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Measurement of heterogeneity in low consistency pulp refining by comminution modeling Heymer, Jens Olaf 2009

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Measurement of Heterogeneity in Low Consistency Pulp Refining by Comminution Modeling  by Jens Olaf Heymer Dipl.-Ing. (TU), The Dresden University of Technology, 2002 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy  in  The Faculty of Graduate Studies (Chemical and Biological Engineering) The University of British Columbia (Vancouver) June, 2009 © Jens Olaf Heymer 2009  Abstract Pulp refining is a key operation in papermaking aimed at improving fibre quality for paper. In pulp refiners, pulp in suspension flows through an annulus between a rotor and stator having surfaces with bars which impose cyclic loading on the pulp during bar crossings. This process has long been known to be heterogeneous in its treatment of pulp. However, there is no method available to measure heterogeneity of treatment in operating refiners. The objective of this thesis was to develop such a method. The approach was based on applying comminution theory and measurements of fibre length before and after refining. Specifically, measurements of fibre shortening were used to obtain two key parameters in the comminution differential equation: the selection function and the breakage function. A procedure was employed to account for breakage at any point along a fibre length instead of an arbitrary point as in previous studies. The selection function was expressed as the product of two factors: a probability of impact and a probability that intensity of an impacted fibre exceeded the rupture intensity. These probabilities were expressed in terms of refiner variables, with an increment of energy being a single bar crossing. Values for the selection and breakage functions were obtained by regression fitting to fibre length data. These length distribution data were obtained for a range of operating conditions in a pilot plant disc refiner, an industrial conical refiner, and a laboratory Escher Wyss refiner. Findings for predicted numbers of impacts and intensities were consistent with the findings of previous studies.  The probabilities derived from the selection and breakage functions gave distributions of numbers and intensities of impacts. The findings showed, for example, that homogeneity of treatment is more dependent on numbers of impacts than on uniformity of intensity ii  during impacts. A single number “homogeneity index” was proposed and defined as the product of the number of impacts on fibres having length-weighted average and the probability of intensity being within +/- 2kJ/kg of the mean divided by the specific energy. Using this definition, it was shown that homogeneity of refining treatment increased with flow rate through the refiner and decreased with increasing groove depth and consistency. It also showed the degree to which increased residence time of fibres in a refiner could compensate for a coarse bar pattern in attaining homogeneity of treatment.  iii  Table of Contents Abstract ............................................................................................................................... ii List of Tables ................................................................................................................... viii List of Figures .................................................................................................................... ix Nomenclature ..................................................................................................................... xi Acknowledgments............................................................................................................. xv 1 Introduction................................................................................................................... 1 2 Review of Literature ..................................................................................................... 3 2.1 Effect on Pulp........................................................................................................... 3 2.1.1 Internal Fibrillation ..................................................................................... 3 2.1.2 External Fibrillation.................................................................................... 4 2.1.3 Fibre Shortening ......................................................................................... 4 2.1.4 Fines Generation ......................................................................................... 4 2.1.5 Fibre Straightening ..................................................................................... 5 2.1.6 Heterogeneity.............................................................................................. 5 2.2 Mechanics of Refining ............................................................................................. 5 2.2.1 Fibrage Theory............................................................................................ 6 2.2.2 Refining as Lubrication Process (Hydrodynamics) .................................... 6 2.2.3 Transport Phenomena in Refiners .............................................................. 7 2.2.4 "Treatment of Flocs" Hypothesis................................................................ 8 2.2.5 Refining as a Fatigue Process ..................................................................... 9 2.2.6 Forces on Fibres during Bar Crossing Events .......................................... 10 2.3 Quantitative Characterization of the Refining Mechanism.................................... 11 iv  2.3.1 Specific Energy......................................................................................... 11 2.3.2 Refining Intensity ..................................................................................... 11 2.3.3 Fibre Intensity........................................................................................... 13 2.4 Heterogeneity ......................................................................................................... 15 2.5 Review of Refining Action as a Comminution Process ......................................... 17 3 Objective of Thesis...................................................................................................... 20 4 Analysis ........................................................................................................................ 21 4.1 Interpretation of Comminution Equation ............................................................... 22 4.2 Link to Refiner Mechanics ..................................................................................... 23 4.2.1 Intensity .................................................................................................... 23 4.2.2 Number of Impacts ................................................................................... 23 4.3 Fibre Capture Factor............................................................................................... 25 4.4 Probability Distribution of Intensity ...................................................................... 25 4.5 Selection Function Expressed in Terms of Refiner Variables ............................... 27 4.6 Relationship between S and B ................................................................................ 28 4.7 Curve -Fitting Procedure........................................................................................ 29 5 Experimental Program............................................................................................... 30 5.1 Validation of Comminution Approach................................................................... 31 5.2 Pilot-scale Refining Trials...................................................................................... 33 5.3 Escher-Wyss Heterogeneity Trial .......................................................................... 35 5.4 Conflo Refining Trial ............................................................................................. 35 6 Results and Discussion................................................................................................ 37 6.1 Validation of Comminution Approach................................................................... 37 6.2 Selection and Breakage Functions ......................................................................... 40 v  6.2.1 Selection Function .................................................................................... 40 6.2.2 Breakage Function .................................................................................... 43 6.3 Determining Parameters Affecting Selection Function ......................................... 44 6.3.1 Determining f(li) ....................................................................................... 45 6.3.2 Determination of K and mT....................................................................... 45 6.4 Discussion of Predicted Probabilities and Intensities ............................................ 46 6.4.1 Probability of Impact ................................................................................ 46 6.4.2 Intensity of Impacts .................................................................................. 48 6.5 Comparisons to Previous Studies........................................................................... 50 6.5.1 Intensities .................................................................................................. 50 6.5.2 Probabilities of Impacts in Compression Refining ................................... 51 6.6 Homogeneity Index ................................................................................................ 53 7 Summary and Conclusion .......................................................................................... 57 8 Recommendations ....................................................................................................... 59 References ........................................................................................................................ 61 Appendices....................................................................................................................... 71 Appendix A: Distributions of Intensity in Refiner Gaps .............................................. 71 Appendix B: Integration of Pressure Distribution ........................................................ 73 Appendix C: Fibre Strength Distributions .................................................................... 75 C.1 Average Rupture Intensity IC ....................................................................... 76 C.2 Distribution of Rupture Intensity ................................................................. 78 Appendix D: Breakage and Selection Functions .......................................................... 80 Appendix E: Validation of Transition Matrix Approach .............................................. 85 E.1 Assessment of the Transition Matrix ........................................................... 85 vi  E.1 Are B and S Matrices unique?...................................................................... 87 Appendix F: Validation of Approach by Tracer Experiment ....................................... 89 Appendix G: Form of Capture Function f(li) ................................................................ 91  vii  List of Tables Table 1: Sprout-Waldron Plate Dimensions ..................................................................... 32 Table 2: Sprout-Waldron Angle Analysis......................................................................... 32 Table 3: Summary of experimental trials from Olson et al./[59]. ...................................... 34 Table 4: Summary of refiner plate geometries. ................................................................ 34 Table 5: Escher-Wyss trial conditions .............................................................................. 35 Table 6: Conflo JC-00 refiner plate geometry. ................................................................. 36 Table 7: Conflo JC-00 trial conditions.............................................................................. 36 Table 8: Comparison of obtained values for K, mT , IA and NA for a representative specific energy input of 120 kWh/t (all are converted into SI-units) ............................ 46 Table 9: Escher-Wyss trails including calculated number of bar crossings, NB and breaking length, BL .......................................................................................... 52 Table 10: Summary of Pilot-scale Refining Trials results................................................ 54 Table E.1: Transition matrix assessed by Bertrand et.al/1999[91] .................................... 85 Table E.2: Calculated evolution of particle size distribution............................................ 86 Table E.3: Assessed Transition Matrix ............................................................................. 86 Table E.4: Particle size distribution .................................................................................. 88 Table E.5: Comparison of assumed and estimated Selection Function values................. 88  viii  List of Figures Figure 1: Pulp flow pattern within the refiner plate by Fox/1979[39] ................................. 8 Figure 2: Same amount of energy imparted into fibres in two ways ................................ 13 Figure 3: A representative refiner bar pattern showing bar and fibre dimensions............ 24 Figure 4: Plate pattern showing mass location ................................................................. 24 Figure 5: Probability that upon being captured, the average intensity IA exceeds the rupture strength IC of two different pulp fibres.............................................. 27 Figure 6: Sprout-Waldron LC-Plate (CEL = 0.81328 km/rev) ......................................... 31 Figure 7: 12” Sprout-Waldron Refiner at UBC (8”inner - and 12”outer diameter) ......... 32 Figure 8: Fibre Length Distribution of tracer fibres (Douglas Fir)................................... 33 Figure 9: Refined Douglas Fir (dark brown) and White Spruce fibres using the 12” Sprout-Waldron refiner .................................................................................. 37 Figure 10: Comparison of the selection function probability, Si, for 4.3 mm long fibres (White Spruce) for various trial conditions and refiners................................ 38 Figure 11: Selection function curve for Douglas Fir and White Spruce fibre fractions after refining at SEL = 3.0 J/m ............................................................................... 39 Figure 12: The experimentally determined selection function for the Escher–Wyss refiner trials using a single refiner plate for increasing SEL. .................................... 41 Figure 13: Selection function values for 4.3 mm long fibres for all 15 trials................... 42 Figure 14: Selection function dependency on Consistency for the same plate pattern on a Double Disc refiner (Trial A-E) for fibre length of 4.3 mm .......................... 43 Figure 15: Breakage probability of White Spruce fibres during trial E............................ 44  ix  Figure 16: Probability of capture for average fibre length lA during one bar crossing for various plate geometries................................................................................. 47 Figure 17: Capture probability for average fibre length lA during one bar crossing for various flow conditions .................................................................................. 48 Figure 18: Fibre-length-reduction characterization as function of intensity. ................... 49 Figure 19: Actual intensity distribution vs. the distribution of rupture strength, IC. ........ 49 Figure 20: Comparisons of Average Intensity IA to intensity levels calculated based on previous studies by Leider and Nissan/1977[36] and Kerekes/1990[55]......... 50 Figure 21: Tensile strength plotted against cumulative probability for P = 0.0021. ........ 52 Figure 22: Comparison of K·lA /(lA+0.5·D) and P for Trials A - Q.................................. 53 Figure 23: Homogeneity development in dependency of throughput and refiner speed using a CONFLO refiner at 1% consistency and SEL = 4 J/m ...................... 55 Figure 24: Relationship between Tear Index and Homogeneity Index, H. ...................... 56 Figure A.1: Pressure Distribution within a fibre flock covering a 8 x 8 mm area, at ≈ 30% consistency measured by Endres et al./2005[63] ............................................ 72 Figure C.1: Tensile strength vs. fibril angle for springwood and summer wood fibre of white spruce, reproduced from Page et al./1972[87]....................................... 75 Figure C.2: Frequency distribution of tensile strength for black spruce acid sulphite fibres, pulped to 63% yield, at 0.42 and 1.50 span length, reproduced from Page et al./1976[86]......................................................................................... 78 Figure C.3: Comparison of the calculated rupture strength distribution of White Spruce fibres using Gamma and Weibull probability distribution functions............. 79 Figure F.1: Stochastic Approach to Batch Refining ......................................................... 89 Figure G.2: Comparison of best fit for Capture Function G.2 and G.5 for Trial M ......... 92 x  Nomenclature  a  Shape parameter for intensity distribution, [kJ/kg]-1  A  Comminution matrix  AF  Cross sectional area fibre, [m2]  b  Scale parameter for intensity distribution, [kJ/kg]-1  B  Breakage function matrix  Bij  Breakage function, [-]  BL  Breaking length of paper, [km]  BEL  Breaking Edge Length, [km]  C  C-Factor, [s-1]  CF  Consistency pulp suspension, [%] or [-]  CEL  Cutting Edge Length, [km/s]  E  Energy input. [kWh/t]  dE  Incremental energy input, [kWh/t]  D  Groove Depth, [mm]  EM  Elastic Modulus (Young’s modulus), [Pa]  f(li)  Probability of capture form of fibres of length li, [-]  FCi  Factor for fibres length i to be captured during one bar crossing event, [-]  Fmax  Maximum force, [N]  G  Groove width, [mm]  H  Homogeneity index, [-]  I  Intensity of impacts, [kJ/kg]  xi  IA  Average intensity of impacts, [kJ/kg]  IC  Critical rupture strength of fibres, [kJ/kg]  ICA  Average critical rupture strength of fibres, [kJ/kg]  In  Identity matrix  K  Empirical constant depending on machine parameter, [-]  lA  Average fibre length, [mm]  m&  Mass throughput of O.D. pulp, [t/h] or [kg/s]  mG  Mass of fibres in the grooves per unit bar length, [kg/m]  mT  Mass of fibres in the gap per unit bar length, [kg/m]  MEL  Modified Edge Load, [J/m]  n(E)  Column vector of fibre length classes at energy input level E  ni  Numbers of fibre length class i, [-]  dni*/ni  Incremental shortening of fibres of length class i, [-]  n*Tracer,j(E) Fraction of the number of tracer fibres in length class j, [-] nt  Initial state vector  N  Number of impacts on fibre, [-]  NA  Average number of impacts, [-]  NB  Number of bar crossings fibre experiences while traveling thr. refiner, [-]  NC  Number of compression cycles, [-]  outi  Estimated values according to Markovian transition matrix approach  P  Average probability of fibre capture for one bar crossing, [-]  pC(IC)  Rupture strength - probability distribution density function, [-]  pI(I)  Intensity probability distribution density function, [-]  xii  PR  Probability of refining intensity exceeding the rupture strength of fibres, [-]  pGamma(I)  Gamma probability distribution density function, [-]  Pnet  Net energy input, [kW] or [kJ/s]  Q  Volume throughput, [m3/s]  R1  Inner radius refiner [m]  R2  Outer radius refiner, [mm]  RBA  Relative Bonded Area [-]  RD(E)  Matrix of eigenvalues  R(NB)  Fraction of fibres that have been captured and therefore treated, [-]  S  Selection function matrix  Si  Selection function, [kWh/t]-1 or [kJ/kg]-1  SEL  Specific Edge Load, [Ws/m] or [J/m]  SRE  Specific Refining Energy, [kWh/t] or [kJ/kg]  SSL  Specific Surface Load, [Ws/m2] or [J/m2]  t  Energy state, [kWh/t], or [kJ/kg]  t(j, i)  Probability transition matrix  T.E.A.  Tensile energy absorption [J/m2]  UT  Matrix of the eigenvectors  vij  Elements of V(E)  V&  Volume flow through the refiner, [m3/s]  V(E)  Transition matrix  w  Fibre coarseness, [mg/m]  WR  Bar Width of the Rotor, [mm]  xiii  WS  Bar Width of the Stator, [mm]  x  Number of Bins of fibre length measurement, [-]  α  Shape parameter for gamma distribution function, [-]  αB  Bar angle, [°]  β  Scale parameter for gamma distribution function, [-]  δE  Small energy input, [kJ/kg]  ε  Elastic strain at failure, [-]  θ  Sector angle refiner plate, [-]  k  Shape parameter for Weibull probability distribution function, [-]  λ  Scale parameter for Weibull probability distribution function, [-]  µ  Learning rate parameter  ρ  Density fibre, [kg/m3]  σ  Stress (tensile strength), [N/m2] or [Pa] or [101 dyne/cm2]  υ  Poisson ratio, [-]  φ  Bar angle, [°]  ω  Rotational speed, [rev/min] or [rev/s]  xiv  Acknowledgments  I would like to thank my supervisors James Olson and Richard Kerekes for providing strong leadership, inspiration, and support over the course of my Ph.D., and for showing me what it takes to bring an academic quest through to completion. I would also like to thank my committee members Mark Martinez and Chad Bennington for their help and advice in this project. I wish also to acknowledge my co-researchers and staff of the pulp and paper centre at UBC for their work ethic and encouragement, namely Sean Delfel, Paul Krochak, Tim Paterson, and Ken Wong. It was absolutely necessary to have a supportive network of friends and family to help me on my journey through this project. My deepest thanks to those in my life who always supported and encouraged me over the past years. Grateful acknowledgement is made for financial and technical support from NSERC, CANFOR, STORA ENSO and FPInnovations, Paprican Division.  xv  Chapter 1: Introduction  1 Introduction Pulp refining is one of the most important unit operations in papermaking. Its purpose is to modify fibre morphology in order to improve paper properties by a technology which has developed from batch beaters in the early 19th century to flow-through disc, conical, and cylindrical refiners in use today. In pulp refiners fibres are trapped in the gaps between bars during bar crossings during pulp flow through an annulus between a rotor and stator which have bars on their surfaces, and subjected to cyclic compression and shear forces which modify fibre properties. The result of refining is changes in physical and optical properties of paper have been studied in detail by various authors over the past decades (Higgins and de Yong/1962[1], Page/1985[2], Seth/1996[3] and Hiltunen et al./2002[4]) to name only a few). It has been generally found that, within the commercial range, refining increases tensile strength (about threefold), burst strength, folding endurance, and sheet density, whereas it reduces tear strength of paper and sometimes fibre length. Although there is extensive literature on changes in fiber morphology produced by refining, there is relatively little on how the refining action takes place, that is, on the refining process itself (Atack/1980[5], Page/1989[6]). Consequently, quantitative characterization of refining action is limited. The most widely used quantitative measure of refining action is net energy expended per unit mass, i.e. specific refining energy (SRE). In addition, it is common to use a “refining intensity” which reflects the energy expended per loading cycle at each bar crossing event. However, although useful, energybased characterizations have limitations because most changes within fibres occur as a  1  Chapter 1: Introduction  result of forces, not energy. The use of force in characterizing refining action has only recently been undertaken (Kerekes and Senger/2006[46]). It has long been known that refining is very non-uniform in its treatment of pulp (Danforth/1986[7]). However, the causes and consequences of this heterogeneity have been little studied. Furthermore, heterogeneity of treatment has not been taken into account in any of the refining theories or characterizations, due in good part to a lack of means to measure heterogeneity. Thus, it is reasonable to expect that a better understanding of the nature of heterogeneity in refining will lead to a better understanding of the process and perhaps offer a basis for improved technology. The objective of this dissertation is to develop a method to measure heterogeneity of treatment in operating refiners.  2  Chapter 2: Review of Literature  2 Review of Literature Several excellent reviews of pulp refining have been produced over the years. Generally they have emphasized changes in fibre morphology caused by refining, for example: Higgins and de Yong/1962[1], Page et al./1962[8], Atack/1977[9], Page/1985[2], Seth/1996[3] and Hiltunen et al./2002[4]). The fundamentals of the refining process have also received study, as shown in the reviews of Ebeling/1980[10], Page/1989[6], Hietanen/1990[11] and Hietanen and Ebeling/1990[12]). The following review of the literature will focus on highlights from past studies relevant to heterogeneity of treatment.  2.1  Effect on Pulp  The main effects of refining on pulp are: internal fibrillation, external fibrillation, fibre shortening and fines generation. More recently, fibre straightening has been recognized as another primary effect by Page/1985[2], Mohlin and Miller/1995[13] and Seth/2006[14]. These outcomes of refining are described below in more detail.  2.1.1 Internal Fibrillation Measurements by Mohlin/1975[15], Kerekes/1985[16] and Paavilainen/1993[17] have shown that refining chemical pulp increases fibre flexibility. This occurs from delamination in the cell wall, which lowers its effective elasticity. Water is drawn into the fibre walls by capillary forces, causing swelling which is often used as an indicator of degree of refining. The weakened cell wall is more flexible and collapsible, thereby giving a larger relative bonded area in paper. This in turn increases bonding and thereby paper strength.  3  Chapter 2: Review of Literature  2.1.2 External Fibrillation Another effect in commercial refining is the delamination of fibre surfaces, called external fibrillation. This can be defined as a peeling off of fibrils from the fiber surface, while leaving them attached to the fiber surface (Page et al./1967[18], Page/1989[6]). Clark/1969[19] emphasized that the fiber surface can be fibrillated even in the early beating stage, and that the external fibrils serve as bonding agents for inter-fiber bonding. The amount of such fibrillation (parts of fibre wall still attached to fibre) can be quantified by measuring the increase in the specific surface of the long fibre fraction (Page/1989[6]). Although external fibrillation was studied from the earliest years of refining because it was easily observable under microscope, and was thought to be the dominant result of refining, several modern authors, i.e. Page/1985[2] and Hartman/1985[20] concluded that external fibrillation has only limited effect on the tensile strength or elastic modulus.  2.1.3 Fibre Shortening Refining causes fibre shortening which is generally undesirable in refining. In some rare applications it is a desired effect to improve formation by decreasing the crowding number (Kerekes/1995[21]). In early work, Page/1989[6] showed that the mode of fibre shortening was tensile failure rather than “cutting” by scissor-like shearing.  2.1.4 Fines Generation Fines, meaning loose fibrous material of size less than 0.3 mm, are produced in refining as a result of fibre shortening or removal of fibrils from fibre walls. According to  4  Chapter 2: Review of Literature  Page/1989[6], generated fines consist mostly of fragments of P1 and S1 layers of the fibre wall due to the abrasion of fibres against each other or against refiner bars. 2.1.5 Fibre Straightening Fibre straightening is considered an important refining effect (Page/1985[2], Mohlin/1991[22], Seth/2001[23]). If fibres remain curly or kinked in a paper network, they must be stretched before they can carry load, meaning that fewer fibres carry load during the initial stretching of paper, and therefore paper is weak. Fibre straightening permits fibre segments between bonded sites in paper to be “active” in carrying load. The mechanism by which refiners straighten fibres is likely due in part to tensile strain in refiner gaps and by causing fibre diameter swelling. In the latter case, upon drying, shrinkage of swollen fibres stretches fibres segments between bonded sites and thereby straightens them.  2.1.6 Heterogeneity It has long been known that refining is a heterogeneous process and that the resulting heterogeneity affects paper properties. One of the earliest workers to show this was Danforth/1986[7] by mixing of fibres subjected to differing levels of refining. Larger heterogeneity gave poorer properties, e.g. lower tensile strength. Others have shown heterogeneity by observations of morphology changes (Page/1989[6]). The literature on heterogeneity will be reviewed extensively later in section 2.6.  2.2  Mechanics of Refining  Current knowledge of the mechanics of refining is far less than is knowledge of the refining result. The design of refiners is largely empirical. In addition to action on pulp, 5  Chapter 2: Review of Literature  design is based on factors such as sufficient groove size to permit flow of the fibre suspension through the machine, bar wear, and material cost. Scientific studies of the process of refining have been carried out by various researchers. Some highlights of this past work are described below.  2.2.1 Fibrage Theory Smith/1922[24] proposed that a refiner bar moving through stock collects a “beard” or fibrage of fibers on the leading edge. This capture of fibre transports fibres into the gap between rotor and stator bars. A study of the compressive and shearing forces exerted on the bars during refining by Goncharov/1971[33] gave some support to the fibrage mechanism.  2.2.2 Refining as Lubrication Process (Hydrodynamics) Rance/1951[25] and Steenberg/1951[26] analyzed refining as a lubrication process. Their experiments offered indirect evidence of the occurrence of (a) fluid (hydrodynamic) lubrication, (b) boundary lubrication, and (c) lubrication breakdown, which produced excessive metal wear in the refiner. Steenberg/1951[26] carried out his experiments with a Valley beater varying the consistency, load and peripheral speed. He concluded that Valley beating is carried out under hydrodynamic (fluid) lubrication conditions and that the work absorption capacity, i.e., coefficient of friction, decreases with beating as does the apparent viscosity of the pulp. More recently Frazier/1988[27] applied the lubrication theory to refiner mechanical pulping and derived an expression for the shear force applied to pulp. Similarly Roux/1997[28] performed a lubrication analysis in order to characterize hydrodynamic forces in refining. Based on experiments Lundin et al./1999[29] propose 6  Chapter 2: Review of Literature  that the power consumption in refining is governed by a lubrication film between the rotor and the stator. Although characterization of the refining process as a hydrodynamic phenomenon has yielded some useful correlations, treating the fibre suspension as a fluid has some serious limitations. Refiner gaps are 5-10 fibre diameters deep and therefore the water-fibre suspension in a gap is not a continuum. Consequently the suspension in the gap does not meet a primary property of a fluid.  2.2.3 Transport Phenomena in Refiners Flow behavior in refiners has been examined by several workers. Complex flow patterns have been observed and residence time distributions have been measured using high speed photography (Halme and Syrjanen/1964[30], Herbert and Marsh/1968[31], Banks/1967[32], Gonchorov/1971[33], Fox et al./1979[34] and Ebeling/1980[10]). Halme and Syrjanen/1964[30], Herbert and Marsh/1968[31] observed that pulp flows primarily in the radial direction outward in the grooves of the rotor and inward in the grooves of the stator. Other recirculation flows, called secondary and tertiary flows, are also present in disc refiners. Fox et al.[34] proposed that the secondary flows arise from the motion of the land area of the bars passing over the grooves and that tertiary flows result from the difference in static pressure between opposing grooves of the rotor and stator (see Figure 1). Using a material and energy balance, Leider and Rihs/1977[35] and Leider and Nissan/1977[36] related the pressure drop across the refiner to its volumetric flow rate. This accounts for the action of the centrifugal and frictional forces and is expressed in terms of operating conditions, fluid properties and plate properties.  7  Chapter 2: Review of Literature  Primary Flow  Secondary Flow  Tertiary Flow  Stator  Rotor ω  Figure 1: Pulp flow pattern within the refiner plate by Fox/1979[34,37]  Arjas et al./1969[38] studied the effect of flow rate and angular velocity on the shape of the residence time distribution curve. They concluded that the flow rate has an effect on the mean residence time, but the effect of the angular speed is not very large.  2.2.4 "Treatment of Flocs" Hypothesis Page et al./1962[8] showed that flocs rather than individual fibres are treated in refining , although it is possible that fibers may be oriented in a floc network to receive fiber treatment. From high-speed photography of a conical refiner, they concluded that a refiner is inefficient since only a small proportion of fibers are subject to treatment. Page et al./1962[8], Hietanen and Ebeling/1990[12] found that flocculation plays an important role in low-consistency refining. Similarly, studying power and gap size relationships, Batchelor & Ludin/2006[39] estimated the fraction of bar length covered with fibres. Banks/1967[32] proposed a detailed picture of refining action based on floc treatment in contrast to the fibrage theory which assumes a uniform coverage of the bar edges by  8  Chapter 2: Review of Literature  individual fibres. He summarized the mechanics of refining based on observations using high-speed photography: 1. Flocs consolidate when they are trapped between approaching tackle elements. 2. Mechanical pressure induced by the tackle elements becomes high enough and causes plastic deformation in the fibers composing the floc. Consolidation continues. 3. The floc compressed between the bars is sheared; flocs (and fibers) are ruptured. 4. Release of mechanical pressure allows absorption of water to take place into the ruptured fibrils and fibers. 5. Turbulent agitation may disperse the floc or its remnants into the general mass flow.  2.2.5 Refining as a Fatigue Process Fibres may be exposed to many bar crossing events during their passage through a refiner. Thus, fibres are likely to experience multiple loading cycles. Page et al./1962[8] and Atack/1977[9] suggested that this repeated straining causes fatigue weakening of the fibres, and this in turn produces internal fibrillation. On the other hand, fibre shortening (cutting) is likely to be caused by one severe loading cycle. Page/1989[6] suggested rupture takes place by tensile failure rather then a scissor-like cutting action. Kerekes and Olson/2003[40] proposed that fibre rupture in refiners is caused by a single event rather than fatigue process. Another approach to account for the cyclical nature of the process was proposed by Steenberg/1979[41]. He considered the refining process to be an irreversible kappa 9  Chapter 2: Review of Literature  process, in which there is a critical path for the formation of a stress concentration chain, which causes a structural breakdown. For another structural breakdown to occur the particles, i.e. fibers, need to be rearranged. Thus cyclicality is needed to overcome heterogeneity. Most recently, Goosen et al./2007[42] examined the role of fibre redistribution in pulp refining by compression. They concluded that the role of cyclic loading is to expose new fibres to loading cycles rather than to produce fatigue weakening. In essence, cyclicality in the process is to overcome heterogeneity rather then to produce fatigue weakening.  2.2.6 Forces on Fibres during Bar Crossing Events A number of studies have examined the manner in which forces are imposed on fibres within refiner gaps. Page/ 1989[6] suggested that refining should be quantified by the stresses and strains applied to fibres. These stresses and strains are caused by bar forces which have been the subjects of several studies. Goncharov/1971[43] was the first to measure pressures on refiner bars during bar crossing events. Later Martinez et al./1997[44] and Batchelor et al./1997[45] analyzed the compressive and shear forces action on single flocs and measured normal and shear forces in a “single bar” refiner apparatus. Kerekes and Senger/[46] developed equations for bar forces and fibre forces from the specific edge load (SEL). Page/1989[6], Martinez et al./1997[44] and Koskenhely et al./2007[47] addressed issues concerning the bar edge force.  10  Chapter 2: Review of Literature  2.3  Quantitative Characterization of the Refining Mechanism  As described earlier, specific refining energy is the most common quantitative characterization of refining action. This energy is factored into the number of impacts imposed on pulp and the intensity of each impact, i.e. loading cycle. Most intensities are based on energy per loading cycle, though in recent work Kerekes and Senger/2006[46] have proposed a force-based intensity. These concepts are described in detail below.  2.3.1 Specific Energy Specific refining energy (SRE) is the useful energy imparted to the pulp suspension, i.e. the total energy less the no-load energy. It is obtained by dividing the net power input into the refiner by the fibre mass throughput.  SRE =  Pnet Pnet = m& C F ⋅ V& ⋅ ρ  (1)  The typical range for SRE in chemical pulp refining is between 80 to 250 kWh/t, depending on the furnish and the desired papermaking property.  2.3.2 Refining Intensity The major means of characterizing refining intensity today is by a “machine intensity”, meaning an intensity which reflects energy expended per bar crossing without direct reference to how this energy is expended pulp. The most common such method is the Specific Edge Load (SEL). This was introduced by Wultsch and Flucher/1958[48] and developed further by Brecht and Siewert/1966[49] who demonstrated the dominant impact of the bar edge. The SEL is determined by the net power divided by the bar edge length (BEL) multiplied by the rotational speed ω. The BEL is determined by multiplying the 11  Chapter 2: Review of Literature  number of bars on stator by the number of bars on rotor by the refining zone length. The result is expressed in Ws/m or J/m. The expression for SEL is: SEL =  BEL =  Pnet P = net BEL ⋅ ω CEL  (2)  R2  (3)  N nr ⋅ n s ⋅ r n r ⋅ n s ∆r ≅ ∫R cosθ ∑1 cosθ 1  An extension of the SEL, the specific surface load (SSL), was suggested by Lumiainen/1990[50] to account for bar width rather than just bar edges. This was accomplished by including bar width W with the SEL, thereby giving the area of the bars and units Ws/m2 or J/m2. SSL =  Pnet  (4)   W R + WS   CEL ⋅ ω ⋅   2 ⋅ cos(α B 2 )   Using this approach Lumiainen distinguished between refiner plates having identical cutting edge length (CEL) but differing bar widths.  Meltzer and Rautenbach/1994[51] used the “crossing speed” – which is a measure of fibre treatment frequency by calculating the crossing speed per dry fibre mass flow – to further modify the SEL-concept. This gave the modified edge load (MEL) which accounts for bar width, W, groove width, G, and the average intersecting angle of rotor and stator bars. For identical rotor and stator geometry the following expression was derived: MEL = SEL ⋅  1 W +G ⋅ 2 ⋅ tan α B W  (5)  The factor W+G / W considers the refining work of the bar surfaces and the probability of fibre treatment received.  12  Chapter 2: Review of Literature  2.3.3 Fibre Intensity The above characterizations based on energy per bar crossing are “machine intensities” that make no direct link to the pulp in the refiner. To make this link, characterizations have been developed that factor SRE into two quantitative parameters: the number of impacts, N, and the intensity of each impact, I on pulp. In this case, I is a “fibre intensity”. The product of these in the SRE are shown in equation (6). SRE = N ⋅ I  (6)  Intensity of impacts, I  Fibre cutting SRE1 = SRE2 SRE1 Fibre straightening Fibrillation  SRE2  Number of impacts, N Figure 2: Same amount of energy imparted into fibres in two ways  As shown in Figure 2 differing combinations of N and I give differing refining results. Because only 2 of the variables in equation (6) are independent, it is common to use SRE and I. Lewis and Danforth/1962[52] and later Danforth/1969[53] defined expressions for N and I, but these were empirical in nature and did not rigorously represent N as a dimensionless variable and I as energy per bar per impact.  13  Chapter 2: Review of Literature  Van Stiphout/1964[54] calculated the probability of a fibre being caught in the refiner gap. Based on this he was able to estimate the forces acting on fibres and the mass of fibres being treated per length of bar. According to his analysis the mass of fibres per bar length was proportional to the mean residence time. He further postulated that the damaging affect of refining was proportional to the surface sheared and that the net power was active on an amount of fibres enclosed in the gap between the refiner bars. However, two empirical parameters of unknown magnitude limited the application and scientific meaning of this work. Leider and Nissan/1977[36] were the first to derive equations for N and I in rigorous terms. They represented the probability of a fibre being stapled on a bar by fibre length and diameter to give an estimate of “number of impacts per single fibre per refiner pass”. Another approach to define values for N and I to overcome some of the shortcomings of Leider and Nissan was made by Kerekes/1990[55] in the form of the C-factor. SRE =  (7)  Pnet C P = N ⋅ I = ⋅ net m& m& C  In this approach, C is defined as the capacity of a refiner to inflict impacts (loading cycles) on fibres. Kerekes/1990[55] derived equations for the C-factor as a function of plate geometry (bar width, depth, length, angle, and edge geometry), speed, consistency, fiber length, and coarseness. The C-factor for a disc refiner for a simplified case (small gap size, similar bar pattern on rotor and stator), is given by: C=  (  8π 2 GDρC F l A ⋅ n 3ω (1 + 2 tan φ ) R23 − R13 3w(l A + D )  )  (8)  A crucial assumption of the analysis is estimating the probability of fiber contact with the leading edge of the bar, which he expresses as l / (l + D+ T). Kerekes showed that this  14  Chapter 2: Review of Literature  assumption led to logical outcomes at the limits of bar size relative to fibre length. The CFactor analysis is perhaps the most rigorous and comprehensive theory developed to date. It has been applied to a variety of refiner geometries including the disk, conical, and PFI mill. It takes into account factors relating to bar and groove geometry as well as fibre length, which provide a more accurate description of refining intensity. However, the Cfactor predicts only average values rather than distributions of intensity.  2.4  Heterogeneity  It has long been accepted that refining is a heterogeneous process (Page/1989[6]). The sources of heterogeneity are numerous, ranging from: the mechanism of transferring mechanical forces into fibers (Steenberg/1963[56]) to the irregular flow pattern through the refiner (Halme/1962[57]). There are also scales of heterogeneity ranging from large to small. Ryti and Arjas/1969[58] studied the large distributions of residence time in refiners, originating from large scale secondary flows. They postulated that greater uniformity in refining treatment gives a better refining result. They concluded that the refining action is related to the residence time inside the refiner and to the probability of treatment during this time. Fox et al./1979[34] studied flow in a small disc refiner by visual observation. They found that after being “released” from a gap, fibres either become part of the stator flow towards the refiner axis and stapled again on a bar, or become part of the outward rotor flow leaving the refiner. In addition fibres may flow back to the stator from the outside annulus of the refiner. This all resulted in a poor pulp circulation and therefore limited the amount of fibres receiving treatment in refiner gaps.  15  Chapter 2: Review of Literature  Additional sources of heterogeneity arise from a finite probability of capture from the groove (Olson et al./2003[59]), fractional bar coverage (Batchelor and Ludin/2006[60]), non-uniform force distribution within flocs (Batchelor and Ouellet/1997[61]), nonuniform SEL along the radius of the refiner (Roux and Joris/2005[62]), non-homogeneous force distribution on fibre networks while being under load (Endres et al./2005[63]), and out-of-tram of rotors (Prairie et al./ 2007[64]). An added complexity is the wide variability of fibre properties in any pulp. It is well known that species, pulp type, bleaching process, and the relative amounts of thick and thin walled fibers can affect refining. Page/1983[65] has suggested that differences in the beatability and refining action of kraft and sulfite pulps are due to differences in their viscoelastic behavior. Yet another factor is fibre flocculation. Hietanen and Ebeling/1990[12], Hietanen/1991[66] and Lumiainen/1995[67] have all suggested that flocculation is a major source of refining heterogeneity.  Various studies have been conducted to measure heterogeneity of treatment in refining. Simons/1950[68]) used a staining technique to identify damaged fibres. Nisser and Brecht/ 1963[69] evaluated refining heterogeneity by calculating the swollen length of fibres before and after refining. Another approach has been to examine physical changes in fibres. Mohlin and Librandt/1980[70] using SEM technique found that one of the major differences between laboratory refining and production refining is the homogeneity of treatment. Dahrni and Kerekes/1997[71] measured deformations in plastic tracer fibres. They found that 30% of the fibre passed through a lab disc refiner with no visible sign of impact whereas other fibres were severely deformed. Dekker/2005[72] measured the changes in RBA of fibres, by quantifying the optical contact between fibres and a glass slide. He found that only 7-20% of fibres were treated. 16  Chapter 2: Review of Literature  Another approach for measuring heterogeneity is based on comparing properties of papers made from mixtures of unrefined and well-refined pulp (Dillen/1980[73]). As a measure of the heterogeneity he used the coefficient of variation of the amount of refining. Lastly, Hietanen/1991[66] found that the homogenizing effect of deflocculation has a positive impact on paper strength properties.  In summary, the sources of heterogeneity in refining are many and there are some methods to measure it, but no existing methods are suitable for measurement of heterogeneity of treatment in operating refiners.  2.5  Review of Refining Action as a Comminution Process  Comminution is defined as the reduction of particle size by mechanical processing. Models for the comminution process models were developed for crushing and grinding in the powder industry 60 years ago to predict particle size distribution changes during processing. The basics were developed by Epstein/1948[74], Reid/1965[75] and Austin/1971[76]. They modeled batch grinding by the introduction of two statistical functions, a Breakage function, Bij, and a Selection function, Si. Basically Si represents the probability of a particle of size i per unit time or energy and has the units of time-1 or energy-1. The breakage function Bij represents the probability of particles larger than i being reduced to size i. As the particle size can only be reduced during grinding, the values of Bij are dimensionless and range between 0 and 1.  The incremental change in the number of fibres of length i, dni, per incremental applied energy dE is given by the comminution equation shown below: 17  Chapter 2: Review of Literature  dn i = − S i n i + ∑ Bij S j n j dE j >i  (9)  Applied to refining, ni represents the number of fibres of length i available for cutting. Si is the selection function for fibres of length i per unit energy. The product of Si and ni; represents the decrease in number of fibres of length i. The term ΣBijSjnj represents the increase in the number of fibres of length i per unit energy from the cutting of fibres of lengths larger than i. For refiners, we may consider incremental energy to be the discrete event of a single bar crossing assuming this to inflict a single loading cycle on fibres and fibres to be ruptured by a single excessive loading cycle.  Steenberg/1963[56] suggested that several sub-processes have a major influence on the refining result during certain phases of refining. Thus a single parameter cannot be used to describe either the process or the state of the product (i.e. refined pulp). He suggested that at least two of the sub processes, namely the “selection process” and the “treatment process”, deserve more attention.  Other workers applied the model directly to describe fibre length distribution changes caused by refining, for example Kane/1957[77] and Meyer et al./1966[78]. Roux and Mayde/1997[79] applied the comminution theory to analyze the change of the mean fibre length during refining. They predicted the potential for fibre cutting under given conditions to be a function of the energy consumed by the solid phase and the average impact intensity, i.e. the ratio between the net machine power and the “cutting” length of  18  Chapter 2: Review of Literature  bars per unit time. However, their equations had inconsistent units, making their results difficult to interpret rigorously.  Corte and Agg/1980[80] used a comminution model to compare the shortening rate of fibres in a disc refiner and a laboratory beater. They found that the disc refiners cut long fibre more rapidly than short fibres, while a laboratory beater cut long and short fibres at the same rate. Olson et al./2003[59], applying a technique developed by Koka and Trass/1988[81], found that the probability of fibres being selected for cutting during refining is proportional to the applied energy and fibre length, and was independent of consistency. Regression fitting that minimizes the difference between measured and predicted fibre length distributions was used to determine the selection function, Si, for various pulp and refiner combinations.  Several other investigators applied the comminution theory to the production of mechanical pulp in chip refiners. Corson/1972[82] modeled the refining of wood chips into individual fibres using a comminution approach. This approach was used more recently in an expanded form by Strand and Mokvist/1989[83] to model the operation of a chip refiner employed in mechanical pulping of wood chips.  19  Chapter 3: Objective of Thesis  3 Objective of Thesis The above review shows that there is a need for a rigorous, practical method to measure heterogeneity in operating refiners. The objective of this thesis is to develop such a method. The approach is based on interpreting measurements of fibre shortening by a model of the physical process of refining described by comminution theory.  20  Chapter 4: Analysis  4 Analysis Comminution modeling is a potentially useful basis for measuring heterogeneity for several reasons. Fibre shortening obviously reflects the probabilistic nature of refining because some fibres are shortened while others are not. The result, decreased fibre length, is a measurable change for which instruments for rapid measurement are now available. Lastly, the best tracer particles to reflect what happens to fibres are fibres themselves.  Fibre shortening is generally not an objective of refining. Indeed, it is to be avoided in most cases. In this study, fibre shortening will be used merely as an event to characterize the mechanics of the process rather that as a desired outcome. We employ comminution theory as the scientific basis of our approach. To do so, we address several individual problems:   interpretation of comminution equation  link to refiner mechanics  probability of fibre capture in a gap  probability of fibre shortening in a gap  link between breakage function and selection function  21  Chapter 4: Analysis  4.1  Interpretation of Comminution Equation  As discussed earlier, the comminution equation reflects the net change in the number of fibres of size i, as the result of the decrease size i caused by shortening and the increase caused by fibre shortening from larger sizes. dn i = − S i n i + ∑ Bij S j n j dE j >i  (10)  However, our specific interest in this work is only the shortening component of comminution. Therefore, we focus on the first two terms of the comminution equation. We define the incremental shortening, as dni*, to differentiate it from the net shortening dni in equation (10). Rearranging terms, we obtain: *  dn i = − S i ⋅ dE ni  (11)  In essence, this equation relates the incremental fraction of fibres of length i, (dni* / ni) that are shortened to the product of the selection function, Si and the incremental energy supplied, dE. We now consider Si·dE in terms of refiner variables, specifically to fibre rupture during a single bar crossing. The probability of fibre rupture is the product of two other probabilities: a factor of fibre capture from the groove, FCi and the probability that once captured, the fibre is ruptured, PR. We consider PR, the probability that the intensity a fibre sees is greater than its rupture strength (intensity), to be independent of fibre length. Thus we may express Si·dE as  S i ⋅ dE = FCi ⋅ PR  (12)  We now link dE, FCi and PR to refiner mechanics and thereby refiner variables.  22  Chapter 4: Analysis  4.2  Link to Refiner Mechanics  We first link energy per mass supplied to a refiner E, to the product of number of impacts imposed on fibres, N, and the intensity per input I, expressed as energy per mass.  E = N ⋅I  (13)  We may express this equation in incremental form as  dE = N ⋅ dI + I ⋅ dN  (14)  Assuming that for a given refiner, the average intensity at bar crossings is constant and represented by IA, and since dI = 0, we obtain  dE = I A ⋅ dN  (15)  4.2.1 Intensity We now interpret N and IA in terms of measurable refiner variables at a single bar crossing. Refining intensity is commonly defined by the Specific Edge Load (SEL). Kerekes and Senger/2006[46] have shown that SEL represents the energy expended per bar crossing per unit bar length. Thus, the average energy expended on fibre mass within the gap of a bar crossing, IA, is given by SEL divided by the mass of fibres in the gap per unit bar length, mT, i.e.  IA =  SEL mT  (16)  4.2.2 Number of Impacts The number of impacts on pulp, N, is a function of the product of two factors: the probability of an impact at a single bar crossing and the number of bar crossings NB that a 23  Chapter 4: Analysis  fibre would experience if it passed along a stator bar during its residence time in the refiner. We can define a mass-average probability of impact at a bar crossing by the ratio of the mass of pulp captured, mT to the total mass associated with a bar crossing (pulp in gap and pulp in adjacent groove, mG), giving mT / (mT + mG). Thus  dN = mT (mT + mG ) ⋅ dN B  W  D  (17)  Stator  li  G  Pulp fibre  T  Rotor Figure 3: A representative refiner bar pattern showing bar and fibre dimensions.  Stator mT mG  Rotor Figure 4: Plate pattern showing mass location  We assume that the consistency in refiner grooves equals the pulp suspension consistency, CF. Thus  mG = 2 ⋅ D ⋅ G x C F x ρ  (18)  We may note here that the above equations give the following equation when combined:  E = N ⋅IA =  mT SEL N B ⋅ SEL ⋅ NB ⋅ = mT + mG mT mT + mG  (19)  These expressions represent averages. We now consider distributions. 24  Chapter 4: Analysis  4.3  Fibre Capture Factor  The above mass-average probability of capture assumes a dependence only on mass ratio. However, other factors also affect fibre capture. For example, the mass in a groove may not reflect eligible fibres because there may be dead zones of inactive fibres stuck in the groove. This would be a source of heterogeneity. In addition, it has long been known (Smith/1922[24]) that longer fibres tend to be more easily captured then short fibres, individually or as flocs. To reflect these factors, we express the factor of capture as a probability based on fibre length, f(li), and a constant K that reflects other unknown factors influential in capture, such as consistency, rotational speed, throughput and plate geometry. Clearly K falls in the range 0 ≤ K ≤ 1 .Thus  FCi = K ⋅ f (l i )  4.4  (20)  Probability Distribution of Intensity  Once in a gap, fibres rupture when they experience a force exceeding their rupture strength. We consider forces to be represented by intensities expressed as energy per unit mass. Thus, the probability of fibre rupture depends on the intensity applied to a fibre, I, and the rupture intensity, IC. We further assume that, once captured, rupture is independent of fibre length. We assume the distribution of intensity, I, on a pad of pulp to be similar to the spatial pressure distribution measured by Endres et al./2005[63]. In this work, Endres measured local pressure distributions on pads of wet pulp in increments of a few mm, i.e. 8 x 8 mm. In a refiner gap, compression forces are large, about 10 times the shear forces (Prairie et al./2007[64]), and therefore are likely to vary in a similar manner to the measurements of Endres along bar length in gaps, even in the presence of shear forces. In the absence of 25  Chapter 4: Analysis  other information of local forces in refiner gaps, we deemed this to be a reasonable assumption. As shown in Appendix B, the intensity density distribution can be described by Weibull function, and more approximately, by a simple form of the Weibull equation, the gamma density distribution, as shown below  p I ( I ) = aIe − bI  (21)  The constants a and b can be determined from two equations, one normalizing the distribution to 1 the other integrating the distribution multiplied by I to give the average intensity IA, as shown in Appendix B. This gives  pI (I ) =  4 ⋅ 1[kJ kg ]  (I A )2  −  ⋅ Ie  2 ⋅I IA  (22) where I A = SEL mT  The strength of fibres, IC, also has a distribution. We represent this distribution as pC(IC). As shown in Appendix C, this too can be represented by a Weibull density distribution. We now relate PR in equation (12) to above probabilities. Clearly, rupture occurs when the applied intensity I equals or exceeds fibre rupture intensity IC, summed over all fibre rupture intensities. This is given by: (23)  ∞ ∞  PR = ∫  ∫  p I (I ) ⋅ p C (I C ) ⋅ dI ⋅ dI C  0 IC  Solving the double integral of equation (23) does not give a simple general analytical solution owing to the complexity of the probability distribution functions in use: pI(I) as a Gamma density distribution function and pC(IC) as a Weibull density distribution function. It has the form: ∞  PR = ∫ p C (I C ) ⋅ 0  (2 ⋅ I C  + I A )⋅ e IA  −  2⋅ I C  (24)  IA  ⋅ dI C  26  Chapter 4: Analysis  ∞  PR = ∫  k  0 λ  k  ⋅ IC  k −1  ⋅e  I  −  C   λ   k  ⋅  (2 ⋅ I C  + I A )⋅ e IA  −  2⋅ I C  (25)  IA  ⋅ dI C  The solution of equation (25) is a hyper geometric function, for which examples for two different rupture strengths are shown Figure 5.  Probability of Rupture, PR [-]  1.00 0.90  I CA = 35 kJ/kg  0.80  I CA = 25 kJ/kg  0.70 0.60 0.50 0.40  ∞ ∞  PR = PR (I , I C ) = ∫  ∫ p (I ) ⋅ p (I ) ⋅ dI ⋅ dI I  C  C  C  0 IC  0.30 0.20  pI (I ) =  0.10  p C (I C ) =  4 ⋅ 1[kJ kg ]  (I A ) k  λ  k  ⋅ IC  2  k −1  ⋅e  −  ⋅ Ie  I  −  C   λ   2 ⋅I IA  k  0.00 0  10  20  30  40  50  60  70  80  90  100  Average Intensity IA [kJ/kg]  Figure 5: Probability that upon being captured, the average intensity IA exceeds the rupture strength IC of two different pulp fibres  4.5  Selection Function Expressed in Terms of Refiner Variables  We now link the above probabilities to the selection function, Si. We do so by substituting FCi from equation (20) and PR from equation (23) into equation (12). We obtain dE by substituting in equation (15) for IA from equation (16) and for dN from equation (17). Assuming mT « mG. (Batchelor and Lundin/2006[60]) and considering 1 bar crossing to be dNB = 1, we obtain  27  Chapter 4: Analysis  S i = K ⋅ f (l i ) ⋅  mG ∞ ∞ ⋅ p I (I ) ⋅ p C (I C ) ⋅ dI ⋅ dI C SEL ∫0 I∫C  (26)  Equation (26) contains three unknowns mT, K, f(li). These parameters are determined from Si of the comminution equation. However, first the relationship between Si and Bij must be established.  4.6  Relationship between S and B  An analysis relating S and B is shown in Appendix D. In brief, we first determined a suitable breakage function to represent probabilities of rupture anywhere along a fibre length (Olson et al./2003[59]). Previous work by others, e.g. Corte et al./1980[80], assumed that fibres shortened into only two parts of varying length. Our analysis has no such restriction. We determined Bij for a rupture at any point by introducing matrix A that links Bij and Si, giving equation (9) in matrix form as shown below: dn = A⋅n dE  where A = (B − I n ) ⋅ S  (27)  Variable n(E) is a column vector whose entries are the fibre mass fraction in each size interval after the energy input dE. We solved equation (9) by diagonalization of A and introducing UT as the matrix of the eigenvectors. Assuming the process to be Markovian, we re-expressed the population balance as ni(after)=V(E)·ni(before), following the procedure of Berthiaux/2000[84]. We thus decompose V into its characteristic roots, which represent the eigenvalues exp(-Si·E). From the matrix of the eigenvectors we obtained A, and from equation (27), the breakage function Bij. Further details on this analysis are given in Appendix D.  28  Chapter 4: Analysis  4.7  Curve -Fitting Procedure  Lastly, we selected an appropriate procedure for regression fitting data in fibre length changes before and after refining to equation (26) to obtain Si. We accomplished this in using Gauss–Newton algorithm method in two steps. First we determined the functional form of the parameter f(li), assuming it to depend on fibre length alone, and to be independent of mT and K. This was accomplished by fitting curve shapes of Si vs. fibre length. Next, having obtained the form of f(li), we determined parameters mT and K simultaneously by regression fitting to equation (26). This was accomplished by the same regression fitting method.  29  Chapter 5: Experimental Program  5 Experimental Program Our experimental program consisted of analyzing fibre length reductions produced by various refiners. We used MatlabTM software coded to calculate simultaneously the selection and breakage function for different refining conditions. Fibre length distributions before and after refining were measured using a Fibre Quality Analyzer (FQATM). The instrument gives fibre length histograms which contain approximately 150 bins with length increments equal to 0.05 mm. In order to reduce the sampling noise and the large time associated with calculating a 150 x 150 element transition matrix, and within it the S and B matrix, the measured fibre length distributions were compressed into 20 bins with an increment of 0.25 mm and then normalized. Fibres shorter than 0.5 mm were included in the measured distributions of this study, in contrast to the approach of Olson et al./2003[59]. This may introduce an error because short fibres and fines can be generated by peeling off pieces of fibre wall rather than by fibre shortening. However, in this study all fibres were treated as being a result of fibre shortening. The influence of fines on the overall outcome of the parameter calculation was investigated.  The experimental work was carried out in the steps shown as below:   Validation of the comminution approach  Evaluation of data from industrial scale pilot plant trials  Evaluation of data from laboratory refiners  30  Chapter 5: Experimental Program  5.1  Validation of Comminution Approach  The aim of this portion of the experimental work was to validate the comminution approach by a simple case of comminution examined by Berthiaux/2000[84] described in Appendix F. This was accomplished by employing an easily detectable tracer fibre in pulp. The tracer fibre was a laboratory-produced Douglas Fir pulp cooked using the “kraft” process to an H-factor of 1600. The resultant pulp was carefully washed and later classified three times in accordance to TAPPI test method T233 cm-95 with the BauerMcNett classifier. Only the R14 long fibre fraction of the original pulp was retained and  further classified to obtain a narrow distribution (3-4 mm) of long fibres as the tracer. As a result of this procedure, we obtained a nearly perfect, defect-free pulp having a narrow length distribution as shown in Figure 8.  Figure 6: Sprout-Waldron LC-Plate (CEL = 0.81328 km/rev)  For the validation, a pulp mixture (2% Douglas Fir tracer fibre and 98% White Spruce) was circulated through a single-disc refiner shown in Figure 7. The flow loop consists of two large, 455 liter plastic tanks, a 20 HP centrifugal pump, and a PVC pipe network that circulated the pulp from one tank, through the refiner and into the second tank (or into the 31  Chapter 5: Experimental Program  same tank for batch refining). The pulp suspension of 3 % was refined at a target SEL of 3 J/m at 600 rpm up to a SRE of 240 kWh/t. Bar Width [mm] 2.9  Groove Width [mm] 2.8  Groove Depth [mm] 2.8  G. Base Radius [°] 0.5  Draft Angle [°] 3  Table 1: Sprout-Waldron Plate Dimensions  Nominal Angle [°] 10  Cluster Angle [°] 22.5  Avg. Int. Angle [°] 46.5  Groove Volume [cu. cm/cm] 0.566  Table 2: Sprout-Waldron Angle Analysis  Figure 7: 12” Sprout-Waldron Refiner at UBC (8”inner - and 12”outer diameter)  32  Chapter 5: Experimental Program  Pulp samples were obtained at energy increments of 30 kWh/t. From the samples of refined pulp, approximately 100 tracer fibres were collected by pipette under a microscope. The sampling procedure was repeated 30 times in order to obtain a sufficient amount of fibres for measurement of fibre length in the Fibre Quality Analyzer (FQA). 12 DF - original R14 (1 x Fractionation) R14 (2 x Fractionation) R14 (3 x Fractionation)  10  Frequency [%]  8  6  4  2  0 0  1  2  3  4  5  6  7  8  9  10  Fibre length [mm]  Figure 8: Fibre Length Distribution of tracer fibres (Douglas Fir)  5.2  Pilot-scale Refining Trials  The major portion of the data in this experimental program were supplied by Domtar Inc. and were the same data used by Olson et al./2003[59]. The data were obtained on a 22” Beloit Double Disc pilot refiner and an Escher–Wyss laboratory refiner. The trials examined five different plate patterns. For each plate pattern, a range of energies and intensities were explored. Table 3 summarizes the experimental conditions of the various 33  Chapter 5: Experimental Program  refiners. The reported power is the net power, Pnet, which is the total power less the noload power. The consistencies are nominal values. The rotational speed for the double disc refiner (22” Beloit Double Disc) is 900 rpm and for the Escher-Wyss refiner is 1000 rpm. The plate dimensions used in this study are summarized in Table 4.  22” Beloit Double Disc  A B C D E F G H I J K L M N O P Q  Refiner  Plate  1  2  3 4  EscherWyss  Trial  5  Consistency [%] 2.84 4.55 1.55 2.48 4.22 4.19 2.54 4.52 2.70 4.40 2.56 4.51 3.15 1.50 3.00 4.00 3.00  m&  [t/d] 14 14 8 7 7 9 6 6 7 7 4 12 13 -  Pnet [kW] 65 64 34 34 34 64 26 26 33 34 16 55 56 0.7 1.4 2.3 2.3  SEL [J/m] 3.25 3.20 1.70 1.70 1.70 2.67 1.08 1.08 0.14 0.14 0.07 7.86 8.00 1.50 3.00 5.00 5.00  Table 3: Summary of experimental trials from Olson et al./[59].  Plate 1 2 3 4 5  Bar Width [mm] 3.0 2.5 1.0 3.0 6.0  Groove Width [mm] 3.0 2.5 2.5 8.0 12.0  Groove Depth [mm] 4.0 5.0 1.0 4.0 8.0  Bar angle [°] 17.3 21.5 21.25S, -26.5R 21.25 16.0  CEL [km/s] 20.05 24.1 233.2 7.0 0.521  Table 4: Summary of refiner plate geometries.  In addition to measurements of fibre length, some key changes in paper properties resulting from refining were measured, namely, sheet density, burst index, tear index,  34  Chapter 5: Experimental Program  breaking length, tensile index, as well as T.E.A. These were measured according to the appropriate TAPPI standard.  5.3  Escher-Wyss Heterogeneity Trial  Another set of tests was carried on the laboratory Escher-Wyss laboratory refiner in the Vancouver Laboratory of FPInnovations (Paprican) with assistance of Phil Allen of FPInnovations. This refiner has a rather coarse bar pattern with pulp circulating many times through the working zone. The following trial plan was employed: Trial # 1 2 2 4  CF [%] 2 3 4 2  SEL [J/m] 4 4 4 2  Energy Increments [kWh/t] 30/60/90/120 6/12/18/24/30/60/90/120 30/60/90/120 30/60/90/120  Table 5: Escher-Wyss trial conditions  The Escher-Wyss refiner was set to 1000 rpm. The plate patterns used in this study were the same as for Plate 5 in Table 4.  5.4  Conflo Refining Trial  In addition to the above tests, a series of refiner trials were carried out using a industrialsize Conflo JC-00 conical refiner (angle of conical refiner is 20°) at various speeds. The tests were carried out with the assistance of Bill Francis and Norman Roberts of FPInnovations. The entry and exit diameters of the rotor of this refiner were 192 mm and 380 mm. The refiner plate geometry data are shown in Table 6.  35  Chapter 5: Experimental Program  Bar Width [mm] 5.65R, 5.67 S  Groove Width [mm] 6.79R, 7.25S  Groove Depth [mm] 8.88R, 9.37S  Bar angle [°] 18  Table 6: Conflo JC-00 refiner plate geometry. The resulting CEL was calculated to be 1,009 m/rev. The pulp in use was a BC Interior SPF with 50% Lodgepole Pine, 40% White Spruce, 6% Sub-Alpine Fir and 4% other species. Table 7 summarizes the trial conditions. The target SEL was 4 J/m at a suspension consistency of 1%. Run # 1 2 3 4 5 6 7 8 9 10 11 12  ω Q Pnet SEL SRE [rpm] [l/min] [kW] [J/m] [kWh/t] 148 256 9.6 3.88 63 147 499 10.2 4.10 34 148 1019 9.7 3.88 16 996 262 67.9 4.05 431 996 527 68.0 4.06 215 996 1018 67.8 4.04 111 510 254 33.4 3.89 219 510 522 33.7 3.93 108 510 994 33.5 3.90 56 242 249 17.0 4.18 114 738 748 51.2 4.13 114 1095 1087 76.1 4.14 117 Table 7: Conflo JC-00 trial conditions  36  Chapter 7: Summary and Conclusion  6 Results and Discussion 6.1  Validation of Comminution Approach  As described earlier, some preliminary tests were carried out in the Sprout Waldron refiner by placing lab-cooked tracer fibres in a standard pulp. After refining, tracer fibres were sampled from the refined pulp and examined visually for physical deformations indicative of impacts. A visual separation of tracer fibres from the commercial chemical pulp was possible due to the colour difference. The tracer fibres were not bleached and therefore appeared brown whereas the White Spruce fibres were bleached and white in appearance. As shown in Figure 9, some fibres showed no evidence at all of having been impacted while others showed external fibrillation, yet others appeared straightened. Some cutting was also evident as shown in Figure 9.  White Spruce  Douglas Fir  Figure 9: Refined Douglas Fir (dark brown) and White Spruce fibres using the 12” Sprout-Waldron refiner  37  Chapter 7: Summary and Conclusion  As expected due to the high SEL of 3 J/m, the number of short fibres increased and the number of long decreased. Based on the resultant fibre length distribution, a simple calculation of the selection function probability for the longest fibre fraction, i.e. 4.3 mm long fibres, was performed using following relationship:  [  ]  n after ,longest = exp(− S longest ⋅ SRE ) ⋅ nbefore ,longest  (28)  0.05 for l i = 4.3 mm  Sprout Waldron DD (Trial A) DD (Trial M) Esher Wyss (Trial N)  Selection Function Value, Si [kWh/t]  -1  0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005 0 0  20  40  60  80  100  120  140  Specific Refining Energy, SRE [kWh/t]  Figure 10: Comparison of the selection function probability, Si, for 4.3 mm long fibres (White Spruce) for various trial conditions and refiners  The result was compared to values obtained with the double disc and Escher-Wyss refiner trials. As seen in Figure 10, for energies greater than 40 kWh/t the selection function was approximately independent of specific energy for the single disc, double disc and Escher Wyss refiners tested, as found by Olson et al./2003[59]). This energy independence of selection function allows comparison of selection functions between refiners operating at  38  Chapter 7: Summary and Conclusion  different energies. It also validates the application of our approach to calculate Si and Bij using Markov chains. However, once again, the new approach is only possible as long as there is a measurable amount of cutting. 0.01  1.2  White Spruce 0.009  Sprout Waldron Refiner Trial at const. SEL = 3.0 J/m  Douglas Fir  Selection Function Values, Si [kWh/t]  -1  1  0.008 0.007 0.8  0.006  Obtained by Transition Matrix Approach  0.005  0.6  0.004 0.4  0.003 0.002  Obtained by Tracer Method  0.2  0.001 0  0  0  0.5  1  1.5  2  2.5  3  3.5  4  4.5  Fibre Length l i [mm]  Figure 11: Selection function curve for Douglas Fir and White Spruce fibre fractions after refining at SEL = 3.0 J/m  The curves in Figure 11 show that the selection function dependence on fibre length for the two cases follows a similar trend but differs in magnitude. This difference may be due to several possible factors. By our estimate, the Douglas Fir tracer fibre is about 25% stronger than the White Spruce pulp (Ellis and Rudie/1991[85]), and likely even stronger because it is an undamaged lab-cooked fibre as opposed to the commercially produced White Spruce. Thus, it is less likely to rupture when captured, thereby giving smaller Si values as shown in Figure 11. Yet another factor is the assumption of an increment of energy being 30kWh/t. This is 1/4 of the total energy applied in this trial, and is therefore 39  Chapter 7: Summary and Conclusion  very large as an incremental energy. In contrast, the incremental energy in the matrixbased estimate is the much smaller energy of an individual bar crossing. Nevertheless, the two curves show a similar trend in that fibre shortening increases sharply with increasing fibre length for small fibres, then increases at a smaller rate for longer fibre lengths.  6.2  Selection and Breakage Functions  The statistical significance of the measured differences among the trials were evaluated by Student's t-Test. Differences emanating from the transition matrix calculation were considered statistically significant if the probability associated with a Student's t-Test was smaller than 0.65 (A value of 0 would mean that the fibres have been cut severely whereas a value of 1 signifies no cutting, meaning a change in arithmetic fibre length less than 15%.)  6.2.1 Selection Function We now examine the relationships between selection function and some refining variables at observed length changes. Figure 12 shows the selection function dependence on fibre length for various SEL’s on the Escher Wyss lab refiner using the same refining plate. The tests were conducted by varying the applied power and pulp consistency. It is apparent that the selection function increases with fibre length, but tends to plateau at longer fibre lengths as found in the tracer fibre studies (Figure 11). For a given fibre length, the selection function increases with increasing SEL, meaning increasing power because the CEL was not changed.  40  Chapter 7: Summary and Conclusion  0.003 0.8  Trial N, SEL 1.5 J/m  Selection Function Values, Si [kJ/kg]  -1  Trial O, SEL 3.0 J/m 0.0025  0.7  Trial P, SEL 5.0 J/m Trial Q, SEL 5.0 J/m  0.6  0.002 0.5  0.0015  0.4  0.3  0.001  0.2  Increasing SEL 0.0005 Escher-Wyss Refiner Trials at const. specific energy of 120 kWh/t 0  0.1  0  0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  Fibre Length l i [mm]  Figure 12: The experimentally determined selection function for the Escher–Wyss refiner trials using a single refiner plate for increasing SEL.  The effect of SEL on selection function is further illustrated in Figure 13 for fibres of 3.3 mm length from all of the tests. The probability of a fibre being selected for cutting clearly increases with SEL, moderately up to SEL = 2 J/m, more in the SEL 3-5 J/m range, and then greatly at the SEL = 8 J/m range. Thus, fibre shortening clearly depends on intensity as well as probability of capture.  41  Chapter 7: Summary and Conclusion  0.035  Trial L  Selection Function Values, Si [kWh/t]-1  Plate 1 (DD) Plate 2 (DD)  0.03  Plate 3 (DD)  Trial M  Plate 4 (DD) 0.025  Plate 5 (EW) for l i = 4.3 mm  0.02  0.015 Trial P,Q  0.01 Trial O 0.005  Low Intensity Plate: 2, 3 Moderate Intensity Plate: 1, 5 High Intensity Plate: 4  Trial N  0 0  1  2  3  4  5  6  7  8  9  SEL [J/m]  Figure 13: Selection function values for 4.3 mm long fibres for all 15 trials  The selection function decreases with consistency as shown in Figure 14 for a fibre length of 4.3 mm in a double disc refiner at a constant rotational speed with the same plate pattern. These data also show that larger consistency gives a smaller probability of fibre capture. In addition, a strong dependency on Specific Edge Load was evident, similar to the laboratory Escher Wyss results. The likely explanation for this decrease in selection function with increasing consistency is the decreasing mobility of fibres relative to one another. In the case of decreasing mobility, long fibres are not preferentially captured relative to short fibres. Consequently the 4.3 mm fibres in a high consistency suspension have a lower value of selection function.  42  Chapter 7: Summary and Conclusion  0.0050 SEL = 1.7 Ws/m SEL = 3.2 Ws/m  Selection Function Value, Si [kWh/t]  -1  0.0045 Trial A  0.0040 0.0035 0.0030  Trial B 0.0025 0.0020  Trial C Trial D  0.0015  Trial E 0.0010 0.0005  Si for l = 4.3 mm  0.0000 0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  Consistency [%]  Figure 14: Selection function dependency on Consistency for the same plate pattern on a Double Disc refiner (Trial A-E) for fibre length of 4.3 mm  6.2.2 Breakage Function As described earlier, unlike previous approaches, this study did not assume a form of the breakage function. By determining the breakage functions, some insight can be gained on the position along fibre lengths where rupture occurs. This is illustrated in Figure 15. For fibres less than 3 mm in length, there is an increasing probability that daughter fibres will be similar in length to the length of original fibres. Clearly in this length range there is no maximum. However, above 3 mm, there is a maximum at about 0.75, meaning that longer fibres are cut to about ¾ of their original length. This trend was found in all the cases tested. In summary Figure 15 suggests that refiners preferentially cut near the ends of fibres.  43  Chapter 7: Summary and Conclusion  1 4.3 4.0 3.8  0.8  mm fibre length l j  Breakage Function Bij [-]  3.5 3.2 0.6  2.9 2.7 2.4  0.4  2.1 1.9 1.6  0.2  1.3 1.0  0 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  Relative Resultant Fibre Length l i /l j [-]  Figure 15: Breakage probability of White Spruce fibres during trial E.  6.3  Determining Parameters Affecting Selection Function  We now consider the variables which govern the selection function in order to obtain the key parameters that describe heterogeneity of refining action. We do so from equation (26), which is reproduced below: S i = K ⋅ f (l i ) ⋅  mG ∞ ∞ ⋅ p I (I ) ⋅ p C (I C ) ⋅ dI ⋅ dI C SEL ∫0 I∫C  (26)  There are three unknowns in this equation: two, K and f(li) describe the probability of impact, and the third unknown, mT, the mass in the gap, gives the average intensity IA where IA = SEL / mT.  44  Chapter 7: Summary and Conclusion  6.3.1 Determining f(li) As described earlier, the functional form of the length dependence of capture factor, f(li), can be determined from the shapes of curves of the type, as shown in Figure 12 (dotted line). This was accomplished by fitting various length functions to the data by regression analyses. This procedure assumed that K and mT did not affect the form of the function. The form of f(li) examined were once reported by earlier workers, e.g. effect of groove depth (Kerekes1990/[55]) and consistency (Batchelor and Ludin/2006[60]). The various forms examined are summarized in Appendix G. The best fit obtained was of the form:  f (l i ) =  li l i + 0 .5 ⋅ D  (29)  6.3.2 Determination of K and mT The parameters K and mT in equation (26) were determined by regression fitting the data obtained for Si as function of li to equation (30) below. In this equation K is a unknown constant and mT appears in the average intensity IA (IA = SEL / mT) in the integrals. li mG ∞ ∞ Si = K ⋅ ⋅ ⋅ p I (I ) ⋅ p C (I C ) ⋅ dI ⋅ dI C l i + 0.5 ⋅ D SEL ∫0 I∫C  (30)  The results from the regressions are shown in Table 8. A representative energy input of SRE = 120 kWh/t was arbitrary chosen for the regression. The average number of impacts, NA, experienced by fibres was calculated from NA = mT /mG ·NB (equation (17)).  45  Chapter 7: Summary and Conclusion  Trial Plate Refiner Plate Intensity (SEL ) A B 1 1 moderate C D E F DD 2 2 low G H I 3 3 ultra-low J K L 4 4 highest M N O 5 EW 5 moderate P Q  SRE [kJ/kg] 450.0 432.0 414.0 435.6 439.2 414.0 414.0 421.2 327.6 446.4 424.8 432.0 414.0 432.4 429.1 428.0 427.0  SEL [kJ/m] 0.00324 0.00319 0.00170 0.00171 0.00171 0.00267 0.00108 0.00108 0.00014 0.00014 0.00007 0.00786 0.00800 0.00150 0.00300 0.00500 0.00500  mG [kg/m] 6.82E-04 1.09E-03 3.72E-04 5.95E-04 1.01E-03 1.05E-03 6.35E-04 1.13E-03 1.35E-04 2.20E-04 1.28E-04 2.89E-03 2.02E-03 2.88E-03 5.76E-03 7.68E-03 9.60E-03  CF [%] 2.84 4.55 1.55 2.48 4.22 4.19 2.54 4.52 2.70 4.40 2.56 4.51 3.15 1.50 3.00 4.00 3.00  K [-] 0.062 0.051 0.040 0.015 0.012 0.023 0.007 0.005 0.069 0.011 0.004 0.068 0.115 0.003 0.022 0.008 0.007  mT m T /m G IA NA [kg/m] [-] [kJ/kg] [-] 1.66E-04 0.244 19.5 23 2.06E-04 0.189 15.5 28 9.63E-05 0.259 17.7 23 8.49E-05 0.143 20.1 22 1.00E-04 0.099 17.0 26 1.76E-04 0.168 15.2 27 6.23E-05 0.098 17.3 24 6.58E-05 0.058 16.4 26 3.60E-05 0.267 3.9 84 6.37E-05 0.289 2.2 203 2.82E-05 0.220 2.5 171 1.98E-04 0.069 39.6 11 2.31E-04 0.114 34.7 12 6.62E-05 0.023 22.7 19 1.23E-04 0.021 24.3 18 1.66E-04 0.022 30.1 14 2.05E-04 0.021 24.4 18  Table 8: Comparison of obtained values for K, mT , IA and NA for a representative specific energy input of 120 kWh/t (all are converted into SI-units)  6.4  Discussion of Predicted Probabilities and Intensities  We now examine and compare the probabilities of impacts (capture) and the average intensity of impacts on fibres obtained from the regression analysis.  6.4.1 Probability of Impact Recalling from equation (29) that the factor of capture of a fibre of length li at a single bar crossing is given by FCi = K / (li + 0.5·D), we now examine factors which affect this probability. As expected, FCi increases with decreasing groove depth, as shown in Figure 16. This occurs because the fibre capture sweep zone by a bar edge occupies a larger fraction of the total groove depth. It is further evident in Figure 16 that FCi increases with increasing groove width up to about 5 mm, and then decreases beyond this level. The likely explanation for increasing FCi is that a wider groove gives a longer sweep zone which enables more fibre capture. However, this increase has a limit which is likely the capacity of the gap to hold fibre, or the maximum mass of fibres which a bar edge can 46  Chapter 7: Summary and Conclusion  capture. Such a limit was observed by Smith/1922[24] in drawing a bar through an infinite reservoir of pulp. For example, he found that at 3% consistency pulp, a bar captured fibres up to a level of about 0.001 kg/m. We may note here that a sweep length of 5 mm over a depth equal to a fibre length of 2.5 mm contains about 0.0004 kg/m pulp. For a sweep length of 12 mm, this would be about 0.001 kg/m, which is about the limit found by Smith/1922[24]. The probability of fibre impact also depends on flow conditions in a refiner. This is shown in Figure 17. As radial flow velocity increases, the probability of fibre impact increases for an average constant tangential velocity of 26.3 m/s (in case of the double disc refiner) or 6.1 m/s (Escher Wyss laboratory refiner) respectively. 0.08 for l A = 2.20 mm  Plate 1 (DD) Plate 2 (DD)  0.07  Plate 3 (DD) Plate 4 (DD)  0.06  Plate 5 (EW)  K *lA / (lA +0.5*D) [-]  D = 1.0 mm D = 4.0 mm  0.05 0.04 0.03  D = 4.0 mm  0.02 0.01  D = 8.0 mm  D = 5.0 mm  0.00 0  2  4  6  8  10  12  14  Groove Width G [mm]  Figure 16: Probability of capture for average fibre length lA during one bar crossing for various plate geometries  47  Chapter 7: Summary and Conclusion  0.08 Plate 1 (DD) 0.07  for l A = 2.20 mm  Plate 2 (DD) Plate 3 (DD)  0.06  Plate 4 (DD)  K *lA / (lA +0.5*D) [-]  Plate 5 (EW) G = 8.0 mm  0.05 0.04 0.03  G = 3.0 mm  0.02 0.01  G = 12.0 mm G = 2.5 mm  0.00 0.00  0.01  0.02  0.03  0.04  0.05  0.06  0.07  Average Radial Velocity vradial [m/s]  Figure 17: Capture probability for average fibre length lA during one bar crossing for various flow conditions  6.4.2 Intensity of Impacts Values for the average intensity of impact IA are shown in Table 8. For a given SEL, IA depends on mT. The values of mT from the regression vary from 2.3·10-4 kg/m to 2.83·10-5 kg/m, a range which appears reasonable. For example, assuming a consistency for the compressed fibre mat in a refiner gap to be of 30%, a gap of size 0.15mm and 3 mm bar width would hold about 10-4 kg/m of fibres. The values of mT obtained give levels of IA in Table 8 from 2.2kJ/kg to 39.6 kJ/kg. This range is also reasonable. Evidence of this is shown by fibre shortening in Figure 18, where fibre length is relatively unaffected up to an average intensity of about 20 kJ/kg, then diminishes significantly as intensity increases beyond this value.  48  Chapter 7: Summary and Conclusion  2.5  Average Fibre Length lLW [mm]  2  1.5  Unrefined 1  R2 = 0.9477  Plate 1 (DD) Plate 2 (DD) Plate 3 (DD)  0.5  Plate 4 (DD) SRE = 120 kWh/t  Plate 5 (EW) 0 0  5  10  15  20  25  30  35  40  45  50  Average Intensity I A [kJ/kg]  Figure 18: Fibre-length-reduction characterization as function of intensity. 0.08 Trial D Rupture Strength Fibres  Proabability Distribution pI(I) [-]  0.07  I CA = 35 kJ/kg  0.06 0.05 0.04 I A = 20 kJ/kg 0.03 0.02 0.01 0 0  5  10  15  20  25  30  35  40  45  50  Intensity I [kJ/kg]  Figure 19: Actual intensity distribution vs. the distribution of rupture strength, IC. 49  Chapter 7: Summary and Conclusion  The estimated fibre rupture strength is 35 kJ/kg, meaning that values in the high end of the distribution about 20 kJ/kg exceed this value. This takes place as shown in Figure 19.  6.5  Comparisons to Previous Studies 6.5.1 Intensities  We now compare findings of this study to other relevant published work. Leider and Nissan/1977[36] and Kerekes/1990[55] predicted intensities by dividing the specific energy by the estimated number of impacts on pulp fibres. Their predicted intensities are compared to values from this study in Figure 20. 45  Average Intensity from this Study, IA [kJ/kg]  DD refiner (Trial A-M) 40 35 30 25 20 15 10 5  C-Factor Analysis (Kerekes) Leider and Nissan Theory  0 0  5  10  15  20  25  30  35  40  45  Estimated Intensity by other Authors, I [kJ/imp/kg]  Figure 20: Comparisons of Average Intensity IA to intensity levels calculated based on previous studies by Leider and Nissan/1977[36] and Kerekes/1990[55].  50  Chapter 7: Summary and Conclusion  Clearly the C-Factor predicts larger values then those of this study whereas Leider and Nissan/1977[36] predict smaller values. Thus, the results of this study fall between the values of these earlier studies. Leider and Nissan predict low intensities because they estimate many more impacts than does the C-Factor. They assumed that the number of impacts per fibre per bar crossing is proportional to the ratio of cross-sectional area of a fibre to the cross-sectional area of a groove as well as to the rotational speed. On the other hand, Kerekes’ C-Factor assumed the probability of impact to depend upon the ratio lA / (lA + D) and suspension consistency, justifying these assumptions by showing that they led to logical outcomes in limiting cases.  6.5.2 Probabilities of Impacts in Compression Refining In recent work, Goosen et al./2007[86] showed that for compression refining, the development of tensile strength in paper was strongly linked to the probability that fibres experience one impact (loading cycle). This observation suggested that the role of multiple loading cycles in refining was to ensure that many fibres experienced at least one impact, as opposed to the conventional view that the purpose of multiple loading cycles was to introduce fatigue weakening. By a simple statistical model, Goosen et al./2007[86] showed that increase in tensile strength R(NC) was related to the cumulative probability of an impact, P, over NC cycles.  R( N C ) = 1 − (1 − P )  NC  (31)  We tested this relationship on the EW refiner by obtaining the tensile strength for given numbers of bar crossings NB (in place of NC), as shown in Table 9 and plotted in Figure 21. By regression fitting to equation (31) to these data, we obtained the probability P that gave the best fit. 51  Chapter 7: Summary and Conclusion  CF SEL [%] [J/m] 2  2  2  4  3  4  4  4  NB [-] BL [km] NB [-] BL [km] NB [-] BL [km] NB [-] BL [km]  6  12  18  31 3.396  62 3.869  93 4.154  SRE [kWh/t] 24 30 207 4.553 104 4.878 124 156 4.376 4.646 207 4.454  60 415 6.105 207 5.918 311 5.65 415 5.805  90 622 7.152 311 7.058 467 6.138 622 6.697  120 829 7.604 415 7.672 622 7.151 829 6.814  Table 9: Escher-Wyss trails including calculated number of bar crossings, NB and breaking length, BL 8.7  1.0 C=2%; SEL=2J/m 0.9  Commulative probability of fibre being refined  0.8 0.7  6.7  0.6 5.7  0.5  R(NB) [-]  Breaking Length [km]  7.7  0.4 4.7  0.3  R ( N B ) = 1 − (1 − P ) N B P = 0.0021  3.7  2.7 0  100  200  300  400  500  600  700  800  0.2 0.1 0.0 900  Number of bar crossings, N B [-]  Figure 21: Tensile strength plotted against cumulative probability for P = 0.0021.  The result for a consistency of 2% and an SEL = 2 J/m was P = 0.0021. This value is close to the value of FCi = 0.0023 obtained from K·lA / (lA+0.5·D) in Table 8 for the Escher Wyss trials of the previous section. For all four combinations of consistency and SEL, probability P ranged from 0.0019 to 0.0045. Though not conclusive, this finding seems to confirm that probabilities of impacts found in this study are sound, as can be 52  Chapter 7: Summary and Conclusion  seen in Figure 22. It further suggests that the role of multiple loading cycles in refining is to ensure that many fibres see at least one impact as opposed to introducing fatigue weakening, in short, that the role of multiple loading cycles is to reduce heterogeneity. 0.07 Trial A - Q 0.06  0.05  P [-]  0.04  0.03  0.02  0.01  0 0  0.01  0.02  0.03  0.04  0.05  0.06  0.07  K*l A /(l A +0.5*D) [-]  Figure 22: Comparison of K·lA /(lA+0.5·D) and P for Trials A - Q  6.6  Homogeneity Index  We now address the question of defining a useful parameter to represent heterogeneity. For convenience, in order to have a larger value represent a better result, we will define heterogeneity in terms of its inverse – homogeneity. A “Homogeneity Index” may be used to characterize operating refiners. From an industrial standpoint, it is desirable to have a single number index. However, in reality this is difficult. Homogeneity is affected by the number and intensity of impacts as well as by the distribution of each of these variables. Moreover, each of these has subset of additional factors. All this gives rise to 53  Chapter 7: Summary and Conclusion  many possible combinations of parameters that could be employed to define a homogeneity index, H. Somewhat arbitrarily, we choose the following: the product of the number of impacts on fibres of average length and the fraction of intensities within +/- 2 kJ/kg of the average intensity, divided by the specific refining energy, SRE. This can be expressed as:  K⋅ H=  (32)  lA ⋅ N B ⋅ P(I A ± 2kJ / kg ) ⋅ I A l A + 0 .5 ⋅ D SRE  Values of H for our tests are summarized in Table 10 along with the key refiner variables and computed parameters. Trial Plate Refiner Plate Intensity (SEL )  A B C D E F G H I J K L M N O P Q  1  2  1  DD  2  3  3  4  4  5  EW  5  SRE  [kJ/kg] 450.0 432.0 moderate 414.0 435.6 439.2 414.0 low 414.0 421.2 327.6 ultra-low 446.4 424.8 432.0 highest 414.0 432.4 429.1 moderate 428.0 427.0  SEL  mG  CF  mT  [kJ/m] 0.00324 0.00319 0.00170 0.00171 0.00171 0.00267 0.00108 0.00108 0.00014 0.00014 0.00007 0.00786 0.00800 0.00150 0.00300 0.00500 0.00500  [kg/m] 6.82E-04 1.09E-03 3.72E-04 5.95E-04 1.01E-03 1.05E-03 6.35E-04 1.13E-03 1.35E-04 2.20E-04 1.28E-04 2.89E-03 2.02E-03 2.88E-03 5.76E-03 7.68E-03 9.60E-03  [%] 2.84 4.55 1.55 2.48 4.22 4.19 2.54 4.52 2.70 4.40 2.56 4.51 3.15 1.50 3.00 4.00 3.00  [kg/m] 1.66E-04 2.06E-04 9.63E-05 8.49E-05 1.00E-04 1.76E-04 6.23E-05 6.58E-05 3.60E-05 6.37E-05 2.82E-05 1.98E-04 2.31E-04 6.62E-05 1.23E-04 1.66E-04 2.05E-04  NB  IA  NA  H  [-] [kJ/kg] [-] [-] 95 19.5 23 0.015 148 15.5 28 0.020 91 17.7 23 0.010 152 20.1 22 0.006 260 17.0 26 0.008 162 15.2 27 0.009 243 17.3 24 0.004 441 16.4 26 0.005 316 3.9 84 0.116 701 2.2 203 0.027 777 2.5 171 0.011 159 39.6 11 0.028 104 34.7 12 0.033 830 22.7 19 0.005 824 24.3 18 0.012 657 30.1 14 0.009 820 24.4 18 0.008  Table 10: Summary of Pilot-scale Refining Trials results  Several observations can be made from Table 10. The value of H for the Escher-Wyss refiner is similar to that for the Double Disc refiners when comparisons are made at a similar consistency, rotational speed and SEL. This is unexpected for the very coarse bar patterns of the EW. However, in this refiner, pulp circulates through it many times whereas pulp passes only once through the Double Disc refiners. Thus, extended 54  Chapter 7: Summary and Conclusion  residence time can compensate for coarse bar pattern. Another unexpected result is the large values of H for the “ultra-low intensity” (Trials I – K) over an SEL range from 0.07 to 0.14 J/m. The low intensity in these trials was compensated for by a large number of impacts to give the same SRE, more than four times the number in other cases studied, to give the same refining energy. This suggests that number of impacts is a much stronger factor than intensity in determining homogeneity of treatment. 0.020 0.018  1000 rpm  Homogeneity Index H [-]  0.016  1100 rpm  0.014  1000 rpm  0.012  510 rpm  750 rpm  1000 rpm  0.010 510 rpm 250 rpm  0.008 0.006  130  0.004  Number of bar crossings, N B  CONFLOW JC-00 SEL=4 Ws/m C = 1%  0.002  250 500  0.000 0  200  400  600  800  1000  1200  Flow Q [l/min]  Figure 23: Homogeneity development in dependency of throughput and refiner speed using a CONFLO refiner at 1% consistency and SEL = 4 J/m  Flow conditions in refiners also affect H. This factor was evaluated in the CONFLO refiner by varying the rotational speed and throughputs while keeping the SEL constant. The findings are shown in Figure 23. It shows that for a given number of bar crossings, higher throughput flow rate increases refining homogeneity. This agrees with Figure 17 which showed that increasing radial velocity through a refiner increased the probability 55  Chapter 7: Summary and Conclusion  of fibre impact at a bar crossing. The higher H may be due to more effective fibre mixing due to larger shear and turbulence. 20 Plate 1 (DD) - SEL = 1.7..3.2 J/m 18  Plate 2 (DD) - SEL = 1.0..2.7 J/m Plate 3 (DD)- SEL = 0.07..0.14 J/m  Tear Index [mN·m2/g]  16  Plate 4 (DD)- SEL = 7.8..8.0 J/m Plate 5 (EW)- SEL = 1.5..5.0 J/m  14  12  10  8  6 0  0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09  0.1  Homogeneity Index, H [-]  Figure 24: Relationship between Tear Index and Homogeneity Index, H.  Relationship between tear index and homogeneity is shown Figure 24. The ultra-low intensity plate (3) had the smallest loss in tear (shown as the largest tear strength). The other plates all caused a significant reduction in tear strength to the 10 – 12 mN·m2/g range, with the largest intensity (SEL) accomplishing this with the greatest homogeneity. In summary, the experimental results described above show that the analysis depicted in Section 4 presents a useful basis for characterizing uniformity of treatment of pulp fibres in refiner. Moreover, the results show that previous estimates of intensities of impact are either too large or too small, and that the probability of impact corresponds with previous measurements of probability of successful loading cycle in compression refining. 56  Chapter 7: Summary and Conclusion  7 Summary and Conclusions A method to measure heterogeneity of treatment in pulp refiners has been developed using comminution theory and measurements of fibre length before and after refining. This was accomplished by interpreting the selection and breakage functions of the comminution equation in terms of a single bar crossing event, namely probability of impact with fibres, intensity distribution of impact, and location of breakage along a fibre length. These probabilities were interpreted in terms of heterogeneity of treatment.  The method was applied to measure heterogeneity of treatment in three different types of refiners: a pilot disc refiner, an industrial disc refiner, and an Escher-Wyss laboratory refiner. This was accomplished by regression fitting the comminution equation to measured fibre length distributions before and after refining under various operating conditions. The measured probabilities and intensities of impact showed some key findings: -  probability of fibre impact at a bar crossing event increases with decreasing groove depth and increasing groove width up to about a groove width of 10 mm, then decreases  -  probability of fibre impact at a bar crossing event increases with radial pulp flow velocity  -  intensity levels were larger than those predicted by Leider and Nissan/1977[36] and smaller than those predicted by Kerekes/1990[55]  -  increase in tensile strength of pulp with refining was consistent with the cumulative probability of experiencing a loading cycle during a bar crossing  57  Chapter 7: Summary and Conclusion  event, in agreement with the findings of Goosen et al./2007[86] for loading cycles in compression refining  The results of this work enable refining heterogeneity to be characterized in a number of different ways. A simple one, chosen somewhat arbitrarily, is a “Homogeneity Index”, defined as the number of impacts on fibres having the average fibre length multiplied by the fraction of intensities within +/- 2 kJ/kg of the average intensity, divided by the specific refining energy. This index provided quantitative measurement of factors affecting homogeneity, postulated qualitatively by earlier workers, such as: -  increased flow rate through the refiner increased homogeneity of treatment  -  large numbers of impacts at low intensities give greater homogeneity at a given specific energy than do smaller numbers of impacts at higher intensities  -  large residence time in refiners can offset heterogeneity caused by very coarse bar pattern  58  Chapter 8: Recommendations  8 Recommendations This thesis has established a foundation upon which future work can build. This work could take several forms, for example further development of the method or applying it to refiners to correlate results with changes in pulp properties.  For further development of the approach, perhaps the greatest need is a measurement of the amount of pulp captured in a refiner gap, the variable called mT in this thesis. Predicted values of intensity from this work seem reasonable, falling between values predicted by other workers. However, a definitive measurement linked to refiner variables which affect mT would be of great value.  Another further development could be the use of force rather than energy as the basis of intensity related to strain and rupture of fibres. To obtain such forces, detailed knowledge of fibre distribution in refiner gaps, as well as the amount of fibres is needed.  On the question of applying this method, a comprehensive set of experiments ought to be carried out in which key pulp properties are linked to homogeneity of treatment defined in several ways.  In applying the method to mill refiners, it is recognized that most refiners are operated to minimize fibre cutting. Consequently the concept of a “bump test” should be explored in which power is increased for a short period to deliberately cut fibres for this test. While the conditions of a bump test differ from the operating condition, the difference can be  59  Chapter 8: Recommendations  taken into account in the analysis to obtain a useful characterization of homogeneity of treatment obtained.  Lastly, there are other potential “homogeneity indices” than the one presented here. These should be explored. Perhaps the benefits of a two-parameter index outweigh the simplicity of a one-parameter index, and this too should be explored.  60  References  References [1]  Higgins, H. 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R., & Nassar, R., Stochastic simulation of residence time distribution curves. Chemical Engineering Science, 40(9), 1743 (1985).  70  Appendices  Appendices Appendix A: Distributions of Intensity in Refiner Gaps  Once fibres are within the gap between refiner bars, there is a distribution of forces acting upon individual fibres. Clearly, this distribution is complex and can only be modeled approximately. We do so here by considering the pulp to be a mat upon which an average force is exerted that produces a distribution of local pressures. We consider the distribution to be similar to that acting on a wet pad of pulp. In recent work, Endres et al./2005[63] measured pressure distributions on pulp pads and found distributions of the type shown in Figure A.1. This can be represented by a Gamma distribution function, i.e. by equation (A.1), which is essence may represent a range of possible distributions by differing values of α, a shape parameter and β, a scale parameter. I  − 1 p Gamma (I ) = α ⋅ I α −1 ⋅ e β β ⋅ Γ(α )  (A.1)  Given the very approximate nature of our estimated pressure distributions in the gap of refiners, we chose a simple form of the gamma distribution (α = 2, β = 11). This gamma function is shown in Figure A.1 superposed on Endres’ measured pressures. For the purpose of this study, we consider local intensity, I, to have the same distribution as local pressure. Therefore we may express the probability distribution of I, pI,(I) as p I ( I ) = aIe − bI  (A.2)  71  Appendices  0.016  0.04 Measurement by Endres  0.014  0.035 0.03  0.01  0.025  0.008  0.02  0.006  0.015  0.004  0.01  0.002  0.005  Relative Frequency [%]  Measured Frequency [%]  Gamma Distribution 0.012  0  0 0  10  20  30  40  50  60  Pressure [Mpa]  Figure A.1: Pressure Distribution within a fibre flock covering a 8 x 8 mm area, at ≈ 30% consistency measured by Endres et al./2005[63] Prairie et al./2007[64] have shown that in a refiner gap, under low consistency conditions, compression forces are large, about 10 times the shear forces, and therefore are likely to vary in a similar manner to the measurements of Endres.  72  Appendices  Appendix B: Integration of Pressure Distribution  Here we determine the parameters in the pressure distribution on pulp mat cited in Appendix A. p I ( I ) = aIe − bI  (B.1)  First, we determine the relationship between a and b by normalizing the distribution ∞  (B.2)  p I (I )dI = 1 [kJ kg ]  ∫ 0  ∞  ∫ aIe  − bI  0  a b  2  (  a − bxe (−bx ) − e (− bx ) dI = b2  )  ∞  = 1 [kJ kg ] 0  (B.4)  = 1[kJ kg ]  ∴b =  (B.3)  (B.5)  a 1[kJ kg ]  Next we determine values of a and b in terms of average intensity IA. ∞  ∫  p I (I )IdI = 1[kJ kg ] ⋅ I A  (B.6)  0  Upon substitution of equation (B.5) into (B.1) we obtain ∞  ∫I  2  −  ⋅a⋅e  a I 1[kJ kg ]  dI = 1[kJ kg ] ⋅ I A  (B.7)  0  2 ⋅ 1[kJ kg ] a 1[kJ kg ]  = 1[kJ kg ] ⋅ I A  2 a = IA 1[kJ kg ]  (B.8)  (B.9)  73  Appendices  ∴a =  4 ⋅ 1[kJ kg ] IA  (B.10)  2  Substituting equation (B.10) into (B.5) gives: b=  4 ⋅ 1[kJ kg ] I A ⋅ 1[kJ kg ] 2  =  (B.11)  2 IA  This gives pI (I ) =  4 ⋅ 1[kJ kg ] IA  2  −  ⋅I ⋅e  2 ⋅I IA  with I A =  SEL mT  (B.4)  For simplicity, the dimension [kJ/kg] of the normalization to 1 is not be shown in subsequent equations.  74  Appendices  Appendix C: Fibre Strength Distributions  Fibres in a sample of pulp to not have equal strength. Many factors contribute to the distribution of strength among fibres, one of which is the fibril angle. The dependence of tensile strength of individual fibres on fibril angle found by Page et al./1972[87] is shown in Figure C.1.  White Spruce, 60% yield kraft Spring wood Summer Wood 15  9  2  Tensile Strength [10 dynes/cm ]  20  10  5  0 0  10  20  30  40  50  Fibril Angle [º]  Figure C.1: Tensile strength vs. fibril angle for springwood and summer wood fibre of white spruce, reproduced from Page et al./1972[87]  In this study, we estimate fibre strength by the energy needed to rupture a fibre in tension, calling this the average rupture intensity IC. We then consider distributions of rupture intensity.  75  Appendices  C.1 Average Rupture Intensity IC From Hooke’s law it is known that:  σ=  (C.1)  Fmax AF with  EM =  σ  stress (tensile strength), [N/m2] = [Pa] = [dyne/cm2]  Fmax  maximum force, [N]  AF  cross sectional area, [m2]  σ ε  (C.2) with  EM  elastic modulus (Young’s modulus), [Pa]  σ  stress (tensile strength), [N/m2]  ε  elastic strain (dimensionless)  Specific energy, IC, required to strain a single fibre is given by: IC =   N m2   N ⋅ m  J  =  =  3   kg m   kg   kg   1 1 ⋅ ⋅ EM ⋅ ε 2 ρ 2 with  (C.3)  ρ is cell wall density, [kg/m3]  If straining occurs under shock conditions, a state of unaxial strain results, and the above energy is increased (Mark et. al/2001[88]), as shown below: IC =  1 1 (1 − υ ) ⋅ ⋅ EM ⋅ ε 2 ⋅ (1 − 2 ⋅ υ ) ⋅ (1 + υ ) ρ 2 with  υ  (C.4)  Poisson ratio  76  Appendices  We estimate a strain of 10% as the level necessary to deform a fibre in tension to the point of fibre rupture. Other parameters are:  υ = 0.35 ρ = 1500 [kg/m3]  Thus, the specific energy for rupture can be expressed as: IC =  (1 − 0.35) 1 1 2 ⋅ ⋅ EM ⋅ (0.1047 ) ⋅ 3 (1 − 2 ⋅ 0.35) ⋅ (1 + 0.35) 2 1500 kg m  [  ]  (C.5)  From Page et al./1976[89] data we can calculate an average Young’s modulus for the white spruce fibres as: 0.659 ⋅ 10 9 [Pa ] = 6.57GPa 0.1047  (C.6)  I C , average = 5.864 ⋅ 10 −6 m 3 kg ⋅ 6.57[GPa ]  [  (C.7)  ∴ I C , average = 38.55[kJ kg ]  (C.8)  EM average =  σ average ε  =  ]  This rupture intensity value may vary from furnish to furnish. For simplicity we estimate the average rupture strength of white spruce fibres is around 35 kJ/kg.  77  Appendices  C.2 Distribution of Rupture Intensity Fibres have a range of fibre strength IC. Little has been published on this. The available information (D.H. Page and F. El-Hosseiny/1976[89]) is shown in Figure C.2:  Figure C.2 has been removed due to copyright restrictions. The information removed is decribed in the Figure description below. The reference source is: Page, D. H., and El-Hosseiny, F., The mechanical properties of single wood pulp fibres. Pt 4: The influence of defects. Svenk. Papperstidn. 14:471-474 (1976)  Figure C.2: Frequency distribution of tensile strength for black spruce acid sulphite fibres, pulped to 63% yield, at 0.42 and 1.50 span length, reproduced from Page et al./1976[89]  We replaced the distribution in Figure C.2 by a Weibull distribution function shown below in terms of a probability of rupture at an intensity equal to the average rapture strength, pC(IC). p C (I C ) =  k  λk  with  ⋅ IC  k −1  ⋅e  I  −  C   λ   k  k  shape parameter  λ  scale parameter  (C.9)  78  Appendices  Considering the lack of knowledge concerning the functional form of rupture distribution functions, the data may be approximated by a Gamma functions as well, as shown in Figure C.3. 0.08 Gamma Distribution Weibul Distribution  0.07  Frequency [-]  0.06 0.05 0.04 0.03 0.02 0.01 0 0  10  20  30  40  50  60  70  80  Rapture Strength IC [kJ/kg]  Figure C.3: Comparison of the calculated rupture strength distribution of White Spruce fibres using Gamma and Weibull probability distribution functions  We may estimate the parameter of pC(IC) for the Weibull distribution function based on the assumed IC,average = 35 [kJ/kg] and a standard deviation of 5.9 [kJ/kg] to be k = 7 and  λ = 37.41 [kJ/kg]. Therefore, equation (C.9) becomes: p C (I C ) = 1[kJ kg ] ⋅  7  6  37.417 [kJ kg ]  7  ⋅ IC ⋅ e    IC  −   37.41[kJ kg ]   7  (C.10)  79  Appendices  Appendix D: Breakage and Selection Functions During comminution, the number of fibres of length i decreases due to shortening of fibres of size i to smaller sizes, and increases its numbers to the shortening of larger sizes j to size i. The Breakage Function Bij reflects the probability of fibres of length j being shortened to size i as shown by Berthiaux/2000[84]. These changes are described in terms of incremental energy input by the comminution equation (equation (9) in text), which can be rewritten as the following matrix form: (D.1)  dn = A⋅n dE where A = (B − I n ) ⋅ S  (D.2)  Here, n(E) is a column vector whose entries are the ni fibre length fraction in each size interval after the energy input, E. When referring to size intervals indexed from 1 (for the longest fibres) to m (for the shortest fibres and fines), Breakage and Selection functions can be represented by matrix notation, denoted by B the breakage matrix (whose elements are Bij) and by S the selection matrix (whose elements are Si). It is assumed here, that fibres can only undergo length reduction during refining; B is a lower triangular matrix. S is a diagonal matrix. Upon multiplication with the identity matrix, In, A becomes a lower triangular matrix. The solution of equation (D.1) can be made by diagonalization of A, introducing matrix UT as the matrix of the eigenvectors. Thus, it can be demonstrated that  n(E ) = V (E ) ⋅ n(0) V (E ) = U T ⋅ R D (E ) ⋅ U T A = −U T ⋅ S ⋅ U T  −1  (D.3) −1  (D.4) (D.5)  V(E), the so called transition matrix, is a lower triangular matrix which depends on the energy of refining through the previous set of equations, in which RD(E) is a diagonal 80  Appendices  matrix whose entries are e(-Si·E). A process is said Markovian if V(E) is only dependent on the current state and not on the previous states. One can assume that V(E) remains constant for a serious of batch refining processes – which can be linked to a continuous refining. In this case, the evolution of the system is given by: n t +1 = V ( E ) ⋅ nt  (D.6)  If V(E) is known, nt may be determined from an initial state vector n0: nt = V ( E ) t ⋅ n0  (D.7)  where V(E)t is the matrix V(E) raised to power t. The estimation of V(E) is not straightforward and is difficult to be achieved by a usual linear regression. The assessment of V(E) from experimental data is still a subject of study. In this case, the simple matrix calculation may be very difficult or unstable. In recent work, Berthiaux and Dodds/1997[90] have illustrated the difficulties in such assessments and have presented a new estimation technique of breakage and selection parameters in batch grinding. This method has been tested (see Appendix E). Their method requires a prior smoothing of the fibre length distribution. To assess the transition matrix for a given refining step, we employ the approach of Bertrand et al./1999[91]. The transition matrix V(E) has some constraints, which are listed below:  1*. As the elements of V(E) are probability values, they are null or positive and less than 1: 0 ≤ v ij ≤ 1 .  2*. If we suppose that there are no losses during refining, the sum of each column of V must be equal to 1:  81  Appendices  (D.8)  x  ∀j , ∑ v ij = 1. i =1  3* The size of a fibre cannot increase during refining. For j>i, we have vij=0.  In our work, we estimate the transition matrix V(E) from experimental obtained sequence of fibre length histograms or distributions respectively, n0, n1, …, nq (see Chapter 5). Following Bertrand et.al/1999[91], we employ a simple version of a monolayer neural network for the transition matrix estimation. This method does not require complex matrix manipulation or numerical assessment of derivatives. Using a sequence of fibre size histograms n0 and nt+1 it is possible to estimate the transition probabilities vij.. The histograms can be obtained from experiments carried out in laboratory and industrial refiner (n0, n1, …,nq). The basis of the approach is to step-by-step slightly modify the values of the transition probabilities in order to reduce the error of the model. When presenting a given vector nt as input of the system, the least square error is given by: m  ∑ (n  error =  − out i )  2  i ,t +1  (D.9)  i =1  where ni,t+1 represents an element of the vector of expected (observed) value and outi the estimated values assessed according to j  out j = ∑ vij ⋅ n i ,t  (D.10)  i =1  The method of steepest descent can be used for stepwise reducing the error. For this purpose, one must assess (∂error)/(∂vij), the change of error as a function of the change in the transition probability vij. It can readily be shown that (∂error)/(∂vij) is given by:  82  Appendices  ∂error = ni ,t (out j − n j ,t +1 ) ∂vij  (D.11)  Decreasing the error value means a transition probability change proportional to (∂error)/(∂vij). The resulting algorithm is very simple and is given below. (1) The weights (probabilities of the transition matrix) are initialized by giving them small positive random values, respecting the constraints (2*). (2) A discrete t value is randomly selected in the range from 0 to q -1. (3) The observed histogram nt is presented as input of the neural network. (4) The elements of the output vector outi is then assessed according to equation (D.10). The expected values are the elements of nt+1. (5) The weight (transition probabilities) is then modified according to: vij ← v ij + µ ⋅ n i ,t (n j ,t +1 − out j )  (D.12)  where µ, the learning rate parameter, is a positive value, less than 1. (6) If a given weight vij is less than 0, it is replaced by 0. (7) In order to respect constraint (2*), the sumj is then assessed, according to: m  sum j = ∑ v ij  (D.13)  i =1  The weights vij are then replaced by (vij)/(sumj). (8) After each number of iterations, the least-square error of the model is assessed from: 1 q m (ni,t − out i,t )2 error = ∑ ∑ m ⋅ q t =1 i =1  (D.14)  If the error is larger than a given threshold, the procedure is iterated by performing again the steps (2) to (8). Otherwise, the procedure is finished. The steps (6) and (7) above are 83  Appendices  specific of the current problem, and are needed in order to respect constraints (1*) and (2*), respectively. Once the weight elements vij are obtained, the resultant transition matrix V(E) is then decomposed into its characteristic eigenvalue and eigenmatrix vectors and within it determine the selection function matrix S. It is then possible to use this information and equations (D.4) and (D.5) in order to determine the breakage probability matrix B by going back to equation (D.2).  84  Appendices  Appendix E: Validation of Transition Matrix Approach To validate our transition matrix method and MatlabTM calculation routines, we conducted a series of computation for which Transition matrix, or the selection function was known and compared them with our determined selection and breakage functions.  E.1 Assessment of the Transition Matrix From a time sequence of particle size histograms Bertrand et.al/1999[91] assessed, using a linear neuron network, the transition matrix as described in Appendix D. Their estimated transition matrix is shown in Table E.1. >2500 2000 1600 1250 1000 800 630 500 400 315 250 200 <200  >2500 0.3882 0.1297 0.1315 0.1068 0.0635 0.0461 0.0355 0.0261 0.0195 0.0136 0.0091 0.0093 0.0212  2000  1600  1250  1000  800  630  500  400  0.0493 0.2008 0.2140 0.1331 0.0880 0.0670 0.0584 0.0438 0.0289 0.0214 0.0348 0.0604  0.2339 0.2581 0.1687 0.0969 0.0606 0.0462 0.0298 0.0230 0.0183 0.0119 0.0524  0.3122 0.2450 0.1557 0.0869 0.0502 0.0294 0.0284 0.0221 0.0039 0.0662  0.3336 0.2587 0.1538 0.0754 0.0375 0.0355 0.0254 0.0015 0.0786  0.3965 0.2680 0.1351 0.0566 0.0394 0.0253 0.0006 0.0786  0.4770 0.2458 0.0975 0.0526 0.0295 0.0061 0.0916  0.5551 0.2003 0.0837 0.0388 0.0031 0.1190  0.6315 0.1493 0.0609 0.0001 0.1582  315  250  200  <200  0.7521 0.0622 0.8722 0.0538 0.0573 1.0000 0.1319 0.0705 0.0000 1.0000  Table E.1: Transition matrix assessed by Bertrand et.al/1999[91]  As previously mentioned, the upper triangle of the matrix includes elements the probability of which is equal to 0. As there were 13 particle size classes, the matrix was dimensioned 13 x 13. The elements V13,13 (bottom-right) on the table corresponded to the finest particle size. When the particles reached this state, they remained in this class and therefore the corresponding probability was equal to 1. From this estimated transition matrix, we obtained an estimation of the particle size distributions. For this purpose, we applied equation (D.6) with V(E) being the estimated transition matrix given in Table E.1, and nt, the particle size distribution for a given 85  Appendices  energy or passage. The initial particle size distribution was represented by a vector n0 the elements of which are equal to 0, except the first one which is equal to 1. The resultant particle size distributions are shown in Table E.2.  >2500 2000 1600 1250 1000 800 630 500 400 315 250 200 <200  n0  n1  n2  n3  n4  n5  1 0 0 0 0 0 0 0 0 0 0 0 0  0.3882 0.1297 0.1315 0.1068 0.0635 0.0461 0.0355 0.0261 0.0195 0.0136 0.0091 0.0093 0.0212  0.1507 0.0567 0.1079 0.1365 0.1115 0.0934 0.0788 0.0634 0.0463 0.0363 0.0259 0.0211 0.0717  0.0585 0.0223 0.0564 0.0987 0.1059 0.1095 0.1073 0.0946 0.0717 0.0614 0.0452 0.0306 0.1378  0.0227 0.0087 0.0254 0.0564 0.0757 0.0963 0.1124 0.1121 0.0916 0.0841 0.0641 0.0401 0.2106  0.0088 0.0034 0.0107 0.0284 0.046 0.0708 0.0989 0.1137 0.1028 0.1014 0.0809 0.0505 0.2839  Table E.2: Calculated evolution of particle size distribution  The estimated transition matrix using our proposed linear neuron network is shown in Table E.3. >2500 2000 1600 1250 1000 800 630 500 400 315 250 200 <200  >2500 0.3885 0.1287 0.1323 0.1066 0.0635 0.0459 0.0355 0.0261 0.0196 0.0136 0.0091 0.0093 0.0213  2000  1600  1250  1000  800  630  500  400  0.0498 0.2014 0.2138 0.1323 0.0884 0.0667 0.0582 0.0439 0.0290 0.0214 0.0352 0.0600  0.2352 0.2584 0.1667 0.0974 0.0609 0.0463 0.0299 0.0227 0.0183 0.0119 0.0522  0.3134 0.2453 0.1557 0.0873 0.0497 0.0293 0.0281 0.0219 0.0039 0.0655  0.3338 0.2598 0.1531 0.0752 0.0376 0.0354 0.0255 0.0015 0.0780  0.3968 0.2678 0.1346 0.0569 0.0395 0.0256 0.0006 0.0783  0.4759 0.2451 0.0982 0.0528 0.0297 0.0062 0.0922  0.5574 0.1995 0.0838 0.0383 0.0031 0.1179  0.6339 0.1489 0.0606 0.0001 0.1565  315  250  200  0.7508 0.0630 0.8713 0.0542 0.0576 0.9235 0.1319 0.0711 0.0765  <200  1  Table E.3: Assessed Transition Matrix  The estimated transition Matrix and within it the computed particle size distribution were generally in good agreement with the given ones from Table E.2. The root mean square  86  Appendices  error between the given and estimated particle size distribution was 0.00142, which showed that the model was rather well-fitted to obtain a transition matrix.  E.1 Are B and S Matrices unique? Pettersen and Sandvik/1992[92] have raised the question whether several sets of S- and Bmatrices exist to form the required A-matrix? Assuming that S’ and B’ are equivalent in terms of a diagonal matrix and a lower triangular matrix respectively – so that they combine into the same A-matrix, but different to absolute values of the elements of the “correct” S and B one can write:  (B ′ − I )S ′ = (B − I )S  (E.1)  one then has:  (B ′ − I ) = (B − I )S ⋅ S ′ −1  (E.2)  B ′ = I + (B − I )S ⋅ S ′ −1  (E.3)  (  B ′ = BS ⋅ S ′ −1 + I − S ⋅ S ′ −1  )  (E.4)  B has all diagonal elements equal to zero (besides for i=j = max: bij = 1), hence the first  term is a matrix with all terms equal to zero (besides for i=j = max: equals to 1). Hence all diagonal terms are given by:  (I − S ⋅ S ′ ) −1  (E.5)  Now S·S-1 is simply a diagonal matrix with all terms equal to si·s’i-1 where si and s’i are the i’th elements of the matrix S and S’. For B’, to have all diagonal terms equal to zero (besides for i=j = max: equals to 1), one clearly requires si·s’i-1 = 1 (besides for i= max: si = 0). This means that the matrix S’ has to be equal to S, and hence that B’ has to be equal  87  Appendices  to B. This is contrary to the initial assumption. Hence one may conclude: There is only one B and one S which together combine in a specific A matrix. To further illustrate the meaning of this we conducted a series of computation for which the selection function was known and compared them with our determined selection. The initial particle size distribution was represented by a vector n0 the elements of which are equal to 0, except the first one which is equal to 1. The resultant particle size distributions are shown in Table E.4. Upon assuming a randomly generated Selection and Breakage Function matrix we obtained the resultant particle size distribution.  1 2 3 4 5 6 7 8  n0  n1  n2  n3  1 0 0 0 0 0 0 0  0.977 0.005 0.005 0.004 0.003 0.002 0.002 0.003  0.955 0.010 0.010 0.008 0.006 0.004 0.004 0.006  0.933 0.014 0.015 0.012 0.009 0.006 0.006 0.009  Table E.4: Particle size distribution  The estimated particle size distributions were computed using the proposed Transition matrix method. The root mean square error between the given and estimated particle size distribution was 5.7·10-4. Using equation (D.2) and equation (D.4) resulted enabled us to compare the given and estimated Selection Function values, which is shown in Table E.5. Si Assumed Estimated  1 0.023 0.024  2 0.018 0.019  3 0.014 0.014  4 0.010 0.010  5 0.007 0.007  6 0.005 0.005  7 0.003 0.003  8 0 0  Table E.5: Comparison of assumed and estimated Selection Function values  The root mean square error between the given and estimated Si was 0.0005, which showed that the model is well-fitted to obtain a selection function. 88  Appendices  Appendix F: Validation of Approach by Tracer Experiment A “tracer experiment” was carried out to validate the Markov chain (transition matrix) approach used in this thesis. This experiment was carried out by placing a small amount of “dyed” fibres all of the same length and defect free – as a result of being lab-cooked, into the pulp and subjected this to a small amount of refining. Refining Energy Input  1  1  2  2  j  Tracer  j  vjj  i  vji Transition Probabilities x-1  x-1  x  x  vjx Number of Fibres in each Length Class at Energy-State E  Number of Fibres in each Length Class at Energy-State E + δE  Figure F.1: Stochastic Approach to Batch Refining  For a “tracer experiment” as show in Figure F.1 one can follow all the tracer fibres that belong to class j at energy input E. Let us note by n*Tracer,j(E) the fibre length fraction of 89  Appendices  the tracer in class j, n*Tracer,i(E+ δE) being the fibre length fraction of the tracer in class i after one refining step of energy input δE. For any size fraction j, the probability transition t(j, i) from state j to state i is t ( j, i) =  * nTracer ,i (E + δE )  (F.1)  * nTracer , j (E )  Based on Bertriaux/2000[84] it can be shown that for this very small energy input, the breakage and selection functions can be estimated: Bij =  Sj =  * nTarcer ,i (E + δE )  (F.2)  * * nTarcer , j (E ) − nTarcer , j (E + δE )  * * nTarcer , j (E ) − nTarcer , j ( E + δE ) * nTarcer , j (E ) ⋅ [E + δE − E ]  =  1 δE  *  nTarcer  , j (E + δE ) ⋅ 1 −  * nTarcer , j (E )    (F.3)  Figure F.1 shows a batch refining process for which the change in fibre length size between energy input t and energy input E + δE is determined. Let us assume that refining occurs by elementary steps representing breakage events, an assumption which is made in determining the breakage matrix. For repetitive refining this hypothesis seems to be realistic as a given refiner revolution can be seen as an elementary event. For continuous refining processes, in which breakage occurs through a high intensity of randomly distributed events (in which fibres do not all get treated at the same time), this assumption will still stand as long as interval δE is chosen: not too small to ensure that at least one refining event occurs during this energy interval, and not too big to avoid lack of precision due to the multiplicity of the refining events during δE. As pointed out by Fan et al./1985[93], reducing the elementary energy input factor leads to an increase in the precision of the calculations as it allows a closer approach to a continuous grinding phenomena. However, it can also dramatically increase the calculation time. 90  Appendices  Appendix G: Form of Capture Function f(li) Below are the various forms of f(li) that were explored as best representing the dependence of Si on fibre length. This was accomplished by least square fitting to minimize differences between the function and length data. f (l i ) = 1 − e  f (l i ) =  (G.1)  li li + D  (G.2)  li l i + k1 ⋅ D  f (l i ) = 1 − e  f (l i ) =  − k1 ⋅  − k1 ⋅C F  ⋅e  −k2 ⋅  li ⋅ 1 − e − k 2 ⋅C F l i + k1 ⋅ D  (  f (l i ) = 1 − e  k1 ⋅C F  ⋅e  k2 ⋅  (     − k1 ⋅  li D  (G.4)  )  (G.5)  li D  f (l i ) = 1 − e − k1li ⋅ e − k 2C F f (l i ) = 1 − e  (G.3)  li li + D  )    ⋅ 1 − e − k 2 ⋅C F    (  (G.6)  )  (G.7)  We tested the different equations (G.1) to (G.7) to find the best fit to our length vs. selection function value data. The method of least squares is used to minimize the distance between the input curve points and the theoretical function. As a criteria to choose the appropriate length depended capture function the absolute error between input and theoretical value was used. This was evaluated for all 15 trials. An example for this is shown in Figure G.2: where Trial M and the corresponding graphs representing equations (G.2) and (G.5) are compared. Considering the given lack of knowledge of the refining process and it influential parameter and the result of the above analysis we deem the capture function to be best represented by equation (G.2).  91  Appendices  0.03  0.85  0.65 0.02  0.55 0.45  0.015 0.35 Trial M G.2 G.5  0.01  0.25  Capture Function, f(li) [-]  Selection Function value, Si [kWh/t]  -1  0.75 0.025  0.15  0.005 0.05 0  -0.05 0.0  0.5  1.0  1.5  2.0  2.5  3.0  3.5  4.0  4.5  5.0  Fibre Length, l i [mm]  Figure G.2: Comparison of best fit for Capture Function G.2 and G.5 for Trial M  92  

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