Slurry Systems Handbook

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SLURRY SYSTEMS HANDBOOK BAHA E. ABULNAGA, P.E. Mazdak International, Inc.

McGRAW-HILL New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto

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Cataloging-in-Publication Data is on file with the Library of Congress.

Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 1 2 3 4 5 7 8 9 0

DOC/DOC

0 7 6 5 4 3 2

ISBN 0-07-137508-2 The sponsoring editor for this book was Larry S. Hager and the production supervisor was Sherri Souffrance. It was set in Times Roman by Ampersand Graphics, Ltd. Printed and bound by R. R. Donnelley and Sons, Co.

This book was printed on recycled, acid-free paper containing a minimum of 50% recycled de-inked fiber.

McGraw-Hill books are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please write to the Director of Special Sales, McGraw-Hill Professional Publishing, Two Penn Plaza, New York, NY 10121-2298. Or contact your local bookstore. Information contained in this work has been obtained by The McGraw-Hill Companies, Inc. (“McGraw-Hill”) from sources believed to be reliable. However, neither McGraw-Hill nor its authors guarantee the accuracy or completeness of any information published herein, and neither McGraw-Hill nor its authors shall be responsible for any errors, omissions, or damages arising out of use of this information. This work is published with the understanding that McGraw-Hill and its authors are supplying information but are not attempting to render engineering or other professional services. If such services are required, the assistance of an appropriate professional should be sought.

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In memory of my father, Dr. Sayed Abul Naga, and in dedication to my mother, Dr. Hiam Aboul Hussein, who devoted their lives to comparative literature as authors and translators. May their efforts contribute to a better understanding among mankind. And to my children Sayed and Alexander for filling my life with joy and happiness.

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BAHA ABULNAGA, P.E., obtained his Bachelor of Aeronautical Engineering in 1980 from the University of London and his Masters in Materials Engineering in 1986 from the American University of Cairo, Egypt. The first years of his professional career were devoted to the adaptation of air cushion platforms to desert environments, as well as the development of renewable energy systems. In 1988, he joined CSIRO (Australia) as a scientist. There he conducted research on complex multiphase flow for the design of smelting furnaces. Since 1990, he has been active in design of rotating equipment, pumps, and slurry pipelines and processing plants. His career has been a balanced mixture of design of equipment and consulting engineering. He has been employed as a design engineer for a number of manufacturers such as Warman Pumps (now part of Weir Pumps), Svedala Pumps and Process (now part of Metso Mineral Systems), Sulzer Pumps North America, and Mazdak Pumps and Mixers. He has also contracted as a slurry and hydraulics specialist for major consulting engineering firms such as ERM, SNC-Lavalin, Fluor, Bateman, Rescan, and Hatch and Associates. His involvement in the design, expansion, and commissioning of projects has included ASARCO Ray Tailings (USA), LTV Steel (USA), Zaldivar Pipeline (Chile), Southern Peru Expansion (Peru), Lomas Bayes (Chile), Escondida (Chile), BHP Diamets (Canada), Muskeg River Oil Sands (Canada), Bajo Alumbrera (Argentina), Homestake Eskay Creek (Canada), and many other engineering projects, feasibility studies, and audits.

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PREFACE

The science of slurry hydraulics started to flourish in the 1950s with simple tests on pumping sand and coal at moderate concentrations. It has evolved gradually to encompass the pumping of pastes in the food and process industries, mixtures of coal and oil as a new fuel, and numerous mixtures of minerals and water. Because of the diversity of minerals pumped, the wide range in sizes [43 ␮m (mesh 325) to 51 mm (2 in)], and the various physical and chemical properties of the materials, the engineering of slurry systems requires various empirical and mathematical models. The engineering of slurry systems and the design of pipelines is therefore fairly complex. This handbook targets the practicing consultant engineer, the maintenance superintendent, and the economist. Numerous solved problems and simplified computer programs have been included to guide the reader. The structure of the book is essentially in two parts. The first six chapters form the first part of the book and focus on the hydraulics of slurry systems. Chapter 1 is a general introduction on the preparation of slurry, the classification of soils, the siltation of dams, and the history of slurry pipelines. Chapter 2 focuses on water as a carrier of solids. Chapter 3 progresses with the mechanics of mixing solids and liquids and the principles of rheology. Chapter 4 presents the various models of heterogeneous flows of settling slurries, whereas Chapter 5 concentrates on non-Newtonian flows. Due to the importance of open channel flows in the design of long-distance tailings systems or slurry plants, Chapter 6 was dedicated to a better understanding of these complex flows, which are seldom mentioned in books on slurry. In Part II, the book focuses on components of slurry systems and their economic aspects. In Chapter 7, the important equipment of slurry processing plants is presented, including grinding circuits, flotation cells, agitators, mixers, and thickeners. Chapter 8 presents the guidelines for the design of centrifugal slurry pumps, and methods of correction of their performance. Chapter 9 reviews the continuous improvements of positive displacement slurry pumps in their different forms, such as plunger, diaphragm, or lockhopper pumps. As slurry causes wear and corrosion, aspects of the selection of metals and rubbers is presented in Chapter 10. To guide the reader to the various aspects of the design of slurry pipelines, Chapter 11 presents practical cases such as coal, phosphate, limestone, and copper concentrate pipelines. This review of historical data is followed by a review of standards of the American Society of Mechanical Engineers and the American Petroleum Institute, as they are extremely useful tools for the design and monitoring of pipelines. Finally, as the big unknown is too often cost, Chapter 12 closes the book by offering guidelines for a complete feasibility study for a tailings disposal system or a slurry pipeline. The author wishes to thank the staff of Mazdak International Inc, particularly Ms. Mary Edwards for providing typing services with great dedication over a period of two years. The author particularly wishes to thank Fluor Daniel Wright Engineers for allowing him to use their excellent library in Vancouver, Canada. The author wishes to thank his former colleagues in a colorful career, particularly Mr. K. Burgess, C.P.Eng. of Warman International; Mr. A. Majorkwiecz, K. Major, and Mr. Peter Wells of Hatch & Associates; Mr. I. Hanks, P.Eng. and W. McRae of Bateman Engineering; Mr. R. Burmeister

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PREFACE

H. Basmajian, and Dr. C. Shook, consultants; Mr. C. Hunker, P.Eng, V. Bryant, D. Bartlett, and W. Li, P.Eng. of Fluor Daniel; and Mr. A. Oak, P.Eng. of AMEC for allowing him to work on very challenging assignments in Australia and South and North America. The author wishes to thank the following firms for their contributions in the form of figures and data to this handbook: The Metso Group (formerly the companies Nordberg and Svedala), Red Valves, Geho Pumps (Weir Pumps), Mobile Pulley and Machine Works, Inc., Wirth Pumps, Hayward Gordon, Mazdak International Inc., the BHR Group, and GIW/KSB Pumps. The author is grateful to the various publishers and associations who allowed him to reproduce valuable materials in the book.

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CONTENTS

Preface

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PART ONE HYDRAULICS OF SLURRY FLOWS 1 General Concepts of Slurry Flows 1-0 1-1

1-2

1-3 1-4 1-5

1-6 1-7 1-8

1-9 1-10

Introduction Properties of Soils for Slurry Mixtures 1-1-1 Classifications of Soils for Slurry Mixtures 1-1-2 Testing of Soils 1-1-3 Textures of Soils 1-1-4 Plasticity of Soils Slurry Flows 1-2-1 Homogeneous Flows 1-2-2 Heterogeneous Flows 1-2-3 Intermediate Flow Regimes 1-2-4 Flows of Emulsions 1-2-5 Flows of Emulsions - Slurry Mixtures Sinking Velocity of Particles, and Critical Velocity of Flow 1-3-1 Sinking or Terminal Velocity of Particles 1-3-2 Critical Velocity of Flows Density of a Slurry Mixture Dynamic Viscosity of a Newtonian Slurry Mixture 1-5-1 Absolute (or Dynamic) Viscosity of Mixtures with Volume Concentration Smaller Than 1% 1-5-2 Absolute (or Dynamic) Viscosity of Mixtures with Solids with Volume Concentration Smaller Than 20% 1-5-3 Absolute (or Dynamic) Viscosity of Mixtures with High Volume Concentration of Solids Specific Heat Thermal Conductivity and Heat Transfer Slurry Circuits in Extractive Metallurgy 1-8-1 Crushing 1-8-2 Milling and Primary Grinding 1-8-3 Classification 1-8-4 Concentration and Separation Circuits 1-8-5 Piping the Concentrate 1-8-6 Disposal of the Tailings Closed and Open Channel Flows, Pipelines Versus Launders Historical Development of Slurry Pipelines

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1.3 1.4 1.5 1.5 1.8 1.13 1.13 1.15 1.16 1.16 1.16 1.16 1.17 1.17 1.17 1.17 1.19 1.21 1.21 1.21 1.22 1.22 1.22 1.24 1.24 1.25 1.26 1.26 1.30 1.30 1.31 1.32

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CONTENTS

1-11 1-12 1-13 1-14

Sedimentation of Dams—A role for the Slurry Engineer Conclusion Nomenclature References

1.33 1.37 1.37 1.38

2 Fundamentals of Water Flows in Pipes

2.1

2-0 2-1 2-2 2-3

2.4 2-5 2-6 2-7 2-8 2-9 2-10 2-11

2-12 2-13 2-14

Introduction Shear Stress of Liquid Flows Reynolds Number and Flow Regimes Friction Factors 2-3-1 Laminar Friction Factors 2-3-2 Transition Flow Friction Factor 2-3-3 Friction Factor in Turbulent Flow 2-3-4 Hazen–Williams Formula The Hydraulic Friction Gradient of Water in Rubber-Lined Steel Pipes Dynamics of the Boundary Layer 2-5-1 Entrance Length 2-5-2 Friction Velocity Pressure Losses Due to Conduits and Fittings Orifice Plates, Nozzles and Valves Head Losses Pressure Losses Through Fittings at Low Reynolds Number The Bernoulli Equation Energy and Hydraulic Grade Lines with Friction Fundamental Heat Transfer in Pipes 2-11-1 Conduction 2-11-2 Thermal Resistance 2-11-3 The R Value 2-11-4 The Specific Heat or Heat Capacity Cp 2-11-5 Characteristic Length 2-11-6 Thermal Diffusivity 2-11-7 Heat Transfer Conclusion Nomenclature References

3 Mechanics of Suspension of Solids in Liquids 3-0 3-1

Introduction Drag Coefficient and Terminal Velocity of Suspended Spheres in a Fluid 3-1-1 The Airplane Analogy 3-1-2 Buoyancy of Floating Objects 3-1-3 Terminal Velocity of Spherical Particles 3-1-3-1 Terminal Velocity of a Sphere Falling in a Vertical Tube 3-1-3-2 Very Fine Spheres 3-1-3-3 Intermediate Spheres 3-1-3-4 Large spheres 3-1-4 Effects of Cylindrical Walls on Terminal Velocity

2.1 2.1 2.3 2.4 2.6 2.8 2.9 2.18 2.19 2.33 2.33 2.35 2.44 2.49 2.54 2.58 2.58 2.58 2.60 2.60 2.60 2.61 2.61 2.61 2.61 2.62 2.62 2.64

3.1 3.1 3.1 3.1 3.3 3.3 3.3 3.5 3.6 3.7 3.8

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3-1-5 Effects of the Volumetric Concentration on the Terminal Velocity 3-2 Generalized Drag Coefficient—The Concept of Shape Factor 3-3 Non-Newtonian Slurries 3-4 Time-Independent Non-Newtonian Mixtures 3-4-1 Bingham Plastics 3-4-2 Pseudoplastic Slurries 3-4-2-1 Homogeneous Pseudoplastics 3-4-2-2 Pseudohomogeneous Pseudoplastics 3-4-3 Dilatant Slurries 3-4-4 Yield Pseudoplastic Slurries 3-5 Time-Dependent Non-Newtonian Mixtures 3-5-1 Thixotropic Mixtures 3-6 Drag Coefficient of Solids Suspended in Non-Newtonian Flows 3-7 Measurement of Rheology 3-7-1 The Capillary-Tube Viscometer 3-7-2 The Coaxial Cylinder Rotary Viscometer 3-8 Conclusion 3-9 Nomenclature 3-10 References

4 Heterogeneous Flows of Settling Slurries 4-0 4-1

4-2 4-3

4-4

4-5 4-6 4-7 4-8 4-9 4-10

Introduction Regimes of Flow of a Heterogeneous Mixture in Horizontal Pipe 4-1-1 Flow with a Stationary Bed 4-1-2 Flow with a Moving Bed 4-1-3 Suspension Maintained by Turbulence 4-1-4 Symmetric Flow at High Speed Hold Up Transitional Velocities 4-3-1 Transitional Velocities V1 and V2 4-3-2 The Transitional Velocity V3 or Speed for Minimum Pressure Gradient 4-3-3 V4: Transition Speed between Heterogeneous and Pseudohomogeneous Flow Hydraulic Friction Gradient of Horizontal Heterogeneous Flows 4-4-1 Methods Based on the Drag Coefficient of Particles 4-4-2 Effect of Lift Forces 4-4-3 Russian Work on Coarse Coal 4-4-4 Equations for Nickel–Water Suspensions 4-4-5 Models Based on Terminal Velocity Distribution of Particle Concentration in Compound Systems Friction Losses for Compound Mixtures in Horizontal Heterogeneous Flows Saltation and Blockage 4-7-1 Pressure Drop Due to Saltation Flows 4-7-2 Restarting Pipelines after Shut-Down or Blockage Pseudohomogeneous or Symmetric Flows Stratified Flows Two-Layer Models

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3.10 3.12 3.17 3.18 3.18 3.25 3.25 3.27 3.28 3.28 3.30 3.30 3.32 3.32 3.33 3.36 3.38 3.38 3.41

4.1 4.1 4.2 4.3 4.3 4.4 4.4 4.5 4.5 4.7 4.8 4.18 4.19 4.21 4.25 4.26 4.28 4.28 4.30 4.33 4.43 4.43 4.45 4.47 4.48 4.50

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4-11 Vertical Flow of Coarse Particles 4-12 Inclined Heterogeneous Flows 4-12-1 Critical Slope of Inclined Pipes 4-12-2 Two-Layer Model for Inclined Flows 4-13 Conclusion 4-14 Nomenclature 4-15 References

5 Homogeneous Flows of Nonsettling Slurries 5-0 5-1

4.57 4.58 4.59 4.61 4.62 4.63 4.66

5.1

Introduction Friction Losses for Bingham Plastics 5-1-1 Start-up Pressure 5-1-2 Friction Factor in Laminar Regime 5-1-3 Transition to Turbulent Flow Regime 5-1-4 Friction Factor in the Turbulent Flow Regime 5-2 Friction Losses for Pseudoplastics 5-2-1 Laminar Flow 5-2-1-1 The Rabinowitsch–Mooney Relations 5-2-1-2 The Metzner and Reed Approach 5-2-1-3 The Tomita Method 5-2-1-3 Heywood Method 5-2-2 Transition Flow Regime 5-2-3 Turbulent Flow 5.3 Friction Losses for Yield Pseudoplastics 5-3-1 The Hanks and Ricks Method 5-3-2 The Heywood Method 5-3-3 The Torrance Method 5-4 Generalized Methods 5-4-1 The Hershel–Bulkley Model 5-4-2 The Chilton and Stainsby Method 5-4-3 The Wilson–Thomas Method 5-4-4 The Darby Method: Taking into Account Particle Distribution 5-5 Time-Dependent Non-Newtonian Slurries 5-6 Emulsions 5-7 Roughness Effects on Friction Coefficients 5-8 Wall Slippage 5-9 Pressure Loss through Pipe Fittings 5-10 Scaling up From Small to Large Pipes 5-11 Practical Cases of Non-Newtonian Slurries 5-11-1 Bauxite Residue 5-11-2 Kaolin Slurries 5-12 Drag Reduction 5-13 Pulp and Paper 5-14 Conclusion 5-15 Nomenclature 5-16 References

5.1 5.2 5.2 5.5 5.8 5.9 5.11 5.11 5.11 5.11 5.13 5.14 5.14 5.14 5.17 5.17 5.18 5.18 5.19 5.19 5.19 5.22 5.24 5.28 5.29 5.29 5.33 5.34 5.35 5.35 5.35 5.38 5.39 5.40 5.41 5.42 5.44

6 Slurry Flow In Open Channels and Drop Boxes

6.1

6-0 6-1

Introduction Friction for Single-Phase Flows in Open Channels

6.1 6.2

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6-2

6-3 6-4 6-5

6-6 6-7 6-8 6-9 6-10 6-11 6-12 6-13 6-14 6-15

Transportation of Sediments in an Open Channel 6-2-1 Measurements of the Concentration of Sediments 6-2-2 Mean Concentrations for Dilute Mixtures (Cv < 0.1) 6-2-3 Magnitude of ␤ Critical Velocity and Critical Shear Stress Deposition Velocity Flow Resistance and Friction Factor for Heterogeneous Slurry Flows 6-5-1 Flow Resistances in Terms of Friction Velocity 6-5-2 Friction Factors 6-5-2-1 Effect of Roughness 6-5-2-2 Effect of Particle Concentration on Slurry Viscosity 6-5-2-3 Effects of Particle Sizes on the Chezy Coefficient 6-5-2-4 Effect of Bed Form on the Friction 6-5-3 The Graf–Acaroglu Relation 6-5-4 Slip of Coarse Materials 6-5-5 Comparison between Different Models Friction Losses and Slope for Homogeneous Slurry Flows 6-6-1 Bingham Plastics Flocculation Launders Froude Number and Stability of Slurry Flows Methodology of Design Slurry Flow in Cascades Hydraulics of the Drop Box and the Plunge Pool Plunge Pools and Drops Followed by Weirs Conclusion Nomenclature References

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6.9 6.12 6.18 6.22 6.23 6.27 6.29 6.30 6.31 6.31 6.31 6.32 6.33 6.33 6.35 6.36 6.39 6.40 6.44 6.45 6.45 6.54 6.56 6.67 6.71 6.71 6.74

PART TWO EQUIPMENT AND PIPELINES 7 Components of Slurry Plants 7-0 7-1

7-2 7-3 7-4

7-5

Introduction Rock Crushing 7-1-1 Primary Crushers 7-1-1-1 Jaw Crushers 7-1-1-2 Gyratory Crushers 7-1-1-3 Impact Crushers Secondary and Tertiary Crushers 7-2-1 Cone Crushers 7-2-2 Roll Crushers Grinding Circuits 7-3-1 Single-Stage Circuits 7-3-2 Double-Stage Circuits Horizontal Tumbling Mills 7-4-1 Rod Mills 7-4-2 Ball Mills 7-4-3 Autogeneous and Semiautogeneous Mills Agitated Grinding 7-5-1 Vertical Tower Mills

7.3 7.3 7.3 7.4 7.5 7.7 7.8 7.9 7.9 7.11 7.11 7.21 7.23 7.23 7.26 7.26 7.26 7.27 7.28

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7-6

7-7

7-8 7-9 7-10 7-11 7-12 7-13

7-5-2 Vertical Spindle Mills 7-5-3 Roller Mills 7.5.4 Vibrating Ball Mills 7.5.5 Hammer Mills Screening Devices 7-6-1 Trommel Screens 7-6-2 Shaking Screens 7-6-3 Vibrating Screens 7-6-4 Banana Screens Slurry Classifiers 7-7-1 Hydraulic Classifiers 7-7-2 Mechanical Classifiers 7-7-3 Hydrocyclones 7-7-4 Magnetic Separators Flotation Circuits Mixers and Agitators Sedimentation 7-10-1 Gravity Sedimentation 7-10-2 Centrifuges Conclusion Nomenclature References

8 The Design of Centrifugal Slurry Pumps 8.0 8.1 8.2

8-3 8-4 8-5

8-6 8-7 8-8 8-9 8-10

Introduction The Centrifugal Slurry Pump Elementary Hydraulics of the Slurry Pump 8.2.1 Vortex Flow 8-2-2 The Ideal Euler Head 8-2-3 Slip of Flow Through Impeller Channels 8-2-4 The Specific Speed 8-2-5 Net Positive Suction Head and Cavitation The Pump Casing The Impeller, the Expeller and the Dynamic Seal Design of the Drive End 8-5-1 The Radial Thrust Due To Total Dynamic Head 8-5-2 The Axial Thrust Due to Pressure 8-5-3 Thread Pull Force 8-5-4 Radial Force on the Drive End 8-5-5 Total Forces from the Wet End 8-5-6 Flange Loads Adjustment of the Wet End Vertical Slurry Pumps Gravel and Dredge Pumps Affinity Laws Performance Corrections for Slurry Pumps 8-10-1 Corrections for Viscosity and Slip 8-10-2 Concepts of Head Ratio and Efficiency Ratio Due to Pumping Solids

7.28 7.28 7.28 7.31 7.31 7.32 7.32 7.32 7.32 7.32 7.32 7.33 7.33 7.38 7.38 7.40 7.59 7.60 7.62 7.64 7.64 7.66

8.1 8.1 8.2 8.6 8.7 8.8 8.11 8.14 8.18 8.25 8.34 8.42 8.43 8.43 8.48 8.51 8.51 8.52 8.53 8.53 8.59 8.60 8.61 8.61 8.64

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8-10-3 Concepts of Head Ratio and Efficiency Ratio Due to Pumping Froth 8-11 Conclusion 8-12 Nomenclature 8-13 References

9 Positive Displacement Pumps 9-0 9-1 9-2 9-3 9-4 9-5 9-6 9-7 9-8 9-9

Introduction Solid Piston Pumps Plunger Pumps Diaphragm Piston Pumps Accessories for Piston and Plunger Pumps Peristaltic Pumps Rotary Lobe Slurry Pumps The Lockhopper Pump Conclusion References

10 Materials Science for Slurry Systems 10.0 Introduction 10-1 The Stress- Strain Relationship of Metals 10-2 Iron and Its Alloys for the Slurry Industry 10-2-1 Grey Iron 10-2-2 Ductile Iron 10.3 White Iron 10-3-1 Malleable Iron 10-3-2 Low-Alloy White Irons 10-3-3 Ni-Hard 10-3-4 High-Chrome–Molybdenum Alloys 10.4 Natural Rubbers 10-4-1 Natural Aashto 10-4-2 Pure Tan Gum 10-4-3 White Food-Grade Natural Rubber 10-4-4 Carbon-Black-Filled Natural Rubber 10-4-5 Carbon-Black- and Silicon-Filled Natural Rubber 10-4-6 Hard Natural Rubber/ Butadiene Styrene Compound Filled with Graphite 10-5 Synthetic Rubbers 10-5-1 Polychlorene (Neoprene) 10-5-2 Ethylene Propylene Terpolymer (EPDM) 10-5-3 Jade Green Armabond 10-5-4 Armadillo 10-5-5 Nitrile 10-5-6 Carboxylic Nitrile 10-5-7 Hypalon 10-5-8 Fluoro-elastomer (Viton) 10-5-9 Polyurethane 10-6 Wear Due to Slurries 10-7 Conclusion 10-8 References

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8.68 8.72 8.72 8.75

9.1 9.1 9.1 9.6 9.8 9.13 9.13 9.14 9.15 9.16 9.17

10.1 10.1 10.1 10.3 10.3 10.4 10.4 10.4 10.5 10.5 10.6 10.11 10.12 10.12 10.12 10.13 10.13 10.13 10.13 10.14 10.15 10.15 10.15 10.15 10.17 10.17 10.18 10.18 10.18 10.21 10.22

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CONTENTS

11 Slurry Pipelines 11.0 11-1 11-2 11-3

11-4 11-5 11-6

11-7 11-8 11-9 11-10 11-11 11-12 11-13 11-14

Introduction Bauxite Pumping Gold Tailings Coal Slurries 11-3-1 Size of Coal Particles 11-3-2 Degradation of Coal During Hydraulic Transport 11-3-3 Coal–Magnetite Mixtures 11-3-4 Chemical Additions to Coal–Water Mixtures. 11-3-5 Coal–Oil Mixtures 11-3-6 Dewatering Coal Slurry 11-3-7 Ship Loading Coarse Coal 11-3-8 Combustion of Coal–Water Mixtures (CWM) 11-3-9 Pumping Coal Slurry Mixtures Limestone Pipelines Iron Ore Slurry Pipelines Phosphate and Phosphoric Acid Slurries 11-6-1 Rheology 11-6-2 Materials Selection for Phosphate 11-6-3 The Chevron Pipeline 11-6-4 The Goiasfertil Phosphate Pipeline 11-6-5 The Hindustan Zinc Phosphate Pipeline Copper Slurry and Concentrate Pipelines Clay and Drilling Muds Oil Sands Backfill Pipelines Uranium Tailings Codes and Standards for Slurry Pipelines Conclusion References

12 Feasibility Study for A Slurry Pipeline and Tailings Disposal System 12-0 12-1 12-2 12-3 12-4 12-5

12-6 12-7 12-8

Introduction Project Definition Rheology, Thickeners Performance, Pipeline Sizing Reclaim Water Pipeline Emergency Pond Tailings Dams 12-5-1 Wall Building by Spigotting 12-5-2 Deposition by Cycloning 12-5-2-1 Mobile Cycloning by the Upstream Method 12-5-2-2 Mobile Cycloning by the Downstream Method 12-5-2-3 Deposition by Centerline 12-5-2-4 Multicellular Construction Submerged Disposal 12-6-1 Subsea Deposition Techniques Tailings Dam Design Seepage Analysis of Tailings Dams

11.1 11.1 11.1 11.2 11.2 11.2 11.3 11.4 11.5 11.5 11.6 11.8 11.8 11.10 11.10 11.12 11.16 11.17 11.18 11.19 11.20 11.21 11.21 11.22 11.23 11.24 11.27 11.27 11.30 11.31

12.1 12.1 12.2 12.5 12.8 12.9 12.11 12.11 12.12 12.14 12.14 12.15 12.15 12.15 12.17 12.17 12.18

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12-9 12-10 12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

Stability Analysis for Tailings Dams Erosion and Corrosion Hydraulics Pump Station Design Electric Power System Telecommunications Tailings Dam Monitoring Choke Stations and Impactors Establishing an Approach for Start-up and Shutdown Closure and Reclamation Plan Access and Service Roads Cost Estimates 12-20-1 Capital Costs 12-20-2 Operation Cost Estimates 12-21 Project Implementation Plan 12-22 Conclusion 12-23 References

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12.18 12.19 12.19 12.19 12.20 12.21 12.21 12.22 12.22 12.23 12.24 12.24 12.24 12.25 12.27 12.27 12.28

Appendix A Specific Gravity and Hardness of Minerals

A.1

Appendix B Units of Measurement

B.1

Index

I.1

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Page 1.1

PART ONE

HYDRAULICS OF SLURRY FLOWS

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CHAPTER 1

GENERAL CONCEPTS OF SLURRY FLOWS

1-0 INTRODUCTION Slurry is essentially a mixture of solids and liquids. Its physical characteristics are dependent on many factors such as size and distribution of particles, concentration of solids in the liquid phase, size of the conduit, level of turbulence, temperature, and absolute (or dynamic) viscosity of the carrier. Nature offers examples of slurry flows such as seasonal floods that carry silt and gravel. Every year during the flood season, the Nile transports massive amounts of silt over thousands of miles to the Saharan desert. To rephrase Herodotus, who once said “Egypt is the gift of the Nile,” one may consider that one of the most ancient civilizations was dependent on natural slurry flows for its survival. Dredging is one of the most common and ancient processes involving slurry flows; the dredged materials contain a wide range of particles, tree debris, rocks, etc. Mining has employed the concept of slurry flows in pipelines since the mid-nineteenth century, when the technique was used to reclaim gold from placers in California. Long-distance slurry pipelines have evolved in all continents since the mid 1950s. Some slurry mixtures consist of very fine solids at high concentration, such as those in the copper concentrate pipelines of Escondida, Chile, and Bajo Alumbrera, Argentina. Other mixtures are based on coarse particles up to a size of 150 mm (6⬙), such as those pumped from fields of phosphate matrix. This chapter introduces some of the basic principles of slurry mixtures and flows. The slurry engineer has to appreciate the properties of the soil to be mined, dredged, or mixed with water. Original rock sizes, hardness, and plasticity play a major role in the selection of the equipment for crushing, milling, flotation, tailings disposal, or soil reclamation. Understanding sinking and critical speeds are essential when sizing the pipeline. A brief introduction to slurry flows in extractive metallurgy serves the purpose of focusing on the essentials of the application of slurry flows to engineering. Natural slurry flows, even in very dilute forms, can have negative effects on the environment if not properly managed. Some of the great dams of the world built in the twentieth century are starting to suffer from siltation. Behind such dams, large lakes are often man-made. The river flow is brought to a sufficiently slow speed for the silt to deposit at the bottom. Engineers in the twenty-first century will have to learn to manage the siltation of large man-made lakes using the science of dredging and piping slurry flows.

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1-1 PROPERTIES OF SOILS FOR SLURRY MIXTURES Slurry flows occur in nature in different ways. They are often associated with the transportation of silt from one region to another. Strong rains lead to soil erosion, mud slides, and the eventual drainage of slurries toward rivers. These are dilute slurries, in the sense that the soils mix naturally at a weight ratio of solids to liquids smaller than 15%. One very interesting river is the Nile. It may be said that during two months of the year it becomes a massive slurry flow. Torrential tropical rains over Lake Victoria in Uganda and Kenya are the source of the White Nile. Torrential tropical rains over the Ethiopian plateau are the source of the Blue Nile. On their way to the Sudan, both branches of this longest river in the world transport silt and soils. The White Nile seems to lose a lot of its water as it enters the swamps of the Bahr El Ghazal in Sudan. What is left of the White Nile joins the Blue Nile near Khartoum in Sudan. The Nile pursues its trip to the north and gradually enters the Saharan desert through Nubia and Egypt. As the flood season terminates, the silt transported by the Nile sediments by gravity. The silt has deposited for thousands of years, creating a narrow strip of rich farmland. Out of this silt grew the towns and states in Nubia and Egypt. The Pharaohs built an advanced civilization on the silt brought to them by the Nile’s natural slurry flows. The “gift of the Nile” was silt that would not have been deposited without a form of natural slurry flow. A simplified flow sheet (Figure 1-1) of the Nile illustrates this natural slurry flow. The steps in the process are: 앫 Water from the rains is the carrier liquid. 앫 The flow of water from the mountains of Uganda and Kenya moves fast enough during the flood season to scour the ground of silt and transport it in the form of a dilute slurry. (This is a step of slurry formation.)

torential rains

Uganda/Kenya

Sedimentation at Bahr El Ghazal

floods

rains

Ethiopia

The Saharan Desert

silt transported by the White Nile

floods

Nubia Sudan

Egypt

sedimentation by gravity of the silt after the flood (Egypt is the Gift of the Nile)

silt transported by the Blue Nile

FIGURE 1-1 There is no better example of the importance of slurry to civilization than the land of Egypt. For thousands of years, the Nile has transported massive quantities of silt over thousands of kilometers to cover by its floods a narrow stretch of land. From these silt layers, a civilization grew.

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앫 As the waters from the rains over the mountains of Uganda and Kenya join, they form the White Nile. (This step is natural hydrotransport.) 앫 As the White Nile enters the Bahr El Ghazal in Sudan, it spreads and stagnates, forming swamps. A nomadic life has long flourished around these swamps. (This step involves partial sedimentation by stagnation in the swamps.) 앫 In another region (in Ethiopia), rains form the Blue Nile. The flow of water from the mountains of Ethiopia move fast enough during the flood season to scour the ground of silt and transport it in the form of a dilute slurry. (This is another step of slurry formation.) 앫 The Blue and White Nile merge near Khartoum, Sudan, and continue their flow to the north. 앫 As the floods enter Nubia and Egypt, they overflow the banks of the Nile and transport speed of the slurry mixture drops. 앫 Sedimentation of silt occurs, with Egypt acting as a massive clarifier for the waters of the Nile, particularly at its delta with the Mediterranean Sea. (This step is natural gravity sedimentation.) For thousands of years the Pyramids and the Sphinx have stared at this immense natural slurry clarifier that is the Valley of the Nile in the middle of the Saharan Desert (Figure 12). Dredging is an important engineering activity in which gravel is moved in the form of slurry into a hopper on a specially constructed boat (Figure 1-4). A special pump is often used in a drag arm (Figure 1-3), and a special suction mouthpiece (Figure 1-5) is used at the tip of the drag arm. To complete dredging and form the slurry, it is essential to cut through the sand layers, rocks, and debris, using special cutters for sand (Figure 1-6a) and for rocks (Figure 1-6b) with very hard, replaceable blades. The composition of a slurry mixture depends on many factors such as particle size and distribution. Particles may be found in nature as soils or may be created by the processes of crushing, milling, and grinding. For applications such as dredging, natural soils are pumped without any crushing or grinding. For mining processes, an understanding of the physical properties of soils is essential for sizing equipment, crushing and milling, slurry preparation, mixing, and pumping (see Figure 1-7).

1-1-1 Classifications of Soils for Slurry Mixtures There are a variety of methods used to classify soils. Two main classes are: 1. Cohesive soils such as certain silts and clays with a median particle diameter smaller than 0.0625 mm (less than 0.0025 in, or mesh 250) 2. Noncohesive soils such as certain silts and clays with a median particle diameter larger than 0.0625 mm (larger than 0.0025 in, or mesh 250) For underwater dredging, the rock’s strength is determined by its core, and this property has a very important effect on the efficiency of dredging. Herbrich (1991) proposed a classification of soils in terms of unconfined compressive strength (see Table 1-1). The Permanent International Association of Navigation Congresses (1972) adopted a system of classification of soils, reviewed by Sargent (1984) and summarized in Tables

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FIGURE 1-2 For five thousand years, the Sphinx and the Pyramids have stared from the Gizeh plateau in the desert at history and at the Nile, which transforms itself every summer into a natural slurry transporter, bringing silt and life to the desert.

1-2, 1-3, and 1-4, that is recommended for use in dredging. In these tables, visual inspection is mentioned as a quick way to determine the nature of soils. This method does not relieve the engineer from the responsibility of conducting a proper size distribution test and rheology test before any design. The Standard D2488 of the American Society for Testing of Materials (ASTM) (1993) also offers a classification of soils, with a range of particle sizes as presented in Table 1-5. This standard is widely used in North America.

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hopper for solids

bottom of lake discharge pipe

pump

electric cable

drag arm column

FIGURE 1-3 Dredging boat and dredge arm.

FIGURE 1-4 Special dredger boat.

1.7

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FIGURE 1-5 Works.

CHAPTER ONE

Suction mouthpiece for boat dredger. Courtesy of Mobile Pulley and Machine

1-1-2 Testing of Soils Various soil tests are recommended before mixing the soil with water in the early stages of designing a dredging or slurry transportation system. Particle size distribution should be established. Table 1-6 presents conversion factors between the three most common scales for measuring particle size. A number of tests are recommended to determine the dredgeability of soils and their behavior in placer mining or slurry mixing (Table 1-7). In nature, silts may be found in association with clays; thus, the parameters for both silts and clays should be assessed. The following testing parameters are accepted by the industry. Composition Tests 앫 Visual inspection: For the purpose of assessment of the rock mass. Such a test indicates the in situ state of the rock mass. Tests may be conducted in situ or under lab conditions in accordance with British Standard Institute Standard BS 5930 (1999). 앫 Section thickness test: A lab test conducted for the purpose of geotechnical identification and as a tool to determine mineral composition of the rock mass. 앫 Bulk density: Wet and dry tests are conducted under laboratory conditions to assess the weight and volume relationship. (International Journal of Rock Mechanics and Mineral Sciences, 1979). 앫 Porosity: This is a calculation of voids as a percentage of total volume and is based on lab tests on bulk density. 앫 Carbonate content: This lab test should be conducted in accordance with American Society for Testing Materials (ASTM) Standard D3155 (1983) to measure lime content, particularly in limestone and chalks.

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(a)

(b) FIGURE 1-6 (a) Special dredging sand cutter. The blades are replaceable. Courtesy of Mobile Pulley and Machine Works. (b) Special dredging rock cutter. Courtesy of Mobile Pulley and Machine Works. 1.9

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FIGURE 1-7 Mineral process plants can reject fairly coarse material that is left after crushing and milling mineral rocks. In this case, the coarse material is transported by piping in the form of a tailings slurry and used to build a tailings dam.

Strength, Hardness, and Stratification Tests 앫 Surface hardness: This lab test should be conducted to determine hardness in terms of the Mohr’s scale (from 0 for talc to 10 for diamonds). Appendix I presents a tabulation of density and Mohr hardness of minerals. The hardness of minerals is critical to the wear life of equipment associated with slurry flows. 앫 Uniaxial compression: This lab test measures ultimate strength under uniaxial stress. These tests should be done on fully saturated samples. The dimensions of the test sample and the directions of stratification influence stress direction. Cylinder samples

TABLE 1-1 Classification of Soils in Terms of Unconfined Compressive Strength. (After Herbrich, 1991) Unconfined compressive strength Characteristic Very weak Weak Moderately weak Moderately strong Strong Very strong Extremely strong

MPa

103 psi

< 1.25 1.25–5.0 5.0–12.5 12.5–50.0 50–100 100–200 > 200

< 0.145 0.15–0.73 0.73–1.8 1.8–7.3 7.3–14.6 14.6–29.2 > 29.2

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TABLE 1-2 Classification of Noncohesive Dredged Soils after the Permanent International Association of Navigation Congresses (1972, 1984) Type of soils

Identification of particle sizes mm

BS sieve units

Identification

Boulders and cobbles

> 200 60–200

6

Visual examination and measurement

Gravel

Fine 2–6 mm Medium 6–20 mm Coarse 20–80 mm

Fine No. 7—1–4 in Medium 1–4–3–4 in Coarse 3–4–3 in

Visual examination

Sands

Fine 0.06–0.2 mm Medium 0.2–0.6 mm Coarse 0.6–2 mm

Fine mesh 72–200 Visual Medium mesh 25–72 examination. No Coarse mesh 7–25 cohesion when dry

앫

앫 앫 앫 앫 앫 앫

Strength and structural properties

May be found loose in some fields, or in cemented beds, or may appear as weak conglomerate beds or hard packed gravel intermixed with sand Strength varies between compacted, loose and cemented. Homogeneous or stratified structures. Intermixture with silt or clay may produce hardpacked sands

should have a length-to-diameter ratio of 2:1, as per The International Society for Rock Mechanics (1978). Brazilian split: This is a lab test to measure strength as derived from uniaxial testing. This procedure is similar to the uniaxial compression test but with a different lengthto-diameter ratio. For further details, consult The International Society for Rock Mechanics (1977). Point load test: This is a quick lab test to measure strength. It should be conducted with the uniaxial compression test as described by Broch and Franklin (1972). Seismic velocity test: This field in situ test is conducted to check on the stratigraphy and fracturing of rock masses. It is useful for extrapolating field and lab measurements to rock mass behavior. Ultrasonic velocity test: This lab test is conducted on cores in the longitudinal direction. Static modulus of elasticity: This lab test measures stress/strain rate and gives an indication of the brittleness of rock. Drillability: This in situ test measures penetration rate, torque, feed force, fluid pressure, depth of layers, etc., and is used to establish the drill techniques and specification for placer mining or dredging. Angularity: This lab test is conducted to assess the shape of particles by visual inspection in accordance with British Standard Institute BS 812 (1999).

The expertise of a geologist is essential for mining or dredging large areas.

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TABLE 1-3 Classification of Cohesive Natural Soils after the Permanent International Association of Navigation Congresses (1972, 1984) Identification of particle sizes

Type of soils

mm

BS sieve

Identification

Silts

Fine 0.002–0.006 Medium 0.006–0.02 Coarse 0.02–0.06

Passing No. 200

Individual particles are invisible. Wet lumps or coarse are visible. Determination by testing for dilatancy*. Silt can be dusted off fingers after drying and dry lumps are powdered by finger pressure

Clays

Finer than 0.002

N/A

Clays are very cohesive and are plastic without dilatancy. Moist samples stick to fingers with smooth, greasy touch. Dry lumps do not powder.

Strength and structural properties Coarse and sandy particles are nonplastic but similar characteristics to sands. Fine silts are plastic and similar to clays. They are often found in nature intermixed with sand and clay. They may be homogeneous or stratified and their consistency may vary from fluid silt to stiff silt or siltstone Strength

Shear strength

Very soft: may < 20kN/m2 be squeezed < 2.9 psi easily between fingers Soft: easily molded by fingers

20–40 kN/m2 2.9–5.8 psi

Clays shrink and crack by drying and develop high strength

Firm: requires 40–75 kN/m2 strong pressure 5.8–10.9 psi to mold by fingers

Structure of clays may be fissured, intact, homogeneous, stratified, or weathered.

Stiff: can not be molded by fingers, dent by thumbnail

75–150 kN/m2 10.9–21.8 psi

Hard: tough, intended with difficulty by thumbnail

Above 150 kN/m2 21.8 psi

*Dilatancy is a property exhibited by silt when shaken, and is due to high permeability of silt. When a moistened sample is shaken in the open hand, water appears on the surface, giving it a glossy appearance.

TABLE 1-4 Classification of Organic Soils after the Permanent International Association of Navigation Congresses (1972, 1984) Type of soils Peat and organic soils

Identification of particle sizes mm

BS sieve

Identification

N/A

N/A

It is generally identified as brown or black with a strong organic smell and contains wood and fibers.

1.12

Strength and structural properties It may be firm or spongy in nature and its strength is different in horizontal and vertical directions.

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1-1-3 Textures of Soils Granular soils are found in nature as a mixture of particles of different sizes. Two coefficients are used to express such texture: 1. The coefficient of curvature, Cc (equation 1-1) 2. The coefficient of uniformity, Cu (equation 1-2) D230 Cc = ᎏ (D60D10)

(1-1)

D60 Cu = ᎏ D10

(1-2)

Where D10, D30, and D60 are defined as the grain size at which 10%, 30%, and 60% of the soil is finer. According to Herbrich (1991) If 1 < Cc < 3, the grain size distribution will be smooth If Cu > 4 for gravels then there is a wide range of sizes If Cu > 6 for sands then there is a wide range of sizes Alternatively, the soil is said to contain very little fines and is well graded.

1-1-4 Plasticity of Soils For clays and silts, an additional test for the liquid limit (LL) and the plastic limit (PL) are recommended. The liquid limit is defined as the moisture content in soil above which it starts to act as a liquid and below which it acts as a plastic. To conduct a test, a sample of clay is thoroughly mixed with water in a brass cup. The number of bumps required to close a groove cut in the pot of clay in the cup is then measured. This test is called the Atterberg test. The plastic limit is defined as the limit below which the clay will stop behaving as a plastic and will start to crumble. To measure such a limit, a sample of the soil is formed into a tubular shape with a diameter of 3.2 mm (0.125 in) and the water content is measured when the cylinder ceases to roll and becomes friable.

TABLE 1-5 Range of Particle Sizes of Soils According to ASTM D2488 (1993) Material Boulders Cobbles Coarse gravel Fine gravel Coarse sand Medium sand Fine sand Silts and clays

Range of sizes in mm > 300 75–300 19–75 4.75–19 2.00–4.75 0.43–2.00 0.08–0.43 < 0.075

Range of sizes in inches > 12 3–12 0.75–3 0.019–0.75 0.08–0.0188 0.017–0.08 0.003–0.017 < 0.003

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TABLE 1-6 Conversion between Scales of Particle Size U.S. no.

2.5 3 3.5 4 5 6 7

Tyler mesh

2.5 3 3.5 4 5 6 7

Sieve opening (micrometers)

Sieve opening (inches)

Grade of soils Screen shingle gravel

26670 22430 18850 15850 13330 11200 9423 7925 6680 5613 4699 3962 3327 2794

3 2 1.50 1.050 0.883 0.742 0.624 0.525 0.441 0.371 0.321 0.263 0.221 0.185 0.156 0.131 0.110

8 9 10 12

8 9 10 12

2362 1981 1651 1397

0.093 0.078 0.065 0.055

Very coarse sand

14 16 20 24

14 16 20 24

1168 991 833 701

0.046 0.039 0.0328 0.0276

Coarse sand

28 32 35 42 50

28 32 35 42 50

589 495 417 351 297

0.0232 0.0195 0.0164 0.0138 0.0117

Medium sand

60 70 80 100 120 140

60 70 80 100 120 140

250 210 177 149 125 105

0.01 0.0823 0.07 0.06 0.05 0.041

Fine sand

170 200 230

170 200 250 270

88 74 63 53

0.034 0.029 0.025 0.02

Silt

325 400 500 625 1250 2500 12500

43 38 25 20 10 5 1

0.017 0.015 0.01 0.008 0.004 0.002 0.0004

Pulverized silt

0.30:

冪莦

2gh Q = Cd A ᎏᎏ4 1 – (d1/d2)

(2-37)

This equation works for liquids with a dynamic viscosity similar to the viscosity of water. The discharge vena contracta and velocity coefficient presented in Figure 2-18 are based on controlled flow conditions upstream. Flow disturbances can affect the magnitude of these coefficients. Manufacturers of valves in North America have developed a valve coefficient to relate flow rate to pressure drop as Cv, which is defined as: ⌬Ppsi Qgpm = Cv ᎏ S.G.

冪莦

(2-38)

This coefficient is not dimensionally homogeneous and is not equal to the discharge coefficient from orifices and nozzles. Although the flow coefficient Cv was developed for control valves, a relationship is often established for other fittings in terms of the K factor: (29.9)(din)2 Cv = ᎏᎏ 兹苶 K

(2-39)

The reader should be very careful not to confuse Cv (the flow coefficient commonly used in North America) with the discharge coefficient Cd more commonly used in the rest of the world. Such a mix-up can lead to serious errors. Cv is not used outside North America and has no relationship to the terms defined in Equations 2-34 to 2-37. The reader should avoid the common confusion that it sometimes creates.

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FUNDAMENTALS OF WATER FLOWS IN PIPES

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FIGURE 2-19 Cross-section of a Series 39 slurry check valve. (Courtesy of Red Valve Company, Carnegie, PA, U.S.A.)

FIGURE 2-20 Front view of a Series 39 slurry check valve. (Courtesy of Red Valve Company, Carnegie, PA, U.S.A.)

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FIGURE 2-21 Slurry knife-gate valve cross-sectional drawing. (Courtesy of Red Valve Company, Carnegie, PA, U.S.A.)

FIGURE 2-22 U.S.A.)

Slurry knife-gate valve. (Courtesy of Red Valve Company, Carnegie, PA,

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FIGURE 2-23 Slurry pinch valve, showing cut through the rubber sleeve. (Courtesy of Red Valve Company, Carnegie, PA, U.S.A.)

Manufacturers of slurry valves have developed very specific designs to meet the requirements of wear and operation without plugging. These include: 앫 앫 앫 앫 앫

Rubber-lined check valves Rubber-lined knife-gate valves Rubber-lined pinch valves Ceramic ball valves Plug valves

Special check valves are available for sewage and slurry flows. The Red Valve Company Series 39 valves (Figures 2-19 and 2-20) feature a special reinforced elastomer check sleeve. The valve check sleeve seals under reverse flow or back-pressure and opens under pressure from the pump. It does not incorporate any discs that may wear on contact with slurry. This type of valve is therefore different in design than the type shown in books on water flows. The consultant engineer should therefore request from the manufacturer of the slurry check valves the estimated K factor for pressure losses. The Red Valve Company Series 39 slurry check valves are available in sizes up to 48⬙ (1220 mm), with a choice of elastomers such as pure gum rubber, neoprene, Hypalon, chlorobutyl, Buna-N, EPDM, and Viton. Knife-gate valves for slurry flows (Figures 2-21 and 2-22) feature a metal gate sand-

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FIGURE 2-24 Principles of operation of a pinch valve, pinched by a roller. (Courtesy of Red Valve Company, Carnegie, PA, U.S.A.)

wiched between two rubber linings (or cartridges). They are often installed on the suction side of slurry pumps to provide a method of isolating them during repairs and maintenance. Most knife-gate valves are rated to a maximum of 1 MPa (150 psi), but some manufacturers offer valves rated at 2 MPa (300 psi). Globe valves are not suitable for slurry applications because they wear rather rapidly. To control slurry flows, a rubber pinch valve is recommended (Figure 2-23). The valve features a special reinforced sleeve. The sleeve is closed by pinching using a special roller (mechanical pressure) (Figure 2-24) or by the use of air pressure (Figure 2-25). Ceramic ball valves are used as shut-off valves for pipelines, particularly to close under high pressure.

2-8 PRESSURE LOSSES THROUGH FITTINGS AT LOW REYNOLDS NUMBERS Certain slurry flows, particularly those of a non-Newtonian regime, do occur at relatively moderate Reynolds numbers and in laminar conditions (Tables 2-13 to 2-14). For many years, the method using the K factor and the equivalent length has been the most widely

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FIGURE 2-25 Principles of operation of a pneumatically actuated pinch valve. (Courtesy of Red Valve Company, Carnegie, PA, U.S.A.)

accepted method. It is based on experimental data obtained usually in steel pipes at very high Reynolds numbers. As the Reynolds number is reduced closer to laminar flow, the K factor becomes inversely proportional to it. Since certain homogeneous slurries are sometimes pumped at relatively low Reynolds numbers, even quite close to the critical value, it is important to emphasize an alternative approach. Hooper (1992) emphasized the limitations of this method and proposed a two-K method: K1 K⬁ K = ᎏ + ᎏᎏ 1 + 1/D1-in Re

(2-40)

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TABLE 2-13 Equivalent Length of Fittings for Friction Loss of Calculations for Single-Phase Turbulent Flow* Fitting Standard threaded elbow Standard threaded elbow Standard threaded elbow Mitre bend

Standard tee

Type

Length/Diameter Ratio

90 degree 45 degree Long radius 90 degree—5 diameter bend as used in slurry plants 15 degree bend 30 degree bend 45 degree bend 60 degree bend 75 degree bend 90 degree bend Through flow Through branch

30 16 16 4 8 15 25 40 60 20 60

*Data from Ingersoll Rand (1977).

where K1 = value of K at a Reynolds number of 1 K⬁ = value of K at high Reynolds numbers DI-in = internal pipe diameter in inches. Values of these two constants are presented in Table 2-15. Regarding the equivalent length method, Hooper (1992) wrote:

TABLE 2-14 Dynamic Loss Factor K for Expansions and Contractions, where Loss = KV2/2g* Fitting Pipe exit Pipe entrance Pipe entrance (flush)

Reentry pipe Sudden enlargements in pipes Sudden contractions in pipes Gradual enlargements in pipes Gradual contractions in pipes

Description Projecting sharp edged, rounded Inward projecting Sharp edged Bellmouth fillet/diameter = 0.02 Bellmouth fillet/diameter = 0.04 Bellmouth fillet/diameter = 0.06 Bellmouth fillet/diameter = 0.10 Bellmouth fillet/diameter = 0.15 and up

␪ Less than 45 degrees ␪ Larger than 45 degrees ␪ Less than 45 degrees ␪ Larger than 45 degrees

*Data from Ingersoll Rand (1977).

Loss factor K 1.0 0.78 0.5 0.28 0.24 0.15 0.09 0.04 L/D = 65 K = (1 – d 21/d 22) K = 0.5(1 – d 21/d 22) K = 2.6 sin (␪/2)(1 – d 21/d 22)2 K = (1 – d 21/d 22)2 K = 0.8 sin ␪(1 – d 21/d 22) K = 0.5(1 – d 21/d 22)兹(s 苶in 苶苶/2 ␪苶)苶

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TABLE 2-15 Constants for the Two-K Method* (after Hooper 1992) Fitting Elbows

Description 90°

45°

180°

Tees

Used as elbows

Run-through tee

Valves

Gate, ball, plug

Globe Globe Diaphragm Butterfly Check

Type Standard R/D = 1, screwed Standard R/D = 1, flanged/welded Long radius (R/D = 1.5), all types Mitered Elbow R/D = 1.5 1 weld 90° 2 welds 45° 3 welds 30° 4 welds 22.5° 5 welds 18° Standard (R/D = 1.0), all types Long radius (R/D = 1.5), all types Mitered, 1 weld, 45° angle Mitered, 2 welds, 22.5° angle Standard R/D = 1, screwed Standard R/D = 1, flanged/welded Long radius (R/D = 1.5), all types Standard, screwed Long radius, screwed standard, flanged/welded Stub-in-type branch Screwed Flanged or welded Stub-in-type branch Full line size, ␤ = 1 Reduced trim, ␤ = 0.9 Reduced trim, ␤ = 0.85 Standard Angle or Y-type Dam type Lift Swing Tilting check

K1 at Re = 1

K⬁ at very high Re

800 800 800 1000 800 800 800 800 500 500 500 500 1000 1000 1000 500 800 800 1000 200 150 100 300 500 1000 1500 1000 1000 800 2000 1500 1000

0.40 0.25 0.20 1.15 0.35 0.30 0.27 0.25 0.20 0.15 0.25 0.15 0.60 0.35 0.30 0.70 0.40 0.80 1.00 0.10 0.05 0.00 0.10 0.15 0.25 4.00 2.00 2.00 0.25 10.0 1.50 0.50

*Use R/D = 1.5 values for R/D = 5 pipe bends, 45° to 180°. Use appropriate tee values for flowthrough crosses.

The equivalent-length method concept contains a booby trap for the unwary. Every equivalent length method has a specific friction factor ( f ) associated with it, because the equivalent lengths were originally developed from the K factor in the formula Le = KD/f. This is why the latest version of the equivalent length method (the 1976 edition of the Crane Technical Paper 410 . . . properly requires the use of two friction factors. The first is the actual friction factor for the pipe ( f ), and the second is a “standard” friction factor for the particular fitting ( fT). Thus the two-K method is as easy to use and as accurate as the updated equivalent-length method. The two-K method will be explored further in Chapter 5.

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2-9 THE BERNOULLI EQUATION The last few sections of this chapter examined the concept of friction and pressure losses. The presence of friction forces, changes in elevation between one point and another along the piping, the presence of a pump to add energy to the fluid, or a turbine to extract energy can all be expressed in terms of the extended Bernoulli’s equation: (Ep + Ev + Ez)1 + EA = (Ep + Ev + Ez)2 + EE + Ef + Em

(2-41)

U 22 P1 U 21 P2 ᎏ + ᎏ + Z2g + EA = ᎏ + ᎏ + Z2g + EE + Ef + Em ␳ 2 ␳ 2 where subscripts 1 and 2 refer to points 1 and 2. Ep = P1/␳ = energy due to static pressure per unit mass U 21/2 = energy due to dynamic pressure per unit mass Z = location of point above a reference datum EA = energy added (e.g., by a pump) per unit mass EE = energy extracted (e.g., by a turbine) per unit mass Ef = Energy per unit mass due to friction losses Em = Energy lost due to fittings, per unit mass In USCS units.

2-10 ENERGY AND HYDRAULIC GRADE LINES WITH FRICTION When the total energy for flow in a pipeline is plotted against distance, a profile called the energy gradient line is obtained. The energy drops with friction or extraction through a turbine, and increases by absorption from a pump. The hydraulic gradient is the sum of the pressure and the potential energies. The hydraulic gradient is therefore smaller than the energy gradient by the dynamic head (Figure 2-26).

2-11 FUNDAMENTAL HEAT TRANSFER IN PIPES In many areas of the world, mining is done in cold climates (Figure 2-27). Long tailing pipelines are exposed to wind, snow, and freezing conditions. In some oil–sand processes, temperature is used to facilitate the pumping or separation of tar from sand. In other processes, hot slurries are fed to autoclave furnaces. The field of heat transfer is immense, but in the following paragraphs, some fundamentals will be reviewed. There are three main phenomena of heat transfer: 1. Conduction 2. Convection 3. Radiation

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EGL EGL

2

V1 /2 HGL

EA

HGL

2

V2 /2

2

E v= V /2 Energy and Hydraulic Gradients

Energy and Hydraulic Gradients

For a pump

For an expansion

FIGURE 2-26 Energy and hydraulic gradients.

FIGURE 2-27 The construction of mines may require pipelines that operate in extremely cold environments. This water pipeline was insulated and heat-traced for an Arctic environment.

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TABLE 2-16 Examples of Conductivity Range Material

Range of conductivity K, W/m °K

Range of conductivity K, Btu-ft/hr-ft2 °F

0.03–0.21 0.09–0.70 0.03–2.6 8.7–78 14–120 52–420

0.02–0.12 0.05–0.40 0.02–1.5 5.0–45 8.0–70 30–240

Insulators Nonmetallic Liquids Nonmetallic Solids Liquid metals Metallic alloys Pure metals

2-11-1 Conduction Heat transfer by conduction occurs essentially by molecular vibration and movement of free electrons. As metals have more free electrons than nonmetals, they are better conductors of heat. Thermal conductivity, also known as thermal conductance, is a measure of the rate of heat transfer per unit thickness. Examples of conductivity range are presented in Table 2-16 Thermal conductivity is a function of temperature. For metals it decreases with temperature, whereas for insulators it increases with temperature. To simplify matters, it is common to assume the thermal conductivity at the average temperature of 1–2(T1 + T2). 2-11-2 Thermal Resistance Defining heat transfer power as Qt, thermal resistance is defined as T1 – T2 Rth = ᎏ Qt

(2-42)

where Qt is expressed in watts or Btu/hr. For a flat plate with a thickness path length L and an area A, and if heat transfer occurs by conduction and kth is the thermal conductivity of the material, the resistance factor Rth is: L Rth = ᎏ kthA

(2-43)

For a layer of insulation around a pipe, this equation is expressed in terms of the inner and outer radius of the insulation layer: ln(RO/RI) Rth = ᎏ 2␲kthL

(2-44)

2-11-3 The R Value One term commonly used by the industry is the thermal resistance per unit area or R value.

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T1 – T2 R Value = ᎏ Rth A Qt /A

2.61

(2-45)

2-11-4 The Specific Heat or Heat Capacity C␳ The specific heat capacity is defined as the energy required to increase the temperature of a unit mass by a unit degree and is calculated as Qt = C␳ m⌬T

(2-46).

2-11-5 Characteristic Length Characteristic length is defined as the ratio of the volume to its surface area and is calculated as V Lc = ᎏ As

(2-47)

2-11-6 Thermal Diffusivity Thermal diffusivity is a measure of the speed of propagation of a specific temperature into a solid. The higher the diffusivity, the faster the material will reach a certain temperature. Thermal diffusivity is calculated as kth ⬀= ᎏ ␳eC␳

(2-48)

where ␳e = thermal resistivity (⍀-cm or ⍀-in) ⬀ = diffusivity (m2/s or ft2/hr) Kth = conductivity (W/m-°K or Btu-ft/hr-ft2-°F) C␳ = specific heat capacity (J/kg°K – Btu/lbm-°F) 2-11-7 Heat Transfer Heat transfer is essentially a transmission of energy from one body to another in a period of time. For this reason, it has the same unit as power in SI units, i.e., the watt. In USCS units Btu/hr is used. However, many equations ignore the time factor. Heat transfer per unit area qta is often used so that the total heat transfer Qt over an area A is calculated as Qt = qta A Qt = mC␳⌬T where m = the mass of the body ⌬T = the temperature change or power or rate of heat transfer The rate of heat transfer or power associated with the flow is expressed as Pwt = ␳QC␳⌬T

(2-49)

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Heat transfer can take different forms when slurry is stored in tanks, varies in thickness, or flows in pipes. In the northern climates, loss of heat can lead to frozen pipelines. In the hot climates, the heat absorbed from the sun leads to expansion of plastic lines and significant pipe stresses.

2-12 CONCLUSION In this chapter, some very important principles regarding water flows were introduced. Since water is the principal carrier of slurry mixtures, the tools developed in this chapter such as hydraulic friction gradients and methods to correlate the friction velocity with the friction factor will be extensively used for pipe flow and open channel flow of heterogeneous mixtures (Chapters 4 and 6). This chapter discussed some specific valve types such pinch, rubber sleeve, and check valves. These valves have their own experimental loss coefficients, which need to be obtained from manufacturers. This chapter presented the conventional K and the new two-K loss factors. The two-K factor as developed by Hooper is of particular importance for slurry flows at low Reynolds numbers. The engineer should therefore avoid the common pitfall of using published data on turbulent water flows for conventional waterworks valves when estimating the losses in a slurry system.

2-13 NOMENCLATURE A As C Cc Cd C␳ Cv Cve din DH Di Dij E EA EE Ef Em Ep Ev Ez fD fN Fr F12 g

Cross-sectional area of the flow Surface area Hazen–Williams roughness factor Coefficient of contraction Discharge coefficient Specific heat or heat capacity Valve coefficient Velocity coefficient Pipe diameter expressed in inches Hydraulic diameter = 4A/P Conduit inner diameter (m) Inner diameter of the pipe j Energy per unit mass Energy added per unit mass Energy extracted per unit mass Energy due to friction loss per unit mass Energy lost due to fittings per unit mass Energy due to static pressure per unit mass Energy due to dynamic pressure per unit mass Potential energy per unit mass due to elevation above a reference point Darcy friction factor Fanning friction factor Friction force Force between points 1 and 2 Acceleration due to gravity (9.8 m/s2)

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gc h Hf Hv kth Kf L Lc Le Lj m P Ppsi Pwt Q Qgpm Qideal qth r RI RH R Ri RO Rth Re S S.G. T TDH u U Uf Umax VO Vth W y+ ZA ␥ d␥/dt ⬀ ␦ ␧ ␮

␲ ␪ ␳ ␳e ␶

2.63

Conversion factor between slugs and lbm or 32.2 ft/sec2 Spacing between plates Head loss due to friction Head loss in the Hazen–Williams formula Conductivity Pressure loss of the fitting f Length of conduit or pipe Characteristic length Entrance length Length of the conduit j The mass of the body Pressure Pressure in psi Rate of heat power transfer Flow rate (m3/s) Flow rate expressed in US gallons per minute Ideal flow rate through an orifice as product of area and velocity Heat transfer per unit area local radius Radius at the inner wall of the pipe, or inner radius in an annular flow Hydraulic radius = area/perimeter Resistance factor for thermal insulation is the pipe inner radius (at the inside wall of the pipe) Outer radius in an anuular flow thermal factor Reynolds number Slope or head per unit length Specific gravity Average temperature Total dynamic head that a pump is required to develop Velocity of the flow at distance y Average speed of a flow outside the boundary layer Friction velocity Maximum speed in the boundary layer Practical velocity across an orifice due to vena contracta Theoretical velocity across an orifice weight The relative distance from the wall in the boundary layer Elevation of a point above a reference grade Shear strain Wall shear rate or rate of shear strain with respect to time Diffusivity The thickness of the boundary layer ␦ Linear roughness (m) Carrier liquid absolute or dynamic viscosity (usually expressed in Pascal-seconds or poise) Pythagoras number (ratio of circumference of a circle to its diameter) Duration of the shear for a time-dependent fluid Density in kg/m3 or slug/ft3 Thermal resistivity Shear stress at a height y or at a radius r

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Wall shear stress kinematic viscosity (defined as absolute viscosity divided by density)

2-14 REFERENCES Hooper W. B. 1992. Fittings, Number and Types. pp. 391–397 of The Piping Design Handbook, Edited by J. J. McKetta. New York: Marcel Dekker. Ingersoll Rand. 1977. The Cameron Hydraulic Handbook. Ner Jersey: The Ingersoll Rand Company. Johnson, M. 1982. Non-Newtonian Fluid System Design. Some Problems and Their Solutions. Paper read at the 8th International Conference on the Hydraulic Transport of Solids in Pipe, Johannesburg, South Africa. Lindeburg, M. R. 1997. Mechanical Engineering Reference Manual. Belmont, CA: Professional Publications Inc. Schlichting, H. 1968. Boundary Layer Theory, 6th ed. New York: McGraw-Hill. The Hydraulic Institute.1990. Engineering Data Book. Cleveland, OH: The Hydraulic Institute. Wasp E., J. Penny, and R. Handy. 1977. Solid–Liquid Flow Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans Tech Publications.

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CHAPTER 3

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS

3-0 INTRODUCTION The physical principles of flow of complex mixtures are based on the interaction between the different phases, which may mix well or move in superimposed layers. In this chapter, the basic concepts of motion of particles in a carrying fluid will be presented, as well as the effect of their concentrations and boundaries. In the previous two chapters, we reviewed the physical properties of solids, single-phase flows, and some aspects of mixtures of both. Concepts of non-Newtonian mixtures are reviewed so the reader can understand the principles used to analyze complex homogeneous flows of very fine particles at high volumetric concentration. The physics of solid–liquid mixtures have been the subject of many publications, particularly by chemical and nuclear engineers. In this chapter, an effort is made to focus on the practical equations that a slurry engineer may use to accomplish his/her tasks. The engineer may have to use more than one equation when assessing a mixture to make an engineering judgment.

3-1 DRAG COEFFICIENT AND TERMINAL VELOCITY OF SUSPENDED SPHERES IN A FLUID One fundamental aspect to the transportation of solids by a liquid is the resistance, called the drag force, that such solids will exert, and the ability of the liquid to lift such solids, called the lift force. Both are complex functions of the speed of the flow, the shape of the solid particles, the degree of turbulence, and the interaction between particles and the pipe. One approach is to look at a vehicle that we have all come to know—the airplane. This distraction from the complex world of slurry flows is justifiable. 3-1-1 The Airplane Analogy When an airplane flies in a horizontal plane, it is subject to the forces of downward gravity, upward lift, and drag opposite to its flight path. To maintain steady flight, its engines 3.1

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must develop sufficient thrust to overcome drag. The airplane must also fly above its stalling speed. The lift and drag are aerodynamic forces (Figure 3-1). They are proportional to the surface area, the density of air, the inclination of the airplane body with respect to speed, and the square of the speed. For the airplane wing, these forces are expressed as L = 0.5 CLV 2Sw

(3-1)

D = 0.5 CDV Sw

(3-2)

2

where = density of the fluid V = cruising speed of airplane CL = lift coefficient of wing airfoil CD = drag coefficient of wing airfoil The aerodynamic drag consists of two components: the profile drag and induced drag. The induced drag is proportional to the square of the lift. Airfoils are designed to maximize the lift-to-drag ratio, or to develop the most lift at the least drag penalty: CD = CD0 + kwC L2

(3-3)

where CD0 = the profile drag kw = a function of the shape of the wing (minimum for an elliptical wing and for a wing flying in ground effect) The value of the drag and lift coefficients are determined by the shape of the flying ob-

Thrust

Buoyancy Drag

Wing lift Drag Stabilizer lift

Weight

Thrust Weight

Weight

Forces on an aircraft in steady horizontal flight

Drag

Forces on a rocket in vertical flight

Forces on a free-falling particle immersed in a fluid

FIGURE 3-1 Lift and drag forces on moving objects.

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3.3

ject, but also by the physical properties of a fluid, particularly the density, viscosity, and speed of motion. Nondimensional analysis, an important branch of fluid dynamics, allows the expression of these relationships by characteristic numbers. The Reynolds Number has already introduced in Chapter 2. For an airplane in a steady horizontal linear flight, the lift must overcome weight and the thrust drag. A rocket flying in a vertical plane must develop sufficient thrust to overcome drag forces as well as weight: L = W and T = D T=W+D

For an Airplane For a rocket in vertical flight

3-1-2 Buoyancy of Floating Objects The principle of Archimedes is well known. It states that the buoyancy force developed by an object static in a fluid is equal to the weight of liquid of equivalent volume occupied by the object. When the density of the object is less than the density of the liquid, the object floats, and in the inverse situation, the object sinks. For a sphere immersed in a fluid of density L, the buoyancy force is calculated from the weight of fluid the particle displaces: FBF = (/6)d g3L g

(3-4)

where FBF = buoyancy force dg = sphere diameter g = acceleration due to gravity (9.78–9.81 m/s2)

3-1-3 Terminal Velocity of Spherical Particles Although most solids are not spherical in shape, the sphere is the point of reference for the analysis of irregularly shaped solids. 3-1-3-1 Terminal Velocity of a Sphere Falling in a Vertical Tube When a sphere is allowed to fall freely in a tube, the buoyancy and the drag forces act vertically upward, whereas the weight force acts downward. At the terminal or free settling velocity, in the absence of any centrifugal, electrostatic, or magnetic forces W = D + FBF

(3-5)

d 2g

冢 6 冣d g = 冢 6 冣d g + 0.5 C V 冢 4 冣 3 g S

3 g L

D L

2 t

(3-6)

The drag coefficient corresponding to free fall of the particle is calculated as 4(S – L)gdg CD = 3LV t2 where dg = sphere diameter g = acceleration due to gravity, typically 9.8 m/s2 or 32.2 ft/sec2

(3-7)

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Vt = the terminal (or free settling) speed s = the density of the solid sphere in kg/m3 or slugs/ft3 L = the density of the liquid The terminal (or sinking) velocity is measured using a visual accumulation tube with a recording drum. Various mathematical models have been derived for the drag coefficient. Turton and Levenspiel (1986) proposed the following equation: 0.413 24 ) CD = (1 + 0.173Re 0.657 p Rep 1 + 1.163 × 104Re –1.09 p

(3-8)

Example 3-1 Using the Turton and Levenspiel equation, write a small computer program in quickbasic to tabulate the drag coefficient of a sphere. LPRINT “ Drag coefficient vs. Reynolds Number based on Turton, R., and O. Levenspiel” RE0= 1 15 FOR I=1 TO 10 RE=I*RE0 CD= (24/RE) * (1+0.173*RE^0.657)*(0.413/(1+11630*RE^-1.09) PRINT USING “RE= ###### ; Cd = ##.#### “; RE,CD NEXT I IF RE>1E6 THEN GOTO 30 RE0=RE

TABLE 3-1 Particle Reynolds Number and Corresponding Drag Coefficient for a Sphere Based on the Equation of Turton and Levenspiel (1986) as per Example 3-1 Particle Reynolds number, Rep 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70

Drag coefficient, CD

Particle Reynolds number, Rep


Particle Reynolds number, Rep


28.1520 15.2735 10.8485 8.5809 7.1908 6.2459 5.5588 5.0349 4.6211 4.2851 2.6866 2.0940 1.7729 1.5670 1.4216 1.3124

80 90 100 200 300 400 500 600 700 800 900 1,000 2,000 3,000 4,000 5,000

1.2266 1.1571 1.0994 0.5025 0.6793 0.6085 0.5617 0.5281 0.5029 0.4832 0.4675 0.4547 0.3990 0.3878 0.3883 0.3927

6000 7000 8,000 9,000 10,000 20,000 30,000 40,000 50,000 60,000 70,000 80,000 90,000 100,000 200,000 300,000

0.3983 0.4042 0.4151 0.4151 0.4200 0.4497 0.4617 0.4671 0.4697 0.4709 0.4713 0.4713 0.4711 0.4707 0.4653 0.4609

Page 3.5

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0.6

4

0.4

2

0.2

8000

2000

80

100

60

40

20

6000

0

0

4000

30

6

CD

CD


Rep

25

4

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Rep 0.6 0.4 0.2

0.6

0.2

Rep

Rep

3X10

1X10

5

0 2 4 6 8 10

0.4 5

Rep

CD

3

10

800

600

400

200

0

0

5

Rep

0.2

5

1X10

8X10

0.4

6X10

2X10

10

4

0

0.6

4

0.8

4X10

1.0

4

15

CD

1.2

4

20

CD

Drag Coefficient CD

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FIGURE 3-2 Drag coefficient of a sphere for Reynolds number smaller than 300,000.

GOTO 15 30 END Results are tabulated in Table 3-1 and presented in Figure 3-2 in a linear scale rather than a logarithmic scale. Linear scales are sometimes more useful to the mine operator who is in a remote area and has little time to waste on difficult logarithmic graphs 3-1-3-2 Very Fine Spheres For small particles in the range of a diameter d50 < 0.15 mm (0.0059 in), the most common equation was created by Stokes and reported by Herbich (1991) and Wasp et al. (1977), who indicate that the main forces are due to the viscosity effect in the laminar flow regime: D = 3dg

(3-9)

In the laminar regime, the drag coefficient is inversely proportional to the Reynolds number, i.e., CD = 24/Rep. The terminal velocity is expressed by Stoke’s equation: (S – L)d g2g Vt = 18L

(3-10)

Stokes’s equation is limited to particle Reynolds numbers smaller than 0.1, but has often been used for particle Reynolds Numbers as large as 1 (based on sphere diameter dg).

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From Equation 3-10, Herbich (1968) pointed out that the radius of particles for which the validity of the equation is in doubt is expressed as 4.52L R= (S – L)

冤

冥

3/2

This equation is not set in stone for all situations. Rubey (1933) demonstrated one example by showing that Stoke’s law does not apply to spherical quartz suspended in water when the particle diameter exceeds 0.014 mm (0.00055 in, mesh 105). 3-1-3-3 Intermediate Spheres For the range of particle Reynolds numbers between 1 and 1000, i.e., when dpV0 1 < < 1000 Govier and Aziz (1972) reported that Allen (1900) derived the following equation: ( – L)g Vt = 0.2 L

冤

冥

0.72

d 1.18 p (/)0.45

(3-11)

Example 3-2 A slurry mixture consists of fine rocks at an average particle diameter of 140 m, with a particle density of 2800 kg/m3. The carrier liquid is water with a dynamic viscosity of 1.5 × 10–3 Pa · s. The volumetric concentration of the solids is 12%. Determine the terminal velocity of the particles. Solution Using Equation 1-9, the dynamic viscosity of the mixture is

m = L[1 + 2.5C + 10.05C 2 + 0.00273 exp(16.6C)] = 1.5 × 10–3[1 + 2.5 × 0.12 + 10.05(0.12)2 + 0.00273 exp (16.6 × 0.12)]

m = 2.197 × 10–3 Pa · s. Let us check the magnitude of the Reynolds number: dV0 140 × 10–6 × 0.02504 × 2800 = = 4.468 2.197 × 10–3 The Allen law applies in a transition regime: (140 × 10–6)1.18 Vt = 0.2 [9.81 × 1.8]0.72 (2.197 × 10–3/2800)0.45 2.83 × 10–5 Vt = 0.2 × 7.903 0.001789 Vt = 0.02504 m/s

Richards (1908) demonstrated that Stokes’s equation is inaccurate for particles with a diameter larger than 0.2 mm (0.00787 in, mesh 70) and conducted extensive tests for

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quartz particles (with a specific gravity of 2.65) in laminar, transitional, and turbulent regimes. He derived the following equation for terminal velocity in mm/s: 8.925 Vt = dg{[1 + 95(S/L – 1)d g3]1/2 – 1}

(3-12)

Where dg, the diameter of the sphere, is expressed in mm. This equation covers the range of particles between 0.15–1.5 mm (0.0059–0.059 in) at particle Reynolds numbers between 10 and 1000. 3-1-3-4 Large Spheres For particles with a diameter in excess of 1.5 mm, Herbrich (1991) expressed the terminal velocity by the following equation: Vt = Kt 兹[d 苶苶苶苶苶L苶–苶苶)] 1苶 g( S/

(3-13)

where Kt = an experimental constant = 5.45 for Rep > 800, according to Govier and Aziz (1972). Equation 3-13 is often called Newton’s law. In the regime of Newton’s law, the drag coefficient of a sphere is approximately 0.44, as shown in Figure 3-2. Newton’s law applies to turbulent flow regimes. Other equations for terminal velocity of particles have been developed by various authors. Four different equations are presented in Table 3-2. Example 3-3 Using the Budyruck equation from Table 3-3, determine the terminal velocity of spheres from 0.1 to1 mm. A simple computer program is written in quickbasic as follows: LPRINT LPRINT “BUDRYCK AND RITTINGER EQUATION FOR TERMINAL VELOCITY OF SPHERES IN WATER” LPRINT LPRINT DP0 = .1 FOR I=1 to 11 DP = I*DP0 VS= (8.925/DP)*(SQR(1+157*DP^30-1) LPRINT USING “PARTICLE DIAMETER = ##.### mm TERMINAL VELOCITY Vs = ##.### mm/s”;DP,VS NEXT I FOR J=12 TO 20 DP = J*DP0

TABLE 3-2 Equations for Terminal Speed of Large Spheres Name

Equation*

Application

Budryck Rittinger

Vt = 8.925[(1 + 157d g3)1/2 – 1]/dg Vt = 87(1.65dg)1/2

For dg < 1.1 mm For 1.2 < dg < 2 mm

*Where Vt is expressed in mm/s and dg in mm.

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TABLE 3-3 Calculation of Terminal Velocity of Spheres in Accordance with Budryck’s Equation Particle diameter dp in mm

Terminal velocity Vs in mm/s

Particle diameter dp in mm

Terminal velocity Vs in mm/s

0.1 0.2 0.3 0.4 0.5 0.6

6.75 22.4 38.34 51.85 63.21 73.02

0.7 0.8 0.9 1.0 1.1

81.63 89.49 96.64 103.26 109.45

VS= 87*SQR(1.65*DP) LPRINT USING “PARTICLE DIAMETER = ##.### mm TERMINAL VELOCITY Vs = ##.### mm/s”;DP,VS NEXT J END The results are shown in Tables 3-3, 3-4, and Figure 3-3 Herbich (1968) measured drag coefficients for ocean nodules to be as high as 0.6 at particle Reynolds numbers of 200. This high value is reached with spheres at a particle Reynolds number of 1000. 3-1-4 Effects of Cylindrical Walls on Terminal Velocity The previous paragraphs focused on the settling velocity of a single particle or widely separated particles. The presence of a vessel or cylindrical walls tends to multiply the interaction between particles and cause some collisions. Extensive tests have been conducted on flows in vertical tubes. Brown and associates (1950) recommended multiplying the terminal speed of a single particle by a wall correction factor Fw. For laminar flows they proposed to use the Francis equation: Fw = 1 – (d/Di)9/4

(3-14a)

They proposed to use the Munroe equation for a turbulent flow regime: Fw = 1 – (d/Di)1.5

(3-14b)

where Di = the inner diameter of the tube

TABLE 3-4 Calculation of Terminal Velocity of Spheres in Accordance with Rittinger’s Equation Particle diameter dp in mm

Terminal velocity Vt in mm/s

Particle diameter dp in mm


1.1 1.2 1.3 1.4 1.5

117.21 122.42 127.42 132.23 136.87

1.6 1.7 1.8 1.9 2.0

141.36 145.71 149.93 154.04 158.04

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0.01

0.02

0.03

0.04

0.05

Sphere diameter d p in inches

120

5

4

100 80

3

60 2 40 1 20 0

0 0

0.2

0.4

0.6

0.8

1.0

Te rminal velocity Vt in inch/sec


0

1.2

Sphere diameter d p in mm (a)

0.04

0.05

0.06

0.07

0.08

160

6

140 5 120 100

4

80

3

60 2 40 1.0

1.2

1.4

1.6

1.8

2.0

1

Terminal velocity Vt in inch /sec

Sphere diameter d p in inches Terminal velocity Vt in mm/s

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Sphere diameter d p in mm (b) FIGURE 3-3 Terminal velocity of spheres (a) in accordance with Budryck’s equation, (b) in accordance with Rittinger’s equation.

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CHAPTER THREE

Example 3-4 The flow described in Example 3-2 occurs in a 63 mm ID pipe. Determine the corrected terminal velocity due to the wall effects. Solution The terminal velocity was determined to be 0.02504 m/s. The flow is in transition. Equation 3-14a for laminar flow is Fw = 1 – (d/DI)9/4 Fw = 1 – (0.140/63)9/4 Fw = 0.999 Equation 3-14b for turbulent flow is Fw = 1 – (0.14/63)1.5 = 0.999. More recently, Prokunin (1998) extended the analysis of the interaction of the wall with the motion of a single particle by considering the angle of inclination and any rotation that the particle may incur. His investigation included immersion in non-Newtonian flows by testing with glycerin and silicone. He noticed from his tests that when the particle approaches the wall, it develops a lift force. The lift force seems to increase with a reduction of the gap that separates the particle from the wall. However, Prokunin could not explain this lift force and recommended further research. 3-1-5 Effects of the Volumetric Concentration on the Terminal Velocity As the volumetric concentration of particles increases, it causes interactions and collisions, and transfers momentum between the different (finer and coarser) units. The distance between particles decreases. For spheres at 1% concentration by volume, the interparticle distance is only 4 diameters. It shrinks to 2.5 diameters at 5% and to 2 diameters at 10% concentration by volume. In an ideal laminar flow, the interaction is much simpler than in a turbulent flow. Worster and Denny (1955) published data on the terminal velocity of coal and gravel particles, as shown in Table 3-5. The effect of the concentration is clearly marked by a difference in terminal velocity between a single particle and a volumetric concentration of 30%. Kearsey and Gill (1963) applied the Carman–Kozeney equation of flow through a porous medium to determine the terminal velocity as

TABLE 3-5 Terminal Velocity for Coal and Gravel after Worster and Denny (1955) Coal with a specific gravity of 1.5 ________________________________ Particle size Single particle 30% Concentration ____________ ______________ ________________ mm Inches (cm/s) (ft/s) (cm/s) (ft/s) 1.59 6.4 12.7 25.4

1/16 1– 4 1– 2

1

4.6 15.2 30.5 51.8

0.15 1.50 1.00 1.70

3.0 10.7 21.3 36.6

0.10 0.35 0.70 1.20

Gravel with a specific gravity of 2.67 ________________________________ Single particle 30% Concentration ______________ ________________ (cm/s) (ft/s) (cm/s) (ft/s) 9.1 30.5 61.0 106.7

0.30 1.00 2.00 3.50

3.0 10.7 21.3 36.6

0.10 0.35 0.70 1.20

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冤

(1 – Cv)3 Vc = KzC v2

P

冥冤 s 冥冤 L 冥 1

2 p

3.11

(3-15)

where sp = the specific surface expressed for as sphere as the surface area to volume ratio:

d 2g = 6/dg sp = (d 3g/6) Kz = the Kozney constant, which is a function of particle shape, porosity, particle orientation, and size distribution. The magnitude of Kz is between 3 and 6, but is commonly assumed to be 5 P/Li = the pressure gradient in the pipe due to the flow of the mixture In the process of sedimentation, the pressure gradient is essentially due to the volumetric concentration of the particles and is expressed as P = Cv(s – L)g Li

(3-16)

In addition, the settling velocity due to a volumetric concentration is expressed as

冤

(1 – Cv)3g Vc = KzCv

(s – L)

冥冤冥 s 2 p

(3-17)

For spheres with sp = 6/dg, the equation reduces to

冤

(1 – Cv)3gd 2g Vc = 36KzCv

(s – L)

冥冤冥

(3-18)

As the volumetric concentration increases from 3% to 30%, the velocity drops drastically. Assuming Kz to be equal to 5.0, the settling velocity for spheres reduces to a simple equation: (1 – Cv)3 Vc = V0 10Cv

(3-19)

where V0 = the terminal velocity at very low volumetric concentration Equation 3-19 does not apply to volumetric concentrations smaller than 8%. Equation 3-18 would apply to smaller concentrations. Example 3-5 Assuming that the terminal velocity at a volumetric concentration of 8% is 100 mm/s, apply Equation 3-18 from a volumetric concentration of 8–30%. Plot the results in Figure 3-4. Thomas (1963) proposed the following empirical equation in the range of Vc/V0 of 0.08–1.0: 2.303 log10(Vc/V0) = –5.9CV

(3-20)

Example 3-6 The free settling speed of solid particles is 22 mm/s at a volumetric concentration of 1%. Using the Thomas equation 3-20, determine the settling speed at 25% volumetric concentration.

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Vc / Vo

CHAPTER THREE

1.0 0.8 0.6 0.4 0.2 0

0

0.2 0.1 Volumetric concentration

0.3

FIGURE 3-4 Effect of the volumetric concentration on the terminal velocity of spheres in accordance with Equation 3-18.

Solution 2.303 log10(Vc/V0) = -5.9 × 0.25 Vc/V0 = 10–0.64 Vc/V0 = 0.2288 Vc = 0.2288 × 22 mm/s = 5.03 mm/s The Kozney-based approach is limited to concentrations where the particles come into contact with each other in a vertical flow. Beyond this point, the pressure gradient is smaller than expressed by Equation 3-16. In the case of hard spheres, the settling process completes when the particles come into contact with each other. In the case of flocculated particles or clusters of flocculated fluid, stress may cause deformation and further settling may occur by compaction. Irregularly shaped particles and flocculates cause the development of a structure with its own yield stress level. As the particles move closer, the yield stress increases until equilibrium is reached. The weight of the overburden is then supported by the saturated fluid and the compacted sediment.

3-2 GENERALIZED DRAG COEFFICIENT— THE CONCEPT OF SHAPE FACTOR Every day the slurry engineer has to deal with particles of all shapes and sizes. Although the sphere represents a shape for reference, it is in the minority in the world of crushed or naturally worn rocks. Albertson (1953) conducted an extensive study on the effect of the shape of gravel particles on the fall velocity in a vertical flow (Figure 3-5). He proposed a definition for a shape factor: c A = 兹(a 苶b苶)苶 where a = the longest of three mutually perpendicular axes b = the third axis c = the shortest of three mutually perpendicular axes

(3-21)

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a

l fal f no tio c e dir

c

b

FIGURE 3-5 The axes of an irregularly shaped particle, according to Albertson.

Particles in a free fall tend to align themselves to expose the largest surface to the flow. In other words, they act as free-falling leaves from a tree on an autumn day, where c is taken as the dimension opposite to the direction of the fall. The projected area of the particle is a function of the dimensions “a” and “b” but is often not equaled to such a product as (ab) because particles are usually not rectangular in shape (see Table 3-6). In a different approach, Clift et al. (1978) decided to compare the projected area of a free-falling, irregularly shaped particle, with a sphere of equal projected area in order to define a diameter: da = 兹(4 苶S 苶苶苶)苶 f /

(3-22)

where Sf = the projected area of the free-falling particle However, Albertson (1953) preferred to define a different diameter base, dp, on the fact that the actual volume of the free-falling particle could be equated to a sphere of the

TABLE 3-6 Clift Shape Factor of Various Particles Isometric ____________________________________ Particle c Sphere Cube Tetrahedron Irregular Rounded Cubic angular Tetrahedral

From Wilson et al. (1992).

0.524 0.694 0.328 0.54 0.47 0.38

Typical mineral particles _______________________________________ Particle c Sand Sillimanite Bituminous Coal Blast Furnace Slag Limestone Talc Plumbago Gypsum Flake Graphite Mica

0.26 0.23 0.23 0.19 0.16 0.16 0.16 0.13 0.023 0.003

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same volume but with a diameter of dn. Albertson (1953) therefore proposed a Reynolds number based on dn: dnVt Ren =

(3-23)

There may be a marked difference between naturally worn gravel and crushed gravel. This is a fact that a slurry engineer should bear in mind when extrapolating data from lab results. Because Clift chose an equivalent diameter da based on the projected area, he proposed a different shape factor:

c = particle volume/d a3

(3-24)

Typical values are shown in Table 3-6. The Albertson and Clift shape factors are about 40 years apart in definition but can be related by a factor E:

c = EA

(3-25)

The logarithmic curves as shown in Figure 3-6 are sometimes difficult to read. Table 3-7 presents values of drag coefficient versus Reynolds number rounded off to the first decimal point. The work of Albertson was developed further by the Inter-Agency Committee on Water Resources (1958), who developed the following two non-dimensional coefficients (Figure 3-7): CN = (s/L – 1)g/V t3

(3-26a)

CN = 0.75CD /Ren

(3-26b)

CS = (s/L – 1)gd 3p/(62)

(3-27a)

CS = 0.125CD Re2n

(3-27b)

and

ALBERTSON SHAPE FACTOR = a/ cb

Drag coefficient CD

10.0 0.3 0.5 0.7

1.0

1.0

0.1

00

10 10

100 100

33

10 10

4

10 10

4

55

10 10

6

10 10

6

Particle Reynolds number Rep

FIGURE 3-6 The drag coefficient versus Reynolds number and shape factor. (After Albertson, 1953.)

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3.15

TABLE 3-7 Drag Coefficient versus Reynolds Number for Different Albertson Shape Factors Drag coefficient Reynolds number 7 8 9 10 15 20 32 40 50 60 70 80 100 150 200 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000

Shape factor = 0.3 7.0 6.5 6.1 5.8 4.64 3.95 3.0 2.7 2.5 2.3 2.25 2.2 2.08 1.87 1.75 1.74 1.8 1.9 1.94 1.988 2.0 2.07 2.1 2.3 2.28 2.48 2.21 2.2 2.19 2.183 2.18

Shape factor = 0.5 Shape factor = 0.7 Shape factor = 1.0 6.0 5.5 5.1 4.74 3.7 3.2 2.6 2.28 2.08 1.94 1.74 1.67 1.62 1.44 1.36 1.33 1.34 1.38 1.42 1.47 1.51 1.54 1.58 1.72 1.73 1.69 1.66 1.62 1.58 1.55 1.53

4.7 4.3 4.0 3.75 3.0 2.55 2.1 1.84 1.67 1.56 1.4 1.35 1.3 1.16 1.11 1.08 1.09 1.1 1.12 1.14 1.15 1.16 1.17 1.22 1.19 1.16 1.14 1.13 1.13 1.14 1.14

4.0 3.7 3.4 3.15 2.4 2.0 1.55 1.3 1.12 1.0 0.94 0.844 0.8 0.68 0.6 0.5 0.44 0.4 0.38 0.36 0.34 0.334 0.33 0.3 0.29 0.294 0.3 0.31 0.31 0.32 0.32

The drag coefficient CD is then plotted against the equivalent Reynolds number Ren to determine the terminal velocity. On a logarithmic scale, CN and CS are superposed as straight lines for reference (Figure 3-7). In order to measure the Albertson shape factor, Wasp et al. (1977) developed a correlation between the sieve diameter and the fall diameter dn (Figure 3-8). The approach proposed by Albertson and Clift is limited to free fall of particles in a fluid. However, turbulence can develop new forces. Whenever an engineering contract requires the drag of particles to be measured, the engineer is well advised to conduct tests in a fluid of similar dynamic viscosity as the one that will be used in the project. In addition to the shape factor and drag coefficient, the slurry engineer must also determine the fluid density, dynamic viscosity at the temperature of pumping, particle density (or specific gravity of solids), nominal (or statistical average) diameter, and fall velocity.

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CHAPTER THREE

0.8

S. F

6 5 4 3

sp

he

2

he

0.4

re

re

s

s

0.6

0.2 0

= 0.9

S .F = 0.3 S. F = 0. 5 S. F = 0.7

Sieve diameter (mm)

S. F S . F = 0.3 =0 .5 S. F= S. F 0.7 =0 .9

1.0

7

sp

Sieve diameter (mm)

FIGURE 3-7 CD and CW versus particle Reynolds number for different shape factors. Adapted from the Inter-Agency Committee on Water Resources (1958).

1 0.2

0.4

0.6

0.8

Fall diameter (mm)

1.0

0

1

2

3

4

Fall diameter (mm)

FIGURE 3-8 Relationship between sieve and fall diameter after Wasp et al. (1977).

5

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3.17

Example 3-7 A naturally worn particle has an Albertson shape factor of 0.7. It has a nominal diameter of 250 m. Its density is 3000 kg/m3. It is allowed to free-fall in water at a temperature of 25° C. Calculate the fall velocity for the single particle and the fall velocity if the volumetric concentration of particles is increased to 20%. Solution Referring to Table 2-7 (or Table 2-8 for USCS units), the kinematic viscosity of water is 0.89 × 10–6 m2/s. We need to determine the coefficient CS = 0.125CD/Ren2. The curves published by Inter-Agency Committee on Water Resources indicate that CS = 0.125CD/Ren2 = 0.167(s/L – 1)gd p3/2 = 203. From Figure 3-6, at a shape factor of 0.7 and CS of 203, the Reynolds number would be 7.2Vt = Re/(dp) = 7.2/(890,000 × 0.00025) = 0.0324 m/s for a single particle. Applying Equation 3-18 for a concentration of 20%, the velocity would be 0.256 × 0.0324 = 0.0083 m/s.

3-3 NON-NEWTONIAN SLURRIES Various models have been developed over the years to classify complex two- and threephase mixtures (Table 3-8). In the case of mining, the following mixtures are often encountered: 앫 A fine dispersion containing small particles of a solid, which are uniformly distributed in a continuous fluid and are found in copper concentrate pipelines and in slurry from grinding after classification, etc.

TABLE 3-8 Regimes of Flows for Newtonian and Non-Newtonian Mixtures after Govier and Aziz (1972) Multiphase flows (gas–liquid, liquid–liquid, Single-phase flows gas–solid, liquid–liquid) ___________________________ ___________________________________________________ Single-phase behavior _____________________________________________________ Multiphase behavior ___________________________ Pseudohomogeneous Heterogeneous _______________________________ __________________ True homogeneous Laminar, transition, and Turbulent flow regime only turbulent flow regime Purely viscous

Newtonian flows

Purely viscous, non-Newtonian, and time-independent

Bingham plastic Dilatant Pseudoplastic Yield pseudoplastic

Purely viscous, non-Newtonian and time-dependent

Thixotropic Rheopectic

Viscoelastic

Many forms

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앫 A coarse dispersion containing large particles distributed in a continuous fluid and encountered in SAG mills, cyclone underflows, and in certain tailings lines, etc. 앫 A macro-mixed flow pattern containing either a frothy or highly turbulent mixture of gas and liquid, or two immiscible liquids under conditions in which neither is continuous. Such patterns are found in flotation circuits in which froth is used to separate concentrate from gangue. 앫 A stratified flow pattern containing a gas, liquid, two slurries of different particle sizes, or two immiscible liquids under conditions in which both phases are continuous. Designing a pipeline to operate in a non-Newtonian flow regime must be based on reliable test data about the rheology and particle sizing (see Table 3-9). The engineer must be cautious before venturing into generalizations about rheological properties. In Figure 1-4 of Chapter 1, the relationship between dynamic viscosity and volumetric concentration was presented. In fact, the industry has accepted the criterion that friction losses are highly dependent on slurry viscosity in cases where the average particle diameter is finer than 40–60 microns, and (depending on the specific gravity) at volumetric concentrations in excess of 30%. Fibrous slurries such as fermentation broths, fruit pulps, crushed meal animal feed, tomato puree, sewage sludge, and paper pulp may not contain a high percentage of solids, but may flow as non-Newtonian regimes. With these materials, the long fibers are flexible and intertwine into a close-packed configuration and entrap the suspending medium. The fibers may be flocculated or may form flocs with an open structure. Based on the volume content of the flocs, the mixture may develop high dynamic viscosity. However, because the flocs are compressible, they may deform with the flow. Flocculated slurries are encountered in flotation cells circuits, thickeners, and various processes in mineral extraction plants. With the formation of flocs, the slurry may develop an internal structure. This structure may develop properties leading to a non-Newtonian flow, shear thinning behavior (pseudoplastic), and sometimes thixotropic time-dependent behavior. When shear stresses are applied to the slurry, the floc sizes may shrink and become less capable of entrapping the carrier slurry. At higher shear stresses, the flocs may shrink to the size of particles, and the flow may lose its non-Newtonian behavior.

3-4 TIME-INDEPENDENT NON-NEWTONIAN MIXTURES Certain slurries require a minimum level of stress before they can flow. An example is fresh concrete that does not flow unless the angle of the chute exceeds a certain minimum. Such a mixture is said to posses a yield stress magnitude that must be exceeded before that flow can commence. A number of flows such as Bingham plastics, pseudoplastics, yield pseudoplastics, and dilatant are classified as time-independent non-Newtonian fluids. The relationship of wall shear stress versus shear rate is of the type shown in Figure 3-9 (a), and the relationship between the apparent viscosity and the shear rate is shown in Figure 3-9 (b). The apparent viscosity is defined as

a = Cw/(d/dt)

(3.28)

3-4-1 Bingham Plastics For a Bingham plastics it is essential to overcome a yield stress 0 before the fluid is set in motion. The shear stress versus shear rate is then expressed as

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TABLE 3-9 Examples of Bingham Slurries Yield Stress, Pa

Coefficient of rigidity,

mPa · s (cP)

Particle size, d50

Density, kg/m3

92% under 74 m

1520

80% under 1 m

1280

59

13.1

80% under 1 m

1207

25

6.7

80% under 1 m

1149

7.8

4.0

1520

34.5

44.7

Aqueous clay suspension III

1440

20

32.8

Aqueous clay suspension V

1360

Slurry 54.3% Aqueous suspension of cement, rock Flocculated aqueous China clay suspension No. 1 Flocculated aqueous China clay suspension No. 4 Flocculated aqueous China clay suspension No. 6 Aqueous clay suspension I

3.8

6.65

6.86

19.4

Fine coal @ 49% CW Fine coal @ 68% CW Coal tails @ 31% CW Copper concentrate @ 48% CW 21.4% Bauxite

50% under 40 m 50% under 40 m 50% under 70 m 50% under 35 m < 200m

1163

8.5

4.1

Gold tails @ 31% CW 18% Iron oxide

50% under 50 m < 50 m

1170

5 0.78

87 4.5

7.5 % Kaolin clay Kaolin @ 32% CW Kaolin @ 53% CW with sodium silicate Kimbelite tails @ 37% CW 58% Limestone

Colloidal 50% under 0.8 m 50% under 0.8 m

52.4% Fine liminite Mineral sands tails @ 58% Cw 13.9% Milicz clay 16.8% Milicz clay 19.6% Milicz clay Phosphate tails @ 37% CW 14% Sewage sludge

< 50 m 50% under 160 m

Red mud @ 39% CW Zinc concentrate @ 75% CW Uranium tails @ 58% CW

50% under 15 m < 160 m

1 8.3 2 19

1103

1530 2435

< 70 m < 70 m < 70 m 85% under 10 m 1060 5% under 150 m 50% under 20 m 50% under 38 m

5 40 60 18

7.5 20 6

5 30 15

11.6 2.5

6 15

30 30

16 250

2.3 5.3 13 28.5 3.1

8.7 13.6 25 14 24.5

23 12 4

30 31 15

Reference Hedstrom (1952) Valentik & Whitmore (1965) Valentik & Whitmore (1965) Valentik & Whitmore (1965) Caldwell & Babitt (1941) Caldwell & Babitt (1941) Caldwell & Babitt (1941) Wells (1991) Wells (1991) Wells (1991) Wells (1991) Boger & Nguyen (1987) Wells (1991) Cheng & Whittaker (1972) Thomas (1981) Wells (1991) Wells (1991) Wells (1991) Cheng & Whittaker (1972) Mun (1988) Wells (1991) Parzonka (1964) Parzonka (1964) Parzonka (1964) Wells (1991) Caldwell & Babitt (1941) Wells (1991) Wells (1991) Wells (1991)

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m ha ng i B

stic pla o d seu ld P Yie

c sti Pla

ian on t w stic Ne opla d u Pse

Di lat an t

Shear Stress

CHAPTER THREE

tic as Pl

Apparent viscosity a

m ha ng Bi

Di lat an t

Rate of shear ( = du/dy)

Newtonian

Pse udo plas tic

Rate of shear ( = du/dy) (b) FIGURE 3-9 (a) Shear stress versus shear rate; (b) viscosity versus shear rate of time-independent non-Newtonian fluids.

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w – 0 = d/dt

3.21

(3-29)

where w = shear stress at the wall 0 = yield stress

= the coefficient of rigidity or non-Newtonian viscosity It is also related to a Bingham plastic limiting viscosity at infinite shear rate by the following equation:

0

= + (d/dt)

(3-30)

The magnitude of the yield stress 0 may be as low as 0.01 Pascal for sewage sludge (Dick and Ewing, 1967) or as high as 1000 MPa for asphalts and Bitumen (Pilpel, 1965). The coefficient of rigidity may be as low as the viscosity of water or as high as 1000 poise (100 Pa · s) for some paints and much higher for asphalts and bitumen. In the case of tarbased emulsions or certain tar sands, it is customary to add certain chemicals to reduce the dynamic viscosity of the emulsion or the coefficient of rigidity of the slurry. Tables 3-9 presents examples of Bingham slurries, magnitudes of yield stress, and coefficients of rigidity values. Example 3-8 Samples of a mineral slurry with Cw = 45% are examined in a lab. From the measurements of the rate of shear () and shear stress ( ), determine the yield stress and viscosity. Rate of Shear [s–1] 100 150 200 300 Shear Stress (Pa) 10.93 12.27 13.49 15.68 – 0 (Pa) 4.11 5.45 6.67 8.87

400 17.66 10.85

500 19.49 12.67

600 700 800 21.2 22.84 24.43 14.39 16.03 17.61

The data is plotted in Figure 3-10. At a low shear rate < 100s – 1, the slope is

= 4.426/100 = 0.0443 Pa · s At high shear rate 4.426 = = 0.0164 Pa · s 270

w – 0

= du/dy Take a point at high shear rate (700 s–1): 16.03

= 700

= 0.0229 Pa · s Check at du/dy = 600 14.394

= = 0.02399 600 at du/dy = 800

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CHAPTER THREE

17.61

= 0.022 800 An average = 0.023 Pa · s is taken. Alternative = 0/(du/dy) + a

= 6.82/700 + 0.0164 = 0.026 Pa · s This example shows that at zero rate of shear the shear stress is 6.82 Pa. The yield stress is therefore 6.82 Pa. The yield stress increases as the concentration of solids augments. Thomas (1961) proposed the following relationships between yield stress 0, coefficient of rigidity , concentration by volume Cv, and viscosity of the suspending medium :

0 = K1C v3

(3-31)

/ = exp(K2Cv)

(3-32)

where K1 and K2 = constants and are characteristics of the particle size, shape, and concentration of the electrolyte concentration. These equations were derived from the work of Thomas (1961) on suspensions of titanium dioxide, graphite, kaolin, and thorium oxide in a range of particle sizes from 0.35–13 micrometers and in volume concentration of 2–23%. Thomas (1961) defined a shape factor T1 for nonspherical particles as

T1 = exp[0.7(sp/s0 – 1)]

(3-33)

where sp = the surface area per unit volume of the actual particles s0 = the surface area per unit volume of a sphere of equivalent dimensions or 6/dg

(Pa)

He indicated that the coefficient K1 might then be expressed as

30

Shear stress

28 24 20 16 12 8 4

0

0 0

100

200

300

400

500

600

700

800

Rate of shear FIGURE 3-10 Plot of data for Example 3-8.

900 -1

(sec )

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uT1 K1 = d 2p

3.23

(3-34)

Where K1 is expressed in Pa (or lbf/ft2 with u = 210 in), and the particle diameter dp is expressed in microns. Thomas defined a second shape factor T2 = (sp/s0)1/2 to derive the equation: 苶苶p K2 = 2.5 + 14T 2/兹d

when 0.4 < dp < 20 microns

(3-35)

Thomas (1963) extended his work to flocculated mixtures with dispersed fine and ultrafine particles with overall dimensions up to 115 microns. He derived the following equations:

/ = exp[(2.5 + )Cv]

(3-36)

= 兹[( 苶d苶f苶 /dap 苶苶 –苶] 1苶 p)苶

(3-37)

where where = the ratio of immobilized dispersing fluid to the volume of solids related approximately to the particle and floc apparent diameter df = the apparent floc diameter dapp = the apparent particle diameter This particle diameter is shown by the following: dapp = dp(s0/sp) exp(–1–2 ln2 )

(3-38)

where

= the logarithmic standard deviation In general, and at a constant temperature, the following equations are applied to Bingham plastic slurries:

/ = A exp(BCv)

(3-39)

0 = E exp(FCv)

(3-40)

The constants A, B, E, and F are derived from tests measuring particle size, shape, and the nature of their surface. Gay et al. (1969) proposed the following correlation for high concentrations of solids:

/ = exp{[2.5 + [Cv/(Cv – Cv)]0.48](Cv/Cv)}

(3-41)

where Cv = the maximum packing concentration of solids For a change in temperature in the order of 27°C (50°F). Parzonka (1964) developed the following power law equation:

= K3T a–n

(3-42a)

where n = an exponent K3 = an exponent Ta = absolute temperature Govier and Aziz (1972) proposed an equation based on an exponential drop of Bingham plastic viscosity with temperature:

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CHAPTER THREE

= A exp(B/T)

(3-42b)

To obtain the viscosity, plot the curve of the shear stress ( – 0) in Pascals against the shear rate (s–1). 3-4-2 Pseudoplastic Slurries Pseudoplastic fluids are time-independent non-Newtonian fluids that are characterized by the following: 앫 An infinitesimal shear stress, which is sufficient to initiate motion 앫 The rate of increase of shear stress with respect to the velocity gradient decreases as the velocity gradient increases This type of flow is encountered when fine particles form loosely bound aggregates that are aligned, stable, and reproducible at a given magnitude of shear rate. The behavior of pseudoplastic fluids is difficult to define accurately. Various empirical equations have been developed over the years and involve at least two empirical factors, one of which is an exponent. For these reasons, pseudoplastic slurries are often called power-law slurries. The shear stress is defined in terms of the shear rate by the following equation:

w = K[(d/dt)n]

(3-43)

where K = the power law consistency factor, expressed in Pa · sn n = the power law behavior index, and is smaller than unity Examples of pseudoplastic slurries are shown in Table 3-10. The apparent viscosity of a pseudoplastic is defined in terms of the ratio of the shear stress to the shear rate:

a = [ w/(d/dt)]

(3-44)

3-4-2-1 Homogeneous Pseudoplastics Pseudoplastic slurries are another category of non-Newtonian slurries. Pseudoplastics are divided into homogeneous and pseudohomogeneous mixtures. Whereas in the case of a Bingham slurry, it was pointed out that the coefficient of rigidity was a linear function of the shear rate, in the case of a pseudoplastic, the coefficient of rigidity is expressed by the following power law:

= K(d/dt)n–1

(3-45)

The shear stress is plotted against the shear rate on a logarithmic scale at various volume fractions. From the slope of such a plot, “K,” the power law consistency factor, and “n,” the power law behavior index (smaller than unity) are derived as plotted in Figure 311. As indicated in Figure 3-12 the magnitude “K,” the power law consistency factor, and the power law factor index n are dependent on the volumetric concentration of solids. Example 3-9 A phosphate slurry mixture is tested using a rheogram. The following data describe the relationship between the wall shear stress and the shear rate:

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d/dt w(Pa)

0 25

50 32

100 43

150 51

200 53

300 56

400 58

500 60

600 62

700 63.2

800 64.3

The mixture is non-Newtonian. If it is considered a power law slurry, derive the power law exponent “n” and the power law coefficient K. Solution The first step is to plot the data on a logarithmic scale. In the equation for a pseudoplastic, the coefficient of rigidity is expressed by equations (3.43) and (3.45), the values of “K” and “n.” By using the logarithmic scale: log w = log K + n log (d/dt) log(d/dt) log( w) n

1.699 1.505 —

2 1.633 0.425

2.176 1.707 0.592

2.301 1.724 0.136

2.477 1.748 0.136

2.602 1.763 0.12

2.669 1.778 0.154

2.778 1.792 0.112

2.845 1.8 0.13

2.903 1.808 0.14

log(d/dt)2 – log(d/dt)1 n = (log w)2 – (log w)1 n ⬇ 0.132 1.8 = log K = 0.132 × 2.843 log K = 1.424 K = 26.5 TABLE 3-10 Examples of Power Law Pseudoplastics

Slurry Cellulose acetate Drilling mud—barite Sand in drilling mud

Particle size, d50

Range of weight concentration, %

Graphite Graphite and magnesium hydroxide

16.1 m 5 m

Flocculated kaolin Deflocculated kaolin Magnesium hydroxide Pulverized fuel ash (PFA-P) Pulverized fuel ash (PFA-P)

0.75 m 0.75 m 5 m 38 m

1.5–7.4 1.0–40.0 1.0–15% sand using drilling mud with 18% barite 0.5–5.0 32.2 total (4.1 graphite and 28.1 magnesium hydroxide) 8.9–36.3 31.3–63.7 8.4–45.3 63–71.8

20 m

70–74.4

14.7 m 180 m

Range of consistency coefficient K, Nsn/m2

Angle of flow behavior index, n

Reference

1.4–34.0 0.8–1.3 0.72–1.21

0.38–0.43 0.43–0.62 0.48–0.57

Heywood (1996) Heywood (1996) Heywood (1996)

Unknown

Probably 1

Heywood (1996)

5.22

0.16

Heywood (1996)

0.3–39 0.011–0.6 0.5–68 3.3–9.3

0.117–0.285 0.82–1.56 0.12–0.16 0.44–0.46

Heywood (1996) Heywood (1996) Heywood (1996) Heywood (1996)

2.12–9.02

0.48–0.57

Heywood (1996)

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Shear stress (in units of pressure)

1 0.1 0.01

pe

=

x y/

slo

0

y

K

n

n = y/x

x

0.001 0.0001 0

1

10

100

Shear rate

1000

10,000

(1/sec)

FIGURE 3-11 Plotting the rheology on a logarithmic scale to obtain the consistency factor “K” and the flow behavior index “n” of Pseudoplastics.

Consider d/dt = 700. Check w = K(d/dt)n. 62.9 = 26.5 × 7000.132 This is close to the measured stress of 63.2 Pa. Therefore, the equation of this phosphate slurry is:

w = 26.5(d/dt)0.132 The coefficient of rigidity is obtained as:

1.0

6

ma gne tite

4 2

Flow Behavior Index "n"

clays

8

tite gne ma

Power Law Consistency Factor K Pa.sn/cm 2

10

0.8 0.6 0.4

clays

0.2 0

0 0

20 40 Volume Fraction of solids, CV

0

20 40 Volume Fraction of solids, CV

FIGURE 3-12 Effect of volumetric concentration on the consistency factor “K” and the flow behavior index “n” of Pseudoplastics (after Aziz and Govier, 1972).

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3.27

= K(d/dt)n–1

= 26.5(d/dt)–0.878 at d/dt = 700

= 26.5 × (700)–0.878

= 0.084 Pa · s at d/dt = 600.

= 26.5 × 600 = 0.096 Pa · s 3-4-2-2 Pseudohomogeneous Pseudoplastics Pseudohomogeneous pseudoplastics behave similarly to their homogeneous counterparts. Clay suspensions and magnetite-based slurries demonstrate an exponential relationship between n and Cv as shown in Figure 3-12. The power law factor K has a more complex relationship with Cv, as shown in Figure 3-12. Various equations have been derived to solve the power law factor of pseudoplastics. These equations are presented to help the reader appreciate the rheological constants that must be determined by testing, as will be described in Section 3-6. The Prandtl–Eyring equation is based on Dahlgreen’s (1958) discussion of the study conducted by Eyring and Prandtl on the kinetic theory of liquids:

= A sinh–1[(d/dt)/B]

(3-46)

where A and B = the rheological constants sinh = the hyperbolic function From Equation 3-44, the apparent viscosity is derived as

a = {A/(d/dt)}{sinh–1[(d/dt)/B]}

(3-47)

The Ellis equation is more flexible but is an empirical equation and uses three rheological constants. Skelland (1967) demonstrates how the equation is based on the work of Ellis and Round and is explicit with respect to the velocity gradient rather than the shear rate: (d/dt) = (A0 + A1 ( –1)) w

(3-48)

where A0, A1, and are the rheological coefficients of the slurry material. The apparent viscosity is expressed as

a = 1/(A0 + A1 w( –1))

(3-49)

When A1 = 0, the equation takes on a Newtonian form where A0 = 1/. The equation reduces to the conventional power law equation with = 1/n and A1 = (1/k)1/n. When > 1, the equation approaches a Newtonian flow at low shear stresses, and when < 1, it tends to approach a Newtonian flow at high shear stress. The Cross equation (Cross, 1965) is a versatile equation that is based on measurements of viscosity, 0 at zero shear rate and at infinite shear rates.

– 0 a = 0 + 1 + (d/dt)2/3

(3-50)

where is a coefficient used to express to the shear stability of the mixture. This equation has been tested and has successfully predicted the behavior of a wide

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variety of pseudoplastic mixtures, such as suspensions of limestone, non-aqueous polymer solutions, and nonaqueous pigment paste.

3-4-3 Dilatant Slurries Dilatant fluids are time-independent non-Newtonian fluids and are characterized by the following: 앫 An infinitesimal shear stress is sufficient to initiate motion. 앫 The rate of increase of shear stress with respect to the velocity gradient increases as the velocity gradient increases. Dilatant fluids, therefore, use similar equations as pseudoplastic fluids. They are much less common than pseudoplastics. Dilatancy is observed under specific conditions such as certain concentrations of solids, shear rates, and the shape of particles. Dilatancy is due to the shift, under shear action, of a close packing of particles to a more open distribution in the liquid. Govier et al. (1957) observed the phenomena of dilatancy in suspensions of magnetite, galena, and ferrosilicon in a range of particle sizes from 5 microns to 70 microns. It is observed that the slope of the shear stress versus the shear rate increases, particularly in the range of shear rates from 80 to 120 sec–1. Metzener and Whitlock (1958) explained the phenomenon of dilatancy as follows. Two mechanisms account for the inflection and subsequent increase in the slope of the curve. Initially, the shear stress approaches a magnitude at which the size of flowing particles and aggregates is at a minimum and a Newtonian behavior develops (at the inflection of the curve). As the level of stress rises, the mixture expands volumetrically, and entire layers of particles start to slide or glide over each other. In the interim, the slurry acts as a pseudoplastic until the shear stress is high enough to cause dilatancy. The phenomenon of dilatancy is not easy to model. According to Metzener and Whitlock (1958), it is observed at volumetric concentration in excess of 27–30% and at shear rates in excess of 100 s–1.

3-4-4 Yield Pseudoplastic Slurries Yield pseudoplastic fluids are time-independent non-Newtonian fluids and are characterized by the following: 앫 An infinitesimal shear stress is sufficient to initiate motion. 앫 The rate of increase of shear stress, with respect to the velocity gradient, decreases as the velocity gradient increases. 앫 A yield stress must be overcome at zero shear rate for motion to occur. Examples of yield pseudoplastics are shown in Table 3-11. Equation 3-44 is then modified to account for the yield stress as follows:

w – 0 = K[(d/dt)n]

(3-51)

Equation 3-51 is known as the Herschel–Buckley equation of yield pseudoplastics and is accepted by most slurry experts to describe the rheology of yield pseudoplastics with

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TABLE 3-11 Examples of Yield Pseudoplastics

Slurry Sewage sludge Sewage sludge Sewage sludge Sewage sludge Kaolin slurry Kaolin slurry Kaolin slurry

Density, kg/m3

Yield stress 0, Pa

Range of consistency coefficient K, Nsn/m2

1024 1011 1013 1016 1071 1061 1105

1.268 0.727 2.827 1.273 1.880 1.040 4.180

0.214 0.069 0.047 0.189 0.010 0.014 0.035

Angle of flow behavior index, n

Reference

0.613 0.664 0.806 0.594 0.843 0.803 0.719

Chilton and Stainsby (1998) Chilton and Stainsby (1998) Chilton and Stainsby (1998) Chilton and Stainsby (1998) Chilton and Stainsby (1998) Chilton and Stainsby (1998) Chilton and Stainsby (1998)

low to moderate concentration of solids. At high shear rates, certain complex phenomena such as dilatancy may develop. Certain bentonite clays develop a yield pseudoplastic rheology at 20% concentration by volume. Krusteva (1998) investigated the rheology of a number of inorganic waste slurries such as drilling fluids in petroleum output, residue mineral materials in tailing ponds, filling of abandoned mine galleries, etc. In the case of clay containing industrial wastes, he indicated that colloidal forces of attraction or repulsion are ever present with Brownian forces and may cause thermodynamic instability. Waste materials such as blast furnace slag, fly ash, and material from mine filling exhibited various forms of a yield pseudoplastic rheology. The behavior of yield pseudoplastics can be expressed by the Carson model as described by Lapasin et al. (1998):

n = n0 +n (d/dt)

(3-52)

By binary system, Lapasin meant a mixture of two sizes of particles above the colloidal range and by ternary, three sizes. Alumina powders with average d50 diameters of 0.9 m, 1.4 m, and 3.9 m, and different specific surface areas (8.23 m2/cm3, 5.74 m2/cm3, and 2.65 m2/cm3) were investigated. A dispersing agent was used. Appreciable time-dependent effects were only noticed at a concentration of the dispersing agent below a critical value. Multicomponent suspensions were found to have a viscosity that was dependent on the total volume concentration of solids Cv and on the composition of the dispersed phase expressed as a volume fraction. It was also dependent on the shear rate of the mixture. Vlasak et al. (1998) investigated the addition of peptizing agents to kaolin–water mixtures. These mixtures were described as yield pseudoplastics that follow the Bulkley–Herschel rheological model (these will be discussed in Chapter 5). The addition of peptizing agents initially achieved a rapid drop of viscosity down to 8–10% of the original value up to an optimum concentration. As the concentration of the peptizing agent is increased beyond an optimum value, its effects are neutralized and the viscosity of the slurry increases again. Soda Water-GlassTM as a peptizing agent seemed to achieve the best reduction in viscosity when added at a concentration of 0.4%. The effect was a drastic drop of viscosity by 92% of its original value (without the peptizing agent). The optimum concentration of sodium carbonate, another peptizing agent, was 0.1%. The viscosity was reduced by 90%. These narrow bands of concentration of peptizing agents can effectively reduce the cost of hydro-transporting kaolin–water mixtures by reducing viscosity and therefore the coefficient of friction.

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3-5 TIME-DEPENDENT NON-NEWTONIAN MIXTURES Because crude oils and slurries of tar sands from certain Canadian mining projects develop a time-dependent non-Newtonian behavior in cold temperatures, a section of this chapter will pay attention to these complex thixotropic properties. In time-dependent non-Newtonian flows, the structure of the mixture and the orientation of particles are sensitive to the shear rates. Due to structural changes and reorientation of particles at a given shear rate, the shear stress becomes time-dependent as the particles realign themselves to the flow. In other words, the shear stress takes time to readjust to the prevailing shear rate. Some of these changes may be reversible when the rate of reformation is the same as the rate of decay. However, in the case of flows in which the deformation is extremely slow, the structural changes or particle reorientation may be irreversible (see Figure 3-13).

3-5-1 Thixotropic Mixtures

Shear Stress (

)

When the shear stress of a fluid decreases with the duration of shear strain, the fluid is called thixotropic. The change is then classified as reversible and structural decay is observed with time under constant shear rate. Certain thixotropic mixtures exhibit aspects of permanent deformation and are called false thixotropic. When the rate of structural reformation exceeds the rate of decay under a constant sustained shear rate, the behavior is classified as rheopexy (or negative thixotropy). One typical example of a thixotropic mixture is a water suspension of bentonitic clays. These difficult slurries are produced by mud drilling associated with the use of positive displacement diaphragm or hose pumps. The reader may find throughout literature considerable discussion about “hysterisis.” This function is used to measure the behavior of the mixture by gradually increasing the shear rate and then by decreasing it back in steps. These curves are interesting but are of limited help to the designer of a pumping system.

Th

ix

ro ot

pi

c

R

p heo

ect

ic

Rate of shear ( = du/dy) FIGURE 3-13 Rheology of time-dependent fluids.

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3.31


Moore (1959) proposed expressing the complex behavior of a thixotropic fluid that does not possess a yield stress value in terms of six parameters:

= (0 + c)(d/dt) d/d = a – (a + bd/dt) where

= duration of the shear for a time-dependent fluid a, b, c, and 0 = materials constants = a structural parameter that has two values (0 and 1) at the limits where the material is fully broken down or fully developed

Fredrickson (1970) discussed the modeling of thixotropic mixtures of suspensions of solids in viscous liquids and proposed that rheological tests be conducted to measure four constants to understand the qualitative nature of the mixture. Ritter and Govier (1970) proposed representing the behavior of thixotropic fluids as follows: 앫 The formation of structures, networks, or agglomerates is similar to a second-order chemical reaction. 앫 The breakdown of the structure is similar to a series of consecutive first-order chemical reactions where formation is meant by behavior that is time-dependent, whereas the breakdown occurs when the viscosity of the fluid acts as a Newtonian mixture that is independent of both the shear rate and the duration of shear (Figure 3-14).

4 Duration of shear, min

2

Shear stress, +0.01, lb /ft

abul-3.qxd

0

10 -1

1

8 6

10 100

4 2

10 -2 10

100 Rate of Shear, d /dt + 10 in sec

1000 -1

FIGURE 3-14 Rheology of Pembina crude oil at 44.5°F at constant duration of shear. (After Govier and Aziz, 1972.)

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CHAPTER THREE

Ritter and Govier (1970) therefore proposed to express the shear stress of the fluid in terms of structural stress s and , a component of shearing stress due to the Newtonian component of the fluid:

= s +

(3-53)

s0 + s s – s log = –KD log – log KDR ( 2s0/ s) – s s0 – s

冤

冥

冤

冥

(3-54)

where s0, s = structural stresses at a given shear rate after zero and infinite duration of shear s0 = 0 – (d/dt) s = – (d/dt) KD = a constant that is independent of shear rate but is related to the first-order structural decay process and is expressed in the minutes–1. KDR = a dimensionless measure of the interaction between the network or structure decay and the reestablishment processes The coefficient KDR is evaluated as

2s0 – s1 s KDR = s1 s – 2s

(3-55)

where s1 is measured after a lapse of 1 minute. In Equations 3-54 and 3-55, KDR, KD, s0, s1, and s are determined from rheology tests. Kherfellah and Bekkour (1998) examined the thixotropy of suspensions of montmorillonite and bentonite clays. Montmorillonite clays are used as thickening agents for drilling fluids, paints, pesticides, cosmetics, pharmaceuticals, etc. Commercial bentonite suspensions exhibited thixotropic properties for concentrations higher than 6% by weight. Rheopectic or negative thixotropic mixtures are not common in mining and will not be examined in this chapter.

3-6 DRAG COEFFICIENT OF SOLIDS SUSPENDED IN NON-NEWTONIAN FLOWS Some solids may be transported by highly viscous fluids in a non-Newtonian flow regime. One such example includes solids transported in the process of drilling a tunnel in a sandy soil rich with clay or bentonite. Other examples of solids suspended in non-Newtonian flows are energy slurries, which are mixtures of fine coal and crude oils. In such circumstances, the drag coefficient of the coarse components is of interest. Brown (1991) reviewed the literature for settling of solids in non-Newtonian flows, but cautioned that the studies have been limited to single particles. Considerably more research is needed in this field.

3-7 MEASUREMENT OF RHEOLOGY In the proceeding sections of this chapter, the concepts of Newtonian and non-Newtonian fluids were explored. Measuring the viscosity of a slurry mixture is recommended for ho-

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mogeneous flows, mixtures with a high concentration of particles, and for fibrous and flocculated slurries. Subsieve particles are defined as particles with an average diameter smaller than 35–70 m (depending on whose reference book you consult). Slurry flows with subsieve particles at a relatively high concentration by volume (Cv 30%) are strongly rheologydependent. Heterogeneous flows, flows without subsieve particles, or flows with subsieve particles at a very low concentrations, are not governed by the rheology of the slurry. Flocculation or the addition of flocculates in the process of mixing slurries tends to result in non-Newtonian rheology. Rheology in simple layman’s terms is the relationship between the shear stress and the shear rate of the slurry under laminar flow conditions. Although this relationship extends to transitional and turbulent flows, most tests are conducted in a laminar regime, often in tubes or between parallel plates. 3-7-1 The Capillary-Tube Viscometer The purpose of the capillary tube viscometer is to measure the rheology of a laminar flow under controlled velocity conditions. Tubes are used in a range of diameters from 0.8–12 mm (1/32–1/2 in). The length of the tube is accurately cut to account for entrance effects and end effects. Typically, the length may be as much as 1000 times the inner diameter. The capillary tube viscometer is used to plot the average rate versus the shear stress at the wall of the tube. This is called the pseudoshear diagram, as defined by the Mooney–Rabinovitch equation:

冦

d[ln(8V/Di)] 8 (du/dr)w = 0.75 + 0.2 Di d[ln(P/4Li)]

冧

(3-56)

where (du/dr)w = rate of shear at the wall P = pressure drop due to friction over a length Li of pipe of inner diameter Di V = average velocity of the flow d = derivative The data is then plotted on a logarithmic scale as per Figure 3-15. The use of capillary-like viscometers is complicated by the “effective slip” of nonNewtonian fluid-suspended material, which tends to move away from the wall, leaving an attached layer of liquid. The result is a reduction in the measurements of effective viscosity. Therefore, it is often recommended to conduct such tests in a number of tubes of different diameters. Measuring the pressure loss between two points well away from the entrance and end effects gives the shear stress at the wall as:

w = RiP/(2Li)

(3-57)

By considering that the velocity profile at a height y above the wall is a function of the shear stress we obtain – (du/dy)w = f ( ) It may be possible to establish a relationship between the flow rate Q and the shear stress as Q 1 3 = 3 w R

冕

w

0

2f ( )d

(3-58)

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CHAPTER THREE

100

D4

increasing tube diameter

shear rate

8V D

ter wa 10

1.0

D3 D2 D1

0 0

1.0

Shear Stress FIGURE 3-15 rheometer.

D P 4L

10

Pseudoshear diagram of a non-Newtonian mixture tested in a capillary tube

For a Newtonian flow: 2V Q w 3 = = Di R 4

(3-59)

or = w/(8V/Di). For a Bingham flow:

= (du/dr)w + 0 for > 0, where 0 is the yield stress. The velocity profile is expressed as 2V

Q 3 = = 3 Di w R

冕 ( – )d w

2

0

(3-60)

0

By integration of this equation and by multiplying by 4, the shear rate is derived as 8V w 4 0 1 40 = 1 – + 4 DI

3 w 3 w

冤

冢冣

冢冣冥

(3-61)

Equation 3-61 is called the Buckingham equation. This equation cannot be solved without long iterations. Many engineers prefer to simplify the Buckingham equation by ignoring the term ( 0/ w)4, as this term is of negligible magnitude compared with the other terms:

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w ⬇ 8V /Di + 4/3 0

(3-62)

The modified equation is plotted in Figure 3-16. For a pseudoplastic slurry or power law fluid, the shear stress is expressed by Equation 3-43. By analogy with the method developed for a Bingham flow in a tube, the following equation is expressed: Q 2V 1 3 = = 3 R Di w

冕 ( /K) 2

1/n

d

(3-63)

0

or Q 3 = R

冕

w

0

(3+1/n) (3 + 1/n)K1/n

(3-64)

which once integrated is expressed as 2V n w1/n = Di 3n + 1 K1/n

冢

冣

(3-65)

The effective viscosity is expressed as

e = w/(8V/Di) = K(8V/Di)(n–1)[4n/(3n + 1)]n

(3-66)

w

Unfortunately, Equation 3-66 is of no value when n < 1.0, which is the case for many power law slurries. It would mean that as the shear rate increases, the effective viscosity decreases to zero. This is contradictory to nature. For power law exponents smaller than 1.0, alternative equipment should be used to measure rheology. It is tricky to avoid errors with the use of capillary effect viscometers. A particular source of errors is the end effect. At the entrance exit of the tube, contraction and expansion of the flow cause additional pressure losses.

Shear Stress

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w 2 r0 Velocity profile

shear rate FIGURE 3-16 Pseudoshear diagram for a Bingham plastic.

dV dU dy dy

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3-7-2 The Coaxial Cylinder Rotary Viscometer A more practical instrument to use when measuring rheology is the coaxial cylinder viscometer. In basic terms, it is a device used to measure the resistance or torque when rotating a cylinder in a viscous fluid (Figure 13-17). The moment of inertia in the cylinder is established by the manufacturer. The torque is due to the force the fluid exerts tangentially to the outside surface of the cylinder: T = 2R0h w R0

(3.67)

where T = (surface area) (shear stress) (radius) R0 = outside radius of the rotating cylinder h = height of the cylinder w = shear stress at the wall The shear stress at any radius r in the fluid can be expressed as du T w = = 2r 2h dy

(3.68)

If the liquid is rotating at an angular velocity , then (du/dy)w = –rd/dr

scale to measure torque

rotation of bob at speed

R0 r Rc

FIGURE 3-17 The rotating concentric viscometer.

slurry

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3.37

and

= –rd/dr –T d = dr 2hr

冕

0

d =

冕

Rc

R0

–T 3 dr 2hr

or

冢

1 1 T = 2 – 2 Rc 4h R 0

冣

(3.69)

where Rc is the radius of the outside cylinder. This is known as the Margulus equation. It is obvious that R 20 can be related to the moment of inertia Ik of the rotating bob cup. Since for a Bingham slurry, the rate of shear is expressed as du/dr = ( – 0); the Margulus equation can be demonstrated as Rc 1 1 T 0 = 2 – 2 – ln Rc R0 4h R 0

冢

冣

冢冣

(3.70)

= n[T/(2R 20hK)]1/n [1 – (R0/Rc)2/n]

(3.71)

w = T/(2R 2b h)

(3.72)

This equation is known as the Reiner–Rivlin equation. For a Pseudoplastic: At the wall: A plot of log w versus log can be constructed. The slope gives the flow index n and, by substituting Equation 3-45, the value of K can be calculated. Heywood (1991) discussed errors with the use of rotating viscometers. Particular sources of errors are the end effects from both cylinders and the possible deformation of the laminar layer under the effect of high rotational speed. Heywood recommended the use of cylinders with a long length to diameter ratio. Wall slip effects can be detected by using cylinders of different radius but same length. The vendors of rheometers publish equations to correct for wall slip and end effects. One important problem about the use of rheometers is that they do not distinguish between Bingham and Carson slurries. This can lead to grave mistakes in the design of a pipeline. Certain slurries have a course of fractions that could also precipitate during a rheometer test. Unfortunately, this would give false readings. When there is doubt, the safest approach is to conduct a proper pump test in a loop. Whorlow (1992) published a book on rheological techniques that includes dynamic tests and wave propagation tests. In the appendix, he listed a number of rheological investigation equipment manufacturers. Some of the techniques apply more to polymers and are not relevant to our discussion. Dynamic vibration tests have been extended to fresh concrete (Teixera et al., 1998). Concord and Tassin (1998) described a method to use rheo-optics for the study of thixotropy in synthetic clay suspensions. A rheometer optical analyzer was used on laponite, a synthetic hectorite clay. Laponite was mixed with water and tests were conducted at various intervals for up to 100 days. Rheo-optics seems to be

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CHAPTER THREE

FIGURE 3-18 Stresstech rheometer, courtesy of ATS Rehosystems. The rheometer was developed for the pharmaceutical and cosmetics industries, where materials consistency may vary from fluid to solid.

a new technique based on the ability of solids to reorient themselves by applying to them a negative electrical charge.

3-8 CONCLUSION In this chapter, it was demonstrated that mixtures of solids and liquids are complex systems. The size of the particles, the diameter of the pipe, the interaction with other particles, the viscosity of the carrier, and the temperature of the flow all interact to yield Newtonian or non-Newtonian flows. In the next three chapters, the principles discussed in the present chapter will be applied to calculate the velocity of deposition, the critical velocity, the stratification ratio, and the friction loss in closed and open conduits for heterogeneous and homogeneous mixtures.

3-9 NOMENCLATURE a A

The longest axis of a particle in Albertson’s model Parameter used to express viscosity of non-Newtonian flows

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A0 A1 b B c C CD CDo CL CN CS Cv Cv Cw da dapp df dg dn d D Di E f( ) FBF Fw g gc h Ik K KD KDR Kt Kz K1, K2, K3 ln L Lc LI n P Q r R Rc Re

3.39

Coefficient Coefficient Axis of a particle in Albertson’s model Parameter used to express viscosity of non-Newtonian flows The shortest axis of a particle in Albertson’s model Parameter used to express viscosity of non-Newtonian flows Drag coefficient of an object moving in a fluid Profile drag coefficient of an object moving in a fluid Lift coefficient of an object moving in a fluid Coefficient based on equivalent Ren Coefficient based on equivalent Ren Concentration by volume of the solid particles in percent Maximum packing concentration of solids Concentration by weight of the solid particles in percent Diameter of a sphere with a surface area equal to the surface area of the irregularly shaped particle Apparent particle diameter Apparent flocculant diameter Sphere diameter Diameter of a sphere with a volume equal to the volume of the irregularly shaped particle in Albertson’s model Particle diameter Drag force Tube or pipe inner diameter Factor between Albertson and Clift shape factors Function of Buoyancy force Wall effect correction factor for free-fall speed of a particle Acceleration due to gravity (9.78–9.81 m/s2) Conversion factor, 32.2ft/s2 if U.S. units between lbms and slugs Height of the cylinder Moment of inertia Consistency index or power law coefficient for a pseudoplastic A constant that is independent of shear rate but is related to the first-order structural decay process and is express in minutes–1 A dimensionless measure of the interaction between the network or structure decay and the reestablishment processes Coefficient for terminal velocity Kozney constant Coefficients natural logarithm Lift force Characteristic length Length of pipe or tube Flow behavior index, or exponent for a pseudoplastic ( 540, a = 1.78, b = –0.019 For 160 < Ar < 540, a = 1.19, b = 0.045 For 80 < Ar < 60, a = 0.197, b = 0.4 For Ar < 80, the Wilson and Judge (1976) equation can be used, which expressed the Froude number as

冦

冢

dp Fr = (兹2 苶) 2.0 + 0.30 log10 ᎏ DiCD

冣冧

(4-7)

This correlation is useful in the range of 10–5 < (dp /DiCD) < 10–3. To determine the drag coefficient, the actual density of the liquid should be used, whereas the viscosity should be corrected for the presence of fines. Example 4-3 Water at a viscosity of 0.0015 Pa · s (0.0000313 slugs/ft-sec) is used to transport sand with an average particle diameter of 300 ␮m (0.0118 inch). The volumetric concentration is 0.27. The pipe’s inner diameter is 717 mm (28.35⬙). Using the Gilles equation (Equation 4-6), determine the deposition velocity if the specific gravity of sand is 2.65. Assume CD = 0.45. Solution in SI Units d50 0.3 ᎏ = ᎏ = 0.4 × 10–3 Di 717 Iteration 1 Assuming CD > 10–3, by the Wilson and Judge correlation (Equation 4-7):

冦

冢

0.003 Fr = (兹2苶) 2.0 + 0.30 log10 ᎏᎏ 0.717 × 0.45 Fr = 1.54 FL = Fr/兹2苶 = 1.54/兹2苶 = 1.09

冣冧

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The specific gravity of the mixture is determined as: Sm = Cv(Ss – SL) + SL = 0.27 (2.65 – 1) + 1 = 1.446 苶g苶D 苶苶苶苶苶苶– 1苶 = 4.82 m/s VD = FL兹[2 i(␳ s/␳ L 苶苶] Iteration 2 4 × 9.81 (3 × 10–4)3 × 1000 (1650) Ar = ᎏᎏᎏᎏ = 258.98 3(1.5 × 10–3)2 for 160 < Ar < 540, a = 1.19, b = 0.045. From equation 4.6: Fr = aArb = 1.19 · 258.980.045 = 1.53 FL = F/兹苶2 = 1.53/兹苶2 = 1.082 VD = FL[2gDi(␳s/␳L – 1)]0.5 VD = 1.082[2 · 9.81 · 0.717 · (1.65)]0.5 = 5.21m/s Solution in USCS Units d50 0.00118 ᎏ = ᎏ = 0.4 × 10–3 Di 28.23 Iteration 1 Assuming CD > 10–3, by the Wilson and Judge correlation (Equation 4-7):

冦

冢

0.00118 Fr = (兹苶2) 2.0 + 0.30 log10 ᎏᎏ 28.23 × 0.45

冣冧

Fr = 1.54 FL = 1.54/2 = 1.09 The specific gravity of the mixture is determined as: Sm = Cv(Ss – SL) +SL = 0.27(2.65 – 1) + 1 = 1.446 VD = 1.09[2 · 32.2 · (28.23/12) (2.65 – 1)]0.5 VD = 17.23 ft/sec Iteration 2 The particles’ diameter is 0.984 · 10–3 ft The density of water is 1.93 slugs/ft3 The density of sand is 5.114 slugs/ft3 Water dynamic viscosity is 0.0000313 slugs/ft-sec 4(0.984 · 10–3)3 × 1.93(5.114 – 1.93) · 32.2 Ar = ᎏᎏᎏᎏᎏᎏ = 259 3(0.0000313)2 for 160 < Ar < 540, a = 1.19, b = 0.045. From equation 4.6, Fr = aArb = 1.19 · 258.980.045 = 1.53 FL = 1.53/兹2苶 = 1.082 VD = FL[2gDi(␳s/␳L – 1)]0.5 VD = 1.082[2 · 32.2 · 2.35 · (1.65)]0.5 = 17.1 ft/s

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

4.13

The solution by the Gilles equation is within the limits set by Schiller in Example 4-2. In these two different examples, we applied two different formulae but obtained consistent results. This demonstrates the sensitivity of approaches to equations derived from empirical equations. It may be necessary sometimes try to solve a problem using two different equations, and to use common sense when similar results are obtained. Table 4-3 presents values of the Archimedean number, the resultant magnitude of the factor FL for particles d50 in the range of 0.08 to 50 mm for a specific gravity of 1.5, which is typical of coal-based mixtures. Most coals may be pumped with different sizes of particles as discussed in Chapter 11. The viscosity may be due to the presence of cer-

TABLE 4-3 The Coefficient FL Based on Gilles Equation for Particles Between 0.080 and 50 mm of Specific Gravity of 1.500 (e.g., Coal) as a Function of Viscosity

d50 (mm) 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 1.00 2.00 3.00 4.00 5.00 6.00 8.00 10.00 20.00 30.00 40.00 50.00

␮ = 1 cP ␮ = 5 cP ␮ = 10 cP _____________________ _______________________ _______________________ Archimedean Archimedean Archimedean number Ar FL number Ar FL number Ar FL 3.35 6.54 11.3 17.9 26.8 38.1 52.3 102 177 280 419 596 818 1088 1413 1796 2243 2579 3348 4016 4768 6540 52320 176580 418560 817500 1415640 3348480 6540000 5.23 × 107 17.7 × 108 41.86 × 108 81.75 × 108

Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 0.89 1.062 1.084 1.104 1.420 1.43 1.437 1.445 1.451 2.457 1.463 1.469 1.474 1.478 1.487 1.547 1.583 1.610 1.63 1.647 1.674 1.696 1.764 1.805 1.835 1.859

0.13 0.26 0.45 0.72 1.07 1.53 2.1 4.1 7.1 11.2 16.75 23.8 32.7 43.5 56.51 72 89.7 110.4 134 161 191 262 2093 7063 16742 32700 56505 133939 261600 2092800 7063202 16742404 32700008

Eqn 4-7 0.033 Eqn 4-7 0.065 Eqn 4-7 0.113 Eqn 4-7 0.18 Eqn 4-7 0.27 Eqn 4-7 0.38 Eqn 4-7 0.52 Eqn 4-7 1.02 Eqn 4-7 1.77 Eqn 4-7 2.80 Eqn 4-7 4.19 Eqn 4-7 5.96 Eqn 4-7 8.18 Eqn 4-7 10.9 Eqn 4-7 14.1 Eqn 4-7 18 0.84 22.4 0.914 27.6 0.99 33.5 1.058 40 1.066 48 1.081 65 1.455 523 1.489 1765 1.514 4185 1.533 8175 1.55 14126 1.575 33485 1.595 65400 1.66 523200 1.698 1765800 1.726 4185601 1.749 81750020

Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-17 1.12 1.45 1.475 1.494 1.51 1.534 1.554 1.616 1.654 1.682 1.703

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tain fines, as with peat coals or degradation of the coal during pumping over long distances, or the use of a heavy medium such as magnetite at high concentration as a carrier for coal in a water mixture. Table 4-4 presents values of the factor FL for particles d50 in the range of 0.08 to 50 mm for a specific gravity of 2.65, which is typical of sand and tar-sand-based mixtures. The largest particles are often found in tar sand applications, with some contribution of the tar or oil to viscosity. In this table, there was no need to present the Archimedean number, as this was demonstrated in the previous table. Newitt et al. (1955) preferred to express the speed of transition between “saltation” flow and heterogeneous flow in terms of the terminal velocity of particles (previously discussed in Chapter 3): V3 = 17 Vt

(4.8)

The reader should refer to Equation 3-18, which corrects the terminal velocity of a single particle to a mass of particles at higher volumetric concentration. Although Equation 4-8 has served as the basis of many models, we will later discuss the recent corrections proposed by Wilson et al. (1992). The approach to obtain the magnitude of V3 is basically to conduct a test and measure pressure drop per unit length of pipe. V3 is considered to occur at the minima, or the point of minimum pressure drop. W. E. Wilson (1942) expressed the pressure gradient of noncolloidal solids by referring to clean water and by proposing a correction to the Darcy–Weisbach equation (discussed in Chapter 2). He expressed the consumed power due to friction by the following equation:

FIGURE 4-8 These taconite tailings must be pumped above a deposit velocity of 13 ft/s in 14⬙ pipe due to the size of the particles.

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TABLE 4-4 The Coefficient FL Based on Gilles Equation for Particles Between 0.080 and 50 mm of Specific Gravity of 2.65 (e.g., Sands and Oil Sands) as a Function of Viscosity d50 (mm) 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 1.00 2.00 3.00 4.00 5.00 6.00 8.00 10.00 20.00 30.00 40.00 50.00

␮ = 1 cP, FL

␮ = 5 cP, FL

␮ = 10 cP, FL

Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 0.837 0.964 1.061 1.093 1.421 1.433 1.444 1.454 1.462 1.470 1.478 1.485 1.491 1.497 1.502 1.507 1.512 1.521 1.583 1.620 1.647 1.668 1.685 1.713 1.735 1.805 1.847 1.877 1.901

Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 0.8 0.906 1.016 1.065 1.076 1.087 1.097 1.107 1.116 1.423 1.431 1.489 1.524 1.549 1.569 1.585 1.611 1.632 1.698 1.737 1.766 1.789

Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-7 Eqn 4-17 0.847 0.915 0.984 1.054 1.072 1.450 1.484 1.509 1.528 1.544 1.569 1.589 1.654 1.692 1.720 1.742

冢

冣

(4-9)

⌬Hf g fDV 2 C1CwVt g ᎏ=ᎏ+ᎏ L 2Di V

(4-10)

CwVt fDV ⌬Hf = L ᎏ + C1 ᎏ 2gDi V where ⌬Hf = head loss due to friction (in units of length) fD = Darcy–Weisbach friction factor C1 = constant Equation 4-9 may also be reexpressed as

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By differentiating this equation with respect to V, we obtain for the minimal value –C1CwVt g 2 fDV ᎏ = ᎏᎏ 2Di V2 or fDV C1CwVt g ᎏ=ᎏ Di V2 C1CwVt gDi V 3 = ᎏᎏ fD at constant friction factor fD, or [C1CwVt gDi]1/3 Vmin = ᎏᎏ f D1/3

(4-11)

The magnitude of the Darcy friction factor for water flow in rubber lined and HDPE pipe was computed for pipes from 2⬙ to 18⬙ and results presented in Chapter 2. Wilson (1942) defined a factor C3 to determine whether the particles will settle to form a bed: 2Vt C3 = ᎏᎏ (⌬Hf fD gDi/L)1/2

(4-12)

If C3 > 1 most particles with a terminal velocity Vt will stay in suspension. If C3 ⱕ 1 most particles with a terminal velocity Vt will settle out. Whereas the equations of Newitt et al. (1955) and Wilson (1942) focused on the terminal velocity, the work of Durand and Condolios (1952) focused on the drag coefficient for sand and gravel. Zandi and Govatos (1967) and Zandi (1971) extended the work of Durand to other solids and to different mixtures. They defined an index number as V 2CD1/2 Ne = ᎏᎏ CvDi g(␳s/␳w – 1)

(4-13)

At the critical value when Ne = 40, the flow transition between saltation and heterogeneous regimes occurs. This statement infers that when Ne < 40 saltation occurs, and when Ne ⱖ 40 heterogeneous flow develops. These results, based on a mixture of different particle sizes, did not apply to the work of Blatch (1906), who concentrated on particles of a uniform size (sand 20–30 mesh in water). Babcock (1967) reinterpreted this work and demonstrated that for finely graded particles the transition occurred at an index number of 10. It is obvious that a complex mixture of particles of different sizes can increase the magnitude of the transition index number. Example 4-4 Tailings from a mine consist of solids at a volumetric concentration of 20%. The specific weight of the solids is 4.2. The pipe diameter is 8⬙ with a wall thickness of 0.375⬙ and rubber lining of 0.5⬙. The particle Albertson shape factor is 0.7. The dynamic viscosity is 3 cP. The average d50 = 0.4 mm. Determine the speed of transition from saltation using the Zandi approach as expressed by Equation 4-13. Solution in SI Units Pipe inner diameter Di = 8⬙ – 2 · (0.5 + 0.375) = 6.25⬙ = 158.75 mm

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Iteration 1 Let us first assume a transition from saltation at 3 m/s and let us determine the drag coefficient of the particles in water at the stated dynamic viscosity: Rep = 1000 · 0.0004 · 3/0.003 = 400 From Table 3.7, CD = 1.09. The transition from saltation occurs when Ne = 40. From Equation 4-13, using SI units: 9 · 兹1 苶.0 苶9苶 Ne = ᎏᎏᎏᎏ 0.2 · 0.15875 · 9.81 · (4.2/1 – 1) Ne = 9.43. Iteration 2 Let us first assume a transition from saltation at 6 m/s and let us determine the drag coefficient of the particles in water at the stated dynamic viscosity: Rep = 1000 · 0.0004 · 6/0.003 = 800 From Table 3.7, CD = 1.15. 苶5苶 36 · 兹1苶.1 Ne = ᎏᎏᎏᎏ 0.2 · 0.15875 · 9.81 · (4.2/1 – 1) Ne = 39. The transition from saltation therefore occurs at a speed of 6.1 m/s. Solution in USCS Units Iteration 1 Pipe diameter = 8⬙ – 2 · (0.375 + 0.5) = 6.25⬙ = 0.521 ft Let us first assume a transition from saltation at 10 ft/s and let us determine the drag coefficient of the particles in water at the stated dynamic viscosity. Particle size = 0.4 mm/304.7 mm = 1.3128 × 10–3 ft

␮ = 0.003/47.88 = 6.265 × 10–5 lbf-sec/ft2 Density of water = 62.3 lbm/ft3/32.2 ft/sec = 1.935 slugs/ft3 1.935 slugs/ft3 × 1.3128 × 10–3 ft × 10 ft/sec Re = ᎏᎏᎏᎏᎏ 6.265 × 10–5 lbf-sec/ft2 = 406 From Table 3.7, CD = 1.09. 苶9苶 100 · 兹1苶.0 Ne = ᎏᎏᎏᎏ 0.2 · 0.5208 ft · 32.2 ft/sec · (4.2/1 – 1) Ne = 9.73. Iteration 2 Let us first assume a transition from saltation at 20 ft/s and let us determine the drag coefficient of the particles in water at the stated dynamic viscosity: Rep = 406 · (20/10) = 804

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CHAPTER FOUR

From Table 3.7, CD = 1.15. 202 · 兹1苶.1 苶5苶 Ne = ᎏᎏᎏᎏ 0.2 · 0.5208 ft · 32.2 ft/sec · (4.2/1 – 1) Ne = 39.97. The transition from saltation therefore occurs at a speed of 20 ft/sec. 4-3-3 V4: Transition Speed Between Heterogeneous and Pseudohomogeneous Flow For the transition to pseudohomogeneous flows, Newitt et al. (1955) expressed the speed in terms of the terminal velocity of particles as V4 = (1800 gDiVt)1/3

(4-14)

Refer to Chapter 3 and Equation 3-18 to calculate terminal velocity. Govier and Aziz (1972) applied Newton’s law (i.e., CD = 0.44) for particles immersed in a fluid to Equation 4-14 to yield 4gdp 1/6 V4 = 38.7D 1/3 i ᎏ (S – 1) 3CD

(4-15)

Govier and Aziz (1972) analyzed the work of Spells (1955) on solid particles with a diameter 80 ␮m < dp < 800 ␮m (mesh 180 < dp < 20) and derived the following equation: V 1.63 V4 = 134CD0.816D 0.633 i t

(4-16)

This equation was derived in USCS units with the diameter expressed in feet and the velocity in feet per seconds. Example 4-5 An ore with a specific gravity of 4.1 is to be pumped in a pseudohomogeneous regime in a 24 in pipe with an ID of 22.23 in. The drag coefficient of the particles is assumed to be 0.44. The estimated flow rate is 12,000 US gpm. The particles have a sphericity of 0.72 and a diameter of 250 ␮m. Solve for V4. Solution in SI Units 12,000 × 3.785 Q = ᎏᎏ = 0.757 m3/s 60,000 Pipe ID = 22.25 × 0.0254 = 0.565 m Cross-sectional area = 0.251 m2 Average speed of flow = 3.02 m/s Sphericity = Asp/Ap = 0.72 苶2 苶苶 ×苶25苶0苶 = 218 ␮m dsp = 兹0苶.7

Vt =

ᎏᎏᎏᎏ 冣冪冢莦莦莦莦莦 3 × 0.44 × 1000 4 × 0.218 × 10–3 × 9.81 (4100 – 1000)

Vt = 0.142 m/s

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4.19

By Newitt’s equation (Equation 4.14): V4 = (1800 × 9.81 × 0.565 × 0.142)1/3 V4 = 11.22 m/s Alternatively using Equation 4.16: Di = 1.854 ft Vt = 0.466 ft/sec V 1.63 V4 = 134C D0.816D 0.633 i t V4 = 134 × 0.440.816 × 1.8540.633 × 0.4661.63 = 29.19 ft/sec or 8.9 m/s Solution in USCS Units Q = 12,000 · 0.002228 = 26.736 ft3/sec Pipe ID = 22.25/12 = 1.854 ft Cross-sectional area = 2.7 ft2 Average speed of flow = 9.9 ft/sec Sphericity = Asp/Ap = 0.72 苶2 苶苶 ×苶25苶0苶 = 218 ␮m = 0.000715 ft dsp = 兹0苶.7

The density of water is 1.93 slugs/ft3 The density of solids is 7.913 slugs/ft3 Vt =

冪冢莦莦莦冣莦 4 × 0.000715 × 32.2 (7.913 – 1.93) ᎏᎏᎏᎏ 3 × 0.44 × 1.93 Vt = 0.465 ft/s

By Newitt’s equation (Equation 4.14): V4 = (1800 × 32.2 × 1.854 × 0.465)1/3 V4 = 36.83 ft/sec Alternatively, using Equation 4.16: V 1.63 V4 = 134C D0.816D 0.633 i t V4 = 134 × 0.440.816 × 1.8540.633 × 0.4661.63 = 29.19 ft/sec

4-4 HYDRAULIC FRICTION GRADIENT OF HORIZONTAL HETEROGENEOUS FLOWS Having been able to determine the speed for transition from one regime to another, the slurry engineer must determine the loss of head per unit length due to friction, called the hydraulic friction gradient (Equation 2-24). The hydraulic friction gradient for the slurry (im) is higher than the hydraulic friction gradient for an equivalent volume of water. Since the first slurry pipelines were built, engineers and scientists have tried to correlate the losses with slurry to those of an equivalent volume of water. It was initially assumed that the friction losses would increase in proportion to the vol-

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CHAPTER FOUR

umetric concentration of solids. A term im was then defined as the friction head of the mixture in equivalent meters (or feet) of the carrier fluid (e.g., water) per unit of pipe length. In Chapter 2, the friction hydraulic gradient was introduced by Equation 2-24 and is defined as: fDV 2 i= ᎏ 2gDi There are a number of models to predict friction losses and they are essentially based on the interaction forces between solids and liquid carrier. Some use the drag coefficient, others use the terminal velocity of the solids, and some consider the solids to be moving as a bed with a layer of liquid and suspended fines above it. To reflect the increase in friction head due to the volumetric concentration of solids, Durand and Condolios (1952) proposed a nondimensional ratio im – iL Z= ᎏ CviL

(4-17)

where Cv = the volumetric concentration of solids im = pressure gradient for the slurry mixture in meters of water iL = pressure gradient for an equivalent volume of water or carrier fluid in meters of water

C V3 C V2 C V1

im iL

w at er

in equivalent (m/m) or (ft/ft)

Head loss per pipe length

The reader should not confuse head of slurry in meters or feet of slurry with meters or feet of water. This is not a barometer or some instrument measuring pressure; for this reason everything is kept consistent by using meters or feet of water. By itself, the term i relates only to clear water having the same velocity as the slurry flow. It is convenient to use water as a reference benchmark. (See Figure 4-9.)

Average velocity of flow FIGURE 4-9 Concepts of the hydraulic friction gradients im and iL for slurry mixture and for water.

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4.4.1 Methods Based on the Drag Coefficient of Particles Based on their analysis of test data from 11 references for sand in particle sizes ranging up to 1 inch (25.4 mm), in pipes with a diameter range from 1.5 inch to 22 inch, and in volumetric concentration up to 22%, Zandi and Govatos (1967) derived an equation for the index number Ne (equation 4-13) in terms of the volumetric concentration, and some empirical parameters: V 2C D1/2 ␺ = ᎏᎏ Dig(␳s/␳w – 1)

(4-18)

␺ Ne = ᎏ Cv

(4-19)

Or from equation 4-13:

or ␺ = CvNe. They plotted this function against a parameter ␾ to express head loss as im – iL ␾ = ᎏ = K(␺)m CviL

(4-20)

where iL = hydraulic gradient in terms of water density for a flow of clean water with a mean velocity V im = hydraulic gradient in terms of water density for a slurry flow with a mean velocity V K, m = constants On a logarithmic scale they obtained: For ␺ > 10, K = 6.3 and m = –0.354 For ␺ < 10, K = 280 and m = –1.93 The data is presented in Figure 4-10. The dramatic change in values of K and m at ␺ = 10 has encouraged researchers to develop more sophisticated models that we shall review in the rest of this chapter. Substituting for the value of 40 of the index coefficient, V3 may be expressed as [40 CvDi g(␳s – ␳w)/␳w]1/2 V3 = ᎏᎏᎏ C D1/4

(4-21)

Equation 4-21 is therefore a modified version of Equation 4-2. Equation 4-4 is a different approach, as it accounts for particle size, which is often easier to measure than the drag coefficient. Example 4-4 has shown that some iteration is necessary to obtain the velocity at which the transition from saltation to asymmetric flow occurs. Despite its simplicity, this method continues to be used by dredging engineers who usually deal with sand and gravel mixtures of less than 20% concentration by volume. The personal experience of the author is that often mines and dredging systems have to be designed in very remote areas where there are no slurry labs to conduct tests. This is an unfortunate fact, and sometimes an “overconservative” approach based on Durand, Zandi, and other authors is the only alternative. However, the author does encourage engineers of slurry systems to plan well ahead and test data to avoid very expensive field corrections.

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CHAPTER FOUR 10000

RANGE OF 1 NUMBER Zandi & Govtes

1000 Durand & Condolios

쐌 0–40 쐌 40–310 왌 310–1550 왖 1550–3100

iL ␾= ᎏ Cv iL

100

10

1

.1

.01 .01

0.1

1.0

10 V 2兹C 苶苶D ␺ = ᎏᎏ Di g( ␳s /␳L – 1)

100

1000

FIGURE 4-10 The Zandi–Govatos factors for heterogeneous slurry flows. (From Zandi and Govatos, 1967, reprinted with permission from ASCE.)

Shook et al. (1981) modified Zandi’s equation by proposing “in-situ concentration of particles” Ct rather than volumetric concentration:

␾t = K␺m im – iL ␾t = ᎏ iLCt They measured a magnitude of m = –1 for one single type of coal in different pipe sizes. They measured different values of K for different coals. The in-situ concentration Ct remained constant with speed, but the volumetric concentration of solids Cv that could be moved increased with V. This concept will be reexamined in Section 4.10 as part of the two-layer models. Example 4-6 Using Equations 4-19 to 4-20, consider the pumping of solids in a 305 mm (12 in) ID pipe at a speed of 3.045 m/s (10 ft/s) and a volumetric concentration of 18%. Assume a drag coefficient of 0.45 for the solid particles and a specific gravity of 2.65. Determine the increase in the pressure gradient for flow in the pipe due to the presence of solids.

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Solution in SI Units V = 3.045 m/s pipe Di = 0.305 m (12 in) 3.0452 × 兹苶 0苶 .4苶5 Ne = ᎏᎏᎏ = 68.67 0.18 × 0.305 × (2.65 – 1) 68.67 Ne ␺ = ᎏ = ᎏ = 381.5 Cv 0.18

␺ > 10 then K = 6.3 and m = –0.345 ␾=

= K ␺–0.345 = 0.81

im – iL ᎏ = 0.81 × 0.18 = 0.145 iL im ᎏ = 1.145 iL The slurry causes an increase of pressure gradient of 14.5% by comparison with water at the same velocity. Using the approach developed by Durand and Condolios, the fanning friction factor for the slurry is correlated with the friction factor for an equivalent volume of water by the following equation: gDi(␳s – ␳L) fDm = fDL 1 + Kf Cv ᎏᎏ 苶 V2␳L兹苶 CD

冦

冤

冥冧 3/2

(4-22)

Wasp et al. (1977) deducted that the coefficient Kf is between 80 and 150, depending on the slurry. The most common value is actually 81 for most sands according to Govier and Aziz (1972) (see Table 4-5). Example 4-7 Using Equation 4-22, determine the correction for the friction factor for the portion of solids in a slurry mixture of uniform size distribution. The slurry is pumped at the rate of 16,000 gpm in a rubber-lined 22.75⬙ ID pipe. The volumetric concentration is 22%. Assume Kf = 85 and CD = 0.45. Use the Swain–Jaime equation to determine fL. The specific gravity of the solids is 2.65. The dynamic viscosity of water is 2.7 × 10–5 lbf-sec/ft2. Solution in SI Units 16,000 (3.785) Q = ᎏᎏ = 1.009 m3/s 60,000 Pipe ID = 22.75 (0.0254) = 0.5778 m Area of pipe = 0.262 m2 Velocity = 3.85 m/s Dynamic viscosity = 0.00129 mPa · s

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TABLE 4-5 Correction of Friction Factor Due to Volumetric Concentration of Solids Based on Equation 4-22 Assuming K = 81 gDi (␳s – ␳L) ᎏᎏ V 2␳L兹苶 C苶 D

fDm – fDL ᎏ CV fDL

gDi (␳s – ␳L) ᎏᎏ V 2␳L兹苶 C苶 D

fDm – fDL ᎏ CV fDL

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.15 0.20 0.25 0.30 0.35 0.40

0.081 0.229 0.421 0.648 0.906 1.190 1.500 1.833 2.187 2.561 4.706 7.245 10.125 13.31 16.77 20.49

0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

24.451 28.638 33.039 37.645 42.448 48.024 52.611 57.959 63.477 69.159 75.002 81.000

For the water: 1,000 (3.85) 0.5778 Re = ᎏᎏᎏ = 1,723,292 0.00129 Absolute roughness of rubber = 0.00015 m. Relative roughness 0.00015 ␧ ᎏ = ᎏ = 0.0002596 DI 0.5778 0.25 = 0.0151 fD = ᎏᎏᎏᎏᎏ [log10{(0.0002596/3.7) + (5.74/1,723,2920.9)}2]

冤

冢

9.81 · 0.578 · 1.65 fm = fL 1 + 85 · 0.22 ᎏᎏ 3.852兹苶0苶 .4苶4苶5 fm = fL · 18.067 = 0.273 Solution in USCS Units Q = 35.63 ft3/sec 22.75 Pipe ID = ᎏ = 1.896 ft 12 Area = 2.823 ft2

冣冥 1.5

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Velocity = 12.62 ft/s Dynamic viscosity = 1.29 cP = 0.0129 · 0.002089 lbf-sec/ft2 = 0.00002695 lbf-sec/ft2

冢

冣

62.3 12.62 (1.896) Re = ᎏ ᎏᎏ = 1.7 × 106 32.2 2.695 × 10–4 Absolute roughness of rubber = 0.00049 ft Relative roughness of rubber = 0.0002596 fD = 0.0151

冤

冢

32.2 × 1.896 × 1.65 ᎏ fm = 0.0151 1 + 85 × 0.22 ᎏᎏᎏ 12.622 兹苶0苶 .4苶5

冣冥 = 0.27 1/5

An increase of the friction factor by 18-fold appears to be very high. The engineer in charge of such a problem should seriously consider redesigning the system. At this stage, the reader is encouraged to become familiar with the basic equations before applying them to compound systems. Equation 4-17 can be expressed in terms of the drag coefficients of the solid particles, the pipe inner diameter, the density of the solid and liquid phases, the speed, and an experimental factor Ke: im – iL Dig(␳s/␳L – 1) 1 ᎏ Z = ᎏ = Ke ᎏᎏ 兹C 苶D 苶 CviL V2

冤

冢

冣冥

3/2

(4-23)

Babcock (1968) was very critical of all equations using pressure gradients based on the work of Durand and Condolios or their followers. Geller and Gray (1986) did not agree with Babock’s criticisms and spelled out some of the misgivings. Govier and Aziz (1972) did confirm that errors of the order on 40% have occurred in predicted values of Z, but for all intents and purposes, these equations were the best available till the early 1970s. Herbich (1991) agreed with the value of 81 for most dredged sands and gravel. Sand and gravel are typically dredged, then pumped at a volumetric concentration smaller than 20%.

4.4.2 Effect of Lift Forces It may be considered that the magnitude of the constant m is based on a very large magnitude of data. In an innovative study at the Canada Center for Mineral and Energy Technology (CANMET), Geller and Gray (1986) conducted an extended analysis that demonstrated that lift forces had an effect on the pressure gradient. This study, rather than dismissing the ideas of Durand, supported the previous work and gave it more importance. Reviewing the work of Babock (1971), Geller and Gray (1986) indicated that for fine to intermediate sizes (80/100 quartz sand with d␳ = 0.16 mm) the value of m was –0.25. In addition, they concluded that lift forces are at a maximum when the volumetric concentration Cv is less than 0.23. For intermediate sands at higher volumetric concentration, the lift forces seem to be minimal. This is an important factor to consider (for an understanding of lift forces review Chapter 3, Section 3.1). Furthermore, there is an important coefficient of mechanical friction ␮p, which results from the sliding displacement between solids in contact, which is distinct from the viscous friction.

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4-4-3 Russian Work on Coarse Coal There are no universally accepted models for coarse coal. Work in the former Soviet Union on coarse coal was reported by Traynis (1970) and reviewed by Faddick (1982). From Russian data, the following two equations were reported. For deposition velocity: [(␳c – ␳hm)/␳c]1/3 ᎏ V3 = [Dig]1/2 ᎏᎏ [ fDLk CD]1/3

(4-24)

For the hydraulic gradient for coal: 兹g苶D 苶苶i Cvc(␳s – ␳hm) ␳s – ␳L im = iL 1 + Cv ᎏ + ᎏ · ᎏᎏ ␳L k CdV␳L

冦

冢

冣冤

冥冧

(4-25)

where Cv = total volumetric concentration of solids Cvc = volumetric concentration of coarse solids K = constant for coarse coal = 1.9 CD = drag coefficient considered to be 0.75 for the coarse coal fraction ␳hm = density of heavy medium produced by the fines For the other terms, see Section 4-14. Example 4-8 Coarse coal is to be pumped in a rubber-lined 18 in pipe steel with an inner diameter of 17 in. A screen analysis of the coal indicates that it has a distribution of 20% passing 200 microns. The velocity of pumping is 4.5 m/s and the total weight concentration is 52%. The specific gravity of the coal is 1.35. Determine the hydraulic gradient due to wall friction in the horizontal pipeline. Assume a water dynamic viscosity of 1.2 cP, but correct for viscosity due to solids using Einstein’s equation. Assume a drag coefficient of 0.75 for the coarse coal. Solution Since the weight concentration is 52%, the specific gravity of the mixture is Sm = SL/(1 – (CW (Ss– SL)/Ss) = 1/(1 – 0.52(1.35 – 1)/1.35) = 1.156 The volumetric concentration is Cv = Cw Sm/Ss = 0.52 · 1.156/1.35 = 0.445 The weight concentration of the fines is 20%. Density of the heavy medium carrying the fines is Smf = SL/(1 – (CWf(Ss– SL)/Ss) = 1/(1 – 0.104(1.35 – 1)/1.35) = 1.028 Volumetric concentration of the fines = 0.2 · 0.445 = 0.089. Calculations in SI Units Pipe ID = 17 (0.0254) = 0.432 m Area of pipe = 0.146 m2 Velocity = 4.5 m/s The dynamic viscosity is corrected to take in account the presence of fines at a volumetric concentration of 0.089. The dynamic viscosity of water is 1.2 cP, the Einstein–Thomas equation is applied:

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␮ = ␮L(1 + (2.5 · 0.089) + (10.05 · 0.0892) + 0.00273 exp (16.6 · 0.089)] = 1.314 ␮L = 1.577cP 1,000(4.5) 0.432 Re = ᎏᎏ = 1,232,720 0.001577 Absolute roughness of rubber = 0.00015 m. Relative roughness:

␧ 0.00015 ᎏ = ᎏ = 0.000368 DI 0.432 0.25 = 0.0162 fDL = ᎏᎏᎏᎏᎏ [log10{(0.000368/3.7) + (5.74/1,232,7200.9)}2] iL = fDV 2/(2gDi) = 0.0162 · 4.52/(2 · 9.81 · 0.432) = 0.0387 m/m Using Equation 4.25:

冦

冢

冣冤

兹(9 苶.8 苶1 苶苶·苶 0.4 苶3 苶2 苶)苶 0.8 · 0.445 · (1350 – 1028) 1350 – 1000 im = 1 + 0.445 ᎏᎏ + ᎏᎏ · ᎏᎏᎏ 1000 1.9 · 0.75 · 4.5 1000

冥冧

im = 0.0387[1 + 0.445(0.35)] + [0.0368)] = 0.0815m/m The presence of coal effectively doubles the head losses. The deposition velocity is expressed from Equation 4-24: [(1350 – 1028)/1350]1/3 V3 = [0.432 · 9.81]1/2 ᎏᎏᎏ [0.0162 · 1.9 · 0.75]1/3 V3 = 4.48 m/s Calculations in USCS Units Pipe ID = 17⬙ = 1.417 ft Area of pipe = 1.576 ft2 Velocity = 4.5 m/s = 14.76 ft/sec The dynamic viscosity is corrected to take in account the presence of fines at a volumetric concentration of 0.089. For the water, dynamic viscosity = 1.2 cP = 0.012 · 0.002089 lbfsec/ft2 = 2.507 × 10–5 lbf-sec/ft2. The Einstein–Thomas equation is applied:

␮ = ␮L(1 + (2.5 · 0.089) + (10.05 · 0.0892) + 0.00273 exp (16.6 · 0.089)] = 1.314 ␮L = 3.294 × 10–5lbf-sec/ft2 . For the water, the density is 1.934 slugs/ft3. 1.934 · 14.76 · 1.417 Re = ᎏᎏᎏ = 1.23 × 106 3.294 × 10–5 Absolute roughness of rubber = 0.000492 ft. Relative roughness: 0.000492 ␧ ᎏ = ᎏ = 0.000368 DI 1.417

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0.25 fDL = ᎏᎏᎏᎏᎏ = 0.0162 [log10{(0.000368/3.7) + (5.74/(1.23 × 106)0.9)}2] iL = fDV 2/(2gDi) = 0.0162 · 14.762/(2 · 32.2 · 1.417) = 0.0387 ft/ft Using Equation 4.25, and substituting density with specific gravity

冦

冢

冣冤

兹苶 (3苶2.2 苶苶·苶 1.4 苶1 苶7 苶)苶 0.8 · 0.445 · (1.350 – 1.028) 1.350 – 1 im = 1 + 0.44 ᎏ + ᎏᎏ · ᎏᎏᎏ 1 1.9 · 0.75 · 14.76 1.0

冥冧

im = 0.0387[1 + 0.445(0.35)] + [0.0368)] = 0.0815ft/ft The presence of coal effectively doubles the head losses. The deposition velocity is expressed from Equation 4-24: [(1.350-1.028)/1.350]1/3 V3 = [1.417 · 32.2]1/2 ᎏᎏᎏ [0.0162 · 1.9 · 0.75]1/3 V3 = 14.71 ft/sec The coal slurry is therefore being pumped just above the deposition speed, and therefore at the minimum pressure gradient for horizontal pipelines.

4-4-4 Equations for Nickel–Water Suspensions Ellis and Round (1963) conducted tests on a mixture of nickel particles and water and derived the following equation: im – iL ␾ = ᎏ = K(␺)m = 385 ␺–1.5 CviL

(4-26)

The constants K and m are therefore different from those reported by Zandi and Govatos (1967) for sand particles, as expressed by Equation 4-20.

4-4-5 Models Based on Terminal Velocity Newitt et al. (1955) conducted tests in pipes smaller than 150 mm (6 in) and proposed to express Z in terms of the terminal velocity (instead of the drag coefficient).

␳s – ␳L gDiVt im – i Z = ᎏ = K2 ᎏ ᎏ Cvi ␳L V m3

冤

冥

(4-27)

where K2 = an experimentally determined constant. For small pipes, K2 = 1100. Vm = mean velocity of mixture For solids of different sizes, Newitt suggested a weighted mean diameter as n

dpm = 冱 dpimi/mt i=1

where mi = the mass of solids with particle diameter of dp mt = total mass of solids

(4.28)

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Hayden and Stelson (1968) proposed a modification of the Durand–Condolios equation using the terminal velocity instead of the drag coefficient: gDi[(␳m – ␳L)/␳L]Vt im – iL ᎏ = 100 ᎏᎏᎏ Cv iL V 2 兹g苶d苶苶苶苶)/ 苶L苶–苶苶) 1苶 p(␳ m苶␳

冤

冥

1.3

(4-29)

Geller and Gray (1986) pointed out that the equations of Durand, Newitt, and Babcock converged when m = –1. Newitt et al. (1955) minimized the importance of lift forces when a bed cannot form because of lift forces on particles. However, the work of Bagnold (1954, 1955, 1957) indicated that the submerged weight of particles separated from the bed was transmitted to the bed or the pipe wall under the same conditions. Thus, mechanical friction can contribute to head loss. It may be argued that sometimes it is easier to measure the terminal velocity rather than the drag coefficient, particularly with oddly shaped particles. As Chapter 3 clearly demonstrated, both parameters are interrelated. Example 4-9 The tailings from a small mine are pumped at a weight concentration of 40%. They consist of crushed rock at a specific gravity of 3.2. The d85 of the particles is 1mm. For a flow rate of 280 m3/hr, a smooth high-density polyethylene pipe with an internal diameter of 138 mm is selected. Using Newitt’s method as expressed By equations 4.27 and 4.29, determine the head loss due to the presence of solids, assuming a dynamic viscosity of 1.8 cP. Solution in SI Units Pipe flow area = 0.25 · ␲ · 0.1382 = 0.01496 m2 Average velocity of flow = Q/A = (280/3600)/0.01496 = 5.2 m/s Particle Reynolds number using the density of water = Rep = 0.001 · 3.71 · 1000/0.0018 = 2063 Since Rep > 800, the flow is turbulent and Newton’s law is used to calculate the terminal velocity: Vt = 1.74(dp · g · (␳p – ␳L)/␳L)1/2 = 1.74(0.001 · 9.81 · 2.1)1/2 = 0.25 m/s By Newitt’s method, the transition between saltation and motion occurs at 17Vt or V3 = 17 · 0.25 = 4.25 m/s Since the weight concentration is 40%, the specific gravity of the mixture is Sm = SL/(1 – (CW (Ss– SL)/Ss) = 1/(1 – 0.4(3.1 – 1)/3.1) = 1.372 The volumetric concentration is Cv = (1.372 – 1)/2.1 = 0.177 Using equation 4.27, and assuming K2 = 1100, Z = 1100 · (2.1) · (9.81 · 0.138 · 0.25/5.23) = 5.563 im/i = 1 + 0.177 · 5.563 = 1.985 Using equation 4.29:

冢

9.81 · 0.138 · 2.1 · 0.25 im – iL ᎏ = 100 ᎏᎏᎏ CviL 5.22[9.81 · 0.001 · 2.2)1/2 im/iL = 1 + 0.177 · 11 = 2.95

冣

1.3

= 11

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This example and the use of these two equations indicates that the empirical coefficients of 1100 in the Newitt method for fine coal and sand, or the empirical coefficient of 100 for sand from the Hayden and Stelson equation, do not converge for similar results. Testing would be recommended to confirm the magnitude of these coefficients.

4.5 DISTRIBUTION OF PARTICLE CONCENTRATION IN COMPOUND SYSTEMS The reader may be familiar with the concepts developed in the 1950s and 1960s on uniformly graded solid particles. In reality, slurries often consist of a wide distribution of particles. The coarser ones tend to move at the bottom of the horizontal pipe, and the finer ones move above these bottom layers. Understanding the distribution of these particles in layers above layers is essential for a correct estimation of the friction losses. Initially, the work was done in the 1930s and 1940s on open channel flows and is discussed in Chapter 6, Section 6-2-3. The distribution of volumetric concentration is shown to be a function of depth of the liquid in an open channel flow, raised to a exponent. The exponent is a function of the relation of the terminal velocity to the friction velocity. Ismail (1952) was the first to extend the approach of Vanoni to closed conduits. He focused initially on rectangular closed conduits. This test work demonstrated that the concentration was an exponential function:

冢冣

C Vt Log10 ᎏ = ᎏ (y – a) CA Es

(4-30)

where Es = the mass transfer coefficient a = height of layer A above bottom of the conduit y = distance from the lower boundary C = volumetric concentration of the particle diameter under consideration CA = volumetric concentration of height “A” For many pipes, C/CA is considered by Wasp et al. (1977) to be 0.08 DI from the top of the pipe. Wasp et al. (1977) examined the distribution of concentration of The Consolidation Coal Company’s Ohio coal pipeline at a height of 8% from the bottom of the conduit and at 8% from the top of the conduit; they reinterpreted the work of Ismail (1951) and devised the following equation:

冢

1.8 Vt C log10 ᎏ = – ᎏ CA ␤KxUf

冣

(4-31)

where Uf is the friction velocity (discussed in Chapter 2) Kx is the Von Karman constant ␤ = constant of proportionality Hsu et al. (1971) reexamined the work of Ismail by proposing a polar coordinate system (r, ␪) for the analysis of the distribution of concentration in a pipe: Vt r cos ␣ cos ␪ C(r, ␪) ᎏ = exp ᎏ ᎏ ᎏᎏ Uf RI me C(0, 0)

冤冢

冣冥

(4-32)

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where ␣ = the angle from the horizontal ␪ = angle from the vertical starting at the lowest quadrant point RI = inner diameter of pipe r = local radius for a point in the flow Equation 4-30 can be reduced to

冤冥

Vt C log10 ᎏ = ᎏ (constant) CA Uf

(4-33)

The extent by which the Von Karman constant Kx is suppressed by turbulence is difficult to assess. Ippen (1971) conducted an analysis of turbulent suspensions in open channel flows. This work showed that the concentration close to the lower boundary was the most important factor suppressing the Von Karman constant. This may not be astonishing when we consider that beds of coarse particles form in this region at low speeds. Hunt (1969) developed an equation for diffusion of heterogeneous flows: d(Cv) ES ᎏ + (1 – Cv)CvVt = 0 d(y)

(4-34)

where Cv is the volumetric concentration of solids. This equation shows that when coarse and fine particles are pumped together under certain conditions, the flows may exhibit an increase in concentration of fine particles with increasing height. Example 4-10 Using Hunt’s equation, prove that the ratio of concentration at 0.08 DI from the top is the concentration at pipe center expressed by

冤

冥

VR log10 ᎏ = –1.8 Z VRa where VR = Cv/1 – Cv a = the reference plane at 0.08 DI It has already been shown in Equation (4-31) that

冤冥

C Vt log10 ᎏ = –1.8 ᎏ CA ␤KxUf Let us confirm that Hunt’s approach applies: dCv Es ᎏ + (1 – Cv)CvVt = 0 dy Cv VR = ᎏ 1 – Cv DCv ᎏ = (1 – Cv)2 dVR

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冢

冣

dC dC dVR dVR ᎏ = ᎏ ᎏ (1 – Cv)2 ᎏ dy dVR dy dy But some of Hunt’s equation shows that –Vt(1 – Cv)Cv dC ᎏ = ᎏᎏ Es dy Then dV –Vt dC dVR ᎏ (1 – Cv) Cv = ᎏ · ᎏ = (1 – Cv)2 ᎏ Es dVR dy dy dVR –Vt ᎏ Cv = ᎏ (1 – Cv) Es dy dVR Es ᎏ (1 – Cv) + VtCv = 0 dy Or Cv dVR Es ᎏ + Vt ᎏ = 0 dy (1 – Cv) This is the same as the Equation 4-34. The approach discussed in the previous paragraph is sometimes classified as the distributed concentration approach. The analysis is based on establishing the plane for reference CA, usually at 0.08 diameter. It has been demonstrated that

冤冥

C Vt log10 ᎏ = – 1.8 ᎏ CA ␤KxUf If ␤ is assumed to be unity and there is no suppression for the Von Karman constant, i.e., Kx = 0.4, then

冤冥

冤冥

C Vt log10 ᎏ = –4.5 ᎏ CA Uf

(4-35)

Thomas (1962) commented that the Durand–Condolios approach was limited to sand and similar solids and proposed a more general criterion of evaluating flow of slurries in terms of the ratio Vt/Uf or ratio of free-fall velocity to friction velocity. He indicated that when Vt ᎏ > 0.2 Uf

(4-36)

the solids would be transported as a heterogeneous slurry. Charles and Stevens (1972) suggested that Equation 4.32 should be modified to correspond to C/CA < 0.13, whereas Charles and Stevens’ criterion corresponds to C/CA < 0.27. The Thomas criterion as expressed by Equation 4-31, corresponds to C/CA < 0.13, whereas the Charles and Stevens’ criterion corresponds to C/CA < 0.27.

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Thomas (1962) indicated that the minimum transport condition for particles depends on a number of factors, and derived the following equation for glass beads:

冢

Vt dpUf 0 ᎏ = 4.90 ᎏ Uf ␯

␯

␳S – ␳L

ᎏ冥冣冢 ᎏ DU 冣冤 ␳ i

0.60

f0

0.23

(4-37)

L

where ␯ = kinematic viscosity of water Uf 0 = friction velocity at deposition for limiting case of infinite dilution Thomas (1962) defined a critical friction velocity at which the slurry starts to deposit for a given concentration as

冦

冤冥冧

Vt 苶V苶) ᎏ Uf C = Uf 0 1 + 2.8 (兹C Uf 0

1/3

(4-38)

The approach of Thomas is implicit. It means that to predict Uf, it is important to measure friction loss as a function of velocity. It is then necessary to establish the deposition velocity using Equations 4-34, 4-35, and 4-36.

4-6 FRICTION LOSSES FOR COMPOUND MIXTURES IN HORIZONTAL HETEROGENEOUS FLOWS Many slurries resulting from dredging, cyclone underflow, and tailings disposal are not pumped with single-sized particles. Some authors such as Newitt et al. (1955) proposed the use of a weighted average particle diameter but Hill et al. (1986) proposed that the particles should be divided. The finer particles would move as a heterogeneous flow, while the coarser particles would move as a bed by saltation. The equations of friction loss for each fraction or size of solids should be calculated as in Sections 4-4-1 and 4-4-3. Hill et al. (1986), Wasp et al. (1977), and Gaesler (1967) demonstrated that this approach worked well when applied to pumping water–coal mixtures. The compound or heterogeneous–homogeneous system is the most important and most common in slurry transportation. It involves coarse and fine particles. The fines move as a homogeneous mixture while the remainder move as a heterogeneous mixture. To conduct this analysis, the rheological and physical properties of the solids must be known. This method was pioneered by Wasp et al. (1977) and in some respects was further developed by the “stratification model” described later on. The heterogeneous mixture or bed motion is based on the method of concentration in relation to a reference layer, as described by Equation 4-30. The method proposed by Wasp et al. (1977) can be summarized as follows: 1. Divide the total size fraction into a homogeneous fraction using Durand’s equation. 2. Calculate the friction losses of the homogeneous fraction based on the rheology of the slurry, assuming Newtonian flow. 3. Calculate the friction losses of the heterogeneous fraction using Durand’s equation. 4. Define a ratio C/CA for the size fraction of solids based on friction losses estimated in steps 2 and 3.

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5. Based on the value of C/CA, determine the fraction size of solids in homogeneous and heterogeneous flows. Re-iterate steps 2 to 5 until convergence of the friction loss. Example 4-11 A nickel ore slurry needs to flow by gravity at a weight concentration of 28%. The design flow rate is 1631 m3/hr. The slurry was tested in a 159 mm pipeline with a roughness coefficient of 0.016 mm at a weight concentration of 26.3%. The results of the pressure drop versus speed are presented in Table 4-2. No data was made available on the drag coefficients or terminal velocity of the solids. The particle size distribution of the originally milled ore is presented in Table 4-3. Special screens would be installed to screen away the coarsest particles (larger than 0.850 mm). Conducting a friction loss for a rubber lined steel pipe would be a better option. (See Tables 4-6 and 4-7.) The solids density was measured as 4074 kg/m3. At a weight concentration of 26.3%, this corresponds to a slurry density of 1244 kg/m3. Volumetric concentration is

␳m CV = CW ᎏ = 0.08% ␳s Using the Thomas–Einstein equation for dynamic viscosity correction:

␮ = ␮L(1 + (2.5 · 0.08) + (10.05 · 0.082) + 0.00273 exp(16.6 · 0.08)] = 1.274 · ␮L Analysis of Test Results Water at a temperature of 20° Celsius has a dynamic viscosity of 1 mPa · s. Slurry viscosity is therefore 1.274 mPa · s, and the Reynolds number is 1244(V)DI Re = ᎏᎏ = 155,256(V) = 294,986 1.274 × 10–3 where V = 1.9 m/s The slurry was tested in a pumping test loop. The lab tests indicated a pressure drop of 270 Pa/m at this velocity. The +0.850 mm solids were screened away prior to pump tests.

TABLE 4-6 Pressure Drop versus Speed in a 159 mm ID Steel Pipe at a Weight Concentration of 26.3% (Example 4-11) Temperature 20°C ______________________________________ Velocity (m/s) Pressure drop (kPa) 1.00 1.5 1.9 2.3 2.7 3.1 3.5 4.0

0.085 0.175 0.270 0.360 0.525 0.688 0.847 1.046

Temperature 35°C ______________________________________ Velocity (m/s) Pressure drop (kPa) 0.61 1.00 1.51 1.91 2.30 2.70 3.11 3.50 4.00

0.063 0.079 0.169 0.259 0.358 0.487 0.628 0.793 0.988

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TABLE 4-7 Particle Size Distribution Prior to Screening the Coarsest Solids (Example 4-11) Size (mm)

Volumetric concentration

+ 0.850 –0.850 to +0.400 –0.400 to +0.200 –0.200 to +0.105 –1.05 to +0.044 –0.044

14.3% 1.61% 1.91% 1.41% 1% 79.8%

Table 4-8 indicates the new volumetric concentration of the solids in the slurry after screening the +0.850 mm solids. The method developed by Wasp et al. (1977) has been used very successfully over the last 25 years for Newtonian slurries and will be used in the present calculations. The roughness of a steel pipe is 0.046 mm. Assuming that the –0.044 mm particles were transported by turbulence above the moving bed of coarser particles, the Swain–Jain equation may be used in the range of 5000 < Re < 100,000,000 to determine the friction coefficient of the homogeneous part of the mixture: 0.25 fD = ᎏᎏᎏᎏ = 0.017 {log10 [(␧/Di)/3.7 + 5.74/Re0.9]}2 where fD = the Darcy friction factor For the density of 1244 kg/m3, the pressure losses of the carrier fluid (including the –0.044 mm) at a first iteration is therefore 0.017(1.92) 1244 Loss = ᎏᎏ = 240 Pa/m (2) 0.159 The lab test measured 270 Pa/m; the losses due to the moving bed are therefore 31 Pa/m. Using Table 4-8, apply the Wasp method for calculating the pressure losses of the moving bed. It will be assumed initially that the –0.044 mm particles are part of the homogeneous liquid layer above the bed. It is essential first to determine the drag coefficient and the particle Reynolds number.

TABLE 4-8 Particle Size versus Volume Concentration in the Slurry (Example 4-11)

Particle size (mm)

Original volumetric concentration CV in the solids

New volumetric concentration CV in the solids (after screening)

Volumetric concentration in the slurry (at overall solids CV of mixture at 8%)

+0.850 –0.850 to + 0.400 –0.400 to + 0.200 –0.200 to +0.105 –1.05 to +0.044 –0.044

14.3% 1.61% 1.91% 1.41% 1% 79.8%

— 1.88% 2.23% 1.65% 1.17% 93.1%

— 0.15% 0.178% 0.132% 0.093% 7.45%

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Two cases will be considered: spheres and particles with an Albertson shape factor of 1.0 for the sake of simplicity. To calculate the particle Reynolds number, the density of 1244 kg/m3, viscosity of 1.3 mPas, and the speed of 1.9 m/s of the carrier fluid are used: Rep = 1,818,154 dp where dp = the average particle size. To calculate the drag coefficient of a sphere, the Turton equation (Equation 3.8a) is used. Results are summarized in Table 4-9. Wasp et al. (1977) recommend using Durand’s equation for each fraction of solids to determine the increase in pressure losses due to the moving bed: gDi(␳s – ␳L)/␳L ⌬Pbed = 82 ⌬PLCvbed ᎏᎏ V 2兹苶 C苶 D

冤

冥

1.5

After determining the Darcy friction factor at the pipe diameter of 0.159 m and the speed of 1.9 m/s at a liquid loss of 219 Pa/m, the loss due to each fraction becomes

冤

1 ⌬Pbed = 17,490 Cvbed ᎏ 兹苶苶 CD

冥

1.5

Results of calculations are presented in Table 4-10. The total friction loss is therefore 240 Pa/m + 151.4 = 391.4. By comparison with the measured 270Pa/m, the calculations for the bed are higher and can be refined by the method of concentration using Equation 4-30:

冤冥

C Vt log10 ᎏ = –1.8 ᎏ CA ␤KxUf At 391.4 Pa/m, the equivalent fanning factor is

␳ 391.4 = 2 ff V 2 ᎏ Di 391.4(0.159) fN = ᎏᎏ = 0.0069 2(1.92)1,244 To calculate Uf, use Equation 2-25 from Chapter 2: /苶)苶 = 1.9兹(0 苶.0 苶0苶6苶/2 苶)苶 = 0.1116 m/s U = Um兹(苶fN苶2 Assuming Kx = 0.4 and ␤ = 1, we can iterate the results.

TABLE 4-9 Drag Coefficient for Particles in Example 4-11, Assuming Spherical Shape Particle size distribution (mm)

Average particle size (mm)

Particle Reynolds number

Drag coefficient for a sphere

Drag coefficient for a particle with shape factor of 1

–0.850 to + 0.400 –0.400 to + 0.200 –0.200 to +0.105 –1.05 to +0.044

0.63 0.3 0.15 0.07

1145 545 272 127

0.395 0.545 0.706 1.02

0.474 0.572 0.7413 1.07

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TABLE 4-10 Calculated Losses for Each Fraction of Solids in the Moving Bed in the Lab Test (Example 4-11) Particle size distribution (mm)


–0.850 to + 0.400 –0.400 to + 0.200 –0.200 to +0.105 –1.05 to +0.044 Total for bed

0.63 0.3 0.15 0.07

Calculated losses for spherical particles (Pa/m)

Calculated losses for particles with Albertson shape factor of 1.0 (Pa/m)

58.31 53.85 32.9 17.56 162.62

50.87 51.93 31.73 16.87 151.4

To determine the terminal velocity, we turn to Chapter 3, Equation 3-7: 4(␳S – ␳L) gdg CD = ᎏᎏ 3␳LV 2t 4 (4.074 – 1.244) 9.81 dg V 2t = ᎏᎏᎏ 3 (1.244) CD 29.76 dg V 2t = ᎏ CD The iterated pressure loss is 349.7 Pa/m, which is still higher than the measured 270 Pa/m. For further iteration, the fanning factor must be recalculated:

␳ 349.7 = 2ff V 2 ᎏ Di 349.7 (0.159) ff = ᎏᎏ = 0.00616 2 (1.92) 1244 Uf = 0.106 m/s With this new iteration we are converging toward 107 + 240 = 347, which is above the measured 270Pa/m. Ellis and Round (1963) indicated that Durand’s equation coefficient of 82 was too high for nickel suspensions. We may therefore divide 270/347 = 0.778 to obtain the new value of 63.8 for K. Pipeline Sizing for the Design Flow Rate of 1631 m3/hr at a Weight Concentration of 28% The weight concentration of 28% corresponds to a volumetric concentration of 8.7% and a mixture density of 1267 kg/m3 using the solids density of 4074 kg/m3. The concentration of solids in the bed is tabulated in Table 4-11. The flow of 1631 m3/hr corresponds to 0.453 m3/s. Consider a 20⬙ OD pipe with a wall thickness of 0.375⬙, rubber lined with a rubber thickness of ¼⬙. The internal diameter of the pipe would be DI = [20 – 2(0.375+0.25)] = 18.75⬙ or 477 mm. The cross-sectional area of the pipe would be 0.178 m2 and the average flow speed of the slurry would be calculated as V = 0.453/0.178 = 2.55 m/s. Applying the Thomas–Einstein equation to the volumetric concentration of 8.7% gives an

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TABLE 4-11 Iteration for Calculated Losses for Each Fraction of Solids in the Moving Bed, Based on the Distribution of Concentration—for Lab Tests (Example 4-11)

Particle size distribution (mm) –0.850 to + 0.400 –0.400 to + 0.200 –0.200 to +0.105 –1.05 to +0.044 Total for bed

Average particle size (mm) 0.63 0.3 0.15 0.07

Drag coefficient Terminal for a velocity sphere (mm/s) 0.395 0.545 0.706 1.02

6.89 4.047 2.515 1.43

–1.8 Vt␤ · Kx ·Uf

Iterated concentration C/CA

–0.27 –0.163 –0.1015 –0.057

0.537 0.687 0.79 0.877

Iterated pressure loss (Pa/m) 31.31 36.97 26 15.4 109.68

effective viscosity of the mixture of 1.305 mPa · s at 20° C. The pipeline Reynolds number is therefore 1267(2.55) 0.477 Re = ᎏᎏ = 1,180,931 1.305 × 10–3 For commercially available rubber-lined pipes, the roughness is 0.00015 m. Considering a 477 mm ID pipe, rubber lined, the relative roughness is therefore 0.000315. Applying the Swamee–Jain equation, the Darcy friction factor is calculated as fD = 0.01578. Loss of carrier fluid is calculated as 0.01578 (2.552) 1,267 ᎏᎏᎏ = 136.3 Pa/m 2 (0.477) Using the Wasp method, and applying the Durand’s equation, the calculations yield

冤

9.81 ⌬Pbed = 63.8 (136.3) ᎏᎏ 2.552兹苶 CD 苶

冤

1 ⌬Pbed = (18,216) Cvbed ᎏ 兹苶 CD 苶

冥

1.5

冥

1.5

The drag coefficient is calculated at the particle Reynolds number using the speed of 2.55 m/s, viscosity of 1.305 mPa · s, and density of 1267 kg/m3. Rep = 2,475,747 (dp). Results are presented in Table 4-12. The Durand equation may then be applied to each fraction of solids. The results are shown in Table 4-13. Total losses for slurry mixture are therefore calculated as 136.3 + 165.9 = 302 Pa/m. At 302 Pa/m, the equivalent fanning factor is

␳ 302 = 2ff V 2 ᎏ Di 302 (0.477) ff = ᎏᎏ = 0.0089 2 (2.552) 1244

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To calculate Uf, we use Equation 2-15 from Chapter 2:

冪莦

冪莦

0.0089 ff Uf = U ᎏ = 2.55 ᎏ 2 2 Uf = 0.170 m/s

Assuming Kx = 0.4 and ␤ = 1, we can iterate the results based on the distribution of concentration, as per Table 4-14. Total friction losses = 136 + 129 = 265 Pa/m or 0.0217 m/m.

TABLE 4-12 Second Iteration for Calculated Losses for Each Fraction of Solids in the Moving Bed, Based on the Distribution of Concentration—Lab Tests (Example 4-11)

Particle size distribution (mm) –0.850 to +0.400 –0.400 to +0.200 –0.200 to +0.105 –1.05 to +0.044 Total for bed


Drag coefficient Terminal for a velocity sphere (mm/s)

0.63 0.3 0.15 0.07

0.395 0.545 0.706 1.02

–1.8 Vt␤ · Kx ·Uf

6.89 4.047 2.515 1.43


–0.287 –0.173 –0.108 –0.061

0.516 0.671 0.78 0.868

Iterated pressure loss (Pa/m) 30.1 36.13 25.66 15.24 107

TABLE 4-13 Drag Coefficient of the Solids in the Pipeline (Example 4-11) Particle size distribution (mm)


Particle Reynolds number

Drag coefficient for a sphere


–0.850 to + 0.400 –0.400 to + 0.200 –0.200 to +0.105 –1.05 to +0.044

0.63 0.3 0.15 0.07

1547 743 384 186

0.414 0.493 0.602 0.827

0.497 0.52 0.632 0.861

TABLE 4-14 Calculated Loss for Each Fraction of Solids in the Moving Bed in the 20⬙ Pipeline (Example 4-11)

Particle size distribution (mm) –0.850 to + 0.400 –0.400 to + 0.200 –0.200 to +0.105 –1.05 to +0.044 Total for bed



Volumetric concentration in the slurry (at overall solids CV of mixture at 8.7%)

Calculated losses for particles (with the Albertson shape factor of 1.0 (Pa/m)

0.63 0.3 0.15 0.07

0.497 0.52 0.632 0.861

0.164% 0.194% 0.144% 0.102%

50.47 57.71 37 20.79 165.97

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TABLE 4-15 Iteration for Calculated Losses for Each Fraction of Solids in the Moving Bed, Based on the Distribution of Concentration—for 20⬙ Pipeline (Example 4-11)

Particle size distribution (mm) –0.850 to +0.400 –0.400 to +0.200 –0.200 to 0.105 –1.05 to +0.044 Total for bed

Average particle size (mm) 0.63 0.3 0.15 0.07

Drag coefficient Terminal for a velocity sphere (mm/s) 0.497 0.52 0.632 0.861

6.14 4.14 2.65 2.42

–1.8 Vt␤ · Kx ·Uf


–0.163 –0.1093 –0.07 –0.063

0.687 0.777 0.851 0.86

Iterated pressure loss (Pa/m) 34.7 44.85 31.5 17.88 128.93

The purpose of Example 4-11 was to demonstrate the method developed by Wasp. A number of pipelines have been constructed around the world using this technique and the practical engineer needs to be familiar with this method as well as with the two-layer model and stratified flow models that we will explore later. The following computer program is based on this methodology. CLS DIM dp(50), cvdp(50), rep(50), vt(50), cvn(50), dpbed(50), cd(50) DIM cvind(50), dpav(50), z(50), cca(50), dpnew(50), dfbed(50) pi = 4 * ATN(1) DEF fnlog10 (X) = LOG(X) * .4342944 INPUT “name of ore and project”; ore$, proj$ INPUT “date “; dat$ INPUT “your name please “; name$ PRINT “ please choose between the following system of units” PRINT “ 1- SI units” PRINT “ 2- US Units” PRINT INPUT “ 1 or 2”; ch 10 PRINT IF rt$ = “Y” OR rt$ = “y” THEN PRINT “ .8) AND (nf < 1.5) THEN GOSUB increase IF (nf > .8) AND (nf < 1.5) THEN GOSUB are IF (nf > .8) AND (nf < 1.5) THEN PRINT “flow is critical” IF nf < 1.5 THEN nff = 1 IF nf > = 1.5 THEN nff = 0 IF nf < 1.5 THEN GOTO 456 30001 GOSUB angle PRINT “perimeter”; per PRINT “area “; area rh = area/per PRINT “hydraulic radius”; rh

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INPUT “HIT ANY KEY TO CONTINUE”; l$ RETURN ushape: RETURN froude: nf = v1/SQR(g * mhd) IF nf < .8 THEN PRINT “flow is subcritical” IF (nf > .8) AND (nf < 1.5) THEN PRINT “flow is critical” PRINT “froude number = “; nf RETURN friction: a = -1.378 * (1 + .146 * EXP(–.000029 * he)) PRINT “reynolds “; re m = 1.7 + 40000/re PRINT USING “factor a = ###.###### and exponent m = ##.###”; a; m PRINT INPUT “hit any key to continue “; kkkkkkk$ FTU = (10 ^ a) * re ^ (-.193) PRINT “ft = “; FTU PRINT fl = (16/re) * (1 + he/(6 * re)) PRINT “fl = “; fl ff = (fl ^ m + FTU ^ m) ^ (1/m) fd = 4 * ff IF c > 1 THEN GOTO 666 PRINT USING “in absence of roughness fanning = #.###### and darcy = #.######”; ff; fd [A section of the program here lists all types of materials and their roughness as explained by table 6-2, it is not reproduced here to save space em refers to absolute roughness in meters and emf in ft] PRINT USING “estimated roughness for new system = ##.##### m ##.### ft”; em; emf 666 FOR i = 1 TO 20 fd2 = fd ro = (em/(3.7 * 4 * rh) + 2.51/(re * SQR(fd))) h = -2 * fnlog10(ro) fd = h ^ -2 NEXT i dg = fd2 - fd PRINT “revised darcy factor to account for roughness”; fd PRINT PRINT “iteration error on darcy “; dg ch2 = SQR(8 * g/fd) n2 = rh ^ (1/6)/ch2 PRINT USING “Chazy No = ###.## and Manning number = #.##### (including roughness)”; ch2; n2 s2 = fd * v1 ^ 2/(8 * rh * 9.81) sm = s2 * 100 PRINT USING “recommended slope = ##.### % “; sm PRINT RETURN settling: REM check for any coarse particles being transported in a Non-New-

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SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

6.53

tonian mixture PRINT “iteration on settling speed for particles using Camp equation” INPUT “particle size (mm) “; dp dp2 = .001 * dp/ft v2 = SQR((8 * .8 * 32 * dp2 * (dens/1000 - 1))/fd) v2m = v2 * ft PRINT USING “SETTLING SPEED = #.## m/s ##.## ft/s”; v2m; v2 IF v1 < (v2m * 2) THEN PRINT “warning settling speed is higher than half of average speed” RETURN gradient: ‘grad = (2 * vu/dens) ^ (–.5) * (((fd/(4 * rh)) ^ .5) * v1 ^ 1.5) grad = (dens * q * 9.81 * s2/(area * vu)) ^ .5 PRINT USING “velocity gradient = ###.## sec-1”; grad RETURN depth1: d2 = .1 * r1 777 LE = r1 - d2 beta = fnacos(LE/r1) PRINT “angle beta”; beta ‘INPUT “hit any key to continue”; lllll$ A3 = r1 ^ 2 * (beta - SIN(beta) * COS(beta)) IF A3 < (.975 * area) THEN d2 = d2 + .01 * r1 IF A3 < (.975 * area) THEN GOTO 777 IF A3 > (1.025 * area) THEN dpf = 1 IF A3 > (1.025 * area) THEN GOSUB depth2 PRINT “DEPTH OF SLURRY”; d2 dep = d2 ‘INPUT “hit any key to continue”; k$ RETURN depth2: IF dpf = 1 THEN GOTO 778 d2 = .9 * r1 778 LE = d2 - r1 beta = FNASN(LE/r1) REM next line changed for rev 1.02 - pi in front of beta removed A3 = pi * r1 ^ 2/2 + beta * r1 ^ 2 + r1 ^ 2 * SIN(beta) * COS(beta) IF A3 > 1.025 * area THEN d2 = d2 - .01 * r1 IF A3 > 1.025 * area THEN GOTO 778 IF A3 < .975 * area THEN GOSUB depth1 dep = d2 depus1 = dep/.0254 PRINT USING “depth = ##.### m ###.### in”; dep; depus1 INPUT “hit any key to continue”; k$ RETURN angle: IF dep < r1 IF dep > r1 IF dep = r1 IF dep < r1 IF dep > r1 per = theta RETURN

THEN THEN THEN THEN THEN * r1

theta theta theta theta theta

= = = = =

fnacos((dep - r1)/r1) FNASN((dep - r1)/r1) pi/2 2 * theta 2 * theta + pi

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Flow may accelerate at bends due to the formation of centrifugal forces. The velocity profile is then distorted (Einstein and Hardner, 1954).

6-10 SLURRY FLOW IN CASCADES Cascades are important mechanisms for the transportation of slurry. They are steep open channels and are associated with a high Froude number and steep gradients. Stricklen (1984) suggested that cascades be used on slopes between 5% and 65% with velocities in excess of 10 m/s (33 ft/sec). At these magnitudes of speed, excessive wear would occur on the walls of the open channel cascade. There are three types of boxes to consider for reducing the speed: 1. Cascade feed box (Figure 6-18) 2. Cascade receiving sump (Figure 6-19) 3. Siphon feed box (Figure 6-20) Stricklen (1984) suggested that under certain conditions the localized solid concentration may exceed 65% by volume and may cause a pattern of “slug” flow with considerable localized wear. To mitigate against this problem, while controlling the speed, he suggested that the launder be designed as wide as possible to reduce the hydraulic radius and depth of the flow, but still narrow enough as to avoid slug flow. Two parameters need to be computed in order to check for localized slug flow. 1. The Vedernikov number Ve: U 2 bw Ve = ᎏ ᎏ ᎏᎏ 3 Pw (gym cos ␬)1/2

Low entry slope

(6-82)

Side ventilation window (recommended for deep drops) Na ppe of slurry

Worn-out mill liner used to absorb wear D Worn-out pump liner used to absorb wear Minimum D/3 Fig 6-19

Steep outlet cascade FIGURE 6-18

Entry into a cascade feed box from a low-slope launder.

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steep cascade at inlet

nappe of slurry (ventilation window not shown) low slope for outlet launder worn out mill liner used to absorb wear worn out pump liner used to absorb wear

FIGURE 6-19

Entry into a cascade receiving sump from a steep launder.

feed pipe pipe tee fitting

discharge pipe

Fig 6-21

FIGURE 6-20

Siphon feed pipe drop box.

2. The Montuori number M: U2 M 2 = ᎏᎏ gSL cos ␬

(6-83)

where bw = bottom width of the channel Pw = wetted perimeter of the channel ␬ = tan–1(h/L) = tan–1 S L = length of the channel Figure 6-21 shows a linear limit between the Vedernikov and the Montuori numbers. Below the line, no slug flow occurs and the flow is stable. Above the line, slug flow occurs.

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FIGURE 6-21 The correlation between the Vedernikov number and the square of the Montuori number squared is used to differentiate between slug and no-slug flows. (From Stricklen, 1984.)

If the calculations of the Vedernikov and Montuori numbers indicate that the flow is of a slug type, it will be necessary to determine the intermediate points from which unstable rolling waves would be generated. Niepelt and Locher (1989) as well as Stricklen (1984) proposed to compute a shape factor for the chute: ym x= ᎏ Pw where Pw = wetted perimeter ym = average depth of the slurry in the channel Steep launders may cause the formation of roll-waves that are associated with instability. The Vedernikov number may be used as a design guide to determine these areas. Niepelt and Locher (1989) extended the analysis to slurries and showed a marked difference with water flows (Figure 6-22).

6-11 HYDRAULICS OF THE DROP BOX AND THE PLUNGE POOL Certain remote mines in mountainous regions have chosen over the years to dispose of their tailings at sea level and sometimes to submerge them in the sea. The drop box has been found to be an effective method to achieve energy dissipation during transportation. There are particular design criteria that the drop box or receiving sump must meet to avoid rapid wear of its walls:

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gsL cos (␬) 1 ᎏ2 = ᎏᎏ M U2 FIGURE 6-22 The Vedernikov number is used as a design guide to determine roll waves associated with steep cascades. There is, however, a marked difference between water and slurries. (From Niepelt and Locher, 1989, reprinted by permission of SME.)

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앫 The incoming liquid or nappe should impact the slurry liquid surface in the drop box and not the bottom surface or walls. 앫 The sump should be sized sufficiently large for its walls to be outside the computed area of impingement or high turbulence. 앫 If slug flows or flows at high Froude numbers are allowed to enter the receiving sump, the sump should be fairly long to cope with the fluctuations of flows. 앫 A weir may be installed in the receiving sump to reduce the length of the hydraulic jump. 앫 Froth arresters are recommended for frothy slurries. 앫 The area of high turbulence or the exit from the receiving sump may have to be covered to avoid overfills. The design of such sumps is far from easy. In the next section, the mathematics of the slurry fall will be presented to the reader in a brief practical approach. Excellent books on the engineering of small dams are available for further reading. One question often asked is what is the recommended depth of a plunge pool. The rule of thumb in the case of water is that the plunge pool should be one-third the depth of the waterfall. That means that for a waterfall drop of 30 m one would need to provide an additional depth of 10 m to absorb all the turbulence. This is not always possible to achieve, and energy dissipaters are then introduced to absorb the turbulence. In mining, these energy dissipaters are often worn-out mill liners, pump liners, or impellers that are put at the bottom of the plunge pool to wear away as they absorb the impact of abrasive slurry fall. In this chapter, we shall consider the more common drop box found in many mining plants. The economics and the size of many projects, as well as wear considerations, often reduce the problem to rectangular or circular drop boxes. Other forms of energy dissipaters such as ogees and ski jumps that are discussed in certain books on civil engineering have not found application in mining because of the problem of lining such complex shapes. For a rectangular entry into the fall, the analysis of this problem is based on dividing flow rate Q by the width of the launder before the fall: Q qb = ᎏ w

(6-84)

The following analysis assumes a constant width of the launder starting well upstream from the fall. If y is the depth of the liquid well upstream of the fall, and V is the velocity of the liquid, as in Figure 6-23, the total energy is V2 H=y+ ᎏ 2g

(6-85)

If the flow is subcritical well upstream from the fall, it will tend to accelerate near the fall. Rubin (1997) demonstrated that the minimum energy head for a waterfall occurs when the flow prior to the drop is in a critical regime with a Froude number of 1.0. Under such conditions, the flow accelerates toward the brink of the fall, thus reducing the depth Yb, which according to Fathy and Shaarawi (1954) would be Yb ᎏ = 0.716 Y0

(6-86)

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flow per unit length q =Q/b b Total Energy Line

2

(V /2g)

Y

Y0 subcritical flow

3

Y0 =

flow Q

2

Y = 0.716 width "b"

Q /b

5 Y0 VENTILATION AIR

FIGURE 6-23

Entering a waterfall with minimum energy gradient.

The Froude number of 1.0 occurs five times the critical depth upstream of the brink. The critical depth is defined as

冢冣

q 2b Y0 = ᎏ g

1/3

(6-87)

For water flow, the critical slope is expressed in terms of the critical depth and the Manning roughness factor as 1 gn2 S0 = ᎏ ᎏ Fr [Y0]1/3

(6-88)

But since Fr = 1.0, Equation 6-88 is also expressed as gn2 S0 = ᎏ [Y0]1/3 Obviously, for slurries with different roughness values due to the deposition of sediments or formation of antidunes, Equation (6-88) is not readily applicable. From the point of view of the designer of a slurry drop box, it is important to determine the area of impingement of the jet, the depth of the backwater, and the area of the still water, in order to provide proper liners and protection from wear. The nappe must be properly ventilated, as in Figure 6-24; otherwise the slurry may tear the structure apart. It may appear strange to the reader that the author is focusing on the case of minimum energy with entry in a subcritical flow, although we have reiterated in previous sections of this chapter the need to maintain a supercritical flow for slurries in launders. The minimum energy entry is a case of reference used to understand more complex flows at high Froude number in which the projection of the nappe is even further away. There are cases in which entry is at minimum energy, such as from a lake into a river, or from a large tailings pond into an open channel, or from a relatively horizontal channel into a large drop box used for sampling the tailings. In fact, entering the fall at minimum energy allows for a better capture of samples for analysis (Figure 6-25).

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FIGURE 6-24 This drop box for a large tailings flow features three 24⬙ ventilation windows in each side wall to permit ventilation under the nappe.

The energy dissipation at the bottom of the fall was discussed in detail by Moore (1943) and Rand (1955). The hydraulics of such a fall will therefore be summarized here for practical design considerations, with focus on the main equations. Rand observed three different flows for a waterfall with a well-ventilated nappe, which are depicted in Figures 6-26 to 6-27. In the first case, Case A (Figure 6-27), the flow approaches the crest of the waterfall in a subcritical regime. The flow is characterized by a nonsubmerged nappe at the point of impingement with the apron. Rand indicated without definite proof that the height of the liquid at the crest is 0.715 of the critical depth. The region between the wall and the nappe is called the under-nappe. It has a depth df which is higher than the flow downstream of the point of impingement. In the undernappe, the flow is recirculating. As the nappe hits the apron, it turns smoothly into supercritical regime at a distance Ld from the wall. This distance Ld is called the drop distance. At the point of impingement, the depth of the stream reaches a minimum with a depth d1 at Ld from the wall. After d1, the flow depth increases smoothly while remaining in a supercritical regime until a certain distance Lj and a depth db, where a stationary hydraulic jump occurs between the supercritical and subcritical flows. The depth of the flow increases until a steady level is reached, d3, called the tail water depth. Case B (Figure 6-28) is described by Rand as a borderline case. By comparison with Case A, the flow is critical or slightly supercritical before the crest of the fall. There is no relative distance between d1 and d3, and the hydraulic jump occurs practically at the region of the impingement with the apron and extends over a distance L until a steady-state d2 is reached for the tail water. The nappe is not submerged, but there is no supercritical flow over the apron, so the distance between the region of impingement and the tail water is considered the shortest of the three cases.

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b

Total Energy Line

Page 6.61

2

(V /2g) Y0

6.61

subcritical flow

3

Y0 =

flow Q

2

Q /b

5 Y0

travel of sampling bucket Ventilation air

Sample of slurry FIGURE 6-25

Sampling tailings with a moving bucket crossing the nappe in a tailings drop box.

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subcritical flow

L y c

j

C

ventilation D

B d

d

df

1

d

A d

Lp L Fig 6-30

dL r b

L d2

Lr > Lb Case (C)

FIGURE 6-29

Free fall with a submerged nappe (after Rand, 1955).

Under the nappe, a region of still water develops to a depth df. The intersection of this rotating water with the nappe is at point B of Figure 6-29. The height is df, expressed as df ᎏ = Dr0.22 Dd

(6-93)

The height of the liquid d1 is expressed as d1 ᎏ = 0.54Dr0.425 Dd

(6-94)

The height of the liquid d2 in case (b) for entry in a critical regime is expressed as d2 ᎏ = 1.66Dr0.27 Dd

(6-95)

And the length to the intersection can be expressed by length Lp or LpB = 1.98[Y0(Dd + 0.357Y0 – df)]1/2

(6-96)

The drop length or the length between the drop wall and the location of minimum depth of the liquid at the jump dj in Figure 6-26 at point A is expressed as Ld 1.98(1 + 0.357 Y0/Dd)兹(Y 苶苶苶苶 0/D d) ᎏ = ᎏᎏᎏᎏ Dd 兹[1 苶苶 +苶0.3 苶5 苶7 苶(Y 苶苶苶苶 –苶(d苶f苶 /D苶苶 0/D d)苶 d)]

(6-97)

Finally, the total length of the hydraulic jump from the point dj to the point where the tail–water has stabilized can be expressed as

冢

d2 Lr d1 ᎏ =6 ᎏ – ᎏ Dd D Dd

冣

(6-98)

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These equations are based on proper ventilation of the nappe. If the nappe is not properly ventilated, it becomes semiattached or totally attached to the drop box wall. This leads to a condition where flows may cause vibration of the drop box, which may tear it apart if it is not structurally designed to handle the vibration. The equations of Walter Rand were developed for waterfalls. They are a good reference for designing drop boxes. Unfortunately, very little has been published over the years to examine the effect of solids on the level of turbulence at the toe of the nappe and on the magnitude of the various parameters. Example 6-12 A mass of liquid approaches a free fall at a Froude number of 1.0. The height of the liquid at the brink is measured to be 1.2 m (3.94 ft). The fall is 6 m (19.48 ft) deep. It is assumed that the width of the channel and drop box remain uniform. Determine the geometry of the hydraulic jump at the apron. Solution in SI Units From Equation 6-86: Yb ᎏ = 0.716 Y0 or Y0 = 1.2/0.716 = 1.676 m (or 5.499 ft). The Froude number of 1.0 occurs five times the critical depth upstream of the brink. The critical depth is defined as Y0 = [q b2/g]1/3, so 3 苶.6 苶7 苶6 苶苶 ·苶9苶.8 苶1 苶)苶= 苶苶 6.8 苶1 苶苶 m2苶/s苶 qb = 兹(1

From Equation 6-88, the drop number Dr is 6.812 q b2 Dr = ᎏ = ᎏᎏ = 0.0219 3 (gDd ) (9.81 · 63) The toe of the nappe is determined from Equation 6-90: Lp ᎏ = 1.98 [Dr1/3 + 0.357 Dr2/3]1/2 = 1.98 [0.02191/3 + 0.357 (0.02192/3)]1/2 = 1.098 Dd Lp = 1.098 × 6 = 6.6 m This point is also called the toe of the nappe. The location of the hydraulic jump is obtained from Equation 6-91: Ld ᎏ = 4.30Dr0.27= 4.3 × 0.02190.27 = 1.533 Dd Ld = 1.533 × 6 = 9.195 m The hydraulic jump occurs after the toe of the nappe. Under the nappe, a region of still water develops to a depth df, expressed by Equation 6-93 as df ᎏ = Dr0.22= 0.02190.22 = 0.4314 Dd df = 0.4314 · 6 = 2.59 m If this were slurry, it would be recommended to line this area to a height of 3 m by the length of Lp (6.59 m). The height of the liquid d1 is expressed by Equation 6-94:

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d1 ᎏ = 0.54Dr0.425= 0.54 × 0.02190.425 = 0.1064 Dd d1 = 0.1064 × 6 = 0.6386 m The height of the liquid d2 is expressed by Equation 6-95: d2 ᎏ = 1.66 Dr0.27= 1.66 × 0.02190.27 = 0.5916 Dd d2 = 0.5916 × 6 = 3.55 m The distance between d1 and d2 or length of the hydraulic jump is Lr ᎏ = 6 (d2/Dd – d1/Dd) = 6(0.5916 – 0.1064) = 2.911 Dd Lb = 2.911 × 6 = 17.47 m This length should be lined to the height of d2 + 10% or approximately 4 m. Solution in USCS Units From Equation 6-86: Yb ᎏ = 0.716 Y0 or Y0 = 3.94/0.716 = 5.499 ft. The Froude number of 1.0 occurs five times the critical depth upstream from the brink. The critical depth is defined as Y0 = [q b2/g]1/3, so qb = (5.4993 · 32.2) = 73.17 ft2/sec From equation 6-89, the drop number Dr is qb2 Dr = ᎏ = 73.172/(32.2 · 19.483) = 0.022 (gD d3) The toe of the nappe is determined from Equation 6-90: Lp ᎏ = 1.98[Dr1/3 + 0.357 Dr2/3]1/2 = 1.98[0.0221/3 + 0.357 (0.0222/3)]1/2 = 1.099 Dd Lp = 1.099 × 19.48 = 21.4 ft This point is also called the toe of the nappe. The location of the hydraulic jump is obtained from Equation 6-91: Ld ᎏ = 4.30Dr0.27 = 4.3 × 0.0220.27 = 1.53 Dd Ld = 1.53 × 19.48 = 29.80 ft The hydraulic jump occurs after the toe of the nappe. Under the nappe, a region of still water develops to a depth df, expressed by Equation 6-93 as df ᎏ = Dr0.22 = 0.0220.22 = 0.432 Dd

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df = 0.432 · 19.48 = 8.42 ft If this were slurry, it would be recommended to line this area to a height of 10 ft by the length of Lp or approximately 21.6 ft. The height of the liquid d1 is computed from Equation 6-94: d1 ᎏ = 0.54Dr0.425 = 0.54 × 0.0220.425 = 0.1064 Dd d1 = 0.1064 × 19.48 = 2.07 ft The height of the liquid d2 is computed from Equation 6-95: d2 ᎏ = 1.66 Dr0.27 = 1.66 × 0.0220.27 = 0.5916 Dd d2 = 0.5916 × 19.42 = 11.49 ft The distance between d1 and d2 or length of the hydraulic jump is Lb ᎏ = 6 (d2/Dd – d1/Dd) = 6(0.5916 – 0.1064) = 2.911 Dd Lb = 2.911 × 19.48 = 56.71 ft This length should be lined to the height of d2 + 10% or approximately 12.6 ft.

6-12 PLUNGE POOLS AND DROPS FOLLOWED BY WEIRS In nature, the scouring depth of a waterfall may be typically one third of the depth of the waterfall. An example of an engineering exercise along these lines was the construction of Mossyrock spillway on the Colwitz River near Tacoma, Washington (U.S.A.). The spillway was created to handle a 183 m (600 ft) drop. In the case of slurries, the wear is accelerated by the very nature of the abrasive and erosive particles. Spent mill liners, spent mill balls, steel grading, and spent pump liners are installed at the bottom of drop boxes to prevent wear. It is not always cost effective to design for a scouring depth equal to one third of the free fall. A drop box can be expensive to construct. One of the largest slurry drop boxes was built by Fluor Daniel for the Caujone mine owned by the Southern Peru Copper Corporation in Peru. It was designed to handle a tailing flow of 7.3 m3/s (116,000 gpm). The drop was 10 m (32 ft) (Figures 6-24 and 6-30) deep and the slurry had to be redirected under an existing truck road. The author was the hydraulic engineer on the project. To reduce the length of the pond, it is recommended to add a weir (Windsor, 1938). This alternative method is included in the discussion of the paper of Moore (1943) by L. S. Hall (1943). On the basis of the work of Blackhmereff (1936), Hall developed an approach to reduce the length of the transition region at the toe of the nappe by adding a weir. The weir raises the water level and causes the nappe to impinge water at a higher point of intersection. Referring to Figure 6-30, the length of the pond can be reduced to L⬘. If Dd is the depth of the drop, an energy line E0 is defined as E0 = Dd + 1.5Y0

(6-99)

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Y0 /2

steep cascade at inlet Y0 Z0

E0 D d

d 2 hw

Dd L'

L'

Fig 6 - 32

2 L'

FIGURE 6-30 A weir to control the flow of slurry from the nappe of a drop box. (After Hall, 1943 in his discussion of Moore, 1943.)

The level of the liquid over the weir Z0 can be expressed graphically as in Figure 6-32 or mathematically as in the following equation: Dd d1 (Y0/d1)2 ᎏ=ᎏ + ᎏ – 1.5 Y0 Y0 2␾2

(6-100)

Dd 3Y0 (Y0/d1)3 ᎏ=ᎏ – ᎏ + 1.0 2 d1 2d1 2␾

(6-101)

冦

Z0 3Y0 d1 ᎏ = 1 + ᎏ – ᎏ –1 + Dd 2Dd 2Dd

+ ᎏ – 1冣冥冧冢ᎏ 冪冤莦1莦+莦16莦␾莦莦莦 d 莦莦莦 2d 莦莦莦莦 2

Dd

3Y0

1

1

(6-102)

where ␾ is determined from the following cubic equation:

冤

冥

Y 30 Y0 2Dd ᎏ – ␾2 ᎏ ᎏ + 3 + 2␾2 = 0 d 31 d1 Y0

(6-103)

Depending on the amount of energy dissipation before the location of d1, ␾ may be assumed to be 1.0 for no dissipation at all (Bakhemeteff, 1932) or as low as 0.95 for some dissipation before the jump (Bobin, 1934): 3Y0 Z0 = Dd + ᎏ – d2 – hw 2

(6-104)

where hw is the height of the weir that controls the plunge pool relative to the apron. The length of the plunge pool is expressed as: + ᎏ 冣Y D 冥冪冤冢莦1莦莦莦莦 D 莦莦莦

L⬘ = C

Y0

0

d

d

(6-105)

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1.0

2.0 Z /D 0 d

0.9 Z /D 0 d

1.8

0.8

1.6

0.7

1.4

0.6

1.2

0.5

d /D 1 d

1.0 d /D 1 d

0.4

0.8

0.3

0.6

0.2

0.4

0.1

0.2

0.0

0.0 0.4

0.8

1.2 Y /D 0 d

1.6

2.0

FIGURE 6-31 Curves to determine the height of the weir in a plunge pool.(After Hall, 1943 in his discussion of Moore, 1943, by permission of ASCE.)

where C can equal 1.7 for low spray but can also equal as high as 2.0 for significant spray. Standish Hall (1943) proposed that length L⬘ be followed by an equal transition. Example 6-13 Referring to Example 6-12, determine the length of the plunge pool if a controlling weir is added. Determine the level of the liquid Z0. Solution in SI Units The critical depth was determined to be 1.676 m. The drop is 6 m. Assuming C = 2.0, 2 苶.6 苶7 苶6 苶苶·苶 6苶 +苶1.6 苶7 苶6 苶苶 ] = 7.17 m L⬘ = 2兹[1

1.676 Y0 ᎏ = ᎏᎏ Dd 6 = 0.279 Referring to Figure 6-25: Z0 ᎏ ⬇ 0.84 or Z0 ⬇ 0.84 × 6 = 5.04 m Dd Since Z0 is measured from E0, and E0 = Dd + 1.5 Y0 = 6 + 1.5 × 1.676 = 8.51 m the liquid level is 8.51 – 5.04 = 3.47 m above the apron. If the engineer builds a weir 2 m high (hw) it will be submerged by a depth of 1.47 m, corresponding to the value of d2.

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FIGURE 6-32

Walls of a weir showing sediment coating.

Solution in USCS Units The critical depth was determined to be 5.5 ft. The drop is 19.48 ft. Assuming C = 2.0, L⬘ = 2兹[5 苶.5 苶苶·苶 19苶.4 苶8 苶苶 +苶5.5 苶2苶] = 23.44 ft 5.5 Y0 ᎏ = ᎏ = 0.279 Dd 19.48 Referring to Figure 6-25, Z0 ᎏ ⬇ 0.84 or Z0 ⬇ 0.84 × 19.48 = 16.36 ft Dd Since Z0 is measured from E0, and E0 = Dd + 1.5 Y0 = 19.48 + 1.5 × 5.5 = 27.73 ft the liquid level is 27.73 – 16.36 = 11.1 ft above the apron. If the engineer builds a weir 6.56 ft high (hw) it will be submerged by a depth of 4.82 ft, corresponding to the value of d2. The flow of slurry in flumes and through drop boxes is fairly complex and under certain conditions hydraulic jumps occur with considerable turbulence. For fairly abrasive slurries, wear is a concern. In other situations such as copper mines, the presence of lime in the slurry may actually end up coating the flume with deposited lime that consolidates with time. This deposition of lime or similar sediments coats the flume, but does completely change the roughness of the wall (Figure 6-33). In some cases the designer must try to avoid break up the transported solids such as coal (Kuhn, 1980).

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+1.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

4.0

+0.5 Values of y/y a

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0.0 -0.5 -1.0 -1.5 Fr=4.12

-2.0

-1.0

0.0

Values of x/ya

1.0

2.0

.02 =3 Fr .18 =2 Fr 1.8 = Fr 1 = Fr

-2.0

4.0

FIGURE 6-33 Effect of the Froude number at the entry to the waterfall on the shape of the nappe. [After Rouse (1943) in his discussion of Moore (1943).]

Special transition areas may be lined with abrasion resistant steel or with rubber. The rubber is glued to steel plates that are bolted to the concrete (see Figure 6-1). The analyses of Hall (1943) and Moore (1941,1943) are based on the assumption that the liquid enters the fall from a subcritical regime, with minimum energy, and accelerates at the brink. The projection of the nappe and contact with the apron is even more complicated when the jet approaches the brink at supercritical flows. Rouse, in his discussion of Moore (1943), discussed the changes in Froude numbers of 1–14 (Figure 6-30).

6-13 CONCLUSION Slurry flows in open channels are fairly complex but they follow many of the principles of closed conduit flows discussed in the previous two chapters. When the speed is insufficient or the Froude number is low, deposition occurs and dunes or a stationary bed form. Since most books on slurry flows are focused on pipe flows, this chapter presented an exhaustive review of the mathematics of open channel slurry flows and design of drop boxes. The practical engineer should find in the worked examples a methodology to apply such complex equations. It is hoped that new generations of academicians and students will enrich the understanding of such complex flows. The design of open channel flows requires frequent iterations for slope, stability (Froude number), roughness, etc. The use of modern personal computers with the appropriate equations allows the engineer to optimize the hydraulic design. On a note of caution, the design engineer should not apply data from small to large flumes. The change of the hydraulic radius and the ratio of particle size to depth of flow affect the magnitude of the slope of the launder.

6-14 NOMENCLATURE a a

Nondimensional parameter and function of Hedstrom number Reference depth for concentration calculations

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Ab b bw C Ca CD Ch CL Cm CT Cv Cw Cy d db df dj dp dt d1 d2 d3 d50 d85 Dd DH DI Dr Er E0 fD fD⬘ fD⬘⬘ fDL fN f1 f2 fNL FN Fr fT FT g G h ha hw He

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Area of the horizontal projection of the lee face of the bed forms Nondimensional parameter Wetted width Time-averaged concentration of suspended solids Concentration at height “a” Drag coefficient of particles for a heterogeneous slurry Chezy number Lift coefficient Depth-averaged concentration of solids Mean transport concentration of solid particles in the slurry mixture Volume fraction of solid particles in the slurry mixture Weight fraction of solid particles in the slurry mixture Volume fraction of solid particles in the slurry mixture at level “y” Depth Depth at which a stationary hydraulic jump occurs between the supercritical and subcritical flows on the apron after a free fall Depth of under nappe liquid between drop wall and nappe Depth at the hydraulic jump on the apron from a free fall Diameter of the particle Final depth of the tail water after the hydraulic jump due to fall Depth at the toe of the nappe for a free fall and drop Reference depth for subcritical tail water after the free fall in the case of a hydraulic jump occurring at the toe of the nappe Depth of supercritical flow at beginning of the hydraulic jump downstream of the nappe Particle diameter passing 50% (m) Particle diameter passing 85% (m) Depth of drop box of free-fall drop Hydraulic diameter Conduit inner diameter (m) Drop number for free fall Coefficient correlating relative roughness to friction and average velocity Total energy level for a free-fall problem of a liquid relative to the apron Darcy friction factor Darcy friction factor for the channel without bed forms Darcy friction factor due to the bed forms Darcy friction factor for liquid Fanning friction factor Mathematical function Mathematical function Laminar component of fanning friction factor fluid force normal to the direction of flow Froude number Turbulent component of fanning friction factor Fluid force tangent to the direction of flow Acceleration due to gravity (9.81 m/s2) Flocculation gradient Head due to friction losses Depth ratio defined by Equation 6-31 Height of weir in a plunge pool with a weir Hedstrom number

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J ks Ke Kx L L⬘ Lb Ld Lj Lmix Lp

6.73

Nondimensional parameter to account for dynamic viscosity in deposit velocity Linear roughness (m) Experimental constant Von Karman coefficient Length of conduit Length of drop pool with a controlling weir Distance between the point of impingement of the nappe and the tail water depth Distance between drop wall and toe of the nappe for a free-fall drop Distance between the wall of the free fall and the hydraulic jump on the apron Mixing length for eddies Theoretical distance to intersection of the center of the nappe and the bottom of the drop box with under-nappe pool (see Figure 6-17) Lr Total length to the stable tail water m Exponent from the Darby equation M Montuori number n Manning roughness number qb Flow rate per unit width of launder (m2/s) qbs Flow rate of sediments per unit width Q Flow rate (m3/s) P Power Patm Atmospheric pressure PL Plasticity number Pw Wetted perimeter R Radius Re Reynolds number Rep Particle Reynolds number RH Hydraulic radius (m) RH⬘ Hydraulic radius due to grain roughness RH⬘⬘ Hydraulic radius due to bedforms S Slope Sm Specific gravity of mixture U Horizontal component of velocity U⬘ Horizontal component of velocity due to turbulence Uav Average speed Ub Bed velocity Ubc Critical velocity to start the motion of the bed Ucr Critical velocity to start the flow of cohesive elements Uf friction velocity Uf⬘ Friction velocity due grain roughness Uf⬘⬘ Friction velocity due to dunes or bedforms Um Average speed Umax Maximum speed V Average velocity of flow (m/s) V⬘ Average vertical velocity due to eddies VC Camp minimum self-cleaning velocity for a sewer (m/s) VD Deposit velocity in a launder (m/s) Ve Verdinokov number Vm Mean vertical velocity component Vsc Self-cleaning velocity of a launder Vt Particle terminal velocity Vol Volume

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w x X0 y ym Y Y0 Z Z0 Z1

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Width of launder Local horizontal ordinate A coefficient of cohesion of the material Local vertical coordinate in the launder Average depth of the slurry in the launder Depth of launder Critical depth of the liquid at Froude number of one Function of the height above the bed of a launder Depth of liquid surface in a plunge pool over the weir Empirical function of grain distribution above bed

Greek letters ␣ Angle of inclination of flow with respect to particle ␤ Constant of proportionality ␹ Constant of proportionality in Celik’s equation ␧m Coefficient of exchange of momentum between neighboring streams of the fluid ␧s Mass transfer coefficient ␾ Angle of slope ␾ Factor of energy dissipation before the hydraulic jump in a free fall ␸A Graf–Acaroglu function ␩ Coefficient of rigidity ⍀ Data about cohesion ␬ tan–1 S ␭ Wavelength of deposited dunes and antidunes ␶ Absolute (or dynamic) viscosity ␮m Absolute (or dynamic) viscosity of mixture ␯ Dynamic viscosity ␶ Shear stress ␶cr Critical shear stress ␶L Fluid shear stress ␶0 Yield stress for Bingham plastics and pseudoplastics ␶w Shear stress at the wall ␳ Density ␳L Density of carrier liquid ␳m Density of slurry mixture (Kg/m3) ␳s Density of solids in mixture (Kg/m3) ␴ Exponent for effective shear stress ⬇ 0.06 ␰ Sedimentation coefficient ␺A Graf–Acaroglu function ␨D Shape factor ␨1 Shape factor ␨2 Shape factor ␨3 Shape factor

6-15 REFERENCES Abulnaga, B. E. 1997. Channel 1.0 Computer Program for Open Channel Slurry Flows. Developed for Fluor Daniel Wright Engineers. Internal report. Acaroglu, E. R. 1968. Sediment Transport in Conveyance Systems. Ph.D. diss., Cornell University.

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Ambrose H. H. 1953. The transportation of sand in pipes with free surface flow. In Proceedings of the Fifth Hydraulics Conference. Ames: State University of Iowa, pp. 77–88. The American Society of Civil Engineers. 1975. Sedimentation Engineering. Manuals and Reports on Engineering Practice. No. 54. New York: ASCE. The American Society of Civil Engineers and the Water Pollution Control Federation. 1977. Wastewater Treatment Plant Design. ASCE Manual and Reports on Engineering Practice No. 36. (Also published as WCF Manual of Practice No. 8.) Apmann, R. P., and R. R. Rumer, Jr. 1967. Diffusion of Sediments in a Non-Uniform Flow Field. Report prepared for the Department of Civil Engineering, Faculty of Engineering and Applied Science, State University of New York at Buffalo. Report No. 16. Bakhmeteff, B. A. 1932. Hydraulics of Open Channels. New York: McGraw-Hill. Blench, T., V. J. Galay, and A. W. Peterson. 1980. Steady fluid-solid flow in flumes. Paper C-1, presented at the 7th Annual Hydrotransport Conference, Sendai, Japan. BHR Group. Bobin, P. M. 1934. The design of stilling basins. Transactions of the Scientific Research Institute of Hydrotechnics, XIII, 79–123. Bogardi, J. L. 1965. European concepts of sediment transportation. Proc. Am. Soc. Civil Engineers, 91, HY1, 29–54. Boussinesq, M. J. 1877. (Ed.). Essai sur la Theorie des Eaux Courantes. [A Study on the Theory of Flowing Waters.] Memoires, Presentèes par Divers Savants—L’Academie de l’Institut de France, 23, 1–680. [Transactions of the French Academy Institute, 23, 1–680.] Brush, L. M., H. W. Ho, and S. R. Singamsetti. 1962. A study of sediment in suspension. Intern. Assoc. Sci. Hydr., Commiss. Land Erosion, No. 59. Camp, T. R. 1955. Flocculation and flocculation basins. Transactions Am. Soc. of Civil Engineers, 120, 1 1–16. Celik, I., and W. Rodi. 1984. A Deposition-Entrainment Model for Suspended Sediment Transport. Internal Report prepared by the University of Karlsruhe, Germany. Report No. SFB210/T/6. Celik, I., and W. Rodi. 1991. Suspended sediment-transport capacity for open channels. Journal of Hydraulic Engineering, 117, 2, 191–204. Chien, N. 1954. The present status of research on sediment transport. Proc. Am. Soc. Civil Engrs., 80, No 565, 33. Cooper, R. H. 1970. A study of bed Material Transport Based on the Analysis of Flume Experiments. PhD. thesis, Department of Civil Engineering, University of Alberta, Canada. Dominguez, B., R. Souyris, and A. Nazer. 1996. Deposit velocity of slurry flow in open channels. Paper read at the symposium, Slurry Handling and Pipeline Transport. Thirteenth annual International Conference of the British Hydromechanic Research Association, Johannesburg, South Africa. Einstein H. A. 1950. The Bed-Load Function for Sediment Transportation in Open Channel Flows. Technical Bulletin No. 1026. U.S. Deptartment of Agriculture Soil Conservation Service. Einstein H. A. and J. A. Hardner, 1954. Velocity distribution and boundary layer at channel bends. Am. Geophysical Union Trans., 35, 114–120. Einstein, H. A., and N. Chien. 1955. Effects of Heavy Sediment Concentration Near the Bed on Velocity and Sediment Distribution. MRD Sed. Ser. Berkeley: University of California. Fathy, A., and M. A. Shaarawi. 1954. Hydraulics of free overfall. Proc. Am. Soc. Civ. Eng, 80, 564, 1–12. Fortier, S., and F. C. Scobey. 1925. Permissible canal velocities. Trans. Am. Soc. Civil Engrs, 51, 7, 1397–1413. Garde, R. J., and J. Dattari. 1963. Investigation of the total sediment load of streams. Res. J. University of Roorkee. Internal report. Graf, W. H. 1971. Hydraulics of Sediment Transport. New York: McGraw-Hill. Graf, W. H., and E. R. Acaroglu. 1968. Sediment transport in conveyance systems. Part I. Bulletin. Intern. Association of Sci. Hydr., 2. Green, H. R., D. H. Lamb, and A. D. Tylor. 1978. A new launder design procedure. Paper read at the Annual Meeting of the Society of Mining Engineers, March, Denver, Colorado. Grim, R. E. 1962. Applied Clay Mineralogy. New York: McGraw-Hill. Guy, H. P., R. E. Rathbun, and E. V. Richardson. 1967. Recirculation and sand-feed flume experiments. Paper 5428. Am. Soc. of Civil Eng., 93 HYS, 97–114, Sept.

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Hall, S. L. 1943. Discussion to paper by W. L. Moore. 1943. Energy loss at the base of the free overfall. Transaction of the A.S.C.E., 108, 1378–1387. Henderson, F. M. 1990. Open Channel Flow. New York: Macmillan. Ismail, H. M. 1952. Turbulent transfer mechanism and suspended sediments in closed channels. Trans. ASCE, 117, 409–447. Julian, Smart and Allan. 1921. Cyaniding Gold and Silver Ores. Internal report presented to J. B. Lippenicott Co., U.S.A. Reported by Tournier and Judd (1945). Karasev, I. F. 1964. The regimes of eroding channels in cohesive materials. Soviet Hydrol. (Am. Geophysics Union), Vol. 6. Kennedy, J. F. 1963. The mechanics of dunes and antidunes in erodible bed channels. Journal Fluid Mech., 16, 4. Keulegan, G. H. 1938. Laws of turbulent flow in open channels. Journal of Research (National Bureau of Standards, U.S. Dept of Commerce), 21, 707–741. Kuhn, M. 1980. Hydraulic Transport of solids in flumes in the mining industry. Paper C3 read at the 7th International Conference of the Hydraulic Transport of Solids in Pipes, Sendai, Japan. Cranfield, UK: BHRA Fluid Engineering, pp. 111–122. Liu, H. K. 1957. Mechanics of sediment—Ripple formation. Proc. Am. Soc. Civil. Eng., 83, HY2, Paper 1197. Lovera, F., and J. F. Kennedy. 1969. Friction factor for flat-bed flows in sand channels. Proc. Am. Soc. Civil Eng., 95, HY4, Paper 6678, pp. 1227–1234. Majumdar, H., and M. R. Carstens. 1967. Diffusion of Particles by Turbulence: Effect of Particle Size. Water Res. Center, Report WRC-0967, Georgia Inst. Techn., Atlanta, U.S.A. Manning R.1895. On the flow of open channels and pipes. Transactions, Institution of Civil Engineers of Ireland, 10, 14, 161–207. Matyukhin, V. J., and O. N. Prokofyev. 1966. Experimental determination of the coefficient of vertical turbulent diffusion in water for settling particles. Soviet Hydrol. (Am. Geophys.Union), No 3. Ministry of Technology of the United Kingdom. 1969. Charts for the Hydraulic Design of Channels and Pipes. London: Ministry of Technology of the United Kingdom. Moore, W. L. 1943. Energy loss at the base of the free overfall. Transaction of the A.S.C.E., 108, 1343–1392. Neil, C. R. 1967. Mean velocity criterion for scour of coarse uniform bed material. In International Association of Hydrology Research, 12th Congress. Fort Collins, CO. Niepelt, W. A., and F. A. Locher. 1989. Instability in high velocity slurry flows. Mining Engineering, 41, 12, 1204–1209. O’Brien, M. P. 1933. Review of the theory of turbulent flow and its relation to sediment transportation. Trans. Am. Geophysics, 14, 487–491. Rand, W. 1955. Flow geometry at straight drop spillways. Transaction of the Am. Soc. Civ. Eng., 81, 791, 1–13. Reynolds, O. 1895. On the Dynamical theory of incompressible viscous fluids and the determination of the criterion. Catalogue of Scientific Papers, compiled by the Royal Society of London, Vol. 2, pp. 535–577. Cambridge, UK: Cambridge University Press. Richardson, E. G. 1937. The suspension of solids in a turbulent stream. Proceedings of the Royal Society of London, 162, Series A, 583–597. Richardson, E. V., and D. B. Simons. 1967. Resistance to flow in sand channels. Paper read at International Association Hydrology Research, 12th Congress, Fort Collins, Colorado. Rouse, H. 1937. Modern conceptions of the mechanics of fluid turbulence. Transactions of the Am. Soc. Of Civil Engrs., 102, 536. Rubin, M. B. 1997. Relationship of critical flow in waterfall to minimum energy head. Journal of Hydraulics, 123, January, 82–84. Silberman, E. 1963. Friction factors in open channels. Proc. Am. Soc. Civil Engrs., 89, no. HY2, Simons, D. B. and M. L. Albertson. 1963. Univorm water conveyance in alluvial channels. Proc. Am. Soc. Civ. Eng., 128, 1. Slatter, P. T., G. S. Thorvaldsen, and F. W. Petersen. 1996. Particle roughness turbulence. Paper read at the 13th International Conference on Slurry Handling and Pipeline Transport, at British Hydromechanic Research Association, Johannesburg, South Africa. Shook, C. A. 1981. Lead Agency Report II For Coarse Coal Transport. MTCH Hydrotransport Cooperative Programme. Saskatoon, Canada: Saskatchewan Research Council.

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Stricklen, R. 1984. Slurry handling considerations. Paper read at the 1984 Annual Meeting of the American Institute of Mining Engineering, Denver, Colorado, U.S.A. Thomas A. D. 1979.The role of laminar/turbulent transition in determining the critical deposit velocity and the operating pressure gradient for long distance slurry pipelines. Paper read at the 6th International Conference of the Hydraulic Transport of Solids in Pipes. Cranfield, UK: BHRA Fluid Engineering, pp. 13–26. Tournier, E. J. and E. K. Judd. 1945. Storage and mill transport. In Handbook of Mineral Dressing— Ore and Industrial Minerals. New York: Wiley. Vanoni, V. A. 1946. Transportation of suspended sediment by water. Paper no. 2267 Trans. Am. Soc. Civ. Eng. Hydraulics Division, 111, 67–133. Vanoni, V. A., and L. S. Hwang. 1967. Relation between bedforms and friction in streams. Proc. Am. Soc. Civil. Engrs. 93, no. HY3, Van Rijn, L. C. 1981. Comparison of Bed-Load Concentration and Bed-Load Transport. Report prepared by the Delft Hydraulic Laboratory, Delft, The Netherlands. Report No. S 487, Part I. Von Karman, T. 1934. Turbulence and skin friction. Journal of Aeronautical Sciences, 1, 1, 1–20. Von Karman, T. 1935. Some aspects of the turbulence problem. Mechanical Engineering, 57, 407–412. Wasp, E., J. Penny, and R. Ghandi. 1977. Solid-Liquid Flow Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans Tech Publications. Whipple, K. X. 1997. Open channel flow of Bingham fluids. Journal of Geology, 105, 243–262. Wilson, K. C. 1991. Slurry transport in flumes. In Slurry Handling, edited by N. P. Brown and N. I. Heywood. New York: Elsevier Applied Sciences. Windsor, L. M. 1938. The barrier system of flood control. Civil Engineering (October), 675. Wood P.A. 1980. Optimization of flume geometry for open channel transport . Paper C2 read at the 7th International Conference of the Hydraulic Transport of Solids in Pipes, Sendai , Japan. Cranfield, UK: BHRA Fluid Engineering, pp. 101–110. Yalin, M. S. 1977. Mechanics of Sediment Transport. 2nd Edition. Toronto: Pergamon Press. Zippe, H. J., and H. Graf. 1983. Turbulent boundary-layer flow over permeable and non-permeable rough surfaces. J. Hydr. Res., 21, 1, 51–65. Further readings Bagnold, R. A. 1955. Some flume experiments on large grains but little denser than the transporting fluid and their implication. Part 3. Proc. Inst. Civil Engrs, 4. 174–205. Gilbert, G. K. 1914. Transportation of Debris by Running Water. Paper no. 86. U.S. Geological Survey. Guy, H. P., D. B. Simons, and E. V. Richardson. 1966. Summary of Alluvial Channel Data From Flume Experiments, 1956–1961. Paper No. 462-I. U.S. Geological Survey. Khurmi, R. S. 1970. Hydraulics and hydraulic machines. Delhi: S. Chand & Co. Lacey, G. 1930. Stable channels in alluvium. Paper no. 4736. Proc. Inst. Civil Engs., 229, 529–384. Lacey, G. 1934. Uniform flow in alluvial rivers and canals. Paper no. 237. Proc. Inst. Civil Engs., 237, 421–544. Lacey, G. 1947. A general theory of flow in alluvium. Paper no. 5518. Journal Inst. Civil Engs., 17, 1, 16–47. Nino, Y., and M. Garcia. 1998. Experiments on saltation of sand in water. Journal of Hydraulics, 124, 10, 1014–1025. Turton, R. K. 1966. Design of slurry distribution manifolds. Engineer, 221, 641–643. Wilson, K. C. 1980. Analysis of slurry flows with a free surface. Paper C4 read at Hydrotransport 7, Sendai, Japan. Cranfield, UK: BHRA Group, pp 123–132.

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PART TWO

EQUIPMENT AND PIPELINES

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COMPONENTS OF SLURRY PLANTS

7-0 INTRODUCTION In Chapter 1, a typical circuit of a mineral process plant was presented. In Chapters 3 through 6, the theory of slurry flows was examined in detail for different rheology and regimes. To achieve such complex flows, a number of important pieces of machinery, such as mills, pumps, and valves, and drop boxes are needed. Together they form the slurry preparation plant at the start of the pipeline and sometimes the slurry dewatering plant when the concentrate or solids must be dried out for shipping, smelting, or burning as a fuel. Their design is often complex and must account for wear and performance. In simple layman’s terms, rocks that contain ores may be delivered in fairly large pieces. These rocks may be obtained by blasting, special hydraulic jack hammers, excavators, etc. (Figure 7-1). These large rocks need to be reduced to sufficiently small particles to extract the ores—from as large as a few hundred millimeters (or dozens of inches) down to a few millimeters or fractions of inches. This is done by a number of steps, such as crushing, milling, grinding, screening, cycloning, vibrating, etc. Milled rocks are then transported in slurry form and treated in different circuits such as flotation, acid or cyanide leaching, and classification circuits. The concentrate may then be thicked further for transportation to its final destination. The tailings are disposed of in dedicated ponds. The design of mineral processing plants has been the subject of numerous books, and specialized books have been written for each piece of equipment. In this chapter, some of the most important components of slurry systems will be introduced, with sufficient information for the slurry engineer to appreciate the discharge from each type of equipment. The next two chapters are devoted to pumps and valves and Chapter 10 is devoted to materials for manufacturing. It would be beyond the scope of this book to dwell on the chemistry of each process.

7-1 ROCK CRUSHING Rock crushing is not part of the slurry circuit but is more of a preparatory step to the formation of slurries. Crushing will therefore be reviewed briefly, as it is outside the scope of this handbook. 7.3

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FIGURE 7-1 Excavation is a primary source of materials for a mineral processing plant. [Courtesy of Metso Minerals (formerly known as the companies Nordberg and Svedala).]

Solid comminution is the process of reducing the size of particles. Two comminution types are considered: 1. Dry comminution generally reduces rocks down to a diameter of 25 mm (1 in), by impact and mechanical compression. This process involves jaw crushing, gyratory crushing, cone crushing, and grinding using rod mills and ball mills. 2. Wet comminution generally reduces 25 mm (1 in) particles down to very fine sizes by grinding and attrition in slurry form. This process involves semiautogenous mills, autogenous mills, ball mills, hydrocyclones, columns, etc. Comminution via a machine is measured by the reduction ratio, defined as 80% of the particle size at the feed (Fe80) to 80% of the particle size at the output (Cr80). The feed to a grinding mill must be crushed to a size appropriate to the grinding process. Semiautogenous mills require little crushing; ball mills require a finer crushing. A method of ore preparation that is now limited to narrow ore seams or veins in underground mines is the so-called “run of the mine milling.” It consists of blasting the rocks into lumps, usually of the order if 300 mm (12 inch) or larger. The most common approach, however, is to crush the mined rock to an acceptable size. 7-1-1 Primary Crushers Primary crushers absorb any size rocks (depending on the opening at the inlet) and reduce their size down to 50–150 mm (2–6 in). Primary crushers are classified as:

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앫 Jaw crushers 앫 Gyratory crushers 앫 Impact crushers Some mines try to reduce the cost of crushing by blasting the rocks from mountains and hills. Crushing is essentially a process of reducing the size of a stone down to 25 mm (1 in) (Figure 7-2). As this is difficult to achieve in a single stage, it is often encompassed in two or three steps. The stones go through a cycle of primary crushing, secondary crushing, and tertiary crushing. Special machines have been developed for each step of crushing (Figure 7-3). 7-1-1-1 Jaw Crushers These machines operate by compressing the rocks between a fixed plate and a moving jaw (Figure 7-4). The rocks are fed from the top of the crusher. The fixed jaw or plate is usually attached to the wall of a cavity. Through an eccentric mechanism or crankshaft, a moving jaw presses the rocks against the walls of the crusher. Generally, the following two types of machines are used: 1. In the overhead eccentric jaw crusher, also known as the single toggle crusher, the moving plate is forced against the stationary plate by an eccentric mechanism driving at its top, as well as by the rocking of a toggle connected to the bottom of the moving plate.

FIGURE 7-2 Crushing is an essential step in handling hard rock, gravel, and mining ores as well as for recycling. [Courtesy of Metso Minerals (formerly known as the companies Nordberg and Svedala).]

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feed Pivoting jaw fixed jaw

feed

bowl

Head or mantle

pitman

out

(a) Jaw crusher

bowl

feed

(c) Impact crusher

(b) Gyratory crusher

Head or mantle

inclined bowl

feed

cone

(b) Cone crusher

FIGURE 7-3 Principles of crushing.

FIGURE 7-4 Cross-sectional representation of a jaw crusher. [Courtesy of Metso Minerals (formerly known as the companies Nordberg and Svedala).]

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2. The blake jaw crusher features a moving plate that pivots at the top but is oscillated at the bottom. The dimensions and shape of the plates affect the performance of the crusher. The smaller the discharge gap, or required output size, the lower the tonnage from the crusher. Jaw crushers work best on rocks that are not flat or slabs. With a feed opening of 1.67 × 2.13 m (66 × 84 in) and a discharge gap of 200 mm (8 in), the crusher can handle a capacity of 800 tph. The walls and moving blade of the crusher are lined with a hard metal such as manganese steel. The liners are removable for repairs once worn out. The liners may be flat, plain, or ribbed. The final output size of crushed particles depend on the setting of the plates (Figure 75). Curves shown in Figure 7-5 indicate, for example, that for a closed setting of 100 mm (4 in) the size particles will be at a maximum of 160 mm (6.375 in) with a significant portion of particles smaller than 50 mm (2 in). 7-1-1-2 Gyratory Crushers These machines operate on the principle of compressing the rocks in a cone (Figure 7-6) The rocks fall into the cavity from the top. The moving part is an eccentric cone. The

FIGURE 7-5 The size of the output from jaw crushers depends on the plate setting. If the closed side setting (c.s.s) is 100 mm (4⬙), the maximum product size is 160 mm (6 3–8⬙) and the portion of fraction under 50 mm (2⬙) is approximately 35%. [Courtesy of Metso Minerals (formerly known as the companies Nordberg and Svedala).]

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Mainshaft sleeve Spider bushing Spider arm guard Head nut Spider Concave fifth row Concave fourth row Concave third row Concave second row Concave first row Inner deflector ring

Spider cap Mainshaft Retainer bar Guide bushing Seal retainer Tie rod nut Top shell Upper mantle Tie rod Lower mantle Floating ring bracket

Arm guard (inner)

Oil deflector ring

Arm guard (outer)

Dust seal bonnet

Bottom shell Tie rod nut

Floating ring Floating ring retainer

Gear housing shield

Outer bushing

Positive air pressure

Pinion

Eccentric

Inner busing

Seal ring

Countershaft box

Eccentric support

Countershaft

Hydraulic cylinder

Balanced gear

Cylinder sleeve

Eccentric thrust washer

Cylinder shield

Eccentric thrust bearing

Piston cap

Swivel plate

Cylinder head

Socket plate

Transmitter

Thrust plate

FIGURE 7-6 Cross-sectional drawing of a primary gyratory crusher. (Courtesy of Sandvik.)

rocks enter on the largest corner of the cavity but are compressed as the eccentric cone rotates. The outside cone is sometimes called the bowl, and the rotating cone is called the mantle. The bowl reduces in diameter toward the bottom, whereas the mantle increases in diameter with depth in the opposite direction. Gyratory crushers are preferred for slabs or flat-shaped rocks as they snap the rock better. Gyratory crushers are manufactured to handle tonnage flows up to 3500 tph. Sandvik purchased the line of Nordberg mobile primary gyratory crushers (Figure 7-7) that can be moved from one site to another as the mine expands. 7-1-1-3 Impact Crushers These machines operate on the principle of a set of rotating hammers hitting against the rocks. The hammers are fixed to a cylinder. The feed is from the top and as the rocks feed in, they fall between a breaker plate and the rotating cylinder. The hammers produce the required impact to chip the rocks. Impact crushers work best on rocks that are neither abrasive nor silica-rich, as these cause rapid wear of the hammers. Metso Minerals manufactures impact crushers (Figure 7-8) for primary and secondary crushing. Figure 7-9 shows typical gradation curves.

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FIGURE 7-7 Large mobile gyratory crushers are designed with a special frame and wheels to permit relocation from one area of the mine to another. (Courtesy of Sandvik.)

7-2 SECONDARY AND TERTIARY CRUSHERS Crushing the rocks is often achieved in two or three stages. The secondary and tertiary crushing machines resemble the machines used during primary crushing. They consist of vertical cone crushers or horizontal cylinder crushers. The former type is the most widespread. 7-2-1 Cone Crushers Cone crushers operate on the same principle as gyratory crushers. This allows a gradual reduction of the area between the two cones. The rotating cone or mantle is inclined, thus providing a combination of impact loads and compression loads. By comparison with the gyratory crusher, the outer bowl is inverted, and the mantle rotates at much higher speeds. There are two types of cone crushers: 1. The standard type (for secondary crushing) 2. The short head type (for tertiary crushing)

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FIGURE 7-8 Cross-sectional cut through an impact crusher. (Courtesy of Sandvik.)

The two types of cone crushers have different bowl shapes. The standard has a wider feed and is used for larger stones. The short head has a more shallow feed and tighter space surrounding the mantle. The short head is therefore used for finer crushing. Because of the continuous wear of the surfaces, adjustment of the cone crusher is essential. By measuring power on a continuous basis, a feedback loop readjusts the mantle. Screens on the output of the crusher facilitate the separation of coarse and fine stones. In a closed circuit, the coarser stones are returned to the crusher. The fine stones could clog the crusher and must be removed. The diameter of cone crushers may be as low as 0.91 m (36 in) for a capacity of 50–80 tph, or as high 2.13 m (84 in) for a capacity of 500–1100 tph. The finer the output, the smaller is the tonnage.

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FIGURE 7-9 Performance curves of an impact crusher. (Courtesy of Sandvik.)

Figure 7-10 presents a cross-sectional drawing of the Metso Minerals cone crusher and Figure 7-11 shows gradation curves of the output from HP cone crushers. Metso Minerals manufactures complete portable cone/screen plants (Figure 7-12) that are relocated from one area of the mine to another. 7-2-2 Roll Crushers Roll crushers consist of two counterrotating cylinders. The gap between the cylinders is adjusted by threaded bolts. Roll crushers can use springs to hold the cylinders in place. Each cylinder is then driven by its own belt drive. Roll crushers are used for less abrasive stones than cone crushers. They are most effective on soft and friable stones, or when a close-sized product is required.

7-3 GRINDING CIRCUITS The dry ore from crushers is stored in a stockpile (see Figure 1-10). The stockpile then feeds the milling circuit (Figure 7-13). It is claimed that grinding accounts for 60% of the power consumption of a mineral process plant. Elliott (1991) indicates that for a typical copper or zinc concentrator, grinding consumes 12 kWh/t, crushing 2–3 kWh/t, and the rest of the plant 2–3 kWh/t. Obviously, the finer the grinding, the higher the energy consumption. There are two main forms of grinding: 1. Dry grinding when the water content is 34% water by volume

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FIGURE 7-10

Cross-sectional cut through a cone crusher.(Courtesy of Sandvik.)

Between 1% and 34%, the slurry is very difficult to handle and grinding is inefficient. In some plants, an initial grinding process may be followed by some form of classification such as flotation or magnetic separation, which in turn is followed by a second grinding process. This approach tends to eliminate at an early stage a good portion of the gangue (see Chapter 1). It is not possible to achieve the particle size needed through a single grinding phase unless coarse output is required. When a coarse product is required, crushed materials are transported to a rod mill via a conveyor belt and the output is delivered from the rod mill. This is essentially an open circuit. Closed circuits (Figures 7-14–7-16) may include SAG and ball mills, hydrocyclones, and centrifuges. Grinding mills are designed with different approaches to feed and discharge (Figure 7-17). The energy required to reduce the size of a particle is usually a function of its diameter raised to an exponent. Holmes (1957) indicated that this exponent

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FIGURE 7-11 Gradation curves of cone crushers. (Courtesy of Sandvik.)

FIGURE 7-12

Mobile cone and screen plants. (Courtesy of Sandvik.)

7.13

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Water Sprays

conveyor

Mill Feed Stockpile Crushers Stockpile Monorail Belt Feeders

Water Sprays reclaim water Mill Feed Conveyor SAG Mill

Auto Sampler

cyclone overflow

To Rougher Flotation

coarse SAG Mill discharge

reclaim water

Cyclone Feed Pumps Ball Mill

Reclaim water

FIGURE 7-13 Flow chart of a grinding circuit. The stockpile of ore feeds the SAG mill, and the ore is processed even further by ball mills.

is not a constant but a variable. His method of iteration is fairly complex and would require a computer program. For wet grinding, which is where the slurry circuit starts, the resistance to comminution is measured by a grindability work index. It is established by test work. Bond (1952) defined the grindability work index ⌫ from the power W (in kWh per ton) required to reduce the feed size F (mm) to the final product size Cr (mm): –1/2 –1/2 – Fe80 ) W = 10⌫(Cr80

(7-1)

Equation 7-1 is based on reduction of the rock size in a 2.44 m (96 in) ball mill. This equation applies in the case of wet grinding, which is often the first step in a slurry circuit. Typical examples of the grindability work index ⌫ are presented in Table 7-1. The feed, its shape, and mechanical properties ultimately influence the performance of the grinding circuit and the degree of efficiency of ore extraction. The performance of the grinding process is dependent on a successful grinding operation. In an autogenous mill, the feed itself is used as a grinding medium. The larger the particles, the more energy they release on impact with each other. A coarse feed (larger than

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gear Conveyor from stock pile 7.15

feed

primary grinding mill

feed

mill feed box mill feed box ball mill

rods

separation of grinding balls

separation of grinding medium

cyclone feed pump or mill discharge pump

mill discharge pump box

FIGURE 7-14

Two-stage closed circuit for grinding and classification of ore.

Page 7.15

coarse cyclone underflow recirculated to ball mill

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FIGURE 7-15 View of a closed circuit grinding copper ore. In the back of the photo is the large 12.2 m (40 ft) diameter SAG mill that receives the ore from the stockpile. In the front, the ball mill grinds the underflow from the hydrocyclone.

FIGURE 7-16 View of the hydrocyclones set at a height of 30 m above the base of the SAG mill. The overflow is diverted to centrifuges to separate the gold ore from the lighter copper ore. The copper ore is then diverted to the ball mill (on the left-hand side of the photo) for secondary grinding. 7.16

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feed

out

balls

feed

out

grate

slurry

(a) Overflow mills (wet grinding only) - Used for rod mills in open circuits and ball mills in closed circuit Grinding with maximum specific area and suitable for very fine output Simple and robust

(b) Diaphragm or grate mills - Not suitable for rod mills, and mostly used for closed circuit - Used for Autogeneous and Semi-Autogeneous Grinding for very fine output - Coarser output than overflow mills

feed

feed

rods

feed

rods

(c) peripheral central port discharge

(d) peripheral discharge at the end

Peripheral discharge mills are essentially reserved for rod mill grinding, wet or dry Used for coarse grind where close control of final feed size is required, either coarse or fine suitable for open or closed circuits

FIGURE 7-17

Schematic representation of different types of grinding mills.

TABLE 7-1 Typical Examples of Grindability Work Indices (For Wet Grinding in a Ball Mill) Material Barite Bauxite Clay Coal Dolomite Feldspar Fluorspar Granite Limestone Magnetite Quartz Quartzite Sandstone Shale Taconite

Grindability work index

Reference

5 9 7 11 11 12 9 15 12 10 13 10 7 16 23

Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991) Elliott (1991)

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150 mm or 6 in) is important for a fully autogenous mill. Typically, the feed has an 80% passing size of 200 mm (8 in). In a semiautogenous (SAG) mill, steel or high chrome white iron balls are added to the circuit as a grinding medium. As they rotate and are carried away by centrifugal forces, they fall by gravity and impact against the feed or crushed rocks. Due to the difference in density between the steel balls (typically 7610 kg/m3 or a specific gravity of 7.61) and rocks (with a range of specific gravity of 1.3 to 4.0), smaller steel balls in a SAG mill have the effect of large rocks in fully autogenous mills. The d80 of the feed, called F80 in SAG mills, is typically 110 mm (4.5 in). In a mineral process plant, the process of comminution is one of the least efficient and highest consumers of power. A number of equations are used to define the process of dry grinding. These are described by Elliott (1991). Equation 7.1 is often called Bond equation. In practice it is modified by multiplying the right hand side of the equation by so-called “inefficiency factors,” E1 to E9. Dry grinding correction factor E1. For dry grinding circuits, without the addition of water, an inefficiency factor, E1 = 1.3, is applied. Product size correction factor E2. Another efficiency factor in terms of the final product size is defined as E2. If the final product is classified at 80% of the passage diameter, then E2 = 1.2. If the final product is classified at 95%, then E2 = 1.57 (see Table 7-2). Diameter correction factor E3. For a mill with the diameter Dm (in meters), a coefficient E3 is defined as E3 = (2.44/Dm)0.2

(7-2a)

If the diameter of the mill is expressed in inches then E3 = (96/Dmus)0.2

(7-2b)

where Dmus is the diameter of the mill in inches. Oversize correction factor E4. The optimum rock size fed into a rod mill is given as Feop = 16,000 (13/⌫)1/2

expressed in ␮m

(7-3)

and for a ball mill: Feop = 4000 (13/⌫)1/2

expressed in ␮m

TABLE 7-2 Inefficiency Factor E2 for Grinding Circuits Product size control reference % passing

E2

50% 60% 70% 80% 90% 92% 95% 98%

1.035 1.05 1.10 1.20 1.40 1.46 1.47 1.70

Source: “The Science of Communition,” Brochure No. 0647-05-98-N-English, Nordberg, Helsinki, Finland, 1998.

(7-4)

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If the size of the feed is larger than the optimum size Feop, (i.e., if Fe80 ⱕ Feop), then E4 = 1 if Fe80 > Feop (the case of oversized feed); then

冢

冣

Fe80 – Feopt Cr80 E4 = 1 + (⌫ – 7) ᎏᎏ ᎏ Feopt* Fe80

(7-5)

When Equation 7-5 yields a result smaller than 1.0, the result should be corrected to E4 =1.0. This equation should not be used in the case of a rod mill used to feed a ball mill, in which case, E4 = 1.0. Fineness correction factor E5. If the crushed output diameter Cr80 is less than 75 ␮m, then it is necessary to calculate a fineness correction factor E5, defined as Cr80 + 10.3 E5 = ᎏᎏ 1.145Cr80

(7-6)

Otherwise E5 = 1. Correction factor for high/low ratio of reduction rod milling E6. For a rod mill, defining the length of the mill as Lm and the diameter as Dm, a ratio Rr0 is defined as Rr0 = 8 + (5Lm/Dm)

(7-7)

The material reduction ratio is defined as Rr = Fe80/Cr80

(7-8)

If Rr > (Rr0 ± 2), then

冤

(Rr – Rr0)2 E6 = 1 + ᎏᎏ 150

冥

(7-9)

Otherwise a correction factor E6 = 1 is assumed. Correction factor for the low reduction ratio for ball mills. If Rr < 6, or when the ratio of the ball mill feed to the product output sizes is smaller than 6.0, a correction factor E7 is defined as 2(Rr – 1.35) + 0.26 E7 = ᎏᎏ 2(Rr – 1.35)

(7-10)

If the computation of Equation 7-10 exceeds the magnitude of 2.0, it is highly recommended to conduct lab tests and to contact the manufacturer of the mills. Correction factor for rod mills E8. The rod milling feed factor is where the material is fed into a rod mill from an open circuit crusher. Elliott (1991) suggested 1.4 as the magnitude of E8. However, if the source is a closed circuit with rod milling followed by ball milling, then E8 is 1.2. Correction factor for rubber-lined mills E9. When grinding balls are smaller than 80 mm or 3.25 in, rubber liners are used to line the inside walls of the mill. When grinding balls are larger than 80 mm or 3.25 in, metal liners are used. Rubber liners (Figure 7-18) are thicker than metal liners, use more space, and absorb more impact energy than their metal counterparts. It is customary to apply a correction factor E9 = 1.07 for rubber liners. The final power required to mill the feed is then obtained after multiplying all the correction factors by Bond’s equation (7-1). Iteration to consumed energy: Wf = W(E1 × E2 × E3 × E4 × E5 × E6 × E7 × E8 × E9)

(7-11)

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FIGURE 7-18 Rubber lining of SAG mills supplied to the Murin–Murin project in Australia to treat nickel-rich laterites. [Courtesy of Metso Minerals (formerly known as the companies Nordberg and Svedala).]

Equation 7-11 is useful to determine the power to grind down rocks. It must be corrected for worn-out liners, ball charges, and slurry density. It is therefore recommended that in the initial phase of the design of a mineral process plant, lab tests be conducted. Some of the empirical coefficients and equations for E1 to E9 were developed assuming a recirculation load of 250%. This means that the charge load of coarse material that is returned to the mill is about 250% of the fresh feed in a closed circuit. This is not always the case. The author was once involved in the design of a copper concentrate plant for a Peruvian mine in which the presence of soft high clay in the ore increased viscosity tremendously at a weight concentration of 50% to 60%. It became necessary to add water, dilute the slurry, and cut down the recirculation load. When the rocks in the feed are large, and milling is dominated by impact loads, Equation 7.1 should not be used to compute the work index load. Some of the empirical coefficients and equations for E1 to E9 were developed for a final output size with 80% passing 100 ␮m. (mesh 140). When Cr80 < 100 ␮m, Equation 7.11 does not give correct results. Example 7-1 An ore with a grindability index ⌫ = 13 is to be ground in a rod mill with feed from a closed-circuit crusher. The feed has a diameter Fe80 of 26 mm (1 in). The final product is required at 80% to be Cr80 of 10 mm (0.4 in) at a mass throughput of 350 tons/hour (770,000 lbs/hour). Estimate the power consumed by the rod mill.

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Solution Using Equation 7-1, the work input to the rod mill is W = 10 × 13(10–1/2 – 26–1/2) = 130(0.3162 – 0.1961) = 15.61 kWh/ton For wet grinding, E1 = 1. For closed-circuit grinding E2 = 1; E3 will be calculated after other factors. The oversize feed factor E4 is obtained from Equation 7.3. Feop = 16,000(13/13)1/2 = 16,000 ␮m or 16 mm Since Feop < Fe80, then E4 = {[(26/10) + (13 – 7)(26 – 16)]/16}/(26/10) = 0.3846(2.6 + 3.75) = 2.442 Since Cr80 > 75 ␮m, then E5 = 1. From Equation 7-8, the reduction ratio of the material Rr = 26/10 = 2.6. Rr0 will be calculated after selecting the rod mill. Since Rr < 6 then E7 = [2(2.6 – 1.35) + 0.26]/2(2.6 – 1.35) = 1.104 E8 = 1.2 since it is a closed circuit crusher. Iteration to consumed energy Wf = W(E1 × E2 × E3 × E4 × E5 × E6 × E7 × E8) Wf = 15.61 × 1 × 1 × E3 × 2.442 × 1 × E6 × 1.104 × 1.2 = 50.5 × E3 × E6 kWh/ton Since the feed is 350 tons per hour, the total energy consumption would be 350 ton/h × 50.5 kWh/ton E3 × E6 = 17,675 kW × E3 × E6 This would require a number of mills in parallel. From Equation 7-2, if the mill diameter of 6 m (19.7 ft) is selected, then E3 = (2.44/6)0.2 = 0.833 Rod mills with a length to diameter ratio of 2 are selected: Rr0 = 18 and since Rr < (Rr0 ± 2), E6 = 1 Final power consumption is 42.067 kWh/ton or total of 14,723 kW (19,736 hp). With modern technology, a SAG mill should be considered as an alternative to the rod mill (see Tables 7-3 and 7-4).

7-3-1 Single-Stage Circuits When finer material is required, a ball mill is used in a closed circuit. The feed is ground and then classified to separate coarse from fine solids. The coarse solids, also called oversized particles, are returned back to the mill for further grinding. This is called the “recirculation load” and the circuit is considered a closed circuit. In a dry circuit, the classifier may be a set of vibrating screens. In a typical copper or zinc circuit, the recirculation load can be as high as 250–350% of the new feed. The mill and mill discharge pumps must then be sized for the combination of recirculation load and new feed.

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TABLE 7-3 Estimates of Bond Energy Consumption per Mass for Grinding Rocks (Wi) Mineral Andesite Barite Basalt Bauxite Cement clinker Clay Coal Coke Copper ore Diorite Dolomite Emery Feldspar Ferro-chrome Ferro-manganese Ferro-silicon Flint Fluospar Gabbro Glass Gneiss Gold ore Granite Graphite Gravel Gypsum rock Iron ore, hematite Iron ore, hematite—specular Iron ore, magnetite Iron ore, oolitic Iron ore, taconite Lead ore Lead–zinc ore Limestone Manganese ore Magnesite Molybdenum Nickel ore Oil shale Phosphate rock Potash ore Pyrite ore Pyrhotite ore Quartzite Quartz Rutile ore

Specific gravity

Wi (kWh/sh.ton)

Wi (kWh/tonne)

2.84 4.50 2.91 2.20 3.15 2.51 1.4 1.31 3.02 2.82 2.74 3.48 2.59 6.66 6.32 4.41 2.65 3.01 2.83 2.58 2.71 2.81 2.66 1.75 2.66 2.69 3.53 3.28 3.88 3.52 3.54 3.35 3.36 2.66 3.53 3.06 2.70 3.28 1.84 2.74 2.40 4.06 4.04 2.68 2.65 2.80

18.25 4.73 17.10 8.78 13.45 6.30 13 15.13 12.72 20.90 11.27 56.70 10.80 7.64 8.30 10.01 26.16 8.91 18.45 12.31 20.13 14.93 15.13 43.56 16.06 6.73 12.84 13.84 9.97 11.33 14.61 11.90 10.93 12.74 12.20 11.13 12.80 13.65 15.84 9.92 8.05 8.93 9.57 9.58 13.57 12.68

20.08 5.20 18.81 9.66 14.80 6.93 14.3 16.84 13.99 22.99 12.40 62.45 11.88 8.40 9.13 11 28.78 9.8 20.3 13.54 22.14 16.42 16.64 47.92 17.67 7.40 14.12 15.22 10.97 12.46 16.07 13.09 12.02 14 13.42 12.24 14.08 15.02 17.43 10.91 8.86 9.83 10.53 10.54 14.93 13.95

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TABLE 7-3 Continued Mineral Shale Silica sand Silicon carbide Slag Slate Sodium silicate Spodumene ore Syenite Tin ore Titanium ore Trap rock Zinc ore

Specific gravity

Wi (kWh/sh.ton)

Wi (kWh/tonne)

2.63 2.67 2.75 2.74 2.57 2.10 2.79 2.73 3.95 4.01 2.87 3.64

15.87 14.10 25.87 10.24 14.30 13.40 10.37 13.13 10.90 12.33 19.32 11.56

17.46 15.51 28.46 11.26 15.73 14.74 11.41 14.44 11.99 13.56 21.25 12.72

From Denver Sala Basic. Selection Guide for Process Equipment. Reproduced by permission of Metso Minerals (formerly known as the companies Nordberg and Svedala).

7-3-2 Double-Stage Circuits A rod mill in an open circuit may be followed by a ball mill in a closed circuit. This is called a double-stage circuit and is often a wet process. The output from the rod mills is a slurry that contains a high proportion of coarse stones. The slurry is pumped via “mill discharge pumps” to a hydrocyclone. The underflow from the cyclone is then fed to a ball mill. From there, the output from the ball mill is fed once again to the hydrocyclone via the pump. In some circuits, the rod mill discharge is fed first to the ball mill before reaching the hydrocyclone. The hydrocyclones then feed the ball mills by gravity. A set of ball mill discharge pumps may then pump the output to a second classification circuit. The ball mill discharge has its own sets of slurry pumps.

7-4 HORIZONTAL TUMBLING MILLS In a horizontal tumbling mill, the actual body of the mill rotates and imparts energy to the grinding medium (balls or rods) and to the slurry. The combination of centrifugal forces and gravity forces from falling media act to create energy transmission by impact against the mineral. There are three categories of horizontal tumbling mills: 1. rod mills 2. ball mills 3. autogenous and semi-autogenous mills Basically a horizontal tumbling mill is a cylinder lined on the inside with wear-resistant alloy liners. The liners are fixed to the shell by T-bolts and nuts on the outside. The cylinder is carried by hollow trunnions running side bearings at each end.

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From Denver Sala Basic. Selection Guide for Process Equipment. Reproduced by permission of Metso Minerals (formerly known as the companies Nordberg and Svedala).

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Ores (ferrous and nonferrous) Preponderance of fine aggregates Talc and ceramic materials Cement raw materials Cement clinker Coal and petrol, coke Silica ceramics, etc. (must be free of iron) Production to a specific particle diameter or mesh Production to a specific surface area Wet grinding Dry grinding Damp feed (1%–15% moisture) Large feed ( 100 THEN PRINT “warning the required stress limit is 100 MPa”

7-10 SEDIMENTATION Sedimentation is a form of separation of solids from liquids by using gravity forces rather than electrostatic, chemical (flotation), or magnetic forces. Sedimentation may be achieved by gravity forces, using thickeners and clarifiers. On the other hand, it may be accomplished by centrifugal forces, as in centrifuges. In gold extraction circuits, an inter-

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mediary centrifuge is sometimes installed between the hydrocyclones and the ball mill feed box. Centrifuges are sometimes called concentrators because they permit the extraction of some of the heavy metals by applying a very high centrifugal force such as 60 times the acceleration due to gravity (60 g).

7-10-1 Gravity Sedimentation Gravity sedimentation is classified as thickening or increasing the concentration of the feed stream, or clarification or the removal of solids from relatively dilute streams. The former is used to prepare the feed for tailings and concentrate pipeline flow, or for the removal of tailings on trucks. The latter is more frequently used in sewage and waste treatment plants, where the volume of solids is considerably smaller than in tailings and concentrate flows. Considerable research on the use of flocculants in the last quarter of the twentieth century has lead to more concentrated sedimentation with less thickener. It would be beyond the scope of this book to discuss all these new flocculants. In simple terms, a clarifier or a thickener is essentially a sedimentation tank. To make the sedimentation uniform, a rake or arm rotates slowly but continuously. A relatively clear layer of liquid forms at the top and is withdrawn through an overflow box feeding a launder. The slurry in the thickener is denser at lower and lower layers. The bottom of the thickener forms a shallow cone with the center feeding into an underflow pipe to a separate launder or pump. The actual feed to the thickener is through a launder to the center. A feed box leads the slurry to a depth lower than the relatively clear water. Some special processes use intermediary mixing chambers where flocculants are added to accelerate the precipitation. The tank itself may be shallow and called a shallow thickener, or deep and called a deep thickener. The decision to choose either is often based on various parameters such as the final weight concentration, the rate of sedimentation, the viscosity, the design of the rake, as well as other parameters. This is at the basis of the design of the thickener (Figure 7-42). The actual process of sedimentation in a tube is based on the settling (or terminal) speed that was discussed at great length in Chapter 3. It is also depicted in Figure 7-43. Initially, the slurry is uniformly mixed. Gradually, the solids sink, forming three layers of liquid: free of solids, a dilute mixture, and a relatively dense layer. Eventually, all the solids in the dilute layer sediment out, leaving only two layers, one of water and one of a dense mixture with solids at minimum void ratio. The use of certain chemicals can accelerate the sedimentation of solids. The correlation between the terminal velocity of a sphere Vt and the sedimentation speed Vs is correlated to the void fraction ␧ (Cheremisinoff, 1984) by the following equation: Vs = Vt ␧2X(␧)

(7-35)

Where X(␧) is a function of the void ratio that must be determined by tests. The void ratio is Volf ␧ = ᎏᎏ Volf + Volp where Volp = volume filled by the particles Volf = volume of liquid filling the space between the particles

(7-36)

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FIGURE 7-42

Schematics of a thickener used for sedimentation of solids.

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height of dense phase

clear water boundary

dense phase boundary time (minutes)

fig 7 43

FIGURE 7-43 Response of gravity sedimentation with time.

For thickened sludges with a void ratio smaller than 0.7, Cheremisinoff (1984) proposed the following correlation:

␧3 Vs = 0.123 Vt ᎏ 1–␧

冢

冣

(7-37)

Spheres can actually compact in a very dense pattern to a minimum void ratio of 0.215, but Cheremisinoff (1984) indicated that the average void ratio from thickeners was 0.6. For nonspherical and coarse particles, the situation becomes more complex because of the shape factor (discussed in Chapter 3), and it is the norm to conduct sedimentation tests on samples of the slurry before designing the thickener.

7-10-2 Centrifuges Centrifuges use centrifugal force as a means to separate solids from liquids. Liquid is fed into the inlet and a rotating bowl is used to apply the centrifugal force, similar to a clothes drier that separates liquid from clothes by continuously rotating the clothes. Obviously, with slurry, it is more complex (Figure 7-44). The centrifugal force is defined as F = mR␻2

(7-38)

where ␻ = 2␲N/60 R = radius of rotation The ratio of the centrifugal force to the weight is called the centrifugal number Nc: Nc = mR␻2/mg = R␻2/g

(7-39)

For liquid-to-liquid separation, the centrifugal number may be as high as 60,000 for certain tubular sedimentation designs. The mining industry is concerned with wear, so slurries are separated at centrifugal numbers smaller than 100. Cheremisinoff (1984) stated that the settling velocity of a particle in turbulent motion (Re > 500) in a centrifuge is Ks times as much as the free settling velocity, where

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Centrifugal separator. (Courtesy of Knelson Concentrators.)

冪莦

R Ks = 2␲N ᎏ g

(7-40)

The Reynolds number for the particle is calculated using the radial velocity:

␳2␲RNdp Re = ᎏᎏ 60␮ For very fine particles with Re < 2, the migration is in laminar flow:

冢冣

R Ks = 4␲ 2N2 ᎏ g

(7-41)

For transition flow with 2 < Re < 500 4␲ 2N2R Ks = ᎏ g

冢

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0.71

(7-42)

Consider a simple vertical centrifuge as in Figure 7-36. The solids in the slurry move toward the wall at a speed us toward the radius Rw, while the liquid moves toward the axial feed tube at a speed uL toward the radius Ra. If the solids are at a volumetric concentration CV with a flow rate Q, the solids move at a speed us as Qs = 2␲R0Hus = CvQ Separation will occur when us > CvQ/2␲R0H.

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Example 7-5 A small centrifuge with a diameter of 150 mm is designed to handle 1.5 tons/hr of solids at a volumetric concentration of 40%. The density of the solids is 3000 kg/m3. The height of the cone is 125 mm. Determine the minimum speed of solids for separation from liquid. Solution Since the density is 3000 kg/m3 and the centrifuge handles 1500 kg/hr, the volume flow rate of solids is 0.5 m3/hr, or 0.139 kg/s. For separation, us > 0.139/(2 × ␲ × 0.15 × 0.125) and us > 1.18 m/s Considering the settling velocity of many particles, it is obvious that this centrifuge can handle the coarse particles found in certain mining systems.

7-11 CONCLUSION To achieve many of the tasks described in this chapter, slurry must be transported from one point to another. This may be done by gravity flow, by open channel flow, or by pumping. The pump is the workhorse of slurry transportation and will be analyzed in the next two chapters. A lot of different equipment is used in the processing of mineral ores. These were reviewed in this chapter more in terms of their place in the slurry circuit. The performance of the equipment depends on many factors such as proper sizing and the characteristics of rocks and soils that too often cause extensive wear. The materials selected for processing by such equipment will be examined in Chapter 10, as they are also used as criteria in the manufacture of pumps.

7-12 NOMENCLATURE A c C1, C2, C3 CD CL CQ Cp Cr80 CVL d50 D Di Din Dimp Dm Dmus DT e E1

Area of flow across the propeller Blade chord Coefficients of a hydrocyclone Drag coefficient Lift coefficient Flow coefficient Power coefficient d80 of the output wet ground rocks Volume fraction of liquid phase in a slurry tank d50 cut point of a hydrocyclone Drag force Conduit diameter (m) Diameter of mixer in inches Mixer impeller diameter Mill diameter in meters Mill diameter in inches Mixer tank diameter Natural number Dry grinding factor

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E2 E3 E4 E5 E6 E7 E8 E9 Fe80 Feop fw g H HP L n N P Q Rc Re ReB Rr S T Uslip V Vt W

Factor for open circuit grinding to be expressed in terms of the final classification of solids Mill diameter factor Oversize feed factor for grinding Fineness factor for ground or crushed particles Reduction ratio factor for ball or rod mills Low reduction ratio factor for ball or rod mills Correction factor for rod mills Correction factor for rubber-lined mills d80 of the feed rocks Optimum size of feed to a ball or rod mill Correlation factor for a mixer between design settling velocity and terminal velocity of solids Acceleration due to gravity (9.8 m/s2) Height of mixer above bottom of tank Horsepower Lift force Number of impellers Rotational speed in rev/min Power Flow rate (m3/s) Recovery of underflow from a cyclone Reynolds number Reynolds number for a Bingham plastic, using the coefficient of rigidity for viscosity material reduction ratio in a grinding circuit Swirling number Thrust force Slip speed between liquid and solids in a mixer Average velocity of flow (m/s) Terminal velocity of solids Consumed power for wet grinding

Greek letters ␣ Angle of incidence ␧ Void fraction ⌫ Wet grinding factor ␳m Density of slurry mixture (kg/m3 or dlugs/ft3) ␳s Density of solids in mixture (kg/m3 or dlugs/ft3) ⌽ Factor of energy dissipation before the hydraulic jump in a free fall ␾ Concentration by volume in decimal points ␥ Shear strain ␲ Pythagoras number (ratio of circumference of a circle to its diameter) ␪ Duration of the shear for a time-dependent fluid ␳ Density ␶0 Yield stress for a Bingham plastic ␮ Kinematic viscosity (usually expressed in Pascal-seconds or poise) ␻ Angular velocity of particle Subscripts L m

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Particle Solids

7-13 REFERENCES Arterburn, R. A. 1982. The sizing and selection of hydrocyclones. In Design and Installation of Communution Circuits, A. L. Mular and G. V. Jergensen (Eds.). New York: Society of Mining Engineers. Bond, F. C. 1952. Third theory of comminution. Trans. AIME, 193, 484. Burgess, K. E. and B. Abulnaga. 1991. The application of finite element analysis of Warman pumps and process equipment. Paper presented at the Fifth International Conference on Finite Element Analysis, University of Sydney, Sydney, Australia. Cheremisinoff, N. P. 1984. Pocket Handbook for Solid–Liquid Separations. Houston: Gulf Publishing. Dickey, D. S. and J. G. Fenic. 1976. Dimensional analysis for fluid agitation systems. Chemical Engineering Elliott, A. J. 1991. Solids, communition, and grading. In Slurry Handling, edited by N. P. Brown and N. I. Heywood. New York: Elsevier Applied Sciences. Gates, L. E., J. R. Morton, and P. L. Fondy. 1976. Selecting agitator systems to suspend solids in liquids. Chemical Engineering, May 24. Holmes, J. A. 1957. A contribution to the study of comminution, a modified form of Kick’s law. Trans. Inst. Chem. Engrs., 35, 125–156. Mular, A. L. and N. A. Jull. 1978. The selection of cyclone classifiers, pumps, and pump boxes for grinding circuits. In Mineral Processing Plant Design, A. L. Mular and R. B. Bhappu (Eds.). New York: Society of Mining Engineers. Oldshue, J. Y. 1983. Fluid Mixing Technology. New York: Chemical Engineering. Stephiewski, W. Z. and C. N. Keys. 1984. Rotary-Wing Aerodynamics. New York: Dover Publications. Stone, R. 1971. Types and costs of grinding equipment for solid waste water carriage. Paper 19 in Advances in Solid–Liquid Flow in Pipes and Its Applications, edited by I. Zandi. New York: Pergamon Press, pp. 261–269: DENVER-SALA. 1995. Selection Guide for Process Equipment. Colorado Springs: Svedala Industries. Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid–Liquid Flow Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans Tech Publications. Weisman, J., and I. E. Efferding. 1960. Suspension of slurries by mechanical mixers. Am. Inst. Chem. Eng. Journal, 6, 419–426. Further readings Su, Y. S., and F. A. Holland. 1968. Agitation and mixing of non-Newtonian fluids. Chem. & Process. Eng., 49, 77–79. Turner, H. E., and H. E. McCarthy. 1965. Fundamental analysis of slurry grinding. AIChE, 15, 581–584.

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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

8-0 INTRODUCTION The centrifugal slurry pump is the workhorse of slurry flows. Chapter 7 briefed the reader about some important slurry circuits, and it was explained that the grinding circuits consume a fair portion of the power of a concentrator. One particular pump at the discharge of the SAG, ball, or other mills is called the mill discharge pump. Wear in these pumps is particularly harsh, leading to frequent replacement of impellers and liners, because a fair portion of the solids remain fairly coarse until recirculated back through the classification circuit. The design of centrifugal pumps involves a combination of mathematical and empirical formulae and models. Although water pumps have been the subject of extensive research in the past, slurry pumps have been designed based on a compromise of what can be cast with hard alloys, molded in rubber, and what can meet the hydraulic criteria. A lot of papers have been published over the years on various aspects of wear in a slurry impeller or volute, performance corrections and derating, etc. The reader of these papers is often left with the impression that the design of these pumps is a combination of science and art. What is often lacking in the literature are guidelines for the design of slurry pumps. Whereas there are hundreds of manufacturers of water pumps on this planet, the number of manufacturers of slurry and dredge pumps has been reduced to a handful. This chapter presents some guidelines for the design of slurry mill discharge pumps. These guidelines were developed by the author on the basis of the analysis of existing pumps in the market, throughout his career as a consultant engineer. The designer can vary the numbers or dimensions presented in the tables of this chapter within a margin of ±15% to design a pump of his or her choice. These guidelines by themselves must be followed by proper testing, prototype development, finite element analysis, and ultimately by fieldtesting. In this chapter, the concepts of expeller, pump-out vanes, and dynamic seal will also be examined. These are very important aspects of slurry pump design that have suffered from a dearth of information in the published literature. Wear remains a concern for the design of a slurry pump. There is no direct correlation between the best hydraulics and the highest wear life. In fact, the whole activity of designing a slurry pump is to find an optimum compromise.

8.1

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8-1 THE CENTRIFUGAL SLURRY PUMP A centrifugal pump is essentially a rotating machine with an impeller to convert shaft power into fluid pressure. The dynamic energy is then converted into pressure or head in a special diffuser or casing. The manufacturers of slurry pumps have developed a number of specialized designs such as 앫 앫 앫 앫 앫 앫 앫 앫

Dredge pumps with impellers as large as 2.6 m (105 in) Mill discharge pumps for milling and grinding circuits Vertical cantilever pumps (without submerged bearings) Froth handling pumps for flotation circuits High-pressure tailings and pipeline pumps General purpose pumps Low-head slurry pumps for flue gas desulfurization or flotation circuits Submersible slurry pumps

The slurry pump may be cased in a hard metal (Figure 8-1) or may be cast in iron, with an internal liner (Figure 8-2), which may be of hard metal or rubber. The components of the slurry pump are divided into two groups: 1. The bearing assembly or cartridge and frame 2. The wetted parts forming the wet end The main components of the wet end are 앫 앫 앫 앫 앫 앫 앫 앫 앫 앫

The pump casing volute The volute liner The front suction plate, or throat bush in large pumps The rear wear plate The impeller The expeller The shaft sleeve The packing rings The stuffing box and gland, greas cup, and associated water connections In very special cases the mechanical seal The drive end of the pump consists of

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The pump shaft Piston rings or alternative protection against solids penetrating the bearing assembly Forsheda seals or O-rings Bearings and bearing nuts Grease retaining plates, grease nipples. or oil cup

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frame back wearplate suction joint impeller FIGURE 8-1 Components of an unlined hard-metal pump. (Courtesy of Mazdak International Inc.)

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discharge companion flange backplate liner backplate coverplate coverplate liner throatbush

expeller Stuffing box shaft sleeve

bearing assembly

pump shaft

suction flange

pump frame impeller

FIGURE 8-2 Components of a rubber-lined slurry pump. (Courtesy of Mazdak International Inc.)

앫 Bearing cartridge and bearing covers 앫 An adjustable bolt or mechanism to adjust the impeller within the casing by moving the shaft 앫 The pump frame 앫 Couplings or pulleys The purpose of the pump is to produce a certain flow against a certain pressure. This is done at a certain efficiency. The optimum point at which the efficiency is at a maximum is called the best efficiency point. For every size or design of pump, there is a best efficiency point at a given speed. The performance of the pump is plotted on a curve of head versus flow (Figures 8-3 and 8-4) By combining different sizes of pumps on a single chart, a pump tomb chart is produced (Figure 8-5). Before dwelling on the design of a slurry pump, it is essential to have a basic understanding of the hydraulics involved. But since the design of slurry pumps must also take in account the wear due to pumping abrasive solids, many other factors enter into the equation, such as the ability to pump large particles and the use of special alloys or polymers for liners or impellers. Practically all slurry pumps are single stage. Multistage pumps are limited to mine dewatering applications. Slurry pumps are rubber lined whenever they are designed to handle particles finer than 6 mm or 1–4⬙. Because rubber is susceptible to thermal degradation when the tip speed of the impeller exceeds 28 m/s or 5500 ft/min, rubber-lined pumps are typically reserved for a maximum head of 30 m (98.5 ft) per stage. White iron is a very hard material. It is used in different forms such as Ni-hard and 28% chrome to cast impellers, casings, and metal liners of slurry pumps. Due to concern about maximum disk stresses, most white iron slurry pumps are limited to an impeller tip speed of 38 m/s or 7480 ft/min. Metal-lined pumps are limited to 55 m or 180 ft per stage.

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8.5


Flow rate (L/s) 5 300

15

10

90

Head vs flow curve

60 Cu r

ve

50

y

150 fic

ie

nc

40

Ef

Head (ft)

200

100

30

Head (m)

70

Efficiency (%)

80

250

20 50 10 0

0

50

150

100

200

250

Flow rate (US gpm) FIGURE 8-3 Performance of a pump showing head versus flow and efficiency versus flow at constant speed.

Flow rate L/s

40%

3900

15

45%

40

ien fic

30

2700 r/min 2400 2100 1800 1500 1200

20

Ef

50

0

50

100

150

200

Head (m)

cu

rv

e

50

cy

3000

100

0

speed or rotation (rev/min)

60

3300

150

best efficiency curve

70

3600

200

90 80

48%

250

10

MINIMUM LIMIT OF USE 45%

4200

20% 30%

5

300

Head (ft)

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10 250

Flow rate in US gpm

FIGURE 8-4 Composite curve for the performance of a pump showing head versus flow and efficiency versus flow at various speeds.

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CHAPTER EIGHT

300

90 80 70

20000

18000

16000

14000

12000

10000

8000

6000

4000

2000

FLOW IN US GALLONS PER MINUTE

250 1105

903

780

691

60

609

200

460

528

6

390

450

340

20

8 X1 100

30 20

RUBBER RANGE

HEAD (FEET)

510

150

X1

X1

X1

4

2

X1 0

575

18

667

16

816

METAL RANGE 14

40

10X8

12

HEAD (METRES)

8X6 50

50

10 0 0

200

400

600

800

1000

1200

FLOW RATE (LITRES/SECONDS)

FIGURE 8-5 “Tomb chart” for pumps showing size of pump versus flow range and head.

White iron should not be confused with steel. Certain grades of steels are used in slurry, dredging, and phosphate matrix pumps. They are cast at a lower hardness than white iron and by being more ductile can withstand higher disk stresses. Impellers cast in steel can be used in slurry pumps up to a tip speed of 45 m/s (8858 ft/min). These are general guidelines, but the consultant engineer should collaborate closely with the manufacturer. For example, certain special anti-thermal-breakdown additives are used with some rubbers to exceed the limit of 28 m/s or 5500 ft/min on tip speed. In certain situations, a metal impeller may be installed with rubber liners, particularly when there are concerns about slurry surges (water hammer) in tailings pipelines.

8-2 ELEMENTARY HYDRAULICS OF THE SLURRY PUMP The correlation between the tip speed and the head per stage is established from basic hydraulics of impeller design. There have been two schools in the past for the design of water pumps—the American school lead by Stepanoff and the European school lead by Anderson. The Stepanoff method is based on the concept that an impeller is designed on the basis of velocity triangles, and that an ideal volute for best efficiency is then found using

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8.7

various empirical factors. The Anderson school is based on the concept that one of the most important parameters in pump design is the ratio between the throat area of the volute and the impeller discharge area, and therefore more than one volute design can be matched to a given impeller. In the case of slurry pumps, passageways are larger than in water pumps to accommodate solids and the Anderson area ratio is difficult but useful to use. Unfortunately, many leading references on slurry pump design written in North America, such as the work of Herbrich (1991) and Wilson et al. (1992), continue to ignore the area ratio methods and focus on the Stepanoff school, which believes that the impeller is the main producer of head and efficiency. The design of a centrifugal slurry pump is complex. Performance depends on the area ratio, impeller tip angle, recirculation patterns, change with wear of the impeller, back vanes, and front pump-out vanes. The flow in an impeller is fairly complex. A review of the hydraulics is essential to appreciate wear. In simple terms, a vortex is formed.

8-2-1 Vortex Flow The vortex creates a pressure field related to the radius from the center of the vortex in accordance with the following equation:

␻ = C × R mv0

(8-1)

where ␻ = angular speed of rotating fluids Rv0 = local radius of vanes m = exponent Stepanoff (1993) described various forms of vortices from a free vortex, with angular velocity ␻ inversely proportional to the square of the radius Rv0, to a super-forced vortex, in which the angular velocity is proportional to the radius, as shown in Table 8-1. The general distribution of pressure through a vortex, according to Stepanoff, is +z 冢 ᎏ␳ 冣 = 冢 ᎏᎏ 2(m + 1)g 冣 P

2(m+1) C 2Rv0

(8-2)

where C = constant P = pressure ␳ = density m = exponent g = acceleration due to gravity z = liquid elevation above the fixed datum For a forced vortex, the angular speed is constant and the liquid revolves as a solid body. Disregarding friction losses, Stepanoff (1993) claims that no power would be needed to maintain the vortex. The pressure distribution of this ideal solid body rotation is a parabolic function of the radius. When the forced vortex is superimposed on a radial outflow, the motion takes the form of a spiral. This is the type of flow encountered in a centrifugal pump. Particles at the periphery are said to carry the total amount of energy applied to the liquid.

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CHAPTER EIGHT

TABLE 8-1 Patterns of Vortex Flow

Case

Angular velocity distribution, ␻ = C1 × Rmv0

Peripheral velocity distribution, V × Rnv0 = C

Pressure distribution, dp = 兰 (␳␻2/g)rdr

1 2 3

–⬁ ⬁ ␻ = C1 × Rv0 V × Rv0 = C1 –5/2 3/2 ␻ = C2 × Rv0 V × Rv0 = C2 ␻ = C3 × R–2 V × Rv0 = C3 v0

P/␳ = C 21 + z1 P/␳ = C 22/(3 · g · R3v0) + z2 P/␳ = C 23/(2 · g · Rv0) + z3

4 5 6

–3/2 ␻ = C4 × Rv0 V × R1/2 v0 = C4 –1 0 ␻ = C5 × Rv0 V × Rv0 = C5 –1/2 ␻ = C6 × Rv0 V × R –1/2 v0 = C6

P/␳ = –C4/(g · r) + z4 P/␳ = [C 25/g] · log Rv0 + z5 P/␳ = C 26 · Rv0/g + z6

7

␻ = C7 × R0v0

–1 V × Rv0 = C7

P/␳ = C 27 · R2v0/(2 · g) + h7

8 9 10

␻ = C8 × R1/2 v0 ␻ = C9 × Rv0 ␻ = C10 × R mv0

V × R –3/2 v0 = C8 V × R–2 v0 = C9 V × R–(m+1) = C10 v0

P/␳ = C82 · R3v0/(3 · g) + h8 4 P/␳ = C9 · R v0 /(4 · g) + h9 P/␳ = [C 2R2(m+1) ]/ v0 [2(m + 1) · g] + h

Type of vortex

␻ = 0, stationary Z3 + (P/␳) + (v2/(2 · g) = constant, free vortex V = constant V2/Rv0 = constant = centrifugal force ␻ = constant, forced vortex Super forced vortex Super forced vortex General form of super forced vortex

Remarks

␻ is higher toward center of the vortex

␻= constant ␻ is higher toward periphery of vortex

After Stepanoff (1992).

The parabola shown in Figure 8-6 is a state of equilibrium for a forced vortex and is similar to a horizontal plane for a stationary fluid. To maintain a flow outward against the applied pressure, the energy gradient must be smaller than the energy gradient for no flow. This is what happens in a pump at near shut-off condition, where maximum static head is obtained without any flow. As flow increases through the impeller, the head drops. In the case of the expeller, the designer tries to reach the parabola for energy gradient without flow. However, as Case 7 in Table 8-1 shows, the pressure gradient is a square function of R and inversely proportional to the square of the angular velocity. And in fact, below a certain angular velocity, there is not enough pressure to overcome the difference between volute and outside atmospheric pressure. The expeller or dynamic seal then stops performing and leakage occurs.

8-2-2 The Ideal Euler Head The ideal pressure that a pump impeller can develop is called the Euler pressure. Consider the flow through a radial impeller between two radii R1 and R2. The impeller is rotating at an angular speed ␻ (in rad/s) so that the peripheral speeds are respectively: U1 = R1 · ␻

(8-3a)

U2 = R2 · ␻

(8-3b)

The liquid flows radially at a meridional velocity Cm, perpendicular to the peripheral velocity U. The value of Cm is determined from continuity equation, It is necessary to take into account the local area of the flow, which is a function of the radius and the width of

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FIGURE 8-6 Pressure distribution in an impeller versus radius for condition of flow and no flow. (From Stepanoff, 1993. Reprinted by permission of Krieger Publishers.)

the channel, minus the blockage area due to the finite thickness and angle of inclination of the blades. The channels between the impeller vanes follow a certain profile. At the intersection with the radius under consideration, the angle between the vane and the tangent to the radius is defined as ␤. A component of velocity is in the direction of ␤ and is called the relative velocity W. The vector addition of U and W result in the absolute velocity V. Both V and W share the same component of meridional speed Cm; a vector representation is shown in Figure 8-8. The Euler “total” head between radii R1 and R2 is defined as (V 22 – V 21) – (U 22 – U 21) + (W 22 – W 12) HE = ᎏᎏᎏᎏ 2g

(8-4)

where (V 22 – V 21) = change in absolute kinetic energy (W 22 – W 12) = change in relative kinetic energy (U 22 – U 21) + (W 22 – W 12) = change in static energy through the impeller It is clear that W = Cm · cot ␤. Static head rise is gHs = (U 22 – U 21) + (Cm2 · cot ␤2)2 – (Cm1 · cot ␤1)2

(8-5)

Furthermore because the curvature of the front and back shrouds of an impeller, are different, the meridional velocity is not uniform and may be higher toward the back shroud. For a linear variation of the meridional velocity between the front and back shrouds (Figure 8-7), Stepanoff (1993) derived the following equation for theoretical head:

冢冣

冢

U 22 U2Cm2 (V2 – V1)2 Ht = ᎏ – ᎏ 1 + ᎏᎏ g tan ␤2 12 Cm2 g

冣

(8-6)

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8.10

CHAPTER EIGHT

FIGURE 8-7 Pressure and velocity distribution for cases shown in Table 8-1. (From Stepanoff, 1993. Reprinted by permission of Krieger Publishers.)

The term 1 + [(V2 – V1)2/12 C m2 ] is greater for the wide impellers encountered in mining slurry pumps. For slurry pumps, the value of ␤1 at the tip diameter of the eye of the impeller is between 14 and 30 degrees. The value of ␤2 at the tip diameter of the vanes is typically between 25 and 35 degrees. Stepanoff (1993) has indicated that inlet angles as high as 50 degrees are used on water pumps. This is, however, not the case with slurry pumps, as prerotation causes tremendous wear of the throat bush. The vast majority of modern pumps have a discharge angle ␤2 smaller than 90 degrees. They are called impellers and have backward curved vanes. Expellers are often designed with radial vanes (i.e., ␤2 = 90 degrees). Forward vanes with ␤2 larger than 90 de-

W2 2

1

V2

Cm2

2

U2

Outlet velocities at R 2

1

2

R1

2

W

W

C

m1

1

V1

Inlet velocities at R 1

1

U1

V1

U

Cm1 1

1

W1

C

m2

2

U 2

V2

n io tat ro

R2

FIGURE 8-8 Ideal velocity profile in an impeller.

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8.11

grees are restricted to very low flow and high-head pumps and to some expellers. Theoretically, an impeller with forward vanes would give a higher static head rise. Unfortunately, it is also the largest consumer of power and is considered to be inefficient. Clay and other slurries can be very viscous. Herbrich (1991) has suggested using discharge angle ␤2 as high as 60 degrees on impellers for very viscous slurries but did not produce data to support such a suggestion. Stepanoff (1993) recommended the following design procedures for special pumps. These pumps would be suited to pump viscous liquids, but their performance may be impaired on water. 1. Use high impeller discharge angles up to 60 degrees to reduce the impeller diameter necessary to produce the same head and effectively reduce disk friction losses. Consequently, the impeller channels become shorter and the impeller hydraulic friction is reduced. 2. Eliminate close-clearance wide sealing rings at the impeller eye and provide knifeedge seals (one or two) similar to those used on blowers. Leakage loss becomes secondary when pumping viscous liquids. 3. Provide a similar axial seal at the impeller outside diameter to confine the liquid between the impeller and casing walls. This in turns raises the temperature of the liquid in the confined space (due to friction) well above the temperature of the remaining liquid passing through the impeller. Due to the temperature effects, viscosity is artificially reduced and disk friction losses are trimmed down. In fact, Stepanoff (1993) goes as far as suggesting injecting a light or heated oil in the confined space to reduce power loss due to friction. 4. Provide an ample gap (twice the normal) between the casing tongue or cutwater and the impeller outside diameter. Otherwise, the shrouds of the impeller would produce head by viscous drag at low capacities, and would decrease the efficiency of pumping. 5. High rotational speed and high specific speed lead to better efficiency and more head capacity output than low specific speed pumps on viscous liquids. These recommendations were written with very viscous fluids in mind. Obviously, points 2 and 3 would not apply to a slurry pump. However, slurry pumps may use pumpout vanes, which effectively are dynamic seals. These recommendations can be modified to suit the design of special pumps for viscous slurries. The field of slurry pumps for very viscous slurries and difficult flotation frothy slurry associated with the oil sands industry is continuously evolving. In some cases of pumping oil sand froth, it has been found that injecting 1% of water or a light oil as a lubricant just at the suction of the pump can improve the efficiency of the pump.

8-2-3 Slip of Flow Through Impeller Channels Due to the curvature of the vane, the flow on the upper surface of a vane is usually faster than the flow on the lower surface of the vane. If we consider the direction of rotation, the upper surface is also called the advancing surface or leading surface. The lower surface is the trailing surface. The pressure being higher on the trailing surface, the fluid leaves tangentially only at the trailing surface. A certain amount of liquid is attracted by the lower surface of the following vane and a pattern of flow recirculation develops as shown in Figure 8-9. To compare this situation with that of an airplane, which many of us have examined, vortices form behind a flying wing, as air tends to roll from the upper pressure

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8.12

R2

rot ati on of im pe lle r

CHAPTER EIGHT

R1

relative recirculation FIGURE 8-9 Recirculation in pump impellers (after Stepanoff, 1993).

zone of the lower surface toward the lower pressure zone above the wing. A vortex sheet, called “horseshoe vortex,” forms behind the airplane wing. The velocities in a real impeller do not follow the ideal “Euler” impeller pattern, and a degree of “slip” occurs. The angles of flow and forces deviate from the theoretical values as shown in Figure 8-10 by a “lag” angle. The slip factor is in fact as the ratio of the measured absolute velocity to the theoretical Euler absolute velocity at the tip diameter of the vanes:

␮ = V2⬘/V2

(8-7)

Since the average meridional velocity is essentially a function of the ratio of flow rate to the discharge area at the tip of the impeller, it is not affected by slip. However, a change in the absolute velocity is accompanied by changes in the relative velocity and of the angles with respect to the peripheral tangential speed. Various equations have been developed over the years to evaluate the slip factor. The most famous is Stodola’s formula:

␲ · sin ␤2 ␮ = 1 – ᎏᎏ Z

冢

冣

(8-8)

where Z = number of vanes. Stodola’s formula was originally developed for zero flow, but has been extensively used for design flows of water pumps even at best efficiency point. Another equation used to determine slip was developed by Pfeiderer (1961):

␤2 R 22 a ␮=1 1+ ᎏ 1+ ᎏ ᎏ Z 60 S

/冦

冢

冣冧

theoretical

V'2 W'2

W2 2

(8-9)

measured (with slip)

V2 2

2

2

U2 FIGURE 8-10

Slip of flow in impellers versus ideal velocity profile.

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8.13

where S=

冕

R2

R dR

(8-10)

R

S is called the static moment and is obtained by graphical integration along the meridional plane of the vanes. In the special case of a cylindrical vane S=

冕

R2

R

R dR = |(R 22 – R 21)

and the slip factor is

␤2 a 2 ␮ = 1 1 + ᎏ 1 + ᎏ ᎏᎏ Z 60 1 – (R 21/R 22)

/冦

冢

冣

冧

(8-11)

In the special case in which R1/R2 is smaller than 0.5, the slip does not increase anymore, and a ratio R1/R2 = 0.5 should be assumed. The magnitude “a” depends on the design of the casing. Pfeiderer (1961) established the following values for the coefficient “a”: Volute Vaned diffuser Vaneless diffuser

a = 0.65–0.85 a = 0.60 a = 0.85 – 1.0

Most slurry pumps use a volute (Figure 8-11). Vaned diffusers are used in certain mine dewatering pumps. Defining the hydraulic efficiency as ␩H, the head developed by the pump is expressed as: H = ␮␩HU2V2

(8-12)

Equation (8-12) establishes the effect of the casing and the impeller on the head developed at the so-called best efficiency point. Because of the rather simplistic Stodala equa-

volute casing

diffuser vane casing

FIGURE 8-11

Volute and vaned diffusers.

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CHAPTER EIGHT

TABLE 8-2 Test Data from Herbich (1991) Velocity Radial Tangential

Theoretical

Measured

4.21 ft/s (1.3 m/s) 55.80 ft/s (17.0 m/s)

15.6 ft/s (4.8 m/s) 39.6 ft/s (5.2 m/s)

tion (8-8), it is sometimes assumed that the impeller is the main contributor to head. The equation for the head is also expressed in terms of the discharge angle from the vanes, the slip factor, and the hydraulic efficiency as: Cm2 · cot ␤2 U 22 H = ␮␩H ᎏ 1 – ᎏᎏ g U2

冢

冣

(8-13)

Herbich (1991) measured extensively the lag angle and deviation from theoretical angles in the case of the Essayon dredge pump and reported two cases of impeller tip vane discharge angle ␤2 (Table 8-2). In the first case, the vane was designed with a physical tip angle at the vanes of 22.5°. This would have been theoretically the angle for the relative speed W. However, test data measured an average angle of 30.5°. In the second case, the vane had a discharge angle of 35° but test data indicated that the relative velocity was effectively inclined at an average angle of 12°. In fact there is no definite value. In the case of the first impeller with a vane angle of 22.5°, at a flow rate of 63 L/s (1000 gpm) the flow between the channels was measured to have streams inclined between 61° on the lower surface and 25° on the forward surface with various values between 21 47°. A different pattern was noticed at 38 L/s (600 gpm). The distortion of the flow is therefore a function of the ratio of flow rate to normalized flow (at best efficiency point). When the experimental angle is higher than the theoretical, Herbich pointed out that it would mean that the particles tend to avoid contact, thus minimizing the possibility of scour. On the other hand, if the measured angle is less than the theoretical, the solids will impact the vanes and cause wear. Because it is difficult to measure slip, an experimental head coefficient is defined as: 2gHBEP ␺SI = ᎏᎏ U 22

(8-14)

For some historical reasons, U.S. books drop the numerator 2: gHBEP ␺US = ᎏ U 22

(8-15)

The reader must therefore be careful when comparing pumps manufactured in North America with those manufactured in Europe.

8-2-4 Specific Speed The steepness of the curve between the best efficiency capacity and the shut-off point of the pump depends on the geometrical design of impeller and casing. With so many different designs of pumps, engineers have used nondimensional specific speeds and other parameters. In the International System of Units, the specific speed is defined as: Q N · 兹苶 Nq = ᎏ H 3/4

(8-16)

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8.15

where N = rotational speed in rev/min Q = capacity in cubic meters per second, at best efficiency capacity H = differential head in meters at best efficiency capacity The specific speed in the United States is defined as: N · 兹苶 Q NUS = ᎏ H 3/4

(8-17)

where N = rotational speed in rev/min Q = capacity in U.S. gallons per minute, at best efficiency capacity H = differential head in feet at best efficiency capacity Some books include the acceleration of gravity g or 32.2 ft/s in the denominator for the sake of consistency, but for historical reasons Equation 8-17 is used. Another term sometimes used in international books is the characteristic number:

␻ · 兹苶 Q Ks = ᎏ 3/4 [gH]

(8-18)

Most slurry pumps operate at a specific speed smaller than 2000 in U.S. units or 39 in SI units. In this range, the tip diameter of the impeller may be between 2 to 3.5 folds of the suction diameter. The shut-off head is then between 150% and 110% of the best efficiency point head at the same speed (Figure 8-12). Addie and Helmly (1989) measured the head coefficient (as defined in the United States) and the efficiency of a number of slurry and dredge pumps. Their results are shown in Figures 8-13 and 8-14. They pointed out that the slurry and dredge pumps were on the average between 5% and 9% less efficient than their water counterparts. Example 8-1 A slurry pump is to be designed for a head at best efficiency of 150 ft at a flow rate of 1200 gpm. Assuming a head coefficient of 0.5 (by U.S. definition), determine the diameter and the speed of rotation if the specific speed is 1100 (in U.S. units). Solution in USCS Units From Equation 8-15: 32.2 × 150 gHBEP ␾US = ᎏ = 0.5 = ᎏᎏ U 22 U 22 U2 = 98.3 ft/s From Equation 8-17, the specific speed in the United States is defined as: Ns = N · Q1/2/H 3/4 = 1100 = N · 12001/2/1503/4 N = 889 rpm = 93.1 rad/sec Since U = R␻, then R = 98.3/93.1 = 1.06 ft. The impeller diameter is therefore 2.11 ft or 25.3 inch (643 mm). Every manufacturer has their proprietary design criteria, and for a given size some manufacturers may have an impeller design that pumps more than others. In the case of

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8.16

FIGURE 8-12 Shape of impeller versus specific speed in USCS units. [From I. Karasik et al. (Eds.), Pump Handbook, reprinted by permission from McGraw-Hill.]

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FIGURE 8-13 Head coefficient versus specific speed from Addie and Helmly (1989) (reproduced by permission of Central Dredging Association, Delft, Netherlands). This plot is somewhat confusing as it uses the U.S. definition of the head coefficient (as per Equation 8-15) against the specific speed in SI units. The reader should multiply the head coefficient by a factor of 2 to use the SI definition of head coefficient as per Equation 8-14.

slurry pumps, attention must be paid to the wear life of the pump. Too little flow in a large pump leads to excessive recirculation, and too much flow would cause rapid wear. The relationship between the volute shape and the impeller plays a major role, too. These parameters are refined through detailed engineering and field-testing. A good starting point for the design of mill discharge pumps is shown in Tables 8-3 and 8-4. These are realistic values that mills expect from pumps. The next step is to define the steepness of the curve. Slurry pumps are designed to be forgiving as processes too often change. Very steep curves are not encouraged, but flat curves do create overloading problems to the drivers. A shut-off head in the range of 125% to 135% of the best efficiency head is recommended. The slurry pump design engineer should then establish what is often referred to as a 5-points curve, as shown in Tables 8-5 and 8-6.and Figure 8-15. As early as 1938, Anderson developed a concept of the ratio of the area of flows between the vanes of the impeller and the throat area (Figure 8-10) that is basic to the performance of the pump. His methodology is called the “area ratio” (Figure 8-16). Worster (1963) demonstrated this to be correct by mathematical derivation. Anderson (1977, 1980, 1984) extended his analysis by statistical analysis of a large number of water pumps and turbines. Unfortunately, no similar work has been done on slurry pumps and because slurry impellers are fairly wider than water pump impellers to allow the passage of rocks and large particles, the Worster curves do not apply well to the design of solids-handling pumps. Not all applications of pump slurries require wide impellers. In fact in the last 20 years, grinding circuits have greatly evolved to the point that very fine ores are pumped. For these applications, narrower and more efficient impellers should designed.

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CHAPTER EIGHT

FIGURE 8-14 Efficiency of large dredge pumps versus specific speed (in SI units). (From Addie and Helmly, 1989. Reproduced by permission of Central Dredging Association, Delft, Netherlands)

8-2-5 Net Positive Suction Head and Cavitation When the pressure on the suction flange of the pump is insufficient, the pump starts to cavitate and becomes very noisy. The net positive suction head (NPSH; see Figure 8-17) is the absolute head above the vapor pressure at the suction flange of the pump: Pe – PD – PV V e2 NPSHA = ᎏᎏ + Z1 – Z2 ᎏ ␳g ␳g

(8-19)

where Pe = pressure at the surface of the liquid in absolute terms on the suction side PD = pressure losses between the surface of the liquid and the pump, due to friction, valves, etc. PV = vapor pressure Z1 = geodetic elevation of liquid surface above the centerline of the pump impeller Ze= geodetic elevation of the centerline of the pump impeller Ve = suction speed at the eye of the impeller

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TABLE 8-3 Recommendations for Design of Rubber-Lined Mill Discharge Pumps Size suction to Flow Head discharge, inch _____________ ___________ (mm/mm) L/s US gpm m ft 8×6 200/150 10 × 8 250 × 200 12 × 10 300 × 250 14 × 12 350/300 16 × 14 400 × 300 18 × 16 450 × 400 20 × 18 500/450

Efficiency %

Suction speed ____________ m/s ft/s

Discharge speed _____________ m/s ft/s

130

2061

30

98.5

70

4.2

13.7

7.2

23.5

220

3487

30

98.5

74

4.5

14.8

6.7

22.1

310

4915

30

98.5

76

4.4

14.4

6.12

20.1

425

6737

30

98.5

79

4.4

14.5

5.86

19.2

560

8877

30

98.5

81

4.3

14.1

5.64

18.5

685

10859

30

98.5

83

4.3

14.1

5.45

17.9

875

13870

30

98.5

84

4.3

14.2

5.33

17.5

From Abulnaga (2001). Courtesy of Mazdak International Inc.

TABLE 8-4 Recommendations for Design of Metal-Lined or Hard Metal Mill Discharge Pumps Size suction to Flow Head discharge, inch _____________ ___________ (mm/mm) L/s US gpm m ft 8×6 200/150 10 × 8 250 × 200 12 × 10 300/250 14 × 12 350/300 16 × 14 400/300 18 × 16 450/400 20 × 18 500/450

Efficiency %

Suction speed ____________ m/s ft/s

Discharge speed _____________ m/s ft/s

176

2797

55

180

70

5.7

18.6

9.7

32

298

4732

55

180

74

6.1

20

9.1

30

421

6670

55

180

76

6

19.5

8.3

27

577

9143

55

180

79

6

19.7

8

27.3

760

12047

55

180

81

5.8

19.3

8

25.1

924

14647

55

180

83

5.8

19.3

7.4

24.1

1188

18823

55

180

84

5.8

19.3

7.2

23.8


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TABLE 8-5 Preliminary Range for Efficiency versus Flow (L/s units) For Mill Discharge Pump—Rubber-Lined Version Pump Size (suction/discharge)

Ratio of Ratio of flows, efficiency, Q/QN ␩/␩BEP 0.25 0.5 0.75 1.00 1.15

0.5 0.8 0.95 1.0 0.97

8 × 6 in 10 × 8 in 12 × 10 in 14 × 12 in 16 × 14 in 18 × 16 in 20 × 18 in 200 × 150 250 × 200 300 × 250 350 × 300 400 × 350 450 × 400 500 × 450 mm mm mm mm mm mm mm Flow in L/s 32.5 65 97.5 130 150

55 110 165 220 253

77.5 155 232.5 310 356.5

106 213 319 425 489

140 280 420 560 644

171 342 523 684 787

219 438 656 875 1006


Each pump has a minimum required NPSH that is established through testing. It is defined as the required NPSH or NPSHR. The suction-specific speed is defined at the best efficiency point as: N · 兹苶 Q NSS = ᎏᎏ NPSHR3/4

(8-19)

The value of NPSH is established at the point where the suction conditions at best efficiency flow suffer a 3% drop of total dynamic head. Solids present in slurry do not contribute to the vapor pressure, but they contribute to the density of the mixture as well as to the friction or pressure losses on the suction. This could be confusing to the inexperienced engineer who has to handle water vapor pressure as well as slurry density. One approach is to calculate the pressure on the suction in units of pressure and then to convert back into units of length.

TABLE 8-6 Preliminary Range for Efficiency versus Flow (L/s units), Metal-Lined or Hard Metal Mill Discharge Pumps Pump Size (suction/discharge)

Ratio of Ratio of flows, efficiency, Q/QN ␩/␩BEP 0.25 0.5 0.75 1.00 1.15

0.5 0.8 0.95 1.0 0.97

8 × 6 in 10 × 8 in 12 × 10 in 14 × 12 in 16 × 14 in 18 × 16 in 20 × 18 in 200 × 150 250 × 200 300 × 250 350 × 300 400 × 350 450 × 400 500 × 450 mm mm mm mm mm mm mm Flow in L/s 44 88 132 176 202

74.5 149 223.5 298 342.7

105 210 316 421 484

144 288.5 315.8 577 664


190 280 420 760 874

231 462 693 924 1063

297 594 891 1188 1366

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8.21


1.0

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0 0.0

0.5

1.0 Q/Q

FIGURE 8-15 point.

N

HEAD

EF FI CI EN CY

N

1.2

H/H

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0.0 1.5

N

Normalized curves of head and efficiency versus values at the best efficiency

It is often recommended that the available NPSH be at least 0.9 m or approximately 3 ft higher than the required NPSH as shown on the pump curve. Example 8-2 Slurry with a specific gravity of 1.48 is to be pumped from a pond 3 m lower than the centerline of the impeller. The pond is situated at a high altitude. The atmospheric pressure is 85 kPa. The friction losses have been determined to be 1.5 m. The vapor pressure of water is 4.24 kPa. The slurry enters the pump at a velocity of 3.5 m/s. Determine the available NPSH. Solution Pressure due to friction losses is:

␳gH = 1480 · 9.81 · 1.5 = 21,778 Pa The geodetic elevation of the centerline of the pump impeller is 3 m higher than the liquid; this results in a negative pressure or

␳g⌬Z = 1480 · 9.81 · (–3) = –43,556 Pa Dynamic head losses due to a velocity of 3.5 m/s are: 1480 · 3.52/2 = 9065 Pa Net positive pressure is: 85,000 – 43,556 – 21,778 – 9065 – 4240 = 24,491 Pa Converting back into head of water: 24,491/(9.81 · 1000) = 2.496 m of water

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FIGURE 8-16 The area ratio curves for water pumps. No similar curves have been published for slurry or dredge pumps. (From Worster, 1963. Reproduced by permission of the Institution of Mechanical Engineers, UK.

This is very low, and since the engineer must avoid cavitations, he or she may consider the use of a submersible slurry pump or a vertical cantilever pump instead of a horizontal pump on the shore. The NPSH can be expressed as the function of suction speed and the eye tip speed at the suction diameter (Turton, 1994): 0.9 C m2 + 0.115 U 21 NPSH = ᎏᎏ 9.81

(8-20)

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P =P + P E A B

9:21 AM

Absolute Atmospheric Press ure P A Liquid at vapor pressure Pv

Pressurized gas at surface at gauge pressure PB

Page 8.23

H 1 8.23

ZS

ZE Pressure due to friction losses PD FIGURE 8-17

Concept of net positive suction head.

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Example 8-3 A pump impeller rotates at 500 rpm to pump 65 L/s through a suction diameter of 200 mm. Using Equation 8-20, determine the required NPSH. Solution The velocity Cm is determined by dividing the flow rate by the suction area: Cm = 0.065/[0.25 · ␲ · 0.22] = 2.07 m/s U = 2␲RN/60 = 2 · ␲ · 0.1 · 500/60 = 5.24 m/s 0.9 · 2.072 + 0.115 · 5.242 NPSH = ᎏᎏᎏ = 0.715 m 9.81 In reality, NPSH depends on many other factors, particularly clearances at the impeller eye, prerotation, the use of inducers, etc. Many empirical studies tend to support that a low NPSH impeller should have a vane entry angle of 14° to 15°. A cavitations parameter ␴ is defined as the ratio of required NPSH to the pump total dynamic head at the best efficiency point at the given speed: NPSH ␴= ᎏ TDH

(8-21)

Addie and Helmly (1989) measured the cavitations parameter against specific speed for a number of dredging pumps. Their work is represented in Figure 8-18. Tables 8-7 and 8-8 also show certain calculations for the design of mill discharge pumps.

FIGURE 8-18 Cavitation factor versus specific speed (in metric units) for slurry and dredge pumps. (From Addie and Helmly, 1989. Reproduced by permission of Central Dredging Association, Delft, Netherlands)

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TABLE 8-7 Recommendations for Impeller Diameter, Speed, Specific Speed Number, and Cavitations Parameter for Rubber-Lined Mill Discharge Pumps (U.S. Units)

Model 8 × 6 in 10 × 8 in 12 × 10 in 14 × 12 in 16 × 14 in 18 × 16 in 20 × 18 in

Head Sigma Vane d2, d2/dS, Speed, Flow, Efficiency, Head, Specific factor, cavitation inch tip/suction rpm US gpm % ft speed ␺US factor 20.87 26.77 31.10 35.04 39.37 45.28 55.12

3.48 3.35 3.11 3.92 2.81 2.8 2.76

816 667 575 510 450 390 340

2061 3487 4915 6763 8877 10859 13870

70 74 76 79 81 83 84

98.5 98.5 98.5 98.5 98.5 98.5 98.5

1186 1261 1290 1340 1357 1300 1281

0.14 0.13 0.13 0.13 0.133 0.133 0.119

0.14 0.16 0.16 0.17 0.15 0.16 0.16


TABLE 8-8 Recommendations for Impeller Diameter, Speed, Specific Speed Number, and Cavitations Parameter Metal-Lined or Hard Metal Mill Discharge Pumps (U.S. Units)

Model 8 × 6 in 10 × 8 in 12 × 10 in 14 × 12 in 16 × 14 in 18 × 16 in 20 × 18 in

Head Sigma Vane d2, d2/dS, Speed, Flow, Efficiency, Head, Specific factor, cavitation inch tip/suction rpm US gpm % ft speed ␾US factor 20.87 26.77 31.10 35.04 39.37 45.28 55.12

3.48 3.35 3.11 3.92 2.81 2.8 2.76

1005 903 779 691 609 528 460

2790 4721 6654 9121 12018 14701 18779

70 74 76 79 81 83 84

180 180 180 180 180 180 180

1186 1261 1290 1340 1357 1300 1281

0.173 0.13 0.13 0.102 0.132 0.133 0.12

0.14 0.16 0.16 0.17 0.15 0.16 0.16


8-3 THE PUMP CASING The pump casing of a slurry pump often takes the shape of a volute. The best hydraulic design calls for a constant momentum design or a linear increase of the cross-sectional area from the tongue to the throat (Figure 8-19). In reality, the profile of the volute is often simplified to two semicircles. The idea is that hard metals are difficult to cast, and if the shape can be simplified, the casting will flow better during solidification. Rc in Figure 8-19 refers to the cutwater radius. The difference between Rc and R2 is effectively the gap at the cutwater. It must be large enough to accommodate the passage of coarse particles or rocks. The head developed by the pump at shut-off is the sum of the head due to the rotation of the impeller and shape of the volute. Turton (1994) summarized the research of Frost and Nilsen (1991), who concluded that the shut-off head was insensitive to the number of blades, the blade outlet geometry, and the channel width of the impeller. They determined that: HSV = HIMP SV + HVOL SV

(8-22)

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CHAPTER EIGHT

FIGURE 8-19 Parameters for the calculations of the shut-off head of a water pump used in Equation 8-24. (From Frost and Nilsen, 1991. Reproduced by permission from the Institution of Mechanical Engineers, UK.)

where HIMP SV = shut-off head due to the impeller HVOL SV = shut-off head due to the volute HSV = total shut-off head R 22␻2 HIMP SV = ᎏ [1 – (Rs/R2)2] 2g

(8-23)

and R2␻ HVOL SV = ᎏ RMD – R2

冢

冣冦 2

冧

R42 – R 22 2 RMD ln(R4/R2) – 2RMD(R4 – R2) + ᎏ /g 2

(8-24)

Equations (8-23) and (8-24) were derived for water pumps, and it is recommended to confirm the results when designing a new family of slurry pumps. Referring to Figure 8-20, the width of the volute is defined by two components, Xv in the x-direction and Yv in the y direction, when the volute is in a position for vertical top discharge. The magnitude of these two components depends on the clearance at the cutwater, the throat area, the tip diameter of the impeller, and the discharge diameter of the pump. These are refined through experimental testing and hydraulic analysis. A good starting point (or rule of thumb) for the design engineer is to use the shroud diameter of the impeller dt as a reference and to establish XV = Kxdt

1.3 < Kx < 1.4

(8-25)

YV = Kydt

1.2 < Ky < 1.3

(8-26)

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8.27


suction diameter cutwater

discharge impeller tip diameter throat area

tL (liner thickness) R R

R 2

3

R

c

3 Yv

R

R

4

4

RM

t

c (casing thickness) X

V

FIGURE 8-20 Volute shape of a slurry pump simplified for the sake of manufacturing and casting of hard metal casing or liners to a minimum number of partial circles.

Having established a profile of the volute, the thickness of the liner and the thickness of the casing are then added before locating the bolts for lined casings. There is no definite rule of thumb for the thickness of rubber or metal liners. The thickness of the liner is established by the manufacturer on the basis on their experience with the application. Table 8-9 recommends a liner thickness in the range of 4% to 6% of the impeller diameter. Having sized the thickness of the liner, a parameter D for the volute is defined using the width XV as D = XV + 2tL

(8-26)

For a single-stage pump designed for a pressure of 1035 kPa (150 psi), with a ribbed casing, the casing thickness is established as tc ⬇ D/41

(8-27)

Equation 8-27 should be complemented by a full finite element analysis, as the ribs have to be placed correctly. Modern computers are very useful for checking on the size of the ribs. Burgess and Abulnaga (1991) have recommended the use of the equivalent thickness approach. It consists of calculating the second moment of area of the ribs and implementing them in a plate model for the casing. An alternative but much more tedious approach is to use brick elements. Since 1991, the science of minicomputers has advanced greatly and it is now possible to implement very sophisticated three-dimensional models.

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CHAPTER EIGHT

TABLE 8-9 Recommended Dimensions for a Single Stage Mill Discharge Pump (metric size example) Size (mm) Impeller d2 Shroud diameter dt Cutwater diameter dC Cutwater gap (dC – dt)/2 XV = 1.3 dt YV = 1.25 dt Liner thickness tL D = XV + 2 · tL Pressure area Ap (m2)* Working pressure kPa Design pressure kPa F = Ap · Pdesign (kN) D/t Casing thickness tc (with ribs) Number of bolts Load/bolt kN Bolt area mm2** Bolt diameter mm Bolt

200 × 150

250 × 200 300 × 250 350 × 300 400 × 350 450 × 400 500 × 450

530 560 657 49

680 720 843 62

790 830 980 75

890 930 1104 87

1000 1050 1240 95

1150 1200 1426 113

1400 1500 1775 138

728 700 34 796 0.503 1035 1380 694 40 20

936 900 38 1012 0.82 1035 1380 1132 40.7 24

1073 1031 41 1155 1.064 1035 1380 1468 41.17 28

1209 1163 45 1299 1.363 1035 1380 1881 40.42 31

1352 1300 48 1448 1.70 1035 1380 2348 41 34

1560 1500 51 1662 2.24 1035 1380 3105 41.07 39

1950 1875 55 2060 3.87 1035 1380 5341 41.2 50

12 58 347 21 M24

12 94 563 27 M30

12 122 731 31 M36

12 157 940 35 M40

12 196 1174 39 M46

12 259 1551 45 M50

12 445 2662 58 M62

*Ap = 0.9[XV + tL][YV + tL] · 10–6. **Allowed stress on bolt 166 Mpa. Note: these calculations are preliminary and must be confirmed by finite element analysis. They are for single stage, 1.38 MPa rating with ductile iron casing.

Having established the thickness of the casing, it is important to establish the size and number of bolts for radial split casings. An equivalent pressure area is then established using the following formula: Ap = 0.9[XV + tL][YV + tL]

(8-28)

The design pressure PD is usually established as the maximum operating pressure times a factor of 1.25. It is then multiplied by Ap to obtain the total force on the casing Fp: Fp = PD · Ap

(8-29)

The size and number of bolts is then established using the yield stress of the bolts. Detailed finite element analysis of multistage tailings pumps has demonstrated that the maximum stress occurs at the cutwater. Some of the very high pressure pumps feature a special bolt at the cutwater that is larger than the other bolts (Burgess and Abulnaga, 1991). Table 8-9 presents some recommendation for average dimensions of a single-stage mill discharge pump designed for a maximum operating pressure of 1035 kPa (150 psi). In this example, it was arbitrarily assumed that the number of bolts is 12, to give the reader an idea of the effect of loads on size of bolts. Obviously, on the larger pumps, the de-

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TABLE 8-10 Recommended Dimensions for a Single Stage Mill Discharge Pump (USCS units size) Size (in)

8×6

Impeller d2 21⬙ Shroud diameter dt 22⬙ Cutwater diameter dC 25.9⬙ Cutwater gap (dC – dt)/2 1.95⬙ XV = 1.3 dt 28.6⬙ YV = 1.25 dt 27.5⬙ Liner thickness tL 1.34⬙ D = XV + 2 · tL 31.3⬙ Pressure area Ap (in2)* 777 Working pressure psi 150 Design pressure psi 200 F = Ap*Pdesign (lbf) 155400 D/t 40 Casing thickness tc 0.78⬙ (with ribs) Number of bolts 12 Load/bolt lbf 12,950 0.539 Bolt area in2** Min Bolt diameter 0.83⬙ Bolt size (in) 7/8⬙

10 × 8

12 × 10

14 × 12

16 × 14

18 × 16

20 × 18

26.8⬙ 28.3⬙ 33.2⬙ 2.45⬙ 36.8⬙ 35.4⬙ 1.5⬙ 39.8⬙ 1272 150 200 254389 40.7 0.95⬙

31⬙ 32.7⬙ 38.6⬙ 2.95⬙ 42.5⬙ 40.9⬙ 1.6⬙ 45.7⬙ 1687 150 200 337400 41.17 1.1⬙

35⬙ 36.6⬙ 43.5⬙ 3.45⬙ 47.6⬙ 45.8⬙ 1.77⬙ 51.1⬙ 2114 150 200 422800 40.42 1.22⬙

39.4⬙ 41.3⬙ 48.8⬙ 3.77⬙ 53.7⬙ 51.6⬙ 1.89⬙ 57.5⬙ 2973 150 200 594600 41 1.34⬙

45.3⬙ 47.25⬙ 56.1⬙ 4.43⬙ 61.43⬙ 59⬙ 2⬙ 65.43⬙ 3482 150 200 696400 41.07 1.54⬙

55⬙ 59⬙ 69.9⬙ 5.45⬙ 76.7⬙ 73.8⬙ 2.16⬙ 81⬙ 5990 150 200 1198000 41.2 2⬙

12 21,199 0.883⬙ 1.06 11/4

12 28,117 1.17 1.22⬙ 1.375

12 35,233 1.47 1.38⬙ 1.5⬙

12 49,550 2.06 1.62⬙ 1.75⬙

12 58,033 2.42 1.75⬙ 2⬙

12 99,833 4.16 2.3⬙ 2.5⬙

*Ap = 0.9[XV + tL][YV + tL]. **Allowed stress on bolt 24,000 psi. Note: these calculations are preliminary and must be confirmed by finite element analysis. They are for single stage, 200 psi rating with ductile iron casing.

signer may increase the number of bolts to keep them within a reasonable size. Table 8-10 is a similar table using USCS units. The casing pump takes the shape of the volute (Figure 8-21). In addition to the volute liner, a front wear plate or throatbush (Figure 8-22) is bolted to the casing. Compared to a water pump, a slurry pump has a much wider gap at the cutwater with respect to the impeller. This is due to the fact that the slurry pump must move solids that should not jam at the cutwater. In certain cases, oversized pumps were sold to mines and recirculation problems developed with excessive wear. Manufacturers have gone back over their designs and extended the cutwater to cut down the flow by creating a sort of throttling effect. They call this sort of volute a low- flow volute (Figure 8-23). The advantage of this approach is that the pattern of the liner can be modified without having to replace the casing of the pump. Installing a so-called “reduced eye” impeller may also complement this approach. A “reduced eye” impeller is an impeller with a suction diameter smaller than the suction diameter of the casing. This provides a way to throttle the suction. The throatbush of the pump must also be modified to accommodate the reduced eye of the impeller. In the case of water pumps, the emphasis is to operate as close as possible to the best efficiency point, where losses are at a minimum. In the case of slurry pumps, the situation is more complex, as the best efficiency point does not necessarily coincide with the minimum wear point. Certain designs of slurry pumps do point to minimum wear at 80% of

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CHAPTER EIGHT

FIGURE 8-21 Casting for the casing and cover plate of a vertical sump pump—clearly showing the volute shape—with an integral cast elbow at the discharge. (Courtesy of Mazdak International Inc.)

the best efficiency point. This point is too often overlooked when sizing pumps. The consultant engineer is encouraged to discuss this point with the manufacturer. Certain manufacturers of pumps have in-house computational fluid dynamics programs to do a wear performance analysis. Unfortunately, too often these give a two-dimensional profile of velocity in the volute, but insufficient data about vortices in the corners where gouging wear may develop.

FIGURE 8-22 Throatbush or suction liner fixed to the pump front casing plate of a horizontal pump. The casing shape indicates the volute shape of the liner.

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Page 8.31


solid passageway

original cutwater extended cutwater for "low flow" volute throat

8.31

modified throatbush reduced eye impeller

FIGURE 8-23 Restraining the flow by extending the cutwater and modifying the throat of the volute or liner, or decreasing the suction diameter of the impeller are methods for correcting oversized pumps.

Example 8-4 A new mine requires a very large pump to handle 1514 L/s (24,000 US gpm), at a total dynamic head of 43 m (141 ft) and a specific gravity of the mixture of 1.5. Establish some preliminary parameters of design for the casing prior to conducting a finite element analysis. The head ratio is assumed to be 0.9 (see Chapter 9). Assume that this is a pump designed for single-stage operation with a design pressure of 1.4 MPa (200 psi). Solution in SI Units The equivalent water head is 43 m/0.9 = 47.8 m. This is therefore an application for an all-metal pump. Table 8-5 suggests an average suction speed of 6 m/s and a discharge speed of 9 m/s at a discharge head of 55 m. Since the pump will operate at 47.8 m, the ratio of tip speeds is 兹苶 (4苶7苶 .8苶 /5苶5苶) = 兹0苶.8 苶6 苶8 苶 = 0.932. The pump will operate at 0.932 of the maximum allowed speed of 38 m/s for all metal impellers, or 35.42 m/s (or 116 ft/s): 9.81 · 47.8 gHBEP ␴SI = ᎏ = ᎏᎏ = 0.187 2U 22 2 · 35.422 The flow suction speed is established as the ratio of tip speed. This ratio is 0.932, and using the suggested maximum speed of 6 m/s for metal impellers, the suction speed Vs at the flow rate of 1514 L/s is then 0.932 × 6 = 5.59 m/s. The suction area = Q/Vs = 1.514/5.59 = 0.271 m2. The corresponding inner diameter is 0.587 m or 23.12⬙. The discharge speed Vd is 0.932 × 9 = 8.4 m/s. The discharge area = Q/Vd = 1.514/8.4 = 0.18 m2. The discharge inner diameter is 0.478 m or 18.8⬙. These values of suction and discharge diameter will be added to the liner thickness and to the casing thickness before calculating suction and discharge flanges and their corresponding bolt circles. Using Table 8-8 as a reference, the tip-to-suction diameter of the impeller ratio is assumed to be 2.75, or the tip diameter of the impeller becomes 0.587 × 2.75 = 1.615 m.

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CHAPTER EIGHT

Since U = 35.42 m/s,

␻ = U/R = 35.42/1.615 = 21.93 rad/s N = 21.93 · 60/(2 · ␲) = 209.4 rpm Let us round it to 210 rpm. From Equation 8-16, the specific speed (in the International System of Units) is N · 兹苶 Q 苶1 苶4 苶 210 · 兹1苶.5 = ᎏᎏ = 14.22 Nq = ᎏ H3/4 47.83/4 Table 8-9 recommends that the shroud diameter dt be about 6% larger than the impeller vane diameter dV or 1.06 · 1.615 = 1.712 m. The next step is to establish a preliminary layout of the volute using Equations 8-25 and 8-26. It is assumed that Kx = 1.35 or XV = 1.35 · 1.71 = 2.31 m and Ky = 1.25 or XV = 1.25 · 1.71 = 2.14 m Table 8-9 recommends a liner thickness in the range of 4% to 6% of the impeller shroud diameter: tL = 0.04 · 1.712 = 0.0685 m Let us assume 69 mm. Having sized the thickness of the liner, a parameter D defined in Equation 8-26 is: D = 2.31 + 2 · 0.069 = 2.45 m For a well-ribbed casing, the thickness of the casing tC in a preliminary stage of analysis is D/40 or 2450/40 = 63.5 mm; let us assume 64 mm. The outer diameter of the suction nozzle is therefore 587 mm + 2 · (69 + 64) = 853 mm or 33.5⬙ This suggests further iteration or the installation of a companion flange to 900 mm for European sizes of pipes or 36⬙ suction pipes for U.S. sizes of pipes. The outer diameter of the discharge nozzle is therefore 478 mm + 2 · (69 + 64) = 744 mm or 29.29⬙ These calculations suggest that the pump is effectively a pump with a discharge flange of 750 mm for metric pipe sizes or 30⬙ for U.S. sizes of pumps. The equivalent pressure area Ap is then established using Equation 8-28: Ap = 0.9[XV + tL][YV + tL] = 0.9[2.31 + 0.069][2.14 + 0.069] = 4.13 m2 At a design pressure of 1.4 MPa, the total force that the bolts must retain is: Fp = Ap · 1.4 MPa = 4.13 · 1.4 = 5.78 MN

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8.33

Since this is a fairly large casing, the design engineer decides to try 24 bolts around the casing. Each bolt will retain 5.78 MN/24 = 0.241 MN or 241 kN, assuming an allowed stress on bolt of the order of 166 Mpa. The cross-sectional area of the bolt at the minimum thread diameter is 0.241/166 = 0.00145 m2 or a diameter of 42 mm. 20 M48 bolts are therefore recommended. Solution in USCS Units The equivalent water head is 141 ft/0.9 = 156.8 ft of water. This is therefore an application for an all-metal pump. Table 8-5 suggests an average suction speed of 19.7 ft/s and a discharge speed of 29.5 ft/s at a discharge head of 180.5 ft. Since the pump will operate at 47.8 m, the ratio of tip speeds is 兹苶 (1苶5苶6苶 .8苶 /1苶8苶0苶 .5) = 兹苶0苶 .8苶6苶8 = 0.932. The pump will operate at 0.932 of the maximum allowed speed of 124.67 ft/sec for all metal impellers, or 116 ft/s: gHBEP 32.2 · 156.8 ␺US = ᎏ = ᎏᎏ = 0.375 U 22 1162 The flow suction speed is established as the ratio of tip speed. This ratio is 0.932, and using the suggested maximum speed of 19.7 ft/s for metal impellers, the suction speed Vs at the flow rate of 24,000 US gpm (53.47 ft3/sec) is then 0.932 × 19.7 = 18.36 ft/s. The suction area = Q/Vs = 53.47 ft3/18.36 = 2.912 ft2. The corresponding inner diameter is 1.926 ft or 23.12⬙. The discharge speed Vd is 0.932 × 29.5 ft/s = 27.5 ft/sec. The discharge area = Q/Vd = 53.47/27.5 = 1.944 ft/sec2. The discharge inner diameter is 1.573 ft or 18.9⬙. These values of suction and discharge diameter will be added to the liner thickness and to the casing thickness before calculating suction and discharge flanges and their corresponding bolt circles. Using Table 8-8 as a reference, the tip-to-suction diameter of the impeller ratio is assumed to be 2.75, or the tip diameter of the impeller becomes 23.12⬙ × 2.75 = 63.6 in or 5.3 ft. Since U = 116 ft/s,

␻ = U/R = 116/5.3 = 21.9 rad/s N = 21.9 · 60/(2 · ␲) = 209.4 rpm Let us round it to 210 rpm. From Equation 8-16, the specific speed (In the International System of Units) is 210 · 兹2苶4苶0苶0苶0苶 N · 兹苶 Q = ᎏᎏ = 734 NUS = ᎏ H 3/4 156.83/4 Table 8-9 recommends that the shroud diameter dt be about 6% larger than the impeller vane diameter dV or 1.06 · 63.6⬙ = 67.42⬙. The next step is to establish a preliminary layout of the volute using Equations 8-25 and 8-26. It is assumed that Kx = 1.35 or Xv = 1.35 · 67.42⬙ = 91⬙ and Ky = 1.25

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or Xv = 1.25 · 67.42⬙ = 84.3⬙ Table 8-9 recommends a liner thickness in the range of 4% to 6% of the impeller shroud diameter: tL = 0.04 · 67.42⬙ = 2.69⬙ Let us assume 2.7⬙. Having sized the thickness of the liner, a parameter D defined in Equation 8-26: D = 91⬙ + 2 · 2.7⬙ = 96.4⬙ For a well-ribbed casing, the thickness of the casing tC in a preliminary stage of analysis is D/40 or 96.4⬙/40 = 2.41⬙. The outer diameter of the suction nozzle is therefore 23.12⬙ + 2 · (2.7⬙ + 2.41⬙) = 33.34⬙ This suggests further iteration or the installation of a companion flange to 36⬙ suction pipes for U.S. sizes. The outer diameter of the discharge nozzle is therefore 18.9⬙ + 2 · (2.7⬙ + 2.41⬙) = 29.12⬙ These calculations suggest that the pump is effectively a pump with a discharge flange of 30⬙ for U.S. sizes of pipes. The equivalent pressure area Ap is then established using Equation 8-28: Ap = 0.9[XV + tL][YV + tL] = 0.9[91 + 2.7][84.4 + 2.7] = 7345.14 in2 At a design pressure of 200 psi, the total force that the bolts must retain is: Fp = Ap · 1.4 MPa = 7345.14 · 200 = 1,469,028 lbf Since this is a fairly large casing, the design engineer decides to try 24 bolts around the casing. Each bolt will retain 1,469,028 lbf/24 = 61,210 lbf, assuming an allowed stress on bolt of the order of 24,000 psi. The cross-sectional area of the bolt at the minimum thread diameter is 61,210 lbf/24,000 = 2.55 in2 or a diameter of 1.8⬙. 20 1.875⬙ bolts are therefore recommended. The design engineer must make allowance for the diameter of washers and the spotfacing diameter while laying down the design of the casing, as explained in Table 8-11. To complete this preliminary design exercise, the engineer needs to calculate the width of the impeller, including the pump-out vanes. This will be the topic of Section 8-4.

8-4 THE IMPELLER, EXPELLER AND DYNAMIC SEAL Slurry, like any liquid, tends to find its way of least resistance. When a pressure difference exists between the volute pressure and the suction pressure at the front of a slurry pump or the gland and stuffing box pressure (leaking to atmosphere) exits, slurry tends to flow back. However, as passageways narrow near the stuffing box or near the suction, solids become entrapped and accelerate abrasive wear. Leakage of slurry at the stuffing box can be dangerous to the environment, and can damage bearings. Various methods have been developed over the years to counteract leaks. One popular method consists of injecting water at the gland. The gland water pres-

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TABLE 8-11 Size of Metric bolts and Allowance for Spot Facing. Suitable for Slurry Pump Casing and Stuffing Box Bolt size M5 M6 M8 M10 M16 M20 M24 M30 M36 M42 M48 M56 M64

Clearance hole diameter (mm)

Washer outside diameter (mm)

Spot facing diameter (mm)

Erix Back Spot facing diameter (mm)

6 7 9 12 18 23 27 33 39 45 51 59 67

10 12.5 17 21 30 37 44 56 66 78 92 105 115

12 14 19 24 33 41 46 60 70 80 96 110 120

15 15 18 24 33 43 48 62 72 82 108 113 122

sure is usually 35–70 kPa (5–10 psi) above the discharge pressure of the pump. The water acts also as a cooling lubricant to the shaft sleeve and packing rings. As time passes, the abradable packing rings wear slowly, and the operator has to readjust the gland. Thus, the gland rings are usually split with tightening bolts (Figure 8-24). Unfortunately, trucking or pumping fresh gland water to remote tailing pump stations is not always the most economical solution. The pumping cost of gland water is not negligible for large pumps. In some cases such as pumping ore concentrate, the process engineer would prefer to avoid diluting the slurry by adding water at the gland. In the mid1960s, slurry pump designers started to investigate the concept of a dynamic seal. A dynamic seal in its most basic concept consists of a ring of vanes on a shroud capable of creating a vortex. The designer of the dynamic seal tries to create a vortex field strong enough to prevent flow to the center of the vortex. In fact, when pressure is sufficiently reduced at the center of the dynamic seal to a magnitude below the outside atmospheric value, air is sucked in through the gland, and an air ring is formed . Despite the appearance of expellers, dynamic seals, and pump-out vanes in the mid1960s, there is a dearth of technical information of their performance. Various claims made in sales brochures are difficult to substantiate. Universities research centers have not paid much attention either. In some respects, the expeller at first look condradicts traditional thinking. It is in fact an impeller whose purpose is to repel or prevent flow. It goes against the logic of rotodynamics. The dynamic seal of a slurry pump consists of: 앫 Pump-out vanes on the back shroud of the impeller (Figure 8-25) 앫 Antiswirl vanes between the impeller and the expeller 앫 one or more expellers with antiswirl vanes between them The dynamic seal operates only when the pump is rotating at a sufficient speed. When the pump is stationary, the dynamic seal ceases to perform and liquid may leak through the stuffing box, unless an additional stationary seal is provided or external water at sufficient pressure is flushing the gland.

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FIGURE 8-24 Stuffing box of the ZJ slurry pump (made in China) showing piping connection to inject water at high pressure and two adjusting bolts.

FIGURE 8-25 Two front pump-out vanes of a slurry pump, before painting and testing (left) and painted with different colors (right), then installed in the pump of a test loop; the discoloration indicates patterns of wear. (Courtesy of Mazdak International, Inc.)

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Flow in an expeller is complex and depends on the difference in relative motion between the stationary surface of the case liner and the rotating disk of the expeller. Consider Figure 8-26 showing a closed impeller with pump-out vanes on the back shroud. The impeller main vane tip radius is R2, but the pump-out vanes extend only to the radius Rr. A shaft sleeve behind the impeller has Rs as a tip radius. In the front shroud of the impeller, another set of pump-out vanes extend to the radius Rf and provide dynamic sealing between the impeller and the throatbush to repel any solids that may tend to slip toward the suction (where the pressure is obviously lower). As the impeller rotates, a pressure field develops on the front shroud of the impeller due to the front pump-out vanes, and another pressure field develops behind the impeller due to the back pump-out vanes. In an ideal world, both fields should balance each other. In reality, wear of these vanes and the difference of clearance between the front and the back vanes with respect to the casing or its liners tend to create an unbalance. In reference to Table 8-1, Case 7 for a forced vortex we have:

␻ = C7 × R0v0 V × R–1 v0 = C7 P/␳ = C72 · R2v0/(2 · g) + h7 Stepanoff (1993) stipulated that when a disk is rotating against a stationary surface, the average angular speed of the liquid between the two is half the angular speed of the disk. However, when vanes are added to the rotating disk, the rotational speed of the liquid is expressed as

冤

1 + t/x ␻liq = ␻imp ᎏ 2

tf

冥

(8-30)

tb

Hvr

Hvf R2 Rf R1

Rsl xf

Rr

xb B 2

FIGURE 8-26 Dynamic pressure distribution due to front and back pump-out vanes of a slurry pump impeller.

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CHAPTER EIGHT

where t is the depth of the pump-out vanes and x is the total gap between the impeller back shroud and the casing wear surface. x = s + t, where s is the gap between the pumpout vanes and the back shroud. Figure 8-27 represents a simplified case of pump-out vanes that extend down to the shaft sleeve diameter dsL. The average rotational speed of the liquid between the rotating impeller and the stationary shroud therefore ␻imp/2. Applying the Euler head to this region, the head at the radius Rr is therefore: U n2 – Un–1 ⌬H = ᎏᎏ 2g

(8-31)

␻ 2imp ⌬H = ᎏ (R 22 – R 2r ) 8g

(8-32)

Because vanes extend from Rr and Rsl,

␻ 2imp(1 + t/x)2 ⌬H = ᎏᎏ (R 2r – R 2sl) 8g

(8-33)

So if H2 is the head at the tip of the impeller vane, then the head at the stuffing box (in the absence of any expeller) is the head at the sleeve, or Hsl. Because a certain percentage of the dynamic pressure is converted to static head in the volute, H2 is often assumed to be 75% of the total dynamic head:

␻ 2imp Hsl = H2 – ᎏ ([R 22 – R 2r ] – (1 + t/x)2 · (R 2r – R 2sl)) 8g

冢

冣

(8-34)

The design engineer establishes H2 as a design criterion. Since the worst condition that a slurry pump may experience happens when it operates at 30% of the B.E.P capacity and at a head H30, some engineers calculate H2 as: H2 = H30 – H1 When Hs > Hatm, the pump-out vanes will be completely flooded and the liquid will flow to the gland. To prevent this effect, some liquid at a higher pressure than the stuffing box pressure may be injected or an additional expeller may be added. When Hs < Hatm, then the pump-out vanes suck in air and the stuffing box is sealed against loss of slurry (Figure 8-26). In the back of the impeller, a second smaller disk with vanes facing the bearing assembly direction is sometimes installed (Figure 8-27). It is called the expeller in the mining industry and the repeller in the pulp and paper industry. Its diameter is usually smaller than 70% of the pump impeller. Its purpose is to reduce further the head between the hub of the impeller Hb and the stuffing box. Equation (8-34) does not describe the effect of the number of vanes, the breadth of the vanes, or the shape of the vanes. Over the years, different manufacturers have developed various shapes such as: 앫 앫 앫 앫 앫

Straight radial vanes Radial vanes but split in the middle with a gap L-shaped vanes, also called hockey sticks J-shaped vanes Radial vanes with an outside ring

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impeller

Page 8.39

expeller area

te he

FIGURE 8-27

he

Ød Exp

LE

Ød

Ød ho

8.39

l ve

c ve

Geometry of an expeller with radial vanes.

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앫 Radial vanes with an outside ring and a middle ring 앫 Lotus-shaped vanes These shapes are represented in Figure 8-28. Equation (8-34) clearly indicates that the head is proportional to the square of the speed. There is therefore a minimum rotational speed before that the dynamic seal starts to function. The consumed power of an expeller is expressed as: P (kW) = constant · ␳ · D5 · N3

(a) backward curved vanes

(c) L-shaped vanes ( hockey sticks)

(e) simple radial

FIGURE 8-28

(8-35)

(b) radial split at mid- radius

(d) radial with ring at mid- radius

(f ) lotus vanes

Different shapes of vanes and rings of expellers and dynamic seals.

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Although various claims have been made in sales brochures about the merits of each vane type, and numerous patents have been filed, there has been no substantial scientific data to confirm the claims. Often, the final shape is a compromise between the requirements for casting in hard metals and the requirements of the hydraulics. Impellers of slurry pumps must accommodate solids, and this means that the vanes must be wide enough. Each manufacturer has their own criteria, with dredge and gravel pumps requiring very wide impellers (Table 8-12). Adding this passageway to the thickness of the shrouds of pump-out vanes results in the impeller overall width b2 (Figure 8-29). In Equation 8-35, it was pointed out that the power consumption from pump-out vanes is proportional to the diameter raised to the power of five. Instead of trimming the pumpout vanes to a diameter smaller than the impeller main vanes, they are sometimes tapered (tb and tf are gradually reduced toward the tip of the impeller; see Figure 8-29). In Figure 8-29, the pump-out vane thickness at the root is (gf + tfv), whereas at the tip it is tfv. In the back of the impeller, the pump-out vanes start at a diameter db, whereas on the front side they start at dr. These values are plugged into Equation 8-34 to obtain Rr in each case and to calculate axial thrust. Because slurry pumps are often cast in brittle alloys such as the high-chrome white iron, it is important to eliminate sharp edges that may act as stress risers. The manufacturers establish the radii R3, R4, Rc, Rr, Rh, and Rsv shown in Figure 8-29 to allow a smooth casting, but also to improve on the hydraulics. The effect of each parameter on the hydraulics as described in sales brochures is not always well proven. The vane diameter d2 shown in Figure 8-29 is smaller than the shroud diameter dt, but it is the reference diameter for all calculations. The shaft sleeve with a diameter dsl is used in all thrust calculations. The sleeve protects the shaft from wear by the packing and solids that may accumulate between the packing rings.

TABLE 8-12 Recommended Maximum Size of Spheres for the Design of the Width of Vanes of Slurry and Dredge Pumps Mill discharge pumps _______________________________________ Discharge Size Sphere diameter, (mm) (inches) mm (in) 25 38 50 75 100 150 200 250 300 350 400 450 500 600 650

1.5 × 1 2×1 3×2 4×3 6×4 8×6 10 × 8 12 × 10 14 × 12 16 × 14 18 × 16 20 × 18 24 × 20 28 × 24 30 × 26

13 (1/2⬙) 18 (11/16⬙) 20 (3/4⬙) 22 (7/8⬙) 38 (⬇1.5⬙) 50 (⬇2⬙) 63 (⬇2.5⬙) 80 (⬇3) 88 (⬇3.5⬙) 100 (⬇4⬙) 115 (⬇4.5⬙) 127 (⬇5⬙) 140 (⬇5.5⬙) 150 (⬇6⬙) 180 (⬇7⬙)

Gravel and dredge pumps _______________________________________ Discharge Size Sphere diameter, (mm) (inches) mm (in)

100 150 200 250 300 350 400 450 500 600 650 915

6×4 8×6 10 × 8 12 × 10 14 × 12 16 × 14 18 × 16 20 × 18 24 × 20 28 × 24 30 × 26 40 × 36

80 (⬇3) 127 (⬇5⬙) 180 (⬇7⬙) 230 (⬇9⬙) 240 (⬇9.5⬙) 250 (⬇10⬙) 280 (⬇11⬙) 305 (⬇12⬙) 360 (⬇14⬙) 380 (⬇15⬙) 450 (⬇18⬙) 530 (⬇21⬙)

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b t

2

t

bs bv

t bv

t fv g f

g b

Ød

R fv

tb

R

Rc Ød1

Ød h Ø dsl

Ød b

R

fs

2

Rt

Ødr

fsv th

Rr

R

h R sv L

th h i

FIGURE 8-29 Cross-section of an impeller for a slurry pump showing different geometrical parameters.

Most slurry pumps use a threaded shaft. The length of the shaft thread Lth is used in calculations of axial load transmitted from the torque. Some pumps use BSW and others use ACME thread, and some manufacturers have also their own thread designs to make it difficult to pirate their impellers. It is important to establish the center of gravity of the impeller. In the absence of data, it is often assumed to be at a distance Lh. It is also assumed in the calculations that the radial thrust force is applied at the same point.

8-5 DESIGN OF THE DRIVE END The hydraulic loads from the pump wet end are ultimately transmitted to the pump shaft and bearings. Because of the need to access all the pump parts for replacement due to wear during maintenance, slurry pumps have standardized cantilever designs, with all bearings well protected from solids ingestion. The main loads that are transmitted to the pump shaft are: 앫 Radial force due to pressure distribution in the volute 앫 Axial force due to the pump-out vanes and expellers

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앫 Weight of the impeller and expeller 앫 Torque due to speed and power consumption 앫 Radial force on the drive end from pulleys

8-5-1 Radial Thrust Due to Total Dynamic Head The radial force is due to the uneven pressure distribution in the pump casing. It is expressed as: FR = K · ␳gHd2 · B2

(8-36)

where d2 = tip diameter of the impeller vanes B2 = width of the pump casing As shown in Figure 8-26, B2 = b2 + xf + xb

(8-37)

Wear can chip at the surface of the impeller or the casing, thus causing an increase of xf and xb and a reduction of b2 through the life of the pump. The value of K may be as high as 0.40 near the shut-off head and as low as 0.10 at the best efficiency point. It is, however, recommended to conduct proper measurements with proximity probes over the envelope of the flow rate during the design of a new pump. The proximity probes are used to measure the deflection at the gland. The magnitude of the force is then calculated from cantilever stress theory. As shown in Figure 8-30, different shapes of volutes give different values for the radial load. Stepanoff (1993) clearly indicated that the direction of the radial force reverses after the best efficiency point, whereas Angle et al. (1997) do not seem to agree with this supposition. A misunderstanding of the direction of this hydraulic radial force leads to totally different estimation of the bearing life. A calculation that assumes a zero radial load near the best efficiency point (following the Stepanoff approach) can lead to a bearing life ten times as high as another calculation that assumes that the same radial load adds to the weight of the impeller, creating a large bending moment on the shaft and reaction loads at the bearings. A smart salesman may try to convince the consultant slurry engineer of the superiority of his product over the competition in terms of the rigidity of the bearing assembly, whereas in reality it is a matter of adding or subtracting loads. Shafts of slurry pumps have broken at the shaft thread, simply because the radial load was too high and caused rapid fatigue failure. It is therefore strongly recommended to limit the minimum flow rate to half the best efficiency flow rate at the given speed. Throttling an oversize pump is not recommended at all. Downsizing or reducing the speed of the pump is essential to avoid excessive radial load on the pump shaft. Each manufacturer has their recommended value of K for the calculation of the radial load and the bearing life.

8-5-2 Axial Thrust Due to Pressure The axial thrust is due to the fact that the pressure on the suction side is different from the pressure on the back of the impeller. There is a difference between plain impellers and impellers with pump-out vanes, but since pump-out vanes wear out with time due to abrasion and erosion, the design engineer should conduct his calculations for both cases of im-

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(b) two semi-circle casing

8.44

Head

(b) circular casing

Head After Angle & Rudonov (1999)

FR

Head After Angle & Rudonov (1999)

FR

F

R

After Stepanoff (1993)

After Stepanoff (1993)

Q

Q

N

Flow rate

N

Flow rate FIGURE 8-30

Radial load for different shapes of casing versus flow rate.

Q

N

Flow rate

Page 8.44

(a) true volute

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pellers with and without pump-out vanes. The presence of an expeller or the addition of pressurized gland water does affect the axial thrust. Consider in Figure 8-31 a closed impeller without pump-out vanes. The pressure on the suction side is Ps and at the suction diameter d1. The pressure on the back of the impeller is P1. The pressure above d1 on both sides of the impeller is equal and balances out. In the back of the shaft sleeve and shaft there is atmospheric pressure PA, so the resultant force based on the shaft sleeve diameter is: TSL = 0.25␲d 2SLPA On the suction side, there is suction pressure Ps, so the thrust force is: TS = 0.25␲d 12Ps The net thrust is: FA = 0.25␲{P1[d 12 – d 2SL] + PA d 2SL – Ps d 12}

(8-38)

For the first stage, PS is calculated in a very similar way to the NPSH. Some manufacturers design the bearing assembly to absorb the axial thrust from a single stage and others standardize on three stages because they anticipate use in a wide range of applications from mill discharge to tailings disposal. Because tailings pumps are often used in series, the bearing assembly may be designed for a suction pressure equal to the discharge pressure of the stage before the last one, i.e., if M is the number of stages: Ps = (M – 1)␳g(TDHst) + PA

(8-39)

where TDHst is total dynamic head per stage. Referring to Figure 8-29, when pump-out vanes are added in the back shroud, Equation 8-34 is then used to calculate the value of Pb at the root of the pump out vanes Rb: Pb = P2 – 0.125␳␻2imp{[R 22 – R 2b] – [(1 + tb/xb)2 · (R 22 – R 2b)]}

(8-40)

where P2 = 0.75␳g(TDH) + PS.

Ps

FIGURE 8-31

R1

P1 dA

d sl

PA

Axial loads on an impeller with plain shrouds.

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The average thrust force on the back shroud of the impeller is T2b = 0.5(P2 + Pb) ␲{[R 22 – R2b]

(8-41)

This value of the pressure Pb is transmitted to the expeller box and becomes the pressure at the expeller tip diameter dexp (Figure 8-27). The pressure at the expeller diameter dhe (which is often equal to the shaft sleeve or the pressure at the gland) is then Phe = Pb – 0.125␳␻2imp{[R 2exp – R2he] – [(1 + te/(te + cve))2 · (R2exp – R 2he)]}

(8-42)

The average thrust force on the back shroud of the expeller is Tbe = 0.5(Phe + Pb) ␲{R2exp – R 2he}

(8-43)

If the expeller hub diameter is larger than the shaft sleeve, there is a component of axial thrust as Tesl = 0.5(Phe + PA) ␲{R2he – R 2SL}

(8-44)

On the back of the sleeve and shaft, the pressure is essentially atmospheric so that the thrust is Tsl = PA␲R 2SL

(8-45)

On the front shroud of the impeller, pump-out vanes are also added with some impellers. Applying Equation 8-34 to Figure 8-29, the pressure at the front hub Rr is therefore: Pr = P2 – 0.125␳␻2imp{[R 22 – R 2r ] – [(1 + tf/xf)2 · (R 22 – R 2r )]}

(8-46)

The average thrust force on the front shroud of the impeller between R2 and Rr is: T2r = 0.5(P2 + Pr) ␲{[R 22 – R 2r ]

(8-47)

If the front shroud hub diameter dr is larger than the suction diameter ds, there is a component of axial thrust as Trs = 0.5 (Pr + Ps) ␲{R 2r – R S2}

(8-48)

The thrust due to the suction pressure is then Ts = Ps␲RS2

(8-49)

Total axial thrust equals total thrust on the back shroud minus total thrust on the suction: FA = [t2b + Tbe + Tsl] – [Ts + Trs + T2r]

(8-50)

In multistage applications with a number of pump in series, the total axial thrust can change direction as the suction pressure is higher than atmospheric pressure, and the expeller and pump-out vanes’ effectiveness in balancing thrust drops with increasing number of stages. Since the flow calculations need to be repeated at various points on the pump curve, a computer program would be useful. The program AXIAL-RADIAL was developed by the author in Qbasic, a language easy to understand by most engineers, but experts may modify it to PASCAL, C+, Fortran, or other languages as it suits their needs. It calculates both hydraulic and axial loads on the pump impeller. COMPUTER PROGRAM “AXIAL-RADIAL” 9 CLS REM calculations of axial and radial loads on a pump impeller pi = 4 * ATN(1) Rem Calculations will be done assuming a specific gravity of 1.7

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sg = 1.7 INPUT “model name “; na$ INPUT “tip shroud diameter dt (mm) “; dt INPUT “vane tip diameter d2 (mm) “; d2 INPUT “suction diameter ds (mm) “; ds INPUT “starting diameter for front pump out vanes dr (mm) “; dr INPUT “starting diameter for back pump out vanes db (mm) “; db INPUT “back hub diameter dh (mm) “; dh INPUT “shaft sleeve o.d dsl (mm) “; dsl INPUT “overall width of impeller B2 (mm) “; bx INPUT “ vane tip width b2 (mm) “; b2 INPUT “thickness of front shroud tfs (mm)”; tfs INPUT “thickness of front pump out vanes tfv (mm) “; tfv INPUT “anticipated front gap (mm)”; gf sf = gf + tfv ‘INPUT “thickness of back shroud tbs (mm)”; tbs INPUT “thickness of back pump out vanes tbv (mm) “; tbv INPUT “anticipated back gap (mm)”; gb sb = gb + tbv INPUT “speed for metal version”; n PRINT “it shall be assumed that pump out vane to gap ratio =0.7” PRINT a1 = .25 * pi * (dr/25.4) ^ 2 a2 = .25 * pi * (d2/25.4) ^ 2 a3 = .25 * pi * (dsl/25.4) ^ 2 a4 = .25 * pi * (ds/25.4) ^ 2 a5 = .25 * pi * (db/25.4) ^ 2 c = 25.4 DIM h(10), fa(10), fan(10), nr(10), Q(10), k(10),fr(10),f(10) Rem assume a typical curve for an all metal impeller h(1) = 64;k(1)=0.4 h(2) = 62.7;k(2)=0.35 h(3) = 60.5;k(3)=0.25 h(4) = 55;k(4)=0.15 h(5) = 49.5;k(5)=0.10 h(6) = 35;k(6)=0.12 h(7) = 34.2;k(7)=0.15 h(8) = 33;k(8)=0.20 h(9) = 30;k(9)=0.22 h(10) = 27,k(10)=0.25 INPUT “best efficiency flow rate for metal version “; qnm Q(1) = .25 * qnm Q(2) = .5 * qnm Q(3) = .75 * qnm Q(4) = 1 * qnm Q(5) = 1.15 * qnm Rem calculation for rubber Q(6) = .25/1.354 * qnm Q(7) = .5/1.354 * qnm Q(8) = .75/1.354 * qnm Q(9) = 1/1.354 * qnm

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Q(10) = 1.15/1.354 * qnm FOR i = 1 TO 10 h = h(i) h2 = .8 * h/.3048 PRINT “h2= “; h2 INPUT “hit any key to continue “; l$ IF h(i) > 35 THEN nr(i) = n IF h(i) 50 mm or >2 in) in a magnetite-based water mixture. The magnetite consists of very fine particles but they

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11.5

are heavier than coal. When magnetite is mixed with water, this mixture becomes the effective carrier liquid in which the >50 mm (>2 in) coal particles can float. In Chapter 4, it was clearly explained that the difference in density (or specific gravity) between the carried solids and the carrier liquid was an important parameter in friction loss calculations. This is, in basic terms, the concept of using a heavy medium (water and magnetite) as a carrier for coarse solids. A circuit that uses magnetite must have a recovery system at the end of the pipeline. Since magnetite has very strong ferromagnetic properties, it is first screened from coarse coal, and then separated from crushed coal (that resulted from deterioration during pumping), using magnetic separators. The recovered magnetite is then mixed with water at a high volumetric concentration and pumped via a dedicated pipeline and a positive displacement back to the starting feed station of the slurry pipeline. It is then stored in special storage tanks with agitator mixers. To avoid the use of many booster stations in long pipelines. The lockhopper may be used for coarse coal (>50mm or >2 in; see Figure 9-17).

11-3-4 Chemical Additions to Coal–Water Mixtures Special chemicals such as xanthan gum in levels of 200–600 ppm can be used as a stabilizer to prevent settling of coal slurries and to prevent the formation of hard-packed beds during hydrotransport. Miller and Hoyt (1988) recommended Pfizer Flocon 4800C as a very economical additive to coal–water mixtures. Morway (1965) obtained a patent for using a hydrocarbon oil with a small percentage of an imidazoline surfactant to coat coal particles uniformly. After adding this mixture, the coated coal can be mixed with water. The water weight concentration can be reduced to 20%. This slurry with low overall moisture is easier to heat at the final discharge point prior to combustion than plain coal–water mixtures. Bomberger (1965) proposed hexametaphosphate and sulfite as corrosion inhibitors for coal slurries in steel pipelines.

11-3-5 Coal–Oil Mixtures The vast majority of slurries consist of water and solids; however, variations on this are being implemented, especially in the transportation of coal. In the case of thermal plants, slurries of water and coal are difficult to burn, and a complete process of dewatering is needed to separate the coal from water. To rectify this situation, proposals have been made to use heavy crude oils instead of water to transport and burn coal. The viscosity of a heavy oil combined with the degradation of coal during pipeline transportation ultimately leads to non-Newtonian flows. The rheology of such mixtures depends on particle size, temperature, concentration, and the quality of the coal and the carrier oil. With the worsening political situation that started in 2001, coal–oil mixtures may see more and more applications. Kreusing and Franke (1979) recommended the use of coal particles smaller than 5 mm (0.2 in) as a fuel for the blast furnace. To maintain the viscosity of a coal-oil mixture in a range that is pumpable, the slurry may have to be warmed to 50° C (122° F) [fig 11-3]. By using heat, Kreusing and Franke (1979) managed to maintain a slurry viscosity in the range of 13–40 Pa·s (whereas the viscosity of water is 0.001 Pa·s). A viscosity of 40 Pa·s is an upper limit allowable for use in centrifugal pumps. In the case of carbonized lignite, Kreusing and Franke (1979) achieved a maximum weight concentration of 34%, and with brown coal they achieved a maximum weight concentration of 41–46%, depending on the solid particle size. Brown coal is often a low-

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calorific coal and is difficult to export. Kreusing and Franke achieved a maximum weight concentration of 60% with mineral coal. They concluded that it was possible to use coal for a maximum energy substitution of oil of 52%. Since coal is cheaper than oil, this is not a negligible result (Figures 11-3 and 11-4).

11-3-6

Dewatering Coal Slurry

At the discharge point of a coal slurry pipeline, water must be removed because coal cannot be burned at such high water contents. It is essential to establish certain criteria for the design of a dewatering station: 앫 앫 앫 앫 앫 앫 앫 앫

Size distribution Water content End use Rheology Volumetric concentration Suitability of recovered water for further use Storage, stockpiling, or ship loading Coal degradation during transport

FIGURE 11-3 Viscosity of coal–oil slurry mixture. (From Kreusing and Franke, 1979. Reprinted with permission of BHR Group.)

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FIGURE 11-4 Viscosity as a function of temperature for coal–oil mixture. (From Kreusing and Franke, 1979. Reprinted with permission of BHR Group.)

Dewatering may be done mechanically via filter presses, centrifuges, and screens. Because the efficiency of dewatering often depends on the particle size distribution, screening is strongly recommended prior to dewatering. Leninger et al. (1978) reported that in the case of coal with a top size of 10 mm (0.4 in), it was very difficult to use mechanical dewatering devices to reduce moisture below 10.6 %, even using steam with vacuum filtering. For coals with a top size of 3.15 mm (0.125 in), Leninger et al. (1978) suggested using two-stage cyclones with the second stage connected to the overflow of the first. The underflow from the first stage as well as from the second stage should be fed into solid-bowl centrifuges to reduce the residual moisture content to 17.3%. The use of hydrocyclones was also discussed by Abbot (1965). Coal with a top size of 2.4 mm (0.09 in) can be dewatered using two-stage cyclones as well as solid-bowl centrifuges, but the overall moisture is only reduced to 19%. For ultra-fine coal with a top size of 1.4 mm (0.055 in), as in the case of the Ohio and the Black Mesa pipelines, mechanical dewatering must be followed by thermal drying. One way to reduce the cost is to use the waste gas of the thermal plant to dry the coal.

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CHAPTER ELEVEN

11-3-7 Ship Loading Coarse Coal The endpoint of a coal slurry pipeline may be a thermal power plant or a ship loading facility (Figure 11-5) for export. Faddick (1982) reviewed some concepts for loading coarse coal onto ships: 앫 앫 앫 앫

Submarine pipeline Vertical riser Single-point mooring system Flexible hose system

The single point mooring (SPM) system is the most economical and feasible concept. In 1971, the Marcona Corporation installed at Walpipe, New Zealand the first successful SPM for ship loading of iron sand slurry. Reaching a large vessel with slurry is not always easy. Many ships require very deep ports or must stay out in deep waters to be loaded from a mooring point, as is the case with oil tankers. In United States ports, large carriers with drafts greater than 18–20 m (65–70 ft) have to be reached at great distances due to the relative shallowness of most American ports. Submarine pipelines are widely used in the oil and gas industry. For slurries, difficult accesses with possible sedimentation of coarse particles tend to be discouraging. Submarine slurry pipelines are typically limited to nonsettling slurries for tailings disposal, or to relatively short distances. High-density polyethylene (HDPE) pipes are lighter than water and can be floated. There are no records of using HDPE pipes with coarse coal and it is unknown whether they can be used as submarine pipes with this slurry. Flexible hoses are very expensive in large diameter sizes in excess of 250 mm (10 in) NB (normal bore). Adams (1986) did indicate that the use of polyethylene pipe is limited to solids with a maximum diameter of 9.5 mm [3/8 in], which would certainly not make these pipes suitable for fairly coarse solids. In Chapter 4, the Russian’s work on coarse coal friction losses and deposition velocity was presented in Section 4-4-3. The Russian equations are useful as an alternative to complex stratified models. The density of coal varies depending on the moisture content. An average specific gravity of 1.35 is often used in calculations.

11-3-8 Combustion of Coal–Water Mixtures (CWM) In the 1980s, considerable research was conducted in the United States on converting diesel engines to burning coal or its derivatives. The interest in coal fired diesel engines died away in the 1990s when the price of oil dwindled to $12 a barrel. With the threat of a new energy crisis at the turn of the 21st century, interest in coal fired diesel engines may revive. The most promising schemes required gasification of coal into a combustible gas. In an effort to bypass the gasification, plant schemes were proposed to burn coal as slurry in a diesel engine. Tests showed that coal would wear out piston rings, engine liners, etc. The concept required special construction materials, as is often the case with slurries. In an effort to bypass the problems of using pulverized coal in a slurry form with solid pistons, researchers have investigated liquid piston engines. The idea of a liquid piston engine was pioneered in the 19th century in the United Kingdom by Humphrey Engines. In an effort to bypass the problems of wear due to exploding a slurry mixture against sol-

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Seal Water Wash System

Product Distribution Unit

Slurry Pipeline End Manifold

Slurry Marine Hoses

Slurry Loading Arm

FIGURE 11-5 Ship loading of coal slurry. (From Faddick, 1982. Reproduced with permission of BHR Group.) 11.9

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id pistons, Abulnaga (1990) developed a liquid piston engine and obtained an Australian patent. The Abulnaga engine features a special Darrieus or Savonius rotor between two cylinders. Essentially, this means that the liquid pistons are formed as a column of liquid that oscillates between two cylinders. The great advantage of the Darrieus or Savonius rotors is that they maintain the same direction of rotation irrespective of the oscillation direction of the column. Typical coal–water mixtures (CWMs) for direct combustion consist of a weight concentration of 70% coal and 30% water. Prior to combustion, it is important to degrade the coal slurry mixture by applying hot air to accelerate the evaporation of water (Garbett and Yiu, 1988). The concept of direct combustion of CWM in gas turbines has been proposed in the literature. There remain many unknowns, particularly as to the erosion of the blades from fly ash.

11-3-9 Pumping Coal Slurry Mixtures Hughes (1986) described the development of positive displacement pumps for the Siberian coal pipeline Belovo-Novosibirsk. This pipeline is 256 km (160 mi) long. It transports heavily concentrated water–coal slurry. This pipeline features one main pump station and two booster stations. Each pump station features single-acting triplex Ingersoll Dresser pumps. Special 100 bar (1,470 psi) gate valves were manufactured in sizes of 200 mm (8 in) and 350 mm (14 in). Vanderpan (1982) recommended the use of Ni-hard as a material to cast the impellers and liners of coal handling slurry pumps. For certain high pH applications due to acidic water, or in the case of high-salt mixtures, special high-alloy irons may be used instead of Ni-hard. The use of centrifugal pumps in series is usually limited to a discharge pressure of 4.2 MPa (or 600 psi). This may be suitable for puming coarse coal up to a distance of 50 km (30 mi).

11-4 LIMESTONE PIPELINES Limestone is an important material. It was used thousands of years ago to build the colossal pyramids of Egypt and is used today to manufacture cement and concrete. Many derivatives of limestone are used as fertilizer, for alkalination of chemical processes, and as a pollution control substance used to absorb sulfur dioxide pollutants in flue gas desulphurization. In Chapter 1, a number of limestone pipelines are listed. The pipeline focused on in this segment is the Gladstone pipeline of the Queensland Cement and Lime Company in Australia, which started operation in 1979. Venton (1982) described the pipeline in great detail and diagramed examples of a practical design for a cement plant. The Gladstone pipeline is 24 km long and is located 400 km north of Brisbane. The reserves of limestone are overlaid by deposits of clay suitable for a clinker cement plant. A ship loading facility was built in the Gladstone harbor in order to transport the lime to Brisbane (Figure 11-6). At the discharge point of the limestone pipeline, the slurry is dewatered by a thermal drying processes. The lime is then transported in a powder form. To reduce the relatively

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FIGURE 11-6 The Gladstone limestone pipeline in Queensland, Australia. (From Venton, 1982. Reprinted with permission of BHR Group.)

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high cost of thermal drying, mechanical dewatering with filter presses is used. The lime is therefore delivered in a semiwet state, but this is acceptable for a cement plant. The filter presses reduce the water content from 36% to 17% moisture. In the actual cement plant, waste heat from the kiln off-gases is then used for further drying. In a limestone pipeline project, the slurry plant is located at the quarry and needs to be fitted with a milling or grinding circuit. Water also must be available on-site to be blended with the ground limestone. Some of this water may be available locally; however, if the pipeline is relatively short, it may be possible to return water from the pipeline discharge point. Processes of slurry preparation are described in Chapter 7. Limestone pipelines typically operate in a range of 50–60% weight concentration. If other ingredients such as clay are in the slurry, a small pump test loop is recommended on-site to monitor the composition and concentration. In the case of the Gladstone pipeline, the viscosity was in the range of 10 mPa·s (1 cP) at a weight concentration of 56%, 20 mPa·s (2 cP) at a weight concentration of 60%, but rose sharply toward a viscosity of 70 mPa·s (7 cP) at a weight concentration of 68%. The yield stress was in a range of 8–14 Pa (Figure 11-7). The laminar to turbulent velocity in a 200 mm (8 in) pipeline was predicted to be in the range of 1–1.3 m/s at a weight concentration of 62–64%. The Gladstone pipeline uses Wilson-Snyder positive displacement pumps. At the weight concentration of 60–64%, the slurry acted as a Bingham mixture, with non-Newtonian viscosity characteristics. However, it did feature clay, sand, and iron, as the materials were formulated for the manufacture of clinker cement. The pipeline operation speed was maintained at 2 m/s and the pressure drop was around 300 kPa/km. API 5LX steel with a high yield strength was used. Corrosion rates as high as 0.25 mm/year were measured during the initial phase of operation of the pipeline. Pertuit (1985) reported that during the first two years of operation problems of operation included: 앫 Severe knocking and vibration of mainline pumps 앫 Short life of gland packing and piston scouring of the positive displacement pumps These problems were eventually solved. The extremely high rate of corrosion was unexpected, since the lab reports suggested a design for a low corrosion rate of 0.076 mm/year. Venton (1982) reported that the operators brought down the rate of corrosion by adding inhibitors to the slurry composition.

11-5 IRON ORE SLURRY PIPELINES Iron ore is critical to our modern economy. A number of iron ore slurry pipelines have been constructed since the 1960s (see Table 1-9 for specific examples). One of the most famous is the SAMARCO pipeline in Brazil, which is 390 km (245 mi) long. In order to understand its rheology, Thomas (1976) conducted tests on iron ore at a volumetric concentration of 24% and with a solids diameter d50 = 40 ␮m. The tests were conducted in 150 mm (6 in) and 200 mm (8 in) pipe. The head loss in meters of water per meter of pipe length was derived as

␶ = KV xDIy

(11-1)

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FIGURE 11-7 Rheology of the Glasdtone limestone slurry. (From Venton, 1982. Reprinted with permission of BHR Group.)

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where K = 5228 x = 1.77 y = –1.18 The equation of Thomas does not agree well with experimental data published by Hayashi et al. (1980). However, this may be due to the difference in particle size distribution. Lokon et al. (1982) conducted further tests at the Melbourne Institute of Technology in Australia and derived the following equation for the pressure loss gradient in Pa/m: im = KV xDIyCvz

(11-2)

where im = friction gradient of the mixture k = 54.9 x = 1.63 y = –1.42 z = 0.35 Lokon et al. indicated that their power law compared well with commercial pipelines. Their data was based on iron ores pumped with solids in a size range of 30–60 ␮m (mesh 325–250). Obviously, this was the range of nonsettling slurries. Pressure losses are presented in Figure 11-8. Example 11-1 Using the Lokon equation, determine the pressure for an iron ore pipeline under the following conditions: 앫 앫 앫 앫

Pipeline inner diameter is 175 mm Volumetric concentration is 33% Flow rate is 48 L/s Pipeline length is 50 km

Solution in SI Units flow area A = 0.25 × 0.1752 = 0.024 m2 flow speed V = Q/A = 0.048/0.024 = 2 m/s im = KV xDIyCvz im = 54.9 × 21.63 × 0.175–1.42 × 0.330.35 im = 54.9 × 3.095 × 11.88 × 0.6784 im = 1369 Pa/m Klose and Mahler (1982) measured the critical speed of iron ore slurry with particles size in the range of 1 to 2 mm (0.04–0.08 in). However, due to the high density of iron oxide (SG = 5.0) critical speed as high as 3.5 m/s (11.5 ft/sec) were recorded (Figure 11-9). To design economic pipelines, Klose and Mahler suggested the addition of special chemical additives that can reduce the critical speed of the mixture, despite a slight rise in the pressure drop at these lower speeds (Figure 11-10).

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PRESSURE GRADIENT (kpa/Km)

5000

쑗 IRON ORE CV = 24.9 왕 IRON ORE CV = 26.6 쑗 IRON ORE CV = 28.1 첸 IRON ORE CV = 30.2 앳 IRON ORE CV = 31.3 왖 WATER

4000

3000

2000

1000 300

600

VELOCITY (m/s)

500 0.5

1

2

3

4

5

FIGURE 11-8 Pressure losses for iron ore oxides in the range of 30–60 ␮m (mesh 325-250). (From Lokon et al., 1982. Reprinted with permission of BHRA Group.)

Taconite is a very important source of iron in the United States. It is a form of iron sand found in the Mesabi range of Minnesota, as well as in Manitoba and Ontario, Canada. The Shilling Mining Review (1981), in an editorial article, reported on the pumping of taconite tailings using 20 in × 18 in (500 mm × 450 mm) Warman tailings pumps sized to a pressure of 350 psi. The pumps were installed in six stages. Taconite tailings are considered coarse sand and must be pumped in a range of speeds of 3.4–4.3 m/s (11–14 ft/s). Rubber-lined pipes are used. HDPE pipes are subject to very fast wear and are not used for tailings disposal. Taconite tailings are typically pumped at a weight concentration of 35%. The use of special flocculants in modern, efficient thickeners allows pumping up to a weight concentration of 45%. The SAMARCO pipeline in Brazil is one of the longest ever built to transport iron ore oxides and features 500 mm (20 in) and 457 mm (18 in) pipe sections over a distance of 400 km (250 miles). Start-up occurred in 1977 and it is expected to remain in operation for 40 years (Weston, 1985). Another long pipeline to transport iron ore oxide is the La Perla-Hercules pipeline in Mexico, with an overall length of 382 km (239 miles). The pipeline features one main and one booster pump station with single-acting triplex plunger pumps (Thompson, 1995).

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Velocity, v (m/s)

CHAPTER ELEVEN

Concentration, cv FIGURE 11-9 Critical speed of iron ore oxides with particle size in the range of 1 to 2 mm (0.04–0.08 in). (After Klose and Mahler, 1982. Reprinted with permission of BHR Group.)

11-6 PHOSPHATE AND PHOSPHORIC ACID SLURRIES Phosphate is a very important source of fertilizer for agriculture and is mined in large quantities in the United States, Morroco, Egypt, South Africa, China, and other countries. Phosphate rock is sometimes transported in a pre-milled state over a relatively short distance—a few kilometers or miles. In Florida, a method of transporting phosphate rock while mining it in a very similar fashion to dredging a river using a pump has been developed. Large phosphate matrix pumps driven by diesel engines are available on baseplates. These are relocated from site to site as the phosphate matrix field is mined out. Tillotson (1953) described the phosphate matrix in Polk and Hillsborough counties. About 5120 km2 (2000 mi2) contained high grades of phosphate. Eight million short tons of saleable phosphate pebbles were produced annually. Tillotson described the phosphate matrix as an unconsolidated mixture of clay smaller than 1 mm (0.04 in), silica sand, and phosphate rock of a much larger size. This mixture ranged in size from rocks as large as 150 mm (6 in) to as small as 38 ␮m (400 mesh). Because phosphate matrix pumps have to handle lumps as large as 150 mm (6 in), they tend to be large with 500 mm (20 in) suction and discharge sizes.

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22 DN DN 200 200 c cv ==0.64 0.64 v

20 18 Pressure loss Dp (105 Pa/1000 ml)

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8

4% 4%

6

3%3%

vvcc

4 vvcc

2

2% 2%

0 0

1.0

2.0 3.0 Velocity, v (m/s)

4.0

5.0

6.0

FIGURE 11-10 Reduction of critical speed of iron ore oxides with particle size in the range of 1 to 2 mm (0.04–0.08 in) by the use of special chemical additives. (After Klose and Mahler, 1982. Reprinted with permission of BHR Group.)

Once the phosphate matrix is pumped or transported, it is processed in a special phosphate rock treatment plant. Nordin (1982), of the Phospnate Development Corporation Ltd., described how each year a South African plant produces approximately three million tons of phosphate rock from foskorite and pyroxenite ores. The ore was then classified through flotation, thickening, and filters before being stockpiled. The result was fine gray-white crystalline powder of mineral apatite, with a 36.5% P2O5 content, a solid specific gravity of 3.17, and d50 ⬇ 106 ␮m. The hardness of the apatite was measured at 5.0 on the Mohr scale. Nordin (1982) reported that milled phosphate rock is easy to pump in a weight concentration of 30–70%. He conducted tests on a 100 mm (4 in) loop and obtained the values of critical velocity shown in Table 11-1, where d70 ⬇ 75 ␮m. He recommended pumping at 0.3 m/s (1 ft/s) above critical speed (Table 11-2).

11-6-1 Rheology Landel et al. (1963) investigated the rheology of a bimodal (fine and coarse) distribution of phosphate ore. They reported that in certain cases the finer particles act as a carrier for

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TABLE 11-1 Critical and Recommended Speed of Pumping Phosphate Rock with d70 ⬇ 75 ␮m (data from Nordin, 1982) Weight concentration, %

Critical velocity m/s Recommended pumping velocity, m/s

30

40

50

63.5

67

70

1.30 1.60

1.10 1.40

0.90 1.20

0.85 1.15

1.1 1.40

1.35 1.65

the coarser solids and that for all intents and purposes the slurry may be considered nonNewtonian. They proposed the following equation for the consistency factor:

冢

Cv K = ␮L 1 – ᎏ Cmax

冣

–2.5

(11-3)

where K = fluid consistency index (Pa·s) Cmax = maximum solids volumetric concentration Cv = volumetric concentration of solids Peterson and Mackie (1996) proposed the following equation for phosphate ore:

冢

Cv3 ␶0 = B ᎏᎏ Cmax – Cv

冣

(11-4)

where B = 13.3. The data presented by Peterson and Mackie (1996) on the critical speed is consistent with the data from Nordin (1982). Anand et al. (1986) indicated that the corrosion rate due to Maton phosphate is of the order of 0.3 mm/year in steel pipes. A total corrosion and wear allowance of 0.4 mm/year is suggested by Peterson and Mackie (1996). It is, however, recommended to assume more wear in the initial dozens of kilometers (miles) in a long pipeline, as particle attrition and degradation often occur in the initial portion of the pipeline. As the particles become less sharp, their abrasion of the pipeline decreases (Table 11-3).

11-6-2 Materials Selection for Phosphate The Miller number of phosphate ore is smaller than 50 (Abulnaga, 2000). This means that phosphate is suitable for pumping with piston reciprocating pumps.

TABLE 11-2 Power Consumption 100 mm (4 in) ID Pipe for Phosphate Rock with d70 ⬇ 75 ␮m Weight concentration, %

Power consumption, kW/km After Nordin (1982).

30

40

50

63.5

67

70

0.196

0.151

0.128

0.112

0.109

0.113

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TABLE 11-3 Example of Phosphate Ore Properties Property

Fines

Coarse

Product

Solids specific gravity Freely settled particles packing (%) Coefficient of sliding friction (␮p) d10 Particle size (␮m) d50 Particle size (␮m) d90 Particle size (␮m)

3.2 40 0.53 60 22 74

3.2 44 0.58 50 92 150

3.2 51 0.58 14 75 145

After Paterson (1996).

Pumping phosphoric acid slurries represents a challenge to the manufacturing of slurry pumps due to the combination of corrosion and wear in some of the critical circuits such as flash cooling, filter feed, and gypsum removal. Walker (1993) reported that the wear life of these pumps can be as low as a few thousand hours. Traditionally, pumps were lined with rubber or manufactured out of stainless steel. Rubber linings proved less than optimal. Tearing problems occurred during flash cooler applications, lowering the life of some components to 3000 hrs. Erratic tearing also decreased the wear life on filter feeds to as low as 1900 hrs, and local holing (formation of holes in the liner) decreased wear life to 2300 hrs. In some respects, installing pumps made out of stainless steel is an attractive option since they can be repaired by welding, but stainless steels are not as hard as abrasionresistant white iron. A special alloy, which offers as good resistance to corrosion as stainless steel and hardness as white iron, is Hyperchrome, developed in Australia. Hyperchrome was derived from hardfacing weld deposit materials defined in the Australian Standard as AS2576-1982 Type 2. Walker (1993) described an important improvement of wear life components over comparable stainless steel components. Tian et al. (1996) reported the development of a new alloy, a white iron with a very high chromium content similar to hyperchrome. The new alloy achieved a service life of 2.5–3 years for an impeller in gypsum tailings service in a phosphoric acid environment, whereas the Cd4MCu material impeller was badly worn out after 3 months of operation. Both Weir-Warman and KSB-GIW, the largest manufacturers of slurry pumps, have experimented with the use of high chrome alloys with chromes in excess of 30% for slurry pumps handling phosphate rock.

11-6-3 The Chevron Pipeline About 280 km (175 mi) southeast of Salt Lake City, U.S.A. at Vernal, Utah, there is a large layer of phosphate ore that covers an area of 38,400 km2 (15,000 mi2) with an estimated reserve of 700 million short tons (640 million metric tons). Between 1961 and 1986, the ore was transported by truck over a distance of 216 km (135 mi) and then loaded onto railroad cars. In 1986, a pipeline for the phosphate concentrate was commissioned. The diameter of the pipeline is 250 mm (10 in). A pipeline feed station with a test loop and a pump station is located at Vernal, and a booster is located at Richard’s Gap in

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Wyoming. After covering an overall distance of 150 km (94.3 mi) the pipeline terminates at Rock Springs, Wyoming. At Vernal, the concentrate is thickened in a special thickener, then conditioned in three agitator tanks. These large tanks are 15.24 m × 15.24 m (50 ft × 50 ft), each holding 2000 tons of phosphate. The slurry has a weight concentration of 53–60%. Weston and Worthen (19xx) indicated that each agitation tank is fit with a 200 hp mixer. The slurry is tested in a 91 m (100 ft) long pump loop prior to being fed to the pipeline. The pipeline was designed for a flow of 86 L/s (1370 US gpm) and a pressure of 17.915 MPa (2600 psi) per pump station. The booster station is located at 77 km (48 mi), halfway along the pipeline. Two positive displacement Wilson-Snyder pumps were installed at the main pump station. These were driven by 746 kW (1000 hp) direct current motors. The booster station is connected to a water pond and draws water on demand to avoid slack flow by providing additional back pressure in low flow conditions. Choke stations are also provided for additional back pressure. According to Weston and Worthen, the pipeline used highyield-strength steel rated at 413 MPa (60,000 psi). An allowance of 2.5 mm (0.1 in) for corrosion/erosion over a lifetime of 25 years was factored into the design. The thickness of the pipeline varied between 6.4–12.7 mm (0.25–0.5 in). The pipeline crossed the Rocky Mountains, so the elevation varied between a low point of 1676 m (5500 ft) and a high point of 2499 m (8200 ft). To minimize freezing problems, the pipeline was buried to a depth of 1.8 m (6 ft). There are 12 monitoring or testing points along the pipeline to monitor for pressure. If freezing or sedimentation develop, the resultant increase in pressure is automatically detected. Slurry is pumped at an average speed of 1.5–1.67 m/s (5–5.5 ft/sec) and it takes about 26 hours for the material to be transported from start to finish. The pipeline was designed to transport 2273 million metric tons (2500 million short tons) of phosphate concentrate per year. Initially, it operated on a special batch mode with 12 hours of water and 12 hours of slurry. To monitor corrosion, the three following methods are used: 1. Corrosion spools (sacrificial thickness loss) 2. Ultrasonic testing (to measure pipe thickness) 3. Corrosion probes (to measure corrosivity) Corrosion of phosphate pipelines is reduced by the use of special inhibitors or by raising the pH to the alkaline range (alkalination).

11-6-4 The Goiasfertil Phosphate Pipeline Pertuit (1985) described the Goiasfertil phosphate pipeline. It was constructed in the state of Goias in Brazil to transport phosphate ore along difficult terrain over a distance of 14.5 km (9 miles) from a mine to an existing railway station in the town of Cataloa. The pipeline was designed to ship 900,000 metric tons of concentrate per year over a period of 6750 hours. The slurry consisted of solids at a weight concentration of 63% to 66%. The particle distribution consists of 20–25% + 150 microns, and about 25–35% minus 45 microns. The start-up was in August 1982. The pump station consists of charging centrifugal pumps, a safety and test loop, and two mainline Wilson-Snyder positive displacement pumps. They are controlled by variable speed drivers and the speed is adjusted according to the flow rate.

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11-6-5 The Hindustan Zinc Phosphate Pipeline Pertuit (1985) described the Hindustan zinc phosphate pipeline, which was commissioned in late 1983 near the town of Udaipar in India. This pipeline was designed to ship on an intermittent basis 400 metric tons of phosphate per year via a 73 mm (2.875 in) diameter pipe over a distance of 10.5 km (6.5 miles). Phosphate is shipped at a weight concentration of 65–68% by weight. The pump station consists of charging centrifugal pumps and Worthington plunger pumps. At the terminal station, thickeners and agitated tanks were installed.

11-7 COPPER SLURRY AND CONCENTRATE PIPELINES Venton and Boss (1996) described the wear of the OK Tedi copper concentrate pipeline in Papua New Guinea. This pipeline is 155 km (95 mi) long. Severe localized wear caused replacement of certain section of the pipeline. The pipeline was commissioned in 1987. The mine is 156 km (96 mi) from the seaport of Kiunga. About 96 km (60 mi) of the pipeline uses gravity flow. A booster station was installed to promote flow over the remaining 60 km (37.5 mi). The normal flow rate was in the range of 85–88 m3/hr (374–388 US gpm) with a weight concentration of 55–60%. The speed of flow was in the range of 1.22–1.4 m/s (4–4.6 ft/sec). The wall thickness was in the range of 5.6–11 mm (0.22–0.433 in) (Table 11-4). The pipeline operated in batches of water and concentrate. The water was not neutralized by an oxygen scavenger. Venton and Boss (1996) indicated that an initial pipeline failure occurred in 1991. They attributed this failure to accentuated wear due to coarser particles, not the design of the valves. The most severe wear occurred at the change of pipe thickness from 5.6–6.4 mm (0.22–0.25 in). Corrosion was also a factor as no oxygen scavenger had been used. The operators installed corrosion-meter probes in 1992 on the top and bottom section of the pipeline to monitor wear as loss of wall thickness. Wear of 0.37 mm/year (0.0145 in/year) was measured for the bottom section of the pipe, and wear of 0.18 mm/year (0.007 in/year) for the top section with continuous slurry water/batching. Venton and Boss (1996) reported that there were four batches of slurries at 500 m3 (17,657 ft3) and four batches of water of 40–50 m3 (1,413–1,766 ft3) per day. To mitigate against wear, the top size (+106 microns) was cut down to 1%, water batching was eliminated altogether, and the pipeline was allowed to shut down and restart with slurry. Unfortunately, 60 km (37.5 mi) of pipe had to be replaced with thicker walled pipe to continue operation over its anticipated life of 15 years.

TABLE 11-4 Velocity of Flow of Copper Concentrate Pipelines Pipeline OK Tedi DN150 (6 in OD) Bonguinville DN 150 (6 in OD) Freeport DN 100 (4 in OD) DN 125 (5 in OD) After Venton and Boss (1996).

Nominal flow rate 85 m3/h 65–109

374 USGPM 227–479

Nominal velocity 1.24 m/s 1.23–2.04 1.1–1.6 1.2–1.5

4.07 ft/s 4.03–6.7 3.6–5.25 3.95–4.1

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Venton and Boss (1996) described in great detail the operating problems of the OK Tedi pipeline. Their recommendations for better operation included the following: 앫 Installing tracking modules on each pig for pigging the pipeline. 앫 Replacing the Rockwell Nordstrom valves, which often fail due to inadequate lubrication in remote valve stations, by other valves. Tests were run on Audco full bore valves, Mogas metal-seated valves, and Larox high-pressure hydraulically activated punch valves. The Mogas valves did not require lubrication and lasted 140 operating cycles whereas the Rockwell Nordstrom plug lasted 35 cycles, the Audco ball lasted 20 cycles, and the Larox pinch valve lasted 45 cycles. The Mogas valve was therefore the most appropriate for this copper concentrate pipeline. One of the largest copper mines in the world is operated by Minera Escondida Ltd. in Chile. The mine is located at an altitude of 3100 m (10,170 ft) above sea level. To transport the copper concentrate a pipeline was constructed. The pipeline uses a single pump station at the beginning of the pipeline and gravity throughout the remainder. The pipeline spans 165 km (103 mi) of mountainous terrain and transports the slurry at a cost of 1–1.5 dollars per metric ton. This style of transportation is considerably cheaper than the alternative option of trucks and railroads. Nordstrom valves were used on the pipeline (Boggan and Buckwalter, 1996). Bajo Alumbrera is located in northwest Argentina near Catamarca. The plant processes 90,000 tons of ore a day. Copper concentrate is shipped to a port via a 152 mm (6 in) pipeline over a distance of 320 km. Geho positive displacement diaphragm pumps in the main pipeline and a couple of booster stations provide the power to pump over such a long distance.

11-8 CLAY AND DRILLING MUDS Sellgrem et al. (2000) conducted tests on sand as well as sand–clay mixtures pumped by centrifugal pumps. The phosphate clays had a diameter d50 between 1 ␮m and 50 ␮m. The sands were much coarser with d50 of 0.64 mm (0.025 in), 1.27 mm (0.05 in), and 2.2 mm (0.09 in). The presence of clay and other particles finer than 75 ␮m and a concentration smaller than 20% had a beneficial effect by reducing the head loss and efficiency derating factor. The data recorded by Sellgrem et al. (2000) should not be applied to a higher concentration of clays because the viscosity effect introduces a new component to the equation. Drilling muds and bentonite are pumped at high concentration in the oil industry using positive displacement pumps. Certain important minerals such as bauxite for the aluminum industry are found in bentonite. Soft high clay is present in certain copper ores. In a diluted form at weight concentration smaller than 40%, it can be handled fairly well. However, particular attention must be paid in milling circuits when the concentration may approach 50%, as the viscosity affects flow and recirculation loads. Codelco in Chile is one of the largest producers of copper in the world. To dispose tailings to the Ovejeria Tailings Dam, slurry had to be piped from an elevation of 4000 m above sea level down to 700 m via a 57 km (36 mile) long pipeline. At the dam, the coarse and fine are separated using a cyclone station. Three Wirth positive displacement pumps are then used to pump the coarse material around a 4 km loop at a flow rate of 140 m3/hr (616 US gpm) and at a pressure of 4.0 MPa (580 psi) (Figure 11-11).

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FIGURE 11-11 Tailings solids segragation station and pumping facility for copper tailings at Codelco Andina, Chile, featuring the use of diaphragm pumps. (Courtesy of Wirth Pumps, Germany.)

11-9 OIL SANDS In 2001, Canadian oil sands were the most pumped slurries in the world. Due to this large amount of slurry (1.9 m3/s) (ranging up to 30,000 US gpm), pump manufacturers developed new technologies, including technology for froth treatment. At the Summit Meeting at Quebec during April 2001, Canada encouraged the United States to invite U.S. corporations to invest billions of dollars in the oil sand fields of Alberta. Even without new U.S. investments, an estimated 20 billion will be invested between 2000 and 2020. This is a continuously growing industry that will require sophisticated slurry systems. The process of extracting oil from sand is a vast topic and only a few aspects will be touched on in this chapter. In basic terms, Alberta, Canada sits on layers of tar-rich sands. The shallowest layers, which are most accessible for open pit mining, are in the north of Alberta near the Athabasca River and Fort McMurray. Between the first discoveries in the 1930s and the end of the century, a number of technologies were developed to the extract oil from the sand. The initial approach was to heat slurries of tar sand to a temperature that reduced the viscosity of the oil and separated it from the sand. Other technologies developed coannular flows that separated the oil from the sand by degradation of the natural lumps of oil and sand. More recently, solvents were developed that dissolve the tar or oils out of the sand. The latest solvent-based technologies use lower temperatures, reducing energy costs. Outside the Athabasca region, the oil sands are located in deeper layers. The proposed extraction method pumps hot steam down approximately 100 m (300 ft) of pipe to the oil sand bed. The steam would then resurface carrying the oil and tar. This technology was

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originally proposed to recover oil from oil shale in Colorado, U.S., and Queensland, Australia. Syncrude and Suncor-Muskeg River Shell in conjunction with a Canadian Research Institute, the Saskachewan Research Institute, the University of Alberta, and the University of Toronto, developed this new technology for oil sand slurries pumping. The concept of stratified two-layer flows was extensively investigated by these companies and institutes to handle 63 mm (2.5 in) lumps. By 1999, the price of synthetic oil produced from processed tar sand by Syncrude and Suncor dropped low enough to compete with natural oil from Texas and the Middle East. A dedicated pipeline from Edmonton, Canada to Chicago, U.S.A. became the longest pipeline for synthetic fuel. In a recent paper, Sanders et al. (2000) discussed the effects of bitumen on sand hydrotransport and conducted tests on a number of grades of oil sands. They reported that pipeline pressure losses due to friction at cold or warm temperatures increased with the length of the pipe. A time dependency developed, which was attributed to the formation of a thin coating of bitumen at the wall of the pipe. They defined an equivalent pipe roughness in the presence of bitumen of 650–1150 ␮m, which is much higher than normal steel at a roughness of 63 ␮m. In a 250 mm (10 in) pipe, the presence of fines in the oil sand slurry reduced the deposition velocity to 1.1 m/s, whereas the absence of fines increased the deposition velocity to 2.7 m/s. Due to the change in speeds, different approaches are used in the designs of pipelines for coarse grade and fine grade ores. The lower-grade ores, with less bitumen, do not necessarily exhibit this phenomenon of wall coating and therefore require higher pressure for pumping. Another pipe coating focused on in numerous tests is the tar coating of pipes in froth treatment plants. Under certain conditions, the injection of water through a ring just a few diameters before the pump suction reduces power consumption and improves the efficiency of pumps. It is not known whether tar deposits on the impeller end causes a degradation of pump performance. The physical properties of oils in oil sand would defy any designer of centrifugal slurry pumps and there are no standard methods to account for derating of performance. McKibben et al (2000) conducted tests on water–oil mixtures (without sand) with crude oils of a viscosity of 5300–11,200 Pa·s. They found evidence against two popular theories. First, they showed how the injection water did not form a layer at the wall to reduce pressure losses as was commonly thought. Instead, it formed slug around the oil and transported at lower pressure losses. Secondly, they also indicated that the viscosity of the oil was of no consequence. Therefore, it must be said that the flow of oil, sand, and water as a mixture is fairly complex. The Canadian oil sand projects have encouraged the manufacturers of slurry pumps to develop special mechanical seals for slurry pumps (Swamanathan et al., 1990).

11-10 BACKFILL PIPELINES A backfill is essentially a mine residue or tailing pumped back to fill excavated or mined pits. A backfill can be mixed with other low-permeability materials such as clays to help seal the area. Particularly in the case of underground backfilling, the water content should be minimized to avoid costly dewatering. Backfill slurries are therefore dense, with a high weight concentration (around 50–65%). Multistage centrifugal pumps or positive displacement pumps are used to transport them (Figure 11-12). Steward (1996) conducted an empirical study of vertical and horizontal pipelines. He demonstrated how backfill consisting of fine and coarse material could be classified

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FIGURE 11-12 Backfilling of very dense slurry using diaphragm pumps. (Courtesy of Wirth Pumps, Germany.)

through cyclones in order to separate them. The cyclone underflow was then drained by gravity. In the South African gold and uranium mines, Steward (1996) reported flow speeds as high as 12 m/s (40 ft/s). These extremely high speeds are the cause of rapid wear and erosion–corrosion. To support and reinforce an underground excavated area after the ore has been removed, the backfill (including both fine and coarse material) is thickened. The product is called full plant tailings. Steward (1996) defined the particle sharpness as the rate of directional change in the particle perimeter. Pipe wear is measured as the rate of mass loss per unit of time (kg/s or slugs/s). Wear in a pipeline is an exponential function of the flow speed: dw/dt = KVn Since it is also a function of other parameters, Steward (1996) proposed the following function: dw/dt = f (Sm, V, DI, d90, SI, M) where f = function of Sm = specific gravity of mixture

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V = velocity of flow DI = pipe inner diameter d90 = 90% passing diameter of particles SI = sharpness index M = mass of solids per unit length of pipe From his tests, Steward (1996) derived the following empirical equation (in SI units) (Table 11-5): log10(dw/dt) = m1Sm + m2V + m3DI + m4d90 + m5SI + m6M + M7 where m1 = –6.31374 m2 = 0.3186193 m3 = –0.131869 m4 = 0.0054758 m5 = 1.7709578 m6 = 0.6162088 m7 = 6.5961888 Coetzee (1990) determined that one-third of the loss of pipeline wall thickness associated with pumping mine water is due to corrosion because mine water is often acidic. Since corrosion is an important contributor to wear of backfilled pipes, it became evident that lining the pipes was necessary. To compare piping materials, tests were conducted by Steward (1996) and indicated that a polyurethane rubber at a Shore hardness 55 Shore A provided the best pipeline protection in a test with slurry pumped at a speed of 3 m/s. By comparison, ASTM steel 106 grade B wore seven times faster than polyurethane 82 Shore A, or high-density polyethylene. Steward (1996) indicated that the mixing of cementitious binders with slurry could reduce wear considerably in backfill applications. Backfill paste is formed by dewatering slurry of tailings (thickening and filtering). Mixing dewatered slurry with cement (3%–5%) produces a stiff backfill (1.5–3.5 MPa strength, or 218–508 psi). Coarse aggregates (50 mm or >2 in) up to distances in excess of 105 km (60 mi). A successful pipeline project depends on proper economics. This will be the topic of the next chapter.

11-14 REFERENCES Abbot, J. 1965. Use of Hydrocyclones for Thickening and Recovery in the National Coal Board. Filtration and Separation, 2, 3, 204–208, 234. Abulnaga, B. E. 1990. An Internal Combustion Engine Featuring the Use of an Oscillating Liquid Column and a Hydraulic Turbine to Convert the Energy of Fuels. Australian Patent AU-B20956/88. Adams, W. I. 1986. Polyethylene Pipelines for Slurry Transportation. In 11th International Conference on Coal Technology. Washington, D.C.: Coal and Slurry Technology Association. Anand, S., S. K. Ghosh, S. Govindan, and D. B. Nayan. 1986. Maton Rock Phosphate Concentrate Pipeline. Working paper, BHRA Group, Hydrotransport 10, Innsbruck. Boggan, J. and R. Buckwalter. 1996. Slurry Pipeline Helps Remedy Corrosion at Record Height. Pipeline and Gas Journal, 223. Bomberger, D. R. 1965. Hexavalant Chromium Reduces Corrosion in a Coal-Slurry Pipeline. Materials Protection, 4, 1, 41–48. Brackebush, F. W. 1994a. Basics of Paste Backfill Systems. Mining Engineering, 46, 1175–1178. Brackebush, F. W. 1994b. Basics of Paste Backfill Systems. Mining Engineering, 47, 1041–1042. Brooks, D. A. and C. H. Dodwell. 1985. The Economic and Technical Evaluation of Slurry Pipeline Transport Techniques in the International Economic Coal Trade. In 10th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association. Buckwalter, R. and A. Walters. 1989. Selection of coal slurry pipeline technologies for gasification combined power cycle plants. In Proceedings of the 14th International Conference on Coal and Slurry Technology. Washington, DC: The Coal and Slurry Association. Burgess, K. E. 2000. Froth Pumping. Technical Bulletin No. 28. Sidney Australia: Warman International. Coetzee, R. 1990. Wear and Corrosion of Mild Steel Tubes in Backfill and Backfill Feed Filtrate. Report 332891. Physical Metallurgy Division. Council for Mineral Technology, South Africa. Ercolani, D., E. Carniani, S. Meli, L. Pelligrini, and M. Primercio. 1988. Shear Degradation of Concentrated Coal–Water Slurries in Pipeline Flows. In 13th International Conference on Coal Technology, Denver, CO. Washington, D.C.: Coal and Slurry Technology Association. Faddick, R. R. 1982. Ship loading Coarse-Coal Slurries. Working paper A-3, in 8th International Conference on Solids in Pipes, Johannesburg, South Africa. Gandhi, R. G. 1985. Fosferil phosphate slurry pipeline. In 10th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association. Garbett, E. S. and S. M. Yiu. 1988. The Effect of Convective Heat on the Disintegration of a Coal–Water Mixture in Pneumatic Atomization. In 13th International Conference on Coal Technology, Denver, CO. Washington, D.C.: Coal and Slurry Technology Association. Gillies, R. G., J. Schaan, R. J. Sumner, M. J. McKibben, and C. A. Shook. 2000. Deposition Veloci-

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ties for Newtonian Slurries in Turbulent Flows. Canadian Journal of Chemical Engineering, 78, 4, 704–708. Hayashi, H. et al. 1980. Some Experimental Studies on Iron Concentrate Slurry Transport in Pilot Plant. Working paper, BHRA Group, Hydrotransport 7, Sendai, Japan. Hughes, C. V. 1986. Coal Slurry Pump Development Update. Mainline Pumps for the BelovoNovosibirsk Pipeline. In 11th International Conference on Coal Technology, Washington, D.C.: Coal and Slurry Technology Association. Klose, R. B. and H. W. Mahler. 1982. Investigations into the hydraulic transportation behaviour of ore and coal suspensions with coarse particles. In Hydrotransport 8, Johannnesburg. Cranfield, UK: BHRA Group. Kreusing, H., and F. H. Franke. 1979. Investigations on the Flow and Pumping Behavior of Coal–Oil Mixtures with Particular Reference to the Injection of Coal–Oil Slurry in the Blast Furnace. Working paper C-2, BHRA Group, Hydrotransport 6, BHRA. Landel, R. F., B. G. Mosen, and A. J. Bauman. 1963. In 4th International Conference on Rheology, Brown University, Part 3, p. 663. New York: Interscience Publishers. Leninger, D., W. Erdmann, and R. Kohling. 1978. Dewatering of Hydraulically Delivered Coal. Working paper E-7, BHRA Group, Hydrotransport 5, Hanover. Lokon, H. B., P. W. Johnson, and R. R. Horsley. 1982. A “Scale-up” Model for Predicting Head Loss Gradients in Iron Ore Slurry Pipelines. Working paper B-2, BHRA Group, Hydrotransport 8. Madsen, B. W., S. D. Cramer, and W. K. Collins. 1995. Corrosion in a Phosphate Pipeline. Materials Performance, 34, 70–73. McKibben, M., R. G. Gilles, and C. A. Shook. 2000. A Laboratory Investigation of Horizontal Well Heavy Oil–Water Flows. Canadian Journal of Chemical Engineering, 78, 734–751. Miller, J. W. and H. L. Hoyt. 1988. Evaluation of Polymers as Suspending Aids for Coal–Water Slurries. In 13th International Conference on Coal Technology. Washington, D.C.: Coal and Slurry Technology Association. Morway, A. J. 1965. Stabilized Oiled Coal Slurry in Water. US Patent 31,201,168 assigned to Esso Research & Engineering Co. N.J., USA. Nordin, M. 1982. Slurry for Sale. Working paper F-2, BHRA Group, Hydrotransport 8. Olofinsky, E. P. 1988. Belovo-Novosibirsk Coal Transportation Pipeline. In 13th International Conference on Coal Technology, Denver, CO. Washington, D.C.: Coal and Slurry Technology Association. Peterson, A. J. C. and K. Mackie. 1996. An Economic and Technical Assessment of the Hydraulic Transport of Phosphate Ore. BHRA Group, Hydrotransport 13. Pertuit, P. 1985. Gladstone Limestone Slurry Pipeline. In 10th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association. Pertuit, P. 1985. Goiasferil Phosphate Pipeline. In 10th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association. Pertuit, P. 1985. Hindustan Zinc Phosphate Pipeline. In 10th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association. Pipelin, A. P., M. Weintraub, and A. A. Orning. 1966. Report of Investigation No. 6743, prepared for the US Bureau of Mines. Sanders, R. S., A. L. Ferre, W. B. Maciejewski, R. Giles, and C. Shook. 2000. Bitumen Effects on Pipeline Hydraulics during Oil–Sand Hydrotransport. Canadian Journal of Chemical Engineering, 78, 4, 731–742. Schaan, J., R. J. Sumner, R. G. Gillies, and C. A. Shook. 2000. The Effect of Particle Shape on Pipeline Friction for Newtonian Slurries of Fine Particles. Canadian Journal of Chemical Engineering, 74, 4, 717–725. Sellgrem, A., G. Addie, and S. Scott. 2000. The Effect of Sand–Clay Slurries on the Performance of Centrifugal Pumps. Canadian Journal of Chemical Engineering, 78, 4, 764–769. Shook, C. A., D. B. Haas, W. H. W. Husband, and M. Smail. 1979. Degradation of Coarse Coal Particles during Hydraulic Transport. Working paper C-1, BHRA Group, Hydrotransport 6. Steward, N. R. 1991. The Determination of Wear Relationships for FORSOC Fillset Binder Modified Classified Tailings at High Relative Density. Report for Gold and Uranium Division of the Anglo American Corporation of South Africa Ltd.

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Swamanthan, S., A. Fair, and J. Wong. 1990. In Search of Mechanical Seals for Slurry Pumps. In 15th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association. Steward, N. B. 1996. An Empirical Evaluation of the Wear of Backfill Transport Pipelines. Working paper, BHRA Group, Hydrotransport 13, Cranfield, England. Thomas, A. D. 1976. Scale-up Methods for Pipeline Transport of Slurries. Int. Journal of Mineral Processing, 3, 51–69. Thompson, T. L. 1985. La Perla/Hercules Pipeline. In 10th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association. Tian, H., G. Addie, and R. S. Hagler. 1996. Development of Corrosion Resistant White Irons for Use in Phos-acid Service. Paper presented at the Annual Conference of Central Florida section of the American Institute of Chemical Engineers. Tillotson, I. S. 1953. Hydraulic Transportation of Solids. M.n. Congress Journal, 39, 1, 41–44. Vanderpan, R. I. 1982. Proper Pump Selection for Coal Preparation Plants. In World Coal. San Francisco: Miller Freeman Publications. Venton, P. B. 1982. The Gladstone Pipeline. Working paper A-4, BHRA Group, Hydrotransport 8. Venton, P. B. and T. J. Boss. 1996. An Analysis of Wear Mechanisms in the 155 km OK Tedi Copper Concentrate Slurry Pipeline. Working paper, BHR Group, Hydrotransport, 533–548. Walker, C. I. 1993. A New Alloy for Phosphoric Acid Slurries. Paper presented at the 1993 Clearwater Convention, American Institute of Chemical Engineers. Weston, M. D. 1985. SAMARCO Pipeline. In 10th International Conference on Coal Technology, Lake Tahoe, Nevada. Washington, D.C.: Coal and Slurry Technology Association. Weston, M. D. and L. Worthen. 1987. Chevron Phosphate Slurry Pipeline commissioning and startup. In Proceedings of the 12th International Conference on Coal and Slurry Technology. Washington, DC: The Coal and Slurry Association. Editorial Articles Moving Mountains through a Slurry Pipeline. Engineering and Mining Journal, 195, 94–95, 1994. A Conductance Based Solids Concentration Sensor for Large Diameter Slurry Pipelines. Journal of Fluid Engineering, 122, 4 Variety of Slurry Pumps in Taconite Processing Plants. Skillings Mining Review, 70, 32, Aug 8, 1981. Further Readings Abulnaga, B. A. 2000. A Review of the Yichang Phophate Pipeline Feasibility Study. HATCH, unpublished. USSR Plans Coal Slurry Pipelines. Oil and Gas Journal, 82, 58–59, 1984. Braca, R. M. 1988. Use Needs Coal Slurry Pipeline. Pipeline and Gas Journal, 215, 32–36. Catalano, L. 1983. Railroads Kill Eminent Domain for Coal-Slurry Pipelines. Power Journal, 127, 9. Harvey, W. W. and Hossain, M. A. 1987. Co-recovery of Chromium from Domestic Nickel Laterites. Journal of Metals, 39, 21–25. Mahr, D. and B. Robert. 1986. Coal Slurry Pipelines Overland Belt Conveyors See Bright Future. Power Engineering, 90, 24–28. Maki, G. A. and D. M. Smith. 1983. Potash Mines and Mining/Saskatchewan/Thickeners/Design. CIM Bulletin, 76, 57–62. Maki, G. A., R. G. Roden, and P. J. Fullman. 1990. Stacking of Potash Mill Tailings. CIM Bulletin, 83, 96–98. Marrey, D. T. 1985. Exporting Colorado Water in Coal Slurry Pipeline. Journal of Water Resources Planning and Management, 111, 207–221. Nalziger, R. H. 1988. Ferrochromium from Domestic Lateritic Chromites. Journal of Metals, 40, 34–37. Nasr-El-Din, H., C. A. Shook, and M. N. Esmail. 1984. Isokinetic Probe Sampling from Slurry Pipelines. Canadian Journal of Chemical Engineering, 62, 179–185. Postlethwaite, J. 1987. The Control of Erosion–Corrosion in Slurry Pipelines. Materials Performance, 26, 41–45. Postlethwaite, J., M. H. Dobbin, and K. Bergevin. 1986. The Role of Oxygen Mass Transfer in the Erosion-Corrosion of Slurry Pipelines. Corrosion, 42, 514–521.

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Schaan, J., and C. A. Shook. 2000. Anomalous Friction in Slurry Flows. Canadian Journal of Chemical Engineering, 78, 4, 726–730. Shvartsburd, V. 1983. Pipelining and Burning Coal, Here are Important Criteria for Designing Coal Slurry Pipelines. Oil and Gas Journal, 81, 91–95. Wasp, E. J. 1983. Slurry Pipelines. Scientific American, 249, 48–55.

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CHAPTER 12

FEASIBILITY STUDY FOR A SLURRY PIPELINE AND TAILINGS DISPOSAL SYSTEM

12-0 INTRODUCTION A consultant engineer has to convince his clients of the merits of a slurry pipeline over alternative methods of transportation, whether it is for tailings disposal or concentrate shipping. One very important step in the design of a slurry pipeline is to appreciate the economics involved in the process. There is no question that this is the effort of a team of engineers, geologists, and accountants. It is therefore very important to appreciate the different and complex facets of a feasibility study. The exercise of a feasibility study or basic engineering should go through a number of steps, or follow a kind of checklist. In this chapter, the different steps are presented for this purpose. A pipeline for the disposal of tailings may be a few kilometers or miles long, whereas a slurry concentrate pipeline may be few hundred kilometers or miles long. The role of the geologist or foundation engineer is critical to the successful construction of a tailings dam. It would be beyond the scope of this book to discuss geophysics. In recent years, there has been a trend toward disposing of tailings in the sea. Whether tailings are disposed over land or in the sea, there are environmental concerns that must be satisfied. The engineer should be aware of these issues. The presence of some corrosion inhibitors, cyanide, or toxic materials in the tailings must be handled carefully. Tailings dams are sometimes within reach of agricultural fields and seepage could have negative effects on the quality of underground water. Environmental concerns may represent hidden costs with particular repercussions on slurry projects. This chapter presents an overview of basic engineering for a feasibility study. The study consists of identifying the components of a pipeline (such as feeding station, main and booster stations, emergency dump ponds, final disposal tailings pond, and area for sub-sea disposal), the size of the pipeline based on the anticipated flow rate, and the material of the pipe based on pressure, chemical attack, erosion, and corrosion. Other aspects of the study outside the scope of the slurry engineer and which cannot be covered in this book involve the geological survey, the cost of excavation, the cost of construction, power lines, power stations, transformer stations, and SCADA or control systems. The specialists involved in these areas rely on the slurry engineer for extensive information and 12.1

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help in basic engineering by providing data on stability of soil, difficulty of the terrain, cost of power transmission, etc. In turn, this collaborative information is fed to the estimators and the project managers. The slurry engineer will be requested to make suggestions, review the feasibility study, and help purchase the equipment.

12-1 PROJECT DEFINITION At an early stage of the feasibility study, the project is defined in the following terms: 앫 Volume of slurry to be transported over the life of the project 앫 Annual pumped flow of slurry 앫 Starting point of the pipeline, such as a smelter or tailings dam, and a final point of the pipeline, such as a port for export of the concentrate or power plant for burning coal 앫 Proposed contour of the pipeline 앫 Existing roads and need for new roads for access to the pipeline, tailings dam, or booster station 앫 Proposed pressure rating of the pipeline and the number of main and booster stations 앫 Rheology of the slurry 앫 Environmental impact of the project 앫 Stability of soil along the contour of the pipeline and possibility of seismic problems or landslides 앫 Need for a dewatering plant at the end of the pipeline 앫 Need for local generation of electricity for booster pump stations or for reclaim water stations 앫 Need for a reclaim water pipeline to return water to the starting point of the slurry pipeline 앫 Estimation of excavation costs if the pipeline is buried or if electric conduits are underground 앫 Estimation of costs for power poles to transmit electricity 앫 Protection of the pipelines from freezing in cold environments 앫 Allowance for water hammer and transients 앫 Allowance for thermal expansion in hot climates 앫 Required modifications to existing thickeners, or filtering and dewatering plants as part of expansions of production and pumped flow rates 앫 Mitigation against erosion, abrasion, and corrosion 앫 Required purchase of land for pipeline contour, tailings dam, dewatering plant, and booster stations 앫 Engineering costs 앫 Construction costs A general schematic diagram for the tailings disposal pipeline (Figure 12-1) or the concentrate pipeline (Figure 12-2) is made at an early stage to define the major components of the pipeline.

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fines for submerged disposal

reclaim water pipeline pump station (usually barge mounted)

clarified water

spigot

thickener corrosion inhibitors

fines for submerged disposal

cyclones

isolation valve

coarse for banks of pond

tailings pipeline feed sump

submerged disposal

Emergency pond

tailings pipeline Tailings Disposal Ponds (Dams)

pipeline feed pump station (from 1 to 9 pumps in series up to 7.7 MPa (1100 psi))

FIGURE 12-1

General schematic for tailings disposal pipeline.

Page 12.3

dilution water

coarse for banks of pond

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slurry

Tailings pond (single or multiple cells) (or sometimes the sea is used instead of man made ponds)

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Mineral Process Plant

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clarified water

thickener

isolation valve

Field pump test Loop to adjust concentration

corrosion inhibitors

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dilution water

concentrate pipeline feed & storage agitator tank

Emergency pond

concentrate pipeline

pipeline feed pump station centrifugal pumps up to 7.7 MPa,1100 psi reciprocating up to 18 MPa (2600psi))

FIGURE 12-2

General schematic for concentrate pipeline.

filter/dewatering plant

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FEASIBILITY STUDY FOR A SLURRY PIPELINE

12-2 RHEOLOGY, THICKENER PERFORMANCE, AND PIPELINE SIZING Thickeners are often located at the starting point of the pipeline. Thickeners are installed for a certain production capacity and can be modified for higher output through the use of flocculants. Certain slurries, particularly those rich in fines, silt, and clay, can prove troublesome for the thickeners. At concentrations in excess of 50–55% by weight, the presence of such fines could dramatically increase the viscosity and yield stress. This in turn could force higher power to be needed for pipeline feed pumps, or could force the operator to dilute the slurry. Pilot plant tests are highly recommended. In fact, in large mines, a local pump test loop is sometimes built at the location of the thickeners. The underflow or the concentrate is pumped through the test loop in order to measure the viscosity and pressure drop. The information from the test loop is then used to adjust the operation of the main pipeline pumps and to feed information to the dewatering plant. During the feasibility study, samples of the ore are sent to a rheology lab. Samples should be taken from different boreholes. Some boreholes may yield coarser material at the higher levels but finer materials at deeper depth. This information is used to predict the performance of the pumps throughout the lifetime of the project. For example, in the earlier years of the project, the slurry may be coarser and of heterogeneous flow. As the life of the project progresses, finer material may be pumped at higher concentrations as non-Newtonian flows. Samples from different boreholes are also mixed for testing. The blended samples are quite important as the thickeners may be handling soils from different excavation points, such as a mixture of sulfides and oxides in different proportions. From the rheology of the slurry and the optimum performance of the thickeners, the slurry engineer decides the range of concentrations needed to pump the slurry. The speed of operation is then decided on the basis of the ratio of coarse to fine particles, the velocity of deposition, and the friction losses. Example 12-1 Samples of tailings from a copper process plant are tested for viscosity and yield stress. Results are plotted in Table 12-1. Determine the maximum concentration for designing the thickness or pumping of slurry It is obvious from the data that the viscosity and yield stress rise sharply above a weight concentration of 55%. The slurry engineer would be wise to consider operations above 55% as unstable. Having decided that the maximum weight concentration is 55%, the information is then given to the process engineer in charge of selecting the thickener. Upon review of the

TABLE 12-1 Combined Fine Tailings Weight concentration 40 44 49.9 55 60

Reduced viscosity (slurry/water) 4.95 7.45 12.5 26.4 45

Yield stress (Pa) 1.5 2.7 6.2 12.4 25

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data, the process engineer notices that the thickeners may perform well without flocculants up to a maximum weight concentration of 50%. Because flocculants are expensive, a trade-off study is conducted on the power consumption of pumping slurry at 55%. The study reveals that there is an important increase in capital cost if investment in special thickeners is made to thicken the slurry at 55%. The amount of expensive flocculants for a weight concentration of 55% increases the operating cost and the slurry is more viscous at 55% weight concentration. Despite the fact that there is an increase in the amount of water pumped at 50% weight concentration, a good compromise is found between cost of operation of the thickeners and the capital costs needed for the pipeline to handle the flow for operation at a weight concentration of 53%. The thickeners, pipeline size, and pumps are then sized to produce slurry at a 53% concentration by weight. Thickeners (Figure 12-3) are considered the starting point of tailings and concentrate slurry pipelines. For tailings pipelines, they feed directly into the tailings sump, but for concentrate pipelines they feed special storage tanks with agitators. The sump for the tailings pipeline (Figure 12-4) may be built of concrete or rubberlined steel. A number of pipes are installed in the feed side such as: 앫 Dilution process water pipes 앫 slurry pipes 앫 pipes from emergency ponds The concentrate storage tanks for slurry pipelines are essentially large tanks with agitators (Figure 12-5). A small pump test loop near these tanks is used to test the concentrate before feeding it to the pipeline pumps. Feed is essentially from the thickeners, but continuous agitation in the tank and addition of viscosity control agents, corrosion inhibitors, and even some dilution water are part of the process. Not every slurry pipeline requires thickeners. Dredging pipelines and phosphate rock pumping are both transport low-concentration slurries. These slurries are pumped over shorter distances and use pipes and pumps that are physically relocated from one point to another. Some pipelines operate totally as an open channel flow, such as the tailings pipeline of Southern Peru Copper in Peru.

FIGURE 12-3

Thickeners. (Courtesy of Geho Pumps.)

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FIGURE 12-4

Sump for tailings pipeline.

FIGURE 12-5

Concentrate storage tank.

12.7

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12-3 RECLAIM WATER PIPELINE Although slurry is pumped from the process plant to a tailings disposal site, reclaim water is often pumped back from the tailings pond to the mine. A popular method of feeding the reclaim water into a pipeline is by installing vertical turbine (mixed flow) pumps on a barge or onshore near a pump station (Figure 12-6). The number of stages of these vertical pumps is set by the total dynamic head and the possibility of installing booster pump stations along the pipeline route. The pipeline material may be constructed of steel or high-density polyethylene. The latter, however, is limited to a pressure rating of 1.4 MPa (200 psi) on large pipe sizes (see Chapter 2 for more details on the pressure rating of HDPE). If the reclaim water pipeline is steel and the tailings have been neutralized for corrosion using lime, the pipeline may gradually suffer from deposition of lime on the inside walls. Over time, this increases the pipe’s roughness; friction losses increase and the penalty could be higher power consumption. To prevent such a problem, polypig launching and receiving stations are installed at the start and end of the pipeline. Polypigs are sponge-filled bullets with brushes sized to the pipe diameter. As they move in the pipe, they clean its surface. The methods of sizing the reclaim pipeline for single-phase water were covered in Chapter 2. Floating pump stations are often designed as a catamaran for adequate stability. The pumps are located in the middle of the barge. The catamaran is built with buoyancy tanks on each side that can be filled. Some catamarans have a false bottom to protect the suction of the pump. Reclaim water enters from the side pump inlet via a proper fish screen. The fish screen prevents any fish or aquatic plants from being pumped back to the mine.

FIGURE 12-6

Pump station.

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12-4 EMERGENCY POND An emergency pond should be carved out or built at the start of the pipeline near the pump station or at the lowest point in the pipeline. The purpose of the emergency pond is to provide a means of draining the slurry pipeline (Figure 12-7). The decision to dig an emergency pond is often based on the ability to restart a pipeline after a shut down. Restarting is often difficult with particularly coarse slurries, taconite, sand, and dredging rocks. With finer slurries and clays, it may be possible to restart the pipeline without draining it, provided that the maximum slope does not exceed the critical value (discussed in Chapter 4). An emergency pond is needed in cold areas to avoid freezing the pipeline after a shutdown. Sometimes a special valve chamber is installed with a valve on a tee branch. This valve automatically opens on power failure to divert slurry to the emergency pond. An emergency pond needs its own pumping system. It can consist of a vertical slurry pump floating on a pontoon or barge (Figure 12-8). The sump pump feeds a booster pump that redirects the slurry back to the pipeline pump box or back to the thickeners. Submersible pumps (Figure 12-9) with augers are also used for emergency ponds near thickeners, particularly with concentrate pipelines. Special water sparges are installed

FIGURE 12-7

Emergency pond.

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motor pump

floats

FIGURE 12-8

Emergency pond pumping system.

FIGURE 12-9

Submersible pump.

12.10

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around the emergency pond to dilute the slurry. Although fairly reliable, submersible pumps require a special shop to rebuild them and replace the seals. In remote mines, they tend to be less popular than vertical cantilever pumps. It is also recommended to install emergency ponds near booster pump stations. These should be connected to the booster station by a drainpipe. A pontoon on the pond with a cantilever pump (Figure 12-8) is recommended to pump back the spill to the booster station pump box. If the tailings dam is to be located in a flood plain, the civil engineer may recommend an emergency spillway. A decant pond may serve as an emergency spillway. Sometimes it is more economical to provide adequate height of the walls of the dam to contain the 1:100 year flood, particularly when purchasing more land is an expensive proposition.

12-5 TAILINGS DAMS Many pipelines are used for pumping tailings. Selecting a site for disposal of tailings is based on many factors: 앫 The tailings dam must be able to be used for the life term of the mine (e.g., 10–20 years). 앫 The site bedrock or foundation must be stable to build the dam walls. These are typically made of sand and coarse rejects and some are built at a rate of 4.6 m (15 ft) per year. 앫 The site must not interfere with future expansion of the mine and must not be on an ore deposit. For this reason, the tailings disposal system is sometimes a long distance away from the mine or surrounding economic centers (towns, cities, and agricultural fields). 앫 The tailings disposal area must be designed to minimize contamination such as seepage of liquids to surrounding areas. 앫 The volume of the tailings containment must be calculated to account for disposal volumes, runoff of snow or rain, and the pumping out of reclaim water. 앫 The process of separation of slimes from coarse materials at the tailings dam must be designed carefully. It can be as simple as a spigot when there is a considerable portion of coarse materials, or as complex as a two-stage cyclone when there are a lot of fines in the tailings. 앫 Accessibility to the site is important for repairs and for construction of the tailings dam. The guidelines for constructing a tailings dam have been established by the International Commission on Large Dams (1982). These are reviewed briefly in the following paragraphs. These are general principles that must be adapted to every site and condition. It is important to be able to separate the coarse from the fine particles when building a tailings dam. The coarse solids are used to build the dam walls, whereas the fines are used to form the beaches (Figure 12-10).

12-5-1 Wall Building by Spigotting One method of constructing the walls of a dam is to use the coarse material in the tailings. The fines or slimes are allowed to sink to the bottom of the tailings pond or to form

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slurry cyclone overflow is used to make beaches of fines

cyclone underflow Rock Toe

(coarse material) beaches of fines

decant water intake

pond toe trench

filter drain bed

FIGURE 12-10

Using tailings to build dams and beaches.

beaches between the water and the dam. When there is a high content of coarse particles, as in the taconite mines of Minnesota (U.S.A.), it is sufficient to use a spigot to separate coarse from fine particles. At the exit from the spigot, the coarse particles separate under gravity and pressure while the finer particles are carried further away. A bulldozer relocates and then compacts the material by rolling over it. The banks are gradually built this way. This operation is difficult in the winter in Canada, Siberia, and northern United States. Therefore, the actual construction of the dam is limited to the summer months. Although the great majority of tailing dams are built on the concept of a single spigot, some use the concept of multiple parallel spigots. In the single-spigot approach, all the tailings are dispensed at one point. After a couple of days or so, the spigot is then moved approximately 15 m (50 ft) away. At each location, the bulldozer is brought in to compact the coarse material. The banks of the dam are thus gradually built. In the multispigot system, the spigots are fixed in place. The diameter of the pipeline is gradually reduced around the tailings dam. This method is particularly interesting in very cold climates when construction of the dam is difficult in the six months of the year when construction is not possible.

12-5-2 Deposition by Cycloning A spigot may not be sufficient to separate coarse from fine particles. More pressure and force may be needed. One particularly useful piece of equipment is the hydrocyclone (which was presented in Chapter 7). The coarse material is diverted to the underflow and the finer material to the overflow. In a certain ratio of coarse and fine particles, a single cyclone is sufficient, but when the coarse material is less than 25% of the tailings, two cyclones in series may be needed. It is strongly recommend that a cyclonability test be conducted in a lab before deciding whether a single-stage or a two-stage cyclone is needed. Example 12-2 Tailings from a mine were tested for cyclonability. The following particle size distribution was obtained: Particle size (microns) Cumulative % passing

152 83.3

110 75.4

74 67

53 61

44 54

37 52

29 50

25 44

22 17 42 40

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It is clear that the fines (

Slurry Systems Handbook

Slurry Systems Handbook

HVAC Systems Design Handbook

Small Computer Systems Handbook

Facility Piping Systems Handbook

Facility Piping Systems Handbook

Facility Piping Systems Handbook

Automotive Embedded Systems Handbook

BOAT MECHANICAL SYSTEMS HANDBOOK

NASA Systems Engineering Handbook

Facility Piping Systems Handbook

Handbook of dynamical systems

Handbook of Dynamical Systems

CDMA Systems Engineering Handbook

Automotive Quality Systems Handbook

The quality systems handbook

AC power systems handbook

Handbook of dynamical systems

Handbook of dynamical systems

Embedded Systems Handbook

NASA Systems Engineering Handbook

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Handbook of Power Systems II (Energy Systems)

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Reference Power Systems Protection Handbook

Handbook of Signal Processing Systems

ISO 9000 Quality Systems Handbook,

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