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The hydrodynamics of individual pulp fibres Wong, Tze Bun 2000

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THE HYDRODYNAMICS OF INDIVIDUAL PULP FIBRES By TZE B U N WONG B.A.Sc, The University of British Columbia, 1998 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as, conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 2000 ©Tze Bun Wong, 2000 UBC Special Collections - Thesis Authorisation Form http://www.library.ubc.ca/spcoll/thesauth.htir In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may"be granted by the head of my department or by h i s or her re p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. The U n i v e r s i t y of B r i t i s h Columbia Vancouver, Canada Date 1 of 1 10/12/2000 11:06 PM Abstract In order to understand how pulp fibres with different properties (e.g. length, diameter, coarseness, etc.) can be fractionated in hydrocyclones, there is a need to understand the hydrodynamic behavior of individual pulp fibres. Measurements of the settling velocity of fibres in fluids of different density and viscosity can be used to give the drag coefficient of fibres as a function of the Reynolds number. Such measurements can yield Reynolds numbers up to about 0.1. In order to achieve the higher fibre Reynolds numbers present in a hydrocyclone, a novel device, called the rotating tank, was constructed to measure the velocity of fibres in water under the influence of a centrifugal field. The tank is a circular cylinder made of plexiglas, 30.5cm in diameter, and 4.45cm high, with its axis vertical. The tank is filled with fluid and is spun at a constant rate until the fluid is in solid body rotation. Fibres or other objects to be tested can be placed in the tank through a hole along the axis of the cylinder, and centrifugal forces cause the fibres to be flung outwards. A fibre inside the rotating tank is subject to a centrifugal force, a drag force, and a pressure force. The radial equation of motion of a fibre in the rotating tank will be exactly the same as during gravitational settling except the gravitational acceleration is replaced by the centrifugal acceleration. Therefore, high Reynolds numbers can be achieved by using a high rotation speed of the tank. Our rotating tank was tested and found to be capable of spinning at 1500 rpm. The tank has been primarily validated by reproducing Stokes' theory for the drag of a sphere. The tank results were in good agreement (4% error) with Stokes' theory. Further validation of the tank was done with copper wires and nylon fibres, which are similar to wood fibres. The results were in good agreement (<15% error) with the results of other researchers i i Tests with hardwood kraft and softwood thermomechanical pulp (TMP) fibres in the tank yielded the following results: • The orientation of a fibre settling in the tank depends strongly on its initial orientation. In contrast with a widely-used assumption, most fibres do not orient themselves with their long axis perpendicular to the direction of radial motion. The fibre orientation can be anything from purely perpendicular to purely parallel to the direction of radial motion with the majority of the fibres oriented in the range from 20 to 40 degrees from the perpendicular. • Fibres do not rotate regardless of their orientation. They will maintain their orientation during the entire settling process in the rotating tank. • The settling velocity of pulp fibres is a strong function of their orientation. Fibres settling at an angle of larger than 45 degrees to their direction of motion settle faster than those at smaller angles. The orientation effect becomes more pronounced as the angular acceleration increases. • For hardwood kraft pulp, the plot shows that the settling velocity increases with fibre length but the plot of softwood TMP does not show the same dependence. • The shape of the fibre does not significantly affect the settling rate. Curved fibres and fibres having sharp corners settle at about the same rate as straight fibres. • The kraft and TMP used have on average much larger drag coefficients than any other objects tested under the same conditions. The apparent density and mean diameter are used to plot the drag coefficient against the Reynolds number. The apparent densities for hardwood kraft and softwood TMP are 1329 kg/m3 and 1265 kg/m 3 respectively whereas the mean diameters for hardwood kraft and softwood TMP are 22.3 and 33.8 microns respectively. A least square fits from the mean values gives the mean drag coefficient as a function of Reynolds number for hardwood kraft and softwood TMP: Hardwood kraft: C 113 for 0.007< Re< 1.0 D Re 0.571 Softwood TMP: C 74 for0.007<Re< 1.0 D Re 0.763 111 Table of Contents Abstract i i List of Figures vi List of Symbols viii Acknowledgement x Chapter 1 Introduction 1 1.1 Motivation 1 1.2 Pulp Fibre Properties 3 1.2.1 Structure of Wood Fibres 3 1.2.2 Effect of Fibre Structure 4 1.2.3 Fibre Fractionation 5 1.3 Literature Review 9 1.3.1 Sphere 9 1.3.2 Cylinder 13 1.3.3 Fibre 26 1.3.4 Wall Effect 29 1.4 Selection of Previous Work 34 1.5 Scope of this Study 36 Chapter 2 Theoretical Analysis 37 2.1 Dimensional Analysis 37 2.2 Fundamental Concept 39 2.3 Oblique Settling 42 2.4 Sensitivity Analysis 43 Chapter 3 Experimental Apparatus and Procedures 45 3.1 Fluid Preparation 45 3.1.1 Temperature Control 45 3.1.2 Fluid Density and Viscosity 46 3.2 Object Dimension 47 3.3 Object Density 48 3.4 Preparation of Objects 51 3.5 Rotating Tank 55 3.6 Free Settling 59 iv Cahpter 4 Validation of Rotating Tank 61 4.1 Glass Sphere 61 4.2 Copper Wire 63 4.3 Nylon Fibre 67 4.4 Aspect Ratio Effect 71 Chapter 5 Results and Discussion 73 5.1 Length Effect on Settling Rate 73 5.2 Orientation Effect on Settling Rate 78 5.3 Shape Effect on Settling Rate 83 5.4 Drag Coefficient Vs Reynolds Number 85 5.5 Diameter Effect on Drag Coefficient 87 5.6 Regression Analysis 89 Chapter 6 Conclusion 92 Chapter 7 Recommendations 94 Bibliography 96 Appendix A Sensitivity Analysis and Sample Calculations 102 Appendix B Gilmont® Viscosimeter Calibration 110 Appendix C Fluid Properties 112 Appendix D Fibre Diameter 114 Appendix E Fibre Length- from Fibre Quality Analyzer 118 Appendix F Fibre Length- from Experiments (HVCR) 122 Appendix G Pulp Fibre Density (kraft and thermomechanical Pulp) 125 Appendix H Fibre Diffusion Model 129 Appendix I Sphere Density 131 Appendix J Design of Rotating Tank 134 Appendix K Object Settling under Gravity 138 Appendix L Glass Sphere Experimental Data 148 Appendix M Copper Wire Experimental Data 155 Appendix N Nylon Fibre Experimental Data 161 Appendix O Kraft Pulp Experimental Data 175 Appendix P Thermomechanical Pulp Experimental Data 185 Appendix Q Regression Data (kraft and thermomechanical Pulp) 197 V List of Figures Figure 1.1. A cross section of Douglas-Fir showing the abrupt change in cell wall thickness at the annual ring and the earlywood-latewood differentiation 4 Figure 1.2. Idealized fibre structures. The thick-walled fibres (A) are less conformable than the thin-walled fibres (B) 5 Figure 1.3. Schematic diagram of a hydrocyclone 8 Figure 1.4. Drag coefficient against Reynolds number for cylinder and fibre 35 Figure 2.1. Forces on a slender body in creeping flow-needle falls at an angle 42 Figure 3.1. Schematic diagram of fibre density measurement 49 Figure 3.2. Micrograph (5x) of nylon fibre (1 mm 8 denier) 52 Figure 3.3. Micrograph (5x) of softwood TMP 53 Figure 3.4. Micrograph (5x) of hardwood kraft pulp fibre 53 Figure 3.5. Rotating Tank A) schematic diagram B) actual set-up 57 Figure 3.6. A schematic diagram of free settling experiment 59 Figure 4.1. Drag coefficient Vs Reynolds number for 2 and 3mm glass sphere 62 Figure 4.2. Drag coefficient Vs Reynolds number for copper wire 65 Figure 4.3. Orientation distribution of copper wire settling 65 Figure 4.4. Orientation effect on copper wire settling 66 vi Figure 4.7. Orientation effect on nylon fibre settling 70 Figure 4.8. Aspect ratio effect on nylon fibre settling 72 Figure 5.1. Length effect on settling rate for hardwood kraft pulp 76 Figure 5.2. Length effect on settling rate for softwood TMP 77 Figure 5.3. Orientation distribution of a) hardwood kraft b) softwood TMP 80 Figure 5.4. Orientation effect on settling rate for hardwood kraft pulp 81 Figure 5.5. Orientation effect on settling rate for softwood TMP 82 Figure 5.6. Shape effect on settling rate a) hardwood kraft b) softwood TMP 84 Figure 5.7. Drag coefficient Vs Reynolds number a) hardwood kraft b) softwood TMP.86 Figure 5.8. Diameter effect on drag coefficient a) hardwood kraft b) softwood TMP 88 V l l List of Symbols p fluid viscosity 8 orientation angle measured from the axis perpendicular to the direction of motion \|/ stream function p fluid density Pcyiinder cylinder density p fibre fibre density p object object density P sphere sphere density Ap object density minus fluid density Ae volume drag ratio defined as the ratio of the drag force on an object to that on a sphere having the same volume and velocity A object characteristic area a sphere radius b half length of a cylinder c cylinder radius D container diameter d object diameter dc cylinder diameter ds sphere diameter E aspect ratio = L/dc F ' drag force per unit length FD,C drag force on a cylinder FD,S drag force on a sphere Fo,stokes Stokes's drag force for a sphere viii F D G g H K k w 1 L m R r Re U U c u , infinity Uobserved Us V T 00 H V C R F Q A TMP A.R. Y drag force acceleration associated the force field to which the object is subjected gravitational acceleration distance between two parallel plates the ratio of settling velocity of an object to the settling velocity of a perfect sphere having the same volume as the object. wall correction constant distance of an object from the boundary length of a cylinder mass of a object container radius radial position of the object in the tank p-U-d Reynolds number based on diameter and defined as object velocity terminal settling velocity of a cylinder velocity of objects in infinite fluid domain velocity of the objects observed terminal settling velocity of a sphere volume of an object drag force per unit area angular speed of the rotating tank High Speed Video Camera Fibre Quality Analyzer Thermo-Mechanical Pulp Aspect Ratio (Length / Diameter) Euler constant = 0.577 I X Acknowledgement The author would like to thank his advisor, Professor Sheldon Green who continuously gave the author valuable suggestions and encouragement throughout the period of this investigation and made this thesis possible. The author would like to express the appreciation to the help of the staffs in Pulp and Paper Center at the University of British Columbia. Peter Taylor (Senior Tradesman) machined the device and all other components necessary for this investigation with highest machining quality and tolerated the endless requests for modifications. Tim Peterson (Administrative Technician) helped collect all the materials and equipment and made sure the orders always on time. Ken Wong (Laboratory Services Coordinator) tirelessly clarified the use of laboratory equipment for pulp fiber preparation. Rita Penco (Former Librarian) painstakingly helped review the literature over almost ten decades and collected over eighty papers and theses regardless of their ages and locations. Many thanks to Dr. James Olson, who gave valuable advise on how to visualize the fibers in the rotating device and other pulp and paper aspects. Most important was the love and support from the author's girl friend, Fiona Tin Lok Lam, without which the author would have never been able to overcome the challenges and to get through this. Chapter 1 Introduction 1.1 Motivation The pulp and paper industry, being one of the largest industries in Canada, has challenges for research and development in many areas of engineering. In particular, mechanical and chemical engineers are required to study and improve the devices and processes which convert the wood chips and recycled paper to daily paper products. This is very important to maintain Canada's leading role in the worldwide pulp and paper market as the total sales of the pulp and paper industry were over a hundred billion dollars in 1995. Statistics show that the demand of paper products continues to grow and increases linearly with population growth. Having the geographical advantage, British Columbia has close access to many developing Asia countries where the largest demand of paper products is occurring. One of the traditional advantages of British Columbia has been the relatively easy access to the high quality softwood trees on the coast and the interior. However, the advantage is being eroded due to restrictions on logging and slow natural regeneration. Consequently, British Columbia can no longer rely on the fibre supply advantage. One of the strategies for British Columbia to overcome this challenge is to produce higher quality paper with lower cost. In fact, British Columbia is capable of producing very high quality paper due to the easy access to the coniferous forests that cover much of the province. The pulp fibres from these trees are long, slender, and flexible which allow the fibres to conform and bond well to produce papers with higher strength and printability. However, these good fibres are mixed with less desirable fibres that are thicker in cell wall and less conformable. A technology that can separate fibres 1 would be of great benefit in manufacturing high quality paper products. In addition, the environmental protection issues serve as the impetus for the paper recycling industry. Although efforts have been made to make sure that the recycled papers are pre-grouped when collected, a substantial amount of separation is required to extract the good fibres to make high quality paper. Pulp and paper technologists have been focusing on improvement of the existing processes to separate fibres with different properties, known as fibre fractionation. Screening is one of the processes to separate fibres according to their length. Hydrocyclones, originally used to remove sand, grit, and, small shives are found capable of separating fibre with different length and coarseness. With the advent of the high-speed computer, the flows in these processes can be modeled numerically. Some researchers approximate the fibre as a rigid circular cylinder while some try more sophisticated model which assume the fibre as a chain of spheroids or ellipsoids. However, in order to model the fibre motion accurately, both methods require an understanding of the hydrodynamics of pulp fibres. 2 1.2 Pulp Fibre Properties The qualities of a paper product are determined by the properties of the pulp fibres from which it is manufactured. The effect of the pulp fibres on paper products can be understood by looking at the fibre structure and packing. 1.2.1 Structure of Wood Fibres The primary source for pulp fibres in Canada is softwood or coniferous trees. By far the largest constituent of conifers is the long thin tracheid cell (95% by volume, 99% by weight). These fibres serve as the structural units of the tree. The strength and stiffness required for this role make them ideal as pulp fibres. Deciduous or hardwood trees are also made up primarily of tracheids (65% by volume, 86% by weight). The shape of softwood tracheid cells can be thought of as a hollow cylinder. Average mature softwood fibres are 4 mm long with a diameter of 30 microns. The growth cycle of trees is evident from the annual rings. These rings are visible due to the abrupt change in the thickness of the cell walls. This is shown in Figure 1.1. Earlywood (springwood) fibres have properties, which are quite different from those of latewood (summerwood) fibres. The diameter of earlywood fibres is larger than latewood fibres. The reasons for the dramatic change of wood cell structure in a tree are changes in the growing conditions. Earlywood fibres are developed during the early part of the growing season (spring) while the latewood fibres are developed during the later part of the growing season (summer). Water supply, temperature, and sunlight are some of the essential factors to development of wood cells. The properties of the whole fibre depend on the cell wall thickness. 3 Figure 1.1. A cross section of Douglas-Fir showing the abrupt change in cell wall thickness at the annual ring and the earlywood-latewood differentiation. Taken from Fig. 2-9 in [64]. 1.2.2 Effect of Fibre Structure The properties of paper are dependent on the structural characteristics of wood fibres. Fibre length and cell wall thickness are the two most important parameters. In general, fibres with thinner cell walls (earlywood) collapse readily into ribbons during sheet formation. Fibres with thicker cell walls (latewood) resist collapse and do not contribute to interfibre bonding to the same extent. The thicker-walled fibres tend to produce an open, absorbent, bulky sheet with low burst/tensile strength and high tearing resistance. To illustrate the principle, Figure 1.2 shows two schematic diagrams of idealized fibre structure. In the top diagram, the thicker-walled fibres are shown as hollow cylinders that are less conformable; in the bottom diagram, the thin-walled fibres are as ribbon-like elements. The numbers of fibres and contact points are set to be same for both cases. The area of contact and potential bonding sites are clearly much greater in the sheet composed of thin-walled fibres (earlywood). Figure 1.2. Idealized fibre structures. The thick-walled fibres (A) are less conformable than the thin-walled fibres (B). Taken from Fig. 2-25 [64]. 1.2.3 Fibre Fractionation Obviously, there is a strong desire for bringing the properties of the pulp fibres closer to those desired for the finished product. One industrial solution is to use fibre fractionation. Fibre fractionation is the mechanical separation of fibres from a mixture to produce at least two fractions that have higher percentages of fibres with certain properties. Pulp mills use screens and cleaners to remove shives and contaminants from papermaking fibres. They can also be used to fractionate fibres. By further understanding the mechanisms that control their operation, refinements may be found which will allow fibres to be separated on the basis of small differences in physical properties. 5 Pressure screens utilize a cylindrical, perforated screen located inside a vessel. The pulp slurry is introduced into the vessel so that it must either pass through the screen (accepts) or remain outside and exit through a discharge (rejects). Pressure screens may operate on either of two mechanisms. In barrier screening, the perforations are smaller than the width of the unwanted particles. The screen serves as the barrier. The disadvantage of this method is the low throughput of slurry caused by the small perforations required to remove all the undesirable particles. Probability screening is more widely used in industry. In this method, the perforations are shaped as slots in the screen. Their dimensions are larger than the smallest diameter of the desired particles. In the case of long particles, such as fibres, it is length and orientation of the particle as it approaches the slot that determines whether it will pass through. The ability of a particle to conform to streamline as it approaches the slot will also determine its chance of passing through. In this way, probability screening fractionates on the basis of particle length and flexibility. The fibre mats that form over the screen perforations complicate screening. These are removed by rotation foils that cause a pressure pulse. The suction pressure generated in the trailing edge of the foils purges the openings of fibres. The influence of the fluid flow on the orientation deformation of the fibre near the slot opening is clear. An understanding of this area of screening is vital for further improvements in fractionation technology. Hydrocyclones are another type of equipment that could be adapted to fibre fractionation. Figure 1.3 is a schematic diagram showing the general principles of operation of a hydrocyclone. Pulp slurry is fed at an angle to the hydrocyclone wall at the top of the inverted conical vessel to impart a rotating motion. As the slurry travels in a spiral down the vessel, the centrifugal acceleration, typically several hundreds times that of gravity, forces the fibres or particles that are denser than the fluid to migrate to the walls of the vessel. The radial settling rate is an important parameter in hydrocyclone 6 fractionation. The fibres with a higher radial settling rate then become entrained in a downward flow that takes them to the reject outlet at the bottom of the vessel. The pulp fibres with lower radial settling rate remain in the swirling flow. Fibres that are close to the centre are picked up by the upward air core where they are removed through an outlet at the centre of the top of the hydrocyclone. A typical Reynolds number in the radial direction in a hydrocyclone would be around five but it totally depends on the geometry of the hydrocyclone, the pulp fibres, and the operating conditions. In addition, shear forces pull apart the fibre networks to enhance fractionation. Since the hydrodynamic properties of pulp fibres in a centrifugal field have never been studied, most numerical models and analytical studies in hydrocyclones tend to simulate the pulp fibre as a rigid or flexible cylinder or chains of ellipsoids or spheroids of which the hydrodynamic properties have been studied extensively. This thesis is to assist the numerical modeling of the motion of pulp fibres, and is aimed at exploring the effects of fibre properties on fibre hydrodynamics. 7 A C C t P T S AIR AWt! R E J E C T S Figure 1.3. Schematic diagram of a hydrocyclone. Taken from Fig. 9-30 in [64]. 1.3 Literature Review Hydrodynamics of particles at low Reynolds number flow are of fundamental importance to many engineering aspects. However, very little work has been done on synthetic fibres, cotton fibres, and pulp fibres. Nonetheless, this section will review the previous work on spheres, cylinders, and some fibres in creeping flow, which is mainly used for the validation of the rotating tank. In addition, boundary effect will also be included as many researchers have pointed out the substantial effect of boundaries on objects in creeping flow. Due to the large content of this section, each sub-section is categorized into analytical and experimental studies. 1.3.1 Sphere Stokes was the pioneer of finding the theoretical prediction of flows around a sphere at low Reynolds numbers. The Navier-Stokes equations were solved with small Reynolds number. As the Reynolds number was small, expansion in powers of Reynolds numbers no longer involved a "singular perturbation" and the nonlinear convection term u • Vu was not of highest order, and it seemed mathematically reasonable simply to drop this term. Thus, Stokes defined a new class of flow now commonly called "creeping flows" and deduced from this approximation, the formula D = 6n/jaU for the resistance encountered by a solid sphere of radius a, moving slowly with speed U through a fluid of viscosity p. After Stokes, a number of researchers were working both analytically and experimentally to improve the Stokes formula. Analytical work involved solving the linearized Navier-Stokes equations with higher order terms. Experimental work usually was to confirm the range of validity of Stokes formula or the boundary effect. 9 Analytical Goldstein (1928) completely solved the Oseen's equations for the flow of a viscous fluid at small Reynolds numbers past a fixed spherical obstacle. His results could be represented as an equation of drag coefficient: 24f , 3 D 19 2 71 _ 3 30179 _ 4 122519 n 5 1 Cn = —<l +— Re Re + Re J Re + Re -...k.(1.1) D Re 16 1280 20480 34406400 560742400 J Tomotika and Aoi (1950) improved the Oseen drag formula, corrected to all terms of order Re. 24 Re 1 + —Re+ 16 160 Re 2 lnRe+...0(Re 2) .(1.2) To obtain higher order approximations of an incompressible flow, Chester and Breach in 1968 proposed that viscous fluid past a sphere at small Reynolds numbers beyond the first term by Stokes (1851) was very difficult because an expansion in terms of the Reynolds number, for the flow in the vicinity of the sphere, was not valid at large distances from the sphere. Kaplun (1957) therefore matched that with a separate expansion that was calculated for the outer flow. Proudman and Pearson (1957) applied the matched asymptotic expansions to carry the analysis as far as the term of order Re2logRe. Chester and breach (1968) continued the analysis of Proudman and Pearson as far as the terms of order Re3logRe. The drag on a sphere in creeping flow could be calculated as follows: F f l =F D , 5 / 0 , J l + ^ R e + ^ R e 2 5 323 logRe+/ + - l o g 2 - — 5 3oU + - ^ R e 3 logRe+0(Re 3)L. .(1.3) where Fo.stokes, the Stokes' drag, = dnpfMU and the above expansion was concluded by Maxworthy (1965) experimental measurements to be practical in the limited range 0 < Re < 0.5 White (1991) then offered a curve-fit formula for sphere. 10 24 6 Cn « — + ^ = + 0.4 for 0 < R e < 2 x l 0 5 Re 1 + VRe (1.4) Experimental Castleman (1926) summarized some experimental work on spheres. Silvey (1916) observed the rate of fall of drops of mercury, of diameters from 0.012 cm to 0.07 cm in caster oil, over a range of Reynolds numbers from 0.000024 to 0.0066. His results agreed with Stokes' law. Arnold (1911) observed the rate of fall of metal spheres, of diameters from 0.013 cm to 0.14 cm, in colza oil in a tube of internal diameter 1.09 cm. The Reynolds numbers ranged from 0.002 to 2.4. Since the proximity of the walls of the fall tube caused a violation of infinite extent of fluid boundary. The observed results must be corrected so that they would prevail in an infinite fluid. Such correction was proposed by Ladenburg (1907) and as follow: Umfinitv=Uobserved(\ + 2A^-) (1.5) Allen (1955) observed the motion of air bubbles and of solid spheres, with Reynolds numbers ranged from 0.009 to 8200. For small values of Reynolds number, for which he used air bubbles in aniline, Allen's results seemed to be quite scattered and to fall, in general, below the theoretical curve. The scattering was to be expected from the difficulty in measuring the diameter of the small bubbles and the fact that the air in the bubble was gradually frittered away by the fluid medium. Therefore, Allen's measurements of the diameters were too small, and if corrected for this effect, the points would fall nearer the theoretical curve. Allen's observation at small values of Re therefore could not be regarded as contradictory to Stokes' law. Liebster and Schiller (1924) observed the rate of fall of steel spheres of diameters 0.1 cm to 0.7 cm, in glycerin, sugar solutions, and water. Their observations covered a range of values of Reynolds numbers from 0.12 to 2000. Nevertheless, no curve fitting was available. 11 Davies (1945) deduced a single formula, fitted by the method of least squares to critically selected data from a number of experimenters: This equation tends to Stokes' law for low values of Reynolds number. Maxworthy (1965) made an accurate measurement of sphere drag at low Reynolds numbers and revealed some of the inaccuracies of previous measurements. His comparison with theoretical studies showed that the Oseen formula was as accurate as any in predicting sphere drag below a Reynolds number of 0.4. Also, the Proudman-Pearson (1957) formula was accurate up to 1.5% for Reynolds number up to 0.65, but represented the overall trends with Reynolds numbers rather poorly; while the Goldstein (1929) formula was accurate up to 1.5% for Reynolds number up to 0.45, but gave a particularly good representation of the behavior of the drag as Re—»0. Thus, it is the inescapable conclusion of this work that the formulae of Oseen and Goldstein represented the drag of a sphere most accurately for Re<0.45, and that the addition of higher-order terms added very little to the accuracy at low Reynolds number. Maxworthy pointed out that the experimentally determined drag was non-dimensionalized with respect to the Stokes' drag, large differences appeared between experiment and theory. Maxworthy, therefore, concluded that it was impossible to determine the drag to better than ±20%. However, Maxworthy's experiments covered only Reynolds numbers between 0.4 and 11 so that his suggestions were not verified completely. Pruppacher and Steinberger (1968) accurately measured the drag on a sphere falling in a viscous medium for Reynolds numbers between 0.001 and 10.0. When the fractional deviation (F D / Fo,stokes -1) was plotted as a function of the Reynolds number, significant inconsistencies among the results of measurements previously reported in literature and significant differences between these and their own results were revealed. Their experimental results also deviated from most theories available; however, they -0.00023363(CD Re 2 ) 2 + 0.000002015\CD Re 2 ) 3 -0.0000000069105(C f l R e 2 ) 4 24 for Re<4orCDRe 2<140. (1.6) 12 were consistent with the theory of Proudman and Pearson for vanishingly small Reynolds numbers and at Reynolds numbers between 0.5 and 10 with Carrier's semi-empirical modification of Oseen's theory. Their results were least-squares fitted to obtain the following expressions: f \ •1 I = 0.102 Re 0 ' 9 5 for 0.001 < Re < 2 (1.7) V D, Stokes V ^ D,Stokes = 0.115ReU 8 Ufor Re>2 (1.8) A comparison of their experimental results with those of Maxworthy showed fair agreement for Reynolds number between 3 and 10. However, at Reynolds numbers smaller than three, Maxworthy's values for the above equations departed increasingly from their values. At Reynolds numbers larger than 0.7, their measurement of the drag represented a lower boundary to all other. 1.3.2 Cylinder The first theoretical treatment of the problem was given by Stokes (1851), who could find no steady flow satisfying his linearized governing equations. This was the famous Stokes' paradox. Stokes explained this as follows: the pressure of the cylinder on the fluid continually tends to increase the quantity of fluid which it carries with it, while the friction of the fluid at a distance from the cylinder continually tends to diminish it. In the case of a sphere, these two causes eventually counteract each other, and the motion becomes uniform. However, in the case of a cylinder, the increase in the quantity of fluid carried continually gains on the decrease due to the friction of the surrounding fluid, and the quantity carried increases indefinitely as the cylinder moves on. The Stokes' paradox remained unresolved until Oseen, in 1910, explained that the failure resulted from the neglect of the non-linear inertia terms, which become dominant far from the body. Hence, 13 as a remedy, he suggested alternative linearized equations that partially account for the inertia terms. From there on, the flow around a cylinder in creeping flow has been studied by a number of researchers. The work includes two and three-dimensional cylinders in finite and infinite domain of fluids. Analytical An approximate solution of Oseen's equations was first given by Lamb (1911). He derived a theoretical equation to predict the drag of a 2-D circular cylinder placed in a stream of fluid with its axis normal to the flow. He stated that: C = — (19) D Re(2.002-ln(Re)) CD is the drag coefficient defined as: c ° = j j k l < U 0 ) Because of simplifying assumptions made in its derivation, the Lamb's equation can be applied only when the Reynolds number is small compared to unity, i.e., at Re<0.1 Solving the Laplace's equation, V 2 ^ = 0 in two dimensions, Bairstow (1921) found the drag of a cylinder between two fixed infinite flat plates in a uniform stream with Reynolds numbers smaller than 0.2. A cylinder of unit radius was fixed relative to the parallel walls of a channel so that its centre was five units away from each wall. Fluid was forced through the channel in quantity sufficient to give unit velocity to the fluid in the center of the channel at infinity. He found that the results of the above flow conditions were different from when the cylinder was moved at uniform velocity through stationary fluid by two percent for y/ and four percent for —I—. Along with dynamical similarity, dn Bairstow was able to generalize the formula for resistance of a cylinder from his results. 14 - ™ - = 7.1-pU2d2 Ud The formula can be re-written with the definition of drag coefficient as: 14.2 C n = -Re (1.11) .(1.12) The formula was compared with the resistance of a cylinder in a viscous fluid of infinite extent worked by Oseen. 8TT Re 1 ,Re. .(1.13) V 2 y is the Euler constant equal to 0.577. He also mentioned that both formulae are subject to the same limitation of Re<0.2. Burgers (1938), neglecting the fluid inertia effects, solved the Navier-Stokes equations to find the force on a circular cylinder of finite length which was held at rest in a uniform stream flowing in the direction of the symmetry axis. For this case, he obtained the value of the force as: AnjjbU l n ( f ) -0 .72 where b is the half-length and c is the radius of the cylinder. Fn = H M f W U (1.14) Broersma (1959) investigated analytically the Oseen-Burger's theory to calculate the viscous force constant of a moving cylinder closed at the ends including first-order effects in width and length. Experimental results, obtained from macroscopic models, were used to verify this. He found that the discrepancy was about 20% for b/a>20 where b is the half-length of the cylinder and c is the radius of the cylinder. It was near 20% for shorter cylinders. The force on a cylinder moving with the body axis perpendicular to the direction of motion is given by: FJJ^}L (1.15) 15 where £ = Inf—1, he found that rfe >2) = 0.35 - 4 ( - - 0.43)2 ± 0.2 5. \c J £ Tomotika and Aoi (1950) refined the derivation of Lamb to extend the prediction of drag coefficient up to Re=4. They also computed the flow patterns around a sphere and a cylinder in detail on the basis of Goldstein's exact analytical solution of Oseen's linearized equations of motion for the steady flow of an incompressible viscous fluid past a body. They found numerically, in accordance with observations, that stationary vortex-ring, though of very weak strength, was formed behind a sphere even when Reynolds number was 0.1 and two standing eddies were always formed behind a cylinder even when Reynolds number was as slow as 0.05. The pressure and frictional drag contributed to the total drag on the sphere is the ratio of 1:2 regardless of Reynolds numbers while the total drag on the cylinder was divided, in exactly the same proportion, into the pressure and frictional drags for any value of the Reynolds numbers. Kaplun (1957) gave what was considered to be the valid correction to Lamb's solution. r \ ° Re 1 + .(1.16) V n=2 where s = {\-y-\og^)~^. This extension was criticized by Proudman and Pearson (1957). The objection was that Lamb's solution was already corrected to the order of approximation necessarily inherent in approximating to the true equations of motion by Oseen's equations, and that nothing was added by obtaining higher order approximations to the solution of the latter. Dennis and Shimshoni (1964), using the numerical methods developed by Dennis and Dunwoody (1964), investigated the steady motion of a viscous, incompressible fluid past a fixed circular cylinder over a large range of Reynolds numbers. Their drag coefficients were found to agree reasonably well with experimental measurements for 16 low Reynolds numbers but started to become higher for values of Re greater than about 30. They also calculated the critical Reynolds number for the non-separated flow to be 5.6 while it was known experimentally that, below a certain critical Reynolds number, no separation took place. Another estimation of critical value of separation by Nisi and Porter (1923), by Homann (1936), and by Taneda (1956) was about Re=3.2. Bowen and Masliyah (1973) solved the Navier-Stokes equations for steady, incompressible, creeping flow past a body of revolution to its axis of symmetry and attempted to find a parameter based on particle geometry which would yield a simple correlation for predicting the Stokes resistance of an arbitrary particle, which Wadell (1934) had tried but failed to completely reduce all of the data points to a single line. Bowen and Masliyah solved the Stokes flow for the following shapes: prolate spheroids, oblate spheroids, p-square deformations, p-deformations, merging spheres, cylinders, cylinders with spherical caps, cylinders with conical caps, double-headed or simple cones, cones with spherical caps, and used a polynomial fit to yield the following relationship: A = 0.244 +1.0352 - 0.712S2 + 0.441S3 (1.17) where A is the drag force on the particle to that on a sphere with equivalent side perimeter and £ is the ratio of the surface area of the particle to that of the sphere with equivalent side perimeter. The above formula gives a value of drag which agrees with all of the computed values within plus and minus 10% and which share an average discrepancy of Cho, Cho, and Park (1992) suggested the following equation for an infinitely long cylinder in a bounded wall. 1.6%. -1 4 (1.18) 1 + V J In a Re 17 where a is the ratio of the diameter of thin cylinder to the diameter of the reservoir. When the above equation is compared to Cox' equation (1970), it can be showed that when the direction of a finite cylinder is perpendicular to the symmetric axis, the drag force becomes approximately twice as large as when it is translating along the axis. In fact, Taylor (1969) showed that this phenomenon was a property of all long axisymmetric bodies in creeping flow, regardless of their cross-sectional shapes. Leal (1975), investigated theoretically the creeping motion of axisymmetric rod-like particle using a perturbation expansion for a Theologically slow flow and suggested that the orientation of the thin cylinder was determined by the initial orientation, provided that the cylinder did not drift too close to the side wall of the reservoir tank. In addition, no intrinsic preference was shown for any orientation. Blumberg and Mohr (1968) studied the effect of the orientation on the settling velocity of cylindrical particles at low Reynolds number. The terminal velocity of a particle at low Reynolds numbers was found to depend on orientation of the particle with respect to the gravity field, and experimental data were collected on cylinders. Using Brenner's Analysis, they mathematically derived an equation for particles settling in cylindrical bounding surface at arbitrary location. U u = 6TTJUC 1 1 sin 2 0 + Aj K<PA  A . .(1.19) where the principal values §A and (|)R can be obtained directly from experimental measurements of u at 0 = 0 and nil. If the characteristic dimension of the particle is taken to be the radius of a sphere of equal volume, u is the ratio of the terminal velocity of the particle as measured to that of an equal-volume sphere in an unbounded fluid. K A is a wall effect term. The value of K A depends on the position of the particle with respect to the upper fluid-air interface and the bottom of the container. The infinite-cylinder wall-effect factor K A was used because Brenner(1962) showed that the theoretical value of K A 18 applying to an infinite cylindrical container fitted the data of Heiss and Coull(1952) taken w — u from a finite cylinder. Their experimental result is as follows: y/ = - r2— where u —u max mm y/ = sin 2 0 and 9 is the angle of orientation of the particle. The application of the results is very limited because one has to know the maximum and minimum settling velocities of a particle beforehand in order to predict other angles. Cox (1970) considered the flow around a long slender solid body which may or may not be straight and developed a general theory to find the hydrodynamic force exerted on such body when it was placed in a undisturbed flow field. If such a body is of half length b and has a cross-sectional radius of c, an expansion of the velocity field about such a body is made in terms of the parameter C = c/b. He also included a specific example of which the body centerline was bent in a circle. Considering a cylinder at rest in a fluid undergoing a uniform translation with its longitudinal axis perpendicular to the direction of motion, Cox determined the resistance force acting on the cylinder: + 0 \ j ^ L \ (1.20) ln( f ) + C l[ln(c)] 3j where C is the constant depending upon the body shape. The constant C is given as: C = + i + i f l n 9 A J + 1 ' l - , 2 ^ 2 4 J , \ X 2 J ds (1.21) and for the case of cylinders, where ?i(s)=l for -1 < s < +1, C = —^ + ln2 = 0.19315 Gluckman, Weinbaum, and Pfeffer (1972) were able to solve numerically the derived integral equation similar to the Stokes creeping-flow equations to provide the first theoretical solutions for low aspect ratio cylinders. Their theoretically predicted drag results were in excellent agreement with experimentally measure values. They defined the settling factor K as the ratio of the terminal settling velocity of the body to the settling velocity of a perfect sphere having the same volume as the object. The Stokes' law drag 19 force acting on the sphere having the same volume as the cylinder is F D S = 67T/uUsa. The drag force on a cylinder can be written as the same form by introducing a multiplication factor X that depends only on the geometry of the cylinder: FD C = 67TjuUccX. Where Uc is the settling velocity of the cylinder. If F D > S equals FD,C, one can write = {®L£± ? a n ( j since the sphere and the cylinder have the X same volume, a = defined as: K = —— v4 , L being the cylinder length. Thus, the settling factor K is (ale) X . Their calculated values of K are compared with Happel and Brenner (1965), and Heiss and Coull (1952). They also showed that the drag on long finite cylinder could be well approximated by a series of prolate spheroids. Their drag correlation factors for long cylinders based on prolate spheroid representation were compared with Burger's approximate solution: InfjU CL .(1.22) In •0.72 From the definition of X, the force on cylinders can be represented by FD = 6TTJUUCCX L which then gives: X = '3r In -0.72 A general solution of the 3-D Stokes equations was developed by Tsay and Weinbaum (1991) for the viscous flow past a square array of circular cylindrical fibres confined between two parallel walls. They solved the entire flow field, and discussed the effects of the spacing between fibres and the channel width on the channel flow resistance. They showed that the viscous layer around the fibres was the critically important for causing the channel resistance to sharply increase. Regardless of the channel width, the friction coefficient started to increase dramatically when the open gap 20 between adjacent fibres became smaller that the scale of channel width. This strongly suggested that the sudden increase of drag coefficient was caused by the overlapping of the viscous layers. However, no effect on the hydrodynamics of fibres was mentioned. Underwood (1969) calculated the steady, two-dimensional, incompressible flow past a circular cylinder for Reynolds numbers between 0.4 and 10.0 by employing the semi-analytical method of series truncation to reduce the governing partial differential equations of motion to a system of ordinary differential equations which can be integrated numerically. The results were compared favorably with available experimental data and numerical results. The results were inadequate to match the asymptotic expansion solution of Kaplun (1957) and Proudman & Pearson (1957) above Re=2 although the formers formulation of the drag coefficient was obviously preferable. Lamb's (1911) approximate solution of the Oseen equations was adequate at low Reynolds numbers, but it failed above Re=2. The full Oseen solution approaches Tritton's (1959) experimental data for low Reynolds numbers; however, the agreement worsens as Reynolds numbers increase. Experimental Finn (1953) was able to verify Lamb's equation experimentally by the measuring the drag coefficients for a cylinder at Reynolds numbers from 0.06 to 6.0. Tungsten wires, diameter of 4.1, 6.1 and 12.6 p inches, were mounted in a flow chamber having air flows ranged from 20 to 730 centimeters per second. He claimed that the wall effect in his experiment could be neglected because the size of the flow chamber was 1500 times the diameter of the largest wire. Happel and Brenner (1965) presented an empirical correlation for calculating the settling rates based on experimental results obtained for a number of axisymmetric bodies, i.e. 21 logK = log a r -<p2 0.25 V. cj a -1 .(1.23) where (p is defined as the ratio of the area of a sphere of the same volume as the particle to the area of the particle itself. They analytically derived an expression for finite cylinders moving parallel to its axis An/uUb F, In V c , .(1.24) 0.72 where b is the length of the cylinder and c is the radius of the cylinder. If a long straight cylinder moves perpendicular to its axis, a similar approach can be employed. Burgers gives only the roughest approximation for this case, requiring the vanishing of the resultant velocity U+u only at the central section of the cylinder. This leads to the result: 87TjuUb In V c ) .(1.25) + 0.5 Jayaweera and Mason (1965) determined the terminal velocity, drag coefficient, and orientation of cylinders falling in a large tank of viscous fluid for Reynolds numbers ranging from smaller than 0.01 to 1000. The experimental terminal velocities of cylinders falling in the tank were compared with values computed from both the Burgers-Broersma and the Lamb formulae as corrected for wall effects by Brenner's correlation. The agreement for long thin cylinders was good in all three cases at Re=0.02; at Re=0.24, the Burgers-Broersma formula gave velocities that were 50% higher. Their experimental results along with those of Relf (1913), Wieselsberger (1922) and Tritton (1959) showed that the effect of the walls became small as Reynolds number beyond unity. The agreement between these experimental results was quite impressive but they all gave coefficients that were systematically lower than the theoretical values calculated by Bairstow et. al. (1922), Thorn (1933), Southwell & Squire (1934), Tomotika & Aoi (1950), Kawaguti (1953) and Allen & Southwell (1955). 22 Heiss and Coull (1952) reported accurate experimental determinations for cylinders, spheroids, and rectangular parallelepipeds, and developed a general correlation for settling factors. In terms of the volume drag ratio A e , their results for motion normal to the axis may be written as: A e =—^=exp[0.576V^TGr 2 - l) ] (1.26) where cp is the sphericity and X2 is the shape factor. For cylinders, these results may be written explicitly in terms of the aspect ratio, E=L/d, using ( l 8 £ 2 ) 3 (D = -i <— (2E +1) Z2 .(1.27) V 16y and the principal resistance may be obtained from the drag ratios as i FD - 2>7id A e (1.28) V * J Pruppacher et. al. (1970) showed a curve fit to the many determinations of CD for steady crossflow past long cylinders given as: CD =9.689Re - 0 7 8(l + 0.147Re0 8 2 ) for0.KRe<5 (1.29) Jayaweera and Cottis (1969) gave similar curves for cylinders of finite length based on data of Jayaweera and Mason (1965). Expressions fitted to these curves are given as follows: 23 log Re = aa +alw + a2w2 + a 3w 3 where 0.81824 - 5 ^ 6 8 9 , , . , , „ E 1.54674 0.53872 a. =2.41277+ ; — a, = -0.50560-E Ez 1.34714 0.65696 E E2 27 9 3 a, = 0.82343 an a, — a , 3 64 ° 16 ' 4 2 White (1945) carried out experiments with wires in various fluids for Reynolds numbers between 10"5 to 102. He found that at very low speeds, the resistance was independent of the density of the fluid when the fluid was finite in extent. In such purely viscous motion, the boundary distance controlled the pattern and Lamb's formula was much more restricted in scope than usually thought; when the Reynolds number was 10"4 the presence of boundaries 500 diameters away multiplied the drag two-fold. At higher speeds, the resistance became independent of this distance and agreed with Lamb's solution for a cylinder in an infinite fluid. White's result of drag force is as follow: T d - F " - 5 M (1.31) Re This is valid only in an infinite or nearly infinite fluid. White also studied the local end-effect of the wires. He compared the settling of wires of different length and came up d with a end-correction: 10— L 1 + 2.4-D-L for Re<l to be subtracted from the observed xd value of . After correcting his results by the end-correction, he looked at the juU boundary nearness ratio d/D. He found that the resistance coefficient increasing threefold 24 as d/D increases from 0.0017 to 0.12. He derived a logarithmic type of curve to represent such a relation: rd = FD = 6.3 This is only valid when Re is very small and it starts to merge to Lamb's solution at Re=0.7. White then combined both equation and generated a general expression: - ^ - - 0 . 1 8 3 1 n ^ = - 0 . 1 8 3 1 n ^ (1.33) pJJ d 7.4// He then experimentally determined an expression for cylinders moving between parallel plates. — — = = __-lL for Re between 0.004 to 0.12 (1.34) fjLU juL2U ln# d Again, the results must be corrected by end-correction factor. The data of Tritton (1959) and Wieselberger (1921) can be curve-fitted with the smooth formula: CD «1 + - ^ - for 1.0<Re<2xl05 (1.35) Re1 Sucker and Brauer (1975) offered an outstanding curve-fit to the same data obtained by Tritton and Wieselberger: „ , , „ 6.8 1.96 0.0004Re , _ T ,, CD * 1.18 + — — + : Y where E=L/d (1.36) D R e 0 8 9 - l + 3.64£-7 Re 2 Re 2 Malaika (1949) experimentally studied the shape effect in the zones of viscous and surface drag. . He used the sphericity and settling factor for comparisons so that all other shape of particles are compared to a sphere of same volume. Although he did not develop any prediction of drag for any geometry, some observations of the interests of 25 this paper are mentioned. He found that particles of symmetry about three mutually perpendicular planes were stable in any settling position in the zone of deformation drag (Re<0.1). For higher Reynolds numbers (0.1<Re<500), a tendency was found for the particles to orient themselves with the largest cross-section in the three mutually perpendicular planes of symmetry in a position normal to the direction of motion. Moreover, it was observed that for low Reynolds number values in this zone (0.1<Re<1.0) a relatively long distance of travel was necessary for a particle to assume its stable position if released in another position. With increasing values of the Reynolds number the stable position described above was more readily assumed by the particles. A l l particles settling in a fluid in the deformation drag zone follow a resistance law similar to the Stokes law for the spheres with a correlation coefficient depending on the shape and orientation of the particle. Sherman (1990) pointed out an interesting idea by comparing two equations for drag on infinite and finite cylinders. As an example, he chose the Reynolds number to be 0.1 and the aspect ratio to be 500 that was considered a long cylinder. He found that the drag calculated from the 3-D equation is almost three times higher than that from 2-D equation. Therefore, people used to think that a very long cylinder could be conveniently idealized as a piece of an infinite cylinder but this is not clearly not so in creeping flow, in which the freedom of the fluid to escape around the ends of a finite cylinder has an essential effect. He also pointed out that the size of the fluid domain can affect the drag significantly. 1.3.3 Fibre Aidun (1956) was the pioneer to study the hydrodynamics of fibres. He carried out free settling experiments with different types of fibres (Nylon, Rayon, Dacron, Orion, and 26 Cotton) and determined a general expression to predict the drag on fibres for aspect ratio L/d>90: CD = —^5- for0.007<Re<0.1, ±4% (1.37) CD =^4^r forO.KRe<2, ±4% (1.38) R e 0 6 8 CD = for 0.007<Re<2, ± 9% (1.39) Re He observed that a fibre while falling through a fluid medium assumes the shape of a catenary depending on its diameter and length. The large diameter fibres fell horizontally with their axes normal to the direction of motion. But that observation was made on fibres with aspect ratio greater than 90. He also found that the rate of fall increased with length and up to a length of half a centimeter, and then became independent of the length. Tritton (1959) experimentally measured the drag on fibres, by observing the deflection upon the bending of quartz fibres, in a stream of viscous incompressible fluid with Reynolds numbers between 0.5 and 100. The experimental results were compared with various theoretical calculations: Lamb, Bairstow, Cave, & Lang, and Tomotika & Aoi , Southwell & Squire, and Kaplun. The experimental points fitted in satisfactorily with the overall picture given by these various theories. At low Reynolds numbers, the experimental curve did not join smoothly to the common curve of the theories, but the disagreement was within the limits of experimental error. Taylor (1977) measured the aerodynamic drag for several cotton fibres. Individual fibres were selected for fibre fineness and installed in a specially designed wind tunnel apparatus. Correlations of the test results showed that aerodynamic drag coefficient for all cotton fibres can be expressed by the relation: CD = for0.07<Re<0.8 (1.40) Re 27 C D = ^ p r f i M 0 . 8 < R e < 3 (1.41) Laminar flow drag theories for smooth circular cylinders in crossflow were compared with the data recorded for cotton fibres. However, cotton fibres have an irregular surface, and individual fibres may have a considerable amount of crimp along their length, further complicating viscous flow around them. However, as cotton samples differing by 206% in fineness the recorded force measurements differed by only 16.7% at the high velocities and 6.6 % at the low velocities. Quartz and nylon fibres were also used to study the effect of geometric irregularities of cotton fibres. He concluded that the surface texture of small fibres does not significantly influence aerodynamic drag. Thus, at low Reynolds numbers, crossflow drag forces are dominated by fluid friction in the surrounding media and only weakly influenced by friction at the fibre surface. His data were consistently higher than Lamb's theory for low velocities (Re<0.1) and were below his theory at higher velocities; however, the test results did agree quite well with the other reported data in these regions. He suggested that the reasons were the end effect and fibre-to-fibre aerodynamic interference effects. Hemstreet and Taylor (1980) measured the axial drag coefficient of individual, long-staple cotton fibres in an electric field at low Reynolds number. The electric force, as a function of field strength E, was measured for each fibre. The fibre was then mounted in a wind tunnel between plane parallel electrodes with field E aligned in the flow direction. When the fibre collapsed in the air stream, the field E and the flow velocity were measured, and the drag force was calculated. The results were in good agreement with Higuchi and Katsu's work (1961), and the drag force was approximately half the value of Taylor's measurement of cross-flow drag:CD = 4.4295 Re~° 8 1 4 5. 28 1.3.4 Wal l Effect Howard Brenner (1961) developed a general theory for the effect of wall proximity on the Stokes resistance of an arbitrary particle. The theory was developed completely for the case where the motion of the particle was parallel to a principal axis of resistance. In this case, the wall correction can be calculated from the known resistance force experienced by the particle in a unbounded fluid, providing 1) that the wall correction is already known for a spherical particle and 2) that the particle is small in comparison to its distance from the boundary. The theory developed by Brenner was confirmed by cited experimental data (Squires & Squires 1937; Heiss & Coull 1952; Pettyjohn & Christiansen 1948). Brenner reported the surprisingly large effects of wall proximity on Stokes resistance of a settling particle were well known by citing a illustration of Birkhoff (1950) ' . . .a sphere falling slowly in a cylindrical tube of viscous liquid, having 100 times the cross-section of the sphere, encounters 20% more resistance than if there were no walls'. The central result of Brenner's analysis is remarkably simple. Let F D denote the drag on the particle when moving in the bounded medium with velocity U, and let FD,oo denote the drag on the particle when moving through the unbounded fluid at the same velocity. The correction to the Stokes law resistance is then of the form, F 1 —— = , in which a and 1 are, respectively the characteristic particle FD,CO l - £ ( f e ) + *(f)3 and wall dimensions, and u. is the viscosity of the fluid. The dimensionless constant k is independent of the shape of the particle, depending solely on the nature of the bounding wall. The value of k can be obtained at once by comparing the above equation to the known solution of the problem for a spherical particle of radius a, in which D o c ^ r c i i aU . Some typical values of k of the interests here are as follows: Particle moving along the axis of a circular cylinder; l=radius of the cylinder (Oseen 1927; Haberman & Sayre 1958): k=2.1044. Particle falling midway between two finite, plane, parallel walls and 29 moving parallel to them; 1= distance from the centre of particle to either wall (Oseen 1927): k= 1.004. Faxen (1921) did not develop a general expression for sphere moving parallel to plane walls at arbitrary distance from walls. He derived for the case that the distance to one wall was three times as great to the other wall. The corresponding resistance formula becomes: 67rjulla • ix FD = 1 - 0.6526 + 0.1475 'a}3 -0.131 fa}* - 0.0644 + o ' a ^  where a is the particle radius and 1 is the closest distance from one wall. .(1.42) McNown (1948) obtained data for the sedimentation of a sphere along the axis of a cylinder, which was well correlated by the relationship developed by Faxen (1923) and Happel et. at. (1957). For sedimentation, this takes the form: 1 67rjuUa 1-2.104 fa} \Rj + 2.087 'a* + R c. • 1 .(1.43) Here CA - actual drag coefficient of a sphere in an unbounded medium, Cs = drag coefficient according to Stokes' law. Thus (CA/CS)-1 is a measure of fractional deviation of the actual drag in an infinite medium from that calculated by Stokes' Law. When the diameter of the particle becomes appreciable with respect to the diameter of the container in which it is settling, the container wall will exert an additional retarding effect. This can be allowed for by introducing a factor in Stokes' equation for frictional drag. FD = 2>7r^Uckw Ladenburg (1932) proposed the following value of K w as modified by Faxen: 1 + 2.1 .(1.44) 30 Francis (1933) showed experimentally that Ladenburg's correction can only be applied for values of (a/D) less than 0.1. His experiment values, covering a range of (a/D) from 0.13 to 0.97 and Re from 0.000015 to 6.9, can be empirically correlated by: k,., = 1 -' a ^ .(1.45) Bohlin (1960), using an extension of the method of reflections as originally developed by Faxen, carried the approximation further for a sphere in the axial position. He gave the formula for the sphere at the cylinder axis: 1 K 1-2.10443 - I+ 2.08877J -6.94813^-J •1.372 R + 3.87 \R •4.19 .(1.46) Haberman and Sayre (1958) determined wall correlation factors for rigid spheres moving within a still liquid inside an infinitely long cylinder by numerically solving the algebraic system: 2YaY fa 1-1 - - -0.20217 -.2j\R) .(1.47) 1-2.105^|j +2.0865^j -1.7068^j +0.72603^ Slezkin (1955) derived an expression for a cylinder moving perpendicular to the axis in a tube whose outer wall was solid, the limiting case of b » a yields: 47UjuU F' = In .(1.48) 1 Alternatively, i f the outside wall is frictionless, 31 F' = ATT/UU In v u J 1 + -2 .(1.49) White (1946) reported the experimental data at very low Reynolds numbers for wires, of length-to-diameter ratios of 50 and less, falling sidewise as they settled in a vertical cylindrical container. White correlated his data by the formula: F' 6.3 juU In 0.8D .(1.50) A circular cylinder moving perpendicular to its axis, with its axis midway parallel to the walls, gives rise to a two-dimensional problem. Two cases are of interests. The drag force of a cylinder moving parallel to the walls was given by Faxen (1946). For a unit length of cylinder, the resistance was ultimately found to be: AnfjJJ F' = In •0.9157 + 1.7244 2 CA -1.7302 3 ) .(1.51) The drag force of a cylinder moving perpendicular to the planes was studied by Westberg (1948). For a unit length of a cylinder, the resistance is found to be: ATTJUU F' In 0.62026 + 1.04207 .(1.52) The drag force of a cylinder moving parallel to a single plane wall was found by Takaisi (1956). Drag per unit length is: AK/UU F' In .(1.53) 32 where 1 is distance from wall, and c is the radius of cylinder. In addition to wall effect, Wadell (1934) pointed out other factors influencing the settling velocity. They are roughness of the surface and the roundness of the corners and edges. The surface roughness can usually be neglected but the roundness of the corners and edges of a solid are recognized as rather important factors influencing the resistance. Roughness on the surface of a solid particle is normally characterized by the "relative roughness," 8 /d, the ratio of the effective roughness height to the mean outside diameter of the particle. The most significant effects of surface roughness on flow past a particle occur in the transition to turbulence. However, the transition to turbulence is of no interest to this study as the maximum Reynolds number in this study is smaller than two. 33 1.4 Selection of Previous Work Previous work on cylinders does not show consistent results as the previous work on spheres does in creeping flow because cylinders have a lower degree of symmetry and the aspect ratio effect. Finite and infinite aspect ratio cylinders show a quite large difference in drag coefficient. Some expressions for cylinders have been selected to illustrate the deviations among previous work. Figure 1.4 plots the drag coefficients against Reynolds numbers for 2 and 3 dimensional cylinders in finite and infinite fluid domains. Lamb's formula derived for 2-D cylinders in an infinite domain is generally the lower bound for others. A l l other predictions are above Lamb's 2-D cylinder at Reynolds number smaller than 0.1 except the work by Aidun who explained the decrease was due to the catenary shape of the fibre. The deviation of others from Lamb could be five times larger at Re=0.0001 and five times smaller at Re=2. A l l theories seem to converge at Reynolds number close to 0.1. In order not to distract the readers by putting all the previous work in the same graph, only a few representative expressions are used for comparison in the following chapters: Lamb's and Heiss & Coull are selected because the former is for 2-D cylinder and the latter is for 3-D, Aidun and Taylor are selected because the work was on fibre. Since there is no exact wall correction for the geometry of the rotating tank, some of the results obtained from the tank can only be corrected by the wall effect on an object moving in the middle of two parallel plates. 34 35 1.5 Scope of this Study There is a clear need to study the hydrodynamics of pulp fibre experimentally. In order to illustrate the impetus for the development of the novel device used here, the most common method for determining fibre hydrodynamics is briefly reviewed here. Measuring the free settling rate of an object of known geometry and density in a fluid of known density and viscosity can yield the drag coefficient at that particular Reynolds number. The expression for drag coefficient can be derived from Newton's second Law and the definition of drag coefficient, Q _ nxdx(P/n>er~Pji„id)xg The drag coefficient is obtained by measuring the terminal velocity, UT, of the object. The Reynolds number is L-J x CM. x O calculated as: R C = t H f l u i d . A range of drag coefficients and Reynolds numbers can be achieved by using different fluids. However, there is a limitation in the range of Reynolds number that can be studied because of the limitation of the fluid density and viscosity. Since the density of pulp fibres is a function of the surrounding fluid (fibres have a cavity that will be filled with surrounding fluid), changing the surrounding fluid will change the fibre apparent density and hence the drag coefficient. Therefore, a device has been developed to achieve a range of Reynolds numbers with one working fluid (water) which is also the carrying fluid in many pulp and paper processes. In this study, a novel device, called the Rotating Tank, has been developed to study the two different types of pulp fibres (softwood thermomechanical and hardwood kraft pulp). The tank was validated with glass spheres, copper wires, and nylon fibres. The density and viscosity of the fluids used were measured experimentally and so were the dimensions and density of all objects. Experiments were performed at angular speeds from 400 to 900 rpm. 36 Chapter 2 Theoretical Analysis 2.1 Dimensional Analysis Buckingham's 7r-theorem was applied to find a general relationship between the drag coefficient and other variables that would be very useful in analyzing the experimental data. In the case of an arbitrary object settling under either gravitational or centrifugal force, the settling rate is dependent on the density difference between the object and the fluid, the particle size, the viscosity, and the acceleration. U = fn(Ay,p,/u,d) v \ (2.1) Ar = a\Pobject ~ P) where a is the acceleration due to a force field (e.g. gravitational in free settling). There are three fundamental units and five variables. Two dimensionless groups are expected. Since Ay and U may be used as dependent variables, the independent variables will be d, p, and u., dimensional analysis yields the following pi groups: d-U-p 1 = Re (2.2) niJ_M_P_ ( 2.3) which implies K2 = fn(7ty) in general, one will assume the function as d 3•Ay • p - C ( R e ) 6 . Using the definition of drag coefficient and re-arranging the left-hand side of the equation yields the following: 37 2 - R e ' - C ° = C ( R e ) > n C / ; = ^ R c - (2.4) CD=K- Re" The above equations indicate that the drag coefficient is proportional to the Reynolds number. The exact value of the constant and the exponential powers of the groups involved have to be found from experimental data. 38 2.2 Fundamental Concept The developments of the boundary layer theory have shown that the total resistance experienced by a body immersed in a streaming fluid is due to one or more of three different types of drag, namely, "deformation", "surface", and "form" drag. In general, the change from the first to the last type comes progressively with increasing Reynolds number. When the Reynolds number is low where viscosity plays a significant role in determining the drag on the body, the deformation drag dominates the others. Its effect extends to the entire fluid domain and therefore the boundary will have significant effect on the drag. In surface drag, the effect of viscosity is restricted to a thin layer near the surface of the body. The form drag is created by the development of eddies and decrease in pressure in the wake of the body. Since the majority of practical problems, including those of fibre separation in hydrocyclones, are cases where the flow velocity is known and it is required to solve the settling rate of the objects, the useful diagram is the customary plot of the coefficient of resistance against the Reynolds number. For an arbitrary object travelling in a fluid under the influence of a force field (gravitational, centrifugal or electrical), The total force acting on that object to cause it to accelerate can be represented by the equation of motion as: m -d = FG -FB -FD. d is the absolute acceleration of the object. FG is the force exerted on the object under a particular force field (e.g. gravitational in free settling and centrifugal in rotating tank). FB is the pressure force that the fluid exerted on the particle. Because of the similarity of the expression, FG and FB are usually combined as follows: FG-FB={pobject-p\v-G (2.5) V is the volume of the object and G is the acceleration due to a particular force field. Newton developed an expression for the resisting force, for any shape of particle falling through a fluid medium. It is given as: 39 where C D is the drag coefficient, A is the frontal area of the particle, U is the velocity of fall, p is the density of fluid. At terminal velocity, the drag force is equal to the net driving force. Equating Equations (2.5) and (2.6) yields: P ' C d 2 A ' U ' =(Pobjec,-p)-V-G (2.7) For a sphere: A = , V F 4 6 _*>-d\psphere-p\G ( 2 g ) 7i • d2 For a cylinder: A - d • L,V = —-— L r _ Vcylinder / (j Q x c o ~ ^ I 772 *'d\p cylinder- P\G 2-prU2 Equation (2.8) and (2.9) can be used for both free settling and rotating tank experiment with the modification for the G. In free settling, the force field is gravitational, hence G equals to 9.81m/s2. In rotating tank, the particle is subjected to the centrifugal force in the radial direction. G is the centripetal acceleration and equals to r-co2, where r is the radial position of the and co is the angular speed of the particle. The Reynolds number at terminal velocity is defined as usual: Re = P d ^t ^ g a r e s m ^ a w j ( j e range of M Reynolds number and drag coefficient can be easily achieved by controlling the angular speed of the rotating tank. The above equations provide a convenient way of representing the data. However, for the cases of pulp fibre, it is sometimes more useful to plot the settling rate as a function of angular acceleration due to the variations of fibre density and diameter. Equation (2.9) with G replaced by r-co2 can be re-written as: 40 prCD-A-Ut , x ~ -XPfibre-PyV-r " coA (2.10) From the dimensional analysis in previous section, CD is shown to be a function of Reynolds number. If we assume CD cc — and by the definition of Re, CD is inversely Re proportional to settling velocity. Equation (2.10) can be re-written as: U, ocr-co 2 (2.11) Although the rotating tank experiments confirm the proportionality of the relation, the equations or expressions given by analytical and dimensional analysis are too simple to predict the settling of pulp fibre in a centrifugal field. In Chapter 5, empirical studies, however, will show the settling rate is not a function of angular speed only. 41 2.3 Oblique Settling In the literature review, a number of researchers have pointed out the importance of orientation on settling rates. An analysis of orientation effect on the path of settling given by White (1991) is included here. He described an interesting behavior of slender body when falling unsymmetrically. Consider the needle in Figure 2.1, oriented with its axis at an angle 6 to the horizontal, it must fall such that its total fluid force F is vertical, to balance the body weight. Since the drag coefficient normal to the direction of motion is exactly twice as large as that parallel to the direction of motion, it falls at an angle 9 with respect to its axis such that Fn/Ft=2sina/cosa=tan(90o-9), or for falling rod or needle: 2 tana=cot9 For example, if 9=20°, then a=54°, or the needle moves along a direction (20°+54°) or 74° from the horizontal. Figure 2.1. Forces on a slender body in creeping flow-needle falls at an angle. Taken from Fig 3-36 [83]. 42 2.4 Sensitivity Analysis In order to understand the effect of each measuring variables on the drag coefficient and Reynolds number, the equations of drag coefficient and Reynolds number are differentiated with respect to each variable. The partial derivatives are evaluated for each data point. The differentials of each variable are experimental errors for every data point. Some error bars on plots of drag coefficient against Reynolds number are calculated from this analysis. The details of sensitivity analysis and sample calculations are given in Appendix A. However, a particular case of sphere in rotating tank is given here as illustration. The drag coefficient derived from previous section is given as: i M ^ ^ 3-p-U2 which is then differentiated with respect to all variables to yield: D dD dpsphere Hsp"ere dp dr dco dU The partial derivatives are listed as follows: dCD = 4-Ap-r-oo2 dD 3-p-U2 dCD _4-Ap-r-co2 ~dD~ 3-p-U 2 (2.15) dCD _ A-Ap-o)2 dr ~ 3-p-U 2 (2-16) SCD = S-Ap-r-6) dco 3-p-U2 dCD = -8-Ap-r-co2 dU 3-p-U3 Likewise, the same procedures for the Reynolds number yield: 43 K i - £ ± ± ( 2 . 1 9 ) M I T , 5Re , dRe I r v 5Re , r T 5Re , J R e = dp + dD + dU + du (2.20) dp dD dU dp ^ l s s £ ± (2.21) dp p ^ = £dL (2.22) dD p d_3l = £_P (2.23) du p <]3! = - P » V ( 2 . 24) dp p} The differentials of each variables are taken as the error associated with the instruments or the deviation among the variables. The reason for the strange scatter of the data point in drag coefficient against Reynolds number plots should be pointed out here. Since the settling velocity is present in the expressions of drag coefficient and Reynolds number, any variation in settling velocity measurement causes the data points in the graph to shift simultaneously in the vertical and horizontal directions. The drag coefficient is inversely proportional to the square of settling velocity and the Reynolds number is directly proportional to the settling velocity. If the settling velocity is measured slightly smaller than the expected value, the Reynolds number will decrease (shifted to the left) and the drag coefficient will increase (shifted up). The shift will be exactly opposite if the settling velocity is measured to be slightly larger than the expected value. Therefore, the data points spread out as a line. 44 Chapter 3 Experimental Apparatus and Procedures 3.1 Fluid Preparation Al l fluids were poured into the apparatus at least twenty-four hours before the experiment to achieve temperature equilibrium with the surroundings and to allow the removal of any dissolved gases in the fluids. Sometimes suction was applied to remove the air. In addition to thermal convection currents that will affect the settling, air bubbles were another problem in settling experiments. Because of the extra buoyant force of the bubbles sticking to the object, the settling rate will be lowered and the object will sometimes move upward. 3.1.1 Temperature Control A l l experiments were carried out in a room in which the temperature was nearly constant. The temperature of the room was monitored by a mercury thermometer and it was found that the fluctuation of temperature during a typical experiment day was less than one degree. The temperature of the fluid was monitored by an alcohol thermometer that had been verified by a digital thermometer. The temperature must be kept constant throughout the fluid to avoid the creation of convection currents due to temperature difference within fluid mass. Although the temperature effect in fluids on density is negligibly small, the viscosity of some fluids (Paraffin oil and Glycerin) is a strong function of temperature. Accurate measurements of the fluid temperature were necessary before and after each experiment. 45 3.1.2 Fluid Density and Viscosity Fluid density was measured with a calibrated cylinder of known weight. The cylinder was calibrated by pouring in 99.8 g of distilled water at 20°C to give a 100 ml reference mark. Fluids with unknown density were filled up to the reference mark. The weight of the 100ml fluid was then measured in a electronic balance. From the measured mass and volume, the density was readily calculated. The viscosity of fluids was calculated with a Gilmont® viscosimeter. It is a falling ball method viscosimeter. It consists of a glass tube with two prescribed marks and a standardized metal sphere. The tube was filled with the unknown fluid. By determining the falling time of a sphere in the glass tube, the viscosity of fluid was calculated. The viscosimeter was calibrated first with fluid of known viscosity. The calibration of the viscosimeter is given in Appendix B. Distilled water at 20°C was used as the standard fluid. The fluids used in the experiments were distilled water, glycerin, paraffin oil and gasoline. The summary of their properties is given in Appendix C. 46 3.2 Object Dimension An Axioplan 2 imaging microscope was used to measure the diameters of nylon and pulp fibres. Presoaked fibres were placed on a slide and covered with another slide so that they were flat and horizontal. A micrograph was captured in a computer in which the diameter of the fibre was measured. The diameter of each fibre was averaged from the measurements at three different locations. One hundred and twenty fibres were measured to give the diameter distribution of each kind of pulp fibre. The statistics and histograms of fibre diameter (Nylon, softwood TMP, and hardwood kraft) are given in Appendix D. The mean fibre length was measured in the Fibre Quality Analyzer. The distribution of the fibre length is represented as a histogram along with a mean arithmetic length. The length distributions of all fibres are given in Appendix E. The length measurements have been done with different sample sizes in order to investigate the effect of sample size on the mean values. From the results of FQA, the length distribution becomes constant when the fibre population is over a thousand. It suggests that sample size is important when estimating pulp fibre properties. Moreover, the length of every fibre that was captured by the high speed video camera was measured to study the length effect on settling rate. The length distributions from experimental measurements are given in Appendix F. The length distributions of pulp fibres with similar sample sizes from both methods are similar. The diameter of each glass sphere was measured by a digital caliper before being introduced into the container. The diameter was measured at the three mutually perpendicular axes to give an averaged value. A meter long copper wire was precisely cut into uniform pieces. The diameter of each short copper wire was measured at three different locations by a digital caliper. The diameter is very consistent at all locations. 47 3.3 Object Density The density of a wet pulp fibre is a function of its surrounding fluids because of the cavity in a fibre. The cavity filled with the surrounding fluid has significant contribution to the fibre weight and causes the fibre to swell because the surrounding fluid would increase the fibre volume. Therefore, instead of dry density, pulp fibres have another property termed apparent density. Neutral buoyancy method was used to measure the fibre apparent density. Mixtures of water and sugar solution were prepared at different densities. A small amount of fibres was put in the solution and mixed well with a agitator. Twenty minutes were allowed for settling. Fibres that are denser than the solution will sink to the bottom, those that are less dense will float on the surface, and those that are neutrally buoyant will suspend in the fluid. The number of fibres was counted in the three different regions. The percent of neutral buoyant fibre will be calculated. The idea is illustrated in Figure 3.1 and the experimental results for hardwood kraft and softwood TMP fibres are given in Appendix G. The apparent density for hardwood kraft is estimated to be 1329 kg/m3 with 86 % of fibres suspended in the body of the fluid. The apparent density for softwood TMP is estimated to be 1265 kg/m3 with 98 % of fibres suspended in the body of the fluid. Nylon fibre, a perfect solid cylinder with no cavity and swelling, was used to confirm the neutral buoyant method. A very sharp result was obtained for nylon fibres that nearly all the fibres sank to the bottom at a density of 1100 kg/m3 and floated on the surface at 1180 kg/m 3. Therefore, the averaged density of the nylon fibre was concluded to be 1140 kg/m 3, which is close to the book value of Nylon 6-6. The results for nylon fibre, hardwood kraft, and Thermomechanical pulp are listed in Appendix G. Dr. E. Statie developed a model to predict the wet densities of pulp fibres from their dry densities. Water fills the cavity of a dry fibre and then swelling takes place. The weight of the water entered can be calculated by knowing the internal diameter of the fibre. The increase in 48 volume due to swelling is estimated from swelling factors, which are unknown for most pulp fibres. The wet density estimated from the neutral buoyant method was confirmed to be reasonable by Dr. E. Static In order to minimize the diffusion of the sugar solution into the pulp fibres, the time allowed for the fibre in the sugar solution should be as small as possible. However, sufficient time is also needed for the fibres to settle. An estimation of the rate of diffusion was done by a one dimensional diffusion model given in Appendix H. The results show that more than an hour is required to fill up the cavity of fibre with glucose solution. The density of glass spheres supplied by Abbey Crafts & Arts Supply was not known. Unlike fibres, the density of glass spheres can be measured in a more direct manner. Exactly 10 ml of water was introduced into a graduated cylinder to give a calibration mark. The cylinder was then dried and filled with glass spheres as close to the mark as possible. The weight of the spheres was measured by an electronic balance. A known volume of water was added to the cylinder up to the 10 ml mark. The volume of the spheres in the cylinder was calculated by subtracting the total volume by the volume of water added. The density of glass spheres was estimated with another experiment as Figure 3.1. Schematic diagram of fibre density measurement. 49 comparisons. Fifty glass spheres were weighed. In addition, the average volume was calculated by measuring the diameter of individual sphere. The two method agreed with each other with less than 2% error. The density for 2 and 3 mm diameter sphere is given in Appendix I. Although copper wire is made of pure copper of which the density is known in the literature, its density was checked experimentally. The diameter of the wire was checked to be uniform across a meter long copper wire. Exactly 10 cm copper wire was cut and weighted in a electronic balance. The volume and weight were readily calculated to give the density. The measured value was within 10% error of the literature value for pure copper. 50 3.4 Preparation of Objects The objects used in this study were glass spheres, copper wires, nylon fibres, thermomechanical pulp (TMP), and kraft pulp. Efforts have been put to identify the species of the kraft and TMP but it was told by the supplier that it was impossible. However, the TMP was identified as softwood pulp and the kraft was identified as hardwood pulp. Due to the variations of pulp fibre properties, only the averaged values of the properties are given. Glass spheres with diameters of 2 and 3 mm were purchased from Abbey Crafts & Arts Supply. Damages and irregularities were found in some spheres: therefore, careful selection of quality sphere was necessary to improve the experiment accuracy. The 3 3 densities of the 3 mm and 2 mm sphere were estimated to be 2458 kg/m and 2504 kg/m respectively. Electrical copper wires manufactured by Omega Engineering, Inc. were used. The plastic coating of the electrical wire was carefully removed in order not to scratch the surface of the copper wire. Scratching may cause surface irregularities, and hence increase the surface roughness. A very sharp razor was used to cut the wire into pieces of 6mm long to minimize the deformation at the two ends. The diameter of the wire was found to be consistent with the claimed value of 0.254 mm. The density of the wire was found to be 8890 kg/m 3. Different nylon fibres, available in the Pulp and Paper Centre (UBC), were used. Nylon fibres (1 mm and 8 denier) were used in both free settling and rotating tank experiments. A picture of the nylon fibre is given in Figure 3.2. The nylon fibres were characterized by the length and the linear density. Denier is the linear density commonly used in the textile industry. Its units are kilograms per nine thousand meters. Nylon fibres 51 of the same denier (15) but different length (3, 5, and 7 mm) were also used in free settling experiment to study the length effect of fibres. Figure 3.2. Micrograph (5x) of nylon fibre (1 mm 8 denier). Softwood thermomechanical Pulp (TMP) provided by the Pulp and Paper Centre in UBC is shown in Figure 3.3. The average diameter and length of this softwood TMP are 33.84 um and 1.257 mm respectively. The averaged apparent density is 1265 kg/m3. Softwood TMP was a market pulp, flash dried and stored at the Pulp and Paper Centre in UBC. 52 Figure 3.3. Micrograph (5x) of softwood TMP. A picture of hardwood kraft Pulp supplied by the Pulp and Paper Centre in UBC is shown in Figure 3.3. The average diameter and length of hardwood kraft pulp are 22.25 um and 0.7954 mm respectively. The averaged apparent density is 1329 kg/m3. Hardwood kraft pulp was a market pulp, flash dried and stored at the Pulp and Paper Centre in UBC. Figure 3.4. Micrograph (5x) of hardwood kraft pulp fibre. 53 A l l objects were washed with distilled water to remove any dirt and contaminants, and soaked for twenty-four hours or longer in the fluid in which their falling rates were to be determined. Pre-soaking eliminated the possibility of changes in physical characteristics of the fibres during experimentation by any solvation or swelling, which was taken care of during pre-soaking. Dimensions of the fibres were measured after soaking in order to get the swollen diameters that they attained during settling. Due to mechanical actions and thermal effects in pulping, and the way they are stocked, fibres are curled and clumped together by stresses. Pulps must be cut out from the block of solid. The removed portion of the pulp was then soaked in distilled water for twelve hours at room temperature. The edges of the portion were removed by hand carefully because the cutting out the sample could chop the fibres thus affecting the length of the fibres at the edges of the sample portion. A portion of the pulp was then converted to pulp slurry in a re-pulper with a large amount of distilled water. The pulp slurry was then heated for forty-five minutes at a temperature above 90°C and disintegrated for thirty minutes to remove any residual stresses. Both nylon and pulp fibres were dyed with Rit® Scarlet dye to increase their visibility in the rotating tank. It was found by comparing the dyed and un-dyed fibres in the Fibre Quality Analyzer and in free settling that the dye would not change the geometry and hydrodynamics of the fibres. 54 3.5 Rotating Tank Apparatus The rotating tank is a circular, transparent container driven by an electric motor. The rotating tank was constructed from two circular plexiglas plates and a plexiglas tube as illustrated in Figure 3.5. The design details are given in Appendix J. The tank is 30.5 cm in diameter and 9.5 cm high. The cavity inside for fluids is 25.4 cm in diameter and 4.45 cm in depth. The tank was driven by an electric motor. The tank was assembled and balanced in a lathe to minimize vibration during high speed rotation. Four dial pins and marks were produced to ensure that the tank could be brought back to the original calibration after each disassembly. A one-inch diameter hole was drilled in the centre of the top plate for the introduction of fluids and objects. Concentric circles (one centimeter apart) were marked on both top and bottom for measuring the distance and alignment of the camera. The tank was capable of spinning at 1500 rpm. In order to freeze the motion of the tank, a strobe light (StroboTac® 1531-AD General Motor) was placed underneath the tank and was aligned with the camera. The strobe light was synchronized with the tank by a Retro-reflective optical sensor (MPL1 Honeywell®). Sometimes, a polarizing disc and a light diffusion box were needed to give the right exposure. The high speed video camera (HSV-1000 N A C , Inc.) with a C-mount macro lenses was placed on the top of the tank and aligned with the strobe. The camera had to be adjusted to be exactly perpendicular to the top surface of the tank in order to measure the distance traveled by an object accurately. After a picture of the top concentric circles was taken, the camera was lowered to such a position that the bottom concentric circles were clearly seen from the screen. The overlapping of the top and bottom concentric circles on the screen of the high speed video camera confirmed this important calibration. The camera was placed to focus at the region that was seven centimeter from the centre of the tank and the lenses were set to focus at the mid-depth of the tank with a depth of field of ±5 millimeters. The camera had to be calibrated for 55 accurate measurement of distance in the screen. A picture of the transparent scale was taken at where the camera was focusing for reference. The actual area that can be viewed by the camera was 1 cm x 1.5 cm. The tank was qualitatively checked to see if there was any turbulence by dropping a dye into the centre of the tank rotating at certain angular speed. With the strobe to freeze the motion of the tank, the pattern of the dye moving in the tank was observed to be a radial straight line. It was also found that fifteen minutes was sufficient for the tank to achieve solid body rotation. 56 Measurement The rotating tank experiments had to be carried out in a dark or dim room in order for the strobe illumination to be clearly visible. The tank was rotated for at least fifteen minutes to achieve solid body rotation. The high-speed camera, strobe light and the optical sensor had to be warmed up for 15 minutes before taking any measurement. Before the introduction of objects into the tank, the angular speed of the tank was measured by a hand-held optical tachometer (Shimpo® DT-205) and checked to make sure that it was constant. The selection of angular speed was to get as wide a range of Reynolds number as possible. However, there was a limitation in the lower bound of Reynolds number because of the action of gravity. Gravity would act to pull the object downward. Ideally, the object should be kept at the mid-depth of the tank. Therefore, the angular acceleration must be large enough that the gravity effect was relatively small and negligible. The displacement of the object inside the tank due to gravity was estimated with MathCAD and given in Appendix K. Different objects have different minimum angular speeds. The high speed video camera was turned on after the object was introduced into the tank by tweezers. As the object passed the region where the camera was viewing, pictures of the object were taken at a rate of 500 frames per second. By measuring the locations of the object at two consecutive frames, the velocity of the object could be calculated. Some of the dimensions, the orientation and the shape of the object during settling could also be measured with the high speed video camera. 58 3.6 Free Settling Apparatus According to literature, the wall effect is substantial in creeping flow. A sphere that is five hundred times smaller than the container will experience almost twice as large as the drag of an object in an infinite domain. Aidun's experimental work on free settling showed that at least six inches of fluid column was necessary for the fibres to attain the terminal velocity. So, the container must be carefully selected to minimize the wall effect. The container chosen for free settling for all objects (glass spheres, copper wires, and fibres) was a graduated Pyrex® glass cylinder (No. 3022) shown in Figure 3.6. Its diameter is 15 centimeters and total length is 45.7 centimeters. 15cm 7.62cm 45.7cm 4 • 15cm Figure 3.6. A schematic diagram of free settling experiment. 59 Measurement From the primary settling experiments, it was found that more than 15 cm of fluid column was necessary for the fibres to attain their terminal velocities. Therefore, the cylinder was graduated in such a way that at least 15 cm of fluid column was above the starting mark. The second mark was set according to the fluids used. If less dense and less viscous fluids were used, the object would move faster so that the second mark was set to give a longer travelling distance to minimize the reaction time error associated with the use of a stop watch. The objects were dropped at the top of the fluid column by means of a small tweezers. The stopwatch was started as soon as the objects reached the starting mark and stopped as soon as they reached the final mark. 6 0 Chapter 4 Validation of Rotating Tank Before the tank could be applied to measure the hydrodynamics of pulp fibres, it had to be validated with other objects for which the hydrodynamic properties had been studied. The sequence of validation was to test the tank with glass spheres, copper wires, and nylon fibres. Glass spheres were used because Stokes' Law for a sphere had been studied analytically and verified experimentally for a wide range of Reynolds number by a large number of researchers. Although glass sphere experiments were sufficient to validate the rotating tank experiment, copper wire and nylon fibres were used to further validate the tank. Due to its uniform circular cross section, copper wire can be modeled as a perfect cylinder, which has also been studied extensively in the literature. The results of the copper wire tests were to confirm some of the cylinder theory. Nylon fibres, having a uniform circular cross section, were more similar to pulp fibres because of their flexibility. 4.1 Glass Sphere Glass spheres with diameters of two and three millimeters have been tested in the rotating tank filled with pure glycerin. The results together with free settling results are given in Appendix L. Experiments were performed at three different angular speeds to produce a range of Reynolds numbers from 0.1 to 0.50 which is in the Stokes' flow regime (Re<l). Besides, the settling results of glass spheres in glycerin are included to give the lowest bound of the Reynolds number. The graphs of drag coefficient against Reynolds number for both 2 and 3 mm spheres are shown in Figure 4.1. Since it is too distracting to include all the data points in the graph, only the trend lines, determined by a least square method, and the equations are shown in the graph. Generally speaking, results with 2 mm spheres 61 are in better agreement with Stokes' Law than those of 3 mm spheres. Both 2 and 3 mm spheres had higher drag coefficient than the theory because the theory applies to a sphere in an infinite fluid domain. The retarding force created by the boundary of the tank caused the sphere to move slower. From the graphs, 3 mm spheres have higher drag coefficients than 2 mm because the larger the object the larger the retarding force. The results have also been corrected with wall effect suggested by Ladernberg (1932). The Ladernberg correction is based on a sphere moving at the middle of two infinite parallel plates separated by a distance. However, the shape of the fluid domain in the rotating tank is a finite circular cylinder. Both corrected results are improved to agree better with theory. The trend line for 3 mm spheres is displaced more than the 2 mm spheres after the correction with wall effect but is still 12% higher while the discrepancy of 2 mm sphere has been improved from 7% to 2%. Experiments with glass spheres with a circular mark were performed to confirm that such objects do not rotate about any of their axes in a fluid of solid body rotation. 0.01 Figure 4.1. Drag coefficient Vs Reynolds number for 2 and 3mm glass sphere. 62 4.2 Copper Wire A long copper wire with uniform diameter of 0.254 mm was cut into pieces with length of 6 mm to give an aspect ratio of twenty-three. The particular aspect ratio was chosen to be similar to the nylon and pulp fibres. They were tested in the rotating tank filled with glycerin. Only two angular speeds could be achieved for copper wire because of its large density difference with glycerin. If the angular speeds are too low, the copper wire will move quickly to the bottom of the tank due to gravity. Therefore, the angular speed should be as large as possible. However, the maximum angular speed of the tank was found to be 1200 rpm due to the electric motor maximum output power and vibrations at high angular speed. The rotating tank and free settling results are given in Appendix M. The plot of drag coefficient against Reynolds number, including the free settling results, is shown in Figure 4.2. The data points at one angular speed spread out from a mean value with a particular slope. This has been explained in section 2.4 that the presence of settling velocity in both drag and Reynolds number expression causes the unusual scatter of data. The trend line, a result of least square estimates, does not agree well with Lamb's (1911) and Aidun's (1956) prediction but agrees well (±12%) with Heiss & Coull (1952). Lamb's analytical result is purely for a two dimensional cylinder in infinite fluid domain while Aidun's experimental results were performed with fibres of aspect ratio greater than 90. The agreement between Lamb and Aidun suggests Aidun's prediction should only be used for 2-D cylinder or at least L/D greater than 90. However, the copper wire having an aspect ratio of 23 cannot be modeled as 2-D cylinder. Therefore, other analytical results for 3-D cylinder are included for comparisons. The results show good agreement with Cox, Heiss & Coull, and Broersma which are all for finite cylinders. The experimental data are correlated with the least square estimate: Cn = ^ T 7 ^ for 0.0007<Re<0.01 (4.1) 63 Although the general trend agrees well with others, the data are actually scattered quite substantially. The spread in the data is partially from experimental error and wall effect. However, the main contribution to data spread is settling orientation. From experimental observations, it was found that some of the wires did not settle with their axes perpendicular to the direction of motion, which is widely-assumed to be the most stable position for cylinders in free settling, but at an angle to that. In free settling experiment, wires maintained the initial tilted orientation for a long distance (at least 25.4 cm). As mentioned in Section 2.3, titled slender bodies do not travel in the direction of the resultant force but at an angle. Nearly all title wires moved towards the container wall with their orientation unchanged until they are eventually rotated by the retarding force of the wall, to the most stable position. The orientation distribution of settling in tank at two different angular speeds is given in Figure 4.3. The angle, horizontal axis, is defined as the angle from the axis that is perpendicular to the direction of motion: zero degrees correspond to purely perpendicular to the direction of motion and 90 degrees correspond to purely parallel to the direction of motion: The orientation does not have any particular distribution or influence from angular acceleration. However, other than zero degrees, 30 and 40 degrees are the most frequent orientations that the wires would attain. It was also observed that the tilted wires did not rotate and persisted with the same orientation during settling. This is expected because nonsymmetrical needle falling in Section 2.3 gives clear explanation and the observation that glass spheres do not rotate supports it. Figure 4.4 shows the effect of orientation on the settling rate. Below 20 degrees, there is no or weak evidence on any orientation effect. However, above 20 degrees, there is strong evidence that fibres tend to move faster if their long axis is oriented towards the direction of motion. Long slender bodies moving with their axes parallel to the free stream experience smaller drag than those perpendicular to the free stream. In the case of infinite prolate spheroids, the difference was found analytically to be a factor of two. 64 Figure 4.2. Drag coefficient Vs Reynolds number for copper wire. 30.0% 25.0% 20.0% 15.0% 10.0% 5.0% 0.0% Frequency g458rpm H362rpm n 0 10 20 30 40 50 60 70 80 90 More Angle Figure 4.3. Orientation distribution of copper wire settling. 65 A) • 0 degree • 1 to 20 degree A 20 to 40 degree x 4 1 to 90 degree 140 160 A n g u l a r Speed (m/sA2) B) Settling Velocity (m/s) • • • • • • • « • . • • • * * * Orientation Angle (degree) 0.025 0.020 Figure 4.4. Orientation effect on copper wire settling. 66 4.3 Nylon Fibre Dyed nylon fibres, 1 mm and 8 denier, were tested in the rotating tank filled with distilled water. Experiments were performed at angular speeds of about 400, 600, and 900 rpm. The results from rotating tank and free settling experiments are given in Appendix N. The drag coefficient against Reynolds number plot, including the free settling results, is shown in Figure 4.5. The data points at each angular speed again spread out as a line. This is due to the occurrence of the settling velocity in both drag coefficient and Reynolds number (see Sec 2.4). The red trend line of the results lies among other theories although it does not fit onto any of the theory. However, there is a trend that the free settling data points (from Re=0.1 to Re=0.001) are generally lower than those from rotating tank experiments. This is possibly due to the wall effect in the tank since the container of free settling is much bigger than the tank. However, there is no exact wall correction available for cylinders in the rotating tank. The closest available correction is the 2-D cylinder moving in the middle of two infinite parallel plane walls given by Faxen (1946). With Faxen correction, the data points are brought closer together but the magnitude is small. With least square estimate, a simple power-fit equation to predict the drag coefficient is given as: CD =-^L for0.0007<Re<1.0 (5.2) D RE0.881 It was also observed that some nylon fibres were settling with their axes tilted at an angle to the direction of motion. With the aid of the camera, the settling orientation of nylon fibres could be recorded. The orientation distribution of nylon fibres during settling is shown in Figure 4.6. The trends for three different angular speeds are similar in that there is a large population of fibre orienting themselves between 0 and 20 degrees. Unlike the copper wires, there was no fibre settling with its axis parallel to the direction of motion. The maximum angle for nylon fibres is 60 degrees and very few fibres attained an angle larger than 40 degrees. The orientation effect on settling rate is shown in Figure 67 4.7. There is a very slight increase in settling velocity with increasing angle. There is also another indication that the orientation effect is greater at high angular acceleration. Obviously, there is not enough data for the orientation above 40 degrees. When the nylon fibre results are compared with those of copper wire, both graphs suggest that the orientation effect is minor for angles below 20 degrees. This is the reason why the nylon fibre chart does not show the orientation effect because most of the fibres (>80%) settled with angle equal to or smaller than 20 degrees. Although the plot of nylon fibres looks different from that of copper wire, they, in fact, agree with each other for angles smaller than 20 degrees. However, there are not enough samples of nylon fibres oriented larger than 40 degrees. 68 70.0% 60.0% 50.0% 40.0% 30.0% 20.0% 10.0% 0.0% Figure 4.6. Orientation distribution of nylon fibre settling. 69 A) 0.025 0.02 0.015 0.01 0.005 Settling Rate (iris) i * Odegree • lto20degree x 21 to30 degree x 31to60degee 100 200 B) 0.02 0.015 0.01 0.005 300 400 500 Angular Speed (mfsA2) 600 -ft t »r~t= 700 800 Settling Velocity (m/s) , 415rpm • 670rpm & 4 900rpm A A A & A A 10 15 20 25 30 35 40 45 Orientation Angle (degree) Figure 4.7. Orientation effect on nylon fibre settling. 70 4.4 Aspect Ratio Effect The aspect ratio of a cylinder is defined as ratio of the length over the diameter. Cylinders with very large aspect ratios usually will be treated as 2-D. Those with small aspect ratio are classified as finite cylinders or fibres. However, no work has been done to give a clear transition from a finite to an infinite cylinder. It is even more difficult to give a clear transition on fibres because of the fibre flexibility. From the micrograph shown in Figure 3.2, it is clear that nylon fibre can be modeled as a cylinder due to its uniform circular cross section. Small aspect ratio fibres remain straight while long fibres take a curved shape during settling: the longer the fibre, the more it will bend during settling. Long fibres, therefore, will experience less drag force than a straight cylinder having the same aspect ratio. Small aspect ratio fibres cannot be modeled as a 2-D cylinder but neither can the long fibre. Aidun (1953) carried out free settling experiments on fibres of aspect ratio larger than 90. The drag coefficients that he obtained for those fibres are smaller than Lamb's analytical results for a 2-D cylinder in an infinite fluid domain. His explanation was the catenary shape of fibre during settling. In order to investigate the aspect ratio effect on settling, free settling experiments with nylon fibres with aspect ratios of 33, 70, 116, 163 were carried out. A l l fibres except those with aspect ratio of 33 assumed the shape of curved with various degrees. The longer the fibre, the more it will bend during settling. The results are shown in Figure 4.8. They are all very close to Aidun's prediction. Unfortunately, the results do not show a clear indication of the aspect ratio effect as some of the lines are overlapping with slightly different slopes. 71 Chapter 5 Results and Discussion Al l experimental data for hardwood kraft pulp and softwood thermomechanical pulp are given in Appendix O and P respectively. The data are grouped in different ways for discussion and are shown either as settling rate plots or drag coefficient and Reynolds number plots. A l l the factors that might affect pulp fibre settling rates will be discussed: angular acceleration, length, orientation, shape, and diameter. When interpreting the results, the readers should be aware that the number of data points for some extreme data is small. For example, the percentage of fibres travelling with their axes parallel to the direction of motion is very small. Plots of drag coefficients against Reynolds numbers will be included for comparisons with other theories. Least square fits are sometimes used to show the trends for some data series. Finally, the data will be analyzed by multiple regression to give the correlation of each factor and a prediction for the settling rates for each pulp. 5.1 Length Effect on Settling Rate From the output of the FQA (Fibre Quality Analyzer) for the two pulps, it was found that the length distribution of pulp fibres covered a range from 0.1 mm to 5 mm. The details of the output are given in Appendix E. Hardwood kraft pulp has a mean length of 0.795 mm with a standard deviation of 0.488 mm and softwood TMP has a mean length of 1.258 mm and a standard deviation of 1.044 mm, both from a sample size of ten thousand fibres. The high speed camera allowed the measurement of fibre length. Therefore, the length distribution of each pulp can be found based on the experimental sample sizes. The detailed statistics of the length distribution obtained from experiments are included in Appendix F. The means and standard deviations, although not the same, are similar to 73 those from the FQA with similar sample sizes. Both measurements suggest that pulp contains a wide distribution of lengths. Although it cannot be verified by nylon experiments, the length effect on settling rate is substantial for cylinders. The flexibility of long fibre will assume curved shape, which will further affect the settling velocity. The plots of settling rate against angular speed and the plots of settling rate against the fibre length for both hardwood kraft and softwood TMP are shown in Figures 5.1 and 5.2. Both types of plots with least square fits are aimed to show the length effect, the angular acceleration effect and the relationship between the two. The data are grouped by fibre length in four categories. For all length groups, there is a clear indication that the settling rate increases with angular velocity. Both softwood TMP and hardwood kraft have the similar patterns and spreads. The similarity of the normalized standard deviation shows that the data have the same spread at different angular speeds. Hardwood kraft and softwood TMP have the same mean settling rate at an angular acceleration of about 150m/s2. However, softwood TMP tends to have higher mean settling rates than hardwood kraft at the two higher angular speeds. If C D is assumed to be inversely proportional to Reynolds number as indicated in Section 2.2, it can be shown that the settling rate is directly proportional to the square of angular speeds for both softwood TMP and hardwood kraft. The mean settling velocities of each fibre length group at different angular speeds are correlated by a least squares method to show the effect of fibre length. For hardwood kraft pulp, there is an indication that the settling rates are proportional to the fibre length at all angular speeds. On the other hand, softwood TMP settling rates show very weak or no dependence on fibre length at different angular speeds. Once again, the readers should be aware of the sample sizes of each length group. Although the number of data points for long fibres is small, it could affect the slope of the trend lines quite substantially as clearly seen in the settling rates against the fibre length plots. 74 On the very bottom left in the graphs, the free settling data are barely visible due to the difference in the magnitude of the settling rate between free settling and in the rotating tank. Since the free settling experiments were carried out without the use of high speed camera, the length could not be measured and nearly all measurements were on fibres that were perpendicular or very close to perpendicular to the direction of motion. The comparisons between free settling and rotating tank results would be best illustrated by a drag coefficient against Reynolds number plot in Section 5.5. The scatter of the data for just one group at one angular speed is many times larger than the experimental error owing to the large variation in other fibre properties. For example, the data of hardwood kraft pulp in the range of 0.5 mm to 1 mm at angular acceleration of 700 m/s2 have a normalized standard deviation of two. That is due to the fact that fibres of similar length might have very different orientation, diameter and density, hence different settling rate. 75 0.014 0.012 . Settling Rate (m/s) 0.010 -0.008 0.006 0.004 0.002 0.000 - L * » 0.0mm<L<0.5mm • 0.5mm<L< 1.0mm A lmm<L<2.0mm A & x 2.0mm<L<4.0mm x Free Settling 1 * 100 200 300 400 500 Angular Acceleration (m/sA2) 600 700 800 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0.5 1.5 2 2.5 Length (mm) Figure 5.1. Length effect on settling rate for hardwood kraft pulp. 76 0.024 . 0.02_ 0.02 0.018 0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 Settling Rate (m/s) L< 1 mm 1 mm<L<2mm • a 4 2mm<L<3mm X 3mm<L<5mm • Free Settling 100 300 400 Angular Acceleration (m/sA2) 8 700 0.03 0.025 0.02 0.015 0.01 0.005 Settling Rate (m/s) A i * A A ft V - «• A » A * A 4 1 * • A * A A, * B g | A B ? * A | j * A§ / | 4 A _ 6 & ft A 4 o * Q •»! ."H « I m H A ; : 0.5 1.5 2 2.5 3 L e n g t h (mm) 3.5 4.5 Figure 5.2. Length effect on settling rate for softwood TMP. 77 5.2 Orientation Effect on Settling Rate Experimental observations showed that most fibres did not settle with their axes perpendicular to the direction of motion. Instead, the fibre orientation during settling can be anything from totally perpendicular to totally parallel, relative to the direction of radial motion. The orientation distributions for hardwood kraft and softwood TMP at different angular speeds are given in Figure 5.3. The histograms show the frequency of different fibre orientation. The horizontal axis is the fibre angle, which is measured between the fibre long axis and the radial axis. Zero degrees correspond to a fibre that is aligned perpendicular to the direction of motion and 90 degrees corresponds to a fibre that is parallel to the direction of motion. The histograms show that the settling orientations have peaks at around 30 degrees for both hardwood kraft and softwood TMP. Most fibres settle at an angle smaller than 45 degrees. In addition, the angular acceleration has no effect on the settling orientation for both hardwood kraft and softwood TMP. The orientation distributions of both hardwood kraft and softwood TMP are similar for all angular speeds. The same plots as Figures 5.1 and 5.2 but grouped to show the orientation effect on settling rate are in Figure 5.4 and 5.5 The data are divided into four groups as indicated in the graphs. Their mean settling rates as a function of the angular acceleration are fitted by a least squares method. Both hardwood kraft and softwood TMP behaved similarly. Fibres having large angles settled faster than those at smaller angles because the former experienced less drag. As other slender bodies, fibres that are parallel to the direction of radial motion will have the fastest settling rate. As the fibre orientation angle increases from the 0°-20° group to the 25°-45° group, the increase in settling rates is small and so is the increase from the 50°-75° group to 80°-90° group. However, there is a bigger gap between 25°-45° group to 50°-75° group. This suggests that a plot of settling rate versus angular orientation would have a smaller slope between roughly 0° and 35°, a steeper 78 slope from 35° to 60°, and then a smaller slope from 60° to 90° group as shown in Figures 5.4 and 5.5. This qualitatively agrees with the results of copper wire and nylon fibres of which the orientation effect is small at angles smaller than 20°. Another interesting point is that the orientation effect is more pronounced at higher angular speeds than at low angular speed, which might indicate a Reynolds number dependence of the orientation effect. 79 a) hardwood kraft 4 0 . 0 % 0 15 30 45 60 75 M o re A n g l e b) softwood TMP 3 0 .0% , 2 5 .0% 2 0 .0% 1 5 . 0 % 1 0 . 0 % S . 0 % J 0 .0% 0.0 H i . 3 2 2.5 3 3.8 4 5.0 5 6.3 6 7.5 7 8.8 M o r e A n g l e Figure 5.3. Orientation distribution of a) hardwood kraft b) softwood TMP. 80 0.014 0.004 0.002 0.000 40 50 60 Orientation Angle (degree) Figure 5.4. Orientation effect on settling rate for hardwood kraft pulp. 0.03 , 0.025 0.02 0.015 . Settling Velocity (m/s) + 0 to 20 degree A A * Q 20 to 45 degree • 4 50 to 75 degree Q A x 80 to 90 degree 9 X A A 0 .A x ts *> 40.01 0.005 700 0.03 0.025 0.02 0.015 0.01 0.005 Settling Velocity (m/s) A • 420rpm a 636rpm 4 873rpm 8 A A A A A * « A . 40 50 60 Orientation Angle (degree) Figure 5.5. Orientation effect on settling rate for softwood TMP. 82 5.3 Shape Effect on Settling Rate As mentioned earlier, fibre flexibility will cause long fibres to settle with a curved shape. The effect was claimed to be significant by Aidun although the free settling experiments on the aspect ratio effect are not able to confirm this. The shape of fibres during settling is also of interest in this study. The shape of every fibre during settling was measured. The shapes of the fibres are classified into two groups: straight and curved. Fibres that have sharp corners are also put in the curved group as the curved fibres. For hardwood kraft pulp, on average seventy percent of fibres were straight. For softwood TMP, the average is about eighty percent. This is in fact of no surprise because of the way that the pulp fibres were prepared. As a result of the uneven distribution, the shape effect may not be captured and shown from this study. Nevertheless, the settling velocity plots in Figure 5.6 show that the shape of the fibre has only a very weak effect on settling rates for both hardwood kraft and softwood TMP. 83 a) hardwood kraft 0.014 0.012 0.01 0.008 0.006 0.004 0.002 Settling Velocity (m/s) • • • s . Catenary v B"® \ ^-s5S= & \ ^ ^ ^ ^ ^ • ^zsff^S} Straight / • • » • Angular Acceleration (m/sA2) o 100 200 300 400 500 600 700 800 b) softwood TMP 0.03 Figure 5.6. Shape effect on settling rate a) hardwood kraft b) softwood TMP. 5.4 Drag Coefficient Vs Reynolds Number In previous settling rate plots, the free settling results are all very close to the origin due to the fact that gravity is between one and two orders of magnitude smaller than the angular acceleration. Plotting the drag coefficient against the Reynolds number eliminates this weakness of settling rate against angular acceleration plots. Due to the fact that the spread of data can be distracting, the mean values with error bars for each pulp are plotted and joined by the least squares method, as shown in Figure 5.7. The equations from least squares method are shown in the graphs. Two types of error bars are shown in the graphs. The blue error bars are the standard deviation of the data and the red error bars represent the error due to the variation of diameter alone. It should be noted that the mean fibre diameter and density are used to produce this plot. Both hardwood kraft and softwood TMP have trend lines with similar slopes and they are above all other predictions. This indicates that pulp fibres in the rotating tank experience much more drag than simple cylinders. The increase in drag force might be due to the presence of other fibres in the tank. However, it was observed from the camera that in a volume of 1 cm x 1.5 cm x 1 cm, there were on average five to ten fibres. Therefore, on average, fibres are five millimeters apart, which is hundreds of fibre diameter. Alternatively, the consistency was calculated to be 0.0001% which is an extremely low value. Therefore, consistency should not be a factor. Another reason could be that wood fibres are simply not cylinders in the sense that wood fibres have substantial structure (e.g. microfibrils) and non-circular cross section (some conifer wood fibres have square cross-section). Both features will cause the drag on pulp fibre to increase. The free settling results of hardwood kraft and softwood TMP are close to the low Reynolds number end of tank results. The agreement between free settling and tank results implies that the increase in drag is not due to the geometry of the rotating tank or the interaction with other fibres. The fact that softwood TMP has higher drag coefficient than hardwood kraft is possibly due to the larger mean diameter of softwood TMP fibres. 85 Since their settling rates at each angular speed are very similar as shown in the previous plots, the increase in drag coefficient is from either mean diameter or apparent density, a) hardwood kraft 0.0001 0.0010 0.0100 0.1000 1.0000 10.0000 b) softwood TMP - R e -Cd 4 o » — Lamb&Newton Aidun — Taylor Heiss&Coull Figure 5.7. Drag coefficient Vs Reynolds number a) hardwood kraft b) softwood TMP. 86 5.5 Diameter Effect on Drag Coefficient The error bars in the previous plots do not show the diameter effect clearly. The effect of diameter variation is best illustrated in Figure 5.8 with the drag coefficients and Reynolds numbers calculated from the minimum, mean, and maximum diameter. The increase in drag coefficients is 30 times from the minimum diameter to maximum diameter for both hardwood kraft and softwood TMP. This effect explains a large part of the variation of drag coefficients from fibre to fibre observed experimentally. The huge effect of diameter suggests that the diameter of each fibre should be measured during settling to obtain more accurate drag coefficient and Reynolds number plots. However, the magnification of the camera is not large enough to measure the diameter of pulp fibres. It should be noted that the variation in fibre wet density has not been taken into consideration. The sensitivity analysis shows that the drag coefficient is extremely sensitive to the object density. For nylon fibres, ten percent change in density can cause almost seventy percent change in drag coefficient. 87 a) hardwood kraft n w ft — Lamb&Newton — Aidun — Taylor — Heiss&Coull = Cd lean Dia me te r 0( )0 0( JQ A 1 ( fl ft Min Diameter ft i * 1-0.001 0.010 0.100 1.000 10.000 b) softwood TMP •101 ftfl ftp :d . a m b & N e w t o n U d u n "aylor l e i s s & C o u l l C Me£ in I Di an IN :t er -1 f)f #1 0- — t f ] Vlnx D amf ter / — 1 / If 00 1 — 1 ^ «-« r ^ ; »- -/— F 0.0001 0.001 0.01 0.1 1 10 Figure 5.8. Diameter effect on drag coefficient a) hardwood kraft b) softwood TMP. 88 5.6 Regression Analysis The large variations in fibre properties (length, diameter, and density) make it very difficult to use the drag equations to make predictions for an individual fibre. Multiple regression method can be used to correlate the settling rate with angular acceleration, orientation, and fibre length. The results of hardwood kraft and softwood TMP were analyzed with multiple regression. The dependent variable is the settling velocity and the independent variables are length, orientation, shape, and angular acceleration that can be measured for each fibre. Although the previous section shows that the diameter also have huge effect on settling rate, the rotating tank was, however, not capable of measuring the diameter for each fibre. The detailed results of the regression analysis for hardwood kraft and softwood TMP are given in Appendix Q. The regression equations, which could be used to predict the mean settling rates in metres per second of pulp fibres in terms of their length and orientation at a particular angular acceleration, are given as follows: hardwood kraft: usetmng = 6.829• IO - 6 • rco2 + 0.001014• L + 2.039• 10"5 -0-0.0001361 softwood TMP: Usealing = 1-593-10"5 -rco1 + 0.0003146• L + 4.346• 10"5 -0-0.00215 rco2 is the angular acceleration in meter per second squared, L is the fibre length in millimeter, and 9 is the angle of the fibre, in degrees measured from the axis perpendicular to the direction of motion. The expressions could be very useful to predict the settling velocity of hardwood kraft and softwood TMP in fractionation. For example, pulp slurry mixed with hardwood kraft and softwood TMP is fed to a hydrocyclone. At a particular region where the angular acceleration (100 times of gravity) is known and the orientation (20°) and fibre length (1 mm) are assumed to be the same for both pulp fibres, 89 the settling rates of hardwood kraft and softwood TMP are estimated as 1.97E-3 m/s and 6.27E-4 m/s. Therefore, hardwood kraft is settling faster than softwood TMP and more likely to be collected at the bottom of a hydrocyclone. However, this estimate ignores the turbulence and secondary flows that are found in hydrocyclones. The significance of each variable was tested with two methods: P-value test and t-test. The smaller the P-value, the higher the dependence. Most mathematicians reject the null hypothesis i f the P-value is smaller than unity. That is exactly opposite to the t-test for which null hypothesis will be rejected for absolute values larger than one. Both t-test and P-value test reject the null hypothesis tests on all variables; therefore, they should be all retained in the expressions. However, their degrees of correlation are not the same. The settling velocity is most strongly correlated with angular acceleration and orientation but much less so with the fibre length except for hardwood kraft pulp of which the length effect is greater than orientation effect. The results of t-test and P-value test are given as follows: hardwood kraft Coefficients t-test P-value Intercept -0.00013609 -0.495302375 0.6206984 Orientation (degree) 2.03881 E-05 4.79812163 2.37779E-06 Fibre length(mm) 0.001014139 5.249416755 2.65617E-07 w2*rm e a n(m/s2) 6.82934E-06 19.03220585 6.65847E-56 softwood TMP Coefficients t-test P-value Intercept -0.002150174 -5.565693814 4.17305E-08 Orientation (degree) 4.34604E-05 7.201614029 2.07944E-12 Fibre length(mm) 0.000314559 2.37881267 0.017725775 w2*rmean(m/s2) 1.5929E-05 28.1794521 4.9146E-107 90 Regression analysis gave a good estimate of the correlation between settling velocity and other variables. However, the actual effect of the variables on settling velocity cannot be simply reflected by the magnitude of correlation. For example, fibre length of softwood TMP has been shown to have no effect on settling velocity in the previous section. However, both t-test and P-value test reject the null hypothesis on fibre length that fibre length is retained in the expression. One should be very careful when interpreting and using the results from this multiple regression analysis. 91 Chapter 6 Conclusion Based on the results obtained from the rotating tank experiments, a number of conclusions can be drawn and are listed as follows: 1. The rotating tank has been successfully developed and validated with spheres, copper wires, and nylon fibres of which the results show excellent agreement with previous work. It can be confidently used to study wood fibres under similar centrifugal force (several hundred times of gravity) as in a hydrocyclone. 2. Specific tests with the hardwood kraft and thermomechanical pulp fibres generate the following observations: • The orientation of a fibre in the tank depends strongly on its initial orientation. In contrast with a widely used assumption, most fibres do not orient themselves with their long axes perpendicular to the direction of radial motion. The fibre orientation can be anything from purely perpendicular to purely parallel to the direction of radial motion with the majority of the fibres oriented in the range of 20 to 40 degrees from the perpendicular. • Fibres do not rotate regardless of their orientation. They will maintain their orientation during the entire settling process in the rotating tank. • The settling velocity of pulp fibres is a strong function of their orientation. Fibres settling at an angle of larger than 45 degrees to their direction of motion settle faster than those at smaller angles. The orientation effect becomes more pronounced as the angular acceleration increases. • For hardwood kraft pulp, the plot shows that the settling velocity increases with fibre length but the plot of softwood TMP does not show the same dependence. 9 2 • The shape of the fibre does not affect the settling rate. Curved fibres or fibres having sharp corners settle at almost the same rate as straight fibres. • Softwood kraft and hardwood TMP have, on average, much larger drag coefficients than any other objects tested under the same conditions. The apparent density and mean diameter are used to plot the drag coefficient against the Reynolds number. The apparent densities for hardwood kraft and softwood TMP are 1329 kg/m3 and 1265 kg/m3 respectively whereas the mean diameters for hardwood kraft and softwood TMP are 22.3 and 33.8 microns respectively. A Least squares fits from the mean values gives the mean drag coefficient as a function of Reynolds number for hardwood kraft and softwood TMP. Hardwood kraft: CD = U* for0.007<Re< 1.0 R e 0 5 7 1 Softwood TMP: CD = ™ for 0.007< Re< 1.0 D R e 0 . 7 6 3 Multiple regression analysis yields two expressions to predict the mean settling rates of hardwood kraft and softwood TMP in terms of angular speed, length, and orientation. Hardwood kraft: usealing =6.829-10 - 6-rco 2 + 0.001014-Z + 2.039-10"5 -0-0.0001361 Softwood TMP: usealing = 1-593-10"s-rco2 + 0.0003146-L + 4.346-10"5 -0-0.00215 rco is the angular acceleration in meter per second squared, L is the fibre length in millimeter, and 9 is the angle of the fibre, in degrees measured from the axis perpendicular to the direction of radial motion. The settling velocity is most strongly correlated with the angular acceleration and the fibre orientation, and much less so with the fibre length. 93 Chapter 7 Recommendations Although the work of this study is fairly complete, there are still a number of areas that require further study. Based on the variability of measurements of some pulp properties (length, diameter, and density), it is reasonable to assume the drag coefficient of pulp fibre is likewise highly variable. As a result, more fibres would be required to give a better estimate of the drag coefficient. To accomplish that with the existing rotating tank set-up, one should use a computer to capture the images from the high speed camera and let the computer analyze the pictures to get the length, orientation, settling speed, and ideally the diameter of each fibre. Once the tank is computerized, different types and species of pulp fibres could be studied much more efficiently. In all pulp and paper processes, pulp fibres are always surrounded by other fibres. The distance from one fibre to another can be expressed in terms of the consistency (weight ratio of fibre to water). From other studies, it has been proven that the proximity of other particles can have a substantial effect on particle settling. It is reasonable to assume the settling rate of pulp fibre is also a function of the consistency. Future work should be done on investigating the effect of consistency on settling rate. This work is particularly important because a pulp suspension in a hydrocyclone is usually at a high enough concentration that fibre-fibre interactions could be significant. However, the rotating tank set-up may not be able to achieve that because it is very hard to measure the settling velocity of an individual fibre if the consistency of the pulp slurry is high. What would be captured in the high speed camera may be floes or continuous fibre network for high consistency pulp suspension. In order to completely understand how pulp fibres with different properties can be separated in a hydrocyclone, experiments should be carried out in a hydrocyclone. Laser 94 Induced Fluorescence with a high speed digital camera, Laser Doppler Velocimetry, and Particle Image Velocimetry are the techniques can be used to study the hydrodynamics of pulp fibre in a hydrocyclone. For example, in order to understand how earlywood and latewood fibre can be separated in a hydrocyclone, the earlywood could be dyed with one color and latewood dyed with another. Using a high speed camera focusing at one cross-section of the hydrocyclone, one can understand how the pulp fibres move in the radial direction. 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Wieselsberger, C , "New data on the laws of fluid resistance", National Advisory Committee for Aeronautics, Technical Note no. 84, 1922. 101 Appendix A Sensitivity Analysis and Sample Calculations 102 Sensitivity Analysis Rotating Tank Sphere: Equation of motion in the radial direction gives the drag coefficient as: 4 •£>• {Psphere ~Pfluid)'r 3-Pn^d -U It is then differentiated with respect to all variables: dCn , n dCD , 8CD , 8CD , dCD , 8CD J r r dCD =^&dD + — 2 - dpsphere + dp ihlid +—ILdr + —±dG> + —fdU dD dpsphere dpfhlid '7 dr See DU The partial derivatives are listed as follows: dcD 4 • Ap • r • co2 3D 3-P fluid -U2 dCD 4-D-ra2 dP sphere 1-Pflu,d-U2 dCD -4-D-r-a2 dp jiuid 3 • P fluid • dCD 4 • Ap • co2 dr ^Pfluid-U2 dcD 8 • Ap -r-co <9co 3-Pfluid -U2 dcD - 8 • Ap • r • co2 dU Cylinder: Equation of motion in the radial direction gives the drag coefficient as: 103 1 1 ^'{p cylinder ~Pfluid)'r" co2 C ° 2'P.fmid-U1 It is then differentiated with respect to all variables: An 8 C D J n d C o > d c D 8CD 8CD 8CD J T T d C ° = ~ ^ r V d D + ^ cylinder + T - £ ~ d P fluid + ^ d r + - ^ d & + ^rfdU $D cylinder 5P fluid ^ SCO dU The partial derivatives are listed as follows: dCD _ n • Ap • r • co2 ~dD~2-pfluid-U2 8Cn % • D • r • co2 dP cylinder 2 • p f h M • U dCD - T I - D T - C O 2 -pcvVmder dp jhlid Z-Pfluid-U2 dCD 7t • Ap • co 2 dCD Tt • Ap • r • co T2 da Pfluid-C 8CD _ - n • Ap • r • co2 ~dU~ Pfluid-U3 Free Settling Sphere: Equation of motion in the direction of gravity gives the drag coefficient as: £ _ ^'D' sphere ~ P fluid )' S ^ P fluid It is then differentiated with respect to all variables: d c D _ d c D . d c n , . e c — - dD + — dpS[ dD d p s p h e r e The partial derivatives are listed as: d C n = ^ - d D - ^ - d p , + - ^ - d p a . d + ^ - d U D ^.r-v sphere ^ r fluid -\T T dD d p s p h e r e dpfluid • dU 104 3CD _ 4 -Ap-g dD 3-p^-U2 8CD _ 4 -Ap-g 3D 3-pfluid-U2 dCD =-4-D-g-psphl d C g = - 8 - A p - g dU 3-pfMd-U' Cylinder: Equation of motion in the direction of gravity gives the drag coefficient as: c = n ' D \ P cylinder ~P fluid)'S 1-P fluid It is then differentiated with respect to all variables: dCD , n dCD , dCD , dCD J T T dCD = —^dD + —-^dpcvlinder +—-e-rfp + — - 5 - dU dD dpcyljnder dpJhid • dU The partial derivatives are listed as: 3CD ^ T t -Ap-g 3D 2-pfluid.U2 3CD = n-D-g dp cylinder 2 ' P fluid ' ^ dCD =~^-D-g-P cylinder dpfluid ~ 2-p)luid-U2 dCD _ - n • Ap • r • co2 ~dU~ P.mid-U' Reynolds Number Re = 'fluid •DU d Re = dp dPjiuid dRe D-U fluid dRe J n dRe „ T dRe , + ——dD + -—dU + ——d\x dD dU d\i dPjiuid r 1 dRe P fluid • U dD dRe D-P fluid dU r1 5 Re ~ P fluid ' d/u P2 Rotating Tank Sample Calculations Glass Sphere diameter of sphere= 0.003 m density of sphere= 2557 kg/m 3 density of fluid= 1250 kg/m 3 radial pos.= 0.06 m angular speed= 35 rad/s mean vel.= 0.05 m/s viscosity of fluid= 0.931441 kg/m*s dCd/dD= 40987.52 dCd/dp s p here= 0.09408 dCd/dpfi u i d= 0.19245 dCd/dr= 2049.376 dCd/dco= 7.026432 dCd/dv= 4918.502 dCd= 7.417229 Plus and minus dD= 0.00005000 m dpsphere= 5.00000000 kg/m3 dpfluid= 5.00000000 kg/m3 dr= 0.00000500 m dco= 0.52359877 rad/s dv= 0.00005000 m/s dn= 0.00500000 kg/m*s dRe/dpfiui= 0.000161 dRe/dD= 67.100332 dRe/dv= 4.026020 dRe/d(i= 0.216118 dRe= 0.00544211 106 Equation of motion in the radial direction gives the drag coefficient as: Cn = 4" D \ P sphere' P fluid)' V~ co2 3-P fluid -U2 R e = ^ - D - U Substituting the data: c _ 4-0.003-(2557-1250)-0.06-35 2 _ 1 2 3 ± ? m D 3-1250-0.05 2 R ^ 1250.0.003.Q.05 = 0 2 ± 0 0 0 5 4 4 0.931441 Stokes Law: D Re 0.2 In order to understand the dependence of Drag and coefficient on, each variable will be investigated individually. The error differential of each variable will be increased by 10% of the measured values while others are kept at zero. Again, the above values are used as illustration in the following table: d C D dRe dD= 9.99% 10.07% dpsphere= 19.56% 0% dpfluid= 19.56% 10.07% dr= 9.99% 0% dco= 19.99% 0% dv= 19.99% 10.07% du= 0% 10.07% 107 Nylon Fiber diameter of fiber= 3.0E-05 m density of fluid= 998 kg/m 3 density of fiber= 1142.25 kg/m 3 mean radial pos.= 0.07 m Angular speed= 55 rad/s mean velocity= 0.006 m/s viscosity of fluid= 0.000899 kg/m*s dCd/dD= 1335445 dCd/dpfib er= 0.399433 dCd/dp f l u i d = 0.457166 dCd/dr= 823.1169 dCd/dco= 2.095207 dCd/dv= 19206.06 dCd= 5.714398 Plus and minus dD= 0.00000012 m dpfiber= 6.45000000 kg/m3 dpfluid= 2.00000000 kg/m3 dr= 0.00000500 m dco= 0.52359877 rad/s dv= 0.00005000 m/s d(i= 0.00000100 kg/m*s dRe/dp f l u i= 0.00029 dRe/dD= 6660.73415 dRe/dv= 47.89656 dRe/dn= 319.66560 dRe= 0.00409968 Equation of motion in the radial direction gives the drag coefficient as: K ' D - ( P cylinder ~ P fluid )" T ' ' Cr • CO 2'Pfluid -U Re = Pfluid -D-U r1 Substituting all the numbers: it • 30E - 5 • (l 142.3 - 998) • 0.07 • 552 Re = 2-998-0.0062 998 -30E - 6 -0.006 40.1±5.714 0.000899 = 0.2 ±0.00410 Lamb's Equation C 8TT D Re- (2.002 -ln(Re)) 108 In order to understand the dependence of Drag and coefficient on, each variable will be investigated individually. The error differential of each variable will be increased by 10% of the measured values while others are kept at zero. Again, the above values are used as illustration in the following table: d C D dRe dD= 9.99% 9.99% dpsphere= 69.13% 0% dpfiuid= 90.54% 11.45% dr= 9.99% 0% dco= 19.98% 0% dv= 19.98% 9.99% du= 0% 9.99% 109 Appendix B Gilmont® Viscosimeter Calibration Gilmont® Viscosimeter Calibration To determine the viscosimeter constant K Reference: water @23°C Viscosity = 1 cp = 1 e-3 kg/m*s According to the formula given by manufacture: K = u/(pt-p)*t where jx = viscosity of reference liquid p t = density of ball used p = density of liquid used Since stainless steel ball size #3 was used, p, = 8.02 g/ml Density of water: weight of cylinder = 312.2 g weight of cylinder + water = 956.1 9 volume of water added = 650 ml mass of water = 643.9 g density of water = 0.991 g/ml Trial Time of falling [s] 1 34.62 2 38.5 3 36.03 4 34.81 5 36.69 6 36.64 7 39.4 Averaged 36.67 Averaged time in minutes = 0.6112 K = 0.233 A p p e n d i x C F l u i d P r o p e r t i e s Fluid Properties The properties of fluids used for the experiments are listed below: Density [kg/mJ] Viscosity [kg/m*s] 1 Distilled Water @20°C 998 0.001013 2 Glycerin @20°C Fisher Scientific Lot no. 982833 G33-4, Class 11 IB, 99.7% 1250* 1.4787* 3 Methanol @23°C Fisher Scientific Lot no. 992743 A412-4, Class IB, 99.9% 783.56* 0.000529* 4 Paraffin Oil, Light @25°C Fisher Scientific Lot no. 982838 0121-1, Class 11 IB Saybolt Viscosity 158 Maximum Kinematic Viscosity not more than 33.5 Centistokes @ 40°C 835* 0.0556* 5 Gasoline @24°C Esso Company Premium 730.6* 0.000491* * values are measured experimentally by the methods mentioned in Section 3.1.2 113 A p p e n d i x D F i b r e D i a m e t e r \ Fibre Diameter Thermo-Mechanical Pulp Diameter(/um) Frequency Frequency Diameter(jum) 9.21 2% 1 17.69 6% 3 Mean 33.83884 26.18 17% 8 Standard Error 1.619544 34.66 28% 13 Median 33.583 43.14 28% 13 Mode 17.075 51.63 13% 6 Standard Deviation 11.10303 More 6% 3 Sample Variance 123.2773 Kurtosis -0.31537 Skewness 0.067246 Range 50.90624 Minimum 9.20676 Maximum 60.113 Sum 1590.425 Count 47 Largest(1) 60.113 Smallest(1) Confidence Level(95.0%) 9.20676 3.259969 T M P ( m e a n = 33.84microns) 3 0 % 2 5 % 2 0 % 1 5 % 1 0 % 5 % 0 % 9 .21 1 7 . 6 9 2 6 . 1 8 3 4 . 6 6 4 3 . 1 4 5 1 . 6 3 Diameter (microns) More 115 Kraft Pulp Diameter(jum) Frequency Frequency Diameter(/um) 11.4 1% 1 14.7 7% 8 Mean 22.24579 18.0 21% 23 Standard Error 0.662019 21.3 21% 22 Median 21.2532 24.6 21% 23 Mode 15.8571 27.9 8% 9 Standard Deviation 6.847973 31.2 11% 12 Sample Variance 46.89474 34.5 4% 4 Kurtosis 1.227109 37.8 1% 1 Skewness 1.04955 41.1 1% 1 Range 32.9587 More 3% 3 Minimum 11.4138 Maximum 44.3725 Sum 2380.3 Count 107 Largest(1) 44.3725 Smallest(1) 11.4138 Confidence Level(95.0%) 1.312517 Kraft Pulp (mean=22.25microns) 20% -15% -10% -5% 0% -, — , i i mm i H S I i v/ /u - i i i i i i i i i i 11.4 14.7 18.0 21.3 24.6 27.9 31.2 34.5 37.8 41.1 More Diameter (microns) 116 Nylon Fibre (1mm 7denier) Diameter(iLim) Frequency Frequency Diameter(jum) 24.09 25.71 27.32 28.94 30.56 32.17 More 2.2% 0.0% 8.7% 4.3% 47.8% 32.6% 4.3% 1 0 4 2 22 15 2 Mean 29.92283 Standard Error 0.246262 Median 30.1699 Mode 30.6608 Standard Deviation 1.688287 Sample Variance 2.850313 "Kurtosis 2.833494 Skewness -1.14505 Range 9.6911 Minimum 24.0946 Maximum 33.7857 Sum 1406.373 Count 47 Largest(1) 33.7857 Smallest(1) 24.0946 Confidence Level(95.0%) 0.495699 60% 50% 40% 30% 20% 10% 0% Nylon Fibre (1mm 8 Denier) Diameter Histogram wt WB 24.09 25.71 27.32 28.94 30.56 Diameter (microns) 32.17 More 117 Appendix E Fibre Length From Fibre Quality Analyzer (FQA) 118 Fibre Length From Fibre Quality Analyzer Thermo-Mechanical Pulp Length (mm) Mean 1.257667 Standard Error 0.010419 Median 0.9285 Mode 0.072 Standard Deviation 1.044319 Sample Variance 1.090602 Kurtosis 1.10559 Skewness 1.103549 Range 9.595 Minimum 0.072 Maximum 9.667 Sum 12634.52 Count 10046 Largest(l) 9.667 Smallest(l) 0.072 Confidence Level(95.0%) 0.020424 2.00% 0.00% 0.2 0.6 1 1.4 1.8 2.2 2.6 3 3.4 3.8 4.2 4.6 5 Length (mm) 119 Kraft Pulp Length (mm) Mean 0.795394 Standard Error 0.004899 Median 0.736 Mode 0.181 Standard Deviation 0.488047 Sample Variance 0.23819 Kurtosis 30.25897 Skewness 4.041147 Range 8.754 Minimum 0.072 Maximum 8.826 Sum 7893.491 Count 9924 Largest(l) 8.826 Smallest(l) 0.072 Confidence Level(95.0%) 0.009603 3 0 . 0 % 2 5 . 0 % 2 0 . 0 % 1 5 . 0 % 4-1 0 . 0 % 5 . 0 % 0 . 0 % 1 K ra f t «3 L e n g t h ( m m ) 120 Nylon (1mm 8 denier) Length (mm) Mean 0.970949 Standard Error 0.003336 Median 1.012 Mode 0.108 Standard Deviation 0.345204 Sample Variance 0.119166 Kurtosis 30.37386 Skewness 1.305473 Range 6.448 Minimum 0.072 Maximum 6.52 Sum 10396.93 Count 10708 Largest(l) 6.52 Smallest 1) 0.072 Confidence Level(95.0%) 0.006539 4 0.0 0 % 3 5 .0 0 % 3 0.0 0 % 2 5 .0 0 % 2 0.0 0 % 1 5 .0 0 % 1 0.0 0 % 5 .0 0 % 0.0 0 % H i N y l o n F ib re 0.2 0.8 1.4 2 2.6 3.2 3.8 4 .4 5 L e n g t h ( m m ) 121 Appendix F Fibre Length From Experiments (HVCR) Fibre Length From Experiments (HVCR) Kraf t Length (mm) Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count Largest(1) Smallest(1) Confidence Level(95 915rpm 0.88858541 0.042386225 0.726072607 0.660066007 0.510397737 0.26050585 11.3261069 2.931010623 3.300330033 0.330033003 3.630363036 128.8448845 145 3.630363036 0.330033003 0%) 0.083779661 636rpm 0.934579439 0.047575673 0.858085809 0.99009901 0.492126584 0.242188574 9.637766657 2.649492804 2.97029703 0.330033003 3.300330033 100 107 3.300330033 0.330033003 0.094323453 390rpm 0.760056204 0.025851414 0.660066007 0.660066007 0.2598035 0.067497859 7.662466595 2.11063231 1.782178218 0.330033003 2.112211221 76.76567657 101 2.112211221 0.330033003 0.051288476 all 0.865752298 0.024012399 0.726072607 0.660066007 0.451152026 0.20353815 12.88752946 3.032607739 3.300330033 0.330033003 3.630363036 305.6105611 353 3.630363036 0.330033003 0.047225852 25% 20% 15% 10% 5% 0% Kraft Pulp (145 fibres, mean=0.8657mm) Length (mr) Cv Cv Cv v N - N - <V V 'b- *b y 'b- fc fc' fc- ^ 123 25% Kraft Pulp (353 fibres) 20% -15% -10% -5% -0% i i i i • , t l n fl Fl i l I fl _ (1 : I in I in. i i i i I I ' I i I I I ! r > I I I i i i i i I I I I I I I I I I I I I I I I I I I I l I I I I I I I I I l I Length (mm) TMP actual length(mm) Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count Largest(1) Smallest(1) Confidence Level(95.0%) 873rpm 1.82121438 0.059933085 1.782178218 1.98019802 0.817378303 0.668107289 -0.659023299 0.366230285 3.432343234 0.528052805 3.96039604 338.7458746 186 3.96039604 0.528052805 0.118240267 636rpm 1.677028168 0.067317125 1.650165017 1.848184818 0.882855813 0.779434386 -0.194648048 0.626179307 3.630363036 0.330033003 3.96039604 288.4488449 172 3.96039604 0.330033003 0.13287966 420rpm 1.502232576 0.070969924 1.320132013 0.660066007 0.925334593 0.856244109 1.213626375 1.19733341 4.290429043 0.330033003 4.620462046 255.379538 170 4.620462046 0.330033003 0.140101668 All 1.671542154 0.038415359 1.584158416 1.98019802 0.882717755 0.779190634 -0.026321415 0.697466965 4.290429043 0.330033003 4.620462046 882.5742574 528 4.620462046 0.330033003 0.075466079 124 12% 10% 8% 6% 4% 2% 0% T M P (186 f ib res , mean=1 .67mm) JUL * ^ ^ N> ^ * 0> & o><? * J ^ Length (mm) T M P ( 5 2 8 f i b r e s ) 9.0% 8.0% 7.0% 6.0% 5.0% 4.0% 3.0% 2.0% 1.0% 0.0% 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4 2.7 3 3.3 3.6 3.9 4.2 4.5 4.8 Mo L e n g t h ( m m ) 125 Appendix G Pulp Fibre Density (kraft and thermomechanical Pulp) 126 Pulp Fibre Density Objective-to prepare fluids of different density by diluting a concentrated sugar solution Assumptions • diffusion rate of sugar solution into fibres is negligible compared to settling rate of fibres Original sugar solution • dissolved 4kg of sugar in 2L distilled water • density measured= 1329.3 kg/m3 Calibration • calibrate a graduated cylinder by pouring 100ml distilled water @20°C into it and mark the 100ml line Fibre density measurement • put a very small amount of fibres(to avoid flocculation) into the solution. • mix them thoroughly by an agitator • let them settle for 30mins • count the fibre on the surface, bottom and suspended in the body the fluids. TMP Kraft Beaker# 1 Beaker# 1 Distilled Water sugar solution (ml) Distilled Water (ml) sugar solution (ml) (ml) 100 400 100 400 Solution weight (100ml)= 106.9 g Solution weight (100ml)= 106.89 g Density= 1068.9 kg/m3 Density= 1068.9 kg/m3 Observations Observations Top= 2 1.0% Top= 5 2.1% Body= 5 2.4% Body= 16 6.6% Bottom= 200 96.6% Bottom= 220 91.3% 127 Beaker# 2 Beaker# 2 Distilled Water sugar solution (ml) Distilled Water (ml) sugar solution (ml) (ml) 200 300 200 300 Solution weight 113.4 g Solution weight (100ml)= 113.35 g (100ml)= Density= 1133.5 kg/m 3 Density= 1133.5 kg/m 3 Observations Observations Top= 0 0.0% Top= 10 5.9% Body= 27 21.3% Body= 30 17.6% Bottom= 100 78.7% Bottom= 130 76.5% Beaker# 3 Beaker# 3 Distilled Water sugar solution (ml) Distilled Water (ml) sugar solution (ml) (ml) 400 100 400 100 Solution weight 126.5 g Solution weight (100ml)= 126.5 g (100ml)= Density= 1265 kg/m 3 Density= 1265 kg/m 3 Observations Observations Top= 0 0.0% Top= 40 10.3% Body= 250 98.0% Body= 300 76.9% Bottom= 5 2.0% Bottom= 50 12.8% Beaker# 4 Beaker# 4 Distilled Water sugar solution (ml) Distilled Water (ml) sugar solution (ml) (ml) 0 500 0 500 Solution weight 132.93 g Solution weight (100ml)= 132.93 g (100ml)= Density= 1329.3 kg/m 3 Density= 1329.3 kg/m 3 Observations Observations Top= 14 8.0% Top= 50 14.3% Body= 160 92.0% Body= 300 85.7% Bottom= 0 0.0% Bottom= 0 0.0% 128 Appendix H Fibre Diffusion Model Fibre Diffusion Model Diffusion Rate (Glycerol-Water) into pulp fiber cavity-One dimensional diffusion model pm := 126 density of surrounding fluid (kg/m-*) pf := 998 initial fluid density inside the fiber (kg/m-*) dp := pf - pm density difference (kg/m-*) L := 0.00 length of the fiber (m) a := 0.0000000009 diffusivity constant [glycerol-water] (m /^s) Tt := 3.14159265 n := 1,3.. 100 Solution: density distribution u(x,t) := pm+ 4- — ( 2 2 \ -n •% -a-t n V -•exp| n sin n-7i-x^ —J) x:= 0,0.00001. L The following graph shows the density distribution in the cavity of a pulp fibre at different time. 1300 r 0.0015 A p p e n d i x I S p h e r e D e n s i t y Average Sphere Density 3mm 1) Measuring the weight of the empty dry cylinder Weight = 26.6 g 2) Calibrating the cylinder by adding water at 20°C Weight of dry cylinder + added water = 36.5 g Wieght of water added = 36.5-26.6= 9.9 g Density of water at 20°C = 998 kg/m3 Volume of Water added = 9.92 ml 3 ) Add spheres to the cylinder and not to exceed the calibration mark Weight of dry cylinder + spheres = 38.9 g Weight of Sphere = 12.3 g 4) Add water to fill the cylinder up to the calibration mark Volume of water added = 5.11 ml Sphere volume = total volume - water added = 9.92 - 5.11 ml = 4.81 ml Density of sphere = mass / volume =2557 kg/m3 2mm 1) Measuring the weight of the empty dry cylinder Weight = 26.6 g 2) Calibrating the cylinder by adding water at 20°C 132 Weight of dry cylinder + added water = 36.58 g Weight of water added = 36.58-26.6= 9.98 g Density of water at 20°C = 998 kg/m3 Volume of Water added = 10.0 ml 3 ) Add spheres to the cylinder and not to exceed the calibration mark Weight of dry cylinder + spheres = 41.0 g Weight of Sphere = 14.4 g 4) Add water to fill the cylinder up to the calibration mark Volume of water added = 4.25 ml Sphere volume = total volume - water added = 10.0 - 4.25 ml = 5.75 ml Density of sphere = mass / volume = 2504.4 kg/m3 133 Appendix J Design of Rotating Tank Design of Rotating Tank Design Variables and Constants co := 3 0 0 — angular speed of the tank sec Ri := 0.127m inner radius of the tank H := 0.0508 m height of the tank kg p := 1300— maximum density of the fluid inside the tank m3 Y := p. g specific gravity of the fluid FS := 1.5 Factor of Safety Pressure Distribution inside the Tank po := 103600Pa atmospheric pressure P(r,z) := Po - yz+ 0.5 P r 2 co 2 p r e ssure distribution z:=0..-H v e r t i c a i distance r := 0.. Ri radial distance Maximum Pressure at Ri and -H Pmax:= P(Ri,-H) Pmax= 1.048x 106Pa Part B - Center Tube Material Properties material selected = Plexiglas oy := 620000000Pa Yield Strength xy := 31000000Pa Shear yield strength ka pm := 1182.63— density of material m3 case 1 - assume thin-walled section Radial Stress Ri al(t) :=Pmax- — 2-t Axial stress Ri a2(t) := Pmax— t Maximum Shear Using Mohr Cirlce Ri xmaj(t) := Pmax- — 2-t . ty Tmax< — FS PmaxRi t := FS t = 3.219m Thickness of the tube Wa :=2-7iRit-Hpm Wa = 0.154kg case 2 - assume thick-walled section define: Ri=inner radius and Ro=outer radius Ro X(Ro) := — Ri Radial Stress ( x Ro 2^ l „ „ \ Pmax on\r,A.,RoJ := < Ro 2^ 1 + X2 - 1 V f Axial stress „„t » \ Pmax a0O(r,A,,Roj := x2 - 1 V Maximum Stress Occurs at r=Ri c r M : = ^ . ( l - r ) X~- 1 2; CTeW:=I^i.(i + x2) r - I | or(x) - GQJx) ix) :=tma: -tmax< — FS Solving for X X:= 1.00 x2-x  F S root(f(A.),A.) = 1.026 Ro:=Ri-root(f(A.),A.) Ro= 13.035cm Wb := n-(Ro2-Ri2)-H-pm Wb = 0.163kg 136 Part A and Part B Part a, the top plate's already given (Circular Plexiglass Plate). Part B, the bottom plate will be exactly the same as Part A with the bottom machined to fit the flange and the shaft. Pressure Distribution on the bottom part 1 2 2 P(r) := po + y-H + ~"P' r 0 3 Total hydrostatic force acting on the bottom plate F = 2.919x 104newton Total force Normal Stress over the bottom plate F F:= CT TI-Ri2 5.76 x 105Pa Safety Factor SF= 1.076x 103 Bolt Design assume no alternating stress, hence no fatique N := 8 number of bolts load carried by each bolt Select a grade for bolts Use SAE Grade No. 5 Sy := 420000000Pa Minimum yield strength dr := dr = 4.073x 10 m 137 Appendix K Object Settling Under Gravity Nylon Fiber Settling under Gravity Density of Fluid@26°C kg pf := 998 — 3 m Viscosity of Fluid@26°C kg u:= 0.00105— ms Density of Fiber kg ps := 1140 — 3 m Height of the Tank dist := 0.04445m Settling Factor = K K:=0.8 Diameter of fiber = r Two Intermediate Variable k:= ' p s - p f \ I PS J K-ir-ps-r2 Velocity Function U ( r , t ) : = J U e R ( r ) t - l ) R(r) Distance Function S(r,t):= — R(r) r e R ( r ) i The results for the vertical distance, the velocity and the Reynolds number are given in the following charts. 139 1.5 10 + U(0.00001,t) U(0.00005, t) 1 10 " 4-5 1 0 J 4-Velocity (m/s) Fibre Diameter = 0.00005m Fibre Diameter = 0.00001m Re(0.00001,t) Re(0.00005, t) h 0 2 10" 0.00824 0.00742 0.006591-0.00577 0.00495 0.00412 U 4 1 0 6 1 0 t 8-i° Time(s) 0.0033 0.00247 0.00165 8.24304 10"4 U 0 1 1 - Re -" - R=0.00005m - .' , R= 0.00001m • 1 •< Time (s) _i 1 1 1 1 0 110" 4 2-10"4 3 10"4 4 10"4 5 10~4 6 1 0 - 4 7 I O - 4 8 10~4 9 10"4 0.001 141 Steel Sphere Falling in Spinning Tank (Glycerin) Density of Fluid@26°C kg m Viscosity of Fluid@26°C u := 0.873621 — ms Density of Sphere ps := 7868.6^-m3 Height of the Tank dist := 0.04445m Two Intermediate Variable k:= ^ps - pf^l V ps J R(r) := -9-u 2-psr u ( r > t ) * ( e * w - ' _ i ) R(r) Velocity Function pf-U(r,t)r Re(r,t):=^ Distance Function k S(r,t) := R(r) 143 i—i—r " i — i — i — i — i — i — i — i — i — i — i — i — i — r 0.08 0.06 S(0.0001,t) 0.04 0.02 J I L L _ _ l I I I I I I I I L J L 0 0.15 0.3 0.45 0.6 0.75 0.9 1.05 1.2 1.35 1.5 1.65 1.8 1.95 2.1 2.25 2.4 2.55 2.7 2.85 3 0.0236 0.0213 h 0.0189 r-U(0.001,t) Re(0.001,t) 0.0024 0.002 0.004 0.006 0.008 0.012 0.014 0.016 0.018 0.02 144 Platinum Fiber Settling under Gravity Density of Fluid@26°C pf := 998- — 3 m Viscosity of Fluid@26°C kg u:= 0.00105— ms Density of Fiber kg ps := 21450 — 3 m Height of the Tank dist := 0.04445m diameter of fiber=r Settling Factor=K K:= 0.8 Two Intermediate Variable ps - pf |^ „ k:= | ps ) -48-^ R(r) := K-7i-ps-r" U ( r , t ) : = ^ - . 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' ~ l ( N l C N l c < J l C N 1 l ' N l < : s ' l co s » cn b k cs n k in co N » N N CN CN M co M co rt M ^ C ^ Appendix P Thermomechanical Pulp Experimental Data E E o o o o T in ID s oo cn CN ro T£ i n eg E CT E e E if CO 0) <u IV CD in rr> oo if) O) " "UjJjVcoLLg T ™ O > < 8 T S : < O 5 T - C O C O C J ^ ^ - ^ O T-" CO c d d CL Q 2i K E £ CD «= c a> a> Q x LLI tn cu '•c cu Q O co ~ -o ±; E co a) oj co 3 E I L » p Q. 0 o 1 E d I 1 1 I I I 186 1864.164631 2139.45208 3302.1581 3761.36791 2519.36872 3092.98576 2775.292241 2495.61802 3974.32308 3392.7142 2221.39483 1876.26657 2044.22552 2469.85789 2348.11551 2980.85799 2886.06851 3417.62392 3241.65768 2761.81573| j 0.010482404 I 0.009784796 0.007875971 0.007379554 0.009016898 0.008137932 0.008591098 0.009059703 0.007179124 0.00777015 0.009602629 0.010448544 0.010010105 0.009106826 0.009339923 0.008289578 0.008424608 0.007741781 0.007949128 | 0.008612033| 0.000274635 0.000256358 | 0.000206347 0.000193341 | 0.000236239 0.000213211 | 0.000225083 | 0.000237361 | 0.00018809 | 0.000203575 0.000251585 0.000273748 | 0.000262261 | 0.000238595 0.000244702 0.000217184 0.000220721 | 0.000202832 | | 0.000208264 j | 0.000225632 CO oS CO oi CO ci CO oi CO oi CO oi CO oi CO oi 00 oi CO oi CO oi CO oi co oi CO O ) CO oi CO oi CO oi CO oi 00 oi CO oi 0.05 I 0.05 | 0.05 0.05 | 0.05 0.05 0.05 | 0.05 0.05 | 0.05 0.05 | 0.05 ] 0.05 | 0.05 0.05 0.05 0.05 0.05 0.05 I 0.05 182.06 | 195.04 | 242.31 258.61 | 211.65 234.51 222.14 | 210.65 | | 265.83 | 245.61 | 198.74 | | 182.65 | | 190.65 | | 209.56 | | 204.33 | 230.22 | 226.53 | | 246.51 ! | 240.08 | | 221.6 j 182.06 | 195.04 | 242.31 I 258.61 | 211.65 234.51 | 222.14 | r 210.65 ] | 265.83 | | 245.61 | 198.74 | 182.65 | 190.65 f 209.56 | 204.33 | 230.22 | 226.53 | 246.51 | 240.08 | 221.6 o o o o o o o o o o o o o o o o o o o o CO CM CO CO CO TT CO m CO CO CO t— CO OD CO cn CO o TT TT CN TT CO TT TT TT cn TT CO TT r~ TT CO O J T T OS I E E E E E E E o o o o o o o -C II II II LL £ 3 S2 o Q. ! l II I i c tc I-o c "E c "5 co ? co m \ 9 9 3 LU LU • n co i o ro co m gj 9 § CN £ 9 £ LU + 7 co UJ rt t ^ MJ co N O JG t2 CD rt o CO >S . T- CD O CO ^ ^ ^ o T-1 rt co d q CD Q X U J 1 * E B S 3 •g s I J ) CO CO CO CO c I 453.71952981 447.0244861 358.1895559 455.9901076 653.4066401 521.762076 ( 524.16030871 86.46050507 83.69713905 304.7987401 287.5301057 266.9363861 306.1652701 327.4636657 94.12123871 126.2704216 94.20722694 247.9634038 711.3745053 718.4286048 1079.187708 242.9523881 223.0472911 279.3320636 457.5739133 403.5982927 502.9470929 178.4101894 348.7985077 329.8750308 1427.993423 2559.858229 849.3012517 084.4104661 696.8551518 1586.983732 1249.972347 1624.056342 182.0694642 365.7885818 521.5034347 565.8188552 388.9619747 619.7872599 445.0369873 405.8937604 824.1543045 427.2853785 462.6097483 273.8313923 225.9741301 240.6512517 244.2154512 275.7432296 30308.00997 4214.983086 337.1690597 244.8501472 2265.313603 329.95054 9821.498663 1112.474745 1438.171815 220093.9006 | 1801.3087631 229.0881438 i 375.5694287 | 0.0810143 0.0822687 0.0925877 0.0825574 0.0672315 0.0761004 0.0767784 0.1861871 0.1918587 0.0995885 0.1035563 0.1039184 0.0979169 0.0952585 0.1775272 0.1555151 0.1818572 0.1125783 0.063358 0.063358 0.0519592 0.1094365 0.1151963 0.1039184 0.0806374 0.0863973 0.0779388 0.1324758 0.0914558 0.0952585 0.0460785 0.0346049 0.0584541 0.0548458 0.0657231 0.0432993 0.0490726 0.0432993 0.1303538 0.0901887 i 0.0763135 0.0738671 0.0867199 0.069376 0.0825574 0.0832511 96968S0 0 I 0.0825574 0.0792551 0.104283 0.1157347 0.1125783 0.1129733 0.1041004 0.0100844| 0.0272439 0.0925877 0.1099479 0.0365923 0.09365751 0.0173486 0.0520502 0.0446703 0.0036327 0.040545 0.1129733 0.0869025 0.0021590941 0.002192527 0.002467536 0.00220022 0.001791773 0.002028136 0.0020462051 0.004962035 0.005113187 0.002654112 0.002759858 0.002769508 0.002609563 0.002538715 0.004731242 0.004144601 0.004846639 0.0030003 0.001688541 0.001688541 0.001384754 0.002016571 0.003070074 0.002769508 0.002149052 0.002302556 0.002077131 0.003530586 0.002437369 0.002538715 0.00122803 0.000922248 0.001557848 0.001461685 0.001751572 0.001153962 0.001307823 0.001153962 0.003474032 0.002403602 0.002033817 0.001968618 0.002311155 0.001848924 0.00220022 0.002218709 0.001571586 0.00220022 0.002112211 | 0.002779225 | 0.003084421 0.0030003 0.003010827 | 0.0027743581 | 0.000268757| 0.000726073 0.002467536 I 0.0029302 | 0.000975214 | 0.0024960481 0.000462354 | 0.0013871791 0.0011905 9.68154E-05 I 0.001080557 0.003010827 | 0.002316021 | 147.707259] 150.06945 152.303954 154.155401 146.494242 149.877921 153.2615991 148.664904 152.814698 149.941764 152.942384 142.982878 145.600441 147.388044 147.132672 151.473995 154.538459 155.879161 141.642175 143.046721 144.51511 144.323581 146.813457 149.622549 147.579573 149.43102 151.537838 155.304574 144.706639 148.473375 150.388664 152.048582 143.940523 146.8773 146.303334 147.579573 149.303334 151.027094 153.453128 147.579573 CO CO o CO ci LO 153.133913 145.089697 147.962631 150.452507 139.535357 142.152919 144.451267 144.132052 I 147.7072591 150.133293 151.282466 154.602302 148.218003 | 152.878541| 155.176888 143.3659361 | 146.813457| 150.452507 143.557465! 146.621928' 149.494863: CO CO c\i 144.068209; 146.8773 145.025854 | 140.684531i 0.076356502| 0.077577624 0.078732739 0.079689835 0.075729439 0.077478614 0.0792277891 0.076851551 0.078996766| 0.077511617 0.079062772 0.073914257 0.075267393 0.076191485 0.076059472 0.078303696 0.079887855| 0.080580924 0.073221188 0.073947261 0.074706337 0.074607327 0.075894455 0.077346601 0.076290495 0.077247591 0.0783367 0.080283894 0.074805347 0.076752541 0.07774264 0.078600726 0.074409307 0.075027459 0.077181584 0.076290495 0.077181584 0.078072673 0.079326799 0.076290405 0.077874653 0.079161782 0.075003366 0.076488515 0.077775644 0.072132079 0.073485215 0.074673333 0.074508317 0.076356502 0.077610627 0.078204686 0.079920858 0.076620528 0.079029769 0.080217888 0.074112277 0.075894455 0.077775644 0.074211287 0.075795446 0.077280594 0.073584224 0.074475314 0.075927459 0.074970363! 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S i Csl £ | £ | E £ £ 194 o o co "ci "CT "oi T: "CTDJI§'CT"§"& a CJ a 01 ^ ct DI cc ra ra n x ^  "(S 2 2 o 2 E 2 "2 E E 2 2 2 E o S "S "2 JS '2 2 o O 555555 55cnoo55555555555555 25555555555) ^ n ^ i i ^ ^ ^ ^ ^ ^ ^ ^ ^ i ^ " g C T i § : < ! 5 l § ; " o : T ; "§"s"ra"CT"CT-ra"CT^'a^,"&|&|CT|g;"c: "c a cn g _g: o: .n .g: .cc .ra _ra g: gi g: _ro £ 2 2 E E E 2 2 2 2 2 2 2 E 2 E 2 2 *2 2 o 2 2 2 2 5 "2 2 J J 2 2 2 2 2 o o 2 2 E 2 2 2 2 2 E E E E 5 2 5555555555555553Scoc/55555555555555555co 5555555555555555255555w55w 5555552555S5>55 55to55255555 fifitioo's fi s fi a fi o o 5 £ z 2 o co 55 coco o o o "CTJ "03 "D) "& "CT "C "c ^ ^ ^ ^ ^ ^ ^ 1 j ^ " r a C T ' g : ~ g : ~ g ] " c : ~c g a t t 5'™S.5.cj.fJ.ra.cn.g;.g;.m 5 E 2 2 2 S | cj o E 2 2 | 2 | | | ^ S 35 « ii ifl ii cntocncotfltowffiSffiS coco co «55«HHfficScofficoco S£S|fejSJ§j£J 196 Appendix Q Regression Data (kraft and thermomechanical Pulp) 197 IS- 00 L O , Tf C O •<-00 C O M -K in (B C M f O C O C O o o d 3 Q . •S2 13 I C O c o CO CO 2, CD CD CD CD cu cr CD W ^£ to a: CD 5 C O ° C N C D CT) o g L U C D C D c o CT> o © O CM O CO Cvi in co i -r— L O c o CM 00 " IS-a> O J C O Tf L O c o c o c .9 ro c o z> c o - o cu ac Tf I CO LU L O C O C O o o o ' T — I L U Tf L U Tf CT) 00 L O C O O T - L O s o n 00 o L O r\i c i s L O L O 9 £ § ^ • * L U C D C O m e © m 22 © C M C M O ^ • co L O C O C O o o o L U 5 L U Tl- CT) 00 L O C O O TT T - I O s o n 00 o m c\i d s C D O • L U 2 © 2 C O § ?! 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I s - Is- C O C O T f I s - I s -co oo C D a> L O I s - Is- I s - I s - Is- C M L O T— T— T— T f I s - \— co C 0 T f T f C D C O 0 0 T f T f T f T f T f C D C M L O L O L O T f I s - C O o o T— C D C D c- C M C M C M C N C M C M 0 0 C O co co I s - C M co T f T f L O L O 0 0 0 0 o C D C D cn cn cn L O C D C O co co C O I s - I s - o o m L O L O L O C D C D co C O C O co Is- Is- oo oo oo oo oo oo C D C D o o O o o o o o o o O o o o o o o o o O o o o o o o o o o o O o o o o o o o o o o d d d d d d d d d d d d d d d d d d d _^ T f Is- L O C O C D Cf) C M L O 0 0 T— T f I s - co co C D C D C M L O C O C D C D a> o o o T— T— T— C M C M C M o co C O co T f T f T f o co C O co C D C D C M L O oo 7— T f Is- C O co C O C D C M L O co C D C D a> o o o i — T — T— C M C M C M o co co C O T f T f T f o co C D co C D cn C M L O 0 0 T— T f I s - co co C D C D C M L O C O C D C D C D o o o T— T— T— CM C M C M o co co co T f T f T f o co C O co C D cn C M L O oo T— T f Is- co C O C D C D C M L O 0 0 C D C D C D o o o T— T — T— C M C M C M o co co co T f T f T f o co C D co C O C D C M L O 0 0 i — T f Is- co co C D C D C M L O 0 0 Tf' T f T f L O L O L O C D C D C D i < l< i< C O cb 0 0 C D C D C D C O L O 0 0 I s - 0 0 T f 0 0 0 0 C O T f L O 0 0 0 0 C M 0 0 0 0 \— T f C D I s - o C D C N C D CD cn C O Is- T f Is- C D C O T f C D C O C M T f co o Is- T f co C D o C M o C M o co C M C D C M co o L O L O co o o T f C D o o 0 0 co C M C O i — C O r- I s - co C O C M C D T f oo a> 0 0 •c— C M co C N T f co C D co co L O T f C D C M C D T f co co L O T f T— o 0 0 T f o C O C O o L O O o r— o C D C D 1— C D o T f C M o L O o C O o C D C D C M C D 0 0 T f 0 3 0 0 C O L O I s - T f C D C O 0 0 T f 0 0 T f T f C D o C D T f C M O C D oo C M T f C D T f X— C M C N 0 0 T— C O C O X— C N T f C M d v - L O o C M co O C M 1 L O d 1 d d d d 1 d d d d • d i i d d 1 d 1 d d d 1 d o d C M L O CD T f 0 0 C D T f L O C M co co T f L O 0 0 C D L O C M C D T f T f co T f o co C D C D L O T— C D Is- N . L O r- o C D o C O C D x— C M C O CO 0 0 C N T f 0 0 C O cn O co L O C O T— 0 0 C O 1111 C M L O o cn o C N T f o C M cn 0 0 C M L O T f 0 0 0 0 T f T— C M UJ Is- L O C D C D L O L O cn C N \— cn C M L O T f C D Is- T f T f o L O O C M C O C O C O C O C M L O L O CM C O C D T f C D T— Is- 1— T f L O T f T f C M I s -o o o o o o o o O O o o o o o 1 O O o o o o o o o o o o O O o o o o o C D i O O o o o o o o o o o o O O o o o o o O O o d • d d d d 1 d d d d 1 d I d • d d I d 1 d d d d d • 0 0 T f T f T— L O L O C M L O 0 0 L O T f oo C M C M I s - I s - L O C M 0 0 L O C O C D C D T— C O T f co cn 0 0 o 0 0 C M co C O T— C D T— O C D C N T— O C O CD CD L O I s - C M co C D o co T f T f o o C O C O T— T f C O C D co co C D C O L O cn L O 0 0 C D T f C M C M T f o o T f L O C O m C O T f C M I s - L O cn C M 0 0 C O T f T f T f co I s - C D C D C D C O C N C M C O C D 0 0 C M C O C O i — C O cn C D o C D C 0 C D C M O C M o C N C M C M 7— o C M C M o C M T— r— C M 1— C M i — co C M o O o o o o o p o O p o o o o o O o o O o Q o o o o o d o O d o o o o o O o o O d d d d d d d d d d d d d d d d d L O C D 0 0 cn o C M co T f L O C D Is- oo C D o C M co T— C M C M C M C M C M C M C M C M C M C M co co co co Is- IS- CD CO co CO CD CM CD T f I s - CO CD CO CD CD CD CD CM C 0 IS- T f CM CM LO LO Is- IS- co CD CD CD CD CD CD CD CD LO T f o IS- T f CD 0 0 LO LO CD o LO CD LO LO LO LO LO LO LO IS- O T f X— co CO IS- T f T f O LO T— I s - LO LO LO LO LO LO LO co CD o X — T f LO LO IS- IS- CO T f IS- CD LO I s -CD CD CD CD CD CD CD o o o o CM CM CM CM CM CM CM CM CM CN CM CM CM O O o o o o O O o o o O O o o O O O o O O O o o o o O O o o o o O o o O O O o O d d d d d d d d d d d d d d d d d d d d LO LO CO CD CD IS- IS- IS- oo 0 0 oo co CD CD CD IS- LO co T— T— T f IS- o co co CD CM LO CO t— T f Is- o CO CD X — o CO CD LO LO LO CD CD CD CD IS- IS- I s - oo oo co CD CD CO CO o O T f IS- o CO CD CD CM LO 0 0 T— T f I s - o CO CD o CO CD LO LO LO CO CO CD CD IS- f - I s - 0 0 oo 0 0 CD CD CD co o O T f IS- o co CD CD CM LO 0 0 1— T f IS- o co CD o CD CD L 0 LO LO CD CD CD CD IS- IS- Is- 0 0 0 0 0 0 CD CD CD CO O O T f IS- q co CO CD CM LO 0 0 T— T f I s - o CO T f LO CO CD d d d CM CM c\i oo co 0 0 T f T f T— LO LO x— T— CO LO CD CD co LO r- I s - T f Is- CD CO CO 0 0 0 0 T f CO LO o oo LO CD CM 0 0 co 0 0 Is- CM LO IS- CD co CM CO T— LO CD CO CM LO T f O CO co CD 0 0 CO LO T— O CD x— LO LO I s -LO IS- T f CO CD CD IS- T— CM CM CD T f oo o IS- IS- T f o oo I s -LO CD co o LO 0 0 CO CM CD CO CD co co 0 0 IS- co I s - o CD T f CO T f T f CO CM CO LO T— CD LO Is- 0 0 CD CO CM CM CO T f CM LO CD IS- co o Is- CD o Is- co CD 0 0 0 0 o CM CM CM 0 0 co CO CM CM CD CO T f CM Is- o Is- o T f CD T f CM CM i — CD 0 0 co o d d d T f CM d T f LO o IS- CN CM LO CO T— CD LO IS- co • 1 d 1 d d d 1 d d d • d d d 1 d • d d 1 o d d co CO co CO 0 0 T f T f CM CD co co 0 0 CD T f LO CO CD 0 0 0 0 IS- CD T f o co o CD o co I s - CD t— T f CO o CD CD CM LO T f co LO T f T f CO CD co co CD CD o T f o CO CO IS- T T f CD CD CM T f IS- CO 0 0 O I s - CO 0 0 IS- CM 1— CM IS- co o CD T f CO CD LO co o CD co CD IS- Is- i — T f CD 0 0 CD a> oo CM T f CO T f o o co T f IS- Is- T— o CO T f 0 0 T f T— o o CM T— o O o p o o o o o T— O o o o o o o o T— o O o o o o o o o o O o o o o o o o o o O o o o 1 o o o o o o O o o o o d p o o d d 1 d d 1 d d d d 1 d d d 1 d • d d 1 d • 1 d d T— CM CD IS- 0 0 LO T f CD CD CM CM CD co _^ CD oo LO CD CO 0 0 o CD CD T f i— CM co CO o O 0 0 T— T f T f o CD CD co IS-Is- co co CM LO 0 0 T f o CO CD CD co X — 0 0 CO CM Is- CO CD CD T f CM CM Is- LO o T f CM CM T— O co LO T f T— T f CD CO T— co CD CD T f CM 0 0 T f CM LO I s - CO LO CO CD CD LO o 1— T f CO T— CO co CD co o CD CD CD i — CD CD CO O o T— CM T - o O CM CM CM CM p T— o T— T— CM CN co CN CM CM CM o O o o o O O o o o o o o O o o o O o O o o o O o o o o d o o o O o o o O o d d d d d d d d d d d d d d d d d T f LO CD Is- CO CD o CM co T f LO CO IS- 0 0 CD o CM co co co co co co co T f T f T f T f T f T f T f T f T f T f LO LO LO LO 199 LO CO CD CD CD CD CO 00 CD CD CD CD CO co co CD T— T — T — T — LO CO CO CO CD CO CD CD O O o CD CD CO CD CD CD CD CO CD CD T f T f T f CD CD IS- CM CM CM IS-CO LO LO CO LO LO CD CO IS- CO CO CO LO CN CM CM CM CO CO CO CO co T f T f T f T f CN CM CM o CM CN CM CM CM CM CM CM CN O O O q O O O O o o O O O O O O d O O O o o o O O O d d d d d d d d d d d d _^ CM CM CM CM CO CO co T f T f T f LO LO CM LO 00 T — T f Is- O CO CD CD CM LO 00 t— CM CM CM CO CO CO co T f T f T f CN LO 00 T f Is- O CO CO CD CM LO 00 1— CM CM CM CO CO co CO T f T f T f CN LO 00 T f Is- O CO CO CD CM LO 00 T— T— CM CM CN CO CO co co T f T f T f CN LO 00 T f Is- O CO CO CD CM LO 00 CO CD CD 1^ •< 00 00 00 00 CD CD CD CD CD Is- 00 CD T f T f T f CN LO IS- CD IS-CD T f o o I s - 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CM CM L O C M L O C M L O C O C O in L O CM CM CO 1 -•<- CM 00 i-•r- CM 00 T -t- CM 00 r-<D S L O L O C M C O Tf Is-C M C N Tf Is-C M C M Tf Is-C M C M Tf Is-Is-; Is-: L O L O co co o co co co o co C O C O L O L O Tf Tf Tf C O CT) C M C O C O Tf C O C D C M C O C O Tf C D CT) C M C O C O Tf C D C D C M cd cd d L O L O L O L O L O L O 00 Tf Tf L O C O Tf Tf L O C O Tf Tf L O C O d d L O L O L O L O T- Tf L O L O i- Tf L O L O T - Tf L O L O T - Tf d d co CD C D C D Is- o L O C D Is- O L O C D Is- O L O C D Is- O d C D C D C O Is- Is-C O C O C D C D C D C D C O C D CO C D C D C D C O C O C D C O C O C D C O C O C D C O C O C O Is- CO C N L O Is- Is-C N in Is- Is-C N L O Is- r-C N L O Cvi CM C O C D C O 00 C O T -I S - C O C O Is-co Is-co C M C O C D C D 00 CD Tf S 00 CO Tf Is-C0 CO Tf Is-CT> C D O C O C O C O C D C D s o n - C D C D O C O CT) C D O C O C O C O C D C D Tf Tf C O C D _^ T— Is- 00 C D C D C M C M C O co C O C D C O C D Tf C N Tf in r- L O L O C D C D C O L O co Tf T- Tf m Tf co C M Is- co Is- Is- C D Is- in 00 o CT> 00 C D Is- C O 1— C O co Tf C M C M L O O o T— Tf o C O Tf in co L O in C N CT) Is- oo 00 C D C O T— co 00 ^ — in Tf 00 C M 00 Is- C M 00 T— CO C D Tf C D C D C D t— O Is- 1— o m o co C D C O C D O C O C0 Tf co o C O C D o Is- C M Is- Tf * — C D C D co Tf Tf Is- Is- C O L O 00 C D Tf C D co C O Is- 00 C D C N C D Tf Is- Tf Tf o Is- i— C N C D co C D C O O C D m C D CT> oo C O C D o C M C D C O 00 C O C O co C M Is- L O co C D Tf 00 Is- L O C O C N co C D C D o o C O O Tf C D Is- C D CT) Is- C D O C D T— C O C D C N co C D 00 Is- Is-T— Tf 00 L O Is- Is- L O co Tf C N Is- co C D Is- C D O o Tf 00 Tf T— 00 C O C O O C N Is- C O L O co C O L O Is- o Is- 00 T— Tf Is- Is- C O C D C O co o Tf C M Tf Tf 00 00 00 f- T— L O C D L O Is- Tf 00 C M Is- C O O T— 00 Tf C O Is- L O in t— in 00 C O co co Is- Is- cp Tf Is- C D 5 Tf L O C O 00 L O 00 co o Tf Is- T— o Tf C M o T— C D C D T— T— T— C O C O oo co in Is- C O C O o d C O q d L O co C M L O co L O co C M T— C D o 00 o cp q Tf Tf Tf o Tf C D C M C O C O Is-d d d d d d d i •J-1 d d d T— d d d d cd 1 d C M •r- d d • d C O T— d 1 d d 1 d i d d 00 CM Is- CO CO Tf t- CD LO CD CO o O t-o o o o CD L O L O CD CT) i-CD 00 CO 00 CO CM O O O O O O d d in co Tf o n s CM O CO CD Is-p. 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O CT) C O L O 95 CM C M CT) L O 1 - o o T - C O L O Is- T - T f ro C M T -Is- C M 00 ro T - C N ro o 00 00 00 00 o o o o CM 00 CD CO CD CO Is- L O L O 00 L O L O ro t - ro T - co ro co ro o o o o o CD s i -CO CM CN 0 0 0 o o p o d d T f ro C O T f L O O T f T -CD L O L O C O C D L O 00 sr-1 - T f 00 o Is- CD ro co CO CM CN 1 -o o o o d d Is- CN CO T -LO CO CN CN CO 00 O CM O CN o o ro 10 00 T -Is- T f 00 CO 00 C N O O O O d d CM CD T f L O C O L O ro 00 ro Is-L O 00 L O T -L O 00 CO Is-co 00 Is- 00 O CO CD L O O o o o CM 00 CN L O 00 1 -CM L O 1 - CM Is- O CD CD O O O O O O O O O T - 00 T f C D T - CT) •<- T f L O Is-00 CM L O CD O O O O O Is- 1 -CM CO Is-ro Is-1 - C O Is- ro L O L O 0 o co C O CO T -L O ro C O T f o 00 co co CD CO o o o o d d CO CM T - ro CM L O L O Is-co ro T - ro CO L O o o o o d d C O L O co o co o CM L O CT) 1 -T - ro C O 10 L O L O Is- C O L O O C O 00 00 ro CM 00 CD L O O O o o T f L O co o C M C O 1 - ro ro C M CD O L O CO o o o o o o o o T - 00 ro L O L O C O Is- 1 -o ro CO CN L O CD o o o o d d T f T -T f s i -CO CM T f L O 00 CO 00 CO CD CO o o o o C O CN Is-T f L O v-O Is- Is-co ro C M T - T f Is-C O L O 00 L O Is- C D O O O O O O O O O O O ro co C N C M L O Is-T f C O T — ro co L O o co ro co C M ro O C D L O O O O d p p o C M C N ro Is-C M CO O T f CM CO CN T -Is- CO LO CD CO LO O O 00 CN T f CO 00 CO ID T - CO — s CO ro L O ro •<-L O Is-O O o o O O O O O O T f Is-V - L O L O C O T f CT) t - 00 00 co T — o C M ro CO CO CO LO o o T f L o c o s o o r o o - r - c M c o T f L o c o s c a r o o - r - c N c o T f i o c D s c o r o ^ ^ s P U s T - ^ C M C N C M t N C N C N C M t N C N C N C N C N C M C N C N C N C N C N C N C N C M C N C N C N C N 204 CO CD CD co co co CM CD LO o o o T — T — T — CD CN CN o o o CO CD CD o Tf CO o o o co LO LO o 00 O CD CD CD T— CM CM CM CO LO O o o o 1— CD CM CM O o o co co co o CD CO d d d o o o o o o o d O O o o d d d d d CM CM CM co co co Tf Tf Tf 00 Tf Is- o co CD CD CM CM CM CM co co co CO Tf co T— Tf Is- o co CD CD CM X — CM CM CM CO co co CO Tf CO T — Tf Is- o co CD CD CM T— CM CM CM co co CO CO Tf 00 T— Tf Is- o CO CO CD CM CD l< l< 00 co 00 00 d Is- Is- Is- IS- Is- IS- Is- Is- Is-LO LO Is- CD 00 Is- Tf 00 CD x— oo IS- oo T — Is- T— CD Tf CD Is- Is- CM CM T— Tf T— Is-Tf LO T — CM o Tf Tf Tf Tf CD Tf CM O CD Is- Is- CO CD O Tf CD CM Tf CO Tf CD O CM CD CO CO Tf CD CD T- CM O Tf LO 00 LO Is- CD CO CO CM LO CO CD LO Tf LO co CM d CM d • d d 1 1 d CM CO CD Is- co Is- CD Tf Is- Tf T — T — co LO co 00 x— Tf x— 00 Is- o co o 00 Tf 00 oo CD co CD Is- o Is-T— Tf 1— o o O Tf LO CD o LO CM CO CD CO o 00 CM Tf LO Tf o CO T— T— o CM CN o co o o o o o o o o o o o o o o o o o o d d d • d 1 d 1 d 1 d 1 d d Tf Is- CD CO Is- CO CD CO CO LO CM CD LO 00 o to LO 00 CD O co LO CO Is- 00 T — Tf CO LO Is- o T— CD LO CD CM Is- CO Tf LO o CD Tf Is-o T— LO o co T — CD Tf LO CD CD LO CD CO CD LO CO CD o o O o o o o o O o o O o o o o o O d d d d d d d d d Tf LO CO Is- 00 CD o t— CN LO LO LO LO LO LO CO CO CD CM CN CM CM CM CM CM CN CM LO I s- CO Tf oo 00 LO CO co LO IS- CD CO CD co CO CO CD CD I s-CO CD CD O Tf CO o o CD co CO CO LO O IS- CD o o LO o T— CO Is - Tf CM co LO CD CM LO CO CO co co Tf Tf CO CD CD IS-CD CD co o CO CD o o CD CD O O o o o o o o o o O O o d o o d d o o d d d d d d d LO LO LO LO CO co CO IS- IS- IS-LO CO T— Tf IS- o CO co CD CM Tf Tf LO LO LO CO CD CD CD Is-LO 00 i— Tf IS- o co CD CD CM Tf Tf LO LO LO CD CO CD CD IS-LO 00 T — Tf Is- o CO CD CD CM Tf Tf LO LO LO CD CD CD CD Is-LO CO T— Tf IS- q CO q q CM d d d d d T— CM Is- Is- oo CO CO oo oo oo 00 00 co CD CM CD IS- CO LO Tf CD I s-LO I s- CD CO LO CN o CO Tf o IS- CD O o CO IS- IS- LO Tf IS-co LO 00 co o o o 00 CD Tf 1— 00 CO CM T— CD T— o CD CM CD o Tf CD Is- 00 CO IS-T— CO LO o CO CM 00 co 00 i— LO CD Tf 00 CM CO o CO CM q co T — CO co CM LO CM co T — d 1 1 d d 1 T — ^ d 1 1 1 LO CO CO IS- CO CD CD CD LO co o o co Tf co LO IS- CM o LO co co CO CD CD LO CD Tf co co 00 CO CD CM CM CO CD I s- CO o CO CO 00 00 IS- CD IS- 00 CO 00 LO 00 CD LO T— T— IS- CD o Tf o O o o o o X— CM CM o o O o o q o o o O o o O o d d o o o O d 1 d • d d 1 1 d 1 d d 1 d 1 T— LO CM IS- I s- CD Tf co T— co Is- Is- o IS- CM LO CM LO CD CM o CM CD IS- CM T— CD CM 00 Tf CO CO CO 03 T— CM Is- o CM 00 Tf co 1— Tf o 1— T— CM 00 o CD 00 00 T— CO 1— IS- CD LO CO CO CD LO CO o CO co O O o o o o o q o o q O o o o o o d o o d d d d d d d d d CO Tf LO CD IS- 00 CD o CM CD CD CD CD co CD CD I s- I s- IS-CM CN CM CM CM CM CM CM CM CM I s- CM IS- IS- IS- IS- I s- IS- CM LO T— CD o o o o o Tf o Is- CM o co o o o o o o o I s-Tf CD o o o o o Tf Tf o o CO LO IS- IS- IS- IS- IS- LO Tf CM LO co CD o o o o o o T— Is- CM CD CD o o o o o Is- Is- o IS-o o O O d d d d d o o o o o d o o d d d d d 00 oo oo 00 CD CD CD IS- LO CO T_ LO 00 T— Tf IS- o co CD 00 o CD IS- I s- co CO 00 CD CD CO co o LO 00 T— Tf IS- o co CD o CD IS- IS- co 00 00 CD CD CD co o LO 00 r— Tf IS- o co CD o CD Is- Is- 00 oo oo CD CD q co o i q 00 Tf IS- o co Tf LO CD CM CM cb cb cb Tf Tf CO 00 LO oo oo oo oo 00 00 oo 00 CO Tf I s- CO CD LO CD T— 00 CD CM CO LO Tf CM Is- LO CM o LO co CO 00 CO CO co Tf oo CM LO CO CD CO CM co IS- oo Tf co o IS- o oo co CD CO LO IS- CO o co 00 CD o CM 00 LO LO IS- CD 00 LO 1— 00 o CD LO CD Is- Tf 1— 00 00 1— Tf T— o co CD CD oo 00 LO IS- Tf CD CD CM CM O IS- IS- IS- o o CM I q q CM d T — d d 1 Tf CD CM CM LO IS- 00 co Tf CM LO I s- CD •*— o o 00 T— CM CN 00 CM CD CD Tf co 1 oo Tf Tf 00 o CO 00 LO CD T— LU o IS- 00 LO LO T — LO CM co CD CM CD I s- CN CD CD CM I s- T — IS- LO CD co 00 LO CO CO CM T— CM Tf o 00 T— x— T— O o O o o T — o o o o o o o O o o o o o o o o d d d d d d d d 1 d 1 d d 1 co CD CD CO T— CM CN Tf CM 00 00 IS- CM 00 CD Tf CD CD o IS-co LO IS- oo IS- Tf o oo T— o CM CD LO CD IS- LO CO co CM LO o LO co CD Tf co 00 T— Is- T— CM CD o CM LO CM CO CM 1— co o Tf LO CD CO LO CD I s- CD CD CD CO CO o o O O o o o o o o o q o O o o o o o o o o d d d d d d d d d d d CO Tf LO CO IS- 00 CD o CM co IS- I s- IS- IS- IS- IS- IS- 00 00 CO CO CM CM CM CM CM CM CM CN CM CM CM LO Tf Is- 00 Is- LO LO CD 00 oo CM I s- CD oo co I s - Is- oo o CD Is- o o Is- oo o o Tf o Is-o CO Tf LO LO o o CD oo LO LO o oo Is- Tf LO IO Tf LO CM CM oo oo 00 Tf Is- Is- LO LO CD Is- Is- Is - Is- N - o o Is- Is- Is-o o o o o q q o o o o o o o o d d o o o d d d d d d d d ^ _^ CM CN CM CM CO oo oo Tf CM LO 00 Tf Is- o oo CD o T — T — T — CM CM CM oo oo co CD CM LO 00 Tf Is- o oo CD o T— CM CM CM CO oo CO CD CM LO 00 T— Tf Is- o oo CD o X— T — 1— CM CM CM co oo oo CD CM LO CO Tf Is- o oo CD LO d d d Is-; 1^ Is-; 00 CO d oo oo oo oo 00 00 oo oo co oo I s- LO CD CD r— 00 CM Tf LO CM Is- T— CM C D Tf CD CO oo CD CM CM Is- CO Is- CO 00 Tf o 00 oo CD 00 LO 1— Tf CD 00 CD oo CD T — CD oo CD o oo T— LO Tf LO CN CO CO Is- CM CM Is - CM X— Tf CD oo CD LO LO Is- T— Tf 00 CM Tf o o CO CO o LO iq T — q CD q CO oq LO CM CM X— 1 d 1 T— T— 1 d 1 d 1 d Tf I s- co CD Tf 00 CD co Tf Is-co oo oo T— Tf T— CN LO oo Is-o CD CD CD CM CD LO CM 3: T — co T— CD 00 CD Is- 00 Is- CO LO CD o 00 CD Tf o CM CD T— 00 co CM Tf CN LO o T— oo CO CO CM o CM o T — CM CM o o o o o o q o o o o o o o o o d o o o o o o d 1 d 1 d d d d 1 d 1 d 1 d Tf Is- ,_ Tf CO Is- CO CD CD oo LO 00 CO Is- LO o CM CO o 00 Tf CM CM t— Is - 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co CD co I s- T t T t o T t CO T t oo T t LO CM o CM CM co CD T t I s- 00 CO I s- cn Is- CO 00 T t o T t 00 T— LO cn T f o O) T t CD co t— CM T t oo CM LO ^— Is- CM T t LO T— OO CO cn d • 1 T— 1 d 1 d CO i 1 d co CM CNJ CO CD co CM 00 T t T t CM 00 00 CM CM Is- T— 00 CM o CM OO 00 LO CM cn CM LO CO CD O Is- CO 00 Is-T t CM o LO CM T— T - CD T t CO CO CO o I s- o LO Is- 00 T— T t Is- LO LO CM o CO Is- Is- CO T— Is- CO Is- oo o LO p CM CM o o o CM T t CM CM 1— O o o o p o O o O o o 1 O o o o d o O o O o o q d d 1 d 1 d i d 1 d d 1 d • d T I - CM LO CD CO LO CO 00 LO T t I s- oo CS co T t T t LO CD O) Is- CO CM co o 00 00 o CM CO CD CO o T— CD o cn LO CM T t CO 00 Is- CD T— CM CD CO CO 00 T f T t CO Is- co T t cn T— LO o Is-CD Is- Is- co CD T t CO o o 00 CM T t o LO LO CO Is- CD CO CD Is- LO CO LO o o o o o O O o o O o o d o o o o O O o o o o o d d d d d d d d d d d T f LO CO Is- 00 cn o CM co T t IO cn CO cn cn cn cn o o O o o o CM CM CM CM CM CM co CO CO CO co CO LO LO LO co LO T t oo cn 00 00 oo co CM CM T t co CD CD o o T— CO CO oo 00 00 T t 00 o T - o o cn 00 CM CO o o LO CM LO o o LO LO 00 Is-LO LO LO Is- LO LO oo oo CM Is- LO co CM CM CO CO T t OO o o 1— oo T t co 00 00 00 co co co o o 00 oo oo oo O o o o o o o o o o O o o o o o o o o o d d d d d d d d d d oo co CO CO 00 00 oo I s- LO co 1— LO oo T— T t Is- o CO a> cn o CD cn Is- Is- CO co oo cn oo CO oo o o LO 00 T t Is- o CO OO o CD cn Is- Is- 5 co co OO 00 CO oo o o LO 00 T t Is- o co cn o CD cn I s- I s- CO 00 00 cn O) cp oo o o LO 00 v— T t r— o co T t LO CD oo CM CM cb cb cb T t T t oo oo LO LO cn cn 00 oo oo O) oo cn oo T t CD IO IO co 00 cn oo CD co T t I s-oo CD T t OO CO o LO LO 00 1— co oo I s- 00 CD CO T f CD CM cn I s- CM co Is-CO LO —^ OO x— 00 T— o o CO o CM CM CO L O LO oo co Is- oo oo oo CM 00 00 co Is- cn T t o oo oo oo Is- T— LO o T t T t T t oo o LO CO o oo o T t co oo 00 I s- o CD T— CM LO CM LO T t T t LO CO o CD LO T t CP 00 d d T— 1 1 d 1 1 d 1 CM d 1 1 d 1 T t CO CO CM LO LO CO CO CO T t CO LO CD 00 T t LO o o CD CD cn T t CO LO co Is- oo T t 1 1— oo LO CD Is-o Is- CM CM o CO LU o T t CM oo T t o LO LO 00 T t LO T t CO 00 co CM a> CM Is- CO CD CM T t 00 o o T— LO CM o o CM CM T— CM T t T t o CM CM 1— o o o o o o T t o o o o o o o o o o o CM o d o o o d d d 1 d 1 d 1 d 1 1 d d 1 d 1 d 00 CD CD CM LO LO 00 CM T t LO 00 CO LO CM co oo co oo t— o 00 T— CD o LO o T t T t T t T t 00 Is- LO o CM LO T— o CM o o CM LO oo o LO CM o o oo T t CM CD 0O T— 00 CO cn oo CM 00 T— 00 T— 00 T t CD CD CM CM LO CD CO CO LO CO LO I s- o LO CO CD o o o o o o o o p o o o o o o o o o o o d o o o d d d d d d d d d d d CO Is- 00 oo o T— CM CO T t LO CO I s-o o o o T— r— T— co CO CO co CO co oo oo co co co co T— T t Is- in oo 00 LO 00 oo CD o o o o T t CD o oo co Is- Is- 00 CM o o o o co CO o Is- o 00 CO T— o o o o Is- T t o cn I s- — CM CD o T— ! — 1— T— Is- oo LO CD co m CO oo CO o p p o o 00 o LO CO Is- T t r— CM o o T— o o o T— T— T— T— p r— o o o d o o o o o d d d d d d d d ^ _ CM CM CM CM oo oo co T t T t T t LO LO CM LO oo T— T t Is- o oo CO cn CM LO 00 T— T— 1— CM CM CM CO 0O oo oo T t T t T t CM LO 00 1— T f Is- o co CO O) CM LO oo CM CM CM co oo 00 oo T t T f T t CM LO CO T t Is- o CO CD cn CM LO CO 1— T— CM CM CM co oo oo oo T t T t T t CM LO 00 T— T t Is- o oo CD CJ) CM LO 00 d d d Is-; l < r-; CO CO CO 06 d d d oo oo oo oo oo cn cn cn 00 coo cn OO cn CD T t LO T t CM CM LO CO CD CM T— 00 LO CM cn T t 00 T— CM co o m T— in 00 00 LO I s- I s- o Is- x— T t oo I s- T t 1— CO 00 oo T— CM oo CD CO oo Is- T— 00 00 cn T— T t Is- CD co CM Is- T t CM T t 00 T LO oo oo T t T t T— CD CM CD 00 T— OO T t T— oo LO CD LO co T t T t co LO CO oo Is-Is- 00 00 CO LO CM T t T t LO oo CO oo CO oo T t co p LO I s- o CM CD m LO Is- T t CM T— 1 1 d d d 1 d d d 1 d 1 Is- LO T t oo CO 00 LO co CD CM CM r— LO LO CM o CM co o 00 T t OO o oo LO o o o 00 LO CM 1 T t o CD 00 co LO CM T f 00 T f T— CO 1X1 Is- CM CM Is- 00 CD o CO CO co LO T— X— o o ^— co LO T— T f CM CM CD 00 1— CM cn o oo 00 I s - 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LO T— LO oo Is- CD oo CD T— T— T— CM CM CO LO 00 T f oo oo 1— o T f T f T f Is- T— CO T f CO 00 LO LO T— Is- LO LO LO LO T f CD Is- Is- T— T— co 00 00 oo oo X— T— CD T f LO LO CD Is- CM CD co oo 00 LO CD T f LO LO LO LO LO CD CD CD CO CD Is- Is-T— T— T — ^— T— T— o o o o o o o o o o o o o o o o o o o o o o o o o o d d d d d d d d d d d d d Is- LO co Is- LO oo Is- LO co Is- LO oo Is-co CM CO CD Is- oo CD CM CO CD Is- CO CD CO T — oo CD oo oo CD CO co CO 00 oo CD CO co oo CO CD oo CD LO co CD T— co CD co Q co CD oo CD co CD CM co CD CO CO x— 00 T— 00 T f o CD LO T f o CD LO T f o CM CM T f LO Is- Is- CD o LO Is- CO Is- CD 0O cp d d d T— CM CM CD CO LO CO CD CN O CD O LO CM Is- T -O 00 T f LO O O O O O O d 00 LO T f Is-Is- O - _ CM 00 00 T f LO - • -CD CD CO T f O CM T f O O CO CD LO - ' oo LO CM oo to CM LO O LO CM CM T - CD CD CM T -n O) CO CM CO V t- O T - 00 d> c6 c6 d> LO oo CO LO LO oo T f LO CD 3 T f T f Is- LO LO - r -LO O °H d o CO LO CD Is- T f CO LO CM T f O CO CD Is- 00 CM 00 CM * O O x - T f O CM O O Is-CD CM d CO T f T - CD CD T -C0 CD 00 CN CM CO O O O O O O CD LO CO CM CN CM Is- CO T - T f O O 1- CM O o o o T f T f LO LO CD CM CD CM O rf CO .;. CO CO LU LU LO o CD LO O CO CM CO Is-O O O O O O T f CM o CO CM OO CD Is-O O O d Is- 00 00 CM LO CM CO CM CM CO Is- T -CM CM O O O O d d T f LO LO CO T f Is-Is- T f CM T -CD O T - CM O O o o T f 1-00 o Is- oo O CM CO •<-LO 00 O O O O O O CD CD T f CO CD CD 00 CO CD LO CO CM CM CM O O O o o o d LO CD CO CM O P LO O CO CM CO CO CO i -CD CM O o T f o 0O T -O LO CM CO O O O o d d T - C M C O T f L O t D S O O C D O T - C M C O 214 C O C O I S - T t T f I S - I S - T f C O T— 00 • s t L O C O T— C D C D in I S - L O C D C O in T— 00 C M T t I S - C M C M C M C M C M I S - C D Is- C M C S I T— C D co T f T t C M T t co T— I S - C D C O o T— o co co T— C O T— C M C M T— L O in C D C M C M C M C M C M C M C D o I S - C D m C D C D C O 00 C M T t co C O L O o I S - C O I S - L O 1— C M 00 C M T— T— T t co C D o o CM o o o L O o o o o o C D I S -T— T t C O C O C O C M C M L O co C D C D C M T— oo C M C D oo C O co C D I S - I S - T t L O C D o o C M o C D C D C M o o o o o I S - oo C D O ) C O T f T f L O L O co L O L O I O I S - C O C O C D I S - o C M o C O T t I S - I S - co C O o T— T— T— T t T t in C D C M C M C M C M C M o 1— Is- I S - oo co 00 co 00 oo co oo oo C D C D C D C D o o C M o o o o o o C M C M C M 1— •*— T— T— 1— C M C M C M C M C M C M C M T— \— T— o T— T— C M C M o C M C M C M C M C M C M o o o C M C M C M C M C M o o o o o C M C M o o o o O o o p o o o o o o o o o o p o o o o o o p o o o o o o o p o o o o o o o o o o o o o d o o o o o o o o o o d o o o o o o d d d o o o o o d d d d d o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d m co CM co •<- co 00 C O co 00 o C M O C O CM Is- L O C O C D Is- C O C D C O C O Is- in CD CM CD CO CD O CD „ : C O Is-C O C O co co C O C D C O C O CO T -O C D Is- Is-C M T t T t T t m co Is- co co co C D C O T t M ^ 00 LO CD 00 Is- L O CD CM co ••-co co CD in C O C D co co C O C D C O C D C O T -O CD CM CM in co I S - co co co CD CO w 8 LO T t Is- L O CD CM CD CD co m CD CD C O Is-C O C D C O C O C O C D C O C D 00 T-O CD Is- Is-Is- CD d d in co Is- CO oo co co ^ 00 L O CD co Is- LO CD CM CD T -CD CO C D » • T t o CO Is-; Is- L O co r— L O co Is- m co Is- L O co Is- L O C D Is- co C O C M co C O Is- co C D C M C O C D Is-C D C O C O C O C D C O co 00 C O C O X— C O C D C O C D T t co C O O co C O C D C O C D co co C D C D C D T— d co oo C O d co 00 C O d co oo t— T t d oo o C D d C D in T t o o> in o C M C M C M T t m Is- Is- C D T— L O Is-C M C O 00 C M T t oo d d d 00 d od d d d T— T— _^ CD co co CD CO Is- 00 CO CM Is- Is- CD CM CD CM CD CM CM LO CO 00 LO 00 CO CD Is- T t CM Is- CO 03 Is- CO CD Is- CD Is-Is- o in Is- T t CD co CD 00 LO Is- co CM CD CM Is- CM CM 00 LO LO CO oo LO o CD LO CM o Is- 00 Is- C0 Is- T t 00 00 oo CM o \ — co 00 00 Is- T t Is- CD o CD T f Is- CM T t oo Is- T t LO CD LO CD CD Is- Is- o 00 o CO CD co CM 00 CM o Is- Is- LO CM CM cn CD CO CD o CM 00 o CO Is- Is- T— oo LO oo CM oo T— 00 T t LO CD CO r— oo o T— o T t CD LO Is- CD T t co oo CD CM CD T t CD O LO co 00 LO T t o oo T t Is- CM 00 m CO o co CD LO CO oo Is- Is- oo 00 T— co T— CD CD oo co CO CD T t T t Is— T t CO CD CD o CD o CO co r-. 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CO Is- Is- I s - CO T t T t T— CD LO CO I s - Is- T— CO CM CO 00 CO T t CO CM CO 00 CM CO 00 CD T t T t CN CM CD i n CD T t 00 Is- 00 00 LO T— I s - 00 00 CO I s - o LO CD •c— CM o T— LO co 00 CD CO CD CD CO i n CM T t o CD T t CO i n CO 00 o o CO i n o CM CN i n X— CD I s - CO o 00 Is- T t Is- CD CM Is- o I s - T— CO o CD CM I s - IO T t CD o i n CM CO T t CO T— T— I s - 0 0 CO o Is- CD oo Is- 00 CO T t I s - 00 CD CM CD CM I s - CM t— T— CM 00 LO T t o CD CD C0 CD o T t LO CM I s - o T— I s - I s - I s -o CM I s - x— CM I s - T f o 00 T— CD o CO CO CM co T t CM 00 CD CD CD CM T t CD co co CD oo o T t 00 T t T t T— CD CO oo T t T t T t T— CN I s - T - LO T t o CD T t CD o CM CM CD CM CO T— CD 1— CM CD CM CD i n LO i n T t CM oo co 00 CM I s - 00 co co LO LO T t LO LO o T t LO I s - CO i n LO LO LO i n CD i n LO o I s - LO Is- i n T t i n LO T t o LO o LO LO CO LO CD i n LO IO i n LO LO i n o o o o o o o o o o o o o o o o o o o o o o o o o p o o o o o o o o o o o o o o o o o Q o o o o o o o o o o o o d o o o o o o o o d o d o o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d T t LO CD I s - 00 CD o CM 00 T t LO CD I s - 00 CD o T— CM oo T t LO CO I s - 00 CD o T— CM 0 0 T t i n CD I s - 00 CD o T— T— CM CM CM CM CM CM CM CM CN CN 00 00 00 oo 00 CO 00 00 00 00 T t T t T t T t T t T t T t T t T t T t m CM CM CM CM CN CM CM CM CM CM CM CM CM CM CN CM CM CM CM CM CM CM CM CM CM CM CM CN CM CM CM CM CM CM CM CN CM 220 I s - to L O C O T— C O C O C O C O C O C D C O T— oo co I s - C M C M I s - L O C M C D T— T f T— cn I s - I s - C O T f T f 00 00 00 T f T f C O C O C O C D T f C O L O C M L O C O 00 C O 00 C M T— C D 00 T f T t I s - o 00 L O T— C M L O T t co oo C M I s - I s - L O 00 00 o o o T— L O co T f oo o T — COO o O o L O L O L O L O o C O C M L O C O C D I s - T — C D T T I s - I s - o co C D r— o L O L O o T f T f T f T f T f 00 T f to o C O C M Is- C O o C O I S - Is- Is- Is- I S - oo I s - 00 oo T T I s - I s - 00 C D L O T — C O oo C M oo C M co co oo 1— T— 00 00 00 C O C O o i L O I s - C D T t L O L O r- cn T— T — i — 1— L O L O I S - o I O L O C M co C D L O co T— I s - 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cn o o o ^ o o o o Is- LO Tf CO I s -00 CO T - O CM LO CO 00 00 00 00 O T - o CO CO LO O LO O Is-co o o o o oo o o o d d d LO T - CO Tf ^ r - Is- Is-T - 05 LO CO i- coo coo o T - O O T -O O o o o d d d OO LO Is- CO C D cn oo cn to o 00 o r - O O O d d cn T -Is- Is-V - LO CD 00 LO Tf 00 Is-cn o o T -o o d d 00 T -00 Is-Tf co 00 CM Tf 00 T f 1-cn cn o o o o d d Is- co Tf CM LO 00 co Is-O i-LO CD 00 OO o o o o I s - LO CO Is-Tf O 00 Tf CO Is-Tf CD 00 00 O o o o 00 00 CM T -cn L O L O CD L O • < -CD CD cn cn o o o o Tf CM CO CO LO Is-co cn CD Is-CD O oo o O T -o o O O O O O O O O Tf Tf CD CM Tf OO 00 CD CM COO 00 1-00 1-O T " O O d d 00 CO 00 O LO 00 CO 00 o o o o d d T - cn cn L O T- Is-oo co 00 00 Tf CD o cn i- o o o ci d c6 c6 CO 0O CO LO Is- 00 CM LO Tf LO CO Is-T- Tf Tf 00 T - 00 C O 00 T - T - O CO o •<- O CO i - O 1- O o « o o CO v-O CM 00 00 O0 LO O 00 COO 00 cn cn O o o o d d Tf I s -LO CD Is- 00 00 o •<— CM 00 Tf LO CO Is- 00 Is- Is- Is- Is- Is- 00 00 CO 00 00 00 00 00 co 00 co co oo co oo oo 0O oo co oo oo oo co 0 v l C 0 T f L 0 C 0 i s - 0 0 C 0 0 O T - i r g c 0 T f L 0 C D r s - 0 0 C 0 0 O ' r - C M C 0 c n r o c D O T r o c D O ) C D o o o o o o o o o o r - T - T - T -C O C O C O O O O O C O O O C O T f T f T f T f T f T f T f T f T f T f T f T f T f T f 224 CO CO CD CO m IO CO M S CD CD O O 01 CO CD 00 O T t - s t CM CO LO LO T t 00 00 T t rs- o o co o o oo i n i - i - s t - » - oo o o o o o o o r; d d 1Z, d d S T— T— T f T t T t T t C M T t C O T t C M C M L O T t T t T t L O L O Is- 00 L O C D 0) C D C D C M C D C D C D L O C M C M T— T— ^ — o o T— C O C M T f co co co o T t T t L O oo Is- T t co C O C M o T— T— L O Is- L O L O 00 L O m o o o L O L O Is- Is- C O OO T t T— C D C O \ — C O co L O 00 C D co C M C D o C D 00 CO C M T— C O C D o coo cn o o o T t T t T t T t T t T t C D L O L O L O o L O L O L O o L O L O C O o C D oo C O C D T t T t Is- | s - T— o o L O L O T— C M r s . 00 C O ^ — T— co oo oo L O oo Is- OO r- T t C O X— T— T— o co L O L O C D Is- Is-! — 1— •*— o o C M C M C O co T t L O T— T— L O co C O oo oo T— C M co C M T t C D oo co T— C O T t T t C O 00 00 o o o T— 1— T— i — o o x— T— X— T— T— C M C M C M T— C M C M C M C M co T— co co T— co co d \ — T— T— T— T— T— p o T— o T— 1— o o o o o O o o o o o o o o o O o o d O O O O o d c o d o o d d d d d d d d d d d d d d d d d d d d d d d d d CO Is- LO CO Is- LO CO r s - i o CO Is- i o co Is- LO co Is- LO CO | s - LO co Is- LO co | s - LO co Is- m CO Is- uo CO r s . LO co t s . co co CM co CD Is- CO co CM CO co | s - CO CD CM CO co h- CO co CM co CD Is- CO co CM co CD t s . co co CM CO CD r s - co CD CO CD CO CO CD 00 co CO co CO co co CO CO co CO 00 CO CD T— co CD co co CO CO co CD co CO CD T— co CD CO co CD co CD m co CD co CD 00 CO CO T t co CD o co CD CO co CD CO CO CD CD co CO LO CO CD T— co CO CO co CD T t co CD 00 LO T t T— T t LO d co 00 o | s -cn ( s . r^  00 00 to oo T— T t o Is-! oo oo o CM coo CM d 00 00 LO T t T t LO d 00 oo o Is-CD | s -d co oo IO CD T t o d CD 00 o CM r— CD CM d oo co LO T t T t LO oo oo o Is-CD Is-CM CD co LO 00 T— T t o CM CD oo o CM CD CM d oo oo LO T t T t LO cq | s - cn CO CD o CM CD oo CM T t oo LO r s . cq Is- cn co CO o CM p ro d d d d Is-; r^  d d d co d d d d d d T - v - T— 1— CM CM d d d d 00 00 00 00 00 oo 00 CO oo co 00 oo oo CD OO CD oo oo oo CD CD oo oo CD CD CD T t OO O CO 00 o Is-C M T t CD L O T t r s - r s . T t C D 00 C M 00 C D T t C O i n 00 C D Is- to | s - CO C D C D ro I s- C O 00 C D 00 C M Is- co 00 L O C D ro T— T t T t | s - C M C M C D ,— N . C M co | s - r— C O C M C O o i n i n C M C D o C D T t L O T— I s- co oo C O o C M 00 L O Is- o o uo T— uo co o uo T t C M 00 T — O C M co 00 uo T— o o | s - o C D T t Is- C M co 00 C O 1— C D o T— co 00 00 ro o C M 00 C M uo oo T t T— 00 00 T t r s . O i s - O C D C M T t T t C M | s - C D 1— C M C O C D T— Is- C M C D T t T t C D C D 00 uo co L O C D | s - T— C M o oo uo C D ro ro \ — Is- 00 C O C O C D co T t T t o T t L O C D T t C O C O C M C D T— T— 00 oo L O to O ro Is- C O T t o oo C D T t T m T t Is- — | s - o C D uo C M C D T t C D co C D oo O C D C O C M C D C O O C M C M Is- O co T— 00 00 Is- | s - C M o L O i s . T— C D o T t co co T— o T— T— t— 00 T t C M I s . O O C M L O to L O C M T t L O C M C D co co UO co | s - Is- C O T— C D co C M C D T t ro 1— C M o Is- T t C D o T— ro C D T t T t T t I s . T— C M C M 00 co T t T t C M 00 C O r- C O C M C O C D oo co T t C M r s . o L O C O C D T— C D oo co T— T t T t C D o o o T— d 00 C O o C M C D L O C O C M d 1 C M T— T— o 1— T t O d T t C D O T t d oo T t p o 00 T t C O Is- 00 T t cq o T t d T™ d d d d d d T - 1 d d d d d o T— • T— d d d 1 d d d T ~ d d d d d d C s i 1 1 C M 1 d 1 CO CD CD 00 CO CO i - CM CM O CD O 00 | s -CO 00 T t CM CD 1- CN CD CM O O O O CM O o o o o 00 CD CO 00 T - CD CN UO o o o o CO rs-m o r-. CM CD CM T t CD CM CO o o CD - s t I s- o O 00 T - T t CO CO LO uo T - CO co co LO CM CD CM CD O T t 00 CD t -O T -o o o o LO Is-00 oo I— 00 o o o o CO CM 00 CD CO CM ~ L O L O co CD m 00 L O L O oo 00 T-00 T t oo ro i s . CM CO o o o o ^ , CM CO 00 CD CO ro CD CO T t D IS - T -00 CD T t T -CD 00 T t CO CO LO 00 CM CM CM CO o o T f CM CM o o o o d d oo in v - O 2 lil £; °> f5 CM i2 p 00 d ^ 00 CD LO Is . O T t CD O C D | s -00 CM UO T t CM CD 00 CD CN T t O T -o o d d o o o o d d CD 1-00 T -00 CO CO CO CD LO CM O CM LO N-CM CD 00 o o o o o o o d d | s -o LO 0O T t ro o C M T t T - CO cn cn T t co co co CO O T -l O r - O O O O O o o o o T t CM CD CM T t 00 T— 00 T — CO | s - 00 LO |s- CD LO CD Is. UO CN o CD CM CD CM CD T t o 00 CD 00 CM T t O CM CO LO CO CO o uo CO T t ro CM CO 00 T t |s-O CO I s - 00 CD |s- 00 CN CO oo 00 T t T t o — T— 00 CO Is. CD LO x— oo CM 00 CO oo T t 1— oo o CM 00 CM co O T - |s- | s - LO o CD o LO |s- oo o T t r- CD CM oo oo CD CD 00 00 CD oo o 00 oo CD 00 CD CD o o o o o o o o 1— o T— o o o o o o T— o o O o o o o o o o o o o o o o o d d d d d d d d d d d d d d d d d T - | S -CD CD | s - CD CD T t T - CD o o T - O ° . d o CO to T- CD ro CM CO LO CD 00 CD | s - CD Is- T— Is- T t 00 T— T— LO T— oo o O uo o | s - 1— CD CO LO CM T— CD o CD LO LO T— T t oo T t | s - T— CD 00 | s - o | s - CM T t T t o 00 00 CD CD 00 T— CN CN o oo CM T t LO CD | s - T t co o T t o o T t r— o oo Is- CO CD CD CO T t CO CO oo oo oo CM 00 | s - CO oo T— CD CD o CO CN CD o oo ro | s - O | s - T— CD CD CD CD CD CD CD 00 CM ro Is- Is- T t CD T— oo 00 CD 00 CD T— CD CD CD CD oo o o CD o 00 00 00 o o oo o o O O O T— o o o O o T— T— o T— o o o 1— T— o o o O O o o o o O O o o o o o o o o o o o d d d d d d d d d d d d d d d d d d d d d T t T t CO Is- co CD o T— CM 00 T t in CD Is- 00 00 o t— CM CO T t LO CD Is- 00 CD O T - CM CO T t LO CO | s - 00 CD O T— CN CM CM CM CM CM CM CM CM CM oo ro 00 CO CO 00 00 00 CO CO T t T t T t T t T t T t T t T t T t T t LO T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t T t 225 Is- Is- Is- Is- Is- 00 00 T f 00 00 00 Is- Is-00 CD CD CD CT> i — T — LO CM CM co LO CD CM T f T f T f T f cn CJ) T f O 1— co Is- Is-oo 00 00 00 00 00 00 T f CD CD CD CM T— co cn CD CJ) CJ) T— ^— 1— IO Is- CO oo in cn o O o o T f T f o T f LO Is- T f T— CO T f T f T f T f T f T f d T— T f T f 1— p \ — T— T— O CD o o o o o o o d O o d d d d d d d d d d r s - m c o i s - m c o r s - m c o i s - i n c o i s -C O i s - C O C D C N C O C O I s - C O C D C M C O C O CO 00 CO CO •»— C O C D C O C O C D C D C O C D C D C D c O C D C O C O C D C D c O C D i n c O C D T - ^ C O i - m l O f ^ C O T - j i B r -0 ) S w t S o f f l S i o t S o o N 0 , 0 ) 0 C " c \ C N 0 ) M - m 0 ' N S T f to r L O ^ T ° ? ^ C D T f T f ' d iriiri d d d d CD CD CD CD O) CD CD C D C D CO T f T— co LO 00 co CD CM T f CM T f CO CO Is- T f CM CD CO Is- o T f Is- co CD o LO T— 00 x— CM CD CO co 00 o 1— Is- oo oo CM 00 m T f CO CD CM Is- T f CD T f CD CO CD T f Is-o CO CO 00 Is- LO o CD T— CD CM o T— co CO CO T— CD T f CD CD oo o oo LO in CD CM in T— o CO Is- r- 00 CD CM o T— T f T f LO i— O • — T— s— T— T— O CM CM CM CM CM o d o o o o o d O o o o o d d d d d d d d d d d l O C O r s - L O C O I s - L O C O I s - L O C O r s - L O C O h - C O C O C M c O C O f — C O C O C N C O C O r — CO 0 0 c O C O - \ — C O C O C O C O C D C D C O C D O O C O T - C O t O O O c O C O T f c O C O O C O C O C D C O I ^ C O i - f / C O T - m C O ' r - 0 ; c O ^ ( 5 J C O K L O T f & O C D S L O T f S o C D S L O C D O C N C N T f L O I s - l — CD c o p P CM P °°. Oj oq s s d d d d d d d CD CD C D C D CDCD CDCD CD co LO Is- co CM 00 T f CD CD Is- T— CO T— CO CD CO T f T— CM T f T— T— LO CD Is-CO T f 00 00 T f o CD Is- 00 00 LO O co CD O CD T f o T— CO o CD Is- Is- CD T f LO o Is- CD ^ — CO T f o 00 T— Is- Is- CO CD T— O CO CO T— CO CM co CM T f Is- o CD co CM 00 CM T f T f CD oo co LO T f Is-CO CO °? CM Is- T f CD CO T f T f CO CD T -CM Is- d 00 CM T ~ T f T f CO LO in 00 T f d • d d r— Tf" LO T— d d I I I I CM \ — CD CO Is- oo CD CD Is- T LO Tf CO T— CM LO o co 00 CM T— CD CO Tf r- o o CD CD CM T— Tf Tf o LO LO LO T— CD CO CO CM O oo CO CM CD CD CD Tf CM LO CO 00 CO CM CO T— CD CM LO CO CM CM 00 CD CD CM CD CO Tf CM Tf 00 CD O T— o Tf CD CD Is- O 1— CD CO CM 00 Is- Tf CD CD T— CD O CM LO Tf in CD CO LO T— 00 O O CD LO CO T— Is- CM LO CD LO Tf co Is- T— d d  d CM d i - 1 d d d d X— T— I I I I I CD CM CM T f co co LO T f T f CO 00 LO T f T f CO CO CD co CO CD LO CM T f T— •*— Is- 1— 00 T — LO CD CD LO CD Is- 00 LO T — Is- CD CD T— CD CM CN CO o Is- 00 Is- 1— o LO LO ^— CN LO CD Is- Is- T— o 00 00 CO CD T f CM Is- o CD CD CN T— LO CO Is-CO T— CM CN O CO T— co T f T f T— CM co o O O O O o 1— o T— o o o o o o O O O o o o o o o o o d d d d d d d d d d d d d I I I I I Is- CM CM Is- CD T f CO CD CM T f LO CO CO co CD LO CD CO in LO o Is- CD CO LO o 5 CN CM CO CN CD CO CO CD 00 00 T f CD LO o Is- X— Is- o Is- CO CO LO Is- CD O o CN T— CM LO CO CD T f in o o T f LO CD Is- CO T— LO in 00 CD o Is- CD o CD CD 00 o o CD CD CD o T— CD CD T— O O o T— X— o o o T— o o T— o O o o o o o o o o o o o d d d d d d d d d d d d d co CM CM CD 1— LO CD CD CD CD Is- CD LO co CO co Is- CM co o CM T— in CD co CO co CO Is- CM CD CM Is- Tf CN •(— Tf CD T— CO CM CD LO Is- o 00 00 CD CO CD Tf Tf CM CD CD t— CM Tf Is- Is- CM CD LO Tf T— CM CD LO Is- CD CD CM o CD o LO T— T— T— o CO CD T— LO T— Is- co CD T— LO o T— v - T— CM CO 1— o o o o o o o o o o o o o o o o o o o o o d o o o o o o d d d d d d d d d d d d d CM CD CO CD CD LO _^ CO T f CO CD CO CO LO CO CD T f CM o CD Is- Is- o CD o CO Is- LO T— co CD CO Is- Is- co T f Is- co CO m CM T f LO o Is- Is- CM co in CM LO T f T— T f 00 T f CO T f Is- 1— CO CO CN T f Is- CD o CO CD CM Is- CO 00 CO CD T f CD CM CD CN Is- o 00 o o CD o CD CO CD 00 CD CD o CD T— o T— o T— o o o o o T— o o o o o o o o o o o o o o o d d d d d d d d d d d d d d T f LO CO Is- 00 CD o T— LO LO LO LO in LO CD CO T f T f T f T f T f T f T f T f CN CO T f LO CO Is- CO CD o CD CD CO CD CD CD CO CO Is-T f T f T f T f T f T f T f T f T f T— CM CO T f LO CO Is- 00 CD o Is- Is- Is- Is- Is- Is- Is- Is- Is- 00 T f T f T f T f T f T f T f T f T f T f • o Q. "ro 3 Lo O) c o ro « c 0 4» • • • • • • • • • • • • • • O -e-c o re c 0> 4R> • • CM O in o o o o —•—> mn mm » • »—•-LO o in o o o o o o S | B n p | S 3 y 227 • LO i t • • • LO • O CL 15 •v CO a> fc ? E c 3 U re • • E E c re 3 •*-< O re • • • «tr«r • • • • „ • • ^ ^^^^^^ • • • • «*| * • • I o 4» • CM O LO O O O O LO O o LO o o o o sienpjsey 228 o I s -229 230 231 232 

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