PIPE FLOW OF HOMOGENEOUS SLURRY by DONALD JOHN HALLBOM A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS OF THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Mining Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2008 ©Donald John Hallbom, 2008 ABSTRACT Designers of early long distance conventional slurry pipelines relied heavily on loop testing to determine key design parameters, such as the pressure gradient and deposition velocity. Over time, a “rheology-based design” methodology was developed that allowed the key design parameters to be determined based on bench scale tests. This methodology is now routinely used to design slurry pipelines hundreds of kilometres long. Unfortunately, the current rheology-based design methodology has been found to be inadequate in predicting the flow behaviour of thickened slurries of fine particles at high solids contents. As a result, loop testing is still heavily used for the design of a thickened slurry transport systems, resulting in high testing costs that create a barrier to the adoption of this technology. Furthermore, there has been limited adoption of rheology-based design in the mineral processing industry beyond piping systems, even though many operating problems are rheology related. It is proposed that the ineffective crossover of rheologybased design is due to the inadequacy of the available consistency models, the “disconnect” between slurry rheology and its physical properties, and the complexity of the available design equations. The objective of this Thesis is to devise a “slurry rheology” that is more conducive to application by practicing engineers without impairing its accuracy or utility for research purposes. The cornerstone is the development of a new rheological model and constitutive equation for homogeneous slurry based on the aggregation/deaggregation of the suspended mineral particles. This “yield plastic” model is shown to describe a family of models that includes the Newtonian, Bingham plastic and Casson models as special cases. It also closely approximates the results of many consistency models, including power law, yield power law, Cross and Carreau-Yasuda. The yield plastic model is then used to develop design equations to determine the pressuregradient of laminar and turbulent pipe flow. A relative energy dissipation criterion is proposed for the laminar-turbulent transition and shown to be consistent with currently used transition models for Newtonian and Bingham fluids. Finally, a new dimensionless group (the “stress number”) is proposed that is directly proportional to the pressure-gradient and ii independent of the velocity. When the design equations are presented graphically in terms of the stress number and the plastic Reynolds number, the resulting “design curve diagram” is shown to be a dimensionless (pressure-gradient vs. velocity) pipe flow curve. The net result is that the hydraulic design of homogeneous slurry systems only requires the use of a single constitutive equation and three engineering design equations. The results are presented in a conceptually easy form that will foster an intuitive understanding of nonNewtonian pipe flow. This will assist engineers to understand the impact of slurry rheology when designing, operating and troubleshooting slurry pipelines and, in the future, other slurry related processes. iii TABLE OF CONTENTS ABSTRACT................................................................................................................ii TABLE OF CONTENTS............................................................................................iv LIST OF TABLES ..................................................................................................... x LIST OF FIGURES ...................................................................................................xi LIST OF SYMBOLS ................................................................................................xiv GLOSSARY ...........................................................................................................xvii PREFACE..............................................................................................................xxv ACKNOWLEDGEMENTS ..................................................................................... xxvi DEDICATION....................................................................................................... xxvii CHAPTER 1: INTRODUCTION ............................................................................. 1 1.1. Mineral Processing is Slurry Processing .................................................... 1 1.2. The Flow Behaviour of Slurry ..................................................................... 1 1.3. Pipeline Hydraulic Design........................................................................... 3 1.3.1 Slurry characterization......................................................................................... 4 1.3.2 Estimation of pressure gradient ........................................................................... 4 1.3.3 Advantages of rheology-based design ................................................................. 7 1.4. Application of Rheology in Mineral Processing......................................... 11 1.5. Thickened Slurry....................................................................................... 13 1.5.1 Thickened tailings.............................................................................................. 14 1.5.2 Paste backfill...................................................................................................... 15 1.5.3 New equipment.................................................................................................. 16 1.5.4 New ore bodies .................................................................................................. 16 1.5.5 Availability of water .......................................................................................... 17 1.5.6 Design of thickened slurry systems ................................................................... 18 1.5.7 Problems with rheological models..................................................................... 18 1.5.8 Virtual rheology................................................................................................. 21 1.6. Thesis Objective ....................................................................................... 22 iv 1.7. Thesis Outline........................................................................................... 22 CHAPTER 2: 2.1. SLURRY RHEOLOGY – LITERATURE REVIEW.......................... 24 Introduction............................................................................................... 24 2.2. Slurry – the Micro-Scale ........................................................................... 24 2.2.1 Liquid phase....................................................................................................... 25 2.2.2 Solid phase......................................................................................................... 25 2.2.3 Dilute non-aggregating slurry............................................................................ 26 2.2.4 Concentrated non-aggregating slurry ................................................................ 26 2.2.5 Particle shape ..................................................................................................... 28 2.2.6 Swelling and porous particles............................................................................ 28 2.2.7 Aggregating particles......................................................................................... 30 2.2.8 Particle interactions ........................................................................................... 30 2.2.9 Inter-particle bonds............................................................................................ 32 2.2.10 Structure......................................................................................................... 33 2.2.11 Build-up and breakdown of structure ............................................................ 34 2.2.12 Limits to aggregation..................................................................................... 36 2.3. Slurry – the Meso-Scale ........................................................................... 36 2.3.1 Equivalent viscosity in turbulent flow............................................................... 37 2.3.2 Shear thinning.................................................................................................... 38 2.3.3 The yield stress .................................................................................................. 39 2.3.4 Loss of strength.................................................................................................. 39 2.3.5 Time-dependent fluids ....................................................................................... 40 2.4. Constitutive Equations.............................................................................. 42 2.4.1 Constitutive equations available ........................................................................ 42 2.4.2 Constitutive equations used in mineral processing............................................ 44 2.4.3 Engineering equations ....................................................................................... 45 2.4.4 The yield power law family of models .............................................................. 47 2.4.5 The Casson model.............................................................................................. 48 2.5. The yield plastic family ............................................................................. 49 2.6. Prior Art .................................................................................................... 50 2.6.1 Heimann-Fincke (Heinz-Casson) ...................................................................... 51 2.6.2 Oka..................................................................................................................... 53 2.6.3 Haake rheometer software ................................................................................. 54 2.7. Summary .................................................................................................. 54 CHAPTER 3: 3.1. NEW RHEOLOGICAL MODEL...................................................... 56 Introduction............................................................................................... 56 3.2. General Assumptions ............................................................................... 56 3.2.1 Description of suspension.................................................................................. 56 3.2.2 Particles move.................................................................................................... 57 3.2.3 Particles may form aggregates........................................................................... 57 v 3.2.4 3.2.5 3.2.6 3.2.7 3.2.8 3.2.9 3.2.10 3.2.11 3.2.12 3.2.13 3.2.14 3.2.15 Bonds and links.................................................................................................. 57 Strength of bonds............................................................................................... 58 Strength of aggregates ....................................................................................... 58 Velocity gradients break aggregates.................................................................. 58 Aggregates reach an equilibrium size................................................................ 59 Bonds form randomly but break preferentially ................................................. 60 Dispersed particles increase apparent viscosity............................................. 60 Aggregates increase the effective volume concentration .............................. 61 Aggregation increases the slurry viscosity .................................................... 61 Minimum viscosity when particles are fully dispersed ................................. 61 Three-part viscosity ....................................................................................... 62 Applied and internal shear rates .................................................................... 62 3.3. The Yield Plastic Model ............................................................................ 63 3.3.1 Derivation of the basic model............................................................................ 63 3.3.2 Alternate forms of the basic model.................................................................... 66 3.3.3 Determination of the yield plastic model parameters ........................................ 66 3.4. Low Shear Rate Behaviour of the Yield Plastic Model.............................. 67 3.4.1 “Zero” shear rate................................................................................................ 67 3.4.2 Alternate form of the extended yield plastic model........................................... 68 3.4.3 Determination of the extended yield plastic model parameters......................... 69 3.5. Summary .................................................................................................. 69 CHAPTER 4: 4.1. COMPARISON WITH EXISTING MODELS .................................. 71 Introduction............................................................................................... 71 4.2. Desirable Characteristics for a Rheological Model ................................... 71 4.2.1 Yield plastic model ............................................................................................ 72 4.3. Commonly Used Rheological Models....................................................... 73 4.3.1 Newtonian.......................................................................................................... 73 4.3.2 Bingham plastic ................................................................................................. 74 4.3.3 Casson and Heinz .............................................................................................. 75 4.3.4 Power law .......................................................................................................... 76 4.3.5 Yield power law................................................................................................. 78 4.3.6 Cross and Carreau-Yasuda ................................................................................ 80 4.4. Other Rheological Models ........................................................................ 83 4.4.1 Sisko .................................................................................................................. 83 4.4.2 Ellis .................................................................................................................... 84 4.4.3 Meter.................................................................................................................. 85 4.4.4 Eyring ................................................................................................................ 85 4.5. Summary .................................................................................................. 86 CHAPTER 5: 5.1. LAMINAR PIPE FLOW .................................................................. 88 Introduction............................................................................................... 88 vi 5.2. Laminar Pipe Flow.................................................................................... 88 5.3. Exact Solutions for Yield Plastics ............................................................. 90 5.3.1 Yield plastic ....................................................................................................... 90 5.3.2 Yield plastic fluids (1/k = an integer) ................................................................ 90 5.3.3 Numerical solutions for yield plastics ............................................................... 91 5.4. General Solutions for Yield Plastics.......................................................... 94 5.4.1 Need for a general solution................................................................................ 94 5.4.2 Engineering approximation ............................................................................... 94 5.4.3 Rheological approximation................................................................................ 96 5.5. Summary .................................................................................................. 99 CHAPTER 6: 6.1. TURBULENT PIPE FLOW........................................................... 101 Introduction............................................................................................. 101 6.2. General Comments on Turbulent Flow................................................... 101 6.2.1 Wilson-Thomas drag reduction model ............................................................ 102 6.2.2 Wilson-Thomas equation for pressure-gradient .............................................. 104 6.3. Modified Wilson-Thomas Model ................................................................. 105 6.4. Turbulent Flow of Yield Plastics ................................................................. 106 6.4.1 General comments ........................................................................................... 106 6.4.2 Viscosity ratio.................................................................................................. 107 6.4.3 Area ratio ......................................................................................................... 107 6.4.4 Drag reduction factor....................................................................................... 111 6.5. Comments on Turbulent Flow................................................................. 112 6.5.1 Equivalent “viscosities”................................................................................... 112 6.5.2 Choice of consistency model ........................................................................... 112 6.5.3 Transitional flow range.................................................................................... 113 6.5.4 Reynolds stresses ............................................................................................. 115 6.5.5 The “pseudo-fluid” approximation .................................................................. 115 6.5.6 Rough wall turbulent flow ............................................................................... 116 6.6. Summary ................................................................................................ 118 CHAPTER 7: 7.1. LAMINAR-TURBULENT TRANSITION........................................ 119 Introduction............................................................................................. 119 7.2. Laminar-Turbulent Transition.................................................................. 120 7.2.1 Newtonian transition........................................................................................ 120 7.2.2 Two transition points ....................................................................................... 121 7.3. Transition Criteria ................................................................................... 122 7.3.1 Intersection criterion........................................................................................ 122 7.3.2 Friction factor criterion.................................................................................... 122 7.3.3 Yield stress criterion ........................................................................................ 123 7.3.4 Local stability criterion.................................................................................... 123 vii 7.3.5 A new transition criterion ................................................................................ 124 7.4. Transition of Yield Plastics ..................................................................... 126 7.4.1 Comparison of equations for Bingham plastics............................................... 128 7.5. Summary ................................................................................................ 129 CHAPTER 8: 8.1. DESIGN CURVE DIAGRAM........................................................ 131 Introduction............................................................................................. 131 8.2. Slurry Hydraulic Engineering .................................................................. 131 8.2.1 The Moody diagram ........................................................................................ 131 8.2.2 Conceptual engineering ................................................................................... 134 8.3. Dimensionless Groups ........................................................................... 135 8.3.1 Standard dimensionless groups........................................................................ 135 8.3.2 Dimensionless group for pressure-gradient ..................................................... 137 8.3.3 The stress number ............................................................................................ 138 8.4. Flow Equations Using the Stress Number .............................................. 139 8.4.1 Newtonian........................................................................................................ 139 8.4.2 Bingham plastic ............................................................................................... 142 8.4.3 Yield plastic ..................................................................................................... 145 8.5. Design Curve Diagram (for Yield Plastics) ............................................. 148 8.6. Summary ................................................................................................ 149 CHAPTER 9: CASE STUDY.............................................................................. 150 9.1. Introduction............................................................................................. 150 9.2. Rotational Rheometry............................................................................. 150 9.3. Kaolin Clay Slurry Rheology ................................................................... 153 9.4. Laminar Pipe Flow.................................................................................. 158 9.5. Turbulent Pipe Flow................................................................................ 159 9.5.1 Wilson-Thomas turbulence equation............................................................... 159 9.5.2 Pseudo-fluid turbulence equation .................................................................... 161 9.6. Laminar-Turbulent Transition in Pipe Flow ............................................. 163 9.7. Pipe Flow Curves ................................................................................... 165 9.8. Design Curve Diagram ........................................................................... 166 9.9. Summary ................................................................................................ 167 CHAPTER 10: SUMMARY............................................................................... 169 10.1. Introduction ......................................................................................... 169 10.2. Review of Thesis................................................................................. 169 viii 10.3. Novel Contributions............................................................................. 172 10.4. Future Research Work........................................................................ 175 10.4.1 Physical-rheological parameter correlations................................................ 175 10.4.2 Shear thickening .......................................................................................... 176 10.4.3 Rotational rheometry ................................................................................... 177 10.4.4 Explicit solution of laminar pipe flow equation .......................................... 177 10.4.5 Laminar pipe flow of extended yield plastics.............................................. 177 10.4.6 Drag reduction factor................................................................................... 177 10.4.7 Collapsing plug............................................................................................ 178 10.4.8 Alternate flow configurations ...................................................................... 178 10.4.9 Rotating equipment...................................................................................... 178 10.4.10 Transition and energy dissipation ratio........................................................ 179 10.4.11 Time-dependent slurry................................................................................. 179 10.5. Conclusions ........................................................................................ 180 REFERENCES ..................................................................................................... 181 APPENDIX A: DERIVATION OF LAMINAR FLOW SOLUTIONS......................... 191 APPENDIX B: DERIVATION OF TURBULENT FLOW SOLUTIONS ................... 203 APPENDIX C: DESIGN CURVE DIAGRAMS....................................................... 221 APPENDIX D: CASE STUDY DATA..................................................................... 226 APPENDIX E: PUBLICATIONS ............................................................................ 231 ix LIST OF TABLES Table 1.1: Rheological parameters for lignite in methanol ................................................... 19 Table 2.1: Approximate iso-electric points (i.e.p.) for common minerals ............................ 31 Table 2.2: Relative force and bond strength of particles with different surface charge ........ 32 Table 4.1: Summary Comparison of Rhelogical Models ...................................................... 87 Table 5.1: Exact Solutions for Selected Yield Plastic Fluids ................................................ 92 Table 5.2: Apparent Shear Rate Multiplier (χ) by Numerical Integration ............................ 93 Table 5.3: Exponent Function: f3n(Z,k) .................................................................................. 97 Table 5.4: Approximate Apparent Shear Rate Multiplier (χ)................................................ 98 Table 6.1: Exact Solutions for Selected Yield Plastic Fluids .............................................. 110 Table 6.2: Area Ratios for Selected Yield Plastic Fluids .................................................... 110 Table 6.3: Drag Reduction Factors for Selected Yield Plastic Fluids ................................. 111 Table 8.1: Newtonian pipe flow equations .......................................................................... 141 Table 9.1: Rheometry Results 10%v/v Kaolin Slurry (No Dispersant)............................... 154 Table 9.2: Rheometry Results 14%v/v Kaolin Slurry (Partially Dispersed) ....................... 154 Table 9.3: Model Parameters 10%v/v Kaolin Slurry (All Data) ......................................... 155 Table 9.4: Model Parameters 14%v/v Kaolin Slurry (All Data) ......................................... 156 Table 9.5: Pipe Flow Loop Data for Kaolin Slurry ............................................................. 157 Table 9.6: Transition velocity in pipe flow 10%v/v kaolin slurry....................................... 164 x LIST OF FIGURES Figure 1.1: Impact of salt and dispersant on kaolin clay slurry pipe flow curves (Data from Litzenberger 2004)........................................................................................................... 9 Figure 1.2: Thickened mineral slurry (paste) ........................................................................ 13 Figure 1.3: Theoretical flow curves for lignite-methanol slurry in a 200 mm pipe .............. 20 Figure 2.1: Dilatency caused by high packing density .......................................................... 27 Figure 2.2: “Trapped” fluid caused by porosity and aggregation ......................................... 29 Figure 2.3: Effect of pH on surface charge of minerals ........................................................ 31 Figure 2.4: The effect of viscous shear stress on particle “structure” ................................... 35 Figure 2.5a/b: Turbulent pipe flow of kaolin slurry dispersed with TSPP (Data from Litzenberger 2004)......................................................................................................... 37 Figure 2.6: Effect of dispersant on the flow of kaolin slurry (Data from Litzenberger 2004) ....................................................................................................................................... 38 Figure 2.7a/b: Time dependent consistency of 68%w red mud (Nguyen 1983) ................... 40 Figure 2.8: High shear rate data for 68%w red mud (after Nguyen 1983)............................ 42 Figure 3.1: Shear breakage and impact attrition of aggregates ............................................. 59 Figure 3.2: Three-part viscosity of yield plastics .................................................................. 62 Figure 3.3: Parameter extraction for yield plastic model ...................................................... 67 Figure 4.1: Effect of changing k for yield plastics ................................................................ 76 Figure 4.2: Power law model fits to yield plastic flow curves .............................................. 77 Figure 4.3: Power law model fits to yield plastic flow curves (low range)........................... 78 Figure 4.4: Yield power law model fits to yield plastic flow curves .................................... 79 Figure 4.5. Flow behaviour of emulsion (Masalova et al. 2005) .......................................... 82 Figure 4.6: Basic and extended yield plastic representation of emulsion data...................... 83 Figure 4.7: Sisko model fits to yield plastic flow curves ...................................................... 84 Figure 5.1: Laminar flow in a cylinder.................................................................................. 89 Figure 5.2: Apparent shear rate multipliers for yield plastics ............................................... 93 Figure 5.3: Apparent shear rate multipliers and approximation ............................................ 95 Figure 6.1: Pipe flow curves for kaolin slurry in laminar and turbulent flow (Data from Litzenberger 2004)....................................................................................................... 102 xi Figure 6.2: Eddy scales in turbulent flow (Wilson and Thomas 2006) ............................... 103 Figure 6.3: Determination of Area Ratio (α)....................................................................... 108 Figure 6.4: Comparison with Bingham plastic data and the theory of Hanks (Wilson and Thomas 1985) .............................................................................................................. 114 Figure 7.1: Pipe flow curves for kaolin slurry - transition (Data from Litzenberger 2004)126 Figure 7.2: Transition Reynolds number for yield plastics ................................................. 128 Figure 7.3: Comparison of transition models for Bingham plastics.................................... 129 Figure 8.1: Simplified Moody diagram for Newtonian fluids............................................. 132 Figure 8.2: Hedström -Moody diagram for yield plastics ................................................... 133 Figure 8.3: Stress number vs. Reynolds number relationship for Newtonian fluids........... 141 Figure 8.4: Comparison of smooth wall turbulent flow equations ...................................... 143 Figure 8.5: Bingham transition relationship ........................................................................ 145 Figure 8.6: Laminar flow relationship (yield plastic).......................................................... 147 Figure 8.7: Design curve diagram for yield plastics............................................................ 148 Figure 9.1: Rheograms for kaolin clay slurry in concentric cylinder rheometer (Data from Litzenberger 2004)....................................................................................................... 155 Figure 9.2: Flow loop data for kaolin slurry (25.825 mm pipe) (Data from Litzenberger 2004) ..................................................................................................................................... 156 Figure 9.3: Comparison of shear rate vs. shear stress relationships: rheometer and flow low (Data from Litzenberger 2004) .................................................................................... 158 Figure 9.4: Comparison of measured and calculated laminar flow data (Data from Litzenberger 2004)....................................................................................................... 159 Figure 9.5: Smooth wall turbulent flow curves based on the Wilson-Thomas model (Data from Litzenberger 2004).............................................................................................. 161 Figure 9.6: Comparison of pseudo-fluid turbulent flow models and measured data (Data from Litzenberger 2004).............................................................................................. 162 Figure 9.7: Comparison of transition criterion for 10% kaolin slurry (Data from Litzenberger 2004) ............................................................................................................................ 163 Figure 9.8: Comparison of measured data for kaolin clay slurry and the pressure gradients predicted using the yield plastic model (Data from Litzenberger 2004) ..................... 165 xii Figure 9.9: Design Curve Diagram representation of kaolin slurry data (Data from Litzenberger 2004)....................................................................................................... 167 Figure A.1: Shear rate in a pipe ........................................................................................... 192 Figure B.1: Turbulent pipe flow velocity profile (Newtonian) ........................................... 207 Figure B.2: Effect of sub-layer thickening on turbulent pipe flow velocity profile (N = Newtonian; NN = Non-Newtonian)............................................................................. 212 xiii LIST OF SYMBOLS (m2) A Area B Yield (Bingham) number Ci Constant used in derivations (i = 0, 1, 2…) CC, CCY, CE Cross, Carreau-Yasuda, Eyring d Particle diameter d80 Diameter that 80% of particles are smaller than (typ.) (m) D Pipe/tube internal diameter fN = ΔPD 2τ w = 2 ρV 2 LρV 2 Fanning friction factor (pipe flow) F Force G Torque on CCR bob Ha = τ w ρD 2 μ∞2 (-) (varies) (s) (m) (m) (-) (N) (N-m) Stress number (-) HaC Critical stress number (transition) (-) τ o ρD 2 He = μ∞2 Hedström number (-) HeB Hedström number (Bingham plastic) (-) k Scaling factor (yield plastic) (-) K Consistency coefficient (PL and YPL fluids) L Length of pipe n Flow behaviour index (PL and YPL fluids) P0, P1 Pressure at points 0 and 1 (Pa) ΔP Pressure drop (between two points) (Pa) ΔP/L Pressure-gradient (Pa/m) q Flow rate through differential area (m3/s) Q Total (bulk) flow rate (m3/s) r Distance along the radius of a pipe (m) rc Radius of unyielded core (plug) (m) R Pipe/tube internal radius (D/2) (m) xiv (Pa-sn) (m) (-) Re = ρVD μ Reynolds number (Newtonian) (-) Re B = ρVD η B Plastic Reynolds number (Bingham) (-) ReC, ReBC, RepC Critical Reynolds number (-) Re p = ρVD μ∞ Plastic Reynolds number (yield plastic) (-) ReC, ReBC, RepC Critical Reynolds number (-) Re MR = 16 f N Metzner-Reed Reynolds number (-) Friction Reynolds number (yield plastic) (-) Re * = Dρu * μ∞ Ri, Ro, Ra Bob, cup, average radius (CCR) (m) t Gap or annulus radius (CCR) (m) u = u(r) Velocity at radius r (m/s) u* = τ w ρ Friction velocity (turbulent flow) (m/s) u , u′ Average, fluctuating axial velocity (m/s) u+ = u u * Friction velocity (turbulent flow) U = umax Maximum velocity pipe (m/s) v , v′ Average, fluctuating radial velocity (m/s) V Bulk velocity in pipe (m/s) VC Critical bulk velocity in pipe (transition) (m/s) VL, VT Laminar, turbulent bulk velocity in pipe (m/s) w , w′ Average, fluctuating rotational velocity (m/s) y Distance from wall (= R – r) y+ = yρu * (-) (m) Dimensionless distance from wall (-) z = τo τ Stress ratio (at any point) (-) Z = τo τw Wall stress ratio (at solid boundary) (-) α Rheogram area ratio (-) β Drag reduction factor (-) χ Apparent shear rate multiplier (-) ε Effective roughness of pipe wall η xv (m) φ Concentration (volume of solids/total volume) (-) φeff Effective volume concentration (-) φm Maximum volume concentration (-) γ& = dγ du = dt dr Shear rate (1/s) γ&o Base shear rate (1/s) γ&a Applied shear rate (1/s) Γ = 8V/D Apparent shear rate in a pipe (1/s) η = τ γ& Apparent viscosity (Pa-s) ηB, ηC Plastic viscosity (Bingham, Casson) (Pa-s) ηo, η∞ Low, high shear viscosity plateau (Cross, Carreau) (Pa-s) κ Karman coefficient (~0.4) (-) λ Degree of structure (-) μ Absolute viscosity (Newtonian fluids) (Pa-s) μ∞ Infinite shear rate viscosity (yield plastic fluids) (Pa-s) ρ Suspension/fluid density (kg/m3) ρl, ρs Liquid, solids density (kg/m3) σ Stress normal to the plane of shear (Pa) τ Shear stress (Pa) τo Yield stress (zero shear rate shear stress) (Pa) τB,τC Yield stress (Bingham, Casson) (Pa) τw Shear stress at the wall (Pa) Ω Rotational or angular velocity (CCR) ψ Particle shape factor xvi (Rad/s) (-) GLOSSARY This glossary defines the main technical terms as they are used in this Thesis. Where practical, the definitions follow NIST 960-3 “The Use of Nomenclature in Dispersion Science and Technology”. Aggregate: A general term used to describe a cohesive mass of particles, which includes agglomerates, coagulates and flocs. The formation of aggregates is called aggregation. Anti-thixotropy: A reversible time-dependent increase in viscosity at a particular shear rate caused by a build-up of structure over time. Apparent viscosity: The ratio of the shear stress to the shear rate at some nominal value of the shear rate and time. The apparent viscosity changes with shear rate for nonNewtonian fluids and with time for time-dependent fluids. Apparent yield stress: A stress determined by extrapolation from high shear rate portion of the flow curve to the stress axis. Used for non-ideal viscoplastic materials, where the yield stress is indefinite. Same as true yield stress for ideal viscoplastic materials. Bond: A general term for the net forces that hold aggregated particles together. Strongly bonded particles require a high shear stress to cause mechanical deaggregation. Bonds may be broken and reformed indefinitely without a significant loss strength (see link) Boundary layer: In turbulent flow, the radius of the pipe where the projected velocity curves of the laminar sub-layer and the turbulent core would intersect. Used in the twopart approximation for turbulent flow calculations. Brownian motion: Random fluctuations of small particles due to thermal energy. The effect is significant for colloidal particles, but becomes negligible for coarse particles. Capillary rheometer: A device for determining the consistency of a fluid by relating the volumetric flow rate through a tube to the applied pressure-gradient. Clay: Hard particles less than 2 μm in size. xvii Coarse: Particles larger than 37 μm that will not pass through a 400# Taylor mesh sieve screen. This includes: gravel, sand, and larger silt particles. Comminution: Breaking down large continuous solids into smaller pieces. Crushing and grinding are comminution processes. Concentric cylinder rheometer: A rotational rheometer where the consistency of a fluid is measured by shearing the material in the gap between an inner cylinder (bob) and an outer cylinder (cup). The shear stress is determined from the induced torque and the shear rate is determined from the rotational speed, the gap dimensions and an inferred rheology of the material. Consistency: A general term for the flow properties of a suspension. For a Newtonian fluid, consistency is equivalent to viscosity. A “higher consistency” means that the suspension has an apparent viscosity in the shear rate range of interest that is greater than a “lower consistency” suspension, although this may not be the case throughout the entire shear rate range. Constitutive equation: An algebraic equation to describe the flow behaviour implied by a rheological model. Generally describes the relationship between two of the following: apparent viscosity, shear stress and shear rate. Deaggregation: A general term used to describe any process by which a cohesive mass or cluster of particles is broken down into individual particles or smaller aggregates. The term does not include attrition of the particles themselves. Deformation: Movement of parts of a material body relative to one another such that continuity of the body is not destroyed, resulting in a change in shape or volume or both. Continuous deformation is flow. Diffusion-limited rate: Refers to a rate of aggregation corresponding to the frequency of encounter (collision rate) of the particles. Dispersant: A chemical substance that prevents aggregation and causes deaggregation. Dispersion: A two-phase system in which discrete particles of one phase (solid, liquid, gas) are dispersed in a continuous medium of another phase. xviii DLVO: An abbreviation for the theory of the stability of colloidal suspensions derived independently by Derjaguin and Landau, and by Verwey and Overbeek. The theory calculates the combined effects of the attractive van der Waals forces and the repulsive (or attractive) electrostatic forces. Dynamic equilibrium: A state in which opposing dynamic forces just balance to obtain a quasi-equilibrium condition. A suspension is in dynamic equilibrium when the breakdown and rebuilding of structure occur at the same rate. Electrostatic stabilization: A mechanism in which aggregation is inhibited by the presence of a mutually repulsive electrostatic potential surrounding each particle. Fine (sub-sieve): Particles smaller than 37 μm that will pass through a 400# Taylor mesh sieve screen. This includes: clay and smaller silt particles. Flow: Continuously increasing deformation of a material under the action of finite forces that does not eventually return to zero when the forces are removed. Flow curve [rheogram]: A graphical representation of fluid flow behaviour under shear in which shear stress is related to shear rate. Flow curve [pipe]: A graphical representation of fluid flow behaviour in which the pressure gradient is related to bulk velocity. Fully dispersed: A term used to describe a suspension where there is no significant aggregation of the particles. Gravel: Hard particles in the 2000 to 40,000 μm size range. Hard sphere interaction: A largely theoretical construct in which particles do not interact until they come into contact. At the moment of contact the particles will bond or rebound off one another. Heteroaggregation: The aggregation of particles of dissimilar materials. Infinite shear rate viscosity: The high shear rate limiting value of the apparent viscosity. Often associated with the second Newtonian region in shear-thinning fluids. Isoelectric point (IEP): The pH at which the zeta potential has a value of zero for a suspension of particles with the same surface material. xix Laminar flow: Streamline flow of a viscous fluid without large irregular fluctuations. Laminar sub-layer: In turbulent flow, the region near the pipe wall where flow behaviour is dominated by viscous effects and where inertial effects are insignificant. Link: An inter-particle bond that, once broken, cannot reform or reforms with a significantly lower strength than the original bond. Liquid phase: The suspending medium that exhibits continuity throughout a solid-liquid or gas-liquid dispersion. Monodisperse: Theoretically, all of the particles in a monodisperse suspension are the same size. Practically, the particles will have a narrow size distribution, such as would be obtained using two successive Taylor mesh sieves. Newtonian: Flow model for fluids in which a linear relationship exists between shear stress and shear rate. This is the same as a shear rate independent viscosity. No-slip: Condition in which fluid adjacent to a surface moves with the velocity of that surface. High consistency suspensions may be considered to “slip” when the bulk of the material slides on a thin layer of the much lower viscosity carrier fluid. Non-interacting: Refers to particles where the bonding forces (e.g., van der Waals or electrostatic) are insignificant compared to hydrodynamic or inertial forces at the shear rate range of interest. Often refers to coarse particle suspensions. Non-Newtonian: Any fluid that does not exhibit a linear relationship between shear stress and shear rate. This is the same as a shear rate dependent viscosity. Normal stress: The component of stress that acts in a direction normal (perpendicular) to the plane of the shear stress. Orthokinetic aggregation: The process of aggregation induced by hydrodynamic motions, such as stirring, sedimentation or convection. Particle: Any discontinuity in a dispersed system. This term will be used in reference to solid materials and aggregates. xx Paste: A semi-solid suspension with a soft and malleable consistency. In mineral processing, paste refers to a suspension with a liquid content above the plastic limit (when it becomes a brittle, crumbly cake) and an apparent yield stress above 100 Pa. Perikinetic aggregation: The process of aggregation induced by Brownian motion. Plastic: A property in which a material behaves like a solid when the stress is below some critical value, the yield stress, but flows when this value is exceeded. Plastic viscosity: The limiting high shear rate viscosity implied by the consistency model for a viscoplastic material (e.g., Bingham plastic). Point of zero charge (PZC): For solid-liquid suspensions, the pH at which hydroxyl and proton adsorption is just balanced to cancel net charge. Poiseuille flow: Laminar flow in a pipe of circular cross section under a constant pressure gradient. Polydisperse: A suspension in which many particle sizes occur. Reaction-limited rate: Refers to a rate of aggregation that is controlled by the reactivity of the particles (i.e., the frequency of collisions resulting in particle bonding.) Reynolds number: A dimensionless number that expresses the ratio of the inertial forces to the viscous forces. Reynolds stress: The stress in a fluid due to erratic turbulent fluctuations that are imposed on the mean stresses due to average motion. The term is used herein to describe all turbulent stresses that can disperse aggregates but do not affect the average axial shear stress at the pipe wall (i.e., the pressure gradient). Rheology: The science of the deformation and flow of matter. The term “rheology” is commonly used to mean consistency. Rheological model: A physical model to explain the flow behaviour (consistency) of a fluid. The resulting behaviour is described by a constitutive equation. In practice, the term “rheological model” is often used to mean “constitutive equation”. Rheomalaxis: An irreversible decrease in viscosity during shearing attributed to permanent break down in the material structure. xxi Rheopexy: The tendency of a material to recover some of its pre-shear viscosity at a faster rate when gently sheared than when it is allowed to stand. Sand: Hard particles in the 63 to 2000 μm size range. Sedimentation volume: The volume of sediment formed after allowing particles to settle. Used to estimate the volume of aggregates in an unstable suspension. Shear: The relative movement of parallel adjacent layers of a material. Shear rate: The rate of change of shear strain with time. Shear strain: The relative in-plain displacement of two parallel layers in a material body divided by their separation distance. Shear stress: The component of stress that causes successive parallel layers of a material body to move relative to each other in their own planes of shear. Shear-thickening: An increase in apparent viscosity with increasing shear rate during steady shear flow. Shear-thinning: A decrease in apparent viscosity with increasing shear rate during steady shear flow. Silt: Hard particles in the 2 to 63 μm size range. Slurry: A general term for a concentrated suspension. Stable suspension: A suspension that does not have significant aggregation as measured over a relevant time frame (e.g., the time it take the suspension to travel through a pipe). Steady shear flow: Condition under which a fluid is sheared continuously in one direction. Steady state [equilibrium] flow: Condition under which a constant stress or shear rate is maintained for a sufficient time to allow dynamic equilibrium to be achieved in a fluid containing time-dependent structure. Steric stabilization: A mechanism in which aggregation is inhibited by the presence of an adsorbed polymer layer surrounding each particle. Stress: Force per unit area. xxii Structure: Aggregation caused by the formation of stable physical bonds between particles in a suspension that affects the rheological behaviour of the material and causes plastic properties. Suspension: A dispersion in which solid particles are suspended in a liquid medium. Thickened slurry: A suspension where the solids concentration is high enough that the non-Newtonian properties significantly affect the flow behaviour on the macro-scale. Thixotropy: A reversible time-dependent decrease in viscosity at a particular shear rate caused by a breakdown of structure over time. Transition zone: In turbulent flow, the region between the laminar sub-layer and the turbulent core, where both viscous and inertial effects are significant. Turbulent core: In turbulent flow, the region taking up most of the center of the pipe where flow dominated by inertial effects and where viscous effects are insignificant. Turbulent flow: Flow of a viscous fluid with large irregular fluctuations in the local velocity and pressure. Ultrasonication: The deaggregation of a suspension by the application of high-energy, high frequency sound. Unstable suspension: A suspension that has significant aggregation as measured over a relevant time frame (e.g., the time it take the suspension to travel through a pipe). Viscoplastic: A property in which a material behaves like a solid when the stress is below some critical value, the yield stress, but flows like a viscous liquid when this value is exceeded. Viscosity [absolute viscosity]: The ratio of the shear stress to the shear rate under simple steady shear. For a Newtonian liquid, the viscosity is independent of the shear rate. For non-Newtonian liquids, the viscosity varies with the shear rate. Viscosity ratio [relative viscosity]: Ratio of the viscosity of a suspension to the viscosity of the suspending medium. xxiii Viscous: The tendency of a liquid to resist flow as a result of internal friction. During viscous flow, mechanical energy is dissipated as heat and the stress developed depends on the rate of deformation. Viscous sub-layer: In turbulent flow, the viscous sub-layer includes the laminar sub- layer and the portion of the portion of the transition zone between the wall and the boundary layer. Yield stress: The critical shear stress value, below which an ideal viscoplastic material will not flow. Once the yield stress is exceeded the material flows like a liquid. Zero shear rate viscosity: The low shear rate limiting value of viscosity. Associated with the low shear rate Newtonian region in some shear thinning materials. Zeta potential: The voltage potential drop across the mobile part of the electrical double layer. Used as a surrogate for surface charge. xxiv PREFACE During the brilliant development of theoretical hydrodynamics in the second half of the last century, contact with reality and with practical engineering problems was more and more lost. This was due to the fact that in this so-called classical hydrodynamics everything was sacrificed to logical construction and no results could be obtained unless they could be deduced from the basic equations. Yet, in order to overcome the mathematical difficulties, these equations were simplified in a manner which often was not permissible even as an approximation. The hydraulics, on the other hand, which tried to answer the multitudinous problems of practice, disintegrated into a collection of unrelated problems. Each individual question was solved by assuming a formula containing some undetermined coefficients and then determining these by experiments. Each problem was treated as a separate case and there was lacking an underlying theory by which the various problems could be correlated. L. Prandtl and O. Tietjens, Fundamentals of Hydro- and Aeromechanics (1932 CE) This summary of the divided state of Newtonian fluid mechanics in the last half of the nineteenth century could be used to describe the divided state of non-Newtonian fluid mechanics in the last half of the twentieth century. In the mineral processing industry, engineers design equipment and systems to handle thousands of tonnes per hour of nonNewtonian “slurry” using, at best, decades old theory or, at worst, ignoring non-Newtonian fluid dynamics all together. The hundreds of papers on “rheology” or “non-Newtonian fluids” published annually are little used by (and often of little use to) practicing engineers. Fortunately, in the first half of the twentieth century, researchers on Newtonian fluids made a concerted effort to bring theoretical and practical back together, there were rapid advances in both. To this day, practicing engineers routinely use some of the results, such as the Colebrook equation. It is with this successful synergy in mind that the research on nonNewtonian fluids described in this Thesis was undertaken. xxv ACKNOWLEDGEMENTS I would like to acknowledge the generous support of Pipeline Systems Incorporated of Concord, California, U.S.A. during the research period for Doctoral Thesis. They showed their support by giving me direct access to many of the world’s leading hydrotransport specialists, 25 years of laboratory results, and data on operating slurry pipelines; by allowing me to focus primarily on projects in my area of research); by opening an office in Vancouver to shorten my commute; by allowing me the flexible work schedule needed to complete my research; and by providing financial assistance that allowed me to present papers around the world (i.e., Canada, U.S.A., Chile, Australia, Italy, Turkey, South Africa, Botswana, and China). I would also like to acknowledge the financial support of Fluor and the National Science and Engineering Research Council of Canada (NSERC) in the first two years of my research while I was narrowing in my research focus. Finally, I would like to thank my supervisor, Dr. Bernhard Klein, for his guidance, advice, and patience. xxvi DEDICATION To those who believe that the role of the theoretical is to explain the practical. xxvii CHAPTER 1: INTRODUCTION First of all I will explain the method of preparing the ore, for since Nature usually creates metals in an impure state, mixed with earth, stones, and “solidified juices”[salts], it is necessary to separate most of these impurities from the ore as far as can be before they are smelted…. Georgius Agricola, De Re Metallica (1556 CE) 1.1. Mineral Processing is Slurry Processing While the technology has changed dramatically since Agricola was writing, in its most basic sense, mineral processing is still the art of breaking ore into small particles (or finding ones that have broken down naturally), separating out the valuable minerals and disposing of the remainder. Another thing that has not changed since the sixteenth century is that most mineral processing operations are carried out on mixtures of solids (minerals) and a carrier liquid (usually water or an aqueous solution) referred to as “slurry”. Unit operations carried out on slurry include: grinding, hydro-cyclone fractionating, wet screening, centrifuging, jigging, leaching, flotation, dense media separating, storing in agitated tanks, filtering of concentrate, open channel “launders”, thickening, heat exchangers, and disposal of tailings in an impoundment. The successful design and operation of a mineral processing facility fundamentally depends on understanding the flow behaviour of slurry. Specifically, it depends on understanding how the slurry will flow through and between operating units. 1.2. The Flow Behaviour of Slurry When slurries are comprised of very coarse particles (gravel), particle inertial effects dominate, the fluid and solid phase largely retain their separate identities, and the increase in the mixture’s apparent viscosity over that of the carrier liquid is usually quite small (Wasp et al. 1977). When flowing in a pipe, the particles settle quickly and tend to slide along the bottom of the pipe, moved along by the viscous drag of the liquid flowing above the bed of solids (Wilson et al. 2006). They are, therefore, refered to as “settling-slurries”. The solids 1 concentration varies significantly across the vertical axis of the pipe, so the flow is “heterogeneous”. When slurries are comprised of very fine particles (clay), the solid particles will settle very slowly and thermal agitation (Brownian motion) may be sufficient to keep the particles suspended indefinitely. The presence of these solids can have a significant effect on the mixture properties, usually resulting in a sharp increase in the mixture viscosity compared to that of the carrier liquid. Often these mixtures exhibit a non-Newtonian consistency (Wasp et al. 1977). When flowing in a pipe, the particles tend to be evenly distributed across the vertical axis of the pipe, even in laminar flow, so the flow is “homogeneous”. These “nonsettling” slurries can be analyzed as a “continuum” or as a continuous fluid with the average properties of the mixture. The flow behaviour of homogeneous slurry can be strongly affected by the presence of chemicals (e.g., dispersants, flocculants, reagents, dissolved species, etc.) in the carrier liquid. Most slurry flow behaviour may be considered to fall between two extremes. The particles in slurries of interest to mineral process engineers are generally of an intermediate size (silt and sand) and the behaviour is intermediate between the extremes above. When these “moderately-settling” slurries flow in a pipe, the particles tend to settle out in laminar flow, but are more-or-less evenly distributed across the vertical axis of the pipe in turbulent flow. The turbulent flow of these slurries is referred to as “pseudohomogeneous”. Almost all industrial mineral slurries have a wide range of particle sizes, often spanning several orders of magnitude. Furthermore, the particle size, material, and concentration will usually change as they move through the plant or even within a given operating unit (e.g., in a grinding mill or a thickener). The flow behaviour tends to become increasingly homogeneous as the particles get smaller and the volume concentration gets higher. Throughout the processing stages, a variety of chemicals are added that can dramatically affect the flow behaviour of suspensions containing fine particles. 2 1.3. Pipeline Hydraulic Design The most common slurry handling “unit” in any mineral processing plant is a pipeline. A given plant will have hundreds of pipelines of various sizes. Most of the operating units mentioned in Section 1 are fed by and/or has its output carried away by a slurry pipeline. Short slurry pipelines have been used for many years in dredging operations, in mineral processing facilities and for tailings disposal (Durand and Condolios 1952). In 1957 the 174 km long, 250 mm diameter Consolidation Coal pipeline in Ohio and the 116 km long, 150 mm diameter American Gilsonite pipeline in Colorado/Utah started operation. Between 1957 and 1977, a dozen pipelines over 25 km in length were built and operated and a variety of materials were being transported including: limestone, phosphate, copper concentrate and zinc concentrate. The 85 km, 225 mm diameter Savage River iron concentrate pipeline in Tasmania (1967) proved the viability of using pipelines to transport abrasive materials. Other major systems include the 439 km long, 450 mm diameter Black Mesa coal pipeline in Arizona (1972) and the 395 km long, 500 mm diameter Samarco iron concentrate pipeline in Brazil (1977) (Wasp et al. 1977) While the use of large safety factors, “rules-of-thumb” and “trial and error” were adequate for designing a 100-metre dredge pipeline or a 1000-metre tailings pipeline; they were unsuitable for a 100,000-metre transport pipeline. The development of long distance transport pipelines required a significant advancement in slurry transport technology. Extensive and expensive test programs were undertaken to prove the design methods, material behaviour and the reliability of the mechanical components. This included determination of such things as: • Optimum grind size • Optimum solids content • Transition and deposition velocities • Hydraulic gradients at various slurry velocities • Practicality of, and methodology for, restarting a pipeline full of slurry 3 1.3.1 Slurry characterization The first step in the design of a slurry pipeline is characterization of the material. A number of standard tests are carried out: • Particle size distribution measurements (by sieve and/or laser) • Dry solids density measurements (by air pycnometer or Le Chatelier flask) • Settling tests to determine the depth of the settled bed • Penetration tests to determine the “hardness” of the settled bed to predict the practicality of re-suspending the solids after a pipeline shutdown • Slide testing (i.e., measuring the angle at which a settled bed in a pipe will start to slide) to determine the maximum slope for long sections of pipe. • Conductivity testing to determine the corrosion potential of bare steel pipes • Abrasion testing to determine the wear potential of moving parts exposed to the slurry (e.g., pistons and cylinders in piston pumps) 1.3.2 Estimation of pressure gradient The critical design parameter for any pipeline is the pressure gradient, or the pressure loss per unit length. This allows the pipe, pumps and motors to be selected and optimized. It also allows the optimum transport solids concentration range to be determined. The original long distance pipelines were designed based on pressure gradients found in loop tests. 1.3.2.1 Loop test-based design A basic loop test facility is: a slurry hopper, a pump, a method for flow measurement, and a device for measuring differential pressure over a known length of straight pipe. Slurry is pumped through the loop at a range of flow rates and the flow and pressure drop are measured, giving the pressure gradient vs. flow rate curve. Two or more test sections with different diameters, or pipe materials (i.e., roughness) may be included to allow more information to be gathered from each test run and correction for wall slip, if necessary. Additional instrumentation may be included to gather information such as the solids content (a movable gamma density gauge) or the distribution of particle sizes (sample ports) across 4 the vertical axis of the pipe. The steps for determining the pressure gradient for homogeneous slurry using a loop test facility are: 1. Estimate the pipe size required to handle the tonnage of slurry required for the project 2. Build a loop test facility with a test sections in the estimated pipe size range 3. Obtain a large representative sample of the solids material to be transported 4. Obtain a large representative sample of the suspending liquid 5. Estimate the solids content range of the slurry to be transported 6. Prepare a slurry sample above the estimated solids content range 7. Measure the pressure gradient (ΔP/L) over a wide range of flow rates (Q) in the loop test facility 8. Dilute the slurry with carrier fluid two (or more) times to obtain pipe flow curves at different solids contents 9. Interpolate between the results to obtain hydraulic gradient vs. flow curves for different diameter pipe diameters and solids contents Degradation of the solids may be measured by taking samples before and after testing. The tests may be run for a long period to see if coarse solids tend to build up on the bottom of the pipe over time (“sanding”), although this is ineffective if the supply of coarse particles is limited. 1.3.2.2 Rheology-based design By the 1980’s, long distance slurry transportation was a mature technology. The results of previous test programs and data from operating systems had been correlated with laboratory scale testing. It became practical to design conventional hydrotransport systems using “rheology-based design” alone. Rheology-based design uses the flow behaviour (“rheology”) of a material measured at the laboratory scale to predict the flow behaviour at an industrial scale, often in a very different configuration. As a result, it was no longer necessary to do loop testing to design most pipelines. The basic steps for the pressure gradient estimation for homogeneous slurry based on the slurry rheology are: 5 1. Obtain a standard rheometer (capillary or concentric cylinder) 2. Obtain a small representative sample of the solids material to be transported 3. Obtain a small representative sample of the suspending liquid 4. Estimate the solids content range of the slurry to be transported 5. Prepare three slurry samples that span the estimated solids content range 6. Measure the shear stress (τ) vs. shear rate ( γ& ) behaviour in rheometer 7. Fit the shear stress-shear rate data for each sample to a constitutive equation, such as Bingham plastic (Bingham 1916). τ = τ B + η Bγ& 8. Fit a semi-empirical physical relationship model to the rheological constants at the different solids contents. For a Bingham plastic, the following physical relationships between yield stress (τB), the plastic viscosity (ηB) and the solids volume concentration (φ) may be used: τ B = C0φ C and η B = μ exp(C2φ ) 1 Where μ is the Newtonian viscosity of the carrier fluid. The values of C0, C1, and C2 are determined from the Bingham fits to the data. 9. Use the physical relationships to estimate the rheological behaviour at any volume concentration in the test range. τ ≈ C0φ C + μ exp(C2φ )γ& 1 10. Use the solids density (ρs) and carrier liquid density (ρl) to calculate the slurry density (ρ). ρ = φρs + (1 − φ )ρl 11. Use the Buckingham (1922) design equation to determine the laminar flow velocity (VL) as a function of the pressure gradient (ΔP/L): 6 VL = Dτ w ⎛ 4 1 4⎞ ⎜1 − Z + Z ⎟ 8η B ⎝ 3 3 ⎠ Where: Z = τB ⎛ ΔP ⎞⎛ D ⎞ and τ w = ⎜ ⎟⎜ ⎟ τw ⎝ L ⎠⎝ 4 ⎠ 12. Use a design equation (e.g., Wilson-Thomas 1985) to determine the turbulent flow velocity (VT) as a function of the pressure gradient: ⎛ ρDu * ⎞ ⎛1− Z ⎞ ⎟⎟ + 2.5u * ln⎜ VT = 2.5u * ln⎜⎜ ⎟ + u * Z (14.1 + 1.25Z ) ⎝1+ Z ⎠ ⎝ ηB ⎠ Where: u* = τw ρ 13. Use a spreadsheet to generate the pressure gradient vs. flow curves for different diameter pipes and different solids contents. The flow regime is assumed to be whichever gives the lowest velocity for a given pressure gradient 1.3.3 Advantages of rheology-based design At first glance, the use of loop test results would appear to be superior to rheology-based design. The pressure gradient curves are measured directly, rather than being estimated using semi-empirical flow models. Secondary effects, such as wall slip and bed build-up, are automatically accounted for, at least in theory. Additional information, such as particle degradation and pump de-rating factors can be obtained in a straightforward manner. However, there are a number of advantages to rheology-based design. 1.3.3.1 Test pipe size Determining the pipe size of the loop’s test section is problematic because the final pipe size is not known in the design phase. In fact, selecting the pipe size is one of the main objectives of the hydraulic design. The designer is forced to guess at the size unless there is some other basis (e.g., the pipeline is existing). The probability of having the correct test pipe is increased if two or three sizes are used. However, there is a practical limit to how wide a size range the pipes can have. 7 Assume a loop test facility is designed with 100 and 200 mm pipe test sections. A typical velocity range for industrial slurry pipelines is 1.0 to 3.0 m/s. To get the design velocity in the small pipe, the velocity range in the large pipe would be 0.25 to 0.75 m/s and, if the slurry settles, it will likely “sand out” (i.e., build up a bed of solids and plug the pipe). To get 1.0 to 3.0 m/s in the large pipe, the velocity in the small pipe would be 4.0 to 12.0 m/s. The pump would need to deliver the slurry over the entire flow range from 1.0 to 12 m/s in the smaller pipe. 1.3.3.2 Samples Large diameter loop test facilities require large quantities of sample. However, there is rarely a large “representative” sample of the solids available in the early design phase. Since the mine does not exist when the slurry pipelines are being designed for a new facility, the samples come from bench scale or pilot plant testing. Furthermore, the samples available may not be “representative” since the process (e.g., grind size, flotation reagents, thickener flocculants, etc.) will still be in development. There may be several processes under consideration, each with different particle sizes and carrier fluid properties. With fine (clay and silt) slurries, the impact of dissolved species in the carrier fluid may be dramatic. This may be observed in Fig. 1.1, which shows the pipe flow curves for a constant solids content kaolin clay slurry with varying concentrations of coagulant (calcium salt) and dispersant TSPP (tetra-sodium pyrophosphate) (data from Litzenberger 2004). Typically multiple samples will need to be assessed. A full scale loop facility with a total length of 100 m of 300 mm diameter pipe would require about 8,000 litres of slurry. To do twelve loop tests, 100,000 litres of slurry would need to be obtained and disposed of. In comparison, twelve tests in a concentric cylinder rheometer take less than 10 litres of sample regardless of the pipe size. 8 Clay Slurry (Cv = 14%v) in 25.8 mm Pipe Pressure Gradient (kPa/m) 6 5 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 Velocity (m/s) Water 0.1%CaCl2; 0%TSPP 0.1%CaCl2; 0.1%TSPP 0.1%CaCl2; 0.27%TSPP +10g CaCl2 0.1%CaCl2; 0.13%TSPP +5g CaCl2 +15g CaCl2 Figure 1.1: Impact of salt and dispersant on kaolin clay slurry pipe flow curves (Data from Litzenberger 2004) 1.3.3.3 Instrumentation The loop test facility needs to be instrumented in order to produce flow curve data. The main instruments are to measure the slurry flow and the pressure differential. Flow meters are calibrated with a Newtonian fluid (water), which can give significant errors with nonNewtonian fluids. Magnetic flow meters will give incorrect readings if the slurry contains magnetite. Pressure transmitters designed to measure small pressure differences tend to be fragile and they may be damaged if subjected to high pressures. More robust pressure transmitters may not be sensitive enough to measure small pressure differentials. Instruments do not have perfect accuracy and the errors tend to be a percentage of the fullscale readings. The pressure gradient of a fluid flowing at a low velocity may only be a few percent of the full-scale reading. This will be of the same order of magnitude as the instrument error. Methods for improving the accuracy include using higher quality instruments and using different instruments for different test ranges. Both options increase the cost and the improved accuracy may not be justifiable for a single use system. 9 1.3.3.4 Extrapolating results These problems may be avoided by using research grade loop-test facilities, such as those that are available at many universities. This eliminates the cost of building a facility, although the cost of transporting tonnes of sample to the test site and disposing of it afterwards can be substantial. These loop test facilities usually have high quality instruments and sophisticated data collection systems. However, existing research facilities will have a limited number of pipe sizes available. As a result, it may be necessary to extrapolate from the results obtained in a 50 mm test pipe to the actual pipe diameter, which may be 300 mm or larger. Clearly, extrapolating between pipe sizes negates the main advantage of loop testing: the ability to measure the pressure gradients directly. Extrapolation requires the use of a theory to estimate the impact of pipe size on the pressure gradient. In effect, the loop test facility is used as a large capillary rheometer for rheology-based design. 1.3.3.5 Particle degradation To limit the amount of sample used, common practice is to dilute the slurry in the loop. As a result, the slurry makes a large number of circuits through the pump and pipe. If centrifugal pumps are used, the high shear stresses near the impeller may cause attrition of the particles. Very fine particles are formed by the breakdown of the large particles, causing the physical properties of the slurry to change between tests and the results of the loop test will likely not be representative. 1.3.3.6 Other equipment While the pressure gradient in a pipeline is important, it is only a small part of the design of a slurry transport system. A flow loop test gives a single piece of information; an understanding of the slurry consistency has a much wider applicability. • The slurry storage tanks have agitators that must be sized (i.e., diameter, speed, power, etc.). Chokes (ceramic orifices) need to be sized. Floor drains need to be sized and sloped. All of these are affected by the slurry consistency and cannot be designed directly from loop test data. 10 • The head and efficiency de-rating of centrifugal slurry pumps need to be estimated. This could be tested directly in a loop test. However, unless the pump used in the test is the same type and size that will be used in the final pipeline, some scaling theory must be used. • In addition to piping systems, the performance of most mineral processing units is affected by the slurry consistency. This includes: grinding, thickening, cyclone sizing, screening, heating, mixing, transporting by launders (open-channel chutes) and dense media separating. 1.4. Application of Rheology in Mineral Processing Since “slurry rheology” affects the behaviour of most unit operations it would be reasonable to assume that rheology-based-design plays a key role in mineral process engineering. Is this actually the case? Consider the book “Mineral Processing Technology, 5th Ed.” (Wills 1992). Over 400 pages of this “introduction to the practical aspects of ore treatment and mineral recovery” focus on the processing of mineral slurry but only one reference to applied rheology was found (page 454). This was one sub-section related to dense medium separation and how, when the medium is a concentrated suspension, the presence of a yield stress may prevent the separation of small particles. Methods to reduce the yield stress (“dilution” and “agitation”) or increase the shearing forces on the particles (“substituting centrifugal force for gravity”) are discussed qualitatively. Consider the proceeding of the “Mineral Processing Plant Design, Practice, and Control” conference (SME, Vancouver, Canada 2002). The proceedings contain 138 papers that “cover all aspects of plant design” written by “the mining industry’s leading engineers, consultants, and operators”. It was intended to be “a new standard text for university-level instruction” and for “practical, quick reference for engineers.” No direct mention of rheology or non-Newtonian behaviour is made until page 1207, where the problems with clay “slimes” in flotation cells is discussed (solution: scalp them off, dilute and treat separately). There are various qualitative references to things that are clearly rheologyrelated problems (e.g. moisture making the ore feeding grinding rolls “sticky”). The only 11 quantitative uses of rheology are in two papers: “Selection and Sizing of Slurry Pumps” and “Slurry Pipeline Transportation”. (Note: The SME apparently intended to have a chapter on “rheology in the mineral processing industry” [not completed], although “discussing” rheology is not the same as “using” rheology.) Consider the proceedings of the XXIII International Mineral Processing Congress (Istanbul, Turkey, 2006). This is the most important conference in the industry and the proceedings contain 453 short papers, ranging from general to technical, on a variety of topics. Only five papers used rheology to explain or solve an engineering issue (e.g., estimating the pressure loss in a pipeline). Of these, four of the papers related to slurry pumping systems. Consider the proceedings of the International Laterite Nickel Symposium (TMS, Charlotte, USA 2004). Nickel and cobalt are found in limonite and smectite clay deposits (Klein and Hallbom 2002). This “nickel laterite” is processed at high temperatures and pressures (~250ºC and ~40 bar). There is a significant economic incentive in pushing the solids content as high as possible: smaller equipment (titanium autoclaves and heat recovery system) and less acidic wastewater to be neutralized and replaced. In some of the papers, it was recognized that “rheology comes into play in a big way” (Curlook 2004) and that this is why the slurry could only be handled at “about 40% solids” by weight. However, only one of 52 papers (excluding Hallbom and Klein 2004) focused on slurry rheology and it just discussed measurement of the yield stress. There have, of course, been papers published that relate to slurry rheology and most mineral process engineers are aware that “rheology” affects the way their process units operate. However, based on these industrial conferences proceedings, it is reasonable to conclude that rheology is not commonly applied in the mineral processing industry. Non-Newtonian behaviour of slurry is, for the most part ignored, allowed for by using “safety factors”, avoided by removing “slimes”, or eliminated using dilution. There are situations where nonNewtonian flow behaviour needs to be considered and the “dilution solution” cannot be used: thickened slurry. 12 1.5. Thickened Slurry Slurry may be considered to be a transitional material spanning the range between liquid (at very low volume concentrations: “turbid water”) and solid (at very high volume concentrations: “cake” or “soil”). The flow behaviour of slurry also transitions between liquid-like to solid-like as the volume concentration increases. Mineral processing has been, and often still is, carried out on rather dilute “conventional” slurry. The problems associated with non-Newtonian flow behaviour of fine slurries can always be reduced or eliminated by adding additional carrier fluid. This allows the designer or operator to work in the wellunderstood Newtonian (or pseudo-Newtonian) flow regime. However, there are many instances where this “dilution solution” is not practical or desirable. Over the last few decades, there has been a gradual progression towards processing and transporting “thickened slurry” (see Fig. 1.2). Figure 1.2: Thickened mineral slurry (paste) Thickened slurries are suspensions where the solids concentration is high enough that the non-Newtonian properties significantly affect the flow behaviour on the macro-scale, but not so high that it becomes brittle or crumbly (i.e., cake). Thickened slurries are primarily composed of fine particles (silt and clay) at relatively high solids concentrations. Thickened slurries may contain coarse particles, but they do not significantly affect the flow behaviour. The dominant feature of thickened slurry is its yield stress. If the yield stress is greater than about 100 Pa and it behaves as a semi-solid with a soft and malleable consistency, thickened slurry is often referred to as “paste”. 13 Several areas of development are moving the mining industry towards the increased use of thickened slurry systems. The more important are discussed below. 1.5.1 Thickened tailings One important slurry pipeline application is the transfer of mine tailings (waste) to surface impoundments, referred to as “tailings ponds” (see Jewell and Fourie 2006 as a general reference). Historically, tailings have been transported and deposited at low solids concentrations. When tailings are discharged into an impoundment as conventional slurry, the particles tend to segregate. The coarse material settles out near the discharge points forming a beach. The fine material is washed into the pond, forming a mud bed. The mud bed has little strength when wet and does not compact well. In the event of a dam breach, the loosely packed tailings can flow out of the impoundment and cause a dangerous mudflow. The mud does not drain well, making reclamation of the area difficult at the end of the mine life. When tailings are thickened, the coarse and fine particles stay intermingled when they flow into the impoundment. These un-segregated tailings self-compact to a higher solids content than the fine material on its own. The compacted un-segregated tailings have a greater shear strength than fine tailings alone, so they are less prone to flow in the event of a dam failure. Furthermore, overtopping of the dam by the decant pond and causing a washout is one of the main risks for tailings impoundments. The smaller volume of stored water reduces this risk. Thickened tailings disposal was first implemented in 1973 at the Kidd Creek mine in Ontario. Since then, more than a dozen thickened tailings systems have been installed, either as new systems or by converting existing conventional systems. Thickened slurry flowing into an impoundment will form an overall slope of 1 to 3% compared to roughly 0.2% to 0.5% for conventional slurry. If the thickened slurry is poured in thin layers and allowed to desiccate between pours, slopes of 5 to 7% may be obtained. Even higher slopes are possible if the material is mechanically compacted between pours. Thickened slurry may be poured from a central discharge point to form “mounds” hundreds of meters across with only a small perimeter berm, significantly reducing the cost of tailings impoundments in areas with a “flat” topography. 14 In 1985, Alcoa developed the “dry stacking” method for red mud (bauxite residue) disposal, where thin layers of thickened slurry are poured into paddocks and allowed to desiccate before the next layer is poured. The resulting stacks have a high shear strength and high density. This allows stacking to a greater height and decreases the final volume per ton of solids. The result is a smaller impoundment footprint for a given capacity. The ability of the stack to support compressive loads (equipment) makes earlier reclamation practical. The operating cost and environmental impact of pipelining thickened slurry to the “dry stack” is significantly less than hauling filter cake to the impoundment by truck as is done at, for example, Alunorte in the Brazilian Amazon. 1.5.2 Paste backfill Underground mining results in the formation of large void spaces that need to be backfilled to stabilize the surrounding ground. One potential source of backfill material is the mine’s tailings, which is readily available and also reduces the amount of tailings to be deposited on the surface (see Potvin et al. 2005 as a general reference). If the tailings are coarse (sand), they may often be used as a free-draining hydraulic fill. However, if the tailings are fine (clay or silt) they will not drain well and will result in a weak fill. A binder (cementing agent) is often added to increase the strength of the fine fill material forming a weak concrete similar to grout. Binder is a major operating cost for a cemented paste backfill system, so considerable effort is made to reduce the binder dosage required. Reducing the slurry water content reduces the amount of binder required to meet the required cured fill strength. The resulting “paste” has a yield stress of several hundred Pascal, compared to approximately one Pascal for conventional slurry. An economical way of transferring the paste to the stopes is by slurry pipeline. While typical paste backfill pipelines are small diameter (100 to 200 mm) and rarely more than a few kilometres long, they are expensive to install if a significant portion of the route requires drilling boreholes through hard rock or developing access ways. Furthermore, the overall pipeline layout constantly changes as the filling operations move from stope to stope. In deep underground mines, the paste systems flow by gravity or have pumps with heads that are only a small portion of the static head. The Kidd Creek paste backfill system will 15 eventually be 3000 m deep and has no paste pumps. These systems are “rheology controlled”, the flow rate is adjusted by increasing or decreasing the slurry consistency. 1.5.3 New equipment Developments in process equipment have made it practical to produce and transport thickened non-settling slurry in large volumes. Some of the more important developments include: • Ultra-fine grinding mills (e.g., stirred mills) allow ore to be ground to smaller sizes than was previously practical, resulting in the production of finer slurries that exhibit strong non-Newtonian effects at relatively low solids contents. • Deep bed (or high compression) thickeners allow high yield stress thickened slurry to be produced more economically than with filters, particularly at high tonnage rates. The ability to transfer the high yield stress underflow by pipeline makes the overall installation more cost effective. • Positive displacement pumps allow thickened slurries to be transported without the significant losses in efficiency observed with centrifugal pumps. Piston diaphragm pumps, can reliably handle slurries with yield stresses over 100 Pa and develop discharge pressures of 25 MPa or higher. Modified concrete pumps can handle very high consistency pastes with yield stresses over 400 Pa, including those that contain large particles (gravel) and develop operating pressures of 12 MPa or higher. 1.5.4 New ore bodies The high-pressure acid leach (HPAL) process has made the recovery of nickel and cobalt from “nickel laterite” economically viable. Nickel laterite is composed of clay and silt sized non-spherical particles and the ore starts to behave as thickened slurry at relatively low solids contents, typically 20% to 30%w. Two nickel laterite mines currently in development (in Madagascar and New Guinea) are located in remote highland regions unsuitable for a complex hydrometallurgical plant. The HPAL plants will be constructed near the port and the ore transported over 100 km from the mine. Both mines are planning to use a pipeline for ore transport. These are both large tonnage systems, requiring large amounts of water to be obtained and either treated for disposal or pumped back to the mine. The ability to 16 increase the slurry solids content from, say, 25% to 40%w would cut the water usage in half and significantly reduce the volume being transferred. Bauxite is another fine laterite ore that needs to be transported at low solids content (~40%w) in a conventional pipeline. The bauxite pipelines under consideration have high tonnage rates, typically 4M to 12M tonnes per year. Increasing the solids content would reduce the amount of water required for transport and the volume being transferred. Hard rock mines are exploiting increasingly disseminated ore bodies. Processing of these ores requires grinding to a very fine particle size (often less than 10 μm) to optimize mineral recovery and concentrate grade. These fine particles can be transported as conventional slurry at low solids contents, but thickened slurry transport would save large quantities of water. High concentration slurries of fine particles tend to have a non-Newtonian consistency. 1.5.5 Availability of water Many important mining areas are in desert and semi-desert regions, such as northern Chile, western Peru, southwest United States, most of Australia, and southern Africa. Obtaining large volumes of water in these areas is problematic. Even when large sources of fresh water exist, there are usually other stakeholders, such as farmers and ranchers, who have claims on the resource. Ground water supplies have a limited refreshment rate and may be of poor quality (e.g., saline). Rainfall tends to be seasonal, resulting in parts of the year when river flows are small or non-existent. Thickened tailings impoundments do not require the large water ponds associated with conventional tailings impoundments. The total excess water entering the impoundment is small and there is no need to decant segregated fine material. As a result, only a small pond to catch bleed water and rainfall is required. Since evaporation from the ponds is a major source of water loss in arid regions, a smaller pond it has been suggested that the use of thickened tailings can reduce net water usage per tonne of processed ore. In arid regions where water availability is the limiting operating input, northern Chile being a notable example, this would effectively increase the viable mine production rate. 17 1.5.6 Design of thickened slurry systems Conventional slurry lines hundreds of kilometres long are currently designed using rheology-based design. The ability to use rheology-based design reduces the cost and time requirements for the testing program by roughly 70 to 90%. The method even works well with new materials. The first long distance bauxite pipeline (230 km), which recently started up in Brazil, was designed without loop testing for the pressure gradient. Furthermore, once an organization has built up a database of test data, conceptual studies may be undertaken on common materials without any testing at all. While the additional time and cost related to loop testing is always a concern, it is during the conceptual and feasibility study stage that it is especially acute. The design of any process starts out with an evaluation of several potential options. For tailings transport, this may include: trucking, conveyors and a pipeline. With a conventional slurry pipeline, the entire conceptual study will cost roughly $50,000. If the option of a thickened slurry pipeline is to be considered, and loop tests are required, the cost of the study can easily escalate to over $250,000. This extra cost acts as a serious barrier for the technology since a thickened tailings or paste system will not be installed if it is not considered in the conceptual stage. Despite the success of rheology-based design for conventional slurry systems, loop test based design is still widely used for thickened slurry systems. There are several potential reasons why adapting the rheology-based design method to thickened slurry has been unsuccessful. However, the major impediment appears to be the rheological models used to describe the consistency of thickened slurry. 1.5.7 Problems with rheological models The consistency of thickened slurries may often be modelled reasonably well using several models. Darby (1984) tested fine lignite-methanol slurry (47%w) in a concentric cylinder rheometer (0.9 to 150 s-1) and a capillary rheometer (10 to 4000 s-1). The data was fit to the constitutive equations for three different rheological models: power law, Bingham plastic, and Casson. Each of the models was fit over two shear rate ranges. The resulting constants are shown in Table 1.1 (converted from the original CGS units). τ = Kγ& n Power law [1.1] 18 τ = τ B + η Bγ& Bingham plastic [1.2] τ = τ C + ηCγ& Casson [1.3] Where τ is the measured shear stress, γ& is the applied shear rate, and K, n, τB, ηB, τC and ηC are the rheological parameters of the three models. Table 1.1: Rheological parameters for lignite in methanol Shear Rate Range 0.9< γ& <150 K (Pa-sn) 19.7 n (-) 0.154 0.9< γ& <4000 16.8 0.214 R2 τB ηB 0.98 (Pa) 24.7 (Pa-s) 0.155 0.95 29.6 0.0317 R2 τC ηC R2 0.90 (Pa) 20.8 (Pa-s) 0.0341 0.94 0.97 23.3 0.0137 0.98 Each of the models and shear rate ranges indicate that the material is thickened slurry of moderate consistency and give a reasonable fit to the data (R2 ≈ 1). However, the rheological constants are not consistent between models nor are they consistent between shear rate ranges for a given model. Consider the yield stress, which is the applied shear stress below which the material will not flow. The Bingham and Casson models indicate yield stresses ranging from 21 to 30 Pa, while the power law model assumes no yield stress at all. Even for a single plastic model, Bingham or Casson, the inferred yield stress varies with the shear rate range of the measurement. The three constitutive equations, with the rheological constants given in Table 1.1, can be used to predict pipe flow curves. Figure 1.3 shows the theoretical laminar and turbulent flow curves for the lignite-methanol slurry flowing in a 200 mm ID smooth pipe. The WilsonThomas (1985) model has been used to estimate the turbulent flow pressure gradients. (Note: The methodology for developing the laminar pipe flow curves is shown in the body of this Thesis.) The density of the slurry was not mentioned (in Darby 1984) but 1000 kg/m3 was assumed. As can be seen in Fig. 1.3, the choice of consistency model and range of shear rate measurement significantly affects the predicted pressure gradient. For example, assume this slurry was going to be pumped at 4.5 m/s. The predicted pressure-gradients vary from 0.9 19 kPa/m to 1.8 kPa/m. Three of the cases predict laminar flow and three of the cases predict turbulent flow. However, in an operating pipeline there would be an actual pressure gradient and the flow would be either laminar or turbulent. 4.0 200 mm ID smooth pipe Pressure Gradient (kPa/m) 3.5 3.0 Bingham Low 2.5 Bingham All Casson Low 2.0 Casson All Power Low 1.5 Power All 1.0 0.5 0.0 0 1 2 3 4 5 6 7 8 Bulk Velocity (m/s) Figure 1.3: Theoretical flow curves for lignite-methanol slurry in a 200 mm pipe It is important to recognise that the flow curves in Fig. 1.3 are all based on data from a single sample. All of the flow curves are based on the same assumptions (i.e., homogeneous time-independent slurry, same density, smooth pipe wall, no slip, Poiseuille laminar flow model, the Wilson and Thomas (1985) turbulent flow model, etc.) In a real system, there will be differences in the materials. There may also be secondary effects due to: wall slip, particle segregation, wall roughness, density variations, elbows, fittings, a settled bed, pulsating flow (PD pump), etc. Given the spread in the flow curves even in this ideal case, it is easy to see why loop testing is used for thickened slurry systems design. Despite the wide spread in the estimated pipe flow curves, there are several observations that may be made about Table 1.1 and Figure 1.3. • They are inconsistent with the assertion that the choice of rheological model depends “largely on the preference of the designer” (Wasp et al. 1977). The different consistency models give significantly different results. Presumably, one consistency 20 model (although not necessarily one of the models above) must give the closest approximation to the actual behaviour. • The shear rate range of the data significantly affects the predicted pressure gradient. This justifies the common warning that data should be fit in the shear rate range of the process. However, the impact of changing the shear rate range is not the same for all of the models. • The low range and full range flow curves of the Casson model are much closer together than the Bingham model in both the laminar and turbulent flow regimes, and much closer together than the power law model in the turbulent regime. The model parameters are less shear rate dependent and, therefore, closer to being material “constants” rather than local curve fitting parameters. 1.5.8 Virtual rheology Researchers in slurry rheology have asked the following question: “Why use a constitutive equation at all if you have a rheogram (i.e., data from a rheometry or a loop test)? The data can be used to calculate the pipe flow behaviour directly.” The answer is that working engineers almost never have rheological data available. Even when tests results are available, they will have been performed on a finite number of small samples, and these may not correspond with whatever material is flowing through a pipe at any given time. Rheology-based-design uses what may be called “virtual rheology” based on physical parameters that can be measured or controlled by the operator. There are many physical parameters that can affect the slurry consistency and these effects become more pronounced as the particle size decreases. For example, the pipe flow behaviour (i.e., the consistency) of the slurry in Fig. 1.1 is strongly dependent on the concentrations of CaCl2 and TSPP. Other parameters that affect the consistency include: temperature, pH, particle size, particle density, particle material(s), slurry density, solids content, and coagulant/dispersant concentration. In Section 3.2.2 a relationship was made between the Bingham plastic parameters (i.e., “yield stress” and “plastic viscosity”) and the solids content. In theory, a similar method may be used to relate the parameters of any consistency model to any physical parameter 21 (e.g., Cerpa et al. 2001). However, the variability of the parameters with shear rate range seen in Table 1.1 is problematic if one is attempting to determine a relationship between the parameter and a fixed physical property (e.g., solids content or pH). It may be noted that, in Table 1.1 the best-fit parameters of the Casson model change less with the shear rate range than the other two models. Based on internal consistency, it would appear that the Casson model gives the best fit of the three models. The question is whether it gives the best fit or whether another model could give an even better fit. 1.6. Thesis Objective The first objective of this Doctoral Thesis is to develop a new rheological model and constitutive equation to describe the consistency of homogeneous non-Newtonian slurry. The second objective is to develop the basic design equations required to apply this new model to slurry pipeline design: laminar and turbulent pressure gradient relationships and the transition velocity. These two objectives, if successfully met, will establish the theoretical basis for rheology-based design of thickened slurry systems. This would decrease the cost and time requirements for the design phase of these systems, removing a major impediment to the advancement of this important technology. The focus of this Thesis will be on mineral slurry, but the resulting model and design equations should be applicable to other non-settling slurries. 1.7. Thesis Outline This Thesis is broken into three main sections: The first section (Chapters 2, 3 and 4) is theoretical: A new rheological model is developed for homogenous mineral slurry. Chapter 2 is a literature review of basic concepts and observations related to the flow behaviour of mineral suspension. This forms the basis for a new rheological model and constitutive equation (“yield plastic”) to describe the flow behaviour of viscoplastic suspensions derived in Chapter 3. In Chapter 4, the yield plastic model is compared with the consistency models commonly used to describe the flow behaviour of viscoplastic suspensions. It is shown that use of the yield plastic model allows the flow behaviour of most shear thinning slurries to be modeled using a single constitutive 22 equation with parameters that may be related to the physical properties of the slurry (e.g., solid content, pH, shear history, etc.). The second section (Chapters 5, 6 and 7) is analytical: The new rheological model is used to derive the standard equations required for pipeline hydraulics engineering. Chapter 5 presents the derivation of the pressure gradient equation for laminar pipe flow based on Buckingham’s method. Chapter 6 presents the derivation of the pressure gradient equation for turbulent pipe flow based on the Wilson-Thomas method. Chapter 7 presents the derivation of a laminar-turbulent transition equation based on a specific energy dissipation ratio model that may be considered to be a modification of the Hedström intersection method. The third section (Chapters 8 and 9) is practical: The pipeline design equations are simplified (analytically and conceptually) and then shown to be consistent with observed behaviour. Chapter 8 presents a new dimensionless group (the “stress number”) and a novel “design curve diagram” that relates the stress number (directly proportional to the pressure gradient) to the plastic Reynolds number (directly proportional to the bulk velocity). The design curve diagram is essentially a non-dimensional flow curve, which makes it conceptually easier than the Moody diagram (i.e., Fanning friction factor vs. Reynolds number). Chapter 9 demonstrates the use of the constitutive equation and the design equations developed in this Thesis to describe the pipe flow of kaolin clay slurry. The predictions made using the yield plastic model are compared to those made using other popular constitutive equation. 23 CHAPTER 2: SLURRY RHEOLOGY – LITERATURE REVIEW We see how quickly through a colander wines will flow, and how, on the other hand, the sluggish olive oil delays: no doubt, because it is wrought of elements more large, or else more crooked and inter-tangled. Lucretius, De Rerum Natura (ca. 55 BCE) 2.1. Introduction This literature review chapter establishes the physical basis for the rheological model and constitutive equation for homogeneous slurry that will be derived in the following Chapter. The inferred behaviour of slurry at the micro-scale (i.e., between the particles and the liquid) and the observed behaviour of slurry at the meso-scale are discussed. Constitutive equations currently used to describe the meso-scale behaviour of non-Newtonian fluids are reviewed, with emphasis on those models that are commonly used for engineering design in the mineral processing industry. The phenomenological “yield plastic” constitutive equation proposed by Hallbom and Klein (2004) is presented and is shown to describe a “family” of constitutive equations that is a superior alternative to the “yield power law” family. Prior art constitutive equations that are similar to the yield plastic model are identified. It is shown that no physical justification was given for the implied flow behaviour, that the usage was limited to describing rheometer flow curves, and that these obscure models have not been developed for engineering design use. 2.2. Slurry – the Micro-Scale This section considers the behaviour of non-settling suspensions of fine particles at a small enough scale that the behaviour of the individual particles is significant. 24 2.2.1 Liquid phase The continuous phase of a suspension is a liquid, often referred to as the “carrier fluid”. In the mineral processing industry the carrier fluid is usually water or an aqueous solution (dilute acid or alkali), although other liquids, such as alcohol (methanol) and hydrocarbons (kerosene), are occasionally used. Carrier fluids used outside the minerals industry include: plasma (blood), water-petroleum emulsions (oil sands slurry) and agricultural oils (chocolate). In a liquid, the molecules are free to change their position but are restricted by cohesive forces such that they maintain a relatively fixed volume. If the volume under consideration is many times the volume of a liquid molecule, the collection may be treated as a “continuum”. For fluid mechanics purposes, the continuum approximation is usually adequate although some liquid behaviour (e.g., Brownian motion) needs to be explained in terms of a collection of moving molecules. The carrier fluid is assumed to be a Newtonian liquid, with a viscosity (μ) that is independent of the shear rate. The viscosity of liquids is essentially independent of the pressure but it decreases with increasing temperature. The density of liquids increases with pressure but may be idealized as incompressible in the pressure ranges normally encountered in hydraulics engineering. The density of liquids changes with temperature but the change is generally small and may be idealized as constant at some average value. These idealizations are not suitable for some analysis (e.g., water hammer and natural convection), but are suitable for hydraulic design. 2.2.2 Solid phase The discontinuous phase of a suspension is a solid, often referred to as the “suspended solids” or the “particles”. The particles in a suspension are often idealized as hard spheres (e.g., Einstein 1906, Krieger 1972). Actual particles are not necessarily “hard” and may be soft and/or flexible. Red blood cells, long chain polymers and cellulose fibres are examples of soft/flexible particles. However, considering the particles as “hard” is usually a good assumption in the mineral processing industry. The assumption that the particles are spheres is more problematic. 25 Mineral particles are rarely spheres and in many cases they are very non-spherical. Clay mineral and mica particles tend to be thin sheets that are better idealized as “plates” or disks. Goethite and asbestos particles are needle-shaped (acicular). Some crushed minerals have an angular or blocky (roughly cubic) form. Many particles have arbitrary shapes. 2.2.3 Dilute non-aggregating slurry When non-aggregating particles are added to a Newtonian liquid, the apparent viscosity of the mixture (η) is found to increase. The apparent viscosity of dilute suspensions is found to be proportional to the viscosity of the carrier liquid (μ) and the viscosity multiplier (η/μ) increases with increasing volume concentration (φ). Einstein (1906) proposed the following viscosity multiplier model for dilute suspensions of identical spheres based on the disruption of flow streamlines. η μ = 1 + 2.5φ + ... Viscosity of dilute suspensions [2.1] The viscosity multiplier is independent of the shear rate, therefore the dilute suspension is Newtonian. Equation [2.1] only holds for suspensions with solids concentrations up to a few percent by volume (Hunter 2002). As φ increases, the particles start to crowd one another and the hydrodynamic interactions cause the viscosity multiplier to increase at a faster rate than would be caused by individual particles. Various suggestions have been made to extend the power series suggested in Eq. [2.1], with the one by Thomas (1965) being widely used for mineral slurries at moderate concentrations. η μ = 1 + 2.5φ + 10.05φ 2 + 0.00273 exp(16.6φ ) 2.2.4 Viscosity of suspensions [2.2] Concentrated non-aggregating slurry The volume concentration of the solid particles can only be increased until they come into physical contact. For identical spheres, the maximum packing density (φm) is in the range 0.524 (simple cubic packing), 0.680 (body centred cubic) and 0.741 (face centered cubic or hexagonal close packed). The random packing density of spheres tends to be roughly 0.64. The maximum packing density of monomodal spherical particles does not depend on the particle size. The maximum packing density will increase when there is a wide particle size distribution. For example, in a box full of 100 mm spheres there will be void spaces between 26 the spheres. Additional 10 mm spheres can be added to the box, filling the spaces between the larger spheres, leaving smaller spaces. Additional 1 mm spheres can fit into these spaces and so on until all of the space is filled. Sphere volume is added without increasing the total volume. The shape of the particles also affects the maximum packing density. Irregular particles will generally have a lower random packing density than spheres, although the actual relationship is ill defined. In practice, φm is typically in the range 0.6 to 0.7 and 0.65 may be used as a first approximation. Once the maximum packing density has been reached, the particles cannot move relative to one another without increasing their overall volume (see Fig. 2.1). Figure 2.1: Dilatency caused by high packing density This volume increases causes “dilatency” or shear-thickening behaviour. If the volume cannot expand (as in a pipeline) the particles are effectively “locked” and the viscosity is assumed to be infinite. This upper limit is allowed for by equations such as that proposed by Krieger and Dougherty (1959) for concentrated suspensions of spheres: η ⎛ φ ⎞ = ⎜⎜1 − ⎟⎟ μ ⎝ φm ⎠ −2.5φ m Concentrated suspensions [2.3] Packing related shear-thickening (dilatency) starts once the loose packing density is reached. Since the particles need to jostle past one another, the apparent viscosity will vary depending on the shear rate, so it is not Newtonian. This will not normally be significant if φ < ~0.8φm so [2.3] may be considered to hold when φ < ~0.5. 27 While there are quantative differences between equations [2.1], [2.2] and [2.3], they are qualitatively the same: the slurry viscosity is always higher than the carrier viscosity and it increases as the volume concentration (φ) increases. 2.2.5 Particle shape If two identical spherical particles are allowed to settle in a quiescent liquid, the terminal fall velocity will be the same. If one particle is a sphere and the other a cube of equal volume and mass, the terminal velocities will differ. The particle fall velocity affects the flow behaviour of a suspension (Wasp et al. 1977). Therefore, it may be concluded that particle shape affects the hydrodynamic behaviour of the particles in a suspension. The fall velocity of a non-spherical particle depends on the orientation of the particle. An extreme case would be a thin disk. If the direction of the fall is normal to the face of the disk, the velocity will be slower than if it were parallel to the face. If a large number of nonspherical particles with different orientations have a lower average fall velocity than spheres of equal volume and mass, this may be interpreted as each of the particles having an effective “hydraulic” volume that is larger by a shape factor (ψ). As a result, the effective hydraulic volume concentration is increased: φh = φψ Hydraulic volume concentration [2.4] The hydraulic shape factor does not directly affect the maximum packing density, so [2.3] can be modified as: η ⎛ φψ = ⎜1 − μ ⎜⎝ φm 2.2.6 ⎞ ⎟⎟ ⎠ −2.5φ m Concentrated suspensions [2.5] Swelling and porous particles If the density of the dry material is known, then the volume of the solids may be deduced from the mass (e.g., 265 g of silica with a density of 2.65 g/mL has a volume of 100 mL). However, this is not necessarily the case as it assumes that the size of the particles in a suspension is identical to the size of the dry particles. Consider a suspension made of activated Carbopol, which can swell up to 1000 times its original volume in water (Lubrizol TDS-222). Carbopol with an unswollen (“dry”) volume 28 concentration of 0.2%v that swells to 300 times its original volume would have an effective volume concentration of 60%v, near the maximum packing density. The result is that adding a small amount of dry Carbopol to a solution can cause a much larger increase in the apparent viscosity than its dry volume concentration can explain. Consider a suspension made of crushed pumice particles. Pumice is composed of siliceous materials with a solids density of ~2.6 g/mL. Therefore, 104 g of the material would have a true solids volume of 40 mL. If 960 mL of water were added to the solids (i.e., 1000 mL total), this would result in a suspension with a nominal volume concentration of 4.0%v. However, pumice is highly vesicular volcanic rock. If it had a porosity of 90% and a dry density of about 0.26 g/mL, 104 g of pumice would have a “rock” volume of 400 mL. If the pores fill with water then only 600 mL of the water is available to act as the carrier fluid, so the effective volume concentration will be 40%v. As a result, the suspension viscosity will be significantly higher than would be expected at 4.0%v. This filling of pores is a significant consideration with coal slurry pipelines and hydraulic analysis is based on the “saturated volume concentration” (see Fig. 2.2). Porous Particle (2.0 S.G.) Porosity: 50%v/v Aggregate (2.0 S.G.) Packing Density: 50%v/v Solid (3.0 S.G.) Fluid (1.0 S.G.) Figure 2.2: “Trapped” fluid caused by porosity and aggregation The viscosity increase due to the swelling of particles and the filling of particle pores is essentially the same phenomenon. The effective volume concentration has increased because a portion of the carrier fluid is “trapped” inside the particles. 29 2.2.7 Aggregating particles An “aggregate” is a number of solid particles that have clustered together and move as a group. In an aggregated suspension there will be a large number of individual aggregates. The packing density of each aggregate must be less than or equal to the maximum packing density. Between the particles there will be void spaces filled with fluid. Since the aggregates can move with the bulk fluid, the interstitial fluid is effectively “trapped” and becomes part of the aggregate (see Fig. 2.2). The net effect of trapped fluid is that the effective volume concentration (φeff) of the aggregates (i.e., solids + trapped fluid) increases and the volume of carrier fluid decreases (Quemeda 1998). The magnitude of the change in φeff is dependent on the degree of aggregation and how loosely packed the aggregates are. The ability to form stable aggregates depends on how the particles interact. 2.2.8 Particle interactions The interaction of small particles is dominated by surface chemistry. These interactions can be described by the theory proposed by Derjaguin and Landau and also by Verwey and Overbeek (the DLVO theory), which is described in detail elsewhere (for example, Hunter 2002; Tadros 1996). While the DLVO theory is complex, the aspect that is pertinent to the present discussion is that van der Waals (VDW) forces and electrostatic (ES) forces dominate the interactions between very small (clay sized or colloidal) particles. The VDW forces are caused by induced and/or permanent dipoles in the particles. Induced dipoles cause dispersion forces in neutrally charged particles. As two neutrally charged particles approach, the charges will migrate such that one particle will have a net negative charge on the near side and the other particle will have a net positive charge on the near side. To maintain neutrality, the far sides of the particles will gain an opposite charge. The VDW forces are short range and are always attractive. The ES forces are caused by the surface charge of the particle. The ES forces are long range in comparison to the VDW forces. The ES forces may be attractive, repulsive, or nonexistent, depending on the particle material and the solution properties. At low pH values the surface charge of most minerals is positive. At high pH values the surface charge is negative. 30 The charge will be zero at some intermediate pH, known as the iso-electric point (i.e.p.), which varies between minerals and is affected by the degree of oxidation of the particle surface and the electrolyte species in solution. For illustrative purposes, the approximate i.e.p. of some common minerals are given in Table 2.1. Table 2.1: Approximate iso-electric points (i.e.p.) for common minerals Mineral Iso-Electric Point Silica ~2-3 Magnetite / Haematite ~6-7 Alumina ~9 If the slurry contains a single mineral then all of the particles will have the same surface charge at any given pH. Many types of slurry (e.g., mine tailings) are made up of a number of different minerals. Each mineral may have a different surface charge at any given pH. Figure 2.3 shows idealized zeta potential curves for three minerals with different i.e.p. values. ζ + 0 0 2 4 6 A – A: + B: + C: + 0 + + 8 10 B – + + – 0 + 12 pH C – – + – – 0 – – – Figure 2.3: Effect of pH on surface charge of minerals If a slurry were made up of the three minerals shown in Fig. 2.3 then in the pH range of ~3 to ~9, there will be particles with both positive and negative surface charges. There would also be particles with a negligible surface charge at pH = ~6. At pH values below ~2, all 31 particles will have a positive charge. At pH values above ~10, all particles will have a negative charge. Other sources of particle interactions may be present including: viscous drag (hydrodynamic coupling of doublets), magnetic (ferromagnetic minerals), a net particle charge (rather than a surface charge) and adhesion to long chain polymer flocculants. 2.2.9 Inter-particle bonds “Bond” will be used as a general term for connections between particles that may be formed, broken and reformed an indefinite number of times without a loss in strength. There are various types of bonds (e.g., viscous, magnetic, etc.) but for aggregated particles the main bonding forces are the VDW and ES forces. Connections that lose strength once they break and reform (e.g., polymer flocculants) or cannot reform at all (e.g., particle attrition) are referred to as “links”. The strength of a bond is determined by the magnitude of the shear force required to separate two bound particles. For the present purpose, it is adequate to classify the bonds qualitatively, such as “weak” and “strong”. If the surface charge on both particles is high and of the same polarity (positive or negative) then the ES repulsion will exceed the VDW attraction and the particles will not be able to bond. If the surface charge is low then the VDW attraction will exceed ES repulsion and the particles will bond and form aggregates. If the charges are opposite, there will be an attractive ES force that augments the VDW attraction causing a strong bond. The actual values will vary, but it is reasonable to assume that they can roughly be grouped as shown in Table 2.2. Table 2.2: Relative force and bond strength of particles with different surface charge Particles: VDW Force ES Force: Bond: +:+ or –:– +:– +:0 or –:0 0:0 Weak Attractive Weak Attractive Weak Attractive Weak Attractive Repulsive Attractive Weak Attractive None None Strong Moderate Weak If there are more than two particles bonding in an aggregate, the presence of the other particles may also affect the net bond strength. If a strongly positive particle is bound to a 32 strongly negative particle, the net charge of the doublet will be roughly neutral. A second strongly negative particle attaching to the doublet will cause the net charge of the triplet to be negative, which will weaken the strength of both bonds (i.e., between each negative particle and the single positive particle). For spheres, the orientation of the particles does not affect the contact area. Most real particles are not spherical, and many are not even nearly so. For disc or needle-shaped particles, the area in contact may change by over an order of magnitude, depending on the particle orientation. The size of the contact area will affect the bond strength. Presumably, the larger the contact area, the stronger the bonds will be. Mustafa et al. (2003) found that agglomerates formed by oblate graphite particles were more difficult to disperse than spherical particles of the same material in the same suspending fluid. The size of the particles also affects the relative strength of the bonds. Usui (2002) found that the inter-particle bonding energy increased with increasing particle size, but was essentially independent of shear rate. However, the mass of the particles increases at a faster rate than the bonding energy. This increases the inertial forces on the particles more than the increase in the bonding force. For coarse particles the effects of surface chemistry related bonds become negligible and sand slurries are considered to be non-aggregating. 2.2.10 Structure The flow behaviour of fine suspensions is often related to a loosely defined property called “structure”. Casson (1959) considered structure to be the formation of chains of needle-like ink particles causing an increase in apparent viscosity in much the same way that increasing molecular weight (i.e., longer molecular chains) causes an increased viscosity in polymers. Scott Blair (1967) idealized slurry as a more-or-less evenly distributed array of particles in an inviscid fluid, with the degree of structure being a function of the number of formed bonds per unit volume and a higher bond density causing a higher apparent viscosity (i.e., more bonds need to be sheared to allow flow). In the present work, the term “structure” has a similar meaning to that used by Usui (2001); specifically, that the structure is related to the average size of the aggregates, which generally decreases with increasing shear rate. Usui (2001) assumed that the aggregates are near spherical clusters of small particles at or very near to the maximum packing density. 33 This may be reasonable when the aggregate is formed by accretion of individual particles to a larger aggregate. However, it does not seem justified when two aggregates (particle clusters) bond together to form a larger aggregate. It also seems unlikely when there slurry is made of particles with strong and opposite charges (heteroaggregation), since there is a limit to how close the similarly charged particles can pack together. 2.2.11 Build-up and breakdown of structure In order for two particles to bond, they must come into close proximity (“contact”), which requires relative movement. In an otherwise quiescent fluid, there will still be some movement due to Brownian motion, inter-particle forces and differential settling. Shearing of the slurry causes relative movement of the particles, increasing the contact rate. With strong potential bonds, ES forces may “catch” passing particles at relatively long distances. With weak bonds, it is necessary for the approach momentum to be high enough to overcome the ES repulsion. The rate of particle contact increases as the average interparticle spacing decreases. If the particle size is fixed, then the inter-particle spacing decreases as the volume concentration increases. If the volume concentration is fixed, then the inter-particle spacing decreases as the particle size decreases. Once a bond has formed, it may or may not remain formed. Bonds can be broken by viscous shear stresses or inter-particle collisions. Brownian motion alone may be sufficient to break very weak bonds. Stresses of sufficient magnitude to break stronger bonds come from external sources. As the applied shear rate increases, both the magnitude of the viscous shear stresses and the energy of the interparticle collisions increase. At very high shear rates, the stresses become so large that any bonds that form are immediately broken. If the shear rate is high enough, the solid particles themselves will eventually be broken down (attrition). The rate of bond formation increases with the number of potential bonding sites available and the relative motion (i.e., the contact rate). The rate of bond breakage increases with the number of existing bonds available to be broken and the shear rate (i.e., viscous shear stress and collision energy). At any given shear rate, there will be a dynamic equilibrium level of structure where the rate of bond formation equals the bond breakage rate. At very high shear rates, the particles would be almost fully dispersed and the free particles would behave as if they were non-interacting. At moderate shear rates, loosely packed aggregates will form and 34 grow into larger aggregates. At very low shear rates, the aggregates may eventually combine to form a network that spans the entire volume. Over some ranges of low shear rate, the structure may build-up with increasing shear rates if the frequency of particle contact increases faster than the breakage rate (Galdala-Maria and Acrivos 1980) so gentle stirring may increase the aggregation rate. However, the degree of aggregation and apparent viscosity generally decrease as the shear stress (and shear rate) increases (see Fig. 2.4). Log(η) Network Increasing “Structure” Large Aggregates Small Aggregates Dispersed Log(τ) Figure 2.4: The effect of viscous shear stress on particle “structure” Most mineral suspensions are made from non-spherical, heterogeneous particles. The strength of the individual bonds will vary with the particle charge, shape, size, and presence of other particles. The strength of individual bonds is less important than the fact that the relative strengths of the bonds will vary. If a hard sphere interaction model is assumed, then the particles do not interact until they come into contact. At the moment of contact, the particles will bond together if there is bonding potential or rebound off one another. Weak and strong bonds will develop in a more-or-less random fashion. If ES forces are present, the attraction between oppositely charged particles may somewhat bias the bond formation towards stronger bonds. 35 Once the slurry starts to undergo shearing, some of the bonds will start to break. By definition, weak bonds are more easily broken by shear and collision stresses than strong bonds. In quiescent slurry, where the particle motion is mainly due to Brownian motion, only the weakest bonds will be broken. Stronger and stronger bonds will start to break as the shear rate increases. The freed particles will contact other particles and reform bonds that will also break if they are weak. Therefore, the average bond strength between particles will increase as shear rate increases. The net result is that a higher incremental increase in the shear rate will be required to break the aggregates down incrementally than would be required if the bond strength stayed the same. 2.2.12 Limits to aggregation In a dilute suspension, aggregation may be limited by a lack of particles or particles capable of forming bonds. For example, assume a volume of fluid contained only one negatively charged particle and one positively charged particle. Aggregation can only continue until these two particles are bonded. In a suspension that has a large number of particles with a high positive surface charge and a single particle with a high negative surface charge, aggregation can only continue until the negatively charged particle is “coated” with positively charged particles. 2.3. Slurry – the Meso-Scale This section considers the bulk behaviour of slurry at a large enough scale that the suspension may be treated as a “continuum” or a continuous fluid with the bulk properties of the suspension. The continuum approximation needs to be applied with care since the volume of the solid particles is many orders of magnitude larger than a water molecule. The diameter of the largest particles should be less than one-third of the width of the flow path, such as the width of a rheometer gap width or the diameter of a capillary/pipe (Schramm 2000), and preferably less than one-tenth of the width (Coussot 1997). It must always be remembered that using the continuum approximation does not change the fact that the slurry is a suspension of particles in a liquid. 36 2.3.1 Equivalent viscosity in turbulent flow When a fully dispersed suspension is in turbulent flow it is observed that the flow curve is similar to a Newtonian turbulent flow curve. Figure 2.5a shows the turbulent flow curves for 7 0.010 6 0.009 Fanning Friction Factor Pressure Gradient (kPa/m) water and kaolin slurry at four volume concentrations (data from Litzenberger 2004). 5 4 3 2 0.008 0.007 0.006 0.005 1 0.004 0 0 0 1 2 Cv = 10% Cv = 14% Cv = 17% 40000 60000 80000 Pseudo-Reynolds Number 3 Bulk Velocity (m/s) Water 20000 Cv = 19% Nikuradse Colebrook Water (0.9 cP) Cv = 10% (2.4 cP) Cv = 19% (5.7 cP) Cv = 14% (3.4 cP) Cv = 17% (4.4 cP) Figure 2.5a/b: Turbulent pipe flow of kaolin slurry dispersed with TSPP (Data from Litzenberger 2004) The kaolin slurry has been dispersed with 0.27%w/w tetra-sodium pyrophosphate (TSPP). All five flow curves exhibit a pressure gradient that increases with the bulk velocity to the power of ~1.8, as would be expected with a Newtonian fluid. Fig. 2.5b shows the friction factor vs. the “best fit” pseudo-Reynolds number relationship for the same test loop flow curves compared to the Nikuradse (1932) equation for smooth wall turbulent flow and the Colebrook (1939) (or Colebrook and White 1937) equation for partially rough wall turbulent flow. When a suspension is not chemically dispersed (or is only partially chemically dispersed) there is a marked difference in the laminar flow behaviour indicating a non-Newtonian consistency (see Fig. 2.6). There is a significant increase in the observed laminar-turbulent transition velocity, but after a relatively short transition period, the data once again follows a pseudo-Newtonian fully turbulent flow curve. In each case shown in Fig. 2.6 (data from 37 Litzenberger 2004), the turbulent flow curve of the partially dispersed kaolin slurry is the same as (or is approaching) the fully dispersed turbulent flow curve at the same volume concentration. This implies that the “equivalent viscosity” in fully turbulent flow is a fundamental property of the suspension and not related to aggregation. Pressure Gradient (kPa/m) 7 6 5 4 3 2 1 0 0 1 2 3 Bulk Velocity (m/s) Water 14%v; 0.27%TSPP 17%v; 0.27% TSPP 10%v; 0.27% TSPP 14%v; 0.1%TSPP 17%v; 0.13% TSPP 10%v; 0% TSPP 14%v; 0%TSPP Figure 2.6: Effect of dispersant on the flow of kaolin slurry (Data from Litzenberger 2004) 2.3.2 Shear thinning The apparent viscosity of non-Newtonian suspensions subjected to an increasing shear rate (e.g., laminar flow at an increasing velocity) may either decrease (shear thinning) or increase (shear thickening). Most mineral suspensions are shear thinning (Cross 1965, Govier and Aziz 1977). Dilatant (shear thickening) fluids are much less common than pseudo-plastic (shear thinning) fluids and dilatency is observed only in certain ranges of concentration in suspensions of irregularly shaped solids in liquids (Govier and Aziz 1977). As discussed in Section 2.4, this normally occurs at solids concentrations near the maximum packing density. Shear thickening is sometimes reported at lower concentrations, although it may be difficult to differentiate between shear thickening and the onset of turbulence in a rheometer. 38 2.3.3 The yield stress Some suspensions are “plastic”, meaning they will not flow appreciably unless the applied shear stress exceeds a minimum value referred to as “yield stress”. Likewise, a plastic suspension will stop flowing (i.e., the shear rate will go to zero) when the stress drops below a certain value, also referred to as the yield stress or the zero shear rate stress (τo). The yield stress stops a plastic material from flowing during a “slump test”, which is a standard test for the workability of concrete (ASTM C143 1996) that has been adapted to thickened slurries (see Fig. 1.2). The yield stress allows pottery clay to be thrown into thin-walled vessels that will hold their shape until they dry. The yield stress of conventional slurry, such as would be transported in a long distance pipeline, will be on the order of 1 Pa. The yield stress of thickened slurry, such as would be used as a mine backfill paste, can be well over 100 Pa and will dominate the material’s flow behaviour. Nevertheless, direct measurement of the yield stress is problematic (Barnes 1999). The measured yield stress value to stop flow is often not the same as the value for flow to commence. This has lead to the use of the terms “dynamic yield stress” and “static yield stress” (Cheng 1985). The yield stress is often found by extrapolation of a rheogram curve back to the “zero shear rate” axis, but the value of yield stress depends on both the consistency model and shear rate range used for extrapolation (see Table 1.1). It has been contended that the yield stress is a “myth” (Barnes and Walters 1985) and that the apparent viscosity reaches a finite value (i.e., Newtonian plateau) at small shear rates. However, the low shear rate viscosity is often many orders of magnitude higher than the apparent viscosity of flowing slurry, so the point is usually moot from an engineering point of view. 2.3.4 Loss of strength A high yield stress suspension, such as mineral paste, will flow down an incline until the sheet is thin enough that the gravitational shear stress equals the yield stress and then the flow will cease or become extremely slow (Coussot 1997). The presence of vibrations from heavy equipment or an earthquake will cause the material to flow again. In a slump test, when the containment cylinder is removed, the mineral paste will collapse until the cross sectional area is large enough that the gravitational shear stress equals the yield stress (see Fig. 1.2) (Hallbom 2005). If the table that the slump test is carried out on is rapped with 39 one’s knuckles or a hammer, the paste will slump further. This property is used when a “vibrator” is used to make wet concrete flow into the formwork and level out. The cause of the loss in strength is due to the breakage of inter-particle bonds. A similar phenomenon occurs in saturated soil when an increase in the pore pressure that separates the particles, breaking the friction “bonds” and causing a loss of strength and stiffness (“liquefaction”). 2.3.5 Time-dependent fluids With time-independent suspensions, there is a fixed correlation between the applied shear rate and the measured shear stress, at least on the time scale of the experiment. With some suspensions, the consistency correlation changes over time. This could be caused by the chemical reactions (curing of the binder in a cemented backfill paste), the dissolution of some of the particles (rock salt in water), destruction of long chain polymer flocculants under shear, or particle attrition. These would be referred to as “irreversible time dependence”. However, in many instances the time dependence is reversible. For example, when concentrated residue from processing bauxite in the Bayer process (red mud) is sheared (agitated), it is found (Nguyen 1983) that the apparent viscosity decreases over time (Fig 2.7a) and the yield stress will “breakdown” (Fig 2.7b). However, if the red mud is allowed to rest it is found that there is yield stress “regrowth”, albeit at a different rate than the breakdown (Fig. 2.7b). Several things may be noted from Fig. 2.7a/b. Figure 2.7a/b: Time dependent consistency of 68%w red mud (Nguyen 1983) The first is that the consistency breakdown does not continue indefinitely, but rather decays exponentially towards an equilibrium value. In Nguyen’s (1983) example, the change is 40 negligible after 118 hours of “agitation” and the suspension reaches an “equilibrium” consistency. If the agitation speed is reduced to zero (the red mud is allowed to rest), then the consistency (yield stress) increases exponentially towards a different (higher) equilibrium. It may, therefore, be concluded that the equilibrium consistency depends on the agitation speed (i.e., shear rate). The second is that both the yield stress and the “viscosity” appear to be reduced over time when the red mud is agitated. If the Caldwell and Babbitt’s (1941) method is used to determine the Bingham plastic constants from the high apparent shear rate data, then after 4 hours of agitation the yield stress is ~246 Pa and the plastic viscosity is ~152 mPa-s. After 118 to 180 hours of agitation the yield stress is ~117 Pa and the plastic viscosity is ~72 mPas. In order to describe the flow array of instantaneous flow curves, a time function of both the yield stress and the plastic viscosity would be required. Furthermore, both plastic viscosities are an order of magnitude higher than would be expected for a 68%w (~41%v) suspension, which according to [2.2] should be ~6.2 mPa-s. Nguyen (1983) proposed that the Casson model better described the red mud data than the Bingham plastic model. Extracting the plastic viscosity for a Casson fluid is similar to the Caldwell-Babbitt method, except that the square root of the plastic viscosity is found from the slope of the square roots of the wall shear stress and apparent shear rates. If the wall shear stress and the apparent shear rate are raised to the power of 0.4 rather than 0.5, it is found that the high shear rate data in Fig. 2.7 describes a series of straight parallel lines (see Fig. 2.8). Furthermore, the average slope of the curves in Fig. 2.8 is 0.132 (mPa-s)0.4. This implies that the “plastic viscosity” is ~6.3 mPa-s, which is comparable to the predicted value above. If this “general Casson” method is used (i.e., using a power factor of ~0.4 rather than 0.5), then time dependence in Fig. 2.7a reduces to a time dependent yield stress and a constant plastic viscosity equal to the viscosity of a fully dispersed suspension. The third is that the timescale of the breakdown (hours) and the regrowth (days) of concentrated red mud is long compared to the time it takes to do a rheology test (minutes). The individual flow curves in Fig. 2.7a may be considered to be time-independent and the array of curves equivalent to the “constant structure” flow curves in the Cheng and Evans 41 (1965) model for time dependence (Klein and Hallbom 2004). However, if there is no change in the structure during a rheology test, it is difficult to explain the non-Newtonian behaviour (NB: the Casson model is a structure breakdown model). 11 (Wall Shear Stress)^0.4 y = 0.1269x + 9.462 10.5 y = 0.1327x + 8.8086 10 y = 0.1313x + 7.8534 9.5 y = 0.1399x + 7.1794 9 8.5 y = 0.1284x + 6.6953 8 7.5 7 6 8 10 12 14 16 18 20 (Pseudo Shear Rate)^0.4 4 hr 7 hr 24 hr 40 hr 180 hr Figure 2.8: High shear rate data for 68%w red mud (after Nguyen 1983) 2.4. Constitutive Equations Unlike with Newtonian fluids, the consistency of a non-Newtonian suspension cannot be defined by a single variable (i.e., the “viscosity”). Instead, a constitutive equation with at least two terms, and often more, is required to relate the apparent viscosity to the shear rate or shear stress. A constitutive equation is a concise, usually algebraic, summary of the flow behaviour of a real material. The flow behaviour of interest is usually the relationship between the apparent viscosity and the shear rate. 2.4.1 Constitutive equations available A large number of constitutive equations have been proposed over the years. The following sources were reviewed for references to constitutive equations that could be used for shearthinning, time-independent, viscoplastic fluids. This selection, while not exhaustive, includes a range of sources that can be considered to represent the state of the art for slurry transport design engineering. 42 • “Paste and Thickened Tailings – A Guide, 2nd Ed.” (Jewell and Fourie 2006), • “Pipeline Hydrotransport with Applications in the Oil Sands Industry” (Shook et al. 2002) • “Introduction to Modern Colloid Science” (Hunter 2002 reprint), • “Slurry Systems Handbook” (Abulnaga 2002) • “Proceedings of the SME Mineral Processing Plant Design, Practice, and Controls Conference– Section 10” Vancouver (2002) • “NIST 960-3 Recommended Practice Guide: The Use of Nomenclature in Dispersion Science and Technology” (Hackley and Ferraris 2001), • “Chemical Engineering Fluid Mechanics” (Darby 1996) • “Rheological Methods in Food Process Engineering 2nd Ed.” (Steffe 1996), • “Hydraulic Design for Flow of Complex Fluids – Course Notes” (Hanks 1981), • “Solid-Liquid Flow – Slurry Pipeline Transportation” (Wasp et al. 1977), • “The Flow of Complex Mixtures in Pipes” (Govier and Aziz 1977) The following constitutive equations, put into apparent viscosity (η) form, were found: • Bingham plastic: η = τ o γ& + η B • Carreau-Yasuda: η= • Casson plastic: η = τ o γ& + ηC • Cross: η= ηo − η∞ + η∞ a 1 + (CC γ& ) • Ellis: η= ηo a −1 1 + (τ τ 2 ) • Generalized Herschel-Bulkley: η = (τ o γ& )b + Kγ& n −1 ηo − η∞ (1 + (C &a CY γ ) ( ( 43 )( 1− n ) / a ) 2 ) a + η∞ • Herschel-Bulkley (Yield power law): η = τ o γ& + Kγ& n −1 • Meter-Bird: η= • Modified Casson: η = τ o γ& + Kγ& n −1 • Newton: η=μ • Powell-Eyring: η= • Power law (Ostwald-de Waele): η = Kγ& n −1 • Prandtl-Eyring: η= ηo sinh −1 (CPγ& ) CPγ& • Reiner-Philippoff: η= ηo − η∞ + η∞ 2 1 + (τ τ 2 ) • Sisko: η = Kγ& n −1 + η∞ • Van Waser: η= • Vocaldo: η = (τ o γ& )a + Kγ& • Whorlow: η = Cw1 + Cw 2γ& 2 + Cw3γ& 4 ... 2.4.2 ηo − η∞ + η∞ a −1 1 + (τ τ 2 ) ( ) 2 (ηo − η∞ )sinh −1 (CPγ& ) + η CPγ& ∞ ηo 1 + Cvw1γ& + Cvw 2γ& a ( ) 1/ a Constitutive equations used in mineral processing Based on the above list, it would appear that there are at least eighteen constitutive equations available that are suitable for use with fine mineral slurry. However, the number of models used in practice is quite small. The only non-Newtonian models mentioned in the 544 papers in the proceedings discussed in Chapter 1 (section 1.2) are: Bingham plastic, power law, Herschel-Bulkley and Casson. 44 The main area where a constitutive equation was applied, rather than merely mentioned or measured, was slurry transport (pumps and pipelines). The following proceedings of recent major international conferences related to the pipelining of viscous slurries were reviewed. • “Proceedings of the 17th International Conference on the Hydraulic Transport of Solids (Hydrotransport 17)”, Cape Town, South Africa, 7-11 May 2007 (44 papers) • “Proceedings of the 10th International Seminar on Paste and Thickened Tailings”, Perth, Australia, 13-15 March 2007 (40 papers) • “Proceedings of the 9th International Seminar on Paste and Thickened Tailings”, Limerick, Ireland, 3-7 April 2006 (36 papers) • “Proceedings of the 8th International Seminar on Paste and Thickened Tailings” Santiago, Chile, 20-22 April 2005 (21 papers) Rheology is discussed in many of the papers in the proceedings and its importance is often emphasized. Various methods of measuring the consistency are given. Some papers give an indication of the consistency of the slurry being described or handled. Often the yield stress (or “slump”) alone is used in papers focused on thickening or estimating deposition slopes. The number of times that a standard non-Newtonian constitutive equation was actually “used” quantitatively for flowing slurries is listed below (some papers used more than one model): • Bingham plastic: 32 • Herschel-Bulkley: 9 • Casson: 3 • Power Law: 2 Where rheology is applied in a quantitative way, the dominant constitutive equation is clearly Bingham plastic with Herschel-Bulkley being the second choice. 2.4.3 Engineering equations While constitutive equations suitable for fitting rheograms are important to rheologists, engineers are interested in constitutive equations as a means to determine a property that 45 will affect the design or operation of a slurry handling system. This may be the speed and power requirement for a slurry tank agitator or the way that mine tailings will flow in an impoundment. In theory, an engineer can use any of the constitutive equations presented in the previous sub-section and work out the flow properties from first principles or using computational fluid dynamics. However, this almost never happens in practice. Engineers use algebraic design equations, preferably ones that have been confirmed experimentally. Several books and “course notes” related to slurry hydraulics were reviewed to determine which non-Newtonian constitutive equations had design equations available to calculate pressure gradient in laminar and turbulent pipe flow: • Shook et al. (2002) “Pipeline Hydrotransport with Applications in the Oil Sands Industry”: o Laminar: Bingham plastic o Turbulent: Bingham plastic and Casson • Steffe (1996) “Rheological Methods in Food Process Engineering 2nd Ed.”: o Laminar: Bingham plastic, power law, Herschel-Bulkley and Casson o Turbulent: Bingham plastic and power law • Darby (1996) “Chemical Engineering Fluid Mechanics”: o Laminar: Bingham plastic and power law o Turbulent: Bingham plastic and power law • Hanks (1981) “Hydraulic Design for Flow of Complex Fluids – Course Notes”: o Laminar: Bingham plastic and power law o Turbulent: Bingham plastic and power law • Govier and Aziz (1977) “The Flow of Complex Mixtures in Pipes”: o Laminar: Bingham plastic, power law, Herschel-Bulkley, Powell-Eyring, Ellis, and Meter o Turbulent: Bingham plastic, power law and Herschel-Bulkley 46 • Wasp et al. (1977) “Solid-Liquid Flow – Slurry Pipeline Transportation”: o Laminar: Bingham plastic and power law o Turbulent: Bingham plastic and power law As can be seen, only four non-Newtonian constitutive equations have readily available design equations for both laminar and turbulent pipe flow: Bingham plastic, Casson, power law, and Herschel-Bulkley. Not surprisingly, these are the same four constitutive equations that are actually used in the mineral processing industry (see Section 2.4.2). 2.4.4 The yield power law family of models The following constitutive equations may be considered to be part of the same “family”: Bingham plastic, Herschel-Bulkley, power law, Newtonian, and true plastic (“yield stress only”). The similarity between these constitutive equations is clear when they are put into shear stress vs. shear rate form: τ = τ o + Kγ& n Herschel-Bulkley [2.6] τ = τ o + Kγ&1 Bingham plastic [2.7] τ = 0 + Kγ& n Power law [2.8] τ = 0 + Kγ&1 Newtonian [2.9] τ = τ o + 0γ& n True plastic [2.10] The most general version of this family is the Herschel-Bulkley model, which has been shown to fit the rheograms of a large number of slurries (Govier and Aziz 1977, Steff 1996). However, the general behaviour implied by the yield power law model is inconsistent with the observed physical behaviour of suspensions. Specifically, the yield power law model for shear-thinning slurries (n < 1) predicts that the apparent viscosity will drop below the carrier fluid viscosity at high shear rates. What is actually observed is that at sufficiently high shear rates, the rate of change in the viscosity becomes zero and the data conforms to the Newtonian model (Govier and Aziz 1977). At very high shear rates, the yield power law model predicts that the apparent viscosity will tend to zero, which is “clearly physically inadmissible” (Hanks 1981). 47 Newtonian fluids have a flow behaviour index of n = 1 and shear-thinning fluids have a flow behaviour index of n < 1. To have a high shear rate Newtonian plateau, the flow behaviour index must increase from, say, 0.4 to 1.0 over a range of shear rates. This increase in the flow behaviour index with shear rate is commonly observed for power law and yield power law fits to real data (see Table 1.1). Furthermore, the changing flow behaviour index means that the consistency index (K) changes its magnitude and its units (Pa-sn). The predicted yield stress will usually change as well. This means that, even if all physical properties (e.g., solids content, pH, etc.) are fixed, the model still has parameters that vary with shear rate or “variable constants”. If one is attempting to generate a pipeline flow curve then the variable flow behaviour index is problematic. As the velocity increases, the shear rate increases and, therefore, the flow behaviour index increases. However, the flow behaviour index appears multiple times in the laminar flow design equation (see [5.43] or Appendix A). In effect, this means that the slurry will follow a different flow curve at each flow rate. One solution to this problem is to measure the consistency in the shear rate range of the application, and this approach is often advised (e.g., Darby 1984). While this approach is pragmatic, it makes having a constitutive equation redundant. 2.4.5 The Casson model Since the yield power law model does not agree with the observed behaviour of suspensions, at least at the extremes, it is reasonable to inquire if there is a better constitutive equation. The only constitutive equation that is commonly used in the mineral processing industry that is not in the yield power law family is the Casson model, which in shear stress vs. shear rate form is: τ 1 / 2 = τ o1 / 2 + (ηC γ& )1 / 2 Casson [2.11] The Casson model has several advantages: • At high shear rates, the apparent viscosity approaches a finite value (ηC) • The units of the parameters are rational (i.e., Pa and Pa-s) • It reduces to the Newtonian model for a yield stress of zero 48 • It reduces to the “true plastic” models for a plastic viscosity of zero • It models the flow of some suspensions accurately (e.g., blood, molten chocolate, printing ink) • The model parameters are easily extracted from a rheogram These advantages are identical to those of the most commonly used non-Newtonian model: Bingham plastic. It may also be noted that the form of the two models is very similar. τ 1 = τ o1 + (η Bγ& )1 Bingham plastic [2.12] However, many shear-thinning suspensions are not particularly well represented by either the Casson model or the Bingham model. While it is possible to use either model to get a good fit over a limited range of data, the phenomenon of “variable constants” is found when measuring the flow behaviour over different ranges (Darby 1984). In some cases, the Casson model will also predict a limiting viscosity that is less than the viscosity of the carrier fluid. 2.5. The yield plastic family To eliminate these problems, it has been proposed (Hallbom and Klein 2004) that the Casson and Bingham models may be generalized using a variable exponent k, which is a dimensionless “scaling factor” (0 < k ≤ 1), and μ∞ which is the limiting high (“infinite”) shear rate Newtonian viscosity. This “yield plastic” model (with a “yield” stress and a “plastic” viscosity) describes an alternative “family” of constitutive equations: τ k = τ ok + (μ∞γ& )k Yield plastic [2.13] τ 1 = τ o1 + (μ∞γ& )1 Bingham plastic [2.14] τ 1 / 2 = τ o1 / 2 + (μ∞γ& )1 / 2 Casson (plastic) [2.15] τ k = 0 + (μ∞γ& )k Newtonian [2.16] τ k = τ ok + (0γ& )k True plastic [2.17] The yield plastic model does not reduce to either the power law model or the HerschelBulkley model. However, it has been noted (Scott Blair 1966) that the apparent success of 49 the Casson model is because it cannot be distinguished from the yield power law model over a limited range of real data. The reverse argument can be made for the apparent success of the yield power law model and the power law model, since the yield plastic model can be used to represent a range of data as well as either (Hallbom and Klein 2004). More importantly, since its limiting conditions are physically admissible (i.e., a finite viscosity), the yield plastic model can represent a wider range of data. The yield plastic model was originally developed phenomenologically. It was proposed to deal with problems associated with analyzing time-dependent suspensions, based on the assumption that the behaviour of fully dispersed particles at high shear rates is not affected by the behaviour at low shear rates. This “fixed” the value of μ∞ leaving only two shearhistory dependent parameters. If the value of the scaling factor is assumed to be constant, time-dependent behaviour reduces to a variable yield stress (see Fig. 2.8). This dramatically simplified the study of time-dependence relative to the Cheng and Evans (1965) model, which used an array of Herschel-Bulkley flow curves, each with three parameters and an equilibrium curve with another three parameters. 2.6. Prior Art The proposed yield plastic model was developed independently and phenomenologically. Given the simplicity and flexibility of the equation it was presumed that it would have been proposed by earlier researchers, so an extensive review of the literature was undertaken. No reference to a yield plastic-like model was found in the following literature: • Major rheology related journals: Journal of Rheology, Journal of Non-Newtonian Fluid Mechanics, Rheologica Acta, Proceedings of the Royal Society. • Rheology related books, including: Govier and Aziz 1977; Steffe 1996, Whorlow 1980, Malkin 1994, Darby 1996, Hunter 2002, and Wasp et al. 1977. • The recommended practice guide the use of nomenclature for dispersion science and technology (NIST-960-3 2001) • Applied rheology course notes: Hanks 1981, Ancey 2005 However, three references to similar equations were found and are discussed below. 50 2.6.1 Heimann-Fincke (Heinz-Casson) Heimann and Fincke (1962) investigated the consistency of melted and dispersed chocolate to see if it was better described using the Heinz (1959) model (originally developed for paints) or the Casson (1959) model (originally developed for ink). They found that some chocolate suspensions fit the Casson model best, while others fit the Heinz model best. None of the chocolate suspensions fit the Bingham model, but it was observed that the three models seemed closely related. This is apparent when put in the form of the yield plastic model: τ 1 = τ o1 + (μ∞γ& )1 Bingham [2.18] τ 2 / 3 = τ o2 / 3 + (μ∞γ& )2 / 3 Heinz [2.19] τ 1 / 2 = τ o1 / 2 + (μ∞γ& )1 / 2 Casson [2.20] The Heinz model is another special case of the yield plastic model (k = 2/3). Heimann and Fincke (1962) proposed the following constitutive equation: ⎛ τ 1 C 0 − C1 ⎞ ⎟⎟ γ& = ⎜⎜ ⎝ C2 ⎠ C0 Heimann and Fincke [2.21] Where C0, C1 and C2 are “materialkonstanten” obtained from rheological measurements, which is to say, from fitting to flow curves. Heimann and Fincke noted that for C0 = 1, 1.5 and 2, their model was equivalent to the Bingham, Heinz and Casson models respectively. This is made clear by rearranging [2.21] into the following form: τ 1 C = C1 + C2γ&1 C 0 [2.22] 0 The physical meaning of the material constants is not stated, but they mention the yield point and infinite shear rate viscosity in reference to the Casson and Heinz models. It may be inferred that they intended a similar meaning in their proposed model. To provide dimensional consistency, this implies that C1 = τ o1 C 0 and C2 = μ∞1 C 0 . If this inference is correct, then equation [2.22] has the same form as [2.13]. The Heimann-Fincke model is essentially a phenomenological model and no physical reason is given for the changing value of the exponent or the value of the infinite shear rate viscosity. 51 Whether the two models are equivalent depends on the interpretation of the values that the exponent can take. Based on the context of the paper, the authors apparently meant that the value of the exponent C0 alternated between 1.5 and 2, while in the paper’s summary they recommend the use of 1, 1.5, or 2 for the exponent. It would, therefore appear that Heimann and Fincke model is a special case of the yield plastic model. Some later researchers assumed that C0 is not confined to integers, making it equivalent to 1/k. For example, Ofoli et al. (1987) refer to the “Heinz-Casson” model (with a reference to Fincke) and tabulate values of the “power index” (i.e., 1/C0) of 0.3 and 0.12. By this interpretation, the HeimannFincke model is equivalent to the phenomenological version the yield plastic model (Hallbom and Klein 2004). Steffe coauthored three papers referencing the Heimann-Fincke (“Heinz-Casson”) model (Olfoli et al. 1987; Mogan et al. 1989 related to “protein dough”; and Dolan et al. 1989 related to “starch gelatinization”). It is notable that Steffe does not mention the “HeinzCasson” model in his book on food rheology (Steffe 1996). Olfoli et al. (1987) presented rheograms for various foods (i.e., Ketchup, mustard, Miracle Whip, apricot puree, and molten milk chocolate coating) fit with the following constitutive equations: Bingham plastic, Casson, Herschel-Bulkley, Heinz-Casson and their own four-parameter “generalized viscosity model”. The Heinz-Casson model fits the test data better than any of the standard constitutive equations used (or, by inspection, the power law model). The generalized viscosity model gives an improved fit in two cases, but the differences are small and only noticeable at very low shear rates (<0.2 s-1). Other references found were one relating to dulce de leche or sweetened condensed milk (Rovendo et al. 1990), one relating to “semi-solid food materials” (Song and Chang 1999), and one relating to sago starch paste (Sopade and Kiaka 2001). All of these references are from the same industry as the original paper (food processing). References published after (Hallbom and Klein 2004) found are: Song et al. (2006) relating to xanthan gum, and Ojo and Akanabi (2006) relating to “soy-ogi” (a suspension of soy-maize flour). The Heimann-Fincke (Heinz-Casson) model appears to have been little used outside the food processing industry. Even in that industry, the published references found only mention it in passing, as one of a variety of constitutive equations that may be used describe food 52 consistency, or used it to fit a rheogram. No evidence was found that the Heimann-Fincke model has been developed for use in engineering applications. 2.6.2 Oka Oka (1971) presented a theoretical study of the rheological behaviour of time-independent, non-Newtonian suspensions based on the breakdown of structure. The term “structure” is used in the sense proposed by Scott Blair (1967) and is explicitly differentiated from the sense proposed by Casson (1959). The flow behaviour is determined by “the number of bonds per unit volume of the suspension at any value of τ” (Oka 1971). Oka started with a modified form of Scott Blair’s (1967) model for the breakdown of structure: λ dλ = −C1 dγ& (γ& + C3 )C0 [2.23] λ dλ = −C2 dτ (τ + C4 )C0 [2.24] Where: C1, C2, C3, and C4 are positive constants. C0 is a dimensionless constant less than or equal to one. The constants C3, and C4 were added to eliminate the “infinite” change in the structure as γ& or τ approaches zero, but no physical explanation was put forward. Oka (1971) derived the constitutive equations for suspensions with a yield stress and where C3 = C4 = 0. Two cases were considered: C0 = 0 and C0 = 1/2. The resulting constitutive equations were equivalent to the Bingham plastic and Casson models, respectively. An arbitrary value of C0, was not considered, but C0 = 1-k would have resulted in a model similar to the yield plastic model. Oka (1971) noted “it seems difficult at present to give a theoretical explanation for these relations”. Oka (1971) also derived the model for suspensions with no yield stress when C0 < 1, with the result being what he referred to as the “generalized Casson model”: (τ + C4 )1− C 0 = C5 + C6 (γ& + C3 )1− C 0 [2.25] He noted that, when τ >> C4 and γ& >> C3 , [2.25] reduces to a form similar to the yield plastic model: 53 τ 1− C = C5 + C6γ&1− C 0 [2.26] 0 There have been at least 10 references to this paper (“Web of Science” database). The papers reviewed generally named it as a theoretical justification for the use of the Casson model (e.g., Hanks 1980). Several were related to the flow of blood, which is often modelled as a Casson fluid. No evidence was found that the Oka model has been developed for use in engineering applications. 2.6.3 Haake rheometer software Schramm (2000) includes a screen shot of the models “of rheological significance” included in the HAAKE software package for rheometer data analysis (Schramm’s Fig 131). In this list, the “Casson” model is shown with a variable (“n”) as the exponent. When the HAAKE software was reviewed, it was found that “n” was defined as 0.50, which is to say that it is the standard Casson model. The software also included the same equation with a variable “n”, which is the same as [2.13]. However, this model was included in the list of “further polynomials” that can be used for “curve fitting”, which implies that the yield plastic model was not recognized to be “of rheological significance”. 2.7. Summary Homogenous mineral slurries are composed of small (clay sized or colloidal) solid particles suspended in a Newtonian liquid. The observed non-Newtonian flow behaviour of homogenous at the meso-scale must be due to the physical interactions of the particles and liquid at the micro-scale. Since the purpose of a rheological model or constitutive equation is to describe the meso-scale flow behaviour over a range of shear rates, it follows that the model parameters ought to be relatable to the micro-scale interactions. This chapter reviewed the observed (or generally accepted) micro-scale behaviour of particles and liquid, with an emphasis on aspects that are assumed to significantly affect the meso-scale slurry behaviour. It is proposed that a consistency model for homogeneous slurries must be consistent with the observed micro-scale behaviour. This chapter reviewed the observed flow behaviour of homogeneous slurry at the meso-scale, such as when it is flowing in a pipeline or in a rheometer. It is proposed that flow behaviour 54 implied by a constitutive equation for homogeneous slurries must be consistent with the observed meso-scale behaviour. (Note: the key micro-scale and meso-scale aspects of slurry behaviour that are related to the derivation of the new rheological model and constitutive equation are summarized in the next chapter.) A large number of rheological models have been proposed for describing the flow curves of time-independent fluids, eighteen of which were presented in this chapter. However, the models used for engineering in the mineral processing industry are mainly limited to: Newtonian, Bingham plastic, power law, Herschel-Bulkley (yield power law), and Casson. The first three of these models are special cases of the yield power law model, which can be considered to describe the entire rheological “family”. The yield power law model does not, however, include the Casson model, which was shown to have several advantages. The phenomenological “yield plastic” consistency model proposed by Hallbom and Klein (2004) was shown to include the Casson, Newtonian and Bingham plastic models as special cases. It was proposed that the yield plastic model describes an alternate family of constitutive equations. A number of advantages of the yield plastic model over the yield power law model are discussed. Three instances were found where models similar to, but not identical to, the yield plastic model have been proposed in the past. These rheological models were phenomenological (i.e., they seemed to fit rheograms) and no physical basis for the model parameters was proposed. One source (Haake) includes it in a list of mathematical functions that are not of “rheological significance”. The use of these obscure models has been limited, essentially confined to comparative material characterization of two types of soft particle suspensions (food and blood). No evidence was found that these models have been developed for use in engineering applications. The objective of the remainder of this Thesis is to propose a physical basis for the yield plastic model and to derive the basic engineering design equations for laminar and turbulent pipe flow. 55 CHAPTER 3: NEW RHEOLOGICAL MODEL Rule 1: We are to admit no more causes of natural things than such as are both true and sufficient to explain their appearances…for Nature is pleased with simplicity, and affects not the pomp of superfluous causes. Isaac Newton, Philosophiae Naturalis Principia Mathemantica (1687 CE) 3.1. Introduction This chapter establishes the physical basis for a new rheological model and constitutive equation to describe the flow behaviour of viscoplastic suspensions. The key observations on slurry behaviour from the previous chapter are summarized and used to formulate a simple physical model for shear thinning behaviour: apparent viscosity increases due to the increased effective volume concentration when water is trapped in aggregates and that aggregates break down under increasing shear stress that varies with the relative interparticle bond strength. A simple three-parameter algebraic consistency model is derived based on the physical model and is shown to be the same as the phenomenological “yield plastic” model proposed by Hallbom and Klein (2004). 3.2. General Assumptions The observations made in Chapter 2 may be used to formulate a physical model for the meso-scale rheological behaviour of time-independent suspensions of fine solids, based on behaviour at the micro-scale. The main observations are summarized below: 3.2.1 Description of suspension A suspension is a multitude of particles suspended in a viscous Newtonian liquid. If the particles are small, the effects of gravity are not significant and the particles will not settle in the time frame of interest (i.e., the suspension is “non-settling”). The particles do not need to be the same size as long as they are all small enough to be considered “non-settling”. The particles need not be an idealized shape nor do they all need to be the same shape. The particles do not need to be made of the same material and an individual particle may be made of more than one material. It is assumed that the particles are rigid and do not break 56 (i.e., there is no particle attrition). The volume concentration of the particles is well below the maximum packing volume concentration, so they can move past one another. 3.2.2 Particles move The particles in a suspension will move relative to one another. With “non-settling” suspensions of fine particles, this relative motion is negligible at the meso-scale (i.e., the particles do not settle to the bottom of a beaker and do not segregate when flowing through an elbow in a pipe), which allows the suspension to be treated as a continuum. In a flowing suspension, the particles move at the local velocity of the flow field (i.e., the differential velocity is less than the very low settling velocity). However, at the micro-scale, thermal agitation (Brownian motion), gravity (differential settling or rising) and inertia forces (in a flowing suspension) do cause small relative particle motion. As a result of this relative movement, the closely spaced particles will come into “contact” (i.e., close proximity). 3.2.3 Particles may form aggregates When two particles contact, they will either rebound off each other or connect. If the particles never connect, the particles remain dispersed and the suspension is “stable”. If two particles do connect, they form an “aggregate”. When this aggregate comes into contact with other particles or aggregates, they will likewise rebound or connect. This process may continue until an aggregate of many particles is formed. Large aggregates will settle and the suspension is “unstable”. In order for aggregation to occur the particles (or aggregates) must contact and there must be a potential to form “bonds” or “links”. 3.2.4 Bonds and links An interparticle connection that can be broken and reformed continuously without a significant change in the connection strength is a “bond”. An interparticle connection that can only be formed and broken small number of times and there is a significant drop in the connection strength after each cycle is a “link”. Often, a link will only be able to form and break a single time. The current discussion focuses on time-independent suspensions that can be sheared cyclically without a significant change in the flow behaviour over time, so it is assumed that all interparticle connections are bonds. 57 3.2.5 Strength of bonds For fine mineral particles in a suspension, the main attractive forces are electrostatic (opposite charge) at long range and van der Waals at short range. The electrostatic forces may also be repulsive (similar charge) or negligible (no charge). The strength of a given bond depends on the net attractive and repulsive forces. When there are particles made of different materials in the suspension, the strength of the individual bonds will vary. When the particles are different shapes, the strength of the bond may depend on the orientation of the particles (e.g., two needles bonded along the edge compared to being bonded at the ends). When electrostatic charges are significant, the strength of the bonds may be affected by the charge of neighbouring particles. There are other possible bonding forces (notably ferromagnetism in magnetite suspensions) but this does not change the main assumption: The strength of the individual bonds in a real suspension will vary. 3.2.6 Strength of aggregates The shearing force required to break an individual particle is assumed to be much higher than the force required to break interparticle bonds. Therefore, the strength of an aggregate depends on the number and the strength of the individual bonds. Tightly packed aggregates will be stronger than loosely packed aggregates. Aggregates with strong bonds will be stronger than aggregates with weak bonds. The strength of different shear planes through randomly packed aggregates will vary and the aggregate will tend to break across the weakest plane (see figure 3.1). Large aggregates formed by the bonding of two or more smaller aggregates will generally be more easily broken than small aggregates. 3.2.7 Velocity gradients break aggregates The shear rate due to velocity gradients in a suspension cause viscous shear stresses in the aggregates. The magnitude of the viscous shear stresses will increase as the shear rate increases and as the apparent viscosity of the suspension increases. If the shear stress exceeds the strength of the aggregate across any plane, the aggregate will break (see Fig.3.1). The shear stress may break off single particles or groups of particles. The magnitude of the shear stress will increase as the size of the aggregate increases because of the increased differential velocity between the sides of the aggregate. Deaggregation may also be cased by 58 aggregate-aggregate collisions. The magnitude of the collision stresses will increase as the mass and the relative velocity of the aggregates increases. Two aggregates moving in the same flow stream will have approximately the same velocity, so most collisions will be glancing. As a result, collisions tend to break off isolated particles at the surface (impact attrition), causing a general rounding of the aggregates (see Fig.3.1). τ V+dV Weak Weak V τ Figure 3.1: Shear breakage and impact attrition of aggregates It is recognized that velocity gradients may also induce aggregation. Weakly repulsive bonds require a minimum collision velocity to force the particles close enough for the van der Waals forces to act. Gentle stirring of a suspension may cause aggregation to occur more quickly or to a greater extent than in a quiescent suspension. However, moderate to high velocity gradients will tend to break down aggregates. 3.2.8 Aggregates reach an equilibrium size When a suspension of potentially aggregating particles is subjected to very high shear rates, particle interactions will be frequent but the viscous stresses will also be very high, much higher than the strength of the interparticle bonds. Any bonds that form will be broken almost immediately; resulting in what is effectively a fully dispersed suspension. At moderate shear rates, particle interactions will still be relatively frequent and bonds that are formed are less likely to break because of the lower viscous shear stresses. Aggregates can form and grow until they are at a size where the viscous stresses can break them. At low shear rates, there will be a low interaction rate but the low viscous shear and collision 59 stresses mean that even weak bonds that do form will be unbroken. Large, loosely packed aggregates are able to form. Therefore, the average degree of aggregation will generally decrease as the shear rate increases. At any given shear rate, there will be a dynamic equilibrium where the aggregate building and breakage rates will be equal. If the aggregation reaches equilibrium quickly relative to the time scale of interest (e.g., the time it takes to do a rheological measurement) it is time-independent. If the aggregation reaches equilibrium slowly it is time-dependent. 3.2.9 Bonds form randomly but break preferentially The particle contacts that cause bonding will be almost random, although they may be skewed towards stronger bonds if the attractive electrostatic forces are significant. When there is a variation in bond strengths, any stresses caused by viscous drag or impact are more likely to break weak bonds than strong bonds. As the weak bonds break, the average strength of the remaining bonds will increase. Furthermore, aggregates will tend to break at their weak points. On average, the aggregates will become more tenacious (harder to break) as their size is reduced by velocity gradients. 3.2.10 Dispersed particles increase apparent viscosity The presence of dispersed solid particles in a liquid disrupts the flow streams, which increases the suspension’s apparent viscosity. The apparent viscosity of dispersed suspensions is Newtonian and proportional to the viscosity of the suspending liquid. The viscosity multiplier of the suspension increases as the volume concentration increases and increases exponentially as the particle volume concentration approaches the maximum volume concentration. The particle density has a negligible effect on the slurry viscosity. The size of the particles normally has a negligible effect on the suspension viscosity (see note below). A wide particle size distribution increases the maximum volume concentration because the small particles can fit in the void spaces between the large particles. For a given volume concentration, a wide particle size distribution will decreases the apparent viscosity of the suspension. When the particles are non-spherical, the particle orientation will impact the effective size of the particles and the maximum volume concentration. 60 (Note: A sub-micron thick layer of liquid will be bound to the surface of the each particle and increases its effective volume. This increase is negligible for silt and sand sized particles but, with clay-sized particles, the volume of the bound liquid layer may be comparable to the solid volume.) 3.2.11 Aggregates increase the effective volume concentration The volumetric packing density of the particles in an aggregate must be less than, or equal to, the maximum packing density. The void spaces between the particles will contain “trapped” liquid that effectively becomes part of the solid fraction rather than the suspending liquid. Since the volume of the liquid/solid aggregates must be greater than the total volume of the individual solid particles, aggregation increases the effective volume concentration. The average packing density of the aggregates in the suspension will vary depending the particle size distribution, the particle shapes, and the way the aggregates are formed. As the aggregates get larger, the average packing density will generally decrease. 3.2.12 Aggregation increases the slurry viscosity The apparent viscosity of a suspension increases as the volume concentration increases. The formation of aggregates increases the effective volume concentration of the slurry. Therefore, the apparent viscosity of a suspension will increase when the degree of aggregation increases. The volume of the aggregates may be several times the volume of the individual particles, so an aggregated suspension with a moderate free particle volume concentration may have an effective volume concentration equal to the maximum packing density of the aggregates. The aggregates would form a “network” that spans the entire volume. For flow to commence, there must be sufficient shear stress to rupture the network (i.e., the yield stress). This aggregation related increase in the apparent viscosity is assumed to be the primary cause of Ostwald’s (1930) structural viscosity. 3.2.13 Minimum viscosity when particles are fully dispersed Since aggregation increases the suspension viscosity, the lowest viscosity occurs when the particles are fully dispersed. It is irrelevant whether the dispersion is due to chemical additives (“dispersants”), ultrasonication or viscous shear stresses. Total viscous dispersion 61 of strongly bonded particles requires very high shear rates, so the fully dispersed viscosity is referred to as the “infinite shear rate viscosity”. 3.2.14 Three-part viscosity The apparent viscosity of a non-Newtonian suspension may be considered to have three separate components (see Fig. 3.2): • The constant Newtonian viscosity of the carrier liquid (μ), • A viscosity increment due to the dispersed particles (μ∞−μ) that is roughly constant, although there will be some variation due to the shape and orientation of the nonspherical particles, and • A structural viscosity increment due to aggregation (η −μ∞) that varies with degree Log (η) of aggregation and is responsible for non-Newtonian behaviour. “Structural” viscosity due to aggregates increasing the volume concentration Increased viscosity due to presence of dispersed particles Viscosity of carrier liquid μ∞ μ Log(γ&) Figure 3.2: Three-part viscosity of yield plastics 3.2.15 Applied and internal shear rates The shearing of the suspension breaks down the structure. The shear rate may be broken down into two components: applied and internal. The applied shear rate is induced by an external source of interest (e.g., a rotating rheometer bob, a pipe wall, a pump impellor or an agitator blade). The internal shear rate comes from all other sources (e.g., thermal energy 62 (Brownian motion), differential settling, electrostatic forces attracting/repelling particles, vibrations from surrounding equipment, ultrasonication, etc.). The internal shear rate may be small but it will always be present. As a result, the total shear rate will always be higher than the applied shear rate. With a Newtonian liquid, the internal shear rate may be ignored because the viscosity is shear rate independent. With a shear rate dependent non-Newtonian suspension the presence of the internal shear rate causes additional structure break down. In most engineering applications, the applied shear rate will be orders of magnitude higher than the internal shear rate and can be assumed to be equal to the total shear rate. However, there are low applied shear rate applications where the internal shear rate will be significant portion of the total shear rate (e.g., when trying to measure the yield stress in a rheometer). Furthermore, there are instances where the internal shear rates are not small, such as when a “vibrator” is used to make concrete flow. 3.3. The Yield Plastic Model 3.3.1 Derivation of the basic model The level of structure will be designated by the scalar variable (λ), where λ = 0 when the particles are fully dispersed and λ increases as aggregation increases (Cheng and Evans 1964). It is proposed that there is a single-valued relationship between λ and the structural viscosity (η −μ∞) and that the structural viscosity increases as λ increases. Assume that a suspension held at a fixed shear rate ( γ& ) until it reaches some equilibrium level of structure (λ). A small relative increase in the shear rate ( dγ& γ& ) will cause a small relative decrease in the structure (dλ/λ). This implies the relationship: dλ λ = −C0 dγ& γ& [3.1] Where C0 is a “local constant” and the negative sign indicates that structure breaks down as the shear rate increases. The same exercise carried out starting from different initial shear rate values would be expected to have the same relationship as Eq. [3.1], but not necessarily the same value of C0. If the value of C0 changes with the shear rate, it is a function of the shear rate. It is also expected to be a function of the slurry properties (e.g., pH, particularly 63 the magnitude and range of bond strengths). It is proposed that this function may be approximated by a simple power relationship: C0 = C1γ& k [3.2] Where C1 is a “material constant” and the exponent (k) is a “scaling constant” related to the tenacity of the aggregates. Physically, a lower value of k implies a faster increase in the average aggregate strength with decreasing λ. The expected value of the scaling factor is: 0 < k ≤ 1. Combining Eq. [3.1] and Eq. [3.2] gives the following structural breakdown model: dλ λ = −(C1γ& k ) = −C1λγ& k −1 dγ& γ& [3.3] By a similar argument, a small relative increase in the shear stress (dτ/τ) will cause a small relative decrease in the structure (dλ/λ). The viscous shear stress on an aggregate will be proportional to the apparent viscosity of the slurry around the aggregate and the effective shear rate in the local flow field. There will be a negligible change in the apparent viscosity of the suspension over a very small change in the shear rate ( dγ& ), so the shear stress (τ) will be proportional to the shear rate ( γ& ). η= τ γ& Apparent viscosity [3.4] Therefore, it is proposed that the exponent (scaling constant) is the same for the shear stress vs. structure relationship. However, the material constant would not be the same. This implies the relationship: dλ λ ( = − C2τ k ) dττ [3.5] dλ = −C2λτ k −1 dτ [3.6] A suspension in dynamic equilibrium at any given shear rate will have a certain average degree or level of structure (λ). Equating Eq. [3.3] and Eq. [3.6] gives: dλ λ = −C1γ& k −1dγ& = −C2τ k −1dτ [3.7] 64 τ k −1dτ = C3γ& k −1dγ& [3.8] Where C3 = C1/C2. The shear stress at a zero shear rate will be τo, which is commonly referred to as the “yield stress”. The shear stress (τ) at an arbitrary value of γ& will depend on how much the structure has broken down from its maximum level. Since the maximum level of structure will be at γ& = 0 , the value of τ may be found by integrating Eq. [3.8] between γ& = 0 and γ& = γ& : τ ∫τ τ k −1 γ& dτ = C3 ∫ γ& k −1dγ& (k ≠ 0) [3.9] 0 o τ γ& ⎡τ k ⎤ ⎡ γ& k ⎤ = C 3⎢ ⎢k⎥ ⎥ + C4 ⎣ ⎦τ o ⎣ k ⎦0 [3.10] τ k − τ ok = C3γ& k + C4 k [3.11] The value of the integration constant C4 in Eq. [3.11] can be determined by considering the end condition where the shear rate approaches zero. As γ& → 0 , τ → τ o so: τ ok − τ ok = C3 (0) + C4 k = 0 Æ C4 k = 0 [3.12] Since k ≠ 0, it follow that C4 = 0 . Therefore, Eq. [3.11] becomes: τ k = τ ok + C3γ& k [3.13] The value of C3 in Eq. [3.13] can be determined by considering the end condition where the shear rate approaches “infinity” and the slurry is fully dispersed. Dividing Eq. [3.13] through by γ& k and using the definition of apparent viscosity Eq. [3.4] gives: τ k τ ok C3γ& k τ ok k = + = + C3 η Æ γ& k γ& k γ& k γ& k [3.14] As the shear rate becomes large ( γ& → ∞ ), the structural viscosity becomes negligible and the apparent viscosity approaches the fully dispersed viscosity ( η → μ∞ ): μ∞k = 0 + C3 Æ C3 = μ∞k [3.15] 65 Substituting the value for C3 into Eq. [3.13] gives the desired constitutive relationship, which may be referred to as the yield plastic model: τ k = τ ok + μ∞k γ& k Yield plastic model [3.16] This is identical to the phenomenological model proposed by Hallbom and Klein (2004). 3.3.2 Alternate forms of the basic model Dividing Eq. [3.16] through by γ& k and using the definition of the apparent viscosity, Eq. [3.4], puts the yield plastic model in terms of the apparent viscosity and shear rate: ηk = τ ok + μ∞k γ& k Yield plastic model [3.17] The definition of the apparent viscosity, Eq. [3.4], may be used to eliminate the shear rate term in Eq. [3.17]: ηk = k τ ok τ ok k k k ⎛τo ⎜ + = + = μ μ η ∞ (τ k η k ) ∞ ⎜⎝ τ k γ& k ⎞ ⎟⎟ + μ∞k ⎠ [3.18] Rearranging Eq. [3.18] gives the yield plastic model in terms of the apparent viscosity and shear stress: μ∞k η = (1− τ ok / τ k ) k Yield plastic model [3.19] It should be noted that equations [3.16], [3.17] and [3.19] are all equivalent forms of the same constitutive model. 3.3.3 Determination of the yield plastic model parameters The parameters of the yield plastic model may be extracted from a rheogram if the measured shear stresses and shear rates are raised to the power of the scaling factor (k) and plotted to give a reduced (i.e., τ k vs. γ& k ) rheogram. As can be seen by inspection of Eq. [3.16], this will give a linear relationship where the y-axis intercept will be τ ok and the slope of the line will be μ∞k . Raising these values to the power of 1/k, gives the model parameters. It may be noted that this is the same method that is used for extracting model parameters for Casson 66 plastics and, implicitly, for Bingham plastics. Figure 3.3 shows a typical reduced rheogram for a yield plastic (τo = 50 Pa, μ∞ = 0.010 Pa-s and k = 0.75). 30 k Shear Stress (Pa ) 25 k 20 η 0.75 15 = 0.010.75 = 0.0316 (Pa-s)0.75 10 τo 0.75 = 500.75 = 18.803 Pa0.75 Case 1 5 y = 0.0316x + 18.803 2 R =1 0 0 50 100 150 k 200 k Shear Rate (1/s ) Figure 3.3: Parameter extraction for yield plastic model When the value of k is not known (or assumed, as is the case with Casson and Bingham plastics) it must be determined using an iterative process. This is a relatively straightforward procedure with a computer spreadsheet. If the assumed value of k is too high, the reduced rheogram flow curve will be concave down. If the value of k is too low, the reduced rheogram flow curve will be concave up. Because the correct reduced rheogram flow curve is a straight line, the extracted parameters will be the same anywhere along the curve. However, there will be a deviation at low shear rates for reasons discussed in the following section. 3.4. Low Shear Rate Behaviour of the Yield Plastic Model 3.4.1 “Zero” shear rate Equation [3.16] was derived using the total shear rate, which include both the applied shear rate and the various internal shear rates (e.g., thermal agitation, differential settling, vibrations). The combined sources of the internal shear rate may be accounted for as a base shear rate ( γ&o ), which is the residual when the applied shear rate is “zero”. The total shear rate ( γ& ) is then equal to the sum of the applied shear rate ( γ&a ) and the base shear rate ( γ&o ): 67 γ& = γ&a + γ&o [3.20] This simply means that the degree of aggregation, and therefore the apparent viscosity, is a function of the total shear rate affecting on the aggregates. Substituting [3.20] into [3.17] gives a version of the yield plastic model “extended” into the low shear rate range: τ ok + μ∞k η = k (γ&a + γ&o ) k Extended yield plastic model [3.21] The base shear rate would normally be low (less than 0.1 s-1) and can usually be ignored at moderate applied shear rates (say, greater than 10 s-1). The base shear rate may be used as a design parameter in certain calculations such as liquefaction of cohesive soils due to an earthquake. The derivation of the yield plastic model assumed that the solids concentration was high enough that a volume spanning aggregate network could form. In a dilute suspension, this is not possible and the “base shear rate” is the shear rate where aggregation terminates due to lack of potential bonding sites. The “yield stress” would be the shear stress value that the rising apparent viscosity trends towards while the aggregates are forming. 3.4.2 Alternate form of the extended yield plastic model The extended yield plastic model may be put in terms of the measured shear stress due to the applied shear rate, which is what is implicitly measured in a rheometer (i.e., as opposed to the total shear stress and shear rate experienced by the slurry): τ k = η k γ&ak = τ ok γ&ak + μ k γ& k (γ&a + γ&o )k ∞ a [3.22] τ ok τ = + μ∞k γ&ak k (1 + γ&o γ&a ) k Extended yield plastic model [3.23] When the applied shear rate is reduced to zero, the suspension is still being agitated by the internal shear rate (i.e., the base shear rate), resulting in a limiting “zero shear rate” viscosity (μo): 68 τ ok μ = + μ∞k Æ τ ok = (μok − μ∞k )γ&ok k (0 + γ&o ) k o [3.24] Substituting Eq. [3.24] back into Eq. [3.21]: ηk = (μ ) μok − μ∞k − μ∞k γ&ok k + μk + μ = ∞ (γ&a + γ&o )k (γ&a / γ&o + 1)k ∞ k o [3.25] Equation [3.25] implies that η ≈ μo as long as γ&a << γ&o . This would appear as a low shear rate Newtonian viscosity plateau if the apparent viscosity were plotted against the applied shear rate. Equation [3.25] can be rearranged to give: η k − μ∞k 1 = k k μo − μ∞ (1 + γ&a / γ&o )k Extended yield plastic model [3.26] This version of the yield plastic model does not contain a yield stress and is therefore a “pseudo-plastic” fluid model. It should be noted that Eq. [3.26] does not imply that there is no yield stress; rather it implies that it is not possible to achieve a true shear rate of zero. The zero shear rate viscosity ( μo ) depends on the value of γ&o , and in most engineering applications (e.g., around vibrating equipment) the internal shear rate will be different than in a carefully isolated rheometer. 3.4.3 Determination of the extended yield plastic model parameters If test data is available at extremely low shear rates, the internal shear rate may be estimated from the measured low shear rate Newtonian viscosity plateau ( μo ) and the yield stress using Eq. [3.22]: γ&o = 3.5. τo k μ −μ k o k ∞ ≈ τo μo [3.27] Summary A new rheological model was developed to describe the flow behaviour of aggregating (“unstable”) suspensions of fine particles. This model was derived using a “structural model”, where the rheological behaviour is caused by the formation and breakage of aggregates. The model accounts for the fact that the bond strengths between dissimilar non69 spherical particles will vary, and that the average bond strength will generally increase as the structure is broken down. A simple power relation was proposed to describe the change in the average bond strength as the shear rate increases. The resulting “yield plastic” model is identical to the phenomenological model proposed by Hallbom and Klein (2004). The yield plastic model has the following advantages: • It is an algebraically simple three-parameter model • The parameters have “real” units (i.e., Pa, Pa-s) • The parameters can be related to physical properties of the suspension • The parameters are easily extracted from a rheogram (i.e., a τ k vs. γ& k graph) The key parameter in the yield plastic model is the “infinite shear rate” viscosity, which is the Newtonian viscosity of the fully dispersed suspension. This is a fixed value for a given set of particles at a given free-particle volume concentration. It does not depend on the pH or ion-content of the carrier fluid, which is expected to simplify the study of suspension surface chemistry. It does not depend on the shear history of the material as long as there is no particle attrition, which is expected to simplify the study of time-dependence. If the internal shear rate (i.e., from sources of agitation not ascribed to the surface applying a shear rate), the yield plastic model predicts the commonly observed low shear rate Newtonian viscosity plateau. 70 CHAPTER 4: COMPARISON WITH EXISTING MODELS “Now, no matter how beautiful a map may be, it is useless to a traveller unless it accurately shows the relationship of places to each other, the structure of the territory.” S.I. Hayakawa, Language in Action (1941 CE) 4.1. Introduction This chapter compares the yield plastic model with various rheological models and constitutive equations that have been proposed for viscoplastic suspensions. The characteristics that are desirable in a “good” constitutive equation are discussed and it is shown that the yield plastic model has most of these characteristics. It is shown the yield plastic model may be used as a general rheological model for shear-thinning homogeneous slurries. 4.2. Desirable Characteristics for a Rheological Model Before discussing existing models, it is useful to consider what characteristics are desirable in a rheological model and constitutive equation. According to Cross (1965) a constitutive equations is “a flow equation relating viscosity to rate of shear. Ideally such an equation should meet the following requirements: 1. It should give an accurate fit of experimental data over a wide range of shear rates. 2. It should involve a minimum number of independent constants [parameters]. 3. The appropriate constants should be readily evaluated. 4. The constants should have a real physical significance.” To this list may be added the following desirable characteristics: 5. It should be algebraically simple. 6. It should be applicable to a wide variety of suspensions. 71 7. It should have parameters that do not vary with the shear rate range of measurement (“constants”). 8. It should have physically consistent units (e.g., Pa and Pa-s). 9. It should be adaptable to engineering design equations of reasonable complexity. Some of these characteristics are compatible: A model with less independent constants (#2) will usually be simpler algebraically (#5) and constants with “real physical significance” (#4) are likely to have physically consistent units (#8). However, some of these characteristics are incompatible: Increasing the number of independent constants (#2) will improve the “fit” to a range of data (#1), but it will increase the complexity of the model (#5), make it harder to determine the parameters (#3), and make it harder to use the model for engineering purposes (#9). A “good” model needs to strike a balance between simplicity, applicability and practicality. 4.2.1 Yield plastic model As shown elsewhere (Hallbom and Klein 2004), the yield plastic model gives an accurate fit to mineral suspensions data at intermediate and high shear rates. In the extended form it will also fit the Newtonian plateau observed in low shear rate data. (#1) The yield plastic model only has three independent parameters. (#2) The independent parameters of the yield plastic model are easily extractable from a linear fit to a reduced rheogram, where the measured shear stresses and shear rates are raised to the power of k, as shown in Fig. 3.3 (Requirement #3) The parameters of the yield plastic model may be related to the fully dispersed apparent viscosity (μ∞), the change in the breakdown rate of aggregates (k) and the apparent yield stress (τo) that the suspension would reach if the true shear rate were to go to zero. ( #4) The yield plastic model is relatively simple algebraically. (#5) Most mineral suspensions are pseudo-Newtonian or shear thinning (Govier and Aziz 1977) and may be modelled with the yield plastic model (Hallbom and Klein 2004). Any fluid or suspension that is currently modelled using the Newtonian, Bingham or Casson models may be modelled using the yield plastic model. (#6) 72 Since the data fit in a reduced rheogram is linear (see Fig. 3.3), the yield plastic model parameters are constant and do not change with shear rate. (#7) The yield plastic model parameters have physically consistent units (Pa, Pa-s, and dimensionless), making them easy to use in the non-dimensional groups that are ubiquitous in fluid dynamics. (#8) The objective of the second section of this Thesis is to develop the basic engineering design equations for pipe flow. It will be shown that relatively simple design equations for laminar and turbulent pressure gradients may be obtained. (#9) The yield plastic model has all of the characteristics that are desirable in a constitutive equation. The objective of the remainder of this Chapter is to compare the yield plastic model with other rheological models that have been proposed over the years. The focus will be on models suitable for suspensions that show a decrease in the apparent viscosity as the shear rate increases. This is “the largest and probably most important class of nonNewtonian fluids” (Cross 1965) including the majority of mineral suspensions. The discussion will be limited to time-independent suspensions. 4.3. Commonly Used Rheological Models This section considers the non-Newtonian rheological models that are currently used with appreciable frequency in the mineral processing industry (see Chapter 2). 4.3.1 Newtonian The single parameter model proposed by Newton (1687) is the quintessential rheological model for fluids. It simply says that the apparent viscosity (or, more correctly, the “fluidity”, which is the inverse of the apparent viscosity) is independent of the shear rate. η=μ Newtonian [4.1] Where μ is the constant absolute viscosity. All suspensions become Newtonian as the volume concentration approaches zero (i.e., they are just the carrier liquid) and even high consistency suspensions appear to approach a Newtonian viscosity plateau at very high shear rates. The Newtonian (“pseudo-fluid”) model is widely used to describe the flow behaviour of low consistency suspensions, even ones that are non-Newtonian. From a 73 practical engineering standpoint, suspensions become “non-Newtonian” when the errors caused by the assumption of Newtonian behaviour get too large to be hidden by various safety and design factors. The yield plastic model reduces to the Newtonian model for τ o = 0 (i.e., no structure forms) and when the shear rate approaches infinity: η k = 0 / γ& k + μ∞k = τ ok / ∞ + μ∞k = μ∞k Æ η = μ∞ 4.3.2 [4.2] Bingham plastic As discussed in Chapter 2, when a non-Newtonian model is used explicitly in the mineral processing industry, it is usually the empirical model proposed by Bingham (1916): η = τ o γ& + η B Bingham plastic [4.3] The Bingham plastic constitutive equation has only two parameters, which have consistent units (i.e., Pa and Pa-s) that are easily determined from a shear stress vs. shear rate rheogram using a linear fit. The “yield stress” can be related to a commonly observed phenomenon: thickened slurry will not flow (or flows extremely slowly) at low shear stresses. A variety of design equations have been developed and tested for the Bingham plastic model over the years. For the design of long distance pipelines for conventional slurries, where the yield stress of the slurry is on the order of 1 Pa, the Bingham model is the main consistency model used to characterize slurry behaviour. The problem with the Bingham plastic model is that, for most slurry, it does not give an accurate fit of experimental data over a wide range of shear rates. As a result, the values of the constants determined by experiment vary depending on the method and shear rate range of measurement (see Table 1.1). The yield plastic model reduces to the Bingham plastic model for k = 1: η 1 = τ o1 / γ&1 + μ∞1 [4.4] 74 4.3.3 Casson and Heinz Casson (1959) proposed a semi-empirical model for aggregating suspensions: η = τ o γ& + ηC Casson [4.5] The relatively simple Casson constitutive equation has two parameters, which have consistent units (i.e., Pa and Pa-s) that are easily determined from a reduced rheogram, where the square root of the shear stress is plotted against the square root of the shear rate. The Casson model accurately describes the flow behaviour of some mineral suspensions and appears to be the “standard” model for blood and molten chocolate. The use of the Casson model for slurry pipeline engineering is becoming increasingly common and a limited selection of design equations has been developed. While the Casson model gives an excellent fit to some rheological data, the fit is usually only approximate and the values of the parameters determined by experiment vary depending on the method and shear rate range of measurement (see Table 1.1). The yield plastic model reduces to the Casson model for k = 1/2: η 1 / 2 = τ o1 / 2 / γ&1 / 2 + μ∞1 / 2 [4.6] For many suspensions, the best fit “switches” between Bingham and Casson models as, say, the solids content increases. To account for the intermediate flow conditions, alternate models of the same form but different exponents may be used. Heinz (1959) proposed one such model. The yield plastic model reduces to the Heinz model for k = 2/3.: η 2 / 3 = τ o2 / 3 / γ& 2 / 3 + μ∞2 / 3 Heinz [4.7] A general class of two-parameters constitutive equations of this form may be obtained by arbitrarily defining different values for the scaling factor (k) in the yield plastic model. This is, in effect, is what is currently done when an engineer chooses between using the Bingham and Casson plastic models (see Fig. 4.1). 75 Yield Plastic Fluids 1000 Bingham Heinz Casson Yield Stress Viscosity 100 η a/ η k = 1/3 k=2 10 1 0.1 0.1 1 10 100 1000 τ/τo Figure 4.1: Effect of changing k for yield plastics 4.3.4 Power law It is a common empirical observation that rheogram data for suspensions often follows a “power law” relationship in the middle shear rate range over a range 10 to 100 fold shear rate range (Govier and Aziz 1977). This relationship can be described using the following constitutive equation (Ostwald 1925, De Waele 1930): η = Kγ& n −1 Power law [4.8] For shear-thinning suspensions, the value of the flow index (n) is less than one. A true power law fluid rheogram would appear as a straight line when plotted on logarithmic coordinates. The consistency index (K) has inconsistent units (i.e., Pa-sn). The power law model once found widespread engineering acceptance because of its simplicity (Hanks 1981) and because design equations are readily available (e.g., Wasp et al. 1977), although its use for engineering seems to be becoming less common. The yield plastic model does not reduce to the power law model. However, over any given range of data (i.e. the practical range of a rheometer) the measured viscosity of a yield 76 plastic fluid is consistent with the power law model. Figure 4.2 shows three cases of the yield plastic model fit with the power law model over a decade of shear rates: • Case 1: τo = 50 Pa, η∞ = 0.01 Pa-s, k = 0.75 • Case 2: τo = 10 Pa, η∞ = 0.01 Pa-s, k = 0.50 • Case 3: τo = 0.5 Pa, η∞ = 0.01 Pa-s, k = 0.25 1 Case 1 Apparent Viscosity (Pa-s) Case 2 Case 3 y = 28.236x -0.8705 2 R = 0.9991 0.1 y = 2.8248x -0.6212 2 R = 0.9965 y = 0.358x -0.3559 2 R = 0.9962 0.01 100 1000 Shear Rate (1/s) Figure 4.2: Power law model fits to yield plastic flow curves The estimated power law parameters are: • Case 1: K = 28.2 Pa-s0.13, n = 0.13 • Case 2: K = 2.82 Pa-s0.38, n = 0.38 • Case 3: K = 0.36 Pa-s0.64, n = 0.64 In each case, the power law fit to the yield plastic “data” is excellent (i.e., R2 > 0.996) and it would not be possible to distinguish between the two based on real (imperfect) data. If the same yield plastic data is fit over the range 100 to 500 1/s, rather than 100 to 1000 1/s, the fit is even better (i.e., higher R2). However, as can be seen below and in Fig. 4.3, the power law parameters have all changed: • Case 1: K = 34.1 Pa-s0.095, n = 0.095 • Case 2: K = 3.72 Pa-s0.33, n = 0.33 77 • Case 3: K = 0.42 Pa-s0.61, n = 0.61 Apparent Viscosity (Pa-s) 1 Case 1 Case 2 Case 3 y = 34.126x -0.9052 2 R = 0.9998 0.1 y = 3.7175x -0.6715 2 R = 0.9987 y = 0.4217x -0.386 2 R = 0.9983 0.01 100 1000 Shear Rate (1/s) Figure 4.3: Power law model fits to yield plastic flow curves (low range) This phenomenon of “variable constants” is observed in power law fits to real data, such as that shown in Table 1.1. The “Newtonian viscosity plateau” behaviour observed at high shear rates requires the flow index to be the same as for a Newtonian fluid (i.e., n = 1). By definition, n ≠ 1 for nonNewtonian fluids, which means that the flow index must change as the shear rate increases. For the shear-thinning fluids, the flow index must increase with increasing shear rate. Both the value and the units of the consistency index (K) also change with shear rate. The power law model [4.8] implies that the apparent viscosity would drop below the carrier liquid’s viscosity at high shear rates and tend towards zero at extremely high shear rates, which “is clearly physically inadmissible” (Hanks 1981). The physically inadmissible limiting conditions reveal the pitfall of a constitutive equation that is not based on an underlying rheological model. 4.3.5 Yield power law The dominant feature of thickened slurry is the presence of a yield stress below, which the material does not flow (see Fig. 1.1), or flows extremely slowly (Barnes and Walters 1985). The Bingham and Casson models contain a yield stress (τo) term and may often be used to 78 model these “plastic” materials. However, the curvature of measured rheograms is usually only roughly approximated by these models. To better describe these materials, Herschel and Bulkley (1926) developed a three-parameter empirical constitutive equation by appending a yield stress to the power law model, giving the yield power law model: η = τ o γ& + Kγ& n −1 Yield power law (Herschel-Bulkley) [4.9] For shear-thinning suspensions the value of the flow index (n) is less than one. The consistency index does not have consistent units (i.e., Pa-sn). The yield power law model contains the Newtonian (τo = 0, n = 1), power law (τo = 0), and Bingham plastic (n = 1) models as special cases and it has been found to give a good fit to the rheograms for a wide variety of suspensions. As a result, researchers commonly use the yield power law model as a general constitutive equation for shear thinning suspensions. Use by non-specialist engineers has been limited due to the complexity of the design equations although its use seems to becoming more common. Figure 4.4 shows three cases of the yield plastic model fit with the yield power law model over a decade of shear rates. 0.1 Reduced Viscosity (Pa-s) Case 1 Case 2 Case 3 y = 0.0943x -0.2195 2 R =1 y = 0.252x -0.3157 2 R = 0.9999 y = 0.1661x -0.2523 2 R = 0.9999 0.01 100 1000 Shear Rate (1/s) Figure 4.4: Yield power law model fits to yield plastic flow curves The estimated yield power law parameters are: • Case 1: τo = 50.15 Pa, K = 0.094 Pa-s0.78, n = 0.78 79 • Case 2: τo = 11.46 Pa, K = 0.252 Pa-s0.68, n = 0.68 • Case 3: τo = 2.08 Pa, K = 0.166 Pa-s0.75, n = 0.75 The fits are essentially perfect (R2 > 0.9999) and it would be impossible to differentiate between the models with a limited range of real data (see Scott Blair 1966). It should be noted that the “best fit” yield stresses found using the yield power law model are higher than the actual yield stress use to generate the “data” (see above). The error is insignificant for Case 1, but it is over 300% for Case 3. The yield power law model has the same general weaknesses as the power law model: no underlying rheological model, inconsistent units, a vanishing high shear rate viscosity, and constants that vary with shear rate range. These weaknesses in the yield power law model were the impetus for the development of the yield plastic model. 4.3.6 Cross and Carreau-Yasuda Careful rheological measurements on concentrated suspensions with an “apparent yield stress” usually reveal that there is a low shear rate Newtonian viscosity plateau (Barnes and Walters 1985). A review paper on the apparent yield stress (Barnes 1999) presented controlled stress (viscosity vs. shear stress) rheograms for 15 materials, none of which showed a Newtonian plateau extending beyond a shear rate of ~10-2 s-1, with most being orders of magnitude below that. This is consistent with the “base shear rate” theory proposed in the previous Chapter. As discussed above, there is also a Newtonian viscosity plateau at very high shear rates. Therefore, the observed rheogram for a shear-thinning suspension with an apparent yield stress over a wide shear rate range would be expected to exhibit high and low shear rate Newtonian “plateau” viscosities and an intermediate region where the viscosity is decreasing over a range of shear rates. Cross (1970) and Carreau (1968) proposed rheological models for this type of suspension behaviour: η − η∞ 1 = ηo − η∞ 1 + (CC γ& )a η − η∞ 1 = ηo − η∞ 1 + (CCY γ& )a ( Cross [4.10] Carreau [4.11] )( 1− n ) / a 80 Cross (1965) originally fixed the value of the dimensionless exponent (a) at 2/3, but later (Cross 1970) changed it to a variable to give the model more flexibility. The four parameter Cross model is a special case of the five parameter Carreau model (a = 1 – n). Both models have consistent units (i.e., Pa-s and s). Neither model is widely used in the minerals industry, but a brief review of recent literature from other fields indicates that the Carreau model is currently more popular among researchers. The Carreau model can be made to fit almost any data set. However, the Carreau model has several setbacks. The five parameters that give the Carreau constitutive equation its flexibility also make the model complicated. This complexity would likely be compounded if the Carreau model were used to develop engineering design equations, although there do not seem to be any such equations available at present. The values of the parameters are difficult to extract from test data, usually requiring numerical curve fitting. The high and low shear rate viscosities would be ambiguous based on the moderate shear rate range data normally available from, say, an industrial grade concentric cylinder rheometer (e.g., 10 to 500 s-1). The effort and/or equipment required to obtain data at a shear rate of, say, 0.0001 s-1 would be difficult to justify for a pipeline where the applied shear rate of interest is on the order 100 s-1. Furthermore, the data for the low shear rate plateau is likely to be inconclusive, such as the data for a concentrated emulsion shown in Fig. 4.5 (Masalova et al. 2005) where increasing shear rate curve exhibits a low shear rate viscosity plateau but the decreasing shear rate curve does not. One option is to model the two curves with separate models, such as the Cross model (increasing shear rate) and the yield power law model (decreasing shear rate). However, using two different rheological models, with seven adjustable parameters, hardly seem justified given that the two data curves are virtually identical for at shear rates covering the five orders of magnitude above ~0.01 s-1. The curvature of the flow curve in Fig. 4.5 is clearly visible (concave up) because of the extremely large data range (the shear rate covers seven orders of magnitude). This is consistent with the yield plastic model. In the previous Chapter, the concept of the applied and base shear rates was proposed, giving the extended version of the yield plastic model. 81 Figure 4.5. Flow behaviour of emulsion (Masalova et al. 2005) η k − μ∞k 1 = k k μo − μ∞ (1 + γ&a / γ&o )k Extended yield plastic model [4.12] Equation [4.13] is of the same general form as the Cross and Carreau models and gives similar results. The yield stress, infinite shear rate viscosity, and scaling factor are easily determined using the method shown above. The base shear rate is determined from the measured low shear rate viscosity plateau. Figure 4.6 shows the effect of adding a small base shear rate to a yield plastic fit to the (scaled) data in Fig. 4.5. The extended yield plastic model parameters are: • Case 1: τo = 42 Pa, μ∞ = 0.012 Pa-s, k = 0.20, γ&o = 0 s-1 • Case 2: τo = 42 Pa, μ∞ = 0.012 Pa-s, k = 0.20, γ&o = 0.012 s-1 With the exception of the tightness of the break in the viscosity at the end of the Newtonian plateau, the rheograms in Figs. 4.6 and 4.7 are virtually identical. The extracted infinite shear rate viscosity parameter is reasonable for an oil-water mixture and the yield stress is similar to the 48-55 Pa estimated by Masalova et al. (2005). For engineering purposes, such as designing an emulsion transfer pipeline, the base shear rate is insignificant and designers would use the basic yield plastic model. 82 1.E+06 Apparent Viscosity (Pa-s) 1.E+05 1.E+04 1.E+03 1.E+02 1.E+01 1.E+00 1.E-05 Case 1 Case 2 1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 Shear Rate (1/s) Figure 4.6: Basic and extended yield plastic representation of emulsion data 4.4. Other Rheological Models This section briefly considers other rheological models for shear-thinning viscoplastic materials that are occasionally used for design purposes. 4.4.1 Sisko One method to eliminate the “vanishing viscosity” problem with the power law model is to append a base viscosity term. This gives the three-parameter Sisko (1958) model: η = Kγ& n −1 + η∞ Sisko [413] Like the power law model, the Sisko model has no underlying rheological model, inconsistent units for the consistency index (i.e., Pa-sn). A laminar pipe flow equation is apparently available, but Govier and Aziz (1977) advise that it is complex and chose not to present it. The author has not seen the Sisko model used for engineering purposes. Figure 4.7 shows three cases of the yield plastic model (see above) fit with the Sisko model over a decade of shear rates. The estimated Sisko parameters are: • Case 1: K = 45.4 Pa-s0.029, n = 0.029, η∞ = 0.0154 Pa-s 83 • Case 2: K = 6.72 Pa-s0.18, n = 0.18, η∞ = 0.0163 Pa-s • Case 3: K = 0.73 Pa-s0.45, n = 0.45, η∞ = 0.0153 Pa-s Reduced Viscosity (Pa-s) 1 Case 1 Case 2 Case 3 y = 45.434x -0.9709 2 R =1 0.1 y = 6.724x -0.8169 2 R =1 y = 0.732x 0.01 100 1000 -0.5539 2 R =1 Shear Rate (1/s) Figure 4.7: Sisko model fits to yield plastic flow curves The Sisko model gives an essentially perfect fit to the yield plastic data over the range considered and the fit is nearly the same for the low range data. It would be impossible to differentiate between the two models based on a limited range of rheometry data. 4.4.2 Ellis The three-parameter Ellis model gives a flow curve similar to the extended yield plastic flow curve “Case 2” in Fig. 4.7. η 1 , = ηo 1 + (τ τ 2 )a −1 Ellis [4.14] The Ellis model is simple and has consistent units (i.e., Pa, Pa-s, and dimensionless), but it predicts a vanishing apparent viscosity at high shear rates. Design equations for laminar pipe flow of Ellis fluids are available (Govier and Aziz 1977), but none were found for turbulent pipe flow. 84 4.4.3 Meter The four-parameter Meter (or Meter-Bird) model may be considered the Ellis model with an additional parameter for the high shear rate viscosity. η − η∞ 1 = ηo − η∞ 1 + (τ τ 2 )a −1 Meter [4.15] The Meter model is relatively simple and has consistent units (i.e., Pa and dimensionless). The Reiner-Philippoff model is a special case of the Meter model: η − η∞ 1 = ηo − η∞ 1 + (τ τ 2 )2 Reiner-Philippoff [4.16] The Meter model gives results equivalent to the Carreau model but they are in terms of the shear stress rather than the shear rate (Hackley and Ferris 2001) so it has the same problems with determining the values of the rheological parameters from limited data as for the Carreau model. Design equations for laminar pipe flow of Meter fluids are available (see Govier and Aziz 1977), but none were found for turbulent pipe flow. Meter fluids can be modelled using the extended yield plastic model. 4.4.4 Eyring The two-parameter Prandtl-Eyring model, based on Eyring’s kinetic theory of liquids, gives results similar to the Ellis model and fits experimental data over limited ranges of shear rates (Govier and Aziz 1977). It has a vanishing high shear rate viscosity, which may be dealt with by adding a infinite shear rate term in the same manner as the Meter model, giving the Powell-Eyring model: η sinh −1 (CEγ& ) = ηo CEγ& Prandtl-Eyring [4.17] η − η∞ sinh −1 (CEγ& ) = ηo − η∞ CEγ& Powell-Eyring [4.18] The two Eyring models have parameters with consistent units (Pa-s and s). The PowellEyring model has the same problems with determining the values of the rheological parameters from limited data as for the Cross and Carreau models. Design equations for 85 laminar pipe flow of Eyring fluids are available (see Govier and Aziz 1977), but none were found for turbulent pipe flow. Powell-Eyring fluids can be modelled using the extended yield plastic model. 4.5. Summary The yield plastic model was shown to have most of the characteristics that are desirable in a rhelogical model. These characteristics are summarized in Table 4.1 (Note: the laminar and turbulent flow equations are presented in the following Chapters). The main exception is the ability to model shear thickening, which is unusual with mineral slurries, except at very high solids contents. The yield plastic model does not predict the Newtonian plateau viscosity observed at low shear rates, but this was shown to be explainable in terms of a small base shear rate (e.g., due to thermal agitation). While the three adjustable parameters make the yield plastic model more complicated than the power law or Bingham plastic models, it is no more complicated than the yield power law model. The yield plastic model may be used as a two-parameter model by fixing one of the parameters, as is implicitly done when the Bingham or Casson models are used. The yield plastic model contains the Newtonian, Bingham plastic, Casson, and Heinz models as special cases. The yield plastic model will give similar results to the other commonly or occasionally used models for shear-thinning fluids, which shows that the past ability of these models to fit some data well, or even extremely well, does not preclude the yield plastic model. More importantly, it means that a single constitutive equation may be used for most slurry, rather than the five or more currently required. 86 Table 4.1: Summary Comparison of Rhelogical Models Model parameters Newtonian 1 Physically consistent units Yes Bingham 2 Yes Yes Yes Yes No Yes No Yes/Yes Yes Casson 2 Yes Yes Yes Yes No Yes No Yes/Yes Increasing Heinz 2 Yes Yes Yes Yes No Yes No No/No No Power law 2 No Yes No No No Yes Yes Yes/Yes Decreasing Yield power law 3 No Yes Yes No No Yes Yes Yes/Yes Increasing Cross 4 (or 3) Yes No No Yes Yes Yes No No/No No Carreau-Yasuda 5 Yes No No Yes Yes Yes No No/No No Sisko 3 No No No No No Yes No Yes/No No Ellis 3 Yes No No No Yes Yes No Yes/No No Meter 4 Yes No No Yes Yes Yes No Yes/No No Reiner-Philippoff 3 Yes No No Yes Yes Yes No Yes/No No Prandtl-Eyring 2 Yes No No No Yes Yes No Yes/No No Powell-Eyring 3 Yes No No Yes Yes Yes No Yes/No No Yield plastic 3 Yes Yes Yes Yes No Yes No Yes/Yes - Ext yield plastic 4 Yes Yes No Yes Yes Yes No No/Yes - Parameters easily determined Yes Yield stress No High shear rate viscosity Yes Low shear rate viscosity Yes Models shear thinning No Models shear thickening No Laminar / turbulent flow eqns Yes/Yes Commonly used for engineering Yes Note: The availability of laminar and/or turbulent flow equations is based onwhat has been found in standard references, such as would be used by practicing hydraulics engineers. 87 CHAPTER 5: LAMINAR PIPE FLOW Everything flows… Heraclitus of Ephesus, (ca. 500 BCE) 5.1. Introduction This chapter presents design equations for the laminar flow of a yield plastic fluid through a pipe or tube. The Buckingham method (for Bingham plastics) is adapted to derive the pressure gradient vs. bulk velocity relationships for yield plastics. The exact general equation is in an integral form that is easily solved by numerical integration. Two general solutions in algebraic form are derived. The first equation is a simplified “engineering approximation” that is always conservative and usually adequate for calculating the design pressure gradient in a pipeline. However, this approximation significantly overestimates the pressure gradient at low velocities. The second equation is a somewhat more complex “rheological approximation” that gives a good fit across the entire velocity range and may also be used to determine the model parameters, if the pressure gradient vs. bulk velocity relationship is known (e.g., from a capillary rheometer). 5.2. Laminar Pipe Flow At low bulk velocities, the flow of a viscous fluid in a cylinder will be laminar, without large irregular fluctuations in the pressure and local velocity. The local velocity of the fluid (u) will vary across the diameter of the pipe. The fluid is continuous but it can be approximated as the flowing in a multitude of concentric thin annuli (or lamella) of fluid flowing at the average velocity at radius r (see Fig. 5.1). A simple force balance between the may be used to show that the shear stress (τ) at radius (r) in the pipe is related to the pressure gradient (ΔP/L) by the following relationship: τ = ΔP ⋅ r 2 L [5.1] The shear stress at the centerline of the pipe (r = 0) will be zero. The maximum shear stress occurs at the pipe wall (r = R) and is referred to as the wall shear stress (τw). 88 V= uavg = Q/πR2 R P0 P1 r dr τ τw L Flow r=0, u=umax r=r, u=u(r) Pipe Wall r=R, u=0 Figure 5.1: Laminar flow in a cylinder τ w = ΔPR 2 L = ΔPD 4 L Wall shear stress [5.2] The relationship between the bulk volumetric flow rate (Q) and the wall shear stress (τw) is given by the following standard equation (e.g., Steffe 1996): τw πR 3 2 Q = 3 ∫τ f (τ )dτ τw 0 [5.3] Calculations are simplified if Eq. [5.3] is defined in terms of the apparent shear rate (Γ): Γ≡ Γ= 8V D 4 τ Apparent shear rate [5.4] τw ∫τ 3 w 0 2 f (τ )dτ Laminar pipe flow [5.5] To use the general apparent shear rate equation [5.5] the following conditions are required: • Steady laminar flow • No rotational or tangential flow velocity • The rheological properties of the fluid/plastic do not vary across the pipe • An incompressible fluid/plastic • A time-independent fluid/plastic • No slip at the pipe wall • No significant end effects Note: A detailed derivation of Eq. [5.5] is given in Appendix A. 89 5.3. Exact Solutions for Yield Plastics 5.3.1 Yield plastic For the yield plastic model, the shear rate is: ⎧(τ k − τ ok )1 / k / μ∞ → τ ≥ τ o & f (τ ) = γ = ⎨ → τ < τo 0 ⎩ Yield plastic [5.6] The shear rate is broken into two regions because the fluid will not flow wherever the shear stress is less than the yield stress (τ ≤ τo), so the shear rate is zero. As a result, when a fluid with a yield stress is in laminar flow there will be a solid core (or “plug”) flowing at the centre of the pipe. Substituting Eq. [5.6] into Eq. [5.5]: Γ= 4 τ τo ∫τ 3 w 0 2 (0)dτ + τw 4 τ μ∞ 3 w ∫τ τ 2 (τ − τ ) k k 1/ k o dτ = o 4 τ μ∞ 3 w τw ∫τ τ 2 (τ k − τ ok )1 / k dτ [5.7] o The radius of the core (rc) can be calculated for yielding fluids in terms of the dimensionless stress ratio (z) and the wall stress ratio (Z), which are always less than unity where there is flow in the pipe. z = τ o /τ = r / R Stress ratio [5.8] Z = τ o / τ w = rc / R Wall stress ratio (at pipe wall) [5.9] The integration of Eq. [5.7] can be simplified by making it a function of the stress ratio changing the upper limits of integration to (τw/τw = 1) and the lower limit to (τo/τw = Z): 1/ k τ 4τ w w τ 2 ⎛ τ k τ ok ⎞ ⎜ − ⎟ Γ= μ∞ τ∫o τ w2 ⎜⎝ τ wk τ wk ⎟⎠ Γ= 5.3.2 4τ w μ∞ dτ [5.10] τw 1 ∫z 2 ( z k − Z k )1 / k dz Laminar flow - yield plastic [5.11] Z Yield plastic fluids (1/k = an integer) Equation [5.11] can be solved explicitly whenever 1/k is an integer by expanding the bracket and integrating. For example, if k = 1/3: 90 Γ= 4τ w μ∞ 1 2 3 ∫ z ( z 3 − Z 3 ) dz = 1 1 4τ w Z μ∞ 1 ∫z 2 2 1 1 2 ( z − 3z 3 Z 3 + 3z 3 Z 3 − Z )dz [5.12] Z 1 4τ w ⎡ z 4 9 z 3 Z 3 9 z 3 Z 3 Zz 3 ⎤ Γ= + − ⎢ − ⎥ μ∞ ⎣⎢ 4 11 10 3 ⎥⎦ Z [5.13] 4τ w ⎛⎜ ⎛⎜ 1 9 Z 3 9 Z 3 Z ⎞⎟ ⎛ Z 4 9 Z 4 9 Z 4 Z 4 ⎞ ⎞⎟ ⎟ −⎜ − + − Γ= − + − μ∞ ⎜⎝ ⎜⎝ 4 11 10 3 ⎟⎠ ⎜⎝ 4 11 10 3 ⎟⎠ ⎟ ⎠ [5.14] 11 10 1 1 2 2 τ ⎛ 36Z 36Z 4Z Z 4 ⎞⎟ Γ = w ⎜1 − + − + μ∞ ⎜⎝ 11 10 3 165 ⎟⎠ 1 3 2 3 k = 1/3 [5.15] The same derivation by expansion and integration has been performed for k = 1/2, 1/4, 1/5, 1/6 and 1/7. The calculations are included in Appendix A. The resulting laminar flow equations are presented in Table 5.1. Note that all of the equations listed in Table 5.1 reduce to the Newtonian solution when the yield stress (and, therefore, Z) goes to zero. 5.3.3 Numerical solutions for yield plastics Equation [5.11] can be easily solved using numerical integration if the definition of the wall shear stress ratio Eq. [5.9] is be used to put Eq. [5.11] in terms of the model parameters (τo, μ∞, and k) and the wall shear stress ratio (Z) alone. ⎞ ⎛ τ ⎞⎛ 1 ⎞ ⎛ τ ⎞⎛ 1 ⎞⎛ 1 Γ = ⎜⎜ o ⎟⎟⎜ ⎟⎜⎜ 4 ∫ z 2 ( z k − Z k )1 / k dz ⎟⎟ = ⎜⎜ o ⎟⎟⎜ ⎟ f n (Z , k ) ⎝ μ∞ ⎠⎝ Z ⎠⎝ Z ⎠ ⎝ μ∞ ⎠⎝ Z ⎠ [5.23] 1 χ = f n (Z , k ) = 4∫ z 2 ( z k − Z k )1 / k dz Apparent shear rate multiplier [5.24] Z For example, [5.24] may be solved with a Casio fx-3950P scientific (pocket) calculator using the following simple program: {Program}: ?ÆA:?ÆB:4∫(X^2(X^A-B^A)^(1/A),B,1) Where: A = k, B = Z, X = z and the result of the integration is χ. The apparent shear rate may be found using Eq. [5.25]: 91 Γ= τw τ χ χ= o μ∞ Z μ∞ [5.25] Table 5.1: Exact Solutions for Selected Yield Plastic Fluids k Newtonian (τo = 0) 1 Bingham 1/2 Casson 1/3 1/4 1/5 1/6 1/7 Laminar Flow Relationship Γ= τw μ∞ [5.17] τ ⎛ 4Z Z 4 ⎞ ⎟ Γ = w ⎜⎜1 − + μ∞ ⎝ 3 3 ⎟⎠ [5.18] τ ⎛ 16Z 4Z Z 4 ⎞⎟ Γ = w ⎜1 − + − μ∞ ⎜⎝ 7 3 21 ⎟⎠ 1 2 [5.15] τ ⎛ 36Z 36Z 4Z Z 4 ⎞⎟ Γ = w ⎜1 − + − + μ∞ ⎜⎝ 11 10 3 165 ⎟⎠ 1 3 2 3 Z 4 ⎞⎟ τ w ⎛⎜ 64Z 96Z 64Z 4Z 1− Γ= + − + − μ∞ ⎜⎝ 15 14 13 3 1365 ⎟⎠ 1 4 3 4 2 4 Z 4 ⎞⎟ τ ⎛ 100Z 200Z 200Z 100Z 4Z Γ = w ⎜1 − + − + − + μ∞ ⎜⎝ 19 18 17 16 3 11628 ⎟⎠ 1 5 Eqn [5.16] 3 5 2 5 4 5 Γ= τ w ⎛ 144 1 / 6 360 2 / 6 480 3 / 7 360 4 / 6 144 5 / 6 4 1 ⎞ Z + Z − Z + Z − Z + Z− Z4⎟ ⎜1 − η∞ ⎝ 23 22 21 20 19 3 100947 ⎠ Γ= τ w ⎛ 196 1 / 7 588 2 / 7 980 3 / 7 980 4 / 7 588 5 / 7 196 6 / 7 4 1 ⎞ Z + Z − Z + Z − Z + Z − Z+ Z4⎟ ⎜1 − η∞ ⎝ 27 26 25 24 23 22 3 888029 ⎠ [5.19] [5.20] [5.21] [5.22] The apparent shear rate multipliers (χ) for various values of k and Zk are listed in Table 5.2. Selected values of the apparent shear rate multiplier (χ) are presented graphically in Fig. 5.2 to show the general relationship 92 Table 5.2: Apparent Shear Rate Multiplier (χ) by Numerical Integration Zk \ k 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9 0.0187 0.0151 0.0115 0.0080 0.0048 0.0024 0.00078 0.00012 0.8 0.0699 0.0605 0.0502 0.0391 0.0276 0.0168 0.0078 0.0021 0.7 0.1467 0.1320 0.1149 0.0953 0.0736 0.0506 0.0284 0.0169 0.6 0.2432 0.2245 0.2019 0.1750 0.1435 0.1078 0.0694 0.0329 0.5 0.3542 0.3332 0.3072 0.2754 0.2367 0.1903 0.1361 0.0772 0.4 0.4752 0.4539 0.4272 0.3937 0.3517 0.2990 0.2333 0.1535 0.3 0.6027 0.5832 0.5586 0.5271 0.4868 0.4328 0.3650 0.2722 0.2 0.7339 0.7185 0.6989 0.6736 0.6406 0.5962 0.5345 0.4448 0.1 0.8667 0.8578 0.8465 0.8317 0.8120 0.7848 0.7452 0.6832 0.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.2 1.0 Bingham 0.8 Casson Χ k = 1/3 0.6 k = 1/4 k = 1/5 0.4 Newton 0.2 0.0 0.0 0.2 0.4 0.6 Z 0.8 1.0 k Figure 5.2: Apparent shear rate multipliers for yield plastics 93 5.4. General Solutions for Yield Plastics 5.4.1 Need for a general solution Engineers will use the laminar flow equation(s) to determine the pressure gradient of the suspension flowing in a pipeline. Their objective will be to pick a pipe size and determine a pump pressure. A solution to the laminar flow equation that requires point-by-point integration or the use of several individual equations is unlikely to be used by design engineers. The objective of this section is to develop a single algebraic equation that gives results to an acceptable level of accuracy and is adaptable to computer spreadsheets. 5.4.2 Engineering approximation Many of the engineers using the laminar flow equation for yield plastics will be nonspecialists who are willing to give up accuracy for simplicity, particularly away from the design point. The approximate engineering solution starts with the observation that all of the exact solutions in Table 5.1 have the same general form: ( χ = 1 − f1n ( Z , k ) Z k + f 2 n ( Z , k ) Z k ) [5.26] The main function (f1n(Z,k)) is the lowest order term. The remainder function (f2n(Z,k)) contains all of the higher order terms. The main and remainder functions for the first three solutions for yield plastic fluids with a yield stress are: n f1 ( Z ,1) = 4 4 = 3 (4 − k )k n f1 ( Z ,1 / 2) = n f1 ( Z ,1 / 3) = n f 2 ( Z ,1) = Z3 3 16 4 = 7 (4 − k )k f 2 ( Z ,1 / 2) = 36 4 = 11 (4 − k )k f 2 ( Z ,1 / 3) = n n k = 1 [5.27] 4Z 1 / 2 Z 7 / 2 − 3 21 k = 1/2 [5.28] 36Z 1 / 3 4Z 2 / 3 Z 11 / 3 − + 10 3 165 k = 1/3 [5.29] This same relationship holds for the main function, f1n(Z,k), in all the solutions presented in Table 5.2. It is proposed that main function holds for all real values of the scaling factor (k) over the range 0 < k < 1. Therefore, Eq. [5.26] becomes: 94 ⎛ ⎞ 4 n χ = ⎜⎜1 − Z k + f 2 ( Z , k ) Z k ⎟⎟ (4 − k )k ⎝ [5.30] ⎠ The approximate value of χ given by Eq. [5.31] is compared to the exact values in Fig. 5.3. 1.2 Bingham 1.0 Casson k = 1/3 0.8 k = 1/4 k = 1/5 Χ 0.6 Newton k = ~1 k = ~1/2 0.4 k = ~1/3 k = ~1/4 0.2 k = ~1/5 0.0 0.0 0.2 0.4 0.6 Z 0.8 1.0 k Figure 5.3: Apparent shear rate multipliers and approximation In order for a suspension to flow in a pipe, the wall shear stress must exceed the yield stress (i.e., τw > τo), which means that Z must be less than one. Since the remainder function contains higher order Z terms, its value decreases as the flow rate increase (i.e., Z decreases) and becomes negligible as Z approaches zero. Using the relationship (1 − ax) ≈ (1 − x) a for x < 1, the value of χ may be approximated by: 1/ k ⎛ ⎞ 4 4 χ ≈1− Z k ≈ ⎜⎜1 − Z k ⎟⎟ (4 − k )k ⎝ (4 − k ) ⎠ [5.31] Substituting Eq. [5.31] into Eq. [5.25] gives the apparent shear rate equation: 1/ k ⎞ 4 τ ⎛ Z k ⎟⎟ Γ ≈ w ⎜⎜1 − μ∞ ⎝ (4 − k ) ⎠ [5.32] Raising Eq. [5.32] to the power of k and rearranging gives the wall shear stress explicitly: 95 τ wk ≈ Γ k μ∞k + 4 τ ok (4 − k ) Laminar flow – yield plastic [5.33] For k = 1, Eq. [5.33] reduces to the Caldwell and Babbitt (1941) approximation for laminar pipe flow of Bingham fluids: 4 3 τ w ≈ Γμ ∞ + τ o [5.34] In an expanded form, suitable for engineering purposes, Eq. [5.34] becomes: 1/ k k ΔP ⎛ 4 ⎞⎪⎧⎛ 8V ⎞ k ⎛ 4 ⎞ k ⎫⎪ ≈ ⎜ ⎟⎨⎜ ⎟ μ∞ + ⎜ ⎟τ o ⎬ L ⎝ D ⎠⎪⎩⎝ D ⎠ ⎝ 4 − k ⎠ ⎪⎭ [5.35] The physical effect of ignoring the remainder function is that Eq. [5.33] will overestimate the pressure gradient at any given flow. For a Bingham plastic, the pressure gradient would be overestimated by 33% at μ∞Γ/τo = ~0 (i.e., near zero flow) and 3.4% at μ∞Γ/τo = 0.5. For a Casson fluid, the equivalent errors are 31% and 2.0%, respectively. Pipelines are generally designed for a maximum flow rate and in most cases τw >> τo (i.e., Z << 1) so the error will be small and conservative. 5.4.3 Rheological approximation In the previous section, it was assumed that the pressure gradient was being determined at a known flow rate, which is the most common engineering application. However, in some situations (e.g., a gravity flow system with a known elevation difference and length), the pressure gradient is known and the flow rate is to be estimated. The use of Eq. [5.33] will lead to a significant underestimation of the flow rate. For example, if the material is a Bingham plastic and the value of Z is greater than 0.75, Eq. [5.33] will predict zero flow although flow always occurs if Z < 1. A better estimate of the low shear rate properties is also required for rheometry (capillary or loop testing), so the improved model developed below will be referred to as the “rheological approximation”. The rheological solution for the laminar flow of yield plastics starts with the observation that for all of the exact solutions in Table 5.1, the limiting values of χ are: • χ = 0 at Z = 1(by definition). 96 • χ = 1 at Z ≈ 0 (reduces to the Newtonian apparent shear rate) This is consistent with the relationship: y = (1 − x) a . Equation [5.26] may be converted into this form as follows: ( ( ) ) χ = 1 − f1n ( Z , k ) − f 2 n ( Z , k ) Z k = (1 − Z k ) f 3n ( Z , k ) [5.36] The exponent function f3n(Z,k) can be extracted by taking the logarithm of [5.36]: ln(1 − f1 ( Z , k ) Z k + f 2 ( Z , k ) Z k ) = f 3 ( Z , k ) ln (1 − Z k ) [5.37] f3 ( Z , k ) = ln(1 − f1 ( Z , k ) Z k + f 2 ( Z , k ) Z k ) / ln (1 − Z k ) [5.38] The numerical values for f3n(Z,k) for the exact solutions in Table 5.1 are listed in Table 5.3. Table 5.3: Exponent Function: f3n(Z,k) f3 Z k k 1.00 0.50 0.33 0.25 0.20 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 1.73 1.65 1.59 1.54 1.50 1.46 1.42 1.39 1.36 2.63 2.54 2.48 2.43 2.39 2.36 2.34 2.32 2.30 3.59 3.50 3.44 3.40 3.37 3.34 3.32 3.30 3.29 4.57 4.48 4.43 4.39 4.35 4.33 4.31 4.29 4.28 5.56 5.47 5.42 5.38 5.35 5.32 5.30 5.29 5.27 3/(3-k)k 1.50 2.40 3.38 4.36 5.36 While it is possible to determine an algebraic expression for f3n(Z,k), there is little point because the resulting equation would be too unwieldy for practical use. Instead, it is noted the value of f3n(Z,k) increases with increasing Z for all values of k, but the it stays within a relatively narrow range of values. Furthermore, the form of Eq. [5.36] forces χ to the correct end points (i.e., χ = 1 at Z = 0; χ = 0 at Z = 1) regardless of the value of f3n(Z,k). The maximum effect of the exponent will be when Zk = ~0.5 and it is proposed that the value of f3n(Z,k) at this value may be used for the entire flow range. 97 As shown in Table 5.3, this may be approximated by the following expression: f3 (Z , k ) ≈ 3 (3 − k )k [5.39] Therefore: χ ≈ (1 − Z k ) 3 (3− k ) k [5.40] The approximate values of χ, based on Eq. [5.40], are given in Table 5.4. Table 5.4: Approximate Apparent Shear Rate Multiplier (χ) Zk \ k 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1.0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9 0.0316 0.0259 0.0197 0.0137 0.0083 0.0040 0.0013 0.0002 0.8 0.0894 0.0777 0.0644 0.0498 0.0350 0.0210 0.0096 0.0026 0.7 0.1643 0.1479 0.1284 0.1061 0.0814 0.0556 0.0310 0.0116 0.6 0.2530 0.2335 0.2097 0.1813 0.1482 0.1109 0.0711 0.0336 0.5 0.3536 0.3328 0.3068 0.2748 0.2360 0.1895 0.1354 0.0767 0.4 0.4648 0.4445 0.4186 0.3860 0.3450 0.2935 0.2291 0.1508 0.3 0.5857 0.5677 0.5445 0.5145 0.4757 0.4248 0.3574 0.2669 0.2 0.7155 0.7017 0.6836 0.6598 0.6282 0.5854 0.5254 0.4376 0.1 0.8538 0.8460 0.8356 0.8217 0.8029 0.7766 0.7379 0.6769 0.0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 Comparison Tables 5.2 and 5.4 shows that Eq. [5.40] is a good approximation throughout the entire range. Substituting Eq. [5.40] into Eq. [5.25] gives the desired relationship: ⎛τ ⎞ Γ ≈ ⎜⎜ w ⎟⎟ 1 − Z k ⎝ μ∞ ⎠ ( ) 3 (3− k ) k Laminar flow – yield plastic [5.41] To appreciate the simplicity of Eq. [5.41] it is useful to compare it to the apparent shear rate equation for the commonly used yield power law model (Herschel and Bulkley 1926). The yield power law consistency model has the form: 98 τ = τ o + Kγ& n Yield power law [5.42] The derivation of the apparent shear rate equation for a yield power law fluid is given in Appendix A. The resulting equation is: 1 n +1 ⎡ n ⎛ τ ⎞ n ⎛ 4n ⎞ ⎛ 2n ⎞ ⎛ ⎞⎤ Γ=⎜ w⎟ ⎜ Z ⎟⎥ ⎟(1 − Z ) n ⎢1 + ⎜ ⎟ Z ⎜1 + ⎝ K ⎠ ⎝ 3n + 1 ⎠ ⎣ ⎝ 2n + 1 ⎠ ⎝ n + 1 ⎠ ⎦ Yield power law [5.43] Using the definition of Z eliminates τw from the right-hand side of [5.62]: ⎛ τ ⎞ (1 − Z k ) Γ = ⎜⎜ o ⎟⎟ Z ⎝ μ∞ ⎠ 3 (3− k ) k [5.44] Equation [5.44] is often easier to use than Eq. [5.41] because it has a single variable (Z) for any given pipe/material system, where the rheological parameters and pipe diameter are fixed. 5.5. Summary In this chapter, the design equations for the laminar pipe flow of yield plastics developed from first principles using a modification of Buckingham’s method. The general equation is given in an integral form that is easily solved by numeric integration. Numerical solutions for the apparent shear rate multiplier (χ) were presented for 88 cases. Exact algebraic solutions were derived for yield plastics with scaling factors where 1/k is an integer (1 through 7). This covers the scaling factor range normally encountered for mineral slurries. An “engineering approximation” of the yield plastic laminar flow equation was developed. This allows the laminar pressure loss gradient of a yield plastic suspension to be estimated directly, if the rheological parameters are known. The simple engineering approximation model is algebraic and gives the pressure gradient results explicitly in terms of the wall stress ratio. The calculated pressure gradient is always conservative, but the difference relative to an exact solution is small except at low shear rates. A “rheological approximation” of the laminar flow equation was also developed. This model gives improved results across the full range of shear stress ratios and much better results at low shear rates. This improved accuracy is of benefit when designing systems for high yield 99 stress suspensions and when doing rheological characterization using a capillary rheometer or a loop test facility. While the rheological approximation of the laminar flow model is more complicated than the engineering approximation, it is considerably simpler than the equivalent yield power law (Herschel-Bulkley) laminar flow equation. 100 CHAPTER 6: TURBULENT PIPE FLOW Big whorls have little whorls, which feed on their velocity, and little whorls have lesser whorls and so on to viscosity. Lewis Richardson, The Supply of Energy to and from Atmospheric Eddies (1920 CE) 6.1. Introduction At very low bulk velocities, the flow of a viscous fluid in a pipe will be laminar with noncrossing streamlines that run parallel to the centre-line of the pipe. As the velocity is increased, flow remains laminar until it reaches a velocity range where it transitions to turbulent flow. Turbulent flow is characterised by large irregular fluctuations in local velocity (turbulent eddies). The rate of increase in the pressure-gradient with increasing velocity is significantly higher in turbulent flow than in laminar flow (see Fig. 6.1). To design a pipeline system for a yield plastic in turbulent flow, an engineer requires: A method to determine the transition velocity and a turbulent flow design equation (pressure gradient vs. velocity). The transition velocity will be addressed in the following Chapter. The objective of this Chapter is to develop the turbulent flow design equation. 6.2. General Comments on Turbulent Flow The flow of slurry in a pipeline will be turbulent if the velocity is sufficiently high. For example Fig. 6.1 shows three pipe flow curves for a 14% by volume (%v) kaolin clay suspension, with 0.1% calcium salt (weight salt/weight clay) and varying amounts of the dispersant tetra-sodium pyrophosphate (TSPP), flowing in a 25.825 mm inside diameter pipe (Litzenberger 2004). The pipe flow curves for the slurry with “0% TSPP” and “0.1% TSPP” show a slowly rising (non-Newtonian) laminar flow pressure-gradient at low velocities. There is a transition point (at ~1.8 m/s and ~2.7 m/s, respectively), after which there is a quickly rising turbulent pressure gradient. (Note: the slurry with “0.27% TSPP” is turbulent throughout the entire range.) 101 Pressure Gradient (kPa/m) 6.0 5.0 4.0 3.0 2.0 1.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Velocity (m/s) 14%w, 0% TSPP 14%w, 0.1% TSPP 14%w, 0.27% TSPP Figure 6.1: Pipe flow curves for kaolin slurry in laminar and turbulent flow (Data from Litzenberger 2004) Because of its chaotic nature, turbulent flow is difficult to analyze from first principles (using the Navier-Stokes equations) even for simple Newtonian liquids. However, researcher have generally used semi-empirical methods, such as Karman-Prandtl velocityprofile based analysis, to develop turbulent flow equations that have been used successfully for many years. Several researchers have proposed semi-empirical methods for the derivation of turbulent pipe flow design equations for non-Newtonian fluids. While none have gained general acceptance, the model proposed by Wilson and Thomas (1985) is increasingly being used for piping system design. For example, it is the recommended model for calculating the turbulent flow pressure gradient of non-settling slurries in “Pipeline Hydrotransport with Applications in the Oil Sand Industry” (Shook, et al. 2002) and “Slurry Transport Using Centrifugal Pumps” (Wilson et al. 2006). 6.2.1 Wilson-Thomas drag reduction model The Karman-Prandtl velocity profile analysis method may be used to determine the pressure gradient of a fluid in turbulent flow. If the fluid is non-Newtonian, then the variable apparent viscosity at the pipe wall (i.e., at the wall shear stress) is used instead of the constant 102 Newtonian viscosity. However, for shear thinning fluids, it is found that the resulting design equations significantly overestimate the observed pressure gradient. In other words, the “effective viscosity” in turbulent flow is significantly less than the “apparent viscosity” implied by the constitutive equation. Wilson and Thomas (1985) proposed a model to explain this lower effective viscosity based on “a conceptual model by Lumley (1973, 1978) to explain the phenomenon of drag reduction in aqueous flows, obtained by the addition of small quantities of certain longchain molecules. These substances act to increase the size of the smallest, dissipative, turbulent eddies” (Wilson et al. 2006). Since the maximum possible eddy size decreases as the wall is approached, increasing the size of the smallest eddies increases the distance from the wall at which the largest eddies and the smallest eddies are the same size. This effectively thickens the viscous sub-layer (see Fig. 6.2). “If other quantities are unaffected the thickened sub-layer will, in turn, produce a higher mean velocity for the same wall shear stress, giving a lower friction factor” (Wilson et al. 2006). Figure 6.2: Eddy scales in turbulent flow (Wilson and Thomas 2006) Wilson and Thomas proposed that the mechanism responsible eddy suppression in nonNewtonian fluids is the rheology of the fluid. They assumed that the rate of energy 103 dissipation in the turbulent energy cascade is proportional to the area under the shear stress vs. shear rate curve (rheogram). The area under the rheogram of a shear thinning fluid will be larger than the area under the rheogram of a Newtonian fluid with the same apparent viscosity. “As the energy available for dissipation is fixed by the ‘turbulent energy cascade’, the result is an increase in the micro-eddy size” (Wilson et al. 2006). They assume that the dissipation process imposes a constant-Reynolds number relation on the smallest eddies, so the eddy size and typical velocity is proportional to αη/ρ. This limits how small the dissipative eddies can be and, therefore, how close they can come the pipe wall. The thickness of the viscous sub-layer would be increased by the ratio α. The term “α” is the ratio of non-Newtonian and Newtonian rheogram areas. 6.2.2 Wilson-Thomas equation for pressure-gradient The derivation of the design equation based on the Wilson-Thomas model is presented cursively in the primary sources (Wilson and Thomas 1985, Thomas and Wilson 1987, and Wilson et al. 2006). The resulting equation for smooth wall (SW) turbulent flow was put in a form that relates the bulk velocity to the friction velocity: ⎛ Dρu * ⎞ V = 2.5u * ln⎜⎜ ⎟⎟ + u * (11.6(α − 1) − 2.5 ln(α )) ⎝ η ⎠ Wilson-Thomas [6.1] u* = τ w ρ Friction velocity [6.2] τw = ΔPD 4L Wall shear stress [6.3] Where V is the bulk velocity, D is the internal diameter of the pipe, ρ is the fluid density, η is the apparent viscosity at the wall shear stress, and ΔP/L is the pressure gradient. WilsonThomas note that “in the part of the flow nearest the pipe axis, some change in the velocity profile would result from a non-zero τo, but the effect on the mean velocity is very small” and “whether such blunting actually occurs remains a moot point” (Wilson et al. 2006). The first term in [6.1] is velocity the fluid would have if it were Newtonian with a viscosity equal to η. The second term accounts for the increase in velocity due to non-Newtonian 104 consistency. In other words, for a given flow rate, the pressure gradient will be lower (i.e., drag reduction). Wilson and Thomas (1985) proposed that, since it is not based on a specific rheological model, the pressure loss equation ([6.1]) is a general equation for smooth wall turbulent pipe flow of homogeneous viscoplastic fluids. All that is required to determine the turbulent pipe flow curve is a determination of η and α as a function of the wall shear stress. This can be done directly from the rheogram, if one is available. However, it is usually more practical to use a constitutive equation. Wilson and Thomas (1985) used their model to derive smooth wall turbulent flow equations for Bingham plastic and power law fluids and later extended it to yield power law fluids (Thomas and Wilson 1987). Shook et al. (2002) presented the equation for smooth wall turbulent flow of Casson fluids, although no derivation was included. 6.3. Modified Wilson-Thomas Model Wilson and Thomas (1985) used Karman’s constants for the turbulent velocity profile. If the values that are consistent with Nikuradse are used, then the general Wilson-Thomas pressure gradient equation becomes: ⎛ 1.12 Dρu * ⎞ V = 2.457u * ln⎜⎜ ⎟⎟ + u * (11.7(α − 1) − 2.457 ln(α )) η ⎝ ⎠ [6.4] (The derivation of Eq. [6.4] is given in Appendix B.) For any rheological model with a limiting high shear rate viscosity (μ∞), Eq. [6.4] may be rearranged as follows: ⎛ ⎛ DρV V = 2.457 ln⎜⎜1.12⎜⎜ u* ⎝ μ∞ ⎝ ⎛ ⎛ μ∞ ⎞⎛ e 4.76 (α −1) ⎞ ⎞ ⎞⎛ u * ⎞ ⎞ ⎟ ⎟ ⎟⎟⎜ ⎟ + 2.457 ln⎜ ⎜⎜ ⎜ η ⎟⎟⎜⎜ α ⎟⎟ ⎟ ⎟ V ⎝ ⎠ ⎠⎝ ⎠⎠ ⎠ ⎝⎝ ⎠ [6.5] Recalling the conventional dimensionless groups for used for hydraulics engineering: 2τ ⎛ u *⎞ f N = w2 = 2⎜ ⎟ ρV ⎝V ⎠ 2 Fanning Friction factor [6.6] 105 Re p = ρVD μ∞ Plastic Reynolds number [6.7] Substituting Eqs. [6.6] and [6.7] into Eq. [6.5]: ⎛ ⎛ 2 = 2.457 ln⎜1.12(Re p )⎜⎜ ⎜ fN ⎝ ⎝ fN 2 ⎞⎞ ⎟ ⎟ + 2.457 ln(β ) ⎟⎟ ⎠⎠ [6.8] Where β is the “drag reduction factor”: ⎛ μ ⎞ e 4.76 (α −1) β = ⎜⎜ ∞ ⎟⎟ ⎝η ⎠ α Drag reduction factor [6.9] Equation [6.51] may be simplified and put into conventional “base-10” engineering form: ⎛ 1.26 1 = −4.0 log⎜ ⎜ Re fN ⎝ p fN ⎞ ⎟ + 4.0 log(β ) ⎟ ⎠ SW turbulent flow [6.10] The first term in Eq. [6.10] is identical to the widely used Nikuradse (1932) equation for turbulent flow of a Newtonian fluid, except that the infinite shear rate (“plastic”) viscosity is used in the Reynolds number. For Newtonian fluids in a hydraulically smooth pipe, it is found that the Nikuradse equation “provides an excellent fit to essentially all reliable data in the range: 3000 < Re < 3,000,000” (Govier and Aziz 1977). The second term in Eq. [6.10] accounts for rheology related drag reduction. 6.4. Turbulent Flow of Yield Plastics In this section, the modified Wilson-Thomas model will be used to derive smooth wall turbulent flow equation for yield plastics. 6.4.1 General comments According to the yield plastic model, the infinite shear rate viscosity is constant (for a given fluid), the non-Newtonian consistency only affects β. When β > 1, there is a reduction in the pressure gradient. When β < 1, there is an increase in the pressure gradient (“drag augmentation”). When β = 1, the second term in Eq. [6.10] becomes zero and there is no drag effect. 106 In order to determine the value of the drag reduction factor, it is necessary to determine the viscosity ratio (μ∞/η) and the area ratio (α), both of which depend on the consistency of the fluid. These relationships are derived in the following Sections. 6.4.2 Viscosity ratio For a yield plastic fluid, the viscosity ratio may be found directly from the apparent viscosity vs. shear stress form of the basic model (see [3.19]): ηk = μ∞k (1− τ ok / τ k ) Yield plastic [6.11] The apparent viscosity ratio is determined at the pipe wall, where τ = τw: ⎛ μ∞ ⎜⎜ ⎝ η k ⎛τ ⎞ ⎛τ ⎞ ⎟⎟ = 1 − ⎜ o ⎟ = 1 − ⎜⎜ o ⎝τ ⎠ ⎠ ⎝τ w k ⎞ ⎟⎟ = 1 − Z k ⎠ [6.12] Where Z = τo/τw is the wall shear stress ratio. Therefore, the viscosity ratio is: 1/ k ⎛ μ∞ ⎞ ⎜⎜ ⎟⎟ = (1 − Z k ) ⎝η ⎠ 6.4.3 Viscosity ratio – yield plastic [6.13] Area ratio The area ratio (α) is the ratio of the area under the rheogram of the fluid in the pipe to the area under a Newtonian rheogram with the same apparent viscosity (see Fig. 6.3). The area ratio is determined up to the wall, where τ = τw and γ& = γ&w . The area for a Newtonian rheogram is the triangle between (0, 0) and ( γ&w , τw): A0 = τ wγ&w Newtonian rheogram area [6.14] 2 The area ratio of a yield plastic may be determined from the shear stress vs. shear rate form of the basic yield plastic model: τ k = τ ok + (μ∞γ& )k Yield plastic [6.15] 107 τ τ k = τ ok + (μ∞γ& )k τo η2 τ w− 2 τ w −1 γ&w − 2 γ& γ&w −1 Figure 6.3: Determination of Area Ratio (α) At the pipe wall: τ wk = τ ok + (μ∞γ&w )k [6.16] Equation [6.16] may be rearranged to give the infinite shear rate viscosity at the wall: ⎛ ⎛ τ ⎞ k ⎞⎛ τ ⎞ k μ = ⎜1 − ⎜⎜ o ⎟⎟ ⎟⎜⎜ w ⎟⎟ = 1 − Z k (τ w γ&w )k ⎜ ⎝ τ w ⎠ ⎟⎝ γ&w ⎠ ⎝ ⎠ ( k ∞ ) [6.17] Substitution of [6.17] into [6.15] and rearranging gives: τ = k τ ok + (1 − Z k )(τ w γ&w )k γ& k = τ w k Z k + (1 − Z k )(γ& γ&w )k [6.18] The area under a rheogram is found by integrating τ from γ& = 0 to γ& = γ&w : γ& w A1 = ∫ τdγ& [6.19] 0 Substituting [6.18] into [6.19]: γ& w k A1 = τ w ∫ ⎛⎜ k Z k + (1 − Z k )(γ& γ&w ) ⎞⎟dγ& 0 ⎝ ⎠ Yield plastic rheogram area [6.20] The area ratio is found by dividing [6.28] by [6.14]: 108 A 2τ γ& w α = 1 = w ∫ ⎛⎜ k Z k + 1 − Z k (γ& γ&w )k A0 τ wγ&w 0 ⎝ ( ) ⎛ ⎞⎟dγ& = 2 γ& w ⎜ Z k + 1 − Z k ∫0 ⎜ ⎠ ⎝ ( ⎛ γ& ⎞ ⎜⎜ ⎟⎟ ⎝ γ&w ⎠ ) k 1/ k ⎞ ⎟ ⎟ ⎠ dγ& γ&w [6.21] Defining x = γ& γ&w ; dx = dγ& γ&w ; and changing the limits of integration: α = 2∫ (Z k + (1 − Z k )x k ) dx 1 1/ k Area ratio [6.22] 0 Equation [6.22] may be solved directly whenever 1/k is an integer by expanding the shear stress term and integrating. For a Casson plastic (i.e., 1/k = 2) the solution is: 1 α = 2∫ 0 ( ( ⎡ 4 α = 2 ⎢ Zx + 3 ⎣ ⎛ ⎝ α = 2⎜ Z + α =1+ ) ) 1 2 ( ( ) ( )) Z + 1 − Z x1 / 2 dx = 2∫ Z + 2 Z − Z x1 / 2 + 1 − 2 Z + Z x dx 4 3 ( ( ) Z −Z x 3/ 2 ) ( 0 ( ) 1 x2 ⎤ + 1− 2 Z + Z 2 ⎥⎦ 0 Z − Z + 1− 2 Z + Z [6.24] ) 12 ⎞⎟ = 1 + ⎛⎜ 83 − 2 ⎞⎟ ⎠ [6.23] ⎝ 8 ⎞ ⎛ Z + ⎜ 2 − + 1⎟ Z 3 ⎠ ⎠ ⎝ 2 1 Z+ Z 3 3 [6.25] Area ratio – Casson [6.26] Table 6.1 presents the resulting area ratio equations for various values of k derived using the method above. (The derivations are presented in Appendix B). An explicit solution for the area ratio was not found for the general case where the scaling factor is any real number in the range 0< k ≤ 1. However, Eq. [6.22] can be easily solved by numerical integration. For example, it can be solved with a Casio fx-3950P scientific (pocket) calculator using the following simple program: {Program}: ?ÆA:?ÆB: 2∫((B+(1-B)X^A)^(1/A),0,1) Where: A = k, B = Zk, X = x and the result of the integration is α. Table 6.2 shows the numerical area ratios for various values of Zk and k. 109 Table 6.1: Exact Solutions for Selected Yield Plastic Fluids k Newtonian (τo = 0) 1 Bingham 1/2 Casson 1/3 1/4 1/5 Eqn [6.27] Area Ratio α =1 [6.28] α =1+ Z 2 3 [6.26] 1 3 α = 1 + Z 1/ 2 + Z [6.29] α =1+ 6 1/ 3 3 2 / 3 1 Z + Z + Z 10 10 10 α = 1+ 20 1 / 4 10 2 / 4 4 3 / 4 1 Z + Z + Z + Z 35 35 35 35 [6.30] [6.31] 70 1 / 5 35 2 / 5 15 3 / 5 5 3/5 1 α =1+ Z + Z + Z + Z + Z 126 126 126 126 126 Table 6.2: Area Ratios for Selected Yield Plastic Fluids Zk \ k 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1.0 2.000 2.000 2.000 2.000 2.0000 2.000 2.000 2.000 0.9 1.900 1.895 1.890 1.884 1.877 1.870 1.862 1.853 0.8 1.800 1.791 1.781 1.771 1.759 1.747 1.733 1.718 0.7 1.700 1.688 1.675 1.661 1.646 1.630 1.613 1.592 0.6 1.600 1.586 1.570 1.554 1.538 1.520 1.502 1.483 0.5 1.500 1.484 1.468 1.451 1.434 1.417 1.399 1.382 0.4 1.400 1.384 1.368 1.352 1.336 1.320 1.304 1.289 0.3 1.300 1.285 1.271 1.257 1.243 1.230 1.218 1.206 0.2 1.200 1.188 1.177 1.166 1.156 1.147 1.138 1.130 0.1 1.100 1.093 1.086 1.080 1.075 1.070 1.066 1.062 0.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 110 Within the accuracy required for the engineering calculations, the area ratio is: α =1+ Zk − 6.4.4 1− k k Z 1− Zk 1+ k ( ) Area ratio – yield plastics [6.32] Drag reduction factor Equations [6.13] and [6.32] may be used in [6.9] to determine the drag reduction factor. However, since both the viscosity ratio and the area ratio are functions of the stress ratio (Z) and the scaling factor (k) alone, the drag reduction factor may be determined directly. Table 6.3 shows the drag reduction factors for various values of Zk and k. Table 6.3: Drag Reduction Factors for Selected Yield Plastic Fluids Zk \ k 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.9 3.817 2.894 2.058 1.330 0.746 0.336 0.103 0.015 0.8 5.007 4.031 3.091 2.224 1.442 0.802 0.338 0.083 0.7 4.940 4.111 3.294 2.507 1.768 1.108 0.565 0.190 0.6 4.348 3.706 3.055 2.482 1.828 1.251 0.735 0.317 0.5 3.602 3.123 2.657 2.191 1.733 1.284 0.844 0.442 0.4 2.877 2.548 2.225 1.904 1.581 1.251 0.909 0.559 0.3 2.246 2.033 1.830 1.624 1.412 1.191 0.950 0.673 0.2 1.727 1.607 1.493 1.374 1.253 1.123 0.970 0.781 0.1 1.313 1.267 1.216 1.166 1.115 1.056 0.987 0.890 0.0 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 Within the accuracy required for engineering calculations, the drag reduction factor for yield plastics (based on the Wilson-Thomas model) is: β ≈ e 4.76 Z (1 − Z k )k k 2 (1− 0.5 Z ) k Drag reduction factor – yield plastics [6.33] 111 6.5. Comments on Turbulent Flow The objective up to this point was to use an accepted theory to derive the smooth wall turbulent pipe flow design equation for yield plastics. Except as noted, the assumptions made by the original researchers were used. This Section considers the observed flow behaviour of homogeneous slurries and the physical basis of the yield plastic model in relation to the underlying assumptions and the design equation. 6.5.1 Equivalent “viscosities” It may be observed in Table 6.3, that the Wilson-Thomas model predicts that the value of β Æ 1 as the value of Zk Æ 0 (i.e., τw >> τo) in all cases. In other words, when the pressure gradient is high, the drag reduction becomes negligible. The “effective fully turbulent viscosity” is equal to the “infinite shear rate viscosity” (μ∞). It may also be observed that the pipe flow curves shown in Fig. 6.1 are markedly different at low velocities but they converge with the dispersed slurry flow curve (“0.27% TSPP”) at high velocities. The “effective fully turbulent viscosity” is equal to the “chemically dispersed viscosity”. It follows that the “effective fully turbulent viscosity”, the “chemically dispersed viscosity” and the yield plastic “infinite shear rate viscosity” are identical. Specifically, they are the Newtonian viscosity of a suspension of fully dispersed and randomly oriented particles. (Note: This hypothesis assumes that there is no physical change in the particles, such as “swelling”, dissolution or attrition. It is also recognized that some variation may occur due to alignment of non-spherical particles.) 6.5.2 Choice of consistency model The implications of the hypothesis above on the choice of consistency model may be demonstrated using following “thought experiment”. Consider an aggregating slurry that is in fully developed smooth wall turbulent flow with a high flow rate pressure gradient that implies an “effective fully turbulent viscosity” of 4 mPa-s. Furthermore, assume that there is no change in the pressure gradient at high flow rates when a dispersant is added, which implies that the “chemically dispersed viscosity” is 112 also 4 mPa-s. (Note: this essentially describes the flow curves presented in Fig. 6.1). Finally, assume that accurate rheological data (from a concentric cylinder or capillary rheometer) is available for the aggregated slurry and that the rheologist “fits” the data with the Bingham plastic (k = 1) and Casson (k = 1/2) models and determines the following infinite shear rate (“plastic”) viscosities (Note: see Table 1.1): • Bingham: 8 mPa-s • Casson: 2 mPa-s If an engineer were to use the Bingham viscosity to calculate the fully turbulent pipe flow curves with a Newtonian design equation (or the Wilson-Thomas equation), it would be found that at very high velocities the predicted pressure gradient was ~15% higher than the measured pressure gradient. Some “mechanism” would be required to explain the observed “drag reduction”. If the Casson viscosity were used instead, the predicted pressure gradient would be ~15% lower than the measured pressure gradient. Some “mechanism” would be required to explain the observed “drag augmentation”. Furthermore, it would be necessary to have another “mechanism” to explain why adding a dispersant would decrease the viscosity in one case (Bingham) and decrease the viscosity in another (Casson). The difficulty with assuming a physical mechanism (e.g., boundary layer thickness changes or wall roughness) is that the different “behaviour” described is not due to a variation in the slurry consistency, rather it is due to the (arbitrary) choice of consistency model used to fit rheological data. With the yield plastic model, the solution to this conundrum is trivial: the viscosity of the fully dispersed slurry is 4 mPa-s and there is no drag effect. Whether or not the particles aggregate when they are at lower velocities is irrelevant. 6.5.3 Transitional flow range The Bingham plastic data originally presented by Wilson and Thomas (1985) only extends up to two or three times the apparent transition velocity (see Fig. 6.4). With a Newtonian fluid, laminar-turbulent transition is assumed to begin when Re = 2100 but fully developed turbulent flow is not reached until Re ≈ 4000 (Streeter and Wylie 1985), which is to say roughly twice the transition velocity. There is no reason to suppose that a non-Newtonian 113 fluid would transition more rapidly. The presented data is, therefore, almost all in the laminar flow and transition range. Figure 6.4: Comparison with Bingham plastic data and the theory of Hanks (Wilson and Thomas 1985) Nevertheless, the transition range data presented showed reasonable agreement with their model and appeared to be both qualitatively and quantitatively superior to the Hanks model. It is plausible that, although their analysis assumes fully turbulent flow, the phenomenon that the Wilson-Thomas model actually addresses is in the transition region where the aggregates are only partially dispersed. However this “structural breakdown” transition would be difficult to separate from what might be called “onset” transition. For Newtonian fluids flowing in the transition range it is found that the average pressure gradient has a (vaguely defined) value between that of laminar flow and that of fully turbulent flow. At any given point along the pipeline (i.e., at an instrument location), when the Reynolds number is “between 2000 and 4000 the flow randomly oscillates between being laminar and turbulent” (Potter and Wiggert 1997). In terms of the fluid flowing in the entire pipeline, some (moving) zones will be turbulent while the remainder of the flow is laminar. Since the laminar flow pressure gradient is substantially lower than the turbulent flow pressure gradient, this means that the average pressure gradient in the transition range will be lower than the fully turbulent pressure gradient. There is no immediately apparent reason that a non-Newtonian fluid would transition simultaneously along the entire length of pipe. In fact, given the inherent variability of slurries (e.g., due to small differences in solids content), it seems less likely than with Newtonian fluids. 114 6.5.4 Reynolds stresses Many researchers, including Wilson and Thomas (1985), calculate the average turbulent flow “shear stress” using the same force balance as is used for laminar flow. They neglect the Reynolds stresses due to the erratic turbulent fluctuations. When the consistency of a suspension is non-Newtonian due to the presence of aggregates (i.e., the yield plastic hypothesis), the Reynolds stress becomes part of the total stress that disperses the particles. The result is that total shear stress will be higher than the average axial shear stress. Using the terminology of Chapter 3, Section 4, the Reynolds stresses increase the base shear rate (see [3.21]). With a Newtonian fluid, the presence of Reynolds stresses does not affect the viscosity since it is shear rate independent (by definition). In laminar flow, the Reynolds stresses are zero. Two predictions may be made based on the inclusion of Reynolds stresses in the total shear stress. The first is that diameter of the plug in turbulent flow will be smaller than it would be predicted to be in laminar flow at the same pressure gradient. The plug would be expected to quickly erode and then collapse as the turbulence increases. The second prediction is that, when a suspension is shear thinning, the apparent viscosity in the boundary will be significantly lower than would be calculated based on the average axial wall shear stress. The effects will be most pronounced in the transition zone, where eddies are present and viscous effects are significant. The pressure gradient would approach that of a “chemically dispersed” suspension when the bulk velocity is a relatively small multiple of the transition velocity. 6.5.5 The “pseudo-fluid” approximation Some form of “drag reduction” may be considered to occur anywhere the pressure gradient is below that of the fully turbulent pressure gradient of the dispersed suspension. In the analysis of loop test data, failure to understand the cause of drag reduction effects may lead to erroneous conclusions about the required pipeline pressure gradient if it is necessary to scale between pipe sizes. An understanding of drag reduction can also explain certain operating anomalies, such as when adding dispersant to “thin out” a suspension causes the pressure gradient to increase (see Fig. 6.1 at ~2.7 m/s). 115 However, engineers performing hydraulic analysis as part of the design of a slurry pipeline system can usually ignore drag reduction (i.e., assume β = 1). This “pseudo-fluid” approximation has long been used for Bingham plastics (e.g., Hedström 1952). Gover and Aziz (1977) noted “on the basis of data of [several research groups], it seems well established from an empirical point of view that, at Rep values above [the transition Reynolds number], and following some transition range, the friction factor for Bingham plastics is dependent only on Rep and the pipe roughness. For smooth pipes the fN- Rep relationship is close to that for Newtonian fluids”. It is proposed that this is true for all yield plastics if μ∞ is equal to the dispersed viscosity. The hydraulic engineer may use the infinite shear rate viscosity with any standard Newtonian turbulent flow equation, such as the Nikuradse equation. ⎛ 1.26 1 = −4.0 log⎜ ⎜ Re fN ⎝ p fN ⎞ ⎟ ⎟ ⎠ SW turbulent flow Pseudo-fluid [6.34] The equation proposed by Knudsen and Katz (1958) for SW turbulent flow gives similar results to the Nikuradse equation in the Reynolds number range of interest to slurry pipeline designers and is easier to use: fN = 0.046 Re 0p.2 [6.35] It should be noted that the Wilson-Thomas model and the pseudo-fluid model both predict that the SW turbulent flow pressure gradient will approach the same limiting flow curve. The difference is how quickly the limiting flow curve is reached. 6.5.6 Rough wall turbulent flow While the current research focuses on smooth wall (SW) turbulent flow, it is possible to make some reasonable assumptions about rough wall turbulent flow. At relatively low velocities (i.e., at a small multiple of the transition velocity) in pipelines of normal roughness (e.g., commercial steel pipe), the pressure gradient of Newtonian fluids is the same as for a smooth walled pipe. At much higher velocities, the pressure gradient is found to be independent of the fluid viscosity, but it is strongly affected by the roughness of 116 pipe. The Nikuradse equation for Newtonian fluids in fully rough wall (FRW) turbulent flow is (Govier and Aziz 1977): 1 ⎛ ε ⎞ = −4.0 log⎜ ⎟ fN ⎝ 3.7 D ⎠ FRW turbulent flow [6.36] Where (ε/D) is the relative roughness. Since FRW turbulent flow is independent of the fluid’s viscosity, it is reasonable to assume that it will be unaffected by a variable (i.e., nonNewtonian) consistency. However, the presence of the solid particles in a suspension would be expected to affect the “hydraulic roughness”. Slatter (1996) suggests using the ~85%w passing diameter of the particles for ε, if it is larger than the hydraulic roughness of the pipe wall. Commonly used values for hydraulic roughness are ~50 μm for commercial steel and rubber-lined pipes and ~20 μm for plastic (HDPE) pipes. With the “fine” (<37 μm) particles typically considered to be non-settling, [6.79] can be used without modification. In the flow range between SW and FRW turbulent flow, there is a region where both the Reynolds number and the relative roughness affect the flow. This intermediate region is referred to as partially rough wall (PRW) turbulent flow. The empirical equation proposed by Colebrook (1939) for PRW flow of Newtonian fluids in naturally rough (commercial) pipe has been found to give a good approximation to measurements. It is the basis for the Moody diagram and most “friction loss” tables. The equivalent equation for yield plastics may be found by combining [6.10] and [6.36] in the same manner as Colebrook (1939) for Newtonian fluids and proposed by Slatter (1996) for non-Newtonian slurries (using a different SW turbulent flow model): ⎛ 1.26 1 ε ⎞⎟ = −4.0 log⎜ + ⎜ β Re f 3.7 D ⎟⎠ fN p N ⎝ PRW turbulent flow [6.37] As discussed in Section 6.5.5, for engineering analysis it should be adequate to use the “pseudo-fluid” assumption (i.e., β = 1), but this needs to be confirmed experimentally 117 6.6. Summary In this Chapter, the turbulent flow analysis method proposed by Wilson and Thomas (1985) was described in detail (as a literature review). The Wilson-Thomas model was then used to determine the fully turbulent flow design equations for yield plastics. The resulting equation was shown to be very close to the widely used Nikuradse equation for smooth wall turbulent flow if the infinite shear rate (or “plastic”) viscosity is used rather than the apparent viscosity. The use of this “plastic Nikuradse” equation requires the inclusion of a “drag reduction” factor (β). The assumptions of the Wilson-Thomas model were used to derive an explicit equation for the drag reduction factor, which was shown to be a function of the stress ratio (Z) and the scaling factor (k) alone. Several comments were made based on the observed flow behaviour of homogeneous slurries and the physical basis of the yield plastic model in relation to the underlying assumptions and the design equation. These include: • The yield plastic “infinite shear rate viscosity” is the same as the “effective fully turbulent viscosity” and the “chemically dispersed viscosity”. Differences are generally due to using the incorrect model (e.g., the wrong scaling factor). • Reynolds stresses are expected to cause the drag reduction effects to be eliminated (i.e., β Æ 1) more quickly than would be predicted by the Wilson-Thomas model. For engineering purposes, it is adequate to use the “pseudo-fluid” approximation • It was proposed that the “pseudo-fluid” version of the Colebrook equation could be used to account for the effects of wall roughness and the solid particles. 118 CHAPTER 7: LAMINAR-TURBULENT TRANSITION The classification of the constituents of a chaos, nothing less is here essayed. Herman Melville, Moby Dick (1851 CE) 7.1. Introduction The pipe flow of any viscous fluid is laminar at low velocities and turbulent at high velocities. It follows that there will be some intermediate velocity where the flow transitions from laminar to turbulent. The transition velocity is often referred to as the “critical velocity” because it is important to most aspects of slurry pipelining: • When doing theoretical research, the interest is mainly the onset of flow instability. The “critical velocity” is when eddies appear anywhere in the flow field. • When characterizing the rheology of a suspension in a capillary rheometer or loop test facility, the accuracy of the characterization is improved by using data over the widest possible range of velocities. However, the relationships used to determine the rheological parameters are based on laminar flow and utilization of data in the turbulent region will lead to misleading conclusions, such as shear thickening. The “critical velocity” is when the pressure gradient deviates from the laminar flow pressure gradient. • When designing/operating a slurry pipeline where all of the particles are non-settling (i.e., clay sized) the most cost effect operating point is near where the specific energy usage (power per tonne transported) is lowest. The “critical velocity” is where the differential pressure gradient is increasing at the same rate as the differential velocity. • When designing/operating a slurry pipeline where there are enough non-settling particles to make the consistency non-Newtonian but there are sufficient settling particles to be a “sanding” risk, the velocity must be high enough that the flow is 119 sufficiently turbulent to keep the coarse particles suspended. The “critical velocity” is when fully developed turbulent flow is established. The objective of this Chapter is to develop a methodology for determining the transition velocity of yield plastics. The difference in the definitions used to determine the velocity at which transition occurs will be discussed. The existing transition criteria that are commonly for Bingham plastics will be compared. A new transition criterion will be proposed and used to develop a design equation for the laminar-turbulent transition of a yield plastic fluid flowing in a pipe or tube. 7.2. Laminar-Turbulent Transition Experimental and theoretical studies have been performed on the transition of nonNewtonian suspensions since at least the early 1950’s and a number of semi-empirical relationships have been proposed for the transition criteria. Despite this, consensus has not been reached even for the well-studied Bingham plastic model (Frigaard and Nouar 2003). Consider the following question: “What would happen to the transition point friction factor in a large diameter pipe if the yield stress of a Bingham plastic is doubled?” According to the commonly used transition criteria discussed below, the answer is that the friction factor would either decrease (e.g., Slatter and Wasp 2000) or increase (e.g., Hanks 1963), unless it stayed the same (e.g., Metzner and Reed 1955). While this is undoubtedly true, it is of little practical use. In order to understand this lack of consensus, after more than a half century of research, it will be useful to consider the transition of Newtonian fluids. 7.2.1 Newtonian transition The generally accepted criterion for laminar-turbulent transition of Newtonian fluids is straightforward: transition occurs when the Re = 2100. This is, of course, an oversimplification. When a Newtonian fluid is flowing in a pipe, deviations from true laminar flow may be observed at Re ≈ 1225, occasional eddies are observed at Re ≈ 2100 and turbulence is usually fully developed by Re ≈ 3000, although under special circumstances laminar flow may be observed at Reynolds numbers as high as 50,000 (Govier and Aziz 1977). 120 It is more reasonable to claim that the majority of transition occurs when 2000 < Re < 3000. Laminar flow equations can be used when Re < 2000. Turbulent flow equations can be used when Re > 3000. In the transition range (2000 < Re < 3000), the flow regime will tend to oscillate between laminar and turbulent flow with the frequency of turbulence increasing as Re increases. Since the turbulent flow pressure-gradient is significantly higher than the laminar flow pressure gradient, there is a rapid rise in the energy dissipation rate (i.e., “head loss”) over a relatively short transition range. 7.2.2 Two transition points Based on the behaviour of Newtonian fluids, two distinct definitions may be made for “transition”. The first is the onset of instability in the laminar flow, characterized by the first appearance of eddies (i.e., the “instability point transition”). The second, at a somewhat higher velocity, is where the pressure gradient deviates significantly from the laminar flow pressure gradient (i.e., the “pressure break point transition”). With Newtonian fluids, the pressure break point transition velocity is a small and relatively constant multiple (~105%) of the instability point transition velocity, so there is little need to differentiate. With a non-Newtonian fluid, particularly one with a significant yield stress, the multiple cannot be assumed to be small nor constant. It is proposed that one reason that “the discrepancy between theoretical and experimental values is still not fully resolved” (Frigaard and Nouar 2003) is a failure to differentiate between the two transition points. In the following discussion, “transition” will refer to the pressure break point transition, unless otherwise noted, as this is the point of interest to pipeline engineers. Another probable reason for the discrepancy is the practice of “force fitting” rheology data to the wrong consistency model. For example, using the Bingham plastic model for suspension that would be better described by the Casson fluid (or something in between). This problem may be addressed by developing a transition equation for a single consistency model that can describe a wide range of suspensions: yield plastic. 121 7.3. Transition Criteria Before considering the yield plastics in general, it will be useful to review the special case of the yield plastic model where the majority of the research has been performed: Bingham plastic (k = 1). While many transition equations have been developed, most are based on criterion that may be included in one of four groups: intersection, friction factor, yield stress, and local stability. These will be considered separately. 7.3.1 Intersection criterion Hedström (1952) proposed: “turbulence sets in when the [laminar flow] curves and the Newtonian [turbulent flow] curve intersect.” The intersection criterion only requires the use of standard non-Newtonian laminar flow and Newtonian turbulent flow pressure gradient equations, but it is found that it underestimates the observed transition velocity. NonNewtonian turbulent flow models may be used and will give somewhat different results for the intersection point (Slatter 2007). 7.3.2 Friction factor criterion Metzner and Reed (1955) proposed: “non-Newtonian fluids should begin to deviate from the laminar [flow] line at approximately the same ratio of viscous shear to inertial forces as do Newtonian data for smooth pipes, namely at fN = 0.008.” If the Newtonian critical Reynolds number is taken to be 2100, then the critical friction factor is: ( f N )C = 0.0076 Metzner-Reed [7.1] One advantage of this method is that the transition friction factor may be directly determined from loop test data without knowledge of the fluid’s rheology (see [6.2]). For a Bingham plastic, the Caldwell-Babbitt equation (see [5.55]) may be used to approximate the laminar flow friction factor giving the following relationship (Thomas D.G. 1963): ⎛ 1 HeB Re BC = 2100⎜⎜1 + ⎝ 6 Re BC ⎞ ⎟⎟ ⎠ [7.2] Where: 122 Re B = ρVD ηB Bingham Reynolds number [7.3] HeB = τ o ρD 2 η B2 Bingham Hedström number [7.4] The observed transition friction factors are often significantly below 0.0076 and the criterion tends to underestimate the break point velocity. 7.3.3 Yield stress criterion The Hedström number has a constant value for any given Bingham plastic flowing in a pipe with a fixed diameter. In industrial sized slurry pipelines handling Bingham plastic slurries, it may be assumed to be in the 105 < HeB < 107 range if transition is an issue. When HeB > ~105, HeB >> ReB and equation [7.4] may be simplified to: Yield stress criterion (He > 105) [7.5] Re BC = C2 HeB0.5 Where C2 is a dimensionless measure of transition. Based on [7.5], C2 is ~19. Equation [7.5] may be rearranged into a form where the critical velocity may be found directly if the yield stress is known: Vc = C2 τ o ρ [7.6] Researchers using various empirical and semi-empirical method have proposed different values for C2 at high values of HeB: ≥20 (GIW Hydraulics Laboratory (ref. Wilson et al. 2006)), 25 (Wilson and Thomas (A.D.) 2006), and 26 (Slatter and Wasp 2000). 7.3.4 Local stability criterion Hanks (1963) proposed a theoretical model based on instability due to rotational momentum transfer that predicts that the instability point transition will occur when the value of Z equals a critical value, which can be found from: Z HeB = 3 (1 − Z ) 16,800 Hanks [7.7] A similar result is found using the instability theory of Ryan and Johnson (1959). The critical plastic Reynolds number is found using the Buckingham equation [5.35]. The results 123 of the Hanks (1963) model may be approximated by assuming that the critical plastic Reynolds number is assumed to be the greater of 2100 or: Re BC ≈ 121HeB0.35 Local stability criterion (ReBC > 2100) [7.8] The local stability method has a stronger theoretical foundation than the other methods and it is not confined to pipe flow. Nevertheless, the Hanks criterion is rarely used by engineers because the predicted transition appears to deviate from the observed transition (i.e., the Hanks criterion predicts a lower ReBC than observed) when the HeB > 105 (Hanks 1963, Slatter and Wasp 2000). One probable cause of this deviation is that the Hanks (1963) theory predicts the instability point transition, while the transition point is generally “observed” as a change in the measured pressure gradient (i.e., the pressure break-point transition). It may be noted that when HeB = 105, Z = 0.55 (see [7.8]), which means the plug is >55% of the pipe diameter in the Hedström range of interest. If instability point transition occurs when the unyielded plug occupies a large fraction of the pipe cross section, then turbulence in the flowing annulus would tend to decrease the size of the plug, limiting the rise in the pressure gradient. Another possible cause of the deviation is that the material has been “force fit” to the Bingham plastic model, when it would be better fit by another model – specifically, the yield plastic model. 7.3.5 A new transition criterion As discussed in Section 7.3.1, Hedström (1952) assumed that transition would occur where the Bingham plastic laminar flow curve and the pseudo-fluid turbulent flow curve intercept. However, this criterion is not correct even for a Newtonian fluid. If the Newtonian fully turbulent flow curve given by the Knudsen-Katz equation ([6.35]) is extended back to the Newtonian laminar flow curve, they intersect at Re = ~1500. This is well below the general accepted transition Reynolds number (2100), although it is consistent with the Reynolds number where non-rotational deviations in the flow are initially observed. When the Reynolds number is 2100, the Newtonian laminar and turbulent friction factors are: • Laminar flow: (fN)lam = 0.0076 • Turbulent flow: (fN)turt ≈ 0.0100 124 Newtonian transition is, therefore, assumed to occur at a velocity when the smooth wall fully turbulent flow friction factor is ~130% of the laminar flow friction factor. At a given flow rate and pipe size, the pressure-gradient is directly proportional to the friction factor. The expected behaviour of a Newtonian fluid flowing in a pipe, in term of a P/L vs. V flow curve, could be described as: The pressure-gradient will follow the laminar flow curve to a velocity where the fully turbulent pressure gradient is about 130% of the laminar flow pressure gradient. As the flow rate is increased further, the pressure-gradient “breaks” (deviates above the laminar flow curve) and increases rapidly towards the fully turbulent flow curve. The pressure-gradient follows the fully turbulent flow curve once the velocity is more than ~150% of the “break-point” velocity. Consider the kaolin slurry flow curves shown in Fig. 6.1. If one were to describe the flow curves for “0%TSPP” and “0.1%TSPP” in relation to the fully turbulent flow curve of the dispersed “0.27%TSPP” sample, the qualitative description would be identical to the one given for Newtonian fluids in. There would, in fact, be little difference in the quantitative description. The measured data in Fig. 6.1 (Litzenberger 2004) is repeated Fig. 7.1 along with a “transition” curve that is the turbulent flow pressure gradient of the dispersed“0.27%TSPP” sample divided by 130%. If the laminar flow data is extrapolated to the “transition” curve, the intersection is at the velocity where the measured pressure gradient deviates significantly (~10%) from what would be expected in laminar flow. Since the energy dissipation rate is directly proportional to the pressure-gradient, the normally assumed Newtonian transition occurs when the smooth wall fully turbulent flow energy dissipation rate is ~130% of the laminar flow energy dissipation rate. The kaolin slurry curves in Fig. 7.1 (with undefined non-Newtonian consistencies) transition at a similar energy dissipation rate ratio. Therefore, it is proposed that this is a general criterion, and that pressure break point transition occurs when: ( f N )turb = 1.3( f N )lam [7.9] 125 Pressure Gradient (kPa/m) 6 5 4 3 2 1 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Velocity (m/s) 14%v; 0%TSPP 14%v; 0.1%TSPP 14%v; 0.27%TSPP Transition Figure 7.1: Pipe flow curves for kaolin slurry - transition (Data from Litzenberger 2004) The smooth wall turbulent friction factor is used for transition because wall roughness does not affect laminar flow. If the “plastic” Knudsen-Katz equation [6.35] is used to describe the turbulent friction factor, then the critical friction factor is: ( f N )C = 0.0460.21.3 = 0.0354 0.2 Re p Transition criteria – yield plastics [7.10] Re p Transition occurs when the laminar flow curve equation for yield plastics [5.41] intercepts the transition curve defined by equation [7.10]. The intercept can be found graphically or by numerical iteration, but it is often useful to be able to estimate the transition velocity using an explicit algebraic equation. 7.4. Transition of Yield Plastics The same dimensionless groups used to describe the transition of Bingham plastics may be generalized for yield plastics by using the infinite shear rate viscosity (μ∞) of the fully dispersed suspension: 126 Re p = He = ρVD μ∞ Plastic Reynolds number [7.11] τ o ρD 2 1 = Zf N Re 2p 2 2 μ∞ Hedström number [7.12] Recalling that the laminar flow curve for a yield plastic is described by: Γ≈ 3 (3− k ) k τw ( 1− Z k ) μ∞ Laminar flow – yield plastics [7.13] In terms of Rep and He, equation [7.13] is: 8 Re p = ( He 1− Z k Z ) 3 (3− k ) k k 1 2 ⎛⎜ ⎛⎜ 2 He ⎞⎟ ⎞⎟ = Re p f N 1 − ⎜ ⎜ f Re 2 ⎟ ⎟ 2 ⎝ ⎝ N p⎠ ⎠ 3 (3− k ) k [7.14] Rearranging [7.14]: ⎛ 16 ⎜ ⎜ Re f ⎝ p N ⎞ ⎟ ⎟ ⎠ (1− k / 3) k ⎛ 2 He ⎞ ⎟ =1− ⎜ ⎜ f Re 2 ⎟ N p ⎝ ⎠ f Re 2 ⎛ ⎛ 16 He = N p ⎜1 − ⎜ 2 ⎜ ⎜⎝ Re p f N ⎝ ⎞ ⎟ ⎟ ⎠ k [7.15] (1− k / 3) k 1/ k ⎞ ⎟ ⎟ ⎠ [7.16] At the transition point, Rep = RepC and fN = (fN)C. Using the criterion given by [7.10]: 0.0354 Re 2pC ⎛⎜ ⎛ 16 Re0pC.2 He = 1− ⎜ 2 Re 0pC.2 ⎜ ⎜⎝ 0.0354 Re pC ⎝ ⎞ ⎟ ⎟ ⎠ (1− k / 3) k 1/ k ⎞ ⎟ ⎟ ⎠ 1/ k (1− k / 3) k .8 ⎛ ⎞ ⎛ 20840.8 ⎞ Re1pC ⎜ ⎟ ⎜ ⎟ 1− = 0.8 ⎟ ⎜ ⎟ ⎜ Re pC ⎠ 56.5 ⎝ ⎝ ⎠ [7.17] Therefore, pressure break point transition of yield plastics is expected to occur when: 1/ k 0.8 (1− k / 3) k .8 ⎛ ⎞ ⎛ 2084 ⎞ Re1pC ⎜ ⎟ ⎜ ⎟ He ≈ 1− ⎟ 56.5 ⎜ ⎜⎝ Re pC ⎟⎠ ⎝ ⎠ Transition – yield plastic [7.18] It may be easily shown that when He = 0 (i.e., a Newtonian fluid), equation [7.18] predicts the correct critical Reynolds number of approximately 2100 (i.e., 2084). The He vs RepC relationship is shown graphically in Figure 7.2. 127 Critical Reynolds Number 100000 10000 1000 100 1000 10000 100000 1000000 10000000 Hedstrom Number YP, k=1 YP, k=1/2 YP, k=1/3 YP, k=1/4 YP, k=1/5 Figure 7.2: Transition Reynolds number for yield plastics 7.4.1 Comparison of equations for Bingham plastics For a Bingham plastic, k = 1, and the yield plastic transition equation [7.18] reduces to: 0.533 .8 ⎛ ⎛ 2084 ⎞ ⎞⎟ Re1pC ⎜ ⎟ He = 1− ⎜ 56.5 ⎜ ⎜⎝ Re pC ⎟⎠ ⎟ ⎝ ⎠ [7.19] In Figure 7.3, the transition points predicted by [7.19] is compared with the results of the D.G. Thomas (1963) [7.2] and Hanks (1963) [7.8] equations. The results from the recent Slatter-Wasp and Wilson-Thomas three-part equations are shown as well. The Slatter-Wasp (2000) equations are: Re pC = 2100 (He < 1700) [7.20] (1700 < He < 105) [7.21] Re pC = 155 He0.35 (He > 105) [7.22] Re pC = 26 He0.5 The Wilson-Thomas (2006) equations are: { ( ) Re pC = 2100 1.0 + 8.3 10−8 [log(He)] 13 } (He < 1700) [7.23] 128 Re pC = 80 He0.4 (1700 < He < 105) [7.24] Re pC = 25 He0.5 (He > 105) [7.25] Critical Reynolds Number 100000 10000 1000 100 1000 10000 100000 1000000 10000000 Hedstrom Number YP, k=1 DG Thomas Slatter-Wasp Wilson-Thomas Hanks Figure 7.3: Comparison of transition models for Bingham plastics The proposed transition model gives the similar results to Slatter-Wasp for He > 100,000 and to Wilson-Thomas for He > 20,000. The proposed transition model gives similar results to the D.G. Thomas for He < ~3000. It should be noted that reduction in the critical Reynolds number near He = 1700 predicted by Wilson-Thomas is contradicted by the data presented (Wilson and Thomas 2006, Fig. 7). 7.5. Summary One of the critical design points for slurry pipeline design is the velocity at which the flow transitions from laminar to turbulent flow. Several commonly used models were reviewed and show to give conflicting results. It was suggested that at least part of this lack of consensus was due to differing definition for what constitutes “transition”. For engineers, transition is the point where the pressure gradient deviates significantly from the laminar flow value (i.e., pressure break transition) rather than the initial onset of eddies. 129 A new transition criterion was proposed based on the observation that pressure break transition occurs at the velocity where the fully turbulent flow pressure gradient would be ~130% of the laminar flow pressure gradient. This criterion is essentially a modification of the Hedström laminar-turbulent intersection criterion. An algebraic equation for the transition velocity of yield plastics was derived. The predictions of this equation for Bingham plastics were shown to be in general agreement with existing models. 130 CHAPTER 8: DESIGN CURVE DIAGRAM I confess that I shall expound many things differently from my predecessors – although with their aid, for it was they who first opened the road of inquiry into these things. Nicholaus Copernicus, De Revolutionibus Orbium Coelestium (1543 CE) 8.1. Introduction This chapter presents a new dimensionless group (the “stress number”) for the analysis of pipe flow for Newtonian and non-Newtonian fluids. When the stress number is used instead of the Fanning friction factor in a Moody-type diagram, the resulting “design curve diagram” is shown to be a non-dimensional version of a pipe flow curve. 8.2. Slurry Hydraulic Engineering In order to solve many engineering problems it is necessary to use experimentation to establish relationships between variables of interest. To minimize the scope of the required experiments the concept of dimensional similitude is used. This allows results from a smallscale experiment to be used to estimate the results on a larger scale by the use of dimensionless groups. 8.2.1 The Moody diagram The relationship between pressure gradient and velocity for Newtonian pipe flow is often presented using a Moody (1944) diagram, which relates the friction factor to the Reynolds number and the relative roughness of the pipe. A simplified version of the Moody diagram is presented in Fig. 8.1. For laminar pipe flow (Re < 2100), which is unaffected by pipe roughness, the relationship is found using the Hagen-Poiseuille equation: fN = 16 Re Laminar [8.1] For turbulent flow (Re > 3000), relationship is found using the Colebrook (1939) equation, which is suitable for smooth and rough pipes: 131 ⎛ 1.26 ε ⎞⎟ 1 = −4.0 log⎜ + ⎜ Re f 3.7 D ⎟⎠ fN N ⎝ Turbulent [8.2] Where ε/D is the relative roughness of the pipe wall. Fanning Friction Factor 1.E-01 1.E-02 Laminar Smooth e = 0.0001 e = 0.001 e = 0.01 1.E-03 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Reynolds Number Figure 8.1: Simplified Moody diagram for Newtonian fluids The Moody diagram was once widely used by engineers because determining the friction factors for hydraulic design only requires calculating the Reynolds number and relative roughness. The friction factor is read directly off the Moody-diagram. This was much easier than iteratively solving the Colebrook equation with a slide-rule. It is, therefore, not surprising that early researchers investigating non-Newtonian fluids generally presented data using an “fN vs. Re” workspace, a practice which continues to this day. The difficulty with a non-Newtonian fluid is that it is not clear what to use for the “viscosity” in the “Reynolds number”. Hedström (1952) presented laminar flow results for Bingham plastics on a Moody diagram using the plastic viscosity in the Reynolds number (i.e., the plastic Reynolds number). Figure 8.2 shows a simplified Hedström-Moody diagram for yield plastics. 132 1.E-01 Fanning Friction Factor He = 10 3 He = 10 5 He = 10 7 He = 10 9 He = 0 Newtonian 1.E-02 Newton Bingham Casson Turbulent YP k = 1/3 YP k = 1/4 Turbulent 1.E-03 1.E+02 Transition (YP) 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Rep Figure 8.2: Hedström -Moody diagram for yield plastics The flow equations for yield plastics are given in Chapters 5 and 6. The turbulent flow is assumed to be in a smooth pipe (ε/D = 0) and that the drag reduction factor is assumed to be negligible (β = 1). The critical (transition) Reynolds number ([7.10]) is also shown. The Hedström-Moody diagram can be adapted for any consistency model with a limiting high shear rate viscosity (e.g., Heinz, Cross, Carreau, Sisko, etc.). Determining a comparable Reynolds number is problematic with consistency models that imply there is no limiting viscosity (e.g., power law, yield power law, etc.). The apparent viscosity varies so a true Reynolds number cannot be used. The “plastic Reynolds number” cannot be used since it would always be infinity or zero, unless n = 1. The exponent in the units for the consistency coefficient (Pa-sn) compounds the problem because it cannot be made dimensionless with a simple ratio of parameters. To get around this problem, Metzner and Reed (1955) proposed a “generalized Reynolds number” for non-Newtonian fluids where: Re MR ≡ 16 fN [8.3] 133 This method may be used for any consistency model. Moody-type diagrams using the Metzner-Reed Reynolds number are common in the literature (e.g., Hanks and Pratt 1967) and there is generally an excellent correlation in the laminar flow region, with all of the data falling on a single straight line. Of course, this is because the laminar flow relationship with the Metzner-Reed Reynolds number is a tautology: fN = 16 16 = = fN Re MR 16 f N [8.4] In the turbulent regime the fN vs. ReMR relationship curves split into individual lines for each value of the flow behaviour index (n). Because the ordinate varies, the impact of, say, doubling the flow rate is obscure because it is not equivalent to doubling the Metzner-Reed Reynolds number. Metszner-Reed-Moody diagrams are of little practical value to engineers. 8.2.2 Conceptual engineering With the advent of personal computers and programmable calculators, the friction factor can be calculated easily and with better accuracy than interpolating off a log-log graph. Since it is now of limited quantitative value, it may seem odd that one would be hard pressed to find a recent fluid mechanics textbook or hydraulics handbook that does not contain a version of the Moody diagram. It is proposed that the main reason for the continued “use” of the Moody diagram is that it is a tool for fostering an intuitive understanding of the friction factor and what affects it. Being a graphical representation of the Newtonian design equations for pipe flow ([8.1] and [8.2]) it immediately shows the important features of pipeline hydraulics: • How the fluid behaves in the laminar flow, partially rough wall turbulent flow, and fully rough wall turbulent flow regimes. • The Reynolds number range where the flow transitions from laminar to turbulent flow and from partially rough wall to fully rough wall turbulent flow. • Where the pipe roughness can be ignored and a smooth pipe approximation used. In the Author’s opinion, engineers can be considered “experienced” when they can solve problems intuitively, with testing and mathematical analysis used to confirm, refine, and 134 support their solution. The primary objective of an engineering design tool ought to be to promote an intuitive understanding of the problem and possible solutions. The Moody diagram meets this objective. Of course, what hydraulics engineers really need to understand is how variables affect the pressure gradient, and the friction factor is just a means to find the pressure-gradient. Consider Fig. 6.1, which shows the measured flow curves for 14%w kaolin slurry with 0%, 0.1%, and 0.27% TSPP. How could one determine what would happen if the system were run at 1.5 m/s with 0.05% TSPP? One option is to do another loop test at the design condition. However, an experienced hydraulics engineer will simply look at the existing flow curves and immediately deduce that the flow will be laminar and that the pressure gradient will be ~2.5 kPa/m. In fact, the entire flow curve could be estimated, with accuracy sufficient for design, without any understanding of the slurry’s rheology. The objective of this Chapter is to develop an alternative to the Moody diagram for the graphical representation of the yield plastic flow relationships that will foster an intuitive understanding of the impact on the flow curve due to changing variables. In effect, this will be a dimensionless flow curve diagram. 8.3. Dimensionless Groups The objective of this Section is to formulate a new dimensionless group for pipeline flow. 8.3.1 Standard dimensionless groups According to the Buckingham π-theorem, the number of dimensionless groups required to define an engineering problem is the total number of independent variables minus the number of basic dimensions (see Streeter and Wylie 1985). The basic yield plastic model is described by: τ k = τ ok + (μ∞γ& )k Yield plastic [8.5] The pressure gradient of a yield plastic flowing in a hydraulically smooth pipe is a function of the following parameters: ΔP L = f n (V , D, ε , ρ , μ∞ ,τ o , k ) 135 • Bulk flow velocity (V): m/s • Pipe inside diameter (D): m • Pipe roughness (ε): m • Fluid density (ρ): kg/m3 = Pa-s2/m2 • Plastic viscosity (μ∞): Pa-s • Yield stress (τo): Pa • Scaling factor (k): - • Pressure gradient (ΔP/L): Pa/m With eight independent variables and three basic dimensions (Pa, m, and s), five dimensionless groups are required to describe the relationship. Since the basic function of a flow curve is to show how the pressure-gradient changes with the flow rate it is preferable to have these two variables be included in one group only. The plastic Reynolds number is directly proportional to the bulk velocity and independent of the pressure gradient. It will be used as a dimensionless surrogate for the bulk velocity: Re p = ρVD μ∞ Plastic Reynolds number [8.6] The relative roughness will be used as a dimensionless surrogate for the wall roughness: ε Relative roughness [8.7] D Three dimensionless groups are commonly used with the Bingham plastic model for the yield stress parameter: the yield (or Bingham) number, the wall stress ratio, and the Hedström number. All three dimensionless groups can be generalized for the yield plastic model by using the yield stress and infinite shear rate viscosity. B= τoD μ∞V Yield number [8.8] ⎛ L4 ⎞ τ o Z = τo⎜ ⎟= ⎝ ΔPD ⎠ τ w Wall stress ratio [8.9] 136 He = τ o ρD 2 μ∞2 Hedström number [8.10] The yield number is a function of the velocity and its value changes along the flow curve. The wall stress ratio is a function of the pressure gradient and its value also changes along the flow curve. The Hedström number is independent of both the velocity and the pressuregradient. Its value is constant for any given (time independent) slurry in a pipe of a given diameter and a flow curve is effectively an iso-Hedström number curve (see Fig. 8.2). The generalized Hedström number will be used as a surrogate for the yield stress. The scaling factor is already a dimensionless number and does not need a surrogate: Scaling factor [8.11] k The groups Rep, ε/D, He, and k make up four of the five dimensionless groups required. The remaining parameter is the pressure gradient. 8.3.2 Dimensionless group for pressure-gradient The conventional dimensionless surrogate for the pressure-gradient is the Fanning friction factor: fN = ΔPD 2τ = w2 2 2 LρV ρV Fanning friction factor [8.12] Although the friction factor is directly proportional to the pressure gradient, it is also an inverse function of the bulk velocity squared. When the velocity goes up, the friction factor will go down or stay the same (in rough pipes at high velocities). A better surrogate is the inverse of the wall stress ratio (1/Z), which is both directly proportional to the pressure gradient and independent of the velocity: 1 ΔP ⎛ D ⎞ τ w ⎜ ⎟= = Z L ⎜⎝ 4τ o ⎟⎠ τ o [8.13] The difficulty with the inverse wall stress ratio is that it includes the yield stress. Most fluids – and therefore most yield plastics – are Newtonian and have a zero yield stress. Even when there is a yield stress, it is found that the yield stress has an insignificant impact on the flow behaviour of yield plastics beyond a relatively short transition zone (see Chapter 6). 137 Furthermore, the true yield stress is notoriously difficult to determine precisely, if it exists at all (Barnes and Walters 1985). According to the physical basis of the yield plastic model, the yield stress and the scaling factor are ephemeral properties that depend on the degree of aggregation and the relative strength of the interparticle bonds. They can be changed by adding a coagulant or dispersant. With time dependent suspension they change with the shear history. There is only one “fixed” rheological parameter: the viscosity of the fully dispersed suspension or the infinite shear rate viscosity (μ∞). 8.3.3 The stress number It is proposed that a new dimensionless group be used as the surrogate for the pressuregradient – the “stress number”: Ha = ΔP ⎛ ρD 3 ⎞ τ w ρD 2 ⎜ ⎟= μ∞2 L ⎜⎝ 4 μ∞2 ⎟⎠ Stress number [8.14] The stress number is directly proportional to the pressure gradient and independent of the bulk velocity. It is also independent of the yield stress and the scaling factor, making it applicable to Newtonian fluids. The stress number may be related to the friction factor and the plastic Reynolds number: Ha = τ w ρD 2 1 ⎛ 2τ w ⎞⎛ ρ 2V 2 D 2 ⎞ 1 ⎟ = f N Re 2p = ⎜⎜ ⎟⎜ 2 ⎝ ρV 2 ⎟⎠⎜⎝ μ∞2 ⎟⎠ 2 μ∞2 [8.15] If the yield plastic has a yield stress, the stress number may be related to the Hedström number and the wall stress ratio: ⎛ τ o ρD 2 ⎞⎛ τ w ⎞ He ⎟⎟⎜⎜ ⎟⎟ = Ha = ⎜⎜ 2 μ ⎝ ∞ ⎠⎝ τ o ⎠ Z [8.16] The stress number may also be related to the friction velocity and the friction Reynolds numbers used in turbulent flow calculations: 2 2 ⎛ ⎛ τ ⎞⎛ ρD ⎞ ⎞ ⎛ u * ρD ⎞ ⎟⎟ ⎟ = ⎜⎜ ⎟⎟ = (Re*)2 Ha = ⎜ ⎜⎜ w ⎟⎟⎜⎜ ⎜ ⎟ ρ μ μ ∞ ⎠ ⎠⎝ ∞ ⎠ ⎠ ⎝ ⎝⎝ 138 [8.17] Although the stress number was developed independently, a similar dimensionless group for Newtonian fluids referred to as “Karman 1”, was later found in a list of dimensionless groups put out by NASA (Land 1972), with a note saying that it was a “reference only” in an earlier text (“Douglas 1969”). 8.4. Flow Equations Using the Stress Number The flow equations developed in Chapters 5, 6, and 7 may be put in terms of the stress number (Ha) and the dimensionless groups Rep, ε/D, He, and k. 8.4.1 Newtonian When the yield stress is zero, the yield plastic model reduces to the Newtonian model for all values of the scaling factor. 8.4.1.1 Laminar flow The conventional equation for laminar flow of a Newtonian fluid (Re < 2100) is given by [8.1]. This may be put in terms of the stress number by multiplying both sides of Eq. [8.1] by one half of the Reynolds number squared: ⎛ Re 2 ⎞ 16 ⎛ Re 2 ⎞ ⎟⎟ = Ha = ⎜ ⎟ = 8 Re f N ⎜⎜ Re ⎜⎝ 2 ⎟⎠ ⎝ 2 ⎠ [8.18] Therefore, the laminar flow equation in terms of the stress number is: Re = Ha 8 8.4.1.2 Laminar flow – Newtonian [8.19] Turbulent flow The Colebrook equation [8.2] for turbulent flow of a Newtonian fluid (Re > 3000) can be put in terms of the stress number by noting that: Ha = f N Re 2 Æ 2 fN = 2 Ha Re [8.20] Substituting Eq. [8.20] into Eq. [8.2]: 139 ε ⎞ Re ⎛ 1.26 = −4.0 log⎜ + ⎟ 2 Ha ⎝ 2 Ha 3.37 D ⎠ [8.21] Therefore, the turbulent flow equation in terms of the stress number is: ε ⎞ ⎛ 0.89 Re = −5.66 Ha log⎜ + ⎟ ⎝ Ha 3.7 D ⎠ Turbulent flow - Newtonian [8.22] Unlike with the friction factor form, the stress number form of the Colebrook equation can be used to determine the Reynolds number directly, without the need to iterate. The Reynolds number (i.e., the velocity) can be found directly if Ha (i.e., the pressure-gradient) and the relative roughness are known or assumed. For turbulent flow in smooth pipes, the Knudsen-Katz equation correlates well with the smooth wall Colebrook equation (i.e., the Nikuradse equation) and is easier to use. Equation [6.3] may be put in terms of the stress number by multiplying both sides of the equation by one half of the Reynolds number squared: ⎛ Re 2 ⎞ 0.046 ⎛ Re 2 ⎞ ⎜ ⎟ ⎜ ⎟⎟ = 0.023 Re1.8 fN ⎜ = Ha = 0.2 ⎜ ⎟ Re ⎝ 2 ⎠ ⎝ 2 ⎠ [8.23] Therefore, the Knudsen-Katz equation in terms of the stress number is: Re1.8 Ha = 43.5 8.4.1.3 SW Turbulent flow – Newtonian [8.24] Transition The estimated transition Reynolds number (i.e., 2100) is unchanged. However, it may be put in terms of the stress number using [8.19]: Ha = 16800 8.4.1.4 Transition – Newtonian [8.25] Comparison of equations Table 8.1 compares the pressure-gradient relationships with the friction factor and stress number design equations. The stress number equations are no more complex than the equivalent friction factor equations. In the case of the Colebrook equation [8.2], the stress 140 number version is more convenient to use. More importantly, the Ha vs. Re relationships are identical to the P/L vs. V relationships. Table 8.1: Newtonian pipe flow equations Flow Regime Pipe flow curve Laminar ΔP = C0V L Turbulent ΔP ≈ C1V 1.8 L (smooth wall) Friction factor fN = fN ≈ ΔP = C3 L Transition Stress number 16 Re 0.046 Re0.2 f N = 0.0076 Ha = 8 Re Ha ≈ Re1.8 43.5 Ha = 16800 The Ha vs. Re equivalent of the standard (Newtonian) Moody diagram, is shown in Fig. 8.3. 1.E+10 1.E+09 Stress Number 1.E+08 1.E+07 1.E+06 1.E+05 Laminar 1.E+04 Smooth e = 0.0001 1.E+03 e = 0.001 e = 0.01 1.E+02 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Reynolds Number Figure 8.3: Stress number vs. Reynolds number relationship for Newtonian fluids 141 8.4.2 Bingham plastic When k = 1, the yield plastic model reduces to the Bingham plastic model. 8.4.2.1 Laminar flow The laminar flow of Bingham plastics is described using [5.3], which may be expanded to: 8V τ w ⎛ 4τ o τ o4 ⎞ ⎜1 − ⎟ = + D η∞ ⎜⎝ 3τ w 3τ w4 ⎟⎠ Laminar flow – Bingham [8.26] Equation [8.26] may be put in terms of the conventional friction factor: ⎛ 8 8He 16 He4 ⎞⎟ = f N ⎜1 − + ⎜ 3 f Re 2 3 f 4 Re8 ⎟ Re p N p N p ⎠ ⎝ fN = [8.27] He 8 ⎛⎜ 2 He4 ⎞ − 3 7⎟ 1+ Re p ⎜⎝ 3 Re p 3 f N Re p ⎟⎠ [8.28] Equation [8.28] needs to be solved iteratively, since neither the friction factor nor the plastic Reynolds number can be separated. The laminar flow equation for Bingham plastics in terms of the stress number is: Ha ⎛⎜ 4 He 1 ⎛ He ⎞ + ⎜ Re p = 1− ⎟ 8 ⎜⎝ 3Ha 3 ⎝ Ha ⎠ 4 ⎞ ⎟ ⎟ ⎠ Laminar flow – Bingham [8.29] The stress number form of the Buckingham equation ([8.29]) is easier to use than the friction factor form ([8.28]) because it gives the plastic Reynolds number directly as a function of Ha. 8.4.2.2 Turbulent flow The Wilson-Thomas smooth wall turbulent flow equation (see [6.57]) in terms of the stress number is: ( Re p = 2.457 Ha ln 1.12β Ha ) SW turbulent flow – Bingham [8.30] In terms of the stress number, the drag reduction factor (β) [6.80] is: 142 ⎛ ⎝ β ≈ e 4.76 He / Ha ⎜1 − He ⎞ ⎟ Ha ⎠ 2 − He / Ha Drag reduction factor – Bingham [8.31] As noted in Chapter 6, it is always conservative to assume that β = 1 for Bingham plastics and [8.30] reduces to the (plastic) Nikuradse smooth wall turbulent flow equation. For most engineering applications, the (plastic) Knudsen-Katz smooth wall turbulent flow equation may be used instead of the Nikuradse equation (see Fig. 8.4): Ha = Re1p.8 SW Turbulent flow – Bingham [8.32] 43.5 1.E+10 1.E+09 1.E+08 1.E+07 Ha 1.E+06 1.E+05 1.E+04 1.E+03 Nikuradse Knudsen and Katz 1.E+02 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Rep Figure 8.4: Comparison of smooth wall turbulent flow equations 8.4.2.3 Transition The “pressure-gradient break point transition” for Bingham plastics may be found in terms of He and Rep using [7.19]. However, from the transition criterion [7.9] it may be shown that transition occurs at: 143 HaC = Re1p.8 1.3(43.5) = Re1p.8 Transition – Bingham [8.33] 56.5 The Hedström (1952) transition criterion predicts that transition will occur when the laminar flow line intersects the turbulent flow line. Therefore: HaC = Re1p.8 Hedström [8.34] 43.5 Metzner and Reed (1955) proposed that transition will occur at a constant friction factor, usually taken to be fN = 0.0076. Using relationship [8.15]: 2 1 0.0076 2 Re p 2 Re p = HaC = f N Re p = 2 2 263 Metzner-Reed [8.35] The Hanks (1963) criterion for the instability point transition [7.7] in stress number form using relationship [8.16]: ⎛ ⎛ 16,800 ⎞ ⎞ ⎟⎟ ⎟ He = HaC ⎜⎜1 − ⎜⎜ ⎟ Ha C ⎠⎠ ⎝ ⎝ HaC Re p = 8 [8.36] ⎛ 4 ⎛ ⎛ 16,800 ⎞ ⎞ 1 ⎛ ⎛ 16,800 ⎞ ⎞ 4 ⎞ ⎜1 − ⎜ 1 − ⎜ ⎟⎟ ⎟ ⎟ ⎟ + ⎜1 − ⎜ ⎜ 3 ⎜ ⎜⎝ HaC ⎟⎠ ⎟ 3 ⎜ ⎜⎝ HaC ⎟⎠ ⎟ ⎟ ⎠ ⎠ ⎠ ⎝ ⎝ ⎝ Hanks [8.37] Slatter and Wasp (2000) proposed a three-part model for the transition of Bingham plastics flowing in a pipe. When He < 1700, the plastic Reynolds number at the critical stress number is the same as Metzner-Reed ([8.34]): Re p = 263HaC = 16.2 HaC0.5 Slatter-Wasp 1 [8.38] When 1700 < He < ~105: HaC Re p = 8 4 1 / 0.35 ⎛ ⎞ 1 ⎛ (Re p 155)1 / 0.35 ⎞ ⎞⎟ ⎜ − 4 ⎛⎜ (Re p 155) ⎟+ ⎜ ⎟ ⎜⎜1 3 ⎜ ⎟ ⎜ ⎟ ⎟⎟ 3 Ha Ha C C ⎝ ⎠ ⎝ ⎠ ⎠ ⎝ When He > ~105: 144 Slatter-Wasp 2 [8.39] HaC Re p = 8 2 2 4⎞ ⎛ ⎜ − 4 ⎛⎜ (Re p 26) ⎞⎟ + 1 ⎛⎜ (Re p 26) ⎞⎟ ⎟ ⎜⎜1 3 ⎜ Ha ⎟ 3 ⎜ HaC ⎟ ⎟⎟ C ⎝ ⎠ ⎝ ⎠ ⎠ ⎝ Slatter-Wasp 3 [8.40] The five Bingham transition relationships are compared in Fig. 8.5: 1.E+10 1.E+09 1.E+08 Ha 1.E+07 1.E+06 Yield plastic Hedstrom 1.E+05 Metzner-Reed Hanks 1.E+04 1.E+03 Slatter-Wasp 1.E+04 1.E+05 1.E+06 1.E+07 Rep Figure 8.5: Bingham transition relationship 8.4.2.4 Comparison of equations As with Newtonian fluids, the stress number forms of the Bingham plastic design equations are no more complex than the friction factor versions and easier to apply. 8.4.3 Yield plastic For the general case of yield plastics the design equations are as follows: 8.4.3.1 Laminar flow The laminar flow of yield plastics is described using [5.62], which may be expanded to: 145 8V τ w ⎛⎜ ⎛ τ o ⎞ ≈ 1− ⎜ ⎟ D μ∞ ⎜ ⎜⎝ τ w ⎟⎠ ⎝ k ⎞ ⎟ ⎟ ⎠ 3 (3− k ) k Laminar flow – yield plastic [8.41] In terms of the friction factor: 3 ⎛ ⎛ 8He ⎞ k ⎞ ( 3− k ) k 8 ⎟ ⎟ ≈ f N ⎜1 − ⎜ 2 ⎟ ⎟ ⎜ ⎜ 3 f Re Re p ⎝ ⎝ N p⎠ ⎠ [8.42] In terms of the stress number: Ha ⎛⎜ ⎛ He ⎞ Re p ≈ 1− ⎜ ⎟ 8 ⎜⎝ ⎝ Ha ⎠ 3 k ⎞ (3− k ) k ⎟ ⎟ ⎠ Laminar flow – yield plastic [8.43] The Ha vs Rep relationships given by [8.43] are shown graphically in Fig. 8.6 for Newtonian, Bingham, and Casson fluids as well as for k = 1/3 (i.e., k = 1, 1/2 and 1/3) at various values of the Hedström number (i.e., He = 0, 103, 105, 107, and 109). Using the stress number makes the relationships between parameters clear. The Hedström number uniquely defines the “family” of laminar flow curves while the scaling factor determines how quickly the non-Newtonian effects break down. All of the flow curves trend asymptotically between Ha = He and the Newtonian laminar flow line (based on the infinite shear rate viscosity) as Rep increases. 8.4.3.2 Turbulent flow The Wilson-Thomas smooth wall turbulent flow equation for yield plastics is identical to the Bingham version [8.30]: ( Re p = 2.457 Ha ln 1.12β Ha ) SW turbulent flow – yield plastic [8.44] In terms of the stress number, the drag reduction factor (β) [6.80] is: 2 ⎛ ⎛ He ⎞ k ⎞ k 4.76 ( He / Ha )k ⎜ ⎟ β ≈e 1− ⎜ ⎜⎝ Ha ⎟⎠ ⎟ ⎝ ⎠ (1− 0.5( He Ha ) ) k Drag reduction factor – yield plastic [8.45] 146 1.E+10 He = 10 9 He = 10 7 He = 10 5 He = 10 3 1.E+09 1.E+08 1.E+07 Ha 1.E+06 1.E+05 1.E+04 Newton Bingham 1.E+03 Casson He = 0 Newtonian 1.E+02 1.E+01 1.E+02 1.E+03 YP k = 1/3 YP k = 1/4 1.E+04 1.E+05 1.E+06 1.E+07 Rep Figure 8.6: Laminar flow relationship (yield plastic) However, it is conservative to assume that β = 1 when k > ~0.45. For most engineering applications, the Knudsen-Katz smooth wall turbulent flow equation is adequate: Ha = Re1p.8 SW Turbulent flow – yield plastic [8.46] 43.5 Equation [8.44] with β = 1 (i.e., the “plastic” Nikuradse equation) is compared to [8.46] in Fig. 8.4. 8.4.3.3 Transition The pressure-gradient break point transition for yield plastics may be found in terms of He and Rep using [7.18]. However, from the transition criterion [7.9]: HaC = Re1p.8 Transition – yield plastic [8.47] 56.5 147 8.5. Design Curve Diagram (for Yield Plastics) The flow of all yield plastics (including Newtonian fluids, Bingham plastics and Casson plastics) may be described by three relatively straightforward equations (i.e., [8.43], [8.46], and [8.47]). These may be presented on a single graph that will be referred to as a “design curve diagram” (DCD). Figure 8.7 show the full design curve diagram for yield plastics. 1.E+10 He = 10 9 1.E+09 1.E+08 He = 10 7 1.E+07 Turbulent Ha 1.E+06 He = 10 5 1.E+05 He = 10 3 Newton Bingham 1.E+04 Casson YP k = 1/3 1.E+03 1.E+02 1.E+01 YP k = 1/4 He = 0 Newtonian Turbulent Transition (YP) 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 Rep Figure 8.7: Design curve diagram for yield plastics Two material-independent lines define the right side of the full DCD: • A fixed Newtonian laminar flow line for the fully dispersed suspension (He = 0). There are no flow conditions to the right of this line. • A fixed fully turbulent flow line representing the condition where turbulent stresses have completely broken down the plug and the particle aggregates (“structure”). • A fixed line through the break points in the flow curves (break point transition). 148 The laminar flow curves depend on the yield stress and the scaling factor. If the suspension has a yield stress, there can be no flow when Ha < He unless there is “wall slip”. This provides a lower limit to the value of Ha. The laminar flow curves all extend asymptotically between the Ha = He line and the Newtonian laminar flow (Ha = 8Rep) line, with the “tightness of the turn” increasing as the scaling factor increases. The Laminar flow curves will generally not extend beyond the transition line. The DCD defines the entire flow range with the exception of the transition region, which is poorly defined even for Newtonian fluids. Detailed versions of the DCD are included in Appendix C. In addition to the general DCD (Fig. 8.7), the DCD’s are presented in the Hedström number range suitable for pipeline engineering (105 < He < 108) and laboratory experiments (0 < He < 106) 8.6. Summary The objective of this Chapter was to present design equations for non-Newtonian pipe flow in a way that is conceptually easier for practicing engineers. A new dimensionless group (the “stress number”) was proposed for the analysis of pipe flow that is suitable for any fluid or slurry with a finite high shear rate viscosity plateau. Five dimensionless groups were shown to define the pipe flow of time independent fluids. • Stress number: Directly proportional to ΔP/L, independent of V • Reynolds number: Directly proportional to V, independent of ΔP/L • Hedström number: Independent of V and ΔP/L (0 for Newtonian fluids) • Scaling factor: Independent of V and ΔP/L (N/A for Newtonian fluids) • Relative roughness: Independent of V and ΔP/L (0 for “smooth” pipe) The pipe flow equations were reformulated in terms of these groups and shown to be less complicated than the friction factor equivalents. It was shown that, when these equations are plotted on a stress number vs. Reynolds number graph, the resulting “design curve diagram” is a dimensionless pipe flow curve, making non-Newtonian effects easier to understand than when using Moody-type (fN vs. Re) diagrams. 149 CHAPTER 9: CASE STUDY The empiricist…thinks he believes only what he sees, but he is much better at believing than at seeing. George Santayana, Scepticism and Animal Faith (1923 CE) 9.1. Introduction The goal of rheology-based design is to be able to design industrial-scale systems based on small-scale laboratory tests. In this Thesis, the “system” is a pipeline handling homogenous slurry and the “test” is consistency measurement using a concentric cylinder rheometer (CCR), which is commonly used for mineral slurries. The objective of this Chapter is to show that the yield plastic model and the design equations developed in this Thesis allow a homogenous slurry pipeline to be designed using concentric cylinder rheometer results. Furthermore, it will be shown that the yield plastic model gives more accurate result than other commonly used consistency models. This case study will be based on the work of Litzenberger (2004), who performed a series of tests on kaolin clay slurry as part of the research for a Masters of Science degree. In these tests calcium salt (CaCl2·2H20) was used as an aggregating agent and tetra-sodium pyrophosphate (TSPP) was used as a dispersant. The tests were carried out with various concentrations of clay, salt, and TSPP. In each case, the consistency was measured in a Haake RV 3 concentric cylinder rheometer (MV/MV1 system) and in a 25.825 mm internal diameter vertical pipe loop. (Note: The results for the tests discussed in this section are included in Appendix D.) 9.2. Rotational Rheometry Before analyzing the kaolin clay test data, the basics of rotational rheometry will be reviewed. (This section may be considered to be a literature review.) Most modern CCR’s have a “Searle” configuration, with a rotating cylinder (bob) of radius Ri inside a stationary cylinder (cup) of radius Ro. The cup and bob make up the “measuring 150 system”. The fluid consistency is measured in the “gap” between the cup and the bob. Some CCR’s apply a controlled torque to the bob and measure the resulting rotational speed (controlled stress rheometer), while others apply a controlled rotation speed to the bob and measure the torque caused by viscous drag (controlled shear rate rheometer). The torque and speed may be used to infer the shear stress and shear rate in the gap based on the dimensions of the measuring system, the material properties, and an allowance for end effects (Whorlow 1992). When using a CCR to test suspensions of hard particles, the width of the gap (t = Ro – Ri) must be several times the diameter of the largest particles to prevent particle jamming and to justify the continuum assumption (i.e., that the size of the particles are negligible) used in rheometry calculations (Coussot 1997). With slurries containing coarse particles, the measuring system must have a “wide gap” configuration. The wide gap causes the variation in the shear stress with radius to become significant. This can cause errors in the estimate of the shear rate with non-Newtonian fluids. Correction factors for wide-gaps are available for Newtonian fluids and Bingham plastics (Whorlow 1992) and Casson plastics (Hanks 1983). No similar correction factor has been developed for yield plastics. Fortunately, with the nonsettling slurries of fine particles under consideration in this Thesis, it is possible to use a CCR with a “narrow gap”. If the annular gap is very small compared with Ri, then the shear stress may be assumed to be constant across the annulus between the two cylinders and the following equations may be used to determine the shear stress and shear rate (Whorlow 1992). τ= G ⎛ 1 ⎞ ⎜ ⎟ Leq ⎜⎝ 2πRa2 ⎟⎠ Shear stress [9.1] ⎛ Ra ⎞ ⎟ ⎝ t ⎠ γ& = Ω⎜ Shear rate [9.2] Where G is the torque on the bob (N-m), Leq is the corrected bob length (m), Ra is the average radius of the gap (m), t is the width of the gap (m), and Ω is angular velocity or rotation speed of the bob (rad/s). The bob length correction takes into account the end effects, particularly the viscous drag on the top and bottom of the cylinder. 151 τ k = τ ok + (μ∞γ& )k Yield plastic [9.3] k k ⎛ G 1 ⎞ ⎛ ΩRa ⎞ k ⎜ ⎟ τ μ ≈ + ⎟ ⎜ o ∞ ⎜ L 2πR 2 ⎟ t ⎠ ⎝ a ⎠ ⎝ eq k [9.4] 3 ⎛ G ⎞ ⎛ ⎞ ⎜ ⎟ ≈ τ ok 2πRa2 k + (μ∞ Ω )k ⎜ 2πRa ⎟ ⎜ ⎟ ⎜L ⎟ ⎝ t ⎠ ⎝ eq ⎠ ( ) k Rheometer relation – yield plastic [9.5] The results of the narrow gap approximation ([9.5]) may be compared with the exact result for Bingham fluids (Whorlow 1992) based on the dimensions of a Haake MV/MV1 measuring system: • Length of spindle (bob): 0.06000 m • Radius of spindle (bob): 0.02004 m • Radius of cup: 0.02100 m • Average radius (mid-gap): 0.02052 m • Gap width: 0.00096 m ⎛ G ⎞ ⎟(17.70 ) − τ o (0.0468) ⎟ L ⎝ eq ⎠ Bingham [9.6] ⎛ G ⎞ ⎟(17.68) − τ o (0.0468) ⎟ L eq ⎝ ⎠ Yield plastic (k = 1) [9.7] μ∞ Ω = ⎜⎜ μ∞ Ω ≈ ⎜⎜ For Casson (k = 1/2) fluids (Hanks 1983) the comparable equations are: 1/ 2 ⎛ G ⎞ ⎛ Gτ ⎞ μ∞ Ω = ⎜⎜ ⎟⎟(17.70 ) − ⎜⎜ o ⎟⎟ ⎝ Leq ⎠ ⎝ Leq ⎠ (1.820) + τ o (0.0468) Casson [9.8] (1.819) + τ o (0.0468) Yield plastic (k = ½) [9.9] 1/ 2 ⎛ G ⎞ ⎛ Gτ ⎞ μ∞ Ω ≈ ⎜⎜ ⎟⎟(17.68) − ⎜⎜ o ⎟⎟ ⎝ Leq ⎠ ⎝ Leq ⎠ In both cases, the values of the constants are the same to three significant figures. It may thus be assumed that the error due to the narrow gap approximation is negligible for the 152 cases k = 1 and k = ½. It is assumed that this will also be the case for yield plastics in the narrow gapped Haake MV/MV1 measuring system. The final consideration is the correction for end effects. For a Newtonian fluid, the end correction is accounted for using calibration oils with standardized viscosity vs. temperature relationships. However, the end correction is not necessarily the same for a non-Newtonian fluid. Whorlow (1992, Fig. 3.11) compares the length correction for a CCR with a 1.2 mm gap and varying bob (cylinder) lengths. For a “Newtonian oil” the correction was 9 mm and for a “clay suspension” the correction was 19 mm. The width of the gap directly affects the shear stress on the measuring surface but it does not affect the shear stress on the end of the bob. The ratio of the gap widths in Whorlow’s rheometer and a MV1/MV measuring system is 1.25. Since the output of the rheometer software was already corrected for Newtonian effects, the equivalent length correction is Leq/L = (1.25*60+19)/(1.25*60+9) = 1.12. An end correction of 12% will be used for non-Newtonian clay slurry. 9.3. Kaolin Clay Slurry Rheology For the purposes of this Case Study it was desirable to have flow loop results with a significant number of data points in both the laminar and fully turbulent flow regimes. The laminar flow should be non-Newtonian (i.e., the pressure gradient does not increase in direct proportion with the bulk velocity). The test runs should be carried out at different solids contents so that the slurries will have different densities and (it is proposed) different fully dispersed viscosities. Finally, there should be loop test results available for fully (chemically) dispersed suspensions at the same solids content. Two of Litenberger’s (2004) test runs that meet these criteria were selected. The first test run (G2000208 or “Data 208”) was carried out on a 10%v/v clay suspension with 0.10%w/w of calcium salt and no TSPP. The equivalent fully dispersed slurry (G2000212) contained 0.27%w/w TSPP (mass TSPP/mass clay). The consistency was measured in a concentric cylinder rheometer. Two rheometry passes (rotation speed up – rotation speed down) were carried out. The averaged rheometer results are presented in Table 9.1 along with the shear stress and shear rate values calculated using [9.1] and [9.2]. 153 The second test run (G2000105 or “Data 105”) was carried out on a 14%v/v clay suspension with 0.10%w/w of calcium salt and 0.1%w/w TSPP. The equivalent fully dispersed slurry (G2000215) contained 0.27%w/w TSPP (mass TSPP/mass clay). Two rheometry passes (rotation speed up – rotation speed down) were carried out. The averaged rheometer results are presented in Table 9.2 along with the shear stress and shear rate values calculated using [9.1] and [9.2]. Table 9.1: Rheometry Results 10%v/v Kaolin Slurry (No Dispersant) Ω (rad/s) G/L (N-m/m) γ& (1/s) τ (Pa) 3.35 0.918E-2 71.6 3.096 4.73 0.964E-2 101.1 3.253 6.70 1.04E-2 143.2 3.493 9.48 1.15E-2 202.6 3.864 13.4 1.23E-2 286.4 4.151 18.95 1.38E-2 405.1 4.657 26.81 1.58E-2 573.1 5.332 37.91 1.86E-2 810.3 6.227 Table 9.2: Rheometry Results 14%v/v Kaolin Slurry (Partially Dispersed) Ω (rad/s) G/L (N-m/m) γ& (1/s) τ (Pa) 4.73 2.02E-2 101.1 6.770 6.70 2.17E-2 143.2 7.274 9.48 2.35E-2 202.6 7.879 13.4 2.56E-2 286.4 8.601 18.95 2.82E-2 405.1 9.457 26.81 3.16E-2 573.1 10.600 37.91 3.58E-2 810.3 12.028 154 The rheogram results in Tables 9.1 (“208”) and 9.2 (“105”) are presented graphically in Fig. 9.1. The data indicate that the slurry is shear thinning with an apparent yield stress. Neither curve appears to transition to turbulence at high shear rates. 14 12 Shear Stress (Pa) 10 8 6 4 2 0 0 200 400 600 800 1000 Shear Rate (1/s) Data 208 (10%) Data 105 (14%) Figure 9.1: Rheograms for kaolin clay slurry in concentric cylinder rheometer (Data from Litzenberger 2004) The rheometry data was fit to the following rheological models: yield plastic, Bingham, Casson, power law and yield power law (Herschel-Bulkley). The resulting model parameters (based on the full range of data) for G2000208 are presented in Table 9.3. The consistency model parameters for G2000105 are presented in Table 9.4. The accuracy of the model “fits” for the various models is shown in terms of R2. Table 9.3: Model Parameters 10%v/v Kaolin Slurry (All Data) Yield Plastic Bingham Casson k 0.72 1 0.5 τo 2.522 Pa 2.886 Pa 2.046 Pa μ∞ 0.00279 Pa-s 0.00426 Pa-s 0.00137 Pa-s Yield Power Law 0 Pa 2.515 Pa 0.287 n 0.766 0.287 0.861 Pa-s K R2 Power Law 0.9990 0.9960 0.9967 155 0.9719 0.0219 Pa-s0.766 0.9986 Table 9.4: Model Parameters 14%v/v Kaolin Slurry (All Data) Yield Plastic Bingham Casson Power Law Yield Power Law k 0.63 1 0.5 τo 5.238 Pa 6.341 Pa 4.584 Pa 0 Pa 4.69 Pa μ∞ 0.00367 Pa-s 0.00734 Pa-s 0.00220 Pa-s n 0.274 0.598 K 1.870 Pa-s0.274 0.1337 Pa-s0.598 0.9902 0.9999 R2 0.9988 0.9894 0.9999 Litzenberger’s (2004) pipe loop test data for runs G2000208 (“Data 208”) and G2000105 (“Data 105”) is given in Table 9.5. The flow curve results are presented in Fig. 9.2. Both flow curves are laminar at low velocities and turbulent at high velocities. They transition at ~1.4 m/s and ~2 m/s respectively. 7 Pressure Gradient (kPa/m) 6 5 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 Bulk Velocity (m/s) Data 208 (10%) Data 105 (14%) Figure 9.2: Flow loop data for kaolin slurry (25.825 mm pipe) (Data from Litzenberger 2004) 156 Table 9.5: Pipe Flow Loop Data for Kaolin Slurry 10%v/v (G2000208) 14%v/v (G2000105) 3 Density: 1228 kg/m3 Density: 1161 kg/m V (m/s) ΔP/L (kPa/m) V (m/s) ΔP/L (kPa/m) 3.19 5.182 3.23 5.845 2.86 4.229 3.00 5.144 2.35 2.947 2.50 3.562 1.75 1.735 2.00 2.123 1.60 1.454 1.60 1.79 1.46 1.071 1.40 1.677 1.30 0.859 1.19 1.582 1.15 0.794 1.00 1.504 1.00 0.754 0.90 1.467 0.84 0.716 0.80 1.422 0.70 0.681 0.70 1.402 0.55 0.641 0.60 1.346 0.40 0.597 0.45 1.281 0.29 0.562 The rheometer and loop test data may be compared by converting the velocity data into apparent shear stress form (Γ = 8V/D) and the pressure gradient into the wall shear stress form (τw = ΔPD/4L). The results are presented graphically in Fig. 9.3. It can be observed that both rheometer flow curves are somewhat below and parallel to the pipe flow curves until the pipe flow goes turbulent. This is exactly what would be expected for a yield plastic suspension (e.g., see Eq. [5.34] for a Bingham plastic). This is consistent with the end correction assumption made above. 157 Wall Shear Stress and Shear Stress (Pa) 40 35 30 25 20 15 10 5 0 0 200 400 600 800 1000 Apparent Shear Rate and Shear Rate (1/s) Data 208 (10%) Pipe 208 Data 105 (14%) Pipe 105 Figure 9.3: Comparison of shear rate vs. shear stress relationships: rheometer and flow low (Data from Litzenberger 2004) 9.4. Laminar Pipe Flow The laminar pipe flow pressure gradient for yield plastic, Bingham and Casson fluids can be calculated using the following equation: ⎛τ ⎞ Γ ≈ ⎜⎜ w ⎟⎟ 1 − Z k ⎝ μ∞ ⎠ ( ) 3 (3− k ) k Laminar flow – yield plastic [5.62] The laminar pipe flow pressure gradients for yield power law, Bingham and power law fluids can be calculated using the following equation: 1 n +1 ⎡ n ⎞⎤ ⎛ 2n ⎞ ⎛ ⎛ τ ⎞ n ⎛ 4n ⎞ Z ⎟⎥ Γ=⎜ w⎟ ⎜ ⎟ Z ⎜1 + ⎟(1 − Z ) n ⎢1 + ⎜ ⎝ K ⎠ ⎝ 3n + 1 ⎠ ⎣ ⎝ 2n + 1 ⎠ ⎝ n + 1 ⎠ ⎦ Yield power law [5.64] Where: P D τ D =τw = o L 4 Z 4 V =Γ Pressure gradient [9.10] D 8 Bulk velocity [9.11] 158 The laminar flow pressure-gradient vs. bulk velocity relationships in a 25.825 mm pipeline were calculated for each of the models above, based on the parameters given in Table 9.1. The results are shown Fig. 9.4, along with the loop test data. 0.9 0.8 Pressure Gradient (kPa/m) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 Bulk Velocity (m/s) Bingham Casson Yield Power Law Power Law Yield Plastic Data 208 Figure 9.4: Comparison of measured and calculated laminar flow data (Data from Litzenberger 2004) At higher velocities the five models give similar results and fit the data well enough for engineering purposes. However, the flow curves differ significantly at low velocities. 9.5. Turbulent Pipe Flow Two turbulent flow models were considered in Chapter 6. The first uses the method proposed by Wilson and Thomas (1985). The second is a “pseudo-fluid” model that uses a standard Newtonian turbulent flow model and the plastic or infinite shear rate viscosity. These models will be considered separately. 9.5.1 Wilson-Thomas turbulence equation The Wilson-Thomas model may be used to calculate the smooth wall turbulent flow pressure-gradients for all of the consistency models included in Tables 9.3 and 9.4. 159 ⎛ 1.12 Dρu * ⎞ V = 2.457 ln⎜⎜ ⎟⎟ + (11.7(α − 1) − 2.457 ln (α )) η u* ⎠ ⎝ [6.49] It may be noted that the expanded form of the Wilson-Thomas equation is used rather than the engineering form [6.54] as the latter form cannot be used with the power law and the yield power law models. The friction velocity can be determined directly from the pressure gradient: u* = τw PD = ρ L4ρ [9.12] The value of α can be calculated using the formulas given in Appendix B. The apparent viscosity (η) can be calculated by putting the shear stress vs. shear rate consistency models into apparent viscosity vs. shear stress form. For the two “families” of consistency models, the relationships are: η= (1 − (τ μ∞ o τ )k Yield plastic [9.13] ) 1/ k 1/ n ⎞ 1−1 / n ⎛ K ⎟⎟ τ η = ⎜⎜ ⎝ 1 − (τ o τ ) ⎠ Yield power law [9.14] The Wilson-Thomas model posits that the shear stress in the boundary layer (τ) is equal to the shear stress at the wall (τw), which is also directly proportional to the pressure gradient. Therefore, the Wilson-Thomas model is best solved in terms of the pressure gradient. The smooth wall pressure gradients calculated using the Wilson-Thomas model and the parameters in Table 9.3 are shown in Fig. 9.5. Using the standard Wilson-Thomas model (i.e., ignoring the dispersion caused by Reynolds stresses) and turbulent flow equation, none of the rheological models gives a particularly good fit to the measured data, although the yield plastic and Bingham plastic models give the closest approximation. It may be noted that the Bingham model fits the data reasonably well in the transitional zone. This is consistent with the data presented by Wilson and Thomas (1985). However, the basis of the Wilson-Thomas model assumes that turbulent flow is fully developed (i.e., it fits where it is not supposed to be applicable). 160 6 Pressure Gradient (kPa/m) 5 4 3 2 1 0 0.5 1 1.5 2 2.5 3 3.5 Bulk Velocity (m/s) Power Law Bingham Yield Power Law Casson Yield Plastic Data 208 Figure 9.5: Smooth wall turbulent flow curves based on the Wilson-Thomas model (Data from Litzenberger 2004) 9.5.2 Pseudo-fluid turbulence equation Two smooth wall Newtonian turbulent flow models will be considered. The first is the Kundsen-Katz (1958) Blasius-type model, which is used for its simplicity. The second is the Nikuradse (1932) model, which is used for its accuracy and its ability to be correlated with velocity profile observations. The Nikuradse model is also equivalent to the Wilson-Thomas model with a drag reduction factor of β = 1. fN = 0.046 Re0.2 ⎛ 1.26 1 = −4.0 log⎜ ⎜ Re f fN N ⎝ [6.3] ⎞ ⎟ ⎟ ⎠ [6.4] The model parameters for yield plastic, Bingham and Casson fluids listed in Table 9.3 were used to calculate the turbulent flow gradient and the results are presented in Fig. 9.6. (Note: the power law and yield power law models do not have a “plastic viscosity” so the pseudo- 161 fluid model cannot be used). The flow loop data in Table 9.5 (G2000208) is presented in Fig. 9.6 along with low data for a similar 10%v/v dispersed with 0.27%w/w TSPP/clay (run G2000212). 6 Pressure Gradient (kPa/m) 5 4 3 2 1 0 0.5 1 1.5 2 2.5 3 3.5 Bulk Velocity (m/s) Knudsen-Katz YP Nikuradse Casson Nikuradse YP Data 208 Nikuradse Bingham Data 212 Figure 9.6: Comparison of pseudo-fluid turbulent flow models and measured data (Data from Litzenberger 2004) The difference between the pressure-gradient of the aggregated slurry (Data 208) and the dispersed slurry (Data 212) are negligible in the fully turbulent flow regime (i.e., V > 1.75 m/s). The pseudo-fluid turbulence model accurately predicts the fully turbulent flow if the yield plastic model is used. The Bingham model overestimates the pressure-gradient. The Casson model underestimates the pressure-gradient. It may also be noted that there is no significant difference between the Nikuradse and Knudsen-Katz predictions. The pseudo-fluid turbulence model using the yield plastic consistency model’s infinite shear rate viscosity gives the best correlation with the measured data. This is consistent with the observations made in Chapter 6 (i.e., that turbulence would disaggregate the particles in the suspension.) 162 9.6. Laminar-Turbulent Transition in Pipe Flow The laminar-turbulent transition velocity may be determined using the methods discussed in Chapter 7. The methods that use the yield stress criterion (e.g., Slatter-Wasp (2000) and Wilson-Thomas (2006) at high Hedström numbers, and [7.18]) calculate the transition velocity directly or via a critical Reynolds number. The remaining methods are determined, graphically or analytically; from the intersection of the laminar flow curve and what may be call a “critical value curve”, which are: • Hedström: Fully turbulent flow curve (pseudo-fluid with plastic viscosity) • Metzner-Reed: Iso-friction factor curve (fN = 0.0078) • Hanks: Critical pressure gradient (where Z is found using [7.7]) Figure 9.7 shows the transition criterion graphically. 1.6 Pressure Gradient (kPa/m) 1.4 1.2 1.0 0.8 0.6 0.4 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Bulk Velocity (m/s) Hedstrom - Bingham Metzner-Reed Slatter-Wasp Wilson-Thomas Hanks Eqn [7.10] Eqn [7.18] Laminar - Yield Plastic Data 208 Figure 9.7: Comparison of transition criterion for 10% kaolin slurry (Data from Litzenberger 2004) 163 Four of the criteria (Hedström, Slatter-Wasp, Wilson-Thomas, and Hanks) are for Bingham plastics and are shown as approximations. (Note: This is commonly done in engineering practice when the Bingham plastic model is assumed). To keep the figure clear, only the yield plastic laminar flow curve is shown because it overlaps the Bingham curve in the flow rate range of interest (see Fig. 9.3). For the Hedström intersection method the Knudsen-Katz equation for smooth wall turbulent flow is used ([6.35]). The calculated transition velocities are listed in Table 9.6. The Hedström intersection velocity and the transition velocities determined from Equations [7.10] and [7.18] were calculated using the infinite shear rate viscosities for k = 1 (“Bingham”) and k = 0.72 (“Yield Plastic”). (Note: Both critical value curves were not shown on Fig. 9.7 for clarity reasons). The Metzner-Reed criterion is not consistency dependent. It may be noted that Equations [7.10] and [7.18] give identical results for both the Bingham and Yield Plastic cases. For the Bingham approximation, the Hedström, Metzner-Reed and Hanks criteria give essentially the same value for the transition velocity. However, the predicted velocity (~1.06 m/s) is well below where the pressure gradient data is observed to “break” significantly from the laminar curve (~1.3 m/s). The Slatter-Wasp, Wilson-Thomas, and proposed criteria cluster in the 1.25 to 1.35 m/s range. This is consistent with the “instability point” and “pressure break-point” transition velocities discussed in Chapter 7. Table 9.6: Transition velocity in pipe flow 10%v/v kaolin slurry Transition Criteria Transition – Yield Plastic Transition – Bingham Hedström (1952) 1.12 m/s ~1.06 m/s Metzner-Reed (1955) 1.07 m/s ~1.07 m/s Slatter-Wasp (2000) - ~1.30 m/s Wilson-Thomas (2006) - ~1.25 m/s Hanks (1963) - ~1.05 m/s Intersection using [7.10] 1.35 m/s ~1.27 m/s Equation [7.18] 1.35 m/s ~1.27 m/s 164 The transition velocity calculated using the yield plastic model is somewhat conservative, as expected (i.e., it is intended to predict the velocity that transition will occur below). It may be possible to tighten the energy dissipation ratio (e.g., from 1.3 to 1.25) but further test work in larger pipes would be required to justify the change. 9.7. Pipe Flow Curves The combined pipe flow curves for the 10%v/v sample (G2000208, “Data 208”), calculated using the yield plastic model are shown in Fig. 9.8 along with the original data. The laminar flow curve is calculated using [5.62]. The turbulent flow curve is calculated using [6.3]. The transition velocity is calculated using [7.18]. The fit is excellent except for a small transition region (~1.3 to ~1.6 m/s). The fully turbulent flow curve gives conservative results in this transition region. 7 Pressure Gradient (kPa/m) 6 5 4 3 2 1 0 0 0.5 1 1.5 2 2.5 3 3.5 Bulk Velocity (m/s) Data 208 (10%) Y.P. 208 Data 212 Y.P. 105 Data 105 (14%) Trans 208 Data 215 Trans 105 Figure 9.8: Comparison of measured data for kaolin clay slurry and the pressure gradients predicted using the yield plastic model (Data from Litzenberger 2004) Figure 9.8 also shows the pipe flow data for a fully dispersed 10%v/v sample (G2000212, “Data 212”). It can be seen that the yield plastic turbulent flow curve fits the data across the entire data range for the dispersed slurry. 165 Similar calculations were made for the second 14%v/v sample (G2000105, “Data 105”) using the same set of equations as for the 10%v/v sample. In this case the transition velocity is calculated to be 2.03 m/s. Once again, the fit is excellent except for a small transition region (~2.0 to ~2.5 m/s). It may be noted that the length of the transition region is roughly the same as for Newtonian fluid (i.e., between Reynolds number of 2100 and ~3000). The pseudo-fluid fully turbulent flow curve gives conservative results in the transition region, making it suitable for engineering design. Figure 9.8 also shows the pipe flow data for a fully dispersed 14%v/v sample (G2000215, Data 215”). It can be seen that the yield plastic turbulent flow curve fits the data across the entire data range for the dispersed slurry. 9.8. Design Curve Diagram The flow loop data for the four kaolin slurries discussed in Section 8 may be put in terms of the stress number (Ha) and the plastic Reynolds number (Rep). Samples G2000208 and G2000212 are made dimensionless using the infinite shear rate viscosity determined for G2000208. Samples G2000105, and G2000215 are made dimensionless using the infinite shear rate viscosity determined for G2000105. The resulting dimensionless data is shown on the Design Curve Diagram (DCD) in Figure 9.9. Both non-Newtonian slurry samples (“Data 208” and “Data 105”) exhibit the same general behaviour. The laminar flow portion is a nearly straight line with a slope intermediate between the Bingham and Casson curves. This is expected since calculated scaling factors are 0.72 and 0.63 respectively. The laminar flow curves cross the fully turbulent flow line (“Turbulent”) but break upward before the transition line (“Transition (YP)”). After transition, the flow curves approach the fully turbulent line over a relatively short transition range and then follow it to the end of the test range. Both dispersed (Newtonian) slurry samples (“Data 212” and “Data 215”) also exhibit the same general behaviour. The data follows the fully turbulent flow line across the entire range. 166 Design Curve Diagram 1.E+07 He = 10 6 Stress Number (Ha) 1.E+06 Turbulent He = 10 5 Newton Bingham 1.E+05 Casson YP k = 1/3 YP k = 1/4 Turbulent He = 10 Transition (YP) Data 208 4 Data 212 Data 105 He = 0 Newtonian 1.E+04 3 He = 10 1.E+03 Data 215 1.E+04 1.E+05 Plastic Reynolds Number (Rep ) Figure 9.9: Design Curve Diagram representation of kaolin slurry data (Data from Litzenberger 2004) 9.9. Summary The accuracy and practicality of the yield plastic consistency model and the related pipeline design equations developed in this Thesis were tested using published data on kaolin clay slurry from the University of Saskatchewan (Litzenberger 2004). The methodology for extracting rheological data from concentric cylinder rheometers (CCR’s) with was reviewed. It was shown that the “narrow gap” approximation would not lead to significant errors with the measuring system used in the original research. The published CCR data for two slurry samples was used to determine the rheological parameters based on the yield plastic model and four other commonly used models. 167 The parameters for the five consistency models (as well as the slurry density) were used to calculate the laminar flow pressure gradients in a 25.825 mm pipeline for a 10%v/v kaolin slurry. While there was significant deviation in the predicted pressure gradients at low bulk velocities, the results were similar in the test range. The models all gave reasonable agreement with the measured data. The parameters for the five consistency models (as well as the slurry density) were used to calculate the turbulent flow pressure gradients in a 25.825 mm pipeline for a 10%v/v kaolin slurry using the Wilson-Thomas model. The five models gave significantly different predictions, none of which appeared to be consistent with the measured data. When the pseudo-fluid turbulent flow method was used with the models that have a limiting high shear rate viscosity. The yield plastic model was shown to accurately predict the flow loop data beyond a short transition region. It also predicted the pressure gradient of the fully dispersed slurry at the same solids content. This supports the hypothesis that the turbulence disaggregates the particles and the slurry acts as a Newtonian fluid in fully turbulent flow so a drag reduction factor of β = 1 may be used in most cases. The transition velocity was calculated for the 10%v/v slurry using seven of the methods described or developed in Chapter 7. The predictions were shown to cluster into two groups, one about 25% higher than the other. This higher “group” coincided with the observed deviation in the pressure gradient from the laminar flow curve. This supports the hypothesis that there are two distinct transition velocities: the “instability point” and “pressure breakpoint”. The latter is of interest to hydraulics engineers. The yield power law model parameters and the developed design equations were also used to predict the pressure gradient for two 14%v/v slurries; one partially dispersed and the other fully dispersed. The predicted results were in excellent agreement with the measured data in both cases. The infinite shear rate viscosities for the non-Newtonian slurries (based on the yield plastic model) were used to put the flow loop data for the four slurries into “stress number” and plastic Reynolds number form. The dimensionless results were plotted onto a Design Curve Diagram. The results showed that the slurries behaved in a consistent and easily understood manner. 168 CHAPTER 10: SUMMARY In nearly every detective novel…there comes a time where the detective has collected all the facts he needs for at least some phase of his problem. These facts often seem quite strange, incoherent, and wholly unrelated…[but]… suddenly, by Jove, he has it! Not only does he have an explanation for the clews at hand, but he knows certain other events must have happened. Since he now knows exactly where to look for it, he may go out, if he likes, to collect further confirmation for his theory. Albert Einstein and Leopold Infeld, The Evolution of Physics (1938 CE) 10.1. Introduction This chapter summarizes the objectives and developments of this Thesis. The contributions to knowledge are listed. Future research opportunities enabled by the current research are discussed. 10.2. Review of Thesis This Section summarizes the key points in the previous Chapters of this Thesis. Chapter 1. The successful design and operation of a mineral processing facility fundamentally depends on understanding slurry rheology. Rheology-based design of slurry pipelines was shown to be more cost effective than the alternative methods. Nevertheless, rheology-based design has not been adopted by practicing engineers to any significant degree. It is proposed that this is mainly due to the plethora of consistency models available, the complexity of the design equations, and the counter-intuitive methods used to present data. The objective of this Thesis is to overcome these obstacles and establish the theoretical basis for rheology-based-design of homogeneous slurry systems. Chapter 2: The inferred behaviour of slurry at the micro-scale and at the observed behaviour at the meso-scale is discussed. The non-Newtonian rheology of suspensions is due to aggregation and deaggregation of particles. In particular, the loosely packed aggregates trap carrier fluid, which increases the effective volume concentration. This was used to develop a three-part viscosity model: the Newtonian viscosity of the carrier fluid, the Newtonian 169 viscosity multiplier due to the dispersed particles, and the non-Newtonian viscosity adder due to the aggregation of the particles. Three constitutive equations that are commonly used for mineral suspensions (Newtonian, Bingham plastic, power law) are part of the “yield power law” family of consistency equations. Several problems with the yield power law model are discussed. Hallbom and Klein (2004) propose a new phenomenological constitutive equation, referred to as “yield plastic”. The yield plastic model was shown to describe an alternate family (Newtonian, Bingham plastic and Casson). This model has the advantage of relative simplicity, accuracy over short and long ranges of data, and consistent units. Although similar models have been proposed, they were never developed and their use has been extremely limited. No engineering design equations appear to have been developed. Chapter 3: The physical basis of the yield plastic model was derived using a “structural model”, where the rheological behaviour is caused by the formation and breakage of particle aggregates. The model assumes that the average bond strength will generally increase following simple power relation as the structure is broken down. The key concept in the yield plastic model is that the “infinite shear rate” viscosity is the Newtonian viscosity of the fully dispersed suspension and it does not depend on factors that cause aggregation. The particle dispersion is caused by the total shear rate, which includes the internal shear rate and shear rate applied by an external source (e.g., a pipe wall). If only the applied shear rate is accounted for, as is usually the case, the yield plastic model predicts the observed low shear rate Newtonian viscosity plateau. Chapter 4: A list of “desirable characteristics” for a rheological model was presented. It was shown that the yield plastic model has all the characteristics. The basic and extended versions of the yield plastic model were compared to the following models: Newtonian, Bingham plastic, Casson, Heinz, power law, yield power law, Cross, Carreau-Yasuda, Sisko, Ellis, Meter, Reiner-Philippoff, Prandtl-Eyring, and Powell-Eyring models. It was shown that the yield plastic model has advantages over each. Chapter 5: The pressure gradient vs. flow rate equation was derived for the laminar pipe flow of yield plastics. The equation was presented in integral form in terms of the pseudoshear rate (Γ), the wall shear stress (τw) and the shear ratio (Z): 170 Γ= 4τ w μ∞ 1 ∫z 2 ( z k − Z k )1 / k dz Laminar pipe flow - yield plastic [10.1] Z The integral was solved explicitly for a number of cases: k = 1 (Bingham), 1/2 (Casson), 1/3, 1/4, 1/5, 1/6, and 1/7 as well as for all values of k when τo = 0 (Newtonian). A general laminar flow equation in algebraic form was developed for yield plastics for real values of the scaling factor in the range ~0.2 < k ≤ 1: 4 τ ok (4 − k ) [10.2] ⎛τ ⎞ 3 (3− k ) k Γ = ⎜⎜ w ⎟⎟(1 − Z k ) ⎝ μ∞ ⎠ [10.3] τ wk ≈ Γ k μ∞k + The second equation is “approximate” but the error in the calculated pressure-gradient would be less than ~1%. The result is a single consistency model and laminar flow design equation suitable for shear-thinning mineral suspensions. Chapter 6: The pressure gradient vs. flow rate equation was derived for the turbulent flow of yield plastics in hydraulically smooth pipes. The equation was developed using velocity the Wilson-Thomas model for drag reduction. The resulting equation was presented in the conventional friction factor vs. (plastic) Reynolds number form: ⎛ 1.26 ⎞ 1 ⎟ = −4.0 log⎜ ⎜ β Re f ⎟ fN p N ⎠ ⎝ SW turbulent flow – yield plastic [10.4] The method for calculating “drag reduction factor” (β) was derived. However, it was shown that β approaches 1 as the pressure-gradient is increased. When Reynolds stresses are taken into consideration, it was suggested that β could be assumed to be one soon after transition. When β = 1, equation [10.4] is equivalent to the “plastic Nikuradse” equation. Chapter 7: A new criterion was proposed for the laminar-turbulent transition of yield plastics. The proposed criterion is that the pressure break transition will occur when the smooth wall turbulent friction factor is ~30% higher than the laminar friction factor. It was shown that, for Bingham plastics flowing in industrial sized pipes, the proposed transition equation is consistent with the Slatter-Wasp (2000) model (He > 100,000) and Wilson-Thomas (2006) 171 (He > 20,000). It is also consistent with the Thomas (1963) model at low values of the Hedström number. The criterion also holds for Newtonian fluids. Chapter 8: A dimensionless group (the “stress number”, Ha) was proposed as an alternative to the conventional friction factors that is independent of the bulk velocity. Ha = τ w ρD 2 μ∞2 Stress number [10.5] The stress number is applicable to all fluids with a limiting high shear rate viscosity (i.e., all real fluids and suspensions). In hydraulically smooth pipes, the pipe flow curves for any yield plastic fluid may be determined by three equations: 3 k Ha ⎛⎜ ⎛ He ⎞ ⎞⎟ (3− k ) k Re p ≈ 1− ⎜ ⎟ 8 ⎜⎝ ⎝ Ha ⎠ ⎟⎠ ( Re p = 2.457 Ha log 1.12β Ha HaC = Laminar flow – yield plastic [8.42] ) SW Turbulent flow – yield plastic [8.45] Re1p.8 Transition – yield plastic [8.46] 56.5 These equations were presented graphically, forming a novel “workspace” for the analysis of pipe flow referred to as the design curve diagram (DCD). The DCD is essentially a nondimensional pipe flow curve, making it conceptually easier to use than the Hedström-Moody diagram. The DCD may be used for Newtonian and non-Newtonian fluids. Chapter 9: The advantages of the yield plastic model and design equations developed in this Thesis were shown by means of a case study. This case study used concentric-cylinder rheometer and pipe loop test data obtained as part of the research for a Masters of Science degree in Chemical Engineering at the University of Saskatchewan in Saskatoon (“Rheological Study of Kaolin Clay Slurries” by Chad Litzenberger 2004). 10.3. Novel Contributions The following items in this thesis are claimed as novel and/or as furthering the art: • A new consistency model for shear-thinning suspensions was developed based on a modified aggregation-deaggregation structural model. The resulting three-parameter 172 “yield plastic” model has consistent units (Pa, Pa-s, -) and a reasonable high shear rate limiting viscosity (i.e., the viscosity of the fully dispersed suspension). • A three-part interpretation of shear-thinning consistency, where the infinite shear rate viscosity is Newtonian and independent of the non-Newtonian and time-dependence effects. • The yield plastic model was shown to describe a general family of consistency models that includes the Newtonian, Bingham, and Casson models as special cases. It is proposed to be an alternative to the currently used three-parameter “yield power law” (Herschel-Bulkley) family that include the Newtonian, Bingham and power law models as special cases. • The total or true shear rate was identified to be a combination of the applied shear rate (i.e., the shear rate caused by a pipe wall in flow) and the base shear rate (i.e., the additional motion due to thermal agitation, differential settling, equipment vibration, turbulent fluctuations, etc.). The base shear rate can be minimized but never eliminated. The finite base shear rate was shown to explain the “Newtonian plateau viscosity” often noted during very low shear rate/stress rheometry on suspensions with an apparent yield stress. • A laminar pipe flow equation for yield plastics, in a form suitable for numerical integration was derived. Exact algebraic equations were derived for k = 1/3, 1/4, 1/5, 1/6, and 1/7 (This had been done by other for k = 1 and k = 1/2) • A Caldwell-Babbitt-type equation, in an algebraic form suitable for most engineering purposes, was derived for the laminar pipe flow of yield plastics. (This had been done by others for k = 1). This equation overestimates the pressure gradient by up to 33% at low velocities, but converges quickly as the velocity increases. • A general algebraic equation was derived for laminar flow of yield plastics that gives pressure gradient results within experimental accuracy for the full velocity range. This equation is suitable for rheological characterization or engineering purposes when increased accuracy at low shear rates is required. 173 • A modification of the Wilson-Thomas drag-reduction model was used to develop a Nikuradse type design equation for the turbulent flow of yield plastics in hydraulically smooth pipes. (This had been done by others for k = 1 and k = 1/2). • A theory for the “break point” (i.e., the velocity in the laminar-turbulent transition range when the actual pressure gradient significantly deviates from the laminar flow pressure gradient) based on the ratio of the specific energy dissipation ratios of fully turbulent smooth wall flow and laminar flow at an given flow rate. Under normal conditions (e.g., in industrial pipelines) the break point for Newtonian fluids will occur when the energy dissipation ratio is approximately 1.3. • It was proposed that the break point energy dissipation ratio for yield plastics will be the same as the Newtonian energy dissipation ratio (i.e., ~1.3) if the dispersed suspension viscosity (μ∞) is used to determine the fully turbulent flow energy dissipation. (This may be considered a modification of the Hedstrom intersection method.) • An approximate equation for directly estimating the break point velocity in industrial slurry pipes (i.e., ~105 < He < ~107) was developed for yield plastics. (This had been done by others for k = 1). • A dimensionless group (the “stress number”, Ha) was proposed as an alternative to the conventional friction factor. The stress number is directly proportional to the pressure gradient in a pipeline and independent of the bulk velocity. • The conventional design equations were reformulated in terms of the stress number and shown to be less mathematically complex. • The “Bingham plastic viscosity” term in the plastic Reynolds number (Rep) was replaced with the fully dispersed suspension viscosity (μ∞) making it generally applicable to all real suspensions. • The “Bingham plastic viscosity” term and the “Bingham yield stress” term in the plastic Hedström number (He) were replaced with the fully dispersed suspension viscosity (μ∞) and the “yield plastic yield stress”, making it generally applicable to all yield plastics. 174 • The stress number and the (modified) plastic Reynolds number were used to develop a “design curve diagram” as an alternative to the conventional Moody-type (fN vs. Rep) diagram for representing pipe flow. The stress number (y-axis) is directly proportional to the pressure gradient in a pipeline and independent of the bulk velocity. The plastic Reynolds number (x-axis) is directly proportional to the bulk velocity and independent of the pressure gradient. A unique Ha vs. Rep flow curve may be plotted for each value of the modified Hedström number and scaling factor. The net result is that the design curve diagram is a dimensionless version of a standard flow curve diagram (i.e., ΔP/L or H/L vs. V or Q). 10.4. Future Research Work The research carried out in this Thesis may be used as the basis for a wide variety of potential research projects in the future. These include: 10.4.1 Physical-rheological parameter correlations As noted in the introduction to this Thesis, the goal of slurry rheologists ought to be a bridge between the parameters that a designer or operator can control or measure and the flow behaviour in a pipe or piece of equipment. An operator running a slurry pipeline handling the kaolin slurry shown in Fig. 1.2 wants to know the affect on the operation of the system of the CaCl2 to TSPP ratio. Specifically, he wants to understand how to deal with immediate operating problems quickly (e.g., if the flow drops out of the desired turbulent flow, he should increase the TSPP addition) rather than waiting several hours or days for a rheology report. There are already a number of tentative physical-rheological parameter correlations. For example, it is commonly assumed that the dispersed viscosity of slurry is directly proportional to the viscosity of the Newtonian carrier fluid, which is a function of the temperature. This allows the behaviour of the slurry flowing through, say, a heat exchanger to be approximated as the temperature changes without having to do rheometry at a multitude of discrete temperatures. The Einstein-type equations are an attempt at a correlation between solids volume concentration and apparent viscosity. A similar, albeit 175 more complex, correlation between solids volume concentration and the yield stress and plastic viscosity of a Bingham plastic was described in the introductory chapter. The constant parameters of the yield plastic model make it practical for researchers to define one-to-one correlations between physical parameters and physical parameters. The relationship between the yield plastic model parameters and the following physical parameters would be of interest to practicing engineers: • Volume concentration • Temperature • Carrier liquid viscosity • Zeta potential and pH for heterogeneous mineral mixtures (e.g., at varying ratios of silica and alumina) • Particle size, particularly for clay sized particles where the liquid bonded to the surface make up a significant portion of the effective volume • Particle shape • Particle size distribution (e.g., the ratio between the P80 and P50 fraction) • Coagulant concentration (e.g., CaCl2) • Dispersant concentration (e.g., TSPP) 10.4.2 Shear thickening The yield plastic model presented is fundamentally a shear-thinning consistency model, in that its physical basis is the breakdown of structure with increased shear. It has been reported that some mineral slurry exhibits time-independent shear-thickening, particularly near the maximum packing concentration. A structure based consistency model for shearthickening, similar to the yield plastic model, is required to be able to address all slurries. It should be noted that shear-thickening is rarely encountered with mineral slurries and the author has never seen the behaviour. 176 10.4.3 Rotational rheometry Characterization of a material’s consistency is often carried out in a rotational rheometer. There are several types of rotational rheometers including: concentric cylinder (Couette or Searle), open cylinder (“Brookfield”), parallel plate, and conical. The presence of relatively large (sand) top-sized particles found in many mineral slurries requires the use of a wide “gap” between the test surfaces (e.g., the inner and outer cylinders). When the slurry exhibits non-Newtonian behaviour the apparent (pseudo-Newtonian) shear rate must be corrected to the true shear rates. A corrected equation, such as is available for Bingham, Casson and power law fluids, is required. 10.4.4 Explicit solution of laminar pipe flow equation The exact solution for the laminar pipe flow equation was given in integral form. While the integral is easily solved numerically, an exact, explicit general algebraic solution would be preferable. However, it should be noted that if the exact solution were too complex practicing engineers would not use it. The approximate general algebraic solutions presented in this Thesis are considered to be an appropriate trade-off between accuracy and complexity. 10.4.5 Laminar pipe flow of extended yield plastics The laminar pipe flow equations developed in this Thesis assumed that the base shear rate was negligible and thus there was no low shear rate Newtonian plateau. The equivalent equation may be developed for the extended yield plastic model. It is expected that the difference will normally be negligible except at very low velocities. At low velocities, the base shear rate should give results similar to “wall slip”. However, there are situations where the base shear rate would not be negligible. For example, a “concrete vibrator” may be installed in the pipe between a hopper and a pump to allow high yield stress paste to flow into the pump suction. 10.4.6 Drag reduction factor The equation for the drag reduction factor (β) presented in this Thesis was based on the assumptions of the consistency related sub-layer thickening model presented by Wilson and Thomas (1985). Several issues with this model were raised; the most important being that 177 the model ignores Reynolds stresses, which would tend to decrease the apparent viscosity of a shear-thinning suspension. It was conjectured that the drag reduction factor should approach unity (i.e., no drag reduction) more quickly than the Wilson-Thomas model implies. This conjecture is supported by the Case Study. It is proposed that this may be investigated by comparative loop tests on aggregating (non-Newtonian) and chemically dispersed suspensions. 10.4.7 Collapsing plug It is often assumed (e.g., by Wilson and Thomas 1985) that in turbulent flow the ratio of diameter of the unyielded plug to the pipeline diameter is equal to the stress ratio (Z), as it is in laminar flow. The basis of the extended yield plastic model implies that once turbulence has been established, the plug diameter will decrease more quickly than the stress ratio would indicate. Furthermore, the plug should collapse completely soon after the pressurebreak transition. It is proposed that this may be confirmed by anemometry or visually if a translucent yielding slurry is used. 10.4.8 Alternate flow configurations While flow through a cylinder is the most common flow configuration, there are others of significant importance in the design and operation of a mineral processing plant. These include: • Annular flow (axial) between two cylinders (e.g., drilling mud) • Sheet flow (e.g., thickened tailings flowing in an impoundment) • Open channel flow in troughs of various sizes and shapes (e.g., launders) In each case, equations for laminar and turbulent (or sub-critical and super-critical) flow are required as well as a method for determining the transition point. 10.4.9 Rotating equipment Slurry with moderate yield stresses may be pumped using centrifugal slurry pumps. It is found that pumps handling slurry produce less head and have a lower efficiency than when pumping clear water. There are standard derating methods for viscosity and the presence of solid particles, but these both assume that the flow behaviour is Newtonian. Attempts have 178 been made to develop a method for “rheological derating” of pumps but there is no generally accepted method and the various methods proposed give inconsistent results. The methods employed usually require forcing rheometry data into a certain model (usually the Bingham plastic model), with the result that the parameters vary with the shear rate range. It is proposed that using the yield plastic model will allow a one-to-one correlation between the model parameters and the derating factors. Mineral slurry is often stored in agitated tanks or processed in agitated vessels (e.g., autoclaves). A method for estimating the agitator size and speed, as well as the power requirement for yield plastics is required for a cost effective design. 10.4.10 Transition and energy dissipation ratio The energy-dissipation ratio transition model proposed in this Thesis is mainly empirical and based on the observed pressure-break transition of slurries. However, its correlation with the accepted transition of Newtonian fluids and the high Hedström number transition models proposed by Slatter-Wasp (2000) and Wilson-Thomas (2006) for Bingham plastics is remarkably close (see Fig. 7.3). The correlation with the Thomas (1963) model for Bingham plastics at low values of the Hedström number is also well within experimental error limits. It is proposed that this justifies further research from a theoretical standpoint. 10.4.11 Time-dependent slurry Although the scope of this Thesis was limited to time-independent slurry, the yield plastic model was originally (Hallbom and Klein 2004) proposed for time-dependent slurry. Analysis of time-dependent slurries is complex than using existing models, such as the one proposed by Cheng and Evans (1965). The Cheng-Evans model has an array of yield power law curves for the instantaneous flow curves and an additional yield power law curve for the equilibrium flow curves. This gives three variable parameters and three fixed parameters. Using the yield plastic model fixes the infinite shear rate viscosity as equal to the dispersed viscosity of the slurry locks in one of the rheological parameters. In at least one case (concentrated red mud) the scaling factor was also found to be essentially constant. Analysis of time-dependence is reduced to a variable yield stress and two fixed parameters. It is 179 proposed that the current research will reduce the complexity of research on timedependence. 10.5. Conclusions This Thesis develops a new theoretical basis for the analysis of slurry mechanics. The physically based yield plastic family of consistency models is an alternative to the phenomenological yield power law family of models. The yield plastic model can be used instead of most of the standard consistency models for shear-thinning slurries. The design equations for pipe flow are less complex than the equivalent equations for other models of equal flexibility. The use of a single consistency model, three design equations, and the conceptual simplicity of the design curve diagram will allow engineers to get a qualitative understanding of nonNewtonian pipe flow, quickly and without loss of quantitative accuracy. This will make help in transforming rheology-based-design for something that mineral process engineers could use and should use to something that they do use. 180 REFERENCES Abulnaga, B.E.; (2002); “Slurry Systems Handbook”; McGraw-Hill Ancey, C.; (2005); “Notebook, Introduction to fluid rheology”; Ecole Polytechnique Federal de Lausanne ASTM C143 (1996); “Standard Test Method for Slump of Hydraulic Cement Concrete”, Annual Book of ASTM Standards, Vol. 04.02, “Concretes and Aggregates”, Designation C143-90a Barnes, H.A.; (1999); “The yield stress – a review or ‘παντα ρει’ – everything flows”; Journal of Non-Newtonian Fluid Mechanics, 81, pp. 133-178 Barnes, H.A.; Walters, K.; (1985); “The yield stress myth?”; Rheologica Acta, 24, pp. 323-326 Bingham, E.C.; (1916); “An investigation of the laws of plastic flow”; Bulletin of the US Bureau of Standards, Vol. 13, p. 309 Blasius, H.; (1913); Mitt. Forschungsarb; No. 131 Buckingham, E.; (1922); “On plastic flow through capillary tubes”; Proceedings of the A.S.T.M., 21, p. 1154 Caldwell, D.H. and Babbitt, H.E.; (1941); “The flow of muds, sludges, and suspensions in circular pipe”; Trans A.I.Ch.E., Vol. 37, p. 237 Carreau, P.J.; (1968); “Rheological equation from molecular network theory”; Doctoral dissertation, University of Wisconsin, Madison Wisconsin Casson, N.; (1959); “A flow equation for pigment-oil suspensions of the printing ink type”; British Society of Rheology Proceedings, Rheology of Disperse Systems, pp. 84- 104 181 Cerpa, A.; García-González, M.T.; Serna, C.J.; Tartaj, P.; (2001); “Relationship between the colloidal and rheological properties of mineral suspensions”; Canadian Journal of Chemical Engineering, 79 (August), pp. 608-611 Cheng, D.C-H.; (1985); “Yield stress: a time-dependent property and how to measure it”; British Society of Rheology Conf. on “New Techniques in Experimental Rheology”, U. of Reading Cheng, D.C-H.; Evans, F.; (1965); “Phenomenological characterization of the rheological behaviour in inelastic reversible thixotropic and antithixotropic fluids”; British Journal of Applied Physics, Vol. 16, pp. 1599-1617 Colebrook, C.J.; (1939); “Turbulent flow in pipes with particular reference to the transition region between the smooth and rough pipe laws”; Journal of the Institute of Civil Engineers, (4), pp. 133-156 Colebrook, C.J.; White, C.M.; (1937) “Experiments with fluid friction in roughened pipes”; Proceedings of the Royal Society of London, Series A, Mathematical and Physical Science, 161 (906), pp. 367-381 Coussot, P.; (1997); “Mudflow Rheology and Dynamics”; IAHR Monograph Series Cross, M.M.; (1965); “Rheology of non-Newtonian fluids: A new flow equation for pseudoplastic systems”; Journal of Colloid Science, 20, pp. 417-437 Cross, M.M.; (1970); “Kinetic interpretation of non-Newtonian flow”; Journal of Colloid and Interface Science, 33 (1), May, pp. 30-35 Curlook, W.; (2004); “Improvement to the acid pressure leaching of nickel laterite ores”; TMS International Nickel Symposium, Charlotte NC, pp. 325-334 Darby, R.; (1984); “Determination and utilization of rheological properties for prediction of flow behavior of pseudohomogeneous slurries”; Proc. of the 9th Inter. Tech. Conference on Slurry Transportation, Lake Tahoe, pp. 107-115 182 Darby, R.; (1996); “Chemical Engineering Fluid Mechanics”; Marcel Dekker, Inc. De Waele, A.; (1930); “Plastic and pseudo-plastic flow”; Journal of Rheology, 1 (2) pp. 139-148 Dolan, K.D.; Steffe, J.F.; Morgan, R.G.; (1989); “Back extrusion and simulation of viscosity development during starch gelatinization”; Journal of Food Process Engineering, 11, pp. 79-101 Durand, R., Condolios, E.; (1952); “Hydraulic transport of coal and solid materials in pipes”; Proceedings of a colloquium on the hydraulic transport of coal, National Coal Board, London, UK, pp. 39-52 Einstein, A.; (1906); “Ein neue Bestimung der Molekul-dimensions”; Annalen Physik, 19, pp. 289-306 Fourie, A.; Jewell, R.; (2007); (editors); Proceedings of the Tenth International Seminar on Paste and Thickened Tailings; Perth, Australia, March Frigaard, I.A.; Nouar, C.; (2003); “Predicting transition to turbulence in well construction flows”; Society of Petroleum Engineers, SPE 81150, pp. 1-9 Galdala-Maria, F.; Acrivos, A.; (1980); “Shear-induced structure in a concentrated suspension of solid spheres”; Journal of Rheology, 24 (6), pp. 799-814 Govier, G.W., Aziz, K.; (1977); “The Flow of Complex Mixtures in Pipes”; Robert E. Krieger Publishing Co., New York Hackley, V.A.; Ferraris, C.F.; (2001); “NIST recommended practice guide to the use of nomenclature in dispersion science and technology”; NIST SP-960-3 Hallbom, D.J.; (2005); “The ‘lump’ test”; Proceedings of the Eighth International Seminar on Paste and Thickened Tailings, Santiago, Chile, April, pp. 73-97 Hallbom, D.J.; Klein, B.; (2004); “Flow array for nickel laterite slurry”; TMS International Nickel Symposium, Charlotte NC, pp. 415-428 183 Hanks, R.W.; (1963); “The laminar-turbulent transition for flow in pipes, concentric annuli, and parallel plates”; A.I.Ch.E. Journal, January, pp. 45-48 Hanks, R.W.; (1980); “Critical velocities in methanol-coal slurries”; ASME Paper No. 80WRK/Pet-4, pp. 14-16 Hanks, R.W.; (1981); “Course notes: Hydraulic design for low of complex fluids”; Richard W. Hanks Associates, Inc., Orem, Utah Hanks, R.W., Pratt, D.R.; (1967);“On the flow of Bingham plastic slurries in pipes and between parallel plates”; Society of Petroleum Engineers Journal, 1 (4), pp.342-346 Hedström, B.O.A.; (1952); “Flow of plastic materials”; Industrial and Engineering Chemistry, 44, pp. 651-656 Heimann, W.; Fincke, A.; (1962); “Beiträge zur Rheometrie der Schkoladen III: Anwndung einer modifizierten Casson-Gleichung auf Milchschokoladen und Kakaomassen”; Z Lebensm Unters Forsch, 117, pp. 297-301 Heinz, W.; (1959); Materialprüf, 1, p. 311 Herschel, W.H.; Bulkley, R.; (1926) “Measurement of consistency as applied to rubber-benzene solutions”; American Society of Testing Material Proc., 26, pp. 621-633 Heywood, N.I.; (2007); (chairman technical committee); Proceedings of the Seventeenth International Conference on the Hydraulic Transport of Solids (Hydrotransport 17); SAIMM and BHR Group, Cape Town, South Africa Hunter, R.J.; (2002); “Introduction to Modern Colloid Science”; Oxford Science Publications Imrie, W.P., Lane, D.M.; (2004); (editors); Proceedings of the International Laterite Nickel Symposium; The Minerals, Metals & Material Society (TMS), Charlotte NC, USA, March 184 Jewell, R., Barrera, S.; (2005); (editors); Proceedings of the Eighth International Seminar on Paste and Thickened Tailings; Santiago, Chile, April Jewell, R.J., Fourie, A.B.; (2006); “Paste and Thickened Tailings – A Guide 2nd Ed.”; Australian Centre for Geomechanics Jewell, R.; Lawson, S.; Newman, P.; (2006); (editors); Proceedings of the Ninth International Seminar on Paste and Thickened Tailings; Limerick, Ireland, April Klein, B.; Hallbom, D. J.; (2002); “Modifying the rheology of nickel laterite suspensions”; Minerals Engineering, 15, pp. 745-749 Knudsen, J.G. and Katz, D.L.; (1958); “Fluid dynamics and heat transfer”; McGraw- Hill Book Co., New York Krieger, I.M.; (1972); “Rheology of monodisperse lattices”; Adv. Colloid Interface Sci., 3, pp. 111-136 Krieger, I.M.; Dougherty, T.J.; (1959); “A mechanism for non-Newtonian flow in suspensions of rigid spheres”; Transactions of the Society of Rheology, 3, pp. 137-152 Land, N.S.; (1972); “A compilation of nondimensional numbers”; National Aeronautics and Space Administration; NASA SP-274 Liang, S.; (2004) “Yield stress of starch-based and gum-based food thickeners”; BE Thesis, University of Queensland, Department of Chemical Engineering Litzenberger, C.G.; (2004) “Rheological study of kaolin clay slurries”; MS Thesis, University of Saskatchewan, Saskatoon, Department of Chemical Engineering Lumley, J.L.; (1973); “Drag reduction in turbulent flow by polymer additives”; J. Poly Sci., Macromol, Rev. 7; A. Peterlin (Ed.), Interscience, New York, pp. 263-290 Lumley, J.L.; (1978); “Two-phase flow and non-Newtonian flow”; Turbulence, P. Bradshaw (Ed.), Topics in Applied Physics, Vol. 12, Springer-Verlag, Berlin, Ch. 7 185 Malkin, A.Y.; (1994); “Rheology fundamentals”; ChemTech Publishing Masalova, I.; Taylor, M.; Kharatiyan, E.; Malkin, A.Y.; (2005); “Rheopexy in highly concentrated emulsions”; Journal of Rheology, 49 (4), pp. 839-849 Metzner, A.B.; Reed, J.C.; (1955); “Flow of non-Newtonian fluids – correlation of the laminar, transition, and turbulent-flow regions”; A.I.Ch.E. Journal, 1 (4), pp. 434-440 Moody, L.F.; (1944); “Fiction factors for pipe flow”; Transactions of the A.S.M.E (November), pp. 671-684 Morgan, R.G.; Steffe, J.F.; Ofoli, R.Y.; (1989); “A generalized viscosity model for extrusion of protein doughs”; Journal of Food Process Engineering, 11, pp. 55-78 Mular, A.L.; Halbe, D.N.; Barratt, D.J.; (2002); (editors) Mineral Processing Plant Design, Practice, and Control Proceedings; Vancouver, Canada Mustafa; Usui, H.; Ishizuki, M.; Shige, I.; Suzuki, H.; (2003); “Rheological characterization of non-spherical graphite suspensions”; Korea-Australia Rheology Journal, 15 (1), pp. 19-25 Newton, I.; (1687); “The mathematical principles of natural philosophy”; (Translated by Motte, A., Prometheus Books, New York, 1995) Nguyen, Q.D.; (1983); “Rheology of Concentrated Bauxite Residue Suspensions”; PhD Thesis, Monash University, Australia Nguyen, Q.D.; Boger, D.V.; (1998); “Application of rheology to solving tailings disposal problems”; International Journal of Mineral Processing, 54, pp. 217-233 Nikuradse, J.; (1932); “Gesetzmässigkeiten der turbulenten Strömung in glatten Rohren”; Forschungsheft No. 356, V.D.I Verlag, Berlin, p. 36 Nouar, C.; Frigaard, I.A.; (2001); “Nonlinear stability of Poiseuille flow of a Bingham fluid: theoretical results and comparison with phenomenological criteria”; Journal of Non-Newtonian Fluid Mechanics, 100, pp. 127-149 186 Ojo, A.; Akanbi, C.T.; (2006); “Rheology of soy-ogi: effect of concentration and temperature”; Journal of Food Engineering; 72, pp. 141-146 Oka, S.; (1971); “An approach to a unified theory of the flow behavior of time- independent non-Newtonian suspension”; Japanese Journal of Applied Physics, 10 (3), pp. 287-291 Olfoi, R.Y.; Morgan, R.G.; Steffe, J.F.; (1987); “A generalized model for inelastic fluid foods”; Journal of Texture Studies, 18, pp. 213-230 Onal, G.; Acarkan, N.; Celik, M.S.; Arslan, F.; Atesok, G.; Guney, A.; Sirkeci, A.A.; Yuce, A.E., Perek, K.T.; (2006); (editors); Proceedings of XXIII International Mineral Processing Congress; Istanbul, Turkey, September Otswald, W.; (1925); “Uber die Geschwin-digkeitfunktion der viskisität disperser systeme”; Koloid Zeistricht, 36, pp. 99-117 Potter, M.C; Wiggert, D.C.; (1997); “Mechanics of Fluids 2nd E.”; Prentice Hall Potvin, Y.; Thomas, E.G.; Fourie, A.B.; (2005); “Handbook on Mine Fill”; Australian Centre for Geomechanics Prandtl, L.; (1921); Z. Angew. Math. Mech., 8, p. 85 Quemada, D.; (1998); “Rheological Modeling of Complex Fluids. I. The concept of effective volume fraction revisited”; The European Physical Journal – Applied Physics, pp. 119-127 Rovedo, C.A.; Viollaz, P.E.; (1991); “The effect of pH and temperature on the rheological behavior of dulce de leche, a typical dairy Argentine product”; Journal of Dairy Science, 74, pp. 1497-1502 Ryan, N.W.; Johnson, M.M.; (1959); “Transition from laminar to turbulent flow in pipes”; A.I.Ch.E Journal (5), pp. 433-435 187 Schramm, G.; (2000); “A practical approach to rheology and rheometry 2nd Ed.”; Gebrueder HAAKE GmbH, Germany Scott Blair, G.W.; (1966); “The success of Casson’s equation”; Rheologica Acta, 5 (3), pp. 184-187 Scott Blair, G.W.; (1967); “A model to describe the flow curves of concentrated suspensions of spherical particles”; Rheologica Acta, 6 (3), pp. 201-202 Shook, C.A.; Gillies, R.G., Sanders, R.S.; (2002); “Pipeline hydrotransport with applications in the oil sands industry”; SRC Publication No. 11508-1E02, SRC Pipe Flow Technology Centre, Saskatoon, Canada Sisko, A.W.; (1958); Ind. Eng. Chem., 50, p.1789 Slatter, P.T.; (1996); “Turbulent flow of non-Newtonian slurries in pipes”; J. Hydrol. Hydromech., 44, 1, pp. 24-38 Slatter, P.T.; (2007); “Transition velocity estimation for flow in large pipes”; Proceedings of the Tenth International Seminar on Paste and Thickened Tailings; Perth, Australia, March Slatter, P.T. and Wasp, E.J.; (2000); “The laminar/turbulent transition in large pipes”; 10th International Conference on Transport and Sedimentation of Solid Particles; Wroclaw, Poland, September, pp. 389-399 Sofrá, F.; Boger, D.V.; (2002); “Environmental rheology for waste minimisation in the minerals industry”; Chemical Engineering Journal, 86, pp. 319-330 Song, K-W.; Kuk, H-Y.; Chang. G-S.; (2006); “Rheology of concentrated xanthan gum solutions: oscillatory shear flow behavior”; Korea-Australia Journal of Rheology, 18 (2), pp. 67-81 188 Song, K-W.; Chang, G-S.; (1999); “Steady shear flow and dynamic viscoelastic properties of semi-solid food materials”; The Korean Journal of Rheology, 11 (2), pp. 143-152 Sopade, P.A.; Kiaka, K.; (2001); “Rheology and microstructure of sago starch from Papua New Guinea”; Journal of Food Engineering, 50, pp. 47-57 Steffe, J.F.; (1996); “Rheological Methods in Food Processing Engineering 2nd Ed.”; Freeman Press Streeter, V.L.; Wylie, E.B.; (1985); “Fluid Mechanics 8th Ed.”; McGraw-Hill Book Company Tadros, Th.F.; (1996); “Correlation of viscoelastic properties of stable and flocculated suspensions with their interparticle interactions”; Journal of Colloid and Interface Science, 68, pp. 97-200 Thomas, A.D.; Wilson, K.C.; (1987); “New analysis of non-Newtonian turbulent flow – yield-power-law fluids”; Canadian Journal of Chemical Engineering, 65, pp. 335-338 Thomas, D.G.; (1963); “Non-Newtonian suspensions. Part 1, physical properties and laminar transport characteristics”; Ind. And Eng. Chem., 55 (11), pp.18-29 Thomas, D.G.; (1964); “Turbulent disruption of flocs in small particle size suspensions”; A.I.Ch.E. Journal, 10 (4), pp. 517-523 Thomas, D.G.; (1965); “Transport characteristics of suspensions: part VII. A note on the viscosity of Newtonian suspensions of uniform spherical particles”; Journal of Colloid Science, 20, pp. 267-277 Usui, H.; (2001); “Viscosity prediction of dense slurries with agglomerative coal particles”; Journal of the Society of Rheology – Japan, 29 (2), pp. 71-76 Usui, H.; (2002); “Prediction of dispersion characteristics and rheology in dense slurries”; Journal of Chemical Engineering of Japan, 35 (9), pp. 815-829 189 Van Olphen, H.; (1963); “An introduction to clay colloid chemistry for clay technologists, geologists, and soil scientists”, Interscience Publishers Wasp, E.J.; Kenny, J.P.; Gandhi, R.L.; (1977); “Solid-Liquid Flow Slurry Pipeline Transportation”;Trans Tech Pub, Series on Bulk Material Handling Vol.1 (1975/77) No.4 Whorlow, R.W.; (1992); “Rheological Techniques”; Ellis Horwood Ltd. Wills, B.A.; (1992); “Mineral Processing Technology 5th Edition”; Pergamon Press Ltd., Oxford Wilson, K.C.; Thomas, A.D.; (1985); “A new analysis of the turbulent flow of non- Newtonian fluids”; Canadian Journal of Chemical Engineering, 63, pp. 539-546 Wilson, K.C.; Thomas, A.D.; (2006); “Analytic model of laminar-turbulent transition for Bingham plastics”; Canadian Journal of Chemical Engineering, 84 (5), pp. 520-526 Wilson, K.C., Addie, G.R., Sellgren, A. and Clift, R.; (2006); “Slurry transport using centrifugal pumps – third edition”, Springer Science+Business Media Inc., New York 190 APPENDIX A: DERIVATION OF LAMINAR FLOW SOLUTIONS A1 Introduction The purpose of this Appendix is to show the derivations of the laminar pipe flow velocity vs. pressure gradient relationship for yield plastics. The equivalent derivations for the power law and yield power law models are included for reference. A2 Laminar Pipe Flow At low bulk velocities, the flow of a viscous fluid in a cylinder will be laminar, without large irregular fluctuations in the pressure or local velocity. If the flow is steady, then the pressure gradient is found as follows. The viscous shear stress (τ) is the defined as the shearing force (F) per unit area (A): τ ≡F/A Shear stress [A.1] A force balance is used to find the shear stress in steady laminar pipe flow. There is no acceleration in steady flow so, from Newton’s first law, there can be no net force. From Fig. 5.1 it can be seen that at radius (r) the pressure (P) is resisted by the shear stress (τ): ∑ F = ( P − P )(πr 0 1 2 ) − τ (2πrL) = 0 τ = ΔP ⋅ r 2 L [A.2] At the pipe wall, r = R and the shear stress equals the wall shear stress (τw). τ w = ΔPR 2 L = ΔPD 4 L Wall shear stress [A.3] Combining [A.2] and [A.3]: r τ = R τw 191 r = (R / τ w )τ [A.4] Equation [A.4] can be differentiated with respect to the shear stress to give: dr = (R τ w )dτ [A.5] The rate of strain due to shear, or shear rate, ( γ& ) is defined as: γ& ≡ dγ / dt = du / dr Shear rate [A.6] The local velocity of the fluid (u) will vary across the diameter of the pipe. The fluid is continuous but it can be approximated as the flowing in a multitude of concentric thin annuli (or lamella) of fluid flowing at the average velocity at radius r (see Fig. A.1). u(0) = umax R r Centerline τ u+du dr τ+dτ du Wall: u(R) = 0 u Figure A.1: Shear rate in a pipe The relative motion of the annuli on either side of a given annulus will cause the fluid element to be strained by the viscous shear stress and to flow if it is above the yield stress. The velocity variation across the pipe is a function of the radius (i.e., u = f(r)). The differential flow rate of the annulus between radii r and r+dr is: dQ = u (2π ⋅ r )dr The entire flow rate through the pipe (Q) is therefore the sum of the flow elements from the centreline (r = 0) to the pipe wall (r = R): Q R R 0 0 0 Q = ∫ dQ = ∫ u (2π ⋅ r )dr = π ∫ u (2r )dr Since d(r2)/dr = 2r, the variable of integration can be changed to r2. The upper limit of integration then becomes r2 = R2. Integrating by parts: 192 R2 [ Q = π ∫ (u )dr = πur 2 ] 2 2 R 0 0 R2 − π ∫ r 2 du 0 If it is assumed that the fluid velocity at the pipe wall is zero (i.e., the “no-slip” boundary condition), then u = 0 at r = R, so: [ R2 ] R2 Q = π 0 R − πu 0 − π ∫ r du = −π ∫ r 2 du 2 2 2 0 [A.7] 0 The shear rate will be some function of the shear stress (f(τ)). Because the shear stress is opposite to the direction of flow (see Fig. A.1), the shear rate function is negative. γ& = du / dr = − f (τ ) Æ du = − f (τ )dr [A.8] Substituting equations [A.4], [A.5] and [A.8] into Eq. [A.7] and changing the limits of integration (τ = 0 at r = 0 and τ = τw at r = R) gives: Q = −π R2 τw ∫ (( R / τ )τ ) (− f (τ )dr ) = +π ∫ ( R / τ 2 w 0 w ) 2τ 2 f (τ )( R / τ w )dτ 0 τw πR 3 2 Q = 3 ∫τ f (τ )dτ τw 0 [A.9] Calculations are simplified if Eq. [A.9] is defined in terms an apparent shear rate (Γ) based on the average (or bulk) velocity (V): V = Q / πR 2 Γ≡ Bulk velocity 8V D Apparent shear rate τ Q = VπR 2 = Γ(2 R) 2 πR3 w 2 πR = 3 ∫τ f (τ )dτ 8 τw 0 [A.10] Equation [A.10] may be reorganized into the general apparent shear rate equation for laminar pipe flow: Γ= 4 τ τw ∫τ 3 w 0 2 f (τ )dτ General apparent shear rate equation [A.11] 193 In Chapter 5 it is shown that, for yield plastic fluids Eq. [A.11] becomes: Γ= 4τ w μ∞ 1 ∫z 2 ( z k − Z k )1 / k dz Laminar flow - yield plastic [A.12] Z Where: A3 Z= τo τw Wall shear ratio (-) z= τo τ Local shear ratio (-) Newtonian Most fluids (e.g., water, oil, mercury) are Newtonian (1687), where the shear stress is directly proportional to the shear rate: τ = μγ& Newtonian [A.13] The shear rate as a function of shear stress for a Newtonian fluid is therefore: f (τ ) = γ& = τ μ Substituting Eq. [A.13] into Eq. [A.11] gives: τw τ τw ⎛τ ⎞ 4 w 3 4 ⎡τ 4 ⎤ τ τ dτ = 3 ⎢ ⎥ = w Γ = 3 ∫τ ⎜⎜ ⎟⎟dτ = 3 ∫ τw 0 ⎝ μ ⎠ μτ w 0 μτ w ⎣ 4 ⎦ 0 μ 4 2 Rearranging gives: Γ= τw μ Newtonian [A.14] Equation [A.14] can be expanded using the definitions of the apparent shear rate (Γ) and the wall shear stress (τw) and rearranged into the well-known Hagen-Poiseuille equation for the laminar flow of a Newtonian fluid (see Streeter and Wylie 1985). ΔPR ΔP 8μQ 128μQ ⎛ 8 ⎞⎛ Q ⎞ 4μ∞Q = μ∞ ⎜ Æ = = ⎟⎜ 2 ⎟ = 3 2L L πR 4 πD 4 ⎝ 2 R ⎠⎝ πR ⎠ πR 194 A4 Yield Plastic (k = 1/1) or Bingham Plastic The Bingham plastic (1916) model is a yield plastic fluid with a scaling factor of one. τ 1 = τ o1 + ( μ∞γ& )1 Bingham [A.15] The laminar flow equation for a Bingham plastic may be derived by substituting k = 1 into Eq. [A.12] and integrating: Γ= 4τ w μ∞ 1 ∫z 2 ( z − Z )dz = Z 4τ w μ∞ 1 ∫ (z 3 − Zz 2 )dz Z 1 4τ ⎡ z 4 Zz 3 ⎤ 4τ w ⎛ 1 Z Z 4 Z 4 ⎞ ⎟ ⎜ − − Γ= w⎢ − = + μ∞ ⎣ 4 3 ⎥⎦ Z μ∞ ⎜⎝ 4 3 4 3 ⎟⎠ Γ= τ w ⎛ 4Z Z 4 ⎞ ⎜1 − ⎟ + μ∞ ⎜⎝ 3 3 ⎟⎠ Bingham [A.15] Equation [A.15] is equivalent to the Buckingham (1922) equation for the laminar flow of a Bingham plastic material with no wall slip. Equation [A.12] may be solved explicitly whenever the inverse of the scaling factor is an integer. Six examples are given in the following Sections: 195 A5 Yield Plastic (k = 1/2) or Casson τ 1 / 2 = τ o1 / 2 + (μ∞γ& )1 / 2 1/ 2 ⎛τ ⎞ ⎜⎜ ⎟⎟ ⎝τw ⎠ γ& = 1/ 2 ⎛τ ⎞ = ⎜⎜ o ⎟⎟ ⎝τw ⎠ (Casson, k = 1/2) 1/ 2 ⎛ μ γ& ⎞ + ⎜⎜ ∞ ⎟⎟ ⎝ τw ⎠ 1/ 2 =z 1/ 2 =Z 1/ 2 ⎛ μ γ& ⎞ + ⎜⎜ ∞ ⎟⎟ ⎝ τw ⎠ 2 τ w 1/ 2 τ ( z − Z 1 / 2 ) = w (z − 2 z1 / 2 Z 1 / 2 + Z ) μ∞ μ∞ 1 Γ = 4 ∫ z 2γ&dz = Z 4τ w μ∞ ∫ z (z − 2 z 1 2 1/ 2 Z ) Z 1 / 2 + Z dz = 1 4τ w μ∞ ∫ (z 1 Z 3 ) − 2 z 5 / 2 Z 1 / 2 + z 2 Z dz 4τ ⎡ z 4 4 z 7 / 2 Z 1 / 2 z 3 Z ⎤ τ ⎛ 16 4 ⎛ 16 4 ⎞ ⎞ Γ= w⎢ − + = w ⎜⎜1 − Z 1 / 2 + Z − ⎜1 − + ⎟ Z 4 ⎟⎟ ⎥ μ∞ ⎣ 4 7 3 ⎦ Z μ∞ ⎝ 7 3 7 3⎠ ⎠ ⎝ Γ= τ w ⎛ 16 1 / 2 4 1 4⎞ ⎜1 − Z + Z − Z ⎟ μ∞ ⎝ 7 3 21 ⎠ 196 A6 Yield Plastic (k = 1/3) τ 1 / 3 = τ o1 / 3 + (μ∞γ& )1 / 3 1/ 3 ⎛τ ⎞ ⎜⎜ ⎟⎟ ⎝τw ⎠ γ& = 1/ 3 ⎛τ ⎞ = ⎜⎜ o ⎟⎟ ⎝τw ⎠ 1/ 3 ⎛ μ γ& ⎞ + ⎜⎜ ∞ ⎟⎟ ⎝ τw ⎠ 1/ 3 =z 1/ 3 =Z 1/ 3 ⎛ μ γ& ⎞ + ⎜⎜ ∞ ⎟⎟ ⎝ τw ⎠ 3 τ w 1/ 3 τ ( z − Z 1 / 3 ) = w (z − 3z 2 / 3Z 1 / 3 + 3 z1 / 3Z 2 / 3 − Z ) μ∞ μ∞ 1 Γ = 4∫ z 2γ&dz = Z Γ= (k = 1/3) 4τ w μ∞ ∫ (z 1 Z 3 4τ w μ∞ ∫ z (z − 3z 1 Z 2 2/3 ) Z 1 / 3 + 3z1 / 3 Z 2 / 3 − Z dz ) − 3z 8 / 3 Z 1 / 3 + 3z 7 / 3 Z 2 / 3 − z 2 Z dz 1 4τ w ⎡ z 4 9 z11 / 3 Z 1 / 3 9 z10 / 3 Z 2 / 3 z 3 Z ⎤ Γ= − + − μ∞ ⎢⎣ 4 11 10 3 ⎥⎦ Z Γ= τ w ⎛ 36 1 / 3 36 2 / 3 4 ⎛ 36 36 4 ⎞ 4 ⎞ + − ⎟ Z ⎟⎟ ⎜⎜1 − Z + Z − Z − ⎜1 − μ∞ ⎝ 11 10 3 ⎝ 11 10 3 ⎠ ⎠ Γ= τ w ⎛ 36 1 / 3 36 2 / 3 4 1 4⎞ Z ⎟ ⎜1 − Z + Z − Z + μ∞ ⎝ 11 10 3 165 ⎠ 197 A7 Yield Plastic (k = 1/4) τ 1 / 4 = τ o1 / 4 + (μ∞γ& )1 / 4 1/ 4 ⎛τ ⎞ ⎜⎜ ⎟⎟ ⎝τ w ⎠ γ& = 1/ 4 ⎛τ ⎞ = ⎜⎜ o ⎟⎟ ⎝τw ⎠ 1/ 4 ⎛ μ γ& ⎞ + ⎜⎜ ∞ ⎟⎟ ⎝ τw ⎠ 1/ 4 =z 1/ 4 =Z 1/ 4 ⎛ μ γ& ⎞ + ⎜⎜ ∞ ⎟⎟ ⎝ τw ⎠ 4 τ w 1/ 4 τ ( z − Z 1 / 4 ) = w (z − 4 z 3 / 4 Z 1 / 4 + 6 z 2 / 4 Z 2 / 4 − 4 z1 / 4 Z 3 / 4 + Z ) μ∞ μ∞ 1 Γ = 4 ∫ z 2γ&dz = Z Γ= (k = 1/4) 4τ w μ∞ ∫ (z 1 Z 3 4τ w μ∞ ∫ z (z − 4 z 1 Z 2 3/ 4 ) Z 1 / 4 + 6 z 2 / 4 Z 2 / 4 − 4 z1 / 4 Z 3 / 4 + Z dz ) − 4 z11 / 4 Z 1 / 4 + 6 z10 / 4 Z 2 / 4 − 4 z 9 / 4 Z 3 / 4 + z 2 Z dz 1 4τ w ⎡ z 4 16 z15 / 4 Z 1 / 4 24 z14 / 4 Z 2 / 4 16 z13 / 4 Z 3 / 4 z 3Z ⎤ Γ= − + − + ⎥ μ∞ ⎢⎣ 4 15 14 13 3 ⎦Z Γ= τ w ⎛ 64 1 / 4 96 2 / 4 64 3 / 4 4 ⎛ 64 96 64 4 ⎞ 4 ⎞ + − + ⎟ Z ⎟⎟ ⎜⎜1 − Z + Z − Z + Z − ⎜1 − μ∞ ⎝ 15 14 13 3 ⎝ 15 14 13 3 ⎠ ⎠ Γ= τ w ⎛ 64 1 / 4 96 2 / 4 64 3 / 4 4 1 ⎞ Z4⎟ ⎜1 − Z + Z − Z + Z − μ∞ ⎝ 15 14 13 3 1365 ⎠ 198 A8 Yield Plastic (k = 1/5) τ 1 / 5 = τ o1 / 5 + (μ∞γ& )1 / 5 1/ 5 ⎛τ ⎞ ⎜⎜ ⎟⎟ ⎝τw ⎠ γ& = 1/ 5 ⎛τ ⎞ = ⎜⎜ o ⎟⎟ ⎝τw ⎠ 1/ 5 ⎛ μ γ& ⎞ + ⎜⎜ ∞ ⎟⎟ ⎝ τw ⎠ 1/ 5 =z 1/ 5 =Z 1/ 5 ⎛ μ γ& ⎞ + ⎜⎜ ∞ ⎟⎟ ⎝ τw ⎠ 5 τ w 1/ 5 τ ( z − Z 1 / 5 ) = w (z − 5 z 4 / 5 Z 1 / 5 + 10 z 3 / 5 Z 2 / 5 − 10 z 2 / 5 Z 3 / 5 + 5 z1 / 5 Z 4 / 5 − Z ) μ∞ μ∞ 1 Γ = 4∫ z 2γ&dz = Z Γ= (k = 1/5) 4τ w μ∞ ∫ (z 1 Z 3 4τ w μ∞ ∫ z (z − 5z 1 Z 2 4/5 ) Z 1 / 5 + 10 z 3 / 5 Z 2 / 5 − 10 z 2 / 5 Z 3 / 5 + 5 z1 / 5 Z 4 / 5 − Z dz ) − 5 z14 / 5 Z 1 / 5 + 10 z13 / 5 Z 2 / 5 − 10 z12 / 5 Z 3 / 5 + 5 z11 / 5 Z 4 / 5 − z 2 Z dz 1 4τ w ⎡ z 4 25 z19 / 5 Z 1 / 5 50 z18 / 5 Z 2 / 5 50 z17 / 5 Z 3 / 5 25 z16 / 5 Z 4 / 5 z 3 Z ⎤ Γ= − + − + − μ∞ ⎢⎣ 4 19 18 17 16 3 ⎥⎦ Z Γ= τ w ⎛ 100 1 / 5 200 2 / 5 200 3 / 5 100 4 / 5 4 ⎛ 100 200 200 100 4 ⎞ 4 ⎞ + − + − ⎟Z ⎟ Z + Z − Z + Z − Z − ⎜1 − ⎜⎜1 − μ∞ ⎝ 19 18 17 16 3 19 18 17 16 3 ⎠ ⎟⎠ ⎝ Γ= τ w ⎛ 100 1 / 5 200 2 / 5 200 3 / 5 100 4 / 5 4 1 ⎞ Z + Z − Z + Z − Z+ Z4⎟ ⎜1 − μ∞ ⎝ 19 18 17 16 3 11628 ⎠ 199 A9 Yield Plastic (k = 1/6) τ 1 / 6 = τ o1 / 6 + (μ∞γ& )1 / 6 Γ= 4τ w μ∞ ∫ (z 1 Z 3 (k = 1/6) ) − 6 z17 / 6 Z 1 / 6 + 15 z16 / 6 Z 2 / 6 − 20 z15 / 6 Z 3 / 6 + 15 z14 / 6 Z 4 / 6 − 6 z13 / 6 Z 5 / 6 + z 2 Z dz 1 4τ ⎡ z 4 36 z 23 / 6 Z 1 / 6 90 z 22 / 6 Z 2 / 6 120 z 21 / 6 Z 3 / 6 90 z 20 / 6 Z 4 / 6 36 z19 / 6 Z 5 / 6 z 3 Z ⎤ Γ= w ⎢ − + − + − + ⎥ μ∞ ⎣ 4 23 22 21 20 19 3 ⎦Z Γ= τ w ⎛ 144 1 / 6 360 2 / 6 480 3 / 7 360 4 / 6 144 5 / 6 4 1 ⎞ Z + Z − Z + Z − Z + Z− Z4⎟ ⎜1 − μ∞ ⎝ 23 22 21 20 19 3 100947 ⎠ A10 Yield Plastic (k = 1/7) τ 1 / 7 = τ o1 / 7 + (μ∞γ& )1 / 7 Γ= 4τ w μ∞ ∫ (z 1 Z 3 (k = 1/7) ) − 7 z 20 / 7 Z 1 / 7 + 21z19 / 7 Z 2 / 7 − 35 z18 / 7 Z 3 / 7 + 35 z17 / 7 Z 4 / 7 − 21z16 / 7 Z 5 / 7 + 7 z17 / 7 Z 6 / 7 − z 2 Z dz 1 Γ= Γ= 4τ w ⎡ z 4 49 z 27 / 7 Z 1/ 7 147 z 26 / 7 Z 2 / 7 245 z 25 / 7 Z 3 / 7 245 z 24 / 7 Z 4 / 7 147 z 23 / 7 Z 5 / 7 49 z 22 / 7 Z 6 / 7 z 3 Z ⎤ − + − + − + − μ ∞ ⎢⎣ 4 27 26 25 24 23 22 3 ⎥⎦ Z τ w ⎛ 196 1 / 7 588 2 / 7 980 3 / 7 980 4 / 7 588 5 / 7 196 6 / 7 4 1 ⎞ Z + Z − Z + Z − Z + Z − Z+ Z4⎟ ⎜1 − μ∞ ⎝ 27 26 25 24 23 22 3 888029 ⎠ 200 A11 Power Law For comparative purposes, the laminar pipe flow solution for power law fluids is shown here. The solution for yield power law (Herschel-Bulkley) fluids is shown in the following Section. τ = Kγ& n Γ= (Power law) 8V 4 τw 4 = 3 ∫ τ 2γ&dτ = 3 0 D τw τw τw ∫ 0 1/ n ⎛τ ⎞ ⎟ ⎝K⎠ τ 2⎜ dτ = τw ⎡ τ 3+1 / n ⎤ Γ = 1/ n 3 ⎢ K τ w ⎣ 3 + 1 / n ⎥⎦ 0 4 1/ n ⎛ 4n ⎞⎛ τ w ⎞ Γ=⎜ ⎟⎜ ⎟ ⎝ 3n + 1 ⎠⎝ K ⎠ 201 4 τ 3 w τw ∫ 0 τ 2 +1 / n K 1/ n dτ = 4 K τ 1/ n 3 w τw ∫ 0 τ 2 +1 / n dτ A12 Yield Power Law (Herschel-Bulkley) τ = τ o + Kγ& n (Herschel-Bulkley) 8V 4 τw 4 Γ= = 3 ∫ τ 2γ&dτ = 3 τ D τw o τw τw ∫τ o 1/ n ⎛τ −τo ⎞ τ ⎜ ⎟ ⎝ K ⎠ Let x = τ − τ o and τ = x + τ o Æ Γ= Γ= 4 K τ 1/ n 3 w τ w −τ o ∫ 0 dτ = 2 (x + τ o )2 x1 / n dx = 4 2 +1 / n τw K τ ∫ 1/ n 3 τ o w τ 2 (τ − τ o )1 / n dτ dx = 1 − 0 = 1 so dx = dτ dτ 4 τ w −τ o K τ 1/ n 3 w ∫ 0 τ w −τ o ⎡ x ⎤ τ x 2τ x + o + ⎢ K τ ⎣ 3 + 1/ n 2 + 1/ n 1 + 1 / n ⎥⎦ 0 3 +1 / n 4 2 1+1 / n o 1/ n 3 w (x 2 +1 / n ) + 2τ o x1+1 / n + τ o2 x1 / n dx τ w −τ o 2 n +1 n +1 ⎡ 3 n +1 ⎤ τ o2 x n ⎥ 4n ⎢ x n 2τ o x n = 1/ n 3 + + K τ w ⎢ 3n + 1 2n + 1 n +1 ⎥ ⎣⎢ ⎦⎥ 0 3 n +1 2 n +1 n +1 τ o2 (τ w − τ o ) n ⎞⎟ 4n ⎛⎜ (τ w − τ o ) n 2τ o (τ w − τ o ) n Γ = 1/ n 3 ⎜ + + ⎟⎟ K τ w ⎜ 3n + 1 2n + 1 n +1 ⎝ ⎠ 1/ n ⎛τ ⎞ Γ = 4 n⎜ w ⎟ ⎝K⎠ 2 n +1 ⎛ (1 − Z ) n ⎜⎜ (1 − Z ) + 2Z (1 − Z ) + ⎝ 3n + 1 1/ n ⎛ 4n ⎞⎛ τ w ⎞ Γ=⎜ ⎟⎜ ⎟ ⎝ 3n + 1 ⎠⎝ K ⎠ (1 − Z ) n ⎛⎜⎜ (1 − Z )2 + ⎛⎜ 6n + 2 ⎞⎟Z (1 − Z ) + ⎛⎜ 3n + 1 ⎞⎟Z 2 ⎞⎟⎟ n +1 ⎝ 1/ n ⎛ 4n ⎞⎛ τ w ⎞ Γ=⎜ ⎟⎜ ⎟ ⎝ 3n + 1 ⎠⎝ K ⎠ 1/ n ⎛ 4n ⎞⎛ τ w ⎞ Γ=⎜ ⎟⎜ ⎟ ⎝ 3n + 1 ⎠⎝ K ⎠ n +1 (1 − Z ) n n +1 (1 − Z ) n 1/ n ⎛ 4n ⎞⎛ τ w ⎞ Γ=⎜ ⎟⎜ ⎟ ⎝ 3n + 1 ⎠⎝ K ⎠ 2n + 1 Z2 ⎞ ⎟ n + 1 ⎟⎠ ⎝ n +1 ⎠ ⎠ ⎛ ⎛ ⎛ 4n + 2 ⎞ ⎛ 6 n + 2 ⎞ ⎞ ⎛ ⎛ 3n + 1 ⎞ ⎛ 6n + 2 ⎞ ⎞ 2 ⎞ ⎜⎜1 − ⎜⎜ ⎜ ⎟−⎜ ⎟ ⎟⎟ Z + ⎜⎜1 − ⎜ ⎟+⎜ ⎟ ⎟⎟ Z ⎟⎟ + + 2 n 1 2 n 1 n 1 2 n 1 + + ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎠ ⎠ ⎝ ⎠ ⎝ ⎝ ⎛ ⎛ 2n ⎞ ⎞ 2⎞ ⎛ 2n 2 ⎜1 + ⎜ ⎟⎟ Z ⎟ ⎜⎜ + Z ⎟ ⎜ ⎝ 2n + 1 ⎠ ⎟ ( ) + + ( n 1 ) 2 n 1 ⎠ ⎠ ⎝ ⎝ n +1 (1 − Z ) n ⎝ 2n + 1 ⎠ ⎛ ⎛ 2n ⎞ ⎛ n ⎞⎞ Z ⎟ ⎟⎟ ⎜⎜1 + ⎜ ⎟ Z ⎜1 + ⎝ ⎝ 2n + 1 ⎠ ⎝ n + 1 ⎠ ⎠ 202 APPENDIX B: DERIVATION OF TURBULENT FLOW SOLUTIONS B1 Introduction The derivation of the turbulent flow design equation based on the Wilson-Thomas model is presented cursively in the primary sources (Wilson and Thomas 1985, Thomas and Wilson 1987, and Wilson et al. 2006). The first purpose of this Appendix is to derive the WilsonThomas model in detail and make it consistent with the Nikuradse equation. To use the Wilson-Thomas model it is necessary to calculate the “ratio of the rheogram areas: non-Newtonian/Newtonian” (α). The area of a rheogram is the area under the shear stress vs. shear rate curve. The second purpose of this Appendix is to show the methodology for calculating α for a yield plastic material. B2 Turbulent Flow of Newtonian Fluids This Section may be considered to be a literature review of several sources, principally: Govier and Aziz (1977), Hanks (1981), Potter and Wiggert (1997), Streeter and Wylie (1985), and Wasp et al. (1977). The following analysis assumes that an incompressible fluid is in fully developed turbulent flow, far from an entrance and well above the laminar-turbulent transition. The fluid is assumed to be flowing in a straight pipe at a steady rate with no bulk rotation. When a fluid is flowing in a pipe, the average magnitude of the axial velocity increases from zero at the pipe wall (the “no slip” assumption) to a maximum value near the pipe’s centreline. Although the average (or bulk) flow is steady, the presence of turbulent eddies causes the local velocity to vary with time. The instantaneous velocity at any point in a turbulent flow field may be separated into the average velocity in the axial, radial, and 203 rotational directions ( u , v , w ) and the erratic turbulent fluctuations in the same directions (u′, v′, w′). The turbulent fluctuations cancel out over a relatively short time interval, so the timeaveraged axial velocity is: u . Since the fluid cannot flow through the pipe wall, the timeaveraged radial velocity is zero. It is assumed that there is no bulk rotation so the timeaveraged rotational velocity is also zero. The Newtonian fluid is assumed to have a constant density (ρ) and viscosity (μ). To simplify calculations, the time-averaged velocity ( u ) and the distance from the pipe wall (y = R-r) may be put into dimensionless form using the “friction velocity”, which is a function of the wall shear stress: τw = ΔPD 4L Wall shear stress [B.1] u* = τ w ρ u+ = y+ = Friction velocity [B.2] ρ u =u τw u* Dimensionless velocity [B.3] yρu * Dimensionless distance from wall [B.4] μ B.2.1 Newtonian velocity profile When a Newtonian fluid is in fully developed turbulent flow, three flow zones may be noted. Taking up most of the central area of the pipe is a region characterized by turbulent eddies. This region is referred to the “turbulent core”. Adjacent to the pipe wall is a narrow annulus where the flow is essentially laminar. This region is referred to as the “laminar sub-layer”. Between the turbulent core and the laminar sub-layer is a relatively narrow “transition zone”. The three zones will be addressed separately. Turbulent core Momentum transfer in the turbulent core is primarily due to inertial mixing; viscous effects play an insignificant role. In the turbulent core there will be a wide range of eddy sizes, from 204 large scale rotations down to the smallest eddies at the energy dissipative scale. The velocity profile in turbulent core may be modelled using Prandtl’s (1921) mixing length theory. u* = κy du = du dy u * dy κ y [B.5] Where κ is the dimensionless Karman constant. With an incompressible liquid, the density is constant so, at any given wall shear stress, the friction velocity (u*) is constant. Therefore, Eq. [B.5] may be integrated to give: u= u* κ ln ( y ) + C0 Dividing through by the friction velocity: u 1 C = u + = ln ( y ) + 0 u* κ u* [B.6] Equation [B.6] may be expanded using definition [B.4] and rearranged to give: u+ = ⎛ y + μ ⎞ C0 1 ⎟⎟ + ln⎜⎜ = ln y + + C1 κ ⎝ ρu * ⎠ u * κ 1 ( ) [B.7] The values for κ and C1 must be found by experimentation and different researchers have proposed different values. The Karman constant is usually taken to be 0.40 or 0.41. The value of C1 is in the range 5.5 to 5.7. Wilson and Thomas (1985) used κ = 0.4 and C1 = 5.5 for the constants in [B.7], which results in the Karman equation for smooth wall turbulent flow. However, the experiments of Nikuradse (1932) imply that κ = 0.407 and C1 = 5.66 for Newtonian fluids. Using these values, the resulting the velocity profile is: u + = 2.457 ln ( y + ) + 5.66 = 2.457 ln (10 y + ) Velocity in turbulent core [B.8] Experimental data indicates that [B.8] holds for y+ > ~30 (Govier and Aziz 1977). Laminar sub-layer 205 It is observed that the maximum eddy size decreases as the pipe wall is approached. This is assumed to be because the presence of the pipe wall restricts the ability of the fluid to rotate. At some small distance from the pipe wall, the scale of the largest and smallest eddies coincide. Inside this interface the fluid is unable to rotate and the flow is essentially laminar. Inertial mixing in this sub-layer is insignificant. The fluid in direct contact with the wall is assumed to be static and there is a large increase in velocity over a rather small distance. This causes a high shear rate and, therefore, a high viscous shear stress. Since the laminar sub-layer is extremely thin, the curvature of the pipe wall may be ignored and the timeaveraged velocity is assumed to be directly proportional to the distance from the wall. The shear stress at the wall is: ⎛ du ⎞ μu τ w = μ ⎜⎜ ⎟⎟ = y ⎝ dy ⎠ wall Shear stress at the pipe wall [B.9] Combining [B.9] with definitions [B.3] and [B.4] gives: u+ = u ρy ρyu ⎛ u ⎞⎛ yρu * ⎞ = = ⎜ ⎟⎜⎜ ⎟⎟ = u + y + * μu μ μ u ⎝ ⎠⎝ ⎠ Therefore, the dimensionless sub-layer velocity profile is: u+ = y+ Velocity in laminar sub-layer [B.10] Experimental data on Newtonian fluids indicates that [B.10] holds for y+ < ~5 (Govier and Aziz 1977). Transition zone The transition zone is located between the laminar sub-layer and the turbulent core, which is to say it is in the range ~5 < y+ < ~30. Within the transition zone, both inertial mixing and viscous effects are significant. As the distance from the wall increases, the inertial mixing effects increase and the viscous effects decrease. While research has been carried out on the velocity profile of the transition zone, for the present purposes it is adequate to apportion the transition zone to the laminar sub-layer and the turbulent core. In this two-region model, the laminar sub-layer and the part of the transition zone that extends out to a “boundary layer” is referred to as the “viscous sub-layer”. The location of the boundary layer is assumed to be 206 the intersection of laminar sub-layer and turbulent core velocity curves (see Fig. B.1). The intersection may be found by equating Eq. [B.8] and [B.10]: y + = 2.457 ln (10 y + ) = 11.7 [B.11] Viscous sub-layer 30 Turbulent core 25 20 u + 15 ( u + = 2.457 ln 10 y + 10 ) u+ = y+ 5 Viscous sub-layer N Turbulent core 11.7 0 1 10 y+ 100 1000 Figure B.1: Turbulent pipe flow velocity profile (Newtonian) This implies that the boundary layer is roughly one-quarter of the way through the transition zone (measured from the pipe wall). The velocity at the boundary layer is proportional to the dimensionless thickness of the laminar sub-layer (see Eq. [B.10]) so: (u ) + sub − layer = 11.7 Boundary layer velocity [B.12] B.2.2 Newtonian Pressure-Gradient The flow through the pipe is calculated in much the same way as for laminar flow. However, the time-averaged velocity ( u ) is used instead of the instantaneous local velocity of the fluid. The differential flow rate of the annulus between radii r and r+dr is: dQ = u (2π ⋅ r )dr The total flow rate through the pipe (Q) is the sum of the flow elements from the centreline (r = 0) to the pipe wall (r = R). Q R Q = ∫ dQ = ∫ u (2π ⋅ r )dr 0 [B.13] 0 207 The bulk velocity through a cylinder is: V= Q πR 2 [B.14] Substituting [B.14] into [B.13] gives the relationship for the bulk velocity, which is not dependent on the fluid’s rheology. R 2 V = 2 ∫ u rdr R 0 [B.15] In fully turbulent flow, most of the fluid flows in the turbulent core because it has a higher average velocity and a very much larger area than the sub-layer. It may be assumed, without significant error, that the measured velocity profile turbulent core extends to the pipe wall. The velocity variation across the pipe is a function of the radius. Using the velocity vs. radius relationship given by [B.8]: u = u +u* = (2.457 ln (10 y + ))u * [B.16] Substituting [B.16] into [B.15]: 2(u * 2.457 ) ⎛ ρu * ⎞ ln⎜⎜10 y ⎟rdr 2 ∫ μ ⎟⎠ R ⎝ 0 R V= Integrating and noting that D = 2R gives the velocity of Newtonian fluids in smooth pipes: ⎞ ⎛ ⎛ Dρu * ⎞ ⎛ 10 Rρu * D ⎞ V = (u * 2.457 )ln⎜⎜ ⎟⎟ = (u * 2.457 )ln⎜⎜ ⎜⎜ ⎟⎟(1.12 )⎟⎟ ⎝ 4.48 μ 2 R ⎠ ⎠ ⎝⎝ μ ⎠ Rearranging: ⎛ ⎛ DρV V = 2.457 ln⎜⎜ ⎜⎜ u* ⎝⎝ μ ⎞ ⎞⎛ u * ⎞ ⎟⎟⎜ ⎟(1.12 )⎟⎟ ⎠⎝ V ⎠ ⎠ [B.17] Recalling the conventional dimensionless groups for Newtonian fluids: fN = 2τ w ⎛ u *⎞ = 2⎜ ⎟ 2 ρV ⎝V ⎠ 2 Fanning Friction factor [B.18] 208 Re = ρVD μ Reynolds number [B.19] Substituting [B.18] and [B.19] into [B.17]: ⎛ ⎛ 2 = 2.457 ln⎜ (Re )⎜⎜ ⎜ fN ⎝ ⎝ fN 2 ⎞ ⎞ ⎟1.12 ⎟ ⎟ ⎟ ⎠ ⎠ Which may be simplified and put into the conventional “base-10” engineering form: ⎛ 1.26 1 = −4.0 log⎜ ⎜ Re f fN N ⎝ ⎞ ⎟ = 4.0 log Re f N − 0.4 ⎟ ⎠ ( ) [B.20] Equation [B.20] is the widely used Nikuradse (1932) equation for turbulent flow of a Newtonian fluid in a hydraulically smooth pipe. For Newtonian fluids, it is found that the Nikuradse equation “provides an excellent fit to essentially all reliable data in the range: 3000 < Re < 3,000,000” (Govier and Aziz 1977). B3 Turbulent Flow of Non-Newtonian Fluids The general assumptions used to develop design equations using the Wilson-Thomas model are the same as for a Newtonian fluid. The suspension is assumed to be a continuum (i.e., a continuous fluid) with the bulk rheological properties of the suspension. The main difference is that the “viscosity” is the apparent viscosity at the local shear conditions. B.3.1 Non-Newtonian velocity profile When a non-Newtonian fluid is in fully turbulent flow, the same three flow zones are noted as with Newtonian fluids: the “turbulent core”, the “laminar sub-layer”, and the “transition zone”. There may also be a fourth flow zone when the fluid has a yield stress: the “plug” or un-yielded zone near the centreline of the pipe. The four zones will be addressed separately. Plug Many suspensions have a “yield stress” below which the fluid will not flow or will flow at a negligible rate. In steady laminar flow, a simple force balance between the pressure differential across the pipe and the viscous drag on the wall of the pipe show that the average axial shear stress will be zero at the pipeline centreline and increase linearly to the 209 wall shear stress at the pipe wall. Wilson and Thomas (1985) assumed that the force balance is not changed by the presence of turbulence, so there would always be an un-yielded plug at the centreline of the pipe that moves with the bulk fluid at the same velocity as the inner boundary of the turbulent core. They included a function to allow for flattening of the flow profile, although “whether such blunting actually occurs remains a moot point” (Wilson and Thomas 2006). Furthermore, the turbulent flow profile is relatively flat at the pipe centreline, whether a plug exists or not. “In the part of the flow nearest the pipe axis, some change in the velocity profile would result from a non-zero τo, but the effect on the mean velocity is very small” (Wilson et al. 2006). Therefore, the profile flattening function proposed by Wilson and Thomas (1985) will not be included in the following analysis. Turbulent core Since viscous effects play an insignificant role in the turbulent core with Newtonian fluids, Wilson and Thomas (1985) assume that a non-Newtonian “viscosity” (consistency) will not change the basic shape of the flow profile given in [B.10]. It will, however, be shifted by the thickened sub-layer. The velocity profile in the turbulent core may be considered to have two parts: the constant velocity of the boundary layer and the variable velocity of the remainder of the core. For a Newtonian fluid, these would be: (u ) + sub − layer (u ) + core = 11.7 Newtonian [B.21] ( ) = 2.457 ln 10 y + − 11.7 Newtonian [B.22] To account for the non-Newtonian consistency, the equation for the dimensionless distance from the wall is put in terms of the apparent viscosity (η) of the fluid at the pipe wall shear stress (τw). y+ = yρu * [B.23] η The value of η is assumed to be constant at any given flow rate, although the value changes with the pressure gradient (i.e., the wall shear stress). Transition zone 210 The presence of cascading eddies in the turbulent core and the transition zone will cause large variations in the local shear rate. With a non-Newtonian fluid, this will also cause variations in the local viscosity. Wilson and Thomas (1985) assumed that the turbulent regions will experience the full range of shear rates so that, “as far as dissipative energy is concerned, the non-Newtonian fluid acts as if it were a Newtonian fluid with viscosity αη.” The residual velocity profile in the turbulent core is therefore: (u ) + core ⎛ y+ ⎞ = 2.457 ln⎜⎜10 ⎟⎟ − 11.7 ⎝ α ⎠ Non-Newtonian [B.24] The value of α is evaluated across the pipe at τ = τw, so it will be constant in steady flow. Viscous sub-layer It follows from the Wilson-Thomas model that the thickness of the viscous sub-layer should be multiplied by the area ratio (α). y + = 11.7α [B.25] Since the sub-layer is very thin (relative to the pipe radius), the shear stress and the viscosity will be roughly constant throughout the sub-layer. Therefore, the non-Newtonian velocity profile in the sub-lay will directly proportional to the distance from the wall. (u ) + sub − layer = y + = 11.7α Non-Newtonian [B.26] Combined velocity profile The combined velocity profile in the turbulent core after thickening of the sub-layer may be found by adding [B.24] and [B.26]: u + = (u + )sub − layer + (u + )core ⎛ y+ ⎞ u + = 2.457 ln⎜⎜10 ⎟⎟ + 11.7(α − 1) ⎝ α ⎠ Non-Newtonian [B.27] The effects of laminar sub-layer thickening of the turbulent velocity profile are shown in Fig. B.2. Since the majority of the velocity curve is shifted up (i.e., has a greater velocity), the total flow rate at any given pressure gradient is increased. 211 Viscous sub-layer 30 Turbulent core ( ) u + = 2.457 ln 10 y + α + 11.7(α − 1) 25 20 u + 15 ( u + = 2.457 ln 10 y + 10 u+ = y+ 5 Viscous sub-layer N Turbulent core NN Turbulent core α 11.7 11.7 0 1 10 y+ ) 100 1000 Figure B.2: Effect of sub-layer thickening on turbulent pipe flow velocity profile (N = Newtonian; NN = Non-Newtonian) B.3.2 Non-Newtonian Pressure-Gradient As is the case with Newtonian fluids, when a non-Newtonian fluid is in turbulent flow, most of the volume flows in the turbulent core. It may again be assumed, without significant error, that the velocity profile of the turbulent core extends to the pipe wall. u+ = ⎛ y+ ⎞ ⎛ u e 4.76(α −1) ⎞ ⎟ = 2.457 ln⎜⎜10 ⎟⎟ + 11.7(α − 1) = 2.457 ln⎜⎜10 y + α ⎟⎠ u* ⎝ α ⎠ ⎝ Substituting [B.28] into [B.15] V= R R 4.76 (α −1) 4.76 (α −1) ⎛ ⎞ ⎞ 2 2(u * 2.457 ) ⎛ + e + e ⎜ ⎟ ⎜ ⎟⎟rdr = u * 2 . 457 ln 10 y rdr ln 10 y 2 ∫ 2 ∫ ⎜ ⎟ ⎜ α ⎠ α R 0 R ⎝ ⎝ ⎠ 0 Using the non-Newtonian definition for y+ and noting that r = R – y: R 2(u * 2.457 ) ⎛ ρu * e 4.76(α −1) ⎞ ⎜ ln 10 V= y ∫0 ⎜⎝ η α ⎟⎟⎠(R − y )dy R2 R ⎛ ρu * e 4.76(α −1) ⎞⎛ y ⎞ dy ⎟⎟⎜1 − ⎟ V = 2(u * 2.457 )∫ ln⎜⎜10 y η α ⎠⎝ R ⎠ R ⎝ 0 Changing the variable of integration to x = y/R: 212 [B.28] 1 ⎛ Rρu * e 4.76(α −1) ⎞ ⎟(1 − x )dx V = 2(u * 2.457 )∫ ln⎜⎜10 x η α ⎟⎠ ⎝ 0 1 V = 2(u * 2.457 )∫ (ln ( xC1 ) − x ln (xC1 ))dx 0 Integrating: 1 ⎡ ⎛ x2 x 2 ⎞⎤ V = 2(u * 2.457 )⎢( x ln (C1 x ) − x ) − ⎜⎜ ln(C1 x ) − ⎟⎟⎥ 4 ⎠⎦ 0 ⎝ 2 ⎣ ⎡ ⎛ C ⎞⎤ ⎡ ⎛ C ⎞⎤ 3⎤ ⎡ V = (u * 2.457 )⎢ln (C1 ) − ⎥ = (u * 2.457 )⎢ln⎜ 3 /12 ⎟⎥ = (u * 2.457 )⎢ln⎜ 1 ⎟⎥ 2⎦ ⎣ ⎣ ⎝ e ⎠⎦ ⎣ ⎝ 4.48 ⎠⎦ Therefore: ⎛ 10 Rρu * D ⎛ e 4.76 (α −1) ⎞ ⎞ ⎟⎟ ⎟⎟ ⎜⎜ V = (u * 2.457 )ln⎜⎜ ⎝ 4.48 η 2 R ⎝ α ⎠ ⎠ Noting that D = 2R, the general equation for the turbulent flow velocity of non-Newtonian fluids in smooth pipes: ⎛ ⎛ Dρu * ⎞⎛ 1.12e 4.76 (α −1) ⎞ ⎞ V ⎟⎟ ⎟⎟ = 2.457 ln⎜⎜ ⎜⎜ ⎟⎟⎜⎜ u* α ⎠⎠ ⎝ ⎝ η ⎠⎝ Expanding: ⎛ 1.12 Dρu * ⎞ V = 2.457 ln⎜⎜ ⎟⎟ + (11.7(α − 1) − 2.457 ln (α )) η u* ⎠ ⎝ [B.29] To use Eq. [B.29] it is necessary to calculate the area ratio (α). The area ratios for special cases of the yield plastic yield plastic are derived in the following section The equivalent derivations for the power law and yield power law models are included for reference. 213 B4 Newtonian τ = μγ& (Newton) τ o = 0 Æ Z = 0 for all k ≠ 0 α = 2∫ (Z k + (1 − Z k )x k ) dx =2∫ (0k + (1 − 0k )x k ) dx 1 1/ k 0 1 1/ k 0 α = 2∫ xdx = [x 2 ]0 = 1 − 0 1 1 0 α =1 (Newtonian) α =1 B5 Yield Plastic (k = 1/1) or Bingham Plastic τ = τ o + μ∞γ& (Bingham, k = 1) α = 2∫ (Z k + (1 − Z k )x k ) dx =2 ∫ (Z + (1 − Z )x )dx 1 1/ k 0 ⎡ 1 0 1 1 ⎤ α = 2 ⎢ Zx + (1 − Z )x 2 ⎥ = (2 Z + 1 − Z ) − (0) 2 ⎦ ⎣ 0 α =1+ Z (Bingham) 214 B6 Yield Plastic (k = 1/2) or Casson τ 1 / 2 = τ o1 / 2 + (μ∞γ& )1 / 2 (Casson, k = 1/2) α = 2 ∫ (Z k + (1 − Z k )x k ) dx =2 ∫ (Z 1 / 2 + (1 − Z 1 / 2 )x1 / 2 ) dx 1 1 1/ k 2 0 0 ( ) α = 2 ∫ Z + 2Z 1 / 2 (1 − Z 1 / 2 )x1 / 2 + (1 − Z 1 / 2 ) x dx 1 0 2 α = 2⎢ Zx + Z 1 / 2 (1 − Z 1 / 2 )x 3 / 2 + (1 − Z 1 / 2 ) x 2 ⎥ 3 2 ⎣ ⎦ ⎡ 4 1 α = 2⎜ Z + Z 1 / 2 (1 − Z 1 / 2 ) + ⎛ ⎝ 4 3 α = 2Z + ( 2 ⎤ 1 0 ) 2⎞ 1 1 − Z 1 / 2 ⎟ − (0) 2 ⎠ 8 1/ 2 ( Z − Z ) + (1 − 2Z 1 / 2 + Z ) 3 ⎛8 ⎝3 ⎞ ⎠ ⎛ ⎝ 8 3 ⎞ ⎠ α = 1 + ⎜ − 2 ⎟ Z 1 / 2 + ⎜ 2 − + 1⎟ Z 2 3 α = 1 + Z 1/ 2 + Z 3 (Casson) 215 B7 Yield Plastic (k = 1/3) τ 1 / 3 = τ o1 / 3 + (μ∞γ& )1 / 3 (k = 1/3) α = 2 ∫ (Z k + (1 − Z k )x k ) dx = 2∫ (Z 1 / 3 + (1 − Z 1 / 3 )x1 / 3 ) dx 1 1 1/ k 3 0 0 ( ) α = 2 ∫ Z + 3Z 2 / 3 (1 − Z 1 / 3 )x1 / 3 + 3Z 1 / 3 (1 − Z 1 / 3 ) x 2 / 3 + (1 − Z 1 / 3 ) x dx 1 0 2 3 α = 2⎢ Zx + Z 2 / 3 (1 − Z 1 / 3 )x 4 / 3 + Z 1 / 3 (1 − Z 1 / 3 ) x 5 / 3 + (1 − Z 1 / 3 ) x 2 ⎥ 4 5 ⎣ ⎦ ⎡ 9 ⎛ ⎝ α = 2⎜ Zx + α = 2Z + 9 ( ) ( 2 ) ( 3 ⎤ 1 0 ) 2 3⎞ 9 9 1 − Z 1 / 3 + Z 1 / 3 1 − Z 1 / 3 + 1 − Z 1 / 3 ⎟ − (0 ) 4 5 ⎠ ( ) ( ) ( 18 2 / 3 18 1 / 3 Z −Z + Z − 2Z 2 / 3 + Z + 1 − 3Z 1 / 3 + 3Z 2 / 3 − Z 4 5 ) 18 18 ⎞ ⎛ 18 ⎞ ⎛ 18 36 ⎞ ⎛ − 3 ⎟Z 1 / 3 + ⎜ − + 3 ⎟ Z 2 / 3 + ⎜ 2 − + − 1⎟ Z 5 4 5 ⎝5 ⎠ ⎝4 ⎠ ⎝ ⎠ α = 1+ ⎜ α =1+ 6 (Z )1 / 3 + 3 (Z )2 / 3 + 1 Z 10 10 10 (k = 1/3) 216 B8 Yield Plastic (k = 1/4) τ 1 / 4 = τ o1 / 4 + (η∞γ& )1 / 4 (k = 1/4) α = 2 ∫ (Z k + (1 − Z k )x k ) dx =2 ∫ (Z 1 / 4 + (1 − Z 1 / 4 )x1 / 4 ) dx 1 1/ k 1 4 0 0 ( ) α = 2∫ Z + 4Z 3 / 4 (1 − Z 1 / 4 )x1 / 4 + 6Z 2 / 4 (1 − Z 1 / 4 ) x 2 / 4 + 4Z 1 / 4 (1 − Z 1 / 4 ) x3 / 4 + (1 − Z 1 / 4 ) x dx 1 0 2 3 4 1 α = 2⎢ Zx + Z 3 / 4 (1 − Z 1 / 4 )x 5 / 4 + Z 2 / 4 (1 − Z 1 / 4 ) x 6 / 4 + Z 1 / 4 (1 − Z 1 / 4 ) x 7 / 4 + (1 − Z 1 / 4 ) x 2 ⎥ 5 6 7 2 ⎦0 ⎣ ⎡ 16 ⎛ ⎝ 2 3 4⎞ 16 1 16 3 / 4 24 Z (1 − Z 1 / 4 ) + Z 2 / 4 (1 − Z 1 / 4 ) + Z 1 / 4 (1 − Z 1 / 4 ) + (1 − Z 1 / 4 ) ⎟ − (0) 6 7 2 5 ⎠ α = 2⎜ Z + ( 24 ) ( 2 ) ( 16 3 1 4 ⎤ ) ( 32 3 / 4 48 2 / 4 32 1 / 4 Z −Z + Z − 2Z 3 / 4 + Z + Z − 3Z 2 / 4 + 3Z 3 / 4 − Z + 1 − 4Z 1 / 4 + 6Z 2 / 4 − 4Z 3 / 4 + Z 5 6 7 α = 2Z + ) 32 48 32 ⎞ ⎞ ⎛ ⎛ 32 ⎞ ⎛ 48 96 ⎞ ⎛ 32 96 96 − 4 ⎟Z 1 / 4 + ⎜ − + 6 ⎟Z 2 / 4 + ⎜ − + − 4 ⎟Z 3 / 4 + ⎜ 2 − + − + 1⎟ Z 7 6 7 5 6 7 ⎝ 7 ⎠ ⎝ 6 ⎠ ⎝ 5 ⎠ ⎝ ⎠ α =1+ ⎜ α =1+ 20 1 / 4 10 2 / 4 4 3 / 4 Z Z + Z + Z + 35 35 35 35 (k = 1/4) 217 B9 Yield Plastic (k = 1/5) τ 1 / 5 = τ o1 / 5 + (η∞γ& )1 / 5 (k = 1/5) α = 2 ∫ (Z k + (1 − Z k )x k ) dx = 2∫ (Z 1 / 5 + (1 − Z 1 / 5 )x1 / 5 ) dx 1 1 1/ k 5 0 0 ( ) α = 2∫ Z + 5Z 4 / 5 (1 − Z 1/ 5 )x1/ 5 + 10Z 3 / 5 (1 − Z 1/ 5 ) x 2 / 5 + 10 Z 2 / 5 (1 − Z 1/ 5 ) x 3 / 5 + 5Z 1/ 5 (1 − Z 1/ 5 ) x 4 / 5 + (1 − Z 1/ 5 ) x dx 1 0 2 3 4 5 1 α = 2⎢Zx + Z 4 / 5 (1 − Z 1/ 5 )x 6 / 5 + Z 3 / 5 (1 − Z 1/ 5 ) x 7 / 5 + Z 2 / 5 (1 − Z 1/ 5 ) x 8 / 5 + Z 1/ 5 (1 − Z 1/ 5 ) x 9 / 5 + (1 − Z 1/ 5 ) x 2 ⎥ 6 7 8 9 2 ⎣ ⎦0 ⎡ 25 50 ( ) 50 2 ( ) 3 ( ) 25 1 4 ( ) ( 5 ⎤ ) ⎛ ⎝ 2 3 4 5⎞ 25 4 / 5 50 50 25 1 Z 1 − Z 1 / 5 + Z 3 / 5 1 − Z 1 / 5 + Z 2 / 5 1 − Z 1 / 5 + Z 1 / 5 1 − Z 1 / 5 + 1 − Z 1 / 5 ⎟ − (0 ) 6 7 8 9 2 ⎠ α = 2Z + 4 4 3 4 2 3 4 1 2 3 4 ⎞ ⎞ ⎛ ⎞ 50 ⎛ 15 ⎞ 100 ⎛ 25 ⎞ 100 ⎛ 53 50 ⎛ 5 5 5 5 5 5 5 5 5 5 5 ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜Z − Z ⎟ + ⎟ 7 ⎜ Z − 2Z + Z ⎟ + 8 ⎜ Z − 3Z + 3Z − Z ⎟ + 9 ⎜ Z − 4 Z + 6Z − 4 Z + Z ⎟ + ⎜1 − 5Z + 10 Z − 10 Z + 5Z − Z ⎟ 6 ⎜⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ α = 2⎜ Z + 50 100 100 50 ⎞ ⎛ ⎛ 50 200 300 200 ⎞ ⎛ 100 300 300 ⎞ ⎛ 100 200 ⎞ ⎛ 50 ⎞ + − + − 1⎟ Z + − + 5 ⎟Z 4 / 5 + ⎜ 2 − − + − 10 ⎟ Z 3 / 5 + ⎜ − − + 10 ⎟ Z 2 / 5 + ⎜ − 5 ⎟Z 1/ 5 + ⎜ 7 8 9 6 7 8 9 9 8 9 ⎝ ⎠ ⎝ 6 ⎠ ⎝ 7 ⎠ ⎝ 8 ⎠ ⎝ 9 ⎠ α =1+ ⎜ α = 1+ 70 1 / 5 35 2 / 5 15 3 / 5 5 3/ 5 Z Z + Z + Z + Z + 126 126 126 126 126 218 (k = 1/5) B10 Power Law τ = Kγ& n (Power law) τ m = Kγ&mn (Shear stress at maximum shear rate) γ& m γ& m 0 0 A1 = ∫ τdγ& = ∫ A0 = α= γ& m ⎡ Kγ& n +1 ⎤ Kγ&mn +1 τ mγ&m & & Kγ dγ = ⎢ ⎥ = n +1 = n +1 ⎣ n + 1 ⎦0 n τ mγ&m (Power law rheogram area) (Newtonian rheogram area) 2 2 A1 ⎛ τ mγ&m ⎞⎛ 2 ⎞ ⎟⎟ = =⎜ ⎟⎜⎜ A0 ⎝ n + 1 ⎠⎝ τ mγ&m ⎠ n + 1 (Power law) 219 B11 Herschel-Bulkley (Yield Power Law) τ = τ o + Kγ& n (Herschel-Bulkley) τ m = τ o + Kγ&mn (Shear stress at maximum shear rate) γ& m γ& m 0 0 A1 = ∫ τdγ& = ∫ γ& m ⎡ Kγ& n +1 ⎤ Kγ&mn +1 & τ o + Kγ& dγ& = ⎢τ oγ& + τ γ = + o m n + 1 ⎥⎦ 0 n +1 ⎣ ( n ) ⎛ ⎛ ⎛ ⎞ Kγ&mn ⎞ ⎛ n ⎞ &n ⎞ & ⎛ n ⎞ ⎜ ⎟⎟γ&m = ⎜⎜τ o + Kγ&mn − ⎜ A1 = ⎜τ o + ⎟ Kγ m ⎟⎟γ m = ⎜⎜τ m − ⎜ ⎟(τ m − τ o )⎟⎟γ&m n +1⎠ ⎝ n +1⎠ ⎝ n + 1⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎛⎛ 1 ⎞ ⎛ n ⎞ ⎞& A1 = ⎜⎜ ⎜ ⎟τ m + ⎜ ⎟τ o ⎟⎟γ m ⎝ n + 1⎠ ⎠ ⎝⎝ n + 1⎠ A0 = (Herschel-Bulkley rheogram area) τ mγ&m (Newtonian rheogram area) 2 α= A1 ⎛ ⎛ ⎛ 1 ⎞ ⎛ n ⎞ ⎞ & ⎞⎟⎛ 2 ⎞ ⎟ = ⎜⎜ ⎜⎜ ⎜ ⎟τ m + ⎜ ⎟τ o ⎟⎟γ m ⎜⎜ A0 ⎝ ⎝ ⎝ n + 1 ⎠ ⎝ n + 1 ⎠ ⎠ ⎟⎠⎝ τ mγ&m ⎟⎠ α= 2 (1 + nZ ) n +1 (Herschel-Bulkley) 220 APPENDIX C: DESIGN CURVE DIAGRAMS C1 Introduction This Appendix presents detailed versions of the Design Curve Diagram described in Chapter 8. Three different Hedström ranges of interest are presented. (Note: The exact equations listed in Table 5.1 were used to generate the laminar flow curves.) C2 General DCD The General DCD is a detailed version of Fig. 8.7 showing the flow curves for a wide range of Hedström numbers. C3 Engineering DCD Slurry is typically transported in large diameter pipelines (0.1 to 0.6 m). The bulk velocity is limited to avoid excessive wear (if coarse particles are present) and to keep pumping pressures reasonable. The yield stress for conventional slurries tends to be ~1 Pa. With pastes, the yield stress may be as high as ~400 Pa. As a result, the Hedström number in industrial pipelines tends to be in the range 105 < He < 108. C4 Laboratory DCD Laboratory experiments on slurry are typically performed using small diameter tubes or capillaries (0.01 to 0.1 m) where the Hedström number is usually less than 106. 221 Design Curve Diagram (General) 1.E+10 He = 10 1.E+09 9 Stress Number (Ha) 1.E+08 He = 10 1.E+07 7 Turbulent 1.E+06 He = 10 1.E+05 5 Newton Bingham 1.E+04 He = 10 1.E+03 Casson 3 1.E+02 1.E+02 YP k = 1/3 YP k = 1/4 He = 0 Newtonian Turbulent Transition (YP) 1.E+03 1.E+04 1.E+05 Plastic Reynolds Number (Rep ) 222 1.E+06 1.E+07 Design Curve Diagram (Engineering) Stress Number (Ha) 1.E+09 He = 10 1.E+08 8 He = 10 1.E+07 7 He = 10 1.E+06 6 Turbulent Newton He = 10 Bingham 5 Casson 1.E+05 YP k = 1/3 YP k = 1/4 He = 0 Newtonian 1.E+04 1.E+03 Turbulent Transition (YP) 1.E+04 1.E+05 Plastic Reynolds Number (Rep ) 223 1.E+06 Design Curve Diagram (Laboratory) 1.E+07 He = 10 6 Stress Number (Ha) 1.E+06 He = 10 5 Turbulent 1.E+05 He = 10 1.E+04 Newton 4 Bingham Casson YP k = 1/3 YP k = 1/4 3 He = 10 1.E+03 1.E+02 He = 0 Newtonian Turbulent Transition (YP) 1.E+03 1.E+04 Plastic Reynolds Number (Rep ) 224 1.E+05 APPENDIX D: CASE STUDY DATA D.1 Introduction The Case Study Chapter 9 (as well as several of the explanatory Figures in this Thesis) used data obtained by Chad Litzenberger as part of his research for a Masters of Science degree in Chemical Engineering at the University of Saskatchewan in Saskatoon. The data was published in the following Thesis: Litzenberger, C.G.; (2004) “Rheological study of kaolin clay slurries”; MS Thesis, University of Saskatchewan, Saskatoon, Department of Chemical Engineering This research project investigated the rheology of kaolin clay slurry at various solids concentration. The clay slurry was aggregated with calcium salt (dehydrated calcium chloride, CaCl2•2H20) and dispersed with tetra-sodium pyrophosphate (TSPP), both of which were added in various quantities. The consistency of the slurry was measured in a Haake RV3 concentric-cylinder rheometer. The pressure gradient of the flowing slurry was measured in a vertical pipe loop. Details of the test program may be found in the source document. The published data sets used in this Thesis are included in this Appendix. (Note: The rheometer measuring system dimensions are listed incorrectly on these data sheets. The correct dimensions for a Haake MV/MV1 are listed elsewhere in the source document and were used for analysis in this Thesis.) 226 D.2 Run Number G2000208 227 D.3 Run Number G2000212 228 D.4 Run Number G2000105 229 D.5 Run Number G2000215 230 APPENDIX E: PUBLICATIONS E.1 Introduction The following is a list of the journal papers and conference papers published by the Author during the research period. The publications have been divided into two groups: those that are directly related to the topic of this Thesis and those that are indirectly related (e.g., design of slurry transport systems). E.2 Directly Related Publications 1. Hallbom, D.J. and Klein, B., (2004), “Flow Array for Nickel Laterite Slurry”, Proceedings of the TMS International Nickel Symposium, Charlotte, USA, 14-18 March, pp. 415-428 2. Hallbom, D.J. and Klein, B., (2006), “Laminar Pipe Flow of a Yield Plastic Fluid”, Proceedings of the 2nd JKMRC International Student Conference, Brisbane, Australia, 7-8 March, pp. 141-165 3. Hallbom, D.J. and Klein, B., (2006), “General Casson, Part 1 – A Phenomenological Model for Slurry Rheology”, Proceedings of the 5th International Conference on Conveying and Handling of Particulate Solids (CHoPS 5), Sorrento, Italy, 27-31 August 4. Hallbom, D.J. and Klein, B., (2006), “General Casson, Part 2 – A Physical Model for Slurry Rheology”, Proceedings of the 5th International Conference on Conveying and Handling of Particulate Solids (CHoPS 5), Sorrento, Italy, 27-31 August 5. Hallbom, D.J. and Klein, B., (2006) “A Comparison of Slurry Rheological Models”, Proceedings of the 1st Southern African Conference on Rheology (SASOR 2006), Cape Town, South Africa, 28-29 September 231 6. Hallbom, D.J., (2007), “Extracting Rheological Parameters from Low Shear Rate Loop Test Data”, Proceedings of the 10th International Seminar on Paste and Thickened Tailings (Paste 2007), Perth, Australia, 13-15 March, pp. 259-268 7. Hallbom, D.J. and Klein, B., (2007), “Design Curve Diagrams for Yield Plastic Fluids”, Proceedings of the 17th International Conference on the Hydraulic Transport of Solids (Hydrotransport 17), Cape Town, South Africa, 7-11 May, pp. 57-76 8. Hallbom, D.J, (2008), “Turbulent Pipe Flow of Yield Plastic Slurries”, Proceedings of the 11th International Seminar on Paste and Thickened Tailings (Paste 2008), Kasane, Botswana, 5-9 May, {Peer reviewed paper accepted} 9. Hallbom, D.J. and Klein, B., (2008) “Laminar Pipe Flow of Yield Plastic Fluids”, Proceedings of the XXIV International Mineral Processing Conference, Beijing, China, 23-28 September {Peer reviewed paper accepted} 10. Hallbom, D.J. and Klein, B., (TBA), “A Physical Model for Yield Plastic Fluids”, Particle Science and Technology – An International Journal {Accepted for publication} E.3 Indirectly Related Publications 11. Klein, B. and Hallbom, D.J., (2004), “Modifying the Rheology of Nickel Laterite Suspensions”, Minerals Engineering, 15, pp. 745-749 12. Hallbom, D.J., (2005), “The ‘Lump’ Test”, Proceedings of the 8th International Seminar on Paste and Thickened Tailings (Paste 2005), Santiago, Chile, 20-22 April, pp. 73-97 13. Hallbom, D. Norwood, W., and Gandhi, R., (2005)“Long-Distance Hydrotransport Systems for Nickel Laterite”, Cape Town, South Africa, (Nickel 2005) 14. Hallbom, D.J., (2006), “Over the Hills and Far Away – Long Distance Slurry Pipelines in Mountainous Regions”, Proceedings of the 38th Annual Canadian Mineral Processors Operators Conference, Ottawa, Canada, 17-19 January 232 15. Hallbom, D.J. and Chapman, J.P., (2006), “The Slide – A Simple Way to Design Paste Lines”, Proceedings of the 9th International Seminar on Paste and Thickened Tailings (Paste 2006), Limerick, Ireland, 3-7 April, pp. 383-393 16. Hallbom, D.J. and Gandhi, R.L., (2006), “Long Distance Hydrotransport Systems for Bauxite Ore”, Proceedings of the XXIII International Mineral Processing Conference, Istanbul, Turkey, 3-8 September, pp. 1966-1971 17. Hallbom, D.J. and Norwood, W.J., (2007), “Fuzzy Rheology and Smooth Running Paste Systems”, Proceedings of the 10th International Seminar on Paste and Thickened Tailings (Paste 2007), Perth, Australia, 13-15 March, pp. 199-210 18. Hallbom, D.J., (2007), “The Operating Range Method for Backfill Hydraulics”, Proceedings of the 9th International Symposium in Mining with Backfill (MineFill 2007), Montreal, Canada, 29 April to 02 May 233
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Pipe flow of homogeneous slurry Hallbom, Donald John 2008
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Title | Pipe flow of homogeneous slurry |
Creator |
Hallbom, Donald John |
Publisher | University of British Columbia |
Date Issued | 2008 |
Description | The objective of this Thesis is to devise a system for the "rheology-based design" of non-settling (homogeneous) slurry pipelines that is more conducive to application by practicing engineers without impairing its accuracy or utility for research purposes. The cornerstone is the development of a new rheological model and constitutive equation for homogeneous slurry based on the aggregation/deaggregation of the suspended mineral particles. This “yield plastic” model is shown to describe a family of models that includes the Newtonian, Bingham plastic and Casson models as special cases. It also closely approximates the results of many consistency models, including power law, yield power law, Cross and Carreau-Yasuda. The yield plastic model is then used to develop design equations to determine the pressure-gradient of laminar and turbulent pipe flow. A relative energy dissipation criterion is proposed for the laminar-turbulent transition and shown to be consistent with currently used transition models for Newtonian and Bingham fluids. Finally, a new dimensionless group (the “stress number”) is proposed that is directly proportional to the pressure-gradient and independent of the velocity. When the design equations are presented graphically in terms of the stress number and the plastic Reynolds number, the resulting “design curve diagram” is shown to be a dimensionless (pressure-gradient vs. velocity) pipe flow curve. The net result is that the hydraulic design of homogeneous slurry systems only requires the use of a single constitutive equation and three engineering design equations. The results are presented in a conceptually easy form that will foster an intuitive understanding of non-Newtonian pipe flow. This will assist engineers to understand the impact of slurry rheology when designing, operating and troubleshooting slurry pipelines and, in the future, other slurry related processes. |
Extent | 2546218 bytes |
Subject |
Rheology Fluid mechanics Pipeline hydraulics Suspensions |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2008-09-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0066641 |
URI | http://hdl.handle.net/2429/2399 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mining Engineering |
Affiliation |
Applied Science, Faculty of Mining Engineering, Keevil Institute of |
Degree Grantor | University of British Columbia |
GraduationDate | 2008-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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