the relationship between net-power, gap and pulp. Ch. 5 implements1the results of Ch. 3 and 4 to develop a framework to analyze refining systems found in TMP lines.Finally, the highlights of this dissertation are presented in Ch. 6 as well as recommendations forfuture work and research.2Chapter 2BackgroundIf you want to make an apple pie from scratch, you must first create the universe— Carl Sagan (American astronomer, 1934 – 1996)2.1 Mechanical PulpingIn a conventional TMP line, wood chips are fed into a primary HC refining stage at consistenciesranging between 20–40%. Inside the refiner, chips are subject to forces imposed by the refiningelements thus gradually grinding them into pulp. Later on, fibres are subject to another HCstage where the main objective is to develop certain structural changes in the fibres’ morphologyto make them more suitable for paper making. Finally the pulp is diluted down to a consistencybetween 3–6 % to undergo a tertiary LC refining stage. The role of this last refining stage hasbeen traditionally limited to reduce shives and adjust freeness levels [1].In addition to refiners, pressure screening is also involved in TMP as a fractionation method;the aim of the fractionation stages is to generate more uniform streams enabling better targetedrefining treatments. With this, two sections within the process can be clearly identified, namelymain line and reject line. The main line consist of mostly fibres after the primary HC stage andthe main objective of this part of the process is to develop fibre properties with a combinationof HC and LC stages. On the other hand, the reject line consists of shives and coarse materialseparated from the main line via pressure screening. This material is not fully defibrated nordeveloped and is in need of further refining. Reject refining is also carried out by a combinationof HC and LC refiners (See Fig. 2.1a and 2.1b).TMP is known for having high yields but also for being energy intensive. Theoreticalestimations claim that a very small part of the energy spent, about 13%, is actually used forwood chip defibration and fibre development [2]. As a consequence, large amounts of energyare needed to produce mechanical pulp where a large fraction of the energy is consumed by HCrefiners; for example, typical TMP for newsprint paper grade takes about 2000–2500 kWh/t toproduce, where 1500–2000 kWh/t is consumed by HC refiners. To put TMP energy consumptioninto a broader perspective, in the province of British Columbia, Canada, TMP mills consume3(a) Symbols used in block diagramsHC Refiner LC Refiner Pressure Screen(b) Block diagram of a conventional TMP(c) Block diagram of a TMP line with LC refining implementationFigure 2.1: (a) Symbols used in the block diagrams. (b) Block diagram of a typicalTMP line showing two HC refiners and a tertiary LC refiner in the main line and a HC andLC refiner in the reject line.(c) Block diagram showing a secondary LC refiner in the mainline and two LC refiners in the reject lineabout 10% of the province’s annual demand and out of that share, HC refiners consume 60% [3].Thus, any electrical saving would have a significant benefit on the sector.Pilot-scale studies have shown that it is possible to reduce energy consumption by replacingthe secondary HC stage with LC for a given tensile strength target. In general, energy savingsbetween 100–200 kWh/t are achieved by using LC either in the main line or reject line [1, 4–9].Fig. 2.1c shows an example of LC implementation in a TMP line. Despite the benefits in energysavings, fibre shortening during LC refining reduces tensile strength, limiting the amount ofenergy that can be saved. Moreover, it has been also observed that LC refining produces pulpwith lower tear index and light scattering (at the same tensile strength index) whereas freenesslevels are similar compared to the HC refined pulps [1].LC refining is becoming more popular and extensively used nowadays at different positionsin TMP production lines. Despite its current popularity, the role of LC refining as part of aTMP system has not been thoroughly understood mainly because there is no systematic way ofstudying refining systems. Perhaps the logistic and economic demand of conceiving a particularsystem at industrial scales discourages researchers and slows down the pace at which meaningfulresults are generated. Additionally, mechanical pulping is carried out according to specific needs4to meet pulp quality specifications that vary from case to case which translates to a large amountof systems to consider. Finally, complex interactions arise when LC refiners and fractionationstages are combined exacerbating the problem from a system-design point of view. Likewise,operating a refining system at pilot-scales is impractical due to the inability of quickly achievingsteady states and the lack of a unlimited source of feedstock.The information available in literature on this specific subject is limited for the reasonsstated before; these are some of the most relevant studies concerning refining system-design atindustrial scale: Sandberg et al. [9] reported their experiences installing a brand-new LC refinerin a TMP line. In general, they observed the electrical energy consumption to be 40–50% lowerthan the conventional HC refining at a certain tensile index. Additionally they observed lowerlight scattering values and increased screening availability. Later on, Sandberg et al. [1] studiedseveral TMP configurations at industrial scale to assess energy efficiency and pulp quality for eachconfiguration. They observed that LC improved the energy efficiency at a given tensile strengthwhen used in both the main and reject line but produced pulps with lower light scattering andfibre length. Recently, Sandberg et al. [10] described modifications done to a conventional TMPline to produce newsprint grade pulp at 1500 kWh/t; following process intensification concepts,they managed to remove all fractionation stages by implementing chip impregnation, improvedchip refining and low consistency refining resulting in less unit operations.There have also been studies done at pilot-scale involving some aspects of system-design.For instance, Andersson et al. [11] studied the effect of long fibre concentration (measured bythe R14+R28 Bauer McNett fractions) on LC refining of mechanical pulps; they observed thatincreasing the long fibre concentration allows better refiner loadability 1 and higher tensile indexincrease. Miller et al. [7] studied the effects of fractionation via pressure screening in multistagerefining; they concluded that fractionation allows a better refiner loadability and enables harshtreatments to fibres without excessive fibre shortening. Lemrini et al. [12] fractionated and HCand LC refined pulp to then further refine the screened pulp in LC–HC stages at different energysplit ratios. They found that fractionation allowed more energy to be applied via LC refiningwithout excessive fibre shortening but lower tensile strength index was observed compared toHC refined pulp.2.2 Low Consistency Refining OverviewLC refining is a complex process involving a heterogeneous raw material undergoing a heteroge-neous treatment where large numbers of variables and parameters are involved. The complexityof this particular process arises from the interactions between variables which ultimately affectthe changes in the pulp properties. Fig. 2.2 shows a scheme of a LC refining operation wherethe variables involved are identified.Industrial LC refiners are equipped with a bar-groove patterned rotor and stator (oftentimes1The term loadability is recurrent in some refining studies and it is used to describe the increase in refiningpower due to an increase on the refiner’s feed mean fibre length.5TemperatureConsistencyPulp typePressureFlow ratePulp propertiesRotational speedPowerGapPlate geometryRefiner sizeMotor RefinerTank Inlet OutletPressureFlow ratePulp propertiesFigure 2.2: Scheme of a LC refining operation showing the variables involved.(a) Lateral-view (b) Cross-viewgap PULPFigure 2.3: (a) Lateral-view of a disc refiner showing a stator section (red), a rotorsection (blue) moving at an angular speed ω. For illustrative purposes, a single bar from arotor section and a complete stator section are forming bar-bar crossings (green areas); Riand Ro are inner and outer radius, respectively. (b) Cross-view of a disc refiner showingpulp (dashed-region) trapped by a bar-bar crossing formed by a stator bar (red) and arotor bar (blue) moving at a linear speed ωr; fN and fs are the normal and shear forces,respectively; gap is the distance between plates; Bw is bar width; Gw is groove width; Dwis groove depth.called plates or refining segments) separated by a gap of 0.1–2.5 mm. Fibres in a pulp suspensionare fed into the gap and are subject to normal and shear forces (see Fig. 2.3 for details). Theforces are responsible for structural changes to fibres. These changes are necessary becauseof the poor performance of untreated fibres to form a network. The extent and nature of theneeded treatments strongly depends on the fibre source and their end use.2.3 Fibre Structural Changes during LCForces applied to fibres during refining cause structural changes which can be grouped as: (1)fibre deformation, (2) internal delamination, (3) external fibrillation, (4) fibre shortening and(5) fines generation.62.3.1 Fibre DeformationFibres in a wood matrix are usually straight, with the exception of certain deformations dueto growth stresses. However, fibres in pulp suspensions are seldom straight because they getdeformed and damaged during treatment stages (e.g. chipping, mid-consistency operations).Non-straight fibres are known to decrease tensile strength index and increase tear and stretchindex [5]. LC refining straightens fibres at low intensity treatments; this has been alwaysconcluded since average fibre length slightly increases with refining.2.3.2 Internal DelaminationInternal delamination is the separation of fibres’ cell wall layers. When fibres are subject tonormal forces, they become more flexible as fibre lumen collapses and cell walls get delaminated.Collapsed fibres have increased bonding area and flexible fibres are more likely to constitutestronger webs, thus increasing tensile strength and decreasing bulk.The degree of internal delamination is normally related to tensile strength increase. Alterna-tively, it can be directly measured by the water retention value technique as delaminated cellwalls are able to allocate more water in the newly created spaces [13]. A more sophisticatedtechnique to measure internal delamination is to use Simons’ stains, which is a two-componentdifferential stain with differences in molecular sizes, allowing to differentiate the accessibility ofthe interior structure of fibres [14]. Fernando et al. [15–17] have used this technique to assessthe degree of development of TMP fibres.Over the years, it was believed that strain cycles weaken fibres by a fatigue process. Thistheory was formally addressed by Goosen et al. [18] who demonstrated that pulp strength isnot modified by imposing multiple compression cycles on the same fibres, meaning that alreadycompressed fibres are not subject to additional structural changes due to further compressiontreatments. The study pointed out fibre redistribution as a key factor in compression refining.Heymer et al. [19] used changes in tensile strength of LC refined pulps to estimate the probabilityof a bar-bar crossing (compression event) to be successful. They found that a few events wereneeded to treat fibres and increase pulp strength supporting Goosen et al. findings. Mohlin andLindbrant [20] and Decker [21] found that only a small fraction of fibres are refined, reinforcingthe idea that multiple cycles in LC refining are needed to treat different fibres rather than modifythem due to fatigue.2.3.3 External FibrillationExternal fibrillation is the generation of new surfaces by peeling fibrils from the outer fibresurface without detaching them from the main body. While external fibrils serve as a bondingagent between fibres, it was thought to be the dominant factor in sheet strength and elasticmodulus, but Page [22] concluded that it has only a limited effect.The degree of external fibrillation is related to the drainability of the pulp measured asCanadian standard freeness (CSF) or Schopper-Riegler (SR). These two test methods are related7to the ease of water flow through the porous network at a given hydraulic pressure gradient,in other words permeability. El-Hosseiny and Yan [23] developed a mathematical model todescribe CSF using a filtration theory where concepts of permeability were involved. Moreover,Lindsay and Bardy [24, 25], Lindsay [26] and Vomhoff [27] measured permeability of pulps butthe studies were more focused on pressing and dewatering studies.2.3.4 Fibre ShorteningFibre shortening is the size reduction of fibres during refining. It is generally an undesiredeffect because it reduces sheet strength. On the other hand, it can increase light scattering,sheet formation, smoothness and decrease porosity. Also shorter fibre length is associated withdecreased bulk and lower freeness.Early studies relied on the use of classifiers (e.g. Bauer McNett) to assess fibre lengthchanges during refining. Yet, classifiers provide limited information since they only generate fivefractions which are not sharply separated [28]. On the other hand, Corte [29] mentioned thatsize reduction of cellulosic fibres can be studied by using a comminution model to describe thecutting mechanisms in terms of probabilities. However at that time, fibre length measurementswere manual and made it difficult to determine fibre length distribution (FLD). The introductionof optical analyzers simplified these measurements enabling FLD determination of pulp samples.Nonetheless, the vast majority of modern studies still use fibre length averages and refiningenergy consumption to assess fibre cutting. This approach provides a limited explanation to acomplex phenomena.There are only a few studies that used comminution models to study fibre shortening andthus the available information about cutting mechanisms is limited. Roux and Mayade [30]described mean fibre length changes due to refining using a comminution model. However, there-distribution of smaller sizes after cutting was not included in the study. Olson et al. [28] useda comminution model to describe FLD changes during refining assuming a random re-distributionof smaller sizes. It was found that the cutting rate was strongly dependent on refining energyand fibre length, independent on the feed consistency and higher at high energy treatments,which suggested that cutting was not a fatigue process. Heymer [31] used a comminution modelto assess the heterogeneity of a refining process analyzing FLD changes. It was found thatheterogeneity was greater in treatments with a small number of impacts at high intensity (orgreater homogeneity with large number of impacts at low intensity) and larger residence timescan decrease the heterogeneity.2.3.5 Fines GenerationFines are generated when fiber wall fragments and fibrillar material are removed from the cellwall body. Fines are known to decrease drainability of pulp, increase sheet strength due to thebridging effect, decrease air permeability and improve formation.The definition of fines is not clearly specified. Some criteria classify fines as objects able to8pass through 100 or 200 mesh. Other criteria define them as any particle shorter than 100 µm.Moreover, fines characterization in the pulp and paper field is challenging as there is no dedicatedcharacterization technique [32]. For instance, gravimetric methods such as Bauer McNett or BritDynamic Drainage Jar have the advantage of accounting for material in sub-micro size rangebut the individual size and shape of particles is not given. On the other hand, the image-basedmethods give direct information about size and shape but only account for particles larger thanthe resolution of the instrument. Techniques developed in other fields could help to characterizefines (e.g. equivalent diameter methods), ideally an image-based method so that fine materialcan be studied [32].2.4 LC Refining CharacterizationSeveral efforts have been made to characterize the degree of refining by relating process parametersand variables to the refining outcome. LC refining theories have their foundations in refiningof chemical pulps because in the early days it was almost exclusively used for chemical pulpswhereas most of the research about TMP focused on HC refining [33].LC refining has been extensively studied from many perspectives. Observations of pressure,temperature, power, vibrations, residence time, normal and shear forces have been made byseveral studies, yet the operation is still carried out without a complete understanding of itsmechanisms.2.4.1 LC Refiner Power ConsumptionA particular feature of LC refiners is that not all the power supplied by the motor is effectivesince part of the power is consumed to overcome mechanical, pumping and hydraulic losses.This is oftentimes called no-load-power (NLP). With this, the total refining power is describedas:Ptotal = Pe + PNL (2.1)where Pe is the effective power used to refine fibres and PNL is NLP. There is no generalagreement on how to define and measure NLP. Some researchers regard it as the threshold powerat which fibres undergo changes. Another definition is the power required to rotate the rotor ina pulp suspension. As for the method of measurement, a common practice is to back the platesoff while the pulp suspension is flowing through the refiner [34]. Furthermore, correlations topredict NLP in general are simply empirical formulas that are not dimensionally correct anddisregard important factors such as gap and plate geometry [34].Herbert and Marsh [35] analyzed a refiner operation by performing energy balances. Theyproposed a formula to calculate the total power consumed by a disc refiner considering that thepower has three components: (1) disc friction, (2) pumping losses and (3) effective work. The9originally published formula has been rewritten in order to present it in dimensionless form as:ηf =Pfρω3Do5 (2.2a)ηp =Ppρω2Do2Q(2.2b)ηe =PeωDi(Do2 −Di2)P(2.2c)where ηi is a dimensionless constant, Pi is power, ρ is density, ω is rotational speed, Di andDo are the inner and outer diameter, respectively. P is contact pressure and Q is volumetricflow rate. The subscripts f , p and e refer to friction losses, pumping losses and effective work,respectively. Moreover, Eq. 2.2a and 2.2b constitute NLP.Refiners are sometimes regarded as pieces of turbo-machinery or very inefficient centrifugalpumps. In that context, Eq. 2.2a and 2.2b are dimensionless numbers used in centrifugal pumpdesign and have proved to be useful in most of the cases. For instance Eq. 2.2a is well knownas the power or Newton number and Eq. 2.2b also called the head coefficient. It is importantto point out that the pumping losses ηp in pulp refiners are about 2% of ηe and are usuallydisregarded [34, 35]. Thus, friction losses ηf are the main contribution to NLP and can be ashigh as 35%.The effective work Pe is also known as net-power Pnet. It is well-known that gap has adirect impact on net-power which can be realized by monitoring net-power at different gapswhile keeping all other variables constant. This is the simplest of the trials in refining and leadsto net-power–gap curves (referred as power–gap curves from now onwards). These power–gapcurves have been subject of study by several authors such as Leider and Nissan [36] who proposeda linear relationship between the power and the inverse of gap. Likewise, Mohlin [37] showed asimilar behaviour in a conical refiner.Although the validity and contribution of these studies are recognized, the resultant power–gap curve is limited to the particular refining conditions used in the trials. For instance, varyingthe rotational speed or plate geometry will affect the power response and generate a differentcurve. This situation was addressed by Elahimehr et al. [38] who used dimensionless analysis torelate power with plate geometry, rotational speed and gap.Power–gap relationships are also affected by the pulp type. This has been observed byMohlin [39] who reported that refined pulps required smaller gaps than unrefined pulp to achievethe same refining power under the same refining conditions. Similarly, Andersson et al. [11] intheir study intentionally changed the feed pulp by means of pressure screening, and although theaim of the study was to investigate the impact of screening in pulp properties development, theyalso observed changes in the power–gap relationships. Although from experience it is knownthat pulp type changes the power–gap relationships, there is not current understanding to whatextent pulp type can affect these relationships.102.4.2 Energy Based TheoriesEnergy based refining theories rely on the net-power to quantify the amount of refining. Forinstance, the ratio between Pnet and the pulp mass flow rate m˙R passing through the refiner givesrise to the most basic notion of energy expenditure known as specific energy consumption (SEC)(see Eq. 2.3).SEC =Pnetm˙R(2.3)It is worth mentioning that SEC is oftentimes confused with specific refining energy (SRE)because in some specific cases these two values are equal (e.g. single refining stage). SREquantifies the energy content of pulp (e.g. the amount of refining that pulp has been subjectto) whereas SEC quantifies the energy supplied to the pulp by the refiner. For instance, in asingle refining stage, the initial flow energy content is zero (SRE0=0) and once it is single stagerefined, its energy content increases by SEC (SRE1=SEC). In general, if pulp undergoes q stagesof refining, the pulp energy content SRE follows:SREq =q∑i=0SECi (2.4)where SEC0 = 0. The SEC concept is a quantitative method to describe refining by quantifyingthe amount of energy transfered to pulp but does not provide any further information about thenature and extent. Two refining operations can have the same energy consumption yet lead todifferent effects on pulp. In order to improve this description, Lewis and Danforth [40] proposedthat energy is described as the product of number of impacts N and intensity of impacts I as:SEC = N I (2.5)It is straight forward to realize that the number of impacts N strongly depends on plategeometry. On the other hand, the notion of intensity I is more complex to define. Attempts toquantify I include the specific edge length (SEL), modified edge load (MEL) and specific surfaceload (SSL) theories just to mention the most well known. Lundin [41] does an outstanding jobin reviewing all of them. In essence, these theories strive to describe the intensity of impactsI by relying on the plate geometry characteristics. In general, the physical interpretation ofthese theories vary according with the assumptions and considerations they are built on. Forinstance, the SEL considers that leading edges of the bar-bar crossings are responsible for therefining treatment, thus the importance of measuring the energy transfer per bar length. TheSSL theory acknowledges the importance of the leading edges but also includes bar-width toprovide an estimate on energy transfer per bar area. Roux [42] proposed a better way to regardintensity; by analyzing plate geometry aspects and using physical considerations, the intensitywas expressed as the net normal force per average number of crossings. These refining theoriesare overlapping to some extent because in the end, their aim is to describe intensity of impacts.11Despite the scientific rigour refining theories have been built upon, their major drawbacksare that (1) they are machine-based-theories and thus completely disregard fibre and pulpproperties, (2) some important machine variables (e.g. gap, groove depth) are not considered atall.In an effort to address these drawbacks, Kerekes [43] proposed a way to characterize pulprefiners by describing their ability to impose impacts on fibres (see Eq. 2.6). This theory isknown as the C-factor and combines refiner variables namely plate geometry, rotational speedand gap, pulp variables such as fibre length, coarseness and consistency. Although this work hasstrong theoretical basis and addresses drawbacks from other refining theories, for some unknownreason, the C-factor (C) is not used at all to characterize refining.SEC = (C/m˙R) (Pnet/C) (2.6a)N = C/m˙R (2.6b)I = Pnet/C (2.6c)More recently, Roux et al. [44] proposed a refining characterization using the concept of impulse2;by regarding the bar-bar crossing in a refiner as a pulp compression event. An expression for theimpulse was derived in terms of common refining variables (plate geometry, rotational speed);the study showed that it is possible to characterize changes in pulp properties such as bulk, fibrelength and tensile among others on a variety of refining conditions.2.4.3 Force Based TheoriesForce based characterization has recently become an important topic in refining. Some authorsare emphatic to highlight the need to characterize refining via forces because bars impose forceson fibres, thus the product of force and bar speed represents the power consumption [45]. Inother words, the energy expenditure is a consequence of forces causing the refining effect onfibres.The following are some examples of works studying forces in refining: theoretical studies[45–48]; forces measurements [49–51]; and characterization via force-based approach [52]. Thesepast studies have made outstanding progress towards a better understanding on the effect offorces on fibres during refining. Yet, there is still the need for a complete characterization ofrefining in terms of forces [46].2.5 Summary of LiteratureRefining is an important operation in the pulp and paper field because: (1) it is the main wayto modify fibres and (2) it is energy intensive. Furthermore, LC refining is becoming broadlyused in TMP lines as a strategy to reduce energy consumption. Although it has been extensively2Impulse defined as the integral of normal force over time12studied, refining is yet not well understood. The following list attempts to summarize some ofthe gaps in understanding of LC refining that are in the scope of this dissertation:• Fibre shortening is a main issue in LC refining of mechanical pulps as it degrades pulpquality. To date there is no complete understanding of this complex process. Althoughit has been pointed that comminution models mechanistically describe fibre shortening,there has been only a hanful of studies using this approach. Moreover, the re-distributionof smaller classes has not been fully studied. For these reasons, it is still unclear howmachine variables and process variables affect FLD changes.• Existing models of LC refining have been developed through the refining of chemicalpulps. This raises the question whether these models can be also applied to LC refining ofmechanical pulps or not.• Current refining models do not accurately characterize refining operations. Firstly, theyhave shortcomings from the theoretical point of view (e.g derived empirically underassumptions not well justified). Secondly, important variables and parameters are notconsidered (e.g. gap, flow rate). Thirdly, there is no collective agreement on how importantterms should be used (e.g. intensity, no-load-power), neither the definition of them; ingeneral, stipulative definitions 3 are used for these terms.• Characterizations of power–gap relationships have not taken into account pulp. Althoughexperimental evidence suggest that pulp changes has an impact on power–gap relationships,the dependency of these relationships with pulp has not been quantitatively described.• The understanding of LC refining’s role in bigger and complex systems is limited as thereis not a systematical method to study and compare them. Moreover, as fractionationstages and LC refiners are used together in TMP lines, it is unclear how these two unitoperations behave when are coupled and the overall impact on the system behaviour.2.6 Objectives of the thesisThis research is focused on the LC refining of mechanical pulps. The specific objectives are:• To develop a framework to mechanistically describe fibre length distribution changes dueto LC refining.• To understand how the net-power is affected by important variables such as refiningconditions, plate geometry, gap and pulp type.• To develop a framework capable of analyzing refining systems using the developed modelsand pressure screening models available in literature.3A stipulative definition is a type of definition in which a new or currently-existing term is given a new specificmeaning for the purposes of argument or discussion in a given context.13Chapter 3Fibre Shortening Analysis Using aComminution ModelGreat things are done by a series of small things brought together – Vincent VanGogh (Dutch painter, 1853 – 1890)The size reduction of particles is a three-dimensional problem, however its rigorous mathe-matical analysis is complex. In the case of size reduction of wood pulp fibres due to LC refining,the problem is simplified to an idealized one-dimensional case since fibres have high aspect ratio(between 30 and 100) and size reduction practically occurs along fibres’ length. Hence, sizereduction of wood pulp fibres is often referred as fibre shortening.The one-dimensional size reduction mechanism was first described by Epstein [53] whointroduced the concepts of probability of breakage and distribution of smaller sizes. This balancepopulation method known as the comminution model has been used in several fields to study themechanisms of particle size reduction due to crushing, grinding, vibration, and other processesincluding LC refining.This chapter focuses on the description of FLD changes of pulp due to LC refining usinga comminution model. In Sec. 3.1 the relevant theoretical background is developed. Sec. 3.2describes the experimental data collected and the methodology used to analyze data. Sec. 3.3and Sec. 3.4 present results and discussion. Finally, Sec. 3.5 summarizes this chapter with theconclusions.3.1 Comminution model equation for LC refiningRoux and Mayade [30], Olson et al. [28] and Heymer [31] used the comminution model equationusing SEC as an independent variable written as:dyid (SEC)= −S¯iyi +p∑j=1BijS¯jyj (3.1)14where yi is the proportion (or relative frequency) of fibres in the bin i, Si is called the selectionfunction and describes the probability of a fibre in the ith to get shortened. Bij is called thebreakage function and describes the conditional probability of a fibre in the ith bin to getshortened to a fibre in the jth bin. The details of the mathematical derivation are presented inApp. A.1. Eq. 3.1 is just an adaptation from the linear batch grinding equation and although it isvalid to analyze fibre shortening during LC refining, it does not allow comparisons between trialsperformed at different conditions because important refining variables such as plate geometry,angular velocity and refiner size are not considered. Many refining studies have emphasizedthat plate geometry and angular velocity are key to understand changes in pulp properties.For example, Elahimehr et al. [54] demonstrated how the number and size of bar-bar crossingschanged with plate geometry and angular velocity and later on, Elahimehr et al. [38] relatedthe effect of plate geometry to pulp properties. Additionally, the use of SEC as the independentvariable in Eq. 3.1 poses another limitation since, as mentioned before, two refining operationscan have the same energy consumption, yet lead to different fibre modifications; for instance, alarge number of impacts of low-energy-per-impact leads to fibrillation, whereas a small numberof impacts of high-energy-per-impact leads to fibre shortening [43].To address these issues, the comminution model equation is written in terms of the refinerradius as (see App. A.2 for details):Qω12piαβ1rdyidr= −Siyi +p∑j=1BijSjyj (3.2)where Q is the volumetric flow rate and ω is the angular velocity. Moreover, α = Bw/(Bw +Gw)and is a dimensionless number that describes the ratio of bar area to total area of the refiner.This particular number has been widely used by several refining studies such as Elahimehr et al.[38, 54, 55] and Rajabi Nasab et al. [34] and is also included in some refining theories suchas the MEL theory. On the other hand, β = 2GwDw/(Bw +Gw) and is a plate characteristicconstant that, coupled with the factor 2pir, describes the free area of flow of disc refiners; theconstant β has been also used by Kerekes [43] and Heymer et al. [19]. Eq. 3.2 was derived for aSingle-Disc refiner as mentioned in App. A.2. Care must be taken when this equation is appliedto other types of refiners (e.g. Twin-Flow disc refiners) since the values of β will change. Asmentioned before, β is closely related to the free area of flow, thus for Twin-Flow disc refiners,this area is doubled due to the characteristics of the refiner itself (two chambers with two setsof rotor/stator plates)4. Finally, Bw is the bar width, Gw is the groove width and Dw is thegroove depth (see Fig. 2.3 for details).In App. A.2 it was assumed that the selection function S∗i was proportional to the angularvelocity ω and the ratio of bar area α. The reason for this assumption is based on the findings4Another way to regard this situation is to assume that the total volumetric flow fed to the refiner is equallydistributed into the two chambers of the Twin-Flow disc refiner, thus the value of β is the same as the Single-Discrefiner case but the volumetric flow (the one used in the equations) is half of the one fed to the refiner.15of Elahimehr et al. [55] who observed that fibre shortening was strongly correlated to productof angular velocity and the bar interaction length (BIL), the latter being a measurement ofthe leading edge of bar crossings. Since the bar-bar crossing shapes are parallelograms, BIL isequal to half the perimeter of the crossing. Additionally, Elahimehr et al. [38] demonstratedthat: (1) the bar-bar crossing area (Ac) is a linear function of α2 and (2) the square root ofbar-bar crossing area is linearly correlated to the crossing perimeter. Therefore the followingassumptions is made: √Ac ∼ BIL ∼ α (3.3)3.1.1 Model ParametrizationFor short periods of grinding, the products of breakage are too small in quantity to be significantlyre-broken making possible the calculation of Bij [56]. This situation applied to an industrialrefiner would require to collect a sample at a radial position close to the inner radius ri. Assumingthis sampling could be achieved, the sample would contain a mixture of other radial positionssince pulp flows inwards in the stator grooves and outwards in the rotor grooves [57]. Industrialrefiners thus impose a significant limitation on the experimental calculation of Bij making itnecessary to reduce the number of parameters in the model in order to assess fibre shorteningusing a comminution model.In literature, comminution models have been parametrized and solved in different waysfollowing heuristic and domain knowledge. For instance, the most basic assumption for thesolution of the batch grinding equation is to consider both the selection function and breakagefunction to be independent of the grinding time [58]. In the case of LC refining it is assumedthat the Si and Bij are independent of the radius r. Moreover, for this study, some of theguidelines stated by Nakajima and Tanaka [59] were followed. Specifically, the selection functionwas assumed to be:Si = K(lilo)n(3.4)where K is a positive value, li is the bin size in mm, n is a real number and lo is definedarbitrarily as 1 mm for convenience. Considering the power-law form of Eq. 3.4, it should bementioned that K is closely related to the cutting rate inside the refiner since it determines theorder of magnitude of the selection function. Furthermore, the parameter n will dictate thenature of the selection function dependency on the bin size. Using this assumption, it is possibleto describe a wide range of behaviours, including those described by Corte and Agg [60] whoassumed the same rate of cutting for long and short fibres, the assumption of Olson et al. [28]where the selection function was almost linear with the fibre length, and the selection functiondependency with fibre length found by Heymer [31] whose shapes are described by a power-lawfunction.On the other hand, the column entries of the breakage function are symmetric. Cuttinga fibre in half produces two equally long fibres; cutting a fibre at certain distance from one16Figure 3.1: Diagram showing how a 4 units long fibre (black, l4 = 4 units) is shorteneddepending on the cutting position. The fibre can be cut at three different positions (markedwith an x). From left to right, cutting at the first position produces one 1 unit long fibre(l1 = 1 unit) and one 3 units long fibre (l3 = 3 units). Cutting at the second positionproduces two equally long fibres (l2 = 2 units each). Lastly, cutting at the third position,produces the same pair of fibres as the first case.end produces the same pair of fibres if it is cut the same distance from the other end. Cuttinga fibre will produce two shorter fibres whose lengths sum to the original fibre length. Theprevious statements are illustrated in Fig. 3.1. These conditions lead to the following definitionof breakage function:Bij =1p∑j=1Bij0 j ≥ ijm j ≤ i2(i− j)m i2 > j < i(3.5)where the parameter m can be any real number. Since fibres are only shortened, Bij = 0 forj ≥ i and the breakage function must be subject to the constraint ∑pj=1Bij = 1 so each of Bijrows are normalized. Note that this reasoning considers the order of the bins to be inverted (e.g.i = 1 contains the longest fibres, i = 2 the second longest fibres, . . . , i = p contains the shortestfibres). There are three general cases depending on the value of m as depicted in Fig. 3.2. Thephysical interpretation of Bij is: for a single fibre of length lj , Bij is the probability of cuttingat location i. Consider a single fibre extended over the x-axis, the probability of rupture at aspecific location is read on the y-axis. Thus when m < 0 fibres are more likely to be cut at thetails; if m = 0 cutting is even (random); if m > 0 cutting will more likely occur at the middlepoints.Using this Si and Bij parametrization, the comminution model described in Eq. 3.2 containingp(p+ 1)/2 parameters in total with p equations is simplified to a model with only (K,m,n) asparameters. Eq. 3.2 can also be written in matrix form as:Qω12piαβ1rdYdr= AY (3.6)where matrix A is defined as A = (B − I)S. The matrix B is formed by all the entries ofthe breakage function Bij leading to a strictly lower triangular matrix. The matrix S is a174 5 6 7 8 9 10Bi,3i  m< 0m= 0m> 0Figure 3.2: Example of shape for the Bi,3 for the three different cases of m. In thisexample the number of bins is p = 10. For m < 0 fibres break predominantly at the tails.For m = 0 fibre rupture is even. For m > 0 fibre cutting is higher at the middle points.diagonal matrix with the selection function elements Si. Lastly, Y is a column vector withall the frequencies yi. Following the assumption that the selection function Si and breakagefunction Bij are independent of the radius r, the analytic solution to the matrix differentialEq. 3.6 is:Y(Ro) = expM[piωQαβA(Ro2 −Ri2)]Y(Ri) (3.7)Note that the solution involves the matrix exponential function denoted as expM (thesubscript M is used to avoid confusion with the exponential of a scalar). Thus, the FLD afterrefining is related to the initial conditions through the matrix A described by (K,m, n).Eq. 3.7 can be rewritten as:Y(Ro) = expM[(1− λ2)2VTVGαA]Y(Ri) (3.8)where VT = ωRo and it is called tip-speed, describing the tangential speed of the rotor tip.VG = Q/(2piβRo) and it is called groove-speed, describing the radial pulp flow speed at the outerradius of the refiner Ro. Lastly, λ = Ri/Ro being the inner-to-outer-radius ratio. These threeconcepts are commonly used in sizing and designing of industrial refiners as well as scaling-upand trials design calculations.Eq. 3.7 is also analogous to:Y(Ro) = expM [αωAΘ] Y(Ri) (3.9)where Θ = piβ(Ro2−Ri2)/Q is the residence time of pulp inside the refiner. Based on Eq. 3.9, thebehaviour of fibre shortening due to refining is described the same way reactants and productsevolve in a chemical reactor; changes in pulp FLD depend on the residence time Θ and a set ofkinetic constants represented by αωA.18Figure 3.3: Diagram showing refining in stages. Each refining outlet feeds a subsequentrefining stage. This refining configuration is typically found in industry.0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 3.4: Evolution of fibre length distribution during refining in stages. Initial pulp(black solid line) plotted together with observed results (colour lines) for the 1st to 5threfining stage; yi is proportion (or relative frequency) of fibres in the bin i. Data correspondsto actual refining experiments.3.1.2 Refining in StagesOne of the most common industrial refining configurations is refining in stages where initialpulp Y0 is refined in sequential refining stages as shown in Fig. 3.3. If Eq. 3.9 is used to analyzethe evolution of the FLD through the stages, it can be deducted that for the qth refining stage,the FLD is described by:Yq = expM [αωAqΘ] Yq−1 (3.10)A common practice is to set the refining stages to operate at the same angular velocity, plategeometry and SEC. Hence, it could be assumed that the matrix A is the same for all stagesgiven the previous considerations. Thus the general equation for a system of q refining stageswill be:Yq = expM [αωqAΘ] Y0 (3.11)Fig. 3.4 shows a typical example of the FLD evolution by stage.3.1.3 Batch RefiningAlthough refining in stages is common in industry, it is laborious to implement on a laboratoryscale. Not having a constant feedstock supply and having just one refiner impose some operational19Figure 3.5: Configuration for batch refining. A single refiner and tank are arranged asshown to achieve high degrees of refining at laboratory scales.difficulties and restrictions. Batch refining is a convenient method to achieve high degrees ofrefining on a laboratory scale (see Fig. 3.5). In batch refining, there are two processes happeningsimultaneously; (1) fibre shortening inside the refiner and (2) mixing in the feed tank. UsingEq. 3.7, the refiner outlet YRR(t) is related to the feed YT (t) as:YRR(t) = expM [αωAΘ] YT (t) (3.12)Note that in this case it is also assumed that the matrix A remains constant over time giventhat the refining operation is carried out keeping the SEC constant. Assuming ideal mixing, themass balance at the tank is written as:VdYT (t)dt= QYRR(t)−QYT (t) (3.13)where Q is the volumetric flow and V is the tank volume. Combining Eq. 3.12 and 3.13 anddenoting Aˆ = αωAΘ:dYT (t)dt=QV[expM(Aˆ)− I]YT (t) (3.14)In doing so, the differential equation can be solved for YT (t). Using C =QV [expM(Aˆ)− I],the integration of Eq. 3.14 yields:YT (t) = expM (Ct) YT (0) (3.15)3.2 Materials and MethodsThe proposed comminution model parametrization was used to analyze FLD changes due torefining. In the following subsections the experimental data, fibre length measurement methodsand parameter estimation procedure is described in detail.3.2.1 Refining TrialsThis study analyzes the data of two sets of refining trials. The first set of trials (1-9) wereconducted at the Andritz R&D facility in Springfield, Ohio, USA. Primary TMP pulp containing20Trial Screen typePower SEC(kW) (kWh/t)1 - 126 802 Holes 0.8 mm 126 803 Holes 1.0 mm 126 804 Holes 1.2 mm 126 805 Slots 0.15 mm 126 806 - 63 407 Holes 0.8 mm 63 408 Holes 1.0 mm 63 409 Holes 1.2 mm 63 40Table 3.1: Summary of refining trials performed at Andritz R&D facility. For the holed-screens, the values correspond to the hole diameter, whereas for the slotted-screen, thevalue corresponds to its width.a mixture of Spruce-Pine-Fir (SPF) species from the interior of British Columbia, Canada wasused for these trials. Pulp was fractionated via pressure screening using four different basketsand each reject fraction was LC refined in stages at constant SEC per stage. Two levels of SECwere examined. The desired SEC level was achieved by adjusting the refiner gap. Un-screenedpulp was also LC refined as a control. A 22” Twin-Flow disc refiner (Ro = 0.28 m, Ri = 0.16 m)was used with the Andritz Durametal T101/102 plates (Bw = 1.65 mm, Gw = 2.29 mm andD = 6.35 mm). The consistency for this set of trials was 4%. The rotational speed and flowrate were kept constant at 1035 RPM and 660 L/min, respectively. The total of 9 refining trialsis summarized in Table 3.1.The second set of trials (A-F) were performed at the University of British Columbia’s Pulpand Paper Centre, Vancouver, Canada. Different primary TMP pulps, rotational speeds andSEC were examined. Pulp was refined in batch-mode using a 16” Single-Disc refiner (Ro = 0.22m, Ri = 0.13 m) with FineBarTMplates provided by Aikawa. The tank volume and consistencywere kept constant at 250 L and 4%, respectively. Details of trials (A-F) are summarized inTable Fibre length measurementsSamples were taken after each refining stage for the first set of trials (1-9) and at different timeintervals for the second set of trials (A-F). Fibre length was measured following the TAPPIT-271 test method [61]. Approximately 20000 single fibres per sample were measured.3.2.3 Parameter EstimationThe maximum likelihood estimation (MLE) method was used to estimate the parameters of thecomminution model. MLE allows to easily obtain the standard errors and confidence intervals21Trial PulpPlate SEC Q ωBw (mm) Gw (mm) Dw (mm) (kWh/t) (L/min) (RPM)A Pine 1.6 3.2 4.8 70 250 1185B SPF 1.6 3.2 4.8 70 250 1188C Spruce 1.6 3.2 4.8 107 261 1441D Spruce 1.6 3.2 4.8 95 250 1436E Spruce 1.0 2.4 4.8 100 265 1440F SPF 2.0 3.6 4.8 70 300 1200Table 3.2: Summary of refining trials (A-F) performed at the UBC Pulp and PaperCentre. Three different primary pulps and plates were used and a wide range of SEC andω were investigatedfor the corresponding parameters. It also allows to perform likelihood ratio tests in order tocompare the fit between models with different parametrization. Finally, since the yi variabilitiesacross the bins are different, an error minimization scheme would have to address this issue byweighing the errors.It was assumed that the observed data follows a multinomial distribution, which is ageneralization of the binomial distribution. Rather than modeling the possible outcomes of anumber of trials (e.g. 100 coin tosses) into two categories (e.g. heads or tails), the multinomialdistribution accommodates the trials outcomes into a certain number of categories. In thiscase, the multinomial distribution gives the probability Ψ of any particular combination ofnumber of successes ni 5 accomodated in p categories (number of bins) with yi probabilities.Mathematically, this is expressed as:Ψ =nt!n1!n2! . . . np!yn11 yn22 . . . ynpp (3.16)where Ψ is the probability mass function of the multinomial distribution, ni refers to the numberof fibres in the ith bin and nt =∑ni which is the total number of fibres. For the multinomialdistribution to be valid, it has to be ensured that each fibre measurement is independent fromthe others within a pulp sample. The latter condition is satisfied considering that the automatedoptical analyzer used to measure fibre length highly dilutes the pulp sample to ensure singlefibre measurements.Since there is a mathematical expression for yi as a function of the parameters (K,m, n),the likelihood function of interest for a given trial is:Ψ(K,m, n) =q∑s=1{yn11 yn22 . . . ynpp }s (3.17)where the subscript s refers to a specific sample during a trial and q is the number of samples5Note the difference between number of fibres ni and the comminution model parameter n22taken during the trial. For instance using refining in stages, q corresponds to the number ofstages, whereas for batch refining q corresponds to the number of samples taken over the timeintervals. Note that the constant term nt!/(n1!n2! . . . np!) was dropped since it does not dependon the parameters of the comminution model. The log likelihood function is:log Ψ(K,m, n) =q∑s=1p∑i=1{ni log yi}s (3.18)So it is desirable to find the (K,m, n) parameters that maximize the log Ψ(K,m, n) function.Later on, the confidence intervals are estimated from the Fisher information matrix once the loglikelihood is maximized.3.2.4 Likelihood Ratio TestThe likelihood ratio test (LRT) was used to compare the fit between an unrestricted model anda restricted model. The test statistic D is defined as:D = 2 (log Ψu − log Ψr) (3.19)where the subscript u is the unrestricted case and r is the restricted case. D is considered tobe asymptotically chi-squared distributed with d degrees of freedom (χ2d) [62]. The degrees offreedom d is the number of parameters of the unrestricted model minus the number of parametersof the restricted model. With this, a pvalue is calculated as follows:pvalue = 1− Φ[χ2d(D)](3.20)where the operator Φ denotes the cumulative density function of the χ2d function. Thus, thepvalue can be used to determine whether to reject the restricted model or not. If pvalue is lessthan a defined significance level (e.g. pvalue < 0.05), there is strong evidence to reject therestricted model. Eq. 3.19 is valid to calculate the test statistic D for a single refining trial.Since the aim is to evaluate the performance of the restricted model among all refining trials incomparison to an unrestricted model, a global test statistic DT for all trials must be calculatedas:DT = 2T∑t=1(log Ψtu − log Ψtr)(3.21)where the superscript t refers to the different refining trials that are being considered.3.3 ResultsHistograms were constructed for fibres in the [0.2, 4.0] mm range with a fixed bin width of 0.2mm resulting in p = 19 bins. Objects smaller than 0.2 mm are generated by different mechanismsthan cutting and are therefore disregarded; in fact, they are generated by removing fractions of230.3 0.5 0.7 0.9 1.1 1.30.0000.0020.0040.0060.0080.010Figure 3.6: K vs gap for the trials (1-9). Markers are color coded according to the SEClevel; red is 80 kWh/t and blue is 40 kWh/t. H corresponds to holed-screen and S toslotted-screens. Open blue-symbols are the noisy trials 6 and 9. Unscreened pulp showedhigher cutting rate than screened pulps at the same SECcellulose from the fibre wall. On the other hand, fibres larger than 4.0 mm were out of the rangeof interest and also almost nonexistent for the type of pulp used. The three parameters (K,m, n)were estimated for trials (1-9) and (A-F) following the MLE methodology described previously.Trials 6 and 9 presented high degree of noise in their FLD (no clear differences between one stageand the next one), therefore these results were excluded from analysis but presented with opensymbols.Trials (1-9) investigated the effect of fractionated pulps on LC refining performance. Usingpressure screening as a fractionation method, the initial pulp was separated into two streamswith different FLD. The power–gap relationship during refining was strongly affected by thepulp fractionation. For example, trials 2-5 (screened pulps) required a wider gap to achieve 80kWh/t than the gap for trial 1 (unscreened pulp). Similarly, trials 7-9 needed wider gaps toachieve 40 kWh/t than the gaps needed for the trial 6. Hence, screened pulp required largergaps compared to unscreened pulp for a given SEC.Fig. 3.6 shows the parameter K plotted against the average refiner gap measured for thetrials (1-9). As mentioned before, K is closely related to the cutting rate. Thus, the cutting ratefor the unscreened pulp was significantly higher compared to the screened pulp. Two possibleexplanations for this phenomena are proposed: (1) Screened pulps experienced less cutting atthe same SEC level because they contain longer fibres compared to the unscreened pulp or (2)Screened pulps experienced a lower cutting rate at the same SEC level because gap was larger.In order to elucidate this situation, K parameters for trials (1-9) and (A-F) were plottedtogether against average measured gap as seen in Fig. 3.7. The results showed how K is relatedto the inverse of gap by a power-law relationship. In contrast to trials (1-9), trials (A-F) explored240.1 0.3 0.5 0.7 0.9 1.1 1.30.0000.0040.0080.0120.016Figure 3.7: K vs gap for trials (1-9) and (A-F). Dashed-line corresponds to a proposedpower-law function fit of the form K = a/gapb with R2 = 0.89. The fitting constantsa = 0.002 and b = 1.19. Open blue-symbols are the noisy trials 6 and 9. A power-lawfunction seems to correlate well K with gap.a wider range of SEC, plate geometries, wood species (and therefore FLD) and rotational speeds.With this in mind and given the behaviour of K vs gap, the hypothesis that cutting rate ismainly a function of the gap seems to hold stronger. To further support this hypothesis, studiesby Kerekes [52] and Harirforoush et. al. [51] demonstrated that gap has a strong influenceon forces applied to fibres and an important number of reports have shown that smaller gapslead to a greater average fibre length decrease (decrease in average fibre length is an indirectindication of cutting rate).For comparison purposes, Eq. 3.1 (comminution model equation in terms of SEC) was usedto estimate the correspondent K¯ parameters in (kWh/t)−1 and were plotted against the averagegap as shown in Fig. 3.8. As mentioned before, Eq. 3.1 is valid to analyze fibre shortening in LCrefining but does not allow comparisons between trials performed at different conditions. It isclear that K¯ show different behaviours which differ from the results depicted in Fig. 3.7, whereall the data points collapsed to a single power-law curve.In App. A.1, the comminution model equation was derived neglecting the contribution ofgap in β under the assumption that gap << 2GwDw/(Bw +Gw). This simplification does notseem to have a major impact in most of the trials (gap=[0.2-0.5] mm) since the contribution ofgap in β is around 5%. However, for trials with large gaps (trials 6-9) the contribution of gap inβ could be up to 15%, thus the simplification might not be sound. This situation was formallyaddressed by comparing K values calculated in two different scenarios: simplified scenario wherethe contribution of gap was neglected and rigorous scenario where gap was taken into account.Table 3.3 shows the results of these two scenarios. As can be seen, there is not a significantdifference between the values of K (relative errors are in average 4%). Interestingly, the relative250.15 0.35 0.55 0.75 0.95 1.15 1.3500.511.522.5 10-3Figure 3.8: K¯ parameters estimated using Eq. 3.1 vs gap for trials (1-9) and (A-F). Openblue-symbols are the noisy trials 6 and 9. The behaviour between trials (1-9) and (A-F) isdifferent in contrast to the results depicted in Fig. 3.7.errors of trials 6 and 9 were high; it is important to recall that these two trials have been labelledas noisy trials.Fig. 3.9 shows cutting location described by parameter m as a function of gap. It is interestingto note that most of the m values were greater than zero, meaning that, fibre rupture was morelikely to occur at the middle points at different degrees depending on the m values, whereascutting towards the fibres’ ends occurred only in a few cases. Moreover, m seems to be affectednot only by the gap but also by the plate geometry described by α. Starting from the upperregion denoted by the dashed line and moving diagonally towards the origin, the values of αdecrease. In other words, coarse plates (plates with low α values) tend to promote a more evenlydistributed cutting whereas fine plates (high α values) are prone to cut fibres at the middlepoints.Fig. 3.10 shows n vs gap. Although the data points appear disperse, there seems to be aminimum around gap=0.6 mm. For smaller and larger gaps, the values of n increase but overall,they fall within the range n = [1.1, 2.3]. The fact that the values are different than 1 meansthat there are deviations from linearity between the fibre length and the selection function. Ifn = 1 it would imply that Si/li = K and therefore the selection function per unit length of fibreis constant and equal for long and short fibres; given that n > 1, the selection function per unitlength of fibre is higher for long fibres than for short fibres because Si/li = K × ln−1i /lno . Thissuggests that the cutting is not entirely linearly proportional to the fibre length but there areother effects taking place that are being described by n. Since fibres inside the refiner remainflocculated [63–66] and are subject to forces when they are captured by bar-bar crossings, it ishypothesized that within a floc, long fibres take more load than short fibres thus resulting inlong fibres getting cut to a greater degree than short fibres.26Trialsimplified rigorouserror (%)β (mm) K × 103 β (mm) K × 1031 15.09 8.99 15.85 8.76 -2.632 15.09 4.71 16.15 4.60 -2.393 15.09 4.99 16.19 4.85 -2.894 15.09 3.22 16.38 3.10 -3.875 15.09 2.81 16.35 2.68 -4.856 15.09 0.61 16.56 0.53 -15.097 15.09 1.15 17.29 1.21 4.968 15.09 0.57 17.14 0.54 -5.569 15.09 3.33 16.84 2.99 -11.37A 6.40 14.04 6.59 13.65 -2.86B 6.40 7.42 6.67 7.13 -4.07C 6.40 13.30 6.64 12.82 -3.74D 6.40 7.21 6.75 6.94 -3.89E 6.78 8.46 7.02 8.16 -3.68F 6.17 15.59 6.38 15.08 -3.38Table 3.3: Comparison of K values for trials (1-9) and (A-F) in two scenarios. In thesimplified scenario, the contribution of gap in β was neglected whereas in the rigorousscenario, the contribution of gap in β taken into consideration. The relative errors of Kare shown in the last column. As can be seen the errors are low, except for the trials 6and 9; this is interesting since these two trials are regarded as noisy trials.0.1 0.3 0.5 0.7 0.9 1.1 1.3-10123456Figure 3.9: m vs gap for trials (1-9) and (A-F). Dashed-line ellipses enclose trialsperformed with the same plate. For the big dashed ellipse, α=0.42; for the green-,α=0.35; for the purple markers, α=0.33; for the black-3, α=0.29; Open blue-symbols arethe noisy trials 6 and 9. Parameter m is affected by plate geometry and gap. Most of mvalues are greater than 0 meaning fibre rupture happens predominantly at the middle ofthe fibre.270.1 0.3 0.5 0.7 0.9 1.1 3.10: n vs gap for trials (1-9) and (A-F). Open blue-symbols are the noisy trials6 and 9. There is no apparent trend in the data points. All n values resulted greater than1 indicating that long-fibres’ cutting rate is higher than short-fibres’ cutting rate.In Sec. 3.1.2 and 3.1.3, the governing equations for refining in series and batch-mode wereobtained assuming that the matrix A was constant, given that refining was carried out at thesame SEC level across the stages (or constant over time for the batch mode case). As mentionedbefore, gap needed to be adjusted accordingly at each stage (or over time for the batch modecase) to keep the SEC at the same level. Since gap was different over the stages for trials(1-9) and over time for trials (A-F), coupled with the results of Fig. 3.7 and 3.10 showing thecorrelation between (K,m, n) with gap, it seems logical to regard the matrix A as dependenton the gap rather than constant based on a constant SEC.With these considerations, the refining trials data was analyzed once more in order toestimate the correspondent parameters per stage and time interval for each trial. Thus acollection of (Kq,mq, nq) parameters per trial were estimated, as opposed to single (K,m, n)values per trial estimated before (the subscript q is used to avoid confusion between one and theother).Fig. 3.11 shows four examples of Kq values as function of gapq and Fig. 3.12 combines allthe Kq values from trials (1-9) and (A-F). As can be seen in Fig. 3.11, Kq values have anequally power-law behaviour with inverse of gap as K, supporting the proposed hypothesisthat cutting rate is mainly a function of gap. Previously, it was demonstrated that changingthe pulp properties via pressure screening did not have a major impact on K; here it is alsodemonstrated that changing pulp properties by refining does not have a major impact on Keither. The parameters mq and nq were also estimated and are presented in Fig. 3.13 and 3.14.In these two aforementioned plots, data points follow similar trends to those depicted in Fig. 3.9and 3.10.Regardless of whether matrix A is considered constant or not, these two approaches yield280 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 3.11: Four examples of Kq vs gapq. The dashed-line corresponds to the same onein Fig. 3.7 and it is plotted as reference.0 0.2 0.4 0.6 0.8 1 1.2 3.12: Kp values obtained considering a non-constant matrix A. The dashed-linecorresponds to the same one in Fig. 3.7 and it is plotted as reference. Open blue-symbolsare the noisy trials 6 and 9. Similar behaviours between K and Kq suggests that bothapproaches are valid.290 0.2 0.4 0.6 0.8 1 1.2 1.4-2-10123456Figure 3.13: mp values obtained considering a non-constant matrix A. The dashed-lineellipses correspond to the ones shown in Fig. 3.9 and are included as reference. Openblue-symbols are the noisy trials 6 and 9. Data points have a similar segregation to theone observed in Fig. 0.3 0.5 0.7 0.9 1.1 1.30.511.522.533.5Figure 3.14: np values obtained considering a non-constant matrix A. Open blue-symbols are the noisy trials 6 and 9. Values of nq are located on the same regions as theircounterparts n values presented in Fig. 3.10.30Case DT pvalue Resultm = 0 & n = 1 9962.01 0 rejectm = 0 477.60 0.0012 rejectn = 1 1473.25 5× 10−16 rejectm→∞ 970.21 1× 10−13 rejectTable 3.4: Model comparisons using likelihood ratio test. All cases were rejected accordingto the statistical test at a 5% significance level.similar results and trends indicating that both ways are equally valid. Nevertheless, estimating(K,m, n) grants some statistical advantages since the number of data points is greater thanwhen estimating (Kq,mq, nq) thus making the results less sensitive to measurement errors andprocess noise. One can consider (K,m, n) as global mean values of (Kq,mq, nq) but in the end,they are describing the phenomena in a similar way.3.3.1 Model Fitting Accuracy and ComparisonsThe accuracy of the fit of the proposed parametrization was assessed using the LRT described inSec. 3.2.4. In the hypothetical situation of being able to estimate all parameters from Eq. 3.2(190 parameters in total for p = 19), the fit between the experimental data yexpi and the estimateddata yesti is supposed to be perfect, meaning yexpi = yesti . Thus, it is possible to calculate theunrestricted model objective function value log Ψtu. Later on, it can be compared with log Ψtrassociated to the restricted model (K,m, n) using LRT. The test statistic calculated usingEq. 3.21 was DT = 2025.5 resulting in a pvalue = 1 which is greater than the significance levelof 0.05. Therefore, there is no statistical evidence to reject the restricted model with (K,m, n)as parameters. For illustrative purposes, App. B shows the experimental and predicted fibrelength distributions for different refining trials. From a qualitatively point of view, it is seenthat the proposed parametrization does a good job predicting experimental values, supportingthe previous statistic test result.Additionally, it was further explored whether it is possible to make other simplifications inthe proposed parametrization. For instance, Roux and Mayade [30] assumed that the fibres arecut only at the midpoints; that is the case when m→∞. Olson et.al.[28] used the comminutionmodel assuming a random cutting; that is when m = 0. It was also suggested that the selectionfunction was linearly dependent on the fibre length; that is when n = 1. Here, the model(K,m, n) becomes the unrestricted model and all the cases described before are restricted models.The comparison results are shown in Table 3.4. In all cases, the statistical test rejected theadditional restrictions proposed. With this, it is deduced that if any of these specific cases iscompared to a more general model than (K,m, n), it will also be rejected.313.4 DiscussionTraditionally, refining studies have used SEC combined with the SEL theory to characterize thenature of a particular refining treatment. The SEL aims to include the plate geometry in theanalysis since it is well known that plate geometry plays an important role in refining. However,the results obtained in this study had no correlation with SEC or SEL.This model parametrization greatly decreased the number of parameters needed to describefibre shortening in LC refining. The resulting model could seem too parsimonious since it reducesthe number of parameters to only 3. For example, for p = 19 the number of parameters isreduced from 190 to only 3, which could seem excessive. However, the parameters had correlationto the refining conditions which allows for simple interpretations of the results and the likelihoodratio test proved that this parametrization describes the behaviour accurately.The bin width was selected taking into consideration the trade-off between a large numberof bins and a small number. For instance, a large number of bins would allow to extract moredetailed information from the results (e.g. breakage function) and also would result in smallstandard errors and narrower confidence intervals. Although it may seem that a large number ofbins is beneficial, there is a point where the number of fibres in each bin becomes noisy, losingall relevance of data representation; with a bin width of 0.2 mm there was no apparent noise(e.g. bins with zero counts).According to Austin et al. [67] non-first-order breakage result from operations where theparticle size is sufficiently large compared to the ball and mill diameter. This can be somehowtranslated to LC refining by comparing the typical fibre areas to the bar-bar area (ratio of 1:17fibre area:bar-bar area). Although this is not a rigorous analysis, this premise can be used atleast to justify the assumption of first order breakage in LC refining.With this parametrization, it is implicitly assumed that Si was a function of fibre lengthwhen its shape was chosen. Under this reasoning, there could be also the situation that theresultant fibres after cutting are distributed differently depending on their length, in otherwords, the Bij could presumably also be dependent on the fibre length. This was not takeninto consideration in this study (a single m was used for all fibres). Additionally, as mentionedbefore, wood fibres are known to be composed of two sub-populations differing in their cell-wallthickness (earlywood and latewood); this supports the idea of proposing at least two sets ofparameters to describe the behaviour of each population separately. Furthermore, this approachis also valid to study fibre shortening during co-refining, where two or more different pulpsare intentionally mixed and refined. App. C describes a few considerations to approach thisproblem.3.5 ConclusionsA simplified framework to describe fibre shortening in low consistency refining using a com-minution model was developed. The framework has the advantage of using a few parameters to32describe the concepts of selection and breakage involved in comminution.Over the range of variables explored, correlations between the parameters and gap werefound, specifically:• The parameter K, which strongly affects the selection function, was found to be mainlydependent on the refiner gap. A power-law relationship describes well the dependencybetween K and inverse of gap.• There is a clear indication that closing the gap favors fibre cutting in the middle pointsof fibres, whereas wider gaps promote a more evenly distributed cutting. There is alsoevidence that plate geometry affects the cutting location. Coarser plates promote a moreevenly distributed cutting whereas with finer plates, cutting more likely occurs in themiddle of fibres.• The selection function as a function of fibre length deviates from a linear behaviour. Theselection function per unit length of fibre is not equal for all fibre classes, it is higher forlong fibres than for short fibres. These deviations depend on the refiner gap.33Chapter 4Power-Gap Relationships in LCRefiningSometimes we can’t see why normal isn’t normal – Dr. Gregory House (fictionalcharacter)LC refining processes have been traditionally controlled and evaluated in terms of pulp andhandsheet properties. The majority of refining studies take this approach to show how pulpproperties are affected by refining. Alternatively, other studies have gone more into detail tounderstand the role of plate geometry, forces and energy in the overall refining operation.In contrast, it is rather seldom to assess the same situation from a different perspective; e.g.to understand the response of the refiner behaviour to common variables such as pulp typeand its changes, plate geometry, rotational speed, refiner diameter. This particular approach isimportant in cases where it is desired to optimize energy consumption for a particular refiningoperation. Moreover, this is an every day situation at industrial scales, where refiners operate ata wide range of conditions due to changes in feedstock properties and process conditions. Thedescription and understanding of the power–gap relationships are the main focus of this chapter.Sec. 4.1 develops theoretical considerations to describe normal forces as a function of refiningvariables. Later on in Sec. 4.2, these considerations are used to develop a correlation thatdescribes refining net-power. Sec. 4.3 describes the methodology followed to test the hypothesismade in Sec. 4.1 and experimental data used to fit the correlation proposed in Sec. 4.2. InSec. 4.3 and Sec. 4.4 results and discussion are presented and Sec. 4.5 finalizes the chapter withconcluding remarks.4.1 Normal ForcesWhen pulp is fed to a refiner, fibres are forced to pass between the rotor and stator plates. Dueto the plates’ bar-groove patterns and the rotational nature of the rotor, fibres are subject tomainly normal and shear forces when they are captured by a bar-bar crossing (see Fig. 2.3).34Normal and shear forces are the responsible for the refining effect on pulp and also for therefiner power consumption. Thus, the description of forces is a step towards the understandingof power.Although studies on forces during refining have made important contributions to involvethem in refining characterization, the results are oftentimes related to mean values and refiningtheories [68] or, in the case of bar-force measurement studies, in the description of forcesdistributions. Yet, there is not much information in the literature about the relationship offorces with gap, radial position, rotational speed and pulp.4.1.1 Clutch Operation Approach - Constant Wear TheoryAccording to Herbert and Marsh [35], the effective work of a disc refiner can be described as thefriction between two discs, similar to a clutch where the constant wear theory applies most ofthe time. The constant wear theory can be written as:f¯N (r)r = constant (4.1)where f¯N is the normal force or contact force between the two discs. As a result, their studyconcluded that there is an optimum inner to outer diameter ratio at Di/Do ≈ 0.58. However thebehaviour of a disc refiner is far from being similar to a clutch operation. In a clutch operation,the discs are in direct and constant contact and wear occurs because of this contact; in a discrefiner, the plates are never in direct contact and there is always a pulp suspension between theplates. Therefore, the constant wear theory does not seem to be valid for pulp refiners.4.1.2 Force Measurements StudiesTo date, there have been just a few studies measuring forces in LC refining. Unfortunately,sensors have only been installed at a single radial position, making it impossible to realizewhether there is a radial dependency of forces or not. In contrast, the force dependency withradius has been previously studied in HC refiners. For instance, Backlund [69] observed thatthe tangential force is highest at the periphery, exhibiting an increasing trend along the plategap. A few years later, Olender et al. [70] installed three bar-force-sensors at different radialpositions and observed an increasing trend of shear and normal forces with radius when rejectpulp was refined. Although there are significant differences between HC and LC refiners, itseems reasonable to assume a similar increasing dependency of forces with radius in LC as well.Additionally, it can be deduced that forces in the periphery of the refiner are greater than thoseclose to the inner radius due to the difference in the linear speed between these two points.4.1.3 Theoretical Analysis - Normal ForcesReviewing previous studies and experimental evidence reported by other authors, it is recurrentto find that forces are, at least, function of rotational speed, plate geometry, gap and pulp.35Moreover, it is clear that normal forces give rise to shear forces which in the end translates toenergy expenditure. In that sense, this theoretical analysis starts by relating the normal stressexerted by a bar-bar crossing on pulp as the product of three functions h1, h2 and h3 as:f¯NAc∼ h1(ωr) h2(gap) h3(pulp) (4.2)where Ac the area of a bar-bar crossing. By performing a simple dimensional analysis, thenature of h1 function is hypothesized to be:h1(ωr) = ρ (ωr)2 (4.3)Note that the dimensions of h1 are the same as the normal stress (pressure in Pa). This firsthypothesis is in agreement with the work done by Olender et al. [70], who observed measurednormal and shear forces increased with radial position. Moreover, the square dependency ofnormal force with bar linear speed described in Eq. 4.3 coincides with the theoretical analysisdone by Eriksen et al. [71, 72].In order to approach the nature of the h2 and h3 functions, the pulp capture by a bar-barcrossing is regarded as a compression event. The compression rheology of suspensions has beensubject of study in several fields, including wood pulp suspensions. Typically, a suspensionsample is placed in a container with a rigid base, and is compressed from above by a porouspiston. As the sample is compressed, the fluid passes through the porous piston and theconcentration of the solid fraction increases. The compressive stress can be described by certainpower-law curves in terms of solidity [73–75]. These relationships are rewritten here in terms ofpulp consistency C as:Py = a1(C − Co)a2 (4.4)where Py is the load (compressive strain) applied to pulp, C and Co are the consistencies atthe compressed and uncompressed states, respectively. a1 and a2 are fitting parameters thatdepend on the material (pulp). Since in this type of experiments mass is conserved, there is arelationship between the consistencies and the piston height (gap) as:Cgap = Cogapo (4.5)where gap and gapo are the piston heights at the compressed and uncompressed state, respectively.Combining Eq. 4.4 and 4.5 yields:Py = a1Coa2(gapogap− 1)a2(4.6)Thus, product of the functions h2 and h3 is hypothesized to be:h2(gap) h3(pulp) = M(pulp)Ga2 (4.7a)36where the function M collects the pulp dependent terms a1Coa2 and G is defined as:G =(gapogap− 1)(4.7b)Since it is rather difficult to elucidate the nature of M as function of pulp without performingtailored experiments, in this study refiner’s feed length-weighed mean fibre length (lw) is used asan indirect measurement of the pulp’s characteristics. Summarizing the theoretical considerationsof this subsection (combining Eq. 4.2, 4.3 and 4.7), the force f¯N exherted by a bar-bar crossingof area Ac is expressed as:f¯NAc∼ ρ (ωr)2M(lw)Ga2 (4.8)where the nature of the function M is yet to be realized.4.2 Refining Net-PowerSo far, the theoretical analysis done in Sec. 4.1 led to a mathematical expression of f¯N as afunction of ωr, G and lw. However, the refining power consumption arises from shear forces,thus the first step towards an expression for the refining net-power is to express the shear forcesas the product of normal forces f¯N and a friction coefficient µf as:f¯s = µf f¯N (4.9)The friction coefficient µf is assumed to be dependent on the nature of the surfaces involved(refiner plate material and pulp). On the other hand, the power consumption of a bar-barcrossing is the product of shear force and the linear speed. The previous statement translatesinto a mathematical equation as:Pc = f¯sωr (4.10)where Pc is the power consumption by an arbitrary bar-bar crossing at a distance r from thecentre of the stator which is rotating at angular speed ω. If this reasoning is extended to allbar-bar crossings in a Single-disc refiner, it is possible to describe the net-refining power as thecontribution of every single bar-bar crossing power consumption as:Nc∑i=1Pc(r) = ωNc∑i=1f¯sr (4.11)where Nc is the total number of bar-bar crossings and the left hand side of the equation is equalto Pnet. Eq. 4.11 is developed for a Single-disc refiner. If this equation needs to be adapted to aTwin-Flow disc refiner, a factor of 2 needs to be included in the right hand side of the equation.The reasoning is that a Twin-Flow disc refiner has twice the amount of bar-bar crossings.Eq. 4.11 can be rewritten in integral form by relating plate geometry characteristics withthe number of crossings Nc. According to Roux [42] and confirmed by to Elahimehr et al. [54],37the number of crossings Nc as a function of the refiner radius r and the plate geometry can beapproximated as:dNc ≈ 2pirdr(Bw +Gw)2sin(γ¯) (4.12)where Bw and Gw are the bar width and groove width, respectively and γ¯ is the average angleof the crossings. Combining Eq. 4.11 and 4.12 and assuming that Bw and Gw do not changewith r, yields:Pnet =2piω sin(γ¯)(Bw +Gw)2∫ RoRif¯sr2dr (4.13)Using the theoretical considerations summarized in Eq. 4.8 into Eq. 4.13 and expressing thearea of a single bar-bar crossing as Ac ≈ Bw2/sin γ¯ yields:Pnet = 2piω3(BwBw +Gw)2µfM(lw)Ga2∫ RoRir4dr (4.14)and solving the integral and rearranging terms one gets:Pdα2= µfM(lw)Ga2 (4.15a)Pd =Pnetρω3(Ro5 −Ri5)(4.15b)where Pd is a dimensionless power number based on the net-power and refiner inner and outerradius. On the other hand, α = Bw/Bw+Gw is the dimensionless number describing the plategeometry used before in Ch. 3. Note that the function M is grouping constant terms in orderto simplify the notation. It is important to highlight that there is still the need to realize thefunctional form of the product µfM ; as mentioned before, this is intended to be describedby a function dependent on lw. Since Eq. 4.15a is a correlation for the net-power, one finalconsideration has to be made: specifically, it must be assumed that gapo is the gap at the NLPregime. With this, if gap = gapo, leads to G = 0 and thus Pd = 0 satisfying the conditions ofno-load-power regime Pnet = 0.4.3 Materials and MethodsThe theoretical framework built in the previous section allowed to derive a correlation to predictrefining net-power. The hypotheses made in Sec. 4.1 around the dependency of normal forceswith ωr and G are tested using bar-force sensor measurements data. Also, the developedframework is tested to realize if it is suitable to describe power–gap relationships by fittingpilot-scale refining data to the proposed correlation and comparing it to industrial-scale data. Inthe following sections, experimental data (bar-force and refining), test methods and numericalmethods used are described.384.3.1 Bar-Force MeasurementsHarirforoush et al. [51] measured forces in a LC refiner using a custom designed and fabricatedbar-force sensor. The sensor was installed in the 16” Single-Disc refiner of the Pulp and PaperCentre at the University of British Columbia, Vancouver, Canada. Forces were measured atdifferent gaps and three rotational speeds. The author of this dissertation had access to thebar-force sensor measurements data.4.3.2 Pilot-Scale Refining TrialsA collection of pilot-scale refining trials data involving LC refining of primary TMP pulps wascompiled and subject to study. A total of 3 sets of refining trials were grouped together. Thedescription of each group is as follows:1. Market TMP pulps with a mixture of SPF species with initial CSF of 110, 330 and 550 mLwere refined in stages at a wide range of refining conditions (plate geometries, rotationalspeeds, net-powers).2. Primary TMP pulp (CSF = 600 mL) containing a SPF mixture from the interior of BritishColumbia, Canada was fractionated using pressure screening. The reject fractions wererefined in stages at a constant net-power per stage. Unscreened pulp was also refined forreference. Two levels of net-powers were examined. Plate geometry, rotational speed andflow rates were kept constant for all trials.3. SPF chips from an RT Pressafiner were impregnated with sulphite (2%) then HC refined toobtain different primary cTMP pulps (CSF 300 and 650 mL). The primary pulps were thenLC refined in batch-mode. Different net-powers, plate geometries and rotational speedswere investigated. Pulp without chemical impregnation was also LC refined as controlFor all trials, net-power Pnet was kept constant at the desired targets by adjusting gapaccordingly. The refining trials from sets (1-2) were performed at Andritz R&D facility (Spring-field, Ohio, USA) using a 22” Twin-Flow disc refiner (Ro = 0.28 m and Ri = 0.16 m). Whereasthe trials from set 3 were carried out at the Pulp and Paper Centre (University of BritishColumbia, Vancouver, Canada) using a 16” Single-Disc refiner (Ro = 0.22 m and Ri = 0.13 m).For all trials, the consistency was kept constant at 4%. In total, data from 31 refining trialswere compiled to yield 151 data points.4.3.3 No-Load-Power and Net-Power CalculationAt the beginning of each refining trial, the no-load-power was measured by backing the plates offcompletely and recording the refining power while the pulp was flowing through the refiner. Lateron, during the refining trials, the net-power was calculated by subtracting the no-load-powerfrom the total refining power. Although differences in the refiners’ design and configurations can39lead to different gaps at the time of measuring the no-load-power, this is a conventional methodto measure it. Moreover, the plates were sufficiently apart from each other (approximately 2.5mm) so that the no-load-power measurements were far away from generating changes in pulpproperties.4.3.4 Fibre Length MeasurementsPulp samples were taken after each refining stage or at different time intervals (for the batch-mode refining). The samples were analyzed using the TAPPI T-271 test method. Later on, thelength-weighted fibre length was calculated from the fibre length distributions measurementsusing the following equation:lw =p∑i=1nil2ip∑i=1nili(4.16)4.3.5 Gap MeasurementsFor the set (1-2) of trials, the gap was estimated using a linear variable differential transformer(LVDT) whose ends were attached to the refiner housing using magnetic clamps. Thus themovement of the housing when gap is adjusted gives a relative position of the gap between thetwo stators of the Twin-Flow disc refiner. The half of this measured distance yields the distancebetween the stator and its adjacent rotor assuming the floating rotor is centred.For the set 3 of trials performed at the Pulp and Paper Centre, the gap is measured bydetermining the shaft movement (attached to the rotor side), thus giving a real plate position.4.3.6 Parameter Estimation and Outliers DetectionThe parameter estimation was performed solving a traditional error minimization scheme. Theproblem was posed as a nonlinear least-squares problem. The advantage of this scheme is thatparameters confidence intervals and prediction error bars and confidence intervals are easilycalculated using the covariance matrix. Additionally, outlier detection is fairly simple. Thereported confidence intervals (parameters and predictions) are at 95% for both cases. Theoutliers were identified finding those data points whose Cook’s distance was greater than 1 [76].4.4 Results4.4.1 Bar-Force-Sensor Data AnalysisThis section is devoted to test the hypothesis presented in Eq. 4.3 and 4.7. To do so, resultsfrom bar-force measurements study performed by Harirforoush et al. [51] are presented herein.Fig. 4.1a shows the measured normal forces f¯N as a function of gap and ω. As expected, the40(a) f¯N vs gap0.2 0.3 0.4 0.5 0.6 0.7 0.805101520800 RPM1000 RPM1200 RPM(b) f¯N/ω2 vs G100 101 10210-410-310-2800 RPM1000 RPM1200 RPMFigure 4.1: (a) Normal forces measured by Harirforoush et al. [51] using a customdesigned and fabricated bar-force-sensor at different gaps and 800 RPM (©), 1000 RPM() and 1200 RPM () as rotational speeds. (b) log-log plot of ¯fN/ω2 vs G of the datapresented on the left hand side plot. These results support the hypothesis that f¯N ∼ ω2 asthe data points show the same linear behaviour. Moreover, the dashed-line’s slope is 1.90suggesting that f¯N ∼ G2latter two have a strong influence in the measured forces. Yet, this particular plot does not helpto clarify the dependency of normal forces with gap and rotational speed. Further data analysisshowed that the relationship between normal forces and rotational speed was indeed f¯N ∝ ω2 ashypothesized in Eq. 4.3. This can be seen in Fig. 4.1b where f¯N/ω2 was plotted against G on alog-log scale. Data points from different rotational speeds showed the same linear behaviourover the range of the x-axis. Interestingly, Fig. 4.1b also suggest that the force dependency withG is almost squared (f¯N ∝ G2) as the slope of the dashed line is 1.90. With this result, thevalue of a2 is realized and approximated to Dimensionless Quantity Pdα2As has been highlighted before, net-power depends on several variables and each power–gaprelationship is specific for a particular set of refining conditions. For illustrative purposes,Fig. 4.2 shows two examples of the usefulness of using Pd/α2 . In Fig. 4.2a and 4.2d data ofnet-power vs gap is presented for cases where lw was between [1.05, 1.15] and [1.50, 1.60] mm,respectively6; the data does not seem to have any correlation nor generalized trend and the Pnetdifferences are evident for the two refiner sizes presented. As seen in Fig. 4.2b and 4.2e, with theuse Pd, the y-axis values are now in the same order of magnitude, yet there is still dispersion inthe data points. In contrast, by using Pd/α2 data from both refiners depict similar behaviours6It is important to mentioning that these data points do not correspond to refining curves. They are acollection to power versus gap data points whose lw fall withing the aforementioned ranges.410  0.2 0.4 0.6 0.80  50 10015016" Single-Disc refiner 22" Twin-Flow disc refiner0  0.2 0.4 0.6 0.80  0.2 0.4 0.6 0.80  0.2 0.4 0.6 0.80  50 1001500  0.2 0.4 0.6 0.80  0.2 0.4 0.6 0.80 (b) (c)(d) (e) (f)Figure 4.2: (a) and (d): Experimental data of net-power Pnet vs. gap. (b) and (e): Plotof dimesionless number Pd vs. gap. (c) and (f): Plot of dimensionless numberPd/α2 vs.gap. For plots (a), (b) and (c) (top row), data points are in the range lw = [1.05, 1.15] mmwhereas for plots (d), (e) and (f) (bottom row), data points are in the range lw = [1.50, 1.60]mm. Plots (a) and (d) show differences in net-power Pnet between the two refiners; datadoes not seem to have a generalized trend. In (b) and (e), y-values of Pd are now in thesame order of magnitude, yet data is somehow disperse. In contrast, (c) and (f) show howthe use Pd/α2 is suitable for assessing experimental data from different refiner sizes andplate geometries.as seen in Fig. 4.2c and 4.2f. Both dimensionless quantities, Pd and α2, are crucial to assessexperimental data from different refiner sizes and plate geometries.4.4.3 Fibre Length DependencyFig. 4.2 gives an idea of the effect of pulp on the power–gap curves; as expected, power-gapcurves are shifted to the right with the increase of lw. In Eq. 4.15a, µfM is describing thedependency of these power-gap curves with pulp where lw is used as an indirect measure of pulpcharacteristics. Hence, the following dimensionless group was arbitrary defined:µfM(lw) =(lwc1)c2(4.17)where c1 and c2 are fitting parameters. Thus, that the correlation to predict net-power is writtenas:Pdα2=(lwc1)c2G2 (4.18)424.4.4 Fitting ResultsFrom a theoretical point of view, refiner gap is an independent variable which affects othervariables (e.g. net-power). However, from a practical point of view, the refiner gap is oftentreated as a dependent variable for many reasons. First and foremost, because it is a commonpractice to set power targets among refiners and achieve them by changing gap. Secondly, almostall refining operations are controlled by SEC and the easiest way to manipulate it is by changingthe gap. Lastly, because it is rather challenging to keep track of gap at industrial scale as itcould be difficult to accurately measure it, especially in Twin-Flow disc refiners. For thesereasons and because of the way the refining trials were designed and executed, it is imperativeto consider the refining power as an independent variable and gap as a dependent variable; inthe end, for each power target, gap was adjusted.Eq. 4.18 was used to fit the experimental data. Fitting the data to this particular equationdid not produce good fitting results (R2 = 0.70). Further data analysis showed that Pd is notexactly proportional to G2. Hence, it was imperative to modify the original fitting equation to:Pdα2=(lwc1)c2G(2−c3) (4.19)which introduces a new fitting parameter c3. Data was fitted once more using Eq. 4.19 and theresults of the fitting are summarized in Table 4.1 where the estimated values and their confidenceintervals are presented. Additionally, Fig. 4.3 shows a parity plot between the predicted gap andthe measured gap where the outliers are marked as red-asterisks. This time there was a good fit(R2 = 0.91 and RMSE = 0.0599) considering the wide range of refining conditions, pulp typesand plate geometries included in the data. Furthermore, error bars are rather short (between2-8%) and confidence intervals of the fitted parameters are significantly narrow.The fitting results also point that Pd is almost proportional to lw3. This is indeed aninteresting result as lw3 resembles a measure of volume. On the other hand, fitting results alsosuggest that the power–gap relationship is of the form Pd ∝ G; this is somehow in agreementwith previous studies that described power–gap relationship as Pnet ∝ 1/gapc with c = [0.7, 1.3][35, 37, 38, 51, 77]. However, the use of G is more accurate to describe net-power because theinverse of gap function does not satisfies the conditions of NLP since it asymptotically approachesto zero at wide gaps, whereas the function G is equals to zero at the NLP regime.Fig. 4.4 plots Pd/α2 as a function gap for different values of lw. This plot was constructedusing the fitting constants (c1, c2, c3) in Eq. 4.18. The developed correlation is able to describeobservations made by other researchers. For instance, refined pulps need a smaller gap to achievethe same net-power than unrefined pulp; as pulp gets refined, fibres get shortened, so if anarbitrary horizontal line (constant power) is drawn in Fig. 4.4, any decrease in lw along thehorizontal line will lead to a decrease in gap. Similarly, reject fractions of screened pulps requirea larger gap to achieve the same net-power compared to un-screened pulps; reject fractions havelonger fibre length, so again, considering an arbitrary horizontal line, any increase in lw leads430.0 0.2 0.4 0.6 0.8 1.0 4.3: Parity plot between predicted and measured gap. R2 = 0.91 and RMSE =0.0599. Error bars of predictions were estimated using the covariance matrix at a 95% ofconfidence. Red-asterisks (∗) are data points identified as outliers according to the Cook’sdistance criteria.Estimate Confidence Interval (95%)c1 4.664 0.3081c2 2.701 0.1569c3 0.955 0.0569R2 0.91RMSE 0.0599Table 4.1: Fitting results and statistics of the fit of experimental data on the power-gapcorrelation presented Eq. 4.18. It is important to mention that c1 has units of mmto an increase in gap. It is worth mentioning once more that this study uses lw as an indirectmeasure of pulp properties changes.4.4.5 Comparisons with Industrial-Scale dataThe correlation was also compared to industrial-scale refining data. It could have been possibleto use the industrial scale data along with the pilot-scale data to develop the correlation, splittingthe data set into two (fitting and validation dataset). Nevertheless, the developed correlation wastested to realize whether it can describe equally well data from both pilot-scale industrial-scalerefiners or not.Power–gap curves from a 72” and 58” Twin-Flow disc refiners were measured using pulpswhose lw was 1.30 and 1.65 mm, respectively. Additionally, power–gap values were collectedfrom the 72” Twin-Flow disc refiner using different operational conditions; for this collection440.50.50.5111.51.5222.530.0 0.2 0.4 0.6 0.8 1.0 4.4: Plot of dimensionless quantity Pd/α2 as a function of gap for different lw.Lines correspond to constant values of lw as indicated. The plot was constructed using thefitted constants (lwo, c1, c2) in Eq. 4.15a. Data points correspond to industrial scale data.of points, the pulp had lw ≈ 1.55 mm. Later on, industrial data of Pd/α2 vs gap was plottedalongside with the contour plot shown in Fig. 4.4. The coefficient of correlation between themeasured industrial data and predicted values was R2 = 0.91 with RSME = 0.0468. Moreover,the correlation was found to predict industrial scale refining data within 5-25% of relative error.4.5 DiscussionSome of the previous works on refining power such as Elahimehr et al. [38], Luukkonen et al.[77] have used a dimensionless power number defined as:Pd∗ =Pnetρω3Do5 (4.20)to assess refining data and develop correlations. By only using the outer diameter Do, it isimplicitly assumed that the refiner’s plates are complete solid discs. This, of course, is not trueand can lead to improper results. Defining a dimensionless power using the inner and outerradii provides a truly detailed information of the effective working area of the refiner plates.In Sec. 4.1, it was hypothesized that the normal forces f¯N vary with (ωr)2. Although usingbar-force-sensor data, the relationship between forces and rotational speed was realized, theforce-radius dependency should be formally addressed in the future by measuring forces in LCrefining at different radial positions. Additionally, if the f¯N ∼ r2 hypothesis is true, it impliesthat pulp would be subject to a distribution of forces as it passes through the refiner, thusfurther increasing the heterogeneity of the treatment. A possible way to solve this issue (if it isa problem at all) would be to design refiner plates similar to a conical viscometer, where the45distance between the plates is wider at the periphery, thus ensuring a constant force profile overthe radius.In the pulp and paper field, there is generalized and strong interest in developing a frameworkcapable of assessing and scaling-up data from pilot-scale refiners to industrial-scale refiners.Researchers have been partially successful in this milestone by developing refining theories.Perhaps the missing link between these two scales is a well-defined dimensionless number suchas Pd/α2 . Moreover with this study, the understanding of power as a function of gap is improvedby the use of G instead of gap alone. Finally, this study also takes a step forward towards aforce based refining characterization since it relates forces with power consumption.Based on the developed correlation, some affinity laws can be derived, similar to the lawsused in centrifugal pump design. Affinity laws are useful to predict performance from an knownbehaviour at different conditions. An example of affinity laws for LC refining is shown in App. DAlthough the correlation developed in this study was exclusively built for LC refining ofprimary mechanical pulps, the framework used here can be extended to develop a similarcorrelation for any type of pulps.4.6 ConclusionsA correlation to predict refiner dimensionless power Pd was developed as a function of twocommonly used variables; one related to pulp (lw) and the other related to the machine side (gap).Pd was almost proportional to lw3 and G. The use of well-defined dimensionless power provedto be adequate to compare refiners of different sizes. The correlation built on pilot-scale datawas compared to industrial-scale refining data showing good agreement between the predictedand measured values. Theoretical assumptions were supported using force measurements data.Experimental evidence suggest that normal forces are dependent on the square of ω and G.46Chapter 5Refining Systems AnalysisMeasure what is measurable, and make measurable what is not so – Galileo Galilei(Italian astronomer, 1564 – 1642)A system composed merely of an LC refiner is a complex system to analyze since it is bynature a heterogeneous process involving a heterogeneous raw material. Additionally, developinga mathematical model capable of accurately describing certain pulp properties changes dueto refining is challenging. Part of the difficulties faced during modeling is that oftentimes theproperties of interest are not conservative (e.g. tensile, bulk, freeness). On the other hand,modeling conservative properties (e.g. fibre length) is an easier task. In the case of fibre length,screening models and comminution models (for LC refining) actually describe FLD changes usingmass balances as a main tool of analysis.In this chapter, three basic refining systems are studied. The systems consist of one LCrefiner and one pressure screen arranged in different configurations namely Reject-refining,Feed-back rejects and Feed-back reject-refining. These systems exhibit recirculation, flow mixingand coupling between screening and refining unit operations as seen in Fig. 5.1. In Sec. 5.1,5.2, 5.3 and 5.4, screening and LC refining comminution models are reviewed, followed by otherimportant definitions and correlations needed for this chapter. Sec. 5.5 describes the approachused to analyze the refining systems and how the reviewed models are used. Sec. 5.6 presentsthe results on the theoretical cases analyzed and also a study case where simulations results andindustrial-scale data is compared. Finally, Sec. 5.7 closes the chapter with a series of concludingremarks and recommendations.5.1 Screening ModelsPressure screening is a fractionation method where a pulp feed stream is separated into twodifferent streams, each with different FLD. In pressure screening fibres are sorted mainly by fibrelength [78] and to some low extent based on fibre flexibility [79]. Fig. 5.2 shows a block diagramof a pressure screening operation.47(a) Reject-refining (b) Feed-back rejects (c) Feed-back reject-refiningFigure 5.1: Block diagrams of the three refining systems studied. (a) Reject-refining :initial pulp is screened and the reject fraction is then refined to be later on re-combinedwith the accept fraction. (b) Feed-back rejects: initial pulp is refined, then fractionatedand the rejects are pumped back to the refiner inlet. (c) Feed-back reject-refining : initialpulp is screened and the reject fraction is then refined to be later on pumped back to thescreen feed.Olson and coworkers [80–83] developed models for pressure screening by performing simplemass balances around a cross section element of a screen assuming plug flow. Thus, performanceequations are derived to describe the accepts and rejects FLD in terms of volumetric reject ratioRv and a passage ratio function P (li). The former is just the ratio between the reject volumetricflow rate Qr and feed volumetric flow rate Qf (Rv =Qr/Qf ). The latter one characterizes thefibre passage through a single screen aperture as a function of fibre length (or bin size). It isworth mentioning that P (li) is dependent on the screen basket geometry, rotor design and speedamong others, and must be experimentally determined. Assuming a constant coarseness for allfibres within the pulp and denoting yif , yir and yia elements in the FLD of the feed, rejects andaccepts, respectively, according to Olson et al. [81] one has:yir = RvP (li)−1p∑i=1yifRvP (li)−1 yif (5.1a)yia = 1−RvP (li)1−p∑i=1yifRvP (li) yfi (5.1b)In addition to sort fibres, pressure screens change the consistency [81]; rejects are thickenedand accepts are diluted. These changes in consistency can also be described using P (li), Rv andyif as:CrCf=p∑i=1yifRvP (li)−1 (5.2a)48Figure 5.2: Block diagram depicting a pressure screening operation. A initial pulp streamis fed to the screen and is fractionated into an accepts stream and rejects stream. Q isvolumetric flow rate; C is consistency; yi are elements of FLD; the superscripts f , a and rrefer to feed, accepts and rejects, respectively.Figure 5.3: Block diagram of a single LC refining stage. yi are elements of FLD; thesuperscripts o, and rr refer to the initial pulp, and refined pulp, respectively.CaCf=1−RvCr/Cf1−Rv (5.2b)The ratio Cr/Cf is usually regarded as the reject thickening ratio T . Thus, by knowing thepassage ratio function P (li), Rv and yif , it is possible to fully describe changes in FLD duringpressure screening.5.2 Comminution ModelIn Ch. 3 a comminution model was developed to describe the FLD changes due to LC refining.By denoting yio and yirr elements in the FLD of the initial and refined pulp, respectively, basedon Fig. 5.3, the comminution model equations can be rewritten as:{yirr} = expm[piαβωRo2Q(1− λ2)A]{yio} (5.3a)A = f(gap, α) (5.3b)λ =RiRo(5.3c)5.3 Refining Size AspectsPulp refiners are usually sized and operated following field domain and heuristic. Manufacturersusually recommend operation conditions and pre-establish upper and lower limits to allow someflexibility during normal operation. As a result, there is a close relationship between the refinersize and the normal operation conditions namely flow rate Q and rotational speed ω. Fig. 5.4490.2 0.3 0.4 0.5 0.6 30 40 50300400500600700800900Figure 5.4: Twin-Flow disc refiner recommended flow rates and rotational speeds forvarious refiner sizes. Bottom x-axis: refiner outer radius Ro in m; top x-axis: refiner outerdiameter Do in inches; left y-axis: flow rate Q in gpm and m3 s−1; right y-axis: rotationalspeed ω in rad s−1 and RPM. Dashed lines correspond to the upper and lower limits forthe flow rate, whereas the solid lines correspond to the recommended values. Data takenfrom AFT Aikawa Fibre Technologies [84].shows the recommended flow rates and rotational speeds for various refiner sizes according toAFT Aikawa Fibre Technologies [84]. Based on this information, the following relationships canbe made:ω =31.47Ro0.75 [rad s−1] (5.4a)Q = 0.37Ro2.37 [m3 s−1] (5.4b)Combining Eq. 5.4a and 5.4bωQ=85.05Ro3.12 [rad m−3] (5.4c)where Ro is in meters. Eq. 5.4c defines the ratioω/Q as a function of the external radius Ro.Thus, for a given refiner size, this relationship has an impact on the comminution model as theratio ω/Q is present in Eq. 5.3a.5.4 Refining Power ConsumptionIn Ch. 4, a correlation to predict refining dimensionless power as a function of plate gap, refiner’sfeed lw and plate geometry was developed. The correlation is here re-shown:Pdα2=(lwc1)c2G(2−c3) (4.18)50Figure 5.5: Block diagram showing an arbitrary model. Input variables: initial pulpFLD {yoi } and consistency Co, gap and Rv; Fixed parameters: screen size, refiner size(Ri and Ro), plate geometry (α and β), Q and ω; Output variables: flows FLD {yki } andconsistencies Ck, dimensionless power Pd and thickening factor T .Pd =Pnetρω3(R5o −R5i )(4.15b)G =(gapogap− 1)(4.7b)where lw is the refiner’s feed length-weighed mean fibre length in mm. gap is distance betweenplates in mm and gapo = 2.5 mm. Finally c1 = 4.664 mm, c2 = 2.701 and c3 = 0.955.5.5 MethodologyFig. 5.5 shows a representative block diagram of an arbitrary system where input and outputvariables are indicated. This arbitrary system can be simulated using the models described inSec. 5.1, 5.2, 5.3 and 5.4. Moreover, ideal mixing was assumed at each mixing point to predictthe mixed-flows FLD. Models were implemented in MATLAB R© Simulink R© toolbox. Thisenvironment was then used to simulate the three different refining systems shown in Fig. 5.1.Although it is possible to derive explicit governing equations for all systems in consideration(see App. E.1 for an example), using a software to aid in the calculations is faster and moreconvenient. Furthermore, there may be cases where it is mathematically impossible to deriveexplicit governing equations (e.g. system with various refiners and screens with intricate flowcombinations and recirculations).For all systems, the input variables were varied between gap = [0.2, 2.0] mm and Rv =[0.1, 0.5]. Additionally, it was set that the initial flow and feed to the refiner were 4% consistency,as the comminution model was built upon refining data at that consistency. Thus, it wasnecessary to implement a dilution flow before each screening operation and adjust it accordinglyto always achieve a reject consistency of 4%. Finally, regarding refiner size, the inner radius wasset to Ri = 0.6Ro. This is a common practice done by manufacturers.The FLD were determined for each single flow of the system. Later on, lw values werecalculated from the FLD and used to construct performance curves for each system.510.2 0.6 1.0 1.4 1.81.501.551.601.651.701.75Figure 5.6: Curves of lw vs. gap for five different refiner sizes. As can be seen, ifmanufacturer recommended volumetric flow rates and rotational speed are followed, smallrefiners are prone to cut to a higher extent than big refiners.5.6 Results and Discussion5.6.1 Refiner size effect on fibre shorteningA system composed of a single refining stage was simulated to realize the impact of the refinersize on the fibre shortening. Five different refiner sizes were considered and simulated using thecomminution model. The resulting lw values were plotted against gap as shown in Fig. 5.6. Asa side note, all figures presented in this section plot lw in the y-axis and, unless otherwise isstated, lw refers to the fibre length leaving the system in consideration.Interestingly, simulated results shows that large refiners produce less fibre shortening thansmall refiners. The reason for this difference in performance is due to the ratio ωRo2/Q which isin the argument of the matrix exponential operator of the comminution model (see Eq. 5.3a).From Eq. 5.3a and 5.4c one can derive the following relationship:ωRo2Q∼ 1Ro1.2 (5.6)The ratio ωRo2/Q can be split into two components: (1) a measure of pulp residence timerepresented in Ro2/Q and (2) frequency of bar-bar crossings represented in ω. Hence,ωRo2/Qaccounts for the probability of fibres being captured by bar-bar crossings during their time spentinside the refiner. Therefore, if manufacturer recommended flow rates and rotational speedare followed, FLD changes due to refining decrease with refiner size as deduced from Eq. 5.6.This is perhaps, one of the reasons why some pilot-scale refining trials results do not matchindustrial-scale results.52Figure 5.7: Scheme of Reject-refining used to derive equations for the ratio E. m˙i andei refer to the mass flow rate and energy content, respectively; eR is the energy suppliedby the refiner (note the dashed-line used to represent an energy flow and to differentiatefrom material flows).5.6.2 Split-flows and recirculation aspectsData presented in this subsection and onwards correspond to simulation results considering asystem with a refiner size Ro = 0.254 m (Do = 20 inches). The idea is to depict the generalbehaviour of each system rather than present specific data for each refiner size considered 7.The systems shown in Fig. 5.1 were simulated following the methodology described in Sec. 5.5.These three systems were compared to the performance of a single refining stage regarded fromnow on as the reference case (black solid-line).Introducing a recirculation flow allows for some share of fibres to get refined more thanonce. This recirculation not only has an impact on the fibre shortening (as will be seen in thefollowing sections) but also in the overall energy that is transferred to pulp. A similar situationhappens in Reject-refining, where the initial pulp is split into accepts and rejects. After refiningthe rejects and re-combining with the accepts, the energy content of the re-combined flow isaffected by the mass reject ratio at the screen. In order to describe the effect of the split ratiosand recirculations on the system behaviour, a ratio E defined as the mass flow rate flowingthrough the refiner over the mass flow rate leaving the system E = m˙R/m˙S is introduced.In a system like Reject-refining, this ratio is fairly straight-forward to realize, however fora system with recirculation this is not done in a single step. To further explain the conceptof E, consider the Fig. 5.7 where Reject-refining is shown. It is fairly easy to conclude thatE = m˙3/m˙5 . Furthermore, if the mass flow rates are expressed as the product of volumetric flowrate and pulp consistency, one gets E = RvT . The product RvT in pressure screening is knownas the mass reject ratio.One can derive an alternative definition of E as shown in Eq. 5.7. Starting by setting theinitial flow to one (m˙1 = m˙5=1) one has:m˙3 = m˙4 = RvT (5.7a)7It was previously stated that the fibre shortening is a function of refiner size if recommended flow rate androtational speed are followed53From an energy balance around the refiner:e4 = e3 + eR (5.7b)Where eR is the energy supplied by the refiner or SEC (eR is used to facilitate the notation) andei refers to the energy content of the i flow or SRE (again, ei is used to facilitate the notation).Performing an energy balance at the mixing point yields:e5 = m˙2e2 + m˙4e4 (5.7c)Now, the initial flow energy content is zero (e1 = 0) since the initial pulp is unrefined. As aconsequence, e1 = e2 = e3 = 0, combining Eq. 5.7a, 5.7b and 5.7c yields:e5eR= RvT = E (5.7d)Thus, E is also the ratio between the energy content of the flow leaving the system over theenergy supplied by the refiner E = eS/eR (alternatively, it can be written as E =SRE/SEC). Itturns out that the ratio E has the following mathematical expressions depending on the systemconfiguration:• Reject-refining : E = RvT• Feed-back rejects: E = 1/1−RvT• Feed-back reject-refining : E = RvT/1−RvTTherefore, E is dependent of Rv and T . The first dependency is quite obvious given that thevolumetric reject ratio has a direct impact on the flows in the system. On the other hand, thedependency of E with T is a bit more complex since T itself has other dependencies as shownin Eq. 5.2a. These dependencies are illustrated in Fig. 5.8 where the ratio E is plotted as afunction of gap for various Rv.54(a) Reject-refining0.0 0.4 0.8 1.2 1.6 Feed-back rejects0.0 0.4 0.8 1.2 1.6 Feed-back reject-refining0.0 0.4 0.8 1.2 1.6 5.8: Ratio E as a function of gap for different values of Rv. E is mass flow rate flowing through the refiner overthe mass flow rate leaving the system. Alternatively, E is the ratio of the energy content in the flow leaving the systemover the energy supplied by the refiner. The reference case is a system of a single refining stage (black solid-line) whereE = 1 always. In principle, the E dependency with Rv is quite obvious since it directly affects the flows in the system.However the E dependency with gap is not straight forward to deduce. The latter dependency is linked to the rejectthickening factor T which describes the mass split ratio in pressure screening and it is function of the feed’s FLD and Rvfor a given screen.55From Fig. 5.8a is seen that in Reject-refining E does not depend on gap but only on Rv.This is natural since screening is totally independent from refining. In contrast, in Feed-backrejects (Fig. 5.8b) and Feed-back reject-refining (Fig. 5.8c), E is indeed dependent of both Rvand gap. This is because screening and refining are coupled via the recirculation flow.Values of E for the Reject-refining system are below the reference case, between the rangeE = [0.3, 0.6], because the initial flow is split and only a portion of the pulp is refined. Conversely,Feed-back rejects and Feed-back reject-refining have E ratios greater than the reference case, inthe range E = [1.5, 4.0] and E = [0.8, 3.0], respectively8.At this point, it is imperative to recall that the refiner size was the same for all systemsand therefore the refiner’s flow rate was also the same. With this in mind, the ratio E helpsto realize the system’s throughput if the first definition is used (E = m˙R/m˙S ). In general, thesystem’s throughput m˙S is the refiner’s flow rate m˙R (i.e. recommendend flow rate or refiner’scapacity) over E. Analogously, E can also be used to characterize a system in terms of energycontents if the second definition is used (E = eS/eR). The energy content of the flow leaving thesystem eS is E times the energy supplied by the refiner eR.For instance, for cases where E < 1 (e.g. Reject-refining), the system’s throughput m˙Sis higher than the refiner’s flow rate but the energy content of the flow leaving the system islower. Conversely, when E > 1, the system’s throughput is lower than the refiner’s capacityand the energy content of the flow leaving the system is higher. The definitions of E and theaforementioned statements will become handy in the next subsection when the systems areassessed in terms of overall energy content.It is worth noting that the previous systems assessments using E are based on havingthe same refiner size across the systems. As explained before, throughputs and flow energycontents depend on their configuration. A different way of assessing systems would be settingthe same throughput for all systems. If this would be the case, the refiners’ flow rates would bevary between systems and therefore refiner sizes would be different. (based on manufacturers’recommendations). Since refiner size has an impact on fibre shortening, the systems wouldbehave differently and comparisons between them would not be possible under the same grounds.For the sake of the discussion, refiner’s flow rate could be set to a specific value so that refinersize do not have an impact on fibre shortening (e.g. same fibre shortening regardless of refinersize). In doing so, it would be possible to assess the systems based on a target throughput usingE. In this hypothetic case, E would give an idea on the system’s refining capacity demands.For instance, if E < 1, the system would require a small refiner whereas for cases where E > 1,the refiner size would be considerable bigger.8These values of E correspond to values of gap in the range gap = [0.10, 0.40] mm which are typical operationvalues found in industry.565.6.3 Performance CurvesFig. 5.9 shows curves of lw vs gap at different Rv values. In these curves, at wide gaps fibrelength remains practically unchanged, but as gap start to decrease changes in fibre length areseen. This is the typical trend seen in single-stage refining trials. Although the behaviours aresimilar, lw changes with Rv and the system configuration. For instance, Reject-refining producesa pulp with higher lw as seen in Fig. 5.9a but the effect of Rv seems to be quite limited. This isbecause only a share of the initial pulp, the long fibre fraction, is refined. This behaviour hasbeen observed by Miller et al. [7] at pilot-scales. Fig. 5.9b and 5.9c differ drastically from theircounterpart Fig. 5.9a because (1) all values of lw are bellow the reference line and (2) the effectof Rv on lw is more pronounced. This is due to the recirculation flow involved in the systemsallowing a share of fibres to get refined more than once which leads to higher fibre shortening.Since the recirculation flow is directly affected by the Rv, it is observed that higher values of Rvproduce lower fibre length in systems with recirculation (again, high Rv values allow a greatershare of fibres get refined more than once due to recirculation).The correlation developed in Ch. 4 was used to transform the curves presented in Fig. 5.9 intocurves that relate lw with a dimensionless quantityPd/α2 . It is worth mentioning that refiningprocesses are usually assessed in terms of the energy supplied by the refiner (eR) which is theratio of the net-power Pnet over the mass flow rate flowing through the refiner m˙R. Therefore,the dimensionless quantity Pd/α2 is directly related to the refining net-power, thus for a givenrefiner size, it is safe to assume that:eR =Pnetm˙R∼ Pdα2(5.8)Where m˙R = CRQR. Thus,Pd/α2 is, in a way, a measure of eRFig. 5.10 shows values of lw as function ofPd/α2 for different values Rv for the three systems.For illustrative purposes, a secondary x-axis of SEC in kWh/t was added. Dashed-lines ofconstant gap were added to the plots as reference and to ease the comparisons between Fig. 5.9and 5.10. Moreover, in Fig. 5.10a and 5.10b, the dashed-lines have a circle indicating where itintersects the reference curve. In Reject-refining, this intersection point corresponds to Rv → 1;at volumetric reject ratio equals one, this system becomes a single stage refining and theperformance curve corresponds to the reference case. Similarly, for the Feed-back rejects system,the intersection point corresponds to Rv → 0; if this is the case, again, the system will behaveas a single stage refining system. This situation does not happen in Feed-back reject-refining. IfRv → 0 the system will behave as a single screening operation. On the other hand, if Rv → 1,the system behaves as batch refining system. For this reason there is no indication of theperformance curves intersecting the reference case line.Although lw decreases as the dimensionless quantityPd/α2 in the x-axis increases as expected,the systems seem to behave differently between one and another. As mentioned before, Reject-refining, which produces higher lw than the reference case, now has more distinctive differences57across Rv as seen in Fig. 5.10a. As for the Feed-back rejects, the curves show a drastic decreasein lw in the first region of the x-axis whereas for Feed-back reject-refining this decrease is lessdrastic.Interestingly, there is an attractive operation point in the Feed-back rejects system; for eachgap curve (more evident at gap = 0.15), there is a point where the maximum power is consumed.This point arises between the perfect trade-off between the enriching in long fibres of the innerflow (feed to the refiner) and fibre shortening due to refining. An increased concentration oflong fibres in the inner flow allows a better loadability [1, 7, 11]. This is desired because itallows to widen the gap to re-establish the desired energy consumption and thus decreasingfibre shortening due to the wider gap. Conversely, large recirculations leads to excessive cuttingand thus a decreased long fibre concentration in the inner flow. Although the Rv value of theseoptimum points seems to be fairly low compared to values in industry, these optimums arethe result of the combination of screen type, initial pulp, refiner size, etc and is unique foreach particular system. Thus, varying the system variables could help to find an operationallyattractive optimum point and ultimately help to achieve the main objective of the LC refiningimplementation in TMP; to consume as much power as possible via LC refining without excessivefibre shortening.The performance curves in terms of Pd/α2 are useful to compare the systems based on SECas explained by Eq. 5.8. However, this metric does not provide information about the energycontent of the flow leaving the system, which is also important. As explained before, split-flowsand recirculation affect the energy content of the flow leaving the system. To account forsplit-flows and recirculation, the previously defined E = m˙R/m˙S ratio is used together withEq. 5.8 to find an expression for the energy content of the flow leaving the system eS as:eS =Pnetm˙S=Pnetm˙Rm˙Rm˙S∼ EPdα2(5.9)Performance curves in terms of the dimensionless quantity E Pdα2are shown in Fig. 5.11where, again for illustrative purposes, secondary x-axis of SREsys in kWh/t was added. Thesecurves are helpful to evaluate the performance of a system since it gives an idea about therefining degree of pulp and allows direct comparisons across systems.As can be seen in Fig. 5.11a, Reject-refining produces the pulp with greater lw than thereference case, yet it produces pulp with the lowest energy content as it is not able to surpassvalues of 1.2 at gap values of 0.15 mm. However, given that lw is higher than the reference,this allows the pulp to possibly undergo a second refining stage to achieve higher energycontents without experiencing excessive fibre shortening. This system configuration should beimplemented if at least two refining stages in series are installed. From Fig. 5.11b is seen thatthe Feed-back rejects system can produce large changes in lw but is also capable of achievinghigh energy contents at relatively wide gaps. This could be particularly useful for applicationswhere the pulp needs to undergo gentle treatments. Lastly, in Fig. 5.11c is seen that Feed-back58reject-refining achieves fairly high energy contents at a reasonable decrease in fibre lengthpartially because only a share of the flow is refined. In some sense, this specific system is ahybrid between Reject-refining and Feed-back rejects. Since only the reject fraction is beingrefined, it would be suitable in cases where the main objective is to develop properties of thelong fibre fraction material or effectively reduce shives.5.6.4 Industrial Scale Data ComparisonsA subsystem of the Holmen Paper TMP mill in Braviken, Sweden was compared to simulationresults. The industrial refining system subject to study was composed of an Andritz 72”Twin-Flow disc refiner (TF-72) and three pressure screening units (two Ahlstom-F4 and oneAhlstom-F5). The system configuration is exactly the one presented in Fig. 5.1b with theparticularity of having the three screens operating in parallel. The pulp feeding the system wasprimary HC refined spruce pulp with lw = 1.40 mm and 4% consistency. The TF-72 operated atω = 325 RPM, Q = 252 L/s and gap = 0.14 mm. The refining plates used had the followingcharacteristics: Ri = 0.405 and Ro = 0.914 m. Bw = 1.60, Gw = 3.20 and Dw = 7.35 mm. Thescreens were provided with 0.15 mm slot AFT macroflow baskets and AFT-GHC rotors andoperated at a passing speed of 20 m/s. Two of the screens (F4(2) and F5) had reject dilutionto prevent plugging. For the screens F4(1), F4(2) and F5, the Rv values were 0.17, 0.15 and0.11, respectively. Once the system achieved a steady state, flow rates and consistencies ofcertain flows were measured together with refining power. Samples were collected and analyzedto measure their FLD. In order to be consistent with the simulations results, the FLD werebounded to fibres within the range l = [0.2; 4.0] mm and values of lw were calculated. Table 5.1summarizes the experimental data, simulation results and relative errors.Relative errors were reasonable low for the fibre length values lw, between 0–11% whichfalls within the error of analysis. However, the simulation over-predicted the consistencies andreject thickening ratio values related to the F4(2) and F5 screens. This discrepancy is associatedto the reject dilution (a water flow) used during normal operation to prevent plugging; thesewater flows were not experimentally measured nor accounted in the simulation. A dilution, ofcourse, leads to lower consistencies which is seen in the experimental data. As a side note, rejectdilutions are usually in the order of 5% of the reject volumetric ratio so it is improbable to affectthe screening performance.59Flowlw (mm) C (%)exp sim error (%) exp sim error (%)feed refiner 1.41 1.44 2.34refined pulp 1.34 1.38 2.99rejects 1 1.49 1.49 0.00 4.39 4.38 0.23rejects 2 1.52 1.51 0.07 3.80 4.38 14.37rejects 3 1.58 1.50 5.19 3.10 3.67 18.87accepts 1 1.15 1.27 11.13 1.34 1.18 11.64ScreenTRefinerPnet (MW)exp sim error (%) exp sim error (%)F4(1) 2.523 2.517 0.23TF72 3.0 2.8 6.00F4(2) 2.360 2.699 14.37F5 2.711 3.222 18.87Table 5.1: Summary of the experimental data (exp) and simulation results (sim). Ingeneral, relative errors were reasonable low, between 0-18%. Simulated thickening factorT values for the F4(2) and F5 screens are higher than the experimental values. Duringoperation, reject dilution were used in these two screens to prevent plugging. Dilutionswere not measured nor accounted in the simulations.60(a) Reject-refining0.0 0.4 0.8 1.2 1.6 Feed-back rejects0.0 0.4 0.8 1.2 1.6 Feed-back reject-refining0.0 0.4 0.8 1.2 1.6 5.9: Performance curves in terms of gap for all systems described in Fig. 5.1. Each system is compared to areference case of a single refiner (black solid-line).61(a) Reject-refining0.0 0.4 0.8 1.2 1.6 25 50 75 100 125(b) Feed-back rejects0.0 0.4 0.8 1.2 1.6 25 50 75 100 125(c) Feed-back reject-refining0.0 0.4 0.8 1.2 1.6 25 50 75 100 125Figure 5.10: Performance curves in terms of dimensionless quantity Pdαfor all systems described in Fig. 5.1. Each systemis compared to a reference case of a single refiner (black solid-line). Dashed-black lines are lines of constant gap. Values ofSEC (top x-axis) were calculated from the dimensionless power Pd for Twin-Flow disc refiner with Ro = 0.2540 m (Do = 20in) and following recommended volumetric flow rate Q and rotational speed ω values from Fig. 5.4.62(a) Reject-refining0.0 0.4 0.8 1.2 1.6 25 50 75 100 125(b) Feed-back rejects0.0 0.4 0.8 1.2 1.6 25 50 75 100 125(c) Feed-back reject-refining0.0 0.4 0.8 1.2 1.6 25 50 75 100 125Figure 5.11: Performance curves in terms of dimensionless quantity E Pdαfor all systems described in Fig. 5.1. Eachsystem is compared to a reference case of a single refiner (black solid-line). Dashed-black lines are lines of constant gap.Values of SRE (top x-axis) were calculated from the dimensionless power Pd for a Twin-Flow disc refiner with Ro = 0.2540m (Do = 20 in) and following recommended volumetric flow rate Q and rotational speed ω values from Fig. 5.4.635.7 ConclusionsRefining systems were theoretically analyzed from a system-design point of view. In general itwas found that:• Refiner size affects fibre shortening if manufacturer recommended flow rates and rotationalspeed are followed. Small refiners cut fibres to a greater extent than large refiners at agiven gap.• Recirculation greatly affects the system performance. It allows to achieve higher energycontents but also could drastically decrease fibre length.• Combining refining and screening allows an increase in the fibre length feeding the refinerthus increasing its loadability.And specifically for each system:• Reject-refining could be beneficial in cases where the pulp can go through at least tworefining stages in series. Moreover, pulp can undergo harsh treatments without excessivefibre shortening as only the long fibre fraction of pulp is refined. This system’s throughputis higher compared to the refiner’s capacity.• Feed-back rejects could be suitable for applications requiring gentle treatments (wide gaps)or where only a single stage of refining is possible since it can achieve high energy contents.However the system throughput is significantly lower compared to the refiner’s capacity.• Feed-back reject-refining could be regarded as a hybrid between the previous two systems.Achieves relatively high energy contents without excessive fibre shortening. This systemcould be suitable to efficiently reduce shives and develop properties of the long fibrefraction.Finally, the models implementation predicted industrial data reasonable well, proving thatsimulation results can aid in system design and evaluation.64Chapter 6SummaryLet us gather facts in order to get ourselves thinking – Georges-Louis Leclerc(French naturalist, 1707 – 1788)6.1 ContributionsThe work done in this dissertation purposely refrained from using current refining theories toanalyze data as they were simply not suitable to achieve the objectives proposed for this research.Instead, the needed theoretical background was derived from mass balances in the case of thecomminution model and from force balances in the case of the power–gap relationships. Duringthis theoretical background derivation process, an interesting combination of variables werefound that deserve to be further studied; this will be explained in the following subsection.Results of this research help to broaden the understanding of three key aspects of LCrefining of mechanical pulps. Specifically: (1) fibre shortening, a crucial factor in pulp quality,(2) understanding refining power consumption, a necessary step towards energy savings andoptimization and (3) describing the coupled behaviour of LC refining and pressure screening inthree basic systems found in TMP lines.A comminution model parametrization was developed to describe the intricate mechanismsof fibre shortening during LC refining. By writing the comminution model equation in terms ofthe refiner radius r, the role of flow rate Q, rotational speed ω and plate geometry (describedby α) is explicitly included in Eq. 3.6. Analysis of fibre length distributions from before andafter refining led to conclude that the selection function was found to be highly correlated tothe inverse of gap (Si ∼ 1/gap) and independent of pulp type. Moreover, the breakage functionwas found to be dependant not only on gap (narrow gaps promote an even cutting pattern) butalso plate geometry characteristics described by the constant α. Using a comminution modelequation with the refiner radius as independent variable allowed to compare refining data fromdifferent refiners under the same grounds.During the study of power–gap relationships, the role of refiner size and rotational speed wasincluded in the developed correlation (Eq. 4.18) by using a dimensionless power number in terms65of the inner and outer refiner radii Pd, whereas the role of plate geometry was described by α2.On the other hand, the role of pulp properties in the power consumption was indirectly describedby the mean fibre length. The analysis of several refining trials data with a wide range ofconditions led to find that refiner power is be linearly dependent of the function G and increasingwith the cube of lw. The theoretical assumptions made during the correlation development weresupported with force measurements data. Additionally, the developed correlation built uponpilot-scale data was validated using industrial-scale refining data proving useful for upscalingcalculations.Refining systems typically found in TMP lines were modelled by combining the developedcomminution model and power–gap correlation and screening models available in literature.Although only three basic systems were studied, with the implementation of the mathematicalmodels in MATLAB R© Simulink R©, more complex systems with numerous unit operations(refiners and screens) can be considered and easily studied. Each considered system characteristicsand behaviour were described in terms of fibre length at different conditions of reject ratios andrefiner gap. From the results, recommendations for potential applications were made aimed toindustry.6.2 Future WorkThe following recommendations are based on findings during the various stages of this research.• It would be interesting to study fibre shortening in co-refining and to realize if simplemixing rules (such as ideal mixing) are enough to predict the refined pulp FLD. Additionally,sub-populations cutting rate results could give insight on how refining energy is distributedamong the sub-populations during co-refining.• From the comminution model, it was found that FLD changes were function of the residencetime of fibres inside the refiner and a couple of functions which ultimately were functionof gap for a given plate geometry (see Eq. 3.9). Thus, in the case of fibre shortening, a LCrefiner can be regarded as a chemical reactor, where the concentration of a reactant of afirst order reaction is described using an exponential decay function in terms of a reactionconstant (dependent on temperature) and a the residence time (dependent on flow rateand reactor volume). With this, it is hypothesized that some pulp properties can alsobe described the same way. Indeed, freeness and bulk curves depict curves which can bedescribed with an exponential decay function. For the case of tensile, a more sophisticatedapproach is needed as it is well known that it reaches a maximum point after some refiningto later decrease if pulp is further refined. Perhaps tensile strength could be modelledsimilar to a set of reaction in series where the aim is to maximize the concentration of theintermediate product.• Typically, refining trials designed to compare industrial-scale and pilot-scale refiners useSEC as a metric. Instead a different approach should be followed. First and foremost, it66must be ensured that the distribution of forces is the same across refiners, hence gap andlinear speed should be the same; these two conditions should lead to operate the refinersat the same dimensionless power Pd. Secondly, the residence time of pulp must be also thesame for all refiners to ensure fibres get the same amount of treatment at different refinerscales. With this, it would be possible to assess the performance of different refiners underthe same grounds.• Although force measurements was not the main topic of this dissertation, future researchshould focus on realizing the force-radius dependency by installing bar-force-sensors atdifferent radial positions. The force dependency was needed to develop the correlation inCh. 4 and was assumed to be f¯N ∼ (ωr)2. Subsequently, force-radius dependency studiesshould lead to formally address the discussion around a theoretical optimum inner radiusri. Alternatively, the latter issue can be also addressed by measuring power–gap curvesusing refining plates with different inner radii ri.• In refiners, as outer radius and rotational speed increase, the power consumption increasesto the fifth power and cube, respectively. Theoretically, it would be possible to drasticallydecrease power consumption by designing a counter-rotating disc LC refiner. The idea isnot new at all; in fact this is seen in HC refiners. For a given refiner size, if the rotationalspeed of each disc is half of the one of a conventional rotor-stator refiner, the energyconsumption would be 25% of the conventional refiner. However, the machine itself wouldhave additional complexities such as having an additional set of moving parts (shaft,bearings, motor) and would occupy more space.67Bibliography[1] C. Sandberg, J. E. Berg, and P. Engstrand. Low consistency refining of mechanical pulp -System design. TAPPI Journal, 16(7):419–429, 2017. → pages 3, 4, 5, 58[2] R. J. 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Factors affecting the shear forces in high-consistency refining.Journal of Pulp and Paper Science, 28(11):364–368, 2002.[51] R. Harirforoush, P. Wild, and J. Olson. The relation between net power, gap, and forceson bars in low consistency refining. Nordic Pulp and Paper Research Journal, 31(1):71–78,2016. → pages 12, 25, 39, 40, 41, 43[52] R. J. Kerekes. Force-based characterization of refining intensity. Nordic Pulp and PaperResearch Journal, 26(1):14–20, 2011. → pages 12, 25[53] B. Epstein. Logarithmico-Normal distribution in breakage of solids. Industrial andEngineering Chemistry, 40:2289–2291, 1948. → page 14[54] A. Elahimehr, J. A. Olson, D. M. Martinez, and J Heymer. Estimating the area andnumber of bar crossings between refiner plates. Nordic Pulp and Paper Research Journal,27(5):836–843, 2012. → pages 15, 37[55] A. Elahimehr, J. A. Olson, D. M. Martinez, and J. Heymer. 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Fibre floc drainage - A possible cause forsubstantial pressure peaks in low-consistency refiners. Nordic Pulp and Paper ResearchJournal, 23(3):321–326, 2008. → page 36[72] O. Eriksen, C. Holmqvist, and U. B. Mohlin. Theoretical outline of the cause for observedcavitation in a low-consistency refiner. Nordic Pulp and Paper Research Journal, 23(3):315–320, 2008. → page 36[73] K. A. Landman, C. Sarikoff, and L. R. White. Dewatering of flocculated suspensions bypressure filtration. Physics of fluids A: Fluid dynamics, 3(6):1495–1509, 1991. → page 36[74] K. A. Landman and L. R. White. Solid/liquid separation of flocculated suspensions.Advances in Colloid and Interface Science, 51(29):175–246, 1994.72[75] R. Bu¨rger, F. Concha, and K. H. Karlsen. Phenomenological model of filtration processes:1. Cake formation and expression. Chemical Engineering Science, 56:4537–4553, 2001. →page 36[76] H. Riazoshams, M. Habshah Jr., and M.B. Adam. On the outlier detection in nonlinearregression. International Journal of Mathematical, Computational, Physical, Electrical andComputer Engineering, 3(12):1105–1111, 2009. → page 40[77] A. Luukkonen, J. A. Olson, and D. M. Martinez. Low consistency refining of mechanicalpulp: Relationships between refiner operating conditions and pulp properties. Nordic Pulpand Paper Research Journal, 27(5):882–885, 2012. → pages 43, 45[78] G. M. Scott and S. Abubakr. Fractionation of secondary fiber - A review. Progress inPaper Recycling, 3(3):50–59, 1994. → page 47[79] E. Young and J. A. Olson. Development of a continuous high-efficiency laboratory fibrefractionator. The Canadian Journal of Chemical Engineering, 82:433–441, 2004. → page 47[80] J. A. Olson. Fibre length fractionation cause by pulp screening, slotted screen plates.Journal of Pulp and Paper Science, 27(8):255–261, 2001. → pages 47, 48[81] J. Olson, N. Roberts, B. Allison, and R. Gooding. Fibre length fractionation caused bypulp screening. Journal of Pulp and Paper Science, 24(12):393–397, 1998. → page 48[82] J. Olson and G. Wherrett. A model of fibre fractionation by slotted screen apertures.Journal of Pulp and Paper Science, 24(12):398–403, 1998.[83] J. Olson, B. Allison, and N. Roberts. Fibre length fractionation cause by pulp screening.Smooth-hole screen plates. Journal of Pulp and Paper Science, 26(1):12–16, 2000. → page48[84] AFT Aikawa Fibre Technologies. Introduction to stock prep refining. URLhttp://www.aikawagroup.com/downloads/Training Manual.pdf. last accessed: 29-Sep-2017.→ page 5073Appendix AGrinding equation for continuoussystemsA.1 Specific Refining EnergyThe batch grinding equation is usually posed as:dwidt= −S∗i wi +p∑j=1bijS∗jwj (A.1)However, the interest is in expressing everything in terms of probabilities yi instead of massfraction wi. Hence, it is done by using the coarseness of fibres Ωi (linear density in mass perunit length):wi =Ωiliyi∑Ωiliyi(A.2)Assuming the coarseness is constant for all fibres within a sample:wi =liyi∑liyi(A.3)Then, its derivative becomes:dwidt=li∑liyidyidt(A.4)Now, by definition bij is the fraction material broken out of particle size j with falls into i. Itcan be written as the probability of breaking a material from size j to size i (Bij) times themass of material i over mass of material j as:bij = BijΩiliΩjlj(A.5)74Assuming a constant coarseness Ωi = Ωj , the previous equation can be further simplified to:bij = Bijlilj(A.6)Combining Eq. A.1, A.3, A.4 and A.6:li∑liyidyidt= −S∗i( liyi∑liyi)+p∑j=1(Bijlilj)S∗j( ljyj∑liyi)(A.7)The term li∑ liyi is common for all terms and thus can be canceled (even in the summationoperator where the index of summation is j and therefore it can be taken out). Eq. A.7 issimplified to:dyidt= −S∗i yi +p∑j=1BijS∗j yj (A.8)Using the definition of total derivative, it is possible to obtain the grinding equation in terms ofspecific refining energy:dyidt=∂yi∂t+∂yi∂(SEC)d(SEC)dt(A.9)Assuming steady state operation, it is straight forward to conclude that ∂yi∂t = 0. Moreover, bydefinition SEC = Ptm , where P is refiner power consumption (kW), m is pulp mass (ton), and tis time. Thus SEC’s time derivative is:d(SEC)dt=Pm(A.10)Combining Eq. A.8, A.9 and A.10 yields:dyid(SEC)=mP(−S∗i yi +p∑j=1BijS∗j yj) (A.11)Finally, the selection function is expressed in the inverse units of SEC (kWh/t)−1 by combiningS∗i (in h−1) with m/P (in t/kW) to yield:dyid(SEC)= −S¯iyi +p∑j=1BijS¯jyj (A.12)Where S¯i is the selection function in (kW h t−1)−1A.2 Refiner radiusIt is also possible to derive the grinding equation for low consistency refining in terms of refinerradius. The following mathematical derivation is done for a Single-Disc refiner. From the75material derivative:dyidt=∂yi∂t+ (u∇)yi (A.13)In a cylindrical coordinates system, the material derivative is:dyidt=∂yi∂t+(∂yi∂rdrdt+∂yi∂θdθdt)(A.14)In a steady state operation of a continuous flow system, the partial derivative ∂yi/∂t = 0. Alsoit is straight forward that dθ/dt = ω. As for the dr/dt term, it is also straight forward that:drdt= ur (A.15)where radial velocity ur is defined as:ur =QAcs(A.16)The cross section area of a Single-Disc refiner Acs in terms of the plate geometry is:Acs(r) = 2pi[ 2GwDwBw +Gw+ gap]r (A.17)The contribution of the gap to the cross sectional area is disregarded since gap << 2GwDw/(Bw+Gw). Combining Eq. A.15 and A.16 yields:drdt=Q2piβr(A.18)where β = 2GwD/(Bw +Gw). Combining Eq. A.8, A.14 and A.18 the initial grinding equationhas been adapted for a refiner in terms of radius to:( Q2piβr∂yi∂r+ ω∂yi∂θ)= −S∗i yi +p∑j=1BijS∗j yj (A.19)At this point the following assumptions are made based on the findings of Elahimeher et.al.[55]:1. Due to symmetry, ∂yi/∂θ = 0.2. Express S∗i as selection function per fraction of bar-bar crossing area Si such as:S∗i ∝BwBw +GwSi3. S∗i is proportional to ω as bar-bar crossings frequency increases with the angular velocity.Thus:S∗i ∝ ωBwBw +GwSi76By denoting α = Bw/(Bw +Gw), the grinding equation for low consistency refining in terms ofprobabilities is finally obtained:Qω12piαβ1rdyidr= −Siyi +p∑j=1BijSjyj (A.20)77Appendix BExperimental and predicted fibrelength distributions for selectedtrials(a) Trial 10.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 Trial 30.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 B.1: Fibre length distribution for trials 1 and 3 (refining in stages). Solid linescorrespond the experimental data whereas square data points correspond to the valuespredicted by the comminution model (Eq. 3.7).78(a) Trial A0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 Trial C0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 B.2: Fibre length distribution for trials A and C (batch refining). Solid linescorrespond the experimental data whereas square data points correspond to the valuespredicted by the comminution model (Eq. 3.15).79Appendix CComminution in Pulp MixturesConsider a pulp composed of z sub-populations (e.g. a mixture of different pulp types). Ifall sub-populations initial FLD are known, it is possible to describe the fibre shortening of themixture (e.g co-refining). Assuming superposition principle (γi as a mixing rule), the mixtureFLD at any given moment is:Ym =z∑i=1γiYi (C.1)From the comminution model, the FLD of each sub-population Yi can be predicted as:Yi = expm [αωAiΘ] Yi(0) (C.2)This approach assumes no interaction between the sub-populations during refining and fibres ofdifferent sub-populations get the same treatment. Thus the mixture FLD is described by:Ym =z∑i=1γi expm [αωAiΘ] Yi(0) (C.3)Thus, by taking samples after refining and measuring their FLD, a set of parameters(Ki,mi, ni) can be estimated. In other words, one comminution model matrix A per eachsub-population. These results could help to realize how the refining energy is split among thesub-populations80Appendix DAffinity Laws in Low ConsistencyRefiningAffinity laws, similar to the ones used in centrifugal pump design, can be developed using thecorrelation developed in Ch. 4. Here, affinity laws for SEC and SEL will be used as examples.From the definition of SEC:SEC =Pnetm˙R=α2ρω3(Ro5 −Ri5)m˙R(lwc1)c2G (D.1)For a given refiner size (Ro) and initial pulp (lw) and a set of refining conditions (subscript1), one can predict the performance for a second set of refining conditions (subscript 2) as:SEC2SEC1=(G2G1)(α2α1)2(ω2ω1)3(m˙1m˙2)(1− λ251− λ15)(D.2)Where λ = Ri/Ro . From the definition of SEL and bar edge length (BEL), one can derive thefollowing equations:BEL =4pi23Ro3 −Ri3(Bw +Gw)2 (D.3a)SEL =PnetωBEL=3B2w4pi2ρω2(Ro5 −Ri5)(Ro3 −Ri3) ( lwc1)c2G (D.3b)Thus, an affinity law for SEL would be:SEL2SEL1=(G2G1)(Bw2Bw1)2(ω2ω1)2(1− λ251− λ23)(1− λ131− λ15)(D.4)The previous equations give insight about the impact of operational variables on SEC andSEL, two commonly used concepts in refining. This can be derived for other refining theoriesdefinition as well. These affinity laws involve operation conditions (gap, flow rate) which aresomewhat easy to manipulate. In contrast, some others are not that straight-forward to modify81(a) SEC2SEC1 ,SEL2SEL1vs. λ20.0 0.2 0.4 0.6 0.8 SEC2SEC1 ,SEL2SEL1vs. ω2ω10.7 0.8 0.9 1.0 1.1 1.2 D.1: Plots of the ratios SEC2SEC1and SEL2SEL1against different parameters. (a) Adecrease in λ2 (decrease in Ri) is beneficial since SEC increases and SEL decreases. (b)Changes in the angular velocities have different impacts, but more pronounced on SEC.(e.g. plate geometry, rotational speed) yet can lead to significant changes in SEC and SEL.Fig. D.1 illustrates the effect of some parameters on SEC and SEL. Fig. D.1a shows that adecrease in λ (by decreasing ri) increases SEC and decreases SEL, both benefiting the refiningoperation. For instance, it is common to have λ1 ≈ 0.6, yet a reasonable decrease down toλ2 = 0.50 can increase the SEC by 5% and decrease the SEL by 6%. However, at least inTwin-Flow disc refiners, the inner radius can not be drastically decreased since there must beenough room to distribute the pulp flow between the two chambers inside the refiner.Fig. D.1b shows how an increase in rotational speed leads to an increase in both SEC and SEL.Although it is a common belief that increasing ω automatically decreases SEL (from Eq. D.3b),an increase in ω increases refining power (both no-load and net power) to a greater extent thanthe SEL decreases. By increasing ω, the power increases and thus SEC; so if the operational SECtarget is meant to be restored to its original value, gap must be increased. With this operationalmanoeuvre in the end (1) power is kept constant, (2) rotational speed is increased and (3) gapis wider resulting in a the same SEC and lower SEL. Interestingly, experience shows that thismanoeuvre leads to less fibre cutting at the same energy consumption. At first glance, less fibrecutting is attributed to the decreased SEL but as demonstrated in Ch. 3, the decreased fibrecutting is due to wider gaps.82Appendix EMathematical Analysis of a RefiningSystemE.1 Flows fibre length distributionsThe following mathematical procedure includes matrix operations. To simplify the notation,matrices for the screening and refining operations and vectors for the fibre length distributionsare defined and presented in bold fonts.For screening:H = diag RP (li)−1vp∑i=1RP (li)−1v (E.1a)For refining:J = expm[piαβωRo2Q(1− λ2)A](E.1b)For the fibre length distribution of the flow k :Yk = {yik} (E.1c)Consider the Feed-back reject-refining system with the notation shown in the Fig. E.1. Therejects are related to the screen feed as:Yr = HYfs (E.2)The accepts are related to the screen feed as:Ya = (I−RvH)Yfs (E.3)83Figure E.1: Feed-back reject-refiningThe refined pulp is related to the screen rejects as:Yrr = JYr (E.4)Combining Eq. E.2 and E.4 yields:Yrr = JHYfs (E.5)Now, from a mass balance at the mixing point:QfsCfsYfs = QfCfYf +QrrCrrYrr (E.6)Dividing Eq. E.6 by QfsCfs, the FLD of the screen feed Yfs is obtained as:Yfs = (1−Rv)TYf +RvTYrr (E.7)Replacing Eq. E.5 in Eq. E.7:Yrr = JH[(1−Rv)TYf +RvTYrr](E.8)And finally, finding an explicit expression for Yrr in terms of the initial pulp Yf :Yrr = (1−Rv)TJH[I−RvTJH]−1Yf (E.9)Where I is the identity matrix. Once Yrr is known, it is used in Eq. E.7 to calculate Yfs.Subsequently it is possible to calculate Yr and Ya using Eq. E.2 and E.3, respectively.E.2 Calculation of EIf the mass flow rate flowing through the screen is set to 1, one has:1 = m˙a + m˙r (E.10a)84Expressing the mass flow rates as the product of volumetric flow rate times consistency:1 = QaCa +QrCr (E.10b)1QaCa= 1 +QrCrQaCa(E.10c)11−RvT − 1 =QrCrQaCa(E.10d)Since the mass flow leaving the system QaCa is the same as the mass flow feeding the systemQfCf :QrCrQfCf=RvT1−RvT (E.10e)On the left-hand side one has the ratio mass flow rate flowing through the refiner over the massflow rate leaving the system which is the definition of E.E =RvT1−RvT (E.10f)The alternate definition of E in terms of energy content is also derived. Setting m˙fs = 1and ef = 0.err = er + eR (E.11a)Where eR is the energy supplied by the refiner (SEC). At the mixing point:efs = m˙rrerr (E.11b)Since m˙rr = m˙r = RvT , combining Eq. E.11a and E.11b leads to:efs = RvT (er + eR) (E.11c)efs − (RvT )er = (RvT )eR (E.11d)As efs = er = ea, the interest is on finding an expression for the ratioea/eR (energy contentof the flow leaving the system over the energy supplied by the refiner), which is the alternatedefinition of E.ea(1−RvT ) = (RvT )eR (E.11e)E =RvT1−RvT (E.11f)85


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