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Observation of transition from laminar plug to well-mixed flow of fibre suspensions in Hagen-Poiseuille… Nikbakht, Abbas 2016

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Observation of Transition fromLaminar Plug to Well-Mixed Flowof Fibre Suspensions inHagen-Poiseuille FlowbyAbbas Nikbakht,B.Sc., K.N.Toosi University of Technology, 2003M.Sc., Shiraz University, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2016c© Abbas Nikbakht, 2016AbstractThe focus of the present work is an experimental study of the transition to turbulentflow of papermaking fibres in a cylindrical pipe. The suspensions used in this studypossess a yield stress. With this class of fluid the radial profile in fully developedslow flow is characterized by an unyielded or plug zone. With increasing flow ratesthe size of the plug diminishes. One of the open remaining questions with these sus-pensions is the role of the plug during transition. In this work we characterize thesize of the plug using ultrasound Doppler velocimetry as a function of flowrate fordilute, i.e. less than 2% consistency papermaking suspensions in a 50mm diameterpipe. The plug size was determined through analysis of local spatial and temporalvariations in velocity, strain-rate, and the fluctuating component of velocity. Withthese we were able to estimate the yield stress of the suspension through knowledgeof the applied pressure gradient and find the yield stress to be in the range of 2-60 Pa,depending upon consistency, fibre type and manufacturing methodology. The yieldstress measurements were benchmarked against measurement methodologies reportedin the literature. During flow, we observe complex behavior with the plug in whichwe found that with increasing velocity the plug diminishes through a densificationmechanism in response to increasing frictional pressure drop. At higher Re, it dimin-ishes through an erosion type mechanism. We estimate the critical Reynolds numberiiAbstractfor the disappearance of the plug to be Rec 105. In perhaps the most unique mea-surements in this work we find that drag reduction begins when rp/R < 0.8, evenwhen dilute long chain polymers are added to the system. For papermaking fibres,drag reduction displays non-monotonic behavior with plug size.iiiPrefaceIn this work, I was responsible for experimental design, experimental procedures anddata analysis. Dr. Olson and Dr. Martinez supervised the research and providedfeedback and reviewed the manuscript. The data for the flow of mono disperse nylonsuspensions in a pipe is that of Chen Wang, a research colleague.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Understanding Flowing Fibre Suspensions . . . . . . . . . . . . . . . 62.1.1 Velocity Profiles of Wood Fibre Suspensions . . . . . . . . . . 162.2 Turbulent Drag Reduction . . . . . . . . . . . . . . . . . . . . . . . . 183 Methods and Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . 22vTable of Contents3.1 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 536 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72A Ultrasonic Doppler Velocimetry . . . . . . . . . . . . . . . . . . . . . 72B Table of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75C Matlab Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113viList of Tables2.1 Yield stress data for soft wood bleached craft suspension obtained usingdifferent methods (From Derakhshandeh et al. (2011)). . . . . . . . . 63.1 A summary of the experimental conditions tested. For each fluid andfor each concentration, several different bulk velocities were chosen tocover all the transition stages indicated in the introduction. . . . . . 263.2 A summary of the experimental conditions tested to study nylon fibresuspensions flow. For each fluid and for each concentration, severaldifferent bulk velocities were chosen to cover all the stages indicated inthe introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.3 A summary of the experimental conditions tested to study the effect ofpolymer additive on the fibre suspension flow (Softwood, NBSK). Foreach fluid and for each concentration, several different bulk velocitieswere chosen to cover all the stages indicated in the introduction. . . . 284.1 Summary of literature and experimental values of the yield stresses(at the onset of the flow) and fitted constants to the power-law model(τy = aCmb). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48viiList of Tables6.1 Estimates of the plug size, yield stress and the ratio of Reynolds stressto yield stress for Series 2-30 at the conditions given in Table 3.1 . . . 766.2 Estimates of the plug size and yield stress of nylon fibre suspensionsfor Series 1-16 at the conditions given in Table 3.2 . . . . . . . . . . . 966.3 Estimates of the plug size, yield stress and the ratio of Reynolds stressto yield stress for Series 3-8 at the conditions given in Table 3.3 . . . 1066.4 Pressure drop per unit length of the pipe for the fibre suspensions givenin Table 3.3, Series 3-5. . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.5 Pressure drop per unit length of the pipe for the fibre suspensionsmixed with polymer (100ppm of Polyacrylamide) given in Table 3.3,Series 6-8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1106.6 Drag reduction for fibre suspensions for three different consistencies asa function of plug size for cases studied in Table 3.3 and presented in4.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1116.7 Drag reduction for fibre suspensions mixed with 100ppm of polymer(Polyacrylamide) for three different consistencies as a function of plugsize for cases studied in Table 3.3 and presented in 4.14. . . . . . . . 112viiiList of Figures2.1 Examination of the deviations of f from 64/ReG for three differentclasses of fluids. Glycerin represents a Newtonian fluid. Xanthan andCarbopol represent shear thinning and viscoplastic fluids, respectively.Reproduced from Guzel et al. (2009b) . . . . . . . . . . . . . . . . . 122.2 Schematic of the stages of transition. Reproduced from Jäsberg (2007) 152.3 Schematic representation of pressure drop versus flow rate for fibresuspension in a flow channel. The water curve is measured in a hy-draulically smooth pipe, and dotted lines divide the flow domain intofive main regimes (labeled I - V) based on the flow behaviour (see Fig.2.2 ). (Reproduced from Jäsberg (2007)). . . . . . . . . . . . . . . . 162.4 The piecewise logarithmic velocity profiles proposed by Jäsberg (2007)for wood fibre suspension flows. (Reproduced from Jäsberg (2007)). . 173.1 Schematic of the experimental set up. PT and FT are defined as apressure and flow transducers, respectively . . . . . . . . . . . . . . . 23ixList of Figures3.2 Representative examples of the pressure drop and time averaged localvelocity profiles acquired using UDV. The UDV measurements are anensemble-average of approximately 2000 instantaneous measurements.The data in (a) are described in Table 3.1 and represent series 1 aswell as Series 27 through 30, triangle symbols represent the pressuredrop as a function of velocity with the Taitel & Dukler (1976) law. In(b), the data are given by series 11. The continuous lines in (b) arethe UDV measurements and dashed line is 1/9th power law relation-ship to represent the turbulent velocity field for water. The shadedarea close to the pipe wall represents the regions of uncertainty in themeasurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3 Representative ensemble-averaged UDV measurements of (a) local ve-locity (Method 1) , (b) strain-rate (Method 2), (c) turbulence intensity(Eq.2.6) (Method 3) and (d) plug size. The data are taken from Series30 (Re = 1.6×105) described in Table 3.1. The shaded regions in eachFigure represent the different features of the uncertainty in the signal. 303.4 SEM micrograph of mechanically refined cellulose pulp suspension atdifferent applied forces. (a) Series 2, (b) Series 7, (c) Series 13, (d)Series 18, (e) Series 23, (f) Series 27. . . . . . . . . . . . . . . . . . . 31xList of Figures4.1 A comparison of the behavior of water to that of a pulp suspension.For this case we compare water (Series 47) to fibre suspension (Series51) as gives in Table 3.3. Figures (a) and (b) are representative ve-locity profiles. Figures (c) and (d) are contour plots of the evolutionof the velocity profiles as a function of the wall stress. Figures (d)also includes additional information regarding the position of the plug.Figures (e) and (f) are estimates of the turbulent intensity. . . . . . 334.2 A comparison of the behavior of water-polymer to that of a pulp-polymer suspension. For this case we compare water with polymeradditive (Series 48) to fibre suspension with polymer additive (Series54) as gives in Table 3.3. Figures (a) and (b) are representative ve-locity profiles. Figures (c) and (d) are contour plots of the evolutionof the velocity profiles as a function of the wall stress. Figures (d)also includes additional information regarding the position of the plug.Figures (e) and (f) are estimates of the turbulent intensity. . . . . . . 414.3 Representative estimates of the plug size. The data shown are for fibresuspensions with and without polymer additive (Series 54  and Series51  from Table 3.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.4 Turbulent intensity of fiber suspension flow (Series 4, Table 3.1) atdifferent Reynolds number of and experimental results of Xu & Aidun(2005) for fibre suspension with the concentration of 1% and Re =12× 103. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42xiList of Figures4.5 Dimensionless velocity profiles of fibre suspension (Series 4, Table 3.1)as a function of dimensionless distance from the tube wall. The bulkvelocity is varied from 1.0 m/s to 6.4 m/s where the flow is in themixed or turbulent flow regimes. . . . . . . . . . . . . . . . . . . . . . 434.6 The slopes of the yield region of the representative cases from Table3.1. (a) Softwood, (b) Softwood @2000kWh/t, (c) Hardwood and (d)Hardwood @900kWh/t . . . . . . . . . . . . . . . . . . . . . . . . . . 444.7 Estimates of the plug size for Series 2-30 at the conditions given inTable 3.1. (a) Softwood, (b) Softwood @2000kWh/t, (c) Hardwood, (d)Hardwood @400kWh/t, (e) Hardwood @600kWh/t and (f) Hardwood@900kWh/t. The uncertainty in the estimates are given by the errorbars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.8 Estimates of the yield stress for Series 2-30 at the conditions given inTable 3.1. (a) Softwood, (b) Softwood @2000kWh/t, (c) Hardwood, (d)Hardwood @400kWh/t, (e) Hardwood @600kWh/t and (f) Hardwood@900kWh/t. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.9 Yield stress of pulp suspension at different applied refining energies.(a) Hardwood (Series 13-18), (b) Hardwood @400kWh/t (Series 18-22),(c) Hardwood @600kWh/t (Series 23-26), (d) Hardwood @900kWh/t(Series 27-30) from Table 3.1. © results from UDV-viscometer and results from UDV-pipe flow at the onset of the flow. . . . . . . . . . . 474.10 Summary of literature and experimental values of fitted constants tothe power-law model (τy = aCmb). . . . . . . . . . . . . . . . . . . . 48xiiList of Figures4.11 (a) Yield stress values measured with UDV-viscometer as a function ofrefining energy for refined hardwood. ©1.0%, 51.5%,♦2.0%,2.5%.(b) Fitted constants to the power-law model (τy = aCmb) for all refinedhard wood. UDV-viscometer:Ba, b. UDV-pipe flow Ia, b. . . . . 494.12 Relationship between yield stress , τy,and fibre morphology for Nylonfibres (Table 3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.13 The ratio of ρu′2 at the edge of the plug to the yield stress as a functionof plug size. The dashed line represents a linear regression of power lawfunction to the entire data set. The solid symbols represent maximaof the τy(Pa) data in Figure 4.8. . . . . . . . . . . . . . . . . . . . . 514.14 Drag Reduction versus (a) Reynolds number based on water property,Rew, and (b) the plug size for cases studied. A legend relating thesymbol to the testing condition is given in Table 3.3. . . . . . . . . . 52A.1 Schematic of a particle passing through the ultrasound field. . . . . . 73xiiiAcknowledgementsI wish to express my gratitude to my co-supervisors; Dr. Mark Martinez and Dr.James Olson for their guidance, suggestions and valuable discussions throughout thecourse of this research.I would like to extend my sincerest thanks and appreciation to the technical staffand students of the Pulp and Paper Centre at the University of British Columbia. Inparticular, I would like to thank George Soong and Ario Madani for their great help.I gratefully acknowledge financial support of the Natural Sciences and EngineeringResearch Council of Canada (NSERC), Aikawa Fiber Technologies (AFT), Domtarand Canfor.Last but not the least, I would like to thank my family: my parents and to mybrothers and sister for supporting me throughout my life in general. And Thank youSoheila for your love and your support.xivDedicationTo my parentsMohammad and FereshtehandTo my wifeSoheilaxvChapter 1IntroductionThere is a need to characterize papermaking fibre suspensions in a simple manner thatreflects its process capability during papermaking operations. A number of years ago,Wahren and his co-workers (Meyer & Wahren (1964), Thalen & Wahren (1964)),attempted this and developed a simple experimental approach to characterize thenetwork capabilities of fibres by measuring its sedimentation concentration. Whileattractive because of its simplicity, this method has not been widely used, in part ofdifficulties in performing a settling experiment, and to some degree, to interpretationdifficulties due to network compressibility. The goal of this research is to extendWahren's vision and use classical rheological parameters, i.e. yield stress, to replacethe sedimentation concentration. We do so by interpreting the flow field of thesesuspensions in a pipe.The focus of the present work is an experimental study of the behavior of semi-dilute, opaque fibre suspensions during fully-developed pressure-driven flow in a cylin-drical pipe. Under certain concentration limits papermaking suspensions are consid-ered to be shear thinning and possess a yield stress τy. With these suspensions, theradial profile in fully developed flow in a pipe is characterized by an unyielded orplug zone. The radius of the plug zone is dictated by a balance between the frictional1Chapter 1. Introductionpressure drop and the yield stress of the suspension. With increasing flow rates thesize of the plug diminishes.In this work, the instantaneous velocity profiles of a number of different fibresuspensions are measured using pulsed ultrasound Doppler velocimetry as a functionof volumetric flowrate and concentration. We developed subsequent data processingalgorithms to estimate the size of the plug from which we could determine the yieldstress directly. A sensitivity study was performed to understand the effect of upstreamprocessing conditions i.e LC refining or fibre length, on yield stress. In subsequentwork we examine two separate, yet complimentary, questions from this data set,namely, when does the plug disappear and what are the mechanisms in play withturbulent drag reduction.This thesis is organized as a monograph. In Chapter 2 the background literatureis reviewed and used to formulate the research questions and objectives. The mainexperimental device, a turbulent flow loop, the measurement protocol and algorithmsare presented in Chapter 3. The results are presented and discussed in Chapter 4.Here we divide the work into three subsections to answer the questions regarding theusefulness of the methodology for yield stress measurement, and the mechanism fortransition and turbulent drag reduction, and the relationship between fibre morphol-ogy and yield stress.2Chapter 2BackgroundBefore summarizing the existing literature on the flow properties of these suspensions,it is instructive to first define the regimes of fibre interaction in the range used inpapermaking. Broadly speaking, natural cellulose fibre suspensions are composed offlexible rod-like particles which have a wide distribution in both length and diameterdepending upon species and growing conditions. This physical make-up leads to acomplex rheological behavior with suspensions showing non-linear stress/strain-raterelationships. The major feature of the stress/strain-rate is that of an inelastic non-Newtonian fluid in the concentration used for papermaking.What is unique and somewhat perplexing with these fiber suspension is that underdilute conditions, these fibers interact with each other, often mechanically, to give riseto a local particle pressure, which is often described as a network. Flocculation, theprecursor to network formation, occurs over all length and concentration scales, whilenetworking occurs above the sediment concentration. The propensity for a suspensionto flocculate has been characterized by a single dimensionless number, termed thecrowding number N (Kerekes & Schell (1992))3Chapter 2. BackgroundN =CmLf22ω(2.1)where Cm is the bulk density of the suspension (kg/m3); Lf is the interaction length,with a dimension of m, based upon the length of the fiber; and w is fiber coarse-ness defined as the mass per unit length (kg/m). It should be noted that as thesesuspensions are polydisperse, having a length distribution p(li), Lf is estimated usingLf =∑p(li)li (2.2)The physical significance of N is given by its definition: it reflects the number of fibersin a spherical volume of diameter equal to Lf . Networking generally does not occurwhen N  1. Martinez et al. (2001) determined that the sedimentation concentrationoccurs at N = 16(±4). Celzard et al. (2009) reported N = 60 as "rigidity threshold"as predicted from percolation theory corresponds to the onset of coherent fibre flocshaving mechanical network strength. Most papermaking operations occur in the range10 < N < 45. Some pre-papermaking operations, such as LC refining, occur withN as high as 200. Indeed, networking is a complex process as there is a couplingbetween the orientation distribution to its rheology. The coupling limits dramaticallythe ability to smoothen concentration variations in the flow. The defining featureis that a micro/mesoscopic length scale plays a key role in determining the intricateproperties of the flow behavior of the macroscopic scale. Determining this remainsone of the open remaining questions in this field.4Chapter 2. BackgroundIndeed, pulp suspensions do not flow until a critical shear stress (or yield stress) isexceeded. Suspension yield stress has been measured either directly using viscometer,or inferred from dynamic and tensile measurements. In perhaps the earliest measure-ments in this area, Thalen & Wahren (1964) correlated the yield stress with thesuspension mass concentration.τy = a(C − Cs)b (2.3)where Cs is the sediment concentration. This proposed power-law indicates a finiteconcentration in which the suspension behaves as a viscoplastic. In the 1990's therewas resurgence in the literature and a number of groups were active in estimatingτy. The key players during this time where Bennington & Kerekes (1996), Swerin(1998) and Jokinen & Ebeling (1985). Bennington and his co-workers were perhapsthe most prolific and reported power law relationships, with either mass or volumeconcentrations, for a number of suspensions at concentration regimes much higherthan the sediment values. We find values of b to range between 1.25-3 in this liter-ature. Few, if any of these studies report the results in terms of crowding number.Perhaps the most recent, and most comprehensive study in this area was performedby Derakhshandeh (2011). A summary of his findings are given in Table 2.1. Inthis work, Derakhshandeh (2011) developed a simple bench top rheometer in whichthe flow field could be determined through ultrasound Doppler velocimetry. By ex-amining the location for fluidization at a known torque, the yield stress could be52.1. Understanding Flowing Fibre SuspensionsReference Measurement method Yield stress (Pa)Cm = 3% Cm = 6%Bennington et al. ( 1990) Baed concentric-cylinder 176 1220Swerin et al. ( 1992) Couette cell 19.3 117Damani et al. ( 1993) Parallel plate geometry 60Wikstrom and Rasmuson (1998) Baed concentric-cylinder 131 1100Ein-Mozaffari et al. (2005) Concentric-cylinder 350Dalpke and Kerekes (2005) Vane in cup geometry 130Derakhshandeh et al. (2010a) Vane in cup geometry 248Derakhshandeh et al. (2010a) Vane in cup geometry 154Derakhshandeh et al. (2010a) Velocimetry-rheometry 137Table 2.1: Yield stress data for soft wood bleached craft suspension obtained using differentmethods (From Derakhshandeh et al. (2011)).determined.2.1 Understanding Flowing Fibre SuspensionsIn the previous section we examined the threshold stress required to initiate motion ina fiber suspension. In this subsection we describe the behavior of these suspensions inpressure driven pipe flow, or Hagen-Poiseuille flow. We note two distinct flow statesand focus our understanding on the bound and transition from the plug-like to thewell mixed (or turbulent) state.Understanding transition from a plug-like to well mixed with these suspensionsis difficult. There are many previous reports of experiments on different materials(Newtonian or non-Newtonian) which mimic aspects of the flow considered here, andinsight can be gained by considering these first. We categorize the papers into 3groups which will be summarized below.62.1. Understanding Flowing Fibre SuspensionsThe first group of papers represent the simplest case, that is, studies of transitionof a Newtonian fluid in Hagen-Poiseuille flow. Since Reynolds' experiment, a largenumber of experimental and theoretical studies have been conducted to characterizetransition. In laminar flow, fluid particles follow straight lines that are parallel to eachother called streamlines. In turbulent flow different sizes of eddies are superimposedon the streamlines. Larger eddies carry the fluid particles across the streamlines andsmaller eddies create mixing that causes dissipation. The onset of turbulence is notimmediate. There is a process of instability that makes laminar flow a turbulentone. In this transitional zone, the flow is neither laminar nor fully turbulent, andthe observed pressure drop are intermediate between those for laminar and turbulentflow.The details of Newtonian transition is still under investigation. From the engi-neering perspective, it is generally accepted that transition can be predicted usingone dimensionless parameter, the Reynolds number Re = UD/ν where U is the bulkvelocity (m/s), D is the diameter of the pipe (m) and ν is the kinematic viscosity(m2/s). When Re exceeds a critical value, even small disturbances, which alwaysexist in a physical system, can cause instability and transition. From a mathematicalperspective, although hydraulic stability theory is capable of predicting instability ofsome flow configurations (transient growth or amplification of small disturbances), itis unable to predict transition (i.e. a critical Reynolds number) for pipe flows becausethe flow is stable at all Reynolds numbers. So nonlinear analysis is a must to be ableto have a predictive mechanism in mathematical fashion. For a Newtonian fluid itis known that the Reynolds number above 2100 is generally accepted as the critical72.1. Understanding Flowing Fibre Suspensionsvalue of practical interest to transition.There are a number of means to characterize the onset of transition. In thesimplest case, the relation between pressure drop and velocity is used to identify theflow regimes. The change from the laminar to the turbulent flow regime results in alarge increase in the flow resistance. The functional relationships and physical flowpatterns are fundamentally different for the two regimes. The Fanning friction factorf can be derived exactly for laminar flow and empirically for turbulent flow. TheFanning friction factor is related to the shear stress at the wall as:τ =fρU22(2.4)The value of f in laminar pipe flows for Newtonian fluids is 64/Re. Measuring thefriction factor departure from 64/Re is an effective way to detect the transition. Inaddition to this, characterization of the point of transition in experiments can alsobe based on the statistics of the time-series of the velocity and pressure, because themotion of turbulent eddies, which are random cause fluctuation. Here, the root-mean-square (rms) of local velocity fluctuations u′urms =√u′2 (2.5)is calculated to measure turbulence strength andI(r) =urms(r)u(r)(2.6)82.1. Understanding Flowing Fibre Suspensionsfor turbulence intensity I. The observation of the velocity and the turbulence intensityat the centerline is a generally accepted method to detect transition for Newtonianfluids.There is a large number of works which attempted to characterize flow in thistransition region (Bandyopadhyay (1986), Eliahou et al. (1998), Hof et al. (2003),Hof (2005), Toonder & Nieuwstadt (1997), Wygnanski & Champagne (1973)).Wygnanski and his coworkers found that flow disturbances evolve into two differentturbulent states called puffs and slugs during transition (Eliahou et al. (1998),Wygnanski & Champagne (1973), Wygnanski et al. (1975)). They observe anddescribe the evolution of the localized turbulent puffs and slugs in details such astheir shape, the way they propagate, their velocity profiles and turbulence intensitiesinside them. The puff is found when the Reynolds number is below Re ∼ 2700; theslug appears when Re > 3000. Both puffs and slugs are characterized by an abruptchange in the local velocity. The flow conditions are laminar outside the structureand turbulent inside. The puff and slug are distinguished from each other by theabruptness of the initial change between the laminar and turbulent states. It hasbeen reported that for a puff, the velocity trace is saw-toothed whilst a slug has asquare form on velocity-time readings.Since this classic study, a number of works have been reported in the literature at-tempting to further characterize transition experimentally. Bandyopadhyay (1986),for example, reports streamwise vortex patterns near the trailing edge of puffs andslugs. Darbyshire & Mullin (1995) indicate that a critical amplitude of the dis-turbance is required to cause transition and this value depends on Re. Toonder &92.1. Understanding Flowing Fibre SuspensionsNieuwstadt (1997) performed LDV profile measurements of a turbulent pipe flowwith water. They found that the rms of the axial velocity fluctuations near the wallare independent of Reynolds number. Eliahou et al. (1998) investigated, experimen-tally, transitional pipe flow by introducing periodic perturbations from the wall andconcluded that amplitude threshold is sensitive to disturbance's azimuthal structure.Han et al. (2000) expanded on the work of Eliahou et al. (1998) and advances theargument that transition is related with the azimuthal distribution of the streamwisevelocity disturbances. Transition starts with the appearance of spikes in the tem-poral traces of the velocity and there is a self-sustaining mechanism responsible forhigh-amplitude streaks. They indicated that spikes not only propagate downstreambut also propagate across the flow, approaching the pipe axis. Hof et al. (2003)performed an experimental investigation of the transition to turbulence in a pipe.Although they did not measure the velocity field, they introduced a scaling law whichindicates that the amplitude of perturbation required to cause transition scales asO(Re−1). Draad et al. (1998) proposed several scaling laws and also Peixinho andMullin (2007). Draad et al. (1998) examined transition from laminar to turbulentby imposing disturbances to a Newtonian fluid (water) flow in a cylindrical pipe fa-cility. They reported a critical disturbance velocity, which is the smallest disturbanceat a given Reynolds number for which transition occurs. They found that for largewavenumbers, i.e. large frequencies, the dimensionless critical disturbance velocityscales according to Re−1 , while for small wavenumbers, i.e. small frequencies, itscales as Re−2/3. Peixinho & Mullin (2007) perturbed the flow using small impul-sive jets and push-pull disturbances from holes in the pipe wall. They reported that102.1. Understanding Flowing Fibre Suspensionsthe critical value required to cause transition scales in proportion to Re−1 for jetsand the threshold scales as Re−1.3 or Re−1.5 for push-pull disturbances with the pre-cise value depending on the orientation of the perturbation. These experimental andnumerical studies for Newtonian fluids within the last few years have improved ourunderstanding of the transition to turbulence.The second (much smaller) group of papers (Group 2) report investigations ofpressure driven flow of general non-Newtonian fluids (including some viscoplasticfluids). Non-Newtonian fluids have been generally treated in similar fashion to thatof a Newtonian fluids. In order to make use of standard Newtonian theory a valuefor the viscosity of the fluid is required. Usually defining a unique value for viscosityis meaningless for a non-Newtonian as the viscosity is not constant for a given fluidand pipe diameter. It must be evaluated at a given value of strain rate. There havebeen a variety of attempts to generalize the Newtonian approach, discussed above,and examples of this are given in the classic works of Metzner & Reed (1955) andDodge & Metzner (1959). This concept has been extended by Govier & Aziz (1972)for Bingham fluids. In perhaps the most recent work in this area, Guzel et al.(2009a,b), also used deviations from the laminar friction factor curve to define theonset of transition. In this case they defined an average Reynolds number ReG byevaluating the local viscosity µ(r) in the pipe through knowledge of local velocityfield u(r), i.e.ReG =4ρR∫ R0u(r)µ(r)rdr =4ρu2cR|Px| (2.7)112.1. Understanding Flowing Fibre SuspensionsFigure 2.1: Examination of the deviations of f from 64/ReG for three different classesof fluids. Glycerin represents a Newtonian fluid. Xanthan and Carbopol represent shearthinning and viscoplastic fluids, respectively. Reproduced from Guzel et al. (2009b)where R is the radius of the pipe (m); uc is the centreline velocity (m/s); ρ is thedensity of the fluid (kg/m3); and Px is the pressure drop per unit length (Pa/m).With this, they report (see Figure 2.1) for three classes of fluids that transition occurswhen the friction factor deviates from 64/ReG and the critical Reynolds numberdepends upon the rheology of the fluid.In recent years there has been a number of increasingly detailed studies of transi-tion of non-Newtonian fluids (Draad et al. (1998), Escudier et al. (2005), Peixinhoet al. (2005, 2008), Pinho & Whitelaw (1990)). One of the key findings in this liter-ature was advanced by Peixinho et al. (2005), who show that transition for the yieldstress fluid takes place in two stages. In the first stage, transition is characterized122.1. Understanding Flowing Fibre Suspensionsby local velocity profiles that deviate from the theoretical laminar solution withoutany observable differences in the urms of the signal. In some cases, like that reportedby Guzel et al. (2009a), the flow profile is asymmetric. In the second stage, withincreasing the Reynolds number, turbulent puffs and slugs appear. With viscoplasticfluids, Guzel et al. (2009a) indicates that the plug is present during the second stageof transition and disappears only when the the Reynolds (turbulent) stress is greaterthan the yield stress.The final group of papers (Group 3) examine transitional flows in fibre suspensions.In Hagen-Poiseuille flow, papermaking fibre suspensions display a number of featuressimilar to that of viscoplastic fluids. Using non-invasive tools such as NMR imaging orultrasound doppler velocimetry (UDV), in Hagen Poiseuille flow before transition, aplug is evident in the central portion of the pipe and diminishes in size with increasingvelocity (Haavisto et al. (2011), Li et al. (1994), Powell et al. (1996), Raiskinmäki& Kataja (2005), Wiklund et al. (2006)). A similar feature has been reported byXu & Aidun (2005) for pressure-driven flow of a dilute suspension in a rectangularchannel. In related literature, Heath et al. (2007) visualized the flow of a fibresuspension in a backward facing step using positron emission tomography. Here theynote that they report plug-like behavior in the pipe, before the step. They find thatover the backward facing step, the plug is stable and densifies in the axial direction.Transition is quite complex and most authors indicate that it occurs in stages. Inearly work, Robertson & Mason (1957),Forgacs et al. (1958) and Daily & Bugliarello(1958) categorized the stages as either plug, mixed, or turbulent. The plug regimeindicates a coherent plug, i.e. particles with correlated velocities, which are present132.1. Understanding Flowing Fibre Suspensionsin all regions of the channel except in the near wall region. The motion of the plug isassisted by a laminar lubricating film of water in the region next to the wall. In themixed region, a plug is present in the central portion of the pipe but in this case, thelubricating film is now turbulent. These authors qualitatively describe that the plugdiminishes when the turbulent stress is larger than the yield stress of the suspensions.In the fully turbulent region, the plug is broken-down and the flow becomes turbulentover the entire cross-section of the pipe. Further insights into this behavior are givenby Duffy & Titchener (1975), Duffy et al. (1976). Of interest is the recent workby Jäsberg (2007), who measured the thickness of the lubricating layer in the plugregime, using a laser technique, as well as the turbulent characteristics of the flowfield in the mixed regime using UDV. Using this data, Jäsberg (2007) indicates thatthere are five stages (shown schematically in Figures 2.2 and 2.3). He explained theseregimes as:1. In regime I there is a plug flow in entire cross section of the pipe. Elasticitybehavior of the fibres' network absorbs part of the turbulent energy of fibers.Reaction of this elastic energy, elastic force, pushes the fibres toward the pipewall. Elastic force in this case is larger than the lift force acting on fibres whichkeeps fibers in a contact with the wall. This contact increases the pressure loss.2. By increasing the velocity, regime II, lift force increases more than elastic forceand makes fibres being away from pipe's wall. In this case a fibreless layer(lubrication layer) appears near the wall. The thickness of this layer increaseswith increasing the flow rate. Since there is no contact between fibre and wallthe loss may decrease.142.1. Understanding Flowing Fibre SuspensionsFigure 2.2: Schematic of the stages of transition. Reproduced from Jäsberg (2007)3. In regime III fluctuations in the layer near the wall disengage fibres on thesurface of the plug flow. These fibres moves randomly in the layer and maycontact the wall. In this case pressure loss increases linearly with increasing theflow rate.4. In regime IV the plug flow gets narrower and the turbulent annulus remainsaround it. Turbulence in this case is high and prevent fibres from making anetwork. Changing of pressure with flow rate is almost quadratic in this regime.5. Increasing the velocity extends turbulence over the entire cross section of thepipe (regime V). The change in pressure loss with flow rate remains quadratic.Jäsberg (2007) opens the possibility that forces such as (inertial) lift or lubricationforces play a significant role in the behavior of the plug, especially in the near wallregion.152.1. Understanding Flowing Fibre SuspensionsFigure 2.3: Schematic representation of pressure drop versus flow rate for fibre suspensionin a flow channel. The water curve is measured in a hydraulically smooth pipe, and dottedlines divide the flow domain into five main regimes (labeled I - V) based on the flow behaviour(see Fig. 2.2 ). (Reproduced from Jäsberg (2007)).2.1.1 Velocity Profiles of Wood Fibre SuspensionsFor wood fibre suspension Jäsberg (2007) found that the velocity profile consists oflogarithmic near wall region, a yield region and a core region (Figure 2.4). From hisexperimental results he reported the following equations for velocity profiles in theseregions:u+ =1klogy+ +B + ∆u+ (2.8)where∆u+ =0 if 0 < y+ ≤ y+Lαklog(y+/y+L ) if y+L < y+ ≤ y+C (≤ y+H)∆u+P − βk log(y+/y+L ) if y+C < y+ ≤ R+(2.9)162.1. Understanding Flowing Fibre SuspensionsFigure 2.4: The piecewise logarithmic velocity profiles proposed by Jäsberg (2007) forwood fibre suspension flows. (Reproduced from Jäsberg (2007)).Which α and β give the slop (relative to Newtonian profile value) of the curve in theyield region and the core region, respectively. Variables with superscript + are given inwall units, i.e. the velocity u+ = u/Uτ and distance to the wall y+ = yUτ/υ are scaledwith solvent kinematic viscosity υ and friction velocity Uτ = ((−dP/dx)(D/)(1/4))1/2,where (-dP/dx) is the pressure gradient which drives the flow, D is the pipe diameterand ρ is the fluid mass density. Figure 2.4 illustrates the simplified profile and themeaning of various parameters. Quantities y+L , y+H and α are constants for a givensuspension. Instead y+C and β depend on flow rate (on τw) in a manner to be found.172.2. Turbulent Drag Reduction2.2 Turbulent Drag ReductionIn the previous section we described the effect of the plug (i.e yield stress) on thetransition to the turbulent state. This is not the only unique feature with thesesuspensions. These suspensions also display turbulent drag reduction (TDR). At thispoint we turn our attention to understanding this phenomenon. Before doing so, wemust define the concept of drag reduction. Drag reduction occurs when the measuredfrictional pressure drop is less than that achieved with water at an equivalent flowrate. Early researchers have noted that small additions of polymers or surfactants,simple solids (such as fine grains or fibres) and combinations of polymers and simplesolids can substantially reduce turbulent drag (Brautlecht & Seth (1933), Burgeret al. (1982), Figueredo & Sabadini (2003), Mysels (1949), Toms (1948), White& Mungal (2008)). Drag reduction obtained by the addition of soluble polymer tothe fluid has received more study than the other types mentioned. Under certainconditions of turbulent flow, drag is drastically reduced by even minute amounts ofsuitable additive. The most effective drag-reducing polymers, in general, possess alinear flexible structure and a very high molecular weight (Hoyt (1972)). The drag-reducing abilities of polymer solutions are known to be triggered by a critical levelof shear stress parameterized by the so-called `onset Reynolds number' (Virk et al.(1970)). Virk and his co-workers (Virk et al. (1970, 1967), Virk (1975)) found thatthere is a limitation for drag reduction with polymer additives and defined this as themaximum drag reduction asymptote (MDR) or the Virk asymptote.Valuable insights into drag reduced flow have been obtained by using visualizationtechniques such as particle image velocimetry (PIV) and laser Doppler anemometry182.2. Turbulent Drag Reduction(LDA) (Durst et al. (1985), Harder & Tiederman (1991), Luchik & Tierderman(1988), Ptasinski et al. (2001), Tiederman (1990), Toonder & Nieuwstadt (1997),Toonder et al. (1997), Warholic et al. (1999), Warholic et al. (2001), Wei &Willmarth (1992), White et al. (2004), Willmarth et al. (1987)). It has been ob-served that the root mean-square (rms) of the fluctuations in the streamwise velocityincreases while the rms of the fluctuations in the wall-normal direction decreases withdrag reduction. Two popular, amongst a backdrop of others, mechanisms are emerg-ing in this literature based upon direct numerical simulations (Benzi et al. (2006),Dubief et al. (2004), Min et al. (2003a,b), Ptasinski et al. (2003), Sureshkumar etal. (1997), Toonder et al. (1997)) . Lumley (1969), on one hand, proposed a `vis-cous' mechanism in which energy is dampened through unraveling of polymer chains.Tabor & deGennes (1986), on the other hand, proposed an `elastic' mechanism inwhich the smallest eddies of the flow are dampened by the presence of the polymer.In the second major category of papers in this literature, we examine drag reduc-tion with the presence of solid particles. Radin et al. (1975), for example, demon-strates that drag reduction can be obtained with fibrous additives which have aspectratio greater than 25 to 35. McComb & Chan (1985) and Kerekes & Douglas (1972)report a 70% drag reduction achieved with two vastly different particle systems. Lee& Duffy (1976) demonstrate TDR with papermaking fibres. Vaseleski & Metzner(1974) propose that the TDR mechanism with particle systems is different than withpolymer system. Here they advance the argument that drag reduction occurs thatthrough a dampening of the turbulent energy by the fibres in the central portion ofthe pipe.192.2. Turbulent Drag ReductionFinally, in what we consider the most relevant literature to the case under study,we find a smaller set of papers in literature in which both polymer and dilute particlesare added to achieve turbulent drag reduction. Sharma et al. (1979), for example,injected hair-like fibres (asbetos) at the boundary and centerline of a turbulent pipeflow of water and polymeric solution. Lee et al. (1974) showed a remarkable synergis-tic effect in which the drag reduction of the combination of additives was greater thanthe individual species. Indeed, they were able to exceed the maximum drag reductionasymptote advanced by Virk et al. (1970). This synergism was also demonstratedin other labs using a variety of polymer-fibre combinations. Dingilian & Ruckenstein(1974), Malhotra et al. (1987), Reddy & Singh (1985). Kale & Metzner (1976)and Metzner (1977) speculate that fibres and polymers reduced turbulent drag bydifferent mechanisms.At this point we summarize this literature review and attempt to frame the ques-tions which will be addressed in this thesis. To begin, we find that in the concentrationregimes relevant to papermaking, papermaking fibres are networked with a rheologycharacterized by a yield stress, τy. There as been tremendous effort in the 1960's aswell as in the 1990's to characterize τy using commercially available or in-house vanerheometers. There has been little progress in understanding the relationship betweenthe fibre characteristics and the resulting yield stress.During flow, a plug exists in the central portion of the pipe and that transition isquite complicated and occurs in a number of stages. Most authors agree that thereare between 3-5 stages. Unlike Newtonian or shear-thinning fluids, the early stagesof transition are not visible by examining the time dependent velocity fluctuations202.2. Turbulent Drag Reductionalong the pipe axis; here velocity fluctuations remain at approximately laminar levels(Guzel et al. (2009a)). Transition-like behavior occurs in regions closer to the wall asthere is a significant increase in turbulent intensity levels. For generalized Newtonianfluids, asymmetrical flow profiles are usually the first indication of transition and mostlikely a genuine effect of a traveling wave structure, a global mode of instability. Thisis not reported in the literature for fibre suspensions. Finally, there is no indicationsin the literature of the effect of the plug during transition. This is especially evidentduring the final stages of transition when the plug is small in comparison to the sizeof the pipe. This is one of the open remaining questions in this literature and servesas the motivation for one aspect of study in this thesis.We also find that small additions of either polymers or high-aspect ratio fibresto a Newtonian fluid can drastically alter the turbulent flow characteristics. Thesefindings strongly hint at the existence of a key mechanism of turbulence momentumtransport with which the polymer or the fibres interferes. We note that synergisticeffects are also present when polymers and fibres are combined and it is speculatedthat these work by different mechanisms. With fibre suspensions at any industriallyrelevant concentration, a plug is present and it is uncertain what role the plug playsduring drag reduction and is there synergistic effects with polymer additions. Thisserves as another question to be addressed in this thesis.21Chapter 3Methods and MaterialsIn this section, we describe the main experimental tool used in this thesis: a Hagen-Poiseuille flow loop. Here, we measure the local field, using Doppler velocimetry, vol-umetric flowrate, and pressure drop for conditions in which the flow is fully developed.With these measured signals we are able to estimate the plug size and correspondingyield stress, through knowledge of the pressure drop. This methodology is not uniqueand closely follows the work of Raiskinmäki & Kataja (2005).All results reported are from a test section of a straight, smooth pipe 10 m longwith an inner diameter 50.8 mm, located over 100 pipe diameters from the last dis-turbance. The setup is illustrated schematically in Figure 3.1. The flow is generatedby a variable-frequency driven (VFD) centrifugal pump from an inlet reservoir ofapproximately 4 m3 capacity to an outlet reservoir of the same capacity. The pumpcan provide a maximum flow rate of 0.1 m3/s, with water as the working fluid. Thisrepresents a Re number of 2.5× 106. Laminar flow with water was not achievable inthis loop. The test section of the flow channel is comprised of one continuous pipewhich was aligned horizontally with the aid of a laser leveling tool.The measuring system used for velocity profile measurements is pulsed UltrasonicDoppler Velocimeter (UDV) (Model 3010, www.signal-processing.com) with a fre-22Chapter 3. Methods and MaterialsFigure 3.1: Schematic of the experimental set up. PT and FT are defined as a pressureand flow transducers, respectivelyquency of 8 MHz. The ultrasound transducer has an active diameter of 5 mm andthe measuring volume is a thin-disc shape element with 5 mm diameter. One thou-sand instantaneous measurements were used to estimate the average and fluctuatingcomponents of the velocity at each radial location. Two pressure transducers (PT )were located near the inlet and outlet of the flow channel, separated by a distanceof 10 m. The accuracy of the transducers were 0.02% of the full scale and they werecalibrated with a digital pressure gauge. Pressure readings later reported through ourpaper are averaged over 10 seconds at a data acquisition rate of 100 Hz to increase theconfidence in the estimate. Flow rates were estimated using two alternative methods:1. Using a magnetic flow meter (Rosemount) installed near the outlet reservoir.2. Radial profiles of the axial velocity are measured via UDV and the flow rate isestimated by integration.The difference between the two measurements varied by at most 8%. Typically thedifference between the measurements was estimated to be 3%.23Chapter 3. Methods and MaterialsBefore we present the methodology to estimate the plug size rp and yield stressτy, we must first discuss the uncertainty of our measurements using UDV. To high-light this discussion, we include representative pressure drop and time-averaged localvelocity fields as a function of increasing flow rate, see Figure 3.2. Because of reflec-tion, it is extremely difficult to resolve the local velocity in the region near the wallusing UDV. We estimate the region of uncertainty to be between 1-2 mm and high-light the region as grey in this Figure. We follow closely the signal analysis routinegiven by Raiskinmäki & Kataja (2005) as well as Jäsberg (2007) and estimate rpby examining the velocity, the strain-rate, and turbulence intensity (Eq.2.6) of thesignals, see Figure 3.3(a)-(c). Here we define the edge of the plug as the region withlargest curvature in the signal. We estimate the lower bound to be the point wherethe signal is 0.99 of the value in the central portion of the pipe.The uncertainty inthe estimate of the plug is given by the shaded region in these Figures. The plugsize is considered as the average of the three estimates and the uncertainty in this isestimated to be approximately 20%. In Figure 3.3(d), an estimate of the plug sizeas a function of Re is shown. The grey region near rp/R→ 1 represents uncertaintycreated though reflection in the UDV measurements. The region near rp/R < 0.1, isour lower detection bound. We cannot detect small plugs with confidence. With rp,τy may be estimated from knowledge of the fully developed pressure drop per unitlength Px using the methodology given by Raiskinmäki & Kataja (2005)τy =12|Px|rp (3.1)243.1. Experimental StudiesWe also compared our results to values of yield stress determined using the equipmentoutlined by Derakhshandeh (2011).3.1 Experimental StudiesThree large studies where undertaken in this thesis. In the first study, see Table 3.1,we characterize the yield stress of a number of different papermaking suspensions. Asshown in the table, we examine yield stress as a function of fiber type, refining en-ergy, concentration and volumetric flowrate. This study was quite exhaustive as 822experiments were conducted. For certain, selected, pulps we compared our measure-ment to that using the methodology outlined by Derakhshandeh (2011). Suspensionsof two different type of fibre, softwood(SW) and hardwood(HW), at different levelof Specific Refining Energy (SRE) and at different concentrations (wt/wt) for eachcase and one Newtonian fluid (water) were tested in this work. The NBSK pulp(SW) (series 2-12 in Table 3.1) was obtained from Canfor Pulp (Prince George, BC)while the Aspen pulp (HW) (series 13-30) was from Domtar Corp (Kamloops, BC).The fibre lengths were determined using a Fibre Quality Analyzer (www.optest.com)following the TAPPI Test Method (T 271 pm-91). The suspensions were mixed inthe reservoirs and then circulated through the flow channel itself in order to ensuretheir homogeneity, prior to conducting any experiment. Dried pulp was diluted withwater in the tank to reach a desired consistency of 4%. Suspensions were refined withthe low consistency refiner in Pulp and Paper Center at UBC. The morphology ofthe refined fibres were examined by a Hitachi S-2600N Scanning Electron Microscopy253.1. Experimental StudiesSeries Fibre SRE Fibre Concentration Bulk No. ofType [kWh/t] Length [mm] %[wt/wt] Velocity[m/s] Experiments1 Water 0   0.1-8.3 282 SW 0 2.6 0.50 0.8-8.8 233 SW 0 2.6 0.75 0.1-9.2 314 SW 0 2.6 1.00 0.2-9.2 245 SW 0 2.6 1.50 0.2-9.0 276 SW 0 2.6 2.00 0.1-9.3 497 SW 2000 0.23 0.50 0.8-8.8 188 SW 2000 0.23 0.75 0.1-9.2 269 SW 2000 0.23 1.00 0.2-9.2 2710 SW 2000 0.23 1.52 0.2-9.0 2611 SW 2000 0.23 1.65 0.2-9.3 2512 SW 2000 0.23 1.95 0.1-8.1 2213 HW 0 1.14 0.80 0.5-6.3 3214 HW 0 1.14 1.10 0.1-7.9 3515 HW 0 1.14 1.50 0.9-7.1 2216 HW 0 1.14 1.90 0.3-8.4 3217 HW 0 1.14 2.45 0.4-9.1 2518 HW 400 0.40 0.93 0.3-6.4 1519 HW 400 0.40 1.13 0.1-7.2 2420 HW 400 0.40 1.70 0.8-10.3 2821 HW 400 0.40 2.20 0.9-9.9 2722 HW 400 0.40 3.21 0.5-10.1 2723 HW 600 0.29 1.00 0.1-7.2 2124 HW 600 0.29 1.60 0.1-8.4 2325 HW 600 0.29 2.00 0.1-8.1 2126 HW 600 0.29 2.75 0.2-7.1 2427 HW 900 0.18 0.75 0.1-9.1 2828 HW 900 0.18 1.10 0.1-9.4 2529 HW 900 0.18 1.50 0.1-9.5 3330 HW 900 0.18 1.75 0.1-9.3 54Table 3.1: A summary of the experimental conditions tested. For each fluid and for eachconcentration, several different bulk velocities were chosen to cover all the transition stagesindicated in the introduction.263.1. Experimental StudiesSeries L D A Cm N Bulk Velocity No. of[mm] [mm] (L/D) %[wt/wt] [m/s] Experiments31 3.048 0.014 224 0.34 100 0.7-6.9 5132 3.048 0.014 224 0.31 90 0.7-7.0 5033 3.048 0.014 224 0.28 80 0.8-7.0 4834 3.048 0.014 224 0.24 70 0.7-7.2 5135 3.048 0.019 159 0.69 100 0.6-7.5 30936 3.048 0.019 159 0.62 90 0.4-7.6 22337 3.048 0.019 159 0.55 80 0.5-7.6 12738 3.048 0.019 159 0.48 70 0.6-7.6 13039 3.048 0.027 112 1.37 100 0.8-7.3 4740 3.048 0.027 112 1.24 90 0.8-7.2 5041 3.048 0.027 112 1.10 80 0.8-7.2 5642 3.048 0.027 112 0.96 70 0.6-7.3 6243 1.016 0.019 53 6.10 100 0.4-8.2 24444 1.016 0.019 53 5.50 90 0.5-8.1 11045 1.016 0.019 53 4.90 80 0.5-8.1 11046 1.016 0.019 53 4.30 70 0.5-8.0 106Table 3.2: A summary of the experimental conditions tested to study nylon fibre suspen-sions flow. For each fluid and for each concentration, several different bulk velocities werechosen to cover all the stages indicated in the introduction.(SEM), shown in Figure 3.4.In the second study we attempted to gain insight into the relationship betweenlength, diameter, and concentration on yield stress on idealized, monodispersed nylonfibre suspensions. Nylon fibers with four different sizes were used in this study. Thedensity of all the nylon fibers is 1.15 g/cm3. The detailed test conditions, fiber sizeand concentration are shown in Table 3.2. The suspensions were mixed in the tankand then circulated through the flow channel in order to ensure their homogeneity,prior to conducting any experiment.In the final study, shown in Table 3.3, we examine the turbulent drag reducing273.1. Experimental StudiesSeries Fibre Conc. Polymer Conc. Symbols Bulk Velocity No. Tests%[wt/wt] (ppm) [m/s]47 0 0 − · − 0.1-8.3 1748 0 100 4 0.1-8.4 1949 0.5 0 © 0.1-6.9 2750 0.75 0  0.2-7.6 2251 1 0  0.3-7.6 2352 0.5 100  0.2-8.1 2253 0.75 100  0.1-8.5 2454 1 100  0.1-8.7 22Table 3.3: A summary of the experimental conditions tested to study the effect of poly-mer additive on the fibre suspension flow (Softwood, NBSK). For each fluid and for eachconcentration, several different bulk velocities were chosen to cover all the stages indicatedin the introduction.properties of these suspensions with the addition of polymer solutions. The polymerthat we used in our experiments is Anionic Polyacrylamide (APAM), Superfoc A-110supplied by Kemira, with a molecular weight of 6 − 8 × 106(g/mol) , according tothe manufacturer. This polymer is resistant to mechanical degradation (Toonder &Nieuwstadt (1997)), which is essential for our re-circulatory experimental set-up. Inthe experiments presented in this work we used a 100 w.p.p.m of Superfloc A110 -fibre suspensions at different concentrations (wt/wt) as shown in Table 3.3. Thatmeans that only 100 g of polymer was dissolved in the suspensions.283.1. Experimental Studies(a)(b)Figure 3.2: Representative examples of the pressure drop and time averaged local velocityprofiles acquired using UDV. The UDV measurements are an ensemble-average of approx-imately 2000 instantaneous measurements. The data in (a) are described in Table 3.1 andrepresent series 1 as well as Series 27 through 30, triangle symbols represent the pressuredrop as a function of velocity with the Taitel & Dukler (1976) law. In (b), the data aregiven by series 11. The continuous lines in (b) are the UDV measurements and dashed line is1/9th power law relationship to represent the turbulent velocity field for water. The shadedarea close to the pipe wall represents the regions of uncertainty in the measurements.293.1. Experimental StudiesFigure 3.3: Representative ensemble-averaged UDV measurements of (a) local velocity(Method 1) , (b) strain-rate (Method 2), (c) turbulence intensity (Eq.2.6) (Method 3) and(d) plug size. The data are taken from Series 30 (Re = 1.6 × 105) described in Table 3.1.The shaded regions in each Figure represent the different features of the uncertainty in thesignal.303.1. Experimental Studies(a) (b)(c) (d)(e) (f)Figure 3.4: SEM micrograph of mechanically refined cellulose pulp suspension at differentapplied forces. (a) Series 2, (b) Series 7, (c) Series 13, (d) Series 18, (e) Series 23, (f) Series27.31Chapter 4Results and DiscussionIn this section the results will be presented and discussed. As mentioned earlier,a large number of studies were conducted. Indeed, approximately 3010 differentconditions were tested. This poses a problem about how to present the results asthe data set is large. If we focus on presenting all results, in detail, the significanceof the work will be diminished. As a result, in this section we present a high - levelsummary focusing on the contributions and the detailed results shown in AppendixA. Hence we will focus on four aspects of the trials:(a) a phenomenological description of transition of fibre suspensions(b) benchmarking the yield stress measurements to standard methodologies(c) understanding the sensitivity of yield stress to upstream processing conditionsand fibre morphology.(d) characterizing turbulent drag reductionWe begin by examining the time averaged velocity and turbulent intensity profilesof the fibre suspensions flows, see Figures 4.1 and 4.2. For each measurement, overone thousand instantaneous velocity measurements were used in the ensemble averageand the confidence interval for each point is small. It should be noted that the velocityprofiles have been made dimensionless by scaling the ensemble average with uc and32Chapter 4. Results and Discussion(a) Water (Series 1) (b) Fibre suspension (Series 5)−1 −0.5 0 0.5 100.20.40.60.81r/Ru/uc  τw = 5.3 Paτw = 20.5 Paτw = 34.8 Pauuc= (1 − rR )19−1 −0.5 0 0.5 100.20.40.60.81r/Ru/uc  τw = 2.6 Paτw = 8.4 Paτw = 25.1 Paτw = 47.1 Pa(c) (d)(i) (ii) (iii)10 20 30−1−0.500.51τw (Pa)r/R  0.70.80.913 10 20 30 40 50 60 70−1−0.500.51τw (Pa)r/R  0.60.70.80.912rp(e) (f)(i) (ii) (iii)10 20 30−1−0.500.51τw (Pa)r/R  0.050.10.1520 40 60−1−0.500.51τw (Pa)r/R  0.020.040.060.082rpFig. 4. A comparison of the behavior of water to that of a fibre suspension. Forthis case we compare Series 1 to Series 5 as gives in Table 1. Figures (a) and (b)are representative velocity profiles. Figures (c) and (d) are contour plots of theevolution of the velocity profiles as a function of the wall stress. Figures (e) and (f)are estimates of the turbulent intensity.16Figure 4.1: A comparison of the behavior of water to that of a pulp suspension. Forthis case we compare water (Series 47) to fibre suspension (Series 51) as gives in Table 3.3.Figures (a) and (b) are represe tative velocity profiles. Figures (c) and (d) are contour plotsof the evolution of the velocity profiles as a function of the wall stress. Figures (d) alsoincludes additional information regarding the position of the plug. Figures (e) and (f) areestimates of the turbulent intensity.33Chapter 4. Results and Discussionthey appear to be symmetric. The graphs are ordered such that we compare thebehavior of water to that of fibre suspension (Figure 4.1) and that of dilute polymersolution to that of a dilute polymer solution with fibres (Figure 4.2). The figuresare arranged in columns for each case considered. Asymmetry has been reported inprevious works for viscoplastic fluids during transition (Escudier et al. (2005), Guzelet al. (2009a), Peixinho et al. (2005)). However, in this study we did not see thisphenomenon.In addition we also include five qualitative features on these contour plots in orderto highlight the steps characterizing transition:(a) The first feature on the graph is the position of the plug. This is shown ar-tistically as the solid white line. The data for this has been taken from themeasurements shown in Figure 4.3.(b) We display the roman numeral (i), as the second feature, to represent the onsetof drag reduction. This point was determined as the point where the frictionalpressure drop equaled that of water.(c) The third feature, shown as (ii), is the point where the ρu′2 approximatelyequals the yield stress of the suspension.(d) The next feature, which is shown as (iii), is the point at which the plug is notdetectable using our methodology.(e) The final feature (iv) is the point at which the drag reduction equals the maxi-mum theoretical value. This only occurred with polymer additives.At this point we examine Jäsberg's qualitative figure and compare this phe-nomenological picture to our measurements. Our measurements indicate that tur-34Chapter 4. Results and Discussionbulent drag reduction occurs when the plug nearly encompasses the entire pipe, i.e.stage III in Jäsberg's image. The majority of our measurements occur in stages IIIand IV. This quantitative behaviour is perfect in both fibre and fibre-polymer sys-tems. The results presented in Figures 4.1-4.2 are presented in a non-traditionalmanner and may be difficult to interpret. Here, we were able to create contour plotsof the results as we have an extremely large data set. Contour plots are not typical.The second feature which is difficult to interpret is that we present the results interms of the wall stress τw instead of the traditional Reynolds number. We do so asthe definition of Re is not as meaningful with suspensions (or suspension-polymer) aswith a Newtonian fluid. To help alleviate this, especially with turbulent intensity wepresent this in a more traditional manner in Figure 4.4 for one representative case.Included in this are literature values from Xu & Aidun (2005) which indicates thatturbulent intensity outside the plug is much higher than inside the plug.Figure 4.5 shows the dimensionless mean velocity profiles for representative cases(Series 4, Table 3.1) for bulk velocity ranging from 1.0 m/s to 6.4 m/s as a function ofdimensionless distance from the pipe's wall. Each curve consists of near wall region,a yield region and a core region where the velocity gradient is zero. The yield regioncan be approximated by a piecewise logarithmic profile of the formu+ = αlogy+ + β, (4.1)It can be seen from Figure 4.5 that slope of the yield region, α, depends on flowrate. Figure 4.6 shows the slopes of the yield region, α, of the suspension flows for35Chapter 4. Results and Discussionsome representative cases. Transition from the plug flow to the turbulent flow can beobserved in this Figure. The slope varies such that at low flow rates, plug flow, theslope is zero and approaches the Newtonian profile slope value ( 2.5 in the logarithmicy+ -scale) as the flow rate increases (turbulent flow). For some cases the slope athigh flowrates is higher than the Newtonian value which shows the drag reduction.Estimates of the evolution of the plug size as a function of τw for all cases testedis given in Figure 4.7. The error bars in the graphs represent the uncertainty inestimates of plug size using the three different methodologies. What is evident in thisFigure is that the plug size diminishes in a complex manner with increasing τw overthe range tested. What is evident is that rp does not decrease linearly with increasingwall stress. Similar findings have been reported by Mih & Parker (1967), Duffy etal. (1976), Powell et al. (1996) and Xu & Aidun (2005). With these measurementswe were able to estimate the yield stress using Equation 3.1.The yield stress of the suspensions as a function of τw is shown in Figure 4.8.A non-monotonic behavior is observed for all cases in which yield stress of the pluginitially increases before diminishing as the plug begins to break. This behaviorhas not been reported previously. We find that the maximum yield stress occurs atdifferent τw for each suspension tested and transition to turbulence across the entirecross-section of the pipe can only occur after the plug has disappeared. We find thatthis occurs at a τw which is much higher than the expected value for a Newtonianfluid in pipe flow (which is τw = 0.02 Pa at Re=2400 in a 50mm pipe). As a resultwe conclude that the plug retards transition.36Chapter 4. Results and DiscussionAt this point we attempt to benchmark these measurements against Derakhshan-deh (2011) methodology. We do so by correlating the estimate of τy in the pipe, atthe onset of the flow to the bench top values. The results are shown in Figure 4.9.What is encouraging is that the results scale with each other.In addition to this we attempted to capture the data by comparison to previousdata in the literature (Figure 4.10). In general we find that our data, at the onsetof the flow, follows power law relationship τy = aCbm with b found to be somewherebetween 2-3. A summary of the curve-fits are shown in Table 4.1.One trend that can report from our data set is that the coefficient "a" seems todepend in the energy applied during LC refining. As shown in Figure 4.11, we find"b" to be effectively independent of specific refining energy (SRE) while "a" increaseswith SRE, measures in both the pipe and Derakhshandeh (2011) methodology.At this point we turn our attention to attempting to relate the fibre morphologyto the yield stress measurements. As described previously, we conducted a similarseries of tests on a monodisperse nylon suspensions to help elucidate or gain insightinto this behaviour. These conditions are described in Table 3.2 and the results arequalitatively similar in form to the papermaking suspensions and are not repeatedhere. We find the correlation between the yield stress, fibre morphology and flowcondition asτy = 0.018τw0.5N2A−1, r2 = 0.94 (4.2)37Chapter 4. Results and Discussionwhich A is the aspect ratio (l/D) and N is the crowding number. The results areshown in Figure 4.12. Other factors like fibre flexibility, orientation or flocculationstate could play a role in determining τy as well.For fibre suspensions, the role of the plug region in retarding transition is largelyunknown. If one interprets the plug region to be fully rigid below the yield stress thenthe flow is analogous to that with the plug replaced by a solid cylinder moving at theappropriate speed. As shown by Sadeghi & Higgins (1991), sliding Couette-Poiseuilleflow, stabilizes the flow. These authors show that when the sliding velocity is 36% ofthe maximum Poiseuille velocity, the neutral curves for symmetric mode vanishes andthe flow becomes linearly stable. Presumably, since the effective viscosity becomesinfinite at the yield surface the flow should be locally stabilized. In the absence ofmulti-dimensional flow measurement, three possible behaviors may be postulated attransition: (i) transition may occur in the yielded annulus around the plug, leavingthe plug region intact; (ii) transition is retarded until the plug region thins to suchan extent that the ρu′2 (in the annular region) can exceed the yield stress; or (iii) acombination of (i) and (ii).In Figure 4.13 we present the ratio of ρu′2 at the edge of the plug to the yield stressas a function of plug size for all Series explained in Table 3.1. The first observationthat can be made from this figure is that data for all twenty nine measurementscollapse onto one curve, indicating the possibility of a scaling law. The dashed line inthis figure shows the function fit (ρu′2τy= 0.04( rpR)−2.3) to the data set with a correlationcoefficient of 0.73.38Chapter 4. Results and DiscussionWhat we see is that initially, i.e. when rp/R→ 1, the values ofρu′2τy< 1 (4.3)and the plug size diminishes somewhat linearly. As the ρu′2 is much smaller thanthe yield stress, region (a), the reduction in size must result from a response to thenetwork to shear, created from the frictional pressure drop. Duffy & Titchener (1975)argue that papermaking fibres suspensions densify under shear. Region (b) in thisfigure, delineates the area where non-monotonic behavior was observed in Figure 4.8.With rp → 0, region (c), ρu′2 is much greater than τy from which we conclude thatthe plug breaks apart through an erosion type mechanism. Mih & Parker (1967)have indicated this as a potential mechanistic step. This figure suggests to us thatthe disappearance of the plug during transition occurs in two steps, i.e. densificationin a response to frictional pressure drop followed by erosion when the ρu′2 greaterthan the yield stress.We now turn to what we believe to be the most unique measurements in thiswork. As shown in Figure 4.14b, drag reduction is plotted as a function of plug size.Again the symbols (i)-(iv) are placed on the graph to aid in understanding. For boththe fibre and fibre-polymer suspensions, we argue that a critical plug size is requiredto initiate drag reduction. In this case, we find that drag reduction occurs withrp/R . 0.8. Once initiated, the fundamental behavior for these two systems differs.For polymer-fibre systems, drag reduction approaches the maximum theoretical valuewith diminishing plug size. For fibre suspensions, the drag reduction displays non-39Chapter 4. Results and Discussionmonotonic behavior with rp.Like Vaseleski & Metzner (1974), we also postulate that dreg reduction occursthrough a dampening of the turbulent energy by the presence of the fibres. In ourcase, we argue that the fibres in the plug are responsible for dampening a portionof the energy, most likely through interfibre friction. Indeed it is difficult to justifythese results from a theoretical basis, but these findings imply two superimposedmechanisms for drag reduction.40Chapter 4. Results and Discussion(a) Polymer solution (Series 2) (b) Fibre-Polymer suspension (Series 8)−1 −0.5 0 0.5 100.20.40.60.81r/Ru/uc  τw = 7.0 Paτw = 11.9 Paτw = 26.2 Pa−1 −0.5 0 0.5 100.20.40.60.81r/Ru/uc  τw = 1.9 Paτw = 10.9 Paτw = 12.2 Paτw = 19.6 Pa(c) (d)(i) (iv) (ii) (iii)5 10 15 20 25−1−0.500.51τw (Pa)r/R  0.50.60.70.80.915 10 15 20−1−0.500.51τw (Pa)r/R  0.60.70.80.912rp(e) (f)(i) (iv) (ii) (iii)5 10 15 20 25−1−0.500.51τw (Pa)r/R  0.050.10.150.25 10 15 20−1−0.500.51τw (Pa)r/R  0.020.030.040.050.060.072rpFig. 5. A comparison of the behavior of water-polymer to that of a fibre-polymersuspension. For this case we compare Series 2 to Series 8 as gives in Table 1. Figures(a) and (b) are representative velocity profiles. Figures (c) and (d) are contour plotsof the evolution of the velocity profiles as a function of the wall stress. Figures (e)and (f) are estimates of the turbulent intensity.17Figure 4.2: A comparison of the behavior of water-polymer to that of a pulp-polymersuspension. For this cas we compare water with polymer additive (Series 48) to fibresuspension with polymer additive (Series 54) as gives in Table 3.3. Figures (a) and (b) arerepresentative velocity profiles. Figures (c) and (d) are contour plots of the evolution ofthe velocity profiles as a function of the wall stress. Figures (d) also includes additionalinformation regarding the position of the plug. Figures (e) and (f) are estimates of theturbulent intensity.41Chapter 4. Results and Discussion0 20 40 60 80 100 = w ( P a ) 0 0.2 0.4 0.6 0.8 1 r p = R Figure 4.3: Representative estimates of the plug size. The data shown are for fibre sus-pensions with and without polymer additive (Series 54  and Series 51  from Table 3.3).0 0.5 100.020.040.060.080.11− r/R√u′2U  Re = 64× 103Re = 158× 103Re = 224× 103Re = 294× 103Xu & Aidun (2005)Figure 4.4: Turbulent intensity of fiber suspension flow (Series 4, Table 3.1) at differentReynolds number of and experimental results of Xu & Aidun (2005) for fibre suspensionwith the concentration of 1% and Re = 12× 103.42Chapter 4. Results and Discussion100 102 104010203040y+u+  u+ = 2.5lny+ + 5.5u+ = y+Ub = 1.0(m/s)Ub = 1.4(m/s)Ub = 2.1(m/s)Ub = 3.6(m/s)Ub = 6.4(m/s)Figure 4.5: Dimensionless velocity profiles of fibre suspension (Series 4, Table 3.1) as afunction of dimensionless distance from the tube wall. The bulk velocity is varied from 1.0m/s to 6.4 m/s where the flow is in the mixed or turbulent flow regimes.43Chapter 4. Results and Discussion0 200 400 600 800 10000246810Q[l/min]α  Series2Series4Series5Series6(a) (b)(c) (d)0 200 400 600 800 10000246810Q[l/min]α  Series7Series8Series9Series100 200 400 600 800 100001234Q[l/min]α  Series13Series14Series15Series160 200 400 600 800 1000051015Q[l/min]α  Series27Series28Series29Series30Figure 4.6: The slopes of the yield region of the representative cases from Table 3.1. (a)Softwood, (b) Softwood @2000kWh/t, (c) Hardwood and (d) Hardwood @900kWh/t44Chapter 4. Results and Discussion0 20 40 60 80 10000.20.40.60.81τw (Pa)r p/R  Series2Series3Series4Series5Series6(a) (b)(c) (d)0 20 40 60 80 10000.20.40.60.81τw (Pa)r p/R  Series 7Series 8Series 9Series 10Series 11Series 120 20 40 60 8000.20.40.60.81τw (Pa)r p/R  Series 13Series 14Series 15Series 16Series 170 20 40 60 80 10000.20.40.60.81τw (Pa)r p/R  Series 18Series 19Series 20Series 21Series 220 20 40 60 8000.20.40.60.81τw (Pa)r p/R  Series 23Series 24Series 25Series 260 50 100 15000.20.40.60.81τw (Pa)r p/R  Series 27Series 28Series 29Series 30(e) (f)Figure 4.7: Estimates of the plug size for Series 2-30 at the conditions given in Table 3.1.(a) Softwood, (b) Softwood @2000kWh/t, (c) Hardwood, (d) Hardwood @400kWh/t, (e)Hardwood @600kWh/t and (f) Hardwood @900kWh/t. The uncertainty in the estimatesare given by the error bars.45Chapter 4. Results and Discussion0 20 40 60 80 100010203040τw (Pa)τy(Pa)  Series 2Series 3Series 4Series 5Series 6(a) (b)(c) (d)0 20 40 60 80 1000102030405060τw (Pa)τy(Pa)  Series 7Series 8Series 9Series 10Series 11Series 120 20 40 60 8005101520τw (Pa)τy(Pa)  Series 13Series 14Series 15Series 16Series 170 20 40 60 80 100020406080τw (Pa)τy(Pa)  Series 18Series 19Series 20Series 21Series 220 20 40 60 80010203040τw (Pa)τy(Pa)  Series 23Series 24Series 25Series 260 50 100 150051015202530τw (Pa)τy(Pa)  Series 27Series 28Series 29Series 30(e) (f)Figure 4.8: Estimates of the yield stress for Series 2-30 at the conditions given in Table3.1. (a) Softwood, (b) Softwood @2000kWh/t, (c) Hardwood, (d) Hardwood @400kWh/t,(e) Hardwood @600kWh/t and (f) Hardwood @900kWh/t.46Chapter 4. Results and Discussion(a) (b)0.5 1 1.5 2 310−1100101102Consistency (%)τy(Pa)  0.5 1 1.5 2 310−1100101102Consistency (%)τy(Pa)  Cm(%) Cm(%)(c) (d)0.5 1 1.5 2 310−1100101102Consistency (%)τy(Pa)  Cm(%) Cm(%)0.5 1 1.5 2 310−1100101102Consistency (%)τy(Pa)  Figure 4.9: Yield stress of pulp suspension at different applied refining energies. (a) Hard-wood (Series 13-18), (b) Hardwood @400kWh/t (Series 18-22), (c) Hardwood @600kWh/t(Series 23-26), (d) Hardwood @900kWh/t (Series 27-30) from Table 3.1. © results fromUDV-viscometer and  results from UDV-pipe flow at the onset of the flow.47Chapter 4. Results and Discussion100 101100101102103104Cm(%)  Bennington et al. (1990)SW Dalpke and Kerekes (2005)HW Dalpke and Kerekes (2005)Derakhshandeh et al. (2010)SW (Series 2− 6)SW@2000kW/t (Series 7− 12)HW (Series 13 − 17)HW@400kW/t (Series 18 − 22)HW@600kW/t (Series 23 − 26)HW@900kW/t (Series 27 − 30)Fibre Length Cm Range τy Range a b ReferenceType (mm) (%) (Pa)SW 2.21 0.40-5.10 0.5-1158 8.21 2.79 Bennington et al. (1990)HW 0.94 2.50-5.10 43.7-285.9 1.50 3.32 Dalpke & Kerekes (2005)SW 2.89 1.40-4.10 71.0-833.8 30.91 2.35 Dalpke & Kerekes (2005)SW 2.96 0.50-5.00 1.6-358.5 8.11 2.51 Derakhshandeh et al. (2010)HW 1.28 0.50-5.00 0.2-181.6 7.25 2.81 Derakhshandeh et al. (2010)SW(Series 2-6) 3.1-32.0 1.51 2.11 SW @2000kWh/t(Series 7-12) 3.1-19.0 7.41 1.71 HW (Series 13-17) 1.9-5.8 0.38 3.11 HW @400kWh/t(Series 18-22) 2.5-16.4 0.61 3.30 HW @600kWh/t(Series 23-26) 3.1-22.2 1.25 3.11 HW @900kWh/t(Series 27-30) 2.8-19.9 2.33 2.91 Figure 4.10 & Table 4.1: Summary of literature and experimental values of fitted con-stants to the power-law model (τy = aCmb).48Chapter 4. Results and Discussion0 200 400 600 800 1000010203040SRE (kWhr/t)τy(Pa)  (a)(b)0 500 100001234SRE [kWhr/t]  baFigure 4.11: (a) Yield stress values measured with UDV-viscometer as a function of refiningenergy for refined hardwood. ©1.0%, 51.5%,♦2.0%,2.5%. (b) Fitted constants to thepower-law model (τy = aCmb) for all refined hard wood. UDV-viscometer:Ba, b. UDV-pipe flow Ia, b.49Chapter 4. Results and Discussion0 0.05 0.1 0.15 0.200.511.522.533.5 x 10−3τw0.5(D/l)τ y/N2  Series31Series32Series33Series34Series35Series36Series37Series38Series39Series40Series41Series42Series43Series44Series45Series46Figure 4.12: Relationship between yield stress , τy,and fibre morphology for Nylon fibres(Table 3.2).50Chapter 4. Results and Discussion10−1 10010−2100102  ρu′2/τyrp/R(b) (a)(c)Figure 4.13: The ratio of ρu′2 at the edge of the plug to the yield stress as a function ofplug size. The dashed line represents a linear regression of power law function to the entiredata set. The solid symbols represent maxima of the τy(Pa) data in Figure 4.8.51Chapter 4. Results and Discussion0 1 2 3 4 5x 105020406080100RewDR(%)  MDR(a)0 0.2 0.4 0.6 0.8 1020406080100rp/RDR(%)  MDR(b)Figure 4.14: Drag Reduction versus (a) Reynolds number based on water property, Rew,and (b) the plug size for cases studied. A legend relating the symbol to the testing conditionis given in Table 3.3.52Chapter 5Summary and ConclusionsIn this work we measured the instantaneous velocity profiles of fully developed Hagen-Poiseuille flow using both nylon and papermaking fibre suspensions. In some cases,dilute long chained polymers were added to the suspensions. The velocity signalswere analyzed to extract the plug size from which we estimated the yield stress. Alarge number of conditions were tested from which we qualitatively characterized theflow during transition. The results indicate that:(a) The onset of drag reduction occurs when the plug exists in the pipe. In fact wefind drag reduction occurs when the plug size rp/R < 0.8(b) The plug diminishes in complex manner with increasing flowrate. Initially wefind the plug diminishes in size by densification and then through an erosiontype mechanism. The transition between the densification and erosion occurswhen ρu′2 ≈ τy. The plug is evident even when the elongational turbulent stressis much greater than the yield stress.(c) We determine a relationship between the yield stress and the fibre morphologyfor monodisperse nylon suspensions. We attempted to derive a scaling law basedupon the crowding number and could achieve a goodness fit of r2 ∼ 0.94.53Chapter 5. Summary and Conclusions(d) We find that with papermaking fibres alone, drag reduction varies non-monotonicallywith plug size. Initially drag reduction quickly increases with decreasing plugsize and then starts to decrease. A contrasting behaviour is observed whenpolymers are added to the systems. Drag reduction approaches its maximumasymptotic.54Chapter 6ContributionThe contributions to knowledge that have resulted from this research work are iden-tified as follows:1. With the instantaneous velocity measurements, we estimated the size of theplug and the yield stress of pulp fibre suspensions was measured using the plugsize and the wall shear with a novel technique developed in this work :UDV-pipeflow technique. This method of measuring the yield stress is the most reliableto provide raw data for pulp fibre suspensions flow.2. The yield stress of the plug flow at the core of the pipe was found to be dependenton the pipe wall shear stress. The yield stress increases due to the contractingof the plug and experiences a peak before the plug breaks down because ofhigh turbulence inside its network. This is the first time that this evolution isobserved and a mechanism is proposed for it in the literature.3. Flow of highly refined fibre suspensions were studied for the first time in theliterature. It is reported that these suspensions are exhibiting very unique flowbehaviour and higher yield stress.4. In this work we shed a light on the transition from plug flow to fully turbulent55Chapter 6. Contributionflow for fibre suspension flows. We find that the plug size scales quite well withthe ratio of the ρu′2 to the yield stress of the suspension.56ReferencesBandyopadhyay, P.R. 1986 Aspects of the equilibrium puff in transitional pipeflow. J. Fluid Mech. 163, 439-458.Bennington, C. P. J., Kerekes, R.J. & Grace, J.R. 1990 The yield stress offibre suspensions. Canadian Journal of Chemical Engineering. 68(10), 748-757.Bennington, C.P.J., Azevedo, G., John, D.A., Birt, S.M. & Wolgast,B.H. 1995 The yield stress of medium and high consistency mechanical pulp fibresuspensions at high gas contents. 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Fluid Mech. 69(2), 283-304.Xu, H. & Aidun C.K. 2005 Characteristics of a fibre suspension in flow in a rect-angular channel. Int. J. Multiphase Flow. 31(3), 318-336.71AppendicesA Ultrasonic Doppler VelocimetryDoppler ultrasound technique, was originally applied in the medical field and datesback more then 30 years. The use of pulsed emissions has extended this techniqueto other fields and has open the way to new measuring techniques in fluid dynamics.The term "Doppler ultrasound velocimetry" implies that the velocity is measured byfinding the Doppler frequency in the received signal, as it is the case in Laser Dopplervelocimetry. In fact, in ultrasonic pulsed Doppler velocimetry, this is never the case.Velocities are derived from shifts in positions between pulses, and the Doppler effectplays a minor role. Unfortunately, many publications, even recent ones, fails to makethe distinction, resulting in erroneous system description and fallacious interpretationof the influence from various physical effects.In pulsed Doppler ultrasound, instead of emitting continuous ultrasonic waves, anemitter sends periodically a short ultrasonic burst and a receiver collects continuouslyechoes issues from targets that may be present in the path of the ultrasonic beam. Bysampling the incoming echoes at the same time relative to the emission of the bursts,the shift of positions of scatters are measured. Let assume a situation, as illustratedin the figure below, where only one particle is present along the ultrasonic beam.72A. Ultrasonic Doppler VelocimetryFigure A.1: Schematic of a particle passing through the ultrasound field.From the knowledge of the time delay Td between an emitted burst and the echoissue from the particle, the depth p of this particle can computed by:P =c.Td2(6.1)where c is the sound velocity of the ultrasonic wave in the liquid. If the particleis moving at an angle θ regarding the axis of the ultrasonic beam, its velocity can bemeasured by computing the variation of its depth between two emissions separatedin time by Tprf , (prf : pulse repetition frequency):P2 − P1 = V.Tprf .cos(θ) = c2.(T2 − T1) (6.2)The time difference (T2-T1) is always very short, most of the time lower than amicrosecond. It is advantageous to replace this time measurement by a measurement73A. Ultrasonic Doppler Velocimetryof the phase shift of the received echo.δ = 2pi.fe(T2 − T1) (6.3)where fe is the emitting frequency. With this information the velocity of the targetis expressed by (fd : Dopplerfrequency):V =c.δ4pi.fe.Tprf .cos(θ)=c.fd2.fe.cos(θ)(6.4)74B. Table of ResultsB Table of ResultsIn this appendix, the results from chapter 4 is presented in data table.75B. Table of ResultsTable 6.1: Estimates of the plug size, yield stress and the ratio of Reynolds stress to yieldstress for Series 2-30 at the conditions given in Table 3.1Series τw (Pa) rp/R τy (Pa) ρu′2/τy2 0.63 0.76(±0.01) 0.47(±0.18) 9.28(±0.40)2 0.63 0.68(±0.01) 0.43(±0.16) 16.11(±3.75)2 3.14 0.56(±0.01) 1.76(±0.14) 1.83(±0.08)2 6.90 0.37(±0.01) 2.57(±0.10) 2.03(±0.03)2 10.67 0.07(±0.01) 0.76(±0.09) 13.78(±1.44)2 16.94 0.02(±0.00) 0.34(±0.07) 47.13(±8.57)2 24.47 0.01(±0.00) 0.30(±0.02) 97.84(±6.18)2 31.58 0.01(±0.00) 0.32(±0.13) 119.37(±55.15)2 39.95 0.01(±0.00) 0.55(±0.05) 67.04(±5.63)2 47.48 0.02(±0.00) 0.73(±0.04) 65.82(±3.32)2 57.93 0.01(±0.00) 0.76(±0.18) 70.45(±15.01)3 0.78 0.93(±0.01) 0.73(±0.22) 0.68(±0.07)3 0.47 0.81(±0.00) 0.38(±0.19) 14.19(±0.05)3 1.10 0.77(±0.02) 0.84(±0.18) 10.08(±0.80)3 0.78 0.75(±0.01) 0.59(±0.18) 6.54(±0.19)3 2.04 0.70(±0.01) 1.44(±0.17) 0.37(±0.04)3 2.98 0.65(±0.03) 1.92(±0.17) 0.40(±0.00)3 4.55 0.58(±0.07) 2.62(±0.35) 0.37(±0.02)3 5.18 0.52(±0.01) 2.67(±0.14) 0.47(±0.00)Continued on next page76B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy3 6.74 0.44(±0.01) 3.00(±0.13) 0.76(±0.00)3 10.20 0.32(±0.01) 3.24(±0.15) 1.07(±0.02)3 12.08 0.24(±0.02) 2.89(±0.25) 1.67(±0.10)3 14.27 0.15(±0.01) 2.19(±0.11) 3.19(±0.09)3 17.72 0.08(±0.01) 1.49(±0.11) 5.09(±0.34)3 23.06 0.02(±0.00) 0.39(±0.07) 40.25(±6.85)3 29.02 0.01(±0.00) 0.36(±0.07) 63.52(±11.25)3 42.82 0.01(±0.00) 0.49(±0.15) 59.01(±18.36)4 1.41 0.96(±0.00) 1.35(±0.23) 0.49(±0.01)4 1.41 0.91(±0.02) 1.28(±0.22) 3.38(±0.12)4 1.41 0.81(±0.01) 1.14(±0.19) 3.43(±0.54)4 2.98 0.73(±0.02) 2.17(±0.18) 0.46(±0.04)4 6.12 0.69(±0.01) 4.24(±0.19) 0.35(±0.01)4 10.51 0.59(±0.02) 6.23(±0.22) 0.28(±0.00)4 12.08 0.53(±0.01) 6.34(±0.16) 0.67(±0.00)4 18.98 0.42(±0.00) 7.97(±0.10) 0.57(±0.00)4 24.00 0.31(±0.01) 7.55(±0.21) 1.22(±0.01)4 30.27 0.19(±0.00) 5.76(±0.15) 2.30(±0.02)4 37.18 0.08(±0.00) 3.01(±0.18) 5.51(±0.29)4 47.53 0.05(±0.01) 2.35(±0.44) 13.52(±2.38)Continued on next page77B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy5 2.65 0.96(±0.01) 2.55(±0.23) 0.07(±0.01)5 2.65 0.86(±0.00) 2.28(±0.21) 0.19(±0.01)5 5.16 0.82(±0.01) 4.23(±0.21) 0.27(±0.03)5 7.11 0.78(±0.02) 5.57(±0.25) 0.26(±0.03)5 10.46 0.75(±0.02) 7.86(±0.24) 0.28(±0.03)5 14.92 0.69(±0.04) 10.34(±0.58) 0.43(±0.05)5 23.28 0.66(±0.03) 15.41(±0.73) 0.31(±0.02)5 29.98 0.62(±0.01) 18.56(±0.26) 0.41(±0.01)5 36.39 0.55(±0.00) 19.85(±0.14) 0.47(±0.00)5 45.31 0.42(±0.01) 19.18(±0.60) 0.62(±0.01)5 53.96 0.30(±0.00) 16.03(±0.10) 0.96(±0.00)5 64.00 0.17(±0.00) 10.81(±0.24) 2.04(±0.02)5 75.99 0.09(±0.01) 7.05(±0.51) 3.93(±0.25)5 86.59 0.04(±0.02) 3.88(±1.79) 10.22(±6.02)6 7.67 0.99(±0.00) 7.61(±0.24) 0.11(±0.01)6 7.95 0.99(±0.00) 7.88(±0.24) 0.21(±0.02)6 9.34 0.98(±0.00) 9.15(±0.24) 0.18(±0.02)6 9.06 0.97(±0.01) 8.76(±0.24) 0.19(±0.08)6 8.23 0.93(±0.02) 7.69(±0.26) 0.15(±0.04)6 7.67 0.91(±0.02) 6.97(±0.26) 0.14(±0.01)Continued on next page78B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy6 10.46 0.90(±0.02) 9.42(±0.31) 0.22(±0.08)6 12.13 0.85(±0.04) 10.26(±0.50) 0.30(±0.13)6 12.41 0.87(±0.00) 10.82(±0.21) 0.15(±0.00)6 14.92 0.85(±0.01) 12.68(±0.25) 0.14(±0.01)6 16.31 0.82(±0.02) 13.37(±0.44) 0.22(±0.02)6 19.94 0.79(±0.04) 15.81(±0.77) 0.25(±0.04)6 24.12 0.79(±0.02) 19.02(±0.53) 0.16(±0.01)6 26.35 0.76(±0.02) 20.14(±0.51) 0.21(±0.01)6 31.65 0.75(±0.01) 23.73(±0.31) 0.19(±0.00)6 37.79 0.70(±0.03) 26.52(±1.23) 0.30(±0.01)6 40.02 0.71(±0.03) 28.60(±1.06) 0.25(±0.02)6 38.06 0.71(±0.02) 26.91(±0.69) 0.22(±0.00)6 50.33 0.63(±0.03) 31.51(±1.55) 0.27(±0.00)6 52.29 0.61(±0.03) 32.08(±1.83) 0.32(±0.01)6 62.05 0.58(±0.01) 35.83(±0.69) 0.30(±0.01)6 64.83 0.54(±0.01) 35.17(±0.66) 0.34(±0.00)6 72.64 0.48(±0.01) 34.64(±0.51) 0.43(±0.00)6 82.12 0.38(±0.03) 30.99(±2.19) 0.49(±0.02)6 88.54 0.33(±0.01) 29.54(±0.47) 0.55(±0.01)7 6.55 0.40(±0.02) 2.64(±0.15) 1.18(±0.38)Continued on next page79B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy7 9.34 0.37(±0.01) 3.44(±0.10) 1.29(±0.02)7 14.92 0.15(±0.00) 2.24(±0.04) 10.86(±0.07)7 20.50 0.05(±0.02) 1.02(±0.45) 32.77(±10.19)7 25.79 0.01(±0.00) 0.31(±0.05) 78.16(±11.89)7 34.72 0.01(±0.00) 0.45(±0.11) 68.12(±15.25)7 41.13 0.01(±0.00) 0.50(±0.08) 63.66(±9.55)7 50.89 0.01(±0.00) 0.54(±0.22) 92.31(±39.55)7 62.05 0.02(±0.01) 1.11(±0.35) 51.41(±18.68)8 5.49 0.84(±0.02) 4.63(±0.22) 0.04(±0.01)8 11.10 0.55(±0.01) 6.06(±0.20) 0.27(±0.03)8 15.25 0.35(±0.02) 5.35(±0.25) 0.41(±0.01)8 18.42 0.32(±0.01) 5.89(±0.26) 0.60(±0.01)8 20.86 0.30(±0.03) 6.17(±0.58) 0.83(±0.08)8 24.28 0.29(±0.04) 7.00(±1.04) 1.17(±0.18)8 32.09 0.19(±0.00) 5.95(±0.13) 9.73(±0.09)8 38.67 0.09(±0.02) 3.50(±0.65) 15.17(±2.16)8 49.65 0.01(±0.00) 0.46(±0.10) 136.93(±25.79)8 54.78 0.01(±0.00) 0.59(±0.13) 110.88(±26.93)8 63.07 0.02(±0.00) 1.08(±0.24) 54.94(±14.02)8 72.83 0.02(±0.01) 1.29(±0.40) 44.37(±15.77)Continued on next page80B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy8 82.84 0.02(±0.00) 1.48(±0.39) 38.80(±11.23)9 5.49 0.98(±0.01) 5.40(±0.24) 0.02(±0.00)9 10.37 0.97(±0.01) 10.04(±0.24) 0.03(±0.00)9 14.52 0.84(±0.02) 12.22(±0.32) 0.02(±0.00)9 22.33 0.50(±0.03) 11.11(±0.63) 0.05(±0.00)9 27.94 0.42(±0.01) 11.73(±0.35) 0.10(±0.00)9 31.11 0.38(±0.01) 11.90(±0.32) 0.18(±0.00)9 34.28 0.32(±0.03) 11.05(±1.06) 0.27(±0.02)9 37.94 0.28(±0.00) 10.56(±0.12) 0.36(±0.00)9 40.38 0.26(±0.00) 10.43(±0.09) 0.42(±0.00)9 44.04 0.21(±0.00) 9.14(±0.07) 0.65(±0.00)9 49.90 0.17(±0.00) 8.65(±0.13) 3.60(±0.03)9 60.15 0.09(±0.01) 5.16(±0.42) 13.39(±0.91)9 68.69 0.03(±0.00) 1.99(±0.32) 34.33(±5.13)10 17.88 0.98(±0.01) 17.57(±0.25) 0.00(±0.00)10 17.25 0.99(±0.00) 17.03(±0.25) 0.02(±0.00)10 24.78 0.96(±0.02) 23.83(±0.48) 0.01(±0.00)10 30.01 0.90(±0.03) 26.92(±0.95) 0.01(±0.00)10 35.66 0.78(±0.05) 27.95(±1.72) 0.01(±0.00)10 47.79 0.74(±0.04) 35.40(±1.82) 0.03(±0.01)Continued on next page81B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy10 57.62 0.56(±0.01) 32.09(±0.34) 0.05(±0.00)10 63.27 0.52(±0.00) 32.81(±0.34) 0.07(±0.00)10 67.03 0.48(±0.01) 32.30(±0.72) 0.12(±0.00)10 69.33 0.46(±0.01) 32.18(±0.57) 0.16(±0.00)10 70.59 0.46(±0.00) 32.12(±0.35) 0.20(±0.00)10 74.35 0.43(±0.01) 32.09(±0.41) 0.26(±0.01)11 22.05 0.99(±0.01) 21.89(±0.28) 0.01(±0.00)11 20.10 0.95(±0.02) 19.18(±0.46) 0.01(±0.00)11 21.32 0.94(±0.03) 20.06(±0.64) 0.01(±0.00)11 22.29 0.93(±0.03) 20.82(±0.71) 0.01(±0.00)11 24.00 0.90(±0.02) 21.68(±0.57) 0.02(±0.00)11 33.52 0.85(±0.01) 28.42(±0.47) 0.02(±0.00)11 46.20 0.75(±0.03) 34.59(±1.29) 0.02(±0.00)11 53.04 0.96(±0.00) 50.86(±0.34) 0.01(±0.00)11 55.72 0.67(±0.01) 37.10(±0.77) 0.03(±0.00)11 59.87 0.63(±0.01) 37.58(±0.55) 0.05(±0.00)11 67.92 0.59(±0.05) 40.33(±3.07) 0.05(±0.00)11 75.73 0.53(±0.04) 40.15(±3.07) 0.08(±0.00)11 79.39 0.53(±0.03) 41.85(±2.59) 0.10(±0.01)11 85.49 0.49(±0.02) 41.82(±1.72) 0.23(±0.01)Continued on next page82B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy12 25.41 0.98(±0.00) 24.90(±0.25) 0.00(±0.00)12 28.55 0.97(±0.00) 27.72(±0.26) 0.00(±0.00)12 27.29 0.86(±0.02) 23.34(±0.62) 0.01(±0.00)12 28.97 0.83(±0.01) 24.13(±0.33) 0.01(±0.00)12 39.63 0.80(±0.01) 31.81(±0.40) 0.09(±0.00)12 52.18 0.78(±0.01) 40.79(±0.78) 0.03(±0.00)12 64.94 0.74(±0.01) 48.30(±0.50) 0.06(±0.00)12 73.93 0.68(±0.02) 50.10(±1.22) 0.06(±0.01)12 80.00 0.61(±0.02) 48.99(±1.58) 0.08(±0.01)12 85.44 0.59(±0.01) 50.41(±0.70) 0.09(±0.00)12 88.57 0.58(±0.01) 51.35(±0.63) 0.14(±0.00)13 0.42 0.70(±0.06) 0.29(±0.17) 23.64(±4.23)13 1.53 0.58(±0.04) 0.90(±0.15) 3.01(±0.24)13 1.25 0.57(±0.03) 0.72(±0.14) 4.30(±1.14)13 1.81 0.52(±0.03) 0.94(±0.13) 3.39(±0.04)13 3.49 0.45(±0.02) 1.55(±0.13) 0.80(±0.02)13 4.04 0.43(±0.02) 1.76(±0.12) 0.64(±0.01)13 4.60 0.33(±0.02) 1.51(±0.13) 0.83(±0.03)13 6.83 0.26(±0.04) 1.79(±0.29) 1.89(±0.25)13 7.95 0.18(±0.03) 1.42(±0.21) 8.10(±0.60)Continued on next page83B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy13 13.52 0.10(±0.05) 1.34(±0.69) 16.98(±5.16)13 16.31 0.05(±0.04) 0.76(±0.66) 46.46(±28.71)13 18.54 0.02(±0.03) 0.46(±0.47) 94.02(±59.95)13 22.45 0.03(±0.00) 0.72(±0.07) 36.15(±3.61)13 24.40 0.02(±0.01) 0.58(±0.13) 51.69(±11.81)13 30.81 0.01(±0.01) 0.43(±0.27) 80.19(±38.54)13 43.64 0.02(±0.01) 0.91(±0.58) 40.63(±23.03)14 0.47 0.88(±0.01) 0.41(±0.21) 0.69(±0.06)14 1.41 0.79(±0.03) 1.11(±0.19) 0.38(±0.02)14 1.41 0.61(±0.02) 0.86(±0.15) 0.51(±0.03)14 1.73 0.58(±0.00) 1.00(±0.14) 23.74(±0.14)14 3.29 0.63(±0.03) 2.07(±0.17) 0.31(±0.02)14 3.61 0.57(±0.02) 2.05(±0.16) 0.29(±0.01)14 3.29 0.52(±0.01) 1.70(±0.13) 0.39(±0.00)14 4.86 0.45(±0.01) 2.16(±0.11) 0.36(±0.02)14 6.12 0.38(±0.02) 2.34(±0.15) 0.40(±0.04)14 8.63 0.34(±0.01) 2.89(±0.11) 0.37(±0.01)14 13.33 0.29(±0.04) 3.87(±0.55) 0.49(±0.05)14 17.41 0.16(±0.01) 2.81(±0.10) 5.60(±0.14)14 24.00 0.05(±0.02) 1.28(±0.53) 22.13(±9.62)Continued on next page84B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy14 32.16 0.03(±0.01) 0.92(±0.40) 37.15(±21.26)14 40.63 0.02(±0.01) 0.91(±0.45) 44.51(±29.60)14 50.04 0.03(±0.01) 1.58(±0.73) 27.26(±11.19)14 59.14 0.03(±0.00) 1.75(±0.22) 28.35(±3.31)14 70.53 0.03(±0.00) 1.77(±0.27) 36.29(±5.23)15 1.59 0.74(±0.03) 1.18(±0.18) 0.58(±0.02)15 1.10 0.63(±0.05) 0.69(±0.16) 0.58(±0.04)15 3.05 0.56(±0.05) 1.72(±0.21) 0.47(±0.04)15 4.03 0.51(±0.03) 2.06(±0.16) 0.42(±0.04)15 6.71 0.45(±0.03) 3.04(±0.22) 0.47(±0.02)15 10.37 0.39(±0.01) 4.02(±0.17) 0.62(±0.03)15 15.74 0.28(±0.02) 4.44(±0.38) 0.92(±0.08)15 20.62 0.19(±0.01) 3.91(±0.24) 1.25(±0.07)15 26.47 0.10(±0.01) 2.58(±0.22) 2.30(±0.20)15 34.53 0.03(±0.01) 1.00(±0.21) 18.28(±4.40)15 43.55 0.02(±0.00) 0.74(±0.16) 30.98(±7.29)16 3.78 0.86(±0.00) 3.25(±0.21) 0.22(±0.01)16 2.56 0.73(±0.05) 1.87(±0.22) 9.28(±3.48)16 3.05 0.57(±0.03) 1.75(±0.17) 12.78(±4.08)16 3.54 0.54(±0.05) 1.91(±0.21) 2.41(±1.75)Continued on next page85B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy16 5.25 0.53(±0.05) 2.79(±0.29) 1.81(±1.17)16 6.95 0.52(±0.07) 3.65(±0.48) 2.09(±1.25)16 9.15 0.51(±0.05) 4.64(±0.51) 1.42(±1.16)16 11.10 0.51(±0.07) 5.69(±0.81) 0.93(±0.24)16 12.57 0.52(±0.08) 6.55(±0.95) 0.87(±0.23)16 15.49 0.51(±0.06) 7.87(±0.90) 0.80(±0.10)16 20.86 0.49(±0.05) 10.25(±1.00) 0.83(±0.17)16 25.74 0.49(±0.10) 12.54(±2.57) 0.72(±0.36)16 31.11 0.34(±0.07) 10.60(±2.17) 0.61(±0.12)16 38.92 0.25(±0.02) 9.57(±0.94) 0.68(±0.05)16 45.99 0.15(±0.03) 6.77(±1.54) 330.23(±178.53)16 54.05 0.05(±0.02) 2.69(±1.12) 0.00(±0.00)17 7.22 0.84(±0.01) 6.04(±0.21) 0.08(±0.00)17 5.33 0.70(±0.03) 3.72(±0.24) 0.31(±0.02)17 6.80 0.63(±0.04) 4.26(±0.30) 0.36(±0.03)17 9.31 0.62(±0.05) 5.80(±0.51) 0.34(±0.03)17 12.65 0.60(±0.06) 7.65(±0.71) 0.48(±0.06)17 20.18 0.59(±0.06) 11.96(±1.28) 0.47(±0.07)17 23.53 0.53(±0.06) 12.45(±1.43) 0.71(±0.16)17 30.01 0.47(±0.09) 14.17(±2.82) 0.49(±0.09)Continued on next page86B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy17 35.87 0.39(±0.08) 13.81(±3.03) 0.75(±0.17)17 41.31 0.32(±0.07) 13.22(±2.79) 0.86(±0.19)17 46.33 0.19(±0.01) 8.62(±0.70) 67.68(±5.64)17 57.83 0.09(±0.01) 5.02(±0.38) 0.00(±0.00)17 64.10 0.04(±0.00) 2.44(±0.25) 9.33(±0.92)18 0.98 0.38(±0.02) 0.37(±0.09) 19.18(±2.49)18 3.24 0.31(±0.02) 0.99(±0.10) 1.54(±0.03)18 5.37 0.19(±0.02) 1.02(±0.11) 1.01(±0.08)18 13.87 0.07(±0.04) 0.90(±0.62) 34.62(±24.59)18 18.78 0.06(±0.03) 1.22(±0.64) 29.06(±12.32)18 24.42 0.06(±0.04) 1.43(±1.02) 40.94(±44.38)18 32.40 0.04(±0.02) 1.24(±0.67) 28.17(±13.96)18 43.86 0.03(±0.01) 1.28(±0.53) 28.70(±13.62)19 1.09 0.79(±0.00) 0.86(±0.19) 0.10(±0.00)19 1.50 0.75(±0.03) 1.12(±0.18) 0.15(±0.02)19 1.92 0.70(±0.03) 1.35(±0.18) 0.96(±0.10)19 3.31 0.70(±0.01) 2.32(±0.17) 1.42(±0.26)19 4.66 0.66(±0.03) 3.10(±0.22) 0.66(±0.02)19 5.48 0.63(±0.05) 3.47(±0.32) 0.53(±0.02)19 8.20 0.52(±0.08) 4.28(±0.66) 0.44(±0.05)Continued on next page87B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy19 10.89 0.37(±0.04) 4.08(±0.41) 0.72(±0.06)19 17.32 0.08(±0.03) 1.42(±0.58) 14.17(±4.69)19 24.42 0.07(±0.03) 1.71(±0.66) 21.40(±6.48)19 30.49 0.03(±0.02) 0.94(±0.54) 36.36(±21.33)19 41.29 0.03(±0.01) 1.26(±0.28) 18.96(±3.86)20 3.87 0.98(±0.01) 3.78(±0.23) 0.05(±0.00)20 4.31 0.93(±0.00) 3.99(±0.22) 0.20(±0.00)20 6.20 0.87(±0.01) 5.37(±0.22) 0.23(±0.01)20 11.07 0.85(±0.00) 9.37(±0.20) 0.19(±0.00)20 15.09 0.80(±0.01) 12.14(±0.29) 0.19(±0.00)20 20.02 0.72(±0.02) 14.42(±0.47) 0.25(±0.01)20 24.17 0.58(±0.02) 13.99(±0.52) 0.29(±0.01)20 26.92 0.39(±0.02) 10.55(±0.50) 0.44(±0.02)20 32.95 0.11(±0.02) 3.71(±0.64) 7.43(±0.80)20 36.70 0.07(±0.02) 2.67(±0.75) 13.32(±2.93)20 43.72 0.06(±0.03) 2.57(±1.33) 23.31(±9.23)20 48.69 0.06(±0.03) 2.68(±1.38) 25.67(±10.42)20 61.21 0.06(±0.03) 3.97(±2.03) 19.31(±6.98)20 72.92 0.06(±0.03) 4.28(±2.07) 15.81(±5.48)21 9.17 0.96(±0.04) 8.81(±0.41) 0.02(±0.00)Continued on next page88B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy21 6.64 0.97(±0.02) 6.43(±0.28) 0.02(±0.00)21 8.69 0.96(±0.03) 8.35(±0.35) 0.06(±0.00)21 12.84 0.92(±0.03) 11.80(±0.45) 0.13(±0.00)21 17.74 0.91(±0.03) 16.07(±0.53) 0.15(±0.00)21 22.30 0.88(±0.04) 19.61(±0.81) 0.15(±0.00)21 26.92 0.84(±0.05) 22.49(±1.33) 0.18(±0.01)21 32.95 0.79(±0.04) 26.11(±1.47) 0.17(±0.01)21 37.19 0.72(±0.02) 26.61(±0.90) 0.21(±0.01)21 40.32 0.58(±0.03) 23.44(±1.36) 0.37(±0.02)21 43.72 0.55(±0.05) 23.88(±2.29) 0.38(±0.03)21 50.02 0.38(±0.02) 19.07(±1.25) 0.54(±0.02)21 55.71 0.17(±0.01) 9.55(±0.54) 5.49(±0.31)21 64.60 0.09(±0.04) 5.74(±2.70) 9.58(±2.89)22 23.22 0.99(±0.01) 22.89(±0.26) 0.00(±0.00)22 19.75 0.96(±0.03) 19.01(±0.65) 0.00(±0.00)22 27.66 0.96(±0.03) 26.42(±0.86) 0.00(±0.00)22 24.50 0.97(±0.02) 23.77(±0.43) 0.03(±0.00)22 26.56 0.96(±0.02) 25.55(±0.48) 0.02(±0.00)22 28.41 0.92(±0.06) 26.20(±1.65) 0.05(±0.00)22 29.58 0.91(±0.04) 26.84(±1.29) 0.06(±0.00)Continued on next page89B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy22 36.21 0.90(±0.03) 32.66(±1.11) 0.07(±0.00)22 42.56 0.89(±0.03) 37.72(±1.39) 0.09(±0.00)22 52.08 0.89(±0.03) 46.23(±1.80) 0.09(±0.00)22 55.71 0.90(±0.03) 49.88(±1.42) 0.10(±0.00)22 63.74 0.88(±0.03) 56.00(±2.17) 0.10(±0.01)22 71.95 0.87(±0.04) 62.44(±2.56) 0.11(±0.00)22 80.13 0.85(±0.02) 67.82(±1.78) 0.12(±0.00)23 2.93 0.94(±0.00) 2.74(±0.22) 0.74(±0.00)23 7.95 0.64(±0.01) 5.06(±0.18) 1.75(±0.03)23 8.51 0.51(±0.03) 4.37(±0.26) 0.58(±0.02)23 10.18 0.43(±0.03) 4.39(±0.30) 0.72(±0.06)23 14.92 0.37(±0.01) 5.51(±0.23) 7.96(±0.28)23 21.05 0.36(±0.02) 7.50(±0.47) 10.07(±0.40)23 29.14 0.29(±0.01) 8.56(±0.31) 6.97(±0.22)23 36.95 0.23(±0.01) 8.63(±0.21) 5.14(±0.03)23 44.48 0.16(±0.02) 7.09(±0.93) 5.28(±0.54)24 7.93 0.94(±0.02) 7.49(±0.29) 0.02(±0.00)24 10.37 0.83(±0.06) 8.62(±0.62) 0.11(±0.00)24 16.47 0.73(±0.04) 12.10(±0.73) 0.16(±0.01)24 21.35 0.67(±0.04) 14.39(±0.78) 0.15(±0.01)Continued on next page90B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy24 23.79 0.44(±0.06) 10.54(±1.51) 0.22(±0.04)24 25.74 0.39(±0.02) 9.91(±0.54) 0.27(±0.02)24 28.67 0.32(±0.01) 9.26(±0.43) 0.35(±0.01)24 30.87 0.26(±0.02) 7.96(±0.68) 0.77(±0.02)24 33.06 0.21(±0.01) 6.90(±0.24) 1.18(±0.03)24 46.24 0.11(±0.01) 5.22(±0.66) 7.01(±0.55)24 63.81 0.05(±0.02) 3.45(±1.25) 18.96(±5.62)25 19.53 0.99(±0.00) 19.30(±0.25) 0.00(±0.00)25 17.02 0.89(±0.05) 15.23(±0.86) 0.06(±0.00)25 23.45 0.84(±0.03) 19.65(±0.65) 0.09(±0.01)25 30.04 0.81(±0.04) 24.37(±1.30) 0.08(±0.01)25 35.53 0.75(±0.04) 26.77(±1.30) 0.07(±0.00)25 39.76 0.71(±0.04) 28.21(±1.79) 0.09(±0.01)25 43.37 0.69(±0.04) 30.09(±1.82) 0.10(±0.00)25 47.76 0.64(±0.05) 30.48(±2.45) 0.17(±0.01)25 52.47 0.56(±0.05) 29.15(±2.40) 0.21(±0.03)25 56.70 0.53(±0.04) 30.07(±2.12) 0.24(±0.01)26 47.27 0.99(±0.00) 46.57(±0.27) 0.02(±0.00)26 40.57 0.92(±0.02) 37.40(±0.72) 0.12(±0.00)26 41.69 0.82(±0.02) 34.19(±0.94) 0.10(±0.00)Continued on next page91B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy26 43.92 0.71(±0.04) 31.24(±1.76) 0.14(±0.01)26 50.61 0.63(±0.03) 32.08(±1.65) 0.09(±0.00)26 57.86 0.60(±0.06) 34.69(±3.33) 0.09(±0.00)26 57.58 0.58(±0.05) 33.41(±2.72) 0.10(±0.00)26 64.28 0.51(±0.06) 32.51(±3.59) 0.12(±0.02)26 69.92 0.46(±0.03) 31.87(±1.94) 0.16(±0.00)26 67.63 0.40(±0.03) 27.16(±1.91) 0.19(±0.01)26 79.09 0.30(±0.01) 23.64(±1.18) 0.24(±0.01)27 2.76 0.65(±0.06) 1.79(±0.22) 0.54(±0.03)27 3.89 0.47(±0.03) 1.81(±0.17) 2.98(±0.16)27 6.84 0.44(±0.02) 3.02(±0.17) 4.37(±0.40)27 10.45 0.40(±0.00) 4.18(±0.10) 1.62(±0.27)27 12.29 0.18(±0.02) 2.20(±0.31) 11.85(±1.22)27 15.04 0.11(±0.02) 1.64(±0.23) 16.88(±1.62)27 18.78 0.10(±0.03) 1.83(±0.50) 21.06(±3.87)27 22.52 0.05(±0.02) 1.03(±0.44) 35.74(±12.57)27 24.92 0.03(±0.02) 0.71(±0.41) 54.60(±24.33)27 31.12 0.02(±0.01) 0.69(±0.22) 64.80(±22.07)27 42.13 0.02(±0.01) 0.97(±0.21) 35.96(±6.77)27 63.10 0.02(±0.02) 1.46(±0.95) 32.28(±15.54)Continued on next page92B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy27 68.41 0.03(±0.01) 1.89(±0.63) 24.85(±6.90)27 90.76 0.03(±0.01) 2.72(±0.93) 19.44(±7.48)28 2.81 0.63(±0.06) 1.77(±0.22) 0.05(±0.00)28 7.57 0.49(±0.08) 3.69(±0.59) 0.07(±0.01)28 11.11 0.32(±0.02) 3.56(±0.19) 0.96(±0.05)28 12.29 0.32(±0.04) 3.88(±0.45) 1.15(±0.12)28 12.54 0.30(±0.03) 3.77(±0.37) 0.46(±0.04)28 13.06 0.28(±0.03) 3.61(±0.42) 59.09(±7.23)28 15.04 0.28(±0.03) 4.18(±0.38) 1.42(±0.13)28 28.70 0.12(±0.06) 3.35(±1.77) 13.36(±6.04)28 42.13 0.04(±0.02) 1.84(±0.68) 22.51(±10.14)28 58.20 0.03(±0.00) 1.95(±0.12) 22.17(±1.23)28 71.23 0.03(±0.00) 1.85(±0.26) 26.63(±3.99)28 85.42 0.03(±0.00) 2.28(±0.22) 23.38(±2.17)28 92.61 0.02(±0.01) 1.90(±0.71) 31.31(±9.51)29 12.39 0.60(±0.02) 7.39(±0.28) 0.02(±0.00)29 16.47 0.56(±0.02) 9.16(±0.38) 0.73(±0.15)29 22.74 0.44(±0.03) 10.08(±0.60) 0.24(±0.17)29 27.76 0.40(±0.06) 11.03(±1.56) 0.20(±0.14)29 33.72 0.35(±0.04) 11.75(±1.40) 0.20(±0.00)Continued on next page93B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy29 34.98 0.34(±0.03) 11.81(±0.98) 0.18(±0.02)29 42.51 0.31(±0.01) 13.12(±0.52) 0.19(±0.01)29 48.16 0.27(±0.02) 13.17(±0.79) 3.44(±0.20)29 54.43 0.19(±0.05) 10.37(±2.61) 7.68(±1.36)29 56.00 0.07(±0.01) 3.70(±0.53) 15.10(±1.90)29 62.59 0.04(±0.01) 2.61(±0.33) 20.96(±2.63)29 70.43 0.03(±0.00) 2.41(±0.19) 26.11(±1.86)29 80.15 0.03(±0.00) 2.14(±0.19) 40.87(±3.54)29 92.08 0.03(±0.00) 2.88(±0.40) 22.72(±3.00)29 99.29 0.02(±0.01) 2.44(±0.76) 25.34(±6.57)29 103.68 0.02(±0.01) 2.23(±0.86) 31.98(±9.63)29 105.88 0.02(±0.00) 1.87(±0.07) 38.29(±1.40)30 23.37 0.58(±0.04) 13.64(±0.95) 0.00(±0.00)30 24.63 0.49(±0.08) 12.01(±2.09) 0.00(±0.00)30 28.39 0.46(±0.07) 13.17(±1.85) 0.03(±0.00)30 27.45 0.46(±0.05) 12.49(±1.49) 0.07(±0.01)30 36.55 0.38(±0.06) 13.88(±2.09) 0.01(±0.00)30 36.23 0.39(±0.07) 14.30(±2.49) 0.01(±0.00)30 40.31 0.36(±0.05) 14.65(±1.99) 0.02(±0.00)30 44.08 0.36(±0.06) 15.74(±2.56) 0.02(±0.00)Continued on next page94B. Table of ResultsTable 6.1  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy30 46.27 0.35(±0.06) 16.15(±2.60) 0.04(±0.00)30 51.61 0.35(±0.04) 17.91(±1.98) 0.05(±0.00)30 55.37 0.34(±0.04) 19.10(±2.22) 0.07(±0.00)30 58.19 0.34(±0.02) 19.57(±1.11) 0.07(±0.00)30 60.70 0.34(±0.03) 20.49(±1.97) 0.11(±0.01)30 62.90 0.32(±0.03) 20.38(±1.85) 0.15(±0.01)30 64.78 0.33(±0.03) 21.29(±1.69) 0.14(±0.00)30 66.04 0.32(±0.03) 21.16(±1.76) 0.17(±0.01)30 67.29 0.42(±0.11) 28.19(±7.68) 0.17(±0.04)30 67.92 0.31(±0.03) 21.31(±1.78) 0.19(±0.02)30 72.94 0.30(±0.04) 21.75(±3.06) 0.18(±0.02)30 79.84 0.31(±0.04) 24.53(±3.09) 0.16(±0.01)30 84.86 0.29(±0.03) 24.75(±2.61) 0.20(±0.02)30 96.15 0.24(±0.02) 23.50(±1.98) 0.47(±0.04)30 98.98 0.16(±0.00) 16.05(±0.37) 7.10(±0.14)30 102.11 0.13(±0.01) 13.56(±1.18) 5.98(±0.53)30 109.64 0.06(±0.02) 6.84(±1.68) 18.22(±3.88)30 117.17 0.02(±0.00) 1.89(±0.36) 40.88(±8.09)30 130.96 0.01(±0.00) 0.75(±0.32) 106.95(±49.60)95B. Table of ResultsTable 6.2: Estimates of the plug size and yield stress of nylon fibre suspensions for Series1-16 at the conditions given in Table 3.2Series τw (Pa) rp/R τy (Pa)31 1.08 0.76 (±0.01) 0.82 (±0.18)31 1.50 0.69 (±0.04) 1.03 (±0.17)31 2.42 0.62 (±0.04) 1.50 (±0.15)31 3.58 0.52 (±0.01) 1.86 (±0.13)31 4.96 0.40 (±0.02) 1.98 (±0.10)31 6.35 0.34 (±0.03) 2.17 (±0.10)31 8.19 0.27 (±0.01) 2.19 (±0.07)31 12.11 0.18 (±0.01) 2.22 (±0.07)31 15.57 0.15 (±0.01) 2.32 (±0.05)31 19.50 0.12 (±0.00) 2.37 (±0.05)31 24.34 0.07 (±0.00) 1.70 (±0.03)32 1.08 0.70 (±0.01) 0.75 (±0.19)32 1.50 0.61 (±0.01) 0.91 (±0.16)32 1.50 0.57 (±0.01) 0.85 (±0.15)32 3.11 0.42 (±0.01) 1.30 (±0.11)32 4.73 0.30 (±0.01) 1.42 (±0.10)32 6.35 0.23 (±0.00) 1.48 (±0.07)32 8.42 0.17 (±0.01) 1.45 (±0.05)32 11.65 0.14 (±0.01) 1.57 (±0.06)Continued on next page96B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)32 15.11 0.11 (±0.01) 1.60 (±0.08)32 19.73 0.08 (±0.02) 1.67 (±0.20)2 23.42 0.07 (±0.00) 1.54 (±0.06)33 1.08 0.66 (±0.01) 0.71 (±0.18)33 1.38 0.54 (±0.00) 0.75 (±0.14)33 1.75 0.45 (±0.02) 0.79 (±0.11)33 3.63 0.29 (±0.00) 1.04 (±0.09)33 4.71 0.23 (±0.00) 1.09 (±0.06)33 5.79 0.20 (±0.00) 1.13 (±0.05)33 8.48 0.14 (±0.00) 1.19 (±0.04)33 11.71 0.11 (±0.00) 1.34 (±0.05)33 14.94 0.09 (±0.00) 1.28 (±0.06)33 19.52 0.06 (±0.00) 1.13 (±0.06)34 1.08 0.51 (±0.01) 0.54 (±0.12)34 1.75 0.39 (±0.00) 0.69 (±0.09)34 2.29 0.33 (±0.01) 0.76 (±0.08)34 3.36 0.25 (±0.01) 0.85 (±0.04)34 5.52 0.16 (±0.02) 0.86 (±0.05)34 9.02 0.11 (±0.01) 1.01 (±0.06)34 12.25 0.09 (±0.03) 1.13 (±0.17)Continued on next page97B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)34 16.29 0.06 (±0.00) 1.05 (±0.04)35 1.21 0.86 (±0.02) 1.04 (±0.20)35 2.02 0.80 (±0.02) 1.62 (±0.19)35 3.10 0.68 (±0.01) 2.10 (±0.16)35 4.44 0.53 (±0.01) 2.36 (±0.13)35 5.52 0.44 (±0.01) 2.46 (±0.09)35 7.40 0.34 (±0.02) 2.53 (±0.09)35 9.29 0.31 (±0.00) 2.91 (±0.06)35 12.25 0.24 (±0.01) 2.89 (±0.07)35 14.67 0.21 (±0.00) 3.14 (±0.04)35 17.90 0.18 (±0.00) 3.29 (±0.06)35 21.13 0.17 (±0.01) 3.50 (±0.12)35 24.09 0.14 (±0.01) 3.34 (±0.14)35 28.40 0.11 (±0.00) 3.20 (±0.05)35 32.17 0.10 (±0.01) 3.08 (±0.15)35 37.01 0.07 (±0.01) 2.47 (±0.30)36 0.94 0.73 (±0.01) 0.69 (±0.19)36 1.21 0.72 (±0.00) 0.87 (±0.19)36 2.02 0.57 (±0.00) 1.16 (±0.12)36 3.10 0.48 (±0.00) 1.47 (±0.10)Continued on next page98B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)36 4.71 0.45 (±0.02) 2.14 (±0.08)36 6.06 0.35 (±0.02) 2.13 (±0.07)36 8.48 0.25 (±0.02) 2.14 (±0.05)36 10.36 0.22 (±0.01) 2.25 (±0.06)36 13.33 0.18 (±0.01) 2.41 (±0.04)36 15.48 0.14 (±0.01) 2.15 (±0.06)36 20.32 0.10 (±0.01) 1.94 (±0.13)36 25.71 0.07 (±0.01) 1.75 (±0.16)37 1.21 0.68 (±0.01) 0.83 (±0.15)37 2.02 0.48 (±0.00) 0.98 (±0.08)37 3.36 0.33 (±0.01) 1.12 (±0.05)37 4.98 0.26 (±0.03) 1.30 (±0.05)37 6.60 0.23 (±0.02) 1.49 (±0.05)37 8.48 0.18 (±0.01) 1.50 (±0.02)37 10.90 0.14 (±0.05) 1.55 (±0.14)37 13.59 0.12 (±0.01) 1.63 (±0.03)37 18.98 0.10 (±0.05) 1.81 (±0.30)37 22.48 0.08 (±0.01) 1.79 (±0.14)37 27.32 0.06 (±0.01) 1.63 (±0.27)38 0.94 0.51 (±0.00) 0.48 (±0.11)Continued on next page99B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)38 2.56 0.35 (±0.01) 0.89 (±0.05)38 3.36 0.26 (±0.00) 0.89 (±0.04)38 4.98 0.19 (±0.01) 0.93 (±0.03)38 6.60 0.16 (±0.05) 1.09 (±0.08)38 8.48 0.13 (±0.01) 1.07 (±0.02)38 10.90 0.11 (±0.02) 1.21 (±0.08)38 14.13 0.10 (±0.00) 1.35 (±0.04)38 18.17 0.08 (±0.02) 1.44 (±0.20)38 23.29 0.05 (±0.00) 1.08 (±0.02)39 2.65 0.77 (±0.01) 2.03 (±0.18)39 3.81 0.65 (±0.02) 2.46 (±0.16)39 4.96 0.60 (±0.01) 3.00 (±0.15)39 6.58 0.51 (±0.02) 3.34 (±0.14)39 8.19 0.50 (±0.01) 4.09 (±0.13)39 10.27 0.47 (±0.02) 4.79 (±0.14)39 12.81 0.38 (±0.01) 4.91 (±0.13)39 13.96 0.35 (±0.03) 4.92 (±0.20)39 18.57 0.28 (±0.02) 5.18 (±0.19)39 23.42 0.26 (±0.02) 6.04 (±0.23)39 28.50 0.25 (±0.01) 7.13 (±0.09)Continued on next page100B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)39 34.03 0.17 (±0.02) 5.63 (±0.33)39 37.96 0.13 (±0.03) 4.94 (±0.56)39 46.72 0.10 (±0.02) 4.62 (±0.35)39 52.03 0.07 (±0.01) 3.84 (±0.28)40 1.69 0.70 (±0.01) 1.19 (±0.16)40 2.29 0.66 (±0.01) 1.51 (±0.16)40 3.36 0.58 (±0.01) 1.94 (±0.14)40 5.25 0.55 (±0.01) 2.87 (±0.13)40 6.33 0.48 (±0.01) 3.03 (±0.12)40 8.48 0.44 (±0.01) 3.72 (±0.11)40 11.71 0.33 (±0.01) 3.91 (±0.10)40 14.67 0.27 (±0.01) 3.98 (±0.10)40 18.98 0.21 (±0.01) 3.95 (±0.09)40 23.55 0.14 (±0.01) 3.25 (±0.15)40 28.67 0.10 (±0.01) 2.92 (±0.19)41 1.38 0.66 (±0.01) 0.92 (±0.16)41 2.02 0.63 (±0.00) 1.27 (±0.15)41 2.56 0.55 (±0.01) 1.40 (±0.13)41 3.36 0.44 (±0.01) 1.50 (±0.11)41 4.71 0.38 (±0.01) 1.81 (±0.09)Continued on next page101B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)41 6.60 0.32 (±0.02) 2.11 (±0.09)41 8.21 0.26 (±0.02) 2.14 (±0.08)41 11.44 0.19 (±0.01) 2.15 (±0.06)41 14.94 0.12 (±0.00) 1.76 (±0.04)41 19.25 0.10 (±0.00) 1.90 (±0.04)41 24.09 0.06 (±0.01) 1.48 (±0.20)42 0.77 0.53 (±0.01) 0.41 (±0.13)42 1.48 0.44 (±0.01) 0.64 (±0.10)42 1.48 0.42 (±0.01) 0.62 (±0.10)42 2.29 0.35 (±0.01) 0.80 (±0.08)42 3.36 0.29 (±0.02) 0.98 (±0.07)42 4.17 0.29 (±0.01) 1.21 (±0.05)42 6.60 0.22 (±0.00) 1.45 (±0.02)42 9.83 0.15 (±0.01) 1.49 (±0.03)42 13.33 0.09 (±0.02) 1.18 (±0.15)42 17.90 0.06 (±0.00) 1.11 (±0.03)43 6.60 0.98 (±0.02) 6.46 (±0.53)43 7.94 0.97 (±0.01) 7.67 (±0.51)43 10.63 0.95 (±0.00) 10.10 (±0.50)43 11.92 0.92 (±0.00) 10.92 (±0.48)Continued on next page102B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)43 16.29 0.90 (±0.00) 14.65 (±0.48)43 20.08 0.83 (±0.01) 16.65 (±0.45)43 25.03 0.77 (±0.01) 19.16 (±0.41)43 35.24 0.66 (±0.02) 23.14 (±0.85)43 41.60 0.63 (±0.01) 26.20 (±0.51)43 49.63 0.54 (±0.05) 26.62 (±1.77)43 55.16 0.51 (±0.00) 28.02 (±0.29)43 60.35 0.48 (±0.02) 28.71 (±1.05)43 62.11 0.47 (±0.05) 29.12 (±2.43)43 66.82 0.43 (±0.02) 29.03 (±1.41)43 71.12 0.42 (±0.04) 30.13 (±2.34)44 4.71 0.85 (±0.03) 4.01 (±0.47)44 5.79 0.82 (±0.00) 4.73 (±0.43)44 7.13 0.78 (±0.01) 5.59 (±0.41)44 9.56 0.77 (±0.00) 7.39 (±0.41)44 10.90 0.78 (±0.00) 8.46 (±0.41)44 12.52 0.76 (±0.00) 9.49 (±0.40)44 14.13 0.69 (±0.00) 9.74 (±0.36)44 16.56 0.65 (±0.01) 10.68 (±0.37)44 20.06 0.67 (±0.00) 13.50 (±0.36)Continued on next page103B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)44 23.55 0.65 (±0.01) 15.36 (±0.40)44 27.05 0.60 (±0.00) 16.33 (±0.33)44 35.12 0.54 (±0.00) 19.08 (±0.31)44 38.69 0.49 (±0.00) 19.10 (±0.29)44 46.55 0.45 (±0.00) 21.01 (±0.30)44 47.95 0.43 (±0.00) 20.56 (±0.34)44 53.56 0.40 (±0.01) 21.34 (±0.89)44 59.26 0.36 (±0.00) 21.59 (±0.25)45 4.17 0.81 (±2.10) 3.38 (±0.43)45 4.98 0.80 (±2.38) 3.96 (±0.42)45 7.13 0.74 (±3.02) 5.30 (±0.39)45 8.48 0.71 (±3.38) 6.05 (±0.38)45 9.83 0.70 (±3.79) 6.88 (±0.37)45 11.71 0.68 (±4.34) 8.00 (±0.36)45 13.33 0.62 (±4.43) 8.25 (±0.41)45 16.29 0.69 (±5.99) 11.29 (±0.41)45 19.79 0.56 (±5.80) 11.05 (±0.30)45 21.94 0.51 (±5.79) 11.08 (±0.43)45 27.32 0.49 (±7.00) 13.50 (±1.23)45 32.71 0.45 (±7.50) 14.56 (±0.28)Continued on next page104B. Table of ResultsTable 6.2  Continued from previous pageSeries τw (Pa) rp/R τy (Pa)45 38.09 0.39 (±7.53) 14.68 (±1.07)45 45.36 0.37 (±8.50) 16.64 (±0.54)45 51.28 0.32 (±8.33) 16.34 (±1.21)45 56.40 0.27 (±7.81) 15.35 (±1.14)45 65.28 0.14 (±4.71) 9.28 (±0.27)46 2.29 0.82 (±0.00) 1.89 (±0.43)46 2.83 0.70 (±0.00) 1.99 (±0.37)46 3.90 0.70 (±0.00) 2.74 (±0.37)46 4.44 0.61 (±0.01) 2.72 (±0.33)46 5.79 0.61 (±0.00) 3.56 (±0.32)46 6.86 0.51 (±0.00) 3.47 (±0.27)46 8.21 0.52 (±0.00) 4.27 (±0.27)46 11.44 0.40 (±0.01) 4.60 (±0.26)46 14.40 0.34 (±0.02) 4.92 (±0.30)46 18.98 0.31 (±0.01) 5.82 (±0.30)46 23.29 0.24 (±0.03) 5.58 (±0.83)46 34.86 0.18 (±0.01) 6.26 (±0.27)46 40.24 0.14 (±0.01) 5.73 (±0.27)105B. Table of ResultsTable 6.3: Estimates of the plug size, yield stress and the ratio of Reynolds stress to yieldstress for Series 3-8 at the conditions given in Table 3.3Series τw (Pa) rp/R τy (Pa) ρu′2/τy49 0.16 0.92(±0.02) 0.14(±0.22) 0.44(±0.02)49 0.47 0.82(±0.01) 0.38(±0.19) 0.78(±0.02)49 1.10 0.62(±0.01) 0.68(±0.15) 0.22(±0.00)49 1.41 0.51(±0.01) 0.72(±0.12) 0.36(±0.00)49 2.67 0.55(±0.03) 1.46(±0.15) 0.25(±0.01)49 4.24 0.42(±0.03) 1.79(±0.17) 0.40(±0.01)49 9.25 0.25(±0.01) 2.29(±0.09) 1.94(±0.04)49 17.10 0.12(±0.01) 2.03(±0.11) 5.24(±0.17)49 27.76 0.07(±0.02) 1.95(±0.57) 9.67(±2.25)49 28.08 0.06(±0.01) 1.56(±0.31) 10.67(±1.72)49 39.37 0.04(±0.01) 1.58(±0.25) 13.04(±1.71)49 48.78 0.01(±0.00) 0.66(±0.17) 35.07(±8.63)49 65.72 0.01(±0.00) 0.80(±0.16) 38.41(±6.81)49 82.66 0.01(±0.00) 1.12(±0.29) 30.76(±7.71)50 0.98 0.77(±0.05) 0.75(±0.19) 2.08(±0.12)50 1.25 0.76(±0.01) 0.95(±0.18) 0.06(±0.00)50 1.53 0.64(±0.02) 0.98(±0.16) 0.12(±0.00)50 3.49 0.50(±0.03) 1.74(±0.16) 0.33(±0.00)50 9.34 0.43(±0.01) 4.00(±0.13) 0.21(±0.01)Continued on next page106B. Table of ResultsTable 6.3  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy50 16.31 0.31(±0.04) 5.09(±0.61) 0.53(±0.05)50 26.07 0.14(±0.02) 3.71(±0.64) 1.45(±0.27)50 36.39 0.03(±0.00) 1.15(±0.09) 11.44(±0.88)50 50.06 0.01(±0.00) 0.46(±0.15) 46.82(±13.64)50 65.11 0.01(±0.00) 0.61(±0.26) 63.30(±33.84)50 84.08 0.01(±0.00) 0.72(±0.27) 58.97(±27.83)51 2.09 0.98(±0.02) 2.04(±0.24) 0.05(±0.03)51 2.65 0.93(±0.03) 2.46(±0.23) 0.17(±0.00)51 2.93 0.87(±0.04) 2.53(±0.23) 0.58(±0.29)51 3.21 0.83(±0.02) 2.66(±0.21) 0.06(±0.00)51 5.16 0.79(±0.01) 4.09(±0.20) 0.06(±0.00)51 8.23 0.68(±0.03) 5.61(±0.31) 0.10(±0.00)51 14.64 0.56(±0.06) 8.25(±0.95) 0.16(±0.00)51 26.63 0.37(±0.07) 9.73(±1.83) 0.31(±0.05)51 35.83 0.25(±0.05) 8.87(±1.70) 0.61(±0.10)51 48.38 0.15(±0.04) 7.08(±1.86) 1.67(±0.36)51 67.07 0.04(±0.02) 2.83(±1.39) 9.04(±4.61)51 82.40 0.01(±0.00) 0.70(±0.14) 56.79(±9.86)52 0.04 0.86(±0.02) 0.04(±0.20) 9.59(±0.94)52 0.65 0.70(±0.01) 0.46(±0.17) 1.93(±0.17)Continued on next page107B. Table of ResultsTable 6.3  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy52 0.92 0.59(±0.00) 0.54(±0.14) 0.77(±0.01)52 1.26 0.49(±0.01) 0.61(±0.12) 0.85(±0.01)52 1.61 0.40(±0.00) 0.65(±0.10) 0.51(±0.00)52 2.92 0.31(±0.03) 0.89(±0.12) 4.22(±0.16)52 4.84 0.15(±0.01) 0.72(±0.06) 20.12(±0.95)52 7.28 0.04(±0.02) 0.30(±0.16) 90.35(±59.73)52 10.15 0.01(±0.00) 0.14(±0.03) 225.21(±48.02)52 14.68 0.02(±0.01) 0.24(±0.10) 117.93(±56.91)52 18.87 0.01(±0.00) 0.27(±0.02) 96.55(±7.81)53 0.37 0.90(±0.03) 0.33(±0.21) 0.18(±0.15)53 0.58 0.77(±0.05) 0.44(±0.19) 5.51(±1.13)53 0.89 0.73(±0.03) 0.65(±0.18) 14.85(±0.18)53 1.41 0.67(±0.02) 0.95(±0.16) 2.80(±0.08)53 1.93 0.61(±0.01) 1.19(±0.15) 0.67(±0.01)53 2.56 0.58(±0.02) 1.50(±0.15) 8.72(±0.66)53 4.03 0.55(±0.01) 2.22(±0.14) 0.48(±0.02)53 5.28 0.47(±0.02) 2.47(±0.15) 0.69(±0.01)53 8.21 0.35(±0.02) 2.88(±0.21) 0.97(±0.06)53 12.71 0.24(±0.01) 3.07(±0.15) 5.26(±0.16)53 15.74 0.12(±0.00) 1.87(±0.06) 21.81(±0.65)Continued on next page108B. Table of ResultsTable 6.3  Continued from previous pageSeries τw (Pa) rp/R τy (Pa) ρu′2/τy53 19.71 0.03(±0.00) 0.66(±0.08) 72.19(±7.61)54 1.09 0.91(±0.05) 0.99(±0.22) 0.05(±0.01)54 1.70 0.84(±0.06) 1.43(±0.22) 1.61(±0.88)54 2.05 0.77(±0.01) 1.58(±0.18) 0.28(±0.00)54 2.57 0.72(±0.00) 1.85(±0.17) 0.22(±0.01)54 3.53 0.65(±0.01) 2.30(±0.16) 4.04(±0.18)54 5.27 0.61(±0.00) 3.22(±0.15) 0.88(±0.01)54 7.71 0.56(±0.00) 4.31(±0.14) 0.35(±0.01)54 11.46 0.46(±0.02) 5.24(±0.21) 0.37(±0.02)54 13.20 0.37(±0.04) 4.84(±0.58) 2.11(±0.29)54 15.73 0.25(±0.02) 3.93(±0.35) 6.18(±0.53)54 21.66 0.10(±0.00) 2.26(±0.07) 19.40(±0.33)109B. Table of Results0.50% 0.75% 1.00%∆P/L (Pa/m) Q (L/min) ∆P/L (Pa/m) Q (L/min) ∆P/L (Pa/m) Q (L/min)36.52 203.42 30.91 112.96 25.58 62.9746.64 206.84 50.14 112.72 37.76 63.1561.60 211.61 64.04 118.86 51.56 68.2378.24 223.35 89.91 151.43 68.09 91.14129.79 282.92 116.13 193.46 90.42 132.39205.94 477.42 160.34 296.50 128.36 230.69255.82 668.62 226.36 529.85 165.78 353.51292.35 842.05 321.34 1017.34 218.93 587.33446.15 1868.09 412.73 1678.25 295.62 1057.96609.18 3482.81 560.42 3094.26 512.70 2959.17850.50 7213.39 677.08 4571.64 669.59 4926.24845.78 7579.03 836.42 7971.62Table 6.4: Pressure drop per unit length of the pipe for the fibre suspensions given in Table3.3, Series 3-5.0.50% 0.75% 1.00%∆P/L (Pa/m) Q (L/min) ∆P/L (Pa/m) Q (L/min) ∆P/L (Pa/m) Q (L/min)26.30 24.94 22.01 49.55 28.75 129.1839.04 33.83 33.60 55.76 45.87 144.2159.58 49.10 52.13 69.77 65.49 150.8799.38 81.12 89.91 102.73 97.74 168.40155.08 132.39 141.87 151.88 141.08 198.50258.68 241.00 235.34 263.63 191.57 245.29381.81 402.19 349.30 413.63 278.08 338.16536.03 652.94 539.02 727.35 353.21 433.28744.22 1114.78 715.80 1095.86 498.64 677.581039.03 1996.64 894.15 1570.64 688.47 1090.36818.02 1346.831056.51 2064.39Table 6.5: Pressure drop per unit length of the pipe for the fibre suspensions mixed withpolymer (100ppm of Polyacrylamide) given in Table 3.3, Series 6-8.110B. Table of Results0.50% 0.75% 1.00%rp/R DR [%] rp/R DR [%] rp/R DR [%]0.68 0.68 0.66 17.21 0.86 7.280.62 0.62 0.64 30.25 0.83 16.080.54 0.54 0.61 31.93 0.82 22.810.53 0.53 0.50 34.62 0.79 30.000.47 0.47 0.47 34.74 0.76 34.210.31 0.31 0.43 33.22 0.68 36.840.25 0.25 0.35 31.13 0.61 37.540.17 0.17 0.31 29.11 0.56 36.990.12 0.12 0.23 27.38 0.48 35.310.08 0.08 0.14 25.84 0.37 33.740.04 0.04 0.08 24.11 0.31 32.670.03 0.03 0.03 22.95 0.25 31.510.01 0.01 0.01 21.39 0.19 30.490.01 0.01 0.01 20.73 0.15 29.470.01 0.01 0.01 16.06 0.07 28.130.01 16.81 0.04 26.920.01 25.37Table 6.6: Drag reduction for fibre suspensions for three different consistencies as a functionof plug size for cases studied in Table 3.3 and presented in 4.14.111B. Table of Results0.50% 0.75% 1.00%rp/R DR [%] rp/R DR [%] rp/R DR [%]0.70 38.76 0.73 20.01 0.80 17.960.66 56.76 0.69 41.08 0.77 24.730.59 61.29 0.67 48.01 0.76 40.600.52 65.34 0.64 55.32 0.72 49.100.49 66.46 0.61 58.23 0.67 56.850.42 68.92 0.58 66.40 0.65 62.810.40 69.91 0.57 71.40 0.63 71.060.34 72.24 0.55 74.97 0.61 75.040.31 75.83 0.53 77.73 0.56 79.740.23 75.99 0.47 82.22 0.51 80.120.15 79.69 0.41 82.37 0.46 80.980.09 80.51 0.35 82.21 0.41 83.190.04 81.56 0.28 81.57 0.37 84.490.01 83.07 0.24 81.57 0.33 83.780.01 83.50 0.16 81.65 0.25 83.740.02 81.45 0.12 81.74 0.17 83.050.01 81.48 0.07 83.62 0.10 83.210.01 80.14 0.03 82.99 0.07 83.460.01 82.58 0.02 83.21Table 6.7: Drag reduction for fibre suspensions mixed with 100ppm of polymer (Polyacry-lamide) for three different consistencies as a function of plug size for cases studied in Table3.3 and presented in 4.14.112C. Matlab CodesC Matlab CodesIn this appendix, the Matlab codes used to read UDV-data, pressure and flowratedata, measure plug size and estimate the yield stress are presented.113C. Matlab Codesclcclearclose allrho=1000;vis=1.004e-6; %Kinematic ViscosityWall=6.2;d = 48.4e-3; % Pipe inside diameter (m), PVC - Schedule 80Area = (d/2)^2*pi; % Cross-sectional areaR=d/2*1e3;L =10.15 ; % Pipe length (m)XpV=[];Vp=[]; XpSD=[];SDp=[]; XpdV=[];dVp=[];XpdSD=[];dSDp=[]; r_V=[];r_SD=[];dVdr_rR=[];dSDdr_rR=[]; TY=[];XpUP=[];VUP=[];PATH='D:\GD\Domtar Pulp 4 UBC\Domtar_Jun2015';[FileName,PathName] = uigetfile('*DP.lvm','Select the Excel file',PATH);filename=[PathName,FileName]; A=load(filename);QDP=[A(:,1),A(:,2),A(:,4) ]';figure (100); plot(QDP(1,:),QDP(2,:),'.')[FileName,PathName] = uigetfile('*stat.add','Select the UDV file',...114C. Matlab CodesPathName,'MultiSelect', 'on');if iscell(FileName) ;LF=length(FileName);else; LF=1; end; FileName=char(FileName);k=0;for ii=1:LFclcr_V=[];r_SD=[];dVdr_rR=[];dSDdr_rR=[];filename=[[PathName,FileName(ii, :)]];% ------------------------------------------------Fname=FileName(ii, :);Fname(find(Fname==' '))=[];Fname(end-8:end)=[];ind=find(Fname=='_'); Fname(1:ind)=[];ind=find(Fname=='_');Fname(1:ind)=[]; ind=find(Fname=='Q'); Fname(1:ind)=[];ind=find(Fname=='l'); Fname(ind)=[];ind=find(Fname=='i');Fname(ind)=[];ind=find(Fname=='t');Fname(ind)=[];ind=find(Fname=='_'); Fname(1:ind)=[];FnameQF=str2num(Fname);% -------------------------------------------------A = importdata(filename, '');[r,V,SD]=RVSD(A,R,Wall);VT=V;115C. Matlab Codesr=abs(r-R);%---------------------------------k=k+1;[FL(k)]=CalcFlowrate(abs(r),abs(V));FileName(ii,:),FL(k),QFDP(k)=interp1(QDP(1,:),QDP(2,:),QF);if isnan(DP(k))DP(k)=max(QDP(2,:));end% [DP(k)]=CalcDP(FL(k),QDP);figure (100);hold on;plot(FL(k),DP(k) ,'ro'); plot(QF,DP(k) ,'s');FL(k)=QF;DR(k)=interp1(QDP(1,:),QDP(3,:),FL(k));VB(k)=(FL(k)/6e4)/(pi*R^2/1e6);Re(k)=VB(k)*2*(R/1e3)/vis;%---------------------------------[dVdr_rR]=GammaDot(dVdr_rR,r,V,R,1);[dSDdr_rR]=GammaDot(dSDdr_rR,r,SD,R,1);%---------------------------------r=r/R;SDV=SD;[rp_V,V_rp]=Rp_by_hist(1-V/max(V),r,50);V_rp=(1-V_rp)*max(V);[rp_SD,SD_rp]=Rp_by_hist(SDV,r,50);116C. Matlab Codes[rp_Ga,Ga_rp]=Rp_by_hist(abs(dVdr_rR(:,2)),dVdr_rR(:,1),50);[rp_GaSD,GaSD_rp]=Rp_by_hist(abs(dSDdr_rR(:,2)),dSDdr_rR(:,1),50);d = 48.4e-3; L=10.15; nu = 1.1e-6;TW(k)=DP(k)*d/4/L; us=(TW(k)/998).^0.5;[r2,V2,SD2]=RVSD(A,R,Wall);y=r2/1e3; yp=y.*us/nu; upp=V2./us; uppW=2.5*log(yp)+5.5;figure(101);plot(yp,upp,'--',yp,uppW,'k-'); [x,y]=ginput(1);xUP=1-(x./us*nu)/(d/2); yUP=y*us;XpUP=[XpUP,xUP];VUP=[VUP,yUP];figk=hgload('SDWater.fig'); figure (figk);hold on;plot(r,SDV,r,V/max(V)*mean(SDV),'--');plot(XpV,Vp/max(V)*mean(SDV),'r--*') ;plot(XpUP,VUP/max(V)*mean(SDV),'k--*') ;plot(XpSD,SDp,'r--*');plot(rp_SD,SD_rp,'go');[xSD,ySD]=ginput(1);plot(xSD,ySD,'ro');XpSD=[XpSD,xSD];SDp=[SDp,ySD];figure (3); clf;hold on117C. Matlab Codesplot(dVdr_rR(:,1),abs(dVdr_rR(:,2)),r,V/max(V)*max(abs(dVdr_rR(:,2))),'--');plot(XpdV,abs(dVp),'r--*');plot(rp_Ga,Ga_rp,'go');plot(xSD,0,'s');[xG,yG]=ginput(1); plot(xG,yG,'ro');XpdV=[XpdV,xG];dVp=[dVp,yG];figure(10);hold on;plot(r,V);plot(XpV,Vp,'r--*') ;plot(rp_V,V_rp,'go');plot(xSD,spline(r,V,xSD),'s');plot(xG,spline(r,V,xG),'rs');[xV,yV]=ginput(1); plot(xV,yV,'ro');XpV=[XpV,xV];Vp=[Vp,yV];TW(k)=DP(k)*d/4/L;TY_V(k)=DP(k)*XpV(k)*d/4/L;TY_UP(k)=DP(k)*XpUP(k)*d/4/L;TY_SD(k)=DP(k)*XpSD(k)*d/4/L;TY_gama(k)=DP(k)*XpdV(k)*d/4/L;Re_Stres_taw_V(k)=1e3*(spline(r,SD,XpV(k)))^2./TY_V(k);Re_Stres_taw_UP(k)=1e3*(spline(r,SD,XpUP(k)))^2./TY_UP(k);Re_Stres_taw_SD(k)=1e3*(spline(r,SD,XpSD(k)))^2./TY_SD(k);Re_Stres_taw_gama(k)=1e3*(spline(r,SD,XpdV(k)))^2./TY_gama(k);118C. Matlab Codesend[Re IRe]=sort(Re);XpV=XpV(IRe);XpSD=XpSD(IRe);XpdV=XpdV(IRe);XpUP=XpUP(IRe);dVp=dVp(IRe);SDp=SDp(IRe);Vp=Vp(IRe);VUP=VUP(IRe);TY_V=TY_V(IRe);TY_SD=TY_SD(IRe);TY_gama=TY_gama(IRe);TY_UP=TY_UP(IRe);Re_Stres_taw_V=Re_Stres_taw_V(IRe);Re_Stres_taw_SD=Re_Stres_taw_SD(IRe);Re_Stres_taw_gama=Re_Stres_taw_gama(IRe);Re_Stres_taw_UP=Re_Stres_taw_UP(IRe);DP=DP(IRe);TW=TW(IRe);DR=DR(IRe);AP=[XpV;XpSD;XpdV;XpUP];SDAP=std(AP);MAP=mean(AP);AT=[TY_V;TY_SD;TY_gama;TY_UP];MT=mean(AT);SDT=std(AT);SDTy=((200./DP).^2+(SDAP./MAP).^2).^0.5.*MT;SDDR=((200./DP).^2).^0.5.*DR;ATI=[Re_Stres_taw_V;Re_Stres_taw_SD;Re_Stres_taw_gama;Re_Stres_taw_UP];SDATI=std(ATI);MTI=mean(ATI);dat.Re=Re; dat.DP=DP; dat.DR=DR;dat.rp=AP; dat.ty=AT; dat.tw=TW; dat.errorbar_ty=SDTy;dat.TI=ATI; dat.SDATI=SDATI;dat.dVp=dVp; dat.SDp=SDp; dat.Vp=Vp;function [dVdr_rR]=GammaDot(dVdr_rR,r,V,R,k)119C. Matlab Codesdr=diff(r);dV=diff(V);rR=r(2:end)/R;[size(dV) size(dr)];dVdr=dV./dr;if k==1dVdr_rR=[rR',dVdr'];elser_ref=dVdr_rR(:,1);Vi=interp1(rR,dVdr,r_ref');dVdr_rR=[dVdr_rR,Vi'];endfunction [r,V,SD]=RVSD(A,R,Wall);r=str2num(A{5});V=-str2num(A{8})./1e3;SD=str2num(A{10})./1e3;V=V(1:length(r));SD=SD(1:length(r));120C. Matlab Codesr=(r-Wall);indr=find(r<=0); V(indr)=[]; r(indr)=[]; SD(indr)=[];indr=find(r>R); V(indr)=[]; r(indr)=[]; SD(indr)=[];function [FL]=CalcFlowrate(r,Vm);FL=0;dr=abs(r(end)-r(end-1));for i=1:length(r)FL=FL+Vm(i)*2*pi*r(i)*dr*6e-2;endfunction [rp,xp]=Rp_by_hist(x,r,histn);[nn,yh]=hist(x,histn);yhn=yh(nn==max(nn)); %% yhn is theif max(nn)<10yhn=yh(1);endxs=x;rs=r;CR2=min(yhn);rs=rs(xs<CR2);xs=xs(xs<CR2);121C. Matlab Codesrp=max(rs); % higher r , closer to the wallxp=xs(rp==rs);122

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