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The energy expended on pulp fibres during low consistency refining Martinez, Mark 1995

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The Energy Expended on Pulp Fibres During Low Consistency RefiningbyMark MartinezB.A.Sc. University of Toronto, 1987M.A.Sc. University of Toronto, 1990A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF CHEMICAL ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAMarch 1995© Mark Martinez, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)_____________________________Department of CdLQThe University of British ColumbiaVancouver, CanadaDate (7 irDE-6 (2/88)AbstractRefining is a process in which the physical structure of the papermaking fibre is modified byrepeated strain. While there is extensive literature on the changes in fibre morphology, thereis little on the nature and magnitude of the forces which impose the strain. The objectiveof this dissertation was to estimate the force and energy expended on papermaking fibres inpulp refining. Both hydrodynamic and mechanical forces were considered.In the first half of the dissertation, a model is formulated to consider the hydrodynamicforce acting on a single fibre trapped and transported on a moving bar edge. Equations weredeveloped to predict the force as a function of fibre properties such as stiffness, length, anddiameter, and fluid variables such as velocity and fluid viscosity. From this, the energyexpended was estimated to vary within the range 10-12 J to 10’J for the cases studied.In the second half of the dissertation, theoretical and experimental estimates of the mechanicalforce acting on fibre flocs were developed. It was found that substantial force on fibrescould only be imposed when the floc was compressed to near its zero void volume.Upon further compression of the floc, force was found to increase linearly with degreeof compression. The theoretical estimates were found to agree reasonably well with theexperimental measurements. The energy expenditure per fibre associated with these forceswas found to be about i05 J. This compares favorably with estimates reported in theliteratureThe energy associated with the mechanical force was approximately 6—7 orders of magnitudegreater than that associated with the hydrodynamic force. This suggests that the lattercontributes little to the refining action in pulp.UTable of ContentsAbstractList of TablesList of FiguresAcknowledgmentsIIVvivifi1 Introduction 12 Literature Review2.1 Qualitative Description of Refining2.2 Quantitative Characterization of Refining2.3 Forces on Fibres during Refining2.4 Summary of Literature Review2.5 Objectives of the Research3 The3.1..41114• . 15Hydrodynamic Force on Fibres During Fibre Transport.AnalysisForce and Torque BalanceHydrodynamic Force on FibresFibre Bending3.2 ResultsStiff Fibreflexible Fibre3.3 Discussion3.4 Conclusionsill161718202428283136404 The Force on Fibres Between Passing Bars4.1 Review of Previous Work4.2 AnalysisForce Balance on FibresEstimating the Normal ForceForces on Fibres in Contact with BarForces on Flocs4.3 ExperimentalObjectives of the Experimental WorkApparatusTesting Method4.4 ResultsPhenomenological BehaviourThe Effect of Gap Size on Forces on Fibres.Working Force Distribution During Bar Crossing4242434345535863636363686868747778807 Nomenclature8 ReferencesA Estimating the Normal Force of Fibre Floes838995Surfaces4.5 Discussion4.6 Summary and Conclusions5 Summary and Conclusions6 Recommendations 82ivList of Tables2.1 Different Classifications of the Effects of Beating on Fibre Morphology.2.2 Postulated Mechanisms of Refining 72.3 Comparative Evaluation of Refiner Performance 92.4 Observations of Refiners with High-Speed Cine-Films 134.5 Description of Strain Gauge and Signal Conditioner 664.6 Typical Values for the Parameters used in Equation (4.46) 73VList of Figures1.1 Simplified representation of a single disc refiner 22.2 Refiner flow pattern showing three main regions of flows 123.3 Schematic of the flow and forces on a fibre in contact with a bar edge 213.4 Force balance on a differential fibre length 263.5 Trapping of fibres: (a) ps=O.2 (stiff); (b)t5=0.4 (stiff);(c) iLs=O.2 (completely flexible).The velocites V and U are shown in Figure 3.2 323.6 Deflection of a single fibre as a function of bending number, Be 353.7 Cross-section of a fibre 364.8 Forces on an individual fibre in a floe in the refining gap 444.9 Fibre distribution in a cross-section of thickness dx of an ideal floe 454.10 Lattice spacing of fibres in floe undergoing compression 474.11 Schematic for the definition of the “ploughing” force 544.12 Schematic of the corner force 574.13 Theoretical estimate of the effect of gap size on peak working force 604.14 Region of the floe under compression and shear during bar passage 614.15 Working force distribution over the initial width of the bar 624.16 Photograph of the single bar refiner4.17 Schematic of single bar refiner showing the bars, floe support, and location of thestrain gauge 654.18 Phenomenological behaviour of a floe between passing bars of the single bar refiner. . . . 694.19 Schematic of the phenomenological behaviour of a floe during bar passage 69vi4.20 Schematic of the observed floc behaviour between passing bars 704.21 The effect of gap size on peak shear (Cm=6—7%, p—6O 7O kg/rn3). The dotted lineshows the abrupt decrease of working force at low GIL0 714.22 The effect of lower consistency on peak shear (Cm=3—4%, Po607°kg/rn3). The dataextrapolate to a level of GIL0 approximately equal to their consistencies at zero force. . . 714.23 The effect of radius of curvature of the bar on the measured and predicted peakworking force 744.24 The effect of bar surface roughness on the peak working force 754.25 Typical working force distribution along the width of the bar surface 76A.26 The normal pressure on Jacquelin flocs in uniaxial compression 96VI’AcknowledgmentsI am deeply indebted to my family, in particular to my parents, Michael and Teresa, mybrother, Patrick, and his family, Celia and Lucas, and my friends, Marilyn, George, Anita,Robert, Gary, and Lisa for their constant support and encouragement. I appreciate the adviceand helpful suggestions of my research supervisor, Dr. R.J. Kerekes.The financial support in the form of scholarships from NSERC, UBC, Paprican, and Dupontis greatly appreciated.VII’Chapter 1: IntroductionChapter 1IntroductionRefiners are mechanical devices employed to modify papermaking fibre morphology. Theyare rotary devices having bar patterns machined on a rotor and stator constructed of chrome-nickel steel (see Figure 1.1). The rotor and stator are either disc or conical in shape and areseparated by a gap of about three fibre diameters (90 tim). Papermaking fibres flow in thegrooves of the disc and are trapped between opposing bars in the narrow gap between therotor and stator. The fibres are “beaten” or “refined” by repeated impacts with the bars.Historically, the research effort in refining has been both substantial and protracted, datingback to the early years of this century. Between 1967 and March 1994, approximately5000 references are cited in the PaperChem data base using the keywords “refiners” or“refining”. Early research recognized that there were a number of different “beating effects”on fibre properties. Traditionally, “beating effects” have been classified into three categories:fibre cutting or splitting of the fibre; external fibrillation; and internal fibrillation. Fibrecutting is a shortening of the fibre while fibre fibrillation is the rupture of bonds betweenmicrofibrillar layers. External fibrillation refers to the loosening of the fibrils and raising orpartial detachment of microfibrils on the surface of the fibre. Internal fibrillation refers tothe less visible disruption of bonds within the fibre (Smook, 1990). These “effects” result inincreased fibre flexibility and paper densification (Page, 1989). This phenomenon enhancesmost paper properties.While much is known about the effects of refining, relatively little is known on how it iscarried out. Clearly, force is imposed on the fibres, but the magnitude of the force applied1Chapter 1: Introduction3tator________InletoutletGrooveBarJLZJF1R1’Refiner Disc Section AA’(Front View)Figure 1.1: Simplified representation of a single disc refiner.to the individual fibres has yet to be theoretically or experimentally determined.A full, rigorous, theoretical analysis of the beating of pulp fibres would be most desirable.However, such an analysis is formidable given our current understanding. In light of thecomplexity, consideration was given in this dissertation to understanding one fundamentalaspect, namely the estimation of the nature and magnitude of the force acting on pulp fibres.DriveInletA A’0)02Chapter 1: IntroductionThe relevant literature is reviewed in this dissertation in Chapter 2. Although a large numberof technical articles have been published, most of the work focused on empirical studiesrelating changes in fibre morphology to the type of refining treatment. The aim here is notto review all of this literature, but to focus on those studies which add insight to the beatingmechanism. After a brief review of the qualitative and quantitative effects of refining onfibre structure (Chapter 2.1—2.2), estimates of both the hydrodynamic and mechanical forcesapplied to fibres are presented (Chapter 2.3). The summary of the literature review sectionfocuses on the subject of this dissertation and concludes with a statement of the objectivesfor this research work (Chapter 2.4—2.5).The theoretical work is presented in Chapters 3 and 4. Consideration was given to understanding the forces applied to the fibres. The analysis considered both the hydrodynamicforces during fibre transport (Chapter 3) and the mechanical forces applied during bar passage (Chapter 4). The utility of the theoretical estimates are demonstrated by comparisonto experimental results.Chapter 5 summarizes the major findings in this work and these findings are reiterated anditemized in Chapter 6. Recommendations for future research work are given in Chapter 7.The appendix is introduced to record information and subsidiary findings which are importantto clarify the main text.3Chapter 2: Literature ReviewChapter 2Literature Review2.1 Qualitative Description of RefiningRefining is a process whereby the physical structure of the papermaking fibre is modifiedby force imposed in a cyclic fashion. Early research on beating recognized that there werea number of different “beating effects”. The older refining literature usually classifies theeffects into three categories: cutting or splitting of the fibre; external fibrillation of the fibresurface; and internal fibrillation of the fibre surface. Internal fibrillation, the loosening of theinternal structure of the fibre, makes fibre swelling possible; this enhances fibre flexibility.External fibrillation changes the external surface of the fibre; it makes the fibre look “hairy”under an optical microscope resulting in increased external surface area.Today, we recognize that there are numerous different beating effects in addition to the onesdescribed above, and all of them are at different times important (Page, 1989). For example,Levlin and Jousunaa (1988) recognize the effect of the dissolution of hemicelluloses andlignin from the cell wall material as being the chief “effect” affecting individual fibre strengthproperties. A comparative list of primary beating effects appearing in the literature has beencompiled and is presented in Table 2.1. Further secondary effects have been summarized ina number of excellent reviews, see Page (1989), Atack (1977), Ebling (1980), Higgins andde Yong (1962), and Clark (1980).4Chapter 2: Literature ReviewTable 2.1 Different Classifications of the Effects of Beating on Fibre Morphology.Historical 1) cutting; 2) external fibrillation; 3) internalfibrillationHiggins and de Yong (1962) 1) fibre shortening; 2) external fibrillation; 3)breaking of intra-fibre hydogen bonds; 4) productionof finesFahey (1970) 1) brealdng of covalent bonds, for example C-C; 2)breaking of intra-fibre hydrogen bondsGiertz (1980) 1) cutting and crushing; 2) successive cleavage ofexternal layers of the cell wall and breaking away ofthese layers; 3) breaking of intra-fibre hydrogenbonds; 4) creation of dislocations i.e. Region of afibre where microfibrils are misaligned from theoriginal to a more transverse directionClark (1980) 1) shortening; 2) external splitting; 3) production ofdebris; 4) logitudinal compressionEbeling (1980) As reviewed by Ebeling the effect of beating on fibremorphology can be summarized into the followinggroups: 1) fibre shortening; 2) successive cleavage ofexternal layers of the cell wall and their subsequentbreaking away; 3) delamination of internal cell walllayers; 4) local dislocations of cell wall structure,dissolution of the chemical components of the cellwall and simultaneous formation of collodialcarbohydrate solution on the surfaces of the affectedfibres5Chapter 2: Literature ReviewThe beating or refining of pulp changes several “end product” paper properties simultaneously. As an empirical rule, paper properties such as tensile strength, burst strength’, foldingendurance, internal bond strength, and density, are increased whereas opacity, permeability, and absorbance are decreased. The literature contains many practical “rules-of-thumb”pertaining to the practice of beating or refining to achieve targeted paper products. It is commonly known that narrow bars at low speed and consistency enhance fibre cutting. Similarly,wide bars at high peripheral speeds and pulp consistencies create fibre fibrillation.While much is known about the effects of refining, relatively little is known about howit is carried out. Indeed, most postulated mechanisms of refining have been inferred fromobserved changes in fibre properties. A summary of several descriptive models is presentedin Table 2.2. Clearly, force is imposed on the fibre in a cyclic fashion but its magnitude andnature, i.e. hydrodynamic (Frazier, 1988) or mechanical (Goncharov, 1971; Page, 1989),has yet to be determined. Alaskevich (1971) states, in an unsubstantiated manner, that the“beating effect” results from fibre fatigue.2.2 Quantitative Characterization of RefiningDespite a lack of fundamental knowledge of the force applied to fibres in refiners, semi-empirical models have been developed to describe the action of refiners. A summary of thedifferent models is presented in Table 2.3. As shown, many authors (Lewis and Danforth,1962; Leider and Nissan, 1977; Kerekes, 1990) describe the refining action in terms of twoquantitative parameters: the number of impacts the fibre are subjected to per unit mass, N(no. impacts/kg); and the intensity of each impact, I (J/impact). Together, these parameters1 Resistance of a paper sheet to deformation by an expanding rubber diaphram, as measured by the hydraulic pressure at the point ofrupture (Smook, 1990)6Chapter 2: Literature ReviewTable 2.2 Postulated Mechanisms of Refining.Author MechanismSteenberg (1951) Steenburg claims that floes instead of fibres are being refined. He observedthat the gap in a valley beater decreased instantaneously to 1/10 of itsoriginal value when 7% CMC was added.Atack and May They speculate that pulp fragments must be compressed between the(1963) revolving plates, and constrained to either roll between the plates or beheld by one plate as the other plate slides over.Banks (1967) Refining is a four step process: 1) localized preliminary removal of waterfrom the floe; 2) mechanical pressure exceeds elastic limit of fibres in floe;3) floe is sheared; and 4) floe is released and reabsorbs water.Epenmiller (1969) Refining is a five step process: 1) fibre wads gather causing preliminarydewatering; 2) the fibre wads experience mechanical pressure in the rangeof 1000 to 5000 psi; 3) fibre wads slide under pressure; 4) mechanicalrelease, water reabsorption; and 5) dispersion.Goncharov (1971) 1) fibres (fibrage) attach to leading edges and are subjected to largecompressive forces; 2) there is mechanical pressure causing pulling on thefibre layers; 3) fibre layers become separated; and 4) rotor element movesoff stator element.Alaskevich (1971) Refining takes place due to the fatigue stresses accumulated.Pearson et aL (1978) “...a compressive force to collapse the lumen (the inside cavity of the fibre)and a second force to roll the fibre when flattened...”.Fox (1979) Fox observed that floes lie across the leading edge of the rotor and aretransported to the refining gap.Steenburg (1980) Speculates that stress transfer proceeds along a limited number of stresschains.Atack (1980) Atack equates refining to a grinding operation; one bar holds the pulpwhilst the other one works it.Blechschmidt (1986) 1) Within the grooves there is a decomposition and reformation of fibrenetworks; 2) At the meeting of refiner bar there is an increase inconsistency; and 3) When the bars slide over each other-there is partialraising of pulp consistency, fibrillation by introduction of shear forces, andforce transmission by either bar-fibre of fibre-fibre interaction.Frazier (1988) Frazier attempts to relate traditional hydrodynamic lubrication theory to therefiner.7Chapter 2: Literature Revieware combined to account for the net expenditure of energy on pulp, E (J/kg)E—NxI (2.1)As stated by Kerekes (1990), the “energy of refining may be applied in different ways toyield different effects. For example, it is well known that a large number of impacts ofsmall intensity leads to fibrillation, while a small number of impacts of high intensity leadsto cutting. Equivalent refining action is achieved when N and I (i.e. N1=N2and 11=12)... areequal, not necessarily when their specific energies, E, are equal”. This observation stronglysuggests that two parameters, a combination of either E, N, or I, is required to characterizethe refining treatment.N and I are related to key refining variables: pulp mass flowrate through the refiner, F; netpower expended on this flow, F; and a third variable, C. This variable represents the capacityof the refiner to inffict impacts upon fibres. It is linked to P. F, and E in the following mannerC PPE=NxI=x?= (2.2)The variables F and P are easily measured, however, C, is not. It must be estimated fromthe geometry and bar configuration of the refiner, fibre properties, and pulp consistency.One of the earliest quantitative expressions for C was given by Brecht (1967)C x nrnswrLr (2.3)where n. and n are the number of bars on the rotor and stator; L,. , the length of the refiningzone; and, Wr, the rotational velocity of the rotor.8Chapter 2: Literature ReviewTable 2.3 Comparative Evaluation of Refiner Performance.Author DescriptionMime 1927 A one-parameter model in which refiner performance wasevaluated by the ratio of cutting length to beatingcapacity.Lewis and Danforth (1962) A qualitative description in which performance can bedescribed by two parameters: the number of impacts, andthe nature of the impacts.Van Stiphout (1964) A quantitative two-parameter model in which theeffective power and the weight of fibres passing througha gap with a length of 1 cm were considered.Brecht (1967) A quantitative one-parameter model which assumes thatthe useful refining energy operates over a length of theleading edge. This widely used model is commonlyreferred to as the specific edge load theory (SEL).Danforth (1969) Two empirical equations were presented to augment thedescription presented in 1962.Tappi Stock Preparation Formalized the concept of Lewis and Danforth that refinerCommittee (1971) performance can be described in terms of two parameters:the intensity of impact (1), and the number of impacts (N).Kline (1976) Kline suggested that SEL theory be modified to considerthe effect of the bar surface area.Leider and Nissan (1977) They used a novel approach based upon a single fibre todefine N and I.Pashinskii and Takhtuev They related refiner efficiency to cutting length and SEL.(1982)Levlin (1984) Related refiner performance to SEL and specific energy.Kerekes (1990) A rigorous derivation of N and I using an individual fibreas frame of reference. Kerekes analyzed the probabilityof a fibre impact in a differential element and integratedthis result over the radius of the refiner.9Chapter 2: Literature ReviewSimilarly, Danforth (1969) defined C usingC (X XrXsWr (2.4)where Xr and X are the length of the bars on the rotor and stator, and Wr is the rotationalvelocity.Both of these equations have proved useful in practice, but have shortcomings from atheoretical standpoint. For example, Brecht’s equation does not consider factors known tobe important to refining such as bar angle, pulp consistency, and fibre properties (Heitanenand Ebling, 1990). Similarly, Danforth’s equation breaks down when the length of the rotorand stator are doubled; the number of impacts on a fibre predicted four-fold increases whena two-fold increase is expected (Kerekes, 1990).Kerekes (1990) addressed these shortcomings and rederived C rigorously. A simple differential equation was used to compute the number of bar crossings from refiner variables, fibrelength (lw) and coarseness (w), and pulp consistency (Cm). Kerekes then assumed an auxillary function to relate the probability of fibre impact to a bar crossing. He then integratedthe expression over the radius of the refiner to get the total number of fibre impacts N. Theresulting expression was then compared to Equation (2.2), i.e. N=CIF, to define C. For adisc refiner, it is given byc —8lr2GwGdpwlwCmn3r(1+ 2 tan 4)(R — R) (25)— 3w(l+Gd)The utility of this approach has been demonstrated by Kerekes et al. (1993). In this work,they estimated the limiting values for I based upon network rupture and fibre rupture. These10Chapter 2: Literature Reviewwere i0 and iO J/fibre respectively. They found values of I for mill refiners to be inthe range of l0 to i0 J/fibre.The usefulness of two-parameter models has, in general, been questioned by Page (1989).Based upon a principal component analysis, he argues that, since there are a large numberof independent beating effects, more than two independent beating parameters are requiredto adequately describe refiner performance. He has shown that at least three variables arerequired to describe the response to a group of different laboratory beaters, emphasizing theinadequacy of any two-parameter description. As a result, present descriptions of a refinerby two-parameter models are by no means comprehensive measures of the energy transferredfrom refiner bar to fibre. In a crude way, they reflect the fact that some form of fibre impactof a given average number and intensity have occurred. This approach, in general, has anengineering value, but is severely limited as a research tool. A more exact knowledge ofwhat happens within the refiner is required.2.3 Forces on Fibres during RefiningHydrodynamic Force: High speed photography has been used for flow visualization studiesin refiners (Banks, 1967; Ebling, 1980; Fox et at., 1979; Goncharov, 1971; Halme andSyrjanen, 1964; Herbert and Marsh, 1968; McKenzie and Prosser, 1981; Page et at., 1962).A number of unrelated observations have been reported which give modest insight into theoperation of the refiner (see Table 2.4). It was found that pulp flows primarily in the radialdirection: outward in the grooves of the rotor tackle, and inward in the grooves of the statortackle (Atack et aL, 1984; Halme and Syrjanen, 1964; Herbert and Marsh, 1968) at about 3.5rn/s (Atack et a!., 1984). Minor recirculation currents, called secondary and tertiary flows,have also been observed (see Figure 2.2). Fox et a!. (1979) assert that secondary flows arise11Chapter 2: Literature ReviewPrimary FlowSecondary Flow Tertiary FlowFigure 2.2: Refiner flow pattern showing three main regions of flowsfrom: 1) friction of the rotor bar sliding over the groove (cavity) of the stator; and 2) frictionof the stator sliding over the groove (cavity) of the rotor . Tertiary flows result from thedifference in static pressure between opposing grooves of the rotor and stator.A crude estimate of the hydrodynamic behavior of a refiner is available. Using a Bemouffiequation, Leider and Rihs (1977) related pressure drop to volumetric flowrate. Theirexpression accounts for the action of centrifugal and frictional forces, and is expressedRotorFlow4rStatorcoO’ InletRotor12Chapter 2: Literature ReviewTable 2.4 Observations of Refiners with High-Speed Cine-Films.Page et al. (1962) “It appears as though the whole action of the beaters may bethe breaking down of floes when the two bars approach eachother”.Halme and Syrjanen Stock has a tendency to flow in grooves (of the stator) towards(1964) the inlet.Herbert and Marsh Observed the flow outward in the groove of the rotor and(1968) inward in the groove of the stator.Banks (1967) 1) fibre floes covered 50-70% of the bar surface with a lengthto width ratio of 2:1; 2) some floes remained on the stator for atleast one revolution of the rotor; and 3) sections of flocs wouldseparate from the master floe and pass across stationary bars.Goncharov (1971) “A fibre layer is formed on the frontal edge of the elements,the width of this layer is 2-3 mm along the width of theworking surface of the element”.McKenzie and Prosser “The basic unit involved in the beating process appears to be a(1981) fibre floe”.in terms of operating conditions, fluid properties, and plate properties. Frazier (1988) usedtraditional lubrication theory and assumed that the pulp water suspension was “fluidized”.He derived an expression for the shear force applied, but its utility is severely limited byassumptions regarding the rheology of the fibre-water slurry. Leider and Nissan (1977)performed a similar analysis and give an expression for the net energy expended. However,none of these studies have linked the predicted force to the force applied on individual fibres.Mechanical Force: Early attempts to measure forces on fibres were largely measurements13Chapter 2: Literature Reviewof pressure on refiner bar surfaces or of thrust on refiner plates. For example, Goncharov(1971) measured the pressure distribution over the bar surface and found peak pressuresof 3.5 MPa in a low consistency (L-C) laboratory refiner operating at 2.5—3% consistency.Moreover, the peak pressure was found to occur over the first 2—3 mm of the bar and beapproximately 13 times greater than the average pressure. In similar work, Nordman et at.(1981) found pressures two orders of magnitude lower than those of Goncharov. In still latermeasurements in chip refiners, Atack (1980) found pressures of the same order of magnitudeas those of Nordman. Atack speculated that the pressure was determined by the thicknessand compressibility of the pulp between refiner bars. In summary, pressure in refiners hasbeen measured in a number of past studies, but the results vary greatly. None of these studieshave linked the findings to forces on individual fibres.24 Summary of Literature Review1. Refining is a process whereby the physical structure of the papermaking fibre is modifiedby stress imposed cyclically. Fibres are typically cut or fibrillated. These phenomenacan enhance paper quality.2. Present descriptions of refiners by two-parameter models are by no means a comprehensive measure of the energy transferred from refiner bar to fibre. They reflect the fact thatsome form of fibre impact of a given number and intensity has, on average, occurred.The approach has an engineering value but is severely limited as a research tool.3. The flow patterns within a refiner are complex and have yet to be fully understood. Ingeneral terms, flow has been observed to occur radially outward in the grooves of therotor and radially inward in the grooves of the stator.4. A large body of information is available in the literature on the changes in fibre14Chapter 2: Literature Reviewmorphology produced by refining. From this, the mechanism of beating has been inferred.Few, if any, of these models specifically address the central issue of this dissertation,that being, a quantitative estimate of the force acting upon individual fibres during thebeating process.2.5 Objectives of the ResearchThe specific objectives of the research are:1. To identify and estimate the hydrodynamic forces acting on individual pulp fibres whichare trapped and transported on the leading edge of a bar (Chapter 3).2. To develop a simple model of the mechanical forces (normal and shear) acting onthe fibres during bar passage; to confirm the model by observation and experimentalmeasurements of the shear forces acting on pulp in a laboratory-scale single-bar refiner(Chapter 4).15Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportChapter 3The Hydrodynamic Force onFibres During Fibre TransportIt has been speculated, in the literature, that the localized beating event occurs in two distinctsteps: first, fibres present in a water suspension as floe (aggregates of fibres) are trapped andtransported to the refining gap (Fox et al., 1979); and second, the fibres are physically heldin the refining gap in some manner and deformed or strained (Atack, 1980; Banks, 1967;Goncharov, 1971). The first step is defined here as the “trapping” phase, and the second stepthe “working” phase. In each step, energy is expended on the fibres.In this chapter, we formulate a model to consider the force distribution acting on singlefibres which are trapped and transported to the refining gap. After a detailed mathematicaldescription of the problem, equations are presented to predict the force acting on fibres as afunction of fibre properties such as stiffness, length, and diameter; and fluid properties suchas velocity and viscosity. The intensity of impact (Ib) during trapping will also be estimated.Intuitively, when a fibre impacts ‘upon a bar edge, the flow field imposes forces that causetranslation and rotation over and around the tip. The degree and direction of rotation isgoverned by the torque created by the force distribution. If the fibre is flexible, the forcesmay also cause the fibre to bend. Fibre bending diminishes the moment arm and therebydecreases the torque causing rotation. Trapping occurs when the translational and rotationalforces are in equilibrium, that is, when both the net applied force and the associated torqueequal zero.16Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportAlthough refining is generally carried out at fibre concentrations at which fibres are in floes,as an initial step we examine here the general problem of the behaviour of a single fibre.It will be shown that the general theory may be modified to consider both stiff and flexiblefibres.3=1 AnalysisThe analysis presented outlines conditions which cause fibre trapping; the fibre is allowed torotate, translate and bend simulataneously. In this case, flexible fibres are considered to be ina driven cavity flow at moderate velocities, i.e. 1—2 m/s [Lumianen, 1995). We consider thefibre to be a slender rod. The fibre moves and bends under the influence of the hydrodynamicforce. Based upon fibre diameter, say 20 jim, at this velocity we would anticipate a Re equalto about 40. This estimate was made with v=2 mis, D=20 x 10, and v=l xl0 m2/s.The estimate of the hydrodynamic force acting on the fibre is based upon the theory givenby Cox (1970) for a slender body. Cox’s theory uses Stokes’s equation and requires Reto be small. As a result, the analysis uses a moderate Reynolds number flow with a forcecalculation based upon a low Reynolds number result. This framework is used as there areno other expresions in the literature for bent, yawed cylinders.It should be noted that the expression given by Cox is valid for an isolated slender body in aninfinite fluid. The formulae is applied, however, to a fibre touching a wall, so the influenceof the wall should be considered. Unfortunately the section in Cox’s paper which deals withthis situation gives no useful results, and hence can not be used within this model.Finally, the estimates presented within this model are correct based upon this framework17Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transportand can be easily updated when accurate expresions for the hydrodynamic force becomeavailable. The usefulness of this model is that it correctly “poses” the problem, and thetrends predicted should be considered qualitatively correct.3.1.1 Force and Torque Balance We begin the analysis by considering a fibre as along slender body of circular cross-section of length, L, and having a characteristic value ofthe cross-sectional radius of Di2. We consider s to be the distance along the fibre centre-linefrom a reference point at the midpoint, hence—4 s 4. The cross-sectional radius atany point of the centre-line is taken to be D(s) where )(s) is a function of s. The fibreis considered placed in a flow field with value U(x,y), this flow field itself satisfying theequation of motion, i.e.U•VU= -VP+V2U (3.1)V.U=O (3.2)with P being a pressure field and , the viscosity. It is also assumed that the velocity ofthe material points on the centre-line of the fibre is given by the function U*(s) and that theentire centre-line is given by r=R(s). The vector r is defined as the position of a generalpoint on the fibre relative to a fixed set of rectangular Cartesian co-ordinates with the originat the corner of the bar. The fibre may be assumed to be bent in any manner whatsoeverso long as the radius of curvature of such bending is at all points of order L. By limitingour scope to the plane parallel to the imposed flow, the fibre is subjected to a hydrodynamicforce per unit length f with components f=(f1, f2) labeled in Figure 3.3. Hence, the totalhydrodynamic force F acting along the length of the fibre and torque G acting relative to18Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transportthe origin is given byLF=ff.t(s)ds (3.3)LG=ffxR(s)ds (3.4)where t is a unit vector in a direction tangential to the fibre.As shown in Figure 3.3, the fibre is in contact with the bar edge and is subjected to anadditional force, namely a frictional force. In dealing with the frictional force, it is importantto realize that the frictional force opposes the tendency of the fibre to move. Furthermore,with static friction, the frictional force balances the force tending to cause motion; when theapplied force reaches the maximum value of the static frictional force, motion impends. Forour case, the maximum static frictional force acting on the fibre is assumed to beLffrnax = f . n(s’)ds (3.5)where is the static coefficient of friction between pulp and the refiner bar; and n(s’), avector normal to the fibre at the point of contact. The integral represents the total forceacting normal to the fibre at the point of contact.Trapping occurs when there is no translation or rotation of the fibre, that is when both theforces and the torques acting on the fibre balance. This occurs when the absolute value ofthe applied force (IFI) does not exceed the maximum static frictional force and when the19Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transportassociated torque (G) equals zero, i.e.I F I —fj. 0 (3.6)G=0 (3.7)It is important to realize that the applied hydrodynamic force can be either positive ofnegative. Since the frictional force always opposes the hydrodynamic force, for simplicityand clarity, the trapping criterion is defined in terms of the absolute values.3.1.2 Hydrodynamic Force on Fibres Cox (1970) obtained an expression for thehydrodynamic force per unit length on a slender body (Re—+0 based upon fibre diameter)using an asymptotic expansion in terms of the ratio of the cross-sectional diameter (D) tobody length (L). The results are valid for curved slender bodies and for cases in which theaxis of the body is not parallel to the direction of motion of the fluid. Cox showed that theresistance to motion is given byf — U — U J + (U — U*)ln(2/A) dRdR2irt lnt (lmic) ds ds(3.8)+ (U — U*) F3 — 211 +01 1(ink) L ds ds j ((ini)3where J is by definition a vector given by[sei]{ }—1 s+e (3.9){oJk_}{uk()_UZ(I)}dI202 hydrodynamicforceper unit lengthF-total hydrodynamicforceactingonthefibreG-moment actingonthefibrefricfrictionalforces=1J2s,=s(pointofcontact)ds-vectors=-L/2ydsUFigure3.3:Schematicoftheflowandforcesonafibreincontactwithabaredge.Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transportwhere i is equal to the ratio of fibre length to diameter, L/D; I, the identity matrix; R, thevalue of r at the point on the centre-line under consideration; and R, the value of r an arbitrarypoint on the centre-line with s = 1. Since the above expressions for the hydrodynamic forceper unit length acting on the body are in vectorial form, one may consider them relative to anyrectangular Cartesian co-ordinate system which one may choose. The value of appearingin the equations is arbitrary, satisfying only the inequality 0 <€ <<< 1. However, it may beshown that as —* 0, the value of J given by Equation (3.9) is of the form (Cox, 1970).1 —{U(R) — U*(s)} in E + 0(1) (3.10)so that when this value of J is substituted in Equation (3.8), the resulting expression is finiteand independent of c in the limit of E —* 0. Thus, the final expression for the force per unitlength acting on the fibre is independent of c.For a smooth cylinder in a fluid undergoing uniform translation, the hydrodynamic force perunit length is given byfi= In (L/D)— U*)(2— cos2 0) — (V — V*) sin & cos e} (3.11)f2= ln (L/D) {(V — V*)(2 — sin2 0) — (U — U*) sin 0 cos 8} (3.12)where U and V are the velocity components in the x and y-directions respectively. Theseexpressions are only first order accurate and apply to cylinders whose centre-lines are eithercurved or straight and where derived by closely following the examples given by Cox (1970).22Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportThe fibre is assumed to be stationary (U*=O) and in a driven cavity flow field as shown inFigure 3.3. The flow field near the edge of the bar is approximated in a simple manner withthe following expressionsu u*_1(IJ,0,0) y>° (313— 1(o,-vo) y<0Hence, the final expression for the hydrodynamic force per unit length acting on a pulp fibreis given byI 1= 27r[L f U(2 — cos2 0) y > 0 (3 14)1 in (L/D) 1 Vsin 0 cos 0 y <02ir f UsinOcosO g > 0I f I ln(L/D) V(2 — sin20) , <o (3.15)In this dissertation, we use this expression to represent the hydrodynamic drag per unitlength acting on a pulp fibre. For clarity we use the absolute value for the magnitude ofthe hydrodynamic force.Under industrial conditions we would anticipate the upper limit of Reynolds numbers basedon fibre diameter to be about 40. Hence, these expressions underestimate the true “drag”force. The literature contains many references for drag coefficients for cylinders in eitherparallel or cross flow at higher Reynolds numbers (Pruppacher et a!., 1970; Jayaweera andMason, 1965). These expressions can also be used for a yawed cylinder when the assumptionis made that the forces can be resolved separately in the normal and tangential directions usingthe velocity components in those directions (Clift et a!., 1978). However, these expressionsare not valid for bent cylinders, the case we are studying here. The literature is bereft ofexpressions for the drag coefficient for a bent yawed cylinder at large Reynolds numbers.23Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportThere is, however, one expression available (Kheyat and Cox, 1989) which is valid for Re<1but its usefulness is limited as it becomes singular under certain flow conditions.3.1.3 Fibre Bending When a fibre impacts upon an edge, the flow field imposes forcesthat cause translation and rotation over and around the tip. If the fibre is flexible, the forcesmay also cause the fibre to bend. Fibre bending diminishes the moment arm and therebydecreases the torque causing rotation. An analysis similar to that performed by Newman(1992) is used to determine fibre deflection. We begin the analysis by assuming the fibre istrapped on the bar edge and subjected to the hydrodynamic forces as illustrated in Figure 3.3.Two force balances will be performed in orthogonal directions (normal and tangential) andare described below.Part 1 — Force Balance in the Normal Direction: A force balance in the normal direction isperformed on a differential element along the length of the fibre (see Figure 3.4).Tsin () + (T + dT)sin () + V8cos () — (V8 + dV8) () (3.16)+ fi sin lids + f2 cos lids = 0where T is the tensile force; and V, the shear force. From this point on we neglectthe “absolute magnitude” brackets from the hydrodynanüc force terms. By neglectingdifferentials which are squared, Equation (3.16) becomes!=T....+fisinO+f2cos (3.17)24Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportThe shear force V5 is related to M by taking moments about one end of the element at s+dsM + vcos ()ds + Tsin ()ds + fi sin (8)(3.18)+f2cos(8) —(M+dM)=oUpon rearrangement and neglecting squared differentialsdM—=V (3.19)dsFrom classical beam bending theoryM = —EI (3.20)where E is the modulus of elasticity; and I, the cross-sectional moment of inertia.Combination of Equations (3.19) and (3.20) yieldsdM d20V8 = — = -EI--- (3.21)ds dsDifferentiation of Equation (3.21) givesdV d30= —EI----. (3.22)ds dsSubstituting Equation (3.22) into (3.17) yieldsd30 dO—EI—. = T— + fi. sin 0 + f2 cos 0 (3.23)25M+dMn\\U4 IU—‘Ft—V+dV\ \\Tf2dsvs\ \\I\/MFigure3.4:Forcebalanceonadifferentialfibrelength.Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportPart 2 — Force Balance in the Tangential Direction: A force balance in the tangentialdirection is performed on a differential element along the length of the fibre (see Figure 3.4)+ dT) cos () — T cos (v-) + (V8 + dV5)sin (2) + 8sin (2) (3.24)+ fi cos Ods— f2 sin 8ds = 0Assuming in the limit that dO—*O, neglecting squared differentials, and using Equation (3.21),Equation (3.24) becomes=— fi cos 0 + f2 sin 0 (3.25)Two other equations are added to complete the fibre geometry. From elementary trigonometrydx—= cos 0 (3.26)ds—= sin (3.27)dsFour boundary conditions are required to solve Equations (3.17) and (3.25) and are presentedbelow:1. At the ends of the fibre(s = ±L/2) the fibre can not support a moment, i.e.dO d28ds ds27Chapter 3: The Hydmdynamic Force on Fibres During Fibre Transport2. At the point of contact with the bar (s=s’): the tensile force (T) is equal to the frictionalforce (ffric)LT= 2. n(s’)ds (3.29)3. At the point of contact with the bar (s=s’), the torque is finite and non-zero and basedupon the curvature of the fibre, i.e.LG=f f x R(s’)ds = 0 (3.30)3.2 Results3.2.1 Stiff Fibre The equations given above will now be used to determine the steady-state behaviour of a fibre for which the body centre-line is straight on a bar edge. Considerthe body in contact with the bar edge at some point, d, between its ends. Let the midpointof the fibre be taken as s=0, so that s’ varies within the range—4 <s’ < 4. The centre-liner=R(s) of the body is given by= 6j1 (s — cos 0 + 6i2 (s — s’) sin 0 (3.31)Hence the normal (n) and the tangential (t) direction components of the fibre are given byn= (sin 8,—cos 0) (3.32)t = (cos 0,sin 0) (3.33)28Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportBy substitution of these expressions into Equations (3.3) and (3.4), one may obtain the totalforce and torque acting on the particle.LF=f f tds (fi cos 0— f2 sin 0)ds (3.34)L k=f f x R(s)ds =ff1(s — s’) sin 0 + f2(s — s’) cos Ods (3.35)If we substitute expressions (3.14) and (3.15), Equations (3.34) and (3.35) may, after amodest amount of calculation, be shown to beLF= in (L/D) { f U cos Ods _f V sin Ods } (3.36)LG= In (L/D) {Usin of (s — s’)ds + Vcos of (s — sl)dS} (3.37)Upon integrationF= in (L/D) [Ucos o( — — Vsin + (3.38)G= in (LID) [Usin e( — )2 — Vcos o( +29Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportCalculating the maximum static frictional force: As stated earlier, the maximum staticfrictional force, as given by Equation (3.5), was assumed to be linearly proportional tothe normal force acting on the fibre. Upon substitution of Equations (3.14) and (3.15),Equation (3.5) becomesLfm itsf (fi sin 9+ f2 cos O)ds(3.40)= 8l (L/D) { Usin Of ds ± Vcosf ds}8lfl (L/D)[ (2 )+ (2+ )]Example — the trapping of stiff fibres: Trapping occurs when the translational forces androtational torques are both in equilibrium, that is, when the applied hydrodynamic force isless than or equal to the maximum frictional force and when the associated torque equals zeroI F I ffmaa 0 (3.41)G = 0 (3.42)For the frictionless case, trapping occurs when G=I F 1=0(1 U \LV) —12 (3.43)2 ((U)±i)Equations (3.41) and (3.42) have also been solved for the case with friction and the resultsare shown in Figures 3.5(a) and (b) for two different coefficients of friction: it=O.2 and30Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportThese assigned values are arbitrary as there is no definitive literature value. Thereare estimates within the literature and they range from 0.11 (Goncharov, 1971) to 0.75 (Milesand May, 1989). It is apparent that the curves shift in magnitude, but their nature remainsthe same. In general, the region in this Figure in which fibres trap increases with increasingcoefficient of friction.This phenomenon results from the fact that the coefficient of friction affects the frictionalforce at the point of contact. Increasing frictional forces causes a greater resistance for fibrestapling.3.2.2 Flexible Fibre As an example of trapping with a fibre whose centre-line is notstraight, we must first consider how the fibre is deflected by the flow field. Equations (3.17)and (3.25) will now be used to determine deflection of a fibre around a bar edge. As anillustrative example, we consider the symmetrical case with the body in contact with thebar edge at the midpoint between its ends, i.e. d=O and with U=V. Hence we only need tosolve the bending equations over the half-length of the fibre and we choose to do so for theportion of the fibre in the domain y>O.As a first order approximation, one may evaluate the force per unit length f acting on fibrefor y>O using Equations (3.14) and (3.15). They are restated here for convenience27rtU1 2fi= 1 IL\ — cos (3.44)In2irUf2= L sin cos 6 (3.45)in ()31Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transport00Coa)2•0013cu00.5C00ci)C00Cci)2013Cu0C-)C0________________________________00)C00Ca)2VCo13CuC0C)C00Figure 3.5: Trapping of fibres: (a) p=O.2 (stiff); (b) (stiff);(c) itO.2(completely flexible). The velocites V and U are shown in Figure 3.2.trapping region70.25 0.50 0.75 1.Velocity Ratio, V/U (dimensionless)0.50-0.25-0.00-0.250.58.o0.50 -0.25-0.00-0.250.50—0.25-0.00-0.25Figure (a)stiff fibre0Figure (b)stiff fibreI.Ls=04Figure (c)flexible fibrern..m.jiiiion0.25 0.50 0.75 1.00Velocity Ratio, V/U (dimensionless)trapping region /0.25 0.50 0.75 1.00Velocity Ratio, V/U (dimensionless)32Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportSubstitution of Equations (3.44) and (3.45) into Equations (3.23) and (3.25) yieldsd36 dO 47r ‘U—El—— = T— + L sin 6 (3.46)ds3 ds in ()= EI-44-— 2irpUcos 0 (3.47)ds ds ds In ()Dimensionless quantities will be used based upon fibre length, L, the fluid viscosity, t, anda characteristic velocity, U. Normalizing s and T usings = s*L (3.48)T =21rUL (3.49)ln()Equations (3.46) and (3.47) become_L _T*+2 sin6 (3.50)Be ds* ds*dT* 1 d20 dO=-. —. a—.— cos 0 (3.51)where Be, defined in this work as the bending number, is a dimensionless parameter given by2 UL3Thn()EIthe ratio of hydrodynamic force to bending rigidity.The equations given above have been solved numerically using a Runge-Kutta procedure(Press et al., 1992) using the boundary conditions given by Equations (3.28)-(?) and the33Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transportresults are plotted in Figure 3.6 for a series of bending numbers. Clearly, fibre bendingincreases with bending number. As a subjective classification, we can see that with Be<1O,the fibre is considered stiff; with 1 1<Be<75, the fibre deflects modestly; and, with Be>75 thefibre is considered flexible and drapes over the rotor bar.As an example, typical values of Be are calculated for a stone-groundwood (SOW) pulp anda semi-bleached kraft pulp (SBK) under conditions of a L-C refiner for values of El givenby Tam Doo and Kerekes (1981)27rILUaveL3 2(10)50(2 X io)Besow= in (L/D)EI = (‘6)7o X 10—12= 8.9 (3.53)27i7LUaveL3 27r(103)50(3 x io)BeSBK= ln (L/D) El= ln (<-6)3 < 1o—= (3.54)Clearly, SBK pulp is flexible and SGW pulp is stiff.As an illustrative example, the trapping behaviour of a completely flexible fibre (Be —* cc)is shown in Figure 3.5(c) . The two extremes are shown, that is a stiff fibre (Be=O) anda completely flexible fibre. It is apparent the curves are similar in shape but the region oftrapping increases substantially with increasing bending number. This phenomenon resultsfrom the fact that the ratio of shear force to normal force and the overall moment acting onthe fibre decreases with increasing fibre bending.34Deflection(Be=1 0)ModestDeflection(Be=22)LargeDeflection(Be=75)IStiff(Be=0)SmallFigure3.6:Deflectionofasinglefibreasafunctionofbendingnumber,Be.Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transport3.3 DiscussionAxlcross-sectionFigure 3.7: Cross-section of a fibreX2Having determined the force distribution acting on individual fibres trapped against bar edges,we may now estimate the energy expended. The work done by external forces in causingdeformation is stored within the fibre in the form of strain energy. In an ideal process in whichbending doesn’t exceed the elastic limit of the fibre no dissipation of energy takes place; allof the energy is recovered upon unloading. The incremental energy, dlb, required to bend afibre through a small distance is simply the product odV where u is the stress applied, E,the strain, and V, the volume of the fibre. Substitution of a linear elastic relationship yieldslb Jff11= 2 E dsdx1x2where dxjdx2 represents the cross-sectional area of the fibre (see Figure 3.7).(3.55)For illustrative purposes, we consider the body in contact with the bar edge at the midpointbetween its ends, i.e. s’=O. Symmetry is assumed so that U=V. Two cases are considered,A’4 436Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transportone being a fibre undergoing small deflection, and the other a fibre which undergoes largedeflection.Case 1 — Small Beam Deflection (Be < 1): In determining the strain energy for this case,we assume from classical small beam bending theory that the applied stress is given byM(s)xia(s)=(3.56)where M(s) is the moment distribution along the length of the fibre; and I, the moment ofinertia. Upon substitution of Equation (3.56) into Equation (3.55) yields+f 1 [MU]2 [ffxdxidx2]ds (3.57)Here, the double integral in the bracket is (by definition) the moment of inertia, I. Thus,the strain energy is= 2f [‘)ds (3.58)where the integration is carried out over the half-length of fibre.Clearly, the moment distribution is given byM(s) =—f (fi sin 0+ f2 cos 0)sds (3.59)37Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportIntegration and substitution of Equations (3.14) and (3.15) into Equation (3.59) yieldsM(s) = f [(2 — cos2 0) sin 0 + sin 0 cos2 0]sdsn () o (3.60)2irtU. 2=— IL’ (sin 6)sin )This equation is further simplified to include the bending numberM(s) = —Be (sin 0)s2 (361)Substitution of Equation (3.61) into Equation (3.58) and integrating givesL= JBe2 sin2L (3.62)— Be2 sin2 0 El— 160 LAs an order of magnitude estimate(5 x 101)2(5 x lo_1)2 5() 10—1210_’2J (3.63)Hence, the energy of bending (Ib) is calculated to be about 1012 J. It should be reiteratedthat this analysis is valid only when Be < 1. The stiffness of the fibre in this case wasgiven by Kumar (1991).Case 2 — Large Beam Deflection (Be— 002: We begin the analysis by assuming that forthis loading the strain in the fibre results from the tangential force (T) given by Equation 3.51T* =— f (cos 8)ds* (3.64)38Chapter 3: The Hydrodynamic Force on Fibres During Fibre Transportfor the completely flexible case. In dimensional form we see that for the frictionless case,T= ln(L/D) (cos O)s (3.65)Normalizing the tensile force to the cross-sectional area (A) of the fibre gives2ir,tU s= in () (cos 8) (3.66)Substitution of Equation (3.66) into Equation (3.55), which has been transformed into cylindrical coordinates, we obtain= j (2riU; cos2rdrdO] dx (3.67)Here, the double integral in the bracket is the cross-sectional area, A and is equal to7r(D2_D?)A= ° (3.68)where D is the diameter of the fibre; and the subscripts o and i refer to the outer diameterand inner diameter respectively. The strain energy is thusL=cos2 /s2ds (3.69)1 2 2 El2cosThe cross-sectional moment of inertia, I, is given by hence, the above equationreduces to=El (D_D)cos2 0 (3.70)39Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportAs an order of magnitude estimate(2 x 102)2 5 x 1012 ((35 x lo_6)2 — (25 x lo_6)2\ 100 10J (3.71)384 3 x lO (3 x 10_3)2,,Hence, the energy of bending for a single fibre is calculated to be about 10.11 J. For boththe stiff and flexible case we see that the energy expended in bending is proportional to thebending number squared, i.e.x Be2 (3.72)3.4 ConclusionsThe theory presented outlines the conditions which cause fibre trapping; the fibre is allowedto rotate, translate, and bend simultaneously. The model equations are based upon threeassumptions: the fibre is placed in a uniform undisturbed flow field; the ratio of cross-sectional diameter of the fibre to its body length is small; and the Reynolds number for thefibre, based on fibre diameter, equals zero. It was found that:1. To classify fibre bending at a bar edge, a non-dimensional number, derived by nondimensionalizing the force balance governing large amplitude bending, was employed.This number, defined here as the “Bending number”, Be, was found to depend uponwater viscosity, t, fibre length, L, fibre diameter, d, flow velocity, U, and fibre stiffness,El. It is given byB— 27r1uUL3373— ln (L/D)EI40Chapter 3: The Hydrodynamic Force on Fibres During Fibre TransportThis number was used as a quantitative criterion for classifying fibres as “stiff’ or“flexible”. The likelihood of fibre trapping was found to be a function of 3 parameters:the velocity ratio, V/U, Be, and the coefficient of friction2. The likelihood of trapping increased with fibre flexibility and coefficient of friction.3. For the cases considered, the energy expenditure associated with bending pulp fibresaround a bar was found to be proportional to Be2 and range from 1012 to 10.11 3.41Chapter 4: The Force on Fibres Between Passing BarsChapter 4The Force on FibresBetween Passing Bars4.1 Review of Previous WorkHaving determined the hydrodynamic forces on fibres stapled on moving bars, we nowexamine the forces exerted on fibres during bar crossings. Although these forces have longbeen linked to the refining process, very few past studies have examined their nature.Early attempts to measure forces on fibres were largely measurements of pressure on refinerbar surfaces or of thrust on refiner plates. For example, Goncharov (1971) measured thepressure distribution over the bar surface and found peak pressures of 3.5 MPa in a lowconsistency (L-C) laboratory refiner operating at 2.5—3% consistency. Moreover, the peakpressure was found to occur over the first 2—3 mm of the bar and was approximately 13 timesgreater than the average pressure. In similar work, Nordman et a!. (1981) found pressurestwo orders of magnitude lower than those of Goncharov. In still later measurements in chiprefiners, Atack (1980) found pressures of the same order of magnitude as those of Nordman.Mack speculated that the pressure was determined by the thickness and compressibility ofthe pulp between refiner bars. In summary, pressure in refiners has been measured in anumber of past studies, but the results vary greatly. None of these studies have linked thefindings to forces on individual fibres.Another approach to characterize refining action on individual fibres has been developed inrecent work (Kerekes, 1990; Kerekes et a!., 1993). A new method of characterizing pulprefiners by the “C-Facto?’ gave an average number of impacts (deformation cycles), N, and42Chapter 4: The Force on Fibres Between Passing Barsan intensity of impact, I. The latter, expressed as energy/impact/fibre, represents the energydissipated in a straining cycle. Since energy is the product of force and distance, I offers abasis for estimating force if the distance over which it is applied is known. However, thereis insufficient data available in the literature to make this link.The objective in this part of the thesis is to obtain order of magnitude estimates of forcesacting upon fibres during a single bar crossing. We do so by considering a force balance onfibres, the compression of floes, and then the forces on fibres in contact with bar surfaces.4.2 Analysis4.2.1 Force Balance on Fibres To create refining action, a force on the fibre must insome way impose strain on the fibre. To accomplish this, rather than simply accelerate thefibres, the fibres must be “held” in some manner. If we consider a fibre in contact with abar surface (see Figure 4.8), the “holding” force comes from surface friction and a cornerforce. The applied “working” force comes from surface friction on the opposing side of thefibre in question, probably induced by other fibres. These other fibres may be restrained byfriction, but it is clear that the fibres in contact with the bar surface are of major importancein determining whether the entire floe is restrained or accelerated. We now examine theforce on these fibres quantitatively.We consider a force balance between the force applied to a fibre, F, and a holding force,Fh. We consider the holding force to be made up of a corner force, F, and surface frictionforces between the bar and the fibre and other fibres both governed by N, the normal forceon the bar surface (see Figure 4.8).43Chapter 4: The Force on Fibres Between Passing Barsindividual fibrewithin a flocNFigure 4.8: Forces on an individual fibre in a fioc in the refining gap.Thus, to strain a fibre, we must havewhereandFwFhF = liO!N(4.1)(4.2)Fh =F+,u3N (4.3)where I’d is the fibre-to-fibre coefficient of friction and ls is the fibre-bar coefficient offriction.It should be noted that if Fw>Fh the fibre is accelerated rather than strained. For this notto occur, the holding force, Fh, must be adequate to balance the applied working force, F.It is clear from the above equations that this depends on the size of the normal force, N,F (interfibre friction)bar surface44Chapter 4: The Force on Fibres Between Passing Bars5,’yFlocFibresL(x)IyCross-SectionzFigure 4.9: Fibre distribution in a cross-section of thickness dx of an ideal floc.and the corner force, F. For this reason, the subsequent analysis will focus on determiningquantitative expressions for N and F.4.2.2 Estimating the Normal Force To derive an expression for N, we must firstconsider compression of the flocs from an uncompressed state, to the level of compressionexpected between bars in a refiner.We begin the analysis by considering a floe, an aggregate of fibres, as a spherical bodyhaving a characteristic cross-sectional radius ofL0/2 and an overall mass consistency of Cm.Fibres are assumed to be uniformly distributed throughout the entire volume of the floc. Asa first approximation, in any cross-section of the floe of thickness dx, fibres are assumed tobe evenly spaced as shown in Figure 4.9.zdxLz (x)Lz (x)45Chapter 4: The Force on Fibres Between Passing BarsRegimes of Hoc Compression: In the uncompressed state at usual refining consistencies(C1—6%), fiocs typically have a porosity (p) approximately equal to 0.92. Thus, whencompression first begins, there is a reduction in pore volume and an increase in the numberof fibres in contact. We call this “regime 1”. However, at some degree of compression, thenumber of fibres in contact is such that further compression of the fioc must compress theindividual fibres themselves. We call this “regime 2”. These two regimes of compression areillustrated in Figure 4.10. The following analysis considers these two compression regimesseparately.In “regime 1” , we assume the floc of fibres to be spherical in shape with diameter, L0. Thechord-lengths of L, and L, as depicted in Figure 4.9, are given by= L01(2)2L(x) =— (2)2Under uniaxial compression in the y-direction, we assume the initial compression to bedensification by decreasing pore size and increasing fibre contact. In the extreme casewhen all fibres come into contact, by assuming an orderly lattice arrangement the typicalinterfibre spacing is approximately where D is the effective diameter of the fibre. Thecompressed y-dfrection size L’(x) is y’(x). We denote all floc dimensions in region 2 asprimed variables.As a simplification, during compression we assume no fibre re-arrangement in the x-direction.2 Porosity is the fraction of void space in the floc. For low consistency flora it can be shown that porosity is related to consistency(fraction) by p=1—CmIl.5.46Chapter 4: The Force on Fibres Between Passing BarsFibres incontactLy(X)= y,(x)Fibres collapsed ontheir lumenL’(x)= y (x)thickness of a collapsed fibre (t)Figure 4.10: Lattice spacing of fibres in floe undergoing compression.Regime 2yFibres not incontact 1z L(x)L(x)yRegime 1q2Di2DL’ (x)47Chapter 4: The Force on Fibres Between Passing BarsAt floc size Y’cfr) and smaller, the compression is in regime 2. Here, compression takesplace by compression of individual fibres and therefore the floe can support much higherloading than regime 1.Calculating the Number of Fibres in a Floe: In “regime 1” the number of fibres in thefloe can be calculated using a differential mass balance where the mass of fibres in anycross-section dx is given by7rL2(x)flf(X)WdX= Po dx (4.6)where n/x) is the average number of fibres in the cross-section at x, w, fibre coarseness3(kg/rn), Po’ bulk density of the fibres in the uncompressed floe (kg fibres/volume of floe),and L(x), the cross-sectional dimension of the uncompressed floe (m). For floes of lowconsistency Po = 1000Cm, where Cm is the mass consistency (fraction) of the floe.Upon rearrangement and substitution of Equation (4.4), the average number of fibres in thecross-section is given by=— (2)2)As an interesting aside, we can now calculate the total number of fibres in a floe (Nf) byintegrating Equation (4.7) over the entire volume of the floe, i.e.N1 = n1(x)dx = Lf(1- ()2)dx (4.8)The weight of fibre wall material per unit length of fibre (Smook, 1990)48Chapter 4: The Force on Fibres Between Passing Barswhere l is the length weigthed fibre length average4. Upon integration we see thatAT Po.Lvf = 6 wlCalculating the First Critical Distance y’f.: For a compressed fioc in “regime 2” (see Figure4.10,) the number of fibres n/x) in any cross-section is simpiy the product of the numberof fibres in a row in the z-direction (flfr(X)) and the number of such rows (nr(x)) in they-direction. Hence,flf(X) = flfr(X) X flr(X) (4.10)As an example, if we consider regime 2 of Figure 4.10, there are four fibres in each row(nfr=4) and there are six rows (nr=6) for a total of 24 fibres.At the onset of regime 2, the fioc size is y(x) in the y-direction. The compressed thicknessof the floc is simply the product of the number of rows and the interfibre spacing. At thispoint the interfibre spacing is simply Hence, )7’c(X) can be estimated to bey(x) = (4.11)Upon compression, the floc increases its width in the x-z plane. The width of the compressedfioc is simply the chord-length L(x) and is approximated by substitution of Equation (4.5)into the following expressionL(x) =7L(x) =— ()2(4.12)This is the fiit moment of the tithe length disthbution.49Chapter 4: The Force on Fibres Between Passing Barswhere y is the ratio of the width of the compressed floc to the diameter of the uncompressedfloe, i.e. -y = The number of fibres spanning the width of the floe (fr) is given by(nfr(X) — L’.(x) (4.13)Rearranging terms and assuming flf,. >> 1, yields=___— (2)2(4.14)At this point we may now calculate the number of rows usingflf(X)m(x) = (4.15)flfr(X)Substitution of Equations (4.7) and (4.14) into the above expression gives= v’•S7r PoLoD/i — (2)2 (4.16)Substitution of this equation into Equation (4.11) yields an expression for the critical distancethe floe has to be uniaxially compressed to have fibres in close contact, and thus to supportsignificant normal loady(x) = pD2/— (fl2 (4.17)As an example of the usefulness of this intermediate result, the ratio can be estimated forthe subsequent experimental work. This estimate was obtained at x=O with: w=O.215 x 1O50Chapter 4: The Force on Fibres Between Passing Barskg/rn; D=20x 10.6 m; and 7=1.1. For floes of low consistency Po = 1000Cm, where Cm isthe mass consistency (fraction) of the floe. This givesP Cm (4.18)The implication of this result is that for fibres to be in contact, and thereby tosupport significant normal load, fiocs have to be compressed to a level of whichis approximately equal to their consistency.Calculating the Second Critical Distance y’df: As the level of compression increases withinregime “2”, the fibres collapse in on their lumen. The fibres are still assumed to be inan orderly lattice with the thickness of each fibre being t (see Figure 4.10). The cell wallthickness is the difference in the outer diameter to the inner diameter of the fibre, i.e. jD—D1From a mass balance on an incremental length of fibre ds we see thatir(D2— D2)WPcw (4.19)where p is the density of the cell wall material; and w, fibre coarseness. Hence, t is given byt = D (i— V/i — 4w 2) (4.20)lrPcw DThis second critical distance Y’d is approximated in a manner as described above, i.e. it isthe product of the cell wall thickness times the number of rowsy(x) = mx) (4.21)51Chapter 4: The Force on Fibres Between Passing BarsSubstitution of Equations (4.16) and (4.20) into the above expression gives= L0 (i —— rpD2) (_ ()2) (4.22)The ratio of Y’c to Y’d is given byy(x) = t=— /1 — (423)y(x) v’D 2 V ‘wpD2)A typical value for this ratio can be estimated for the subsequent experimental work to be0.2. This estimate was obtained with the values for w and D given above, and, withp,=l50O kg/m3 (Panshin and DeZeeuw, 1980; Stone et a!., 1966).Estimating the Strain The strain E(x) acting on a differential element of a floe compressedfrom y to a smaller size G is given by(x) = 1—y’(x) = 1— PD2Ly _()2 (4.24)The upper limit of strain is the second critical distance y’(x), i.e. G varies within the limitsy<G<y.Estimating the Normal Force: By assuming linear elastic behavior, the incremental normalforce acting on the element of a floe compressed from size L0 to a smaller size G is given bydN = Ee(x)dA (4.25)52Chapter 4: The Force on Fibres Between Passing Barswhere E is the bulk modulus of elasticity of the floe (Pa); and d, the incremental area ofthe floe in compression. The incremental area for the compressed floe is given bydA = L’(x)dx =7LoJ1 — ()2dx (4.26)Substitution of Equations (4.24) and (4.26) in Equation (4.25) givesdN = E(1—D2ry 2)70 — ()2dx (4.27)Integration within the limits of x and x0 (see Figure?) yieldsN = E2—p0D2(z xo)] (4.28)2/ — (2)2+sin_i () (4.29)=2xo/i — (2)2+ () (4.30)where x and x0 vary between L0/2 to -L012 and represent the fraction of the floe undercompression4.2.3 Forces on Fibres in Contact with Bar Surfaces As stated earlier, the normalforces compressing floes and thereby acting on individual fibres create a “holding” force at53Chapter 4: The Force on Fibres Between Passing BarsIFigure 4.11: Schematic for the definition of the “ploughing” force.the bar surface. These were described earlier as a corner force, F, and a friction force. Herewe will examine the nature of the corner force and derive an expression for it.The corner force, identified by Page (1989) as a “ploughing force”, is one that occurs asa result of localized normal force that creates an indentation in the material in question.Corner forces (Fe) are traditionally treated as a localized point force protruding into thematerial creating a “plough” that is resisted by surface friction (Ff) and a compressive force(Fdeform) required to displace the material as the “plough” passes over it (see Figure 4.11)(Bowden and Tabor, 1950). In summary,F = + Fdeform (4.31)The frictional force is linearly related to the normal force. The deformation force is relatedto the elasticity of the fibre and the depth of penetration of the “plough”. In the case of54FCF.tncChapter 4: The Force on Fibres Between Passing Barspulp fibres and steel bars, we have what may be regarded as a special case of a “ploughing”force. We have a very soft material in contact with a very hard one with the radius ofcurvature of the bar approximately equal to the radius of curvature of fibre. The bar maycause large indentations in the fibre, perhaps flattening it. The problem becomes a complexone of non-linear deformation. In addition to this, this problem is unique as the localizednormal force is transmitted through a fibrous medium composed of randomly oriented pulpfibres. No relevant work could be found in the literature. However, for the purposes ofthis thesis we will carry out a simplified analysis in order to obtain an order-of-magnitudeestimate of the corner force.We consider the problem to be one of two curved surfaces of nearly equal radius in contactwith one another under force, N. The rigid bar, with a rounded corner, creates an indentationin the more flexible fibre on the surface and therefore there is a pressure acting over the surfaceof this indentation area. We consider: (1) the frictional force (ffj) to be linearly proportionalto the applied normal force; and (2) the compressive force (Fdefo) to be the force exerted bythis maximum contact pressure multiplied by the projected area of indentation (Ar). Hence,F = /L8N + PeAp (4.32)To obtain an expression for the maximum pressure, c, we use the analysis of Hertz(summarized by Timoshenko and Goodier, 1951) of the contact stress between two cylinders.This givesP = (4.33)55Chapter 4: The Force on Fibres Between Passing Barswhere A is the area of contact between the bar and the fibre and is given byA2mn(2Ecw(11)) (434where Rj is the radius of curvature of the fibre5, Rb, the radius of curvature of the bar, E,the modulus of elasticity of the fibre’s cell wall, ii, Poisson’s ratio of the cell wall material,and m and n are geometrical constants whose product is approximately equal to unity. Uponsubstitution we see thatPC— (4.35)The pressure, P, creates an indentation of depth 6 as illustrated in Figure 4.12. Fromgeometry, it can readily be shown that the depth of indentation is6 = Rb(1—(4.36)The contact area, A, must approximately equal 2RfRbO. Hence, we can obtain an expressionfor 0 upon substitution of Equation (4.34)mn (31_v2 RfRb (437)37r RfRb s\2 E Rf + Rb)Clearly, for this expression6only 0 is bounded between O<0<ir. We can now estimate Fdeformas the product of the normal pressure, P, multiplied by the projected area (Ar) normal toThe radius of curvature is not necessarily the radius of the fibre as the fibre flattens under the normal load6 This expression is valid for small penetration depths. For the case of large penetration B is bounded betwen O<B<3ir12.56Chapter 4: The Force on Fibres Between Passing BarsCornerForceNormal ForceFigure 4.12: Schematic of the corner force.N, i.e.Fdeform PA = P6(2Rf) (4.38)This gives2RfRb(1O)(2 E / 1 1 \‘Fdeform37r —— 3(1_v2)Rf+Rb)) N5 (4.39)mnSubstitution of Equation (4.11) into Equation (4.32) yields an expression for the corner forceacting on an individual fibre2_____12 /1 1F = ,uN +3’f (‘ — 3(1 — v2) + N (4.40)mm57Chapter 4: The Force on Fibres Between Passing Bars4.2.4 Forces on Flocs The “working” force on a hoc, F, can be determined from theforce on individual fibres multiplied by the number of fibres in such contact.F = flfrFc + jt3N (4.41)where flfr are the number of fibres in a row of the floc in contact with the bar edge. Uponsubstitution of Equations (4.14) and (4.40) into Equation (4.41)F = TN (—+ (2T + 1)8N (4.42)where—3lrRfRb (2 E Rf + Rb”mn \3(1—v) RfRb I= 2/D\/’— (2)2and the normal force, N, is given by Equations (4.28)—(4.30). The variable “x” in theseequations refers to the location along the length of the floe in contact with the corner ofthe bar. This equation is further simplified by using a power series to represent the cosinefunction, i.e.=1—2)2•.. (4.45)58Chapter 4: The Force on Fibres Between Passing BarsAs (is typically two orders of magnitude greater than N, the truncation of the series aftertwo terms does not represent a great sacrifice in accuracy. Hence,F = aN +a3N (4.46)where a, is defined here as the “corner force coefficient” and is equal to a = and a3,the “shear force coefficient” and is equal to a3 = 4i8(T + 1). This equation is used to predictthe peak working force and the working force distribution acting on a floc. It is only validover the initial period of the bar crossing when the floe is in contact with the bar edge.Prediction 1 —Estimating the Peak Working Force on a Floe: As an example, these equationsare used to predict the peak working force acting on a floe between passing bars as a functionof the gap size G. There are many ways of calculating this value. The first method wouldbe an analytical one in which the maximum value is estimated from the first derivative ofEquation (4.46). The second method, which is employed here, is one in which the maximumis estimated by searching the function numerically. The results are shown Figure 4.13. Itis apparent that peak shear increases with decreasing GIL0 in an approximately linearmanner.Prediction 2 — Working Force Distribution along the Length of the Bar: As another example,these equations were also used to predict the working force distribution acting on a floc overthe initial width of the refiner bar. We begin this prediction by assuming that at t=O the barsbegin to pass over one another and the floc is pinched at x0 between the leading edges ofthe rotor and stator. With t> 0, the rotor bar passes over the stator bar and the region of thefloe being compressed increases from x0 to x (see Figure 4.14). If we assume that at t=O the59Chapter 4: The Force on Fibres Between Passing Bars15.012.510.07.55.0-2.5-00—’0.000GIL0 (dimensionless)Figure 4.13: Theoretical estimate of the effect of gap size on peak working force.floe is pinched at x=-L012, then x varies with time according to the following expression(4.47)where Ub is the velocity of the bar, and t, the time. This equation applies only when Ubt < Li,.Upon substitution of this relationship into Equations (4.28)—(4.30), the normal force actingon the floe is given by[b(t) 167 W G (Ubt)1— p0D2L L0 ]— 2(— + Ubt)—+ Ubt) (4.49)zci)0LI0a)00.025 0.050 0.075N(t)=E2-4 (4.48)60Chapter 4: The Force on Fibres Between Passing BarsRegion undergoing_—/compression and /shear X0xStatorFigure 4.14: Region of the floc under compression and shear during bar passage.The total working force acting on the floc is given bywhereF(t) = a(t)N(t) +a8(t)N(t)2mn L0(3(l_v2)’\3a(t) = 96/rD 2E } (R1Rb)(4.50)(4.51)Rotor1 L_ f2(42+Ubt) ) (4.52)1 261Chapter 4: The Force on Fibres Between Passing BarszG)20UCI—0Figure 4.15: Working force distribution over the initial width of the bar.a8(t) = s(21-(2(_± Ubt))2+ 1) (4.53)This equation applies only when Ubt < L0. After this point the floe is no longer in contactwith the corner of the stator bar. We used this equation to predict the initial shape ofthe working force distribution curve for a floe passing between two bars and is shown inFigure 4.15. The shape of this curve is similar in shape to the normal distribution curveas reported by Goncharov (1971).t4t (mm)62Chapter 4: The Force on Fibres Between Passing Bars4.3 Experimental4.3.1 Objectives of the Experimental Work The objective of this experimental workis to confirm by observation and experimental measurement the predictions in the previoussection of the force acting on single flocs. Specifically, we will measure the peak workingforce and the working force distribution and compare the results to predicted values.4.3.2 Apparatus This study was carried out on a “single bar refiner” constructed forthis purpose. The apparatus, shown in Figures 4.16 and 4.17, consists of two horizontaldiscs, approximately 25 cm in diameter, with one radial bar on each surface. The lowerdisc rotates at a low speed (30 rpm maximum). The upper stationary disc is coupled to ashaft instrumented with a strain gauge to measure torque. The bars on the discs are radially-oriented and approximately 10 mm in width and height from the disc surface. The cornersof the bars have been profiled to an exact radius of curvature.A small support bracket was secured to the leading edge of the rotor bar to hold individualfloes, enabling reproducible positioning and reliable transport of the floes to the refining gap.The “floc holder”, shown in Figure 4.17, was recessed 2 mm below the plane of the barsurface to avoid interference with the refining action itself.4.3.3 Testing Method Individual flocs of fibres were prepared in an inclined rotatingcylinder using the method of Jacquelin (1966, 1968). In this method, a suspension of fibres inwater of known mass consistency is placed in a 2 L beaker, inclined at a 45° angle and rotatedat 30 rpm for 24 hours. This process produces coherent individual flocs which were thenrandomly selected and their diameter (L0) measured using a micrometer. Most floes were63Chapter 4: The Force on Fibres Between Passing BarsFigure 4.16: Photograph of the single bar refiner64//________________\ \Figure4.17:Schematicofsinglebarrefinershowingthebars,fiocsupport,andlocationofthestraingauge.I.(/Floc77HoIder7//L4StrainGuagemountedtoShaftStationaryPlate(stator)RotatingPlate(rotor)Table4.5DescriptionofStrainGaugeandSignalConditioner.DescriptionTheoverall torqueactingwasmeasuredusingastraingaugereactiontorquetransducer,type2102—100,whichwasobtainedfromtheEatonCorporationofTroy,Michigan(tel.13136430220).Thetorquesensingelementisa4armbondedWheatstonebridgewitharesistanceof350ohms.Nominaloutputis2.236mVIV,withamaximumallowabletorqueof10Nm.Thetransducerwascalibratedin-situbyapplyingaknowntorquetothestator.StrainGauge0ThevoltageoutputfromthetorquetransducerwasamplifiedusinganAriesAl1100SignalConditionerobtainedfromAriesInstrumentsofPickering,Ontario(tel14168397611).SignalConditionerTheunithasagainwhichisfullyadjustablewitharangefrom10to11000andazerooffsetupto±5mV/V.Theexcitationvoltageisvariableovertherangeof0.25Vto15V.Theoutputis10VDC(fullrange)at20mA.ThedatawasloggedonaIBMATusingLABTECHDataLogger.NOTEBOOK, acommerciallyavailablesoftwarepackage.Chapter 4: The Force on Fibres Between Passing Barsfound to have an approximately spherical shape and therefore their size was determinedby averaging their diameters along two orthogonal axes. Flocs were removed from thesuspension and tested in the refiner apparatus in air rather than water. The floes employedwere from a hemlock balsam mixture of semi-bleached kraft pulp. The length-weightedaverage fibre length was 2.29 mm and the average coarseness was 0.215 mg/mm.The force measurements on individual flocs were carried out by first adjusting the gapbetween the rotor and stator to a size, G, using a feeler gauge. G was typically set between70 and 200 ILm. The rotor bar was positioned approximately 900 ahead of the stator bar andthen rotation started at approximately 1 rpm (tip speed 8 mm/s). As the bars approachedeach other, the floes were compressed and sheared in the gap. Torque (T) was recorded asa function of time at a sampling rate of 100 Hz. From this torque, the shear or workingforce, F, was calculated usingW Rwhere R is the radial distance to the floe.67Chapter 4: The Force on Fibres Between Passing Bars4.4 Results4.4.1 Phenomenological Behaviour The behaviour of a typical floe during bar passageis shown in Figure 4.18. In this example, a 5 mm diameter floe at 3% consistency (Cm=O.03)is compressed and sheared in a gap spacing of approximately 0.07 mm. Clearly, as the barsapproached (Figure 4.18(a)), the floe was pinched between the leading edges of the rotor andstator. During the bar crossing, compression and dewatering of the floe occurred. When thebar surfaces were superimposed, most of the floe was trapped within the gap, but a smallfraction of it remained draped over the leading edge of the bar, as shown in Figure 4.18(b).The floe appeared “pinched” between the sharp corner of the rotor bar and the surface of thestator. This behaviour is illustrated schematically in Figure 4.19.These observations agree with earlier work by others using high speed cinematography inwhich floes were found to behave in one of two ways during bar crossing: either theyremained intact while being compressed and sheared or they were pulled apart. Both typesof behaviour were observed in this work. Floes were found to compress and remain intact atlarger gap sizes, but ruptured into two parts at very small gap sizes, as illustrated schematicallyin Figure 4.20. The latter phenomenon may well be that which causes “pad collapse” andconsequent plate clash in refiners.4.4.2 The Effect of Gap Size on Forces on Fibres The effect of gap size on theworking force on floes, F, is strongly determined by the mode of behaviour as describedabove. It is also determined by the floe size, L0, and consistency, Cm. The effects of thesevariables can be shown by normalizing the gap size with floe diameter (G/L0)and comparingthe results at different consistencies, radius of curvature, and surface roughness of the bar.68Chapter 4: The Force on Fibres Between Passing BarsFigure 4.18: Phenomenological behaviour of a floe between passing bars of the single bar refiner.Figure 4.19: Schematic of the phenomenological behaviour of a floc during bar passage.69Chapter 4: The Force on Fibres Between Passing BarspFigure 4.20: Schematic of the observed floe behaviour between passing bars.The effect of consistency: The findings for the consistency range 6—7% are shown inFigure 4.21. It is apparent that, within an upper and lower limit of GILO, F increaseswith decreasing GIL0 in an approximately linear manner.Above the upper limit of y(O)/L0 0.07 — 0.08, F is zero. This is consistent with thetheory, i.e. equation 4.18, which states that the floe has to be compressed to the ratioY0) Cm (4.55)before it can support any significant normal force.70Chapter 4: The Force on Fibres Between Passing Bars15.0 :12.5 :.• • Theory1floc ruptureoo0 0.025 0.050 0.075G/L(dimensionless)Figure 4.21: The effect of gap size on peak shear (Cm=6—7%, p=60—7Okg/rn3).The dotted line shows the abrupt decrease of working force at low GIL0.15.0zU)2LE1O.07.5TheoryOOJ 0.025 0.050 0.075GIL,(dimensionless)Figure 4.22: The effect of lower consistency on peak shear (Cm34%, Po607°kg/rn3). Thedata extrapolate to a level of GIL, approximately equal to their consistencies at zero force.71Chapter 4: The Force on Fibres Between Passing BarsBelow the lower limit, in this case approximately at G/LQ 0.0 15, F3 decreases abruptly,as shown by the dotted line in Figure 4.21. This abrupt decrease corresponds to thetransition in fioc behaviour from compression and shear to the rupture described aboveand occurs at a gap size equal to Y’d/Y’c as defined in Equation 4.23. The absolute valueof the measurements are estimated to vary within 10% as shown on the graphs. For fiocswhich remained intact during a bar crossing the data strongly suggests (p=O.OOl) that theworking force varies linearly with G/L0, i.e. the slope is statistically significant. However,only two-thirds (r2=0.67) of this variation can be accounted for by the ratio G/L0. Allexperimental points were correlated with visual observation.At a lower concentration of approximately 3—4%, a similar linear behaviour is apparent(Figure 4.22). However, in this case, the sudden decrease at the lower limit was not found,for reasons that are not evident. It is speculated that the rupture point would occur at a verysmall gap size, which is experimentally unattainable in this apparatus. The upper limit wasfound by extrapolation to occur at approximately G/LO 0.035.At both fioc consistencies tested the upper limit occurred when GIL0 approximately equalledthe consistency of the uncompressed fioc. This signifies that compression to near zero voidvolume took place. This level of interfibre contact appears necessary to support significantforce on passing bars. In contrast the lower limit occurs when the bar gap is sufficientlysmall to shear the floe into two parts.Theoretical predictions are also shown in Figures 4.21 and 4.22. The predictions were madeusing Equation (4.46) using assumed values for the parameters required in the equation (seeTable 4.6). The peak values were obtained by searching the equation numerically for variousvalues of GILO. At both consistencies, the dependence of peak working force on GIL0 was72Chapter 4: The Force on Fibres Between Passing BarsTable 4.6 Typical Values for the Parameters used in Equation (4.46).Radius of curvature of the lOx 10-6 see Appendix Afibre Rf (m)Radius of curvature of the bar Rf AssumedRb (m)Modulus of elasticity of the 10 xl Assumed (equal to the moduluscell wall (Pa) of elasticity of a fibre in uniaxialtension - Jayne, 1959)Poisson’s ratio ii 0.25 AssumedModulus of Elasticity of the 2x 106 see Appendix Afloe E (Pa)Parameter 1857 see Appendix A(kg/rn3)Size of the floe L0 (m) 2.2x rn-3 Assumed (equal to the averagefibre length)Bulk density of the floe Po 1000Cm Applicable to low consistency(kg/rn3) floes onlyCoefficient of friction 0.11 Goncharov (1971)similar to the dependence predicted from the theoretical analysis. In both cases the slopeof the theoretical line was lower in value than the slope of the experimental points. Thediscrepancy between the two most likely results from the simplicity of the model and thelack of good literature values for the model parameters.The Effect of Radius of Curvature: The effect of varying radius of curvature of the bar onshear is shown in Figure 4.23 for a “sharp” bar (Rb=O) and a “dull” bar (Rb=0.39 mm). Eachstudy was perfonned with floes between 3—4% consistency. One observes that sharp barsgenerally impose twice as much force on a floe than dull bars. The measured working forceincreased essentially linearly with decreasing GIL0 and was similar in shape and magnitudeto the theoretical predictions..73Chapter 4: The Force on Fibres Between Passing Barsza,0U0)0a,0Figure 4.23: The effect of radius of curvature of thebar on the measured and predicted peak working force.The Effect of Surface Roughness: The effect of surface roughness of the bar on the peakworking force is shown in Figure 4.24. Two different bar materials were tested: a 316 SSbar with a surface roughness of 0.577 (±0.019) m and a brass bar with a surface roughnessof 0.484 (±0.082) jim. Surface roughness was measured using a Talisurf. Clearly, thereare no significant differences in the peak working force measured for the roughness rangeexperimentally tested.4.4.3 Working Force Distribution During Bar Crossing In addition to the peakforces described above, we measured the force distribution as a function of time as thebars passed over one another. A typical force distribution from shearing a floe is shown inFigure 4.25. As the bars crossed one another, the force increased to a maximum at a timeof 0.35 s, which corresponded to a distance of approximately 3—5 mm. The working forcethen decreased gradually. The shape of this distribution appears to be similar to the normal0 0.025 0.050 0.075G/L(dimensionIess)74Chapter 4: The Force on Fibres Between Passing Bars15• L577 (±0.019) j.tm2b00)o5 V7%7V400.484 (±0.082) j.tm0 0.025 0.050 0.075G/Lb(dimensionless)Figure 4.24: The effect of bar surface roughness on the peak working force.force distribution curve reported by Goncharov (1971), although in some cases Goncharovreported a second peak which was not observed here.The theoretical predictions are also shown in this Figure. Clearly, the theory predicts theinitial shape but underestimates both the magnitude and location of the peak. In this casethe peak working force was experimentally measured to occur at 0.35 s with a magnitudeof 15 N. The theoretical equations predict the peak at 0.25 s with a magnitude of 8 N. Asstated earlier this discrepancy most likely results from the simplicity of the model and lackof good literature values for the model parameters. It should be noted that the theory can be“curve-fit” to the experimental data using non-linear parameter estimation algorithms.Energy Expended on Fibres: The area under the experimental curve in Figure 4.25 gives theenergy expended in shearing the fioc through the single bar crossing. The energy/impact/fibre,I, may be determined by dividing this energy among the total number of fibres in the fioc,75Chapter 4: The Force on Fibres Between Passing Barszci)00LLc,)C0Figure 4.25: Typical working force distribution along the width of the bar surface.Nf (see Equation 4.9),1 fFwubdt (4.56)where F represents working force, Ub, bar velocity, and t, time. For the distribution shown,I was found to be about iO J/impactlfibre. This agrees reasonably well with values ofI of iO to 10-6 J/impactJfibre calculated for mill refiners using the C-Factor (Kerekes eta!., 1993).150 0.25 0.50 0.75 1.00Time (sec)76Chapter 4: The Force on Fibres Between Passing Bars45 DiscussionThe Energy Expended on Individual Fibres: Having determined the force distribution andenergy expenditure on individual fiocs during a bar crossing, we may now estimate the samefor individual fibres. To do so, we must make some assumptions.If we assume that forces during bar passage are distributed evenly among fibres, we obtainfrom the peak force (Fe) in Figure 4.25 a value of i03 N/fibre. Assuming this producesstrain in uniaxial tension, the peak stress is estimated by dividing the force by the cross-sectional area of the cell wall material (p-). Hence, the maximum strain induced in thefibre , , is given bye= (4.57)wEand estimated to be about 0.1%, assuming a modulus of elasticity of 10 GPa (Jayne, 1959),a fibre coarsness of O.2x 10 kg/rn (measured experimentally, see Appendix A), and a cellwall density of 1500 kg/rn3 (Panshin and DeZeeuw, 1980; Stone et al., 1966). This value ofc may be compared to estimated values of E in refining. Leider and Nissan (1977) estimatethat tensile fibre strain just into the plastic range was required for refining and that this wasapproximately 3%. This value was also the approximate level of strain required for fibrerupture and was used by Kerekes et a!. (1992) to estimate an upper limit of i in refining.The energies associated with the strains may be estimated from the approximate area underthe stress-strain curve (Ec2V) where V is the volume of the fibre (l f-). For E0.1%,I 10—8 .1/fibre, and for e=3%, I i0 J/fibre.The above observations give insight into the energy expenditure in refining. Although thelevels of I measured in the study are of the same order of magnitude as those estimated by77Chapter 4: The Force on Fibres Between Passing BarsLeider and Nissan (1977) and those predicted by mill refiners by Kerekes et a!. (1992) usingthe C-Factor, the force associated with these energies are far less than the values consideredby the previous workers. They are levels of force associated with tensile strain of 0.1%as opposed to 3%. The energies associated with these smaller strains are three orders ofmagnitude less than the rupture energy of a fibre. This suggests that the energy in a refiningstrain cycle is expended on other modes of strain (shear, compression) and quite probably,on friction that does not contribute to fibre strain and is therefore an energy loss.4.6 Summary and ConclusionsThe general theory presented outlines the working force distribution along the width ofthe bar. The model equations are based upon three assumptions: a fioc is spherical inshape with fibre-centers arranged in an orderly lattice along any cross-section; a fioc cannotsupport significant load until the fibre-centres are in intimate contact; and that the effectof the bar-edge is analogous to a classic contact stress problem. Albeit ideal, the theorypredicts outcomes dictated by logic. The utility of the model was tested by comparison toexperimental data. It was found that:1) Measureable forces could be imposed on fibre floes only over a range of gap size. Theupper limit of this range corresponded to floc compression to near zero void volume, whereasthe lower limit occurred at a gap size that ruptured the floes into two segments.2) The average energy/fibre expended in a bar crossing was found to be of the same orderof magnitude as those calculated for pulp refiners by the C-Factor, i.e. i0 to 10-6 J/fibre.3) Assuming all forces to be distributed evenly among fibres, the peak force associated withthis energy expenditure gave approximately 0.1% strain in uniaxial tension.78Chapter 4: The Force on Fibres Between Passing BarsIn summary, the results of this work confirm the approximate order of magnitude ofenergy/impact on fibres measured by the C-Factor. However, the results also suggest thatonly a small fraction of this energy is expended in tensile strain of the fibre. The remainderis assumed to be expended on other forms of strain and probably a great part is lost infrictional dissipation.It was further shown that the working force distribution displayed a steep increase over thefirst 2—3 mm as the bars crossed and then decreased slowly over the remaining width of thebar. The results were similar in shape and magnitude to the experimental data of Goncharov(1971). Further, peak working force increased with fioc consistency and decreased withradius of curvature of the bar. The dependence of peak working force on GIL0 was similarto the dependence predicted from the theoretical analysis. The peak working force exertedon the floc was found to be less than 20 N for the experimental conditions tested. For thisforce acting over the distance of a typical refiner bar, the intensity of impact was found tobe about the same level as estimated by the C-Factor analysis.79Chapter 5: Summary and ConclusionsChapter 5Summary and ConclusionsRefining is a process in which the physical structure of the papermaking fibre is modifiedby repeated strain. While there is extensive literature on the changes in fibre morphologyproduced by this strain, there is little on how the strain is imposed, that is, on the refiningprocess itself. Of particular importance are the forces that impose the strain. With this inmind, the objective of this dissertation was to identify and estimate the magnitude of theforces, both hydrodynamic and mechanical, acting on papermaking fibres.Analysis of the Hydrodynamic Forces: The analysis of the hydrodynamic force (3) considered a number of force and moment balances acting on a single fibre in contact with arefiner bar. The fibre was allowed to translate, rotate and bend.To classify fibre bending at a bar edge, a non-dimensional number, derived by nondimensionalizing the force balance governing large amplitude bending, was employed. Thisnumber, defined here as the “Bending number”, Be, was found to depend upon water viscosity, JL, fibre length, L, fibre diameter, d, a characteristic flow velocity, U, and fibre stiffness,El. It is given byB— 27ruUL351e— ln(L/D)EIThis number was used as a quantitative criterion for classifying fibres as “stiff’ or “flexible”.The likelihood of fibre trapping was found to be a function of parameters such as the velocityratio, Be, and the coefficient of friction,.It was found that the probability of trapping80Chapter 5: Summary and Conclusionsincreased with fibre flexibility and coefficient of friction. The energy expended on the fibreduring trapping was estimated to vary within the range 10.12 J to 10.11 JAnalysis of the Mechanical Forces: A simple analysis of the force applied to a single fibrein a floe was presented. It was found that the fibre was subjected to three types of force:normal force — through floe compression; shear force through interfibre friction and contactwith the bar surface; and a corner force induced by the sharp corner of the bar digging intothe softer fibre. Simple expressions were derived to quantitatively estimate each type offorce. It was found that both shear and normal force were dependent on the normal force.The expression was integrated over the entire volume of the floe and tested in a single barrefiner built for this purpose. It was found that the ratio of gap size to floe size (GIL0)had to be less than a critical level for any measurable force to be exerted on the fibre. AsG/L4,, decreased the measured force increased approximately linearly. Peak working forcewas found to increase with floe consistency and decrease with radius of curvature of the bar.The dependence of peak working force on GIL0 was similar to the dependence predictedfrom the theoretical analysis. The energy dissipation estimated was approximately the levelestimated using the C-Factor analysis, i.e. i0 J.The energies associated with the mechanical forces were approximately 6—7 orders ofmagnitude greater than those associated with hydrodynamic forces. This suggests that thelatter contribute little to the refining action in pulp.Although the theory and experiments in this study represent much simpler cases than existsin a commercial refiner, these results are a useful first step in identifying the type of forcesthat exist in refiners and quantifying their order of magnitude.81Chapter 6: RecommendationsChapter 6Recommendations1. Hydrodynamic Force: Extend the work to cover the case of trapping of floes. Thework can then be used directly to extend “C-Factor Theory” to define scientifically aprobability of trapping.2. Mechanical Force: More experimental work is in order. To begin with, a new testrig should be devised and built so that run-out and out-of-tram errors are eliminated.The apparatus should be instrumented further so that position, both separation distancebetween the bars and degree of superposition, are easily determined. The jig shouldbe sealed so that testing could be performed in a water medium. A more extensiveexperimental programme should be undertaken in which variables such as bar profile,surface roughness, and pulp fibre species are tested.82Chapter 7: NomenclatureChapter 7NomenclatureChapter 2C C-Factor (s’)E Energy expended on pulp (J)F Mass flow of pulp through refiner (kgs”)Gd Groove depth (m)G Groove width (m)I Intensity of Impact (1)1 Average fibre length (m)Lr Length of refining zone (m)n Bar number density (m1)r Number of bars of the rotorn Number of bars on the statorN Number of impacts per unit mass(kg’)P Net power applied (Jr’)R1 Inner radius of refining zone (m)R2 Outer radius of refining zone (m)Xr Length of rotor bars (m)X Length of stator bars (m)Pw Density of water (kgm3)q Average bar angle (rad.)w Fibre Coarseness (kgni’)83Chapter 7: NomenclatureWr Rotational velocity (s1)Chapter 3A Cross-sectional area of the fibre (m2)Be Bending number EIli(L/D)D Nominal fibre diameter (m)B Modulus of elasticity of the fibre (Pa)f Hydrodynamic force per unit length acting on the fibre (Nm1)fi Hydrodynamic force per unit length acting on the fibre in the x-direction (Nm’)f2 Hydrodynamic force per unit length acting on the fibre in the y-direction (Nm’)fmax Maximum frictional force between the fibre and the bar (N)F Total hydrodynamic force acting on the fibre (N)G Torque (Nm)I Cross sectional moment of inertia (m4)I Indentity matrix‘b Energy expended by bending the fibre (J)L Fibre length (m)M Moment (Nm)n Unit vector in the normal directionP Hydrodynamic pressure fieldr Position vector of the fibre (m)R(s) The value of r at the point on the centreline under consideration (m)s Distance along the length of the fibre (m)s’ Distance along the length of the fibre in contact with the bar edge (m)84Chapter 7: Nomenclaturet Unit vector in the tangential directionT Tensile force (N)U Hydrodynamic velocity component in the x-direction (m sU Velocity of the fibre in the x-direction (m s1)U Hydrodynamic flow field (m s1)V Hydrodynamic velocity component in the y-direction (m s-I:’V* Velocity of the fibre in the x-direction (m s1)V Shear force (N)x The x-direction in a Cartesian coordinate systemy The y-direction in a Cartesian coordinate systemKronecker deltae Arbitrary constant0 Angle of incidence between fibre and fluid (rad.)i Ratio of fibre length to fibre diameter)(s) A dimensionless function which describes the variance of fibre diameter as a functionof st Viscosity of water (Pa s)a Tensile stress (Pa)Chapter 4a Corner force coefficient ()a Shear force coefficient (18T + 1))A Area of the floc in compression (m2)A Projected area of contact between the fibre and the corner of the bar (m2)85Chapter 7: NomenclatureA Actual area of contact between the fibre and the corner of the bar (m2)Cm Floe consistency (fraction)D Fibre diameter (m)D Inner diameter of the fibre (m)E Bulk modulus of elasticity of the floe (Pa)Modulus of elasticity of the cell wall materialF Corner force (N)FdefoDefOflflation force (N)h Holding force (N)Ff Working force applied to a floe (N)FfSurface frcition between fibre and the corner of the bar (N)F Working force (N)G Gap size (m)L0 Nominal diameter of a floe (m)L Chord-length of the floe in the x-y plane (m)L’ Thickness of the compressed floe in the x-y plane (m)L Chord-length of the floe in the x-y plane (m)m Geometrical constantn Geometrical constantflf Number of fibre in a cross-section of the floeflf r Number of fibres in a row11r Number of rowsN Normal force (N)Nf Totoal number of fibres in a floe86Chapter 7: NomenclatureP Maximum contact pressure (Pa)R Moment arm (m)Rf Radius of curvature of the fibre (m)Rb Radius of curvature of the bar (m)T Tensile force (N)Ub Linear velocity of the bar (ms’)x Arbitrary limit of integration where—<x < 4. (m)x0 Arbitrary limit of integration where — <x < 412. (m)Y’c Critical distance for floe to support significant normal load (M)?d Critical distance when all fibres in floc have collapsed upon their lumen (m)Ratio of compressed diameter to uncompressed diameter of the floe6 Depth of penetration of bar into the cell wall of the fibre (m)Arbitrary constant defined by Equation 4.41I-’d Coefficient of friction between two fibresCoefficient of friction between fibre and barv Poisson’s ratioArbitrary constant defined by Equation 4.28Pcw Density of the cell wall material (kgm3)Po Bulk density of the floe (kgm3)T Arbitrary constant defined by Equation 4.42b Arbitrary constant defined by Equation 4.27w Fibre coarseness (kgm1)87Chapter 7: NomenclatureChapter 52ir4uUL3Be Bending number Elm (LID)G Gap size (m)L0 Nominal diameter of the floe (m)88Chapter 8: ReferencesChapter 8References1. Alaskevich, Yu D. (1971). Study of Pressure in the liquid layer between knives of abeater. Bum. Promst. 46(10) 16—17 (Engi. Trans.)2. Atack, D., May, D. (1963). Mechanical reduction of chips by double disc refining. PulpPap. Can 64[C]:T75—T833. Atack, D. (1977). Advances in beating and refining. In: (F. Bolam ed.), Trans. ofFundamental Research Symposium, pp. 26 1—97, Tech. Sect. BP+BMA Oxford4. Atack, D. (1980). Towards a theory of refiner mechanical pulping Appita J.34[3]:223—2275. Atack, D., Stationwala, M.I., Karnis, A. (1984). What happens in refining. Pulp Pap.Can. 85[12]:T303—.T3086. Atack, D., Stationwala, M.I., Fontebasso, J., Huusari, B., Perkola, M., Ahlqvist, P.(1989). High speed photography of pulp flow patterns in a 5 MW pressurized refiner,International Mechanical Pulping Conference,pp. 280—293, Helsinki.7. Attalla, W. (1980). On the energy requirement in refining Tappi J. 63(6): 121—1228. Banks, W.A. (1967). Design considerations and engineering characteristics of discrefiners. Pap. Technol. 8(4):363—3699. Blechschmidt, J., Naujock, H.T. (1986). New findings in the beating of pulp fibre andin particular of waste paper. Eucepa Conference Proceedings— Development trends inthe science and technology ofpulp and paper making, pp. 14—22, Florence.10. Bowden, F.P., Tabor, D. (1950). The Friction and Lubrication of Solids. ClaredonPress, Oxford.89Chapter 8: References11. Brecht, W. (1967). A method for the comparative evaluation of bar equipped beatingdevices. Tappi .1. 50[8]:40A-44A12. Clark, J d’A. (1980). Pulp Technology and Treatment for Paper. Miller Freeman,San Francisco.13. Clift, R., Grace, J.R., Weber, M.E. (1978). Bubbles, Drops, and Particles. pp 71—73,Academic Press, Inc. San Diego14. Cox, R.G. (1970). The Motion of Long Slender Bodies in a Viscous Fluid. Part 1.General Theory. J. Fluid Mech. 44:791—81015. Danforth, D.W. (1969). Theory/Practice. Southern Pulp Pap. Manuf 32(7):52—5316. Ebeling, K. (1980). A critical review of current theories for refining chemical pulps.IPC Conference on Refining, pp. 1—33, Appleton, WI.17. Espenmiller, H.P. (1969). The theory and practice of refining. Southern Pulp Pap.Manuf 32(4):50—57.18. Fahey, M.D. (1970). Mechanical treatment of chemical pulps. Tappi J.53[11]:2050—206419. Fox, T.S., Brodkey, R.S., Nissan, A.H. (1979). High speed photography of stocktransport in a disc refiner. Tappi J. 62[3]:55—5820. Frazier, W.C. (1988). Applying hydrodynamic lubrication theory to predict refinerbehavior. J. Pulp Pap. Sci. 14[1]:1—5.21. Goncharov, V.N. (1971). Force factors in a disc refiner and their effect on the beatingprocess. Bum. Promst. 5:12-1422. Giertz, H.W. (1980). The influence of beating on individual fibres and the causal effectson paper properties. IPC Conference on Refining, pp. 87—92, Appleton, WI.23. Halme, M. (1964). Use of formulas in the study of beating equipment. Pap. Trade J.90Chapter 8: References140[45]:32—35.24. Halme, M., Syrjanen, A. (1964). Flow of stock in a conical refiner as observed by highspeed film camera. Atti. Cong. Europ. Tech. Carteuria, pp. 273—277, Venice.25. Heitanen, S., Ebling, K. (1990). A New hypothesis for the mechanism of refining. Pap.Puu 72(2):l72—179.26. Herbeit, W., Marsh, P.G. (1968). Mechanics and fluid mechanics of a disc refiner.Tappi J. 51[5]:235—239.27. Higgins, H.G., de Yong J. (1962). The beating process: Primary Effects and TheirInfluence on Pulp and Paper Properties, In: (F. Bolam ed.), Trans. of FundamentalResearch Symposium, pp. 651—691, Tech. Sect. BP+BMA Oxford28. Jacquelin, G. (1966). Papermaking fibrous networks in an aqueous mediumA.T.I.P Rev.20[4]: 153—16329. Jacquelin, G. (1968). Surface properties of fibres and cohesion of fibrous networks.A.T.I.P Rev. 22[2]:129—13430. Jayaweera, K.O.L.F., Mason, B.J. (1965). The behaviour of freely falling cylinders andcones in a viscous fluid. J. Fluid Mech. 22[4]:709—72031. Jayne, B.A. (1959). Mechanical properties of wood fibres. Tappi .1. 42(6):461—46732. Kerekes, R.J. (1990). Characterization of pulp refiners by a C-Factor. Nor. Pulp Pap.Res. J. 5(1):3—833. Kerekes, R.J., Clara, M., Dharni, S., Martinez, M. (1993). Application of the C-Factorto characterize pulp refiners. .1. Pulp Pap. Sci. 19[3]:125—130.34. Khayat, R.E., Cox, R.G. (1989). Inertia effects on the motion of long slender bodies.J. Fluid Mech. 20:435—46235. Kline, R.E. (1976). A practical approach to refining. Pap. Trade J. 12:41-46.91Chapter 8: References36. Kumar, A. (1991). Passage of fibres through screen apertures. Ph.D. Thesis, Dept. ofChem. Eng., University of British Columbia.37. Leask, R.A. (1981). The theory of chip refining — A status report. Svensk Papperstidn.84[14]:28—35.38. Leider, P.J., Nissan, A.H., (1977). Understanding disc refiners-The mechanical treatmentof fibres. Tappi J. 60[10]:85—8939. Leider, P.J., Rihs, J. (1977). Understanding the disc refiner. Tappi .1. 60[9]:98—10240. Levlin, J.E. (1984). Oppurtunities for energy savings in L-C refining, French-EnglishSeminar on Refining, Grenoble.41. Levlin, J.E., Jousunaa, T. (1988). New puips require new refining techniques. PTI10:304—31242. Lewis, J., Danforth, D.W. (1962). Stock preparation analysis. Tappi J. 45[3]:185—18843. McKenzie, A.W., Prosser, N.A. (1981). The beating action of a PFI mill. AppitaJ.34[4]:293—297.44. Miles, K.B., May, W.D. (1989). The flow of Pulp in chip refiners. CPPA AnnualMeeting. Jan. 1989. Montreal pg A177—A18945. Newman, B. (1992). Personal communications, Dept. of Mech. Eng., McGill University.46. Nordman, L., Levlin, J.E., Makkonen, T., Jokisalo, H. (1981). Conditions in a L-Crefiner as observed by physical measurements. Pap. Puu 173[4]:171—178.47. Page, D.H., Ksky, J., Booth, D. (1962). Some initial observations on the action of thebeater. What are we doing? BP+BMA Bulletin 28:1—7.48. Page, D.H. (1989). The beating of chemical puips-The action and effects. In: (F. Bolamed.), Trans. of Fundamental Research Symposium, pp. 1—38, Tech. Sect. BP+BMAOxford.92Chapter 8: References49. Panshin, A.J., deZeeuw, C. (1980). Textbook of Wood Technology 4th Ed., McGraw-Hill, New York50. Pashinskii, V.F., Takhtuev, B.G. (1982). Method for calculating beating systems ofpulp. Bum. Promst. 11:26 (Engi. Trans.)51. Pearson, A.J., Trout, G.J., Sibly, P.R., Tyler, A.G. (1978). Refiner pulp from P. Radiata.Pap.Puu 4a:241—254.52. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery (1992). Numerical Reciepiesin C: The art of scientific computing. (2nd ed.) Cambridge University Press. (pg71 1)Pruppacher, H.R., Le Clair, B.P., Hamielec, A.E. (1970) Some relations betweendrag and flow pattern of viscous flow past a sphere and a cylinder at low and intermediateReynolds numbers. J. Fluid Mech 44[4]:781—79053. Smook, G.A. (1990). Handbook of Pulp and Paper Technology. Angus WildePublications, Vancouver54. Steenberg, B.K. (1951) BP+BMA32 388—39555. Steenberg, B.K. (1980). A Model of refining as a special case of milling. IPC Conferenceon Refining, pp. 107—120, Appleton, WI.56. Stone, J.E., Scaflan, A.M, Aberson G.M.A. (1966). The wall density of native cellulosefibres. Pulp Pap. Can. 67[5]:T263—26757. Tam Doo, P.A., Kerekes, R.J. (1981). A method to measure wet fibre flexibility. TappiJ. 64[3]:113—11658. Tappi Stock Preparation Committee (1971). An Introduction to Refining Variables. TappiJ. 54[1O]:1738—1741.59. Timoshenko, S.P., Goodier, J.N. (1951). Theory of Elasticity (2nd ed.) pp. 372—384,McGraw Hill, New York93Chapter 8: References60. Van Den Akker, J.A. (1958). Fundamentals of papermaking fibres. In: (F. Bolam ed.),Trans. of Fundamental Research Symposium, pp. 435—446, Tech. Sect. BPi-BMACambridge.61. Van Stiphout, J.M.J. (1964). A Preliminary study of the refining action on cellulosefibres. Tappi J. 47[2]:189A-191A.94Appendix A Estimating the NormalForce of Fibre FlocsA small experiment was conducted to confirm the validity of the equation predicting thenormal force on fibre floes. The relevant equations, Equations (4.28)— (4.30), are reproducedhere for convienience.N = E1 [ — — pD2L _xo)] (A.1)Ji — (2)2+ sin()= 2xo,/i — (2)2 (-)The utility of this equation is demonstrated by comparison to experimental data. In this workindividual Jacquelin floes were prepared using the technique described in Section 4.3.2 andcompressed in air in an strength testing apparatus. These were compressed from size L0 toa smaller size G uniaxially at a rate of approxiamtely 1 mm/mm. For this case, the entirefloe is under compression, i.e. x0—-L12 and x=L0/2, henceN—E°1l 167 w A4— 4 L 2 p0D2L ( . )Peak force was measured by a 100 N load cell. The results are shown in Figure A.2695c..JEz.—cPFigure A.26: The normal pressure on Jacquelin flocs in uniaxial compressionThe first observation that can be made is that the ratio of gap size to floe size (GIL0)had tobe less than about 0.075 for any force to be measured. As G/LO decreased below this levelthe normal force increased with approximately linearly7GIL0. This observation is consistentwith equation A.4Second, the intercept, predicted by equation A.4, was found to be equal toG__— =Po= 0.075 (A.5)L0 167wfor the experimental conditions tested. For floes with a bulk density of Po=77 kg/rn3, thisGILD (dimensionless)The data stmsigly suggests (p=0.001) that the normal force varies linearly with GIL0. However only approximately two-thirds(12=0.67) of the variation can be accounted for by the ratio of GIL096implies= 633 (A.6)Further, both y and w were measured independently and found to be 1.1 and O.215x106kg/rn respectively. This implies that the diameter of the fibre equals 20x106 m; this valueis within the expected range of typical fibre diameters.Finally, the modulus of elasticity E was estimated from the slope of the line and was foundto be approxiamtely 2 MPa.97

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