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The use of mixing-sensitive chemical reactions to characterize mixing in the liquid phase of fibre suspensions Mmbaga, Joseph Philemon 1999

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The Use of Mixing-Sensitive Chemical Reactions to Characterize Mixing in The Liquid Phase of Fibre Suspensions By Joseph Philemon Mmbaga BSc. Eng. (Hons.), University of Dar es Salaam, 1986 MSc. (Chem Eng.), New Jersey Institute of Technology, 1989 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In THE FACULTY OF GRADUATE STUDIES (Chemical and Bio-Resource Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1999 © Joseph Philemon Mmbaga, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Chemical and Bio-Resource Engineering The University of British Columbia Vancouver, Canada Date: 14"" September 1999 11 ABSTRACT Local energy dissipation rates in the liquid phase of fibre suspensions have been determined in a medium (Save = 10 - 200W/kg) and high-intensity mixer (Save= 1000 -5000W/kg) by using mixing-sensitive chemical reactions and the Engulfment model of micromixing. The presence of fibre was found to significantly reduce the amount of energy dissipated at the smallest scales, with the reduction mainly dependent on fibre concentration (Cv) and fibre aspect ratio (L/d). Local energy dissipation in the presence of fibres (s) was related to energy dissipation without fibres (SQ) through an exponential function, s = So exp(-aCv) where a is the damping factor that was found to be 52 ± 6 for FBK and 63 + 7 for polyethylene fibres. For spherical particles, the ratio of particle diameter (dp) to the length scale of energy containing eddies (le) was found useful in delienating particle influence on turbulence. Particles with dp/le < 0.1 did not influence local energy dissipation for mass concentration up to Cm = 0.1, whereas, particles with dp/le > 0.1 reduced the local dissipation rates at all concentrations tested. The presence of gas was found to reduce the overall power consumption as well as the local energy dissipation. This was attributed to the diminished ability to transfer momentum from the rotor to suspension as a result of lower density and viscosity around the rotor tip caused by accumulation of gas. Flow visualization with the aid of high-speed video revealed three different types of flow pattern which had direct impact on the overall power consumption. The flow field was also well predicted using computational fluid dynamics (CFD) with the k-e turbulence model. Profiles of dimensionless energy dissipation from measured product distribution were found to reflect the flow pattern. I l l The reduction in energy dissipation rates in low and medium consistency (MC) suspensions has been accounted by a fibre-fibre friction model based on statistical geometry and fluid mechanics. Also, a unified approach based on the apparent suspension viscosity has been shown to correctly predict the change in local energy dissipation rates. IV TABLE OF CONTENTS ABSTRACT ii LIST OF FIGURES xiv LIST OF TABLES xxiv ACKNOWLEDGEMENTS xxix DEDICATION XXX 1 INTRODUCTION 1 2 LITERATURE REVIEW 5 2.1 INTRODUCTION 5 2.2 TURBULENCE AND MIXING 6 2.2.1 The Turbulent Flow Field in Agitated Vessels 6 2.2.2 Energy Dissipation in Turbulent Flows 8 2.2.2.1 Spectra for Turbulent Energy and its Dissipation 9 2.2.2.2 Spectra for Concentration Fluctuations 11 2.2.3 The Mixing Process 13 2.2.3.1 Macromixing 13 2.2.3.2 Micromixing 13 2.2.4 The Turbulent Mixer 15 2.2.4.1 Kinetics of Mixing 16 2.3 MODELING OF TURBULENT MIXING AND REACTION 20 2.3.1 Background 20 2.3.2 A Statistical Approach 21 V 2.3.3 A Deterministic Approach 22 2.3.3.1 Multi-Environment Models 22 2.3.3.2 Coalescence-Redispersion Models 23 2.3.3.3 Cell Balance (Network of Zones) Models 23 2.3.3.4 Slab Diffusion Models 23 2.3.3.5 Interaction by Exchange with Mean (lEM) Model 24 2.3.3.6 EDD Model and its Simplification - The Engulfment Model 25 2.3.4 Mixing-Sensitive Chemical Reactions 26 2.3.4.1 Micromixing Test Reactions 27 2.3.4.2 Bourne Reactions 30 2.3.4.3 Extended Bourne Reaction 30 2.3.5 Estimation of Energy Dissipation Rates from Micromixing Experiments.... 32 2.4 TURBULENCE AND MIXING PULP FIBRE SUSPENSIONS 35 2.4.1 Introduction 35 2.4.2 Pulp Suspension Characteristics 35 2.4.2.1 Yield Stress 39 2.4.3 Pulp Suspension Flow 40 2.4.3.1 Pulp Flow in Pipes 40 2.4.3.2 Flow Regimes in Rotary Devices 43 2.4.3.3 Pulp Fluidization 45 2.4.3.4 Apparent Viscosity 47 2.4.4 Turbulence Parameters in Fibre Suspensions 49 2.4.4.1 Drag Reduction in Fibre Suspensions 49 2.4.4.2 Flocculation in Pulp Suspensions 52 VI 2.4.4.3 Turbulent Diffosivity in Pulp Suspensions 53 2.4.5 Mixing Pulp Suspensions 54 2.4.5.1 Pulp Mixers 56 2.4.5.2 Mixer Characterization by Power Dissipation 58 2.4.5.3 Mixing Assessment in Bleaching Operations 58 2.5 TURBULENCE MODULATION IN OTHER DISPERSED SYSTEMS 59 2.6 SCOPE OF WORK 63 EXPERIMENTAL 64 3.1 MIXERS 64 3.1.1 The Medium-Intensity Mixer 64 3.1.2 High-Shear Mixer 70 3.2 MATERL\LS 73 3.2.1 Fibres 73 3.2.2 Other Dispersed Systems....: 74 3.2.3 Gas as Dispersed Phase 74 3.2.4 Source of Chemical Reagents and Analysis 75 3.2.4.1 Chemical Reagents 75 3.2.4.2 Sampling and Analysis of Reactant and Dye Products 75 3.3 MiCROMixiNG TEST PROCEDURES 76 3.3.1 Medium-Intensity Reactor 76 3.3.2 Critical Feed Time Analysis 77 3.3.3 Baseline Characterization of Micromixing 78 3.3.4 High-Shear Mixer with Flow Loop 78 V l l 3.4 OVERALL ENERGY DISSIPATION 80 3.5 FLOW VISUALIZATION 81 3.5.1 Bulk Flow Characteristics 81 3.5.2 Flow and Reaction Zone VisuaHzation 81 3.5.3 CFD Analysis 81 4 ADSORPTION OF REACTANT AND PRODUCT DYES ON THE DISPERSED PHASE 82 4.1 INTRODUCTION 82 4.2 ADSORPTION PHENOMENA 83 4.3 ADSORPTION TESTS 83 4.4 RESULTS AND DISCUSSION 84 4.4.1 Adsorption of 1-naphthol and 2-naphthol on FBK Fibre 84 4.4.2 Diffiisional Limitations 89 4.4.3 Adsorption of Product Dyes on FBK Fibres 90 4.4.3.1 loniclnteractions 90 4.4.3.2 London Forces (van der Waal) 92 4.4.3.3 Hydrophobic Interactions 92 4.4.3.4 Accesibility of Cellulose 92 4.4.4 Correlations to Account for the Adsorption of Dyes on FBK Fibre 93 4.4.4.1 Composite Correlation 94 4.4.5 Other Fibres 94 4.4.5.1 Non-adsorbing Fibres (Polyethylene) 96 4.4.5.2 Nylon Fibres 97 Vl l l 4.4.6 Test Accuracy 98 4.5 SUMMARY 99 MACROMIXING IN THE MEDIUM-INTENSITY MIXER.... 101 5.1 INTRODUCTION 101 5.2 OVERALL ENERGY DISSIPATION 101 5.2.1 Water Systems 101 5.2.2 Fibre Suspensions 102 5.2.3 Fibre Suspensions with Gas 105 5.2.4 Other Dispersed Phases 107 5.2.4.1 Glass Beads 107 5.2.4.2 Polyethylene Beads and Polyethylene Fibres 112 5.2.4.3 Polyethylene Beads and Glass Beads 112 5.3 FLOW PATTERN 114 5.3.1 Visual Observations 114 5.3.2 C F D Analysis 121 5.4 SUMMARY 124 MICROMIXING IN THE MEDIUM-INTENSITY MIXER 127 6.1 INTRODUCTION 127 6.2 DETERMINATION OF CRITICAL FEED TIME 127 6.3 TEST REPRODUCIBILITY 129 6.4 REACTION ZONE VISUALIZATION 129 6.5 SINGLE POINT TESTS 133 IX 6.5.1 Tests with Water Alone (baseline characteristics) 134 6.6 PROFILES OF DIMENSIONLESS ENERGY DISSIPATION RATE 137 6.6.1 Construction of Energy Dissipation Profiles 137 6.6.2 Profiles in Axial Planes 140 6.6.3 Profiles in Radial Plane 144 6.6.4 Energy Dissipation in the Rotor Swept Area 154 6.7 SUMMARY ; 158 7 MICROMIXING IN FIBRE SUSPENSIONS: THE MEDIUM-INTENSITY MIXER 160 7.1 INTRODUCTION 160 7.2 SINGLE POINT ANALYSIS 160 7.2.1 Effect of Fibre Concentration 160 7.2.2 Effect of L/d Ratio 162 7.2.3 Effect of Fibre Flexibility 163 7.2.4 Relative Energy Dissipation in the Presence of Fibres 166 7.2.5 Effect of Gas Void Fraction 168 7.2.5.1 Pulp Suspensions with Gas 169 7.3 PROFILES OF ENERGY DISSIPATION RATES 172 7.3.1 Fibre Suspensions (fibre + water) 172 7.3.2 Gas-Liquid Suspensions 175 7.3.3 Fibre Suspensions with Gas 176 7.4 SUMMARY 180 X 8 MEDIUM-INTENSITY MIXER WITH GLASS AND POLYETHYLENE BEADS 182 8.1 INTRODUCTION 182 8.2 RESULTS AND DISCUSSION 182 8.2.1 Effect of Particle Concentration 182 8.2.2 The Effect of Particle Size 183 8.2.3 Effect of Particle Density 186 8.2.4 Effect of Particle Reynolds Number (Rcp) 187 8.3 PROFILES OF ENERGY DISSIPATION 188 8.3.1 Suspensions Without Gas 189 8.3.2 Suspensions in the Presence of Gas 189 8.4 SUMMARY 195 9 HIGH-INTENSITY MIXER WITH PULP SUSPENSIONS 197 9.1 INTRODUCTION 197 9.2 OVERALL FLOW CHARACTERISTICS IN THE HIGH-SHEAR MIXER 198 9.2.1 Flow Pattern 198 9.2.2 Flow Lx)op Characteristics 199 9.2.3 Reaction Zone Visualization 201 9.2.4 Overall Energy Dissipation 201 9.3 MiCROMixiNG RESULTS 204 9.3.1 Experiments with Flow Loop 204 9.3.2 Effect of Recirculation Rate on Micromixing 205 XI 9.3.3 Effect of Fibre Concentration on Micromixing in Flow Loop 206 9.4 EXPERIMENTS WITHOUT FLOW LOOP 208 9.4.1 Comparison of Experiments with and without Flow Loop 211 9.5 CORRELATIONS FOR DAMPING OF LOCAL ENERGY DISSIPATION RATE 212 9.5.1 Correlation Through a Fibre Concentration and Damping Factor 212 9.5.1.1 Comparison with Mass Transfer Correlation 214 9.6 MODEL FOR FIBRE AND FLOC INTERACTIONS 215 9.6.1 Fibre-Fibre Friction 216 9.6.2 Comparison 219 9.7 CORRELATION USING THE APPARENT VISCOSITY 225 9.7.1 E-Model and Viscosity 226 9.8 SUMMARY 231 10 CONCLUSIONS 236 10.1 THE ROLE OF FIBRES ON TURBULENCE 236 10.2 EFFECT OF OTHER DISPERSED PARTICLES ON TURBULENCE 237 10.3 DISTRIBUTION OF LOCAL ENERGY DISSIPATION RATE 237 10.4 THE USE OF MIXING-SENSITIVE CHEMICAL REACTIONS 238 10.5 IMPLICATIONS OF PRESENT FINDINGS 239 11 RECOMMENDATIONS FOR FUTURE WORK 241 12 NOMENCLATURE 242 13 REFERENCES 247 Xl l 14 APPENDICES 266 A REVIEW SUMMARY 267 B FIBRE STRUCTURE AND PROPERTIES 275 B.l FIBRE LENGTH 277 B.2 FIBRE STIFFNESS 280 B.3 FIBRE SWELLING 280 B.4 FIBRE SUSPENSION FREENESS 280 B.5 OTHER PROPERTIES/MATERIALS 282 B.5.1 Maximum Packing Fraction 282 B.5.2 Classification of Glass and Polyethylene Beads 282 C CHEMICALS USED AND PREPARATION 285 C.l CHEMICALS 285 C.1.1 Preparation of 1- and 2-naphthol 285 C.1.2 Diazotization 286 C.2 PREPARATION OF PRODUCT DYES 288 C.3 QUANTIFICATION OF DYE PRODUCTS 288 D ADSORPTION EFFECTS IN DISPERSED SYSTEMS 290 D.l ADSORPTION MODELS 290 D.2 CORRELATIONS TO ACCOUNT FOR ADSORPTION ON FBK FIBRE 291 D.2.1 Individual Component Correlation 291 D.2.2 Composite Correlation 292 D.3 DiFFUsioNAL LIMITATIONS 294 E FLOW AND REACTION ZONE VISUALIZATION 297 E.l FLOW VISUALIZATION USING HIGH-SPEED VIDEO 297 XIU E.2 DISCRETIZATION OF FLOW ZONES 301 F COMPUTATIONAL FLUID DYNAMICS (CFD) 305 F.l THEORETICAL ASPECTS 305 F.Ll Turbulence Modeling and the k-z Model 306 F.2 COMPUTATIONAL SUMMARY 308 G IMPLICATIONS FOR MASS TRANSFER 313 H DATA TABLES 314 H.l MEDIUM-INTENSITY DATA 314 H.2 HIGH-INTENSITY MIXER DATA 322 I ENERGY DISSIPATION DUE TO FIBRE-FIBRE CONTACTS 326 1.1 BACKGROUND 326 1.2 ENERGY DISSIPATION DUE TO FIBRE-FIBRE CONTACTS 328 1.3 ENERGY DISSIPATION DUE TO FIBRE-FLUID INTERACTION 330 J ESTIMATION OF TURBULENCE QUANTITIES BY OTHER MEANS.... 333 J.l RELATIVE ENERGY DISSIPATION RATES IN SIR 334 K MODELING OF MICROMIXING 336 K.l EDD MODEL 336 K.2 SIMPLIFICATION OF EDD MODEL 339 K.3 MODIFICATIONS FOR SEMI-BATCH OPERATIONS WITH FLOW LOOP 340 XIV LIST OF FIGURES Figure 2.1: Energy and energy dissipation spectra 10 Figure 2.2: Concentration fluctuation and concentration fluctuation dissipation spectra ( S c » l ) 11 Figure 2.3: Fluid deformation and vorticity (a) large and small scale deformations in the inertial subrange, (b) fine scale laminar deformations in viscous subrange, (c) formation of laminated structures through the action of vorticity 17 Figure 2.4: Product distribution vs. energy dissipation rate for normal and extended Bourne reaction system 31 Figure 2.5: Schematic for procedure to estimate local energy dissipation rate from experiment and simulation 34 Figure 2.6: Schematic representation of number of fibres in a volume swept out by length of a single fibre (Kerekes and Schell, 1992) 37 Figure 2.7: (a) Schematic representation of the three basic flow mechanisms of pulp suspensions in pipes, (b) velocity profiles, (c) Pressure drop for the flow of Cm = 0.03 pulp in a 150 mm pipe (Brecht and Heller, 1950). A-D: Plug flow; D-F: Mixed flow; F-H: Turbulent flow 41 Figure 2.8: (a) High-shear rotary device (b) Typical response of pulp fibre suspension in a high-shear rotary device (GuUichsen and Harkonen, 1981) 44 Figure 3.1: Schematic of experimental setup for semi-batch operation in a medium-intensity mixer 65 Figure 3.2: Schematic of medium-intensity mixer showing rotor and tank dimensions ..66 Figure 3.3: Photograph of mixer showing feed point location and rotor 67 Figure 3.4: Cross-section of mixing tank showing working section and feed point locations 69 Figure 3.5: Schematic of experimental setup for high-intensity mixer with flow loop ....71 Figure 3.6: Schematic of high-shear chamber showing feed arrangement 72 Figure 4.1: Plot of adsorption vs. contact time for FBK suspension for different mass concentrations at 298K (a) 1-naphthol (b) 2-naphthol 86 XV Figure 4.2: Equilibrium adsorption of 1-naphthol and 2-naphthol mixture (1:2) on FBK suspension as a function of mass consistency 87 Figure 4.3: Plot showing equilibrium adsorption of 1- and 2- naphthol on FBK suspension (Cm = 0.03) determined with each naphthol separately. Solid lines are fitted with Langmuir model. 87 Figure 4.4: Change in the ratio of naphthols with fibre concentration 89 Figure 4.5: Adsorption of product dyes on FBK vs. contact time at 298K, Cm = 0.05 91 Figure 4.6: Product distribution vs. mass concentration for FBK fibre. The solid lines for each curve are given by equation (4.1) 95 Figure 4.7: Comparison of product distribution vs. mass concentration for polyethylene, fibreglass, FBK and nylon fibre (^g =0.41) 96 Figure 4.8: Equilibrium adsorption of 1-naphthol and 2-naphthol mixture (1:2) on Nylon suspension as a function of mass consistency 97 Figure 4.9: Product distribution vs. mass concentration for Nylon fibre. The solid lines for each curve are given by Equation (4.2) 98 Figure 5.1: Average energy dissipation versus rotor speed for water and FBK suspension at different mass concentrations 103 Figure 5.2: Average energy dissipation versus rotor speed for water and Nylon fibres (L = 1.1mm) 104 Figure 5.3: Average energy dissipation versus rotor speed for water with gas void fraction from 0 to 0.40 106 Figure 5.4: Overall energy dissipation for O.SOmm glass bead suspension at different mass concentrations. Dotted line represent data for water alone 108 Figure 5.5: Overall energy dissipation for 0.11mm glass beads. Dotted line represent data for water alone 109 Figure 5.6: Comparison of the overall energy dissipation for glass beads of different sizes at a fixed mass concentration (Cm = 0.05). Dotted line give data for water alone 109 Figure 5.7: Plot of power number vs. dispersed phase concentration for a suspension of glass bead. Dotted line gives data for water alone I l l XVI Figure 5.8: "B-I Type" flow pattern in medium-intensity mixer observed for all cases with water (Xg < 0.20) 117 Figure 5.9: "B-II Type" pattern in medium-intensity mixer. Transitional type of flow observed in water + gas or water + gas + fibre suspensions 118 Figure 5.10: "D Type" flow pattern in medium-intensity mixer when operated with fibre at high concentrations. Also observed when small glass beads were used at high rotor speeds (N > 15 rev/s) 119 Figure 5.11: Photograph showing the medium-intensity mixer when operated with a Cv = 0.021 (Cm = 0.016) FBK suspension at N = 17.3 rev/s and Xg = 0.20 (a) top view showing large gas cavities behind rotor, (b) side view showing "B-I Type" flow pattern 120 Figure 5.12: Photograph showing medium-intensity mixer when operated with a Cv = 0.021 (Cm = 0.02) polyethylene beads suspension at N = 17.3 rev/s. (a) top view (b) side view 121 Figure 5.14: Flow pattern firom CFD simulation on medium-intensity mixer-at 17.3 rev/s (a) radial plane at z/H = 0.5 (b) vertical plane midway between baffles 123 Figure 5.15: CFD simulation of the flow pattern in the medium-intensity mixer with water (N=17.3 rev/s.). (a) Horizontal plane at z/H = 0.75, (b) horizontal plane at the middle of the vessel (z/H = 0.50). Note the higher concentration of vectors and their outward circumferential direction in the middle plane. 124 Figure 6.1: Plot of feed time vs. product distribution for water in the medium-intensity mixer (N = 7.3rev/s, feed point F13, CA2/C AI = 2, C^ - 21.54 mol/m^) 128 Figure 6.2: Plot of feed time vs. product distribution for water case (N=7.3 rev/s, feed point Fi3, A2/A1 = 2, CBO = 21.54 mol/m^ Xg= 0.20) 129 Figure 6.3: Plot of feed time vs. product distribution for 0.05Cm Polyethylene fibre suspension (N = 20.3 rev/s, feed point F13, A2/Ai=2, CBO= 21.54 mol/m^) 129 Figure 6.4: Visualization of reaction plume using high-speed video. Plume color intensified for ease of identification. Feed point F33, N = 7.3 rev/s. Elapsed times (a) 0 ms, (b) 4 ms, (c) 28 ms, (d) 32 ms 132 Figure 6.5: Visualization of reaction plume using high-speed video. Plume color intensified for ease of identification. Feed point F52, N = 7.3 rev/s. Elapsed times (a) 0 ms, (b) 12 ms, (c) 20 ms 133 XVl l Figure 6.6: Product distribution in the medium-intensity reactor for different feed points at N = 17.3 rev/s. Axial positions z/H = 0.05, 0.25, 0.5, 0.75 and 0.95. Standard test conditions 134 Figure 6.7: Product distribution in the medium-intensity reactor for different feed points along impeller radial direction (r/R = 0.579) at N = 17.3 rev/s 135 Figure 6.8: Product distribution versus rotor speed for medium-intensity mixer. Solid lines represent E-model predictions with cp = 1 while dotted lines represents predictions with cp =6 137 Figure 6.9: Discretization of different zones of energy dissipation fi-om reaction zone visualization (all dimensions in mm) 139 Figure 6.10: Dimensionless energy dissipation rate for water. Duplicate tests done on separate days under same conditions, (a) and (b) N = 7.3 rev/s, (c) and (d) N = 17.3 rev/s 142 Figure 6.11: Dimensionless energy dissipation for aqueous system (single phase) at different rotational speeds: (a) N = 7.3rev/s, (b) N = 12.3rev/s, (c) N = 17.3 rev/s, (d) N = 19.5 rev/s 144 Figure 6.12: Schematic layout of feed point location for radial plane mapping (coordinate positions shown in Table 6.3) 145 Figure 6.13: Contour profile of dimensionless energy dissipation rate in a radial plane across mixer for N=17.3 rev/s. (a) z/H = 0.05 (b) z/H = 0.25 147 Figure 6.14: Contour profile of dimensionless energy dissipation rate in a radial plane across mixer for N=17.3 rev/s. (a) z/H = 0.5 (b) z/H = 0.75 148 Figure 6.15: Contour profile of dimensionless energy dissipation rate in a radial plane across mixer for N= 17.3 rev/s. z/H = 0.95 149 Figure 6.16: 3D reconstruction of the distribution of energy dissipation rate in the medium-intensity mixer with water at N = 12.3 rev/s (a) Full quadrant view, (b) slice at z/H=0.5, (c) slice at z/H = 0.25 150 Figure 6.17: 3D reconstruction of the distribution of energy dissipation rate in the medium-intensity mixer with water at N = 17.3 rev/s (a) Full quadrant view, (b) slice at z/H=0.5, (c) slice at z/H = 0.25 151 XVIll Figure 6.18: 3D reconstruction of the distribution of energy dissipation rate in the medium-intensity mixer with polyethylene fibres (Cy = 0.021) at N = 17.3 rev/s (a) Full quadrant view, (b) slice at z/H=0.5, (c) slice at z/H = 0.25. ..152 Figure 6.19: Rotor design and feed point location for determination of micromixing behind rotor vanes 155 Figure 7.1: Micromixing index (XQ) VS rotor speed for FBK suspension (Cm = 0.0 -0.016, Xg=0.0, feed point F32, €"^210°^,^ 2, VAA^B = 50, C^,= 0.52 mol/m^ C°=21.54mol/m^) 161 Figure 7.2: XQ versus rotor speed for a polyethylene fibre suspension (L = 3.2mm, Cm = 0.0 - 0.04, Xg= 0.0; C^, = 0.52 mol/m^ C^^/C",^ 2, VAA^B = 50, C° =21.54 mol/m , tf= 180 s, feed point F32) 161 Figure 7.3: XQ versus mixing intensity for Cm = 0.02 polyethylene fibre suspension for aspect ratio (L/d) 84 and 167 (Feedpoint F32, C°,= 0.052 molW, C;2/C°,= 2, VAA^B = 50, C; =21.54 mol/m^ tf = 180 s) 162 Figure 7.4: Product distribution in fibre suspensions of different fibre aspect ratio (N = 17.3 rev/s, feed point F32 , C°,= 0.52 mo\lm\ C°JC°^,= 2, VAA^B = 50, C° =21.54 molW, tf = 180 s ) 163 Figure 7.5: Product distribution in fibre suspensions of different fibre aspect ratio at various volum concentrations (N = 17.3 rev/s, feed point F32 , C^, = 0.52 mol/m^ C^2/C;, = 2, VA/VB = 50 , q=21.54mol/m^ tf= 180 s ) 164 Figure 7.6: Relative energy dissipation rate in the presence of fibres for different L/d ratios (lSf = 7.3-19.5 rev/s, feed point F32, standard conditions) 166 Figure 7.7: Product distribution in medium-intensity mixer in the presence of gas at different void fractions (Cm = 0.0, Xg = 0.0 - 0.40, feedpoint F32, C°i = 0.52 mol/m^ C\^ /C°,= 2, VA/VB = 50, C°=21.54 mol/m^ tf = 110 - 180 s) ....168 Figure 7.8: Comparison of relative energy dissipation rate in the presence of gas 169 Figure 7.9: Micromixing characteristics in a polyethylene fibre suspension (Xg = 0.20, Cv = 0.0 - 0.034, feedpoint F32) 170 Figure 7.10: Relative energy dissipation in a suspension of polyethylene fibre with gas (Xg = 0.20, Cv = 0.0 - 0.034, feedpoint F32, standard conditions) 170 XIX Figure 7.11: Dimensionless energy dissipation for FBK suspension at Cy = 0.0215: (a) N = 7.3rev/s, (b) N = 12.3rev/s, (c) N = 17.3rev/s, (d) N = 19.5rev/s 174 Figure 7.12: Dimensionless energy dissipation for polyethylene suspensions at N = 17.3 rev/s. (a) Cv = 0.0215, (b)Cv= 0.043 174 Figure 7.13: Dimensionless energy dissipation for two-phase system (water+gas Xg = 0.20) at different rotational speeds (a) N - 13 rev/s, (b) N = 12.3 rev/s, (c) N =17.3 rev/s, (d)N = 20.7 rev/s 177 Figure 7.14: Dimensionless energy dissipation for three-phase system (FBK Q = 0.017, Xg = 0.20) at different rotational speeds (a) N = 7.3 rev/s, (b) N = 12.3 rev/s, (c) N = 17.3 rev/s, (d) N = 20.7rev/s 178 Figure 8.1: XQ in a glass bead suspension as a function of volumetric fraction for different particle diameters (N = 12.3 rev/s, standard chemical test conditions) 183 Figure 8.2: Product distribution in the presence of glass beads of different sizes (Cv = 0.021, feedpoint F32, standard chemical conditions) 184 Figure 8.3: Relative local energy dissipation rate for glass beads suspension. (Feed point F32, standard conditions) 185 Figure 8.4: Comparison of XQ for glass and polyethylene beads (standard chemical conditions) 186 Figure 8.5: Comparison of local energy dissipation for glass and polyethylene beads (standard chemical conditions) 186 Figure 8.6: Particle Reynolds numbers for different glass beads 188 Figure 8.7: Distribution of dimensionless energy dissipation rate. Glass beads suspension (0.80mm, 0.043 Cy, Xg = 0.0). (a) N=12.3 rev/s, (b) N= 17.3 rev/s 191 Figure 8.8: Dimensionless local energy dissipation rate for 0.1 Imm (Cy = 0.043, Xg = 0.0). (a) N = 12.3 rev/s, (b) N = 17.3 rev/s 192 Figure 8.9: Distribution of dimensionless energy dissipation rates for polyethylene beads (dp = 3 mm) atN=17.3 rev/s, Xg=0.0 (a) Cy = 0.021 (b)Cy = 0.043 193 Figure 8.10: Distribution of dimensionless energy dissipation rates for 0.85mm glass beads (Xg= 0.20, Cy=0.021). (a) N=12.3 rev/s, (b) N=17.3 rev/s 194 Figure 9.1: Fluidizer pumping capacity for different initial pump flow rates 199 XX Figure 9.2: Comparison of centrally located feed vs. off-centre for two initial flow rates ( Q \ - 20 and 124 L/min) 200 Figure 9.3: Reaction zone visualization in high-shear mixer with flow loop: QA =100 L/min, N = 0 rev/s, fed point location F32(r/R = 0.727, z/H = 0.05). Reaction plume follows the external flow when the rotor is not moving 202 Figure 9.4: Reaction zone visualization in high-shear mixer with flow loop: QA "= 100 L/min, N = 6 rev/s, fed point location F32(r/R = 0.727, z/H = 0.05). Reaction plume re-directed towards the rotor even at very low rotational speeds 202 Figure 9.5: Overall energy dissipation rate for FBK suspension as a fiinction of rotor speed. Solid line computed for water alone using Np = 3.4 203 Figure 9.6: Overall energy dissipation rate versus fibre volume concentration for high-shear mixer with flow loop (0.0 < Cy < 0.08; QA = 70 - 200 L/min), and without loop (0.0 < Cv< 0.14; QA = 0 L/min) symbols with dot centre 204 Figure 9.7: Product distribution versus rotor speed for high-shear mixer with flow loop. Solid line represents E-model prediction with (p = 1, dashed line 9 = 0.4 and dotted line (p = 4. Two feed nozzles at 0 = 0° and 0 = 180° F31 (r/R = 0.50, z/H = 0.5) F32 (r/R = 0.727, z/H =0.5), F33 (r/R = 0.955, z/H = 0.5) (C°, = 0.52, C°2/C°,= 2; VA/VB = 100, C° = 43.08 mol/m^ tf = 200s, T = 298+lK) 206 Figure 9.8: The effect of recirculation rate on product distribution at constant rotor speed in the high-shear mixer. (Cv = 0.0, C°2/C^i= 2, VA/VB = 100, C°,= 0.52 mol/m^ C° = 43.08 mol/m^ tf = 200s, T = 298±0.5°K). Two feed nozzles at 0 = 0° and 0 = 180° F32 (r/R = 0.727, z/H = 0.5) 207 Figure 9.9: Product distribution versus FBK fibre suspension concentration in high-shear mixer wdth flow loop (QA = 180 - 200 L/min). (Conditions: C°2 /C°i = 2, VA/VB = 100, C;,= 0.52 mol/m^ C;= 43.08 mol/m\ tf = 200s, T = 298±1°K). Two feed nozzles at 0 = 0° and 0 = 180° F32 (r/R = 0.727, z/H = 0.5) 208 Figure 9.10: Local energy dissipation rate in FBK suspension. (Estimated from E-Model for the experimental conditions: C^2^^AI~ 2, VA/VB = 100, C'^^= 0.52 mol/m^ C;= 43.08 mol/m^ tf = 200s, T = 298±1°K). Two feed nozzles at 0 = 0° and 0 = 180° F32 (r/R = 0.727, z/H = 0.5) 209 XXI Figure 9.11: Critical feed time for high-intensity mixer without flow loop at 16.7 rev/s and 50 rev/s,.(Cm= 0.0, C^J0"^,= 2, VAAVB = 100, C°,= 0.52 molW, C° = 43.08mol/m^T=298±l°K)) 210 Figure 9.12: Product distribution in high-intensity mixer without flow loop (JQA == 0 : ; 2 / C ; = 2 , V A A ^ B = IOO, c;,^ ,3 X -L/min, N = 66.7 rev/s, tf =21s, C° I ° = 2, V A^B  100, C° = 0.52 molW, Cl= 43.08 molW, T = 295 - 301 °K). Six feed nozzles equi-spaced at 9 = 60° F32 (r/R = 0.727, z/H = 0.5) 212 Figure 9.13: Local energy dissipation for semi-batch fluidizer operated without flow loop. Conditions as in Figure 9.12 213 Figure 9.14: Comparison of product distribution high-shear mixer with flow loop and without flow loop (water only). Solid line represents E-model prediction with (p = 0.4 and dotted line cp = 4. Flow loop: Two feed nozzles at 6 = 0° and 9 = 180° F32 (r/R = 0.727, z/H = 0.5) tf = 200s, T - 298±1K; Without flow loop: six feed nozzles equi-spaced by 60° (r/R = 0.727, z/H = 0.5), tf = 21s, T = 298+IK (C°, = 0.52, CJC^^ = 2; VA/VB = 100, C° = 43.08 mol/m^) 214 Figure 9.15: Relative energy dissipation rate for FBK suspension in the high-shear with flow loop, QA = 180- 200 L/min 215 Figure 9.16: Schematic representation of the cascade process in the presence of fibres 218 Figure 9.17: Comparison of energy dissipation due to fibre-fibre fi"iction using different estimates for number of contacts per fibre. Experimental data for high-shear mixer atN = 66.7 rev/s 223 Figure 9.18: Comparison of energy dissipation due to fibre-fibre friction with additional adhesive force, FQ (FQ fi"om Andersson and Rasmusson, 1997) 224 Figure 9.19: Estimation of energy dissipation due to fibre-fibre friction using different relative velocities. Number of contacts calculated firom Meyer and Wahren (1964) 225 Figure 9.20: Estimation of energy dissipation due to fibre-fibre friction using velocity estimates from Andersson and experimentally determined velocity profile from Figure 9.19 226 Figure 9.21: Comparison of experimental XQ values with E-model prediction using the apparent viscosity of fibre suspension. High intensity mixer with flow loop. xxu Flow loop: tf = 200s, T = 298±1K (C^i = 0.52, C°JC°^,^ 2; VAA^B = 100, = 4 1 Hfi Yr^rMm\ CI = 43.08 molW) 230 Figure 9.22: Correlation for relative energy dissipation rate using effective viscosity (equation 9.20) 225 Figure B-1: The structure of fibre wall 276 Figure B-2: Length weighted distribution of FBK fibre used 278 Figure B-3: Population distribution of FBK fibre used 278 Figure B-4: Length weighted distribution of nylon fibre 279 Figure B-5: Population distribution of nylon fibre 279 Figure B-6: Particle size distribution for 0.11 mm glass beads 284 Figure B-7: Particle size distribution for 0.55 mm glass beads 284 Figure C-1: Representation of the extended reaction scheme: Azo coupling of 1-naphthol and 2-naphthol with diazotized sulphanilic acid 287 Figure C-2: Standard spectra for Azo dyestuffs (CQ-R = Cp.R = Cs = 0.03 mol/m'^ ; CQ = 0.04 molW) 289 Figure C-3: Standard Spectra for 1 - and 2- naphthol (CAI = CA2 = 0.04 molW) 289 Figure E-1: Photograph of setup for reaction and flow visualization 297 Figure E-2: Photograph of reaction zone in high shear mixer showing independent reaction zones (a) two feed points for experiments with flow loop (b) 2 out of six feed points in experiments without flow loop 298 Figure E-3: Flow visualization in high-shear mixer with external recirculation Q(^= 100 L/min (a) N = 0 rev/s reaction zone directed outwards following the path of incoming fluid (b) N = 6 rev/s - flow towards rotor 299 Figure E-4: Sketch of locus of reaction zone for different radial and axial feed point positions in the medium-intensity mixer 300 Figure E-5: Averaging of local energy dissipation rate in a cylindrical geometry 301 Figure F-1: Computational grid used and sliding mesh specifications 309 Figure F-2: Flow patterns in radial plane atz/H = 0.95 310 XXl l l Figure F-3: Flow patterns in a radial plane at z/H = 0.75 showing axial motion 311 Figure F-4: Flow patterns in a radial plane at z/H = 0.50 312 Figure G-1: Gas-Liquid mass transfer coefficient in FBK suspension 313 Figure I-l: Normal and frictional forces at fibre-fibre contact points 327 Figure K-1: Diffusion and reaction equations solved over laminated regions with initial thickness of 26o and 25 fort> 0 336 Figure K-2: Features of the Engulfment model, (a) mass transfer by engulfinent of local surroundings showing growth of mixed zone, (b) concentration profile of B for mixing without reaction, (c) concentration profile of B for mixing with instantaneous reaction (t2 > ti > to) 339 Figure K-2: Flow loop simulation 341 XXIV LIST OF TABLES Table 2.1: Micromixing test reactions 29 Table 2.2: Fibre suspension regimes (Kerekes and Schell, 1992) 38 Table 2.3: Range of mixing scales in fibre suspensions 52 Table 3.1: Dimensions of the medium-intensity mixer 68 Table 3.2: Dimensions of the high-intensity mixer 72 Table 3.4: Dimensions and physical properties of fibres 74 Table 3.5: Dimensions and physical properties of other particles 75 Table 3.6: Experimental conditions for medium-intensity mixer 77 Table 3.7: Experimental conditions for high-intensity mixer 79 Table 4.1: Correlation accuracy for FBK suspension (Equation 4.1) 99 Table 5.1: Power number for fibre suspensions (Xg= 0.0) 104 Table 5.2: Power number and bulk density at different gas void fractions 105 Table 5.3: Power number for fibre suspensions in medivim-intensity mixer (Xg = 0.20) 74 Table 5.4: Power number for glass beads (Xg = 0.0) I l l Table 5.5: Power number for glass beads, Xg = 0.20 - 0.10 112 Table 5.6: Power number for polyethylene fibres and beads 112 Table 6.1: Estimates of the size of reaction zone 131 Table 6.2: Weighting factor for inhomogeneous distribution of energy dissipation 139 Table 6.3: The average energy dissipation rate in medium-intensity mixer 143 Table 6.4: Radial and circumferential feedpoint locations in medium-intensity mixer (XQ values for N = 12.3rev/s,z/H = 0.05) 146 Table 6.5: Local energy dissipation rates from E-model for 3D mapping in medium-intensity mixer (N = 12.3 rev/s) 153 Table 6.6: Local energy dissipation rates in the rotor swept volume in medium-intensity mixer (N = 17.3 rev/s) 155 _ _ _ _ _ _ _ ^ XXV Table 7.1: Crowding factor for different fibre suspensions 164 Table 7.2: Damping factor for energy dissipation due to the presence of fibres 167 Table 7.3: Damping factor for energy dissipation rates due to the presence of fibres .171 Table 7.4: Energy dissipation rates in fibre suspension for different rotor speeds 175 Table 7.5: Energy dissipation rates in fibre suspension with gas (Xg = 0.20) for different rotor speeds 179 Table 8.1: Particle relaxation time and relative size for Save - 43 W/kg 185 Table 9.1: Maximxun suspension concentration handled at a given rotational speed in high-shear mixer 205 Table 9.2: Local and overall energy dissipation rates in an FBK suspension (Cv = 0.034) at different rotor speeds 209 Table 9.3: Product distribution from E-model at mean reaction temperature and different temperature ranges 212 Table 9.4: Damping factor for energy dissipation rate due to the presence of fibres 217 Table A-1: Parameters of homogeneous turbulence 267 Table A-2: Summary of micromixing models 268 Table A-3: Particle - turbulence interactions 270 Table A-4: Investigations in pulp fibre suspension flow characteristics 271 Table A-5: Determination of turbulence parameters in pulp/fibre suspensions 272 Table A-6: Measurement of macroscale uniformity in mill situations (Beimington et al, 1989) 274 Table A-7: Measurement of fibre-scale uniformity in mill chlorination mixers (Patterson andKerekes, 1984) 274 Table A-8: Comparison of mixers used in pulp bleaching (Bennington et al, 1989) 274 Table B-l:Kajaani results for FBK fibre 277 Table B-5: Fibre freeness values after different treatments 281 Table B-6: Maximum packing fraction for different material 282 Table B-7: Glass beads classification 283 XXVI Table B-8: Polyethylene beads classification 283 Table C-1: Chemicals used, specifications and suppliers 285 Table C-2: Typical formulation for 1- and 2-naphthol for 40L solution 285 Table C-3: Typical formulation for diazotization of 21.54 mol/m^ sulphanilic acid 286 Table D-1: Adsorption model pargimeters for 1- and 2- naphthol on FBK 293 Table D-2: Adsorption model parameters for R (o-R + p-R) on FBK 293 Table D-3: Adsorption model parameters for S on FBK 293 Table D-4: Adsorption parameters for Q on FBK 294 Table D-5: Calculation of the Weisz modulus for diffiisional limitations in primary coupling reactions 295 Table D-6: Calculation of the Weisz modulus for diffiisional limitations in primary coupling reactions 296 Table E-1: The effect of different weighing factors on average energy dissipation rate (using data for axial plane midway between baffles) 303 Table E-2: Weighting factors for the distribution of energy dissipation rates in the rotor swept volume 304 Table F-1: Constants used for the k-s model 306 Table F-2: Power number from CFD and torque measurements 307 Table H-1: Analytical measurements of products and micromixing index (XQ) (T=298K, C ,^ = 0.52mol/m^; CJC^^ =2.0, C° = 23.6molW, VAAVB = 50) 313 Table H-2: Micromixing index in water at 17.3 rev/s, (feed point F32, standard reaction conditions) 313 Table H-3: Micromixing in a suspension of polyethylene fibres Xg = 0.0 314 Table H-4: Micromixing in a suspension of polyethylene fibres, Xg = 0.2 314 Table H-5: Micromixing in a suspension of FBK fibres 314 Table H-6: Product distribution in water with different gas void fractions, (feed point F32, standard reaction conditions) .315 XXVl l Table H-7: Product distribution in Cy = 0.021 glass beads suspension (feed point F32, standard reaction conditions) 315 Table H-8: Product distribution in Cy = 0.021 polyethylene beads (feed point F32, standard reaction conditions) 315 Table H-9: Energy dissipation rates in fibre suspensions at different rotational speeds and concentrations (standard reaction conditions) 315 Table H-10: Energy dissipation rates in glass and polyethylene bead suspensions at different rotational speeds and concentrations (standard reaction conditions) 316 Table H-11: Energy dissipation rates in fibre suspensions at different rotational speeds and concentrations. (Xg = 0.20) (standard reaction conditions) 316 Table H-12: Energy dissipation rates in glass and polyethylene bead suspensions at different rotational speeds and concentrations (Xg = 0.20) (standard reaction conditions) 317 Table H-13: Local energy dissipation from E-model for 3-D mapping in medium-intensity mixer with water (N =12.3 3 rev/s) (standard reaction conditions) 318 Table H-14: Local energy dissipation fi-om E-model for 3-D mapping in medium-intensity mixer with water (N = 17.3 rev/s) (standard reaction conditions). 319 Table H-15: Local energy dissipation from E-model for 3-D mapping in medium-intensity mixer with Cv = 0.021 polyethylene suspension (N = 17.3 rev/s) (standard reaction conditions) 320 Table H-16: Overall energy dissipation rate in high-shear mixer with flow Loop 321 Table H-17a: Micromixing in an FBK suspension of different concentrations N= 50.0 rev/s 321 Table H-17b: Micromixing in an FBK suspension of different concentrations. N= 66.7 rev/s 322 Table H-17c: Micromixing in an FBK suspension of different concentrations N = 83.3 rev/s 322 Table H-18a: Temperature scheduling for high-shear mixer (N = 50 rev/s) 322 Table H-18b: Temperature scheduling for high-shear mixer (N = 66.7 rev/s) 322 xxvm Table H-18c: Temperature scheduling for high-shear mixer (N = 83.3 rev/s) 323 Table H-19: N = 50 rev/s high-intensity mixer without flow loop 323 Table H-20: N = 66.7 rev/s high-intensity mixer without flow loop 324 Table H-21: N = 83.3 rev/s high-intensity mixer without loop 324 Table H-22: Suspension properties and Kolmogorov microscales at N = 66.7 rev/s 324 Table I-l: Estimation of energy dissipation by fibre-fibre and fibre-fluid friction 332 Table J-1: Maximum relative energy dissipation rate (s/Save) in the discharge zone of Rushton Turbine in a standard baffled vessel unless otherwise stated 335 XXIX ACKNOWLEDGMENT I would like to express my sincere gratitude to those who have assisted me to complete this work. First and foremost, thanks and praise be to the Almighty Lord, my Saviour and Redeemer, Jesus Christ, whose mercy is infinite and the source of all wisdom and knowledge. I like to thank my supervisor, Dr Chad Bennington for giving me the opportunity to perform this research. His guidance as well as financial support was invaluable in the course of this thesis. I appreciate the valuable contribution of my supervisory committee: Professors Richard Branion, Sawas Hatzikiriakos and Paul Watkinson of Chemical and Bio-Resource Engineering. Also Professor Ian Gartshore of the department of Mechanical Engineering. Your suggestions and constructive criticism is highly appreciated. Many special thanks Dr. Vilas Rewatkar for many hours of discussions, during the write-up of this thesis. I can't thank you enough for all you have done for me. I would like to acknowledge an invaluable assistance from Professor Richard Kerekes, especially in drag reduction, flocculation and viscosity properties of pulp suspensions. I have gained a lot from your deep insights and great wisdom. Special thanks to Professor Sheldon Green of the department of Mechanical Engineering for access to CFD codes and his PhD student, Ali Roshanzamir for his assistance in computations. This part of my study would not have been possible without your generosity. The assistance of Professor Jerzy Baldyga of Warsaw Technical University through numerous e-mails and faxes is deeply appreciated. Also many special thanks to Professor John Bourne of ETH Zurich providing the foundation for micromixing experiments, numerical codes as well as valuable suggestions throughout this thesis. The friendship of the former and present members of the Mixing Group is acknowledged. Special thanks to Drs. Mohammad Shaharuzaman, Vilas Rewatkar, Min-Hua Wang and George Owusu for the lively exchange of ideas and words of courage. You have been a great inspiration to me. The help of the technical and administrative staff of the Pulp and Paper Center at the University of British Columbia was highly essential to the successful completion of XXX the present study. Peter Taylor and Tim Paterson, I owe a lot to you guys. You have treated me in a very special way, and only God can repay you for your kindness. Rita Penco - you were very prompt with literature search and document retrieval, a true treasure to the information age. Technical assistance from Ken Wong and administrative assistance from Georgina White, Lisa Brandly and Brenda Dutka is highly appreciated. Also the wonderful technical and administrative staff at the Chemical and Bio-Resource Engineering Department: Horace Lam, Helsa Leong, Lori Tanaka and Shelagh Penty -may God bless all of you. Last, but not the least, to my extended family back in Tanzania who have been behind me throughout this long journey. To my wife and daughter for being and bearing with me. I can't thank them enough for their understanding during all this period. I cannot forget members of the St. Anselm's Anglican Church for providing the spiritual support that has sustained me, and my friends for social and moral support. Finally, I would like to acknowledge the financial support from TAN047 Project, University of Dar Salaam that enabled me to undertake this study. DEDICATION I dedicate this thesis in loving memory of my mother, Salome Petro Sameji, my father, Mzee Philemon Eliaza Mmbaga, and my brother, Peter Mmbaga who passed away during this period. Chapter 1: Introduction Chapter 1 1 Introduction The creation of fluid-like motion by the application of high shear in fibre suspensions has enabled pulp and paper unit operations traditionally carried out at low consistency (mass concentration, Cm < 0.04) to be carried out at medium-consistency (MC) in the range (0.06 < Cm < 0.15). This has led to significant benefits through reduction of amount of water handled in unit processes and equipment size. High power dissipation is usually applied to disrupt fibre networks and create fluid-like motion under turbulent conditions. However, very little is known about local flow behaviour of fibre suspensions under such conditions. This is because of the difficulty in resolving turbulent quantities at conditions where two or more phases are present with application of high power dissipation rates. Most industrial operations are, however, carried out at these conditions, and thus the need for more information. The use of chemical reactions sensitive to the smallest scales of turbulence to measure micromixing in liquid phase reactors is well documented in the literature (Patterson, 1981; Baldyga and Bourne, 1984; 1986; Villermaux, 1986; Li and Toor, 1986). Micromixing effects are studied by using a well-defined reaction system that displays mixing-sensitivity, which can be related to the local energy dissipation rate (Baldyga and Bourne, 1984; 1989). Recently, this technique has been used to study micromixing in the presence of pulp fibre suspensions (Bennington and Bourne, 1990; Bennington and Thangavel, 1993). In the present work, we use mixing sensitive chemical reactions to obtain information about the local flow behaviour in the presence of fibres. Chapter 1: Introduction The aim of this study is to determine the effects of fibres on the energy dissipation rate and mixing in pulp fibre suspensions. This is carried out by measuring the local energy dissipation rate in the liquid phase of pulp suspensions at different volume concentrations and comparing it with those measured in an aqueous system, hi addition to increasing the understanding of small-scale mixing in dispersed fibre systems, the study also seeks to explore implications in particle-turbulence interactions in other dispersed systems. The second chapter is divided into four parts. In the first part, the interrelation between the fluid mechanical view of mixing and the systems approach of macro- and micromixing is reviewed. Mixing at the smallest scales (micromixing) is examined in the light of turbulent theory. In the second part, the solution methods for micromixing in liquid-phase systems are presented. The basis for use of chemical reactions as mixing probes and the mathematical models of micromixing are identified. A method to estimate the turbulent energy dissipation rate from the product distribution of a mixing-sensitive chemical reaction is also presented. In the third part, parameters that determine the behaviour of fibre suspension in motion are discussed and the various methods used to determine turbulence parameters reviewed. We also compile literature studies on how the fluid turbulence may be affected by the presence of other dispersed particles. The last part of this review chapter gives the objectives and scope of this thesis. In the third chapter, experimental methods, equipment and procedures used in this study are described. Chapter Four evaluates the potential problems of adsorption in the dispersed system. The characterization of the adsorption of reactant and product dyes from the coupling reaction between 1-naphthol and 2-naphthol with diazotized sulphanilic acid in alkaline conditions is carried out. This allows correction for adsorption and proper interpretation of micromixing results when adsorption effects are involved. Chapter 1: Introduction Chapter Five examines the overall mixing characteristics (macromixing) in a rotary device in the presence of fibre suspensions as well as other dispersed phases. The use of power number as a comparative measure of the overall power dissipation in dispersed systems is introduced. This will be useful in comparing with local mixing characteristics obtained in a medium-intensity rotor stator device in subsequent chapters. We also identify different types of flow regimes and show how they relate to overall energy dissipation. Macromixing is also explored using computational fluid dynamics. Chapter Six examines micromixing interactions in a medium-intensity reactor with water alone. The effect of various operating parameters are compared at fixed feed point locations. This is examined in relation to single point measurements as well as planar measurements in order to re-construct the local energy dissipation profile in the medium-intensity mixer. Chapter Seven is devoted to micromixing in the liquid phase of fibre suspensions in the medium-intensity mixer. The Engulfment model of micromixing is implemented to evaluate the local energy dissipation rate and relative energy dissipation in the presence of fibres. The effect of the presence of gas in a pulp suspension is also investigated in this chapter. Similar measurements for other dispersed systems (glass and polyethylene beads) are reported in Chapter Eight. In Chapter Nine, macroscale and microscale interactions in a high-shear mixing device are explored. Pulp suspensions of up to Cm = 0.10 are mixed under high shear rates. The effects of recirculation rate and fibre concentration on local energy dissipation rate are examined. An empirical model for the prediction of the damping of local energy dissipation in fibre suspensions is proposed and the decay rate compared with damping observed for mass transfer coefficient measurements in an industrial type mixer. Modeling and correlation aspects are presented in this chapter. Chapter 1: Introduction Chapter ten summarizes the overall conclusions of this thesis: the effects of fibres on local energy dissipation in the liquid phase of fibre suspensions; the overall energy dissipation in multiphase systems and the effect of other dispersed systems. Construction of relative energy dissipation profiles from micromixing experiments as a tool to characterizing mixers; and implications for mixing and mass transfer correlations are presented. The implications of the dimensionless local energy dissipation profiles in mixers to correlate mass transfer are presented. In the last chapter, recommendations for future work are given. Chapter 2: Literature Review Chapter 2 2 LITERATURE REVIEW 2.1 Introduction Mixing phenomena in chemical reactors may be considered from either the point of view of fluid mechanics using the theory of turbulence or from chemical engineering with a systems approach of macro- and micromixing theory. In a way, both approaches are complementary as turbulent mixing affects chemical reactions depending on the kinetics involved as well as the way in which reactants are transported to the reaction zone. For non-linear reaction kinetics, the residence time distribution (RTD) is not sufficient to predict the yield and conversion of a chemical reaction. In such cases, it is necessary to consider the degree and intensity of segregation experienced by fluid elements (Dankwertz 1958; Zwietering, 1959). The ultimate mixing at the molecular level is achieved by small scale turbulent eddies, whose size and intensity depend on the local energy dissipation rate. This phenomenon becomes important particularly in the case of multiple simultaneous reactions where it can affect the selectivity of a desired product. Turbulence is important in fluid mixing operations as it determines to a large extent the microscale or molecular scale mixing that takes place within the suspension. Turbulence accelerates reaction and promotes uniformity by ensuring that small-scale homogeneity is achieved within the suspension. It also influences the rate at which chemical reactions occur, and can contribute to the distribution of reaction products formed. By using well-characterized mixing-sensitive reactions, we can determine the local energy dissipation within a mixer. This allows us to measure energy dissipation in Chapter 2: Literature Review the liquid phase of dispersed (fibre) systems where traditional methods of measurement are limited. This chapter reviews the current understanding of micromixing in the light of turbulence theory, solution methods for turbulent mixing and reaction, turbulent flow characteristics of fibre suspensions and mixing in pulp bleaching. It also identifies an improved mathematical model of micromixing that relates to the key physical processes contributing to mixing on the molecular scale and describes how it will be used to determine the local energy dissipation rates in the liquid phase of fibre suspensions. 2.2 Turbulence and Mixing 2.2.1 The Turbulent Flow Field in Agitated Vessels Turbulent motion is characterized by unsteady movements in parts of the fluid, the result of which is a superposition of a spectrum of velocity fluctuations on an overall mean flow. Large fluid elements (also called eddies) produced in turbulent flow correspond to velocity fluctuations of low frequency and are of a size comparable to the physical dimension of the system (tank or impeller diameter). Smaller eddies result from interactions between large eddies, and correspond to velocity fluctuations of high frequency. Energy is transferred from large scales to progressively smaller scales (energy cascade) until a limiting scale is reached where energy is dissipated into heat by the action of viscosity. According to Kolmogorov's theory, at sufficiently high Reynolds numbers, there is a range of high wave numbers where turbulence is statistically in equilibrium and uniquely determined by the energy dissipation rate (s) and kinematic viscosity (v) (Hinze, 1975). In this state of universal equilibrium, otherwise known as isotropic turbulence, a Chapter 2: Literature Review length scale (XK), a time scale (T^:) and a velocity scale (VA:) may be defined respectively as:' \ -r^^\l (2.1) ^ ) (2.2) ^v^ v,^{vsf (2.3) Another characteristic of turbulent flow is the Reynolds number (Re). This may be regarded at the ratio between forces that impart energy at a given scale to those which dissipate it. Thus at the largest scales where energy is extracted from the mean flow with characteristic velocity U and length scale L, the Reynolds number must be very high, i.e Re = — >10^ (2.4) V On the other hand, at the smallest scales where energy is dissipated (Kolmogorov scale), the Reynolds number is very small Re = i ^ i « i (2.5) V As more energy is added to the fluid than can be dissipated directly at a given scale, the flow will disintegrate to form shorter length scales at which more energy can be dissipated. The transfer of energy from large eddies to smaller eddies is not influenced by viscosity, as long as the Reynolds number is sufficiently large (Re > 10 ). However, the effects of viscosity finally dominate in the high wave number range and most of the Parameters of homogeneous isotropic turbulence shown in Appendix A-1 Chapter 2: Literature Review 8 energy received from the low wave numbers is dissipated as heat. If there is no continued supply of external energy source to the system, turbulence will decay. 2.2.2 Energy dissipation in Turbulent Flows Turbulent flows are dissipative. The kinetic energy of flow is defined as k-—U: 2 ' (2.6) where u\ is the r.m.s fluctuating velocity. The production of turbulent kinetic energy from the mean flow is given by P = -U'^j-Sy (2.7) where s-^ is the mean strain rate tensor given by _ 1 f ^-du. du, ' +- -, dxj fix,. (2.8) and the dissipation of turbulent kinetic energy by the work of viscous deformation is given by where s',, is the strain rate fluctuations tensor defined as (2.9) ^ . = 2 du] du'j • + • , dXj Sr,. (2.10) For isotropic turbulence, all components of the strain tensor can be expressed by du[ I dx^ thus we may re-write equation (2.9) according to Taylor (Hinze, 1975) s = \5v fdu'^' ydx.j ISv-T-(2.11) Chapter 2: Literature Review where X,g is the Taylor microscale and u is the root mean square (rms) velocity. We can therefore estimate the rate of strain characterizing viscous dissipation of energy as - ^ = 0.26 - I (2.12) An expression relating all length scales of flow (L > A,g > XK) to energy dissipation is given by eoc— (2.13) There is no doubt, therefore, that local energy dissipation is a very important parameter in turbulent flows. By examining local energy dissipation rates we can learn a lot about the local flow processes that take place at the smallest scales. We first look at the distinction of scales though the energy spectrum, E{k). 2.2.2.1 Spectra for Turbulent Energy and its Dissipation The distinction between small scales and large scales can be best visualized when we look at the energy spectrum of turbulence, E(k), and its dissipation spectrum, D(k) = I^E(k). The energy spectrum represents the distribution of the energy content of the turbulent flow over various length scales. A,, in terms of wave numbers, k = li^/k (Lumley, 1965; Tennekes and Lumley, 1972). Likewise, the dissipation spectrum represents the dissipation of energy at various length scales. Figure 2.1 shows a schematic of the spectra for isotropic turbulence, where it can be seen that energy is introduced at large scales and dissipated at the Kolmogorov scale. For the high wave number region, two zones may be identified, namely, the inertial subrange in which the vortex elements are statistically independent of primary vortices; and the viscous subrange in which kinetic energy is converted into heat. Chapter 2: Literature Review 10 _ .,_, , 1 / 1 • 1 '• 1 : I :* 1 ; 1 —1 1 — • • ! I 1 | -inertial subrange I . I . I \ \ \ 1 _,—,,...,., _ , — , — \nlVI\r\\ in(L(K)) - - - ln(k'E(k)) -• VISCOUS subrange Figure 2.1 Schematic of energy and energy dissipation spectra (Hinze, 1975) The process of formation of smaller scales from larger scales constitutes the essence of mixing. The key parameter to achieving good mixing is the supply of energy and its dissipation at the smallest scales. If the ultimate goal of mixing is to achieve chemical reactions, a much smaller scale than the Kolmogorov scale is considered. Mixing at the molecular level is characterized by the Batchelor scale (A,B). Batchelor (1959) defined this scale as: -Ag -^vD^r^ V ^ J = X^I^ (2.14) For liquid mixtures where the Schmidt number (Sc) is much greater than unity, the Batchelor scale is smaller than the Kolmogorov scale, which implies that molecular mixing is slower than viscous mixing. We shall now look at scales with respect to the concentration fluctuation spectrum. Chapter 2: Literature Review 11 2.2.2.2 Spectra for Concentration Fluctuations The simplest statistical measure of the departure from uniformity of a mixture is the root-mean-square fluctuation of concentration (\inmixedness') and is defined as: c\=A^i-c^ (2.15) The structure of the concentration field in a turbulent mixer is characterized by the three dimensional spectrum of concentration fluctuation, G{k) (Batchelor, 1959; Corrsin, 1964; Rosenweig, 1964; Pao, 1965). This contains contributions to "unmixedness" from eddies of all sizes and is related to the concentation variance through: - J = G{k)dk (2.16) The concentration spectrum may be divided into three main subranges, namely: inertial-convective, viscous-convective and viscous-diffusive subranges. These regimes can be distinguished in the wave number spectrums for scalar fluctuations as shown in Figure 2.2 for liquid mixtures with high Schmidt numbers (Tennekes and Lumley, 1972; Hinze, 1975). 1 1 1 ^ 1 —' 1 '— T" / : / \ . / : ' inertial convective subrange 1 ; I 1 1 -viscous convective subrange ~r • 1 ' \n(r\l\A\ -in^u^K)) ln(k'G(l<)) •s \ s \. \ viscous diffusive N^subrange I . I . k, k. In (k) Figure 2.2: Concentration fluctuation and concentration fluctuation dissipation spectra (Sc » 1) Chapter 2: Literature Review 12 In the inertial convective subrange (ICS) {k^^ <k <kf,), the size of completely segregated concentration eddies are reduced by inertial action. Any transport in this range takes place through convection and there is no diffusion. For any mixing process, time and volume are required to reduce the scale of segregation as eddies pass through this subrange and no molecular dissipation takes place. The rate of decay of concentration variance in this subrange was obtained by Rosenweig (1964) as: G{k) = Ba dt £ 'k (2.17) where Ba is constant that has been found to be equal to 0.4 (Hinze, 1975). In the viscous-convective sub-range (VCS) (A:^  <k<kg), the dissipation of concentration variance is mainly due to the deformations of fluid elements. In this sub-range, eddies are subjected to laminar deformation by elongation (Batchelor, 1959; Ottino et al., 1980; Bolzern and Bourne, 1983) or one-dimensional shear (Angst at al, 1982). Molecular mixing proceeds by deformation of eddies, engulfment and by molecular diffusion. Molecular diffusion becomes increasingly important as the size of eddies approaches the Batchelor scale. In the viscous-diffusive sub-range (VDS) wave numbers, (k > kg), laminar strain and molecular diffusion contribute to the micromixing. When k = kg, laminar strain and molecular diffusion contribute equally to the spectral transfer, but at much higher wave numbers (k>kg), concentration variance is rapidly dissipated by molecular diffusion. Batchelor (1959) proposed the following equation for the spectrum: G{k) ~ CI dt k ' exp -CI ^ v (2.18) where CI is as constant of proportionality. Chapter 2: Literature Review 13 Any model of micromixing should in principle account for this cascade of energy and scales to faithfully represent the phenomena that take place. However, physical properties of the mixer determine the significance of each process. In the case of liquid phase mixing, both convective and diffusive mixing may be important, with convective mixing being more important in non-viscous liquids, and diffusive mixing being more important for viscous liquids. Accordingly, the existing models of micromixing in literature deal with one or more sub-ranges in the concentration spectrum, depending on physical properties of the mixture. Appendix A-2 gives a summary of micromixing models and the sub-ranges that they describe. 2,2.3 The Mixing Process The process of mixing of two streams of miscible liquid can be described from the Eulerian or Lagrangian points of view. In the Lagrangian frame perspective, mixing is considered in time, whereby the history of a fluid element is followed, thus allowing the identification and description of elementary processes that constitute mixing. The main concept in this approach is that of the division of mixing into two sub-processes namely, macromixing and micromixing. 2.2.3.1 Macromixing Macromixing refers to large-scale flow characteristics (e.g. convection, turbulent dispersion) that are responsible for large-scale distributions in the system (e.g. residence time distribution, age distribution, etc.). These features are characterized by flow pattern, power input and circulation time. 2.2.3.2 Micromixing Micromixing theory is concerned with all features of mixing that cause attainment of homogeneity at the molecular level. Mixing occurs in three successive or simultaneous stages, as proposed by Beek and Miller (1959) and corroborated by different Chapter 2: Literature Review X4 investigators, e.g. Brodkey (1981), David and Villeurmaux (1983) and Baldyga and Bourne (1984). These steps are: -• Inertial-convective disintegration of large eddies that lead to reduction of segregation scale, • A viscous-convective process of formation of laminated structures within energy dissipating vortices caused by fluid engulfment, and • Molecular diffusion within the deforming laminated structures Danckwerts (1958) and Zwietering (1959) introduced the main concepts of micromixing theory relating the degree to which material has been spread out by turbulent action (scale of segregation) and the approach to uniformity by the action of molecular difflisivity (intensity of segregation). The scale of segregation (Ls) is the average distance between eddies/clumps of the same component in a mixture. It is therefore a measure of large-scale break-up processes and is defined as: CO 4 = \R,{r)dr (2.19) 0 where Re is the correlation coefficient defined by: \p'i) Here Ci,x and Ci,x+r are local instantaneous concentration values of substance / at positions separated by a distance r from each other and c\ the fluctuating component of concentration i. In liquids (very small molecular diffusivity) the scale would decrease to some limiting value depending on the distribution of eddy sizes caused by the turbulent field. Chapter 2: Literature Review 15 The intensity of segregation. Is is a measure of the difference in concentration between neighbouring clumps of fluid. This is measured at a point for a long enough time to obtain a true average. Is=^ (2.21) These measures are, however, only good for the characterization of extreme conditions of mixedness, i.e., complete segregation (a state whereby no mass exchange takes place between points in a fluid system) and maximum mixedness (mass exchange between points is instantaneous). For conditions where partial segregation exists, more details must be known. Considering the size of molecules, even at the condition of maximum mixedness, two species will still be distinguishable unless there is molecular diffusion. Thus, turbulent mixing is only useful in as much as it hastens the approach to molecular mixing (since the microscales that are reached by turbulence are still larger than molecular scale). The final stage of mixing can only be achieved by molecular diffusion. 2.2.4 The Turbulent Mixer Corrsin (1964) derived an expression for the time dependence of the variance of concentration inhomogeneity (a^) in a batch mixer for isotropic turbulence as: 4 = ^ = e x p ( - r / r J (2.22) where Xm is the time scale for turbulent micromixing and QQ is the initial concentration variance. According to Corrsin, the turbulent micromixing time for liquids can be estimated as: T„ = 2 (4 / s} + 0.5(v/ / sf In Sc (2.23) Chapter 2: Literature Review 15 Equation (2.22) also applies to plug flow mixers, where x is the mean residence time. For the other extreme state of macromixing - backmixing, Rosensweig (1964) derived an expression for the intensity of segregation by considering the conservation of segregation in a CSTR as: ^ ^ = 7 T T V ^ (2.24) Here again Xm is the characteristic turbulent mixing time scale, which may be estimated as r.=[L]lsj (2.25) Both of these theories show that Xm = K(Z-5 I sf with K = 0(1) and independent of the state of mixing present. This reflects the isotropic nature of the fine scale turbulent processes. 2.2.4.1 Kinetics of Mixing If we assign variance contribution to the ICS, VCS and VDS parts of concentration spectrum as (cr,^ j, yp'l) ^^'^ K^I) respectively, the total concentration variance (cr^  jean written as: 0 - ' = a ' + 0 - 2 + 0 - 3 (2.26) The rate of decay of concentration variance in the ICS can be evaluated using Corrsin's model for a turbulent mixer (Eq. 2.22): dt r„ The disintegration of large eddies brings about the formation of finer scale inhomogeneity (from VCS wave numbers), thus the rate of dissipation of af is equal to the rate of production of cy]. The role of vorticity becomes important at these scales. Vortices provide the mechanism to form laminated structures by engulfing the fluid in their immediate surroundings. Figure 2.3 shows a schematic representation of disintegration of ' (2.27) Chapter 2: Literature Review 17 (a) : ^ t^ A (b) p /B (c) A I Figure 2.3: Fluid deformation and vorticity (a) large and small scale deformations in the inertial subrange, (b) fine scale laminar deformations in the viscous subrange, (c) formation of laminated structures through the action of vorticity (after Baldyga and Bourne, 1984). Chapter 2: Literature Review 18 fluid elements by inertial convective subrange, and also the action of vorticity in forming laminated structures at Kolmogorov's scales. The importance of vortices in fluid mixing has been reviewed recently by Bakker (Bakker and Van den Akker, 1994; Bakker, 1996). Vorticity is distributed over eddies of various sizes. However, we are only interested in the most energetically active eddies, with the highest vorticity, which are responsible for the engulfment process leading to mixing on the molecular scale. The mean lifetime of these eddies (time taken for an eddy to return to isotropy) was estimated by Baldyga and Bourne (1984) as T =12 ^v^ (2.28) While these vortices incorporate their surroundings, unsteady state diffusion and mass transfer in shrinking laminae proceed with a time constant characterized by shear and diffusion (Baldyga and Bourne, 1984) '^DS ~ ^ Sinh-\0.05Sc) (2.29) V In the engulfment process, the active fluid elements of high vorticity increases their volume, decreasing the volumes of fluid elements that are not active. Such a process brings about the coarse scale dilution of material resulting in the dissipation of a]. Baldyga and Rohani (1987a) obtained the decay equation for (T] as: d<j 2 ^=-Eal (2.30) dt where E is the engulfment rate parameter which is calculated from the eddy lifetime as: ln2 (e^ E^ = 0.0578 \v) (2.31) Chapter 2: Literature Review 19 The full kinetics governing the production and dissipation of al is given by: d<7^ <T, 1 —^ = ^-Eal (2.32) at T„ Finally, the rate of dissipation of a^ should be equal to the rate of production of (T3 . Assuming that the mixing process in all slabs forming laminated structures is independent of flows at larger scales, we can describe molecular diffusion term G such that: ^ = -Gal (2.33) at By combining equation (2.30) and (2.33), the production and dissipation terms in a^ balance out, giving da 2 dt Combining (2.22), (2.32) and (2.34) gives ^-ECTI-GO-] (2.34) da: ^^-Ga; (2.35) dt This means that the only mechanism of mixing on the molecular scale is molecular diffusion. However, it should be kept in mind that the value of CTJ is strongly dependent on the rates of mixing in the inertial convective and viscous convective subranges. The effect of turbulence, therefore, is to speed up molecular diffusion by increasing available surface area (through vortex stretching), reducing diffusion distances and increasing concentration gradients. Chapter 2: Literature Review 20 2.3 Modeling of Turbulent Mixing and Reaction 2.3.1 Background As shown earlier, the problem of turbulent mixing and reaction involves not only different length scales, but also different time scales. Moreover, further complications are brought about by the non-linear terms from convective accelerations and also due to chemical reactions that must be accounted for. The most complete approach to modeling of turbulent mixing flow and reaction problem is the solution of the overall differential equations for momentum and mass balances. For an incompressible Newtonian fluid, the momentum equation is du, du, 1 dp d^u, ^ + w . - ^ = ^ + 1/ •— (2.36) dt duj p Sx:,. dxjdxj The mass balance for a species with reaction rate R ^ + . ^ = Z ) ^ + i?, (2.37) dt dxj dxj and the continuity equation is du ^ = 0 (2.38) 2x, Direct numerical simulation (DNS) solution attempts to solve these equations for all scales. This approach has been applied to study reacting flows (Leonard and Hill, 1988; Gao and Obrien, 1991; Chakrabarty and Hill, 1995). However, DNS is currently limited to low Reynolds numbers and simple flows, because for large Reynolds numbers the fine scales are too small to be resolved. Chapter 2: Literature Review 21 2.3.2 A Statistical Approach The other approach is to use statistical methods to characterize turbulent mixing, similar to those used for the statistical theory of turbulent motion. Here, the basic equation for turbulent mixing is that of mass conservation, i.e.: ^^u^^ = D^^R, (2.37) dt dxj dxj The Reynolds averaged mass balance equation then becomes: 2-^ + ^ , ^ = Z ) ^ - A ^ + ^  (2.39) dt ' dXj dx] dXj ' ' ' For a second order irreversible reaction, e.g. A+B -> products, and we can write the averaged reaction term as RA ^ ^2c^Cfi = ^2{^A -CB-cyB) (2-40) The added terms in equations (2.39 - 2.40) are the source of the closure problem for reactive mixing. The correlation between the small-scale velocity fluctuations and concentration fluctuations and multiple concentration fluctuations (arising from the reaction term) needs to be determined. Simple closure theories (e.g. Toor, 1969) ignore the correlation between the velocity and concentration fields, and thereby close the equations for maximum mixedness or complete segregation, in the sense that c'^ c'g in the reaction term can be related to the intensity of segregation of a passive tracer. Full closure approaches employ probability density function (PDF) methods (Fox, 1995) and treat the effects of turbulent convection and chemical reaction exactly while modelling the effects of molecular diffusion. For this matter, PDF methods are computationally intensive and not easily adaptable to complex geometries. Chapter 2: Literature Review 22 2.3.3 A Deterministic Approach Simple but effective models bypass the closure problem by replacing the convective diffusion equations with simplified equations that can be solved by separating reaction from mass transfer with lumped parameters (Interaction models) or assuming simultaneous diffusion and reaction takes place in a hypothetical fluid element having simple geometry and motion. The basis for such models lies in the fact that turbulent flow at small scales is strongly laminar and can be described in a deterministic way. The advantage of such models is that virtually any reaction occurring in the liquid phase can be can be described. This is the approach adopted in this study. Examples of such physical models include the multi-environment models, coalescence-redispersion, mass transfer between two slabs, lamellar stretching and interaction with the mean. Appendix A-2 gives a tabular summary of different physical models that have been proposed to describe the turbulent mixing and reaction in liquid systems. The major models will be briefly described in the next sections. 2.3.3.1 Multi-environment Models The idea of multi-environment models consists of division of a mixer into two or more environments, representing fluid entering and leaving a reactor. The entering environment is considered totally segregated while the leaving environment is perfectly micro-mixed, with an exchange parameter relating transfer rates between the environments. There are several variations of the environment models - two-environment models (e.g. Weinstein and Adler, 1967), three-environment models (Richie and Trobgy, 1979) or the four-environment model (Mehta and Tarbel, 1986). All these are segregated models based on the RTD concept, and therefore system specific and empirical in nature. They merely differ in either in the structure of the environment interaction or in the transfer rate. They can only give information as far as the inertial convective subrange of concentration fluctuations is concerned. Chapter 2: Literature Review 23 2.3.3.2 Coalescence-Redispersion Models The coalescence-redispersion model was proposed by Curl (1963) to describe mixing and chemical reaction. In this approach, packets of fluid entering a vessel are moved through it in such a way as to approximate the observed convection pattern. In contrast to segregated models based on RTD concepts (Levenspiel, 1972; Carmon et al., 1977), the packets are allowed to mix with one another using rules designed to simulate mixing that would actually occur. This analysis, however, is limited to describing macroscale effects only. 2.3.3.3 Cell Balance (Network of Zones) Models Mann and Mavros (1982) introduced the network-of-zones model. Here, the mixer volume is viewed as a network of perfectly mixed cells (ideal reactors). The cells are connected in such a way as to mimic an experimentally obtained RTD. Mass exchange between the cells can be specified from experimental data or computed using turbulence models. The turbulence level in each cell is characterized by the local values of energy dissipation, s, and segregation macroscale, Ls. The model is therefore best suited for the description of macroscale effects. 2.3.3.4 Slab Diffusion Models Mao and Toor (1970) originally proposed a slab-diffusion model that was based on plug flow RTD. Here, the reactor fluid is made up of two planar slabs with a common interface. Mixing is controlled by molecular diffusion between alternate slabs of thickness 5, that enter the reactor completely segregated and proceed to intermix by molecular diffusion and undergo chemical reaction as they move through the reactor at the mean fluid velocity. In the case of chemical reaction, a single constant diffusion coefficient can be used for all components, i.e. Chapter 2: Literature Review 24 '- = 0 f + i?. (2.41) The symmetry imposed by the alternate slab makes all fluxes zero at x = 0 and x = 5, thus equation (2.41) can be solved to give the dependence of reaction products and mixing' parameters: C, -C, ( r ,Da, /? ) (2.42) where T = Dt/6 , Da = Mixing modulus, (3 = kinetic ratio. A major shortcoming of the slab diffusion model is the fact that reactions cannot proceed without fluid deformations. Ottino (1980) has generalized the model to include fluid deformation (lamelar models). Nevertheless, the model covers only the viscous convective part of the concentration spectrum. 2.3.3.5 Interaction by Exchange with Mean (lEM) Model Villermaux and Devillon (1972) and Costa and Trevissoi (1972) introduced the lEM model. The model relates the concentration change at a point due to chemical reaction and the mass exchange between the point and its surrounding environment. In its original formulation, the lEM model accounts for interactions in the inertial convective sub-range only. Here, the engulfment rate is given by e/k, where k is the kinetic energy of turbulence, and 8 the rate of dissipation of k. In this case, fluid engulfment is caused by vortices of size equal to energy containing eddies, and thus does not depend on viscosity. It is unlikely that such a model represents mixing at the smallest scales, although it has been used frequently. The lEM model has been extended to include incorporation parameters, making it similar to the E-Model (to be described later) in some respects (Villermaux and Falk, 1994). In this work, we use the E-model because of its strict but sound formulation based on turbulence theory. Chapter 2: Literature Review 25 2.3.3.6 EDD Model and its Simplification - The Engulfment Model In 1984, Bourne and Baldyga introduced a comprehensive model that was able to describe local mixing and reaction at the smallest mixing scales (Baldyga and Bourne, 1984). As its name implies, the EDD (engulfment-deformation-diffiision) model describes the last two stages of mixing described earlier (viscous convective and viscous diffusion) a fluid element undergoes in order to complete a fluid-phase reaction. The model was later simplified for the frequently encountered situation in liquid-phase reactors where the engulfment step is the only limiting stage, hence the name Engulfment Model (E-Model). The theoretical and mathematical development of these models is described in Appendix K. The E-model relates the product distribution for fast competitive reactions to the local turbulent energy dissipation rate. It has been used to study mixing and reaction effects in simulated microgravity conditions (Wenger et al., 1993), influence of pipe diameter in mesomixing (Baldyga et. al., 1993), agitated tanks (Baldyga and Bourne, 1988; Bourne et al., 1989; Bourne and Hibler, 1990; Bourne and Thoma, 1991; Balydga et al., 1993; Bourne and Yu, 1994), influence of viscosity (Bourne et al., 1989; Gholap et al., 1991; Bourne et al., 1995), influence of feed distribution (Bourne and Hibler, 1990; Rice and Baud, 1990), static mixers (Bourne and Yu, 1992; Jo et al, 1994), rotor-stator mixers (Bourne and Garcia-Rosas, 1986; Bourne et al., 1992; Bourne and Studer, 1992), grid generated turbulence (Bourne and Lips, 1991), tubular reactors (Bourne and Tovstiga, 1988), impingement mixing (Demyanovich and Bourne, 1989), bioreactors (Dunlop and Ye, 1990; Wenger et. al., 1993), centrifugal pumps (Bolzern and Bourne, 1985), jet reactors (Balydga et al. 1994), etc. This model is selected for this study because of its widespread use, and its solid support from a fluid mechanics view of mixing as outlined in Appendix K. Chapter 2: Literature Review 26 2.3.4 Mixing-Sensitive Chemical Reactions Experimental studies (e.g. Paul and Treybal, 1971; Truong and Methot, 1976; Cannon et al., 1977; Angst et al., 1981; Bourne, 1983; Pohorecki and Balydga, 1983; David and ViUeurmaux, 1983; Li and Toor, 1986) have demonstrated that the yield of desired product for fast competitive series or parallel consecutive reactions can be enhanced by increasing the homogeneity of reagents at the molecular scale. Partial segregation can therefore be studied through its influence on conversion and yield of chemical reactions. Such chemical reactions may thus be considered as molecular probes capable of measuring segregation. When coupled with the mathematical models described earlier, micromixing parameters may be extracted from experimental data. When two liquid streams, one containing reactant A, and the other containing reactant B are brought together in a turbulent environment, chemical reactions begin to take place as molecular diffusion brings reactants into contact. Consider a competitive-consecutive reaction of the type: A + B^^^^R (2.43) R + B^^^^S (2.44) If the reaction rates are slow enough that concentrations are uniform throughout the mixture before reaction takes place, the maximum amount of R that can be formed depends on the ratio k\lk2, the conversion and the initial mole ratio of the reagents. If, however, the reactions are very fast, the product distribution is influenced by the degree of mixedness on the molecular scale in the reaction zone, in addition to the kinetic and stoichiometric factors. This is because the rate of consumption of the reagents is sufficiently high that their transport to and away from the reaction zone causes steep concentration gradients between the segregated A-rich and B-rich regions and the reaction occurs in the narrow zones between these regions. A micromixing index may be defined Chapter 2: Literature Review 27 based on the quantity of the limiting reagent that ends up in the secondary product, and for the reaction system above is: X, = "-^^ (2.45) When mixing has no effect on the reactions, Xs is determined by kinetic parameters such as CBO, ^I/^2, and initial stoichiometric ratio. On the other hand, when mixing and reaction rates are comparable, Xs will depend on mixing intensity as well. We can therefore use Xs as a qualitative indicator (micromixing index) of the state of mixedness in a reactor. In addition to the qualitative character of the micromixing index, the use of mathematical models that relate product distribution to local energy dissipation provides a powerful tool for investigating micromixing in different reactor geometry and conditions. Thus we have the basis for using chemical reactions as micromixing probes. If we can have a reaction system that is completely characterized (kinetics) we can use the product distribution to obtain information about the state of mixedness of a system at the smallest scales. Thus by using chemical reactions one can obtain the information about local flow characteristics. In particular, we can estimate the rate of local turbulent energy dissipation from micromixing measurements. 2.3.4.1 Micromixing Test Reactions Over the last decade, several different reactions have been proposed and used to study micromixing in chemical reactors. Fast reactions that are sensitive to micromixing require competition between possible reaction pathways. Such reactions can be series-parallel (competitive consecutive, as in equations 2.43 and 2.44) or parallel (equation set 2 in table 2.1). The characteristics necessary for micromixing test reactions include: 1. Rapid, irreversible, second order kinetics; with rates capable of responding to the mixing intensity over a desired range. Chapter 2: Literature Review 28 2. Fast, accurate, and inexpensive method of analysis. 3. Highly reactive substances that can be used in low concentrations, with water as the preferred solvent. 4. Considerations for different hazards (fire, explosion, toxicity, corrosion, volatility) must also be taken into account. Since a chemical reaction becomes mixing-controlled if its half-life is of the order of, or smaller than, that of the mixing process, the product distribution of multiple reactions will depend upon the mixing intensity over a particular range of energy dissipation. The choice of a test reaction to study a specific system is dictated by, among other things, the range of energy dissipation rates that the reactions are sensitive to. The maximum rate of energy dissipation measurable can be easily determined by simulation. Table 2.1 shows some test reactions that have been widely used. Table 2.1: Micromixing Test Reactions Set 1 2 3 Reaction System i^ ' Bourne Reaction (Baldyga and Bourne, 1984) A +B ^' )i? R +B^^S A = 1-naphthol; B = sulphanilic acid R = monoazo dye; S = bisazo dye 2'"'Bourne Reaction (Bourne and Yu, 1994) A +B ^' >P C + B^^Q A=HCI; B = NaOH; C = CH2CICOOC2H5 P = NaCl; Q = CH2ClC00Na Extended Bourne Reaction (Bourne et al., 1992) A, +B ^ ^ o - i ? A, +B ^-^^p-R 0 R + B ^^^ >S L p-R + B ^-^^S A^+B—^Q Al = 1-Naphthol; A2 = 2-naphthol B = sulphanilic acid o-R = ortho-monoazo dye p-R = para-monoazo dye S = bisazo dye; Q = monoazo dye Rate Constants, ki m^/(mol.s) ki = 7300 ^2=1.6 /ti = 1.3x10^ ^2=0.0306 ^10=12238 A:ip = 921 A:2o= 1.835 A:2p - 22.25 ^3=124.5 Micromixing Index 2C s ~ C^ + ZLg Co Y - -^~C +c Co Y - -^ C +C +C +2C IC ^^^ C +C +C +2C Typical Range* (W/kg) 0.001-50 0.001-10 0.5-10^ Determined for product distributions between 0.05 and 0.45 Chapter 2: Literature Review 30 2.3.4.2 Bourne Reactions The original reaction used by Bourne and co-workers (Baldyga and Bourne, 1984; 1988) involves the coupling of 1-naphthol with diazotized sulphanilic acid under alkaline buffered conditions (reaction set 1 in Table 2.1). It has been widely used for studying micromixing in many different mixing configurations (see references on page 20). Most micromixing experiments made over the last two decades have used this system. Although the primary coupling between 1-naphthol and diazotized sulphanilic acid is very fast {k\ = 7.3 x 10 m /mol.s), the secondary coupling is too slow {k2 = 1.6 m /mol.s) to be able to work with high-intensity mixers. Consequently, most studies have been made at low energy dissipation ranges (s = 0.001 to 100 W/kg). Likewise, the acid-base neutralization of HCl and NaOH (2"^ * Bourne Reaction) is extremely fast ( i^ = 1.3 x 10^ m^/mol.s) but the secondary hydrolysis reaction is slow (kj = 0.0306 m^/mol.s) and therefore the maximum limit for energy dissipation is only 10 W/kg. In this work, we are interested in energy dissipation rates higher than these values and use an enhanced version of Bourne's original reaction. 2.3.4.3 Extended Bourne Reaction While the first Bourne reaction is adequate for determining micromixing at low energy dissipation rates (< 100 W/kg), the addition of 2-naphthol to the scheme (Table 2.1, reaction set 3) extends the range of energy dissipation that can be investigated to lO^W/kg as shown in Figure 2.4. This extension was first proposed by Bourne and co-workers (Bourne et al., 1992) and is suitable for study of mixing in medium- and high-intensity devices. Another major advantage of this reaction system selected is the flexibility offered by varying the initial stoichiometric ratio of 2-naphthol to 1-naphthol iN°^2 ^^°A\^ 0-This gives an extra degree of freedom as shown in Figure 2.4. Chapter 2: Literature Review 31 W 10^ 10^-10^ 10^ 10" 10-^  --j • • 1 1 1 V 1 1 i P^T— \ ^ ^=0 " \ ^ ^ ^ " ^ 1 1 1 1 1 — : \ ^ = 2 : \ \ " \ \ • \ \ • \ :\l-1 • 1 1 1 0.01 0.1 Product distribution.X Figure 2.4: Product distribution vs. energy dissipation rate for extended Bourne reaction (medium-intensity mixer conditions: C^, ICl= 0.0241, VAA'^B = 50, C ,^ = 0.52 molW, C; - 21.54 molW, N"^^ IN"^, = ^, N"^, INl = \ .2) Chapter 2: Literature Review 3 2 This scheme will be used throughout this study. Additional details and preparation methods for the chemicals are detailed in Appendix C. The relative yield of the dyes are given by product distributions - for the competitive-consecutive part of the reaction by: ^ ^ ^ - C +C '-fc +2C ^2.46) and, for the parallel part by: Co X , - - 2 _ (2.47) ^ C +C +C +2C ^Q ^ ^oR ^ ^pR ^ ^^S at high mixing intensities, the parallel competitive part dominates and negligible amounts of S are formed. 2.3.5 Estimation of Energy Dissipation Rates from Micromixing Experiments The engulfment model of micromixing may be applied when Sc < 4000 and relates product distribution {XQ OVX'^ ) to the local turbulent energy dissipation rate (s) for a given reaction. Thus, for the extended Bourne reaction: Xo=fn j^l" ]^° — V \ kinetics, - ^ , - ^ , Da, -^, Temperature (2.48) rs g yv ,^ Vg J When the reaction conditions are fixed (choosing reactor and selecting reactant concentrations, volumetric ratios and operating temperature) the product distribution depends on the Damkohler number, which is the ratio of the characteristic time for micromixing by eddy engulfment to chemical reaction, given by: Xs without prime refers to product distribution of the original Bourne reaction (1" reaction set in Table 2.1) Chapter 2: Literature Review 33 E V Da= '/" "" ^ (2.49) 1 + -A V ^ s , and related to the turbulent energy dissipation through the engulfment rate parameter, E, as estimated by Baldyga and Bourne (1984): E = 0.05S\-\ (2.50) Micromixing proceeds according to the engulfment model (see Appendix K.2 for the formulation according to Baldyga and Bourne, 1989), where the concentration of species is obtained from the solution of ^ = £(<c, >-c , ) + i?, (2.51) at Here, Cj and <c,> are concentration of / in the reaction zone and its surrounding respectively, and Rj is the chemical reaction source term. E-model simulation therefore involves solution of equation (2.51) for all components involved in a reaction scheme and calculating the product distribution at the end of reaction. By comparing the experimentally measured product distribution with that computed for the test conditions using the E- model, the local energy dissipation rate can be estimated. The energy dissipation rate estimated in this way is averaged over the reaction zone, which varies in size with mixing intensity. Figure 2.5 summarizes the procedure for estimating the local energy dissipation rate from micromixing experiments. Chapter 2: Literature Review 34 EXPERIMENT Geometry Test reaction ANALYSIS spectrophotometry (experimental) KINETICS Test reactions Rate constants SIMULATION Reaction zone s, exDt. conditions Xe (model) Change s value Local energy dissipation rate (e) Figure 2.5: Schematic for procedure to estimate the local energy dissipation rate from experiment and simulation Chapter 2: Literature Review 35 2.4 Turbulence and Mixing Pulp Fibre Suspensions 2.4.1 Introduction Turbulence is important in pulp mixing operations as it determines the degree and level of small-scale mixing that take place within the suspension. Turbulence accelerates reaction and promotes uniformity by ensxiring that small-scale homogeneity is achieved within the suspension. Since energy dissipation occurs through small-scale fluid motions, it can be used to characterize the level of small-scale motion and thus the degree of fluid-like behaviour created within a suspension. The generation of high intensity and small-scale turbulence is often considered the most efficient way to attain suspension uniformity. However, the concept of small-scale turbulence or microscale requirements in pulp suspension processing is still unclear. Oldshue (1994) suggests that the presence of fibres in pulp suspension may not affect small-scale eddies. This study will specifically examine the energy dissipation rate during this state of turbulent motion. Fibre properties and their measurement are described in Appendix B. Here, we review the characterization of pulp fibre suspensions. 2.4.2 Pulp Suspension Characteristics The addition of fibre to a liquid produces a suspension for which the rheological properties differ from those of the original liquid. Studies have shown behaviour characteristic of viscoelastic, pseudoplastic and yield bearing fluids (Wahren, 1980; Ganani, 1985; Bermington et al., 1990; Damani et al., 1993). Fibre suspensions form coherent fibre networks which posseses measurable strength resulting from interactions between neighbouring fibres (Kerekes et al., 1985; Soszynski and Kerekes, 1988a,b). Chapter 2: Literature Review 36 The unique rheology of fibre suspensions derives from the high number of contacts among fibres, even at low concentrations. Fibres moving in shear flow sweep out a much larger volume than their own. Mason and co-workers (1954) observed that fibre-fibre interactions became important when a "critical concentration" was exceeded. This is the concentration of fibre that gave one fibre in a spherical volume of diameter equal to fibre length, and was found to depend on fibre aspect ratio (L/d), fibre concentration (Cy) and fibre stiffness. Cylindrical particles rotating freely in a shear field transfer momentum from regions of high velocity to regions of lower velocity. The viscosity of suspension thus increases with concentration and aspect ratio until a critical concentration is reached. Above this concentration, fibres interlock and form rotating aggregates called floes, which in turn rotate freely. A further increase in concentration causes the formation of progressively more continuous networks. Thalen and Wahren (1964) observed that fibre suspensions adopt network strength when the suspension concentration exceeded a certain level, which they termed as "sediment concentration". Meyer and Wahren (1964) showed that the concentration that gave three contact points per fibre was sufficient for a continuous network to be built up, and closely approximated the "sediment concentration". The volumetric concentration, fibre aspect ratio and number of fibre/fibre contacts in a homogeneous suspension are related through the following equation: C. ' ^ < ^ , (2 .2) which can be simplified for 1 « nc < L/d to give the number of contacts per fibre as Chapter 2: Literature Review 37 Kerekes and co-workers (Kerekes et al, 1985, Soszynski and Kerekes, 1988; Kerekes and Schell, 1992,) have linked the concepts of Mason's "critical concentration" and Wahren's "sediment concentration" through the use of "crowding factor" (Ncr). The crowding factor represents the number of fibres within the rotational sphere of influence of a single fibre, and the concept is illustrated in Figure 2.6. This number is calculated from volume fraction Cy, the fibre length, L and the fibre diameter, d as: r r\ N=-a (2.54) v«y Figure 2.6: Schematic representation of number of fibres in a volume swept out by length of a single fibre (Kerekes and Schell, 1992). By simplifying Meyer and Wahren's expression (2.52) for the case L/d » 1 we can link Ncr and nc through: N^=-47m: 3(«.-l) (2.55) It is clear that the "critical concentration" and "sediment concentration" are special cases of the crowding factor (Ncr = 1 and Ncr = 60 respectively). Chapter 2: Literature Review 38 Let us consider suspension behaviour at various levels of Ncr. When Ncr < 1, fibres are free to move relative to one another in translation. They occasionally collide, and may remain temporarily together. As Ncr increases, more collisions take place through translation, and eventually through rotation. When Ncr reaches 60 where Uc s 3, fibres form a continuous network between the bounding walls of the suspension. In this condition, fibres lock into a network, and through the frictional forces between fibres create mechanical strength in the network (Mason, 1954). Based on this type of inter-fibre contact, fibre suspensions can be classified into different regimes as shown in Table 2.2 (Kerekes and Schell, 1992). Table 2.2: Fibre Suspension Regimes (after Kerekes and Schell, 1992) Regime Dilute Semi-Concentrated Concentrated Crowding factor, Ncr Ncr< l l < N c r < 6 0 Ncr > 60 Number of contacts per fibre, nc nc< 1 1 < nc < 3 nc>3 Type of contact between fibres Chance collision Forced collision Continuous contact Pulp suspensions are routinely characterized by their mass concentration, which is readily determined. However, the volume concentration of the suspension is required in order to calculate fibre network properties and to compare the rheology of pulp suspension. For a pulp fibre suspension where gas is excluded and the lumen is assumed to be fully collapsed, Bennington (1988) derived a useful expression for estimation of volume concentration from measured mass concentration (Cm) and the amount of water adsorbed in fibre wall (Xw) as: Cy = \+x,. yP.j f 1 + V V 1-C„ C m J Pf \yw J (2.56) Chapter 2: Literature Review 39 When a significant amount of gas is included in the suspension, the bulk characteristics of the suspension must also be included for conversion of mass concentration to volume concentration, i.e. C =C Pf Pw Pt (2.57) where pb is the bulk density of the suspension. A pulp fibre sample can contain fibres of varying lengths, diameters and cell wall thicknesses, depending on the species and age of tree, and the pulping process. Characterization of pulp fibre suspensions is accomplished by using suitable averages to represent the population. As suspension properties are affected primarily by the long fibre fraction, the length-weighted fibre length is used to characterize the suspension and for calculation of the fibre aspect ratio. Figure B-2 and B-3 in Appendix B show the length distribution and population distribution of FBK fibre used in this study. Because of the diversity of physical dimensions of natural fibres, synthetic fibres with uniform properties are sometimes used as model fibres in order to be able to relate suspension characteristics and rheological behaviour. Figure B-4 and B-5 in Appendix B shows the length and population distribution for nylon fibre. 2.4.2.1 Yield Stress When the number of contact per fibre is sufficiently large, fibre networks develop mechanical strength from forces that act at fibre contact points. Kerekes and co workers (Kerekes et al , 1985; Soszynski and Kerekes, 1988a, Kerekes, 1996) have identified that forces from fibre bending produce normal forces, which in turn give frictional forces that inhibit relative motion among fibres. When the external force applied to a suspension does not exceed these frictional forces, the suspension has properties of a solid (e.g. elastic modulus). However, when the force is large enough, a suspension adopts the Chapter 2: Literature Review 40 properties of a fluid, such as flowing when the stress is sustained. The shear stress at the point when the suspension begins to flow is termed as the yield stress of the suspension (xy). Bennington et al. (1990, 1995) has reported extensive measurements on the yield stress of different pulp suspensions. 2.4.3 Pulp Suspension Flow We shall first review the flow of pulp suspensions in pipes because this is where most of the fundamental data for pulp suspension behaviour has been obtained, probably due to geometrical simplicity and a sound foundation in flow engineering. Then we shall review the motion of fibres in rotary devices that are used to determine the fluid dynamic behaviour of flbre suspensions. This geometry is also important because many mixers for medium-consistency use the design. 2.4.3.1 Pulp Flow in Pipes Basically, there are three main flow regimes of flow in pulp suspensions, namely plug, mixed and turbulent flow as shown in Figure 2.7(a). Plug flow regime In this regime, the flow of fibres consists of a plug of fibres moving as a solid through the pipe as shown in Figure 2.7(a). Plug flow is a direct result of the tendency of fibres in suspensions to form networks/entanglements. Such networks have well defined mechanical strength properties and the shear stress applied as a result of flow is proportional to the pressure drop per unit length of pipe and the distance from pipe axis (T = (AP/L).r/2). Most of the pressure drop in this regime comes fiom mechanical friction between fibres and wall (Brecht and Heller, 1950). As the flow rate increases, there is a rolling regime in which fibres move away from the plug surface and roll along between the plug and pipe wall (B-C). Chapter 2: Literature Review 41 Rolling friction B Laminar armulus C Turbulent annulus D E Mixed Turbulence G :^(^u(? ( ^ v > ^ Plug " • increasing velocity (a) R (b) 10^  o CD (C) 10' 0.1 1 10 Velocity, [m/s] Figure 2.7: (a) Schematic representation of the three basic flow mechanisms of pulp suspensions in pipes (adapted from Gullichsen and Harkonen, 1981), (b) velocity profiles, (c) Pressure drop for the flow of Cm = 0.03 pulp in a 150 mm pipe (Brecht and Heller, 1950). A-D: Plug flow; D-F: Mixed flow; F-H: Turbulent flow. Chapter 2: Literature Review 42 A further increase in flow rate causes fibres to migrate away from the wall and form a sub-layer. This leads to a clear water armulus between the fibre plug and wall as shown in the Figure 2.7 (C-D). The shear stress in this regime is laminar and velocity distribution is linear (couette flow). The size of this annulus increases as velocity is increased and all shear takes place in this zone. Mixed Flow Regime As the flow rate increases, the clear water annulus eventually turns turbulent (D-E). Once turbulence occurs in the annulus, fibres peel away from the plug surface and are mixed in a random fashion in the annulus. The transition from plug to mixed flow occurs at a point where the shear stress imposed on the plug is equal to the strength of the fibre network (Xy/Xw = fp/R). As the flow velocity increases, the size of the turbulent annulus grows, which decreases the size of the plug (E-F). Turbulent Flow Regime Further increases in flow rate cause the turbulent annulus to cover the entire pipe section as shown in Figure 2.7 (F-H). While there are some doubts if the plug can be eliminated completely while shear at the pipe axis is zero (r = 0, x = 0), it is fair to assume that the plug has disappeared at sufficiently high flow velocities (Kerekes, 1994). In the turbulent regime, fibres move relative to each other and the suspension may be considered as a conventional fluid. While floes may form in this regime, they are continuously formed and disrupted, contrary to the plug flow regime where floes may exist as coherent entities (Kerekes et al, 1985). Considering the flow in a pipe. Figure 2.7(b) shows how the pressure drop changes in a pulp suspension as the different regimes are encountered. At low velocity (A to B) the pressure drop increases with velocity. This continues until the clear annulus of water forms (B). This annulus then grows and increases in size with increasing velocity, Chapter 2: Literature Review 43 resulting in decrease in pressure drop. This occurs because the velocity imposed by the pulp plug creates a gradient over a large distance, and thus wall shear stress decreases. Finally, transition to turbulent flow in the annulus occurs (C), giving the mixed regime described earlier. As the velocity is increased beyond this point, the pressure drop increases. The curve eventually crosses the water line (D). Beyond this point, the suspension exhibits a phenomenon termed as drag reduction, whereby the pressure drop of the suspension is less than that of water (further discussion on drag reduction is discussed in section 2.4.4.1). At very high velocities, drag reduction is diminished, and the pressure drop loss curve approaches that of water. 2.4.3.2 Flow Regimes in Rotary Devices Consider a pulp suspension placed in a chamber between rotor and housing as shown in Figure 2.8 (a), with the rotor speed being increased gradually from zero to higher velocities. In order to create relative motion within a pulp suspension, sufficient energy must be applied to overcome the network strength. Before that, only the part of the pulp that is in direct contact with the rotor will flow, thus forming a cavity of flowing pulp in a radial zone beyond the tips of the vanes of the rotor. As the rotor speed is increased, the intensity of motion in the cavern increases, and so does the size of the cavern. As the speed of the rotor is increased further, the extent of the radial cavity zone extends outward to the housing of the vessel. If this housing has vanes that hold the plug in place, the strong tangential motion induced by the rotor will adopt a radial component. This brings the entire vessel contents into intense motion and eventually complete turbulent flow. Chapter 2: Literature Review 44 (a) c o H 2 — • Complete turbulence 7 ^ A p ^ L - ^ '^ 8% A - ^ 1 i 'M 0 1 1 ^ (b) 0 1000 2000 Rotor speed, rpm Figure 2.8: (a) High-shear rotary device (b) Typical response of pulp fibre suspensions in a high-shear rotary device (GuUichsen and Harkonen, 1981). The first comprehensive work on rotary devices was carried out by GuUichsen and Harkonen (1981) and had been a starting point for the revolutionary advances in medium-consistency (MC) processing operations in pulp and paper industry. Once it was determined that fibre suspensions at medium-consistency could be made to flow just like water, operations that were traditionally carried out at low consistencies could now be Chapter 2: Literature Review 45 carried out at medium consistencies (e.g. pumping and mixing). This resulted in considerable savings through reduced water use, storage facilities and improved mixing (to be discussed later in this chapter). Figure 2.8(b) shows a typical shear response as obtained by Gullichsen and Harkonen in their experiments. All in all, rotary devices can be considered to have flow regimes that are somewhat similar to pipe flow, namely a mixed regime of turbulent flow adjacent to plug flow and a turbulent flow in which the entire contents are in relative motion. We should therefore be able to draw from some of the experiences in pipe flow when studying rotary devices where helpful. We now describe a phenomenon that has been associated with this motion and the ensuing suspension properties. 2.4.3.3 Pulp Fluidization Pulp suspensions in turbulent flow regime display a range of fluid-like behaviour, which has sometimes been referred to as "fluidized". One way to determine if a pulp suspension is fluidized is to determine the power dissipation per unit volume (or mass). The concept behind this is that if fibres are in a high degree of relative motion, then considerable energy is dissipated through the action of viscosity. Thus a measurement of the power dissipation is an indicator of the relative motion among fibres. However, the power dissipation considered here is the overall power, and not local power as has always been assumed. Thus one of the aims of this study is to be able to measure, or at least estimate the local energy dissipation in pulp suspension at these conditions. The onset of fluidization was observed when vessel contents came into a vigorous state of turbulence. Accordingly, Wahren (1980) was the first to employ the concept of "power dissipation per unit volume" to signify the onset of fluidization. He estimated Chapter 2: Literature Review 46 values for the onset of fluidization from the pulp yield stress of a fibre network and obtained the following correlation: T' £p= — = \.2x\0'C„ (2.58) /^  where Cm is the suspension mass concentration given and Sp the power dissipation at the onset of fluidization (W/m ),Xy is the yield stress and |j, is the viscosity of water. From GuUichsen and Harkonen's work (1981), fluidization was defined as the onset of turbulent motion throughout the vessel. An expression for Sp was obtained from their data by Bennington (1988) as: Ep=l>Ax\Q'Cl' (2.59) In a more recent work, Bennington and Kerekes (1996) have developed an expression to estimate the energy dissipation necessary for fluidization by considering the power necessary to overcome the yield stress of the suspension. The power dissipation necessary to achieve a state of complete turbulence {Sp) in a rotary device is given by: (D V'' 4 / - . 2.5 Sp = 4.5x10'C r 'T \D J (2.60) for the range (1.3 < Dj/D < 3.1; 1 < Cm < 12.6), where D is the rotor diameter and Dj is the housing diameter as shovra in Figure 2.8(a). The local value of Sp was estimated by extrapolating equation (2.60) to zero gap, i.e. D ^ - D T £^ =4.5xlO'C^' (0.01<Cm<0.12) (2.61) The above expressions give values for power dissipation that can differ for about three orders of magnitude. There is obviously a need to reconcile these estimates as they refer to the same phenomena. One of the main discrepancies here arises from the value of Chapter 2: Literature Review 47 viscosity used. Let's address this issue using arguments put forward by Bennington and Kerekes (1996) with regard to viscosity. 2.4.3.4 Apparent Viscosity It is common practice to associate changes in suspension behaviour to its physical properties. When a pulp suspension is fluidized, it must have its own apparent viscosity. Bennington and Kerekes (1996) determined the apparent viscosity of pulp suspension from yield stress measurements. Power dissipation is linked to suspension yield stress through: £,=-^ (2.62) where Xy is the yield stress, /u^ is the apparent viscosity of the suspension. Using yield stress for the SBK pulp suspension of Bennington and Kerekes (1996), the following expression was obtained for ju^ H^ = 1.5JC10-'C„'' (0.01 < Cm < 0.12) (2.63) Newtonian fluids are characterized by viscosity that is independent of shear rate. Non-Newtonian behaviour may be characterized by the non-linearity of the stress-strain relationship or visco-elasticity (display of solid-like and liquid like properties). These non-linearities may arise from orientation of molecules with increasing shear rate, thus giving a shear dependent viscosity. When such re-orientation of molecules leads to lower apparent viscosity, we have a shear-thinning fluid and when it leads to increase in viscosity we have a shear thickening fluid. Both types of behaviour has been observed in fibre suspensions depending on flow geometry (Wilkstrom and Rasmuson, 1998). For dilute suspensions (crowding factor less than unity), viscosity can be predicted from Einstein's equation (1907): CItapter 2: Literature Review 48 /.„=//„(l + 2.5C,) (2.64) where |a,ni is the viscosity of the mixture (suspension) and |u,o is the viscosity of the suspending medium. Several attempts have been made to include the effect of particle interaction by including higher order terms in the Einstein's equation. It is unlikely that such an equation can describe viscosity of fibre suspensions. A reasonable match with experimental data has been obtained from a theoretical expression (Fedor, 1974; Ishii and Zuber, 1979; Barnes et al, 1989; Ferguson and Kemblowski, 1991) of the form: Mn, = Mo \ ^v J (2.65) where C^^is the maximum solid phase concentration, and [|J.] is an intrinsic constant, depending on the solid. A value of [|J,] = 2.5 has been successfrilly used for particles with aspect ratio of 1 (spherical and nearly spherical particles), whereas [|a.] = 0.07A^ ^ has been successful for rod-like particles (Barnes et al, 1989). A value of [|J.] = 0.02A was found to match the empirical correlation developed by Bennington et al (1989) with the FBK fibre used in this study. Another property to be used in these investigations is the suspension density, which, together with viscosity gives the kinematic viscosity (v). The mixture density (pm) may be estimated from the respective volume concentrations of the solid and suspending medium as: / . „ , - ( 1 - C , ) p , + C , / . , (2.66) where pw is the density of the suspending medium and ps is the density of the solid phase. The kinematic viscosity of the mixture is therefore given by v^ „, = — (2.67) Pm Chapter 2: Literature Review 49 2.4.4 Turbulence Parameters in Fibre Suspensions Different techniques have been used to determine turbulence parameters in fibre suspensions. Turbulence measurement using impact probes were carried out by earlier investigators (Daily et al., 1961; Mih and Parker, 1967; Duffy and Titchner, 1976; Wahren, 1980) but spatial resolution and interference with flow pattern development limited their reliability. The dispersive properties of turbulent motion have also been used to infer turbulence parameters (Bobkolwicz and Gauvin, 1965, 1967; Anderson, 1966; Luthi, 1987; Louettgen et al., 1991) either by use of hot wire anemometry or thermistor probes. Optical means of measuring turbulence such as LDA (Ek, 1979; Kerekes and Garner, 1982; d'lncau, 1983a; McComb and Chan, 1985; Steen, 1989) have also been used, although the range of application is limited to low suspension concentrations due to the opacity of the pulp suspensions. Kerekes and Garner measured turbulence intensities in channel flow of a Cm = 0.005 suspension following a grid using LDA. They found that the average turbulence intensity was reduced by up to 28% when compared to water tests. They also noted an apparent increase in turbulence intensity at higher wave numbers. The increase in turbulence at high wave numbers was interpreted as an artifact, due to inability of the laser used to discriminate between seeding particles and fibres at high wave numbers. Most of these studies point to the reduction of turbulence intensity in the presence of fibre. However, some results have indicated increase in turbulence at scales comparable to fibre length (Ek, 1979; McComb and Chan, 1985). The apparent change observed by Kerekes and Garner was also found at lengths corresponding to fibre length (Kerekes and Garner, 1982). 2.4.4.1 Drag Reduction in Fibre Suspensions The flow of fibre suspensions in pipes has been found to exhibit drag reduction (Kerekes, 1970; Hoyt, 1972; Kerekes and Douglas, 1972; Radin et al., 1975; Lee and Duffy, 1976). Drag reduction occurs in a fibre suspension when the skin friction loss of Chapter 2: Literature Review 50 the suspension is lower than the skin friction loss for the suspending medium at the same volumetric flow rate. Fibre variables that affect the flow in turbulent drag reducing regime have been studied for dilute suspensions using synthetic and natural fibres (Daily and Bugliarello, 1961; Bobkowicz and Gauvin, 1965; Kerekes 1970; Hoyt, 1972; Radin et al., 1975). It is evident from these studies that for fully developed turbulent flow, the amount of drag reduction increases with volumetric concentration (Cy), length to diameter ratio (L/d) and fibre fiexibility. Furthermore, no drag reduction was observed in the case of non-fibrous material (Daily and Chen, 1964; Radin et. al., 1975). When compared to the phenomena of drag reduction observed in polymers, the drag reduction in the presence of fibres takes place in the turbulent core flow (Lee and Duffy, 1976) while that of polymer solutions takes place near the wall (Virk, 1975; McComb and Rabie, 1982a,b,c). This was demonstrated by noting that, when a polymer is injected at the centre of a pipe, drag reduction was observed downstream after the dye has reached the wall region, while when the dye is injected at the wall, the onset of drag reduction was much earlier. It is further evidenced that there is a maximum (asymptotic) value for drag reduction that can be reached for a given flow. Kerekes and Douglas (1972) confirmed this for the case of fibre suspensions and Lumley (1977) for the case of polymer solutions. Noting the drag reduction potential of fibres and polymer solutions, some investigators have studied the combined effect of polymer solutions and fibres on drag reduction (Valeski and Metzner, 1974; Lee et al, 1974; Quraishi et al, 1977; Hattori et al, 1988; Virk et al., 1997). When fibres and polymer solutions are combined, they reduce drag by an amount greater than the two independent additives, thus indicating a synergistic effect. This again goes to suggest that the only difference between fibre drag reduction and polymer drag reduction is where it takes place and its scale. Although there are many studies on drag reduction, recent effort has concentrated mainly on correlating drag reduction with flow parameters (Darby and Pivsa-Art, 1991; Chapter 2: Literature Review 51 Virk et al., 1997; Sood and Rhodes, 1998). Fewer studies have considered the mechanistic aspects and the relationship between drag reduction and local energy dissipation. According to Kerekes and Douglas (1972), the interaction between fibres and surrounding fluid suppresses the radial turbulent movement of the fluid, and thereby decreases drag through a reduction in radial momentum transfer. At the same time, fibres may also contribute to increase in radial momentum transfer through fibre-fibre contact, and thereby increase drag. In order to observe a net effect of drag reduction, Kerekes (1970) concluded that particle inertia must exert a significant though not overwhelming effect on the flow. Increasing concentration increases the magnitude of drag reduction, therefore, a limiting condition exists whereby the influence of increased viscosity (through increased concentration) will entirely supercede drag reduction. Such conditions were established and verified by Kerekes and seem to agree with preposition put forward by Lumley (1977) with regard to drag reduction observed in polymer solutions. For the case of drag reduction observed in polymers solutions, Lumley (1977) suggests that drag reduction in polymers occurs as a result of expansion of polymer molecules near bounding surfaces (in the buffer layer vorticity is equal to the strain rate), thus causing a difference between buffer layer and viscous sublayer leading to dampening of small-scale eddies. Since molecular expansion will also cause increase in elongational viscosity, and increase in viscosity reduces the strain rate, and thereby stops the expansion. This leads to an equilibrium condition where partially expanded molecules increase the viscosity just enough to hold them at that expansion. This lead to a saturation of drag reduction with concentration, analogous to what Kerekes and Douglas found experimentally for fibre suspensions. While this study will not look at drag reduction per se, it is obvious that our results for local energy dissipation and the effect of fibres are expected to reflect this situation, and possibly offer more insights to these complex phenomena. Chapter 2: Literature Review 52 2.4.4.2 Flocculation in Pulp Suspensions This is the tendency of fibres to aggregate and form regions of locally denser networks as pointed out earlier. Floes are regions of greater strength in suspension and may behave as independent entities in a flowing suspension (coherent floes). This phenomenon has been investigated for a long time (Forgacs et al, 1958, Robertson and Mason, 1954; Mason, 1950; Kerekes and co-workers, 1983, 1985, 1992, 1995; Steen, 1989, 1991). Turbulence and flocculation in pulp fibre suspensions are interdependent because turbulent shear gradients in a flow deform and disrupt fibre floes while at the same time fibres exert a damping effect on turbulent velocity fluctuations. Robertson and Mason (1957) were among the first investigators to appreciate the role of fibre flocculation and the resulting formation of structure in the suspensions. They suggested that the presence of floes might inhibit the development of small-scale turbulence, thus reducing the rate of energy dissipation. However, these suggestions were based on speculations from pressure-drop measurements; no specific measurements of turbulence parameters were undertaken. Parker (1961) observed that higher turbulence intensities resulted in smaller floes. Anderson (1966) evaluated flocculation from photographs taken in transmitted light and turbulence from the difftision of a dyestuff. He observed that turbulence intensity was inversely proportional to the degree of flocculation. Our current understanding of the flocculation behaviour is based on extensive investigations carried out by Professor Kerekes and co-workers at the University of British Columbia, spanning from early 80's to-date. The main parameters that govern mechanical flocculation are fibre interlocking forces (Kerekes and Soszynski, 1988a,b). Zhao and Kerekes (1993) have shown that by increasing the viscosity of the suspending medium, the concentration at which floes formed could be increased. According to Kerekes and Schell (1995), fibre length and fibre coarseness exerts their influence on Chapter 2: Literature Review 53 suspension uniformity through the number of contacts per fibre and size of floes formed. This phenomena is therefore governed by the same fibre variables namely, Cy, L/d and flexibility. It is important to note that this phenomenon has been investigated predominantly with respect to paper forming operations where flocculation is a nuisance, and thus concentrations of no more than Cy = 0.02% and also lower intensities than those found in high-shear mixing devices. It is therefore not very clear how the state of flocculation is under high mixing intensities - especially the number of contacts. Lee and Brodkey (1987) have characterized the mechanism of floe disruption by a simple model comprising of large-scale global phenomena that includes deformation, breakage and fragmentation of entire floe and small-scale phenomenon of surface erosion involving individual fibres. Steen (1991) presented a numerical model for flocculation in turbulent flow. Mechanism for rupture and build-up are discussed in terms of interaction between turbulent structure and fibres. These mechanisms seem to point out the same observations that floe formation and disruption are based on hydrodynamic interactions. Floe build-up or disruption is mainly due to turbulence induced fluctuations. Therefore, energy dissipation rates should reflect the state of flocculation. 2.4.4.3 Turbulent Diffusivity in Pulp Suspensions Bobkowicz and Gauvin (1965) injected a stream of hot water at the centerline of a turbulent flowing suspension and measured the rate of dispersion from temperature measurements across the radial direction of a vertical pipe. When compared to the case of fibre-free flow, they observed a decrease in dispersion with increasing fibre concentration. Other experiments were limited to visual observation of dye dispersion in the presence of fibre suspensions (Anderson, 1966; Duffy et al., 1976 and Luthi, 1987). These have generally yielded qualitative information indicating a decrease in turbulent Chapter 2: Literature Review 54 dispersion with increased fibre concentration. Louttgen et al., (1991) injected a saline solution in a vertical pipe and measured the axial and radial spreading of the solution by monitoring the conductivity. Conductivity profiles indicated that turbulent dispersion could increase or decrease depending on position in the flow system. 2.4.5 Mixing Pulp Suspensions Key requirement for good mixing is that all pulp in the mixer flow. This means that the yield stress of the pulp suspension must be exceeded everywhere in the vessel. If this is not accomplished, dead zones appear where there is no shear, and no mixing occurs. A major unit operation in pulp and paper where mixing is of critical importance is pulp bleaching. Bleaching involves multiple simultaneous reactions between bleaching reagents and lignin (other components of pulp may also be involved to a lesser or greater extent depending on the nature of the bleaching reagent used and its selectivity). Therefore, the rate and quality of mixing will affect the efficiency of the bleaching process. For example, in chlorination, the dissolution, diffusion and chemical reaction determine the overall bleaching rate. Studies in pulp bleaching have shown that improved mixing enhances efficient use of bleaching chemicals and production of more uniformly bleached pulp (Torregrossa, 1983; Kolmodin, 1984; Bergor et al., 1985; Helander et al. 1994). Nevertheless, good mixing in pulp bleaching and its characterization remains a difficult task (Bennington et al , 1989). The goal of mixing processes in pulp bleaching is to ensure that each fibre is contacted with the correct amount of bleaching chemical. This requires mixing on all scales relevant to the system. One useful way of defining the non-uniformities is the distance or scale of relative motion required to remove them. Chapter 2: Literature Review 55 Large-scale (macroscale) non-uniformities are defined as those that require substantial bulk movement of fluid over relatively large distances to even out. In a stirred vessel, the largest macroscale distances may be of the order of vessel diameter (DT). If the size of non-uniformity is considered as the mixing load, Ls, then the number of turnovers needed to remove is n = LS/DT. When the mixing of tw^ o uniform streams takes place, the distance between the two streams entering the mixer largely determines the mixing load. Therefore, we can sometimes achieve better macroscale mixing by simply altering the flow entry points in a mixer. On the other extreme, non-uniformities approaching the molecular level (microscale) require molecular diffusion to even them out. The order of magnitude for this scale is therefore the smallest scale of turbulence, i.e. the Kolmogoroff length (^K)- The presence of fibres in suspension imposes an intermediate scale of non-uniformity having the size comparable to that of the length of fibres (floes are usually 2 - 3 times the length of fibre (L)) and floes - the fibre-scale. Local relative fluid movement induced by velocity gradients or turbulence removes fibre-scale non-uniformities provided that the yield stress has been overcome. Table 2.3 shows these different scales of mixing in pulp fibre suspensions. Table 2.3: Range of mixing scales in fibre suspensions (after Bennington et al, 1989) Scale Macroscale Fibre-scale Microscale Order of Magnitude ^ D «L «XK Achieved by Bulk motion Turbulent shear Diffiision There are many challenges to achieving uniformity at all these scales due to the complex rheology of fibre suspensions. First, fibre suspensions have a yield stress that Chapter 2: Literature Review 56 can produce zones of reduced motion and stagnation in the suspension. This phenomenon has been studied by Bennington and co-workers (1990) and shown to be significant with increasing fibre concentration. The second issue concerns the flocculated nature of pulp suspensions as described earlier. Floes must be disrupted in order to facilitate contacting of fibres with chemicals during passage through mixers. 2.4.5.1 Pulp Mixers Devices for pulp bleaching have evolved from STR's to static mixers to high intensity mixers, with an increase in typical overall energy dissipation rates firom 20W/kg to 1000 - 5000W/kg respectively. Table A-8 in Appendix A shows an example of the range of mixers used in pulp bleaching. At low mass concentrations (Cm ^ 0.04), the mixing of liquid chemicals into pulp suspensions is a relatively easy task and can be carried out in continuous stirred vessels (STR). These mixers provide needed backmixing, in addition to creating turbulence in the discharge flow of impeller. However, better mixing could be achieved when the power dissipation within the suspension was increased. This led to the development of dynamic mixers where the emphasis was placed on the generation of turbulence within the suspension at the expense of backmixing. Where necessary, a series of mixers was used to provide additional residence time. Static mixers were also found to provide sufficient mixing at low suspension concentrations although backmixing was eliminated as the suspension moves in plug flow through these mixers. At medium mass concentrations (0.06 < Cm < 0.15), mixing is conventionally accomplished using peg mixers. These are tubular vessels having one or two shafts, with attached pegs, that rotate between stationary elements attached to the mixer casing. As the pulp suspension is conveyed through the mixer, the rotating pegs shear the suspension against stationary elements. This shearing action exposes new fibre surface to the Chapter 2: Literature Review 57 bleaching chemical. However, these mixers are not designed to create turbulence within the suspension, but rather distribute chemicals uniformly. Dynamic high-shear or high-intensity mixers were developed when it was realized that turbulence could be induced by dissipation of sufficient energy within the suspension. Several mixers have been designed to operate in the high-energy dissipation range, which is effective for dispersing floes and inducing good fibre-scale mixing. In order to keep the power requirement at acceptable levels, the mixer volumes are usually kept low, and consequently, the residence time is small (as short at 0.05 seconds) so little macroscale mixing occurs. Medium- and high-intensity mixers are increasingly applied in pulp bleaching operations (GuUichsen and Harkonen, 1981; Reeve et al., 1984; Bennington et al., 1989; Bennington, 1996), which has led to improved pulp bleaching. For example, ozone bleaching at medium consistency has been possible because of improvements in mixing technology (Torregrossa, 1983; Kolmodin, 1984; Bergor et al., 1985; Liebergott, 1990, White et al., 1993; Hurst, 1993; Helander et al., 1994). Table A-8 in Appendix A shows a comparison of different pulp mixers for a fixed production rate. It is evident that the most distinctive feature is the energy dissipation. In high-shear mixers, chemicals and pulp are mixed during passage through zones of intense shear. The high shear is achieved by imposing high rotational speeds across narrow gaps through which the pulp suspension flows. Despite different design variations among the high-shear mixers, they all attempt to fluidize the suspension in the mixer-working zone. The high shear rate ensures floe disruption and good fibre scale mixing but also leads to high power dissipation per unit volume. Small mixing zones are therefore used to keep the power dissipation at acceptable levels, although the size and power requirements of these mixers have always been a concern (Bermington, 1996). Chapter 2: Literature Review 58 2.4.5.2 Mixer Characterization by Power Dissipation Power dissipation is used to characterize mixers because it indicates the level of relative suspension motion achievable. Since power expended over time yields energy, the total energy treatment per unit mass of fibre, which indicates the total mixing action imparted in a mixer, has also be used to characterize pulp mixers. Table A-8 in Appendix A shows some typical values for industrial pulp bleaching mixers. In general, the required level of suspension motion determines the power dissipation. At low suspension concentration lower energy dissipation is required to induce movement in suspension and consequently longer residence times are possible. At medium suspension concentration, greater power dissipation is necessary in order to induce motion. However, as the power required to impart turbulence is so large, extremely short residence times are used in high-intensity mixers. The concept of power dissipation needs further development and/or clarification as to the actual effective power that is utilized to generate small-scale motion. Although some studies have indicated that the level of turbulence can determine to some extent the distribution of products created in bleaching (Earl, 1990), it is not clear how this energy is utilized or what fraction of the total expended energy is actually useful. One of the objectives of this thesis is to answer this question. 2.4.5.3 Mixing Assessment in Bleaching Operations The assessment of mixing quality in pulp bleaching is mostly done indirectly, by measuring the product quality at the end - e.g. measurement of such properties as residual chemical content, pulp viscosity or pulp brightness. Other indicators of mixing quality include - temperature variation, charge deviation, etc. (see Table A-6 in Appendix A). Measurements of bleaching uniformity by Paterson and Kerekes (1984) represents fibre-scale uniformity at best (see Table A-7 in Appendix A) since the micro-sampler used was limited to 2 mm separation. Nevertheless, these techniques were useful Chapter 2: Literature Review 59 for comparison of different mills or mixer operations. Bennington and Bourne (1989) and Bennington and Thangavel (1993) have investigated the effects of microscale-mixing in fibre suspensions by use of mixing sensitive reactions. No other measurements of microscale uniformity in pulp bleaching devices are found in the literature. There is clearly a need for further understanding of fibre suspension behaviour at the smallest scales, which is of relevance to industry. Specifically, what is the effect of fibres on the local energy dissipation rate? Before we conclude this review, we shall take a look at studies that have been done in other dispersed systems to try to elucidate the role of particles on turbulence and energy dissipation. 2.5 Turbulence Modulation in Other Dispersed Systems Insights of the effect of fibres on turbulence can be obtained by looking at parallel studies in fluid flow when particles of different shapes and sizes are introduced in a turbulent flow field. This problem has been studied both experimentally and theoretically by a number of investigators, mostly working with low particle concentrations in pipes or free jets. Systems studied varied, including gas-solid, gas-liquid, glass beads or spherical balls over a wide range of particle diameters (0.04 to 4mm), densities (0.012 to 2500 kg/m^) and volume concentrations (10'^ to 0.20). A review of experimental data by Gore and Crowe (1989) concluded that no consensus has been reached yet as to the effect of particles in changing turbulent intensity of the liquid phase of dispersed suspensions. They proposed a simple physical model to explain the decrease in turbulent intensity caused by addition of particles based on the ratio between particle diameter (dp) and the characteristic length of the most energetic eddies in the absence of particles (U). The critical value (dp/le = 0.1) provided a demarcation of particle sizes that caused turbulent intensity of liquid phase to either Chapter 2: Literature Review 60 increase or decrease with the addition of particles to flow. The length scale is not expected to change dramatically with the addition of particles (Gore and Crowe) although there are those who believe otherwise (Tsuji and Morikawa, 1982; Tsuji et al, 1984; Hetsroni and Sokolov, 1971). The presence of solid particles affects the turbulence of the carrier fluid by not only modulating the intensity but also changing the energy spectra -mostly by decreasing the spectra components at high frequency with a corresponding change in the distribution of the fluctuation energy (Hetsroni and Sokolov, 1971; Faeth, 1987; Tsuji et al., 1988). However, Tsuji and Morikawa (1982) did not observe appreciable differences in the frequency power spectrum for particles with large dp/le, whereas for smaller dp/U, the power spectrum increased at lower frequencies and increased at higher values. This was similar to that observed by Kerekes and Gamer (1982) for pulp fibre suspensions. Mizukami et al. (1992) also found that streamwise power spectra densities increased at high frequencies, but they interpreted this as noise and suggested that high frequency data should be ignored. Observations by Kerekes and Gamer (1982) found the same evidence, and were also dismissed as laser ambiguity. However, this time the two phases were discriminated. On the analytical side, a demarcation based on particle Reynolds number (Rep) is suggested (Hetsroni, 1989; Rashidi et al., 1990; Tsuji et al., 1984; Parthasarathy and Faeth, 1990; Yuan and Michaelides, 1992). In this case, the disturbance of flow velocity due to wake or vortices shed by large particles is the predominant mechanism for enhancement of turbulence, while the dissipation of energy from the flow for acceleration of small particles which follows the fluid motion is the main mechanism for suppression of turbulence. Using the data compiled by Gore and Crowe (1989), Hetsroni (1989) estimated the limits for particle Reynolds number as Rcp > 400 and Rcp < 110 for enhancement and suppression respectively, where Chapter 2: Literature Review 61 RQ^=-£1 i i (2.68) where (u-us) is the fluctuating relative velocity between fluid and particles - slip velocity, dp the particle size and v fluid viscosity. Rashidi et al, (1990) found that large (but moderately light) polystyrene particles (dp - 1.1 mm, pp = 1030 kg/m ) increase turbulence intensity and Reynolds stresses, whereas small particles (dp/U < 0.1) decrease the same parameters. These effects were found to increase with particle concentration. Heavier particles (in this case glass beads, dp = 0.088 mm, - pp = 2500 kg/m^) did not show significant modulation of turbulence. Measurements of burst frequency and mean streak spacing showed no significant change with increase in particle concentration, implying no change in dissipation scale. Schreck and Kleis (1993) observed a monotonic increase in turbulence intensity with increasing particle concentration. The increase was found to be larger than what would have been expected from slip velocity between two phases. They speculated that particles enhanced the transfer of energy to the smaller eddies, extending the dissipation spectrum to the smaller scales. However, the modification was only apparent in the stream-wise direction while transverse data was inconclusive. Another criteria used to signify the effect of particles on the liquid phase turbulence is the particle relaxation time, t* (time taken for a particle to be accelerated to within 63% of its terminal velocity) defined (Hetsroni, 1989) as r * = - ^ ^ (2.69) 18// and the characteristic turnover time for energetic eddies 4 defined as Chapter 2: Literature Review 62 /, = ^ (2.70) " U where U is the characteristic velocity and U the length scale of the most energetic eddies. The length scale of the most energetic eddies in pipeline flow may be taken as 0.2 x pipe diameter (Gore and Crowe, 1989; Hetsroni, 1989). In stirred vessels, U may be taken as 0.1 X impeller diameter (Kresta and Wood, 1993; Mavros et al., 1987; Kresta, 1998). Small particles {t* « te) increase the rate of turbulent energy dissipation compared to single phase flow by extracting energy from the flow and dissipating it. When t* = te, particle response is mixed, i.e particles can increase or decrease the energy dissipation rate. For larger particles, where t* » te, the particles are sensitive to turbulent fluctuations of low frequency only (high wave number) and possess a higher mean velocity relative to the fluid. In a recent investigation, Yarin and Hetsroni (1994), using a method based Prandtl's mixing length theory and kinetic energy balance, have suggested the limits for particle-turbulence interactions. In the limit of fine particles (dp/le < 0.1), the turbulence depends only on the total mass content of the suspension; whereas for coarse particles ( dp/le > 0.1), it is determined by total mass content, density ratio of the phase and the aerodynamic properties of the particles. This corroborates the conclusion by Faeth and others (Faeth, 1987; Parthasarathy, 1990) for the existence of small particle and coarse particle limits. In summary, different factors contribute to turbulence modification in dispersed systems. Yuan and Michaelides (1992) provide a good summary, namely: (a) Dissipation of turbulent kinetic energy by the particles (b) Shedding of vortices or the presence of wakes behind the particles (c) Fluid moving with the particle as added fluid mass to the particle Chapter 2: Literature Review 63 (d) Enhancement of the velocity gradient between two rigid particles (e) Increase of the apparent viscosity due to the presence of particles (f) Deformation of the dispersed phase While these studies offer insight into interactions between dispersed phase and the continuous phase, the spherical particles do not interact with each other in the same manner as fibrous suspensions. An analysis with fibre suspension will, nevertheless, benefit from these insights. 2.6 Scope of Work The objective of the present research is to study mixing in pulp fibre suspensions and investigate the effect of fibres on turbulent energy dissipation rate. The scope of this work includes a study of the magnitude and distribution of local energy dissipation in the liquid phase of pulp fibre suspensions in medium- and high-intensity mixing devices. The mixing-sensitive reaction schemes and mathematical model developed by Bourne and co-workers will be used for this study. Findings from this study would lead to further understanding of how pulp fibres modify liquid phase turbulence and subsequently affect mixing under conditions used in industrial mixers. While major attention will be given to fibre suspensions, extension to other dispersed systems will also be carried out. This should enable us to make a general comparison on the effects of various dispersed systems under turbulent mixing condition. Since the processing of pulp suspension always contain a certain amount of gas, this thesis will also look at the effect of gas in the presence of fibres as well as other dispersed systems Chapter 3: Experimental 64 Chapter 3 3 EXPERIMENTAL 3.1 Mixers Two rotor-stator mixers, where the rotor extends the full length of the vessel, were used in this study. The mixers had nominal volumes of 2.5 and 3.4L and were capable of imparting a maximum of power 200 and 5000 W/kg, respectively. 3.1.1 The Medium-Intensity Mixer The 2.5L medium intensity mixer was constructed from a cylindrical tank made of Plexiglas. Four baffles spaced at 90 degrees on the side of the vessel were provided to eliminate swirling effects in the vessel. The rotor consisted of four-blades mounted on a cylindrical hub at 90°, and extended the full length of the vessel as shown in Figures 3.1 and 3.2. Both ends were supported by bearings to minimize frictional losses. Dimensions of the mixer are given in Table 3.2. The mixing power was provided by a 0.25kW motor (G.K. Heller, Floral Park, NY) and the power input to the mixer was measured using a reaction torque transducer (model # QWFK-8M; Sensotec, Columbus, Ohio) mounted on the rotor shaft. The impeller rotational speed measured using a digital tachometer (model # 08212; Cole Palmer, Chicago, IL). The cylindrical Plexiglas tank was enclosed in a square tank connected to a temperature-controlled bath to maintain constant temperature during the coupling reactions. The square tank also minimized diffraction effects during high-speed video visualization. Chapter 3: Experimental 65 (A) Figure 3.1: Schematic of experimental setup for semi-batch operation in medium-intensity mixer: (A) mixing chamber, (B) micro-pump, (C) feed reservoir, (D) motor, (E) torque transducer, (F) feed pipe, and (G) overflow reservoir Chapter 3: Experimental 66 Figure 3.2 Schematic of the medium-intensity mixer showing rotor and tank dimensions (Ail dimensions in mm). Chapter 3: Experimental 67 The top lid of the mixer was provided with feed ports at r = 55, 70 and 90 mm from the rotor axis as shown in Figure 3.3. In this way, the feed tube could be positioned at at three different radial positions. The feed line consisted of a 120 mm feed tube of 1mm internal diameter. When not in use, the feed ports were blocked with stainless steel plugs. In the axial direction, the feed point was determined by the depth of the feed tube inside the tank. This could be adjusted to any distance within the span (height) of the tank. For mapping experiments, a total of fifteen injection points were used to map the mixer cross-section in a plane mid-way between baffles. These correspond to axial positions (z/H) of 0.05, 0.25, 0.50, 0.75 and 0.95, and radial positions (r/R) of 0.579, 0.737 and 0.947, as shown in Figure 3.4. Figure 3.3: Photograph of the mixer plan showing feed point locations and the rotor. Chapter 3: Experimental 68 Table 3.1: Dimensions of the Medium-Intensity Mixer PARAMETER Tank diameter, DT (mm) Tank height, H (mm) Number of baffles Baffle dimensions (mm) Rotor diameter, D (mm) Rotor height, (mm) Number of rotor vanes Vane dimensions (mm) Vessel volume, VT (m ) D/DT Value 190 100 4 10x10x100 100 100 4 25x10x100 2.5x10"^ 0.53 Chapter 3: Experimental 69 H=100 r = 90 70 55 Rotor 'Fn '51 Ri=50 Ro=95 '19 Fn* •F„ 'F , , F,,» F:i1 *FT,2 F ^ S * 'F41 •F4. . F4 .» • r-^?. r .s^* t 5 20 25 25 20 Figure 3.4: Cross-section of mixing tank showing working section and feed point locations for profile measurements. All dimensions in millimeters. Chapter 3: Experimental 70 3.1.2 High-Shear Mixer A high-shear mixer designed earlier (Bennington, 1988) was used for the high-intensity micromixing experiments. The mixer consisted of a 3.4L mixing chamber formed between two concentric cylinders, with the inner rotor capable of achieving rotational speeds of up to 83.3 rev/s (5000 rpm), while imparting up to an average of 5000 W/kg in the chamber. The mixer was powered with a 22.5 kW variable speed DC motor. Torque was measured through a built-in strain gauge that covered the range 0 -43±0.3 N.m. and rotational speeds were determined using remote a optical sensor over the range 1 - 83.3±0.1 rev/s. The plexiglas plate enclosing the mixing chamber was provided with multiple feed points at the same radius (r = 80 mm) at 60° intervals midway between the wall baffles as shown in Figure 3.4. This allowed for faster feeding, and hence shorter time for experimentation (Bourne and Thoma, 1991). Details of the mixing chamber are shown in Figure 3.6, and a schematic arrangement for semi-batch operation in Figure 3.5. A flow loop consisting of a flexible tube of 50 mm internal diameter and 4m total length connecting the high-shear mixer to a 270L polyethylene tank as shown in Figure 3.5. A centrifugal pump located below the tank and powered by a 7.5HP motor (Leeson Electric, Ontario) was used to circulate the suspension through the loop. Temperature was kept constant using a cooling coil connected to HAABCE temperature control system. The flowrate through the loop was measured using a flow-tube mounted on the return line and connected to a Rosemount magnetic flow transmitter. By using the recycle loop, the temperature rise due to high energy dissipation rates was kept to a minimum (± 1 °C) as the reactants were cooled before being recirculated to the mixer. For example, in one Figure 3.5: Schematic of experimental setup for high-intensity mixer with flow loop: ((Ml) motor, (P) centrifugal pump, (C) high-shear chamber, (Tl) storage/recirculation tank, (D) rotor drive, (M2) secondary mixer, (2) standby tank, (F) flowmeter, (CC) cooling coil. Flow direction shown by arrows. Chapter 3: Experimental 72 turnover time, the reactants spend an average of 1 to 3 seconds in the high shear zone and 15 to 40 seconds respectively in the 270L buffer tank (corresponding to QA = 70 to 200 L/min). This enabled the mixer to be operated for longer times without accumulating excessive heat in the mixing chamber. Feed Distributor Location of feed points Feed Tube (7^0 L-100-100 220 Figure 3.6: Schematic of high-shear mixing chamber showing feed arrangement (all dimensions in mm). Chapter 3: Experimental 73 Table 3.2: Dimensions of High-Intensity Mixer used PARAMETER Tank diameter, DT (mm) Tank height, H (mm) Number of baffles Baffle dimensions (mm) Rotor diameter, D (mm) Rotor height, mm Number of rotor vanes Vane dimensions (mm) Vessel volume, V (L) D/DT Values 220 100 6 lOxlOx 100 100 100 6 25x10x100 3.4 0.45 3.2 Materials 3.2.1 Fibres A fully bleached kraft (FBK) pulp was obtained from a western Canadian mill (Howe Sound Paper, BC). The pulp, obtained in sheet form, was soaked in distilled water and defibred in a standard British disintegrator for 15,000 revs, to obtain a pulp freeness of about 650 ml CSF. The water in the pulp was then extracted using a centrifugal extractor to reach Cm = 0.45, packed in an airtight container, and refrigerated (4°C) until needed. Nylon fibres were obtained from Cellusuade Products (Illinois, MI). Polyethylene fibre was obtained from Allied Fibre and Plastics (Petersburg, VA) and fibreglass from Fibreglas Canada Inc. (Guelph, ON). All synthetic fibres were first washed with mild acid to remove surface impurities, then rinsed with distilled water and dried at 105°C The dimensions and physical properties of the fibres used are summarized in Table 3.3. Chapter 3: Experimental 74 Table 3.3: Dimensions and Physical Properties of Fibres used Fibre Type FBK Nylon Polyethylene Fiberglass Dimensions L (mm) 2.3 1.1 3.2 3.2 6.2 7.5 d (mm) 0.022 0.045 0.045 0.038 0.038 0.09 L/d 104 24 71 84 164 83 Density (kgW) 1500 1140 1140 970 970 1770 Physical Properties Xw 0.62 0.08 0.08 0.01 0.01 ND Elastic Modulus (Pa) 3.5x10^ 1.8x10^ 1.8x10^ 6.5x10'" 6.5xl0'° 6.9x10'" 3.2.2 Other Dispersed Systems To compare results obtained with fibres with other dispersed phase studies, the effects of spherical particles were investigated. Glass beads were obtained from Potter Industries Ltd. (Montreal, QC), and polyethylene beads were obtained from Novacor Chemicals (Edmonton, Alberta). Dimensions and physical properties of these materials are shown in Table 3.4. With glass particles, it was also possible to examine the effects of particle size and density. With the polyethylene beads, it was possible to examine the effect of particle shape for the same material (c.f. polyethylene fibres). In all cases, water was used as the liquid phase. 3.2.3 Gas as Dispersed Phase A number of runs were conducted with gas as the dispersed phase. In this case, only sufficient fluid was used to create the required void volume, with nitrogen gas used to prevent oxidation of reactants or products. The gas void fraction was varied from 0 to 40%, with the majority of gassed experiments being carried out at 20%. The presence of gas was also examined in conjunction with fibres and water (three-phase systems). Chapter 3: Experimental 75 Table 3.4: Dimensions and Physical Properties of Spherical Particles used Material Polyethylene beads Glass beads Diameter, dp (mm) 2.7 0.11 0.55 0.80 1.20 2.40 Density (kg/m') 970 2520 3.2.4 Source of Chemical Reagents and Analysis 3.2.4.1 Chemical Reagents 1-naphthol and 2-naphthol used in the coupling experiments need to be of very high purity (99+%). Some difficulties in finding a source of 1- and 2-naphthol were encountered, despite claims of high purity. We found that the crystal form of the naphthols varied widely - from a fine white crystalline powder form, to dull brown crystalline clumps (dp = 2 - 3 mm). The most consistent and preferred source of 1- and 2-naphthol was Sigma Chemicals. There ware no problems with the rest of chemicals used in this study. Appendix C describes the preparation and source of the coupling components used in this study. 3.2.4.2 Sampling and Analysis of Reactant and Dye Products At the end of each experimental run, the limiting reagent is fully consumed and we have a uniform concentration of dye products throughout the mixer. A 2mL sample of dye products is drawn from the mixer and pipetted into a 40-mL beaker. 20mL of Na2C03/NaHC03 buffer solution (pH = 9.9, ionic strength = 444.4 molW) is added to • all values shown are averages. Table B-7 in Appendix shows the size distribution for glass beads Chapter 3: Experimental 76 dilute the sample to obtain concentrations within the range of spectrophotometer. Part of the diluted sample was then drawn into a 1-cm path length, quartz cuvette and scarmed in the spectrophotometer (UV-Visible Diode Array Spectrophotometer - Model HP5284A, Hewlett Packard, Waldbronn, Germany). The amount of each dye component in the mixture was quantified through multi-component linear regression analysis using published spectra (Bourne et al., 1992). Figures C-2 and C-3 in Appendix C show the standard spectra for the product dyes and 1- and 2-naphthols respectively. The accuracy with which experiments were performed could be evaluated by comparing the total moles of dye produced (VA+VB)(CQ+CO-R+CP.R+2CS) with the quantity of limiting reagent (B) used in the test (VBCBO)- For tests made in aqueous solution, mass balances were found have a mean and standard deviation of 98.2+1.3% (based on 10 measurements). The analytical accuracy (standard deviation) with which the product distribution (XQ) could be measured was ±0.003, with the experimental reproducibility for both aqueous and fibre tests measured as ±0.009. All values were obtained for 10 repeated measurements as shown in Tables H-1 and H-2 in Appendix H. 3.3 Micromixing Test Procedures 3.3.1 Medium-Intensity Reactor The experimental conditions used for the tests done at medium-intensity are given in Table 3.5. The naphthols were buffered with 111.1 mol/m each of Na2C03 and NaHCOa just prior to reaction giving pH = 9.9 and an ionic strength of 444.4 mol/m as detailed in Appendix C. Coupling reactions were carried out in a semi-batch mode. Freshly prepared diazotized sulphanilic acid (B) was metered gravimetrically at a selected injection point with the aid of constant-flow micro-pump (model 7617-70, Ismatec Instruments, Switzerland). The overflow from added B was returned to the mixer using a Chapter 3: Experimental 77 peristaltic pump (Cole Palmer model 9013) at the same flowrate. Figure 3.1 shows the schematic of experimental setup for semibatch operation. When tested with a dispersed phase, the required amount of the dispersed phase was weighed and added to the aqueous solution mixture of 1- and 2-naphthol to give the desired mass concentration prior to reaction. For the case when the dispersed phase consisted of FBK fibres or nylon fibres, twenty minutes were allowed for the fibres to allow adsorption to reach completion before running the coupling reactions. Table 3.6: Experimental Conditions for Medium-Intensity Mixer Component C° C° IC" ^ A2 ' ^ A\ CI PH VA N° IN" Uf VAA^B tf T Concentration 0.52 molW 2.0 21.54 mol/m^ 9.9 2.5 L 1.2 1 50 180s 25+0.5°C 3.3.2 Critical Feed Time Analysis A number of experimental conditions must be met to ensure that the product distribution can be correctly interpreted in terms of local energy dissipation. For example, the product distribution can be affected by the bulk transport of A to the reaction zone. If the feed time (tf) of B is so short that B arrives at the reaction zone faster than A can be transported there by circulation within the vessel, a stoichiometric imbalance occurs, the reaction no longer takes place in an A-rich environment and the secondary reaction increases. However, by keeping the feed time sufficiently long we can maintain the reaction in an A-rich environment ensuring that XQ measures microscale effects only. Chapter 3: Experimental 78 This is confirmed when an increase in feed times no longer influences XQ. Experiments were conducted to determine the critical feed time (tc) for all reactor systems as described in section 6.2. It was found that a feed time of 180 seconds was sufficient for operation in the microscale regime in the medium-intensity mixer. 3.3.3 Baseline Characterization of Micromixing Baseline characteristics were obtained when the medium-intensity mixer was operated with water alone with rotor speed varied from 4 to 25 rev/s. This would then be used for comparison when the reactor is operated with different dispersed phases. Power dissipation characteristics were also obtained for mixer operation in water and with the dispersed systems. For micromixing experiments, baseline characteristics were obtained for a fixed feed point while varying the rotor speed. They were also obtained for a fixed rotor speed with different injection positions across the mixer - along the rotor axis and at different radial positions. Most experiments for single point were carried out at position F32 (see Figure 3.4). 3.3.4 High-Shear Mixer with Flow Loop The High-Shear mixer was operated in two modes. First, experiments were carried out with recirculation. In this case, SOL of 1-naphthol and 2-naphthol solution were initially placed in the 270L tank. Then the amount of fibre needed to obtain a desired mass concentration was added. The mixture was then recirculated in the loop using a centrifugal pump. Once the desired operating conditions were established in the system, injection of the limiting reagent at selected feed-points initiated the micromixing experiments. The same micropump used for medium-intensity mixer was used to feed the limiting reagent (diazotized suphanilic acid) in the high-shear mixer. Experiments were carried out with fibre concentrations of Cm = 0.0 to 0.06. Mass concentrations above Cm = 0.06 could not be carried out in this semi-batch mode due to the inability of the Chapter 3: Experimental 79 centrifugal pump to maintain recirculation in the system and inadequate mixing in the 270L tank. Only FBK fibre was used in the flow loop tests. In order to reach the investigative target of Cm = 0.10, it was found necessary to operate the high-shear mixer without the flow loop. This necessitated changes in the operating conditions. The number of feed nozzles was increased from two to six, thus enabling the feed time to be reduced to 21 seconds. It was confirmed that this feed time was still in micromixing regime by performing critical feed time experiments at 50 rev/s (3000 rpm). For the flow loop, samples were taken from a pipe outlet below the re-circulation tank and analyzed for reaction products using the diode array spectrophotometer. For experiments without the flow loop, samples were taken after opening the mixer cover plate and removing the mixer contents. The concentration of dye products was determined using the spectrophotometer as described earlier. Table 3.7 shows the experimental conditions for high-shear mixer with and without the flow loop. Table 3.7: Experimental conditions for high-shear mixer. Parameter c /c° CI PH VcAl+Al") N^JNl Hf VAA^B Recirculation rate tf T Units MolW -Mol/m^ _ L . . . L/min s °C With flow loop 0.52 2.0 43.08 9.9 50 1.2 2 100 20-210 200 25+1 Without flow loop 0.52 2.0 43.08 9.9 3.4 1.2 6 100 _ 21 25±5 Chapter 3: Experimental 80 3.4 Overall Energy Dissipation The mean overall power consumption is obtained from torque (M) and rotor speed (N) measurements using: P = 2KNM (3.1) The overall energy dissipation per unit mass (also called specific power consumption or power input per unit mass) can then be obtained: pV where V = volume of the mixer and p = density of the fluid or suspension mixed. From dimensional analysis, the power number (Np) for a given rotor is constant for fully turbulent flow and is given by: A^  = ^ , (3.3) Although equation 3.3 was derived for a single-phase liquid, the presence of a second phase can be accounted for by substituting the average suspension density for p. If we combine equation (3.2) and (3.3) we obtain an expression for average power dissipation: We can therefore obtain the power number by taking a least square regression of the average energy dissipation against the cube of impeller rotational speed. This method has been found to be useful in obtaining the power number under turbulent conditions (Mmbaga and Bennington, 1997) and has been used recently by other investigators (Bubicco et al., 1997). Chapter 3: Experimental 81 3.5 Flow Visualization 3.5.1 Bulk Flow Characteristics The mean flow characteristics examined were the mean power consumption and the flow pattern. Overall power characteristics were examined in the presence of water (baseline) and in the presence fibre suspensions at different mass concentrations. This consisted of measuring the power dissipation at different rotational speeds. For each set of experiments, the required amount of the dispersed phase was added to the aqueous solution mixture to give a desired mass concentration prior to reaction. 3.5.2 Flow and Reaction Zone Visualization Flow visualization was carried out with the aid of a high-speed video (HSV 1000, NAC Videosystem) with which events taking place over a time span of one millisecond could be captured and stored for analysis. The video was operated at 500 frames per second for most visualization experiments. The reaction zone visualization was carried out with the aid of a fast reaction (acid-base neutralization) using phenolphthalein as an indicator. Further details of the technique and results are given in section 6.2 in Chapter six. 3.5.3 CFD Analysis Computational fluid dynamics (CFD) was used to verify the flow pattern and the overall flow characteristics in the medium intensity mixer. This was done for the water (base case) only. Information regarding the flow patterns and overall characteristics will be obtained for comparison with those observed experimentally. Appendix F gives the principles involved in the computation of macro-flow characteristics in turbulent field by using a commercial CFD code (Fluent 4.5). Chapter 4: Adsorption of Reactant and Product Dyes on the Dispersed Phase 82 Chapter 4 4 Adsorption of Reactant and Product Dyes on the Dispersed Phase 4.1 Introduction The application of mixing-sensitive reactions in dispersed systems is complicated if the reactants and/or product dyes adsorb onto the dispersed phase. In order to ensure correct interpretation of the experimental data, it is necessary to account for the differential adsorption of product dyes, when it occurs. It is also important to know the actual initial reactant concentrations, and predict any concentration changes that may not be due micromixing interactions. Adsorption can affect interpretation of the results in two ways. First, if the reactants are adsorbed on the fibre and are not available for reaction, the initial molar ratio of reactants (N°A/N°B) can be changed. Mathematical modeling of the reaction system can quantify the magnitude of this effect. Second, the product distribution could be altered by the preferential adsorption of one or more of the product dyes. This would undoubtedly mask the micromixing effects being investigated. Bennington and Thangavel (1993) correlated the changes in product distribution with the dispersed solids loading to account for adsorption, thus permitting correct interpretation of the results using the micromixing model of Balydga and Bourne (1989). In the past, failure to account for adsorption on the solid phase (Villermaux et. al., 1994) led to erroneous conclusions regarding micromixing effects. A clear understanding of the adsorption characteristics is essential for reliable interpretations in dispersed systems. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 83 4.2 Adsorption Phenomena The adsorption of a solute onto a solid surface occurs as a result of both the solvent and adsorbent interactions. The primary force may relate to the solvophobicity of adsorbate for the solute or to a particular affinity of the adsorbate for the surface of the adsorbent. Parameters that determine the adsorption characteristics of a given system include concentration, molecular weight, size, polarity and structure of the adsorbate. Also, the presence of other competing adsorbates in the system, and the physical characteristics of the adsorbent material require consideration. Usually, a combination of these factors is responsible for adsorption in any particular system. A search for the complete mechanism of dye adsorption is therefore beyond the scope of this study. Instead, we shall use correlation of experimental data with models to predict and correct for adsorption effects. Some common models for adsorption that can be used to relate adsorbate and equilibrium concentration have been reviewed (Mmbaga and Bennington, 1998) and are presented in Appendix D.l In the next section, the adsorption of 1-naphthol and 2-naphthols (Ai and A2) is studied over the range (0 - 5.25 mol/m ), while the reaction products (R, Q, and S) were studied over the concentration range obtainable under the test conditions chosen (0.05 - 1.0 molW). The dispersed phase (FBK, nylon, and polyethylene fibres) concentration was varied from zero to ten percent by mass for the tests. 4.3 Adsorption Tests The adsorption experiments were carried out under identical conditions to those used for the mixing tests, namely, T = 298K, buffered to a pH 9.9 and ionic strength of 444.4 molW using 111.1 molW of NaHCOs and Na2C03. Working concentrations for the extended coupling reaction were set as 0.52 mol/m for 1-naphthol and 1.04 mol/m Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 84 for 2-naphthol. Coupling of this mixture with 21.54 mol/m'^  of diazotized sulphanilic acid in an aqueous alkaline environment at the different mixing intensities available in a medium-intensity mixer produced a mixture of different product dyes having product distributions in the range, XQ, ranging from 0.07 - 0.42. Also, tests were conducted where the solution was dominated by only one dye. These solutions were prepared by varying mixing conditions during the generation as outlined in Appendix C.2. Adsorption experiments were carried out individually for 1-naphthol and 2-naphthol, and jointly in a 2:1 ratio mixture. Adsorption tests were conducted by contacting a known weight of fibre with 450mL of a previously prepared sample solution of known composition. Since the addition of fibre in wet form (unavoidable for FBK samples) introduces a certain amount of water to the system, the total amount of water in the sample was fixed at 500 mL, with the balance of the water (50 mL) made up by the water carried with the pulp and additional buffer solution. This ensured that all samples received constant dilution, and had the same initial dye concentration. The suspension was mixed in a laboratory mixer/shaker (Eberbach, Aim Arbor, MI) at 400 rpm. Samples were taken at 5-minute intervals for kinetic studies. After equilibrium was attained, the aqueous phase was separated from suspension using a sintered glass flannel, and then analyzed for concentration using the spectrophotometer. The amount of solute adsorbed was obtained by material balance. 4.4 Results and Discussion 4.4.1 Adsorption of 1-naphthoI and 2-naphthol on FBK Fibre Figure 4.1 shows the adsorption of 1- and 2-naphthol vs. contact time. Equilibrium adsorption of these components occurs in approximately 30 minutes and does not decrease by more than 5% over the range of suspension mass concentrations Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 85 tested in this study. The adsorption of 1-naphthol and 2-naphthol increases as the FBK suspension concentration is decreased. For example, for the adsorption of 1-naphthol, at Cm = 0.01 qe = 1.5|a-mol/g, while for Cm = 0.10, qe = 0.4 |j,-mol/g (here Cm refers to the mass of fibres in a unit mass suspension (fibre + water); whereas qe refers to mols of solute adsorbed per unit mass of fibre) . For an equal quantity of fibre, the suspension at the lower mass concentration can accommodate more Ai, i.e 1.5 vs 0.4|a.-mol/g. This may be interpreted in terms of the porous gel structure of the cellulose substrate, where the increase in amount of adsorbent per unit volume decreases the accessibility of cellulose (Hedberg and Lindstrom, 1993). This phenomenon has been observed in other systems (Dada and Wenzel, 1991). The naphthol adsorption data were fitted to the adsorption models described in Appendix D.l using non-linear least squares fitting routine (NSLF Microcal Origin, Northampton, MA). Reasonable agreement was found for all models as shown in Table D-1. We chose the Langmuir model because it gave high and consistent x^ values for all tests. The competition between the naphthols was examined by conducting adsorption tests in a solution containing 0.52 mol/m 1-naphthol and 1.04 mol/m of 2-naphthol. Under these conditions, a decrease in the adsorption of each component was observed. For example at Cm = 0.03, adsorption tests conducted independently (Figs. 4.1(a) and (b)) give adsorption of Ai of 1.2 \i-n\o\lg and A2 of 1.5 |^-mol/g. With A1/A2 in a 1:2 mixture (Figure 4.2) we find that the adsorption decreases to 0.8 |j,-mol/g and 0.58 |a,-mol/g respectively. This implies that there may be difficulties when one attempts to correlate the adsorption effects by using models developed using individual component isotherms (see appendix D.2 one such attempt). Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 86 (a) 2 -.PJ D 0 A V c. 0.01 0.03 0.05 0.10 l-naphthd t, min (b) .CD D 0 A V c. 0.01 0.03 0.05 0.10 2-naphthol t, min Figure 4.1 : Plot of adsorption vs. contact time for FBK suspension for different mass concentrations at 298K. (a) 1-naphthol (b) 2-naphthol. Adsorption tests conducted independently. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 87 -2^  o zL "tt) 1.5-1.0-0.5-0.0-' I • 1 ' 1 ' 1 — ' 1 • 1 ' 1 ' 1 — n — 1-naphthol —0—2-naphthol 1 ' 1 -~ 0.00 0.02 0.04 0.06 0.08 0.10 Figure 4.2 Equilibrium adsorption from a 1:2 mixture of 1-naphthol and 2-naphthol on FBK fibre as a function of suspension mass fraction. 5-• 1-naphthol o 2-naphthol 2000 3000 C ,^ \i-mo\IL 5000 Figure 4.3: Plot showing the equilibrium adsorption of 1- and 2-naphthol on FBK suspension (Cm = 0.03) determined with each naphthol independently. Solid lines fitted using Langmuir model. Chapter 4: Adsorption ofReactant arid Product Dyes on the Dispersed Phase 33 For adsorption experiments carried out on the naphthols separately (Figure 4.3), 2-naphthol is adsorbed more than 1-naphthol at the same concentration. The greater affinity of 2-naphthol over 1-naphthol can be explained in terms of its solvophobicity for water. Since the bonding between a substance and the solvent in which it is dissolved has to be broken before adsorption onto the solid phase can occur, the greater the solubility of a compound, the stronger the bond and hence the smaller the extent of adsorption (Weber, 1985). This was found to be true for the present system. Here, the solubility of 2-naphthol is 3.25 mol/m^ while that of 1-naphthol is 5.25 molW. Despite adsorption from our FBK reaction systems, the 1-naphthol and 2-naphthol concentration in solution is only slightly reduced as compared to the non-adsorbing cases. Also, the initial ratio of the two naphthols (A2/A1) remains constant over a wide range as shown in Figure 4.4. The degree to which adsorption would affect the product distribution can be readily estimated. For example, for the case considered above (FBK suspension at Cm - 0.03), if the solution added to the fibre suspension has C^, = 0.521 mol/m'' and C 2^ "^  ^ -^6 mol/m^, the equilibrium naphthol concentration would be reduced to 0.496 and 1.04 mol/m^ respectively. Modeling shows that this has negligible effect on the product distribution, with XQ decreasing by 0.003 to 0.005, which is below the level of reproducibility of ±0.01. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 89 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 .10 Figure 4.4 Change in the ratio of 2-naphthol to 1-naphthol (A2/A1) with increasing fibre concentration 4.4.2 Diffusional Limitations A certain amount of water is held in pulp fibres - both in the cell wall and in the fibre lumen. For the FBK fibre used in this study, 0.62 g of water is held in the fibres for every gram of fibre (i.e., Xw - 0.62 g/g). Part of this water may not be available during mixing, thus changing the concentration of the reactants in suspension. For a suspension of Cm = 0.01, only 0.6% of water resides in fibre wall, whereas for Cm = 0.03, the amount of water increases to 1.9%. If the reactants associated with this water are not readily available for reaction, the product distribution, and consequently the interpretation of mixing tests may be affected. A check on the Weisz modulus (Mw) (Levenspiel, 1984) will determine if there are any diffusional limitations for the coupling reactions. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 90 Calculations of the Weisz modulus (Appendix D.3) shows that all primary reactions involving 1-naphthol and 2-naphthol have Mw > 7 (see Table D.5 in Appendix D). This implies that there is strong diffusion resistance for part of chemicals that are held in the fibre wall. Let's take an extreme case and assume that all adsorbed chemicals are not available for reaction. Since the amount of material adsorbed has been determined, the effect on product can be predicted using the engulfment model of micromixing. For example, if we assume that the total amount of solution inside a fibre (lumen and fibre wall) is twice the weight of dry fibre, at Cm = 0.05, for a worst case scenario, about 5% of the 1-naphthol and 2-naphthol will not be available for reaction. This will reduce the initial molar ratio from 1.2 to 1.14. The E-model shows that this will decrease the product distribution (XQ) by 0.1 - 0.5%, which is less than the error associated with micromixing experiments. Furthermore, the flexing action imparted on the fibre during mixing would probably pump the fluid in and out of the fibre matrix, thus all components may still participate in the reaction (Reeve and Earl, 1986). 4.4.3 Adsorption of Product Dyes on FBK Fibres FBK fibres have some affinity for all the product dyes, but especially for mono-substituted dyes {Q and p-R and o-R) as shown in Figure 4.7. A number of mechanisms may contribute to the adsorption behaviour observed, including the establishment of an equilibrium between the bulk water and water in the fibre, direct dyeing of cellulose, surface charge on the cellulose fibrils and van der Waal forces (Rys and Zollinger, 1972; Arson and Stratton, 1983; Larsson and Stenius, 1987; Zhang et al., 1994). Some of the likely mechanisms are discussed briefly in the next subsections. 4.4.3.1 Ionic Interactions The azo dyestuffs are negatively charged. Likewise, the pulp fibre consists of micro-fibrils of cellulose, which are negatively charged throughout the whole pH range. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 91 The hydrophilic properties of cellulose are due to the chemical constituents, namely the hydroxyl groups (0H-) located along the cellulose chains. The dissociation of ionic groups is the main source of surface charge in the fibre (Zhang et al., 1994). An increase in pH or carboxylic content increases the surface charge on the cellulose. At the high pH used in this study (pH = 9.9) it is highly unlikely for direct chemical bonding between the negatively charged species to account for the observed adsorption. The possibility of hydrogen bonding being the mechanism for attachment between the cellulose fibre and dye is also not probable as the affinity due to hydrogen bonding of water molecules of the amorphous areas of the fibres is so strong that the dye molecules are not able to displace these water molecules (Abrahart, 1977). The affinity of the dyes for cellulose must therefore be explained on other grounds, namely van der Waals forces. 6 - 1 5 -4 -D) 1 '-D- 2 -0-^ A n r 1 ^ A /^-"^ Ik-— 1 x ait o 1 n o V 1 6 o V 1 S o V 1 D Q O pR A oR V S 6 <-> , ---10 20 30 t, min Figure 4.5: Adsorption of product dyes on FBK vs. contact time at 298K (Cni= 0.05, C' 'Q= 0.4165 molW, C V R = 0.1186 molW, C V R = 0.3684 molW, C°s = 0.0793 molW) Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 92 4.4.3.2 London Forces (van der Waal) The most plausible explanation for the observed dye adsorption is the van der Waals forces. Two chemically inert molecules can attract one another with London forces (van der Waal) whereby the energy of attraction decreases by the inverse power of distance. This observation is corroborated by the fact that bis-substituted product, S (a relatively large size molecule) is not adsorbed to as great an extent as the mono substituted products, o-R, p-R and Q (relatively small size molecules). Accordingly, as many dye molecules as sterically possible can approach the fibre molecules. The planar structure of the dye ions and the inner surface of cellulose favour this. 4.4.3.3 Hydrophobic Interactions These forces may be construed as the attraction between hydrophobic parts of a dye molecule and the hydrophobic parts of the cellulose chains. Cellulose is less hydrophobic when compared to nylon (Murray, 1996), and this explains the greater adsorption in nylon than in FBK fibre suspension (to be shown later). 4.4.3.4 Accessibility of Cellulose Cellulose has a two-phase morphology containing both crystalline and non-crystalline material (Roberts, 1996). The existence of regions with differing degrees of molecular disorder means that the extent to which a reagent can penetrate the cellulose structure will depend both upon the molecular size of the reagent, and also the extent to which the cellulose is swollen under experimental conditions. Most chemical reagents (including water) penetrate the non-crystalline regions only. The accessibility of the non-crystalline fraction under conditions of use has more significance to adsorption than crytstallinity (Tripp, 1971; Roberts, 1996). This would also explain the relatively higher adsorption rates for the smaller molecules of Ai and A2 both in size and cellulose accesibility. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 93 4.4.4 Correlations to Account for the Adsorption of Dyes on FBK Fibre In order to correctly evaluate the product distribution for a range of mixing conditions, it was necessary to find means of accounting for dye adsorption onto fibres. Two approaches have been evaluated, namely, following the adsorption of individual components separately, or following a composite correlation to account for changes in the product distribution (see Appendix D for details). In the first case, the adsorption behaviour of each individual dye component is followed independently to establish their isotherm. In this case, a single model may be used for all components, provided it gives the best correlation. One may also follow the suggestion given by Yun et al., (1996) to select the best model for each of the individual components that gives best fit to single-component equilibrium data. In our case, the Langmuir model gave the best correlation for all components (see Table D-1 to D-4 in Appendix D). Obviously, this would work best if the individual components adsorb independently on the fibres. The other approach is to develop a composite correlation that relates XQ values with and without adsorption. In this case, a solution of known product distribution, XQ"' , is prepared and contacted with different fibre concentrations. This procedure is repeated for different initial concentrations, thus covering the expected range of XQ. From all the data, a single correlation is developed that relates the final/measured concentration with the initial concentration. The first approach (individual component correction) was not successful in this study due to interactions and competition between the different dyes in solution. We therefore followed the second approach, which seems to be the best method so far given the complexity of the mixtures is involved in this study. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 94 4.4.4.1 Composite Correlation The change in product distribution, XQ, due to adsorption of the dyes can be correlated with the amount of fibre in suspension as shown in Figure 4.6. The solid lines in the figure are drawn using equation (4.1) which is developed from multiple non-linear regression of adsorption data for a range of initial product distribution (0.07 < XQ< 0.42) and fibre concentrations (0.0 < Cm < 0.10). Xg =X™' *(1-8.62C„+47.33C^) r^=0.97 (4.1) here, XQ is the product distribution measured in the system and XQ"' is the product distribution in the absence of adsorption. The corrected product distribution (XQ'"') is obtained from equation (4.1) as: Xo"-=i T\ (4.2) 2 (l-8.62C„,+47.33C^) 4.4.5 Other Fibres Other types of fibres were also tested to see if we could find the one that could be used without the effects of adsorption. In the following section, synthetic (model) fibres are characterized for use with mixing sensitive chemical reactions. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 95 n A D V a 0.421 0.330 0.265 0.228 0.141 0.106 0 ® o X e + ® 0.361 0.276 0.209 0.165 0.128 0.075 1 r 0.06 0.08 0.10 Figure 4.6: Product distribution vs. mass concentration for FBK fibre. The solid lines for each curve are given by equation (4.1) Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 96 0.00 Figure 4.7: Comparison of product distribution vs. mass concentration for polyethylene, fibreglas, FBK and nylon fibre (XQ°= 0.41) 4.4.5.1 Non-adsorbing Fibres (Polyethylene) Figure 4.7 shows a comparison of the different fibres used in this study. Polyethylene fibres did not adsorb any of the naphthols or product dyes. This observation lends more credibility to the mechanisms postulated previously for the adsorption of dye products onto fibres. The polyethylene fibres have a very low water retention value, and the individual fibres are highly non-porous. Polyethylene was therefore used for subsequent experiments where the effects of fibre adsorption were completely eliminated. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 97 4.4.5.2 Nylon Fibres For nylon fibres, the adsorption of 1- and 2-naphthol is quite significant (10 to 40 times that of FBK). Figure 4.8 shows a plot of adsorption for a mixture of 1- and 2-naphthol (1:2) in nylon fibre for €„ = 0.0 - 0.10. 50-4 0 -^ 3 0 -o E I ^ 20-1 10--" r 0.00 0.02 0.04 0.06 C„ —n— 1 -naphthol —O— 2-naphthol 0.08 0.10 Figure 4.8: Equilibrium adsorption of 1- and 2-naphthol on nylon fibre as a fiinction of suspension mass concentration. C°AI = 0.52 mollvc?, C°A2 = 1 -04 molW, T = 298°K Despite these observations, however, within the concentration values used for this study, both 1- and 2-naphthol are adsorbed at the same rate, which implies that their ratio will not change significantiy. When the adsorption of the product dyes was examined, nylon was found to have significant affinity for the product dyes as shown in Figure 4.9. Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 98 0.5-0.4-T 1 r o X o o A V O © ^ 0.3861 0.3105 0.2661 0.2216 0.1701 0.1105 0.0821 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Figure 4.9: Product distribution vs. mass concentration for nylon fibre. Solid lines for each curve are given by equation (4.3). The solid lines in Figure 4.9 are computed using equation (4.3), which correlates the product distribution and mass concentrations for different dye concentrations. XQ=X'^"*(\-21.9C„+195.2CI) r^-0.95 (4.3) It should be noted that correlation given by equation (4.3) can only be used to correct for adsorption effects in the range (0 < Cm < 0.04) and (0.08 < X^ " < 0.39). Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase 99 4.4.6 Test Accuracy When the product distribution for FBK is corrected using equation (4.1), the errors introduced by the correction lie between 0.9% and 30.2%) for the lowest and highest mass concentrations tested respectively. In Table 4.1, the minimum and maximum for XQ values that can be corrected within the range tested (0.01 < Cm < 0.1) are shown. Also shovm are the 95%) confidence interval computed for different aqueous phase product distribution and suspension mass concentration. Table 4.1: Correlation Accuracy for FBK (Equation 4.2) Parameters ' ^ m 0.01 0.03 0.05 0.07 0.1 XQ 0.35 0.05 0.32 0.05 0.28 0.05 0.26 0.05 0.25 0.05 Estimate \r corr. 0.381 0.054 0.408 0.064 0.407 0.073 0.414 0.080 0.409 0.082 95%) Confidence interval \r corr. 0.385-0.377 0.055-0.054 0.443- 0.392 0.066-0.061 0.448-0.374 0.080-0.067 0.416-0.347 0.095-0.069 0.567-0.320 0.113-0.064 4.5 Summary 1. The azo coupling reactions between mixtures of 1- and 2-naphthol and diazotized sulphanilic acid have been characterized for FBK and model fibres. Nylon and wood pulp adsorbed both reactants and product dyes, with the adsorption best described by Chapter 4: Adsorption ofReactant and Product Dyes on the Dispersed Phase JOO Langmuir isotherms. Correlation methods based on regression analysis on specific adsorbates are necessary. The following expression can be used to correct the measured micromixing index in FBK suspensions in the range (0 < Cm ^ 0.1) and (0.075 < X J " < 0.42): X'^''- = XQ /(I -8.62C„ + 47.33C^), r^  = 0.97. 2. Nylon fibres were also found to adsorb the azo dyes. The following expression can be used to correct for adsorption effects in the range (0 < Cm < 0.04) and (0.08 < XQ''-< 0.38): Xg""' =X2/(l-27.9Cm+195.2Cm^), r^  = 0.95. 3. By accounting for dye adsorption on the dispersed phase, correct interpretation of mixing in the aqueous phase of the suspension can be made. The error in the corrected product distribution increased with mass concentration. For suspensions having Cm = 0.01 it is approximately ±0.9% (at 95% confidence level), whereas when the consistency is increased to Cm = 0.10, the error increases to approximately ±30.2%. This would increase the uncertainty of energy dissipation. 4. There is strong difflisional resistance for the primary coupling reactions when carried out in the presence of fibres. However, because most of the reactions take place within the liquid phase, this does not affect the product distribution significantly. 5. Polyethylene fibres did not adsorb any of the products or reactants and can therefore be used as ideal model fibres for studies involving azo coupling reactions. However, specific fibre properties should be taken into account when using model fibres. 6. Fiberglas cannot be used in such experiments, not only because of adsorption effects but also because of its brittleness that causes comminution of the fibres when mixed even at medium intensities. Chapter 5: Macromixing in the Medium-Intensity Mixer JQl Chapter 5 5 Macromixing in the Medium-Intensity Mixer 5.1 Introduction In general, the flow pattern inside a mixer depends on the nature of the suspension being mixed, in particular, it changes drastically in the presence of gas. A a result of change in flow conditions, overall energy dissipation rate may also change. A detailed investigation of flow pattern and the overall energy dissipation in the presence of dispersed phase will be carried out in this chapter. Macromixing characteristics will be examined in terms of overall energy dissipation rate (Save = P/pV) and the resulting flow patterns observed in the medium-intensity mixer. 5.2 Overall Energy Dissipation The overall energy dissipation rate in the medium-intensity mixer was evaluated for liquid (water) and for liquid-fibre, gas-liquid, and fibre-gas-liquid systems. It was also determined in the presence of other dispersed systems (glass beads and polyethylene beads) again as a function of mass concentration. This is the first step in understanding how much power is required/consumed in a system in order to achieve a given level of macroscale motion in the mixer. 5.2.1 Water Systems The medium-intensity mixer was operated with water and the rotor speed varied from 4 to 25 rev/s. The power consumed by the mixer was calculated from torque measurements as described in the experimental section (Chapter 3). The power number of the four-blade rotor used in this study was found to be Np = 5.0±0.2 under turbulent Chapter 5: Macromixing in the Medium-Intensity Mixer 102 conditions. In subsequent operations, the power number (Np) will be used to compare the overall energy dissipation for different loading conditions. In all cases, the suspension density is used in calculating the power number. 5.2.2 Fibre Suspensions Figure 5.1 shows the plot of average energy dissipation in the medium-intensity mixer when operated with water and FBK pulp fibre of Cy = 0.0 - 0.065 (Cm = 0 - 0.05). The solid line in Figure 5.1, is fitted using Equation 3.4 (Chapter 3) with water alone. As the rotor speed (N) was increased, the power drawn by the rotor increased as the cube of N, as expected. The fibre suspension was in a state of turbulent motion and the power number remained at 5.0, as with the case with water. However, at high fibre concentrations (Cy > 0.039 or Cm > 0.03) and lower speeds (N < 12 rev/s), complete motion was not achieved in the entire vessel until a minimum speed was attained that increased with suspension concentration. Once the point of complete motion was reached, the energy dissipation was similar to that drawn in the presence of water alone. For example, for Cy = 0.065, the pulp fibre suspension was not in complete motion for rotational speeds less than 13 rev/s, thus the shear applied in the suspension was not transmitted throughout the vessel. Energy dissipation rate below this speed is higher than that for water case alone, as shown in Figure 5.1. Chapter 5: Macromixing in the Medium-Intensity Mixer 103 200-150-100-50-0-9 N, rev/s Figure 5.1: Average energy dissipation versus rotor speed for water and FBK suspension at different mass concentrations. Solid line for water Np = 5.0. Figure 5.2 shows average energy dissipation in the presence of nylon fibres (L= 1.1 mm) for concentrations up to Cy = 0.065. In this case, the suspension was in complete motion for the range of speed and concentrations tested. All points lie on the curve for water alone. The difference between Figure 5.1 and 5.2 at the lower rotational speeds would appears to be due to the difference in suspension yield stress for Nylon and FBK fibre. For nylon fibre suspension, yield stress was overcome at all concentrations even for low rotational speeds (Bennington and Bourne, 1989). The higher yield stress in fibre suspension can also be explained by the wide length distribution and also less bending stiffness. Chapter 5: Macromixing in the Medium-Intensity Mixer 104 200 150-100-co N, rev/s Figure 5.2: Average energy dissipation versus rotor speed for water and nylon fibre ( L = 1.1 mm) suspension at different mass concentrations. Solid line is for water Np = 5.0. Computing the power number at different fibre concentrations can also be used to compare the power dissipation in the mixer under turbulent conditions. Table 5.1 shows power number computed for polyethylene, FBK and nylon fibres. For all types of fibres studied, no significant difference in power number was observed in the range of concentrations used (0.0 < Cy ^ 0.065). Table 5.1: Power number for various fibre suspensions (Xg = 0.0) Cv 0.0 0.013 0.039 0.065 Water 5.010.2 ---Polyethylene L - 3.2 mm -5.0±0.3 4.9±0.3 5.0±0.3 FBK L = 2.3 mm -5.0+0.3 5.0±0.3 5.0±0.3 Nylon L= 1.1 mm -5.0±0.2 5.0+0.3 5.0±0.3 Chapter 5: Macromixing in the Medium-Intensity Mixer 105 5.2.3 Fibre Suspensions with Gas Figure 5.3 shows overall energy dissipation for the mixing system in the presence of gas. As expected (see Bates et al., 1966), the overall energy dissipation decreases with increasing gas void fraction. The reduction in energy dissipation in the presence of gas is likely due to the formation of clinging cavities behind rotor blades as well as reduction in the gas-liquid mixture density. The reduction in power number for gas-liquid mixtures and mixture densities are reported in Table 5.2. We note that the reduction in power number for Xg < 0.20 corresponds to reduction in mixture density. However, for Xg > 0.32, the decrease in power number is greater than that would have been expected to correspond with decrease in bulk density. This drastic reduction in power corresponds to flooding of the rotor swept volume with gas. This change is also evidenced in the flow pattern to be discussed in section 5.3. Since the volume swept out by the rotor is 26%, for gas void fractions greater than the rotor swept volume, large gas cavities are formed in the rotor volume, leading to poor mixing. Table 5.2: Power number and bulk density for different gas void fractions Xg 0 0.12 0.20 0.32 0.40 Power number Np 5.0+0.1 4.2±0.2 3.8±0.2 2.9±0.2 2.110.2 Bulk Density, kg/m^ Pb 1000 880 800 680 600 Change % ANp -16% 24% 42% 58% Apb -12% 20% 32% 40% Chapter 5: Macromixing in the Medium-Intensity Mixer 106 400 300 S 200-100-N, rev/s Figure 5.3: Overall energy dissipation versus rotor speed for water with gas void fractions from 0 to 0.40 Table 5.3 shows the power number for the different fibre systems tested in the presence of gas at a fixed void fraction of Xg = 0.20. The overall power dissipation decreased with increasing fibre concentrations in the presence of gas. This phenomenon is observed for the three different types of fibres tested. While the observed trend is decreasing power number with increasing concentration, the change in power number is within experimental scatter, except for polyethylene fibres Cm = 0.05 (Cv = 0.052). The greater decrease in Np for polyethylene corresponds to a change in flow pattern. This phenomenon will be discussed fiarther in section 5.3. Chapter 5: Macromixing in the Medium-Intensity Mixer 107 Table 5.3: Power number for fibre suspensions in medium-intensity mixer (Xg = 0.20) '-^m 0.00 0.01 0.03 0.05 Np Polyethylene L = 3.2mm 3.8+0.2 3.6±0.2 3.3+0.2 3.0+0.3 FBK L = 2.3mm 3.8+0.2 3.8±0.4 3.5±0.4 3.3±0.4 Nylon L = 1.1 mm 3.8±0.2 3.8±0.3 3.7+0.3 3.510.3 Np change Polyethylene --5% -13% -21% FBK -0.1% 8% -13% Nylon -0.1% 2.8% 2.8% Pb change -+1.0% +3.1% +5.3% 5.2.4 Other Dispersed Phases In order to elucidate the effects of particle size and density on the overall power dissipation, glass beads (dp = 0.11 - 2.4 mm, density = 2520 kgW) and polyethylene beads (dp = 2.7 mm, density = 970 kg/m ) were used. Comparison of polyethylene beads with polyethylene fibres used earlier will also help elucidate the effect of particle shape. 5.2.4.1 Glass Beads Figure 5.4 shows the overall energy dissipation for glass beads (dp = 0.80 mm) at different mass concentrations. It is evident that the energy dissipation curves lie close to or above the baseline case for water alone. As the particle size or concentration is increased, more energy is required to achieve the same level of bulk movement of the suspension. The additional power draw signifies an increase in fluid resistance to rotational motion. On the other hand, for the smallest particles tested (dp =0.11 mm), Figure 5.5 shows that there is a consistent decrease in the power draw for speeds greater than 7 rev/s. The deviation becomes even larger as either the rotational speed or dispersed phase concentration is increased. This finding suggests that there is greater slippage in the suspension of small beads that helps to reduce the fluid resistance to motion. Figure 5.6 Chapter 5: Macromixing in the Medium-Intensity Mixer 108 compares the overall energy dissipation for glass beads of different sizes at a fixed volume concentration (Cv = 0.021). 150-D ) § 100-> W 5 0 -D 0.043 O 0.091 A 0.147 V 0.211 1 - 1 a 1 ' ^ / 1 / v • /n 1 ' 10 15 20 25 N, rev/s Figure 5.4 Overall energy dissipation for dp = 0.80 mm glass bead suspensions at different mass concentrations. The solid line gives data for water alone. Chapter 5: Macromixing in the Medium-Intensity Mixer 109 CO <iuu -150-100-5 0 -Glass 0.11 mm D o A V 0.043 0.091 0.147 0.211 • • 1 • 1 y / O ' 1 /l • M D a 1 • 10 15 N, rev/s 20 25 Figure 5.5: Overall energy dissipation for dp = 0.11 mm glass beads. The solid line gives data for water alone. 200-150-. 100-5 0 -Figure 5.6: Comparison of the overall energy dissipation for glass beads of different sizes at a fixed mass concentration (Cv= 0.021). The solid line gives data for water alone. Chapter 5: Macromixing in the Medium-Intensity Mixer WQ Table 5.4 shows power numbers calculated for glass beads at different volume concentrations. Also shown is the percent change in bulk density and apparent viscosity as calculated from equation (3.67) and (3.66) respectively in chapter 3. Considering an experimental scatter of about 6 - 8%, there is no change in power number over the concentration tested for dp= 0.55 and 0.80 mm. For dp = 1.2 and dp = 2.4 mm there is no change in power number up to Cy = 0.043. However, for Cy = 0.091, the power number is larger for large sized beads, probably due to increase in suspension resistance and energy dissipation due to particle-particle interactions. From Table 5.4 we also see that the change in bulk density is not substantially large, but the viscosity seem to have changes quite a bit. However, the effects of viscosity are not expected to be substantial, when there is a large density difference between the suspending medium and dispersed particles (Kikuchi et al., 1987). Tables 5.5 shows the power numbers determined for suspensions of glass beads (dp = 0.11mm and 0.80mm) in the presence of gas. Here, the volume of water in the suspension was kept constant (2L) while different weights of glass were added. This resulted in decrease of gas void fraction from Xg = 0.20 at Gy = 0.0 to Xg = 0.10 at Cy = 0.107. In the same table, the percent change in bulk density and apparent viscosity are also shown. Figure 5.7 shows that for speeds above 16 rev/s, the power curve drops drastically. This corresponds to changes in flow pattern from "B-I Type" flow to "D Type" flow (flow regimes to be explained later). The effects of flow pattern on overall power consumption in a mixer are well documented in the literature (Oldshue, 1983; Tatterson, 1991, Ibrahim and Nienow, 1995). The values reported in Table 5.5 were obtained for points below the drop. Chapter 5: Macromixing in the Medium-Intensity Mixer 111 Table 5.4: Power number for glass beads, Xg = 0.0 Cv 0.00 0.021 0.043 0.091 0.147 0.211 Power number, Np dp (mm) 0.11 5.010.2 3.910.3 3.610.2 3.410.2 3.210.2 3.210.2 0.55 5.010.2 4.610.3 4.810.3 5.010.1 5.010.3 5.110.3 0.80 5.010.2 5.010.3 5.010.4 5.110.3 5.110.6 5.310.3 1.20 5.010.2 5.010.3 5.110.3 5.410.3 5.710.3 6.010.4 2.40 5.010.2 5.110.3 5.010.3 ---Change in properties Apb 0 1% 2% 5% 7% 11% Al^a 0 5.4% 12% 28% 52% 91% 200' 150-100-5 0 -0<^ o A V O 0.017 0.036 0.080 0.107 o I 20 A O 30 N, rev/s Figure 5.7: Average energy dissipation rate for glass beads (dp = 0.80 mm, Xg different mass concentrations. Solid line gives data for water alone. 0.20) at Chapter 5: Macromixing in the Medium-Intensity Mixer 112 Table 5.5: Power number for glass beads, Xg=^ 0.20-0.10 Cv 0.00 0.017 0.036 0.080 0.107 doCmm) 0.11 3.6±0.2 3.5±0.3 3.1±0.2 2.3±0.2 2.0±0.1 0.80 3.6±0.2 3.5±0.3 3.7±0.3. 4.2±0.2 4.4+0.2 No change 0.11 --3±1% -14±1% -36±2% -44±3% 0.80 --3% +3% +16% +22% Change Pb 1% 3% 8% 10% l^ a 4% 10% 26% 39% 5.2.4.2 Polyethylene Beads and Polyethylene Fibres Table 5.6 show a comparison of different shapes (polyethylene beads dp = 2.7 mm and fibres L = 3.2 mm, density = 970 kg/m ). No difference in the overall energy dissipation was observed for the range of concentrations tested in the medium-intensity mixer. There was also no difference in the overall flow pattern observed for both suspensions. When the suspensions contain gas, however, the power number decreases slightly with increasing fibre concentration, but increases for the case of beads (when compared to the baseline case with Xg = 0.20). These changes are still within the experimental scatter of data. Table 5.6: Power number for polyethylene fibres (L =3.2 mm) and beads (dp = 2.7 mm) Cv 0 0.01 0.021 0.031 0.042 0.052 0.104 0.155 Fibres Xe = 5.0±0.2 4.9±0.3 5.010.3 4.910.3 5.010.3 5.010.3 --Beads 0.0 5.0+0.2 -5.010.3 -4.910.3 -5.010.3 5.110.3 Fibres Beads Xg - 0.20 3.810.2 3.610.3 3.410.3 3.310.3 3.210.3 3.010.3 --3.810.2 -3.710.4 -4.310.5 ---Chapter 5: Macromixing in the Medium-Intensity Mixer 113 5.2.4.3 Polyethylene Beads and Glass Beads When we compare the glass beads (dp = 2.4 mm, density = 2520 kg/m^) and polyethylene beads (dp = 2.7 mm, density = 970 kg/m ), a larger increase in overall energy dissipation for glass beads was observed. Therefore we can conclude that the increase in overall energy dissipation was mainly due to density difference and not the size of the particles alone. These findings are in agreement with some reported results in the literature (Bubbico et al., 1997). In their study, when working with PVC particles (dp = 4 mm, density = 1270 kg/m'^ ) and sand particles (dp = 1.3 mm, density = 2590 kg/m^) they observed an increase in power dissipation with increasing solids content. It is usually reported that so long as the appropriate density of suspension is used, the presence of particles should not effect the overall power dissipation. This study shows that this is only true when considering small particle size (dp < 1 mm) for mass concentrations less than Cm = 0.10 (Cy- 0.043). With an increase in either particle size or volume concentration, additional energy is dissipated due to particle-particle interactions. As for the smallest particles tested (dp = 0.11 mm), there seems to be a consistent decrease in overall energy dissipation with increasing Cy and/or rotor speed. Here, we speculate that the glass beads may actually be acting as zones of slippage between the rotor and the suspension and/or the suspension and the vessel walls, helping reduce the overall resistance to the rotary motion applied. It is also possible that small particles at higher concentrations change the fluid rheology or flow pattern and affect the overall energy dissipation rate. When the flow pattern was examined, it was found that the flow pattern of the smallest beads was different from that of larger particles. For all particle sizes covered (dp = 0 .11-2 .4 mm) it was observed that the flow pattern for dp = 0.55 - 2.4 mm, was Chapter 5: Macromixing in the Medium-Intensity Mixer 114 similar to that of water alone, i.e "B-I Type" of flow. For the smallest particle size (dp =0.11 mm) the flow pattern was different. While the rest exhibited a "B shaped" flow pattern (to be described later in this chapter) throughout the speeds tested, the smallest beads changed from cellular to radial flow- (forming a single loop that has been termed "D Type" flow in this thesis). This aspect will be discussed later to show how it can account for reduction overall energy dissipation. 5.3 Flow Pattern The flow pattern is another means that can be used to characterize a mixing system on the macroscale level. It is also related to the power draw characteristics of a given suspension/geometry conflguration as shall be demonstrated in this thesis. Here, the flow pattern is visualized with the aid of a high-speed video and also computed using a computational fluid dynamic (CFD) simulation. 5.3.1 Visual Observations The pattern of flow in the medium-intensity mixer was obtained by visual observation with the aid of high-speed video image analysis. In general, three main flow patterns were observed when operating the mixer at different conditions as shown in Figures 5.8 to 5.12. The "B-I Type" flow pattern (Figure 5.8) consisted of two circulation loops when viewed from the side of the vessel. The upper loop consists of radial outflow at the rotor discharge, with flow returning inward at the top and bottom of the vessel. The lower loop is a reflection of the top and they interact at the middle of the vessel. When viewed from the top, the flow formed four cells, corresponding to the number of baffles in the vessel. This flow pattern was observed for most operating conditions in the medium-intensity mixer. The pattern was more visible during experiments in the presence of colored dye Chapter 5: Macromixing in the Medium-Intensity Mixer 115 products. This pattern was also observed for gas-liquid, liquid-fibre, liquid-gas-fibre and the other dispersed phases used under conditions of complete motion in the mixer. The flow pattern is similar to the eight-shape pattern formed by a disk turbine in a standard stirred vessel. This observation raises the possibility for considering the rotor as an extended disk blade turbine that covers the height of a mixing tank. Figure 5.9 shows the second type of pattern that was observed. This pattern was observed when the mixer was operated in the presence of gas at large gas void ratios (Xg > 0.20) and higher mixing intensities. We call this "B-II Type" because the flow pattern is similar to "B-I", with the only difference being the direction of the fluid streamlines. The flow direction for "B-I Type" is radially inward while that of "B-II Type" is radially outward (when viewed from the top or bottom of the mixer). There is also no clear delineation of cells as in "B-I Type" flow. This may be due to accumulation of gas in the vicinity of the rotor at high gas void fraction as shown in Figure 5.10. This type of flow corresponds to a decrease in power number and the appearance of large cavities behind in the rotor swept volume. The appearance of a large cavity in the rotor region is hard to see by the unaided eye, but high-speed video clearly showed zones of high gas accumulation. A third type of flow pattern that was observed had a "D shape" as shown in Figure 5.10. The flow consisted of a single loop from the bottom to the top of the vessel. At given Xg > 0.2 if we decrease the rotor speed, flow conditions shifts to "B-II Type". In general, for water with low gas void fractions (Xg), the double-loop flow pattern with radial inward flow ("B-I Type") was observed. With increasing solids concentration, the flow pattern changed from "B-I Type" to "D Type " which resulted in a decrease in power consumption. The "B-II Type" flow pattern was very unstable and the zone where the two loops interact shifts up and down around the vessel mid-plane. The instabilities increase with increasing gas void fraction. At higher rotor speed, this flow Chapter 5: Macromixing in the Medium-Intensity Mixer X16 pattern shifts to "D Type" which is a more stable flow condition. For example, when operating the vessel with Xg = 0.32 we first had flow pattern of "B-II Type ", but as the speed was increased to 19.5 rev/s, the flow changed to "D Type" where power consumption was even lower. Chapter 5: Macromixing in the Medium Intensity Mixer 117 W i 1 . ~ i Figure 5.8: "B-I Type" flow pattern in medium intensity mixer obtained for all cases with water (Xg < 0.20) Chapter 5: Macromixing in the Medium Intensity Mixer 118 ^ / . ^ Figure 5.9: "B-II Type" in medium-intensity mixer. Transient type of flow observed in water + gas or water + gas + fibre transitions. Chapter 5: Macromixing in the Medium Intensity Mixer 119 m Figure 5.10: "D Type" flow pattern in medium-intensity mixer when operated with a fibre at high concentrations. Also obtained when smallest glass beads were used at higher speeds (N > 15 rev/s). Chapter 5: Macromixing in the Medium-Intensity Mixer 120 (a) Figure 5.11: Photographs showing the medium-intensity mixer when operated with a Cv =0.021 (Cm = 0.016) FBK suspension at N = 17.3 rev/s. and Xg = 0.20 (a) Top view showing large gas cavities behind rotor (b) side view showing "B-I Type" flow pattern. Chapter 5: Macromixing in the Medium-Intensity Mixer 121 Figure 5.12: Photograph showing medium-intensity mixer when operated with a Cy = 0.021 (Cm = 0.02) polyethylene beads suspension at 17.3 rev/s. (a) top view, (b) side view. Chapter 5: Macromixing in the Medium-Intensity Mixer 122 5.3.2 CFD Analysis A commercial computational fluid dynamic code (Fluent 4.5) was used to obtain the flow pattern and the overall flow characteristics in the medium-intensity mixer. This was done for water only as the equations necessary for use with fibre suspensions are still in development stage (Wherrett et al, 1997). CFD gives usefljl information regarding the flow patterns and overall characteristics for comparison with experimentally observed case. Appendix F gives the principles involved in the computation of macro-flow characteristics in a turbulent field. Figure 5.14 shows the velocity vectors in an axial plane midway between baffles (45°). The visually observed B-shaped pattern is well reproduced by CFD simulation. The outward and inward motions are clearly shown together with the axial motion that completed the circulation loops. Figure 5.15 compares of flow pattern for the top horizontal plane with a plane that is just above the middle of the mixer for N = 17.3 rev/s. Here, we see that for the middle plane (Figure 5.15(b)), velocity vectors are aligned away from the rotor, whereas velocity vectors at the top plane (Figure 5.15(a)) are pointed inwards. This corresponds to the observed pattern, whereby the flow is moving outwards at the mid-plane and inwards at the top plane to give one half of the "B shape". Furthermore, zones of reduced motion behind baffles are clearly seen, as expected. Other simulation results are shown in Appendix F. With the knowledge about the macro-flow characteristics of the medium intensity mixer, we are now ready to investigate the local flow characteristics. Chapter 5: Macromixing in the Medium-Intensity Mixer 123 Figure 5.14: CFD simulation of the flow pattern in the medium-intensity mixer with water (N = 17.3 rev/s) showing "B-I Type" flow pattern in an axial plane between baffles. :ing in the Medium-Intensity Mixer 124 (a) (b) Figure 5.15: CFD simulation of the flow pattern in the medium-intensity mixer with water (N = 17.3 rev/s.). (a) inward radial flow at the upper part of the vessel of the vessel (z/H = 0.95), (b) outward radial flow at the middle of the vessel (z/H = 0.50). Chapter 5: Macromixing in the Medium-Intensity Mixer 125 5.4 SUMMARY The following is the summary of macroscale behaviour (overall power dissipation and flow pattern) in a medium intensity mixer when operated with fibrous and particulate suspensions of different physical properties: 1. In the case of fibrous material (Xg = 0.0) at a fixed rotor speed, the overall energy dissipation remains constant with increasing fibre concentration up to Cv = 0.065 (Cm = 0.05). 2. For the case of fibrous material at a fixed gas void fraction (e.g Xg = 0.20), the overall energy dissipation decreases with increasing fibre concentration. This has been found to correspond to the decrease in suspension (bulk) density. 3. In the case of glass beads, the power dissipation remained the same only at low rotational speeds and low volume concentrations (Cy < 0.043). However, for large diameter particles (dp > 1.0 mm (corresponding to dp/le > 0.1)) at high concentrations (Cy > 0.043), the overall average dissipation increases with increasing volume concentration and particle size. This has been interpreted as evidence that particles dissipate energy supplied by the impeller through collisions 4. For the smallest glass beads tested (dp < 0.11 mm) at high concentrations (Cy > 0.043), the average power dissipation decreased with increasing volume concentration for both gassed and un-gassed conditions. We attribute this to lubrication or slippage effects induced by the beads at high relative velocities. 5. When particles of similar size but different densities are compared (c.f glass beads: p = 2520 kgW, dp = 2.4 mm; polyethylene beads: p = 970 kgW, dp = 2.7 mm), the power draw for neutrally buoyant polyethylene beads remained the same as that for Chapter 5: Macromixing in the Medium-Intensity Mixer 126 water, while it increased for the heavier glass beads. This clearly shows the effects of density are more dominant. 6. Three types of flow pattern have been observed in a rotor-stator mixer. The most common type consists of interacting loops and four cells when viewed from the top of the mixer. The number of cells formed corresponded to the number of baffles inside the vessel. The appearance of the "B Type" flow pattern was more apparent in the presence of gas. However, at large gas content or higher rotational speeds changed the flow pattern to "D Type" consisting of single loop, with large gas accumulations in the rotor area. This change in flow pattern causes a large drop in overall energy dissipation rate. 7. The existence of "B-I Type" flow pattern (for water alone) was successfully predicted by CFD simulation. 8. Apart from the main flow, small regions of recirculation were observed behind baffles and these formed relatively slow or stagnant zones when the vessel was operated with high concentrations of the dispersed phase and low rotor speeds. 9. There were considerable flow/torque instabilities when the vessel was operated with high gas void fraction (Xg > 0.32). This corresponded to changes in flow pattern from "B-Il Type" to "D Type" and formation of large cavities behind rotor blades. 10. The reduction in the overall energy dissipation in the presence of gas is due to the reduced drag on the rotor blade as a result of the reduction in gas-liquid mixture density as well as gas cavities formed behind the rotor blades . Chapter 6: Micromixing in the Medium Intensity Mixer 127 Chapter 6 6 Micromixing in the Medium-Intensity Mixer 6.1 Introduction In this Chapter we examine micromixing in the aqueous phase in the medium-intensity mixer which will be used to compare the micromixing studies in dispersed media. 6.2 Determination of Critical Feed Time As explained in the experimental section, the critical feed time delineates the mixer operation regime where macromixing effects do not influence the product distribution. Figure 6.1 shows that in water, at the lowest impeller speed tested (N = 7.3 rev/s), feed times greater than 40 seconds gave a constant product distribution for our test conditions. The critical feed time (tc) for the medium-intensity reactor was therefore 40 seconds when operated with water. When a FBK pulp suspension at Cm = 0.016 (Cy = 0.021) was tested under identical conditions, the critical feed time increased to 60 seconds. However, the critical feed time was not affected by the presence of gas, although the value of XQ increased as shown in Figure 6.2. For polyethylene fibre at Cm = 0.05 (Cv = 0.052) critical feed time increased to 80 seconds for N = 20.6 rev/s as shown in Figure 6.3. In this case, complete suspension motion could not be achieved at lower rotational speeds. A critical feed time of 180 seconds was selected for all experiments made in the medium-intensity mixer for all feed positions. Tests were conducted only where complete motion existed in the vessel. Chapter 6: Micromixing in the Medium Intensity Mixer 128 Feed time, s Figure 6.1: Plot of feed time vs. product distribution for water in the medium-intensity mixer (feed point F13, N = 7.3 rev/s, C''^^IC''^,=- 2, VAA^B = 50, C^,= 0.52 mol/m^ CI =21.54 mol/m^ T = 298°K). Chapter 6: Micromixing in the Medium Intensity Mixer 129 t,,s Figure 6.2: Plot of feed time vs. product distribution for water (N = 7.3 rev/s, feed point Fi3, C°, = 0.52 mol/m^ C^^ IC^x^X VA/VB - 50, C^ =21.54 mol/m\ Xg = 0.20, T = 298°K) X 50 100 150 Feed time, s Figure 6.3: Plot of feed time vs. product distribution for Cy = 0.052 polyethylene fibre suspension (N = 20.3 rev/s, feed point F^, C^, = 0.52 mol/m'', C\-^ IC°^^ = 2, VA/VB = 50, C° = 21.54 mol/m^ T = 298°K) Chapter 6: Micromixing in the Medium Intensity Mixer 130 6.3 Test Reproducibility Test accuracy was evaluated by comparing the total quantity of dye products produced (VA+VB)(CQ+CO-R+CP.R+2CS) with the quantity of limiting reagent (B) used in a test (VBCBO). In this case, we found closure for up to 98.2+1.3% based on 10 repeated measurements. The micromixing index (XQ) could be measured to within ±0.003 analytically and to ±0.009 (s.d.) in replicate tests as shown in Appendix H. Analytical measurements were repeated at least two times and averaged. 6.4 Reaction Zone Visualization The product distribution measured from micromixing reactions directly relates to energy dissipation experienced in the reaction zone. The reaction zone has a finite size and can be observed in aqueous systems using acid-base neutralization with phenolphthalein as an indicator (Rice et al, 1964; Bourne and Dell'ava 1987). Figure 6.4 shows images of the reaction plume as recorded using high-speed video system (HSV 1000, NAC Video systems) in water for feed location F33 and rotor speed (N = 7.3 rev/s). Here, 0.05M HCl was used as the bulk fluid and O.IM NaOH solution was added through the feed pipe at the same feed rate (QB =16 mL/min) as that selected for micromixing experiments. Phenolphthalein indicator was added in both solutions. The concentrations are higher than those used in azo coupling reactions in order to make the plume visible, which will also increase the size of the reaction zone. Thus the estimates given here are only approximate and are on the high side. The reaction plume, which has been enhanced in these images, could be seen to rapidly change orientation shape and size as the feed location or rotor speed was changed. From these images, it was possible to estimate the effective volume of reaction zone, assuming a cylindrical shape, as shown in Table 6.1. The reaction zone varied from 0.3 to 3.1cm^ depending on the rotational speed and feed location in the medium-intensity reactor. When compared with a typical volume of LDV (0.005 cm'^  - Schaffer et al., 1997), this zone may seem large. However, the effects that Chapter 6: Micromixing in the Medium Intensity Mixer 131 are measured are a result of interactions that take place at even much smaller scales (Kolmogorov scales). Table 6.1: Estimates of the size of reaction zone Feedpoint F33 F32 F31 F12 Impeller Speed (rev/s) 7.3 12.3 12.3 7.3 7.3 Reaction Zone Volume (cm'') 2.8-3.1 1.5 0.7 0.3 0.4-0.5 Figure 6.5 shows video images of the reaction plume during acid-base neutralization when viewed from the bottom of the tank. The video images confirm that the reaction is localized and does not take place over the whole reactor volume but in a small volume within the reactor. In the analysis for distribution of energy dissipation rates (to be shown later), we fix the local energy dissipation to the location of the feed point. This is clearly a simplification of the real situation as the energy dissipation is averaged over a finite volume of the reaction zone. Chapter 6: Micromixing in the Medium Intensity Mixer 132 (a) (b) (C) m Figure 6.4: Visualization of reaction plume using high-speed video. Acid-base neutralization. The plume color has been intensified for ease of identification. Feed point F33, N = 7.3 rev/s. Elapsed times (a) 0 ms, (b) 4 ms, (c) 28 ms, (d) 34 ms. Chapter 6: Micromixing in the Medium Intensity Mixer 133 (a) (b) (c) Figure 6.5: Visualization of reaction plume using high-speed video. The plume color has been intensified for ease of identification. Feed point F52, N = 7.3 rev/s. Elapsed times: (a) 0 ms, (b) 12 ms, (c) 20 ms. Chapter 6: Micromixing in the Medium Intensity Mixer 134 6.5 Single Point Tests In this section we analyze the product distribution measured in the medium-intensity mixer at different rotational speeds and different feed-point locations. Single point analysis draws conclusions based on measurements conducted at one fixed feed-point location in a mixer. This is the interpretation that has been done most often in mixing-sensitive reactions for single-phase systems. All tests were conducted in the micromixing regime where the feed time (tf) was greater than the critical feed time (tc). Figure 6.6 shows the product distribution for different feed locations across the medium-intensity mixer at a constant rotational speed (N = 17.3 rev/s). It is evident that the product distribution varies significantly with feed-point location across the mixer. 0.3' 0 .2-X 0.1-0.0-Axial position, z/H —D—0.05 —O—0.25 —AT-0.50 —V—0.75 0.95 0.5 1 0.6 0.7 0.8 0.9 1.0 r/R Figure 6.6: Product distribution in the medium-intensity reactor for different feed points at N = 17.3 rev/s. Axial positions z/H = 0.05, 0.25, 0.5, 0.75 and 0.95. (C°, = 0.52 mol/m^ C^^i^C^^r 2, VA/VB = 50, C" = 21.54 mol/m^ tf = 180s, T = 298°K). Chapter 6: Micromixing in the Medium Intensity Mixer 135 Likewise, Figure 6.7 shows the product distribution across one axial direction in the medium-intensity reactor. Both Figures shows a significant variation in the micromixing index (XQ) and therefore local energy dissipation across a mixer plane. Adjacent to the rotor (r/R = 0.579; z/H = 0.5), the product distribution is lowest, indicating the highest level of energy dissipation rate. As we move to regions remote from the rotor, the product distribution increases markedly, indicating a significant drop in energy dissipation. As it is shown that the product distribution depends on the point of injection, it is imperative to note the location of feed point when making comparisons in mixers. All fiirther comparisons for single-point measurements in this section were performed at feed point F32 (r/R = 0.737, z/H = 0.5) unless otherwise stated. z/H Figure 6.7: Product distribution in the medium-intensity reactor for different feed points along impeller axis (r/R = 0.579) at N = 17.3 rev/s. (C°,= 0.52 mol/m^ C^j/C^i^ 2, VA/VB = 50, CI =21.54 mo\lm\ tf = 180s, T = 298°K). Chapter 6: Micromixing in the Medium Intensity Mixer 136 6.5.1 Tests with Water Alone (baseline characteristics) The product distributions (XQ) measured in aqueous solution for impeller rotational speeds of 7.3 to 20.6 rev/s with feed point at F31, F32 and F33 are shown in Figure 6.8. It is evident, as expected, that the product distribution decreases as mixing intensity is increased. For point F31 (closer to the rotor), XQ is lower as the energy dissipation is high as expected, while for point F13 (remote from rotor) XQ is higher, which means energy dissipation rate is lower. Figure 6.8 shows the product distribution obtained experimentally and using the E-model simulation assuming the local energy dissipation rate in the reaction zone is equal to the average energy input to the mixer. It is evident that the model agrees well with the trend of experimental data, but that the local energy dissipation is greater than the average energy dissipation. The solid line was obtained using the E-model with energy dissipation set equal to the average energy dissipation rate measured. The dotted line is obtained by setting the local energy dissipation rate to be six times the average dissipation rate {e - 6s^^^). By considering three different feed points shown in Figure 6.8, the local energy dissipation rate at a point can therefore be related to the overall average dissipation through a position dependent parameter cp such that £ = (P£are (6-1) Feed point F32 was particularly special in this case because the local energy dissipation calculated here was about the same as the average energy dissipation rate calculated using mixing-sensitive chemical reactions over the entire mixer volume (to be shown later). Since it was noted that the dissipation rate is position dependent, the next set of experiments were designed to determine the local energy dissipation in a complete vertical plane of the mixer as described in the experimental section. Chapter 6: Micromixing in the Medium Intensity Mixer 137 u.o -, 0.4-0.3-0.2-0 . 1 -nn-\ \ . 8^  \ . ^ O 1 1 ° F3, ° F3, ^ F,3 E-model (9 = 6) , ^ ^ ^ - 1 1 10 20 N, rev/s Figure 6.8: Product distribution versus rotor speed for medium-intensity mixer. Solid lines represent E-model predictions with cp = 1 while dotted lines represents predictions with 9 = 6. (Feed points F31: r/R = 0.5, z/H = 0.5; F32: r/R = 0.737, z/H = 0.5; F13: r/R = 0.947, z/H = 0.95). (C;,= 0.52 mol/m^ 0^2/C°,- 2, VA/VB = 50, CI =21.54 mol/m^ tf- 180 s, T = 298°K) Chapter 6: Micromixing in the Medium Intensity Mixer 138 6.6 Profiles of Dimensionless Energy Dissipation Rate From the single-point measurements, we found that the energy dissipation varied significantly across the mixer, hence it was thought desirable to carry out identical measurements in a given mixer plane and analyze the dissipation profile across it. Many investigators have recognized this effect when working with traditional stirred tank reactors (STR). In the case of STR, the difference between the region closest to the rotor and that remote from it can be quite large (e.g. Zhou and Kresta (1996) reported differences over 100 times). The maximum dissipation rate is usually located closest to the rotor tips while the minimum lies at a point remote from the rotor. In this part of study, we exploit this fact in order to capture the distribution of energy dissipation rates in the mixer. 6.6.1 Construction of Energy Dissipation Profiles XQ profiles were measured across one tank cross-section mid-way between the wall baffles using fifteen different locations corresponding to dimensionless axial positions of z/H = 0.05, 0.25, 0.50, 0.75 and 0.95; and dimensionless radial positions, r/R = 0.579, 0.737 and 0.947 as shown in Figure 6.9. From the measurements of product distribution, the local energy dissipation rate corresponding to each feed-point was evaluated using the E-model. In order to obtain the average energy dissipation, the energy dissipation rate determined at each point has to be weighted according to its position in the mixer. From flow visualization, we divide the region between rotor and vessel wall into three annular zones. Zone 1 is the immediate rotor zone covering a relatively smaller volume (r = 50 to r = 60 mm) but with the highest dissipation rate. Zone 2 is the middle zone extending from r = 60 to r = 75 mm, and zone 3 extends from r = 75 to r =95 mm (vessel wall) as shown in Figure 6.9. Chapter 6: Micromixing in the Medium Intensity Mixer 139 9 5 7 5 60 , 5 0 Figure 6.9: Discretization of different zones of energy dissipation from reaction zone visualization (all dimensions in mm). Table 6.2: Weighting factor for inhomogeneous distribution of energy dissipation Zone 1 2 3 Radial limits, mm 50 -60 60 - 75 7 5 - 9 5 Radial weighting factor, wj 0.1686 0.3103 0.5211 Chapter 6: Micromixing in the Medium Intensity Mixer 140 The average energy dissipation in the evaluation plane was then computed by weighting each feed point according to its radial distance from the mixer axis, i.e. ^ = Z Z^/^y^// (6-2) where Wj is the radial weighting factor given in Table 6.1 and Wj is the axial weighting factor equal to 1/5 (all points weighed equally in axial direction). Details of the selection of weight factors can be found in Appendix E.2. The local dissipation rates were normalized with respect to the average dissipation rate in the evaluation plane to obtain the dimensionless energy dissipation rate (^^ ): <l>,=^4- (6.3) s Contour plots of dimensionless energy dissipation rate {(f), we shall omit the indices when not referring to a particular position) were created using Tecplot® 7.0 Contour software. The dimensionless values were then read into an array of 5 rows and 3 columns representing the physical positions of the feed points. The contour lines connect points of equal energy dissipation from each grid points while the space between successive contour lines contain only points whose values fall within the interval defined by the contour lines. Interpolation of values to cover entire vessel plane was done using the Kriging method that uses a combination of weights to minimize estimation error (For more details on the Kriging method see Davis, 1986). The method takes the average value from four nearest data points in test, ensuring that points closest to the destination point are given the greater weighing than that remote from it. Since each data value actually represents the local energy dissipation rate averaged within a small reaction zone (volume over which reaction takes place), only the closest data points in the source zone are used. Although values are obtained with reference to discrete points, we may consider the dimensionless energy dissipation rate is spatially continuous. By taking values of ^ at Chapter 6: Micromixing in the Medium Intensity Mixer 141 specific locations, we reconstruct the spatially continuous nature of the dissipation. It has also been shown in literature (Bourne and Dell'ava, 1987) that the average energy dissipation in the reaction zone is independent of its profile, i.e. regardless of the fact that the dissipation profile may be increasing, decreasing or oscillating, the product distribution will still reflect the average energy dissipation in the reaction zone. 6.6.2 Profiles in Axial Planes Figure 6.10(a) shows a contour plot for the dimensionless energy distribution for the medium-intensity mixer at a constant rotor speed (water at N = 7.3 rev/s) made using the reaction conditions given in Table 3.5. The profile is consistent with expectations, namely higher dissipation rates closer to the rotor, and maximum dissipation rates along the mid-plane of the vessel where the circulation from top £ind bottom of the tank interact. The contours also show symmetry along the vessel mid-plane. The reproducibility of the contour plots for a given test condition was evaluated for a number of runs. Figure 6.10(b) shows test results for water made on a different day under identical conditions to that in Figure 6.10(a). The profiles are similar within the errors inherent in the evaluation technique used. The average energy dissipation rate, e , for the two tests was found to be 88 and 75 W/kg, respectively. Comparable repeatability was found for different speeds (e.g Figure 6.10 (c) and (d) shows repeated tests atN = 17.3 rev/s). Similar energy dissipation profiles with increasing rotational speed as shown in Figure 6.11. Overall, the distribution is the same and the maximum relative dissipation (^ax) did not change significantly as the rotational speed was increased. The average energy dissipation rate in the sampling plane at various rotational speeds calculated based on product distribution (£^) is compared with the overall mean energy dissipation based on power input to the mixer (save) in Table 6.3. Chapter 6: Micromixing in the Medium Intensity Mixer 142 (a) (c) 0.6 0.8 r/R (d) 0.6 0.8 r/R Figure 6.10: Dimensionless energy dissipation rate for water. Duplicate tests done on separate days under same conditions (C^, = 0.52 molW, C°^2IC''AX = 2, VAA^B = 50, CI = 21.54 molW, tf - 180 s, T - 298°K). (a) and (b) N = 7.3 rev/s, (c) and (d)N= 17.3 rev/s. Chapter 6: Micromixing in the Medium Intensity Mixer 143 Table 6.3: The average energy dissipation rate in Medium-Intensity mixer N rev/s 7.3 12.3 17.3 19.5 Energy Dissipation Rate (W/kg) local average £ 88±8 410±30 632±45 907±92 Overall average £ave 11 + 1 43+3 105+6 141±8 Ratio g /Save 8.2 9.6 6.2 6.4 Table 6.3 shows that for the aqueous tests, the overall energy dissipation (save) is significantly lower than the local average measured using chemical reactions (£^). It is not surprising to see that the local energy dissipation calculated for the sampling plane midway between baffles gives higher values than the average calculated from total power consumption. Detailed comparison is made at the end of this chapter. Chapter 6: Micromixing in the Medium Intensity Mixer 144 (a) 0.6 0.8 r/R (b) (c) (d) 0.6 0.8 r/R 0.6 0.8 r/R Figure 6.11 Dimensionless energy dissipation for aqueous system (single phase) at different rotational speeds: (a) N = 7.3rev/s, (b) N = 12.3 rev/s, (c) N = 17.3 rev/s, and (d) N = 19.5 rev/s. (C°, = 0.52 mol/m^ C°2/C°,= 2, VA/VB = 50, C° =21.54 mol/m^ tf= 180 s, T = 298°K). Chapter 6: Micromixing in the Medium Intensity Mixer 145 6.6.3 Profiles in Radial Plane Since the mixer can be divided into four geometrically identical quadrants, we made measurements in a single quadrant. Figure 6.12 shows feed point locations and quadrant division for averaging purposes. The aim here was to see if we could obtain better closure for energy dissipation, i.e. s = Save- Here, seventeen points along the quadrant covered between two baffles were used as feed points. The radial locations of the feed points were the same as the one used in the single plane tests. Table 6.4 shows the locations of the feed points used. r I" Figure 6.12: Schematic layout of feed point location for radial plane mapping (coordinate positions shown in Table 6.3). Zone I comprises of F and F ' Chapter 6: Micromixing in the Medium Intensity Mixer 146 Table 6.4: Radial and circumferential positions of feed-points in the medium-intensity mixer. (XQ values shown for N=12.3 rev/s, axial position z/H = 0.05) Point designation 1 2 3 . 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Radial position r/R 0.579 0.737 0.895 0.947 0.579 0.737 0.947 0.579 0.737 0.947 0.579 0.737 0.947 0.579 0.737 0.895 0.947 Angle 0(°) 0 0 0 6 22.5 22.5 22.5 45 45 45 62.5 62.5 62.5 90 90 90 84 Radial weight factors 0.1686 0.3103 0.2452 0.2759 0.1686 0.3103 0.5211 0.1686 0.3103 0.5211 0.1686 0.3103 0.5211 0.1686 0.3103 0.2452 0.2759 XQ 0.202 0.226 0.301 0.207 0.208 0.256 0.228 0.214 0.243 0.232 0.208 0.237 0.223 0.196 0.226 0.301 0.227 Chapter 6: Micromixing in the Medium Intensity Mixer Ul (a) z/H = 0.05 (b) z/H = 0.25 Figure 6.13: Contour profile of dimensionless energy dissipation rate in a radial plane across mixer for N = 17.3 rev/s. (a) z/H = 0.05, (b) z/H = 0.25. Chapter 6: Micromixing in the Medium Intensity Mixer 148 (a) z/H = 0.50 (b) z/H = 0.75 Figure 6.14: Contour profile of dimensionless energy dissipation rate in a radial plane across mixer for N = 17.3 rev/s. (a) z/H = 0.5, (b) z/H = 0.75. Chapter 6: Micromixing in the Medium Intensity Mixer 149 Figure 6.15: Contour profile of dimensionless energy dissipation rate in a radial plane across mixer for N = 17.3 rev/s. z/H = 0.95. From the radial profiles, it can be seen that plane 1 (z/H = 0.05) and plane 5 (z/H = 0.95) have very similar patterns, and that plane 2 (z/H = 0.25) and plane 4 (z/H = 0.75) have similar profiles. The central plane (z/H = 0.5) has a slightly different pattern, with notably larger area with (()max- This concurs with observations made on the axial plane midway between baffles where it was found that the highest relative dissipation takes place along this plane. It was further observed that the regions behind the baffles displayed lower dissipation rates than the rest of the points observed in any given plane. This is in agreement with what we would expect from less active zones behind the baffles. The data for all planes can be combined into a 3-D plots for better visualization of the energy distribution. Figures 6.16 and 6.17 shows the distribution in the medium-intensity mixer for N = 12.3 and 17.3 rev/s, where as Figure 6.18 shows the distribution in the presence of polyethylene fibres at N = 17.3 rev/s and Cy = 0.021. Chapter 6: Micromixing in the Medium-Intensity Mixer 150 (a) (b) .X 3 2.5 2 1.5 1 0.5 (c) Figure 6.16: 3D reconstruction of the distribution of energy dissipation rate in medium-intensity mixer with water at N = 12.3 rev/s. (a) full quadrant view, (b) slice at 9 = 45° rotated to show mid plane measured in single-plane studies. Chapter 6: Micromixing in the Medium-Intensity Mixer 151 (a) (b) 3 2.5 2 1.5 0.5 (c) ^J\__J - •y- '^ i i ^* '^ ^s^ [^ '^\\\'"\^^^ Figure 6.17: 3D reconstruction of the distribution of energy dissipation rate in medium-intensity mixer with water at N = 17.3 rev/s. (a) full quadrant view, (b) slice at 6 = 45° rotated to show mid plane measured in single-plane studies. Chapter 6: Micromixing in the Medium-Intensity Mixer 152 (a) (b) Y (c) 1 yS-IS^Cj" 3 2.5 2 1.5 1 0.5 Figure 6.18: 3D reconstruction of the distribution of energy dissipation rate in medium-intensity mixer with polyethylene (Cy = 0.021) at N = 17.3 rev/s. (a) full quadrant view, (b) slice at 0 = 45° rotated to show mid plane measured in single-plane studies. Chapter 6: Micromixing in the Medium Intensity Mixer 153 > CO II Si CO l-l Id o W o 00 (U ; - ( d o CO a u o o i-l (U <u M ^ . — 1 >-( CL, 4) > • y T 3 C (D o -•=; N - d •n .5? O O hn ^ t^ ^ i n o o >o t ^ C 3 >n o >o l-^ o i n OS o o CD ^ =tt: ^ r--MD • — ( m m o <N ro r-m 0 0 r-VO o i n C3N O O N r -i n o -oo ( N ( N C 5 SO o O 1—1 1—1 CO o so O r--m r-o ( N F—H oo so r--( N OO '^ t <N ^ OO i n o i n CTs OO o ro oo m r--o oo ^ i n oo O ^ • ^ ( N OS SO I ^ ^ OS o -* oo OS i n m oo OS i n so r -i n o so (-0 m i n i n O N r--i n o i n rsi o oo ^ T—i ( N oo i n so 1 — . so ( N ( N O i n ^ i n r~~ CO r-o so oo i n m r~-m ( N r~-^ SO ( N O C 5 ro i n ( N r--' ^ c^ o r-m m so 1-^ t-H 1-H Tl-r-<N ( N i n oo C3S (N m i n m ^ i n vt ^ (J\ t-~-i n <z> oo so r~-^ o i n <N oo so as so <N OO OS m i n r-~ ro r~~ o OS oo m CTs oo so m i n i n CTs r-^ m ro m CN i n r~ 'ct-CDs o CO OO oo so > CO oo 1—1 T T so en . — ( i n o so m m i n i n so C3S r-i n o r—. <N o oo <N 1—1 CN oo i n so r—< SO CN CN ^ SO ^ i n CN so r-~ m t> o CN ( N r-~ m <N m •^ 1—i r—4 C^ (7s m <N oo CN i n CN so r-^ c^ o en i n m ^ V-H m m C3S r^ so CN m i n oo r-so o i n CJS OS (7s r-i n o XT OO CN en » — I y—* SO o CN o 1 — ( m <j} so C_) C7\ r-~ m t> < 0 i n i n r~-^ so i—t *—. CN SO i n <N CN "sf m as i n OS oo o so so f—4 i n CTs <N ^ o i n (N ^ ^ CN »—* (50 r-'^ CJs o r-~ m 1—1 t>-^ m en OS OO CN en CN CN ^ (U d) 5 P ^ cs3 d "O O (U "x:! •*:! •fci -S, (U tJB >- ' o ^ h -CO CD (L) h n a X ) S l - c O Chapter 6: Micromixing in the Medium Intensity Mixer 154 Table 6.5 gives the total energy dissipation along different planes calculated by taking account of all measured points. Since points (1,2 and 3) represent the same zone as points (14, 15 and 16), these points were given Vi the weight as the other points. The average energy dissipation in the radial plane can vary from 328 to 1391 W/kg depending on the location of the measuring plane. We also note that at axial plane z/H = 0.5 we have the highest energy dissipation rate compared to other locations. It is interesting to note that the total energy dissipation rate for the entire quadrant is not very different from that which was calculated from the axial plane midway between baffles (c.f quadrant average = 637 W/kg and mid-plane (plane III) average = 633 W/kg). Measurements from the mid-plane were therefore used for comparison in most of the experiments conducted. 6.6.4 Energy Dissipation in the Rotor Swept Area In order to determine the energy dissipation inside the rotor swept volume (« 26% total liquid volume in the mixer) micromixing reactions were carried out by feeding the limiting reagent through the rotor. Figure 6.19 shows the rotor used and arrangement for feeding though the rotor. The position of the feed point was adjusted by using 1 mm feed tubes of different lengths. Table 6.6 shows the estimated energy dissipation values from E-model at different radial and axial positions. Values vary from 415 to 1525 W/kg for N = 17.3 rev/s depending on the location of the feed point. This clearly indicates inhomogeneity in the rotor swept volume. The weighted average of all data points in the rotor swept area was found to be 1010 W/kg for N = 17.3 and 406 W/kg for N = 12.3 rev/s. Chapter 6: Micromixing in the Medium Intensity Mixer 155 R o t o r_ shaft ReQctant_ chamber R o t o r_ hub R o t a t i n g seal R o t o r vane B in Figure 6.19: Rotor design and feed point location for determination of micromixing behind rotor vanes. Table 6.6: Local energy dissipation rates in the rotor swept volume N (rev/s) 17.3 12.3 r/R 0.316 0.389 0.474 0.316 0.389 0.474 z/H 0.95 795 415 610 262 144 215 0.75 1525 1245 1218 648 642 512 0.5 925 1111 1404 220 306 647 0.25 1525 1245 1218 648 642 512 0.05 795 415 610 262 144 215 Chapter 6: Micromixing in the Medium Intensity Mixer 156 For the sake of comparison, we compute the average energy dissipation rate, taking account of dissipation inside the rotor swept volume (26% of total mixer volume) as well as the rest of the vessel volume (74% of mixer volume) by the following procedure: For N = 17.3 rev/s (Save = 105 W/kg) 8 = (8 in rotor swept volume) x vol. fraction + (s outside rotor volume) x vol. fraction 8 = 1010 X 0.26 + 632 x 0.74 = 730 W/kg For N = 12.3 rev/s (save = 43 W/kg) 8 = 406 X 0.26 + 410 x 0.74 = 409 W/kg This indicates that the energy dissipation obtained chemically is about six to ten times higher than the actual energy dissipation. This can be attributed to one or a combination of the following factors: (1) the feed pipe imparts additional energy in the reaction zone, (2) the E-model over-estimates the local energy dissipation rate under conditions used in this study (high energy dissipation rates), (3) the method used for averaging does not capture the distribution of energy dissipation rates in the vessel adequately. In micromixing regime, the energy imparted by the flow through feed tube is usually considered negligible as compared to the energy imparted by the rotor. Any contribution from the feed pipe is therefore not included in the E-model. However, the amount of energy imparted by the flow through feed tube is dissipated in the reaction zone and may affect the product distribution in some cases ( Bourne and Lip, 1991). Micromixing studies carried out in homogeneous grid turbulence using azo coupling reactions by Bourne and Lips (1991) showed higher energy dissipation rates than Chapter 6: Micromixing in the Medium Intensity Mixer 157 predicted by theoretical expressions for grid turbulence. The difference was attributed to the feed pipe contribution. It may be possible that the E-model over estimates the local energy dissipation rate at high mixing intensities. Bakker and Van den Akker (1994, 1996) are of the opinion that the reduction of scale from feed pipe diameter down to Kolmogorov scales needs to be included in micromixing models, even when operating in micromixing controlled regime. The method of averaging and weighting for the different positions may also be inadequate to capture the distribution of energy dissipation completely. Table E-1 in Appendix E shows that different weighting factors would give different average dissipation rates. In the present study we use weighting factors based on observed reaction zone limits (Figure 9.6). In the literature, most micromixing experiments are carried out at a couple of feed point locations, either closest to the impeller or away from it. The averaging of the entire vessel has not been confronted in the manner followed in this study. It is therefore difficult to compare the factor found in the present study with other micromixing experiments in the literature. The approach taken in this study is to compute the energy dissipation from E-model and normalize it with a factor to acount for the difference between estimated values and measured power input. We also compare energy dissipation in the presence of dispersed phase with that of the suspending medium (water) alone at identical conditions. 6.7 Summary 1. The critical feed time determined in the presence of fibres is higher than that for water under similar operating conditions. A feed time of 180 seconds was found to be Chapter 6: Micrpmixing in the Medium.Intensity Mixer 158 sufficient for all tests of micromixing in the medium-intensity reactor (Cv < 0.05, Xg < 0.20, N > 7.3 rev/s). The presence of gas did not influence the critical feed time. 2. The reaction zone is localized and its estimated size varies between 0.3 and 3cm^ depending on the local energy dissipation rate, which depends on feed point location and impeller rotational speed. 3. The product distribution (XQ) varies for different feed locations in the mixer. Positions closer to the rotor had the lowest XQ value while those close to the outer wall had the highest values. The minimum values of XQ were always found at the vessel mid-height. 4. The local energy dissipation rate in the liquid phase can be obtained using the E-model of micromixing provided an appropriate relationship is established between the mean overall energy dissipation based on overall power input and the local average energy dissipation. The proportionality factor necessary to close the difference was found to vary from is 6 to 10. 5. Profiles of energy dissipation rate can be reconstructed in a mixing device provided that enough points are taken into consideration. From these profiles, the spatial nature of energy dissipation in a mixing device is obtained. The profiles were subsequently used to compare the hydrodynamic behaviour of the medium-intensity mixer when operated with different dispersed systems. 6. By mapping the mixer both in the axial and radial direction we have found that the local average energy dissipation rate computed from the chemical method and E-model is about six times that measured using the power input. However, the average found from a single plane axial plane midway between baffles (45°) gives reasonable representation of the overall average energy dissipation with a difference of only 10 -Chapter 6: Micromixing in the Medium Intensity Mixer 159 15 %. It is therefore sufficient to map over a single plane if we are only interested in the average value. 7. The rotor-stator mixer behaves quite differently from the traditional STR. In the rotor-stator mixer, the difference between maximum and minimum dissipation zones is not as high as that found in STR (Table J-1 in Appendix J shows different values determined for the standard STR in the literature for comparison). Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 160 Chapter 7 7 Micromixing in Fibre Suspensions: IVIedium-Intensity Mixer 7.1 Introduction In this Chapter, micromixing in the liquid phase of fibre suspensions is studied. The local energy dissipation rate in the liquid phase is determined using the E-Model, from which the relative energy dissipation rates in the presence of fibres are determined. 7.2 Single Point Analysis Single point tests were carried out for feed point location at F32 in all comparison tests. The effect of different fibre properties on the micromixing index and local energy dissipation rates are examined. 7.2.1 Effect of Fibre Concentration Figure 7.1 shows the variation of micromixing index {XQ) with increasing rotational speed for FBK suspension and Figure 7.2 show the same for polyethylene fibres. All values reported for FBK fibre were corrected for adsorption using respective correlations developed in Chapter 4 and denoted as XQ'''''. The presence of fibres increased XQ values indicating a reduction in available energy at the smallest scales. XQ increased as fibre mass concentration increased for all fibres tested. It should be remembered that the overall energy dissipation rate (Save) was found to be unaffected by the presence of fibres as discussed in chapter 5. When we compare polyethylene and FBK fibres at the same volumetric concentration, we see that XQ values are slightly higher for FBK suspension than for the polyethylene suspension. Therefore, there must be other factors, in addition to concentration, that contribute to the observed trend. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 161 0.5 • 0.4-fc 0.3 • X 0.2 0 . 1 -0.0' 10 N, rev/s FBK, Xg=0.0 —r-20 Figure 7.1: XQ''''' versus rotor speed for FBK suspension (Cy = 0.0 - 0.021 or Cm = 0 -0.16, Xg= 0.0; feed point F32, C''^2lC''^r 2, VAA^B = 50, C;,= 0.52 molW, CI = 21.54 molW, tf = 180s, T = 298°K) 0.5 0 .4-0.3-0 .2-0.1 0.0 —a—0.00 —0—0.010 —'5^0.021 —V--0.032 —0—0.042 Polyethylene fibres, L=3.2mm, Xg=0.0 —r-10 I 20 N, rev/s Figure 7.2; XQ versus rotor speed for a polyethylene fibre suspension (Cy = 0.0 - 0.042 or Cm = 0 - 0.04, Xg= 0.0; feed point F32, C^JC^.^ 2, VAAVB = 50, C^,= 0.52 molW, CI =21.54 mo\lm\ tf = 180s, T = 298°K) Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 162 0.5 • 0 .4-0.3 • a X 0.2' 0 . 1 -0.0-—•— L/d= 84 L/d = 163 Polyethylene, C^  = 0.021 10 —r-15 I 20 25 N, rev/s Figure 7.3: XQ versus mixing intensity for Cy = 0.021 (Cm = 0.02) polyethylene fibre suspension for aspect ratio (L/d) 84 and 163 (Feed point F32, standard chemical conditions). 7.2.2 Effect of L/d Ratio The effect of L/d ratio was examined using polyethylene fibres that were available in two different lengths (3.2 and 6.2 mm) giving L/d = 84 and L/d =163 respectively. It is apparent that it is not just the suspension concentration that affects the micromixing index (XQ) but also the fibre aspect ratio as shown in Figure 7.3. The longer fibres result in higher XQ values than shorter fibres at the same mass concentration. This indicates a greater reduction of local energy dissipation rate in the presence of longer fibres. Figure 7.4 shows the effects of L/d ratio and concentration on the product distribution. Here we note that the effect of increasing concentration or L/d ratio is to increase the product distribution, hence decrease micromixing efficiency. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 163 u.o -0.4-0.3-0.2-0 . 1 -0 .0-• 1 D-I 1 -1 - I-- L/d = 84 - L / d = 163 1 1 1 1 1 1 ^-'-''''''^^ ^ --- ' - - i - I • 1 1 0.00 0.01 0.02 0.03 0.04 0.05 Figure 7.4: Micromixing index (XQ) in fibre suspensions of different fibre aspect ratio at various volume concentrations. (N = 17.3 rev/s, feed point F32, standard reaction conditions). 7.2.3 Effect of Fibre Flexibility In order to elucidate the effect of fibre flexibility on XQ, we need to compare the different fibre types used here at the same volume concentration and L/d ratio. The effect of fibre aspect ratio and fibre concentration can be accounted for by the crowding factor (Kerekes and Schell, 1992). The crowding (Ncr) factor relates to the ability of fibre suspension to form floes. Table 7.1 gives the calculated crowding factors for FBK, Nylon and polyethylene fibre suspensions of different concentrations. According to Kerekes, for Ncr > 60, fibres in a suspension are in continuous contact, while below this regime (Ncr < 60) they only come in contact occasionally (chance collision). Thus if we could compare these three fibres at the same crowding conditions, we may see the difference in XQ as a result of the flexibility variations. In this case, we would expect that FBK is more flexible than Nylon, which is more flexible than polyethylene (according to their modulus of Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 164 elasticity - Table 3.4, chapter 3). However, we could not isolate flexibility effects as shown in Figure 7.5. Here, polyethylene, FBK and nylon give the same product distribution at Cy = 0.01. At Cy = 0.02, FBK fibres and polyethylene still show the same XQ, but nylon fibres gives higher XQ. This suggests that flexibility is the least of the three and its influence is hidden by the other factors, including fibre length distribution. Table 7.1: Crowding factor (Ncr) for different fibre suspensions Fibre Nylon Polyethylene FBK Dimensions d, (mm) 0.045 0.038 0.038 0.022 L, (mm) 3.2 3.2 6.2 2.3 Cv L/d 71 84 163 105 0.010 0.021 0.031 0.042 Crowding factor 35 49 197 76 70 98 393 152 105 147 590 227 140 197 786 303 0.5 o 0.2 -0.00 0.01 0.02 0.03 0.04 0.05 Figure 7.5: Micromixing index (XQ) in fibre suspensions of different fibre aspect ratio at various volume concentrations. (N = 17.3 rev/s, feed point F32, standard reaction conditions). Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 165 7.2.4 Relative Energy Dissipation in the Presence of Fibres Having established the effect of fibres on XQ, we estimate the local energy dissipation rate using the E-model. An increase in XQ implies a decrease in local energy dissipation rate if everything else remains the same. From the previous section we conclude that the presence of fibres and the resulting floes and entanglements leads to loss of energy that would have otherwise reached the smallest scales of turbulence. This energy is lost through fibre-fibre interaction as earlier suggested by Bennington and co-workers (Bennington and Bourne, 1989; Bennington and Thangavel, 1993). We can therefore say that the local energy dissipation rate is affected most by volume concentration, followed by the L/d ratio, and then fibre flexibility. The effects of these three factors are manifested in the ability of suspension to form fibre entanglements and floes. This is the main factor that distinguishes fibres from other dispersed systems (to be discussed in Chapter 8), i.e. the ability to form entanglements. Part of the energy that is imparted in the suspension is used up in overcoming the fi-ictional interactions between fibre-fibre entanglements. For comparison, we define the relative energy dissipation rate as ratio of energy dissipation in the presence of fibres (e) to energy dissipation in the absence of fibres (So). Figure 7.6 summarizes the effect of fibres on relative energy dissipation. Data for polyethylene beads (L/d = 1) is superimposed to show the effect of L/d. It is evident that there are large differences between L/d = 1 and L/d = 84, and these differences increase as the volume concentration is increased. For example, for Cy = 0.03, the relative energy dissipation is about 0.7 for L/d = 1 whereas for L/d = 84, it decreases to about 0.1. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 166 0.00 0.01 0.02 0.03 0.04 0.05 Figure 7.6: Relative energy dissipation rate in the presence of fibres for different L/d ratios (feed point F32, N = 7.3 - 19.5 rev/s, standard reaction conditions) The decrease in local energy dissipation rate due to the presence of fibres can be correlated with the volumetric concentration using an exponential function of the form = exp(-aC^) (7.1) where e is the energy dissipation in the presence of fibres and So is the energy dissipation rate in the absence of fibres, and a is the damping factor, which is a function of the factors (other than Cy) that affect the energy dissipation rate. Table 7.2 shows the a values for FBK and polyethylene fibres. Also included in the table is the damping factor for polyethylene beads (dp = 2.7 mm, L/d = 1) for comparison. It is clear that while the propensity to form fibre entanglements accounts for most of the reduction, other factors may also be at play. These factors will be investigated in chapter 8 where we look at Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 167 different type of beads and bead sizes. Further discussion on this type of modeling for the reduction of energy dissipation rate will be given in chapter 9 where a wide range of FBK suspension concentration is tested. Table 7.2: Damping factor for energy dissipation due to the presence of fibres Dispersed Phase FBK fibres (L = 2.3 mm) Polyethylene fibres (L = 3.2 mm) Polyethylene fibres (L = 6.2 mm) Polyethylene beads (dp= 2.7 mm) Aspect ratio, L/d 105 84 163 1 a value 55±6 63±7 94±6 14±4 7.2.5 Effect of Gas Void Fraction The processing of pulp suspensions usually involves the presence of a gas phase, either as part of processing application (e.g. bleaching with a gaseous reactant) or simply trapped in interstitial spaces within the pulp suspension (Reeve and Earl, 1986; Bennington, 1993; Bennington and Smith, 1994). It is therefore important to investigate the behaviour of the local energy dissipation in the presence of gas. Figure 7.7 shows XQ values in the medium-intensity mixer for different gas void fractions (Xg = 0.0 - 0.40) at various rotational speeds for water. Here, XQ increases with increasing gas void fi:action, signifying a decrease in local energy dissipation rate. We compare the reduction in local energy dissipation with the reduction in overall energy dissipation. Figure 7.8 shows that the relative decrease in local energy dissipation is much larger than the relative decrease in overall energy dissipation. This suggests that the presence of gas has more effect than just reducing overall energy dissipation (Save)- It Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 168 shows that the hydrodynamic conditions inside the mixer change significantly in the presence of gas. The presence of gas in the form of bubbles increases the heterogeneity of the suspension, as reflected through the contours of dimensionless energy dissipation rates. The interaction between gas bubbles and small eddies also tends to reduce the local energy dissipation. o 10 u.a --0 .4 -0 .3-• 0.2-0 . 1 -no-1 • 1 <^v.^^^ ^ ^ ^ ^ ^ ^ ^ ^ " - ^ ^ $ ^ ^ > - ^ ^ ^ > ^ ^ ^ ^ ^ ^ ^ ^ ^ \ ^ —n—0.00 —o—0.12 —A—0.20 —V—0.32 —O—0.40 ^^ ^^ §^ 1 1 1 -" • " --20 N, rev/s Figure 7.7: Micromixing index (XQ) in medium-intensity mixer in the presence of gas at different void fractions (Cy = 0.0, Xg = 0.0 - 0.40, feed point F32, standard chemical conditions). Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 169 1.2 ^_o 0.8-0.6-0.4-0.2-0.0-0.0 - Overall relative energy dissipation rate ( E , „ / ( £ . , , ) „ - relative energy dissipation rate (E/E ) 0.1 I 0.2 0.3 i 0.4 0.5 Figure 7.8: Comparison of the relative energy dissipation in the presence of gas If we now look at fibre suspensions in the presence of gas, a more complex picture emerges, which can still be characterized by using mixing-sensitive chemical reactions. Figure 7.9 shows the micromixing index (XQ) in polyethylene fibre suspensions in the presence of gas at Xg = 0.2. Here again, as fibre concentration is increased, product distribution increased. When the local energy dissipation in the absence of fibres (8o) is compared with the local energy dissipation in the presence of fibres (s) under the same gassed conditions (Xg = 0.20) as shown in Figure 7.10, several points can be noted. First, for a given fibre concentration, the ratio (e/so) increases as the rotor speed is increased. This is in contrast to the situation where there was no gas in the system (Xg = 0). In the absence of gas we found that the ratio was independent of rotor speed and decreased with increasing fibre concentration. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 11Q N, rev/s Figure 7.9: Micromixing index in a suspension of polyethylene fibre with gas (L = 3.2 mm, Cv = 0.0 - 0.034, Xg = 0.20, feed point F32, standard reaction conditions). 1.2 1.0 0.8 -0.6 -0.4 -0.2 0.0 -o V 0 . 0 0 0 . 0 0 8 0 . 0 17 0 . 0 2 5 0 . 0 3 4 N , rev/s Figure 7.10: Relative energy dissipation in a suspension of polyethylene fibre with gas (Cv = 0.0 - 0.034, Xg = 0.20, feed point F32, standard reaction conditions). Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 111 A larger fraction of energy is able to reach the lowest scales with increasing intensity. We also note that all the lines for different consistencies are parallel to each other, which implies that the rate of increase of the parameter is the same for a given gas void fraction. Table 7.3 compares the damping factor for polyethylene beds in the presence of gas. Provided we compare the relative dissipation at the same conditions, the effect of the presence of gas is insignificant at the 95% confidence level. Because of the presence of the gas, however, there is greater variability in the measured power dissipation value. This agrees with literature findings with regard to the effect of gas on micromixing (Bourne, 1993). Table 7.3: Damping factor for energy dissipation due to the presence of fibres Polyethylene fibres (L = 3.2mm,L/d = 84) Xg = 0.0 63±8 Xg=0.20 69±18 Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 172 7.3 Profiles of Energy Dissipation Rates 7.3.1 Fibre Suspensions (fibre + water) Figure 7.11 shows contour plots for the dimensionless energy distribution for a FBK fibre suspension at different speeds. The profiles do not differ significantly over the range of speeds tested. However, the maximum relative dissipation rate is lower in the presence of fibres than without fibres (see Figure 6.10). Higher energy dissipation rates are found closest to the rotor with a maximum along the vessel mid-plane of the vessel where the interaction between the top and bottom loops takes place. The profile shows symmetry along the vessel mid-plane. Figure 7.12 shows the dimensionless energy dissipation for polyethylene suspension at two concentrations. First, when we compare polyethylene fibres and FBK fibres at the same concentration and rotor speed, there is no difference in their contour profiles. However, while the energy distribution profiles are similar, the ultimate energy that reaches the smallest scales is significantly different. Table 7.4 shows that FBK gives slightly less reduction in the local energy dissipation than polyethylene. This is in agreement with what was observed earlier (section 7.2.3). At higher concentrations, the flow profiles for polyethylene changes. In this case, the dissipation profile is more uniform, and the symmetry on the vessel mid-plane is no longer present. This corresponds to the "D Type" flow pattern described in Chapter 5. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 173 X N 1 0.8 0.6 0.4 0.2 0 --\ Ys 1 1 1 1 0.6 ) / t 1 1 1 1 1 0.8 1 r/R (a) (c) 0.6 0.8 r/R (b) (d) 0.6 0.8 r/R Figure 7.11 Dimensionless energy dissipation for FBK suspension at Cy = 0.0215, Xg = 0.0, standard reaction conditions: (a) N = 7.3rev/s, (b) N = 12.3rev/s, (c) N = 17.3rev/s, (d) N = 19.5rev/s. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 174 (a) 0.8 -0 . 6 -I N 0.4 -0.2 -0 :l ' , 1 , 0.6 ^ 0 . 5 ^ / / / , , ,/ 1 J / / 0.8 1 (b) r/R Figure 7.12 Dimensionless energy dissipation for polyethylene suspensions at N = 17.3 rev/s. Xg = 0.0, standard reaction conditions (a) Cv= 0.0215 (b) Cv= 0.043. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 175 Table 7.4: Energy dissipation rates in fibre suspension (Xg = 0.0) for different rotor speeds Water FBK fibre Polyethylene Cv 0 0.0215 0.0215 0.0215 0.0215 0.043 0.043 L 2.3 3.2 N, rev/s 7.3 12.3 17.3 19.5 7.3 12.3 17.3 19.5 12.3 17.3 19.5 17.3 19.5 e 88±8 410±30 632±45 907±92 22±4 174±30 187±40 207+56 146+18 174±24 284±28 125±25 386+30 Save? 11±1 43±2 103±5 141+7 12±1 38±2 101+5 141±7 57±3 104+5 145±7 83±5 117±6 S /£ave 9.4 10.8 6.7 8.4 2.1 3.6 2.1 2.1 2.5 2.4 2.4 2.4 2.4 Table 7.4 summarizes the results for the local average energy dissipation rates determined for the plane 45° between baffles. We note that the ratio e/eave is about 8 for water, whereas it is reduced to about 2.4 for FBK and polyethylene fibre at the same concentration. 7.3.2 Gas-Liquid Suspensions Figure 7.13 shows the dimensionless energy dissipation profile for a two-phase suspension of gas and water without fibre. Here we observe that in the presence of gas, there is appreciable difference between the contour profiles compared to the case where there was no gas (Figure 6.10 in Chapter 6). While the profiles are much sharper in the absence of gas, here with Xg = 0.20, the profiles are more uniform (blunt). With increasing rotor speed, the dissipation profiles become even more uniform. As a result of the change in flow pattern from inward "B-I Type" to outward "B-11" flow pattern, the zone of maximum energy dissipation is still at the vessel mid-plane. It is also observed that in the presence of gas, there is a significant difference between Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 176 contour profiles for the dimensionless energy dissipation profile when compared to the case with water alone (without gas). 7.3.3 Fibre Suspensions with Gas When both fibres and gas are present in a suspension (a three-phase system), the hydrodynamics become more complex as shown in Figure 7.14. The maximum relative energy dissipation is higher in the three-phase system and it increases as the rotational speed is increased. There is also a concentration of high (f) at the mid-plane close to the rotor. The appearance of this region of intense activity seems to be at the vessel mid-plane, just as in the case of single or two-phase suspensions. This is the same plane that was observed to have more intense activities from flow visualization. When the fibre concentration is doubled from Cy = 0.021 to 0.042 the contour profile observed was significantly different as shown in Figure 7.12. There is no clear demarcation at the vessel mid-plane, which is usually observed for "B-I Type" flow pattern. This indicates that the flow generated inside the vessel was not "B-I Type" which is in agreement with flow visualization. The visual observations suggest that the overall flow pattern under such experimental conditions was "D Type", which is accompanied with reduction in overall power consumption, which leads to further reduction in local energy dissipation rates and decreased and ^ax values. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 111 (a) 0.6 0.8 r/R (b) 0.6 0,8 r/R (c) 0.6 0.8 r/R (d) 0.6 0.8 r/R Figure 7.13: Dimensionless energy dissipation for two-phase system (water+gas, Xg = 0.20) at different rotational speeds (a) N = 7.3 rev/s, (b) N = 12.3 rev/s, (c) N = 17.3 rev/s, (d) N = 20.7 rev/s. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 178 (a) 0.6 0.8 r/R (b) (c) (d) 0.6 0.8 r/R Figure 7.14: Dimensionless energy dissipation for three-phase system (FBK Cy 0.017, Xg = 0.20) at different rotational speeds (a) N = 7.3 rev/s, (b) N 12.3 rev/s, (c) N = 17.3 rev/s, (d) N = 20.7 rev/s. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 179 Table 7.5: Energy dissipation rates in fibre suspensions with gas (Xg = 0.2) Water FBK fibre Polyethylene Cv 0 0.017 0.017 0.017 0.034 0.034 L 2.3 3.2 N, rev/s 7.3 12.3 17.3 20.6 7.3 12.3 17.3 20.6 17.3 20.6 17.3 20.6 £ 46±13 165±18 460+51 382±55 4.2±2 31+8 78±10 88±13 152±8 261+13 47±3 168±13 Save? 8.6±1 23±2 79±4 107+6 4±0.6 32±2 80±4 127±6 94±5 127+6 74±4 87±4 ^ /^ave 6.4 1.1 7.3 5.0 1.5 1.4 1.6 1.4 1.5 1.7 1.6 1.8 Table 7.5 gives a summary of the energy dissipation rates determined in fibre suspensions in the presence of gas (Xg = 0.20). Here we note that when the rotor speed (N) is increased, the ratio s l&a\e. for FBK and polyethylene fibres for all concentration values lies between 1.4 and 1.8. Similar observations were made in the absence of gas with a slightly higher ratio of 2.5. We can conclude that the two fibre suspensions behave in the same way in the presence of gas as well as in the absence of gas. Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 180 7.4 SUMMARY 1. Factors that affect the local energy dissipation rate in the liquid phase of fibre suspensions have been identified as:- fibre concentration (Cv), fibre aspect ratio (L/d) and fibre flexibility. These parameters are linked to the ability of fibre suspensions to flocculate. The formation of floes in fibre suspensions has been quantified as the main phenomenon that leads to reduction of small-scale turbulent activities (microscale). 2. The local energy dissipation rate in the liquid phase of a fibre suspension decreased in the presence of fibres. The overall average dissipation rate remained constant over the range of concentrations tested. It is therefore clear that the mean energy dissipation based on overall power input to a mixer is not sufficient to describe overall mixing performance. 3. Dimensionless energy profiles constructed from micromixing data show that there is a spatial distribution of energy dissipation rates in the mixer, with higher relative dissipation values being experienced for feed positions closest to the rotor and midway between the top and bottom of the mixer. Under a complete state of turbulent motion and for the same flow type ("B-I Type"), the energy dissipation profiles for fibre suspensions do not differ from that of water alone. The presence of fibres also seems to have a streamlining effect on the energy distribution profiles. 4. The profiles of energy dissipation rate observed for polyethylene fibre at concentration Cv = 0.043 and at 17.5 rev/s are in agreement with flow visualization, which suggest a "D Type" flow pattern. 5. Comparisons between fibres and beads of the same material (polyethylene) revealed that the shape of the dispersed phase does have a significant effect in the ultimate dissipation of energy at the smallest scales. In general, polyethylene fibres reduced the Chapter 7: Micromixing in Fibre Suspensions: Medium-Intensity Mixer 181 local energy dissipation to a greater extent than polyethylene beads at the same concentration. This further confirms the role of fibre entanglement and flocculation as most important factor that affects flow at the smallest scales. 6. Hydrodynamic conditions of fibre suspensions with gas are more complex than fibre suspensions alone. Here, the flow is characterized by much reduced local dissipation rates. This is due in part to the reduction in power consumption in the presence of gas. The reduction in energy dissipation rate at the smallest scales is more than the reduction in overall dissipation. The presence of gas in the form of bubbles after being dispersed increases the inhomogeneity of the suspension, as reflected through the contours of dimensionless energy dissipation rates. The interaction between gas bubbles and small eddies also tends to reduce the local energy dissipation. However, over the range of tests carried out in this study, no significant change was observed on overall micromixing efficiency. 7. Changes in flow pattern that were observed on the macroscale level are also reflected in the distribution of dimensionless energy dissipation profiles. The double loop ("B-I and B-II Type") pattern results in higher energy dissipation at the smallest scales than the single loop ("D Type") pattern. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 182 Chapter 8 8. Medium-Intensity Mixer with Glass and Polyethylene beads 8.1 Introduction The medium-intensity mixer (described in Chapter 3) was used to study local mixing characteristics in the liquid phase of suspensions of glass beads (dp = 0.11 - 2.4 mm) in order to explore the effect of dispersed phase density and particle size. Polyethylene beads (dp = 2.7 mm) were also investigated, and this enabled us to compare the effects of particle shape (c.f polyethylene fibres). Both glass and polyethylene beads did not adsorb the product dyes or naphthols, thus no correction for adsorption was necessary. Experimental conditions used were the same as described in Chapter 7 for fibres. 8.2 Results and Discussion 8.2.1 Effect of Particle Concentration As was shown earlier in Chapter 5, at a given rotor speed, increasing the concentration of glass beads increased the overall energy dissipation rate. What we want to answer here is whether this increased energy ends up at the smallest scales for micromixing or is dissipated before reaching the smallest scales. Figure 8.1 shows the plot of XQ against volume concentration for glass beads at N = 12.3 rev/s. From the figure, we see that the product distribution is not influenced by bead concentration for dp = 0.1 mm, 0.55 mm and 0.80 mm. All the XQ values plotted in this particle range are scattered around the baseline case without beads (Cv = 0.0) over the range of concentrations tested (0 < Cv < 0.15). Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 183 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Figure 8.1: XQ in a glass bead suspension as a function of volumetric fraction for different particle diameters (N = 12.3 rev/s, Feed point F32, standard reaction conditions). On the other hand, for large size beads (dp =1.2 and 2.4mm (dp/le > 0.1) XQ values are larger than water. These differences increase with increasing concentration. 8.2.2 The Effect of Particle Size Figure 8.2 shows a plot of XQ against rotor speed for four particle sizes (glass beads). For smaller particles (dp/le < 0.1) XQ values are close to that found in water alone. However, in general as particle size increased, XQ values increased. In other words, the local energy dissipation decreases with increasing particle size, which is in contradiction to what we saw on the macro-scale where we found that the overall energy dissipation rate increased as particle size increased. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 184 0.5 • 0.4-0.3' 0.2 0 . 1 -0.0 (glass beads: C^ = 0.021, X =0.0) — I — 10 N, rev/s —o—0.11 mm —A—0.55 mm —V—0.80 mm —0~-1.20 mm —+—2.40 mm Aqueous 20 Figure 8.2: Product distribution in the presence of glass beads of different sizes (Cv 0.021, Xg = 0.0, feed point F32, standard reaction conditions). Therefore the increased power draw in the presence of particles does not automatically increase the energy that is dissipated at microscales. This indicates that the increase in power consumption is lost by frictional collisions between particles. The effect of particle size on energy dissipation can be explained using criteria proposed in literature. We can use the criteria employed by Gore and Crowe (1989) to compare the particle size to the size of most energetic eddies (dp/le). The length scale of the energy containing eddies is estimated as le - 0.1 x D where D is the rotor diameter). Table 8.1 shows the ratio for various particle sizes at N = 12.3 rev/s (8ave = 43 W/kg). From Gore's criteria, dp/le < 0.1 for particles with diameter less than 1 mm, and dp/le > 0.1 for particles with diameter greater than 1.0 mm. This is consistent with what we have observed. The smallest particles used (dp = 0.11 mm) are distinguished from the rest of particles by their relaxation time. The particle relaxation time (t*) is calculated using equation (2.69). The relaxation time for dp = 0.11mm, 1.7ms, is much smaller than the rest of particles. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 185 whereas for dp = 1.2 mm it is 201.6ms; The smallest particles therefore respond differently from the rest. 2.0 1.6-1.2--0.11 mm •0.55 mm -0.80 mm •1.20 mm -2.40 mm 0.04 0.05 0.02 0.03 Figure 8.3: Relative local energy dissipation rate for glass bead suspensions. (Feed point F32, standard conditions). Table 8.1: Particle relaxation time and particle relative size for Save = 43 W/kg dp,niin 0.11 0.55 0.80 1.20 2.40 dA 0.01 0.06 0.09 0.12 0.24 t*, ms^ 1.7 42.2 89.6 201.6 806.4 8.2.3 Effect of Particle Density For the effects of particle density, we compare glass beads (density = 2520 kg/m^) and polyethylene beads (density = 970 kg/m^). Figure 8.4 shows a comparison of the product distribution for glass beads (dp = 2.4 mm) and polyethylene beads (dp = 2.7 mm) ^ These values are strictly for order of magnitude comparison only as equation 2.69 over-estimates the relaxation time when applied outside the Stokes regime (RCp < 1) Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 186 both at Cv = 0.021. While the size of the two particles are nearly the same, their densities are different. 0.4 0,3-0 .2-X 0.1-0.0' —D—polyethylene beads (d^ = 2.7 mm) —O—glass beads (d„ = 2.4 mm) I 10 20 N, rev/s Figure 8.4: Comparison of product distribution for glass and polyethylene beads suspensions (Cv = 0.021, feed point F32, standard reaction conditions). 1000' 800' 600-400' 200 -- polyethylene beads (d^ = 2.7 mm) •glass beads (d =2.4 mm) I 20 N, rev/s Figure 8.5: Comparison of local dissipation rates in glass and polyethylene beads suspension (Cv = 0.021 feed point F32, standard reaction conditions) Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 187 The heavier glass beads reduce the product distribution (XQ) much more than polyethylene beads. When this is interpreted by the Engulfment model, there is greater reduction in energy dissipation at the smallest scales in glass bead suspensions (Figure 8.5). Previous studies by Villermaux et al., (1994) with small glass beads (dp = 0.04, 0.201 and 1.25 mm) concluded that particles had no effect on micromixing efficiency. Considering the range of concentrations that they used (Cv = 0.08 for dp = 0.04mm and Cv = 0.03 for dp = 0.201 and 1.25 mm), their conclusions are in agreement with what we have found in this study. Barresi (1997) found a reduction in energy dissipation rate using chemical methods in glass beads (dp = 0.46 mm, Cv = 0.13). However, they found no change for glass beads of smaller size (dp =0.138 mm, Cv = 0.13), and polyethylene beads (dp = 3 mm, Cv = 0.06). He attributed the decrease in power dissipation to increased suspension density. All these findings are in agreement with that found in the present study. At relatively low concentrations (Cv < 0.043), the increase in power transmitted to the fluid due to the increase in suspension density is compensated by the damping effects of particles. At higher concentrations, however, the interaction between particles becomes the dominant means of dissipating energy. According to Buyevich (1994), dissipation of energy due to particle collision results from: surface friction that impedes the sliding of spheres at their points of contact; the in-elasticity of the collisions, and the work of viscous forces which arise as a consequence of jump-like changes in fluid velocity that accompany any collision. 8.2,4 Effect of Particle Reynolds Number (Rcp) The above effects can be incorporated using the particle Reynolds number. The particle Reynolds number was estimated from power number group according to Kuboi et Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 188 al. (1974). Kuboi and co-workers have shown that there exists a relationship between the specific power group (sdpVv )^'^ ^ and the particle Reynolds number (Rcp) in the form: Re^=0.5 ^sd'.y^ v ^ y (8.1) Figure 8.6 shows a plot of particle Reynolds number for the different particles used. According to Hetsroni (1989), there would be turbulence reduction in the suspension if Rcp < 110 and turbulence generation if Rcp > 400, and in this case, only particles with dp = 2.4 had Reynolds numbers above 400. Thus conditions for the other particles are mostly mixed because they fall in the intermediate range. It is therefore clear that at lower speeds, most particles lie in the range where no generation of turbulence is expected, while at rotor higher speeds (N > 10 rev/s), the generation of turbulence due to wakes behind the particles is much smaller than dissipation due to particle-particle interactions. 0). a: 800-600-4 0 0 -200 -0-1 -1 —clp = 0.11 mm — •dp = 0.55 mm • • dp = 0.80 mm •— dp = 1.2 mm • - dp = 2.4 mm .^.'.. 1 y' y' y' y' ^y' 1 ' 1 Rep = 400 Rep = 110 10 20 N, rev/s Figure 8.6: Particle Reynolds number for different glass bead sizes. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 189 Figure 8.6 also demonstrates that the effects of particles on energy dissipation rate are more pronounced at higher rotational speeds. It should be noted that most studies have been conducted at low rotational speeds (Nf < 7 rev/s) and low average energy dissipation rates (8ave< 5W/kg), where some effects are not seen even for large dp. 8.3 Profiles of Energy Dissipation Energy profiles were obtained in the same manner as described in Chapter 7.5, i.e. at a plane midway between baffles (45°) for fifteen injection points (Figure 3.4 in chapter 3). 8.3.1 Suspensions Without Gas Figure 8.7 shows the distribution of dimensionless energy dissipation for a glass bead suspension of Cy = 0.043 (dp = 0.80 mm) at two rotational speeds. Essentially, no difference was observed in the contour profiles. When a suspension of the same concentration but having smaller beads (dp = 0.11 mm) was tested, the contour profiles differed from the rest (Figure 8.8.). In this case, contour profiles for smaller beads were flattened, indicating more uniform energy dissipation throughout. This also corresponded to a change in flow pattern from type "B-I Type" to "D Type", which is in agreement with flow visualization. For particle sizes greater than 0.11 mm, the profiles for energy dissipation rate correspond to "B-I type" flow patterns. The profiles of polyethylene beads were similar to that observed for fibrous suspension of the same concentration range as shown in Figure 8.9. Here, the profile remained the same, but the maximum dimensionless energy dissipation decreased with increasing concentration. The flow pattern for all these cases was "B-I Type". Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 190 8.3.2 Suspensions in the Presence of Gas Similar profiles were obtained when a three-phase suspension of glass beads, water and gas was used. Figure 8.10 shows the profile for a 0.80 mm glass bead suspension with Xg = 0.2. The following observations were made with regard to glass bead suspensions with gas: • The contour profiles for dimensionless energy dissipation rate (^ do not differ significantly over the range of speed (N = 7.3 - 19.5 rev/s) and particle concentrations used (Cm = 0-0.10). • ^ax values do not vary significantly when N is increased over the range tested (N = 7.3 - 19.5 rev/s) • There is noticeable decrease in ^ value due to the presence of glass particles • There is no significant difference in the contour profile of ^ when the mass concentration is doubled from Cy — 0.021 to Cy = 0.043 at constant impeller speed. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 191 (a) (b) Figure 8.7: Distribution of dimensionless energy dissipation rate. Glass beads suspension (dp = 0.80 mm, Cy = 0.043, Xg = 0.0, standard reaction conditions). (a) N = 12.3 rev/s. (h) N= 17.3 rev/s. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 192 (b) 0.8 r/R Figure 8.8: Distribution of dimensionless energy dissipation rates for a 0.1 Imm glass beads suspension (Cv= 0.043, Xg = 0.0). (a) N =12.3 rev/s, (b) N = 17.3 rev/s. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 193 (a) 0.6 0.8 r/R (b) Figure 8.9: Distribution of dimensionless energy dissipation rates for polyethylene beads (dp - 2.7 mm) at N=17.3 rev/s, Xg= 0.0 (a) Cv = 0.021, (b)Cv = 0.043. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 194 (b) Figure 8.10: Distribution of dimensionless energy dissipation rates for 0.80 mm glass beads (Xg= 0.20, Cv= 0.021). (a) N = 12.3 rev/s, (b) N = 17.3 rev/s. Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 195 8.4 Summary 1. Liquid turbulence can be damped by increased particle concentration. 2. A reduction in the local energy dissipation was observed in suspensions of glass beads. When the particle diameter is large and its density high (high particle Reynolds numbers) , there is large relative velocity between particles and liquid because of particle's inability to follow the fluid motion. As a result, there is an increase in overall power draw. 3. While it has been reported in the literature that the presence of glass beads does not influence micromixing efficiency, we find that this statement is true only for small size particles (dp/le < 0.1) at low concentrations (0 < Cy < 0.043). 4. The reduction of turbulence depends not only on solid concentration but also on particle size and density. This is in agreement with the literature where an increase in particle Reynolds number leads to a greater reduction in local energy dissipation. 5. For dense suspensions of large size (Cv > 0.043, dp/le > 0.1) particles, the reduction of energy dissipation in the liquid phase is large and has been attributed to energy dissipation due to particle-particle interactions. 6. Previous attempts to use mixing-sensitive reactions to study micromixing in glass suspensions were limited to single point comparisons at low energy dissipation rates (Save < 2W/kg). From the present study, it is evident that differences become more apparent at higher energy dissipation rates. The present study has used the distribution of energy dissipation rates from Chapter 8: Medium-Intensity Mixer with Glass and Polyethylene Beads 196 micromixing experiments as a means of examining local mixing characteristics. 7. The presence of glass particles produces more blunt profiles of energy dissipation rate. It is further shown that increasing the concentration of the dispersed phase decreases the maximum dissipation rate significantly. Chapter 9: High-Intensity Mixer with Pulp Suspensions 197 Chapter 9 9. High-Intensity IVIixer with Pulp Suspensions 9.1 Introduction The high-intensity mixer provides high-energy dissipation rates (Sgve up to 5000 W/kg) in the mixing zone. This is a desirable feature to achieve small-scale mixing, but it leads to a rise in temperature during batch operation. Therefore, in practice, such mixers are either operated for very short batch times (e.g. for the commercial laboratory Quantum"^" mixer, the batch operation time is only 4 to 10 seconds) or in continuous mode with a short residence time (e.g. the typical residence time for a Sunds"^" high-shear (industrial) mixer is 0.2 seconds (Bennington, 1996). By incorporating the high-shear mixer in a flow loop, temperature could be maintained at 25°C by providing an external cooling unit. In the flow loop, the reactants were subjected to the high-shear zone for a short time ( 1 - 3 seconds), while spending 15 - 45 seconds in the external tank where cooling was provided. We also increased the volume ratio (VAA^B) from 50 to 100 and doubled the concentration of diazotized sulphanilic acid to CI = 43.08 mol/m^. The total batch time was reduced from six minutes to three minutes. In this way, we were able to maintain the temperature of the reactants at 25 ± 1°C even for the highest speed achievable in the mixer (N = 83.3 rev/s). In this way, it was possible to use mixing sensitive chemical reactions to study microscale mixing in the high-intensity mixer. The details of experimental procedure for this section have been described in chapter 3. Following are the reaction conditions for all tests reported in this chapter: C°,= 0.52 molW, C°2/C°,= 2, Cg= 43.08 molW, and volumetric ratio between Chapter 9: High-Intensity Mixer with Pulp Suspensions 198 naphthols solution (A1+A2) and diazotized sulphanilic acid solution (B) of 100. A molar excess of 20% based on 1-naphthol was maintained. 9.2 Overall Flow Characteristics in the High-Shear Mixer 9.2.1 Flow Pattern Flow visualization in the high-shear mixer was limited to viewing the front of the mixer only as the rest of mixer was made of stainless steel. Extensive investigations of the flow conditions in the mixer for both batch and continuous processing of pulp suspensions with and without gas were reported in earlier studies (Bennington, 1988; 1993). Here we shall note some additional characteristics due to flow loop operation and concentrate on the local flow behaviour. The flow patterns observed are similar to previously reported patterns. Mainly, the cellular flow pattern ("B-I Type") was observed for the range of speeds tested in this study (N = 16.5 - 83.3 rev/s). No gas was used in any of the experiments in the high-shear mixer and for all cases complete turbulent motion was maintained. Thus, other patterns were not evident in this part of the study. The cellular flow consisted of six cells, corresponding to the six baffles. The flow was radially inward. We can therefore speculate that the side pattern would be "B-I Type". Since the power number was the same for all speeds tested, the flow pattern did not change throughout the experiments. Likewise, the overall energy dissipation rate did not change, whether we were running the mixer with or without flow loop, as will be shown later. Chapter 9: High-Intensity Mixer with Pulp Suspensions 199 9.2.2 Flow Loop Characteristics The flow loop was first operated with water in order to understand its flow behaviour. At a fixed initial recirculation rate ( Q°^), the rotor speed was increased in steps from 0 to 83.3 rev/s. At each increment point, the flow rate was measured at a point downstream of the mixer using a magnetic flow meter as described in the experimental section (3.1.2). From Figure 9.1 it can be seen that the flow rate increased as the rotor speed increased. In fact, the centrifugal pump could be shut-off once complete circulation of suspension in the flow loop had been established, and flow still maintained. The high-shear mixer behaved as a pump when connected to the flow loop, however, the rotor could not initiate flow on its own. Figure 9.1: Fluidizer pumping capacity for different initial pump flow rates It is interesting to note that when the location of the inlet at the mixer faceplate was changed it affected the total flow rate. Figure 9.2 compares two inlet locations, one at Chapter 9: High-Intensity Mixer with Pulp Suspensions 200 the centre and the other off-centre. For the off-centre position (80 mm from mixer centerline), a sharp decrease in flow appears at high flow rates. The point at which the drop was observed increased with increasing initial flow rate (Q°^). The drop is likely due to a decrease in rotor pumping capacity as gas accumulated at the centre of the rotor. For the centre position, flow is directed to the suction side of the rotor, resulting in an increased total flow throughout the operation range. The centrally located feed was used throughout the experiments. a 200' 150-100 5 0 -off-centre: — • O— centre: —S -S— 20 40 N, rev/s I 60 —r-80 Figure 9.2: Comparison of centrally located feed vs. off-centre feed for two initial flow rates (g° =20 and 124 L/min) Chapter 9: High-Intensity Mixer with Pulp Suspensions 201 9.2.3 Reaction Zone Visualization For the high-shear mixer operated with the flow loop, under all experimental conditions, reaction zone visualization indicated that the reaction was always completed well within the mixer. In the presence of the flow loop, it was necessary to check how the reaction zone behaved, especially at the entry to the mixer. Figure 9.3 shows the reaction zone visualization using the NaOH/HCl technique described earlier. When the flow loop is running with QA = 100 L/min and the rotor is not moving (N = 0 rev/s), the reaction zone moves outwards, following the path of the incoming stream. However, when the rotor is running as low as 6 rev/s (300 rpm), with the same external flow (100 L/min) the reaction zone is directed inward following the flow pattern generated by the rotor (Figure 9.4). This shows again that the rotor flow is the dominant flow in the mixer. 9.2.4 Overall Energy Dissipation Figure 9.5 shows the curve for energy dissipation versus rotor speed for a FBK suspension is close to that of water (Cv = 0.013 - 0.133; Cm = 0.01 - 0.10), except at low rotor speeds and high mass concentrations. This is also observed in the plot of power number against suspension concentration for the range of mass concentrations tested. At higher mass concentrations, the yield stress of the suspension is high, and therefore before the suspension has attained complete motion, only part of the suspension responds to rotor movement (i.e., cavern formation, as pointed out by Bennington et al., 1991). Consequently, the energy imparted per unit mass of suspension is higher than when complete turbulent motion exists throughout the mixer. Chapter 9: High-Intensity Mixer with Pulp Suspension 202 Figure 9.3: Reaction zone visualization in high-shear mixer with flow loop: Q\ = 100 L/min, N = 0 rev/s, feed point location F32 (r/R = 0.727, z/H = 0.05). Reaction plume follows the external flow when the rotor is not moving. Figure 9.4: Reaction zone visualization in high-shear mixer with flow loop: QK= 100 L/min, N = 6 rev/s, feed point location F32 (r/R = 0.727, z/H = 0.05). Reaction plume re-directed towards the rotor even at very low speeds. Chapter 9: High-Intensity Mixer with Pulp Suspensions 203 3000 2000-CO 1000-0<^ • ^ ^ t ^ . D O A V •c> 0.013 0.039 0.065 0.092 0.133 FBK 75 N, rev/s Figure 9.5: Overall mean energy dissipation rate for FBK suspension as a function of rotor speed (batch operation). The Solid line is computed for water using Np== 3.4. Figure 9.6 shows that the average power input per unit mass remained constant for a given rotational speed even when fibre was added to the mixer. There was also no difference in overall energy dissipation in the mixing chamber with or without (QA = 0.0 L/min)the flow loop. This was also reflected by the power numbers which remained constant for the range of suspension mass concentrations tested in this study (Cv = 0.0 -0.133). The maximum concentration that could be handled in the flow loop was limited to Cv = 0.079 (Cm = 0.06). This is because of limitations with the external pump used to circulate the suspension. Chapter 9: Hi^h-Intensity Mixer with Pulp Suspensions 204 CO 10000' 9000-8000-7000-6000-5000 A-4000-3000 A . Q Q—-Q-2000-1000 5 — s — n — B -0 ^ — , — I — . — r •H • -0.00 0.02 0.04 — I — 0.06 — I — 0.08 T T T —n—N = 50.0rev/s - 0 - N = 66.7rev/s — A - N = 83.3rev/s -o--• 0.10 — I — 0.12 — I — 0.14 Figure 9.6: Overall energy dissipation rate versus fibre volume concentration for the high-shear mixer with flow loop (0.0 < Cy < 0.08; QA = 70 - 200 L/min), and without flow loop (0.0 < Cy < 0.133; Q A = 0 L/min) symbols with dotted centre. When the high-shear mixer was operated without the flow loop, much higher consistencies could be handled although there were some limitations for the minimum energy required to achieve complete turbulent motion in the mixer content ("fluidization"). For example, at 50 rev/s (save = 1230 ± 28.2 W/kg) complete vessel motion could not be obtained above Cm = 0.08 (Cv = 0.109). When we calculate the minimum fluidization energy from the Bennington and Kerekes equation, EF = 1180 W/kg. Table 9.1 shows a comparison of experimentally observed limits with minimum fluidization energy as calculated fi"om equation 2.60. These experimentally observed limits are in excellent agreement with predictions based on literature correlation for minimum fluidization energy required for the rotor stator mixer (Bennington and Kerekes, 1996). Chapter 9: High-Intensity Mixer with Pulp Suspensions 205 Table 9.1: Maximum suspension concentration handled at given rotational speed in the high-shear mixer (from experimental observations). N, rev/s 50 66.7 83.3 Overall energy dissipation rate Save, W/kg 1230±28 2960±59 5000±60 Maximum suspension concentration Cm (Cv) 0.08(0.109) 0.11(0.147) 0.14(0.189) Minimum fluidization energy (eq. 2.60) EF, W/kg 1180 2617 4780 9.3 Micromixing Results 9.3.1 Experiments with Flow Loop Figure 9.7 shows the product distribution for rotor speeds N = 50-83.3 rev/s for three different feed points: F31 (r/R = 0.50, z/H = 0.5), F32 (r/R = 0.727, Z/H =0.5), F33 (r/R = 0.955, z/H = 0.5) along the mixer radial plane. First, we note that the product distribution decreases with increasing rotor speed, as expected. We also note that the feeds at three different positions give different XQ values, indicating that the reaction experiences different energy dissipation rates. The point closest to the rotor (F31) experiences the highest energy dissipation rates and gives the lowest XQ values as shown in Figure 9.7. The point midway between rotor and wall (F32) gives the highest XQ values, and thus experiences the least energy dissipation. This is not surprising, considering that flow visualization indicated that the incoming fluid is incorporated into the rotor flow, thus shifting the reaction zone as compared to the case without the flow loop. The lines in Figure 9.8 are drawn using the E-model simulation with the position parameter (9) taking values from 0.4 to 4. Point F32 was used for all fiirther experiments with flow loop. Chapter 9: Hi^h-Intensity Mixer with Pulp Suspensions 206 o 0 .4 -0 . 3 -0 .2 -0 . 1 -0 .0 -1 . 1 . s * • . , ' 1 ' 1 — 1 - . . 1 0 F3, (p = 0.4 _ — -1 ip - 1 • - - - (p = 4 " • - ^ - - - . - - -. A —1 ' 20 40 60 N, rev/s 80 100 Figure 9.7: Product distribution versus rotor speed for high-shear mixer with flow loop. The solid line represents E-model predictions with (p = 1, dashed line cp = 0.4 and dotted line cp - 4. Two feed nozzles at 0 = 0° and 6 = 180° F31 (r/R = 0.50, z/H = 0.5), F32 (r/R = 0.727, Z/H =0.5), F33 (r/R = 0.955, z/H = 0.5). (C°, = 0.52, CJC^, = 2; VA/VB = 100, C^ = 43.08 mo^m^ tf = 200s, T = 298±1K) Chapter 9: High-Intensity Mixer with Pulp Suspensions 207 9.3.2 Effect of Recirculation Rate on Micromixing Figure 9.8 shows a plot of XQ versus recirculation rate for water tests made at different rotational speeds. For all rotational speeds tested, there was no change in the product distribution at a constant rotor speed even with increasing recirculation rate. This indicates that the energy imparted by the rotor is the dominant factor in the mixer. 0.4' 0.3' o • 0.2-0 . 1 -0.0' I — r T — ^ T — ' — r •N = 16.3rev/s • N = 50.0 rev/s • N = 66.7 rev/s 25 50 75 —I— 100 — I — I — r -125 150 175 Recirculation rate, Q^, L/min —r-200 Figure 9.8: The effect of recirculation rate on product distribution at constant rotor speed in the high-shear mixer. (Cv = 0.0, €"^21^^^= 2, VAA^B = 100, C°,= 0.52 molW, CI - 43.08 mol/m^ tf = 200s, T = 298±0.5°K). Two feed nozzles at 9 = 0° and 0 = 180° F32 (r/R = 0.727, z/H = 0.5). Chapter 9: High-Intensity Mixer with Pulp Suspensions 208 9.3.3 Effect of Fibre Concentration on Micromixing in Flow Loop Figure 9.9 shows XQ'''' values as a function of fibre concentration for three rotor speeds. Here we can see that at any given rotational speed, an increase in fibre concentration leads to increase in XQ'''''. When interpreted by the E-model, this translates to a decrease in the energy dissipation rate with increasing fibre concentration. 0.5 • 0 . 1 -0.0 0.00 0.02 N = 50.0 rev/s —o— N = 66.7 rev/s —A—N = 83.3 rev/s 0.04 0.06 0.08 0.10 Figure 9.9: Product distribution versus FBK fibre suspension concentration in high-shear mixer operated with the flow loop (QA = 180 - 200 L/min). (Conditions: C^^IC^x^ X VAA^B = 100, C°,= 0.52 mol/m^ C°= 43.08 mol/m^ tf = 200s, T = 298±1°K). Two feed nozzles at 9 = 0° and 9 = 180° F32 (r/R = 0.727, z/H = 0.5). In Figure 9.10 it is seen that there is a considerable decrease in the energy dissipated locally in the presence of fibres, despite constant overall energy dissipation (Figure 9.6). Chapter 9: High-Intensity Mixer with Pulp Suspensions 209 5000 4000-3000-" 2000 -1000-0.00 —D— N = 50.0 rev/s - V - N = 66.7 rev/s - O - N = 83.3 rev/s 0.02 0.04 0.06 0.08 Figure 9.10: Local energy dissipation rate in FBK suspension. (Estimated from E-Model for the experimental conditions: C°2/C°,= 2, VAA^B = 100, C;,= 0.52 molW, C°= 43.08 molW, tf = 200s, T = 298±1°K). Two feed nozzles at 8 = 0° and 9 = 180° F32 (r/R = 0.727, z/H = 0.5). Table 9.2 shows the local energy dissipation measured in the liquid phase for a Cv = 0.039 (Cm = 0.03) suspension at N = 50 rev/s (3000 rpm). Only 60 W/kg is dissipated at the smallest scales. When the rotational speed is increased to 66.7 rev/s (4000 rpm), 180 W/kg reaches the smallest scales, and when the rotational speed is increased to 83.3 rev/s (5000 rpm), the amount that reaches the smallest scales increases to 400 W/kg. Table 9.2: Local and overall energy dissipation rates in an FBK suspension (Cy = 0.039 or Cm = 0.03) at different rotor speeds. N, rev/s 50 66.7 83.3 Energy dissipation rate, W/kg Overall (input) 1230 2960 5020 Local 60 180 400 % dissipation at microscale 5 6 8 Chapter 9: High-Intensity Mixer with Pulp Suspensions 210 9.4 Experiments Without Flow Loop In order to increase the fibre suspension concentration to Cm = 0.10, it was found necessary to operate without recirculation. This necessitated some changes in the operating conditions. The number of feed nozzles was increased from two to six, thus enabling the critical feed time to be reduced to 6 seconds. It was confirmed that this feed rate was still in the micromixing regime by plotting the product distribution against feed time at 16.7 and 50 rev/s as shown in Figure 9.11. At 16.7 rev/s XQ is independent of feed rate for feed times greater than 55s. When the rotor speed was increased to 50 rev/s, XQ became constant after a feed time of only 6 seconds. Therefore a feed time of 21 seconds (three times the critical feed time) was chosen for all experiments made without the flow loop for N > 50 rev/s. Reaction zone visualization using acid-base neutralization showed that the different reaction zones did not overlap with each other, even when using six feed nozzles (Appendix E-2). 0.3-0.2-a X 0.1 0.0' o N = 50 rev/s • N = 16.6 rev/s —r-20 40 —I-60 80 t ,s Figure 9.11: Critical feed time for high-intensity mixer without flow loop at 16.7 rev/s and 50 rev/s,.(Cm= 0.0, C°^2^C°i= 2, VA/VB = 100, C;,= 0.52 mol/m^ C° = 43.08 mol/m^ T = 298±1°K). Chapter 9: High-Intensity Mixer with Pulp Suspensions 111 By operating the high-shear mixer without the flow loop, it was no longer possible to limit the temperature increase. We therefore had to change the way reactions were carried out. An operations table was established to determine the starting temperature of each experimental condition such that the average reaction temperature was kept at 298°K. Table H-16 - H-18 in Appendix H shows the temperature scheduling for different rotor speeds. We could account for temperature variation in the E-model simulation by updating the rate constants as well as fluid viscosity as a function of extent of chemical addition. The effect of this is shown in Table 9.3. Here, the fluid (water) viscosity was updated using an Arrhenius type relationship while the rate constants were updated using the rule of thumb (Rys and Zollinger, 1972) that coupling reaction rate doubled for every 10°C temperature rise. It is evident that the values do not differ significantly from estimation with T = 25°C throughout the reaction, and consequently we assumed that the temperature was T = 25°C for our calculations. In all cases, the average reaction temperature was maintained at T = 25±1°C throughout the reaction. Table 9.3: Product distribution from E-Model at mean reaction temperature and different temperature ranges N, rev/s 33.3 50.0 66.7 83.3 To=25,AT = 0 0.2488 0.1884 0.1506 0.1252 To =24.5, AT =1 0.2430 0.1833 0.1461 0.1213 To =24, AT =2 0.2432 0.1834 0.1462 0.1214 To=20,AT=10 0.2443 0.1846 0.1475 0.1226 Figure 9.12 shows values of the product distribution {XQ'^'') as a function of fibre concentration in the high-shear mixer for batch operation at an average energy dissipation rate of 3000 W/kg (N = 66.7 rev/s). Over the whole range of concentration (0.0 < Cy ^ 0.133), the XQ'''' values increased with increasing fibre concentration. Chapter 9: High-Intensity Mixer with Pulp Suspensions 212 0 .4 -0.14 Figure 9.12: Product distribution in high-intensity mixer without flow loop {QA - 0 L/min, N = 66.7 rev/s, tf -21s, C^-.IC^r 2, VAA^B = 100, C;,= 0.52 molW, C;= 43.08 molW, T = 295 - 301 °K). Six feed nozzles equi-spaced at e = 60° F32 (r/R = 0.727, z/H = 0.5). Figure 9.13 shows the corresponding local energy dissipation. It is evident that at higher fibre concentrations, very little energy shows up at the lowest scales. For example, at Cv = 0.133, less than 1% of the energy supplied is dissipated in water at the smallest scales. The rest of the energy dissipation takes place at higher length scales as shown earlier. Chapter 9: High-Intensity Mixer with Pulp Suspensions 213 Jit: 03 1000-100 •: 10-'—•—•—I 1 1 1 1 • ---• 0.01 0.1 Figure 9.13: Local energy dissipation for semi-batch fluidizer operated without flow loop. Conditions as in Figure 9.12. 9.4.1 Comparison of Experiments with and without Flow Loop Figure 9.14 compares the product distribution when the high shear mixer was operated with the flow loop (two feed nozzles) and when it was operated without the flow loop (six feed nozzles) all equi-spaced on a pitch-circle diameter of 80 mm. We see that Chapter 9: High-Intensity Mixer with Pulp Suspensions 214 according to the E-model simulation, the point F32 experiences energy dissipation rates that are about 40% lower than the average energy dissipation rate while in experiments without the flow loop, the same feed point experiences about 4 times the average energy dissipation. The possibility of temperature being the cause of this effect is not likely as it was shown earlier that the product distribution is not affected significantly provided the reactions are carried out at the average designated temperature. 0.5 • 0.4-0.3-0.2-0 . 1 -0.0' 40 O with flow loop, n, = 2 D without flow loop, n, = 6 (p = 0.4 9 = 4 60 80 100 N, rev/s Figure 9.14: Comparison of product distribution high-shear mixer with flow loop and without flow loop (water only). Solid line represents E-model prediction with (p = 0.4 and dotted line 9 = 4. Flow loop: Two feed nozzles at 9 = 0° and e = 180° F32 (r/R = 0.727, z/H = 0.5) tf = 200s, T = 298±1K; Without flow loop: six feed nozzles equi-spaced by 60° (r/R = 0.727, z/H = 0.5), tf = 21s, T = 298±1K (C°, = 0.52, C"^2/C°^^= 2; VA/VB = 100, C° = 43.08 mol/m^). Chapter 9: High-Intensity Mixer with Pulp Suspensions 215 9.5 Correlations for Damping of Local Energy Dissipation Rate It is clear that only a small portion of the energy imparted in the mixer ends up in the liquid phase at the smallest scales. In chapter 7, we have already shown that this difference is related to pulp characteristics, namely, Cv, L/d and fibre flexibility. It has long been speculated that the presence of fibres and floes inhibit the development of small-scale turbulence (Mason, 1954, Daily and Bugliarello, 1961; Norman et al, 1978). Since all the energy that is imparted in a mixer will ultimately be dissipated as heat, we can account for the energy input in a mixer though the different dissipation mechanisms. If we assume that three methods of dissipation exist in pulp suspensions, namely the dissipation through fibre-fibre interactions, fibre-fluid interactions and fluid interactions (eddies) that lead to viscous dissipation. In the absence of fibres, the transfer of energy fi-om largest to smallest scales follows the cascade as described in chapter 2. In the presence of fibres, additional dissipation is due to fibre-fibre interactions and fibre-fluid interactions. This takes place at the expense of energy that would have otherwise been transferred to the smallest scales, and thus the observed reduction in liquid dissipation as fibre concentration is increased. In this section, we develop a correlation that can be used to account for the decrease in local energy dissipation due to the presence of fibres. 9.5.1 Correlation Through a Fibre Concentration and Damping Factor Figure 9.15 shows a plot for the reduction of energy dissipation as a function of fibre concentration at various rotor speeds. It is clear that fibre concentration is the most dominant factor in damping of local energy dissipation. The decrease in energy dissipation rate can be correlated to the fibre concentration through an exponential damping function Chapter 9: High-Intensity Mixer with Pulp Suspensions 216 £ = So exp(-aCv) (9.1) where £ is the local energy dissipation in the presence of fibres, SQ is the local energy dissipation without fibres. The damping factor (a) depends on fibre aspect ratio and fibre type (fibre flexibility). Although the damping factor seems to vary slightly with shear rate (rotational speed), all data for N = 50 - 83.3 rev/s can be correlated by a single factor, a = 52+6 (at 95% confidence level). Table 9.4 shows a values for different fibre types. Using this type of unified approach we should be able to correlate the effects of fibre on transport processes that are affected by energy dissipation at the smallest scales, e.g. mass transfer. l O l S r 0.8-• o A N = 50 rev/s N = 66.7 rev/s N = 83.3 rev/s - s / s „ -exp ( -aC^ ) a = 52±6 -> , , 1 , 1 — - T — L j ' • •• • • -•y^""'""r • • r • 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Figure 9.15: Relative energy dissipation rate for FBK suspension in the high-shear with flow loop, QA = 180- 200 L/min. Chapter 9: High-Intensity Mixer with Pulp Suspensions 111 Table 9.4: Damping factors for energy dissipation due to the presence of fibres (medium-and high intensity) Fibre type FBK Polyethylene L/d 105 84 162 a (95% Conf. Level) 52±6 63+7 94±6 9.5.1.1 Comparison with Mass Transfer Correlation We have compared these results with results obtained by Rewatkar and Bennington (1999) in a similar rotary device of commercial design that is used for mass transfer studies (Appendix G). They found that the gas-liquid mass transfer coefficient decreased exponentially with increase in fibre concentration. In their case, the damping factor for gas-liquid mass transfer coefficient was found to be a = 50±13. It is interesting to note that the damping of local energy dissipation rate with fibre corresponds to the reduction in mass transfer rate at the smallest scales. This implies that with knowledge of the reduction of local energy dissipation rate due to the presence of fibres, we can predict what will happen to local transport processes. We can therefore use the information on local energy dissipation to correlate various transport processes that take place at the smallest scales (e.g. mass transfer). Chapter 9: High-Intensity Mixer with Pulp Suspensions 218 9.6 Model for Fibre and Floe Interactions In this section, we use the knowledge gained about the reduction in local energy dissipation measured in the liquid phase to see if fibre-fibre friction can account for the observed change. The main assumption here is that the energy that is imparted in a mixer will be dissipated through (a) fibre-fibre friction, (b) fibre-fluid drag and (c) fluid-fluid (eddy) interaction. The chemical method used in this study accounts for (c). Contributions from fibre-fluid interaction are estimated from drag calculations (and are expected to be small - see Appendix 1.2). Fibre-fibre friction contribution is derived below. ROTOR Save Fibre Suspension Cv, L/d, E Fibre- Fibre Fluid-Fluid Fibre-Fluid Cf.f Sd EXQ Figure 9.16: Schematic representation of the cascade process in the presence of fibres. Chapter 9: High-Intensity Mixer with Pulp Suspensions 219 9.6.1 Fibre-Fibre Friction The energy dissipation due to fibre-fibre fi"iction can be estimated firom the normal forces that act at fibre contact points per unit mass of suspension with a fluctuating relative (fibre-fibre) motion of Ur (m/s) is given by Sf_j=HpF„NcU^ (9.1) where Fn = normal force per contact, N/contact Nc = number of contacts per unit mass of fibre, (contacts/kg) Ur = fluctuating relative velocity between fibres, m/s HF - coefficient of fi-iction between fibres here taken as |XF = 0.7 (Andersson and Rasmuson, 1997) The normal force acting on a fibre-fibre contact point can be estimated from beam bending theory (Famood, 1994, Kerekes, 1995) see Appendix I.l for the derivation which gives the normal force as F = ^ ^ (9 2) where 5 = maximum bending deflection which is proportional to fibre diameter (d) and may be set as 6 = 0.6d following Famood at al. (1994). El = fibre stiffness in N.m AL = the distance between two contact points in meters, and is obtained by dividing the length of fibre with the number of contacts per fibre (uc), i.e. Chapter 9: High-Intensity Mixer with Pulp Suspensions 220 AL = — (9.3) to obtain the number of contacts per fibre (Uc), we consider a homogeneous random network of fibres with vmiform fibre length. In such a network, the number of contacts per fibre is given by a number of expressions from the literature, for example Meyer and Wahren (1964) (equation 2.52 in chapter 2 after simplification) " ' - i ^ (9.4) Pan (1993) gives and Dodson (1996) n ^ = ^ ^ (9.5) 2 + TlCy «, = 2ACy (9.6) The total number contacts that are actively engaged in providing the frictional resistance in a fibre network is not readily apparent. It may be computed from the total number of fibres in a unit mass of suspension as N,=N^xn: (9.7) where Nf = number of fibres per unit mass of suspension «* = number of independent contacts per fibre The total number of fibres (Nf) is readily calculated using the mass concentration of the suspension as Chapter 9: High-Intensity Mixer with Pulp Suspensions 221 i V / = - % (9.8) coL where Cm = fibre mass fraction, kg fibre/kg suspension (0 = fibre weight per unit length (coarseness), kg/m L = fibre length, m Now we need to know how many independent contacts we have in a suspension. Kerekes' crowding factor (Kerekes and Schell, 1992) gives the number of fibres in the volume swept out by a single fibre length (L) based on fibre volume concentration (Cv) and aspect ratio (A = L/d) as N.r=\CyA' (9.9) For a giyen fiber with Uc contacts, the probability (Pc) of this single fibre interacting with any other fibre in its swept volume is given by Pc=^ (9-10) cr Thus the number of independent contacts per fibre is given by i V e = i V ^ x « : = A ^ ^ x P c X « , = - ^ (9.11) Finally, the fluctuating relative velocity between fibres (Ur) due to hydrodynamic forces necessary to disrupt a contact is estimated from the average energy dissipation (£ave) and the minimum distance between two points in relative motion in the network (here taken as fibre diameter, d) by using an empirical correlation (see appendix I) as «,=0.5(^„,,^)' (9.16) Chapter 9: High-Intensity Mixer with Pulp Suspensions 222 9.6.2 Comparison Figure 9.17 shows a comparison of the energy dissipation using different estimates for number of contacts according to theoretical models based on statistical geometry. At low concentrations, all models under-predict the energy dissipation by about two orders of magnitude. As the concentration is increased, the estimates of Dodson and Pan increases much faster than Meyer and Wahren and give values closer to experimentally determined values for concentrations above Cy = 0.05. The reason for the different predictions is the fact that Dodson and Pan's models have a direct relationship between nc and Cy while for Wahren's relationship, nc is proportional to the square root of Cy, thus the number of contacts does not increase as fast as predicted by Pan or Dodson's expression. We proceed with our comparisons using Meyer and Wahrens expression. It should be borne in mind that this model assumes that the only forces to be overcome are the frictional forces that result from normal load at contact points. Recently, Andersson and Rasmusson (1997) have measured an additional adhesive force (beyond the normal force) when determining the friction coefficient of fibres. This force has been attributed to fibre hooking (Kerekes et al., 1985; Soszynski, 1987; Andersson and Rasmusson, 1997) and it is the first time it has been measured for fibres. From their measurements, the maximum additional force (FQ) for kraft fibre was found to be about 0.06 - 0.10 mN (Andersson and Rasmusson, 1997). If we include the additional force, the calculated Sf.f is a bit higher as shown in Figure 9.18. However, the model does not agree with experimentally observed data and we only have a qualitative agreement. The only other parameter that we do not have a handle on is the relative velocity with which the fibres move during turbulent rupture of flocs/contacts. This would depend on the velocity field around fibres/flocs. If we assume that all other parameters have been estimated reasonably, the fluctuating velocity at fibre contacts is the only unknown. Figure 9.19 shows estimates for energy dissipation for a range of values from 0.2m/s to 2m/s. Chapter 9: Hi^h-Intensity Mixer with Pulp Suspensions 223 • 10^1 J ^ 3 10 -o 10'1 10^1 10°-"r ' 1 =0-5(B3ve n y y / ' 1 o / r 1 d) '^' o / . • / / 1 o ^ ^ r ^ ' V 1 1 5 ^ , # V ' 1 ' 1 ' 1 ' 1 ^ ^ ^ ^ ' ^ , - * * ^ ^ -x: O ^ ^ - ' O o / - ' ^ ^ ^ — - ' ^ ' • • ^ y ^ Meyer and Wahren (1964) Pan (1993) Dodson (1996) ' 1 ' 1 ' 1 ' 1 ; • •t j • -": • •: . -: \ . ^^  0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Figure 9.17: Comparison of energy dissipation due to fibre-fibre friction using different estimates for number of contacts. Experimental data for high-shear mixer at N = 66.7 rev/s. Chapter 9: High-Intensity Mixer with Pulp Suspensions 224 '. u=0.5(e d) r \ ave ' 10^1 10'-: 10^-10^-10" T ^ 1/3 n o •» r T ' 1 « r o o 5 o o F=n,F +F f r'F n o — -o - F . -F„ •F ^ 0 - ^ = 0 = 0.06mN = 0.10mN XQ — I — 0.06 — 1 — 0.10 —T— 0.12 0.00 0.02 0.04 0.08 0.14 Figure 9.18: Comparison of energy dissipation due to fibre-fibre friction using different estimates of the additional force of friction. Effect of additional adhesive force on energy dissipation rate. Fo from Andersson and Rasmusson, 1997. Chapter 9: High-Intensity Mixer with Pulp Suspensions 225 — I 1 1 1 1 r • n (Meyer and Wahren) 10^  i o w 10'-co ^Q'-^ -*—r 0 XQ — I 1 1 1 1 1 1 1 1 1 1 1 1 r — 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Figure 9.19: Estimation of energy dissipation due to fibre-fibre friction using different fluctuating velocities. Uc calculated from Meyer and Wahren, 1964. Chapter 9: Hi^h-Intensity Mixer with Pulp Suspensions 226 10^ T — ' — r 1 — 1 — I — I — r 10^ ^ CD ^ 1 0 ' -10^-10" o Fo= 0.06mN u. estimated O 8 -Syp. 0 XQ J i .1 I I I I I I L J L. 0.00 0.01 0.02 0.03 0.04 0.05 C„ 0.06 0.07 0.08 Figure 9.20: Estimation of energy dissipation due to fibre-fibre friction using estimates from Andersson (1998) and experimentally determined velocity profile from Figure 9.19 Chapter 9: High-Intensity Mixer with Pulp Suspensions 227 First, we note that for a given velocity, the curve crosses the experimental data only once. Thus, if velocity is the only parameter left to match the model with experimental data, the velocity must be decreasing as we increase the volume concentration. This trend seems correct intuitively. For example, for a given speed, the addition of fibres will ultimately lead to complete stagnation of the suspension. If we proceed with data from Figure 9.19 and develop a correlation that relates volume concentration to the change in velocity we obtain: u^ -u[ exp(-20.6C^) where u[ is the rms velocity in the absence of fibres (see Appendix 1.3 for further discussion of velocity estimates). Figure 9.20 shows estimated energy dissipation due to fibre-fibre friction using the suggested velocity profile and data from Andersson (1996). We see that the estimated values fall within the experimental data. While this does not validate Andersson's data, or our own, it demonstrates the self-consistency in the hypothesized relationship. It should be noted that few studies have actually studied the flocculation state under turbulent conditions. It is not clear from these considerations how many contacts are actually available. Soszynski (1987) counted the number of contacts in a floe and found fewer contacts than expected from isotropic random network models based on statistical geometry. In addition, the number of active contacts decreases as the fibre concentration is increased. How do we explain the flocculation concept in a turbulent flow? According to Steen (1989), in a turbulent fibre suspension, floes are contained within large eddies whereas the smallest eddies contain only single fibres or no fibre at all. During turbulent transfer of energy from large scales to smaller scales, the internal fibre network is deformed. Floe breakup is probable if the deformation forces (turbulent stresses) are larger than the floe internal strength. The length scale of the straining vortices will be of Chapter 9: High-Intensity Mixer with Pulp Suspensions 228 the same order as the floe scale. Eddies of smaller scale will agitate the outer part of the floe, making the floe weaker thus increasing the probability of fibre dislodgment. The rupture of floes is linked to the turbulent energy cascade, whereby eddies interact with other eddies of adjacent scale. On the other hand, floe aggregation process is composed of a small-scale activation and large-scale collision and network settling. These processes take place at the same location but at different time scales. Thus even under turbulent conditions, a fibre suspension consists of floes that move as coherent entities. These entities interact with turbulent eddies, thus further reducing the energy that reaches the smallest scales. In conclusion, the flow of fibre suspensions at the smallest scales is very complex. Energy has to be spent in order to unlock fibres in the network, and these fibres will interlock again as soon as they reach regions of insufficient energy. The distribution of energy dissipation has been shown to vary in the mixer, therefore floes moving in turbulent motion will experience different energy dissipation rates in moving fi-om one region of the mixer to another. As a result, there is a constant formation and disruption of fibre networks. 9.7 Correlation Using the Apparent Viscosity In turbulent flow where momentum transport is primarily fi-om large scales of turbulence, one can consider the pulp suspension to be a fluid and assign it an apparent viscosity. The energy cascade fi-om large-scale to small-scales is dissipated by fibre-fibre fiiction, relative motion between fluid and individual fibres or floes and shear in the fluid elements. Viscous and turbulent momentum transfer can be combined together to give an "effective" viscosity which, multiplied by the mean velocity gradient, gives the momentum transfer of the fluid in the flow. The effect of viscosity alone can be considered by an "apparent" viscosity which combines the effect of momentum transfer Chapter 9: High-Intensity Mixer with Pulp Suspensions 229 from the fibre-fibre interactions and molecular interactions. The local energy dissipation can be related to the apparent viscosity. 9.7.1 E-Model and Viscosity If we revisit the E-Model, we see that the engulfment coefficient (E) relates the local energy dissipation with kinematic viscosity, i.e 1 E = 0.058 (equation 2.38) Using the viscosity estimated from literature (Chapter 2) we calculate the apparent viscosity of fibre suspension under turbulent condition using an empirical relationship developed by Bennington and Kerekes (1996). //„ =1.5 10-^C„/' (equation 2.52) where |ia is the apparent viscosity, and Cm is the fibre mass concentration. We can also use an adaptation of a theoretical expression from literature (Ferguson and Kemblowski, 1991) J^a=Mo f c V^ '^ inax 1--C, (equation 2.54) ' ' y where [ia is the apparent viscosity, |io is the viscosity of water, C"" is the maximum solid concentration. Using equation 2.52 and 2.54 with our FBK data we found that a value of [pi] = 0.02A^^ ^ gave a good match for the two estimates. Barnes (1989) suggested [\i] = 0.01 A^'^ for a suspension of rod-like particles, thus our fit is fairly close to Barnes theoretical estimate. Hence, we use equation 2.54 with [\i] = 0.02A to obtain a general equation for suspension viscosity. Using estimates from the above equation and the E-model, we can estimate the product distribution in the presence of fibres. Figure 9.21 shows a comparison of the product distribution estimated by E-model with experimentally measured values at the same conditions (Feed location F32, e=(peave)-Chapter 9: High-Intensity Mixer with Pulp Suspensions 230 > ^ 0.4-0.3-0.2-0 . 1 -nn-1 1 1 1 1 1 1 1 1 70.4 .. 33.0 t....'"""- ^.. 15.8 0 . . J- " •*• -. 7.7 y _ ^ X 1 , . , , , . , . D 0.0 0 0.013 A 0.026 Y 0.039 0 0.052 + 0.065 X 0.079 40 50 60 70 80 N, rev/s 90 Figure 9.21: Comparison of experimental XQ values with E-model prediction using the apparent viscosity of fibre suspension. High intensity mixer with flow loop. Flow loop: tf = 200s, T = 298±1K, Feed location^ F32; e = cpeave- (C°, = 0.52, CJC^, = 2; VAA^B = 100, C° 43.08 mol/m^). Chapter 9: High-Intensity Mixer with Pulp Suspensions 231 It is remarkable that E-model prediction is in such good agreement with the experimental data. The effect of fibres in decreasing the energy dissipation at the smallest scales can therefore be accounted for using the apparent viscosity of suspension in the E-Model. In the literature, the E-model has also been used to predict product distribution in liquids with high viscosity (Gholap et al, 1991; Baldyga et al, 1994; Bourne at al, 1995; Guichardon et al, 1997). In this case, increased viscosity resulted in increased product distribution (decreased micromixing efficiency). This is in agreement with what we have shown here. Recall that for a Newtonian fluid in laminar flow the average shear rate (velocity gradient) G, is related to the shear stress through x=HG (9.17) But the local strain rate at a point is given by / , G = C1 v^y (9.18) where CI is a constant. If we combine (9.17) and (9.18) we can write an expression for energy dissipation rate as - 2 -Z" .2 e = Cr'^- (9.19) Thus we obtain a relationship between the power input per unit mass of fluid and viscosity, similar to the expression used by Wahren (1980), with constant CI = 1. In this case therefore, we can use the same argument put forward by Bennington and Kerekes (1996) in replacing the laminar viscosity in equation 9.19 with the apparent viscosity for the pulp suspension. Hietaniemi and Gullichsen (1995) used similar arguments. In this Chapter 9: High-Intensity Mixer with Pulp Suspensions 232 case, momentum transport takes place mainly by inertial turbulent forces rather than liquid viscosity which is associated with molecular transport. If we look at the way Wahren (1980) estimated the dissipation rate (Equation 2.43 in chapter 2), the role of viscosity is apparent. Originally, he had used the viscosity of water in the formula and subsequently, the values calculated were much higher (three orders of magnitude) than those found by GuUichsen (1981) and Bermington et al., (1991). Bermington and Kerekes (1996) noted that the main cause of discrepancy in Wahren's values was the use of the viscosity of water. When they re-calculated Wahren's data using the apparent viscosity, they found much closer agreement with the experimental data. The above discussion leads to the possibility of correlating the change in local energy dissipation rate with the apparent viscosity. The following correlation was therefore developed to account for the reduction of local energy dissipation rate in the presence of fibres: £ £„ (9.20) where P is a coefficient to be determined from experimental data, jio is the viscosity of water, |ia is the apparent viscosity of suspension under turbulent conditions. By fitting equation (9.20) to experimental data we obtain p = -1.0±0.1 for FBK suspension in the range tested (N = 50 - 83.3 rev/s. Figure 9.22). From the present study, we clearly see that we can replace the liquid viscosity with an apparent viscosity that also takes in the effect of fibres to obtain correct predictions for product distribution (XQ) fin the E-model. For engineering purposes therefore, we can use apparent viscosity data of fibre suspension to predict changes on transport processes in the presence of fibres as shown earlier for the case of mass transfer. Chapter 9: High-Intensity Mixer with Pulp Suspensions 233 CO I . U S 0.8-0.6-0.4-0.2-0.0-P 1 \« « 1 • 1 • 1 1 {• -1 / e/e„ = (n/n/-^ * X:A D O A T ' 1 ' 1 " N = 50.0 rev/s N = 66.7 rev/s N = 83.3 rev/s -_ I T ' • - " J'^ fwiifcMfcii II III 1 fai^iii 0.00 0.02 0.04 0.06 0.08 0.10 Figure 9.22: Correlation for relative energy dissipation rate using effective viscosity (equation 9.20) Chapter 9: High-Intensity Mixer with Pulp Suspensions 234 9.8 Summary 1. The overall and local energy dissipation rates have been determined in a high-shear mixer with pulp fibre suspensions up to Cm = 0.10 (Cy = 0.133). Under turbulent conditions, the overall energy dissipation rate (8ave = P/pV) in high-shear mixer remains constant, even with increasing fibre concentration. However, local energy dissipation rate decreases with increasing fibre concentration. 2. Operation of the high-shear mixer with a flow loop has enabled the control of temperature to within ±1°C allowing mixing sensitive reactions to be used to estimate local energy dissipation rate. 3. The decrease in local energy dissipation rate can be correlated to the fibre concentration through e = EQ exp(-aCv) where 8 is the local energy dissipation in the presence of fibres, EQ local energy dissipation without fibres and a = 52±6 is the damping factor for FBK (a also depends on L/d and flexibility) 4. The energy imparted by the rotor in high-shear mixer is the dominant factor determining micromixing when flow loop is used. External recirculation/flow does not influence micromixing provided that reactions are completed inside the mixer. 5. By increasing the number of feed points from two to six, the time required to carry out the reaction in semi-batch mixer can be reduced by a factor of three. Thus, provided that the reaction zones do not overlap, operation time can be reduced in batch studies by increasing the number of feed points. 6. From the results presented in this study, it is evident that small-scale turbulence in the liquid phase is greatly reduced due to the presence of fibres, and that the reduction is increased as fibre concentration is increased. Here, we have been able to quantify the relative amount of energy dissipation that ended up at the smallest scales in pulp suspension in low- and medium-consistency processing conditions. Chapter 9: High-Intensity Mixer with Pulp Suspensions 235 7. The reduction of energy dissipation in the presence of fibres can be accounted for by an increase in fibre-fibre fiiction provided an appropriate estimate of number of fibre contacts in a suspension under turbulent conditions is used. There is clearly a need to carry out further studies on the state of flocculation under turbulent conditions. 8. The reduction in local energy dissipation rate can also be correlated by use of an effective viscosity. In the present study, we have been able to predict the product distribution in the presence of fibres by including an effective viscosity in the Engulfinent model. This has led to the development of correlation for relative local energy dissipation rate based of suspension relative viscosity in the form £ = £„ i/l^ / /"o )"^ where p = 1.0±0.1 for FBK fibres. Chapter 10: Conclusions 236 Chapter 10 10. Conclusions This thesis has investigated micromixing in the liquid phase of fibre suspensions at concentrations relevant for mixing in low- and medium-consistency operations in the pulp and paper industry. The study was also extended to cover the liquid phase of other dispersed systems for comparison and the clarification of the mechanism by which energy is dissipated at the smallest scales in the presence of dispersed phases. In this Chapter, the overall conclusions based on their studies are presented. 10.1 The Role of Fibres on Turbulence. It has been found that at fixed rotor speeds, the overall energy dissipation (Save = P/pV) remains constant with increasing fibre concentrations, however, the local energy dissipation decreases at the same conditions. The overall energy dissipation rate is therefore not sufficient to describe mixing in the liquid phase when two or more phases are involved, as in the case of fibre suspensions in MC processing operations. The mechanical energy that is imparted though the action of rotor reaches the smallest scales through the energy cascade. The transfer of energy through the cascade is inhibited by the presence of fibres. Most of the energy is lost through fibre-fibre and fibre-liquid interactions and as a result, very little reaches the smallest scales. As a consequence, only part of the supplied energy contributes to the local scale activities (e.g. mass transfer and reaction). The effective energy dissipation rate can be correlated using an exponential decay fiinction, £ = £„ ex^{-aCy), where a is a measure of damping and has been found to be equal to 52 ± 6 for FBK and 63 ± 7 for polyethylene fibres. The effective energy dissipation can also be deduced if one knows the turbulent viscosity properties of Chapter 10: Conclusions 237 suspension under consideration, hi this case, a correlation of the form e = £^{in^ /ju^y^where p = 1.0±0.1 for FBK can be used to predict the reduction in local transport processes. 10.2 Effect of other Dispersed Particles on Turbulence For the case of dispersed particles that do not form floes or entanglements, the energy dissipation is mainly due to particle collision and fluid drag. While some studies have inferred an increase in power dissipation in the presence of dispersed phase, this does not necessarily translate to an increase in local energy dissipation in the liquid phase. When the diameter of a particle suspended is large and its density high, gravity and inertial effects cause a large relative movement between the particles and the fluid, which contribute to an increase in the overall energy dissipation rate (specific power consumption). On the other hand, when the density of particles is low, the effect of generation of turbulence is negligibly small in comparison to that of turbulence damping. In the present study, small particles (corresponding to dimensionless particle size (dp/le < 0.1) showed little or no influence in the local energy dissipation rate when present in low concentrations (Cv < 0.043). Coarse particles (corresponding to dimensionless particle size (dp/le > 0.1) dissipate energy through particle collisions, particle-liquid interactions and fluctuating relative velocity that results between particles and fluid. 10.3 Distribution of Local Energy Dissipation Rate The energy dissipation rate estimated fi-om the micromixing model is indicative of the local flow behaviour in the liquid phase. The information about local behaviour is best illustrated by the dimensionless energy dissipation profiles. It has been shown that local energy dissipation varies across mixers, even at high-energy dissipation rates. While different investigators have recognized this effect when working with traditional stirred tank reactors, this is the first study to attempt a Chapter 10: Conclusions 238 reconstruction of the energy dissipation profile in mixer fi-om micromixing experiments. Maximum energy dissipation rate is closest to the rotor tips while the minimum lies close to the mixer walls. Mixer design should therefore exploit this fact in order determine the best feed-point configuration. While the estimated values of local energy dissipation obtained by this method have some limitations, there is no doubt that the profile of energy dissipation could be captured with a reasonable degree of confidence. Therefore, usefiil inferences regarding local flow behaviour in the presence of dispersed phases can be made. The energy profiles were found to be more uniform in the presence of fibres, while a more heterogeneous field was found in the presence of gas. Profiles give overall indication of flow pattern or type of flow that is in agreement with flow visualization. 10.4 The Use of Mixing-Sensitive Chiemicai Reactions One major factor to be considered in this application of mixing sensitive reactions in dispersed systems is the possibility of affecting the product distribution by means other than turbulence intensity, hi this study, the adsorption of chemicals on certain fibres was found to be a major obstacle that had to be overcome before meaningful interpretation of micromixing experiments could be made. The best approach would be to use chemicals that do not adsorb on the fibres. For the particular system used in this study (FBK), it was necessary to use a complex system of dye products that unfortunately adsorbed on FBK fibre. It was therefore necessary to develop a correlation in order to account for dye adsorption. Mixing-sensitive chemical reactions can be used to determine the best location for chemical injection where fast reactions are involved like pulp bleaching. In the present study, the best location was found to be close to the rotor and in the vessel mid-plane. Chapter 10: Conclusions 239 This location has been found to be independent of the presence of fibres, gas or other dispersed phases, provided there is complete turbulent motion in the suspension 10.5 Implications of Present Findings • By measuring local energy dissipation rates we can predict the effect of the presence of fibres or particles on local phenomena like mass transfer. It is interesting to note that the local energy dissipation rate in the presence of fibres varies in the same way as gas liquid mass transfer rate measured in the presence of fibres (Rewatkar and Bennington, 1999). • • We have increased knowledge about local flow characteristics in the liquid-phase of fibre suspensions under turbulent conditions. This should help in flow management/design of contacting equipment in pulp and paper industries. The relative dissipation profile reconstruction method can be used to compare the performance of a given mixer under different operating conditions or different mixers for a given duty. The data generated fi-om this work will be useful for validating models for simulations in high-intensity mixing devices. In this case, the E-model has been used to estimate local energy dissipation rate and found that a factor is required to close the difference between overall energy dissipation and estimated local dissipation rates. The factor necessary to close varied fi-om 3 -10. This has been likened to the factor used in estimation of energy dissipation from fluid velocity measurements in the literature. Chapter 10: Conclusions 240 • Three flow regimes ("Type B-I, B-II and D") have been identified in mixing of fibre suspensions under different conditions. These have a direct impact on the overall energy dissipation. "Type B-I" gives the highest overall energy dissipation as compared to "Type B-II" and "Type D". • The presence of gas in fibre suspensions has been found to affect both the macroscale and microscale flow behaviour. The overall energy dissipation is reduced by a diminished ability to transfer momentum fi-om the rotor to the suspension as a result of the lower density and viscosity at the rotor tip caused by the accumulation of gas there. It is also possible that the presence of dispersed phase gas bubbles may interact with eddies, leading to the reduction in small-scale turbulence. • Using the apparent viscosity of pulp suspension in the E-model allows correct prediction of the product distribution in the presence of fibres. Therefore by adopting an apparent viscosity, we account for the damping of turbulence in fibre suspensions. The apparent viscosity must be determined at the same turbulent conditions. If we know the mixing conditions without dispersed phase, the local mixing conditions in the presence of fibres can be estimated using viscosity correlation developed (e/eo = {\ij\i^'^). Chapter 10: Conclusions 241 11. Recommendations for future work This thesis has touched a number of diverse areas. Future work should concentrate on specific issues that have been raised here, notably; 1. An invesfigation on viscosity properties of fibre suspensions under turbulent conditions. The aim of such a study would be to improve and standardize apparent viscosity measurements in order to be able to consolidate measurements made in rotary devices with measurements made in pipe flow. This would give predictive methods for local scale transport processes in industrial equipment. 2. An investigation on the state of flocculation under turbulent conditions. Such a study should be aimed at improving/advancing knowledge in the field of paper making (formation) at high consistencies. 3. Computafional modeling (CFD) of fibre suspension behaviour with a rheological model developed in (1). 4. Investigation on energy dissipation in fibre suspensions, with the aim of accounting for all possible dissipation rates. An attempt should be made to design a mixer in which viscous dissipation can be measured fi^om temperature rise, power input, and chemical methods. 5. An experimental and/or numerical study to determine number of contacts in fibre suspensions under turbulent conditions would also be helpfijl. 6. Use of heterogeneous reactions (gas-liquid or liquid-solid reactions) in modeling of micromixing in fibre suspensions. Chapter 12: Nomenclature 242 12. NOMENCLATURE a orientation factor in equation D. 12 A fibre aspect ratio (L/d) Ai 1-naphthol Aa 2-naphthol b Langmuir adsorption parameter, L/|i-mol B diazotized sulphanilic acid C1 constant in equation 2.19 Cbuik bulk concentration, mol/m'' c„ C, concentration of compenent /, mol.m"'' <Ci> average concentration of species / , mol.m" Cj concentration fluctuation, mol/m c,. mean concentration, mol/m^ Ce equilibrium concentration, |J,-mol/L Cm mass concentration, fraction Cv volume concentration, fraction d fibre diameter, m dp particle diameter, m D rotor diameter, m D diffusion coefficient, m .s" Deff effective difflisivity, m^.s'' D T tank diameter, m D(k) dissipation spectra Da Damkohler number E(k) energy spectra E engulfhient rate coefficient, s"' E elastic modulus, Pa G shear rate, s" Ge elongational strain rate, s"' G(k) concentration spectra Is intensity of segregation Chapter 12: Nomenclature 243 k wave number ki reaction rate constant, m .mof .s KF distribution coefficient (Freundlich model), L/g KL distribution coefficient (Langmuir model), L/|j.-mol Kf distribution coefficient (Extended Freundlich model), L/g le length scale of the most energetic eddies, m L length, m Ls segregation macroscale Z,^  length weighted average length, m L^ weight weighted average length, m m mass, g mv amount of fibre per unit volume, g/L M mixing modulus, equation (K.7) M torque, N.m N number of moles N rotor speed, rev/s Ncr Kerekes crowding factor Nc total number of contacts per unit mass of suspension N number of eddy generations Np power number Nf number of fibres nf number of feed points ric number of contacts per fibre n number of turnovers P power, W Qo maximum solid phase concentration, |i-moI/g Q 2-(14 sulphophenyl-azo)-2-naphthol or its ion in aqueous alkaline solution (monoazo dyestuff) Qp, recirculation rate, L/min Rn reaction rate, mol/m^.s Re autocorrelation coefficient Chapter 12: Nomenclature 244 Re Reynolds number (Re = NDVV) Rep particle Reynolds number (Rep = u,d/v) o-R 2-(l ,4 sulphophenyl-azo)-1 -naphthol or its ion in aqueous alkaline solution (ortho monoazo dyestuff) p-R 4-(l,4 sulphophenyl-azo)-1-naphthol or its ion in aqueous alkaline solution (para monoazo stuff) S 2,4-(l,4 sulphophenyl-azo)-1-naphthol or its ion in aqueous alkaline solution (bisazo dyestuff) Sc Schmidt number (Sc = v/D) t time, s tc critical feed time, s to diffusion time, s tos diffusion-shear time, s te characteristic time of most energetic eddies, s t* particle relaxation time, s tf feed time, s tR reaction time, s tM mixing time, s T temperature, °K T dimensionless t ime for reaction and mixing, q adsorption, | i-mol/g qe equilibrium adsorption, )J,-mol/g u velocity, m/s u' velocity fluctuation, m/s Ur fluctuating relative velocity, m/s U characteristic velocity, m/s Xg gas void fraction X Q , , X S product distribution (equations 2.34 and 2.35) Xw water retention value, kg/kg Greek a volume ratio, damping factor Chapter 12: Nomenclature 245 p 5 e Save 1^  ^a M-eff | lF X XK Xc IB h VK X Xm \ "TDS a o' e p <t) 1^ Tl ^ 9 V CO fi kinetic ratio, exponent in viscosity correlation slab thickness, fibre half width, m energy dissipation rate, W/kg (m^s'^ ) average energy dissipation rate, W/kg dynamic viscosity, Pa.s apparent viscosity, Pa.s effective viscosity, Pa.s coefficient of friction length scale, m Kolmogorov microscale, m Corrsin microscale, m Batchelor microscale, m Taylor microscale, m Kolmogorof velocity scale, m/s time scale, s micromixing time, s eddy lifetime, s time for diffusion and shear, s number of feed discretizations in E-model variance dimensionless time density, kg/m'' dimensionless energy dissipation rate porosity tortuosity factor ratio of 1 -naphthol to 2-naphthol concentration position dependent parameter kinematic viscosity, m^.s"' fibre weight per unit length, kg/m angular velocity, rad/s Chapter 12: Nomenclature 246 Superscripts ave. average corr. corrected m measured max maximum o initial, condition without fibres Subscripts eff effective f fibre f-f g P s w Abbi fibre-fibre gas particle solid water reviations CFD computational fluid dynamics CSF Canadian Standard Freeness, ml DNS direct numerical simulation EDD engulfinent, diffusion, deformation FBK fiiUy bleached kraft ICS inetrial convective subrange lEM interaction by exchange with mean LDV laser doppler velocimery PDF probability density function RTD residence time distribution STR stirred tank reactor VCS viscous convective subrange VDS viscous diffusive subrange Chapter 13: References 247 13. 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RMS fluctuations Autocorelation Macroscale ke of turbulent motion Turbulent energy dissipation Energy Spectra E(k) Dissipation spectra Taylor microscale Taylor time constant Viscous dissipation Kolmogorov microscale Viscous dissipation time constant Velocity ^ = {i/f\ 00 Lf = \nr)dr 0 q = iu'' ^~ dt ~ 2 dt « ' ' = ! \E{k)dk = 2\E,{k,)dk, 0 0 £ = 2v \eE{k)dk = 30v \k^E,{k{)dk^ 0 0 0 2 _ 30w/'2 1/ ^ / ~ 10 v i,=(.'/.r ^K ={vlsf^ Segregation intensity Segregation dissipation Concentration spectra G(k) Dissipation spectra Corrsin microscale Corrsin time constant Corrsin microscale Batchelor microscale Concentration d = {cy c{x)c{x + r) sir) = ,2 00 Ls = jg(r)dr 0 c"/c\' ^s ~ dt CO c'' = JG(k)dk. 0 •0 £^=2D^k^G(k)dk 0 0 2 _ I2Dc'^ •s 120 \ £ / A,=(D^fsr t TO I ON 1 2. 3. Category Hydrodynamic models Multi-environment models Diffusion models a.k.a. Coalescence-redispersion model Cell balance (network-of-zones) Rigid slab Lamellar stretch Table A-2: Summary of Micromixing Models Study Curl (1963) Levenspiel (1972) Kattan & Adler (1972) Cannon et al., (1977) Mann and Mavros 1982 Patterson, 1981 Rippin(1966) Ritchie & Trogby (1979) Mehta & Tarbell (1983) Mao &Toor (1970) Truong & Methot (1976) Ottino (1982) Main Principles Fluid packets in a vessel are moved in such a way as to approximate the convection pattern in real case monte carlo simulation Mixer volume viewed as a network of perfectly mixed cells crossed by the same flowrate. Turbulence level in each cell characterized by Ls and e Simulation of intermediate levels of mixing in continuous flow systems -micromixing takes place in a small fraction of time/region Diffusion in rigid fluid elements Diffusion with reaction in deforming lamellae Equations dp dpR p-po dt dc T c J = 4\p(c')pi2c-c')-p(c) 0 1 CO dc, da dt dx, " D dc. dC: _ dt dx. \dc,' ^d^ ^ + R, Regime Macro Macro Macro ICS VCS t I !^ >3 s s ON 00 4 5 Category Mass transfer models Engulfment Model 3.*K*M* Interaction by exchange with mean (lEM) Erosive mixing Shrinking aggregate model inetrial spectral mixing Engulfment-Deformation-Diffiision (EDD) E-Model Table A-2 (Contd.):.Summary of Mixin Study Costa & Trevisoi (1972) David and Villeurmaux(1983) Klein etal (1980) Pohorecki and Baldyga (1983) Bourne and Baldyga (1984) (1989) Main Principles Micromixing follows 1^ ' order mass transfer between each eddy and the mean environment Peeling off smaller fragments from segregated lumps of fluid by turbulent action at their external surface Reduction of completely segregated elements as transfer through ICS Micromixing in a shrinking laminated structure within an energy dissipating vortex Engulfment rate controlling g^odels Equations dc. (<c, > - c , ) dt t„ D / = /«(l-f) ' e * 7 r>-2/3„-l/4 dt C dt Sx. dx. ^ = E({c,>-c,>i?, £ = 0.058(f/vy" Regime ICS VCS ICS VCS& VDS t >3 3 S Appendix A 270 Table A-3: Particle - Turbulence Interactions Investigator(s) Gore & Crowe (1989) Hetsroni(1989) Rashidi, Hetsroni and Banerjee (1990) Yuan & Michaelides (1992) Schreck & Kleis (1993) Yarin & Hetsroni (1994) System used, Conditions and parameters measured Compilation of previous experimental data by various authors using gas-solid, liquid-solid and gas-liquid systems Turbulent jets, Newtonian fluids with monodisperse particles monodisperse particles polystyrene (dp= 0.12 and \.\mm;pp =1030kg/m^), glass beads (dp=0.088mm, p^ = 2500 kg/m^) based on interactions of single particles with eddies • dilute gas-solid systems • mainly in pipe and jet flows Neutrally buoyant particles Grid turbulence Polydisperse system Model based on Prandtl mixing length theory to describe the process of supression and enhancement of turbulence in particle laden flows Remarks 1. Small particles reduce turbulence intensity while larger particles enhance it. 2. dp/le> 0.1 enhancement dp/le < 0.1 reduction 1. Particles with low Rcp (<110) cause suppression of turbulence while particles with higher RCp (> 400) numbers enhances it. 1. Large particles increase turbulence and Reynold stresses while small particles decreases 2. above effects are enhanced by particle concentration 1. Dissipation of power from an eddy (for particle acceleration) is the predominant mechanism for turbulence reduction 2. Velocity disturbances due to wake/vortex shedding is the predominant mechanism for turbulence enhancement 1. Monotonia increase in dissipation rate of turbulence energy with particle concentration 2. Speculates that particles enhance the transfer of energy to smaller eddies, extending the dissipation spectrum to small scales 3. Particles increase the isotropy of the flow field and modify the high wave-number end of the turbulence energy spectrum. 1. Turbulence modulation depends on total mass content, phase density ratio, particle Reynolds number and particle size/turbulent ratio 2. Turbulent intensity lower in polydispersed system Appendix A 211 Table A-4: Investigations in Pulp Fibre Suspension Flow Characteristics Investigator(s) Brecht & Heller (1935,1950) Head & Durst (1957) Robertson & Mason (1954,1957) Daily &Chou (1961) Bobkolwicz & Gauvin (1965,1967) Gullichsen & Harkonen (1981) Bennington (1988) Middisetal.,(1994) system used, conditions and parameters measured Horizontal 5.9 -9.84" pipe 0.005 < Cn^< 0.05 bleached & unbleached sulfite, groundwood 12" pipe; 0<Ci„< 0.06 stock:bleached sulfite, bleached sulfate and groundwood 7/8" glass tube; 0 < C^< 0.01 optical determination of flocculation Rigid spherical particles d = 0.64 mm Temperature profile in a 2" vertical pipe by thermistor probe after imposing a hot point source Stock: nylon fibers L/d= 11.7 -51 .1 , 0.005 < C „ < 0.04 Concentric cylinder LC (0 < C„ < 0.06) and MC (0.08 <Cn,< 0.15) • bleached, semi-bleached & unbleached pine kraft • spruce groundwood Concentric cylinder LC,MC, HC (0 < C„, < 0.35) 25.4 mm brass pipe • nylon fibers • 0 < Cm < 0.05 • l<u<10m/s • 20 < L/d < 120 observations & comments remarkable reduction in head-loss compared to water curve 1. reduction in head loss 2. yield stress 1. reduction in head loss 2. three flow regimes (plug, mixed and turbulent) identified above C„ = 0.002 Frictional factors higher for flows with suspensions of solid spherical Particles eddy diffiisivity, turbulence intensity increased with L/d and Cm same shear response for low (0-0.06) and medium (0.08-0.15) consistency suspensions Yield stress identified flow regimes from high speed visualization Drag reduction strongly dependent on L/d, Cm, and flow velocity. Reduction in heat transfer coefficient corresponds to drag reduction Appendix A 111 Table A-5: Determination of Turbulence Parameters in Pulp/Fibre Suspensions Investigator(s) Daily & Burgliarero (1961) Anderson (1966) Bobkolwicz & Gauvin (1965,1967) Mih& Parker (1967) Hoyt(1972) Ek(1979) Measurement technique and conditions "Turbulence gauge" Stock: poplar groundwood 0<Cm<0.01 • energy spectra and velocity profile measurement dye dispersion tall channel with isotropic grid 0.0011< Cm < 0.0047 Injection of hot water from a point source and temperature measurement by thermistor probe downstream. 0<C™<0.04 Turbulent velocity profile using an annular purge impact tube Rayon fibers: 0.002< €„< 0.005 Hardwood pulp: 0.005 < €„< 0.01 Asbestos, glass fibers and acrylic fibres Single component LDV Cm = 0.005 Sulfite pulp 10mm pipe observations & comments • rms velocity increased with Rg and decreasing Cj^ from centerline to wall • energy spectra in water showed same trend as in air decrease in dye dispersion in the presence of fibers. Further decrease with increasing consistency eddy diffusivity, radial turbulent intensity and Lagrangian length scales increase with L/d and C ^ Overall frictional loss reduced considerably analogy between turbulent flow of fiber suspension to turbulent flow of New1:onian fluids in rough pipes reduction of turbulent flow resistance significance of L/d longitudinal turbulence increase near wall (1 <r/R<0.5) and decreased near the center Appendix A 273 Table A-5 Contd.: Turbulence Parameters in Pulp Fibre Suspensions Investigator(s) Kerekes & Gamer (1982) d'Incau (1983a) McComb& Chan (1985) Luthi(1987) Steen(1989) Lee & Duffy (1976) Louttgenetal. (1991) Measurement technique and conditions Single component LDA Channel grid (0 < C„ < 0.005) LDA Channel grid C„ = 0.005 Single component LDA Asbestos high L/d dye dispersion C„, = 0.06 LDA Index matching 1.2 &12g/l,L= 1,3mm 23.8mm vertical pipe L = 2.7, d = 0.03 mm 100 mm pipe annular purge impact probe 0 < C„, < 0.06 3" pipe temperature contour determination observations & comments • damping of turbulence intensities • increased turbulence at higher frequencies reduction of turbulent intensity at low suspension concentration • up to 70% drag reduction • u' decrease, w' increase • transition from polymer-like to fiber-like drag reduction dye inserted at the center stays long while at side quickly dispersed short fibers increased turbulent intensity at scales comparable to fibre length. Long fibers at high C^i decreased turbulence at all scales drag reduction attributed to the turbulent core region floc-turbulence interactions result in restructuring of turbulence - increased turbulent dispersion and decreased momentum transfer Appendix A 274 Table A-6: Measurement of Macroscale Uniformity In Mill Mixers (Bennington et al. 1989) Investigator Torregrossa (1983) Kolmodin(1984) Sinn (1984) Breed (1985) Bergor et al (1985) Kuoppamaki (1985) Measurement Technique Inert tracer (LiCl) Inert tracer (LiCl) Temperature profile Reactive tracer Inert tracer (LiCl) Radioactive tracer (Ba-137) Mixer Type Single shaft peg Tower Static Fluidizer Tower Mixers Kamyr MC Peg Mixer Kamyr MC Ingersoll Rand MC Tower mixer Kamyr MC Static mixer Mixing Quality 20 -40 % 40% 40 - 60% 5 - 1 0 % 40 - 60% 2 -5% 4Cdev uniform 56% 6% 14-74% 6 - 8% 0.5 % axial 12% radial Table A-7: Measurement of Fibre-Scale Uniformity In Mill Chlorination Mixers (Patterson and Kerekes, 1984) Mill A B C El E2 Laboratory Mixer Type Inline static Double chamber CST Inline static mixer Two CST's Two in-line mixers Good mixing Bad Mixing Mixing indent Z , '=1 T • T^ — a, -a [ n ) 2 I ,1 — a' 0.132 0.162 0.197 0.123 0.051 0.00 0.67 Table A-8: Comparison of Mixers used in Pulp Bleaching (Bennington et al, 1989) Mixer Type Fluidizer Peg Mixer In-Tower Static Mixing zone (m )^ 0.03 0.8 0.7 0.7 Power Cons. (kW) 103.6 138.4 138.4 11.1 Rotor Speed rpm 1000 325 220 -Residence Time (s) 0.35 9 8 8 Energy Expenditure Kwh/ADT 2.9 3.2 3.2 0.25 Energy dissipation KWh/m^ 3500 150 175 14 Appendix B: Fibre Properties 275 Appendix B B Fibre Structure and Properties Pulp fibres are hollow filament-wound composite structures, consisting of cellulosic fibrils surrounded by a matrix of lignin and hemicellulose, as shown in Figure B-1. The cellulose and hemicellulose give fibres their hydrophilic properties. In a cell wall, cellulose chains aggregate to form long threads called microfibrils. The cell wall consists of several layers of microfibrils with different orientations. Lignin is present mainly between fibres and has a complex, irregular structure. Further details about the microstructure of pulp fibres can be found in Casey (1980). Properties of pulp fibres depend on the origin of wood and the pulping process used to separate the fibres. There is no wonder therefore, that pulp fibres are very heterogeneous, and no two fibres may be alike. There is a major distinction between softwoods and hardwoods woods. In softwoods, the tracheid cells, with typical dimensions L = 3 - 5mm, d = 0.03 mm account for more than 90% of the volume, with the balance being Ray cells and small amounts of epithelial cells. Hardwood fibres consist of wood cells with average length of 1 mm and diameter of 0.02mm. Vessel cells are usually 10% of hardwood, but may be as high as 60%. Chemical pulping (e.g. the kraft process) dissolves the lignin and leaves fibres intact, thus producing smooth, long fibres. On the other hand, mechanical pulping separates fibres by a refining or grinding action, thus producing short and fragmented fibres. Because of the diverse population, methods of characterization of fibre length involve the use of suitable averages to represent the population. Fibres used in this study were of Hemlock species and their length distribution was characterized using Kajaani-FS200. Nylon fibres were characterized using the Optex Fibre analyzer as they clogged the small capillary of the Kajaani. Polyethylene fibres were obtained from earlier measurements (Bennington, 1988). Appendix B: Fibre Properties 276 Figure B-1: The structure of a fibre wall. ML is the middle lamella (the region where lignin bonds adjacent fibres); P is the primary fibre wall; SI, S2, S3 are the three layers of the secondary wall; W is the lumen (central canal of the fibre which may be empty, collapsed or filled with water in the case of wet fibre). Figure taken from Roberts, 1996. Appendix B: Fibre Properties in B. 1 Fibre Length Fibre length is an important property that influences the flocculation and formation of entanglements in fibre suspensions. The weighted average fibre length is a more relevant measure of fibre length due to the influence of length on pulp and sheet properties. The average fibre length was determined using Kajaani FS-200 Fibre Analyser which measures their lengths and population distribution. The length weighted fibre L, length is defined by Equation (B.l) and and the weight weighted fibre length L^ defined by equation (B.2). n L.. = (=1 (=1 (B.l) (B.2) Table B-I: Kajaani results for FBK fibre Total number of fibres counted Arithmetic average length Length weighted average length Weight weighted average length Coarseness 10913 0.85 mm 2.27 mm 2.99 mm 0.236 mg/m Figure B-2 shows the length weighted distribution of fibres used in this study whereas Figure B-3 shows their population distribution. Table B-1 shows a summary of the Kajaani analysis for the Kraft fibre used in this study. Figures B-4 and B-5 show the length weighted length and respective population distribution for nylon fibre. Appendix B: Fibre Properties 278 3 4 5 Fibre Length, mm Figure B-2: Length weighted distribution of FBK fibre used. 2 3 4 5 Fibre Length, mm Figure B-3: Population distribution of FBK fibre used Appendix B: Fibre Properties 279 2 3 Fibre Length, mm Figure B-4: Length weighted distribution of nylon fibre Fibre Length,mm Figure B-5: Population distribution of nylon fibre Appendix B: Fibre Properties 280 B.2 Fibre Stiffness Fibre stiffness (product of elastic modulus and moment of inertia) is an important property that affects network formation and strength. The stiffness can be determined fibre deflection (Kerekes and Tam Doo, 1981), deflection and microscope (Samuelssonn, 1964), cell swelling and osmotic pressure (Scallan and Tigerstrom, 1992). Values for elastic modulus for the different fibres used in this study were obtained from literature and are presented in Table 3.3. B.3 Fibre Swelling The interaction of fibre with water is of both physical and chemical nature. Water is adsorbed onto the hydrophobic surfaces of pulp. Lumen is also capable of holding significant amounts of water. This water is responsible for the swelling of fibres (Xw = kg of water per kg of OD fibre) i.e. water in the interior of the cell wall. Fibre saturation point (FSP) and water retention value (WKV) are used to denote this water. Hydrodynamically, water adsorbed in fibre is part of the fibre, not the suspending medium and the effective volume is the swelled volume. FSP determination is time consuming and is done using solute exclusion methods (Scallan, 1972), WRV can be obtained by centrifiaging the sample at 3000 g for 15 min or weighing in a hygroscopic chamber (Soszynski, 1987). Xw value for the FBK fibres used were determined by FSP methods by Paprican (Pte Claire laboratories). Values for the rest of the fibres used in this study were obtained firom literature. The Xw values for fibres used in this study are reported in Table 3.3. B.4 Fibre Suspension Freeness The fi-eeness of a fibre suspension is a measure of its ability to retain water in a network. It is therefore usually interpreted as its drainage capacity or porosity of fibre network. Freeness of pulp fibre used in this study was measured using the Canadian Standard Freeness Tester and the average values are reported in Table B-5. Note that the freeness of suspension decreased after passage through the high-intensity mixer. Appendix B: Fibre Properties 281 Table B-5: Fibre freeness values after different treatments Initial pulp 3 min in M-I 3 min in H-I with loop 3 min in H-I with loop 3 min in H-I with loop 25 s in H-I without loop 25 s in H-I without loop 25 s in H-I without loop 3 min in H-I with loop 3 min in H-I with loop 3 min in H-I with loop 25 s in H-I without loop 25 s in H-I without loop 25 s in H-I without loop 25 s in H-I without loop 25 s in H-I without loop N, rev/s 0 19.5 50 66.7 83.3 50 66.7 83.3 50 66.7 83.3 50 66.7 83.3 66.7 83.3 ^m -0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.05 0.05 0.05 0.05 0.05 0.05 0.1 0.1 Freeness, niL * 650 645 640 625 620 648 590 538 635 630 625 620 588 538 624 544 B.5 Other Properties/Materials B.5.1 Maximum Packing Fraction The maximum packing factor is the maximum amount of material that can be packed in a unit volume. This was determined by filling ajar of known volume (25 mL) ' Freeness determined according CPPA standard C.l (1994) Appendix B: Fibre Properties 282 with material and performing mass balance to determine the weight of material that can be packed. This value is used in the determination of suspension viscosity. Table B-6 shows values used. Table B-6: Maximum Material Glass dp = 0.11 mm Glass dp= 0.55 mm Glass dp = 0.80 mm Glass dp = 1.20 mm Glass dp = 2.40 mm Fibreglass Polyethylene beads Polyethylene fibres Nylon fibres FBK fibres (Cm = 0.45) Wi g 35.07 37.27 37.30 35.61 35.35 22.86 14.13 4.87 5.52 19.45 packing fraction for different material W2 g 35.22 37.21 37.10 35.4 35.55 22.41 14.31 4.83 5.18 19.21 Wave g 35.15 37.24 37.20 35.51 35.45 22.63 14.22 4.85 5.35 19.33 density g/cc 2.54 2.54 2.53 2.50 2.50 1.77 0.97 0.97 1.14 1.1 volume cc 13.84 14.66 14.70 14.20 14.18 12.79 14.66 5.0 4.69 17.57 0.55 0.59 0.59 0.57 0.57 0.51 0.59 0.20 0.19 0.31 B.5.2 Classification of Glass and Polyethylene Beads The glass beads used in this study were of technical grade. Table B-7 shows the glass beads size range according to supplier specification and the particle diameter for the different the ranges. The particle size distribution for each range was characterized by a particle size analyzer (Malvern Mastersizer'^" 2000, Malvern Instruments, UK). Figure B-6 shows the size distribution for dp = 0.11 mm and Figure B-7 shows the distribution for dp = 0.55 mm. All particle sizes have a narrow distribution about the mean volume average diameter. It is clear that manufacturer specifications are quite accurate. The equivalent diameter of the polyethylene beads (beyond the Mastersizer range) was determined manually by measuring the length and diameter of 100 beads using a Appendix B: Fibre Properties 283 vernier caliper. Table B-8 shows the size classification for the polyethylene beads. The equivalent diameter corresponds to the diameter of a sphere of the same volume as the beads whose shape was taken to be cylindrical. Table B-7: Glass beads classification Particle designation AOll A055 A080 A120 A240 Size range (Supplier) |am 149-88 600 - 500 850-710 1400-1000 2800 - 2000 US mesh 100-170 3 0 - 3 5 2 0 - 2 5 14-18 7-10 Average diameter, fim (specification) dp 115 550 800 1200 2400 volume average (Mastersizer) 113 553 805 1210 * Out of range for the Malvern Mastersizer™ 2000 Table B-8: Polyethylene beads classification parameter mean height mean diameter mm 2.3 2.5 Count 100 100 equivalent diameter, mm 2.7 Appendix B: Fibre Properties 284 30-25' 20' 15' 0) E > 0.01 0.1 1 10 100 Particle size, ^m 1000 Figure B-6: Particle size distribution for 0.11 mm glass beads (D E o > 0 -n_ 0.01 0.1 1 10 100 Particle size, \\r(\ 1000 Figure B-7: Particle size distribution for 0.55 mm glass beads Appendix C: Chemicals Used and Preparation 285 Appendix C C Chemicals Used and Preparation C.1 Chemicals Table C-1: Chemicals used, Specifications and Suppliers Chemical Sulfamic acid Sulfanilic acid 1 -naphthol 2-naphthol Sodium Carbonate Sodium Bicarbonate Sodium Nitrite Hydrochloric Acid Formula NH2-SO3H NH2-C6H4-SO3H C10H7OH C10H7OH Na2C03 NaHCOs NaN02 HCl Mw 97.09 173.19 144.19 144.19 105.99 84.01 69 36.46 Supplier Fisher Scientific Fisher Scientific Sigma Chemicals/ Acros Organics Sigma Chemicals/ Acros Organics Fisher Scientific Fisher Scientific Fisher Scientific Fisher Scientific Product No. 62799203 974149 EEC 201969 12819 EEC 2051827 180825000 030697 942698 980566 292329 C.1.1 Preparation of 1- and 2-naplithol Required amounts of 1- and 2-naphthol were dissolved in distilled water overnight under constant stirring. Table C-2 shows typical formulation for preparation of 40L of 1-naphthol 2-naphthol mixture. Buffering by addition of Na2C03 and NaHCOs to give an ionic strength of I = 444.4 mol/m was done just prior to reactions. Table C-2: Typical formulation for 1- and 2-naphthol for 40L solution Chemical 1 -naphthol 2-naphthol Na2C03 NaHCOs Weight, g 2.9818 5.9037 471.02 373.34 Concentration, mol/m 0.52 1.04 111.1 111.1 Appendix C: Chemicals Used and Preparation 286 C.1.2 Diazotization HO,S - (C,H,) - NH^ + 2HCI + NaNO^ > HO,S - (C,H, )-N = NCl + NaCl + IH^O Procedure: 1. Add and dissolve in 2/3 volume of distilled water Yi mole sodium carbonate for every mole of sulphanilic acid used. 2. Add and dissolve the sulphanilic acid 3. Add remaining 1/3 water volume as ice. 4. Add and dissolve 1.04 moles NaN02 per mole of sulphanilic acid added (4% excess). 5. Add HCl (37%) one mole per mole NaCOa, 2 moles per mole of sulfanilic acid added. 6. React for I/2 an hour. 7. Add sulphamic acid or urea to neutralize the 4% excess sodium nitrite. Table C-3 shows typical component formulation for preparation of 1 litre diazotized sulphanilic acid. Figure C.l shows a schematic of the reactions taking place in simultaneous coupling of 1- and 2-naphthol with diazotized sulphanilic acid. Table C-3: Typical formulation for diazotization of 21.54 mol/m^ sulphanilic acid (IL) Chemical Sulphanilic acid Sodium nitrite HCl Sodium carbonate Sulphamic acid Weight, g 3.7308 1.5458 2.3281 (5.29mL) 1.1416 0.0837 Appendix C: Chemicals Used and Preparation 287 0" "N-\ / -SO, -H" (A2) (B) (Q) Figure C-1: Representation of the extended reaction scheme: Azo coupling of 1-naphthol and 2-naphthol with diazotized sulphanilic acid Appendix C: Chemicals Used and Preparation 288 C.2 Preparation of Product Dyes In order to study the adsorption characteristics of the different dye products separately, the azo products were prepared under reaction conditions that favoured the formation of a desired dye product. No attempt was made to purify the samples. 4-[l,4-(Sulphophenyl)Azo]-l-Naphthol (p-R) and 2-[l,4'-(Sulphophenyl)Azo] -1-Naphthol (o-R) Equimolar amounts of Ai and B are coupled in aqueous environment under as intense mixing condition as possible to obtain product mixtures where p-R and o-R dominate. It was observed that the close overlap between the o-R and p-R spectra sometimes resulted in negative concentrations being measured for o-R and higher than expected concentrations for p-R. However, the sum of the two was found to correspond to the total R formed. Thus, the adsorption of o-R and p-R was therefore followed in combined form. This technique was also used Bourne and co-workers (Bourne et al., 1992;Wengeretal., 1992). 2,4-[l,4-(Sulphophenyl)Azo]-l-Naphthol (S) A mixture containing primarily S was prepared by first coupling equimolar amounts of 1-naphthol (Ai) and diazotized sulphanilic acid under high-intensity conditions to obtain R exclusively (as above). The mixture was then coupled with more B under mild mixing conditions to ensure that all R was converted to S. Since excessive diazonium ions may further degrade the formed bis-azo (S), it was not possible to obtain a solution with greater than 0.35mol/m^ of S. l-[4 '-(Sulphophenyl)Azo]-2-Naphthol (Q) Q is prepared by coupling equimolar amount of 2-naphthol and diazotized sulphanilic acid. No other products are formed in this coupling reaction. C.3 Quantification of Dye Products Figures C.2 and C.3 shows the standard spectra for 1- and 2-naphthol and the four dyestuffs at 298°K and I = 444.4 molW (Bourne et al., 1992). Appendix C: Chemicals used and Preparations 289 1.0' Wavelength, nm Figure C-2: Standard spectra for Azo dyestuffs (CQ-R = Cp.R =Cs =0.03 mol/m CQ = 0.04 molW) 1.0-0.8-0.6-— 1-naphthol - •2-naphthol — I ' r -260 280 —I ' 1 • 1 300 320 340 Wavelength, nm —r 360 380 400 Figure C-3: Standard spectra for 1- and 2- naphthol (CM = CA2 = 0.04 mol/m ) Appendix D: Adsorption Effects in Dispersed Systems 290 Appendix D D ADSORPTION EFFECTS IN DISPERSED SYSTEMS D.1 Adsorption Models Adsorption in liquid-solid systems involves preferential partitioning of a solute from the liquid phase onto the solid surface. The relationship between the amount of adsorbate associated with a unit weight of solid adsorbent (qe) and the residual concentration of the adsorbate in the liquid phase (Ce) from which it is partially removed at a constant temperature is referred as the adsorption isotherm. A number of isotherms have been used to explain adsorption phenomena. The simplest isotherm is a linear relationship (constant partitioning) between the adsorbate and the equilibrium concentration, given by: ^e-K^C^ (D.l) where Kp is the partition coefficient between the phases. This relationship, also known as Henry's law, is valid for many systems at low solute concentrations. The Langmuir isotherm assumes that adsorption occurs on localized sites on a homogeneous surface with no interaction between adsorbate molecules. The maximum adsorption occurs when the surface is covered by a monolayer of adsorbate, with adsorption falling off as this limit is approached asymptotically. The Langmuir isotherm may be represented as: (1 + ^.Cj where XL is the partition coefficient (measure of the affinity) and Qo is the solid phase concentration of solute corresponding to complete saturation of all available adsorption sites (measure of the saturation capacity). The Freundlich model accounts for changes in partition with changing equilibrium solute concentration by introducing an exponent to the concentration term. The model Appendix D: A dsorption Effects in Dispersed Systems 291 corresponds to an exponential distribution of site energies which is characteristic of heterogeneous surfaces (Sips, 1952; Weber, 1985) and may be represented as: q.=K,Ct (D.3) Here K^: is the partition coefficient and b is an exponential term that varies between zero and 1. Although the Freundlich model has been extensively used for adsorption, the fitting parameters have no physical basis (Weber, 1985; Esposito et al., 1996). More complex adsorption models have been developed to describe other observed adsorption phenomena. Such models are often equivalent to the standard models for the limiting conditions of high or low Ce (Sips, 1952; Hill, 1949). An example of these equations is the extended Freundlich model where the distribution is a complex fimction of concentration and given by equation (D.4). q,=K^C':^' (D.4) Parameters for the different models are determined from non-linear regression of a given adsorption data set (qe vs Ce). Tables D-1 to D-4 shows the parameters for different systems. D.2 Correlations to Account for Adsorption on FBK Fibre D.2.1 Individual Component Correlation In order to obtain the correction for adsorption of each individual dye, the isotherms for adsorption individual dyes were determined and corresponding parameters obtained by fitting the data to the models as shown in Tables D-2 to D-4. The data was then used to correct for each dye in the mixture. The corrected concentration of an individual component is then obtained a the sum of measured concentration in equilibrium, plus the amount that was adsorbed, i.e.. Appendix D: Adsorption Effects in Dispersed Systems 292 C""'=C'" +q^m^ (D.5) / ^ corr. Xr^l \ (D-6) •» corr. 'S therefore XT- = 7 ^ \ i r - (D.7) [Cl + C ; + 2C; )+ (^g + 9 , + 2^, }n. For a multicomponent system composed of solute that individually follow Langmuir isotherms and do not compete for adsorption sites, the equilibrium adsorption can be predicted using: ?, = !:p'^' (D.8) with the K\ and Q\ being Langmuir adsorption parameters measured for each component separately (Perry, 1997). We could not obtain satisfactory correlation based on equation D.8, probably due to the interaction among adsorbates. D.2.2 Composite Correlation A composite correlation was developed from multiple linear regression of adsorption data for FBK. The initial XQ is obtained from the expression that was determined for FBK fibre used in this study. XQ = X'Q"- * (1 -8.62C„ + 4733CI) r^  = 0.97 (D.9) Appendix D: Adsorption Effects in Dispersed Systems 293 Table D-1: Adsorption Model Parameters for 1- and 2- naphthol on FBK Model Linear Langmuir Freundlich Extended Freundlich Parameter Kp (L/g) KL (L//d-mol) Qo (/d-mol/g) KpfL/g) h Kf(L/g) b (Uju-mol) '^' 1-naphthol 1.21 ±0.09x10"^ 0.19±0.07xl0"^ 10.6±2.6 0.012+0.011 0.72±0.11 21.7±4.0 -11.8+1.2 r^ 0.913 0.977 0.951 0.985 2-naphthol 1.74±0.09xl0"^ 0.1410.09x10'^ 16.5±8.4 5.8±0.44xl0"^ 0.84±0.10 28.0±6.5 -12.1±1.3 r" 0.949 0.966 0.957 0.975 Table D-2: Adsorption Model Parameters for R (o-R + p-R) on FBK Model Linear Langmuir Freundlich Extended Freundlich Parameter Kp(y^ KL (U^-mol) Qo (ju-mol/^ . KF(L/g) b Kf(L/g) b (U^-mol) '^' R 8.5±0.5xlO'^ 0.16±04xl0"^ 74.21±13.3 0.029±0.01 0.84+.05 124.7±8.0 -11.2±0.3 ? 0.982 0.997 0.992 0.937 Table D-3: Adsorption Model Parameters for S on FBK Model Linear Langmuir Freundlich Extended Freundlich Parameter Kp(L/g) Kl (L/n-mol) Qo (u-mol/g) KF(L/g) b Kf(L/g) b QJ^-mol) '^' S 0.0192±0.002 1.32±0.5 20.1 ±6.1 0.0516±0.015 0.8237±.054 57.2±12.1 -7.00±0.6 ? 0.986 0.993 0.994 0.946 Appendix D: Adsorption Effects in Dispersed Systems 294 Table D-4: Adsorption Parameters for Q on FBK Model Linear Langmuir Freundlich Extended Freundlich Parameter Kp (L/g) Ki (U^-mol) Qo (ju-mol/g) KF(L/g) b Kf(L/g) b (U/u-mol) '^' Q 9.11 ±0.6x10-^ 0.1±0.01xlO"^ 101.98±18.6 0.0101+0.001 0.9810.06 79.83±8.6 -10.1 ±4.2 ? 0.992 0.995 0.993 0.954 D.3 Diffusional Limitations The possibility of diffusional limitation is examined by computing the Weisz modulus, Mw, (Levenspiel, 1984). Values of Mw less than 0.15 indicate that there are no diffusional limitations, while a value greater than 7 imply strong diffusional limitations. The Weisz modulus is given by (Levenspiel, 1984): M = • •ir:)s' C D (D.IO) where 5 Cbulk Deff •5 = rate of reaction, mols/(m .s) = characteristic length (here taken as fibre half width or radius), m = concentration of coupling component, mol/m = the effective diffiisivity, m /s (its estimation is explained below) For a bed of fibrous materials, the effective diffiisivity depends on the orientation of fibres, porosity of the fibre structure and tortuosity (Tomadakis and Sotirchos, 1993). For a three-dimensional random network of fibres we may use Deff=a/Tl.Db, ulk (D.ll) Appendix D: Adsorption Effects in Dispersed Systems 295 where 9 is the porosity of the fibre network, r\ is the tortuosity factor and Dbuik the bulk diffusion coefficient. We adapt the relationship for S and r| from Tomadakis and Sotirchos (1993) who give 7] = ^ 1 - ^ / 3-& (D.12) p y where "a" is a directional (orientation) factor and Sp is the percolation threshold. For a 3-D random network of fibres, a = 0.661 and 9p = 0.037 (see Table 1 in Tomadakis and Sotirchos (1993) for other orientations). Therefore for a bed porosity of 0 = 0.6 in our case, r| = 1.42 giving B/r\ = 0.42. Based on the above estimates, Table D-5 shows modulus calculations for the primary coupling reactions. Table D-5: Calculation of the Weisz modulus for difftisional limitations for the primary coupling reactions Component Rate constant (m /mol.s) Concentration (mol/m'^ ) -1 Concentration (mol/m ) Reaction, ki*CB*Ci, mol/m^.s Fibre half-width, 5 (m) DiffUsivity, (m /s) Difussion factor Effective Diffiisivity, (m^/s) Weisz Modulus i ki Ci CB ri 5 D S/71 Deff Mw Ai kio 921 0.52 21.54 10315 7.3E-06 7.80E-10 0.42 3.276E-10 78.62 Ai kip 12238 0.52 21.54 137075 7.3E-06 7.80E-10 0.42 3.276E-10 1044.65 A2 k3 124.5 1.04 21.54 2789 7.3E-06 7.80E-10 0.42 3.276E-10 21.26 Appendix D: Adsorption Effects in Dispersed Systems 296 It is evident that all primary coupling reactions may be difusionally limited because M^ > 7. To asses the effect of this limitation on product distribution (XQ), lets assume all chemical that adsorbs on fibre will not be available for reaction. Thus for a Cm = 0.05 suspension 5% of chemicals may be unavailable for reaction. This means that the initial molar ratio NAI/NBO is reduced from 1.2 to 1.14. Computational modeling using the E-model with these values shows that the product distribution will be reduced by 0.1% to 0.5%. This is lower than our experimental errors, thus the limitation will not curtail our deductions. As for the secondary coupling reactions. Table D-6 shows that reactions are only marginally limited by diffusion. We do not anticipate any problem with secondary coupling reactions. Table D-6: Calculation of the Weisz modulus for diffusional limitations for the secondary coupling reactions Component Rate constant (m /mol.s) Concentration (molW) Concentration (mol/m^) -> Reaction rate ki*CB*Ci, mol/m .s Fibre half-vddth, 6 (m) Diffusivity, (m Is) Diffusion factor Effective Diffusivity, (m Is) Weisz Modulus i ki Ci CB r. 5 D S/r| Deff Mw o-R k2D 22.25 0.1185 21.54 56.79 7.3E-06 7.80E-10 0.42 3.28E-10 0.43 p-R k2o 1.835 0.3684 21.54 176.56 7.3E-06 7.80E-10 0.42 3.276E-10 1.35 Appendix E: Flow and Reaction Zone Visualization 297 Appendix E E Flow and Reaction Zone Visualization E. 1 Flow visualization using high-speed cine-photography Flow visualization using high-speed cine-photography was employed to locate and determine the extent of the reaction zone using the instantaneous reaction between HCL and NaOH solutions (Rice et al, 1964; Bourne and Dell'ava 1987). O.IM NaOH solution containing phenolphthalein indicator was added to 0.05M HCl solution in the mixing chamber, also containing phenolphthalein indicator. The O.IM NaOH was fed at 16 ml/min, equivalent to tf = 180 s for standard reaction conditions. Apart from giving an estimate of the local reaction volume, this technique also affords an opportunity to examine the independence of the reaction zones in case of multiple feed points. Figure E-1: Photograph of setup for reaction and flow visualization Appendix E: Flow and Reaction Zone Visualization 298 (a) (b) Figure E-2: Photograph of reaction zone in high-shear mixer showing independent reaction zones (a) two feed points for experiments with flow loop (b) two out of six points for experiments without flow loop. Appendix E: Flow and Reaction Zone Visualization 299 (a) (b) Figure E-3: Flow visualization in high-shear mixer with external recirculation Q^ = 100 L/min (a) N = 0 rev/s reaction zone directed outwards following the path of incoming fluid (b) N = 6 rev/s reaction zone towards the rotor. Appendix E: Flow and Reaction Zone Visualization 300 Figure E.3: Sketch of the locus of reaction zone for different feed point locations in the medium intensity mixer. Feed point (a) near rotor tip (r = 55 mm), (b) at the middle (r = 70 mm), (c) near vessel wall (r = 90 mm). Appendix E: Flow and Reaction Zone Visualization 301 E.2 Discretization of Flow Zones From reaction zone visualization, the reaction flame behaved differently for different feed points in the mixer. This is illustrated in Figures E-4 where schematic sketches of he reaction plume is shown. It is obvious that the average energy dissipation calculated from different points in the vessel needs to take this fact into account. Consider a cylindrical volume where the reaction takes place as shown in Figure E-5. We may average the energy dissipation based on the volume that is swept by the reaction plume Figure E.5: Averaging of local energy dissipation in a cylindrical geometry The average energy dissipation in the zone is given by I-£ = Inr-s-jdr Wt^ (E.l) where Ri = inner rotor edge and Ro = outer wall (vessel). To correct for the curvature of the reaction zone equation 1 needs to be discretized to consider the radial interval where Appendix E: Flow and Reaction Zone Visualization 302 experimental points are obtained. Each Sij value is the multiplied by a weighting factor according to its radial distance. If we define / as the distance between a given zone, such that h+h+h = Ro-Ri, then the radial weighting factors are obtained from equation E-1 as Zone 1 Zone 2 (R,+h+i,y-{R,+i,y (E.2) Zone 3 R'„-R^ Rl-iR,+i,+i,y Rl-Rj (E.3) (E.4) Four cases are considered. Case 1: Zones of equal weight {w\ = ^2 = ^3) Case 2: Zones of equal length (l\ = h^ h) Case 3: Zone with length based on location of feed point (1\ = Rl-Ro; h = R2-R1; I2 = R3-R2) Case 4: Zone allocation based on reaction zone visualization From flow visualization, we divide the region between rotor and vessel wall into three annular zones. Zone 1 is the immediate rotor zone covering a relatively smaller volume (r = 50 to r = 60 mm) but with the highest dissipation rate. Zone 2 is the middle zone extending from r = 60 to r = 75 mm, and zone 3 extends from r = 75 to r =95 mm (vessel wall). Table E. 1 shows a comparison of different weighting factors for the four cases described. It is obvious that different weighting will give different values for the average energy dissipation rate. We used weighting based of reaction zone visualization (case 4). Appendix E: Flow and Reaction Zone Visualization 303 Table E. 1: The effect different weighting factors on average energy dissipation (using data for axial plane midway between baffles (Plane III in Table 6.5, chapter 6). Zone R mm radial weight factor s Case 1: equal weights (wi = W2 = W3) 1 2 3 Ri Rl R2 R3 50 55 70 90 Wj 0.3333 0.3333 0.3333 SwjWjSij 682 Case 1: equal zone length (li = I2 = I3 ) 1 2 3 Ri Rl R2 R3 = Ro 50 65 80 95 Wj 0.2644 0.3333 0.4023 SwiWjSij 662 Case 3: zone length based on feed point coordinates 1 2 3 RI Rl R2 R3 50 55 70 90 Wj 0.0805 0.2874 0.6322 EwjWjSjj 605 Case 4: zone length based on reaction zone visualization 1 2 3 RI Rl R2 R3=Ro 50 60 75 95 Wj 0.1686 0.3103 0.5211 SwjWjSjj 633 Appendix E: Flow and Reaction Zone Visualization 304 For determination of energy dissipation in the rotor zone, however, we did not have flow visualization data and therefore used weights based on equal length (case 2). Table E.2 shows the weighting factors used for rotor swept zone. Table E.2: Weighting factors for distribution of energy dissipation rates in rotor swept zone Rotor Zone 1 2 3 Radial limits, mm 25-33 33-42 42-50 Radial weighting factor, wj 0.2593 0.3333 0.4074 Appendix F: Computational Fluid Dynamics 305 Appendix F F Computational Fluid Dynamics (CFD) Computational Fluid Dynamics (CFD) can be used to get a quantitative description of the turbulent flow field in a mixer. CFD involves solution of the Navier-stokes equations describing transport of fluid momentum. Sliding mesh does not require any experimentally determined input, and is therefore best suited for the current problem. We use CFD to predict the average energy dissipation rate, the bulk flow pattern and to predict impeller power number. The sliding mesh model is ideally suited for the rotor stator mixer because it involves two regions, one attached to the rotor and the other attached to the stationary vessel wall. The two meshes slide relative to one another along slipping planes within the fluid domain. The flow patterns in rotor-stator is very complex and the presence of stationary baffles makes the flow inherently unsteady. Mean quantities exhibit periodic unsteadiness. The solution is therefore computationally intensive. A 333 mHz Pentium computer with 256 MB memory was able to give satisfactory results within reasonable time (under 10 hours). F. 1 Theoretical Aspects The basic idea in sliding mesh approach is the application of two grids: one moving with the rotating geometry (rotor) while the other is fixed to the stationary geometry (vessel walls and baffles). The two meshes interact along a surface of slip. Arbitrary Lagrangian and Eulerian methods are used to describe the general transport equations in both of the grid regions. Since the flow is unsteady, a time dependent solution procedure must be used. Appendix F: Computational Fluid Dynamics 306 The equations governing flow in the frame of reference of the moving mesh may be written in Cartesian tensor form as: ^P+^p{u,-vJ=0 (F.l) d P'''+^P^^r'')^>=-^+^k+\,)+\ (F-2) dt '^  ' axj ^ J " ' ax. 5Xj d where — is the total derivative representing the time rate change of a variable as seen by an observer riding on the moving mesh, p is the fluid density, Wj is the flow velocity component, Vj is the grid velocity component arising from mesh motion, xy is the molecular stress tensor for Newtonian fluids, ttjj is the Reynolds stress tensor and Sui is thesource term for the momentum equation. The Reynolds stress term is the source of closure problems in turbulence modeling. F.1.1 Turbulence Modeling and the k-s Model The purpose of turbulence modeling is to close the Reynolds stress terms that result from averaging the Navier-Stokes equation. One of the common methods is to model the Reynolds stress term as a term with a different viscosity (vr) that depends on the flow properties instead of fluid properties (Bossinesque hypothesis), thus " ' " ; = -Vj —'- + —- (F.3) with the k-z model used v , = C ^ — (F.4) E and the equations for kinetic energy (A:) and dissipation are respectively: Appendix F: Computational Fluid Dynamics 307 Dk _ d Dt ~ dXj Ds _ d Dt ~ ax, Vj dk Vj ok + P,-8 +C —P -C — ''k ' '' k where Pk is the production of kinetic energy by deformation ac/, ' ' dx. (F.5) (F.6) (F.7) Other constants are given in Table F-1. The values of the constants in Table F-1 have been determined empirically (Fluent 4.3 Manual, 1996). Table F-1: Constants used for the k-s model c. 0.09 ClE 1.44 C28 1.92 CTk 1.0 CTE 1.3 Appendix F: Computational Fluid Dynamics 308 F.2 Computational Summary The convergence criterion used was achievement of minimum residual of less than 0.001 for all calculated parameters. Six hours were required for simulation of a typical case. Figure F-1 shows the computational grid used for medium-intensity mixer simulations. Only one eighth of the mixer was computed due to symmetry. The total number of computational cells used was 56 x 27 x 19 == 28728. Symmetry boundary condition was used for the mid-plane of the mixer and the top wall was set for no-slip boundary conditions. The rotor was set as moving wall and the baffles as stationary walls. Radial slip plane was set at j = 16 as shown in Figure F-1. Table F-2 compares the power number computed from CFD and that obtained from direct torque measurements in the medium-intensity mixer. Values from CFD are about 40% lower than those calculated from direct measurements using torque transducer. A refined grid around rotor vanes and baffles would probably give much closer agreement. In a recent comparison of CFD and LDV measurements, Andersson (1998) obtained CFD values that were 25%) lower than those measured by LDV. The use of sliding mesh technique in computational fluid mixing to obtain the macroscale flow parameters should be encouraged by these results. Table F-2: Power number from CFD and Torque measurements N, rev/s 12.3 17.3 CFD ^psdV N =^ ' pN'D' 2.9 3.0 Torque InNM ' pN'D' 5.0 5.0 Appendix F: Computational Fluid Dynamics 309 no slip plane symmetry plane baffle area radial slip plane, j=16 cyclic planes Figure F-1: Computational grid used and sliding mesh specifications Appendix F: Computational Fluid Dynamics 310 Figure F-2: Flow patterns in radial plane at z/H == 0.95 depicting radial inward motion. Appendix F: Computational Fluid Dynamics 311 Figure F-3: Flow patterns in radial plane at z/H = 0.75 depicting radial inward motion. Appendix F: Computational Fluid Dynamics 312 Figure F-4: Flow patterns in radial plane at z/H = 0.50 depicting radial inward motion Appendix G: Implications for Mass Transfer 313 Appendix G G Implications for Mass Transfer Experiments have been carried out in a high pressure PHT reactor (PAFRICAN), to determine gas-liquid mass transfer coefficient (kta) for the catalytic oxidation of sodium sulfite in the presence of FBK fibres (Rewatkar and Bennington, 1999). These values are compared with the same coefficient determined in the absence of fibres (kLa)o as shown in Figure G-1 below. It is interesting to note that the reduction in mass transfer coefficient follows the same exponential decay reduction in local energy dissipation and may be correlated by an expression similar to one found in this study. 1.0# -1 1 — I 1 — 1 1 1 — I 1 1 1 1 1 r exp(-aC^) a = 50±13 ® N = 2400 rpm m N = 3000 rpm 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 Figure G. 1: Gas-Liquid mass transfer coefficient in FBK suspension Appendix H: Data Tables 314 Appendix H H Data Tables H.1 Medium-Intensity Data Table H-1: Analytical measurements of products and micromixing index (XQ) (T=298K, C°, = 0.52 mol/m' C°2/C°, = 2.0, CI = 23.6mol/m', VAA^B = 50) # 1 2 3 4 5 6 7 8 9 10 MEAN STDEV. 95% CONF. [Q] mol/m 0.0986 0.1026 0.1054 0.1025 0.1036 0.1049 0.1051 0.1043 0.1035 0.1021 0.103 0.002 0.001 [o-R] mol/m 0.1909 0.2047 0.1917 0.2133 0.2034 0.1931 0.1928 0.1971 0.2011 0.2110 0.200 0.008 0.006 [p-R] mol/m 0.2352 0.2406 0.2507 0.2355 0.2424 0.2496 0.2497 0.2466 0.2439 0.2369 0.243 0.006 0.004 [S] mol/m^ 0.0061 0.0064 0.0057 0.0070 0.0064 0.0057 0.0061 0.0063 0.0066 0.0073 0.006 0.001 0.0004 XQ 0.1836 0.1830 0.1884 0.1813 0.1843 0.1877 0.1877 0.1860 0.1843 0.1808 0.185 0.003 0.002 Table H-2: Micromixing index in water at 19.3 rev/s, (feed point F32, standard reaction conditions). # 1 2 3 4 5 6 7 8 9 10 [Q] mol/m 0.0645 0.0677 0.0701 0.0646 0.0684 0.0619 0.0599 0.0688 0.0633 0.0691 [o-R] mol/m^ 0.2562 0.2195 0.2187 0.2404 0.2069 0.2428 0.2352 0.2304 0.2337 0.2187 [p-R] -5 mol/m 0.1245 0.1569 0.1525 0.1376 0.1649 0.1520 0.1495 0.1396 0.1514 0.1523 [S] mol/m 0.0072 0.0042 0.0042 0.0059 0.0042 0.0054 0.0051 0.0056 0.0050 0.0047 Mean STDEV XQ 0.1403 0.1496 0.1559 0.1422 0.1525 0.1324 0.1316 0.1529 0.1381 0.1537 0.1449 0.0091 MB% 99.32 97.79 97.18 98.18 96.94 101.03 98.27 97.25 99.03 97.12 98.21 1.3 Appendix H: Data Tables 315 Table H-3: Micromixing in a suspension of polyethylene fibres Xg = 0.0 N rev/s 7.3 12.3 17.3 19.5 ^m 0.0 0.283 0.218 0.165 0.145 0.01 0.319 0.241 0.192 0.172 0.02 0.369 0.281 0.221 0.202 0.03 0.350 0.271 0.251 0.04 0.332 0.310 Table H-4: Micromixing in a suspension of polyethylene fibres, Xg = 0.2 N rev/s 7.3 12.3 17.3 20.6 23.7 ^m 0 0.320 0.239 0.185 0.171 0.159 0.01 0.354 0.266 0.205 0.183 0.169 0.02 0.401 0.310 0.240 0.216 0.205 0.03 -0.396 0.320 0.270 0.263 0.04 --0.371 0.323 0.299 0.05 ---0.414 0.392 Table H-5: Micromixing in a suspension of FBK fibres (Xg = 0.0, Feed point F32, standard reaction conditions). N rev/s 7.3 12.3 17.3 19.5 »-^ni 0 0.283 0.218 0.165 0.145 0.01 0.339 0.250 0.195 0.174 0.016 0.384 0.294 0.223 0.193 Table H-6: Product distribution in water with different gas void fractions. (Cn point F32, standard reaction conditions) 0.0, Feed N, rev/s 7.3 12.3 17.3 19.5 X, 0 0.283 0.218 0.165 0.145 0.12 0.302 0.226 0.174 0.166 0.20 0.32 0.239 0.185 0.171 0.32 0.341 0.254 0.201 0.185 0.40 0.389 0.297 0.225 0.207 Appendix H: Data Tables 316 Table H-7: Product distribution in Cy = 0.021 glass beads suspension (Xg point F32, standard reaction conditions). 0.0, Feed 7.3 12.3 17.3 19.5 Water 0.283 0.218 0.165 0.145 dp=0.11mm 0.281 0.207 0.159 0.137 dp=0.55mm 0.288 0.218 0.167 0.153 dp=1.2min 0.301 0.227 0.191 0.178 dp=2.4mm 0.327 0.243 0.192 0.179 Table H-8: Product distribution in Cy standard reaction conditions). 0.021 Polyethylene beads (Feed point F32, N rev/s 7.3 12.3 17.3 19.5 '^m 0.0 0.283 0.218 0.165 0.145 0.02 0.295 0.251 0.181 0.151 0.04 0.314 0.258 0.186 0.150 Table H-9: Energy dissipation rates in fibre suspensions at different rotational speeds and concentrations (standard reaction conditions). System Water FBK fibre Polyethylene Nylon Dispersed Phase Cy 0 0.0215 0.0215 0.0215 0.0215 0.043 0.043 0.0215 L (mm) 2.3 3.2 1.1 N (rev/s) 7.3 12.3 17.3 19.5 7.3 12.3 17.3 19.5 12.3 17.3 19.5 17.3 19.5 17.3 Energy Dissipation (W/kg) Chemical £|oc 101 497 683 1187 25 180 211 301 142 245 319 266 410 201 input Save 11 43 102 141 12 38 101 141 57 104 145 83 117 100 Appendix H: Data Tables 317 Table H-10: Energy dissipation rates in glass and polyethylene bead suspensions at different rotational speeds and concentrations (Xg = 0.0, standard reaction conditions). System Glass beads Glass beads Polyethylene beads Polyethylene beads Dispersed Phase Cv 0.043 0.043 0.021 0.043 0.021 0.043 dp ( m m ) 0.80 0.11 3 3 N (rev/s) 7.3 12.3 17.3 7.3 12.3 17.3 12.3 17.3 Energy Dissipation (W/kg) Chemical Sloe 130 587 821 250 260 721 240 300 420 415 input Save 9.6 34 82 9 40 109 35 42 81 88 Table H-11: Energy dissipation rates in fibre suspensions at different rotational speeds and concentrations. (Xg = 0.20, standard reaction conditions). System Water FBK fibre Polyethylene Dispersed Phase Cv 0 0.0215 0.0215 0.0215 0.043 0.043 L (mm) 2.3 3.2 N (rev/s) 7.3 12.3 17.3 20.6 7.3 12.3 17.3 20.6 17.3 20.6 17.3 20.6 Energy Dissipation (W/kg) Chemical E|oc 63 176 490 540 6.1 45.7 126 108 170 288 66 331 Input Save 10 23 67 107 6 37 101 153 114 168 87 127 Appendix H: Data Tables 318 Table H-12: Energy dissipation rates in glass and polyethylene bead suspensions at different rotational speeds and concentrations (Xg = 0.20, standard reaction conditions). System Glass beads Polyethylene beads Dispersed Phase Cv 0.021 0.021 0.021 0.043 0.043 0.043 0.021 0.021 0.021 0.043 0.043 0.043 dp (mm) 0.80 2.7 N (rev/s) 12.3 17.3 20.6 12.3 17.3 20.6 7.3 12.3 17.3 7.3 12.3 17.3 Energy Dissipation (W/kg) Chemical 58 139 171 145 401 610 11 33 91 44 122 314 Input ^ave 33 67 92 35 91 106 11 23 84 21 48 103 Appendix H: Data Tables 319 Table H-13: Local energy dissipation from E-model for 3-D mapping in medium-intensity mixer with water (N =12.3 rev/s, Xg = 0.0, standard reaction conditions). # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 r/R 0.579 0.737 0.895 0.947 0.579 0.737 0.947 0.579 0.737 0.947 0.579 0.737 0.947 0.579 0.737 0.895 0.947 Plane average Quadrant average e 0 0 0 6 22.5 22.5 22.5 45 45 45 62.5 62.5 62.5 90 90 90 84 z/H 0.95 531 466 716 485 525 447 293 519 350 267 440 490 250 531 466 716 235 418 0.75 635 413 115 587 571 244 399 511 306 371 571 337 437 721 413 115 407 400 0.5 372 374 460 469 252 1305 900 499 532 728 1171 1187 1197 1128 1202 830 900 834 0.25 640 405 121 611 553 239 411 531 294 367 582 333 423 706 405 121 396 398 0.05 648 461 695 509 509 442 296 493 353 280 418 514 243 510 442 695 243 419 494 Plane Average 477 487 410 551 424 Appendix H: Data Tables 320 Table H-14: Local energy dissipation from E-model for 3-D mapping in medium-intensity mixer with water (N = 17.3 rev/s, Xg= 0.0, standard reaction conditions). # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 r/R 0.579 0.737 0.895 0.947 0.579 0.737 0.947 0.579 0.737 0.947 0.579 0.737 0.947 0.579 0.737 0.895 0.947 Plane average Quadrant average 8 0 0 0 6 22.5 22.5 22.5 45 45 45 62.5 62.5 62.5 90 90 90 84 z/H 0.95 950 603 584 192 533 450 300 543 398 233 533 464 282 950 603 342 111 425 0.75 678 110 124 440 605 226 264 532 269 347 605 226 393 678 110 125 442 344 0.5 1532 2076 784 851 1734 1658 1171 1982 1684 955 1734 1658 1711 1532 2076 462 501 1384 0.25 732 102 127 480 659 214 237 522 250 368 641 212 432 671 113 116 429 353 0.05 933 1128 681 738 831 802 358 741 476 938 831 802 372 933 1128 475 516 680 637 Plane Average 674 598 633 688 435 Appendix H: Data Tables 321 Table H-15: Local energy dissipation from E-model for 3-D mapping in medium-intensity mixer with Cy = 0.021 polyethylene suspension (N = 17.3 rev/s, Xg = 0.0, standard reaction conditions). # 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 r/R 0.579 0.737 0.895 0.947 0.579 0.737 0.947 0.579 0.737 0.947 0.579 0.737 0.947 0.579 0.737 0.895 0.947 Plane average Quadrant average 9 0 0 0 6 22.5 22.5 22.5 45 45 45 62.5 62.5 62.5 90 90 90 84 z/H 0.95 400 369 448 405 292 168 217 255 118 212 292 140 126 400 369 448 447 276 0.75 213 118 294 79 366 87 72 200 27 105 167 90 102 213 118 294 117 135 0.5 0.25 1126 594 318 40 623 387 155 596 345 164 623 1018 166 1126 594 318 82 409 233 111 285 74 362 94 83 216 31 110 163 96 104 213 106 282 140 138 Plane 0.05 Average 388 350 443 393 289 165 226 242 121 209 298 146 132 416 369 466 438 276 247 318 200 174 210 265 Appendix H: Data Tables 322 H.2 High-Intensity Mixer Data Table H-16: Overall energy dissipation rate in high-shear mixer with flow Loop '-'m 0 0.01 0.02 0.03 0.04 0.05 0.06 Cv 0 0.013 0.026 0.039 0.052 0.065 0.079 MEAN STDEV 95% Conf. N, rev/s 50 1217 1198 1240 1197 1275 1250 1225 1229 ±28.2 ±26.1 66.7 3048 2918 2880 3009 2982 2940 2917 2956 ±59.2 ±54.8 83.3 5005 5081 5044 4889 4975 4981 4999 4996 ±60.4 ±55.9 Table H-17(a): Micromixing in FBK suspension at different concentrations. High-shear mixer with flow loop, N = 50.0 rev/s. '-^ m 0 0.01 0.02 0.03 0.04 0.05 Cv 0.000 0.013 0.026 0.039 0.052 0.065 [Q] mol/m 0.0857 0.0815 0.0609 0.0809 0.0779 0.0726 [o-R] mol/m 0.1749 0.1581 0.2141 0.1297 0.1253 0.0837 [p-R] mol/m^ 0.1727 0.1434 0.0724 0.1180 0.1080 0.0937 [S] mol/m^ 0.0004 0.0011 0.0055 0.0015 0.0011 0.0000 ^Q 0.197 0.212 0.220 0.244 0.249 0.291 \rCorr 0.197 0.230 0.260 0.311 0.340 0.423 Appendix H: Data Tables 323 Table H-17(b): Micromixing in FBK suspension at different concentrations. High-shear mixer with flow loop, N = 66.7 rev/s '-rn 0 0.01 0.02 0.03 0.04 0.05 0.06 Cv 0.000 0.013 0.026 0.039 0.052 0.065 0.079 [Q] mol/m^ 0.0754 0.0703 0.0733 0.0740 0.0774 0.0630 0.0809 [o-R] mol/m 0.1689 0.1645 0.1460 0.1243 0.1113 0.0978 0.0835 [p-R] mol/m 0.1818 0.1671 0.1405 0.1312 0.1374 0.1214 0.1335 IS] mol/m 0.0000 0.0002 0.0004 0.0001 0.0000 0.0000 0.0000 XQ o.in 0.175 0.203 0.224 0.237 0.223 0.271 \rCorr 0.177 0.190 0.240 0.286 0.325 0.325 0.416 Table H-17(c): Micromixing in FBK suspension of different concentrations. High-shear mixer with flow loop, N = 83.3 rev/s. '-'m 0 0.01 0.02 0.03 0.04 0.05 0.06 Cy 0.000 0.013 0.026 0.039 0.052 0.065 0.079 [Q] mol/m'' 0.052 0.057 0.056 0.061 0.059 0.062 0.061 [o-R] mol/m'' 0.1992 0.1848 0.2863 0.1616 0.2184 0.2208 0.1910 [p-R] mol/m'' 0.1585 0.1850 0.0668 0.1319 0.0653 0.0751 0.0604 [S] mol/m^ 0.0010 -0.0002 0.0097 0.0010 0.0000 0.0000 0.0000 ^Q 0.1271 0.1343 0.1309 0.1700 0.1729 0.1763 0.1982 •ycorr 0.1271 0.1462 0.1546 0.2168 0.2366 0.2565 0.3034 Table H-18(a): Temperature scheduling for high-shear mixer without loop (N = 50 rev/s) ^ - ' H l 0 0.01 0.02 0.03 0.04 0.06 0.08 Cv 0.0000 0.0129 0.0259 0.0390 0.0522 0.0788 0.1058 M,N.m 14.5 14.7 14.1 14.4 14.9 14.5 14.4 N, rpm 3001 3001 3001 3002 3019 2990 3001 MEAN STDEV To 23.8 23.2 23.7 23.3 23.5 23.7 23.5 24 ±0.2 Tf 26.0 25.5 25.8 25.5 26.1 26.3 26.1 26 ±0.3 l a v e 24.9 24.4 24.7 24.4 24.8 25.0 24.8 25 ±0.2 Appendix H: Data Tables 324 Table H-18(b): Temperature scheduling for high-shear mixer without loop (N = 66.7 rev/s) ^ m 0 0.01 0.02 0.03 0.04 0.06 0.08 0.1 Cv 0.0000 0.0129 0.0259 0.0390 0.0522 0.0788 0.1058 0.1331 M,N.m 22.7 23.3 23.5 24.2 25.2 26.2 26.9 27.1 N, rpm 4003 3992 4003 4005 4002 4001 4001 4001 MEAN STDEV To 22.6 21.6 22.5 22.6 22.3 21.9 22.2 22.5 22 ±0.4 Tf 27.9 28.5 28.0 28.8 28.3 27.9 28.9 29.1 28 ±0.5 *ave 25.2 25.0 25.2 25.7 25.3 24.9 25.5 25.8 25 ±0.3 Table H-18(c): Temperature scheduling for high-shear mixer without loop (N = 83.3 rev/s) v^m 0 0.02 0.04 0.06 0.08 0.1 Cv 0.0000 0.0259 0.0522 0.0788 0.1058 0.1331 M,N.in 35.2 36.1 37.9 38.6 39.5 40.0 N, rpm 4969 4962 4963 4922 4857 4814 MEAN STDEV To 20.1 20.5 20.2 18.2 18.5 18.8 19 ±1.0 Tf 31.5 32.5 32.2 33.5 34.3 34.8 33 ±1.3 l a v e 25.8 26.5 26.2 25.9 26.4 26.8 26 ±0.4 Table H-19: Micromixing in FBK suspension of different concentrations. High-intensity mixer without flow loop, N = 50 rev/s *-^ m 0 0.01 0.02 0.03 0.06 0.08 Cv 0 0.013 0.026 0.039 0.079 0.109 [Q] mol/m 0.048 0.049 0.065 0.064 0.063 0.050 [o-R] mol/m 0.165 0.161 0.149 0.114 0.092 0.082 [p-R] mol/m 0.204 0.177 0.146 0.128 0.094 0.082 [S] mol/m -0.0010 -0.0003 0.0004 0.0002 0.0012 0.0012 XQ 0.116 0.127 0.181 0.208 0.251 0.230 ycorr 0.116 0.139 0.213 0.265 0.385 0.374 MB% 99.8% 92.8% 86.6% 73.6% 60.4% 51.9% Appendix H: Data Tables 325 Table H-20: Micromixing in FBK suspension of different concentrations. High-intensity mixer without flow loop, N = 66.7 rev/s '-^m 0 0.01 0.02 0.03 0.06 0.08 0.10 Cv 0 0.013 0.026 0.039 0.079 0.109 0.133 [Q] mol/m^ 0.056 0.052 0.065 0.063 0.064 0.085 0.058 [o-R] mol/m 0.122 0.076 0.071 0.125 0.052 0.125 0.078 [p-R] mol/m 0.234 0.233 0.215 0.160 0.195 0.167 0.088 [S] mol/m^ 0 0 0 0.001 0 0 0.001 XQ 0.095 0.144 0.186 0.179 0.207 0.225 0.257 •y corr 0.095 0.157 0.220 0.229 0.316 0.367 0.420 - MB% 99.0% 86.6% 84.4% 84.1% 74.7% 64.4% 57.0% Table H-21: Micromixing in FBK suspension of different concentrations. High-intensity mixer without flow loop, N = 83.3 rev/s. »-^ m 0 0.02 0.04 0.06 0.08 0.1 Cv 0.000 0.026 0.052 0.079 0.106 0.133 [Q] mol/m^ 0.036 0.051 0.077 0.057 0.058 0.067 [o-R] mol/m^ 0.216 0.2 0.146 0.082 0.071 0.099 [p-R] mol/m^ 0.17 0.128 0.151 0.087 0.098 0.08 [S] mol/m 0.001 0.003 0.001 0.002 0.003 0.004 XQ 0.086 0.132 0.204 0.247 0.248 0.264 \r corr 0.086 0.155 0.279 0.379 0.405 0.432 MB% 100.8% 91.6% 89.2% 60.5% 55.4% 60.7% Table H-22: FBK Suspension Properties and Kolmogorov microscales at N = 66.7 rev/s ^m -0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Cv -0 0.013 0.026 0.039 0.052 0.065 0.079 0.092 0.106 0.119 0.133 P kg/m^ 1000 1006 1013 1019 1026 1033 1039 1046 1053 1060 1067 Using equation (2.54) M'a Pa.s l.OE-03 2.0E-03 3.9E-03 7.9E-03 1.6E-02 3.4E-02 7.3E-02 1.6E-01 3.6E-01 8.3E-01 2.0E+00 Va m^/s l.OOE-06 1.94E-06 3.83E-06 7.71E-06 1.58E-05 3.30E-05 7.04E-05 1.53E-04 3.42E-04 7.82E-04 1.83E-03 Using equation (2.52) ^a Pa.s l.OE-03 1.5E-03 1.3E-02 4.5E-02 l.lE-01 2.2E-01 3.9E-01 6.3E-01 9.5E-01 1.4E+00 1.9E+00 Va m^/s l.OE-06 1.5E-06 1.3E-05 4.4E-05 l.lE-04 2.1E-04 3.7E-04 6.0E-04 9.0E-04 1.3E-03 1.8E-03 IK m 4.3E-06 5.8E-06 2.9E-05 7.3E-05 1.4E-04 2.4E-04 3.6E-04 5.2E-04 7.0E-04 9.2E-04 1.2E-03 Appendix I: Energy Dissipation Due to Fibre-Fibre Contacts 326 Appendix I I Energy Dissipation due to Fibre-Fibre Contacts /. 1 Background Consider fibres in a random, homogeneous network extending to the boundary of the mixing vessel without any motion. In this state, fibres are interlocked in a manner consistent to what was proposed by Meyer and Wahren (1964). In this case, it is assumed that every fibre is locked in position by contact with at least three fibres so as to transmit forces (fibre network). We can therefore estimate the number of contacts per fibre using the expression developed by Meyer and Wahren (1964). First, let's define some terms. A contact point is a point of contact between two fibres. An active fibre is a constituent element in a fibre network that contributes to interlocking forces. When the fibre network is subjected to shear, every active constituent is acted upon by axial forces that give rise to elongation (or deformation), and normal forces giving rise to bending (deflection) at contact points (see Figure I-l). Bending forces are the most dominant in network strength and will be considered as the sole contribution to be overcome (Meyer and Wahren, 1964, Soszynski and Kerekes, 1988, Farnood et al, 1993; Kerekes, 1995). Being the dominant forces in fibre suspension flow, hydrodynamic forces perform work on fibres to disperse local fibre networks (floes). This energy is partially stored in the fibres as internal energy, and partially dissipated as heat. If sufficient shear is applied in this network, the large network will disintegrate into smaller fi-agments (floes) that constitute the fluid-like flow behaviour observed in the vessel. If we assume that sufficient shear has been applied to the fibre suspension such that all fibre contacts are broken, this would represent the maximum amount of fibre-fibre frictional energy that can be dissipated in a suspension. On the other hand, if the fragments exist as coherent entities moving with no relative motion within them, then only fibres in the periphery of the floes are subjected to the fluctuating relative motion. The frictional energy dissipated is proportionately smaller. In the case of coherent floes, a certain amount of energy is contained as internal energy of the constrained fibres. On the Appendix J: Energy Dissipation Due to Fibre-Fibre Contacts 327 other extreme, if no contact is broken in the network the fibres exist as a single network, we are back to the original state of no motion. Thus at least the scenario considered is self-consistent. An intermediate state is the most probable where fibres exist as coherent floes with a certain fraction of fibre contacts still intact while the contacts at the periphery of floes are being continuously formed and disrupted by hydrodynamic forces. The number of floes in a suspension can be estimated from Kerekes crowding factor. To obtain the number of floes, the total number of fibres is divided by the crowding factor, i.e. Nfiocs = Nf/Ncr Fn/2 Fn/2 Figure I-l: Normal and fiictional forces at fibre-fibre contact points Appendix I: Energy Dissipation Due to Fibre-Fibre Contacts 328 1.2 Energy dissipation due to fibre-fibre contacts Consider frictional forces acting at fibre contacts, Ff that is induced by normal forces at fibre contacts, Fp, as shown in Figure I-l. The frictional force acting at fibre contact point (fibre-fibre friction) is given by: Ff^MfK (i.i) where /JF if coefficient of friction and F„ normal force acting on fibre contact. If we consider the fibre to bend as a simple beam as a result of normal force imposed by fibre contact, the resulting deflection is given by (Rerekes, 1995): S = ^ ^—^ (1.2) 4SEI where EI is the beam (fibre) stiffness and AL the distance between two contact points. The normal force is therefore obtained from (1.1) as Taking AL«iL/nc, where L is the the length of fibre and Uc the number of contacts per fibre, and combining (1.1) and (1.3) we obtain the frictional force per unit contact as F^=^^tfE!i (1.4) In order to inactivate a fibre contact, work must be done to overcome the frictional force. Work done per unit time gives energy, therefore the energy dissipation in fibre-fibre friction can be obtained by multiplying the frictional force with the relative motion induced between fibre contacts, Ur, i.e Appendix I: Energy Dissipation Due to Fibre-Fibre Contacts 329 s^_^=F^xu^ (1.5) The fluctuating relative velocity between two points in a turbulent field (ur) can be estimated from literature correlation established by Kuboi et al. (1974) between particle (fibre) Reynolds number and specific power group, namely Re, = ^ = 0.5 Ud'^ K - ' , (1.6) where d is the distance between two points in relative motion, here taken as fibre diameter, E is the overall energy dissipation rate, and v the kinematic viscosity of the suspending medium. Thus the relative velocity is obtained as w, = 0.5{£d)' (1.7) We can also estimate the relative velocity following an approach proposed by Kerekes (1995). If we take the distance over which the hydrodynamic drag takes place as AL, and assume an average shear rate, G, the minimum relative velocity may be estimated as: u^=GAL = GL/n^ (1.8) G = the average shear rate which we estimate from the velocity gradient between rotor tip (R|) and vessel wall (Ro) and the gap between them (Ro-Ri) as OR, G = '— (1.9) where Q is the angular velocity in rad/s. If we set the inter-contact distance as AL and combine (1.8) and (1.9), the fluctuating relative velocity (minimum) is given by u=GAL = '- '- (LIO) The total number contacts that are actively engaged in providing the frictional resistance in a flbre network may be computed from the total number of fibres in a unit mass of suspension as N,=N^xn: (Ml) Appendix I: Energy Dissipation Due to Fibre-Fibre Contacts 330 where Nf is the number of fibres per unit mass of suspension which can be calculated from fibre coarseness (©, kg/m) and suspension mass concentration (Cm) as Nf=^ (1.12) For a given fiber with ric contacts, the probability (Pc) of this single fibre interacting with any other fibre in its swept volume is given by P c = - ^ (1.13) N Thus the number of independent contacts per fibre is given by Nc=N^xn:=N^xP,xn^=.^f^ (1.14) Thus the energy dissipation per unit mass of fibre suspension is given by: s^_f=/UpF„NcU^ (1.15) The friction coefficient for fibres can be from recent measurements (Andersson and Rasmuson, 1997; Andersson, 1998) as \iv = 0.6 - 0.7. The maximum deflection is of the order of fibre diameter (d). Famood et al., (1994) have shown that for a stable network of unfolded fibres, the maximum deflection can be taken as 5 = 0.6d. 1.3 Energy dissipation due to fibre-fluid interaction The energy dissipated as a result of the fluctuating relative velocity between phases is given by the drag force (FD) X relative velocity (Ur), i.e s,=F^xu^xN^ (1.16) where the drag force is given by Appendix I: Energy Dissipation Due to Fibre-Fibre Contacts 331 FD=\C^PUIA^ (1.17) and CD the drag coefficient and Ap the projected area in the direction of flow. The maximum projected area of a fibre undergoing translation and rotation in shear flow is given L X d, where L is length of fibre and d the fibre diameter. Aidun (1956) obtained correlation for drag coefficient for free settling fibres as: CD = 7.7Rep"°'^^ (0.007 <Rep<. l ) (1.18) CD = 10.78Rep-''^ ^ (0.1< Rcp < 2) (1.19) In order to account for the fact that fibres could exist as floes, one may use drag coefficient for spheres. The drag coefficient used by Ishii and Zuber, (1979) and also Momii et al., (1986) for spherical particles may be used, replacing particle Reynolds number with fibre Reynolds number (Rcf): 24 Cn = (l +0.15 Re}) (1.20) The fibre Reynolds number, which is the ratio of inertia forces to viscous forces acting upon a fibre, was defined by Kerekes (1995) as: R e , = ^ ^ (1.21) We obtain the following expression for energy dissipation due to fibre-fluid as s,=^C^pul{Ld)N^ (1.22) Table I-l compares the estimated energy dissipation due to fibre-fibre friction, fibre-fluid drag and fluid-fluid interactions (from chemical measurements) at N = 66.7 rev/s. The energy dissipation due to fibre-fluid drag is smaller that that due to fibre-fibre friction (less than one percent). Drag coefficients calculated using equation 1.20, are twice the values calculated by equation 1.18. Table I-l: Estimation of energy dissipation due to fibre-fibre and fibre -fluid friction at N = 66.7 rev/s in high shear mixer with flow loop. Different relative velocity estimates. Cv 0 0.013 0.026 0.039 0.052 0.065 0.079 Cv 0.013 0.026 0.039 0.052 0.065 0.079 Relative velocity Ur 0 0.20 0.20 0.20 0.20 0.20 0.20 Ur 0.17 0.12 0.10 0.08 0.08 0.07 Crowding factor Ncr 0 94.7 189.4 284.2 378.9 473.6 575.6 Ner 94.7 . 189.4 284.2 378.9 473.6 575.6 Number of fibres Per kg Nf 0 1.84E+07 3.68E+07 5.53E+07 7.37E+07 9.21E+07 l.llE+08 Nf 1.84E+07 3.68E+07 5.53E+07 7.37E+07 9.21E+07 l.llE+08 Number of contacts He 0 2.7 5.2 7.7 10.1 12.3 14.7 ric 2.7 5.2 7.7 10.1 12.3 14.7 Reynolds number Ref 0 4.4 4.4 4.4 4.4 4.4 4.4 Ref 3.7 2.6 2.1 1.9 1.7 1.5 Drag coefficient CD 0 7.6 7.6 7.6 7.6 7.6 7.6 CD 8.8 11.7 14.0 15.9 17.5 19.1 Energy dissipation, W/kg Ed 0 29 58 88 117 146 175 £d 20 19 18 18 18 17 Sf.f 0 43 347 1172 2779 5427 9620 Sf.f 36 205 565 1160 2027 3260 £xo * 3000 1570 789 389 192 94 45 £xo 1570 789 389 192 94 45 s TO o' 3 b K TO O TO TO :3i to ' normalized for total dissipation Appendix J: Estimation of Turbulence Quantities by Other Means 333 Appendix J J Estimation of Turbulence Quantities by Other IVIeans Most approaches to quantifying mixing are mainly concerned with resolving the turbulent fluid mechanics parameters namely, the Reynolds stresses, turbulent kinetic energy and the turbulent energy dissipation rate. While the first two parameters can be measured, the rate of energy dissipation can only be estimated. This is accomplished mostly by using LDV. Recently, CFD is being used to provide detailed spatial predictions of impeller generated flow. However, LDV is still required to validate the CFD predictions. It is important to emphasize that the local energy dissipation rate, s, is not measured directly, but calculated even when LDV is used. It can either be calculated by integration of the energy dissipation spectrum (Nishikawa et al., 1976; Okamoto et al., 1981) or by dimensional arguments using characteristic velocity and length scales of turbulent eddies (Laufliutte and Mersmann, 1985; Wu and Patterson, 1989; Kresta and Wood, 1993; Zhou and Kresta, 1996). Kresta (1998) has provided an in-depth summary of the different methods for estimation of 8. According to Townsend (1976) the local energy dissipation rate is related to the rms fluctuating velocity (M') and the macroscale of turbulence (A) through £ O Z ^ (J.l) A According to Brodkey (1975), the macroscale can be substituted with the impeller diameter (D) as a reasonable measure of the largest eddies in a mixer, thus s = C ^ (J.2) D Appendix J: Estimation of Turbulence Quantities by Other Means 334 The constant C (Brodkey constant) has been shown experimentally by Laufhutte and Mersmann (1985) and theoretically (Werner and Mersmann, 1994) to lie between 6 and 10. Badou et al., (1997) has compared the energy dissipation calculated from dimensional arguments and that calculated from energy spectrum. In their case, s is calculated as s = A ^ (J.3) D where k is the turbulent kinetic energy and D impeller diameter. The constant A is calculated aposteori in such a way that the total dissipation is in agreement with power consumption, i.e Kresta and Wood (1993) used A = 1, and le = D/10, whereas Wu and patterson used A = 0.85. Mavros et al., (1997) have compared the energy dissipation rate for different impellers by assuming U - D/10 and computing the microscale /^  =UT^, where Xg is the integral time-scale. They found only slight discrepancies in dissipation from the two methods. JA Relative energy dissipation Rates in STR Table J-1 shows a summary of different investigations showing the relative energy dissipation rates in standard stirred vessel and an estimate of percent contribution to different vessel regions. Appendix J: Estimation of Turbulence Quantities by Other Means 335 Table J-1: Maximum relative energy dissipation rate (s/save) in the discharge zone of Rushton Turbine in a standard baffled vessel vmless otherwise stated Investigation Cutter (1966) Okamotoet al., (1981) Laufliutte & Mersmann (1987) Geisler & Mersmann (1988) Costes and Cauderc (1988) Wu and Patterson (1989) Kresta and Wood (1991) Zhou and Kresta (1996) Schaferetal.,(1997) Wemesson & Tragarth, (1998) Derksen and Van den Akker (1999) Relativ dissi (^E/Eavejinin 0.25 0.21 0.08 0.2 0.06 0.086 0.2 e energy pation (8/8ave)max 70 6.5 8 10 10 22 92 100 6 75 100 % contribution to total dissipation Impeller swept vol. 20 0 30 54 Impeller discharge 50 65 30 35 80 18 60 Rest of tank 30 35 40 11 20 22 * blank values = not estimated Appendix K: Modeling of Micromixing 336 Appendix K K Modeling of Micromixing K.1 EDD MODEL Formulation and Solution of Diffusion-Convective Reaction (Ref. Baldyga and Bourne, 1984) Consider the laminated structure within the smallest scales energy dissipating eddies as shown in Figure K-1. I • I 26o I I +6 Figure K-1: Diffusion and reaction equations solved over laminated regions with initial thickness of 25o and 25 for t > 0. Molecular diffusion and chemical reaction in a shrinking laminated structure within a small energy-dissipating vortex can be represented as: 5c, 5c, 5^c, dt dx dx^ where the local instantaneous rate of shrinkage is given by: (s/vyx (K.1) u = {4 + st^/v} (K.2) Appendix K: Modeling of Micromixing 337 From symmetry, boundary conditions are x = ±5, 5c, /dx = 0. The initial conditions for the first vortex are: t = 0, -6o < x < 0; Ci(x,o) = Ciis(x); t = 0, 0<x<5o , Ci(x,o) = Ciie(x) where Cus and Cue are the initial distributions of B-rich and A-rich solutions respectively. After time TCO=12(V/£')^ (the mean lifetime of an energy dissipating vortex), two second generation vortices are formed. Their initial conditions are t = Xa,, -5o< x < 0, Ci(x, Tea) = Ci2s(x), 0 < X < 60 , Ci(x, Tjo) = Ci2e(x), with Ci2s the Concentration in the first vortex when t = Xo,, and Ci2e the concentration profile in the environment as it forms eddies of the second generation. Similar considerations apply to later generations. Using dimensionless forms: X = x/5; 9 = tD/5o^ ; Ci = Ci/cjo, equation (K.l) becomes dC. ( s:\ de ^ = D ySoj ,Dc, (K.3) Each slab forming the vortex is assumed to have an initial thickness of A,K, thus 25o = \y^ I sj ,h and 60 being related through: 5_ 1 + 32 1 + 0'Sc' 32 -1 (K.4) For the consecutive competitive reaction A + B- -^R R + B "' >S (K.5) if we choose B as reference material (j = B), the reaction term in equation (K.3) becomes: R,=-(k,/k,)MC,C, R,=-{kJk,)MC,{C,+C,) (K.6) R,^{kJk,)MC,{C,-C,) where the mixing modulus, M is defined as D (K.7) and the boundary conditions become X = ±1; dCjdX = 0. The solution of the non-linear partial differential equations (PDE's) is then required over the interval -5 < x < 5, and for a long reaction time (t-^00), so that the concentrations are obtained when the limiting reagent (B) has been fully consumed. The concentrations then depends on: Appendix K: Modeling of Micromtxing 338 (k N" V C, = fn —^•,——;—^:M:Sc;reactortype k N° V (K.8) For semi-batch operation, A-rich solution is initially charged to the reactor, while B-rich solution is slowly fed in. The solution of (K.3) involves discretization of the feed into a parts, whereby each part reacts with N generations of eddies. After one part has been added in the reactor and reaction run to completion, reactor contents are updated. A second feed part of B-rich feed enters the reactor, it encounters an environment with CM of A concentration and R and S. CAI is valid for all generations of vortices for the second part of B, after which, the concentrations A, R, and S are again updated to give CA2- The procedure is repeated until the whole amount of B is consumed. Appendix K: Modeling of Micromixing 339 K.2 Simplification of EDD Model It has been shown by Baldyga and Bourne (1989) that for Sc < 4000, the effects of molecular diffusion are negligible when compared to the effects of viscous-convective stretching. When this is so, the diffusive term in EDD formulation can be omitted, thus the set of coupled partial differential equations can be reduced to ordinary differential equations for micromixing calculations. Physically, this means that diffusion between layers wdthin a vortex is much faster than engulfment and therefore not the rate-determining step. Thus, the engulfment model is formulated with the assumption that viscous-convective stretching controls mixing at scales comparable to the Kolmogorov dissipative scales. The simplest formulation of the model reflects a situation where a small volume of concentrated reactant (B) is added to a large volume of a second reactant (A) (see Figure F-2). CE CBO (t = to) 1 CB(t = ti) CB(t = t2) CB (t = <») — • CBO (t - to) CB(t = ti) CB(t = t2) ^CB(t = a)) Figure K-2: Features of the Engulfment model, (a) mass transfer by engulfment of local surroundings showing growth of mixed zone, (b) concentration profile of B for mixing without reaction, (c) concentration profile of B for mixing with instantaneous reaction (t2 > ti > to) Appendix K: Modeling of Micromixing 340 What is referred as reaction zone is the region where B is present. For VBO « VAC, the reaction zone grows according to dV - ^ = EV, (K.9) at where E is the engulfment rate coefficient and applies to the viscous-convective (VCS) part of concentration spectrum. A mass balance on substance i in the growing reaction zone gives ^ f c ^ = £F,<c,>+i?,K, (K.10) at where <Cj> is the concentration of / in the local environment of the growing eddy and the first RHS term in equation (K.IO) is the rate of addition of i by engulfment, and the second term it's production by reaction. Since diffusion is rapid relative to engulfment, Cj refers to uniform concentration within the growing reaction zone. Introducing equation (K.9) to (K.10) gives the basic E model dc -L = E{<c,>-c,) + R, (K.11) dt or using dimensionless parameters: 9 = tE; Cj = CJ/CBO; Da = kaCso/E; Py = k^lki dC, ;- = (<C,. >-C,) + DaY^PyC,Cj (K.12) d9 K.3 Modifications for Semi-batch Operations witii Flow Loop The overall operation mode of the reactor used in this study was semi-batch, even when operated with flow loop. However, in flow loop operation, the method by which concentrations are updated is modified. This is because, as shown in Figure K-3 below, although we are feeding the limiting reactant (B) in semi-batch mode, we are also feeding (Ai and A2) simultaneously. As a result, the possibility of self-engulfment, i.e. fluid of the same stream mixing with itself is increased, and must be accounted for. The volume of solution B added to the semi-batch reactor is discretized into a equal parts, each of with volume VB and concentration CBO. When any part has been added, the volume of the reaction zone grows according to the following equation (K.9). Appendix K: Modeling of Micromixing 341 Q(A1+A2) B Figure K-3: Flow loop simulation Engulfments take place with the same environment, having the same concentration <Ci>. During the application of equation (K. 15) to any part of feed, <Ci> is constant. In going to the next eddy, however, the concentration of the surroundings is updated by carrying out mass balance, which represents mixing in the uniform reaction zone with the surroundings. Updating of <Ci> is as follows. The j * part of the feed completely reacts in time 0, and the concentration of i is then Cj^ e. The volume of this solution started at VBO/CJ and is finally (VBo/cr)e .^ The rest of the solution has a volume VA+JVBQ/CV - (VBJ^)Q^, and concentration <Ci>j.i. The surroundings for the G+1)* part have concentrations {C,)Jaa + j-e')+C,y n CCCT + J (K.13) To account for self-engulfinent, we follow the approach given by Baldyga and Bourne (1989). Define parameter Zo as is the fi-action of B-solution in the local environment where self-engulfinent may take place 2o = Q. Bo QAO+QBO (K.14) Appendix K: Modeling of Micromixing 342 For self-engulfment of B, the zone grows as dt E{\-X,)X, (K.15) therefore the local environment for self engulfinent is given by X Zj ' \-ZS\-e') (K.16) Equation (K.17) is therefore modified to account for self-engulfment as {c)r-{c.), ( X„ j-\ a<7 + j -aa + j + C, X. (K.17) 

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